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

% Refs     : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source   : [Des23]
% Names    : FOL_Seq_Calc2/0018_EPathHintikka/prob_00267_010819__13086082_1 [Des23]

% Status   : Theorem
% Rating   : ? v8.2.0
% Syntax   : Number of formulae    : 1813 ( 676 unt; 528 typ;   0 def)
%            Number of atoms       : 3663 (1829 equ;   0 cnn)
%            Maximal formula atoms :    8 (   2 avg)
%            Number of connectives : 11222 ( 454   ~;  43   |; 296   &;9189   @)
%                                         (   0 <=>;1240  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   17 (   5 avg)
%            Number of types       :   72 (  71 usr)
%            Number of type conns  : 1773 (1773   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  460 ( 457 usr;  27 con; 0-4 aty)
%            Number of variables   : 3636 ( 678   ^;2866   !;  92   ?;3636   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            2023-01-19 08:45:18.378
%------------------------------------------------------------------------------
% Could-be-implicit typings (71)
thf(ty_n_t__Stream__Ostream_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_J_J,type,
    stream6210534828274662995m_rule: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_J,type,
    produc8828831911945107917m_rule: $tType ).

thf(ty_n_t__Stream__Ostream_It__Product____Type__Oprod_It__Prover__Orule_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_J_J,type,
    stream8953843411776101167m_rule: $tType ).

thf(ty_n_t__Stream__Ostream_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_Mt__Prover__Orule_J_J,type,
    stream8099677779113257519e_rule: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Prover__Orule_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_Mt__Prover__Orule_J,type,
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thf(ty_n_t__Stream__Ostream_It__Stream__Ostream_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_J_J,type,
    stream3752074346242807894m_rule: $tType ).

thf(ty_n_t__Set__Oset_It__Stream__Ostream_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_J_J,type,
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thf(ty_n_t__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_J_J,type,
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thf(ty_n_t__Stream__Ostream_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_J,type,
    stream2709947120125613254m_rule: $tType ).

thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_J,type,
    list_P2774625669004309958m_rule: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J,type,
    produc340336539035504054m_rule: $tType ).

thf(ty_n_t__Stream__Ostream_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_J,type,
    stream4408948924543953275ist_fm: $tType ).

thf(ty_n_t__FSet__Ofset_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_J,type,
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thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J,type,
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thf(ty_n_t__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_It__SeCaV__Ofm_Mt__SeCaV__Ofm_J_J_J,type,
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thf(ty_n_t__Stream__Ostream_It__Product____Type__Oprod_It__Prover__Orule_Mt__Prover__Orule_J_J,type,
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thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Prover__Orule_Mt__Prover__Orule_J_J,type,
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thf(ty_n_t__Stream__Ostream_It__Stream__Ostream_It__List__Olist_It__SeCaV__Ofm_J_J_J,type,
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thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__SeCaV__Ofm_Mt__SeCaV__Ofm_J_J,type,
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thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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thf(ty_n_t__List__Olist_It__List__Olist_It__List__Olist_It__SeCaV__Ofm_J_J_J,type,
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thf(ty_n_t__Set__Oset_It__List__Olist_It__List__Olist_It__SeCaV__Ofm_J_J_J,type,
    set_list_list_fm: $tType ).

thf(ty_n_t__Set__Oset_It__Set__Oset_It__List__Olist_It__SeCaV__Ofm_J_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Prover__Orule_Mt__Prover__Orule_J,type,
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thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_It__SeCaV__Ofm_J_J_J,type,
    set_set_set_fm: $tType ).

thf(ty_n_t__Stream__Ostream_It__Stream__Ostream_It__Prover__Orule_J_J,type,
    stream_stream_rule: $tType ).

thf(ty_n_t__Set__Oset_I_062_It__List__Olist_It__SeCaV__Ofm_J_M_Eo_J_J,type,
    set_list_fm_o: $tType ).

thf(ty_n_t__Stream__Ostream_It__Stream__Ostream_It__SeCaV__Ofm_J_J,type,
    stream_stream_fm: $tType ).

thf(ty_n_t__Stream__Ostream_It__Stream__Ostream_It__Nat__Onat_J_J,type,
    stream_stream_nat: $tType ).

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

thf(ty_n_t__Set__Oset_It__Stream__Ostream_It__Prover__Orule_J_J,type,
    set_stream_rule: $tType ).

thf(ty_n_t__Stream__Ostream_It__List__Olist_It__SeCaV__Ofm_J_J,type,
    stream_list_fm: $tType ).

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

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

thf(ty_n_t__Set__Oset_It__Set__Oset_It__Prover__Orule_J_J,type,
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thf(ty_n_t__Stream__Ostream_It__Stream__Ostream_I_Eo_J_J,type,
    stream_stream_o: $tType ).

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

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

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

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

thf(ty_n_t__Set__Oset_I_062_It__Prover__Orule_M_Eo_J_J,type,
    set_rule_o: $tType ).

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

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

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

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

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

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

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

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

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

thf(ty_n_t__List__Olist_It__Prover__Orule_J,type,
    list_rule: $tType ).

thf(ty_n_t__Set__Oset_It__Prover__Orule_J,type,
    set_rule: $tType ).

thf(ty_n_t__Set__Oset_I_062_I_Eo_M_Eo_J_J,type,
    set_o_o: $tType ).

thf(ty_n_t__List__Olist_It__SeCaV__Otm_J,type,
    list_tm: $tType ).

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

thf(ty_n_t__List__Olist_It__Nat__Onat_J,type,
    list_nat: $tType ).

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

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

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

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

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

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

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

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

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

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

% Explicit typings (457)
thf(sy_c_Abstract__Completeness_OPersistentRuleSystem_001t__Prover__Orule_001t__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J,type,
    abstra3967921542344661089ist_fm: ( rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o ) > stream_rule > set_Pr5202636777678657877ist_fm > $o ).

thf(sy_c_Abstract__Completeness_ORuleSystem_001t__Prover__Orule_001t__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J,type,
    abstra5221733350967904376ist_fm: ( rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o ) > stream_rule > set_Pr5202636777678657877ist_fm > $o ).

thf(sy_c_Abstract__Completeness_ORuleSystem_OminWait_001t__Prover__Orule_001t__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J,type,
    abstra1963046427795717555ist_fm: ( rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o ) > stream_rule > produc6018962875968178549ist_fm > nat ).

thf(sy_c_Abstract__Completeness_ORuleSystem_Oper_001t__Prover__Orule_001t__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J,type,
    abstra5255361903751151037ist_fm: ( rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o ) > stream_rule > set_Pr5202636777678657877ist_fm > rule > $o ).

thf(sy_c_Abstract__Completeness_ORuleSystem_Opos_001t__Prover__Orule,type,
    abstract_pos_rule: stream_rule > rule > nat ).

thf(sy_c_Abstract__Completeness_ORuleSystem__Defs_OSaturated_001t__Prover__Orule_001t__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J,type,
    abstra6097777249025082867ist_fm: ( rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o ) > stream_rule > stream2709947120125613254m_rule > $o ).

thf(sy_c_Abstract__Completeness_ORuleSystem__Defs_Oenabled_001t__Prover__Orule_001t__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J,type,
    abstra1707737187183579335ist_fm: ( rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o ) > rule > produc6018962875968178549ist_fm > $o ).

thf(sy_c_Abstract__Completeness_ORuleSystem__Defs_Oepath_001t__Prover__Orule_001t__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J,type,
    abstra6789711989322986974ist_fm: ( rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o ) > stream_rule > stream2709947120125613254m_rule > $o ).

thf(sy_c_Abstract__Completeness_ORuleSystem__Defs_Ofair_001t__Prover__Orule,type,
    abstra3799686578551160190r_rule: stream_rule > stream_rule > $o ).

thf(sy_c_Abstract__Completeness_ORuleSystem__Defs_Osaturated_001t__Prover__Orule_001t__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J,type,
    abstra2533313685312581075ist_fm: ( rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o ) > rule > stream2709947120125613254m_rule > $o ).

thf(sy_c_Abstract__Completeness_ORuleSystem__Defs_Otrim_001t__Prover__Orule_001t__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J,type,
    abstra7806236140005899928ist_fm: ( rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o ) > stream_rule > produc6018962875968178549ist_fm > stream_rule ).

thf(sy_c_Basic__BNFs_Osndsp_001t__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_001t__Prover__Orule,type,
    basic_6091292966480414684m_rule: produc340336539035504054m_rule > rule > $o ).

thf(sy_c_Complete__Lattices_OSup__class_OSup_001_062_I_Eo_M_Eo_J,type,
    complete_Sup_Sup_o_o: set_o_o > $o > $o ).

thf(sy_c_Complete__Lattices_OSup__class_OSup_001_062_It__List__Olist_It__SeCaV__Ofm_J_M_Eo_J,type,
    comple173717950668161943t_fm_o: set_list_fm_o > list_fm > $o ).

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

thf(sy_c_Complete__Lattices_OSup__class_OSup_001_062_It__Prover__Orule_M_Eo_J,type,
    comple579058054629451178rule_o: set_rule_o > rule > $o ).

thf(sy_c_Complete__Lattices_OSup__class_OSup_001_062_It__SeCaV__Ofm_M_Eo_J,type,
    comple3443612784043876893p_fm_o: set_fm_o > fm > $o ).

thf(sy_c_Complete__Lattices_OSup__class_OSup_001_Eo,type,
    complete_Sup_Sup_o: set_o > $o ).

thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Nat__Onat,type,
    complete_Sup_Sup_nat: set_nat > nat ).

thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_I_Eo_J,type,
    comple90263536869209701_set_o: set_set_o > set_o ).

thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__List__Olist_It__SeCaV__Ofm_J_J,type,
    comple8784269564784259782ist_fm: set_set_list_fm > set_list_fm ).

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

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

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

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

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

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

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

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

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thf(sy_c_List_Obind_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_001t__SeCaV__Ofm,type,
    bind_P95098981060663284ule_fm: list_P2774625669004309958m_rule > ( produc340336539035504054m_rule > list_fm ) > list_fm ).

thf(sy_c_List_Obind_001t__SeCaV__Ofm_001t__List__Olist_It__SeCaV__Ofm_J,type,
    bind_fm_list_fm: list_fm > ( fm > list_list_fm ) > list_list_fm ).

thf(sy_c_List_Obind_001t__SeCaV__Ofm_001t__SeCaV__Ofm,type,
    bind_fm_fm: list_fm > ( fm > list_fm ) > list_fm ).

thf(sy_c_List_Obind_001t__SeCaV__Ofm_001t__SeCaV__Otm,type,
    bind_fm_tm: list_fm > ( fm > list_tm ) > list_tm ).

thf(sy_c_List_Obind_001t__SeCaV__Otm_001t__List__Olist_It__SeCaV__Ofm_J,type,
    bind_tm_list_fm: list_tm > ( tm > list_list_fm ) > list_list_fm ).

thf(sy_c_List_Obind_001t__SeCaV__Otm_001t__SeCaV__Ofm,type,
    bind_tm_fm: list_tm > ( tm > list_fm ) > list_fm ).

thf(sy_c_List_Obind_001t__SeCaV__Otm_001t__SeCaV__Otm,type,
    bind_tm_tm: list_tm > ( tm > list_tm ) > list_tm ).

thf(sy_c_List_Oconcat_001t__List__Olist_It__SeCaV__Ofm_J,type,
    concat_list_fm: list_list_list_fm > list_list_fm ).

thf(sy_c_List_Oconcat_001t__SeCaV__Ofm,type,
    concat_fm: list_list_fm > list_fm ).

thf(sy_c_List_Oconcat_001t__SeCaV__Otm,type,
    concat_tm: list_list_tm > list_tm ).

thf(sy_c_List_Olist_Olist__all_001_Eo,type,
    list_all_o: ( $o > $o ) > list_o > $o ).

thf(sy_c_List_Olist_Olist__all_001t__List__Olist_It__SeCaV__Ofm_J,type,
    list_all_list_fm: ( list_fm > $o ) > list_list_fm > $o ).

thf(sy_c_List_Olist_Olist__all_001t__Nat__Onat,type,
    list_all_nat: ( nat > $o ) > list_nat > $o ).

thf(sy_c_List_Olist_Olist__all_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J,type,
    list_a9038620553111681026m_rule: ( produc340336539035504054m_rule > $o ) > list_P2774625669004309958m_rule > $o ).

thf(sy_c_List_Olist_Olist__all_001t__Prover__Orule,type,
    list_all_rule: ( rule > $o ) > list_rule > $o ).

thf(sy_c_List_Olist_Olist__all_001t__SeCaV__Ofm,type,
    list_all_fm: ( fm > $o ) > list_fm > $o ).

thf(sy_c_List_Olist_Olist__all_001t__SeCaV__Otm,type,
    list_all_tm: ( tm > $o ) > list_tm > $o ).

thf(sy_c_List_Olist_Orec__list_001t__Set__Oset_I_Eo_J_001_Eo,type,
    rec_list_set_o_o: set_o > ( $o > list_o > set_o > set_o ) > list_o > set_o ).

thf(sy_c_List_Olist_Orec__list_001t__Set__Oset_It__List__Olist_It__SeCaV__Ofm_J_J_001t__List__Olist_It__SeCaV__Ofm_J,type,
    rec_li3562514433299826070ist_fm: set_list_fm > ( list_fm > list_list_fm > set_list_fm > set_list_fm ) > list_list_fm > set_list_fm ).

thf(sy_c_List_Olist_Orec__list_001t__Set__Oset_It__Nat__Onat_J_001t__Nat__Onat,type,
    rec_list_set_nat_nat: set_nat > ( nat > list_nat > set_nat > set_nat ) > list_nat > set_nat ).

thf(sy_c_List_Olist_Orec__list_001t__Set__Oset_It__Prover__Orule_J_001t__Prover__Orule,type,
    rec_li2005092410820253458e_rule: set_rule > ( rule > list_rule > set_rule > set_rule ) > list_rule > set_rule ).

thf(sy_c_List_Olist_Orec__list_001t__Set__Oset_It__SeCaV__Ofm_J_001t__SeCaV__Ofm,type,
    rec_list_set_fm_fm: set_fm > ( fm > list_fm > set_fm > set_fm ) > list_fm > set_fm ).

thf(sy_c_List_Olist_Orec__list_001t__Set__Oset_It__SeCaV__Otm_J_001t__SeCaV__Otm,type,
    rec_list_set_tm_tm: set_tm > ( tm > list_tm > set_tm > set_tm ) > list_tm > set_tm ).

thf(sy_c_List_Olist_Oset_001_Eo,type,
    set_o2: list_o > set_o ).

thf(sy_c_List_Olist_Oset_001t__List__Olist_It__List__Olist_It__SeCaV__Ofm_J_J,type,
    set_list_list_fm2: list_list_list_fm > set_list_list_fm ).

thf(sy_c_List_Olist_Oset_001t__List__Olist_It__SeCaV__Ofm_J,type,
    set_list_fm2: list_list_fm > set_list_fm ).

thf(sy_c_List_Olist_Oset_001t__List__Olist_It__SeCaV__Otm_J,type,
    set_list_tm2: list_list_tm > set_list_tm ).

thf(sy_c_List_Olist_Oset_001t__Nat__Onat,type,
    set_nat2: list_nat > set_nat ).

thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J,type,
    set_Pr4534715572506550497m_rule: list_P2774625669004309958m_rule > set_Pr1822751329126368876m_rule ).

thf(sy_c_List_Olist_Oset_001t__Prover__Orule,type,
    set_rule2: list_rule > set_rule ).

thf(sy_c_List_Olist_Oset_001t__SeCaV__Ofm,type,
    set_fm2: list_fm > set_fm ).

thf(sy_c_List_Olist_Oset_001t__SeCaV__Otm,type,
    set_tm2: list_tm > set_tm ).

thf(sy_c_List_Omember_001_Eo,type,
    member_o: list_o > $o > $o ).

thf(sy_c_List_Omember_001t__List__Olist_It__SeCaV__Ofm_J,type,
    member_list_fm: list_list_fm > list_fm > $o ).

thf(sy_c_List_Omember_001t__Nat__Onat,type,
    member_nat: list_nat > nat > $o ).

thf(sy_c_List_Omember_001t__Prover__Orule,type,
    member_rule: list_rule > rule > $o ).

thf(sy_c_List_Omember_001t__SeCaV__Ofm,type,
    member_fm: list_fm > fm > $o ).

thf(sy_c_List_Omember_001t__SeCaV__Otm,type,
    member_tm: list_tm > tm > $o ).

thf(sy_c_List_Onull_001t__List__Olist_It__SeCaV__Ofm_J,type,
    null_list_fm: list_list_fm > $o ).

thf(sy_c_List_Onull_001t__SeCaV__Ofm,type,
    null_fm: list_fm > $o ).

thf(sy_c_List_Onull_001t__SeCaV__Otm,type,
    null_tm: list_tm > $o ).

thf(sy_c_List_Ounion_001t__List__Olist_It__SeCaV__Ofm_J,type,
    union_list_fm: list_list_fm > list_list_fm > list_list_fm ).

thf(sy_c_List_Ounion_001t__Nat__Onat,type,
    union_nat: list_nat > list_nat > list_nat ).

thf(sy_c_List_Ounion_001t__SeCaV__Ofm,type,
    union_fm: list_fm > list_fm > list_fm ).

thf(sy_c_List_Ounion_001t__SeCaV__Otm,type,
    union_tm: list_tm > list_tm > list_tm ).

thf(sy_c_Nat_OSuc,type,
    suc: nat > nat ).

thf(sy_c_Orderings_Obot__class_Obot_001_062_I_Eo_M_Eo_J,type,
    bot_bot_o_o: $o > $o ).

thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Nat__Onat_M_Eo_J,type,
    bot_bot_nat_o: nat > $o ).

thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Prover__Orule_M_Eo_J,type,
    bot_bot_rule_o: rule > $o ).

thf(sy_c_Orderings_Obot__class_Obot_001_Eo,type,
    bot_bot_o: $o ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_Eo_J,type,
    bot_bot_set_o: set_o ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__List__Olist_It__SeCaV__Ofm_J_J,type,
    bot_bot_set_list_fm: set_list_fm ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
    bot_bot_set_nat: set_nat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    bot_bo2099793752762293965at_nat: set_Pr1261947904930325089at_nat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_J,type,
    bot_bo3426410157013319936m_rule: set_Pr1822751329126368876m_rule ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Prover__Orule_J,type,
    bot_bot_set_rule: set_rule ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__SeCaV__Ofm_J,type,
    bot_bot_set_fm: set_fm ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__SeCaV__Otm_J,type,
    bot_bot_set_tm: set_tm ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_I_Eo_J_J,type,
    bot_bot_set_set_o: set_set_o ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
    bot_bot_set_set_nat: set_set_nat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Prover__Orule_J_J,type,
    bot_bot_set_set_rule: set_set_rule ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__SeCaV__Ofm_J_J,type,
    bot_bot_set_set_fm: set_set_fm ).

thf(sy_c_Orderings_Oord__class_OLeast_001t__Nat__Onat,type,
    ord_Least_nat: ( nat > $o ) > nat ).

thf(sy_c_Orderings_Oord__class_Oless_001_062_I_Eo_M_Eo_J,type,
    ord_less_o_o: ( $o > $o ) > ( $o > $o ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001_062_It__List__Olist_It__SeCaV__Ofm_J_M_Eo_J,type,
    ord_less_list_fm_o: ( list_fm > $o ) > ( list_fm > $o ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001_062_It__Nat__Onat_M_Eo_J,type,
    ord_less_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001_062_It__Prover__Orule_M_Eo_J,type,
    ord_less_rule_o: ( rule > $o ) > ( rule > $o ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001_062_It__SeCaV__Ofm_M_Eo_J,type,
    ord_less_fm_o: ( fm > $o ) > ( fm > $o ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001_Eo,type,
    ord_less_o: $o > $o > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
    ord_less_nat: nat > nat > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_I_Eo_J,type,
    ord_less_set_o: set_o > set_o > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__List__Olist_It__SeCaV__Ofm_J_J,type,
    ord_less_set_list_fm: set_list_fm > set_list_fm > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J,type,
    ord_less_set_nat: set_nat > set_nat > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Prover__Orule_J,type,
    ord_less_set_rule: set_rule > set_rule > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__SeCaV__Ofm_J,type,
    ord_less_set_fm: set_fm > set_fm > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
    ord_less_eq_nat: nat > nat > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__SeCaV__Ofm_J,type,
    ord_less_eq_set_fm: set_fm > set_fm > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__SeCaV__Otm_J,type,
    ord_less_eq_set_tm: set_tm > set_tm > $o ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_Eo_J,type,
    top_top_set_o: set_o ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Nat__Onat_J,type,
    top_top_set_nat: set_nat ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_J,type,
    top_to1730629564055774885ist_fm: set_Pr5202636777678657877ist_fm ).

thf(sy_c_Product__Type_OPair_001_Eo_001_Eo,type,
    product_Pair_o_o: $o > $o > product_prod_o_o ).

thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Nat__Onat,type,
    product_Pair_nat_nat: nat > nat > product_prod_nat_nat ).

thf(sy_c_Product__Type_OPair_001t__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_001t__Prover__Orule,type,
    produc1733806532565653680m_rule: produc6018962875968178549ist_fm > rule > produc340336539035504054m_rule ).

thf(sy_c_Product__Type_OPair_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J,type,
    produc6261311607089640965m_rule: produc340336539035504054m_rule > produc340336539035504054m_rule > produc8828831911945107917m_rule ).

thf(sy_c_Product__Type_OPair_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_001t__Prover__Orule,type,
    produc4831648765031708129e_rule: produc340336539035504054m_rule > rule > produc9112364199808626345e_rule ).

thf(sy_c_Product__Type_OPair_001t__Prover__Orule_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J,type,
    produc97041875958848737m_rule: rule > produc340336539035504054m_rule > produc7920154866009513897m_rule ).

thf(sy_c_Product__Type_OPair_001t__Prover__Orule_001t__Prover__Orule,type,
    produc5849431337705160893e_rule: rule > rule > produc7694839378271647877e_rule ).

thf(sy_c_Product__Type_Oprod_Osnd_001t__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_001t__Prover__Orule,type,
    produc6514711208036230658m_rule: produc340336539035504054m_rule > rule ).

thf(sy_c_ProverLemmas_Oaffects,type,
    affects: rule > fm > $o ).

thf(sy_c_Prover_Ochildren,type,
    children: list_tm > rule > list_fm > list_list_fm ).

thf(sy_c_Prover_Oeff,type,
    eff: rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o ).

thf(sy_c_Prover_Oparts,type,
    parts: list_tm > rule > fm > list_list_fm ).

thf(sy_c_Prover_Orule_OAlphaDis,type,
    alphaDis: rule ).

thf(sy_c_Prover_Orules,type,
    rules: stream_rule ).

thf(sy_c_Relation_OField_001_Eo,type,
    field_o: set_Product_prod_o_o > set_o ).

thf(sy_c_Relation_OField_001t__Nat__Onat,type,
    field_nat: set_Pr1261947904930325089at_nat > set_nat ).

thf(sy_c_Relation_OField_001t__Prover__Orule,type,
    field_rule: set_Pr7340090144965549541e_rule > set_rule ).

thf(sy_c_Relation_OField_001t__SeCaV__Ofm,type,
    field_fm: set_Pr4706815898642364871_fm_fm > set_fm ).

thf(sy_c_Relation_OId__on_001t__Nat__Onat,type,
    id_on_nat: set_nat > set_Pr1261947904930325089at_nat ).

thf(sy_c_Relation_Orefl__on_001t__Nat__Onat,type,
    refl_on_nat: set_nat > set_Pr1261947904930325089at_nat > $o ).

thf(sy_c_Relation_Orefl__on_001t__SeCaV__Ofm,type,
    refl_on_fm: set_fm > set_Pr4706815898642364871_fm_fm > $o ).

thf(sy_c_SeCaV_Ofm_ODis,type,
    dis: fm > fm > fm ).

thf(sy_c_SeCaV_Omember_001_Eo,type,
    member_o2: $o > list_o > $o ).

thf(sy_c_SeCaV_Omember_001t__List__Olist_It__SeCaV__Ofm_J,type,
    member_list_fm2: list_fm > list_list_fm > $o ).

thf(sy_c_SeCaV_Omember_001t__Nat__Onat,type,
    member_nat2: nat > list_nat > $o ).

thf(sy_c_SeCaV_Omember_001t__Prover__Orule,type,
    member_rule2: rule > list_rule > $o ).

thf(sy_c_SeCaV_Omember_001t__SeCaV__Ofm,type,
    member_fm2: fm > list_fm > $o ).

thf(sy_c_SeCaV_Omember_001t__SeCaV__Otm,type,
    member_tm2: tm > list_tm > $o ).

thf(sy_c_Set_OBall_001_Eo,type,
    ball_o: set_o > ( $o > $o ) > $o ).

thf(sy_c_Set_OBall_001t__List__Olist_It__SeCaV__Ofm_J,type,
    ball_list_fm: set_list_fm > ( list_fm > $o ) > $o ).

thf(sy_c_Set_OBall_001t__Nat__Onat,type,
    ball_nat: set_nat > ( nat > $o ) > $o ).

thf(sy_c_Set_OBall_001t__Prover__Orule,type,
    ball_rule: set_rule > ( rule > $o ) > $o ).

thf(sy_c_Set_OBall_001t__SeCaV__Ofm,type,
    ball_fm: set_fm > ( fm > $o ) > $o ).

thf(sy_c_Set_OCollect_001_Eo,type,
    collect_o: ( $o > $o ) > set_o ).

thf(sy_c_Set_OCollect_001t__List__Olist_It__SeCaV__Ofm_J,type,
    collect_list_fm: ( list_fm > $o ) > set_list_fm ).

thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
    collect_nat: ( nat > $o ) > set_nat ).

thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J,type,
    collec1009276759030335883m_rule: ( produc340336539035504054m_rule > $o ) > set_Pr1822751329126368876m_rule ).

thf(sy_c_Set_OCollect_001t__Prover__Orule,type,
    collect_rule: ( rule > $o ) > set_rule ).

thf(sy_c_Set_OCollect_001t__SeCaV__Ofm,type,
    collect_fm: ( fm > $o ) > set_fm ).

thf(sy_c_Set_OCollect_001t__Set__Oset_It__SeCaV__Ofm_J,type,
    collect_set_fm: ( set_fm > $o ) > set_set_fm ).

thf(sy_c_Set_OPow_001_Eo,type,
    pow_o: set_o > set_set_o ).

thf(sy_c_Set_OPow_001t__Nat__Onat,type,
    pow_nat: set_nat > set_set_nat ).

thf(sy_c_Set_OPow_001t__Prover__Orule,type,
    pow_rule: set_rule > set_set_rule ).

thf(sy_c_Set_OPow_001t__SeCaV__Ofm,type,
    pow_fm: set_fm > set_set_fm ).

thf(sy_c_Set_Ofilter_001_Eo,type,
    filter_o: ( $o > $o ) > set_o > set_o ).

thf(sy_c_Set_Ofilter_001t__List__Olist_It__SeCaV__Ofm_J,type,
    filter_list_fm: ( list_fm > $o ) > set_list_fm > set_list_fm ).

thf(sy_c_Set_Ofilter_001t__Nat__Onat,type,
    filter_nat: ( nat > $o ) > set_nat > set_nat ).

thf(sy_c_Set_Ofilter_001t__Prover__Orule,type,
    filter_rule: ( rule > $o ) > set_rule > set_rule ).

thf(sy_c_Set_Ofilter_001t__SeCaV__Ofm,type,
    filter_fm: ( fm > $o ) > set_fm > set_fm ).

thf(sy_c_Set_Oimage_001_062_I_Eo_M_Eo_J_001t__Set__Oset_I_Eo_J,type,
    image_o_o_set_o: ( ( $o > $o ) > set_o ) > set_o_o > set_set_o ).

thf(sy_c_Set_Oimage_001_062_It__List__Olist_It__SeCaV__Ofm_J_M_Eo_J_001t__Set__Oset_It__List__Olist_It__SeCaV__Ofm_J_J,type,
    image_4471151405024504270ist_fm: ( ( list_fm > $o ) > set_list_fm ) > set_list_fm_o > set_set_list_fm ).

thf(sy_c_Set_Oimage_001_062_It__Nat__Onat_M_Eo_J_001t__Set__Oset_It__Nat__Onat_J,type,
    image_nat_o_set_nat: ( ( nat > $o ) > set_nat ) > set_nat_o > set_set_nat ).

thf(sy_c_Set_Oimage_001_062_It__Prover__Orule_M_Eo_J_001t__Set__Oset_It__Prover__Orule_J,type,
    image_7644734524678384448t_rule: ( ( rule > $o ) > set_rule ) > set_rule_o > set_set_rule ).

thf(sy_c_Set_Oimage_001_062_It__SeCaV__Ofm_M_Eo_J_001t__Set__Oset_It__SeCaV__Ofm_J,type,
    image_fm_o_set_fm: ( ( fm > $o ) > set_fm ) > set_fm_o > set_set_fm ).

thf(sy_c_Set_Oimage_001_Eo_001_Eo,type,
    image_o_o: ( $o > $o ) > set_o > set_o ).

thf(sy_c_Set_Oimage_001_Eo_001t__Nat__Onat,type,
    image_o_nat: ( $o > nat ) > set_o > set_nat ).

thf(sy_c_Set_Oimage_001_Eo_001t__Prover__Orule,type,
    image_o_rule: ( $o > rule ) > set_o > set_rule ).

thf(sy_c_Set_Oimage_001_Eo_001t__SeCaV__Ofm,type,
    image_o_fm: ( $o > fm ) > set_o > set_fm ).

thf(sy_c_Set_Oimage_001_Eo_001t__Set__Oset_I_Eo_J,type,
    image_o_set_o: ( $o > set_o ) > set_o > set_set_o ).

thf(sy_c_Set_Oimage_001_Eo_001t__Set__Oset_It__Nat__Onat_J,type,
    image_o_set_nat: ( $o > set_nat ) > set_o > set_set_nat ).

thf(sy_c_Set_Oimage_001_Eo_001t__Set__Oset_It__Prover__Orule_J,type,
    image_o_set_rule: ( $o > set_rule ) > set_o > set_set_rule ).

thf(sy_c_Set_Oimage_001_Eo_001t__Set__Oset_It__SeCaV__Ofm_J,type,
    image_o_set_fm: ( $o > set_fm ) > set_o > set_set_fm ).

thf(sy_c_Set_Oimage_001t__List__Olist_It__List__Olist_It__SeCaV__Ofm_J_J_001t__Set__Oset_It__List__Olist_It__SeCaV__Ofm_J_J,type,
    image_3687226712311829663ist_fm: ( list_list_fm > set_list_fm ) > set_list_list_fm > set_set_list_fm ).

thf(sy_c_Set_Oimage_001t__List__Olist_It__SeCaV__Ofm_J_001_Eo,type,
    image_list_fm_o: ( list_fm > $o ) > set_list_fm > set_o ).

thf(sy_c_Set_Oimage_001t__List__Olist_It__SeCaV__Ofm_J_001t__Nat__Onat,type,
    image_list_fm_nat: ( list_fm > nat ) > set_list_fm > set_nat ).

thf(sy_c_Set_Oimage_001t__List__Olist_It__SeCaV__Ofm_J_001t__Set__Oset_It__List__Olist_It__SeCaV__Ofm_J_J,type,
    image_1803486656538583717ist_fm: ( list_fm > set_list_fm ) > set_list_fm > set_set_list_fm ).

thf(sy_c_Set_Oimage_001t__List__Olist_It__SeCaV__Ofm_J_001t__Set__Oset_It__Nat__Onat_J,type,
    image_3154564810619571642et_nat: ( list_fm > set_nat ) > set_list_fm > set_set_nat ).

thf(sy_c_Set_Oimage_001t__List__Olist_It__SeCaV__Ofm_J_001t__Set__Oset_It__SeCaV__Ofm_J,type,
    image_list_fm_set_fm: ( list_fm > set_fm ) > set_list_fm > set_set_fm ).

thf(sy_c_Set_Oimage_001t__List__Olist_It__SeCaV__Ofm_J_001t__Set__Oset_It__SeCaV__Otm_J,type,
    image_list_fm_set_tm: ( list_fm > set_tm ) > set_list_fm > set_set_tm ).

thf(sy_c_Set_Oimage_001t__List__Olist_It__SeCaV__Otm_J_001t__Set__Oset_It__SeCaV__Otm_J,type,
    image_list_tm_set_tm: ( list_tm > set_tm ) > set_list_tm > set_set_tm ).

thf(sy_c_Set_Oimage_001t__Nat__Onat_001_Eo,type,
    image_nat_o: ( nat > $o ) > set_nat > set_o ).

thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Nat__Onat,type,
    image_nat_nat: ( nat > nat ) > set_nat > set_nat ).

thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Prover__Orule,type,
    image_nat_rule: ( nat > rule ) > set_nat > set_rule ).

thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__SeCaV__Ofm,type,
    image_nat_fm: ( nat > fm ) > set_nat > set_fm ).

thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_I_Eo_J,type,
    image_nat_set_o: ( nat > set_o ) > set_nat > set_set_o ).

thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
    image_nat_set_nat: ( nat > set_nat ) > set_nat > set_set_nat ).

thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_It__Prover__Orule_J,type,
    image_nat_set_rule: ( nat > set_rule ) > set_nat > set_set_rule ).

thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_It__SeCaV__Ofm_J,type,
    image_nat_set_fm: ( nat > set_fm ) > set_nat > set_set_fm ).

thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_001_062_It__SeCaV__Ofm_M_Eo_J,type,
    image_5627174782981912967e_fm_o: ( produc340336539035504054m_rule > fm > $o ) > set_Pr1822751329126368876m_rule > set_fm_o ).

thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_001_Eo,type,
    image_8969478372357302939rule_o: ( produc340336539035504054m_rule > $o ) > set_Pr1822751329126368876m_rule > set_o ).

thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J,type,
    image_2498997284783760755m_rule: ( produc340336539035504054m_rule > produc340336539035504054m_rule ) > set_Pr1822751329126368876m_rule > set_Pr1822751329126368876m_rule ).

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    image_2303374963288105462ule_fm: ( produc340336539035504054m_rule > fm ) > set_Pr1822751329126368876m_rule > set_fm ).

thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_001t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_J,type,
    image_2683796897779739177m_rule: ( produc340336539035504054m_rule > set_Pr1822751329126368876m_rule ) > set_Pr1822751329126368876m_rule > set_se4309406418413213730m_rule ).

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    image_377115208147135190set_fm: ( produc340336539035504054m_rule > set_fm ) > set_Pr1822751329126368876m_rule > set_set_fm ).

thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_001t__Set__Oset_It__Set__Oset_It__SeCaV__Ofm_J_J,type,
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thf(sy_c_Set_Oimage_001t__Prover__Orule_001_Eo,type,
    image_rule_o: ( rule > $o ) > set_rule > set_o ).

thf(sy_c_Set_Oimage_001t__Prover__Orule_001t__Nat__Onat,type,
    image_rule_nat: ( rule > nat ) > set_rule > set_nat ).

thf(sy_c_Set_Oimage_001t__Prover__Orule_001t__Prover__Orule,type,
    image_rule_rule: ( rule > rule ) > set_rule > set_rule ).

thf(sy_c_Set_Oimage_001t__Prover__Orule_001t__SeCaV__Ofm,type,
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thf(sy_c_Set_Oimage_001t__Prover__Orule_001t__Set__Oset_I_Eo_J,type,
    image_rule_set_o: ( rule > set_o ) > set_rule > set_set_o ).

thf(sy_c_Set_Oimage_001t__Prover__Orule_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Set_Oimage_001t__Prover__Orule_001t__Set__Oset_It__Prover__Orule_J,type,
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thf(sy_c_Set_Oimage_001t__Prover__Orule_001t__Set__Oset_It__SeCaV__Ofm_J,type,
    image_rule_set_fm: ( rule > set_fm ) > set_rule > set_set_fm ).

thf(sy_c_Set_Oimage_001t__SeCaV__Ofm_001_Eo,type,
    image_fm_o: ( fm > $o ) > set_fm > set_o ).

thf(sy_c_Set_Oimage_001t__SeCaV__Ofm_001t__Nat__Onat,type,
    image_fm_nat: ( fm > nat ) > set_fm > set_nat ).

thf(sy_c_Set_Oimage_001t__SeCaV__Ofm_001t__Prover__Orule,type,
    image_fm_rule: ( fm > rule ) > set_fm > set_rule ).

thf(sy_c_Set_Oimage_001t__SeCaV__Ofm_001t__SeCaV__Ofm,type,
    image_fm_fm: ( fm > fm ) > set_fm > set_fm ).

thf(sy_c_Set_Oimage_001t__SeCaV__Ofm_001t__Set__Oset_I_Eo_J,type,
    image_fm_set_o: ( fm > set_o ) > set_fm > set_set_o ).

thf(sy_c_Set_Oimage_001t__SeCaV__Ofm_001t__Set__Oset_It__List__Olist_It__SeCaV__Ofm_J_J,type,
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thf(sy_c_Set_Oimage_001t__SeCaV__Ofm_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Set_Oimage_001t__SeCaV__Ofm_001t__Set__Oset_It__Prover__Orule_J,type,
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thf(sy_c_Set_Oimage_001t__SeCaV__Ofm_001t__Set__Oset_It__SeCaV__Ofm_J,type,
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thf(sy_c_Set_Oimage_001t__SeCaV__Ofm_001t__Set__Oset_It__SeCaV__Otm_J,type,
    image_fm_set_tm: ( fm > set_tm ) > set_fm > set_set_tm ).

thf(sy_c_Set_Oimage_001t__SeCaV__Otm_001t__Set__Oset_It__List__Olist_It__SeCaV__Ofm_J_J,type,
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thf(sy_c_Set_Oimage_001t__SeCaV__Otm_001t__Set__Oset_It__SeCaV__Ofm_J,type,
    image_tm_set_fm: ( tm > set_fm ) > set_tm > set_set_fm ).

thf(sy_c_Set_Oimage_001t__SeCaV__Otm_001t__Set__Oset_It__SeCaV__Otm_J,type,
    image_tm_set_tm: ( tm > set_tm ) > set_tm > set_set_tm ).

thf(sy_c_Set_Oimage_001t__Set__Oset_I_Eo_J_001_062_I_Eo_M_Eo_J,type,
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thf(sy_c_Set_Oimage_001t__Set__Oset_I_Eo_J_001_Eo,type,
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thf(sy_c_Set_Oimage_001t__Set__Oset_I_Eo_J_001t__Set__Oset_I_Eo_J,type,
    image_set_o_set_o: ( set_o > set_o ) > set_set_o > set_set_o ).

thf(sy_c_Set_Oimage_001t__Set__Oset_It__List__Olist_It__SeCaV__Ofm_J_J_001_062_It__List__Olist_It__SeCaV__Ofm_J_M_Eo_J,type,
    image_4048089684711339992t_fm_o: ( set_list_fm > list_fm > $o ) > set_set_list_fm > set_list_fm_o ).

thf(sy_c_Set_Oimage_001t__Set__Oset_It__List__Olist_It__SeCaV__Ofm_J_J_001_Eo,type,
    image_set_list_fm_o: ( set_list_fm > $o ) > set_set_list_fm > set_o ).

thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001_062_It__Nat__Onat_M_Eo_J,type,
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thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001_Eo,type,
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thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
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    image_4873481958868436096set_fm: ( set_Pr1822751329126368876m_rule > set_set_fm ) > set_se4309406418413213730m_rule > set_set_set_fm ).

