TPTP Problem File: SLH0867^1.p

%------------------------------------------------------------------------------
% 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_Calc2/0018_EPathHintikka/prob_00376_015236__13151782_1 [Des23]

% Status   : Theorem
% Rating   : ? v8.2.0
% Syntax   : Number of formulae    : 2025 ( 645 unt; 744 typ;   0 def)
%            Number of atoms       : 3841 (1439 equ;   0 cnn)
%            Maximal formula atoms :    7 (   2 avg)
%            Number of connectives : 12377 ( 433   ~;  38   |; 243   &;10456   @)
%                                         (   0 <=>;1207  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   17 (   6 avg)
%            Number of types       :  128 ( 127 usr)
%            Number of type conns  : 2283 (2283   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  620 ( 617 usr;  26 con; 0-12 aty)
%            Number of variables   : 4596 (1161   ^;3367   !;  68   ?;4596   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            2023-01-19 08:45:26.728
%------------------------------------------------------------------------------
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thf(ty_n_t__Product____Type__Oprod_It__Prover__Orule_Mt__SeCaV__Ofm_J,type,
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thf(ty_n_t__Stream__Ostream_It__Stream__Ostream_It__SeCaV__Ofm_J_J,type,
    stream_stream_fm: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__SeCaV__Ofm_Mt__SeCaV__Ofm_J,type,
    product_prod_fm_fm: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    product_prod_nat_nat: $tType ).

thf(ty_n_t__Stream__Ostream_It__List__Olist_It__SeCaV__Ofm_J_J,type,
    stream_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__Stream__Ostream_It__SeCaV__Ofm_J_J,type,
    set_stream_fm: $tType ).

thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_I_Eo_J_J_J,type,
    set_set_set_o: $tType ).

thf(ty_n_t__List__Olist_It__List__Olist_It__SeCaV__Otm_J_J,type,
    list_list_tm: $tType ).

thf(ty_n_t__List__Olist_It__List__Olist_It__SeCaV__Ofm_J_J,type,
    list_list_fm: $tType ).

thf(ty_n_t__Set__Oset_It__Set__Oset_It__Prover__Orule_J_J,type,
    set_set_rule: $tType ).

thf(ty_n_t__Set__Oset_It__List__Olist_It__SeCaV__Otm_J_J,type,
    set_list_tm: $tType ).

thf(ty_n_t__Set__Oset_It__List__Olist_It__SeCaV__Ofm_J_J,type,
    set_list_fm: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__SeCaV__Ofm_M_Eo_J,type,
    product_prod_fm_o: $tType ).

thf(ty_n_t__Product____Type__Oprod_I_Eo_Mt__SeCaV__Ofm_J,type,
    product_prod_o_fm: $tType ).

thf(ty_n_t__Product____Type__Oprod_I_Eo_Mt__Nat__Onat_J,type,
    product_prod_o_nat: $tType ).

thf(ty_n_t__Set__Oset_It__Set__Oset_It__SeCaV__Otm_J_J,type,
    set_set_tm: $tType ).

thf(ty_n_t__Set__Oset_It__Set__Oset_It__SeCaV__Ofm_J_J,type,
    set_set_fm: $tType ).

thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
    set_set_nat: $tType ).

thf(ty_n_t__Set__Oset_I_062_It__SeCaV__Ofm_M_Eo_J_J,type,
    set_fm_o: $tType ).

thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_M_Eo_J_J,type,
    set_nat_o: $tType ).

thf(ty_n_t__Stream__Ostream_It__Prover__Orule_J,type,
    stream_rule: $tType ).

thf(ty_n_t__Product____Type__Oprod_I_Eo_M_Eo_J,type,
    product_prod_o_o: $tType ).

thf(ty_n_t__Stream__Ostream_It__SeCaV__Otm_J,type,
    stream_tm: $tType ).

thf(ty_n_t__Stream__Ostream_It__SeCaV__Ofm_J,type,
    stream_fm: $tType ).

thf(ty_n_t__Set__Oset_It__Set__Oset_I_Eo_J_J,type,
    set_set_o: $tType ).

thf(ty_n_t__Stream__Ostream_It__Nat__Onat_J,type,
    stream_nat: $tType ).

thf(ty_n_t__Set__Oset_It__Prover__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__SeCaV__Otm_J,type,
    list_tm: $tType ).

thf(ty_n_t__List__Olist_It__SeCaV__Ofm_J,type,
    list_fm: $tType ).

thf(ty_n_t__Set__Oset_It__SeCaV__Otm_J,type,
    set_tm: $tType ).

thf(ty_n_t__Set__Oset_It__SeCaV__Ofm_J,type,
    set_fm: $tType ).

thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
    set_nat: $tType ).

thf(ty_n_t__Stream__Ostream_I_Eo_J,type,
    stream_o: $tType ).

thf(ty_n_t__List__Olist_I_Eo_J,type,
    list_o: $tType ).

thf(ty_n_t__Set__Oset_I_Eo_J,type,
    set_o: $tType ).

thf(ty_n_t__Prover__Orule,type,
    rule: $tType ).

thf(ty_n_t__SeCaV__Otm,type,
    tm: $tType ).

thf(ty_n_t__SeCaV__Ofm,type,
    fm: $tType ).

thf(ty_n_t__Nat__Onat,type,
    nat: $tType ).

% Explicit typings (617)
thf(sy_c_BNF__Cardinal__Order__Relation_OnatLess,type,
    bNF_Ca8459412986667044542atLess: set_Pr1261947904930325089at_nat ).

thf(sy_c_Basic__BNFs_Ofsts_001t__Nat__Onat_001t__Nat__Onat,type,
    basic_fsts_nat_nat: product_prod_nat_nat > set_nat ).

thf(sy_c_Basic__BNFs_Ofsts_001t__Prover__Orule_001t__SeCaV__Ofm,type,
    basic_fsts_rule_fm: product_prod_rule_fm > set_rule ).

thf(sy_c_Basic__BNFs_Osnds_001t__Nat__Onat_001t__Nat__Onat,type,
    basic_snds_nat_nat: product_prod_nat_nat > set_nat ).

thf(sy_c_Basic__BNFs_Osnds_001t__Prover__Orule_001t__SeCaV__Ofm,type,
    basic_snds_rule_fm: product_prod_rule_fm > set_fm ).

thf(sy_c_Complete__Lattices_OInf__class_OInf_001_Eo,type,
    complete_Inf_Inf_o: set_o > $o ).

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__SeCaV__Ofm_J_M_Eo_J,type,
    comple173717950668161943t_fm_o: set_list_fm_o > list_fm > $o ).

thf(sy_c_Complete__Lattices_OSup__class_OSup_001_062_It__Nat__Onat_M_062_It__Nat__Onat_M_Eo_J_J,type,
    comple3592611370556534995_nat_o: set_nat_nat_o > nat > nat > $o ).

thf(sy_c_Complete__Lattices_OSup__class_OSup_001_062_It__Nat__Onat_M_Eo_J,type,
    comple8317665133742190828_nat_o: set_nat_o > nat > $o ).

thf(sy_c_Complete__Lattices_OSup__class_OSup_001_062_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_M_062_It__Prover__Orule_M_Eo_J_J,type,
    comple5956027139101513408rule_o: set_Pr7982104872110991011rule_o > produc6018962875968178549ist_fm > rule > $o ).

thf(sy_c_Complete__Lattices_OSup__class_OSup_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
    comple1901448594430626575_nat_o: set_Pr5582243495563764594_nat_o > product_prod_nat_nat > $o ).

thf(sy_c_Complete__Lattices_OSup__class_OSup_001_062_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_M_Eo_J,type,
    comple3597286212584221062rule_o: set_Pr3586676963299879929rule_o > produc340336539035504054m_rule > $o ).

thf(sy_c_Complete__Lattices_OSup__class_OSup_001_062_It__Prover__Orule_M_062_It__SeCaV__Ofm_M_Eo_J_J,type,
    comple8604870961538653134e_fm_o: set_rule_fm_o > rule > fm > $o ).

thf(sy_c_Complete__Lattices_OSup__class_OSup_001_062_It__SeCaV__Ofm_M_Eo_J,type,
    comple3443612784043876893p_fm_o: set_fm_o > fm > $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__Nat__Onat,type,
    complete_Sup_Sup_nat: set_nat > nat ).

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__List__Olist_It__SeCaV__Ofm_J_J,type,
    comple8784269564784259782ist_fm: set_set_list_fm > set_list_fm ).

thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Nat__Onat_J,type,
    comple7399068483239264473et_nat: set_set_nat > set_nat ).

thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Product____Type__Oprod_I_Eo_M_Eo_J_J,type,
    comple2567785453016644468od_o_o: set_se491199571785101031od_o_o > set_Product_prod_o_o ).

thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    comple5685304695842803022at_nat: set_se7855581050983116737at_nat > set_Pr1261947904930325089at_nat ).

thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_J,type,
    comple5648871819318716351m_rule: set_se4309406418413213730m_rule > set_Pr1822751329126368876m_rule ).

thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Product____Type__Oprod_It__Prover__Orule_Mt__SeCaV__Ofm_J_J,type,
    comple3749034766840985439ule_fm: set_se6349671006166185922ule_fm > set_Pr8969532344284305932ule_fm ).

thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Product____Type__Oprod_It__SeCaV__Ofm_Mt__SeCaV__Ofm_J_J,type,
    comple3052221160759688756_fm_fm: set_se7864838648521419175_fm_fm > set_Pr4706815898642364871_fm_fm ).

thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Prover__Orule_J,type,
    comple7665822446944348827t_rule: set_set_rule > set_rule ).

thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__SeCaV__Ofm_J,type,
    comple2134933779557159616set_fm: set_set_fm > set_fm ).

thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__SeCaV__Otm_J,type,
    comple2138885804642794802set_tm: set_set_tm > set_tm ).

thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Set__Oset_I_Eo_J_J,type,
    comple4436988014476444997_set_o: set_set_set_o > set_set_o ).

thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Set__Oset_It__SeCaV__Ofm_J_J,type,
    comple4326961365146087328set_fm: set_set_set_fm > set_set_fm ).

thf(sy_c_Conditionally__Complete__Lattices_Opreorder__class_Obdd__above_001_Eo,type,
    condit5488710616941104124bove_o: set_o > $o ).

thf(sy_c_Conditionally__Complete__Lattices_Opreorder__class_Obdd__above_001t__Nat__Onat,type,
    condit2214826472909112428ve_nat: set_nat > $o ).

thf(sy_c_Conditionally__Complete__Lattices_Opreorder__class_Obdd__above_001t__Set__Oset_It__SeCaV__Ofm_J,type,
    condit7632555770478454199set_fm: set_set_fm > $o ).

thf(sy_c_Countable__Set_Ocountable_001_Eo,type,
    counta5976203206615340371able_o: set_o > $o ).

thf(sy_c_Countable__Set_Ocountable_001t__List__Olist_It__SeCaV__Ofm_J,type,
    counta8303570107911152436ist_fm: set_list_fm > $o ).

thf(sy_c_Countable__Set_Ocountable_001t__Nat__Onat,type,
    counta1168086296615599829le_nat: set_nat > $o ).

thf(sy_c_Countable__Set_Ocountable_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    counta3401040167773629372at_nat: set_Pr1261947904930325089at_nat > $o ).

thf(sy_c_Countable__Set_Ocountable_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J,type,
    counta1649821642195540155m_rule: set_Pr1822751329126368876m_rule > $o ).

thf(sy_c_Countable__Set_Ocountable_001t__SeCaV__Ofm,type,
    counta2008398395262058670ble_fm: set_fm > $o ).

thf(sy_c_Countable__Set_Ocountable_001t__Set__Oset_It__SeCaV__Ofm_J,type,
    counta6288402088370727310set_fm: set_set_fm > $o ).

thf(sy_c_EPathHintikka_Opseq,type,
    pseq: produc340336539035504054m_rule > list_fm ).

thf(sy_c_EPathHintikka_Otree__fms,type,
    tree_fms: stream2709947120125613254m_rule > set_fm ).

thf(sy_c_Equiv__Relations_Oequiv_001_Eo,type,
    equiv_equiv_o: set_o > set_Product_prod_o_o > $o ).

thf(sy_c_Equiv__Relations_Oequiv_001t__List__Olist_It__SeCaV__Ofm_J,type,
    equiv_equiv_list_fm: set_list_fm > set_Pr1916137733696349511ist_fm > $o ).

thf(sy_c_Equiv__Relations_Oequiv_001t__Nat__Onat,type,
    equiv_equiv_nat: set_nat > set_Pr1261947904930325089at_nat > $o ).

thf(sy_c_Equiv__Relations_Oequiv_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J,type,
    equiv_6275699240365100160m_rule: set_Pr1822751329126368876m_rule > set_Pr4971326047967503661m_rule > $o ).

thf(sy_c_Equiv__Relations_Oequiv_001t__SeCaV__Ofm,type,
    equiv_equiv_fm: set_fm > set_Pr4706815898642364871_fm_fm > $o ).

thf(sy_c_Equiv__Relations_Oquotient_001_Eo,type,
    equiv_quotient_o: set_o > set_Product_prod_o_o > set_set_o ).

thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
    plus_plus_nat: nat > nat > nat ).

thf(sy_c_HOL_OThe_001_Eo,type,
    the_o: ( $o > $o ) > $o ).

thf(sy_c_HOL_OThe_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    the_Pr7557018466319803784at_nat: ( product_prod_nat_nat > $o ) > product_prod_nat_nat ).

thf(sy_c_HOL_OThe_001t__Product____Type__Oprod_It__Prover__Orule_Mt__SeCaV__Ofm_J,type,
    the_Pr8424379526464080271ule_fm: ( product_prod_rule_fm > $o ) > product_prod_rule_fm ).

thf(sy_c_If_001t__Set__Oset_It__SeCaV__Ofm_J,type,
    if_set_fm: $o > set_fm > set_fm > set_fm ).

thf(sy_c_Lattices_Oinf__class_Oinf_001_062_I_Eo_M_Eo_J,type,
    inf_inf_o_o: ( $o > $o ) > ( $o > $o ) > $o > $o ).

thf(sy_c_Lattices_Oinf__class_Oinf_001_062_It__List__Olist_It__SeCaV__Ofm_J_M_Eo_J,type,
    inf_inf_list_fm_o: ( list_fm > $o ) > ( list_fm > $o ) > list_fm > $o ).

thf(sy_c_Lattices_Oinf__class_Oinf_001_062_It__Nat__Onat_M_062_It__Nat__Onat_M_Eo_J_J,type,
    inf_inf_nat_nat_o: ( nat > nat > $o ) > ( nat > nat > $o ) > nat > nat > $o ).

thf(sy_c_Lattices_Oinf__class_Oinf_001_062_It__Nat__Onat_M_Eo_J,type,
    inf_inf_nat_o: ( nat > $o ) > ( nat > $o ) > nat > $o ).

thf(sy_c_Lattices_Oinf__class_Oinf_001_062_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_M_062_It__Prover__Orule_M_Eo_J_J,type,
    inf_in7523237975460508159rule_o: ( produc6018962875968178549ist_fm > rule > $o ) > ( produc6018962875968178549ist_fm > rule > $o ) > produc6018962875968178549ist_fm > rule > $o ).

thf(sy_c_Lattices_Oinf__class_Oinf_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
    inf_in5163264567034779214_nat_o: ( product_prod_nat_nat > $o ) > ( product_prod_nat_nat > $o ) > product_prod_nat_nat > $o ).

thf(sy_c_Lattices_Oinf__class_Oinf_001_062_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_M_Eo_J,type,
    inf_in1825323639692802311rule_o: ( produc340336539035504054m_rule > $o ) > ( produc340336539035504054m_rule > $o ) > produc340336539035504054m_rule > $o ).

thf(sy_c_Lattices_Oinf__class_Oinf_001_062_It__Prover__Orule_M_062_It__SeCaV__Ofm_M_Eo_J_J,type,
    inf_inf_rule_fm_o: ( rule > fm > $o ) > ( rule > fm > $o ) > rule > fm > $o ).

thf(sy_c_Lattices_Oinf__class_Oinf_001_062_It__SeCaV__Ofm_M_Eo_J,type,
    inf_inf_fm_o: ( fm > $o ) > ( fm > $o ) > fm > $o ).

thf(sy_c_Lattices_Oinf__class_Oinf_001_Eo,type,
    inf_inf_o: $o > $o > $o ).

thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_I_Eo_J,type,
    inf_inf_set_o: set_o > set_o > set_o ).

thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__List__Olist_It__SeCaV__Ofm_J_J,type,
    inf_inf_set_list_fm: set_list_fm > set_list_fm > set_list_fm ).

thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Nat__Onat_J,type,
    inf_inf_set_nat: set_nat > set_nat > set_nat ).

thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    inf_in2572325071724192079at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).

thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_J,type,
    inf_in2520071923633375486m_rule: set_Pr1822751329126368876m_rule > set_Pr1822751329126368876m_rule > set_Pr1822751329126368876m_rule ).

thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Product____Type__Oprod_It__Prover__Orule_Mt__SeCaV__Ofm_J_J,type,
    inf_in356324440158061726ule_fm: set_Pr8969532344284305932ule_fm > set_Pr8969532344284305932ule_fm > set_Pr8969532344284305932ule_fm ).

thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__SeCaV__Ofm_J,type,
    inf_inf_set_fm: set_fm > set_fm > set_fm ).

thf(sy_c_Lattices_Osup__class_Osup_001_062_I_Eo_M_Eo_J,type,
    sup_sup_o_o: ( $o > $o ) > ( $o > $o ) > $o > $o ).

thf(sy_c_Lattices_Osup__class_Osup_001_062_It__List__Olist_It__SeCaV__Ofm_J_M_Eo_J,type,
    sup_sup_list_fm_o: ( list_fm > $o ) > ( list_fm > $o ) > list_fm > $o ).

thf(sy_c_Lattices_Osup__class_Osup_001_062_It__Nat__Onat_M_062_It__Nat__Onat_M_Eo_J_J,type,
    sup_sup_nat_nat_o: ( nat > nat > $o ) > ( nat > nat > $o ) > nat > nat > $o ).

thf(sy_c_Lattices_Osup__class_Osup_001_062_It__Nat__Onat_M_Eo_J,type,
    sup_sup_nat_o: ( nat > $o ) > ( nat > $o ) > nat > $o ).

thf(sy_c_Lattices_Osup__class_Osup_001_062_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_M_062_It__Prover__Orule_M_Eo_J_J,type,
    sup_su1495437568605221017rule_o: ( produc6018962875968178549ist_fm > rule > $o ) > ( produc6018962875968178549ist_fm > rule > $o ) > produc6018962875968178549ist_fm > rule > $o ).

thf(sy_c_Lattices_Osup__class_Osup_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
    sup_su798857527126471912_nat_o: ( product_prod_nat_nat > $o ) > ( product_prod_nat_nat > $o ) > product_prod_nat_nat > $o ).

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    szip_rule_fm: stream_rule > stream_fm > stream8658065904927289446ule_fm ).

thf(sy_c_Wellfounded_Olex__prod_001t__Nat__Onat_001t__Nat__Onat,type,
    lex_prod_nat_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr8693737435421807431at_nat ).

thf(sy_c_Wellfounded_Olex__prod_001t__Prover__Orule_001t__SeCaV__Ofm,type,
    lex_prod_rule_fm: set_Pr7340090144965549541e_rule > set_Pr4706815898642364871_fm_fm > set_Pr8271737078293475437ule_fm ).

thf(sy_c_Wellfounded_Omeasure_001t__Nat__Onat,type,
    measure_nat: ( nat > nat ) > set_Pr1261947904930325089at_nat ).

thf(sy_c_Wellfounded_Owf_001_Eo,type,
    wf_o: set_Product_prod_o_o > $o ).

thf(sy_c_Wellfounded_Owf_001t__Nat__Onat,type,
    wf_nat: set_Pr1261947904930325089at_nat > $o ).

thf(sy_c_Wfrec_Osame__fst_001t__Nat__Onat_001t__Nat__Onat,type,
    same_fst_nat_nat: ( nat > $o ) > ( nat > set_Pr1261947904930325089at_nat ) > set_Pr8693737435421807431at_nat ).

thf(sy_c_Wfrec_Osame__fst_001t__Prover__Orule_001t__SeCaV__Ofm,type,
    same_fst_rule_fm: ( rule > $o ) > ( rule > set_Pr4706815898642364871_fm_fm ) > set_Pr8271737078293475437ule_fm ).

thf(sy_c_member_001_Eo,type,
    member_o2: $o > set_o > $o ).

thf(sy_c_member_001t__List__Olist_It__SeCaV__Ofm_J,type,
    member_list_fm2: list_fm > set_list_fm > $o ).

thf(sy_c_member_001t__Nat__Onat,type,
    member_nat: nat > set_nat > $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_I_Eo_Mt__List__Olist_It__SeCaV__Ofm_J_J,type,
    member7814064844939679665ist_fm: produc7435837740378433416ist_fm > set_Pr4658132521726357736ist_fm > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_I_Eo_Mt__Nat__Onat_J,type,
    member2802428098988154798_o_nat: product_prod_o_nat > set_Pr2101469702781467981_o_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_I_Eo_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    member6694634564379471929at_nat: produc5146536252030154256at_nat > set_Pr2390076351701138800at_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_I_Eo_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_J,type,
    member3169346647243466388m_rule: produc5731340206564399485m_rule > set_Pr5066739526756200755m_rule > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_I_Eo_Mt__SeCaV__Ofm_J,type,
    member6824368184199080235d_o_fm: product_prod_o_fm > set_Pr3863124289995447138d_o_fm > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__SeCaV__Ofm_J_M_Eo_J,type,
    member8238630628690094383t_fm_o: produc7860403524128848134t_fm_o > set_Pr4615000397952541542t_fm_o > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__SeCaV__Ofm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J,type,
    member1926098302810316688ist_fm: produc3245234490656042599ist_fm > set_Pr1916137733696349511ist_fm > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__SeCaV__Ofm_J_Mt__SeCaV__Ofm_J,type,
    member7999069721096432778_fm_fm: produc7449846734227509985_fm_fm > set_Pr1501096078598818625_fm_fm > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J,type,
    member4699826688122452638ist_fm: produc6018962875968178549ist_fm > set_Pr5202636777678657877ist_fm > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    member8440522571783428010at_nat: product_prod_nat_nat > set_Pr1261947904930325089at_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J,type,
    member7231649785386036813m_rule: produc340336539035504054m_rule > set_Pr1822751329126368876m_rule > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    member8206827879206165904at_nat: produc859450856879609959at_nat > set_Pr8693737435421807431at_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_J,type,
    member1411645983481675894m_rule: produc8828831911945107917m_rule > set_Pr4971326047967503661m_rule > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Prover__Orule_Mt__SeCaV__Ofm_J_Mt__Product____Type__Oprod_It__Prover__Orule_Mt__SeCaV__Ofm_J_J,type,
    member6853779175081790390ule_fm: produc501962496949067021ule_fm > set_Pr8271737078293475437ule_fm > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Prover__Orule_Mt__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_J,type,
    member21886688540021743ist_fm: produc2353945479044264792ist_fm > set_Pr3602992621978123022ist_fm > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Prover__Orule_Mt__Prover__Orule_J,type,
    member1444691403335511342e_rule: produc7694839378271647877e_rule > set_Pr7340090144965549541e_rule > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Prover__Orule_Mt__SeCaV__Ofm_J,type,
    member9168179392308330989ule_fm: product_prod_rule_fm > set_Pr8969532344284305932ule_fm > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__SeCaV__Ofm_M_Eo_J,type,
    member1731323461681951669d_fm_o: product_prod_fm_o > set_Pr2813250270173872364d_fm_o > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__SeCaV__Ofm_Mt__List__Olist_It__SeCaV__Ofm_J_J,type,
    member8098592186287407126ist_fm: produc7549369199418484333ist_fm > set_Pr937226627985004749ist_fm > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__SeCaV__Ofm_Mt__Prover__Orule_J,type,
    member6041693456836784459m_rule: product_prod_fm_rule > set_Pr7067330020180114026m_rule > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__SeCaV__Ofm_Mt__SeCaV__Ofm_J,type,
    member7780952600467998736_fm_fm: product_prod_fm_fm > set_Pr4706815898642364871_fm_fm > $o ).

thf(sy_c_member_001t__Prover__Orule,type,
    member_rule: rule > set_rule > $o ).

thf(sy_c_member_001t__SeCaV__Ofm,type,
    member_fm2: fm > set_fm > $o ).

thf(sy_c_member_001t__SeCaV__Otm,type,
    member_tm2: tm > set_tm > $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__List__Olist_It__SeCaV__Ofm_J_J,type,
    member_set_list_fm: set_list_fm > set_set_list_fm > $o ).

thf(sy_c_member_001t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_J,type,
    member5802698409304902403m_rule: set_Pr1822751329126368876m_rule > set_se4309406418413213730m_rule > $o ).

thf(sy_c_member_001t__Set__Oset_It__SeCaV__Ofm_J,type,
    member_set_fm: set_fm > set_set_fm > $o ).

thf(sy_v_n____,type,
    n: nat ).

thf(sy_v_p____,type,
    p: fm ).

thf(sy_v_pre____,type,
    pre: list_P2774625669004309958m_rule ).

thf(sy_v_q____,type,
    q: fm ).

thf(sy_v_r____,type,
    r: rule ).

thf(sy_v_steps,type,
    steps: stream2709947120125613254m_rule ).

thf(sy_v_suf____,type,
    suf: stream2709947120125613254m_rule ).

% Relevant facts (1277)
thf(fact_0__092_060open_062Con_Ap_Aq_A_092_060in_062_Atree__fms_Asteps_092_060close_062,axiom,
    member_fm2 @ ( con @ p @ q ) @ ( tree_fms @ steps ) ).

% \<open>Con p q \<in> tree_fms steps\<close>
thf(fact_1_case__prodI,axiom,
    ! [F: rule > fm > $o,A: rule,B: fm] :
      ( ( F @ A @ B )
     => ( produc3561889649859641891e_fm_o @ F @ ( product_Pair_rule_fm @ A @ B ) ) ) ).

% case_prodI
thf(fact_2_case__prodI,axiom,
    ! [F: nat > nat > $o,A: nat,B: nat] :
      ( ( F @ A @ B )
     => ( produc6081775807080527818_nat_o @ F @ ( product_Pair_nat_nat @ A @ B ) ) ) ).

% case_prodI
thf(fact_3_case__prodI2,axiom,
    ! [P: product_prod_rule_fm,C: rule > fm > $o] :
      ( ! [A2: rule,B2: fm] :
          ( ( P
            = ( product_Pair_rule_fm @ A2 @ B2 ) )
         => ( C @ A2 @ B2 ) )
     => ( produc3561889649859641891e_fm_o @ C @ P ) ) ).

% case_prodI2
thf(fact_4_case__prodI2,axiom,
    ! [P: product_prod_nat_nat,C: nat > nat > $o] :
      ( ! [A2: nat,B2: nat] :
          ( ( P
            = ( product_Pair_nat_nat @ A2 @ B2 ) )
         => ( C @ A2 @ B2 ) )
     => ( produc6081775807080527818_nat_o @ C @ P ) ) ).

% case_prodI2
thf(fact_5_case__prod__conv,axiom,
    ! [F: rule > fm > $o,A: rule,B: fm] :
      ( ( produc3561889649859641891e_fm_o @ F @ ( product_Pair_rule_fm @ A @ B ) )
      = ( F @ A @ B ) ) ).

% case_prod_conv
thf(fact_6_case__prod__conv,axiom,
    ! [F: nat > nat > $o,A: nat,B: nat] :
      ( ( produc6081775807080527818_nat_o @ F @ ( product_Pair_nat_nat @ A @ B ) )
      = ( F @ A @ B ) ) ).

% case_prod_conv
thf(fact_7_affects__def,axiom,
    ( affects
    = ( ^ [R: rule,P2: fm] :
          ( produc3561889649859641891e_fm_o
          @ ^ [X: rule,Xa: fm] :
              ( case_rule_o
              @ ( case_fm_o
                @ ^ [Nat: nat,List: list_tm] : $false
                @ ^ [Fm1: fm,Fm2: fm] : $false
                @ ^ [Y: fm,Xb: fm] : $true
                @ ^ [Fm1: fm,Fm2: fm] : $false
                @ ^ [Fm: fm] : $false
                @ ^ [Fm: fm] : $false
                @ ^ [Fm: fm] : $false
                @ Xa )
              @ ( case_fm_o
                @ ^ [Nat: nat,List: list_tm] : $false
                @ ^ [Y: fm,Xb: fm] : $true
                @ ^ [Fm1: fm,Fm2: fm] : $false
                @ ^ [Fm1: fm,Fm2: fm] : $false
                @ ^ [Fm: fm] : $false
                @ ^ [Fm: fm] : $false
                @ ^ [Fm: fm] : $false
                @ Xa )
              @ ( case_fm_o
                @ ^ [Nat: nat,List: list_tm] : $false
                @ ^ [Fm1: fm,Fm2: fm] : $false
                @ ^ [Fm1: fm,Fm2: fm] : $false
                @ ^ [Fm1: fm,Fm2: fm] : $false
                @ ^ [Fm: fm] : $false
                @ ^ [Fm: fm] : $false
                @ ( case_fm_o
                  @ ^ [Nat: nat,List: list_tm] : $false
                  @ ^ [Fm1: fm,Fm2: fm] : $false
                  @ ^ [Fm1: fm,Fm2: fm] : $false
                  @ ^ [Y: fm,Xb: fm] : $true
                  @ ^ [Fma: fm] : $false
                  @ ^ [Fma: fm] : $false
                  @ ^ [Fma: fm] : $false )
                @ Xa )
              @ ( case_fm_o
                @ ^ [Nat: nat,List: list_tm] : $false
                @ ^ [Fm1: fm,Fm2: fm] : $false
                @ ^ [Fm1: fm,Fm2: fm] : $false
                @ ^ [Y: fm,Xb: fm] : $true
                @ ^ [Fm: fm] : $false
                @ ^ [Fm: fm] : $false
                @ ^ [Fm: fm] : $false
                @ Xa )
              @ ( case_fm_o
                @ ^ [Nat: nat,List: list_tm] : $false
                @ ^ [Fm1: fm,Fm2: fm] : $false
                @ ^ [Fm1: fm,Fm2: fm] : $false
                @ ^ [Fm1: fm,Fm2: fm] : $false
                @ ^ [Fm: fm] : $false
                @ ^ [Fm: fm] : $false
                @ ( case_fm_o
                  @ ^ [Nat: nat,List: list_tm] : $false
                  @ ^ [Y: fm,Xb: fm] : $true
                  @ ^ [Fm1: fm,Fm2: fm] : $false
                  @ ^ [Fm1: fm,Fm2: fm] : $false
                  @ ^ [Fma: fm] : $false
                  @ ^ [Fma: fm] : $false
                  @ ^ [Fma: fm] : $false )
                @ Xa )
              @ ( case_fm_o
                @ ^ [Nat: nat,List: list_tm] : $false
                @ ^ [Fm1: fm,Fm2: fm] : $false
                @ ^ [Fm1: fm,Fm2: fm] : $false
                @ ^ [Fm1: fm,Fm2: fm] : $false
                @ ^ [Fm: fm] : $false
                @ ^ [Fm: fm] : $false
                @ ( case_fm_o
                  @ ^ [Nat: nat,List: list_tm] : $false
                  @ ^ [Fm1: fm,Fm2: fm] : $false
                  @ ^ [Y: fm,Xb: fm] : $true
                  @ ^ [Fm1: fm,Fm2: fm] : $false
                  @ ^ [Fma: fm] : $false
                  @ ^ [Fma: fm] : $false
                  @ ^ [Fma: fm] : $false )
                @ Xa )
              @ ( case_fm_o
                @ ^ [Nat: nat,List: list_tm] : $false
                @ ^ [Fm1: fm,Fm2: fm] : $false
                @ ^ [Fm1: fm,Fm2: fm] : $false
                @ ^ [Fm1: fm,Fm2: fm] : $false
                @ ^ [Fm: fm] : $false
                @ ^ [Y: fm] : $true
                @ ^ [Fm: fm] : $false
                @ Xa )
              @ ( case_fm_o
                @ ^ [Nat: nat,List: list_tm] : $false
                @ ^ [Fm1: fm,Fm2: fm] : $false
                @ ^ [Fm1: fm,Fm2: fm] : $false
                @ ^ [Fm1: fm,Fm2: fm] : $false
                @ ^ [Fm: fm] : $false
                @ ^ [Fm: fm] : $false
                @ ( case_fm_o
                  @ ^ [Nat: nat,List: list_tm] : $false
                  @ ^ [Fm1: fm,Fm2: fm] : $false
                  @ ^ [Fm1: fm,Fm2: fm] : $false
                  @ ^ [Fm1: fm,Fm2: fm] : $false
                  @ ^ [Y: fm] : $true
                  @ ^ [Fma: fm] : $false
                  @ ^ [Fma: fm] : $false )
                @ Xa )
              @ ( case_fm_o
                @ ^ [Nat: nat,List: list_tm] : $false
                @ ^ [Fm1: fm,Fm2: fm] : $false
                @ ^ [Fm1: fm,Fm2: fm] : $false
                @ ^ [Fm1: fm,Fm2: fm] : $false
                @ ^ [Fm: fm] : $false
                @ ^ [Fm: fm] : $false
                @ ( case_fm_o
                  @ ^ [Nat: nat,List: list_tm] : $false
                  @ ^ [Fm1: fm,Fm2: fm] : $false
                  @ ^ [Fm1: fm,Fm2: fm] : $false
                  @ ^ [Fm1: fm,Fm2: fm] : $false
                  @ ^ [Fma: fm] : $false
                  @ ^ [Fma: fm] : $false
                  @ ^ [Y: fm] : $true )
                @ Xa )
              @ ( case_fm_o
                @ ^ [Nat: nat,List: list_tm] : $false
                @ ^ [Fm1: fm,Fm2: fm] : $false
                @ ^ [Fm1: fm,Fm2: fm] : $false
                @ ^ [Fm1: fm,Fm2: fm] : $false
                @ ^ [Y: fm] : $false
                @ ^ [Fm: fm] : $false
                @ ^ [Fm: fm] : $false
                @ Xa )
              @ ( case_fm_o
                @ ^ [Nat: nat,List: list_tm] : $false
                @ ^ [Fm1: fm,Fm2: fm] : $false
                @ ^ [Fm1: fm,Fm2: fm] : $false
                @ ^ [Fm1: fm,Fm2: fm] : $false
                @ ^ [Fm: fm] : $false
                @ ^ [Fm: fm] : $false
                @ ( case_fm_o
                  @ ^ [Nat: nat,List: list_tm] : $false
                  @ ^ [Fm1: fm,Fm2: fm] : $false
                  @ ^ [Fm1: fm,Fm2: fm] : $false
                  @ ^ [Fm1: fm,Fm2: fm] : $false
                  @ ^ [Fma: fm] : $false
                  @ ^ [Y: fm] : $false
                  @ ^ [Fma: fm] : $false )
                @ Xa )
              @ X )
          @ ( product_Pair_rule_fm @ R @ P2 ) ) ) ) ).

% affects_def
thf(fact_8_split__part,axiom,
    ! [P3: $o,Q: rule > fm > $o] :
      ( ( produc3561889649859641891e_fm_o
        @ ^ [A3: rule,B3: fm] :
            ( P3
            & ( Q @ A3 @ B3 ) ) )
      = ( ^ [Ab: product_prod_rule_fm] :
            ( P3
            & ( produc3561889649859641891e_fm_o @ Q @ Ab ) ) ) ) ).

% split_part
thf(fact_9_split__part,axiom,
    ! [P3: $o,Q: nat > nat > $o] :
      ( ( produc6081775807080527818_nat_o
        @ ^ [A3: nat,B3: nat] :
            ( P3
            & ( Q @ A3 @ B3 ) ) )
      = ( ^ [Ab: product_prod_nat_nat] :
            ( P3
            & ( produc6081775807080527818_nat_o @ Q @ Ab ) ) ) ) ).

% split_part
thf(fact_10_fm_Oinject_I4_J,axiom,
    ! [X41: fm,X42: fm,Y41: fm,Y42: fm] :
      ( ( ( con @ X41 @ X42 )
        = ( con @ Y41 @ Y42 ) )
      = ( ( X41 = Y41 )
        & ( X42 = Y42 ) ) ) ).

% fm.inject(4)
thf(fact_11_rule_Osimps_I114_J,axiom,
    ! [F1: $o,F2: $o,F3: $o,F4: $o,F5: $o,F6: $o,F7: $o,F8: $o,F9: $o,F10: $o,F11: $o] :
      ( ( case_rule_o @ F1 @ F2 @ F3 @ F4 @ F5 @ F6 @ F7 @ F8 @ F9 @ F10 @ F11 @ betaCon )
      = F4 ) ).

% rule.simps(114)
thf(fact_12_prod_Oinject,axiom,
    ! [X1: nat,X2: nat,Y1: nat,Y2: nat] :
      ( ( ( product_Pair_nat_nat @ X1 @ X2 )
        = ( product_Pair_nat_nat @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_13_prod_Oinject,axiom,
    ! [X1: rule,X2: fm,Y1: rule,Y2: fm] :
      ( ( ( product_Pair_rule_fm @ X1 @ X2 )
        = ( product_Pair_rule_fm @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_14_old_Oprod_Oinject,axiom,
    ! [A: nat,B: nat,A4: nat,B4: nat] :
      ( ( ( product_Pair_nat_nat @ A @ B )
        = ( product_Pair_nat_nat @ A4 @ B4 ) )
      = ( ( A = A4 )
        & ( B = B4 ) ) ) ).

% old.prod.inject
thf(fact_15_old_Oprod_Oinject,axiom,
    ! [A: rule,B: fm,A4: rule,B4: fm] :
      ( ( ( product_Pair_rule_fm @ A @ B )
        = ( product_Pair_rule_fm @ A4 @ B4 ) )
      = ( ( A = A4 )
        & ( B = B4 ) ) ) ).

% old.prod.inject
thf(fact_16_case__prodD,axiom,
    ! [F: rule > fm > $o,A: rule,B: fm] :
      ( ( produc3561889649859641891e_fm_o @ F @ ( product_Pair_rule_fm @ A @ B ) )
     => ( F @ A @ B ) ) ).

% case_prodD
thf(fact_17_case__prodD,axiom,
    ! [F: nat > nat > $o,A: nat,B: nat] :
      ( ( produc6081775807080527818_nat_o @ F @ ( product_Pair_nat_nat @ A @ B ) )
     => ( F @ A @ B ) ) ).

% case_prodD
thf(fact_18_case__prodE,axiom,
    ! [C: rule > fm > $o,P: product_prod_rule_fm] :
      ( ( produc3561889649859641891e_fm_o @ C @ P )
     => ~ ! [X3: rule,Y3: fm] :
            ( ( P
              = ( product_Pair_rule_fm @ X3 @ Y3 ) )
           => ~ ( C @ X3 @ Y3 ) ) ) ).

% case_prodE
thf(fact_19_case__prodE,axiom,
    ! [C: nat > nat > $o,P: product_prod_nat_nat] :
      ( ( produc6081775807080527818_nat_o @ C @ P )
     => ~ ! [X3: nat,Y3: nat] :
            ( ( P
              = ( product_Pair_nat_nat @ X3 @ Y3 ) )
           => ~ ( C @ X3 @ Y3 ) ) ) ).

% case_prodE
thf(fact_20_case__prodE2,axiom,
    ! [Q: $o > $o,P3: rule > fm > $o,Z: product_prod_rule_fm] :
      ( ( Q @ ( produc3561889649859641891e_fm_o @ P3 @ Z ) )
     => ~ ! [X3: rule,Y3: fm] :
            ( ( Z
              = ( product_Pair_rule_fm @ X3 @ Y3 ) )
           => ~ ( Q @ ( P3 @ X3 @ Y3 ) ) ) ) ).

% case_prodE2
thf(fact_21_case__prodE2,axiom,
    ! [Q: $o > $o,P3: nat > nat > $o,Z: product_prod_nat_nat] :
      ( ( Q @ ( produc6081775807080527818_nat_o @ P3 @ Z ) )
     => ~ ! [X3: nat,Y3: nat] :
            ( ( Z
              = ( product_Pair_nat_nat @ X3 @ Y3 ) )
           => ~ ( Q @ ( P3 @ X3 @ Y3 ) ) ) ) ).

% case_prodE2
thf(fact_22_mem__case__prodI2,axiom,
    ! [P: product_prod_rule_fm,Z: fm,C: rule > fm > set_fm] :
      ( ! [A2: rule,B2: fm] :
          ( ( P
            = ( product_Pair_rule_fm @ A2 @ B2 ) )
         => ( member_fm2 @ Z @ ( C @ A2 @ B2 ) ) )
     => ( member_fm2 @ Z @ ( produc8514419448307895902set_fm @ C @ P ) ) ) ).

% mem_case_prodI2
thf(fact_23_mem__case__prodI2,axiom,
    ! [P: product_prod_rule_fm,Z: list_fm,C: rule > fm > set_list_fm] :
      ( ! [A2: rule,B2: fm] :
          ( ( P
            = ( product_Pair_rule_fm @ A2 @ B2 ) )
         => ( member_list_fm2 @ Z @ ( C @ A2 @ B2 ) ) )
     => ( member_list_fm2 @ Z @ ( produc8148429868913231716ist_fm @ C @ P ) ) ) ).

% mem_case_prodI2
thf(fact_24_mem__case__prodI2,axiom,
    ! [P: product_prod_rule_fm,Z: produc340336539035504054m_rule,C: rule > fm > set_Pr1822751329126368876m_rule] :
      ( ! [A2: rule,B2: fm] :
          ( ( P
            = ( product_Pair_rule_fm @ A2 @ B2 ) )
         => ( member7231649785386036813m_rule @ Z @ ( C @ A2 @ B2 ) ) )
     => ( member7231649785386036813m_rule @ Z @ ( produc8735870174858355873m_rule @ C @ P ) ) ) ).

% mem_case_prodI2
thf(fact_25_mem__case__prodI2,axiom,
    ! [P: product_prod_rule_fm,Z: $o,C: rule > fm > set_o] :
      ( ! [A2: rule,B2: fm] :
          ( ( P
            = ( product_Pair_rule_fm @ A2 @ B2 ) )
         => ( member_o2 @ Z @ ( C @ A2 @ B2 ) ) )
     => ( member_o2 @ Z @ ( produc3402564150753746179_set_o @ C @ P ) ) ) ).

% mem_case_prodI2
thf(fact_26_mem__case__prodI2,axiom,
    ! [P: product_prod_nat_nat,Z: fm,C: nat > nat > set_fm] :
      ( ! [A2: nat,B2: nat] :
          ( ( P
            = ( product_Pair_nat_nat @ A2 @ B2 ) )
         => ( member_fm2 @ Z @ ( C @ A2 @ B2 ) ) )
     => ( member_fm2 @ Z @ ( produc2762169475385161349set_fm @ C @ P ) ) ) ).

% mem_case_prodI2
thf(fact_27_mem__case__prodI2,axiom,
    ! [P: product_prod_nat_nat,Z: list_fm,C: nat > nat > set_list_fm] :
      ( ! [A2: nat,B2: nat] :
          ( ( P
            = ( product_Pair_nat_nat @ A2 @ B2 ) )
         => ( member_list_fm2 @ Z @ ( C @ A2 @ B2 ) ) )
     => ( member_list_fm2 @ Z @ ( produc8363348377050837003ist_fm @ C @ P ) ) ) ).

% mem_case_prodI2
thf(fact_28_mem__case__prodI2,axiom,
    ! [P: product_prod_nat_nat,Z: produc340336539035504054m_rule,C: nat > nat > set_Pr1822751329126368876m_rule] :
      ( ! [A2: nat,B2: nat] :
          ( ( P
            = ( product_Pair_nat_nat @ A2 @ B2 ) )
         => ( member7231649785386036813m_rule @ Z @ ( C @ A2 @ B2 ) ) )
     => ( member7231649785386036813m_rule @ Z @ ( produc6364813917282460346m_rule @ C @ P ) ) ) ).

% mem_case_prodI2
thf(fact_29_mem__case__prodI2,axiom,
    ! [P: product_prod_nat_nat,Z: $o,C: nat > nat > set_o] :
      ( ! [A2: nat,B2: nat] :
          ( ( P
            = ( product_Pair_nat_nat @ A2 @ B2 ) )
         => ( member_o2 @ Z @ ( C @ A2 @ B2 ) ) )
     => ( member_o2 @ Z @ ( produc59986286002894506_set_o @ C @ P ) ) ) ).

% mem_case_prodI2
thf(fact_30_mem__case__prodI,axiom,
    ! [Z: fm,C: rule > fm > set_fm,A: rule,B: fm] :
      ( ( member_fm2 @ Z @ ( C @ A @ B ) )
     => ( member_fm2 @ Z @ ( produc8514419448307895902set_fm @ C @ ( product_Pair_rule_fm @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_31_mem__case__prodI,axiom,
    ! [Z: list_fm,C: rule > fm > set_list_fm,A: rule,B: fm] :
      ( ( member_list_fm2 @ Z @ ( C @ A @ B ) )
     => ( member_list_fm2 @ Z @ ( produc8148429868913231716ist_fm @ C @ ( product_Pair_rule_fm @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_32_mem__case__prodI,axiom,
    ! [Z: produc340336539035504054m_rule,C: rule > fm > set_Pr1822751329126368876m_rule,A: rule,B: fm] :
      ( ( member7231649785386036813m_rule @ Z @ ( C @ A @ B ) )
     => ( member7231649785386036813m_rule @ Z @ ( produc8735870174858355873m_rule @ C @ ( product_Pair_rule_fm @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_33_mem__case__prodI,axiom,
    ! [Z: $o,C: rule > fm > set_o,A: rule,B: fm] :
      ( ( member_o2 @ Z @ ( C @ A @ B ) )
     => ( member_o2 @ Z @ ( produc3402564150753746179_set_o @ C @ ( product_Pair_rule_fm @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_34_mem__case__prodI,axiom,
    ! [Z: fm,C: nat > nat > set_fm,A: nat,B: nat] :
      ( ( member_fm2 @ Z @ ( C @ A @ B ) )
     => ( member_fm2 @ Z @ ( produc2762169475385161349set_fm @ C @ ( product_Pair_nat_nat @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_35_mem__case__prodI,axiom,
    ! [Z: list_fm,C: nat > nat > set_list_fm,A: nat,B: nat] :
      ( ( member_list_fm2 @ Z @ ( C @ A @ B ) )
     => ( member_list_fm2 @ Z @ ( produc8363348377050837003ist_fm @ C @ ( product_Pair_nat_nat @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_36_mem__case__prodI,axiom,
    ! [Z: produc340336539035504054m_rule,C: nat > nat > set_Pr1822751329126368876m_rule,A: nat,B: nat] :
      ( ( member7231649785386036813m_rule @ Z @ ( C @ A @ B ) )
     => ( member7231649785386036813m_rule @ Z @ ( produc6364813917282460346m_rule @ C @ ( product_Pair_nat_nat @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_37_mem__case__prodI,axiom,
    ! [Z: $o,C: nat > nat > set_o,A: nat,B: nat] :
      ( ( member_o2 @ Z @ ( C @ A @ B ) )
     => ( member_o2 @ Z @ ( produc59986286002894506_set_o @ C @ ( product_Pair_nat_nat @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_38_mem__case__prodE,axiom,
    ! [Z: fm,C: rule > fm > set_fm,P: product_prod_rule_fm] :
      ( ( member_fm2 @ Z @ ( produc8514419448307895902set_fm @ C @ P ) )
     => ~ ! [X3: rule,Y3: fm] :
            ( ( P
              = ( product_Pair_rule_fm @ X3 @ Y3 ) )
           => ~ ( member_fm2 @ Z @ ( C @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_39_mem__case__prodE,axiom,
    ! [Z: list_fm,C: rule > fm > set_list_fm,P: product_prod_rule_fm] :
      ( ( member_list_fm2 @ Z @ ( produc8148429868913231716ist_fm @ C @ P ) )
     => ~ ! [X3: rule,Y3: fm] :
            ( ( P
              = ( product_Pair_rule_fm @ X3 @ Y3 ) )
           => ~ ( member_list_fm2 @ Z @ ( C @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_40_mem__case__prodE,axiom,
    ! [Z: produc340336539035504054m_rule,C: rule > fm > set_Pr1822751329126368876m_rule,P: product_prod_rule_fm] :
      ( ( member7231649785386036813m_rule @ Z @ ( produc8735870174858355873m_rule @ C @ P ) )
     => ~ ! [X3: rule,Y3: fm] :
            ( ( P
              = ( product_Pair_rule_fm @ X3 @ Y3 ) )
           => ~ ( member7231649785386036813m_rule @ Z @ ( C @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_41_mem__case__prodE,axiom,
    ! [Z: $o,C: rule > fm > set_o,P: product_prod_rule_fm] :
      ( ( member_o2 @ Z @ ( produc3402564150753746179_set_o @ C @ P ) )
     => ~ ! [X3: rule,Y3: fm] :
            ( ( P
              = ( product_Pair_rule_fm @ X3 @ Y3 ) )
           => ~ ( member_o2 @ Z @ ( C @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_42_mem__case__prodE,axiom,
    ! [Z: fm,C: nat > nat > set_fm,P: product_prod_nat_nat] :
      ( ( member_fm2 @ Z @ ( produc2762169475385161349set_fm @ C @ P ) )
     => ~ ! [X3: nat,Y3: nat] :
            ( ( P
              = ( product_Pair_nat_nat @ X3 @ Y3 ) )
           => ~ ( member_fm2 @ Z @ ( C @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_43_mem__case__prodE,axiom,
    ! [Z: list_fm,C: nat > nat > set_list_fm,P: product_prod_nat_nat] :
      ( ( member_list_fm2 @ Z @ ( produc8363348377050837003ist_fm @ C @ P ) )
     => ~ ! [X3: nat,Y3: nat] :
            ( ( P
              = ( product_Pair_nat_nat @ X3 @ Y3 ) )
           => ~ ( member_list_fm2 @ Z @ ( C @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_44_mem__case__prodE,axiom,
    ! [Z: produc340336539035504054m_rule,C: nat > nat > set_Pr1822751329126368876m_rule,P: product_prod_nat_nat] :
      ( ( member7231649785386036813m_rule @ Z @ ( produc6364813917282460346m_rule @ C @ P ) )
     => ~ ! [X3: nat,Y3: nat] :
            ( ( P
              = ( product_Pair_nat_nat @ X3 @ Y3 ) )
           => ~ ( member7231649785386036813m_rule @ Z @ ( C @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_45_mem__case__prodE,axiom,
    ! [Z: $o,C: nat > nat > set_o,P: product_prod_nat_nat] :
      ( ( member_o2 @ Z @ ( produc59986286002894506_set_o @ C @ P ) )
     => ~ ! [X3: nat,Y3: nat] :
            ( ( P
              = ( product_Pair_nat_nat @ X3 @ Y3 ) )
           => ~ ( member_o2 @ Z @ ( C @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_46_Pair__inject,axiom,
    ! [A: rule,B: fm,A4: rule,B4: fm] :
      ( ( ( product_Pair_rule_fm @ A @ B )
        = ( product_Pair_rule_fm @ A4 @ B4 ) )
     => ~ ( ( A = A4 )
         => ( B != B4 ) ) ) ).

% Pair_inject
thf(fact_47_Pair__inject,axiom,
    ! [A: nat,B: nat,A4: nat,B4: nat] :
      ( ( ( product_Pair_nat_nat @ A @ B )
        = ( product_Pair_nat_nat @ A4 @ B4 ) )
     => ~ ( ( A = A4 )
         => ( B != B4 ) ) ) ).

% Pair_inject
thf(fact_48_prod__cases,axiom,
    ! [P3: product_prod_rule_fm > $o,P: product_prod_rule_fm] :
      ( ! [A2: rule,B2: fm] : ( P3 @ ( product_Pair_rule_fm @ A2 @ B2 ) )
     => ( P3 @ P ) ) ).

% prod_cases
thf(fact_49_prod__cases,axiom,
    ! [P3: product_prod_nat_nat > $o,P: product_prod_nat_nat] :
      ( ! [A2: nat,B2: nat] : ( P3 @ ( product_Pair_nat_nat @ A2 @ B2 ) )
     => ( P3 @ P ) ) ).

% prod_cases
thf(fact_50_surj__pair,axiom,
    ! [P: product_prod_rule_fm] :
    ? [X3: rule,Y3: fm] :
      ( P
      = ( product_Pair_rule_fm @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_51_surj__pair,axiom,
    ! [P: product_prod_nat_nat] :
    ? [X3: nat,Y3: nat] :
      ( P
      = ( product_Pair_nat_nat @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_52_old_Oprod_Oexhaust,axiom,
    ! [Y4: product_prod_rule_fm] :
      ~ ! [A2: rule,B2: fm] :
          ( Y4
         != ( product_Pair_rule_fm @ A2 @ B2 ) ) ).

% old.prod.exhaust
thf(fact_53_old_Oprod_Oexhaust,axiom,
    ! [Y4: product_prod_nat_nat] :
      ~ ! [A2: nat,B2: nat] :
          ( Y4
         != ( product_Pair_nat_nat @ A2 @ B2 ) ) ).

% old.prod.exhaust
thf(fact_54_rule_Ocase__distrib,axiom,
    ! [H: $o > $o,F1: $o,F2: $o,F3: $o,F4: $o,F5: $o,F6: $o,F7: $o,F8: $o,F9: $o,F10: $o,F11: $o,Rule: rule] :
      ( ( H @ ( case_rule_o @ F1 @ F2 @ F3 @ F4 @ F5 @ F6 @ F7 @ F8 @ F9 @ F10 @ F11 @ Rule ) )
      = ( case_rule_o @ ( H @ F1 ) @ ( H @ F2 ) @ ( H @ F3 ) @ ( H @ F4 ) @ ( H @ F5 ) @ ( H @ F6 ) @ ( H @ F7 ) @ ( H @ F8 ) @ ( H @ F9 ) @ ( H @ F10 ) @ ( H @ F11 ) @ Rule ) ) ).

% rule.case_distrib
thf(fact_55_prod_Ocase__distrib,axiom,
    ! [H: $o > $o,F: rule > fm > $o,Prod: product_prod_rule_fm] :
      ( ( H @ ( produc3561889649859641891e_fm_o @ F @ Prod ) )
      = ( produc3561889649859641891e_fm_o
        @ ^ [X12: rule,X22: fm] : ( H @ ( F @ X12 @ X22 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_56_prod_Ocase__distrib,axiom,
    ! [H: $o > $o,F: nat > nat > $o,Prod: product_prod_nat_nat] :
      ( ( H @ ( produc6081775807080527818_nat_o @ F @ Prod ) )
      = ( produc6081775807080527818_nat_o
        @ ^ [X12: nat,X22: nat] : ( H @ ( F @ X12 @ X22 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_57_prod_Odisc__eq__case,axiom,
    ! [Prod: product_prod_rule_fm] :
      ( produc3561889649859641891e_fm_o
      @ ^ [Uu: rule,Uv: fm] : $true
      @ Prod ) ).

% prod.disc_eq_case
thf(fact_58_prod_Odisc__eq__case,axiom,
    ! [Prod: product_prod_nat_nat] :
      ( produc6081775807080527818_nat_o
      @ ^ [Uu: nat,Uv: nat] : $true
      @ Prod ) ).

% prod.disc_eq_case
thf(fact_59_fm_Ocase__distrib,axiom,
    ! [H: $o > $o,F1: nat > list_tm > $o,F2: fm > fm > $o,F3: fm > fm > $o,F4: fm > fm > $o,F5: fm > $o,F6: fm > $o,F7: fm > $o,Fm3: fm] :
      ( ( H @ ( case_fm_o @ F1 @ F2 @ F3 @ F4 @ F5 @ F6 @ F7 @ Fm3 ) )
      = ( case_fm_o
        @ ^ [X12: nat,X22: list_tm] : ( H @ ( F1 @ X12 @ X22 ) )
        @ ^ [X12: fm,X22: fm] : ( H @ ( F2 @ X12 @ X22 ) )
        @ ^ [X12: fm,X22: fm] : ( H @ ( F3 @ X12 @ X22 ) )
        @ ^ [X12: fm,X22: fm] : ( H @ ( F4 @ X12 @ X22 ) )
        @ ^ [X: fm] : ( H @ ( F5 @ X ) )
        @ ^ [X: fm] : ( H @ ( F6 @ X ) )
        @ ^ [X: fm] : ( H @ ( F7 @ X ) )
        @ Fm3 ) ) ).

% fm.case_distrib
thf(fact_60_old_Oprod_Ocase,axiom,
    ! [F: rule > fm > $o,X1: rule,X2: fm] :
      ( ( produc3561889649859641891e_fm_o @ F @ ( product_Pair_rule_fm @ X1 @ X2 ) )
      = ( F @ X1 @ X2 ) ) ).

% old.prod.case
thf(fact_61_old_Oprod_Ocase,axiom,
    ! [F: nat > nat > $o,X1: nat,X2: nat] :
      ( ( produc6081775807080527818_nat_o @ F @ ( product_Pair_nat_nat @ X1 @ X2 ) )
      = ( F @ X1 @ X2 ) ) ).

% old.prod.case
thf(fact_62_fm_Osimps_I53_J,axiom,
    ! [F1: nat > list_tm > $o,F2: fm > fm > $o,F3: fm > fm > $o,F4: fm > fm > $o,F5: fm > $o,F6: fm > $o,F7: fm > $o,X41: fm,X42: fm] :
      ( ( case_fm_o @ F1 @ F2 @ F3 @ F4 @ F5 @ F6 @ F7 @ ( con @ X41 @ X42 ) )
      = ( F4 @ X41 @ X42 ) ) ).

% fm.simps(53)
thf(fact_63_cond__case__prod__eta,axiom,
    ! [F: rule > fm > $o,G: product_prod_rule_fm > $o] :
      ( ! [X3: rule,Y3: fm] :
          ( ( F @ X3 @ Y3 )
          = ( G @ ( product_Pair_rule_fm @ X3 @ Y3 ) ) )
     => ( ( produc3561889649859641891e_fm_o @ F )
        = G ) ) ).

% cond_case_prod_eta
thf(fact_64_cond__case__prod__eta,axiom,
    ! [F: nat > nat > $o,G: product_prod_nat_nat > $o] :
      ( ! [X3: nat,Y3: nat] :
          ( ( F @ X3 @ Y3 )
          = ( G @ ( product_Pair_nat_nat @ X3 @ Y3 ) ) )
     => ( ( produc6081775807080527818_nat_o @ F )
        = G ) ) ).

% cond_case_prod_eta
thf(fact_65_case__prod__eta,axiom,
    ! [F: product_prod_rule_fm > $o] :
      ( ( produc3561889649859641891e_fm_o
        @ ^ [X: rule,Y: fm] : ( F @ ( product_Pair_rule_fm @ X @ Y ) ) )
      = F ) ).

% case_prod_eta
thf(fact_66_case__prod__eta,axiom,
    ! [F: product_prod_nat_nat > $o] :
      ( ( produc6081775807080527818_nat_o
        @ ^ [X: nat,Y: nat] : ( F @ ( product_Pair_nat_nat @ X @ Y ) ) )
      = F ) ).

% case_prod_eta
thf(fact_67_split__cong,axiom,
    ! [Q2: product_prod_rule_fm,F: rule > fm > $o,G: rule > fm > $o,P: product_prod_rule_fm] :
      ( ! [X3: rule,Y3: fm] :
          ( ( ( product_Pair_rule_fm @ X3 @ Y3 )
            = Q2 )
         => ( ( F @ X3 @ Y3 )
            = ( G @ X3 @ Y3 ) ) )
     => ( ( P = Q2 )
       => ( ( produc3561889649859641891e_fm_o @ F @ P )
          = ( produc3561889649859641891e_fm_o @ G @ Q2 ) ) ) ) ).

% split_cong
thf(fact_68_split__cong,axiom,
    ! [Q2: product_prod_nat_nat,F: nat > nat > $o,G: nat > nat > $o,P: product_prod_nat_nat] :
      ( ! [X3: nat,Y3: nat] :
          ( ( ( product_Pair_nat_nat @ X3 @ Y3 )
            = Q2 )
         => ( ( F @ X3 @ Y3 )
            = ( G @ X3 @ Y3 ) ) )
     => ( ( P = Q2 )
       => ( ( produc6081775807080527818_nat_o @ F @ P )
          = ( produc6081775807080527818_nat_o @ G @ Q2 ) ) ) ) ).

% split_cong
thf(fact_69_mem__Collect__eq,axiom,
    ! [A: fm,P3: fm > $o] :
      ( ( member_fm2 @ A @ ( collect_fm @ P3 ) )
      = ( P3 @ A ) ) ).

% mem_Collect_eq
thf(fact_70_mem__Collect__eq,axiom,
    ! [A: list_fm,P3: list_fm > $o] :
      ( ( member_list_fm2 @ A @ ( collect_list_fm @ P3 ) )
      = ( P3 @ A ) ) ).

% mem_Collect_eq
thf(fact_71_mem__Collect__eq,axiom,
    ! [A: produc340336539035504054m_rule,P3: produc340336539035504054m_rule > $o] :
      ( ( member7231649785386036813m_rule @ A @ ( collec1009276759030335883m_rule @ P3 ) )
      = ( P3 @ A ) ) ).

% mem_Collect_eq
thf(fact_72_mem__Collect__eq,axiom,
    ! [A: $o,P3: $o > $o] :
      ( ( member_o2 @ A @ ( collect_o @ P3 ) )
      = ( P3 @ A ) ) ).

% mem_Collect_eq
thf(fact_73_mem__Collect__eq,axiom,
    ! [A: product_prod_nat_nat,P3: product_prod_nat_nat > $o] :
      ( ( member8440522571783428010at_nat @ A @ ( collec3392354462482085612at_nat @ P3 ) )
      = ( P3 @ A ) ) ).

% mem_Collect_eq
thf(fact_74_mem__Collect__eq,axiom,
    ! [A: nat,P3: nat > $o] :
      ( ( member_nat @ A @ ( collect_nat @ P3 ) )
      = ( P3 @ A ) ) ).

% mem_Collect_eq
thf(fact_75_Collect__mem__eq,axiom,
    ! [A5: set_fm] :
      ( ( collect_fm
        @ ^ [X: fm] : ( member_fm2 @ X @ A5 ) )
      = A5 ) ).

% Collect_mem_eq
thf(fact_76_Collect__mem__eq,axiom,
    ! [A5: set_list_fm] :
      ( ( collect_list_fm
        @ ^ [X: list_fm] : ( member_list_fm2 @ X @ A5 ) )
      = A5 ) ).

% Collect_mem_eq
thf(fact_77_Collect__mem__eq,axiom,
    ! [A5: set_Pr1822751329126368876m_rule] :
      ( ( collec1009276759030335883m_rule
        @ ^ [X: produc340336539035504054m_rule] : ( member7231649785386036813m_rule @ X @ A5 ) )
      = A5 ) ).

% Collect_mem_eq
thf(fact_78_Collect__mem__eq,axiom,
    ! [A5: set_o] :
      ( ( collect_o
        @ ^ [X: $o] : ( member_o2 @ X @ A5 ) )
      = A5 ) ).

% Collect_mem_eq
thf(fact_79_Collect__mem__eq,axiom,
    ! [A5: set_Pr1261947904930325089at_nat] :
      ( ( collec3392354462482085612at_nat
        @ ^ [X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ A5 ) )
      = A5 ) ).

% Collect_mem_eq
thf(fact_80_Collect__mem__eq,axiom,
    ! [A5: set_nat] :
      ( ( collect_nat
        @ ^ [X: nat] : ( member_nat @ X @ A5 ) )
      = A5 ) ).

% Collect_mem_eq
thf(fact_81_Collect__cong,axiom,
    ! [P3: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
      ( ! [X3: product_prod_nat_nat] :
          ( ( P3 @ X3 )
          = ( Q @ X3 ) )
     => ( ( collec3392354462482085612at_nat @ P3 )
        = ( collec3392354462482085612at_nat @ Q ) ) ) ).

% Collect_cong
thf(fact_82_Collect__cong,axiom,
    ! [P3: nat > $o,Q: nat > $o] :
      ( ! [X3: nat] :
          ( ( P3 @ X3 )
          = ( Q @ X3 ) )
     => ( ( collect_nat @ P3 )
        = ( collect_nat @ Q ) ) ) ).

% Collect_cong
thf(fact_83_internal__case__prod__def,axiom,
    produc8345629890154448889e_fm_o = produc3561889649859641891e_fm_o ).

% internal_case_prod_def
thf(fact_84_internal__case__prod__def,axiom,
    produc9139807813499881076_nat_o = produc6081775807080527818_nat_o ).

% internal_case_prod_def
thf(fact_85_case__prod__Pair__iden,axiom,
    ! [P: product_prod_rule_fm] :
      ( ( produc3450399560216472459ule_fm @ product_Pair_rule_fm @ P )
      = P ) ).

% case_prod_Pair_iden
thf(fact_86_case__prod__Pair__iden,axiom,
    ! [P: product_prod_nat_nat] :
      ( ( produc2626176000494625587at_nat @ product_Pair_nat_nat @ P )
      = P ) ).

% case_prod_Pair_iden
thf(fact_87_pred__equals__eq2,axiom,
    ! [R2: set_Pr1822751329126368876m_rule,S: set_Pr1822751329126368876m_rule] :
      ( ( ( ^ [X: produc6018962875968178549ist_fm,Y: rule] : ( member7231649785386036813m_rule @ ( produc1733806532565653680m_rule @ X @ Y ) @ R2 ) )
        = ( ^ [X: produc6018962875968178549ist_fm,Y: rule] : ( member7231649785386036813m_rule @ ( produc1733806532565653680m_rule @ X @ Y ) @ S ) ) )
      = ( R2 = S ) ) ).

% pred_equals_eq2
thf(fact_88_pred__equals__eq2,axiom,
    ! [R2: set_Pr8969532344284305932ule_fm,S: set_Pr8969532344284305932ule_fm] :
      ( ( ( ^ [X: rule,Y: fm] : ( member9168179392308330989ule_fm @ ( product_Pair_rule_fm @ X @ Y ) @ R2 ) )
        = ( ^ [X: rule,Y: fm] : ( member9168179392308330989ule_fm @ ( product_Pair_rule_fm @ X @ Y ) @ S ) ) )
      = ( R2 = S ) ) ).

% pred_equals_eq2
thf(fact_89_pred__equals__eq2,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( ( ^ [X: nat,Y: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R2 ) )
        = ( ^ [X: nat,Y: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ S ) ) )
      = ( R2 = S ) ) ).

% pred_equals_eq2
thf(fact_90_same__fst__def,axiom,
    ( same_fst_rule_fm
    = ( ^ [P4: rule > $o,R3: rule > set_Pr4706815898642364871_fm_fm] :
          ( collec4002201796035967736ule_fm
          @ ( produc1764452498099688894e_fm_o
            @ ( produc8261155500375466884e_fm_o
              @ ^ [X4: rule,Y5: fm] :
                  ( produc3561889649859641891e_fm_o
                  @ ^ [X: rule,Y: fm] :
                      ( ( X4 = X )
                      & ( P4 @ X )
                      & ( member7780952600467998736_fm_fm @ ( product_Pair_fm_fm @ Y5 @ Y ) @ ( R3 @ X ) ) ) ) ) ) ) ) ) ).

% same_fst_def
thf(fact_91_same__fst__def,axiom,
    ( same_fst_nat_nat
    = ( ^ [P4: nat > $o,R3: nat > set_Pr1261947904930325089at_nat] :
          ( collec7088162979684241874at_nat
          @ ( produc6590410687421337004_nat_o
            @ ( produc8739625826339149834_nat_o
              @ ^ [X4: nat,Y5: nat] :
                  ( produc6081775807080527818_nat_o
                  @ ^ [X: nat,Y: nat] :
                      ( ( X4 = X )
                      & ( P4 @ X )
                      & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y5 @ Y ) @ ( R3 @ X ) ) ) ) ) ) ) ) ) ).

% same_fst_def
thf(fact_92_curry__case__prod,axiom,
    ! [F: rule > fm > $o] :
      ( ( produc592185131678301890e_fm_o @ ( produc3561889649859641891e_fm_o @ F ) )
      = F ) ).

% curry_case_prod
thf(fact_93_curry__case__prod,axiom,
    ! [F: nat > nat > $o] :
      ( ( produc1310100445399344235_nat_o @ ( produc6081775807080527818_nat_o @ F ) )
      = F ) ).

% curry_case_prod
thf(fact_94_case__prod__curry,axiom,
    ! [F: product_prod_rule_fm > $o] :
      ( ( produc3561889649859641891e_fm_o @ ( produc592185131678301890e_fm_o @ F ) )
      = F ) ).

% case_prod_curry
thf(fact_95_case__prod__curry,axiom,
    ! [F: product_prod_nat_nat > $o] :
      ( ( produc6081775807080527818_nat_o @ ( produc1310100445399344235_nat_o @ F ) )
      = F ) ).

% case_prod_curry
thf(fact_96_The__split__eq,axiom,
    ! [X5: rule,Y4: fm] :
      ( ( the_Pr8424379526464080271ule_fm
        @ ( produc3561889649859641891e_fm_o
          @ ^ [X4: rule,Y5: fm] :
              ( ( X5 = X4 )
              & ( Y4 = Y5 ) ) ) )
      = ( product_Pair_rule_fm @ X5 @ Y4 ) ) ).

% The_split_eq
thf(fact_97_The__split__eq,axiom,
    ! [X5: nat,Y4: nat] :
      ( ( the_Pr7557018466319803784at_nat
        @ ( produc6081775807080527818_nat_o
          @ ^ [X4: nat,Y5: nat] :
              ( ( X5 = X4 )
              & ( Y4 = Y5 ) ) ) )
      = ( product_Pair_nat_nat @ X5 @ Y4 ) ) ).

% The_split_eq
thf(fact_98_ssubst__Pair__rhs,axiom,
    ! [R4: produc6018962875968178549ist_fm,S2: rule,R2: set_Pr1822751329126368876m_rule,S3: rule] :
      ( ( member7231649785386036813m_rule @ ( produc1733806532565653680m_rule @ R4 @ S2 ) @ R2 )
     => ( ( S3 = S2 )
       => ( member7231649785386036813m_rule @ ( produc1733806532565653680m_rule @ R4 @ S3 ) @ R2 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_99_ssubst__Pair__rhs,axiom,
    ! [R4: rule,S2: fm,R2: set_Pr8969532344284305932ule_fm,S3: fm] :
      ( ( member9168179392308330989ule_fm @ ( product_Pair_rule_fm @ R4 @ S2 ) @ R2 )
     => ( ( S3 = S2 )
       => ( member9168179392308330989ule_fm @ ( product_Pair_rule_fm @ R4 @ S3 ) @ R2 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_100_ssubst__Pair__rhs,axiom,
    ! [R4: nat,S2: nat,R2: set_Pr1261947904930325089at_nat,S3: nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ R4 @ S2 ) @ R2 )
     => ( ( S3 = S2 )
       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ R4 @ S3 ) @ R2 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_101_curryI,axiom,
    ! [F: product_prod_rule_fm > $o,A: rule,B: fm] :
      ( ( F @ ( product_Pair_rule_fm @ A @ B ) )
     => ( produc592185131678301890e_fm_o @ F @ A @ B ) ) ).

% curryI
thf(fact_102_curryI,axiom,
    ! [F: product_prod_nat_nat > $o,A: nat,B: nat] :
      ( ( F @ ( product_Pair_nat_nat @ A @ B ) )
     => ( produc1310100445399344235_nat_o @ F @ A @ B ) ) ).

% curryI
thf(fact_103_same__fstI,axiom,
    ! [P3: rule > $o,X5: rule,Y6: fm,Y4: fm,R2: rule > set_Pr4706815898642364871_fm_fm] :
      ( ( P3 @ X5 )
     => ( ( member7780952600467998736_fm_fm @ ( product_Pair_fm_fm @ Y6 @ Y4 ) @ ( R2 @ X5 ) )
       => ( member6853779175081790390ule_fm @ ( produc3578872663747121477ule_fm @ ( product_Pair_rule_fm @ X5 @ Y6 ) @ ( product_Pair_rule_fm @ X5 @ Y4 ) ) @ ( same_fst_rule_fm @ P3 @ R2 ) ) ) ) ).

% same_fstI
thf(fact_104_same__fstI,axiom,
    ! [P3: nat > $o,X5: nat,Y6: nat,Y4: nat,R2: nat > set_Pr1261947904930325089at_nat] :
      ( ( P3 @ X5 )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y6 @ Y4 ) @ ( R2 @ X5 ) )
       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ X5 @ Y6 ) @ ( product_Pair_nat_nat @ X5 @ Y4 ) ) @ ( same_fst_nat_nat @ P3 @ R2 ) ) ) ) ).

% same_fstI
thf(fact_105_curryD,axiom,
    ! [F: product_prod_rule_fm > $o,A: rule,B: fm] :
      ( ( produc592185131678301890e_fm_o @ F @ A @ B )
     => ( F @ ( product_Pair_rule_fm @ A @ B ) ) ) ).

% curryD
thf(fact_106_curryD,axiom,
    ! [F: product_prod_nat_nat > $o,A: nat,B: nat] :
      ( ( produc1310100445399344235_nat_o @ F @ A @ B )
     => ( F @ ( product_Pair_nat_nat @ A @ B ) ) ) ).

% curryD
thf(fact_107_curryE,axiom,
    ! [F: product_prod_rule_fm > $o,A: rule,B: fm] :
      ( ( produc592185131678301890e_fm_o @ F @ A @ B )
     => ( F @ ( product_Pair_rule_fm @ A @ B ) ) ) ).

% curryE
thf(fact_108_curryE,axiom,
    ! [F: product_prod_nat_nat > $o,A: nat,B: nat] :
      ( ( produc1310100445399344235_nat_o @ F @ A @ B )
     => ( F @ ( product_Pair_nat_nat @ A @ B ) ) ) ).

% curryE
thf(fact_109_lex__prod__def,axiom,
    ( lex_prod_rule_fm
    = ( ^ [Ra: set_Pr7340090144965549541e_rule,Rb: set_Pr4706815898642364871_fm_fm] :
          ( collec4002201796035967736ule_fm
          @ ( produc1764452498099688894e_fm_o
            @ ( produc8261155500375466884e_fm_o
              @ ^ [A3: rule,B3: fm] :
                  ( produc3561889649859641891e_fm_o
                  @ ^ [A6: rule,B5: fm] :
                      ( ( member1444691403335511342e_rule @ ( produc5849431337705160893e_rule @ A3 @ A6 ) @ Ra )
                      | ( ( A3 = A6 )
                        & ( member7780952600467998736_fm_fm @ ( product_Pair_fm_fm @ B3 @ B5 ) @ Rb ) ) ) ) ) ) ) ) ) ).

