%------------------------------------------------------------------------------
% File     : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain   : Archive of Formal Proofs
% Problem  :
% Version  : Especial.
% English  :

% Refs     : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source   : [Des23]
% Names    : FOL_Seq_Calc3/0011_Completeness/prob_00100_003315__12090900_1 [Des23]

% Status   : Theorem
% Rating   : ? v8.2.0
% Syntax   : Number of formulae    : 2452 ( 882 unt;1165 typ;   0 def)
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%            Maximal formula atoms :   11 (   2 avg)
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%            Maximal formula depth :   21 (   5 avg)
%            Number of types       :  136 ( 135 usr)
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% SPC      : TH0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            2023-01-19 15:28:04.278
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    bNF_cs8045896581061604419ist_fm: set_Pr7058068377845519745ist_fm > ( produc1511181096255975734ist_fm > produc1996495991257130529ist_fm ) > ( produc225247741881318192ist_fm > produc1996495991257130529ist_fm ) > ( produc1996495991257130529ist_fm > produc1511181096255975734ist_fm ) > ( produc1996495991257130529ist_fm > produc225247741881318192ist_fm ) > $o ).

thf(sy_c_BNF__Def_Ocsquare_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_J_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Syntax__Orule_J_001t__Syntax__Orule_001t__Product____Type__Oprod_It__Syntax__Orule_Mt__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_J,type,
    bNF_cs5859241137876923659ist_fm: set_Pr5209464977148653383ist_fm > ( produc164195504107695125m_rule > rule ) > ( produc4630866025709511057ist_fm > rule ) > ( produc1754732471204517991ist_fm > produc164195504107695125m_rule ) > ( produc1754732471204517991ist_fm > produc4630866025709511057ist_fm ) > $o ).

thf(sy_c_BNF__Def_Ocsquare_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Syntax__Orule_J_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Syntax__Orule_J_001t__Syntax__Orule_001t__Product____Type__Oprod_It__Syntax__Orule_Mt__Syntax__Orule_J,type,
    bNF_cs1207223144858359663e_rule: set_Pr1008144964186165195m_rule > ( produc164195504107695125m_rule > rule ) > ( produc780299800278082923e_rule > rule ) > ( produc164195504107695125m_rule > produc164195504107695125m_rule ) > ( produc164195504107695125m_rule > produc780299800278082923e_rule ) > $o ).

thf(sy_c_BNF__Def_Ocsquare_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Syntax__Orule_J_Mt__Syntax__Ofm_J_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Syntax__Orule_J_Mt__Syntax__Ofm_J_001t__Syntax__Ofm_001t__Product____Type__Oprod_It__Syntax__Ofm_Mt__Syntax__Ofm_J,type,
    bNF_cs5705172639096540902_fm_fm: set_Pr8819451114087901350ule_fm > ( produc7564083655194549958ule_fm > fm ) > ( product_prod_fm_fm > fm ) > ( produc7564083655194549958ule_fm > produc7564083655194549958ule_fm ) > ( produc7564083655194549958ule_fm > product_prod_fm_fm ) > $o ).

thf(sy_c_BNF__Def_Ocsquare_001t__Product____Type__Oprod_It__Syntax__Ofm_Mt__Syntax__Ofm_J_001t__Product____Type__Oprod_It__Syntax__Ofm_Mt__Syntax__Ofm_J_001t__Syntax__Ofm_001t__Product____Type__Oprod_It__Syntax__Ofm_Mt__Syntax__Ofm_J,type,
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thf(sy_c_BNF__Def_Ocsquare_001t__Product____Type__Oprod_It__Syntax__Orule_Mt__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_J_001t__Product____Type__Oprod_It__Syntax__Orule_Mt__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_J_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_J,type,
    bNF_cs8058160078142677555ist_fm: set_Pr2241365836748572487ist_fm > ( produc4630866025709511057ist_fm > produc1996495991257130529ist_fm ) > ( produc1754732471204517991ist_fm > produc1996495991257130529ist_fm ) > ( produc4630866025709511057ist_fm > produc4630866025709511057ist_fm ) > ( produc4630866025709511057ist_fm > produc1754732471204517991ist_fm ) > $o ).

thf(sy_c_BNF__Def_Ocsquare_001t__Product____Type__Oprod_It__Syntax__Orule_Mt__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_J_001t__Product____Type__Oprod_It__Syntax__Orule_Mt__Syntax__Orule_J_001t__Syntax__Orule_001t__Product____Type__Oprod_It__Syntax__Orule_Mt__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_J,type,
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thf(sy_c_BNF__Def_Ocsquare_001t__Product____Type__Oprod_It__Syntax__Orule_Mt__Syntax__Orule_J_001t__Product____Type__Oprod_It__Syntax__Orule_Mt__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_J_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Syntax__Orule_J,type,
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thf(sy_c_BNF__Def_Ocsquare_001t__Product____Type__Oprod_It__Syntax__Orule_Mt__Syntax__Orule_J_001t__Product____Type__Oprod_It__Syntax__Orule_Mt__Syntax__Orule_J_001t__Syntax__Orule_001t__Product____Type__Oprod_It__Syntax__Orule_Mt__Syntax__Orule_J,type,
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    bNF_pi8146509907818838_fm_fm: ( fm > fm > $o ) > ( fm > fm > $o ) > fm > fm > fm ).

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    bNF_pi3835482382339038057ist_fm: ( rule > produc1996495991257130529ist_fm > $o ) > ( produc1996495991257130529ist_fm > produc1996495991257130529ist_fm > $o ) > rule > produc1996495991257130529ist_fm > produc1996495991257130529ist_fm ).

thf(sy_c_BNF__Def_Opick__middlep_001t__Syntax__Orule_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_001t__Syntax__Orule,type,
    bNF_pi7614133673861551101m_rule: ( rule > produc1996495991257130529ist_fm > $o ) > ( produc1996495991257130529ist_fm > rule > $o ) > rule > rule > produc1996495991257130529ist_fm ).

thf(sy_c_BNF__Def_Opick__middlep_001t__Syntax__Orule_001t__Syntax__Orule_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J,type,
    bNF_pi4963049444341131953ist_fm: ( rule > rule > $o ) > ( rule > produc1996495991257130529ist_fm > $o ) > rule > produc1996495991257130529ist_fm > rule ).

thf(sy_c_BNF__Def_Opick__middlep_001t__Syntax__Orule_001t__Syntax__Orule_001t__Syntax__Orule,type,
    bNF_pi5293732350936020661e_rule: ( rule > rule > $o ) > ( rule > rule > $o ) > rule > rule > rule ).

thf(sy_c_BNF__Def_Orel__fun_001_062_It__List__Olist_It__Syntax__Ofm_J_M_Eo_J_001_062_It__List__Olist_It__Syntax__Ofm_J_M_Eo_J_001_062_I_062_It__List__Olist_It__Syntax__Ofm_J_M_Eo_J_M_062_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_M_Eo_J_J_001_062_I_062_It__List__Olist_It__Syntax__Ofm_J_M_Eo_J_M_062_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_M_Eo_J_J,type,
    bNF_re3600363984329575581t_fm_o: ( ( list_fm > $o ) > ( list_fm > $o ) > $o ) > ( ( ( list_fm > $o ) > produc1996495991257130529ist_fm > $o ) > ( ( list_fm > $o ) > produc1996495991257130529ist_fm > $o ) > $o ) > ( ( list_fm > $o ) > ( list_fm > $o ) > produc1996495991257130529ist_fm > $o ) > ( ( list_fm > $o ) > ( list_fm > $o ) > produc1996495991257130529ist_fm > $o ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001_062_It__List__Olist_It__Syntax__Ofm_J_M_Eo_J_001_062_It__List__Olist_It__Syntax__Ofm_J_M_Eo_J_001_062_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_M_Eo_J_001_062_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_M_Eo_J,type,
    bNF_re6700684614245656919t_fm_o: ( ( list_fm > $o ) > ( list_fm > $o ) > $o ) > ( ( produc1996495991257130529ist_fm > $o ) > ( produc1996495991257130529ist_fm > $o ) > $o ) > ( ( list_fm > $o ) > produc1996495991257130529ist_fm > $o ) > ( ( list_fm > $o ) > produc1996495991257130529ist_fm > $o ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001_062_It__List__Olist_It__Syntax__Ofm_J_M_Eo_J_001_062_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_M_Eo_J_001_062_I_062_It__List__Olist_It__Syntax__Ofm_J_M_Eo_J_M_062_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_M_Eo_J_J_001_062_I_062_It__Syntax__Orule_M_Eo_J_M_062_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Syntax__Orule_J_M_Eo_J_J,type,
    bNF_re6390700563477671843rule_o: ( ( list_fm > $o ) > ( produc1996495991257130529ist_fm > $o ) > $o ) > ( ( ( list_fm > $o ) > produc1996495991257130529ist_fm > $o ) > ( ( rule > $o ) > produc164195504107695125m_rule > $o ) > $o ) > ( ( list_fm > $o ) > ( list_fm > $o ) > produc1996495991257130529ist_fm > $o ) > ( ( produc1996495991257130529ist_fm > $o ) > ( rule > $o ) > produc164195504107695125m_rule > $o ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001_062_It__List__Olist_It__Syntax__Ofm_J_M_Eo_J_001_062_It__Syntax__Orule_M_Eo_J_001_062_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_M_Eo_J_001_062_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Syntax__Orule_J_M_Eo_J,type,
    bNF_re8866850227915748580rule_o: ( ( list_fm > $o ) > ( rule > $o ) > $o ) > ( ( produc1996495991257130529ist_fm > $o ) > ( produc164195504107695125m_rule > $o ) > $o ) > ( ( list_fm > $o ) > produc1996495991257130529ist_fm > $o ) > ( ( rule > $o ) > produc164195504107695125m_rule > $o ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001_062_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_M_Eo_J_001_062_It__List__Olist_It__Syntax__Ofm_J_M_Eo_J_001_062_I_062_It__Syntax__Orule_M_Eo_J_M_062_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Syntax__Orule_J_M_Eo_J_J_001_062_I_062_It__List__Olist_It__Syntax__Ofm_J_M_Eo_J_M_062_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_M_Eo_J_J,type,
    bNF_re7217375449204646129t_fm_o: ( ( produc1996495991257130529ist_fm > $o ) > ( list_fm > $o ) > $o ) > ( ( ( rule > $o ) > produc164195504107695125m_rule > $o ) > ( ( list_fm > $o ) > produc1996495991257130529ist_fm > $o ) > $o ) > ( ( produc1996495991257130529ist_fm > $o ) > ( rule > $o ) > produc164195504107695125m_rule > $o ) > ( ( list_fm > $o ) > ( list_fm > $o ) > produc1996495991257130529ist_fm > $o ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001_062_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_M_Eo_J_001_062_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_M_Eo_J_001_062_I_062_It__Syntax__Orule_M_Eo_J_M_062_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Syntax__Orule_J_M_Eo_J_J_001_062_I_062_It__Syntax__Orule_M_Eo_J_M_062_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Syntax__Orule_J_M_Eo_J_J,type,
    bNF_re2806322975267763907rule_o: ( ( produc1996495991257130529ist_fm > $o ) > ( produc1996495991257130529ist_fm > $o ) > $o ) > ( ( ( rule > $o ) > produc164195504107695125m_rule > $o ) > ( ( rule > $o ) > produc164195504107695125m_rule > $o ) > $o ) > ( ( produc1996495991257130529ist_fm > $o ) > ( rule > $o ) > produc164195504107695125m_rule > $o ) > ( ( produc1996495991257130529ist_fm > $o ) > ( rule > $o ) > produc164195504107695125m_rule > $o ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001_062_It__Syntax__Orule_M_Eo_J_001_062_It__List__Olist_It__Syntax__Ofm_J_M_Eo_J_001_062_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Syntax__Orule_J_M_Eo_J_001_062_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_M_Eo_J,type,
    bNF_re7886380955999696916t_fm_o: ( ( rule > $o ) > ( list_fm > $o ) > $o ) > ( ( produc164195504107695125m_rule > $o ) > ( produc1996495991257130529ist_fm > $o ) > $o ) > ( ( rule > $o ) > produc164195504107695125m_rule > $o ) > ( ( list_fm > $o ) > produc1996495991257130529ist_fm > $o ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001_062_It__Syntax__Orule_M_Eo_J_001_062_It__Syntax__Orule_M_Eo_J_001_062_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Syntax__Orule_J_M_Eo_J_001_062_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Syntax__Orule_J_M_Eo_J,type,
    bNF_re5822392312811141597rule_o: ( ( rule > $o ) > ( rule > $o ) > $o ) > ( ( produc164195504107695125m_rule > $o ) > ( produc164195504107695125m_rule > $o ) > $o ) > ( ( rule > $o ) > produc164195504107695125m_rule > $o ) > ( ( rule > $o ) > produc164195504107695125m_rule > $o ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__List__Olist_It__Syntax__Ofm_J_001t__List__Olist_It__Syntax__Ofm_J_001_062_It__List__Olist_It__Syntax__Ofm_J_M_Eo_J_001_062_It__List__Olist_It__Syntax__Ofm_J_M_Eo_J,type,
    bNF_re6571772915969521119t_fm_o: ( list_fm > list_fm > $o ) > ( ( list_fm > $o ) > ( list_fm > $o ) > $o ) > ( list_fm > list_fm > $o ) > ( list_fm > list_fm > $o ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__List__Olist_It__Syntax__Ofm_J_001t__List__Olist_It__Syntax__Ofm_J_001_062_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_M_Eo_J_001_062_It__Syntax__Orule_M_Eo_J,type,
    bNF_re6226240235925790797rule_o: ( list_fm > list_fm > $o ) > ( ( produc1996495991257130529ist_fm > $o ) > ( rule > $o ) > $o ) > ( list_fm > produc1996495991257130529ist_fm > $o ) > ( list_fm > rule > $o ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__List__Olist_It__Syntax__Ofm_J_001t__List__Olist_It__Syntax__Ofm_J_001_062_It__Syntax__Orule_M_Eo_J_001_062_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_M_Eo_J,type,
    bNF_re8084688274390359943t_fm_o: ( list_fm > list_fm > $o ) > ( ( rule > $o ) > ( produc1996495991257130529ist_fm > $o ) > $o ) > ( list_fm > rule > $o ) > ( list_fm > produc1996495991257130529ist_fm > $o ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__List__Olist_It__Syntax__Ofm_J_001t__List__Olist_It__Syntax__Ofm_J_001_Eo_001_Eo,type,
    bNF_re2260024341499725215fm_o_o: ( list_fm > list_fm > $o ) > ( $o > $o > $o ) > ( list_fm > $o ) > ( list_fm > $o ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__List__Olist_It__Syntax__Ofm_J_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_001_Eo_001_Eo,type,
    bNF_re2963754933915346338fm_o_o: ( list_fm > produc1996495991257130529ist_fm > $o ) > ( $o > $o > $o ) > ( list_fm > $o ) > ( produc1996495991257130529ist_fm > $o ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__List__Olist_It__Syntax__Ofm_J_001t__Syntax__Orule_001_Eo_001_Eo,type,
    bNF_re2504994000271363972le_o_o: ( list_fm > rule > $o ) > ( $o > $o > $o ) > ( list_fm > $o ) > ( rule > $o ) > $o ).

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    bNF_re3089193160343016676fm_o_o: ( produc1996495991257130529ist_fm > list_fm > $o ) > ( $o > $o > $o ) > ( produc1996495991257130529ist_fm > $o ) > ( list_fm > $o ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_001_Eo_001_Eo,type,
    bNF_re188606575539848477fm_o_o: ( produc1996495991257130529ist_fm > produc1996495991257130529ist_fm > $o ) > ( $o > $o > $o ) > ( produc1996495991257130529ist_fm > $o ) > ( produc1996495991257130529ist_fm > $o ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_001t__List__Olist_It__Syntax__Ofm_J_001t__List__Olist_It__Syntax__Ofm_J,type,
    bNF_re4823320336450819679ist_fm: ( produc1996495991257130529ist_fm > produc1996495991257130529ist_fm > $o ) > ( list_fm > list_fm > $o ) > ( produc1996495991257130529ist_fm > list_fm ) > ( produc1996495991257130529ist_fm > list_fm ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Syntax__Orule_J_001_Eo_001_Eo,type,
    bNF_re356467991940469145le_o_o: ( produc1996495991257130529ist_fm > produc164195504107695125m_rule > $o ) > ( $o > $o > $o ) > ( produc1996495991257130529ist_fm > $o ) > ( produc164195504107695125m_rule > $o ) > $o ).

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    bNF_re2379315951688697920m_rule: ( produc1996495991257130529ist_fm > produc164195504107695125m_rule > $o ) > ( list_fm > rule > $o ) > ( produc1996495991257130529ist_fm > list_fm ) > ( produc164195504107695125m_rule > rule ) > $o ).

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thf(sy_c_member_001t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Syntax__Orule_J_J,type,
    member7979558945778333922m_rule: set_Pr1008144964186165195m_rule > set_se6866588098022362241m_rule > $o ).

thf(sy_c_member_001t__Set__Oset_It__Syntax__Ofm_J,type,
    member_set_fm: set_fm > set_set_fm > $o ).

thf(sy_c_member_001t__Stream__Ostream_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_J,type,
    member5006235884692811216ist_fm: stream8299795917829157543ist_fm > set_st6731251264560614279ist_fm > $o ).

thf(sy_c_member_001t__Stream__Ostream_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Syntax__Orule_J_J,type,
    member6117863459850561468m_rule: stream727092118206550309m_rule > set_st6921181922238103643m_rule > $o ).

thf(sy_c_member_001t__Stream__Ostream_It__Product____Type__Oprod_It__Syntax__Orule_Mt__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_J_J,type,
    member7351084332412968760ist_fm: stream1960312990768957601ist_fm > set_st1928155158715136471ist_fm > $o ).

thf(sy_c_member_001t__Syntax__Ofm,type,
    member_fm: fm > set_fm > $o ).

thf(sy_c_member_001t__Syntax__Orule,type,
    member_rule: rule > set_rule > $o ).

thf(sy_v_p,type,
    p: fm ).

thf(sy_v_steps,type,
    steps: stream727092118206550309m_rule ).

% Relevant facts (1276)
thf(fact_0_prod_Oexpand,axiom,
    ! [Prod: produc1754732471204517991ist_fm,Prod2: produc1754732471204517991ist_fm] :
      ( ( ( ( produc5894184038225292483ist_fm @ Prod )
          = ( produc5894184038225292483ist_fm @ Prod2 ) )
        & ( ( produc1639249265332130309ist_fm @ Prod )
          = ( produc1639249265332130309ist_fm @ Prod2 ) ) )
     => ( Prod = Prod2 ) ) ).

% prod.expand
thf(fact_1_prod_Oexpand,axiom,
    ! [Prod: produc780299800278082923e_rule,Prod2: produc780299800278082923e_rule] :
      ( ( ( ( produc7264528503596239631e_rule @ Prod )
          = ( produc7264528503596239631e_rule @ Prod2 ) )
        & ( ( produc6725306161561501521e_rule @ Prod )
          = ( produc6725306161561501521e_rule @ Prod2 ) ) )
     => ( Prod = Prod2 ) ) ).

% prod.expand
thf(fact_2_prod_Oexpand,axiom,
    ! [Prod: product_prod_fm_fm,Prod2: product_prod_fm_fm] :
      ( ( ( ( product_fst_fm_fm @ Prod )
          = ( product_fst_fm_fm @ Prod2 ) )
        & ( ( product_snd_fm_fm @ Prod )
          = ( product_snd_fm_fm @ Prod2 ) ) )
     => ( Prod = Prod2 ) ) ).

% prod.expand
thf(fact_3_prod_Oexpand,axiom,
    ! [Prod: produc7564083655194549958ule_fm,Prod2: produc7564083655194549958ule_fm] :
      ( ( ( ( produc1758149581174257386ule_fm @ Prod )
          = ( produc1758149581174257386ule_fm @ Prod2 ) )
        & ( ( produc8694862632519904556ule_fm @ Prod )
          = ( produc8694862632519904556ule_fm @ Prod2 ) ) )
     => ( Prod = Prod2 ) ) ).

% prod.expand
thf(fact_4_prod_Oexpand,axiom,
    ! [Prod: produc4630866025709511057ist_fm,Prod2: produc4630866025709511057ist_fm] :
      ( ( ( ( produc4228417144610596823ist_fm @ Prod )
          = ( produc4228417144610596823ist_fm @ Prod2 ) )
        & ( ( produc4514744107061996309ist_fm @ Prod )
          = ( produc4514744107061996309ist_fm @ Prod2 ) ) )
     => ( Prod = Prod2 ) ) ).

% prod.expand
thf(fact_5_prod_Oexpand,axiom,
    ! [Prod: produc164195504107695125m_rule,Prod2: produc164195504107695125m_rule] :
      ( ( ( ( produc6879501374131015971m_rule @ Prod )
          = ( produc6879501374131015971m_rule @ Prod2 ) )
        & ( ( produc7165828336582415457m_rule @ Prod )
          = ( produc7165828336582415457m_rule @ Prod2 ) ) )
     => ( Prod = Prod2 ) ) ).

% prod.expand
thf(fact_6_prod_Oexpand,axiom,
    ! [Prod: produc1996495991257130529ist_fm,Prod2: produc1996495991257130529ist_fm] :
      ( ( ( ( produc1501393135466168645ist_fm @ Prod )
          = ( produc1501393135466168645ist_fm @ Prod2 ) )
        & ( ( produc4588648349897876871ist_fm @ Prod )
          = ( produc4588648349897876871ist_fm @ Prod2 ) ) )
     => ( Prod = Prod2 ) ) ).

% prod.expand
thf(fact_7_prod__eqI,axiom,
    ! [P: produc1754732471204517991ist_fm,Q: produc1754732471204517991ist_fm] :
      ( ( ( produc5894184038225292483ist_fm @ P )
        = ( produc5894184038225292483ist_fm @ Q ) )
     => ( ( ( produc1639249265332130309ist_fm @ P )
          = ( produc1639249265332130309ist_fm @ Q ) )
       => ( P = Q ) ) ) ).

% prod_eqI
thf(fact_8_prod__eqI,axiom,
    ! [P: produc780299800278082923e_rule,Q: produc780299800278082923e_rule] :
      ( ( ( produc7264528503596239631e_rule @ P )
        = ( produc7264528503596239631e_rule @ Q ) )
     => ( ( ( produc6725306161561501521e_rule @ P )
          = ( produc6725306161561501521e_rule @ Q ) )
       => ( P = Q ) ) ) ).

% prod_eqI
thf(fact_9_prod__eqI,axiom,
    ! [P: product_prod_fm_fm,Q: product_prod_fm_fm] :
      ( ( ( product_fst_fm_fm @ P )
        = ( product_fst_fm_fm @ Q ) )
     => ( ( ( product_snd_fm_fm @ P )
          = ( product_snd_fm_fm @ Q ) )
       => ( P = Q ) ) ) ).

% prod_eqI
thf(fact_10_prod__eqI,axiom,
    ! [P: produc7564083655194549958ule_fm,Q: produc7564083655194549958ule_fm] :
      ( ( ( produc1758149581174257386ule_fm @ P )
        = ( produc1758149581174257386ule_fm @ Q ) )
     => ( ( ( produc8694862632519904556ule_fm @ P )
          = ( produc8694862632519904556ule_fm @ Q ) )
       => ( P = Q ) ) ) ).

% prod_eqI
thf(fact_11_prod__eqI,axiom,
    ! [P: produc4630866025709511057ist_fm,Q: produc4630866025709511057ist_fm] :
      ( ( ( produc4228417144610596823ist_fm @ P )
        = ( produc4228417144610596823ist_fm @ Q ) )
     => ( ( ( produc4514744107061996309ist_fm @ P )
          = ( produc4514744107061996309ist_fm @ Q ) )
       => ( P = Q ) ) ) ).

% prod_eqI
thf(fact_12_prod__eqI,axiom,
    ! [P: produc164195504107695125m_rule,Q: produc164195504107695125m_rule] :
      ( ( ( produc6879501374131015971m_rule @ P )
        = ( produc6879501374131015971m_rule @ Q ) )
     => ( ( ( produc7165828336582415457m_rule @ P )
          = ( produc7165828336582415457m_rule @ Q ) )
       => ( P = Q ) ) ) ).

% prod_eqI
thf(fact_13_prod__eqI,axiom,
    ! [P: produc1996495991257130529ist_fm,Q: produc1996495991257130529ist_fm] :
      ( ( ( produc1501393135466168645ist_fm @ P )
        = ( produc1501393135466168645ist_fm @ Q ) )
     => ( ( ( produc4588648349897876871ist_fm @ P )
          = ( produc4588648349897876871ist_fm @ Q ) )
       => ( P = Q ) ) ) ).

% prod_eqI
thf(fact_14_exE__realizer_H,axiom,
    ! [P2: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm > $o,P: produc1754732471204517991ist_fm] :
      ( ( P2 @ ( produc1639249265332130309ist_fm @ P ) @ ( produc5894184038225292483ist_fm @ P ) )
     => ~ ! [X: produc1996495991257130529ist_fm,Y: produc1996495991257130529ist_fm] :
            ~ ( P2 @ Y @ X ) ) ).

% exE_realizer'
thf(fact_15_exE__realizer_H,axiom,
    ! [P2: rule > rule > $o,P: produc780299800278082923e_rule] :
      ( ( P2 @ ( produc6725306161561501521e_rule @ P ) @ ( produc7264528503596239631e_rule @ P ) )
     => ~ ! [X: rule,Y: rule] :
            ~ ( P2 @ Y @ X ) ) ).

% exE_realizer'
thf(fact_16_exE__realizer_H,axiom,
    ! [P2: fm > fm > $o,P: product_prod_fm_fm] :
      ( ( P2 @ ( product_snd_fm_fm @ P ) @ ( product_fst_fm_fm @ P ) )
     => ~ ! [X: fm,Y: fm] :
            ~ ( P2 @ Y @ X ) ) ).

% exE_realizer'
thf(fact_17_exE__realizer_H,axiom,
    ! [P2: fm > produc164195504107695125m_rule > $o,P: produc7564083655194549958ule_fm] :
      ( ( P2 @ ( produc8694862632519904556ule_fm @ P ) @ ( produc1758149581174257386ule_fm @ P ) )
     => ~ ! [X: produc164195504107695125m_rule,Y: fm] :
            ~ ( P2 @ Y @ X ) ) ).

% exE_realizer'
thf(fact_18_exE__realizer_H,axiom,
    ! [P2: produc1996495991257130529ist_fm > rule > $o,P: produc4630866025709511057ist_fm] :
      ( ( P2 @ ( produc4514744107061996309ist_fm @ P ) @ ( produc4228417144610596823ist_fm @ P ) )
     => ~ ! [X: rule,Y: produc1996495991257130529ist_fm] :
            ~ ( P2 @ Y @ X ) ) ).

% exE_realizer'
thf(fact_19_exE__realizer_H,axiom,
    ! [P2: rule > produc1996495991257130529ist_fm > $o,P: produc164195504107695125m_rule] :
      ( ( P2 @ ( produc7165828336582415457m_rule @ P ) @ ( produc6879501374131015971m_rule @ P ) )
     => ~ ! [X: produc1996495991257130529ist_fm,Y: rule] :
            ~ ( P2 @ Y @ X ) ) ).

% exE_realizer'
thf(fact_20_exE__realizer_H,axiom,
    ! [P2: list_fm > list_fm > $o,P: produc1996495991257130529ist_fm] :
      ( ( P2 @ ( produc4588648349897876871ist_fm @ P ) @ ( produc1501393135466168645ist_fm @ P ) )
     => ~ ! [X: list_fm,Y: list_fm] :
            ~ ( P2 @ Y @ X ) ) ).

% exE_realizer'
thf(fact_21_prod__eq__iff,axiom,
    ( ( ^ [Y2: produc1754732471204517991ist_fm,Z: produc1754732471204517991ist_fm] : ( Y2 = Z ) )
    = ( ^ [S: produc1754732471204517991ist_fm,T: produc1754732471204517991ist_fm] :
          ( ( ( produc5894184038225292483ist_fm @ S )
            = ( produc5894184038225292483ist_fm @ T ) )
          & ( ( produc1639249265332130309ist_fm @ S )
            = ( produc1639249265332130309ist_fm @ T ) ) ) ) ) ).

% prod_eq_iff
thf(fact_22_prod__eq__iff,axiom,
    ( ( ^ [Y2: produc780299800278082923e_rule,Z: produc780299800278082923e_rule] : ( Y2 = Z ) )
    = ( ^ [S: produc780299800278082923e_rule,T: produc780299800278082923e_rule] :
          ( ( ( produc7264528503596239631e_rule @ S )
            = ( produc7264528503596239631e_rule @ T ) )
          & ( ( produc6725306161561501521e_rule @ S )
            = ( produc6725306161561501521e_rule @ T ) ) ) ) ) ).

% prod_eq_iff
thf(fact_23_prod__eq__iff,axiom,
    ( ( ^ [Y2: product_prod_fm_fm,Z: product_prod_fm_fm] : ( Y2 = Z ) )
    = ( ^ [S: product_prod_fm_fm,T: product_prod_fm_fm] :
          ( ( ( product_fst_fm_fm @ S )
            = ( product_fst_fm_fm @ T ) )
          & ( ( product_snd_fm_fm @ S )
            = ( product_snd_fm_fm @ T ) ) ) ) ) ).

% prod_eq_iff
thf(fact_24_prod__eq__iff,axiom,
    ( ( ^ [Y2: produc7564083655194549958ule_fm,Z: produc7564083655194549958ule_fm] : ( Y2 = Z ) )
    = ( ^ [S: produc7564083655194549958ule_fm,T: produc7564083655194549958ule_fm] :
          ( ( ( produc1758149581174257386ule_fm @ S )
            = ( produc1758149581174257386ule_fm @ T ) )
          & ( ( produc8694862632519904556ule_fm @ S )
            = ( produc8694862632519904556ule_fm @ T ) ) ) ) ) ).

% prod_eq_iff
thf(fact_25_prod__eq__iff,axiom,
    ( ( ^ [Y2: produc4630866025709511057ist_fm,Z: produc4630866025709511057ist_fm] : ( Y2 = Z ) )
    = ( ^ [S: produc4630866025709511057ist_fm,T: produc4630866025709511057ist_fm] :
          ( ( ( produc4228417144610596823ist_fm @ S )
            = ( produc4228417144610596823ist_fm @ T ) )
          & ( ( produc4514744107061996309ist_fm @ S )
            = ( produc4514744107061996309ist_fm @ T ) ) ) ) ) ).

% prod_eq_iff
thf(fact_26_prod__eq__iff,axiom,
    ( ( ^ [Y2: produc164195504107695125m_rule,Z: produc164195504107695125m_rule] : ( Y2 = Z ) )
    = ( ^ [S: produc164195504107695125m_rule,T: produc164195504107695125m_rule] :
          ( ( ( produc6879501374131015971m_rule @ S )
            = ( produc6879501374131015971m_rule @ T ) )
          & ( ( produc7165828336582415457m_rule @ S )
            = ( produc7165828336582415457m_rule @ T ) ) ) ) ) ).

% prod_eq_iff
thf(fact_27_prod__eq__iff,axiom,
    ( ( ^ [Y2: produc1996495991257130529ist_fm,Z: produc1996495991257130529ist_fm] : ( Y2 = Z ) )
    = ( ^ [S: produc1996495991257130529ist_fm,T: produc1996495991257130529ist_fm] :
          ( ( ( produc1501393135466168645ist_fm @ S )
            = ( produc1501393135466168645ist_fm @ T ) )
          & ( ( produc4588648349897876871ist_fm @ S )
            = ( produc4588648349897876871ist_fm @ T ) ) ) ) ) ).

% prod_eq_iff
thf(fact_28_treeA__snth,axiom,
    ! [P: fm,Steps: stream727092118206550309m_rule] :
      ( ( member_fm @ P @ ( treeA @ Steps ) )
     => ? [N: nat] : ( member_fm @ P @ ( set_fm2 @ ( produc1501393135466168645ist_fm @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ ( sdrop_7224736112439592940m_rule @ N @ Steps ) ) ) ) ) ) ) ).

% treeA_snth
thf(fact_29_sdrop__szip,axiom,
    ! [N2: nat,S1: stream8299795917829157543ist_fm,S2: stream_rule] :
      ( ( sdrop_7224736112439592940m_rule @ N2 @ ( szip_P1977448745965526924m_rule @ S1 @ S2 ) )
      = ( szip_P1977448745965526924m_rule @ ( sdrop_4442373711808556042ist_fm @ N2 @ S1 ) @ ( sdrop_rule @ N2 @ S2 ) ) ) ).

% sdrop_szip
thf(fact_30_sdrop__szip,axiom,
    ! [N2: nat,S1: stream727092118206550309m_rule,S2: stream727092118206550309m_rule] :
      ( ( sdrop_7373388980841566196m_rule @ N2 @ ( szip_P2499414959592755846m_rule @ S1 @ S2 ) )
      = ( szip_P2499414959592755846m_rule @ ( sdrop_7224736112439592940m_rule @ N2 @ S1 ) @ ( sdrop_7224736112439592940m_rule @ N2 @ S2 ) ) ) ).

% sdrop_szip
thf(fact_31_sdrop__szip,axiom,
    ! [N2: nat,S1: stream_rule,S2: stream_rule] :
      ( ( sdrop_2199339672054592340e_rule @ N2 @ ( szip_rule_rule @ S1 @ S2 ) )
      = ( szip_rule_rule @ ( sdrop_rule @ N2 @ S1 ) @ ( sdrop_rule @ N2 @ S2 ) ) ) ).

% sdrop_szip
thf(fact_32_sdrop__szip,axiom,
    ! [N2: nat,S1: stream_list_fm,S2: stream_list_fm] :
      ( ( sdrop_4442373711808556042ist_fm @ N2 @ ( szip_list_fm_list_fm @ S1 @ S2 ) )
      = ( szip_list_fm_list_fm @ ( sdrop_list_fm @ N2 @ S1 ) @ ( sdrop_list_fm @ N2 @ S2 ) ) ) ).

% sdrop_szip
thf(fact_33_sdrop__szip,axiom,
    ! [N2: nat,S1: stream_rule,S2: stream8299795917829157543ist_fm] :
      ( ( sdrop_2468034597186633064ist_fm @ N2 @ ( szip_r8549736553299883584ist_fm @ S1 @ S2 ) )
      = ( szip_r8549736553299883584ist_fm @ ( sdrop_rule @ N2 @ S1 ) @ ( sdrop_4442373711808556042ist_fm @ N2 @ S2 ) ) ) ).

% sdrop_szip
thf(fact_34_sdrop__szip,axiom,
    ! [N2: nat,S1: stream727092118206550309m_rule,S2: stream_rule] :
      ( ( sdrop_6392754500180887844e_rule @ N2 @ ( szip_P553291425819358518e_rule @ S1 @ S2 ) )
      = ( szip_P553291425819358518e_rule @ ( sdrop_7224736112439592940m_rule @ N2 @ S1 ) @ ( sdrop_rule @ N2 @ S2 ) ) ) ).

% sdrop_szip
thf(fact_35_sdrop__szip,axiom,
    ! [N2: nat,S1: stream_rule,S2: stream727092118206550309m_rule] :
      ( ( sdrop_1764812713384251940m_rule @ N2 @ ( szip_r2254114822203693110m_rule @ S1 @ S2 ) )
      = ( szip_r2254114822203693110m_rule @ ( sdrop_rule @ N2 @ S1 ) @ ( sdrop_7224736112439592940m_rule @ N2 @ S2 ) ) ) ).

% sdrop_szip
thf(fact_36_sdrop__szip,axiom,
    ! [N2: nat,S1: stream8299795917829157543ist_fm,S2: stream8299795917829157543ist_fm] :
      ( ( sdrop_1261538545438237392ist_fm @ N2 @ ( szip_P6003557782763710490ist_fm @ S1 @ S2 ) )
      = ( szip_P6003557782763710490ist_fm @ ( sdrop_4442373711808556042ist_fm @ N2 @ S1 ) @ ( sdrop_4442373711808556042ist_fm @ N2 @ S2 ) ) ) ).

% sdrop_szip
thf(fact_37_sdrop__szip,axiom,
    ! [N2: nat,S1: stream727092118206550309m_rule,S2: stream8299795917829157543ist_fm] :
      ( ( sdrop_3792585760419738520ist_fm @ N2 @ ( szip_P5077708527527104496ist_fm @ S1 @ S2 ) )
      = ( szip_P5077708527527104496ist_fm @ ( sdrop_7224736112439592940m_rule @ N2 @ S1 ) @ ( sdrop_4442373711808556042ist_fm @ N2 @ S2 ) ) ) ).

% sdrop_szip
thf(fact_38_sdrop__szip,axiom,
    ! [N2: nat,S1: stream8299795917829157543ist_fm,S2: stream727092118206550309m_rule] :
      ( ( sdrop_9094843686636319420m_rule @ N2 @ ( szip_P5379742314337339356m_rule @ S1 @ S2 ) )
      = ( szip_P5379742314337339356m_rule @ ( sdrop_4442373711808556042ist_fm @ N2 @ S1 ) @ ( sdrop_7224736112439592940m_rule @ N2 @ S2 ) ) ) ).

% sdrop_szip
thf(fact_39_fst__swap,axiom,
    ! [X2: produc1754732471204517991ist_fm] :
      ( ( produc5894184038225292483ist_fm @ ( produc3697025406284169335ist_fm @ X2 ) )
      = ( produc1639249265332130309ist_fm @ X2 ) ) ).

% fst_swap
thf(fact_40_fst__swap,axiom,
    ! [X2: produc780299800278082923e_rule] :
      ( ( produc7264528503596239631e_rule @ ( produc2320433982106674371e_rule @ X2 ) )
      = ( produc6725306161561501521e_rule @ X2 ) ) ).

% fst_swap
thf(fact_41_fst__swap,axiom,
    ! [X2: product_prod_fm_fm] :
      ( ( product_fst_fm_fm @ ( product_swap_fm_fm @ X2 ) )
      = ( product_snd_fm_fm @ X2 ) ) ).

% fst_swap
thf(fact_42_fst__swap,axiom,
    ! [X2: produc7564083655194549958ule_fm] :
      ( ( produc767568915850739050m_rule @ ( produc9128311380267582622ule_fm @ X2 ) )
      = ( produc8694862632519904556ule_fm @ X2 ) ) ).

% fst_swap
thf(fact_43_fst__swap,axiom,
    ! [X2: produc164195504107695125m_rule] :
      ( ( produc4228417144610596823ist_fm @ ( produc2641608828089375343m_rule @ X2 ) )
      = ( produc7165828336582415457m_rule @ X2 ) ) ).

% fst_swap
thf(fact_44_fst__swap,axiom,
    ! [X2: produc4630866025709511057ist_fm] :
      ( ( produc6879501374131015971m_rule @ ( produc9213896635423732003ist_fm @ X2 ) )
      = ( produc4514744107061996309ist_fm @ X2 ) ) ).

% fst_swap
thf(fact_45_fst__swap,axiom,
    ! [X2: produc1996495991257130529ist_fm] :
      ( ( produc1501393135466168645ist_fm @ ( produc7936585717479832313ist_fm @ X2 ) )
      = ( produc4588648349897876871ist_fm @ X2 ) ) ).

% fst_swap
thf(fact_46_snd__swap,axiom,
    ! [X2: produc1754732471204517991ist_fm] :
      ( ( produc1639249265332130309ist_fm @ ( produc3697025406284169335ist_fm @ X2 ) )
      = ( produc5894184038225292483ist_fm @ X2 ) ) ).

% snd_swap
thf(fact_47_snd__swap,axiom,
    ! [X2: produc780299800278082923e_rule] :
      ( ( produc6725306161561501521e_rule @ ( produc2320433982106674371e_rule @ X2 ) )
      = ( produc7264528503596239631e_rule @ X2 ) ) ).

% snd_swap
thf(fact_48_snd__swap,axiom,
    ! [X2: product_prod_fm_fm] :
      ( ( product_snd_fm_fm @ ( product_swap_fm_fm @ X2 ) )
      = ( product_fst_fm_fm @ X2 ) ) ).

% snd_swap
thf(fact_49_snd__swap,axiom,
    ! [X2: produc4153787046269522630m_rule] :
      ( ( produc8694862632519904556ule_fm @ ( produc8137730714944064286m_rule @ X2 ) )
      = ( produc767568915850739050m_rule @ X2 ) ) ).

% snd_swap
thf(fact_50_snd__swap,axiom,
    ! [X2: produc4630866025709511057ist_fm] :
      ( ( produc7165828336582415457m_rule @ ( produc9213896635423732003ist_fm @ X2 ) )
      = ( produc4228417144610596823ist_fm @ X2 ) ) ).

% snd_swap
thf(fact_51_snd__swap,axiom,
    ! [X2: produc164195504107695125m_rule] :
      ( ( produc4514744107061996309ist_fm @ ( produc2641608828089375343m_rule @ X2 ) )
      = ( produc6879501374131015971m_rule @ X2 ) ) ).

% snd_swap
thf(fact_52_snd__swap,axiom,
    ! [X2: produc1996495991257130529ist_fm] :
      ( ( produc4588648349897876871ist_fm @ ( produc7936585717479832313ist_fm @ X2 ) )
      = ( produc1501393135466168645ist_fm @ X2 ) ) ).

% snd_swap
thf(fact_53_sdrop__simps_I1_J,axiom,
    ! [N2: nat,S3: stream727092118206550309m_rule] :
      ( ( shd_Pr7235097944458474089m_rule @ ( sdrop_7224736112439592940m_rule @ N2 @ S3 ) )
      = ( snth_P1712579846118164081m_rule @ S3 @ N2 ) ) ).

% sdrop_simps(1)
thf(fact_54_sdrop__simps_I1_J,axiom,
    ! [N2: nat,S3: stream_rule] :
      ( ( shd_rule @ ( sdrop_rule @ N2 @ S3 ) )
      = ( snth_rule @ S3 @ N2 ) ) ).

% sdrop_simps(1)
thf(fact_55_sdrop__simps_I1_J,axiom,
    ! [N2: nat,S3: stream8299795917829157543ist_fm] :
      ( ( shd_Pr772355297128350925ist_fm @ ( sdrop_4442373711808556042ist_fm @ N2 @ S3 ) )
      = ( snth_P2624749845746815685ist_fm @ S3 @ N2 ) ) ).

% sdrop_simps(1)
thf(fact_56_sdrop__simps_I1_J,axiom,
    ! [N2: nat,S3: stream6494289010434245521m_rule] :
      ( ( shd_Pr2340755956080993207m_rule @ ( sdrop_7373388980841566196m_rule @ N2 @ S3 ) )
      = ( snth_P8950227884416379311m_rule @ S3 @ N2 ) ) ).

% sdrop_simps(1)
thf(fact_57_sdrop__simps_I1_J,axiom,
    ! [N2: nat,S3: stream_fm] :
      ( ( shd_fm @ ( sdrop_fm @ N2 @ S3 ) )
      = ( snth_fm @ S3 @ N2 ) ) ).

% sdrop_simps(1)
thf(fact_58_sdrop__simps_I1_J,axiom,
    ! [N2: nat,S3: stream_stream_fm] :
      ( ( shd_stream_fm @ ( sdrop_stream_fm @ N2 @ S3 ) )
      = ( snth_stream_fm @ S3 @ N2 ) ) ).

% sdrop_simps(1)
thf(fact_59_sdrop__simps_I1_J,axiom,
    ! [N2: nat,S3: stream_list_fm] :
      ( ( shd_list_fm @ ( sdrop_list_fm @ N2 @ S3 ) )
      = ( snth_list_fm @ S3 @ N2 ) ) ).

% sdrop_simps(1)
thf(fact_60_sdrop__simps_I1_J,axiom,
    ! [N2: nat,S3: stream3588992565182678257e_rule] :
      ( ( shd_Pr4573454438745646103e_rule @ ( sdrop_2199339672054592340e_rule @ N2 @ S3 ) )
      = ( snth_P8487267234472522255e_rule @ S3 @ N2 ) ) ).

% sdrop_simps(1)
thf(fact_61_sdrop__simps_I1_J,axiom,
    ! [N2: nat,S3: stream1960312990768957601ist_fm] :
      ( ( shd_Pr2478396429205514213ist_fm @ ( sdrop_2468034597186633064ist_fm @ N2 @ S3 ) )
      = ( snth_P6179250367719980013ist_fm @ S3 @ N2 ) ) ).

% sdrop_simps(1)
thf(fact_62_sdrop__simps_I1_J,axiom,
    ! [N2: nat,S3: stream3016090547575141045m_rule] :
      ( ( shd_st3931407289205852537m_rule @ ( sdrop_8061686475400732284m_rule @ N2 @ S3 ) )
      = ( snth_s8878883179916095873m_rule @ S3 @ N2 ) ) ).

% sdrop_simps(1)
thf(fact_63_pred__prod__beta,axiom,
    ( basic_1736577632341059187ist_fm
    = ( ^ [P3: produc1996495991257130529ist_fm > $o,Q2: produc1996495991257130529ist_fm > $o,Xy: produc1754732471204517991ist_fm] :
          ( ( P3 @ ( produc5894184038225292483ist_fm @ Xy ) )
          & ( Q2 @ ( produc1639249265332130309ist_fm @ Xy ) ) ) ) ) ).

% pred_prod_beta
thf(fact_64_pred__prod__beta,axiom,
    ( basic_5896771853912976575e_rule
    = ( ^ [P3: rule > $o,Q2: rule > $o,Xy: produc780299800278082923e_rule] :
          ( ( P3 @ ( produc7264528503596239631e_rule @ Xy ) )
          & ( Q2 @ ( produc6725306161561501521e_rule @ Xy ) ) ) ) ) ).

% pred_prod_beta
thf(fact_65_pred__prod__beta,axiom,
    ( basic_6733234405827732053_fm_fm
    = ( ^ [P3: fm > $o,Q2: fm > $o,Xy: product_prod_fm_fm] :
          ( ( P3 @ ( product_fst_fm_fm @ Xy ) )
          & ( Q2 @ ( product_snd_fm_fm @ Xy ) ) ) ) ) ).

% pred_prod_beta
thf(fact_66_pred__prod__beta,axiom,
    ( basic_7801200691956009114ule_fm
    = ( ^ [P3: produc164195504107695125m_rule > $o,Q2: fm > $o,Xy: produc7564083655194549958ule_fm] :
          ( ( P3 @ ( produc1758149581174257386ule_fm @ Xy ) )
          & ( Q2 @ ( produc8694862632519904556ule_fm @ Xy ) ) ) ) ) ).

% pred_prod_beta
thf(fact_67_pred__prod__beta,axiom,
    ( basic_3520275952944978471ist_fm
    = ( ^ [P3: rule > $o,Q2: produc1996495991257130529ist_fm > $o,Xy: produc4630866025709511057ist_fm] :
          ( ( P3 @ ( produc4228417144610596823ist_fm @ Xy ) )
          & ( Q2 @ ( produc4514744107061996309ist_fm @ Xy ) ) ) ) ) ).

% pred_prod_beta
thf(fact_68_pred__prod__beta,axiom,
    ( basic_6171360182465397619m_rule
    = ( ^ [P3: produc1996495991257130529ist_fm > $o,Q2: rule > $o,Xy: produc164195504107695125m_rule] :
          ( ( P3 @ ( produc6879501374131015971m_rule @ Xy ) )
          & ( Q2 @ ( produc7165828336582415457m_rule @ Xy ) ) ) ) ) ).

% pred_prod_beta
thf(fact_69_pred__prod__beta,axiom,
    ( basic_5093365364719461109ist_fm
    = ( ^ [P3: list_fm > $o,Q2: list_fm > $o,Xy: produc1996495991257130529ist_fm] :
          ( ( P3 @ ( produc1501393135466168645ist_fm @ Xy ) )
          & ( Q2 @ ( produc4588648349897876871ist_fm @ Xy ) ) ) ) ) ).

% pred_prod_beta
thf(fact_70_prod_Ocollapse,axiom,
    ! [Prod: produc1316288955051664109set_fm] :
      ( ( produc3030414577503514149set_fm @ ( produc1719312957141167633set_fm @ Prod ) @ ( produc6202189304966566995set_fm @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_71_prod_Ocollapse,axiom,
    ! [Prod: produc1754732471204517991ist_fm] :
      ( ( produc2592314732767339223ist_fm @ ( produc5894184038225292483ist_fm @ Prod ) @ ( produc1639249265332130309ist_fm @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_72_prod_Ocollapse,axiom,
    ! [Prod: produc780299800278082923e_rule] :
      ( ( produc1769597910952761123e_rule @ ( produc7264528503596239631e_rule @ Prod ) @ ( produc6725306161561501521e_rule @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_73_prod_Ocollapse,axiom,
    ! [Prod: product_prod_fm_fm] :
      ( ( product_Pair_fm_fm @ ( product_fst_fm_fm @ Prod ) @ ( product_snd_fm_fm @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_74_prod_Ocollapse,axiom,
    ! [Prod: produc7564083655194549958ule_fm] :
      ( ( produc803751574383482622ule_fm @ ( produc1758149581174257386ule_fm @ Prod ) @ ( produc8694862632519904556ule_fm @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_75_prod_Ocollapse,axiom,
    ! [Prod: produc1996495991257130529ist_fm] :
      ( ( produc381145313068854617ist_fm @ ( produc1501393135466168645ist_fm @ Prod ) @ ( produc4588648349897876871ist_fm @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_76_prod_Ocollapse,axiom,
    ! [Prod: produc164195504107695125m_rule] :
      ( ( produc491467635432902671m_rule @ ( produc6879501374131015971m_rule @ Prod ) @ ( produc7165828336582415457m_rule @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_77_prod_Ocollapse,axiom,
    ! [Prod: produc4630866025709511057ist_fm] :
      ( ( produc7063755442767259331ist_fm @ ( produc4228417144610596823ist_fm @ Prod ) @ ( produc4514744107061996309ist_fm @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_78_split__beta,axiom,
    ( produc2632167636358383921ist_fm
    = ( ^ [F: rule > produc1996495991257130529ist_fm > produc1996495991257130529ist_fm,Prod3: produc4630866025709511057ist_fm] : ( F @ ( produc4228417144610596823ist_fm @ Prod3 ) @ ( produc4514744107061996309ist_fm @ Prod3 ) ) ) ) ).

% split_beta
thf(fact_79_split__beta,axiom,
    ( produc1628407760719572648t_fm_o
    = ( ^ [F: rule > produc1996495991257130529ist_fm > $o,Prod3: produc4630866025709511057ist_fm] : ( F @ ( produc4228417144610596823ist_fm @ Prod3 ) @ ( produc4514744107061996309ist_fm @ Prod3 ) ) ) ) ).

% split_beta
thf(fact_80_split__beta,axiom,
    ( produc4544959197559615795ist_fm
    = ( ^ [F: list_fm > list_fm > produc1996495991257130529ist_fm,Prod3: produc1996495991257130529ist_fm] : ( F @ ( produc1501393135466168645ist_fm @ Prod3 ) @ ( produc4588648349897876871ist_fm @ Prod3 ) ) ) ) ).

% split_beta
thf(fact_81_split__beta,axiom,
    ( produc4154610405322092878ist_fm
    = ( ^ [F: list_fm > list_fm > list_fm,Prod3: produc1996495991257130529ist_fm] : ( F @ ( produc1501393135466168645ist_fm @ Prod3 ) @ ( produc4588648349897876871ist_fm @ Prod3 ) ) ) ) ).

% split_beta
thf(fact_82_split__beta,axiom,
    ( produc2011568048573023402t_fm_o
    = ( ^ [F: list_fm > list_fm > $o,Prod3: produc1996495991257130529ist_fm] : ( F @ ( produc1501393135466168645ist_fm @ Prod3 ) @ ( produc4588648349897876871ist_fm @ Prod3 ) ) ) ) ).

% split_beta
thf(fact_83_prod_Oinject,axiom,
    ! [X1: rule,X22: produc1996495991257130529ist_fm,Y1: rule,Y22: produc1996495991257130529ist_fm] :
      ( ( ( produc7063755442767259331ist_fm @ X1 @ X22 )
        = ( produc7063755442767259331ist_fm @ Y1 @ Y22 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y22 ) ) ) ).

% prod.inject
thf(fact_84_prod_Oinject,axiom,
    ! [X1: rule,X22: rule,Y1: rule,Y22: rule] :
      ( ( ( produc1769597910952761123e_rule @ X1 @ X22 )
        = ( produc1769597910952761123e_rule @ Y1 @ Y22 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y22 ) ) ) ).

% prod.inject
thf(fact_85_prod_Oinject,axiom,
    ! [X1: fm,X22: fm,Y1: fm,Y22: fm] :
      ( ( ( product_Pair_fm_fm @ X1 @ X22 )
        = ( product_Pair_fm_fm @ Y1 @ Y22 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y22 ) ) ) ).

% prod.inject
thf(fact_86_prod_Oinject,axiom,
    ! [X1: list_fm,X22: list_fm,Y1: list_fm,Y22: list_fm] :
      ( ( ( produc381145313068854617ist_fm @ X1 @ X22 )
        = ( produc381145313068854617ist_fm @ Y1 @ Y22 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y22 ) ) ) ).

% prod.inject
thf(fact_87_prod_Oinject,axiom,
    ! [X1: set_fm,X22: set_fm,Y1: set_fm,Y22: set_fm] :
      ( ( ( produc3030414577503514149set_fm @ X1 @ X22 )
        = ( produc3030414577503514149set_fm @ Y1 @ Y22 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y22 ) ) ) ).

% prod.inject
thf(fact_88_old_Oprod_Oinject,axiom,
    ! [A: rule,B: produc1996495991257130529ist_fm,A2: rule,B2: produc1996495991257130529ist_fm] :
      ( ( ( produc7063755442767259331ist_fm @ A @ B )
        = ( produc7063755442767259331ist_fm @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_89_old_Oprod_Oinject,axiom,
    ! [A: rule,B: rule,A2: rule,B2: rule] :
      ( ( ( produc1769597910952761123e_rule @ A @ B )
        = ( produc1769597910952761123e_rule @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_90_old_Oprod_Oinject,axiom,
    ! [A: fm,B: fm,A2: fm,B2: fm] :
      ( ( ( product_Pair_fm_fm @ A @ B )
        = ( product_Pair_fm_fm @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_91_old_Oprod_Oinject,axiom,
    ! [A: list_fm,B: list_fm,A2: list_fm,B2: list_fm] :
      ( ( ( produc381145313068854617ist_fm @ A @ B )
        = ( produc381145313068854617ist_fm @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_92_old_Oprod_Oinject,axiom,
    ! [A: set_fm,B: set_fm,A2: set_fm,B2: set_fm] :
      ( ( ( produc3030414577503514149set_fm @ A @ B )
        = ( produc3030414577503514149set_fm @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_93_swap__swap,axiom,
    ! [P: produc1996495991257130529ist_fm] :
      ( ( produc7936585717479832313ist_fm @ ( produc7936585717479832313ist_fm @ P ) )
      = P ) ).

% swap_swap
thf(fact_94_swap__swap,axiom,
    ! [P: produc164195504107695125m_rule] :
      ( ( produc9213896635423732003ist_fm @ ( produc2641608828089375343m_rule @ P ) )
      = P ) ).

% swap_swap
thf(fact_95_swap__swap,axiom,
    ! [P: produc4630866025709511057ist_fm] :
      ( ( produc2641608828089375343m_rule @ ( produc9213896635423732003ist_fm @ P ) )
      = P ) ).

% swap_swap
thf(fact_96_swap__simp,axiom,
    ! [X2: rule,Y3: rule] :
      ( ( produc2320433982106674371e_rule @ ( produc1769597910952761123e_rule @ X2 @ Y3 ) )
      = ( produc1769597910952761123e_rule @ Y3 @ X2 ) ) ).

% swap_simp
thf(fact_97_swap__simp,axiom,
    ! [X2: fm,Y3: fm] :
      ( ( product_swap_fm_fm @ ( product_Pair_fm_fm @ X2 @ Y3 ) )
      = ( product_Pair_fm_fm @ Y3 @ X2 ) ) ).

% swap_simp
thf(fact_98_swap__simp,axiom,
    ! [X2: set_fm,Y3: set_fm] :
      ( ( produc7775435602796846021set_fm @ ( produc3030414577503514149set_fm @ X2 @ Y3 ) )
      = ( produc3030414577503514149set_fm @ Y3 @ X2 ) ) ).

% swap_simp
thf(fact_99_swap__simp,axiom,
    ! [X2: list_fm,Y3: list_fm] :
      ( ( produc7936585717479832313ist_fm @ ( produc381145313068854617ist_fm @ X2 @ Y3 ) )
      = ( produc381145313068854617ist_fm @ Y3 @ X2 ) ) ).

% swap_simp
thf(fact_100_swap__simp,axiom,
    ! [X2: rule,Y3: produc1996495991257130529ist_fm] :
      ( ( produc9213896635423732003ist_fm @ ( produc7063755442767259331ist_fm @ X2 @ Y3 ) )
      = ( produc491467635432902671m_rule @ Y3 @ X2 ) ) ).

% swap_simp
thf(fact_101_swap__simp,axiom,
    ! [X2: produc1996495991257130529ist_fm,Y3: rule] :
      ( ( produc2641608828089375343m_rule @ ( produc491467635432902671m_rule @ X2 @ Y3 ) )
      = ( produc7063755442767259331ist_fm @ Y3 @ X2 ) ) ).

% swap_simp
thf(fact_102_snth__szip,axiom,
    ! [S1: stream8299795917829157543ist_fm,S2: stream_rule,N2: nat] :
      ( ( snth_P1712579846118164081m_rule @ ( szip_P1977448745965526924m_rule @ S1 @ S2 ) @ N2 )
      = ( produc491467635432902671m_rule @ ( snth_P2624749845746815685ist_fm @ S1 @ N2 ) @ ( snth_rule @ S2 @ N2 ) ) ) ).

% snth_szip
thf(fact_103_snth__szip,axiom,
    ! [S1: stream727092118206550309m_rule,S2: stream727092118206550309m_rule,N2: nat] :
      ( ( snth_P8950227884416379311m_rule @ ( szip_P2499414959592755846m_rule @ S1 @ S2 ) @ N2 )
      = ( produc5927390650430071747m_rule @ ( snth_P1712579846118164081m_rule @ S1 @ N2 ) @ ( snth_P1712579846118164081m_rule @ S2 @ N2 ) ) ) ).

% snth_szip
thf(fact_104_snth__szip,axiom,
    ! [S1: stream_rule,S2: stream_fm,N2: nat] :
      ( ( snth_P5391786766372275226ule_fm @ ( szip_rule_fm @ S1 @ S2 ) @ N2 )
      = ( product_Pair_rule_fm @ ( snth_rule @ S1 @ N2 ) @ ( snth_fm @ S2 @ N2 ) ) ) ).

% snth_szip
thf(fact_105_snth__szip,axiom,
    ! [S1: stream_fm,S2: stream_rule,N2: nat] :
      ( ( snth_P4328312469133779354m_rule @ ( szip_fm_rule @ S1 @ S2 ) @ N2 )
      = ( product_Pair_fm_rule @ ( snth_fm @ S1 @ N2 ) @ ( snth_rule @ S2 @ N2 ) ) ) ).

% snth_szip
thf(fact_106_snth__szip,axiom,
    ! [S1: stream_rule,S2: stream_rule,N2: nat] :
      ( ( snth_P8487267234472522255e_rule @ ( szip_rule_rule @ S1 @ S2 ) @ N2 )
      = ( produc1769597910952761123e_rule @ ( snth_rule @ S1 @ N2 ) @ ( snth_rule @ S2 @ N2 ) ) ) ).

% snth_szip
thf(fact_107_snth__szip,axiom,
    ! [S1: stream_fm,S2: stream_fm,N2: nat] :
      ( ( snth_P8653008792276846885_fm_fm @ ( szip_fm_fm @ S1 @ S2 ) @ N2 )
      = ( product_Pair_fm_fm @ ( snth_fm @ S1 @ N2 ) @ ( snth_fm @ S2 @ N2 ) ) ) ).

% snth_szip
thf(fact_108_snth__szip,axiom,
    ! [S1: stream_stream_fm,S2: stream_rule,N2: nat] :
      ( ( snth_P3285027868766484906m_rule @ ( szip_stream_fm_rule @ S1 @ S2 ) @ N2 )
      = ( produc858280020141504446m_rule @ ( snth_stream_fm @ S1 @ N2 ) @ ( snth_rule @ S2 @ N2 ) ) ) ).

% snth_szip
thf(fact_109_snth__szip,axiom,
    ! [S1: stream_stream_fm,S2: stream_fm,N2: nat] :
      ( ( snth_P4090355645105449269_fm_fm @ ( szip_stream_fm_fm @ S1 @ S2 ) @ N2 )
      = ( produc7919197687345228873_fm_fm @ ( snth_stream_fm @ S1 @ N2 ) @ ( snth_fm @ S2 @ N2 ) ) ) ).

% snth_szip
thf(fact_110_snth__szip,axiom,
    ! [S1: stream_rule,S2: stream_stream_fm,N2: nat] :
      ( ( snth_P7026095387229993258eam_fm @ ( szip_rule_stream_fm @ S1 @ S2 ) @ N2 )
      = ( produc3062392735954935998eam_fm @ ( snth_rule @ S1 @ N2 ) @ ( snth_stream_fm @ S2 @ N2 ) ) ) ).

% snth_szip
thf(fact_111_snth__szip,axiom,
    ! [S1: stream_fm,S2: stream_stream_fm,N2: nat] :
      ( ( snth_P989435127820226997eam_fm @ ( szip_fm_stream_fm @ S1 @ S2 ) @ N2 )
      = ( produc8595645164685244105eam_fm @ ( snth_fm @ S1 @ N2 ) @ ( snth_stream_fm @ S2 @ N2 ) ) ) ).

% snth_szip
thf(fact_112_old_Oprod_Ocase,axiom,
    ! [F2: rule > produc1996495991257130529ist_fm > produc1996495991257130529ist_fm,X1: rule,X22: produc1996495991257130529ist_fm] :
      ( ( produc2632167636358383921ist_fm @ F2 @ ( produc7063755442767259331ist_fm @ X1 @ X22 ) )
      = ( F2 @ X1 @ X22 ) ) ).

% old.prod.case
thf(fact_113_old_Oprod_Ocase,axiom,
    ! [F2: rule > produc1996495991257130529ist_fm > $o,X1: rule,X22: produc1996495991257130529ist_fm] :
      ( ( produc1628407760719572648t_fm_o @ F2 @ ( produc7063755442767259331ist_fm @ X1 @ X22 ) )
      = ( F2 @ X1 @ X22 ) ) ).

% old.prod.case
thf(fact_114_old_Oprod_Ocase,axiom,
    ! [F2: list_fm > list_fm > produc1996495991257130529ist_fm,X1: list_fm,X22: list_fm] :
      ( ( produc4544959197559615795ist_fm @ F2 @ ( produc381145313068854617ist_fm @ X1 @ X22 ) )
      = ( F2 @ X1 @ X22 ) ) ).

% old.prod.case
thf(fact_115_old_Oprod_Ocase,axiom,
    ! [F2: list_fm > list_fm > list_fm,X1: list_fm,X22: list_fm] :
      ( ( produc4154610405322092878ist_fm @ F2 @ ( produc381145313068854617ist_fm @ X1 @ X22 ) )
      = ( F2 @ X1 @ X22 ) ) ).

% old.prod.case
thf(fact_116_old_Oprod_Ocase,axiom,
    ! [F2: list_fm > list_fm > $o,X1: list_fm,X22: list_fm] :
      ( ( produc2011568048573023402t_fm_o @ F2 @ ( produc381145313068854617ist_fm @ X1 @ X22 ) )
      = ( F2 @ X1 @ X22 ) ) ).

% old.prod.case
thf(fact_117_old_Oprod_Oexhaust,axiom,
    ! [Y3: produc4630866025709511057ist_fm] :
      ~ ! [A3: rule,B3: produc1996495991257130529ist_fm] :
          ( Y3
         != ( produc7063755442767259331ist_fm @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_118_old_Oprod_Oexhaust,axiom,
    ! [Y3: produc780299800278082923e_rule] :
      ~ ! [A3: rule,B3: rule] :
          ( Y3
         != ( produc1769597910952761123e_rule @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_119_old_Oprod_Oexhaust,axiom,
    ! [Y3: product_prod_fm_fm] :
      ~ ! [A3: fm,B3: fm] :
          ( Y3
         != ( product_Pair_fm_fm @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_120_old_Oprod_Oexhaust,axiom,
    ! [Y3: produc1996495991257130529ist_fm] :
      ~ ! [A3: list_fm,B3: list_fm] :
          ( Y3
         != ( produc381145313068854617ist_fm @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_121_old_Oprod_Oexhaust,axiom,
    ! [Y3: produc1316288955051664109set_fm] :
      ~ ! [A3: set_fm,B3: set_fm] :
          ( Y3
         != ( produc3030414577503514149set_fm @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_122_surj__pair,axiom,
    ! [P: produc4630866025709511057ist_fm] :
    ? [X: rule,Y: produc1996495991257130529ist_fm] :
      ( P
      = ( produc7063755442767259331ist_fm @ X @ Y ) ) ).

% surj_pair
thf(fact_123_surj__pair,axiom,
    ! [P: produc780299800278082923e_rule] :
    ? [X: rule,Y: rule] :
      ( P
      = ( produc1769597910952761123e_rule @ X @ Y ) ) ).

% surj_pair
thf(fact_124_surj__pair,axiom,
    ! [P: product_prod_fm_fm] :
    ? [X: fm,Y: fm] :
      ( P
      = ( product_Pair_fm_fm @ X @ Y ) ) ).

% surj_pair
thf(fact_125_surj__pair,axiom,
    ! [P: produc1996495991257130529ist_fm] :
    ? [X: list_fm,Y: list_fm] :
      ( P
      = ( produc381145313068854617ist_fm @ X @ Y ) ) ).

% surj_pair
thf(fact_126_surj__pair,axiom,
    ! [P: produc1316288955051664109set_fm] :
    ? [X: set_fm,Y: set_fm] :
      ( P
      = ( produc3030414577503514149set_fm @ X @ Y ) ) ).

% surj_pair
thf(fact_127_pred__prod__split,axiom,
    ! [P2: $o > $o,Q3: rule > $o,R: rule > $o,Xy2: produc780299800278082923e_rule] :
      ( ( P2 @ ( basic_5896771853912976575e_rule @ Q3 @ R @ Xy2 ) )
      = ( ! [X3: rule,Y4: rule] :
            ( ( Xy2
              = ( produc1769597910952761123e_rule @ X3 @ Y4 ) )
           => ( P2
              @ ( ( Q3 @ X3 )
                & ( R @ Y4 ) ) ) ) ) ) ).

% pred_prod_split
thf(fact_128_pred__prod__split,axiom,
    ! [P2: $o > $o,Q3: set_fm > $o,R: set_fm > $o,Xy2: produc1316288955051664109set_fm] :
      ( ( P2 @ ( basic_677064363366268865set_fm @ Q3 @ R @ Xy2 ) )
      = ( ! [X3: set_fm,Y4: set_fm] :
            ( ( Xy2
              = ( produc3030414577503514149set_fm @ X3 @ Y4 ) )
           => ( P2
              @ ( ( Q3 @ X3 )
                & ( R @ Y4 ) ) ) ) ) ) ).

% pred_prod_split
thf(fact_129_pred__prod__split,axiom,
    ! [P2: $o > $o,Q3: rule > $o,R: produc1996495991257130529ist_fm > $o,Xy2: produc4630866025709511057ist_fm] :
      ( ( P2 @ ( basic_3520275952944978471ist_fm @ Q3 @ R @ Xy2 ) )
      = ( ! [X3: rule,Y4: produc1996495991257130529ist_fm] :
            ( ( Xy2
              = ( produc7063755442767259331ist_fm @ X3 @ Y4 ) )
           => ( P2
              @ ( ( Q3 @ X3 )
                & ( R @ Y4 ) ) ) ) ) ) ).

% pred_prod_split
thf(fact_130_pred__prod__split,axiom,
    ! [P2: $o > $o,Q3: fm > $o,R: fm > $o,Xy2: product_prod_fm_fm] :
      ( ( P2 @ ( basic_6733234405827732053_fm_fm @ Q3 @ R @ Xy2 ) )
      = ( ! [X3: fm,Y4: fm] :
            ( ( Xy2
              = ( product_Pair_fm_fm @ X3 @ Y4 ) )
           => ( P2
              @ ( ( Q3 @ X3 )
                & ( R @ Y4 ) ) ) ) ) ) ).

% pred_prod_split
thf(fact_131_pred__prod__split,axiom,
    ! [P2: $o > $o,Q3: list_fm > $o,R: list_fm > $o,Xy2: produc1996495991257130529ist_fm] :
      ( ( P2 @ ( basic_5093365364719461109ist_fm @ Q3 @ R @ Xy2 ) )
      = ( ! [X3: list_fm,Y4: list_fm] :
            ( ( Xy2
              = ( produc381145313068854617ist_fm @ X3 @ Y4 ) )
           => ( P2
              @ ( ( Q3 @ X3 )
                & ( R @ Y4 ) ) ) ) ) ) ).

% pred_prod_split
thf(fact_132_pred__prod__split,axiom,
    ! [P2: $o > $o,Q3: produc1996495991257130529ist_fm > $o,R: rule > $o,Xy2: produc164195504107695125m_rule] :
      ( ( P2 @ ( basic_6171360182465397619m_rule @ Q3 @ R @ Xy2 ) )
      = ( ! [X3: produc1996495991257130529ist_fm,Y4: rule] :
            ( ( Xy2
              = ( produc491467635432902671m_rule @ X3 @ Y4 ) )
           => ( P2
              @ ( ( Q3 @ X3 )
                & ( R @ Y4 ) ) ) ) ) ) ).

% pred_prod_split
thf(fact_133_prod__cases,axiom,
    ! [P2: produc4630866025709511057ist_fm > $o,P: produc4630866025709511057ist_fm] :
      ( ! [A3: rule,B3: produc1996495991257130529ist_fm] : ( P2 @ ( produc7063755442767259331ist_fm @ A3 @ B3 ) )
     => ( P2 @ P ) ) ).

% prod_cases
thf(fact_134_prod__cases,axiom,
    ! [P2: produc780299800278082923e_rule > $o,P: produc780299800278082923e_rule] :
      ( ! [A3: rule,B3: rule] : ( P2 @ ( produc1769597910952761123e_rule @ A3 @ B3 ) )
     => ( P2 @ P ) ) ).

% prod_cases
thf(fact_135_prod__cases,axiom,
    ! [P2: product_prod_fm_fm > $o,P: product_prod_fm_fm] :
      ( ! [A3: fm,B3: fm] : ( P2 @ ( product_Pair_fm_fm @ A3 @ B3 ) )
     => ( P2 @ P ) ) ).

% prod_cases
thf(fact_136_prod__cases,axiom,
    ! [P2: produc1996495991257130529ist_fm > $o,P: produc1996495991257130529ist_fm] :
      ( ! [A3: list_fm,B3: list_fm] : ( P2 @ ( produc381145313068854617ist_fm @ A3 @ B3 ) )
     => ( P2 @ P ) ) ).

% prod_cases
thf(fact_137_prod__cases,axiom,
    ! [P2: produc1316288955051664109set_fm > $o,P: produc1316288955051664109set_fm] :
      ( ! [A3: set_fm,B3: set_fm] : ( P2 @ ( produc3030414577503514149set_fm @ A3 @ B3 ) )
     => ( P2 @ P ) ) ).

% prod_cases
thf(fact_138_Pair__inject,axiom,
    ! [A: rule,B: produc1996495991257130529ist_fm,A2: rule,B2: produc1996495991257130529ist_fm] :
      ( ( ( produc7063755442767259331ist_fm @ A @ B )
        = ( produc7063755442767259331ist_fm @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_139_Pair__inject,axiom,
    ! [A: rule,B: rule,A2: rule,B2: rule] :
      ( ( ( produc1769597910952761123e_rule @ A @ B )
        = ( produc1769597910952761123e_rule @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_140_Pair__inject,axiom,
    ! [A: fm,B: fm,A2: fm,B2: fm] :
      ( ( ( product_Pair_fm_fm @ A @ B )
        = ( product_Pair_fm_fm @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_141_Pair__inject,axiom,
    ! [A: list_fm,B: list_fm,A2: list_fm,B2: list_fm] :
      ( ( ( produc381145313068854617ist_fm @ A @ B )
        = ( produc381145313068854617ist_fm @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_142_Pair__inject,axiom,
    ! [A: set_fm,B: set_fm,A2: set_fm,B2: set_fm] :
      ( ( ( produc3030414577503514149set_fm @ A @ B )
        = ( produc3030414577503514149set_fm @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_143_prod__cases3,axiom,
    ! [Y3: produc4630866025709511057ist_fm] :
      ~ ! [A3: rule,B3: list_fm,C: list_fm] :
          ( Y3
         != ( produc7063755442767259331ist_fm @ A3 @ ( produc381145313068854617ist_fm @ B3 @ C ) ) ) ).

% prod_cases3
thf(fact_144_prod__induct3,axiom,
    ! [P2: produc4630866025709511057ist_fm > $o,X2: produc4630866025709511057ist_fm] :
      ( ! [A3: rule,B3: list_fm,C: list_fm] : ( P2 @ ( produc7063755442767259331ist_fm @ A3 @ ( produc381145313068854617ist_fm @ B3 @ C ) ) )
     => ( P2 @ X2 ) ) ).

% prod_induct3
thf(fact_145_mem__case__prodE,axiom,
    ! [Z2: fm,C2: rule > rule > set_fm,P: produc780299800278082923e_rule] :
      ( ( member_fm @ Z2 @ ( produc6871130354493381310set_fm @ C2 @ P ) )
     => ~ ! [X: rule,Y: rule] :
            ( ( P
              = ( produc1769597910952761123e_rule @ X @ Y ) )
           => ~ ( member_fm @ Z2 @ ( C2 @ X @ Y ) ) ) ) ).

% mem_case_prodE
thf(fact_146_mem__case__prodE,axiom,
    ! [Z2: fm,C2: fm > fm > set_fm,P: product_prod_fm_fm] :
      ( ( member_fm @ Z2 @ ( produc9176972916100549972set_fm @ C2 @ P ) )
     => ~ ! [X: fm,Y: fm] :
            ( ( P
              = ( product_Pair_fm_fm @ X @ Y ) )
           => ~ ( member_fm @ Z2 @ ( C2 @ X @ Y ) ) ) ) ).

% mem_case_prodE
thf(fact_147_mem__case__prodE,axiom,
    ! [Z2: list_fm,C2: rule > rule > set_list_fm,P: produc780299800278082923e_rule] :
      ( ( member_list_fm @ Z2 @ ( produc6150216243082989006ist_fm @ C2 @ P ) )
     => ~ ! [X: rule,Y: rule] :
            ( ( P
              = ( produc1769597910952761123e_rule @ X @ Y ) )
           => ~ ( member_list_fm @ Z2 @ ( C2 @ X @ Y ) ) ) ) ).

% mem_case_prodE
thf(fact_148_mem__case__prodE,axiom,
    ! [Z2: set_fm,C2: rule > rule > set_set_fm,P: produc780299800278082923e_rule] :
      ( ( member_set_fm @ Z2 @ ( produc3470636841925650932set_fm @ C2 @ P ) )
     => ~ ! [X: rule,Y: rule] :
            ( ( P
              = ( produc1769597910952761123e_rule @ X @ Y ) )
           => ~ ( member_set_fm @ Z2 @ ( C2 @ X @ Y ) ) ) ) ).

% mem_case_prodE
thf(fact_149_mem__case__prodE,axiom,
    ! [Z2: list_fm,C2: fm > fm > set_list_fm,P: product_prod_fm_fm] :
      ( ( member_list_fm @ Z2 @ ( produc4770878748755606884ist_fm @ C2 @ P ) )
     => ~ ! [X: fm,Y: fm] :
            ( ( P
              = ( product_Pair_fm_fm @ X @ Y ) )
           => ~ ( member_list_fm @ Z2 @ ( C2 @ X @ Y ) ) ) ) ).

% mem_case_prodE
thf(fact_150_mem__case__prodE,axiom,
    ! [Z2: set_fm,C2: fm > fm > set_set_fm,P: product_prod_fm_fm] :
      ( ( member_set_fm @ Z2 @ ( produc74870073573622922set_fm @ C2 @ P ) )
     => ~ ! [X: fm,Y: fm] :
            ( ( P
              = ( product_Pair_fm_fm @ X @ Y ) )
           => ~ ( member_set_fm @ Z2 @ ( C2 @ X @ Y ) ) ) ) ).

% mem_case_prodE
thf(fact_151_mem__case__prodE,axiom,
    ! [Z2: fm,C2: list_fm > list_fm > set_fm,P: produc1996495991257130529ist_fm] :
      ( ( member_fm @ Z2 @ ( produc4802205653939627508set_fm @ C2 @ P ) )
     => ~ ! [X: list_fm,Y: list_fm] :
            ( ( P
              = ( produc381145313068854617ist_fm @ X @ Y ) )
           => ~ ( member_fm @ Z2 @ ( C2 @ X @ Y ) ) ) ) ).

% mem_case_prodE
thf(fact_152_mem__case__prodE,axiom,
    ! [Z2: fm,C2: set_fm > set_fm > set_fm,P: produc1316288955051664109set_fm] :
      ( ( member_fm @ Z2 @ ( produc8077690151231853760set_fm @ C2 @ P ) )
     => ~ ! [X: set_fm,Y: set_fm] :
            ( ( P
              = ( produc3030414577503514149set_fm @ X @ Y ) )
           => ~ ( member_fm @ Z2 @ ( C2 @ X @ Y ) ) ) ) ).

% mem_case_prodE
thf(fact_153_mem__case__prodE,axiom,
    ! [Z2: list_fm,C2: list_fm > list_fm > set_list_fm,P: produc1996495991257130529ist_fm] :
      ( ( member_list_fm @ Z2 @ ( produc492036894160897540ist_fm @ C2 @ P ) )
     => ~ ! [X: list_fm,Y: list_fm] :
            ( ( P
              = ( produc381145313068854617ist_fm @ X @ Y ) )
           => ~ ( member_list_fm @ Z2 @ ( C2 @ X @ Y ) ) ) ) ).

% mem_case_prodE
thf(fact_154_mem__case__prodE,axiom,
    ! [Z2: set_fm,C2: list_fm > list_fm > set_set_fm,P: produc1996495991257130529ist_fm] :
      ( ( member_set_fm @ Z2 @ ( produc5469387797198216490set_fm @ C2 @ P ) )
     => ~ ! [X: list_fm,Y: list_fm] :
            ( ( P
              = ( produc381145313068854617ist_fm @ X @ Y ) )
           => ~ ( member_set_fm @ Z2 @ ( C2 @ X @ Y ) ) ) ) ).

% mem_case_prodE
thf(fact_155_fst__conv,axiom,
    ! [X1: set_fm,X22: set_fm] :
      ( ( produc1719312957141167633set_fm @ ( produc3030414577503514149set_fm @ X1 @ X22 ) )
      = X1 ) ).

% fst_conv
thf(fact_156_fst__conv,axiom,
    ! [X1: rule,X22: produc1996495991257130529ist_fm] :
      ( ( produc4228417144610596823ist_fm @ ( produc7063755442767259331ist_fm @ X1 @ X22 ) )
      = X1 ) ).

% fst_conv
thf(fact_157_fst__conv,axiom,
    ! [X1: produc1996495991257130529ist_fm,X22: produc1996495991257130529ist_fm] :
      ( ( produc5894184038225292483ist_fm @ ( produc2592314732767339223ist_fm @ X1 @ X22 ) )
      = X1 ) ).

% fst_conv
thf(fact_158_fst__conv,axiom,
    ! [X1: rule,X22: rule] :
      ( ( produc7264528503596239631e_rule @ ( produc1769597910952761123e_rule @ X1 @ X22 ) )
      = X1 ) ).

% fst_conv
thf(fact_159_fst__conv,axiom,
    ! [X1: fm,X22: fm] :
      ( ( product_fst_fm_fm @ ( product_Pair_fm_fm @ X1 @ X22 ) )
      = X1 ) ).

% fst_conv
thf(fact_160_fst__conv,axiom,
    ! [X1: produc1996495991257130529ist_fm,X22: rule] :
      ( ( produc6879501374131015971m_rule @ ( produc491467635432902671m_rule @ X1 @ X22 ) )
      = X1 ) ).

% fst_conv
thf(fact_161_fst__conv,axiom,
    ! [X1: list_fm,X22: list_fm] :
      ( ( produc1501393135466168645ist_fm @ ( produc381145313068854617ist_fm @ X1 @ X22 ) )
      = X1 ) ).

% fst_conv
thf(fact_162_fst__eqD,axiom,
    ! [X2: set_fm,Y3: set_fm,A: set_fm] :
      ( ( ( produc1719312957141167633set_fm @ ( produc3030414577503514149set_fm @ X2 @ Y3 ) )
        = A )
     => ( X2 = A ) ) ).

% fst_eqD
thf(fact_163_fst__eqD,axiom,
    ! [X2: rule,Y3: produc1996495991257130529ist_fm,A: rule] :
      ( ( ( produc4228417144610596823ist_fm @ ( produc7063755442767259331ist_fm @ X2 @ Y3 ) )
        = A )
     => ( X2 = A ) ) ).

% fst_eqD
thf(fact_164_fst__eqD,axiom,
    ! [X2: produc1996495991257130529ist_fm,Y3: produc1996495991257130529ist_fm,A: produc1996495991257130529ist_fm] :
      ( ( ( produc5894184038225292483ist_fm @ ( produc2592314732767339223ist_fm @ X2 @ Y3 ) )
        = A )
     => ( X2 = A ) ) ).

% fst_eqD
thf(fact_165_fst__eqD,axiom,
    ! [X2: rule,Y3: rule,A: rule] :
      ( ( ( produc7264528503596239631e_rule @ ( produc1769597910952761123e_rule @ X2 @ Y3 ) )
        = A )
     => ( X2 = A ) ) ).