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thf(sy_c_Set_Oimage_001t__Set__Oset_It__Prover__Orule_J_001_062_It__Prover__Orule_M_Eo_J,type,
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thf(sy_c_Set_Oimage_001t__Set__Oset_It__Prover__Orule_J_001_Eo,type,
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thf(sy_c_Set_Oimage_001t__Set__Oset_It__Prover__Orule_J_001t__Set__Oset_It__Prover__Orule_J,type,
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thf(sy_c_Set_Oimage_001t__Set__Oset_It__SeCaV__Ofm_J_001_062_It__SeCaV__Ofm_M_Eo_J,type,
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thf(sy_c_Set_Oimage_001t__Set__Oset_It__SeCaV__Ofm_J_001_Eo,type,
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thf(sy_c_Set_Oimage_001t__Set__Oset_It__SeCaV__Ofm_J_001t__Set__Oset_It__SeCaV__Ofm_J,type,
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    image_1703615012366172913t_rule: ( stream_rule > set_rule ) > set_stream_rule > set_set_rule ).

thf(sy_c_Set_Oimage_001t__Stream__Ostream_It__SeCaV__Ofm_J_001t__Set__Oset_It__SeCaV__Ofm_J,type,
    image_7285378133382788895set_fm: ( stream_fm > set_fm ) > set_stream_fm > set_set_fm ).

thf(sy_c_Set_Oinsert_001_Eo,type,
    insert_o: $o > set_o > set_o ).

thf(sy_c_Set_Oinsert_001t__List__Olist_It__SeCaV__Ofm_J,type,
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thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
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thf(sy_c_Set_Oinsert_001t__Product____Type__Oprod_I_Eo_M_Eo_J,type,
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thf(sy_c_Set_Oinsert_001t__Prover__Orule,type,
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thf(sy_c_Set_Oinsert_001t__SeCaV__Ofm,type,
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thf(sy_c_Set_Oinsert_001t__SeCaV__Otm,type,
    insert_tm: tm > set_tm > set_tm ).

thf(sy_c_Set_Oinsert_001t__Set__Oset_I_Eo_J,type,
    insert_set_o: set_o > set_set_o > set_set_o ).

thf(sy_c_Set_Oinsert_001t__Set__Oset_It__Nat__Onat_J,type,
    insert_set_nat: set_nat > set_set_nat > set_set_nat ).

thf(sy_c_Set_Oinsert_001t__Set__Oset_It__Prover__Orule_J,type,
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thf(sy_c_Set_Oinsert_001t__Set__Oset_It__SeCaV__Ofm_J,type,
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    q: fm ).

thf(sy_v_steps,type,
    steps: stream2709947120125613254m_rule ).

thf(sy_v_suf____,type,
    suf: stream2709947120125613254m_rule ).

% Relevant facts (1277)
thf(fact_0_pre,axiom,
    ( list_a9038620553111681026m_rule
    @ ^ [Xs: produc340336539035504054m_rule] :
        ( ( produc6514711208036230658m_rule @ Xs )
       != alphaDis )
    @ pre ) ).

% pre
thf(fact_1__092_060open_062_092_060And_062r_O_Aaffects_Ar_A_IDis_Ap_Aq_J_A_061_A_Ir_A_061_AAlphaDis_J_092_060close_062,axiom,
    ! [R: rule] :
      ( ( affects @ R @ ( dis @ p @ q ) )
      = ( R = alphaDis ) ) ).

% \<open>\<And>r. affects r (Dis p q) = (r = AlphaDis)\<close>
thf(fact_2__092_060open_062Dis_Ap_Aq_A_092_060in_062_Atree__fms_Asteps_092_060close_062,axiom,
    member_fm3 @ ( dis @ p @ q ) @ ( tree_fms @ steps ) ).

% \<open>Dis p q \<in> tree_fms steps\<close>
thf(fact_3_fm_Oinject_I3_J,axiom,
    ! [X31: fm,X32: fm,Y31: fm,Y32: fm] :
      ( ( ( dis @ X31 @ X32 )
        = ( dis @ Y31 @ Y32 ) )
      = ( ( X31 = Y31 )
        & ( X32 = Y32 ) ) ) ).

% fm.inject(3)
thf(fact_4_list_Opred__True,axiom,
    ( ( list_a9038620553111681026m_rule
      @ ^ [Uu: produc340336539035504054m_rule] : $true )
    = ( ^ [Uu: list_P2774625669004309958m_rule] : $true ) ) ).

% list.pred_True
thf(fact_5_suf,axiom,
    ( linear7925859179493631106m_rule
    @ ^ [Step: produc340336539035504054m_rule] :
        ( ( produc6514711208036230658m_rule @ Step )
        = alphaDis )
    @ suf ) ).

% suf
thf(fact_6_n,axiom,
    member_fm3 @ ( dis @ p @ q ) @ ( set_fm2 @ ( pseq @ ( shd_Pr4562317740776619530m_rule @ ( sdrop_8169176516188972301m_rule @ n @ steps ) ) ) ) ).

% n
thf(fact_7__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062n_O_ADis_Ap_Aq_A_092_060in_062_Aset_A_Ipseq_A_Ishd_A_Isdrop_An_Asteps_J_J_J_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
    ~ ! [N: nat] :
        ~ ( member_fm3 @ ( dis @ p @ q ) @ ( set_fm2 @ ( pseq @ ( shd_Pr4562317740776619530m_rule @ ( sdrop_8169176516188972301m_rule @ N @ steps ) ) ) ) ) ).

% \<open>\<And>thesis. (\<And>n. Dis p q \<in> set (pseq (shd (sdrop n steps))) \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_8_ori,axiom,
    ( ( sdrop_8169176516188972301m_rule @ n @ steps )
    = ( shift_2334844276748245581m_rule @ pre @ suf ) ) ).

% ori
thf(fact_9_sndsp_Ocases,axiom,
    ! [P: produc340336539035504054m_rule,A: rule] :
      ( ( basic_6091292966480414684m_rule @ P @ A )
     => ( A
        = ( produc6514711208036230658m_rule @ P ) ) ) ).

% sndsp.cases
thf(fact_10_sndsp_Osimps,axiom,
    ( basic_6091292966480414684m_rule
    = ( ^ [P2: produc340336539035504054m_rule,A2: rule] :
          ( A2
          = ( produc6514711208036230658m_rule @ P2 ) ) ) ) ).

% sndsp.simps
thf(fact_11_sndsp_Ointros,axiom,
    ! [P: produc340336539035504054m_rule] : ( basic_6091292966480414684m_rule @ P @ ( produc6514711208036230658m_rule @ P ) ) ).

% sndsp.intros
thf(fact_12_Hintikka_OAlphaDis,axiom,
    ! [H: set_fm,P: fm,Q: fm] :
      ( ( hintikka @ H )
     => ( ( member_fm3 @ ( dis @ P @ Q ) @ H )
       => ( ( member_fm3 @ P @ H )
          & ( member_fm3 @ Q @ H ) ) ) ) ).

% Hintikka.AlphaDis
thf(fact_13__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062pre_Asuf_O_A_092_060lbrakk_062list__all_A_I_092_060lambda_062xs_O_Asnd_Axs_A_092_060noteq_062_AAlphaDis_J_Apre_059_Aholds_A_Iis__rule_AAlphaDis_J_Asuf_059_Asdrop_An_Asteps_A_061_Apre_A_064_N_Asuf_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
    ~ ! [Pre: list_P2774625669004309958m_rule] :
        ( ( list_a9038620553111681026m_rule
          @ ^ [Xs: produc340336539035504054m_rule] :
              ( ( produc6514711208036230658m_rule @ Xs )
             != alphaDis )
          @ Pre )
       => ! [Suf: stream2709947120125613254m_rule] :
            ( ( linear7925859179493631106m_rule
              @ ^ [Step: produc340336539035504054m_rule] :
                  ( ( produc6514711208036230658m_rule @ Step )
                  = alphaDis )
              @ Suf )
           => ( ( sdrop_8169176516188972301m_rule @ n @ steps )
             != ( shift_2334844276748245581m_rule @ Pre @ Suf ) ) ) ) ).

% \<open>\<And>thesis. (\<And>pre suf. \<lbrakk>list_all (\<lambda>xs. snd xs \<noteq> AlphaDis) pre; holds (is_rule AlphaDis) suf; sdrop n steps = pre @- suf\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_14_list__all__cong,axiom,
    ! [X: list_rule,Ya: list_rule,P3: rule > $o,Pa: rule > $o] :
      ( ( X = Ya )
     => ( ! [Z: rule] :
            ( ( member_rule3 @ Z @ ( set_rule2 @ Ya ) )
           => ( ( P3 @ Z )
              = ( Pa @ Z ) ) )
       => ( ( list_all_rule @ P3 @ X )
          = ( list_all_rule @ Pa @ Ya ) ) ) ) ).

% list_all_cong
thf(fact_15_list__all__cong,axiom,
    ! [X: list_o,Ya: list_o,P3: $o > $o,Pa: $o > $o] :
      ( ( X = Ya )
     => ( ! [Z: $o] :
            ( ( member_o3 @ Z @ ( set_o2 @ Ya ) )
           => ( ( P3 @ Z )
              = ( Pa @ Z ) ) )
       => ( ( list_all_o @ P3 @ X )
          = ( list_all_o @ Pa @ Ya ) ) ) ) ).

% list_all_cong
thf(fact_16_list__all__cong,axiom,
    ! [X: list_nat,Ya: list_nat,P3: nat > $o,Pa: nat > $o] :
      ( ( X = Ya )
     => ( ! [Z: nat] :
            ( ( member_nat3 @ Z @ ( set_nat2 @ Ya ) )
           => ( ( P3 @ Z )
              = ( Pa @ Z ) ) )
       => ( ( list_all_nat @ P3 @ X )
          = ( list_all_nat @ Pa @ Ya ) ) ) ) ).

% list_all_cong
thf(fact_17_list__all__cong,axiom,
    ! [X: list_fm,Ya: list_fm,P3: fm > $o,Pa: fm > $o] :
      ( ( X = Ya )
     => ( ! [Z: fm] :
            ( ( member_fm3 @ Z @ ( set_fm2 @ Ya ) )
           => ( ( P3 @ Z )
              = ( Pa @ Z ) ) )
       => ( ( list_all_fm @ P3 @ X )
          = ( list_all_fm @ Pa @ Ya ) ) ) ) ).

% list_all_cong
thf(fact_18_list__all__cong,axiom,
    ! [X: list_list_fm,Ya: list_list_fm,P3: list_fm > $o,Pa: list_fm > $o] :
      ( ( X = Ya )
     => ( ! [Z: list_fm] :
            ( ( member_list_fm3 @ Z @ ( set_list_fm2 @ Ya ) )
           => ( ( P3 @ Z )
              = ( Pa @ Z ) ) )
       => ( ( list_all_list_fm @ P3 @ X )
          = ( list_all_list_fm @ Pa @ Ya ) ) ) ) ).

% list_all_cong
thf(fact_19_list__all__cong,axiom,
    ! [X: list_tm,Ya: list_tm,P3: tm > $o,Pa: tm > $o] :
      ( ( X = Ya )
     => ( ! [Z: tm] :
            ( ( member_tm3 @ Z @ ( set_tm2 @ Ya ) )
           => ( ( P3 @ Z )
              = ( Pa @ Z ) ) )
       => ( ( list_all_tm @ P3 @ X )
          = ( list_all_tm @ Pa @ Ya ) ) ) ) ).

% list_all_cong
thf(fact_20_list__all__cong,axiom,
    ! [X: list_P2774625669004309958m_rule,Ya: list_P2774625669004309958m_rule,P3: produc340336539035504054m_rule > $o,Pa: produc340336539035504054m_rule > $o] :
      ( ( X = Ya )
     => ( ! [Z: produc340336539035504054m_rule] :
            ( ( member7231649785386036813m_rule @ Z @ ( set_Pr4534715572506550497m_rule @ Ya ) )
           => ( ( P3 @ Z )
              = ( Pa @ Z ) ) )
       => ( ( list_a9038620553111681026m_rule @ P3 @ X )
          = ( list_a9038620553111681026m_rule @ Pa @ Ya ) ) ) ) ).

% list_all_cong
thf(fact_21_list_Opred__mono__strong,axiom,
    ! [P3: rule > $o,X: list_rule,Pa: rule > $o] :
      ( ( list_all_rule @ P3 @ X )
     => ( ! [Z: rule] :
            ( ( member_rule3 @ Z @ ( set_rule2 @ X ) )
           => ( ( P3 @ Z )
             => ( Pa @ Z ) ) )
       => ( list_all_rule @ Pa @ X ) ) ) ).

% list.pred_mono_strong
thf(fact_22_list_Opred__mono__strong,axiom,
    ! [P3: $o > $o,X: list_o,Pa: $o > $o] :
      ( ( list_all_o @ P3 @ X )
     => ( ! [Z: $o] :
            ( ( member_o3 @ Z @ ( set_o2 @ X ) )
           => ( ( P3 @ Z )
             => ( Pa @ Z ) ) )
       => ( list_all_o @ Pa @ X ) ) ) ).

% list.pred_mono_strong
thf(fact_23_list_Opred__mono__strong,axiom,
    ! [P3: nat > $o,X: list_nat,Pa: nat > $o] :
      ( ( list_all_nat @ P3 @ X )
     => ( ! [Z: nat] :
            ( ( member_nat3 @ Z @ ( set_nat2 @ X ) )
           => ( ( P3 @ Z )
             => ( Pa @ Z ) ) )
       => ( list_all_nat @ Pa @ X ) ) ) ).

% list.pred_mono_strong
thf(fact_24_list_Opred__mono__strong,axiom,
    ! [P3: fm > $o,X: list_fm,Pa: fm > $o] :
      ( ( list_all_fm @ P3 @ X )
     => ( ! [Z: fm] :
            ( ( member_fm3 @ Z @ ( set_fm2 @ X ) )
           => ( ( P3 @ Z )
             => ( Pa @ Z ) ) )
       => ( list_all_fm @ Pa @ X ) ) ) ).

% list.pred_mono_strong
thf(fact_25_list_Opred__mono__strong,axiom,
    ! [P3: list_fm > $o,X: list_list_fm,Pa: list_fm > $o] :
      ( ( list_all_list_fm @ P3 @ X )
     => ( ! [Z: list_fm] :
            ( ( member_list_fm3 @ Z @ ( set_list_fm2 @ X ) )
           => ( ( P3 @ Z )
             => ( Pa @ Z ) ) )
       => ( list_all_list_fm @ Pa @ X ) ) ) ).

% list.pred_mono_strong
thf(fact_26_list_Opred__mono__strong,axiom,
    ! [P3: tm > $o,X: list_tm,Pa: tm > $o] :
      ( ( list_all_tm @ P3 @ X )
     => ( ! [Z: tm] :
            ( ( member_tm3 @ Z @ ( set_tm2 @ X ) )
           => ( ( P3 @ Z )
             => ( Pa @ Z ) ) )
       => ( list_all_tm @ Pa @ X ) ) ) ).

% list.pred_mono_strong
thf(fact_27_list_Opred__mono__strong,axiom,
    ! [P3: produc340336539035504054m_rule > $o,X: list_P2774625669004309958m_rule,Pa: produc340336539035504054m_rule > $o] :
      ( ( list_a9038620553111681026m_rule @ P3 @ X )
     => ( ! [Z: produc340336539035504054m_rule] :
            ( ( member7231649785386036813m_rule @ Z @ ( set_Pr4534715572506550497m_rule @ X ) )
           => ( ( P3 @ Z )
             => ( Pa @ Z ) ) )
       => ( list_a9038620553111681026m_rule @ Pa @ X ) ) ) ).

% list.pred_mono_strong
thf(fact_28__092_060open_062ev_A_Iholds_A_Iis__rule_AAlphaDis_J_J_A_Isdrop_An_Asteps_J_092_060close_062,axiom,
    ( linear1816559245158234865m_rule
    @ ( linear7925859179493631106m_rule
      @ ^ [Step: produc340336539035504054m_rule] :
          ( ( produc6514711208036230658m_rule @ Step )
          = alphaDis ) )
    @ ( sdrop_8169176516188972301m_rule @ n @ steps ) ) ).

% \<open>ev (holds (is_rule AlphaDis)) (sdrop n steps)\<close>
thf(fact_29_shift__left__inj,axiom,
    ! [Xs2: list_P2774625669004309958m_rule,S1: stream2709947120125613254m_rule,S2: stream2709947120125613254m_rule] :
      ( ( ( shift_2334844276748245581m_rule @ Xs2 @ S1 )
        = ( shift_2334844276748245581m_rule @ Xs2 @ S2 ) )
      = ( S1 = S2 ) ) ).

% shift_left_inj
thf(fact_30_holds_Oelims_I3_J,axiom,
    ! [X: rule > $o,Xa: stream_rule] :
      ( ~ ( linear7354821883049305950s_rule @ X @ Xa )
     => ~ ( X @ ( shd_rule @ Xa ) ) ) ).

% holds.elims(3)
thf(fact_31_holds_Oelims_I3_J,axiom,
    ! [X: produc340336539035504054m_rule > $o,Xa: stream2709947120125613254m_rule] :
      ( ~ ( linear7925859179493631106m_rule @ X @ Xa )
     => ~ ( X @ ( shd_Pr4562317740776619530m_rule @ Xa ) ) ) ).

% holds.elims(3)
thf(fact_32_holds_Oelims_I2_J,axiom,
    ! [X: rule > $o,Xa: stream_rule] :
      ( ( linear7354821883049305950s_rule @ X @ Xa )
     => ( X @ ( shd_rule @ Xa ) ) ) ).

% holds.elims(2)
thf(fact_33_holds_Oelims_I2_J,axiom,
    ! [X: produc340336539035504054m_rule > $o,Xa: stream2709947120125613254m_rule] :
      ( ( linear7925859179493631106m_rule @ X @ Xa )
     => ( X @ ( shd_Pr4562317740776619530m_rule @ Xa ) ) ) ).

% holds.elims(2)
thf(fact_34_holds_Oelims_I1_J,axiom,
    ! [X: rule > $o,Xa: stream_rule,Y: $o] :
      ( ( ( linear7354821883049305950s_rule @ X @ Xa )
        = Y )
     => ( Y
        = ( X @ ( shd_rule @ Xa ) ) ) ) ).

% holds.elims(1)
thf(fact_35_holds_Oelims_I1_J,axiom,
    ! [X: produc340336539035504054m_rule > $o,Xa: stream2709947120125613254m_rule,Y: $o] :
      ( ( ( linear7925859179493631106m_rule @ X @ Xa )
        = Y )
     => ( Y
        = ( X @ ( shd_Pr4562317740776619530m_rule @ Xa ) ) ) ) ).

% holds.elims(1)
thf(fact_36_holds_Osimps,axiom,
    ( linear7354821883049305950s_rule
    = ( ^ [P4: rule > $o,Xs: stream_rule] : ( P4 @ ( shd_rule @ Xs ) ) ) ) ).

% holds.simps
thf(fact_37_holds_Osimps,axiom,
    ( linear7925859179493631106m_rule
    = ( ^ [P4: produc340336539035504054m_rule > $o,Xs: stream2709947120125613254m_rule] : ( P4 @ ( shd_Pr4562317740776619530m_rule @ Xs ) ) ) ) ).

% holds.simps
thf(fact_38_holds__aand,axiom,
    ! [P3: produc340336539035504054m_rule > $o,Steps: stream2709947120125613254m_rule,Q2: produc340336539035504054m_rule > $o] :
      ( ( ( linear7925859179493631106m_rule @ P3 @ Steps )
        & ( linear7925859179493631106m_rule @ Q2 @ Steps ) )
      = ( linear7925859179493631106m_rule
        @ ^ [Step: produc340336539035504054m_rule] :
            ( ( P3 @ Step )
            & ( Q2 @ Step ) )
        @ Steps ) ) ).

% holds_aand
thf(fact_39_member,axiom,
    ( member_rule2
    = ( ^ [P2: rule,Z2: list_rule] : ( member_rule3 @ P2 @ ( set_rule2 @ Z2 ) ) ) ) ).

% member
thf(fact_40_member,axiom,
    ( member_o2
    = ( ^ [P2: $o,Z2: list_o] : ( member_o3 @ P2 @ ( set_o2 @ Z2 ) ) ) ) ).

% member
thf(fact_41_member,axiom,
    ( member_nat2
    = ( ^ [P2: nat,Z2: list_nat] : ( member_nat3 @ P2 @ ( set_nat2 @ Z2 ) ) ) ) ).

% member
thf(fact_42_member,axiom,
    ( member_fm2
    = ( ^ [P2: fm,Z2: list_fm] : ( member_fm3 @ P2 @ ( set_fm2 @ Z2 ) ) ) ) ).

% member
thf(fact_43_member,axiom,
    ( member_list_fm2
    = ( ^ [P2: list_fm,Z2: list_list_fm] : ( member_list_fm3 @ P2 @ ( set_list_fm2 @ Z2 ) ) ) ) ).

% member
thf(fact_44_member,axiom,
    ( member_tm2
    = ( ^ [P2: tm,Z2: list_tm] : ( member_tm3 @ P2 @ ( set_tm2 @ Z2 ) ) ) ) ).

% member
thf(fact_45_tree__fms__in__pseq,axiom,
    ! [P: fm,Steps: stream2709947120125613254m_rule] :
      ( ( member_fm3 @ P @ ( tree_fms @ Steps ) )
     => ? [N: nat] : ( member_fm3 @ P @ ( set_fm2 @ ( pseq @ ( snth_P6679518042731451922m_rule @ Steps @ N ) ) ) ) ) ).

% tree_fms_in_pseq
thf(fact_46_pseq__in__tree__fms,axiom,
    ! [X: produc340336539035504054m_rule,Steps: stream2709947120125613254m_rule,P: fm] :
      ( ( member7231649785386036813m_rule @ X @ ( sset_P4484857331586881186m_rule @ Steps ) )
     => ( ( member_fm3 @ P @ ( set_fm2 @ ( pseq @ X ) ) )
       => ( member_fm3 @ P @ ( tree_fms @ Steps ) ) ) ) ).

% pseq_in_tree_fms
thf(fact_47_holds__mono,axiom,
    ! [P3: produc340336539035504054m_rule > $o,Xs2: stream2709947120125613254m_rule,Q2: produc340336539035504054m_rule > $o] :
      ( ( linear7925859179493631106m_rule @ P3 @ Xs2 )
     => ( ! [X2: produc340336539035504054m_rule] :
            ( ( P3 @ X2 )
           => ( Q2 @ X2 ) )
       => ( linear7925859179493631106m_rule @ Q2 @ Xs2 ) ) ) ).

% holds_mono
thf(fact_48_stream__all__shift,axiom,
    ! [P3: produc340336539035504054m_rule > $o,Xs2: list_P2774625669004309958m_rule,S: stream2709947120125613254m_rule] :
      ( ( stream2134157564947672044m_rule @ P3 @ ( shift_2334844276748245581m_rule @ Xs2 @ S ) )
      = ( ( list_a9038620553111681026m_rule @ P3 @ Xs2 )
        & ( stream2134157564947672044m_rule @ P3 @ S ) ) ) ).

% stream_all_shift
thf(fact_49_ev__ev,axiom,
    ! [Phi: stream2709947120125613254m_rule > $o] :
      ( ( linear1816559245158234865m_rule @ ( linear1816559245158234865m_rule @ Phi ) )
      = ( linear1816559245158234865m_rule @ Phi ) ) ).

% ev_ev
thf(fact_50_sdrop__simps_I1_J,axiom,
    ! [N2: nat,S: stream2709947120125613254m_rule] :
      ( ( shd_Pr4562317740776619530m_rule @ ( sdrop_8169176516188972301m_rule @ N2 @ S ) )
      = ( snth_P6679518042731451922m_rule @ S @ N2 ) ) ).

% sdrop_simps(1)
thf(fact_51_sdrop__simps_I1_J,axiom,
    ! [N2: nat,S: stream_rule] :
      ( ( shd_rule @ ( sdrop_rule @ N2 @ S ) )
      = ( snth_rule @ S @ N2 ) ) ).

% sdrop_simps(1)
thf(fact_52_ev__holds__sset,axiom,
    ! [P3: rule > $o,Xs2: stream_rule] :
      ( ( linear292482573288640205v_rule @ ( linear7354821883049305950s_rule @ P3 ) @ Xs2 )
      = ( ? [X3: rule] :
            ( ( member_rule3 @ X3 @ ( sset_rule @ Xs2 ) )
            & ( P3 @ X3 ) ) ) ) ).

% ev_holds_sset
thf(fact_53_ev__holds__sset,axiom,
    ! [P3: produc340336539035504054m_rule > $o,Xs2: stream2709947120125613254m_rule] :
      ( ( linear1816559245158234865m_rule @ ( linear7925859179493631106m_rule @ P3 ) @ Xs2 )
      = ( ? [X3: produc340336539035504054m_rule] :
            ( ( member7231649785386036813m_rule @ X3 @ ( sset_P4484857331586881186m_rule @ Xs2 ) )
            & ( P3 @ X3 ) ) ) ) ).

% ev_holds_sset
thf(fact_54_ev__False,axiom,
    ! [Omega: stream2709947120125613254m_rule] :
      ~ ( linear1816559245158234865m_rule
        @ ^ [X3: stream2709947120125613254m_rule] : $false
        @ Omega ) ).

% ev_False
thf(fact_55_ev__mono,axiom,
    ! [Phi: stream2709947120125613254m_rule > $o,Xs2: stream2709947120125613254m_rule,Psi: stream2709947120125613254m_rule > $o] :
      ( ( linear1816559245158234865m_rule @ Phi @ Xs2 )
     => ( ! [Xs3: stream2709947120125613254m_rule] :
            ( ( Phi @ Xs3 )
           => ( Psi @ Xs3 ) )
       => ( linear1816559245158234865m_rule @ Psi @ Xs2 ) ) ) ).

% ev_mono
thf(fact_56_ev__or,axiom,
    ! [Phi: stream2709947120125613254m_rule > $o,Psi: stream2709947120125613254m_rule > $o] :
      ( ( linear1816559245158234865m_rule
        @ ^ [Xs: stream2709947120125613254m_rule] :
            ( ( Phi @ Xs )
            | ( Psi @ Xs ) ) )
      = ( ^ [Xs: stream2709947120125613254m_rule] :
            ( ( linear1816559245158234865m_rule @ Phi @ Xs )
            | ( linear1816559245158234865m_rule @ Psi @ Xs ) ) ) ) ).

% ev_or
thf(fact_57_stream__all__def,axiom,
    ( stream2134157564947672044m_rule
    = ( ^ [P4: produc340336539035504054m_rule > $o,S3: stream2709947120125613254m_rule] :
        ! [P2: nat] : ( P4 @ ( snth_P6679518042731451922m_rule @ S3 @ P2 ) ) ) ) ).

% stream_all_def
thf(fact_58_stream__all__def,axiom,
    ( stream_all_rule
    = ( ^ [P4: rule > $o,S3: stream_rule] :
        ! [P2: nat] : ( P4 @ ( snth_rule @ S3 @ P2 ) ) ) ) ).

% stream_all_def
thf(fact_59_snth__sset,axiom,
    ! [S: stream_list_fm,N2: nat] : ( member_list_fm3 @ ( snth_list_fm @ S @ N2 ) @ ( sset_list_fm @ S ) ) ).

% snth_sset
thf(fact_60_snth__sset,axiom,
    ! [S: stream_fm,N2: nat] : ( member_fm3 @ ( snth_fm @ S @ N2 ) @ ( sset_fm @ S ) ) ).

% snth_sset
thf(fact_61_snth__sset,axiom,
    ! [S: stream_o,N2: nat] : ( member_o3 @ ( snth_o @ S @ N2 ) @ ( sset_o @ S ) ) ).

% snth_sset
thf(fact_62_snth__sset,axiom,
    ! [S: stream_nat,N2: nat] : ( member_nat3 @ ( snth_nat @ S @ N2 ) @ ( sset_nat @ S ) ) ).

% snth_sset
thf(fact_63_snth__sset,axiom,
    ! [S: stream2709947120125613254m_rule,N2: nat] : ( member7231649785386036813m_rule @ ( snth_P6679518042731451922m_rule @ S @ N2 ) @ ( sset_P4484857331586881186m_rule @ S ) ) ).

% snth_sset
thf(fact_64_snth__sset,axiom,
    ! [S: stream_rule,N2: nat] : ( member_rule3 @ ( snth_rule @ S @ N2 ) @ ( sset_rule @ S ) ) ).

% snth_sset
thf(fact_65_ev_Obase,axiom,
    ! [Phi: stream2709947120125613254m_rule > $o,Xs2: stream2709947120125613254m_rule] :
      ( ( Phi @ Xs2 )
     => ( linear1816559245158234865m_rule @ Phi @ Xs2 ) ) ).

% ev.base
thf(fact_66_shd__sset,axiom,
    ! [A: stream_list_fm] : ( member_list_fm3 @ ( shd_list_fm @ A ) @ ( sset_list_fm @ A ) ) ).

% shd_sset
thf(fact_67_shd__sset,axiom,
    ! [A: stream_fm] : ( member_fm3 @ ( shd_fm @ A ) @ ( sset_fm @ A ) ) ).

% shd_sset
thf(fact_68_shd__sset,axiom,
    ! [A: stream_o] : ( member_o3 @ ( shd_o @ A ) @ ( sset_o @ A ) ) ).

% shd_sset
thf(fact_69_shd__sset,axiom,
    ! [A: stream_nat] : ( member_nat3 @ ( shd_nat @ A ) @ ( sset_nat @ A ) ) ).

% shd_sset
thf(fact_70_shd__sset,axiom,
    ! [A: stream2709947120125613254m_rule] : ( member7231649785386036813m_rule @ ( shd_Pr4562317740776619530m_rule @ A ) @ ( sset_P4484857331586881186m_rule @ A ) ) ).

% shd_sset
thf(fact_71_shd__sset,axiom,
    ! [A: stream_rule] : ( member_rule3 @ ( shd_rule @ A ) @ ( sset_rule @ A ) ) ).

% shd_sset
thf(fact_72_ev__iff__sdrop,axiom,
    ( linear292482573288640205v_rule
    = ( ^ [P4: stream_rule > $o,Omega2: stream_rule] :
        ? [M: nat] : ( P4 @ ( sdrop_rule @ M @ Omega2 ) ) ) ) ).

% ev_iff_sdrop
thf(fact_73_ev__iff__sdrop,axiom,
    ( linear1816559245158234865m_rule
    = ( ^ [P4: stream2709947120125613254m_rule > $o,Omega2: stream2709947120125613254m_rule] :
        ? [M: nat] : ( P4 @ ( sdrop_8169176516188972301m_rule @ M @ Omega2 ) ) ) ) ).

% ev_iff_sdrop
thf(fact_74_ev__imp__shift,axiom,
    ! [Phi: stream2709947120125613254m_rule > $o,Xs2: stream2709947120125613254m_rule] :
      ( ( linear1816559245158234865m_rule @ Phi @ Xs2 )
     => ? [Xl: list_P2774625669004309958m_rule,Xs22: stream2709947120125613254m_rule] :
          ( ( Xs2
            = ( shift_2334844276748245581m_rule @ Xl @ Xs22 ) )
          & ( Phi @ Xs22 ) ) ) ).

% ev_imp_shift
thf(fact_75_ev__shift,axiom,
    ! [Phi: stream2709947120125613254m_rule > $o,Xs2: stream2709947120125613254m_rule,Xl2: list_P2774625669004309958m_rule] :
      ( ( linear1816559245158234865m_rule @ Phi @ Xs2 )
     => ( linear1816559245158234865m_rule @ Phi @ ( shift_2334844276748245581m_rule @ Xl2 @ Xs2 ) ) ) ).

% ev_shift
thf(fact_76_ev__prefix,axiom,
    ! [P3: produc340336539035504054m_rule > $o,Xs2: stream2709947120125613254m_rule] :
      ( ( linear1816559245158234865m_rule @ ( linear7925859179493631106m_rule @ P3 ) @ Xs2 )
     => ? [Pre: list_P2774625669004309958m_rule,Suf: stream2709947120125613254m_rule] :
          ( ( list_a9038620553111681026m_rule
            @ ^ [Xs: produc340336539035504054m_rule] :
                ~ ( P3 @ Xs )
            @ Pre )
          & ( linear7925859179493631106m_rule @ P3 @ Suf )
          & ( Xs2
            = ( shift_2334844276748245581m_rule @ Pre @ Suf ) ) ) ) ).

% ev_prefix
thf(fact_77_mem__Collect__eq,axiom,
    ! [A: rule,P3: rule > $o] :
      ( ( member_rule3 @ A @ ( collect_rule @ P3 ) )
      = ( P3 @ A ) ) ).

% mem_Collect_eq
thf(fact_78_mem__Collect__eq,axiom,
    ! [A: list_fm,P3: list_fm > $o] :
      ( ( member_list_fm3 @ A @ ( collect_list_fm @ P3 ) )
      = ( P3 @ A ) ) ).

% mem_Collect_eq
thf(fact_79_mem__Collect__eq,axiom,
    ! [A: fm,P3: fm > $o] :
      ( ( member_fm3 @ A @ ( collect_fm @ P3 ) )
      = ( P3 @ A ) ) ).

% mem_Collect_eq
thf(fact_80_mem__Collect__eq,axiom,
    ! [A: $o,P3: $o > $o] :
      ( ( member_o3 @ A @ ( collect_o @ P3 ) )
      = ( P3 @ A ) ) ).

% mem_Collect_eq
thf(fact_81_mem__Collect__eq,axiom,
    ! [A: nat,P3: nat > $o] :
      ( ( member_nat3 @ A @ ( collect_nat @ P3 ) )
      = ( P3 @ A ) ) ).

% mem_Collect_eq
thf(fact_82_Collect__mem__eq,axiom,
    ! [A3: set_rule] :
      ( ( collect_rule
        @ ^ [X3: rule] : ( member_rule3 @ X3 @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_83_Collect__mem__eq,axiom,
    ! [A3: set_list_fm] :
      ( ( collect_list_fm
        @ ^ [X3: list_fm] : ( member_list_fm3 @ X3 @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_84_Collect__mem__eq,axiom,
    ! [A3: set_fm] :
      ( ( collect_fm
        @ ^ [X3: fm] : ( member_fm3 @ X3 @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_85_Collect__mem__eq,axiom,
    ! [A3: set_o] :
      ( ( collect_o
        @ ^ [X3: $o] : ( member_o3 @ X3 @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_86_Collect__mem__eq,axiom,
    ! [A3: set_nat] :
      ( ( collect_nat
        @ ^ [X3: nat] : ( member_nat3 @ X3 @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_87_Collect__cong,axiom,
    ! [P3: nat > $o,Q2: nat > $o] :
      ( ! [X2: nat] :
          ( ( P3 @ X2 )
          = ( Q2 @ X2 ) )
     => ( ( collect_nat @ P3 )
        = ( collect_nat @ Q2 ) ) ) ).

% Collect_cong
thf(fact_88_snth__sset__smerge,axiom,
    ! [Ss: stream6017534608192929797ist_fm,N2: nat,M2: nat] : ( member_list_fm3 @ ( snth_list_fm @ ( snth_stream_list_fm @ Ss @ N2 ) @ M2 ) @ ( sset_list_fm @ ( smerge_list_fm @ Ss ) ) ) ).

% snth_sset_smerge
thf(fact_89_snth__sset__smerge,axiom,
    ! [Ss: stream_stream_fm,N2: nat,M2: nat] : ( member_fm3 @ ( snth_fm @ ( snth_stream_fm @ Ss @ N2 ) @ M2 ) @ ( sset_fm @ ( smerge_fm @ Ss ) ) ) ).

% snth_sset_smerge
thf(fact_90_snth__sset__smerge,axiom,
    ! [Ss: stream_stream_o,N2: nat,M2: nat] : ( member_o3 @ ( snth_o @ ( snth_stream_o @ Ss @ N2 ) @ M2 ) @ ( sset_o @ ( smerge_o @ Ss ) ) ) ).

% snth_sset_smerge
thf(fact_91_snth__sset__smerge,axiom,
    ! [Ss: stream_stream_nat,N2: nat,M2: nat] : ( member_nat3 @ ( snth_nat @ ( snth_stream_nat @ Ss @ N2 ) @ M2 ) @ ( sset_nat @ ( smerge_nat @ Ss ) ) ) ).

% snth_sset_smerge
thf(fact_92_snth__sset__smerge,axiom,
    ! [Ss: stream3752074346242807894m_rule,N2: nat,M2: nat] : ( member7231649785386036813m_rule @ ( snth_P6679518042731451922m_rule @ ( snth_s6182113952396108578m_rule @ Ss @ N2 ) @ M2 ) @ ( sset_P4484857331586881186m_rule @ ( smerge193809993764105000m_rule @ Ss ) ) ) ).

% snth_sset_smerge
thf(fact_93_snth__sset__smerge,axiom,
    ! [Ss: stream_stream_rule,N2: nat,M2: nat] : ( member_rule3 @ ( snth_rule @ ( snth_stream_rule @ Ss @ N2 ) @ M2 ) @ ( sset_rule @ ( smerge_rule @ Ss ) ) ) ).

% snth_sset_smerge
thf(fact_94_ev__prefix__sdrop,axiom,
    ! [P3: rule > $o,Xs2: stream_rule] :
      ( ( linear292482573288640205v_rule @ ( linear7354821883049305950s_rule @ P3 ) @ Xs2 )
     => ? [N: nat] :
          ( ( list_all_rule
            @ ^ [Xs: rule] :
                ~ ( P3 @ Xs )
            @ ( stake_rule @ N @ Xs2 ) )
          & ( linear7354821883049305950s_rule @ P3 @ ( sdrop_rule @ N @ Xs2 ) ) ) ) ).

% ev_prefix_sdrop
thf(fact_95_ev__prefix__sdrop,axiom,
    ! [P3: produc340336539035504054m_rule > $o,Xs2: stream2709947120125613254m_rule] :
      ( ( linear1816559245158234865m_rule @ ( linear7925859179493631106m_rule @ P3 ) @ Xs2 )
     => ? [N: nat] :
          ( ( list_a9038620553111681026m_rule
            @ ^ [Xs: produc340336539035504054m_rule] :
                ~ ( P3 @ Xs )
            @ ( stake_5421812949518764133m_rule @ N @ Xs2 ) )
          & ( linear7925859179493631106m_rule @ P3 @ ( sdrop_8169176516188972301m_rule @ N @ Xs2 ) ) ) ) ).

% ev_prefix_sdrop
thf(fact_96_sdrop__szip,axiom,
    ! [N2: nat,S1: stream2709947120125613254m_rule,S2: stream2709947120125613254m_rule] :
      ( ( sdrop_7192298464603511222m_rule @ N2 @ ( szip_P811719526838699976m_rule @ S1 @ S2 ) )
      = ( szip_P811719526838699976m_rule @ ( sdrop_8169176516188972301m_rule @ N2 @ S1 ) @ ( sdrop_8169176516188972301m_rule @ N2 @ S2 ) ) ) ).

% sdrop_szip
thf(fact_97_sdrop__szip,axiom,
    ! [N2: nat,S1: stream2709947120125613254m_rule,S2: stream_rule] :
      ( ( sdrop_1938960342593938834e_rule @ N2 @ ( szip_P2044787555563963556e_rule @ S1 @ S2 ) )
      = ( szip_P2044787555563963556e_rule @ ( sdrop_8169176516188972301m_rule @ N2 @ S1 ) @ ( sdrop_rule @ N2 @ S2 ) ) ) ).