% lex_prod_def
thf(fact_110_lex__prod__def,axiom,
    ( lex_prod_nat_nat
    = ( ^ [Ra: set_Pr1261947904930325089at_nat,Rb: set_Pr1261947904930325089at_nat] :
          ( collec7088162979684241874at_nat
          @ ( produc6590410687421337004_nat_o
            @ ( produc8739625826339149834_nat_o
              @ ^ [A3: nat,B3: nat] :
                  ( produc6081775807080527818_nat_o
                  @ ^ [A6: nat,B5: nat] :
                      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A3 @ A6 ) @ Ra )
                      | ( ( A3 = A6 )
                        & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B3 @ B5 ) @ Rb ) ) ) ) ) ) ) ) ) ).

% lex_prod_def
thf(fact_111_in__lex__prod,axiom,
    ! [A: rule,B: fm,A4: rule,B4: fm,R4: set_Pr7340090144965549541e_rule,S2: set_Pr4706815898642364871_fm_fm] :
      ( ( member6853779175081790390ule_fm @ ( produc3578872663747121477ule_fm @ ( product_Pair_rule_fm @ A @ B ) @ ( product_Pair_rule_fm @ A4 @ B4 ) ) @ ( lex_prod_rule_fm @ R4 @ S2 ) )
      = ( ( member1444691403335511342e_rule @ ( produc5849431337705160893e_rule @ A @ A4 ) @ R4 )
        | ( ( A = A4 )
          & ( member7780952600467998736_fm_fm @ ( product_Pair_fm_fm @ B @ B4 ) @ S2 ) ) ) ) ).

% in_lex_prod
thf(fact_112_in__lex__prod,axiom,
    ! [A: nat,B: nat,A4: nat,B4: nat,R4: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( product_Pair_nat_nat @ A4 @ B4 ) ) @ ( lex_prod_nat_nat @ R4 @ S2 ) )
      = ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ A4 ) @ R4 )
        | ( ( A = A4 )
          & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ B4 ) @ S2 ) ) ) ) ).

% in_lex_prod
thf(fact_113_inv__image__def,axiom,
    ( inv_image_nat_nat
    = ( ^ [R: set_Pr1261947904930325089at_nat,F12: nat > nat] :
          ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ^ [X: nat,Y: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F12 @ X ) @ ( F12 @ Y ) ) @ R ) ) ) ) ) ).

% inv_image_def
thf(fact_114_case__swap,axiom,
    ! [F: rule > fm > $o,P: product_prod_rule_fm] :
      ( ( produc8778389512312809213rule_o
        @ ^ [Y: fm,X: rule] : ( F @ X @ Y )
        @ ( product_swap_rule_fm @ P ) )
      = ( produc3561889649859641891e_fm_o @ F @ P ) ) ).

% case_swap
thf(fact_115_case__swap,axiom,
    ! [F: fm > rule > $o,P: product_prod_fm_rule] :
      ( ( produc3561889649859641891e_fm_o
        @ ^ [Y: rule,X: fm] : ( F @ X @ Y )
        @ ( product_swap_fm_rule @ P ) )
      = ( produc8778389512312809213rule_o @ F @ P ) ) ).

% case_swap
thf(fact_116_case__swap,axiom,
    ! [F: nat > nat > $o,P: product_prod_nat_nat] :
      ( ( produc6081775807080527818_nat_o
        @ ^ [Y: nat,X: nat] : ( F @ X @ Y )
        @ ( product_swap_nat_nat @ P ) )
      = ( produc6081775807080527818_nat_o @ F @ P ) ) ).

% case_swap
thf(fact_117_pair__imageI,axiom,
    ! [A: rule,B: fm,A5: set_Pr8969532344284305932ule_fm,F: rule > fm > fm] :
      ( ( member9168179392308330989ule_fm @ ( product_Pair_rule_fm @ A @ B ) @ A5 )
     => ( member_fm2 @ ( F @ A @ B ) @ ( image_4470303875010677334_fm_fm @ ( produc3754595766265636222_fm_fm @ F ) @ A5 ) ) ) ).

% pair_imageI
thf(fact_118_pair__imageI,axiom,
    ! [A: nat,B: nat,A5: set_Pr1261947904930325089at_nat,F: nat > nat > fm] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ A5 )
     => ( member_fm2 @ ( F @ A @ B ) @ ( image_952686206022208695nat_fm @ ( produc7024581997620079013nat_fm @ F ) @ A5 ) ) ) ).

% pair_imageI
thf(fact_119_pair__imageI,axiom,
    ! [A: rule,B: fm,A5: set_Pr8969532344284305932ule_fm,F: rule > fm > $o] :
      ( ( member9168179392308330989ule_fm @ ( product_Pair_rule_fm @ A @ B ) @ A5 )
     => ( member_o2 @ ( F @ A @ B ) @ ( image_5123236402052875003e_fm_o @ ( produc3561889649859641891e_fm_o @ F ) @ A5 ) ) ) ).

% pair_imageI
thf(fact_120_pair__imageI,axiom,
    ! [A: nat,B: nat,A5: set_Pr1261947904930325089at_nat,F: nat > nat > $o] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ A5 )
     => ( member_o2 @ ( F @ A @ B ) @ ( image_3693632289388996572_nat_o @ ( produc6081775807080527818_nat_o @ F ) @ A5 ) ) ) ).

% pair_imageI
thf(fact_121_pair__imageI,axiom,
    ! [A: rule,B: fm,A5: set_Pr8969532344284305932ule_fm,F: rule > fm > list_fm] :
      ( ( member9168179392308330989ule_fm @ ( product_Pair_rule_fm @ A @ B ) @ A5 )
     => ( member_list_fm2 @ ( F @ A @ B ) @ ( image_5033969083099691228ist_fm @ ( produc5377714713960095748ist_fm @ F ) @ A5 ) ) ) ).

% pair_imageI
thf(fact_122_pair__imageI,axiom,
    ! [A: nat,B: nat,A5: set_Pr1261947904930325089at_nat,F: nat > nat > list_fm] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ A5 )
     => ( member_list_fm2 @ ( F @ A @ B ) @ ( image_7662018698615822013ist_fm @ ( produc4460857912617378987ist_fm @ F ) @ A5 ) ) ) ).

% pair_imageI
thf(fact_123_pair__imageI,axiom,
    ! [A: produc6018962875968178549ist_fm,B: rule,A5: set_Pr1822751329126368876m_rule,F: produc6018962875968178549ist_fm > rule > fm] :
      ( ( member7231649785386036813m_rule @ ( produc1733806532565653680m_rule @ A @ B ) @ A5 )
     => ( member_fm2 @ ( F @ A @ B ) @ ( image_2303374963288105462ule_fm @ ( produc3408830948553821398ule_fm @ F ) @ A5 ) ) ) ).

% pair_imageI
thf(fact_124_pair__imageI,axiom,
    ! [A: produc6018962875968178549ist_fm,B: rule,A5: set_Pr1822751329126368876m_rule,F: produc6018962875968178549ist_fm > rule > $o] :
      ( ( member7231649785386036813m_rule @ ( produc1733806532565653680m_rule @ A @ B ) @ A5 )
     => ( member_o2 @ ( F @ A @ B ) @ ( image_8969478372357302939rule_o @ ( produc1247074552912856443rule_o @ F ) @ A5 ) ) ) ).

% pair_imageI
thf(fact_125_pair__imageI,axiom,
    ! [A: produc6018962875968178549ist_fm,B: rule,A5: set_Pr1822751329126368876m_rule,F: produc6018962875968178549ist_fm > rule > set_fm] :
      ( ( member7231649785386036813m_rule @ ( produc1733806532565653680m_rule @ A @ B ) @ A5 )
     => ( member_set_fm @ ( F @ A @ B ) @ ( image_377115208147135190set_fm @ ( produc271619708354958774set_fm @ F ) @ A5 ) ) ) ).

% pair_imageI
thf(fact_126_pair__imageI,axiom,
    ! [A: produc6018962875968178549ist_fm,B: rule,A5: set_Pr1822751329126368876m_rule,F: produc6018962875968178549ist_fm > rule > list_fm] :
      ( ( member7231649785386036813m_rule @ ( produc1733806532565653680m_rule @ A @ B ) @ A5 )
     => ( member_list_fm2 @ ( F @ A @ B ) @ ( image_1907216182867722364ist_fm @ ( produc1426620797128653660ist_fm @ F ) @ A5 ) ) ) ).

% pair_imageI
thf(fact_127_n,axiom,
    member_fm2 @ ( con @ p @ q ) @ ( set_fm2 @ ( pseq @ ( shd_Pr4562317740776619530m_rule @ ( sdrop_8169176516188972301m_rule @ n @ steps ) ) ) ) ).

% n
thf(fact_128__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062n_O_ACon_Ap_Aq_A_092_060in_062_Aset_A_Ipseq_A_Ishd_A_Isdrop_An_Asteps_J_J_J_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
    ~ ! [N: nat] :
        ~ ( member_fm2 @ ( con @ p @ q ) @ ( set_fm2 @ ( pseq @ ( shd_Pr4562317740776619530m_rule @ ( sdrop_8169176516188972301m_rule @ N @ steps ) ) ) ) ) ).

% \<open>\<And>thesis. (\<And>n. Con p q \<in> set (pseq (shd (sdrop n steps))) \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_129_pair__in__swap__image,axiom,
    ! [Y4: rule,X5: produc6018962875968178549ist_fm,A5: set_Pr1822751329126368876m_rule] :
      ( ( member21886688540021743ist_fm @ ( produc4260130291714670858ist_fm @ Y4 @ X5 ) @ ( image_4512606224792521493ist_fm @ produc3121145899861168912m_rule @ A5 ) )
      = ( member7231649785386036813m_rule @ ( produc1733806532565653680m_rule @ X5 @ Y4 ) @ A5 ) ) ).

% pair_in_swap_image
thf(fact_130_pair__in__swap__image,axiom,
    ! [Y4: produc6018962875968178549ist_fm,X5: rule,A5: set_Pr3602992621978123022ist_fm] :
      ( ( member7231649785386036813m_rule @ ( produc1733806532565653680m_rule @ Y4 @ X5 ) @ ( image_1469007562577134357m_rule @ produc5647469659010186090ist_fm @ A5 ) )
      = ( member21886688540021743ist_fm @ ( produc4260130291714670858ist_fm @ X5 @ Y4 ) @ A5 ) ) ).

% pair_in_swap_image
thf(fact_131_pair__in__swap__image,axiom,
    ! [Y4: fm,X5: rule,A5: set_Pr8969532344284305932ule_fm] :
      ( ( member6041693456836784459m_rule @ ( product_Pair_fm_rule @ Y4 @ X5 ) @ ( image_7705133132229287953m_rule @ product_swap_rule_fm @ A5 ) )
      = ( member9168179392308330989ule_fm @ ( product_Pair_rule_fm @ X5 @ Y4 ) @ A5 ) ) ).

% pair_in_swap_image
thf(fact_132_pair__in__swap__image,axiom,
    ! [Y4: rule,X5: fm,A5: set_Pr7067330020180114026m_rule] :
      ( ( member9168179392308330989ule_fm @ ( product_Pair_rule_fm @ Y4 @ X5 ) @ ( image_2394372996319723537ule_fm @ product_swap_fm_rule @ A5 ) )
      = ( member6041693456836784459m_rule @ ( product_Pair_fm_rule @ X5 @ Y4 ) @ A5 ) ) ).

% pair_in_swap_image
thf(fact_133_pair__in__swap__image,axiom,
    ! [Y4: nat,X5: nat,A5: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y4 @ X5 ) @ ( image_5168914502847457605at_nat @ product_swap_nat_nat @ A5 ) )
      = ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X5 @ Y4 ) @ A5 ) ) ).

% pair_in_swap_image
thf(fact_134_swap__simp,axiom,
    ! [X5: fm,Y4: rule] :
      ( ( product_swap_fm_rule @ ( product_Pair_fm_rule @ X5 @ Y4 ) )
      = ( product_Pair_rule_fm @ Y4 @ X5 ) ) ).

% swap_simp
thf(fact_135_swap__simp,axiom,
    ! [X5: rule,Y4: fm] :
      ( ( product_swap_rule_fm @ ( product_Pair_rule_fm @ X5 @ Y4 ) )
      = ( product_Pair_fm_rule @ Y4 @ X5 ) ) ).

% swap_simp
thf(fact_136_swap__simp,axiom,
    ! [X5: nat,Y4: nat] :
      ( ( product_swap_nat_nat @ ( product_Pair_nat_nat @ X5 @ Y4 ) )
      = ( product_Pair_nat_nat @ Y4 @ X5 ) ) ).

% swap_simp
thf(fact_137_in__inv__image,axiom,
    ! [X5: nat,Y4: nat,R4: set_Pr1261947904930325089at_nat,F: nat > nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X5 @ Y4 ) @ ( inv_image_nat_nat @ R4 @ F ) )
      = ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) @ R4 ) ) ).

% in_inv_image
thf(fact_138_ori,axiom,
    ( ( sdrop_8169176516188972301m_rule @ n @ steps )
    = ( shift_2334844276748245581m_rule @ pre @ suf ) ) ).

% ori
thf(fact_139_image__eqI,axiom,
    ! [B: fm,F: fm > fm,X5: fm,A5: set_fm] :
      ( ( B
        = ( F @ X5 ) )
     => ( ( member_fm2 @ X5 @ A5 )
       => ( member_fm2 @ B @ ( image_fm_fm2 @ F @ A5 ) ) ) ) ).

% image_eqI
thf(fact_140_image__eqI,axiom,
    ! [B: $o,F: fm > $o,X5: fm,A5: set_fm] :
      ( ( B
        = ( F @ X5 ) )
     => ( ( member_fm2 @ X5 @ A5 )
       => ( member_o2 @ B @ ( image_fm_o2 @ F @ A5 ) ) ) ) ).

% image_eqI
thf(fact_141_image__eqI,axiom,
    ! [B: fm,F: $o > fm,X5: $o,A5: set_o] :
      ( ( B
        = ( F @ X5 ) )
     => ( ( member_o2 @ X5 @ A5 )
       => ( member_fm2 @ B @ ( image_o_fm2 @ F @ A5 ) ) ) ) ).

% image_eqI
thf(fact_142_image__eqI,axiom,
    ! [B: $o,F: $o > $o,X5: $o,A5: set_o] :
      ( ( B
        = ( F @ X5 ) )
     => ( ( member_o2 @ X5 @ A5 )
       => ( member_o2 @ B @ ( image_o_o2 @ F @ A5 ) ) ) ) ).

% image_eqI
thf(fact_143_image__eqI,axiom,
    ! [B: list_fm,F: fm > list_fm,X5: fm,A5: set_fm] :
      ( ( B
        = ( F @ X5 ) )
     => ( ( member_fm2 @ X5 @ A5 )
       => ( member_list_fm2 @ B @ ( image_fm_list_fm2 @ F @ A5 ) ) ) ) ).

% image_eqI
thf(fact_144_image__eqI,axiom,
    ! [B: fm,F: list_fm > fm,X5: list_fm,A5: set_list_fm] :
      ( ( B
        = ( F @ X5 ) )
     => ( ( member_list_fm2 @ X5 @ A5 )
       => ( member_fm2 @ B @ ( image_list_fm_fm2 @ F @ A5 ) ) ) ) ).

% image_eqI
thf(fact_145_image__eqI,axiom,
    ! [B: $o,F: list_fm > $o,X5: list_fm,A5: set_list_fm] :
      ( ( B
        = ( F @ X5 ) )
     => ( ( member_list_fm2 @ X5 @ A5 )
       => ( member_o2 @ B @ ( image_list_fm_o2 @ F @ A5 ) ) ) ) ).

% image_eqI
thf(fact_146_image__eqI,axiom,
    ! [B: list_fm,F: $o > list_fm,X5: $o,A5: set_o] :
      ( ( B
        = ( F @ X5 ) )
     => ( ( member_o2 @ X5 @ A5 )
       => ( member_list_fm2 @ B @ ( image_o_list_fm2 @ F @ A5 ) ) ) ) ).

% image_eqI
thf(fact_147_image__eqI,axiom,
    ! [B: list_fm,F: list_fm > list_fm,X5: list_fm,A5: set_list_fm] :
      ( ( B
        = ( F @ X5 ) )
     => ( ( member_list_fm2 @ X5 @ A5 )
       => ( member_list_fm2 @ B @ ( image_9148017957442633541ist_fm @ F @ A5 ) ) ) ) ).

% image_eqI
thf(fact_148_image__eqI,axiom,
    ! [B: produc340336539035504054m_rule,F: fm > produc340336539035504054m_rule,X5: fm,A5: set_fm] :
      ( ( B
        = ( F @ X5 ) )
     => ( ( member_fm2 @ X5 @ A5 )
       => ( member7231649785386036813m_rule @ B @ ( image_6079799016930204900m_rule @ F @ A5 ) ) ) ) ).

% image_eqI
thf(fact_149_member,axiom,
    ( member4200870186495857963m_rule
    = ( ^ [P2: produc340336539035504054m_rule,Z2: list_P2774625669004309958m_rule] : ( member7231649785386036813m_rule @ P2 @ ( set_Pr4534715572506550497m_rule @ Z2 ) ) ) ) ).

% member
thf(fact_150_member,axiom,
    ( member_o
    = ( ^ [P2: $o,Z2: list_o] : ( member_o2 @ P2 @ ( set_o2 @ Z2 ) ) ) ) ).

% member
thf(fact_151_member,axiom,
    ( member_fm
    = ( ^ [P2: fm,Z2: list_fm] : ( member_fm2 @ P2 @ ( set_fm2 @ Z2 ) ) ) ) ).

% member
thf(fact_152_member,axiom,
    ( member_list_fm
    = ( ^ [P2: list_fm,Z2: list_list_fm] : ( member_list_fm2 @ P2 @ ( set_list_fm2 @ Z2 ) ) ) ) ).

% member
thf(fact_153_member,axiom,
    ( member_tm
    = ( ^ [P2: tm,Z2: list_tm] : ( member_tm2 @ P2 @ ( set_tm2 @ Z2 ) ) ) ) ).

% member
thf(fact_154_Compr__image__eq,axiom,
    ! [F: fm > fm,A5: set_fm,P3: fm > $o] :
      ( ( collect_fm
        @ ^ [X: fm] :
            ( ( member_fm2 @ X @ ( image_fm_fm2 @ F @ A5 ) )
            & ( P3 @ X ) ) )
      = ( image_fm_fm2 @ F
        @ ( collect_fm
          @ ^ [X: fm] :
              ( ( member_fm2 @ X @ A5 )
              & ( P3 @ ( F @ X ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_155_Compr__image__eq,axiom,
    ! [F: $o > fm,A5: set_o,P3: fm > $o] :
      ( ( collect_fm
        @ ^ [X: fm] :
            ( ( member_fm2 @ X @ ( image_o_fm2 @ F @ A5 ) )
            & ( P3 @ X ) ) )
      = ( image_o_fm2 @ F
        @ ( collect_o
          @ ^ [X: $o] :
              ( ( member_o2 @ X @ A5 )
              & ( P3 @ ( F @ X ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_156_Compr__image__eq,axiom,
    ! [F: fm > $o,A5: set_fm,P3: $o > $o] :
      ( ( collect_o
        @ ^ [X: $o] :
            ( ( member_o2 @ X @ ( image_fm_o2 @ F @ A5 ) )
            & ( P3 @ X ) ) )
      = ( image_fm_o2 @ F
        @ ( collect_fm
          @ ^ [X: fm] :
              ( ( member_fm2 @ X @ A5 )
              & ( P3 @ ( F @ X ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_157_Compr__image__eq,axiom,
    ! [F: $o > $o,A5: set_o,P3: $o > $o] :
      ( ( collect_o
        @ ^ [X: $o] :
            ( ( member_o2 @ X @ ( image_o_o2 @ F @ A5 ) )
            & ( P3 @ X ) ) )
      = ( image_o_o2 @ F
        @ ( collect_o
          @ ^ [X: $o] :
              ( ( member_o2 @ X @ A5 )
              & ( P3 @ ( F @ X ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_158_Compr__image__eq,axiom,
    ! [F: nat > fm,A5: set_nat,P3: fm > $o] :
      ( ( collect_fm
        @ ^ [X: fm] :
            ( ( member_fm2 @ X @ ( image_nat_fm @ F @ A5 ) )
            & ( P3 @ X ) ) )
      = ( image_nat_fm @ F
        @ ( collect_nat
          @ ^ [X: nat] :
              ( ( member_nat @ X @ A5 )
              & ( P3 @ ( F @ X ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_159_Compr__image__eq,axiom,
    ! [F: nat > $o,A5: set_nat,P3: $o > $o] :
      ( ( collect_o
        @ ^ [X: $o] :
            ( ( member_o2 @ X @ ( image_nat_o @ F @ A5 ) )
            & ( P3 @ X ) ) )
      = ( image_nat_o @ F
        @ ( collect_nat
          @ ^ [X: nat] :
              ( ( member_nat @ X @ A5 )
              & ( P3 @ ( F @ X ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_160_Compr__image__eq,axiom,
    ! [F: fm > nat,A5: set_fm,P3: nat > $o] :
      ( ( collect_nat
        @ ^ [X: nat] :
            ( ( member_nat @ X @ ( image_fm_nat @ F @ A5 ) )
            & ( P3 @ X ) ) )
      = ( image_fm_nat @ F
        @ ( collect_fm
          @ ^ [X: fm] :
              ( ( member_fm2 @ X @ A5 )
              & ( P3 @ ( F @ X ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_161_Compr__image__eq,axiom,
    ! [F: $o > nat,A5: set_o,P3: nat > $o] :
      ( ( collect_nat
        @ ^ [X: nat] :
            ( ( member_nat @ X @ ( image_o_nat2 @ F @ A5 ) )
            & ( P3 @ X ) ) )
      = ( image_o_nat2 @ F
        @ ( collect_o
          @ ^ [X: $o] :
              ( ( member_o2 @ X @ A5 )
              & ( P3 @ ( F @ X ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_162_Compr__image__eq,axiom,
    ! [F: nat > nat,A5: set_nat,P3: nat > $o] :
      ( ( collect_nat
        @ ^ [X: nat] :
            ( ( member_nat @ X @ ( image_nat_nat2 @ F @ A5 ) )
            & ( P3 @ X ) ) )
      = ( image_nat_nat2 @ F
        @ ( collect_nat
          @ ^ [X: nat] :
              ( ( member_nat @ X @ A5 )
              & ( P3 @ ( F @ X ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_163_Compr__image__eq,axiom,
    ! [F: list_fm > fm,A5: set_list_fm,P3: fm > $o] :
      ( ( collect_fm
        @ ^ [X: fm] :
            ( ( member_fm2 @ X @ ( image_list_fm_fm2 @ F @ A5 ) )
            & ( P3 @ X ) ) )
      = ( image_list_fm_fm2 @ F
        @ ( collect_list_fm
          @ ^ [X: list_fm] :
              ( ( member_list_fm2 @ X @ A5 )
              & ( P3 @ ( F @ X ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_164_image__image,axiom,
    ! [F: set_fm > set_fm,G: produc340336539035504054m_rule > set_fm,A5: set_Pr1822751329126368876m_rule] :
      ( ( image_set_fm_set_fm @ F @ ( image_377115208147135190set_fm @ G @ A5 ) )
      = ( image_377115208147135190set_fm
        @ ^ [X: produc340336539035504054m_rule] : ( F @ ( G @ X ) )
        @ A5 ) ) ).

% image_image
thf(fact_165_image__image,axiom,
    ! [F: produc340336539035504054m_rule > set_fm,G: produc340336539035504054m_rule > produc340336539035504054m_rule,A5: set_Pr1822751329126368876m_rule] :
      ( ( image_377115208147135190set_fm @ F @ ( image_2498997284783760755m_rule @ G @ A5 ) )
      = ( image_377115208147135190set_fm
        @ ^ [X: produc340336539035504054m_rule] : ( F @ ( G @ X ) )
        @ A5 ) ) ).

% image_image
thf(fact_166_imageE,axiom,
    ! [B: fm,F: fm > fm,A5: set_fm] :
      ( ( member_fm2 @ B @ ( image_fm_fm2 @ F @ A5 ) )
     => ~ ! [X3: fm] :
            ( ( B
              = ( F @ X3 ) )
           => ~ ( member_fm2 @ X3 @ A5 ) ) ) ).

% imageE
thf(fact_167_imageE,axiom,
    ! [B: fm,F: $o > fm,A5: set_o] :
      ( ( member_fm2 @ B @ ( image_o_fm2 @ F @ A5 ) )
     => ~ ! [X3: $o] :
            ( ( B
              = ( F @ X3 ) )
           => ~ ( member_o2 @ X3 @ A5 ) ) ) ).

% imageE
thf(fact_168_imageE,axiom,
    ! [B: $o,F: fm > $o,A5: set_fm] :
      ( ( member_o2 @ B @ ( image_fm_o2 @ F @ A5 ) )
     => ~ ! [X3: fm] :
            ( ( B
              = ( F @ X3 ) )
           => ~ ( member_fm2 @ X3 @ A5 ) ) ) ).

% imageE
thf(fact_169_imageE,axiom,
    ! [B: $o,F: $o > $o,A5: set_o] :
      ( ( member_o2 @ B @ ( image_o_o2 @ F @ A5 ) )
     => ~ ! [X3: $o] :
            ( ( B
              = ( F @ X3 ) )
           => ~ ( member_o2 @ X3 @ A5 ) ) ) ).

% imageE
thf(fact_170_imageE,axiom,
    ! [B: fm,F: list_fm > fm,A5: set_list_fm] :
      ( ( member_fm2 @ B @ ( image_list_fm_fm2 @ F @ A5 ) )
     => ~ ! [X3: list_fm] :
            ( ( B
              = ( F @ X3 ) )
           => ~ ( member_list_fm2 @ X3 @ A5 ) ) ) ).

% imageE
thf(fact_171_imageE,axiom,
    ! [B: list_fm,F: fm > list_fm,A5: set_fm] :
      ( ( member_list_fm2 @ B @ ( image_fm_list_fm2 @ F @ A5 ) )
     => ~ ! [X3: fm] :
            ( ( B
              = ( F @ X3 ) )
           => ~ ( member_fm2 @ X3 @ A5 ) ) ) ).

% imageE
thf(fact_172_imageE,axiom,
    ! [B: list_fm,F: $o > list_fm,A5: set_o] :
      ( ( member_list_fm2 @ B @ ( image_o_list_fm2 @ F @ A5 ) )
     => ~ ! [X3: $o] :
            ( ( B
              = ( F @ X3 ) )
           => ~ ( member_o2 @ X3 @ A5 ) ) ) ).

% imageE
thf(fact_173_imageE,axiom,
    ! [B: $o,F: list_fm > $o,A5: set_list_fm] :
      ( ( member_o2 @ B @ ( image_list_fm_o2 @ F @ A5 ) )
     => ~ ! [X3: list_fm] :
            ( ( B
              = ( F @ X3 ) )
           => ~ ( member_list_fm2 @ X3 @ A5 ) ) ) ).

% imageE
thf(fact_174_imageE,axiom,
    ! [B: list_fm,F: list_fm > list_fm,A5: set_list_fm] :
      ( ( member_list_fm2 @ B @ ( image_9148017957442633541ist_fm @ F @ A5 ) )
     => ~ ! [X3: list_fm] :
            ( ( B
              = ( F @ X3 ) )
           => ~ ( member_list_fm2 @ X3 @ A5 ) ) ) ).

% imageE
thf(fact_175_imageE,axiom,
    ! [B: fm,F: produc340336539035504054m_rule > fm,A5: set_Pr1822751329126368876m_rule] :
      ( ( member_fm2 @ B @ ( image_2303374963288105462ule_fm @ F @ A5 ) )
     => ~ ! [X3: produc340336539035504054m_rule] :
            ( ( B
              = ( F @ X3 ) )
           => ~ ( member7231649785386036813m_rule @ X3 @ A5 ) ) ) ).

% imageE
thf(fact_176_Inf_OINF__cong,axiom,
    ! [A5: set_Pr1822751329126368876m_rule,B6: set_Pr1822751329126368876m_rule,C2: produc340336539035504054m_rule > set_fm,D: produc340336539035504054m_rule > set_fm,Inf: set_set_fm > set_fm] :
      ( ( A5 = B6 )
     => ( ! [X3: produc340336539035504054m_rule] :
            ( ( member7231649785386036813m_rule @ X3 @ B6 )
           => ( ( C2 @ X3 )
              = ( D @ X3 ) ) )
       => ( ( Inf @ ( image_377115208147135190set_fm @ C2 @ A5 ) )
          = ( Inf @ ( image_377115208147135190set_fm @ D @ B6 ) ) ) ) ) ).

% Inf.INF_cong
thf(fact_177_Sup_OSUP__cong,axiom,
    ! [A5: set_Pr1822751329126368876m_rule,B6: set_Pr1822751329126368876m_rule,C2: produc340336539035504054m_rule > set_fm,D: produc340336539035504054m_rule > set_fm,Sup: set_set_fm > set_fm] :
      ( ( A5 = B6 )
     => ( ! [X3: produc340336539035504054m_rule] :
            ( ( member7231649785386036813m_rule @ X3 @ B6 )
           => ( ( C2 @ X3 )
              = ( D @ X3 ) ) )
       => ( ( Sup @ ( image_377115208147135190set_fm @ C2 @ A5 ) )
          = ( Sup @ ( image_377115208147135190set_fm @ D @ B6 ) ) ) ) ) ).

% Sup.SUP_cong
thf(fact_178_imageI,axiom,
    ! [X5: fm,A5: set_fm,F: fm > fm] :
      ( ( member_fm2 @ X5 @ A5 )
     => ( member_fm2 @ ( F @ X5 ) @ ( image_fm_fm2 @ F @ A5 ) ) ) ).

% imageI
thf(fact_179_imageI,axiom,
    ! [X5: fm,A5: set_fm,F: fm > $o] :
      ( ( member_fm2 @ X5 @ A5 )
     => ( member_o2 @ ( F @ X5 ) @ ( image_fm_o2 @ F @ A5 ) ) ) ).

% imageI
thf(fact_180_imageI,axiom,
    ! [X5: $o,A5: set_o,F: $o > fm] :
      ( ( member_o2 @ X5 @ A5 )
     => ( member_fm2 @ ( F @ X5 ) @ ( image_o_fm2 @ F @ A5 ) ) ) ).

% imageI
thf(fact_181_imageI,axiom,
    ! [X5: $o,A5: set_o,F: $o > $o] :
      ( ( member_o2 @ X5 @ A5 )
     => ( member_o2 @ ( F @ X5 ) @ ( image_o_o2 @ F @ A5 ) ) ) ).

% imageI
thf(fact_182_imageI,axiom,
    ! [X5: fm,A5: set_fm,F: fm > list_fm] :
      ( ( member_fm2 @ X5 @ A5 )
     => ( member_list_fm2 @ ( F @ X5 ) @ ( image_fm_list_fm2 @ F @ A5 ) ) ) ).

% imageI
thf(fact_183_imageI,axiom,
    ! [X5: list_fm,A5: set_list_fm,F: list_fm > fm] :
      ( ( member_list_fm2 @ X5 @ A5 )
     => ( member_fm2 @ ( F @ X5 ) @ ( image_list_fm_fm2 @ F @ A5 ) ) ) ).

% imageI
thf(fact_184_imageI,axiom,
    ! [X5: list_fm,A5: set_list_fm,F: list_fm > $o] :
      ( ( member_list_fm2 @ X5 @ A5 )
     => ( member_o2 @ ( F @ X5 ) @ ( image_list_fm_o2 @ F @ A5 ) ) ) ).

% imageI
thf(fact_185_imageI,axiom,
    ! [X5: $o,A5: set_o,F: $o > list_fm] :
      ( ( member_o2 @ X5 @ A5 )
     => ( member_list_fm2 @ ( F @ X5 ) @ ( image_o_list_fm2 @ F @ A5 ) ) ) ).

% imageI
thf(fact_186_imageI,axiom,
    ! [X5: list_fm,A5: set_list_fm,F: list_fm > list_fm] :
      ( ( member_list_fm2 @ X5 @ A5 )
     => ( member_list_fm2 @ ( F @ X5 ) @ ( image_9148017957442633541ist_fm @ F @ A5 ) ) ) ).

% imageI
thf(fact_187_imageI,axiom,
    ! [X5: fm,A5: set_fm,F: fm > produc340336539035504054m_rule] :
      ( ( member_fm2 @ X5 @ A5 )
     => ( member7231649785386036813m_rule @ ( F @ X5 ) @ ( image_6079799016930204900m_rule @ F @ A5 ) ) ) ).

% imageI
thf(fact_188_image__iff,axiom,
    ! [Z: set_fm,F: produc340336539035504054m_rule > set_fm,A5: set_Pr1822751329126368876m_rule] :
      ( ( member_set_fm @ Z @ ( image_377115208147135190set_fm @ F @ A5 ) )
      = ( ? [X: produc340336539035504054m_rule] :
            ( ( member7231649785386036813m_rule @ X @ A5 )
            & ( Z
              = ( F @ X ) ) ) ) ) ).

% image_iff
thf(fact_189_bex__imageD,axiom,
    ! [F: produc340336539035504054m_rule > set_fm,A5: set_Pr1822751329126368876m_rule,P3: set_fm > $o] :
      ( ? [X6: set_fm] :
          ( ( member_set_fm @ X6 @ ( image_377115208147135190set_fm @ F @ A5 ) )
          & ( P3 @ X6 ) )
     => ? [X3: produc340336539035504054m_rule] :
          ( ( member7231649785386036813m_rule @ X3 @ A5 )
          & ( P3 @ ( F @ X3 ) ) ) ) ).

% bex_imageD
thf(fact_190_image__cong,axiom,
    ! [M: set_Pr1822751329126368876m_rule,N2: set_Pr1822751329126368876m_rule,F: produc340336539035504054m_rule > set_fm,G: produc340336539035504054m_rule > set_fm] :
      ( ( M = N2 )
     => ( ! [X3: produc340336539035504054m_rule] :
            ( ( member7231649785386036813m_rule @ X3 @ N2 )
           => ( ( F @ X3 )
              = ( G @ X3 ) ) )
       => ( ( image_377115208147135190set_fm @ F @ M )
          = ( image_377115208147135190set_fm @ G @ N2 ) ) ) ) ).

% image_cong
thf(fact_191_ball__imageD,axiom,
    ! [F: produc340336539035504054m_rule > set_fm,A5: set_Pr1822751329126368876m_rule,P3: set_fm > $o] :
      ( ! [X3: set_fm] :
          ( ( member_set_fm @ X3 @ ( image_377115208147135190set_fm @ F @ A5 ) )
         => ( P3 @ X3 ) )
     => ! [X6: produc340336539035504054m_rule] :
          ( ( member7231649785386036813m_rule @ X6 @ A5 )
         => ( P3 @ ( F @ X6 ) ) ) ) ).

% ball_imageD
thf(fact_192_rev__image__eqI,axiom,
    ! [X5: fm,A5: set_fm,B: fm,F: fm > fm] :
      ( ( member_fm2 @ X5 @ A5 )
     => ( ( B
          = ( F @ X5 ) )
       => ( member_fm2 @ B @ ( image_fm_fm2 @ F @ A5 ) ) ) ) ).

% rev_image_eqI
thf(fact_193_rev__image__eqI,axiom,
    ! [X5: fm,A5: set_fm,B: $o,F: fm > $o] :
      ( ( member_fm2 @ X5 @ A5 )
     => ( ( B
          = ( F @ X5 ) )
       => ( member_o2 @ B @ ( image_fm_o2 @ F @ A5 ) ) ) ) ).

% rev_image_eqI
thf(fact_194_rev__image__eqI,axiom,
    ! [X5: $o,A5: set_o,B: fm,F: $o > fm] :
      ( ( member_o2 @ X5 @ A5 )
     => ( ( B
          = ( F @ X5 ) )
       => ( member_fm2 @ B @ ( image_o_fm2 @ F @ A5 ) ) ) ) ).

% rev_image_eqI
thf(fact_195_rev__image__eqI,axiom,
    ! [X5: $o,A5: set_o,B: $o,F: $o > $o] :
      ( ( member_o2 @ X5 @ A5 )
     => ( ( B
          = ( F @ X5 ) )
       => ( member_o2 @ B @ ( image_o_o2 @ F @ A5 ) ) ) ) ).

% rev_image_eqI
thf(fact_196_rev__image__eqI,axiom,
    ! [X5: fm,A5: set_fm,B: list_fm,F: fm > list_fm] :
      ( ( member_fm2 @ X5 @ A5 )
     => ( ( B
          = ( F @ X5 ) )
       => ( member_list_fm2 @ B @ ( image_fm_list_fm2 @ F @ A5 ) ) ) ) ).

% rev_image_eqI
thf(fact_197_rev__image__eqI,axiom,
    ! [X5: list_fm,A5: set_list_fm,B: fm,F: list_fm > fm] :
      ( ( member_list_fm2 @ X5 @ A5 )
     => ( ( B
          = ( F @ X5 ) )
       => ( member_fm2 @ B @ ( image_list_fm_fm2 @ F @ A5 ) ) ) ) ).

% rev_image_eqI
thf(fact_198_rev__image__eqI,axiom,
    ! [X5: list_fm,A5: set_list_fm,B: $o,F: list_fm > $o] :
      ( ( member_list_fm2 @ X5 @ A5 )
     => ( ( B
          = ( F @ X5 ) )
       => ( member_o2 @ B @ ( image_list_fm_o2 @ F @ A5 ) ) ) ) ).

% rev_image_eqI
thf(fact_199_rev__image__eqI,axiom,
    ! [X5: $o,A5: set_o,B: list_fm,F: $o > list_fm] :
      ( ( member_o2 @ X5 @ A5 )
     => ( ( B
          = ( F @ X5 ) )
       => ( member_list_fm2 @ B @ ( image_o_list_fm2 @ F @ A5 ) ) ) ) ).

% rev_image_eqI
thf(fact_200_rev__image__eqI,axiom,
    ! [X5: list_fm,A5: set_list_fm,B: list_fm,F: list_fm > list_fm] :
      ( ( member_list_fm2 @ X5 @ A5 )
     => ( ( B
          = ( F @ X5 ) )
       => ( member_list_fm2 @ B @ ( image_9148017957442633541ist_fm @ F @ A5 ) ) ) ) ).

% rev_image_eqI
thf(fact_201_rev__image__eqI,axiom,
    ! [X5: fm,A5: set_fm,B: produc340336539035504054m_rule,F: fm > produc340336539035504054m_rule] :
      ( ( member_fm2 @ X5 @ A5 )
     => ( ( B
          = ( F @ X5 ) )
       => ( member7231649785386036813m_rule @ B @ ( image_6079799016930204900m_rule @ F @ A5 ) ) ) ) ).

% rev_image_eqI
thf(fact_202_shift__left__inj,axiom,
    ! [Xs: list_P2774625669004309958m_rule,S1: stream2709947120125613254m_rule,S22: stream2709947120125613254m_rule] :
      ( ( ( shift_2334844276748245581m_rule @ Xs @ S1 )
        = ( shift_2334844276748245581m_rule @ Xs @ S22 ) )
      = ( S1 = S22 ) ) ).

% shift_left_inj
thf(fact_203_pre,axiom,
    ( list_a9038620553111681026m_rule
    @ ^ [Xs2: produc340336539035504054m_rule] :
        ( ( produc6514711208036230658m_rule @ Xs2 )
       != betaCon )
    @ pre ) ).

% pre
thf(fact_204_parts__preserves__unaffected,axiom,
    ! [R4: rule,P: fm,Z3: list_fm,A5: list_tm] :
      ( ~ ( affects @ R4 @ P )
     => ( ( member_list_fm2 @ Z3 @ ( set_list_fm2 @ ( parts @ A5 @ R4 @ P ) ) )
       => ( member_fm2 @ P @ ( set_fm2 @ Z3 ) ) ) ) ).

% parts_preserves_unaffected
thf(fact_205_suf,axiom,
    ( linear7925859179493631106m_rule
    @ ^ [Step: produc340336539035504054m_rule] :
        ( ( produc6514711208036230658m_rule @ Step )
        = betaCon )
    @ suf ) ).

% suf
thf(fact_206_children__preserves__unaffected,axiom,
    ! [P: fm,Z: list_fm,R4: rule,Z3: list_fm,A5: list_tm] :
      ( ( member_fm2 @ P @ ( set_fm2 @ Z ) )
     => ( ~ ( affects @ R4 @ P )
       => ( ( member_list_fm2 @ Z3 @ ( set_list_fm2 @ ( children @ A5 @ R4 @ Z ) ) )
         => ( member_fm2 @ P @ ( set_fm2 @ Z3 ) ) ) ) ) ).

% children_preserves_unaffected
thf(fact_207_tree__fms__in__pseq,axiom,
    ! [P: fm,Steps: stream2709947120125613254m_rule] :
      ( ( member_fm2 @ P @ ( tree_fms @ Steps ) )
     => ? [N: nat] : ( member_fm2 @ P @ ( set_fm2 @ ( pseq @ ( snth_P6679518042731451922m_rule @ Steps @ N ) ) ) ) ) ).

% tree_fms_in_pseq
thf(fact_208_image__paired__Times,axiom,
    ! [F: produc340336539035504054m_rule > set_fm,G: produc340336539035504054m_rule > set_fm,A5: set_Pr1822751329126368876m_rule,B6: set_Pr1822751329126368876m_rule] :
      ( ( image_5641574685687872095set_fm
        @ ( produc5064645917040859725set_fm
          @ ^ [X: produc340336539035504054m_rule,Y: produc340336539035504054m_rule] : ( produc5928365999827228247set_fm @ ( F @ X ) @ ( G @ Y ) ) )
        @ ( produc1849974139216714806m_rule @ A5
          @ ^ [Uu: produc340336539035504054m_rule] : B6 ) )
      = ( produc5276883515181919240set_fm @ ( image_377115208147135190set_fm @ F @ A5 )
        @ ^ [Uu: set_fm] : ( image_377115208147135190set_fm @ G @ B6 ) ) ) ).

% image_paired_Times
thf(fact_209_pseq__in__tree__fms,axiom,
    ! [X5: produc340336539035504054m_rule,Steps: stream2709947120125613254m_rule,P: fm] :
      ( ( member7231649785386036813m_rule @ X5 @ ( sset_P4484857331586881186m_rule @ Steps ) )
     => ( ( member_fm2 @ P @ ( set_fm2 @ ( pseq @ X5 ) ) )
       => ( member_fm2 @ P @ ( tree_fms @ Steps ) ) ) ) ).

% pseq_in_tree_fms
thf(fact_210_SigmaI,axiom,
    ! [A: fm,A5: set_fm,B: fm,B6: fm > set_fm] :
      ( ( member_fm2 @ A @ A5 )
     => ( ( member_fm2 @ B @ ( B6 @ A ) )
       => ( member7780952600467998736_fm_fm @ ( product_Pair_fm_fm @ A @ B ) @ ( product_Sigma_fm_fm @ A5 @ B6 ) ) ) ) ).

% SigmaI
thf(fact_211_SigmaI,axiom,
    ! [A: fm,A5: set_fm,B: $o,B6: fm > set_o] :
      ( ( member_fm2 @ A @ A5 )
     => ( ( member_o2 @ B @ ( B6 @ A ) )
       => ( member1731323461681951669d_fm_o @ ( product_Pair_fm_o @ A @ B ) @ ( product_Sigma_fm_o @ A5 @ B6 ) ) ) ) ).

% SigmaI
thf(fact_212_SigmaI,axiom,
    ! [A: $o,A5: set_o,B: fm,B6: $o > set_fm] :
      ( ( member_o2 @ A @ A5 )
     => ( ( member_fm2 @ B @ ( B6 @ A ) )
       => ( member6824368184199080235d_o_fm @ ( product_Pair_o_fm @ A @ B ) @ ( product_Sigma_o_fm @ A5 @ B6 ) ) ) ) ).

% SigmaI
thf(fact_213_SigmaI,axiom,
    ! [A: $o,A5: set_o,B: $o,B6: $o > set_o] :
      ( ( member_o2 @ A @ A5 )
     => ( ( member_o2 @ B @ ( B6 @ A ) )
       => ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ B ) @ ( product_Sigma_o_o @ A5 @ B6 ) ) ) ) ).

% SigmaI
thf(fact_214_SigmaI,axiom,
    ! [A: rule,A5: set_rule,B: fm,B6: rule > set_fm] :
      ( ( member_rule @ A @ A5 )
     => ( ( member_fm2 @ B @ ( B6 @ A ) )
       => ( member9168179392308330989ule_fm @ ( product_Pair_rule_fm @ A @ B ) @ ( produc7224384929229372695ule_fm @ A5 @ B6 ) ) ) ) ).

% SigmaI
thf(fact_215_SigmaI,axiom,
    ! [A: nat,A5: set_nat,B: nat,B6: nat > set_nat] :
      ( ( member_nat @ A @ A5 )
     => ( ( member_nat @ B @ ( B6 @ A ) )
       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( produc457027306803732586at_nat @ A5 @ B6 ) ) ) ) ).

% SigmaI
thf(fact_216_SigmaI,axiom,
    ! [A: fm,A5: set_fm,B: list_fm,B6: fm > set_list_fm] :
      ( ( member_fm2 @ A @ A5 )
     => ( ( member_list_fm2 @ B @ ( B6 @ A ) )
       => ( member8098592186287407126ist_fm @ ( produc5674586226468309085ist_fm @ A @ B ) @ ( produc6795173406573533582ist_fm @ A5 @ B6 ) ) ) ) ).

% SigmaI
thf(fact_217_SigmaI,axiom,
    ! [A: list_fm,A5: set_list_fm,B: fm,B6: list_fm > set_fm] :
      ( ( member_list_fm2 @ A @ A5 )
     => ( ( member_fm2 @ B @ ( B6 @ A ) )
       => ( member7999069721096432778_fm_fm @ ( produc3468876765965275089_fm_fm @ A @ B ) @ ( produc4589463946070499586_fm_fm @ A5 @ B6 ) ) ) ) ).

% SigmaI
thf(fact_218_SigmaI,axiom,
    ! [A: list_fm,A5: set_list_fm,B: $o,B6: list_fm > set_o] :
      ( ( member_list_fm2 @ A @ A5 )
     => ( ( member_o2 @ B @ ( B6 @ A ) )
       => ( member8238630628690094383t_fm_o @ ( produc8850898633069939702t_fm_o @ A @ B ) @ ( produc2902879935907792807t_fm_o @ A5 @ B6 ) ) ) ) ).

% SigmaI
thf(fact_219_SigmaI,axiom,
    ! [A: $o,A5: set_o,B: list_fm,B6: $o > set_list_fm] :
      ( ( member_o2 @ A @ A5 )
     => ( ( member_list_fm2 @ B @ ( B6 @ A ) )
       => ( member7814064844939679665ist_fm @ ( produc3646295350333167352ist_fm @ A @ B ) @ ( produc6921648690025796265ist_fm @ A5 @ B6 ) ) ) ) ).

% SigmaI
thf(fact_220_mem__Sigma__iff,axiom,
    ! [A: fm,B: fm,A5: set_fm,B6: fm > set_fm] :
      ( ( member7780952600467998736_fm_fm @ ( product_Pair_fm_fm @ A @ B ) @ ( product_Sigma_fm_fm @ A5 @ B6 ) )
      = ( ( member_fm2 @ A @ A5 )
        & ( member_fm2 @ B @ ( B6 @ A ) ) ) ) ).

% mem_Sigma_iff
thf(fact_221_mem__Sigma__iff,axiom,
    ! [A: fm,B: $o,A5: set_fm,B6: fm > set_o] :
      ( ( member1731323461681951669d_fm_o @ ( product_Pair_fm_o @ A @ B ) @ ( product_Sigma_fm_o @ A5 @ B6 ) )
      = ( ( member_fm2 @ A @ A5 )
        & ( member_o2 @ B @ ( B6 @ A ) ) ) ) ).

% mem_Sigma_iff
thf(fact_222_mem__Sigma__iff,axiom,
    ! [A: $o,B: fm,A5: set_o,B6: $o > set_fm] :
      ( ( member6824368184199080235d_o_fm @ ( product_Pair_o_fm @ A @ B ) @ ( product_Sigma_o_fm @ A5 @ B6 ) )
      = ( ( member_o2 @ A @ A5 )
        & ( member_fm2 @ B @ ( B6 @ A ) ) ) ) ).

% mem_Sigma_iff
thf(fact_223_mem__Sigma__iff,axiom,
    ! [A: $o,B: $o,A5: set_o,B6: $o > set_o] :
      ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ B ) @ ( product_Sigma_o_o @ A5 @ B6 ) )
      = ( ( member_o2 @ A @ A5 )
        & ( member_o2 @ B @ ( B6 @ A ) ) ) ) ).

% mem_Sigma_iff
thf(fact_224_mem__Sigma__iff,axiom,
    ! [A: rule,B: fm,A5: set_rule,B6: rule > set_fm] :
      ( ( member9168179392308330989ule_fm @ ( product_Pair_rule_fm @ A @ B ) @ ( produc7224384929229372695ule_fm @ A5 @ B6 ) )
      = ( ( member_rule @ A @ A5 )
        & ( member_fm2 @ B @ ( B6 @ A ) ) ) ) ).

% mem_Sigma_iff
thf(fact_225_mem__Sigma__iff,axiom,
    ! [A: nat,B: nat,A5: set_nat,B6: nat > set_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( produc457027306803732586at_nat @ A5 @ B6 ) )
      = ( ( member_nat @ A @ A5 )
        & ( member_nat @ B @ ( B6 @ A ) ) ) ) ).

% mem_Sigma_iff
thf(fact_226_mem__Sigma__iff,axiom,
    ! [A: fm,B: list_fm,A5: set_fm,B6: fm > set_list_fm] :
      ( ( member8098592186287407126ist_fm @ ( produc5674586226468309085ist_fm @ A @ B ) @ ( produc6795173406573533582ist_fm @ A5 @ B6 ) )
      = ( ( member_fm2 @ A @ A5 )
        & ( member_list_fm2 @ B @ ( B6 @ A ) ) ) ) ).

% mem_Sigma_iff
thf(fact_227_mem__Sigma__iff,axiom,
    ! [A: list_fm,B: fm,A5: set_list_fm,B6: list_fm > set_fm] :
      ( ( member7999069721096432778_fm_fm @ ( produc3468876765965275089_fm_fm @ A @ B ) @ ( produc4589463946070499586_fm_fm @ A5 @ B6 ) )
      = ( ( member_list_fm2 @ A @ A5 )
        & ( member_fm2 @ B @ ( B6 @ A ) ) ) ) ).

% mem_Sigma_iff
thf(fact_228_mem__Sigma__iff,axiom,
    ! [A: list_fm,B: $o,A5: set_list_fm,B6: list_fm > set_o] :
      ( ( member8238630628690094383t_fm_o @ ( produc8850898633069939702t_fm_o @ A @ B ) @ ( produc2902879935907792807t_fm_o @ A5 @ B6 ) )
      = ( ( member_list_fm2 @ A @ A5 )
        & ( member_o2 @ B @ ( B6 @ A ) ) ) ) ).

% mem_Sigma_iff
thf(fact_229_mem__Sigma__iff,axiom,
    ! [A: $o,B: list_fm,A5: set_o,B6: $o > set_list_fm] :
      ( ( member7814064844939679665ist_fm @ ( produc3646295350333167352ist_fm @ A @ B ) @ ( produc6921648690025796265ist_fm @ A5 @ B6 ) )
      = ( ( member_o2 @ A @ A5 )
        & ( member_list_fm2 @ B @ ( B6 @ A ) ) ) ) ).

% mem_Sigma_iff
thf(fact_230_Collect__case__prod,axiom,
    ! [P3: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
      ( ( collec7088162979684241874at_nat
        @ ( produc6590410687421337004_nat_o
          @ ^ [A3: product_prod_nat_nat,B3: product_prod_nat_nat] :
              ( ( P3 @ A3 )
              & ( Q @ B3 ) ) ) )
      = ( produc2761391749766926216at_nat @ ( collec3392354462482085612at_nat @ P3 )
        @ ^ [Uu: product_prod_nat_nat] : ( collec3392354462482085612at_nat @ Q ) ) ) ).

% Collect_case_prod
thf(fact_231_Collect__case__prod,axiom,
    ! [P3: product_prod_nat_nat > $o,Q: nat > $o] :
      ( ( collec7029190964493513045at_nat
        @ ( produc8758269395934548017_nat_o
          @ ^ [A3: product_prod_nat_nat,B3: nat] :
              ( ( P3 @ A3 )
              & ( Q @ B3 ) ) ) )
      = ( produc7672662199629908489at_nat @ ( collec3392354462482085612at_nat @ P3 )
        @ ^ [Uu: product_prod_nat_nat] : ( collect_nat @ Q ) ) ) ).

% Collect_case_prod
thf(fact_232_Collect__case__prod,axiom,
    ! [P3: nat > $o,Q: product_prod_nat_nat > $o] :
      ( ( collec5903703980526211963at_nat
        @ ( produc5864757623865647827_nat_o
          @ ^ [A3: nat,B3: product_prod_nat_nat] :
              ( ( P3 @ A3 )
              & ( Q @ B3 ) ) ) )
      = ( produc1809337555817847783at_nat @ ( collect_nat @ P3 )
        @ ^ [Uu: nat] : ( collec3392354462482085612at_nat @ Q ) ) ) ).

% Collect_case_prod
thf(fact_233_Collect__case__prod,axiom,
    ! [P3: rule > $o,Q: fm > $o] :
      ( ( collec4199261512444720427ule_fm
        @ ( produc3561889649859641891e_fm_o
          @ ^ [A3: rule,B3: fm] :
              ( ( P3 @ A3 )
              & ( Q @ B3 ) ) ) )
      = ( produc7224384929229372695ule_fm @ ( collect_rule @ P3 )
        @ ^ [Uu: rule] : ( collect_fm @ Q ) ) ) ).

% Collect_case_prod
thf(fact_234_Collect__case__prod,axiom,
    ! [P3: nat > $o,Q: nat > $o] :
      ( ( collec3392354462482085612at_nat
        @ ( produc6081775807080527818_nat_o
          @ ^ [A3: nat,B3: nat] :
              ( ( P3 @ A3 )
              & ( Q @ B3 ) ) ) )
      = ( produc457027306803732586at_nat @ ( collect_nat @ P3 )
        @ ^ [Uu: nat] : ( collect_nat @ Q ) ) ) ).

% Collect_case_prod
thf(fact_235_sdrop__simps_I1_J,axiom,
    ! [N3: nat,S2: stream2709947120125613254m_rule] :
      ( ( shd_Pr4562317740776619530m_rule @ ( sdrop_8169176516188972301m_rule @ N3 @ S2 ) )
      = ( snth_P6679518042731451922m_rule @ S2 @ N3 ) ) ).

% sdrop_simps(1)
thf(fact_236__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062pre_Asuf_O_A_092_060lbrakk_062list__all_A_I_092_060lambda_062xs_O_Asnd_Axs_A_092_060noteq_062_ABetaCon_J_Apre_059_Aholds_A_Iis__rule_ABetaCon_J_Asuf_059_Asdrop_An_Asteps_A_061_Apre_A_064_N_Asuf_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
    ~ ! [Pre: list_P2774625669004309958m_rule] :
        ( ( list_a9038620553111681026m_rule
          @ ^ [Xs2: produc340336539035504054m_rule] :
              ( ( produc6514711208036230658m_rule @ Xs2 )
             != betaCon )
          @ Pre )
       => ! [Suf: stream2709947120125613254m_rule] :
            ( ( linear7925859179493631106m_rule
              @ ^ [Step: produc340336539035504054m_rule] :
                  ( ( produc6514711208036230658m_rule @ Step )
                  = betaCon )
              @ Suf )
           => ( ( sdrop_8169176516188972301m_rule @ n @ steps )
             != ( shift_2334844276748245581m_rule @ Pre @ Suf ) ) ) ) ).

% \<open>\<And>thesis. (\<And>pre suf. \<lbrakk>list_all (\<lambda>xs. snd xs \<noteq> BetaCon) pre; holds (is_rule BetaCon) suf; sdrop n steps = pre @- suf\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_237_shd__sset,axiom,
    ! [A: stream_fm] : ( member_fm2 @ ( shd_fm @ A ) @ ( sset_fm @ A ) ) ).

% shd_sset
thf(fact_238_shd__sset,axiom,
    ! [A: stream_list_fm] : ( member_list_fm2 @ ( shd_list_fm @ A ) @ ( sset_list_fm @ A ) ) ).

% shd_sset
thf(fact_239_shd__sset,axiom,
    ! [A: stream_o] : ( member_o2 @ ( shd_o @ A ) @ ( sset_o @ A ) ) ).

% shd_sset
thf(fact_240_shd__sset,axiom,
    ! [A: stream2709947120125613254m_rule] : ( member7231649785386036813m_rule @ ( shd_Pr4562317740776619530m_rule @ A ) @ ( sset_P4484857331586881186m_rule @ A ) ) ).

% shd_sset
thf(fact_241_SigmaE,axiom,
    ! [C: product_prod_fm_fm,A5: set_fm,B6: fm > set_fm] :
      ( ( member7780952600467998736_fm_fm @ C @ ( product_Sigma_fm_fm @ A5 @ B6 ) )
     => ~ ! [X3: fm] :
            ( ( member_fm2 @ X3 @ A5 )
           => ! [Y3: fm] :
                ( ( member_fm2 @ Y3 @ ( B6 @ X3 ) )
               => ( C
                 != ( product_Pair_fm_fm @ X3 @ Y3 ) ) ) ) ) ).

% SigmaE
thf(fact_242_SigmaE,axiom,
    ! [C: product_prod_fm_o,A5: set_fm,B6: fm > set_o] :
      ( ( member1731323461681951669d_fm_o @ C @ ( product_Sigma_fm_o @ A5 @ B6 ) )
     => ~ ! [X3: fm] :
            ( ( member_fm2 @ X3 @ A5 )
           => ! [Y3: $o] :
                ( ( member_o2 @ Y3 @ ( B6 @ X3 ) )
               => ( C
                 != ( product_Pair_fm_o @ X3 @ Y3 ) ) ) ) ) ).

% SigmaE
thf(fact_243_SigmaE,axiom,
    ! [C: product_prod_o_fm,A5: set_o,B6: $o > set_fm] :
      ( ( member6824368184199080235d_o_fm @ C @ ( product_Sigma_o_fm @ A5 @ B6 ) )
     => ~ ! [X3: $o] :
            ( ( member_o2 @ X3 @ A5 )
           => ! [Y3: fm] :
                ( ( member_fm2 @ Y3 @ ( B6 @ X3 ) )
               => ( C
                 != ( product_Pair_o_fm @ X3 @ Y3 ) ) ) ) ) ).

% SigmaE
thf(fact_244_SigmaE,axiom,
    ! [C: product_prod_o_o,A5: set_o,B6: $o > set_o] :
      ( ( member7466972457876170832od_o_o @ C @ ( product_Sigma_o_o @ A5 @ B6 ) )
     => ~ ! [X3: $o] :
            ( ( member_o2 @ X3 @ A5 )
           => ! [Y3: $o] :
                ( ( member_o2 @ Y3 @ ( B6 @ X3 ) )
               => ( C
                 != ( product_Pair_o_o @ X3 @ Y3 ) ) ) ) ) ).

% SigmaE
thf(fact_245_SigmaE,axiom,
    ! [C: product_prod_rule_fm,A5: set_rule,B6: rule > set_fm] :
      ( ( member9168179392308330989ule_fm @ C @ ( produc7224384929229372695ule_fm @ A5 @ B6 ) )
     => ~ ! [X3: rule] :
            ( ( member_rule @ X3 @ A5 )
           => ! [Y3: fm] :
                ( ( member_fm2 @ Y3 @ ( B6 @ X3 ) )
               => ( C
                 != ( product_Pair_rule_fm @ X3 @ Y3 ) ) ) ) ) ).

% SigmaE
thf(fact_246_SigmaE,axiom,
    ! [C: product_prod_nat_nat,A5: set_nat,B6: nat > set_nat] :
      ( ( member8440522571783428010at_nat @ C @ ( produc457027306803732586at_nat @ A5 @ B6 ) )
     => ~ ! [X3: nat] :
            ( ( member_nat @ X3 @ A5 )
           => ! [Y3: nat] :
                ( ( member_nat @ Y3 @ ( B6 @ X3 ) )
               => ( C
                 != ( product_Pair_nat_nat @ X3 @ Y3 ) ) ) ) ) ).

% SigmaE
thf(fact_247_SigmaE,axiom,
    ! [C: produc7549369199418484333ist_fm,A5: set_fm,B6: fm > set_list_fm] :
      ( ( member8098592186287407126ist_fm @ C @ ( produc6795173406573533582ist_fm @ A5 @ B6 ) )
     => ~ ! [X3: fm] :
            ( ( member_fm2 @ X3 @ A5 )
           => ! [Y3: list_fm] :
                ( ( member_list_fm2 @ Y3 @ ( B6 @ X3 ) )
               => ( C
                 != ( produc5674586226468309085ist_fm @ X3 @ Y3 ) ) ) ) ) ).

% SigmaE
thf(fact_248_SigmaE,axiom,
    ! [C: produc7449846734227509985_fm_fm,A5: set_list_fm,B6: list_fm > set_fm] :
      ( ( member7999069721096432778_fm_fm @ C @ ( produc4589463946070499586_fm_fm @ A5 @ B6 ) )
     => ~ ! [X3: list_fm] :
            ( ( member_list_fm2 @ X3 @ A5 )
           => ! [Y3: fm] :
                ( ( member_fm2 @ Y3 @ ( B6 @ X3 ) )
               => ( C
                 != ( produc3468876765965275089_fm_fm @ X3 @ Y3 ) ) ) ) ) ).

% SigmaE
thf(fact_249_SigmaE,axiom,
    ! [C: produc7860403524128848134t_fm_o,A5: set_list_fm,B6: list_fm > set_o] :
      ( ( member8238630628690094383t_fm_o @ C @ ( produc2902879935907792807t_fm_o @ A5 @ B6 ) )
     => ~ ! [X3: list_fm] :
            ( ( member_list_fm2 @ X3 @ A5 )
           => ! [Y3: $o] :
                ( ( member_o2 @ Y3 @ ( B6 @ X3 ) )
               => ( C
                 != ( produc8850898633069939702t_fm_o @ X3 @ Y3 ) ) ) ) ) ).

% SigmaE
thf(fact_250_SigmaE,axiom,
    ! [C: produc7435837740378433416ist_fm,A5: set_o,B6: $o > set_list_fm] :
      ( ( member7814064844939679665ist_fm @ C @ ( produc6921648690025796265ist_fm @ A5 @ B6 ) )
     => ~ ! [X3: $o] :
            ( ( member_o2 @ X3 @ A5 )
           => ! [Y3: list_fm] :
                ( ( member_list_fm2 @ Y3 @ ( B6 @ X3 ) )
               => ( C
                 != ( produc3646295350333167352ist_fm @ X3 @ Y3 ) ) ) ) ) ).

% SigmaE
thf(fact_251_SigmaD1,axiom,
    ! [A: produc6018962875968178549ist_fm,B: rule,A5: set_Pr5202636777678657877ist_fm,B6: produc6018962875968178549ist_fm > set_rule] :
      ( ( member7231649785386036813m_rule @ ( produc1733806532565653680m_rule @ A @ B ) @ ( produc4664469595690820543m_rule @ A5 @ B6 ) )
     => ( member4699826688122452638ist_fm @ A @ A5 ) ) ).

% SigmaD1
thf(fact_252_SigmaD1,axiom,
    ! [A: rule,B: fm,A5: set_rule,B6: rule > set_fm] :
      ( ( member9168179392308330989ule_fm @ ( product_Pair_rule_fm @ A @ B ) @ ( produc7224384929229372695ule_fm @ A5 @ B6 ) )
     => ( member_rule @ A @ A5 ) ) ).

% SigmaD1
thf(fact_253_SigmaD1,axiom,
    ! [A: nat,B: nat,A5: set_nat,B6: nat > set_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( produc457027306803732586at_nat @ A5 @ B6 ) )
     => ( member_nat @ A @ A5 ) ) ).

% SigmaD1
thf(fact_254_SigmaD2,axiom,
    ! [A: produc6018962875968178549ist_fm,B: rule,A5: set_Pr5202636777678657877ist_fm,B6: produc6018962875968178549ist_fm > set_rule] :
      ( ( member7231649785386036813m_rule @ ( produc1733806532565653680m_rule @ A @ B ) @ ( produc4664469595690820543m_rule @ A5 @ B6 ) )
     => ( member_rule @ B @ ( B6 @ A ) ) ) ).

% SigmaD2
thf(fact_255_SigmaD2,axiom,
    ! [A: rule,B: fm,A5: set_rule,B6: rule > set_fm] :
      ( ( member9168179392308330989ule_fm @ ( product_Pair_rule_fm @ A @ B ) @ ( produc7224384929229372695ule_fm @ A5 @ B6 ) )
     => ( member_fm2 @ B @ ( B6 @ A ) ) ) ).

% SigmaD2
thf(fact_256_SigmaD2,axiom,
    ! [A: nat,B: nat,A5: set_nat,B6: nat > set_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( produc457027306803732586at_nat @ A5 @ B6 ) )
     => ( member_nat @ B @ ( B6 @ A ) ) ) ).

% SigmaD2
thf(fact_257_SigmaE2,axiom,
    ! [A: fm,B: fm,A5: set_fm,B6: fm > set_fm] :
      ( ( member7780952600467998736_fm_fm @ ( product_Pair_fm_fm @ A @ B ) @ ( product_Sigma_fm_fm @ A5 @ B6 ) )
     => ~ ( ( member_fm2 @ A @ A5 )
         => ~ ( member_fm2 @ B @ ( B6 @ A ) ) ) ) ).

% SigmaE2
thf(fact_258_SigmaE2,axiom,
    ! [A: fm,B: $o,A5: set_fm,B6: fm > set_o] :
      ( ( member1731323461681951669d_fm_o @ ( product_Pair_fm_o @ A @ B ) @ ( product_Sigma_fm_o @ A5 @ B6 ) )
     => ~ ( ( member_fm2 @ A @ A5 )
         => ~ ( member_o2 @ B @ ( B6 @ A ) ) ) ) ).

% SigmaE2
thf(fact_259_SigmaE2,axiom,
    ! [A: $o,B: fm,A5: set_o,B6: $o > set_fm] :
      ( ( member6824368184199080235d_o_fm @ ( product_Pair_o_fm @ A @ B ) @ ( product_Sigma_o_fm @ A5 @ B6 ) )
     => ~ ( ( member_o2 @ A @ A5 )
         => ~ ( member_fm2 @ B @ ( B6 @ A ) ) ) ) ).

% SigmaE2
thf(fact_260_SigmaE2,axiom,
    ! [A: $o,B: $o,A5: set_o,B6: $o > set_o] :
      ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ B ) @ ( product_Sigma_o_o @ A5 @ B6 ) )
     => ~ ( ( member_o2 @ A @ A5 )
         => ~ ( member_o2 @ B @ ( B6 @ A ) ) ) ) ).

% SigmaE2
thf(fact_261_SigmaE2,axiom,
    ! [A: rule,B: fm,A5: set_rule,B6: rule > set_fm] :
      ( ( member9168179392308330989ule_fm @ ( product_Pair_rule_fm @ A @ B ) @ ( produc7224384929229372695ule_fm @ A5 @ B6 ) )
     => ~ ( ( member_rule @ A @ A5 )
         => ~ ( member_fm2 @ B @ ( B6 @ A ) ) ) ) ).

% SigmaE2
thf(fact_262_SigmaE2,axiom,
    ! [A: nat,B: nat,A5: set_nat,B6: nat > set_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( produc457027306803732586at_nat @ A5 @ B6 ) )
     => ~ ( ( member_nat @ A @ A5 )
         => ~ ( member_nat @ B @ ( B6 @ A ) ) ) ) ).

% SigmaE2
thf(fact_263_SigmaE2,axiom,
    ! [A: fm,B: list_fm,A5: set_fm,B6: fm > set_list_fm] :
      ( ( member8098592186287407126ist_fm @ ( produc5674586226468309085ist_fm @ A @ B ) @ ( produc6795173406573533582ist_fm @ A5 @ B6 ) )
     => ~ ( ( member_fm2 @ A @ A5 )
         => ~ ( member_list_fm2 @ B @ ( B6 @ A ) ) ) ) ).

% SigmaE2
thf(fact_264_SigmaE2,axiom,
    ! [A: list_fm,B: fm,A5: set_list_fm,B6: list_fm > set_fm] :
      ( ( member7999069721096432778_fm_fm @ ( produc3468876765965275089_fm_fm @ A @ B ) @ ( produc4589463946070499586_fm_fm @ A5 @ B6 ) )
     => ~ ( ( member_list_fm2 @ A @ A5 )
         => ~ ( member_fm2 @ B @ ( B6 @ A ) ) ) ) ).

% SigmaE2
thf(fact_265_SigmaE2,axiom,
    ! [A: list_fm,B: $o,A5: set_list_fm,B6: list_fm > set_o] :
      ( ( member8238630628690094383t_fm_o @ ( produc8850898633069939702t_fm_o @ A @ B ) @ ( produc2902879935907792807t_fm_o @ A5 @ B6 ) )
     => ~ ( ( member_list_fm2 @ A @ A5 )
         => ~ ( member_o2 @ B @ ( B6 @ A ) ) ) ) ).

% SigmaE2
thf(fact_266_SigmaE2,axiom,
    ! [A: $o,B: list_fm,A5: set_o,B6: $o > set_list_fm] :
      ( ( member7814064844939679665ist_fm @ ( produc3646295350333167352ist_fm @ A @ B ) @ ( produc6921648690025796265ist_fm @ A5 @ B6 ) )
     => ~ ( ( member_o2 @ A @ A5 )
         => ~ ( member_list_fm2 @ B @ ( B6 @ A ) ) ) ) ).

% SigmaE2
thf(fact_267_snd__eqD,axiom,
    ! [X5: rule,Y4: fm,A: fm] :
      ( ( ( product_snd_rule_fm @ ( product_Pair_rule_fm @ X5 @ Y4 ) )
        = A )
     => ( Y4 = A ) ) ).

% snd_eqD
thf(fact_268_snd__eqD,axiom,
    ! [X5: nat,Y4: nat,A: nat] :
      ( ( ( product_snd_nat_nat @ ( product_Pair_nat_nat @ X5 @ Y4 ) )
        = A )
     => ( Y4 = A ) ) ).

% snd_eqD
thf(fact_269_snd__eqD,axiom,
    ! [X5: produc6018962875968178549ist_fm,Y4: rule,A: rule] :
      ( ( ( produc6514711208036230658m_rule @ ( produc1733806532565653680m_rule @ X5 @ Y4 ) )
        = A )
     => ( Y4 = A ) ) ).

% snd_eqD
thf(fact_270_snd__conv,axiom,
    ! [X1: rule,X2: fm] :
      ( ( product_snd_rule_fm @ ( product_Pair_rule_fm @ X1 @ X2 ) )
      = X2 ) ).

% snd_conv
thf(fact_271_snd__conv,axiom,
    ! [X1: nat,X2: nat] :
      ( ( product_snd_nat_nat @ ( product_Pair_nat_nat @ X1 @ X2 ) )
      = X2 ) ).

% snd_conv
thf(fact_272_snd__conv,axiom,
    ! [X1: produc6018962875968178549ist_fm,X2: rule] :
      ( ( produc6514711208036230658m_rule @ ( produc1733806532565653680m_rule @ X1 @ X2 ) )
      = X2 ) ).

% snd_conv
thf(fact_273_Collect__case__prod__Sigma,axiom,
    ! [P3: product_prod_nat_nat > $o,Q: product_prod_nat_nat > product_prod_nat_nat > $o] :
      ( ( collec7088162979684241874at_nat
        @ ( produc6590410687421337004_nat_o
          @ ^ [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
              ( ( P3 @ X )
              & ( Q @ X @ Y ) ) ) )
      = ( produc2761391749766926216at_nat @ ( collec3392354462482085612at_nat @ P3 )
        @ ^ [X: product_prod_nat_nat] : ( collec3392354462482085612at_nat @ ( Q @ X ) ) ) ) ).

% Collect_case_prod_Sigma
thf(fact_274_Collect__case__prod__Sigma,axiom,
    ! [P3: product_prod_nat_nat > $o,Q: product_prod_nat_nat > nat > $o] :
      ( ( collec7029190964493513045at_nat
        @ ( produc8758269395934548017_nat_o
          @ ^ [X: product_prod_nat_nat,Y: nat] :
              ( ( P3 @ X )
              & ( Q @ X @ Y ) ) ) )
      = ( produc7672662199629908489at_nat @ ( collec3392354462482085612at_nat @ P3 )
        @ ^ [X: product_prod_nat_nat] : ( collect_nat @ ( Q @ X ) ) ) ) ).

% Collect_case_prod_Sigma
thf(fact_275_Collect__case__prod__Sigma,axiom,
    ! [P3: nat > $o,Q: nat > product_prod_nat_nat > $o] :
      ( ( collec5903703980526211963at_nat
        @ ( produc5864757623865647827_nat_o
          @ ^ [X: nat,Y: product_prod_nat_nat] :
              ( ( P3 @ X )
              & ( Q @ X @ Y ) ) ) )
      = ( produc1809337555817847783at_nat @ ( collect_nat @ P3 )
        @ ^ [X: nat] : ( collec3392354462482085612at_nat @ ( Q @ X ) ) ) ) ).

% Collect_case_prod_Sigma
thf(fact_276_Collect__case__prod__Sigma,axiom,
    ! [P3: rule > $o,Q: rule > fm > $o] :
      ( ( collec4199261512444720427ule_fm
        @ ( produc3561889649859641891e_fm_o
          @ ^ [X: rule,Y: fm] :
              ( ( P3 @ X )
              & ( Q @ X @ Y ) ) ) )
      = ( produc7224384929229372695ule_fm @ ( collect_rule @ P3 )
        @ ^ [X: rule] : ( collect_fm @ ( Q @ X ) ) ) ) ).

% Collect_case_prod_Sigma
thf(fact_277_Collect__case__prod__Sigma,axiom,
    ! [P3: nat > $o,Q: nat > nat > $o] :
      ( ( collec3392354462482085612at_nat
        @ ( produc6081775807080527818_nat_o
          @ ^ [X: nat,Y: nat] :
              ( ( P3 @ X )
              & ( Q @ X @ Y ) ) ) )
      = ( produc457027306803732586at_nat @ ( collect_nat @ P3 )
        @ ^ [X: nat] : ( collect_nat @ ( Q @ X ) ) ) ) ).

% Collect_case_prod_Sigma
thf(fact_278_snd__def,axiom,
    ( produc6514711208036230658m_rule
    = ( produc1291517305509475439e_rule
      @ ^ [X12: produc6018962875968178549ist_fm,X22: rule] : X22 ) ) ).

% snd_def
thf(fact_279_swap__product,axiom,
    ! [A5: set_fm,B6: set_rule] :
      ( ( image_2394372996319723537ule_fm
        @ ( produc7388818202471328049ule_fm
          @ ^ [I: fm,J: rule] : ( product_Pair_rule_fm @ J @ I ) )
        @ ( produc751277574191962813m_rule @ A5
          @ ^ [Uu: fm] : B6 ) )
      = ( produc7224384929229372695ule_fm @ B6
        @ ^ [Uu: rule] : A5 ) ) ).