% fst_eqD
thf(fact_166_fst__eqD,axiom,
    ! [X2: fm,Y3: fm,A: fm] :
      ( ( ( product_fst_fm_fm @ ( product_Pair_fm_fm @ X2 @ Y3 ) )
        = A )
     => ( X2 = A ) ) ).

% fst_eqD
thf(fact_167_fst__eqD,axiom,
    ! [X2: produc1996495991257130529ist_fm,Y3: rule,A: produc1996495991257130529ist_fm] :
      ( ( ( produc6879501374131015971m_rule @ ( produc491467635432902671m_rule @ X2 @ Y3 ) )
        = A )
     => ( X2 = A ) ) ).

% fst_eqD
thf(fact_168_fst__eqD,axiom,
    ! [X2: list_fm,Y3: list_fm,A: list_fm] :
      ( ( ( produc1501393135466168645ist_fm @ ( produc381145313068854617ist_fm @ X2 @ Y3 ) )
        = A )
     => ( X2 = A ) ) ).

% fst_eqD
thf(fact_169_snd__conv,axiom,
    ! [X1: set_fm,X22: set_fm] :
      ( ( produc6202189304966566995set_fm @ ( produc3030414577503514149set_fm @ X1 @ X22 ) )
      = X22 ) ).

% snd_conv
thf(fact_170_snd__conv,axiom,
    ! [X1: produc1996495991257130529ist_fm,X22: produc1996495991257130529ist_fm] :
      ( ( produc1639249265332130309ist_fm @ ( produc2592314732767339223ist_fm @ X1 @ X22 ) )
      = X22 ) ).

% snd_conv
thf(fact_171_snd__conv,axiom,
    ! [X1: rule,X22: rule] :
      ( ( produc6725306161561501521e_rule @ ( produc1769597910952761123e_rule @ X1 @ X22 ) )
      = X22 ) ).

% snd_conv
thf(fact_172_snd__conv,axiom,
    ! [X1: fm,X22: fm] :
      ( ( product_snd_fm_fm @ ( product_Pair_fm_fm @ X1 @ X22 ) )
      = X22 ) ).

% snd_conv
thf(fact_173_snd__conv,axiom,
    ! [X1: produc164195504107695125m_rule,X22: fm] :
      ( ( produc8694862632519904556ule_fm @ ( produc803751574383482622ule_fm @ X1 @ X22 ) )
      = X22 ) ).

% snd_conv
thf(fact_174_snd__conv,axiom,
    ! [X1: list_fm,X22: list_fm] :
      ( ( produc4588648349897876871ist_fm @ ( produc381145313068854617ist_fm @ X1 @ X22 ) )
      = X22 ) ).

% snd_conv
thf(fact_175_snd__conv,axiom,
    ! [X1: produc1996495991257130529ist_fm,X22: rule] :
      ( ( produc7165828336582415457m_rule @ ( produc491467635432902671m_rule @ X1 @ X22 ) )
      = X22 ) ).

% snd_conv
thf(fact_176_snd__conv,axiom,
    ! [X1: rule,X22: produc1996495991257130529ist_fm] :
      ( ( produc4514744107061996309ist_fm @ ( produc7063755442767259331ist_fm @ X1 @ X22 ) )
      = X22 ) ).

% snd_conv
thf(fact_177_snd__eqD,axiom,
    ! [X2: set_fm,Y3: set_fm,A: set_fm] :
      ( ( ( produc6202189304966566995set_fm @ ( produc3030414577503514149set_fm @ X2 @ Y3 ) )
        = A )
     => ( Y3 = A ) ) ).

% snd_eqD
thf(fact_178_snd__eqD,axiom,
    ! [X2: produc1996495991257130529ist_fm,Y3: produc1996495991257130529ist_fm,A: produc1996495991257130529ist_fm] :
      ( ( ( produc1639249265332130309ist_fm @ ( produc2592314732767339223ist_fm @ X2 @ Y3 ) )
        = A )
     => ( Y3 = A ) ) ).

% snd_eqD
thf(fact_179_snd__eqD,axiom,
    ! [X2: rule,Y3: rule,A: rule] :
      ( ( ( produc6725306161561501521e_rule @ ( produc1769597910952761123e_rule @ X2 @ Y3 ) )
        = A )
     => ( Y3 = A ) ) ).

% snd_eqD
thf(fact_180_snd__eqD,axiom,
    ! [X2: fm,Y3: fm,A: fm] :
      ( ( ( product_snd_fm_fm @ ( product_Pair_fm_fm @ X2 @ Y3 ) )
        = A )
     => ( Y3 = A ) ) ).

% snd_eqD
thf(fact_181_snd__eqD,axiom,
    ! [X2: produc164195504107695125m_rule,Y3: fm,A: fm] :
      ( ( ( produc8694862632519904556ule_fm @ ( produc803751574383482622ule_fm @ X2 @ Y3 ) )
        = A )
     => ( Y3 = A ) ) ).

% snd_eqD
thf(fact_182_snd__eqD,axiom,
    ! [X2: list_fm,Y3: list_fm,A: list_fm] :
      ( ( ( produc4588648349897876871ist_fm @ ( produc381145313068854617ist_fm @ X2 @ Y3 ) )
        = A )
     => ( Y3 = A ) ) ).

% snd_eqD
thf(fact_183_snd__eqD,axiom,
    ! [X2: produc1996495991257130529ist_fm,Y3: rule,A: rule] :
      ( ( ( produc7165828336582415457m_rule @ ( produc491467635432902671m_rule @ X2 @ Y3 ) )
        = A )
     => ( Y3 = A ) ) ).

% snd_eqD
thf(fact_184_snd__eqD,axiom,
    ! [X2: rule,Y3: produc1996495991257130529ist_fm,A: produc1996495991257130529ist_fm] :
      ( ( ( produc4514744107061996309ist_fm @ ( produc7063755442767259331ist_fm @ X2 @ Y3 ) )
        = A )
     => ( Y3 = A ) ) ).

% snd_eqD
thf(fact_185_prod_Osplit__sel__asm,axiom,
    ! [P2: produc1996495991257130529ist_fm > $o,F2: rule > produc1996495991257130529ist_fm > produc1996495991257130529ist_fm,Prod: produc4630866025709511057ist_fm] :
      ( ( P2 @ ( produc2632167636358383921ist_fm @ F2 @ Prod ) )
      = ( ~ ( ( Prod
              = ( produc7063755442767259331ist_fm @ ( produc4228417144610596823ist_fm @ Prod ) @ ( produc4514744107061996309ist_fm @ Prod ) ) )
            & ~ ( P2 @ ( F2 @ ( produc4228417144610596823ist_fm @ Prod ) @ ( produc4514744107061996309ist_fm @ Prod ) ) ) ) ) ) ).

% prod.split_sel_asm
thf(fact_186_prod_Osplit__sel__asm,axiom,
    ! [P2: $o > $o,F2: rule > produc1996495991257130529ist_fm > $o,Prod: produc4630866025709511057ist_fm] :
      ( ( P2 @ ( produc1628407760719572648t_fm_o @ F2 @ Prod ) )
      = ( ~ ( ( Prod
              = ( produc7063755442767259331ist_fm @ ( produc4228417144610596823ist_fm @ Prod ) @ ( produc4514744107061996309ist_fm @ Prod ) ) )
            & ~ ( P2 @ ( F2 @ ( produc4228417144610596823ist_fm @ Prod ) @ ( produc4514744107061996309ist_fm @ Prod ) ) ) ) ) ) ).

% prod.split_sel_asm
thf(fact_187_prod_Osplit__sel__asm,axiom,
    ! [P2: produc1996495991257130529ist_fm > $o,F2: list_fm > list_fm > produc1996495991257130529ist_fm,Prod: produc1996495991257130529ist_fm] :
      ( ( P2 @ ( produc4544959197559615795ist_fm @ F2 @ Prod ) )
      = ( ~ ( ( Prod
              = ( produc381145313068854617ist_fm @ ( produc1501393135466168645ist_fm @ Prod ) @ ( produc4588648349897876871ist_fm @ Prod ) ) )
            & ~ ( P2 @ ( F2 @ ( produc1501393135466168645ist_fm @ Prod ) @ ( produc4588648349897876871ist_fm @ Prod ) ) ) ) ) ) ).

% prod.split_sel_asm
thf(fact_188_prod_Osplit__sel__asm,axiom,
    ! [P2: list_fm > $o,F2: list_fm > list_fm > list_fm,Prod: produc1996495991257130529ist_fm] :
      ( ( P2 @ ( produc4154610405322092878ist_fm @ F2 @ Prod ) )
      = ( ~ ( ( Prod
              = ( produc381145313068854617ist_fm @ ( produc1501393135466168645ist_fm @ Prod ) @ ( produc4588648349897876871ist_fm @ Prod ) ) )
            & ~ ( P2 @ ( F2 @ ( produc1501393135466168645ist_fm @ Prod ) @ ( produc4588648349897876871ist_fm @ Prod ) ) ) ) ) ) ).

% prod.split_sel_asm
thf(fact_189_prod_Osplit__sel__asm,axiom,
    ! [P2: $o > $o,F2: list_fm > list_fm > $o,Prod: produc1996495991257130529ist_fm] :
      ( ( P2 @ ( produc2011568048573023402t_fm_o @ F2 @ Prod ) )
      = ( ~ ( ( Prod
              = ( produc381145313068854617ist_fm @ ( produc1501393135466168645ist_fm @ Prod ) @ ( produc4588648349897876871ist_fm @ Prod ) ) )
            & ~ ( P2 @ ( F2 @ ( produc1501393135466168645ist_fm @ Prod ) @ ( produc4588648349897876871ist_fm @ Prod ) ) ) ) ) ) ).

% prod.split_sel_asm
thf(fact_190_prod_Osplit__sel,axiom,
    ! [P2: produc1996495991257130529ist_fm > $o,F2: rule > produc1996495991257130529ist_fm > produc1996495991257130529ist_fm,Prod: produc4630866025709511057ist_fm] :
      ( ( P2 @ ( produc2632167636358383921ist_fm @ F2 @ Prod ) )
      = ( ( Prod
          = ( produc7063755442767259331ist_fm @ ( produc4228417144610596823ist_fm @ Prod ) @ ( produc4514744107061996309ist_fm @ Prod ) ) )
       => ( P2 @ ( F2 @ ( produc4228417144610596823ist_fm @ Prod ) @ ( produc4514744107061996309ist_fm @ Prod ) ) ) ) ) ).

% prod.split_sel
thf(fact_191_prod_Osplit__sel,axiom,
    ! [P2: $o > $o,F2: rule > produc1996495991257130529ist_fm > $o,Prod: produc4630866025709511057ist_fm] :
      ( ( P2 @ ( produc1628407760719572648t_fm_o @ F2 @ Prod ) )
      = ( ( Prod
          = ( produc7063755442767259331ist_fm @ ( produc4228417144610596823ist_fm @ Prod ) @ ( produc4514744107061996309ist_fm @ Prod ) ) )
       => ( P2 @ ( F2 @ ( produc4228417144610596823ist_fm @ Prod ) @ ( produc4514744107061996309ist_fm @ Prod ) ) ) ) ) ).

% prod.split_sel
thf(fact_192_prod_Osplit__sel,axiom,
    ! [P2: produc1996495991257130529ist_fm > $o,F2: list_fm > list_fm > produc1996495991257130529ist_fm,Prod: produc1996495991257130529ist_fm] :
      ( ( P2 @ ( produc4544959197559615795ist_fm @ F2 @ Prod ) )
      = ( ( Prod
          = ( produc381145313068854617ist_fm @ ( produc1501393135466168645ist_fm @ Prod ) @ ( produc4588648349897876871ist_fm @ Prod ) ) )
       => ( P2 @ ( F2 @ ( produc1501393135466168645ist_fm @ Prod ) @ ( produc4588648349897876871ist_fm @ Prod ) ) ) ) ) ).

% prod.split_sel
thf(fact_193_prod_Osplit__sel,axiom,
    ! [P2: list_fm > $o,F2: list_fm > list_fm > list_fm,Prod: produc1996495991257130529ist_fm] :
      ( ( P2 @ ( produc4154610405322092878ist_fm @ F2 @ Prod ) )
      = ( ( Prod
          = ( produc381145313068854617ist_fm @ ( produc1501393135466168645ist_fm @ Prod ) @ ( produc4588648349897876871ist_fm @ Prod ) ) )
       => ( P2 @ ( F2 @ ( produc1501393135466168645ist_fm @ Prod ) @ ( produc4588648349897876871ist_fm @ Prod ) ) ) ) ) ).

% prod.split_sel
thf(fact_194_prod_Osplit__sel,axiom,
    ! [P2: $o > $o,F2: list_fm > list_fm > $o,Prod: produc1996495991257130529ist_fm] :
      ( ( P2 @ ( produc2011568048573023402t_fm_o @ F2 @ Prod ) )
      = ( ( Prod
          = ( produc381145313068854617ist_fm @ ( produc1501393135466168645ist_fm @ Prod ) @ ( produc4588648349897876871ist_fm @ Prod ) ) )
       => ( P2 @ ( F2 @ ( produc1501393135466168645ist_fm @ Prod ) @ ( produc4588648349897876871ist_fm @ Prod ) ) ) ) ) ).

% prod.split_sel
thf(fact_195_prod_Oswap__def,axiom,
    ( produc7775435602796846021set_fm
    = ( ^ [P4: produc1316288955051664109set_fm] : ( produc3030414577503514149set_fm @ ( produc6202189304966566995set_fm @ P4 ) @ ( produc1719312957141167633set_fm @ P4 ) ) ) ) ).

% prod.swap_def
thf(fact_196_prod_Oswap__def,axiom,
    ( produc3697025406284169335ist_fm
    = ( ^ [P4: produc1754732471204517991ist_fm] : ( produc2592314732767339223ist_fm @ ( produc1639249265332130309ist_fm @ P4 ) @ ( produc5894184038225292483ist_fm @ P4 ) ) ) ) ).

% prod.swap_def
thf(fact_197_prod_Oswap__def,axiom,
    ( produc2320433982106674371e_rule
    = ( ^ [P4: produc780299800278082923e_rule] : ( produc1769597910952761123e_rule @ ( produc6725306161561501521e_rule @ P4 ) @ ( produc7264528503596239631e_rule @ P4 ) ) ) ) ).

% prod.swap_def
thf(fact_198_prod_Oswap__def,axiom,
    ( product_swap_fm_fm
    = ( ^ [P4: product_prod_fm_fm] : ( product_Pair_fm_fm @ ( product_snd_fm_fm @ P4 ) @ ( product_fst_fm_fm @ P4 ) ) ) ) ).

% prod.swap_def
thf(fact_199_prod_Oswap__def,axiom,
    ( produc9128311380267582622ule_fm
    = ( ^ [P4: produc7564083655194549958ule_fm] : ( produc9036542945914740094m_rule @ ( produc8694862632519904556ule_fm @ P4 ) @ ( produc1758149581174257386ule_fm @ P4 ) ) ) ) ).

% prod.swap_def
thf(fact_200_prod_Oswap__def,axiom,
    ( produc7936585717479832313ist_fm
    = ( ^ [P4: produc1996495991257130529ist_fm] : ( produc381145313068854617ist_fm @ ( produc4588648349897876871ist_fm @ P4 ) @ ( produc1501393135466168645ist_fm @ P4 ) ) ) ) ).

% prod.swap_def
thf(fact_201_prod_Oswap__def,axiom,
    ( produc9213896635423732003ist_fm
    = ( ^ [P4: produc4630866025709511057ist_fm] : ( produc491467635432902671m_rule @ ( produc4514744107061996309ist_fm @ P4 ) @ ( produc4228417144610596823ist_fm @ P4 ) ) ) ) ).

% prod.swap_def
thf(fact_202_prod_Oswap__def,axiom,
    ( produc2641608828089375343m_rule
    = ( ^ [P4: produc164195504107695125m_rule] : ( produc7063755442767259331ist_fm @ ( produc7165828336582415457m_rule @ P4 ) @ ( produc6879501374131015971m_rule @ P4 ) ) ) ) ).

% prod.swap_def
thf(fact_203_szip_Osimps_I1_J,axiom,
    ! [S1: stream8299795917829157543ist_fm,S2: stream_rule] :
      ( ( shd_Pr7235097944458474089m_rule @ ( szip_P1977448745965526924m_rule @ S1 @ S2 ) )
      = ( produc491467635432902671m_rule @ ( shd_Pr772355297128350925ist_fm @ S1 ) @ ( shd_rule @ S2 ) ) ) ).

% szip.simps(1)
thf(fact_204_szip_Osimps_I1_J,axiom,
    ! [S1: stream727092118206550309m_rule,S2: stream727092118206550309m_rule] :
      ( ( shd_Pr2340755956080993207m_rule @ ( szip_P2499414959592755846m_rule @ S1 @ S2 ) )
      = ( produc5927390650430071747m_rule @ ( shd_Pr7235097944458474089m_rule @ S1 ) @ ( shd_Pr7235097944458474089m_rule @ S2 ) ) ) ).

% szip.simps(1)
thf(fact_205_szip_Osimps_I1_J,axiom,
    ! [S1: stream_rule,S2: stream_rule] :
      ( ( shd_Pr4573454438745646103e_rule @ ( szip_rule_rule @ S1 @ S2 ) )
      = ( produc1769597910952761123e_rule @ ( shd_rule @ S1 ) @ ( shd_rule @ S2 ) ) ) ).

% szip.simps(1)
thf(fact_206_szip_Osimps_I1_J,axiom,
    ! [S1: stream_rule,S2: stream_fm] :
      ( ( shd_Pr2491830098381126178ule_fm @ ( szip_rule_fm @ S1 @ S2 ) )
      = ( product_Pair_rule_fm @ ( shd_rule @ S1 ) @ ( shd_fm @ S2 ) ) ) ).

% szip.simps(1)
thf(fact_207_szip_Osimps_I1_J,axiom,
    ! [S1: stream_fm,S2: stream_rule] :
      ( ( shd_Pr1428355801142630306m_rule @ ( szip_fm_rule @ S1 @ S2 ) )
      = ( product_Pair_fm_rule @ ( shd_fm @ S1 ) @ ( shd_rule @ S2 ) ) ) ).

% szip.simps(1)
thf(fact_208_szip_Osimps_I1_J,axiom,
    ! [S1: stream_fm,S2: stream_fm] :
      ( ( shd_Pr8144422112483312429_fm_fm @ ( szip_fm_fm @ S1 @ S2 ) )
      = ( product_Pair_fm_fm @ ( shd_fm @ S1 ) @ ( shd_fm @ S2 ) ) ) ).

% szip.simps(1)
thf(fact_209_szip_Osimps_I1_J,axiom,
    ! [S1: stream_set_fm,S2: stream_set_fm] :
      ( ( shd_Pr8744048338668924313set_fm @ ( szip_set_fm_set_fm @ S1 @ S2 ) )
      = ( produc3030414577503514149set_fm @ ( shd_set_fm @ S1 ) @ ( shd_set_fm @ S2 ) ) ) ).

% szip.simps(1)
thf(fact_210_szip_Osimps_I1_J,axiom,
    ! [S1: stream_list_fm,S2: stream_list_fm] :
      ( ( shd_Pr772355297128350925ist_fm @ ( szip_list_fm_list_fm @ S1 @ S2 ) )
      = ( produc381145313068854617ist_fm @ ( shd_list_fm @ S1 ) @ ( shd_list_fm @ S2 ) ) ) ).

% szip.simps(1)
thf(fact_211_szip_Osimps_I1_J,axiom,
    ! [S1: stream_rule,S2: stream8299795917829157543ist_fm] :
      ( ( shd_Pr2478396429205514213ist_fm @ ( szip_r8549736553299883584ist_fm @ S1 @ S2 ) )
      = ( produc7063755442767259331ist_fm @ ( shd_rule @ S1 ) @ ( shd_Pr772355297128350925ist_fm @ S2 ) ) ) ).

% szip.simps(1)
thf(fact_212_szip_Osimps_I1_J,axiom,
    ! [S1: stream8299795917829157543ist_fm,S2: stream_fm] :
      ( ( shd_Pr7265661315679113588_fm_fm @ ( szip_P3148547612769835799_fm_fm @ S1 @ S2 ) )
      = ( produc1957533461433998618_fm_fm @ ( shd_Pr772355297128350925ist_fm @ S1 ) @ ( shd_fm @ S2 ) ) ) ).

% szip.simps(1)
thf(fact_213_Product__Type_OCollect__case__prodD,axiom,
    ! [X2: produc1754732471204517991ist_fm,A4: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm > $o] :
      ( ( member60238275747228048ist_fm @ X2 @ ( collec888080998205339090ist_fm @ ( produc2796214937669505452t_fm_o @ A4 ) ) )
     => ( A4 @ ( produc5894184038225292483ist_fm @ X2 ) @ ( produc1639249265332130309ist_fm @ X2 ) ) ) ).

% Product_Type.Collect_case_prodD
thf(fact_214_Product__Type_OCollect__case__prodD,axiom,
    ! [X2: produc780299800278082923e_rule,A4: rule > rule > $o] :
      ( ( member3753523862196722196e_rule @ X2 @ ( collec5224624948449408342e_rule @ ( produc8052491652537269344rule_o @ A4 ) ) )
     => ( A4 @ ( produc7264528503596239631e_rule @ X2 ) @ ( produc6725306161561501521e_rule @ X2 ) ) ) ).

% Product_Type.Collect_case_prodD
thf(fact_215_Product__Type_OCollect__case__prodD,axiom,
    ! [X2: product_prod_fm_fm,A4: fm > fm > $o] :
      ( ( member8474499337054950954_fm_fm @ X2 @ ( collec4425076577877259116_fm_fm @ ( produc2811014937302177098m_fm_o @ A4 ) ) )
     => ( A4 @ ( product_fst_fm_fm @ X2 ) @ ( product_snd_fm_fm @ X2 ) ) ) ).

% Product_Type.Collect_case_prodD
thf(fact_216_Product__Type_OCollect__case__prodD,axiom,
    ! [X2: produc7564083655194549958ule_fm,A4: produc164195504107695125m_rule > fm > $o] :
      ( ( member7952525889772502127ule_fm @ X2 @ ( collec7776316114922985905ule_fm @ ( produc3554709660730922949e_fm_o @ A4 ) ) )
     => ( A4 @ ( produc1758149581174257386ule_fm @ X2 ) @ ( produc8694862632519904556ule_fm @ X2 ) ) ) ).

% Product_Type.Collect_case_prodD
thf(fact_217_Product__Type_OCollect__case__prodD,axiom,
    ! [X2: produc1996495991257130529ist_fm,A4: list_fm > list_fm > $o] :
      ( ( member8102475879199740618ist_fm @ X2 @ ( collec7231575776666803724ist_fm @ ( produc2011568048573023402t_fm_o @ A4 ) ) )
     => ( A4 @ ( produc1501393135466168645ist_fm @ X2 ) @ ( produc4588648349897876871ist_fm @ X2 ) ) ) ).

% Product_Type.Collect_case_prodD
thf(fact_218_Product__Type_OCollect__case__prodD,axiom,
    ! [X2: produc164195504107695125m_rule,A4: produc1996495991257130529ist_fm > rule > $o] :
      ( ( member4220325220686508332m_rule @ X2 @ ( collec2069506698588392810m_rule @ ( produc7369492558480804444rule_o @ A4 ) ) )
     => ( A4 @ ( produc6879501374131015971m_rule @ X2 ) @ ( produc7165828336582415457m_rule @ X2 ) ) ) ).

% Product_Type.Collect_case_prodD
thf(fact_219_Product__Type_OCollect__case__prodD,axiom,
    ! [X2: produc4630866025709511057ist_fm,A4: rule > produc1996495991257130529ist_fm > $o] :
      ( ( member8686995742288324264ist_fm @ X2 @ ( collec6536177220190208742ist_fm @ ( produc1628407760719572648t_fm_o @ A4 ) ) )
     => ( A4 @ ( produc4228417144610596823ist_fm @ X2 ) @ ( produc4514744107061996309ist_fm @ X2 ) ) ) ).

% Product_Type.Collect_case_prodD
thf(fact_220_surjective__pairing,axiom,
    ! [T2: produc1316288955051664109set_fm] :
      ( T2
      = ( produc3030414577503514149set_fm @ ( produc1719312957141167633set_fm @ T2 ) @ ( produc6202189304966566995set_fm @ T2 ) ) ) ).

% surjective_pairing
thf(fact_221_surjective__pairing,axiom,
    ! [T2: produc1754732471204517991ist_fm] :
      ( T2
      = ( produc2592314732767339223ist_fm @ ( produc5894184038225292483ist_fm @ T2 ) @ ( produc1639249265332130309ist_fm @ T2 ) ) ) ).

% surjective_pairing
thf(fact_222_surjective__pairing,axiom,
    ! [T2: produc780299800278082923e_rule] :
      ( T2
      = ( produc1769597910952761123e_rule @ ( produc7264528503596239631e_rule @ T2 ) @ ( produc6725306161561501521e_rule @ T2 ) ) ) ).

% surjective_pairing
thf(fact_223_surjective__pairing,axiom,
    ! [T2: product_prod_fm_fm] :
      ( T2
      = ( product_Pair_fm_fm @ ( product_fst_fm_fm @ T2 ) @ ( product_snd_fm_fm @ T2 ) ) ) ).

% surjective_pairing
thf(fact_224_surjective__pairing,axiom,
    ! [T2: produc7564083655194549958ule_fm] :
      ( T2
      = ( produc803751574383482622ule_fm @ ( produc1758149581174257386ule_fm @ T2 ) @ ( produc8694862632519904556ule_fm @ T2 ) ) ) ).

% surjective_pairing
thf(fact_225_surjective__pairing,axiom,
    ! [T2: produc1996495991257130529ist_fm] :
      ( T2
      = ( produc381145313068854617ist_fm @ ( produc1501393135466168645ist_fm @ T2 ) @ ( produc4588648349897876871ist_fm @ T2 ) ) ) ).

% surjective_pairing
thf(fact_226_surjective__pairing,axiom,
    ! [T2: produc164195504107695125m_rule] :
      ( T2
      = ( produc491467635432902671m_rule @ ( produc6879501374131015971m_rule @ T2 ) @ ( produc7165828336582415457m_rule @ T2 ) ) ) ).

% surjective_pairing
thf(fact_227_surjective__pairing,axiom,
    ! [T2: produc4630866025709511057ist_fm] :
      ( T2
      = ( produc7063755442767259331ist_fm @ ( produc4228417144610596823ist_fm @ T2 ) @ ( produc4514744107061996309ist_fm @ T2 ) ) ) ).

% surjective_pairing
thf(fact_228_mem__Collect__eq,axiom,
    ! [A: set_Pr1008144964186165195m_rule,P2: set_Pr1008144964186165195m_rule > $o] :
      ( ( member7979558945778333922m_rule @ A @ ( collec6561902996944018720m_rule @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_229_mem__Collect__eq,axiom,
    ! [A: set_fm,P2: set_fm > $o] :
      ( ( member_set_fm @ A @ ( collect_set_fm @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_230_mem__Collect__eq,axiom,
    ! [A: list_P8332459875677254181m_rule,P2: list_P8332459875677254181m_rule > $o] :
      ( ( member1784032718715961276m_rule @ A @ ( collec5373559803083089146m_rule @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_231_mem__Collect__eq,axiom,
    ! [A: list_P342308711384885665ist_fm,P2: list_P342308711384885665ist_fm > $o] :
      ( ( member3017253591278368568ist_fm @ A @ ( collec6606780675645496438ist_fm @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_232_mem__Collect__eq,axiom,
    ! [A: list_P5616295576739893671ist_fm,P2: list_P5616295576739893671ist_fm > $o] :
      ( ( member8448403643636553168ist_fm @ A @ ( collec2128064864322541586ist_fm @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_233_mem__Collect__eq,axiom,
    ! [A: list_fm,P2: list_fm > $o] :
      ( ( member_list_fm @ A @ ( collect_list_fm @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_234_mem__Collect__eq,axiom,
    ! [A: fm,P2: fm > $o] :
      ( ( member_fm @ A @ ( collect_fm @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_235_Collect__mem__eq,axiom,
    ! [A4: set_se6866588098022362241m_rule] :
      ( ( collec6561902996944018720m_rule
        @ ^ [X3: set_Pr1008144964186165195m_rule] : ( member7979558945778333922m_rule @ X3 @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_236_Collect__mem__eq,axiom,
    ! [A4: set_set_fm] :
      ( ( collect_set_fm
        @ ^ [X3: set_fm] : ( member_set_fm @ X3 @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_237_Collect__mem__eq,axiom,
    ! [A4: set_li5339030221936397403m_rule] :
      ( ( collec5373559803083089146m_rule
        @ ^ [X3: list_P8332459875677254181m_rule] : ( member1784032718715961276m_rule @ X3 @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_238_Collect__mem__eq,axiom,
    ! [A4: set_li346003458413430231ist_fm] :
      ( ( collec6606780675645496438ist_fm
        @ ^ [X3: list_P342308711384885665ist_fm] : ( member3017253591278368568ist_fm @ X3 @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_239_Collect__mem__eq,axiom,
    ! [A4: set_li5537492308219346823ist_fm] :
      ( ( collec2128064864322541586ist_fm
        @ ^ [X3: list_P5616295576739893671ist_fm] : ( member8448403643636553168ist_fm @ X3 @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_240_Collect__mem__eq,axiom,
    ! [A4: set_list_fm] :
      ( ( collect_list_fm
        @ ^ [X3: list_fm] : ( member_list_fm @ X3 @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_241_Collect__mem__eq,axiom,
    ! [A4: set_fm] :
      ( ( collect_fm
        @ ^ [X3: fm] : ( member_fm @ X3 @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_242_Collect__cong,axiom,
    ! [P2: fm > $o,Q3: fm > $o] :
      ( ! [X: fm] :
          ( ( P2 @ X )
          = ( Q3 @ X ) )
     => ( ( collect_fm @ P2 )
        = ( collect_fm @ Q3 ) ) ) ).

% Collect_cong
thf(fact_243_Collect__cong,axiom,
    ! [P2: list_P8332459875677254181m_rule > $o,Q3: list_P8332459875677254181m_rule > $o] :
      ( ! [X: list_P8332459875677254181m_rule] :
          ( ( P2 @ X )
          = ( Q3 @ X ) )
     => ( ( collec5373559803083089146m_rule @ P2 )
        = ( collec5373559803083089146m_rule @ Q3 ) ) ) ).

% Collect_cong
thf(fact_244_Collect__cong,axiom,
    ! [P2: list_P342308711384885665ist_fm > $o,Q3: list_P342308711384885665ist_fm > $o] :
      ( ! [X: list_P342308711384885665ist_fm] :
          ( ( P2 @ X )
          = ( Q3 @ X ) )
     => ( ( collec6606780675645496438ist_fm @ P2 )
        = ( collec6606780675645496438ist_fm @ Q3 ) ) ) ).

% Collect_cong
thf(fact_245_Collect__cong,axiom,
    ! [P2: list_P5616295576739893671ist_fm > $o,Q3: list_P5616295576739893671ist_fm > $o] :
      ( ! [X: list_P5616295576739893671ist_fm] :
          ( ( P2 @ X )
          = ( Q3 @ X ) )
     => ( ( collec2128064864322541586ist_fm @ P2 )
        = ( collec2128064864322541586ist_fm @ Q3 ) ) ) ).

% Collect_cong
thf(fact_246_Collect__cong,axiom,
    ! [P2: list_fm > $o,Q3: list_fm > $o] :
      ( ! [X: list_fm] :
          ( ( P2 @ X )
          = ( Q3 @ X ) )
     => ( ( collect_list_fm @ P2 )
        = ( collect_list_fm @ Q3 ) ) ) ).

% Collect_cong
thf(fact_247_conjI__realizer,axiom,
    ! [P2: set_fm > $o,P: set_fm,Q3: set_fm > $o,Q: set_fm] :
      ( ( P2 @ P )
     => ( ( Q3 @ Q )
       => ( ( P2 @ ( produc1719312957141167633set_fm @ ( produc3030414577503514149set_fm @ P @ Q ) ) )
          & ( Q3 @ ( produc6202189304966566995set_fm @ ( produc3030414577503514149set_fm @ P @ Q ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_248_conjI__realizer,axiom,
    ! [P2: produc1996495991257130529ist_fm > $o,P: produc1996495991257130529ist_fm,Q3: produc1996495991257130529ist_fm > $o,Q: produc1996495991257130529ist_fm] :
      ( ( P2 @ P )
     => ( ( Q3 @ Q )
       => ( ( P2 @ ( produc5894184038225292483ist_fm @ ( produc2592314732767339223ist_fm @ P @ Q ) ) )
          & ( Q3 @ ( produc1639249265332130309ist_fm @ ( produc2592314732767339223ist_fm @ P @ Q ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_249_conjI__realizer,axiom,
    ! [P2: rule > $o,P: rule,Q3: rule > $o,Q: rule] :
      ( ( P2 @ P )
     => ( ( Q3 @ Q )
       => ( ( P2 @ ( produc7264528503596239631e_rule @ ( produc1769597910952761123e_rule @ P @ Q ) ) )
          & ( Q3 @ ( produc6725306161561501521e_rule @ ( produc1769597910952761123e_rule @ P @ Q ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_250_conjI__realizer,axiom,
    ! [P2: fm > $o,P: fm,Q3: fm > $o,Q: fm] :
      ( ( P2 @ P )
     => ( ( Q3 @ Q )
       => ( ( P2 @ ( product_fst_fm_fm @ ( product_Pair_fm_fm @ P @ Q ) ) )
          & ( Q3 @ ( product_snd_fm_fm @ ( product_Pair_fm_fm @ P @ Q ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_251_conjI__realizer,axiom,
    ! [P2: produc164195504107695125m_rule > $o,P: produc164195504107695125m_rule,Q3: fm > $o,Q: fm] :
      ( ( P2 @ P )
     => ( ( Q3 @ Q )
       => ( ( P2 @ ( produc1758149581174257386ule_fm @ ( produc803751574383482622ule_fm @ P @ Q ) ) )
          & ( Q3 @ ( produc8694862632519904556ule_fm @ ( produc803751574383482622ule_fm @ P @ Q ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_252_conjI__realizer,axiom,
    ! [P2: list_fm > $o,P: list_fm,Q3: list_fm > $o,Q: list_fm] :
      ( ( P2 @ P )
     => ( ( Q3 @ Q )
       => ( ( P2 @ ( produc1501393135466168645ist_fm @ ( produc381145313068854617ist_fm @ P @ Q ) ) )
          & ( Q3 @ ( produc4588648349897876871ist_fm @ ( produc381145313068854617ist_fm @ P @ Q ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_253_conjI__realizer,axiom,
    ! [P2: produc1996495991257130529ist_fm > $o,P: produc1996495991257130529ist_fm,Q3: rule > $o,Q: rule] :
      ( ( P2 @ P )
     => ( ( Q3 @ Q )
       => ( ( P2 @ ( produc6879501374131015971m_rule @ ( produc491467635432902671m_rule @ P @ Q ) ) )
          & ( Q3 @ ( produc7165828336582415457m_rule @ ( produc491467635432902671m_rule @ P @ Q ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_254_conjI__realizer,axiom,
    ! [P2: rule > $o,P: rule,Q3: produc1996495991257130529ist_fm > $o,Q: produc1996495991257130529ist_fm] :
      ( ( P2 @ P )
     => ( ( Q3 @ Q )
       => ( ( P2 @ ( produc4228417144610596823ist_fm @ ( produc7063755442767259331ist_fm @ P @ Q ) ) )
          & ( Q3 @ ( produc4514744107061996309ist_fm @ ( produc7063755442767259331ist_fm @ P @ Q ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_255_exI__realizer,axiom,
    ! [P2: set_fm > set_fm > $o,Y3: set_fm,X2: set_fm] :
      ( ( P2 @ Y3 @ X2 )
     => ( P2 @ ( produc6202189304966566995set_fm @ ( produc3030414577503514149set_fm @ X2 @ Y3 ) ) @ ( produc1719312957141167633set_fm @ ( produc3030414577503514149set_fm @ X2 @ Y3 ) ) ) ) ).

% exI_realizer
thf(fact_256_exI__realizer,axiom,
    ! [P2: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm > $o,Y3: produc1996495991257130529ist_fm,X2: produc1996495991257130529ist_fm] :
      ( ( P2 @ Y3 @ X2 )
     => ( P2 @ ( produc1639249265332130309ist_fm @ ( produc2592314732767339223ist_fm @ X2 @ Y3 ) ) @ ( produc5894184038225292483ist_fm @ ( produc2592314732767339223ist_fm @ X2 @ Y3 ) ) ) ) ).

% exI_realizer
thf(fact_257_exI__realizer,axiom,
    ! [P2: rule > rule > $o,Y3: rule,X2: rule] :
      ( ( P2 @ Y3 @ X2 )
     => ( P2 @ ( produc6725306161561501521e_rule @ ( produc1769597910952761123e_rule @ X2 @ Y3 ) ) @ ( produc7264528503596239631e_rule @ ( produc1769597910952761123e_rule @ X2 @ Y3 ) ) ) ) ).

% exI_realizer
thf(fact_258_exI__realizer,axiom,
    ! [P2: fm > fm > $o,Y3: fm,X2: fm] :
      ( ( P2 @ Y3 @ X2 )
     => ( P2 @ ( product_snd_fm_fm @ ( product_Pair_fm_fm @ X2 @ Y3 ) ) @ ( product_fst_fm_fm @ ( product_Pair_fm_fm @ X2 @ Y3 ) ) ) ) ).

% exI_realizer
thf(fact_259_exI__realizer,axiom,
    ! [P2: fm > produc164195504107695125m_rule > $o,Y3: fm,X2: produc164195504107695125m_rule] :
      ( ( P2 @ Y3 @ X2 )
     => ( P2 @ ( produc8694862632519904556ule_fm @ ( produc803751574383482622ule_fm @ X2 @ Y3 ) ) @ ( produc1758149581174257386ule_fm @ ( produc803751574383482622ule_fm @ X2 @ Y3 ) ) ) ) ).

% exI_realizer
thf(fact_260_exI__realizer,axiom,
    ! [P2: list_fm > list_fm > $o,Y3: list_fm,X2: list_fm] :
      ( ( P2 @ Y3 @ X2 )
     => ( P2 @ ( produc4588648349897876871ist_fm @ ( produc381145313068854617ist_fm @ X2 @ Y3 ) ) @ ( produc1501393135466168645ist_fm @ ( produc381145313068854617ist_fm @ X2 @ Y3 ) ) ) ) ).

% exI_realizer
thf(fact_261_exI__realizer,axiom,
    ! [P2: rule > produc1996495991257130529ist_fm > $o,Y3: rule,X2: produc1996495991257130529ist_fm] :
      ( ( P2 @ Y3 @ X2 )
     => ( P2 @ ( produc7165828336582415457m_rule @ ( produc491467635432902671m_rule @ X2 @ Y3 ) ) @ ( produc6879501374131015971m_rule @ ( produc491467635432902671m_rule @ X2 @ Y3 ) ) ) ) ).

% exI_realizer
thf(fact_262_exI__realizer,axiom,
    ! [P2: produc1996495991257130529ist_fm > rule > $o,Y3: produc1996495991257130529ist_fm,X2: rule] :
      ( ( P2 @ Y3 @ X2 )
     => ( P2 @ ( produc4514744107061996309ist_fm @ ( produc7063755442767259331ist_fm @ X2 @ Y3 ) ) @ ( produc4228417144610596823ist_fm @ ( produc7063755442767259331ist_fm @ X2 @ Y3 ) ) ) ) ).

% exI_realizer
thf(fact_263_prod_Oexhaust__sel,axiom,
    ! [Prod: produc1316288955051664109set_fm] :
      ( Prod
      = ( produc3030414577503514149set_fm @ ( produc1719312957141167633set_fm @ Prod ) @ ( produc6202189304966566995set_fm @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_264_prod_Oexhaust__sel,axiom,
    ! [Prod: produc1754732471204517991ist_fm] :
      ( Prod
      = ( produc2592314732767339223ist_fm @ ( produc5894184038225292483ist_fm @ Prod ) @ ( produc1639249265332130309ist_fm @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_265_prod_Oexhaust__sel,axiom,
    ! [Prod: produc780299800278082923e_rule] :
      ( Prod
      = ( produc1769597910952761123e_rule @ ( produc7264528503596239631e_rule @ Prod ) @ ( produc6725306161561501521e_rule @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_266_prod_Oexhaust__sel,axiom,
    ! [Prod: product_prod_fm_fm] :
      ( Prod
      = ( product_Pair_fm_fm @ ( product_fst_fm_fm @ Prod ) @ ( product_snd_fm_fm @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_267_prod_Oexhaust__sel,axiom,
    ! [Prod: produc7564083655194549958ule_fm] :
      ( Prod
      = ( produc803751574383482622ule_fm @ ( produc1758149581174257386ule_fm @ Prod ) @ ( produc8694862632519904556ule_fm @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_268_prod_Oexhaust__sel,axiom,
    ! [Prod: produc1996495991257130529ist_fm] :
      ( Prod
      = ( produc381145313068854617ist_fm @ ( produc1501393135466168645ist_fm @ Prod ) @ ( produc4588648349897876871ist_fm @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_269_prod_Oexhaust__sel,axiom,
    ! [Prod: produc164195504107695125m_rule] :
      ( Prod
      = ( produc491467635432902671m_rule @ ( produc6879501374131015971m_rule @ Prod ) @ ( produc7165828336582415457m_rule @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_270_prod_Oexhaust__sel,axiom,
    ! [Prod: produc4630866025709511057ist_fm] :
      ( Prod
      = ( produc7063755442767259331ist_fm @ ( produc4228417144610596823ist_fm @ Prod ) @ ( produc4514744107061996309ist_fm @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_271_case__prod__beta,axiom,
    ( produc2632167636358383921ist_fm
    = ( ^ [F: rule > produc1996495991257130529ist_fm > produc1996495991257130529ist_fm,P4: produc4630866025709511057ist_fm] : ( F @ ( produc4228417144610596823ist_fm @ P4 ) @ ( produc4514744107061996309ist_fm @ P4 ) ) ) ) ).

% case_prod_beta
thf(fact_272_case__prod__beta,axiom,
    ( produc1628407760719572648t_fm_o
    = ( ^ [F: rule > produc1996495991257130529ist_fm > $o,P4: produc4630866025709511057ist_fm] : ( F @ ( produc4228417144610596823ist_fm @ P4 ) @ ( produc4514744107061996309ist_fm @ P4 ) ) ) ) ).

% case_prod_beta
thf(fact_273_case__prod__beta,axiom,
    ( produc4544959197559615795ist_fm
    = ( ^ [F: list_fm > list_fm > produc1996495991257130529ist_fm,P4: produc1996495991257130529ist_fm] : ( F @ ( produc1501393135466168645ist_fm @ P4 ) @ ( produc4588648349897876871ist_fm @ P4 ) ) ) ) ).

% case_prod_beta
thf(fact_274_case__prod__beta,axiom,
    ( produc4154610405322092878ist_fm
    = ( ^ [F: list_fm > list_fm > list_fm,P4: produc1996495991257130529ist_fm] : ( F @ ( produc1501393135466168645ist_fm @ P4 ) @ ( produc4588648349897876871ist_fm @ P4 ) ) ) ) ).

% case_prod_beta
thf(fact_275_case__prod__beta,axiom,
    ( produc2011568048573023402t_fm_o
    = ( ^ [F: list_fm > list_fm > $o,P4: produc1996495991257130529ist_fm] : ( F @ ( produc1501393135466168645ist_fm @ P4 ) @ ( produc4588648349897876871ist_fm @ P4 ) ) ) ) ).

% case_prod_beta
thf(fact_276_pred__prod__inject,axiom,
    ! [P1: rule > $o,P22: rule > $o,A: rule,B: rule] :
      ( ( basic_5896771853912976575e_rule @ P1 @ P22 @ ( produc1769597910952761123e_rule @ A @ B ) )
      = ( ( P1 @ A )
        & ( P22 @ B ) ) ) ).

% pred_prod_inject
thf(fact_277_pred__prod__inject,axiom,
    ! [P1: set_fm > $o,P22: set_fm > $o,A: set_fm,B: set_fm] :
      ( ( basic_677064363366268865set_fm @ P1 @ P22 @ ( produc3030414577503514149set_fm @ A @ B ) )
      = ( ( P1 @ A )
        & ( P22 @ B ) ) ) ).

% pred_prod_inject
thf(fact_278_pred__prod__inject,axiom,
    ! [P1: rule > $o,P22: produc1996495991257130529ist_fm > $o,A: rule,B: produc1996495991257130529ist_fm] :
      ( ( basic_3520275952944978471ist_fm @ P1 @ P22 @ ( produc7063755442767259331ist_fm @ A @ B ) )
      = ( ( P1 @ A )
        & ( P22 @ B ) ) ) ).

% pred_prod_inject
thf(fact_279_pred__prod__inject,axiom,
    ! [P1: fm > $o,P22: fm > $o,A: fm,B: fm] :
      ( ( basic_6733234405827732053_fm_fm @ P1 @ P22 @ ( product_Pair_fm_fm @ A @ B ) )
      = ( ( P1 @ A )
        & ( P22 @ B ) ) ) ).

% pred_prod_inject
thf(fact_280_pred__prod__inject,axiom,
    ! [P1: list_fm > $o,P22: list_fm > $o,A: list_fm,B: list_fm] :
      ( ( basic_5093365364719461109ist_fm @ P1 @ P22 @ ( produc381145313068854617ist_fm @ A @ B ) )
      = ( ( P1 @ A )
        & ( P22 @ B ) ) ) ).

% pred_prod_inject
thf(fact_281_pred__prod__inject,axiom,
    ! [P1: produc1996495991257130529ist_fm > $o,P22: rule > $o,A: produc1996495991257130529ist_fm,B: rule] :
      ( ( basic_6171360182465397619m_rule @ P1 @ P22 @ ( produc491467635432902671m_rule @ A @ B ) )
      = ( ( P1 @ A )
        & ( P22 @ B ) ) ) ).

% pred_prod_inject
thf(fact_282_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P2: set_fm > set_fm > $o,X2: set_fm,Y3: set_fm,A: produc1316288955051664109set_fm] :
      ( ( P2 @ X2 @ Y3 )
     => ( ( A
          = ( produc3030414577503514149set_fm @ X2 @ Y3 ) )
       => ( P2 @ ( produc1719312957141167633set_fm @ A ) @ ( produc6202189304966566995set_fm @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_283_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P2: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm > $o,X2: produc1996495991257130529ist_fm,Y3: produc1996495991257130529ist_fm,A: produc1754732471204517991ist_fm] :
      ( ( P2 @ X2 @ Y3 )
     => ( ( A
          = ( produc2592314732767339223ist_fm @ X2 @ Y3 ) )
       => ( P2 @ ( produc5894184038225292483ist_fm @ A ) @ ( produc1639249265332130309ist_fm @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_284_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P2: rule > rule > $o,X2: rule,Y3: rule,A: produc780299800278082923e_rule] :
      ( ( P2 @ X2 @ Y3 )
     => ( ( A
          = ( produc1769597910952761123e_rule @ X2 @ Y3 ) )
       => ( P2 @ ( produc7264528503596239631e_rule @ A ) @ ( produc6725306161561501521e_rule @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_285_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P2: fm > fm > $o,X2: fm,Y3: fm,A: product_prod_fm_fm] :
      ( ( P2 @ X2 @ Y3 )
     => ( ( A
          = ( product_Pair_fm_fm @ X2 @ Y3 ) )
       => ( P2 @ ( product_fst_fm_fm @ A ) @ ( product_snd_fm_fm @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_286_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P2: produc164195504107695125m_rule > fm > $o,X2: produc164195504107695125m_rule,Y3: fm,A: produc7564083655194549958ule_fm] :
      ( ( P2 @ X2 @ Y3 )
     => ( ( A
          = ( produc803751574383482622ule_fm @ X2 @ Y3 ) )
       => ( P2 @ ( produc1758149581174257386ule_fm @ A ) @ ( produc8694862632519904556ule_fm @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_287_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P2: list_fm > list_fm > $o,X2: list_fm,Y3: list_fm,A: produc1996495991257130529ist_fm] :
      ( ( P2 @ X2 @ Y3 )
     => ( ( A
          = ( produc381145313068854617ist_fm @ X2 @ Y3 ) )
       => ( P2 @ ( produc1501393135466168645ist_fm @ A ) @ ( produc4588648349897876871ist_fm @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_288_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P2: produc1996495991257130529ist_fm > rule > $o,X2: produc1996495991257130529ist_fm,Y3: rule,A: produc164195504107695125m_rule] :
      ( ( P2 @ X2 @ Y3 )
     => ( ( A
          = ( produc491467635432902671m_rule @ X2 @ Y3 ) )
       => ( P2 @ ( produc6879501374131015971m_rule @ A ) @ ( produc7165828336582415457m_rule @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_289_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P2: rule > produc1996495991257130529ist_fm > $o,X2: rule,Y3: produc1996495991257130529ist_fm,A: produc4630866025709511057ist_fm] :
      ( ( P2 @ X2 @ Y3 )
     => ( ( A
          = ( produc7063755442767259331ist_fm @ X2 @ Y3 ) )
       => ( P2 @ ( produc4228417144610596823ist_fm @ A ) @ ( produc4514744107061996309ist_fm @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_290_pred__prod_Ocases,axiom,
    ! [P1: rule > $o,P22: rule > $o,A: produc780299800278082923e_rule] :
      ( ( basic_5896771853912976575e_rule @ P1 @ P22 @ A )
     => ~ ! [A3: rule,B3: rule] :
            ( ( A
              = ( produc1769597910952761123e_rule @ A3 @ B3 ) )
           => ( ( P1 @ A3 )
             => ~ ( P22 @ B3 ) ) ) ) ).

% pred_prod.cases
thf(fact_291_pred__prod_Ocases,axiom,
    ! [P1: set_fm > $o,P22: set_fm > $o,A: produc1316288955051664109set_fm] :
      ( ( basic_677064363366268865set_fm @ P1 @ P22 @ A )
     => ~ ! [A3: set_fm,B3: set_fm] :
            ( ( A
              = ( produc3030414577503514149set_fm @ A3 @ B3 ) )
           => ( ( P1 @ A3 )
             => ~ ( P22 @ B3 ) ) ) ) ).

% pred_prod.cases
thf(fact_292_pred__prod_Ocases,axiom,
    ! [P1: rule > $o,P22: produc1996495991257130529ist_fm > $o,A: produc4630866025709511057ist_fm] :
      ( ( basic_3520275952944978471ist_fm @ P1 @ P22 @ A )
     => ~ ! [A3: rule,B3: produc1996495991257130529ist_fm] :
            ( ( A
              = ( produc7063755442767259331ist_fm @ A3 @ B3 ) )
           => ( ( P1 @ A3 )
             => ~ ( P22 @ B3 ) ) ) ) ).

% pred_prod.cases
thf(fact_293_pred__prod_Ocases,axiom,
    ! [P1: fm > $o,P22: fm > $o,A: product_prod_fm_fm] :
      ( ( basic_6733234405827732053_fm_fm @ P1 @ P22 @ A )
     => ~ ! [A3: fm,B3: fm] :
            ( ( A
              = ( product_Pair_fm_fm @ A3 @ B3 ) )
           => ( ( P1 @ A3 )
             => ~ ( P22 @ B3 ) ) ) ) ).

% pred_prod.cases
thf(fact_294_pred__prod_Ocases,axiom,
    ! [P1: list_fm > $o,P22: list_fm > $o,A: produc1996495991257130529ist_fm] :
      ( ( basic_5093365364719461109ist_fm @ P1 @ P22 @ A )
     => ~ ! [A3: list_fm,B3: list_fm] :
            ( ( A
              = ( produc381145313068854617ist_fm @ A3 @ B3 ) )
           => ( ( P1 @ A3 )
             => ~ ( P22 @ B3 ) ) ) ) ).

% pred_prod.cases
thf(fact_295_pred__prod_Ocases,axiom,
    ! [P1: produc1996495991257130529ist_fm > $o,P22: rule > $o,A: produc164195504107695125m_rule] :
      ( ( basic_6171360182465397619m_rule @ P1 @ P22 @ A )
     => ~ ! [A3: produc1996495991257130529ist_fm,B3: rule] :
            ( ( A
              = ( produc491467635432902671m_rule @ A3 @ B3 ) )
           => ( ( P1 @ A3 )
             => ~ ( P22 @ B3 ) ) ) ) ).

% pred_prod.cases
thf(fact_296_pred__prod_Osimps,axiom,
    ( basic_5896771853912976575e_rule
    = ( ^ [P12: rule > $o,P23: rule > $o,A5: produc780299800278082923e_rule] :
        ? [B4: rule,C3: rule] :
          ( ( A5
            = ( produc1769597910952761123e_rule @ B4 @ C3 ) )
          & ( P12 @ B4 )
          & ( P23 @ C3 ) ) ) ) ).

% pred_prod.simps
thf(fact_297_pred__prod_Osimps,axiom,
    ( basic_677064363366268865set_fm
    = ( ^ [P12: set_fm > $o,P23: set_fm > $o,A5: produc1316288955051664109set_fm] :
        ? [B4: set_fm,C3: set_fm] :
          ( ( A5
            = ( produc3030414577503514149set_fm @ B4 @ C3 ) )
          & ( P12 @ B4 )
          & ( P23 @ C3 ) ) ) ) ).

% pred_prod.simps
thf(fact_298_pred__prod_Osimps,axiom,
    ( basic_3520275952944978471ist_fm
    = ( ^ [P12: rule > $o,P23: produc1996495991257130529ist_fm > $o,A5: produc4630866025709511057ist_fm] :
        ? [B4: rule,C3: produc1996495991257130529ist_fm] :
          ( ( A5
            = ( produc7063755442767259331ist_fm @ B4 @ C3 ) )
          & ( P12 @ B4 )
          & ( P23 @ C3 ) ) ) ) ).

% pred_prod.simps
thf(fact_299_pred__prod_Osimps,axiom,
    ( basic_6733234405827732053_fm_fm
    = ( ^ [P12: fm > $o,P23: fm > $o,A5: product_prod_fm_fm] :
        ? [B4: fm,C3: fm] :
          ( ( A5
            = ( product_Pair_fm_fm @ B4 @ C3 ) )
          & ( P12 @ B4 )
          & ( P23 @ C3 ) ) ) ) ).

% pred_prod.simps
thf(fact_300_pred__prod_Osimps,axiom,
    ( basic_5093365364719461109ist_fm
    = ( ^ [P12: list_fm > $o,P23: list_fm > $o,A5: produc1996495991257130529ist_fm] :
        ? [B4: list_fm,C3: list_fm] :
          ( ( A5
            = ( produc381145313068854617ist_fm @ B4 @ C3 ) )
          & ( P12 @ B4 )
          & ( P23 @ C3 ) ) ) ) ).

% pred_prod.simps
thf(fact_301_pred__prod_Osimps,axiom,
    ( basic_6171360182465397619m_rule
    = ( ^ [P12: produc1996495991257130529ist_fm > $o,P23: rule > $o,A5: produc164195504107695125m_rule] :
        ? [B4: produc1996495991257130529ist_fm,C3: rule] :
          ( ( A5
            = ( produc491467635432902671m_rule @ B4 @ C3 ) )
          & ( P12 @ B4 )
          & ( P23 @ C3 ) ) ) ) ).

% pred_prod.simps
thf(fact_302_pred__prod_Ointros,axiom,
    ! [P1: rule > $o,A: rule,P22: rule > $o,B: rule] :
      ( ( P1 @ A )
     => ( ( P22 @ B )
       => ( basic_5896771853912976575e_rule @ P1 @ P22 @ ( produc1769597910952761123e_rule @ A @ B ) ) ) ) ).

% pred_prod.intros
thf(fact_303_pred__prod_Ointros,axiom,
    ! [P1: set_fm > $o,A: set_fm,P22: set_fm > $o,B: set_fm] :
      ( ( P1 @ A )
     => ( ( P22 @ B )
       => ( basic_677064363366268865set_fm @ P1 @ P22 @ ( produc3030414577503514149set_fm @ A @ B ) ) ) ) ).

% pred_prod.intros
thf(fact_304_pred__prod_Ointros,axiom,
    ! [P1: rule > $o,A: rule,P22: produc1996495991257130529ist_fm > $o,B: produc1996495991257130529ist_fm] :
      ( ( P1 @ A )
     => ( ( P22 @ B )
       => ( basic_3520275952944978471ist_fm @ P1 @ P22 @ ( produc7063755442767259331ist_fm @ A @ B ) ) ) ) ).

% pred_prod.intros
thf(fact_305_pred__prod_Ointros,axiom,
    ! [P1: fm > $o,A: fm,P22: fm > $o,B: fm] :
      ( ( P1 @ A )
     => ( ( P22 @ B )
       => ( basic_6733234405827732053_fm_fm @ P1 @ P22 @ ( product_Pair_fm_fm @ A @ B ) ) ) ) ).

% pred_prod.intros
thf(fact_306_pred__prod_Ointros,axiom,
    ! [P1: list_fm > $o,A: list_fm,P22: list_fm > $o,B: list_fm] :
      ( ( P1 @ A )
     => ( ( P22 @ B )
       => ( basic_5093365364719461109ist_fm @ P1 @ P22 @ ( produc381145313068854617ist_fm @ A @ B ) ) ) ) ).

% pred_prod.intros
thf(fact_307_pred__prod_Ointros,axiom,
    ! [P1: produc1996495991257130529ist_fm > $o,A: produc1996495991257130529ist_fm,P22: rule > $o,B: rule] :
      ( ( P1 @ A )
     => ( ( P22 @ B )
       => ( basic_6171360182465397619m_rule @ P1 @ P22 @ ( produc491467635432902671m_rule @ A @ B ) ) ) ) ).

% pred_prod.intros
thf(fact_308_eq__snd__iff,axiom,
    ! [B: set_fm,P: produc1316288955051664109set_fm] :
      ( ( B
        = ( produc6202189304966566995set_fm @ P ) )
      = ( ? [A5: set_fm] :
            ( P
            = ( produc3030414577503514149set_fm @ A5 @ B ) ) ) ) ).

% eq_snd_iff
thf(fact_309_eq__snd__iff,axiom,
    ! [B: produc1996495991257130529ist_fm,P: produc1754732471204517991ist_fm] :
      ( ( B
        = ( produc1639249265332130309ist_fm @ P ) )
      = ( ? [A5: produc1996495991257130529ist_fm] :
            ( P
            = ( produc2592314732767339223ist_fm @ A5 @ B ) ) ) ) ).

% eq_snd_iff
thf(fact_310_eq__snd__iff,axiom,
    ! [B: rule,P: produc780299800278082923e_rule] :
      ( ( B
        = ( produc6725306161561501521e_rule @ P ) )
      = ( ? [A5: rule] :
            ( P
            = ( produc1769597910952761123e_rule @ A5 @ B ) ) ) ) ).

% eq_snd_iff
thf(fact_311_eq__snd__iff,axiom,
    ! [B: fm,P: product_prod_fm_fm] :
      ( ( B
        = ( product_snd_fm_fm @ P ) )
      = ( ? [A5: fm] :
            ( P
            = ( product_Pair_fm_fm @ A5 @ B ) ) ) ) ).

% eq_snd_iff
thf(fact_312_eq__snd__iff,axiom,
    ! [B: fm,P: produc7564083655194549958ule_fm] :
      ( ( B
        = ( produc8694862632519904556ule_fm @ P ) )
      = ( ? [A5: produc164195504107695125m_rule] :
            ( P
            = ( produc803751574383482622ule_fm @ A5 @ B ) ) ) ) ).

% eq_snd_iff
thf(fact_313_eq__snd__iff,axiom,
    ! [B: list_fm,P: produc1996495991257130529ist_fm] :
      ( ( B
        = ( produc4588648349897876871ist_fm @ P ) )
      = ( ? [A5: list_fm] :
            ( P
            = ( produc381145313068854617ist_fm @ A5 @ B ) ) ) ) ).

% eq_snd_iff
thf(fact_314_eq__snd__iff,axiom,
    ! [B: rule,P: produc164195504107695125m_rule] :
      ( ( B
        = ( produc7165828336582415457m_rule @ P ) )
      = ( ? [A5: produc1996495991257130529ist_fm] :
            ( P
            = ( produc491467635432902671m_rule @ A5 @ B ) ) ) ) ).

% eq_snd_iff
thf(fact_315_eq__snd__iff,axiom,
    ! [B: produc1996495991257130529ist_fm,P: produc4630866025709511057ist_fm] :
      ( ( B
        = ( produc4514744107061996309ist_fm @ P ) )
      = ( ? [A5: rule] :
            ( P
            = ( produc7063755442767259331ist_fm @ A5 @ B ) ) ) ) ).

% eq_snd_iff
thf(fact_316_sndI,axiom,
    ! [X2: produc1316288955051664109set_fm,Y3: set_fm,Z2: set_fm] :
      ( ( X2
        = ( produc3030414577503514149set_fm @ Y3 @ Z2 ) )
     => ( ( produc6202189304966566995set_fm @ X2 )
        = Z2 ) ) ).

% sndI
thf(fact_317_sndI,axiom,
    ! [X2: produc1754732471204517991ist_fm,Y3: produc1996495991257130529ist_fm,Z2: produc1996495991257130529ist_fm] :
      ( ( X2
        = ( produc2592314732767339223ist_fm @ Y3 @ Z2 ) )
     => ( ( produc1639249265332130309ist_fm @ X2 )
        = Z2 ) ) ).

% sndI
thf(fact_318_sndI,axiom,
    ! [X2: produc780299800278082923e_rule,Y3: rule,Z2: rule] :
      ( ( X2
        = ( produc1769597910952761123e_rule @ Y3 @ Z2 ) )
     => ( ( produc6725306161561501521e_rule @ X2 )
        = Z2 ) ) ).

% sndI
thf(fact_319_sndI,axiom,
    ! [X2: product_prod_fm_fm,Y3: fm,Z2: fm] :
      ( ( X2
        = ( product_Pair_fm_fm @ Y3 @ Z2 ) )
     => ( ( product_snd_fm_fm @ X2 )
        = Z2 ) ) ).

% sndI
thf(fact_320_sndI,axiom,
    ! [X2: produc7564083655194549958ule_fm,Y3: produc164195504107695125m_rule,Z2: fm] :
      ( ( X2
        = ( produc803751574383482622ule_fm @ Y3 @ Z2 ) )
     => ( ( produc8694862632519904556ule_fm @ X2 )
        = Z2 ) ) ).

% sndI
thf(fact_321_sndI,axiom,
    ! [X2: produc1996495991257130529ist_fm,Y3: list_fm,Z2: list_fm] :
      ( ( X2
        = ( produc381145313068854617ist_fm @ Y3 @ Z2 ) )
     => ( ( produc4588648349897876871ist_fm @ X2 )
        = Z2 ) ) ).

% sndI
thf(fact_322_sndI,axiom,
    ! [X2: produc164195504107695125m_rule,Y3: produc1996495991257130529ist_fm,Z2: rule] :
      ( ( X2
        = ( produc491467635432902671m_rule @ Y3 @ Z2 ) )
     => ( ( produc7165828336582415457m_rule @ X2 )
        = Z2 ) ) ).

% sndI
thf(fact_323_sndI,axiom,
    ! [X2: produc4630866025709511057ist_fm,Y3: rule,Z2: produc1996495991257130529ist_fm] :
      ( ( X2
        = ( produc7063755442767259331ist_fm @ Y3 @ Z2 ) )
     => ( ( produc4514744107061996309ist_fm @ X2 )
        = Z2 ) ) ).

% sndI
thf(fact_324_eq__fst__iff,axiom,
    ! [A: set_fm,P: produc1316288955051664109set_fm] :
      ( ( A
        = ( produc1719312957141167633set_fm @ P ) )
      = ( ? [B4: set_fm] :
            ( P
            = ( produc3030414577503514149set_fm @ A @ B4 ) ) ) ) ).

% eq_fst_iff
thf(fact_325_eq__fst__iff,axiom,
    ! [A: rule,P: produc4630866025709511057ist_fm] :
      ( ( A
        = ( produc4228417144610596823ist_fm @ P ) )
      = ( ? [B4: produc1996495991257130529ist_fm] :
            ( P
            = ( produc7063755442767259331ist_fm @ A @ B4 ) ) ) ) ).

% eq_fst_iff
thf(fact_326_eq__fst__iff,axiom,
    ! [A: produc1996495991257130529ist_fm,P: produc1754732471204517991ist_fm] :
      ( ( A
        = ( produc5894184038225292483ist_fm @ P ) )
      = ( ? [B4: produc1996495991257130529ist_fm] :
            ( P
            = ( produc2592314732767339223ist_fm @ A @ B4 ) ) ) ) ).

% eq_fst_iff
thf(fact_327_eq__fst__iff,axiom,
    ! [A: rule,P: produc780299800278082923e_rule] :
      ( ( A
        = ( produc7264528503596239631e_rule @ P ) )
      = ( ? [B4: rule] :
            ( P
            = ( produc1769597910952761123e_rule @ A @ B4 ) ) ) ) ).

% eq_fst_iff
thf(fact_328_eq__fst__iff,axiom,
    ! [A: fm,P: product_prod_fm_fm] :
      ( ( A
        = ( product_fst_fm_fm @ P ) )
      = ( ? [B4: fm] :
            ( P
            = ( product_Pair_fm_fm @ A @ B4 ) ) ) ) ).

% eq_fst_iff
thf(fact_329_eq__fst__iff,axiom,
    ! [A: produc1996495991257130529ist_fm,P: produc164195504107695125m_rule] :
      ( ( A
        = ( produc6879501374131015971m_rule @ P ) )
      = ( ? [B4: rule] :
            ( P
            = ( produc491467635432902671m_rule @ A @ B4 ) ) ) ) ).

% eq_fst_iff
thf(fact_330_eq__fst__iff,axiom,
    ! [A: list_fm,P: produc1996495991257130529ist_fm] :
      ( ( A
        = ( produc1501393135466168645ist_fm @ P ) )
      = ( ? [B4: list_fm] :
            ( P
            = ( produc381145313068854617ist_fm @ A @ B4 ) ) ) ) ).

% eq_fst_iff
thf(fact_331_fstI,axiom,
    ! [X2: produc1316288955051664109set_fm,Y3: set_fm,Z2: set_fm] :
      ( ( X2
        = ( produc3030414577503514149set_fm @ Y3 @ Z2 ) )
     => ( ( produc1719312957141167633set_fm @ X2 )
        = Y3 ) ) ).

% fstI
thf(fact_332_fstI,axiom,
    ! [X2: produc4630866025709511057ist_fm,Y3: rule,Z2: produc1996495991257130529ist_fm] :
      ( ( X2
        = ( produc7063755442767259331ist_fm @ Y3 @ Z2 ) )
     => ( ( produc4228417144610596823ist_fm @ X2 )
        = Y3 ) ) ).

% fstI
thf(fact_333_fstI,axiom,
    ! [X2: produc1754732471204517991ist_fm,Y3: produc1996495991257130529ist_fm,Z2: produc1996495991257130529ist_fm] :
      ( ( X2
        = ( produc2592314732767339223ist_fm @ Y3 @ Z2 ) )
     => ( ( produc5894184038225292483ist_fm @ X2 )
        = Y3 ) ) ).

% fstI
thf(fact_334_fstI,axiom,
    ! [X2: produc780299800278082923e_rule,Y3: rule,Z2: rule] :
      ( ( X2
        = ( produc1769597910952761123e_rule @ Y3 @ Z2 ) )
     => ( ( produc7264528503596239631e_rule @ X2 )
        = Y3 ) ) ).

% fstI
thf(fact_335_fstI,axiom,
    ! [X2: product_prod_fm_fm,Y3: fm,Z2: fm] :
      ( ( X2
        = ( product_Pair_fm_fm @ Y3 @ Z2 ) )
     => ( ( product_fst_fm_fm @ X2 )
        = Y3 ) ) ).

% fstI
thf(fact_336_fstI,axiom,
    ! [X2: produc164195504107695125m_rule,Y3: produc1996495991257130529ist_fm,Z2: rule] :
      ( ( X2
        = ( produc491467635432902671m_rule @ Y3 @ Z2 ) )
     => ( ( produc6879501374131015971m_rule @ X2 )
        = Y3 ) ) ).

% fstI
thf(fact_337_fstI,axiom,
    ! [X2: produc1996495991257130529ist_fm,Y3: list_fm,Z2: list_fm] :
      ( ( X2
        = ( produc381145313068854617ist_fm @ Y3 @ Z2 ) )
     => ( ( produc1501393135466168645ist_fm @ X2 )
        = Y3 ) ) ).

% fstI
thf(fact_338_split__cong,axiom,
    ! [Q: produc4630866025709511057ist_fm,F2: rule > produc1996495991257130529ist_fm > produc1996495991257130529ist_fm,G: rule > produc1996495991257130529ist_fm > produc1996495991257130529ist_fm,P: produc4630866025709511057ist_fm] :
      ( ! [X: rule,Y: produc1996495991257130529ist_fm] :
          ( ( ( produc7063755442767259331ist_fm @ X @ Y )
            = Q )
         => ( ( F2 @ X @ Y )
            = ( G @ X @ Y ) ) )
     => ( ( P = Q )
       => ( ( produc2632167636358383921ist_fm @ F2 @ P )
          = ( produc2632167636358383921ist_fm @ G @ Q ) ) ) ) ).

% split_cong
thf(fact_339_split__cong,axiom,
    ! [Q: produc4630866025709511057ist_fm,F2: rule > produc1996495991257130529ist_fm > $o,G: rule > produc1996495991257130529ist_fm > $o,P: produc4630866025709511057ist_fm] :
      ( ! [X: rule,Y: produc1996495991257130529ist_fm] :
          ( ( ( produc7063755442767259331ist_fm @ X @ Y )
            = Q )
         => ( ( F2 @ X @ Y )
            = ( G @ X @ Y ) ) )
     => ( ( P = Q )
       => ( ( produc1628407760719572648t_fm_o @ F2 @ P )
          = ( produc1628407760719572648t_fm_o @ G @ Q ) ) ) ) ).

% split_cong
thf(fact_340_split__cong,axiom,
    ! [Q: produc1996495991257130529ist_fm,F2: list_fm > list_fm > produc1996495991257130529ist_fm,G: list_fm > list_fm > produc1996495991257130529ist_fm,P: produc1996495991257130529ist_fm] :
      ( ! [X: list_fm,Y: list_fm] :
          ( ( ( produc381145313068854617ist_fm @ X @ Y )
            = Q )
         => ( ( F2 @ X @ Y )
            = ( G @ X @ Y ) ) )
     => ( ( P = Q )
       => ( ( produc4544959197559615795ist_fm @ F2 @ P )
          = ( produc4544959197559615795ist_fm @ G @ Q ) ) ) ) ).

% split_cong
thf(fact_341_split__cong,axiom,
    ! [Q: produc1996495991257130529ist_fm,F2: list_fm > list_fm > list_fm,G: list_fm > list_fm > list_fm,P: produc1996495991257130529ist_fm] :
      ( ! [X: list_fm,Y: list_fm] :
          ( ( ( produc381145313068854617ist_fm @ X @ Y )
            = Q )
         => ( ( F2 @ X @ Y )
            = ( G @ X @ Y ) ) )
     => ( ( P = Q )
       => ( ( produc4154610405322092878ist_fm @ F2 @ P )
          = ( produc4154610405322092878ist_fm @ G @ Q ) ) ) ) ).

% split_cong
thf(fact_342_split__cong,axiom,
    ! [Q: produc1996495991257130529ist_fm,F2: list_fm > list_fm > $o,G: list_fm > list_fm > $o,P: produc1996495991257130529ist_fm] :
      ( ! [X: list_fm,Y: list_fm] :
          ( ( ( produc381145313068854617ist_fm @ X @ Y )
            = Q )
         => ( ( F2 @ X @ Y )
            = ( G @ X @ Y ) ) )
     => ( ( P = Q )
       => ( ( produc2011568048573023402t_fm_o @ F2 @ P )
          = ( produc2011568048573023402t_fm_o @ G @ Q ) ) ) ) ).

% split_cong
thf(fact_343_internal__case__prod__def,axiom,
    produc7989377663358243335ist_fm = produc2632167636358383921ist_fm ).

% internal_case_prod_def
thf(fact_344_internal__case__prod__def,axiom,
    produc9204273236492169598t_fm_o = produc1628407760719572648t_fm_o ).

% internal_case_prod_def
thf(fact_345_internal__case__prod__def,axiom,
    produc6109684375392212445ist_fm = produc4544959197559615795ist_fm ).

% internal_case_prod_def
thf(fact_346_internal__case__prod__def,axiom,
    produc341677611771467556ist_fm = produc4154610405322092878ist_fm ).

% internal_case_prod_def
thf(fact_347_internal__case__prod__def,axiom,
    produc7663040964597641044t_fm_o = produc2011568048573023402t_fm_o ).

% internal_case_prod_def
thf(fact_348_fstOp__def,axiom,
    ( bNF_fs6361025107161210196ist_fm
    = ( ^ [P3: list_fm > list_fm > $o,Q2: list_fm > list_fm > $o,Ac: produc1996495991257130529ist_fm] : ( produc381145313068854617ist_fm @ ( produc1501393135466168645ist_fm @ Ac ) @ ( bNF_pi3539088406881798022ist_fm @ P3 @ Q2 @ ( produc1501393135466168645ist_fm @ Ac ) @ ( produc4588648349897876871ist_fm @ Ac ) ) ) ) ) ).

% fstOp_def
thf(fact_349_fstOp__def,axiom,
    ( bNF_fs2143400664727635963e_rule
    = ( ^ [P3: produc1996495991257130529ist_fm > rule > $o,Q2: rule > rule > $o,Ac: produc164195504107695125m_rule] : ( produc491467635432902671m_rule @ ( produc6879501374131015971m_rule @ Ac ) @ ( bNF_pi580457772512444745e_rule @ P3 @ Q2 @ ( produc6879501374131015971m_rule @ Ac ) @ ( produc7165828336582415457m_rule @ Ac ) ) ) ) ) ).

% fstOp_def
thf(fact_350_fstOp__def,axiom,
    ( bNF_fs3887825027587321911ist_fm
    = ( ^ [P3: rule > produc1996495991257130529ist_fm > $o,Q2: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm > $o,Ac: produc4630866025709511057ist_fm] : ( produc7063755442767259331ist_fm @ ( produc4228417144610596823ist_fm @ Ac ) @ ( bNF_pi3835482382339038057ist_fm @ P3 @ Q2 @ ( produc4228417144610596823ist_fm @ Ac ) @ ( produc4514744107061996309ist_fm @ Ac ) ) ) ) ) ).

% fstOp_def
thf(fact_351_fstOp__def,axiom,
    ( bNF_fs6525992336556323171ist_fm
    = ( ^ [P3: rule > rule > $o,Q2: rule > produc1996495991257130529ist_fm > $o,Ac: produc4630866025709511057ist_fm] : ( produc1769597910952761123e_rule @ ( produc4228417144610596823ist_fm @ Ac ) @ ( bNF_pi4963049444341131953ist_fm @ P3 @ Q2 @ ( produc4228417144610596823ist_fm @ Ac ) @ ( produc4514744107061996309ist_fm @ Ac ) ) ) ) ) ).

% fstOp_def
thf(fact_352_fstOp__def,axiom,
    ( bNF_fs9177076566076742319m_rule
    = ( ^ [P3: rule > produc1996495991257130529ist_fm > $o,Q2: produc1996495991257130529ist_fm > rule > $o,Ac: produc780299800278082923e_rule] : ( produc7063755442767259331ist_fm @ ( produc7264528503596239631e_rule @ Ac ) @ ( bNF_pi7614133673861551101m_rule @ P3 @ Q2 @ ( produc7264528503596239631e_rule @ Ac ) @ ( produc6725306161561501521e_rule @ Ac ) ) ) ) ) ).

% fstOp_def
thf(fact_353_fstOp__def,axiom,
    ( bNF_fs6019504843140265091e_rule
    = ( ^ [P3: rule > rule > $o,Q2: rule > rule > $o,Ac: produc780299800278082923e_rule] : ( produc1769597910952761123e_rule @ ( produc7264528503596239631e_rule @ Ac ) @ ( bNF_pi5293732350936020661e_rule @ P3 @ Q2 @ ( produc7264528503596239631e_rule @ Ac ) @ ( produc6725306161561501521e_rule @ Ac ) ) ) ) ) ).

% fstOp_def
thf(fact_354_fstOp__def,axiom,
    ( bNF_fstOp_fm_fm_fm
    = ( ^ [P3: fm > fm > $o,Q2: fm > fm > $o,Ac: product_prod_fm_fm] : ( product_Pair_fm_fm @ ( product_fst_fm_fm @ Ac ) @ ( bNF_pi8146509907818838_fm_fm @ P3 @ Q2 @ ( product_fst_fm_fm @ Ac ) @ ( product_snd_fm_fm @ Ac ) ) ) ) ) ).

% fstOp_def
thf(fact_355_sndOp__def,axiom,
    ( bNF_sn5292590578203848342ist_fm
    = ( ^ [P3: list_fm > list_fm > $o,Q2: list_fm > list_fm > $o,Ac: produc1996495991257130529ist_fm] : ( produc381145313068854617ist_fm @ ( bNF_pi3539088406881798022ist_fm @ P3 @ Q2 @ ( produc1501393135466168645ist_fm @ Ac ) @ ( produc4588648349897876871ist_fm @ Ac ) ) @ ( produc4588648349897876871ist_fm @ Ac ) ) ) ) ).

% sndOp_def
thf(fact_356_sndOp__def,axiom,
    ( bNF_sn6617447237779055481m_rule
    = ( ^ [P3: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm > $o,Q2: produc1996495991257130529ist_fm > rule > $o,Ac: produc164195504107695125m_rule] : ( produc491467635432902671m_rule @ ( bNF_pi3438825987790746729m_rule @ P3 @ Q2 @ ( produc6879501374131015971m_rule @ Ac ) @ ( produc7165828336582415457m_rule @ Ac ) ) @ ( produc7165828336582415457m_rule @ Ac ) ) ) ) ).

% sndOp_def
thf(fact_357_sndOp__def,axiom,
    ( bNF_sn6445580637211383353e_rule
    = ( ^ [P3: produc1996495991257130529ist_fm > rule > $o,Q2: rule > rule > $o,Ac: produc164195504107695125m_rule] : ( produc1769597910952761123e_rule @ ( bNF_pi580457772512444745e_rule @ P3 @ Q2 @ ( produc6879501374131015971m_rule @ Ac ) @ ( produc7165828336582415457m_rule @ Ac ) ) @ ( produc7165828336582415457m_rule @ Ac ) ) ) ) ).

% sndOp_def
thf(fact_358_sndOp__def,axiom,
    ( bNF_sn1604800272185294753ist_fm
    = ( ^ [P3: rule > rule > $o,Q2: rule > produc1996495991257130529ist_fm > $o,Ac: produc4630866025709511057ist_fm] : ( produc7063755442767259331ist_fm @ ( bNF_pi4963049444341131953ist_fm @ P3 @ Q2 @ ( produc4228417144610596823ist_fm @ Ac ) @ ( produc4514744107061996309ist_fm @ Ac ) ) @ ( produc4514744107061996309ist_fm @ Ac ) ) ) ) ).

% sndOp_def
thf(fact_359_sndOp__def,axiom,
    ( bNF_sn3966363008258636333ist_fm
    = ( ^ [P3: produc1996495991257130529ist_fm > rule > $o,Q2: rule > produc1996495991257130529ist_fm > $o,Ac: produc1754732471204517991ist_fm] : ( produc7063755442767259331ist_fm @ ( bNF_pi787741758270327581ist_fm @ P3 @ Q2 @ ( produc5894184038225292483ist_fm @ Ac ) @ ( produc1639249265332130309ist_fm @ Ac ) ) @ ( produc1639249265332130309ist_fm @ Ac ) ) ) ) ).

% sndOp_def
thf(fact_360_sndOp__def,axiom,
    ( bNF_sn4255884501705713901m_rule
    = ( ^ [P3: rule > produc1996495991257130529ist_fm > $o,Q2: produc1996495991257130529ist_fm > rule > $o,Ac: produc780299800278082923e_rule] : ( produc491467635432902671m_rule @ ( bNF_pi7614133673861551101m_rule @ P3 @ Q2 @ ( produc7264528503596239631e_rule @ Ac ) @ ( produc6725306161561501521e_rule @ Ac ) ) @ ( produc6725306161561501521e_rule @ Ac ) ) ) ) ).

% sndOp_def
thf(fact_361_sndOp__def,axiom,
    ( bNF_sn5573932575237119429e_rule
    = ( ^ [P3: rule > rule > $o,Q2: rule > rule > $o,Ac: produc780299800278082923e_rule] : ( produc1769597910952761123e_rule @ ( bNF_pi5293732350936020661e_rule @ P3 @ Q2 @ ( produc7264528503596239631e_rule @ Ac ) @ ( produc6725306161561501521e_rule @ Ac ) ) @ ( produc6725306161561501521e_rule @ Ac ) ) ) ) ).

% sndOp_def
thf(fact_362_sndOp__def,axiom,
    ( bNF_sndOp_fm_fm_fm
    = ( ^ [P3: fm > fm > $o,Q2: fm > fm > $o,Ac: product_prod_fm_fm] : ( product_Pair_fm_fm @ ( bNF_pi8146509907818838_fm_fm @ P3 @ Q2 @ ( product_fst_fm_fm @ Ac ) @ ( product_snd_fm_fm @ Ac ) ) @ ( product_snd_fm_fm @ Ac ) ) ) ) ).

% sndOp_def
thf(fact_363_sndOp__def,axiom,
    ( bNF_sn1891237298393626667_fm_fm
    = ( ^ [P3: produc164195504107695125m_rule > fm > $o,Q2: fm > fm > $o,Ac: produc7564083655194549958ule_fm] : ( product_Pair_fm_fm @ ( bNF_pi9059978157863958811_fm_fm @ P3 @ Q2 @ ( produc1758149581174257386ule_fm @ Ac ) @ ( produc8694862632519904556ule_fm @ Ac ) ) @ ( produc8694862632519904556ule_fm @ Ac ) ) ) ) ).