% sdrop_szip
thf(fact_98_sdrop__szip,axiom,
    ! [N2: nat,S1: stream_rule,S2: stream2709947120125613254m_rule] :
      ( ( sdrop_746751008794826386m_rule @ N2 @ ( szip_r6533552703345879972m_rule @ S1 @ S2 ) )
      = ( szip_r6533552703345879972m_rule @ ( sdrop_rule @ N2 @ S1 ) @ ( sdrop_8169176516188972301m_rule @ N2 @ S2 ) ) ) ).

% sdrop_szip
thf(fact_99_sdrop__szip,axiom,
    ! [N2: nat,S1: stream_rule,S2: stream_rule] :
      ( ( sdrop_9113879250048157294e_rule @ N2 @ ( szip_rule_rule @ S1 @ S2 ) )
      = ( szip_rule_rule @ ( sdrop_rule @ N2 @ S1 ) @ ( sdrop_rule @ N2 @ S2 ) ) ) ).

% sdrop_szip
thf(fact_100_sdrop__szip,axiom,
    ! [N2: nat,S1: stream4408948924543953275ist_fm,S2: stream_rule] :
      ( ( sdrop_8169176516188972301m_rule @ N2 @ ( szip_P2924820683901490861m_rule @ S1 @ S2 ) )
      = ( szip_P2924820683901490861m_rule @ ( sdrop_9176333610110415838ist_fm @ N2 @ S1 ) @ ( sdrop_rule @ N2 @ S2 ) ) ) ).

% sdrop_szip
thf(fact_101_sdrop__wait,axiom,
    ! [Phi: stream_rule > $o,Xs2: stream_rule] :
      ( ( linear292482573288640205v_rule @ Phi @ Xs2 )
     => ( Phi @ ( sdrop_rule @ ( linear2710171048243638257t_rule @ Phi @ Xs2 ) @ Xs2 ) ) ) ).

% sdrop_wait
thf(fact_102_sdrop__wait,axiom,
    ! [Phi: stream2709947120125613254m_rule > $o,Xs2: stream2709947120125613254m_rule] :
      ( ( linear1816559245158234865m_rule @ Phi @ Xs2 )
     => ( Phi @ ( sdrop_8169176516188972301m_rule @ ( linear8417479883454578709m_rule @ Phi @ Xs2 ) @ Xs2 ) ) ) ).

% sdrop_wait
thf(fact_103_Stream_Osmember__def,axiom,
    ( smember_list_fm
    = ( ^ [X3: list_fm,S3: stream_list_fm] : ( member_list_fm3 @ X3 @ ( sset_list_fm @ S3 ) ) ) ) ).

% Stream.smember_def
thf(fact_104_Stream_Osmember__def,axiom,
    ( smember_fm
    = ( ^ [X3: fm,S3: stream_fm] : ( member_fm3 @ X3 @ ( sset_fm @ S3 ) ) ) ) ).

% Stream.smember_def
thf(fact_105_Stream_Osmember__def,axiom,
    ( smember_o
    = ( ^ [X3: $o,S3: stream_o] : ( member_o3 @ X3 @ ( sset_o @ S3 ) ) ) ) ).

% Stream.smember_def
thf(fact_106_Stream_Osmember__def,axiom,
    ( smember_nat
    = ( ^ [X3: nat,S3: stream_nat] : ( member_nat3 @ X3 @ ( sset_nat @ S3 ) ) ) ) ).

% Stream.smember_def
thf(fact_107_Stream_Osmember__def,axiom,
    ( smembe4439892024482649336m_rule
    = ( ^ [X3: produc340336539035504054m_rule,S3: stream2709947120125613254m_rule] : ( member7231649785386036813m_rule @ X3 @ ( sset_P4484857331586881186m_rule @ S3 ) ) ) ) ).

% Stream.smember_def
thf(fact_108_Stream_Osmember__def,axiom,
    ( smember_rule
    = ( ^ [X3: rule,S3: stream_rule] : ( member_rule3 @ X3 @ ( sset_rule @ S3 ) ) ) ) ).

% Stream.smember_def
thf(fact_109_sset__shift,axiom,
    ! [Xs2: list_fm,S: stream_fm] :
      ( ( sset_fm @ ( shift_fm @ Xs2 @ S ) )
      = ( sup_sup_set_fm @ ( set_fm2 @ Xs2 ) @ ( sset_fm @ S ) ) ) ).

% sset_shift
thf(fact_110_sset__shift,axiom,
    ! [Xs2: list_list_fm,S: stream_list_fm] :
      ( ( sset_list_fm @ ( shift_list_fm @ Xs2 @ S ) )
      = ( sup_sup_set_list_fm @ ( set_list_fm2 @ Xs2 ) @ ( sset_list_fm @ S ) ) ) ).

% sset_shift
thf(fact_111_sset__shift,axiom,
    ! [Xs2: list_tm,S: stream_tm] :
      ( ( sset_tm @ ( shift_tm @ Xs2 @ S ) )
      = ( sup_sup_set_tm @ ( set_tm2 @ Xs2 ) @ ( sset_tm @ S ) ) ) ).

% sset_shift
thf(fact_112_sset__shift,axiom,
    ! [Xs2: list_P2774625669004309958m_rule,S: stream2709947120125613254m_rule] :
      ( ( sset_P4484857331586881186m_rule @ ( shift_2334844276748245581m_rule @ Xs2 @ S ) )
      = ( sup_su6946459741510085528m_rule @ ( set_Pr4534715572506550497m_rule @ Xs2 ) @ ( sset_P4484857331586881186m_rule @ S ) ) ) ).

% sset_shift
thf(fact_113_sset__shift,axiom,
    ! [Xs2: list_rule,S: stream_rule] :
      ( ( sset_rule @ ( shift_rule @ Xs2 @ S ) )
      = ( sup_sup_set_rule @ ( set_rule2 @ Xs2 ) @ ( sset_rule @ S ) ) ) ).

% sset_shift
thf(fact_114_sset__shift,axiom,
    ! [Xs2: list_nat,S: stream_nat] :
      ( ( sset_nat @ ( shift_nat @ Xs2 @ S ) )
      = ( sup_sup_set_nat @ ( set_nat2 @ Xs2 ) @ ( sset_nat @ S ) ) ) ).

% sset_shift
thf(fact_115_in__set__member,axiom,
    ! [X: rule,Xs2: list_rule] :
      ( ( member_rule3 @ X @ ( set_rule2 @ Xs2 ) )
      = ( member_rule @ Xs2 @ X ) ) ).

% in_set_member
thf(fact_116_in__set__member,axiom,
    ! [X: $o,Xs2: list_o] :
      ( ( member_o3 @ X @ ( set_o2 @ Xs2 ) )
      = ( member_o @ Xs2 @ X ) ) ).

% in_set_member
thf(fact_117_in__set__member,axiom,
    ! [X: nat,Xs2: list_nat] :
      ( ( member_nat3 @ X @ ( set_nat2 @ Xs2 ) )
      = ( member_nat @ Xs2 @ X ) ) ).

% in_set_member
thf(fact_118_in__set__member,axiom,
    ! [X: fm,Xs2: list_fm] :
      ( ( member_fm3 @ X @ ( set_fm2 @ Xs2 ) )
      = ( member_fm @ Xs2 @ X ) ) ).

% in_set_member
thf(fact_119_in__set__member,axiom,
    ! [X: list_fm,Xs2: list_list_fm] :
      ( ( member_list_fm3 @ X @ ( set_list_fm2 @ Xs2 ) )
      = ( member_list_fm @ Xs2 @ X ) ) ).

% in_set_member
thf(fact_120_in__set__member,axiom,
    ! [X: tm,Xs2: list_tm] :
      ( ( member_tm3 @ X @ ( set_tm2 @ Xs2 ) )
      = ( member_tm @ Xs2 @ X ) ) ).

% in_set_member
thf(fact_121_epath__preserves__unaffected,axiom,
    ! [P: fm,Steps: stream2709947120125613254m_rule,Pre2: list_P2774625669004309958m_rule,Suf2: stream2709947120125613254m_rule] :
      ( ( member_fm3 @ P @ ( set_fm2 @ ( pseq @ ( shd_Pr4562317740776619530m_rule @ Steps ) ) ) )
     => ( ( abstra6789711989322986974ist_fm @ eff @ rules @ Steps )
       => ( ( Steps
            = ( shift_2334844276748245581m_rule @ Pre2 @ Suf2 ) )
         => ( ( list_a9038620553111681026m_rule
              @ ^ [Xs: produc340336539035504054m_rule] :
                  ~ ( affects @ ( produc6514711208036230658m_rule @ Xs ) @ P )
              @ Pre2 )
           => ( member_fm3 @ P @ ( set_fm2 @ ( pseq @ ( shd_Pr4562317740776619530m_rule @ Suf2 ) ) ) ) ) ) ) ) ).

% epath_preserves_unaffected
thf(fact_122_tree__fms__def,axiom,
    ( tree_fms
    = ( ^ [Steps2: stream2709947120125613254m_rule] :
          ( comple2134933779557159616set_fm
          @ ( image_377115208147135190set_fm
            @ ^ [Ss2: produc340336539035504054m_rule] : ( set_fm2 @ ( pseq @ Ss2 ) )
            @ ( sset_P4484857331586881186m_rule @ Steps2 ) ) ) ) ) ).

% tree_fms_def
thf(fact_123_stream__all__iff,axiom,
    ( stream2134157564947672044m_rule
    = ( ^ [P4: produc340336539035504054m_rule > $o,S3: stream2709947120125613254m_rule] :
        ! [X3: produc340336539035504054m_rule] :
          ( ( member7231649785386036813m_rule @ X3 @ ( sset_P4484857331586881186m_rule @ S3 ) )
         => ( P4 @ X3 ) ) ) ) ).

% stream_all_iff
thf(fact_124_stream__all__iff,axiom,
    ( stream_all_rule
    = ( ^ [P4: rule > $o,S3: stream_rule] :
        ! [X3: rule] :
          ( ( member_rule3 @ X3 @ ( sset_rule @ S3 ) )
         => ( P4 @ X3 ) ) ) ) ).

% stream_all_iff
thf(fact_125_parts__preserves__unaffected,axiom,
    ! [R: rule,P: fm,Z3: list_fm,A3: list_tm] :
      ( ~ ( affects @ R @ P )
     => ( ( member_list_fm3 @ Z3 @ ( set_list_fm2 @ ( parts @ A3 @ R @ P ) ) )
       => ( member_fm3 @ P @ ( set_fm2 @ Z3 ) ) ) ) ).

% parts_preserves_unaffected
thf(fact_126_assms_I1_J,axiom,
    abstra6789711989322986974ist_fm @ eff @ rules @ steps ).

% assms(1)
thf(fact_127_assms_I2_J,axiom,
    abstra6097777249025082867ist_fm @ eff @ rules @ steps ).

% assms(2)
thf(fact_128_rules__repeat__sdrop,axiom,
    ! [K: nat,R: rule] :
    ? [N: nat] :
      ( ( snth_rule @ ( sdrop_rule @ K @ rules ) @ N )
      = R ) ).

% rules_repeat_sdrop
thf(fact_129_epath__sdrop,axiom,
    ! [Steps: stream2709947120125613254m_rule,N2: nat] :
      ( ( abstra6789711989322986974ist_fm @ eff @ rules @ Steps )
     => ( abstra6789711989322986974ist_fm @ eff @ rules @ ( sdrop_8169176516188972301m_rule @ N2 @ Steps ) ) ) ).

% epath_sdrop
thf(fact_130_list_Opred__set,axiom,
    ( list_all_fm
    = ( ^ [P4: fm > $o,X3: list_fm] :
        ! [Y2: fm] :
          ( ( member_fm3 @ Y2 @ ( set_fm2 @ X3 ) )
         => ( P4 @ Y2 ) ) ) ) ).

% list.pred_set
thf(fact_131_list_Opred__set,axiom,
    ( list_all_list_fm
    = ( ^ [P4: list_fm > $o,X3: list_list_fm] :
        ! [Y2: list_fm] :
          ( ( member_list_fm3 @ Y2 @ ( set_list_fm2 @ X3 ) )
         => ( P4 @ Y2 ) ) ) ) ).

% list.pred_set
thf(fact_132_list_Opred__set,axiom,
    ( list_all_tm
    = ( ^ [P4: tm > $o,X3: list_tm] :
        ! [Y2: tm] :
          ( ( member_tm3 @ Y2 @ ( set_tm2 @ X3 ) )
         => ( P4 @ Y2 ) ) ) ) ).

% list.pred_set
thf(fact_133_list_Opred__set,axiom,
    ( list_a9038620553111681026m_rule
    = ( ^ [P4: produc340336539035504054m_rule > $o,X3: list_P2774625669004309958m_rule] :
        ! [Y2: produc340336539035504054m_rule] :
          ( ( member7231649785386036813m_rule @ Y2 @ ( set_Pr4534715572506550497m_rule @ X3 ) )
         => ( P4 @ Y2 ) ) ) ) ).

% list.pred_set
thf(fact_134_list__all__iff,axiom,
    ( list_all_fm
    = ( ^ [P4: fm > $o,X3: list_fm] :
        ! [Y2: fm] :
          ( ( member_fm3 @ Y2 @ ( set_fm2 @ X3 ) )
         => ( P4 @ Y2 ) ) ) ) ).

% list_all_iff
thf(fact_135_list__all__iff,axiom,
    ( list_all_list_fm
    = ( ^ [P4: list_fm > $o,X3: list_list_fm] :
        ! [Y2: list_fm] :
          ( ( member_list_fm3 @ Y2 @ ( set_list_fm2 @ X3 ) )
         => ( P4 @ Y2 ) ) ) ) ).

% list_all_iff
thf(fact_136_list__all__iff,axiom,
    ( list_all_tm
    = ( ^ [P4: tm > $o,X3: list_tm] :
        ! [Y2: tm] :
          ( ( member_tm3 @ Y2 @ ( set_tm2 @ X3 ) )
         => ( P4 @ Y2 ) ) ) ) ).

% list_all_iff
thf(fact_137_list__all__iff,axiom,
    ( list_a9038620553111681026m_rule
    = ( ^ [P4: produc340336539035504054m_rule > $o,X3: list_P2774625669004309958m_rule] :
        ! [Y2: produc340336539035504054m_rule] :
          ( ( member7231649785386036813m_rule @ Y2 @ ( set_Pr4534715572506550497m_rule @ X3 ) )
         => ( P4 @ Y2 ) ) ) ) ).

% list_all_iff
thf(fact_138_Ball__set__list__all,axiom,
    ! [Xs2: list_fm,P3: fm > $o] :
      ( ( ! [X3: fm] :
            ( ( member_fm3 @ X3 @ ( set_fm2 @ Xs2 ) )
           => ( P3 @ X3 ) ) )
      = ( list_all_fm @ P3 @ Xs2 ) ) ).

% Ball_set_list_all
thf(fact_139_Ball__set__list__all,axiom,
    ! [Xs2: list_list_fm,P3: list_fm > $o] :
      ( ( ! [X3: list_fm] :
            ( ( member_list_fm3 @ X3 @ ( set_list_fm2 @ Xs2 ) )
           => ( P3 @ X3 ) ) )
      = ( list_all_list_fm @ P3 @ Xs2 ) ) ).

% Ball_set_list_all
thf(fact_140_Ball__set__list__all,axiom,
    ! [Xs2: list_tm,P3: tm > $o] :
      ( ( ! [X3: tm] :
            ( ( member_tm3 @ X3 @ ( set_tm2 @ Xs2 ) )
           => ( P3 @ X3 ) ) )
      = ( list_all_tm @ P3 @ Xs2 ) ) ).

% Ball_set_list_all
thf(fact_141_Ball__set__list__all,axiom,
    ! [Xs2: list_P2774625669004309958m_rule,P3: produc340336539035504054m_rule > $o] :
      ( ( ! [X3: produc340336539035504054m_rule] :
            ( ( member7231649785386036813m_rule @ X3 @ ( set_Pr4534715572506550497m_rule @ Xs2 ) )
           => ( P3 @ X3 ) ) )
      = ( list_a9038620553111681026m_rule @ P3 @ Xs2 ) ) ).

% Ball_set_list_all
thf(fact_142_stake__sdrop,axiom,
    ! [N2: nat,S: stream_rule] :
      ( ( shift_rule @ ( stake_rule @ N2 @ S ) @ ( sdrop_rule @ N2 @ S ) )
      = S ) ).

% stake_sdrop
thf(fact_143_stake__sdrop,axiom,
    ! [N2: nat,S: stream2709947120125613254m_rule] :
      ( ( shift_2334844276748245581m_rule @ ( stake_5421812949518764133m_rule @ N2 @ S ) @ ( sdrop_8169176516188972301m_rule @ N2 @ S ) )
      = S ) ).

% stake_sdrop
thf(fact_144_UN__Un,axiom,
    ! [M3: nat > set_nat,A3: set_nat,B: set_nat] :
      ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ M3 @ ( sup_sup_set_nat @ A3 @ B ) ) )
      = ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ M3 @ A3 ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ M3 @ B ) ) ) ) ).

% UN_Un
thf(fact_145_UN__Un,axiom,
    ! [M3: produc340336539035504054m_rule > set_fm,A3: set_Pr1822751329126368876m_rule,B: set_Pr1822751329126368876m_rule] :
      ( ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ M3 @ ( sup_su6946459741510085528m_rule @ A3 @ B ) ) )
      = ( sup_sup_set_fm @ ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ M3 @ A3 ) ) @ ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ M3 @ B ) ) ) ) ).

% UN_Un
thf(fact_146_UN__Un,axiom,
    ! [M3: nat > set_fm,A3: set_nat,B: set_nat] :
      ( ( comple2134933779557159616set_fm @ ( image_nat_set_fm @ M3 @ ( sup_sup_set_nat @ A3 @ B ) ) )
      = ( sup_sup_set_fm @ ( comple2134933779557159616set_fm @ ( image_nat_set_fm @ M3 @ A3 ) ) @ ( comple2134933779557159616set_fm @ ( image_nat_set_fm @ M3 @ B ) ) ) ) ).

% UN_Un
thf(fact_147_UN__I,axiom,
    ! [A: rule,A3: set_rule,B2: rule,B: rule > set_rule] :
      ( ( member_rule3 @ A @ A3 )
     => ( ( member_rule3 @ B2 @ ( B @ A ) )
       => ( member_rule3 @ B2 @ ( comple7665822446944348827t_rule @ ( image_rule_set_rule @ B @ A3 ) ) ) ) ) ).

% UN_I
thf(fact_148_UN__I,axiom,
    ! [A: rule,A3: set_rule,B2: $o,B: rule > set_o] :
      ( ( member_rule3 @ A @ A3 )
     => ( ( member_o3 @ B2 @ ( B @ A ) )
       => ( member_o3 @ B2 @ ( comple90263536869209701_set_o @ ( image_rule_set_o @ B @ A3 ) ) ) ) ) ).

% UN_I
thf(fact_149_UN__I,axiom,
    ! [A: rule,A3: set_rule,B2: nat,B: rule > set_nat] :
      ( ( member_rule3 @ A @ A3 )
     => ( ( member_nat3 @ B2 @ ( B @ A ) )
       => ( member_nat3 @ B2 @ ( comple7399068483239264473et_nat @ ( image_rule_set_nat @ B @ A3 ) ) ) ) ) ).

% UN_I
thf(fact_150_UN__I,axiom,
    ! [A: fm,A3: set_fm,B2: rule,B: fm > set_rule] :
      ( ( member_fm3 @ A @ A3 )
     => ( ( member_rule3 @ B2 @ ( B @ A ) )
       => ( member_rule3 @ B2 @ ( comple7665822446944348827t_rule @ ( image_fm_set_rule @ B @ A3 ) ) ) ) ) ).

% UN_I
thf(fact_151_UN__I,axiom,
    ! [A: fm,A3: set_fm,B2: $o,B: fm > set_o] :
      ( ( member_fm3 @ A @ A3 )
     => ( ( member_o3 @ B2 @ ( B @ A ) )
       => ( member_o3 @ B2 @ ( comple90263536869209701_set_o @ ( image_fm_set_o @ B @ A3 ) ) ) ) ) ).

% UN_I
thf(fact_152_UN__I,axiom,
    ! [A: fm,A3: set_fm,B2: nat,B: fm > set_nat] :
      ( ( member_fm3 @ A @ A3 )
     => ( ( member_nat3 @ B2 @ ( B @ A ) )
       => ( member_nat3 @ B2 @ ( comple7399068483239264473et_nat @ ( image_fm_set_nat @ B @ A3 ) ) ) ) ) ).

% UN_I
thf(fact_153_UN__I,axiom,
    ! [A: $o,A3: set_o,B2: rule,B: $o > set_rule] :
      ( ( member_o3 @ A @ A3 )
     => ( ( member_rule3 @ B2 @ ( B @ A ) )
       => ( member_rule3 @ B2 @ ( comple7665822446944348827t_rule @ ( image_o_set_rule @ B @ A3 ) ) ) ) ) ).

% UN_I
thf(fact_154_UN__I,axiom,
    ! [A: $o,A3: set_o,B2: $o,B: $o > set_o] :
      ( ( member_o3 @ A @ A3 )
     => ( ( member_o3 @ B2 @ ( B @ A ) )
       => ( member_o3 @ B2 @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B @ A3 ) ) ) ) ) ).

% UN_I
thf(fact_155_UN__I,axiom,
    ! [A: $o,A3: set_o,B2: nat,B: $o > set_nat] :
      ( ( member_o3 @ A @ A3 )
     => ( ( member_nat3 @ B2 @ ( B @ A ) )
       => ( member_nat3 @ B2 @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ B @ A3 ) ) ) ) ) ).

% UN_I
thf(fact_156_UN__I,axiom,
    ! [A: nat,A3: set_nat,B2: rule,B: nat > set_rule] :
      ( ( member_nat3 @ A @ A3 )
     => ( ( member_rule3 @ B2 @ ( B @ A ) )
       => ( member_rule3 @ B2 @ ( comple7665822446944348827t_rule @ ( image_nat_set_rule @ B @ A3 ) ) ) ) ) ).

% UN_I
thf(fact_157_UN__iff,axiom,
    ! [B2: fm,B: produc340336539035504054m_rule > set_fm,A3: set_Pr1822751329126368876m_rule] :
      ( ( member_fm3 @ B2 @ ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ B @ A3 ) ) )
      = ( ? [X3: produc340336539035504054m_rule] :
            ( ( member7231649785386036813m_rule @ X3 @ A3 )
            & ( member_fm3 @ B2 @ ( B @ X3 ) ) ) ) ) ).

% UN_iff
thf(fact_158_SUP__identity__eq,axiom,
    ! [A3: set_set_fm] :
      ( ( comple2134933779557159616set_fm
        @ ( image_set_fm_set_fm
          @ ^ [X3: set_fm] : X3
          @ A3 ) )
      = ( comple2134933779557159616set_fm @ A3 ) ) ).

% SUP_identity_eq
thf(fact_159_SUP__identity__eq,axiom,
    ! [A3: set_o] :
      ( ( complete_Sup_Sup_o
        @ ( image_o_o
          @ ^ [X3: $o] : X3
          @ A3 ) )
      = ( complete_Sup_Sup_o @ A3 ) ) ).

% SUP_identity_eq
thf(fact_160_SUP__identity__eq,axiom,
    ! [A3: set_nat] :
      ( ( complete_Sup_Sup_nat
        @ ( image_nat_nat
          @ ^ [X3: nat] : X3
          @ A3 ) )
      = ( complete_Sup_Sup_nat @ A3 ) ) ).

% SUP_identity_eq
thf(fact_161_UN__ball__bex__simps_I4_J,axiom,
    ! [B: produc340336539035504054m_rule > set_fm,A3: set_Pr1822751329126368876m_rule,P3: fm > $o] :
      ( ( ? [X3: fm] :
            ( ( member_fm3 @ X3 @ ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ B @ A3 ) ) )
            & ( P3 @ X3 ) ) )
      = ( ? [X3: produc340336539035504054m_rule] :
            ( ( member7231649785386036813m_rule @ X3 @ A3 )
            & ? [Y2: fm] :
                ( ( member_fm3 @ Y2 @ ( B @ X3 ) )
                & ( P3 @ Y2 ) ) ) ) ) ).

% UN_ball_bex_simps(4)
thf(fact_162_UN__ball__bex__simps_I2_J,axiom,
    ! [B: produc340336539035504054m_rule > set_fm,A3: set_Pr1822751329126368876m_rule,P3: fm > $o] :
      ( ( ! [X3: fm] :
            ( ( member_fm3 @ X3 @ ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ B @ A3 ) ) )
           => ( P3 @ X3 ) ) )
      = ( ! [X3: produc340336539035504054m_rule] :
            ( ( member7231649785386036813m_rule @ X3 @ A3 )
           => ! [Y2: fm] :
                ( ( member_fm3 @ Y2 @ ( B @ X3 ) )
               => ( P3 @ Y2 ) ) ) ) ) ).

% UN_ball_bex_simps(2)
thf(fact_163_bex__UN,axiom,
    ! [B: produc340336539035504054m_rule > set_fm,A3: set_Pr1822751329126368876m_rule,P3: fm > $o] :
      ( ( ? [X3: fm] :
            ( ( member_fm3 @ X3 @ ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ B @ A3 ) ) )
            & ( P3 @ X3 ) ) )
      = ( ? [X3: produc340336539035504054m_rule] :
            ( ( member7231649785386036813m_rule @ X3 @ A3 )
            & ? [Y2: fm] :
                ( ( member_fm3 @ Y2 @ ( B @ X3 ) )
                & ( P3 @ Y2 ) ) ) ) ) ).

% bex_UN
thf(fact_164_ball__UN,axiom,
    ! [B: produc340336539035504054m_rule > set_fm,A3: set_Pr1822751329126368876m_rule,P3: fm > $o] :
      ( ( ! [X3: fm] :
            ( ( member_fm3 @ X3 @ ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ B @ A3 ) ) )
           => ( P3 @ X3 ) ) )
      = ( ! [X3: produc340336539035504054m_rule] :
            ( ( member7231649785386036813m_rule @ X3 @ A3 )
           => ! [Y2: fm] :
                ( ( member_fm3 @ Y2 @ ( B @ X3 ) )
               => ( P3 @ Y2 ) ) ) ) ) ).

% ball_UN
thf(fact_165_Union__Un__distrib,axiom,
    ! [A3: set_set_nat,B: set_set_nat] :
      ( ( comple7399068483239264473et_nat @ ( sup_sup_set_set_nat @ A3 @ B ) )
      = ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ A3 ) @ ( comple7399068483239264473et_nat @ B ) ) ) ).

% Union_Un_distrib
thf(fact_166_Union__Un__distrib,axiom,
    ! [A3: set_set_fm,B: set_set_fm] :
      ( ( comple2134933779557159616set_fm @ ( sup_sup_set_set_fm @ A3 @ B ) )
      = ( sup_sup_set_fm @ ( comple2134933779557159616set_fm @ A3 ) @ ( comple2134933779557159616set_fm @ B ) ) ) ).

% Union_Un_distrib
thf(fact_167_SUP__UNION,axiom,
    ! [F: produc340336539035504054m_rule > set_fm,G: produc340336539035504054m_rule > set_Pr1822751329126368876m_rule,A3: set_Pr1822751329126368876m_rule] :
      ( ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ F @ ( comple5648871819318716351m_rule @ ( image_2683796897779739177m_rule @ G @ A3 ) ) ) )
      = ( comple2134933779557159616set_fm
        @ ( image_377115208147135190set_fm
          @ ^ [Y2: produc340336539035504054m_rule] : ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ F @ ( G @ Y2 ) ) )
          @ A3 ) ) ) ).

% SUP_UNION
thf(fact_168_SUP__UNION,axiom,
    ! [F: fm > set_fm,G: produc340336539035504054m_rule > set_fm,A3: set_Pr1822751329126368876m_rule] :
      ( ( comple2134933779557159616set_fm @ ( image_fm_set_fm @ F @ ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ G @ A3 ) ) ) )
      = ( comple2134933779557159616set_fm
        @ ( image_377115208147135190set_fm
          @ ^ [Y2: produc340336539035504054m_rule] : ( comple2134933779557159616set_fm @ ( image_fm_set_fm @ F @ ( G @ Y2 ) ) )
          @ A3 ) ) ) ).

% SUP_UNION
thf(fact_169_SUP__UNION,axiom,
    ! [F: fm > $o,G: produc340336539035504054m_rule > set_fm,A3: set_Pr1822751329126368876m_rule] :
      ( ( complete_Sup_Sup_o @ ( image_fm_o @ F @ ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ G @ A3 ) ) ) )
      = ( complete_Sup_Sup_o
        @ ( image_8969478372357302939rule_o
          @ ^ [Y2: produc340336539035504054m_rule] : ( complete_Sup_Sup_o @ ( image_fm_o @ F @ ( G @ Y2 ) ) )
          @ A3 ) ) ) ).

% SUP_UNION
thf(fact_170_Union__iff,axiom,
    ! [A3: rule,C: set_set_rule] :
      ( ( member_rule3 @ A3 @ ( comple7665822446944348827t_rule @ C ) )
      = ( ? [X3: set_rule] :
            ( ( member_set_rule @ X3 @ C )
            & ( member_rule3 @ A3 @ X3 ) ) ) ) ).

% Union_iff
thf(fact_171_Union__iff,axiom,
    ! [A3: list_fm,C: set_set_list_fm] :
      ( ( member_list_fm3 @ A3 @ ( comple8784269564784259782ist_fm @ C ) )
      = ( ? [X3: set_list_fm] :
            ( ( member_set_list_fm @ X3 @ C )
            & ( member_list_fm3 @ A3 @ X3 ) ) ) ) ).

% Union_iff
thf(fact_172_Union__iff,axiom,
    ! [A3: $o,C: set_set_o] :
      ( ( member_o3 @ A3 @ ( comple90263536869209701_set_o @ C ) )
      = ( ? [X3: set_o] :
            ( ( member_set_o @ X3 @ C )
            & ( member_o3 @ A3 @ X3 ) ) ) ) ).

% Union_iff
thf(fact_173_Union__iff,axiom,
    ! [A3: nat,C: set_set_nat] :
      ( ( member_nat3 @ A3 @ ( comple7399068483239264473et_nat @ C ) )
      = ( ? [X3: set_nat] :
            ( ( member_set_nat @ X3 @ C )
            & ( member_nat3 @ A3 @ X3 ) ) ) ) ).

% Union_iff
thf(fact_174_Union__iff,axiom,
    ! [A3: fm,C: set_set_fm] :
      ( ( member_fm3 @ A3 @ ( comple2134933779557159616set_fm @ C ) )
      = ( ? [X3: set_fm] :
            ( ( member_set_fm @ X3 @ C )
            & ( member_fm3 @ A3 @ X3 ) ) ) ) ).

% Union_iff
thf(fact_175_UnionI,axiom,
    ! [X4: set_rule,C: set_set_rule,A3: rule] :
      ( ( member_set_rule @ X4 @ C )
     => ( ( member_rule3 @ A3 @ X4 )
       => ( member_rule3 @ A3 @ ( comple7665822446944348827t_rule @ C ) ) ) ) ).

% UnionI
thf(fact_176_UnionI,axiom,
    ! [X4: set_list_fm,C: set_set_list_fm,A3: list_fm] :
      ( ( member_set_list_fm @ X4 @ C )
     => ( ( member_list_fm3 @ A3 @ X4 )
       => ( member_list_fm3 @ A3 @ ( comple8784269564784259782ist_fm @ C ) ) ) ) ).

% UnionI
thf(fact_177_UnionI,axiom,
    ! [X4: set_o,C: set_set_o,A3: $o] :
      ( ( member_set_o @ X4 @ C )
     => ( ( member_o3 @ A3 @ X4 )
       => ( member_o3 @ A3 @ ( comple90263536869209701_set_o @ C ) ) ) ) ).

% UnionI
thf(fact_178_UnionI,axiom,
    ! [X4: set_nat,C: set_set_nat,A3: nat] :
      ( ( member_set_nat @ X4 @ C )
     => ( ( member_nat3 @ A3 @ X4 )
       => ( member_nat3 @ A3 @ ( comple7399068483239264473et_nat @ C ) ) ) ) ).

% UnionI
thf(fact_179_UnionI,axiom,
    ! [X4: set_fm,C: set_set_fm,A3: fm] :
      ( ( member_set_fm @ X4 @ C )
     => ( ( member_fm3 @ A3 @ X4 )
       => ( member_fm3 @ A3 @ ( comple2134933779557159616set_fm @ C ) ) ) ) ).

% UnionI
thf(fact_180_UN__ball__bex__simps_I1_J,axiom,
    ! [A3: set_set_fm,P3: fm > $o] :
      ( ( ! [X3: fm] :
            ( ( member_fm3 @ X3 @ ( comple2134933779557159616set_fm @ A3 ) )
           => ( P3 @ X3 ) ) )
      = ( ! [X3: set_fm] :
            ( ( member_set_fm @ X3 @ A3 )
           => ! [Y2: fm] :
                ( ( member_fm3 @ Y2 @ X3 )
               => ( P3 @ Y2 ) ) ) ) ) ).

% UN_ball_bex_simps(1)
thf(fact_181_UN__ball__bex__simps_I3_J,axiom,
    ! [A3: set_set_fm,P3: fm > $o] :
      ( ( ? [X3: fm] :
            ( ( member_fm3 @ X3 @ ( comple2134933779557159616set_fm @ A3 ) )
            & ( P3 @ X3 ) ) )
      = ( ? [X3: set_fm] :
            ( ( member_set_fm @ X3 @ A3 )
            & ? [Y2: fm] :
                ( ( member_fm3 @ Y2 @ X3 )
                & ( P3 @ Y2 ) ) ) ) ) ).

% UN_ball_bex_simps(3)
thf(fact_182_Sup__set__def,axiom,
    ( comple7665822446944348827t_rule
    = ( ^ [A4: set_set_rule] :
          ( collect_rule
          @ ^ [X3: rule] : ( complete_Sup_Sup_o @ ( image_set_rule_o @ ( member_rule3 @ X3 ) @ A4 ) ) ) ) ) ).

% Sup_set_def
thf(fact_183_Sup__set__def,axiom,
    ( comple8784269564784259782ist_fm
    = ( ^ [A4: set_set_list_fm] :
          ( collect_list_fm
          @ ^ [X3: list_fm] : ( complete_Sup_Sup_o @ ( image_set_list_fm_o @ ( member_list_fm3 @ X3 ) @ A4 ) ) ) ) ) ).

% Sup_set_def
thf(fact_184_Sup__set__def,axiom,
    ( comple90263536869209701_set_o
    = ( ^ [A4: set_set_o] :
          ( collect_o
          @ ^ [X3: $o] : ( complete_Sup_Sup_o @ ( image_set_o_o @ ( member_o3 @ X3 ) @ A4 ) ) ) ) ) ).

% Sup_set_def
thf(fact_185_Sup__set__def,axiom,
    ( comple7399068483239264473et_nat
    = ( ^ [A4: set_set_nat] :
          ( collect_nat
          @ ^ [X3: nat] : ( complete_Sup_Sup_o @ ( image_set_nat_o @ ( member_nat3 @ X3 ) @ A4 ) ) ) ) ) ).

% Sup_set_def
thf(fact_186_Sup__set__def,axiom,
    ( comple2134933779557159616set_fm
    = ( ^ [A4: set_set_fm] :
          ( collect_fm
          @ ^ [X3: fm] : ( complete_Sup_Sup_o @ ( image_set_fm_o @ ( member_fm3 @ X3 ) @ A4 ) ) ) ) ) ).

% Sup_set_def
thf(fact_187_Sup_OSUP__cong,axiom,
    ! [A3: set_Pr1822751329126368876m_rule,B: set_Pr1822751329126368876m_rule,C: produc340336539035504054m_rule > set_fm,D: produc340336539035504054m_rule > set_fm,Sup: set_set_fm > set_fm] :
      ( ( A3 = B )
     => ( ! [X2: produc340336539035504054m_rule] :
            ( ( member7231649785386036813m_rule @ X2 @ B )
           => ( ( C @ X2 )
              = ( D @ X2 ) ) )
       => ( ( Sup @ ( image_377115208147135190set_fm @ C @ A3 ) )
          = ( Sup @ ( image_377115208147135190set_fm @ D @ B ) ) ) ) ) ).

% Sup.SUP_cong
thf(fact_188_Sup_OSUP__cong,axiom,
    ! [A3: set_nat,B: set_nat,C: nat > nat,D: nat > nat,Sup: set_nat > nat] :
      ( ( A3 = B )
     => ( ! [X2: nat] :
            ( ( member_nat3 @ X2 @ B )
           => ( ( C @ X2 )
              = ( D @ X2 ) ) )
       => ( ( Sup @ ( image_nat_nat @ C @ A3 ) )
          = ( Sup @ ( image_nat_nat @ D @ B ) ) ) ) ) ).

% Sup.SUP_cong
thf(fact_189_Inf_OINF__cong,axiom,
    ! [A3: set_Pr1822751329126368876m_rule,B: set_Pr1822751329126368876m_rule,C: produc340336539035504054m_rule > set_fm,D: produc340336539035504054m_rule > set_fm,Inf: set_set_fm > set_fm] :
      ( ( A3 = B )
     => ( ! [X2: produc340336539035504054m_rule] :
            ( ( member7231649785386036813m_rule @ X2 @ B )
           => ( ( C @ X2 )
              = ( D @ X2 ) ) )
       => ( ( Inf @ ( image_377115208147135190set_fm @ C @ A3 ) )
          = ( Inf @ ( image_377115208147135190set_fm @ D @ B ) ) ) ) ) ).

% Inf.INF_cong
thf(fact_190_Inf_OINF__cong,axiom,
    ! [A3: set_nat,B: set_nat,C: nat > nat,D: nat > nat,Inf: set_nat > nat] :
      ( ( A3 = B )
     => ( ! [X2: nat] :
            ( ( member_nat3 @ X2 @ B )
           => ( ( C @ X2 )
              = ( D @ X2 ) ) )
       => ( ( Inf @ ( image_nat_nat @ C @ A3 ) )
          = ( Inf @ ( image_nat_nat @ D @ B ) ) ) ) ) ).

% Inf.INF_cong
thf(fact_191_UnionE,axiom,
    ! [A3: rule,C: set_set_rule] :
      ( ( member_rule3 @ A3 @ ( comple7665822446944348827t_rule @ C ) )
     => ~ ! [X5: set_rule] :
            ( ( member_rule3 @ A3 @ X5 )
           => ~ ( member_set_rule @ X5 @ C ) ) ) ).

% UnionE
thf(fact_192_UnionE,axiom,
    ! [A3: list_fm,C: set_set_list_fm] :
      ( ( member_list_fm3 @ A3 @ ( comple8784269564784259782ist_fm @ C ) )
     => ~ ! [X5: set_list_fm] :
            ( ( member_list_fm3 @ A3 @ X5 )
           => ~ ( member_set_list_fm @ X5 @ C ) ) ) ).

% UnionE
thf(fact_193_UnionE,axiom,
    ! [A3: $o,C: set_set_o] :
      ( ( member_o3 @ A3 @ ( comple90263536869209701_set_o @ C ) )
     => ~ ! [X5: set_o] :
            ( ( member_o3 @ A3 @ X5 )
           => ~ ( member_set_o @ X5 @ C ) ) ) ).