% swap_product
thf(fact_280_swap__product,axiom,
    ! [A5: set_nat,B6: set_nat] :
      ( ( image_5168914502847457605at_nat
        @ ( produc2626176000494625587at_nat
          @ ^ [I: nat,J: nat] : ( product_Pair_nat_nat @ J @ I ) )
        @ ( produc457027306803732586at_nat @ A5
          @ ^ [Uu: nat] : B6 ) )
      = ( produc457027306803732586at_nat @ B6
        @ ^ [Uu: nat] : A5 ) ) ).

% swap_product
thf(fact_281__092_060open_062ev_A_Iholds_A_Iis__rule_ABetaCon_J_J_A_Isdrop_An_Asteps_J_092_060close_062,axiom,
    ( linear1816559245158234865m_rule
    @ ( linear7925859179493631106m_rule
      @ ^ [Step: produc340336539035504054m_rule] :
          ( ( produc6514711208036230658m_rule @ Step )
          = betaCon ) )
    @ ( sdrop_8169176516188972301m_rule @ n @ steps ) ) ).

% \<open>ev (holds (is_rule BetaCon)) (sdrop n steps)\<close>
thf(fact_282_sset__sproduct,axiom,
    ! [S1: stream2709947120125613254m_rule,S22: stream2709947120125613254m_rule] :
      ( ( sset_P3768201806174902753m_rule @ ( sprodu7455803463837168214m_rule @ S1 @ S22 ) )
      = ( produc1849974139216714806m_rule @ ( sset_P4484857331586881186m_rule @ S1 )
        @ ^ [Uu: produc340336539035504054m_rule] : ( sset_P4484857331586881186m_rule @ S22 ) ) ) ).

% sset_sproduct
thf(fact_283_sset__sproduct,axiom,
    ! [S1: stream4408948924543953275ist_fm,S22: stream_rule] :
      ( ( sset_P4484857331586881186m_rule @ ( sprodu2217682051239556511m_rule @ S1 @ S22 ) )
      = ( produc4664469595690820543m_rule @ ( sset_P5379989128061332361ist_fm @ S1 )
        @ ^ [Uu: produc6018962875968178549ist_fm] : ( sset_rule @ S22 ) ) ) ).

% sset_sproduct
thf(fact_284_snth__szip,axiom,
    ! [S1: stream2709947120125613254m_rule,S22: stream2709947120125613254m_rule,N3: nat] :
      ( ( snth_P8853764340393315953m_rule @ ( szip_P811719526838699976m_rule @ S1 @ S22 ) @ N3 )
      = ( produc6261311607089640965m_rule @ ( snth_P6679518042731451922m_rule @ S1 @ N3 ) @ ( snth_P6679518042731451922m_rule @ S22 @ N3 ) ) ) ).

% snth_szip
thf(fact_285_snth__szip,axiom,
    ! [S1: stream4408948924543953275ist_fm,S22: stream_rule,N3: nat] :
      ( ( snth_P6679518042731451922m_rule @ ( szip_P2924820683901490861m_rule @ S1 @ S22 ) @ N3 )
      = ( produc1733806532565653680m_rule @ ( snth_P7093566783922538521ist_fm @ S1 @ N3 ) @ ( snth_rule @ S22 @ N3 ) ) ) ).

% snth_szip
thf(fact_286_snth__szip,axiom,
    ! [S1: stream_rule,S22: stream_fm,N3: nat] :
      ( ( snth_P5527445687894232498ule_fm @ ( szip_rule_fm @ S1 @ S22 ) @ N3 )
      = ( product_Pair_rule_fm @ ( snth_rule @ S1 @ N3 ) @ ( snth_fm @ S22 @ N3 ) ) ) ).

% snth_szip
thf(fact_287_snth__szip,axiom,
    ! [S1: stream_nat,S22: stream_nat,N3: nat] :
      ( ( snth_P9042041890044241317at_nat @ ( szip_nat_nat @ S1 @ S22 ) @ N3 )
      = ( product_Pair_nat_nat @ ( snth_nat @ S1 @ N3 ) @ ( snth_nat @ S22 @ N3 ) ) ) ).

% snth_szip
thf(fact_288_stream__all__shift,axiom,
    ! [P3: produc340336539035504054m_rule > $o,Xs: list_P2774625669004309958m_rule,S2: stream2709947120125613254m_rule] :
      ( ( stream2134157564947672044m_rule @ P3 @ ( shift_2334844276748245581m_rule @ Xs @ S2 ) )
      = ( ( list_a9038620553111681026m_rule @ P3 @ Xs )
        & ( stream2134157564947672044m_rule @ P3 @ S2 ) ) ) ).

% stream_all_shift
thf(fact_289_holds_Osimps,axiom,
    ( linear7925859179493631106m_rule
    = ( ^ [P4: produc340336539035504054m_rule > $o,Xs2: stream2709947120125613254m_rule] : ( P4 @ ( shd_Pr4562317740776619530m_rule @ Xs2 ) ) ) ) ).

% holds.simps
thf(fact_290_holds_Oelims_I1_J,axiom,
    ! [X5: produc340336539035504054m_rule > $o,Xa2: stream2709947120125613254m_rule,Y4: $o] :
      ( ( ( linear7925859179493631106m_rule @ X5 @ Xa2 )
        = Y4 )
     => ( Y4
        = ( X5 @ ( shd_Pr4562317740776619530m_rule @ Xa2 ) ) ) ) ).

% holds.elims(1)
thf(fact_291_holds_Oelims_I2_J,axiom,
    ! [X5: produc340336539035504054m_rule > $o,Xa2: stream2709947120125613254m_rule] :
      ( ( linear7925859179493631106m_rule @ X5 @ Xa2 )
     => ( X5 @ ( shd_Pr4562317740776619530m_rule @ Xa2 ) ) ) ).

% holds.elims(2)
thf(fact_292_holds_Oelims_I3_J,axiom,
    ! [X5: produc340336539035504054m_rule > $o,Xa2: stream2709947120125613254m_rule] :
      ( ~ ( linear7925859179493631106m_rule @ X5 @ Xa2 )
     => ~ ( X5 @ ( shd_Pr4562317740776619530m_rule @ Xa2 ) ) ) ).

% holds.elims(3)
thf(fact_293_list__all__cong,axiom,
    ! [X5: list_o,Ya: list_o,P3: $o > $o,Pa: $o > $o] :
      ( ( X5 = Ya )
     => ( ! [Z4: $o] :
            ( ( member_o2 @ Z4 @ ( set_o2 @ Ya ) )
           => ( ( P3 @ Z4 )
              = ( Pa @ Z4 ) ) )
       => ( ( list_all_o @ P3 @ X5 )
          = ( list_all_o @ Pa @ Ya ) ) ) ) ).

% list_all_cong
thf(fact_294_list__all__cong,axiom,
    ! [X5: list_fm,Ya: list_fm,P3: fm > $o,Pa: fm > $o] :
      ( ( X5 = Ya )
     => ( ! [Z4: fm] :
            ( ( member_fm2 @ Z4 @ ( set_fm2 @ Ya ) )
           => ( ( P3 @ Z4 )
              = ( Pa @ Z4 ) ) )
       => ( ( list_all_fm @ P3 @ X5 )
          = ( list_all_fm @ Pa @ Ya ) ) ) ) ).

% list_all_cong
thf(fact_295_list__all__cong,axiom,
    ! [X5: list_list_fm,Ya: list_list_fm,P3: list_fm > $o,Pa: list_fm > $o] :
      ( ( X5 = Ya )
     => ( ! [Z4: list_fm] :
            ( ( member_list_fm2 @ Z4 @ ( set_list_fm2 @ Ya ) )
           => ( ( P3 @ Z4 )
              = ( Pa @ Z4 ) ) )
       => ( ( list_all_list_fm @ P3 @ X5 )
          = ( list_all_list_fm @ Pa @ Ya ) ) ) ) ).

% list_all_cong
thf(fact_296_list__all__cong,axiom,
    ! [X5: list_tm,Ya: list_tm,P3: tm > $o,Pa: tm > $o] :
      ( ( X5 = Ya )
     => ( ! [Z4: tm] :
            ( ( member_tm2 @ Z4 @ ( set_tm2 @ Ya ) )
           => ( ( P3 @ Z4 )
              = ( Pa @ Z4 ) ) )
       => ( ( list_all_tm @ P3 @ X5 )
          = ( list_all_tm @ Pa @ Ya ) ) ) ) ).

% list_all_cong
thf(fact_297_list__all__cong,axiom,
    ! [X5: list_P2774625669004309958m_rule,Ya: list_P2774625669004309958m_rule,P3: produc340336539035504054m_rule > $o,Pa: produc340336539035504054m_rule > $o] :
      ( ( X5 = Ya )
     => ( ! [Z4: produc340336539035504054m_rule] :
            ( ( member7231649785386036813m_rule @ Z4 @ ( set_Pr4534715572506550497m_rule @ Ya ) )
           => ( ( P3 @ Z4 )
              = ( Pa @ Z4 ) ) )
       => ( ( list_a9038620553111681026m_rule @ P3 @ X5 )
          = ( list_a9038620553111681026m_rule @ Pa @ Ya ) ) ) ) ).

% list_all_cong
thf(fact_298_sdrop__szip,axiom,
    ! [N3: nat,S1: stream2709947120125613254m_rule,S22: stream2709947120125613254m_rule] :
      ( ( sdrop_7192298464603511222m_rule @ N3 @ ( szip_P811719526838699976m_rule @ S1 @ S22 ) )
      = ( szip_P811719526838699976m_rule @ ( sdrop_8169176516188972301m_rule @ N3 @ S1 ) @ ( sdrop_8169176516188972301m_rule @ N3 @ S22 ) ) ) ).

% sdrop_szip
thf(fact_299_sdrop__szip,axiom,
    ! [N3: nat,S1: stream4408948924543953275ist_fm,S22: stream_rule] :
      ( ( sdrop_8169176516188972301m_rule @ N3 @ ( szip_P2924820683901490861m_rule @ S1 @ S22 ) )
      = ( szip_P2924820683901490861m_rule @ ( sdrop_9176333610110415838ist_fm @ N3 @ S1 ) @ ( sdrop_rule @ N3 @ S22 ) ) ) ).

% sdrop_szip
thf(fact_300_ev__False,axiom,
    ! [Omega: stream2709947120125613254m_rule] :
      ~ ( linear1816559245158234865m_rule
        @ ^ [X: stream2709947120125613254m_rule] : $false
        @ Omega ) ).

% ev_False
thf(fact_301_ev__or,axiom,
    ! [Phi: stream2709947120125613254m_rule > $o,Psi: stream2709947120125613254m_rule > $o] :
      ( ( linear1816559245158234865m_rule
        @ ^ [Xs2: stream2709947120125613254m_rule] :
            ( ( Phi @ Xs2 )
            | ( Psi @ Xs2 ) ) )
      = ( ^ [Xs2: stream2709947120125613254m_rule] :
            ( ( linear1816559245158234865m_rule @ Phi @ Xs2 )
            | ( linear1816559245158234865m_rule @ Psi @ Xs2 ) ) ) ) ).

% ev_or
thf(fact_302_ev__iff__sdrop,axiom,
    ( linear1816559245158234865m_rule
    = ( ^ [P4: stream2709947120125613254m_rule > $o,Omega2: stream2709947120125613254m_rule] :
        ? [M2: nat] : ( P4 @ ( sdrop_8169176516188972301m_rule @ M2 @ Omega2 ) ) ) ) ).

% ev_iff_sdrop
thf(fact_303_ev__shift,axiom,
    ! [Phi: stream2709947120125613254m_rule > $o,Xs: stream2709947120125613254m_rule,Xl: list_P2774625669004309958m_rule] :
      ( ( linear1816559245158234865m_rule @ Phi @ Xs )
     => ( linear1816559245158234865m_rule @ Phi @ ( shift_2334844276748245581m_rule @ Xl @ Xs ) ) ) ).

% ev_shift
thf(fact_304_ev__imp__shift,axiom,
    ! [Phi: stream2709947120125613254m_rule > $o,Xs: stream2709947120125613254m_rule] :
      ( ( linear1816559245158234865m_rule @ Phi @ Xs )
     => ? [Xl2: list_P2774625669004309958m_rule,Xs22: stream2709947120125613254m_rule] :
          ( ( Xs
            = ( shift_2334844276748245581m_rule @ Xl2 @ Xs22 ) )
          & ( Phi @ Xs22 ) ) ) ).

% ev_imp_shift
thf(fact_305_ev__prefix,axiom,
    ! [P3: produc340336539035504054m_rule > $o,Xs: stream2709947120125613254m_rule] :
      ( ( linear1816559245158234865m_rule @ ( linear7925859179493631106m_rule @ P3 ) @ Xs )
     => ? [Pre: list_P2774625669004309958m_rule,Suf: stream2709947120125613254m_rule] :
          ( ( list_a9038620553111681026m_rule
            @ ^ [Xs2: produc340336539035504054m_rule] :
                ~ ( P3 @ Xs2 )
            @ Pre )
          & ( linear7925859179493631106m_rule @ P3 @ Suf )
          & ( Xs
            = ( shift_2334844276748245581m_rule @ Pre @ Suf ) ) ) ) ).

% ev_prefix
thf(fact_306_szip_Osimps_I1_J,axiom,
    ! [S1: stream_rule,S22: stream_fm] :
      ( ( shd_Pr3686495215551550890ule_fm @ ( szip_rule_fm @ S1 @ S22 ) )
      = ( product_Pair_rule_fm @ ( shd_rule @ S1 ) @ ( shd_fm @ S22 ) ) ) ).

% szip.simps(1)
thf(fact_307_szip_Osimps_I1_J,axiom,
    ! [S1: stream_nat,S22: stream_nat] :
      ( ( shd_Pr4260400998323988397at_nat @ ( szip_nat_nat @ S1 @ S22 ) )
      = ( product_Pair_nat_nat @ ( shd_nat @ S1 ) @ ( shd_nat @ S22 ) ) ) ).

% szip.simps(1)
thf(fact_308_szip_Osimps_I1_J,axiom,
    ! [S1: stream2709947120125613254m_rule,S22: stream2709947120125613254m_rule] :
      ( ( shd_Pr4461660664618831993m_rule @ ( szip_P811719526838699976m_rule @ S1 @ S22 ) )
      = ( produc6261311607089640965m_rule @ ( shd_Pr4562317740776619530m_rule @ S1 ) @ ( shd_Pr4562317740776619530m_rule @ S22 ) ) ) ).

% szip.simps(1)
thf(fact_309_szip_Osimps_I1_J,axiom,
    ! [S1: stream4408948924543953275ist_fm,S22: stream_rule] :
      ( ( shd_Pr4562317740776619530m_rule @ ( szip_P2924820683901490861m_rule @ S1 @ S22 ) )
      = ( produc1733806532565653680m_rule @ ( shd_Pr3211216682057661985ist_fm @ S1 ) @ ( shd_rule @ S22 ) ) ) ).

% szip.simps(1)
thf(fact_310_holds__aand,axiom,
    ! [P3: produc340336539035504054m_rule > $o,Steps: stream2709947120125613254m_rule,Q: produc340336539035504054m_rule > $o] :
      ( ( ( linear7925859179493631106m_rule @ P3 @ Steps )
        & ( linear7925859179493631106m_rule @ Q @ Steps ) )
      = ( linear7925859179493631106m_rule
        @ ^ [Step: produc340336539035504054m_rule] :
            ( ( P3 @ Step )
            & ( Q @ Step ) )
        @ Steps ) ) ).

% holds_aand
thf(fact_311_list_Opred__True,axiom,
    ( ( list_a9038620553111681026m_rule
      @ ^ [Uu: produc340336539035504054m_rule] : $true )
    = ( ^ [Uu: list_P2774625669004309958m_rule] : $true ) ) ).

% list.pred_True
thf(fact_312_list_Opred__mono__strong,axiom,
    ! [P3: $o > $o,X5: list_o,Pa: $o > $o] :
      ( ( list_all_o @ P3 @ X5 )
     => ( ! [Z4: $o] :
            ( ( member_o2 @ Z4 @ ( set_o2 @ X5 ) )
           => ( ( P3 @ Z4 )
             => ( Pa @ Z4 ) ) )
       => ( list_all_o @ Pa @ X5 ) ) ) ).

% list.pred_mono_strong
thf(fact_313_list_Opred__mono__strong,axiom,
    ! [P3: fm > $o,X5: list_fm,Pa: fm > $o] :
      ( ( list_all_fm @ P3 @ X5 )
     => ( ! [Z4: fm] :
            ( ( member_fm2 @ Z4 @ ( set_fm2 @ X5 ) )
           => ( ( P3 @ Z4 )
             => ( Pa @ Z4 ) ) )
       => ( list_all_fm @ Pa @ X5 ) ) ) ).

% list.pred_mono_strong
thf(fact_314_list_Opred__mono__strong,axiom,
    ! [P3: list_fm > $o,X5: list_list_fm,Pa: list_fm > $o] :
      ( ( list_all_list_fm @ P3 @ X5 )
     => ( ! [Z4: list_fm] :
            ( ( member_list_fm2 @ Z4 @ ( set_list_fm2 @ X5 ) )
           => ( ( P3 @ Z4 )
             => ( Pa @ Z4 ) ) )
       => ( list_all_list_fm @ Pa @ X5 ) ) ) ).

% list.pred_mono_strong
thf(fact_315_list_Opred__mono__strong,axiom,
    ! [P3: tm > $o,X5: list_tm,Pa: tm > $o] :
      ( ( list_all_tm @ P3 @ X5 )
     => ( ! [Z4: tm] :
            ( ( member_tm2 @ Z4 @ ( set_tm2 @ X5 ) )
           => ( ( P3 @ Z4 )
             => ( Pa @ Z4 ) ) )
       => ( list_all_tm @ Pa @ X5 ) ) ) ).

% list.pred_mono_strong
thf(fact_316_list_Opred__mono__strong,axiom,
    ! [P3: produc340336539035504054m_rule > $o,X5: list_P2774625669004309958m_rule,Pa: produc340336539035504054m_rule > $o] :
      ( ( list_a9038620553111681026m_rule @ P3 @ X5 )
     => ( ! [Z4: produc340336539035504054m_rule] :
            ( ( member7231649785386036813m_rule @ Z4 @ ( set_Pr4534715572506550497m_rule @ X5 ) )
           => ( ( P3 @ Z4 )
             => ( Pa @ Z4 ) ) )
       => ( list_a9038620553111681026m_rule @ Pa @ X5 ) ) ) ).

% list.pred_mono_strong
thf(fact_317_sndI,axiom,
    ! [X5: product_prod_rule_fm,Y4: rule,Z: fm] :
      ( ( X5
        = ( product_Pair_rule_fm @ Y4 @ Z ) )
     => ( ( product_snd_rule_fm @ X5 )
        = Z ) ) ).

% sndI
thf(fact_318_sndI,axiom,
    ! [X5: product_prod_nat_nat,Y4: nat,Z: nat] :
      ( ( X5
        = ( product_Pair_nat_nat @ Y4 @ Z ) )
     => ( ( product_snd_nat_nat @ X5 )
        = Z ) ) ).

% sndI
thf(fact_319_sndI,axiom,
    ! [X5: produc340336539035504054m_rule,Y4: produc6018962875968178549ist_fm,Z: rule] :
      ( ( X5
        = ( produc1733806532565653680m_rule @ Y4 @ Z ) )
     => ( ( produc6514711208036230658m_rule @ X5 )
        = Z ) ) ).

% sndI
thf(fact_320_eq__snd__iff,axiom,
    ! [B: fm,P: product_prod_rule_fm] :
      ( ( B
        = ( product_snd_rule_fm @ P ) )
      = ( ? [A3: rule] :
            ( P
            = ( product_Pair_rule_fm @ A3 @ B ) ) ) ) ).

% eq_snd_iff
thf(fact_321_eq__snd__iff,axiom,
    ! [B: nat,P: product_prod_nat_nat] :
      ( ( B
        = ( product_snd_nat_nat @ P ) )
      = ( ? [A3: nat] :
            ( P
            = ( product_Pair_nat_nat @ A3 @ B ) ) ) ) ).

% eq_snd_iff
thf(fact_322_eq__snd__iff,axiom,
    ! [B: rule,P: produc340336539035504054m_rule] :
      ( ( B
        = ( produc6514711208036230658m_rule @ P ) )
      = ( ? [A3: produc6018962875968178549ist_fm] :
            ( P
            = ( produc1733806532565653680m_rule @ A3 @ B ) ) ) ) ).

% eq_snd_iff
thf(fact_323_tree__fms__def,axiom,
    ( tree_fms
    = ( ^ [Steps2: stream2709947120125613254m_rule] :
          ( comple2134933779557159616set_fm
          @ ( image_377115208147135190set_fm
            @ ^ [Ss: produc340336539035504054m_rule] : ( set_fm2 @ ( pseq @ Ss ) )
            @ ( sset_P4484857331586881186m_rule @ Steps2 ) ) ) ) ) ).

% tree_fms_def
thf(fact_324_set__product,axiom,
    ! [Xs: list_fm,Ys: list_fm] :
      ( ( set_Pr5149718152543245948_fm_fm @ ( product_fm_fm @ Xs @ Ys ) )
      = ( product_Sigma_fm_fm @ ( set_fm2 @ Xs )
        @ ^ [Uu: fm] : ( set_fm2 @ Ys ) ) ) ).

% set_product
thf(fact_325_set__product,axiom,
    ! [Xs: list_fm,Ys: list_list_fm] :
      ( ( set_Pr2319238409469995138ist_fm @ ( product_fm_list_fm @ Xs @ Ys ) )
      = ( produc6795173406573533582ist_fm @ ( set_fm2 @ Xs )
        @ ^ [Uu: fm] : ( set_list_fm2 @ Ys ) ) ) ).

% set_product
thf(fact_326_set__product,axiom,
    ! [Xs: list_fm,Ys: list_tm] :
      ( ( set_Pr5153670177628881134_fm_tm @ ( product_fm_tm @ Xs @ Ys ) )
      = ( product_Sigma_fm_tm @ ( set_fm2 @ Xs )
        @ ^ [Uu: fm] : ( set_tm2 @ Ys ) ) ) ).

% set_product
thf(fact_327_set__product,axiom,
    ! [Xs: list_list_fm,Ys: list_fm] :
      ( ( set_Pr2219715944279020790_fm_fm @ ( product_list_fm_fm @ Xs @ Ys ) )
      = ( produc4589463946070499586_fm_fm @ ( set_list_fm2 @ Xs )
        @ ^ [Uu: list_fm] : ( set_fm2 @ Ys ) ) ) ).

% set_product
thf(fact_328_set__product,axiom,
    ! [Xs: list_list_fm,Ys: list_list_fm] :
      ( ( set_Pr2277559400926618108ist_fm @ ( produc2951761921913218181ist_fm @ Xs @ Ys ) )
      = ( produc1166773336316955016ist_fm @ ( set_list_fm2 @ Xs )
        @ ^ [Uu: list_fm] : ( set_list_fm2 @ Ys ) ) ) ).

% set_product
thf(fact_329_set__product,axiom,
    ! [Xs: list_list_fm,Ys: list_tm] :
      ( ( set_Pr2223667969364655976_fm_tm @ ( product_list_fm_tm @ Xs @ Ys ) )
      = ( produc4589463946071417972_fm_tm @ ( set_list_fm2 @ Xs )
        @ ^ [Uu: list_fm] : ( set_tm2 @ Ys ) ) ) ).

% set_product
thf(fact_330_set__product,axiom,
    ! [Xs: list_tm,Ys: list_fm] :
      ( ( set_Pr486430433484093322_tm_fm @ ( product_tm_fm @ Xs @ Ys ) )
      = ( product_Sigma_tm_fm @ ( set_tm2 @ Xs )
        @ ^ [Uu: tm] : ( set_fm2 @ Ys ) ) ) ).

% set_product
thf(fact_331_set__product,axiom,
    ! [Xs: list_tm,Ys: list_list_fm] :
      ( ( set_Pr2432459030173936272ist_fm @ ( product_tm_list_fm @ Xs @ Ys ) )
      = ( produc6908394027277474716ist_fm @ ( set_tm2 @ Xs )
        @ ^ [Uu: tm] : ( set_list_fm2 @ Ys ) ) ) ).

% set_product
thf(fact_332_set__product,axiom,
    ! [Xs: list_tm,Ys: list_tm] :
      ( ( set_Pr490382458569728508_tm_tm @ ( product_tm_tm @ Xs @ Ys ) )
      = ( product_Sigma_tm_tm @ ( set_tm2 @ Xs )
        @ ^ [Uu: tm] : ( set_tm2 @ Ys ) ) ) ).

% set_product
thf(fact_333_ev__prefix__sdrop,axiom,
    ! [P3: produc340336539035504054m_rule > $o,Xs: stream2709947120125613254m_rule] :
      ( ( linear1816559245158234865m_rule @ ( linear7925859179493631106m_rule @ P3 ) @ Xs )
     => ? [N: nat] :
          ( ( list_a9038620553111681026m_rule
            @ ^ [Xs2: produc340336539035504054m_rule] :
                ~ ( P3 @ Xs2 )
            @ ( stake_5421812949518764133m_rule @ N @ Xs ) )
          & ( linear7925859179493631106m_rule @ P3 @ ( sdrop_8169176516188972301m_rule @ N @ Xs ) ) ) ) ).

% ev_prefix_sdrop
thf(fact_334_sdrop__wait,axiom,
    ! [Phi: stream2709947120125613254m_rule > $o,Xs: stream2709947120125613254m_rule] :
      ( ( linear1816559245158234865m_rule @ Phi @ Xs )
     => ( Phi @ ( sdrop_8169176516188972301m_rule @ ( linear8417479883454578709m_rule @ Phi @ Xs ) @ Xs ) ) ) ).

% sdrop_wait
thf(fact_335_member__product,axiom,
    ! [X5: produc340336539035504054m_rule,A5: set_Pr5202636777678657877ist_fm,B6: set_rule] :
      ( ( member7231649785386036813m_rule @ X5 @ ( produc8488632834096016437m_rule @ A5 @ B6 ) )
      = ( member7231649785386036813m_rule @ X5
        @ ( produc4664469595690820543m_rule @ A5
          @ ^ [Uu: produc6018962875968178549ist_fm] : B6 ) ) ) ).

% member_product
thf(fact_336_snd__image__times,axiom,
    ! [A5: set_Pr5202636777678657877ist_fm,B6: set_rule] :
      ( ( ( A5 = bot_bo1798318783834493889ist_fm )
       => ( ( image_4399899049665951567e_rule @ produc6514711208036230658m_rule
            @ ( produc4664469595690820543m_rule @ A5
              @ ^ [Uu: produc6018962875968178549ist_fm] : B6 ) )
          = bot_bot_set_rule ) )
      & ( ( A5 != bot_bo1798318783834493889ist_fm )
       => ( ( image_4399899049665951567e_rule @ produc6514711208036230658m_rule
            @ ( produc4664469595690820543m_rule @ A5
              @ ^ [Uu: produc6018962875968178549ist_fm] : B6 ) )
          = B6 ) ) ) ).

% snd_image_times
thf(fact_337_snd__image__times,axiom,
    ! [A5: set_o,B6: set_o] :
      ( ( ( A5 = bot_bot_set_o )
       => ( ( image_7896445794123959606_o_o_o @ product_snd_o_o
            @ ( product_Sigma_o_o @ A5
              @ ^ [Uu: $o] : B6 ) )
          = bot_bot_set_o ) )
      & ( ( A5 != bot_bot_set_o )
       => ( ( image_7896445794123959606_o_o_o @ product_snd_o_o
            @ ( product_Sigma_o_o @ A5
              @ ^ [Uu: $o] : B6 ) )
          = B6 ) ) ) ).

% snd_image_times
thf(fact_338_empty__iff,axiom,
    ! [C: fm] :
      ~ ( member_fm2 @ C @ bot_bot_set_fm ) ).

% empty_iff
thf(fact_339_empty__iff,axiom,
    ! [C: list_fm] :
      ~ ( member_list_fm2 @ C @ bot_bot_set_list_fm ) ).

% empty_iff
thf(fact_340_empty__iff,axiom,
    ! [C: produc340336539035504054m_rule] :
      ~ ( member7231649785386036813m_rule @ C @ bot_bo3426410157013319936m_rule ) ).

% empty_iff
thf(fact_341_empty__iff,axiom,
    ! [C: $o] :
      ~ ( member_o2 @ C @ bot_bot_set_o ) ).

% empty_iff
thf(fact_342_all__not__in__conv,axiom,
    ! [A5: set_fm] :
      ( ( ! [X: fm] :
            ~ ( member_fm2 @ X @ A5 ) )
      = ( A5 = bot_bot_set_fm ) ) ).

% all_not_in_conv
thf(fact_343_all__not__in__conv,axiom,
    ! [A5: set_list_fm] :
      ( ( ! [X: list_fm] :
            ~ ( member_list_fm2 @ X @ A5 ) )
      = ( A5 = bot_bot_set_list_fm ) ) ).

% all_not_in_conv
thf(fact_344_all__not__in__conv,axiom,
    ! [A5: set_Pr1822751329126368876m_rule] :
      ( ( ! [X: produc340336539035504054m_rule] :
            ~ ( member7231649785386036813m_rule @ X @ A5 ) )
      = ( A5 = bot_bo3426410157013319936m_rule ) ) ).

% all_not_in_conv
thf(fact_345_all__not__in__conv,axiom,
    ! [A5: set_o] :
      ( ( ! [X: $o] :
            ~ ( member_o2 @ X @ A5 ) )
      = ( A5 = bot_bot_set_o ) ) ).

% all_not_in_conv
thf(fact_346_Collect__empty__eq,axiom,
    ! [P3: product_prod_nat_nat > $o] :
      ( ( ( collec3392354462482085612at_nat @ P3 )
        = bot_bo2099793752762293965at_nat )
      = ( ! [X: product_prod_nat_nat] :
            ~ ( P3 @ X ) ) ) ).

% Collect_empty_eq
thf(fact_347_Collect__empty__eq,axiom,
    ! [P3: nat > $o] :
      ( ( ( collect_nat @ P3 )
        = bot_bot_set_nat )
      = ( ! [X: nat] :
            ~ ( P3 @ X ) ) ) ).

% Collect_empty_eq
thf(fact_348_Collect__empty__eq,axiom,
    ! [P3: $o > $o] :
      ( ( ( collect_o @ P3 )
        = bot_bot_set_o )
      = ( ! [X: $o] :
            ~ ( P3 @ X ) ) ) ).

% Collect_empty_eq
thf(fact_349_empty__Collect__eq,axiom,
    ! [P3: product_prod_nat_nat > $o] :
      ( ( bot_bo2099793752762293965at_nat
        = ( collec3392354462482085612at_nat @ P3 ) )
      = ( ! [X: product_prod_nat_nat] :
            ~ ( P3 @ X ) ) ) ).

% empty_Collect_eq
thf(fact_350_empty__Collect__eq,axiom,
    ! [P3: nat > $o] :
      ( ( bot_bot_set_nat
        = ( collect_nat @ P3 ) )
      = ( ! [X: nat] :
            ~ ( P3 @ X ) ) ) ).

% empty_Collect_eq
thf(fact_351_empty__Collect__eq,axiom,
    ! [P3: $o > $o] :
      ( ( bot_bot_set_o
        = ( collect_o @ P3 ) )
      = ( ! [X: $o] :
            ~ ( P3 @ X ) ) ) ).

% empty_Collect_eq
thf(fact_352_UN__ball__bex__simps_I3_J,axiom,
    ! [A5: set_set_fm,P3: fm > $o] :
      ( ( ? [X: fm] :
            ( ( member_fm2 @ X @ ( comple2134933779557159616set_fm @ A5 ) )
            & ( P3 @ X ) ) )
      = ( ? [X: set_fm] :
            ( ( member_set_fm @ X @ A5 )
            & ? [Y: fm] :
                ( ( member_fm2 @ Y @ X )
                & ( P3 @ Y ) ) ) ) ) ).

% UN_ball_bex_simps(3)
thf(fact_353_UN__ball__bex__simps_I1_J,axiom,
    ! [A5: set_set_fm,P3: fm > $o] :
      ( ( ! [X: fm] :
            ( ( member_fm2 @ X @ ( comple2134933779557159616set_fm @ A5 ) )
           => ( P3 @ X ) ) )
      = ( ! [X: set_fm] :
            ( ( member_set_fm @ X @ A5 )
           => ! [Y: fm] :
                ( ( member_fm2 @ Y @ X )
               => ( P3 @ Y ) ) ) ) ) ).

% UN_ball_bex_simps(1)
thf(fact_354_UnionI,axiom,
    ! [X7: set_list_fm,C2: set_set_list_fm,A5: list_fm] :
      ( ( member_set_list_fm @ X7 @ C2 )
     => ( ( member_list_fm2 @ A5 @ X7 )
       => ( member_list_fm2 @ A5 @ ( comple8784269564784259782ist_fm @ C2 ) ) ) ) ).

% UnionI
thf(fact_355_UnionI,axiom,
    ! [X7: set_Pr1822751329126368876m_rule,C2: set_se4309406418413213730m_rule,A5: produc340336539035504054m_rule] :
      ( ( member5802698409304902403m_rule @ X7 @ C2 )
     => ( ( member7231649785386036813m_rule @ A5 @ X7 )
       => ( member7231649785386036813m_rule @ A5 @ ( comple5648871819318716351m_rule @ C2 ) ) ) ) ).

% UnionI
thf(fact_356_UnionI,axiom,
    ! [X7: set_o,C2: set_set_o,A5: $o] :
      ( ( member_set_o @ X7 @ C2 )
     => ( ( member_o2 @ A5 @ X7 )
       => ( member_o2 @ A5 @ ( comple90263536869209701_set_o @ C2 ) ) ) ) ).

% UnionI
thf(fact_357_UnionI,axiom,
    ! [X7: set_fm,C2: set_set_fm,A5: fm] :
      ( ( member_set_fm @ X7 @ C2 )
     => ( ( member_fm2 @ A5 @ X7 )
       => ( member_fm2 @ A5 @ ( comple2134933779557159616set_fm @ C2 ) ) ) ) ).

% UnionI
thf(fact_358_Union__iff,axiom,
    ! [A5: list_fm,C2: set_set_list_fm] :
      ( ( member_list_fm2 @ A5 @ ( comple8784269564784259782ist_fm @ C2 ) )
      = ( ? [X: set_list_fm] :
            ( ( member_set_list_fm @ X @ C2 )
            & ( member_list_fm2 @ A5 @ X ) ) ) ) ).

% Union_iff
thf(fact_359_Union__iff,axiom,
    ! [A5: produc340336539035504054m_rule,C2: set_se4309406418413213730m_rule] :
      ( ( member7231649785386036813m_rule @ A5 @ ( comple5648871819318716351m_rule @ C2 ) )
      = ( ? [X: set_Pr1822751329126368876m_rule] :
            ( ( member5802698409304902403m_rule @ X @ C2 )
            & ( member7231649785386036813m_rule @ A5 @ X ) ) ) ) ).

% Union_iff
thf(fact_360_Union__iff,axiom,
    ! [A5: $o,C2: set_set_o] :
      ( ( member_o2 @ A5 @ ( comple90263536869209701_set_o @ C2 ) )
      = ( ? [X: set_o] :
            ( ( member_set_o @ X @ C2 )
            & ( member_o2 @ A5 @ X ) ) ) ) ).

% Union_iff
thf(fact_361_Union__iff,axiom,
    ! [A5: fm,C2: set_set_fm] :
      ( ( member_fm2 @ A5 @ ( comple2134933779557159616set_fm @ C2 ) )
      = ( ? [X: set_fm] :
            ( ( member_set_fm @ X @ C2 )
            & ( member_fm2 @ A5 @ X ) ) ) ) ).

% Union_iff
thf(fact_362_image__is__empty,axiom,
    ! [F: produc340336539035504054m_rule > set_fm,A5: set_Pr1822751329126368876m_rule] :
      ( ( ( image_377115208147135190set_fm @ F @ A5 )
        = bot_bot_set_set_fm )
      = ( A5 = bot_bo3426410157013319936m_rule ) ) ).

% image_is_empty
thf(fact_363_image__is__empty,axiom,
    ! [F: $o > $o,A5: set_o] :
      ( ( ( image_o_o2 @ F @ A5 )
        = bot_bot_set_o )
      = ( A5 = bot_bot_set_o ) ) ).

% image_is_empty
thf(fact_364_empty__is__image,axiom,
    ! [F: produc340336539035504054m_rule > set_fm,A5: set_Pr1822751329126368876m_rule] :
      ( ( bot_bot_set_set_fm
        = ( image_377115208147135190set_fm @ F @ A5 ) )
      = ( A5 = bot_bo3426410157013319936m_rule ) ) ).

% empty_is_image
thf(fact_365_empty__is__image,axiom,
    ! [F: $o > $o,A5: set_o] :
      ( ( bot_bot_set_o
        = ( image_o_o2 @ F @ A5 ) )
      = ( A5 = bot_bot_set_o ) ) ).

% empty_is_image
thf(fact_366_image__empty,axiom,
    ! [F: produc340336539035504054m_rule > set_fm] :
      ( ( image_377115208147135190set_fm @ F @ bot_bo3426410157013319936m_rule )
      = bot_bot_set_set_fm ) ).

% image_empty
thf(fact_367_image__empty,axiom,
    ! [F: $o > $o] :
      ( ( image_o_o2 @ F @ bot_bot_set_o )
      = bot_bot_set_o ) ).

% image_empty
thf(fact_368_Sup__bot__conv_I2_J,axiom,
    ! [A5: set_set_o] :
      ( ( bot_bot_set_o
        = ( comple90263536869209701_set_o @ A5 ) )
      = ( ! [X: set_o] :
            ( ( member_set_o @ X @ A5 )
           => ( X = bot_bot_set_o ) ) ) ) ).

% Sup_bot_conv(2)
thf(fact_369_Sup__bot__conv_I2_J,axiom,
    ! [A5: set_set_fm] :
      ( ( bot_bot_set_fm
        = ( comple2134933779557159616set_fm @ A5 ) )
      = ( ! [X: set_fm] :
            ( ( member_set_fm @ X @ A5 )
           => ( X = bot_bot_set_fm ) ) ) ) ).

% Sup_bot_conv(2)
thf(fact_370_Sup__bot__conv_I2_J,axiom,
    ! [A5: set_o] :
      ( ( bot_bot_o
        = ( complete_Sup_Sup_o @ A5 ) )
      = ( ! [X: $o] :
            ( ( member_o2 @ X @ A5 )
           => ( X = bot_bot_o ) ) ) ) ).

% Sup_bot_conv(2)
thf(fact_371_Sup__bot__conv_I1_J,axiom,
    ! [A5: set_set_o] :
      ( ( ( comple90263536869209701_set_o @ A5 )
        = bot_bot_set_o )
      = ( ! [X: set_o] :
            ( ( member_set_o @ X @ A5 )
           => ( X = bot_bot_set_o ) ) ) ) ).

% Sup_bot_conv(1)
thf(fact_372_Sup__bot__conv_I1_J,axiom,
    ! [A5: set_set_fm] :
      ( ( ( comple2134933779557159616set_fm @ A5 )
        = bot_bot_set_fm )
      = ( ! [X: set_fm] :
            ( ( member_set_fm @ X @ A5 )
           => ( X = bot_bot_set_fm ) ) ) ) ).

% Sup_bot_conv(1)
thf(fact_373_Sup__bot__conv_I1_J,axiom,
    ! [A5: set_o] :
      ( ( ( complete_Sup_Sup_o @ A5 )
        = bot_bot_o )
      = ( ! [X: $o] :
            ( ( member_o2 @ X @ A5 )
           => ( X = bot_bot_o ) ) ) ) ).

% Sup_bot_conv(1)
thf(fact_374_ball__UN,axiom,
    ! [B6: produc340336539035504054m_rule > set_fm,A5: set_Pr1822751329126368876m_rule,P3: fm > $o] :
      ( ( ! [X: fm] :
            ( ( member_fm2 @ X @ ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ B6 @ A5 ) ) )
           => ( P3 @ X ) ) )
      = ( ! [X: produc340336539035504054m_rule] :
            ( ( member7231649785386036813m_rule @ X @ A5 )
           => ! [Y: fm] :
                ( ( member_fm2 @ Y @ ( B6 @ X ) )
               => ( P3 @ Y ) ) ) ) ) ).

% ball_UN
thf(fact_375_bex__UN,axiom,
    ! [B6: produc340336539035504054m_rule > set_fm,A5: set_Pr1822751329126368876m_rule,P3: fm > $o] :
      ( ( ? [X: fm] :
            ( ( member_fm2 @ X @ ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ B6 @ A5 ) ) )
            & ( P3 @ X ) ) )
      = ( ? [X: produc340336539035504054m_rule] :
            ( ( member7231649785386036813m_rule @ X @ A5 )
            & ? [Y: fm] :
                ( ( member_fm2 @ Y @ ( B6 @ X ) )
                & ( P3 @ Y ) ) ) ) ) ).

% bex_UN
thf(fact_376_UN__ball__bex__simps_I2_J,axiom,
    ! [B6: produc340336539035504054m_rule > set_fm,A5: set_Pr1822751329126368876m_rule,P3: fm > $o] :
      ( ( ! [X: fm] :
            ( ( member_fm2 @ X @ ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ B6 @ A5 ) ) )
           => ( P3 @ X ) ) )
      = ( ! [X: produc340336539035504054m_rule] :
            ( ( member7231649785386036813m_rule @ X @ A5 )
           => ! [Y: fm] :
                ( ( member_fm2 @ Y @ ( B6 @ X ) )
               => ( P3 @ Y ) ) ) ) ) ).

% UN_ball_bex_simps(2)
thf(fact_377_UN__ball__bex__simps_I4_J,axiom,
    ! [B6: produc340336539035504054m_rule > set_fm,A5: set_Pr1822751329126368876m_rule,P3: fm > $o] :
      ( ( ? [X: fm] :
            ( ( member_fm2 @ X @ ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ B6 @ A5 ) ) )
            & ( P3 @ X ) ) )
      = ( ? [X: produc340336539035504054m_rule] :
            ( ( member7231649785386036813m_rule @ X @ A5 )
            & ? [Y: fm] :
                ( ( member_fm2 @ Y @ ( B6 @ X ) )
                & ( P3 @ Y ) ) ) ) ) ).

% UN_ball_bex_simps(4)
thf(fact_378_SUP__identity__eq,axiom,
    ! [A5: set_set_fm] :
      ( ( comple2134933779557159616set_fm
        @ ( image_set_fm_set_fm
          @ ^ [X: set_fm] : X
          @ A5 ) )
      = ( comple2134933779557159616set_fm @ A5 ) ) ).

% SUP_identity_eq
thf(fact_379_SUP__identity__eq,axiom,
    ! [A5: set_o] :
      ( ( complete_Sup_Sup_o
        @ ( image_o_o2
          @ ^ [X: $o] : X
          @ A5 ) )
      = ( complete_Sup_Sup_o @ A5 ) ) ).

% SUP_identity_eq
thf(fact_380_Times__empty,axiom,
    ! [A5: set_o,B6: set_o] :
      ( ( ( product_Sigma_o_o @ A5
          @ ^ [Uu: $o] : B6 )
        = bot_bo7073875226086086771od_o_o )
      = ( ( A5 = bot_bot_set_o )
        | ( B6 = bot_bot_set_o ) ) ) ).

% Times_empty
thf(fact_381_UN__I,axiom,
    ! [A: fm,A5: set_fm,B: $o,B6: fm > set_o] :
      ( ( member_fm2 @ A @ A5 )
     => ( ( member_o2 @ B @ ( B6 @ A ) )
       => ( member_o2 @ B @ ( comple90263536869209701_set_o @ ( image_fm_set_o @ B6 @ A5 ) ) ) ) ) ).

% UN_I
thf(fact_382_UN__I,axiom,
    ! [A: $o,A5: set_o,B: $o,B6: $o > set_o] :
      ( ( member_o2 @ A @ A5 )
     => ( ( member_o2 @ B @ ( B6 @ A ) )
       => ( member_o2 @ B @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B6 @ A5 ) ) ) ) ) ).

% UN_I
thf(fact_383_UN__I,axiom,
    ! [A: fm,A5: set_fm,B: fm,B6: fm > set_fm] :
      ( ( member_fm2 @ A @ A5 )
     => ( ( member_fm2 @ B @ ( B6 @ A ) )
       => ( member_fm2 @ B @ ( comple2134933779557159616set_fm @ ( image_fm_set_fm @ B6 @ A5 ) ) ) ) ) ).

% UN_I
thf(fact_384_UN__I,axiom,
    ! [A: $o,A5: set_o,B: fm,B6: $o > set_fm] :
      ( ( member_o2 @ A @ A5 )
     => ( ( member_fm2 @ B @ ( B6 @ A ) )
       => ( member_fm2 @ B @ ( comple2134933779557159616set_fm @ ( image_o_set_fm @ B6 @ A5 ) ) ) ) ) ).

% UN_I
thf(fact_385_UN__I,axiom,
    ! [A: fm,A5: set_fm,B: list_fm,B6: fm > set_list_fm] :
      ( ( member_fm2 @ A @ A5 )
     => ( ( member_list_fm2 @ B @ ( B6 @ A ) )
       => ( member_list_fm2 @ B @ ( comple8784269564784259782ist_fm @ ( image_fm_set_list_fm @ B6 @ A5 ) ) ) ) ) ).

% UN_I
thf(fact_386_UN__I,axiom,
    ! [A: list_fm,A5: set_list_fm,B: $o,B6: list_fm > set_o] :
      ( ( member_list_fm2 @ A @ A5 )
     => ( ( member_o2 @ B @ ( B6 @ A ) )
       => ( member_o2 @ B @ ( comple90263536869209701_set_o @ ( image_list_fm_set_o @ B6 @ A5 ) ) ) ) ) ).

% UN_I
thf(fact_387_UN__I,axiom,
    ! [A: $o,A5: set_o,B: list_fm,B6: $o > set_list_fm] :
      ( ( member_o2 @ A @ A5 )
     => ( ( member_list_fm2 @ B @ ( B6 @ A ) )
       => ( member_list_fm2 @ B @ ( comple8784269564784259782ist_fm @ ( image_o_set_list_fm @ B6 @ A5 ) ) ) ) ) ).

% UN_I
thf(fact_388_UN__I,axiom,
    ! [A: list_fm,A5: set_list_fm,B: fm,B6: list_fm > set_fm] :
      ( ( member_list_fm2 @ A @ A5 )
     => ( ( member_fm2 @ B @ ( B6 @ A ) )
       => ( member_fm2 @ B @ ( comple2134933779557159616set_fm @ ( image_list_fm_set_fm @ B6 @ A5 ) ) ) ) ) ).

% UN_I
thf(fact_389_UN__I,axiom,
    ! [A: list_fm,A5: set_list_fm,B: list_fm,B6: list_fm > set_list_fm] :
      ( ( member_list_fm2 @ A @ A5 )
     => ( ( member_list_fm2 @ B @ ( B6 @ A ) )
       => ( member_list_fm2 @ B @ ( comple8784269564784259782ist_fm @ ( image_1803486656538583717ist_fm @ B6 @ A5 ) ) ) ) ) ).

% UN_I
thf(fact_390_UN__I,axiom,
    ! [A: fm,A5: set_fm,B: produc340336539035504054m_rule,B6: fm > set_Pr1822751329126368876m_rule] :
      ( ( member_fm2 @ A @ A5 )
     => ( ( member7231649785386036813m_rule @ B @ ( B6 @ A ) )
       => ( member7231649785386036813m_rule @ B @ ( comple5648871819318716351m_rule @ ( image_7929583154892365210m_rule @ B6 @ A5 ) ) ) ) ) ).

% UN_I
thf(fact_391_UN__iff,axiom,
    ! [B: fm,B6: produc340336539035504054m_rule > set_fm,A5: set_Pr1822751329126368876m_rule] :
      ( ( member_fm2 @ B @ ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ B6 @ A5 ) ) )
      = ( ? [X: produc340336539035504054m_rule] :
            ( ( member7231649785386036813m_rule @ X @ A5 )
            & ( member_fm2 @ B @ ( B6 @ X ) ) ) ) ) ).

% UN_iff
thf(fact_392_Sup__empty,axiom,
    ( ( comple90263536869209701_set_o @ bot_bot_set_set_o )
    = bot_bot_set_o ) ).

% Sup_empty
thf(fact_393_Sup__empty,axiom,
    ( ( comple2134933779557159616set_fm @ bot_bot_set_set_fm )
    = bot_bot_set_fm ) ).

% Sup_empty
thf(fact_394_Sup__empty,axiom,
    ( ( complete_Sup_Sup_o @ bot_bot_set_o )
    = bot_bot_o ) ).

% Sup_empty
thf(fact_395_SUP__bot__conv_I2_J,axiom,
    ! [B6: produc340336539035504054m_rule > set_fm,A5: set_Pr1822751329126368876m_rule] :
      ( ( bot_bot_set_fm
        = ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ B6 @ A5 ) ) )
      = ( ! [X: produc340336539035504054m_rule] :
            ( ( member7231649785386036813m_rule @ X @ A5 )
           => ( ( B6 @ X )
              = bot_bot_set_fm ) ) ) ) ).

% SUP_bot_conv(2)
thf(fact_396_SUP__bot__conv_I1_J,axiom,
    ! [B6: produc340336539035504054m_rule > set_fm,A5: set_Pr1822751329126368876m_rule] :
      ( ( ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ B6 @ A5 ) )
        = bot_bot_set_fm )
      = ( ! [X: produc340336539035504054m_rule] :
            ( ( member7231649785386036813m_rule @ X @ A5 )
           => ( ( B6 @ X )
              = bot_bot_set_fm ) ) ) ) ).

% SUP_bot_conv(1)
thf(fact_397_SUP__bot,axiom,
    ! [A5: set_Pr1822751329126368876m_rule] :
      ( ( comple2134933779557159616set_fm
        @ ( image_377115208147135190set_fm
          @ ^ [X: produc340336539035504054m_rule] : bot_bot_set_fm
          @ A5 ) )
      = bot_bot_set_fm ) ).

% SUP_bot
thf(fact_398_SUP__const,axiom,
    ! [A5: set_Pr1822751329126368876m_rule,F: set_fm] :
      ( ( A5 != bot_bo3426410157013319936m_rule )
     => ( ( comple2134933779557159616set_fm
          @ ( image_377115208147135190set_fm
            @ ^ [I: produc340336539035504054m_rule] : F
            @ A5 ) )
        = F ) ) ).

% SUP_const
thf(fact_399_SUP__const,axiom,
    ! [A5: set_o,F: set_fm] :
      ( ( A5 != bot_bot_set_o )
     => ( ( comple2134933779557159616set_fm
          @ ( image_o_set_fm
            @ ^ [I: $o] : F
            @ A5 ) )
        = F ) ) ).

% SUP_const
thf(fact_400_SUP__const,axiom,
    ! [A5: set_o,F: $o] :
      ( ( A5 != bot_bot_set_o )
     => ( ( complete_Sup_Sup_o
          @ ( image_o_o2
            @ ^ [I: $o] : F
            @ A5 ) )
        = F ) ) ).

% SUP_const
thf(fact_401_UN__constant,axiom,
    ! [A5: set_o,C: set_o] :
      ( ( ( A5 = bot_bot_set_o )
       => ( ( comple90263536869209701_set_o
            @ ( image_o_set_o
              @ ^ [Y: $o] : C
              @ A5 ) )
          = bot_bot_set_o ) )
      & ( ( A5 != bot_bot_set_o )
       => ( ( comple90263536869209701_set_o
            @ ( image_o_set_o
              @ ^ [Y: $o] : C
              @ A5 ) )
          = C ) ) ) ).

% UN_constant
thf(fact_402_UN__constant,axiom,
    ! [A5: set_Pr1822751329126368876m_rule,C: set_fm] :
      ( ( ( A5 = bot_bo3426410157013319936m_rule )
       => ( ( comple2134933779557159616set_fm
            @ ( image_377115208147135190set_fm
              @ ^ [Y: produc340336539035504054m_rule] : C
              @ A5 ) )
          = bot_bot_set_fm ) )
      & ( ( A5 != bot_bo3426410157013319936m_rule )
       => ( ( comple2134933779557159616set_fm
            @ ( image_377115208147135190set_fm
              @ ^ [Y: produc340336539035504054m_rule] : C
              @ A5 ) )
          = C ) ) ) ).

% UN_constant
thf(fact_403_UN__constant,axiom,
    ! [A5: set_o,C: set_fm] :
      ( ( ( A5 = bot_bot_set_o )
       => ( ( comple2134933779557159616set_fm
            @ ( image_o_set_fm
              @ ^ [Y: $o] : C
              @ A5 ) )
          = bot_bot_set_fm ) )
      & ( ( A5 != bot_bot_set_o )
       => ( ( comple2134933779557159616set_fm
            @ ( image_o_set_fm
              @ ^ [Y: $o] : C
              @ A5 ) )
          = C ) ) ) ).

% UN_constant
thf(fact_404_empty__def,axiom,
    ( bot_bo2099793752762293965at_nat
    = ( collec3392354462482085612at_nat
      @ ^ [X: product_prod_nat_nat] : $false ) ) ).

% empty_def
thf(fact_405_empty__def,axiom,
    ( bot_bot_set_nat
    = ( collect_nat
      @ ^ [X: nat] : $false ) ) ).

% empty_def
thf(fact_406_empty__def,axiom,
    ( bot_bot_set_o
    = ( collect_o
      @ ^ [X: $o] : $false ) ) ).

% empty_def
thf(fact_407_UN__empty2,axiom,
    ! [A5: set_Pr1822751329126368876m_rule] :
      ( ( comple2134933779557159616set_fm
        @ ( image_377115208147135190set_fm
          @ ^ [X: produc340336539035504054m_rule] : bot_bot_set_fm
          @ A5 ) )
      = bot_bot_set_fm ) ).

% UN_empty2
thf(fact_408_UN__empty,axiom,
    ! [B6: $o > set_o] :
      ( ( comple90263536869209701_set_o @ ( image_o_set_o @ B6 @ bot_bot_set_o ) )
      = bot_bot_set_o ) ).

% UN_empty
thf(fact_409_UN__empty,axiom,
    ! [B6: produc340336539035504054m_rule > set_fm] :
      ( ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ B6 @ bot_bo3426410157013319936m_rule ) )
      = bot_bot_set_fm ) ).

% UN_empty
thf(fact_410_UN__empty,axiom,
    ! [B6: $o > set_fm] :
      ( ( comple2134933779557159616set_fm @ ( image_o_set_fm @ B6 @ bot_bot_set_o ) )
      = bot_bot_set_fm ) ).

% UN_empty
thf(fact_411_SUP__eq__const,axiom,
    ! [I2: set_fm,F: fm > set_fm,X5: set_fm] :
      ( ( I2 != bot_bot_set_fm )
     => ( ! [I3: fm] :
            ( ( member_fm2 @ I3 @ I2 )
           => ( ( F @ I3 )
              = X5 ) )
       => ( ( comple2134933779557159616set_fm @ ( image_fm_set_fm @ F @ I2 ) )
          = X5 ) ) ) ).

% SUP_eq_const
thf(fact_412_SUP__eq__const,axiom,
    ! [I2: set_list_fm,F: list_fm > set_fm,X5: set_fm] :
      ( ( I2 != bot_bot_set_list_fm )
     => ( ! [I3: list_fm] :
            ( ( member_list_fm2 @ I3 @ I2 )
           => ( ( F @ I3 )
              = X5 ) )
       => ( ( comple2134933779557159616set_fm @ ( image_list_fm_set_fm @ F @ I2 ) )
          = X5 ) ) ) ).

% SUP_eq_const
thf(fact_413_SUP__eq__const,axiom,
    ! [I2: set_Pr1822751329126368876m_rule,F: produc340336539035504054m_rule > set_fm,X5: set_fm] :
      ( ( I2 != bot_bo3426410157013319936m_rule )
     => ( ! [I3: produc340336539035504054m_rule] :
            ( ( member7231649785386036813m_rule @ I3 @ I2 )
           => ( ( F @ I3 )
              = X5 ) )
       => ( ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ F @ I2 ) )
          = X5 ) ) ) ).

% SUP_eq_const
thf(fact_414_SUP__eq__const,axiom,
    ! [I2: set_o,F: $o > set_fm,X5: set_fm] :
      ( ( I2 != bot_bot_set_o )
     => ( ! [I3: $o] :
            ( ( member_o2 @ I3 @ I2 )
           => ( ( F @ I3 )
              = X5 ) )
       => ( ( comple2134933779557159616set_fm @ ( image_o_set_fm @ F @ I2 ) )
          = X5 ) ) ) ).

% SUP_eq_const
thf(fact_415_SUP__eq__const,axiom,
    ! [I2: set_fm,F: fm > $o,X5: $o] :
      ( ( I2 != bot_bot_set_fm )
     => ( ! [I3: fm] :
            ( ( member_fm2 @ I3 @ I2 )
           => ( ( F @ I3 )
              = X5 ) )
       => ( ( complete_Sup_Sup_o @ ( image_fm_o2 @ F @ I2 ) )
          = X5 ) ) ) ).

% SUP_eq_const
thf(fact_416_SUP__eq__const,axiom,
    ! [I2: set_list_fm,F: list_fm > $o,X5: $o] :
      ( ( I2 != bot_bot_set_list_fm )
     => ( ! [I3: list_fm] :
            ( ( member_list_fm2 @ I3 @ I2 )
           => ( ( F @ I3 )
              = X5 ) )
       => ( ( complete_Sup_Sup_o @ ( image_list_fm_o2 @ F @ I2 ) )
          = X5 ) ) ) ).

% SUP_eq_const
thf(fact_417_SUP__eq__const,axiom,
    ! [I2: set_Pr1822751329126368876m_rule,F: produc340336539035504054m_rule > $o,X5: $o] :
      ( ( I2 != bot_bo3426410157013319936m_rule )
     => ( ! [I3: produc340336539035504054m_rule] :
            ( ( member7231649785386036813m_rule @ I3 @ I2 )
           => ( ( F @ I3 )
              = X5 ) )
       => ( ( complete_Sup_Sup_o @ ( image_8969478372357302939rule_o @ F @ I2 ) )
          = X5 ) ) ) ).

% SUP_eq_const
thf(fact_418_SUP__eq__const,axiom,
    ! [I2: set_o,F: $o > $o,X5: $o] :
      ( ( I2 != bot_bot_set_o )
     => ( ! [I3: $o] :
            ( ( member_o2 @ I3 @ I2 )
           => ( ( F @ I3 )
              = X5 ) )
       => ( ( complete_Sup_Sup_o @ ( image_o_o2 @ F @ I2 ) )
          = X5 ) ) ) ).

% SUP_eq_const
thf(fact_419_SUP__constant,axiom,
    ! [A5: set_o,C: set_o] :
      ( ( ( A5 = bot_bot_set_o )
       => ( ( comple90263536869209701_set_o
            @ ( image_o_set_o
              @ ^ [Y: $o] : C
              @ A5 ) )
          = bot_bot_set_o ) )
      & ( ( A5 != bot_bot_set_o )
       => ( ( comple90263536869209701_set_o
            @ ( image_o_set_o
              @ ^ [Y: $o] : C
              @ A5 ) )
          = C ) ) ) ).

% SUP_constant
thf(fact_420_SUP__constant,axiom,
    ! [A5: set_Pr1822751329126368876m_rule,C: set_fm] :
      ( ( ( A5 = bot_bo3426410157013319936m_rule )
       => ( ( comple2134933779557159616set_fm
            @ ( image_377115208147135190set_fm
              @ ^ [Y: produc340336539035504054m_rule] : C
              @ A5 ) )
          = bot_bot_set_fm ) )
      & ( ( A5 != bot_bo3426410157013319936m_rule )
       => ( ( comple2134933779557159616set_fm
            @ ( image_377115208147135190set_fm
              @ ^ [Y: produc340336539035504054m_rule] : C
              @ A5 ) )
          = C ) ) ) ).

% SUP_constant
thf(fact_421_SUP__constant,axiom,
    ! [A5: set_o,C: set_fm] :
      ( ( ( A5 = bot_bot_set_o )
       => ( ( comple2134933779557159616set_fm
            @ ( image_o_set_fm
              @ ^ [Y: $o] : C
              @ A5 ) )
          = bot_bot_set_fm ) )
      & ( ( A5 != bot_bot_set_o )
       => ( ( comple2134933779557159616set_fm
            @ ( image_o_set_fm
              @ ^ [Y: $o] : C
              @ A5 ) )
          = C ) ) ) ).

% SUP_constant
thf(fact_422_SUP__constant,axiom,
    ! [C: $o,A5: set_o] :
      ( ( complete_Sup_Sup_o
        @ ( image_o_o2
          @ ^ [Y: $o] : C
          @ A5 ) )
      = ( ( ( A5 = bot_bot_set_o )
         => bot_bot_o )
        & ( ( A5 != bot_bot_set_o )
         => C ) ) ) ).

% SUP_constant
thf(fact_423_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_424_SUP__empty,axiom,
    ! [F: produc340336539035504054m_rule > set_fm] :
      ( ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ F @ bot_bo3426410157013319936m_rule ) )
      = bot_bot_set_fm ) ).

% SUP_empty
thf(fact_425_SUP__empty,axiom,
    ! [F: $o > set_fm] :
      ( ( comple2134933779557159616set_fm @ ( image_o_set_fm @ F @ bot_bot_set_o ) )
      = bot_bot_set_fm ) ).

% SUP_empty
thf(fact_426_SUP__empty,axiom,
    ! [F: $o > $o] :
      ( ( complete_Sup_Sup_o @ ( image_o_o2 @ F @ bot_bot_set_o ) )
      = bot_bot_o ) ).

% SUP_empty
thf(fact_427_UNION__empty__conv_I1_J,axiom,
    ! [B6: produc340336539035504054m_rule > set_fm,A5: set_Pr1822751329126368876m_rule] :
      ( ( bot_bot_set_fm
        = ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ B6 @ A5 ) ) )
      = ( ! [X: produc340336539035504054m_rule] :
            ( ( member7231649785386036813m_rule @ X @ A5 )
           => ( ( B6 @ X )
              = bot_bot_set_fm ) ) ) ) ).

% UNION_empty_conv(1)
thf(fact_428_UNION__empty__conv_I2_J,axiom,
    ! [B6: produc340336539035504054m_rule > set_fm,A5: set_Pr1822751329126368876m_rule] :
      ( ( ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ B6 @ A5 ) )
        = bot_bot_set_fm )
      = ( ! [X: produc340336539035504054m_rule] :
            ( ( member7231649785386036813m_rule @ X @ A5 )
           => ( ( B6 @ X )
              = bot_bot_set_fm ) ) ) ) ).

% UNION_empty_conv(2)
thf(fact_429_emptyE,axiom,
    ! [A: fm] :
      ~ ( member_fm2 @ A @ bot_bot_set_fm ) ).

% emptyE
thf(fact_430_emptyE,axiom,
    ! [A: list_fm] :
      ~ ( member_list_fm2 @ A @ bot_bot_set_list_fm ) ).

% emptyE
thf(fact_431_emptyE,axiom,
    ! [A: produc340336539035504054m_rule] :
      ~ ( member7231649785386036813m_rule @ A @ bot_bo3426410157013319936m_rule ) ).

% emptyE
thf(fact_432_emptyE,axiom,
    ! [A: $o] :
      ~ ( member_o2 @ A @ bot_bot_set_o ) ).

% emptyE
thf(fact_433_equals0D,axiom,
    ! [A5: set_fm,A: fm] :
      ( ( A5 = bot_bot_set_fm )
     => ~ ( member_fm2 @ A @ A5 ) ) ).

% equals0D
thf(fact_434_equals0D,axiom,
    ! [A5: set_list_fm,A: list_fm] :
      ( ( A5 = bot_bot_set_list_fm )
     => ~ ( member_list_fm2 @ A @ A5 ) ) ).

% equals0D
thf(fact_435_equals0D,axiom,
    ! [A5: set_Pr1822751329126368876m_rule,A: produc340336539035504054m_rule] :
      ( ( A5 = bot_bo3426410157013319936m_rule )
     => ~ ( member7231649785386036813m_rule @ A @ A5 ) ) ).

% equals0D
thf(fact_436_equals0D,axiom,
    ! [A5: set_o,A: $o] :
      ( ( A5 = bot_bot_set_o )
     => ~ ( member_o2 @ A @ A5 ) ) ).

% equals0D
thf(fact_437_equals0I,axiom,
    ! [A5: set_fm] :
      ( ! [Y3: fm] :
          ~ ( member_fm2 @ Y3 @ A5 )
     => ( A5 = bot_bot_set_fm ) ) ).

% equals0I
thf(fact_438_equals0I,axiom,
    ! [A5: set_list_fm] :
      ( ! [Y3: list_fm] :
          ~ ( member_list_fm2 @ Y3 @ A5 )
     => ( A5 = bot_bot_set_list_fm ) ) ).

% equals0I
thf(fact_439_equals0I,axiom,
    ! [A5: set_Pr1822751329126368876m_rule] :
      ( ! [Y3: produc340336539035504054m_rule] :
          ~ ( member7231649785386036813m_rule @ Y3 @ A5 )
     => ( A5 = bot_bo3426410157013319936m_rule ) ) ).

% equals0I
thf(fact_440_equals0I,axiom,
    ! [A5: set_o] :
      ( ! [Y3: $o] :
          ~ ( member_o2 @ Y3 @ A5 )
     => ( A5 = bot_bot_set_o ) ) ).

% equals0I
thf(fact_441_ex__in__conv,axiom,
    ! [A5: set_fm] :
      ( ( ? [X: fm] : ( member_fm2 @ X @ A5 ) )
      = ( A5 != bot_bot_set_fm ) ) ).

% ex_in_conv
thf(fact_442_ex__in__conv,axiom,
    ! [A5: set_list_fm] :
      ( ( ? [X: list_fm] : ( member_list_fm2 @ X @ A5 ) )
      = ( A5 != bot_bot_set_list_fm ) ) ).

% ex_in_conv
thf(fact_443_ex__in__conv,axiom,
    ! [A5: set_Pr1822751329126368876m_rule] :
      ( ( ? [X: produc340336539035504054m_rule] : ( member7231649785386036813m_rule @ X @ A5 ) )
      = ( A5 != bot_bo3426410157013319936m_rule ) ) ).

% ex_in_conv
thf(fact_444_ex__in__conv,axiom,
    ! [A5: set_o] :
      ( ( ? [X: $o] : ( member_o2 @ X @ A5 ) )
      = ( A5 != bot_bot_set_o ) ) ).

% ex_in_conv
thf(fact_445_UnionE,axiom,
    ! [A5: list_fm,C2: set_set_list_fm] :
      ( ( member_list_fm2 @ A5 @ ( comple8784269564784259782ist_fm @ C2 ) )
     => ~ ! [X8: set_list_fm] :
            ( ( member_list_fm2 @ A5 @ X8 )
           => ~ ( member_set_list_fm @ X8 @ C2 ) ) ) ).

% UnionE
thf(fact_446_UnionE,axiom,
    ! [A5: produc340336539035504054m_rule,C2: set_se4309406418413213730m_rule] :
      ( ( member7231649785386036813m_rule @ A5 @ ( comple5648871819318716351m_rule @ C2 ) )
     => ~ ! [X8: set_Pr1822751329126368876m_rule] :
            ( ( member7231649785386036813m_rule @ A5 @ X8 )
           => ~ ( member5802698409304902403m_rule @ X8 @ C2 ) ) ) ).

% UnionE
thf(fact_447_UnionE,axiom,
    ! [A5: $o,C2: set_set_o] :
      ( ( member_o2 @ A5 @ ( comple90263536869209701_set_o @ C2 ) )
     => ~ ! [X8: set_o] :
            ( ( member_o2 @ A5 @ X8 )
           => ~ ( member_set_o @ X8 @ C2 ) ) ) ).

% UnionE
thf(fact_448_UnionE,axiom,
    ! [A5: fm,C2: set_set_fm] :
      ( ( member_fm2 @ A5 @ ( comple2134933779557159616set_fm @ C2 ) )
     => ~ ! [X8: set_fm] :
            ( ( member_fm2 @ A5 @ X8 )
           => ~ ( member_set_fm @ X8 @ C2 ) ) ) ).

% UnionE
thf(fact_449_Union__empty,axiom,
    ( ( comple90263536869209701_set_o @ bot_bot_set_set_o )
    = bot_bot_set_o ) ).

% Union_empty
thf(fact_450_Union__empty,axiom,
    ( ( comple2134933779557159616set_fm @ bot_bot_set_set_fm )
    = bot_bot_set_fm ) ).

% Union_empty
thf(fact_451_Union__empty__conv,axiom,
    ! [A5: set_set_o] :
      ( ( ( comple90263536869209701_set_o @ A5 )
        = bot_bot_set_o )
      = ( ! [X: set_o] :
            ( ( member_set_o @ X @ A5 )
           => ( X = bot_bot_set_o ) ) ) ) ).

% Union_empty_conv
thf(fact_452_Union__empty__conv,axiom,
    ! [A5: set_set_fm] :
      ( ( ( comple2134933779557159616set_fm @ A5 )
        = bot_bot_set_fm )
      = ( ! [X: set_fm] :
            ( ( member_set_fm @ X @ A5 )
           => ( X = bot_bot_set_fm ) ) ) ) ).

% Union_empty_conv
thf(fact_453_empty__Union__conv,axiom,
    ! [A5: set_set_o] :
      ( ( bot_bot_set_o
        = ( comple90263536869209701_set_o @ A5 ) )
      = ( ! [X: set_o] :
            ( ( member_set_o @ X @ A5 )
           => ( X = bot_bot_set_o ) ) ) ) ).

% empty_Union_conv
thf(fact_454_empty__Union__conv,axiom,
    ! [A5: set_set_fm] :
      ( ( bot_bot_set_fm
        = ( comple2134933779557159616set_fm @ A5 ) )
      = ( ! [X: set_fm] :
            ( ( member_set_fm @ X @ A5 )
           => ( X = bot_bot_set_fm ) ) ) ) ).

% empty_Union_conv
thf(fact_455_UN__UN__flatten,axiom,
    ! [C2: produc340336539035504054m_rule > set_fm,B6: produc340336539035504054m_rule > set_Pr1822751329126368876m_rule,A5: set_Pr1822751329126368876m_rule] :
      ( ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ C2 @ ( comple5648871819318716351m_rule @ ( image_2683796897779739177m_rule @ B6 @ A5 ) ) ) )
      = ( comple2134933779557159616set_fm
        @ ( image_377115208147135190set_fm
          @ ^ [Y: produc340336539035504054m_rule] : ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ C2 @ ( B6 @ Y ) ) )
          @ A5 ) ) ) ).

% UN_UN_flatten
thf(fact_456_UN__UN__flatten,axiom,
    ! [C2: fm > set_fm,B6: produc340336539035504054m_rule > set_fm,A5: set_Pr1822751329126368876m_rule] :
      ( ( comple2134933779557159616set_fm @ ( image_fm_set_fm @ C2 @ ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ B6 @ A5 ) ) ) )
      = ( comple2134933779557159616set_fm
        @ ( image_377115208147135190set_fm
          @ ^ [Y: produc340336539035504054m_rule] : ( comple2134933779557159616set_fm @ ( image_fm_set_fm @ C2 @ ( B6 @ Y ) ) )
          @ A5 ) ) ) ).

% UN_UN_flatten
thf(fact_457_UN__E,axiom,
    ! [B: $o,B6: fm > set_o,A5: set_fm] :
      ( ( member_o2 @ B @ ( comple90263536869209701_set_o @ ( image_fm_set_o @ B6 @ A5 ) ) )
     => ~ ! [X3: fm] :
            ( ( member_fm2 @ X3 @ A5 )
           => ~ ( member_o2 @ B @ ( B6 @ X3 ) ) ) ) ).

% UN_E
thf(fact_458_UN__E,axiom,
    ! [B: $o,B6: $o > set_o,A5: set_o] :
      ( ( member_o2 @ B @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B6 @ A5 ) ) )
     => ~ ! [X3: $o] :
            ( ( member_o2 @ X3 @ A5 )
           => ~ ( member_o2 @ B @ ( B6 @ X3 ) ) ) ) ).

% UN_E
thf(fact_459_UN__E,axiom,
    ! [B: fm,B6: fm > set_fm,A5: set_fm] :
      ( ( member_fm2 @ B @ ( comple2134933779557159616set_fm @ ( image_fm_set_fm @ B6 @ A5 ) ) )
     => ~ ! [X3: fm] :
            ( ( member_fm2 @ X3 @ A5 )
           => ~ ( member_fm2 @ B @ ( B6 @ X3 ) ) ) ) ).

% UN_E
thf(fact_460_UN__E,axiom,
    ! [B: fm,B6: $o > set_fm,A5: set_o] :
      ( ( member_fm2 @ B @ ( comple2134933779557159616set_fm @ ( image_o_set_fm @ B6 @ A5 ) ) )
     => ~ ! [X3: $o] :
            ( ( member_o2 @ X3 @ A5 )
           => ~ ( member_fm2 @ B @ ( B6 @ X3 ) ) ) ) ).

% UN_E
thf(fact_461_UN__E,axiom,
    ! [B: list_fm,B6: fm > set_list_fm,A5: set_fm] :
      ( ( member_list_fm2 @ B @ ( comple8784269564784259782ist_fm @ ( image_fm_set_list_fm @ B6 @ A5 ) ) )
     => ~ ! [X3: fm] :
            ( ( member_fm2 @ X3 @ A5 )
           => ~ ( member_list_fm2 @ B @ ( B6 @ X3 ) ) ) ) ).

% UN_E
thf(fact_462_UN__E,axiom,
    ! [B: list_fm,B6: $o > set_list_fm,A5: set_o] :
      ( ( member_list_fm2 @ B @ ( comple8784269564784259782ist_fm @ ( image_o_set_list_fm @ B6 @ A5 ) ) )
     => ~ ! [X3: $o] :
            ( ( member_o2 @ X3 @ A5 )
           => ~ ( member_list_fm2 @ B @ ( B6 @ X3 ) ) ) ) ).

% UN_E
thf(fact_463_UN__E,axiom,
    ! [B: $o,B6: list_fm > set_o,A5: set_list_fm] :
      ( ( member_o2 @ B @ ( comple90263536869209701_set_o @ ( image_list_fm_set_o @ B6 @ A5 ) ) )
     => ~ ! [X3: list_fm] :
            ( ( member_list_fm2 @ X3 @ A5 )
           => ~ ( member_o2 @ B @ ( B6 @ X3 ) ) ) ) ).

% UN_E
thf(fact_464_UN__E,axiom,
    ! [B: fm,B6: list_fm > set_fm,A5: set_list_fm] :
      ( ( member_fm2 @ B @ ( comple2134933779557159616set_fm @ ( image_list_fm_set_fm @ B6 @ A5 ) ) )
     => ~ ! [X3: list_fm] :
            ( ( member_list_fm2 @ X3 @ A5 )
           => ~ ( member_fm2 @ B @ ( B6 @ X3 ) ) ) ) ).