% sndOp_def
thf(fact_364_image2__eqI,axiom,
    ! [B: rule,F2: fm > rule,X2: fm,C2: rule,G: fm > rule,A4: set_fm] :
      ( ( B
        = ( F2 @ X2 ) )
     => ( ( C2
          = ( G @ X2 ) )
       => ( ( member_fm @ X2 @ A4 )
         => ( member3753523862196722196e_rule @ ( produc1769597910952761123e_rule @ B @ C2 ) @ ( bNF_Gr8478660977794743862e_rule @ A4 @ F2 @ G ) ) ) ) ) ).

% image2_eqI
thf(fact_365_image2__eqI,axiom,
    ! [B: fm,F2: fm > fm,X2: fm,C2: fm,G: fm > fm,A4: set_fm] :
      ( ( B
        = ( F2 @ X2 ) )
     => ( ( C2
          = ( G @ X2 ) )
       => ( ( member_fm @ X2 @ A4 )
         => ( member8474499337054950954_fm_fm @ ( product_Pair_fm_fm @ B @ C2 ) @ ( bNF_Gr1203782223388445388_fm_fm @ A4 @ F2 @ G ) ) ) ) ) ).

% image2_eqI
thf(fact_366_image2__eqI,axiom,
    ! [B: rule,F2: list_fm > rule,X2: list_fm,C2: rule,G: list_fm > rule,A4: set_list_fm] :
      ( ( B
        = ( F2 @ X2 ) )
     => ( ( C2
          = ( G @ X2 ) )
       => ( ( member_list_fm @ X2 @ A4 )
         => ( member3753523862196722196e_rule @ ( produc1769597910952761123e_rule @ B @ C2 ) @ ( bNF_Gr5332805151242961350e_rule @ A4 @ F2 @ G ) ) ) ) ) ).

% image2_eqI
thf(fact_367_image2__eqI,axiom,
    ! [B: rule,F2: set_fm > rule,X2: set_fm,C2: rule,G: set_fm > rule,A4: set_set_fm] :
      ( ( B
        = ( F2 @ X2 ) )
     => ( ( C2
          = ( G @ X2 ) )
       => ( ( member_set_fm @ X2 @ A4 )
         => ( member3753523862196722196e_rule @ ( produc1769597910952761123e_rule @ B @ C2 ) @ ( bNF_Gr4980373826170425452e_rule @ A4 @ F2 @ G ) ) ) ) ) ).

% image2_eqI
thf(fact_368_image2__eqI,axiom,
    ! [B: fm,F2: list_fm > fm,X2: list_fm,C2: fm,G: list_fm > fm,A4: set_list_fm] :
      ( ( B
        = ( F2 @ X2 ) )
     => ( ( C2
          = ( G @ X2 ) )
       => ( ( member_list_fm @ X2 @ A4 )
         => ( member8474499337054950954_fm_fm @ ( product_Pair_fm_fm @ B @ C2 ) @ ( bNF_Gr8523389835497889372_fm_fm @ A4 @ F2 @ G ) ) ) ) ) ).

% image2_eqI
thf(fact_369_image2__eqI,axiom,
    ! [B: fm,F2: set_fm > fm,X2: set_fm,C2: fm,G: set_fm > fm,A4: set_set_fm] :
      ( ( B
        = ( F2 @ X2 ) )
     => ( ( C2
          = ( G @ X2 ) )
       => ( ( member_set_fm @ X2 @ A4 )
         => ( member8474499337054950954_fm_fm @ ( product_Pair_fm_fm @ B @ C2 ) @ ( bNF_Gr2527083199176774402_fm_fm @ A4 @ F2 @ G ) ) ) ) ) ).

% image2_eqI
thf(fact_370_image2__eqI,axiom,
    ! [B: list_fm,F2: fm > list_fm,X2: fm,C2: list_fm,G: fm > list_fm,A4: set_fm] :
      ( ( B
        = ( F2 @ X2 ) )
     => ( ( C2
          = ( G @ X2 ) )
       => ( ( member_fm @ X2 @ A4 )
         => ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ B @ C2 ) @ ( bNF_Gr2680958839322950764ist_fm @ A4 @ F2 @ G ) ) ) ) ) ).

% image2_eqI
thf(fact_371_image2__eqI,axiom,
    ! [B: set_fm,F2: fm > set_fm,X2: fm,C2: set_fm,G: fm > set_fm,A4: set_fm] :
      ( ( B
        = ( F2 @ X2 ) )
     => ( ( C2
          = ( G @ X2 ) )
       => ( ( member_fm @ X2 @ A4 )
         => ( member6828610780098320534set_fm @ ( produc3030414577503514149set_fm @ B @ C2 ) @ ( bNF_Gr6461146187696025656set_fm @ A4 @ F2 @ G ) ) ) ) ) ).

% image2_eqI
thf(fact_372_image2__eqI,axiom,
    ! [B: list_fm,F2: list_fm > list_fm,X2: list_fm,C2: list_fm,G: list_fm > list_fm,A4: set_list_fm] :
      ( ( B
        = ( F2 @ X2 ) )
     => ( ( C2
          = ( G @ X2 ) )
       => ( ( member_list_fm @ X2 @ A4 )
         => ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ B @ C2 ) @ ( bNF_Gr1661195949116424188ist_fm @ A4 @ F2 @ G ) ) ) ) ) ).

% image2_eqI
thf(fact_373_image2__eqI,axiom,
    ! [B: list_fm,F2: set_fm > list_fm,X2: set_fm,C2: list_fm,G: set_fm > list_fm,A4: set_set_fm] :
      ( ( B
        = ( F2 @ X2 ) )
     => ( ( C2
          = ( G @ X2 ) )
       => ( ( member_set_fm @ X2 @ A4 )
         => ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ B @ C2 ) @ ( bNF_Gr6790864367800852130ist_fm @ A4 @ F2 @ G ) ) ) ) ) ).

% image2_eqI
thf(fact_374_curry__case__prod,axiom,
    ! [F2: rule > produc1996495991257130529ist_fm > produc1996495991257130529ist_fm] :
      ( ( produc5297400653133107536ist_fm @ ( produc2632167636358383921ist_fm @ F2 ) )
      = F2 ) ).

% curry_case_prod
thf(fact_375_curry__case__prod,axiom,
    ! [F2: rule > produc1996495991257130529ist_fm > $o] :
      ( ( produc2127844983229472711t_fm_o @ ( produc1628407760719572648t_fm_o @ F2 ) )
      = F2 ) ).

% curry_case_prod
thf(fact_376_curry__case__prod,axiom,
    ! [F2: list_fm > list_fm > produc1996495991257130529ist_fm] :
      ( ( produc2602884778838818004ist_fm @ ( produc4544959197559615795ist_fm @ F2 ) )
      = F2 ) ).

% curry_case_prod
thf(fact_377_curry__case__prod,axiom,
    ! [F2: list_fm > list_fm > list_fm] :
      ( ( produc3843686865886723565ist_fm @ ( produc4154610405322092878ist_fm @ F2 ) )
      = F2 ) ).

% curry_case_prod
thf(fact_378_curry__case__prod,axiom,
    ! [F2: list_fm > list_fm > $o] :
      ( ( produc4804894507120027467t_fm_o @ ( produc2011568048573023402t_fm_o @ F2 ) )
      = F2 ) ).

% curry_case_prod
thf(fact_379_case__prod__curry,axiom,
    ! [F2: produc4630866025709511057ist_fm > produc1996495991257130529ist_fm] :
      ( ( produc2632167636358383921ist_fm @ ( produc5297400653133107536ist_fm @ F2 ) )
      = F2 ) ).

% case_prod_curry
thf(fact_380_case__prod__curry,axiom,
    ! [F2: produc4630866025709511057ist_fm > $o] :
      ( ( produc1628407760719572648t_fm_o @ ( produc2127844983229472711t_fm_o @ F2 ) )
      = F2 ) ).

% case_prod_curry
thf(fact_381_case__prod__curry,axiom,
    ! [F2: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm] :
      ( ( produc4544959197559615795ist_fm @ ( produc2602884778838818004ist_fm @ F2 ) )
      = F2 ) ).

% case_prod_curry
thf(fact_382_case__prod__curry,axiom,
    ! [F2: produc1996495991257130529ist_fm > list_fm] :
      ( ( produc4154610405322092878ist_fm @ ( produc3843686865886723565ist_fm @ F2 ) )
      = F2 ) ).

% case_prod_curry
thf(fact_383_case__prod__curry,axiom,
    ! [F2: produc1996495991257130529ist_fm > $o] :
      ( ( produc2011568048573023402t_fm_o @ ( produc4804894507120027467t_fm_o @ F2 ) )
      = F2 ) ).

% case_prod_curry
thf(fact_384_ssubst__Pair__rhs,axiom,
    ! [R2: rule,S3: produc1996495991257130529ist_fm,R: set_Pr2241365836748572487ist_fm,S4: produc1996495991257130529ist_fm] :
      ( ( member8686995742288324264ist_fm @ ( produc7063755442767259331ist_fm @ R2 @ S3 ) @ R )
     => ( ( S4 = S3 )
       => ( member8686995742288324264ist_fm @ ( produc7063755442767259331ist_fm @ R2 @ S4 ) @ R ) ) ) ).

% ssubst_Pair_rhs
thf(fact_385_ssubst__Pair__rhs,axiom,
    ! [R2: rule,S3: rule,R: set_Pr6543236023296505803e_rule,S4: rule] :
      ( ( member3753523862196722196e_rule @ ( produc1769597910952761123e_rule @ R2 @ S3 ) @ R )
     => ( ( S4 = S3 )
       => ( member3753523862196722196e_rule @ ( produc1769597910952761123e_rule @ R2 @ S4 ) @ R ) ) ) ).

% ssubst_Pair_rhs
thf(fact_386_ssubst__Pair__rhs,axiom,
    ! [R2: fm,S3: fm,R: set_Pr4463079037648049377_fm_fm,S4: fm] :
      ( ( member8474499337054950954_fm_fm @ ( product_Pair_fm_fm @ R2 @ S3 ) @ R )
     => ( ( S4 = S3 )
       => ( member8474499337054950954_fm_fm @ ( product_Pair_fm_fm @ R2 @ S4 ) @ R ) ) ) ).

% ssubst_Pair_rhs
thf(fact_387_ssubst__Pair__rhs,axiom,
    ! [R2: list_fm,S3: list_fm,R: set_Pr7058068377845519745ist_fm,S4: list_fm] :
      ( ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ R2 @ S3 ) @ R )
     => ( ( S4 = S3 )
       => ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ R2 @ S4 ) @ R ) ) ) ).

% ssubst_Pair_rhs
thf(fact_388_ssubst__Pair__rhs,axiom,
    ! [R2: set_fm,S3: set_fm,R: set_Pr8693920393607151949set_fm,S4: set_fm] :
      ( ( member6828610780098320534set_fm @ ( produc3030414577503514149set_fm @ R2 @ S3 ) @ R )
     => ( ( S4 = S3 )
       => ( member6828610780098320534set_fm @ ( produc3030414577503514149set_fm @ R2 @ S4 ) @ R ) ) ) ).

% ssubst_Pair_rhs
thf(fact_389_stream__all__def,axiom,
    ( stream720971363120138347ist_fm
    = ( ^ [P3: produc1996495991257130529ist_fm > $o,S: stream8299795917829157543ist_fm] :
        ! [P4: nat] : ( P3 @ ( snth_P2624749845746815685ist_fm @ S @ P4 ) ) ) ) ).

% stream_all_def
thf(fact_390_stream__all__def,axiom,
    ( stream8109186452630104923m_rule
    = ( ^ [P3: stream727092118206550309m_rule > $o,S: stream3016090547575141045m_rule] :
        ! [P4: nat] : ( P3 @ ( snth_s8878883179916095873m_rule @ S @ P4 ) ) ) ) ).

% stream_all_def
thf(fact_391_stream__all__def,axiom,
    ( stream_all_stream_fm
    = ( ^ [P3: stream_fm > $o,S: stream_stream_fm] :
        ! [P4: nat] : ( P3 @ ( snth_stream_fm @ S @ P4 ) ) ) ) ).

% stream_all_def
thf(fact_392_stream__all__def,axiom,
    ( stream_all_rule
    = ( ^ [P3: rule > $o,S: stream_rule] :
        ! [P4: nat] : ( P3 @ ( snth_rule @ S @ P4 ) ) ) ) ).

% stream_all_def
thf(fact_393_stream__all__def,axiom,
    ( stream_all_fm
    = ( ^ [P3: fm > $o,S: stream_fm] :
        ! [P4: nat] : ( P3 @ ( snth_fm @ S @ P4 ) ) ) ) ).

% stream_all_def
thf(fact_394_stream__all__def,axiom,
    ( stream456966007795584843m_rule
    = ( ^ [P3: produc164195504107695125m_rule > $o,S: stream727092118206550309m_rule] :
        ! [P4: nat] : ( P3 @ ( snth_P1712579846118164081m_rule @ S @ P4 ) ) ) ) ).

% stream_all_def
thf(fact_395_curryI,axiom,
    ! [F2: produc4630866025709511057ist_fm > $o,A: rule,B: produc1996495991257130529ist_fm] :
      ( ( F2 @ ( produc7063755442767259331ist_fm @ A @ B ) )
     => ( produc2127844983229472711t_fm_o @ F2 @ A @ B ) ) ).

% curryI
thf(fact_396_curryI,axiom,
    ! [F2: produc780299800278082923e_rule > $o,A: rule,B: rule] :
      ( ( F2 @ ( produc1769597910952761123e_rule @ A @ B ) )
     => ( produc7279490986317915905rule_o @ F2 @ A @ B ) ) ).

% curryI
thf(fact_397_curryI,axiom,
    ! [F2: product_prod_fm_fm > $o,A: fm,B: fm] :
      ( ( F2 @ ( product_Pair_fm_fm @ A @ B ) )
     => ( produc5605581514536698091m_fm_o @ F2 @ A @ B ) ) ).

% curryI
thf(fact_398_curryI,axiom,
    ! [F2: produc1996495991257130529ist_fm > $o,A: list_fm,B: list_fm] :
      ( ( F2 @ ( produc381145313068854617ist_fm @ A @ B ) )
     => ( produc4804894507120027467t_fm_o @ F2 @ A @ B ) ) ).

% curryI
thf(fact_399_curryI,axiom,
    ! [F2: produc1316288955051664109set_fm > $o,A: set_fm,B: set_fm] :
      ( ( F2 @ ( produc3030414577503514149set_fm @ A @ B ) )
     => ( produc4499628267607578751t_fm_o @ F2 @ A @ B ) ) ).

% curryI
thf(fact_400_curryD,axiom,
    ! [F2: produc4630866025709511057ist_fm > $o,A: rule,B: produc1996495991257130529ist_fm] :
      ( ( produc2127844983229472711t_fm_o @ F2 @ A @ B )
     => ( F2 @ ( produc7063755442767259331ist_fm @ A @ B ) ) ) ).

% curryD
thf(fact_401_curryD,axiom,
    ! [F2: produc780299800278082923e_rule > $o,A: rule,B: rule] :
      ( ( produc7279490986317915905rule_o @ F2 @ A @ B )
     => ( F2 @ ( produc1769597910952761123e_rule @ A @ B ) ) ) ).

% curryD
thf(fact_402_curryD,axiom,
    ! [F2: product_prod_fm_fm > $o,A: fm,B: fm] :
      ( ( produc5605581514536698091m_fm_o @ F2 @ A @ B )
     => ( F2 @ ( product_Pair_fm_fm @ A @ B ) ) ) ).

% curryD
thf(fact_403_curryD,axiom,
    ! [F2: produc1996495991257130529ist_fm > $o,A: list_fm,B: list_fm] :
      ( ( produc4804894507120027467t_fm_o @ F2 @ A @ B )
     => ( F2 @ ( produc381145313068854617ist_fm @ A @ B ) ) ) ).

% curryD
thf(fact_404_curryD,axiom,
    ! [F2: produc1316288955051664109set_fm > $o,A: set_fm,B: set_fm] :
      ( ( produc4499628267607578751t_fm_o @ F2 @ A @ B )
     => ( F2 @ ( produc3030414577503514149set_fm @ A @ B ) ) ) ).

% curryD
thf(fact_405_curryE,axiom,
    ! [F2: produc4630866025709511057ist_fm > $o,A: rule,B: produc1996495991257130529ist_fm] :
      ( ( produc2127844983229472711t_fm_o @ F2 @ A @ B )
     => ( F2 @ ( produc7063755442767259331ist_fm @ A @ B ) ) ) ).

% curryE
thf(fact_406_curryE,axiom,
    ! [F2: produc780299800278082923e_rule > $o,A: rule,B: rule] :
      ( ( produc7279490986317915905rule_o @ F2 @ A @ B )
     => ( F2 @ ( produc1769597910952761123e_rule @ A @ B ) ) ) ).

% curryE
thf(fact_407_curryE,axiom,
    ! [F2: product_prod_fm_fm > $o,A: fm,B: fm] :
      ( ( produc5605581514536698091m_fm_o @ F2 @ A @ B )
     => ( F2 @ ( product_Pair_fm_fm @ A @ B ) ) ) ).

% curryE
thf(fact_408_curryE,axiom,
    ! [F2: produc1996495991257130529ist_fm > $o,A: list_fm,B: list_fm] :
      ( ( produc4804894507120027467t_fm_o @ F2 @ A @ B )
     => ( F2 @ ( produc381145313068854617ist_fm @ A @ B ) ) ) ).

% curryE
thf(fact_409_curryE,axiom,
    ! [F2: produc1316288955051664109set_fm > $o,A: set_fm,B: set_fm] :
      ( ( produc4499628267607578751t_fm_o @ F2 @ A @ B )
     => ( F2 @ ( produc3030414577503514149set_fm @ A @ B ) ) ) ).

% curryE
thf(fact_410_csquare__fstOp__sndOp,axiom,
    ! [F2: ( list_fm > list_fm > $o ) > produc1996495991257130529ist_fm > $o,P2: list_fm > list_fm > $o,Q3: list_fm > list_fm > $o] : ( bNF_cs3500245479755196598ist_fm @ ( collec7231575776666803724ist_fm @ ( F2 @ ( relcom6651562906911586515ist_fm @ P2 @ Q3 ) ) ) @ produc4588648349897876871ist_fm @ produc1501393135466168645ist_fm @ ( bNF_fs6361025107161210196ist_fm @ P2 @ Q3 ) @ ( bNF_sn5292590578203848342ist_fm @ P2 @ Q3 ) ) ).

% csquare_fstOp_sndOp
thf(fact_411_csquare__fstOp__sndOp,axiom,
    ! [F2: ( rule > rule > $o ) > produc780299800278082923e_rule > $o,P2: rule > produc1996495991257130529ist_fm > $o,Q3: produc1996495991257130529ist_fm > rule > $o] : ( bNF_cs8798692456506898627m_rule @ ( collec5224624948449408342e_rule @ ( F2 @ ( relcom4184253850255763312m_rule @ P2 @ Q3 ) ) ) @ produc4514744107061996309ist_fm @ produc6879501374131015971m_rule @ ( bNF_fs9177076566076742319m_rule @ P2 @ Q3 ) @ ( bNF_sn4255884501705713901m_rule @ P2 @ Q3 ) ) ).

% csquare_fstOp_sndOp
thf(fact_412_csquare__fstOp__sndOp,axiom,
    ! [F2: ( rule > rule > $o ) > produc780299800278082923e_rule > $o,P2: rule > rule > $o,Q3: rule > rule > $o] : ( bNF_cs2512365784567753947e_rule @ ( collec5224624948449408342e_rule @ ( F2 @ ( relcom6206771563960779778e_rule @ P2 @ Q3 ) ) ) @ produc6725306161561501521e_rule @ produc7264528503596239631e_rule @ ( bNF_fs6019504843140265091e_rule @ P2 @ Q3 ) @ ( bNF_sn5573932575237119429e_rule @ P2 @ Q3 ) ) ).

% csquare_fstOp_sndOp
thf(fact_413_csquare__fstOp__sndOp,axiom,
    ! [F2: ( fm > fm > $o ) > product_prod_fm_fm > $o,P2: fm > fm > $o,Q3: fm > fm > $o] : ( bNF_cs8148956814858235302_fm_fm @ ( collec4425076577877259116_fm_fm @ ( F2 @ ( relcompp_fm_fm_fm @ P2 @ Q3 ) ) ) @ product_snd_fm_fm @ product_fst_fm_fm @ ( bNF_fstOp_fm_fm_fm @ P2 @ Q3 ) @ ( bNF_sndOp_fm_fm_fm @ P2 @ Q3 ) ) ).

% csquare_fstOp_sndOp
thf(fact_414_csquare__fstOp__sndOp,axiom,
    ! [F2: ( produc1996495991257130529ist_fm > rule > $o ) > produc164195504107695125m_rule > $o,P2: produc1996495991257130529ist_fm > rule > $o,Q3: rule > rule > $o] : ( bNF_cs1207223144858359663e_rule @ ( collec2069506698588392810m_rule @ ( F2 @ ( relcom6373949985761432764e_rule @ P2 @ Q3 ) ) ) @ produc7165828336582415457m_rule @ produc7264528503596239631e_rule @ ( bNF_fs2143400664727635963e_rule @ P2 @ Q3 ) @ ( bNF_sn6445580637211383353e_rule @ P2 @ Q3 ) ) ).

% csquare_fstOp_sndOp
thf(fact_415_csquare__fstOp__sndOp,axiom,
    ! [F2: ( rule > produc1996495991257130529ist_fm > $o ) > produc4630866025709511057ist_fm > $o,P2: rule > rule > $o,Q3: rule > produc1996495991257130529ist_fm > $o] : ( bNF_cs7190002734011119527ist_fm @ ( collec6536177220190208742ist_fm @ ( F2 @ ( relcom1533169620735344164ist_fm @ P2 @ Q3 ) ) ) @ produc6725306161561501521e_rule @ produc4228417144610596823ist_fm @ ( bNF_fs6525992336556323171ist_fm @ P2 @ Q3 ) @ ( bNF_sn1604800272185294753ist_fm @ P2 @ Q3 ) ) ).

% csquare_fstOp_sndOp
thf(fact_416_csquare__fstOp__sndOp,axiom,
    ! [F2: ( list_fm > list_fm > $o ) > produc1996495991257130529ist_fm > $o,P2: list_fm > produc1996495991257130529ist_fm > $o,Q3: produc1996495991257130529ist_fm > list_fm > $o] : ( bNF_cs8045896581061604419ist_fm @ ( collec7231575776666803724ist_fm @ ( F2 @ ( relcom4917561651046858416ist_fm @ P2 @ Q3 ) ) ) @ produc163170820371279546ist_fm @ produc4691395870518712766ist_fm @ ( bNF_fs4096963587915184623ist_fm @ P2 @ Q3 ) @ ( bNF_sn1937034955891054637ist_fm @ P2 @ Q3 ) ) ).

% csquare_fstOp_sndOp
thf(fact_417_csquare__fstOp__sndOp,axiom,
    ! [F2: ( produc164195504107695125m_rule > fm > $o ) > produc7564083655194549958ule_fm > $o,P2: produc164195504107695125m_rule > fm > $o,Q3: fm > fm > $o] : ( bNF_cs5705172639096540902_fm_fm @ ( collec7776316114922985905ule_fm @ ( F2 @ ( relcom8913627028627433960_fm_fm @ P2 @ Q3 ) ) ) @ produc8694862632519904556ule_fm @ product_fst_fm_fm @ ( bNF_fs31265614210943977_fm_fm @ P2 @ Q3 ) @ ( bNF_sn1891237298393626667_fm_fm @ P2 @ Q3 ) ) ).

% csquare_fstOp_sndOp
thf(fact_418_csquare__fstOp__sndOp,axiom,
    ! [F2: ( produc1996495991257130529ist_fm > produc1996495991257130529ist_fm > $o ) > produc1754732471204517991ist_fm > $o,P2: produc1996495991257130529ist_fm > rule > $o,Q3: rule > produc1996495991257130529ist_fm > $o] : ( bNF_cs5859241137876923659ist_fm @ ( collec888080998205339090ist_fm @ ( F2 @ ( relcom7011510337094526698ist_fm @ P2 @ Q3 ) ) ) @ produc7165828336582415457m_rule @ produc4228417144610596823ist_fm @ ( bNF_fs840084403518611435ist_fm @ P2 @ Q3 ) @ ( bNF_sn3966363008258636333ist_fm @ P2 @ Q3 ) ) ).

% csquare_fstOp_sndOp
thf(fact_419_csquare__fstOp__sndOp,axiom,
    ! [F2: ( rule > produc1996495991257130529ist_fm > $o ) > produc4630866025709511057ist_fm > $o,P2: rule > produc1996495991257130529ist_fm > $o,Q3: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm > $o] : ( bNF_cs8058160078142677555ist_fm @ ( collec6536177220190208742ist_fm @ ( F2 @ ( relcom835878924308461366ist_fm @ P2 @ Q3 ) ) ) @ produc4514744107061996309ist_fm @ produc5894184038225292483ist_fm @ ( bNF_fs3887825027587321911ist_fm @ P2 @ Q3 ) @ ( bNF_sn7014103632327346809ist_fm @ P2 @ Q3 ) ) ).

% csquare_fstOp_sndOp
thf(fact_420_sndsp_Ointros,axiom,
    ! [P: produc1754732471204517991ist_fm] : ( basic_8543261809493307051ist_fm @ P @ ( produc1639249265332130309ist_fm @ P ) ) ).

% sndsp.intros
thf(fact_421_sndsp_Ointros,axiom,
    ! [P: produc780299800278082923e_rule] : ( basic_7986476677951641847e_rule @ P @ ( produc6725306161561501521e_rule @ P ) ) ).

% sndsp.intros
thf(fact_422_sndsp_Ointros,axiom,
    ! [P: product_prod_fm_fm] : ( basic_sndsp_fm_fm @ P @ ( product_snd_fm_fm @ P ) ) ).

% sndsp.intros
thf(fact_423_sndsp_Ointros,axiom,
    ! [P: produc7564083655194549958ule_fm] : ( basic_2198947812288802002ule_fm @ P @ ( produc8694862632519904556ule_fm @ P ) ) ).

% sndsp.intros
thf(fact_424_sndsp_Ointros,axiom,
    ! [P: produc1996495991257130529ist_fm] : ( basic_7394098686315630381ist_fm @ P @ ( produc4588648349897876871ist_fm @ P ) ) ).

% sndsp.intros
thf(fact_425_sndsp_Ointros,axiom,
    ! [P: produc164195504107695125m_rule] : ( basic_5782820696336509243m_rule @ P @ ( produc7165828336582415457m_rule @ P ) ) ).

% sndsp.intros
thf(fact_426_sndsp_Ointros,axiom,
    ! [P: produc4630866025709511057ist_fm] : ( basic_3131736466816090095ist_fm @ P @ ( produc4514744107061996309ist_fm @ P ) ) ).

% sndsp.intros
thf(fact_427_sndsp_Osimps,axiom,
    ( basic_8543261809493307051ist_fm
    = ( ^ [P4: produc1754732471204517991ist_fm,A5: produc1996495991257130529ist_fm] :
          ( A5
          = ( produc1639249265332130309ist_fm @ P4 ) ) ) ) ).

% sndsp.simps
thf(fact_428_sndsp_Osimps,axiom,
    ( basic_7986476677951641847e_rule
    = ( ^ [P4: produc780299800278082923e_rule,A5: rule] :
          ( A5
          = ( produc6725306161561501521e_rule @ P4 ) ) ) ) ).

% sndsp.simps
thf(fact_429_sndsp_Osimps,axiom,
    ( basic_sndsp_fm_fm
    = ( ^ [P4: product_prod_fm_fm,A5: fm] :
          ( A5
          = ( product_snd_fm_fm @ P4 ) ) ) ) ).

% sndsp.simps
thf(fact_430_sndsp_Osimps,axiom,
    ( basic_2198947812288802002ule_fm
    = ( ^ [P4: produc7564083655194549958ule_fm,A5: fm] :
          ( A5
          = ( produc8694862632519904556ule_fm @ P4 ) ) ) ) ).

% sndsp.simps
thf(fact_431_sndsp_Osimps,axiom,
    ( basic_7394098686315630381ist_fm
    = ( ^ [P4: produc1996495991257130529ist_fm,A5: list_fm] :
          ( A5
          = ( produc4588648349897876871ist_fm @ P4 ) ) ) ) ).

% sndsp.simps
thf(fact_432_sndsp_Osimps,axiom,
    ( basic_5782820696336509243m_rule
    = ( ^ [P4: produc164195504107695125m_rule,A5: rule] :
          ( A5
          = ( produc7165828336582415457m_rule @ P4 ) ) ) ) ).

% sndsp.simps
thf(fact_433_sndsp_Osimps,axiom,
    ( basic_3131736466816090095ist_fm
    = ( ^ [P4: produc4630866025709511057ist_fm,A5: produc1996495991257130529ist_fm] :
          ( A5
          = ( produc4514744107061996309ist_fm @ P4 ) ) ) ) ).

% sndsp.simps
thf(fact_434_sndsp_Ocases,axiom,
    ! [P: produc1754732471204517991ist_fm,A: produc1996495991257130529ist_fm] :
      ( ( basic_8543261809493307051ist_fm @ P @ A )
     => ( A
        = ( produc1639249265332130309ist_fm @ P ) ) ) ).

% sndsp.cases
thf(fact_435_sndsp_Ocases,axiom,
    ! [P: produc780299800278082923e_rule,A: rule] :
      ( ( basic_7986476677951641847e_rule @ P @ A )
     => ( A
        = ( produc6725306161561501521e_rule @ P ) ) ) ).

% sndsp.cases
thf(fact_436_sndsp_Ocases,axiom,
    ! [P: product_prod_fm_fm,A: fm] :
      ( ( basic_sndsp_fm_fm @ P @ A )
     => ( A
        = ( product_snd_fm_fm @ P ) ) ) ).

% sndsp.cases
thf(fact_437_sndsp_Ocases,axiom,
    ! [P: produc7564083655194549958ule_fm,A: fm] :
      ( ( basic_2198947812288802002ule_fm @ P @ A )
     => ( A
        = ( produc8694862632519904556ule_fm @ P ) ) ) ).

% sndsp.cases
thf(fact_438_sndsp_Ocases,axiom,
    ! [P: produc1996495991257130529ist_fm,A: list_fm] :
      ( ( basic_7394098686315630381ist_fm @ P @ A )
     => ( A
        = ( produc4588648349897876871ist_fm @ P ) ) ) ).

% sndsp.cases
thf(fact_439_sndsp_Ocases,axiom,
    ! [P: produc164195504107695125m_rule,A: rule] :
      ( ( basic_5782820696336509243m_rule @ P @ A )
     => ( A
        = ( produc7165828336582415457m_rule @ P ) ) ) ).

% sndsp.cases
thf(fact_440_sndsp_Ocases,axiom,
    ! [P: produc4630866025709511057ist_fm,A: produc1996495991257130529ist_fm] :
      ( ( basic_3131736466816090095ist_fm @ P @ A )
     => ( A
        = ( produc4514744107061996309ist_fm @ P ) ) ) ).

% sndsp.cases
thf(fact_441_fstsp_Ointros,axiom,
    ! [P: produc4630866025709511057ist_fm] : ( basic_2738206861472250801ist_fm @ P @ ( produc4228417144610596823ist_fm @ P ) ) ).

% fstsp.intros
thf(fact_442_fstsp_Ointros,axiom,
    ! [P: produc1754732471204517991ist_fm] : ( basic_345610455209085545ist_fm @ P @ ( produc5894184038225292483ist_fm @ P ) ) ).

% fstsp.intros
thf(fact_443_fstsp_Ointros,axiom,
    ! [P: produc780299800278082923e_rule] : ( basic_8243437378216732085e_rule @ P @ ( produc7264528503596239631e_rule @ P ) ) ).

% fstsp.intros
thf(fact_444_fstsp_Ointros,axiom,
    ! [P: product_prod_fm_fm] : ( basic_fstsp_fm_fm @ P @ ( product_fst_fm_fm @ P ) ) ).

% fstsp.intros
thf(fact_445_fstsp_Ointros,axiom,
    ! [P: produc164195504107695125m_rule] : ( basic_5389291090992669949m_rule @ P @ ( produc6879501374131015971m_rule @ P ) ) ).

% fstsp.intros
thf(fact_446_fstsp_Ointros,axiom,
    ! [P: produc1996495991257130529ist_fm] : ( basic_1563062696195910635ist_fm @ P @ ( produc1501393135466168645ist_fm @ P ) ) ).

% fstsp.intros
thf(fact_447_fstsp_Osimps,axiom,
    ( basic_2738206861472250801ist_fm
    = ( ^ [P4: produc4630866025709511057ist_fm,A5: rule] :
          ( A5
          = ( produc4228417144610596823ist_fm @ P4 ) ) ) ) ).

% fstsp.simps
thf(fact_448_fstsp_Osimps,axiom,
    ( basic_345610455209085545ist_fm
    = ( ^ [P4: produc1754732471204517991ist_fm,A5: produc1996495991257130529ist_fm] :
          ( A5
          = ( produc5894184038225292483ist_fm @ P4 ) ) ) ) ).

% fstsp.simps
thf(fact_449_fstsp_Osimps,axiom,
    ( basic_8243437378216732085e_rule
    = ( ^ [P4: produc780299800278082923e_rule,A5: rule] :
          ( A5
          = ( produc7264528503596239631e_rule @ P4 ) ) ) ) ).

% fstsp.simps
thf(fact_450_fstsp_Osimps,axiom,
    ( basic_fstsp_fm_fm
    = ( ^ [P4: product_prod_fm_fm,A5: fm] :
          ( A5
          = ( product_fst_fm_fm @ P4 ) ) ) ) ).

% fstsp.simps
thf(fact_451_fstsp_Osimps,axiom,
    ( basic_5389291090992669949m_rule
    = ( ^ [P4: produc164195504107695125m_rule,A5: produc1996495991257130529ist_fm] :
          ( A5
          = ( produc6879501374131015971m_rule @ P4 ) ) ) ) ).

% fstsp.simps
thf(fact_452_fstsp_Osimps,axiom,
    ( basic_1563062696195910635ist_fm
    = ( ^ [P4: produc1996495991257130529ist_fm,A5: list_fm] :
          ( A5
          = ( produc1501393135466168645ist_fm @ P4 ) ) ) ) ).

% fstsp.simps
thf(fact_453_fstsp_Ocases,axiom,
    ! [P: produc4630866025709511057ist_fm,A: rule] :
      ( ( basic_2738206861472250801ist_fm @ P @ A )
     => ( A
        = ( produc4228417144610596823ist_fm @ P ) ) ) ).

% fstsp.cases
thf(fact_454_fstsp_Ocases,axiom,
    ! [P: produc1754732471204517991ist_fm,A: produc1996495991257130529ist_fm] :
      ( ( basic_345610455209085545ist_fm @ P @ A )
     => ( A
        = ( produc5894184038225292483ist_fm @ P ) ) ) ).

% fstsp.cases
thf(fact_455_fstsp_Ocases,axiom,
    ! [P: produc780299800278082923e_rule,A: rule] :
      ( ( basic_8243437378216732085e_rule @ P @ A )
     => ( A
        = ( produc7264528503596239631e_rule @ P ) ) ) ).

% fstsp.cases
thf(fact_456_fstsp_Ocases,axiom,
    ! [P: product_prod_fm_fm,A: fm] :
      ( ( basic_fstsp_fm_fm @ P @ A )
     => ( A
        = ( product_fst_fm_fm @ P ) ) ) ).

% fstsp.cases
thf(fact_457_fstsp_Ocases,axiom,
    ! [P: produc164195504107695125m_rule,A: produc1996495991257130529ist_fm] :
      ( ( basic_5389291090992669949m_rule @ P @ A )
     => ( A
        = ( produc6879501374131015971m_rule @ P ) ) ) ).

% fstsp.cases
thf(fact_458_fstsp_Ocases,axiom,
    ! [P: produc1996495991257130529ist_fm,A: list_fm] :
      ( ( basic_1563062696195910635ist_fm @ P @ A )
     => ( A
        = ( produc1501393135466168645ist_fm @ P ) ) ) ).

% fstsp.cases
thf(fact_459_pick__middlep,axiom,
    ! [P2: list_P5616295576739893671ist_fm > list_P8332459875677254181m_rule > $o,Q3: list_P8332459875677254181m_rule > list_rule > $o,A: list_P5616295576739893671ist_fm,C2: list_rule] :
      ( ( relcom6924352960690907314t_rule @ P2 @ Q3 @ A @ C2 )
     => ( ( P2 @ A @ ( bNF_pi7150913733026245695t_rule @ P2 @ Q3 @ A @ C2 ) )
        & ( Q3 @ ( bNF_pi7150913733026245695t_rule @ P2 @ Q3 @ A @ C2 ) @ C2 ) ) ) ).

% pick_middlep
thf(fact_460_pick__middlep,axiom,
    ! [P2: list_rule > list_P342308711384885665ist_fm > $o,Q3: list_P342308711384885665ist_fm > list_P5616295576739893671ist_fm > $o,A: list_rule,C2: list_P5616295576739893671ist_fm] :
      ( ( relcom2364676781511777726ist_fm @ P2 @ Q3 @ A @ C2 )
     => ( ( P2 @ A @ ( bNF_pi2591237553847116107ist_fm @ P2 @ Q3 @ A @ C2 ) )
        & ( Q3 @ ( bNF_pi2591237553847116107ist_fm @ P2 @ Q3 @ A @ C2 ) @ C2 ) ) ) ).

% pick_middlep
thf(fact_461_pick__middlep,axiom,
    ! [P2: list_fm > produc1996495991257130529ist_fm > $o,Q3: produc1996495991257130529ist_fm > list_fm > $o,A: list_fm,C2: list_fm] :
      ( ( relcom4917561651046858416ist_fm @ P2 @ Q3 @ A @ C2 )
     => ( ( P2 @ A @ ( bNF_pi4641893260285395773ist_fm @ P2 @ Q3 @ A @ C2 ) )
        & ( Q3 @ ( bNF_pi4641893260285395773ist_fm @ P2 @ Q3 @ A @ C2 ) @ C2 ) ) ) ).

% pick_middlep
thf(fact_462_pick__middlep,axiom,
    ! [P2: list_fm > list_fm > $o,Q3: list_fm > list_fm > $o,A: list_fm,C2: list_fm] :
      ( ( relcom6651562906911586515ist_fm @ P2 @ Q3 @ A @ C2 )
     => ( ( P2 @ A @ ( bNF_pi3539088406881798022ist_fm @ P2 @ Q3 @ A @ C2 ) )
        & ( Q3 @ ( bNF_pi3539088406881798022ist_fm @ P2 @ Q3 @ A @ C2 ) @ C2 ) ) ) ).

% pick_middlep
thf(fact_463_pick__middlep,axiom,
    ! [P2: list_list_fm > list_P5616295576739893671ist_fm > $o,Q3: list_P5616295576739893671ist_fm > list_list_fm > $o,A: list_list_fm,C2: list_list_fm] :
      ( ( relcom3701813194287065782ist_fm @ P2 @ Q3 @ A @ C2 )
     => ( ( P2 @ A @ ( bNF_pi1903759897840381891ist_fm @ P2 @ Q3 @ A @ C2 ) )
        & ( Q3 @ ( bNF_pi1903759897840381891ist_fm @ P2 @ Q3 @ A @ C2 ) @ C2 ) ) ) ).

% pick_middlep
thf(fact_464_csquare__def,axiom,
    ( bNF_cs8798692456506898627m_rule
    = ( ^ [A6: set_Pr6543236023296505803e_rule,F1: produc4630866025709511057ist_fm > produc1996495991257130529ist_fm,F22: produc164195504107695125m_rule > produc1996495991257130529ist_fm,P13: produc780299800278082923e_rule > produc4630866025709511057ist_fm,P24: produc780299800278082923e_rule > produc164195504107695125m_rule] :
        ! [X3: produc780299800278082923e_rule] :
          ( ( member3753523862196722196e_rule @ X3 @ A6 )
         => ( ( F1 @ ( P13 @ X3 ) )
            = ( F22 @ ( P24 @ X3 ) ) ) ) ) ) ).

% csquare_def
thf(fact_465_csquare__def,axiom,
    ( bNF_cs3500245479755196598ist_fm
    = ( ^ [A6: set_Pr7058068377845519745ist_fm,F1: produc1996495991257130529ist_fm > list_fm,F22: produc1996495991257130529ist_fm > list_fm,P13: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm,P24: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm] :
        ! [X3: produc1996495991257130529ist_fm] :
          ( ( member8102475879199740618ist_fm @ X3 @ A6 )
         => ( ( F1 @ ( P13 @ X3 ) )
            = ( F22 @ ( P24 @ X3 ) ) ) ) ) ) ).

% csquare_def
thf(fact_466_fstOp__in,axiom,
    ! [Ac2: produc164195504107695125m_rule,P2: produc1996495991257130529ist_fm > rule > $o,Q3: rule > rule > $o] :
      ( ( member4220325220686508332m_rule @ Ac2 @ ( collec2069506698588392810m_rule @ ( produc7369492558480804444rule_o @ ( relcom6373949985761432764e_rule @ P2 @ Q3 ) ) ) )
     => ( member4220325220686508332m_rule @ ( bNF_fs2143400664727635963e_rule @ P2 @ Q3 @ Ac2 ) @ ( collec2069506698588392810m_rule @ ( produc7369492558480804444rule_o @ P2 ) ) ) ) ).

% fstOp_in
thf(fact_467_fstOp__in,axiom,
    ! [Ac2: produc4658268289943239723t_rule,P2: list_P5616295576739893671ist_fm > list_P8332459875677254181m_rule > $o,Q3: list_P8332459875677254181m_rule > list_rule > $o] :
      ( ( member1298209074523820610t_rule @ Ac2 @ ( collec1444192142781333632t_rule @ ( produc5542464114451386310rule_o @ ( relcom6924352960690907314t_rule @ P2 @ Q3 ) ) ) )
     => ( member5377009771577760402m_rule @ ( bNF_fs3174262682535962865t_rule @ P2 @ Q3 @ Ac2 ) @ ( collec8771865392286484432m_rule @ ( produc8924637759787548278rule_o @ P2 ) ) ) ) ).

% fstOp_in
thf(fact_468_fstOp__in,axiom,
    ! [Ac2: produc2692489293269318791ist_fm,P2: list_rule > list_P342308711384885665ist_fm > $o,Q3: list_P342308711384885665ist_fm > list_P5616295576739893671ist_fm > $o] :
      ( ( member8555802114704675486ist_fm @ Ac2 @ ( collec8701785182962188508ist_fm @ ( produc7252589092299362866t_fm_o @ ( relcom2364676781511777726ist_fm @ P2 @ Q3 ) ) ) )
     => ( member7429862044449542400ist_fm @ ( bNF_fs7837958540211609085ist_fm @ P2 @ Q3 @ Ac2 ) @ ( collec6796678497650154306ist_fm @ ( produc1631752735954852212t_fm_o @ P2 ) ) ) ) ).

% fstOp_in
thf(fact_469_fstOp__in,axiom,
    ! [Ac2: produc8337847990479660737ist_fm,P2: list_list_fm > list_P5616295576739893671ist_fm > $o,Q3: list_P5616295576739893671ist_fm > list_list_fm > $o] :
      ( ( member197921522497012330ist_fm @ Ac2 @ ( collec8284558004976384940ist_fm @ ( produc3590760558050001162t_fm_o @ ( relcom3701813194287065782ist_fm @ P2 @ Q3 ) ) ) )
     => ( member1793045311909708099ist_fm @ ( bNF_fs6508523473449001845ist_fm @ P2 @ Q3 @ Ac2 ) @ ( collec5453685843627390849ist_fm @ ( produc902864307769019469t_fm_o @ P2 ) ) ) ) ).

% fstOp_in
thf(fact_470_fstOp__in,axiom,
    ! [Ac2: produc780299800278082923e_rule,P2: rule > produc1996495991257130529ist_fm > $o,Q3: produc1996495991257130529ist_fm > rule > $o] :
      ( ( member3753523862196722196e_rule @ Ac2 @ ( collec5224624948449408342e_rule @ ( produc8052491652537269344rule_o @ ( relcom4184253850255763312m_rule @ P2 @ Q3 ) ) ) )
     => ( member8686995742288324264ist_fm @ ( bNF_fs9177076566076742319m_rule @ P2 @ Q3 @ Ac2 ) @ ( collec6536177220190208742ist_fm @ ( produc1628407760719572648t_fm_o @ P2 ) ) ) ) ).

% fstOp_in
thf(fact_471_fstOp__in,axiom,
    ! [Ac2: produc4630866025709511057ist_fm,P2: rule > produc1996495991257130529ist_fm > $o,Q3: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm > $o] :
      ( ( member8686995742288324264ist_fm @ Ac2 @ ( collec6536177220190208742ist_fm @ ( produc1628407760719572648t_fm_o @ ( relcom835878924308461366ist_fm @ P2 @ Q3 ) ) ) )
     => ( member8686995742288324264ist_fm @ ( bNF_fs3887825027587321911ist_fm @ P2 @ Q3 @ Ac2 ) @ ( collec6536177220190208742ist_fm @ ( produc1628407760719572648t_fm_o @ P2 ) ) ) ) ).

% fstOp_in
thf(fact_472_fstOp__in,axiom,
    ! [Ac2: produc1996495991257130529ist_fm,P2: list_fm > produc1996495991257130529ist_fm > $o,Q3: produc1996495991257130529ist_fm > list_fm > $o] :
      ( ( member8102475879199740618ist_fm @ Ac2 @ ( collec7231575776666803724ist_fm @ ( produc2011568048573023402t_fm_o @ ( relcom4917561651046858416ist_fm @ P2 @ Q3 ) ) ) )
     => ( member8482595077848144461ist_fm @ ( bNF_fs4096963587915184623ist_fm @ P2 @ Q3 @ Ac2 ) @ ( collec7064939129013829259ist_fm @ ( produc6456183123267691459t_fm_o @ P2 ) ) ) ) ).

% fstOp_in
thf(fact_473_fstOp__in,axiom,
    ! [Ac2: produc1996495991257130529ist_fm,P2: list_fm > list_fm > $o,Q3: list_fm > list_fm > $o] :
      ( ( member8102475879199740618ist_fm @ Ac2 @ ( collec7231575776666803724ist_fm @ ( produc2011568048573023402t_fm_o @ ( relcom6651562906911586515ist_fm @ P2 @ Q3 ) ) ) )
     => ( member8102475879199740618ist_fm @ ( bNF_fs6361025107161210196ist_fm @ P2 @ Q3 @ Ac2 ) @ ( collec7231575776666803724ist_fm @ ( produc2011568048573023402t_fm_o @ P2 ) ) ) ) ).

% fstOp_in
thf(fact_474_sndOp__in,axiom,
    ! [Ac2: produc780299800278082923e_rule,P2: rule > produc1996495991257130529ist_fm > $o,Q3: produc1996495991257130529ist_fm > rule > $o] :
      ( ( member3753523862196722196e_rule @ Ac2 @ ( collec5224624948449408342e_rule @ ( produc8052491652537269344rule_o @ ( relcom4184253850255763312m_rule @ P2 @ Q3 ) ) ) )
     => ( member4220325220686508332m_rule @ ( bNF_sn4255884501705713901m_rule @ P2 @ Q3 @ Ac2 ) @ ( collec2069506698588392810m_rule @ ( produc7369492558480804444rule_o @ Q3 ) ) ) ) ).

% sndOp_in
thf(fact_475_sndOp__in,axiom,
    ! [Ac2: produc164195504107695125m_rule,P2: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm > $o,Q3: produc1996495991257130529ist_fm > rule > $o] :
      ( ( member4220325220686508332m_rule @ Ac2 @ ( collec2069506698588392810m_rule @ ( produc7369492558480804444rule_o @ ( relcom439222529760170038m_rule @ P2 @ Q3 ) ) ) )
     => ( member4220325220686508332m_rule @ ( bNF_sn6617447237779055481m_rule @ P2 @ Q3 @ Ac2 ) @ ( collec2069506698588392810m_rule @ ( produc7369492558480804444rule_o @ Q3 ) ) ) ) ).

% sndOp_in
thf(fact_476_sndOp__in,axiom,
    ! [Ac2: produc4658268289943239723t_rule,P2: list_P5616295576739893671ist_fm > list_P8332459875677254181m_rule > $o,Q3: list_P8332459875677254181m_rule > list_rule > $o] :
      ( ( member1298209074523820610t_rule @ Ac2 @ ( collec1444192142781333632t_rule @ ( produc5542464114451386310rule_o @ ( relcom6924352960690907314t_rule @ P2 @ Q3 ) ) ) )
     => ( member6491770712979037828t_rule @ ( bNF_sn1086680112216707375t_rule @ P2 @ Q3 @ Ac2 ) @ ( collec5858587166179649734t_rule @ ( produc2090550080120670960rule_o @ Q3 ) ) ) ) ).

% sndOp_in
thf(fact_477_sndOp__in,axiom,
    ! [Ac2: produc2692489293269318791ist_fm,P2: list_rule > list_P342308711384885665ist_fm > $o,Q3: list_P342308711384885665ist_fm > list_P5616295576739893671ist_fm > $o] :
      ( ( member8555802114704675486ist_fm @ Ac2 @ ( collec8701785182962188508ist_fm @ ( produc7252589092299362866t_fm_o @ ( relcom2364676781511777726ist_fm @ P2 @ Q3 ) ) ) )
     => ( member3096056441066211794ist_fm @ ( bNF_sn5750375969892353595ist_fm @ P2 @ Q3 @ Ac2 ) @ ( collec6490912061774935824ist_fm @ ( produc294104671633537790t_fm_o @ Q3 ) ) ) ) ).

% sndOp_in
thf(fact_478_sndOp__in,axiom,
    ! [Ac2: produc8337847990479660737ist_fm,P2: list_list_fm > list_P5616295576739893671ist_fm > $o,Q3: list_P5616295576739893671ist_fm > list_list_fm > $o] :
      ( ( member197921522497012330ist_fm @ Ac2 @ ( collec8284558004976384940ist_fm @ ( produc3590760558050001162t_fm_o @ ( relcom3701813194287065782ist_fm @ P2 @ Q3 ) ) ) )
     => ( member4192383230844517341ist_fm @ ( bNF_sn6419741859600833715ist_fm @ P2 @ Q3 @ Ac2 ) @ ( collec7853023762562200091ist_fm @ ( produc2973715795501690859t_fm_o @ Q3 ) ) ) ) ).

% sndOp_in
thf(fact_479_sndOp__in,axiom,
    ! [Ac2: produc4630866025709511057ist_fm,P2: rule > rule > $o,Q3: rule > produc1996495991257130529ist_fm > $o] :
      ( ( member8686995742288324264ist_fm @ Ac2 @ ( collec6536177220190208742ist_fm @ ( produc1628407760719572648t_fm_o @ ( relcom1533169620735344164ist_fm @ P2 @ Q3 ) ) ) )
     => ( member8686995742288324264ist_fm @ ( bNF_sn1604800272185294753ist_fm @ P2 @ Q3 @ Ac2 ) @ ( collec6536177220190208742ist_fm @ ( produc1628407760719572648t_fm_o @ Q3 ) ) ) ) ).

% sndOp_in
thf(fact_480_sndOp__in,axiom,
    ! [Ac2: produc1996495991257130529ist_fm,P2: list_fm > produc1996495991257130529ist_fm > $o,Q3: produc1996495991257130529ist_fm > list_fm > $o] :
      ( ( member8102475879199740618ist_fm @ Ac2 @ ( collec7231575776666803724ist_fm @ ( produc2011568048573023402t_fm_o @ ( relcom4917561651046858416ist_fm @ P2 @ Q3 ) ) ) )
     => ( member7196661723473486919ist_fm @ ( bNF_sn1937034955891054637ist_fm @ P2 @ Q3 @ Ac2 ) @ ( collec5779005774639171717ist_fm @ ( produc3115149902393377025t_fm_o @ Q3 ) ) ) ) ).

% sndOp_in
thf(fact_481_sndOp__in,axiom,
    ! [Ac2: produc1996495991257130529ist_fm,P2: list_fm > list_fm > $o,Q3: list_fm > list_fm > $o] :
      ( ( member8102475879199740618ist_fm @ Ac2 @ ( collec7231575776666803724ist_fm @ ( produc2011568048573023402t_fm_o @ ( relcom6651562906911586515ist_fm @ P2 @ Q3 ) ) ) )
     => ( member8102475879199740618ist_fm @ ( bNF_sn5292590578203848342ist_fm @ P2 @ Q3 @ Ac2 ) @ ( collec7231575776666803724ist_fm @ ( produc2011568048573023402t_fm_o @ Q3 ) ) ) ) ).

% sndOp_in
thf(fact_482_relcompp__assoc,axiom,
    ! [R2: list_fm > list_fm > $o,S3: list_fm > list_fm > $o,T2: list_fm > list_fm > $o] :
      ( ( relcom6651562906911586515ist_fm @ ( relcom6651562906911586515ist_fm @ R2 @ S3 ) @ T2 )
      = ( relcom6651562906911586515ist_fm @ R2 @ ( relcom6651562906911586515ist_fm @ S3 @ T2 ) ) ) ).

% relcompp_assoc
thf(fact_483_relcompp__assoc,axiom,
    ! [R2: list_fm > list_fm > $o,S3: list_fm > produc1996495991257130529ist_fm > $o,T2: produc1996495991257130529ist_fm > list_fm > $o] :
      ( ( relcom4917561651046858416ist_fm @ ( relcom8878007651405842542ist_fm @ R2 @ S3 ) @ T2 )
      = ( relcom6651562906911586515ist_fm @ R2 @ ( relcom4917561651046858416ist_fm @ S3 @ T2 ) ) ) ).

% relcompp_assoc
thf(fact_484_relcompp__assoc,axiom,
    ! [R2: list_fm > produc1996495991257130529ist_fm > $o,S3: produc1996495991257130529ist_fm > list_fm > $o,T2: list_fm > list_fm > $o] :
      ( ( relcom6651562906911586515ist_fm @ ( relcom4917561651046858416ist_fm @ R2 @ S3 ) @ T2 )
      = ( relcom4917561651046858416ist_fm @ R2 @ ( relcom1701028264527405810ist_fm @ S3 @ T2 ) ) ) ).

% relcompp_assoc
thf(fact_485_relcompp__assoc,axiom,
    ! [R2: list_fm > produc1996495991257130529ist_fm > $o,S3: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm > $o,T2: produc1996495991257130529ist_fm > list_fm > $o] :
      ( ( relcom4917561651046858416ist_fm @ ( relcom3274822705743940433ist_fm @ R2 @ S3 ) @ T2 )
      = ( relcom4917561651046858416ist_fm @ R2 @ ( relcom7948118786659503313ist_fm @ S3 @ T2 ) ) ) ).

% relcompp_assoc
thf(fact_486_relcompp__assoc,axiom,
    ! [R2: list_list_fm > list_P5616295576739893671ist_fm > $o,S3: list_P5616295576739893671ist_fm > list_list_fm > $o,T2: list_list_fm > list_list_fm > $o] :
      ( ( relcom484072178968148995ist_fm @ ( relcom3701813194287065782ist_fm @ R2 @ S3 ) @ T2 )
      = ( relcom3701813194287065782ist_fm @ R2 @ ( relcom1573290599090104216ist_fm @ S3 @ T2 ) ) ) ).

% relcompp_assoc
thf(fact_487_relcompp__assoc,axiom,
    ! [R2: list_list_fm > list_list_fm > $o,S3: list_list_fm > list_P5616295576739893671ist_fm > $o,T2: list_P5616295576739893671ist_fm > list_list_fm > $o] :
      ( ( relcom3701813194287065782ist_fm @ ( relcom3513675634381398740ist_fm @ R2 @ S3 ) @ T2 )
      = ( relcom484072178968148995ist_fm @ R2 @ ( relcom3701813194287065782ist_fm @ S3 @ T2 ) ) ) ).

% relcompp_assoc
thf(fact_488_relcompp__assoc,axiom,
    ! [R2: list_P5616295576739893671ist_fm > list_P8332459875677254181m_rule > $o,S3: list_P8332459875677254181m_rule > list_rule > $o,T2: list_rule > list_rule > $o] :
      ( ( relcom6681704288941848034t_rule @ ( relcom6924352960690907314t_rule @ R2 @ S3 ) @ T2 )
      = ( relcom6924352960690907314t_rule @ R2 @ ( relcom9143888689469128706t_rule @ S3 @ T2 ) ) ) ).

% relcompp_assoc
thf(fact_489_relcompp__assoc,axiom,
    ! [R2: list_rule > list_rule > $o,S3: list_rule > list_P342308711384885665ist_fm > $o,T2: list_P342308711384885665ist_fm > list_P5616295576739893671ist_fm > $o] :
      ( ( relcom2364676781511777726ist_fm @ ( relcom504160833975393662ist_fm @ R2 @ S3 ) @ T2 )
      = ( relcom1034014924007176458ist_fm @ R2 @ ( relcom2364676781511777726ist_fm @ S3 @ T2 ) ) ) ).

% relcompp_assoc
thf(fact_490_relcompp__assoc,axiom,
    ! [R2: list_list_fm > list_P5616295576739893671ist_fm > $o,S3: list_P5616295576739893671ist_fm > list_P5616295576739893671ist_fm > $o,T2: list_P5616295576739893671ist_fm > list_list_fm > $o] :
      ( ( relcom3701813194287065782ist_fm @ ( relcom8260208622964639585ist_fm @ R2 @ S3 ) @ T2 )
      = ( relcom3701813194287065782ist_fm @ R2 @ ( relcom8491732736688609761ist_fm @ S3 @ T2 ) ) ) ).

% relcompp_assoc
thf(fact_491_relcompp__assoc,axiom,
    ! [R2: list_rule > list_P342308711384885665ist_fm > $o,S3: list_P342308711384885665ist_fm > list_P5616295576739893671ist_fm > $o,T2: list_P5616295576739893671ist_fm > list_P5616295576739893671ist_fm > $o] :
      ( ( relcom165759644607821766ist_fm @ ( relcom2364676781511777726ist_fm @ R2 @ S3 ) @ T2 )
      = ( relcom2364676781511777726ist_fm @ R2 @ ( relcom726526821484939538ist_fm @ S3 @ T2 ) ) ) ).

% relcompp_assoc
thf(fact_492_relcompp__apply,axiom,
    ( relcom6924352960690907314t_rule
    = ( ^ [R3: list_P5616295576739893671ist_fm > list_P8332459875677254181m_rule > $o,S5: list_P8332459875677254181m_rule > list_rule > $o,A5: list_P5616295576739893671ist_fm,C3: list_rule] :
        ? [B4: list_P8332459875677254181m_rule] :
          ( ( R3 @ A5 @ B4 )
          & ( S5 @ B4 @ C3 ) ) ) ) ).

% relcompp_apply
thf(fact_493_relcompp__apply,axiom,
    ( relcom2364676781511777726ist_fm
    = ( ^ [R3: list_rule > list_P342308711384885665ist_fm > $o,S5: list_P342308711384885665ist_fm > list_P5616295576739893671ist_fm > $o,A5: list_rule,C3: list_P5616295576739893671ist_fm] :
        ? [B4: list_P342308711384885665ist_fm] :
          ( ( R3 @ A5 @ B4 )
          & ( S5 @ B4 @ C3 ) ) ) ) ).

% relcompp_apply
thf(fact_494_relcompp__apply,axiom,
    ( relcom4917561651046858416ist_fm
    = ( ^ [R3: list_fm > produc1996495991257130529ist_fm > $o,S5: produc1996495991257130529ist_fm > list_fm > $o,A5: list_fm,C3: list_fm] :
        ? [B4: produc1996495991257130529ist_fm] :
          ( ( R3 @ A5 @ B4 )
          & ( S5 @ B4 @ C3 ) ) ) ) ).

% relcompp_apply
thf(fact_495_relcompp__apply,axiom,
    ( relcom6651562906911586515ist_fm
    = ( ^ [R3: list_fm > list_fm > $o,S5: list_fm > list_fm > $o,A5: list_fm,C3: list_fm] :
        ? [B4: list_fm] :
          ( ( R3 @ A5 @ B4 )
          & ( S5 @ B4 @ C3 ) ) ) ) ).

% relcompp_apply
thf(fact_496_relcompp__apply,axiom,
    ( relcom3701813194287065782ist_fm
    = ( ^ [R3: list_list_fm > list_P5616295576739893671ist_fm > $o,S5: list_P5616295576739893671ist_fm > list_list_fm > $o,A5: list_list_fm,C3: list_list_fm] :
        ? [B4: list_P5616295576739893671ist_fm] :
          ( ( R3 @ A5 @ B4 )
          & ( S5 @ B4 @ C3 ) ) ) ) ).

% relcompp_apply
thf(fact_497_relcomppI,axiom,
    ! [R2: list_P5616295576739893671ist_fm > list_P8332459875677254181m_rule > $o,A: list_P5616295576739893671ist_fm,B: list_P8332459875677254181m_rule,S3: list_P8332459875677254181m_rule > list_rule > $o,C2: list_rule] :
      ( ( R2 @ A @ B )
     => ( ( S3 @ B @ C2 )
       => ( relcom6924352960690907314t_rule @ R2 @ S3 @ A @ C2 ) ) ) ).

% relcomppI
thf(fact_498_relcomppI,axiom,
    ! [R2: list_rule > list_P342308711384885665ist_fm > $o,A: list_rule,B: list_P342308711384885665ist_fm,S3: list_P342308711384885665ist_fm > list_P5616295576739893671ist_fm > $o,C2: list_P5616295576739893671ist_fm] :
      ( ( R2 @ A @ B )
     => ( ( S3 @ B @ C2 )
       => ( relcom2364676781511777726ist_fm @ R2 @ S3 @ A @ C2 ) ) ) ).

% relcomppI
thf(fact_499_relcomppI,axiom,
    ! [R2: list_fm > produc1996495991257130529ist_fm > $o,A: list_fm,B: produc1996495991257130529ist_fm,S3: produc1996495991257130529ist_fm > list_fm > $o,C2: list_fm] :
      ( ( R2 @ A @ B )
     => ( ( S3 @ B @ C2 )
       => ( relcom4917561651046858416ist_fm @ R2 @ S3 @ A @ C2 ) ) ) ).

% relcomppI
thf(fact_500_relcomppI,axiom,
    ! [R2: list_fm > list_fm > $o,A: list_fm,B: list_fm,S3: list_fm > list_fm > $o,C2: list_fm] :
      ( ( R2 @ A @ B )
     => ( ( S3 @ B @ C2 )
       => ( relcom6651562906911586515ist_fm @ R2 @ S3 @ A @ C2 ) ) ) ).

% relcomppI
thf(fact_501_relcomppI,axiom,
    ! [R2: list_list_fm > list_P5616295576739893671ist_fm > $o,A: list_list_fm,B: list_P5616295576739893671ist_fm,S3: list_P5616295576739893671ist_fm > list_list_fm > $o,C2: list_list_fm] :
      ( ( R2 @ A @ B )
     => ( ( S3 @ B @ C2 )
       => ( relcom3701813194287065782ist_fm @ R2 @ S3 @ A @ C2 ) ) ) ).

% relcomppI
thf(fact_502_relcomppE,axiom,
    ! [R2: list_P5616295576739893671ist_fm > list_P8332459875677254181m_rule > $o,S3: list_P8332459875677254181m_rule > list_rule > $o,A: list_P5616295576739893671ist_fm,C2: list_rule] :
      ( ( relcom6924352960690907314t_rule @ R2 @ S3 @ A @ C2 )
     => ~ ! [B3: list_P8332459875677254181m_rule] :
            ( ( R2 @ A @ B3 )
           => ~ ( S3 @ B3 @ C2 ) ) ) ).

% relcomppE
thf(fact_503_relcomppE,axiom,
    ! [R2: list_rule > list_P342308711384885665ist_fm > $o,S3: list_P342308711384885665ist_fm > list_P5616295576739893671ist_fm > $o,A: list_rule,C2: list_P5616295576739893671ist_fm] :
      ( ( relcom2364676781511777726ist_fm @ R2 @ S3 @ A @ C2 )
     => ~ ! [B3: list_P342308711384885665ist_fm] :
            ( ( R2 @ A @ B3 )
           => ~ ( S3 @ B3 @ C2 ) ) ) ).

% relcomppE
thf(fact_504_relcomppE,axiom,
    ! [R2: list_fm > produc1996495991257130529ist_fm > $o,S3: produc1996495991257130529ist_fm > list_fm > $o,A: list_fm,C2: list_fm] :
      ( ( relcom4917561651046858416ist_fm @ R2 @ S3 @ A @ C2 )
     => ~ ! [B3: produc1996495991257130529ist_fm] :
            ( ( R2 @ A @ B3 )
           => ~ ( S3 @ B3 @ C2 ) ) ) ).

% relcomppE
thf(fact_505_relcomppE,axiom,
    ! [R2: list_fm > list_fm > $o,S3: list_fm > list_fm > $o,A: list_fm,C2: list_fm] :
      ( ( relcom6651562906911586515ist_fm @ R2 @ S3 @ A @ C2 )
     => ~ ! [B3: list_fm] :
            ( ( R2 @ A @ B3 )
           => ~ ( S3 @ B3 @ C2 ) ) ) ).

% relcomppE
thf(fact_506_relcomppE,axiom,
    ! [R2: list_list_fm > list_P5616295576739893671ist_fm > $o,S3: list_P5616295576739893671ist_fm > list_list_fm > $o,A: list_list_fm,C2: list_list_fm] :
      ( ( relcom3701813194287065782ist_fm @ R2 @ S3 @ A @ C2 )
     => ~ ! [B3: list_P5616295576739893671ist_fm] :
            ( ( R2 @ A @ B3 )
           => ~ ( S3 @ B3 @ C2 ) ) ) ).

% relcomppE
thf(fact_507_eq__comp__r,axiom,
    ! [R: list_fm > list_fm > $o] :
      ( ( relcom6651562906911586515ist_fm
        @ ^ [Y2: list_fm,Z: list_fm] : ( Y2 = Z )
        @ ( relcom6651562906911586515ist_fm @ R
          @ ^ [Y2: list_fm,Z: list_fm] : ( Y2 = Z ) ) )
      = R ) ).

% eq_comp_r
thf(fact_508_eq__OO,axiom,
    ! [R: list_fm > list_fm > $o] :
      ( ( relcom6651562906911586515ist_fm
        @ ^ [Y2: list_fm,Z: list_fm] : ( Y2 = Z )
        @ R )
      = R ) ).

% eq_OO
thf(fact_509_OO__eq,axiom,
    ! [R: list_fm > list_fm > $o] :
      ( ( relcom6651562906911586515ist_fm @ R
        @ ^ [Y2: list_fm,Z: list_fm] : ( Y2 = Z ) )
      = R ) ).

% OO_eq
thf(fact_510_relcompp_Osimps,axiom,
    ( relcom6924352960690907314t_rule
    = ( ^ [R4: list_P5616295576739893671ist_fm > list_P8332459875677254181m_rule > $o,S: list_P8332459875677254181m_rule > list_rule > $o,A1: list_P5616295576739893671ist_fm,A22: list_rule] :
        ? [A5: list_P5616295576739893671ist_fm,B4: list_P8332459875677254181m_rule,C3: list_rule] :
          ( ( A1 = A5 )
          & ( A22 = C3 )
          & ( R4 @ A5 @ B4 )
          & ( S @ B4 @ C3 ) ) ) ) ).

% relcompp.simps
thf(fact_511_relcompp_Osimps,axiom,
    ( relcom2364676781511777726ist_fm
    = ( ^ [R4: list_rule > list_P342308711384885665ist_fm > $o,S: list_P342308711384885665ist_fm > list_P5616295576739893671ist_fm > $o,A1: list_rule,A22: list_P5616295576739893671ist_fm] :
        ? [A5: list_rule,B4: list_P342308711384885665ist_fm,C3: list_P5616295576739893671ist_fm] :
          ( ( A1 = A5 )
          & ( A22 = C3 )
          & ( R4 @ A5 @ B4 )
          & ( S @ B4 @ C3 ) ) ) ) ).

% relcompp.simps
thf(fact_512_relcompp_Osimps,axiom,
    ( relcom4917561651046858416ist_fm
    = ( ^ [R4: list_fm > produc1996495991257130529ist_fm > $o,S: produc1996495991257130529ist_fm > list_fm > $o,A1: list_fm,A22: list_fm] :
        ? [A5: list_fm,B4: produc1996495991257130529ist_fm,C3: list_fm] :
          ( ( A1 = A5 )
          & ( A22 = C3 )
          & ( R4 @ A5 @ B4 )
          & ( S @ B4 @ C3 ) ) ) ) ).

% relcompp.simps
thf(fact_513_relcompp_Osimps,axiom,
    ( relcom6651562906911586515ist_fm
    = ( ^ [R4: list_fm > list_fm > $o,S: list_fm > list_fm > $o,A1: list_fm,A22: list_fm] :
        ? [A5: list_fm,B4: list_fm,C3: list_fm] :
          ( ( A1 = A5 )
          & ( A22 = C3 )
          & ( R4 @ A5 @ B4 )
          & ( S @ B4 @ C3 ) ) ) ) ).

% relcompp.simps
thf(fact_514_relcompp_Osimps,axiom,
    ( relcom3701813194287065782ist_fm
    = ( ^ [R4: list_list_fm > list_P5616295576739893671ist_fm > $o,S: list_P5616295576739893671ist_fm > list_list_fm > $o,A1: list_list_fm,A22: list_list_fm] :
        ? [A5: list_list_fm,B4: list_P5616295576739893671ist_fm,C3: list_list_fm] :
          ( ( A1 = A5 )
          & ( A22 = C3 )
          & ( R4 @ A5 @ B4 )
          & ( S @ B4 @ C3 ) ) ) ) ).

% relcompp.simps
thf(fact_515_relcompp_Ocases,axiom,
    ! [R2: list_P5616295576739893671ist_fm > list_P8332459875677254181m_rule > $o,S3: list_P8332459875677254181m_rule > list_rule > $o,A12: list_P5616295576739893671ist_fm,A23: list_rule] :
      ( ( relcom6924352960690907314t_rule @ R2 @ S3 @ A12 @ A23 )
     => ~ ! [B3: list_P8332459875677254181m_rule] :
            ( ( R2 @ A12 @ B3 )
           => ~ ( S3 @ B3 @ A23 ) ) ) ).

% relcompp.cases
thf(fact_516_relcompp_Ocases,axiom,
    ! [R2: list_rule > list_P342308711384885665ist_fm > $o,S3: list_P342308711384885665ist_fm > list_P5616295576739893671ist_fm > $o,A12: list_rule,A23: list_P5616295576739893671ist_fm] :
      ( ( relcom2364676781511777726ist_fm @ R2 @ S3 @ A12 @ A23 )
     => ~ ! [B3: list_P342308711384885665ist_fm] :
            ( ( R2 @ A12 @ B3 )
           => ~ ( S3 @ B3 @ A23 ) ) ) ).

% relcompp.cases
thf(fact_517_relcompp_Ocases,axiom,
    ! [R2: list_fm > produc1996495991257130529ist_fm > $o,S3: produc1996495991257130529ist_fm > list_fm > $o,A12: list_fm,A23: list_fm] :
      ( ( relcom4917561651046858416ist_fm @ R2 @ S3 @ A12 @ A23 )
     => ~ ! [B3: produc1996495991257130529ist_fm] :
            ( ( R2 @ A12 @ B3 )
           => ~ ( S3 @ B3 @ A23 ) ) ) ).

% relcompp.cases
thf(fact_518_relcompp_Ocases,axiom,
    ! [R2: list_fm > list_fm > $o,S3: list_fm > list_fm > $o,A12: list_fm,A23: list_fm] :
      ( ( relcom6651562906911586515ist_fm @ R2 @ S3 @ A12 @ A23 )
     => ~ ! [B3: list_fm] :
            ( ( R2 @ A12 @ B3 )
           => ~ ( S3 @ B3 @ A23 ) ) ) ).

% relcompp.cases
thf(fact_519_relcompp_Ocases,axiom,
    ! [R2: list_list_fm > list_P5616295576739893671ist_fm > $o,S3: list_P5616295576739893671ist_fm > list_list_fm > $o,A12: list_list_fm,A23: list_list_fm] :
      ( ( relcom3701813194287065782ist_fm @ R2 @ S3 @ A12 @ A23 )
     => ~ ! [B3: list_P5616295576739893671ist_fm] :
            ( ( R2 @ A12 @ B3 )
           => ~ ( S3 @ B3 @ A23 ) ) ) ).

% relcompp.cases
thf(fact_520_pair__in__swap__image,axiom,
    ! [Y3: rule,X2: rule,A4: set_Pr6543236023296505803e_rule] :
      ( ( member3753523862196722196e_rule @ ( produc1769597910952761123e_rule @ Y3 @ X2 ) @ ( image_3377832706447680709e_rule @ produc2320433982106674371e_rule @ A4 ) )
      = ( member3753523862196722196e_rule @ ( produc1769597910952761123e_rule @ X2 @ Y3 ) @ A4 ) ) ).

% pair_in_swap_image
thf(fact_521_pair__in__swap__image,axiom,
    ! [Y3: fm,X2: fm,A4: set_Pr4463079037648049377_fm_fm] :
      ( ( member8474499337054950954_fm_fm @ ( product_Pair_fm_fm @ Y3 @ X2 ) @ ( image_7218310488025046853_fm_fm @ product_swap_fm_fm @ A4 ) )
      = ( member8474499337054950954_fm_fm @ ( product_Pair_fm_fm @ X2 @ Y3 ) @ A4 ) ) ).

% pair_in_swap_image
thf(fact_522_pair__in__swap__image,axiom,
    ! [Y3: set_fm,X2: set_fm,A4: set_Pr8693920393607151949set_fm] :
      ( ( member6828610780098320534set_fm @ ( produc3030414577503514149set_fm @ Y3 @ X2 ) @ ( image_5801568681535411269set_fm @ produc7775435602796846021set_fm @ A4 ) )
      = ( member6828610780098320534set_fm @ ( produc3030414577503514149set_fm @ X2 @ Y3 ) @ A4 ) ) ).

% pair_in_swap_image
thf(fact_523_pair__in__swap__image,axiom,
    ! [Y3: list_fm,X2: list_fm,A4: set_Pr7058068377845519745ist_fm] :
      ( ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ Y3 @ X2 ) @ ( image_6982201141141708101ist_fm @ produc7936585717479832313ist_fm @ A4 ) )
      = ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ X2 @ Y3 ) @ A4 ) ) ).

% pair_in_swap_image
thf(fact_524_pair__in__swap__image,axiom,
    ! [Y3: produc1996495991257130529ist_fm,X2: rule,A4: set_Pr2241365836748572487ist_fm] :
      ( ( member4220325220686508332m_rule @ ( produc491467635432902671m_rule @ Y3 @ X2 ) @ ( image_4130355632178425005m_rule @ produc9213896635423732003ist_fm @ A4 ) )
      = ( member8686995742288324264ist_fm @ ( produc7063755442767259331ist_fm @ X2 @ Y3 ) @ A4 ) ) ).

% pair_in_swap_image
thf(fact_525_pair__in__swap__image,axiom,
    ! [Y3: rule,X2: produc1996495991257130529ist_fm,A4: set_Pr1008144964186165195m_rule] :
      ( ( member8686995742288324264ist_fm @ ( produc7063755442767259331ist_fm @ Y3 @ X2 ) @ ( image_6659396422799841965ist_fm @ produc2641608828089375343m_rule @ A4 ) )
      = ( member4220325220686508332m_rule @ ( produc491467635432902671m_rule @ X2 @ Y3 ) @ A4 ) ) ).

% pair_in_swap_image
thf(fact_526_in__inv__image,axiom,
    ! [X2: rule,Y3: rule,R2: set_Pr6543236023296505803e_rule,F2: rule > rule] :
      ( ( member3753523862196722196e_rule @ ( produc1769597910952761123e_rule @ X2 @ Y3 ) @ ( inv_image_rule_rule @ R2 @ F2 ) )
      = ( member3753523862196722196e_rule @ ( produc1769597910952761123e_rule @ ( F2 @ X2 ) @ ( F2 @ Y3 ) ) @ R2 ) ) ).

% in_inv_image
thf(fact_527_in__inv__image,axiom,
    ! [X2: rule,Y3: rule,R2: set_Pr4463079037648049377_fm_fm,F2: rule > fm] :
      ( ( member3753523862196722196e_rule @ ( produc1769597910952761123e_rule @ X2 @ Y3 ) @ ( inv_image_fm_rule @ R2 @ F2 ) )
      = ( member8474499337054950954_fm_fm @ ( product_Pair_fm_fm @ ( F2 @ X2 ) @ ( F2 @ Y3 ) ) @ R2 ) ) ).

% in_inv_image
thf(fact_528_in__inv__image,axiom,
    ! [X2: fm,Y3: fm,R2: set_Pr6543236023296505803e_rule,F2: fm > rule] :
      ( ( member8474499337054950954_fm_fm @ ( product_Pair_fm_fm @ X2 @ Y3 ) @ ( inv_image_rule_fm @ R2 @ F2 ) )
      = ( member3753523862196722196e_rule @ ( produc1769597910952761123e_rule @ ( F2 @ X2 ) @ ( F2 @ Y3 ) ) @ R2 ) ) ).

% in_inv_image
thf(fact_529_in__inv__image,axiom,
    ! [X2: fm,Y3: fm,R2: set_Pr4463079037648049377_fm_fm,F2: fm > fm] :
      ( ( member8474499337054950954_fm_fm @ ( product_Pair_fm_fm @ X2 @ Y3 ) @ ( inv_image_fm_fm @ R2 @ F2 ) )
      = ( member8474499337054950954_fm_fm @ ( product_Pair_fm_fm @ ( F2 @ X2 ) @ ( F2 @ Y3 ) ) @ R2 ) ) ).

% in_inv_image
thf(fact_530_in__inv__image,axiom,
    ! [X2: rule,Y3: rule,R2: set_Pr7058068377845519745ist_fm,F2: rule > list_fm] :
      ( ( member3753523862196722196e_rule @ ( produc1769597910952761123e_rule @ X2 @ Y3 ) @ ( inv_im1924138480472522489m_rule @ R2 @ F2 ) )
      = ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ ( F2 @ X2 ) @ ( F2 @ Y3 ) ) @ R2 ) ) ).

% in_inv_image
thf(fact_531_in__inv__image,axiom,
    ! [X2: rule,Y3: rule,R2: set_Pr8693920393607151949set_fm,F2: rule > set_fm] :
      ( ( member3753523862196722196e_rule @ ( produc1769597910952761123e_rule @ X2 @ Y3 ) @ ( inv_im9195035823407958047m_rule @ R2 @ F2 ) )
      = ( member6828610780098320534set_fm @ ( produc3030414577503514149set_fm @ ( F2 @ X2 ) @ ( F2 @ Y3 ) ) @ R2 ) ) ).

% in_inv_image
thf(fact_532_in__inv__image,axiom,
    ! [X2: fm,Y3: fm,R2: set_Pr7058068377845519745ist_fm,F2: fm > list_fm] :
      ( ( member8474499337054950954_fm_fm @ ( product_Pair_fm_fm @ X2 @ Y3 ) @ ( inv_image_list_fm_fm @ R2 @ F2 ) )
      = ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ ( F2 @ X2 ) @ ( F2 @ Y3 ) ) @ R2 ) ) ).

% in_inv_image
thf(fact_533_in__inv__image,axiom,
    ! [X2: fm,Y3: fm,R2: set_Pr8693920393607151949set_fm,F2: fm > set_fm] :
      ( ( member8474499337054950954_fm_fm @ ( product_Pair_fm_fm @ X2 @ Y3 ) @ ( inv_image_set_fm_fm @ R2 @ F2 ) )
      = ( member6828610780098320534set_fm @ ( produc3030414577503514149set_fm @ ( F2 @ X2 ) @ ( F2 @ Y3 ) ) @ R2 ) ) ).

% in_inv_image
thf(fact_534_in__inv__image,axiom,
    ! [X2: list_fm,Y3: list_fm,R2: set_Pr6543236023296505803e_rule,F2: list_fm > rule] :
      ( ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ X2 @ Y3 ) @ ( inv_im8274583003813727737ist_fm @ R2 @ F2 ) )
      = ( member3753523862196722196e_rule @ ( produc1769597910952761123e_rule @ ( F2 @ X2 ) @ ( F2 @ Y3 ) ) @ R2 ) ) ).

% in_inv_image
thf(fact_535_in__inv__image,axiom,
    ! [X2: list_fm,Y3: list_fm,R2: set_Pr4463079037648049377_fm_fm,F2: list_fm > fm] :
      ( ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ X2 @ Y3 ) @ ( inv_image_fm_list_fm @ R2 @ F2 ) )
      = ( member8474499337054950954_fm_fm @ ( product_Pair_fm_fm @ ( F2 @ X2 ) @ ( F2 @ Y3 ) ) @ R2 ) ) ).

% in_inv_image
thf(fact_536_snd__sndOp,axiom,
    ! [P2: list_fm > list_fm > $o,Q3: list_fm > list_fm > $o] :
      ( produc4588648349897876871ist_fm
      = ( comp_P1426762558556845121ist_fm @ produc4588648349897876871ist_fm @ ( bNF_sn5292590578203848342ist_fm @ P2 @ Q3 ) ) ) ).

% snd_sndOp
thf(fact_537_snd__sndOp,axiom,
    ! [P2: rule > rule > $o,Q3: rule > produc1996495991257130529ist_fm > $o] :
      ( produc4514744107061996309ist_fm
      = ( comp_P1889728861619134526ist_fm @ produc4514744107061996309ist_fm @ ( bNF_sn1604800272185294753ist_fm @ P2 @ Q3 ) ) ) ).