% UnionE
thf(fact_194_UnionE,axiom,
    ! [A3: nat,C: set_set_nat] :
      ( ( member_nat3 @ A3 @ ( comple7399068483239264473et_nat @ C ) )
     => ~ ! [X5: set_nat] :
            ( ( member_nat3 @ A3 @ X5 )
           => ~ ( member_set_nat @ X5 @ C ) ) ) ).

% UnionE
thf(fact_195_UnionE,axiom,
    ! [A3: fm,C: set_set_fm] :
      ( ( member_fm3 @ A3 @ ( comple2134933779557159616set_fm @ C ) )
     => ~ ! [X5: set_fm] :
            ( ( member_fm3 @ A3 @ X5 )
           => ~ ( member_set_fm @ X5 @ C ) ) ) ).

% UnionE
thf(fact_196_sset__smerge,axiom,
    ! [Ss: stream3752074346242807894m_rule] :
      ( ( sset_P4484857331586881186m_rule @ ( smerge193809993764105000m_rule @ Ss ) )
      = ( comple5648871819318716351m_rule @ ( image_3045026045495828537m_rule @ sset_P4484857331586881186m_rule @ ( sset_s1746085050419710898m_rule @ Ss ) ) ) ) ).

% sset_smerge
thf(fact_197_sset__smerge,axiom,
    ! [Ss: stream_stream_rule] :
      ( ( sset_rule @ ( smerge_rule @ Ss ) )
      = ( comple7665822446944348827t_rule @ ( image_1703615012366172913t_rule @ sset_rule @ ( sset_stream_rule @ Ss ) ) ) ) ).

% sset_smerge
thf(fact_198_sset__smerge,axiom,
    ! [Ss: stream_stream_fm] :
      ( ( sset_fm @ ( smerge_fm @ Ss ) )
      = ( comple2134933779557159616set_fm @ ( image_7285378133382788895set_fm @ sset_fm @ ( sset_stream_fm @ Ss ) ) ) ) ).

% sset_smerge
thf(fact_199_Sup_OSUP__identity__eq,axiom,
    ! [Sup: set_nat > nat,A3: set_nat] :
      ( ( Sup
        @ ( image_nat_nat
          @ ^ [X3: nat] : X3
          @ A3 ) )
      = ( Sup @ A3 ) ) ).

% Sup.SUP_identity_eq
thf(fact_200_Inf_OINF__identity__eq,axiom,
    ! [Inf: set_nat > nat,A3: set_nat] :
      ( ( Inf
        @ ( image_nat_nat
          @ ^ [X3: nat] : X3
          @ A3 ) )
      = ( Inf @ A3 ) ) ).

% Inf.INF_identity_eq
thf(fact_201_Saturated__sdrop,axiom,
    ! [Steps: stream2709947120125613254m_rule,N2: nat] :
      ( ( abstra6097777249025082867ist_fm @ eff @ rules @ Steps )
     => ( abstra6097777249025082867ist_fm @ eff @ rules @ ( sdrop_8169176516188972301m_rule @ N2 @ Steps ) ) ) ).

% Saturated_sdrop
thf(fact_202_SUP__cong,axiom,
    ! [A3: set_rule,B: set_rule,C: rule > $o,D: rule > $o] :
      ( ( A3 = B )
     => ( ! [X2: rule] :
            ( ( member_rule3 @ X2 @ B )
           => ( ( C @ X2 )
              = ( D @ X2 ) ) )
       => ( ( complete_Sup_Sup_o @ ( image_rule_o @ C @ A3 ) )
          = ( complete_Sup_Sup_o @ ( image_rule_o @ D @ B ) ) ) ) ) ).

% SUP_cong
thf(fact_203_SUP__cong,axiom,
    ! [A3: set_fm,B: set_fm,C: fm > $o,D: fm > $o] :
      ( ( A3 = B )
     => ( ! [X2: fm] :
            ( ( member_fm3 @ X2 @ B )
           => ( ( C @ X2 )
              = ( D @ X2 ) ) )
       => ( ( complete_Sup_Sup_o @ ( image_fm_o @ C @ A3 ) )
          = ( complete_Sup_Sup_o @ ( image_fm_o @ D @ B ) ) ) ) ) ).

% SUP_cong
thf(fact_204_SUP__cong,axiom,
    ! [A3: set_o,B: set_o,C: $o > $o,D: $o > $o] :
      ( ( A3 = B )
     => ( ! [X2: $o] :
            ( ( member_o3 @ X2 @ B )
           => ( ( C @ X2 )
              = ( D @ X2 ) ) )
       => ( ( complete_Sup_Sup_o @ ( image_o_o @ C @ A3 ) )
          = ( complete_Sup_Sup_o @ ( image_o_o @ D @ B ) ) ) ) ) ).

% SUP_cong
thf(fact_205_SUP__cong,axiom,
    ! [A3: set_nat,B: set_nat,C: nat > $o,D: nat > $o] :
      ( ( A3 = B )
     => ( ! [X2: nat] :
            ( ( member_nat3 @ X2 @ B )
           => ( ( C @ X2 )
              = ( D @ X2 ) ) )
       => ( ( complete_Sup_Sup_o @ ( image_nat_o @ C @ A3 ) )
          = ( complete_Sup_Sup_o @ ( image_nat_o @ D @ B ) ) ) ) ) ).

% SUP_cong
thf(fact_206_SUP__cong,axiom,
    ! [A3: set_rule,B: set_rule,C: rule > nat,D: rule > nat] :
      ( ( A3 = B )
     => ( ! [X2: rule] :
            ( ( member_rule3 @ X2 @ B )
           => ( ( C @ X2 )
              = ( D @ X2 ) ) )
       => ( ( complete_Sup_Sup_nat @ ( image_rule_nat @ C @ A3 ) )
          = ( complete_Sup_Sup_nat @ ( image_rule_nat @ D @ B ) ) ) ) ) ).

% SUP_cong
thf(fact_207_SUP__cong,axiom,
    ! [A3: set_fm,B: set_fm,C: fm > nat,D: fm > nat] :
      ( ( A3 = B )
     => ( ! [X2: fm] :
            ( ( member_fm3 @ X2 @ B )
           => ( ( C @ X2 )
              = ( D @ X2 ) ) )
       => ( ( complete_Sup_Sup_nat @ ( image_fm_nat @ C @ A3 ) )
          = ( complete_Sup_Sup_nat @ ( image_fm_nat @ D @ B ) ) ) ) ) ).

% SUP_cong
thf(fact_208_SUP__cong,axiom,
    ! [A3: set_o,B: set_o,C: $o > nat,D: $o > nat] :
      ( ( A3 = B )
     => ( ! [X2: $o] :
            ( ( member_o3 @ X2 @ B )
           => ( ( C @ X2 )
              = ( D @ X2 ) ) )
       => ( ( complete_Sup_Sup_nat @ ( image_o_nat @ C @ A3 ) )
          = ( complete_Sup_Sup_nat @ ( image_o_nat @ D @ B ) ) ) ) ) ).

% SUP_cong
thf(fact_209_SUP__cong,axiom,
    ! [A3: set_nat,B: set_nat,C: nat > nat,D: nat > nat] :
      ( ( A3 = B )
     => ( ! [X2: nat] :
            ( ( member_nat3 @ X2 @ B )
           => ( ( C @ X2 )
              = ( D @ X2 ) ) )
       => ( ( complete_Sup_Sup_nat @ ( image_nat_nat @ C @ A3 ) )
          = ( complete_Sup_Sup_nat @ ( image_nat_nat @ D @ B ) ) ) ) ) ).

% SUP_cong
thf(fact_210_SUP__cong,axiom,
    ! [A3: set_rule,B: set_rule,C: rule > set_fm,D: rule > set_fm] :
      ( ( A3 = B )
     => ( ! [X2: rule] :
            ( ( member_rule3 @ X2 @ B )
           => ( ( C @ X2 )
              = ( D @ X2 ) ) )
       => ( ( comple2134933779557159616set_fm @ ( image_rule_set_fm @ C @ A3 ) )
          = ( comple2134933779557159616set_fm @ ( image_rule_set_fm @ D @ B ) ) ) ) ) ).

% SUP_cong
thf(fact_211_SUP__cong,axiom,
    ! [A3: set_fm,B: set_fm,C: fm > set_fm,D: fm > set_fm] :
      ( ( A3 = B )
     => ( ! [X2: fm] :
            ( ( member_fm3 @ X2 @ B )
           => ( ( C @ X2 )
              = ( D @ X2 ) ) )
       => ( ( comple2134933779557159616set_fm @ ( image_fm_set_fm @ C @ A3 ) )
          = ( comple2134933779557159616set_fm @ ( image_fm_set_fm @ D @ B ) ) ) ) ) ).

% SUP_cong
thf(fact_212_SUP__commute,axiom,
    ! [F: produc340336539035504054m_rule > produc340336539035504054m_rule > set_fm,B: set_Pr1822751329126368876m_rule,A3: set_Pr1822751329126368876m_rule] :
      ( ( comple2134933779557159616set_fm
        @ ( image_377115208147135190set_fm
          @ ^ [I: produc340336539035504054m_rule] : ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ ( F @ I ) @ B ) )
          @ A3 ) )
      = ( comple2134933779557159616set_fm
        @ ( image_377115208147135190set_fm
          @ ^ [J: produc340336539035504054m_rule] :
              ( comple2134933779557159616set_fm
              @ ( image_377115208147135190set_fm
                @ ^ [I: produc340336539035504054m_rule] : ( F @ I @ J )
                @ A3 ) )
          @ B ) ) ) ).

% SUP_commute
thf(fact_213_image__Union,axiom,
    ! [F: produc340336539035504054m_rule > set_fm,S4: set_se4309406418413213730m_rule] :
      ( ( image_377115208147135190set_fm @ F @ ( comple5648871819318716351m_rule @ S4 ) )
      = ( comple4326961365146087328set_fm @ ( image_4873481958868436096set_fm @ ( image_377115208147135190set_fm @ F ) @ S4 ) ) ) ).

% image_Union
thf(fact_214_image__Union,axiom,
    ! [F: nat > nat,S4: set_set_nat] :
      ( ( image_nat_nat @ F @ ( comple7399068483239264473et_nat @ S4 ) )
      = ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ ( image_nat_nat @ F ) @ S4 ) ) ) ).

% image_Union
thf(fact_215_image__Union,axiom,
    ! [F: fm > fm,S4: set_set_fm] :
      ( ( image_fm_fm @ F @ ( comple2134933779557159616set_fm @ S4 ) )
      = ( comple2134933779557159616set_fm @ ( image_set_fm_set_fm @ ( image_fm_fm @ F ) @ S4 ) ) ) ).

% image_Union
thf(fact_216_UN__UN__flatten,axiom,
    ! [C: produc340336539035504054m_rule > set_fm,B: produc340336539035504054m_rule > set_Pr1822751329126368876m_rule,A3: set_Pr1822751329126368876m_rule] :
      ( ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ C @ ( comple5648871819318716351m_rule @ ( image_2683796897779739177m_rule @ B @ A3 ) ) ) )
      = ( comple2134933779557159616set_fm
        @ ( image_377115208147135190set_fm
          @ ^ [Y2: produc340336539035504054m_rule] : ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ C @ ( B @ Y2 ) ) )
          @ A3 ) ) ) ).

% UN_UN_flatten
thf(fact_217_UN__UN__flatten,axiom,
    ! [C: fm > set_fm,B: produc340336539035504054m_rule > set_fm,A3: set_Pr1822751329126368876m_rule] :
      ( ( comple2134933779557159616set_fm @ ( image_fm_set_fm @ C @ ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ B @ A3 ) ) ) )
      = ( comple2134933779557159616set_fm
        @ ( image_377115208147135190set_fm
          @ ^ [Y2: produc340336539035504054m_rule] : ( comple2134933779557159616set_fm @ ( image_fm_set_fm @ C @ ( B @ Y2 ) ) )
          @ A3 ) ) ) ).

% UN_UN_flatten
thf(fact_218_UN__E,axiom,
    ! [B2: rule,B: rule > set_rule,A3: set_rule] :
      ( ( member_rule3 @ B2 @ ( comple7665822446944348827t_rule @ ( image_rule_set_rule @ B @ A3 ) ) )
     => ~ ! [X2: rule] :
            ( ( member_rule3 @ X2 @ A3 )
           => ~ ( member_rule3 @ B2 @ ( B @ X2 ) ) ) ) ).

% UN_E
thf(fact_219_UN__E,axiom,
    ! [B2: rule,B: fm > set_rule,A3: set_fm] :
      ( ( member_rule3 @ B2 @ ( comple7665822446944348827t_rule @ ( image_fm_set_rule @ B @ A3 ) ) )
     => ~ ! [X2: fm] :
            ( ( member_fm3 @ X2 @ A3 )
           => ~ ( member_rule3 @ B2 @ ( B @ X2 ) ) ) ) ).

% UN_E
thf(fact_220_UN__E,axiom,
    ! [B2: rule,B: $o > set_rule,A3: set_o] :
      ( ( member_rule3 @ B2 @ ( comple7665822446944348827t_rule @ ( image_o_set_rule @ B @ A3 ) ) )
     => ~ ! [X2: $o] :
            ( ( member_o3 @ X2 @ A3 )
           => ~ ( member_rule3 @ B2 @ ( B @ X2 ) ) ) ) ).

% UN_E
thf(fact_221_UN__E,axiom,
    ! [B2: rule,B: nat > set_rule,A3: set_nat] :
      ( ( member_rule3 @ B2 @ ( comple7665822446944348827t_rule @ ( image_nat_set_rule @ B @ A3 ) ) )
     => ~ ! [X2: nat] :
            ( ( member_nat3 @ X2 @ A3 )
           => ~ ( member_rule3 @ B2 @ ( B @ X2 ) ) ) ) ).

% UN_E
thf(fact_222_UN__E,axiom,
    ! [B2: $o,B: rule > set_o,A3: set_rule] :
      ( ( member_o3 @ B2 @ ( comple90263536869209701_set_o @ ( image_rule_set_o @ B @ A3 ) ) )
     => ~ ! [X2: rule] :
            ( ( member_rule3 @ X2 @ A3 )
           => ~ ( member_o3 @ B2 @ ( B @ X2 ) ) ) ) ).

% UN_E
thf(fact_223_UN__E,axiom,
    ! [B2: $o,B: fm > set_o,A3: set_fm] :
      ( ( member_o3 @ B2 @ ( comple90263536869209701_set_o @ ( image_fm_set_o @ B @ A3 ) ) )
     => ~ ! [X2: fm] :
            ( ( member_fm3 @ X2 @ A3 )
           => ~ ( member_o3 @ B2 @ ( B @ X2 ) ) ) ) ).

% UN_E
thf(fact_224_UN__E,axiom,
    ! [B2: $o,B: $o > set_o,A3: set_o] :
      ( ( member_o3 @ B2 @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B @ A3 ) ) )
     => ~ ! [X2: $o] :
            ( ( member_o3 @ X2 @ A3 )
           => ~ ( member_o3 @ B2 @ ( B @ X2 ) ) ) ) ).

% UN_E
thf(fact_225_UN__E,axiom,
    ! [B2: $o,B: nat > set_o,A3: set_nat] :
      ( ( member_o3 @ B2 @ ( comple90263536869209701_set_o @ ( image_nat_set_o @ B @ A3 ) ) )
     => ~ ! [X2: nat] :
            ( ( member_nat3 @ X2 @ A3 )
           => ~ ( member_o3 @ B2 @ ( B @ X2 ) ) ) ) ).

% UN_E
thf(fact_226_UN__E,axiom,
    ! [B2: nat,B: rule > set_nat,A3: set_rule] :
      ( ( member_nat3 @ B2 @ ( comple7399068483239264473et_nat @ ( image_rule_set_nat @ B @ A3 ) ) )
     => ~ ! [X2: rule] :
            ( ( member_rule3 @ X2 @ A3 )
           => ~ ( member_nat3 @ B2 @ ( B @ X2 ) ) ) ) ).

% UN_E
thf(fact_227_UN__E,axiom,
    ! [B2: nat,B: fm > set_nat,A3: set_fm] :
      ( ( member_nat3 @ B2 @ ( comple7399068483239264473et_nat @ ( image_fm_set_nat @ B @ A3 ) ) )
     => ~ ! [X2: fm] :
            ( ( member_fm3 @ X2 @ A3 )
           => ~ ( member_nat3 @ B2 @ ( B @ X2 ) ) ) ) ).

% UN_E
thf(fact_228_UN__extend__simps_I8_J,axiom,
    ! [B: produc340336539035504054m_rule > set_fm,A3: set_se4309406418413213730m_rule] :
      ( ( comple2134933779557159616set_fm
        @ ( image_338663757272164256set_fm
          @ ^ [Y2: set_Pr1822751329126368876m_rule] : ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ B @ Y2 ) )
          @ A3 ) )
      = ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ B @ ( comple5648871819318716351m_rule @ A3 ) ) ) ) ).

% UN_extend_simps(8)
thf(fact_229_UN__extend__simps_I8_J,axiom,
    ! [B: fm > set_fm,A3: set_set_fm] :
      ( ( comple2134933779557159616set_fm
        @ ( image_set_fm_set_fm
          @ ^ [Y2: set_fm] : ( comple2134933779557159616set_fm @ ( image_fm_set_fm @ B @ Y2 ) )
          @ A3 ) )
      = ( comple2134933779557159616set_fm @ ( image_fm_set_fm @ B @ ( comple2134933779557159616set_fm @ A3 ) ) ) ) ).

% UN_extend_simps(8)
thf(fact_230_UN__extend__simps_I9_J,axiom,
    ! [C: produc340336539035504054m_rule > set_fm,B: produc340336539035504054m_rule > set_Pr1822751329126368876m_rule,A3: set_Pr1822751329126368876m_rule] :
      ( ( comple2134933779557159616set_fm
        @ ( image_377115208147135190set_fm
          @ ^ [X3: produc340336539035504054m_rule] : ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ C @ ( B @ X3 ) ) )
          @ A3 ) )
      = ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ C @ ( comple5648871819318716351m_rule @ ( image_2683796897779739177m_rule @ B @ A3 ) ) ) ) ) ).

% UN_extend_simps(9)
thf(fact_231_UN__extend__simps_I9_J,axiom,
    ! [C: fm > set_fm,B: produc340336539035504054m_rule > set_fm,A3: set_Pr1822751329126368876m_rule] :
      ( ( comple2134933779557159616set_fm
        @ ( image_377115208147135190set_fm
          @ ^ [X3: produc340336539035504054m_rule] : ( comple2134933779557159616set_fm @ ( image_fm_set_fm @ C @ ( B @ X3 ) ) )
          @ A3 ) )
      = ( comple2134933779557159616set_fm @ ( image_fm_set_fm @ C @ ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ B @ A3 ) ) ) ) ) ).

% UN_extend_simps(9)
thf(fact_232_Sup__union__distrib,axiom,
    ! [A3: set_set_nat,B: set_set_nat] :
      ( ( comple7399068483239264473et_nat @ ( sup_sup_set_set_nat @ A3 @ B ) )
      = ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ A3 ) @ ( comple7399068483239264473et_nat @ B ) ) ) ).

% Sup_union_distrib
thf(fact_233_Sup__union__distrib,axiom,
    ! [A3: set_set_fm,B: set_set_fm] :
      ( ( comple2134933779557159616set_fm @ ( sup_sup_set_set_fm @ A3 @ B ) )
      = ( sup_sup_set_fm @ ( comple2134933779557159616set_fm @ A3 ) @ ( comple2134933779557159616set_fm @ B ) ) ) ).

% Sup_union_distrib
thf(fact_234_Sup__union__distrib,axiom,
    ! [A3: set_o,B: set_o] :
      ( ( complete_Sup_Sup_o @ ( sup_sup_set_o @ A3 @ B ) )
      = ( sup_sup_o @ ( complete_Sup_Sup_o @ A3 ) @ ( complete_Sup_Sup_o @ B ) ) ) ).

% Sup_union_distrib
thf(fact_235_complete__lattice__class_OSUP__sup__distrib,axiom,
    ! [F: produc340336539035504054m_rule > set_fm,A3: set_Pr1822751329126368876m_rule,G: produc340336539035504054m_rule > set_fm] :
      ( ( sup_sup_set_fm @ ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ F @ A3 ) ) @ ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ G @ A3 ) ) )
      = ( comple2134933779557159616set_fm
        @ ( image_377115208147135190set_fm
          @ ^ [A2: produc340336539035504054m_rule] : ( sup_sup_set_fm @ ( F @ A2 ) @ ( G @ A2 ) )
          @ A3 ) ) ) ).

% complete_lattice_class.SUP_sup_distrib
thf(fact_236_SUP__absorb,axiom,
    ! [K: rule,I2: set_rule,A3: rule > $o] :
      ( ( member_rule3 @ K @ I2 )
     => ( ( sup_sup_o @ ( A3 @ K ) @ ( complete_Sup_Sup_o @ ( image_rule_o @ A3 @ I2 ) ) )
        = ( complete_Sup_Sup_o @ ( image_rule_o @ A3 @ I2 ) ) ) ) ).

% SUP_absorb
thf(fact_237_SUP__absorb,axiom,
    ! [K: fm,I2: set_fm,A3: fm > $o] :
      ( ( member_fm3 @ K @ I2 )
     => ( ( sup_sup_o @ ( A3 @ K ) @ ( complete_Sup_Sup_o @ ( image_fm_o @ A3 @ I2 ) ) )
        = ( complete_Sup_Sup_o @ ( image_fm_o @ A3 @ I2 ) ) ) ) ).

% SUP_absorb
thf(fact_238_SUP__absorb,axiom,
    ! [K: $o,I2: set_o,A3: $o > $o] :
      ( ( member_o3 @ K @ I2 )
     => ( ( sup_sup_o @ ( A3 @ K ) @ ( complete_Sup_Sup_o @ ( image_o_o @ A3 @ I2 ) ) )
        = ( complete_Sup_Sup_o @ ( image_o_o @ A3 @ I2 ) ) ) ) ).

% SUP_absorb
thf(fact_239_SUP__absorb,axiom,
    ! [K: nat,I2: set_nat,A3: nat > $o] :
      ( ( member_nat3 @ K @ I2 )
     => ( ( sup_sup_o @ ( A3 @ K ) @ ( complete_Sup_Sup_o @ ( image_nat_o @ A3 @ I2 ) ) )
        = ( complete_Sup_Sup_o @ ( image_nat_o @ A3 @ I2 ) ) ) ) ).

% SUP_absorb
thf(fact_240_SUP__absorb,axiom,
    ! [K: rule,I2: set_rule,A3: rule > set_nat] :
      ( ( member_rule3 @ K @ I2 )
     => ( ( sup_sup_set_nat @ ( A3 @ K ) @ ( comple7399068483239264473et_nat @ ( image_rule_set_nat @ A3 @ I2 ) ) )
        = ( comple7399068483239264473et_nat @ ( image_rule_set_nat @ A3 @ I2 ) ) ) ) ).

% SUP_absorb
thf(fact_241_SUP__absorb,axiom,
    ! [K: fm,I2: set_fm,A3: fm > set_nat] :
      ( ( member_fm3 @ K @ I2 )
     => ( ( sup_sup_set_nat @ ( A3 @ K ) @ ( comple7399068483239264473et_nat @ ( image_fm_set_nat @ A3 @ I2 ) ) )
        = ( comple7399068483239264473et_nat @ ( image_fm_set_nat @ A3 @ I2 ) ) ) ) ).

% SUP_absorb
thf(fact_242_SUP__absorb,axiom,
    ! [K: $o,I2: set_o,A3: $o > set_nat] :
      ( ( member_o3 @ K @ I2 )
     => ( ( sup_sup_set_nat @ ( A3 @ K ) @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ A3 @ I2 ) ) )
        = ( comple7399068483239264473et_nat @ ( image_o_set_nat @ A3 @ I2 ) ) ) ) ).

% SUP_absorb
thf(fact_243_SUP__absorb,axiom,
    ! [K: nat,I2: set_nat,A3: nat > set_nat] :
      ( ( member_nat3 @ K @ I2 )
     => ( ( sup_sup_set_nat @ ( A3 @ K ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A3 @ I2 ) ) )
        = ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A3 @ I2 ) ) ) ) ).

% SUP_absorb
thf(fact_244_SUP__absorb,axiom,
    ! [K: rule,I2: set_rule,A3: rule > set_fm] :
      ( ( member_rule3 @ K @ I2 )
     => ( ( sup_sup_set_fm @ ( A3 @ K ) @ ( comple2134933779557159616set_fm @ ( image_rule_set_fm @ A3 @ I2 ) ) )
        = ( comple2134933779557159616set_fm @ ( image_rule_set_fm @ A3 @ I2 ) ) ) ) ).

% SUP_absorb
thf(fact_245_SUP__absorb,axiom,
    ! [K: fm,I2: set_fm,A3: fm > set_fm] :
      ( ( member_fm3 @ K @ I2 )
     => ( ( sup_sup_set_fm @ ( A3 @ K ) @ ( comple2134933779557159616set_fm @ ( image_fm_set_fm @ A3 @ I2 ) ) )
        = ( comple2134933779557159616set_fm @ ( image_fm_set_fm @ A3 @ I2 ) ) ) ) ).

% SUP_absorb
thf(fact_246_UN__extend__simps_I10_J,axiom,
    ! [B: nat > set_fm,F: nat > nat,A3: set_nat] :
      ( ( comple2134933779557159616set_fm
        @ ( image_nat_set_fm
          @ ^ [A2: nat] : ( B @ ( F @ A2 ) )
          @ A3 ) )
      = ( comple2134933779557159616set_fm @ ( image_nat_set_fm @ B @ ( image_nat_nat @ F @ A3 ) ) ) ) ).

% UN_extend_simps(10)
thf(fact_247_UN__extend__simps_I10_J,axiom,
    ! [B: set_fm > set_fm,F: produc340336539035504054m_rule > set_fm,A3: set_Pr1822751329126368876m_rule] :
      ( ( comple2134933779557159616set_fm
        @ ( image_377115208147135190set_fm
          @ ^ [A2: produc340336539035504054m_rule] : ( B @ ( F @ A2 ) )
          @ A3 ) )
      = ( comple2134933779557159616set_fm @ ( image_set_fm_set_fm @ B @ ( image_377115208147135190set_fm @ F @ A3 ) ) ) ) ).

% UN_extend_simps(10)
thf(fact_248_UN__extend__simps_I10_J,axiom,
    ! [B: produc340336539035504054m_rule > set_fm,F: produc340336539035504054m_rule > produc340336539035504054m_rule,A3: set_Pr1822751329126368876m_rule] :
      ( ( comple2134933779557159616set_fm
        @ ( image_377115208147135190set_fm
          @ ^ [A2: produc340336539035504054m_rule] : ( B @ ( F @ A2 ) )
          @ A3 ) )
      = ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ B @ ( image_2498997284783760755m_rule @ F @ A3 ) ) ) ) ).

% UN_extend_simps(10)
thf(fact_249_image__UN,axiom,
    ! [F: fm > fm,B: produc340336539035504054m_rule > set_fm,A3: set_Pr1822751329126368876m_rule] :
      ( ( image_fm_fm @ F @ ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ B @ A3 ) ) )
      = ( comple2134933779557159616set_fm
        @ ( image_377115208147135190set_fm
          @ ^ [X3: produc340336539035504054m_rule] : ( image_fm_fm @ F @ ( B @ X3 ) )
          @ A3 ) ) ) ).

% image_UN
thf(fact_250_Un__Union__image,axiom,
    ! [A3: produc340336539035504054m_rule > set_fm,B: produc340336539035504054m_rule > set_fm,C: set_Pr1822751329126368876m_rule] :
      ( ( comple2134933779557159616set_fm
        @ ( image_377115208147135190set_fm
          @ ^ [X3: produc340336539035504054m_rule] : ( sup_sup_set_fm @ ( A3 @ X3 ) @ ( B @ X3 ) )
          @ C ) )
      = ( sup_sup_set_fm @ ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ A3 @ C ) ) @ ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ B @ C ) ) ) ) ).

% Un_Union_image
thf(fact_251_UN__Un__distrib,axiom,
    ! [A3: produc340336539035504054m_rule > set_fm,B: produc340336539035504054m_rule > set_fm,I2: set_Pr1822751329126368876m_rule] :
      ( ( comple2134933779557159616set_fm
        @ ( image_377115208147135190set_fm
          @ ^ [I: produc340336539035504054m_rule] : ( sup_sup_set_fm @ ( A3 @ I ) @ ( B @ I ) )
          @ I2 ) )
      = ( sup_sup_set_fm @ ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ A3 @ I2 ) ) @ ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ B @ I2 ) ) ) ) ).

% UN_Un_distrib
thf(fact_252_UN__absorb,axiom,
    ! [K: rule,I2: set_rule,A3: rule > set_nat] :
      ( ( member_rule3 @ K @ I2 )
     => ( ( sup_sup_set_nat @ ( A3 @ K ) @ ( comple7399068483239264473et_nat @ ( image_rule_set_nat @ A3 @ I2 ) ) )
        = ( comple7399068483239264473et_nat @ ( image_rule_set_nat @ A3 @ I2 ) ) ) ) ).

% UN_absorb
thf(fact_253_UN__absorb,axiom,
    ! [K: fm,I2: set_fm,A3: fm > set_nat] :
      ( ( member_fm3 @ K @ I2 )
     => ( ( sup_sup_set_nat @ ( A3 @ K ) @ ( comple7399068483239264473et_nat @ ( image_fm_set_nat @ A3 @ I2 ) ) )
        = ( comple7399068483239264473et_nat @ ( image_fm_set_nat @ A3 @ I2 ) ) ) ) ).

% UN_absorb
thf(fact_254_UN__absorb,axiom,
    ! [K: $o,I2: set_o,A3: $o > set_nat] :
      ( ( member_o3 @ K @ I2 )
     => ( ( sup_sup_set_nat @ ( A3 @ K ) @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ A3 @ I2 ) ) )
        = ( comple7399068483239264473et_nat @ ( image_o_set_nat @ A3 @ I2 ) ) ) ) ).

% UN_absorb
thf(fact_255_UN__absorb,axiom,
    ! [K: nat,I2: set_nat,A3: nat > set_nat] :
      ( ( member_nat3 @ K @ I2 )
     => ( ( sup_sup_set_nat @ ( A3 @ K ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A3 @ I2 ) ) )
        = ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A3 @ I2 ) ) ) ) ).

% UN_absorb
thf(fact_256_UN__absorb,axiom,
    ! [K: rule,I2: set_rule,A3: rule > set_fm] :
      ( ( member_rule3 @ K @ I2 )
     => ( ( sup_sup_set_fm @ ( A3 @ K ) @ ( comple2134933779557159616set_fm @ ( image_rule_set_fm @ A3 @ I2 ) ) )
        = ( comple2134933779557159616set_fm @ ( image_rule_set_fm @ A3 @ I2 ) ) ) ) ).

% UN_absorb
thf(fact_257_UN__absorb,axiom,
    ! [K: fm,I2: set_fm,A3: fm > set_fm] :
      ( ( member_fm3 @ K @ I2 )
     => ( ( sup_sup_set_fm @ ( A3 @ K ) @ ( comple2134933779557159616set_fm @ ( image_fm_set_fm @ A3 @ I2 ) ) )
        = ( comple2134933779557159616set_fm @ ( image_fm_set_fm @ A3 @ I2 ) ) ) ) ).

% UN_absorb
thf(fact_258_UN__absorb,axiom,
    ! [K: $o,I2: set_o,A3: $o > set_fm] :
      ( ( member_o3 @ K @ I2 )
     => ( ( sup_sup_set_fm @ ( A3 @ K ) @ ( comple2134933779557159616set_fm @ ( image_o_set_fm @ A3 @ I2 ) ) )
        = ( comple2134933779557159616set_fm @ ( image_o_set_fm @ A3 @ I2 ) ) ) ) ).

% UN_absorb
thf(fact_259_UN__absorb,axiom,
    ! [K: nat,I2: set_nat,A3: nat > set_fm] :
      ( ( member_nat3 @ K @ I2 )
     => ( ( sup_sup_set_fm @ ( A3 @ K ) @ ( comple2134933779557159616set_fm @ ( image_nat_set_fm @ A3 @ I2 ) ) )
        = ( comple2134933779557159616set_fm @ ( image_nat_set_fm @ A3 @ I2 ) ) ) ) ).

% UN_absorb
thf(fact_260_UN__absorb,axiom,
    ! [K: list_fm,I2: set_list_fm,A3: list_fm > set_nat] :
      ( ( member_list_fm3 @ K @ I2 )
     => ( ( sup_sup_set_nat @ ( A3 @ K ) @ ( comple7399068483239264473et_nat @ ( image_3154564810619571642et_nat @ A3 @ I2 ) ) )
        = ( comple7399068483239264473et_nat @ ( image_3154564810619571642et_nat @ A3 @ I2 ) ) ) ) ).

% UN_absorb
thf(fact_261_UN__absorb,axiom,
    ! [K: list_fm,I2: set_list_fm,A3: list_fm > set_fm] :
      ( ( member_list_fm3 @ K @ I2 )
     => ( ( sup_sup_set_fm @ ( A3 @ K ) @ ( comple2134933779557159616set_fm @ ( image_list_fm_set_fm @ A3 @ I2 ) ) )
        = ( comple2134933779557159616set_fm @ ( image_list_fm_set_fm @ A3 @ I2 ) ) ) ) ).

% UN_absorb
thf(fact_262_Saturated__ev__rule,axiom,
    ! [Steps: stream2709947120125613254m_rule,R: rule,N2: nat] :
      ( ( abstra6097777249025082867ist_fm @ eff @ rules @ Steps )
     => ( linear1816559245158234865m_rule
        @ ( linear7925859179493631106m_rule
          @ ^ [Step: produc340336539035504054m_rule] :
              ( ( produc6514711208036230658m_rule @ Step )
              = R ) )
        @ ( sdrop_8169176516188972301m_rule @ N2 @ Steps ) ) ) ).

% Saturated_ev_rule
thf(fact_263_SUP__union,axiom,
    ! [M3: nat > set_nat,A3: set_nat,B: set_nat] :
      ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ M3 @ ( sup_sup_set_nat @ A3 @ B ) ) )
      = ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ M3 @ A3 ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ M3 @ B ) ) ) ) ).

% SUP_union
thf(fact_264_SUP__union,axiom,
    ! [M3: produc340336539035504054m_rule > set_fm,A3: set_Pr1822751329126368876m_rule,B: set_Pr1822751329126368876m_rule] :
      ( ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ M3 @ ( sup_su6946459741510085528m_rule @ A3 @ B ) ) )
      = ( sup_sup_set_fm @ ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ M3 @ A3 ) ) @ ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ M3 @ B ) ) ) ) ).

% SUP_union
thf(fact_265_SUP__union,axiom,
    ! [M3: nat > set_fm,A3: set_nat,B: set_nat] :
      ( ( comple2134933779557159616set_fm @ ( image_nat_set_fm @ M3 @ ( sup_sup_set_nat @ A3 @ B ) ) )
      = ( sup_sup_set_fm @ ( comple2134933779557159616set_fm @ ( image_nat_set_fm @ M3 @ A3 ) ) @ ( comple2134933779557159616set_fm @ ( image_nat_set_fm @ M3 @ B ) ) ) ) ).

% SUP_union
thf(fact_266_SUP__union,axiom,
    ! [M3: nat > $o,A3: set_nat,B: set_nat] :
      ( ( complete_Sup_Sup_o @ ( image_nat_o @ M3 @ ( sup_sup_set_nat @ A3 @ B ) ) )
      = ( sup_sup_o @ ( complete_Sup_Sup_o @ ( image_nat_o @ M3 @ A3 ) ) @ ( complete_Sup_Sup_o @ ( image_nat_o @ M3 @ B ) ) ) ) ).

% SUP_union
thf(fact_267_set__union,axiom,
    ! [Xs2: list_fm,Ys: list_fm] :
      ( ( set_fm2 @ ( union_fm @ Xs2 @ Ys ) )
      = ( sup_sup_set_fm @ ( set_fm2 @ Xs2 ) @ ( set_fm2 @ Ys ) ) ) ).

% set_union
thf(fact_268_set__union,axiom,
    ! [Xs2: list_list_fm,Ys: list_list_fm] :
      ( ( set_list_fm2 @ ( union_list_fm @ Xs2 @ Ys ) )
      = ( sup_sup_set_list_fm @ ( set_list_fm2 @ Xs2 ) @ ( set_list_fm2 @ Ys ) ) ) ).

% set_union
thf(fact_269_set__union,axiom,
    ! [Xs2: list_tm,Ys: list_tm] :
      ( ( set_tm2 @ ( union_tm @ Xs2 @ Ys ) )
      = ( sup_sup_set_tm @ ( set_tm2 @ Xs2 ) @ ( set_tm2 @ Ys ) ) ) ).

% set_union
thf(fact_270_set__union,axiom,
    ! [Xs2: list_nat,Ys: list_nat] :
      ( ( set_nat2 @ ( union_nat @ Xs2 @ Ys ) )
      = ( sup_sup_set_nat @ ( set_nat2 @ Xs2 ) @ ( set_nat2 @ Ys ) ) ) ).

% set_union
thf(fact_271_image__ident,axiom,
    ! [Y3: set_nat] :
      ( ( image_nat_nat
        @ ^ [X3: nat] : X3
        @ Y3 )
      = Y3 ) ).

% image_ident
thf(fact_272_Un__iff,axiom,
    ! [C2: rule,A3: set_rule,B: set_rule] :
      ( ( member_rule3 @ C2 @ ( sup_sup_set_rule @ A3 @ B ) )
      = ( ( member_rule3 @ C2 @ A3 )
        | ( member_rule3 @ C2 @ B ) ) ) ).

% Un_iff
thf(fact_273_Un__iff,axiom,
    ! [C2: list_fm,A3: set_list_fm,B: set_list_fm] :
      ( ( member_list_fm3 @ C2 @ ( sup_sup_set_list_fm @ A3 @ B ) )
      = ( ( member_list_fm3 @ C2 @ A3 )
        | ( member_list_fm3 @ C2 @ B ) ) ) ).

% Un_iff
thf(fact_274_Un__iff,axiom,
    ! [C2: fm,A3: set_fm,B: set_fm] :
      ( ( member_fm3 @ C2 @ ( sup_sup_set_fm @ A3 @ B ) )
      = ( ( member_fm3 @ C2 @ A3 )
        | ( member_fm3 @ C2 @ B ) ) ) ).

% Un_iff
thf(fact_275_Un__iff,axiom,
    ! [C2: $o,A3: set_o,B: set_o] :
      ( ( member_o3 @ C2 @ ( sup_sup_set_o @ A3 @ B ) )
      = ( ( member_o3 @ C2 @ A3 )
        | ( member_o3 @ C2 @ B ) ) ) ).

% Un_iff
thf(fact_276_Un__iff,axiom,
    ! [C2: nat,A3: set_nat,B: set_nat] :
      ( ( member_nat3 @ C2 @ ( sup_sup_set_nat @ A3 @ B ) )
      = ( ( member_nat3 @ C2 @ A3 )
        | ( member_nat3 @ C2 @ B ) ) ) ).

% Un_iff
thf(fact_277_UnCI,axiom,
    ! [C2: rule,B: set_rule,A3: set_rule] :
      ( ( ~ ( member_rule3 @ C2 @ B )
       => ( member_rule3 @ C2 @ A3 ) )
     => ( member_rule3 @ C2 @ ( sup_sup_set_rule @ A3 @ B ) ) ) ).