% UN_E
thf(fact_465_UN__E,axiom,
    ! [B: list_fm,B6: list_fm > set_list_fm,A5: set_list_fm] :
      ( ( member_list_fm2 @ B @ ( comple8784269564784259782ist_fm @ ( image_1803486656538583717ist_fm @ B6 @ A5 ) ) )
     => ~ ! [X3: list_fm] :
            ( ( member_list_fm2 @ X3 @ A5 )
           => ~ ( member_list_fm2 @ B @ ( B6 @ X3 ) ) ) ) ).

% UN_E
thf(fact_466_UN__E,axiom,
    ! [B: produc340336539035504054m_rule,B6: fm > set_Pr1822751329126368876m_rule,A5: set_fm] :
      ( ( member7231649785386036813m_rule @ B @ ( comple5648871819318716351m_rule @ ( image_7929583154892365210m_rule @ B6 @ A5 ) ) )
     => ~ ! [X3: fm] :
            ( ( member_fm2 @ X3 @ A5 )
           => ~ ( member7231649785386036813m_rule @ B @ ( B6 @ X3 ) ) ) ) ).

% UN_E
thf(fact_467_UN__extend__simps_I8_J,axiom,
    ! [B6: produc340336539035504054m_rule > set_fm,A5: set_se4309406418413213730m_rule] :
      ( ( comple2134933779557159616set_fm
        @ ( image_338663757272164256set_fm
          @ ^ [Y: set_Pr1822751329126368876m_rule] : ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ B6 @ Y ) )
          @ A5 ) )
      = ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ B6 @ ( comple5648871819318716351m_rule @ A5 ) ) ) ) ).

% UN_extend_simps(8)
thf(fact_468_UN__extend__simps_I8_J,axiom,
    ! [B6: fm > set_fm,A5: set_set_fm] :
      ( ( comple2134933779557159616set_fm
        @ ( image_set_fm_set_fm
          @ ^ [Y: set_fm] : ( comple2134933779557159616set_fm @ ( image_fm_set_fm @ B6 @ Y ) )
          @ A5 ) )
      = ( comple2134933779557159616set_fm @ ( image_fm_set_fm @ B6 @ ( comple2134933779557159616set_fm @ A5 ) ) ) ) ).

% UN_extend_simps(8)
thf(fact_469_UN__extend__simps_I9_J,axiom,
    ! [C2: produc340336539035504054m_rule > set_fm,B6: produc340336539035504054m_rule > set_Pr1822751329126368876m_rule,A5: set_Pr1822751329126368876m_rule] :
      ( ( comple2134933779557159616set_fm
        @ ( image_377115208147135190set_fm
          @ ^ [X: produc340336539035504054m_rule] : ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ C2 @ ( B6 @ X ) ) )
          @ A5 ) )
      = ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ C2 @ ( comple5648871819318716351m_rule @ ( image_2683796897779739177m_rule @ B6 @ A5 ) ) ) ) ) ).

% UN_extend_simps(9)
thf(fact_470_UN__extend__simps_I9_J,axiom,
    ! [C2: fm > set_fm,B6: produc340336539035504054m_rule > set_fm,A5: set_Pr1822751329126368876m_rule] :
      ( ( comple2134933779557159616set_fm
        @ ( image_377115208147135190set_fm
          @ ^ [X: produc340336539035504054m_rule] : ( comple2134933779557159616set_fm @ ( image_fm_set_fm @ C2 @ ( B6 @ X ) ) )
          @ A5 ) )
      = ( comple2134933779557159616set_fm @ ( image_fm_set_fm @ C2 @ ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ B6 @ A5 ) ) ) ) ) ).

% UN_extend_simps(9)
thf(fact_471_SUP__UNION,axiom,
    ! [F: produc340336539035504054m_rule > set_fm,G: produc340336539035504054m_rule > set_Pr1822751329126368876m_rule,A5: set_Pr1822751329126368876m_rule] :
      ( ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ F @ ( comple5648871819318716351m_rule @ ( image_2683796897779739177m_rule @ G @ A5 ) ) ) )
      = ( comple2134933779557159616set_fm
        @ ( image_377115208147135190set_fm
          @ ^ [Y: produc340336539035504054m_rule] : ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ F @ ( G @ Y ) ) )
          @ A5 ) ) ) ).

% SUP_UNION
thf(fact_472_SUP__UNION,axiom,
    ! [F: fm > set_fm,G: produc340336539035504054m_rule > set_fm,A5: set_Pr1822751329126368876m_rule] :
      ( ( comple2134933779557159616set_fm @ ( image_fm_set_fm @ F @ ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ G @ A5 ) ) ) )
      = ( comple2134933779557159616set_fm
        @ ( image_377115208147135190set_fm
          @ ^ [Y: produc340336539035504054m_rule] : ( comple2134933779557159616set_fm @ ( image_fm_set_fm @ F @ ( G @ Y ) ) )
          @ A5 ) ) ) ).

% SUP_UNION
thf(fact_473_SUP__UNION,axiom,
    ! [F: fm > $o,G: produc340336539035504054m_rule > set_fm,A5: set_Pr1822751329126368876m_rule] :
      ( ( complete_Sup_Sup_o @ ( image_fm_o2 @ F @ ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ G @ A5 ) ) ) )
      = ( complete_Sup_Sup_o
        @ ( image_8969478372357302939rule_o
          @ ^ [Y: produc340336539035504054m_rule] : ( complete_Sup_Sup_o @ ( image_fm_o2 @ F @ ( G @ Y ) ) )
          @ A5 ) ) ) ).

% SUP_UNION
thf(fact_474_UN__Times__distrib,axiom,
    ! [E: produc340336539035504054m_rule > set_fm,F13: produc340336539035504054m_rule > set_fm,A5: set_Pr1822751329126368876m_rule,B6: set_Pr1822751329126368876m_rule] :
      ( ( comple3052221160759688756_fm_fm
        @ ( image_4059705542473616959_fm_fm
          @ ( produc616892727416661549_fm_fm
            @ ^ [A3: produc340336539035504054m_rule,B3: produc340336539035504054m_rule] :
                ( product_Sigma_fm_fm @ ( E @ A3 )
                @ ^ [Uu: fm] : ( F13 @ B3 ) ) )
          @ ( produc1849974139216714806m_rule @ A5
            @ ^ [Uu: produc340336539035504054m_rule] : B6 ) ) )
      = ( product_Sigma_fm_fm @ ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ E @ A5 ) )
        @ ^ [Uu: fm] : ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ F13 @ B6 ) ) ) ) ).

% UN_Times_distrib
thf(fact_475_SUP__cong,axiom,
    ! [A5: set_fm,B6: set_fm,C2: fm > set_fm,D: fm > set_fm] :
      ( ( A5 = B6 )
     => ( ! [X3: fm] :
            ( ( member_fm2 @ X3 @ B6 )
           => ( ( C2 @ X3 )
              = ( D @ X3 ) ) )
       => ( ( comple2134933779557159616set_fm @ ( image_fm_set_fm @ C2 @ A5 ) )
          = ( comple2134933779557159616set_fm @ ( image_fm_set_fm @ D @ B6 ) ) ) ) ) ).

% SUP_cong
thf(fact_476_SUP__cong,axiom,
    ! [A5: set_list_fm,B6: set_list_fm,C2: list_fm > set_fm,D: list_fm > set_fm] :
      ( ( A5 = B6 )
     => ( ! [X3: list_fm] :
            ( ( member_list_fm2 @ X3 @ B6 )
           => ( ( C2 @ X3 )
              = ( D @ X3 ) ) )
       => ( ( comple2134933779557159616set_fm @ ( image_list_fm_set_fm @ C2 @ A5 ) )
          = ( comple2134933779557159616set_fm @ ( image_list_fm_set_fm @ D @ B6 ) ) ) ) ) ).

% SUP_cong
thf(fact_477_SUP__cong,axiom,
    ! [A5: set_Pr1822751329126368876m_rule,B6: set_Pr1822751329126368876m_rule,C2: produc340336539035504054m_rule > set_fm,D: produc340336539035504054m_rule > set_fm] :
      ( ( A5 = B6 )
     => ( ! [X3: produc340336539035504054m_rule] :
            ( ( member7231649785386036813m_rule @ X3 @ B6 )
           => ( ( C2 @ X3 )
              = ( D @ X3 ) ) )
       => ( ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ C2 @ A5 ) )
          = ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ D @ B6 ) ) ) ) ) ).

% SUP_cong
thf(fact_478_SUP__cong,axiom,
    ! [A5: set_o,B6: set_o,C2: $o > set_fm,D: $o > set_fm] :
      ( ( A5 = B6 )
     => ( ! [X3: $o] :
            ( ( member_o2 @ X3 @ B6 )
           => ( ( C2 @ X3 )
              = ( D @ X3 ) ) )
       => ( ( comple2134933779557159616set_fm @ ( image_o_set_fm @ C2 @ A5 ) )
          = ( comple2134933779557159616set_fm @ ( image_o_set_fm @ D @ B6 ) ) ) ) ) ).

% SUP_cong
thf(fact_479_SUP__cong,axiom,
    ! [A5: set_fm,B6: set_fm,C2: fm > $o,D: fm > $o] :
      ( ( A5 = B6 )
     => ( ! [X3: fm] :
            ( ( member_fm2 @ X3 @ B6 )
           => ( ( C2 @ X3 )
              = ( D @ X3 ) ) )
       => ( ( complete_Sup_Sup_o @ ( image_fm_o2 @ C2 @ A5 ) )
          = ( complete_Sup_Sup_o @ ( image_fm_o2 @ D @ B6 ) ) ) ) ) ).

% SUP_cong
thf(fact_480_SUP__cong,axiom,
    ! [A5: set_list_fm,B6: set_list_fm,C2: list_fm > $o,D: list_fm > $o] :
      ( ( A5 = B6 )
     => ( ! [X3: list_fm] :
            ( ( member_list_fm2 @ X3 @ B6 )
           => ( ( C2 @ X3 )
              = ( D @ X3 ) ) )
       => ( ( complete_Sup_Sup_o @ ( image_list_fm_o2 @ C2 @ A5 ) )
          = ( complete_Sup_Sup_o @ ( image_list_fm_o2 @ D @ B6 ) ) ) ) ) ).

% SUP_cong
thf(fact_481_SUP__cong,axiom,
    ! [A5: set_Pr1822751329126368876m_rule,B6: set_Pr1822751329126368876m_rule,C2: produc340336539035504054m_rule > $o,D: produc340336539035504054m_rule > $o] :
      ( ( A5 = B6 )
     => ( ! [X3: produc340336539035504054m_rule] :
            ( ( member7231649785386036813m_rule @ X3 @ B6 )
           => ( ( C2 @ X3 )
              = ( D @ X3 ) ) )
       => ( ( complete_Sup_Sup_o @ ( image_8969478372357302939rule_o @ C2 @ A5 ) )
          = ( complete_Sup_Sup_o @ ( image_8969478372357302939rule_o @ D @ B6 ) ) ) ) ) ).

% SUP_cong
thf(fact_482_SUP__cong,axiom,
    ! [A5: set_o,B6: set_o,C2: $o > $o,D: $o > $o] :
      ( ( A5 = B6 )
     => ( ! [X3: $o] :
            ( ( member_o2 @ X3 @ B6 )
           => ( ( C2 @ X3 )
              = ( D @ X3 ) ) )
       => ( ( complete_Sup_Sup_o @ ( image_o_o2 @ C2 @ A5 ) )
          = ( complete_Sup_Sup_o @ ( image_o_o2 @ D @ B6 ) ) ) ) ) ).

% SUP_cong
thf(fact_483_SUP__commute,axiom,
    ! [F: produc340336539035504054m_rule > produc340336539035504054m_rule > set_fm,B6: set_Pr1822751329126368876m_rule,A5: set_Pr1822751329126368876m_rule] :
      ( ( comple2134933779557159616set_fm
        @ ( image_377115208147135190set_fm
          @ ^ [I: produc340336539035504054m_rule] : ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ ( F @ I ) @ B6 ) )
          @ A5 ) )
      = ( comple2134933779557159616set_fm
        @ ( image_377115208147135190set_fm
          @ ^ [J: produc340336539035504054m_rule] :
              ( comple2134933779557159616set_fm
              @ ( image_377115208147135190set_fm
                @ ^ [I: produc340336539035504054m_rule] : ( F @ I @ J )
                @ A5 ) )
          @ B6 ) ) ) ).

% SUP_commute
thf(fact_484_image__Union,axiom,
    ! [F: produc340336539035504054m_rule > set_fm,S: set_se4309406418413213730m_rule] :
      ( ( image_377115208147135190set_fm @ F @ ( comple5648871819318716351m_rule @ S ) )
      = ( comple4326961365146087328set_fm @ ( image_4873481958868436096set_fm @ ( image_377115208147135190set_fm @ F ) @ S ) ) ) ).

% image_Union
thf(fact_485_image__Union,axiom,
    ! [F: fm > fm,S: set_set_fm] :
      ( ( image_fm_fm2 @ F @ ( comple2134933779557159616set_fm @ S ) )
      = ( comple2134933779557159616set_fm @ ( image_set_fm_set_fm @ ( image_fm_fm2 @ F ) @ S ) ) ) ).

% image_Union
thf(fact_486_image__UN,axiom,
    ! [F: fm > fm,B6: produc340336539035504054m_rule > set_fm,A5: set_Pr1822751329126368876m_rule] :
      ( ( image_fm_fm2 @ F @ ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ B6 @ A5 ) ) )
      = ( comple2134933779557159616set_fm
        @ ( image_377115208147135190set_fm
          @ ^ [X: produc340336539035504054m_rule] : ( image_fm_fm2 @ F @ ( B6 @ X ) )
          @ A5 ) ) ) ).

% image_UN
thf(fact_487_UN__extend__simps_I10_J,axiom,
    ! [B6: set_fm > set_fm,F: produc340336539035504054m_rule > set_fm,A5: set_Pr1822751329126368876m_rule] :
      ( ( comple2134933779557159616set_fm
        @ ( image_377115208147135190set_fm
          @ ^ [A3: produc340336539035504054m_rule] : ( B6 @ ( F @ A3 ) )
          @ A5 ) )
      = ( comple2134933779557159616set_fm @ ( image_set_fm_set_fm @ B6 @ ( image_377115208147135190set_fm @ F @ A5 ) ) ) ) ).

% UN_extend_simps(10)
thf(fact_488_UN__extend__simps_I10_J,axiom,
    ! [B6: produc340336539035504054m_rule > set_fm,F: produc340336539035504054m_rule > produc340336539035504054m_rule,A5: set_Pr1822751329126368876m_rule] :
      ( ( comple2134933779557159616set_fm
        @ ( image_377115208147135190set_fm
          @ ^ [A3: produc340336539035504054m_rule] : ( B6 @ ( F @ A3 ) )
          @ A5 ) )
      = ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ B6 @ ( image_2498997284783760755m_rule @ F @ A5 ) ) ) ) ).

% UN_extend_simps(10)
thf(fact_489_times__eq__iff,axiom,
    ! [A5: set_o,B6: set_o,C2: set_o,D: set_o] :
      ( ( ( product_Sigma_o_o @ A5
          @ ^ [Uu: $o] : B6 )
        = ( product_Sigma_o_o @ C2
          @ ^ [Uu: $o] : D ) )
      = ( ( ( A5 = C2 )
          & ( B6 = D ) )
        | ( ( ( A5 = bot_bot_set_o )
            | ( B6 = bot_bot_set_o ) )
          & ( ( C2 = bot_bot_set_o )
            | ( D = bot_bot_set_o ) ) ) ) ) ).

% times_eq_iff
thf(fact_490_stake__sdrop,axiom,
    ! [N3: nat,S2: stream2709947120125613254m_rule] :
      ( ( shift_2334844276748245581m_rule @ ( stake_5421812949518764133m_rule @ N3 @ S2 ) @ ( sdrop_8169176516188972301m_rule @ N3 @ S2 ) )
      = S2 ) ).

% stake_sdrop
thf(fact_491_snd__image__Sigma,axiom,
    ! [A5: set_Pr5202636777678657877ist_fm,B6: produc6018962875968178549ist_fm > set_rule] :
      ( ( image_4399899049665951567e_rule @ produc6514711208036230658m_rule @ ( produc4664469595690820543m_rule @ A5 @ B6 ) )
      = ( comple7665822446944348827t_rule @ ( image_7679748087749036664t_rule @ B6 @ A5 ) ) ) ).

% snd_image_Sigma
thf(fact_492_snd__image__Sigma,axiom,
    ! [A5: set_Pr1822751329126368876m_rule,B6: produc340336539035504054m_rule > set_fm] :
      ( ( image_2218355382418627706_fm_fm @ produc2872832660204944694ule_fm @ ( produc4849014715371551731ule_fm @ A5 @ B6 ) )
      = ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ B6 @ A5 ) ) ) ).

% snd_image_Sigma
thf(fact_493_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_494_ccSUP__empty,axiom,
    ! [F: produc340336539035504054m_rule > set_fm] :
      ( ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ F @ bot_bo3426410157013319936m_rule ) )
      = bot_bot_set_fm ) ).

% ccSUP_empty
thf(fact_495_ccSUP__empty,axiom,
    ! [F: $o > set_fm] :
      ( ( comple2134933779557159616set_fm @ ( image_o_set_fm @ F @ bot_bot_set_o ) )
      = bot_bot_set_fm ) ).

% ccSUP_empty
thf(fact_496_ccSUP__empty,axiom,
    ! [F: $o > $o] :
      ( ( complete_Sup_Sup_o @ ( image_o_o2 @ F @ bot_bot_set_o ) )
      = bot_bot_o ) ).

% ccSUP_empty
thf(fact_497_cSUP__const,axiom,
    ! [A5: set_Pr1822751329126368876m_rule,C: set_fm] :
      ( ( A5 != bot_bo3426410157013319936m_rule )
     => ( ( comple2134933779557159616set_fm
          @ ( image_377115208147135190set_fm
            @ ^ [X: produc340336539035504054m_rule] : C
            @ A5 ) )
        = C ) ) ).

% cSUP_const
thf(fact_498_cSUP__const,axiom,
    ! [A5: set_o,C: set_fm] :
      ( ( A5 != bot_bot_set_o )
     => ( ( comple2134933779557159616set_fm
          @ ( image_o_set_fm
            @ ^ [X: $o] : C
            @ A5 ) )
        = C ) ) ).

% cSUP_const
thf(fact_499_cSUP__const,axiom,
    ! [A5: set_o,C: $o] :
      ( ( A5 != bot_bot_set_o )
     => ( ( complete_Sup_Sup_o
          @ ( image_o_o2
            @ ^ [X: $o] : C
            @ A5 ) )
        = C ) ) ).

% cSUP_const
thf(fact_500_ccSUP__const,axiom,
    ! [A5: set_Pr1822751329126368876m_rule,F: set_fm] :
      ( ( A5 != bot_bo3426410157013319936m_rule )
     => ( ( comple2134933779557159616set_fm
          @ ( image_377115208147135190set_fm
            @ ^ [I: produc340336539035504054m_rule] : F
            @ A5 ) )
        = F ) ) ).

% ccSUP_const
thf(fact_501_ccSUP__const,axiom,
    ! [A5: set_o,F: set_fm] :
      ( ( A5 != bot_bot_set_o )
     => ( ( comple2134933779557159616set_fm
          @ ( image_o_set_fm
            @ ^ [I: $o] : F
            @ A5 ) )
        = F ) ) ).

% ccSUP_const
thf(fact_502_ccSUP__const,axiom,
    ! [A5: set_o,F: $o] :
      ( ( A5 != bot_bot_set_o )
     => ( ( complete_Sup_Sup_o
          @ ( image_o_o2
            @ ^ [I: $o] : F
            @ A5 ) )
        = F ) ) ).

% ccSUP_const
thf(fact_503_ccSUP__bot,axiom,
    ! [A5: set_Pr1822751329126368876m_rule] :
      ( ( comple2134933779557159616set_fm
        @ ( image_377115208147135190set_fm
          @ ^ [X: produc340336539035504054m_rule] : bot_bot_set_fm
          @ A5 ) )
      = bot_bot_set_fm ) ).

% ccSUP_bot
thf(fact_504_UN__UN__split__split__eq,axiom,
    ! [A5: produc6018962875968178549ist_fm > rule > produc6018962875968178549ist_fm > rule > set_fm,Y7: set_Pr1822751329126368876m_rule,X7: set_Pr1822751329126368876m_rule] :
      ( ( comple2134933779557159616set_fm
        @ ( image_377115208147135190set_fm
          @ ( produc271619708354958774set_fm
            @ ^ [X12: produc6018962875968178549ist_fm,X22: rule] : ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ ( produc271619708354958774set_fm @ ( A5 @ X12 @ X22 ) ) @ Y7 ) ) )
          @ X7 ) )
      = ( comple2134933779557159616set_fm
        @ ( image_377115208147135190set_fm
          @ ^ [X: produc340336539035504054m_rule] :
              ( comple2134933779557159616set_fm
              @ ( image_377115208147135190set_fm
                @ ^ [Y: produc340336539035504054m_rule] :
                    ( produc271619708354958774set_fm
                    @ ^ [X12: produc6018962875968178549ist_fm,X22: rule] : ( produc271619708354958774set_fm @ ( A5 @ X12 @ X22 ) @ Y )
                    @ X )
                @ Y7 ) )
          @ X7 ) ) ) ).

% UN_UN_split_split_eq
thf(fact_505_UN__constant__eq,axiom,
    ! [A: fm,A5: set_fm,F: fm > set_fm,C: set_fm] :
      ( ( member_fm2 @ A @ A5 )
     => ( ! [X3: fm] :
            ( ( member_fm2 @ X3 @ A5 )
           => ( ( F @ X3 )
              = C ) )
       => ( ( comple2134933779557159616set_fm @ ( image_fm_set_fm @ F @ A5 ) )
          = C ) ) ) ).

% UN_constant_eq
thf(fact_506_UN__constant__eq,axiom,
    ! [A: list_fm,A5: set_list_fm,F: list_fm > set_fm,C: set_fm] :
      ( ( member_list_fm2 @ A @ A5 )
     => ( ! [X3: list_fm] :
            ( ( member_list_fm2 @ X3 @ A5 )
           => ( ( F @ X3 )
              = C ) )
       => ( ( comple2134933779557159616set_fm @ ( image_list_fm_set_fm @ F @ A5 ) )
          = C ) ) ) ).

% UN_constant_eq
thf(fact_507_UN__constant__eq,axiom,
    ! [A: produc340336539035504054m_rule,A5: set_Pr1822751329126368876m_rule,F: produc340336539035504054m_rule > set_fm,C: set_fm] :
      ( ( member7231649785386036813m_rule @ A @ A5 )
     => ( ! [X3: produc340336539035504054m_rule] :
            ( ( member7231649785386036813m_rule @ X3 @ A5 )
           => ( ( F @ X3 )
              = C ) )
       => ( ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ F @ A5 ) )
          = C ) ) ) ).

% UN_constant_eq
thf(fact_508_UN__constant__eq,axiom,
    ! [A: $o,A5: set_o,F: $o > set_fm,C: set_fm] :
      ( ( member_o2 @ A @ A5 )
     => ( ! [X3: $o] :
            ( ( member_o2 @ X3 @ A5 )
           => ( ( F @ X3 )
              = C ) )
       => ( ( comple2134933779557159616set_fm @ ( image_o_set_fm @ F @ A5 ) )
          = C ) ) ) ).

% UN_constant_eq
thf(fact_509_sset__smerge,axiom,
    ! [Ss2: stream3752074346242807894m_rule] :
      ( ( sset_P4484857331586881186m_rule @ ( smerge193809993764105000m_rule @ Ss2 ) )
      = ( comple5648871819318716351m_rule @ ( image_3045026045495828537m_rule @ sset_P4484857331586881186m_rule @ ( sset_s1746085050419710898m_rule @ Ss2 ) ) ) ) ).

% sset_smerge
thf(fact_510_sset__smerge,axiom,
    ! [Ss2: stream_stream_fm] :
      ( ( sset_fm @ ( smerge_fm @ Ss2 ) )
      = ( comple2134933779557159616set_fm @ ( image_7285378133382788895set_fm @ sset_fm @ ( sset_stream_fm @ Ss2 ) ) ) ) ).

% sset_smerge
thf(fact_511_set__list__bind,axiom,
    ! [Xs: list_fm,F: fm > list_list_fm] :
      ( ( set_list_fm2 @ ( bind_fm_list_fm @ Xs @ F ) )
      = ( comple8784269564784259782ist_fm
        @ ( image_fm_set_list_fm
          @ ^ [X: fm] : ( set_list_fm2 @ ( F @ X ) )
          @ ( set_fm2 @ Xs ) ) ) ) ).

% set_list_bind
thf(fact_512_set__list__bind,axiom,
    ! [Xs: list_list_fm,F: list_fm > list_list_fm] :
      ( ( set_list_fm2 @ ( bind_list_fm_list_fm @ Xs @ F ) )
      = ( comple8784269564784259782ist_fm
        @ ( image_1803486656538583717ist_fm
          @ ^ [X: list_fm] : ( set_list_fm2 @ ( F @ X ) )
          @ ( set_list_fm2 @ Xs ) ) ) ) ).

% set_list_bind
thf(fact_513_set__list__bind,axiom,
    ! [Xs: list_tm,F: tm > list_list_fm] :
      ( ( set_list_fm2 @ ( bind_tm_list_fm @ Xs @ F ) )
      = ( comple8784269564784259782ist_fm
        @ ( image_tm_set_list_fm
          @ ^ [X: tm] : ( set_list_fm2 @ ( F @ X ) )
          @ ( set_tm2 @ Xs ) ) ) ) ).

% set_list_bind
thf(fact_514_set__list__bind,axiom,
    ! [Xs: list_fm,F: fm > list_tm] :
      ( ( set_tm2 @ ( bind_fm_tm @ Xs @ F ) )
      = ( comple2138885804642794802set_tm
        @ ( image_fm_set_tm
          @ ^ [X: fm] : ( set_tm2 @ ( F @ X ) )
          @ ( set_fm2 @ Xs ) ) ) ) ).

% set_list_bind
thf(fact_515_set__list__bind,axiom,
    ! [Xs: list_list_fm,F: list_fm > list_tm] :
      ( ( set_tm2 @ ( bind_list_fm_tm @ Xs @ F ) )
      = ( comple2138885804642794802set_tm
        @ ( image_list_fm_set_tm
          @ ^ [X: list_fm] : ( set_tm2 @ ( F @ X ) )
          @ ( set_list_fm2 @ Xs ) ) ) ) ).

% set_list_bind
thf(fact_516_set__list__bind,axiom,
    ! [Xs: list_tm,F: tm > list_tm] :
      ( ( set_tm2 @ ( bind_tm_tm @ Xs @ F ) )
      = ( comple2138885804642794802set_tm
        @ ( image_tm_set_tm
          @ ^ [X: tm] : ( set_tm2 @ ( F @ X ) )
          @ ( set_tm2 @ Xs ) ) ) ) ).

% set_list_bind
thf(fact_517_set__list__bind,axiom,
    ! [Xs: list_P2774625669004309958m_rule,F: produc340336539035504054m_rule > list_fm] :
      ( ( set_fm2 @ ( bind_P95098981060663284ule_fm @ Xs @ F ) )
      = ( comple2134933779557159616set_fm
        @ ( image_377115208147135190set_fm
          @ ^ [X: produc340336539035504054m_rule] : ( set_fm2 @ ( F @ X ) )
          @ ( set_Pr4534715572506550497m_rule @ Xs ) ) ) ) ).

% set_list_bind
thf(fact_518_set__list__bind,axiom,
    ! [Xs: list_fm,F: fm > list_fm] :
      ( ( set_fm2 @ ( bind_fm_fm @ Xs @ F ) )
      = ( comple2134933779557159616set_fm
        @ ( image_fm_set_fm
          @ ^ [X: fm] : ( set_fm2 @ ( F @ X ) )
          @ ( set_fm2 @ Xs ) ) ) ) ).

% set_list_bind
thf(fact_519_set__list__bind,axiom,
    ! [Xs: list_list_fm,F: list_fm > list_fm] :
      ( ( set_fm2 @ ( bind_list_fm_fm @ Xs @ F ) )
      = ( comple2134933779557159616set_fm
        @ ( image_list_fm_set_fm
          @ ^ [X: list_fm] : ( set_fm2 @ ( F @ X ) )
          @ ( set_list_fm2 @ Xs ) ) ) ) ).

% set_list_bind
thf(fact_520_set__list__bind,axiom,
    ! [Xs: list_tm,F: tm > list_fm] :
      ( ( set_fm2 @ ( bind_tm_fm @ Xs @ F ) )
      = ( comple2134933779557159616set_fm
        @ ( image_tm_set_fm
          @ ^ [X: tm] : ( set_fm2 @ ( F @ X ) )
          @ ( set_tm2 @ Xs ) ) ) ) ).

% set_list_bind
thf(fact_521_SUP__Sup__eq2,axiom,
    ! [S: set_se4309406418413213730m_rule] :
      ( ( comple5956027139101513408rule_o
        @ ( image_4328092588686559712rule_o
          @ ^ [I: set_Pr1822751329126368876m_rule,X: produc6018962875968178549ist_fm,Y: rule] : ( member7231649785386036813m_rule @ ( produc1733806532565653680m_rule @ X @ Y ) @ I )
          @ S ) )
      = ( ^ [X: produc6018962875968178549ist_fm,Y: rule] : ( member7231649785386036813m_rule @ ( produc1733806532565653680m_rule @ X @ Y ) @ ( comple5648871819318716351m_rule @ S ) ) ) ) ).

% SUP_Sup_eq2
thf(fact_522_SUP__Sup__eq2,axiom,
    ! [S: set_se6349671006166185922ule_fm] :
      ( ( comple8604870961538653134e_fm_o
        @ ( image_7649516631907998350e_fm_o
          @ ^ [I: set_Pr8969532344284305932ule_fm,X: rule,Y: fm] : ( member9168179392308330989ule_fm @ ( product_Pair_rule_fm @ X @ Y ) @ I )
          @ S ) )
      = ( ^ [X: rule,Y: fm] : ( member9168179392308330989ule_fm @ ( product_Pair_rule_fm @ X @ Y ) @ ( comple3749034766840985439ule_fm @ S ) ) ) ) ).

% SUP_Sup_eq2
thf(fact_523_SUP__Sup__eq2,axiom,
    ! [S: set_se7855581050983116737at_nat] :
      ( ( comple3592611370556534995_nat_o
        @ ( image_6923511907955291850_nat_o
          @ ^ [I: set_Pr1261947904930325089at_nat,X: nat,Y: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ I )
          @ S ) )
      = ( ^ [X: nat,Y: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( comple5685304695842803022at_nat @ S ) ) ) ) ).

% SUP_Sup_eq2
thf(fact_524_bot__empty__eq,axiom,
    ( bot_bot_fm_o
    = ( ^ [X: fm] : ( member_fm2 @ X @ bot_bot_set_fm ) ) ) ).

% bot_empty_eq
thf(fact_525_bot__empty__eq,axiom,
    ( bot_bot_list_fm_o
    = ( ^ [X: list_fm] : ( member_list_fm2 @ X @ bot_bot_set_list_fm ) ) ) ).

% bot_empty_eq
thf(fact_526_bot__empty__eq,axiom,
    ( bot_bo4917340991370493829rule_o
    = ( ^ [X: produc340336539035504054m_rule] : ( member7231649785386036813m_rule @ X @ bot_bo3426410157013319936m_rule ) ) ) ).

% bot_empty_eq
thf(fact_527_bot__empty__eq,axiom,
    ( bot_bot_o_o
    = ( ^ [X: $o] : ( member_o2 @ X @ bot_bot_set_o ) ) ) ).

% bot_empty_eq
thf(fact_528_bot__set__def,axiom,
    ( bot_bo2099793752762293965at_nat
    = ( collec3392354462482085612at_nat @ bot_bo482883023278783056_nat_o ) ) ).

% bot_set_def
thf(fact_529_bot__set__def,axiom,
    ( bot_bot_set_nat
    = ( collect_nat @ bot_bot_nat_o ) ) ).

% bot_set_def
thf(fact_530_bot__set__def,axiom,
    ( bot_bot_set_o
    = ( collect_o @ bot_bot_o_o ) ) ).

% bot_set_def
thf(fact_531_Sup__SUP__eq,axiom,
    ( comple173717950668161943t_fm_o
    = ( ^ [S4: set_list_fm_o,X: list_fm] : ( member_list_fm2 @ X @ ( comple8784269564784259782ist_fm @ ( image_4471151405024504270ist_fm @ collect_list_fm @ S4 ) ) ) ) ) ).

% Sup_SUP_eq
thf(fact_532_Sup__SUP__eq,axiom,
    ( comple3597286212584221062rule_o
    = ( ^ [S4: set_Pr3586676963299879929rule_o,X: produc340336539035504054m_rule] : ( member7231649785386036813m_rule @ X @ ( comple5648871819318716351m_rule @ ( image_3007582100768677440m_rule @ collec1009276759030335883m_rule @ S4 ) ) ) ) ) ).

% Sup_SUP_eq
thf(fact_533_Sup__SUP__eq,axiom,
    ( complete_Sup_Sup_o_o
    = ( ^ [S4: set_o_o,X: $o] : ( member_o2 @ X @ ( comple90263536869209701_set_o @ ( image_o_o_set_o @ collect_o @ S4 ) ) ) ) ) ).

% Sup_SUP_eq
thf(fact_534_Sup__SUP__eq,axiom,
    ( comple1901448594430626575_nat_o
    = ( ^ [S4: set_Pr5582243495563764594_nat_o,X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ ( comple5685304695842803022at_nat @ ( image_7124889717316225246at_nat @ collec3392354462482085612at_nat @ S4 ) ) ) ) ) ).

% Sup_SUP_eq
thf(fact_535_Sup__SUP__eq,axiom,
    ( comple8317665133742190828_nat_o
    = ( ^ [S4: set_nat_o,X: nat] : ( member_nat @ X @ ( comple7399068483239264473et_nat @ ( image_nat_o_set_nat @ collect_nat @ S4 ) ) ) ) ) ).

% Sup_SUP_eq
thf(fact_536_Sup__SUP__eq,axiom,
    ( comple3443612784043876893p_fm_o
    = ( ^ [S4: set_fm_o,X: fm] : ( member_fm2 @ X @ ( comple2134933779557159616set_fm @ ( image_fm_o_set_fm @ collect_fm @ S4 ) ) ) ) ) ).

% Sup_SUP_eq
thf(fact_537_Sup__SUP__eq2,axiom,
    ( comple5956027139101513408rule_o
    = ( ^ [S4: set_Pr7982104872110991011rule_o,X: produc6018962875968178549ist_fm,Y: rule] : ( member7231649785386036813m_rule @ ( produc1733806532565653680m_rule @ X @ Y ) @ ( comple5648871819318716351m_rule @ ( image_3007582100768677440m_rule @ collec1009276759030335883m_rule @ ( image_485512386923725989rule_o @ produc1247074552912856443rule_o @ S4 ) ) ) ) ) ) ).

% Sup_SUP_eq2
thf(fact_538_Sup__SUP__eq2,axiom,
    ( comple8604870961538653134e_fm_o
    = ( ^ [S4: set_rule_fm_o,X: rule,Y: fm] : ( member9168179392308330989ule_fm @ ( product_Pair_rule_fm @ X @ Y ) @ ( comple3749034766840985439ule_fm @ ( image_4034295225727782464ule_fm @ collec4199261512444720427ule_fm @ ( image_2502822993840964983e_fm_o @ produc3561889649859641891e_fm_o @ S4 ) ) ) ) ) ) ).

% Sup_SUP_eq2
thf(fact_539_Sup__SUP__eq2,axiom,
    ( comple3592611370556534995_nat_o
    = ( ^ [S4: set_nat_nat_o,X: nat,Y: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( comple5685304695842803022at_nat @ ( image_7124889717316225246at_nat @ collec3392354462482085612at_nat @ ( image_7429393840292777309_nat_o @ produc6081775807080527818_nat_o @ S4 ) ) ) ) ) ) ).

% Sup_SUP_eq2
thf(fact_540_bot__empty__eq2,axiom,
    ( bot_bo7194962629631833601rule_o
    = ( ^ [X: produc6018962875968178549ist_fm,Y: rule] : ( member7231649785386036813m_rule @ ( produc1733806532565653680m_rule @ X @ Y ) @ bot_bo3426410157013319936m_rule ) ) ) ).

% bot_empty_eq2
thf(fact_541_bot__empty__eq2,axiom,
    ( bot_bot_rule_fm_o
    = ( ^ [X: rule,Y: fm] : ( member9168179392308330989ule_fm @ ( product_Pair_rule_fm @ X @ Y ) @ bot_bo7376986642222277792ule_fm ) ) ) ).

% bot_empty_eq2
thf(fact_542_bot__empty__eq2,axiom,
    ( bot_bot_nat_nat_o
    = ( ^ [X: nat,Y: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ bot_bo2099793752762293965at_nat ) ) ) ).

% bot_empty_eq2
thf(fact_543_SUP__Sup__eq,axiom,
    ! [S: set_set_list_fm] :
      ( ( comple173717950668161943t_fm_o
        @ ( image_4048089684711339992t_fm_o
          @ ^ [I: set_list_fm,X: list_fm] : ( member_list_fm2 @ X @ I )
          @ S ) )
      = ( ^ [X: list_fm] : ( member_list_fm2 @ X @ ( comple8784269564784259782ist_fm @ S ) ) ) ) ).

% SUP_Sup_eq
thf(fact_544_SUP__Sup__eq,axiom,
    ! [S: set_se4309406418413213730m_rule] :
      ( ( comple3597286212584221062rule_o
        @ ( image_3805248563727310438rule_o
          @ ^ [I: set_Pr1822751329126368876m_rule,X: produc340336539035504054m_rule] : ( member7231649785386036813m_rule @ X @ I )
          @ S ) )
      = ( ^ [X: produc340336539035504054m_rule] : ( member7231649785386036813m_rule @ X @ ( comple5648871819318716351m_rule @ S ) ) ) ) ).

% SUP_Sup_eq
thf(fact_545_SUP__Sup__eq,axiom,
    ! [S: set_set_o] :
      ( ( complete_Sup_Sup_o_o
        @ ( image_set_o_o_o
          @ ^ [I: set_o,X: $o] : ( member_o2 @ X @ I )
          @ S ) )
      = ( ^ [X: $o] : ( member_o2 @ X @ ( comple90263536869209701_set_o @ S ) ) ) ) ).

% SUP_Sup_eq
thf(fact_546_SUP__Sup__eq,axiom,
    ! [S: set_set_fm] :
      ( ( comple3443612784043876893p_fm_o
        @ ( image_set_fm_fm_o
          @ ^ [I: set_fm,X: fm] : ( member_fm2 @ X @ I )
          @ S ) )
      = ( ^ [X: fm] : ( member_fm2 @ X @ ( comple2134933779557159616set_fm @ S ) ) ) ) ).

% SUP_Sup_eq
thf(fact_547_Sup__set__def,axiom,
    ( comple8784269564784259782ist_fm
    = ( ^ [A7: set_set_list_fm] :
          ( collect_list_fm
          @ ^ [X: list_fm] : ( complete_Sup_Sup_o @ ( image_set_list_fm_o @ ( member_list_fm2 @ X ) @ A7 ) ) ) ) ) ).

% Sup_set_def
thf(fact_548_Sup__set__def,axiom,
    ( comple5648871819318716351m_rule
    = ( ^ [A7: set_se4309406418413213730m_rule] :
          ( collec1009276759030335883m_rule
          @ ^ [X: produc340336539035504054m_rule] : ( complete_Sup_Sup_o @ ( image_4506595978750863461rule_o @ ( member7231649785386036813m_rule @ X ) @ A7 ) ) ) ) ) ).

% Sup_set_def
thf(fact_549_Sup__set__def,axiom,
    ( comple90263536869209701_set_o
    = ( ^ [A7: set_set_o] :
          ( collect_o
          @ ^ [X: $o] : ( complete_Sup_Sup_o @ ( image_set_o_o @ ( member_o2 @ X ) @ A7 ) ) ) ) ) ).

% Sup_set_def
thf(fact_550_Sup__set__def,axiom,
    ( comple5685304695842803022at_nat
    = ( ^ [A7: set_se7855581050983116737at_nat] :
          ( collec3392354462482085612at_nat
          @ ^ [X: product_prod_nat_nat] : ( complete_Sup_Sup_o @ ( image_7457375456213313148_nat_o @ ( member8440522571783428010at_nat @ X ) @ A7 ) ) ) ) ) ).

% Sup_set_def
thf(fact_551_Sup__set__def,axiom,
    ( comple7399068483239264473et_nat
    = ( ^ [A7: set_set_nat] :
          ( collect_nat
          @ ^ [X: nat] : ( complete_Sup_Sup_o @ ( image_set_nat_o @ ( member_nat @ X ) @ A7 ) ) ) ) ) ).

% Sup_set_def
thf(fact_552_Sup__set__def,axiom,
    ( comple2134933779557159616set_fm
    = ( ^ [A7: set_set_fm] :
          ( collect_fm
          @ ^ [X: fm] : ( complete_Sup_Sup_o @ ( image_set_fm_o @ ( member_fm2 @ X ) @ A7 ) ) ) ) ) ).

% Sup_set_def
thf(fact_553_SUP__UN__eq,axiom,
    ! [R4: produc340336539035504054m_rule > set_fm,S: set_Pr1822751329126368876m_rule] :
      ( ( comple3443612784043876893p_fm_o
        @ ( image_5627174782981912967e_fm_o
          @ ^ [I: produc340336539035504054m_rule,X: fm] : ( member_fm2 @ X @ ( R4 @ I ) )
          @ S ) )
      = ( ^ [X: fm] : ( member_fm2 @ X @ ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ R4 @ S ) ) ) ) ) ).

% SUP_UN_eq
thf(fact_554_Sigma__def,axiom,
    ( produc7224384929229372695ule_fm
    = ( ^ [A7: set_rule,B7: rule > set_fm] :
          ( comple3749034766840985439ule_fm
          @ ( image_1602722741251586725ule_fm
            @ ^ [X: rule] :
                ( comple3749034766840985439ule_fm
                @ ( image_5928080754618206522ule_fm
                  @ ^ [Y: fm] : ( insert684836149095756678ule_fm @ ( product_Pair_rule_fm @ X @ Y ) @ bot_bo7376986642222277792ule_fm )
                  @ ( B7 @ X ) ) )
            @ A7 ) ) ) ) ).

% Sigma_def
thf(fact_555_Sigma__def,axiom,
    ( produc457027306803732586at_nat
    = ( ^ [A7: set_nat,B7: nat > set_nat] :
          ( comple5685304695842803022at_nat
          @ ( image_7178329752028323786at_nat
            @ ^ [X: nat] :
                ( comple5685304695842803022at_nat
                @ ( image_7178329752028323786at_nat
                  @ ^ [Y: nat] : ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ bot_bo2099793752762293965at_nat )
                  @ ( B7 @ X ) ) )
            @ A7 ) ) ) ) ).

% Sigma_def
thf(fact_556_cSUP__UNION,axiom,
    ! [A5: set_fm,B6: fm > set_o,F: $o > $o] :
      ( ( A5 != bot_bot_set_fm )
     => ( ! [X3: fm] :
            ( ( member_fm2 @ X3 @ A5 )
           => ( ( B6 @ X3 )
             != bot_bot_set_o ) )
       => ( ( condit5488710616941104124bove_o
            @ ( comple90263536869209701_set_o
              @ ( image_fm_set_o
                @ ^ [X: fm] : ( image_o_o2 @ F @ ( B6 @ X ) )
                @ A5 ) ) )
         => ( ( complete_Sup_Sup_o @ ( image_o_o2 @ F @ ( comple90263536869209701_set_o @ ( image_fm_set_o @ B6 @ A5 ) ) ) )
            = ( complete_Sup_Sup_o
              @ ( image_fm_o2
                @ ^ [X: fm] : ( complete_Sup_Sup_o @ ( image_o_o2 @ F @ ( B6 @ X ) ) )
                @ A5 ) ) ) ) ) ) ).

% cSUP_UNION
thf(fact_557_cSUP__UNION,axiom,
    ! [A5: set_o,B6: $o > set_o,F: $o > $o] :
      ( ( A5 != bot_bot_set_o )
     => ( ! [X3: $o] :
            ( ( member_o2 @ X3 @ A5 )
           => ( ( B6 @ X3 )
             != bot_bot_set_o ) )
       => ( ( condit5488710616941104124bove_o
            @ ( comple90263536869209701_set_o
              @ ( image_o_set_o
                @ ^ [X: $o] : ( image_o_o2 @ F @ ( B6 @ X ) )
                @ A5 ) ) )
         => ( ( complete_Sup_Sup_o @ ( image_o_o2 @ F @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B6 @ A5 ) ) ) )
            = ( complete_Sup_Sup_o
              @ ( image_o_o2
                @ ^ [X: $o] : ( complete_Sup_Sup_o @ ( image_o_o2 @ F @ ( B6 @ X ) ) )
                @ A5 ) ) ) ) ) ) ).

% cSUP_UNION
thf(fact_558_cSUP__UNION,axiom,
    ! [A5: set_fm,B6: fm > set_fm,F: fm > $o] :
      ( ( A5 != bot_bot_set_fm )
     => ( ! [X3: fm] :
            ( ( member_fm2 @ X3 @ A5 )
           => ( ( B6 @ X3 )
             != bot_bot_set_fm ) )
       => ( ( condit5488710616941104124bove_o
            @ ( comple90263536869209701_set_o
              @ ( image_fm_set_o
                @ ^ [X: fm] : ( image_fm_o2 @ F @ ( B6 @ X ) )
                @ A5 ) ) )
         => ( ( complete_Sup_Sup_o @ ( image_fm_o2 @ F @ ( comple2134933779557159616set_fm @ ( image_fm_set_fm @ B6 @ A5 ) ) ) )
            = ( complete_Sup_Sup_o
              @ ( image_fm_o2
                @ ^ [X: fm] : ( complete_Sup_Sup_o @ ( image_fm_o2 @ F @ ( B6 @ X ) ) )
                @ A5 ) ) ) ) ) ) ).

% cSUP_UNION
thf(fact_559_cSUP__UNION,axiom,
    ! [A5: set_o,B6: $o > set_fm,F: fm > $o] :
      ( ( A5 != bot_bot_set_o )
     => ( ! [X3: $o] :
            ( ( member_o2 @ X3 @ A5 )
           => ( ( B6 @ X3 )
             != bot_bot_set_fm ) )
       => ( ( condit5488710616941104124bove_o
            @ ( comple90263536869209701_set_o
              @ ( image_o_set_o
                @ ^ [X: $o] : ( image_fm_o2 @ F @ ( B6 @ X ) )
                @ A5 ) ) )
         => ( ( complete_Sup_Sup_o @ ( image_fm_o2 @ F @ ( comple2134933779557159616set_fm @ ( image_o_set_fm @ B6 @ A5 ) ) ) )
            = ( complete_Sup_Sup_o
              @ ( image_o_o2
                @ ^ [X: $o] : ( complete_Sup_Sup_o @ ( image_fm_o2 @ F @ ( B6 @ X ) ) )
                @ A5 ) ) ) ) ) ) ).

% cSUP_UNION
thf(fact_560_cSUP__UNION,axiom,
    ! [A5: set_fm,B6: fm > set_o,F: $o > set_fm] :
      ( ( A5 != bot_bot_set_fm )
     => ( ! [X3: fm] :
            ( ( member_fm2 @ X3 @ A5 )
           => ( ( B6 @ X3 )
             != bot_bot_set_o ) )
       => ( ( condit7632555770478454199set_fm
            @ ( comple4326961365146087328set_fm
              @ ( image_fm_set_set_fm
                @ ^ [X: fm] : ( image_o_set_fm @ F @ ( B6 @ X ) )
                @ A5 ) ) )
         => ( ( comple2134933779557159616set_fm @ ( image_o_set_fm @ F @ ( comple90263536869209701_set_o @ ( image_fm_set_o @ B6 @ A5 ) ) ) )
            = ( comple2134933779557159616set_fm
              @ ( image_fm_set_fm
                @ ^ [X: fm] : ( comple2134933779557159616set_fm @ ( image_o_set_fm @ F @ ( B6 @ X ) ) )
                @ A5 ) ) ) ) ) ) ).

% cSUP_UNION
thf(fact_561_cSUP__UNION,axiom,
    ! [A5: set_o,B6: $o > set_o,F: $o > set_fm] :
      ( ( A5 != bot_bot_set_o )
     => ( ! [X3: $o] :
            ( ( member_o2 @ X3 @ A5 )
           => ( ( B6 @ X3 )
             != bot_bot_set_o ) )
       => ( ( condit7632555770478454199set_fm
            @ ( comple4326961365146087328set_fm
              @ ( image_o_set_set_fm
                @ ^ [X: $o] : ( image_o_set_fm @ F @ ( B6 @ X ) )
                @ A5 ) ) )
         => ( ( comple2134933779557159616set_fm @ ( image_o_set_fm @ F @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B6 @ A5 ) ) ) )
            = ( comple2134933779557159616set_fm
              @ ( image_o_set_fm
                @ ^ [X: $o] : ( comple2134933779557159616set_fm @ ( image_o_set_fm @ F @ ( B6 @ X ) ) )
                @ A5 ) ) ) ) ) ) ).

% cSUP_UNION
thf(fact_562_cSUP__UNION,axiom,
    ! [A5: set_fm,B6: fm > set_fm,F: fm > set_fm] :
      ( ( A5 != bot_bot_set_fm )
     => ( ! [X3: fm] :
            ( ( member_fm2 @ X3 @ A5 )
           => ( ( B6 @ X3 )
             != bot_bot_set_fm ) )
       => ( ( condit7632555770478454199set_fm
            @ ( comple4326961365146087328set_fm
              @ ( image_fm_set_set_fm
                @ ^ [X: fm] : ( image_fm_set_fm @ F @ ( B6 @ X ) )
                @ A5 ) ) )
         => ( ( comple2134933779557159616set_fm @ ( image_fm_set_fm @ F @ ( comple2134933779557159616set_fm @ ( image_fm_set_fm @ B6 @ A5 ) ) ) )
            = ( comple2134933779557159616set_fm
              @ ( image_fm_set_fm
                @ ^ [X: fm] : ( comple2134933779557159616set_fm @ ( image_fm_set_fm @ F @ ( B6 @ X ) ) )
                @ A5 ) ) ) ) ) ) ).

% cSUP_UNION
thf(fact_563_cSUP__UNION,axiom,
    ! [A5: set_o,B6: $o > set_fm,F: fm > set_fm] :
      ( ( A5 != bot_bot_set_o )
     => ( ! [X3: $o] :
            ( ( member_o2 @ X3 @ A5 )
           => ( ( B6 @ X3 )
             != bot_bot_set_fm ) )
       => ( ( condit7632555770478454199set_fm
            @ ( comple4326961365146087328set_fm
              @ ( image_o_set_set_fm
                @ ^ [X: $o] : ( image_fm_set_fm @ F @ ( B6 @ X ) )
                @ A5 ) ) )
         => ( ( comple2134933779557159616set_fm @ ( image_fm_set_fm @ F @ ( comple2134933779557159616set_fm @ ( image_o_set_fm @ B6 @ A5 ) ) ) )
            = ( comple2134933779557159616set_fm
              @ ( image_o_set_fm
                @ ^ [X: $o] : ( comple2134933779557159616set_fm @ ( image_fm_set_fm @ F @ ( B6 @ X ) ) )
                @ A5 ) ) ) ) ) ) ).

% cSUP_UNION
thf(fact_564_cSUP__UNION,axiom,
    ! [A5: set_list_fm,B6: list_fm > set_o,F: $o > $o] :
      ( ( A5 != bot_bot_set_list_fm )
     => ( ! [X3: list_fm] :
            ( ( member_list_fm2 @ X3 @ A5 )
           => ( ( B6 @ X3 )
             != bot_bot_set_o ) )
       => ( ( condit5488710616941104124bove_o
            @ ( comple90263536869209701_set_o
              @ ( image_list_fm_set_o
                @ ^ [X: list_fm] : ( image_o_o2 @ F @ ( B6 @ X ) )
                @ A5 ) ) )
         => ( ( complete_Sup_Sup_o @ ( image_o_o2 @ F @ ( comple90263536869209701_set_o @ ( image_list_fm_set_o @ B6 @ A5 ) ) ) )
            = ( complete_Sup_Sup_o
              @ ( image_list_fm_o2
                @ ^ [X: list_fm] : ( complete_Sup_Sup_o @ ( image_o_o2 @ F @ ( B6 @ X ) ) )
                @ A5 ) ) ) ) ) ) ).

% cSUP_UNION
thf(fact_565_cSUP__UNION,axiom,
    ! [A5: set_list_fm,B6: list_fm > set_fm,F: fm > $o] :
      ( ( A5 != bot_bot_set_list_fm )
     => ( ! [X3: list_fm] :
            ( ( member_list_fm2 @ X3 @ A5 )
           => ( ( B6 @ X3 )
             != bot_bot_set_fm ) )
       => ( ( condit5488710616941104124bove_o
            @ ( comple90263536869209701_set_o
              @ ( image_list_fm_set_o
                @ ^ [X: list_fm] : ( image_fm_o2 @ F @ ( B6 @ X ) )
                @ A5 ) ) )
         => ( ( complete_Sup_Sup_o @ ( image_fm_o2 @ F @ ( comple2134933779557159616set_fm @ ( image_list_fm_set_fm @ B6 @ A5 ) ) ) )
            = ( complete_Sup_Sup_o
              @ ( image_list_fm_o2
                @ ^ [X: list_fm] : ( complete_Sup_Sup_o @ ( image_fm_o2 @ F @ ( B6 @ X ) ) )
                @ A5 ) ) ) ) ) ) ).

% cSUP_UNION
thf(fact_566_Set_Ois__empty__def,axiom,
    ( is_empty_o
    = ( ^ [A7: set_o] : ( A7 = bot_bot_set_o ) ) ) ).

% Set.is_empty_def
thf(fact_567_refl__on__UNION,axiom,
    ! [S: set_Pr1822751329126368876m_rule,A5: produc340336539035504054m_rule > set_fm,R4: produc340336539035504054m_rule > set_Pr4706815898642364871_fm_fm] :
      ( ! [X3: produc340336539035504054m_rule] :
          ( ( member7231649785386036813m_rule @ X3 @ S )
         => ( refl_on_fm @ ( A5 @ X3 ) @ ( R4 @ X3 ) ) )
     => ( refl_on_fm @ ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ A5 @ S ) ) @ ( comple3052221160759688756_fm_fm @ ( image_9204174774680167498_fm_fm @ R4 @ S ) ) ) ) ).

% refl_on_UNION
thf(fact_568_set__concat,axiom,
    ! [Xs: list_list_list_fm] :
      ( ( set_list_fm2 @ ( concat_list_fm @ Xs ) )
      = ( comple8784269564784259782ist_fm @ ( image_3687226712311829663ist_fm @ set_list_fm2 @ ( set_list_list_fm2 @ Xs ) ) ) ) ).

% set_concat
thf(fact_569_set__concat,axiom,
    ! [Xs: list_list_tm] :
      ( ( set_tm2 @ ( concat_tm @ Xs ) )
      = ( comple2138885804642794802set_tm @ ( image_list_tm_set_tm @ set_tm2 @ ( set_list_tm2 @ Xs ) ) ) ) ).

% set_concat
thf(fact_570_set__concat,axiom,
    ! [Xs: list_list_fm] :
      ( ( set_fm2 @ ( concat_fm @ Xs ) )
      = ( comple2134933779557159616set_fm @ ( image_list_fm_set_fm @ set_fm2 @ ( set_list_fm2 @ Xs ) ) ) ) ).

% set_concat
thf(fact_571_UNION__singleton__eq__range,axiom,
    ! [F: produc340336539035504054m_rule > set_fm,A5: set_Pr1822751329126368876m_rule] :
      ( ( comple4326961365146087328set_fm
        @ ( image_1174956300053327798set_fm
          @ ^ [X: produc340336539035504054m_rule] : ( insert_set_fm @ ( F @ X ) @ bot_bot_set_set_fm )
          @ A5 ) )
      = ( image_377115208147135190set_fm @ F @ A5 ) ) ).

% UNION_singleton_eq_range
thf(fact_572_UNION__singleton__eq__range,axiom,
    ! [F: produc340336539035504054m_rule > fm,A5: set_Pr1822751329126368876m_rule] :
      ( ( comple2134933779557159616set_fm
        @ ( image_377115208147135190set_fm
          @ ^ [X: produc340336539035504054m_rule] : ( insert_fm2 @ ( F @ X ) @ bot_bot_set_fm )
          @ A5 ) )
      = ( image_2303374963288105462ule_fm @ F @ A5 ) ) ).

% UNION_singleton_eq_range
thf(fact_573_insertCI,axiom,
    ! [A: fm,B6: set_fm,B: fm] :
      ( ( ~ ( member_fm2 @ A @ B6 )
       => ( A = B ) )
     => ( member_fm2 @ A @ ( insert_fm2 @ B @ B6 ) ) ) ).

% insertCI
thf(fact_574_insertCI,axiom,
    ! [A: list_fm,B6: set_list_fm,B: list_fm] :
      ( ( ~ ( member_list_fm2 @ A @ B6 )
       => ( A = B ) )
     => ( member_list_fm2 @ A @ ( insert_list_fm2 @ B @ B6 ) ) ) ).

% insertCI
thf(fact_575_insertCI,axiom,
    ! [A: produc340336539035504054m_rule,B6: set_Pr1822751329126368876m_rule,B: produc340336539035504054m_rule] :
      ( ( ~ ( member7231649785386036813m_rule @ A @ B6 )
       => ( A = B ) )
     => ( member7231649785386036813m_rule @ A @ ( insert775845832097999846m_rule @ B @ B6 ) ) ) ).

% insertCI
thf(fact_576_insertCI,axiom,
    ! [A: $o,B6: set_o,B: $o] :
      ( ( ~ ( member_o2 @ A @ B6 )
       => ( A = B ) )
     => ( member_o2 @ A @ ( insert_o2 @ B @ B6 ) ) ) ).

% insertCI
thf(fact_577_insert__iff,axiom,
    ! [A: fm,B: fm,A5: set_fm] :
      ( ( member_fm2 @ A @ ( insert_fm2 @ B @ A5 ) )
      = ( ( A = B )
        | ( member_fm2 @ A @ A5 ) ) ) ).

% insert_iff
thf(fact_578_insert__iff,axiom,
    ! [A: list_fm,B: list_fm,A5: set_list_fm] :
      ( ( member_list_fm2 @ A @ ( insert_list_fm2 @ B @ A5 ) )
      = ( ( A = B )
        | ( member_list_fm2 @ A @ A5 ) ) ) ).

% insert_iff
thf(fact_579_insert__iff,axiom,
    ! [A: produc340336539035504054m_rule,B: produc340336539035504054m_rule,A5: set_Pr1822751329126368876m_rule] :
      ( ( member7231649785386036813m_rule @ A @ ( insert775845832097999846m_rule @ B @ A5 ) )
      = ( ( A = B )
        | ( member7231649785386036813m_rule @ A @ A5 ) ) ) ).

% insert_iff
thf(fact_580_insert__iff,axiom,
    ! [A: $o,B: $o,A5: set_o] :
      ( ( member_o2 @ A @ ( insert_o2 @ B @ A5 ) )
      = ( ( A = B )
        | ( member_o2 @ A @ A5 ) ) ) ).

% insert_iff
thf(fact_581_insert__absorb2,axiom,
    ! [X5: $o,A5: set_o] :
      ( ( insert_o2 @ X5 @ ( insert_o2 @ X5 @ A5 ) )
      = ( insert_o2 @ X5 @ A5 ) ) ).

% insert_absorb2
thf(fact_582_insert__image,axiom,
    ! [X5: fm,A5: set_fm,F: fm > $o] :
      ( ( member_fm2 @ X5 @ A5 )
     => ( ( insert_o2 @ ( F @ X5 ) @ ( image_fm_o2 @ F @ A5 ) )
        = ( image_fm_o2 @ F @ A5 ) ) ) ).

% insert_image
thf(fact_583_insert__image,axiom,
    ! [X5: list_fm,A5: set_list_fm,F: list_fm > $o] :
      ( ( member_list_fm2 @ X5 @ A5 )
     => ( ( insert_o2 @ ( F @ X5 ) @ ( image_list_fm_o2 @ F @ A5 ) )
        = ( image_list_fm_o2 @ F @ A5 ) ) ) ).

% insert_image
thf(fact_584_insert__image,axiom,
    ! [X5: produc340336539035504054m_rule,A5: set_Pr1822751329126368876m_rule,F: produc340336539035504054m_rule > set_fm] :
      ( ( member7231649785386036813m_rule @ X5 @ A5 )
     => ( ( insert_set_fm @ ( F @ X5 ) @ ( image_377115208147135190set_fm @ F @ A5 ) )
        = ( image_377115208147135190set_fm @ F @ A5 ) ) ) ).

% insert_image
thf(fact_585_insert__image,axiom,
    ! [X5: produc340336539035504054m_rule,A5: set_Pr1822751329126368876m_rule,F: produc340336539035504054m_rule > $o] :
      ( ( member7231649785386036813m_rule @ X5 @ A5 )
     => ( ( insert_o2 @ ( F @ X5 ) @ ( image_8969478372357302939rule_o @ F @ A5 ) )
        = ( image_8969478372357302939rule_o @ F @ A5 ) ) ) ).

% insert_image
thf(fact_586_insert__image,axiom,
    ! [X5: $o,A5: set_o,F: $o > $o] :
      ( ( member_o2 @ X5 @ A5 )
     => ( ( insert_o2 @ ( F @ X5 ) @ ( image_o_o2 @ F @ A5 ) )
        = ( image_o_o2 @ F @ A5 ) ) ) ).

% insert_image
thf(fact_587_image__insert,axiom,
    ! [F: produc340336539035504054m_rule > set_fm,A: produc340336539035504054m_rule,B6: set_Pr1822751329126368876m_rule] :
      ( ( image_377115208147135190set_fm @ F @ ( insert775845832097999846m_rule @ A @ B6 ) )
      = ( insert_set_fm @ ( F @ A ) @ ( image_377115208147135190set_fm @ F @ B6 ) ) ) ).

% image_insert
thf(fact_588_image__insert,axiom,
    ! [F: $o > $o,A: $o,B6: set_o] :
      ( ( image_o_o2 @ F @ ( insert_o2 @ A @ B6 ) )
      = ( insert_o2 @ ( F @ A ) @ ( image_o_o2 @ F @ B6 ) ) ) ).

% image_insert
thf(fact_589_singletonI,axiom,
    ! [A: fm] : ( member_fm2 @ A @ ( insert_fm2 @ A @ bot_bot_set_fm ) ) ).

% singletonI
thf(fact_590_singletonI,axiom,
    ! [A: list_fm] : ( member_list_fm2 @ A @ ( insert_list_fm2 @ A @ bot_bot_set_list_fm ) ) ).

% singletonI
thf(fact_591_singletonI,axiom,
    ! [A: produc340336539035504054m_rule] : ( member7231649785386036813m_rule @ A @ ( insert775845832097999846m_rule @ A @ bot_bo3426410157013319936m_rule ) ) ).

% singletonI
thf(fact_592_singletonI,axiom,
    ! [A: $o] : ( member_o2 @ A @ ( insert_o2 @ A @ bot_bot_set_o ) ) ).

% singletonI
thf(fact_593_ImageI,axiom,
    ! [A: fm,B: fm,R4: set_Pr4706815898642364871_fm_fm,A5: set_fm] :
      ( ( member7780952600467998736_fm_fm @ ( product_Pair_fm_fm @ A @ B ) @ R4 )
     => ( ( member_fm2 @ A @ A5 )
       => ( member_fm2 @ B @ ( image_fm_fm @ R4 @ A5 ) ) ) ) ).

% ImageI
thf(fact_594_ImageI,axiom,
    ! [A: fm,B: $o,R4: set_Pr2813250270173872364d_fm_o,A5: set_fm] :
      ( ( member1731323461681951669d_fm_o @ ( product_Pair_fm_o @ A @ B ) @ R4 )
     => ( ( member_fm2 @ A @ A5 )
       => ( member_o2 @ B @ ( image_fm_o @ R4 @ A5 ) ) ) ) ).

% ImageI
thf(fact_595_ImageI,axiom,
    ! [A: $o,B: fm,R4: set_Pr3863124289995447138d_o_fm,A5: set_o] :
      ( ( member6824368184199080235d_o_fm @ ( product_Pair_o_fm @ A @ B ) @ R4 )
     => ( ( member_o2 @ A @ A5 )
       => ( member_fm2 @ B @ ( image_o_fm @ R4 @ A5 ) ) ) ) ).

% ImageI
thf(fact_596_ImageI,axiom,
    ! [A: $o,B: $o,R4: set_Product_prod_o_o,A5: set_o] :
      ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ B ) @ R4 )
     => ( ( member_o2 @ A @ A5 )
       => ( member_o2 @ B @ ( image_o_o @ R4 @ A5 ) ) ) ) ).

% ImageI
thf(fact_597_ImageI,axiom,
    ! [A: rule,B: fm,R4: set_Pr8969532344284305932ule_fm,A5: set_rule] :
      ( ( member9168179392308330989ule_fm @ ( product_Pair_rule_fm @ A @ B ) @ R4 )
     => ( ( member_rule @ A @ A5 )
       => ( member_fm2 @ B @ ( image_rule_fm @ R4 @ A5 ) ) ) ) ).

% ImageI
thf(fact_598_ImageI,axiom,
    ! [A: nat,B: nat,R4: set_Pr1261947904930325089at_nat,A5: set_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R4 )
     => ( ( member_nat @ A @ A5 )
       => ( member_nat @ B @ ( image_nat_nat @ R4 @ A5 ) ) ) ) ).

% ImageI
thf(fact_599_ImageI,axiom,
    ! [A: fm,B: list_fm,R4: set_Pr937226627985004749ist_fm,A5: set_fm] :
      ( ( member8098592186287407126ist_fm @ ( produc5674586226468309085ist_fm @ A @ B ) @ R4 )
     => ( ( member_fm2 @ A @ A5 )
       => ( member_list_fm2 @ B @ ( image_fm_list_fm @ R4 @ A5 ) ) ) ) ).

% ImageI
thf(fact_600_ImageI,axiom,
    ! [A: list_fm,B: fm,R4: set_Pr1501096078598818625_fm_fm,A5: set_list_fm] :
      ( ( member7999069721096432778_fm_fm @ ( produc3468876765965275089_fm_fm @ A @ B ) @ R4 )
     => ( ( member_list_fm2 @ A @ A5 )
       => ( member_fm2 @ B @ ( image_list_fm_fm @ R4 @ A5 ) ) ) ) ).

% ImageI
thf(fact_601_ImageI,axiom,
    ! [A: list_fm,B: $o,R4: set_Pr4615000397952541542t_fm_o,A5: set_list_fm] :
      ( ( member8238630628690094383t_fm_o @ ( produc8850898633069939702t_fm_o @ A @ B ) @ R4 )
     => ( ( member_list_fm2 @ A @ A5 )
       => ( member_o2 @ B @ ( image_list_fm_o @ R4 @ A5 ) ) ) ) ).

% ImageI
thf(fact_602_ImageI,axiom,
    ! [A: $o,B: list_fm,R4: set_Pr4658132521726357736ist_fm,A5: set_o] :
      ( ( member7814064844939679665ist_fm @ ( produc3646295350333167352ist_fm @ A @ B ) @ R4 )
     => ( ( member_o2 @ A @ A5 )
       => ( member_list_fm2 @ B @ ( image_o_list_fm @ R4 @ A5 ) ) ) ) ).

% ImageI
thf(fact_603_Image__empty2,axiom,
    ! [R2: set_Product_prod_o_o] :
      ( ( image_o_o @ R2 @ bot_bot_set_o )
      = bot_bot_set_o ) ).

% Image_empty2
thf(fact_604_singleton__conv,axiom,
    ! [A: product_prod_nat_nat] :
      ( ( collec3392354462482085612at_nat
        @ ^ [X: product_prod_nat_nat] : ( X = A ) )
      = ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) ).

% singleton_conv
thf(fact_605_singleton__conv,axiom,
    ! [A: nat] :
      ( ( collect_nat
        @ ^ [X: nat] : ( X = A ) )
      = ( insert_nat @ A @ bot_bot_set_nat ) ) ).

% singleton_conv
thf(fact_606_singleton__conv,axiom,
    ! [A: $o] :
      ( ( collect_o
        @ ^ [X: $o] : ( X = A ) )
      = ( insert_o2 @ A @ bot_bot_set_o ) ) ).

% singleton_conv
thf(fact_607_singleton__conv2,axiom,
    ! [A: product_prod_nat_nat] :
      ( ( collec3392354462482085612at_nat
        @ ( ^ [Y8: product_prod_nat_nat,Z5: product_prod_nat_nat] : ( Y8 = Z5 )
          @ A ) )
      = ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) ).

% singleton_conv2
thf(fact_608_singleton__conv2,axiom,
    ! [A: nat] :
      ( ( collect_nat
        @ ( ^ [Y8: nat,Z5: nat] : ( Y8 = Z5 )
          @ A ) )
      = ( insert_nat @ A @ bot_bot_set_nat ) ) ).

% singleton_conv2
thf(fact_609_singleton__conv2,axiom,
    ! [A: $o] :
      ( ( collect_o
        @ ( ^ [Y8: $o,Z5: $o] : ( Y8 = Z5 )
          @ A ) )
      = ( insert_o2 @ A @ bot_bot_set_o ) ) ).

% singleton_conv2
thf(fact_610_Image__singleton__iff,axiom,
    ! [B: rule,R4: set_Pr1822751329126368876m_rule,A: produc6018962875968178549ist_fm] :
      ( ( member_rule @ B @ ( image_8271336862079040884m_rule @ R4 @ ( insert4380798862274524997ist_fm @ A @ bot_bo1798318783834493889ist_fm ) ) )
      = ( member7231649785386036813m_rule @ ( produc1733806532565653680m_rule @ A @ B ) @ R4 ) ) ).

% Image_singleton_iff
thf(fact_611_Image__singleton__iff,axiom,
    ! [B: fm,R4: set_Pr8969532344284305932ule_fm,A: rule] :
      ( ( member_fm2 @ B @ ( image_rule_fm @ R4 @ ( insert_rule @ A @ bot_bot_set_rule ) ) )
      = ( member9168179392308330989ule_fm @ ( product_Pair_rule_fm @ A @ B ) @ R4 ) ) ).

% Image_singleton_iff
thf(fact_612_Image__singleton__iff,axiom,
    ! [B: nat,R4: set_Pr1261947904930325089at_nat,A: nat] :
      ( ( member_nat @ B @ ( image_nat_nat @ R4 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
      = ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R4 ) ) ).

% Image_singleton_iff
thf(fact_613_Image__singleton__iff,axiom,
    ! [B: fm,R4: set_Pr3863124289995447138d_o_fm,A: $o] :
      ( ( member_fm2 @ B @ ( image_o_fm @ R4 @ ( insert_o2 @ A @ bot_bot_set_o ) ) )
      = ( member6824368184199080235d_o_fm @ ( product_Pair_o_fm @ A @ B ) @ R4 ) ) ).

% Image_singleton_iff
thf(fact_614_Image__singleton__iff,axiom,
    ! [B: list_fm,R4: set_Pr4658132521726357736ist_fm,A: $o] :
      ( ( member_list_fm2 @ B @ ( image_o_list_fm @ R4 @ ( insert_o2 @ A @ bot_bot_set_o ) ) )
      = ( member7814064844939679665ist_fm @ ( produc3646295350333167352ist_fm @ A @ B ) @ R4 ) ) ).

% Image_singleton_iff
thf(fact_615_Image__singleton__iff,axiom,
    ! [B: produc340336539035504054m_rule,R4: set_Pr5066739526756200755m_rule,A: $o] :
      ( ( member7231649785386036813m_rule @ B @ ( image_4378791091492168891m_rule @ R4 @ ( insert_o2 @ A @ bot_bot_set_o ) ) )
      = ( member3169346647243466388m_rule @ ( produc1145568171119641079m_rule @ A @ B ) @ R4 ) ) ).

% Image_singleton_iff
thf(fact_616_Image__singleton__iff,axiom,
    ! [B: $o,R4: set_Product_prod_o_o,A: $o] :
      ( ( member_o2 @ B @ ( image_o_o @ R4 @ ( insert_o2 @ A @ bot_bot_set_o ) ) )
      = ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ B ) @ R4 ) ) ).

% Image_singleton_iff
thf(fact_617_UN__singleton,axiom,
    ! [A5: set_o] :
      ( ( comple90263536869209701_set_o
        @ ( image_o_set_o
          @ ^ [X: $o] : ( insert_o2 @ X @ bot_bot_set_o )
          @ A5 ) )
      = A5 ) ).

% UN_singleton
thf(fact_618_UN__singleton,axiom,
    ! [A5: set_fm] :
      ( ( comple2134933779557159616set_fm
        @ ( image_fm_set_fm
          @ ^ [X: fm] : ( insert_fm2 @ X @ bot_bot_set_fm )
          @ A5 ) )
      = A5 ) ).

% UN_singleton
thf(fact_619_UN__simps_I1_J,axiom,
    ! [C2: set_o,A: $o,B6: $o > set_o] :
      ( ( ( C2 = bot_bot_set_o )
       => ( ( comple90263536869209701_set_o
            @ ( image_o_set_o
              @ ^ [X: $o] : ( insert_o2 @ A @ ( B6 @ X ) )
              @ C2 ) )
          = bot_bot_set_o ) )
      & ( ( C2 != bot_bot_set_o )
       => ( ( comple90263536869209701_set_o
            @ ( image_o_set_o
              @ ^ [X: $o] : ( insert_o2 @ A @ ( B6 @ X ) )
              @ C2 ) )
          = ( insert_o2 @ A @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B6 @ C2 ) ) ) ) ) ) ).

% UN_simps(1)
thf(fact_620_UN__simps_I1_J,axiom,
    ! [C2: set_Pr1822751329126368876m_rule,A: fm,B6: produc340336539035504054m_rule > set_fm] :
      ( ( ( C2 = bot_bo3426410157013319936m_rule )
       => ( ( comple2134933779557159616set_fm
            @ ( image_377115208147135190set_fm
              @ ^ [X: produc340336539035504054m_rule] : ( insert_fm2 @ A @ ( B6 @ X ) )
              @ C2 ) )
          = bot_bot_set_fm ) )
      & ( ( C2 != bot_bo3426410157013319936m_rule )
       => ( ( comple2134933779557159616set_fm
            @ ( image_377115208147135190set_fm
              @ ^ [X: produc340336539035504054m_rule] : ( insert_fm2 @ A @ ( B6 @ X ) )
              @ C2 ) )
          = ( insert_fm2 @ A @ ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ B6 @ C2 ) ) ) ) ) ) ).