% snd_sndOp
thf(fact_538_snd__sndOp,axiom,
    ! [P2: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm > $o,Q3: produc1996495991257130529ist_fm > rule > $o] :
      ( produc7165828336582415457m_rule
      = ( comp_P422132551275928404m_rule @ produc7165828336582415457m_rule @ ( bNF_sn6617447237779055481m_rule @ P2 @ Q3 ) ) ) ).

% snd_sndOp
thf(fact_539_snd__sndOp,axiom,
    ! [P2: rule > rule > $o,Q3: rule > rule > $o] :
      ( produc6725306161561501521e_rule
      = ( comp_P5356484029182387740e_rule @ produc6725306161561501521e_rule @ ( bNF_sn5573932575237119429e_rule @ P2 @ Q3 ) ) ) ).

% snd_sndOp
thf(fact_540_snd__sndOp,axiom,
    ! [P2: fm > fm > $o,Q3: fm > fm > $o] :
      ( product_snd_fm_fm
      = ( comp_P3218056037334583377_fm_fm @ product_snd_fm_fm @ ( bNF_sndOp_fm_fm_fm @ P2 @ Q3 ) ) ) ).

% snd_sndOp
thf(fact_541_snd__sndOp,axiom,
    ! [P2: produc1996495991257130529ist_fm > rule > $o,Q3: rule > rule > $o] :
      ( produc7165828336582415457m_rule
      = ( comp_P2548450393677710116m_rule @ produc6725306161561501521e_rule @ ( bNF_sn6445580637211383353e_rule @ P2 @ Q3 ) ) ) ).

% snd_sndOp
thf(fact_542_snd__sndOp,axiom,
    ! [P2: rule > produc1996495991257130529ist_fm > $o,Q3: produc1996495991257130529ist_fm > rule > $o] :
      ( produc6725306161561501521e_rule
      = ( comp_P682981744566163948e_rule @ produc7165828336582415457m_rule @ ( bNF_sn4255884501705713901m_rule @ P2 @ Q3 ) ) ) ).

% snd_sndOp
thf(fact_543_snd__sndOp,axiom,
    ! [P2: fm > produc164195504107695125m_rule > $o,Q3: produc164195504107695125m_rule > fm > $o] :
      ( product_snd_fm_fm
      = ( comp_P3262442148557739852_fm_fm @ produc8694862632519904556ule_fm @ ( bNF_sn2716719905591852715ule_fm @ P2 @ Q3 ) ) ) ).

% snd_sndOp
thf(fact_544_snd__sndOp,axiom,
    ! [P2: produc164195504107695125m_rule > fm > $o,Q3: fm > fm > $o] :
      ( produc8694862632519904556ule_fm
      = ( comp_P6388136761583423126ule_fm @ product_snd_fm_fm @ ( bNF_sn1891237298393626667_fm_fm @ P2 @ Q3 ) ) ) ).

% snd_sndOp
thf(fact_545_snd__sndOp,axiom,
    ! [P2: rule > produc1996495991257130529ist_fm > $o,Q3: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm > $o] :
      ( produc4514744107061996309ist_fm
      = ( comp_P5552026984502743150ist_fm @ produc1639249265332130309ist_fm @ ( bNF_sn7014103632327346809ist_fm @ P2 @ Q3 ) ) ) ).

% snd_sndOp
thf(fact_546_fst__fstOp,axiom,
    ! [P2: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm > $o,Q3: produc1996495991257130529ist_fm > rule > $o] :
      ( produc6879501374131015971m_rule
      = ( comp_P1085356462900927218m_rule @ produc5894184038225292483ist_fm @ ( bNF_fs3491168633039030583m_rule @ P2 @ Q3 ) ) ) ).

% fst_fstOp
thf(fact_547_fst__fstOp,axiom,
    ! [P2: rule > produc1996495991257130529ist_fm > $o,Q3: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm > $o] :
      ( produc4228417144610596823ist_fm
      = ( comp_P7879787540529867852ist_fm @ produc4228417144610596823ist_fm @ ( bNF_fs3887825027587321911ist_fm @ P2 @ Q3 ) ) ) ).

% fst_fstOp
thf(fact_548_fst__fstOp,axiom,
    ! [P2: rule > rule > $o,Q3: rule > produc1996495991257130529ist_fm > $o] :
      ( produc4228417144610596823ist_fm
      = ( comp_P7015120915279526048ist_fm @ produc7264528503596239631e_rule @ ( bNF_fs6525992336556323171ist_fm @ P2 @ Q3 ) ) ) ).

% fst_fstOp
thf(fact_549_fst__fstOp,axiom,
    ! [P2: produc1996495991257130529ist_fm > rule > $o,Q3: rule > produc1996495991257130529ist_fm > $o] :
      ( produc5894184038225292483ist_fm
      = ( comp_P5570111842168596734ist_fm @ produc6879501374131015971m_rule @ ( bNF_fs840084403518611435ist_fm @ P2 @ Q3 ) ) ) ).

% fst_fstOp
thf(fact_550_fst__fstOp,axiom,
    ! [P2: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm > $o,Q3: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm > $o] :
      ( produc5894184038225292483ist_fm
      = ( comp_P6475754874365093450ist_fm @ produc5894184038225292483ist_fm @ ( bNF_fs9036858065860919215ist_fm @ P2 @ Q3 ) ) ) ).

% fst_fstOp
thf(fact_551_fst__fstOp,axiom,
    ! [P2: rule > produc1996495991257130529ist_fm > $o,Q3: produc1996495991257130529ist_fm > rule > $o] :
      ( produc7264528503596239631e_rule
      = ( comp_P4809662347396705648e_rule @ produc4228417144610596823ist_fm @ ( bNF_fs9177076566076742319m_rule @ P2 @ Q3 ) ) ) ).

% fst_fstOp
thf(fact_552_fst__fstOp,axiom,
    ! [P2: rule > rule > $o,Q3: rule > rule > $o] :
      ( produc7264528503596239631e_rule
      = ( comp_P5356484029182387740e_rule @ produc7264528503596239631e_rule @ ( bNF_fs6019504843140265091e_rule @ P2 @ Q3 ) ) ) ).

% fst_fstOp
thf(fact_553_fst__fstOp,axiom,
    ! [P2: fm > fm > $o,Q3: fm > fm > $o] :
      ( product_fst_fm_fm
      = ( comp_P3218056037334583377_fm_fm @ product_fst_fm_fm @ ( bNF_fstOp_fm_fm_fm @ P2 @ Q3 ) ) ) ).

% fst_fstOp
thf(fact_554_fst__fstOp,axiom,
    ! [P2: produc1996495991257130529ist_fm > rule > $o,Q3: rule > rule > $o] :
      ( produc6879501374131015971m_rule
      = ( comp_P7463969011078018366m_rule @ produc6879501374131015971m_rule @ ( bNF_fs2143400664727635963e_rule @ P2 @ Q3 ) ) ) ).

% fst_fstOp
thf(fact_555_fst__fstOp,axiom,
    ! [P2: list_fm > list_fm > $o,Q3: list_fm > list_fm > $o] :
      ( produc1501393135466168645ist_fm
      = ( comp_P1426762558556845121ist_fm @ produc1501393135466168645ist_fm @ ( bNF_fs6361025107161210196ist_fm @ P2 @ Q3 ) ) ) ).

% fst_fstOp
thf(fact_556_snth__smap2,axiom,
    ! [F2: produc164195504107695125m_rule > produc164195504107695125m_rule > produc164195504107695125m_rule,S1: stream727092118206550309m_rule,S2: stream727092118206550309m_rule,N2: nat] :
      ( ( snth_P1712579846118164081m_rule @ ( smap2_3865418324013074943m_rule @ F2 @ S1 @ S2 ) @ N2 )
      = ( F2 @ ( snth_P1712579846118164081m_rule @ S1 @ N2 ) @ ( snth_P1712579846118164081m_rule @ S2 @ N2 ) ) ) ).

% snth_smap2
thf(fact_557_snth__smap2,axiom,
    ! [F2: rule > fm > rule,S1: stream_rule,S2: stream_fm,N2: nat] :
      ( ( snth_rule @ ( smap2_rule_fm_rule @ F2 @ S1 @ S2 ) @ N2 )
      = ( F2 @ ( snth_rule @ S1 @ N2 ) @ ( snth_fm @ S2 @ N2 ) ) ) ).

% snth_smap2
thf(fact_558_snth__smap2,axiom,
    ! [F2: fm > rule > rule,S1: stream_fm,S2: stream_rule,N2: nat] :
      ( ( snth_rule @ ( smap2_fm_rule_rule @ F2 @ S1 @ S2 ) @ N2 )
      = ( F2 @ ( snth_fm @ S1 @ N2 ) @ ( snth_rule @ S2 @ N2 ) ) ) ).

% snth_smap2
thf(fact_559_snth__smap2,axiom,
    ! [F2: fm > fm > rule,S1: stream_fm,S2: stream_fm,N2: nat] :
      ( ( snth_rule @ ( smap2_fm_fm_rule @ F2 @ S1 @ S2 ) @ N2 )
      = ( F2 @ ( snth_fm @ S1 @ N2 ) @ ( snth_fm @ S2 @ N2 ) ) ) ).

% snth_smap2
thf(fact_560_snth__smap2,axiom,
    ! [F2: rule > rule > fm,S1: stream_rule,S2: stream_rule,N2: nat] :
      ( ( snth_fm @ ( smap2_rule_rule_fm @ F2 @ S1 @ S2 ) @ N2 )
      = ( F2 @ ( snth_rule @ S1 @ N2 ) @ ( snth_rule @ S2 @ N2 ) ) ) ).

% snth_smap2
thf(fact_561_snth__smap2,axiom,
    ! [F2: rule > fm > fm,S1: stream_rule,S2: stream_fm,N2: nat] :
      ( ( snth_fm @ ( smap2_rule_fm_fm @ F2 @ S1 @ S2 ) @ N2 )
      = ( F2 @ ( snth_rule @ S1 @ N2 ) @ ( snth_fm @ S2 @ N2 ) ) ) ).

% snth_smap2
thf(fact_562_snth__smap2,axiom,
    ! [F2: fm > rule > fm,S1: stream_fm,S2: stream_rule,N2: nat] :
      ( ( snth_fm @ ( smap2_fm_rule_fm @ F2 @ S1 @ S2 ) @ N2 )
      = ( F2 @ ( snth_fm @ S1 @ N2 ) @ ( snth_rule @ S2 @ N2 ) ) ) ).

% snth_smap2
thf(fact_563_snth__smap2,axiom,
    ! [F2: fm > fm > fm,S1: stream_fm,S2: stream_fm,N2: nat] :
      ( ( snth_fm @ ( smap2_fm_fm_fm @ F2 @ S1 @ S2 ) @ N2 )
      = ( F2 @ ( snth_fm @ S1 @ N2 ) @ ( snth_fm @ S2 @ N2 ) ) ) ).

% snth_smap2
thf(fact_564_snth__smap2,axiom,
    ! [F2: rule > rule > rule,S1: stream_rule,S2: stream_rule,N2: nat] :
      ( ( snth_rule @ ( smap2_rule_rule_rule @ F2 @ S1 @ S2 ) @ N2 )
      = ( F2 @ ( snth_rule @ S1 @ N2 ) @ ( snth_rule @ S2 @ N2 ) ) ) ).

% snth_smap2
thf(fact_565_snth__smap2,axiom,
    ! [F2: rule > rule > stream_fm,S1: stream_rule,S2: stream_rule,N2: nat] :
      ( ( snth_stream_fm @ ( smap2_2936232061632879402eam_fm @ F2 @ S1 @ S2 ) @ N2 )
      = ( F2 @ ( snth_rule @ S1 @ N2 ) @ ( snth_rule @ S2 @ N2 ) ) ) ).

% snth_smap2
thf(fact_566_snth_Osimps_I1_J,axiom,
    ! [S3: stream3016090547575141045m_rule] :
      ( ( snth_s8878883179916095873m_rule @ S3 @ zero_zero_nat )
      = ( shd_st3931407289205852537m_rule @ S3 ) ) ).

% snth.simps(1)
thf(fact_567_snth_Osimps_I1_J,axiom,
    ! [S3: stream_stream_fm] :
      ( ( snth_stream_fm @ S3 @ zero_zero_nat )
      = ( shd_stream_fm @ S3 ) ) ).

% snth.simps(1)
thf(fact_568_snth_Osimps_I1_J,axiom,
    ! [S3: stream6494289010434245521m_rule] :
      ( ( snth_P8950227884416379311m_rule @ S3 @ zero_zero_nat )
      = ( shd_Pr2340755956080993207m_rule @ S3 ) ) ).

% snth.simps(1)
thf(fact_569_snth_Osimps_I1_J,axiom,
    ! [S3: stream_rule] :
      ( ( snth_rule @ S3 @ zero_zero_nat )
      = ( shd_rule @ S3 ) ) ).

% snth.simps(1)
thf(fact_570_snth_Osimps_I1_J,axiom,
    ! [S3: stream8299795917829157543ist_fm] :
      ( ( snth_P2624749845746815685ist_fm @ S3 @ zero_zero_nat )
      = ( shd_Pr772355297128350925ist_fm @ S3 ) ) ).

% snth.simps(1)
thf(fact_571_snth_Osimps_I1_J,axiom,
    ! [S3: stream_fm] :
      ( ( snth_fm @ S3 @ zero_zero_nat )
      = ( shd_fm @ S3 ) ) ).

% snth.simps(1)
thf(fact_572_snth_Osimps_I1_J,axiom,
    ! [S3: stream5223287678556985525m_rule] :
      ( ( snth_l1529753283359469185m_rule @ S3 @ zero_zero_nat )
      = ( shd_li6864325268781923961m_rule @ S3 ) ) ).

% snth.simps(1)
thf(fact_573_snth_Osimps_I1_J,axiom,
    ! [S3: stream727092118206550309m_rule] :
      ( ( snth_P1712579846118164081m_rule @ S3 @ zero_zero_nat )
      = ( shd_Pr7235097944458474089m_rule @ S3 ) ) ).

% snth.simps(1)
thf(fact_574_sdrop__smap2,axiom,
    ! [N2: nat,F2: rule > rule > rule,S1: stream_rule,S2: stream_rule] :
      ( ( sdrop_rule @ N2 @ ( smap2_rule_rule_rule @ F2 @ S1 @ S2 ) )
      = ( smap2_rule_rule_rule @ F2 @ ( sdrop_rule @ N2 @ S1 ) @ ( sdrop_rule @ N2 @ S2 ) ) ) ).

% sdrop_smap2
thf(fact_575_sdrop__smap2,axiom,
    ! [N2: nat,F2: rule > rule > produc1996495991257130529ist_fm,S1: stream_rule,S2: stream_rule] :
      ( ( sdrop_4442373711808556042ist_fm @ N2 @ ( smap2_1292763952458720599ist_fm @ F2 @ S1 @ S2 ) )
      = ( smap2_1292763952458720599ist_fm @ F2 @ ( sdrop_rule @ N2 @ S1 ) @ ( sdrop_rule @ N2 @ S2 ) ) ) ).

% sdrop_smap2
thf(fact_576_sdrop__smap2,axiom,
    ! [N2: nat,F2: produc1996495991257130529ist_fm > rule > rule,S1: stream8299795917829157543ist_fm,S2: stream_rule] :
      ( ( sdrop_rule @ N2 @ ( smap2_6133544317484809199e_rule @ F2 @ S1 @ S2 ) )
      = ( smap2_6133544317484809199e_rule @ F2 @ ( sdrop_4442373711808556042ist_fm @ N2 @ S1 ) @ ( sdrop_rule @ N2 @ S2 ) ) ) ).

% sdrop_smap2
thf(fact_577_sdrop__smap2,axiom,
    ! [N2: nat,F2: rule > produc1996495991257130529ist_fm > rule,S1: stream_rule,S2: stream8299795917829157543ist_fm] :
      ( ( sdrop_rule @ N2 @ ( smap2_3943848181979139747m_rule @ F2 @ S1 @ S2 ) )
      = ( smap2_3943848181979139747m_rule @ F2 @ ( sdrop_rule @ N2 @ S1 ) @ ( sdrop_4442373711808556042ist_fm @ N2 @ S2 ) ) ) ).

% sdrop_smap2
thf(fact_578_sdrop__smap2,axiom,
    ! [N2: nat,F2: rule > rule > produc164195504107695125m_rule,S1: stream_rule,S2: stream_rule] :
      ( ( sdrop_7224736112439592940m_rule @ N2 @ ( smap2_4438185555641665375m_rule @ F2 @ S1 @ S2 ) )
      = ( smap2_4438185555641665375m_rule @ F2 @ ( sdrop_rule @ N2 @ S1 ) @ ( sdrop_rule @ N2 @ S2 ) ) ) ).

% sdrop_smap2
thf(fact_579_sdrop__smap2,axiom,
    ! [N2: nat,F2: produc164195504107695125m_rule > rule > rule,S1: stream727092118206550309m_rule,S2: stream_rule] :
      ( ( sdrop_rule @ N2 @ ( smap2_190827247425732959e_rule @ F2 @ S1 @ S2 ) )
      = ( smap2_190827247425732959e_rule @ F2 @ ( sdrop_7224736112439592940m_rule @ N2 @ S1 ) @ ( sdrop_rule @ N2 @ S2 ) ) ) ).

% sdrop_smap2
thf(fact_580_sdrop__smap2,axiom,
    ! [N2: nat,F2: rule > produc164195504107695125m_rule > rule,S1: stream_rule,S2: stream727092118206550309m_rule] :
      ( ( sdrop_rule @ N2 @ ( smap2_2737362159257330783e_rule @ F2 @ S1 @ S2 ) )
      = ( smap2_2737362159257330783e_rule @ F2 @ ( sdrop_rule @ N2 @ S1 ) @ ( sdrop_7224736112439592940m_rule @ N2 @ S2 ) ) ) ).

% sdrop_smap2
thf(fact_581_sdrop__smap2,axiom,
    ! [N2: nat,F2: produc1996495991257130529ist_fm > rule > produc1996495991257130529ist_fm,S1: stream8299795917829157543ist_fm,S2: stream_rule] :
      ( ( sdrop_4442373711808556042ist_fm @ N2 @ ( smap2_4725336451129573111ist_fm @ F2 @ S1 @ S2 ) )
      = ( smap2_4725336451129573111ist_fm @ F2 @ ( sdrop_4442373711808556042ist_fm @ N2 @ S1 ) @ ( sdrop_rule @ N2 @ S2 ) ) ) ).

% sdrop_smap2
thf(fact_582_sdrop__smap2,axiom,
    ! [N2: nat,F2: rule > produc1996495991257130529ist_fm > produc1996495991257130529ist_fm,S1: stream_rule,S2: stream8299795917829157543ist_fm] :
      ( ( sdrop_4442373711808556042ist_fm @ N2 @ ( smap2_7773077075198283587ist_fm @ F2 @ S1 @ S2 ) )
      = ( smap2_7773077075198283587ist_fm @ F2 @ ( sdrop_rule @ N2 @ S1 ) @ ( sdrop_4442373711808556042ist_fm @ N2 @ S2 ) ) ) ).

% sdrop_smap2
thf(fact_583_sdrop__smap2,axiom,
    ! [N2: nat,F2: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm > rule,S1: stream8299795917829157543ist_fm,S2: stream8299795917829157543ist_fm] :
      ( ( sdrop_rule @ N2 @ ( smap2_7376420680649992259m_rule @ F2 @ S1 @ S2 ) )
      = ( smap2_7376420680649992259m_rule @ F2 @ ( sdrop_4442373711808556042ist_fm @ N2 @ S1 ) @ ( sdrop_4442373711808556042ist_fm @ N2 @ S2 ) ) ) ).

% sdrop_smap2
thf(fact_584_sdrop__snth,axiom,
    ! [N2: nat,S3: stream_rule,M: nat] :
      ( ( snth_rule @ ( sdrop_rule @ N2 @ S3 ) @ M )
      = ( snth_rule @ S3 @ ( plus_plus_nat @ N2 @ M ) ) ) ).

% sdrop_snth
thf(fact_585_sdrop__snth,axiom,
    ! [N2: nat,S3: stream8299795917829157543ist_fm,M: nat] :
      ( ( snth_P2624749845746815685ist_fm @ ( sdrop_4442373711808556042ist_fm @ N2 @ S3 ) @ M )
      = ( snth_P2624749845746815685ist_fm @ S3 @ ( plus_plus_nat @ N2 @ M ) ) ) ).

% sdrop_snth
thf(fact_586_sdrop__snth,axiom,
    ! [N2: nat,S3: stream727092118206550309m_rule,M: nat] :
      ( ( snth_P1712579846118164081m_rule @ ( sdrop_7224736112439592940m_rule @ N2 @ S3 ) @ M )
      = ( snth_P1712579846118164081m_rule @ S3 @ ( plus_plus_nat @ N2 @ M ) ) ) ).

% sdrop_snth
thf(fact_587_sdrop__snth,axiom,
    ! [N2: nat,S3: stream6494289010434245521m_rule,M: nat] :
      ( ( snth_P8950227884416379311m_rule @ ( sdrop_7373388980841566196m_rule @ N2 @ S3 ) @ M )
      = ( snth_P8950227884416379311m_rule @ S3 @ ( plus_plus_nat @ N2 @ M ) ) ) ).

% sdrop_snth
thf(fact_588_sdrop__snth,axiom,
    ! [N2: nat,S3: stream_fm,M: nat] :
      ( ( snth_fm @ ( sdrop_fm @ N2 @ S3 ) @ M )
      = ( snth_fm @ S3 @ ( plus_plus_nat @ N2 @ M ) ) ) ).

% sdrop_snth
thf(fact_589_sdrop__snth,axiom,
    ! [N2: nat,S3: stream_stream_fm,M: nat] :
      ( ( snth_stream_fm @ ( sdrop_stream_fm @ N2 @ S3 ) @ M )
      = ( snth_stream_fm @ S3 @ ( plus_plus_nat @ N2 @ M ) ) ) ).

% sdrop_snth
thf(fact_590_sdrop__snth,axiom,
    ! [N2: nat,S3: stream_list_fm,M: nat] :
      ( ( snth_list_fm @ ( sdrop_list_fm @ N2 @ S3 ) @ M )
      = ( snth_list_fm @ S3 @ ( plus_plus_nat @ N2 @ M ) ) ) ).

% sdrop_snth
thf(fact_591_sdrop__snth,axiom,
    ! [N2: nat,S3: stream3588992565182678257e_rule,M: nat] :
      ( ( snth_P8487267234472522255e_rule @ ( sdrop_2199339672054592340e_rule @ N2 @ S3 ) @ M )
      = ( snth_P8487267234472522255e_rule @ S3 @ ( plus_plus_nat @ N2 @ M ) ) ) ).

% sdrop_snth
thf(fact_592_sdrop__snth,axiom,
    ! [N2: nat,S3: stream1960312990768957601ist_fm,M: nat] :
      ( ( snth_P6179250367719980013ist_fm @ ( sdrop_2468034597186633064ist_fm @ N2 @ S3 ) @ M )
      = ( snth_P6179250367719980013ist_fm @ S3 @ ( plus_plus_nat @ N2 @ M ) ) ) ).

% sdrop_snth
thf(fact_593_sdrop__snth,axiom,
    ! [N2: nat,S3: stream3016090547575141045m_rule,M: nat] :
      ( ( snth_s8878883179916095873m_rule @ ( sdrop_8061686475400732284m_rule @ N2 @ S3 ) @ M )
      = ( snth_s8878883179916095873m_rule @ S3 @ ( plus_plus_nat @ N2 @ M ) ) ) ).

% sdrop_snth
thf(fact_594_Collect__case__prod__Grp__in,axiom,
    ! [Z2: produc1805012608956077659t_rule,A4: set_li5339030221936397403m_rule,F2: list_P8332459875677254181m_rule > list_rule] :
      ( ( member6491770712979037828t_rule @ Z2 @ ( collec5858587166179649734t_rule @ ( produc2090550080120670960rule_o @ ( bNF_Gr7726341041440550858t_rule @ A4 @ F2 ) ) ) )
     => ( member1784032718715961276m_rule @ ( produc47317219786245247t_rule @ Z2 ) @ A4 ) ) ).

% Collect_case_prod_Grp_in
thf(fact_595_Collect__case__prod__Grp__in,axiom,
    ! [Z2: produc6986050496780531771ist_fm,A4: set_li346003458413430231ist_fm,F2: list_P342308711384885665ist_fm > list_P5616295576739893671ist_fm] :
      ( ( member3096056441066211794ist_fm @ Z2 @ ( collec6490912061774935824ist_fm @ ( produc294104671633537790t_fm_o @ ( bNF_Gr7947322218390040310ist_fm @ A4 @ F2 ) ) ) )
     => ( member3017253591278368568ist_fm @ ( produc120016142364770561ist_fm @ Z2 ) @ A4 ) ) ).

% Collect_case_prod_Grp_in
thf(fact_596_Collect__case__prod__Grp__in,axiom,
    ! [Z2: produc2148452214272472023t_rule,A4: set_li346003458413430231ist_fm,F2: list_P342308711384885665ist_fm > list_rule] :
      ( ( member6835210318295432192t_rule @ Z2 @ ( collec6202026771496044098t_rule @ ( produc5652714768248047220rule_o @ ( bNF_Gr8069780646756945222t_rule @ A4 @ F2 ) ) ) )
     => ( member3017253591278368568ist_fm @ ( produc390756825102639611t_rule @ Z2 ) @ A4 ) ) ).

% Collect_case_prod_Grp_in
thf(fact_597_Collect__case__prod__Grp__in,axiom,
    ! [Z2: produc4719849802655908294ist_fm,A4: set_li5537492308219346823ist_fm,F2: list_P5616295576739893671ist_fm > list_list_fm] :
      ( ( member4192383230844517341ist_fm @ Z2 @ ( collec7853023762562200091ist_fm @ ( produc2973715795501690859t_fm_o @ ( bNF_Gr5899615463480776905ist_fm @ A4 @ F2 ) ) ) )
     => ( member8448403643636553168ist_fm @ ( produc6245774423584770388ist_fm @ Z2 ) @ A4 ) ) ).

% Collect_case_prod_Grp_in
thf(fact_598_Collect__case__prod__Grp__in,axiom,
    ! [Z2: produc1754732471204517991ist_fm,A4: set_Pr7058068377845519745ist_fm,F2: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm] :
      ( ( member60238275747228048ist_fm @ Z2 @ ( collec888080998205339090ist_fm @ ( produc2796214937669505452t_fm_o @ ( bNF_Gr854411304184578830ist_fm @ A4 @ F2 ) ) ) )
     => ( member8102475879199740618ist_fm @ ( produc5894184038225292483ist_fm @ Z2 ) @ A4 ) ) ).

% Collect_case_prod_Grp_in
thf(fact_599_Collect__case__prod__Grp__in,axiom,
    ! [Z2: produc780299800278082923e_rule,A4: set_rule,F2: rule > rule] :
      ( ( member3753523862196722196e_rule @ Z2 @ ( collec5224624948449408342e_rule @ ( produc8052491652537269344rule_o @ ( bNF_Grp_rule_rule @ A4 @ F2 ) ) ) )
     => ( member_rule @ ( produc7264528503596239631e_rule @ Z2 ) @ A4 ) ) ).

% Collect_case_prod_Grp_in
thf(fact_600_Collect__case__prod__Grp__in,axiom,
    ! [Z2: product_prod_fm_fm,A4: set_fm,F2: fm > fm] :
      ( ( member8474499337054950954_fm_fm @ Z2 @ ( collec4425076577877259116_fm_fm @ ( produc2811014937302177098m_fm_o @ ( bNF_Grp_fm_fm @ A4 @ F2 ) ) ) )
     => ( member_fm @ ( product_fst_fm_fm @ Z2 ) @ A4 ) ) ).

% Collect_case_prod_Grp_in
thf(fact_601_Collect__case__prod__Grp__in,axiom,
    ! [Z2: produc4630866025709511057ist_fm,A4: set_rule,F2: rule > produc1996495991257130529ist_fm] :
      ( ( member8686995742288324264ist_fm @ Z2 @ ( collec6536177220190208742ist_fm @ ( produc1628407760719572648t_fm_o @ ( bNF_Gr7657091880239761484ist_fm @ A4 @ F2 ) ) ) )
     => ( member_rule @ ( produc4228417144610596823ist_fm @ Z2 ) @ A4 ) ) ).

% Collect_case_prod_Grp_in
thf(fact_602_Collect__case__prod__Grp__in,axiom,
    ! [Z2: produc164195504107695125m_rule,A4: set_Pr7058068377845519745ist_fm,F2: produc1996495991257130529ist_fm > rule] :
      ( ( member4220325220686508332m_rule @ Z2 @ ( collec2069506698588392810m_rule @ ( produc7369492558480804444rule_o @ ( bNF_Gr1084804072905404824m_rule @ A4 @ F2 ) ) ) )
     => ( member8102475879199740618ist_fm @ ( produc6879501374131015971m_rule @ Z2 ) @ A4 ) ) ).

% Collect_case_prod_Grp_in
thf(fact_603_Collect__case__prod__Grp__in,axiom,
    ! [Z2: produc1996495991257130529ist_fm,A4: set_list_fm,F2: list_fm > list_fm] :
      ( ( member8102475879199740618ist_fm @ Z2 @ ( collec7231575776666803724ist_fm @ ( produc2011568048573023402t_fm_o @ ( bNF_Gr5437506445679690256ist_fm @ A4 @ F2 ) ) ) )
     => ( member_list_fm @ ( produc1501393135466168645ist_fm @ Z2 ) @ A4 ) ) ).

% Collect_case_prod_Grp_in
thf(fact_604_sdrop__add,axiom,
    ! [N2: nat,M: nat,S3: stream3856641574645492619m_rule] :
      ( ( sdrop_4762382306744956754m_rule @ N2 @ ( sdrop_4762382306744956754m_rule @ M @ S3 ) )
      = ( sdrop_4762382306744956754m_rule @ ( plus_plus_nat @ M @ N2 ) @ S3 ) ) ).

% sdrop_add
thf(fact_605_sdrop__add,axiom,
    ! [N2: nat,M: nat,S3: stream1878305877988316353m_rule] :
      ( ( sdrop_1764812713384251940m_rule @ N2 @ ( sdrop_1764812713384251940m_rule @ M @ S3 ) )
      = ( sdrop_1764812713384251940m_rule @ ( plus_plus_nat @ M @ N2 ) @ S3 ) ) ).

% sdrop_add
thf(fact_606_sdrop__add,axiom,
    ! [N2: nat,M: nat,S3: stream1960312990768957601ist_fm] :
      ( ( sdrop_2468034597186633064ist_fm @ N2 @ ( sdrop_2468034597186633064ist_fm @ M @ S3 ) )
      = ( sdrop_2468034597186633064ist_fm @ ( plus_plus_nat @ M @ N2 ) @ S3 ) ) ).

% sdrop_add
thf(fact_607_sdrop__add,axiom,
    ! [N2: nat,M: nat,S3: stream3588992565182678257e_rule] :
      ( ( sdrop_2199339672054592340e_rule @ N2 @ ( sdrop_2199339672054592340e_rule @ M @ S3 ) )
      = ( sdrop_2199339672054592340e_rule @ ( plus_plus_nat @ M @ N2 ) @ S3 ) ) ).

% sdrop_add
thf(fact_608_sdrop__add,axiom,
    ! [N2: nat,M: nat,S3: stream_list_fm] :
      ( ( sdrop_list_fm @ N2 @ ( sdrop_list_fm @ M @ S3 ) )
      = ( sdrop_list_fm @ ( plus_plus_nat @ M @ N2 ) @ S3 ) ) ).

% sdrop_add
thf(fact_609_sdrop__add,axiom,
    ! [N2: nat,M: nat,S3: stream727092118206550309m_rule] :
      ( ( sdrop_7224736112439592940m_rule @ N2 @ ( sdrop_7224736112439592940m_rule @ M @ S3 ) )
      = ( sdrop_7224736112439592940m_rule @ ( plus_plus_nat @ M @ N2 ) @ S3 ) ) ).

% sdrop_add
thf(fact_610_sdrop__add,axiom,
    ! [N2: nat,M: nat,S3: stream8299795917829157543ist_fm] :
      ( ( sdrop_4442373711808556042ist_fm @ N2 @ ( sdrop_4442373711808556042ist_fm @ M @ S3 ) )
      = ( sdrop_4442373711808556042ist_fm @ ( plus_plus_nat @ M @ N2 ) @ S3 ) ) ).

% sdrop_add
thf(fact_611_sdrop__add,axiom,
    ! [N2: nat,M: nat,S3: stream_rule] :
      ( ( sdrop_rule @ N2 @ ( sdrop_rule @ M @ S3 ) )
      = ( sdrop_rule @ ( plus_plus_nat @ M @ N2 ) @ S3 ) ) ).

% sdrop_add
thf(fact_612_sdrop__add,axiom,
    ! [N2: nat,M: nat,S3: stream6494289010434245521m_rule] :
      ( ( sdrop_7373388980841566196m_rule @ N2 @ ( sdrop_7373388980841566196m_rule @ M @ S3 ) )
      = ( sdrop_7373388980841566196m_rule @ ( plus_plus_nat @ M @ N2 ) @ S3 ) ) ).

% sdrop_add
thf(fact_613_fun_Omap__comp,axiom,
    ! [G: $o > $o,F2: list_fm > $o,V: list_fm > list_fm] :
      ( ( comp_o_o_list_fm @ G @ ( comp_l3888558907433981941ist_fm @ F2 @ V ) )
      = ( comp_l3888558907433981941ist_fm @ ( comp_o_o_list_fm @ G @ F2 ) @ V ) ) ).

% fun.map_comp
thf(fact_614_fun_Omap__comp,axiom,
    ! [G: list_fm > $o,F2: list_fm > list_fm,V: list_fm > list_fm] :
      ( ( comp_l3888558907433981941ist_fm @ G @ ( comp_l3338043728524549509ist_fm @ F2 @ V ) )
      = ( comp_l3888558907433981941ist_fm @ ( comp_l3888558907433981941ist_fm @ G @ F2 ) @ V ) ) ).

% fun.map_comp
thf(fact_615_fun_Omap__comp,axiom,
    ! [G: set_fm > set_fm > set_fm,F2: fm > set_fm,V: fm > fm] :
      ( ( comp_s2169399591454099046_fm_fm @ G @ ( comp_fm_set_fm_fm @ F2 @ V ) )
      = ( comp_f9192711903063529008_fm_fm @ ( comp_s2169399591454099046_fm_fm @ G @ F2 ) @ V ) ) ).

% fun.map_comp
thf(fact_616_fun_Omap__comp,axiom,
    ! [G: set_fm > set_fm > set_fm,F2: set_fm > set_fm,V: fm > set_fm] :
      ( ( comp_s2169399591454099046_fm_fm @ G @ ( comp_s7832666443790905665_fm_fm @ F2 @ V ) )
      = ( comp_s2169399591454099046_fm_fm @ ( comp_s6810799292195144860set_fm @ G @ F2 ) @ V ) ) ).

% fun.map_comp
thf(fact_617_fun_Omap__comp,axiom,
    ! [G: list_fm > set_list_fm,F2: list_fm > list_fm,V: produc1996495991257130529ist_fm > list_fm] :
      ( ( comp_l987677038997058310ist_fm @ G @ ( comp_l1868218281415653180ist_fm @ F2 @ V ) )
      = ( comp_l987677038997058310ist_fm @ ( comp_l1827647928735265851ist_fm @ G @ F2 ) @ V ) ) ).

% fun.map_comp
thf(fact_618_fun_Omap__comp,axiom,
    ! [G: set_list_fm > set_list_fm,F2: list_fm > set_list_fm,V: produc1996495991257130529ist_fm > list_fm] :
      ( ( comp_s6733258352739592400ist_fm @ G @ ( comp_l987677038997058310ist_fm @ F2 @ V ) )
      = ( comp_l987677038997058310ist_fm @ ( comp_s581824039803048177ist_fm @ G @ F2 ) @ V ) ) ).

% fun.map_comp
thf(fact_619_fun_Omap__comp,axiom,
    ! [G: ( set_fm > set_fm ) > set_fm > set_fm,F2: set_fm > set_fm > set_fm,V: fm > set_fm] :
      ( ( comp_s8874546009865274635_fm_fm @ G @ ( comp_s2169399591454099046_fm_fm @ F2 @ V ) )
      = ( comp_s2169399591454099046_fm_fm @ ( comp_s7748757910166073153set_fm @ G @ F2 ) @ V ) ) ).

% fun.map_comp
thf(fact_620_fun_Omap__comp,axiom,
    ! [G: set_fm > set_fm,F2: set_fm > set_fm,V: produc164195504107695125m_rule > set_fm] :
      ( ( comp_s4688689533955628550m_rule @ G @ ( comp_s4688689533955628550m_rule @ F2 @ V ) )
      = ( comp_s4688689533955628550m_rule @ ( comp_s51484990237361015set_fm @ G @ F2 ) @ V ) ) ).

% fun.map_comp
thf(fact_621_fun_Omap__comp,axiom,
    ! [G: list_fm > set_list_fm,F2: produc1996495991257130529ist_fm > list_fm,V: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm] :
      ( ( comp_l987677038997058310ist_fm @ G @ ( comp_P1426762558556845121ist_fm @ F2 @ V ) )
      = ( comp_P6578477294586436619ist_fm @ ( comp_l987677038997058310ist_fm @ G @ F2 ) @ V ) ) ).

% fun.map_comp
thf(fact_622_fun_Omap__comp,axiom,
    ! [G: set_fm > set_fm > set_fm,F2: fm > set_fm,V: produc164195504107695125m_rule > fm] :
      ( ( comp_s7100926763290858283m_rule @ G @ ( comp_f1552047388348977616m_rule @ F2 @ V ) )
      = ( comp_f6656709422951064309m_rule @ ( comp_s2169399591454099046_fm_fm @ G @ F2 ) @ V ) ) ).

% fun.map_comp
thf(fact_623_GrpE,axiom,
    ! [A4: set_li5339030221936397403m_rule,F2: list_P8332459875677254181m_rule > list_rule,X2: list_P8332459875677254181m_rule,Y3: list_rule] :
      ( ( bNF_Gr7726341041440550858t_rule @ A4 @ F2 @ X2 @ Y3 )
     => ~ ( ( ( F2 @ X2 )
            = Y3 )
         => ~ ( member1784032718715961276m_rule @ X2 @ A4 ) ) ) ).

% GrpE
thf(fact_624_GrpE,axiom,
    ! [A4: set_li346003458413430231ist_fm,F2: list_P342308711384885665ist_fm > list_P5616295576739893671ist_fm,X2: list_P342308711384885665ist_fm,Y3: list_P5616295576739893671ist_fm] :
      ( ( bNF_Gr7947322218390040310ist_fm @ A4 @ F2 @ X2 @ Y3 )
     => ~ ( ( ( F2 @ X2 )
            = Y3 )
         => ~ ( member3017253591278368568ist_fm @ X2 @ A4 ) ) ) ).

% GrpE
thf(fact_625_GrpE,axiom,
    ! [A4: set_li346003458413430231ist_fm,F2: list_P342308711384885665ist_fm > list_rule,X2: list_P342308711384885665ist_fm,Y3: list_rule] :
      ( ( bNF_Gr8069780646756945222t_rule @ A4 @ F2 @ X2 @ Y3 )
     => ~ ( ( ( F2 @ X2 )
            = Y3 )
         => ~ ( member3017253591278368568ist_fm @ X2 @ A4 ) ) ) ).

% GrpE
thf(fact_626_GrpE,axiom,
    ! [A4: set_li5537492308219346823ist_fm,F2: list_P5616295576739893671ist_fm > list_list_fm,X2: list_P5616295576739893671ist_fm,Y3: list_list_fm] :
      ( ( bNF_Gr5899615463480776905ist_fm @ A4 @ F2 @ X2 @ Y3 )
     => ~ ( ( ( F2 @ X2 )
            = Y3 )
         => ~ ( member8448403643636553168ist_fm @ X2 @ A4 ) ) ) ).

% GrpE
thf(fact_627_GrpE,axiom,
    ! [A4: set_list_fm,F2: list_fm > list_fm,X2: list_fm,Y3: list_fm] :
      ( ( bNF_Gr5437506445679690256ist_fm @ A4 @ F2 @ X2 @ Y3 )
     => ~ ( ( ( F2 @ X2 )
            = Y3 )
         => ~ ( member_list_fm @ X2 @ A4 ) ) ) ).

% GrpE
thf(fact_628_GrpI,axiom,
    ! [F2: list_P8332459875677254181m_rule > list_rule,X2: list_P8332459875677254181m_rule,Y3: list_rule,A4: set_li5339030221936397403m_rule] :
      ( ( ( F2 @ X2 )
        = Y3 )
     => ( ( member1784032718715961276m_rule @ X2 @ A4 )
       => ( bNF_Gr7726341041440550858t_rule @ A4 @ F2 @ X2 @ Y3 ) ) ) ).

% GrpI
thf(fact_629_GrpI,axiom,
    ! [F2: list_P342308711384885665ist_fm > list_P5616295576739893671ist_fm,X2: list_P342308711384885665ist_fm,Y3: list_P5616295576739893671ist_fm,A4: set_li346003458413430231ist_fm] :
      ( ( ( F2 @ X2 )
        = Y3 )
     => ( ( member3017253591278368568ist_fm @ X2 @ A4 )
       => ( bNF_Gr7947322218390040310ist_fm @ A4 @ F2 @ X2 @ Y3 ) ) ) ).

% GrpI
thf(fact_630_GrpI,axiom,
    ! [F2: list_P342308711384885665ist_fm > list_rule,X2: list_P342308711384885665ist_fm,Y3: list_rule,A4: set_li346003458413430231ist_fm] :
      ( ( ( F2 @ X2 )
        = Y3 )
     => ( ( member3017253591278368568ist_fm @ X2 @ A4 )
       => ( bNF_Gr8069780646756945222t_rule @ A4 @ F2 @ X2 @ Y3 ) ) ) ).

% GrpI
thf(fact_631_GrpI,axiom,
    ! [F2: list_P5616295576739893671ist_fm > list_list_fm,X2: list_P5616295576739893671ist_fm,Y3: list_list_fm,A4: set_li5537492308219346823ist_fm] :
      ( ( ( F2 @ X2 )
        = Y3 )
     => ( ( member8448403643636553168ist_fm @ X2 @ A4 )
       => ( bNF_Gr5899615463480776905ist_fm @ A4 @ F2 @ X2 @ Y3 ) ) ) ).

% GrpI
thf(fact_632_GrpI,axiom,
    ! [F2: list_fm > list_fm,X2: list_fm,Y3: list_fm,A4: set_list_fm] :
      ( ( ( F2 @ X2 )
        = Y3 )
     => ( ( member_list_fm @ X2 @ A4 )
       => ( bNF_Gr5437506445679690256ist_fm @ A4 @ F2 @ X2 @ Y3 ) ) ) ).

% GrpI
thf(fact_633_comp__cong,axiom,
    ! [F2: list_fm > $o,G: list_fm > list_fm,X2: list_fm,F3: list_fm > $o,G2: list_fm > list_fm,X4: list_fm] :
      ( ( ( F2 @ ( G @ X2 ) )
        = ( F3 @ ( G2 @ X4 ) ) )
     => ( ( comp_l3888558907433981941ist_fm @ F2 @ G @ X2 )
        = ( comp_l3888558907433981941ist_fm @ F3 @ G2 @ X4 ) ) ) ).

% comp_cong
thf(fact_634_comp__cong,axiom,
    ! [F2: list_fm > set_list_fm,G: produc1996495991257130529ist_fm > list_fm,X2: produc1996495991257130529ist_fm,F3: list_fm > set_list_fm,G2: produc1996495991257130529ist_fm > list_fm,X4: produc1996495991257130529ist_fm] :
      ( ( ( F2 @ ( G @ X2 ) )
        = ( F3 @ ( G2 @ X4 ) ) )
     => ( ( comp_l987677038997058310ist_fm @ F2 @ G @ X2 )
        = ( comp_l987677038997058310ist_fm @ F3 @ G2 @ X4 ) ) ) ).

% comp_cong
thf(fact_635_comp__cong,axiom,
    ! [F2: set_fm > set_fm,G: produc164195504107695125m_rule > set_fm,X2: produc164195504107695125m_rule,F3: set_fm > set_fm,G2: produc164195504107695125m_rule > set_fm,X4: produc164195504107695125m_rule] :
      ( ( ( F2 @ ( G @ X2 ) )
        = ( F3 @ ( G2 @ X4 ) ) )
     => ( ( comp_s4688689533955628550m_rule @ F2 @ G @ X2 )
        = ( comp_s4688689533955628550m_rule @ F3 @ G2 @ X4 ) ) ) ).

% comp_cong
thf(fact_636_comp__cong,axiom,
    ! [F2: set_fm > set_fm > set_fm,G: produc164195504107695125m_rule > set_fm,X2: produc164195504107695125m_rule,F3: set_fm > set_fm > set_fm,G2: produc164195504107695125m_rule > set_fm,X4: produc164195504107695125m_rule] :
      ( ( ( F2 @ ( G @ X2 ) )
        = ( F3 @ ( G2 @ X4 ) ) )
     => ( ( comp_s7100926763290858283m_rule @ F2 @ G @ X2 )
        = ( comp_s7100926763290858283m_rule @ F3 @ G2 @ X4 ) ) ) ).

% comp_cong
thf(fact_637_comp__cong,axiom,
    ! [F2: set_fm > set_fm > set_fm,G: produc164195504107695125m_rule > set_fm,X2: produc164195504107695125m_rule,F3: set_fm > set_fm > set_fm,G2: fm > set_fm,X4: fm] :
      ( ( ( F2 @ ( G @ X2 ) )
        = ( F3 @ ( G2 @ X4 ) ) )
     => ( ( comp_s7100926763290858283m_rule @ F2 @ G @ X2 )
        = ( comp_s2169399591454099046_fm_fm @ F3 @ G2 @ X4 ) ) ) ).

% comp_cong
thf(fact_638_comp__cong,axiom,
    ! [F2: set_fm > set_fm > set_fm,G: fm > set_fm,X2: fm,F3: set_fm > set_fm > set_fm,G2: produc164195504107695125m_rule > set_fm,X4: produc164195504107695125m_rule] :
      ( ( ( F2 @ ( G @ X2 ) )
        = ( F3 @ ( G2 @ X4 ) ) )
     => ( ( comp_s2169399591454099046_fm_fm @ F2 @ G @ X2 )
        = ( comp_s7100926763290858283m_rule @ F3 @ G2 @ X4 ) ) ) ).

% comp_cong
thf(fact_639_comp__cong,axiom,
    ! [F2: set_fm > set_fm > set_fm,G: fm > set_fm,X2: fm,F3: set_fm > set_fm > set_fm,G2: fm > set_fm,X4: fm] :
      ( ( ( F2 @ ( G @ X2 ) )
        = ( F3 @ ( G2 @ X4 ) ) )
     => ( ( comp_s2169399591454099046_fm_fm @ F2 @ G @ X2 )
        = ( comp_s2169399591454099046_fm_fm @ F3 @ G2 @ X4 ) ) ) ).

% comp_cong
thf(fact_640_comp__apply__eq,axiom,
    ! [F2: list_fm > $o,G: list_fm > list_fm,X2: list_fm,H: list_fm > $o,K: list_fm > list_fm] :
      ( ( ( F2 @ ( G @ X2 ) )
        = ( H @ ( K @ X2 ) ) )
     => ( ( comp_l3888558907433981941ist_fm @ F2 @ G @ X2 )
        = ( comp_l3888558907433981941ist_fm @ H @ K @ X2 ) ) ) ).

% comp_apply_eq
thf(fact_641_comp__apply__eq,axiom,
    ! [F2: list_fm > set_list_fm,G: produc1996495991257130529ist_fm > list_fm,X2: produc1996495991257130529ist_fm,H: list_fm > set_list_fm,K: produc1996495991257130529ist_fm > list_fm] :
      ( ( ( F2 @ ( G @ X2 ) )
        = ( H @ ( K @ X2 ) ) )
     => ( ( comp_l987677038997058310ist_fm @ F2 @ G @ X2 )
        = ( comp_l987677038997058310ist_fm @ H @ K @ X2 ) ) ) ).

% comp_apply_eq
thf(fact_642_comp__apply__eq,axiom,
    ! [F2: set_fm > set_fm,G: produc164195504107695125m_rule > set_fm,X2: produc164195504107695125m_rule,H: set_fm > set_fm,K: produc164195504107695125m_rule > set_fm] :
      ( ( ( F2 @ ( G @ X2 ) )
        = ( H @ ( K @ X2 ) ) )
     => ( ( comp_s4688689533955628550m_rule @ F2 @ G @ X2 )
        = ( comp_s4688689533955628550m_rule @ H @ K @ X2 ) ) ) ).

% comp_apply_eq
thf(fact_643_comp__apply__eq,axiom,
    ! [F2: set_fm > set_fm > set_fm,G: produc164195504107695125m_rule > set_fm,X2: produc164195504107695125m_rule,H: set_fm > set_fm > set_fm,K: produc164195504107695125m_rule > set_fm] :
      ( ( ( F2 @ ( G @ X2 ) )
        = ( H @ ( K @ X2 ) ) )
     => ( ( comp_s7100926763290858283m_rule @ F2 @ G @ X2 )
        = ( comp_s7100926763290858283m_rule @ H @ K @ X2 ) ) ) ).

% comp_apply_eq
thf(fact_644_comp__apply__eq,axiom,
    ! [F2: set_fm > set_fm > set_fm,G: fm > set_fm,X2: fm,H: set_fm > set_fm > set_fm,K: fm > set_fm] :
      ( ( ( F2 @ ( G @ X2 ) )
        = ( H @ ( K @ X2 ) ) )
     => ( ( comp_s2169399591454099046_fm_fm @ F2 @ G @ X2 )
        = ( comp_s2169399591454099046_fm_fm @ H @ K @ X2 ) ) ) ).

% comp_apply_eq
thf(fact_645_sdrop_Osimps_I1_J,axiom,
    ! [S3: stream3856641574645492619m_rule] :
      ( ( sdrop_4762382306744956754m_rule @ zero_zero_nat @ S3 )
      = S3 ) ).

% sdrop.simps(1)
thf(fact_646_sdrop_Osimps_I1_J,axiom,
    ! [S3: stream1878305877988316353m_rule] :
      ( ( sdrop_1764812713384251940m_rule @ zero_zero_nat @ S3 )
      = S3 ) ).

% sdrop.simps(1)
thf(fact_647_sdrop_Osimps_I1_J,axiom,
    ! [S3: stream1960312990768957601ist_fm] :
      ( ( sdrop_2468034597186633064ist_fm @ zero_zero_nat @ S3 )
      = S3 ) ).

% sdrop.simps(1)
thf(fact_648_sdrop_Osimps_I1_J,axiom,
    ! [S3: stream3588992565182678257e_rule] :
      ( ( sdrop_2199339672054592340e_rule @ zero_zero_nat @ S3 )
      = S3 ) ).

% sdrop.simps(1)
thf(fact_649_sdrop_Osimps_I1_J,axiom,
    ! [S3: stream_list_fm] :
      ( ( sdrop_list_fm @ zero_zero_nat @ S3 )
      = S3 ) ).

% sdrop.simps(1)
thf(fact_650_sdrop_Osimps_I1_J,axiom,
    ! [S3: stream727092118206550309m_rule] :
      ( ( sdrop_7224736112439592940m_rule @ zero_zero_nat @ S3 )
      = S3 ) ).

% sdrop.simps(1)
thf(fact_651_sdrop_Osimps_I1_J,axiom,
    ! [S3: stream8299795917829157543ist_fm] :
      ( ( sdrop_4442373711808556042ist_fm @ zero_zero_nat @ S3 )
      = S3 ) ).

% sdrop.simps(1)
thf(fact_652_sdrop_Osimps_I1_J,axiom,
    ! [S3: stream_rule] :
      ( ( sdrop_rule @ zero_zero_nat @ S3 )
      = S3 ) ).

% sdrop.simps(1)
thf(fact_653_sdrop_Osimps_I1_J,axiom,
    ! [S3: stream6494289010434245521m_rule] :
      ( ( sdrop_7373388980841566196m_rule @ zero_zero_nat @ S3 )
      = S3 ) ).

% sdrop.simps(1)
thf(fact_654_Collect__case__prod__Grp__eqD,axiom,
    ! [Z2: produc1996495991257130529ist_fm,A4: set_list_fm,F2: list_fm > list_fm] :
      ( ( member8102475879199740618ist_fm @ Z2 @ ( collec7231575776666803724ist_fm @ ( produc2011568048573023402t_fm_o @ ( bNF_Gr5437506445679690256ist_fm @ A4 @ F2 ) ) ) )
     => ( ( comp_l1868218281415653180ist_fm @ F2 @ produc1501393135466168645ist_fm @ Z2 )
        = ( produc4588648349897876871ist_fm @ Z2 ) ) ) ).

% Collect_case_prod_Grp_eqD
thf(fact_655_Collect__case__prod__Grp__eqD,axiom,
    ! [Z2: produc164195504107695125m_rule,A4: set_Pr7058068377845519745ist_fm,F2: produc1996495991257130529ist_fm > rule] :
      ( ( member4220325220686508332m_rule @ Z2 @ ( collec2069506698588392810m_rule @ ( produc7369492558480804444rule_o @ ( bNF_Gr1084804072905404824m_rule @ A4 @ F2 ) ) ) )
     => ( ( comp_P7754060113139186394m_rule @ F2 @ produc6879501374131015971m_rule @ Z2 )
        = ( produc7165828336582415457m_rule @ Z2 ) ) ) ).

% Collect_case_prod_Grp_eqD
thf(fact_656_Collect__case__prod__Grp__eqD,axiom,
    ! [Z2: produc4630866025709511057ist_fm,A4: set_rule,F2: rule > produc1996495991257130529ist_fm] :
      ( ( member8686995742288324264ist_fm @ Z2 @ ( collec6536177220190208742ist_fm @ ( produc1628407760719572648t_fm_o @ ( bNF_Gr7657091880239761484ist_fm @ A4 @ F2 ) ) ) )
     => ( ( comp_r7873396011713486090ist_fm @ F2 @ produc4228417144610596823ist_fm @ Z2 )
        = ( produc4514744107061996309ist_fm @ Z2 ) ) ) ).

% Collect_case_prod_Grp_eqD
thf(fact_657_Collect__case__prod__Grp__eqD,axiom,
    ! [Z2: produc780299800278082923e_rule,A4: set_rule,F2: rule > rule] :
      ( ( member3753523862196722196e_rule @ Z2 @ ( collec5224624948449408342e_rule @ ( produc8052491652537269344rule_o @ ( bNF_Grp_rule_rule @ A4 @ F2 ) ) ) )
     => ( ( comp_r2807213283240928316e_rule @ F2 @ produc7264528503596239631e_rule @ Z2 )
        = ( produc6725306161561501521e_rule @ Z2 ) ) ) ).

% Collect_case_prod_Grp_eqD
thf(fact_658_Collect__case__prod__Grp__eqD,axiom,
    ! [Z2: product_prod_fm_fm,A4: set_fm,F2: fm > fm] :
      ( ( member8474499337054950954_fm_fm @ Z2 @ ( collec4425076577877259116_fm_fm @ ( produc2811014937302177098m_fm_o @ ( bNF_Grp_fm_fm @ A4 @ F2 ) ) ) )
     => ( ( comp_f628634143932988988_fm_fm @ F2 @ product_fst_fm_fm @ Z2 )
        = ( product_snd_fm_fm @ Z2 ) ) ) ).

% Collect_case_prod_Grp_eqD
thf(fact_659_Collect__case__prod__Grp__eqD,axiom,
    ! [Z2: produc7564083655194549958ule_fm,A4: set_Pr1008144964186165195m_rule,F2: produc164195504107695125m_rule > fm] :
      ( ( member7952525889772502127ule_fm @ Z2 @ ( collec7776316114922985905ule_fm @ ( produc3554709660730922949e_fm_o @ ( bNF_Gr8430601450198604469ule_fm @ A4 @ F2 ) ) ) )
     => ( ( comp_P2656501846579916028ule_fm @ F2 @ produc1758149581174257386ule_fm @ Z2 )
        = ( produc8694862632519904556ule_fm @ Z2 ) ) ) ).

% Collect_case_prod_Grp_eqD
thf(fact_660_Collect__case__prod__Grp__eqD,axiom,
    ! [Z2: produc4719849802655908294ist_fm,A4: set_li5537492308219346823ist_fm,F2: list_P5616295576739893671ist_fm > list_list_fm] :
      ( ( member4192383230844517341ist_fm @ Z2 @ ( collec7853023762562200091ist_fm @ ( produc2973715795501690859t_fm_o @ ( bNF_Gr5899615463480776905ist_fm @ A4 @ F2 ) ) ) )
     => ( ( comp_l5403618335446233404ist_fm @ F2 @ produc6245774423584770388ist_fm @ Z2 )
        = ( produc8600084660692051602ist_fm @ Z2 ) ) ) ).

% Collect_case_prod_Grp_eqD
thf(fact_661_Collect__case__prod__Grp__eqD,axiom,
    ! [Z2: produc1805012608956077659t_rule,A4: set_li5339030221936397403m_rule,F2: list_P8332459875677254181m_rule > list_rule] :
      ( ( member6491770712979037828t_rule @ Z2 @ ( collec5858587166179649734t_rule @ ( produc2090550080120670960rule_o @ ( bNF_Gr7726341041440550858t_rule @ A4 @ F2 ) ) ) )
     => ( ( comp_l2052433623810477756t_rule @ F2 @ produc47317219786245247t_rule @ Z2 )
        = ( produc9172570333982009281t_rule @ Z2 ) ) ) ).

% Collect_case_prod_Grp_eqD
thf(fact_662_Collect__case__prod__Grp__eqD,axiom,
    ! [Z2: produc2148452214272472023t_rule,A4: set_li346003458413430231ist_fm,F2: list_P342308711384885665ist_fm > list_rule] :
      ( ( member6835210318295432192t_rule @ Z2 @ ( collec6202026771496044098t_rule @ ( produc5652714768248047220rule_o @ ( bNF_Gr8069780646756945222t_rule @ A4 @ F2 ) ) ) )
     => ( ( comp_l2780630628754590140t_rule @ F2 @ produc390756825102639611t_rule @ Z2 )
        = ( produc292637902443627837t_rule @ Z2 ) ) ) ).

% Collect_case_prod_Grp_eqD
thf(fact_663_Collect__case__prod__Grp__eqD,axiom,
    ! [Z2: produc1754732471204517991ist_fm,A4: set_Pr7058068377845519745ist_fm,F2: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm] :
      ( ( member60238275747228048ist_fm @ Z2 @ ( collec888080998205339090ist_fm @ ( produc2796214937669505452t_fm_o @ ( bNF_Gr854411304184578830ist_fm @ A4 @ F2 ) ) ) )
     => ( ( comp_P8293428446158863236ist_fm @ F2 @ produc5894184038225292483ist_fm @ Z2 )
        = ( produc1639249265332130309ist_fm @ Z2 ) ) ) ).

% Collect_case_prod_Grp_eqD
thf(fact_664_smap2_Osimps_I1_J,axiom,
    ! [F2: produc164195504107695125m_rule > produc164195504107695125m_rule > produc164195504107695125m_rule,S1: stream727092118206550309m_rule,S2: stream727092118206550309m_rule] :
      ( ( shd_Pr7235097944458474089m_rule @ ( smap2_3865418324013074943m_rule @ F2 @ S1 @ S2 ) )
      = ( F2 @ ( shd_Pr7235097944458474089m_rule @ S1 ) @ ( shd_Pr7235097944458474089m_rule @ S2 ) ) ) ).

% smap2.simps(1)
thf(fact_665_smap2_Osimps_I1_J,axiom,
    ! [F2: rule > rule > rule,S1: stream_rule,S2: stream_rule] :
      ( ( shd_rule @ ( smap2_rule_rule_rule @ F2 @ S1 @ S2 ) )
      = ( F2 @ ( shd_rule @ S1 ) @ ( shd_rule @ S2 ) ) ) ).

% smap2.simps(1)
thf(fact_666_smap2_Osimps_I1_J,axiom,
    ! [F2: rule > fm > rule,S1: stream_rule,S2: stream_fm] :
      ( ( shd_rule @ ( smap2_rule_fm_rule @ F2 @ S1 @ S2 ) )
      = ( F2 @ ( shd_rule @ S1 ) @ ( shd_fm @ S2 ) ) ) ).

% smap2.simps(1)
thf(fact_667_smap2_Osimps_I1_J,axiom,
    ! [F2: fm > rule > rule,S1: stream_fm,S2: stream_rule] :
      ( ( shd_rule @ ( smap2_fm_rule_rule @ F2 @ S1 @ S2 ) )
      = ( F2 @ ( shd_fm @ S1 ) @ ( shd_rule @ S2 ) ) ) ).

% smap2.simps(1)
thf(fact_668_smap2_Osimps_I1_J,axiom,
    ! [F2: fm > fm > rule,S1: stream_fm,S2: stream_fm] :
      ( ( shd_rule @ ( smap2_fm_fm_rule @ F2 @ S1 @ S2 ) )
      = ( F2 @ ( shd_fm @ S1 ) @ ( shd_fm @ S2 ) ) ) ).

% smap2.simps(1)
thf(fact_669_smap2_Osimps_I1_J,axiom,
    ! [F2: rule > rule > fm,S1: stream_rule,S2: stream_rule] :
      ( ( shd_fm @ ( smap2_rule_rule_fm @ F2 @ S1 @ S2 ) )
      = ( F2 @ ( shd_rule @ S1 ) @ ( shd_rule @ S2 ) ) ) ).

% smap2.simps(1)
thf(fact_670_smap2_Osimps_I1_J,axiom,
    ! [F2: rule > fm > fm,S1: stream_rule,S2: stream_fm] :
      ( ( shd_fm @ ( smap2_rule_fm_fm @ F2 @ S1 @ S2 ) )
      = ( F2 @ ( shd_rule @ S1 ) @ ( shd_fm @ S2 ) ) ) ).

% smap2.simps(1)
thf(fact_671_smap2_Osimps_I1_J,axiom,
    ! [F2: fm > rule > fm,S1: stream_fm,S2: stream_rule] :
      ( ( shd_fm @ ( smap2_fm_rule_fm @ F2 @ S1 @ S2 ) )
      = ( F2 @ ( shd_fm @ S1 ) @ ( shd_rule @ S2 ) ) ) ).

% smap2.simps(1)
thf(fact_672_smap2_Osimps_I1_J,axiom,
    ! [F2: fm > fm > fm,S1: stream_fm,S2: stream_fm] :
      ( ( shd_fm @ ( smap2_fm_fm_fm @ F2 @ S1 @ S2 ) )
      = ( F2 @ ( shd_fm @ S1 ) @ ( shd_fm @ S2 ) ) ) ).

% smap2.simps(1)
thf(fact_673_smap2_Osimps_I1_J,axiom,
    ! [F2: rule > produc1996495991257130529ist_fm > rule,S1: stream_rule,S2: stream8299795917829157543ist_fm] :
      ( ( shd_rule @ ( smap2_3943848181979139747m_rule @ F2 @ S1 @ S2 ) )
      = ( F2 @ ( shd_rule @ S1 ) @ ( shd_Pr772355297128350925ist_fm @ S2 ) ) ) ).

% smap2.simps(1)
thf(fact_674_smap2__alt,axiom,
    ! [F2: produc164195504107695125m_rule > produc164195504107695125m_rule > produc164195504107695125m_rule,S1: stream727092118206550309m_rule,S2: stream727092118206550309m_rule,S3: stream727092118206550309m_rule] :
      ( ( ( smap2_3865418324013074943m_rule @ F2 @ S1 @ S2 )
        = S3 )
      = ( ! [N3: nat] :
            ( ( F2 @ ( snth_P1712579846118164081m_rule @ S1 @ N3 ) @ ( snth_P1712579846118164081m_rule @ S2 @ N3 ) )
            = ( snth_P1712579846118164081m_rule @ S3 @ N3 ) ) ) ) ).

% smap2_alt
thf(fact_675_smap2__alt,axiom,
    ! [F2: rule > rule > fm,S1: stream_rule,S2: stream_rule,S3: stream_fm] :
      ( ( ( smap2_rule_rule_fm @ F2 @ S1 @ S2 )
        = S3 )
      = ( ! [N3: nat] :
            ( ( F2 @ ( snth_rule @ S1 @ N3 ) @ ( snth_rule @ S2 @ N3 ) )
            = ( snth_fm @ S3 @ N3 ) ) ) ) ).

% smap2_alt
thf(fact_676_smap2__alt,axiom,
    ! [F2: rule > fm > rule,S1: stream_rule,S2: stream_fm,S3: stream_rule] :
      ( ( ( smap2_rule_fm_rule @ F2 @ S1 @ S2 )
        = S3 )
      = ( ! [N3: nat] :
            ( ( F2 @ ( snth_rule @ S1 @ N3 ) @ ( snth_fm @ S2 @ N3 ) )
            = ( snth_rule @ S3 @ N3 ) ) ) ) ).

% smap2_alt
thf(fact_677_smap2__alt,axiom,
    ! [F2: rule > fm > fm,S1: stream_rule,S2: stream_fm,S3: stream_fm] :
      ( ( ( smap2_rule_fm_fm @ F2 @ S1 @ S2 )
        = S3 )
      = ( ! [N3: nat] :
            ( ( F2 @ ( snth_rule @ S1 @ N3 ) @ ( snth_fm @ S2 @ N3 ) )
            = ( snth_fm @ S3 @ N3 ) ) ) ) ).

% smap2_alt
thf(fact_678_smap2__alt,axiom,
    ! [F2: fm > rule > rule,S1: stream_fm,S2: stream_rule,S3: stream_rule] :
      ( ( ( smap2_fm_rule_rule @ F2 @ S1 @ S2 )
        = S3 )
      = ( ! [N3: nat] :
            ( ( F2 @ ( snth_fm @ S1 @ N3 ) @ ( snth_rule @ S2 @ N3 ) )
            = ( snth_rule @ S3 @ N3 ) ) ) ) ).

% smap2_alt
thf(fact_679_smap2__alt,axiom,
    ! [F2: fm > rule > fm,S1: stream_fm,S2: stream_rule,S3: stream_fm] :
      ( ( ( smap2_fm_rule_fm @ F2 @ S1 @ S2 )
        = S3 )
      = ( ! [N3: nat] :
            ( ( F2 @ ( snth_fm @ S1 @ N3 ) @ ( snth_rule @ S2 @ N3 ) )
            = ( snth_fm @ S3 @ N3 ) ) ) ) ).

% smap2_alt
thf(fact_680_smap2__alt,axiom,
    ! [F2: fm > fm > rule,S1: stream_fm,S2: stream_fm,S3: stream_rule] :
      ( ( ( smap2_fm_fm_rule @ F2 @ S1 @ S2 )
        = S3 )
      = ( ! [N3: nat] :
            ( ( F2 @ ( snth_fm @ S1 @ N3 ) @ ( snth_fm @ S2 @ N3 ) )
            = ( snth_rule @ S3 @ N3 ) ) ) ) ).

% smap2_alt
thf(fact_681_smap2__alt,axiom,
    ! [F2: fm > fm > fm,S1: stream_fm,S2: stream_fm,S3: stream_fm] :
      ( ( ( smap2_fm_fm_fm @ F2 @ S1 @ S2 )
        = S3 )
      = ( ! [N3: nat] :
            ( ( F2 @ ( snth_fm @ S1 @ N3 ) @ ( snth_fm @ S2 @ N3 ) )
            = ( snth_fm @ S3 @ N3 ) ) ) ) ).

% smap2_alt
thf(fact_682_smap2__alt,axiom,
    ! [F2: rule > rule > rule,S1: stream_rule,S2: stream_rule,S3: stream_rule] :
      ( ( ( smap2_rule_rule_rule @ F2 @ S1 @ S2 )
        = S3 )
      = ( ! [N3: nat] :
            ( ( F2 @ ( snth_rule @ S1 @ N3 ) @ ( snth_rule @ S2 @ N3 ) )
            = ( snth_rule @ S3 @ N3 ) ) ) ) ).

% smap2_alt
thf(fact_683_smap2__alt,axiom,
    ! [F2: stream_fm > rule > rule,S1: stream_stream_fm,S2: stream_rule,S3: stream_rule] :
      ( ( ( smap2_2038757848429906218e_rule @ F2 @ S1 @ S2 )
        = S3 )
      = ( ! [N3: nat] :
            ( ( F2 @ ( snth_stream_fm @ S1 @ N3 ) @ ( snth_rule @ S2 @ N3 ) )
            = ( snth_rule @ S3 @ N3 ) ) ) ) ).

% smap2_alt
thf(fact_684_snd__comp__apsnd,axiom,
    ! [F2: list_fm > list_fm] :
      ( ( comp_P1426762558556845121ist_fm @ produc4588648349897876871ist_fm @ ( produc9047611296996456864ist_fm @ F2 ) )
      = ( comp_l1868218281415653180ist_fm @ F2 @ produc4588648349897876871ist_fm ) ) ).

% snd_comp_apsnd
thf(fact_685_snd__comp__apsnd,axiom,
    ! [F2: rule > rule] :
      ( ( comp_P422132551275928404m_rule @ produc7165828336582415457m_rule @ ( produc3063135074001183255ist_fm @ F2 ) )
      = ( comp_r641804198332660996m_rule @ F2 @ produc7165828336582415457m_rule ) ) ).

% snd_comp_apsnd
thf(fact_686_snd__comp__apsnd,axiom,
    ! [F2: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm] :
      ( ( comp_P1889728861619134526ist_fm @ produc4514744107061996309ist_fm @ ( produc4332487942798490499m_rule @ F2 ) )
      = ( comp_P3488392109545816500ist_fm @ F2 @ produc4514744107061996309ist_fm ) ) ).

% snd_comp_apsnd
thf(fact_687_snd__comp__apsnd,axiom,
    ! [F2: rule > rule] :
      ( ( comp_P5356484029182387740e_rule @ produc6725306161561501521e_rule @ ( produc6058821816588017871e_rule @ F2 ) )
      = ( comp_r2807213283240928316e_rule @ F2 @ produc6725306161561501521e_rule ) ) ).

% snd_comp_apsnd
thf(fact_688_snd__comp__apsnd,axiom,
    ! [F2: fm > fm] :
      ( ( comp_P3218056037334583377_fm_fm @ product_snd_fm_fm @ ( produc1127837598810172016_fm_fm @ F2 ) )
      = ( comp_f628634143932988988_fm_fm @ F2 @ product_snd_fm_fm ) ) ).

% snd_comp_apsnd
thf(fact_689_snd__comp__apsnd,axiom,
    ! [F2: list_fm > set_list_fm] :
      ( ( comp_P4975598443037789397ist_fm @ produc1818303792603354685ist_fm @ ( produc8615751935427109974ist_fm @ F2 ) )
      = ( comp_l987677038997058310ist_fm @ F2 @ produc4588648349897876871ist_fm ) ) ).

% snd_comp_apsnd
thf(fact_690_snd__comp__apsnd,axiom,
    ! [F2: rule > produc1996495991257130529ist_fm] :
      ( ( comp_P736715915757637118e_rule @ produc4514744107061996309ist_fm @ ( produc5714219303521602403m_rule @ F2 ) )
      = ( comp_r7188775132017022002e_rule @ F2 @ produc6725306161561501521e_rule ) ) ).

% snd_comp_apsnd
thf(fact_691_snd__comp__apsnd,axiom,
    ! [F2: produc1996495991257130529ist_fm > rule] :
      ( ( comp_P7015120915279526048ist_fm @ produc6725306161561501521e_rule @ ( produc7903915439027271855e_rule @ F2 ) )
      = ( comp_P2997358597886226518ist_fm @ F2 @ produc4514744107061996309ist_fm ) ) ).

% snd_comp_apsnd
thf(fact_692_snd__comp__apsnd,axiom,
    ! [F2: fm > fm] :
      ( ( comp_P8409632166029999505ule_fm @ produc8694862632519904556ule_fm @ ( produc4214510056655550261m_rule @ F2 ) )
      = ( comp_f1859728840441439361ule_fm @ F2 @ produc8694862632519904556ule_fm ) ) ).

% snd_comp_apsnd
thf(fact_693_snd__comp__apsnd,axiom,
    ! [F2: produc1996495991257130529ist_fm > rule] :
      ( ( comp_P4097307840597955944ist_fm @ produc7165828336582415457m_rule @ ( produc1681403713278071351ist_fm @ F2 ) )
      = ( comp_P4878432454318713954ist_fm @ F2 @ produc1639249265332130309ist_fm ) ) ).

% snd_comp_apsnd
thf(fact_694_fst__comp__apfst,axiom,
    ! [F2: list_fm > list_fm] :
      ( ( comp_P1426762558556845121ist_fm @ produc1501393135466168645ist_fm @ ( produc1130293176790411102ist_fm @ F2 ) )
      = ( comp_l1868218281415653180ist_fm @ F2 @ produc1501393135466168645ist_fm ) ) ).

% fst_comp_apfst
thf(fact_695_fst__comp__apfst,axiom,
    ! [F2: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm] :
      ( ( comp_P7463969011078018366m_rule @ produc6879501374131015971m_rule @ ( produc3731403639160201281m_rule @ F2 ) )
      = ( comp_P8245093624798776376m_rule @ F2 @ produc6879501374131015971m_rule ) ) ).

% fst_comp_apfst
thf(fact_696_fst__comp__apfst,axiom,
    ! [F2: rule > rule] :
      ( ( comp_P5356484029182387740e_rule @ produc7264528503596239631e_rule @ ( produc6865027128820921997e_rule @ F2 ) )
      = ( comp_r2807213283240928316e_rule @ F2 @ produc7264528503596239631e_rule ) ) ).

% fst_comp_apfst
thf(fact_697_fst__comp__apfst,axiom,
    ! [F2: fm > fm] :
      ( ( comp_P3218056037334583377_fm_fm @ product_fst_fm_fm @ ( produc5431712386250947886_fm_fm @ F2 ) )
      = ( comp_f628634143932988988_fm_fm @ F2 @ product_fst_fm_fm ) ) ).

% fst_comp_apfst
thf(fact_698_fst__comp__apfst,axiom,
    ! [F2: list_fm > set_list_fm] :
      ( ( comp_P6183667238511117269ist_fm @ produc4725182812612597755ist_fm @ ( produc3450070183611816468ist_fm @ F2 ) )
      = ( comp_l987677038997058310ist_fm @ F2 @ produc1501393135466168645ist_fm ) ) ).

% fst_comp_apfst
thf(fact_699_fst__comp__apfst,axiom,
    ! [F2: rule > produc1996495991257130529ist_fm] :
      ( ( comp_P6969103484960289410e_rule @ produc6879501374131015971m_rule @ ( produc8185590484082982437m_rule @ F2 ) )
      = ( comp_r7188775132017022002e_rule @ F2 @ produc7264528503596239631e_rule ) ) ).

% fst_comp_apfst
thf(fact_700_fst__comp__apfst,axiom,
    ! [F2: rule > rule] :
      ( ( comp_P7879787540529867852ist_fm @ produc4228417144610596823ist_fm @ ( produc5534506254562563289ist_fm @ F2 ) )
      = ( comp_r5108474719934476928ist_fm @ F2 @ produc4228417144610596823ist_fm ) ) ).

% fst_comp_apfst
thf(fact_701_fst__comp__apfst,axiom,
    ! [F2: produc1996495991257130529ist_fm > rule] :
      ( ( comp_P2548450393677710116m_rule @ produc7264528503596239631e_rule @ ( produc1151914582733876081e_rule @ F2 ) )
      = ( comp_P7754060113139186394m_rule @ F2 @ produc6879501374131015971m_rule ) ) ).

% fst_comp_apfst
thf(fact_702_fst__comp__apfst,axiom,
    ! [F2: produc1996495991257130529ist_fm > rule] :
      ( ( comp_P3279769206392031980ist_fm @ produc4228417144610596823ist_fm @ ( produc1080319409639782133ist_fm @ F2 ) )
      = ( comp_P4878432454318713954ist_fm @ F2 @ produc5894184038225292483ist_fm ) ) ).

% fst_comp_apfst
thf(fact_703_fst__comp__apfst,axiom,
    ! [F2: rule > produc1996495991257130529ist_fm] :
      ( ( comp_P5552026984502743150ist_fm @ produc5894184038225292483ist_fm @ ( produc4128060033708492609ist_fm @ F2 ) )
      = ( comp_r7873396011713486090ist_fm @ F2 @ produc4228417144610596823ist_fm ) ) ).

% fst_comp_apfst
thf(fact_704_Collect__split__mono__strong,axiom,
    ! [X5: set_Pr7058068377845519745ist_fm,A4: set_Pr5209464977148653383ist_fm,Y5: set_Pr7058068377845519745ist_fm,P2: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm > $o,Q3: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm > $o] :
      ( ( X5
        = ( image_415097795833635455ist_fm @ produc5894184038225292483ist_fm @ A4 ) )
     => ( ( Y5
          = ( image_415097795833635455ist_fm @ produc1639249265332130309ist_fm @ A4 ) )
       => ( ! [X: produc1996495991257130529ist_fm] :
              ( ( member8102475879199740618ist_fm @ X @ X5 )
             => ! [Xa: produc1996495991257130529ist_fm] :
                  ( ( member8102475879199740618ist_fm @ Xa @ Y5 )
                 => ( ( P2 @ X @ Xa )
                   => ( Q3 @ X @ Xa ) ) ) )
         => ( ( ord_le5613060152082810599ist_fm @ A4 @ ( collec888080998205339090ist_fm @ ( produc2796214937669505452t_fm_o @ P2 ) ) )
           => ( ord_le5613060152082810599ist_fm @ A4 @ ( collec888080998205339090ist_fm @ ( produc2796214937669505452t_fm_o @ Q3 ) ) ) ) ) ) ) ).

% Collect_split_mono_strong
thf(fact_705_Collect__split__mono__strong,axiom,
    ! [X5: set_rule,A4: set_Pr6543236023296505803e_rule,Y5: set_rule,P2: rule > rule > $o,Q3: rule > rule > $o] :
      ( ( X5
        = ( image_8452519424085915243e_rule @ produc7264528503596239631e_rule @ A4 ) )
     => ( ( Y5
          = ( image_8452519424085915243e_rule @ produc6725306161561501521e_rule @ A4 ) )
       => ( ! [X: rule] :
              ( ( member_rule @ X @ X5 )
             => ! [Xa: rule] :
                  ( ( member_rule @ Xa @ Y5 )
                 => ( ( P2 @ X @ Xa )
                   => ( Q3 @ X @ Xa ) ) ) )
         => ( ( ord_le4515388646595166059e_rule @ A4 @ ( collec5224624948449408342e_rule @ ( produc8052491652537269344rule_o @ P2 ) ) )
           => ( ord_le4515388646595166059e_rule @ A4 @ ( collec5224624948449408342e_rule @ ( produc8052491652537269344rule_o @ Q3 ) ) ) ) ) ) ) ).

% Collect_split_mono_strong
thf(fact_706_Collect__split__mono__strong,axiom,
    ! [X5: set_fm,A4: set_Pr4463079037648049377_fm_fm,Y5: set_fm,P2: fm > fm > $o,Q3: fm > fm > $o] :
      ( ( X5
        = ( image_5888559327631113100_fm_fm @ product_fst_fm_fm @ A4 ) )
     => ( ( Y5
          = ( image_5888559327631113100_fm_fm @ product_snd_fm_fm @ A4 ) )
       => ( ! [X: fm] :
              ( ( member_fm @ X @ X5 )
             => ! [Xa: fm] :
                  ( ( member_fm @ Xa @ Y5 )
                 => ( ( P2 @ X @ Xa )
                   => ( Q3 @ X @ Xa ) ) ) )
         => ( ( ord_le7532831136298237057_fm_fm @ A4 @ ( collec4425076577877259116_fm_fm @ ( produc2811014937302177098m_fm_o @ P2 ) ) )
           => ( ord_le7532831136298237057_fm_fm @ A4 @ ( collec4425076577877259116_fm_fm @ ( produc2811014937302177098m_fm_o @ Q3 ) ) ) ) ) ) ) ).

% Collect_split_mono_strong
thf(fact_707_Collect__split__mono__strong,axiom,
    ! [X5: set_Pr1008144964186165195m_rule,A4: set_Pr8819451114087901350ule_fm,Y5: set_fm,P2: produc164195504107695125m_rule > fm > $o,Q3: produc164195504107695125m_rule > fm > $o] :
      ( ( X5
        = ( image_2084494183218120726m_rule @ produc1758149581174257386ule_fm @ A4 ) )
     => ( ( Y5
          = ( image_2608076899697204049_fm_fm @ produc8694862632519904556ule_fm @ A4 ) )
       => ( ! [X: produc164195504107695125m_rule] :
              ( ( member4220325220686508332m_rule @ X @ X5 )
             => ! [Xa: fm] :
                  ( ( member_fm @ Xa @ Y5 )
                 => ( ( P2 @ X @ Xa )
                   => ( Q3 @ X @ Xa ) ) ) )
         => ( ( ord_le6302689129981005894ule_fm @ A4 @ ( collec7776316114922985905ule_fm @ ( produc3554709660730922949e_fm_o @ P2 ) ) )
           => ( ord_le6302689129981005894ule_fm @ A4 @ ( collec7776316114922985905ule_fm @ ( produc3554709660730922949e_fm_o @ Q3 ) ) ) ) ) ) ) ).