% UnCI
thf(fact_278_UnCI,axiom,
    ! [C2: list_fm,B: set_list_fm,A3: set_list_fm] :
      ( ( ~ ( member_list_fm3 @ C2 @ B )
       => ( member_list_fm3 @ C2 @ A3 ) )
     => ( member_list_fm3 @ C2 @ ( sup_sup_set_list_fm @ A3 @ B ) ) ) ).

% UnCI
thf(fact_279_UnCI,axiom,
    ! [C2: fm,B: set_fm,A3: set_fm] :
      ( ( ~ ( member_fm3 @ C2 @ B )
       => ( member_fm3 @ C2 @ A3 ) )
     => ( member_fm3 @ C2 @ ( sup_sup_set_fm @ A3 @ B ) ) ) ).

% UnCI
thf(fact_280_UnCI,axiom,
    ! [C2: $o,B: set_o,A3: set_o] :
      ( ( ~ ( member_o3 @ C2 @ B )
       => ( member_o3 @ C2 @ A3 ) )
     => ( member_o3 @ C2 @ ( sup_sup_set_o @ A3 @ B ) ) ) ).

% UnCI
thf(fact_281_UnCI,axiom,
    ! [C2: nat,B: set_nat,A3: set_nat] :
      ( ( ~ ( member_nat3 @ C2 @ B )
       => ( member_nat3 @ C2 @ A3 ) )
     => ( member_nat3 @ C2 @ ( sup_sup_set_nat @ A3 @ B ) ) ) ).

% UnCI
thf(fact_282_sup_Oright__idem,axiom,
    ! [A: set_nat,B2: set_nat] :
      ( ( sup_sup_set_nat @ ( sup_sup_set_nat @ A @ B2 ) @ B2 )
      = ( sup_sup_set_nat @ A @ B2 ) ) ).

% sup.right_idem
thf(fact_283_sup__left__idem,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ X @ Y ) )
      = ( sup_sup_set_nat @ X @ Y ) ) ).

% sup_left_idem
thf(fact_284_sup_Oleft__idem,axiom,
    ! [A: set_nat,B2: set_nat] :
      ( ( sup_sup_set_nat @ A @ ( sup_sup_set_nat @ A @ B2 ) )
      = ( sup_sup_set_nat @ A @ B2 ) ) ).

% sup.left_idem
thf(fact_285_sup__idem,axiom,
    ! [X: set_nat] :
      ( ( sup_sup_set_nat @ X @ X )
      = X ) ).

% sup_idem
thf(fact_286_sup_Oidem,axiom,
    ! [A: set_nat] :
      ( ( sup_sup_set_nat @ A @ A )
      = A ) ).

% sup.idem
thf(fact_287_image__eqI,axiom,
    ! [B2: rule,F: rule > rule,X: rule,A3: set_rule] :
      ( ( B2
        = ( F @ X ) )
     => ( ( member_rule3 @ X @ A3 )
       => ( member_rule3 @ B2 @ ( image_rule_rule @ F @ A3 ) ) ) ) ).

% image_eqI
thf(fact_288_image__eqI,axiom,
    ! [B2: fm,F: rule > fm,X: rule,A3: set_rule] :
      ( ( B2
        = ( F @ X ) )
     => ( ( member_rule3 @ X @ A3 )
       => ( member_fm3 @ B2 @ ( image_rule_fm @ F @ A3 ) ) ) ) ).

% image_eqI
thf(fact_289_image__eqI,axiom,
    ! [B2: $o,F: rule > $o,X: rule,A3: set_rule] :
      ( ( B2
        = ( F @ X ) )
     => ( ( member_rule3 @ X @ A3 )
       => ( member_o3 @ B2 @ ( image_rule_o @ F @ A3 ) ) ) ) ).

% image_eqI
thf(fact_290_image__eqI,axiom,
    ! [B2: nat,F: rule > nat,X: rule,A3: set_rule] :
      ( ( B2
        = ( F @ X ) )
     => ( ( member_rule3 @ X @ A3 )
       => ( member_nat3 @ B2 @ ( image_rule_nat @ F @ A3 ) ) ) ) ).

% image_eqI
thf(fact_291_image__eqI,axiom,
    ! [B2: rule,F: fm > rule,X: fm,A3: set_fm] :
      ( ( B2
        = ( F @ X ) )
     => ( ( member_fm3 @ X @ A3 )
       => ( member_rule3 @ B2 @ ( image_fm_rule @ F @ A3 ) ) ) ) ).

% image_eqI
thf(fact_292_image__eqI,axiom,
    ! [B2: fm,F: fm > fm,X: fm,A3: set_fm] :
      ( ( B2
        = ( F @ X ) )
     => ( ( member_fm3 @ X @ A3 )
       => ( member_fm3 @ B2 @ ( image_fm_fm @ F @ A3 ) ) ) ) ).

% image_eqI
thf(fact_293_image__eqI,axiom,
    ! [B2: $o,F: fm > $o,X: fm,A3: set_fm] :
      ( ( B2
        = ( F @ X ) )
     => ( ( member_fm3 @ X @ A3 )
       => ( member_o3 @ B2 @ ( image_fm_o @ F @ A3 ) ) ) ) ).

% image_eqI
thf(fact_294_image__eqI,axiom,
    ! [B2: nat,F: fm > nat,X: fm,A3: set_fm] :
      ( ( B2
        = ( F @ X ) )
     => ( ( member_fm3 @ X @ A3 )
       => ( member_nat3 @ B2 @ ( image_fm_nat @ F @ A3 ) ) ) ) ).

% image_eqI
thf(fact_295_image__eqI,axiom,
    ! [B2: rule,F: $o > rule,X: $o,A3: set_o] :
      ( ( B2
        = ( F @ X ) )
     => ( ( member_o3 @ X @ A3 )
       => ( member_rule3 @ B2 @ ( image_o_rule @ F @ A3 ) ) ) ) ).

% image_eqI
thf(fact_296_image__eqI,axiom,
    ! [B2: fm,F: $o > fm,X: $o,A3: set_o] :
      ( ( B2
        = ( F @ X ) )
     => ( ( member_o3 @ X @ A3 )
       => ( member_fm3 @ B2 @ ( image_o_fm @ F @ A3 ) ) ) ) ).

% image_eqI
thf(fact_297_Sup__bool__def,axiom,
    ( complete_Sup_Sup_o
    = ( member_o3 @ $true ) ) ).

% Sup_bool_def
thf(fact_298_sup__set__def,axiom,
    ( sup_sup_set_rule
    = ( ^ [A4: set_rule,B3: set_rule] :
          ( collect_rule
          @ ( sup_sup_rule_o
            @ ^ [X3: rule] : ( member_rule3 @ X3 @ A4 )
            @ ^ [X3: rule] : ( member_rule3 @ X3 @ B3 ) ) ) ) ) ).

% sup_set_def
thf(fact_299_sup__set__def,axiom,
    ( sup_sup_set_list_fm
    = ( ^ [A4: set_list_fm,B3: set_list_fm] :
          ( collect_list_fm
          @ ( sup_sup_list_fm_o
            @ ^ [X3: list_fm] : ( member_list_fm3 @ X3 @ A4 )
            @ ^ [X3: list_fm] : ( member_list_fm3 @ X3 @ B3 ) ) ) ) ) ).

% sup_set_def
thf(fact_300_sup__set__def,axiom,
    ( sup_sup_set_fm
    = ( ^ [A4: set_fm,B3: set_fm] :
          ( collect_fm
          @ ( sup_sup_fm_o
            @ ^ [X3: fm] : ( member_fm3 @ X3 @ A4 )
            @ ^ [X3: fm] : ( member_fm3 @ X3 @ B3 ) ) ) ) ) ).

% sup_set_def
thf(fact_301_sup__set__def,axiom,
    ( sup_sup_set_o
    = ( ^ [A4: set_o,B3: set_o] :
          ( collect_o
          @ ( sup_sup_o_o
            @ ^ [X3: $o] : ( member_o3 @ X3 @ A4 )
            @ ^ [X3: $o] : ( member_o3 @ X3 @ B3 ) ) ) ) ) ).

% sup_set_def
thf(fact_302_sup__set__def,axiom,
    ( sup_sup_set_nat
    = ( ^ [A4: set_nat,B3: set_nat] :
          ( collect_nat
          @ ( sup_sup_nat_o
            @ ^ [X3: nat] : ( member_nat3 @ X3 @ A4 )
            @ ^ [X3: nat] : ( member_nat3 @ X3 @ B3 ) ) ) ) ) ).

% sup_set_def
thf(fact_303_imageI,axiom,
    ! [X: rule,A3: set_rule,F: rule > rule] :
      ( ( member_rule3 @ X @ A3 )
     => ( member_rule3 @ ( F @ X ) @ ( image_rule_rule @ F @ A3 ) ) ) ).

% imageI
thf(fact_304_imageI,axiom,
    ! [X: rule,A3: set_rule,F: rule > fm] :
      ( ( member_rule3 @ X @ A3 )
     => ( member_fm3 @ ( F @ X ) @ ( image_rule_fm @ F @ A3 ) ) ) ).

% imageI
thf(fact_305_imageI,axiom,
    ! [X: rule,A3: set_rule,F: rule > $o] :
      ( ( member_rule3 @ X @ A3 )
     => ( member_o3 @ ( F @ X ) @ ( image_rule_o @ F @ A3 ) ) ) ).

% imageI
thf(fact_306_imageI,axiom,
    ! [X: rule,A3: set_rule,F: rule > nat] :
      ( ( member_rule3 @ X @ A3 )
     => ( member_nat3 @ ( F @ X ) @ ( image_rule_nat @ F @ A3 ) ) ) ).

% imageI
thf(fact_307_imageI,axiom,
    ! [X: fm,A3: set_fm,F: fm > rule] :
      ( ( member_fm3 @ X @ A3 )
     => ( member_rule3 @ ( F @ X ) @ ( image_fm_rule @ F @ A3 ) ) ) ).

% imageI
thf(fact_308_imageI,axiom,
    ! [X: fm,A3: set_fm,F: fm > fm] :
      ( ( member_fm3 @ X @ A3 )
     => ( member_fm3 @ ( F @ X ) @ ( image_fm_fm @ F @ A3 ) ) ) ).

% imageI
thf(fact_309_imageI,axiom,
    ! [X: fm,A3: set_fm,F: fm > $o] :
      ( ( member_fm3 @ X @ A3 )
     => ( member_o3 @ ( F @ X ) @ ( image_fm_o @ F @ A3 ) ) ) ).

% imageI
thf(fact_310_imageI,axiom,
    ! [X: fm,A3: set_fm,F: fm > nat] :
      ( ( member_fm3 @ X @ A3 )
     => ( member_nat3 @ ( F @ X ) @ ( image_fm_nat @ F @ A3 ) ) ) ).

% imageI
thf(fact_311_imageI,axiom,
    ! [X: $o,A3: set_o,F: $o > rule] :
      ( ( member_o3 @ X @ A3 )
     => ( member_rule3 @ ( F @ X ) @ ( image_o_rule @ F @ A3 ) ) ) ).

% imageI
thf(fact_312_imageI,axiom,
    ! [X: $o,A3: set_o,F: $o > fm] :
      ( ( member_o3 @ X @ A3 )
     => ( member_fm3 @ ( F @ X ) @ ( image_o_fm @ F @ A3 ) ) ) ).

% imageI
thf(fact_313_image__iff,axiom,
    ! [Z4: set_fm,F: produc340336539035504054m_rule > set_fm,A3: set_Pr1822751329126368876m_rule] :
      ( ( member_set_fm @ Z4 @ ( image_377115208147135190set_fm @ F @ A3 ) )
      = ( ? [X3: produc340336539035504054m_rule] :
            ( ( member7231649785386036813m_rule @ X3 @ A3 )
            & ( Z4
              = ( F @ X3 ) ) ) ) ) ).

% image_iff
thf(fact_314_image__iff,axiom,
    ! [Z4: nat,F: nat > nat,A3: set_nat] :
      ( ( member_nat3 @ Z4 @ ( image_nat_nat @ F @ A3 ) )
      = ( ? [X3: nat] :
            ( ( member_nat3 @ X3 @ A3 )
            & ( Z4
              = ( F @ X3 ) ) ) ) ) ).

% image_iff
thf(fact_315_bex__imageD,axiom,
    ! [F: produc340336539035504054m_rule > set_fm,A3: set_Pr1822751329126368876m_rule,P3: set_fm > $o] :
      ( ? [X6: set_fm] :
          ( ( member_set_fm @ X6 @ ( image_377115208147135190set_fm @ F @ A3 ) )
          & ( P3 @ X6 ) )
     => ? [X2: produc340336539035504054m_rule] :
          ( ( member7231649785386036813m_rule @ X2 @ A3 )
          & ( P3 @ ( F @ X2 ) ) ) ) ).

% bex_imageD
thf(fact_316_bex__imageD,axiom,
    ! [F: nat > nat,A3: set_nat,P3: nat > $o] :
      ( ? [X6: nat] :
          ( ( member_nat3 @ X6 @ ( image_nat_nat @ F @ A3 ) )
          & ( P3 @ X6 ) )
     => ? [X2: nat] :
          ( ( member_nat3 @ X2 @ A3 )
          & ( P3 @ ( F @ X2 ) ) ) ) ).

% bex_imageD
thf(fact_317_image__cong,axiom,
    ! [M3: set_Pr1822751329126368876m_rule,N3: set_Pr1822751329126368876m_rule,F: produc340336539035504054m_rule > set_fm,G: produc340336539035504054m_rule > set_fm] :
      ( ( M3 = N3 )
     => ( ! [X2: produc340336539035504054m_rule] :
            ( ( member7231649785386036813m_rule @ X2 @ N3 )
           => ( ( F @ X2 )
              = ( G @ X2 ) ) )
       => ( ( image_377115208147135190set_fm @ F @ M3 )
          = ( image_377115208147135190set_fm @ G @ N3 ) ) ) ) ).

% image_cong
thf(fact_318_image__cong,axiom,
    ! [M3: set_nat,N3: set_nat,F: nat > nat,G: nat > nat] :
      ( ( M3 = N3 )
     => ( ! [X2: nat] :
            ( ( member_nat3 @ X2 @ N3 )
           => ( ( F @ X2 )
              = ( G @ X2 ) ) )
       => ( ( image_nat_nat @ F @ M3 )
          = ( image_nat_nat @ G @ N3 ) ) ) ) ).

% image_cong
thf(fact_319_ball__imageD,axiom,
    ! [F: produc340336539035504054m_rule > set_fm,A3: set_Pr1822751329126368876m_rule,P3: set_fm > $o] :
      ( ! [X2: set_fm] :
          ( ( member_set_fm @ X2 @ ( image_377115208147135190set_fm @ F @ A3 ) )
         => ( P3 @ X2 ) )
     => ! [X6: produc340336539035504054m_rule] :
          ( ( member7231649785386036813m_rule @ X6 @ A3 )
         => ( P3 @ ( F @ X6 ) ) ) ) ).

% ball_imageD
thf(fact_320_ball__imageD,axiom,
    ! [F: nat > nat,A3: set_nat,P3: nat > $o] :
      ( ! [X2: nat] :
          ( ( member_nat3 @ X2 @ ( image_nat_nat @ F @ A3 ) )
         => ( P3 @ X2 ) )
     => ! [X6: nat] :
          ( ( member_nat3 @ X6 @ A3 )
         => ( P3 @ ( F @ X6 ) ) ) ) ).

% ball_imageD
thf(fact_321_rev__image__eqI,axiom,
    ! [X: rule,A3: set_rule,B2: rule,F: rule > rule] :
      ( ( member_rule3 @ X @ A3 )
     => ( ( B2
          = ( F @ X ) )
       => ( member_rule3 @ B2 @ ( image_rule_rule @ F @ A3 ) ) ) ) ).

% rev_image_eqI
thf(fact_322_rev__image__eqI,axiom,
    ! [X: rule,A3: set_rule,B2: fm,F: rule > fm] :
      ( ( member_rule3 @ X @ A3 )
     => ( ( B2
          = ( F @ X ) )
       => ( member_fm3 @ B2 @ ( image_rule_fm @ F @ A3 ) ) ) ) ).

% rev_image_eqI
thf(fact_323_rev__image__eqI,axiom,
    ! [X: rule,A3: set_rule,B2: $o,F: rule > $o] :
      ( ( member_rule3 @ X @ A3 )
     => ( ( B2
          = ( F @ X ) )
       => ( member_o3 @ B2 @ ( image_rule_o @ F @ A3 ) ) ) ) ).

% rev_image_eqI
thf(fact_324_rev__image__eqI,axiom,
    ! [X: rule,A3: set_rule,B2: nat,F: rule > nat] :
      ( ( member_rule3 @ X @ A3 )
     => ( ( B2
          = ( F @ X ) )
       => ( member_nat3 @ B2 @ ( image_rule_nat @ F @ A3 ) ) ) ) ).

% rev_image_eqI
thf(fact_325_rev__image__eqI,axiom,
    ! [X: fm,A3: set_fm,B2: rule,F: fm > rule] :
      ( ( member_fm3 @ X @ A3 )
     => ( ( B2
          = ( F @ X ) )
       => ( member_rule3 @ B2 @ ( image_fm_rule @ F @ A3 ) ) ) ) ).

% rev_image_eqI
thf(fact_326_rev__image__eqI,axiom,
    ! [X: fm,A3: set_fm,B2: fm,F: fm > fm] :
      ( ( member_fm3 @ X @ A3 )
     => ( ( B2
          = ( F @ X ) )
       => ( member_fm3 @ B2 @ ( image_fm_fm @ F @ A3 ) ) ) ) ).

% rev_image_eqI
thf(fact_327_rev__image__eqI,axiom,
    ! [X: fm,A3: set_fm,B2: $o,F: fm > $o] :
      ( ( member_fm3 @ X @ A3 )
     => ( ( B2
          = ( F @ X ) )
       => ( member_o3 @ B2 @ ( image_fm_o @ F @ A3 ) ) ) ) ).

% rev_image_eqI
thf(fact_328_rev__image__eqI,axiom,
    ! [X: fm,A3: set_fm,B2: nat,F: fm > nat] :
      ( ( member_fm3 @ X @ A3 )
     => ( ( B2
          = ( F @ X ) )
       => ( member_nat3 @ B2 @ ( image_fm_nat @ F @ A3 ) ) ) ) ).

% rev_image_eqI
thf(fact_329_rev__image__eqI,axiom,
    ! [X: $o,A3: set_o,B2: rule,F: $o > rule] :
      ( ( member_o3 @ X @ A3 )
     => ( ( B2
          = ( F @ X ) )
       => ( member_rule3 @ B2 @ ( image_o_rule @ F @ A3 ) ) ) ) ).

% rev_image_eqI
thf(fact_330_rev__image__eqI,axiom,
    ! [X: $o,A3: set_o,B2: fm,F: $o > fm] :
      ( ( member_o3 @ X @ A3 )
     => ( ( B2
          = ( F @ X ) )
       => ( member_fm3 @ B2 @ ( image_o_fm @ F @ A3 ) ) ) ) ).

% rev_image_eqI
thf(fact_331_inf__sup__aci_I8_J,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ X @ Y ) )
      = ( sup_sup_set_nat @ X @ Y ) ) ).

% inf_sup_aci(8)
thf(fact_332_inf__sup__aci_I7_J,axiom,
    ! [X: set_nat,Y: set_nat,Z4: set_nat] :
      ( ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ Y @ Z4 ) )
      = ( sup_sup_set_nat @ Y @ ( sup_sup_set_nat @ X @ Z4 ) ) ) ).

% inf_sup_aci(7)
thf(fact_333_inf__sup__aci_I6_J,axiom,
    ! [X: set_nat,Y: set_nat,Z4: set_nat] :
      ( ( sup_sup_set_nat @ ( sup_sup_set_nat @ X @ Y ) @ Z4 )
      = ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ Y @ Z4 ) ) ) ).

% inf_sup_aci(6)
thf(fact_334_inf__sup__aci_I5_J,axiom,
    ( sup_sup_set_nat
    = ( ^ [X3: set_nat,Y2: set_nat] : ( sup_sup_set_nat @ Y2 @ X3 ) ) ) ).

% inf_sup_aci(5)
thf(fact_335_sup_Oassoc,axiom,
    ! [A: set_nat,B2: set_nat,C2: set_nat] :
      ( ( sup_sup_set_nat @ ( sup_sup_set_nat @ A @ B2 ) @ C2 )
      = ( sup_sup_set_nat @ A @ ( sup_sup_set_nat @ B2 @ C2 ) ) ) ).

% sup.assoc
thf(fact_336_sup__assoc,axiom,
    ! [X: set_nat,Y: set_nat,Z4: set_nat] :
      ( ( sup_sup_set_nat @ ( sup_sup_set_nat @ X @ Y ) @ Z4 )
      = ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ Y @ Z4 ) ) ) ).

% sup_assoc
thf(fact_337_sup_Ocommute,axiom,
    ( sup_sup_set_nat
    = ( ^ [A2: set_nat,B4: set_nat] : ( sup_sup_set_nat @ B4 @ A2 ) ) ) ).

% sup.commute
thf(fact_338_sup__commute,axiom,
    ( sup_sup_set_nat
    = ( ^ [X3: set_nat,Y2: set_nat] : ( sup_sup_set_nat @ Y2 @ X3 ) ) ) ).

% sup_commute
thf(fact_339_sup_Oleft__commute,axiom,
    ! [B2: set_nat,A: set_nat,C2: set_nat] :
      ( ( sup_sup_set_nat @ B2 @ ( sup_sup_set_nat @ A @ C2 ) )
      = ( sup_sup_set_nat @ A @ ( sup_sup_set_nat @ B2 @ C2 ) ) ) ).

% sup.left_commute
thf(fact_340_sup__left__commute,axiom,
    ! [X: set_nat,Y: set_nat,Z4: set_nat] :
      ( ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ Y @ Z4 ) )
      = ( sup_sup_set_nat @ Y @ ( sup_sup_set_nat @ X @ Z4 ) ) ) ).

% sup_left_commute
thf(fact_341_Ball__def,axiom,
    ( ball_rule
    = ( ^ [A4: set_rule,P4: rule > $o] :
        ! [X3: rule] :
          ( ( member_rule3 @ X3 @ A4 )
         => ( P4 @ X3 ) ) ) ) ).

% Ball_def
thf(fact_342_Ball__def,axiom,
    ( ball_list_fm
    = ( ^ [A4: set_list_fm,P4: list_fm > $o] :
        ! [X3: list_fm] :
          ( ( member_list_fm3 @ X3 @ A4 )
         => ( P4 @ X3 ) ) ) ) ).

% Ball_def
thf(fact_343_Ball__def,axiom,
    ( ball_fm
    = ( ^ [A4: set_fm,P4: fm > $o] :
        ! [X3: fm] :
          ( ( member_fm3 @ X3 @ A4 )
         => ( P4 @ X3 ) ) ) ) ).

% Ball_def
thf(fact_344_Ball__def,axiom,
    ( ball_o
    = ( ^ [A4: set_o,P4: $o > $o] :
        ! [X3: $o] :
          ( ( member_o3 @ X3 @ A4 )
         => ( P4 @ X3 ) ) ) ) ).

% Ball_def
thf(fact_345_Ball__def,axiom,
    ( ball_nat
    = ( ^ [A4: set_nat,P4: nat > $o] :
        ! [X3: nat] :
          ( ( member_nat3 @ X3 @ A4 )
         => ( P4 @ X3 ) ) ) ) ).

% Ball_def
thf(fact_346_UnE,axiom,
    ! [C2: rule,A3: set_rule,B: set_rule] :
      ( ( member_rule3 @ C2 @ ( sup_sup_set_rule @ A3 @ B ) )
     => ( ~ ( member_rule3 @ C2 @ A3 )
       => ( member_rule3 @ C2 @ B ) ) ) ).

% UnE
thf(fact_347_UnE,axiom,
    ! [C2: list_fm,A3: set_list_fm,B: set_list_fm] :
      ( ( member_list_fm3 @ C2 @ ( sup_sup_set_list_fm @ A3 @ B ) )
     => ( ~ ( member_list_fm3 @ C2 @ A3 )
       => ( member_list_fm3 @ C2 @ B ) ) ) ).

% UnE
thf(fact_348_UnE,axiom,
    ! [C2: fm,A3: set_fm,B: set_fm] :
      ( ( member_fm3 @ C2 @ ( sup_sup_set_fm @ A3 @ B ) )
     => ( ~ ( member_fm3 @ C2 @ A3 )
       => ( member_fm3 @ C2 @ B ) ) ) ).

% UnE
thf(fact_349_UnE,axiom,
    ! [C2: $o,A3: set_o,B: set_o] :
      ( ( member_o3 @ C2 @ ( sup_sup_set_o @ A3 @ B ) )
     => ( ~ ( member_o3 @ C2 @ A3 )
       => ( member_o3 @ C2 @ B ) ) ) ).

% UnE
thf(fact_350_UnE,axiom,
    ! [C2: nat,A3: set_nat,B: set_nat] :
      ( ( member_nat3 @ C2 @ ( sup_sup_set_nat @ A3 @ B ) )
     => ( ~ ( member_nat3 @ C2 @ A3 )
       => ( member_nat3 @ C2 @ B ) ) ) ).

% UnE
thf(fact_351_UnI1,axiom,
    ! [C2: rule,A3: set_rule,B: set_rule] :
      ( ( member_rule3 @ C2 @ A3 )
     => ( member_rule3 @ C2 @ ( sup_sup_set_rule @ A3 @ B ) ) ) ).

% UnI1
thf(fact_352_UnI1,axiom,
    ! [C2: list_fm,A3: set_list_fm,B: set_list_fm] :
      ( ( member_list_fm3 @ C2 @ A3 )
     => ( member_list_fm3 @ C2 @ ( sup_sup_set_list_fm @ A3 @ B ) ) ) ).

% UnI1
thf(fact_353_UnI1,axiom,
    ! [C2: fm,A3: set_fm,B: set_fm] :
      ( ( member_fm3 @ C2 @ A3 )
     => ( member_fm3 @ C2 @ ( sup_sup_set_fm @ A3 @ B ) ) ) ).

% UnI1
thf(fact_354_UnI1,axiom,
    ! [C2: $o,A3: set_o,B: set_o] :
      ( ( member_o3 @ C2 @ A3 )
     => ( member_o3 @ C2 @ ( sup_sup_set_o @ A3 @ B ) ) ) ).

% UnI1
thf(fact_355_UnI1,axiom,
    ! [C2: nat,A3: set_nat,B: set_nat] :
      ( ( member_nat3 @ C2 @ A3 )
     => ( member_nat3 @ C2 @ ( sup_sup_set_nat @ A3 @ B ) ) ) ).

% UnI1
thf(fact_356_UnI2,axiom,
    ! [C2: rule,B: set_rule,A3: set_rule] :
      ( ( member_rule3 @ C2 @ B )
     => ( member_rule3 @ C2 @ ( sup_sup_set_rule @ A3 @ B ) ) ) ).

% UnI2
thf(fact_357_UnI2,axiom,
    ! [C2: list_fm,B: set_list_fm,A3: set_list_fm] :
      ( ( member_list_fm3 @ C2 @ B )
     => ( member_list_fm3 @ C2 @ ( sup_sup_set_list_fm @ A3 @ B ) ) ) ).

% UnI2
thf(fact_358_UnI2,axiom,
    ! [C2: fm,B: set_fm,A3: set_fm] :
      ( ( member_fm3 @ C2 @ B )
     => ( member_fm3 @ C2 @ ( sup_sup_set_fm @ A3 @ B ) ) ) ).

% UnI2
thf(fact_359_UnI2,axiom,
    ! [C2: $o,B: set_o,A3: set_o] :
      ( ( member_o3 @ C2 @ B )
     => ( member_o3 @ C2 @ ( sup_sup_set_o @ A3 @ B ) ) ) ).

% UnI2
thf(fact_360_UnI2,axiom,
    ! [C2: nat,B: set_nat,A3: set_nat] :
      ( ( member_nat3 @ C2 @ B )
     => ( member_nat3 @ C2 @ ( sup_sup_set_nat @ A3 @ B ) ) ) ).

% UnI2
thf(fact_361_bex__Un,axiom,
    ! [A3: set_nat,B: set_nat,P3: nat > $o] :
      ( ( ? [X3: nat] :
            ( ( member_nat3 @ X3 @ ( sup_sup_set_nat @ A3 @ B ) )
            & ( P3 @ X3 ) ) )
      = ( ? [X3: nat] :
            ( ( member_nat3 @ X3 @ A3 )
            & ( P3 @ X3 ) )
        | ? [X3: nat] :
            ( ( member_nat3 @ X3 @ B )
            & ( P3 @ X3 ) ) ) ) ).

% bex_Un
thf(fact_362_ball__Un,axiom,
    ! [A3: set_nat,B: set_nat,P3: nat > $o] :
      ( ( ! [X3: nat] :
            ( ( member_nat3 @ X3 @ ( sup_sup_set_nat @ A3 @ B ) )
           => ( P3 @ X3 ) ) )
      = ( ! [X3: nat] :
            ( ( member_nat3 @ X3 @ A3 )
           => ( P3 @ X3 ) )
        & ! [X3: nat] :
            ( ( member_nat3 @ X3 @ B )
           => ( P3 @ X3 ) ) ) ) ).

% ball_Un
thf(fact_363_Un__assoc,axiom,
    ! [A3: set_nat,B: set_nat,C: set_nat] :
      ( ( sup_sup_set_nat @ ( sup_sup_set_nat @ A3 @ B ) @ C )
      = ( sup_sup_set_nat @ A3 @ ( sup_sup_set_nat @ B @ C ) ) ) ).

% Un_assoc
thf(fact_364_Un__absorb,axiom,
    ! [A3: set_nat] :
      ( ( sup_sup_set_nat @ A3 @ A3 )
      = A3 ) ).

% Un_absorb
thf(fact_365_Un__commute,axiom,
    ( sup_sup_set_nat
    = ( ^ [A4: set_nat,B3: set_nat] : ( sup_sup_set_nat @ B3 @ A4 ) ) ) ).

% Un_commute
thf(fact_366_Un__left__absorb,axiom,
    ! [A3: set_nat,B: set_nat] :
      ( ( sup_sup_set_nat @ A3 @ ( sup_sup_set_nat @ A3 @ B ) )
      = ( sup_sup_set_nat @ A3 @ B ) ) ).

% Un_left_absorb
thf(fact_367_Un__left__commute,axiom,
    ! [A3: set_nat,B: set_nat,C: set_nat] :
      ( ( sup_sup_set_nat @ A3 @ ( sup_sup_set_nat @ B @ C ) )
      = ( sup_sup_set_nat @ B @ ( sup_sup_set_nat @ A3 @ C ) ) ) ).

% Un_left_commute
thf(fact_368_imageE,axiom,
    ! [B2: rule,F: rule > rule,A3: set_rule] :
      ( ( member_rule3 @ B2 @ ( image_rule_rule @ F @ A3 ) )
     => ~ ! [X2: rule] :
            ( ( B2
              = ( F @ X2 ) )
           => ~ ( member_rule3 @ X2 @ A3 ) ) ) ).

% imageE
thf(fact_369_imageE,axiom,
    ! [B2: rule,F: fm > rule,A3: set_fm] :
      ( ( member_rule3 @ B2 @ ( image_fm_rule @ F @ A3 ) )
     => ~ ! [X2: fm] :
            ( ( B2
              = ( F @ X2 ) )
           => ~ ( member_fm3 @ X2 @ A3 ) ) ) ).

% imageE
thf(fact_370_imageE,axiom,
    ! [B2: rule,F: $o > rule,A3: set_o] :
      ( ( member_rule3 @ B2 @ ( image_o_rule @ F @ A3 ) )
     => ~ ! [X2: $o] :
            ( ( B2
              = ( F @ X2 ) )
           => ~ ( member_o3 @ X2 @ A3 ) ) ) ).

% imageE
thf(fact_371_imageE,axiom,
    ! [B2: rule,F: nat > rule,A3: set_nat] :
      ( ( member_rule3 @ B2 @ ( image_nat_rule @ F @ A3 ) )
     => ~ ! [X2: nat] :
            ( ( B2
              = ( F @ X2 ) )
           => ~ ( member_nat3 @ X2 @ A3 ) ) ) ).

% imageE
thf(fact_372_imageE,axiom,
    ! [B2: fm,F: rule > fm,A3: set_rule] :
      ( ( member_fm3 @ B2 @ ( image_rule_fm @ F @ A3 ) )
     => ~ ! [X2: rule] :
            ( ( B2
              = ( F @ X2 ) )
           => ~ ( member_rule3 @ X2 @ A3 ) ) ) ).

% imageE
thf(fact_373_imageE,axiom,
    ! [B2: fm,F: fm > fm,A3: set_fm] :
      ( ( member_fm3 @ B2 @ ( image_fm_fm @ F @ A3 ) )
     => ~ ! [X2: fm] :
            ( ( B2
              = ( F @ X2 ) )
           => ~ ( member_fm3 @ X2 @ A3 ) ) ) ).

% imageE
thf(fact_374_imageE,axiom,
    ! [B2: fm,F: $o > fm,A3: set_o] :
      ( ( member_fm3 @ B2 @ ( image_o_fm @ F @ A3 ) )
     => ~ ! [X2: $o] :
            ( ( B2
              = ( F @ X2 ) )
           => ~ ( member_o3 @ X2 @ A3 ) ) ) ).

% imageE
thf(fact_375_imageE,axiom,
    ! [B2: fm,F: nat > fm,A3: set_nat] :
      ( ( member_fm3 @ B2 @ ( image_nat_fm @ F @ A3 ) )
     => ~ ! [X2: nat] :
            ( ( B2
              = ( F @ X2 ) )
           => ~ ( member_nat3 @ X2 @ A3 ) ) ) ).

% imageE
thf(fact_376_imageE,axiom,
    ! [B2: $o,F: rule > $o,A3: set_rule] :
      ( ( member_o3 @ B2 @ ( image_rule_o @ F @ A3 ) )
     => ~ ! [X2: rule] :
            ( ( B2
              = ( F @ X2 ) )
           => ~ ( member_rule3 @ X2 @ A3 ) ) ) ).

% imageE
thf(fact_377_imageE,axiom,
    ! [B2: $o,F: fm > $o,A3: set_fm] :
      ( ( member_o3 @ B2 @ ( image_fm_o @ F @ A3 ) )
     => ~ ! [X2: fm] :
            ( ( B2
              = ( F @ X2 ) )
           => ~ ( member_fm3 @ X2 @ A3 ) ) ) ).

% imageE
thf(fact_378_image__image,axiom,
    ! [F: set_fm > set_fm,G: produc340336539035504054m_rule > set_fm,A3: set_Pr1822751329126368876m_rule] :
      ( ( image_set_fm_set_fm @ F @ ( image_377115208147135190set_fm @ G @ A3 ) )
      = ( image_377115208147135190set_fm
        @ ^ [X3: produc340336539035504054m_rule] : ( F @ ( G @ X3 ) )
        @ A3 ) ) ).

% image_image
thf(fact_379_image__image,axiom,
    ! [F: produc340336539035504054m_rule > set_fm,G: produc340336539035504054m_rule > produc340336539035504054m_rule,A3: set_Pr1822751329126368876m_rule] :
      ( ( image_377115208147135190set_fm @ F @ ( image_2498997284783760755m_rule @ G @ A3 ) )
      = ( image_377115208147135190set_fm
        @ ^ [X3: produc340336539035504054m_rule] : ( F @ ( G @ X3 ) )
        @ A3 ) ) ).

% image_image
thf(fact_380_image__image,axiom,
    ! [F: nat > nat,G: nat > nat,A3: set_nat] :
      ( ( image_nat_nat @ F @ ( image_nat_nat @ G @ A3 ) )
      = ( image_nat_nat
        @ ^ [X3: nat] : ( F @ ( G @ X3 ) )
        @ A3 ) ) ).

% image_image
thf(fact_381_Compr__image__eq,axiom,
    ! [F: rule > rule,A3: set_rule,P3: rule > $o] :
      ( ( collect_rule
        @ ^ [X3: rule] :
            ( ( member_rule3 @ X3 @ ( image_rule_rule @ F @ A3 ) )
            & ( P3 @ X3 ) ) )
      = ( image_rule_rule @ F
        @ ( collect_rule
          @ ^ [X3: rule] :
              ( ( member_rule3 @ X3 @ A3 )
              & ( P3 @ ( F @ X3 ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_382_Compr__image__eq,axiom,
    ! [F: fm > rule,A3: set_fm,P3: rule > $o] :
      ( ( collect_rule
        @ ^ [X3: rule] :
            ( ( member_rule3 @ X3 @ ( image_fm_rule @ F @ A3 ) )
            & ( P3 @ X3 ) ) )
      = ( image_fm_rule @ F
        @ ( collect_fm
          @ ^ [X3: fm] :
              ( ( member_fm3 @ X3 @ A3 )
              & ( P3 @ ( F @ X3 ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_383_Compr__image__eq,axiom,
    ! [F: $o > rule,A3: set_o,P3: rule > $o] :
      ( ( collect_rule
        @ ^ [X3: rule] :
            ( ( member_rule3 @ X3 @ ( image_o_rule @ F @ A3 ) )
            & ( P3 @ X3 ) ) )
      = ( image_o_rule @ F
        @ ( collect_o
          @ ^ [X3: $o] :
              ( ( member_o3 @ X3 @ A3 )
              & ( P3 @ ( F @ X3 ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_384_Compr__image__eq,axiom,
    ! [F: rule > fm,A3: set_rule,P3: fm > $o] :
      ( ( collect_fm
        @ ^ [X3: fm] :
            ( ( member_fm3 @ X3 @ ( image_rule_fm @ F @ A3 ) )
            & ( P3 @ X3 ) ) )
      = ( image_rule_fm @ F
        @ ( collect_rule
          @ ^ [X3: rule] :
              ( ( member_rule3 @ X3 @ A3 )
              & ( P3 @ ( F @ X3 ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_385_Compr__image__eq,axiom,
    ! [F: fm > fm,A3: set_fm,P3: fm > $o] :
      ( ( collect_fm
        @ ^ [X3: fm] :
            ( ( member_fm3 @ X3 @ ( image_fm_fm @ F @ A3 ) )
            & ( P3 @ X3 ) ) )
      = ( image_fm_fm @ F
        @ ( collect_fm
          @ ^ [X3: fm] :
              ( ( member_fm3 @ X3 @ A3 )
              & ( P3 @ ( F @ X3 ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_386_Compr__image__eq,axiom,
    ! [F: $o > fm,A3: set_o,P3: fm > $o] :
      ( ( collect_fm
        @ ^ [X3: fm] :
            ( ( member_fm3 @ X3 @ ( image_o_fm @ F @ A3 ) )
            & ( P3 @ X3 ) ) )
      = ( image_o_fm @ F
        @ ( collect_o
          @ ^ [X3: $o] :
              ( ( member_o3 @ X3 @ A3 )
              & ( P3 @ ( F @ X3 ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_387_Compr__image__eq,axiom,
    ! [F: rule > $o,A3: set_rule,P3: $o > $o] :
      ( ( collect_o
        @ ^ [X3: $o] :
            ( ( member_o3 @ X3 @ ( image_rule_o @ F @ A3 ) )
            & ( P3 @ X3 ) ) )
      = ( image_rule_o @ F
        @ ( collect_rule
          @ ^ [X3: rule] :
              ( ( member_rule3 @ X3 @ A3 )
              & ( P3 @ ( F @ X3 ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_388_Compr__image__eq,axiom,
    ! [F: fm > $o,A3: set_fm,P3: $o > $o] :
      ( ( collect_o
        @ ^ [X3: $o] :
            ( ( member_o3 @ X3 @ ( image_fm_o @ F @ A3 ) )
            & ( P3 @ X3 ) ) )
      = ( image_fm_o @ F
        @ ( collect_fm
          @ ^ [X3: fm] :
              ( ( member_fm3 @ X3 @ A3 )
              & ( P3 @ ( F @ X3 ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_389_Compr__image__eq,axiom,
    ! [F: $o > $o,A3: set_o,P3: $o > $o] :
      ( ( collect_o
        @ ^ [X3: $o] :
            ( ( member_o3 @ X3 @ ( image_o_o @ F @ A3 ) )
            & ( P3 @ X3 ) ) )
      = ( image_o_o @ F
        @ ( collect_o
          @ ^ [X3: $o] :
              ( ( member_o3 @ X3 @ A3 )
              & ( P3 @ ( F @ X3 ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_390_Compr__image__eq,axiom,
    ! [F: nat > rule,A3: set_nat,P3: rule > $o] :
      ( ( collect_rule
        @ ^ [X3: rule] :
            ( ( member_rule3 @ X3 @ ( image_nat_rule @ F @ A3 ) )
            & ( P3 @ X3 ) ) )
      = ( image_nat_rule @ F
        @ ( collect_nat
          @ ^ [X3: nat] :
              ( ( member_nat3 @ X3 @ A3 )
              & ( P3 @ ( F @ X3 ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_391_Un__def,axiom,
    ( sup_sup_set_rule
    = ( ^ [A4: set_rule,B3: set_rule] :
          ( collect_rule
          @ ^ [X3: rule] :
              ( ( member_rule3 @ X3 @ A4 )
              | ( member_rule3 @ X3 @ B3 ) ) ) ) ) ).