% UN_simps(1)
thf(fact_621_UN__simps_I1_J,axiom,
    ! [C2: set_o,A: fm,B6: $o > set_fm] :
      ( ( ( C2 = bot_bot_set_o )
       => ( ( comple2134933779557159616set_fm
            @ ( image_o_set_fm
              @ ^ [X: $o] : ( insert_fm2 @ A @ ( B6 @ X ) )
              @ C2 ) )
          = bot_bot_set_fm ) )
      & ( ( C2 != bot_bot_set_o )
       => ( ( comple2134933779557159616set_fm
            @ ( image_o_set_fm
              @ ^ [X: $o] : ( insert_fm2 @ A @ ( B6 @ X ) )
              @ C2 ) )
          = ( insert_fm2 @ A @ ( comple2134933779557159616set_fm @ ( image_o_set_fm @ B6 @ C2 ) ) ) ) ) ) ).

% UN_simps(1)
thf(fact_622_Sup__bool__def,axiom,
    ( complete_Sup_Sup_o
    = ( member_o2 @ $true ) ) ).

% Sup_bool_def
thf(fact_623_insertE,axiom,
    ! [A: fm,B: fm,A5: set_fm] :
      ( ( member_fm2 @ A @ ( insert_fm2 @ B @ A5 ) )
     => ( ( A != B )
       => ( member_fm2 @ A @ A5 ) ) ) ).

% insertE
thf(fact_624_insertE,axiom,
    ! [A: list_fm,B: list_fm,A5: set_list_fm] :
      ( ( member_list_fm2 @ A @ ( insert_list_fm2 @ B @ A5 ) )
     => ( ( A != B )
       => ( member_list_fm2 @ A @ A5 ) ) ) ).

% insertE
thf(fact_625_insertE,axiom,
    ! [A: produc340336539035504054m_rule,B: produc340336539035504054m_rule,A5: set_Pr1822751329126368876m_rule] :
      ( ( member7231649785386036813m_rule @ A @ ( insert775845832097999846m_rule @ B @ A5 ) )
     => ( ( A != B )
       => ( member7231649785386036813m_rule @ A @ A5 ) ) ) ).

% insertE
thf(fact_626_insertE,axiom,
    ! [A: $o,B: $o,A5: set_o] :
      ( ( member_o2 @ A @ ( insert_o2 @ B @ A5 ) )
     => ( ( A = (~ B) )
       => ( member_o2 @ A @ A5 ) ) ) ).

% insertE
thf(fact_627_insertI1,axiom,
    ! [A: fm,B6: set_fm] : ( member_fm2 @ A @ ( insert_fm2 @ A @ B6 ) ) ).

% insertI1
thf(fact_628_insertI1,axiom,
    ! [A: list_fm,B6: set_list_fm] : ( member_list_fm2 @ A @ ( insert_list_fm2 @ A @ B6 ) ) ).

% insertI1
thf(fact_629_insertI1,axiom,
    ! [A: produc340336539035504054m_rule,B6: set_Pr1822751329126368876m_rule] : ( member7231649785386036813m_rule @ A @ ( insert775845832097999846m_rule @ A @ B6 ) ) ).

% insertI1
thf(fact_630_insertI1,axiom,
    ! [A: $o,B6: set_o] : ( member_o2 @ A @ ( insert_o2 @ A @ B6 ) ) ).

% insertI1
thf(fact_631_insertI2,axiom,
    ! [A: fm,B6: set_fm,B: fm] :
      ( ( member_fm2 @ A @ B6 )
     => ( member_fm2 @ A @ ( insert_fm2 @ B @ B6 ) ) ) ).

% insertI2
thf(fact_632_insertI2,axiom,
    ! [A: list_fm,B6: set_list_fm,B: list_fm] :
      ( ( member_list_fm2 @ A @ B6 )
     => ( member_list_fm2 @ A @ ( insert_list_fm2 @ B @ B6 ) ) ) ).

% insertI2
thf(fact_633_insertI2,axiom,
    ! [A: produc340336539035504054m_rule,B6: set_Pr1822751329126368876m_rule,B: produc340336539035504054m_rule] :
      ( ( member7231649785386036813m_rule @ A @ B6 )
     => ( member7231649785386036813m_rule @ A @ ( insert775845832097999846m_rule @ B @ B6 ) ) ) ).

% insertI2
thf(fact_634_insertI2,axiom,
    ! [A: $o,B6: set_o,B: $o] :
      ( ( member_o2 @ A @ B6 )
     => ( member_o2 @ A @ ( insert_o2 @ B @ B6 ) ) ) ).

% insertI2
thf(fact_635_Set_Oset__insert,axiom,
    ! [X5: fm,A5: set_fm] :
      ( ( member_fm2 @ X5 @ A5 )
     => ~ ! [B8: set_fm] :
            ( ( A5
              = ( insert_fm2 @ X5 @ B8 ) )
           => ( member_fm2 @ X5 @ B8 ) ) ) ).

% Set.set_insert
thf(fact_636_Set_Oset__insert,axiom,
    ! [X5: list_fm,A5: set_list_fm] :
      ( ( member_list_fm2 @ X5 @ A5 )
     => ~ ! [B8: set_list_fm] :
            ( ( A5
              = ( insert_list_fm2 @ X5 @ B8 ) )
           => ( member_list_fm2 @ X5 @ B8 ) ) ) ).

% Set.set_insert
thf(fact_637_Set_Oset__insert,axiom,
    ! [X5: produc340336539035504054m_rule,A5: set_Pr1822751329126368876m_rule] :
      ( ( member7231649785386036813m_rule @ X5 @ A5 )
     => ~ ! [B8: set_Pr1822751329126368876m_rule] :
            ( ( A5
              = ( insert775845832097999846m_rule @ X5 @ B8 ) )
           => ( member7231649785386036813m_rule @ X5 @ B8 ) ) ) ).

% Set.set_insert
thf(fact_638_Set_Oset__insert,axiom,
    ! [X5: $o,A5: set_o] :
      ( ( member_o2 @ X5 @ A5 )
     => ~ ! [B8: set_o] :
            ( ( A5
              = ( insert_o2 @ X5 @ B8 ) )
           => ( member_o2 @ X5 @ B8 ) ) ) ).

% Set.set_insert
thf(fact_639_insert__ident,axiom,
    ! [X5: fm,A5: set_fm,B6: set_fm] :
      ( ~ ( member_fm2 @ X5 @ A5 )
     => ( ~ ( member_fm2 @ X5 @ B6 )
       => ( ( ( insert_fm2 @ X5 @ A5 )
            = ( insert_fm2 @ X5 @ B6 ) )
          = ( A5 = B6 ) ) ) ) ).

% insert_ident
thf(fact_640_insert__ident,axiom,
    ! [X5: list_fm,A5: set_list_fm,B6: set_list_fm] :
      ( ~ ( member_list_fm2 @ X5 @ A5 )
     => ( ~ ( member_list_fm2 @ X5 @ B6 )
       => ( ( ( insert_list_fm2 @ X5 @ A5 )
            = ( insert_list_fm2 @ X5 @ B6 ) )
          = ( A5 = B6 ) ) ) ) ).

% insert_ident
thf(fact_641_insert__ident,axiom,
    ! [X5: produc340336539035504054m_rule,A5: set_Pr1822751329126368876m_rule,B6: set_Pr1822751329126368876m_rule] :
      ( ~ ( member7231649785386036813m_rule @ X5 @ A5 )
     => ( ~ ( member7231649785386036813m_rule @ X5 @ B6 )
       => ( ( ( insert775845832097999846m_rule @ X5 @ A5 )
            = ( insert775845832097999846m_rule @ X5 @ B6 ) )
          = ( A5 = B6 ) ) ) ) ).

% insert_ident
thf(fact_642_insert__ident,axiom,
    ! [X5: $o,A5: set_o,B6: set_o] :
      ( ~ ( member_o2 @ X5 @ A5 )
     => ( ~ ( member_o2 @ X5 @ B6 )
       => ( ( ( insert_o2 @ X5 @ A5 )
            = ( insert_o2 @ X5 @ B6 ) )
          = ( A5 = B6 ) ) ) ) ).

% insert_ident
thf(fact_643_insert__absorb,axiom,
    ! [A: fm,A5: set_fm] :
      ( ( member_fm2 @ A @ A5 )
     => ( ( insert_fm2 @ A @ A5 )
        = A5 ) ) ).

% insert_absorb
thf(fact_644_insert__absorb,axiom,
    ! [A: list_fm,A5: set_list_fm] :
      ( ( member_list_fm2 @ A @ A5 )
     => ( ( insert_list_fm2 @ A @ A5 )
        = A5 ) ) ).

% insert_absorb
thf(fact_645_insert__absorb,axiom,
    ! [A: produc340336539035504054m_rule,A5: set_Pr1822751329126368876m_rule] :
      ( ( member7231649785386036813m_rule @ A @ A5 )
     => ( ( insert775845832097999846m_rule @ A @ A5 )
        = A5 ) ) ).

% insert_absorb
thf(fact_646_insert__absorb,axiom,
    ! [A: $o,A5: set_o] :
      ( ( member_o2 @ A @ A5 )
     => ( ( insert_o2 @ A @ A5 )
        = A5 ) ) ).

% insert_absorb
thf(fact_647_insert__eq__iff,axiom,
    ! [A: fm,A5: set_fm,B: fm,B6: set_fm] :
      ( ~ ( member_fm2 @ A @ A5 )
     => ( ~ ( member_fm2 @ B @ B6 )
       => ( ( ( insert_fm2 @ A @ A5 )
            = ( insert_fm2 @ B @ B6 ) )
          = ( ( ( A = B )
             => ( A5 = B6 ) )
            & ( ( A != B )
             => ? [C3: set_fm] :
                  ( ( A5
                    = ( insert_fm2 @ B @ C3 ) )
                  & ~ ( member_fm2 @ B @ C3 )
                  & ( B6
                    = ( insert_fm2 @ A @ C3 ) )
                  & ~ ( member_fm2 @ A @ C3 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_648_insert__eq__iff,axiom,
    ! [A: list_fm,A5: set_list_fm,B: list_fm,B6: set_list_fm] :
      ( ~ ( member_list_fm2 @ A @ A5 )
     => ( ~ ( member_list_fm2 @ B @ B6 )
       => ( ( ( insert_list_fm2 @ A @ A5 )
            = ( insert_list_fm2 @ B @ B6 ) )
          = ( ( ( A = B )
             => ( A5 = B6 ) )
            & ( ( A != B )
             => ? [C3: set_list_fm] :
                  ( ( A5
                    = ( insert_list_fm2 @ B @ C3 ) )
                  & ~ ( member_list_fm2 @ B @ C3 )
                  & ( B6
                    = ( insert_list_fm2 @ A @ C3 ) )
                  & ~ ( member_list_fm2 @ A @ C3 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_649_insert__eq__iff,axiom,
    ! [A: produc340336539035504054m_rule,A5: set_Pr1822751329126368876m_rule,B: produc340336539035504054m_rule,B6: set_Pr1822751329126368876m_rule] :
      ( ~ ( member7231649785386036813m_rule @ A @ A5 )
     => ( ~ ( member7231649785386036813m_rule @ B @ B6 )
       => ( ( ( insert775845832097999846m_rule @ A @ A5 )
            = ( insert775845832097999846m_rule @ B @ B6 ) )
          = ( ( ( A = B )
             => ( A5 = B6 ) )
            & ( ( A != B )
             => ? [C3: set_Pr1822751329126368876m_rule] :
                  ( ( A5
                    = ( insert775845832097999846m_rule @ B @ C3 ) )
                  & ~ ( member7231649785386036813m_rule @ B @ C3 )
                  & ( B6
                    = ( insert775845832097999846m_rule @ A @ C3 ) )
                  & ~ ( member7231649785386036813m_rule @ A @ C3 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_650_insert__eq__iff,axiom,
    ! [A: $o,A5: set_o,B: $o,B6: set_o] :
      ( ~ ( member_o2 @ A @ A5 )
     => ( ~ ( member_o2 @ B @ B6 )
       => ( ( ( insert_o2 @ A @ A5 )
            = ( insert_o2 @ B @ B6 ) )
          = ( ( ( A = B )
             => ( A5 = B6 ) )
            & ( ( A = (~ B) )
             => ? [C3: set_o] :
                  ( ( A5
                    = ( insert_o2 @ B @ C3 ) )
                  & ~ ( member_o2 @ B @ C3 )
                  & ( B6
                    = ( insert_o2 @ A @ C3 ) )
                  & ~ ( member_o2 @ A @ C3 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_651_insert__commute,axiom,
    ! [X5: $o,Y4: $o,A5: set_o] :
      ( ( insert_o2 @ X5 @ ( insert_o2 @ Y4 @ A5 ) )
      = ( insert_o2 @ Y4 @ ( insert_o2 @ X5 @ A5 ) ) ) ).

% insert_commute
thf(fact_652_mk__disjoint__insert,axiom,
    ! [A: fm,A5: set_fm] :
      ( ( member_fm2 @ A @ A5 )
     => ? [B8: set_fm] :
          ( ( A5
            = ( insert_fm2 @ A @ B8 ) )
          & ~ ( member_fm2 @ A @ B8 ) ) ) ).

% mk_disjoint_insert
thf(fact_653_mk__disjoint__insert,axiom,
    ! [A: list_fm,A5: set_list_fm] :
      ( ( member_list_fm2 @ A @ A5 )
     => ? [B8: set_list_fm] :
          ( ( A5
            = ( insert_list_fm2 @ A @ B8 ) )
          & ~ ( member_list_fm2 @ A @ B8 ) ) ) ).

% mk_disjoint_insert
thf(fact_654_mk__disjoint__insert,axiom,
    ! [A: produc340336539035504054m_rule,A5: set_Pr1822751329126368876m_rule] :
      ( ( member7231649785386036813m_rule @ A @ A5 )
     => ? [B8: set_Pr1822751329126368876m_rule] :
          ( ( A5
            = ( insert775845832097999846m_rule @ A @ B8 ) )
          & ~ ( member7231649785386036813m_rule @ A @ B8 ) ) ) ).

% mk_disjoint_insert
thf(fact_655_mk__disjoint__insert,axiom,
    ! [A: $o,A5: set_o] :
      ( ( member_o2 @ A @ A5 )
     => ? [B8: set_o] :
          ( ( A5
            = ( insert_o2 @ A @ B8 ) )
          & ~ ( member_o2 @ A @ B8 ) ) ) ).

% mk_disjoint_insert
thf(fact_656_insert__compr,axiom,
    ( insert_fm2
    = ( ^ [A3: fm,B7: set_fm] :
          ( collect_fm
          @ ^ [X: fm] :
              ( ( X = A3 )
              | ( member_fm2 @ X @ B7 ) ) ) ) ) ).

% insert_compr
thf(fact_657_insert__compr,axiom,
    ( insert_list_fm2
    = ( ^ [A3: list_fm,B7: set_list_fm] :
          ( collect_list_fm
          @ ^ [X: list_fm] :
              ( ( X = A3 )
              | ( member_list_fm2 @ X @ B7 ) ) ) ) ) ).

% insert_compr
thf(fact_658_insert__compr,axiom,
    ( insert775845832097999846m_rule
    = ( ^ [A3: produc340336539035504054m_rule,B7: set_Pr1822751329126368876m_rule] :
          ( collec1009276759030335883m_rule
          @ ^ [X: produc340336539035504054m_rule] :
              ( ( X = A3 )
              | ( member7231649785386036813m_rule @ X @ B7 ) ) ) ) ) ).

% insert_compr
thf(fact_659_insert__compr,axiom,
    ( insert_o2
    = ( ^ [A3: $o,B7: set_o] :
          ( collect_o
          @ ^ [X: $o] :
              ( ( X = A3 )
              | ( member_o2 @ X @ B7 ) ) ) ) ) ).

% insert_compr
thf(fact_660_insert__compr,axiom,
    ( insert8211810215607154385at_nat
    = ( ^ [A3: product_prod_nat_nat,B7: set_Pr1261947904930325089at_nat] :
          ( collec3392354462482085612at_nat
          @ ^ [X: product_prod_nat_nat] :
              ( ( X = A3 )
              | ( member8440522571783428010at_nat @ X @ B7 ) ) ) ) ) ).

% insert_compr
thf(fact_661_insert__compr,axiom,
    ( insert_nat
    = ( ^ [A3: nat,B7: set_nat] :
          ( collect_nat
          @ ^ [X: nat] :
              ( ( X = A3 )
              | ( member_nat @ X @ B7 ) ) ) ) ) ).

% insert_compr
thf(fact_662_insert__Collect,axiom,
    ! [A: $o,P3: $o > $o] :
      ( ( insert_o2 @ A @ ( collect_o @ P3 ) )
      = ( collect_o
        @ ^ [U: $o] :
            ( ( U != A )
           => ( P3 @ U ) ) ) ) ).

% insert_Collect
thf(fact_663_insert__Collect,axiom,
    ! [A: product_prod_nat_nat,P3: product_prod_nat_nat > $o] :
      ( ( insert8211810215607154385at_nat @ A @ ( collec3392354462482085612at_nat @ P3 ) )
      = ( collec3392354462482085612at_nat
        @ ^ [U: product_prod_nat_nat] :
            ( ( U != A )
           => ( P3 @ U ) ) ) ) ).

% insert_Collect
thf(fact_664_insert__Collect,axiom,
    ! [A: nat,P3: nat > $o] :
      ( ( insert_nat @ A @ ( collect_nat @ P3 ) )
      = ( collect_nat
        @ ^ [U: nat] :
            ( ( U != A )
           => ( P3 @ U ) ) ) ) ).

% insert_Collect
thf(fact_665_refl__on__singleton,axiom,
    ! [X5: nat] : ( refl_on_nat @ ( insert_nat @ X5 @ bot_bot_set_nat ) @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ X5 @ X5 ) @ bot_bo2099793752762293965at_nat ) ) ).

% refl_on_singleton
thf(fact_666_refl__on__singleton,axiom,
    ! [X5: $o] : ( refl_on_o @ ( insert_o2 @ X5 @ bot_bot_set_o ) @ ( insert6201435330877294327od_o_o @ ( product_Pair_o_o @ X5 @ X5 ) @ bot_bo7073875226086086771od_o_o ) ) ).

% refl_on_singleton
thf(fact_667_ImageE,axiom,
    ! [B: fm,R4: set_Pr4706815898642364871_fm_fm,A5: set_fm] :
      ( ( member_fm2 @ B @ ( image_fm_fm @ R4 @ A5 ) )
     => ~ ! [X3: fm] :
            ( ( member7780952600467998736_fm_fm @ ( product_Pair_fm_fm @ X3 @ B ) @ R4 )
           => ~ ( member_fm2 @ X3 @ A5 ) ) ) ).

% ImageE
thf(fact_668_ImageE,axiom,
    ! [B: fm,R4: set_Pr3863124289995447138d_o_fm,A5: set_o] :
      ( ( member_fm2 @ B @ ( image_o_fm @ R4 @ A5 ) )
     => ~ ! [X3: $o] :
            ( ( member6824368184199080235d_o_fm @ ( product_Pair_o_fm @ X3 @ B ) @ R4 )
           => ~ ( member_o2 @ X3 @ A5 ) ) ) ).

% ImageE
thf(fact_669_ImageE,axiom,
    ! [B: $o,R4: set_Pr2813250270173872364d_fm_o,A5: set_fm] :
      ( ( member_o2 @ B @ ( image_fm_o @ R4 @ A5 ) )
     => ~ ! [X3: fm] :
            ( ( member1731323461681951669d_fm_o @ ( product_Pair_fm_o @ X3 @ B ) @ R4 )
           => ~ ( member_fm2 @ X3 @ A5 ) ) ) ).

% ImageE
thf(fact_670_ImageE,axiom,
    ! [B: $o,R4: set_Product_prod_o_o,A5: set_o] :
      ( ( member_o2 @ B @ ( image_o_o @ R4 @ A5 ) )
     => ~ ! [X3: $o] :
            ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X3 @ B ) @ R4 )
           => ~ ( member_o2 @ X3 @ A5 ) ) ) ).

% ImageE
thf(fact_671_ImageE,axiom,
    ! [B: fm,R4: set_Pr8969532344284305932ule_fm,A5: set_rule] :
      ( ( member_fm2 @ B @ ( image_rule_fm @ R4 @ A5 ) )
     => ~ ! [X3: rule] :
            ( ( member9168179392308330989ule_fm @ ( product_Pair_rule_fm @ X3 @ B ) @ R4 )
           => ~ ( member_rule @ X3 @ A5 ) ) ) ).

% ImageE
thf(fact_672_ImageE,axiom,
    ! [B: nat,R4: set_Pr1261947904930325089at_nat,A5: set_nat] :
      ( ( member_nat @ B @ ( image_nat_nat @ R4 @ A5 ) )
     => ~ ! [X3: nat] :
            ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ B ) @ R4 )
           => ~ ( member_nat @ X3 @ A5 ) ) ) ).

% ImageE
thf(fact_673_ImageE,axiom,
    ! [B: fm,R4: set_Pr1501096078598818625_fm_fm,A5: set_list_fm] :
      ( ( member_fm2 @ B @ ( image_list_fm_fm @ R4 @ A5 ) )
     => ~ ! [X3: list_fm] :
            ( ( member7999069721096432778_fm_fm @ ( produc3468876765965275089_fm_fm @ X3 @ B ) @ R4 )
           => ~ ( member_list_fm2 @ X3 @ A5 ) ) ) ).

% ImageE
thf(fact_674_ImageE,axiom,
    ! [B: list_fm,R4: set_Pr937226627985004749ist_fm,A5: set_fm] :
      ( ( member_list_fm2 @ B @ ( image_fm_list_fm @ R4 @ A5 ) )
     => ~ ! [X3: fm] :
            ( ( member8098592186287407126ist_fm @ ( produc5674586226468309085ist_fm @ X3 @ B ) @ R4 )
           => ~ ( member_fm2 @ X3 @ A5 ) ) ) ).

% ImageE
thf(fact_675_ImageE,axiom,
    ! [B: list_fm,R4: set_Pr4658132521726357736ist_fm,A5: set_o] :
      ( ( member_list_fm2 @ B @ ( image_o_list_fm @ R4 @ A5 ) )
     => ~ ! [X3: $o] :
            ( ( member7814064844939679665ist_fm @ ( produc3646295350333167352ist_fm @ X3 @ B ) @ R4 )
           => ~ ( member_o2 @ X3 @ A5 ) ) ) ).

% ImageE
thf(fact_676_ImageE,axiom,
    ! [B: $o,R4: set_Pr4615000397952541542t_fm_o,A5: set_list_fm] :
      ( ( member_o2 @ B @ ( image_list_fm_o @ R4 @ A5 ) )
     => ~ ! [X3: list_fm] :
            ( ( member8238630628690094383t_fm_o @ ( produc8850898633069939702t_fm_o @ X3 @ B ) @ R4 )
           => ~ ( member_list_fm2 @ X3 @ A5 ) ) ) ).

% ImageE
thf(fact_677_Image__iff,axiom,
    ! [B: rule,R4: set_Pr1822751329126368876m_rule,A5: set_Pr5202636777678657877ist_fm] :
      ( ( member_rule @ B @ ( image_8271336862079040884m_rule @ R4 @ A5 ) )
      = ( ? [X: produc6018962875968178549ist_fm] :
            ( ( member4699826688122452638ist_fm @ X @ A5 )
            & ( member7231649785386036813m_rule @ ( produc1733806532565653680m_rule @ X @ B ) @ R4 ) ) ) ) ).

% Image_iff
thf(fact_678_Image__iff,axiom,
    ! [B: fm,R4: set_Pr8969532344284305932ule_fm,A5: set_rule] :
      ( ( member_fm2 @ B @ ( image_rule_fm @ R4 @ A5 ) )
      = ( ? [X: rule] :
            ( ( member_rule @ X @ A5 )
            & ( member9168179392308330989ule_fm @ ( product_Pair_rule_fm @ X @ B ) @ R4 ) ) ) ) ).

% Image_iff
thf(fact_679_Image__iff,axiom,
    ! [B: nat,R4: set_Pr1261947904930325089at_nat,A5: set_nat] :
      ( ( member_nat @ B @ ( image_nat_nat @ R4 @ A5 ) )
      = ( ? [X: nat] :
            ( ( member_nat @ X @ A5 )
            & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ B ) @ R4 ) ) ) ) ).

% Image_iff
thf(fact_680_rev__ImageI,axiom,
    ! [A: fm,A5: set_fm,B: fm,R4: set_Pr4706815898642364871_fm_fm] :
      ( ( member_fm2 @ A @ A5 )
     => ( ( member7780952600467998736_fm_fm @ ( product_Pair_fm_fm @ A @ B ) @ R4 )
       => ( member_fm2 @ B @ ( image_fm_fm @ R4 @ A5 ) ) ) ) ).

% rev_ImageI
thf(fact_681_rev__ImageI,axiom,
    ! [A: fm,A5: set_fm,B: $o,R4: set_Pr2813250270173872364d_fm_o] :
      ( ( member_fm2 @ A @ A5 )
     => ( ( member1731323461681951669d_fm_o @ ( product_Pair_fm_o @ A @ B ) @ R4 )
       => ( member_o2 @ B @ ( image_fm_o @ R4 @ A5 ) ) ) ) ).

% rev_ImageI
thf(fact_682_rev__ImageI,axiom,
    ! [A: $o,A5: set_o,B: fm,R4: set_Pr3863124289995447138d_o_fm] :
      ( ( member_o2 @ A @ A5 )
     => ( ( member6824368184199080235d_o_fm @ ( product_Pair_o_fm @ A @ B ) @ R4 )
       => ( member_fm2 @ B @ ( image_o_fm @ R4 @ A5 ) ) ) ) ).

% rev_ImageI
thf(fact_683_rev__ImageI,axiom,
    ! [A: $o,A5: set_o,B: $o,R4: set_Product_prod_o_o] :
      ( ( member_o2 @ A @ A5 )
     => ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ B ) @ R4 )
       => ( member_o2 @ B @ ( image_o_o @ R4 @ A5 ) ) ) ) ).

% rev_ImageI
thf(fact_684_rev__ImageI,axiom,
    ! [A: rule,A5: set_rule,B: fm,R4: set_Pr8969532344284305932ule_fm] :
      ( ( member_rule @ A @ A5 )
     => ( ( member9168179392308330989ule_fm @ ( product_Pair_rule_fm @ A @ B ) @ R4 )
       => ( member_fm2 @ B @ ( image_rule_fm @ R4 @ A5 ) ) ) ) ).

% rev_ImageI
thf(fact_685_rev__ImageI,axiom,
    ! [A: nat,A5: set_nat,B: nat,R4: set_Pr1261947904930325089at_nat] :
      ( ( member_nat @ A @ A5 )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R4 )
       => ( member_nat @ B @ ( image_nat_nat @ R4 @ A5 ) ) ) ) ).

% rev_ImageI
thf(fact_686_rev__ImageI,axiom,
    ! [A: fm,A5: set_fm,B: list_fm,R4: set_Pr937226627985004749ist_fm] :
      ( ( member_fm2 @ A @ A5 )
     => ( ( member8098592186287407126ist_fm @ ( produc5674586226468309085ist_fm @ A @ B ) @ R4 )
       => ( member_list_fm2 @ B @ ( image_fm_list_fm @ R4 @ A5 ) ) ) ) ).

% rev_ImageI
thf(fact_687_rev__ImageI,axiom,
    ! [A: list_fm,A5: set_list_fm,B: fm,R4: set_Pr1501096078598818625_fm_fm] :
      ( ( member_list_fm2 @ A @ A5 )
     => ( ( member7999069721096432778_fm_fm @ ( produc3468876765965275089_fm_fm @ A @ B ) @ R4 )
       => ( member_fm2 @ B @ ( image_list_fm_fm @ R4 @ A5 ) ) ) ) ).

% rev_ImageI
thf(fact_688_rev__ImageI,axiom,
    ! [A: list_fm,A5: set_list_fm,B: $o,R4: set_Pr4615000397952541542t_fm_o] :
      ( ( member_list_fm2 @ A @ A5 )
     => ( ( member8238630628690094383t_fm_o @ ( produc8850898633069939702t_fm_o @ A @ B ) @ R4 )
       => ( member_o2 @ B @ ( image_list_fm_o @ R4 @ A5 ) ) ) ) ).

% rev_ImageI
thf(fact_689_rev__ImageI,axiom,
    ! [A: $o,A5: set_o,B: list_fm,R4: set_Pr4658132521726357736ist_fm] :
      ( ( member_o2 @ A @ A5 )
     => ( ( member7814064844939679665ist_fm @ ( produc3646295350333167352ist_fm @ A @ B ) @ R4 )
       => ( member_list_fm2 @ B @ ( image_o_list_fm @ R4 @ A5 ) ) ) ) ).

% rev_ImageI
thf(fact_690_Image__singleton,axiom,
    ! [R4: set_Pr1822751329126368876m_rule,A: produc6018962875968178549ist_fm] :
      ( ( image_8271336862079040884m_rule @ R4 @ ( insert4380798862274524997ist_fm @ A @ bot_bo1798318783834493889ist_fm ) )
      = ( collect_rule
        @ ^ [B3: rule] : ( member7231649785386036813m_rule @ ( produc1733806532565653680m_rule @ A @ B3 ) @ R4 ) ) ) ).

% Image_singleton
thf(fact_691_Image__singleton,axiom,
    ! [R4: set_Pr8969532344284305932ule_fm,A: rule] :
      ( ( image_rule_fm @ R4 @ ( insert_rule @ A @ bot_bot_set_rule ) )
      = ( collect_fm
        @ ^ [B3: fm] : ( member9168179392308330989ule_fm @ ( product_Pair_rule_fm @ A @ B3 ) @ R4 ) ) ) ).

% Image_singleton
thf(fact_692_Image__singleton,axiom,
    ! [R4: set_Pr1261947904930325089at_nat,A: nat] :
      ( ( image_nat_nat @ R4 @ ( insert_nat @ A @ bot_bot_set_nat ) )
      = ( collect_nat
        @ ^ [B3: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B3 ) @ R4 ) ) ) ).

% Image_singleton
thf(fact_693_Image__singleton,axiom,
    ! [R4: set_Pr2390076351701138800at_nat,A: $o] :
      ( ( image_5697180853172723644at_nat @ R4 @ ( insert_o2 @ A @ bot_bot_set_o ) )
      = ( collec3392354462482085612at_nat
        @ ^ [B3: product_prod_nat_nat] : ( member6694634564379471929at_nat @ ( produc1017754141990422400at_nat @ A @ B3 ) @ R4 ) ) ) ).

% Image_singleton
thf(fact_694_Image__singleton,axiom,
    ! [R4: set_Pr2101469702781467981_o_nat,A: $o] :
      ( ( image_o_nat @ R4 @ ( insert_o2 @ A @ bot_bot_set_o ) )
      = ( collect_nat
        @ ^ [B3: nat] : ( member2802428098988154798_o_nat @ ( product_Pair_o_nat @ A @ B3 ) @ R4 ) ) ) ).

% Image_singleton
thf(fact_695_singletonD,axiom,
    ! [B: fm,A: fm] :
      ( ( member_fm2 @ B @ ( insert_fm2 @ A @ bot_bot_set_fm ) )
     => ( B = A ) ) ).

% singletonD
thf(fact_696_singletonD,axiom,
    ! [B: list_fm,A: list_fm] :
      ( ( member_list_fm2 @ B @ ( insert_list_fm2 @ A @ bot_bot_set_list_fm ) )
     => ( B = A ) ) ).

% singletonD
thf(fact_697_singletonD,axiom,
    ! [B: produc340336539035504054m_rule,A: produc340336539035504054m_rule] :
      ( ( member7231649785386036813m_rule @ B @ ( insert775845832097999846m_rule @ A @ bot_bo3426410157013319936m_rule ) )
     => ( B = A ) ) ).

% singletonD
thf(fact_698_singletonD,axiom,
    ! [B: $o,A: $o] :
      ( ( member_o2 @ B @ ( insert_o2 @ A @ bot_bot_set_o ) )
     => ( B = A ) ) ).

% singletonD
thf(fact_699_singleton__iff,axiom,
    ! [B: fm,A: fm] :
      ( ( member_fm2 @ B @ ( insert_fm2 @ A @ bot_bot_set_fm ) )
      = ( B = A ) ) ).

% singleton_iff
thf(fact_700_singleton__iff,axiom,
    ! [B: list_fm,A: list_fm] :
      ( ( member_list_fm2 @ B @ ( insert_list_fm2 @ A @ bot_bot_set_list_fm ) )
      = ( B = A ) ) ).

% singleton_iff
thf(fact_701_singleton__iff,axiom,
    ! [B: produc340336539035504054m_rule,A: produc340336539035504054m_rule] :
      ( ( member7231649785386036813m_rule @ B @ ( insert775845832097999846m_rule @ A @ bot_bo3426410157013319936m_rule ) )
      = ( B = A ) ) ).

% singleton_iff
thf(fact_702_singleton__iff,axiom,
    ! [B: $o,A: $o] :
      ( ( member_o2 @ B @ ( insert_o2 @ A @ bot_bot_set_o ) )
      = ( B = A ) ) ).

% singleton_iff
thf(fact_703_doubleton__eq__iff,axiom,
    ! [A: $o,B: $o,C: $o,D2: $o] :
      ( ( ( insert_o2 @ A @ ( insert_o2 @ B @ bot_bot_set_o ) )
        = ( insert_o2 @ C @ ( insert_o2 @ D2 @ bot_bot_set_o ) ) )
      = ( ( ( A = C )
          & ( B = D2 ) )
        | ( ( A = D2 )
          & ( B = C ) ) ) ) ).

% doubleton_eq_iff
thf(fact_704_insert__not__empty,axiom,
    ! [A: $o,A5: set_o] :
      ( ( insert_o2 @ A @ A5 )
     != bot_bot_set_o ) ).

% insert_not_empty
thf(fact_705_singleton__inject,axiom,
    ! [A: $o,B: $o] :
      ( ( ( insert_o2 @ A @ bot_bot_set_o )
        = ( insert_o2 @ B @ bot_bot_set_o ) )
     => ( A = B ) ) ).

% singleton_inject
thf(fact_706_refl__onD,axiom,
    ! [A5: set_fm,R4: set_Pr4706815898642364871_fm_fm,A: fm] :
      ( ( refl_on_fm @ A5 @ R4 )
     => ( ( member_fm2 @ A @ A5 )
       => ( member7780952600467998736_fm_fm @ ( product_Pair_fm_fm @ A @ A ) @ R4 ) ) ) ).

% refl_onD
thf(fact_707_refl__onD,axiom,
    ! [A5: set_list_fm,R4: set_Pr1916137733696349511ist_fm,A: list_fm] :
      ( ( refl_on_list_fm @ A5 @ R4 )
     => ( ( member_list_fm2 @ A @ A5 )
       => ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ A @ A ) @ R4 ) ) ) ).

% refl_onD
thf(fact_708_refl__onD,axiom,
    ! [A5: set_Pr1822751329126368876m_rule,R4: set_Pr4971326047967503661m_rule,A: produc340336539035504054m_rule] :
      ( ( refl_o3542783858073070071m_rule @ A5 @ R4 )
     => ( ( member7231649785386036813m_rule @ A @ A5 )
       => ( member1411645983481675894m_rule @ ( produc6261311607089640965m_rule @ A @ A ) @ R4 ) ) ) ).

% refl_onD
thf(fact_709_refl__onD,axiom,
    ! [A5: set_o,R4: set_Product_prod_o_o,A: $o] :
      ( ( refl_on_o @ A5 @ R4 )
     => ( ( member_o2 @ A @ A5 )
       => ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ A ) @ R4 ) ) ) ).

% refl_onD
thf(fact_710_refl__onD,axiom,
    ! [A5: set_nat,R4: set_Pr1261947904930325089at_nat,A: nat] :
      ( ( refl_on_nat @ A5 @ R4 )
     => ( ( member_nat @ A @ A5 )
       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ A ) @ R4 ) ) ) ).

% refl_onD
thf(fact_711_refl__onD1,axiom,
    ! [A5: set_fm,R4: set_Pr4706815898642364871_fm_fm,X5: fm,Y4: fm] :
      ( ( refl_on_fm @ A5 @ R4 )
     => ( ( member7780952600467998736_fm_fm @ ( product_Pair_fm_fm @ X5 @ Y4 ) @ R4 )
       => ( member_fm2 @ X5 @ A5 ) ) ) ).

% refl_onD1
thf(fact_712_refl__onD1,axiom,
    ! [A5: set_list_fm,R4: set_Pr1916137733696349511ist_fm,X5: list_fm,Y4: list_fm] :
      ( ( refl_on_list_fm @ A5 @ R4 )
     => ( ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ X5 @ Y4 ) @ R4 )
       => ( member_list_fm2 @ X5 @ A5 ) ) ) ).

% refl_onD1
thf(fact_713_refl__onD1,axiom,
    ! [A5: set_Pr1822751329126368876m_rule,R4: set_Pr4971326047967503661m_rule,X5: produc340336539035504054m_rule,Y4: produc340336539035504054m_rule] :
      ( ( refl_o3542783858073070071m_rule @ A5 @ R4 )
     => ( ( member1411645983481675894m_rule @ ( produc6261311607089640965m_rule @ X5 @ Y4 ) @ R4 )
       => ( member7231649785386036813m_rule @ X5 @ A5 ) ) ) ).

% refl_onD1
thf(fact_714_refl__onD1,axiom,
    ! [A5: set_o,R4: set_Product_prod_o_o,X5: $o,Y4: $o] :
      ( ( refl_on_o @ A5 @ R4 )
     => ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X5 @ Y4 ) @ R4 )
       => ( member_o2 @ X5 @ A5 ) ) ) ).

% refl_onD1
thf(fact_715_refl__onD1,axiom,
    ! [A5: set_nat,R4: set_Pr1261947904930325089at_nat,X5: nat,Y4: nat] :
      ( ( refl_on_nat @ A5 @ R4 )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X5 @ Y4 ) @ R4 )
       => ( member_nat @ X5 @ A5 ) ) ) ).

% refl_onD1
thf(fact_716_refl__onD2,axiom,
    ! [A5: set_fm,R4: set_Pr4706815898642364871_fm_fm,X5: fm,Y4: fm] :
      ( ( refl_on_fm @ A5 @ R4 )
     => ( ( member7780952600467998736_fm_fm @ ( product_Pair_fm_fm @ X5 @ Y4 ) @ R4 )
       => ( member_fm2 @ Y4 @ A5 ) ) ) ).

% refl_onD2
thf(fact_717_refl__onD2,axiom,
    ! [A5: set_list_fm,R4: set_Pr1916137733696349511ist_fm,X5: list_fm,Y4: list_fm] :
      ( ( refl_on_list_fm @ A5 @ R4 )
     => ( ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ X5 @ Y4 ) @ R4 )
       => ( member_list_fm2 @ Y4 @ A5 ) ) ) ).

% refl_onD2
thf(fact_718_refl__onD2,axiom,
    ! [A5: set_Pr1822751329126368876m_rule,R4: set_Pr4971326047967503661m_rule,X5: produc340336539035504054m_rule,Y4: produc340336539035504054m_rule] :
      ( ( refl_o3542783858073070071m_rule @ A5 @ R4 )
     => ( ( member1411645983481675894m_rule @ ( produc6261311607089640965m_rule @ X5 @ Y4 ) @ R4 )
       => ( member7231649785386036813m_rule @ Y4 @ A5 ) ) ) ).

% refl_onD2
thf(fact_719_refl__onD2,axiom,
    ! [A5: set_o,R4: set_Product_prod_o_o,X5: $o,Y4: $o] :
      ( ( refl_on_o @ A5 @ R4 )
     => ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X5 @ Y4 ) @ R4 )
       => ( member_o2 @ Y4 @ A5 ) ) ) ).

% refl_onD2
thf(fact_720_refl__onD2,axiom,
    ! [A5: set_nat,R4: set_Pr1261947904930325089at_nat,X5: nat,Y4: nat] :
      ( ( refl_on_nat @ A5 @ R4 )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X5 @ Y4 ) @ R4 )
       => ( member_nat @ Y4 @ A5 ) ) ) ).

% refl_onD2
thf(fact_721_Collect__conv__if,axiom,
    ! [P3: product_prod_nat_nat > $o,A: product_prod_nat_nat] :
      ( ( ( P3 @ A )
       => ( ( collec3392354462482085612at_nat
            @ ^ [X: product_prod_nat_nat] :
                ( ( X = A )
                & ( P3 @ X ) ) )
          = ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) )
      & ( ~ ( P3 @ A )
       => ( ( collec3392354462482085612at_nat
            @ ^ [X: product_prod_nat_nat] :
                ( ( X = A )
                & ( P3 @ X ) ) )
          = bot_bo2099793752762293965at_nat ) ) ) ).

% Collect_conv_if
thf(fact_722_Collect__conv__if,axiom,
    ! [P3: nat > $o,A: nat] :
      ( ( ( P3 @ A )
       => ( ( collect_nat
            @ ^ [X: nat] :
                ( ( X = A )
                & ( P3 @ X ) ) )
          = ( insert_nat @ A @ bot_bot_set_nat ) ) )
      & ( ~ ( P3 @ A )
       => ( ( collect_nat
            @ ^ [X: nat] :
                ( ( X = A )
                & ( P3 @ X ) ) )
          = bot_bot_set_nat ) ) ) ).

% Collect_conv_if
thf(fact_723_Collect__conv__if,axiom,
    ! [P3: $o > $o,A: $o] :
      ( ( ( P3 @ A )
       => ( ( collect_o
            @ ^ [X: $o] :
                ( ( X = A )
                & ( P3 @ X ) ) )
          = ( insert_o2 @ A @ bot_bot_set_o ) ) )
      & ( ~ ( P3 @ A )
       => ( ( collect_o
            @ ^ [X: $o] :
                ( ( X = A )
                & ( P3 @ X ) ) )
          = bot_bot_set_o ) ) ) ).

% Collect_conv_if
thf(fact_724_Collect__conv__if2,axiom,
    ! [P3: product_prod_nat_nat > $o,A: product_prod_nat_nat] :
      ( ( ( P3 @ A )
       => ( ( collec3392354462482085612at_nat
            @ ^ [X: product_prod_nat_nat] :
                ( ( A = X )
                & ( P3 @ X ) ) )
          = ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) )
      & ( ~ ( P3 @ A )
       => ( ( collec3392354462482085612at_nat
            @ ^ [X: product_prod_nat_nat] :
                ( ( A = X )
                & ( P3 @ X ) ) )
          = bot_bo2099793752762293965at_nat ) ) ) ).

% Collect_conv_if2
thf(fact_725_Collect__conv__if2,axiom,
    ! [P3: nat > $o,A: nat] :
      ( ( ( P3 @ A )
       => ( ( collect_nat
            @ ^ [X: nat] :
                ( ( A = X )
                & ( P3 @ X ) ) )
          = ( insert_nat @ A @ bot_bot_set_nat ) ) )
      & ( ~ ( P3 @ A )
       => ( ( collect_nat
            @ ^ [X: nat] :
                ( ( A = X )
                & ( P3 @ X ) ) )
          = bot_bot_set_nat ) ) ) ).

% Collect_conv_if2
thf(fact_726_Collect__conv__if2,axiom,
    ! [P3: $o > $o,A: $o] :
      ( ( ( P3 @ A )
       => ( ( collect_o
            @ ^ [X: $o] :
                ( ( A = X )
                & ( P3 @ X ) ) )
          = ( insert_o2 @ A @ bot_bot_set_o ) ) )
      & ( ~ ( P3 @ A )
       => ( ( collect_o
            @ ^ [X: $o] :
                ( ( A = X )
                & ( P3 @ X ) ) )
          = bot_bot_set_o ) ) ) ).

% Collect_conv_if2
thf(fact_727_Image__eq__UN,axiom,
    ( image_6824901344838659752ule_fm
    = ( ^ [R: set_Pr6561289912211498472ule_fm,B7: set_Pr1822751329126368876m_rule] :
          ( comple2134933779557159616set_fm
          @ ( image_377115208147135190set_fm
            @ ^ [Y: produc340336539035504054m_rule] : ( image_6824901344838659752ule_fm @ R @ ( insert775845832097999846m_rule @ Y @ bot_bo3426410157013319936m_rule ) )
            @ B7 ) ) ) ) ).

% Image_eq_UN
thf(fact_728_Image__eq__UN,axiom,
    ( image_o_fm
    = ( ^ [R: set_Pr3863124289995447138d_o_fm,B7: set_o] :
          ( comple2134933779557159616set_fm
          @ ( image_o_set_fm
            @ ^ [Y: $o] : ( image_o_fm @ R @ ( insert_o2 @ Y @ bot_bot_set_o ) )
            @ B7 ) ) ) ) ).

% Image_eq_UN
thf(fact_729_Image__UN,axiom,
    ! [R4: set_Pr4706815898642364871_fm_fm,B6: produc340336539035504054m_rule > set_fm,A5: set_Pr1822751329126368876m_rule] :
      ( ( image_fm_fm @ R4 @ ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ B6 @ A5 ) ) )
      = ( comple2134933779557159616set_fm
        @ ( image_377115208147135190set_fm
          @ ^ [X: produc340336539035504054m_rule] : ( image_fm_fm @ R4 @ ( B6 @ X ) )
          @ A5 ) ) ) ).

% Image_UN
thf(fact_730_refl__on__empty,axiom,
    refl_on_o @ bot_bot_set_o @ bot_bo7073875226086086771od_o_o ).

% refl_on_empty
thf(fact_731_image__constant,axiom,
    ! [X5: produc340336539035504054m_rule,A5: set_Pr1822751329126368876m_rule,C: set_fm] :
      ( ( member7231649785386036813m_rule @ X5 @ A5 )
     => ( ( image_377115208147135190set_fm
          @ ^ [X: produc340336539035504054m_rule] : C
          @ A5 )
        = ( insert_set_fm @ C @ bot_bot_set_set_fm ) ) ) ).

% image_constant
thf(fact_732_image__constant,axiom,
    ! [X5: fm,A5: set_fm,C: $o] :
      ( ( member_fm2 @ X5 @ A5 )
     => ( ( image_fm_o2
          @ ^ [X: fm] : C
          @ A5 )
        = ( insert_o2 @ C @ bot_bot_set_o ) ) ) ).

% image_constant
thf(fact_733_image__constant,axiom,
    ! [X5: list_fm,A5: set_list_fm,C: $o] :
      ( ( member_list_fm2 @ X5 @ A5 )
     => ( ( image_list_fm_o2
          @ ^ [X: list_fm] : C
          @ A5 )
        = ( insert_o2 @ C @ bot_bot_set_o ) ) ) ).

% image_constant
thf(fact_734_image__constant,axiom,
    ! [X5: produc340336539035504054m_rule,A5: set_Pr1822751329126368876m_rule,C: $o] :
      ( ( member7231649785386036813m_rule @ X5 @ A5 )
     => ( ( image_8969478372357302939rule_o
          @ ^ [X: produc340336539035504054m_rule] : C
          @ A5 )
        = ( insert_o2 @ C @ bot_bot_set_o ) ) ) ).

% image_constant
thf(fact_735_image__constant,axiom,
    ! [X5: $o,A5: set_o,C: $o] :
      ( ( member_o2 @ X5 @ A5 )
     => ( ( image_o_o2
          @ ^ [X: $o] : C
          @ A5 )
        = ( insert_o2 @ C @ bot_bot_set_o ) ) ) ).

% image_constant
thf(fact_736_image__constant__conv,axiom,
    ! [A5: set_Pr1822751329126368876m_rule,C: set_fm] :
      ( ( ( A5 = bot_bo3426410157013319936m_rule )
       => ( ( image_377115208147135190set_fm
            @ ^ [X: produc340336539035504054m_rule] : C
            @ A5 )
          = bot_bot_set_set_fm ) )
      & ( ( A5 != bot_bo3426410157013319936m_rule )
       => ( ( image_377115208147135190set_fm
            @ ^ [X: produc340336539035504054m_rule] : C
            @ A5 )
          = ( insert_set_fm @ C @ bot_bot_set_set_fm ) ) ) ) ).

% image_constant_conv
thf(fact_737_image__constant__conv,axiom,
    ! [A5: set_o,C: $o] :
      ( ( ( A5 = bot_bot_set_o )
       => ( ( image_o_o2
            @ ^ [X: $o] : C
            @ A5 )
          = bot_bot_set_o ) )
      & ( ( A5 != bot_bot_set_o )
       => ( ( image_o_o2
            @ ^ [X: $o] : C
            @ A5 )
          = ( insert_o2 @ C @ bot_bot_set_o ) ) ) ) ).

% image_constant_conv
thf(fact_738_UN__insert__distrib,axiom,
    ! [U2: fm,A5: set_fm,A: $o,B6: fm > set_o] :
      ( ( member_fm2 @ U2 @ A5 )
     => ( ( comple90263536869209701_set_o
          @ ( image_fm_set_o
            @ ^ [X: fm] : ( insert_o2 @ A @ ( B6 @ X ) )
            @ A5 ) )
        = ( insert_o2 @ A @ ( comple90263536869209701_set_o @ ( image_fm_set_o @ B6 @ A5 ) ) ) ) ) ).

% UN_insert_distrib
thf(fact_739_UN__insert__distrib,axiom,
    ! [U2: list_fm,A5: set_list_fm,A: $o,B6: list_fm > set_o] :
      ( ( member_list_fm2 @ U2 @ A5 )
     => ( ( comple90263536869209701_set_o
          @ ( image_list_fm_set_o
            @ ^ [X: list_fm] : ( insert_o2 @ A @ ( B6 @ X ) )
            @ A5 ) )
        = ( insert_o2 @ A @ ( comple90263536869209701_set_o @ ( image_list_fm_set_o @ B6 @ A5 ) ) ) ) ) ).

% UN_insert_distrib
thf(fact_740_UN__insert__distrib,axiom,
    ! [U2: produc340336539035504054m_rule,A5: set_Pr1822751329126368876m_rule,A: $o,B6: produc340336539035504054m_rule > set_o] :
      ( ( member7231649785386036813m_rule @ U2 @ A5 )
     => ( ( comple90263536869209701_set_o
          @ ( image_2025595867862115707_set_o
            @ ^ [X: produc340336539035504054m_rule] : ( insert_o2 @ A @ ( B6 @ X ) )
            @ A5 ) )
        = ( insert_o2 @ A @ ( comple90263536869209701_set_o @ ( image_2025595867862115707_set_o @ B6 @ A5 ) ) ) ) ) ).

% UN_insert_distrib
thf(fact_741_UN__insert__distrib,axiom,
    ! [U2: $o,A5: set_o,A: $o,B6: $o > set_o] :
      ( ( member_o2 @ U2 @ A5 )
     => ( ( comple90263536869209701_set_o
          @ ( image_o_set_o
            @ ^ [X: $o] : ( insert_o2 @ A @ ( B6 @ X ) )
            @ A5 ) )
        = ( insert_o2 @ A @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B6 @ A5 ) ) ) ) ) ).

% UN_insert_distrib
thf(fact_742_UN__insert__distrib,axiom,
    ! [U2: fm,A5: set_fm,A: fm,B6: fm > set_fm] :
      ( ( member_fm2 @ U2 @ A5 )
     => ( ( comple2134933779557159616set_fm
          @ ( image_fm_set_fm
            @ ^ [X: fm] : ( insert_fm2 @ A @ ( B6 @ X ) )
            @ A5 ) )
        = ( insert_fm2 @ A @ ( comple2134933779557159616set_fm @ ( image_fm_set_fm @ B6 @ A5 ) ) ) ) ) ).

% UN_insert_distrib
thf(fact_743_UN__insert__distrib,axiom,
    ! [U2: list_fm,A5: set_list_fm,A: fm,B6: list_fm > set_fm] :
      ( ( member_list_fm2 @ U2 @ A5 )
     => ( ( comple2134933779557159616set_fm
          @ ( image_list_fm_set_fm
            @ ^ [X: list_fm] : ( insert_fm2 @ A @ ( B6 @ X ) )
            @ A5 ) )
        = ( insert_fm2 @ A @ ( comple2134933779557159616set_fm @ ( image_list_fm_set_fm @ B6 @ A5 ) ) ) ) ) ).

% UN_insert_distrib
thf(fact_744_UN__insert__distrib,axiom,
    ! [U2: produc340336539035504054m_rule,A5: set_Pr1822751329126368876m_rule,A: fm,B6: produc340336539035504054m_rule > set_fm] :
      ( ( member7231649785386036813m_rule @ U2 @ A5 )
     => ( ( comple2134933779557159616set_fm
          @ ( image_377115208147135190set_fm
            @ ^ [X: produc340336539035504054m_rule] : ( insert_fm2 @ A @ ( B6 @ X ) )
            @ A5 ) )
        = ( insert_fm2 @ A @ ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ B6 @ A5 ) ) ) ) ) ).

% UN_insert_distrib
thf(fact_745_UN__insert__distrib,axiom,
    ! [U2: $o,A5: set_o,A: fm,B6: $o > set_fm] :
      ( ( member_o2 @ U2 @ A5 )
     => ( ( comple2134933779557159616set_fm
          @ ( image_o_set_fm
            @ ^ [X: $o] : ( insert_fm2 @ A @ ( B6 @ X ) )
            @ A5 ) )
        = ( insert_fm2 @ A @ ( comple2134933779557159616set_fm @ ( image_o_set_fm @ B6 @ A5 ) ) ) ) ) ).

% UN_insert_distrib
thf(fact_746_UN__extend__simps_I1_J,axiom,
    ! [C2: set_o,A: $o,B6: $o > set_o] :
      ( ( ( C2 = bot_bot_set_o )
       => ( ( insert_o2 @ A @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B6 @ C2 ) ) )
          = ( insert_o2 @ A @ bot_bot_set_o ) ) )
      & ( ( C2 != bot_bot_set_o )
       => ( ( insert_o2 @ A @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B6 @ C2 ) ) )
          = ( comple90263536869209701_set_o
            @ ( image_o_set_o
              @ ^ [X: $o] : ( insert_o2 @ A @ ( B6 @ X ) )
              @ C2 ) ) ) ) ) ).

% UN_extend_simps(1)
thf(fact_747_UN__extend__simps_I1_J,axiom,
    ! [C2: set_Pr1822751329126368876m_rule,A: fm,B6: produc340336539035504054m_rule > set_fm] :
      ( ( ( C2 = bot_bo3426410157013319936m_rule )
       => ( ( insert_fm2 @ A @ ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ B6 @ C2 ) ) )
          = ( insert_fm2 @ A @ bot_bot_set_fm ) ) )
      & ( ( C2 != bot_bo3426410157013319936m_rule )
       => ( ( insert_fm2 @ A @ ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ B6 @ C2 ) ) )
          = ( comple2134933779557159616set_fm
            @ ( image_377115208147135190set_fm
              @ ^ [X: produc340336539035504054m_rule] : ( insert_fm2 @ A @ ( B6 @ X ) )
              @ C2 ) ) ) ) ) ).

% UN_extend_simps(1)
thf(fact_748_UN__extend__simps_I1_J,axiom,
    ! [C2: set_o,A: fm,B6: $o > set_fm] :
      ( ( ( C2 = bot_bot_set_o )
       => ( ( insert_fm2 @ A @ ( comple2134933779557159616set_fm @ ( image_o_set_fm @ B6 @ C2 ) ) )
          = ( insert_fm2 @ A @ bot_bot_set_fm ) ) )
      & ( ( C2 != bot_bot_set_o )
       => ( ( insert_fm2 @ A @ ( comple2134933779557159616set_fm @ ( image_o_set_fm @ B6 @ C2 ) ) )
          = ( comple2134933779557159616set_fm
            @ ( image_o_set_fm
              @ ^ [X: $o] : ( insert_fm2 @ A @ ( B6 @ X ) )
              @ C2 ) ) ) ) ) ).

% UN_extend_simps(1)
thf(fact_749_Id__on__def,axiom,
    ( id_on_nat
    = ( ^ [A7: set_nat] :
          ( comple5685304695842803022at_nat
          @ ( image_7178329752028323786at_nat
            @ ^ [X: nat] : ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ X @ X ) @ bot_bo2099793752762293965at_nat )
            @ A7 ) ) ) ) ).

% Id_on_def
thf(fact_750_is__singletonI,axiom,
    ! [X5: $o] : ( is_singleton_o @ ( insert_o2 @ X5 @ bot_bot_set_o ) ) ).

% is_singletonI
thf(fact_751_refl__on__domain,axiom,
    ! [A5: set_fm,R4: set_Pr4706815898642364871_fm_fm,A: fm,B: fm] :
      ( ( refl_on_fm @ A5 @ R4 )
     => ( ( member7780952600467998736_fm_fm @ ( product_Pair_fm_fm @ A @ B ) @ R4 )
       => ( ( member_fm2 @ A @ A5 )
          & ( member_fm2 @ B @ A5 ) ) ) ) ).

% refl_on_domain
thf(fact_752_refl__on__domain,axiom,
    ! [A5: set_list_fm,R4: set_Pr1916137733696349511ist_fm,A: list_fm,B: list_fm] :
      ( ( refl_on_list_fm @ A5 @ R4 )
     => ( ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ A @ B ) @ R4 )
       => ( ( member_list_fm2 @ A @ A5 )
          & ( member_list_fm2 @ B @ A5 ) ) ) ) ).

% refl_on_domain
thf(fact_753_refl__on__domain,axiom,
    ! [A5: set_Pr1822751329126368876m_rule,R4: set_Pr4971326047967503661m_rule,A: produc340336539035504054m_rule,B: produc340336539035504054m_rule] :
      ( ( refl_o3542783858073070071m_rule @ A5 @ R4 )
     => ( ( member1411645983481675894m_rule @ ( produc6261311607089640965m_rule @ A @ B ) @ R4 )
       => ( ( member7231649785386036813m_rule @ A @ A5 )
          & ( member7231649785386036813m_rule @ B @ A5 ) ) ) ) ).

% refl_on_domain
thf(fact_754_refl__on__domain,axiom,
    ! [A5: set_o,R4: set_Product_prod_o_o,A: $o,B: $o] :
      ( ( refl_on_o @ A5 @ R4 )
     => ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ B ) @ R4 )
       => ( ( member_o2 @ A @ A5 )
          & ( member_o2 @ B @ A5 ) ) ) ) ).

% refl_on_domain
thf(fact_755_refl__on__domain,axiom,
    ! [A5: set_nat,R4: set_Pr1261947904930325089at_nat,A: nat,B: nat] :
      ( ( refl_on_nat @ A5 @ R4 )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R4 )
       => ( ( member_nat @ A @ A5 )
          & ( member_nat @ B @ A5 ) ) ) ) ).

% refl_on_domain
thf(fact_756_is__empty__set,axiom,
    ! [Xs: list_fm] :
      ( ( is_empty_fm @ ( set_fm2 @ Xs ) )
      = ( null_fm @ Xs ) ) ).

% is_empty_set
thf(fact_757_is__empty__set,axiom,
    ! [Xs: list_list_fm] :
      ( ( is_empty_list_fm @ ( set_list_fm2 @ Xs ) )
      = ( null_list_fm @ Xs ) ) ).

% is_empty_set
thf(fact_758_is__empty__set,axiom,
    ! [Xs: list_tm] :
      ( ( is_empty_tm @ ( set_tm2 @ Xs ) )
      = ( null_tm @ Xs ) ) ).

% is_empty_set
thf(fact_759_holds__eq2,axiom,
    ! [X5: $o] :
      ( ( linear7433082607169829324olds_o
        @ ^ [Y: $o] : ( Y = X5 ) )
      = ( linear3076992658408248000_HLD_o @ ( insert_o2 @ X5 @ bot_bot_set_o ) ) ) ).

% holds_eq2
thf(fact_760_holds__eq2,axiom,
    ! [X5: produc340336539035504054m_rule] :
      ( ( linear7925859179493631106m_rule
        @ ^ [Y: produc340336539035504054m_rule] : ( Y = X5 ) )
      = ( linear2885739836753817742m_rule @ ( insert775845832097999846m_rule @ X5 @ bot_bo3426410157013319936m_rule ) ) ) ).

% holds_eq2
thf(fact_761_the__elem__def,axiom,
    ( the_elem_o
    = ( ^ [X9: set_o] :
          ( the_o
          @ ^ [X: $o] :
              ( X9
              = ( insert_o2 @ X @ bot_bot_set_o ) ) ) ) ) ).

% the_elem_def
thf(fact_762_set__rec,axiom,
    ( set_fm2
    = ( rec_list_set_fm_fm @ bot_bot_set_fm
      @ ^ [X: fm,Uu: list_fm] : ( insert_fm2 @ X ) ) ) ).

% set_rec
thf(fact_763_set__rec,axiom,
    ( set_list_fm2
    = ( rec_li3562514433299826070ist_fm @ bot_bot_set_list_fm
      @ ^ [X: list_fm,Uu: list_list_fm] : ( insert_list_fm2 @ X ) ) ) ).

% set_rec
thf(fact_764_set__rec,axiom,
    ( set_tm2
    = ( rec_list_set_tm_tm @ bot_bot_set_tm
      @ ^ [X: tm,Uu: list_tm] : ( insert_tm2 @ X ) ) ) ).

% set_rec
thf(fact_765_set__rec,axiom,
    ( set_o2
    = ( rec_list_set_o_o @ bot_bot_set_o
      @ ^ [X: $o,Uu: list_o] : ( insert_o2 @ X ) ) ) ).

% set_rec
thf(fact_766_Id__onI,axiom,
    ! [A: fm,A5: set_fm] :
      ( ( member_fm2 @ A @ A5 )
     => ( member7780952600467998736_fm_fm @ ( product_Pair_fm_fm @ A @ A ) @ ( id_on_fm @ A5 ) ) ) ).

% Id_onI
thf(fact_767_Id__onI,axiom,
    ! [A: list_fm,A5: set_list_fm] :
      ( ( member_list_fm2 @ A @ A5 )
     => ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ A @ A ) @ ( id_on_list_fm @ A5 ) ) ) ).

% Id_onI
thf(fact_768_Id__onI,axiom,
    ! [A: produc340336539035504054m_rule,A5: set_Pr1822751329126368876m_rule] :
      ( ( member7231649785386036813m_rule @ A @ A5 )
     => ( member1411645983481675894m_rule @ ( produc6261311607089640965m_rule @ A @ A ) @ ( id_on_597746392252987865m_rule @ A5 ) ) ) ).

% Id_onI
thf(fact_769_Id__onI,axiom,
    ! [A: $o,A5: set_o] :
      ( ( member_o2 @ A @ A5 )
     => ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ A ) @ ( id_on_o @ A5 ) ) ) ).

% Id_onI
thf(fact_770_Id__onI,axiom,
    ! [A: nat,A5: set_nat] :
      ( ( member_nat @ A @ A5 )
     => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ A ) @ ( id_on_nat @ A5 ) ) ) ).

% Id_onI
thf(fact_771_Id__on__empty,axiom,
    ( ( id_on_o @ bot_bot_set_o )
    = bot_bo7073875226086086771od_o_o ) ).

% Id_on_empty
thf(fact_772_the__elem__eq,axiom,
    ! [X5: $o] :
      ( ( the_elem_o @ ( insert_o2 @ X5 @ bot_bot_set_o ) )
      = X5 ) ).

% the_elem_eq
thf(fact_773_Id__on__iff,axiom,
    ! [X5: fm,Y4: fm,A5: set_fm] :
      ( ( member7780952600467998736_fm_fm @ ( product_Pair_fm_fm @ X5 @ Y4 ) @ ( id_on_fm @ A5 ) )
      = ( ( X5 = Y4 )
        & ( member_fm2 @ X5 @ A5 ) ) ) ).

% Id_on_iff
thf(fact_774_Id__on__iff,axiom,
    ! [X5: list_fm,Y4: list_fm,A5: set_list_fm] :
      ( ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ X5 @ Y4 ) @ ( id_on_list_fm @ A5 ) )
      = ( ( X5 = Y4 )
        & ( member_list_fm2 @ X5 @ A5 ) ) ) ).

% Id_on_iff
thf(fact_775_Id__on__iff,axiom,
    ! [X5: produc340336539035504054m_rule,Y4: produc340336539035504054m_rule,A5: set_Pr1822751329126368876m_rule] :
      ( ( member1411645983481675894m_rule @ ( produc6261311607089640965m_rule @ X5 @ Y4 ) @ ( id_on_597746392252987865m_rule @ A5 ) )
      = ( ( X5 = Y4 )
        & ( member7231649785386036813m_rule @ X5 @ A5 ) ) ) ).

% Id_on_iff
thf(fact_776_Id__on__iff,axiom,
    ! [X5: $o,Y4: $o,A5: set_o] :
      ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X5 @ Y4 ) @ ( id_on_o @ A5 ) )
      = ( ( X5 = Y4 )
        & ( member_o2 @ X5 @ A5 ) ) ) ).

% Id_on_iff
thf(fact_777_Id__on__iff,axiom,
    ! [X5: nat,Y4: nat,A5: set_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X5 @ Y4 ) @ ( id_on_nat @ A5 ) )
      = ( ( X5 = Y4 )
        & ( member_nat @ X5 @ A5 ) ) ) ).

% Id_on_iff
thf(fact_778_Id__on__eqI,axiom,
    ! [A: fm,B: fm,A5: set_fm] :
      ( ( A = B )
     => ( ( member_fm2 @ A @ A5 )
       => ( member7780952600467998736_fm_fm @ ( product_Pair_fm_fm @ A @ B ) @ ( id_on_fm @ A5 ) ) ) ) ).

% Id_on_eqI
thf(fact_779_Id__on__eqI,axiom,
    ! [A: list_fm,B: list_fm,A5: set_list_fm] :
      ( ( A = B )
     => ( ( member_list_fm2 @ A @ A5 )
       => ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ A @ B ) @ ( id_on_list_fm @ A5 ) ) ) ) ).

% Id_on_eqI
thf(fact_780_Id__on__eqI,axiom,
    ! [A: produc340336539035504054m_rule,B: produc340336539035504054m_rule,A5: set_Pr1822751329126368876m_rule] :
      ( ( A = B )
     => ( ( member7231649785386036813m_rule @ A @ A5 )
       => ( member1411645983481675894m_rule @ ( produc6261311607089640965m_rule @ A @ B ) @ ( id_on_597746392252987865m_rule @ A5 ) ) ) ) ).

% Id_on_eqI
thf(fact_781_Id__on__eqI,axiom,
    ! [A: $o,B: $o,A5: set_o] :
      ( ( A = B )
     => ( ( member_o2 @ A @ A5 )
       => ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ B ) @ ( id_on_o @ A5 ) ) ) ) ).

% Id_on_eqI
thf(fact_782_Id__on__eqI,axiom,
    ! [A: nat,B: nat,A5: set_nat] :
      ( ( A = B )
     => ( ( member_nat @ A @ A5 )
       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( id_on_nat @ A5 ) ) ) ) ).

% Id_on_eqI
thf(fact_783_Id__onE,axiom,
    ! [C: product_prod_fm_fm,A5: set_fm] :
      ( ( member7780952600467998736_fm_fm @ C @ ( id_on_fm @ A5 ) )
     => ~ ! [X3: fm] :
            ( ( member_fm2 @ X3 @ A5 )
           => ( C
             != ( product_Pair_fm_fm @ X3 @ X3 ) ) ) ) ).

% Id_onE
thf(fact_784_Id__onE,axiom,
    ! [C: produc3245234490656042599ist_fm,A5: set_list_fm] :
      ( ( member1926098302810316688ist_fm @ C @ ( id_on_list_fm @ A5 ) )
     => ~ ! [X3: list_fm] :
            ( ( member_list_fm2 @ X3 @ A5 )
           => ( C
             != ( produc7863996417982153943ist_fm @ X3 @ X3 ) ) ) ) ).

% Id_onE
thf(fact_785_Id__onE,axiom,
    ! [C: produc8828831911945107917m_rule,A5: set_Pr1822751329126368876m_rule] :
      ( ( member1411645983481675894m_rule @ C @ ( id_on_597746392252987865m_rule @ A5 ) )
     => ~ ! [X3: produc340336539035504054m_rule] :
            ( ( member7231649785386036813m_rule @ X3 @ A5 )
           => ( C
             != ( produc6261311607089640965m_rule @ X3 @ X3 ) ) ) ) ).

% Id_onE
thf(fact_786_Id__onE,axiom,
    ! [C: product_prod_o_o,A5: set_o] :
      ( ( member7466972457876170832od_o_o @ C @ ( id_on_o @ A5 ) )
     => ~ ! [X3: $o] :
            ( ( member_o2 @ X3 @ A5 )
           => ( C
             != ( product_Pair_o_o @ X3 @ X3 ) ) ) ) ).

% Id_onE
thf(fact_787_Id__onE,axiom,
    ! [C: product_prod_nat_nat,A5: set_nat] :
      ( ( member8440522571783428010at_nat @ C @ ( id_on_nat @ A5 ) )
     => ~ ! [X3: nat] :
            ( ( member_nat @ X3 @ A5 )
           => ( C
             != ( product_Pair_nat_nat @ X3 @ X3 ) ) ) ) ).

% Id_onE
thf(fact_788_HLD__iff,axiom,
    ( linear2973384439519985051HLD_fm
    = ( ^ [S5: set_fm,Omega2: stream_fm] : ( member_fm2 @ ( shd_fm @ Omega2 ) @ S5 ) ) ) ).

% HLD_iff
thf(fact_789_HLD__iff,axiom,
    ( linear4005088938741646753ist_fm
    = ( ^ [S5: set_list_fm,Omega2: stream_list_fm] : ( member_list_fm2 @ ( shd_list_fm @ Omega2 ) @ S5 ) ) ) ).

% HLD_iff
thf(fact_790_HLD__iff,axiom,
    ( linear3076992658408248000_HLD_o
    = ( ^ [S5: set_o,Omega2: stream_o] : ( member_o2 @ ( shd_o @ Omega2 ) @ S5 ) ) ) ).

% HLD_iff
thf(fact_791_HLD__iff,axiom,
    ( linear2885739836753817742m_rule
    = ( ^ [S5: set_Pr1822751329126368876m_rule,Omega2: stream2709947120125613254m_rule] : ( member7231649785386036813m_rule @ ( shd_Pr4562317740776619530m_rule @ Omega2 ) @ S5 ) ) ) ).

% HLD_iff
thf(fact_792_is__singleton__the__elem,axiom,
    ( is_singleton_o
    = ( ^ [A7: set_o] :
          ( A7
          = ( insert_o2 @ ( the_elem_o @ A7 ) @ bot_bot_set_o ) ) ) ) ).

% is_singleton_the_elem
thf(fact_793_Id__on__def_H,axiom,
    ! [A5: product_prod_nat_nat > $o] :
      ( ( id_on_2554058798563519774at_nat @ ( collec3392354462482085612at_nat @ A5 ) )
      = ( collec7088162979684241874at_nat
        @ ( produc6590410687421337004_nat_o
          @ ^ [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
              ( ( X = Y )
              & ( A5 @ X ) ) ) ) ) ).

% Id_on_def'
thf(fact_794_Id__on__def_H,axiom,
    ! [A5: nat > $o] :
      ( ( id_on_nat @ ( collect_nat @ A5 ) )
      = ( collec3392354462482085612at_nat
        @ ( produc6081775807080527818_nat_o
          @ ^ [X: nat,Y: nat] :
              ( ( X = Y )
              & ( A5 @ X ) ) ) ) ) ).

% Id_on_def'
thf(fact_795_HLD__def,axiom,
    ( linear2973384439519985051HLD_fm
    = ( ^ [S5: set_fm] :
          ( linear3702449374084777639lds_fm
          @ ^ [X: fm] : ( member_fm2 @ X @ S5 ) ) ) ) ).

% HLD_def
thf(fact_796_HLD__def,axiom,
    ( linear4005088938741646753ist_fm
    = ( ^ [S5: set_list_fm] :
          ( linear6347412853745146029ist_fm
          @ ^ [X: list_fm] : ( member_list_fm2 @ X @ S5 ) ) ) ) ).

% HLD_def
thf(fact_797_HLD__def,axiom,
    ( linear3076992658408248000_HLD_o
    = ( ^ [S5: set_o] :
          ( linear7433082607169829324olds_o
          @ ^ [X: $o] : ( member_o2 @ X @ S5 ) ) ) ) ).

% HLD_def
thf(fact_798_HLD__def,axiom,
    ( linear2885739836753817742m_rule
    = ( ^ [S5: set_Pr1822751329126368876m_rule] :
          ( linear7925859179493631106m_rule
          @ ^ [X: produc340336539035504054m_rule] : ( member7231649785386036813m_rule @ X @ S5 ) ) ) ) ).

% HLD_def
thf(fact_799_is__singletonI_H,axiom,
    ! [A5: set_fm] :
      ( ( A5 != bot_bot_set_fm )
     => ( ! [X3: fm,Y3: fm] :
            ( ( member_fm2 @ X3 @ A5 )
           => ( ( member_fm2 @ Y3 @ A5 )
             => ( X3 = Y3 ) ) )
       => ( is_singleton_fm @ A5 ) ) ) ).

% is_singletonI'
thf(fact_800_is__singletonI_H,axiom,
    ! [A5: set_list_fm] :
      ( ( A5 != bot_bot_set_list_fm )
     => ( ! [X3: list_fm,Y3: list_fm] :
            ( ( member_list_fm2 @ X3 @ A5 )
           => ( ( member_list_fm2 @ Y3 @ A5 )
             => ( X3 = Y3 ) ) )
       => ( is_singleton_list_fm @ A5 ) ) ) ).

% is_singletonI'
thf(fact_801_is__singletonI_H,axiom,
    ! [A5: set_Pr1822751329126368876m_rule] :
      ( ( A5 != bot_bo3426410157013319936m_rule )
     => ( ! [X3: produc340336539035504054m_rule,Y3: produc340336539035504054m_rule] :
            ( ( member7231649785386036813m_rule @ X3 @ A5 )
           => ( ( member7231649785386036813m_rule @ Y3 @ A5 )
             => ( X3 = Y3 ) ) )
       => ( is_sin5529023878435517890m_rule @ A5 ) ) ) ).

% is_singletonI'
thf(fact_802_is__singletonI_H,axiom,
    ! [A5: set_o] :
      ( ( A5 != bot_bot_set_o )
     => ( ! [X3: $o,Y3: $o] :
            ( ( member_o2 @ X3 @ A5 )
           => ( ( member_o2 @ Y3 @ A5 )
             => ( X3 = Y3 ) ) )
       => ( is_singleton_o @ A5 ) ) ) ).

% is_singletonI'
thf(fact_803_the__elem__image__unique,axiom,
    ! [A5: set_Pr1822751329126368876m_rule,F: produc340336539035504054m_rule > set_fm,X5: produc340336539035504054m_rule] :
      ( ( A5 != bot_bo3426410157013319936m_rule )
     => ( ! [Y3: produc340336539035504054m_rule] :
            ( ( member7231649785386036813m_rule @ Y3 @ A5 )
           => ( ( F @ Y3 )
              = ( F @ X5 ) ) )
       => ( ( the_elem_set_fm @ ( image_377115208147135190set_fm @ F @ A5 ) )
          = ( F @ X5 ) ) ) ) ).

% the_elem_image_unique
thf(fact_804_is__singletonE,axiom,
    ! [A5: set_o] :
      ( ( is_singleton_o @ A5 )
     => ~ ! [X3: $o] :
            ( A5
           != ( insert_o2 @ X3 @ bot_bot_set_o ) ) ) ).