% Collect_split_mono_strong
thf(fact_708_Collect__split__mono__strong,axiom,
    ! [X5: set_list_fm,A4: set_Pr7058068377845519745ist_fm,Y5: set_list_fm,P2: list_fm > list_fm > $o,Q3: list_fm > list_fm > $o] :
      ( ( X5
        = ( image_6369949096775035324ist_fm @ produc1501393135466168645ist_fm @ A4 ) )
     => ( ( Y5
          = ( image_6369949096775035324ist_fm @ produc4588648349897876871ist_fm @ A4 ) )
       => ( ! [X: list_fm] :
              ( ( member_list_fm @ X @ X5 )
             => ! [Xa: list_fm] :
                  ( ( member_list_fm @ Xa @ Y5 )
                 => ( ( P2 @ X @ Xa )
                   => ( Q3 @ X @ Xa ) ) ) )
         => ( ( ord_le2055072402069232929ist_fm @ A4 @ ( collec7231575776666803724ist_fm @ ( produc2011568048573023402t_fm_o @ P2 ) ) )
           => ( ord_le2055072402069232929ist_fm @ A4 @ ( collec7231575776666803724ist_fm @ ( produc2011568048573023402t_fm_o @ Q3 ) ) ) ) ) ) ) ).

% Collect_split_mono_strong
thf(fact_709_Collect__split__mono__strong,axiom,
    ! [X5: set_Pr7058068377845519745ist_fm,A4: set_Pr1008144964186165195m_rule,Y5: set_rule,P2: produc1996495991257130529ist_fm > rule > $o,Q3: produc1996495991257130529ist_fm > rule > $o] :
      ( ( X5
        = ( image_3223098899990291461ist_fm @ produc6879501374131015971m_rule @ A4 ) )
     => ( ( Y5
          = ( image_8178078329501008481e_rule @ produc7165828336582415457m_rule @ A4 ) )
       => ( ! [X: produc1996495991257130529ist_fm] :
              ( ( member8102475879199740618ist_fm @ X @ X5 )
             => ! [Xa: rule] :
                  ( ( member_rule @ Xa @ Y5 )
                 => ( ( P2 @ X @ Xa )
                   => ( Q3 @ X @ Xa ) ) ) )
         => ( ( ord_le5052251716133888043m_rule @ A4 @ ( collec2069506698588392810m_rule @ ( produc7369492558480804444rule_o @ P2 ) ) )
           => ( ord_le5052251716133888043m_rule @ A4 @ ( collec2069506698588392810m_rule @ ( produc7369492558480804444rule_o @ Q3 ) ) ) ) ) ) ) ).

% Collect_split_mono_strong
thf(fact_710_Collect__split__mono__strong,axiom,
    ! [X5: set_rule,A4: set_Pr2241365836748572487ist_fm,Y5: set_Pr7058068377845519745ist_fm,P2: rule > produc1996495991257130529ist_fm > $o,Q3: rule > produc1996495991257130529ist_fm > $o] :
      ( ( X5
        = ( image_952663707700710365m_rule @ produc4228417144610596823ist_fm @ A4 ) )
     => ( ( Y5
          = ( image_3681455213663431561ist_fm @ produc4514744107061996309ist_fm @ A4 ) )
       => ( ! [X: rule] :
              ( ( member_rule @ X @ X5 )
             => ! [Xa: produc1996495991257130529ist_fm] :
                  ( ( member8102475879199740618ist_fm @ Xa @ Y5 )
                 => ( ( P2 @ X @ Xa )
                   => ( Q3 @ X @ Xa ) ) ) )
         => ( ( ord_le6285472588696295335ist_fm @ A4 @ ( collec6536177220190208742ist_fm @ ( produc1628407760719572648t_fm_o @ P2 ) ) )
           => ( ord_le6285472588696295335ist_fm @ A4 @ ( collec6536177220190208742ist_fm @ ( produc1628407760719572648t_fm_o @ Q3 ) ) ) ) ) ) ) ).

% Collect_split_mono_strong
thf(fact_711_snd__comp__map__prod,axiom,
    ! [F2: list_fm > list_fm,G: list_fm > list_fm] :
      ( ( comp_P1426762558556845121ist_fm @ produc4588648349897876871ist_fm @ ( produc3421166768339006330ist_fm @ F2 @ G ) )
      = ( comp_l1868218281415653180ist_fm @ G @ produc4588648349897876871ist_fm ) ) ).

% snd_comp_map_prod
thf(fact_712_snd__comp__map__prod,axiom,
    ! [F2: produc1996495991257130529ist_fm > list_fm,G: rule > list_fm] :
      ( ( comp_P8020416857309042165m_rule @ produc4588648349897876871ist_fm @ ( produc1965614570530558770ist_fm @ F2 @ G ) )
      = ( comp_r4098878985462706207m_rule @ G @ produc7165828336582415457m_rule ) ) ).

% snd_comp_map_prod
thf(fact_713_snd__comp__map__prod,axiom,
    ! [F2: rule > list_fm,G: produc1996495991257130529ist_fm > list_fm] :
      ( ( comp_P3263715342056082289ist_fm @ produc4588648349897876871ist_fm @ ( produc1976628786251050148ist_fm @ F2 @ G ) )
      = ( comp_P3263715342056082289ist_fm @ G @ produc4514744107061996309ist_fm ) ) ).

% snd_comp_map_prod
thf(fact_714_snd__comp__map__prod,axiom,
    ! [F2: list_fm > produc1996495991257130529ist_fm,G: list_fm > rule] :
      ( ( comp_P855734935590900386ist_fm @ produc7165828336582415457m_rule @ ( produc8124276167727720944m_rule @ F2 @ G ) )
      = ( comp_l6170393434051197847ist_fm @ G @ produc4588648349897876871ist_fm ) ) ).

% snd_comp_map_prod
thf(fact_715_snd__comp__map__prod,axiom,
    ! [F2: list_fm > rule,G: list_fm > produc1996495991257130529ist_fm] :
      ( ( comp_P3021031701871720116ist_fm @ produc4514744107061996309ist_fm @ ( produc3613848521340044642ist_fm @ F2 @ G ) )
      = ( comp_l2016392514137073923ist_fm @ G @ produc4588648349897876871ist_fm ) ) ).

% snd_comp_map_prod
thf(fact_716_snd__comp__map__prod,axiom,
    ! [F2: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm,G: rule > rule] :
      ( ( comp_P422132551275928404m_rule @ produc7165828336582415457m_rule @ ( produc7680103964935106114e_rule @ F2 @ G ) )
      = ( comp_r641804198332660996m_rule @ G @ produc7165828336582415457m_rule ) ) ).

% snd_comp_map_prod
thf(fact_717_snd__comp__map__prod,axiom,
    ! [F2: rule > produc1996495991257130529ist_fm,G: produc1996495991257130529ist_fm > rule] :
      ( ( comp_P4888803072877744336ist_fm @ produc7165828336582415457m_rule @ ( produc1545012558808995138m_rule @ F2 @ G ) )
      = ( comp_P2997358597886226518ist_fm @ G @ produc4514744107061996309ist_fm ) ) ).

% snd_comp_map_prod
thf(fact_718_snd__comp__map__prod,axiom,
    ! [F2: produc1996495991257130529ist_fm > rule,G: rule > produc1996495991257130529ist_fm] :
      ( ( comp_P6646430376872094402m_rule @ produc4514744107061996309ist_fm @ ( produc2839323599909017514ist_fm @ F2 @ G ) )
      = ( comp_r3406725490111670158m_rule @ G @ produc7165828336582415457m_rule ) ) ).

% snd_comp_map_prod
thf(fact_719_snd__comp__map__prod,axiom,
    ! [F2: rule > rule,G: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm] :
      ( ( comp_P1889728861619134526ist_fm @ produc4514744107061996309ist_fm @ ( produc1941668953357286466ist_fm @ F2 @ G ) )
      = ( comp_P3488392109545816500ist_fm @ G @ produc4514744107061996309ist_fm ) ) ).

% snd_comp_map_prod
thf(fact_720_snd__comp__map__prod,axiom,
    ! [F2: rule > rule,G: rule > rule] :
      ( ( comp_P5356484029182387740e_rule @ produc6725306161561501521e_rule @ ( produc8646833859711509262e_rule @ F2 @ G ) )
      = ( comp_r2807213283240928316e_rule @ G @ produc6725306161561501521e_rule ) ) ).

% snd_comp_map_prod
thf(fact_721_fst__comp__map__prod,axiom,
    ! [F2: list_fm > list_fm,G: list_fm > list_fm] :
      ( ( comp_P1426762558556845121ist_fm @ produc1501393135466168645ist_fm @ ( produc3421166768339006330ist_fm @ F2 @ G ) )
      = ( comp_l1868218281415653180ist_fm @ F2 @ produc1501393135466168645ist_fm ) ) ).

% fst_comp_map_prod
thf(fact_722_fst__comp__map__prod,axiom,
    ! [F2: list_fm > produc1996495991257130529ist_fm,G: list_fm > rule] :
      ( ( comp_P695203452023487288ist_fm @ produc6879501374131015971m_rule @ ( produc8124276167727720944m_rule @ F2 @ G ) )
      = ( comp_l2016392514137073923ist_fm @ F2 @ produc1501393135466168645ist_fm ) ) ).

% fst_comp_map_prod
thf(fact_723_fst__comp__map__prod,axiom,
    ! [F2: produc1996495991257130529ist_fm > list_fm,G: rule > list_fm] :
      ( ( comp_P8020416857309042165m_rule @ produc1501393135466168645ist_fm @ ( produc1965614570530558770ist_fm @ F2 @ G ) )
      = ( comp_P8020416857309042165m_rule @ F2 @ produc6879501374131015971m_rule ) ) ).

% fst_comp_map_prod
thf(fact_724_fst__comp__map__prod,axiom,
    ! [F2: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm,G: rule > rule] :
      ( ( comp_P7463969011078018366m_rule @ produc6879501374131015971m_rule @ ( produc7680103964935106114e_rule @ F2 @ G ) )
      = ( comp_P8245093624798776376m_rule @ F2 @ produc6879501374131015971m_rule ) ) ).

% fst_comp_map_prod
thf(fact_725_fst__comp__map__prod,axiom,
    ! [F2: rule > rule,G: rule > rule] :
      ( ( comp_P5356484029182387740e_rule @ produc7264528503596239631e_rule @ ( produc8646833859711509262e_rule @ F2 @ G ) )
      = ( comp_r2807213283240928316e_rule @ F2 @ produc7264528503596239631e_rule ) ) ).

% fst_comp_map_prod
thf(fact_726_fst__comp__map__prod,axiom,
    ! [F2: fm > rule,G: fm > rule] :
      ( ( comp_P4103961532875582002_fm_fm @ produc7264528503596239631e_rule @ ( produc4377082874624412068m_rule @ F2 @ G ) )
      = ( comp_f4230005197555880711_fm_fm @ F2 @ product_fst_fm_fm ) ) ).

% fst_comp_map_prod
thf(fact_727_fst__comp__map__prod,axiom,
    ! [F2: rule > fm,G: rule > fm] :
      ( ( comp_P6142150318389986619e_rule @ product_fst_fm_fm @ ( produc173016795980717476ule_fm @ F2 @ G ) )
      = ( comp_r3962787682557505521e_rule @ F2 @ produc7264528503596239631e_rule ) ) ).

% fst_comp_map_prod
thf(fact_728_fst__comp__map__prod,axiom,
    ! [F2: fm > fm,G: fm > fm] :
      ( ( comp_P3218056037334583377_fm_fm @ product_fst_fm_fm @ ( produc8016985811614767674_fm_fm @ F2 @ G ) )
      = ( comp_f628634143932988988_fm_fm @ F2 @ product_fst_fm_fm ) ) ).

% fst_comp_map_prod
thf(fact_729_fst__comp__map__prod,axiom,
    ! [F2: rule > list_fm,G: rule > list_fm] :
      ( ( comp_P1962423623166686091e_rule @ produc1501393135466168645ist_fm @ ( produc1880821772452811332ist_fm @ F2 @ G ) )
      = ( comp_r8225966605575789537e_rule @ F2 @ produc7264528503596239631e_rule ) ) ).

% fst_comp_map_prod
thf(fact_730_fst__comp__map__prod,axiom,
    ! [F2: fm > list_fm,G: fm > list_fm] :
      ( ( comp_P105494546137269921_fm_fm @ produc1501393135466168645ist_fm @ ( produc2380606312956990042ist_fm @ F2 @ G ) )
      = ( comp_f8555098094366227372_fm_fm @ F2 @ product_fst_fm_fm ) ) ).

% fst_comp_map_prod
thf(fact_731_fst__comp__apsnd,axiom,
    ! [F2: produc1996495991257130529ist_fm > rule] :
      ( ( comp_P5570111842168596734ist_fm @ produc6879501374131015971m_rule @ ( produc1681403713278071351ist_fm @ F2 ) )
      = produc5894184038225292483ist_fm ) ).

% fst_comp_apsnd
thf(fact_732_fst__comp__apsnd,axiom,
    ! [F2: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm] :
      ( ( comp_P7879787540529867852ist_fm @ produc4228417144610596823ist_fm @ ( produc4332487942798490499m_rule @ F2 ) )
      = produc4228417144610596823ist_fm ) ).

% fst_comp_apsnd
thf(fact_733_fst__comp__apsnd,axiom,
    ! [F2: rule > produc1996495991257130529ist_fm] :
      ( ( comp_P4809662347396705648e_rule @ produc4228417144610596823ist_fm @ ( produc5714219303521602403m_rule @ F2 ) )
      = produc7264528503596239631e_rule ) ).

% fst_comp_apsnd
thf(fact_734_fst__comp__apsnd,axiom,
    ! [F2: rule > produc1996495991257130529ist_fm] :
      ( ( comp_P1085356462900927218m_rule @ produc5894184038225292483ist_fm @ ( produc4729144337346781827ist_fm @ F2 ) )
      = produc6879501374131015971m_rule ) ).

% fst_comp_apsnd
thf(fact_735_fst__comp__apsnd,axiom,
    ! [F2: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm] :
      ( ( comp_P6475754874365093450ist_fm @ produc5894184038225292483ist_fm @ ( produc5106821411131517539ist_fm @ F2 ) )
      = produc5894184038225292483ist_fm ) ).

% fst_comp_apsnd
thf(fact_736_fst__comp__apsnd,axiom,
    ! [F2: produc1996495991257130529ist_fm > rule] :
      ( ( comp_P7015120915279526048ist_fm @ produc7264528503596239631e_rule @ ( produc7903915439027271855e_rule @ F2 ) )
      = produc4228417144610596823ist_fm ) ).

% fst_comp_apsnd
thf(fact_737_fst__comp__apsnd,axiom,
    ! [F2: rule > rule] :
      ( ( comp_P5356484029182387740e_rule @ produc7264528503596239631e_rule @ ( produc6058821816588017871e_rule @ F2 ) )
      = produc7264528503596239631e_rule ) ).

% fst_comp_apsnd
thf(fact_738_fst__comp__apsnd,axiom,
    ! [F2: fm > fm] :
      ( ( comp_P3218056037334583377_fm_fm @ product_fst_fm_fm @ ( produc1127837598810172016_fm_fm @ F2 ) )
      = product_fst_fm_fm ) ).

% fst_comp_apsnd
thf(fact_739_fst__comp__apsnd,axiom,
    ! [F2: rule > rule] :
      ( ( comp_P7463969011078018366m_rule @ produc6879501374131015971m_rule @ ( produc3063135074001183255ist_fm @ F2 ) )
      = produc6879501374131015971m_rule ) ).

% fst_comp_apsnd
thf(fact_740_fst__comp__apsnd,axiom,
    ! [F2: list_fm > list_fm] :
      ( ( comp_P1426762558556845121ist_fm @ produc1501393135466168645ist_fm @ ( produc9047611296996456864ist_fm @ F2 ) )
      = produc1501393135466168645ist_fm ) ).

% fst_comp_apsnd
thf(fact_741_divides__aux__eq,axiom,
    ! [Q: nat,R2: nat] :
      ( ( unique5332122412489317741ux_nat @ ( product_Pair_nat_nat @ Q @ R2 ) )
      = ( R2 = zero_zero_nat ) ) ).

% divides_aux_eq
thf(fact_742_snd__comp__apfst,axiom,
    ! [F2: list_fm > list_fm] :
      ( ( comp_P1426762558556845121ist_fm @ produc4588648349897876871ist_fm @ ( produc1130293176790411102ist_fm @ F2 ) )
      = produc4588648349897876871ist_fm ) ).

% snd_comp_apfst
thf(fact_743_snd__comp__apfst,axiom,
    ! [F2: rule > rule] :
      ( ( comp_P1889728861619134526ist_fm @ produc4514744107061996309ist_fm @ ( produc5534506254562563289ist_fm @ F2 ) )
      = produc4514744107061996309ist_fm ) ).

% snd_comp_apfst
thf(fact_744_snd__comp__apfst,axiom,
    ! [F2: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm] :
      ( ( comp_P422132551275928404m_rule @ produc7165828336582415457m_rule @ ( produc3731403639160201281m_rule @ F2 ) )
      = produc7165828336582415457m_rule ) ).

% snd_comp_apfst
thf(fact_745_snd__comp__apfst,axiom,
    ! [F2: rule > rule] :
      ( ( comp_P5356484029182387740e_rule @ produc6725306161561501521e_rule @ ( produc6865027128820921997e_rule @ F2 ) )
      = produc6725306161561501521e_rule ) ).

% snd_comp_apfst
thf(fact_746_snd__comp__apfst,axiom,
    ! [F2: fm > fm] :
      ( ( comp_P3218056037334583377_fm_fm @ product_snd_fm_fm @ ( produc5431712386250947886_fm_fm @ F2 ) )
      = product_snd_fm_fm ) ).

% snd_comp_apfst
thf(fact_747_snd__comp__apfst,axiom,
    ! [F2: rule > produc1996495991257130529ist_fm] :
      ( ( comp_P682981744566163948e_rule @ produc7165828336582415457m_rule @ ( produc8185590484082982437m_rule @ F2 ) )
      = produc6725306161561501521e_rule ) ).

% snd_comp_apfst
thf(fact_748_snd__comp__apfst,axiom,
    ! [F2: produc1996495991257130529ist_fm > rule] :
      ( ( comp_P2548450393677710116m_rule @ produc6725306161561501521e_rule @ ( produc1151914582733876081e_rule @ F2 ) )
      = produc7165828336582415457m_rule ) ).

% snd_comp_apfst
thf(fact_749_snd__comp__apfst,axiom,
    ! [F2: produc164195504107695125m_rule > fm] :
      ( ( comp_P6388136761583423126ule_fm @ product_snd_fm_fm @ ( produc7685345254987873523_fm_fm @ F2 ) )
      = produc8694862632519904556ule_fm ) ).

% snd_comp_apfst
thf(fact_750_snd__comp__apfst,axiom,
    ! [F2: fm > produc164195504107695125m_rule] :
      ( ( comp_P3262442148557739852_fm_fm @ produc8694862632519904556ule_fm @ ( produc8510827862186099571ule_fm @ F2 ) )
      = product_snd_fm_fm ) ).

% snd_comp_apfst
thf(fact_751_snd__comp__apfst,axiom,
    ! [F2: produc1996495991257130529ist_fm > rule] :
      ( ( comp_P3433142862794662522ist_fm @ produc4514744107061996309ist_fm @ ( produc1080319409639782133ist_fm @ F2 ) )
      = produc1639249265332130309ist_fm ) ).

% snd_comp_apfst
thf(fact_752_surj__swap,axiom,
    ( ( image_6982201141141708101ist_fm @ produc7936585717479832313ist_fm @ top_to728902761071950545ist_fm )
    = top_to728902761071950545ist_fm ) ).

% surj_swap
thf(fact_753_surj__swap,axiom,
    ( ( image_4130355632178425005m_rule @ produc9213896635423732003ist_fm @ top_to2837172092223966711ist_fm )
    = top_to1603951219661559419m_rule ) ).

% surj_swap
thf(fact_754_surj__swap,axiom,
    ( ( image_6659396422799841965ist_fm @ produc2641608828089375343m_rule @ top_to1603951219661559419m_rule )
    = top_to2837172092223966711ist_fm ) ).

% surj_swap
thf(fact_755_map__prod__simp,axiom,
    ! [F2: list_fm > rule,G: list_fm > produc1996495991257130529ist_fm,A: list_fm,B: list_fm] :
      ( ( produc3613848521340044642ist_fm @ F2 @ G @ ( produc381145313068854617ist_fm @ A @ B ) )
      = ( produc7063755442767259331ist_fm @ ( F2 @ A ) @ ( G @ B ) ) ) ).

% map_prod_simp
thf(fact_756_map__prod__simp,axiom,
    ! [F2: list_fm > list_fm,G: list_fm > list_fm,A: list_fm,B: list_fm] :
      ( ( produc3421166768339006330ist_fm @ F2 @ G @ ( produc381145313068854617ist_fm @ A @ B ) )
      = ( produc381145313068854617ist_fm @ ( F2 @ A ) @ ( G @ B ) ) ) ).

% map_prod_simp
thf(fact_757_fst__map__prod,axiom,
    ! [F2: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm,G: rule > rule,X2: produc164195504107695125m_rule] :
      ( ( produc6879501374131015971m_rule @ ( produc7680103964935106114e_rule @ F2 @ G @ X2 ) )
      = ( F2 @ ( produc6879501374131015971m_rule @ X2 ) ) ) ).

% fst_map_prod
thf(fact_758_fst__map__prod,axiom,
    ! [F2: list_fm > produc1996495991257130529ist_fm,G: list_fm > rule,X2: produc1996495991257130529ist_fm] :
      ( ( produc6879501374131015971m_rule @ ( produc8124276167727720944m_rule @ F2 @ G @ X2 ) )
      = ( F2 @ ( produc1501393135466168645ist_fm @ X2 ) ) ) ).

% fst_map_prod
thf(fact_759_fst__map__prod,axiom,
    ! [F2: produc1996495991257130529ist_fm > list_fm,G: rule > list_fm,X2: produc164195504107695125m_rule] :
      ( ( produc1501393135466168645ist_fm @ ( produc1965614570530558770ist_fm @ F2 @ G @ X2 ) )
      = ( F2 @ ( produc6879501374131015971m_rule @ X2 ) ) ) ).

% fst_map_prod
thf(fact_760_fst__map__prod,axiom,
    ! [F2: list_fm > list_fm,G: list_fm > list_fm,X2: produc1996495991257130529ist_fm] :
      ( ( produc1501393135466168645ist_fm @ ( produc3421166768339006330ist_fm @ F2 @ G @ X2 ) )
      = ( F2 @ ( produc1501393135466168645ist_fm @ X2 ) ) ) ).

% fst_map_prod
thf(fact_761_snd__map__prod,axiom,
    ! [F2: list_fm > list_fm,G: list_fm > list_fm,X2: produc1996495991257130529ist_fm] :
      ( ( produc4588648349897876871ist_fm @ ( produc3421166768339006330ist_fm @ F2 @ G @ X2 ) )
      = ( G @ ( produc4588648349897876871ist_fm @ X2 ) ) ) ).

% snd_map_prod
thf(fact_762_snd__map__prod,axiom,
    ! [F2: produc1996495991257130529ist_fm > list_fm,G: rule > list_fm,X2: produc164195504107695125m_rule] :
      ( ( produc4588648349897876871ist_fm @ ( produc1965614570530558770ist_fm @ F2 @ G @ X2 ) )
      = ( G @ ( produc7165828336582415457m_rule @ X2 ) ) ) ).

% snd_map_prod
thf(fact_763_snd__map__prod,axiom,
    ! [F2: rule > list_fm,G: produc1996495991257130529ist_fm > list_fm,X2: produc4630866025709511057ist_fm] :
      ( ( produc4588648349897876871ist_fm @ ( produc1976628786251050148ist_fm @ F2 @ G @ X2 ) )
      = ( G @ ( produc4514744107061996309ist_fm @ X2 ) ) ) ).

% snd_map_prod
thf(fact_764_snd__map__prod,axiom,
    ! [F2: list_fm > produc1996495991257130529ist_fm,G: list_fm > rule,X2: produc1996495991257130529ist_fm] :
      ( ( produc7165828336582415457m_rule @ ( produc8124276167727720944m_rule @ F2 @ G @ X2 ) )
      = ( G @ ( produc4588648349897876871ist_fm @ X2 ) ) ) ).

% snd_map_prod
thf(fact_765_snd__map__prod,axiom,
    ! [F2: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm,G: rule > rule,X2: produc164195504107695125m_rule] :
      ( ( produc7165828336582415457m_rule @ ( produc7680103964935106114e_rule @ F2 @ G @ X2 ) )
      = ( G @ ( produc7165828336582415457m_rule @ X2 ) ) ) ).

% snd_map_prod
thf(fact_766_snd__map__prod,axiom,
    ! [F2: rule > produc1996495991257130529ist_fm,G: produc1996495991257130529ist_fm > rule,X2: produc4630866025709511057ist_fm] :
      ( ( produc7165828336582415457m_rule @ ( produc1545012558808995138m_rule @ F2 @ G @ X2 ) )
      = ( G @ ( produc4514744107061996309ist_fm @ X2 ) ) ) ).

% snd_map_prod
thf(fact_767_snd__map__prod,axiom,
    ! [F2: list_fm > rule,G: list_fm > produc1996495991257130529ist_fm,X2: produc1996495991257130529ist_fm] :
      ( ( produc4514744107061996309ist_fm @ ( produc3613848521340044642ist_fm @ F2 @ G @ X2 ) )
      = ( G @ ( produc4588648349897876871ist_fm @ X2 ) ) ) ).

% snd_map_prod
thf(fact_768_snd__map__prod,axiom,
    ! [F2: produc1996495991257130529ist_fm > rule,G: rule > produc1996495991257130529ist_fm,X2: produc164195504107695125m_rule] :
      ( ( produc4514744107061996309ist_fm @ ( produc2839323599909017514ist_fm @ F2 @ G @ X2 ) )
      = ( G @ ( produc7165828336582415457m_rule @ X2 ) ) ) ).

% snd_map_prod
thf(fact_769_snd__map__prod,axiom,
    ! [F2: rule > rule,G: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm,X2: produc4630866025709511057ist_fm] :
      ( ( produc4514744107061996309ist_fm @ ( produc1941668953357286466ist_fm @ F2 @ G @ X2 ) )
      = ( G @ ( produc4514744107061996309ist_fm @ X2 ) ) ) ).

% snd_map_prod
thf(fact_770_fst__apfst,axiom,
    ! [F2: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm,X2: produc164195504107695125m_rule] :
      ( ( produc6879501374131015971m_rule @ ( produc3731403639160201281m_rule @ F2 @ X2 ) )
      = ( F2 @ ( produc6879501374131015971m_rule @ X2 ) ) ) ).

% fst_apfst
thf(fact_771_fst__apfst,axiom,
    ! [F2: list_fm > list_fm,X2: produc1996495991257130529ist_fm] :
      ( ( produc1501393135466168645ist_fm @ ( produc1130293176790411102ist_fm @ F2 @ X2 ) )
      = ( F2 @ ( produc1501393135466168645ist_fm @ X2 ) ) ) ).

% fst_apfst
thf(fact_772_snd__apfst,axiom,
    ! [F2: list_fm > list_fm,X2: produc1996495991257130529ist_fm] :
      ( ( produc4588648349897876871ist_fm @ ( produc1130293176790411102ist_fm @ F2 @ X2 ) )
      = ( produc4588648349897876871ist_fm @ X2 ) ) ).

% snd_apfst
thf(fact_773_snd__apfst,axiom,
    ! [F2: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm,X2: produc164195504107695125m_rule] :
      ( ( produc7165828336582415457m_rule @ ( produc3731403639160201281m_rule @ F2 @ X2 ) )
      = ( produc7165828336582415457m_rule @ X2 ) ) ).

% snd_apfst
thf(fact_774_snd__apfst,axiom,
    ! [F2: rule > rule,X2: produc4630866025709511057ist_fm] :
      ( ( produc4514744107061996309ist_fm @ ( produc5534506254562563289ist_fm @ F2 @ X2 ) )
      = ( produc4514744107061996309ist_fm @ X2 ) ) ).

% snd_apfst
thf(fact_775_fst__apsnd,axiom,
    ! [F2: rule > rule,X2: produc164195504107695125m_rule] :
      ( ( produc6879501374131015971m_rule @ ( produc3063135074001183255ist_fm @ F2 @ X2 ) )
      = ( produc6879501374131015971m_rule @ X2 ) ) ).

% fst_apsnd
thf(fact_776_fst__apsnd,axiom,
    ! [F2: list_fm > list_fm,X2: produc1996495991257130529ist_fm] :
      ( ( produc1501393135466168645ist_fm @ ( produc9047611296996456864ist_fm @ F2 @ X2 ) )
      = ( produc1501393135466168645ist_fm @ X2 ) ) ).

% fst_apsnd
thf(fact_777_snd__apsnd,axiom,
    ! [F2: list_fm > list_fm,X2: produc1996495991257130529ist_fm] :
      ( ( produc4588648349897876871ist_fm @ ( produc9047611296996456864ist_fm @ F2 @ X2 ) )
      = ( F2 @ ( produc4588648349897876871ist_fm @ X2 ) ) ) ).

% snd_apsnd
thf(fact_778_snd__apsnd,axiom,
    ! [F2: rule > rule,X2: produc164195504107695125m_rule] :
      ( ( produc7165828336582415457m_rule @ ( produc3063135074001183255ist_fm @ F2 @ X2 ) )
      = ( F2 @ ( produc7165828336582415457m_rule @ X2 ) ) ) ).

% snd_apsnd
thf(fact_779_snd__apsnd,axiom,
    ! [F2: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm,X2: produc4630866025709511057ist_fm] :
      ( ( produc4514744107061996309ist_fm @ ( produc4332487942798490499m_rule @ F2 @ X2 ) )
      = ( F2 @ ( produc4514744107061996309ist_fm @ X2 ) ) ) ).

% snd_apsnd
thf(fact_780_range__fst,axiom,
    ( ( image_3223098899990291461ist_fm @ produc6879501374131015971m_rule @ top_to1603951219661559419m_rule )
    = top_to728902761071950545ist_fm ) ).

% range_fst
thf(fact_781_range__fst,axiom,
    ( ( image_6369949096775035324ist_fm @ produc1501393135466168645ist_fm @ top_to728902761071950545ist_fm )
    = top_top_set_list_fm ) ).

% range_fst
thf(fact_782_range__snd,axiom,
    ( ( image_6369949096775035324ist_fm @ produc4588648349897876871ist_fm @ top_to728902761071950545ist_fm )
    = top_top_set_list_fm ) ).

% range_snd
thf(fact_783_range__snd,axiom,
    ( ( image_8178078329501008481e_rule @ produc7165828336582415457m_rule @ top_to1603951219661559419m_rule )
    = top_top_set_rule ) ).

% range_snd
thf(fact_784_range__snd,axiom,
    ( ( image_3681455213663431561ist_fm @ produc4514744107061996309ist_fm @ top_to2837172092223966711ist_fm )
    = top_to728902761071950545ist_fm ) ).

% range_snd
thf(fact_785_prod_Opred__map,axiom,
    ! [Q1: list_fm > $o,Q22: list_fm > $o,F12: list_fm > list_fm,F23: list_fm > list_fm,X2: produc1996495991257130529ist_fm] :
      ( ( basic_5093365364719461109ist_fm @ Q1 @ Q22 @ ( produc3421166768339006330ist_fm @ F12 @ F23 @ X2 ) )
      = ( basic_5093365364719461109ist_fm @ ( comp_l3888558907433981941ist_fm @ Q1 @ F12 ) @ ( comp_l3888558907433981941ist_fm @ Q22 @ F23 ) @ X2 ) ) ).

% prod.pred_map
thf(fact_786_prod_Opred__map,axiom,
    ! [Q1: list_fm > $o,Q22: list_fm > $o,F12: produc1996495991257130529ist_fm > list_fm,F23: rule > list_fm,X2: produc164195504107695125m_rule] :
      ( ( basic_5093365364719461109ist_fm @ Q1 @ Q22 @ ( produc1965614570530558770ist_fm @ F12 @ F23 @ X2 ) )
      = ( basic_6171360182465397619m_rule @ ( comp_l5105733579298676940ist_fm @ Q1 @ F12 ) @ ( comp_list_fm_o_rule @ Q22 @ F23 ) @ X2 ) ) ).

% prod.pred_map
thf(fact_787_prod_Opred__map,axiom,
    ! [Q1: produc1996495991257130529ist_fm > $o,Q22: rule > $o,F12: list_fm > produc1996495991257130529ist_fm,F23: list_fm > rule,X2: produc1996495991257130529ist_fm] :
      ( ( basic_6171360182465397619m_rule @ Q1 @ Q22 @ ( produc8124276167727720944m_rule @ F12 @ F23 @ X2 ) )
      = ( basic_5093365364719461109ist_fm @ ( comp_P4464886101349820410ist_fm @ Q1 @ F12 ) @ ( comp_rule_o_list_fm @ Q22 @ F23 ) @ X2 ) ) ).

% prod.pred_map
thf(fact_788_prod_Opred__map,axiom,
    ! [Q1: produc1996495991257130529ist_fm > $o,Q22: rule > $o,F12: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm,F23: rule > rule,X2: produc164195504107695125m_rule] :
      ( ( basic_6171360182465397619m_rule @ Q1 @ Q22 @ ( produc7680103964935106114e_rule @ F12 @ F23 @ X2 ) )
      = ( basic_6171360182465397619m_rule @ ( comp_P1005906153719353223ist_fm @ Q1 @ F12 ) @ ( comp_rule_o_rule @ Q22 @ F23 ) @ X2 ) ) ).

% prod.pred_map
thf(fact_789_map__prod__surj,axiom,
    ! [F2: produc164195504107695125m_rule > set_fm,G: produc164195504107695125m_rule > set_fm] :
      ( ( ( image_5698427363735666338set_fm @ F2 @ top_to1603951219661559419m_rule )
        = top_top_set_set_fm )
     => ( ( ( image_5698427363735666338set_fm @ G @ top_to1603951219661559419m_rule )
          = top_top_set_set_fm )
       => ( ( image_479386260607784615set_fm @ ( produc8521236829880132656set_fm @ F2 @ G ) @ top_to3650589660821865915m_rule )
          = top_to5621610097446368925set_fm ) ) ) ).

% map_prod_surj
thf(fact_790_fun_Omap__ident__strong,axiom,
    ! [T2: produc164195504107695125m_rule > set_fm,F2: set_fm > set_fm] :
      ( ! [Z3: set_fm] :
          ( ( member_set_fm @ Z3 @ ( image_5698427363735666338set_fm @ T2 @ top_to1603951219661559419m_rule ) )
         => ( ( F2 @ Z3 )
            = Z3 ) )
     => ( ( comp_s4688689533955628550m_rule @ F2 @ T2 )
        = T2 ) ) ).

% fun.map_ident_strong
thf(fact_791_fun_Oset__map,axiom,
    ! [F2: set_fm > set_fm,V: produc164195504107695125m_rule > set_fm] :
      ( ( image_5698427363735666338set_fm @ ( comp_s4688689533955628550m_rule @ F2 @ V ) @ top_to1603951219661559419m_rule )
      = ( image_set_fm_set_fm @ F2 @ ( image_5698427363735666338set_fm @ V @ top_to1603951219661559419m_rule ) ) ) ).

% fun.set_map
thf(fact_792_fun_Oset__map,axiom,
    ! [F2: produc164195504107695125m_rule > set_fm,V: produc164195504107695125m_rule > produc164195504107695125m_rule] :
      ( ( image_5698427363735666338set_fm @ ( comp_P2766130934352412053m_rule @ F2 @ V ) @ top_to1603951219661559419m_rule )
      = ( image_5698427363735666338set_fm @ F2 @ ( image_2192725901198026033m_rule @ V @ top_to1603951219661559419m_rule ) ) ) ).

% fun.set_map
thf(fact_793_subset__code_I1_J,axiom,
    ! [Xs: list_fm,B5: set_fm] :
      ( ( ord_less_eq_set_fm @ ( set_fm2 @ Xs ) @ B5 )
      = ( ! [X3: fm] :
            ( ( member_fm @ X3 @ ( set_fm2 @ Xs ) )
           => ( member_fm @ X3 @ B5 ) ) ) ) ).

% subset_code(1)
thf(fact_794_stream_Omap__sel_I1_J,axiom,
    ! [F2: produc164195504107695125m_rule > produc164195504107695125m_rule,A: stream727092118206550309m_rule] :
      ( ( shd_Pr7235097944458474089m_rule @ ( smap_P743327730374515377m_rule @ F2 @ A ) )
      = ( F2 @ ( shd_Pr7235097944458474089m_rule @ A ) ) ) ).

% stream.map_sel(1)
thf(fact_795_sdrop__smap,axiom,
    ! [N2: nat,F2: rule > rule,S3: stream_rule] :
      ( ( sdrop_rule @ N2 @ ( smap_rule_rule @ F2 @ S3 ) )
      = ( smap_rule_rule @ F2 @ ( sdrop_rule @ N2 @ S3 ) ) ) ).

% sdrop_smap
thf(fact_796_sdrop__smap,axiom,
    ! [N2: nat,F2: rule > produc1996495991257130529ist_fm,S3: stream_rule] :
      ( ( sdrop_4442373711808556042ist_fm @ N2 @ ( smap_r7244373527995750997ist_fm @ F2 @ S3 ) )
      = ( smap_r7244373527995750997ist_fm @ F2 @ ( sdrop_rule @ N2 @ S3 ) ) ) ).

% sdrop_smap
thf(fact_797_sdrop__smap,axiom,
    ! [N2: nat,F2: produc1996495991257130529ist_fm > rule,S3: stream8299795917829157543ist_fm] :
      ( ( sdrop_rule @ N2 @ ( smap_P672085720661394337m_rule @ F2 @ S3 ) )
      = ( smap_P672085720661394337m_rule @ F2 @ ( sdrop_4442373711808556042ist_fm @ N2 @ S3 ) ) ) ).

% sdrop_smap
thf(fact_798_sdrop__smap,axiom,
    ! [N2: nat,F2: rule > produc164195504107695125m_rule,S3: stream_rule] :
      ( ( sdrop_7224736112439592940m_rule @ N2 @ ( smap_r3539241683680467681m_rule @ F2 @ S3 ) )
      = ( smap_r3539241683680467681m_rule @ F2 @ ( sdrop_rule @ N2 @ S3 ) ) ) ).

% sdrop_smap
thf(fact_799_sdrop__smap,axiom,
    ! [N2: nat,F2: produc164195504107695125m_rule > rule,S3: stream727092118206550309m_rule] :
      ( ( sdrop_rule @ N2 @ ( smap_P1838418287296133089e_rule @ F2 @ S3 ) )
      = ( smap_P1838418287296133089e_rule @ F2 @ ( sdrop_7224736112439592940m_rule @ N2 @ S3 ) ) ) ).

% sdrop_smap
thf(fact_800_sdrop__smap,axiom,
    ! [N2: nat,F2: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm,S3: stream8299795917829157543ist_fm] :
      ( ( sdrop_4442373711808556042ist_fm @ N2 @ ( smap_P2116677042904502981ist_fm @ F2 @ S3 ) )
      = ( smap_P2116677042904502981ist_fm @ F2 @ ( sdrop_4442373711808556042ist_fm @ N2 @ S3 ) ) ) ).

% sdrop_smap
thf(fact_801_sdrop__smap,axiom,
    ! [N2: nat,F2: produc1996495991257130529ist_fm > produc164195504107695125m_rule,S3: stream8299795917829157543ist_fm] :
      ( ( sdrop_7224736112439592940m_rule @ N2 @ ( smap_P3231557233864197745m_rule @ F2 @ S3 ) )
      = ( smap_P3231557233864197745m_rule @ F2 @ ( sdrop_4442373711808556042ist_fm @ N2 @ S3 ) ) ) ).

% sdrop_smap
thf(fact_802_sdrop__smap,axiom,
    ! [N2: nat,F2: produc164195504107695125m_rule > produc1996495991257130529ist_fm,S3: stream727092118206550309m_rule] :
      ( ( sdrop_4442373711808556042ist_fm @ N2 @ ( smap_P2929523447053962885ist_fm @ F2 @ S3 ) )
      = ( smap_P2929523447053962885ist_fm @ F2 @ ( sdrop_7224736112439592940m_rule @ N2 @ S3 ) ) ) ).

% sdrop_smap
thf(fact_803_sdrop__smap,axiom,
    ! [N2: nat,F2: produc164195504107695125m_rule > produc164195504107695125m_rule,S3: stream727092118206550309m_rule] :
      ( ( sdrop_7224736112439592940m_rule @ N2 @ ( smap_P743327730374515377m_rule @ F2 @ S3 ) )
      = ( smap_P743327730374515377m_rule @ F2 @ ( sdrop_7224736112439592940m_rule @ N2 @ S3 ) ) ) ).

% sdrop_smap
thf(fact_804_sdrop__smap,axiom,
    ! [N2: nat,F2: produc3793571666044074251m_rule > rule,S3: stream6494289010434245521m_rule] :
      ( ( sdrop_rule @ N2 @ ( smap_P4345194815984152203e_rule @ F2 @ S3 ) )
      = ( smap_P4345194815984152203e_rule @ F2 @ ( sdrop_7373388980841566196m_rule @ N2 @ S3 ) ) ) ).

% sdrop_smap
thf(fact_805_snth__smap,axiom,
    ! [F2: produc164195504107695125m_rule > produc164195504107695125m_rule,S3: stream727092118206550309m_rule,N2: nat] :
      ( ( snth_P1712579846118164081m_rule @ ( smap_P743327730374515377m_rule @ F2 @ S3 ) @ N2 )
      = ( F2 @ ( snth_P1712579846118164081m_rule @ S3 @ N2 ) ) ) ).

% snth_smap
thf(fact_806_prod_Opred__mono,axiom,
    ! [P1: list_fm > $o,P1a: list_fm > $o,P22: list_fm > $o,P2a: list_fm > $o] :
      ( ( ord_le7698008633383072527t_fm_o @ P1 @ P1a )
     => ( ( ord_le7698008633383072527t_fm_o @ P22 @ P2a )
       => ( ord_le8518180823940269308t_fm_o @ ( basic_5093365364719461109ist_fm @ P1 @ P22 ) @ ( basic_5093365364719461109ist_fm @ P1a @ P2a ) ) ) ) ).

% prod.pred_mono
thf(fact_807_prod_Opred__mono,axiom,
    ! [P1: produc1996495991257130529ist_fm > $o,P1a: produc1996495991257130529ist_fm > $o,P22: rule > $o,P2a: rule > $o] :
      ( ( ord_le8518180823940269308t_fm_o @ P1 @ P1a )
     => ( ( ord_less_eq_rule_o @ P22 @ P2a )
       => ( ord_le37115029089170458rule_o @ ( basic_6171360182465397619m_rule @ P1 @ P22 ) @ ( basic_6171360182465397619m_rule @ P1a @ P2a ) ) ) ) ).

% prod.pred_mono
thf(fact_808_smap__alt,axiom,
    ! [F2: produc164195504107695125m_rule > produc164195504107695125m_rule,S3: stream727092118206550309m_rule,S4: stream727092118206550309m_rule] :
      ( ( ( smap_P743327730374515377m_rule @ F2 @ S3 )
        = S4 )
      = ( ! [N3: nat] :
            ( ( F2 @ ( snth_P1712579846118164081m_rule @ S3 @ N3 ) )
            = ( snth_P1712579846118164081m_rule @ S4 @ N3 ) ) ) ) ).

% smap_alt
thf(fact_809_fun_Orel__refl__strong,axiom,
    ! [X2: produc164195504107695125m_rule > set_fm,Ra: set_fm > set_fm > $o] :
      ( ! [Z3: set_fm] :
          ( ( member_set_fm @ Z3 @ ( image_5698427363735666338set_fm @ X2 @ top_to1603951219661559419m_rule ) )
         => ( Ra @ Z3 @ Z3 ) )
     => ( bNF_re8160694294360540439set_fm
        @ ^ [Y2: produc164195504107695125m_rule,Z: produc164195504107695125m_rule] : ( Y2 = Z )
        @ Ra
        @ X2
        @ X2 ) ) ).

% fun.rel_refl_strong
thf(fact_810_fun_Orel__mono__strong,axiom,
    ! [R: set_fm > set_fm > $o,X2: produc164195504107695125m_rule > set_fm,Y3: produc164195504107695125m_rule > set_fm,Ra: set_fm > set_fm > $o] :
      ( ( bNF_re8160694294360540439set_fm
        @ ^ [Y2: produc164195504107695125m_rule,Z: produc164195504107695125m_rule] : ( Y2 = Z )
        @ R
        @ X2
        @ Y3 )
     => ( ! [Z3: set_fm,Yb: set_fm] :
            ( ( member_set_fm @ Z3 @ ( image_5698427363735666338set_fm @ X2 @ top_to1603951219661559419m_rule ) )
           => ( ( member_set_fm @ Yb @ ( image_5698427363735666338set_fm @ Y3 @ top_to1603951219661559419m_rule ) )
             => ( ( R @ Z3 @ Yb )
               => ( Ra @ Z3 @ Yb ) ) ) )
       => ( bNF_re8160694294360540439set_fm
          @ ^ [Y2: produc164195504107695125m_rule,Z: produc164195504107695125m_rule] : ( Y2 = Z )
          @ Ra
          @ X2
          @ Y3 ) ) ) ).

% fun.rel_mono_strong
thf(fact_811_fun_Orel__mono__strong,axiom,
    ! [R: set_fm > fm > $o,X2: produc164195504107695125m_rule > set_fm,Y3: produc164195504107695125m_rule > fm,Ra: set_fm > fm > $o] :
      ( ( bNF_re118521385745904353_fm_fm
        @ ^ [Y2: produc164195504107695125m_rule,Z: produc164195504107695125m_rule] : ( Y2 = Z )
        @ R
        @ X2
        @ Y3 )
     => ( ! [Z3: set_fm,Yb: fm] :
            ( ( member_set_fm @ Z3 @ ( image_5698427363735666338set_fm @ X2 @ top_to1603951219661559419m_rule ) )
           => ( ( member_fm @ Yb @ ( image_372001859017726060ule_fm @ Y3 @ top_to1603951219661559419m_rule ) )
             => ( ( R @ Z3 @ Yb )
               => ( Ra @ Z3 @ Yb ) ) ) )
       => ( bNF_re118521385745904353_fm_fm
          @ ^ [Y2: produc164195504107695125m_rule,Z: produc164195504107695125m_rule] : ( Y2 = Z )
          @ Ra
          @ X2
          @ Y3 ) ) ) ).

% fun.rel_mono_strong
thf(fact_812_fun_Orel__mono__strong,axiom,
    ! [R: fm > set_fm > $o,X2: produc164195504107695125m_rule > fm,Y3: produc164195504107695125m_rule > set_fm,Ra: fm > set_fm > $o] :
      ( ( bNF_re1875054133227635169set_fm
        @ ^ [Y2: produc164195504107695125m_rule,Z: produc164195504107695125m_rule] : ( Y2 = Z )
        @ R
        @ X2
        @ Y3 )
     => ( ! [Z3: fm,Yb: set_fm] :
            ( ( member_fm @ Z3 @ ( image_372001859017726060ule_fm @ X2 @ top_to1603951219661559419m_rule ) )
           => ( ( member_set_fm @ Yb @ ( image_5698427363735666338set_fm @ Y3 @ top_to1603951219661559419m_rule ) )
             => ( ( R @ Z3 @ Yb )
               => ( Ra @ Z3 @ Yb ) ) ) )
       => ( bNF_re1875054133227635169set_fm
          @ ^ [Y2: produc164195504107695125m_rule,Z: produc164195504107695125m_rule] : ( Y2 = Z )
          @ Ra
          @ X2
          @ Y3 ) ) ) ).

% fun.rel_mono_strong
thf(fact_813_fun_Orel__cong,axiom,
    ! [X2: produc164195504107695125m_rule > set_fm,Ya: produc164195504107695125m_rule > set_fm,Y3: produc164195504107695125m_rule > set_fm,Xa2: produc164195504107695125m_rule > set_fm,R: set_fm > set_fm > $o,Ra: set_fm > set_fm > $o] :
      ( ( X2 = Ya )
     => ( ( Y3 = Xa2 )
       => ( ! [Z3: set_fm,Yb: set_fm] :
              ( ( member_set_fm @ Z3 @ ( image_5698427363735666338set_fm @ Ya @ top_to1603951219661559419m_rule ) )
             => ( ( member_set_fm @ Yb @ ( image_5698427363735666338set_fm @ Xa2 @ top_to1603951219661559419m_rule ) )
               => ( ( R @ Z3 @ Yb )
                  = ( Ra @ Z3 @ Yb ) ) ) )
         => ( ( bNF_re8160694294360540439set_fm
              @ ^ [Y2: produc164195504107695125m_rule,Z: produc164195504107695125m_rule] : ( Y2 = Z )
              @ R
              @ X2
              @ Y3 )
            = ( bNF_re8160694294360540439set_fm
              @ ^ [Y2: produc164195504107695125m_rule,Z: produc164195504107695125m_rule] : ( Y2 = Z )
              @ Ra
              @ Ya
              @ Xa2 ) ) ) ) ) ).

% fun.rel_cong
thf(fact_814_fun_Orel__cong,axiom,
    ! [X2: produc164195504107695125m_rule > set_fm,Ya: produc164195504107695125m_rule > set_fm,Y3: produc164195504107695125m_rule > fm,Xa2: produc164195504107695125m_rule > fm,R: set_fm > fm > $o,Ra: set_fm > fm > $o] :
      ( ( X2 = Ya )
     => ( ( Y3 = Xa2 )
       => ( ! [Z3: set_fm,Yb: fm] :
              ( ( member_set_fm @ Z3 @ ( image_5698427363735666338set_fm @ Ya @ top_to1603951219661559419m_rule ) )
             => ( ( member_fm @ Yb @ ( image_372001859017726060ule_fm @ Xa2 @ top_to1603951219661559419m_rule ) )
               => ( ( R @ Z3 @ Yb )
                  = ( Ra @ Z3 @ Yb ) ) ) )
         => ( ( bNF_re118521385745904353_fm_fm
              @ ^ [Y2: produc164195504107695125m_rule,Z: produc164195504107695125m_rule] : ( Y2 = Z )
              @ R
              @ X2
              @ Y3 )
            = ( bNF_re118521385745904353_fm_fm
              @ ^ [Y2: produc164195504107695125m_rule,Z: produc164195504107695125m_rule] : ( Y2 = Z )
              @ Ra
              @ Ya
              @ Xa2 ) ) ) ) ) ).

% fun.rel_cong
thf(fact_815_fun_Orel__cong,axiom,
    ! [X2: produc164195504107695125m_rule > fm,Ya: produc164195504107695125m_rule > fm,Y3: produc164195504107695125m_rule > set_fm,Xa2: produc164195504107695125m_rule > set_fm,R: fm > set_fm > $o,Ra: fm > set_fm > $o] :
      ( ( X2 = Ya )
     => ( ( Y3 = Xa2 )
       => ( ! [Z3: fm,Yb: set_fm] :
              ( ( member_fm @ Z3 @ ( image_372001859017726060ule_fm @ Ya @ top_to1603951219661559419m_rule ) )
             => ( ( member_set_fm @ Yb @ ( image_5698427363735666338set_fm @ Xa2 @ top_to1603951219661559419m_rule ) )
               => ( ( R @ Z3 @ Yb )
                  = ( Ra @ Z3 @ Yb ) ) ) )
         => ( ( bNF_re1875054133227635169set_fm
              @ ^ [Y2: produc164195504107695125m_rule,Z: produc164195504107695125m_rule] : ( Y2 = Z )
              @ R
              @ X2
              @ Y3 )
            = ( bNF_re1875054133227635169set_fm
              @ ^ [Y2: produc164195504107695125m_rule,Z: produc164195504107695125m_rule] : ( Y2 = Z )
              @ Ra
              @ Ya
              @ Xa2 ) ) ) ) ) ).

% fun.rel_cong
thf(fact_816_size__prod__simp,axiom,
    ( basic_7103594793270586102ist_fm
    = ( ^ [F: list_fm > nat,G3: list_fm > nat,P4: produc1996495991257130529ist_fm] : ( plus_plus_nat @ ( plus_plus_nat @ ( F @ ( produc1501393135466168645ist_fm @ P4 ) ) @ ( G3 @ ( produc4588648349897876871ist_fm @ P4 ) ) ) @ ( suc @ zero_zero_nat ) ) ) ) ).

% size_prod_simp
thf(fact_817_size__prod__simp,axiom,
    ( basic_7266071961737426738m_rule
    = ( ^ [F: produc1996495991257130529ist_fm > nat,G3: rule > nat,P4: produc164195504107695125m_rule] : ( plus_plus_nat @ ( plus_plus_nat @ ( F @ ( produc6879501374131015971m_rule @ P4 ) ) @ ( G3 @ ( produc7165828336582415457m_rule @ P4 ) ) ) @ ( suc @ zero_zero_nat ) ) ) ) ).

% size_prod_simp
thf(fact_818_size__prod__simp,axiom,
    ( basic_4614987732217007590ist_fm
    = ( ^ [F: rule > nat,G3: produc1996495991257130529ist_fm > nat,P4: produc4630866025709511057ist_fm] : ( plus_plus_nat @ ( plus_plus_nat @ ( F @ ( produc4228417144610596823ist_fm @ P4 ) ) @ ( G3 @ ( produc4514744107061996309ist_fm @ P4 ) ) ) @ ( suc @ zero_zero_nat ) ) ) ) ).

% size_prod_simp
thf(fact_819_count__notin,axiom,
    ! [X2: fm,Xs: list_fm] :
      ( ~ ( member_fm @ X2 @ ( set_fm2 @ Xs ) )
     => ( ( count_list_fm @ Xs @ X2 )
        = zero_zero_nat ) ) ).

% count_notin
thf(fact_820_count__list__0__iff,axiom,
    ! [Xs: list_fm,X2: fm] :
      ( ( ( count_list_fm @ Xs @ X2 )
        = zero_zero_nat )
      = ( ~ ( member_fm @ X2 @ ( set_fm2 @ Xs ) ) ) ) ).

% count_list_0_iff
thf(fact_821_swap__comp__swap,axiom,
    ( ( comp_P1170188041075780286ist_fm @ produc7936585717479832313ist_fm @ produc7936585717479832313ist_fm )
    = id_Pro7921797020193973010ist_fm ) ).

% swap_comp_swap
thf(fact_822_swap__comp__swap,axiom,
    ( ( comp_P3637575036371394080m_rule @ produc9213896635423732003ist_fm @ produc2641608828089375343m_rule )
    = id_Pro7923832310103879396m_rule ) ).

% swap_comp_swap
thf(fact_823_swap__comp__swap,axiom,
    ( ( comp_P9090089137346658716ist_fm @ produc2641608828089375343m_rule @ produc9213896635423732003ist_fm )
    = id_Pro3167130794850919520ist_fm ) ).

% swap_comp_swap
thf(fact_824_stream__smap__nats,axiom,
    ! [S3: stream727092118206550309m_rule] :
      ( S3
      = ( smap_n3801058020469267436m_rule @ ( snth_P1712579846118164081m_rule @ S3 ) @ ( siterate_nat @ suc @ zero_zero_nat ) ) ) ).

% stream_smap_nats
thf(fact_825_pair__imageI,axiom,
    ! [A: produc1996495991257130529ist_fm,B: rule,A4: set_Pr1008144964186165195m_rule,F2: produc1996495991257130529ist_fm > rule > set_fm] :
      ( ( member4220325220686508332m_rule @ ( produc491467635432902671m_rule @ A @ B ) @ A4 )
     => ( member_set_fm @ ( F2 @ A @ B ) @ ( image_5698427363735666338set_fm @ ( produc966423478193074626set_fm @ F2 ) @ A4 ) ) ) ).

% pair_imageI
thf(fact_826_subset__Collect__iff,axiom,
    ! [B5: set_fm,A4: set_fm,P2: fm > $o] :
      ( ( ord_less_eq_set_fm @ B5 @ A4 )
     => ( ( ord_less_eq_set_fm @ B5
          @ ( collect_fm
            @ ^ [X3: fm] :
                ( ( member_fm @ X3 @ A4 )
                & ( P2 @ X3 ) ) ) )
        = ( ! [X3: fm] :
              ( ( member_fm @ X3 @ B5 )
             => ( P2 @ X3 ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_827_subset__CollectI,axiom,
    ! [B5: set_fm,A4: set_fm,Q3: fm > $o,P2: fm > $o] :
      ( ( ord_less_eq_set_fm @ B5 @ A4 )
     => ( ! [X: fm] :
            ( ( member_fm @ X @ B5 )
           => ( ( Q3 @ X )
             => ( P2 @ X ) ) )
       => ( ord_less_eq_set_fm
          @ ( collect_fm
            @ ^ [X3: fm] :
                ( ( member_fm @ X3 @ B5 )
                & ( Q3 @ X3 ) ) )
          @ ( collect_fm
            @ ^ [X3: fm] :
                ( ( member_fm @ X3 @ A4 )
                & ( P2 @ X3 ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_828_Collect__restrict,axiom,
    ! [X5: set_fm,P2: fm > $o] :
      ( ord_less_eq_set_fm
      @ ( collect_fm
        @ ^ [X3: fm] :
            ( ( member_fm @ X3 @ X5 )
            & ( P2 @ X3 ) ) )
      @ X5 ) ).

% Collect_restrict
thf(fact_829_prop__restrict,axiom,
    ! [X2: fm,Z4: set_fm,X5: set_fm,P2: fm > $o] :
      ( ( member_fm @ X2 @ Z4 )
     => ( ( ord_less_eq_set_fm @ Z4
          @ ( collect_fm
            @ ^ [X3: fm] :
                ( ( member_fm @ X3 @ X5 )
                & ( P2 @ X3 ) ) ) )
       => ( P2 @ X2 ) ) ) ).

% prop_restrict
thf(fact_830_prod_Opred__True,axiom,
    ( ( basic_5093365364719461109ist_fm
      @ ^ [Uu: list_fm] : $true
      @ ^ [Uu: list_fm] : $true )
    = ( ^ [Uu: produc1996495991257130529ist_fm] : $true ) ) ).

% prod.pred_True
thf(fact_831_prod_Opred__True,axiom,
    ( ( basic_6171360182465397619m_rule
      @ ^ [Uu: produc1996495991257130529ist_fm] : $true
      @ ^ [Uu: rule] : $true )
    = ( ^ [Uu: produc164195504107695125m_rule] : $true ) ) ).

% prod.pred_True
thf(fact_832_image__Collect__subsetI,axiom,
    ! [P2: produc164195504107695125m_rule > $o,F2: produc164195504107695125m_rule > set_fm,B5: set_set_fm] :
      ( ! [X: produc164195504107695125m_rule] :
          ( ( P2 @ X )
         => ( member_set_fm @ ( F2 @ X ) @ B5 ) )
     => ( ord_le1461404734466536732set_fm @ ( image_5698427363735666338set_fm @ F2 @ ( collec2069506698588392810m_rule @ P2 ) ) @ B5 ) ) ).

% image_Collect_subsetI
thf(fact_833_fst__diag__id,axiom,
    ! [Z2: list_fm] :
      ( ( comp_P3914610931391992256ist_fm @ produc1501393135466168645ist_fm
        @ ^ [X3: list_fm] : ( produc381145313068854617ist_fm @ X3 @ X3 )
        @ Z2 )
      = ( id_list_fm @ Z2 ) ) ).

% fst_diag_id
thf(fact_834_snd__diag__id,axiom,
    ! [Z2: list_fm] :
      ( ( comp_P3914610931391992256ist_fm @ produc4588648349897876871ist_fm
        @ ^ [X3: list_fm] : ( produc381145313068854617ist_fm @ X3 @ X3 )
        @ Z2 )
      = ( id_list_fm @ Z2 ) ) ).

% snd_diag_id
thf(fact_835_fst__def,axiom,
    ( produc6879501374131015971m_rule
    = ( produc8807799049144449253ist_fm
      @ ^ [X12: produc1996495991257130529ist_fm,X23: rule] : X12 ) ) ).

% fst_def
thf(fact_836_fst__def,axiom,
    ( produc1501393135466168645ist_fm
    = ( produc4154610405322092878ist_fm
      @ ^ [X12: list_fm,X23: list_fm] : X12 ) ) ).

% fst_def
thf(fact_837_snd__def,axiom,
    ( produc4588648349897876871ist_fm
    = ( produc4154610405322092878ist_fm
      @ ^ [X12: list_fm,X23: list_fm] : X23 ) ) ).

% snd_def
thf(fact_838_snd__def,axiom,
    ( produc7165828336582415457m_rule
    = ( produc1213517383330063745e_rule
      @ ^ [X12: produc1996495991257130529ist_fm,X23: rule] : X23 ) ) ).

% snd_def
thf(fact_839_snd__def,axiom,
    ( produc4514744107061996309ist_fm
    = ( produc2632167636358383921ist_fm
      @ ^ [X12: rule,X23: produc1996495991257130529ist_fm] : X23 ) ) ).

% snd_def
thf(fact_840_pred__subset__eq,axiom,
    ! [R: set_fm,S6: set_fm] :
      ( ( ord_less_eq_fm_o
        @ ^ [X3: fm] : ( member_fm @ X3 @ R )
        @ ^ [X3: fm] : ( member_fm @ X3 @ S6 ) )
      = ( ord_less_eq_set_fm @ R @ S6 ) ) ).

% pred_subset_eq
thf(fact_841_siterate_Osimps_I1_J,axiom,
    ! [F2: produc164195504107695125m_rule > produc164195504107695125m_rule,X2: produc164195504107695125m_rule] :
      ( ( shd_Pr7235097944458474089m_rule @ ( sitera7638444125321546697m_rule @ F2 @ X2 ) )
      = X2 ) ).

% siterate.simps(1)
thf(fact_842_snd__fst__flip,axiom,
    ( produc4588648349897876871ist_fm
    = ( comp_P1426762558556845121ist_fm @ produc1501393135466168645ist_fm
      @ ( produc4544959197559615795ist_fm
        @ ^ [X3: list_fm,Y4: list_fm] : ( produc381145313068854617ist_fm @ Y4 @ X3 ) ) ) ) ).

% snd_fst_flip
thf(fact_843_snd__fst__flip,axiom,
    ( produc7165828336582415457m_rule
    = ( comp_P3413117018928051920m_rule @ produc4228417144610596823ist_fm
      @ ( produc1775668667622460877ist_fm
        @ ^ [X3: produc1996495991257130529ist_fm,Y4: rule] : ( produc7063755442767259331ist_fm @ Y4 @ X3 ) ) ) ) ).

% snd_fst_flip
thf(fact_844_snd__fst__flip,axiom,
    ( produc4514744107061996309ist_fm
    = ( comp_P2707267495825058490ist_fm @ produc6879501374131015971m_rule
      @ ( produc2185035559847904517m_rule
        @ ^ [X3: rule,Y4: produc1996495991257130529ist_fm] : ( produc491467635432902671m_rule @ Y4 @ X3 ) ) ) ) ).

% snd_fst_flip
thf(fact_845_fst__snd__flip,axiom,
    ( produc1501393135466168645ist_fm
    = ( comp_P1426762558556845121ist_fm @ produc4588648349897876871ist_fm
      @ ( produc4544959197559615795ist_fm
        @ ^ [X3: list_fm,Y4: list_fm] : ( produc381145313068854617ist_fm @ Y4 @ X3 ) ) ) ) ).

% fst_snd_flip
thf(fact_846_fst__snd__flip,axiom,
    ( produc4228417144610596823ist_fm
    = ( comp_P4888803072877744336ist_fm @ produc7165828336582415457m_rule
      @ ( produc2185035559847904517m_rule
        @ ^ [X3: rule,Y4: produc1996495991257130529ist_fm] : ( produc491467635432902671m_rule @ Y4 @ X3 ) ) ) ) ).

% fst_snd_flip
thf(fact_847_fst__snd__flip,axiom,
    ( produc6879501374131015971m_rule
    = ( comp_P6646430376872094402m_rule @ produc4514744107061996309ist_fm
      @ ( produc1775668667622460877ist_fm
        @ ^ [X3: produc1996495991257130529ist_fm,Y4: rule] : ( produc7063755442767259331ist_fm @ Y4 @ X3 ) ) ) ) ).

% fst_snd_flip
thf(fact_848_fst__diag__fst,axiom,
    ( ( comp_P1085356462900927218m_rule @ produc5894184038225292483ist_fm
      @ ( comp_P2017078995931194366m_rule
        @ ^ [X3: produc1996495991257130529ist_fm] : ( produc2592314732767339223ist_fm @ X3 @ X3 )
        @ produc6879501374131015971m_rule ) )
    = produc6879501374131015971m_rule ) ).

% fst_diag_fst
thf(fact_849_fst__diag__fst,axiom,
    ( ( comp_P1426762558556845121ist_fm @ produc1501393135466168645ist_fm
      @ ( comp_l2016392514137073923ist_fm
        @ ^ [X3: list_fm] : ( produc381145313068854617ist_fm @ X3 @ X3 )
        @ produc1501393135466168645ist_fm ) )
    = produc1501393135466168645ist_fm ) ).

% fst_diag_fst
thf(fact_850_snd__diag__snd,axiom,
    ( ( comp_P2548450393677710116m_rule @ produc6725306161561501521e_rule
      @ ( comp_r8929483813444710872m_rule
        @ ^ [X3: rule] : ( produc1769597910952761123e_rule @ X3 @ X3 )
        @ produc7165828336582415457m_rule ) )
    = produc7165828336582415457m_rule ) ).

% snd_diag_snd
thf(fact_851_snd__diag__snd,axiom,
    ( ( comp_P5552026984502743150ist_fm @ produc1639249265332130309ist_fm
      @ ( comp_P6483749517533010298ist_fm
        @ ^ [X3: produc1996495991257130529ist_fm] : ( produc2592314732767339223ist_fm @ X3 @ X3 )
        @ produc4514744107061996309ist_fm ) )
    = produc4514744107061996309ist_fm ) ).

% snd_diag_snd
thf(fact_852_snd__diag__snd,axiom,
    ( ( comp_P1426762558556845121ist_fm @ produc4588648349897876871ist_fm
      @ ( comp_l2016392514137073923ist_fm
        @ ^ [X3: list_fm] : ( produc381145313068854617ist_fm @ X3 @ X3 )
        @ produc4588648349897876871ist_fm ) )
    = produc4588648349897876871ist_fm ) ).

% snd_diag_snd
thf(fact_853_snd__diag__fst,axiom,
    ( ( comp_P1085356462900927218m_rule @ produc1639249265332130309ist_fm
      @ ( comp_P2017078995931194366m_rule
        @ ^ [X3: produc1996495991257130529ist_fm] : ( produc2592314732767339223ist_fm @ X3 @ X3 )
        @ produc6879501374131015971m_rule ) )
    = produc6879501374131015971m_rule ) ).

% snd_diag_fst
thf(fact_854_snd__diag__fst,axiom,
    ( ( comp_P1426762558556845121ist_fm @ produc4588648349897876871ist_fm
      @ ( comp_l2016392514137073923ist_fm
        @ ^ [X3: list_fm] : ( produc381145313068854617ist_fm @ X3 @ X3 )
        @ produc1501393135466168645ist_fm ) )
    = produc1501393135466168645ist_fm ) ).

% snd_diag_fst
thf(fact_855_fst__diag__snd,axiom,
    ( ( comp_P1426762558556845121ist_fm @ produc1501393135466168645ist_fm
      @ ( comp_l2016392514137073923ist_fm
        @ ^ [X3: list_fm] : ( produc381145313068854617ist_fm @ X3 @ X3 )
        @ produc4588648349897876871ist_fm ) )
    = produc4588648349897876871ist_fm ) ).

% fst_diag_snd
thf(fact_856_fst__diag__snd,axiom,
    ( ( comp_P2548450393677710116m_rule @ produc7264528503596239631e_rule
      @ ( comp_r8929483813444710872m_rule
        @ ^ [X3: rule] : ( produc1769597910952761123e_rule @ X3 @ X3 )
        @ produc7165828336582415457m_rule ) )
    = produc7165828336582415457m_rule ) ).

% fst_diag_snd
thf(fact_857_fst__diag__snd,axiom,
    ( ( comp_P5552026984502743150ist_fm @ produc5894184038225292483ist_fm
      @ ( comp_P6483749517533010298ist_fm
        @ ^ [X3: produc1996495991257130529ist_fm] : ( produc2592314732767339223ist_fm @ X3 @ X3 )
        @ produc4514744107061996309ist_fm ) )
    = produc4514744107061996309ist_fm ) ).

% fst_diag_snd
thf(fact_858_Grp__fst__snd,axiom,
    ! [R: list_fm > list_fm > $o] :
      ( ( relcom4917561651046858416ist_fm @ ( conver3212958163267315148ist_fm @ ( bNF_Gr3980516808638899763ist_fm @ ( collec7231575776666803724ist_fm @ ( produc2011568048573023402t_fm_o @ R ) ) @ produc1501393135466168645ist_fm ) ) @ ( bNF_Gr3980516808638899763ist_fm @ ( collec7231575776666803724ist_fm @ ( produc2011568048573023402t_fm_o @ R ) ) @ produc4588648349897876871ist_fm ) )
      = R ) ).

% Grp_fst_snd
thf(fact_859_Grp__fst__snd,axiom,
    ! [R: produc1996495991257130529ist_fm > rule > $o] :
      ( ( relcom7320010395573596684e_rule @ ( conver8136034010823108629ist_fm @ ( bNF_Gr2501199136280717308ist_fm @ ( collec2069506698588392810m_rule @ ( produc7369492558480804444rule_o @ R ) ) @ produc6879501374131015971m_rule ) ) @ ( bNF_Gr2146954950777597994e_rule @ ( collec2069506698588392810m_rule @ ( produc7369492558480804444rule_o @ R ) ) @ produc7165828336582415457m_rule ) )
      = R ) ).

% Grp_fst_snd
thf(fact_860_Grp__fst__snd,axiom,
    ! [R: rule > produc1996495991257130529ist_fm > $o] :
      ( ( relcom4829821695793320280ist_fm @ ( conver2604404730345294797m_rule @ ( bNF_Gr4144912365832075686m_rule @ ( collec6536177220190208742ist_fm @ ( produc1628407760719572648t_fm_o @ R ) ) @ produc4228417144610596823ist_fm ) ) @ ( bNF_Gr2959555449953857408ist_fm @ ( collec6536177220190208742ist_fm @ ( produc1628407760719572648t_fm_o @ R ) ) @ produc4514744107061996309ist_fm ) )
      = R ) ).

% Grp_fst_snd
thf(fact_861_predicate2__transferD,axiom,
    ! [R1: list_fm > list_fm > $o,R22: list_fm > list_fm > $o,P2: list_fm > list_fm > $o,Q3: list_fm > list_fm > $o,A: produc1996495991257130529ist_fm,A4: set_Pr7058068377845519745ist_fm,B: produc1996495991257130529ist_fm,B5: set_Pr7058068377845519745ist_fm] :
      ( ( bNF_re6571772915969521119t_fm_o @ R1
        @ ( bNF_re2260024341499725215fm_o_o @ R22
          @ ^ [Y2: $o,Z: $o] : ( Y2 = Z ) )
        @ P2
        @ Q3 )
     => ( ( member8102475879199740618ist_fm @ A @ A4 )
       => ( ( member8102475879199740618ist_fm @ B @ B5 )
         => ( ( ord_le2055072402069232929ist_fm @ A4 @ ( collec7231575776666803724ist_fm @ ( produc2011568048573023402t_fm_o @ R1 ) ) )
           => ( ( ord_le2055072402069232929ist_fm @ B5 @ ( collec7231575776666803724ist_fm @ ( produc2011568048573023402t_fm_o @ R22 ) ) )
             => ( ( P2 @ ( produc1501393135466168645ist_fm @ A ) @ ( produc1501393135466168645ist_fm @ B ) )
                = ( Q3 @ ( produc4588648349897876871ist_fm @ A ) @ ( produc4588648349897876871ist_fm @ B ) ) ) ) ) ) ) ) ).

% predicate2_transferD
thf(fact_862_predicate2__transferD,axiom,
    ! [R1: list_fm > list_fm > $o,R22: produc1996495991257130529ist_fm > rule > $o,P2: list_fm > produc1996495991257130529ist_fm > $o,Q3: list_fm > rule > $o,A: produc1996495991257130529ist_fm,A4: set_Pr7058068377845519745ist_fm,B: produc164195504107695125m_rule,B5: set_Pr1008144964186165195m_rule] :
      ( ( bNF_re6226240235925790797rule_o @ R1
        @ ( bNF_re2724311144721052361le_o_o @ R22
          @ ^ [Y2: $o,Z: $o] : ( Y2 = Z ) )
        @ P2
        @ Q3 )
     => ( ( member8102475879199740618ist_fm @ A @ A4 )
       => ( ( member4220325220686508332m_rule @ B @ B5 )
         => ( ( ord_le2055072402069232929ist_fm @ A4 @ ( collec7231575776666803724ist_fm @ ( produc2011568048573023402t_fm_o @ R1 ) ) )
           => ( ( ord_le5052251716133888043m_rule @ B5 @ ( collec2069506698588392810m_rule @ ( produc7369492558480804444rule_o @ R22 ) ) )
             => ( ( P2 @ ( produc1501393135466168645ist_fm @ A ) @ ( produc6879501374131015971m_rule @ B ) )
                = ( Q3 @ ( produc4588648349897876871ist_fm @ A ) @ ( produc7165828336582415457m_rule @ B ) ) ) ) ) ) ) ) ).

% predicate2_transferD
thf(fact_863_predicate2__transferD,axiom,
    ! [R1: list_fm > list_fm > $o,R22: rule > produc1996495991257130529ist_fm > $o,P2: list_fm > rule > $o,Q3: list_fm > produc1996495991257130529ist_fm > $o,A: produc1996495991257130529ist_fm,A4: set_Pr7058068377845519745ist_fm,B: produc4630866025709511057ist_fm,B5: set_Pr2241365836748572487ist_fm] :
      ( ( bNF_re8084688274390359943t_fm_o @ R1
        @ ( bNF_re5113994537810809725fm_o_o @ R22
          @ ^ [Y2: $o,Z: $o] : ( Y2 = Z ) )
        @ P2
        @ Q3 )
     => ( ( member8102475879199740618ist_fm @ A @ A4 )
       => ( ( member8686995742288324264ist_fm @ B @ B5 )
         => ( ( ord_le2055072402069232929ist_fm @ A4 @ ( collec7231575776666803724ist_fm @ ( produc2011568048573023402t_fm_o @ R1 ) ) )
           => ( ( ord_le6285472588696295335ist_fm @ B5 @ ( collec6536177220190208742ist_fm @ ( produc1628407760719572648t_fm_o @ R22 ) ) )
             => ( ( P2 @ ( produc1501393135466168645ist_fm @ A ) @ ( produc4228417144610596823ist_fm @ B ) )
                = ( Q3 @ ( produc4588648349897876871ist_fm @ A ) @ ( produc4514744107061996309ist_fm @ B ) ) ) ) ) ) ) ) ).

% predicate2_transferD
thf(fact_864_predicate2__transferD,axiom,
    ! [R1: produc1996495991257130529ist_fm > rule > $o,R22: list_fm > list_fm > $o,P2: produc1996495991257130529ist_fm > list_fm > $o,Q3: rule > list_fm > $o,A: produc164195504107695125m_rule,A4: set_Pr1008144964186165195m_rule,B: produc1996495991257130529ist_fm,B5: set_Pr7058068377845519745ist_fm] :
      ( ( bNF_re1899679218045841673t_fm_o @ R1
        @ ( bNF_re2260024341499725215fm_o_o @ R22
          @ ^ [Y2: $o,Z: $o] : ( Y2 = Z ) )
        @ P2
        @ Q3 )
     => ( ( member4220325220686508332m_rule @ A @ A4 )
       => ( ( member8102475879199740618ist_fm @ B @ B5 )
         => ( ( ord_le5052251716133888043m_rule @ A4 @ ( collec2069506698588392810m_rule @ ( produc7369492558480804444rule_o @ R1 ) ) )
           => ( ( ord_le2055072402069232929ist_fm @ B5 @ ( collec7231575776666803724ist_fm @ ( produc2011568048573023402t_fm_o @ R22 ) ) )
             => ( ( P2 @ ( produc6879501374131015971m_rule @ A ) @ ( produc1501393135466168645ist_fm @ B ) )
                = ( Q3 @ ( produc7165828336582415457m_rule @ A ) @ ( produc4588648349897876871ist_fm @ B ) ) ) ) ) ) ) ) ).

% predicate2_transferD
thf(fact_865_predicate2__transferD,axiom,
    ! [R1: produc1996495991257130529ist_fm > rule > $o,R22: produc1996495991257130529ist_fm > rule > $o,P2: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm > $o,Q3: rule > rule > $o,A: produc164195504107695125m_rule,A4: set_Pr1008144964186165195m_rule,B: produc164195504107695125m_rule,B5: set_Pr1008144964186165195m_rule] :
      ( ( bNF_re7876943729309676195rule_o @ R1
        @ ( bNF_re2724311144721052361le_o_o @ R22
          @ ^ [Y2: $o,Z: $o] : ( Y2 = Z ) )
        @ P2
        @ Q3 )
     => ( ( member4220325220686508332m_rule @ A @ A4 )
       => ( ( member4220325220686508332m_rule @ B @ B5 )
         => ( ( ord_le5052251716133888043m_rule @ A4 @ ( collec2069506698588392810m_rule @ ( produc7369492558480804444rule_o @ R1 ) ) )
           => ( ( ord_le5052251716133888043m_rule @ B5 @ ( collec2069506698588392810m_rule @ ( produc7369492558480804444rule_o @ R22 ) ) )
             => ( ( P2 @ ( produc6879501374131015971m_rule @ A ) @ ( produc6879501374131015971m_rule @ B ) )
                = ( Q3 @ ( produc7165828336582415457m_rule @ A ) @ ( produc7165828336582415457m_rule @ B ) ) ) ) ) ) ) ) ).

% predicate2_transferD
thf(fact_866_predicate2__transferD,axiom,
    ! [R1: produc1996495991257130529ist_fm > rule > $o,R22: rule > produc1996495991257130529ist_fm > $o,P2: produc1996495991257130529ist_fm > rule > $o,Q3: rule > produc1996495991257130529ist_fm > $o,A: produc164195504107695125m_rule,A4: set_Pr1008144964186165195m_rule,B: produc4630866025709511057ist_fm,B5: set_Pr2241365836748572487ist_fm] :
      ( ( bNF_re512019730919469533t_fm_o @ R1
        @ ( bNF_re5113994537810809725fm_o_o @ R22
          @ ^ [Y2: $o,Z: $o] : ( Y2 = Z ) )
        @ P2
        @ Q3 )
     => ( ( member4220325220686508332m_rule @ A @ A4 )
       => ( ( member8686995742288324264ist_fm @ B @ B5 )
         => ( ( ord_le5052251716133888043m_rule @ A4 @ ( collec2069506698588392810m_rule @ ( produc7369492558480804444rule_o @ R1 ) ) )
           => ( ( ord_le6285472588696295335ist_fm @ B5 @ ( collec6536177220190208742ist_fm @ ( produc1628407760719572648t_fm_o @ R22 ) ) )
             => ( ( P2 @ ( produc6879501374131015971m_rule @ A ) @ ( produc4228417144610596823ist_fm @ B ) )
                = ( Q3 @ ( produc7165828336582415457m_rule @ A ) @ ( produc4514744107061996309ist_fm @ B ) ) ) ) ) ) ) ) ).

% predicate2_transferD
thf(fact_867_predicate2__transferD,axiom,
    ! [R1: rule > produc1996495991257130529ist_fm > $o,R22: list_fm > list_fm > $o,P2: rule > list_fm > $o,Q3: produc1996495991257130529ist_fm > list_fm > $o,A: produc4630866025709511057ist_fm,A4: set_Pr2241365836748572487ist_fm,B: produc1996495991257130529ist_fm,B5: set_Pr7058068377845519745ist_fm] :
      ( ( bNF_re2710157489986801085t_fm_o @ R1
        @ ( bNF_re2260024341499725215fm_o_o @ R22
          @ ^ [Y2: $o,Z: $o] : ( Y2 = Z ) )
        @ P2
        @ Q3 )
     => ( ( member8686995742288324264ist_fm @ A @ A4 )
       => ( ( member8102475879199740618ist_fm @ B @ B5 )
         => ( ( ord_le6285472588696295335ist_fm @ A4 @ ( collec6536177220190208742ist_fm @ ( produc1628407760719572648t_fm_o @ R1 ) ) )
           => ( ( ord_le2055072402069232929ist_fm @ B5 @ ( collec7231575776666803724ist_fm @ ( produc2011568048573023402t_fm_o @ R22 ) ) )
             => ( ( P2 @ ( produc4228417144610596823ist_fm @ A ) @ ( produc1501393135466168645ist_fm @ B ) )
                = ( Q3 @ ( produc4514744107061996309ist_fm @ A ) @ ( produc4588648349897876871ist_fm @ B ) ) ) ) ) ) ) ) ).

% predicate2_transferD
thf(fact_868_predicate2__transferD,axiom,
    ! [R1: rule > produc1996495991257130529ist_fm > $o,R22: produc1996495991257130529ist_fm > rule > $o,P2: rule > produc1996495991257130529ist_fm > $o,Q3: produc1996495991257130529ist_fm > rule > $o,A: produc4630866025709511057ist_fm,A4: set_Pr2241365836748572487ist_fm,B: produc164195504107695125m_rule,B5: set_Pr1008144964186165195m_rule] :
      ( ( bNF_re5098013070747490543rule_o @ R1
        @ ( bNF_re2724311144721052361le_o_o @ R22
          @ ^ [Y2: $o,Z: $o] : ( Y2 = Z ) )
        @ P2
        @ Q3 )
     => ( ( member8686995742288324264ist_fm @ A @ A4 )
       => ( ( member4220325220686508332m_rule @ B @ B5 )
         => ( ( ord_le6285472588696295335ist_fm @ A4 @ ( collec6536177220190208742ist_fm @ ( produc1628407760719572648t_fm_o @ R1 ) ) )
           => ( ( ord_le5052251716133888043m_rule @ B5 @ ( collec2069506698588392810m_rule @ ( produc7369492558480804444rule_o @ R22 ) ) )
             => ( ( P2 @ ( produc4228417144610596823ist_fm @ A ) @ ( produc6879501374131015971m_rule @ B ) )
                = ( Q3 @ ( produc4514744107061996309ist_fm @ A ) @ ( produc7165828336582415457m_rule @ B ) ) ) ) ) ) ) ) ).