% Un_def
thf(fact_392_Un__def,axiom,
    ( sup_sup_set_list_fm
    = ( ^ [A4: set_list_fm,B3: set_list_fm] :
          ( collect_list_fm
          @ ^ [X3: list_fm] :
              ( ( member_list_fm3 @ X3 @ A4 )
              | ( member_list_fm3 @ X3 @ B3 ) ) ) ) ) ).

% Un_def
thf(fact_393_Un__def,axiom,
    ( sup_sup_set_fm
    = ( ^ [A4: set_fm,B3: set_fm] :
          ( collect_fm
          @ ^ [X3: fm] :
              ( ( member_fm3 @ X3 @ A4 )
              | ( member_fm3 @ X3 @ B3 ) ) ) ) ) ).

% Un_def
thf(fact_394_Un__def,axiom,
    ( sup_sup_set_o
    = ( ^ [A4: set_o,B3: set_o] :
          ( collect_o
          @ ^ [X3: $o] :
              ( ( member_o3 @ X3 @ A4 )
              | ( member_o3 @ X3 @ B3 ) ) ) ) ) ).

% Un_def
thf(fact_395_Un__def,axiom,
    ( sup_sup_set_nat
    = ( ^ [A4: set_nat,B3: set_nat] :
          ( collect_nat
          @ ^ [X3: nat] :
              ( ( member_nat3 @ X3 @ A4 )
              | ( member_nat3 @ X3 @ B3 ) ) ) ) ) ).

% Un_def
thf(fact_396_Collect__disj__eq,axiom,
    ! [P3: nat > $o,Q2: nat > $o] :
      ( ( collect_nat
        @ ^ [X3: nat] :
            ( ( P3 @ X3 )
            | ( Q2 @ X3 ) ) )
      = ( sup_sup_set_nat @ ( collect_nat @ P3 ) @ ( collect_nat @ Q2 ) ) ) ).

% Collect_disj_eq
thf(fact_397_image__Un,axiom,
    ! [F: produc340336539035504054m_rule > set_fm,A3: set_Pr1822751329126368876m_rule,B: set_Pr1822751329126368876m_rule] :
      ( ( image_377115208147135190set_fm @ F @ ( sup_su6946459741510085528m_rule @ A3 @ B ) )
      = ( sup_sup_set_set_fm @ ( image_377115208147135190set_fm @ F @ A3 ) @ ( image_377115208147135190set_fm @ F @ B ) ) ) ).

% image_Un
thf(fact_398_image__Un,axiom,
    ! [F: nat > nat,A3: set_nat,B: set_nat] :
      ( ( image_nat_nat @ F @ ( sup_sup_set_nat @ A3 @ B ) )
      = ( sup_sup_set_nat @ ( image_nat_nat @ F @ A3 ) @ ( image_nat_nat @ F @ B ) ) ) ).

% image_Un
thf(fact_399_UN__constant__eq,axiom,
    ! [A: produc340336539035504054m_rule,A3: set_Pr1822751329126368876m_rule,F: produc340336539035504054m_rule > set_fm,C2: set_fm] :
      ( ( member7231649785386036813m_rule @ A @ A3 )
     => ( ! [X2: produc340336539035504054m_rule] :
            ( ( member7231649785386036813m_rule @ X2 @ A3 )
           => ( ( F @ X2 )
              = C2 ) )
       => ( ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ F @ A3 ) )
          = C2 ) ) ) ).

% UN_constant_eq
thf(fact_400_UN__constant__eq,axiom,
    ! [A: rule,A3: set_rule,F: rule > set_fm,C2: set_fm] :
      ( ( member_rule3 @ A @ A3 )
     => ( ! [X2: rule] :
            ( ( member_rule3 @ X2 @ A3 )
           => ( ( F @ X2 )
              = C2 ) )
       => ( ( comple2134933779557159616set_fm @ ( image_rule_set_fm @ F @ A3 ) )
          = C2 ) ) ) ).

% UN_constant_eq
thf(fact_401_UN__constant__eq,axiom,
    ! [A: list_fm,A3: set_list_fm,F: list_fm > set_fm,C2: set_fm] :
      ( ( member_list_fm3 @ A @ A3 )
     => ( ! [X2: list_fm] :
            ( ( member_list_fm3 @ X2 @ A3 )
           => ( ( F @ X2 )
              = C2 ) )
       => ( ( comple2134933779557159616set_fm @ ( image_list_fm_set_fm @ F @ A3 ) )
          = C2 ) ) ) ).

% UN_constant_eq
thf(fact_402_UN__constant__eq,axiom,
    ! [A: fm,A3: set_fm,F: fm > set_fm,C2: set_fm] :
      ( ( member_fm3 @ A @ A3 )
     => ( ! [X2: fm] :
            ( ( member_fm3 @ X2 @ A3 )
           => ( ( F @ X2 )
              = C2 ) )
       => ( ( comple2134933779557159616set_fm @ ( image_fm_set_fm @ F @ A3 ) )
          = C2 ) ) ) ).

% UN_constant_eq
thf(fact_403_UN__constant__eq,axiom,
    ! [A: $o,A3: set_o,F: $o > set_fm,C2: set_fm] :
      ( ( member_o3 @ A @ A3 )
     => ( ! [X2: $o] :
            ( ( member_o3 @ X2 @ A3 )
           => ( ( F @ X2 )
              = C2 ) )
       => ( ( comple2134933779557159616set_fm @ ( image_o_set_fm @ F @ A3 ) )
          = C2 ) ) ) ).

% UN_constant_eq
thf(fact_404_UN__constant__eq,axiom,
    ! [A: nat,A3: set_nat,F: nat > set_fm,C2: set_fm] :
      ( ( member_nat3 @ A @ A3 )
     => ( ! [X2: nat] :
            ( ( member_nat3 @ X2 @ A3 )
           => ( ( F @ X2 )
              = C2 ) )
       => ( ( comple2134933779557159616set_fm @ ( image_nat_set_fm @ F @ A3 ) )
          = C2 ) ) ) ).

% UN_constant_eq
thf(fact_405_SUP__UN__eq,axiom,
    ! [R: produc340336539035504054m_rule > set_fm,S4: set_Pr1822751329126368876m_rule] :
      ( ( comple3443612784043876893p_fm_o
        @ ( image_5627174782981912967e_fm_o
          @ ^ [I: produc340336539035504054m_rule,X3: fm] : ( member_fm3 @ X3 @ ( R @ I ) )
          @ S4 ) )
      = ( ^ [X3: fm] : ( member_fm3 @ X3 @ ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ R @ S4 ) ) ) ) ) ).

% SUP_UN_eq
thf(fact_406_children__preserves__unaffected,axiom,
    ! [P: fm,Z4: list_fm,R: rule,Z3: list_fm,A3: list_tm] :
      ( ( member_fm3 @ P @ ( set_fm2 @ Z4 ) )
     => ( ~ ( affects @ R @ P )
       => ( ( member_list_fm3 @ Z3 @ ( set_list_fm2 @ ( children @ A3 @ R @ Z4 ) ) )
         => ( member_fm3 @ P @ ( set_fm2 @ Z3 ) ) ) ) ) ).

% children_preserves_unaffected
thf(fact_407_set__list__bind,axiom,
    ! [Xs2: list_fm,F: fm > list_list_fm] :
      ( ( set_list_fm2 @ ( bind_fm_list_fm @ Xs2 @ F ) )
      = ( comple8784269564784259782ist_fm
        @ ( image_fm_set_list_fm
          @ ^ [X3: fm] : ( set_list_fm2 @ ( F @ X3 ) )
          @ ( set_fm2 @ Xs2 ) ) ) ) ).

% set_list_bind
thf(fact_408_set__list__bind,axiom,
    ! [Xs2: list_list_fm,F: list_fm > list_list_fm] :
      ( ( set_list_fm2 @ ( bind_list_fm_list_fm @ Xs2 @ F ) )
      = ( comple8784269564784259782ist_fm
        @ ( image_1803486656538583717ist_fm
          @ ^ [X3: list_fm] : ( set_list_fm2 @ ( F @ X3 ) )
          @ ( set_list_fm2 @ Xs2 ) ) ) ) ).

% set_list_bind
thf(fact_409_set__list__bind,axiom,
    ! [Xs2: list_tm,F: tm > list_list_fm] :
      ( ( set_list_fm2 @ ( bind_tm_list_fm @ Xs2 @ F ) )
      = ( comple8784269564784259782ist_fm
        @ ( image_tm_set_list_fm
          @ ^ [X3: tm] : ( set_list_fm2 @ ( F @ X3 ) )
          @ ( set_tm2 @ Xs2 ) ) ) ) ).

% set_list_bind
thf(fact_410_set__list__bind,axiom,
    ! [Xs2: list_fm,F: fm > list_tm] :
      ( ( set_tm2 @ ( bind_fm_tm @ Xs2 @ F ) )
      = ( comple2138885804642794802set_tm
        @ ( image_fm_set_tm
          @ ^ [X3: fm] : ( set_tm2 @ ( F @ X3 ) )
          @ ( set_fm2 @ Xs2 ) ) ) ) ).

% set_list_bind
thf(fact_411_set__list__bind,axiom,
    ! [Xs2: list_list_fm,F: list_fm > list_tm] :
      ( ( set_tm2 @ ( bind_list_fm_tm @ Xs2 @ F ) )
      = ( comple2138885804642794802set_tm
        @ ( image_list_fm_set_tm
          @ ^ [X3: list_fm] : ( set_tm2 @ ( F @ X3 ) )
          @ ( set_list_fm2 @ Xs2 ) ) ) ) ).

% set_list_bind
thf(fact_412_set__list__bind,axiom,
    ! [Xs2: list_tm,F: tm > list_tm] :
      ( ( set_tm2 @ ( bind_tm_tm @ Xs2 @ F ) )
      = ( comple2138885804642794802set_tm
        @ ( image_tm_set_tm
          @ ^ [X3: tm] : ( set_tm2 @ ( F @ X3 ) )
          @ ( set_tm2 @ Xs2 ) ) ) ) ).

% set_list_bind
thf(fact_413_set__list__bind,axiom,
    ! [Xs2: list_P2774625669004309958m_rule,F: produc340336539035504054m_rule > list_fm] :
      ( ( set_fm2 @ ( bind_P95098981060663284ule_fm @ Xs2 @ F ) )
      = ( comple2134933779557159616set_fm
        @ ( image_377115208147135190set_fm
          @ ^ [X3: produc340336539035504054m_rule] : ( set_fm2 @ ( F @ X3 ) )
          @ ( set_Pr4534715572506550497m_rule @ Xs2 ) ) ) ) ).

% set_list_bind
thf(fact_414_set__list__bind,axiom,
    ! [Xs2: list_fm,F: fm > list_fm] :
      ( ( set_fm2 @ ( bind_fm_fm @ Xs2 @ F ) )
      = ( comple2134933779557159616set_fm
        @ ( image_fm_set_fm
          @ ^ [X3: fm] : ( set_fm2 @ ( F @ X3 ) )
          @ ( set_fm2 @ Xs2 ) ) ) ) ).

% set_list_bind
thf(fact_415_set__list__bind,axiom,
    ! [Xs2: list_list_fm,F: list_fm > list_fm] :
      ( ( set_fm2 @ ( bind_list_fm_fm @ Xs2 @ F ) )
      = ( comple2134933779557159616set_fm
        @ ( image_list_fm_set_fm
          @ ^ [X3: list_fm] : ( set_fm2 @ ( F @ X3 ) )
          @ ( set_list_fm2 @ Xs2 ) ) ) ) ).

% set_list_bind
thf(fact_416_set__list__bind,axiom,
    ! [Xs2: list_tm,F: tm > list_fm] :
      ( ( set_fm2 @ ( bind_tm_fm @ Xs2 @ F ) )
      = ( comple2134933779557159616set_fm
        @ ( image_tm_set_fm
          @ ^ [X3: tm] : ( set_fm2 @ ( F @ X3 ) )
          @ ( set_tm2 @ Xs2 ) ) ) ) ).

% set_list_bind
thf(fact_417_snth__szip,axiom,
    ! [S1: stream2709947120125613254m_rule,S2: stream2709947120125613254m_rule,N2: nat] :
      ( ( snth_P8853764340393315953m_rule @ ( szip_P811719526838699976m_rule @ S1 @ S2 ) @ N2 )
      = ( produc6261311607089640965m_rule @ ( snth_P6679518042731451922m_rule @ S1 @ N2 ) @ ( snth_P6679518042731451922m_rule @ S2 @ N2 ) ) ) ).

% snth_szip
thf(fact_418_snth__szip,axiom,
    ! [S1: stream2709947120125613254m_rule,S2: stream_rule,N2: nat] :
      ( ( snth_P6395297581347168077e_rule @ ( szip_P2044787555563963556e_rule @ S1 @ S2 ) @ N2 )
      = ( produc4831648765031708129e_rule @ ( snth_P6679518042731451922m_rule @ S1 @ N2 ) @ ( snth_rule @ S2 @ N2 ) ) ) ).

% snth_szip
thf(fact_419_snth__szip,axiom,
    ! [S1: stream_rule,S2: stream2709947120125613254m_rule,N2: nat] :
      ( ( snth_P5203088247548055629m_rule @ ( szip_r6533552703345879972m_rule @ S1 @ S2 ) @ N2 )
      = ( produc97041875958848737m_rule @ ( snth_rule @ S1 @ N2 ) @ ( snth_P6679518042731451922m_rule @ S2 @ N2 ) ) ) ).

% snth_szip
thf(fact_420_snth__szip,axiom,
    ! [S1: stream_rule,S2: stream_rule,N2: nat] :
      ( ( snth_P6178434775611311401e_rule @ ( szip_rule_rule @ S1 @ S2 ) @ N2 )
      = ( produc5849431337705160893e_rule @ ( snth_rule @ S1 @ N2 ) @ ( snth_rule @ S2 @ N2 ) ) ) ).

% snth_szip
thf(fact_421_snth__szip,axiom,
    ! [S1: stream4408948924543953275ist_fm,S2: stream_rule,N2: nat] :
      ( ( snth_P6679518042731451922m_rule @ ( szip_P2924820683901490861m_rule @ S1 @ S2 ) @ N2 )
      = ( produc1733806532565653680m_rule @ ( snth_P7093566783922538521ist_fm @ S1 @ N2 ) @ ( snth_rule @ S2 @ N2 ) ) ) ).

% snth_szip
thf(fact_422_sup__Un__eq,axiom,
    ! [R2: set_rule,S4: set_rule] :
      ( ( sup_sup_rule_o
        @ ^ [X3: rule] : ( member_rule3 @ X3 @ R2 )
        @ ^ [X3: rule] : ( member_rule3 @ X3 @ S4 ) )
      = ( ^ [X3: rule] : ( member_rule3 @ X3 @ ( sup_sup_set_rule @ R2 @ S4 ) ) ) ) ).

% sup_Un_eq
thf(fact_423_sup__Un__eq,axiom,
    ! [R2: set_list_fm,S4: set_list_fm] :
      ( ( sup_sup_list_fm_o
        @ ^ [X3: list_fm] : ( member_list_fm3 @ X3 @ R2 )
        @ ^ [X3: list_fm] : ( member_list_fm3 @ X3 @ S4 ) )
      = ( ^ [X3: list_fm] : ( member_list_fm3 @ X3 @ ( sup_sup_set_list_fm @ R2 @ S4 ) ) ) ) ).

% sup_Un_eq
thf(fact_424_sup__Un__eq,axiom,
    ! [R2: set_fm,S4: set_fm] :
      ( ( sup_sup_fm_o
        @ ^ [X3: fm] : ( member_fm3 @ X3 @ R2 )
        @ ^ [X3: fm] : ( member_fm3 @ X3 @ S4 ) )
      = ( ^ [X3: fm] : ( member_fm3 @ X3 @ ( sup_sup_set_fm @ R2 @ S4 ) ) ) ) ).

% sup_Un_eq
thf(fact_425_sup__Un__eq,axiom,
    ! [R2: set_o,S4: set_o] :
      ( ( sup_sup_o_o
        @ ^ [X3: $o] : ( member_o3 @ X3 @ R2 )
        @ ^ [X3: $o] : ( member_o3 @ X3 @ S4 ) )
      = ( ^ [X3: $o] : ( member_o3 @ X3 @ ( sup_sup_set_o @ R2 @ S4 ) ) ) ) ).

% sup_Un_eq
thf(fact_426_sup__Un__eq,axiom,
    ! [R2: set_nat,S4: set_nat] :
      ( ( sup_sup_nat_o
        @ ^ [X3: nat] : ( member_nat3 @ X3 @ R2 )
        @ ^ [X3: nat] : ( member_nat3 @ X3 @ S4 ) )
      = ( ^ [X3: nat] : ( member_nat3 @ X3 @ ( sup_sup_set_nat @ R2 @ S4 ) ) ) ) ).

% sup_Un_eq
thf(fact_427_SUP__Sup__eq,axiom,
    ! [S4: set_set_rule] :
      ( ( comple579058054629451178rule_o
        @ ( image_2488564359484039598rule_o
          @ ^ [I: set_rule,X3: rule] : ( member_rule3 @ X3 @ I )
          @ S4 ) )
      = ( ^ [X3: rule] : ( member_rule3 @ X3 @ ( comple7665822446944348827t_rule @ S4 ) ) ) ) ).

% SUP_Sup_eq
thf(fact_428_SUP__Sup__eq,axiom,
    ! [S4: set_set_list_fm] :
      ( ( comple173717950668161943t_fm_o
        @ ( image_4048089684711339992t_fm_o
          @ ^ [I: set_list_fm,X3: list_fm] : ( member_list_fm3 @ X3 @ I )
          @ S4 ) )
      = ( ^ [X3: list_fm] : ( member_list_fm3 @ X3 @ ( comple8784269564784259782ist_fm @ S4 ) ) ) ) ).

% SUP_Sup_eq
thf(fact_429_SUP__Sup__eq,axiom,
    ! [S4: set_set_o] :
      ( ( complete_Sup_Sup_o_o
        @ ( image_set_o_o_o
          @ ^ [I: set_o,X3: $o] : ( member_o3 @ X3 @ I )
          @ S4 ) )
      = ( ^ [X3: $o] : ( member_o3 @ X3 @ ( comple90263536869209701_set_o @ S4 ) ) ) ) ).

% SUP_Sup_eq
thf(fact_430_SUP__Sup__eq,axiom,
    ! [S4: set_set_nat] :
      ( ( comple8317665133742190828_nat_o
        @ ( image_set_nat_nat_o
          @ ^ [I: set_nat,X3: nat] : ( member_nat3 @ X3 @ I )
          @ S4 ) )
      = ( ^ [X3: nat] : ( member_nat3 @ X3 @ ( comple7399068483239264473et_nat @ S4 ) ) ) ) ).

% SUP_Sup_eq
thf(fact_431_SUP__Sup__eq,axiom,
    ! [S4: set_set_fm] :
      ( ( comple3443612784043876893p_fm_o
        @ ( image_set_fm_fm_o
          @ ^ [I: set_fm,X3: fm] : ( member_fm3 @ X3 @ I )
          @ S4 ) )
      = ( ^ [X3: fm] : ( member_fm3 @ X3 @ ( comple2134933779557159616set_fm @ S4 ) ) ) ) ).

% SUP_Sup_eq
thf(fact_432_set__concat,axiom,
    ! [Xs2: list_list_list_fm] :
      ( ( set_list_fm2 @ ( concat_list_fm @ Xs2 ) )
      = ( comple8784269564784259782ist_fm @ ( image_3687226712311829663ist_fm @ set_list_fm2 @ ( set_list_list_fm2 @ Xs2 ) ) ) ) ).

% set_concat
thf(fact_433_set__concat,axiom,
    ! [Xs2: list_list_tm] :
      ( ( set_tm2 @ ( concat_tm @ Xs2 ) )
      = ( comple2138885804642794802set_tm @ ( image_list_tm_set_tm @ set_tm2 @ ( set_list_tm2 @ Xs2 ) ) ) ) ).

% set_concat
thf(fact_434_set__concat,axiom,
    ! [Xs2: list_list_fm] :
      ( ( set_fm2 @ ( concat_fm @ Xs2 ) )
      = ( comple2134933779557159616set_fm @ ( image_list_fm_set_fm @ set_fm2 @ ( set_list_fm2 @ Xs2 ) ) ) ) ).

% set_concat
thf(fact_435_UN__simps_I3_J,axiom,
    ! [C: set_rule,A3: set_rule,B: rule > set_rule] :
      ( ( ( C = bot_bot_set_rule )
       => ( ( comple7665822446944348827t_rule
            @ ( image_rule_set_rule
              @ ^ [X3: rule] : ( sup_sup_set_rule @ A3 @ ( B @ X3 ) )
              @ C ) )
          = bot_bot_set_rule ) )
      & ( ( C != bot_bot_set_rule )
       => ( ( comple7665822446944348827t_rule
            @ ( image_rule_set_rule
              @ ^ [X3: rule] : ( sup_sup_set_rule @ A3 @ ( B @ X3 ) )
              @ C ) )
          = ( sup_sup_set_rule @ A3 @ ( comple7665822446944348827t_rule @ ( image_rule_set_rule @ B @ C ) ) ) ) ) ) ).

% UN_simps(3)
thf(fact_436_UN__simps_I3_J,axiom,
    ! [C: set_rule,A3: set_o,B: rule > set_o] :
      ( ( ( C = bot_bot_set_rule )
       => ( ( comple90263536869209701_set_o
            @ ( image_rule_set_o
              @ ^ [X3: rule] : ( sup_sup_set_o @ A3 @ ( B @ X3 ) )
              @ C ) )
          = bot_bot_set_o ) )
      & ( ( C != bot_bot_set_rule )
       => ( ( comple90263536869209701_set_o
            @ ( image_rule_set_o
              @ ^ [X3: rule] : ( sup_sup_set_o @ A3 @ ( B @ X3 ) )
              @ C ) )
          = ( sup_sup_set_o @ A3 @ ( comple90263536869209701_set_o @ ( image_rule_set_o @ B @ C ) ) ) ) ) ) ).

% UN_simps(3)
thf(fact_437_UN__simps_I3_J,axiom,
    ! [C: set_rule,A3: set_nat,B: rule > set_nat] :
      ( ( ( C = bot_bot_set_rule )
       => ( ( comple7399068483239264473et_nat
            @ ( image_rule_set_nat
              @ ^ [X3: rule] : ( sup_sup_set_nat @ A3 @ ( B @ X3 ) )
              @ C ) )
          = bot_bot_set_nat ) )
      & ( ( C != bot_bot_set_rule )
       => ( ( comple7399068483239264473et_nat
            @ ( image_rule_set_nat
              @ ^ [X3: rule] : ( sup_sup_set_nat @ A3 @ ( B @ X3 ) )
              @ C ) )
          = ( sup_sup_set_nat @ A3 @ ( comple7399068483239264473et_nat @ ( image_rule_set_nat @ B @ C ) ) ) ) ) ) ).

% UN_simps(3)
thf(fact_438_UN__simps_I3_J,axiom,
    ! [C: set_o,A3: set_rule,B: $o > set_rule] :
      ( ( ( C = bot_bot_set_o )
       => ( ( comple7665822446944348827t_rule
            @ ( image_o_set_rule
              @ ^ [X3: $o] : ( sup_sup_set_rule @ A3 @ ( B @ X3 ) )
              @ C ) )
          = bot_bot_set_rule ) )
      & ( ( C != bot_bot_set_o )
       => ( ( comple7665822446944348827t_rule
            @ ( image_o_set_rule
              @ ^ [X3: $o] : ( sup_sup_set_rule @ A3 @ ( B @ X3 ) )
              @ C ) )
          = ( sup_sup_set_rule @ A3 @ ( comple7665822446944348827t_rule @ ( image_o_set_rule @ B @ C ) ) ) ) ) ) ).

% UN_simps(3)
thf(fact_439_UN__simps_I3_J,axiom,
    ! [C: set_o,A3: set_o,B: $o > set_o] :
      ( ( ( C = bot_bot_set_o )
       => ( ( comple90263536869209701_set_o
            @ ( image_o_set_o
              @ ^ [X3: $o] : ( sup_sup_set_o @ A3 @ ( B @ X3 ) )
              @ C ) )
          = bot_bot_set_o ) )
      & ( ( C != bot_bot_set_o )
       => ( ( comple90263536869209701_set_o
            @ ( image_o_set_o
              @ ^ [X3: $o] : ( sup_sup_set_o @ A3 @ ( B @ X3 ) )
              @ C ) )
          = ( sup_sup_set_o @ A3 @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B @ C ) ) ) ) ) ) ).

% UN_simps(3)
thf(fact_440_UN__simps_I3_J,axiom,
    ! [C: set_o,A3: set_nat,B: $o > set_nat] :
      ( ( ( C = bot_bot_set_o )
       => ( ( comple7399068483239264473et_nat
            @ ( image_o_set_nat
              @ ^ [X3: $o] : ( sup_sup_set_nat @ A3 @ ( B @ X3 ) )
              @ C ) )
          = bot_bot_set_nat ) )
      & ( ( C != bot_bot_set_o )
       => ( ( comple7399068483239264473et_nat
            @ ( image_o_set_nat
              @ ^ [X3: $o] : ( sup_sup_set_nat @ A3 @ ( B @ X3 ) )
              @ C ) )
          = ( sup_sup_set_nat @ A3 @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ B @ C ) ) ) ) ) ) ).

% UN_simps(3)
thf(fact_441_UN__simps_I3_J,axiom,
    ! [C: set_nat,A3: set_rule,B: nat > set_rule] :
      ( ( ( C = bot_bot_set_nat )
       => ( ( comple7665822446944348827t_rule
            @ ( image_nat_set_rule
              @ ^ [X3: nat] : ( sup_sup_set_rule @ A3 @ ( B @ X3 ) )
              @ C ) )
          = bot_bot_set_rule ) )
      & ( ( C != bot_bot_set_nat )
       => ( ( comple7665822446944348827t_rule
            @ ( image_nat_set_rule
              @ ^ [X3: nat] : ( sup_sup_set_rule @ A3 @ ( B @ X3 ) )
              @ C ) )
          = ( sup_sup_set_rule @ A3 @ ( comple7665822446944348827t_rule @ ( image_nat_set_rule @ B @ C ) ) ) ) ) ) ).

% UN_simps(3)
thf(fact_442_UN__simps_I3_J,axiom,
    ! [C: set_nat,A3: set_o,B: nat > set_o] :
      ( ( ( C = bot_bot_set_nat )
       => ( ( comple90263536869209701_set_o
            @ ( image_nat_set_o
              @ ^ [X3: nat] : ( sup_sup_set_o @ A3 @ ( B @ X3 ) )
              @ C ) )
          = bot_bot_set_o ) )
      & ( ( C != bot_bot_set_nat )
       => ( ( comple90263536869209701_set_o
            @ ( image_nat_set_o
              @ ^ [X3: nat] : ( sup_sup_set_o @ A3 @ ( B @ X3 ) )
              @ C ) )
          = ( sup_sup_set_o @ A3 @ ( comple90263536869209701_set_o @ ( image_nat_set_o @ B @ C ) ) ) ) ) ) ).

% UN_simps(3)
thf(fact_443_UN__simps_I3_J,axiom,
    ! [C: set_nat,A3: set_nat,B: nat > set_nat] :
      ( ( ( C = bot_bot_set_nat )
       => ( ( comple7399068483239264473et_nat
            @ ( image_nat_set_nat
              @ ^ [X3: nat] : ( sup_sup_set_nat @ A3 @ ( B @ X3 ) )
              @ C ) )
          = bot_bot_set_nat ) )
      & ( ( C != bot_bot_set_nat )
       => ( ( comple7399068483239264473et_nat
            @ ( image_nat_set_nat
              @ ^ [X3: nat] : ( sup_sup_set_nat @ A3 @ ( B @ X3 ) )
              @ C ) )
          = ( sup_sup_set_nat @ A3 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B @ C ) ) ) ) ) ) ).

% UN_simps(3)
thf(fact_444_UN__simps_I3_J,axiom,
    ! [C: set_rule,A3: set_fm,B: rule > set_fm] :
      ( ( ( C = bot_bot_set_rule )
       => ( ( comple2134933779557159616set_fm
            @ ( image_rule_set_fm
              @ ^ [X3: rule] : ( sup_sup_set_fm @ A3 @ ( B @ X3 ) )
              @ C ) )
          = bot_bot_set_fm ) )
      & ( ( C != bot_bot_set_rule )
       => ( ( comple2134933779557159616set_fm
            @ ( image_rule_set_fm
              @ ^ [X3: rule] : ( sup_sup_set_fm @ A3 @ ( B @ X3 ) )
              @ C ) )
          = ( sup_sup_set_fm @ A3 @ ( comple2134933779557159616set_fm @ ( image_rule_set_fm @ B @ C ) ) ) ) ) ) ).

% UN_simps(3)
thf(fact_445_empty__iff,axiom,
    ! [C2: list_fm] :
      ~ ( member_list_fm3 @ C2 @ bot_bot_set_list_fm ) ).

% empty_iff
thf(fact_446_empty__iff,axiom,
    ! [C2: fm] :
      ~ ( member_fm3 @ C2 @ bot_bot_set_fm ) ).

% empty_iff
thf(fact_447_empty__iff,axiom,
    ! [C2: rule] :
      ~ ( member_rule3 @ C2 @ bot_bot_set_rule ) ).

% empty_iff
thf(fact_448_empty__iff,axiom,
    ! [C2: $o] :
      ~ ( member_o3 @ C2 @ bot_bot_set_o ) ).

% empty_iff
thf(fact_449_empty__iff,axiom,
    ! [C2: nat] :
      ~ ( member_nat3 @ C2 @ bot_bot_set_nat ) ).

% empty_iff
thf(fact_450_all__not__in__conv,axiom,
    ! [A3: set_list_fm] :
      ( ( ! [X3: list_fm] :
            ~ ( member_list_fm3 @ X3 @ A3 ) )
      = ( A3 = bot_bot_set_list_fm ) ) ).

% all_not_in_conv
thf(fact_451_all__not__in__conv,axiom,
    ! [A3: set_fm] :
      ( ( ! [X3: fm] :
            ~ ( member_fm3 @ X3 @ A3 ) )
      = ( A3 = bot_bot_set_fm ) ) ).

% all_not_in_conv
thf(fact_452_all__not__in__conv,axiom,
    ! [A3: set_rule] :
      ( ( ! [X3: rule] :
            ~ ( member_rule3 @ X3 @ A3 ) )
      = ( A3 = bot_bot_set_rule ) ) ).

% all_not_in_conv
thf(fact_453_all__not__in__conv,axiom,
    ! [A3: set_o] :
      ( ( ! [X3: $o] :
            ~ ( member_o3 @ X3 @ A3 ) )
      = ( A3 = bot_bot_set_o ) ) ).

% all_not_in_conv
thf(fact_454_all__not__in__conv,axiom,
    ! [A3: set_nat] :
      ( ( ! [X3: nat] :
            ~ ( member_nat3 @ X3 @ A3 ) )
      = ( A3 = bot_bot_set_nat ) ) ).

% all_not_in_conv
thf(fact_455_Collect__empty__eq,axiom,
    ! [P3: rule > $o] :
      ( ( ( collect_rule @ P3 )
        = bot_bot_set_rule )
      = ( ! [X3: rule] :
            ~ ( P3 @ X3 ) ) ) ).

% Collect_empty_eq
thf(fact_456_Collect__empty__eq,axiom,
    ! [P3: $o > $o] :
      ( ( ( collect_o @ P3 )
        = bot_bot_set_o )
      = ( ! [X3: $o] :
            ~ ( P3 @ X3 ) ) ) ).

% Collect_empty_eq
thf(fact_457_Collect__empty__eq,axiom,
    ! [P3: nat > $o] :
      ( ( ( collect_nat @ P3 )
        = bot_bot_set_nat )
      = ( ! [X3: nat] :
            ~ ( P3 @ X3 ) ) ) ).

% Collect_empty_eq
thf(fact_458_empty__Collect__eq,axiom,
    ! [P3: rule > $o] :
      ( ( bot_bot_set_rule
        = ( collect_rule @ P3 ) )
      = ( ! [X3: rule] :
            ~ ( P3 @ X3 ) ) ) ).

% empty_Collect_eq
thf(fact_459_empty__Collect__eq,axiom,
    ! [P3: $o > $o] :
      ( ( bot_bot_set_o
        = ( collect_o @ P3 ) )
      = ( ! [X3: $o] :
            ~ ( P3 @ X3 ) ) ) ).

% empty_Collect_eq
thf(fact_460_empty__Collect__eq,axiom,
    ! [P3: nat > $o] :
      ( ( bot_bot_set_nat
        = ( collect_nat @ P3 ) )
      = ( ! [X3: nat] :
            ~ ( P3 @ X3 ) ) ) ).

% empty_Collect_eq
thf(fact_461_image__is__empty,axiom,
    ! [F: produc340336539035504054m_rule > set_fm,A3: set_Pr1822751329126368876m_rule] :
      ( ( ( image_377115208147135190set_fm @ F @ A3 )
        = bot_bot_set_set_fm )
      = ( A3 = bot_bo3426410157013319936m_rule ) ) ).

% image_is_empty
thf(fact_462_image__is__empty,axiom,
    ! [F: rule > rule,A3: set_rule] :
      ( ( ( image_rule_rule @ F @ A3 )
        = bot_bot_set_rule )
      = ( A3 = bot_bot_set_rule ) ) ).

% image_is_empty
thf(fact_463_image__is__empty,axiom,
    ! [F: $o > rule,A3: set_o] :
      ( ( ( image_o_rule @ F @ A3 )
        = bot_bot_set_rule )
      = ( A3 = bot_bot_set_o ) ) ).

% image_is_empty
thf(fact_464_image__is__empty,axiom,
    ! [F: nat > rule,A3: set_nat] :
      ( ( ( image_nat_rule @ F @ A3 )
        = bot_bot_set_rule )
      = ( A3 = bot_bot_set_nat ) ) ).

% image_is_empty
thf(fact_465_image__is__empty,axiom,
    ! [F: rule > $o,A3: set_rule] :
      ( ( ( image_rule_o @ F @ A3 )
        = bot_bot_set_o )
      = ( A3 = bot_bot_set_rule ) ) ).

% image_is_empty
thf(fact_466_image__is__empty,axiom,
    ! [F: $o > $o,A3: set_o] :
      ( ( ( image_o_o @ F @ A3 )
        = bot_bot_set_o )
      = ( A3 = bot_bot_set_o ) ) ).

% image_is_empty
thf(fact_467_image__is__empty,axiom,
    ! [F: nat > $o,A3: set_nat] :
      ( ( ( image_nat_o @ F @ A3 )
        = bot_bot_set_o )
      = ( A3 = bot_bot_set_nat ) ) ).

% image_is_empty
thf(fact_468_image__is__empty,axiom,
    ! [F: rule > nat,A3: set_rule] :
      ( ( ( image_rule_nat @ F @ A3 )
        = bot_bot_set_nat )
      = ( A3 = bot_bot_set_rule ) ) ).

% image_is_empty
thf(fact_469_image__is__empty,axiom,
    ! [F: $o > nat,A3: set_o] :
      ( ( ( image_o_nat @ F @ A3 )
        = bot_bot_set_nat )
      = ( A3 = bot_bot_set_o ) ) ).

% image_is_empty
thf(fact_470_image__is__empty,axiom,
    ! [F: nat > nat,A3: set_nat] :
      ( ( ( image_nat_nat @ F @ A3 )
        = bot_bot_set_nat )
      = ( A3 = bot_bot_set_nat ) ) ).

% image_is_empty
thf(fact_471_empty__is__image,axiom,
    ! [F: produc340336539035504054m_rule > set_fm,A3: set_Pr1822751329126368876m_rule] :
      ( ( bot_bot_set_set_fm
        = ( image_377115208147135190set_fm @ F @ A3 ) )
      = ( A3 = bot_bo3426410157013319936m_rule ) ) ).

% empty_is_image
thf(fact_472_empty__is__image,axiom,
    ! [F: rule > rule,A3: set_rule] :
      ( ( bot_bot_set_rule
        = ( image_rule_rule @ F @ A3 ) )
      = ( A3 = bot_bot_set_rule ) ) ).

% empty_is_image
thf(fact_473_empty__is__image,axiom,
    ! [F: $o > rule,A3: set_o] :
      ( ( bot_bot_set_rule
        = ( image_o_rule @ F @ A3 ) )
      = ( A3 = bot_bot_set_o ) ) ).

% empty_is_image
thf(fact_474_empty__is__image,axiom,
    ! [F: nat > rule,A3: set_nat] :
      ( ( bot_bot_set_rule
        = ( image_nat_rule @ F @ A3 ) )
      = ( A3 = bot_bot_set_nat ) ) ).

% empty_is_image
thf(fact_475_empty__is__image,axiom,
    ! [F: rule > $o,A3: set_rule] :
      ( ( bot_bot_set_o
        = ( image_rule_o @ F @ A3 ) )
      = ( A3 = bot_bot_set_rule ) ) ).

% empty_is_image
thf(fact_476_empty__is__image,axiom,
    ! [F: $o > $o,A3: set_o] :
      ( ( bot_bot_set_o
        = ( image_o_o @ F @ A3 ) )
      = ( A3 = bot_bot_set_o ) ) ).

% empty_is_image
thf(fact_477_empty__is__image,axiom,
    ! [F: nat > $o,A3: set_nat] :
      ( ( bot_bot_set_o
        = ( image_nat_o @ F @ A3 ) )
      = ( A3 = bot_bot_set_nat ) ) ).