% is_singletonE
thf(fact_805_is__singleton__def,axiom,
    ( is_singleton_o
    = ( ^ [A7: set_o] :
        ? [X: $o] :
          ( A7
          = ( insert_o2 @ X @ bot_bot_set_o ) ) ) ) ).

% is_singleton_def
thf(fact_806_linear__order__on__singleton,axiom,
    ! [X5: nat] : ( order_4473980167227706203on_nat @ ( insert_nat @ X5 @ bot_bot_set_nat ) @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ X5 @ X5 ) @ bot_bo2099793752762293965at_nat ) ) ).

% linear_order_on_singleton
thf(fact_807_linear__order__on__singleton,axiom,
    ! [X5: $o] : ( order_6238756238976269133r_on_o @ ( insert_o2 @ X5 @ bot_bot_set_o ) @ ( insert6201435330877294327od_o_o @ ( product_Pair_o_o @ X5 @ X5 ) @ bot_bo7073875226086086771od_o_o ) ) ).

% linear_order_on_singleton
thf(fact_808_relation__of__def,axiom,
    ( order_1962693441708344834of_nat
    = ( ^ [P4: nat > nat > $o,A7: set_nat] :
          ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ^ [A3: nat,B3: nat] :
                ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A3 @ B3 )
                  @ ( produc457027306803732586at_nat @ A7
                    @ ^ [Uu: nat] : A7 ) )
                & ( P4 @ A3 @ B3 ) ) ) ) ) ) ).

% relation_of_def
thf(fact_809_insert__Times__insert,axiom,
    ! [A: $o,A5: set_o,B: $o,B6: set_o] :
      ( ( product_Sigma_o_o @ ( insert_o2 @ A @ A5 )
        @ ^ [Uu: $o] : ( insert_o2 @ B @ B6 ) )
      = ( insert6201435330877294327od_o_o @ ( product_Pair_o_o @ A @ B )
        @ ( sup_su5769328420594410459od_o_o
          @ ( product_Sigma_o_o @ A5
            @ ^ [Uu: $o] : ( insert_o2 @ B @ B6 ) )
          @ ( product_Sigma_o_o @ ( insert_o2 @ A @ A5 )
            @ ^ [Uu: $o] : B6 ) ) ) ) ).

% insert_Times_insert
thf(fact_810_insert__Times__insert,axiom,
    ! [A: rule,A5: set_rule,B: fm,B6: set_fm] :
      ( ( produc7224384929229372695ule_fm @ ( insert_rule @ A @ A5 )
        @ ^ [Uu: rule] : ( insert_fm2 @ B @ B6 ) )
      = ( insert684836149095756678ule_fm @ ( product_Pair_rule_fm @ A @ B )
        @ ( sup_su1866406141038604088ule_fm
          @ ( produc7224384929229372695ule_fm @ A5
            @ ^ [Uu: rule] : ( insert_fm2 @ B @ B6 ) )
          @ ( produc7224384929229372695ule_fm @ ( insert_rule @ A @ A5 )
            @ ^ [Uu: rule] : B6 ) ) ) ) ).

% insert_Times_insert
thf(fact_811_insert__Times__insert,axiom,
    ! [A: nat,A5: set_nat,B: nat,B6: set_nat] :
      ( ( produc457027306803732586at_nat @ ( insert_nat @ A @ A5 )
        @ ^ [Uu: nat] : ( insert_nat @ B @ B6 ) )
      = ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ A @ B )
        @ ( sup_su6327502436637775413at_nat
          @ ( produc457027306803732586at_nat @ A5
            @ ^ [Uu: nat] : ( insert_nat @ B @ B6 ) )
          @ ( produc457027306803732586at_nat @ ( insert_nat @ A @ A5 )
            @ ^ [Uu: nat] : B6 ) ) ) ) ).

% insert_Times_insert
thf(fact_812_quotient__def,axiom,
    ( equiv_quotient_o
    = ( ^ [A7: set_o,R: set_Product_prod_o_o] :
          ( comple4436988014476444997_set_o
          @ ( image_o_set_set_o
            @ ^ [X: $o] : ( insert_set_o @ ( image_o_o @ R @ ( insert_o2 @ X @ bot_bot_set_o ) ) @ bot_bot_set_set_o )
            @ A7 ) ) ) ) ).

% quotient_def
thf(fact_813_prod__set__simps_I2_J,axiom,
    ! [X5: rule,Y4: fm] :
      ( ( basic_snds_rule_fm @ ( product_Pair_rule_fm @ X5 @ Y4 ) )
      = ( insert_fm2 @ Y4 @ bot_bot_set_fm ) ) ).

% prod_set_simps(2)
thf(fact_814_prod__set__simps_I2_J,axiom,
    ! [X5: nat,Y4: nat] :
      ( ( basic_snds_nat_nat @ ( product_Pair_nat_nat @ X5 @ Y4 ) )
      = ( insert_nat @ Y4 @ bot_bot_set_nat ) ) ).

% prod_set_simps(2)
thf(fact_815_prod__set__simps_I1_J,axiom,
    ! [X5: rule,Y4: fm] :
      ( ( basic_fsts_rule_fm @ ( product_Pair_rule_fm @ X5 @ Y4 ) )
      = ( insert_rule @ X5 @ bot_bot_set_rule ) ) ).

% prod_set_simps(1)
thf(fact_816_prod__set__simps_I1_J,axiom,
    ! [X5: nat,Y4: nat] :
      ( ( basic_fsts_nat_nat @ ( product_Pair_nat_nat @ X5 @ Y4 ) )
      = ( insert_nat @ X5 @ bot_bot_set_nat ) ) ).

% prod_set_simps(1)
thf(fact_817_UnCI,axiom,
    ! [C: fm,B6: set_fm,A5: set_fm] :
      ( ( ~ ( member_fm2 @ C @ B6 )
       => ( member_fm2 @ C @ A5 ) )
     => ( member_fm2 @ C @ ( sup_sup_set_fm @ A5 @ B6 ) ) ) ).

% UnCI
thf(fact_818_UnCI,axiom,
    ! [C: list_fm,B6: set_list_fm,A5: set_list_fm] :
      ( ( ~ ( member_list_fm2 @ C @ B6 )
       => ( member_list_fm2 @ C @ A5 ) )
     => ( member_list_fm2 @ C @ ( sup_sup_set_list_fm @ A5 @ B6 ) ) ) ).

% UnCI
thf(fact_819_UnCI,axiom,
    ! [C: produc340336539035504054m_rule,B6: set_Pr1822751329126368876m_rule,A5: set_Pr1822751329126368876m_rule] :
      ( ( ~ ( member7231649785386036813m_rule @ C @ B6 )
       => ( member7231649785386036813m_rule @ C @ A5 ) )
     => ( member7231649785386036813m_rule @ C @ ( sup_su6946459741510085528m_rule @ A5 @ B6 ) ) ) ).

% UnCI
thf(fact_820_UnCI,axiom,
    ! [C: $o,B6: set_o,A5: set_o] :
      ( ( ~ ( member_o2 @ C @ B6 )
       => ( member_o2 @ C @ A5 ) )
     => ( member_o2 @ C @ ( sup_sup_set_o @ A5 @ B6 ) ) ) ).

% UnCI
thf(fact_821_Un__iff,axiom,
    ! [C: fm,A5: set_fm,B6: set_fm] :
      ( ( member_fm2 @ C @ ( sup_sup_set_fm @ A5 @ B6 ) )
      = ( ( member_fm2 @ C @ A5 )
        | ( member_fm2 @ C @ B6 ) ) ) ).

% Un_iff
thf(fact_822_Un__iff,axiom,
    ! [C: list_fm,A5: set_list_fm,B6: set_list_fm] :
      ( ( member_list_fm2 @ C @ ( sup_sup_set_list_fm @ A5 @ B6 ) )
      = ( ( member_list_fm2 @ C @ A5 )
        | ( member_list_fm2 @ C @ B6 ) ) ) ).

% Un_iff
thf(fact_823_Un__iff,axiom,
    ! [C: produc340336539035504054m_rule,A5: set_Pr1822751329126368876m_rule,B6: set_Pr1822751329126368876m_rule] :
      ( ( member7231649785386036813m_rule @ C @ ( sup_su6946459741510085528m_rule @ A5 @ B6 ) )
      = ( ( member7231649785386036813m_rule @ C @ A5 )
        | ( member7231649785386036813m_rule @ C @ B6 ) ) ) ).

% Un_iff
thf(fact_824_Un__iff,axiom,
    ! [C: $o,A5: set_o,B6: set_o] :
      ( ( member_o2 @ C @ ( sup_sup_set_o @ A5 @ B6 ) )
      = ( ( member_o2 @ C @ A5 )
        | ( member_o2 @ C @ B6 ) ) ) ).

% Un_iff
thf(fact_825_Un__empty,axiom,
    ! [A5: set_o,B6: set_o] :
      ( ( ( sup_sup_set_o @ A5 @ B6 )
        = bot_bot_set_o )
      = ( ( A5 = bot_bot_set_o )
        & ( B6 = bot_bot_set_o ) ) ) ).

% Un_empty
thf(fact_826_Un__insert__left,axiom,
    ! [A: $o,B6: set_o,C2: set_o] :
      ( ( sup_sup_set_o @ ( insert_o2 @ A @ B6 ) @ C2 )
      = ( insert_o2 @ A @ ( sup_sup_set_o @ B6 @ C2 ) ) ) ).

% Un_insert_left
thf(fact_827_Un__insert__right,axiom,
    ! [A5: set_o,A: $o,B6: set_o] :
      ( ( sup_sup_set_o @ A5 @ ( insert_o2 @ A @ B6 ) )
      = ( insert_o2 @ A @ ( sup_sup_set_o @ A5 @ B6 ) ) ) ).

% Un_insert_right
thf(fact_828_Union__Un__distrib,axiom,
    ! [A5: set_set_fm,B6: set_set_fm] :
      ( ( comple2134933779557159616set_fm @ ( sup_sup_set_set_fm @ A5 @ B6 ) )
      = ( sup_sup_set_fm @ ( comple2134933779557159616set_fm @ A5 ) @ ( comple2134933779557159616set_fm @ B6 ) ) ) ).

% Union_Un_distrib
thf(fact_829_Sup__insert,axiom,
    ! [A: set_fm,A5: set_set_fm] :
      ( ( comple2134933779557159616set_fm @ ( insert_set_fm @ A @ A5 ) )
      = ( sup_sup_set_fm @ A @ ( comple2134933779557159616set_fm @ A5 ) ) ) ).

% Sup_insert
thf(fact_830_Sup__insert,axiom,
    ! [A: $o,A5: set_o] :
      ( ( complete_Sup_Sup_o @ ( insert_o2 @ A @ A5 ) )
      = ( sup_sup_o @ A @ ( complete_Sup_Sup_o @ A5 ) ) ) ).

% Sup_insert
thf(fact_831_UN__Un,axiom,
    ! [M: produc340336539035504054m_rule > set_fm,A5: set_Pr1822751329126368876m_rule,B6: set_Pr1822751329126368876m_rule] :
      ( ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ M @ ( sup_su6946459741510085528m_rule @ A5 @ B6 ) ) )
      = ( sup_sup_set_fm @ ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ M @ A5 ) ) @ ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ M @ B6 ) ) ) ) ).

% UN_Un
thf(fact_832_bdd__above__image__sup,axiom,
    ! [F: produc340336539035504054m_rule > set_fm,G: produc340336539035504054m_rule > set_fm,A5: set_Pr1822751329126368876m_rule] :
      ( ( condit7632555770478454199set_fm
        @ ( image_377115208147135190set_fm
          @ ^ [X: produc340336539035504054m_rule] : ( sup_sup_set_fm @ ( F @ X ) @ ( G @ X ) )
          @ A5 ) )
      = ( ( condit7632555770478454199set_fm @ ( image_377115208147135190set_fm @ F @ A5 ) )
        & ( condit7632555770478454199set_fm @ ( image_377115208147135190set_fm @ G @ A5 ) ) ) ) ).

% bdd_above_image_sup
thf(fact_833_sset__shift,axiom,
    ! [Xs: list_fm,S2: stream_fm] :
      ( ( sset_fm @ ( shift_fm @ Xs @ S2 ) )
      = ( sup_sup_set_fm @ ( set_fm2 @ Xs ) @ ( sset_fm @ S2 ) ) ) ).

% sset_shift
thf(fact_834_sset__shift,axiom,
    ! [Xs: list_list_fm,S2: stream_list_fm] :
      ( ( sset_list_fm @ ( shift_list_fm @ Xs @ S2 ) )
      = ( sup_sup_set_list_fm @ ( set_list_fm2 @ Xs ) @ ( sset_list_fm @ S2 ) ) ) ).

% sset_shift
thf(fact_835_sset__shift,axiom,
    ! [Xs: list_tm,S2: stream_tm] :
      ( ( sset_tm @ ( shift_tm @ Xs @ S2 ) )
      = ( sup_sup_set_tm @ ( set_tm2 @ Xs ) @ ( sset_tm @ S2 ) ) ) ).

% sset_shift
thf(fact_836_sset__shift,axiom,
    ! [Xs: list_P2774625669004309958m_rule,S2: stream2709947120125613254m_rule] :
      ( ( sset_P4484857331586881186m_rule @ ( shift_2334844276748245581m_rule @ Xs @ S2 ) )
      = ( sup_su6946459741510085528m_rule @ ( set_Pr4534715572506550497m_rule @ Xs ) @ ( sset_P4484857331586881186m_rule @ S2 ) ) ) ).

% sset_shift
thf(fact_837_UN__simps_I2_J,axiom,
    ! [C2: set_o,A5: $o > set_o,B6: set_o] :
      ( ( ( C2 = bot_bot_set_o )
       => ( ( comple90263536869209701_set_o
            @ ( image_o_set_o
              @ ^ [X: $o] : ( sup_sup_set_o @ ( A5 @ X ) @ B6 )
              @ C2 ) )
          = bot_bot_set_o ) )
      & ( ( C2 != bot_bot_set_o )
       => ( ( comple90263536869209701_set_o
            @ ( image_o_set_o
              @ ^ [X: $o] : ( sup_sup_set_o @ ( A5 @ X ) @ B6 )
              @ C2 ) )
          = ( sup_sup_set_o @ ( comple90263536869209701_set_o @ ( image_o_set_o @ A5 @ C2 ) ) @ B6 ) ) ) ) ).

% UN_simps(2)
thf(fact_838_UN__simps_I2_J,axiom,
    ! [C2: set_Pr1822751329126368876m_rule,A5: produc340336539035504054m_rule > set_fm,B6: set_fm] :
      ( ( ( C2 = bot_bo3426410157013319936m_rule )
       => ( ( comple2134933779557159616set_fm
            @ ( image_377115208147135190set_fm
              @ ^ [X: produc340336539035504054m_rule] : ( sup_sup_set_fm @ ( A5 @ X ) @ B6 )
              @ C2 ) )
          = bot_bot_set_fm ) )
      & ( ( C2 != bot_bo3426410157013319936m_rule )
       => ( ( comple2134933779557159616set_fm
            @ ( image_377115208147135190set_fm
              @ ^ [X: produc340336539035504054m_rule] : ( sup_sup_set_fm @ ( A5 @ X ) @ B6 )
              @ C2 ) )
          = ( sup_sup_set_fm @ ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ A5 @ C2 ) ) @ B6 ) ) ) ) ).

% UN_simps(2)
thf(fact_839_UN__simps_I2_J,axiom,
    ! [C2: set_o,A5: $o > set_fm,B6: set_fm] :
      ( ( ( C2 = bot_bot_set_o )
       => ( ( comple2134933779557159616set_fm
            @ ( image_o_set_fm
              @ ^ [X: $o] : ( sup_sup_set_fm @ ( A5 @ X ) @ B6 )
              @ C2 ) )
          = bot_bot_set_fm ) )
      & ( ( C2 != bot_bot_set_o )
       => ( ( comple2134933779557159616set_fm
            @ ( image_o_set_fm
              @ ^ [X: $o] : ( sup_sup_set_fm @ ( A5 @ X ) @ B6 )
              @ C2 ) )
          = ( sup_sup_set_fm @ ( comple2134933779557159616set_fm @ ( image_o_set_fm @ A5 @ C2 ) ) @ B6 ) ) ) ) ).

% UN_simps(2)
thf(fact_840_UN__simps_I3_J,axiom,
    ! [C2: set_o,A5: set_o,B6: $o > set_o] :
      ( ( ( C2 = bot_bot_set_o )
       => ( ( comple90263536869209701_set_o
            @ ( image_o_set_o
              @ ^ [X: $o] : ( sup_sup_set_o @ A5 @ ( B6 @ X ) )
              @ C2 ) )
          = bot_bot_set_o ) )
      & ( ( C2 != bot_bot_set_o )
       => ( ( comple90263536869209701_set_o
            @ ( image_o_set_o
              @ ^ [X: $o] : ( sup_sup_set_o @ A5 @ ( B6 @ X ) )
              @ C2 ) )
          = ( sup_sup_set_o @ A5 @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B6 @ C2 ) ) ) ) ) ) ).

% UN_simps(3)
thf(fact_841_UN__simps_I3_J,axiom,
    ! [C2: set_Pr1822751329126368876m_rule,A5: set_fm,B6: produc340336539035504054m_rule > set_fm] :
      ( ( ( C2 = bot_bo3426410157013319936m_rule )
       => ( ( comple2134933779557159616set_fm
            @ ( image_377115208147135190set_fm
              @ ^ [X: produc340336539035504054m_rule] : ( sup_sup_set_fm @ A5 @ ( B6 @ X ) )
              @ C2 ) )
          = bot_bot_set_fm ) )
      & ( ( C2 != bot_bo3426410157013319936m_rule )
       => ( ( comple2134933779557159616set_fm
            @ ( image_377115208147135190set_fm
              @ ^ [X: produc340336539035504054m_rule] : ( sup_sup_set_fm @ A5 @ ( B6 @ X ) )
              @ C2 ) )
          = ( sup_sup_set_fm @ A5 @ ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ B6 @ C2 ) ) ) ) ) ) ).

% UN_simps(3)
thf(fact_842_UN__simps_I3_J,axiom,
    ! [C2: set_o,A5: set_fm,B6: $o > set_fm] :
      ( ( ( C2 = bot_bot_set_o )
       => ( ( comple2134933779557159616set_fm
            @ ( image_o_set_fm
              @ ^ [X: $o] : ( sup_sup_set_fm @ A5 @ ( B6 @ X ) )
              @ C2 ) )
          = bot_bot_set_fm ) )
      & ( ( C2 != bot_bot_set_o )
       => ( ( comple2134933779557159616set_fm
            @ ( image_o_set_fm
              @ ^ [X: $o] : ( sup_sup_set_fm @ A5 @ ( B6 @ X ) )
              @ C2 ) )
          = ( sup_sup_set_fm @ A5 @ ( comple2134933779557159616set_fm @ ( image_o_set_fm @ B6 @ C2 ) ) ) ) ) ) ).

% UN_simps(3)
thf(fact_843_UN__insert,axiom,
    ! [B6: produc340336539035504054m_rule > set_fm,A: produc340336539035504054m_rule,A5: set_Pr1822751329126368876m_rule] :
      ( ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ B6 @ ( insert775845832097999846m_rule @ A @ A5 ) ) )
      = ( sup_sup_set_fm @ ( B6 @ A ) @ ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ B6 @ A5 ) ) ) ) ).

% UN_insert
thf(fact_844_UN__insert,axiom,
    ! [B6: $o > set_fm,A: $o,A5: set_o] :
      ( ( comple2134933779557159616set_fm @ ( image_o_set_fm @ B6 @ ( insert_o2 @ A @ A5 ) ) )
      = ( sup_sup_set_fm @ ( B6 @ A ) @ ( comple2134933779557159616set_fm @ ( image_o_set_fm @ B6 @ A5 ) ) ) ) ).

% UN_insert
thf(fact_845_UnE,axiom,
    ! [C: fm,A5: set_fm,B6: set_fm] :
      ( ( member_fm2 @ C @ ( sup_sup_set_fm @ A5 @ B6 ) )
     => ( ~ ( member_fm2 @ C @ A5 )
       => ( member_fm2 @ C @ B6 ) ) ) ).

% UnE
thf(fact_846_UnE,axiom,
    ! [C: list_fm,A5: set_list_fm,B6: set_list_fm] :
      ( ( member_list_fm2 @ C @ ( sup_sup_set_list_fm @ A5 @ B6 ) )
     => ( ~ ( member_list_fm2 @ C @ A5 )
       => ( member_list_fm2 @ C @ B6 ) ) ) ).

% UnE
thf(fact_847_UnE,axiom,
    ! [C: produc340336539035504054m_rule,A5: set_Pr1822751329126368876m_rule,B6: set_Pr1822751329126368876m_rule] :
      ( ( member7231649785386036813m_rule @ C @ ( sup_su6946459741510085528m_rule @ A5 @ B6 ) )
     => ( ~ ( member7231649785386036813m_rule @ C @ A5 )
       => ( member7231649785386036813m_rule @ C @ B6 ) ) ) ).

% UnE
thf(fact_848_UnE,axiom,
    ! [C: $o,A5: set_o,B6: set_o] :
      ( ( member_o2 @ C @ ( sup_sup_set_o @ A5 @ B6 ) )
     => ( ~ ( member_o2 @ C @ A5 )
       => ( member_o2 @ C @ B6 ) ) ) ).

% UnE
thf(fact_849_UnI1,axiom,
    ! [C: fm,A5: set_fm,B6: set_fm] :
      ( ( member_fm2 @ C @ A5 )
     => ( member_fm2 @ C @ ( sup_sup_set_fm @ A5 @ B6 ) ) ) ).

% UnI1
thf(fact_850_UnI1,axiom,
    ! [C: list_fm,A5: set_list_fm,B6: set_list_fm] :
      ( ( member_list_fm2 @ C @ A5 )
     => ( member_list_fm2 @ C @ ( sup_sup_set_list_fm @ A5 @ B6 ) ) ) ).

% UnI1
thf(fact_851_UnI1,axiom,
    ! [C: produc340336539035504054m_rule,A5: set_Pr1822751329126368876m_rule,B6: set_Pr1822751329126368876m_rule] :
      ( ( member7231649785386036813m_rule @ C @ A5 )
     => ( member7231649785386036813m_rule @ C @ ( sup_su6946459741510085528m_rule @ A5 @ B6 ) ) ) ).

% UnI1
thf(fact_852_UnI1,axiom,
    ! [C: $o,A5: set_o,B6: set_o] :
      ( ( member_o2 @ C @ A5 )
     => ( member_o2 @ C @ ( sup_sup_set_o @ A5 @ B6 ) ) ) ).

% UnI1
thf(fact_853_UnI2,axiom,
    ! [C: fm,B6: set_fm,A5: set_fm] :
      ( ( member_fm2 @ C @ B6 )
     => ( member_fm2 @ C @ ( sup_sup_set_fm @ A5 @ B6 ) ) ) ).

% UnI2
thf(fact_854_UnI2,axiom,
    ! [C: list_fm,B6: set_list_fm,A5: set_list_fm] :
      ( ( member_list_fm2 @ C @ B6 )
     => ( member_list_fm2 @ C @ ( sup_sup_set_list_fm @ A5 @ B6 ) ) ) ).

% UnI2
thf(fact_855_UnI2,axiom,
    ! [C: produc340336539035504054m_rule,B6: set_Pr1822751329126368876m_rule,A5: set_Pr1822751329126368876m_rule] :
      ( ( member7231649785386036813m_rule @ C @ B6 )
     => ( member7231649785386036813m_rule @ C @ ( sup_su6946459741510085528m_rule @ A5 @ B6 ) ) ) ).

% UnI2
thf(fact_856_UnI2,axiom,
    ! [C: $o,B6: set_o,A5: set_o] :
      ( ( member_o2 @ C @ B6 )
     => ( member_o2 @ C @ ( sup_sup_set_o @ A5 @ B6 ) ) ) ).

% UnI2
thf(fact_857_Un__def,axiom,
    ( sup_sup_set_fm
    = ( ^ [A7: set_fm,B7: set_fm] :
          ( collect_fm
          @ ^ [X: fm] :
              ( ( member_fm2 @ X @ A7 )
              | ( member_fm2 @ X @ B7 ) ) ) ) ) ).

% Un_def
thf(fact_858_Un__def,axiom,
    ( sup_sup_set_list_fm
    = ( ^ [A7: set_list_fm,B7: set_list_fm] :
          ( collect_list_fm
          @ ^ [X: list_fm] :
              ( ( member_list_fm2 @ X @ A7 )
              | ( member_list_fm2 @ X @ B7 ) ) ) ) ) ).

% Un_def
thf(fact_859_Un__def,axiom,
    ( sup_su6946459741510085528m_rule
    = ( ^ [A7: set_Pr1822751329126368876m_rule,B7: set_Pr1822751329126368876m_rule] :
          ( collec1009276759030335883m_rule
          @ ^ [X: produc340336539035504054m_rule] :
              ( ( member7231649785386036813m_rule @ X @ A7 )
              | ( member7231649785386036813m_rule @ X @ B7 ) ) ) ) ) ).

% Un_def
thf(fact_860_Un__def,axiom,
    ( sup_sup_set_o
    = ( ^ [A7: set_o,B7: set_o] :
          ( collect_o
          @ ^ [X: $o] :
              ( ( member_o2 @ X @ A7 )
              | ( member_o2 @ X @ B7 ) ) ) ) ) ).

% Un_def
thf(fact_861_Un__def,axiom,
    ( sup_su6327502436637775413at_nat
    = ( ^ [A7: set_Pr1261947904930325089at_nat,B7: set_Pr1261947904930325089at_nat] :
          ( collec3392354462482085612at_nat
          @ ^ [X: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X @ A7 )
              | ( member8440522571783428010at_nat @ X @ B7 ) ) ) ) ) ).

% Un_def
thf(fact_862_Un__def,axiom,
    ( sup_sup_set_nat
    = ( ^ [A7: set_nat,B7: set_nat] :
          ( collect_nat
          @ ^ [X: nat] :
              ( ( member_nat @ X @ A7 )
              | ( member_nat @ X @ B7 ) ) ) ) ) ).

% Un_def
thf(fact_863_Collect__disj__eq,axiom,
    ! [P3: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
      ( ( collec3392354462482085612at_nat
        @ ^ [X: product_prod_nat_nat] :
            ( ( P3 @ X )
            | ( Q @ X ) ) )
      = ( sup_su6327502436637775413at_nat @ ( collec3392354462482085612at_nat @ P3 ) @ ( collec3392354462482085612at_nat @ Q ) ) ) ).

% Collect_disj_eq
thf(fact_864_Collect__disj__eq,axiom,
    ! [P3: nat > $o,Q: nat > $o] :
      ( ( collect_nat
        @ ^ [X: nat] :
            ( ( P3 @ X )
            | ( Q @ X ) ) )
      = ( sup_sup_set_nat @ ( collect_nat @ P3 ) @ ( collect_nat @ Q ) ) ) ).

% Collect_disj_eq
thf(fact_865_Sup__union__distrib,axiom,
    ! [A5: set_set_fm,B6: set_set_fm] :
      ( ( comple2134933779557159616set_fm @ ( sup_sup_set_set_fm @ A5 @ B6 ) )
      = ( sup_sup_set_fm @ ( comple2134933779557159616set_fm @ A5 ) @ ( comple2134933779557159616set_fm @ B6 ) ) ) ).

% Sup_union_distrib
thf(fact_866_Sup__union__distrib,axiom,
    ! [A5: set_o,B6: set_o] :
      ( ( complete_Sup_Sup_o @ ( sup_sup_set_o @ A5 @ B6 ) )
      = ( sup_sup_o @ ( complete_Sup_Sup_o @ A5 ) @ ( complete_Sup_Sup_o @ B6 ) ) ) ).

% Sup_union_distrib
thf(fact_867_SUP__union,axiom,
    ! [M: produc340336539035504054m_rule > set_fm,A5: set_Pr1822751329126368876m_rule,B6: set_Pr1822751329126368876m_rule] :
      ( ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ M @ ( sup_su6946459741510085528m_rule @ A5 @ B6 ) ) )
      = ( sup_sup_set_fm @ ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ M @ A5 ) ) @ ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ M @ B6 ) ) ) ) ).

% SUP_union
thf(fact_868_image__Un,axiom,
    ! [F: produc340336539035504054m_rule > set_fm,A5: set_Pr1822751329126368876m_rule,B6: set_Pr1822751329126368876m_rule] :
      ( ( image_377115208147135190set_fm @ F @ ( sup_su6946459741510085528m_rule @ A5 @ B6 ) )
      = ( sup_sup_set_set_fm @ ( image_377115208147135190set_fm @ F @ A5 ) @ ( image_377115208147135190set_fm @ F @ B6 ) ) ) ).

% image_Un
thf(fact_869_Un__empty__left,axiom,
    ! [B6: set_o] :
      ( ( sup_sup_set_o @ bot_bot_set_o @ B6 )
      = B6 ) ).

% Un_empty_left
thf(fact_870_Un__empty__right,axiom,
    ! [A5: set_o] :
      ( ( sup_sup_set_o @ A5 @ bot_bot_set_o )
      = A5 ) ).

% Un_empty_right
thf(fact_871_Union__insert,axiom,
    ! [A: set_fm,B6: set_set_fm] :
      ( ( comple2134933779557159616set_fm @ ( insert_set_fm @ A @ B6 ) )
      = ( sup_sup_set_fm @ A @ ( comple2134933779557159616set_fm @ B6 ) ) ) ).

% Union_insert
thf(fact_872_insert__def,axiom,
    ( insert_o2
    = ( ^ [A3: $o] :
          ( sup_sup_set_o
          @ ( collect_o
            @ ^ [X: $o] : ( X = A3 ) ) ) ) ) ).

% insert_def
thf(fact_873_insert__def,axiom,
    ( insert8211810215607154385at_nat
    = ( ^ [A3: product_prod_nat_nat] :
          ( sup_su6327502436637775413at_nat
          @ ( collec3392354462482085612at_nat
            @ ^ [X: product_prod_nat_nat] : ( X = A3 ) ) ) ) ) ).

% insert_def
thf(fact_874_insert__def,axiom,
    ( insert_nat
    = ( ^ [A3: nat] :
          ( sup_sup_set_nat
          @ ( collect_nat
            @ ^ [X: nat] : ( X = A3 ) ) ) ) ) ).

% insert_def
thf(fact_875_cSUP__union,axiom,
    ! [A5: set_Pr1822751329126368876m_rule,F: produc340336539035504054m_rule > set_fm,B6: set_Pr1822751329126368876m_rule] :
      ( ( A5 != bot_bo3426410157013319936m_rule )
     => ( ( condit7632555770478454199set_fm @ ( image_377115208147135190set_fm @ F @ A5 ) )
       => ( ( B6 != bot_bo3426410157013319936m_rule )
         => ( ( condit7632555770478454199set_fm @ ( image_377115208147135190set_fm @ F @ B6 ) )
           => ( ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ F @ ( sup_su6946459741510085528m_rule @ A5 @ B6 ) ) )
              = ( sup_sup_set_fm @ ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ F @ A5 ) ) @ ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ F @ B6 ) ) ) ) ) ) ) ) ).

% cSUP_union
thf(fact_876_cSUP__union,axiom,
    ! [A5: set_o,F: $o > set_fm,B6: set_o] :
      ( ( A5 != bot_bot_set_o )
     => ( ( condit7632555770478454199set_fm @ ( image_o_set_fm @ F @ A5 ) )
       => ( ( B6 != bot_bot_set_o )
         => ( ( condit7632555770478454199set_fm @ ( image_o_set_fm @ F @ B6 ) )
           => ( ( comple2134933779557159616set_fm @ ( image_o_set_fm @ F @ ( sup_sup_set_o @ A5 @ B6 ) ) )
              = ( sup_sup_set_fm @ ( comple2134933779557159616set_fm @ ( image_o_set_fm @ F @ A5 ) ) @ ( comple2134933779557159616set_fm @ ( image_o_set_fm @ F @ B6 ) ) ) ) ) ) ) ) ).

% cSUP_union
thf(fact_877_cSUP__union,axiom,
    ! [A5: set_o,F: $o > $o,B6: set_o] :
      ( ( A5 != bot_bot_set_o )
     => ( ( condit5488710616941104124bove_o @ ( image_o_o2 @ F @ A5 ) )
       => ( ( B6 != bot_bot_set_o )
         => ( ( condit5488710616941104124bove_o @ ( image_o_o2 @ F @ B6 ) )
           => ( ( complete_Sup_Sup_o @ ( image_o_o2 @ F @ ( sup_sup_set_o @ A5 @ B6 ) ) )
              = ( sup_sup_o @ ( complete_Sup_Sup_o @ ( image_o_o2 @ F @ A5 ) ) @ ( complete_Sup_Sup_o @ ( image_o_o2 @ F @ B6 ) ) ) ) ) ) ) ) ).

% cSUP_union
thf(fact_878_insert__is__Un,axiom,
    ( insert_o2
    = ( ^ [A3: $o] : ( sup_sup_set_o @ ( insert_o2 @ A3 @ bot_bot_set_o ) ) ) ) ).

% insert_is_Un
thf(fact_879_Un__singleton__iff,axiom,
    ! [A5: set_o,B6: set_o,X5: $o] :
      ( ( ( sup_sup_set_o @ A5 @ B6 )
        = ( insert_o2 @ X5 @ bot_bot_set_o ) )
      = ( ( ( A5 = bot_bot_set_o )
          & ( B6
            = ( insert_o2 @ X5 @ bot_bot_set_o ) ) )
        | ( ( A5
            = ( insert_o2 @ X5 @ bot_bot_set_o ) )
          & ( B6 = bot_bot_set_o ) )
        | ( ( A5
            = ( insert_o2 @ X5 @ bot_bot_set_o ) )
          & ( B6
            = ( insert_o2 @ X5 @ bot_bot_set_o ) ) ) ) ) ).

% Un_singleton_iff
thf(fact_880_singleton__Un__iff,axiom,
    ! [X5: $o,A5: set_o,B6: set_o] :
      ( ( ( insert_o2 @ X5 @ bot_bot_set_o )
        = ( sup_sup_set_o @ A5 @ B6 ) )
      = ( ( ( A5 = bot_bot_set_o )
          & ( B6
            = ( insert_o2 @ X5 @ bot_bot_set_o ) ) )
        | ( ( A5
            = ( insert_o2 @ X5 @ bot_bot_set_o ) )
          & ( B6 = bot_bot_set_o ) )
        | ( ( A5
            = ( insert_o2 @ X5 @ bot_bot_set_o ) )
          & ( B6
            = ( insert_o2 @ X5 @ bot_bot_set_o ) ) ) ) ) ).

% singleton_Un_iff
thf(fact_881_SUP__absorb,axiom,
    ! [K: fm,I2: set_fm,A5: fm > set_fm] :
      ( ( member_fm2 @ K @ I2 )
     => ( ( sup_sup_set_fm @ ( A5 @ K ) @ ( comple2134933779557159616set_fm @ ( image_fm_set_fm @ A5 @ I2 ) ) )
        = ( comple2134933779557159616set_fm @ ( image_fm_set_fm @ A5 @ I2 ) ) ) ) ).

% SUP_absorb
thf(fact_882_SUP__absorb,axiom,
    ! [K: list_fm,I2: set_list_fm,A5: list_fm > set_fm] :
      ( ( member_list_fm2 @ K @ I2 )
     => ( ( sup_sup_set_fm @ ( A5 @ K ) @ ( comple2134933779557159616set_fm @ ( image_list_fm_set_fm @ A5 @ I2 ) ) )
        = ( comple2134933779557159616set_fm @ ( image_list_fm_set_fm @ A5 @ I2 ) ) ) ) ).

% SUP_absorb
thf(fact_883_SUP__absorb,axiom,
    ! [K: produc340336539035504054m_rule,I2: set_Pr1822751329126368876m_rule,A5: produc340336539035504054m_rule > set_fm] :
      ( ( member7231649785386036813m_rule @ K @ I2 )
     => ( ( sup_sup_set_fm @ ( A5 @ K ) @ ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ A5 @ I2 ) ) )
        = ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ A5 @ I2 ) ) ) ) ).

% SUP_absorb
thf(fact_884_SUP__absorb,axiom,
    ! [K: $o,I2: set_o,A5: $o > set_fm] :
      ( ( member_o2 @ K @ I2 )
     => ( ( sup_sup_set_fm @ ( A5 @ K ) @ ( comple2134933779557159616set_fm @ ( image_o_set_fm @ A5 @ I2 ) ) )
        = ( comple2134933779557159616set_fm @ ( image_o_set_fm @ A5 @ I2 ) ) ) ) ).

% SUP_absorb
thf(fact_885_SUP__absorb,axiom,
    ! [K: fm,I2: set_fm,A5: fm > $o] :
      ( ( member_fm2 @ K @ I2 )
     => ( ( sup_sup_o @ ( A5 @ K ) @ ( complete_Sup_Sup_o @ ( image_fm_o2 @ A5 @ I2 ) ) )
        = ( complete_Sup_Sup_o @ ( image_fm_o2 @ A5 @ I2 ) ) ) ) ).

% SUP_absorb
thf(fact_886_SUP__absorb,axiom,
    ! [K: list_fm,I2: set_list_fm,A5: list_fm > $o] :
      ( ( member_list_fm2 @ K @ I2 )
     => ( ( sup_sup_o @ ( A5 @ K ) @ ( complete_Sup_Sup_o @ ( image_list_fm_o2 @ A5 @ I2 ) ) )
        = ( complete_Sup_Sup_o @ ( image_list_fm_o2 @ A5 @ I2 ) ) ) ) ).

% SUP_absorb
thf(fact_887_SUP__absorb,axiom,
    ! [K: produc340336539035504054m_rule,I2: set_Pr1822751329126368876m_rule,A5: produc340336539035504054m_rule > $o] :
      ( ( member7231649785386036813m_rule @ K @ I2 )
     => ( ( sup_sup_o @ ( A5 @ K ) @ ( complete_Sup_Sup_o @ ( image_8969478372357302939rule_o @ A5 @ I2 ) ) )
        = ( complete_Sup_Sup_o @ ( image_8969478372357302939rule_o @ A5 @ I2 ) ) ) ) ).

% SUP_absorb
thf(fact_888_SUP__absorb,axiom,
    ! [K: $o,I2: set_o,A5: $o > $o] :
      ( ( member_o2 @ K @ I2 )
     => ( ( sup_sup_o @ ( A5 @ K ) @ ( complete_Sup_Sup_o @ ( image_o_o2 @ A5 @ I2 ) ) )
        = ( complete_Sup_Sup_o @ ( image_o_o2 @ A5 @ I2 ) ) ) ) ).

% SUP_absorb
thf(fact_889_complete__lattice__class_OSUP__sup__distrib,axiom,
    ! [F: produc340336539035504054m_rule > set_fm,A5: set_Pr1822751329126368876m_rule,G: produc340336539035504054m_rule > set_fm] :
      ( ( sup_sup_set_fm @ ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ F @ A5 ) ) @ ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ G @ A5 ) ) )
      = ( comple2134933779557159616set_fm
        @ ( image_377115208147135190set_fm
          @ ^ [A3: produc340336539035504054m_rule] : ( sup_sup_set_fm @ ( F @ A3 ) @ ( G @ A3 ) )
          @ A5 ) ) ) ).

% complete_lattice_class.SUP_sup_distrib
thf(fact_890_Un__Union__image,axiom,
    ! [A5: produc340336539035504054m_rule > set_fm,B6: produc340336539035504054m_rule > set_fm,C2: set_Pr1822751329126368876m_rule] :
      ( ( comple2134933779557159616set_fm
        @ ( image_377115208147135190set_fm
          @ ^ [X: produc340336539035504054m_rule] : ( sup_sup_set_fm @ ( A5 @ X ) @ ( B6 @ X ) )
          @ C2 ) )
      = ( sup_sup_set_fm @ ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ A5 @ C2 ) ) @ ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ B6 @ C2 ) ) ) ) ).

% Un_Union_image
thf(fact_891_UN__Un__distrib,axiom,
    ! [A5: produc340336539035504054m_rule > set_fm,B6: produc340336539035504054m_rule > set_fm,I2: set_Pr1822751329126368876m_rule] :
      ( ( comple2134933779557159616set_fm
        @ ( image_377115208147135190set_fm
          @ ^ [I: produc340336539035504054m_rule] : ( sup_sup_set_fm @ ( A5 @ I ) @ ( B6 @ I ) )
          @ I2 ) )
      = ( sup_sup_set_fm @ ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ A5 @ I2 ) ) @ ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ B6 @ I2 ) ) ) ) ).

% UN_Un_distrib
thf(fact_892_UN__absorb,axiom,
    ! [K: fm,I2: set_fm,A5: fm > set_fm] :
      ( ( member_fm2 @ K @ I2 )
     => ( ( sup_sup_set_fm @ ( A5 @ K ) @ ( comple2134933779557159616set_fm @ ( image_fm_set_fm @ A5 @ I2 ) ) )
        = ( comple2134933779557159616set_fm @ ( image_fm_set_fm @ A5 @ I2 ) ) ) ) ).

% UN_absorb
thf(fact_893_UN__absorb,axiom,
    ! [K: list_fm,I2: set_list_fm,A5: list_fm > set_fm] :
      ( ( member_list_fm2 @ K @ I2 )
     => ( ( sup_sup_set_fm @ ( A5 @ K ) @ ( comple2134933779557159616set_fm @ ( image_list_fm_set_fm @ A5 @ I2 ) ) )
        = ( comple2134933779557159616set_fm @ ( image_list_fm_set_fm @ A5 @ I2 ) ) ) ) ).

% UN_absorb
thf(fact_894_UN__absorb,axiom,
    ! [K: produc340336539035504054m_rule,I2: set_Pr1822751329126368876m_rule,A5: produc340336539035504054m_rule > set_fm] :
      ( ( member7231649785386036813m_rule @ K @ I2 )
     => ( ( sup_sup_set_fm @ ( A5 @ K ) @ ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ A5 @ I2 ) ) )
        = ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ A5 @ I2 ) ) ) ) ).

% UN_absorb
thf(fact_895_UN__absorb,axiom,
    ! [K: $o,I2: set_o,A5: $o > set_fm] :
      ( ( member_o2 @ K @ I2 )
     => ( ( sup_sup_set_fm @ ( A5 @ K ) @ ( comple2134933779557159616set_fm @ ( image_o_set_fm @ A5 @ I2 ) ) )
        = ( comple2134933779557159616set_fm @ ( image_o_set_fm @ A5 @ I2 ) ) ) ) ).

% UN_absorb
thf(fact_896_SUP__insert,axiom,
    ! [F: produc340336539035504054m_rule > set_fm,A: produc340336539035504054m_rule,A5: set_Pr1822751329126368876m_rule] :
      ( ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ F @ ( insert775845832097999846m_rule @ A @ A5 ) ) )
      = ( sup_sup_set_fm @ ( F @ A ) @ ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ F @ A5 ) ) ) ) ).

% SUP_insert
thf(fact_897_SUP__insert,axiom,
    ! [F: $o > set_fm,A: $o,A5: set_o] :
      ( ( comple2134933779557159616set_fm @ ( image_o_set_fm @ F @ ( insert_o2 @ A @ A5 ) ) )
      = ( sup_sup_set_fm @ ( F @ A ) @ ( comple2134933779557159616set_fm @ ( image_o_set_fm @ F @ A5 ) ) ) ) ).

% SUP_insert
thf(fact_898_SUP__insert,axiom,
    ! [F: $o > $o,A: $o,A5: set_o] :
      ( ( complete_Sup_Sup_o @ ( image_o_o2 @ F @ ( insert_o2 @ A @ A5 ) ) )
      = ( sup_sup_o @ ( F @ A ) @ ( complete_Sup_Sup_o @ ( image_o_o2 @ F @ A5 ) ) ) ) ).

% SUP_insert
thf(fact_899_UN__extend__simps_I2_J,axiom,
    ! [C2: set_Pr1822751329126368876m_rule,A5: produc340336539035504054m_rule > set_fm,B6: set_fm] :
      ( ( ( C2 = bot_bo3426410157013319936m_rule )
       => ( ( sup_sup_set_fm @ ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ A5 @ C2 ) ) @ B6 )
          = B6 ) )
      & ( ( C2 != bot_bo3426410157013319936m_rule )
       => ( ( sup_sup_set_fm @ ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ A5 @ C2 ) ) @ B6 )
          = ( comple2134933779557159616set_fm
            @ ( image_377115208147135190set_fm
              @ ^ [X: produc340336539035504054m_rule] : ( sup_sup_set_fm @ ( A5 @ X ) @ B6 )
              @ C2 ) ) ) ) ) ).

% UN_extend_simps(2)
thf(fact_900_UN__extend__simps_I2_J,axiom,
    ! [C2: set_o,A5: $o > set_fm,B6: set_fm] :
      ( ( ( C2 = bot_bot_set_o )
       => ( ( sup_sup_set_fm @ ( comple2134933779557159616set_fm @ ( image_o_set_fm @ A5 @ C2 ) ) @ B6 )
          = B6 ) )
      & ( ( C2 != bot_bot_set_o )
       => ( ( sup_sup_set_fm @ ( comple2134933779557159616set_fm @ ( image_o_set_fm @ A5 @ C2 ) ) @ B6 )
          = ( comple2134933779557159616set_fm
            @ ( image_o_set_fm
              @ ^ [X: $o] : ( sup_sup_set_fm @ ( A5 @ X ) @ B6 )
              @ C2 ) ) ) ) ) ).

% UN_extend_simps(2)
thf(fact_901_UN__extend__simps_I3_J,axiom,
    ! [C2: set_Pr1822751329126368876m_rule,A5: set_fm,B6: produc340336539035504054m_rule > set_fm] :
      ( ( ( C2 = bot_bo3426410157013319936m_rule )
       => ( ( sup_sup_set_fm @ A5 @ ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ B6 @ C2 ) ) )
          = A5 ) )
      & ( ( C2 != bot_bo3426410157013319936m_rule )
       => ( ( sup_sup_set_fm @ A5 @ ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ B6 @ C2 ) ) )
          = ( comple2134933779557159616set_fm
            @ ( image_377115208147135190set_fm
              @ ^ [X: produc340336539035504054m_rule] : ( sup_sup_set_fm @ A5 @ ( B6 @ X ) )
              @ C2 ) ) ) ) ) ).

% UN_extend_simps(3)
thf(fact_902_UN__extend__simps_I3_J,axiom,
    ! [C2: set_o,A5: set_fm,B6: $o > set_fm] :
      ( ( ( C2 = bot_bot_set_o )
       => ( ( sup_sup_set_fm @ A5 @ ( comple2134933779557159616set_fm @ ( image_o_set_fm @ B6 @ C2 ) ) )
          = A5 ) )
      & ( ( C2 != bot_bot_set_o )
       => ( ( sup_sup_set_fm @ A5 @ ( comple2134933779557159616set_fm @ ( image_o_set_fm @ B6 @ C2 ) ) )
          = ( comple2134933779557159616set_fm
            @ ( image_o_set_fm
              @ ^ [X: $o] : ( sup_sup_set_fm @ A5 @ ( B6 @ X ) )
              @ C2 ) ) ) ) ) ).

% UN_extend_simps(3)
thf(fact_903_conditionally__complete__lattice__class_OSUP__sup__distrib,axiom,
    ! [A5: set_Pr1822751329126368876m_rule,F: produc340336539035504054m_rule > set_fm,G: produc340336539035504054m_rule > set_fm] :
      ( ( A5 != bot_bo3426410157013319936m_rule )
     => ( ( condit7632555770478454199set_fm @ ( image_377115208147135190set_fm @ F @ A5 ) )
       => ( ( condit7632555770478454199set_fm @ ( image_377115208147135190set_fm @ G @ A5 ) )
         => ( ( sup_sup_set_fm @ ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ F @ A5 ) ) @ ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ G @ A5 ) ) )
            = ( comple2134933779557159616set_fm
              @ ( image_377115208147135190set_fm
                @ ^ [A3: produc340336539035504054m_rule] : ( sup_sup_set_fm @ ( F @ A3 ) @ ( G @ A3 ) )
                @ A5 ) ) ) ) ) ) ).

% conditionally_complete_lattice_class.SUP_sup_distrib
thf(fact_904_conditionally__complete__lattice__class_OSUP__sup__distrib,axiom,
    ! [A5: set_o,F: $o > set_fm,G: $o > set_fm] :
      ( ( A5 != bot_bot_set_o )
     => ( ( condit7632555770478454199set_fm @ ( image_o_set_fm @ F @ A5 ) )
       => ( ( condit7632555770478454199set_fm @ ( image_o_set_fm @ G @ A5 ) )
         => ( ( sup_sup_set_fm @ ( comple2134933779557159616set_fm @ ( image_o_set_fm @ F @ A5 ) ) @ ( comple2134933779557159616set_fm @ ( image_o_set_fm @ G @ A5 ) ) )
            = ( comple2134933779557159616set_fm
              @ ( image_o_set_fm
                @ ^ [A3: $o] : ( sup_sup_set_fm @ ( F @ A3 ) @ ( G @ A3 ) )
                @ A5 ) ) ) ) ) ) ).

% conditionally_complete_lattice_class.SUP_sup_distrib
thf(fact_905_conditionally__complete__lattice__class_OSUP__sup__distrib,axiom,
    ! [A5: set_o,F: $o > $o,G: $o > $o] :
      ( ( A5 != bot_bot_set_o )
     => ( ( condit5488710616941104124bove_o @ ( image_o_o2 @ F @ A5 ) )
       => ( ( condit5488710616941104124bove_o @ ( image_o_o2 @ G @ A5 ) )
         => ( ( sup_sup_o @ ( complete_Sup_Sup_o @ ( image_o_o2 @ F @ A5 ) ) @ ( complete_Sup_Sup_o @ ( image_o_o2 @ G @ A5 ) ) )
            = ( complete_Sup_Sup_o
              @ ( image_o_o2
                @ ^ [A3: $o] : ( sup_sup_o @ ( F @ A3 ) @ ( G @ A3 ) )
                @ A5 ) ) ) ) ) ) ).

% conditionally_complete_lattice_class.SUP_sup_distrib
thf(fact_906_cSUP__insert,axiom,
    ! [A5: set_Pr1822751329126368876m_rule,F: produc340336539035504054m_rule > set_fm,A: produc340336539035504054m_rule] :
      ( ( A5 != bot_bo3426410157013319936m_rule )
     => ( ( condit7632555770478454199set_fm @ ( image_377115208147135190set_fm @ F @ A5 ) )
       => ( ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ F @ ( insert775845832097999846m_rule @ A @ A5 ) ) )
          = ( sup_sup_set_fm @ ( F @ A ) @ ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ F @ A5 ) ) ) ) ) ) ).

% cSUP_insert
thf(fact_907_cSUP__insert,axiom,
    ! [A5: set_o,F: $o > set_fm,A: $o] :
      ( ( A5 != bot_bot_set_o )
     => ( ( condit7632555770478454199set_fm @ ( image_o_set_fm @ F @ A5 ) )
       => ( ( comple2134933779557159616set_fm @ ( image_o_set_fm @ F @ ( insert_o2 @ A @ A5 ) ) )
          = ( sup_sup_set_fm @ ( F @ A ) @ ( comple2134933779557159616set_fm @ ( image_o_set_fm @ F @ A5 ) ) ) ) ) ) ).

% cSUP_insert
thf(fact_908_cSUP__insert,axiom,
    ! [A5: set_o,F: $o > $o,A: $o] :
      ( ( A5 != bot_bot_set_o )
     => ( ( condit5488710616941104124bove_o @ ( image_o_o2 @ F @ A5 ) )
       => ( ( complete_Sup_Sup_o @ ( image_o_o2 @ F @ ( insert_o2 @ A @ A5 ) ) )
          = ( sup_sup_o @ ( F @ A ) @ ( complete_Sup_Sup_o @ ( image_o_o2 @ F @ A5 ) ) ) ) ) ) ).

% cSUP_insert
thf(fact_909_Union__image__insert,axiom,
    ! [F: produc340336539035504054m_rule > set_fm,A: produc340336539035504054m_rule,B6: set_Pr1822751329126368876m_rule] :
      ( ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ F @ ( insert775845832097999846m_rule @ A @ B6 ) ) )
      = ( sup_sup_set_fm @ ( F @ A ) @ ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ F @ B6 ) ) ) ) ).

% Union_image_insert
thf(fact_910_Union__image__insert,axiom,
    ! [F: $o > set_fm,A: $o,B6: set_o] :
      ( ( comple2134933779557159616set_fm @ ( image_o_set_fm @ F @ ( insert_o2 @ A @ B6 ) ) )
      = ( sup_sup_set_fm @ ( F @ A ) @ ( comple2134933779557159616set_fm @ ( image_o_set_fm @ F @ B6 ) ) ) ) ).

% Union_image_insert
thf(fact_911_Union__image__empty,axiom,
    ! [A5: set_fm,F: produc340336539035504054m_rule > set_fm] :
      ( ( sup_sup_set_fm @ A5 @ ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ F @ bot_bo3426410157013319936m_rule ) ) )
      = A5 ) ).

% Union_image_empty
thf(fact_912_Union__image__empty,axiom,
    ! [A5: set_fm,F: $o > set_fm] :
      ( ( sup_sup_set_fm @ A5 @ ( comple2134933779557159616set_fm @ ( image_o_set_fm @ F @ bot_bot_set_o ) ) )
      = A5 ) ).

% Union_image_empty
thf(fact_913_set__union,axiom,
    ! [Xs: list_fm,Ys: list_fm] :
      ( ( set_fm2 @ ( union_fm @ Xs @ Ys ) )
      = ( sup_sup_set_fm @ ( set_fm2 @ Xs ) @ ( set_fm2 @ Ys ) ) ) ).

% set_union
thf(fact_914_set__union,axiom,
    ! [Xs: list_list_fm,Ys: list_list_fm] :
      ( ( set_list_fm2 @ ( union_list_fm @ Xs @ Ys ) )
      = ( sup_sup_set_list_fm @ ( set_list_fm2 @ Xs ) @ ( set_list_fm2 @ Ys ) ) ) ).

% set_union
thf(fact_915_set__union,axiom,
    ! [Xs: list_tm,Ys: list_tm] :
      ( ( set_tm2 @ ( union_tm @ Xs @ Ys ) )
      = ( sup_sup_set_tm @ ( set_tm2 @ Xs ) @ ( set_tm2 @ Ys ) ) ) ).

% set_union
thf(fact_916_Field__insert,axiom,
    ! [A: nat,B: nat,R4: set_Pr1261947904930325089at_nat] :
      ( ( field_nat @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R4 ) )
      = ( sup_sup_set_nat @ ( insert_nat @ A @ ( insert_nat @ B @ bot_bot_set_nat ) ) @ ( field_nat @ R4 ) ) ) ).

% Field_insert
thf(fact_917_Field__insert,axiom,
    ! [A: $o,B: $o,R4: set_Product_prod_o_o] :
      ( ( field_o @ ( insert6201435330877294327od_o_o @ ( product_Pair_o_o @ A @ B ) @ R4 ) )
      = ( sup_sup_set_o @ ( insert_o2 @ A @ ( insert_o2 @ B @ bot_bot_set_o ) ) @ ( field_o @ R4 ) ) ) ).

% Field_insert
thf(fact_918_List_Oset__insert,axiom,
    ! [X5: $o,Xs: list_o] :
      ( ( set_o2 @ ( insert_o @ X5 @ Xs ) )
      = ( insert_o2 @ X5 @ ( set_o2 @ Xs ) ) ) ).

% List.set_insert
thf(fact_919_List_Oset__insert,axiom,
    ! [X5: fm,Xs: list_fm] :
      ( ( set_fm2 @ ( insert_fm @ X5 @ Xs ) )
      = ( insert_fm2 @ X5 @ ( set_fm2 @ Xs ) ) ) ).

% List.set_insert
thf(fact_920_List_Oset__insert,axiom,
    ! [X5: list_fm,Xs: list_list_fm] :
      ( ( set_list_fm2 @ ( insert_list_fm @ X5 @ Xs ) )
      = ( insert_list_fm2 @ X5 @ ( set_list_fm2 @ Xs ) ) ) ).

% List.set_insert
thf(fact_921_List_Oset__insert,axiom,
    ! [X5: tm,Xs: list_tm] :
      ( ( set_tm2 @ ( insert_tm @ X5 @ Xs ) )
      = ( insert_tm2 @ X5 @ ( set_tm2 @ Xs ) ) ) ).

% List.set_insert
thf(fact_922_ccSUP__insert,axiom,
    ! [A5: set_Pr1822751329126368876m_rule,F: produc340336539035504054m_rule > set_fm,A: produc340336539035504054m_rule] :
      ( ( counta1649821642195540155m_rule @ A5 )
     => ( ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ F @ ( insert775845832097999846m_rule @ A @ A5 ) ) )
        = ( sup_sup_set_fm @ ( F @ A ) @ ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ F @ A5 ) ) ) ) ) ).

% ccSUP_insert
thf(fact_923_ccSUP__insert,axiom,
    ! [A5: set_o,F: $o > set_fm,A: $o] :
      ( ( counta5976203206615340371able_o @ A5 )
     => ( ( comple2134933779557159616set_fm @ ( image_o_set_fm @ F @ ( insert_o2 @ A @ A5 ) ) )
        = ( sup_sup_set_fm @ ( F @ A ) @ ( comple2134933779557159616set_fm @ ( image_o_set_fm @ F @ A5 ) ) ) ) ) ).

% ccSUP_insert
thf(fact_924_ccSUP__insert,axiom,
    ! [A5: set_o,F: $o > $o,A: $o] :
      ( ( counta5976203206615340371able_o @ A5 )
     => ( ( complete_Sup_Sup_o @ ( image_o_o2 @ F @ ( insert_o2 @ A @ A5 ) ) )
        = ( sup_sup_o @ ( F @ A ) @ ( complete_Sup_Sup_o @ ( image_o_o2 @ F @ A5 ) ) ) ) ) ).

% ccSUP_insert
thf(fact_925_in__set__insert,axiom,
    ! [X5: produc340336539035504054m_rule,Xs: list_P2774625669004309958m_rule] :
      ( ( member7231649785386036813m_rule @ X5 @ ( set_Pr4534715572506550497m_rule @ Xs ) )
     => ( ( insert8371175628248280474m_rule @ X5 @ Xs )
        = Xs ) ) ).

% in_set_insert
thf(fact_926_in__set__insert,axiom,
    ! [X5: $o,Xs: list_o] :
      ( ( member_o2 @ X5 @ ( set_o2 @ Xs ) )
     => ( ( insert_o @ X5 @ Xs )
        = Xs ) ) ).

% in_set_insert
thf(fact_927_in__set__insert,axiom,
    ! [X5: fm,Xs: list_fm] :
      ( ( member_fm2 @ X5 @ ( set_fm2 @ Xs ) )
     => ( ( insert_fm @ X5 @ Xs )
        = Xs ) ) ).

% in_set_insert
thf(fact_928_in__set__insert,axiom,
    ! [X5: list_fm,Xs: list_list_fm] :
      ( ( member_list_fm2 @ X5 @ ( set_list_fm2 @ Xs ) )
     => ( ( insert_list_fm @ X5 @ Xs )
        = Xs ) ) ).

% in_set_insert
thf(fact_929_in__set__insert,axiom,
    ! [X5: tm,Xs: list_tm] :
      ( ( member_tm2 @ X5 @ ( set_tm2 @ Xs ) )
     => ( ( insert_tm @ X5 @ Xs )
        = Xs ) ) ).

% in_set_insert
thf(fact_930_Field__square,axiom,
    ! [X5: set_nat] :
      ( ( field_nat
        @ ( produc457027306803732586at_nat @ X5
          @ ^ [Uu: nat] : X5 ) )
      = X5 ) ).

% Field_square
thf(fact_931_Field__empty,axiom,
    ( ( field_nat @ bot_bo2099793752762293965at_nat )
    = bot_bot_set_nat ) ).

% Field_empty
thf(fact_932_Field__empty,axiom,
    ( ( field_o @ bot_bo7073875226086086771od_o_o )
    = bot_bot_set_o ) ).

% Field_empty
thf(fact_933_Field__Un,axiom,
    ! [R4: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
      ( ( field_nat @ ( sup_su6327502436637775413at_nat @ R4 @ S2 ) )
      = ( sup_sup_set_nat @ ( field_nat @ R4 ) @ ( field_nat @ S2 ) ) ) ).

% Field_Un
thf(fact_934_Field__Union,axiom,
    ! [R2: set_se7855581050983116737at_nat] :
      ( ( field_nat @ ( comple5685304695842803022at_nat @ R2 ) )
      = ( comple7399068483239264473et_nat @ ( image_5853224409075402146et_nat @ field_nat @ R2 ) ) ) ).

% Field_Union
thf(fact_935_Field__Union,axiom,
    ! [R2: set_se7864838648521419175_fm_fm] :
      ( ( field_fm @ ( comple3052221160759688756_fm_fm @ R2 ) )
      = ( comple2134933779557159616set_fm @ ( image_1125575614967635793set_fm @ field_fm @ R2 ) ) ) ).

% Field_Union
thf(fact_936_FieldI2,axiom,
    ! [I4: fm,J2: fm,R2: set_Pr4706815898642364871_fm_fm] :
      ( ( member7780952600467998736_fm_fm @ ( product_Pair_fm_fm @ I4 @ J2 ) @ R2 )
     => ( member_fm2 @ J2 @ ( field_fm @ R2 ) ) ) ).

% FieldI2
thf(fact_937_FieldI2,axiom,
    ! [I4: list_fm,J2: list_fm,R2: set_Pr1916137733696349511ist_fm] :
      ( ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ I4 @ J2 ) @ R2 )
     => ( member_list_fm2 @ J2 @ ( field_list_fm @ R2 ) ) ) ).

% FieldI2
thf(fact_938_FieldI2,axiom,
    ! [I4: produc340336539035504054m_rule,J2: produc340336539035504054m_rule,R2: set_Pr4971326047967503661m_rule] :
      ( ( member1411645983481675894m_rule @ ( produc6261311607089640965m_rule @ I4 @ J2 ) @ R2 )
     => ( member7231649785386036813m_rule @ J2 @ ( field_6869660707945823529m_rule @ R2 ) ) ) ).

% FieldI2
thf(fact_939_FieldI2,axiom,
    ! [I4: $o,J2: $o,R2: set_Product_prod_o_o] :
      ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ I4 @ J2 ) @ R2 )
     => ( member_o2 @ J2 @ ( field_o @ R2 ) ) ) ).

% FieldI2
thf(fact_940_FieldI2,axiom,
    ! [I4: nat,J2: nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I4 @ J2 ) @ R2 )
     => ( member_nat @ J2 @ ( field_nat @ R2 ) ) ) ).

% FieldI2
thf(fact_941_FieldI1,axiom,
    ! [I4: fm,J2: fm,R2: set_Pr4706815898642364871_fm_fm] :
      ( ( member7780952600467998736_fm_fm @ ( product_Pair_fm_fm @ I4 @ J2 ) @ R2 )
     => ( member_fm2 @ I4 @ ( field_fm @ R2 ) ) ) ).

% FieldI1
thf(fact_942_FieldI1,axiom,
    ! [I4: list_fm,J2: list_fm,R2: set_Pr1916137733696349511ist_fm] :
      ( ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ I4 @ J2 ) @ R2 )
     => ( member_list_fm2 @ I4 @ ( field_list_fm @ R2 ) ) ) ).

% FieldI1
thf(fact_943_FieldI1,axiom,
    ! [I4: produc340336539035504054m_rule,J2: produc340336539035504054m_rule,R2: set_Pr4971326047967503661m_rule] :
      ( ( member1411645983481675894m_rule @ ( produc6261311607089640965m_rule @ I4 @ J2 ) @ R2 )
     => ( member7231649785386036813m_rule @ I4 @ ( field_6869660707945823529m_rule @ R2 ) ) ) ).

% FieldI1
thf(fact_944_FieldI1,axiom,
    ! [I4: $o,J2: $o,R2: set_Product_prod_o_o] :
      ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ I4 @ J2 ) @ R2 )
     => ( member_o2 @ I4 @ ( field_o @ R2 ) ) ) ).

% FieldI1
thf(fact_945_FieldI1,axiom,
    ! [I4: nat,J2: nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I4 @ J2 ) @ R2 )
     => ( member_nat @ I4 @ ( field_nat @ R2 ) ) ) ).

% FieldI1
thf(fact_946_sup__set__def,axiom,
    ( sup_sup_set_fm
    = ( ^ [A7: set_fm,B7: set_fm] :
          ( collect_fm
          @ ( sup_sup_fm_o
            @ ^ [X: fm] : ( member_fm2 @ X @ A7 )
            @ ^ [X: fm] : ( member_fm2 @ X @ B7 ) ) ) ) ) ).

% sup_set_def
thf(fact_947_sup__set__def,axiom,
    ( sup_sup_set_list_fm
    = ( ^ [A7: set_list_fm,B7: set_list_fm] :
          ( collect_list_fm
          @ ( sup_sup_list_fm_o
            @ ^ [X: list_fm] : ( member_list_fm2 @ X @ A7 )
            @ ^ [X: list_fm] : ( member_list_fm2 @ X @ B7 ) ) ) ) ) ).

% sup_set_def
thf(fact_948_sup__set__def,axiom,
    ( sup_su6946459741510085528m_rule
    = ( ^ [A7: set_Pr1822751329126368876m_rule,B7: set_Pr1822751329126368876m_rule] :
          ( collec1009276759030335883m_rule
          @ ( sup_su8497378972038406893rule_o
            @ ^ [X: produc340336539035504054m_rule] : ( member7231649785386036813m_rule @ X @ A7 )
            @ ^ [X: produc340336539035504054m_rule] : ( member7231649785386036813m_rule @ X @ B7 ) ) ) ) ) ).

% sup_set_def
thf(fact_949_sup__set__def,axiom,
    ( sup_sup_set_o
    = ( ^ [A7: set_o,B7: set_o] :
          ( collect_o
          @ ( sup_sup_o_o
            @ ^ [X: $o] : ( member_o2 @ X @ A7 )
            @ ^ [X: $o] : ( member_o2 @ X @ B7 ) ) ) ) ) ).

% sup_set_def
thf(fact_950_sup__set__def,axiom,
    ( sup_su6327502436637775413at_nat
    = ( ^ [A7: set_Pr1261947904930325089at_nat,B7: set_Pr1261947904930325089at_nat] :
          ( collec3392354462482085612at_nat
          @ ( sup_su798857527126471912_nat_o
            @ ^ [X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ A7 )
            @ ^ [X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ B7 ) ) ) ) ) ).

% sup_set_def
thf(fact_951_sup__set__def,axiom,
    ( sup_sup_set_nat
    = ( ^ [A7: set_nat,B7: set_nat] :
          ( collect_nat
          @ ( sup_sup_nat_o
            @ ^ [X: nat] : ( member_nat @ X @ A7 )
            @ ^ [X: nat] : ( member_nat @ X @ B7 ) ) ) ) ) ).

% sup_set_def
thf(fact_952_sup__Un__eq,axiom,
    ! [R2: set_fm,S: set_fm] :
      ( ( sup_sup_fm_o
        @ ^ [X: fm] : ( member_fm2 @ X @ R2 )
        @ ^ [X: fm] : ( member_fm2 @ X @ S ) )
      = ( ^ [X: fm] : ( member_fm2 @ X @ ( sup_sup_set_fm @ R2 @ S ) ) ) ) ).

% sup_Un_eq
thf(fact_953_sup__Un__eq,axiom,
    ! [R2: set_list_fm,S: set_list_fm] :
      ( ( sup_sup_list_fm_o
        @ ^ [X: list_fm] : ( member_list_fm2 @ X @ R2 )
        @ ^ [X: list_fm] : ( member_list_fm2 @ X @ S ) )
      = ( ^ [X: list_fm] : ( member_list_fm2 @ X @ ( sup_sup_set_list_fm @ R2 @ S ) ) ) ) ).

% sup_Un_eq
thf(fact_954_sup__Un__eq,axiom,
    ! [R2: set_Pr1822751329126368876m_rule,S: set_Pr1822751329126368876m_rule] :
      ( ( sup_su8497378972038406893rule_o
        @ ^ [X: produc340336539035504054m_rule] : ( member7231649785386036813m_rule @ X @ R2 )
        @ ^ [X: produc340336539035504054m_rule] : ( member7231649785386036813m_rule @ X @ S ) )
      = ( ^ [X: produc340336539035504054m_rule] : ( member7231649785386036813m_rule @ X @ ( sup_su6946459741510085528m_rule @ R2 @ S ) ) ) ) ).

% sup_Un_eq
thf(fact_955_sup__Un__eq,axiom,
    ! [R2: set_o,S: set_o] :
      ( ( sup_sup_o_o
        @ ^ [X: $o] : ( member_o2 @ X @ R2 )
        @ ^ [X: $o] : ( member_o2 @ X @ S ) )
      = ( ^ [X: $o] : ( member_o2 @ X @ ( sup_sup_set_o @ R2 @ S ) ) ) ) ).

% sup_Un_eq
thf(fact_956_ccSUP__commute,axiom,
    ! [A5: set_Pr1822751329126368876m_rule,B6: set_Pr1822751329126368876m_rule,F: produc340336539035504054m_rule > produc340336539035504054m_rule > set_fm] :
      ( ( counta1649821642195540155m_rule @ A5 )
     => ( ( counta1649821642195540155m_rule @ B6 )
       => ( ( comple2134933779557159616set_fm
            @ ( image_377115208147135190set_fm
              @ ^ [I: produc340336539035504054m_rule] : ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ ( F @ I ) @ B6 ) )
              @ A5 ) )
          = ( comple2134933779557159616set_fm
            @ ( image_377115208147135190set_fm
              @ ^ [J: produc340336539035504054m_rule] :
                  ( comple2134933779557159616set_fm
                  @ ( image_377115208147135190set_fm
                    @ ^ [I: produc340336539035504054m_rule] : ( F @ I @ J )
                    @ A5 ) )
              @ B6 ) ) ) ) ) ).

% ccSUP_commute
thf(fact_957_sup__Un__eq2,axiom,
    ! [R2: set_Pr1822751329126368876m_rule,S: set_Pr1822751329126368876m_rule] :
      ( ( sup_su1495437568605221017rule_o
        @ ^ [X: produc6018962875968178549ist_fm,Y: rule] : ( member7231649785386036813m_rule @ ( produc1733806532565653680m_rule @ X @ Y ) @ R2 )
        @ ^ [X: produc6018962875968178549ist_fm,Y: rule] : ( member7231649785386036813m_rule @ ( produc1733806532565653680m_rule @ X @ Y ) @ S ) )
      = ( ^ [X: produc6018962875968178549ist_fm,Y: rule] : ( member7231649785386036813m_rule @ ( produc1733806532565653680m_rule @ X @ Y ) @ ( sup_su6946459741510085528m_rule @ R2 @ S ) ) ) ) ).

% sup_Un_eq2
thf(fact_958_sup__Un__eq2,axiom,
    ! [R2: set_Pr8969532344284305932ule_fm,S: set_Pr8969532344284305932ule_fm] :
      ( ( sup_sup_rule_fm_o
        @ ^ [X: rule,Y: fm] : ( member9168179392308330989ule_fm @ ( product_Pair_rule_fm @ X @ Y ) @ R2 )
        @ ^ [X: rule,Y: fm] : ( member9168179392308330989ule_fm @ ( product_Pair_rule_fm @ X @ Y ) @ S ) )
      = ( ^ [X: rule,Y: fm] : ( member9168179392308330989ule_fm @ ( product_Pair_rule_fm @ X @ Y ) @ ( sup_su1866406141038604088ule_fm @ R2 @ S ) ) ) ) ).

% sup_Un_eq2
thf(fact_959_sup__Un__eq2,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( sup_sup_nat_nat_o
        @ ^ [X: nat,Y: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R2 )
        @ ^ [X: nat,Y: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ S ) )
      = ( ^ [X: nat,Y: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( sup_su6327502436637775413at_nat @ R2 @ S ) ) ) ) ).

% sup_Un_eq2
thf(fact_960_ccSUP__sup__distrib,axiom,
    ! [A5: set_Pr1822751329126368876m_rule,F: produc340336539035504054m_rule > set_fm,G: produc340336539035504054m_rule > set_fm] :
      ( ( counta1649821642195540155m_rule @ A5 )
     => ( ( sup_sup_set_fm @ ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ F @ A5 ) ) @ ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ G @ A5 ) ) )
        = ( comple2134933779557159616set_fm
          @ ( image_377115208147135190set_fm
            @ ^ [A3: produc340336539035504054m_rule] : ( sup_sup_set_fm @ ( F @ A3 ) @ ( G @ A3 ) )
            @ A5 ) ) ) ) ).

% ccSUP_sup_distrib
thf(fact_961_ccSUP__union,axiom,
    ! [A5: set_Pr1822751329126368876m_rule,B6: set_Pr1822751329126368876m_rule,M: produc340336539035504054m_rule > set_fm] :
      ( ( counta1649821642195540155m_rule @ A5 )
     => ( ( counta1649821642195540155m_rule @ B6 )
       => ( ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ M @ ( sup_su6946459741510085528m_rule @ A5 @ B6 ) ) )
          = ( sup_sup_set_fm @ ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ M @ A5 ) ) @ ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ M @ B6 ) ) ) ) ) ) ).

% ccSUP_union
thf(fact_962_countable__UN,axiom,
    ! [I2: set_fm,A5: fm > set_fm] :
      ( ( counta2008398395262058670ble_fm @ I2 )
     => ( ! [I3: fm] :
            ( ( member_fm2 @ I3 @ I2 )
           => ( counta2008398395262058670ble_fm @ ( A5 @ I3 ) ) )
       => ( counta2008398395262058670ble_fm @ ( comple2134933779557159616set_fm @ ( image_fm_set_fm @ A5 @ I2 ) ) ) ) ) ).

% countable_UN
thf(fact_963_countable__UN,axiom,
    ! [I2: set_list_fm,A5: list_fm > set_fm] :
      ( ( counta8303570107911152436ist_fm @ I2 )
     => ( ! [I3: list_fm] :
            ( ( member_list_fm2 @ I3 @ I2 )
           => ( counta2008398395262058670ble_fm @ ( A5 @ I3 ) ) )
       => ( counta2008398395262058670ble_fm @ ( comple2134933779557159616set_fm @ ( image_list_fm_set_fm @ A5 @ I2 ) ) ) ) ) ).

% countable_UN
thf(fact_964_countable__UN,axiom,
    ! [I2: set_Pr1822751329126368876m_rule,A5: produc340336539035504054m_rule > set_fm] :
      ( ( counta1649821642195540155m_rule @ I2 )
     => ( ! [I3: produc340336539035504054m_rule] :
            ( ( member7231649785386036813m_rule @ I3 @ I2 )
           => ( counta2008398395262058670ble_fm @ ( A5 @ I3 ) ) )
       => ( counta2008398395262058670ble_fm @ ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ A5 @ I2 ) ) ) ) ) ).