% predicate2_transferD
thf(fact_869_predicate2__transferD,axiom,
    ! [R1: rule > produc1996495991257130529ist_fm > $o,R22: rule > produc1996495991257130529ist_fm > $o,P2: rule > rule > $o,Q3: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm > $o,A: produc4630866025709511057ist_fm,A4: set_Pr2241365836748572487ist_fm,B: produc4630866025709511057ist_fm,B5: set_Pr2241365836748572487ist_fm] :
      ( ( bNF_re6956461109212059689t_fm_o @ R1
        @ ( bNF_re5113994537810809725fm_o_o @ R22
          @ ^ [Y2: $o,Z: $o] : ( Y2 = Z ) )
        @ P2
        @ Q3 )
     => ( ( member8686995742288324264ist_fm @ A @ A4 )
       => ( ( member8686995742288324264ist_fm @ B @ B5 )
         => ( ( ord_le6285472588696295335ist_fm @ A4 @ ( collec6536177220190208742ist_fm @ ( produc1628407760719572648t_fm_o @ R1 ) ) )
           => ( ( ord_le6285472588696295335ist_fm @ B5 @ ( collec6536177220190208742ist_fm @ ( produc1628407760719572648t_fm_o @ R22 ) ) )
             => ( ( P2 @ ( produc4228417144610596823ist_fm @ A ) @ ( produc4228417144610596823ist_fm @ B ) )
                = ( Q3 @ ( produc4514744107061996309ist_fm @ A ) @ ( produc4514744107061996309ist_fm @ B ) ) ) ) ) ) ) ) ).

% predicate2_transferD
thf(fact_870_stream__smap__fromN,axiom,
    ! [S3: stream727092118206550309m_rule,N2: nat] :
      ( S3
      = ( smap_n3801058020469267436m_rule
        @ ^ [J: nat] : ( snth_P1712579846118164081m_rule @ S3 @ ( minus_minus_nat @ J @ N2 ) )
        @ ( siterate_nat @ suc @ N2 ) ) ) ).

% stream_smap_fromN
thf(fact_871_Gr__incl,axiom,
    ! [A4: set_Pr1008144964186165195m_rule,F2: produc164195504107695125m_rule > set_fm,B5: set_set_fm] :
      ( ( ord_le8665424805890289788set_fm @ ( bNF_Gr3605759588368728511set_fm @ A4 @ F2 )
        @ ( produc3629056906227541605set_fm @ A4
          @ ^ [Uu: produc164195504107695125m_rule] : B5 ) )
      = ( ord_le1461404734466536732set_fm @ ( image_5698427363735666338set_fm @ F2 @ A4 ) @ B5 ) ) ).

% Gr_incl
thf(fact_872_subset__fst__snd,axiom,
    ! [A4: set_Pr7058068377845519745ist_fm] :
      ( ord_le2055072402069232929ist_fm @ A4
      @ ( produc7866297306142304650ist_fm @ ( image_6369949096775035324ist_fm @ produc1501393135466168645ist_fm @ A4 )
        @ ^ [Uu: list_fm] : ( image_6369949096775035324ist_fm @ produc4588648349897876871ist_fm @ A4 ) ) ) ).

% subset_fst_snd
thf(fact_873_subset__fst__snd,axiom,
    ! [A4: set_Pr1008144964186165195m_rule] :
      ( ord_le5052251716133888043m_rule @ A4
      @ ( produc1083232162533131166m_rule @ ( image_3223098899990291461ist_fm @ produc6879501374131015971m_rule @ A4 )
        @ ^ [Uu: produc1996495991257130529ist_fm] : ( image_8178078329501008481e_rule @ produc7165828336582415457m_rule @ A4 ) ) ) ).

% subset_fst_snd
thf(fact_874_subset__fst__snd,axiom,
    ! [A4: set_Pr2241365836748572487ist_fm] :
      ( ord_le6285472588696295335ist_fm @ A4
      @ ( produc7655519969867487826ist_fm @ ( image_952663707700710365m_rule @ produc4228417144610596823ist_fm @ A4 )
        @ ^ [Uu: rule] : ( image_3681455213663431561ist_fm @ produc4514744107061996309ist_fm @ A4 ) ) ) ).

% subset_fst_snd
thf(fact_875_SigmaI,axiom,
    ! [A: fm,A4: set_fm,B: fm,B5: fm > set_fm] :
      ( ( member_fm @ A @ A4 )
     => ( ( member_fm @ B @ ( B5 @ A ) )
       => ( member8474499337054950954_fm_fm @ ( product_Pair_fm_fm @ A @ B ) @ ( product_Sigma_fm_fm @ A4 @ B5 ) ) ) ) ).

% SigmaI
thf(fact_876_mem__Sigma__iff,axiom,
    ! [A: fm,B: fm,A4: set_fm,B5: fm > set_fm] :
      ( ( member8474499337054950954_fm_fm @ ( product_Pair_fm_fm @ A @ B ) @ ( product_Sigma_fm_fm @ A4 @ B5 ) )
      = ( ( member_fm @ A @ A4 )
        & ( member_fm @ B @ ( B5 @ A ) ) ) ) ).

% mem_Sigma_iff
thf(fact_877_SigmaE,axiom,
    ! [C2: product_prod_fm_fm,A4: set_fm,B5: fm > set_fm] :
      ( ( member8474499337054950954_fm_fm @ C2 @ ( product_Sigma_fm_fm @ A4 @ B5 ) )
     => ~ ! [X: fm] :
            ( ( member_fm @ X @ A4 )
           => ! [Y: fm] :
                ( ( member_fm @ Y @ ( B5 @ X ) )
               => ( C2
                 != ( product_Pair_fm_fm @ X @ Y ) ) ) ) ) ).

% SigmaE
thf(fact_878_SigmaE2,axiom,
    ! [A: fm,B: fm,A4: set_fm,B5: fm > set_fm] :
      ( ( member8474499337054950954_fm_fm @ ( product_Pair_fm_fm @ A @ B ) @ ( product_Sigma_fm_fm @ A4 @ B5 ) )
     => ~ ( ( member_fm @ A @ A4 )
         => ~ ( member_fm @ B @ ( B5 @ A ) ) ) ) ).

% SigmaE2
thf(fact_879_mem__Times__iff,axiom,
    ! [X2: product_prod_fm_fm,A4: set_fm,B5: set_fm] :
      ( ( member8474499337054950954_fm_fm @ X2
        @ ( product_Sigma_fm_fm @ A4
          @ ^ [Uu: fm] : B5 ) )
      = ( ( member_fm @ ( product_fst_fm_fm @ X2 ) @ A4 )
        & ( member_fm @ ( product_snd_fm_fm @ X2 ) @ B5 ) ) ) ).

% mem_Times_iff
thf(fact_880_mem__Times__iff,axiom,
    ! [X2: produc1996495991257130529ist_fm,A4: set_list_fm,B5: set_list_fm] :
      ( ( member8102475879199740618ist_fm @ X2
        @ ( produc7866297306142304650ist_fm @ A4
          @ ^ [Uu: list_fm] : B5 ) )
      = ( ( member_list_fm @ ( produc1501393135466168645ist_fm @ X2 ) @ A4 )
        & ( member_list_fm @ ( produc4588648349897876871ist_fm @ X2 ) @ B5 ) ) ) ).

% mem_Times_iff
thf(fact_881_mem__Times__iff,axiom,
    ! [X2: produc164195504107695125m_rule,A4: set_Pr7058068377845519745ist_fm,B5: set_rule] :
      ( ( member4220325220686508332m_rule @ X2
        @ ( produc1083232162533131166m_rule @ A4
          @ ^ [Uu: produc1996495991257130529ist_fm] : B5 ) )
      = ( ( member8102475879199740618ist_fm @ ( produc6879501374131015971m_rule @ X2 ) @ A4 )
        & ( member_rule @ ( produc7165828336582415457m_rule @ X2 ) @ B5 ) ) ) ).

% mem_Times_iff
thf(fact_882_mem__Times__iff,axiom,
    ! [X2: produc4630866025709511057ist_fm,A4: set_rule,B5: set_Pr7058068377845519745ist_fm] :
      ( ( member8686995742288324264ist_fm @ X2
        @ ( produc7655519969867487826ist_fm @ A4
          @ ^ [Uu: rule] : B5 ) )
      = ( ( member_rule @ ( produc4228417144610596823ist_fm @ X2 ) @ A4 )
        & ( member8102475879199740618ist_fm @ ( produc4514744107061996309ist_fm @ X2 ) @ B5 ) ) ) ).

% mem_Times_iff
thf(fact_883_product__swap,axiom,
    ! [A4: set_list_fm,B5: set_list_fm] :
      ( ( image_6982201141141708101ist_fm @ produc7936585717479832313ist_fm
        @ ( produc7866297306142304650ist_fm @ A4
          @ ^ [Uu: list_fm] : B5 ) )
      = ( produc7866297306142304650ist_fm @ B5
        @ ^ [Uu: list_fm] : A4 ) ) ).

% product_swap
thf(fact_884_product__swap,axiom,
    ! [A4: set_rule,B5: set_Pr7058068377845519745ist_fm] :
      ( ( image_4130355632178425005m_rule @ produc9213896635423732003ist_fm
        @ ( produc7655519969867487826ist_fm @ A4
          @ ^ [Uu: rule] : B5 ) )
      = ( produc1083232162533131166m_rule @ B5
        @ ^ [Uu: produc1996495991257130529ist_fm] : A4 ) ) ).

% product_swap
thf(fact_885_product__swap,axiom,
    ! [A4: set_Pr7058068377845519745ist_fm,B5: set_rule] :
      ( ( image_6659396422799841965ist_fm @ produc2641608828089375343m_rule
        @ ( produc1083232162533131166m_rule @ A4
          @ ^ [Uu: produc1996495991257130529ist_fm] : B5 ) )
      = ( produc7655519969867487826ist_fm @ B5
        @ ^ [Uu: rule] : A4 ) ) ).

% product_swap
thf(fact_886_fun_Opred__cong,axiom,
    ! [X2: produc164195504107695125m_rule > set_fm,Ya: produc164195504107695125m_rule > set_fm,P2: set_fm > $o,Pa: set_fm > $o] :
      ( ( X2 = Ya )
     => ( ! [Z3: set_fm] :
            ( ( member_set_fm @ Z3 @ ( image_5698427363735666338set_fm @ Ya @ top_to1603951219661559419m_rule ) )
           => ( ( P2 @ Z3 )
              = ( Pa @ Z3 ) ) )
       => ( ( basic_8395401525575979840set_fm
            @ ^ [Uu: produc164195504107695125m_rule] : $true
            @ P2
            @ X2 )
          = ( basic_8395401525575979840set_fm
            @ ^ [Uu: produc164195504107695125m_rule] : $true
            @ Pa
            @ Ya ) ) ) ) ).

% fun.pred_cong
thf(fact_887_fun_Opred__mono__strong,axiom,
    ! [P2: set_fm > $o,X2: produc164195504107695125m_rule > set_fm,Pa: set_fm > $o] :
      ( ( basic_8395401525575979840set_fm
        @ ^ [Uu: produc164195504107695125m_rule] : $true
        @ P2
        @ X2 )
     => ( ! [Z3: set_fm] :
            ( ( member_set_fm @ Z3 @ ( image_5698427363735666338set_fm @ X2 @ top_to1603951219661559419m_rule ) )
           => ( ( P2 @ Z3 )
             => ( Pa @ Z3 ) ) )
       => ( basic_8395401525575979840set_fm
          @ ^ [Uu: produc164195504107695125m_rule] : $true
          @ Pa
          @ X2 ) ) ) ).

% fun.pred_mono_strong
thf(fact_888_image__paired__Times,axiom,
    ! [F2: produc164195504107695125m_rule > set_fm,G: produc164195504107695125m_rule > set_fm,A4: set_Pr1008144964186165195m_rule,B5: set_Pr1008144964186165195m_rule] :
      ( ( image_479386260607784615set_fm
        @ ( produc4357883083655834389set_fm
          @ ^ [X3: produc164195504107695125m_rule,Y4: produc164195504107695125m_rule] : ( produc3030414577503514149set_fm @ ( F2 @ X3 ) @ ( G @ Y4 ) ) )
        @ ( produc7959129826089203956m_rule @ A4
          @ ^ [Uu: produc164195504107695125m_rule] : B5 ) )
      = ( produc7186806611370130262set_fm @ ( image_5698427363735666338set_fm @ F2 @ A4 )
        @ ^ [Uu: set_fm] : ( image_5698427363735666338set_fm @ G @ B5 ) ) ) ).

% image_paired_Times
thf(fact_889_map__prod__surj__on,axiom,
    ! [F2: produc164195504107695125m_rule > set_fm,A4: set_Pr1008144964186165195m_rule,A7: set_set_fm,G: produc164195504107695125m_rule > set_fm,B5: set_Pr1008144964186165195m_rule,B6: set_set_fm] :
      ( ( ( image_5698427363735666338set_fm @ F2 @ A4 )
        = A7 )
     => ( ( ( image_5698427363735666338set_fm @ G @ B5 )
          = B6 )
       => ( ( image_479386260607784615set_fm @ ( produc8521236829880132656set_fm @ F2 @ G )
            @ ( produc7959129826089203956m_rule @ A4
              @ ^ [Uu: produc164195504107695125m_rule] : B5 ) )
          = ( produc7186806611370130262set_fm @ A7
            @ ^ [Uu: set_fm] : B6 ) ) ) ) ).

% map_prod_surj_on
thf(fact_890_subset__snd__imageI,axiom,
    ! [A4: set_list_fm,B5: set_list_fm,S6: set_Pr7058068377845519745ist_fm,X2: list_fm] :
      ( ( ord_le2055072402069232929ist_fm
        @ ( produc7866297306142304650ist_fm @ A4
          @ ^ [Uu: list_fm] : B5 )
        @ S6 )
     => ( ( member_list_fm @ X2 @ A4 )
       => ( ord_le3173885194822255862ist_fm @ B5 @ ( image_6369949096775035324ist_fm @ produc4588648349897876871ist_fm @ S6 ) ) ) ) ).

% subset_snd_imageI
thf(fact_891_subset__snd__imageI,axiom,
    ! [A4: set_Pr7058068377845519745ist_fm,B5: set_rule,S6: set_Pr1008144964186165195m_rule,X2: produc1996495991257130529ist_fm] :
      ( ( ord_le5052251716133888043m_rule
        @ ( produc1083232162533131166m_rule @ A4
          @ ^ [Uu: produc1996495991257130529ist_fm] : B5 )
        @ S6 )
     => ( ( member8102475879199740618ist_fm @ X2 @ A4 )
       => ( ord_less_eq_set_rule @ B5 @ ( image_8178078329501008481e_rule @ produc7165828336582415457m_rule @ S6 ) ) ) ) ).

% subset_snd_imageI
thf(fact_892_subset__snd__imageI,axiom,
    ! [A4: set_rule,B5: set_Pr7058068377845519745ist_fm,S6: set_Pr2241365836748572487ist_fm,X2: rule] :
      ( ( ord_le6285472588696295335ist_fm
        @ ( produc7655519969867487826ist_fm @ A4
          @ ^ [Uu: rule] : B5 )
        @ S6 )
     => ( ( member_rule @ X2 @ A4 )
       => ( ord_le2055072402069232929ist_fm @ B5 @ ( image_3681455213663431561ist_fm @ produc4514744107061996309ist_fm @ S6 ) ) ) ) ).

% subset_snd_imageI
thf(fact_893_subset__fst__imageI,axiom,
    ! [A4: set_Pr7058068377845519745ist_fm,B5: set_rule,S6: set_Pr1008144964186165195m_rule,Y3: rule] :
      ( ( ord_le5052251716133888043m_rule
        @ ( produc1083232162533131166m_rule @ A4
          @ ^ [Uu: produc1996495991257130529ist_fm] : B5 )
        @ S6 )
     => ( ( member_rule @ Y3 @ B5 )
       => ( ord_le2055072402069232929ist_fm @ A4 @ ( image_3223098899990291461ist_fm @ produc6879501374131015971m_rule @ S6 ) ) ) ) ).

% subset_fst_imageI
thf(fact_894_subset__fst__imageI,axiom,
    ! [A4: set_list_fm,B5: set_list_fm,S6: set_Pr7058068377845519745ist_fm,Y3: list_fm] :
      ( ( ord_le2055072402069232929ist_fm
        @ ( produc7866297306142304650ist_fm @ A4
          @ ^ [Uu: list_fm] : B5 )
        @ S6 )
     => ( ( member_list_fm @ Y3 @ B5 )
       => ( ord_le3173885194822255862ist_fm @ A4 @ ( image_6369949096775035324ist_fm @ produc1501393135466168645ist_fm @ S6 ) ) ) ) ).

% subset_fst_imageI
thf(fact_895_set__product,axiom,
    ! [Xs: list_fm,Ys: list_fm] :
      ( ( set_Pr6624747810323803798_fm_fm @ ( product_fm_fm @ Xs @ Ys ) )
      = ( product_Sigma_fm_fm @ ( set_fm2 @ Xs )
        @ ^ [Uu: fm] : ( set_fm2 @ Ys ) ) ) ).

% set_product
thf(fact_896_snd__image__times,axiom,
    ! [A4: set_list_fm,B5: set_list_fm] :
      ( ( ( A4 = bot_bot_set_list_fm )
       => ( ( image_6369949096775035324ist_fm @ produc4588648349897876871ist_fm
            @ ( produc7866297306142304650ist_fm @ A4
              @ ^ [Uu: list_fm] : B5 ) )
          = bot_bot_set_list_fm ) )
      & ( ( A4 != bot_bot_set_list_fm )
       => ( ( image_6369949096775035324ist_fm @ produc4588648349897876871ist_fm
            @ ( produc7866297306142304650ist_fm @ A4
              @ ^ [Uu: list_fm] : B5 ) )
          = B5 ) ) ) ).

% snd_image_times
thf(fact_897_snd__image__times,axiom,
    ! [A4: set_Pr7058068377845519745ist_fm,B5: set_rule] :
      ( ( ( A4 = bot_bo4530011010606209517ist_fm )
       => ( ( image_8178078329501008481e_rule @ produc7165828336582415457m_rule
            @ ( produc1083232162533131166m_rule @ A4
              @ ^ [Uu: produc1996495991257130529ist_fm] : B5 ) )
          = bot_bot_set_rule ) )
      & ( ( A4 != bot_bo4530011010606209517ist_fm )
       => ( ( image_8178078329501008481e_rule @ produc7165828336582415457m_rule
            @ ( produc1083232162533131166m_rule @ A4
              @ ^ [Uu: produc1996495991257130529ist_fm] : B5 ) )
          = B5 ) ) ) ).

% snd_image_times
thf(fact_898_snd__image__times,axiom,
    ! [A4: set_rule,B5: set_Pr7058068377845519745ist_fm] :
      ( ( ( A4 = bot_bot_set_rule )
       => ( ( image_3681455213663431561ist_fm @ produc4514744107061996309ist_fm
            @ ( produc7655519969867487826ist_fm @ A4
              @ ^ [Uu: rule] : B5 ) )
          = bot_bo4530011010606209517ist_fm ) )
      & ( ( A4 != bot_bot_set_rule )
       => ( ( image_3681455213663431561ist_fm @ produc4514744107061996309ist_fm
            @ ( produc7655519969867487826ist_fm @ A4
              @ ^ [Uu: rule] : B5 ) )
          = B5 ) ) ) ).

% snd_image_times
thf(fact_899_fst__image__times,axiom,
    ! [B5: set_rule,A4: set_Pr7058068377845519745ist_fm] :
      ( ( ( B5 = bot_bot_set_rule )
       => ( ( image_3223098899990291461ist_fm @ produc6879501374131015971m_rule
            @ ( produc1083232162533131166m_rule @ A4
              @ ^ [Uu: produc1996495991257130529ist_fm] : B5 ) )
          = bot_bo4530011010606209517ist_fm ) )
      & ( ( B5 != bot_bot_set_rule )
       => ( ( image_3223098899990291461ist_fm @ produc6879501374131015971m_rule
            @ ( produc1083232162533131166m_rule @ A4
              @ ^ [Uu: produc1996495991257130529ist_fm] : B5 ) )
          = A4 ) ) ) ).

% fst_image_times
thf(fact_900_fst__image__times,axiom,
    ! [B5: set_list_fm,A4: set_list_fm] :
      ( ( ( B5 = bot_bot_set_list_fm )
       => ( ( image_6369949096775035324ist_fm @ produc1501393135466168645ist_fm
            @ ( produc7866297306142304650ist_fm @ A4
              @ ^ [Uu: list_fm] : B5 ) )
          = bot_bot_set_list_fm ) )
      & ( ( B5 != bot_bot_set_list_fm )
       => ( ( image_6369949096775035324ist_fm @ produc1501393135466168645ist_fm
            @ ( produc7866297306142304650ist_fm @ A4
              @ ^ [Uu: list_fm] : B5 ) )
          = A4 ) ) ) ).

% fst_image_times
thf(fact_901_fst__image__Sigma,axiom,
    ! [A4: set_Pr7058068377845519745ist_fm,B5: produc1996495991257130529ist_fm > set_rule] :
      ( ( image_3223098899990291461ist_fm @ produc6879501374131015971m_rule @ ( produc1083232162533131166m_rule @ A4 @ B5 ) )
      = ( collec7231575776666803724ist_fm
        @ ^ [X3: produc1996495991257130529ist_fm] :
            ( ( member8102475879199740618ist_fm @ X3 @ A4 )
            & ( ( B5 @ X3 )
             != bot_bot_set_rule ) ) ) ) ).

% fst_image_Sigma
thf(fact_902_fst__image__Sigma,axiom,
    ! [A4: set_list_fm,B5: list_fm > set_list_fm] :
      ( ( image_6369949096775035324ist_fm @ produc1501393135466168645ist_fm @ ( produc7866297306142304650ist_fm @ A4 @ B5 ) )
      = ( collect_list_fm
        @ ^ [X3: list_fm] :
            ( ( member_list_fm @ X3 @ A4 )
            & ( ( B5 @ X3 )
             != bot_bot_set_list_fm ) ) ) ) ).

% fst_image_Sigma
thf(fact_903_The__case__prod,axiom,
    ! [P2: list_fm > list_fm > $o] :
      ( ( the_Pr8646916617578001576ist_fm @ ( produc2011568048573023402t_fm_o @ P2 ) )
      = ( the_Pr8646916617578001576ist_fm
        @ ^ [Xy: produc1996495991257130529ist_fm] : ( P2 @ ( produc1501393135466168645ist_fm @ Xy ) @ ( produc4588648349897876871ist_fm @ Xy ) ) ) ) ).

% The_case_prod
thf(fact_904_The__case__prod,axiom,
    ! [P2: produc1996495991257130529ist_fm > rule > $o] :
      ( ( the_Pr4182551778311249870m_rule @ ( produc7369492558480804444rule_o @ P2 ) )
      = ( the_Pr4182551778311249870m_rule
        @ ^ [Xy: produc164195504107695125m_rule] : ( P2 @ ( produc6879501374131015971m_rule @ Xy ) @ ( produc7165828336582415457m_rule @ Xy ) ) ) ) ).

% The_case_prod
thf(fact_905_The__case__prod,axiom,
    ! [P2: rule > produc1996495991257130529ist_fm > $o] :
      ( ( the_Pr8649222299913065802ist_fm @ ( produc1628407760719572648t_fm_o @ P2 ) )
      = ( the_Pr8649222299913065802ist_fm
        @ ^ [Xy: produc4630866025709511057ist_fm] : ( P2 @ ( produc4228417144610596823ist_fm @ Xy ) @ ( produc4514744107061996309ist_fm @ Xy ) ) ) ) ).

% The_case_prod
thf(fact_906_subset__emptyI,axiom,
    ! [A4: set_fm] :
      ( ! [X: fm] :
          ~ ( member_fm @ X @ A4 )
     => ( ord_less_eq_set_fm @ A4 @ bot_bot_set_fm ) ) ).

% subset_emptyI
thf(fact_907_refl__onI,axiom,
    ! [R2: set_Pr4463079037648049377_fm_fm,A4: set_fm] :
      ( ( ord_le7532831136298237057_fm_fm @ R2
        @ ( product_Sigma_fm_fm @ A4
          @ ^ [Uu: fm] : A4 ) )
     => ( ! [X: fm] :
            ( ( member_fm @ X @ A4 )
           => ( member8474499337054950954_fm_fm @ ( product_Pair_fm_fm @ X @ X ) @ R2 ) )
       => ( refl_on_fm @ A4 @ R2 ) ) ) ).

% refl_onI
thf(fact_908_refl__onD2,axiom,
    ! [A4: set_fm,R2: set_Pr4463079037648049377_fm_fm,X2: fm,Y3: fm] :
      ( ( refl_on_fm @ A4 @ R2 )
     => ( ( member8474499337054950954_fm_fm @ ( product_Pair_fm_fm @ X2 @ Y3 ) @ R2 )
       => ( member_fm @ Y3 @ A4 ) ) ) ).

% refl_onD2
thf(fact_909_refl__onD1,axiom,
    ! [A4: set_fm,R2: set_Pr4463079037648049377_fm_fm,X2: fm,Y3: fm] :
      ( ( refl_on_fm @ A4 @ R2 )
     => ( ( member8474499337054950954_fm_fm @ ( product_Pair_fm_fm @ X2 @ Y3 ) @ R2 )
       => ( member_fm @ X2 @ A4 ) ) ) ).

% refl_onD1
thf(fact_910_refl__onD,axiom,
    ! [A4: set_fm,R2: set_Pr4463079037648049377_fm_fm,A: fm] :
      ( ( refl_on_fm @ A4 @ R2 )
     => ( ( member_fm @ A @ A4 )
       => ( member8474499337054950954_fm_fm @ ( product_Pair_fm_fm @ A @ A ) @ R2 ) ) ) ).

% refl_onD
thf(fact_911_refl__on__domain,axiom,
    ! [A4: set_fm,R2: set_Pr4463079037648049377_fm_fm,A: fm,B: fm] :
      ( ( refl_on_fm @ A4 @ R2 )
     => ( ( member8474499337054950954_fm_fm @ ( product_Pair_fm_fm @ A @ B ) @ R2 )
       => ( ( member_fm @ A @ A4 )
          & ( member_fm @ B @ A4 ) ) ) ) ).

% refl_on_domain
thf(fact_912_snd__image__Sigma,axiom,
    ! [A4: set_list_fm,B5: list_fm > set_list_fm] :
      ( ( image_6369949096775035324ist_fm @ produc4588648349897876871ist_fm @ ( produc7866297306142304650ist_fm @ A4 @ B5 ) )
      = ( comple8664233329159923433ist_fm @ ( image_6665506237851550077ist_fm @ B5 @ A4 ) ) ) ).

% snd_image_Sigma
thf(fact_913_snd__image__Sigma,axiom,
    ! [A4: set_Pr7058068377845519745ist_fm,B5: produc1996495991257130529ist_fm > set_rule] :
      ( ( image_8178078329501008481e_rule @ produc7165828336582415457m_rule @ ( produc1083232162533131166m_rule @ A4 @ B5 ) )
      = ( comple2146307154184993742t_rule @ ( image_2008118025656205143t_rule @ B5 @ A4 ) ) ) ).

% snd_image_Sigma
thf(fact_914_snd__image__Sigma,axiom,
    ! [A4: set_rule,B5: rule > set_Pr7058068377845519745ist_fm] :
      ( ( image_3681455213663431561ist_fm @ produc4514744107061996309ist_fm @ ( produc7655519969867487826ist_fm @ A4 @ B5 ) )
      = ( comple1649742678916152942ist_fm @ ( image_3897101447128207157ist_fm @ B5 @ A4 ) ) ) ).

% snd_image_Sigma
thf(fact_915_snd__image__Sigma,axiom,
    ! [A4: set_Pr1008144964186165195m_rule,B5: produc164195504107695125m_rule > set_fm] :
      ( ( image_2608076899697204049_fm_fm @ produc8694862632519904556ule_fm @ ( produc7179657805070879791ule_fm @ A4 @ B5 ) )
      = ( comple7677669044296858969set_fm @ ( image_5698427363735666338set_fm @ B5 @ A4 ) ) ) ).

% snd_image_Sigma
thf(fact_916_Id__onI,axiom,
    ! [A: fm,A4: set_fm] :
      ( ( member_fm @ A @ A4 )
     => ( member8474499337054950954_fm_fm @ ( product_Pair_fm_fm @ A @ A ) @ ( id_on_fm @ A4 ) ) ) ).

% Id_onI
thf(fact_917_refl__on__UNION,axiom,
    ! [S6: set_Pr1008144964186165195m_rule,A4: produc164195504107695125m_rule > set_fm,R2: produc164195504107695125m_rule > set_Pr4463079037648049377_fm_fm] :
      ( ! [X: produc164195504107695125m_rule] :
          ( ( member4220325220686508332m_rule @ X @ S6 )
         => ( refl_on_fm @ ( A4 @ X ) @ ( R2 @ X ) ) )
     => ( refl_on_fm @ ( comple7677669044296858969set_fm @ ( image_5698427363735666338set_fm @ A4 @ S6 ) ) @ ( comple8500535872550171342_fm_fm @ ( image_7922547816601700037_fm_fm @ R2 @ S6 ) ) ) ) ).

% refl_on_UNION
thf(fact_918_UN__Times__distrib,axiom,
    ! [E: produc164195504107695125m_rule > set_fm,F4: produc164195504107695125m_rule > set_fm,A4: set_Pr1008144964186165195m_rule,B5: set_Pr1008144964186165195m_rule] :
      ( ( comple8500535872550171342_fm_fm
        @ ( image_1150248419239391387_fm_fm
          @ ( produc1078678291668483337_fm_fm
            @ ^ [A5: produc164195504107695125m_rule,B4: produc164195504107695125m_rule] :
                ( product_Sigma_fm_fm @ ( E @ A5 )
                @ ^ [Uu: fm] : ( F4 @ B4 ) ) )
          @ ( produc7959129826089203956m_rule @ A4
            @ ^ [Uu: produc164195504107695125m_rule] : B5 ) ) )
      = ( product_Sigma_fm_fm @ ( comple7677669044296858969set_fm @ ( image_5698427363735666338set_fm @ E @ A4 ) )
        @ ^ [Uu: fm] : ( comple7677669044296858969set_fm @ ( image_5698427363735666338set_fm @ F4 @ B5 ) ) ) ) ).

% UN_Times_distrib
thf(fact_919_Id__onE,axiom,
    ! [C2: product_prod_fm_fm,A4: set_fm] :
      ( ( member8474499337054950954_fm_fm @ C2 @ ( id_on_fm @ A4 ) )
     => ~ ! [X: fm] :
            ( ( member_fm @ X @ A4 )
           => ( C2
             != ( product_Pair_fm_fm @ X @ X ) ) ) ) ).

% Id_onE
thf(fact_920_Id__on__eqI,axiom,
    ! [A: fm,B: fm,A4: set_fm] :
      ( ( A = B )
     => ( ( member_fm @ A @ A4 )
       => ( member8474499337054950954_fm_fm @ ( product_Pair_fm_fm @ A @ B ) @ ( id_on_fm @ A4 ) ) ) ) ).

% Id_on_eqI
thf(fact_921_Id__on__iff,axiom,
    ! [X2: fm,Y3: fm,A4: set_fm] :
      ( ( member8474499337054950954_fm_fm @ ( product_Pair_fm_fm @ X2 @ Y3 ) @ ( id_on_fm @ A4 ) )
      = ( ( X2 = Y3 )
        & ( member_fm @ X2 @ A4 ) ) ) ).

% Id_on_iff
thf(fact_922_UN__UN__split__split__eq,axiom,
    ! [A4: produc1996495991257130529ist_fm > rule > produc1996495991257130529ist_fm > rule > set_fm,Y5: set_Pr1008144964186165195m_rule,X5: set_Pr1008144964186165195m_rule] :
      ( ( comple7677669044296858969set_fm
        @ ( image_5698427363735666338set_fm
          @ ( produc966423478193074626set_fm
            @ ^ [X12: produc1996495991257130529ist_fm,X23: rule] : ( comple7677669044296858969set_fm @ ( image_5698427363735666338set_fm @ ( produc966423478193074626set_fm @ ( A4 @ X12 @ X23 ) ) @ Y5 ) ) )
          @ X5 ) )
      = ( comple7677669044296858969set_fm
        @ ( image_5698427363735666338set_fm
          @ ^ [X3: produc164195504107695125m_rule] :
              ( comple7677669044296858969set_fm
              @ ( image_5698427363735666338set_fm
                @ ^ [Y4: produc164195504107695125m_rule] :
                    ( produc966423478193074626set_fm
                    @ ^ [X12: produc1996495991257130529ist_fm,X23: rule] : ( produc966423478193074626set_fm @ ( A4 @ X12 @ X23 ) @ Y4 )
                    @ X3 )
                @ Y5 ) )
          @ X5 ) ) ) ).

% UN_UN_split_split_eq
thf(fact_923_set__list__bind,axiom,
    ! [Xs: list_P8332459875677254181m_rule,F2: produc164195504107695125m_rule > list_fm] :
      ( ( set_fm2 @ ( bind_P3612422529591132398ule_fm @ Xs @ F2 ) )
      = ( comple7677669044296858969set_fm
        @ ( image_5698427363735666338set_fm
          @ ^ [X3: produc164195504107695125m_rule] : ( set_fm2 @ ( F2 @ X3 ) )
          @ ( set_Pr8505323785428441536m_rule @ Xs ) ) ) ) ).

% set_list_bind
thf(fact_924_set__list__bind,axiom,
    ! [Xs: list_fm,F2: fm > list_fm] :
      ( ( set_fm2 @ ( bind_fm_fm @ Xs @ F2 ) )
      = ( comple7677669044296858969set_fm
        @ ( image_fm_set_fm
          @ ^ [X3: fm] : ( set_fm2 @ ( F2 @ X3 ) )
          @ ( set_fm2 @ Xs ) ) ) ) ).

% set_list_bind
thf(fact_925_treeB__def,axiom,
    ( treeB
    = ( ^ [Steps2: stream727092118206550309m_rule] :
          ( comple7677669044296858969set_fm
          @ ( image_5698427363735666338set_fm
            @ ^ [S: produc164195504107695125m_rule] : ( set_fm2 @ ( produc4588648349897876871ist_fm @ ( produc6879501374131015971m_rule @ S ) ) )
            @ ( sset_P4979403222752902913m_rule @ Steps2 ) ) ) ) ) ).

% treeB_def
thf(fact_926_treeA__def,axiom,
    ( treeA
    = ( ^ [Steps2: stream727092118206550309m_rule] :
          ( comple7677669044296858969set_fm
          @ ( image_5698427363735666338set_fm
            @ ^ [S: produc164195504107695125m_rule] : ( set_fm2 @ ( produc1501393135466168645ist_fm @ ( produc6879501374131015971m_rule @ S ) ) )
            @ ( sset_P4979403222752902913m_rule @ Steps2 ) ) ) ) ) ).

% treeA_def
thf(fact_927_sset__range,axiom,
    ( sset_P4979403222752902913m_rule
    = ( ^ [S: stream727092118206550309m_rule] : ( image_3980074273869850220m_rule @ ( snth_P1712579846118164081m_rule @ S ) @ top_top_set_nat ) ) ) ).

% sset_range
thf(fact_928_snth__sset,axiom,
    ! [S3: stream_fm,N2: nat] : ( member_fm @ ( snth_fm @ S3 @ N2 ) @ ( sset_fm @ S3 ) ) ).

% snth_sset
thf(fact_929_snth__sset,axiom,
    ! [S3: stream727092118206550309m_rule,N2: nat] : ( member4220325220686508332m_rule @ ( snth_P1712579846118164081m_rule @ S3 @ N2 ) @ ( sset_P4979403222752902913m_rule @ S3 ) ) ).

% snth_sset
thf(fact_930_stream_Omap__ident__strong,axiom,
    ! [T2: stream_fm,F2: fm > fm] :
      ( ! [Z3: fm] :
          ( ( member_fm @ Z3 @ ( sset_fm @ T2 ) )
         => ( ( F2 @ Z3 )
            = Z3 ) )
     => ( ( smap_fm_fm @ F2 @ T2 )
        = T2 ) ) ).

% stream.map_ident_strong
thf(fact_931_stream_Omap__ident__strong,axiom,
    ! [T2: stream727092118206550309m_rule,F2: produc164195504107695125m_rule > produc164195504107695125m_rule] :
      ( ! [Z3: produc164195504107695125m_rule] :
          ( ( member4220325220686508332m_rule @ Z3 @ ( sset_P4979403222752902913m_rule @ T2 ) )
         => ( ( F2 @ Z3 )
            = Z3 ) )
     => ( ( smap_P743327730374515377m_rule @ F2 @ T2 )
        = T2 ) ) ).

% stream.map_ident_strong
thf(fact_932_shd__sset,axiom,
    ! [A: stream_fm] : ( member_fm @ ( shd_fm @ A ) @ ( sset_fm @ A ) ) ).

% shd_sset
thf(fact_933_shd__sset,axiom,
    ! [A: stream727092118206550309m_rule] : ( member4220325220686508332m_rule @ ( shd_Pr7235097944458474089m_rule @ A ) @ ( sset_P4979403222752902913m_rule @ A ) ) ).

% shd_sset
thf(fact_934_stream_Oset__map,axiom,
    ! [F2: produc164195504107695125m_rule > set_fm,V: stream727092118206550309m_rule] :
      ( ( sset_set_fm @ ( smap_P9035516689485476898set_fm @ F2 @ V ) )
      = ( image_5698427363735666338set_fm @ F2 @ ( sset_P4979403222752902913m_rule @ V ) ) ) ).

% stream.set_map
thf(fact_935_stream_Oset__map,axiom,
    ! [F2: produc164195504107695125m_rule > produc164195504107695125m_rule,V: stream727092118206550309m_rule] :
      ( ( sset_P4979403222752902913m_rule @ ( smap_P743327730374515377m_rule @ F2 @ V ) )
      = ( image_2192725901198026033m_rule @ F2 @ ( sset_P4979403222752902913m_rule @ V ) ) ) ).

% stream.set_map
thf(fact_936_sset__sdrop,axiom,
    ! [N2: nat,S3: stream8299795917829157543ist_fm] : ( ord_le2055072402069232929ist_fm @ ( sset_P3431374123360452149ist_fm @ ( sdrop_4442373711808556042ist_fm @ N2 @ S3 ) ) @ ( sset_P3431374123360452149ist_fm @ S3 ) ) ).

% sset_sdrop
thf(fact_937_sset__sdrop,axiom,
    ! [N2: nat,S3: stream_rule] : ( ord_less_eq_set_rule @ ( sset_rule @ ( sdrop_rule @ N2 @ S3 ) ) @ ( sset_rule @ S3 ) ) ).

% sset_sdrop
thf(fact_938_sset__sdrop,axiom,
    ! [N2: nat,S3: stream6494289010434245521m_rule] : ( ord_le3684904531561281547m_rule @ ( sset_P270791984657859871m_rule @ ( sdrop_7373388980841566196m_rule @ N2 @ S3 ) ) @ ( sset_P270791984657859871m_rule @ S3 ) ) ).

% sset_sdrop
thf(fact_939_sset__sdrop,axiom,
    ! [N2: nat,S3: stream727092118206550309m_rule] : ( ord_le5052251716133888043m_rule @ ( sset_P4979403222752902913m_rule @ ( sdrop_7224736112439592940m_rule @ N2 @ S3 ) ) @ ( sset_P4979403222752902913m_rule @ S3 ) ) ).

% sset_sdrop
thf(fact_940_sset__sproduct,axiom,
    ! [S1: stream727092118206550309m_rule,S2: stream727092118206550309m_rule] :
      ( ( sset_P270791984657859871m_rule @ ( sprodu3347975263877181204m_rule @ S1 @ S2 ) )
      = ( produc7959129826089203956m_rule @ ( sset_P4979403222752902913m_rule @ S1 )
        @ ^ [Uu: produc164195504107695125m_rule] : ( sset_P4979403222752902913m_rule @ S2 ) ) ) ).

% sset_sproduct
thf(fact_941_sset__sproduct,axiom,
    ! [S1: stream8299795917829157543ist_fm,S2: stream_rule] :
      ( ( sset_P4979403222752902913m_rule @ ( sprodu1224558879542060414m_rule @ S1 @ S2 ) )
      = ( produc1083232162533131166m_rule @ ( sset_P3431374123360452149ist_fm @ S1 )
        @ ^ [Uu: produc1996495991257130529ist_fm] : ( sset_rule @ S2 ) ) ) ).

% sset_sproduct
thf(fact_942_snth__sset__smerge,axiom,
    ! [Ss: stream_stream_fm,N2: nat,M: nat] : ( member_fm @ ( snth_fm @ ( snth_stream_fm @ Ss @ N2 ) @ M ) @ ( sset_fm @ ( smerge_fm @ Ss ) ) ) ).

% snth_sset_smerge
thf(fact_943_snth__sset__smerge,axiom,
    ! [Ss: stream3016090547575141045m_rule,N2: nat,M: nat] : ( member4220325220686508332m_rule @ ( snth_P1712579846118164081m_rule @ ( snth_s8878883179916095873m_rule @ Ss @ N2 ) @ M ) @ ( sset_P4979403222752902913m_rule @ ( smerge3869575255966183559m_rule @ Ss ) ) ) ).

% snth_sset_smerge
thf(fact_944_stream_Orel__compp__Grp,axiom,
    ( stream4404486423654466191ist_fm
    = ( ^ [R3: list_fm > list_fm > $o] :
          ( relcom1428920957874198710ist_fm
          @ ( conver1488781145210796898ist_fm
            @ ( bNF_Gr4037694065301483209ist_fm
              @ ( collec2577108820723035666ist_fm
                @ ^ [X3: stream8299795917829157543ist_fm] : ( ord_le2055072402069232929ist_fm @ ( sset_P3431374123360452149ist_fm @ X3 ) @ ( collec7231575776666803724ist_fm @ ( produc2011568048573023402t_fm_o @ R3 ) ) ) )
              @ ( smap_P9179663279415759932ist_fm @ produc1501393135466168645ist_fm ) ) )
          @ ( bNF_Gr4037694065301483209ist_fm
            @ ( collec2577108820723035666ist_fm
              @ ^ [X3: stream8299795917829157543ist_fm] : ( ord_le2055072402069232929ist_fm @ ( sset_P3431374123360452149ist_fm @ X3 ) @ ( collec7231575776666803724ist_fm @ ( produc2011568048573023402t_fm_o @ R3 ) ) ) )
            @ ( smap_P9179663279415759932ist_fm @ produc4588648349897876871ist_fm ) ) ) ) ) ).

% stream.rel_compp_Grp
thf(fact_945_stream_Orel__compp__Grp,axiom,
    ( stream7405667064201925977m_rule
    = ( ^ [R3: produc1996495991257130529ist_fm > rule > $o] :
          ( relcom7314577030940746930m_rule
          @ ( conver1931248174859038347ist_fm
            @ ( bNF_Gr7884858696775901554ist_fm
              @ ( collec3014615263049862906m_rule
                @ ^ [X3: stream727092118206550309m_rule] : ( ord_le5052251716133888043m_rule @ ( sset_P4979403222752902913m_rule @ X3 ) @ ( collec2069506698588392810m_rule @ ( produc7369492558480804444rule_o @ R3 ) ) ) )
              @ ( smap_P2929523447053962885ist_fm @ produc6879501374131015971m_rule ) ) )
          @ ( bNF_Gr274536878997047754m_rule
            @ ( collec3014615263049862906m_rule
              @ ^ [X3: stream727092118206550309m_rule] : ( ord_le5052251716133888043m_rule @ ( sset_P4979403222752902913m_rule @ X3 ) @ ( collec2069506698588392810m_rule @ ( produc7369492558480804444rule_o @ R3 ) ) ) )
            @ ( smap_P1838418287296133089e_rule @ produc7165828336582415457m_rule ) ) ) ) ) ).

% stream.rel_compp_Grp
thf(fact_946_stream_Orel__compp__Grp,axiom,
    ( stream4754582834681506829ist_fm
    = ( ^ [R3: rule > produc1996495991257130529ist_fm > $o] :
          ( relcom3194607263313142206ist_fm
          @ ( conver3801398595996610669m_rule
            @ ( bNF_Gr5681816557576858438m_rule
              @ ( collec4247836135612270198ist_fm
                @ ^ [X3: stream1960312990768957601ist_fm] : ( ord_le6285472588696295335ist_fm @ ( sset_P222701707499943037ist_fm @ X3 ) @ ( collec6536177220190208742ist_fm @ ( produc1628407760719572648t_fm_o @ R3 ) ) ) )
              @ ( smap_P3836375702350610781m_rule @ produc4228417144610596823ist_fm ) ) )
          @ ( bNF_Gr6802862692733128950ist_fm
            @ ( collec4247836135612270198ist_fm
              @ ^ [X3: stream1960312990768957601ist_fm] : ( ord_le6285472588696295335ist_fm @ ( sset_P222701707499943037ist_fm @ X3 ) @ ( collec6536177220190208742ist_fm @ ( produc1628407760719572648t_fm_o @ R3 ) ) ) )
            @ ( smap_P3387879760727102985ist_fm @ produc4514744107061996309ist_fm ) ) ) ) ) ).

% stream.rel_compp_Grp
thf(fact_947_stream_Oin__rel,axiom,
    ( stream4404486423654466191ist_fm
    = ( ^ [R3: list_fm > list_fm > $o,A5: stream_list_fm,B4: stream_list_fm] :
        ? [Z5: stream8299795917829157543ist_fm] :
          ( ( member5006235884692811216ist_fm @ Z5
            @ ( collec2577108820723035666ist_fm
              @ ^ [X3: stream8299795917829157543ist_fm] : ( ord_le2055072402069232929ist_fm @ ( sset_P3431374123360452149ist_fm @ X3 ) @ ( collec7231575776666803724ist_fm @ ( produc2011568048573023402t_fm_o @ R3 ) ) ) ) )
          & ( ( smap_P9179663279415759932ist_fm @ produc1501393135466168645ist_fm @ Z5 )
            = A5 )
          & ( ( smap_P9179663279415759932ist_fm @ produc4588648349897876871ist_fm @ Z5 )
            = B4 ) ) ) ) ).

% stream.in_rel
thf(fact_948_stream_Oin__rel,axiom,
    ( stream7405667064201925977m_rule
    = ( ^ [R3: produc1996495991257130529ist_fm > rule > $o,A5: stream8299795917829157543ist_fm,B4: stream_rule] :
        ? [Z5: stream727092118206550309m_rule] :
          ( ( member6117863459850561468m_rule @ Z5
            @ ( collec3014615263049862906m_rule
              @ ^ [X3: stream727092118206550309m_rule] : ( ord_le5052251716133888043m_rule @ ( sset_P4979403222752902913m_rule @ X3 ) @ ( collec2069506698588392810m_rule @ ( produc7369492558480804444rule_o @ R3 ) ) ) ) )
          & ( ( smap_P2929523447053962885ist_fm @ produc6879501374131015971m_rule @ Z5 )
            = A5 )
          & ( ( smap_P1838418287296133089e_rule @ produc7165828336582415457m_rule @ Z5 )
            = B4 ) ) ) ) ).

% stream.in_rel
thf(fact_949_stream_Oin__rel,axiom,
    ( stream4754582834681506829ist_fm
    = ( ^ [R3: rule > produc1996495991257130529ist_fm > $o,A5: stream_rule,B4: stream8299795917829157543ist_fm] :
        ? [Z5: stream1960312990768957601ist_fm] :
          ( ( member7351084332412968760ist_fm @ Z5
            @ ( collec4247836135612270198ist_fm
              @ ^ [X3: stream1960312990768957601ist_fm] : ( ord_le6285472588696295335ist_fm @ ( sset_P222701707499943037ist_fm @ X3 ) @ ( collec6536177220190208742ist_fm @ ( produc1628407760719572648t_fm_o @ R3 ) ) ) ) )
          & ( ( smap_P3836375702350610781m_rule @ produc4228417144610596823ist_fm @ Z5 )
            = A5 )
          & ( ( smap_P3387879760727102985ist_fm @ produc4514744107061996309ist_fm @ Z5 )
            = B4 ) ) ) ) ).

% stream.in_rel
thf(fact_950_stream_Orel__cong,axiom,
    ! [X2: stream_fm,Ya: stream_fm,Y3: stream_fm,Xa2: stream_fm,R: fm > fm > $o,Ra: fm > fm > $o] :
      ( ( X2 = Ya )
     => ( ( Y3 = Xa2 )
       => ( ! [Z3: fm,Yb: fm] :
              ( ( member_fm @ Z3 @ ( sset_fm @ Ya ) )
             => ( ( member_fm @ Yb @ ( sset_fm @ Xa2 ) )
               => ( ( R @ Z3 @ Yb )
                  = ( Ra @ Z3 @ Yb ) ) ) )
         => ( ( stream_all2_fm_fm @ R @ X2 @ Y3 )
            = ( stream_all2_fm_fm @ Ra @ Ya @ Xa2 ) ) ) ) ) ).

% stream.rel_cong
thf(fact_951_stream_Orel__cong,axiom,
    ! [X2: stream_fm,Ya: stream_fm,Y3: stream727092118206550309m_rule,Xa2: stream727092118206550309m_rule,R: fm > produc164195504107695125m_rule > $o,Ra: fm > produc164195504107695125m_rule > $o] :
      ( ( X2 = Ya )
     => ( ( Y3 = Xa2 )
       => ( ! [Z3: fm,Yb: produc164195504107695125m_rule] :
              ( ( member_fm @ Z3 @ ( sset_fm @ Ya ) )
             => ( ( member4220325220686508332m_rule @ Yb @ ( sset_P4979403222752902913m_rule @ Xa2 ) )
               => ( ( R @ Z3 @ Yb )
                  = ( Ra @ Z3 @ Yb ) ) ) )
         => ( ( stream7789294416889223348m_rule @ R @ X2 @ Y3 )
            = ( stream7789294416889223348m_rule @ Ra @ Ya @ Xa2 ) ) ) ) ) ).

% stream.rel_cong
thf(fact_952_stream_Orel__cong,axiom,
    ! [X2: stream727092118206550309m_rule,Ya: stream727092118206550309m_rule,Y3: stream_fm,Xa2: stream_fm,R: produc164195504107695125m_rule > fm > $o,Ra: produc164195504107695125m_rule > fm > $o] :
      ( ( X2 = Ya )
     => ( ( Y3 = Xa2 )
       => ( ! [Z3: produc164195504107695125m_rule,Yb: fm] :
              ( ( member4220325220686508332m_rule @ Z3 @ ( sset_P4979403222752902913m_rule @ Ya ) )
             => ( ( member_fm @ Yb @ ( sset_fm @ Xa2 ) )
               => ( ( R @ Z3 @ Yb )
                  = ( Ra @ Z3 @ Yb ) ) ) )
         => ( ( stream8779875082212741684ule_fm @ R @ X2 @ Y3 )
            = ( stream8779875082212741684ule_fm @ Ra @ Ya @ Xa2 ) ) ) ) ) ).

% stream.rel_cong
thf(fact_953_stream_Orel__cong,axiom,
    ! [X2: stream727092118206550309m_rule,Ya: stream727092118206550309m_rule,Y3: stream727092118206550309m_rule,Xa2: stream727092118206550309m_rule,R: produc164195504107695125m_rule > produc164195504107695125m_rule > $o,Ra: produc164195504107695125m_rule > produc164195504107695125m_rule > $o] :
      ( ( X2 = Ya )
     => ( ( Y3 = Xa2 )
       => ( ! [Z3: produc164195504107695125m_rule,Yb: produc164195504107695125m_rule] :
              ( ( member4220325220686508332m_rule @ Z3 @ ( sset_P4979403222752902913m_rule @ Ya ) )
             => ( ( member4220325220686508332m_rule @ Yb @ ( sset_P4979403222752902913m_rule @ Xa2 ) )
               => ( ( R @ Z3 @ Yb )
                  = ( Ra @ Z3 @ Yb ) ) ) )
         => ( ( stream1316448123751145209m_rule @ R @ X2 @ Y3 )
            = ( stream1316448123751145209m_rule @ Ra @ Ya @ Xa2 ) ) ) ) ) ).

% stream.rel_cong
thf(fact_954_stream_Orel__mono__strong,axiom,
    ! [R: fm > fm > $o,X2: stream_fm,Y3: stream_fm,Ra: fm > fm > $o] :
      ( ( stream_all2_fm_fm @ R @ X2 @ Y3 )
     => ( ! [Z3: fm,Yb: fm] :
            ( ( member_fm @ Z3 @ ( sset_fm @ X2 ) )
           => ( ( member_fm @ Yb @ ( sset_fm @ Y3 ) )
             => ( ( R @ Z3 @ Yb )
               => ( Ra @ Z3 @ Yb ) ) ) )
       => ( stream_all2_fm_fm @ Ra @ X2 @ Y3 ) ) ) ).

% stream.rel_mono_strong
thf(fact_955_stream_Orel__mono__strong,axiom,
    ! [R: fm > produc164195504107695125m_rule > $o,X2: stream_fm,Y3: stream727092118206550309m_rule,Ra: fm > produc164195504107695125m_rule > $o] :
      ( ( stream7789294416889223348m_rule @ R @ X2 @ Y3 )
     => ( ! [Z3: fm,Yb: produc164195504107695125m_rule] :
            ( ( member_fm @ Z3 @ ( sset_fm @ X2 ) )
           => ( ( member4220325220686508332m_rule @ Yb @ ( sset_P4979403222752902913m_rule @ Y3 ) )
             => ( ( R @ Z3 @ Yb )
               => ( Ra @ Z3 @ Yb ) ) ) )
       => ( stream7789294416889223348m_rule @ Ra @ X2 @ Y3 ) ) ) ).

% stream.rel_mono_strong
thf(fact_956_stream_Orel__mono__strong,axiom,
    ! [R: produc164195504107695125m_rule > fm > $o,X2: stream727092118206550309m_rule,Y3: stream_fm,Ra: produc164195504107695125m_rule > fm > $o] :
      ( ( stream8779875082212741684ule_fm @ R @ X2 @ Y3 )
     => ( ! [Z3: produc164195504107695125m_rule,Yb: fm] :
            ( ( member4220325220686508332m_rule @ Z3 @ ( sset_P4979403222752902913m_rule @ X2 ) )
           => ( ( member_fm @ Yb @ ( sset_fm @ Y3 ) )
             => ( ( R @ Z3 @ Yb )
               => ( Ra @ Z3 @ Yb ) ) ) )
       => ( stream8779875082212741684ule_fm @ Ra @ X2 @ Y3 ) ) ) ).

% stream.rel_mono_strong
thf(fact_957_stream_Orel__mono__strong,axiom,
    ! [R: produc164195504107695125m_rule > produc164195504107695125m_rule > $o,X2: stream727092118206550309m_rule,Y3: stream727092118206550309m_rule,Ra: produc164195504107695125m_rule > produc164195504107695125m_rule > $o] :
      ( ( stream1316448123751145209m_rule @ R @ X2 @ Y3 )
     => ( ! [Z3: produc164195504107695125m_rule,Yb: produc164195504107695125m_rule] :
            ( ( member4220325220686508332m_rule @ Z3 @ ( sset_P4979403222752902913m_rule @ X2 ) )
           => ( ( member4220325220686508332m_rule @ Yb @ ( sset_P4979403222752902913m_rule @ Y3 ) )
             => ( ( R @ Z3 @ Yb )
               => ( Ra @ Z3 @ Yb ) ) ) )
       => ( stream1316448123751145209m_rule @ Ra @ X2 @ Y3 ) ) ) ).

% stream.rel_mono_strong
thf(fact_958_stream_Orel__refl__strong,axiom,
    ! [X2: stream_fm,Ra: fm > fm > $o] :
      ( ! [Z3: fm] :
          ( ( member_fm @ Z3 @ ( sset_fm @ X2 ) )
         => ( Ra @ Z3 @ Z3 ) )
     => ( stream_all2_fm_fm @ Ra @ X2 @ X2 ) ) ).

% stream.rel_refl_strong
thf(fact_959_stream_Orel__refl__strong,axiom,
    ! [X2: stream727092118206550309m_rule,Ra: produc164195504107695125m_rule > produc164195504107695125m_rule > $o] :
      ( ! [Z3: produc164195504107695125m_rule] :
          ( ( member4220325220686508332m_rule @ Z3 @ ( sset_P4979403222752902913m_rule @ X2 ) )
         => ( Ra @ Z3 @ Z3 ) )
     => ( stream1316448123751145209m_rule @ Ra @ X2 @ X2 ) ) ).

% stream.rel_refl_strong
thf(fact_960_stream_Osel__transfer_I1_J,axiom,
    ! [R: produc164195504107695125m_rule > produc164195504107695125m_rule > $o] : ( bNF_re6841268034828329429m_rule @ ( stream1316448123751145209m_rule @ R ) @ R @ shd_Pr7235097944458474089m_rule @ shd_Pr7235097944458474089m_rule ) ).

% stream.sel_transfer(1)
thf(fact_961_sset__smerge,axiom,
    ! [Ss: stream3016090547575141045m_rule] :
      ( ( sset_P4979403222752902913m_rule @ ( smerge3869575255966183559m_rule @ Ss ) )
      = ( comple2712794622220498846m_rule @ ( image_1704168677691122935m_rule @ sset_P4979403222752902913m_rule @ ( sset_s15734764229120529m_rule @ Ss ) ) ) ) ).

% sset_smerge
thf(fact_962_sset__smerge,axiom,
    ! [Ss: stream_stream_fm] :
      ( ( sset_fm @ ( smerge_fm @ Ss ) )
      = ( comple7677669044296858969set_fm @ ( image_2997072598145739501set_fm @ sset_fm @ ( sset_stream_fm @ Ss ) ) ) ) ).

% sset_smerge
thf(fact_963_sset__sconst,axiom,
    ! [X2: produc164195504107695125m_rule] :
      ( ( sset_P4979403222752902913m_rule @ ( sitera7638444125321546697m_rule @ id_Pro7923832310103879396m_rule @ X2 ) )
      = ( insert8332210346540479045m_rule @ X2 @ bot_bo2182007826548605023m_rule ) ) ).

% sset_sconst
thf(fact_964_insert__subsetI,axiom,
    ! [X2: fm,A4: set_fm,X5: set_fm] :
      ( ( member_fm @ X2 @ A4 )
     => ( ( ord_less_eq_set_fm @ X5 @ A4 )
       => ( ord_less_eq_set_fm @ ( insert_fm2 @ X2 @ X5 ) @ A4 ) ) ) ).

% insert_subsetI
thf(fact_965_sconst__alt,axiom,
    ! [S3: stream727092118206550309m_rule,X2: produc164195504107695125m_rule] :
      ( ( S3
        = ( sitera7638444125321546697m_rule @ id_Pro7923832310103879396m_rule @ X2 ) )
      = ( ( sset_P4979403222752902913m_rule @ S3 )
        = ( insert8332210346540479045m_rule @ X2 @ bot_bo2182007826548605023m_rule ) ) ) ).

% sconst_alt
thf(fact_966_set__rec,axiom,
    ( set_fm2
    = ( rec_list_set_fm_fm @ bot_bot_set_fm
      @ ^ [X3: fm,Uu: list_fm] : ( insert_fm2 @ X3 ) ) ) ).

% set_rec
thf(fact_967_stream_Opred__mono__strong,axiom,
    ! [P2: fm > $o,X2: stream_fm,Pa: fm > $o] :
      ( ( pred_stream_fm @ P2 @ X2 )
     => ( ! [Z3: fm] :
            ( ( member_fm @ Z3 @ ( sset_fm @ X2 ) )
           => ( ( P2 @ Z3 )
             => ( Pa @ Z3 ) ) )
       => ( pred_stream_fm @ Pa @ X2 ) ) ) ).

% stream.pred_mono_strong
thf(fact_968_stream_Opred__mono__strong,axiom,
    ! [P2: produc164195504107695125m_rule > $o,X2: stream727092118206550309m_rule,Pa: produc164195504107695125m_rule > $o] :
      ( ( pred_s4066988841477964019m_rule @ P2 @ X2 )
     => ( ! [Z3: produc164195504107695125m_rule] :
            ( ( member4220325220686508332m_rule @ Z3 @ ( sset_P4979403222752902913m_rule @ X2 ) )
           => ( ( P2 @ Z3 )
             => ( Pa @ Z3 ) ) )
       => ( pred_s4066988841477964019m_rule @ Pa @ X2 ) ) ) ).

% stream.pred_mono_strong
thf(fact_969_stream_Opred__cong,axiom,
    ! [X2: stream_fm,Ya: stream_fm,P2: fm > $o,Pa: fm > $o] :
      ( ( X2 = Ya )
     => ( ! [Z3: fm] :
            ( ( member_fm @ Z3 @ ( sset_fm @ Ya ) )
           => ( ( P2 @ Z3 )
              = ( Pa @ Z3 ) ) )
       => ( ( pred_stream_fm @ P2 @ X2 )
          = ( pred_stream_fm @ Pa @ Ya ) ) ) ) ).

% stream.pred_cong
thf(fact_970_stream_Opred__cong,axiom,
    ! [X2: stream727092118206550309m_rule,Ya: stream727092118206550309m_rule,P2: produc164195504107695125m_rule > $o,Pa: produc164195504107695125m_rule > $o] :
      ( ( X2 = Ya )
     => ( ! [Z3: produc164195504107695125m_rule] :
            ( ( member4220325220686508332m_rule @ Z3 @ ( sset_P4979403222752902913m_rule @ Ya ) )
           => ( ( P2 @ Z3 )
              = ( Pa @ Z3 ) ) )
       => ( ( pred_s4066988841477964019m_rule @ P2 @ X2 )
          = ( pred_s4066988841477964019m_rule @ Pa @ Ya ) ) ) ) ).

% stream.pred_cong
thf(fact_971_shd__def,axiom,
    ( shd_Pr7235097944458474089m_rule
    = ( case_s4599053733867249880m_rule
      @ ^ [X12: produc164195504107695125m_rule,X23: stream727092118206550309m_rule] : X12 ) ) ).

% shd_def
thf(fact_972_stream_Oset,axiom,
    ! [X1: produc164195504107695125m_rule,X22: stream727092118206550309m_rule] :
      ( ( sset_P4979403222752902913m_rule @ ( sCons_5731567480799343806m_rule @ X1 @ X22 ) )
      = ( insert8332210346540479045m_rule @ X1 @ ( sset_P4979403222752902913m_rule @ X22 ) ) ) ).

% stream.set
thf(fact_973_prod_Opred__cong,axiom,
    ! [X2: product_prod_fm_fm,Ya: product_prod_fm_fm,P1: fm > $o,P1a: fm > $o,P22: fm > $o,P2a: fm > $o] :
      ( ( X2 = Ya )
     => ( ! [Z1: fm] :
            ( ( member_fm @ Z1 @ ( basic_fsts_fm_fm @ Ya ) )
           => ( ( P1 @ Z1 )
              = ( P1a @ Z1 ) ) )
       => ( ! [Z22: fm] :
              ( ( member_fm @ Z22 @ ( basic_snds_fm_fm @ Ya ) )
             => ( ( P22 @ Z22 )
                = ( P2a @ Z22 ) ) )
         => ( ( basic_6733234405827732053_fm_fm @ P1 @ P22 @ X2 )
            = ( basic_6733234405827732053_fm_fm @ P1a @ P2a @ Ya ) ) ) ) ) ).

% prod.pred_cong
thf(fact_974_prod_Opred__cong,axiom,
    ! [X2: produc1996495991257130529ist_fm,Ya: produc1996495991257130529ist_fm,P1: list_fm > $o,P1a: list_fm > $o,P22: list_fm > $o,P2a: list_fm > $o] :
      ( ( X2 = Ya )
     => ( ! [Z1: list_fm] :
            ( ( member_list_fm @ Z1 @ ( basic_2778179344159109705ist_fm @ Ya ) )
           => ( ( P1 @ Z1 )
              = ( P1a @ Z1 ) ) )
       => ( ! [Z22: list_fm] :
              ( ( member_list_fm @ Z22 @ ( basic_6053177629474610823ist_fm @ Ya ) )
             => ( ( P22 @ Z22 )
                = ( P2a @ Z22 ) ) )
         => ( ( basic_5093365364719461109ist_fm @ P1 @ P22 @ X2 )
            = ( basic_5093365364719461109ist_fm @ P1a @ P2a @ Ya ) ) ) ) ) ).

% prod.pred_cong
thf(fact_975_prod_Opred__cong,axiom,
    ! [X2: produc164195504107695125m_rule,Ya: produc164195504107695125m_rule,P1: produc1996495991257130529ist_fm > $o,P1a: produc1996495991257130529ist_fm > $o,P22: rule > $o,P2a: rule > $o] :
      ( ( X2 = Ya )
     => ( ! [Z1: produc1996495991257130529ist_fm] :
            ( ( member8102475879199740618ist_fm @ Z1 @ ( basic_4306050201691295519m_rule @ Ya ) )
           => ( ( P1 @ Z1 )
              = ( P1a @ Z1 ) ) )
       => ( ! [Z22: rule] :
              ( ( member_rule @ Z22 @ ( basic_8282993014722632545m_rule @ Ya ) )
             => ( ( P22 @ Z22 )
                = ( P2a @ Z22 ) ) )
         => ( ( basic_6171360182465397619m_rule @ P1 @ P22 @ X2 )
            = ( basic_6171360182465397619m_rule @ P1a @ P2a @ Ya ) ) ) ) ) ).

% prod.pred_cong
thf(fact_976_prod_Opred__mono__strong,axiom,
    ! [P1: fm > $o,P22: fm > $o,X2: product_prod_fm_fm,P1a: fm > $o,P2a: fm > $o] :
      ( ( basic_6733234405827732053_fm_fm @ P1 @ P22 @ X2 )
     => ( ! [Z1: fm] :
            ( ( member_fm @ Z1 @ ( basic_fsts_fm_fm @ X2 ) )
           => ( ( P1 @ Z1 )
             => ( P1a @ Z1 ) ) )
       => ( ! [Z22: fm] :
              ( ( member_fm @ Z22 @ ( basic_snds_fm_fm @ X2 ) )
             => ( ( P22 @ Z22 )
               => ( P2a @ Z22 ) ) )
         => ( basic_6733234405827732053_fm_fm @ P1a @ P2a @ X2 ) ) ) ) ).

% prod.pred_mono_strong
thf(fact_977_prod_Opred__mono__strong,axiom,
    ! [P1: list_fm > $o,P22: list_fm > $o,X2: produc1996495991257130529ist_fm,P1a: list_fm > $o,P2a: list_fm > $o] :
      ( ( basic_5093365364719461109ist_fm @ P1 @ P22 @ X2 )
     => ( ! [Z1: list_fm] :
            ( ( member_list_fm @ Z1 @ ( basic_2778179344159109705ist_fm @ X2 ) )
           => ( ( P1 @ Z1 )
             => ( P1a @ Z1 ) ) )
       => ( ! [Z22: list_fm] :
              ( ( member_list_fm @ Z22 @ ( basic_6053177629474610823ist_fm @ X2 ) )
             => ( ( P22 @ Z22 )
               => ( P2a @ Z22 ) ) )
         => ( basic_5093365364719461109ist_fm @ P1a @ P2a @ X2 ) ) ) ) ).

% prod.pred_mono_strong
thf(fact_978_prod_Opred__mono__strong,axiom,
    ! [P1: produc1996495991257130529ist_fm > $o,P22: rule > $o,X2: produc164195504107695125m_rule,P1a: produc1996495991257130529ist_fm > $o,P2a: rule > $o] :
      ( ( basic_6171360182465397619m_rule @ P1 @ P22 @ X2 )
     => ( ! [Z1: produc1996495991257130529ist_fm] :
            ( ( member8102475879199740618ist_fm @ Z1 @ ( basic_4306050201691295519m_rule @ X2 ) )
           => ( ( P1 @ Z1 )
             => ( P1a @ Z1 ) ) )
       => ( ! [Z22: rule] :
              ( ( member_rule @ Z22 @ ( basic_8282993014722632545m_rule @ X2 ) )
             => ( ( P22 @ Z22 )
               => ( P2a @ Z22 ) ) )
         => ( basic_6171360182465397619m_rule @ P1a @ P2a @ X2 ) ) ) ) ).

% prod.pred_mono_strong
thf(fact_979_prod_Omap__ident__strong,axiom,
    ! [T2: product_prod_fm_fm,F12: fm > fm,F23: fm > fm] :
      ( ! [Z1: fm] :
          ( ( member_fm @ Z1 @ ( basic_fsts_fm_fm @ T2 ) )
         => ( ( F12 @ Z1 )
            = Z1 ) )
     => ( ! [Z22: fm] :
            ( ( member_fm @ Z22 @ ( basic_snds_fm_fm @ T2 ) )
           => ( ( F23 @ Z22 )
              = Z22 ) )
       => ( ( produc8016985811614767674_fm_fm @ F12 @ F23 @ T2 )
          = T2 ) ) ) ).

% prod.map_ident_strong
thf(fact_980_stream_Osel_I1_J,axiom,
    ! [X1: produc164195504107695125m_rule,X22: stream727092118206550309m_rule] :
      ( ( shd_Pr7235097944458474089m_rule @ ( sCons_5731567480799343806m_rule @ X1 @ X22 ) )
      = X1 ) ).

% stream.sel(1)
thf(fact_981_szip__unfold,axiom,
    ! [A: produc1996495991257130529ist_fm,S1: stream8299795917829157543ist_fm,B: rule,S2: stream_rule] :
      ( ( szip_P1977448745965526924m_rule @ ( sCons_307821682964077496ist_fm @ A @ S1 ) @ ( sCons_rule @ B @ S2 ) )
      = ( sCons_5731567480799343806m_rule @ ( produc491467635432902671m_rule @ A @ B ) @ ( szip_P1977448745965526924m_rule @ S1 @ S2 ) ) ) ).

% szip_unfold
thf(fact_982_szip__unfold,axiom,
    ! [A: produc164195504107695125m_rule,S1: stream727092118206550309m_rule,B: produc164195504107695125m_rule,S2: stream727092118206550309m_rule] :
      ( ( szip_P2499414959592755846m_rule @ ( sCons_5731567480799343806m_rule @ A @ S1 ) @ ( sCons_5731567480799343806m_rule @ B @ S2 ) )
      = ( sCons_4186928284053167522m_rule @ ( produc5927390650430071747m_rule @ A @ B ) @ ( szip_P2499414959592755846m_rule @ S1 @ S2 ) ) ) ).

% szip_unfold
thf(fact_983_stream_Oset__induct,axiom,
    ! [X2: fm,A: stream_fm,P2: fm > stream_fm > $o] :
      ( ( member_fm @ X2 @ ( sset_fm @ A ) )
     => ( ! [Z1: fm,Z22: stream_fm] : ( P2 @ Z1 @ ( sCons_fm @ Z1 @ Z22 ) )
       => ( ! [Z1: fm,Z22: stream_fm,Xa: fm] :
              ( ( member_fm @ Xa @ ( sset_fm @ Z22 ) )
             => ( ( P2 @ Xa @ Z22 )
               => ( P2 @ Xa @ ( sCons_fm @ Z1 @ Z22 ) ) ) )
         => ( P2 @ X2 @ A ) ) ) ) ).

% stream.set_induct
thf(fact_984_stream_Oset__induct,axiom,
    ! [X2: produc164195504107695125m_rule,A: stream727092118206550309m_rule,P2: produc164195504107695125m_rule > stream727092118206550309m_rule > $o] :
      ( ( member4220325220686508332m_rule @ X2 @ ( sset_P4979403222752902913m_rule @ A ) )
     => ( ! [Z1: produc164195504107695125m_rule,Z22: stream727092118206550309m_rule] : ( P2 @ Z1 @ ( sCons_5731567480799343806m_rule @ Z1 @ Z22 ) )
       => ( ! [Z1: produc164195504107695125m_rule,Z22: stream727092118206550309m_rule,Xa: produc164195504107695125m_rule] :
              ( ( member4220325220686508332m_rule @ Xa @ ( sset_P4979403222752902913m_rule @ Z22 ) )
             => ( ( P2 @ Xa @ Z22 )
               => ( P2 @ Xa @ ( sCons_5731567480799343806m_rule @ Z1 @ Z22 ) ) ) )
         => ( P2 @ X2 @ A ) ) ) ) ).

% stream.set_induct
thf(fact_985_stream_Oset__cases,axiom,
    ! [E2: fm,A: stream_fm] :
      ( ( member_fm @ E2 @ ( sset_fm @ A ) )
     => ( ! [Z22: stream_fm] :
            ( A
           != ( sCons_fm @ E2 @ Z22 ) )
       => ~ ! [Z1: fm,Z22: stream_fm] :
              ( ( A
                = ( sCons_fm @ Z1 @ Z22 ) )
             => ~ ( member_fm @ E2 @ ( sset_fm @ Z22 ) ) ) ) ) ).

% stream.set_cases
thf(fact_986_stream_Oset__cases,axiom,
    ! [E2: produc164195504107695125m_rule,A: stream727092118206550309m_rule] :
      ( ( member4220325220686508332m_rule @ E2 @ ( sset_P4979403222752902913m_rule @ A ) )
     => ( ! [Z22: stream727092118206550309m_rule] :
            ( A
           != ( sCons_5731567480799343806m_rule @ E2 @ Z22 ) )
       => ~ ! [Z1: produc164195504107695125m_rule,Z22: stream727092118206550309m_rule] :
              ( ( A
                = ( sCons_5731567480799343806m_rule @ Z1 @ Z22 ) )
             => ~ ( member4220325220686508332m_rule @ E2 @ ( sset_P4979403222752902913m_rule @ Z22 ) ) ) ) ) ).

% stream.set_cases
thf(fact_987_stream_Oset__intros_I1_J,axiom,
    ! [X1: fm,X22: stream_fm] : ( member_fm @ X1 @ ( sset_fm @ ( sCons_fm @ X1 @ X22 ) ) ) ).

% stream.set_intros(1)
thf(fact_988_stream_Oset__intros_I1_J,axiom,
    ! [X1: produc164195504107695125m_rule,X22: stream727092118206550309m_rule] : ( member4220325220686508332m_rule @ X1 @ ( sset_P4979403222752902913m_rule @ ( sCons_5731567480799343806m_rule @ X1 @ X22 ) ) ) ).

% stream.set_intros(1)
thf(fact_989_stream_Oset__intros_I2_J,axiom,
    ! [Y3: fm,X22: stream_fm,X1: fm] :
      ( ( member_fm @ Y3 @ ( sset_fm @ X22 ) )
     => ( member_fm @ Y3 @ ( sset_fm @ ( sCons_fm @ X1 @ X22 ) ) ) ) ).

% stream.set_intros(2)
thf(fact_990_stream_Oset__intros_I2_J,axiom,
    ! [Y3: produc164195504107695125m_rule,X22: stream727092118206550309m_rule,X1: produc164195504107695125m_rule] :
      ( ( member4220325220686508332m_rule @ Y3 @ ( sset_P4979403222752902913m_rule @ X22 ) )
     => ( member4220325220686508332m_rule @ Y3 @ ( sset_P4979403222752902913m_rule @ ( sCons_5731567480799343806m_rule @ X1 @ X22 ) ) ) ) ).

% stream.set_intros(2)
thf(fact_991_fsts_Ocases,axiom,
    ! [A: produc1996495991257130529ist_fm,P: produc164195504107695125m_rule] :
      ( ( member8102475879199740618ist_fm @ A @ ( basic_4306050201691295519m_rule @ P ) )
     => ( A
        = ( produc6879501374131015971m_rule @ P ) ) ) ).

% fsts.cases
thf(fact_992_fsts_Ocases,axiom,
    ! [A: list_fm,P: produc1996495991257130529ist_fm] :
      ( ( member_list_fm @ A @ ( basic_2778179344159109705ist_fm @ P ) )
     => ( A
        = ( produc1501393135466168645ist_fm @ P ) ) ) ).

% fsts.cases
thf(fact_993_fsts_Osimps,axiom,
    ! [A: produc1996495991257130529ist_fm,P: produc164195504107695125m_rule] :
      ( ( member8102475879199740618ist_fm @ A @ ( basic_4306050201691295519m_rule @ P ) )
      = ( A
        = ( produc6879501374131015971m_rule @ P ) ) ) ).

% fsts.simps
thf(fact_994_fsts_Osimps,axiom,
    ! [A: list_fm,P: produc1996495991257130529ist_fm] :
      ( ( member_list_fm @ A @ ( basic_2778179344159109705ist_fm @ P ) )
      = ( A
        = ( produc1501393135466168645ist_fm @ P ) ) ) ).

% fsts.simps
thf(fact_995_fsts_Ointros,axiom,
    ! [P: produc164195504107695125m_rule] : ( member8102475879199740618ist_fm @ ( produc6879501374131015971m_rule @ P ) @ ( basic_4306050201691295519m_rule @ P ) ) ).

% fsts.intros
thf(fact_996_fsts_Ointros,axiom,
    ! [P: produc1996495991257130529ist_fm] : ( member_list_fm @ ( produc1501393135466168645ist_fm @ P ) @ ( basic_2778179344159109705ist_fm @ P ) ) ).

% fsts.intros
thf(fact_997_snds_Ocases,axiom,
    ! [A: list_fm,P: produc1996495991257130529ist_fm] :
      ( ( member_list_fm @ A @ ( basic_6053177629474610823ist_fm @ P ) )
     => ( A
        = ( produc4588648349897876871ist_fm @ P ) ) ) ).

% snds.cases
thf(fact_998_snds_Ocases,axiom,
    ! [A: rule,P: produc164195504107695125m_rule] :
      ( ( member_rule @ A @ ( basic_8282993014722632545m_rule @ P ) )
     => ( A
        = ( produc7165828336582415457m_rule @ P ) ) ) ).

% snds.cases
thf(fact_999_snds_Ocases,axiom,
    ! [A: produc1996495991257130529ist_fm,P: produc4630866025709511057ist_fm] :
      ( ( member8102475879199740618ist_fm @ A @ ( basic_5631908785202213397ist_fm @ P ) )
     => ( A
        = ( produc4514744107061996309ist_fm @ P ) ) ) ).

% snds.cases
thf(fact_1000_snds_Osimps,axiom,
    ! [A: list_fm,P: produc1996495991257130529ist_fm] :
      ( ( member_list_fm @ A @ ( basic_6053177629474610823ist_fm @ P ) )
      = ( A
        = ( produc4588648349897876871ist_fm @ P ) ) ) ).

% snds.simps
thf(fact_1001_snds_Osimps,axiom,
    ! [A: rule,P: produc164195504107695125m_rule] :
      ( ( member_rule @ A @ ( basic_8282993014722632545m_rule @ P ) )
      = ( A
        = ( produc7165828336582415457m_rule @ P ) ) ) ).

% snds.simps
thf(fact_1002_snds_Osimps,axiom,
    ! [A: produc1996495991257130529ist_fm,P: produc4630866025709511057ist_fm] :
      ( ( member8102475879199740618ist_fm @ A @ ( basic_5631908785202213397ist_fm @ P ) )
      = ( A
        = ( produc4514744107061996309ist_fm @ P ) ) ) ).

% snds.simps
thf(fact_1003_snds_Ointros,axiom,
    ! [P: produc1996495991257130529ist_fm] : ( member_list_fm @ ( produc4588648349897876871ist_fm @ P ) @ ( basic_6053177629474610823ist_fm @ P ) ) ).

% snds.intros
thf(fact_1004_snds_Ointros,axiom,
    ! [P: produc164195504107695125m_rule] : ( member_rule @ ( produc7165828336582415457m_rule @ P ) @ ( basic_8282993014722632545m_rule @ P ) ) ).

% snds.intros
thf(fact_1005_snds_Ointros,axiom,
    ! [P: produc4630866025709511057ist_fm] : ( member8102475879199740618ist_fm @ ( produc4514744107061996309ist_fm @ P ) @ ( basic_5631908785202213397ist_fm @ P ) ) ).

% snds.intros
thf(fact_1006_snth__Stream,axiom,
    ! [X2: produc164195504107695125m_rule,S3: stream727092118206550309m_rule,I: nat] :
      ( ( snth_P1712579846118164081m_rule @ ( sCons_5731567480799343806m_rule @ X2 @ S3 ) @ ( suc @ I ) )
      = ( snth_P1712579846118164081m_rule @ S3 @ I ) ) ).

% snth_Stream
thf(fact_1007_prod__set__defs_I1_J,axiom,
    ( basic_4306050201691295519m_rule
    = ( ^ [P4: produc164195504107695125m_rule] : ( insert589278167076416497ist_fm @ ( produc6879501374131015971m_rule @ P4 ) @ bot_bo4530011010606209517ist_fm ) ) ) ).

% prod_set_defs(1)
thf(fact_1008_prod__set__defs_I1_J,axiom,
    ( basic_2778179344159109705ist_fm
    = ( ^ [P4: produc1996495991257130529ist_fm] : ( insert_list_fm @ ( produc1501393135466168645ist_fm @ P4 ) @ bot_bot_set_list_fm ) ) ) ).

% prod_set_defs(1)
thf(fact_1009_prod__set__defs_I2_J,axiom,
    ( basic_6053177629474610823ist_fm
    = ( ^ [P4: produc1996495991257130529ist_fm] : ( insert_list_fm @ ( produc4588648349897876871ist_fm @ P4 ) @ bot_bot_set_list_fm ) ) ) ).

% prod_set_defs(2)
thf(fact_1010_prod__set__defs_I2_J,axiom,
    ( basic_8282993014722632545m_rule
    = ( ^ [P4: produc164195504107695125m_rule] : ( insert_rule @ ( produc7165828336582415457m_rule @ P4 ) @ bot_bot_set_rule ) ) ) ).