% empty_is_image
thf(fact_478_empty__is__image,axiom,
    ! [F: rule > nat,A3: set_rule] :
      ( ( bot_bot_set_nat
        = ( image_rule_nat @ F @ A3 ) )
      = ( A3 = bot_bot_set_rule ) ) ).

% empty_is_image
thf(fact_479_empty__is__image,axiom,
    ! [F: $o > nat,A3: set_o] :
      ( ( bot_bot_set_nat
        = ( image_o_nat @ F @ A3 ) )
      = ( A3 = bot_bot_set_o ) ) ).

% empty_is_image
thf(fact_480_empty__is__image,axiom,
    ! [F: nat > nat,A3: set_nat] :
      ( ( bot_bot_set_nat
        = ( image_nat_nat @ F @ A3 ) )
      = ( A3 = bot_bot_set_nat ) ) ).

% empty_is_image
thf(fact_481_image__empty,axiom,
    ! [F: produc340336539035504054m_rule > set_fm] :
      ( ( image_377115208147135190set_fm @ F @ bot_bo3426410157013319936m_rule )
      = bot_bot_set_set_fm ) ).

% image_empty
thf(fact_482_image__empty,axiom,
    ! [F: rule > rule] :
      ( ( image_rule_rule @ F @ bot_bot_set_rule )
      = bot_bot_set_rule ) ).

% image_empty
thf(fact_483_image__empty,axiom,
    ! [F: rule > $o] :
      ( ( image_rule_o @ F @ bot_bot_set_rule )
      = bot_bot_set_o ) ).

% image_empty
thf(fact_484_image__empty,axiom,
    ! [F: rule > nat] :
      ( ( image_rule_nat @ F @ bot_bot_set_rule )
      = bot_bot_set_nat ) ).

% image_empty
thf(fact_485_image__empty,axiom,
    ! [F: $o > rule] :
      ( ( image_o_rule @ F @ bot_bot_set_o )
      = bot_bot_set_rule ) ).

% image_empty
thf(fact_486_image__empty,axiom,
    ! [F: $o > $o] :
      ( ( image_o_o @ F @ bot_bot_set_o )
      = bot_bot_set_o ) ).

% image_empty
thf(fact_487_image__empty,axiom,
    ! [F: $o > nat] :
      ( ( image_o_nat @ F @ bot_bot_set_o )
      = bot_bot_set_nat ) ).

% image_empty
thf(fact_488_image__empty,axiom,
    ! [F: nat > rule] :
      ( ( image_nat_rule @ F @ bot_bot_set_nat )
      = bot_bot_set_rule ) ).

% image_empty
thf(fact_489_image__empty,axiom,
    ! [F: nat > $o] :
      ( ( image_nat_o @ F @ bot_bot_set_nat )
      = bot_bot_set_o ) ).

% image_empty
thf(fact_490_image__empty,axiom,
    ! [F: nat > nat] :
      ( ( image_nat_nat @ F @ bot_bot_set_nat )
      = bot_bot_set_nat ) ).

% image_empty
thf(fact_491_sup__bot_Oright__neutral,axiom,
    ! [A: set_rule] :
      ( ( sup_sup_set_rule @ A @ bot_bot_set_rule )
      = A ) ).

% sup_bot.right_neutral
thf(fact_492_sup__bot_Oright__neutral,axiom,
    ! [A: set_o] :
      ( ( sup_sup_set_o @ A @ bot_bot_set_o )
      = A ) ).

% sup_bot.right_neutral
thf(fact_493_sup__bot_Oright__neutral,axiom,
    ! [A: set_nat] :
      ( ( sup_sup_set_nat @ A @ bot_bot_set_nat )
      = A ) ).

% sup_bot.right_neutral
thf(fact_494_sup__bot_Oneutr__eq__iff,axiom,
    ! [A: set_rule,B2: set_rule] :
      ( ( bot_bot_set_rule
        = ( sup_sup_set_rule @ A @ B2 ) )
      = ( ( A = bot_bot_set_rule )
        & ( B2 = bot_bot_set_rule ) ) ) ).

% sup_bot.neutr_eq_iff
thf(fact_495_sup__bot_Oneutr__eq__iff,axiom,
    ! [A: set_o,B2: set_o] :
      ( ( bot_bot_set_o
        = ( sup_sup_set_o @ A @ B2 ) )
      = ( ( A = bot_bot_set_o )
        & ( B2 = bot_bot_set_o ) ) ) ).

% sup_bot.neutr_eq_iff
thf(fact_496_sup__bot_Oneutr__eq__iff,axiom,
    ! [A: set_nat,B2: set_nat] :
      ( ( bot_bot_set_nat
        = ( sup_sup_set_nat @ A @ B2 ) )
      = ( ( A = bot_bot_set_nat )
        & ( B2 = bot_bot_set_nat ) ) ) ).

% sup_bot.neutr_eq_iff
thf(fact_497_sup__bot_Oleft__neutral,axiom,
    ! [A: set_rule] :
      ( ( sup_sup_set_rule @ bot_bot_set_rule @ A )
      = A ) ).

% sup_bot.left_neutral
thf(fact_498_sup__bot_Oleft__neutral,axiom,
    ! [A: set_o] :
      ( ( sup_sup_set_o @ bot_bot_set_o @ A )
      = A ) ).

% sup_bot.left_neutral
thf(fact_499_sup__bot_Oleft__neutral,axiom,
    ! [A: set_nat] :
      ( ( sup_sup_set_nat @ bot_bot_set_nat @ A )
      = A ) ).

% sup_bot.left_neutral
thf(fact_500_sup__bot_Oeq__neutr__iff,axiom,
    ! [A: set_rule,B2: set_rule] :
      ( ( ( sup_sup_set_rule @ A @ B2 )
        = bot_bot_set_rule )
      = ( ( A = bot_bot_set_rule )
        & ( B2 = bot_bot_set_rule ) ) ) ).

% sup_bot.eq_neutr_iff
thf(fact_501_sup__bot_Oeq__neutr__iff,axiom,
    ! [A: set_o,B2: set_o] :
      ( ( ( sup_sup_set_o @ A @ B2 )
        = bot_bot_set_o )
      = ( ( A = bot_bot_set_o )
        & ( B2 = bot_bot_set_o ) ) ) ).

% sup_bot.eq_neutr_iff
thf(fact_502_sup__bot_Oeq__neutr__iff,axiom,
    ! [A: set_nat,B2: set_nat] :
      ( ( ( sup_sup_set_nat @ A @ B2 )
        = bot_bot_set_nat )
      = ( ( A = bot_bot_set_nat )
        & ( B2 = bot_bot_set_nat ) ) ) ).

% sup_bot.eq_neutr_iff
thf(fact_503_sup__eq__bot__iff,axiom,
    ! [X: set_rule,Y: set_rule] :
      ( ( ( sup_sup_set_rule @ X @ Y )
        = bot_bot_set_rule )
      = ( ( X = bot_bot_set_rule )
        & ( Y = bot_bot_set_rule ) ) ) ).

% sup_eq_bot_iff
thf(fact_504_sup__eq__bot__iff,axiom,
    ! [X: set_o,Y: set_o] :
      ( ( ( sup_sup_set_o @ X @ Y )
        = bot_bot_set_o )
      = ( ( X = bot_bot_set_o )
        & ( Y = bot_bot_set_o ) ) ) ).

% sup_eq_bot_iff
thf(fact_505_sup__eq__bot__iff,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( ( sup_sup_set_nat @ X @ Y )
        = bot_bot_set_nat )
      = ( ( X = bot_bot_set_nat )
        & ( Y = bot_bot_set_nat ) ) ) ).

% sup_eq_bot_iff
thf(fact_506_bot__eq__sup__iff,axiom,
    ! [X: set_rule,Y: set_rule] :
      ( ( bot_bot_set_rule
        = ( sup_sup_set_rule @ X @ Y ) )
      = ( ( X = bot_bot_set_rule )
        & ( Y = bot_bot_set_rule ) ) ) ).

% bot_eq_sup_iff
thf(fact_507_bot__eq__sup__iff,axiom,
    ! [X: set_o,Y: set_o] :
      ( ( bot_bot_set_o
        = ( sup_sup_set_o @ X @ Y ) )
      = ( ( X = bot_bot_set_o )
        & ( Y = bot_bot_set_o ) ) ) ).

% bot_eq_sup_iff
thf(fact_508_bot__eq__sup__iff,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( bot_bot_set_nat
        = ( sup_sup_set_nat @ X @ Y ) )
      = ( ( X = bot_bot_set_nat )
        & ( Y = bot_bot_set_nat ) ) ) ).

% bot_eq_sup_iff
thf(fact_509_sup__bot__right,axiom,
    ! [X: set_rule] :
      ( ( sup_sup_set_rule @ X @ bot_bot_set_rule )
      = X ) ).

% sup_bot_right
thf(fact_510_sup__bot__right,axiom,
    ! [X: set_o] :
      ( ( sup_sup_set_o @ X @ bot_bot_set_o )
      = X ) ).

% sup_bot_right
thf(fact_511_sup__bot__right,axiom,
    ! [X: set_nat] :
      ( ( sup_sup_set_nat @ X @ bot_bot_set_nat )
      = X ) ).

% sup_bot_right
thf(fact_512_sup__bot__left,axiom,
    ! [X: set_rule] :
      ( ( sup_sup_set_rule @ bot_bot_set_rule @ X )
      = X ) ).

% sup_bot_left
thf(fact_513_sup__bot__left,axiom,
    ! [X: set_o] :
      ( ( sup_sup_set_o @ bot_bot_set_o @ X )
      = X ) ).

% sup_bot_left
thf(fact_514_sup__bot__left,axiom,
    ! [X: set_nat] :
      ( ( sup_sup_set_nat @ bot_bot_set_nat @ X )
      = X ) ).

% sup_bot_left
thf(fact_515_Sup__bot__conv_I1_J,axiom,
    ! [A3: set_set_rule] :
      ( ( ( comple7665822446944348827t_rule @ A3 )
        = bot_bot_set_rule )
      = ( ! [X3: set_rule] :
            ( ( member_set_rule @ X3 @ A3 )
           => ( X3 = bot_bot_set_rule ) ) ) ) ).

% Sup_bot_conv(1)
thf(fact_516_Sup__bot__conv_I1_J,axiom,
    ! [A3: set_set_o] :
      ( ( ( comple90263536869209701_set_o @ A3 )
        = bot_bot_set_o )
      = ( ! [X3: set_o] :
            ( ( member_set_o @ X3 @ A3 )
           => ( X3 = bot_bot_set_o ) ) ) ) ).

% Sup_bot_conv(1)
thf(fact_517_Sup__bot__conv_I1_J,axiom,
    ! [A3: set_set_nat] :
      ( ( ( comple7399068483239264473et_nat @ A3 )
        = bot_bot_set_nat )
      = ( ! [X3: set_nat] :
            ( ( member_set_nat @ X3 @ A3 )
           => ( X3 = bot_bot_set_nat ) ) ) ) ).

% Sup_bot_conv(1)
thf(fact_518_Sup__bot__conv_I1_J,axiom,
    ! [A3: set_set_fm] :
      ( ( ( comple2134933779557159616set_fm @ A3 )
        = bot_bot_set_fm )
      = ( ! [X3: set_fm] :
            ( ( member_set_fm @ X3 @ A3 )
           => ( X3 = bot_bot_set_fm ) ) ) ) ).

% Sup_bot_conv(1)
thf(fact_519_Sup__bot__conv_I1_J,axiom,
    ! [A3: set_o] :
      ( ( ( complete_Sup_Sup_o @ A3 )
        = bot_bot_o )
      = ( ! [X3: $o] :
            ( ( member_o3 @ X3 @ A3 )
           => ( X3 = bot_bot_o ) ) ) ) ).

% Sup_bot_conv(1)
thf(fact_520_Sup__bot__conv_I2_J,axiom,
    ! [A3: set_set_rule] :
      ( ( bot_bot_set_rule
        = ( comple7665822446944348827t_rule @ A3 ) )
      = ( ! [X3: set_rule] :
            ( ( member_set_rule @ X3 @ A3 )
           => ( X3 = bot_bot_set_rule ) ) ) ) ).

% Sup_bot_conv(2)
thf(fact_521_Sup__bot__conv_I2_J,axiom,
    ! [A3: set_set_o] :
      ( ( bot_bot_set_o
        = ( comple90263536869209701_set_o @ A3 ) )
      = ( ! [X3: set_o] :
            ( ( member_set_o @ X3 @ A3 )
           => ( X3 = bot_bot_set_o ) ) ) ) ).

% Sup_bot_conv(2)
thf(fact_522_Sup__bot__conv_I2_J,axiom,
    ! [A3: set_set_nat] :
      ( ( bot_bot_set_nat
        = ( comple7399068483239264473et_nat @ A3 ) )
      = ( ! [X3: set_nat] :
            ( ( member_set_nat @ X3 @ A3 )
           => ( X3 = bot_bot_set_nat ) ) ) ) ).

% Sup_bot_conv(2)
thf(fact_523_Sup__bot__conv_I2_J,axiom,
    ! [A3: set_set_fm] :
      ( ( bot_bot_set_fm
        = ( comple2134933779557159616set_fm @ A3 ) )
      = ( ! [X3: set_fm] :
            ( ( member_set_fm @ X3 @ A3 )
           => ( X3 = bot_bot_set_fm ) ) ) ) ).

% Sup_bot_conv(2)
thf(fact_524_Sup__bot__conv_I2_J,axiom,
    ! [A3: set_o] :
      ( ( bot_bot_o
        = ( complete_Sup_Sup_o @ A3 ) )
      = ( ! [X3: $o] :
            ( ( member_o3 @ X3 @ A3 )
           => ( X3 = bot_bot_o ) ) ) ) ).

% Sup_bot_conv(2)
thf(fact_525_ball__empty,axiom,
    ! [P3: rule > $o,X6: rule] :
      ( ( member_rule3 @ X6 @ bot_bot_set_rule )
     => ( P3 @ X6 ) ) ).

% ball_empty
thf(fact_526_ball__empty,axiom,
    ! [P3: $o > $o,X6: $o] :
      ( ( member_o3 @ X6 @ bot_bot_set_o )
     => ( P3 @ X6 ) ) ).

% ball_empty
thf(fact_527_ball__empty,axiom,
    ! [P3: nat > $o,X6: nat] :
      ( ( member_nat3 @ X6 @ bot_bot_set_nat )
     => ( P3 @ X6 ) ) ).

% ball_empty
thf(fact_528_Un__empty,axiom,
    ! [A3: set_rule,B: set_rule] :
      ( ( ( sup_sup_set_rule @ A3 @ B )
        = bot_bot_set_rule )
      = ( ( A3 = bot_bot_set_rule )
        & ( B = bot_bot_set_rule ) ) ) ).

% Un_empty
thf(fact_529_Un__empty,axiom,
    ! [A3: set_o,B: set_o] :
      ( ( ( sup_sup_set_o @ A3 @ B )
        = bot_bot_set_o )
      = ( ( A3 = bot_bot_set_o )
        & ( B = bot_bot_set_o ) ) ) ).

% Un_empty
thf(fact_530_Un__empty,axiom,
    ! [A3: set_nat,B: set_nat] :
      ( ( ( sup_sup_set_nat @ A3 @ B )
        = bot_bot_set_nat )
      = ( ( A3 = bot_bot_set_nat )
        & ( B = bot_bot_set_nat ) ) ) ).

% Un_empty
thf(fact_531_Sup__empty,axiom,
    ( ( comple7665822446944348827t_rule @ bot_bot_set_set_rule )
    = bot_bot_set_rule ) ).

% Sup_empty
thf(fact_532_Sup__empty,axiom,
    ( ( comple90263536869209701_set_o @ bot_bot_set_set_o )
    = bot_bot_set_o ) ).

% Sup_empty
thf(fact_533_Sup__empty,axiom,
    ( ( comple7399068483239264473et_nat @ bot_bot_set_set_nat )
    = bot_bot_set_nat ) ).

% Sup_empty
thf(fact_534_Sup__empty,axiom,
    ( ( comple2134933779557159616set_fm @ bot_bot_set_set_fm )
    = bot_bot_set_fm ) ).

% Sup_empty
thf(fact_535_Sup__empty,axiom,
    ( ( complete_Sup_Sup_o @ bot_bot_set_o )
    = bot_bot_o ) ).

% Sup_empty
thf(fact_536_SUP__bot,axiom,
    ! [A3: set_Pr1822751329126368876m_rule] :
      ( ( comple2134933779557159616set_fm
        @ ( image_377115208147135190set_fm
          @ ^ [X3: produc340336539035504054m_rule] : bot_bot_set_fm
          @ A3 ) )
      = bot_bot_set_fm ) ).

% SUP_bot
thf(fact_537_SUP__bot__conv_I1_J,axiom,
    ! [B: produc340336539035504054m_rule > set_fm,A3: set_Pr1822751329126368876m_rule] :
      ( ( ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ B @ A3 ) )
        = bot_bot_set_fm )
      = ( ! [X3: produc340336539035504054m_rule] :
            ( ( member7231649785386036813m_rule @ X3 @ A3 )
           => ( ( B @ X3 )
              = bot_bot_set_fm ) ) ) ) ).

% SUP_bot_conv(1)
thf(fact_538_SUP__bot__conv_I2_J,axiom,
    ! [B: produc340336539035504054m_rule > set_fm,A3: set_Pr1822751329126368876m_rule] :
      ( ( bot_bot_set_fm
        = ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ B @ A3 ) ) )
      = ( ! [X3: produc340336539035504054m_rule] :
            ( ( member7231649785386036813m_rule @ X3 @ A3 )
           => ( ( B @ X3 )
              = bot_bot_set_fm ) ) ) ) ).

% SUP_bot_conv(2)
thf(fact_539_SUP__const,axiom,
    ! [A3: set_Pr1822751329126368876m_rule,F: set_fm] :
      ( ( A3 != bot_bo3426410157013319936m_rule )
     => ( ( comple2134933779557159616set_fm
          @ ( image_377115208147135190set_fm
            @ ^ [I: produc340336539035504054m_rule] : F
            @ A3 ) )
        = F ) ) ).

% SUP_const
thf(fact_540_SUP__const,axiom,
    ! [A3: set_rule,F: set_fm] :
      ( ( A3 != bot_bot_set_rule )
     => ( ( comple2134933779557159616set_fm
          @ ( image_rule_set_fm
            @ ^ [I: rule] : F
            @ A3 ) )
        = F ) ) ).

% SUP_const
thf(fact_541_SUP__const,axiom,
    ! [A3: set_o,F: set_fm] :
      ( ( A3 != bot_bot_set_o )
     => ( ( comple2134933779557159616set_fm
          @ ( image_o_set_fm
            @ ^ [I: $o] : F
            @ A3 ) )
        = F ) ) ).

% SUP_const
thf(fact_542_SUP__const,axiom,
    ! [A3: set_nat,F: set_fm] :
      ( ( A3 != bot_bot_set_nat )
     => ( ( comple2134933779557159616set_fm
          @ ( image_nat_set_fm
            @ ^ [I: nat] : F
            @ A3 ) )
        = F ) ) ).

% SUP_const
thf(fact_543_SUP__const,axiom,
    ! [A3: set_rule,F: $o] :
      ( ( A3 != bot_bot_set_rule )
     => ( ( complete_Sup_Sup_o
          @ ( image_rule_o
            @ ^ [I: rule] : F
            @ A3 ) )
        = F ) ) ).

% SUP_const
thf(fact_544_SUP__const,axiom,
    ! [A3: set_o,F: $o] :
      ( ( A3 != bot_bot_set_o )
     => ( ( complete_Sup_Sup_o
          @ ( image_o_o
            @ ^ [I: $o] : F
            @ A3 ) )
        = F ) ) ).

% SUP_const
thf(fact_545_SUP__const,axiom,
    ! [A3: set_nat,F: $o] :
      ( ( A3 != bot_bot_set_nat )
     => ( ( complete_Sup_Sup_o
          @ ( image_nat_o
            @ ^ [I: nat] : F
            @ A3 ) )
        = F ) ) ).

% SUP_const
thf(fact_546_UN__constant,axiom,
    ! [A3: set_rule,C2: set_rule] :
      ( ( ( A3 = bot_bot_set_rule )
       => ( ( comple7665822446944348827t_rule
            @ ( image_rule_set_rule
              @ ^ [Y2: rule] : C2
              @ A3 ) )
          = bot_bot_set_rule ) )
      & ( ( A3 != bot_bot_set_rule )
       => ( ( comple7665822446944348827t_rule
            @ ( image_rule_set_rule
              @ ^ [Y2: rule] : C2
              @ A3 ) )
          = C2 ) ) ) ).

% UN_constant
thf(fact_547_UN__constant,axiom,
    ! [A3: set_rule,C2: set_o] :
      ( ( ( A3 = bot_bot_set_rule )
       => ( ( comple90263536869209701_set_o
            @ ( image_rule_set_o
              @ ^ [Y2: rule] : C2
              @ A3 ) )
          = bot_bot_set_o ) )
      & ( ( A3 != bot_bot_set_rule )
       => ( ( comple90263536869209701_set_o
            @ ( image_rule_set_o
              @ ^ [Y2: rule] : C2
              @ A3 ) )
          = C2 ) ) ) ).

% UN_constant
thf(fact_548_UN__constant,axiom,
    ! [A3: set_rule,C2: set_nat] :
      ( ( ( A3 = bot_bot_set_rule )
       => ( ( comple7399068483239264473et_nat
            @ ( image_rule_set_nat
              @ ^ [Y2: rule] : C2
              @ A3 ) )
          = bot_bot_set_nat ) )
      & ( ( A3 != bot_bot_set_rule )
       => ( ( comple7399068483239264473et_nat
            @ ( image_rule_set_nat
              @ ^ [Y2: rule] : C2
              @ A3 ) )
          = C2 ) ) ) ).

% UN_constant
thf(fact_549_UN__constant,axiom,
    ! [A3: set_o,C2: set_rule] :
      ( ( ( A3 = bot_bot_set_o )
       => ( ( comple7665822446944348827t_rule
            @ ( image_o_set_rule
              @ ^ [Y2: $o] : C2
              @ A3 ) )
          = bot_bot_set_rule ) )
      & ( ( A3 != bot_bot_set_o )
       => ( ( comple7665822446944348827t_rule
            @ ( image_o_set_rule
              @ ^ [Y2: $o] : C2
              @ A3 ) )
          = C2 ) ) ) ).

% UN_constant
thf(fact_550_UN__constant,axiom,
    ! [A3: set_o,C2: set_o] :
      ( ( ( A3 = bot_bot_set_o )
       => ( ( comple90263536869209701_set_o
            @ ( image_o_set_o
              @ ^ [Y2: $o] : C2
              @ A3 ) )
          = bot_bot_set_o ) )
      & ( ( A3 != bot_bot_set_o )
       => ( ( comple90263536869209701_set_o
            @ ( image_o_set_o
              @ ^ [Y2: $o] : C2
              @ A3 ) )
          = C2 ) ) ) ).

% UN_constant
thf(fact_551_UN__constant,axiom,
    ! [A3: set_o,C2: set_nat] :
      ( ( ( A3 = bot_bot_set_o )
       => ( ( comple7399068483239264473et_nat
            @ ( image_o_set_nat
              @ ^ [Y2: $o] : C2
              @ A3 ) )
          = bot_bot_set_nat ) )
      & ( ( A3 != bot_bot_set_o )
       => ( ( comple7399068483239264473et_nat
            @ ( image_o_set_nat
              @ ^ [Y2: $o] : C2
              @ A3 ) )
          = C2 ) ) ) ).

% UN_constant
thf(fact_552_UN__constant,axiom,
    ! [A3: set_nat,C2: set_rule] :
      ( ( ( A3 = bot_bot_set_nat )
       => ( ( comple7665822446944348827t_rule
            @ ( image_nat_set_rule
              @ ^ [Y2: nat] : C2
              @ A3 ) )
          = bot_bot_set_rule ) )
      & ( ( A3 != bot_bot_set_nat )
       => ( ( comple7665822446944348827t_rule
            @ ( image_nat_set_rule
              @ ^ [Y2: nat] : C2
              @ A3 ) )
          = C2 ) ) ) ).

% UN_constant
thf(fact_553_UN__constant,axiom,
    ! [A3: set_nat,C2: set_o] :
      ( ( ( A3 = bot_bot_set_nat )
       => ( ( comple90263536869209701_set_o
            @ ( image_nat_set_o
              @ ^ [Y2: nat] : C2
              @ A3 ) )
          = bot_bot_set_o ) )
      & ( ( A3 != bot_bot_set_nat )
       => ( ( comple90263536869209701_set_o
            @ ( image_nat_set_o
              @ ^ [Y2: nat] : C2
              @ A3 ) )
          = C2 ) ) ) ).

% UN_constant
thf(fact_554_UN__constant,axiom,
    ! [A3: set_nat,C2: set_nat] :
      ( ( ( A3 = bot_bot_set_nat )
       => ( ( comple7399068483239264473et_nat
            @ ( image_nat_set_nat
              @ ^ [Y2: nat] : C2
              @ A3 ) )
          = bot_bot_set_nat ) )
      & ( ( A3 != bot_bot_set_nat )
       => ( ( comple7399068483239264473et_nat
            @ ( image_nat_set_nat
              @ ^ [Y2: nat] : C2
              @ A3 ) )
          = C2 ) ) ) ).

% UN_constant
thf(fact_555_UN__constant,axiom,
    ! [A3: set_rule,C2: set_fm] :
      ( ( ( A3 = bot_bot_set_rule )
       => ( ( comple2134933779557159616set_fm
            @ ( image_rule_set_fm
              @ ^ [Y2: rule] : C2
              @ A3 ) )
          = bot_bot_set_fm ) )
      & ( ( A3 != bot_bot_set_rule )
       => ( ( comple2134933779557159616set_fm
            @ ( image_rule_set_fm
              @ ^ [Y2: rule] : C2
              @ A3 ) )
          = C2 ) ) ) ).

% UN_constant
thf(fact_556_UN__simps_I2_J,axiom,
    ! [C: set_rule,A3: rule > set_rule,B: set_rule] :
      ( ( ( C = bot_bot_set_rule )
       => ( ( comple7665822446944348827t_rule
            @ ( image_rule_set_rule
              @ ^ [X3: rule] : ( sup_sup_set_rule @ ( A3 @ X3 ) @ B )
              @ C ) )
          = bot_bot_set_rule ) )
      & ( ( C != bot_bot_set_rule )
       => ( ( comple7665822446944348827t_rule
            @ ( image_rule_set_rule
              @ ^ [X3: rule] : ( sup_sup_set_rule @ ( A3 @ X3 ) @ B )
              @ C ) )
          = ( sup_sup_set_rule @ ( comple7665822446944348827t_rule @ ( image_rule_set_rule @ A3 @ C ) ) @ B ) ) ) ) ).

% UN_simps(2)
thf(fact_557_UN__simps_I2_J,axiom,
    ! [C: set_rule,A3: rule > set_o,B: set_o] :
      ( ( ( C = bot_bot_set_rule )
       => ( ( comple90263536869209701_set_o
            @ ( image_rule_set_o
              @ ^ [X3: rule] : ( sup_sup_set_o @ ( A3 @ X3 ) @ B )
              @ C ) )
          = bot_bot_set_o ) )
      & ( ( C != bot_bot_set_rule )
       => ( ( comple90263536869209701_set_o
            @ ( image_rule_set_o
              @ ^ [X3: rule] : ( sup_sup_set_o @ ( A3 @ X3 ) @ B )
              @ C ) )
          = ( sup_sup_set_o @ ( comple90263536869209701_set_o @ ( image_rule_set_o @ A3 @ C ) ) @ B ) ) ) ) ).

% UN_simps(2)
thf(fact_558_UN__simps_I2_J,axiom,
    ! [C: set_rule,A3: rule > set_nat,B: set_nat] :
      ( ( ( C = bot_bot_set_rule )
       => ( ( comple7399068483239264473et_nat
            @ ( image_rule_set_nat
              @ ^ [X3: rule] : ( sup_sup_set_nat @ ( A3 @ X3 ) @ B )
              @ C ) )
          = bot_bot_set_nat ) )
      & ( ( C != bot_bot_set_rule )
       => ( ( comple7399068483239264473et_nat
            @ ( image_rule_set_nat
              @ ^ [X3: rule] : ( sup_sup_set_nat @ ( A3 @ X3 ) @ B )
              @ C ) )
          = ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ ( image_rule_set_nat @ A3 @ C ) ) @ B ) ) ) ) ).

% UN_simps(2)
thf(fact_559_UN__simps_I2_J,axiom,
    ! [C: set_o,A3: $o > set_rule,B: set_rule] :
      ( ( ( C = bot_bot_set_o )
       => ( ( comple7665822446944348827t_rule
            @ ( image_o_set_rule
              @ ^ [X3: $o] : ( sup_sup_set_rule @ ( A3 @ X3 ) @ B )
              @ C ) )
          = bot_bot_set_rule ) )
      & ( ( C != bot_bot_set_o )
       => ( ( comple7665822446944348827t_rule
            @ ( image_o_set_rule
              @ ^ [X3: $o] : ( sup_sup_set_rule @ ( A3 @ X3 ) @ B )
              @ C ) )
          = ( sup_sup_set_rule @ ( comple7665822446944348827t_rule @ ( image_o_set_rule @ A3 @ C ) ) @ B ) ) ) ) ).

% UN_simps(2)
thf(fact_560_UN__simps_I2_J,axiom,
    ! [C: set_o,A3: $o > set_o,B: set_o] :
      ( ( ( C = bot_bot_set_o )
       => ( ( comple90263536869209701_set_o
            @ ( image_o_set_o
              @ ^ [X3: $o] : ( sup_sup_set_o @ ( A3 @ X3 ) @ B )
              @ C ) )
          = bot_bot_set_o ) )
      & ( ( C != bot_bot_set_o )
       => ( ( comple90263536869209701_set_o
            @ ( image_o_set_o
              @ ^ [X3: $o] : ( sup_sup_set_o @ ( A3 @ X3 ) @ B )
              @ C ) )
          = ( sup_sup_set_o @ ( comple90263536869209701_set_o @ ( image_o_set_o @ A3 @ C ) ) @ B ) ) ) ) ).

% UN_simps(2)
thf(fact_561_UN__simps_I2_J,axiom,
    ! [C: set_o,A3: $o > set_nat,B: set_nat] :
      ( ( ( C = bot_bot_set_o )
       => ( ( comple7399068483239264473et_nat
            @ ( image_o_set_nat
              @ ^ [X3: $o] : ( sup_sup_set_nat @ ( A3 @ X3 ) @ B )
              @ C ) )
          = bot_bot_set_nat ) )
      & ( ( C != bot_bot_set_o )
       => ( ( comple7399068483239264473et_nat
            @ ( image_o_set_nat
              @ ^ [X3: $o] : ( sup_sup_set_nat @ ( A3 @ X3 ) @ B )
              @ C ) )
          = ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ A3 @ C ) ) @ B ) ) ) ) ).

% UN_simps(2)
thf(fact_562_UN__simps_I2_J,axiom,
    ! [C: set_nat,A3: nat > set_rule,B: set_rule] :
      ( ( ( C = bot_bot_set_nat )
       => ( ( comple7665822446944348827t_rule
            @ ( image_nat_set_rule
              @ ^ [X3: nat] : ( sup_sup_set_rule @ ( A3 @ X3 ) @ B )
              @ C ) )
          = bot_bot_set_rule ) )
      & ( ( C != bot_bot_set_nat )
       => ( ( comple7665822446944348827t_rule
            @ ( image_nat_set_rule
              @ ^ [X3: nat] : ( sup_sup_set_rule @ ( A3 @ X3 ) @ B )
              @ C ) )
          = ( sup_sup_set_rule @ ( comple7665822446944348827t_rule @ ( image_nat_set_rule @ A3 @ C ) ) @ B ) ) ) ) ).

% UN_simps(2)
thf(fact_563_UN__simps_I2_J,axiom,
    ! [C: set_nat,A3: nat > set_o,B: set_o] :
      ( ( ( C = bot_bot_set_nat )
       => ( ( comple90263536869209701_set_o
            @ ( image_nat_set_o
              @ ^ [X3: nat] : ( sup_sup_set_o @ ( A3 @ X3 ) @ B )
              @ C ) )
          = bot_bot_set_o ) )
      & ( ( C != bot_bot_set_nat )
       => ( ( comple90263536869209701_set_o
            @ ( image_nat_set_o
              @ ^ [X3: nat] : ( sup_sup_set_o @ ( A3 @ X3 ) @ B )
              @ C ) )
          = ( sup_sup_set_o @ ( comple90263536869209701_set_o @ ( image_nat_set_o @ A3 @ C ) ) @ B ) ) ) ) ).

% UN_simps(2)
thf(fact_564_UN__simps_I2_J,axiom,
    ! [C: set_nat,A3: nat > set_nat,B: set_nat] :
      ( ( ( C = bot_bot_set_nat )
       => ( ( comple7399068483239264473et_nat
            @ ( image_nat_set_nat
              @ ^ [X3: nat] : ( sup_sup_set_nat @ ( A3 @ X3 ) @ B )
              @ C ) )
          = bot_bot_set_nat ) )
      & ( ( C != bot_bot_set_nat )
       => ( ( comple7399068483239264473et_nat
            @ ( image_nat_set_nat
              @ ^ [X3: nat] : ( sup_sup_set_nat @ ( A3 @ X3 ) @ B )
              @ C ) )
          = ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A3 @ C ) ) @ B ) ) ) ) ).

% UN_simps(2)
thf(fact_565_UN__simps_I2_J,axiom,
    ! [C: set_rule,A3: rule > set_fm,B: set_fm] :
      ( ( ( C = bot_bot_set_rule )
       => ( ( comple2134933779557159616set_fm
            @ ( image_rule_set_fm
              @ ^ [X3: rule] : ( sup_sup_set_fm @ ( A3 @ X3 ) @ B )
              @ C ) )
          = bot_bot_set_fm ) )
      & ( ( C != bot_bot_set_rule )
       => ( ( comple2134933779557159616set_fm
            @ ( image_rule_set_fm
              @ ^ [X3: rule] : ( sup_sup_set_fm @ ( A3 @ X3 ) @ B )
              @ C ) )
          = ( sup_sup_set_fm @ ( comple2134933779557159616set_fm @ ( image_rule_set_fm @ A3 @ C ) ) @ B ) ) ) ) ).

% UN_simps(2)
thf(fact_566_emptyE,axiom,
    ! [A: list_fm] :
      ~ ( member_list_fm3 @ A @ bot_bot_set_list_fm ) ).

% emptyE
thf(fact_567_emptyE,axiom,
    ! [A: fm] :
      ~ ( member_fm3 @ A @ bot_bot_set_fm ) ).

% emptyE
thf(fact_568_emptyE,axiom,
    ! [A: rule] :
      ~ ( member_rule3 @ A @ bot_bot_set_rule ) ).

% emptyE
thf(fact_569_emptyE,axiom,
    ! [A: $o] :
      ~ ( member_o3 @ A @ bot_bot_set_o ) ).

% emptyE
thf(fact_570_emptyE,axiom,
    ! [A: nat] :
      ~ ( member_nat3 @ A @ bot_bot_set_nat ) ).

% emptyE
thf(fact_571_equals0D,axiom,
    ! [A3: set_list_fm,A: list_fm] :
      ( ( A3 = bot_bot_set_list_fm )
     => ~ ( member_list_fm3 @ A @ A3 ) ) ).

% equals0D
thf(fact_572_equals0D,axiom,
    ! [A3: set_fm,A: fm] :
      ( ( A3 = bot_bot_set_fm )
     => ~ ( member_fm3 @ A @ A3 ) ) ).

% equals0D
thf(fact_573_equals0D,axiom,
    ! [A3: set_rule,A: rule] :
      ( ( A3 = bot_bot_set_rule )
     => ~ ( member_rule3 @ A @ A3 ) ) ).

% equals0D
thf(fact_574_equals0D,axiom,
    ! [A3: set_o,A: $o] :
      ( ( A3 = bot_bot_set_o )
     => ~ ( member_o3 @ A @ A3 ) ) ).

% equals0D
thf(fact_575_equals0D,axiom,
    ! [A3: set_nat,A: nat] :
      ( ( A3 = bot_bot_set_nat )
     => ~ ( member_nat3 @ A @ A3 ) ) ).

% equals0D
thf(fact_576_equals0I,axiom,
    ! [A3: set_list_fm] :
      ( ! [Y4: list_fm] :
          ~ ( member_list_fm3 @ Y4 @ A3 )
     => ( A3 = bot_bot_set_list_fm ) ) ).

% equals0I
thf(fact_577_equals0I,axiom,
    ! [A3: set_fm] :
      ( ! [Y4: fm] :
          ~ ( member_fm3 @ Y4 @ A3 )
     => ( A3 = bot_bot_set_fm ) ) ).

% equals0I
thf(fact_578_equals0I,axiom,
    ! [A3: set_rule] :
      ( ! [Y4: rule] :
          ~ ( member_rule3 @ Y4 @ A3 )
     => ( A3 = bot_bot_set_rule ) ) ).

% equals0I
thf(fact_579_equals0I,axiom,
    ! [A3: set_o] :
      ( ! [Y4: $o] :
          ~ ( member_o3 @ Y4 @ A3 )
     => ( A3 = bot_bot_set_o ) ) ).

% equals0I
thf(fact_580_equals0I,axiom,
    ! [A3: set_nat] :
      ( ! [Y4: nat] :
          ~ ( member_nat3 @ Y4 @ A3 )
     => ( A3 = bot_bot_set_nat ) ) ).

% equals0I
thf(fact_581_ex__in__conv,axiom,
    ! [A3: set_list_fm] :
      ( ( ? [X3: list_fm] : ( member_list_fm3 @ X3 @ A3 ) )
      = ( A3 != bot_bot_set_list_fm ) ) ).

% ex_in_conv
thf(fact_582_ex__in__conv,axiom,
    ! [A3: set_fm] :
      ( ( ? [X3: fm] : ( member_fm3 @ X3 @ A3 ) )
      = ( A3 != bot_bot_set_fm ) ) ).

% ex_in_conv
thf(fact_583_ex__in__conv,axiom,
    ! [A3: set_rule] :
      ( ( ? [X3: rule] : ( member_rule3 @ X3 @ A3 ) )
      = ( A3 != bot_bot_set_rule ) ) ).

% ex_in_conv
thf(fact_584_ex__in__conv,axiom,
    ! [A3: set_o] :
      ( ( ? [X3: $o] : ( member_o3 @ X3 @ A3 ) )
      = ( A3 != bot_bot_set_o ) ) ).

% ex_in_conv
thf(fact_585_ex__in__conv,axiom,
    ! [A3: set_nat] :
      ( ( ? [X3: nat] : ( member_nat3 @ X3 @ A3 ) )
      = ( A3 != bot_bot_set_nat ) ) ).

% ex_in_conv
thf(fact_586_Union__empty,axiom,
    ( ( comple7665822446944348827t_rule @ bot_bot_set_set_rule )
    = bot_bot_set_rule ) ).

% Union_empty
thf(fact_587_Union__empty,axiom,
    ( ( comple90263536869209701_set_o @ bot_bot_set_set_o )
    = bot_bot_set_o ) ).

% Union_empty
thf(fact_588_Union__empty,axiom,
    ( ( comple7399068483239264473et_nat @ bot_bot_set_set_nat )
    = bot_bot_set_nat ) ).

% Union_empty
thf(fact_589_Union__empty,axiom,
    ( ( comple2134933779557159616set_fm @ bot_bot_set_set_fm )
    = bot_bot_set_fm ) ).