% countable_UN
thf(fact_965_countable__UN,axiom,
    ! [I2: set_o,A5: $o > set_fm] :
      ( ( counta5976203206615340371able_o @ I2 )
     => ( ! [I3: $o] :
            ( ( member_o2 @ I3 @ I2 )
           => ( counta2008398395262058670ble_fm @ ( A5 @ I3 ) ) )
       => ( counta2008398395262058670ble_fm @ ( comple2134933779557159616set_fm @ ( image_o_set_fm @ A5 @ I2 ) ) ) ) ) ).

% countable_UN
thf(fact_966_countable__image,axiom,
    ! [A5: set_Pr1822751329126368876m_rule,F: produc340336539035504054m_rule > set_fm] :
      ( ( counta1649821642195540155m_rule @ A5 )
     => ( counta6288402088370727310set_fm @ ( image_377115208147135190set_fm @ F @ A5 ) ) ) ).

% countable_image
thf(fact_967_aboveS__def,axiom,
    ( order_4940263126897443436at_nat
    = ( ^ [R: set_Pr8693737435421807431at_nat,A3: product_prod_nat_nat] :
          ( collec3392354462482085612at_nat
          @ ^ [B3: product_prod_nat_nat] :
              ( ( B3 != A3 )
              & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A3 @ B3 ) @ R ) ) ) ) ) ).

% aboveS_def
thf(fact_968_aboveS__def,axiom,
    ( order_aboveS_nat
    = ( ^ [R: set_Pr1261947904930325089at_nat,A3: nat] :
          ( collect_nat
          @ ^ [B3: nat] :
              ( ( B3 != A3 )
              & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A3 @ B3 ) @ R ) ) ) ) ) ).

% aboveS_def
thf(fact_969_countable__Collect,axiom,
    ! [A5: set_fm,Phi: fm > $o] :
      ( ( counta2008398395262058670ble_fm @ A5 )
     => ( counta2008398395262058670ble_fm
        @ ( collect_fm
          @ ^ [A3: fm] :
              ( ( member_fm2 @ A3 @ A5 )
              & ( Phi @ A3 ) ) ) ) ) ).

% countable_Collect
thf(fact_970_countable__Collect,axiom,
    ! [A5: set_list_fm,Phi: list_fm > $o] :
      ( ( counta8303570107911152436ist_fm @ A5 )
     => ( counta8303570107911152436ist_fm
        @ ( collect_list_fm
          @ ^ [A3: list_fm] :
              ( ( member_list_fm2 @ A3 @ A5 )
              & ( Phi @ A3 ) ) ) ) ) ).

% countable_Collect
thf(fact_971_countable__Collect,axiom,
    ! [A5: set_Pr1822751329126368876m_rule,Phi: produc340336539035504054m_rule > $o] :
      ( ( counta1649821642195540155m_rule @ A5 )
     => ( counta1649821642195540155m_rule
        @ ( collec1009276759030335883m_rule
          @ ^ [A3: produc340336539035504054m_rule] :
              ( ( member7231649785386036813m_rule @ A3 @ A5 )
              & ( Phi @ A3 ) ) ) ) ) ).

% countable_Collect
thf(fact_972_countable__Collect,axiom,
    ! [A5: set_o,Phi: $o > $o] :
      ( ( counta5976203206615340371able_o @ A5 )
     => ( counta5976203206615340371able_o
        @ ( collect_o
          @ ^ [A3: $o] :
              ( ( member_o2 @ A3 @ A5 )
              & ( Phi @ A3 ) ) ) ) ) ).

% countable_Collect
thf(fact_973_countable__Collect,axiom,
    ! [A5: set_Pr1261947904930325089at_nat,Phi: product_prod_nat_nat > $o] :
      ( ( counta3401040167773629372at_nat @ A5 )
     => ( counta3401040167773629372at_nat
        @ ( collec3392354462482085612at_nat
          @ ^ [A3: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ A3 @ A5 )
              & ( Phi @ A3 ) ) ) ) ) ).

% countable_Collect
thf(fact_974_countable__Collect,axiom,
    ! [A5: set_nat,Phi: nat > $o] :
      ( ( counta1168086296615599829le_nat @ A5 )
     => ( counta1168086296615599829le_nat
        @ ( collect_nat
          @ ^ [A3: nat] :
              ( ( member_nat @ A3 @ A5 )
              & ( Phi @ A3 ) ) ) ) ) ).

% countable_Collect
thf(fact_975_less__cSUP__iff,axiom,
    ! [A5: set_o,F: $o > nat,A: nat] :
      ( ( A5 != bot_bot_set_o )
     => ( ( condit2214826472909112428ve_nat @ ( image_o_nat2 @ F @ A5 ) )
       => ( ( ord_less_nat @ A @ ( complete_Sup_Sup_nat @ ( image_o_nat2 @ F @ A5 ) ) )
          = ( ? [X: $o] :
                ( ( member_o2 @ X @ A5 )
                & ( ord_less_nat @ A @ ( F @ X ) ) ) ) ) ) ) ).

% less_cSUP_iff
thf(fact_976_Sigma__Image,axiom,
    ! [A5: set_Pr1822751329126368876m_rule,B6: produc340336539035504054m_rule > set_fm,X7: set_Pr1822751329126368876m_rule] :
      ( ( image_6824901344838659752ule_fm @ ( produc4849014715371551731ule_fm @ A5 @ B6 ) @ X7 )
      = ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ B6 @ ( inf_in2520071923633375486m_rule @ X7 @ A5 ) ) ) ) ).

% Sigma_Image
thf(fact_977_Int__iff,axiom,
    ! [C: fm,A5: set_fm,B6: set_fm] :
      ( ( member_fm2 @ C @ ( inf_inf_set_fm @ A5 @ B6 ) )
      = ( ( member_fm2 @ C @ A5 )
        & ( member_fm2 @ C @ B6 ) ) ) ).

% Int_iff
thf(fact_978_Int__iff,axiom,
    ! [C: list_fm,A5: set_list_fm,B6: set_list_fm] :
      ( ( member_list_fm2 @ C @ ( inf_inf_set_list_fm @ A5 @ B6 ) )
      = ( ( member_list_fm2 @ C @ A5 )
        & ( member_list_fm2 @ C @ B6 ) ) ) ).

% Int_iff
thf(fact_979_Int__iff,axiom,
    ! [C: produc340336539035504054m_rule,A5: set_Pr1822751329126368876m_rule,B6: set_Pr1822751329126368876m_rule] :
      ( ( member7231649785386036813m_rule @ C @ ( inf_in2520071923633375486m_rule @ A5 @ B6 ) )
      = ( ( member7231649785386036813m_rule @ C @ A5 )
        & ( member7231649785386036813m_rule @ C @ B6 ) ) ) ).

% Int_iff
thf(fact_980_Int__iff,axiom,
    ! [C: $o,A5: set_o,B6: set_o] :
      ( ( member_o2 @ C @ ( inf_inf_set_o @ A5 @ B6 ) )
      = ( ( member_o2 @ C @ A5 )
        & ( member_o2 @ C @ B6 ) ) ) ).

% Int_iff
thf(fact_981_IntI,axiom,
    ! [C: fm,A5: set_fm,B6: set_fm] :
      ( ( member_fm2 @ C @ A5 )
     => ( ( member_fm2 @ C @ B6 )
       => ( member_fm2 @ C @ ( inf_inf_set_fm @ A5 @ B6 ) ) ) ) ).

% IntI
thf(fact_982_IntI,axiom,
    ! [C: list_fm,A5: set_list_fm,B6: set_list_fm] :
      ( ( member_list_fm2 @ C @ A5 )
     => ( ( member_list_fm2 @ C @ B6 )
       => ( member_list_fm2 @ C @ ( inf_inf_set_list_fm @ A5 @ B6 ) ) ) ) ).

% IntI
thf(fact_983_IntI,axiom,
    ! [C: produc340336539035504054m_rule,A5: set_Pr1822751329126368876m_rule,B6: set_Pr1822751329126368876m_rule] :
      ( ( member7231649785386036813m_rule @ C @ A5 )
     => ( ( member7231649785386036813m_rule @ C @ B6 )
       => ( member7231649785386036813m_rule @ C @ ( inf_in2520071923633375486m_rule @ A5 @ B6 ) ) ) ) ).

% IntI
thf(fact_984_IntI,axiom,
    ! [C: $o,A5: set_o,B6: set_o] :
      ( ( member_o2 @ C @ A5 )
     => ( ( member_o2 @ C @ B6 )
       => ( member_o2 @ C @ ( inf_inf_set_o @ A5 @ B6 ) ) ) ) ).

% IntI
thf(fact_985_sdrop__add,axiom,
    ! [N3: nat,M3: nat,S2: stream2709947120125613254m_rule] :
      ( ( sdrop_8169176516188972301m_rule @ N3 @ ( sdrop_8169176516188972301m_rule @ M3 @ S2 ) )
      = ( sdrop_8169176516188972301m_rule @ ( plus_plus_nat @ M3 @ N3 ) @ S2 ) ) ).

% sdrop_add
thf(fact_986_Int__insert__right__if1,axiom,
    ! [A: fm,A5: set_fm,B6: set_fm] :
      ( ( member_fm2 @ A @ A5 )
     => ( ( inf_inf_set_fm @ A5 @ ( insert_fm2 @ A @ B6 ) )
        = ( insert_fm2 @ A @ ( inf_inf_set_fm @ A5 @ B6 ) ) ) ) ).

% Int_insert_right_if1
thf(fact_987_Int__insert__right__if1,axiom,
    ! [A: list_fm,A5: set_list_fm,B6: set_list_fm] :
      ( ( member_list_fm2 @ A @ A5 )
     => ( ( inf_inf_set_list_fm @ A5 @ ( insert_list_fm2 @ A @ B6 ) )
        = ( insert_list_fm2 @ A @ ( inf_inf_set_list_fm @ A5 @ B6 ) ) ) ) ).

% Int_insert_right_if1
thf(fact_988_Int__insert__right__if1,axiom,
    ! [A: produc340336539035504054m_rule,A5: set_Pr1822751329126368876m_rule,B6: set_Pr1822751329126368876m_rule] :
      ( ( member7231649785386036813m_rule @ A @ A5 )
     => ( ( inf_in2520071923633375486m_rule @ A5 @ ( insert775845832097999846m_rule @ A @ B6 ) )
        = ( insert775845832097999846m_rule @ A @ ( inf_in2520071923633375486m_rule @ A5 @ B6 ) ) ) ) ).

% Int_insert_right_if1
thf(fact_989_Int__insert__right__if1,axiom,
    ! [A: $o,A5: set_o,B6: set_o] :
      ( ( member_o2 @ A @ A5 )
     => ( ( inf_inf_set_o @ A5 @ ( insert_o2 @ A @ B6 ) )
        = ( insert_o2 @ A @ ( inf_inf_set_o @ A5 @ B6 ) ) ) ) ).

% Int_insert_right_if1
thf(fact_990_Int__insert__right__if0,axiom,
    ! [A: fm,A5: set_fm,B6: set_fm] :
      ( ~ ( member_fm2 @ A @ A5 )
     => ( ( inf_inf_set_fm @ A5 @ ( insert_fm2 @ A @ B6 ) )
        = ( inf_inf_set_fm @ A5 @ B6 ) ) ) ).

% Int_insert_right_if0
thf(fact_991_Int__insert__right__if0,axiom,
    ! [A: list_fm,A5: set_list_fm,B6: set_list_fm] :
      ( ~ ( member_list_fm2 @ A @ A5 )
     => ( ( inf_inf_set_list_fm @ A5 @ ( insert_list_fm2 @ A @ B6 ) )
        = ( inf_inf_set_list_fm @ A5 @ B6 ) ) ) ).

% Int_insert_right_if0
thf(fact_992_Int__insert__right__if0,axiom,
    ! [A: produc340336539035504054m_rule,A5: set_Pr1822751329126368876m_rule,B6: set_Pr1822751329126368876m_rule] :
      ( ~ ( member7231649785386036813m_rule @ A @ A5 )
     => ( ( inf_in2520071923633375486m_rule @ A5 @ ( insert775845832097999846m_rule @ A @ B6 ) )
        = ( inf_in2520071923633375486m_rule @ A5 @ B6 ) ) ) ).

% Int_insert_right_if0
thf(fact_993_Int__insert__right__if0,axiom,
    ! [A: $o,A5: set_o,B6: set_o] :
      ( ~ ( member_o2 @ A @ A5 )
     => ( ( inf_inf_set_o @ A5 @ ( insert_o2 @ A @ B6 ) )
        = ( inf_inf_set_o @ A5 @ B6 ) ) ) ).

% Int_insert_right_if0
thf(fact_994_insert__inter__insert,axiom,
    ! [A: $o,A5: set_o,B6: set_o] :
      ( ( inf_inf_set_o @ ( insert_o2 @ A @ A5 ) @ ( insert_o2 @ A @ B6 ) )
      = ( insert_o2 @ A @ ( inf_inf_set_o @ A5 @ B6 ) ) ) ).

% insert_inter_insert
thf(fact_995_Int__insert__left__if1,axiom,
    ! [A: fm,C2: set_fm,B6: set_fm] :
      ( ( member_fm2 @ A @ C2 )
     => ( ( inf_inf_set_fm @ ( insert_fm2 @ A @ B6 ) @ C2 )
        = ( insert_fm2 @ A @ ( inf_inf_set_fm @ B6 @ C2 ) ) ) ) ).

% Int_insert_left_if1
thf(fact_996_Int__insert__left__if1,axiom,
    ! [A: list_fm,C2: set_list_fm,B6: set_list_fm] :
      ( ( member_list_fm2 @ A @ C2 )
     => ( ( inf_inf_set_list_fm @ ( insert_list_fm2 @ A @ B6 ) @ C2 )
        = ( insert_list_fm2 @ A @ ( inf_inf_set_list_fm @ B6 @ C2 ) ) ) ) ).

% Int_insert_left_if1
thf(fact_997_Int__insert__left__if1,axiom,
    ! [A: produc340336539035504054m_rule,C2: set_Pr1822751329126368876m_rule,B6: set_Pr1822751329126368876m_rule] :
      ( ( member7231649785386036813m_rule @ A @ C2 )
     => ( ( inf_in2520071923633375486m_rule @ ( insert775845832097999846m_rule @ A @ B6 ) @ C2 )
        = ( insert775845832097999846m_rule @ A @ ( inf_in2520071923633375486m_rule @ B6 @ C2 ) ) ) ) ).

% Int_insert_left_if1
thf(fact_998_Int__insert__left__if1,axiom,
    ! [A: $o,C2: set_o,B6: set_o] :
      ( ( member_o2 @ A @ C2 )
     => ( ( inf_inf_set_o @ ( insert_o2 @ A @ B6 ) @ C2 )
        = ( insert_o2 @ A @ ( inf_inf_set_o @ B6 @ C2 ) ) ) ) ).

% Int_insert_left_if1
thf(fact_999_Int__insert__left__if0,axiom,
    ! [A: fm,C2: set_fm,B6: set_fm] :
      ( ~ ( member_fm2 @ A @ C2 )
     => ( ( inf_inf_set_fm @ ( insert_fm2 @ A @ B6 ) @ C2 )
        = ( inf_inf_set_fm @ B6 @ C2 ) ) ) ).

% Int_insert_left_if0
thf(fact_1000_Int__insert__left__if0,axiom,
    ! [A: list_fm,C2: set_list_fm,B6: set_list_fm] :
      ( ~ ( member_list_fm2 @ A @ C2 )
     => ( ( inf_inf_set_list_fm @ ( insert_list_fm2 @ A @ B6 ) @ C2 )
        = ( inf_inf_set_list_fm @ B6 @ C2 ) ) ) ).

% Int_insert_left_if0
thf(fact_1001_Int__insert__left__if0,axiom,
    ! [A: produc340336539035504054m_rule,C2: set_Pr1822751329126368876m_rule,B6: set_Pr1822751329126368876m_rule] :
      ( ~ ( member7231649785386036813m_rule @ A @ C2 )
     => ( ( inf_in2520071923633375486m_rule @ ( insert775845832097999846m_rule @ A @ B6 ) @ C2 )
        = ( inf_in2520071923633375486m_rule @ B6 @ C2 ) ) ) ).

% Int_insert_left_if0
thf(fact_1002_Int__insert__left__if0,axiom,
    ! [A: $o,C2: set_o,B6: set_o] :
      ( ~ ( member_o2 @ A @ C2 )
     => ( ( inf_inf_set_o @ ( insert_o2 @ A @ B6 ) @ C2 )
        = ( inf_inf_set_o @ B6 @ C2 ) ) ) ).

% Int_insert_left_if0
thf(fact_1003_insert__disjoint_I1_J,axiom,
    ! [A: fm,A5: set_fm,B6: set_fm] :
      ( ( ( inf_inf_set_fm @ ( insert_fm2 @ A @ A5 ) @ B6 )
        = bot_bot_set_fm )
      = ( ~ ( member_fm2 @ A @ B6 )
        & ( ( inf_inf_set_fm @ A5 @ B6 )
          = bot_bot_set_fm ) ) ) ).

% insert_disjoint(1)
thf(fact_1004_insert__disjoint_I1_J,axiom,
    ! [A: list_fm,A5: set_list_fm,B6: set_list_fm] :
      ( ( ( inf_inf_set_list_fm @ ( insert_list_fm2 @ A @ A5 ) @ B6 )
        = bot_bot_set_list_fm )
      = ( ~ ( member_list_fm2 @ A @ B6 )
        & ( ( inf_inf_set_list_fm @ A5 @ B6 )
          = bot_bot_set_list_fm ) ) ) ).

% insert_disjoint(1)
thf(fact_1005_insert__disjoint_I1_J,axiom,
    ! [A: produc340336539035504054m_rule,A5: set_Pr1822751329126368876m_rule,B6: set_Pr1822751329126368876m_rule] :
      ( ( ( inf_in2520071923633375486m_rule @ ( insert775845832097999846m_rule @ A @ A5 ) @ B6 )
        = bot_bo3426410157013319936m_rule )
      = ( ~ ( member7231649785386036813m_rule @ A @ B6 )
        & ( ( inf_in2520071923633375486m_rule @ A5 @ B6 )
          = bot_bo3426410157013319936m_rule ) ) ) ).

% insert_disjoint(1)
thf(fact_1006_insert__disjoint_I1_J,axiom,
    ! [A: $o,A5: set_o,B6: set_o] :
      ( ( ( inf_inf_set_o @ ( insert_o2 @ A @ A5 ) @ B6 )
        = bot_bot_set_o )
      = ( ~ ( member_o2 @ A @ B6 )
        & ( ( inf_inf_set_o @ A5 @ B6 )
          = bot_bot_set_o ) ) ) ).

% insert_disjoint(1)
thf(fact_1007_insert__disjoint_I2_J,axiom,
    ! [A: fm,A5: set_fm,B6: set_fm] :
      ( ( bot_bot_set_fm
        = ( inf_inf_set_fm @ ( insert_fm2 @ A @ A5 ) @ B6 ) )
      = ( ~ ( member_fm2 @ A @ B6 )
        & ( bot_bot_set_fm
          = ( inf_inf_set_fm @ A5 @ B6 ) ) ) ) ).

% insert_disjoint(2)
thf(fact_1008_insert__disjoint_I2_J,axiom,
    ! [A: list_fm,A5: set_list_fm,B6: set_list_fm] :
      ( ( bot_bot_set_list_fm
        = ( inf_inf_set_list_fm @ ( insert_list_fm2 @ A @ A5 ) @ B6 ) )
      = ( ~ ( member_list_fm2 @ A @ B6 )
        & ( bot_bot_set_list_fm
          = ( inf_inf_set_list_fm @ A5 @ B6 ) ) ) ) ).

% insert_disjoint(2)
thf(fact_1009_insert__disjoint_I2_J,axiom,
    ! [A: produc340336539035504054m_rule,A5: set_Pr1822751329126368876m_rule,B6: set_Pr1822751329126368876m_rule] :
      ( ( bot_bo3426410157013319936m_rule
        = ( inf_in2520071923633375486m_rule @ ( insert775845832097999846m_rule @ A @ A5 ) @ B6 ) )
      = ( ~ ( member7231649785386036813m_rule @ A @ B6 )
        & ( bot_bo3426410157013319936m_rule
          = ( inf_in2520071923633375486m_rule @ A5 @ B6 ) ) ) ) ).

% insert_disjoint(2)
thf(fact_1010_insert__disjoint_I2_J,axiom,
    ! [A: $o,A5: set_o,B6: set_o] :
      ( ( bot_bot_set_o
        = ( inf_inf_set_o @ ( insert_o2 @ A @ A5 ) @ B6 ) )
      = ( ~ ( member_o2 @ A @ B6 )
        & ( bot_bot_set_o
          = ( inf_inf_set_o @ A5 @ B6 ) ) ) ) ).

% insert_disjoint(2)
thf(fact_1011_disjoint__insert_I1_J,axiom,
    ! [B6: set_fm,A: fm,A5: set_fm] :
      ( ( ( inf_inf_set_fm @ B6 @ ( insert_fm2 @ A @ A5 ) )
        = bot_bot_set_fm )
      = ( ~ ( member_fm2 @ A @ B6 )
        & ( ( inf_inf_set_fm @ B6 @ A5 )
          = bot_bot_set_fm ) ) ) ).

% disjoint_insert(1)
thf(fact_1012_disjoint__insert_I1_J,axiom,
    ! [B6: set_list_fm,A: list_fm,A5: set_list_fm] :
      ( ( ( inf_inf_set_list_fm @ B6 @ ( insert_list_fm2 @ A @ A5 ) )
        = bot_bot_set_list_fm )
      = ( ~ ( member_list_fm2 @ A @ B6 )
        & ( ( inf_inf_set_list_fm @ B6 @ A5 )
          = bot_bot_set_list_fm ) ) ) ).

% disjoint_insert(1)
thf(fact_1013_disjoint__insert_I1_J,axiom,
    ! [B6: set_Pr1822751329126368876m_rule,A: produc340336539035504054m_rule,A5: set_Pr1822751329126368876m_rule] :
      ( ( ( inf_in2520071923633375486m_rule @ B6 @ ( insert775845832097999846m_rule @ A @ A5 ) )
        = bot_bo3426410157013319936m_rule )
      = ( ~ ( member7231649785386036813m_rule @ A @ B6 )
        & ( ( inf_in2520071923633375486m_rule @ B6 @ A5 )
          = bot_bo3426410157013319936m_rule ) ) ) ).

% disjoint_insert(1)
thf(fact_1014_disjoint__insert_I1_J,axiom,
    ! [B6: set_o,A: $o,A5: set_o] :
      ( ( ( inf_inf_set_o @ B6 @ ( insert_o2 @ A @ A5 ) )
        = bot_bot_set_o )
      = ( ~ ( member_o2 @ A @ B6 )
        & ( ( inf_inf_set_o @ B6 @ A5 )
          = bot_bot_set_o ) ) ) ).

% disjoint_insert(1)
thf(fact_1015_disjoint__insert_I2_J,axiom,
    ! [A5: set_fm,B: fm,B6: set_fm] :
      ( ( bot_bot_set_fm
        = ( inf_inf_set_fm @ A5 @ ( insert_fm2 @ B @ B6 ) ) )
      = ( ~ ( member_fm2 @ B @ A5 )
        & ( bot_bot_set_fm
          = ( inf_inf_set_fm @ A5 @ B6 ) ) ) ) ).

% disjoint_insert(2)
thf(fact_1016_disjoint__insert_I2_J,axiom,
    ! [A5: set_list_fm,B: list_fm,B6: set_list_fm] :
      ( ( bot_bot_set_list_fm
        = ( inf_inf_set_list_fm @ A5 @ ( insert_list_fm2 @ B @ B6 ) ) )
      = ( ~ ( member_list_fm2 @ B @ A5 )
        & ( bot_bot_set_list_fm
          = ( inf_inf_set_list_fm @ A5 @ B6 ) ) ) ) ).

% disjoint_insert(2)
thf(fact_1017_disjoint__insert_I2_J,axiom,
    ! [A5: set_Pr1822751329126368876m_rule,B: produc340336539035504054m_rule,B6: set_Pr1822751329126368876m_rule] :
      ( ( bot_bo3426410157013319936m_rule
        = ( inf_in2520071923633375486m_rule @ A5 @ ( insert775845832097999846m_rule @ B @ B6 ) ) )
      = ( ~ ( member7231649785386036813m_rule @ B @ A5 )
        & ( bot_bo3426410157013319936m_rule
          = ( inf_in2520071923633375486m_rule @ A5 @ B6 ) ) ) ) ).

% disjoint_insert(2)
thf(fact_1018_disjoint__insert_I2_J,axiom,
    ! [A5: set_o,B: $o,B6: set_o] :
      ( ( bot_bot_set_o
        = ( inf_inf_set_o @ A5 @ ( insert_o2 @ B @ B6 ) ) )
      = ( ~ ( member_o2 @ B @ A5 )
        & ( bot_bot_set_o
          = ( inf_inf_set_o @ A5 @ B6 ) ) ) ) ).

% disjoint_insert(2)
thf(fact_1019_if__image__distrib,axiom,
    ! [P3: produc340336539035504054m_rule > $o,F: produc340336539035504054m_rule > set_fm,G: produc340336539035504054m_rule > set_fm,S: set_Pr1822751329126368876m_rule] :
      ( ( image_377115208147135190set_fm
        @ ^ [X: produc340336539035504054m_rule] : ( if_set_fm @ ( P3 @ X ) @ ( F @ X ) @ ( G @ X ) )
        @ S )
      = ( sup_sup_set_set_fm @ ( image_377115208147135190set_fm @ F @ ( inf_in2520071923633375486m_rule @ S @ ( collec1009276759030335883m_rule @ P3 ) ) )
        @ ( image_377115208147135190set_fm @ G
          @ ( inf_in2520071923633375486m_rule @ S
            @ ( collec1009276759030335883m_rule
              @ ^ [X: produc340336539035504054m_rule] :
                  ~ ( P3 @ X ) ) ) ) ) ) ).

% if_image_distrib
thf(fact_1020_Domain__insert,axiom,
    ! [A: rule,B: fm,R4: set_Pr8969532344284305932ule_fm] :
      ( ( domain_rule_fm @ ( insert684836149095756678ule_fm @ ( product_Pair_rule_fm @ A @ B ) @ R4 ) )
      = ( insert_rule @ A @ ( domain_rule_fm @ R4 ) ) ) ).

% Domain_insert
thf(fact_1021_Domain__insert,axiom,
    ! [A: nat,B: nat,R4: set_Pr1261947904930325089at_nat] :
      ( ( domain_nat_nat @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R4 ) )
      = ( insert_nat @ A @ ( domain_nat_nat @ R4 ) ) ) ).

% Domain_insert
thf(fact_1022_Range__insert,axiom,
    ! [A: rule,B: fm,R4: set_Pr8969532344284305932ule_fm] :
      ( ( range_rule_fm @ ( insert684836149095756678ule_fm @ ( product_Pair_rule_fm @ A @ B ) @ R4 ) )
      = ( insert_fm2 @ B @ ( range_rule_fm @ R4 ) ) ) ).

% Range_insert
thf(fact_1023_Range__insert,axiom,
    ! [A: nat,B: nat,R4: set_Pr1261947904930325089at_nat] :
      ( ( range_nat_nat @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R4 ) )
      = ( insert_nat @ B @ ( range_nat_nat @ R4 ) ) ) ).

% Range_insert
thf(fact_1024_Field__def,axiom,
    ( field_nat
    = ( ^ [R: set_Pr1261947904930325089at_nat] : ( sup_sup_set_nat @ ( domain_nat_nat @ R ) @ ( range_nat_nat @ R ) ) ) ) ).

% Field_def
thf(fact_1025_not__psubset__empty,axiom,
    ! [A5: set_o] :
      ~ ( ord_less_set_o @ A5 @ bot_bot_set_o ) ).

% not_psubset_empty
thf(fact_1026_Collect__conj__eq,axiom,
    ! [P3: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
      ( ( collec3392354462482085612at_nat
        @ ^ [X: product_prod_nat_nat] :
            ( ( P3 @ X )
            & ( Q @ X ) ) )
      = ( inf_in2572325071724192079at_nat @ ( collec3392354462482085612at_nat @ P3 ) @ ( collec3392354462482085612at_nat @ Q ) ) ) ).

% Collect_conj_eq
thf(fact_1027_Collect__conj__eq,axiom,
    ! [P3: nat > $o,Q: nat > $o] :
      ( ( collect_nat
        @ ^ [X: nat] :
            ( ( P3 @ X )
            & ( Q @ X ) ) )
      = ( inf_inf_set_nat @ ( collect_nat @ P3 ) @ ( collect_nat @ Q ) ) ) ).

% Collect_conj_eq
thf(fact_1028_Int__Collect,axiom,
    ! [X5: fm,A5: set_fm,P3: fm > $o] :
      ( ( member_fm2 @ X5 @ ( inf_inf_set_fm @ A5 @ ( collect_fm @ P3 ) ) )
      = ( ( member_fm2 @ X5 @ A5 )
        & ( P3 @ X5 ) ) ) ).

% Int_Collect
thf(fact_1029_Int__Collect,axiom,
    ! [X5: list_fm,A5: set_list_fm,P3: list_fm > $o] :
      ( ( member_list_fm2 @ X5 @ ( inf_inf_set_list_fm @ A5 @ ( collect_list_fm @ P3 ) ) )
      = ( ( member_list_fm2 @ X5 @ A5 )
        & ( P3 @ X5 ) ) ) ).

% Int_Collect
thf(fact_1030_Int__Collect,axiom,
    ! [X5: produc340336539035504054m_rule,A5: set_Pr1822751329126368876m_rule,P3: produc340336539035504054m_rule > $o] :
      ( ( member7231649785386036813m_rule @ X5 @ ( inf_in2520071923633375486m_rule @ A5 @ ( collec1009276759030335883m_rule @ P3 ) ) )
      = ( ( member7231649785386036813m_rule @ X5 @ A5 )
        & ( P3 @ X5 ) ) ) ).

% Int_Collect
thf(fact_1031_Int__Collect,axiom,
    ! [X5: $o,A5: set_o,P3: $o > $o] :
      ( ( member_o2 @ X5 @ ( inf_inf_set_o @ A5 @ ( collect_o @ P3 ) ) )
      = ( ( member_o2 @ X5 @ A5 )
        & ( P3 @ X5 ) ) ) ).

% Int_Collect
thf(fact_1032_Int__Collect,axiom,
    ! [X5: product_prod_nat_nat,A5: set_Pr1261947904930325089at_nat,P3: product_prod_nat_nat > $o] :
      ( ( member8440522571783428010at_nat @ X5 @ ( inf_in2572325071724192079at_nat @ A5 @ ( collec3392354462482085612at_nat @ P3 ) ) )
      = ( ( member8440522571783428010at_nat @ X5 @ A5 )
        & ( P3 @ X5 ) ) ) ).

% Int_Collect
thf(fact_1033_Int__Collect,axiom,
    ! [X5: nat,A5: set_nat,P3: nat > $o] :
      ( ( member_nat @ X5 @ ( inf_inf_set_nat @ A5 @ ( collect_nat @ P3 ) ) )
      = ( ( member_nat @ X5 @ A5 )
        & ( P3 @ X5 ) ) ) ).

% Int_Collect
thf(fact_1034_Int__def,axiom,
    ( inf_inf_set_fm
    = ( ^ [A7: set_fm,B7: set_fm] :
          ( collect_fm
          @ ^ [X: fm] :
              ( ( member_fm2 @ X @ A7 )
              & ( member_fm2 @ X @ B7 ) ) ) ) ) ).

% Int_def
thf(fact_1035_Int__def,axiom,
    ( inf_inf_set_list_fm
    = ( ^ [A7: set_list_fm,B7: set_list_fm] :
          ( collect_list_fm
          @ ^ [X: list_fm] :
              ( ( member_list_fm2 @ X @ A7 )
              & ( member_list_fm2 @ X @ B7 ) ) ) ) ) ).

% Int_def
thf(fact_1036_Int__def,axiom,
    ( inf_in2520071923633375486m_rule
    = ( ^ [A7: set_Pr1822751329126368876m_rule,B7: set_Pr1822751329126368876m_rule] :
          ( collec1009276759030335883m_rule
          @ ^ [X: produc340336539035504054m_rule] :
              ( ( member7231649785386036813m_rule @ X @ A7 )
              & ( member7231649785386036813m_rule @ X @ B7 ) ) ) ) ) ).

% Int_def
thf(fact_1037_Int__def,axiom,
    ( inf_inf_set_o
    = ( ^ [A7: set_o,B7: set_o] :
          ( collect_o
          @ ^ [X: $o] :
              ( ( member_o2 @ X @ A7 )
              & ( member_o2 @ X @ B7 ) ) ) ) ) ).

% Int_def
thf(fact_1038_Int__def,axiom,
    ( inf_in2572325071724192079at_nat
    = ( ^ [A7: set_Pr1261947904930325089at_nat,B7: set_Pr1261947904930325089at_nat] :
          ( collec3392354462482085612at_nat
          @ ^ [X: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X @ A7 )
              & ( member8440522571783428010at_nat @ X @ B7 ) ) ) ) ) ).

% Int_def
thf(fact_1039_Int__def,axiom,
    ( inf_inf_set_nat
    = ( ^ [A7: set_nat,B7: set_nat] :
          ( collect_nat
          @ ^ [X: nat] :
              ( ( member_nat @ X @ A7 )
              & ( member_nat @ X @ B7 ) ) ) ) ) ).

% Int_def
thf(fact_1040_IntD2,axiom,
    ! [C: fm,A5: set_fm,B6: set_fm] :
      ( ( member_fm2 @ C @ ( inf_inf_set_fm @ A5 @ B6 ) )
     => ( member_fm2 @ C @ B6 ) ) ).

% IntD2
thf(fact_1041_IntD2,axiom,
    ! [C: list_fm,A5: set_list_fm,B6: set_list_fm] :
      ( ( member_list_fm2 @ C @ ( inf_inf_set_list_fm @ A5 @ B6 ) )
     => ( member_list_fm2 @ C @ B6 ) ) ).

% IntD2
thf(fact_1042_IntD2,axiom,
    ! [C: produc340336539035504054m_rule,A5: set_Pr1822751329126368876m_rule,B6: set_Pr1822751329126368876m_rule] :
      ( ( member7231649785386036813m_rule @ C @ ( inf_in2520071923633375486m_rule @ A5 @ B6 ) )
     => ( member7231649785386036813m_rule @ C @ B6 ) ) ).

% IntD2
thf(fact_1043_IntD2,axiom,
    ! [C: $o,A5: set_o,B6: set_o] :
      ( ( member_o2 @ C @ ( inf_inf_set_o @ A5 @ B6 ) )
     => ( member_o2 @ C @ B6 ) ) ).

% IntD2
thf(fact_1044_IntD1,axiom,
    ! [C: fm,A5: set_fm,B6: set_fm] :
      ( ( member_fm2 @ C @ ( inf_inf_set_fm @ A5 @ B6 ) )
     => ( member_fm2 @ C @ A5 ) ) ).

% IntD1
thf(fact_1045_IntD1,axiom,
    ! [C: list_fm,A5: set_list_fm,B6: set_list_fm] :
      ( ( member_list_fm2 @ C @ ( inf_inf_set_list_fm @ A5 @ B6 ) )
     => ( member_list_fm2 @ C @ A5 ) ) ).

% IntD1
thf(fact_1046_IntD1,axiom,
    ! [C: produc340336539035504054m_rule,A5: set_Pr1822751329126368876m_rule,B6: set_Pr1822751329126368876m_rule] :
      ( ( member7231649785386036813m_rule @ C @ ( inf_in2520071923633375486m_rule @ A5 @ B6 ) )
     => ( member7231649785386036813m_rule @ C @ A5 ) ) ).

% IntD1
thf(fact_1047_IntD1,axiom,
    ! [C: $o,A5: set_o,B6: set_o] :
      ( ( member_o2 @ C @ ( inf_inf_set_o @ A5 @ B6 ) )
     => ( member_o2 @ C @ A5 ) ) ).

% IntD1
thf(fact_1048_IntE,axiom,
    ! [C: fm,A5: set_fm,B6: set_fm] :
      ( ( member_fm2 @ C @ ( inf_inf_set_fm @ A5 @ B6 ) )
     => ~ ( ( member_fm2 @ C @ A5 )
         => ~ ( member_fm2 @ C @ B6 ) ) ) ).

% IntE
thf(fact_1049_IntE,axiom,
    ! [C: list_fm,A5: set_list_fm,B6: set_list_fm] :
      ( ( member_list_fm2 @ C @ ( inf_inf_set_list_fm @ A5 @ B6 ) )
     => ~ ( ( member_list_fm2 @ C @ A5 )
         => ~ ( member_list_fm2 @ C @ B6 ) ) ) ).

% IntE
thf(fact_1050_IntE,axiom,
    ! [C: produc340336539035504054m_rule,A5: set_Pr1822751329126368876m_rule,B6: set_Pr1822751329126368876m_rule] :
      ( ( member7231649785386036813m_rule @ C @ ( inf_in2520071923633375486m_rule @ A5 @ B6 ) )
     => ~ ( ( member7231649785386036813m_rule @ C @ A5 )
         => ~ ( member7231649785386036813m_rule @ C @ B6 ) ) ) ).

% IntE
thf(fact_1051_IntE,axiom,
    ! [C: $o,A5: set_o,B6: set_o] :
      ( ( member_o2 @ C @ ( inf_inf_set_o @ A5 @ B6 ) )
     => ~ ( ( member_o2 @ C @ A5 )
         => ~ ( member_o2 @ C @ B6 ) ) ) ).

% IntE
thf(fact_1052_Int__insert__left,axiom,
    ! [A: fm,C2: set_fm,B6: set_fm] :
      ( ( ( member_fm2 @ A @ C2 )
       => ( ( inf_inf_set_fm @ ( insert_fm2 @ A @ B6 ) @ C2 )
          = ( insert_fm2 @ A @ ( inf_inf_set_fm @ B6 @ C2 ) ) ) )
      & ( ~ ( member_fm2 @ A @ C2 )
       => ( ( inf_inf_set_fm @ ( insert_fm2 @ A @ B6 ) @ C2 )
          = ( inf_inf_set_fm @ B6 @ C2 ) ) ) ) ).

% Int_insert_left
thf(fact_1053_Int__insert__left,axiom,
    ! [A: list_fm,C2: set_list_fm,B6: set_list_fm] :
      ( ( ( member_list_fm2 @ A @ C2 )
       => ( ( inf_inf_set_list_fm @ ( insert_list_fm2 @ A @ B6 ) @ C2 )
          = ( insert_list_fm2 @ A @ ( inf_inf_set_list_fm @ B6 @ C2 ) ) ) )
      & ( ~ ( member_list_fm2 @ A @ C2 )
       => ( ( inf_inf_set_list_fm @ ( insert_list_fm2 @ A @ B6 ) @ C2 )
          = ( inf_inf_set_list_fm @ B6 @ C2 ) ) ) ) ).

% Int_insert_left
thf(fact_1054_Int__insert__left,axiom,
    ! [A: produc340336539035504054m_rule,C2: set_Pr1822751329126368876m_rule,B6: set_Pr1822751329126368876m_rule] :
      ( ( ( member7231649785386036813m_rule @ A @ C2 )
       => ( ( inf_in2520071923633375486m_rule @ ( insert775845832097999846m_rule @ A @ B6 ) @ C2 )
          = ( insert775845832097999846m_rule @ A @ ( inf_in2520071923633375486m_rule @ B6 @ C2 ) ) ) )
      & ( ~ ( member7231649785386036813m_rule @ A @ C2 )
       => ( ( inf_in2520071923633375486m_rule @ ( insert775845832097999846m_rule @ A @ B6 ) @ C2 )
          = ( inf_in2520071923633375486m_rule @ B6 @ C2 ) ) ) ) ).

% Int_insert_left
thf(fact_1055_Int__insert__left,axiom,
    ! [A: $o,C2: set_o,B6: set_o] :
      ( ( ( member_o2 @ A @ C2 )
       => ( ( inf_inf_set_o @ ( insert_o2 @ A @ B6 ) @ C2 )
          = ( insert_o2 @ A @ ( inf_inf_set_o @ B6 @ C2 ) ) ) )
      & ( ~ ( member_o2 @ A @ C2 )
       => ( ( inf_inf_set_o @ ( insert_o2 @ A @ B6 ) @ C2 )
          = ( inf_inf_set_o @ B6 @ C2 ) ) ) ) ).

% Int_insert_left
thf(fact_1056_Int__insert__right,axiom,
    ! [A: fm,A5: set_fm,B6: set_fm] :
      ( ( ( member_fm2 @ A @ A5 )
       => ( ( inf_inf_set_fm @ A5 @ ( insert_fm2 @ A @ B6 ) )
          = ( insert_fm2 @ A @ ( inf_inf_set_fm @ A5 @ B6 ) ) ) )
      & ( ~ ( member_fm2 @ A @ A5 )
       => ( ( inf_inf_set_fm @ A5 @ ( insert_fm2 @ A @ B6 ) )
          = ( inf_inf_set_fm @ A5 @ B6 ) ) ) ) ).

% Int_insert_right
thf(fact_1057_Int__insert__right,axiom,
    ! [A: list_fm,A5: set_list_fm,B6: set_list_fm] :
      ( ( ( member_list_fm2 @ A @ A5 )
       => ( ( inf_inf_set_list_fm @ A5 @ ( insert_list_fm2 @ A @ B6 ) )
          = ( insert_list_fm2 @ A @ ( inf_inf_set_list_fm @ A5 @ B6 ) ) ) )
      & ( ~ ( member_list_fm2 @ A @ A5 )
       => ( ( inf_inf_set_list_fm @ A5 @ ( insert_list_fm2 @ A @ B6 ) )
          = ( inf_inf_set_list_fm @ A5 @ B6 ) ) ) ) ).

% Int_insert_right
thf(fact_1058_Int__insert__right,axiom,
    ! [A: produc340336539035504054m_rule,A5: set_Pr1822751329126368876m_rule,B6: set_Pr1822751329126368876m_rule] :
      ( ( ( member7231649785386036813m_rule @ A @ A5 )
       => ( ( inf_in2520071923633375486m_rule @ A5 @ ( insert775845832097999846m_rule @ A @ B6 ) )
          = ( insert775845832097999846m_rule @ A @ ( inf_in2520071923633375486m_rule @ A5 @ B6 ) ) ) )
      & ( ~ ( member7231649785386036813m_rule @ A @ A5 )
       => ( ( inf_in2520071923633375486m_rule @ A5 @ ( insert775845832097999846m_rule @ A @ B6 ) )
          = ( inf_in2520071923633375486m_rule @ A5 @ B6 ) ) ) ) ).

% Int_insert_right
thf(fact_1059_Int__insert__right,axiom,
    ! [A: $o,A5: set_o,B6: set_o] :
      ( ( ( member_o2 @ A @ A5 )
       => ( ( inf_inf_set_o @ A5 @ ( insert_o2 @ A @ B6 ) )
          = ( insert_o2 @ A @ ( inf_inf_set_o @ A5 @ B6 ) ) ) )
      & ( ~ ( member_o2 @ A @ A5 )
       => ( ( inf_inf_set_o @ A5 @ ( insert_o2 @ A @ B6 ) )
          = ( inf_inf_set_o @ A5 @ B6 ) ) ) ) ).

% Int_insert_right
thf(fact_1060_Int__emptyI,axiom,
    ! [A5: set_fm,B6: set_fm] :
      ( ! [X3: fm] :
          ( ( member_fm2 @ X3 @ A5 )
         => ~ ( member_fm2 @ X3 @ B6 ) )
     => ( ( inf_inf_set_fm @ A5 @ B6 )
        = bot_bot_set_fm ) ) ).

% Int_emptyI
thf(fact_1061_Int__emptyI,axiom,
    ! [A5: set_list_fm,B6: set_list_fm] :
      ( ! [X3: list_fm] :
          ( ( member_list_fm2 @ X3 @ A5 )
         => ~ ( member_list_fm2 @ X3 @ B6 ) )
     => ( ( inf_inf_set_list_fm @ A5 @ B6 )
        = bot_bot_set_list_fm ) ) ).

% Int_emptyI
thf(fact_1062_Int__emptyI,axiom,
    ! [A5: set_Pr1822751329126368876m_rule,B6: set_Pr1822751329126368876m_rule] :
      ( ! [X3: produc340336539035504054m_rule] :
          ( ( member7231649785386036813m_rule @ X3 @ A5 )
         => ~ ( member7231649785386036813m_rule @ X3 @ B6 ) )
     => ( ( inf_in2520071923633375486m_rule @ A5 @ B6 )
        = bot_bo3426410157013319936m_rule ) ) ).

% Int_emptyI
thf(fact_1063_Int__emptyI,axiom,
    ! [A5: set_o,B6: set_o] :
      ( ! [X3: $o] :
          ( ( member_o2 @ X3 @ A5 )
         => ~ ( member_o2 @ X3 @ B6 ) )
     => ( ( inf_inf_set_o @ A5 @ B6 )
        = bot_bot_set_o ) ) ).

% Int_emptyI
thf(fact_1064_disjoint__iff,axiom,
    ! [A5: set_fm,B6: set_fm] :
      ( ( ( inf_inf_set_fm @ A5 @ B6 )
        = bot_bot_set_fm )
      = ( ! [X: fm] :
            ( ( member_fm2 @ X @ A5 )
           => ~ ( member_fm2 @ X @ B6 ) ) ) ) ).

% disjoint_iff
thf(fact_1065_disjoint__iff,axiom,
    ! [A5: set_list_fm,B6: set_list_fm] :
      ( ( ( inf_inf_set_list_fm @ A5 @ B6 )
        = bot_bot_set_list_fm )
      = ( ! [X: list_fm] :
            ( ( member_list_fm2 @ X @ A5 )
           => ~ ( member_list_fm2 @ X @ B6 ) ) ) ) ).

% disjoint_iff
thf(fact_1066_disjoint__iff,axiom,
    ! [A5: set_Pr1822751329126368876m_rule,B6: set_Pr1822751329126368876m_rule] :
      ( ( ( inf_in2520071923633375486m_rule @ A5 @ B6 )
        = bot_bo3426410157013319936m_rule )
      = ( ! [X: produc340336539035504054m_rule] :
            ( ( member7231649785386036813m_rule @ X @ A5 )
           => ~ ( member7231649785386036813m_rule @ X @ B6 ) ) ) ) ).

% disjoint_iff
thf(fact_1067_disjoint__iff,axiom,
    ! [A5: set_o,B6: set_o] :
      ( ( ( inf_inf_set_o @ A5 @ B6 )
        = bot_bot_set_o )
      = ( ! [X: $o] :
            ( ( member_o2 @ X @ A5 )
           => ~ ( member_o2 @ X @ B6 ) ) ) ) ).

% disjoint_iff
thf(fact_1068_Int__empty__left,axiom,
    ! [B6: set_o] :
      ( ( inf_inf_set_o @ bot_bot_set_o @ B6 )
      = bot_bot_set_o ) ).

% Int_empty_left
thf(fact_1069_Int__empty__right,axiom,
    ! [A5: set_o] :
      ( ( inf_inf_set_o @ A5 @ bot_bot_set_o )
      = bot_bot_set_o ) ).

% Int_empty_right
thf(fact_1070_disjoint__iff__not__equal,axiom,
    ! [A5: set_o,B6: set_o] :
      ( ( ( inf_inf_set_o @ A5 @ B6 )
        = bot_bot_set_o )
      = ( ! [X: $o] :
            ( ( member_o2 @ X @ A5 )
           => ! [Y: $o] :
                ( ( member_o2 @ Y @ B6 )
               => ( X = (~ Y) ) ) ) ) ) ).

% disjoint_iff_not_equal
thf(fact_1071_Domain__iff,axiom,
    ! [A: produc6018962875968178549ist_fm,R4: set_Pr1822751329126368876m_rule] :
      ( ( member4699826688122452638ist_fm @ A @ ( domain3391028601666924027m_rule @ R4 ) )
      = ( ? [Y: rule] : ( member7231649785386036813m_rule @ ( produc1733806532565653680m_rule @ A @ Y ) @ R4 ) ) ) ).

% Domain_iff
thf(fact_1072_Domain__iff,axiom,
    ! [A: rule,R4: set_Pr8969532344284305932ule_fm] :
      ( ( member_rule @ A @ ( domain_rule_fm @ R4 ) )
      = ( ? [Y: fm] : ( member9168179392308330989ule_fm @ ( product_Pair_rule_fm @ A @ Y ) @ R4 ) ) ) ).

% Domain_iff
thf(fact_1073_Domain__iff,axiom,
    ! [A: nat,R4: set_Pr1261947904930325089at_nat] :
      ( ( member_nat @ A @ ( domain_nat_nat @ R4 ) )
      = ( ? [Y: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ Y ) @ R4 ) ) ) ).

% Domain_iff
thf(fact_1074_DomainE,axiom,
    ! [A: produc6018962875968178549ist_fm,R4: set_Pr1822751329126368876m_rule] :
      ( ( member4699826688122452638ist_fm @ A @ ( domain3391028601666924027m_rule @ R4 ) )
     => ~ ! [B2: rule] :
            ~ ( member7231649785386036813m_rule @ ( produc1733806532565653680m_rule @ A @ B2 ) @ R4 ) ) ).

% DomainE
thf(fact_1075_DomainE,axiom,
    ! [A: rule,R4: set_Pr8969532344284305932ule_fm] :
      ( ( member_rule @ A @ ( domain_rule_fm @ R4 ) )
     => ~ ! [B2: fm] :
            ~ ( member9168179392308330989ule_fm @ ( product_Pair_rule_fm @ A @ B2 ) @ R4 ) ) ).

% DomainE
thf(fact_1076_DomainE,axiom,
    ! [A: nat,R4: set_Pr1261947904930325089at_nat] :
      ( ( member_nat @ A @ ( domain_nat_nat @ R4 ) )
     => ~ ! [B2: nat] :
            ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B2 ) @ R4 ) ) ).

% DomainE
thf(fact_1077_Domain_ODomainI,axiom,
    ! [A: produc6018962875968178549ist_fm,B: rule,R4: set_Pr1822751329126368876m_rule] :
      ( ( member7231649785386036813m_rule @ ( produc1733806532565653680m_rule @ A @ B ) @ R4 )
     => ( member4699826688122452638ist_fm @ A @ ( domain3391028601666924027m_rule @ R4 ) ) ) ).

% Domain.DomainI
thf(fact_1078_Domain_ODomainI,axiom,
    ! [A: rule,B: fm,R4: set_Pr8969532344284305932ule_fm] :
      ( ( member9168179392308330989ule_fm @ ( product_Pair_rule_fm @ A @ B ) @ R4 )
     => ( member_rule @ A @ ( domain_rule_fm @ R4 ) ) ) ).

% Domain.DomainI
thf(fact_1079_Domain_ODomainI,axiom,
    ! [A: nat,B: nat,R4: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R4 )
     => ( member_nat @ A @ ( domain_nat_nat @ R4 ) ) ) ).

% Domain.DomainI
thf(fact_1080_Domain_Osimps,axiom,
    ! [A: produc6018962875968178549ist_fm,R4: set_Pr1822751329126368876m_rule] :
      ( ( member4699826688122452638ist_fm @ A @ ( domain3391028601666924027m_rule @ R4 ) )
      = ( ? [A3: produc6018962875968178549ist_fm,B3: rule] :
            ( ( A = A3 )
            & ( member7231649785386036813m_rule @ ( produc1733806532565653680m_rule @ A3 @ B3 ) @ R4 ) ) ) ) ).

% Domain.simps
thf(fact_1081_Domain_Osimps,axiom,
    ! [A: rule,R4: set_Pr8969532344284305932ule_fm] :
      ( ( member_rule @ A @ ( domain_rule_fm @ R4 ) )
      = ( ? [A3: rule,B3: fm] :
            ( ( A = A3 )
            & ( member9168179392308330989ule_fm @ ( product_Pair_rule_fm @ A3 @ B3 ) @ R4 ) ) ) ) ).

% Domain.simps
thf(fact_1082_Domain_Osimps,axiom,
    ! [A: nat,R4: set_Pr1261947904930325089at_nat] :
      ( ( member_nat @ A @ ( domain_nat_nat @ R4 ) )
      = ( ? [A3: nat,B3: nat] :
            ( ( A = A3 )
            & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A3 @ B3 ) @ R4 ) ) ) ) ).

% Domain.simps
thf(fact_1083_Domain_Ocases,axiom,
    ! [A: produc6018962875968178549ist_fm,R4: set_Pr1822751329126368876m_rule] :
      ( ( member4699826688122452638ist_fm @ A @ ( domain3391028601666924027m_rule @ R4 ) )
     => ~ ! [B2: rule] :
            ~ ( member7231649785386036813m_rule @ ( produc1733806532565653680m_rule @ A @ B2 ) @ R4 ) ) ).

% Domain.cases
thf(fact_1084_Domain_Ocases,axiom,
    ! [A: rule,R4: set_Pr8969532344284305932ule_fm] :
      ( ( member_rule @ A @ ( domain_rule_fm @ R4 ) )
     => ~ ! [B2: fm] :
            ~ ( member9168179392308330989ule_fm @ ( product_Pair_rule_fm @ A @ B2 ) @ R4 ) ) ).

% Domain.cases
thf(fact_1085_Domain_Ocases,axiom,
    ! [A: nat,R4: set_Pr1261947904930325089at_nat] :
      ( ( member_nat @ A @ ( domain_nat_nat @ R4 ) )
     => ~ ! [B2: nat] :
            ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B2 ) @ R4 ) ) ).

% Domain.cases
thf(fact_1086_Range__iff,axiom,
    ! [A: rule,R4: set_Pr1822751329126368876m_rule] :
      ( ( member_rule @ A @ ( range_3331163552529868882m_rule @ R4 ) )
      = ( ? [Y: produc6018962875968178549ist_fm] : ( member7231649785386036813m_rule @ ( produc1733806532565653680m_rule @ Y @ A ) @ R4 ) ) ) ).

% Range_iff
thf(fact_1087_Range__iff,axiom,
    ! [A: fm,R4: set_Pr8969532344284305932ule_fm] :
      ( ( member_fm2 @ A @ ( range_rule_fm @ R4 ) )
      = ( ? [Y: rule] : ( member9168179392308330989ule_fm @ ( product_Pair_rule_fm @ Y @ A ) @ R4 ) ) ) ).

% Range_iff
thf(fact_1088_Range__iff,axiom,
    ! [A: nat,R4: set_Pr1261947904930325089at_nat] :
      ( ( member_nat @ A @ ( range_nat_nat @ R4 ) )
      = ( ? [Y: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ A ) @ R4 ) ) ) ).

% Range_iff
thf(fact_1089_RangeE,axiom,
    ! [B: rule,R4: set_Pr1822751329126368876m_rule] :
      ( ( member_rule @ B @ ( range_3331163552529868882m_rule @ R4 ) )
     => ~ ! [A2: produc6018962875968178549ist_fm] :
            ~ ( member7231649785386036813m_rule @ ( produc1733806532565653680m_rule @ A2 @ B ) @ R4 ) ) ).

% RangeE
thf(fact_1090_RangeE,axiom,
    ! [B: fm,R4: set_Pr8969532344284305932ule_fm] :
      ( ( member_fm2 @ B @ ( range_rule_fm @ R4 ) )
     => ~ ! [A2: rule] :
            ~ ( member9168179392308330989ule_fm @ ( product_Pair_rule_fm @ A2 @ B ) @ R4 ) ) ).

% RangeE
thf(fact_1091_RangeE,axiom,
    ! [B: nat,R4: set_Pr1261947904930325089at_nat] :
      ( ( member_nat @ B @ ( range_nat_nat @ R4 ) )
     => ~ ! [A2: nat] :
            ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ R4 ) ) ).

% RangeE
thf(fact_1092_Range_Ointros,axiom,
    ! [A: produc6018962875968178549ist_fm,B: rule,R4: set_Pr1822751329126368876m_rule] :
      ( ( member7231649785386036813m_rule @ ( produc1733806532565653680m_rule @ A @ B ) @ R4 )
     => ( member_rule @ B @ ( range_3331163552529868882m_rule @ R4 ) ) ) ).

% Range.intros
thf(fact_1093_Range_Ointros,axiom,
    ! [A: rule,B: fm,R4: set_Pr8969532344284305932ule_fm] :
      ( ( member9168179392308330989ule_fm @ ( product_Pair_rule_fm @ A @ B ) @ R4 )
     => ( member_fm2 @ B @ ( range_rule_fm @ R4 ) ) ) ).

% Range.intros
thf(fact_1094_Range_Ointros,axiom,
    ! [A: nat,B: nat,R4: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R4 )
     => ( member_nat @ B @ ( range_nat_nat @ R4 ) ) ) ).

% Range.intros
thf(fact_1095_Range_Osimps,axiom,
    ! [A: rule,R4: set_Pr1822751329126368876m_rule] :
      ( ( member_rule @ A @ ( range_3331163552529868882m_rule @ R4 ) )
      = ( ? [A3: produc6018962875968178549ist_fm,B3: rule] :
            ( ( A = B3 )
            & ( member7231649785386036813m_rule @ ( produc1733806532565653680m_rule @ A3 @ B3 ) @ R4 ) ) ) ) ).

% Range.simps
thf(fact_1096_Range_Osimps,axiom,
    ! [A: fm,R4: set_Pr8969532344284305932ule_fm] :
      ( ( member_fm2 @ A @ ( range_rule_fm @ R4 ) )
      = ( ? [A3: rule,B3: fm] :
            ( ( A = B3 )
            & ( member9168179392308330989ule_fm @ ( product_Pair_rule_fm @ A3 @ B3 ) @ R4 ) ) ) ) ).

% Range.simps
thf(fact_1097_Range_Osimps,axiom,
    ! [A: nat,R4: set_Pr1261947904930325089at_nat] :
      ( ( member_nat @ A @ ( range_nat_nat @ R4 ) )
      = ( ? [A3: nat,B3: nat] :
            ( ( A = B3 )
            & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A3 @ B3 ) @ R4 ) ) ) ) ).

% Range.simps
thf(fact_1098_Range_Ocases,axiom,
    ! [A: rule,R4: set_Pr1822751329126368876m_rule] :
      ( ( member_rule @ A @ ( range_3331163552529868882m_rule @ R4 ) )
     => ~ ! [A2: produc6018962875968178549ist_fm] :
            ~ ( member7231649785386036813m_rule @ ( produc1733806532565653680m_rule @ A2 @ A ) @ R4 ) ) ).

% Range.cases
thf(fact_1099_Range_Ocases,axiom,
    ! [A: fm,R4: set_Pr8969532344284305932ule_fm] :
      ( ( member_fm2 @ A @ ( range_rule_fm @ R4 ) )
     => ~ ! [A2: rule] :
            ~ ( member9168179392308330989ule_fm @ ( product_Pair_rule_fm @ A2 @ A ) @ R4 ) ) ).

% Range.cases
thf(fact_1100_Range_Ocases,axiom,
    ! [A: nat,R4: set_Pr1261947904930325089at_nat] :
      ( ( member_nat @ A @ ( range_nat_nat @ R4 ) )
     => ~ ! [A2: nat] :
            ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ A ) @ R4 ) ) ).

% Range.cases
thf(fact_1101_Union__disjoint,axiom,
    ! [C2: set_set_o,A5: set_o] :
      ( ( ( inf_inf_set_o @ ( comple90263536869209701_set_o @ C2 ) @ A5 )
        = bot_bot_set_o )
      = ( ! [X: set_o] :
            ( ( member_set_o @ X @ C2 )
           => ( ( inf_inf_set_o @ X @ A5 )
              = bot_bot_set_o ) ) ) ) ).

% Union_disjoint
thf(fact_1102_Union__disjoint,axiom,
    ! [C2: set_set_fm,A5: set_fm] :
      ( ( ( inf_inf_set_fm @ ( comple2134933779557159616set_fm @ C2 ) @ A5 )
        = bot_bot_set_fm )
      = ( ! [X: set_fm] :
            ( ( member_set_fm @ X @ C2 )
           => ( ( inf_inf_set_fm @ X @ A5 )
              = bot_bot_set_fm ) ) ) ) ).

% Union_disjoint
thf(fact_1103_sdrop__snth,axiom,
    ! [N3: nat,S2: stream2709947120125613254m_rule,M3: nat] :
      ( ( snth_P6679518042731451922m_rule @ ( sdrop_8169176516188972301m_rule @ N3 @ S2 ) @ M3 )
      = ( snth_P6679518042731451922m_rule @ S2 @ ( plus_plus_nat @ N3 @ M3 ) ) ) ).

% sdrop_snth
thf(fact_1104_SUP__lessD,axiom,
    ! [F: fm > set_fm,A5: set_fm,Y4: set_fm,I4: fm] :
      ( ( ord_less_set_fm @ ( comple2134933779557159616set_fm @ ( image_fm_set_fm @ F @ A5 ) ) @ Y4 )
     => ( ( member_fm2 @ I4 @ A5 )
       => ( ord_less_set_fm @ ( F @ I4 ) @ Y4 ) ) ) ).

% SUP_lessD
thf(fact_1105_SUP__lessD,axiom,
    ! [F: list_fm > set_fm,A5: set_list_fm,Y4: set_fm,I4: list_fm] :
      ( ( ord_less_set_fm @ ( comple2134933779557159616set_fm @ ( image_list_fm_set_fm @ F @ A5 ) ) @ Y4 )
     => ( ( member_list_fm2 @ I4 @ A5 )
       => ( ord_less_set_fm @ ( F @ I4 ) @ Y4 ) ) ) ).

% SUP_lessD
thf(fact_1106_SUP__lessD,axiom,
    ! [F: produc340336539035504054m_rule > set_fm,A5: set_Pr1822751329126368876m_rule,Y4: set_fm,I4: produc340336539035504054m_rule] :
      ( ( ord_less_set_fm @ ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ F @ A5 ) ) @ Y4 )
     => ( ( member7231649785386036813m_rule @ I4 @ A5 )
       => ( ord_less_set_fm @ ( F @ I4 ) @ Y4 ) ) ) ).

% SUP_lessD
thf(fact_1107_SUP__lessD,axiom,
    ! [F: $o > set_fm,A5: set_o,Y4: set_fm,I4: $o] :
      ( ( ord_less_set_fm @ ( comple2134933779557159616set_fm @ ( image_o_set_fm @ F @ A5 ) ) @ Y4 )
     => ( ( member_o2 @ I4 @ A5 )
       => ( ord_less_set_fm @ ( F @ I4 ) @ Y4 ) ) ) ).

% SUP_lessD
thf(fact_1108_SUP__lessD,axiom,
    ! [F: fm > $o,A5: set_fm,Y4: $o,I4: fm] :
      ( ( ord_less_o @ ( complete_Sup_Sup_o @ ( image_fm_o2 @ F @ A5 ) ) @ Y4 )
     => ( ( member_fm2 @ I4 @ A5 )
       => ( ord_less_o @ ( F @ I4 ) @ Y4 ) ) ) ).

% SUP_lessD
thf(fact_1109_SUP__lessD,axiom,
    ! [F: list_fm > $o,A5: set_list_fm,Y4: $o,I4: list_fm] :
      ( ( ord_less_o @ ( complete_Sup_Sup_o @ ( image_list_fm_o2 @ F @ A5 ) ) @ Y4 )
     => ( ( member_list_fm2 @ I4 @ A5 )
       => ( ord_less_o @ ( F @ I4 ) @ Y4 ) ) ) ).

% SUP_lessD
thf(fact_1110_SUP__lessD,axiom,
    ! [F: produc340336539035504054m_rule > $o,A5: set_Pr1822751329126368876m_rule,Y4: $o,I4: produc340336539035504054m_rule] :
      ( ( ord_less_o @ ( complete_Sup_Sup_o @ ( image_8969478372357302939rule_o @ F @ A5 ) ) @ Y4 )
     => ( ( member7231649785386036813m_rule @ I4 @ A5 )
       => ( ord_less_o @ ( F @ I4 ) @ Y4 ) ) ) ).

% SUP_lessD
thf(fact_1111_SUP__lessD,axiom,
    ! [F: $o > $o,A5: set_o,Y4: $o,I4: $o] :
      ( ( ord_less_o @ ( complete_Sup_Sup_o @ ( image_o_o2 @ F @ A5 ) ) @ Y4 )
     => ( ( member_o2 @ I4 @ A5 )
       => ( ord_less_o @ ( F @ I4 ) @ Y4 ) ) ) ).

% SUP_lessD
thf(fact_1112_UN__extend__simps_I5_J,axiom,
    ! [A5: set_fm,B6: produc340336539035504054m_rule > set_fm,C2: set_Pr1822751329126368876m_rule] :
      ( ( inf_inf_set_fm @ A5 @ ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ B6 @ C2 ) ) )
      = ( comple2134933779557159616set_fm
        @ ( image_377115208147135190set_fm
          @ ^ [X: produc340336539035504054m_rule] : ( inf_inf_set_fm @ A5 @ ( B6 @ X ) )
          @ C2 ) ) ) ).

% UN_extend_simps(5)
thf(fact_1113_UN__extend__simps_I4_J,axiom,
    ! [A5: produc340336539035504054m_rule > set_fm,C2: set_Pr1822751329126368876m_rule,B6: set_fm] :
      ( ( inf_inf_set_fm @ ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ A5 @ C2 ) ) @ B6 )
      = ( comple2134933779557159616set_fm
        @ ( image_377115208147135190set_fm
          @ ^ [X: produc340336539035504054m_rule] : ( inf_inf_set_fm @ ( A5 @ X ) @ B6 )
          @ C2 ) ) ) ).

% UN_extend_simps(4)
thf(fact_1114_Int__UN__distrib,axiom,
    ! [B6: set_fm,A5: produc340336539035504054m_rule > set_fm,I2: set_Pr1822751329126368876m_rule] :
      ( ( inf_inf_set_fm @ B6 @ ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ A5 @ I2 ) ) )
      = ( comple2134933779557159616set_fm
        @ ( image_377115208147135190set_fm
          @ ^ [I: produc340336539035504054m_rule] : ( inf_inf_set_fm @ B6 @ ( A5 @ I ) )
          @ I2 ) ) ) ).

% Int_UN_distrib
thf(fact_1115_Int__UN__distrib2,axiom,
    ! [A5: produc340336539035504054m_rule > set_fm,I2: set_Pr1822751329126368876m_rule,B6: produc340336539035504054m_rule > set_fm,J3: set_Pr1822751329126368876m_rule] :
      ( ( inf_inf_set_fm @ ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ A5 @ I2 ) ) @ ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ B6 @ J3 ) ) )
      = ( comple2134933779557159616set_fm
        @ ( image_377115208147135190set_fm
          @ ^ [I: produc340336539035504054m_rule] :
              ( comple2134933779557159616set_fm
              @ ( image_377115208147135190set_fm
                @ ^ [J: produc340336539035504054m_rule] : ( inf_inf_set_fm @ ( A5 @ I ) @ ( B6 @ J ) )
                @ J3 ) )
          @ I2 ) ) ) ).

% Int_UN_distrib2
thf(fact_1116_snd__eq__Range,axiom,
    ! [R2: set_Pr1822751329126368876m_rule] :
      ( ( image_4399899049665951567e_rule @ produc6514711208036230658m_rule @ R2 )
      = ( range_3331163552529868882m_rule @ R2 ) ) ).

% snd_eq_Range
thf(fact_1117_Range__snd,axiom,
    ( range_3331163552529868882m_rule
    = ( image_4399899049665951567e_rule @ produc6514711208036230658m_rule ) ) ).

% Range_snd
thf(fact_1118_Int__Union,axiom,
    ! [A5: set_fm,B6: set_set_fm] :
      ( ( inf_inf_set_fm @ A5 @ ( comple2134933779557159616set_fm @ B6 ) )
      = ( comple2134933779557159616set_fm @ ( image_set_fm_set_fm @ ( inf_inf_set_fm @ A5 ) @ B6 ) ) ) ).

% Int_Union
thf(fact_1119_Int__Union2,axiom,
    ! [B6: set_set_fm,A5: set_fm] :
      ( ( inf_inf_set_fm @ ( comple2134933779557159616set_fm @ B6 ) @ A5 )
      = ( comple2134933779557159616set_fm
        @ ( image_set_fm_set_fm
          @ ^ [C3: set_fm] : ( inf_inf_set_fm @ C3 @ A5 )
          @ B6 ) ) ) ).

% Int_Union2
thf(fact_1120_ccSUP__inf,axiom,
    ! [B6: set_Pr1822751329126368876m_rule,F: produc340336539035504054m_rule > set_fm,A: set_fm] :
      ( ( counta1649821642195540155m_rule @ B6 )
     => ( ( inf_inf_set_fm @ ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ F @ B6 ) ) @ A )
        = ( comple2134933779557159616set_fm
          @ ( image_377115208147135190set_fm
            @ ^ [B3: produc340336539035504054m_rule] : ( inf_inf_set_fm @ ( F @ B3 ) @ A )
            @ B6 ) ) ) ) ).

% ccSUP_inf
thf(fact_1121_ccSup__inf,axiom,
    ! [B6: set_set_fm,A: set_fm] :
      ( ( counta6288402088370727310set_fm @ B6 )
     => ( ( inf_inf_set_fm @ ( comple2134933779557159616set_fm @ B6 ) @ A )
        = ( comple2134933779557159616set_fm
          @ ( image_set_fm_set_fm
            @ ^ [B3: set_fm] : ( inf_inf_set_fm @ B3 @ A )
            @ B6 ) ) ) ) ).

% ccSup_inf
thf(fact_1122_ccSup__inf,axiom,
    ! [B6: set_o,A: $o] :
      ( ( counta5976203206615340371able_o @ B6 )
     => ( ( inf_inf_o @ ( complete_Sup_Sup_o @ B6 ) @ A )
        = ( complete_Sup_Sup_o
          @ ( image_o_o2
            @ ^ [B3: $o] : ( inf_inf_o @ B3 @ A )
            @ B6 ) ) ) ) ).

% ccSup_inf
thf(fact_1123_inf__ccSUP,axiom,
    ! [B6: set_Pr1822751329126368876m_rule,A: set_fm,F: produc340336539035504054m_rule > set_fm] :
      ( ( counta1649821642195540155m_rule @ B6 )
     => ( ( inf_inf_set_fm @ A @ ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ F @ B6 ) ) )
        = ( comple2134933779557159616set_fm
          @ ( image_377115208147135190set_fm
            @ ^ [B3: produc340336539035504054m_rule] : ( inf_inf_set_fm @ A @ ( F @ B3 ) )
            @ B6 ) ) ) ) ).

% inf_ccSUP
thf(fact_1124_inf__ccSup,axiom,
    ! [B6: set_set_fm,A: set_fm] :
      ( ( counta6288402088370727310set_fm @ B6 )
     => ( ( inf_inf_set_fm @ A @ ( comple2134933779557159616set_fm @ B6 ) )
        = ( comple2134933779557159616set_fm @ ( image_set_fm_set_fm @ ( inf_inf_set_fm @ A ) @ B6 ) ) ) ) ).

% inf_ccSup
thf(fact_1125_inf__ccSup,axiom,
    ! [B6: set_o,A: $o] :
      ( ( counta5976203206615340371able_o @ B6 )
     => ( ( inf_inf_o @ A @ ( complete_Sup_Sup_o @ B6 ) )
        = ( complete_Sup_Sup_o @ ( image_o_o2 @ ( inf_inf_o @ A ) @ B6 ) ) ) ) ).

% inf_ccSup
thf(fact_1126_ccSUP__inf__distrib2,axiom,
    ! [A5: set_Pr1822751329126368876m_rule,B6: set_Pr1822751329126368876m_rule,F: produc340336539035504054m_rule > set_fm,G: produc340336539035504054m_rule > set_fm] :
      ( ( counta1649821642195540155m_rule @ A5 )
     => ( ( counta1649821642195540155m_rule @ B6 )
       => ( ( inf_inf_set_fm @ ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ F @ A5 ) ) @ ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ G @ B6 ) ) )
          = ( comple2134933779557159616set_fm
            @ ( image_377115208147135190set_fm
              @ ^ [A3: produc340336539035504054m_rule] :
                  ( comple2134933779557159616set_fm
                  @ ( image_377115208147135190set_fm
                    @ ^ [B3: produc340336539035504054m_rule] : ( inf_inf_set_fm @ ( F @ A3 ) @ ( G @ B3 ) )
                    @ B6 ) )
              @ A5 ) ) ) ) ) ).

% ccSUP_inf_distrib2
thf(fact_1127_cSUP__lessD,axiom,
    ! [F: fm > nat,A5: set_fm,Y4: nat,I4: fm] :
      ( ( condit2214826472909112428ve_nat @ ( image_fm_nat @ F @ A5 ) )
     => ( ( ord_less_nat @ ( complete_Sup_Sup_nat @ ( image_fm_nat @ F @ A5 ) ) @ Y4 )
       => ( ( member_fm2 @ I4 @ A5 )
         => ( ord_less_nat @ ( F @ I4 ) @ Y4 ) ) ) ) ).

% cSUP_lessD
thf(fact_1128_cSUP__lessD,axiom,
    ! [F: $o > nat,A5: set_o,Y4: nat,I4: $o] :
      ( ( condit2214826472909112428ve_nat @ ( image_o_nat2 @ F @ A5 ) )
     => ( ( ord_less_nat @ ( complete_Sup_Sup_nat @ ( image_o_nat2 @ F @ A5 ) ) @ Y4 )
       => ( ( member_o2 @ I4 @ A5 )
         => ( ord_less_nat @ ( F @ I4 ) @ Y4 ) ) ) ) ).

% cSUP_lessD
thf(fact_1129_cSUP__lessD,axiom,
    ! [F: fm > $o,A5: set_fm,Y4: $o,I4: fm] :
      ( ( condit5488710616941104124bove_o @ ( image_fm_o2 @ F @ A5 ) )
     => ( ( ord_less_o @ ( complete_Sup_Sup_o @ ( image_fm_o2 @ F @ A5 ) ) @ Y4 )
       => ( ( member_fm2 @ I4 @ A5 )
         => ( ord_less_o @ ( F @ I4 ) @ Y4 ) ) ) ) ).

% cSUP_lessD
thf(fact_1130_cSUP__lessD,axiom,
    ! [F: $o > $o,A5: set_o,Y4: $o,I4: $o] :
      ( ( condit5488710616941104124bove_o @ ( image_o_o2 @ F @ A5 ) )
     => ( ( ord_less_o @ ( complete_Sup_Sup_o @ ( image_o_o2 @ F @ A5 ) ) @ Y4 )
       => ( ( member_o2 @ I4 @ A5 )
         => ( ord_less_o @ ( F @ I4 ) @ Y4 ) ) ) ) ).

% cSUP_lessD
thf(fact_1131_cSUP__lessD,axiom,
    ! [F: list_fm > nat,A5: set_list_fm,Y4: nat,I4: list_fm] :
      ( ( condit2214826472909112428ve_nat @ ( image_list_fm_nat @ F @ A5 ) )
     => ( ( ord_less_nat @ ( complete_Sup_Sup_nat @ ( image_list_fm_nat @ F @ A5 ) ) @ Y4 )
       => ( ( member_list_fm2 @ I4 @ A5 )
         => ( ord_less_nat @ ( F @ I4 ) @ Y4 ) ) ) ) ).