% prod_set_defs(2)
thf(fact_1011_prod__set__defs_I2_J,axiom,
    ( basic_5631908785202213397ist_fm
    = ( ^ [P4: produc4630866025709511057ist_fm] : ( insert589278167076416497ist_fm @ ( produc4514744107061996309ist_fm @ P4 ) @ bot_bo4530011010606209517ist_fm ) ) ) ).

% prod_set_defs(2)
thf(fact_1012_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_1013_SUP__set__fold,axiom,
    ! [F2: produc164195504107695125m_rule > set_fm,Xs: list_P8332459875677254181m_rule] :
      ( ( comple7677669044296858969set_fm @ ( image_5698427363735666338set_fm @ F2 @ ( set_Pr8505323785428441536m_rule @ Xs ) ) )
      = ( fold_P2789900844533598888set_fm @ ( comp_s7100926763290858283m_rule @ sup_sup_set_fm @ F2 ) @ Xs @ bot_bot_set_fm ) ) ).

% SUP_set_fold
thf(fact_1014_SUP__set__fold,axiom,
    ! [F2: fm > set_fm,Xs: list_fm] :
      ( ( comple7677669044296858969set_fm @ ( image_fm_set_fm @ F2 @ ( set_fm2 @ Xs ) ) )
      = ( fold_fm_set_fm @ ( comp_s2169399591454099046_fm_fm @ sup_sup_set_fm @ F2 ) @ Xs @ bot_bot_set_fm ) ) ).

% SUP_set_fold
thf(fact_1015_Field__Union,axiom,
    ! [R: set_se5029697570998406465_fm_fm] :
      ( ( field_fm @ ( comple8500535872550171342_fm_fm @ R ) )
      = ( comple7677669044296858969set_fm @ ( image_8325389782290646562set_fm @ field_fm @ R ) ) ) ).

% Field_Union
thf(fact_1016_sup__Un__eq,axiom,
    ! [R: set_fm,S6: set_fm] :
      ( ( sup_sup_fm_o
        @ ^ [X3: fm] : ( member_fm @ X3 @ R )
        @ ^ [X3: fm] : ( member_fm @ X3 @ S6 ) )
      = ( ^ [X3: fm] : ( member_fm @ X3 @ ( sup_sup_set_fm @ R @ S6 ) ) ) ) ).

% sup_Un_eq
thf(fact_1017_Sup__SUP__eq,axiom,
    ( comple3343679655671442028p_fm_o
    = ( ^ [S5: set_fm_o,X3: fm] : ( member_fm @ X3 @ ( comple7677669044296858969set_fm @ ( image_fm_o_set_fm @ collect_fm @ S5 ) ) ) ) ) ).

% Sup_SUP_eq
thf(fact_1018_SUP__Sup__eq,axiom,
    ! [S6: set_set_fm] :
      ( ( comple3343679655671442028p_fm_o
        @ ( image_set_fm_fm_o
          @ ^ [I2: set_fm,X3: fm] : ( member_fm @ X3 @ I2 )
          @ S6 ) )
      = ( ^ [X3: fm] : ( member_fm @ X3 @ ( comple7677669044296858969set_fm @ S6 ) ) ) ) ).

% SUP_Sup_eq
thf(fact_1019_bot__empty__eq,axiom,
    ( bot_bot_fm_o
    = ( ^ [X3: fm] : ( member_fm @ X3 @ bot_bot_set_fm ) ) ) ).

% bot_empty_eq
thf(fact_1020_FieldI2,axiom,
    ! [I: fm,J2: fm,R: set_Pr4463079037648049377_fm_fm] :
      ( ( member8474499337054950954_fm_fm @ ( product_Pair_fm_fm @ I @ J2 ) @ R )
     => ( member_fm @ J2 @ ( field_fm @ R ) ) ) ).

% FieldI2
thf(fact_1021_FieldI1,axiom,
    ! [I: fm,J2: fm,R: set_Pr4463079037648049377_fm_fm] :
      ( ( member8474499337054950954_fm_fm @ ( product_Pair_fm_fm @ I @ J2 ) @ R )
     => ( member_fm @ I @ ( field_fm @ R ) ) ) ).

% FieldI1
thf(fact_1022_SUP__UN__eq,axiom,
    ! [R2: produc164195504107695125m_rule > set_fm,S6: set_Pr1008144964186165195m_rule] :
      ( ( comple3343679655671442028p_fm_o
        @ ( image_1843934041337696867e_fm_o
          @ ^ [I2: produc164195504107695125m_rule,X3: fm] : ( member_fm @ X3 @ ( R2 @ I2 ) )
          @ S6 ) )
      = ( ^ [X3: fm] : ( member_fm @ X3 @ ( comple7677669044296858969set_fm @ ( image_5698427363735666338set_fm @ R2 @ S6 ) ) ) ) ) ).

% SUP_UN_eq
thf(fact_1023_union__set__fold,axiom,
    ! [Xs: list_fm,A4: set_fm] :
      ( ( sup_sup_set_fm @ ( set_fm2 @ Xs ) @ A4 )
      = ( fold_fm_set_fm @ insert_fm2 @ Xs @ A4 ) ) ).

% union_set_fold
thf(fact_1024_Stream_Osmember__def,axiom,
    ( smember_fm
    = ( ^ [X3: fm,S: stream_fm] : ( member_fm @ X3 @ ( sset_fm @ S ) ) ) ) ).

% Stream.smember_def
thf(fact_1025_Stream_Osmember__def,axiom,
    ( smembe6335822699099550295m_rule
    = ( ^ [X3: produc164195504107695125m_rule,S: stream727092118206550309m_rule] : ( member4220325220686508332m_rule @ X3 @ ( sset_P4979403222752902913m_rule @ S ) ) ) ) ).

% Stream.smember_def
thf(fact_1026_Sup__set__fold,axiom,
    ! [Xs: list_set_fm] :
      ( ( comple7677669044296858969set_fm @ ( set_set_fm2 @ Xs ) )
      = ( fold_set_fm_set_fm @ sup_sup_set_fm @ Xs @ bot_bot_set_fm ) ) ).

% Sup_set_fold
thf(fact_1027_szip_Ocode,axiom,
    ( szip_P1977448745965526924m_rule
    = ( ^ [S12: stream8299795917829157543ist_fm,S22: stream_rule] : ( sCons_5731567480799343806m_rule @ ( produc491467635432902671m_rule @ ( shd_Pr772355297128350925ist_fm @ S12 ) @ ( shd_rule @ S22 ) ) @ ( szip_P1977448745965526924m_rule @ ( stl_Pr5027778045650968273ist_fm @ S12 ) @ ( stl_rule @ S22 ) ) ) ) ) ).

% szip.code
thf(fact_1028_szip_Ocode,axiom,
    ( szip_P2499414959592755846m_rule
    = ( ^ [S12: stream727092118206550309m_rule,S22: stream727092118206550309m_rule] : ( sCons_4186928284053167522m_rule @ ( produc5927390650430071747m_rule @ ( shd_Pr7235097944458474089m_rule @ S12 ) @ ( shd_Pr7235097944458474089m_rule @ S22 ) ) @ ( szip_P2499414959592755846m_rule @ ( stl_Pr950425576149878629m_rule @ S12 ) @ ( stl_Pr950425576149878629m_rule @ S22 ) ) ) ) ) ).

% szip.code
thf(fact_1029_List_Oset__insert,axiom,
    ! [X2: fm,Xs: list_fm] :
      ( ( set_fm2 @ ( insert_fm @ X2 @ Xs ) )
      = ( insert_fm2 @ X2 @ ( set_fm2 @ Xs ) ) ) ).

% List.set_insert
thf(fact_1030_in__set__insert,axiom,
    ! [X2: fm,Xs: list_fm] :
      ( ( member_fm @ X2 @ ( set_fm2 @ Xs ) )
     => ( ( insert_fm @ X2 @ Xs )
        = Xs ) ) ).

% in_set_insert
thf(fact_1031_sdrop__simps_I2_J,axiom,
    ! [N2: nat,S3: stream727092118206550309m_rule] :
      ( ( stl_Pr950425576149878629m_rule @ ( sdrop_7224736112439592940m_rule @ N2 @ S3 ) )
      = ( sdrop_7224736112439592940m_rule @ ( suc @ N2 ) @ S3 ) ) ).

% sdrop_simps(2)
thf(fact_1032_sdrop__simps_I2_J,axiom,
    ! [N2: nat,S3: stream8299795917829157543ist_fm] :
      ( ( stl_Pr5027778045650968273ist_fm @ ( sdrop_4442373711808556042ist_fm @ N2 @ S3 ) )
      = ( sdrop_4442373711808556042ist_fm @ ( suc @ N2 ) @ S3 ) ) ).

% sdrop_simps(2)
thf(fact_1033_sdrop__simps_I2_J,axiom,
    ! [N2: nat,S3: stream_rule] :
      ( ( stl_rule @ ( sdrop_rule @ N2 @ S3 ) )
      = ( sdrop_rule @ ( suc @ N2 ) @ S3 ) ) ).

% sdrop_simps(2)
thf(fact_1034_sdrop__simps_I2_J,axiom,
    ! [N2: nat,S3: stream6494289010434245521m_rule] :
      ( ( stl_Pr2200982372225425851m_rule @ ( sdrop_7373388980841566196m_rule @ N2 @ S3 ) )
      = ( sdrop_7373388980841566196m_rule @ ( suc @ N2 ) @ S3 ) ) ).

% sdrop_simps(2)
thf(fact_1035_stream_Ocollapse,axiom,
    ! [Stream: stream727092118206550309m_rule] :
      ( ( sCons_5731567480799343806m_rule @ ( shd_Pr7235097944458474089m_rule @ Stream ) @ ( stl_Pr950425576149878629m_rule @ Stream ) )
      = Stream ) ).

% stream.collapse
thf(fact_1036_szip_Osimps_I2_J,axiom,
    ! [S1: stream8299795917829157543ist_fm,S2: stream_rule] :
      ( ( stl_Pr950425576149878629m_rule @ ( szip_P1977448745965526924m_rule @ S1 @ S2 ) )
      = ( szip_P1977448745965526924m_rule @ ( stl_Pr5027778045650968273ist_fm @ S1 ) @ ( stl_rule @ S2 ) ) ) ).

% szip.simps(2)
thf(fact_1037_szip_Osimps_I2_J,axiom,
    ! [S1: stream727092118206550309m_rule,S2: stream727092118206550309m_rule] :
      ( ( stl_Pr2200982372225425851m_rule @ ( szip_P2499414959592755846m_rule @ S1 @ S2 ) )
      = ( szip_P2499414959592755846m_rule @ ( stl_Pr950425576149878629m_rule @ S1 ) @ ( stl_Pr950425576149878629m_rule @ S2 ) ) ) ).

% szip.simps(2)
thf(fact_1038_sdrop__stl,axiom,
    ! [N2: nat,S3: stream727092118206550309m_rule] :
      ( ( sdrop_7224736112439592940m_rule @ N2 @ ( stl_Pr950425576149878629m_rule @ S3 ) )
      = ( stl_Pr950425576149878629m_rule @ ( sdrop_7224736112439592940m_rule @ N2 @ S3 ) ) ) ).

% sdrop_stl
thf(fact_1039_sdrop__stl,axiom,
    ! [N2: nat,S3: stream8299795917829157543ist_fm] :
      ( ( sdrop_4442373711808556042ist_fm @ N2 @ ( stl_Pr5027778045650968273ist_fm @ S3 ) )
      = ( stl_Pr5027778045650968273ist_fm @ ( sdrop_4442373711808556042ist_fm @ N2 @ S3 ) ) ) ).

% sdrop_stl
thf(fact_1040_sdrop__stl,axiom,
    ! [N2: nat,S3: stream_rule] :
      ( ( sdrop_rule @ N2 @ ( stl_rule @ S3 ) )
      = ( stl_rule @ ( sdrop_rule @ N2 @ S3 ) ) ) ).

% sdrop_stl
thf(fact_1041_sdrop__stl,axiom,
    ! [N2: nat,S3: stream6494289010434245521m_rule] :
      ( ( sdrop_7373388980841566196m_rule @ N2 @ ( stl_Pr2200982372225425851m_rule @ S3 ) )
      = ( stl_Pr2200982372225425851m_rule @ ( sdrop_7373388980841566196m_rule @ N2 @ S3 ) ) ) ).

% sdrop_stl
thf(fact_1042_stream_Ocoinduct__strong,axiom,
    ! [R: stream727092118206550309m_rule > stream727092118206550309m_rule > $o,Stream: stream727092118206550309m_rule,Stream2: stream727092118206550309m_rule] :
      ( ( R @ Stream @ Stream2 )
     => ( ! [Stream3: stream727092118206550309m_rule,Stream4: stream727092118206550309m_rule] :
            ( ( R @ Stream3 @ Stream4 )
           => ( ( ( shd_Pr7235097944458474089m_rule @ Stream3 )
                = ( shd_Pr7235097944458474089m_rule @ Stream4 ) )
              & ( ( R @ ( stl_Pr950425576149878629m_rule @ Stream3 ) @ ( stl_Pr950425576149878629m_rule @ Stream4 ) )
                | ( ( stl_Pr950425576149878629m_rule @ Stream3 )
                  = ( stl_Pr950425576149878629m_rule @ Stream4 ) ) ) ) )
       => ( Stream = Stream2 ) ) ) ).

% stream.coinduct_strong
thf(fact_1043_stream_Ocoinduct,axiom,
    ! [R: stream727092118206550309m_rule > stream727092118206550309m_rule > $o,Stream: stream727092118206550309m_rule,Stream2: stream727092118206550309m_rule] :
      ( ( R @ Stream @ Stream2 )
     => ( ! [Stream3: stream727092118206550309m_rule,Stream4: stream727092118206550309m_rule] :
            ( ( R @ Stream3 @ Stream4 )
           => ( ( ( shd_Pr7235097944458474089m_rule @ Stream3 )
                = ( shd_Pr7235097944458474089m_rule @ Stream4 ) )
              & ( R @ ( stl_Pr950425576149878629m_rule @ Stream3 ) @ ( stl_Pr950425576149878629m_rule @ Stream4 ) ) ) )
       => ( Stream = Stream2 ) ) ) ).

% stream.coinduct
thf(fact_1044_stream_Oexpand,axiom,
    ! [Stream: stream727092118206550309m_rule,Stream2: stream727092118206550309m_rule] :
      ( ( ( ( shd_Pr7235097944458474089m_rule @ Stream )
          = ( shd_Pr7235097944458474089m_rule @ Stream2 ) )
        & ( ( stl_Pr950425576149878629m_rule @ Stream )
          = ( stl_Pr950425576149878629m_rule @ Stream2 ) ) )
     => ( Stream = Stream2 ) ) ).

% stream.expand
thf(fact_1045_stl__sset,axiom,
    ! [X2: fm,A: stream_fm] :
      ( ( member_fm @ X2 @ ( sset_fm @ ( stl_fm @ A ) ) )
     => ( member_fm @ X2 @ ( sset_fm @ A ) ) ) ).

% stl_sset
thf(fact_1046_stl__sset,axiom,
    ! [X2: produc164195504107695125m_rule,A: stream727092118206550309m_rule] :
      ( ( member4220325220686508332m_rule @ X2 @ ( sset_P4979403222752902913m_rule @ ( stl_Pr950425576149878629m_rule @ A ) ) )
     => ( member4220325220686508332m_rule @ X2 @ ( sset_P4979403222752902913m_rule @ A ) ) ) ).

% stl_sset
thf(fact_1047_fst__transfer,axiom,
    ! [A4: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm > $o,B5: rule > rule > $o] : ( bNF_re3591205218739347657ist_fm @ ( basic_3609040256092172161e_rule @ A4 @ B5 ) @ A4 @ produc6879501374131015971m_rule @ produc6879501374131015971m_rule ) ).

% fst_transfer
thf(fact_1048_fst__transfer,axiom,
    ! [A4: produc1996495991257130529ist_fm > list_fm > $o,B5: rule > list_fm > $o] : ( bNF_re7610816408743474068ist_fm @ ( basic_7329884221201608115ist_fm @ A4 @ B5 ) @ A4 @ produc6879501374131015971m_rule @ produc1501393135466168645ist_fm ) ).

% fst_transfer
thf(fact_1049_fst__transfer,axiom,
    ! [A4: list_fm > produc1996495991257130529ist_fm > $o,B5: list_fm > rule > $o] : ( bNF_re1086642384569246310ist_fm @ ( basic_4265173781543994481m_rule @ A4 @ B5 ) @ A4 @ produc1501393135466168645ist_fm @ produc6879501374131015971m_rule ) ).

% fst_transfer
thf(fact_1050_fst__transfer,axiom,
    ! [A4: list_fm > list_fm > $o,B5: list_fm > list_fm > $o] : ( bNF_re4823320336450819679ist_fm @ ( basic_1286064880750865721ist_fm @ A4 @ B5 ) @ A4 @ produc1501393135466168645ist_fm @ produc1501393135466168645ist_fm ) ).

% fst_transfer
thf(fact_1051_snd__transfer,axiom,
    ! [A4: list_fm > list_fm > $o,B5: list_fm > list_fm > $o] : ( bNF_re4823320336450819679ist_fm @ ( basic_1286064880750865721ist_fm @ A4 @ B5 ) @ B5 @ produc4588648349897876871ist_fm @ produc4588648349897876871ist_fm ) ).

% snd_transfer
thf(fact_1052_snd__transfer,axiom,
    ! [A4: list_fm > produc1996495991257130529ist_fm > $o,B5: list_fm > rule > $o] : ( bNF_re2379315951688697920m_rule @ ( basic_4265173781543994481m_rule @ A4 @ B5 ) @ B5 @ produc4588648349897876871ist_fm @ produc7165828336582415457m_rule ) ).

% snd_transfer
thf(fact_1053_snd__transfer,axiom,
    ! [A4: list_fm > rule > $o,B5: list_fm > produc1996495991257130529ist_fm > $o] : ( bNF_re7078097413856527850ist_fm @ ( basic_8978118172011093987ist_fm @ A4 @ B5 ) @ B5 @ produc4588648349897876871ist_fm @ produc4514744107061996309ist_fm ) ).

% snd_transfer
thf(fact_1054_snd__transfer,axiom,
    ! [A4: produc1996495991257130529ist_fm > list_fm > $o,B5: rule > list_fm > $o] : ( bNF_re8002563879577945940ist_fm @ ( basic_7329884221201608115ist_fm @ A4 @ B5 ) @ B5 @ produc7165828336582415457m_rule @ produc4588648349897876871ist_fm ) ).

% snd_transfer
thf(fact_1055_snd__transfer,axiom,
    ! [A4: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm > $o,B5: rule > rule > $o] : ( bNF_re1986445207066496021e_rule @ ( basic_3609040256092172161e_rule @ A4 @ B5 ) @ B5 @ produc7165828336582415457m_rule @ produc7165828336582415457m_rule ) ).

% snd_transfer
thf(fact_1056_snd__transfer,axiom,
    ! [A4: produc1996495991257130529ist_fm > rule > $o,B5: rule > produc1996495991257130529ist_fm > $o] : ( bNF_re4779642490101428437ist_fm @ ( basic_7991631927920859369ist_fm @ A4 @ B5 ) @ B5 @ produc7165828336582415457m_rule @ produc4514744107061996309ist_fm ) ).

% snd_transfer
thf(fact_1057_snd__transfer,axiom,
    ! [A4: rule > list_fm > $o,B5: produc1996495991257130529ist_fm > list_fm > $o] : ( bNF_re827765162060501776ist_fm @ ( basic_7340898436922099493ist_fm @ A4 @ B5 ) @ B5 @ produc4514744107061996309ist_fm @ produc4588648349897876871ist_fm ) ).

% snd_transfer
thf(fact_1058_snd__transfer,axiom,
    ! [A4: rule > produc1996495991257130529ist_fm > $o,B5: produc1996495991257130529ist_fm > rule > $o] : ( bNF_re8992732486024710177m_rule @ ( basic_6697320886820836993m_rule @ A4 @ B5 ) @ B5 @ produc4514744107061996309ist_fm @ produc7165828336582415457m_rule ) ).

% snd_transfer
thf(fact_1059_snd__transfer,axiom,
    ! [A4: rule > rule > $o,B5: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm > $o] : ( bNF_re5970660450855479745ist_fm @ ( basic_7093977281369128321ist_fm @ A4 @ B5 ) @ B5 @ produc4514744107061996309ist_fm @ produc4514744107061996309ist_fm ) ).

% snd_transfer
thf(fact_1060_sdrop_Osimps_I2_J,axiom,
    ! [N2: nat,S3: stream727092118206550309m_rule] :
      ( ( sdrop_7224736112439592940m_rule @ ( suc @ N2 ) @ S3 )
      = ( sdrop_7224736112439592940m_rule @ N2 @ ( stl_Pr950425576149878629m_rule @ S3 ) ) ) ).

% sdrop.simps(2)
thf(fact_1061_sdrop_Osimps_I2_J,axiom,
    ! [N2: nat,S3: stream8299795917829157543ist_fm] :
      ( ( sdrop_4442373711808556042ist_fm @ ( suc @ N2 ) @ S3 )
      = ( sdrop_4442373711808556042ist_fm @ N2 @ ( stl_Pr5027778045650968273ist_fm @ S3 ) ) ) ).

% sdrop.simps(2)
thf(fact_1062_sdrop_Osimps_I2_J,axiom,
    ! [N2: nat,S3: stream_rule] :
      ( ( sdrop_rule @ ( suc @ N2 ) @ S3 )
      = ( sdrop_rule @ N2 @ ( stl_rule @ S3 ) ) ) ).

% sdrop.simps(2)
thf(fact_1063_sdrop_Osimps_I2_J,axiom,
    ! [N2: nat,S3: stream6494289010434245521m_rule] :
      ( ( sdrop_7373388980841566196m_rule @ ( suc @ N2 ) @ S3 )
      = ( sdrop_7373388980841566196m_rule @ N2 @ ( stl_Pr2200982372225425851m_rule @ S3 ) ) ) ).

% sdrop.simps(2)
thf(fact_1064_rel__prod__sel,axiom,
    ( basic_1286064880750865721ist_fm
    = ( ^ [R12: list_fm > list_fm > $o,R23: list_fm > list_fm > $o,P4: produc1996495991257130529ist_fm,Q4: produc1996495991257130529ist_fm] :
          ( ( R12 @ ( produc1501393135466168645ist_fm @ P4 ) @ ( produc1501393135466168645ist_fm @ Q4 ) )
          & ( R23 @ ( produc4588648349897876871ist_fm @ P4 ) @ ( produc4588648349897876871ist_fm @ Q4 ) ) ) ) ) ).

% rel_prod_sel
thf(fact_1065_rel__prod__sel,axiom,
    ( basic_4265173781543994481m_rule
    = ( ^ [R12: list_fm > produc1996495991257130529ist_fm > $o,R23: list_fm > rule > $o,P4: produc1996495991257130529ist_fm,Q4: produc164195504107695125m_rule] :
          ( ( R12 @ ( produc1501393135466168645ist_fm @ P4 ) @ ( produc6879501374131015971m_rule @ Q4 ) )
          & ( R23 @ ( produc4588648349897876871ist_fm @ P4 ) @ ( produc7165828336582415457m_rule @ Q4 ) ) ) ) ) ).

% rel_prod_sel
thf(fact_1066_rel__prod__sel,axiom,
    ( basic_8978118172011093987ist_fm
    = ( ^ [R12: list_fm > rule > $o,R23: list_fm > produc1996495991257130529ist_fm > $o,P4: produc1996495991257130529ist_fm,Q4: produc4630866025709511057ist_fm] :
          ( ( R12 @ ( produc1501393135466168645ist_fm @ P4 ) @ ( produc4228417144610596823ist_fm @ Q4 ) )
          & ( R23 @ ( produc4588648349897876871ist_fm @ P4 ) @ ( produc4514744107061996309ist_fm @ Q4 ) ) ) ) ) ).

% rel_prod_sel
thf(fact_1067_rel__prod__sel,axiom,
    ( basic_7329884221201608115ist_fm
    = ( ^ [R12: produc1996495991257130529ist_fm > list_fm > $o,R23: rule > list_fm > $o,P4: produc164195504107695125m_rule,Q4: produc1996495991257130529ist_fm] :
          ( ( R12 @ ( produc6879501374131015971m_rule @ P4 ) @ ( produc1501393135466168645ist_fm @ Q4 ) )
          & ( R23 @ ( produc7165828336582415457m_rule @ P4 ) @ ( produc4588648349897876871ist_fm @ Q4 ) ) ) ) ) ).

% rel_prod_sel
thf(fact_1068_rel__prod__sel,axiom,
    ( basic_3609040256092172161e_rule
    = ( ^ [R12: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm > $o,R23: rule > rule > $o,P4: produc164195504107695125m_rule,Q4: produc164195504107695125m_rule] :
          ( ( R12 @ ( produc6879501374131015971m_rule @ P4 ) @ ( produc6879501374131015971m_rule @ Q4 ) )
          & ( R23 @ ( produc7165828336582415457m_rule @ P4 ) @ ( produc7165828336582415457m_rule @ Q4 ) ) ) ) ) ).

% rel_prod_sel
thf(fact_1069_rel__prod__sel,axiom,
    ( basic_7991631927920859369ist_fm
    = ( ^ [R12: produc1996495991257130529ist_fm > rule > $o,R23: rule > produc1996495991257130529ist_fm > $o,P4: produc164195504107695125m_rule,Q4: produc4630866025709511057ist_fm] :
          ( ( R12 @ ( produc6879501374131015971m_rule @ P4 ) @ ( produc4228417144610596823ist_fm @ Q4 ) )
          & ( R23 @ ( produc7165828336582415457m_rule @ P4 ) @ ( produc4514744107061996309ist_fm @ Q4 ) ) ) ) ) ).

% rel_prod_sel
thf(fact_1070_rel__prod__sel,axiom,
    ( basic_7340898436922099493ist_fm
    = ( ^ [R12: rule > list_fm > $o,R23: produc1996495991257130529ist_fm > list_fm > $o,P4: produc4630866025709511057ist_fm,Q4: produc1996495991257130529ist_fm] :
          ( ( R12 @ ( produc4228417144610596823ist_fm @ P4 ) @ ( produc1501393135466168645ist_fm @ Q4 ) )
          & ( R23 @ ( produc4514744107061996309ist_fm @ P4 ) @ ( produc4588648349897876871ist_fm @ Q4 ) ) ) ) ) ).

% rel_prod_sel
thf(fact_1071_rel__prod__sel,axiom,
    ( basic_6697320886820836993m_rule
    = ( ^ [R12: rule > produc1996495991257130529ist_fm > $o,R23: produc1996495991257130529ist_fm > rule > $o,P4: produc4630866025709511057ist_fm,Q4: produc164195504107695125m_rule] :
          ( ( R12 @ ( produc4228417144610596823ist_fm @ P4 ) @ ( produc6879501374131015971m_rule @ Q4 ) )
          & ( R23 @ ( produc4514744107061996309ist_fm @ P4 ) @ ( produc7165828336582415457m_rule @ Q4 ) ) ) ) ) ).

% rel_prod_sel
thf(fact_1072_rel__prod__sel,axiom,
    ( basic_7093977281369128321ist_fm
    = ( ^ [R12: rule > rule > $o,R23: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm > $o,P4: produc4630866025709511057ist_fm,Q4: produc4630866025709511057ist_fm] :
          ( ( R12 @ ( produc4228417144610596823ist_fm @ P4 ) @ ( produc4228417144610596823ist_fm @ Q4 ) )
          & ( R23 @ ( produc4514744107061996309ist_fm @ P4 ) @ ( produc4514744107061996309ist_fm @ Q4 ) ) ) ) ) ).

% rel_prod_sel
thf(fact_1073_snth_Osimps_I2_J,axiom,
    ! [S3: stream727092118206550309m_rule,N2: nat] :
      ( ( snth_P1712579846118164081m_rule @ S3 @ ( suc @ N2 ) )
      = ( snth_P1712579846118164081m_rule @ ( stl_Pr950425576149878629m_rule @ S3 ) @ N2 ) ) ).

% snth.simps(2)
thf(fact_1074_sset__induct,axiom,
    ! [Y3: fm,S3: stream_fm,P2: fm > stream_fm > $o] :
      ( ( member_fm @ Y3 @ ( sset_fm @ S3 ) )
     => ( ! [S7: stream_fm] : ( P2 @ ( shd_fm @ S7 ) @ S7 )
       => ( ! [S7: stream_fm,Y: fm] :
              ( ( member_fm @ Y @ ( sset_fm @ ( stl_fm @ S7 ) ) )
             => ( ( P2 @ Y @ ( stl_fm @ S7 ) )
               => ( P2 @ Y @ S7 ) ) )
         => ( P2 @ Y3 @ S3 ) ) ) ) ).

% sset_induct
thf(fact_1075_sset__induct,axiom,
    ! [Y3: produc164195504107695125m_rule,S3: stream727092118206550309m_rule,P2: produc164195504107695125m_rule > stream727092118206550309m_rule > $o] :
      ( ( member4220325220686508332m_rule @ Y3 @ ( sset_P4979403222752902913m_rule @ S3 ) )
     => ( ! [S7: stream727092118206550309m_rule] : ( P2 @ ( shd_Pr7235097944458474089m_rule @ S7 ) @ S7 )
       => ( ! [S7: stream727092118206550309m_rule,Y: produc164195504107695125m_rule] :
              ( ( member4220325220686508332m_rule @ Y @ ( sset_P4979403222752902913m_rule @ ( stl_Pr950425576149878629m_rule @ S7 ) ) )
             => ( ( P2 @ Y @ ( stl_Pr950425576149878629m_rule @ S7 ) )
               => ( P2 @ Y @ S7 ) ) )
         => ( P2 @ Y3 @ S3 ) ) ) ) ).

% sset_induct
thf(fact_1076_stream_Oexhaust__sel,axiom,
    ! [Stream: stream727092118206550309m_rule] :
      ( Stream
      = ( sCons_5731567480799343806m_rule @ ( shd_Pr7235097944458474089m_rule @ Stream ) @ ( stl_Pr950425576149878629m_rule @ Stream ) ) ) ).

% stream.exhaust_sel
thf(fact_1077_stream_Orel__sel,axiom,
    ( stream1316448123751145209m_rule
    = ( ^ [R3: produc164195504107695125m_rule > produc164195504107695125m_rule > $o,A5: stream727092118206550309m_rule,B4: stream727092118206550309m_rule] :
          ( ( R3 @ ( shd_Pr7235097944458474089m_rule @ A5 ) @ ( shd_Pr7235097944458474089m_rule @ B4 ) )
          & ( stream1316448123751145209m_rule @ R3 @ ( stl_Pr950425576149878629m_rule @ A5 ) @ ( stl_Pr950425576149878629m_rule @ B4 ) ) ) ) ) ).

% stream.rel_sel
thf(fact_1078_stream_Orel__coinduct,axiom,
    ! [P2: stream727092118206550309m_rule > stream727092118206550309m_rule > $o,X2: stream727092118206550309m_rule,Y3: stream727092118206550309m_rule,R: produc164195504107695125m_rule > produc164195504107695125m_rule > $o] :
      ( ( P2 @ X2 @ Y3 )
     => ( ! [Stream3: stream727092118206550309m_rule,Stream4: stream727092118206550309m_rule] :
            ( ( P2 @ Stream3 @ Stream4 )
           => ( ( R @ ( shd_Pr7235097944458474089m_rule @ Stream3 ) @ ( shd_Pr7235097944458474089m_rule @ Stream4 ) )
              & ( P2 @ ( stl_Pr950425576149878629m_rule @ Stream3 ) @ ( stl_Pr950425576149878629m_rule @ Stream4 ) ) ) )
       => ( stream1316448123751145209m_rule @ R @ X2 @ Y3 ) ) ) ).

% stream.rel_coinduct
thf(fact_1079_prod_Opred__transfer,axiom,
    ! [R1: list_fm > list_fm > $o,R22: list_fm > list_fm > $o] :
      ( bNF_re3600363984329575581t_fm_o
      @ ( bNF_re2260024341499725215fm_o_o @ R1
        @ ^ [Y2: $o,Z: $o] : ( Y2 = Z ) )
      @ ( bNF_re6700684614245656919t_fm_o
        @ ( bNF_re2260024341499725215fm_o_o @ R22
          @ ^ [Y2: $o,Z: $o] : ( Y2 = Z ) )
        @ ( bNF_re188606575539848477fm_o_o @ ( basic_1286064880750865721ist_fm @ R1 @ R22 )
          @ ^ [Y2: $o,Z: $o] : ( Y2 = Z ) ) )
      @ basic_5093365364719461109ist_fm
      @ basic_5093365364719461109ist_fm ) ).

% prod.pred_transfer
thf(fact_1080_prod_Opred__transfer,axiom,
    ! [R1: list_fm > produc1996495991257130529ist_fm > $o,R22: list_fm > rule > $o] :
      ( bNF_re6390700563477671843rule_o
      @ ( bNF_re2963754933915346338fm_o_o @ R1
        @ ^ [Y2: $o,Z: $o] : ( Y2 = Z ) )
      @ ( bNF_re8866850227915748580rule_o
        @ ( bNF_re2504994000271363972le_o_o @ R22
          @ ^ [Y2: $o,Z: $o] : ( Y2 = Z ) )
        @ ( bNF_re356467991940469145le_o_o @ ( basic_4265173781543994481m_rule @ R1 @ R22 )
          @ ^ [Y2: $o,Z: $o] : ( Y2 = Z ) ) )
      @ basic_5093365364719461109ist_fm
      @ basic_6171360182465397619m_rule ) ).

% prod.pred_transfer
thf(fact_1081_prod_Opred__transfer,axiom,
    ! [R1: produc1996495991257130529ist_fm > list_fm > $o,R22: rule > list_fm > $o] :
      ( bNF_re7217375449204646129t_fm_o
      @ ( bNF_re3089193160343016676fm_o_o @ R1
        @ ^ [Y2: $o,Z: $o] : ( Y2 = Z ) )
      @ ( bNF_re7886380955999696916t_fm_o
        @ ( bNF_re1730353633762483844fm_o_o @ R22
          @ ^ [Y2: $o,Z: $o] : ( Y2 = Z ) )
        @ ( bNF_re3354485303118697389fm_o_o @ ( basic_7329884221201608115ist_fm @ R1 @ R22 )
          @ ^ [Y2: $o,Z: $o] : ( Y2 = Z ) ) )
      @ basic_6171360182465397619m_rule
      @ basic_5093365364719461109ist_fm ) ).

% prod.pred_transfer
thf(fact_1082_prod_Opred__transfer,axiom,
    ! [R1: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm > $o,R22: rule > rule > $o] :
      ( bNF_re2806322975267763907rule_o
      @ ( bNF_re188606575539848477fm_o_o @ R1
        @ ^ [Y2: $o,Z: $o] : ( Y2 = Z ) )
      @ ( bNF_re5822392312811141597rule_o
        @ ( bNF_re4368755934245895273le_o_o @ R22
          @ ^ [Y2: $o,Z: $o] : ( Y2 = Z ) )
        @ ( bNF_re3729014388983930633le_o_o @ ( basic_3609040256092172161e_rule @ R1 @ R22 )
          @ ^ [Y2: $o,Z: $o] : ( Y2 = Z ) ) )
      @ basic_6171360182465397619m_rule
      @ basic_6171360182465397619m_rule ) ).

% prod.pred_transfer
thf(fact_1083_prod__pred__parametric,axiom,
    ! [A4: list_fm > list_fm > $o,B5: list_fm > list_fm > $o] :
      ( bNF_re3600363984329575581t_fm_o
      @ ( bNF_re2260024341499725215fm_o_o @ A4
        @ ^ [Y2: $o,Z: $o] : ( Y2 = Z ) )
      @ ( bNF_re6700684614245656919t_fm_o
        @ ( bNF_re2260024341499725215fm_o_o @ B5
          @ ^ [Y2: $o,Z: $o] : ( Y2 = Z ) )
        @ ( bNF_re188606575539848477fm_o_o @ ( basic_1286064880750865721ist_fm @ A4 @ B5 )
          @ ^ [Y2: $o,Z: $o] : ( Y2 = Z ) ) )
      @ basic_5093365364719461109ist_fm
      @ basic_5093365364719461109ist_fm ) ).

% prod_pred_parametric
thf(fact_1084_prod__pred__parametric,axiom,
    ! [A4: list_fm > produc1996495991257130529ist_fm > $o,B5: list_fm > rule > $o] :
      ( bNF_re6390700563477671843rule_o
      @ ( bNF_re2963754933915346338fm_o_o @ A4
        @ ^ [Y2: $o,Z: $o] : ( Y2 = Z ) )
      @ ( bNF_re8866850227915748580rule_o
        @ ( bNF_re2504994000271363972le_o_o @ B5
          @ ^ [Y2: $o,Z: $o] : ( Y2 = Z ) )
        @ ( bNF_re356467991940469145le_o_o @ ( basic_4265173781543994481m_rule @ A4 @ B5 )
          @ ^ [Y2: $o,Z: $o] : ( Y2 = Z ) ) )
      @ basic_5093365364719461109ist_fm
      @ basic_6171360182465397619m_rule ) ).

% prod_pred_parametric
thf(fact_1085_prod__pred__parametric,axiom,
    ! [A4: produc1996495991257130529ist_fm > list_fm > $o,B5: rule > list_fm > $o] :
      ( bNF_re7217375449204646129t_fm_o
      @ ( bNF_re3089193160343016676fm_o_o @ A4
        @ ^ [Y2: $o,Z: $o] : ( Y2 = Z ) )
      @ ( bNF_re7886380955999696916t_fm_o
        @ ( bNF_re1730353633762483844fm_o_o @ B5
          @ ^ [Y2: $o,Z: $o] : ( Y2 = Z ) )
        @ ( bNF_re3354485303118697389fm_o_o @ ( basic_7329884221201608115ist_fm @ A4 @ B5 )
          @ ^ [Y2: $o,Z: $o] : ( Y2 = Z ) ) )
      @ basic_6171360182465397619m_rule
      @ basic_5093365364719461109ist_fm ) ).

% prod_pred_parametric
thf(fact_1086_prod__pred__parametric,axiom,
    ! [A4: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm > $o,B5: rule > rule > $o] :
      ( bNF_re2806322975267763907rule_o
      @ ( bNF_re188606575539848477fm_o_o @ A4
        @ ^ [Y2: $o,Z: $o] : ( Y2 = Z ) )
      @ ( bNF_re5822392312811141597rule_o
        @ ( bNF_re4368755934245895273le_o_o @ B5
          @ ^ [Y2: $o,Z: $o] : ( Y2 = Z ) )
        @ ( bNF_re3729014388983930633le_o_o @ ( basic_3609040256092172161e_rule @ A4 @ B5 )
          @ ^ [Y2: $o,Z: $o] : ( Y2 = Z ) ) )
      @ basic_6171360182465397619m_rule
      @ basic_6171360182465397619m_rule ) ).

% prod_pred_parametric
thf(fact_1087_prod_Orel__cong,axiom,
    ! [X2: product_prod_fm_fm,Ya: product_prod_fm_fm,Y3: product_prod_fm_fm,Xa2: product_prod_fm_fm,R1: fm > fm > $o,R1a: fm > fm > $o,R22: fm > fm > $o,R2a: fm > fm > $o] :
      ( ( X2 = Ya )
     => ( ( Y3 = Xa2 )
       => ( ! [Z1: fm,Y12: fm] :
              ( ( member_fm @ Z1 @ ( basic_fsts_fm_fm @ Ya ) )
             => ( ( member_fm @ Y12 @ ( basic_fsts_fm_fm @ Xa2 ) )
               => ( ( R1 @ Z1 @ Y12 )
                  = ( R1a @ Z1 @ Y12 ) ) ) )
         => ( ! [Z22: fm,Y23: fm] :
                ( ( member_fm @ Z22 @ ( basic_snds_fm_fm @ Ya ) )
               => ( ( member_fm @ Y23 @ ( basic_snds_fm_fm @ Xa2 ) )
                 => ( ( R22 @ Z22 @ Y23 )
                    = ( R2a @ Z22 @ Y23 ) ) ) )
           => ( ( basic_1661250094485422585_fm_fm @ R1 @ R22 @ X2 @ Y3 )
              = ( basic_1661250094485422585_fm_fm @ R1a @ R2a @ Ya @ Xa2 ) ) ) ) ) ) ).

% prod.rel_cong
thf(fact_1088_prod_Orel__mono__strong,axiom,
    ! [R1: fm > fm > $o,R22: fm > fm > $o,X2: product_prod_fm_fm,Y3: product_prod_fm_fm,R1a: fm > fm > $o,R2a: fm > fm > $o] :
      ( ( basic_1661250094485422585_fm_fm @ R1 @ R22 @ X2 @ Y3 )
     => ( ! [Z1: fm,Y12: fm] :
            ( ( member_fm @ Z1 @ ( basic_fsts_fm_fm @ X2 ) )
           => ( ( member_fm @ Y12 @ ( basic_fsts_fm_fm @ Y3 ) )
             => ( ( R1 @ Z1 @ Y12 )
               => ( R1a @ Z1 @ Y12 ) ) ) )
       => ( ! [Z22: fm,Y23: fm] :
              ( ( member_fm @ Z22 @ ( basic_snds_fm_fm @ X2 ) )
             => ( ( member_fm @ Y23 @ ( basic_snds_fm_fm @ Y3 ) )
               => ( ( R22 @ Z22 @ Y23 )
                 => ( R2a @ Z22 @ Y23 ) ) ) )
         => ( basic_1661250094485422585_fm_fm @ R1a @ R2a @ X2 @ Y3 ) ) ) ) ).

% prod.rel_mono_strong
thf(fact_1089_prod_Orel__refl__strong,axiom,
    ! [X2: product_prod_fm_fm,R1a: fm > fm > $o,R2a: fm > fm > $o] :
      ( ! [Z1: fm] :
          ( ( member_fm @ Z1 @ ( basic_fsts_fm_fm @ X2 ) )
         => ( R1a @ Z1 @ Z1 ) )
     => ( ! [Z22: fm] :
            ( ( member_fm @ Z22 @ ( basic_snds_fm_fm @ X2 ) )
           => ( R2a @ Z22 @ Z22 ) )
       => ( basic_1661250094485422585_fm_fm @ R1a @ R2a @ X2 @ X2 ) ) ) ).

% prod.rel_refl_strong
thf(fact_1090_prod_Orel__compp__Grp,axiom,
    ( basic_1286064880750865721ist_fm
    = ( ^ [R12: list_fm > list_fm > $o,R23: list_fm > list_fm > $o] :
          ( relcom2979441344136134570ist_fm
          @ ( conver4242428939633515119ist_fm
            @ ( bNF_Gr6521490611699154120ist_fm
              @ ( collec888080998205339090ist_fm
                @ ^ [X3: produc1754732471204517991ist_fm] :
                    ( ( ord_le2055072402069232929ist_fm @ ( basic_6510150186836533191ist_fm @ X3 ) @ ( collec7231575776666803724ist_fm @ ( produc2011568048573023402t_fm_o @ R12 ) ) )
                    & ( ord_le2055072402069232929ist_fm @ ( basic_4728562467516582661ist_fm @ X3 ) @ ( collec7231575776666803724ist_fm @ ( produc2011568048573023402t_fm_o @ R23 ) ) ) ) )
              @ ( produc7565992903848568886ist_fm @ produc1501393135466168645ist_fm @ produc1501393135466168645ist_fm ) ) )
          @ ( bNF_Gr6521490611699154120ist_fm
            @ ( collec888080998205339090ist_fm
              @ ^ [X3: produc1754732471204517991ist_fm] :
                  ( ( ord_le2055072402069232929ist_fm @ ( basic_6510150186836533191ist_fm @ X3 ) @ ( collec7231575776666803724ist_fm @ ( produc2011568048573023402t_fm_o @ R12 ) ) )
                  & ( ord_le2055072402069232929ist_fm @ ( basic_4728562467516582661ist_fm @ X3 ) @ ( collec7231575776666803724ist_fm @ ( produc2011568048573023402t_fm_o @ R23 ) ) ) ) )
            @ ( produc7565992903848568886ist_fm @ produc4588648349897876871ist_fm @ produc4588648349897876871ist_fm ) ) ) ) ) ).

% prod.rel_compp_Grp
thf(fact_1091_prod_Orel__compp__Grp,axiom,
    ( basic_4795398152100796463m_rule
    = ( ^ [R12: list_fm > list_fm > $o,R23: produc1996495991257130529ist_fm > rule > $o] :
          ( relcom7150133098416953912m_rule
          @ ( conver4827328049888797842ist_fm
            @ ( bNF_Gr705829324403074923ist_fm
              @ ( collec3296238535900206906m_rule
                @ ^ [X3: produc466554394152024933m_rule] :
                    ( ( ord_le2055072402069232929ist_fm @ ( basic_6200111124537405679m_rule @ X3 ) @ ( collec7231575776666803724ist_fm @ ( produc2011568048573023402t_fm_o @ R12 ) ) )
                    & ( ord_le5052251716133888043m_rule @ ( basic_8636800664448319025m_rule @ X3 ) @ ( collec2069506698588392810m_rule @ ( produc7369492558480804444rule_o @ R23 ) ) ) ) )
              @ ( produc7676726057136416639ist_fm @ produc1501393135466168645ist_fm @ produc6879501374131015971m_rule ) ) )
          @ ( bNF_Gr6600196990217702929m_rule
            @ ( collec3296238535900206906m_rule
              @ ^ [X3: produc466554394152024933m_rule] :
                  ( ( ord_le2055072402069232929ist_fm @ ( basic_6200111124537405679m_rule @ X3 ) @ ( collec7231575776666803724ist_fm @ ( produc2011568048573023402t_fm_o @ R12 ) ) )
                  & ( ord_le5052251716133888043m_rule @ ( basic_8636800664448319025m_rule @ X3 ) @ ( collec2069506698588392810m_rule @ ( produc7369492558480804444rule_o @ R23 ) ) ) ) )
            @ ( produc8614446829993874791e_rule @ produc4588648349897876871ist_fm @ produc7165828336582415457m_rule ) ) ) ) ) ).

% prod.rel_compp_Grp
thf(fact_1092_prod_Orel__compp__Grp,axiom,
    ( basic_2144313922580377315ist_fm
    = ( ^ [R12: list_fm > list_fm > $o,R23: rule > produc1996495991257130529ist_fm > $o] :
          ( relcom7779093774021480238ist_fm
          @ ( conver3231474389345297454m_rule
            @ ( bNF_Gr3436162941030196117m_rule
              @ ( collec4529459408462614198ist_fm
                @ ^ [X3: produc1699775266714432225ist_fm] :
                    ( ( ord_le2055072402069232929ist_fm @ ( basic_1443409609284445803ist_fm @ X3 ) @ ( collec7231575776666803724ist_fm @ ( produc2011568048573023402t_fm_o @ R12 ) ) )
                    & ( ord_le6285472588696295335ist_fm @ ( basic_3880099149195359149ist_fm @ X3 ) @ ( collec6536177220190208742ist_fm @ ( produc1628407760719572648t_fm_o @ R23 ) ) ) ) )
              @ ( produc1389032208193576675m_rule @ produc1501393135466168645ist_fm @ produc4228417144610596823ist_fm ) ) )
          @ ( bNF_Gr2294656957428269287ist_fm
            @ ( collec4529459408462614198ist_fm
              @ ^ [X3: produc1699775266714432225ist_fm] :
                  ( ( ord_le2055072402069232929ist_fm @ ( basic_1443409609284445803ist_fm @ X3 ) @ ( collec7231575776666803724ist_fm @ ( produc2011568048573023402t_fm_o @ R12 ) ) )
                  & ( ord_le6285472588696295335ist_fm @ ( basic_3880099149195359149ist_fm @ X3 ) @ ( collec6536177220190208742ist_fm @ ( produc1628407760719572648t_fm_o @ R23 ) ) ) ) )
            @ ( produc8135082370809556739ist_fm @ produc4588648349897876871ist_fm @ produc4514744107061996309ist_fm ) ) ) ) ) ).

% prod.rel_compp_Grp
thf(fact_1093_prod_Orel__compp__Grp,axiom,
    ( basic_882330199258250419ist_fm
    = ( ^ [R12: produc1996495991257130529ist_fm > rule > $o,R23: list_fm > list_fm > $o] :
          ( relcom2375301643232424290ist_fm
          @ ( conver3268217794181079400ist_fm
            @ ( bNF_Gr8370091105550132289ist_fm
              @ ( collec7217352646538401814ist_fm
                @ ^ [X3: produc4387668504790219841ist_fm] :
                    ( ( ord_le5052251716133888043m_rule @ ( basic_5898077337727170819ist_fm @ X3 ) @ ( collec2069506698588392810m_rule @ ( produc7369492558480804444rule_o @ R12 ) ) )
                    & ( ord_le2055072402069232929ist_fm @ ( basic_8334766877638084165ist_fm @ X3 ) @ ( collec7231575776666803724ist_fm @ ( produc2011568048573023402t_fm_o @ R23 ) ) ) ) )
              @ ( produc6619974131263629869ist_fm @ produc6879501374131015971m_rule @ produc1501393135466168645ist_fm ) ) )
          @ ( bNF_Gr929122299688919989ist_fm
            @ ( collec7217352646538401814ist_fm
              @ ^ [X3: produc4387668504790219841ist_fm] :
                  ( ( ord_le5052251716133888043m_rule @ ( basic_5898077337727170819ist_fm @ X3 ) @ ( collec2069506698588392810m_rule @ ( produc7369492558480804444rule_o @ R12 ) ) )
                  & ( ord_le2055072402069232929ist_fm @ ( basic_8334766877638084165ist_fm @ X3 ) @ ( collec7231575776666803724ist_fm @ ( produc2011568048573023402t_fm_o @ R23 ) ) ) ) )
            @ ( produc3181521936017884591ist_fm @ produc7165828336582415457m_rule @ produc4588648349897876871ist_fm ) ) ) ) ) ).

% prod.rel_compp_Grp
thf(fact_1094_prod_Orel__compp__Grp,axiom,
    ( basic_1419344120586502709m_rule
    = ( ^ [R12: produc1996495991257130529ist_fm > rule > $o,R23: produc1996495991257130529ist_fm > rule > $o] :
          ( relcom938509241273469078e_rule
          @ ( conver6869347575380784913ist_fm
            @ ( bNF_Gr290380328903957098ist_fm
              @ ( collec8730436809561838582m_rule
                @ ^ [X3: produc3793571666044074251m_rule] :
                    ( ( ord_le5052251716133888043m_rule @ ( basic_8177019974233741491m_rule @ X3 ) @ ( collec2069506698588392810m_rule @ ( produc7369492558480804444rule_o @ R12 ) ) )
                    & ( ord_le5052251716133888043m_rule @ ( basic_1757413314321953521m_rule @ X3 ) @ ( collec2069506698588392810m_rule @ ( produc7369492558480804444rule_o @ R23 ) ) ) ) )
              @ ( produc602106178381476086ist_fm @ produc6879501374131015971m_rule @ produc6879501374131015971m_rule ) ) )
          @ ( bNF_Gr1055224040921665006e_rule
            @ ( collec8730436809561838582m_rule
              @ ^ [X3: produc3793571666044074251m_rule] :
                  ( ( ord_le5052251716133888043m_rule @ ( basic_8177019974233741491m_rule @ X3 ) @ ( collec2069506698588392810m_rule @ ( produc7369492558480804444rule_o @ R12 ) ) )
                  & ( ord_le5052251716133888043m_rule @ ( basic_1757413314321953521m_rule @ X3 ) @ ( collec2069506698588392810m_rule @ ( produc7369492558480804444rule_o @ R23 ) ) ) ) )
            @ ( produc5273097181486698926e_rule @ produc7165828336582415457m_rule @ produc7165828336582415457m_rule ) ) ) ) ) ).

% prod.rel_compp_Grp
thf(fact_1095_prod_Orel__compp__Grp,axiom,
    ( basic_7991631927920859369ist_fm
    = ( ^ [R12: produc1996495991257130529ist_fm > rule > $o,R23: rule > produc1996495991257130529ist_fm > $o] :
          ( relcom1593757755746491986ist_fm
          @ ( conver70429133927872871m_rule
            @ ( bNF_Gr5409266173092018254m_rule
              @ ( collec740285645269470066ist_fm
                @ ^ [X3: produc5026792538606481543ist_fm] :
                    ( ( ord_le5052251716133888043m_rule @ ( basic_3420318458980781615ist_fm @ X3 ) @ ( collec2069506698588392810m_rule @ ( produc7369492558480804444rule_o @ R12 ) ) )
                    & ( ord_le6285472588696295335ist_fm @ ( basic_6224083835923769453ist_fm @ X3 ) @ ( collec6536177220190208742ist_fm @ ( produc1628407760719572648t_fm_o @ R23 ) ) ) ) )
              @ ( produc1386733058306926380m_rule @ produc6879501374131015971m_rule @ produc4228417144610596823ist_fm ) ) )
          @ ( bNF_Gr652564657839058378ist_fm
            @ ( collec740285645269470066ist_fm
              @ ^ [X3: produc5026792538606481543ist_fm] :
                  ( ( ord_le5052251716133888043m_rule @ ( basic_3420318458980781615ist_fm @ X3 ) @ ( collec2069506698588392810m_rule @ ( produc7369492558480804444rule_o @ R12 ) ) )
                  & ( ord_le6285472588696295335ist_fm @ ( basic_6224083835923769453ist_fm @ X3 ) @ ( collec6536177220190208742ist_fm @ ( produc1628407760719572648t_fm_o @ R23 ) ) ) ) )
            @ ( produc6121958980326948092ist_fm @ produc7165828336582415457m_rule @ produc4514744107061996309ist_fm ) ) ) ) ) ).

% prod.rel_compp_Grp
thf(fact_1096_prod_Orel__compp__Grp,axiom,
    ( basic_4124365050402646887ist_fm
    = ( ^ [R12: rule > produc1996495991257130529ist_fm > $o,R23: list_fm > list_fm > $o] :
          ( relcom7281857817583229324ist_fm
          @ ( conver3421711799201255370ist_fm
            @ ( bNF_Gr3626400350886154033ist_fm
              @ ( collec7675708960211541914ist_fm
                @ ^ [X3: produc4846024818463359941ist_fm] :
                    ( ( ord_le6285472588696295335ist_fm @ ( basic_6356433651400310919ist_fm @ X3 ) @ ( collec6536177220190208742ist_fm @ ( produc1628407760719572648t_fm_o @ R12 ) ) )
                    & ( ord_le2055072402069232929ist_fm @ ( basic_8793123191311224265ist_fm @ X3 ) @ ( collec7231575776666803724ist_fm @ ( produc2011568048573023402t_fm_o @ R23 ) ) ) ) )
              @ ( produc8587653452645586227ist_fm @ produc4228417144610596823ist_fm @ produc1501393135466168645ist_fm ) ) )
          @ ( bNF_Gr6741358174424549829ist_fm
            @ ( collec7675708960211541914ist_fm
              @ ^ [X3: produc4846024818463359941ist_fm] :
                  ( ( ord_le6285472588696295335ist_fm @ ( basic_6356433651400310919ist_fm @ X3 ) @ ( collec6536177220190208742ist_fm @ ( produc1628407760719572648t_fm_o @ R12 ) ) )
                  & ( ord_le2055072402069232929ist_fm @ ( basic_8793123191311224265ist_fm @ X3 ) @ ( collec7231575776666803724ist_fm @ ( produc2011568048573023402t_fm_o @ R23 ) ) ) ) )
            @ ( produc9060294921435433385ist_fm @ produc4514744107061996309ist_fm @ produc4588648349897876871ist_fm ) ) ) ) ) ).

% prod.rel_compp_Grp
thf(fact_1097_prod_Orel__compp__Grp,axiom,
    ( basic_6697320886820836993m_rule
    = ( ^ [R12: rule > produc1996495991257130529ist_fm > $o,R23: produc1996495991257130529ist_fm > rule > $o] :
          ( relcom110133155400602202m_rule
          @ ( conver3143134706523012835ist_fm
            @ ( bNF_Gr8481971745687158218ist_fm
              @ ( collec1444694503687461746m_rule
                @ ^ [X3: produc5731201397024473223m_rule] :
                    ( ( ord_le6285472588696295335ist_fm @ ( basic_891277668359364655m_rule @ X3 ) @ ( collec6536177220190208742ist_fm @ ( produc1628407760719572648t_fm_o @ R12 ) ) )
                    & ( ord_le5052251716133888043m_rule @ ( basic_3695043045302352493m_rule @ X3 ) @ ( collec2069506698588392810m_rule @ ( produc7369492558480804444rule_o @ R23 ) ) ) ) )
              @ ( produc1428366661664003836ist_fm @ produc4228417144610596823ist_fm @ produc6879501374131015971m_rule ) ) )
          @ ( bNF_Gr4015301224085342286m_rule
            @ ( collec1444694503687461746m_rule
              @ ^ [X3: produc5731201397024473223m_rule] :
                  ( ( ord_le6285472588696295335ist_fm @ ( basic_891277668359364655m_rule @ X3 ) @ ( collec6536177220190208742ist_fm @ ( produc1628407760719572648t_fm_o @ R12 ) ) )
                  & ( ord_le5052251716133888043m_rule @ ( basic_3695043045302352493m_rule @ X3 ) @ ( collec2069506698588392810m_rule @ ( produc7369492558480804444rule_o @ R23 ) ) ) ) )
            @ ( produc4376911675117420340e_rule @ produc4514744107061996309ist_fm @ produc7165828336582415457m_rule ) ) ) ) ) ).

% prod.rel_compp_Grp
thf(fact_1098_prod_Orel__compp__Grp,axiom,
    ( basic_4046236657300417845ist_fm
    = ( ^ [R12: rule > produc1996495991257130529ist_fm > $o,R23: rule > produc1996495991257130529ist_fm > $o] :
          ( relcom3074558381543601566ist_fm
          @ ( conver4081636798531099165e_rule
            @ ( bNF_Gr7506805509491411190e_rule
              @ ( collec2677915376249869038ist_fm
                @ ^ [X3: produc6964422269586880515ist_fm] :
                    ( ( ord_le6285472588696295335ist_fm @ ( basic_5357948189961180587ist_fm @ X3 ) @ ( collec6536177220190208742ist_fm @ ( produc1628407760719572648t_fm_o @ R12 ) ) )
                    & ( ord_le6285472588696295335ist_fm @ ( basic_8161713566904168425ist_fm @ X3 ) @ ( collec6536177220190208742ist_fm @ ( produc1628407760719572648t_fm_o @ R23 ) ) ) ) )
              @ ( produc447026873236687526m_rule @ produc4228417144610596823ist_fm @ produc4228417144610596823ist_fm ) ) )
          @ ( bNF_Gr2889200263857068402ist_fm
            @ ( collec2677915376249869038ist_fm
              @ ^ [X3: produc6964422269586880515ist_fm] :
                  ( ( ord_le6285472588696295335ist_fm @ ( basic_5357948189961180587ist_fm @ X3 ) @ ( collec6536177220190208742ist_fm @ ( produc1628407760719572648t_fm_o @ R12 ) ) )
                  & ( ord_le6285472588696295335ist_fm @ ( basic_8161713566904168425ist_fm @ X3 ) @ ( collec6536177220190208742ist_fm @ ( produc1628407760719572648t_fm_o @ R23 ) ) ) ) )
            @ ( produc1114089474322515446ist_fm @ produc4514744107061996309ist_fm @ produc4514744107061996309ist_fm ) ) ) ) ) ).

% prod.rel_compp_Grp
thf(fact_1099_prod_Oin__rel,axiom,
    ( basic_1286064880750865721ist_fm
    = ( ^ [R12: list_fm > list_fm > $o,R23: list_fm > list_fm > $o,A5: produc1996495991257130529ist_fm,B4: produc1996495991257130529ist_fm] :
        ? [Z5: produc1754732471204517991ist_fm] :
          ( ( member60238275747228048ist_fm @ Z5
            @ ( collec888080998205339090ist_fm
              @ ^ [X3: produc1754732471204517991ist_fm] :
                  ( ( ord_le2055072402069232929ist_fm @ ( basic_6510150186836533191ist_fm @ X3 ) @ ( collec7231575776666803724ist_fm @ ( produc2011568048573023402t_fm_o @ R12 ) ) )
                  & ( ord_le2055072402069232929ist_fm @ ( basic_4728562467516582661ist_fm @ X3 ) @ ( collec7231575776666803724ist_fm @ ( produc2011568048573023402t_fm_o @ R23 ) ) ) ) ) )
          & ( ( produc7565992903848568886ist_fm @ produc1501393135466168645ist_fm @ produc1501393135466168645ist_fm @ Z5 )
            = A5 )
          & ( ( produc7565992903848568886ist_fm @ produc4588648349897876871ist_fm @ produc4588648349897876871ist_fm @ Z5 )
            = B4 ) ) ) ) ).

% prod.in_rel
thf(fact_1100_prod_Oin__rel,axiom,
    ( basic_4795398152100796463m_rule
    = ( ^ [R12: list_fm > list_fm > $o,R23: produc1996495991257130529ist_fm > rule > $o,A5: produc1511181096255975734ist_fm,B4: produc8057652614547790598m_rule] :
        ? [Z5: produc466554394152024933m_rule] :
          ( ( member4427830712138218492m_rule @ Z5
            @ ( collec3296238535900206906m_rule
              @ ^ [X3: produc466554394152024933m_rule] :
                  ( ( ord_le2055072402069232929ist_fm @ ( basic_6200111124537405679m_rule @ X3 ) @ ( collec7231575776666803724ist_fm @ ( produc2011568048573023402t_fm_o @ R12 ) ) )
                  & ( ord_le5052251716133888043m_rule @ ( basic_8636800664448319025m_rule @ X3 ) @ ( collec2069506698588392810m_rule @ ( produc7369492558480804444rule_o @ R23 ) ) ) ) ) )
          & ( ( produc7676726057136416639ist_fm @ produc1501393135466168645ist_fm @ produc6879501374131015971m_rule @ Z5 )
            = A5 )
          & ( ( produc8614446829993874791e_rule @ produc4588648349897876871ist_fm @ produc7165828336582415457m_rule @ Z5 )
            = B4 ) ) ) ) ).

% prod.in_rel
thf(fact_1101_prod_Oin__rel,axiom,
    ( basic_2144313922580377315ist_fm
    = ( ^ [R12: list_fm > list_fm > $o,R23: rule > produc1996495991257130529ist_fm > $o,A5: produc8057652614547790598m_rule,B4: produc1511181096255975734ist_fm] :
        ? [Z5: produc1699775266714432225ist_fm] :
          ( ( member5661051584700625784ist_fm @ Z5
            @ ( collec4529459408462614198ist_fm
              @ ^ [X3: produc1699775266714432225ist_fm] :
                  ( ( ord_le2055072402069232929ist_fm @ ( basic_1443409609284445803ist_fm @ X3 ) @ ( collec7231575776666803724ist_fm @ ( produc2011568048573023402t_fm_o @ R12 ) ) )
                  & ( ord_le6285472588696295335ist_fm @ ( basic_3880099149195359149ist_fm @ X3 ) @ ( collec6536177220190208742ist_fm @ ( produc1628407760719572648t_fm_o @ R23 ) ) ) ) ) )
          & ( ( produc1389032208193576675m_rule @ produc1501393135466168645ist_fm @ produc4228417144610596823ist_fm @ Z5 )
            = A5 )
          & ( ( produc8135082370809556739ist_fm @ produc4588648349897876871ist_fm @ produc4514744107061996309ist_fm @ Z5 )
            = B4 ) ) ) ) ).

% prod.in_rel
thf(fact_1102_prod_Oin__rel,axiom,
    ( basic_882330199258250419ist_fm
    = ( ^ [R12: produc1996495991257130529ist_fm > rule > $o,R23: list_fm > list_fm > $o,A5: produc225247741881318192ist_fm,B4: produc5753620461286005894ist_fm] :
        ? [Z5: produc4387668504790219841ist_fm] :
          ( ( member8348944822776413400ist_fm @ Z5
            @ ( collec7217352646538401814ist_fm
              @ ^ [X3: produc4387668504790219841ist_fm] :
                  ( ( ord_le5052251716133888043m_rule @ ( basic_5898077337727170819ist_fm @ X3 ) @ ( collec2069506698588392810m_rule @ ( produc7369492558480804444rule_o @ R12 ) ) )
                  & ( ord_le2055072402069232929ist_fm @ ( basic_8334766877638084165ist_fm @ X3 ) @ ( collec7231575776666803724ist_fm @ ( produc2011568048573023402t_fm_o @ R23 ) ) ) ) ) )
          & ( ( produc6619974131263629869ist_fm @ produc6879501374131015971m_rule @ produc1501393135466168645ist_fm @ Z5 )
            = A5 )
          & ( ( produc3181521936017884591ist_fm @ produc7165828336582415457m_rule @ produc4588648349897876871ist_fm @ Z5 )
            = B4 ) ) ) ) ).

% prod.in_rel
thf(fact_1103_prod_Oin__rel,axiom,
    ( basic_1419344120586502709m_rule
    = ( ^ [R12: produc1996495991257130529ist_fm > rule > $o,R23: produc1996495991257130529ist_fm > rule > $o,A5: produc1754732471204517991ist_fm,B4: produc780299800278082923e_rule] :
        ? [Z5: produc3793571666044074251m_rule] :
          ( ( member2434505771962676916m_rule @ Z5
            @ ( collec8730436809561838582m_rule
              @ ^ [X3: produc3793571666044074251m_rule] :
                  ( ( ord_le5052251716133888043m_rule @ ( basic_8177019974233741491m_rule @ X3 ) @ ( collec2069506698588392810m_rule @ ( produc7369492558480804444rule_o @ R12 ) ) )
                  & ( ord_le5052251716133888043m_rule @ ( basic_1757413314321953521m_rule @ X3 ) @ ( collec2069506698588392810m_rule @ ( produc7369492558480804444rule_o @ R23 ) ) ) ) ) )
          & ( ( produc602106178381476086ist_fm @ produc6879501374131015971m_rule @ produc6879501374131015971m_rule @ Z5 )
            = A5 )
          & ( ( produc5273097181486698926e_rule @ produc7165828336582415457m_rule @ produc7165828336582415457m_rule @ Z5 )
            = B4 ) ) ) ) ).

% prod.in_rel
thf(fact_1104_prod_Oin__rel,axiom,
    ( basic_7991631927920859369ist_fm
    = ( ^ [R12: produc1996495991257130529ist_fm > rule > $o,R23: rule > produc1996495991257130529ist_fm > $o,A5: produc164195504107695125m_rule,B4: produc4630866025709511057ist_fm] :
        ? [Z5: produc5026792538606481543ist_fm] :
          ( ( member3667726644525084208ist_fm @ Z5
            @ ( collec740285645269470066ist_fm
              @ ^ [X3: produc5026792538606481543ist_fm] :
                  ( ( ord_le5052251716133888043m_rule @ ( basic_3420318458980781615ist_fm @ X3 ) @ ( collec2069506698588392810m_rule @ ( produc7369492558480804444rule_o @ R12 ) ) )
                  & ( ord_le6285472588696295335ist_fm @ ( basic_6224083835923769453ist_fm @ X3 ) @ ( collec6536177220190208742ist_fm @ ( produc1628407760719572648t_fm_o @ R23 ) ) ) ) ) )
          & ( ( produc1386733058306926380m_rule @ produc6879501374131015971m_rule @ produc4228417144610596823ist_fm @ Z5 )
            = A5 )
          & ( ( produc6121958980326948092ist_fm @ produc7165828336582415457m_rule @ produc4514744107061996309ist_fm @ Z5 )
            = B4 ) ) ) ) ).

% prod.in_rel
thf(fact_1105_prod_Oin__rel,axiom,
    ( basic_4124365050402646887ist_fm
    = ( ^ [R12: rule > produc1996495991257130529ist_fm > $o,R23: list_fm > list_fm > $o,A5: produc5753620461286005894ist_fm,B4: produc225247741881318192ist_fm] :
        ? [Z5: produc4846024818463359941ist_fm] :
          ( ( member8807301136449553500ist_fm @ Z5
            @ ( collec7675708960211541914ist_fm
              @ ^ [X3: produc4846024818463359941ist_fm] :
                  ( ( ord_le6285472588696295335ist_fm @ ( basic_6356433651400310919ist_fm @ X3 ) @ ( collec6536177220190208742ist_fm @ ( produc1628407760719572648t_fm_o @ R12 ) ) )
                  & ( ord_le2055072402069232929ist_fm @ ( basic_8793123191311224265ist_fm @ X3 ) @ ( collec7231575776666803724ist_fm @ ( produc2011568048573023402t_fm_o @ R23 ) ) ) ) ) )
          & ( ( produc8587653452645586227ist_fm @ produc4228417144610596823ist_fm @ produc1501393135466168645ist_fm @ Z5 )
            = A5 )
          & ( ( produc9060294921435433385ist_fm @ produc4514744107061996309ist_fm @ produc4588648349897876871ist_fm @ Z5 )
            = B4 ) ) ) ) ).

% prod.in_rel
thf(fact_1106_prod_Oin__rel,axiom,
    ( basic_6697320886820836993m_rule
    = ( ^ [R12: rule > produc1996495991257130529ist_fm > $o,R23: produc1996495991257130529ist_fm > rule > $o,A5: produc4630866025709511057ist_fm,B4: produc164195504107695125m_rule] :
        ? [Z5: produc5731201397024473223m_rule] :
          ( ( member4372135502943075888m_rule @ Z5
            @ ( collec1444694503687461746m_rule
              @ ^ [X3: produc5731201397024473223m_rule] :
                  ( ( ord_le6285472588696295335ist_fm @ ( basic_891277668359364655m_rule @ X3 ) @ ( collec6536177220190208742ist_fm @ ( produc1628407760719572648t_fm_o @ R12 ) ) )
                  & ( ord_le5052251716133888043m_rule @ ( basic_3695043045302352493m_rule @ X3 ) @ ( collec2069506698588392810m_rule @ ( produc7369492558480804444rule_o @ R23 ) ) ) ) ) )
          & ( ( produc1428366661664003836ist_fm @ produc4228417144610596823ist_fm @ produc6879501374131015971m_rule @ Z5 )
            = A5 )
          & ( ( produc4376911675117420340e_rule @ produc4514744107061996309ist_fm @ produc7165828336582415457m_rule @ Z5 )
            = B4 ) ) ) ) ).

% prod.in_rel
thf(fact_1107_prod_Oin__rel,axiom,
    ( basic_4046236657300417845ist_fm
    = ( ^ [R12: rule > produc1996495991257130529ist_fm > $o,R23: rule > produc1996495991257130529ist_fm > $o,A5: produc780299800278082923e_rule,B4: produc1754732471204517991ist_fm] :
        ? [Z5: produc6964422269586880515ist_fm] :
          ( ( member5605356375505483180ist_fm @ Z5
            @ ( collec2677915376249869038ist_fm
              @ ^ [X3: produc6964422269586880515ist_fm] :
                  ( ( ord_le6285472588696295335ist_fm @ ( basic_5357948189961180587ist_fm @ X3 ) @ ( collec6536177220190208742ist_fm @ ( produc1628407760719572648t_fm_o @ R12 ) ) )
                  & ( ord_le6285472588696295335ist_fm @ ( basic_8161713566904168425ist_fm @ X3 ) @ ( collec6536177220190208742ist_fm @ ( produc1628407760719572648t_fm_o @ R23 ) ) ) ) ) )
          & ( ( produc447026873236687526m_rule @ produc4228417144610596823ist_fm @ produc4228417144610596823ist_fm @ Z5 )
            = A5 )
          & ( ( produc1114089474322515446ist_fm @ produc4514744107061996309ist_fm @ produc4514744107061996309ist_fm @ Z5 )
            = B4 ) ) ) ) ).

% prod.in_rel
thf(fact_1108_sinterleave_Ocode,axiom,
    ( sinter8084399408526207096m_rule
    = ( ^ [S12: stream727092118206550309m_rule,S22: stream727092118206550309m_rule] : ( sCons_5731567480799343806m_rule @ ( shd_Pr7235097944458474089m_rule @ S12 ) @ ( sinter8084399408526207096m_rule @ S22 @ ( stl_Pr950425576149878629m_rule @ S12 ) ) ) ) ) ).

% sinterleave.code
thf(fact_1109_sinterleave_Osimps_I1_J,axiom,
    ! [S1: stream727092118206550309m_rule,S2: stream727092118206550309m_rule] :
      ( ( shd_Pr7235097944458474089m_rule @ ( sinter8084399408526207096m_rule @ S1 @ S2 ) )
      = ( shd_Pr7235097944458474089m_rule @ S1 ) ) ).

% sinterleave.simps(1)
thf(fact_1110_Domain__fst,axiom,
    ( domain3604227379074985946m_rule
    = ( image_3223098899990291461ist_fm @ produc6879501374131015971m_rule ) ) ).

% Domain_fst
thf(fact_1111_Domain__fst,axiom,
    ( domain2949914093644414030ist_fm
    = ( image_6369949096775035324ist_fm @ produc1501393135466168645ist_fm ) ) ).

% Domain_fst
thf(fact_1112_fst__eq__Domain,axiom,
    ! [R: set_Pr1008144964186165195m_rule] :
      ( ( image_3223098899990291461ist_fm @ produc6879501374131015971m_rule @ R )
      = ( domain3604227379074985946m_rule @ R ) ) ).

% fst_eq_Domain
thf(fact_1113_fst__eq__Domain,axiom,
    ! [R: set_Pr7058068377845519745ist_fm] :
      ( ( image_6369949096775035324ist_fm @ produc1501393135466168645ist_fm @ R )
      = ( domain2949914093644414030ist_fm @ R ) ) ).

% fst_eq_Domain
thf(fact_1114_Range__snd,axiom,
    ( range_8771097359499353911ist_fm
    = ( image_6369949096775035324ist_fm @ produc4588648349897876871ist_fm ) ) ).

% Range_snd
thf(fact_1115_Range__snd,axiom,
    ( range_5278118250590514865m_rule
    = ( image_8178078329501008481e_rule @ produc7165828336582415457m_rule ) ) ).

% Range_snd
thf(fact_1116_Range__snd,axiom,
    ( range_2627034021070095717ist_fm
    = ( image_3681455213663431561ist_fm @ produc4514744107061996309ist_fm ) ) ).

% Range_snd
thf(fact_1117_snd__eq__Range,axiom,
    ! [R: set_Pr7058068377845519745ist_fm] :
      ( ( image_6369949096775035324ist_fm @ produc4588648349897876871ist_fm @ R )
      = ( range_8771097359499353911ist_fm @ R ) ) ).

% snd_eq_Range
thf(fact_1118_snd__eq__Range,axiom,
    ! [R: set_Pr1008144964186165195m_rule] :
      ( ( image_8178078329501008481e_rule @ produc7165828336582415457m_rule @ R )
      = ( range_5278118250590514865m_rule @ R ) ) ).

% snd_eq_Range
thf(fact_1119_snd__eq__Range,axiom,
    ! [R: set_Pr2241365836748572487ist_fm] :
      ( ( image_3681455213663431561ist_fm @ produc4514744107061996309ist_fm @ R )
      = ( range_2627034021070095717ist_fm @ R ) ) ).

% snd_eq_Range
thf(fact_1120_sset__sinterleave,axiom,
    ! [S1: stream727092118206550309m_rule,S2: stream727092118206550309m_rule] :
      ( ( sset_P4979403222752902913m_rule @ ( sinter8084399408526207096m_rule @ S1 @ S2 ) )
      = ( sup_su3887047029712641783m_rule @ ( sset_P4979403222752902913m_rule @ S1 ) @ ( sset_P4979403222752902913m_rule @ S2 ) ) ) ).

% sset_sinterleave
thf(fact_1121_set__replicate,axiom,
    ! [N2: nat,X2: fm] :
      ( ( N2 != zero_zero_nat )
     => ( ( set_fm2 @ ( replicate_fm @ N2 @ X2 ) )
        = ( insert_fm2 @ X2 @ bot_bot_set_fm ) ) ) ).

% set_replicate
thf(fact_1122_in__set__replicate,axiom,
    ! [X2: fm,N2: nat,Y3: fm] :
      ( ( member_fm @ X2 @ ( set_fm2 @ ( replicate_fm @ N2 @ Y3 ) ) )
      = ( ( X2 = Y3 )
        & ( N2 != zero_zero_nat ) ) ) ).

% in_set_replicate
thf(fact_1123_Bex__set__replicate,axiom,
    ! [N2: nat,A: fm,P2: fm > $o] :
      ( ( ? [X3: fm] :
            ( ( member_fm @ X3 @ ( set_fm2 @ ( replicate_fm @ N2 @ A ) ) )
            & ( P2 @ X3 ) ) )
      = ( ( P2 @ A )
        & ( N2 != zero_zero_nat ) ) ) ).

% Bex_set_replicate
thf(fact_1124_Ball__set__replicate,axiom,
    ! [N2: nat,A: fm,P2: fm > $o] :
      ( ( ! [X3: fm] :
            ( ( member_fm @ X3 @ ( set_fm2 @ ( replicate_fm @ N2 @ A ) ) )
           => ( P2 @ X3 ) ) )
      = ( ( P2 @ A )
        | ( N2 = zero_zero_nat ) ) ) ).

% Ball_set_replicate
thf(fact_1125_well__order__on__domain,axiom,
    ! [A4: set_fm,R2: set_Pr4463079037648049377_fm_fm,A: fm,B: fm] :
      ( ( order_2053106022633971762_on_fm @ A4 @ R2 )
     => ( ( member8474499337054950954_fm_fm @ ( product_Pair_fm_fm @ A @ B ) @ R2 )
       => ( ( member_fm @ A @ A4 )
          & ( member_fm @ B @ A4 ) ) ) ) ).

% well_order_on_domain
thf(fact_1126_set__replicate__Suc,axiom,
    ! [N2: nat,X2: fm] :
      ( ( set_fm2 @ ( replicate_fm @ ( suc @ N2 ) @ X2 ) )
      = ( insert_fm2 @ X2 @ bot_bot_set_fm ) ) ).

% set_replicate_Suc
thf(fact_1127_set__replicate__conv__if,axiom,
    ! [N2: nat,X2: fm] :
      ( ( ( N2 = zero_zero_nat )
       => ( ( set_fm2 @ ( replicate_fm @ N2 @ X2 ) )
          = bot_bot_set_fm ) )
      & ( ( N2 != zero_zero_nat )
       => ( ( set_fm2 @ ( replicate_fm @ N2 @ X2 ) )
          = ( insert_fm2 @ X2 @ bot_bot_set_fm ) ) ) ) ).

% set_replicate_conv_if
thf(fact_1128_underS__incl__iff,axiom,
    ! [R2: set_Pr4463079037648049377_fm_fm,A: fm,B: fm] :
      ( ( order_2400927927029563739_on_fm @ ( field_fm @ R2 ) @ R2 )
     => ( ( member_fm @ A @ ( field_fm @ R2 ) )
       => ( ( member_fm @ B @ ( field_fm @ R2 ) )
         => ( ( ord_less_eq_set_fm @ ( order_underS_fm @ R2 @ A ) @ ( order_underS_fm @ R2 @ B ) )
            = ( member8474499337054950954_fm_fm @ ( product_Pair_fm_fm @ A @ B ) @ R2 ) ) ) ) ) ).

% underS_incl_iff
thf(fact_1129_sset__shift,axiom,
    ! [Xs: list_fm,S3: stream_fm] :
      ( ( sset_fm @ ( shift_fm @ Xs @ S3 ) )
      = ( sup_sup_set_fm @ ( set_fm2 @ Xs ) @ ( sset_fm @ S3 ) ) ) ).

% sset_shift
thf(fact_1130_sset__shift,axiom,
    ! [Xs: list_P8332459875677254181m_rule,S3: stream727092118206550309m_rule] :
      ( ( sset_P4979403222752902913m_rule @ ( shift_5008521991580687660m_rule @ Xs @ S3 ) )
      = ( sup_su3887047029712641783m_rule @ ( set_Pr8505323785428441536m_rule @ Xs ) @ ( sset_P4979403222752902913m_rule @ S3 ) ) ) ).

% sset_shift
thf(fact_1131_underS__I,axiom,
    ! [I: fm,J2: fm,R: set_Pr4463079037648049377_fm_fm] :
      ( ( I != J2 )
     => ( ( member8474499337054950954_fm_fm @ ( product_Pair_fm_fm @ I @ J2 ) @ R )
       => ( member_fm @ I @ ( order_underS_fm @ R @ J2 ) ) ) ) ).

% underS_I
thf(fact_1132_underS__E,axiom,
    ! [I: fm,R: set_Pr4463079037648049377_fm_fm,J2: fm] :
      ( ( member_fm @ I @ ( order_underS_fm @ R @ J2 ) )
     => ( ( I != J2 )
        & ( member8474499337054950954_fm_fm @ ( product_Pair_fm_fm @ I @ J2 ) @ R ) ) ) ).

% underS_E
thf(fact_1133_BNF__Least__Fixpoint_OunderS__Field,axiom,
    ! [I: fm,R: set_Pr4463079037648049377_fm_fm,J2: fm] :
      ( ( member_fm @ I @ ( order_underS_fm @ R @ J2 ) )
     => ( member_fm @ I @ ( field_fm @ R ) ) ) ).

% BNF_Least_Fixpoint.underS_Field
thf(fact_1134_id__stake__snth__sdrop,axiom,
    ! [S3: stream727092118206550309m_rule,I: nat] :
      ( S3
      = ( shift_5008521991580687660m_rule @ ( stake_1447931197033250628m_rule @ I @ S3 ) @ ( sCons_5731567480799343806m_rule @ ( snth_P1712579846118164081m_rule @ S3 @ I ) @ ( sdrop_7224736112439592940m_rule @ ( suc @ I ) @ S3 ) ) ) ) ).

% id_stake_snth_sdrop
thf(fact_1135_id__stake__snth__sdrop,axiom,
    ! [S3: stream8299795917829157543ist_fm,I: nat] :
      ( S3
      = ( shift_5848029981951141578ist_fm @ ( stake_2930890243849202354ist_fm @ I @ S3 ) @ ( sCons_307821682964077496ist_fm @ ( snth_P2624749845746815685ist_fm @ S3 @ I ) @ ( sdrop_4442373711808556042ist_fm @ ( suc @ I ) @ S3 ) ) ) ) ).