% cSUP_lessD
thf(fact_1132_cSUP__lessD,axiom,
    ! [F: fm > set_fm,A5: set_fm,Y4: set_fm,I4: fm] :
      ( ( condit7632555770478454199set_fm @ ( image_fm_set_fm @ F @ A5 ) )
     => ( ( ord_less_set_fm @ ( comple2134933779557159616set_fm @ ( image_fm_set_fm @ F @ A5 ) ) @ Y4 )
       => ( ( member_fm2 @ I4 @ A5 )
         => ( ord_less_set_fm @ ( F @ I4 ) @ Y4 ) ) ) ) ).

% cSUP_lessD
thf(fact_1133_cSUP__lessD,axiom,
    ! [F: $o > set_fm,A5: set_o,Y4: set_fm,I4: $o] :
      ( ( condit7632555770478454199set_fm @ ( image_o_set_fm @ F @ A5 ) )
     => ( ( ord_less_set_fm @ ( comple2134933779557159616set_fm @ ( image_o_set_fm @ F @ A5 ) ) @ Y4 )
       => ( ( member_o2 @ I4 @ A5 )
         => ( ord_less_set_fm @ ( F @ I4 ) @ Y4 ) ) ) ) ).

% cSUP_lessD
thf(fact_1134_cSUP__lessD,axiom,
    ! [F: list_fm > $o,A5: set_list_fm,Y4: $o,I4: list_fm] :
      ( ( condit5488710616941104124bove_o @ ( image_list_fm_o2 @ F @ A5 ) )
     => ( ( ord_less_o @ ( complete_Sup_Sup_o @ ( image_list_fm_o2 @ F @ A5 ) ) @ Y4 )
       => ( ( member_list_fm2 @ I4 @ A5 )
         => ( ord_less_o @ ( F @ I4 ) @ Y4 ) ) ) ) ).

% cSUP_lessD
thf(fact_1135_cSUP__lessD,axiom,
    ! [F: list_fm > set_fm,A5: set_list_fm,Y4: set_fm,I4: list_fm] :
      ( ( condit7632555770478454199set_fm @ ( image_list_fm_set_fm @ F @ A5 ) )
     => ( ( ord_less_set_fm @ ( comple2134933779557159616set_fm @ ( image_list_fm_set_fm @ F @ A5 ) ) @ Y4 )
       => ( ( member_list_fm2 @ I4 @ A5 )
         => ( ord_less_set_fm @ ( F @ I4 ) @ Y4 ) ) ) ) ).

% cSUP_lessD
thf(fact_1136_cSUP__lessD,axiom,
    ! [F: produc340336539035504054m_rule > nat,A5: set_Pr1822751329126368876m_rule,Y4: nat,I4: produc340336539035504054m_rule] :
      ( ( condit2214826472909112428ve_nat @ ( image_5344542309044177805le_nat @ F @ A5 ) )
     => ( ( ord_less_nat @ ( complete_Sup_Sup_nat @ ( image_5344542309044177805le_nat @ F @ A5 ) ) @ Y4 )
       => ( ( member7231649785386036813m_rule @ I4 @ A5 )
         => ( ord_less_nat @ ( F @ I4 ) @ Y4 ) ) ) ) ).

% cSUP_lessD
thf(fact_1137_Refl__Field__Restr,axiom,
    ! [R4: set_Pr1261947904930325089at_nat,A5: set_nat] :
      ( ( refl_on_nat @ ( field_nat @ R4 ) @ R4 )
     => ( ( field_nat
          @ ( inf_in2572325071724192079at_nat @ R4
            @ ( produc457027306803732586at_nat @ A5
              @ ^ [Uu: nat] : A5 ) ) )
        = ( inf_inf_set_nat @ ( field_nat @ R4 ) @ A5 ) ) ) ).

% Refl_Field_Restr
thf(fact_1138_Linear__order__Restr,axiom,
    ! [R4: set_Pr1261947904930325089at_nat,A5: set_nat] :
      ( ( order_4473980167227706203on_nat @ ( field_nat @ R4 ) @ R4 )
     => ( order_4473980167227706203on_nat
        @ ( field_nat
          @ ( inf_in2572325071724192079at_nat @ R4
            @ ( produc457027306803732586at_nat @ A5
              @ ^ [Uu: nat] : A5 ) ) )
        @ ( inf_in2572325071724192079at_nat @ R4
          @ ( produc457027306803732586at_nat @ A5
            @ ^ [Uu: nat] : A5 ) ) ) ) ).

% Linear_order_Restr
thf(fact_1139_Refl__Restr,axiom,
    ! [R4: set_Pr1261947904930325089at_nat,A5: set_nat] :
      ( ( refl_on_nat @ ( field_nat @ R4 ) @ R4 )
     => ( refl_on_nat
        @ ( field_nat
          @ ( inf_in2572325071724192079at_nat @ R4
            @ ( produc457027306803732586at_nat @ A5
              @ ^ [Uu: nat] : A5 ) ) )
        @ ( inf_in2572325071724192079at_nat @ R4
          @ ( produc457027306803732586at_nat @ A5
            @ ^ [Uu: nat] : A5 ) ) ) ) ).

% Refl_Restr
thf(fact_1140_SUP__inf,axiom,
    ! [F: produc340336539035504054m_rule > set_fm,B6: set_Pr1822751329126368876m_rule,A: set_fm] :
      ( ( inf_inf_set_fm @ ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ F @ B6 ) ) @ A )
      = ( comple2134933779557159616set_fm
        @ ( image_377115208147135190set_fm
          @ ^ [B3: produc340336539035504054m_rule] : ( inf_inf_set_fm @ ( F @ B3 ) @ A )
          @ B6 ) ) ) ).

% SUP_inf
thf(fact_1141_Sup__inf,axiom,
    ! [B6: set_set_fm,A: set_fm] :
      ( ( inf_inf_set_fm @ ( comple2134933779557159616set_fm @ B6 ) @ A )
      = ( comple2134933779557159616set_fm
        @ ( image_set_fm_set_fm
          @ ^ [B3: set_fm] : ( inf_inf_set_fm @ B3 @ A )
          @ B6 ) ) ) ).

% Sup_inf
thf(fact_1142_Sup__inf,axiom,
    ! [B6: set_o,A: $o] :
      ( ( inf_inf_o @ ( complete_Sup_Sup_o @ B6 ) @ A )
      = ( complete_Sup_Sup_o
        @ ( image_o_o2
          @ ^ [B3: $o] : ( inf_inf_o @ B3 @ A )
          @ B6 ) ) ) ).

% Sup_inf
thf(fact_1143_less__set__def,axiom,
    ( ord_less_set_fm
    = ( ^ [A7: set_fm,B7: set_fm] :
          ( ord_less_fm_o
          @ ^ [X: fm] : ( member_fm2 @ X @ A7 )
          @ ^ [X: fm] : ( member_fm2 @ X @ B7 ) ) ) ) ).

% less_set_def
thf(fact_1144_less__set__def,axiom,
    ( ord_less_set_list_fm
    = ( ^ [A7: set_list_fm,B7: set_list_fm] :
          ( ord_less_list_fm_o
          @ ^ [X: list_fm] : ( member_list_fm2 @ X @ A7 )
          @ ^ [X: list_fm] : ( member_list_fm2 @ X @ B7 ) ) ) ) ).

% less_set_def
thf(fact_1145_less__set__def,axiom,
    ( ord_le4093866177137961152m_rule
    = ( ^ [A7: set_Pr1822751329126368876m_rule,B7: set_Pr1822751329126368876m_rule] :
          ( ord_le8273235179611064261rule_o
          @ ^ [X: produc340336539035504054m_rule] : ( member7231649785386036813m_rule @ X @ A7 )
          @ ^ [X: produc340336539035504054m_rule] : ( member7231649785386036813m_rule @ X @ B7 ) ) ) ) ).

% less_set_def
thf(fact_1146_less__set__def,axiom,
    ( ord_less_set_o
    = ( ^ [A7: set_o,B7: set_o] :
          ( ord_less_o_o
          @ ^ [X: $o] : ( member_o2 @ X @ A7 )
          @ ^ [X: $o] : ( member_o2 @ X @ B7 ) ) ) ) ).

% less_set_def
thf(fact_1147_psubsetD,axiom,
    ! [A5: set_fm,B6: set_fm,C: fm] :
      ( ( ord_less_set_fm @ A5 @ B6 )
     => ( ( member_fm2 @ C @ A5 )
       => ( member_fm2 @ C @ B6 ) ) ) ).

% psubsetD
thf(fact_1148_psubsetD,axiom,
    ! [A5: set_list_fm,B6: set_list_fm,C: list_fm] :
      ( ( ord_less_set_list_fm @ A5 @ B6 )
     => ( ( member_list_fm2 @ C @ A5 )
       => ( member_list_fm2 @ C @ B6 ) ) ) ).

% psubsetD
thf(fact_1149_psubsetD,axiom,
    ! [A5: set_Pr1822751329126368876m_rule,B6: set_Pr1822751329126368876m_rule,C: produc340336539035504054m_rule] :
      ( ( ord_le4093866177137961152m_rule @ A5 @ B6 )
     => ( ( member7231649785386036813m_rule @ C @ A5 )
       => ( member7231649785386036813m_rule @ C @ B6 ) ) ) ).

% psubsetD
thf(fact_1150_psubsetD,axiom,
    ! [A5: set_o,B6: set_o,C: $o] :
      ( ( ord_less_set_o @ A5 @ B6 )
     => ( ( member_o2 @ C @ A5 )
       => ( member_o2 @ C @ B6 ) ) ) ).

% psubsetD
thf(fact_1151_inf__Int__eq2,axiom,
    ! [R2: set_Pr1822751329126368876m_rule,S: set_Pr1822751329126368876m_rule] :
      ( ( inf_in7523237975460508159rule_o
        @ ^ [X: produc6018962875968178549ist_fm,Y: rule] : ( member7231649785386036813m_rule @ ( produc1733806532565653680m_rule @ X @ Y ) @ R2 )
        @ ^ [X: produc6018962875968178549ist_fm,Y: rule] : ( member7231649785386036813m_rule @ ( produc1733806532565653680m_rule @ X @ Y ) @ S ) )
      = ( ^ [X: produc6018962875968178549ist_fm,Y: rule] : ( member7231649785386036813m_rule @ ( produc1733806532565653680m_rule @ X @ Y ) @ ( inf_in2520071923633375486m_rule @ R2 @ S ) ) ) ) ).

% inf_Int_eq2
thf(fact_1152_inf__Int__eq2,axiom,
    ! [R2: set_Pr8969532344284305932ule_fm,S: set_Pr8969532344284305932ule_fm] :
      ( ( inf_inf_rule_fm_o
        @ ^ [X: rule,Y: fm] : ( member9168179392308330989ule_fm @ ( product_Pair_rule_fm @ X @ Y ) @ R2 )
        @ ^ [X: rule,Y: fm] : ( member9168179392308330989ule_fm @ ( product_Pair_rule_fm @ X @ Y ) @ S ) )
      = ( ^ [X: rule,Y: fm] : ( member9168179392308330989ule_fm @ ( product_Pair_rule_fm @ X @ Y ) @ ( inf_in356324440158061726ule_fm @ R2 @ S ) ) ) ) ).

% inf_Int_eq2
thf(fact_1153_inf__Int__eq2,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( inf_inf_nat_nat_o
        @ ^ [X: nat,Y: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R2 )
        @ ^ [X: nat,Y: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ S ) )
      = ( ^ [X: nat,Y: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( inf_in2572325071724192079at_nat @ R2 @ S ) ) ) ) ).

% inf_Int_eq2
thf(fact_1154_inf__Int__eq,axiom,
    ! [R2: set_fm,S: set_fm] :
      ( ( inf_inf_fm_o
        @ ^ [X: fm] : ( member_fm2 @ X @ R2 )
        @ ^ [X: fm] : ( member_fm2 @ X @ S ) )
      = ( ^ [X: fm] : ( member_fm2 @ X @ ( inf_inf_set_fm @ R2 @ S ) ) ) ) ).

% inf_Int_eq
thf(fact_1155_inf__Int__eq,axiom,
    ! [R2: set_list_fm,S: set_list_fm] :
      ( ( inf_inf_list_fm_o
        @ ^ [X: list_fm] : ( member_list_fm2 @ X @ R2 )
        @ ^ [X: list_fm] : ( member_list_fm2 @ X @ S ) )
      = ( ^ [X: list_fm] : ( member_list_fm2 @ X @ ( inf_inf_set_list_fm @ R2 @ S ) ) ) ) ).

% inf_Int_eq
thf(fact_1156_inf__Int__eq,axiom,
    ! [R2: set_Pr1822751329126368876m_rule,S: set_Pr1822751329126368876m_rule] :
      ( ( inf_in1825323639692802311rule_o
        @ ^ [X: produc340336539035504054m_rule] : ( member7231649785386036813m_rule @ X @ R2 )
        @ ^ [X: produc340336539035504054m_rule] : ( member7231649785386036813m_rule @ X @ S ) )
      = ( ^ [X: produc340336539035504054m_rule] : ( member7231649785386036813m_rule @ X @ ( inf_in2520071923633375486m_rule @ R2 @ S ) ) ) ) ).

% inf_Int_eq
thf(fact_1157_inf__Int__eq,axiom,
    ! [R2: set_o,S: set_o] :
      ( ( inf_inf_o_o
        @ ^ [X: $o] : ( member_o2 @ X @ R2 )
        @ ^ [X: $o] : ( member_o2 @ X @ S ) )
      = ( ^ [X: $o] : ( member_o2 @ X @ ( inf_inf_set_o @ R2 @ S ) ) ) ) ).

% inf_Int_eq
thf(fact_1158_inf__set__def,axiom,
    ( inf_inf_set_fm
    = ( ^ [A7: set_fm,B7: set_fm] :
          ( collect_fm
          @ ( inf_inf_fm_o
            @ ^ [X: fm] : ( member_fm2 @ X @ A7 )
            @ ^ [X: fm] : ( member_fm2 @ X @ B7 ) ) ) ) ) ).

% inf_set_def
thf(fact_1159_inf__set__def,axiom,
    ( inf_inf_set_list_fm
    = ( ^ [A7: set_list_fm,B7: set_list_fm] :
          ( collect_list_fm
          @ ( inf_inf_list_fm_o
            @ ^ [X: list_fm] : ( member_list_fm2 @ X @ A7 )
            @ ^ [X: list_fm] : ( member_list_fm2 @ X @ B7 ) ) ) ) ) ).

% inf_set_def
thf(fact_1160_inf__set__def,axiom,
    ( inf_in2520071923633375486m_rule
    = ( ^ [A7: set_Pr1822751329126368876m_rule,B7: set_Pr1822751329126368876m_rule] :
          ( collec1009276759030335883m_rule
          @ ( inf_in1825323639692802311rule_o
            @ ^ [X: produc340336539035504054m_rule] : ( member7231649785386036813m_rule @ X @ A7 )
            @ ^ [X: produc340336539035504054m_rule] : ( member7231649785386036813m_rule @ X @ B7 ) ) ) ) ) ).

% inf_set_def
thf(fact_1161_inf__set__def,axiom,
    ( inf_inf_set_o
    = ( ^ [A7: set_o,B7: set_o] :
          ( collect_o
          @ ( inf_inf_o_o
            @ ^ [X: $o] : ( member_o2 @ X @ A7 )
            @ ^ [X: $o] : ( member_o2 @ X @ B7 ) ) ) ) ) ).

% inf_set_def
thf(fact_1162_inf__set__def,axiom,
    ( inf_in2572325071724192079at_nat
    = ( ^ [A7: set_Pr1261947904930325089at_nat,B7: set_Pr1261947904930325089at_nat] :
          ( collec3392354462482085612at_nat
          @ ( inf_in5163264567034779214_nat_o
            @ ^ [X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ A7 )
            @ ^ [X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ B7 ) ) ) ) ) ).

% inf_set_def
thf(fact_1163_inf__set__def,axiom,
    ( inf_inf_set_nat
    = ( ^ [A7: set_nat,B7: set_nat] :
          ( collect_nat
          @ ( inf_inf_nat_o
            @ ^ [X: nat] : ( member_nat @ X @ A7 )
            @ ^ [X: nat] : ( member_nat @ X @ B7 ) ) ) ) ) ).

% inf_set_def
thf(fact_1164_Restr__Field,axiom,
    ! [R4: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ R4
        @ ( produc457027306803732586at_nat @ ( field_nat @ R4 )
          @ ^ [Uu: nat] : ( field_nat @ R4 ) ) )
      = R4 ) ).

% Restr_Field
thf(fact_1165_inf__Sup,axiom,
    ! [A: set_fm,B6: set_set_fm] :
      ( ( inf_inf_set_fm @ A @ ( comple2134933779557159616set_fm @ B6 ) )
      = ( comple2134933779557159616set_fm @ ( image_set_fm_set_fm @ ( inf_inf_set_fm @ A ) @ B6 ) ) ) ).

% inf_Sup
thf(fact_1166_inf__Sup,axiom,
    ! [A: $o,B6: set_o] :
      ( ( inf_inf_o @ A @ ( complete_Sup_Sup_o @ B6 ) )
      = ( complete_Sup_Sup_o @ ( image_o_o2 @ ( inf_inf_o @ A ) @ B6 ) ) ) ).

% inf_Sup
thf(fact_1167_SUP__inf__distrib2,axiom,
    ! [F: produc340336539035504054m_rule > set_fm,A5: set_Pr1822751329126368876m_rule,G: produc340336539035504054m_rule > set_fm,B6: set_Pr1822751329126368876m_rule] :
      ( ( inf_inf_set_fm @ ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ F @ A5 ) ) @ ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ G @ B6 ) ) )
      = ( comple2134933779557159616set_fm
        @ ( image_377115208147135190set_fm
          @ ^ [A3: produc340336539035504054m_rule] :
              ( comple2134933779557159616set_fm
              @ ( image_377115208147135190set_fm
                @ ^ [B3: produc340336539035504054m_rule] : ( inf_inf_set_fm @ ( F @ A3 ) @ ( G @ B3 ) )
                @ B6 ) )
          @ A5 ) ) ) ).

% SUP_inf_distrib2
thf(fact_1168_inf__SUP,axiom,
    ! [A: set_fm,F: produc340336539035504054m_rule > set_fm,B6: set_Pr1822751329126368876m_rule] :
      ( ( inf_inf_set_fm @ A @ ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ F @ B6 ) ) )
      = ( comple2134933779557159616set_fm
        @ ( image_377115208147135190set_fm
          @ ^ [B3: produc340336539035504054m_rule] : ( inf_inf_set_fm @ A @ ( F @ B3 ) )
          @ B6 ) ) ) ).

% inf_SUP
thf(fact_1169_in__measure,axiom,
    ! [X5: nat,Y4: nat,F: nat > nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X5 @ Y4 ) @ ( measure_nat @ F ) )
      = ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) ) ).

% in_measure
thf(fact_1170_image__split__eq__Sigma,axiom,
    ! [F: produc340336539035504054m_rule > set_fm,G: produc340336539035504054m_rule > set_fm,A5: set_Pr1822751329126368876m_rule] :
      ( ( image_4560547394020074602set_fm
        @ ^ [X: produc340336539035504054m_rule] : ( produc5928365999827228247set_fm @ ( F @ X ) @ ( G @ X ) )
        @ A5 )
      = ( produc5276883515181919240set_fm @ ( image_377115208147135190set_fm @ F @ A5 )
        @ ^ [X: set_fm] : ( image_377115208147135190set_fm @ G @ ( inf_in2520071923633375486m_rule @ ( vimage8241997723160606624set_fm @ F @ ( insert_set_fm @ X @ bot_bot_set_set_fm ) ) @ A5 ) ) ) ) ).

% image_split_eq_Sigma
thf(fact_1171_image__split__eq__Sigma,axiom,
    ! [F: produc340336539035504054m_rule > $o,G: produc340336539035504054m_rule > set_fm,A5: set_Pr1822751329126368876m_rule] :
      ( ( image_4937524179753822053set_fm
        @ ^ [X: produc340336539035504054m_rule] : ( produc2575683835231190866set_fm @ ( F @ X ) @ ( G @ X ) )
        @ A5 )
      = ( produc3091822685562438531set_fm @ ( image_8969478372357302939rule_o @ F @ A5 )
        @ ^ [X: $o] : ( image_377115208147135190set_fm @ G @ ( inf_in2520071923633375486m_rule @ ( vimage2775786369866102885rule_o @ F @ ( insert_o2 @ X @ bot_bot_set_o ) ) @ A5 ) ) ) ) ).

% image_split_eq_Sigma
thf(fact_1172_vimage__eq,axiom,
    ! [A: fm,F: fm > fm,B6: set_fm] :
      ( ( member_fm2 @ A @ ( vimage_fm_fm @ F @ B6 ) )
      = ( member_fm2 @ ( F @ A ) @ B6 ) ) ).

% vimage_eq
thf(fact_1173_vimage__eq,axiom,
    ! [A: fm,F: fm > $o,B6: set_o] :
      ( ( member_fm2 @ A @ ( vimage_fm_o @ F @ B6 ) )
      = ( member_o2 @ ( F @ A ) @ B6 ) ) ).

% vimage_eq
thf(fact_1174_vimage__eq,axiom,
    ! [A: $o,F: $o > fm,B6: set_fm] :
      ( ( member_o2 @ A @ ( vimage_o_fm @ F @ B6 ) )
      = ( member_fm2 @ ( F @ A ) @ B6 ) ) ).

% vimage_eq
thf(fact_1175_vimage__eq,axiom,
    ! [A: $o,F: $o > $o,B6: set_o] :
      ( ( member_o2 @ A @ ( vimage_o_o @ F @ B6 ) )
      = ( member_o2 @ ( F @ A ) @ B6 ) ) ).

% vimage_eq
thf(fact_1176_vimage__eq,axiom,
    ! [A: fm,F: fm > list_fm,B6: set_list_fm] :
      ( ( member_fm2 @ A @ ( vimage_fm_list_fm @ F @ B6 ) )
      = ( member_list_fm2 @ ( F @ A ) @ B6 ) ) ).

% vimage_eq
thf(fact_1177_vimage__eq,axiom,
    ! [A: list_fm,F: list_fm > fm,B6: set_fm] :
      ( ( member_list_fm2 @ A @ ( vimage_list_fm_fm @ F @ B6 ) )
      = ( member_fm2 @ ( F @ A ) @ B6 ) ) ).

% vimage_eq
thf(fact_1178_vimage__eq,axiom,
    ! [A: list_fm,F: list_fm > $o,B6: set_o] :
      ( ( member_list_fm2 @ A @ ( vimage_list_fm_o @ F @ B6 ) )
      = ( member_o2 @ ( F @ A ) @ B6 ) ) ).

% vimage_eq
thf(fact_1179_vimage__eq,axiom,
    ! [A: $o,F: $o > list_fm,B6: set_list_fm] :
      ( ( member_o2 @ A @ ( vimage_o_list_fm @ F @ B6 ) )
      = ( member_list_fm2 @ ( F @ A ) @ B6 ) ) ).

% vimage_eq
thf(fact_1180_vimage__eq,axiom,
    ! [A: list_fm,F: list_fm > list_fm,B6: set_list_fm] :
      ( ( member_list_fm2 @ A @ ( vimage901937310047459067ist_fm @ F @ B6 ) )
      = ( member_list_fm2 @ ( F @ A ) @ B6 ) ) ).

% vimage_eq
thf(fact_1181_vimage__eq,axiom,
    ! [A: fm,F: fm > produc340336539035504054m_rule,B6: set_Pr1822751329126368876m_rule] :
      ( ( member_fm2 @ A @ ( vimage141910618784141742m_rule @ F @ B6 ) )
      = ( member7231649785386036813m_rule @ ( F @ A ) @ B6 ) ) ).

% vimage_eq
thf(fact_1182_vimageI,axiom,
    ! [F: fm > fm,A: fm,B: fm,B6: set_fm] :
      ( ( ( F @ A )
        = B )
     => ( ( member_fm2 @ B @ B6 )
       => ( member_fm2 @ A @ ( vimage_fm_fm @ F @ B6 ) ) ) ) ).

% vimageI
thf(fact_1183_vimageI,axiom,
    ! [F: $o > fm,A: $o,B: fm,B6: set_fm] :
      ( ( ( F @ A )
        = B )
     => ( ( member_fm2 @ B @ B6 )
       => ( member_o2 @ A @ ( vimage_o_fm @ F @ B6 ) ) ) ) ).

% vimageI
thf(fact_1184_vimageI,axiom,
    ! [F: fm > $o,A: fm,B: $o,B6: set_o] :
      ( ( ( F @ A )
        = B )
     => ( ( member_o2 @ B @ B6 )
       => ( member_fm2 @ A @ ( vimage_fm_o @ F @ B6 ) ) ) ) ).

% vimageI
thf(fact_1185_vimageI,axiom,
    ! [F: $o > $o,A: $o,B: $o,B6: set_o] :
      ( ( ( F @ A )
        = B )
     => ( ( member_o2 @ B @ B6 )
       => ( member_o2 @ A @ ( vimage_o_o @ F @ B6 ) ) ) ) ).

% vimageI
thf(fact_1186_vimageI,axiom,
    ! [F: list_fm > fm,A: list_fm,B: fm,B6: set_fm] :
      ( ( ( F @ A )
        = B )
     => ( ( member_fm2 @ B @ B6 )
       => ( member_list_fm2 @ A @ ( vimage_list_fm_fm @ F @ B6 ) ) ) ) ).

% vimageI
thf(fact_1187_vimageI,axiom,
    ! [F: fm > list_fm,A: fm,B: list_fm,B6: set_list_fm] :
      ( ( ( F @ A )
        = B )
     => ( ( member_list_fm2 @ B @ B6 )
       => ( member_fm2 @ A @ ( vimage_fm_list_fm @ F @ B6 ) ) ) ) ).

% vimageI
thf(fact_1188_vimageI,axiom,
    ! [F: $o > list_fm,A: $o,B: list_fm,B6: set_list_fm] :
      ( ( ( F @ A )
        = B )
     => ( ( member_list_fm2 @ B @ B6 )
       => ( member_o2 @ A @ ( vimage_o_list_fm @ F @ B6 ) ) ) ) ).

% vimageI
thf(fact_1189_vimageI,axiom,
    ! [F: list_fm > $o,A: list_fm,B: $o,B6: set_o] :
      ( ( ( F @ A )
        = B )
     => ( ( member_o2 @ B @ B6 )
       => ( member_list_fm2 @ A @ ( vimage_list_fm_o @ F @ B6 ) ) ) ) ).

% vimageI
thf(fact_1190_vimageI,axiom,
    ! [F: list_fm > list_fm,A: list_fm,B: list_fm,B6: set_list_fm] :
      ( ( ( F @ A )
        = B )
     => ( ( member_list_fm2 @ B @ B6 )
       => ( member_list_fm2 @ A @ ( vimage901937310047459067ist_fm @ F @ B6 ) ) ) ) ).

% vimageI
thf(fact_1191_vimageI,axiom,
    ! [F: produc340336539035504054m_rule > fm,A: produc340336539035504054m_rule,B: fm,B6: set_fm] :
      ( ( ( F @ A )
        = B )
     => ( ( member_fm2 @ B @ B6 )
       => ( member7231649785386036813m_rule @ A @ ( vimage5588858601996818112ule_fm @ F @ B6 ) ) ) ) ).

% vimageI
thf(fact_1192_vimage__Collect__eq,axiom,
    ! [F: product_prod_nat_nat > product_prod_nat_nat,P3: product_prod_nat_nat > $o] :
      ( ( vimage2449269961533847803at_nat @ F @ ( collec3392354462482085612at_nat @ P3 ) )
      = ( collec3392354462482085612at_nat
        @ ^ [Y: product_prod_nat_nat] : ( P3 @ ( F @ Y ) ) ) ) ).

% vimage_Collect_eq
thf(fact_1193_vimage__Collect__eq,axiom,
    ! [F: nat > product_prod_nat_nat,P3: product_prod_nat_nat > $o] :
      ( ( vimage8013328719654469172at_nat @ F @ ( collec3392354462482085612at_nat @ P3 ) )
      = ( collect_nat
        @ ^ [Y: nat] : ( P3 @ ( F @ Y ) ) ) ) ).

% vimage_Collect_eq
thf(fact_1194_vimage__Collect__eq,axiom,
    ! [F: product_prod_nat_nat > nat,P3: nat > $o] :
      ( ( vimage4653281326611754070at_nat @ F @ ( collect_nat @ P3 ) )
      = ( collec3392354462482085612at_nat
        @ ^ [Y: product_prod_nat_nat] : ( P3 @ ( F @ Y ) ) ) ) ).

% vimage_Collect_eq
thf(fact_1195_vimage__Collect__eq,axiom,
    ! [F: nat > nat,P3: nat > $o] :
      ( ( vimage_nat_nat @ F @ ( collect_nat @ P3 ) )
      = ( collect_nat
        @ ^ [Y: nat] : ( P3 @ ( F @ Y ) ) ) ) ).

% vimage_Collect_eq
thf(fact_1196_vimage__empty,axiom,
    ! [F: $o > $o] :
      ( ( vimage_o_o @ F @ bot_bot_set_o )
      = bot_bot_set_o ) ).

% vimage_empty
thf(fact_1197_vimage__Collect,axiom,
    ! [P3: product_prod_nat_nat > $o,F: product_prod_nat_nat > product_prod_nat_nat,Q: product_prod_nat_nat > $o] :
      ( ! [X3: product_prod_nat_nat] :
          ( ( P3 @ ( F @ X3 ) )
          = ( Q @ X3 ) )
     => ( ( vimage2449269961533847803at_nat @ F @ ( collec3392354462482085612at_nat @ P3 ) )
        = ( collec3392354462482085612at_nat @ Q ) ) ) ).

% vimage_Collect
thf(fact_1198_vimage__Collect,axiom,
    ! [P3: product_prod_nat_nat > $o,F: nat > product_prod_nat_nat,Q: nat > $o] :
      ( ! [X3: nat] :
          ( ( P3 @ ( F @ X3 ) )
          = ( Q @ X3 ) )
     => ( ( vimage8013328719654469172at_nat @ F @ ( collec3392354462482085612at_nat @ P3 ) )
        = ( collect_nat @ Q ) ) ) ).

% vimage_Collect
thf(fact_1199_vimage__Collect,axiom,
    ! [P3: nat > $o,F: product_prod_nat_nat > nat,Q: product_prod_nat_nat > $o] :
      ( ! [X3: product_prod_nat_nat] :
          ( ( P3 @ ( F @ X3 ) )
          = ( Q @ X3 ) )
     => ( ( vimage4653281326611754070at_nat @ F @ ( collect_nat @ P3 ) )
        = ( collec3392354462482085612at_nat @ Q ) ) ) ).

% vimage_Collect
thf(fact_1200_vimage__Collect,axiom,
    ! [P3: nat > $o,F: nat > nat,Q: nat > $o] :
      ( ! [X3: nat] :
          ( ( P3 @ ( F @ X3 ) )
          = ( Q @ X3 ) )
     => ( ( vimage_nat_nat @ F @ ( collect_nat @ P3 ) )
        = ( collect_nat @ Q ) ) ) ).

% vimage_Collect
thf(fact_1201_vimage__def,axiom,
    ( vimage7594493593656161133nat_fm
    = ( ^ [F12: product_prod_nat_nat > fm,B7: set_fm] :
          ( collec3392354462482085612at_nat
          @ ^ [X: product_prod_nat_nat] : ( member_fm2 @ ( F12 @ X ) @ B7 ) ) ) ) ).

% vimage_def
thf(fact_1202_vimage__def,axiom,
    ( vimage7064930452948413043ist_fm
    = ( ^ [F12: product_prod_nat_nat > list_fm,B7: set_list_fm] :
          ( collec3392354462482085612at_nat
          @ ^ [X: product_prod_nat_nat] : ( member_list_fm2 @ ( F12 @ X ) @ B7 ) ) ) ) ).

% vimage_def
thf(fact_1203_vimage__def,axiom,
    ( vimage7365293508715718204m_rule
    = ( ^ [F12: product_prod_nat_nat > produc340336539035504054m_rule,B7: set_Pr1822751329126368876m_rule] :
          ( collec3392354462482085612at_nat
          @ ^ [X: product_prod_nat_nat] : ( member7231649785386036813m_rule @ ( F12 @ X ) @ B7 ) ) ) ) ).

% vimage_def
thf(fact_1204_vimage__def,axiom,
    ( vimage2444243245536687506_nat_o
    = ( ^ [F12: product_prod_nat_nat > $o,B7: set_o] :
          ( collec3392354462482085612at_nat
          @ ^ [X: product_prod_nat_nat] : ( member_o2 @ ( F12 @ X ) @ B7 ) ) ) ) ).

% vimage_def
thf(fact_1205_vimage__def,axiom,
    ( vimage_nat_fm
    = ( ^ [F12: nat > fm,B7: set_fm] :
          ( collect_nat
          @ ^ [X: nat] : ( member_fm2 @ ( F12 @ X ) @ B7 ) ) ) ) ).

% vimage_def
thf(fact_1206_vimage__def,axiom,
    ( vimage_nat_list_fm
    = ( ^ [F12: nat > list_fm,B7: set_list_fm] :
          ( collect_nat
          @ ^ [X: nat] : ( member_list_fm2 @ ( F12 @ X ) @ B7 ) ) ) ) ).

% vimage_def
thf(fact_1207_vimage__def,axiom,
    ( vimage6841172051168148547m_rule
    = ( ^ [F12: nat > produc340336539035504054m_rule,B7: set_Pr1822751329126368876m_rule] :
          ( collect_nat
          @ ^ [X: nat] : ( member7231649785386036813m_rule @ ( F12 @ X ) @ B7 ) ) ) ) ).

% vimage_def
thf(fact_1208_vimage__def,axiom,
    ( vimage_nat_o
    = ( ^ [F12: nat > $o,B7: set_o] :
          ( collect_nat
          @ ^ [X: nat] : ( member_o2 @ ( F12 @ X ) @ B7 ) ) ) ) ).

% vimage_def
thf(fact_1209_vimageI2,axiom,
    ! [F: fm > fm,A: fm,A5: set_fm] :
      ( ( member_fm2 @ ( F @ A ) @ A5 )
     => ( member_fm2 @ A @ ( vimage_fm_fm @ F @ A5 ) ) ) ).

% vimageI2
thf(fact_1210_vimageI2,axiom,
    ! [F: $o > fm,A: $o,A5: set_fm] :
      ( ( member_fm2 @ ( F @ A ) @ A5 )
     => ( member_o2 @ A @ ( vimage_o_fm @ F @ A5 ) ) ) ).

% vimageI2
thf(fact_1211_vimageI2,axiom,
    ! [F: fm > $o,A: fm,A5: set_o] :
      ( ( member_o2 @ ( F @ A ) @ A5 )
     => ( member_fm2 @ A @ ( vimage_fm_o @ F @ A5 ) ) ) ).

% vimageI2
thf(fact_1212_vimageI2,axiom,
    ! [F: $o > $o,A: $o,A5: set_o] :
      ( ( member_o2 @ ( F @ A ) @ A5 )
     => ( member_o2 @ A @ ( vimage_o_o @ F @ A5 ) ) ) ).

% vimageI2
thf(fact_1213_vimageI2,axiom,
    ! [F: list_fm > fm,A: list_fm,A5: set_fm] :
      ( ( member_fm2 @ ( F @ A ) @ A5 )
     => ( member_list_fm2 @ A @ ( vimage_list_fm_fm @ F @ A5 ) ) ) ).

% vimageI2
thf(fact_1214_vimageI2,axiom,
    ! [F: fm > list_fm,A: fm,A5: set_list_fm] :
      ( ( member_list_fm2 @ ( F @ A ) @ A5 )
     => ( member_fm2 @ A @ ( vimage_fm_list_fm @ F @ A5 ) ) ) ).

% vimageI2
thf(fact_1215_vimageI2,axiom,
    ! [F: $o > list_fm,A: $o,A5: set_list_fm] :
      ( ( member_list_fm2 @ ( F @ A ) @ A5 )
     => ( member_o2 @ A @ ( vimage_o_list_fm @ F @ A5 ) ) ) ).

% vimageI2
thf(fact_1216_vimageI2,axiom,
    ! [F: list_fm > $o,A: list_fm,A5: set_o] :
      ( ( member_o2 @ ( F @ A ) @ A5 )
     => ( member_list_fm2 @ A @ ( vimage_list_fm_o @ F @ A5 ) ) ) ).

% vimageI2
thf(fact_1217_vimageI2,axiom,
    ! [F: list_fm > list_fm,A: list_fm,A5: set_list_fm] :
      ( ( member_list_fm2 @ ( F @ A ) @ A5 )
     => ( member_list_fm2 @ A @ ( vimage901937310047459067ist_fm @ F @ A5 ) ) ) ).

% vimageI2
thf(fact_1218_vimageI2,axiom,
    ! [F: produc340336539035504054m_rule > fm,A: produc340336539035504054m_rule,A5: set_fm] :
      ( ( member_fm2 @ ( F @ A ) @ A5 )
     => ( member7231649785386036813m_rule @ A @ ( vimage5588858601996818112ule_fm @ F @ A5 ) ) ) ).

% vimageI2
thf(fact_1219_vimageE,axiom,
    ! [A: fm,F: fm > fm,B6: set_fm] :
      ( ( member_fm2 @ A @ ( vimage_fm_fm @ F @ B6 ) )
     => ( member_fm2 @ ( F @ A ) @ B6 ) ) ).

% vimageE
thf(fact_1220_vimageE,axiom,
    ! [A: fm,F: fm > $o,B6: set_o] :
      ( ( member_fm2 @ A @ ( vimage_fm_o @ F @ B6 ) )
     => ( member_o2 @ ( F @ A ) @ B6 ) ) ).

% vimageE
thf(fact_1221_vimageE,axiom,
    ! [A: $o,F: $o > fm,B6: set_fm] :
      ( ( member_o2 @ A @ ( vimage_o_fm @ F @ B6 ) )
     => ( member_fm2 @ ( F @ A ) @ B6 ) ) ).

% vimageE
thf(fact_1222_vimageE,axiom,
    ! [A: $o,F: $o > $o,B6: set_o] :
      ( ( member_o2 @ A @ ( vimage_o_o @ F @ B6 ) )
     => ( member_o2 @ ( F @ A ) @ B6 ) ) ).

% vimageE
thf(fact_1223_vimageE,axiom,
    ! [A: fm,F: fm > list_fm,B6: set_list_fm] :
      ( ( member_fm2 @ A @ ( vimage_fm_list_fm @ F @ B6 ) )
     => ( member_list_fm2 @ ( F @ A ) @ B6 ) ) ).

% vimageE
thf(fact_1224_vimageE,axiom,
    ! [A: list_fm,F: list_fm > fm,B6: set_fm] :
      ( ( member_list_fm2 @ A @ ( vimage_list_fm_fm @ F @ B6 ) )
     => ( member_fm2 @ ( F @ A ) @ B6 ) ) ).

% vimageE
thf(fact_1225_vimageE,axiom,
    ! [A: list_fm,F: list_fm > $o,B6: set_o] :
      ( ( member_list_fm2 @ A @ ( vimage_list_fm_o @ F @ B6 ) )
     => ( member_o2 @ ( F @ A ) @ B6 ) ) ).

% vimageE
thf(fact_1226_vimageE,axiom,
    ! [A: $o,F: $o > list_fm,B6: set_list_fm] :
      ( ( member_o2 @ A @ ( vimage_o_list_fm @ F @ B6 ) )
     => ( member_list_fm2 @ ( F @ A ) @ B6 ) ) ).

% vimageE
thf(fact_1227_vimageE,axiom,
    ! [A: list_fm,F: list_fm > list_fm,B6: set_list_fm] :
      ( ( member_list_fm2 @ A @ ( vimage901937310047459067ist_fm @ F @ B6 ) )
     => ( member_list_fm2 @ ( F @ A ) @ B6 ) ) ).

% vimageE
thf(fact_1228_vimageE,axiom,
    ! [A: fm,F: fm > produc340336539035504054m_rule,B6: set_Pr1822751329126368876m_rule] :
      ( ( member_fm2 @ A @ ( vimage141910618784141742m_rule @ F @ B6 ) )
     => ( member7231649785386036813m_rule @ ( F @ A ) @ B6 ) ) ).

% vimageE
thf(fact_1229_vimageD,axiom,
    ! [A: fm,F: fm > fm,A5: set_fm] :
      ( ( member_fm2 @ A @ ( vimage_fm_fm @ F @ A5 ) )
     => ( member_fm2 @ ( F @ A ) @ A5 ) ) ).

% vimageD
thf(fact_1230_vimageD,axiom,
    ! [A: fm,F: fm > $o,A5: set_o] :
      ( ( member_fm2 @ A @ ( vimage_fm_o @ F @ A5 ) )
     => ( member_o2 @ ( F @ A ) @ A5 ) ) ).

% vimageD
thf(fact_1231_vimageD,axiom,
    ! [A: $o,F: $o > fm,A5: set_fm] :
      ( ( member_o2 @ A @ ( vimage_o_fm @ F @ A5 ) )
     => ( member_fm2 @ ( F @ A ) @ A5 ) ) ).

% vimageD
thf(fact_1232_vimageD,axiom,
    ! [A: $o,F: $o > $o,A5: set_o] :
      ( ( member_o2 @ A @ ( vimage_o_o @ F @ A5 ) )
     => ( member_o2 @ ( F @ A ) @ A5 ) ) ).

% vimageD
thf(fact_1233_vimageD,axiom,
    ! [A: fm,F: fm > list_fm,A5: set_list_fm] :
      ( ( member_fm2 @ A @ ( vimage_fm_list_fm @ F @ A5 ) )
     => ( member_list_fm2 @ ( F @ A ) @ A5 ) ) ).

% vimageD
thf(fact_1234_vimageD,axiom,
    ! [A: list_fm,F: list_fm > fm,A5: set_fm] :
      ( ( member_list_fm2 @ A @ ( vimage_list_fm_fm @ F @ A5 ) )
     => ( member_fm2 @ ( F @ A ) @ A5 ) ) ).

% vimageD
thf(fact_1235_vimageD,axiom,
    ! [A: list_fm,F: list_fm > $o,A5: set_o] :
      ( ( member_list_fm2 @ A @ ( vimage_list_fm_o @ F @ A5 ) )
     => ( member_o2 @ ( F @ A ) @ A5 ) ) ).

% vimageD
thf(fact_1236_vimageD,axiom,
    ! [A: $o,F: $o > list_fm,A5: set_list_fm] :
      ( ( member_o2 @ A @ ( vimage_o_list_fm @ F @ A5 ) )
     => ( member_list_fm2 @ ( F @ A ) @ A5 ) ) ).

% vimageD
thf(fact_1237_vimageD,axiom,
    ! [A: list_fm,F: list_fm > list_fm,A5: set_list_fm] :
      ( ( member_list_fm2 @ A @ ( vimage901937310047459067ist_fm @ F @ A5 ) )
     => ( member_list_fm2 @ ( F @ A ) @ A5 ) ) ).

% vimageD
thf(fact_1238_vimageD,axiom,
    ! [A: fm,F: fm > produc340336539035504054m_rule,A5: set_Pr1822751329126368876m_rule] :
      ( ( member_fm2 @ A @ ( vimage141910618784141742m_rule @ F @ A5 ) )
     => ( member7231649785386036813m_rule @ ( F @ A ) @ A5 ) ) ).

% vimageD
thf(fact_1239_vimage__singleton__eq,axiom,
    ! [A: fm,F: fm > $o,B: $o] :
      ( ( member_fm2 @ A @ ( vimage_fm_o @ F @ ( insert_o2 @ B @ bot_bot_set_o ) ) )
      = ( ( F @ A )
        = B ) ) ).

% vimage_singleton_eq
thf(fact_1240_vimage__singleton__eq,axiom,
    ! [A: list_fm,F: list_fm > $o,B: $o] :
      ( ( member_list_fm2 @ A @ ( vimage_list_fm_o @ F @ ( insert_o2 @ B @ bot_bot_set_o ) ) )
      = ( ( F @ A )
        = B ) ) ).

% vimage_singleton_eq
thf(fact_1241_vimage__singleton__eq,axiom,
    ! [A: produc340336539035504054m_rule,F: produc340336539035504054m_rule > $o,B: $o] :
      ( ( member7231649785386036813m_rule @ A @ ( vimage2775786369866102885rule_o @ F @ ( insert_o2 @ B @ bot_bot_set_o ) ) )
      = ( ( F @ A )
        = B ) ) ).

% vimage_singleton_eq
thf(fact_1242_vimage__singleton__eq,axiom,
    ! [A: $o,F: $o > $o,B: $o] :
      ( ( member_o2 @ A @ ( vimage_o_o @ F @ ( insert_o2 @ B @ bot_bot_set_o ) ) )
      = ( ( F @ A )
        = B ) ) ).

% vimage_singleton_eq
thf(fact_1243_vimage__UN,axiom,
    ! [F: fm > fm,B6: produc340336539035504054m_rule > set_fm,A5: set_Pr1822751329126368876m_rule] :
      ( ( vimage_fm_fm @ F @ ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ B6 @ A5 ) ) )
      = ( comple2134933779557159616set_fm
        @ ( image_377115208147135190set_fm
          @ ^ [X: produc340336539035504054m_rule] : ( vimage_fm_fm @ F @ ( B6 @ X ) )
          @ A5 ) ) ) ).

% vimage_UN
thf(fact_1244_vimage__Union,axiom,
    ! [F: fm > fm,A5: set_set_fm] :
      ( ( vimage_fm_fm @ F @ ( comple2134933779557159616set_fm @ A5 ) )
      = ( comple2134933779557159616set_fm @ ( image_set_fm_set_fm @ ( vimage_fm_fm @ F ) @ A5 ) ) ) ).

% vimage_Union
thf(fact_1245_Pair__vimage__Sigma,axiom,
    ! [X5: rule,A5: set_rule,F: rule > set_fm] :
      ( ( ( member_rule @ X5 @ A5 )
       => ( ( vimage6423823316857616718ule_fm @ ( product_Pair_rule_fm @ X5 ) @ ( produc7224384929229372695ule_fm @ A5 @ F ) )
          = ( F @ X5 ) ) )
      & ( ~ ( member_rule @ X5 @ A5 )
       => ( ( vimage6423823316857616718ule_fm @ ( product_Pair_rule_fm @ X5 ) @ ( produc7224384929229372695ule_fm @ A5 @ F ) )
          = bot_bot_set_fm ) ) ) ).

% Pair_vimage_Sigma
thf(fact_1246_Pair__vimage__Sigma,axiom,
    ! [X5: nat,A5: set_nat,F: nat > set_nat] :
      ( ( ( member_nat @ X5 @ A5 )
       => ( ( vimage8013328719654469172at_nat @ ( product_Pair_nat_nat @ X5 ) @ ( produc457027306803732586at_nat @ A5 @ F ) )
          = ( F @ X5 ) ) )
      & ( ~ ( member_nat @ X5 @ A5 )
       => ( ( vimage8013328719654469172at_nat @ ( product_Pair_nat_nat @ X5 ) @ ( produc457027306803732586at_nat @ A5 @ F ) )
          = bot_bot_set_nat ) ) ) ).

% Pair_vimage_Sigma
thf(fact_1247_Pair__vimage__Sigma,axiom,
    ! [X5: fm,A5: set_fm,F: fm > set_o] :
      ( ( ( member_fm2 @ X5 @ A5 )
       => ( ( vimage3135397528315170223d_fm_o @ ( product_Pair_fm_o @ X5 ) @ ( product_Sigma_fm_o @ A5 @ F ) )
          = ( F @ X5 ) ) )
      & ( ~ ( member_fm2 @ X5 @ A5 )
       => ( ( vimage3135397528315170223d_fm_o @ ( product_Pair_fm_o @ X5 ) @ ( product_Sigma_fm_o @ A5 @ F ) )
          = bot_bot_set_o ) ) ) ).

% Pair_vimage_Sigma
thf(fact_1248_Pair__vimage__Sigma,axiom,
    ! [X5: list_fm,A5: set_list_fm,F: list_fm > set_o] :
      ( ( ( member_list_fm2 @ X5 @ A5 )
       => ( ( vimage934036171275271721t_fm_o @ ( produc8850898633069939702t_fm_o @ X5 ) @ ( produc2902879935907792807t_fm_o @ A5 @ F ) )
          = ( F @ X5 ) ) )
      & ( ~ ( member_list_fm2 @ X5 @ A5 )
       => ( ( vimage934036171275271721t_fm_o @ ( produc8850898633069939702t_fm_o @ X5 ) @ ( produc2902879935907792807t_fm_o @ A5 @ F ) )
          = bot_bot_set_o ) ) ) ).

% Pair_vimage_Sigma
thf(fact_1249_Pair__vimage__Sigma,axiom,
    ! [X5: produc340336539035504054m_rule,A5: set_Pr1822751329126368876m_rule,F: produc340336539035504054m_rule > set_o] :
      ( ( ( member7231649785386036813m_rule @ X5 @ A5 )
       => ( ( vimage6473838429648664948rule_o @ ( produc7990273388919370121rule_o @ X5 ) @ ( produc5570680280418397464rule_o @ A5 @ F ) )
          = ( F @ X5 ) ) )
      & ( ~ ( member7231649785386036813m_rule @ X5 @ A5 )
       => ( ( vimage6473838429648664948rule_o @ ( produc7990273388919370121rule_o @ X5 ) @ ( produc5570680280418397464rule_o @ A5 @ F ) )
          = bot_bot_set_o ) ) ) ).

% Pair_vimage_Sigma
thf(fact_1250_Pair__vimage__Sigma,axiom,
    ! [X5: $o,A5: set_o,F: $o > set_o] :
      ( ( ( member_o2 @ X5 @ A5 )
       => ( ( vimage8945963521958007626od_o_o @ ( product_Pair_o_o @ X5 ) @ ( product_Sigma_o_o @ A5 @ F ) )
          = ( F @ X5 ) ) )
      & ( ~ ( member_o2 @ X5 @ A5 )
       => ( ( vimage8945963521958007626od_o_o @ ( product_Pair_o_o @ X5 ) @ ( product_Sigma_o_o @ A5 @ F ) )
          = bot_bot_set_o ) ) ) ).

% Pair_vimage_Sigma
thf(fact_1251_vimage__eq__UN,axiom,
    ( vimage141910618784141742m_rule
    = ( ^ [F12: fm > produc340336539035504054m_rule,B7: set_Pr1822751329126368876m_rule] :
          ( comple2134933779557159616set_fm
          @ ( image_377115208147135190set_fm
            @ ^ [Y: produc340336539035504054m_rule] : ( vimage141910618784141742m_rule @ F12 @ ( insert775845832097999846m_rule @ Y @ bot_bo3426410157013319936m_rule ) )
            @ B7 ) ) ) ) ).

% vimage_eq_UN
thf(fact_1252_vimage__eq__UN,axiom,
    ( vimage_fm_o
    = ( ^ [F12: fm > $o,B7: set_o] :
          ( comple2134933779557159616set_fm
          @ ( image_o_set_fm
            @ ^ [Y: $o] : ( vimage_fm_o @ F12 @ ( insert_o2 @ Y @ bot_bot_set_o ) )
            @ B7 ) ) ) ) ).

% vimage_eq_UN
thf(fact_1253_above__def,axiom,
    ( order_793026671032835073at_nat
    = ( ^ [R: set_Pr8693737435421807431at_nat,A3: product_prod_nat_nat] :
          ( collec3392354462482085612at_nat
          @ ^ [B3: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A3 @ B3 ) @ R ) ) ) ) ).

% above_def
thf(fact_1254_above__def,axiom,
    ( order_above_nat
    = ( ^ [R: set_Pr1261947904930325089at_nat,A3: nat] :
          ( collect_nat
          @ ^ [B3: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A3 @ B3 ) @ R ) ) ) ) ).

% above_def
thf(fact_1255_wf__UN,axiom,
    ! [I2: set_fm,R4: fm > set_Pr1261947904930325089at_nat] :
      ( ! [I3: fm] :
          ( ( member_fm2 @ I3 @ I2 )
         => ( wf_nat @ ( R4 @ I3 ) ) )
     => ( ! [I3: fm,J4: fm] :
            ( ( member_fm2 @ I3 @ I2 )
           => ( ( member_fm2 @ J4 @ I2 )
             => ( ( ( R4 @ I3 )
                 != ( R4 @ J4 ) )
               => ( ( inf_inf_set_nat @ ( domain_nat_nat @ ( R4 @ I3 ) ) @ ( range_nat_nat @ ( R4 @ J4 ) ) )
                  = bot_bot_set_nat ) ) ) )
       => ( wf_nat @ ( comple5685304695842803022at_nat @ ( image_576625962483964339at_nat @ R4 @ I2 ) ) ) ) ) ).

% wf_UN
thf(fact_1256_wf__UN,axiom,
    ! [I2: set_list_fm,R4: list_fm > set_Pr1261947904930325089at_nat] :
      ( ! [I3: list_fm] :
          ( ( member_list_fm2 @ I3 @ I2 )
         => ( wf_nat @ ( R4 @ I3 ) ) )
     => ( ! [I3: list_fm,J4: list_fm] :
            ( ( member_list_fm2 @ I3 @ I2 )
           => ( ( member_list_fm2 @ J4 @ I2 )
             => ( ( ( R4 @ I3 )
                 != ( R4 @ J4 ) )
               => ( ( inf_inf_set_nat @ ( domain_nat_nat @ ( R4 @ I3 ) ) @ ( range_nat_nat @ ( R4 @ J4 ) ) )
                  = bot_bot_set_nat ) ) ) )
       => ( wf_nat @ ( comple5685304695842803022at_nat @ ( image_5703941223934821165at_nat @ R4 @ I2 ) ) ) ) ) ).

% wf_UN
thf(fact_1257_wf__UN,axiom,
    ! [I2: set_Pr1822751329126368876m_rule,R4: produc340336539035504054m_rule > set_Pr1261947904930325089at_nat] :
      ( ! [I3: produc340336539035504054m_rule] :
          ( ( member7231649785386036813m_rule @ I3 @ I2 )
         => ( wf_nat @ ( R4 @ I3 ) ) )
     => ( ! [I3: produc340336539035504054m_rule,J4: produc340336539035504054m_rule] :
            ( ( member7231649785386036813m_rule @ I3 @ I2 )
           => ( ( member7231649785386036813m_rule @ J4 @ I2 )
             => ( ( ( R4 @ I3 )
                 != ( R4 @ J4 ) )
               => ( ( inf_inf_set_nat @ ( domain_nat_nat @ ( R4 @ I3 ) ) @ ( range_nat_nat @ ( R4 @ J4 ) ) )
                  = bot_bot_set_nat ) ) ) )
       => ( wf_nat @ ( comple5685304695842803022at_nat @ ( image_216278812042616420at_nat @ R4 @ I2 ) ) ) ) ) ).

% wf_UN
thf(fact_1258_wf__UN,axiom,
    ! [I2: set_o,R4: $o > set_Pr1261947904930325089at_nat] :
      ( ! [I3: $o] :
          ( ( member_o2 @ I3 @ I2 )
         => ( wf_nat @ ( R4 @ I3 ) ) )
     => ( ! [I3: $o,J4: $o] :
            ( ( member_o2 @ I3 @ I2 )
           => ( ( member_o2 @ J4 @ I2 )
             => ( ( ( R4 @ I3 )
                 != ( R4 @ J4 ) )
               => ( ( inf_inf_set_nat @ ( domain_nat_nat @ ( R4 @ I3 ) ) @ ( range_nat_nat @ ( R4 @ J4 ) ) )
                  = bot_bot_set_nat ) ) ) )
       => ( wf_nat @ ( comple5685304695842803022at_nat @ ( image_7077428525161401678at_nat @ R4 @ I2 ) ) ) ) ) ).

% wf_UN
thf(fact_1259_wf__UN,axiom,
    ! [I2: set_fm,R4: fm > set_Product_prod_o_o] :
      ( ! [I3: fm] :
          ( ( member_fm2 @ I3 @ I2 )
         => ( wf_o @ ( R4 @ I3 ) ) )
     => ( ! [I3: fm,J4: fm] :
            ( ( member_fm2 @ I3 @ I2 )
           => ( ( member_fm2 @ J4 @ I2 )
             => ( ( ( R4 @ I3 )
                 != ( R4 @ J4 ) )
               => ( ( inf_inf_set_o @ ( domain_o_o @ ( R4 @ I3 ) ) @ ( range_o_o @ ( R4 @ J4 ) ) )
                  = bot_bot_set_o ) ) ) )
       => ( wf_o @ ( comple2567785453016644468od_o_o @ ( image_4023287440801267289od_o_o @ R4 @ I2 ) ) ) ) ) ).

% wf_UN
thf(fact_1260_wf__UN,axiom,
    ! [I2: set_list_fm,R4: list_fm > set_Product_prod_o_o] :
      ( ! [I3: list_fm] :
          ( ( member_list_fm2 @ I3 @ I2 )
         => ( wf_o @ ( R4 @ I3 ) ) )
     => ( ! [I3: list_fm,J4: list_fm] :
            ( ( member_list_fm2 @ I3 @ I2 )
           => ( ( member_list_fm2 @ J4 @ I2 )
             => ( ( ( R4 @ I3 )
                 != ( R4 @ J4 ) )
               => ( ( inf_inf_set_o @ ( domain_o_o @ ( R4 @ I3 ) ) @ ( range_o_o @ ( R4 @ J4 ) ) )
                  = bot_bot_set_o ) ) ) )
       => ( wf_o @ ( comple2567785453016644468od_o_o @ ( image_2640289744062131411od_o_o @ R4 @ I2 ) ) ) ) ) ).

% wf_UN
thf(fact_1261_wf__UN,axiom,
    ! [I2: set_Pr1822751329126368876m_rule,R4: produc340336539035504054m_rule > set_Product_prod_o_o] :
      ( ! [I3: produc340336539035504054m_rule] :
          ( ( member7231649785386036813m_rule @ I3 @ I2 )
         => ( wf_o @ ( R4 @ I3 ) ) )
     => ( ! [I3: produc340336539035504054m_rule,J4: produc340336539035504054m_rule] :
            ( ( member7231649785386036813m_rule @ I3 @ I2 )
           => ( ( member7231649785386036813m_rule @ J4 @ I2 )
             => ( ( ( R4 @ I3 )
                 != ( R4 @ J4 ) )
               => ( ( inf_inf_set_o @ ( domain_o_o @ ( R4 @ I3 ) ) @ ( range_o_o @ ( R4 @ J4 ) ) )
                  = bot_bot_set_o ) ) ) )
       => ( wf_o @ ( comple2567785453016644468od_o_o @ ( image_4527242629906780042od_o_o @ R4 @ I2 ) ) ) ) ) ).

% wf_UN
thf(fact_1262_wf__UN,axiom,
    ! [I2: set_o,R4: $o > set_Product_prod_o_o] :
      ( ! [I3: $o] :
          ( ( member_o2 @ I3 @ I2 )
         => ( wf_o @ ( R4 @ I3 ) ) )
     => ( ! [I3: $o,J4: $o] :
            ( ( member_o2 @ I3 @ I2 )
           => ( ( member_o2 @ J4 @ I2 )
             => ( ( ( R4 @ I3 )
                 != ( R4 @ J4 ) )
               => ( ( inf_inf_set_o @ ( domain_o_o @ ( R4 @ I3 ) ) @ ( range_o_o @ ( R4 @ J4 ) ) )
                  = bot_bot_set_o ) ) ) )
       => ( wf_o @ ( comple2567785453016644468od_o_o @ ( image_5520585345982086004od_o_o @ R4 @ I2 ) ) ) ) ) ).

% wf_UN
thf(fact_1263_Preorder__Restr,axiom,
    ! [R4: set_Pr1261947904930325089at_nat,A5: set_nat] :
      ( ( order_4861654808422542329on_nat @ ( field_nat @ R4 ) @ R4 )
     => ( order_4861654808422542329on_nat
        @ ( field_nat
          @ ( inf_in2572325071724192079at_nat @ R4
            @ ( produc457027306803732586at_nat @ A5
              @ ^ [Uu: nat] : A5 ) ) )
        @ ( inf_in2572325071724192079at_nat @ R4
          @ ( produc457027306803732586at_nat @ A5
            @ ^ [Uu: nat] : A5 ) ) ) ) ).

% Preorder_Restr
thf(fact_1264_equiv__class__nondisjoint,axiom,
    ! [A5: set_fm,R4: set_Pr4706815898642364871_fm_fm,X5: fm,A: fm,B: fm] :
      ( ( equiv_equiv_fm @ A5 @ R4 )
     => ( ( member_fm2 @ X5 @ ( inf_inf_set_fm @ ( image_fm_fm @ R4 @ ( insert_fm2 @ A @ bot_bot_set_fm ) ) @ ( image_fm_fm @ R4 @ ( insert_fm2 @ B @ bot_bot_set_fm ) ) ) )
       => ( member7780952600467998736_fm_fm @ ( product_Pair_fm_fm @ A @ B ) @ R4 ) ) ) ).

% equiv_class_nondisjoint
thf(fact_1265_equiv__class__nondisjoint,axiom,
    ! [A5: set_list_fm,R4: set_Pr1916137733696349511ist_fm,X5: list_fm,A: list_fm,B: list_fm] :
      ( ( equiv_equiv_list_fm @ A5 @ R4 )
     => ( ( member_list_fm2 @ X5 @ ( inf_inf_set_list_fm @ ( image_2708806783351283987ist_fm @ R4 @ ( insert_list_fm2 @ A @ bot_bot_set_list_fm ) ) @ ( image_2708806783351283987ist_fm @ R4 @ ( insert_list_fm2 @ B @ bot_bot_set_list_fm ) ) ) )
       => ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ A @ B ) @ R4 ) ) ) ).

% equiv_class_nondisjoint
thf(fact_1266_equiv__class__nondisjoint,axiom,
    ! [A5: set_Pr1822751329126368876m_rule,R4: set_Pr4971326047967503661m_rule,X5: produc340336539035504054m_rule,A: produc340336539035504054m_rule,B: produc340336539035504054m_rule] :
      ( ( equiv_6275699240365100160m_rule @ A5 @ R4 )
     => ( ( member7231649785386036813m_rule @ X5 @ ( inf_in2520071923633375486m_rule @ ( image_7630142287376862273m_rule @ R4 @ ( insert775845832097999846m_rule @ A @ bot_bo3426410157013319936m_rule ) ) @ ( image_7630142287376862273m_rule @ R4 @ ( insert775845832097999846m_rule @ B @ bot_bo3426410157013319936m_rule ) ) ) )
       => ( member1411645983481675894m_rule @ ( produc6261311607089640965m_rule @ A @ B ) @ R4 ) ) ) ).

% equiv_class_nondisjoint
thf(fact_1267_equiv__class__nondisjoint,axiom,
    ! [A5: set_nat,R4: set_Pr1261947904930325089at_nat,X5: nat,A: nat,B: nat] :
      ( ( equiv_equiv_nat @ A5 @ R4 )
     => ( ( member_nat @ X5 @ ( inf_inf_set_nat @ ( image_nat_nat @ R4 @ ( insert_nat @ A @ bot_bot_set_nat ) ) @ ( image_nat_nat @ R4 @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) )
       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R4 ) ) ) ).

% equiv_class_nondisjoint
thf(fact_1268_equiv__class__nondisjoint,axiom,
    ! [A5: set_o,R4: set_Product_prod_o_o,X5: $o,A: $o,B: $o] :
      ( ( equiv_equiv_o @ A5 @ R4 )
     => ( ( member_o2 @ X5 @ ( inf_inf_set_o @ ( image_o_o @ R4 @ ( insert_o2 @ A @ bot_bot_set_o ) ) @ ( image_o_o @ R4 @ ( insert_o2 @ B @ bot_bot_set_o ) ) ) )
       => ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ B ) @ R4 ) ) ) ).

% equiv_class_nondisjoint
thf(fact_1269_ccINF__insert,axiom,
    ! [A5: set_o,F: $o > $o,A: $o] :
      ( ( counta5976203206615340371able_o @ A5 )
     => ( ( complete_Inf_Inf_o @ ( image_o_o2 @ F @ ( insert_o2 @ A @ A5 ) ) )
        = ( inf_inf_o @ ( F @ A ) @ ( complete_Inf_Inf_o @ ( image_o_o2 @ F @ A5 ) ) ) ) ) ).

% ccINF_insert
thf(fact_1270_wf__less,axiom,
    wf_nat @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ ord_less_nat ) ) ).

% wf_less
thf(fact_1271_Inf__bool__def,axiom,
    ( complete_Inf_Inf_o
    = ( ^ [A7: set_o] :
          ~ ( member_o2 @ $false @ A7 ) ) ) ).

% Inf_bool_def
thf(fact_1272_parts__in__children,axiom,
    ! [P: fm,Z: list_fm,Z3: list_fm,A5: list_tm,R4: rule] :
      ( ( member_fm2 @ P @ ( set_fm2 @ Z ) )
     => ( ( member_list_fm2 @ Z3 @ ( set_list_fm2 @ ( children @ A5 @ R4 @ Z ) ) )
       => ? [B8: list_tm,Xs3: list_fm] :
            ( ( ord_less_eq_set_tm @ ( set_tm2 @ A5 ) @ ( set_tm2 @ B8 ) )
            & ( member_list_fm2 @ Xs3 @ ( set_list_fm2 @ ( parts @ B8 @ R4 @ P ) ) )
            & ( ord_less_eq_set_fm @ ( set_fm2 @ Xs3 ) @ ( set_fm2 @ Z3 ) ) ) ) ) ).

% parts_in_children
thf(fact_1273_nat__descend__induct,axiom,
    ! [N3: nat,P3: nat > $o,M3: nat] :
      ( ! [K2: nat] :
          ( ( ord_less_nat @ N3 @ K2 )
         => ( P3 @ K2 ) )
     => ( ! [K2: nat] :
            ( ( ord_less_eq_nat @ K2 @ N3 )
           => ( ! [I5: nat] :
                  ( ( ord_less_nat @ K2 @ I5 )
                 => ( P3 @ I5 ) )
             => ( P3 @ K2 ) ) )
       => ( P3 @ M3 ) ) ) ).

% nat_descend_induct
thf(fact_1274_UNIV__bool,axiom,
    ( top_top_set_o
    = ( insert_o2 @ $false @ ( insert_o2 @ $true @ bot_bot_set_o ) ) ) ).

% UNIV_bool
thf(fact_1275_Field__natLeq__on,axiom,
    ! [N3: nat] :
      ( ( field_nat
        @ ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ^ [X: nat,Y: nat] :
                ( ( ord_less_nat @ X @ N3 )
                & ( ord_less_nat @ Y @ N3 )
                & ( ord_less_eq_nat @ X @ Y ) ) ) ) )
      = ( collect_nat
        @ ^ [X: nat] : ( ord_less_nat @ X @ N3 ) ) ) ).

% Field_natLeq_on
thf(fact_1276_natLess__def,axiom,
    ( bNF_Ca8459412986667044542atLess
    = ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ ord_less_nat ) ) ) ).

% natLess_def

% Helper facts (3)
thf(help_If_3_1_If_001t__Set__Oset_It__SeCaV__Ofm_J_T,axiom,
    ! [P3: $o] :
      ( ( P3 = $true )
      | ( P3 = $false ) ) ).

thf(help_If_2_1_If_001t__Set__Oset_It__SeCaV__Ofm_J_T,axiom,
    ! [X5: set_fm,Y4: set_fm] :
      ( ( if_set_fm @ $false @ X5 @ Y4 )
      = Y4 ) ).

thf(help_If_1_1_If_001t__Set__Oset_It__SeCaV__Ofm_J_T,axiom,
    ! [X5: set_fm,Y4: set_fm] :
      ( ( if_set_fm @ $true @ X5 @ Y4 )
      = X5 ) ).

% Conjectures (1)
thf(conj_0,conjecture,
    ( ( produc3561889649859641891e_fm_o
      @ ^ [X: rule,Xa: fm] :
          ( case_rule_o
          @ ( case_fm_o
            @ ^ [Nat: nat,List: list_tm] : $false
            @ ^ [Fm1: fm,Fm2: fm] : $false
            @ ^ [Y: fm,Xb: fm] : $true
            @ ^ [Fm1: fm,Fm2: fm] : $false
            @ ^ [Fm: fm] : $false
            @ ^ [Fm: fm] : $false
            @ ^ [Fm: fm] : $false
            @ Xa )
          @ ( case_fm_o
            @ ^ [Nat: nat,List: list_tm] : $false
            @ ^ [Y: fm,Xb: fm] : $true
            @ ^ [Fm1: fm,Fm2: fm] : $false
            @ ^ [Fm1: fm,Fm2: fm] : $false
            @ ^ [Fm: fm] : $false
            @ ^ [Fm: fm] : $false
            @ ^ [Fm: fm] : $false
            @ Xa )
          @ ( case_fm_o
            @ ^ [Nat: nat,List: list_tm] : $false
            @ ^ [Fm1: fm,Fm2: fm] : $false
            @ ^ [Fm1: fm,Fm2: fm] : $false
            @ ^ [Fm1: fm,Fm2: fm] : $false
            @ ^ [Fm: fm] : $false
            @ ^ [Fm: fm] : $false
            @ ( case_fm_o
              @ ^ [Nat: nat,List: list_tm] : $false
              @ ^ [Fm1: fm,Fm2: fm] : $false
              @ ^ [Fm1: fm,Fm2: fm] : $false
              @ ^ [Y: fm,Xb: fm] : $true
              @ ^ [Fma: fm] : $false
              @ ^ [Fma: fm] : $false
              @ ^ [Fma: fm] : $false )
            @ Xa )
          @ ( case_fm_o
            @ ^ [Nat: nat,List: list_tm] : $false
            @ ^ [Fm1: fm,Fm2: fm] : $false
            @ ^ [Fm1: fm,Fm2: fm] : $false
            @ ^ [Y: fm,Xb: fm] : $true
            @ ^ [Fm: fm] : $false
            @ ^ [Fm: fm] : $false
            @ ^ [Fm: fm] : $false
            @ Xa )
          @ ( case_fm_o
            @ ^ [Nat: nat,List: list_tm] : $false
            @ ^ [Fm1: fm,Fm2: fm] : $false
            @ ^ [Fm1: fm,Fm2: fm] : $false
            @ ^ [Fm1: fm,Fm2: fm] : $false
            @ ^ [Fm: fm] : $false
            @ ^ [Fm: fm] : $false
            @ ( case_fm_o
              @ ^ [Nat: nat,List: list_tm] : $false
              @ ^ [Y: fm,Xb: fm] : $true
              @ ^ [Fm1: fm,Fm2: fm] : $false
              @ ^ [Fm1: fm,Fm2: fm] : $false
              @ ^ [Fma: fm] : $false
              @ ^ [Fma: fm] : $false
              @ ^ [Fma: fm] : $false )
            @ Xa )
          @ ( case_fm_o
            @ ^ [Nat: nat,List: list_tm] : $false
            @ ^ [Fm1: fm,Fm2: fm] : $false
            @ ^ [Fm1: fm,Fm2: fm] : $false
            @ ^ [Fm1: fm,Fm2: fm] : $false
            @ ^ [Fm: fm] : $false
            @ ^ [Fm: fm] : $false
            @ ( case_fm_o
              @ ^ [Nat: nat,List: list_tm] : $false
              @ ^ [Fm1: fm,Fm2: fm] : $false
              @ ^ [Y: fm,Xb: fm] : $true
              @ ^ [Fm1: fm,Fm2: fm] : $false
              @ ^ [Fma: fm] : $false
              @ ^ [Fma: fm] : $false
              @ ^ [Fma: fm] : $false )
            @ Xa )
          @ ( case_fm_o
            @ ^ [Nat: nat,List: list_tm] : $false
            @ ^ [Fm1: fm,Fm2: fm] : $false
            @ ^ [Fm1: fm,Fm2: fm] : $false
            @ ^ [Fm1: fm,Fm2: fm] : $false
            @ ^ [Fm: fm] : $false
            @ ^ [Y: fm] : $true
            @ ^ [Fm: fm] : $false
            @ Xa )
          @ ( case_fm_o
            @ ^ [Nat: nat,List: list_tm] : $false
            @ ^ [Fm1: fm,Fm2: fm] : $false
            @ ^ [Fm1: fm,Fm2: fm] : $false
            @ ^ [Fm1: fm,Fm2: fm] : $false
            @ ^ [Fm: fm] : $false
            @ ^ [Fm: fm] : $false
            @ ( case_fm_o
              @ ^ [Nat: nat,List: list_tm] : $false
              @ ^ [Fm1: fm,Fm2: fm] : $false
              @ ^ [Fm1: fm,Fm2: fm] : $false
              @ ^ [Fm1: fm,Fm2: fm] : $false
              @ ^ [Y: fm] : $true
              @ ^ [Fma: fm] : $false
              @ ^ [Fma: fm] : $false )
            @ Xa )
          @ ( case_fm_o
            @ ^ [Nat: nat,List: list_tm] : $false
            @ ^ [Fm1: fm,Fm2: fm] : $false
            @ ^ [Fm1: fm,Fm2: fm] : $false
            @ ^ [Fm1: fm,Fm2: fm] : $false
            @ ^ [Fm: fm] : $false
            @ ^ [Fm: fm] : $false
            @ ( case_fm_o
              @ ^ [Nat: nat,List: list_tm] : $false
              @ ^ [Fm1: fm,Fm2: fm] : $false
              @ ^ [Fm1: fm,Fm2: fm] : $false
              @ ^ [Fm1: fm,Fm2: fm] : $false
              @ ^ [Fma: fm] : $false
              @ ^ [Fma: fm] : $false
              @ ^ [Y: fm] : $true )
            @ Xa )
          @ ( case_fm_o
            @ ^ [Nat: nat,List: list_tm] : $false
            @ ^ [Fm1: fm,Fm2: fm] : $false
            @ ^ [Fm1: fm,Fm2: fm] : $false
            @ ^ [Fm1: fm,Fm2: fm] : $false
            @ ^ [Y: fm] : $false
            @ ^ [Fm: fm] : $false
            @ ^ [Fm: fm] : $false
            @ Xa )
          @ ( case_fm_o
            @ ^ [Nat: nat,List: list_tm] : $false
            @ ^ [Fm1: fm,Fm2: fm] : $false
            @ ^ [Fm1: fm,Fm2: fm] : $false
            @ ^ [Fm1: fm,Fm2: fm] : $false
            @ ^ [Fm: fm] : $false
            @ ^ [Fm: fm] : $false
            @ ( case_fm_o
              @ ^ [Nat: nat,List: list_tm] : $false
              @ ^ [Fm1: fm,Fm2: fm] : $false
              @ ^ [Fm1: fm,Fm2: fm] : $false
              @ ^ [Fm1: fm,Fm2: fm] : $false
              @ ^ [Fma: fm] : $false
              @ ^ [Y: fm] : $false
              @ ^ [Fma: fm] : $false )
            @ Xa )
          @ X )
      @ ( product_Pair_rule_fm @ r @ ( con @ p @ q ) ) )
   != ( r != betaCon ) ) ).

%------------------------------------------------------------------------------