% id_stake_snth_sdrop
thf(fact_1136_id__stake__snth__sdrop,axiom,
    ! [S3: stream_rule,I: nat] :
      ( S3
      = ( shift_rule @ ( stake_rule @ I @ S3 ) @ ( sCons_rule @ ( snth_rule @ S3 @ I ) @ ( sdrop_rule @ ( suc @ I ) @ S3 ) ) ) ) ).

% id_stake_snth_sdrop
thf(fact_1137_id__stake__snth__sdrop,axiom,
    ! [S3: stream6494289010434245521m_rule,I: nat] :
      ( S3
      = ( shift_3194717208791370420m_rule @ ( stake_8908532221384226460m_rule @ I @ S3 ) @ ( sCons_4186928284053167522m_rule @ ( snth_P8950227884416379311m_rule @ S3 @ I ) @ ( sdrop_7373388980841566196m_rule @ ( suc @ I ) @ S3 ) ) ) ) ).

% id_stake_snth_sdrop
thf(fact_1138_Id__on__set,axiom,
    ! [Xs: list_fm] :
      ( ( id_on_fm @ ( set_fm2 @ Xs ) )
      = ( set_Pr6624747810323803798_fm_fm
        @ ( map_fm8969396669541242987_fm_fm
          @ ^ [X3: fm] : ( product_Pair_fm_fm @ X3 @ X3 )
          @ Xs ) ) ) ).

% Id_on_set
thf(fact_1139_ImageI,axiom,
    ! [A: fm,B: fm,R2: set_Pr4463079037648049377_fm_fm,A4: set_fm] :
      ( ( member8474499337054950954_fm_fm @ ( product_Pair_fm_fm @ A @ B ) @ R2 )
     => ( ( member_fm @ A @ A4 )
       => ( member_fm @ B @ ( image_fm_fm @ R2 @ A4 ) ) ) ) ).

% ImageI
thf(fact_1140_list_Oset__map,axiom,
    ! [F2: produc164195504107695125m_rule > set_fm,V: list_P8332459875677254181m_rule] :
      ( ( set_set_fm2 @ ( map_Pr304503626113623905set_fm @ F2 @ V ) )
      = ( image_5698427363735666338set_fm @ F2 @ ( set_Pr8505323785428441536m_rule @ V ) ) ) ).

% list.set_map
thf(fact_1141_list_Oset__map,axiom,
    ! [F2: fm > fm,V: list_fm] :
      ( ( set_fm2 @ ( map_fm_fm @ F2 @ V ) )
      = ( image_fm_fm2 @ F2 @ ( set_fm2 @ V ) ) ) ).

% list.set_map
thf(fact_1142_list_Omap__ident__strong,axiom,
    ! [T2: list_fm,F2: fm > fm] :
      ( ! [Z3: fm] :
          ( ( member_fm @ Z3 @ ( set_fm2 @ T2 ) )
         => ( ( F2 @ Z3 )
            = Z3 ) )
     => ( ( map_fm_fm @ F2 @ T2 )
        = T2 ) ) ).

% list.map_ident_strong
thf(fact_1143_map__idI,axiom,
    ! [Xs: list_fm,F2: fm > fm] :
      ( ! [X: fm] :
          ( ( member_fm @ X @ ( set_fm2 @ Xs ) )
         => ( ( F2 @ X )
            = X ) )
     => ( ( map_fm_fm @ F2 @ Xs )
        = Xs ) ) ).

% map_idI
thf(fact_1144_image__set,axiom,
    ! [F2: produc164195504107695125m_rule > set_fm,Xs: list_P8332459875677254181m_rule] :
      ( ( image_5698427363735666338set_fm @ F2 @ ( set_Pr8505323785428441536m_rule @ Xs ) )
      = ( set_set_fm2 @ ( map_Pr304503626113623905set_fm @ F2 @ Xs ) ) ) ).

% image_set
thf(fact_1145_image__set,axiom,
    ! [F2: fm > fm,Xs: list_fm] :
      ( ( image_fm_fm2 @ F2 @ ( set_fm2 @ Xs ) )
      = ( set_fm2 @ ( map_fm_fm @ F2 @ Xs ) ) ) ).

% image_set
thf(fact_1146_rev__ImageI,axiom,
    ! [A: fm,A4: set_fm,B: fm,R2: set_Pr4463079037648049377_fm_fm] :
      ( ( member_fm @ A @ A4 )
     => ( ( member8474499337054950954_fm_fm @ ( product_Pair_fm_fm @ A @ B ) @ R2 )
       => ( member_fm @ B @ ( image_fm_fm @ R2 @ A4 ) ) ) ) ).

% rev_ImageI
thf(fact_1147_ImageE,axiom,
    ! [B: fm,R2: set_Pr4463079037648049377_fm_fm,A4: set_fm] :
      ( ( member_fm @ B @ ( image_fm_fm @ R2 @ A4 ) )
     => ~ ! [X: fm] :
            ( ( member8474499337054950954_fm_fm @ ( product_Pair_fm_fm @ X @ B ) @ R2 )
           => ~ ( member_fm @ X @ A4 ) ) ) ).

% ImageE
thf(fact_1148_Image__UN,axiom,
    ! [R2: set_Pr4463079037648049377_fm_fm,B5: produc164195504107695125m_rule > set_fm,A4: set_Pr1008144964186165195m_rule] :
      ( ( image_fm_fm @ R2 @ ( comple7677669044296858969set_fm @ ( image_5698427363735666338set_fm @ B5 @ A4 ) ) )
      = ( comple7677669044296858969set_fm
        @ ( image_5698427363735666338set_fm
          @ ^ [X3: produc164195504107695125m_rule] : ( image_fm_fm @ R2 @ ( B5 @ X3 ) )
          @ A4 ) ) ) ).

% Image_UN
thf(fact_1149_stake__sdrop,axiom,
    ! [N2: nat,S3: stream727092118206550309m_rule] :
      ( ( shift_5008521991580687660m_rule @ ( stake_1447931197033250628m_rule @ N2 @ S3 ) @ ( sdrop_7224736112439592940m_rule @ N2 @ S3 ) )
      = S3 ) ).

% stake_sdrop
thf(fact_1150_stake__sdrop,axiom,
    ! [N2: nat,S3: stream8299795917829157543ist_fm] :
      ( ( shift_5848029981951141578ist_fm @ ( stake_2930890243849202354ist_fm @ N2 @ S3 ) @ ( sdrop_4442373711808556042ist_fm @ N2 @ S3 ) )
      = S3 ) ).

% stake_sdrop
thf(fact_1151_stake__sdrop,axiom,
    ! [N2: nat,S3: stream_rule] :
      ( ( shift_rule @ ( stake_rule @ N2 @ S3 ) @ ( sdrop_rule @ N2 @ S3 ) )
      = S3 ) ).

% stake_sdrop
thf(fact_1152_stake__sdrop,axiom,
    ! [N2: nat,S3: stream6494289010434245521m_rule] :
      ( ( shift_3194717208791370420m_rule @ ( stake_8908532221384226460m_rule @ N2 @ S3 ) @ ( sdrop_7373388980841566196m_rule @ N2 @ S3 ) )
      = S3 ) ).

% stake_sdrop
thf(fact_1153_Image__eq__UN,axiom,
    ( image_7747676986113175866ule_fm
    = ( ^ [R4: set_Pr8819451114087901350ule_fm,B7: set_Pr1008144964186165195m_rule] :
          ( comple7677669044296858969set_fm
          @ ( image_5698427363735666338set_fm
            @ ^ [Y4: produc164195504107695125m_rule] : ( image_7747676986113175866ule_fm @ R4 @ ( insert8332210346540479045m_rule @ Y4 @ bot_bo2182007826548605023m_rule ) )
            @ B7 ) ) ) ) ).

% Image_eq_UN
thf(fact_1154_subset__Image1__Image1__iff,axiom,
    ! [R2: set_Pr4463079037648049377_fm_fm,A: fm,B: fm] :
      ( ( order_preorder_on_fm @ ( field_fm @ R2 ) @ R2 )
     => ( ( member_fm @ A @ ( field_fm @ R2 ) )
       => ( ( member_fm @ B @ ( field_fm @ R2 ) )
         => ( ( ord_less_eq_set_fm @ ( image_fm_fm @ R2 @ ( insert_fm2 @ A @ bot_bot_set_fm ) ) @ ( image_fm_fm @ R2 @ ( insert_fm2 @ B @ bot_bot_set_fm ) ) )
            = ( member8474499337054950954_fm_fm @ ( product_Pair_fm_fm @ B @ A ) @ R2 ) ) ) ) ) ).

% subset_Image1_Image1_iff
thf(fact_1155_pair__list__eqI,axiom,
    ! [Xs: list_P5616295576739893671ist_fm,Ys: list_P5616295576739893671ist_fm] :
      ( ( ( map_Pr3743373011763743805ist_fm @ produc1501393135466168645ist_fm @ Xs )
        = ( map_Pr3743373011763743805ist_fm @ produc1501393135466168645ist_fm @ Ys ) )
     => ( ( ( map_Pr3743373011763743805ist_fm @ produc4588648349897876871ist_fm @ Xs )
          = ( map_Pr3743373011763743805ist_fm @ produc4588648349897876871ist_fm @ Ys ) )
       => ( Xs = Ys ) ) ) ).

% pair_list_eqI
thf(fact_1156_pair__list__eqI,axiom,
    ! [Xs: list_P8332459875677254181m_rule,Ys: list_P8332459875677254181m_rule] :
      ( ( ( map_Pr1951777954582829830ist_fm @ produc6879501374131015971m_rule @ Xs )
        = ( map_Pr1951777954582829830ist_fm @ produc6879501374131015971m_rule @ Ys ) )
     => ( ( ( map_Pr7678108474426652576e_rule @ produc7165828336582415457m_rule @ Xs )
          = ( map_Pr7678108474426652576e_rule @ produc7165828336582415457m_rule @ Ys ) )
       => ( Xs = Ys ) ) ) ).

% pair_list_eqI
thf(fact_1157_pair__list__eqI,axiom,
    ! [Xs: list_P342308711384885665ist_fm,Ys: list_P342308711384885665ist_fm] :
      ( ( ( map_Pr452693852626354460m_rule @ produc4228417144610596823ist_fm @ Xs )
        = ( map_Pr452693852626354460m_rule @ produc4228417144610596823ist_fm @ Ys ) )
     => ( ( ( map_Pr2410134268255969930ist_fm @ produc4514744107061996309ist_fm @ Xs )
          = ( map_Pr2410134268255969930ist_fm @ produc4514744107061996309ist_fm @ Ys ) )
       => ( Xs = Ys ) ) ) ).

% pair_list_eqI
thf(fact_1158_equiv__class__eq__iff,axiom,
    ! [A4: set_fm,R2: set_Pr4463079037648049377_fm_fm,X2: fm,Y3: fm] :
      ( ( equiv_equiv_fm @ A4 @ R2 )
     => ( ( member8474499337054950954_fm_fm @ ( product_Pair_fm_fm @ X2 @ Y3 ) @ R2 )
        = ( ( ( image_fm_fm @ R2 @ ( insert_fm2 @ X2 @ bot_bot_set_fm ) )
            = ( image_fm_fm @ R2 @ ( insert_fm2 @ Y3 @ bot_bot_set_fm ) ) )
          & ( member_fm @ X2 @ A4 )
          & ( member_fm @ Y3 @ A4 ) ) ) ) ).

% equiv_class_eq_iff
thf(fact_1159_eq__equiv__class__iff,axiom,
    ! [A4: set_fm,R2: set_Pr4463079037648049377_fm_fm,X2: fm,Y3: fm] :
      ( ( equiv_equiv_fm @ A4 @ R2 )
     => ( ( member_fm @ X2 @ A4 )
       => ( ( member_fm @ Y3 @ A4 )
         => ( ( ( image_fm_fm @ R2 @ ( insert_fm2 @ X2 @ bot_bot_set_fm ) )
              = ( image_fm_fm @ R2 @ ( insert_fm2 @ Y3 @ bot_bot_set_fm ) ) )
            = ( member8474499337054950954_fm_fm @ ( product_Pair_fm_fm @ X2 @ Y3 ) @ R2 ) ) ) ) ) ).

% eq_equiv_class_iff
thf(fact_1160_eq__equiv__class,axiom,
    ! [R2: set_Pr4463079037648049377_fm_fm,A: fm,B: fm,A4: set_fm] :
      ( ( ( image_fm_fm @ R2 @ ( insert_fm2 @ A @ bot_bot_set_fm ) )
        = ( image_fm_fm @ R2 @ ( insert_fm2 @ B @ bot_bot_set_fm ) ) )
     => ( ( equiv_equiv_fm @ A4 @ R2 )
       => ( ( member_fm @ B @ A4 )
         => ( member8474499337054950954_fm_fm @ ( product_Pair_fm_fm @ A @ B ) @ R2 ) ) ) ) ).

% eq_equiv_class
thf(fact_1161_subset__equiv__class,axiom,
    ! [A4: set_fm,R2: set_Pr4463079037648049377_fm_fm,B: fm,A: fm] :
      ( ( equiv_equiv_fm @ A4 @ R2 )
     => ( ( ord_less_eq_set_fm @ ( image_fm_fm @ R2 @ ( insert_fm2 @ B @ bot_bot_set_fm ) ) @ ( image_fm_fm @ R2 @ ( insert_fm2 @ A @ bot_bot_set_fm ) ) )
       => ( ( member_fm @ B @ A4 )
         => ( member8474499337054950954_fm_fm @ ( product_Pair_fm_fm @ A @ B ) @ R2 ) ) ) ) ).

% subset_equiv_class
thf(fact_1162_product__code,axiom,
    ! [Xs: list_fm,Ys: list_fm] :
      ( ( produc8271468480929761268_fm_fm @ ( set_fm2 @ Xs ) @ ( set_fm2 @ Ys ) )
      = ( set_Pr6624747810323803798_fm_fm
        @ ( concat1056723825482393666_fm_fm
          @ ( map_fm2780228255251704305_fm_fm
            @ ^ [X3: fm] : ( map_fm8969396669541242987_fm_fm @ ( product_Pair_fm_fm @ X3 ) @ Ys )
            @ Xs ) ) ) ) ).

% product_code
thf(fact_1163_set__concat,axiom,
    ! [Xs: list_list_fm] :
      ( ( set_fm2 @ ( concat_fm @ Xs ) )
      = ( comple7677669044296858969set_fm @ ( image_list_fm_set_fm @ set_fm2 @ ( set_list_fm2 @ Xs ) ) ) ) ).

% set_concat
thf(fact_1164_equiv__proj,axiom,
    ! [A4: set_list_fm,R: set_Pr7058068377845519745ist_fm,Z2: produc1996495991257130529ist_fm] :
      ( ( equiv_equiv_list_fm @ A4 @ R )
     => ( ( member8102475879199740618ist_fm @ Z2 @ R )
       => ( ( comp_l987677038997058310ist_fm @ ( equiv_506656931107403212ist_fm @ R ) @ produc1501393135466168645ist_fm @ Z2 )
          = ( comp_l987677038997058310ist_fm @ ( equiv_506656931107403212ist_fm @ R ) @ produc4588648349897876871ist_fm @ Z2 ) ) ) ) ).

% equiv_proj
thf(fact_1165_list_Oin__rel,axiom,
    ( list_a5708887152077601097ist_fm
    = ( ^ [R3: list_fm > list_fm > $o,A5: list_list_fm,B4: list_list_fm] :
        ? [Z5: list_P5616295576739893671ist_fm] :
          ( ( member8448403643636553168ist_fm @ Z5
            @ ( collec2128064864322541586ist_fm
              @ ^ [X3: list_P5616295576739893671ist_fm] : ( ord_le2055072402069232929ist_fm @ ( set_Pr8767716839810916150ist_fm @ X3 ) @ ( collec7231575776666803724ist_fm @ ( produc2011568048573023402t_fm_o @ R3 ) ) ) ) )
          & ( ( map_Pr3743373011763743805ist_fm @ produc1501393135466168645ist_fm @ Z5 )
            = A5 )
          & ( ( map_Pr3743373011763743805ist_fm @ produc4588648349897876871ist_fm @ Z5 )
            = B4 ) ) ) ) ).

% list.in_rel
thf(fact_1166_list_Oin__rel,axiom,
    ( list_a4958931515539933215m_rule
    = ( ^ [R3: produc1996495991257130529ist_fm > rule > $o,A5: list_P5616295576739893671ist_fm,B4: list_rule] :
        ? [Z5: list_P8332459875677254181m_rule] :
          ( ( member1784032718715961276m_rule @ Z5
            @ ( collec5373559803083089146m_rule
              @ ^ [X3: list_P8332459875677254181m_rule] : ( ord_le5052251716133888043m_rule @ ( set_Pr8505323785428441536m_rule @ X3 ) @ ( collec2069506698588392810m_rule @ ( produc7369492558480804444rule_o @ R3 ) ) ) ) )
          & ( ( map_Pr1951777954582829830ist_fm @ produc6879501374131015971m_rule @ Z5 )
            = A5 )
          & ( ( map_Pr7678108474426652576e_rule @ produc7165828336582415457m_rule @ Z5 )
            = B4 ) ) ) ) ).

% list.in_rel
thf(fact_1167_list_Oin__rel,axiom,
    ( list_a2307847286019514067ist_fm
    = ( ^ [R3: rule > produc1996495991257130529ist_fm > $o,A5: list_rule,B4: list_P5616295576739893671ist_fm] :
        ? [Z5: list_P342308711384885665ist_fm] :
          ( ( member3017253591278368568ist_fm @ Z5
            @ ( collec6606780675645496438ist_fm
              @ ^ [X3: list_P342308711384885665ist_fm] : ( ord_le6285472588696295335ist_fm @ ( set_Pr3748622270175481660ist_fm @ X3 ) @ ( collec6536177220190208742ist_fm @ ( produc1628407760719572648t_fm_o @ R3 ) ) ) ) )
          & ( ( map_Pr452693852626354460m_rule @ produc4228417144610596823ist_fm @ Z5 )
            = A5 )
          & ( ( map_Pr2410134268255969930ist_fm @ produc4514744107061996309ist_fm @ Z5 )
            = B4 ) ) ) ) ).

% list.in_rel
thf(fact_1168_list_Orel__compp__Grp,axiom,
    ( list_a5708887152077601097ist_fm
    = ( ^ [R3: list_fm > list_fm > $o] :
          ( relcom3701813194287065782ist_fm
          @ ( conver7199315329239816034ist_fm
            @ ( bNF_Gr5899615463480776905ist_fm
              @ ( collec2128064864322541586ist_fm
                @ ^ [X3: list_P5616295576739893671ist_fm] : ( ord_le2055072402069232929ist_fm @ ( set_Pr8767716839810916150ist_fm @ X3 ) @ ( collec7231575776666803724ist_fm @ ( produc2011568048573023402t_fm_o @ R3 ) ) ) )
              @ ( map_Pr3743373011763743805ist_fm @ produc1501393135466168645ist_fm ) ) )
          @ ( bNF_Gr5899615463480776905ist_fm
            @ ( collec2128064864322541586ist_fm
              @ ^ [X3: list_P5616295576739893671ist_fm] : ( ord_le2055072402069232929ist_fm @ ( set_Pr8767716839810916150ist_fm @ X3 ) @ ( collec7231575776666803724ist_fm @ ( produc2011568048573023402t_fm_o @ R3 ) ) ) )
            @ ( map_Pr3743373011763743805ist_fm @ produc4588648349897876871ist_fm ) ) ) ) ) ).

% list.rel_compp_Grp
thf(fact_1169_list_Orel__compp__Grp,axiom,
    ( list_a4958931515539933215m_rule
    = ( ^ [R3: produc1996495991257130529ist_fm > rule > $o] :
          ( relcom6924352960690907314t_rule
          @ ( conver6851233329379293835ist_fm
            @ ( bNF_Gr4521191952927931250ist_fm
              @ ( collec5373559803083089146m_rule
                @ ^ [X3: list_P8332459875677254181m_rule] : ( ord_le5052251716133888043m_rule @ ( set_Pr8505323785428441536m_rule @ X3 ) @ ( collec2069506698588392810m_rule @ ( produc7369492558480804444rule_o @ R3 ) ) ) )
              @ ( map_Pr1951777954582829830ist_fm @ produc6879501374131015971m_rule ) ) )
          @ ( bNF_Gr7726341041440550858t_rule
            @ ( collec5373559803083089146m_rule
              @ ^ [X3: list_P8332459875677254181m_rule] : ( ord_le5052251716133888043m_rule @ ( set_Pr8505323785428441536m_rule @ X3 ) @ ( collec2069506698588392810m_rule @ ( produc7369492558480804444rule_o @ R3 ) ) ) )
            @ ( map_Pr7678108474426652576e_rule @ produc7165828336582415457m_rule ) ) ) ) ) ).

% list.rel_compp_Grp
thf(fact_1170_list_Orel__compp__Grp,axiom,
    ( list_a2307847286019514067ist_fm
    = ( ^ [R3: rule > produc1996495991257130529ist_fm > $o] :
          ( relcom2364676781511777726ist_fm
          @ ( conver1203800790389350509t_rule
            @ ( bNF_Gr8069780646756945222t_rule
              @ ( collec6606780675645496438ist_fm
                @ ^ [X3: list_P342308711384885665ist_fm] : ( ord_le6285472588696295335ist_fm @ ( set_Pr3748622270175481660ist_fm @ X3 ) @ ( collec6536177220190208742ist_fm @ ( produc1628407760719572648t_fm_o @ R3 ) ) ) )
              @ ( map_Pr452693852626354460m_rule @ produc4228417144610596823ist_fm ) ) )
          @ ( bNF_Gr7947322218390040310ist_fm
            @ ( collec6606780675645496438ist_fm
              @ ^ [X3: list_P342308711384885665ist_fm] : ( ord_le6285472588696295335ist_fm @ ( set_Pr3748622270175481660ist_fm @ X3 ) @ ( collec6536177220190208742ist_fm @ ( produc1628407760719572648t_fm_o @ R3 ) ) ) )
            @ ( map_Pr2410134268255969930ist_fm @ produc4514744107061996309ist_fm ) ) ) ) ) ).

% list.rel_compp_Grp
thf(fact_1171_list__all2__same,axiom,
    ! [P2: fm > fm > $o,Xs: list_fm] :
      ( ( list_all2_fm_fm @ P2 @ Xs @ Xs )
      = ( ! [X3: fm] :
            ( ( member_fm @ X3 @ ( set_fm2 @ Xs ) )
           => ( P2 @ X3 @ X3 ) ) ) ) ).

% list_all2_same
thf(fact_1172_list_Orel__refl__strong,axiom,
    ! [X2: list_fm,Ra: fm > fm > $o] :
      ( ! [Z3: fm] :
          ( ( member_fm @ Z3 @ ( set_fm2 @ X2 ) )
         => ( Ra @ Z3 @ Z3 ) )
     => ( list_all2_fm_fm @ Ra @ X2 @ X2 ) ) ).

% list.rel_refl_strong
thf(fact_1173_list_Orel__mono__strong,axiom,
    ! [R: fm > fm > $o,X2: list_fm,Y3: list_fm,Ra: fm > fm > $o] :
      ( ( list_all2_fm_fm @ R @ X2 @ Y3 )
     => ( ! [Z3: fm,Yb: fm] :
            ( ( member_fm @ Z3 @ ( set_fm2 @ X2 ) )
           => ( ( member_fm @ Yb @ ( set_fm2 @ Y3 ) )
             => ( ( R @ Z3 @ Yb )
               => ( Ra @ Z3 @ Yb ) ) ) )
       => ( list_all2_fm_fm @ Ra @ X2 @ Y3 ) ) ) ).

% list.rel_mono_strong
thf(fact_1174_list_Orel__cong,axiom,
    ! [X2: list_fm,Ya: list_fm,Y3: list_fm,Xa2: list_fm,R: fm > fm > $o,Ra: fm > fm > $o] :
      ( ( X2 = Ya )
     => ( ( Y3 = Xa2 )
       => ( ! [Z3: fm,Yb: fm] :
              ( ( member_fm @ Z3 @ ( set_fm2 @ Ya ) )
             => ( ( member_fm @ Yb @ ( set_fm2 @ Xa2 ) )
               => ( ( R @ Z3 @ Yb )
                  = ( Ra @ Z3 @ Yb ) ) ) )
         => ( ( list_all2_fm_fm @ R @ X2 @ Y3 )
            = ( list_all2_fm_fm @ Ra @ Ya @ Xa2 ) ) ) ) ) ).

% list.rel_cong
thf(fact_1175_subset__subseqs,axiom,
    ! [X5: set_fm,Xs: list_fm] :
      ( ( ord_less_eq_set_fm @ X5 @ ( set_fm2 @ Xs ) )
     => ( member_set_fm @ X5 @ ( image_list_fm_set_fm @ set_fm2 @ ( set_list_fm2 @ ( subseqs_fm @ Xs ) ) ) ) ) ).

% subset_subseqs
thf(fact_1176_smerge__def,axiom,
    ( smerge3869575255966183559m_rule
    = ( ^ [Ss2: stream3016090547575141045m_rule] :
          ( flat_P3556391191990922267m_rule
          @ ( smap_n5591844505413028476m_rule
            @ ^ [N3: nat] :
                ( append1085632490384598586m_rule
                @ ( map_st2996440461923888000m_rule
                  @ ^ [S: stream727092118206550309m_rule] : ( snth_P1712579846118164081m_rule @ S @ N3 )
                  @ ( stake_5102249147582673364m_rule @ ( suc @ N3 ) @ Ss2 ) )
                @ ( stake_1447931197033250628m_rule @ N3 @ ( snth_s8878883179916095873m_rule @ Ss2 @ N3 ) ) )
            @ ( siterate_nat @ suc @ zero_zero_nat ) ) ) ) ) ).

% smerge_def
thf(fact_1177_set__append,axiom,
    ! [Xs: list_fm,Ys: list_fm] :
      ( ( set_fm2 @ ( append_fm @ Xs @ Ys ) )
      = ( sup_sup_set_fm @ ( set_fm2 @ Xs ) @ ( set_fm2 @ Ys ) ) ) ).

% set_append
thf(fact_1178_stake__add,axiom,
    ! [M: nat,S3: stream727092118206550309m_rule,N2: nat] :
      ( ( append1085632490384598586m_rule @ ( stake_1447931197033250628m_rule @ M @ S3 ) @ ( stake_1447931197033250628m_rule @ N2 @ ( sdrop_7224736112439592940m_rule @ M @ S3 ) ) )
      = ( stake_1447931197033250628m_rule @ ( plus_plus_nat @ M @ N2 ) @ S3 ) ) ).

% stake_add
thf(fact_1179_stake__add,axiom,
    ! [M: nat,S3: stream8299795917829157543ist_fm,N2: nat] :
      ( ( append2226227739694184764ist_fm @ ( stake_2930890243849202354ist_fm @ M @ S3 ) @ ( stake_2930890243849202354ist_fm @ N2 @ ( sdrop_4442373711808556042ist_fm @ M @ S3 ) ) )
      = ( stake_2930890243849202354ist_fm @ ( plus_plus_nat @ M @ N2 ) @ S3 ) ) ).

% stake_add
thf(fact_1180_stake__add,axiom,
    ! [M: nat,S3: stream_rule,N2: nat] :
      ( ( append_rule @ ( stake_rule @ M @ S3 ) @ ( stake_rule @ N2 @ ( sdrop_rule @ M @ S3 ) ) )
      = ( stake_rule @ ( plus_plus_nat @ M @ N2 ) @ S3 ) ) ).

% stake_add
thf(fact_1181_stake__add,axiom,
    ! [M: nat,S3: stream6494289010434245521m_rule,N2: nat] :
      ( ( append6999072698098818854m_rule @ ( stake_8908532221384226460m_rule @ M @ S3 ) @ ( stake_8908532221384226460m_rule @ N2 @ ( sdrop_7373388980841566196m_rule @ M @ S3 ) ) )
      = ( stake_8908532221384226460m_rule @ ( plus_plus_nat @ M @ N2 ) @ S3 ) ) ).

% stake_add
thf(fact_1182_quotient__eqI,axiom,
    ! [A4: set_fm,R2: set_Pr4463079037648049377_fm_fm,X5: set_fm,Y5: set_fm,X2: fm,Y3: fm] :
      ( ( equiv_equiv_fm @ A4 @ R2 )
     => ( ( member_set_fm @ X5 @ ( equiv_quotient_fm @ A4 @ R2 ) )
       => ( ( member_set_fm @ Y5 @ ( equiv_quotient_fm @ A4 @ R2 ) )
         => ( ( member_fm @ X2 @ X5 )
           => ( ( member_fm @ Y3 @ Y5 )
             => ( ( member8474499337054950954_fm_fm @ ( product_Pair_fm_fm @ X2 @ Y3 ) @ R2 )
               => ( X5 = Y5 ) ) ) ) ) ) ) ).

% quotient_eqI
thf(fact_1183_quotient__eq__iff,axiom,
    ! [A4: set_fm,R2: set_Pr4463079037648049377_fm_fm,X5: set_fm,Y5: set_fm,X2: fm,Y3: fm] :
      ( ( equiv_equiv_fm @ A4 @ R2 )
     => ( ( member_set_fm @ X5 @ ( equiv_quotient_fm @ A4 @ R2 ) )
       => ( ( member_set_fm @ Y5 @ ( equiv_quotient_fm @ A4 @ R2 ) )
         => ( ( member_fm @ X2 @ X5 )
           => ( ( member_fm @ Y3 @ Y5 )
             => ( ( X5 = Y5 )
                = ( member8474499337054950954_fm_fm @ ( product_Pair_fm_fm @ X2 @ Y3 ) @ R2 ) ) ) ) ) ) ) ).

% quotient_eq_iff
thf(fact_1184_in__quotient__imp__closed,axiom,
    ! [A4: set_fm,R2: set_Pr4463079037648049377_fm_fm,X5: set_fm,X2: fm,Y3: fm] :
      ( ( equiv_equiv_fm @ A4 @ R2 )
     => ( ( member_set_fm @ X5 @ ( equiv_quotient_fm @ A4 @ R2 ) )
       => ( ( member_fm @ X2 @ X5 )
         => ( ( member8474499337054950954_fm_fm @ ( product_Pair_fm_fm @ X2 @ Y3 ) @ R2 )
           => ( member_fm @ Y3 @ X5 ) ) ) ) ) ).

% in_quotient_imp_closed
thf(fact_1185_eq__equiv__class__iff2,axiom,
    ! [A4: set_fm,R2: set_Pr4463079037648049377_fm_fm,X2: fm,Y3: fm] :
      ( ( equiv_equiv_fm @ A4 @ R2 )
     => ( ( member_fm @ X2 @ A4 )
       => ( ( member_fm @ Y3 @ A4 )
         => ( ( ( equiv_quotient_fm @ ( insert_fm2 @ X2 @ bot_bot_set_fm ) @ R2 )
              = ( equiv_quotient_fm @ ( insert_fm2 @ Y3 @ bot_bot_set_fm ) @ R2 ) )
            = ( member8474499337054950954_fm_fm @ ( product_Pair_fm_fm @ X2 @ Y3 ) @ R2 ) ) ) ) ) ).

% eq_equiv_class_iff2
thf(fact_1186_UN__equiv__class__inject,axiom,
    ! [A4: set_Pr1008144964186165195m_rule,R2: set_Pr5090152306159784043m_rule,F2: produc164195504107695125m_rule > set_fm,X5: set_Pr1008144964186165195m_rule,Y5: set_Pr1008144964186165195m_rule] :
      ( ( equiv_5668253230753244639m_rule @ A4 @ R2 )
     => ( ( equiv_1588529185009735475set_fm @ R2 @ F2 )
       => ( ( ( comple7677669044296858969set_fm @ ( image_5698427363735666338set_fm @ F2 @ X5 ) )
            = ( comple7677669044296858969set_fm @ ( image_5698427363735666338set_fm @ F2 @ Y5 ) ) )
         => ( ( member7979558945778333922m_rule @ X5 @ ( equiv_1921610036520034574m_rule @ A4 @ R2 ) )
           => ( ( member7979558945778333922m_rule @ Y5 @ ( equiv_1921610036520034574m_rule @ A4 @ R2 ) )
             => ( ! [X: produc164195504107695125m_rule,Y: produc164195504107695125m_rule] :
                    ( ( member4220325220686508332m_rule @ X @ A4 )
                   => ( ( member4220325220686508332m_rule @ Y @ A4 )
                     => ( ( ( F2 @ X )
                          = ( F2 @ Y ) )
                       => ( member2434505771962676916m_rule @ ( produc5927390650430071747m_rule @ X @ Y ) @ R2 ) ) ) )
               => ( X5 = Y5 ) ) ) ) ) ) ) ).

% UN_equiv_class_inject
thf(fact_1187_UN__equiv__class__inject,axiom,
    ! [A4: set_fm,R2: set_Pr4463079037648049377_fm_fm,F2: fm > set_fm,X5: set_fm,Y5: set_fm] :
      ( ( equiv_equiv_fm @ A4 @ R2 )
     => ( ( equiv_5061330122100942062set_fm @ R2 @ F2 )
       => ( ( ( comple7677669044296858969set_fm @ ( image_fm_set_fm @ F2 @ X5 ) )
            = ( comple7677669044296858969set_fm @ ( image_fm_set_fm @ F2 @ Y5 ) ) )
         => ( ( member_set_fm @ X5 @ ( equiv_quotient_fm @ A4 @ R2 ) )
           => ( ( member_set_fm @ Y5 @ ( equiv_quotient_fm @ A4 @ R2 ) )
             => ( ! [X: fm,Y: fm] :
                    ( ( member_fm @ X @ A4 )
                   => ( ( member_fm @ Y @ A4 )
                     => ( ( ( F2 @ X )
                          = ( F2 @ Y ) )
                       => ( member8474499337054950954_fm_fm @ ( product_Pair_fm_fm @ X @ Y ) @ R2 ) ) ) )
               => ( X5 = Y5 ) ) ) ) ) ) ) ).

% UN_equiv_class_inject
thf(fact_1188_sset__flat,axiom,
    ! [S3: stream5223287678556985525m_rule] :
      ( ! [X: list_P8332459875677254181m_rule] :
          ( ( member1784032718715961276m_rule @ X @ ( sset_l1470757290599833361m_rule @ S3 ) )
         => ( X != nil_Pr474642538098133391m_rule ) )
     => ( ( sset_P4979403222752902913m_rule @ ( flat_P3556391191990922267m_rule @ S3 ) )
        = ( comple2712794622220498846m_rule @ ( image_1027408785512475639m_rule @ set_Pr8505323785428441536m_rule @ ( sset_l1470757290599833361m_rule @ S3 ) ) ) ) ) ).

% sset_flat
thf(fact_1189_sset__flat,axiom,
    ! [S3: stream_list_fm] :
      ( ! [X: list_fm] :
          ( ( member_list_fm @ X @ ( sset_list_fm @ S3 ) )
         => ( X != nil_fm ) )
     => ( ( sset_fm @ ( flat_fm @ S3 ) )
        = ( comple7677669044296858969set_fm @ ( image_list_fm_set_fm @ set_fm2 @ ( sset_list_fm @ S3 ) ) ) ) ) ).

% sset_flat
thf(fact_1190_set__empty,axiom,
    ! [Xs: list_fm] :
      ( ( ( set_fm2 @ Xs )
        = bot_bot_set_fm )
      = ( Xs = nil_fm ) ) ).

% set_empty
thf(fact_1191_set__empty2,axiom,
    ! [Xs: list_fm] :
      ( ( bot_bot_set_fm
        = ( set_fm2 @ Xs ) )
      = ( Xs = nil_fm ) ) ).

% set_empty2
thf(fact_1192_empty__set,axiom,
    ( bot_bot_set_fm
    = ( set_fm2 @ nil_fm ) ) ).

% empty_set
thf(fact_1193_list_Oset__sel_I2_J,axiom,
    ! [A: list_fm,X2: fm] :
      ( ( A != nil_fm )
     => ( ( member_fm @ X2 @ ( set_fm2 @ ( tl_fm @ A ) ) )
       => ( member_fm @ X2 @ ( set_fm2 @ A ) ) ) ) ).

% list.set_sel(2)
thf(fact_1194_relImage__relInvImage,axiom,
    ! [R: set_Pr8693920393607151949set_fm,F2: produc164195504107695125m_rule > set_fm,A4: set_Pr1008144964186165195m_rule] :
      ( ( ord_le4425813521181927661set_fm @ R
        @ ( produc7186806611370130262set_fm @ ( image_5698427363735666338set_fm @ F2 @ A4 )
          @ ^ [Uu: set_fm] : ( image_5698427363735666338set_fm @ F2 @ A4 ) ) )
     => ( ( bNF_Gr5805612712483453313set_fm @ ( bNF_Gr1639403061603056734set_fm @ A4 @ R @ F2 ) @ F2 )
        = R ) ) ).

% relImage_relInvImage
thf(fact_1195_shift__simps_I1_J,axiom,
    ! [Xs: list_P8332459875677254181m_rule,S3: stream727092118206550309m_rule] :
      ( ( ( Xs = nil_Pr474642538098133391m_rule )
       => ( ( shd_Pr7235097944458474089m_rule @ ( shift_5008521991580687660m_rule @ Xs @ S3 ) )
          = ( shd_Pr7235097944458474089m_rule @ S3 ) ) )
      & ( ( Xs != nil_Pr474642538098133391m_rule )
       => ( ( shd_Pr7235097944458474089m_rule @ ( shift_5008521991580687660m_rule @ Xs @ S3 ) )
          = ( hd_Pro3588770950400194538m_rule @ Xs ) ) ) ) ).

% shift_simps(1)
thf(fact_1196_hd__in__set,axiom,
    ! [Xs: list_fm] :
      ( ( Xs != nil_fm )
     => ( member_fm @ ( hd_fm @ Xs ) @ ( set_fm2 @ Xs ) ) ) ).

% hd_in_set
thf(fact_1197_list_Oset__sel_I1_J,axiom,
    ! [A: list_fm] :
      ( ( A != nil_fm )
     => ( member_fm @ ( hd_fm @ A ) @ ( set_fm2 @ A ) ) ) ).

% list.set_sel(1)
thf(fact_1198_flat_Osimps_I1_J,axiom,
    ! [Ws: stream5223287678556985525m_rule] :
      ( ( shd_Pr7235097944458474089m_rule @ ( flat_P3556391191990922267m_rule @ Ws ) )
      = ( hd_Pro3588770950400194538m_rule @ ( shd_li6864325268781923961m_rule @ Ws ) ) ) ).

% flat.simps(1)
thf(fact_1199_sset__cycle,axiom,
    ! [Xs: list_fm] :
      ( ( Xs != nil_fm )
     => ( ( sset_fm @ ( cycle_fm @ Xs ) )
        = ( set_fm2 @ Xs ) ) ) ).

% sset_cycle
thf(fact_1200_sset__cycle,axiom,
    ! [Xs: list_P8332459875677254181m_rule] :
      ( ( Xs != nil_Pr474642538098133391m_rule )
     => ( ( sset_P4979403222752902913m_rule @ ( cycle_1017677951696034864m_rule @ Xs ) )
        = ( set_Pr8505323785428441536m_rule @ Xs ) ) ) ).

% sset_cycle
thf(fact_1201_list_Osimps_I15_J,axiom,
    ! [X21: fm,X222: list_fm] :
      ( ( set_fm2 @ ( cons_fm @ X21 @ X222 ) )
      = ( insert_fm2 @ X21 @ ( set_fm2 @ X222 ) ) ) ).

% list.simps(15)
thf(fact_1202_not__in__set__insert,axiom,
    ! [X2: fm,Xs: list_fm] :
      ( ~ ( member_fm @ X2 @ ( set_fm2 @ Xs ) )
     => ( ( insert_fm @ X2 @ Xs )
        = ( cons_fm @ X2 @ Xs ) ) ) ).

% not_in_set_insert
thf(fact_1203_split__list,axiom,
    ! [X2: fm,Xs: list_fm] :
      ( ( member_fm @ X2 @ ( set_fm2 @ Xs ) )
     => ? [Ys2: list_fm,Zs: list_fm] :
          ( Xs
          = ( append_fm @ Ys2 @ ( cons_fm @ X2 @ Zs ) ) ) ) ).

% split_list
thf(fact_1204_split__list__last,axiom,
    ! [X2: fm,Xs: list_fm] :
      ( ( member_fm @ X2 @ ( set_fm2 @ Xs ) )
     => ? [Ys2: list_fm,Zs: list_fm] :
          ( ( Xs
            = ( append_fm @ Ys2 @ ( cons_fm @ X2 @ Zs ) ) )
          & ~ ( member_fm @ X2 @ ( set_fm2 @ Zs ) ) ) ) ).

% split_list_last
thf(fact_1205_split__list__prop,axiom,
    ! [Xs: list_fm,P2: fm > $o] :
      ( ? [X6: fm] :
          ( ( member_fm @ X6 @ ( set_fm2 @ Xs ) )
          & ( P2 @ X6 ) )
     => ? [Ys2: list_fm,X: fm] :
          ( ? [Zs: list_fm] :
              ( Xs
              = ( append_fm @ Ys2 @ ( cons_fm @ X @ Zs ) ) )
          & ( P2 @ X ) ) ) ).

% split_list_prop
thf(fact_1206_split__list__first,axiom,
    ! [X2: fm,Xs: list_fm] :
      ( ( member_fm @ X2 @ ( set_fm2 @ Xs ) )
     => ? [Ys2: list_fm,Zs: list_fm] :
          ( ( Xs
            = ( append_fm @ Ys2 @ ( cons_fm @ X2 @ Zs ) ) )
          & ~ ( member_fm @ X2 @ ( set_fm2 @ Ys2 ) ) ) ) ).

% split_list_first
thf(fact_1207_split__list__propE,axiom,
    ! [Xs: list_fm,P2: fm > $o] :
      ( ? [X6: fm] :
          ( ( member_fm @ X6 @ ( set_fm2 @ Xs ) )
          & ( P2 @ X6 ) )
     => ~ ! [Ys2: list_fm,X: fm] :
            ( ? [Zs: list_fm] :
                ( Xs
                = ( append_fm @ Ys2 @ ( cons_fm @ X @ Zs ) ) )
           => ~ ( P2 @ X ) ) ) ).

% split_list_propE
thf(fact_1208_append__Cons__eq__iff,axiom,
    ! [X2: fm,Xs: list_fm,Ys: list_fm,Xs2: list_fm,Ys3: list_fm] :
      ( ~ ( member_fm @ X2 @ ( set_fm2 @ Xs ) )
     => ( ~ ( member_fm @ X2 @ ( set_fm2 @ Ys ) )
       => ( ( ( append_fm @ Xs @ ( cons_fm @ X2 @ Ys ) )
            = ( append_fm @ Xs2 @ ( cons_fm @ X2 @ Ys3 ) ) )
          = ( ( Xs = Xs2 )
            & ( Ys = Ys3 ) ) ) ) ) ).

% append_Cons_eq_iff
thf(fact_1209_in__set__conv__decomp,axiom,
    ! [X2: fm,Xs: list_fm] :
      ( ( member_fm @ X2 @ ( set_fm2 @ Xs ) )
      = ( ? [Ys4: list_fm,Zs2: list_fm] :
            ( Xs
            = ( append_fm @ Ys4 @ ( cons_fm @ X2 @ Zs2 ) ) ) ) ) ).

% in_set_conv_decomp
thf(fact_1210_split__list__last__prop,axiom,
    ! [Xs: list_fm,P2: fm > $o] :
      ( ? [X6: fm] :
          ( ( member_fm @ X6 @ ( set_fm2 @ Xs ) )
          & ( P2 @ X6 ) )
     => ? [Ys2: list_fm,X: fm,Zs: list_fm] :
          ( ( Xs
            = ( append_fm @ Ys2 @ ( cons_fm @ X @ Zs ) ) )
          & ( P2 @ X )
          & ! [Xa3: fm] :
              ( ( member_fm @ Xa3 @ ( set_fm2 @ Zs ) )
             => ~ ( P2 @ Xa3 ) ) ) ) ).

% split_list_last_prop
thf(fact_1211_split__list__first__prop,axiom,
    ! [Xs: list_fm,P2: fm > $o] :
      ( ? [X6: fm] :
          ( ( member_fm @ X6 @ ( set_fm2 @ Xs ) )
          & ( P2 @ X6 ) )
     => ? [Ys2: list_fm,X: fm] :
          ( ? [Zs: list_fm] :
              ( Xs
              = ( append_fm @ Ys2 @ ( cons_fm @ X @ Zs ) ) )
          & ( P2 @ X )
          & ! [Xa3: fm] :
              ( ( member_fm @ Xa3 @ ( set_fm2 @ Ys2 ) )
             => ~ ( P2 @ Xa3 ) ) ) ) ).

% split_list_first_prop
thf(fact_1212_split__list__last__propE,axiom,
    ! [Xs: list_fm,P2: fm > $o] :
      ( ? [X6: fm] :
          ( ( member_fm @ X6 @ ( set_fm2 @ Xs ) )
          & ( P2 @ X6 ) )
     => ~ ! [Ys2: list_fm,X: fm,Zs: list_fm] :
            ( ( Xs
              = ( append_fm @ Ys2 @ ( cons_fm @ X @ Zs ) ) )
           => ( ( P2 @ X )
             => ~ ! [Xa3: fm] :
                    ( ( member_fm @ Xa3 @ ( set_fm2 @ Zs ) )
                   => ~ ( P2 @ Xa3 ) ) ) ) ) ).

% split_list_last_propE
thf(fact_1213_split__list__first__propE,axiom,
    ! [Xs: list_fm,P2: fm > $o] :
      ( ? [X6: fm] :
          ( ( member_fm @ X6 @ ( set_fm2 @ Xs ) )
          & ( P2 @ X6 ) )
     => ~ ! [Ys2: list_fm,X: fm] :
            ( ? [Zs: list_fm] :
                ( Xs
                = ( append_fm @ Ys2 @ ( cons_fm @ X @ Zs ) ) )
           => ( ( P2 @ X )
             => ~ ! [Xa3: fm] :
                    ( ( member_fm @ Xa3 @ ( set_fm2 @ Ys2 ) )
                   => ~ ( P2 @ Xa3 ) ) ) ) ) ).

% split_list_first_propE
thf(fact_1214_in__set__conv__decomp__last,axiom,
    ! [X2: fm,Xs: list_fm] :
      ( ( member_fm @ X2 @ ( set_fm2 @ Xs ) )
      = ( ? [Ys4: list_fm,Zs2: list_fm] :
            ( ( Xs
              = ( append_fm @ Ys4 @ ( cons_fm @ X2 @ Zs2 ) ) )
            & ~ ( member_fm @ X2 @ ( set_fm2 @ Zs2 ) ) ) ) ) ).

% in_set_conv_decomp_last
thf(fact_1215_in__set__conv__decomp__first,axiom,
    ! [X2: fm,Xs: list_fm] :
      ( ( member_fm @ X2 @ ( set_fm2 @ Xs ) )
      = ( ? [Ys4: list_fm,Zs2: list_fm] :
            ( ( Xs
              = ( append_fm @ Ys4 @ ( cons_fm @ X2 @ Zs2 ) ) )
            & ~ ( member_fm @ X2 @ ( set_fm2 @ Ys4 ) ) ) ) ) ).

% in_set_conv_decomp_first
thf(fact_1216_split__list__last__prop__iff,axiom,
    ! [Xs: list_fm,P2: fm > $o] :
      ( ( ? [X3: fm] :
            ( ( member_fm @ X3 @ ( set_fm2 @ Xs ) )
            & ( P2 @ X3 ) ) )
      = ( ? [Ys4: list_fm,X3: fm,Zs2: list_fm] :
            ( ( Xs
              = ( append_fm @ Ys4 @ ( cons_fm @ X3 @ Zs2 ) ) )
            & ( P2 @ X3 )
            & ! [Y4: fm] :
                ( ( member_fm @ Y4 @ ( set_fm2 @ Zs2 ) )
               => ~ ( P2 @ Y4 ) ) ) ) ) ).

% split_list_last_prop_iff
thf(fact_1217_split__list__first__prop__iff,axiom,
    ! [Xs: list_fm,P2: fm > $o] :
      ( ( ? [X3: fm] :
            ( ( member_fm @ X3 @ ( set_fm2 @ Xs ) )
            & ( P2 @ X3 ) ) )
      = ( ? [Ys4: list_fm,X3: fm] :
            ( ? [Zs2: list_fm] :
                ( Xs
                = ( append_fm @ Ys4 @ ( cons_fm @ X3 @ Zs2 ) ) )
            & ( P2 @ X3 )
            & ! [Y4: fm] :
                ( ( member_fm @ Y4 @ ( set_fm2 @ Ys4 ) )
               => ~ ( P2 @ Y4 ) ) ) ) ) ).

% split_list_first_prop_iff
thf(fact_1218_list_Oset__intros_I2_J,axiom,
    ! [Y3: fm,X222: list_fm,X21: fm] :
      ( ( member_fm @ Y3 @ ( set_fm2 @ X222 ) )
     => ( member_fm @ Y3 @ ( set_fm2 @ ( cons_fm @ X21 @ X222 ) ) ) ) ).

% list.set_intros(2)
thf(fact_1219_list_Oset__intros_I1_J,axiom,
    ! [X21: fm,X222: list_fm] : ( member_fm @ X21 @ ( set_fm2 @ ( cons_fm @ X21 @ X222 ) ) ) ).

% list.set_intros(1)
thf(fact_1220_list_Oset__cases,axiom,
    ! [E2: fm,A: list_fm] :
      ( ( member_fm @ E2 @ ( set_fm2 @ A ) )
     => ( ! [Z22: list_fm] :
            ( A
           != ( cons_fm @ E2 @ Z22 ) )
       => ~ ! [Z1: fm,Z22: list_fm] :
              ( ( A
                = ( cons_fm @ Z1 @ Z22 ) )
             => ~ ( member_fm @ E2 @ ( set_fm2 @ Z22 ) ) ) ) ) ).

% list.set_cases
thf(fact_1221_set__ConsD,axiom,
    ! [Y3: fm,X2: fm,Xs: list_fm] :
      ( ( member_fm @ Y3 @ ( set_fm2 @ ( cons_fm @ X2 @ Xs ) ) )
     => ( ( Y3 = X2 )
        | ( member_fm @ Y3 @ ( set_fm2 @ Xs ) ) ) ) ).

% set_ConsD
thf(fact_1222_List_Oinsert__def,axiom,
    ( insert_fm
    = ( ^ [X3: fm,Xs3: list_fm] : ( if_list_fm @ ( member_fm @ X3 @ ( set_fm2 @ Xs3 ) ) @ Xs3 @ ( cons_fm @ X3 @ Xs3 ) ) ) ) ).

% List.insert_def
thf(fact_1223_set__subset__Cons,axiom,
    ! [Xs: list_fm,X2: fm] : ( ord_less_eq_set_fm @ ( set_fm2 @ Xs ) @ ( set_fm2 @ ( cons_fm @ X2 @ Xs ) ) ) ).

% set_subset_Cons
thf(fact_1224_cycle_Osimps_I1_J,axiom,
    ! [Xs: list_P8332459875677254181m_rule] :
      ( ( shd_Pr7235097944458474089m_rule @ ( cycle_1017677951696034864m_rule @ Xs ) )
      = ( hd_Pro3588770950400194538m_rule @ Xs ) ) ).

% cycle.simps(1)
thf(fact_1225_stake_Osimps_I2_J,axiom,
    ! [N2: nat,S3: stream727092118206550309m_rule] :
      ( ( stake_1447931197033250628m_rule @ ( suc @ N2 ) @ S3 )
      = ( cons_P768676612401224159m_rule @ ( shd_Pr7235097944458474089m_rule @ S3 ) @ ( stake_1447931197033250628m_rule @ N2 @ ( stl_Pr950425576149878629m_rule @ S3 ) ) ) ) ).

% stake.simps(2)
thf(fact_1226_stake__Suc,axiom,
    ! [N2: nat,S3: stream727092118206550309m_rule] :
      ( ( stake_1447931197033250628m_rule @ ( suc @ N2 ) @ S3 )
      = ( append1085632490384598586m_rule @ ( stake_1447931197033250628m_rule @ N2 @ S3 ) @ ( cons_P768676612401224159m_rule @ ( snth_P1712579846118164081m_rule @ S3 @ N2 ) @ nil_Pr474642538098133391m_rule ) ) ) ).

% stake_Suc
thf(fact_1227_SuccD,axiom,
    ! [K: fm,Kl: set_list_fm,Kl2: list_fm] :
      ( ( member_fm @ K @ ( bNF_Greatest_Succ_fm @ Kl @ Kl2 ) )
     => ( member_list_fm @ ( append_fm @ Kl2 @ ( cons_fm @ K @ nil_fm ) ) @ Kl ) ) ).

% SuccD
thf(fact_1228_SuccI,axiom,
    ! [Kl2: list_fm,K: fm,Kl: set_list_fm] :
      ( ( member_list_fm @ ( append_fm @ Kl2 @ ( cons_fm @ K @ nil_fm ) ) @ Kl )
     => ( member_fm @ K @ ( bNF_Greatest_Succ_fm @ Kl @ Kl2 ) ) ) ).

% SuccI
thf(fact_1229_empty__Shift,axiom,
    ! [Kl: set_list_fm,K: fm] :
      ( ( member_list_fm @ nil_fm @ Kl )
     => ( ( member_fm @ K @ ( bNF_Greatest_Succ_fm @ Kl @ nil_fm ) )
       => ( member_list_fm @ nil_fm @ ( bNF_Gr3717417623296551025ift_fm @ Kl @ K ) ) ) ) ).

% empty_Shift
thf(fact_1230_set__relcomp,axiom,
    ! [Xys: list_P5616295576739893671ist_fm,Yzs: list_P5616295576739893671ist_fm] :
      ( ( relcom1422440059698906787ist_fm @ ( set_Pr8767716839810916150ist_fm @ Xys ) @ ( set_Pr8767716839810916150ist_fm @ Yzs ) )
      = ( set_Pr8767716839810916150ist_fm
        @ ( concat7976216156076608738ist_fm
          @ ( map_Pr8824440416164001930ist_fm
            @ ^ [Xy: produc1996495991257130529ist_fm] :
                ( concat7976216156076608738ist_fm
                @ ( map_Pr8824440416164001930ist_fm
                  @ ^ [Yz: produc1996495991257130529ist_fm] :
                      ( if_lis8669769223177106541ist_fm
                      @ ( ( produc4588648349897876871ist_fm @ Xy )
                        = ( produc1501393135466168645ist_fm @ Yz ) )
                      @ ( cons_P7841146678257726167ist_fm @ ( produc381145313068854617ist_fm @ ( produc1501393135466168645ist_fm @ Xy ) @ ( produc4588648349897876871ist_fm @ Yz ) ) @ nil_Pr6600767949821390631ist_fm )
                      @ nil_Pr6600767949821390631ist_fm )
                  @ Yzs ) )
            @ Xys ) ) ) ) ).

% set_relcomp
thf(fact_1231_set__relcomp,axiom,
    ! [Xys: list_P8332459875677254181m_rule,Yzs: list_P342308711384885665ist_fm] :
      ( ( relcom6537966159191678138ist_fm @ ( set_Pr8505323785428441536m_rule @ Xys ) @ ( set_Pr3748622270175481660ist_fm @ Yzs ) )
      = ( set_Pr3989678446260612092ist_fm
        @ ( concat5991451154419593128ist_fm
          @ ( map_Pr2231524137340215122ist_fm
            @ ^ [Xy: produc164195504107695125m_rule] :
                ( concat5991451154419593128ist_fm
                @ ( map_Pr9024115949047551702ist_fm
                  @ ^ [Yz: produc4630866025709511057ist_fm] :
                      ( if_lis4464993736004109363ist_fm
                      @ ( ( produc7165828336582415457m_rule @ Xy )
                        = ( produc4228417144610596823ist_fm @ Yz ) )
                      @ ( cons_P4725796697801531421ist_fm @ ( produc2592314732767339223ist_fm @ ( produc6879501374131015971m_rule @ Xy ) @ ( produc4514744107061996309ist_fm @ Yz ) ) @ nil_Pr8777681598161732717ist_fm )
                      @ nil_Pr8777681598161732717ist_fm )
                  @ Yzs ) )
            @ Xys ) ) ) ) ).

% set_relcomp
thf(fact_1232_set__relcomp,axiom,
    ! [Xys: list_P342308711384885665ist_fm,Yzs: list_P8332459875677254181m_rule] :
      ( ( relcom2847087027301246368m_rule @ ( set_Pr3748622270175481660ist_fm @ Xys ) @ ( set_Pr8505323785428441536m_rule @ Yzs ) )
      = ( set_Pr8154260935348078208e_rule
        @ ( concat764978673452799020e_rule
          @ ( map_Pr4122339249896234970e_rule
            @ ^ [Xy: produc4630866025709511057ist_fm] :
                ( concat764978673452799020e_rule
                @ ( map_Pr7286373299083741782e_rule
                  @ ^ [Yz: produc164195504107695125m_rule] :
                      ( if_lis3233299126996416183e_rule
                      @ ( ( produc4514744107061996309ist_fm @ Xy )
                        = ( produc6879501374131015971m_rule @ Yz ) )
                      @ ( cons_P2268320488107778593e_rule @ ( produc1769597910952761123e_rule @ ( produc4228417144610596823ist_fm @ Xy ) @ ( produc7165828336582415457m_rule @ Yz ) ) @ nil_Pr6050872761112600177e_rule )
                      @ nil_Pr6050872761112600177e_rule )
                  @ Yzs ) )
            @ Xys ) ) ) ) ).

% set_relcomp
thf(fact_1233_the__elem__set,axiom,
    ! [X2: fm] :
      ( ( the_elem_fm @ ( set_fm2 @ ( cons_fm @ X2 @ nil_fm ) ) )
      = X2 ) ).

% the_elem_set
thf(fact_1234_equiv__class__nondisjoint,axiom,
    ! [A4: set_fm,R2: set_Pr4463079037648049377_fm_fm,X2: fm,A: fm,B: fm] :
      ( ( equiv_equiv_fm @ A4 @ R2 )
     => ( ( member_fm @ X2 @ ( inf_inf_set_fm @ ( image_fm_fm @ R2 @ ( insert_fm2 @ A @ bot_bot_set_fm ) ) @ ( image_fm_fm @ R2 @ ( insert_fm2 @ B @ bot_bot_set_fm ) ) ) )
       => ( member8474499337054950954_fm_fm @ ( product_Pair_fm_fm @ A @ B ) @ R2 ) ) ) ).

% equiv_class_nondisjoint
thf(fact_1235_Sigma__Image,axiom,
    ! [A4: set_Pr1008144964186165195m_rule,B5: produc164195504107695125m_rule > set_fm,X5: set_Pr1008144964186165195m_rule] :
      ( ( image_7747676986113175866ule_fm @ ( produc7179657805070879791ule_fm @ A4 @ B5 ) @ X5 )
      = ( comple7677669044296858969set_fm @ ( image_5698427363735666338set_fm @ B5 @ ( inf_in6908352629942102365m_rule @ X5 @ A4 ) ) ) ) ).

% Sigma_Image
thf(fact_1236_image__split__eq__Sigma,axiom,
    ! [F2: produc164195504107695125m_rule > set_fm,G: produc164195504107695125m_rule > set_fm,A4: set_Pr1008144964186165195m_rule] :
      ( ( image_125568711718110417set_fm
        @ ^ [X3: produc164195504107695125m_rule] : ( produc3030414577503514149set_fm @ ( F2 @ X3 ) @ ( G @ X3 ) )
        @ A4 )
      = ( produc7186806611370130262set_fm @ ( image_5698427363735666338set_fm @ F2 @ A4 )
        @ ^ [X3: set_fm] : ( image_5698427363735666338set_fm @ G @ ( inf_in6908352629942102365m_rule @ ( vimage7048576402205911384set_fm @ F2 @ ( insert_set_fm @ X3 @ bot_bot_set_set_fm ) ) @ A4 ) ) ) ) ).

% image_split_eq_Sigma
thf(fact_1237_INF__set__fold,axiom,
    ! [F2: produc164195504107695125m_rule > set_fm,Xs: list_P8332459875677254181m_rule] :
      ( ( comple3647237725801400255set_fm @ ( image_5698427363735666338set_fm @ F2 @ ( set_Pr8505323785428441536m_rule @ Xs ) ) )
      = ( fold_P2789900844533598888set_fm @ ( comp_s7100926763290858283m_rule @ inf_inf_set_fm @ F2 ) @ Xs @ top_top_set_fm ) ) ).

% INF_set_fold
thf(fact_1238_inf__Int__eq,axiom,
    ! [R: set_fm,S6: set_fm] :
      ( ( inf_inf_fm_o
        @ ^ [X3: fm] : ( member_fm @ X3 @ R )
        @ ^ [X3: fm] : ( member_fm @ X3 @ S6 ) )
      = ( ^ [X3: fm] : ( member_fm @ X3 @ ( inf_inf_set_fm @ R @ S6 ) ) ) ) ).

% inf_Int_eq
thf(fact_1239_top__empty__eq,axiom,
    ( top_top_fm_o
    = ( ^ [X3: fm] : ( member_fm @ X3 @ top_top_set_fm ) ) ) ).

% top_empty_eq
thf(fact_1240_Inf__INT__eq,axiom,
    ( comple7708544496286765958f_fm_o
    = ( ^ [S5: set_fm_o,X3: fm] : ( member_fm @ X3 @ ( comple3647237725801400255set_fm @ ( image_fm_o_set_fm @ collect_fm @ S5 ) ) ) ) ) ).

% Inf_INT_eq
thf(fact_1241_INF__INT__eq,axiom,
    ! [R2: produc164195504107695125m_rule > set_fm,S6: set_Pr1008144964186165195m_rule] :
      ( ( comple7708544496286765958f_fm_o
        @ ( image_1843934041337696867e_fm_o
          @ ^ [I2: produc164195504107695125m_rule,X3: fm] : ( member_fm @ X3 @ ( R2 @ I2 ) )
          @ S6 ) )
      = ( ^ [X3: fm] : ( member_fm @ X3 @ ( comple3647237725801400255set_fm @ ( image_5698427363735666338set_fm @ R2 @ S6 ) ) ) ) ) ).

% INF_INT_eq
thf(fact_1242_INF__Int__eq,axiom,
    ! [S6: set_set_fm] :
      ( ( comple7708544496286765958f_fm_o
        @ ( image_set_fm_fm_o
          @ ^ [I2: set_fm,X3: fm] : ( member_fm @ X3 @ I2 )
          @ S6 ) )
      = ( ^ [X3: fm] : ( member_fm @ X3 @ ( comple3647237725801400255set_fm @ S6 ) ) ) ) ).

% INF_Int_eq
thf(fact_1243_refl__on__INTER,axiom,
    ! [S6: set_Pr1008144964186165195m_rule,A4: produc164195504107695125m_rule > set_fm,R2: produc164195504107695125m_rule > set_Pr4463079037648049377_fm_fm] :
      ( ! [X: produc164195504107695125m_rule] :
          ( ( member4220325220686508332m_rule @ X @ S6 )
         => ( refl_on_fm @ ( A4 @ X ) @ ( R2 @ X ) ) )
     => ( refl_on_fm @ ( comple3647237725801400255set_fm @ ( image_5698427363735666338set_fm @ A4 @ S6 ) ) @ ( comple432188930606310376_fm_fm @ ( image_7922547816601700037_fm_fm @ R2 @ S6 ) ) ) ) ).

% refl_on_INTER
thf(fact_1244_fst__vimage__eq__Times,axiom,
    ! [S6: set_Pr7058068377845519745ist_fm] :
      ( ( vimage7509990688292144847ist_fm @ produc6879501374131015971m_rule @ S6 )
      = ( produc1083232162533131166m_rule @ S6
        @ ^ [Uu: produc1996495991257130529ist_fm] : top_top_set_rule ) ) ).

% fst_vimage_eq_Times
thf(fact_1245_fst__vimage__eq__Times,axiom,
    ! [S6: set_list_fm] :
      ( ( vimage5720363820015494ist_fm @ produc1501393135466168645ist_fm @ S6 )
      = ( produc7866297306142304650ist_fm @ S6
        @ ^ [Uu: list_fm] : top_top_set_list_fm ) ) ).

% fst_vimage_eq_Times
thf(fact_1246_vimage__fst,axiom,
    ! [A4: set_Pr7058068377845519745ist_fm] :
      ( ( vimage7509990688292144847ist_fm @ produc6879501374131015971m_rule @ A4 )
      = ( produc1083232162533131166m_rule @ A4
        @ ^ [Uu: produc1996495991257130529ist_fm] : top_top_set_rule ) ) ).

% vimage_fst
thf(fact_1247_vimage__fst,axiom,
    ! [A4: set_list_fm] :
      ( ( vimage5720363820015494ist_fm @ produc1501393135466168645ist_fm @ A4 )
      = ( produc7866297306142304650ist_fm @ A4
        @ ^ [Uu: list_fm] : top_top_set_list_fm ) ) ).

% vimage_fst
thf(fact_1248_snd__vimage__eq__Times,axiom,
    ! [S6: set_list_fm] :
      ( ( vimage5720363820015494ist_fm @ produc4588648349897876871ist_fm @ S6 )
      = ( produc7866297306142304650ist_fm @ top_top_set_list_fm
        @ ^ [Uu: list_fm] : S6 ) ) ).

% snd_vimage_eq_Times
thf(fact_1249_snd__vimage__eq__Times,axiom,
    ! [S6: set_rule] :
      ( ( vimage3527056447469308439e_rule @ produc7165828336582415457m_rule @ S6 )
      = ( produc1083232162533131166m_rule @ top_to728902761071950545ist_fm
        @ ^ [Uu: produc1996495991257130529ist_fm] : S6 ) ) ).

% snd_vimage_eq_Times
thf(fact_1250_snd__vimage__eq__Times,axiom,
    ! [S6: set_Pr7058068377845519745ist_fm] :
      ( ( vimage7968347001965284947ist_fm @ produc4514744107061996309ist_fm @ S6 )
      = ( produc7655519969867487826ist_fm @ top_top_set_rule
        @ ^ [Uu: rule] : S6 ) ) ).

% snd_vimage_eq_Times
thf(fact_1251_vimage__snd,axiom,
    ! [A4: set_list_fm] :
      ( ( vimage5720363820015494ist_fm @ produc4588648349897876871ist_fm @ A4 )
      = ( produc7866297306142304650ist_fm @ top_top_set_list_fm
        @ ^ [Uu: list_fm] : A4 ) ) ).

% vimage_snd
thf(fact_1252_vimage__snd,axiom,
    ! [A4: set_rule] :
      ( ( vimage3527056447469308439e_rule @ produc7165828336582415457m_rule @ A4 )
      = ( produc1083232162533131166m_rule @ top_to728902761071950545ist_fm
        @ ^ [Uu: produc1996495991257130529ist_fm] : A4 ) ) ).

% vimage_snd
thf(fact_1253_vimage__snd,axiom,
    ! [A4: set_Pr7058068377845519745ist_fm] :
      ( ( vimage7968347001965284947ist_fm @ produc4514744107061996309ist_fm @ A4 )
      = ( produc7655519969867487826ist_fm @ top_top_set_rule
        @ ^ [Uu: rule] : A4 ) ) ).

% vimage_snd
thf(fact_1254_Image__INT__subset,axiom,
    ! [R2: set_Pr4463079037648049377_fm_fm,B5: produc164195504107695125m_rule > set_fm,A4: set_Pr1008144964186165195m_rule] :
      ( ord_less_eq_set_fm @ ( image_fm_fm @ R2 @ ( comple3647237725801400255set_fm @ ( image_5698427363735666338set_fm @ B5 @ A4 ) ) )
      @ ( comple3647237725801400255set_fm
        @ ( image_5698427363735666338set_fm
          @ ^ [X3: produc164195504107695125m_rule] : ( image_fm_fm @ R2 @ ( B5 @ X3 ) )
          @ A4 ) ) ) ).

% Image_INT_subset
thf(fact_1255_Image__INT__eq,axiom,
    ! [R2: set_Pr4463079037648049377_fm_fm,A4: set_Pr1008144964186165195m_rule,B5: produc164195504107695125m_rule > set_fm] :
      ( ( single_valued_fm_fm @ ( converse_fm_fm @ R2 ) )
     => ( ( A4 != bot_bo2182007826548605023m_rule )
       => ( ( image_fm_fm @ R2 @ ( comple3647237725801400255set_fm @ ( image_5698427363735666338set_fm @ B5 @ A4 ) ) )
          = ( comple3647237725801400255set_fm
            @ ( image_5698427363735666338set_fm
              @ ^ [X3: produc164195504107695125m_rule] : ( image_fm_fm @ R2 @ ( B5 @ X3 ) )
              @ A4 ) ) ) ) ) ).

% Image_INT_eq
thf(fact_1256_subseqs__powset,axiom,
    ! [Xs: list_fm] :
      ( ( image_list_fm_set_fm @ set_fm2 @ ( set_list_fm2 @ ( subseqs_fm @ Xs ) ) )
      = ( pow_fm @ ( set_fm2 @ Xs ) ) ) ).

% subseqs_powset
thf(fact_1257_Pow__set_I1_J,axiom,
    ( ( pow_fm @ ( set_fm2 @ nil_fm ) )
    = ( insert_set_fm @ bot_bot_set_fm @ bot_bot_set_set_fm ) ) ).

% Pow_set(1)
thf(fact_1258_Pow__set_I2_J,axiom,
    ! [X2: fm,Xs: list_fm] :
      ( ( pow_fm @ ( set_fm2 @ ( cons_fm @ X2 @ Xs ) ) )
      = ( sup_sup_set_set_fm @ ( pow_fm @ ( set_fm2 @ Xs ) ) @ ( image_set_fm_set_fm @ ( insert_fm2 @ X2 ) @ ( pow_fm @ ( set_fm2 @ Xs ) ) ) ) ) ).

% Pow_set(2)
thf(fact_1259_wf__eq__minimal,axiom,
    ( wf_fm
    = ( ^ [R4: set_Pr4463079037648049377_fm_fm] :
        ! [Q2: set_fm] :
          ( ? [X3: fm] : ( member_fm @ X3 @ Q2 )
         => ? [X3: fm] :
              ( ( member_fm @ X3 @ Q2 )
              & ! [Y4: fm] :
                  ( ( member8474499337054950954_fm_fm @ ( product_Pair_fm_fm @ Y4 @ X3 ) @ R4 )
                 => ~ ( member_fm @ Y4 @ Q2 ) ) ) ) ) ) ).

% wf_eq_minimal
thf(fact_1260_wfI__min,axiom,
    ! [R: set_Pr4463079037648049377_fm_fm] :
      ( ! [X: fm,Q5: set_fm] :
          ( ( member_fm @ X @ Q5 )
         => ? [Xa3: fm] :
              ( ( member_fm @ Xa3 @ Q5 )
              & ! [Y: fm] :
                  ( ( member8474499337054950954_fm_fm @ ( product_Pair_fm_fm @ Y @ Xa3 ) @ R )
                 => ~ ( member_fm @ Y @ Q5 ) ) ) )
     => ( wf_fm @ R ) ) ).

% wfI_min
thf(fact_1261_wfE__min,axiom,
    ! [R: set_Pr4463079037648049377_fm_fm,X2: fm,Q3: set_fm] :
      ( ( wf_fm @ R )
     => ( ( member_fm @ X2 @ Q3 )
       => ~ ! [Z3: fm] :
              ( ( member_fm @ Z3 @ Q3 )
             => ~ ! [Y6: fm] :
                    ( ( member8474499337054950954_fm_fm @ ( product_Pair_fm_fm @ Y6 @ Z3 ) @ R )
                   => ~ ( member_fm @ Y6 @ Q3 ) ) ) ) ) ).

% wfE_min
thf(fact_1262_wfE__min_H,axiom,
    ! [R: set_Pr4463079037648049377_fm_fm,Q3: set_fm] :
      ( ( wf_fm @ R )
     => ( ( Q3 != bot_bot_set_fm )
       => ~ ! [Z3: fm] :
              ( ( member_fm @ Z3 @ Q3 )
             => ~ ! [Y6: fm] :
                    ( ( member8474499337054950954_fm_fm @ ( product_Pair_fm_fm @ Y6 @ Z3 ) @ R )
                   => ~ ( member_fm @ Y6 @ Q3 ) ) ) ) ) ).

% wfE_min'
thf(fact_1263_wfI,axiom,
    ! [R2: set_Pr4463079037648049377_fm_fm,A4: set_fm,B5: set_fm] :
      ( ( ord_le7532831136298237057_fm_fm @ R2
        @ ( product_Sigma_fm_fm @ A4
          @ ^ [Uu: fm] : B5 ) )
     => ( ! [X: fm,P5: fm > $o] :
            ( ! [Xa3: fm] :
                ( ! [Y: fm] :
                    ( ( member8474499337054950954_fm_fm @ ( product_Pair_fm_fm @ Y @ Xa3 ) @ R2 )
                   => ( P5 @ Y ) )
               => ( P5 @ Xa3 ) )
           => ( ( member_fm @ X @ A4 )
             => ( ( member_fm @ X @ B5 )
               => ( P5 @ X ) ) ) )
       => ( wf_fm @ R2 ) ) ) ).

% wfI
thf(fact_1264_stake__szip,axiom,
    ! [N2: nat,S1: stream8299795917829157543ist_fm,S2: stream_rule] :
      ( ( stake_1447931197033250628m_rule @ N2 @ ( szip_P1977448745965526924m_rule @ S1 @ S2 ) )
      = ( zip_Pr2138331976196620527m_rule @ ( stake_2930890243849202354ist_fm @ N2 @ S1 ) @ ( stake_rule @ N2 @ S2 ) ) ) ).

% stake_szip
thf(fact_1265_stake__szip,axiom,
    ! [N2: nat,S1: stream727092118206550309m_rule,S2: stream727092118206550309m_rule] :
      ( ( stake_8908532221384226460m_rule @ N2 @ ( szip_P2499414959592755846m_rule @ S1 @ S2 ) )
      = ( zip_Pr1780431989191363299m_rule @ ( stake_1447931197033250628m_rule @ N2 @ S1 ) @ ( stake_1447931197033250628m_rule @ N2 @ S2 ) ) ) ).

% stake_szip
thf(fact_1266_zip__map__fst__snd,axiom,
    ! [Zs3: list_P5616295576739893671ist_fm] :
      ( ( zip_list_fm_list_fm @ ( map_Pr3743373011763743805ist_fm @ produc1501393135466168645ist_fm @ Zs3 ) @ ( map_Pr3743373011763743805ist_fm @ produc4588648349897876871ist_fm @ Zs3 ) )
      = Zs3 ) ).

% zip_map_fst_snd
thf(fact_1267_zip__map__fst__snd,axiom,
    ! [Zs3: list_P8332459875677254181m_rule] :
      ( ( zip_Pr2138331976196620527m_rule @ ( map_Pr1951777954582829830ist_fm @ produc6879501374131015971m_rule @ Zs3 ) @ ( map_Pr7678108474426652576e_rule @ produc7165828336582415457m_rule @ Zs3 ) )
      = Zs3 ) ).

% zip_map_fst_snd
thf(fact_1268_zip__map__fst__snd,axiom,
    ! [Zs3: list_P342308711384885665ist_fm] :
      ( ( zip_ru8710619783530977187ist_fm @ ( map_Pr452693852626354460m_rule @ produc4228417144610596823ist_fm @ Zs3 ) @ ( map_Pr2410134268255969930ist_fm @ produc4514744107061996309ist_fm @ Zs3 ) )
      = Zs3 ) ).

% zip_map_fst_snd
thf(fact_1269_zip__same,axiom,
    ! [A: fm,B: fm,Xs: list_fm] :
      ( ( member8474499337054950954_fm_fm @ ( product_Pair_fm_fm @ A @ B ) @ ( set_Pr6624747810323803798_fm_fm @ ( zip_fm_fm @ Xs @ Xs ) ) )
      = ( ( member_fm @ A @ ( set_fm2 @ Xs ) )
        & ( A = B ) ) ) ).

% zip_same
thf(fact_1270_in__set__zipE,axiom,
    ! [X2: fm,Y3: fm,Xs: list_fm,Ys: list_fm] :
      ( ( member8474499337054950954_fm_fm @ ( product_Pair_fm_fm @ X2 @ Y3 ) @ ( set_Pr6624747810323803798_fm_fm @ ( zip_fm_fm @ Xs @ Ys ) ) )
     => ~ ( ( member_fm @ X2 @ ( set_fm2 @ Xs ) )
         => ~ ( member_fm @ Y3 @ ( set_fm2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_1271_product__lists__set,axiom,
    ! [Xss: list_list_fm] :
      ( ( set_list_fm2 @ ( product_lists_fm @ Xss ) )
      = ( collect_list_fm
        @ ^ [Xs3: list_fm] :
            ( list_all2_fm_list_fm
            @ ^ [X3: fm,Ys4: list_fm] : ( member_fm @ X3 @ ( set_fm2 @ Ys4 ) )
            @ Xs3
            @ Xss ) ) ) ).

% product_lists_set
thf(fact_1272_subset__code_I3_J,axiom,
    ~ ( ord_less_eq_set_fm @ ( coset_fm @ nil_fm ) @ ( set_fm2 @ nil_fm ) ) ).

% subset_code(3)
thf(fact_1273_subset__code_I2_J,axiom,
    ! [A4: set_fm,Ys: list_fm] :
      ( ( ord_less_eq_set_fm @ A4 @ ( coset_fm @ Ys ) )
      = ( ! [X3: fm] :
            ( ( member_fm @ X3 @ ( set_fm2 @ Ys ) )
           => ~ ( member_fm @ X3 @ A4 ) ) ) ) ).

% subset_code(2)
thf(fact_1274_distinct__append,axiom,
    ! [Xs: list_fm,Ys: list_fm] :
      ( ( distinct_fm @ ( append_fm @ Xs @ Ys ) )
      = ( ( distinct_fm @ Xs )
        & ( distinct_fm @ Ys )
        & ( ( inf_inf_set_fm @ ( set_fm2 @ Xs ) @ ( set_fm2 @ Ys ) )
          = bot_bot_set_fm ) ) ) ).

% distinct_append
thf(fact_1275_distinct__concat,axiom,
    ! [Xs: list_list_fm] :
      ( ( distinct_list_fm @ Xs )
     => ( ! [Ys2: list_fm] :
            ( ( member_list_fm @ Ys2 @ ( set_list_fm2 @ Xs ) )
           => ( distinct_fm @ Ys2 ) )
       => ( ! [Ys2: list_fm,Zs: list_fm] :
              ( ( member_list_fm @ Ys2 @ ( set_list_fm2 @ Xs ) )
             => ( ( member_list_fm @ Zs @ ( set_list_fm2 @ Xs ) )
               => ( ( Ys2 != Zs )
                 => ( ( inf_inf_set_fm @ ( set_fm2 @ Ys2 ) @ ( set_fm2 @ Zs ) )
                    = bot_bot_set_fm ) ) ) )
         => ( distinct_fm @ ( concat_fm @ Xs ) ) ) ) ) ).

% distinct_concat

% Helper facts (9)
thf(help_If_2_1_If_001t__List__Olist_It__Syntax__Ofm_J_T,axiom,
    ! [X2: list_fm,Y3: list_fm] :
      ( ( if_list_fm @ $false @ X2 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__List__Olist_It__Syntax__Ofm_J_T,axiom,
    ! [X2: list_fm,Y3: list_fm] :
      ( ( if_list_fm @ $true @ X2 @ Y3 )
      = X2 ) ).

thf(help_If_2_1_If_001t__List__Olist_It__Product____Type__Oprod_It__Syntax__Orule_Mt__Syntax__Orule_J_J_T,axiom,
    ! [X2: list_P6494679522757010417e_rule,Y3: list_P6494679522757010417e_rule] :
      ( ( if_lis3233299126996416183e_rule @ $false @ X2 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__List__Olist_It__Product____Type__Oprod_It__Syntax__Orule_Mt__Syntax__Orule_J_J_T,axiom,
    ! [X2: list_P6494679522757010417e_rule,Y3: list_P6494679522757010417e_rule] :
      ( ( if_lis3233299126996416183e_rule @ $true @ X2 @ Y3 )
      = X2 ) ).

thf(help_If_2_1_If_001t__List__Olist_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_J_T,axiom,
    ! [X2: list_P5616295576739893671ist_fm,Y3: list_P5616295576739893671ist_fm] :
      ( ( if_lis8669769223177106541ist_fm @ $false @ X2 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__List__Olist_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_J_T,axiom,
    ! [X2: list_P5616295576739893671ist_fm,Y3: list_P5616295576739893671ist_fm] :
      ( ( if_lis8669769223177106541ist_fm @ $true @ X2 @ Y3 )
      = X2 ) ).

thf(help_If_3_1_If_001t__List__Olist_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_J_J_T,axiom,
    ! [P2: $o] :
      ( ( P2 = $true )
      | ( P2 = $false ) ) ).

thf(help_If_2_1_If_001t__List__Olist_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_J_J_T,axiom,
    ! [X2: list_P3779035596438412909ist_fm,Y3: list_P3779035596438412909ist_fm] :
      ( ( if_lis4464993736004109363ist_fm @ $false @ X2 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__List__Olist_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_J_J_T,axiom,
    ! [X2: list_P3779035596438412909ist_fm,Y3: list_P3779035596438412909ist_fm] :
      ( ( if_lis4464993736004109363ist_fm @ $true @ X2 @ Y3 )
      = X2 ) ).

% Conjectures (2)
thf(conj_0,hypothesis,
    member_fm @ p @ ( treeB @ steps ) ).

thf(conj_1,conjecture,
    ? [N4: nat] : ( member_fm @ p @ ( set_fm2 @ ( produc4588648349897876871ist_fm @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ ( sdrop_7224736112439592940m_rule @ N4 @ steps ) ) ) ) ) ) ).

%------------------------------------------------------------------------------