% Union_empty
thf(fact_590_empty__def,axiom,
    ( bot_bot_set_rule
    = ( collect_rule
      @ ^ [X3: rule] : $false ) ) ).

% empty_def
thf(fact_591_empty__def,axiom,
    ( bot_bot_set_o
    = ( collect_o
      @ ^ [X3: $o] : $false ) ) ).

% empty_def
thf(fact_592_empty__def,axiom,
    ( bot_bot_set_nat
    = ( collect_nat
      @ ^ [X3: nat] : $false ) ) ).

% empty_def
thf(fact_593_SUP__constant,axiom,
    ! [C2: $o,A3: set_rule] :
      ( ( complete_Sup_Sup_o
        @ ( image_rule_o
          @ ^ [Y2: rule] : C2
          @ A3 ) )
      = ( ( ( A3 = bot_bot_set_rule )
         => bot_bot_o )
        & ( ( A3 != bot_bot_set_rule )
         => C2 ) ) ) ).

% SUP_constant
thf(fact_594_SUP__constant,axiom,
    ! [C2: $o,A3: set_o] :
      ( ( complete_Sup_Sup_o
        @ ( image_o_o
          @ ^ [Y2: $o] : C2
          @ A3 ) )
      = ( ( ( A3 = bot_bot_set_o )
         => bot_bot_o )
        & ( ( A3 != bot_bot_set_o )
         => C2 ) ) ) ).

% SUP_constant
thf(fact_595_SUP__constant,axiom,
    ! [C2: $o,A3: set_nat] :
      ( ( complete_Sup_Sup_o
        @ ( image_nat_o
          @ ^ [Y2: nat] : C2
          @ A3 ) )
      = ( ( ( A3 = bot_bot_set_nat )
         => bot_bot_o )
        & ( ( A3 != bot_bot_set_nat )
         => C2 ) ) ) ).

% SUP_constant
thf(fact_596_SUP__constant,axiom,
    ! [A3: set_rule,C2: set_rule] :
      ( ( ( A3 = bot_bot_set_rule )
       => ( ( comple7665822446944348827t_rule
            @ ( image_rule_set_rule
              @ ^ [Y2: rule] : C2
              @ A3 ) )
          = bot_bot_set_rule ) )
      & ( ( A3 != bot_bot_set_rule )
       => ( ( comple7665822446944348827t_rule
            @ ( image_rule_set_rule
              @ ^ [Y2: rule] : C2
              @ A3 ) )
          = C2 ) ) ) ).

% SUP_constant
thf(fact_597_SUP__constant,axiom,
    ! [A3: set_rule,C2: set_o] :
      ( ( ( A3 = bot_bot_set_rule )
       => ( ( comple90263536869209701_set_o
            @ ( image_rule_set_o
              @ ^ [Y2: rule] : C2
              @ A3 ) )
          = bot_bot_set_o ) )
      & ( ( A3 != bot_bot_set_rule )
       => ( ( comple90263536869209701_set_o
            @ ( image_rule_set_o
              @ ^ [Y2: rule] : C2
              @ A3 ) )
          = C2 ) ) ) ).

% SUP_constant
thf(fact_598_SUP__constant,axiom,
    ! [A3: set_rule,C2: set_nat] :
      ( ( ( A3 = bot_bot_set_rule )
       => ( ( comple7399068483239264473et_nat
            @ ( image_rule_set_nat
              @ ^ [Y2: rule] : C2
              @ A3 ) )
          = bot_bot_set_nat ) )
      & ( ( A3 != bot_bot_set_rule )
       => ( ( comple7399068483239264473et_nat
            @ ( image_rule_set_nat
              @ ^ [Y2: rule] : C2
              @ A3 ) )
          = C2 ) ) ) ).

% SUP_constant
thf(fact_599_SUP__constant,axiom,
    ! [A3: set_o,C2: set_rule] :
      ( ( ( A3 = bot_bot_set_o )
       => ( ( comple7665822446944348827t_rule
            @ ( image_o_set_rule
              @ ^ [Y2: $o] : C2
              @ A3 ) )
          = bot_bot_set_rule ) )
      & ( ( A3 != bot_bot_set_o )
       => ( ( comple7665822446944348827t_rule
            @ ( image_o_set_rule
              @ ^ [Y2: $o] : C2
              @ A3 ) )
          = C2 ) ) ) ).

% SUP_constant
thf(fact_600_SUP__constant,axiom,
    ! [A3: set_o,C2: set_o] :
      ( ( ( A3 = bot_bot_set_o )
       => ( ( comple90263536869209701_set_o
            @ ( image_o_set_o
              @ ^ [Y2: $o] : C2
              @ A3 ) )
          = bot_bot_set_o ) )
      & ( ( A3 != bot_bot_set_o )
       => ( ( comple90263536869209701_set_o
            @ ( image_o_set_o
              @ ^ [Y2: $o] : C2
              @ A3 ) )
          = C2 ) ) ) ).

% SUP_constant
thf(fact_601_SUP__constant,axiom,
    ! [A3: set_o,C2: set_nat] :
      ( ( ( A3 = bot_bot_set_o )
       => ( ( comple7399068483239264473et_nat
            @ ( image_o_set_nat
              @ ^ [Y2: $o] : C2
              @ A3 ) )
          = bot_bot_set_nat ) )
      & ( ( A3 != bot_bot_set_o )
       => ( ( comple7399068483239264473et_nat
            @ ( image_o_set_nat
              @ ^ [Y2: $o] : C2
              @ A3 ) )
          = C2 ) ) ) ).

% SUP_constant
thf(fact_602_SUP__constant,axiom,
    ! [A3: set_nat,C2: set_rule] :
      ( ( ( A3 = bot_bot_set_nat )
       => ( ( comple7665822446944348827t_rule
            @ ( image_nat_set_rule
              @ ^ [Y2: nat] : C2
              @ A3 ) )
          = bot_bot_set_rule ) )
      & ( ( A3 != bot_bot_set_nat )
       => ( ( comple7665822446944348827t_rule
            @ ( image_nat_set_rule
              @ ^ [Y2: nat] : C2
              @ A3 ) )
          = C2 ) ) ) ).

% SUP_constant
thf(fact_603_SUP__empty,axiom,
    ! [F: rule > $o] :
      ( ( complete_Sup_Sup_o @ ( image_rule_o @ F @ bot_bot_set_rule ) )
      = bot_bot_o ) ).

% SUP_empty
thf(fact_604_SUP__empty,axiom,
    ! [F: $o > $o] :
      ( ( complete_Sup_Sup_o @ ( image_o_o @ F @ bot_bot_set_o ) )
      = bot_bot_o ) ).

% SUP_empty
thf(fact_605_SUP__empty,axiom,
    ! [F: nat > $o] :
      ( ( complete_Sup_Sup_o @ ( image_nat_o @ F @ bot_bot_set_nat ) )
      = bot_bot_o ) ).

% SUP_empty
thf(fact_606_SUP__empty,axiom,
    ! [F: rule > set_rule] :
      ( ( comple7665822446944348827t_rule @ ( image_rule_set_rule @ F @ bot_bot_set_rule ) )
      = bot_bot_set_rule ) ).

% SUP_empty
thf(fact_607_SUP__empty,axiom,
    ! [F: rule > set_o] :
      ( ( comple90263536869209701_set_o @ ( image_rule_set_o @ F @ bot_bot_set_rule ) )
      = bot_bot_set_o ) ).

% SUP_empty
thf(fact_608_SUP__empty,axiom,
    ! [F: rule > set_nat] :
      ( ( comple7399068483239264473et_nat @ ( image_rule_set_nat @ F @ bot_bot_set_rule ) )
      = bot_bot_set_nat ) ).

% SUP_empty
thf(fact_609_SUP__empty,axiom,
    ! [F: $o > set_rule] :
      ( ( comple7665822446944348827t_rule @ ( image_o_set_rule @ F @ bot_bot_set_o ) )
      = bot_bot_set_rule ) ).

% SUP_empty
thf(fact_610_SUP__empty,axiom,
    ! [F: $o > set_o] :
      ( ( comple90263536869209701_set_o @ ( image_o_set_o @ F @ bot_bot_set_o ) )
      = bot_bot_set_o ) ).

% SUP_empty
thf(fact_611_SUP__empty,axiom,
    ! [F: $o > set_nat] :
      ( ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ bot_bot_set_o ) )
      = bot_bot_set_nat ) ).

% SUP_empty
thf(fact_612_SUP__empty,axiom,
    ! [F: nat > set_rule] :
      ( ( comple7665822446944348827t_rule @ ( image_nat_set_rule @ F @ bot_bot_set_nat ) )
      = bot_bot_set_rule ) ).

% SUP_empty
thf(fact_613_empty__Union__conv,axiom,
    ! [A3: set_set_rule] :
      ( ( bot_bot_set_rule
        = ( comple7665822446944348827t_rule @ A3 ) )
      = ( ! [X3: set_rule] :
            ( ( member_set_rule @ X3 @ A3 )
           => ( X3 = bot_bot_set_rule ) ) ) ) ).

% empty_Union_conv
thf(fact_614_empty__Union__conv,axiom,
    ! [A3: set_set_o] :
      ( ( bot_bot_set_o
        = ( comple90263536869209701_set_o @ A3 ) )
      = ( ! [X3: set_o] :
            ( ( member_set_o @ X3 @ A3 )
           => ( X3 = bot_bot_set_o ) ) ) ) ).

% empty_Union_conv
thf(fact_615_empty__Union__conv,axiom,
    ! [A3: set_set_nat] :
      ( ( bot_bot_set_nat
        = ( comple7399068483239264473et_nat @ A3 ) )
      = ( ! [X3: set_nat] :
            ( ( member_set_nat @ X3 @ A3 )
           => ( X3 = bot_bot_set_nat ) ) ) ) ).

% empty_Union_conv
thf(fact_616_empty__Union__conv,axiom,
    ! [A3: set_set_fm] :
      ( ( bot_bot_set_fm
        = ( comple2134933779557159616set_fm @ A3 ) )
      = ( ! [X3: set_fm] :
            ( ( member_set_fm @ X3 @ A3 )
           => ( X3 = bot_bot_set_fm ) ) ) ) ).

% empty_Union_conv
thf(fact_617_Union__empty__conv,axiom,
    ! [A3: set_set_rule] :
      ( ( ( comple7665822446944348827t_rule @ A3 )
        = bot_bot_set_rule )
      = ( ! [X3: set_rule] :
            ( ( member_set_rule @ X3 @ A3 )
           => ( X3 = bot_bot_set_rule ) ) ) ) ).

% Union_empty_conv
thf(fact_618_Union__empty__conv,axiom,
    ! [A3: set_set_o] :
      ( ( ( comple90263536869209701_set_o @ A3 )
        = bot_bot_set_o )
      = ( ! [X3: set_o] :
            ( ( member_set_o @ X3 @ A3 )
           => ( X3 = bot_bot_set_o ) ) ) ) ).

% Union_empty_conv
thf(fact_619_Union__empty__conv,axiom,
    ! [A3: set_set_nat] :
      ( ( ( comple7399068483239264473et_nat @ A3 )
        = bot_bot_set_nat )
      = ( ! [X3: set_nat] :
            ( ( member_set_nat @ X3 @ A3 )
           => ( X3 = bot_bot_set_nat ) ) ) ) ).

% Union_empty_conv
thf(fact_620_Union__empty__conv,axiom,
    ! [A3: set_set_fm] :
      ( ( ( comple2134933779557159616set_fm @ A3 )
        = bot_bot_set_fm )
      = ( ! [X3: set_fm] :
            ( ( member_set_fm @ X3 @ A3 )
           => ( X3 = bot_bot_set_fm ) ) ) ) ).

% Union_empty_conv
thf(fact_621_Un__empty__right,axiom,
    ! [A3: set_rule] :
      ( ( sup_sup_set_rule @ A3 @ bot_bot_set_rule )
      = A3 ) ).

% Un_empty_right
thf(fact_622_Un__empty__right,axiom,
    ! [A3: set_o] :
      ( ( sup_sup_set_o @ A3 @ bot_bot_set_o )
      = A3 ) ).

% Un_empty_right
thf(fact_623_Un__empty__right,axiom,
    ! [A3: set_nat] :
      ( ( sup_sup_set_nat @ A3 @ bot_bot_set_nat )
      = A3 ) ).

% Un_empty_right
thf(fact_624_Un__empty__left,axiom,
    ! [B: set_rule] :
      ( ( sup_sup_set_rule @ bot_bot_set_rule @ B )
      = B ) ).

% Un_empty_left
thf(fact_625_Un__empty__left,axiom,
    ! [B: set_o] :
      ( ( sup_sup_set_o @ bot_bot_set_o @ B )
      = B ) ).

% Un_empty_left
thf(fact_626_Un__empty__left,axiom,
    ! [B: set_nat] :
      ( ( sup_sup_set_nat @ bot_bot_set_nat @ B )
      = B ) ).

% Un_empty_left
thf(fact_627_SUP__eq__const,axiom,
    ! [I2: set_fm,F: fm > $o,X: $o] :
      ( ( I2 != bot_bot_set_fm )
     => ( ! [I3: fm] :
            ( ( member_fm3 @ I3 @ I2 )
           => ( ( F @ I3 )
              = X ) )
       => ( ( complete_Sup_Sup_o @ ( image_fm_o @ F @ I2 ) )
          = X ) ) ) ).

% SUP_eq_const
thf(fact_628_SUP__eq__const,axiom,
    ! [I2: set_rule,F: rule > $o,X: $o] :
      ( ( I2 != bot_bot_set_rule )
     => ( ! [I3: rule] :
            ( ( member_rule3 @ I3 @ I2 )
           => ( ( F @ I3 )
              = X ) )
       => ( ( complete_Sup_Sup_o @ ( image_rule_o @ F @ I2 ) )
          = X ) ) ) ).

% SUP_eq_const
thf(fact_629_SUP__eq__const,axiom,
    ! [I2: set_o,F: $o > $o,X: $o] :
      ( ( I2 != bot_bot_set_o )
     => ( ! [I3: $o] :
            ( ( member_o3 @ I3 @ I2 )
           => ( ( F @ I3 )
              = X ) )
       => ( ( complete_Sup_Sup_o @ ( image_o_o @ F @ I2 ) )
          = X ) ) ) ).

% SUP_eq_const
thf(fact_630_SUP__eq__const,axiom,
    ! [I2: set_nat,F: nat > $o,X: $o] :
      ( ( I2 != bot_bot_set_nat )
     => ( ! [I3: nat] :
            ( ( member_nat3 @ I3 @ I2 )
           => ( ( F @ I3 )
              = X ) )
       => ( ( complete_Sup_Sup_o @ ( image_nat_o @ F @ I2 ) )
          = X ) ) ) ).

% SUP_eq_const
thf(fact_631_SUP__eq__const,axiom,
    ! [I2: set_fm,F: fm > set_fm,X: set_fm] :
      ( ( I2 != bot_bot_set_fm )
     => ( ! [I3: fm] :
            ( ( member_fm3 @ I3 @ I2 )
           => ( ( F @ I3 )
              = X ) )
       => ( ( comple2134933779557159616set_fm @ ( image_fm_set_fm @ F @ I2 ) )
          = X ) ) ) ).

% SUP_eq_const
thf(fact_632_SUP__eq__const,axiom,
    ! [I2: set_rule,F: rule > set_fm,X: set_fm] :
      ( ( I2 != bot_bot_set_rule )
     => ( ! [I3: rule] :
            ( ( member_rule3 @ I3 @ I2 )
           => ( ( F @ I3 )
              = X ) )
       => ( ( comple2134933779557159616set_fm @ ( image_rule_set_fm @ F @ I2 ) )
          = X ) ) ) ).

% SUP_eq_const
thf(fact_633_SUP__eq__const,axiom,
    ! [I2: set_o,F: $o > set_fm,X: set_fm] :
      ( ( I2 != bot_bot_set_o )
     => ( ! [I3: $o] :
            ( ( member_o3 @ I3 @ I2 )
           => ( ( F @ I3 )
              = X ) )
       => ( ( comple2134933779557159616set_fm @ ( image_o_set_fm @ F @ I2 ) )
          = X ) ) ) ).

% SUP_eq_const
thf(fact_634_SUP__eq__const,axiom,
    ! [I2: set_nat,F: nat > set_fm,X: set_fm] :
      ( ( I2 != bot_bot_set_nat )
     => ( ! [I3: nat] :
            ( ( member_nat3 @ I3 @ I2 )
           => ( ( F @ I3 )
              = X ) )
       => ( ( comple2134933779557159616set_fm @ ( image_nat_set_fm @ F @ I2 ) )
          = X ) ) ) ).

% SUP_eq_const
thf(fact_635_SUP__eq__const,axiom,
    ! [I2: set_list_fm,F: list_fm > $o,X: $o] :
      ( ( I2 != bot_bot_set_list_fm )
     => ( ! [I3: list_fm] :
            ( ( member_list_fm3 @ I3 @ I2 )
           => ( ( F @ I3 )
              = X ) )
       => ( ( complete_Sup_Sup_o @ ( image_list_fm_o @ F @ I2 ) )
          = X ) ) ) ).

% SUP_eq_const
thf(fact_636_SUP__eq__const,axiom,
    ! [I2: set_list_fm,F: list_fm > set_fm,X: set_fm] :
      ( ( I2 != bot_bot_set_list_fm )
     => ( ! [I3: list_fm] :
            ( ( member_list_fm3 @ I3 @ I2 )
           => ( ( F @ I3 )
              = X ) )
       => ( ( comple2134933779557159616set_fm @ ( image_list_fm_set_fm @ F @ I2 ) )
          = X ) ) ) ).

% SUP_eq_const
thf(fact_637_UN__empty2,axiom,
    ! [A3: set_Pr1822751329126368876m_rule] :
      ( ( comple2134933779557159616set_fm
        @ ( image_377115208147135190set_fm
          @ ^ [X3: produc340336539035504054m_rule] : bot_bot_set_fm
          @ A3 ) )
      = bot_bot_set_fm ) ).

% UN_empty2
thf(fact_638_UN__empty,axiom,
    ! [B: rule > set_rule] :
      ( ( comple7665822446944348827t_rule @ ( image_rule_set_rule @ B @ bot_bot_set_rule ) )
      = bot_bot_set_rule ) ).

% UN_empty
thf(fact_639_UN__empty,axiom,
    ! [B: rule > set_o] :
      ( ( comple90263536869209701_set_o @ ( image_rule_set_o @ B @ bot_bot_set_rule ) )
      = bot_bot_set_o ) ).

% UN_empty
thf(fact_640_UN__empty,axiom,
    ! [B: rule > set_nat] :
      ( ( comple7399068483239264473et_nat @ ( image_rule_set_nat @ B @ bot_bot_set_rule ) )
      = bot_bot_set_nat ) ).

% UN_empty
thf(fact_641_UN__empty,axiom,
    ! [B: $o > set_rule] :
      ( ( comple7665822446944348827t_rule @ ( image_o_set_rule @ B @ bot_bot_set_o ) )
      = bot_bot_set_rule ) ).

% UN_empty
thf(fact_642_UN__empty,axiom,
    ! [B: $o > set_o] :
      ( ( comple90263536869209701_set_o @ ( image_o_set_o @ B @ bot_bot_set_o ) )
      = bot_bot_set_o ) ).

% UN_empty
thf(fact_643_UN__empty,axiom,
    ! [B: $o > set_nat] :
      ( ( comple7399068483239264473et_nat @ ( image_o_set_nat @ B @ bot_bot_set_o ) )
      = bot_bot_set_nat ) ).

% UN_empty
thf(fact_644_UN__empty,axiom,
    ! [B: nat > set_rule] :
      ( ( comple7665822446944348827t_rule @ ( image_nat_set_rule @ B @ bot_bot_set_nat ) )
      = bot_bot_set_rule ) ).

% UN_empty
thf(fact_645_UN__empty,axiom,
    ! [B: nat > set_o] :
      ( ( comple90263536869209701_set_o @ ( image_nat_set_o @ B @ bot_bot_set_nat ) )
      = bot_bot_set_o ) ).

% UN_empty
thf(fact_646_UN__empty,axiom,
    ! [B: nat > set_nat] :
      ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B @ bot_bot_set_nat ) )
      = bot_bot_set_nat ) ).

% UN_empty
thf(fact_647_UN__empty,axiom,
    ! [B: rule > set_fm] :
      ( ( comple2134933779557159616set_fm @ ( image_rule_set_fm @ B @ bot_bot_set_rule ) )
      = bot_bot_set_fm ) ).

% UN_empty
thf(fact_648_UNION__empty__conv_I1_J,axiom,
    ! [B: produc340336539035504054m_rule > set_fm,A3: set_Pr1822751329126368876m_rule] :
      ( ( bot_bot_set_fm
        = ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ B @ A3 ) ) )
      = ( ! [X3: produc340336539035504054m_rule] :
            ( ( member7231649785386036813m_rule @ X3 @ A3 )
           => ( ( B @ X3 )
              = bot_bot_set_fm ) ) ) ) ).

% UNION_empty_conv(1)
thf(fact_649_UNION__empty__conv_I2_J,axiom,
    ! [B: produc340336539035504054m_rule > set_fm,A3: set_Pr1822751329126368876m_rule] :
      ( ( ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ B @ A3 ) )
        = bot_bot_set_fm )
      = ( ! [X3: produc340336539035504054m_rule] :
            ( ( member7231649785386036813m_rule @ X3 @ A3 )
           => ( ( B @ X3 )
              = bot_bot_set_fm ) ) ) ) ).

% UNION_empty_conv(2)
thf(fact_650_Sup__SUP__eq,axiom,
    ( comple579058054629451178rule_o
    = ( ^ [S5: set_rule_o,X3: rule] : ( member_rule3 @ X3 @ ( comple7665822446944348827t_rule @ ( image_7644734524678384448t_rule @ collect_rule @ S5 ) ) ) ) ) ).

% Sup_SUP_eq
thf(fact_651_Sup__SUP__eq,axiom,
    ( comple173717950668161943t_fm_o
    = ( ^ [S5: set_list_fm_o,X3: list_fm] : ( member_list_fm3 @ X3 @ ( comple8784269564784259782ist_fm @ ( image_4471151405024504270ist_fm @ collect_list_fm @ S5 ) ) ) ) ) ).

% Sup_SUP_eq
thf(fact_652_Sup__SUP__eq,axiom,
    ( complete_Sup_Sup_o_o
    = ( ^ [S5: set_o_o,X3: $o] : ( member_o3 @ X3 @ ( comple90263536869209701_set_o @ ( image_o_o_set_o @ collect_o @ S5 ) ) ) ) ) ).

% Sup_SUP_eq
thf(fact_653_Sup__SUP__eq,axiom,
    ( comple8317665133742190828_nat_o
    = ( ^ [S5: set_nat_o,X3: nat] : ( member_nat3 @ X3 @ ( comple7399068483239264473et_nat @ ( image_nat_o_set_nat @ collect_nat @ S5 ) ) ) ) ) ).

% Sup_SUP_eq
thf(fact_654_Sup__SUP__eq,axiom,
    ( comple3443612784043876893p_fm_o
    = ( ^ [S5: set_fm_o,X3: fm] : ( member_fm3 @ X3 @ ( comple2134933779557159616set_fm @ ( image_fm_o_set_fm @ collect_fm @ S5 ) ) ) ) ) ).

% Sup_SUP_eq
thf(fact_655_szip_Osimps_I1_J,axiom,
    ! [S1: stream2709947120125613254m_rule,S2: stream2709947120125613254m_rule] :
      ( ( shd_Pr4461660664618831993m_rule @ ( szip_P811719526838699976m_rule @ S1 @ S2 ) )
      = ( produc6261311607089640965m_rule @ ( shd_Pr4562317740776619530m_rule @ S1 ) @ ( shd_Pr4562317740776619530m_rule @ S2 ) ) ) ).

% szip.simps(1)
thf(fact_656_szip_Osimps_I1_J,axiom,
    ! [S1: stream2709947120125613254m_rule,S2: stream_rule] :
      ( ( shd_Pr2400962586966563157e_rule @ ( szip_P2044787555563963556e_rule @ S1 @ S2 ) )
      = ( produc4831648765031708129e_rule @ ( shd_Pr4562317740776619530m_rule @ S1 ) @ ( shd_rule @ S2 ) ) ) ).

% szip.simps(1)
thf(fact_657_szip_Osimps_I1_J,axiom,
    ! [S1: stream_rule,S2: stream2709947120125613254m_rule] :
      ( ( shd_Pr1208753253167450709m_rule @ ( szip_r6533552703345879972m_rule @ S1 @ S2 ) )
      = ( produc97041875958848737m_rule @ ( shd_rule @ S1 ) @ ( shd_Pr4562317740776619530m_rule @ S2 ) ) ) ).

% szip.simps(1)
thf(fact_658_szip_Osimps_I1_J,axiom,
    ! [S1: stream_rule,S2: stream_rule] :
      ( ( shd_Pr2264621979884435249e_rule @ ( szip_rule_rule @ S1 @ S2 ) )
      = ( produc5849431337705160893e_rule @ ( shd_rule @ S1 ) @ ( shd_rule @ S2 ) ) ) ).

% szip.simps(1)
thf(fact_659_szip_Osimps_I1_J,axiom,
    ! [S1: stream4408948924543953275ist_fm,S2: stream_rule] :
      ( ( shd_Pr4562317740776619530m_rule @ ( szip_P2924820683901490861m_rule @ S1 @ S2 ) )
      = ( produc1733806532565653680m_rule @ ( shd_Pr3211216682057661985ist_fm @ S1 ) @ ( shd_rule @ S2 ) ) ) ).

% szip.simps(1)
thf(fact_660_UN__extend__simps_I2_J,axiom,
    ! [C: set_rule,A3: rule > set_nat,B: set_nat] :
      ( ( ( C = bot_bot_set_rule )
       => ( ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ ( image_rule_set_nat @ A3 @ C ) ) @ B )
          = B ) )
      & ( ( C != bot_bot_set_rule )
       => ( ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ ( image_rule_set_nat @ A3 @ C ) ) @ B )
          = ( comple7399068483239264473et_nat
            @ ( image_rule_set_nat
              @ ^ [X3: rule] : ( sup_sup_set_nat @ ( A3 @ X3 ) @ B )
              @ C ) ) ) ) ) ).

% UN_extend_simps(2)
thf(fact_661_UN__extend__simps_I2_J,axiom,
    ! [C: set_o,A3: $o > set_nat,B: set_nat] :
      ( ( ( C = bot_bot_set_o )
       => ( ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ A3 @ C ) ) @ B )
          = B ) )
      & ( ( C != bot_bot_set_o )
       => ( ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ A3 @ C ) ) @ B )
          = ( comple7399068483239264473et_nat
            @ ( image_o_set_nat
              @ ^ [X3: $o] : ( sup_sup_set_nat @ ( A3 @ X3 ) @ B )
              @ C ) ) ) ) ) ).

% UN_extend_simps(2)
thf(fact_662_UN__extend__simps_I2_J,axiom,
    ! [C: set_nat,A3: nat > set_nat,B: set_nat] :
      ( ( ( C = bot_bot_set_nat )
       => ( ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A3 @ C ) ) @ B )
          = B ) )
      & ( ( C != bot_bot_set_nat )
       => ( ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A3 @ C ) ) @ B )
          = ( comple7399068483239264473et_nat
            @ ( image_nat_set_nat
              @ ^ [X3: nat] : ( sup_sup_set_nat @ ( A3 @ X3 ) @ B )
              @ C ) ) ) ) ) ).

% UN_extend_simps(2)
thf(fact_663_UN__extend__simps_I2_J,axiom,
    ! [C: set_Pr1822751329126368876m_rule,A3: produc340336539035504054m_rule > set_fm,B: set_fm] :
      ( ( ( C = bot_bo3426410157013319936m_rule )
       => ( ( sup_sup_set_fm @ ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ A3 @ C ) ) @ B )
          = B ) )
      & ( ( C != bot_bo3426410157013319936m_rule )
       => ( ( sup_sup_set_fm @ ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ A3 @ C ) ) @ B )
          = ( comple2134933779557159616set_fm
            @ ( image_377115208147135190set_fm
              @ ^ [X3: produc340336539035504054m_rule] : ( sup_sup_set_fm @ ( A3 @ X3 ) @ B )
              @ C ) ) ) ) ) ).

% UN_extend_simps(2)
thf(fact_664_UN__extend__simps_I2_J,axiom,
    ! [C: set_rule,A3: rule > set_fm,B: set_fm] :
      ( ( ( C = bot_bot_set_rule )
       => ( ( sup_sup_set_fm @ ( comple2134933779557159616set_fm @ ( image_rule_set_fm @ A3 @ C ) ) @ B )
          = B ) )
      & ( ( C != bot_bot_set_rule )
       => ( ( sup_sup_set_fm @ ( comple2134933779557159616set_fm @ ( image_rule_set_fm @ A3 @ C ) ) @ B )
          = ( comple2134933779557159616set_fm
            @ ( image_rule_set_fm
              @ ^ [X3: rule] : ( sup_sup_set_fm @ ( A3 @ X3 ) @ B )
              @ C ) ) ) ) ) ).

% UN_extend_simps(2)
thf(fact_665_UN__extend__simps_I2_J,axiom,
    ! [C: set_o,A3: $o > set_fm,B: set_fm] :
      ( ( ( C = bot_bot_set_o )
       => ( ( sup_sup_set_fm @ ( comple2134933779557159616set_fm @ ( image_o_set_fm @ A3 @ C ) ) @ B )
          = B ) )
      & ( ( C != bot_bot_set_o )
       => ( ( sup_sup_set_fm @ ( comple2134933779557159616set_fm @ ( image_o_set_fm @ A3 @ C ) ) @ B )
          = ( comple2134933779557159616set_fm
            @ ( image_o_set_fm
              @ ^ [X3: $o] : ( sup_sup_set_fm @ ( A3 @ X3 ) @ B )
              @ C ) ) ) ) ) ).

% UN_extend_simps(2)
thf(fact_666_UN__extend__simps_I2_J,axiom,
    ! [C: set_nat,A3: nat > set_fm,B: set_fm] :
      ( ( ( C = bot_bot_set_nat )
       => ( ( sup_sup_set_fm @ ( comple2134933779557159616set_fm @ ( image_nat_set_fm @ A3 @ C ) ) @ B )
          = B ) )
      & ( ( C != bot_bot_set_nat )
       => ( ( sup_sup_set_fm @ ( comple2134933779557159616set_fm @ ( image_nat_set_fm @ A3 @ C ) ) @ B )
          = ( comple2134933779557159616set_fm
            @ ( image_nat_set_fm
              @ ^ [X3: nat] : ( sup_sup_set_fm @ ( A3 @ X3 ) @ B )
              @ C ) ) ) ) ) ).

% UN_extend_simps(2)
thf(fact_667_UN__extend__simps_I3_J,axiom,
    ! [C: set_rule,A3: set_nat,B: rule > set_nat] :
      ( ( ( C = bot_bot_set_rule )
       => ( ( sup_sup_set_nat @ A3 @ ( comple7399068483239264473et_nat @ ( image_rule_set_nat @ B @ C ) ) )
          = A3 ) )
      & ( ( C != bot_bot_set_rule )
       => ( ( sup_sup_set_nat @ A3 @ ( comple7399068483239264473et_nat @ ( image_rule_set_nat @ B @ C ) ) )
          = ( comple7399068483239264473et_nat
            @ ( image_rule_set_nat
              @ ^ [X3: rule] : ( sup_sup_set_nat @ A3 @ ( B @ X3 ) )
              @ C ) ) ) ) ) ).

% UN_extend_simps(3)
thf(fact_668_UN__extend__simps_I3_J,axiom,
    ! [C: set_o,A3: set_nat,B: $o > set_nat] :
      ( ( ( C = bot_bot_set_o )
       => ( ( sup_sup_set_nat @ A3 @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ B @ C ) ) )
          = A3 ) )
      & ( ( C != bot_bot_set_o )
       => ( ( sup_sup_set_nat @ A3 @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ B @ C ) ) )
          = ( comple7399068483239264473et_nat
            @ ( image_o_set_nat
              @ ^ [X3: $o] : ( sup_sup_set_nat @ A3 @ ( B @ X3 ) )
              @ C ) ) ) ) ) ).

% UN_extend_simps(3)
thf(fact_669_UN__extend__simps_I3_J,axiom,
    ! [C: set_nat,A3: set_nat,B: nat > set_nat] :
      ( ( ( C = bot_bot_set_nat )
       => ( ( sup_sup_set_nat @ A3 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B @ C ) ) )
          = A3 ) )
      & ( ( C != bot_bot_set_nat )
       => ( ( sup_sup_set_nat @ A3 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B @ C ) ) )
          = ( comple7399068483239264473et_nat
            @ ( image_nat_set_nat
              @ ^ [X3: nat] : ( sup_sup_set_nat @ A3 @ ( B @ X3 ) )
              @ C ) ) ) ) ) ).

% UN_extend_simps(3)
thf(fact_670_UN__extend__simps_I3_J,axiom,
    ! [C: set_Pr1822751329126368876m_rule,A3: set_fm,B: produc340336539035504054m_rule > set_fm] :
      ( ( ( C = bot_bo3426410157013319936m_rule )
       => ( ( sup_sup_set_fm @ A3 @ ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ B @ C ) ) )
          = A3 ) )
      & ( ( C != bot_bo3426410157013319936m_rule )
       => ( ( sup_sup_set_fm @ A3 @ ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ B @ C ) ) )
          = ( comple2134933779557159616set_fm
            @ ( image_377115208147135190set_fm
              @ ^ [X3: produc340336539035504054m_rule] : ( sup_sup_set_fm @ A3 @ ( B @ X3 ) )
              @ C ) ) ) ) ) ).

% UN_extend_simps(3)
thf(fact_671_UN__extend__simps_I3_J,axiom,
    ! [C: set_rule,A3: set_fm,B: rule > set_fm] :
      ( ( ( C = bot_bot_set_rule )
       => ( ( sup_sup_set_fm @ A3 @ ( comple2134933779557159616set_fm @ ( image_rule_set_fm @ B @ C ) ) )
          = A3 ) )
      & ( ( C != bot_bot_set_rule )
       => ( ( sup_sup_set_fm @ A3 @ ( comple2134933779557159616set_fm @ ( image_rule_set_fm @ B @ C ) ) )
          = ( comple2134933779557159616set_fm
            @ ( image_rule_set_fm
              @ ^ [X3: rule] : ( sup_sup_set_fm @ A3 @ ( B @ X3 ) )
              @ C ) ) ) ) ) ).

% UN_extend_simps(3)
thf(fact_672_UN__extend__simps_I3_J,axiom,
    ! [C: set_o,A3: set_fm,B: $o > set_fm] :
      ( ( ( C = bot_bot_set_o )
       => ( ( sup_sup_set_fm @ A3 @ ( comple2134933779557159616set_fm @ ( image_o_set_fm @ B @ C ) ) )
          = A3 ) )
      & ( ( C != bot_bot_set_o )
       => ( ( sup_sup_set_fm @ A3 @ ( comple2134933779557159616set_fm @ ( image_o_set_fm @ B @ C ) ) )
          = ( comple2134933779557159616set_fm
            @ ( image_o_set_fm
              @ ^ [X3: $o] : ( sup_sup_set_fm @ A3 @ ( B @ X3 ) )
              @ C ) ) ) ) ) ).

% UN_extend_simps(3)
thf(fact_673_UN__extend__simps_I3_J,axiom,
    ! [C: set_nat,A3: set_fm,B: nat > set_fm] :
      ( ( ( C = bot_bot_set_nat )
       => ( ( sup_sup_set_fm @ A3 @ ( comple2134933779557159616set_fm @ ( image_nat_set_fm @ B @ C ) ) )
          = A3 ) )
      & ( ( C != bot_bot_set_nat )
       => ( ( sup_sup_set_fm @ A3 @ ( comple2134933779557159616set_fm @ ( image_nat_set_fm @ B @ C ) ) )
          = ( comple2134933779557159616set_fm
            @ ( image_nat_set_fm
              @ ^ [X3: nat] : ( sup_sup_set_fm @ A3 @ ( B @ X3 ) )
              @ C ) ) ) ) ) ).

% UN_extend_simps(3)
thf(fact_674_ccSUP__empty,axiom,
    ! [F: rule > $o] :
      ( ( complete_Sup_Sup_o @ ( image_rule_o @ F @ bot_bot_set_rule ) )
      = bot_bot_o ) ).

% ccSUP_empty
thf(fact_675_ccSUP__empty,axiom,
    ! [F: $o > $o] :
      ( ( complete_Sup_Sup_o @ ( image_o_o @ F @ bot_bot_set_o ) )
      = bot_bot_o ) ).

% ccSUP_empty
thf(fact_676_ccSUP__empty,axiom,
    ! [F: nat > $o] :
      ( ( complete_Sup_Sup_o @ ( image_nat_o @ F @ bot_bot_set_nat ) )
      = bot_bot_o ) ).

% ccSUP_empty
thf(fact_677_ccSUP__empty,axiom,
    ! [F: rule > set_rule] :
      ( ( comple7665822446944348827t_rule @ ( image_rule_set_rule @ F @ bot_bot_set_rule ) )
      = bot_bot_set_rule ) ).

% ccSUP_empty
thf(fact_678_ccSUP__empty,axiom,
    ! [F: rule > set_o] :
      ( ( comple90263536869209701_set_o @ ( image_rule_set_o @ F @ bot_bot_set_rule ) )
      = bot_bot_set_o ) ).

% ccSUP_empty
thf(fact_679_ccSUP__empty,axiom,
    ! [F: rule > set_nat] :
      ( ( comple7399068483239264473et_nat @ ( image_rule_set_nat @ F @ bot_bot_set_rule ) )
      = bot_bot_set_nat ) ).

% ccSUP_empty
thf(fact_680_ccSUP__empty,axiom,
    ! [F: $o > set_rule] :
      ( ( comple7665822446944348827t_rule @ ( image_o_set_rule @ F @ bot_bot_set_o ) )
      = bot_bot_set_rule ) ).

% ccSUP_empty
thf(fact_681_ccSUP__empty,axiom,
    ! [F: $o > set_o] :
      ( ( comple90263536869209701_set_o @ ( image_o_set_o @ F @ bot_bot_set_o ) )
      = bot_bot_set_o ) ).

% ccSUP_empty
thf(fact_682_ccSUP__empty,axiom,
    ! [F: $o > set_nat] :
      ( ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ bot_bot_set_o ) )
      = bot_bot_set_nat ) ).

% ccSUP_empty
thf(fact_683_ccSUP__empty,axiom,
    ! [F: nat > set_rule] :
      ( ( comple7665822446944348827t_rule @ ( image_nat_set_rule @ F @ bot_bot_set_nat ) )
      = bot_bot_set_rule ) ).

% ccSUP_empty
thf(fact_684_cSUP__const,axiom,
    ! [A3: set_Pr1822751329126368876m_rule,C2: set_fm] :
      ( ( A3 != bot_bo3426410157013319936m_rule )
     => ( ( comple2134933779557159616set_fm
          @ ( image_377115208147135190set_fm
            @ ^ [X3: produc340336539035504054m_rule] : C2
            @ A3 ) )
        = C2 ) ) ).

% cSUP_const
thf(fact_685_cSUP__const,axiom,
    ! [A3: set_rule,C2: set_fm] :
      ( ( A3 != bot_bot_set_rule )
     => ( ( comple2134933779557159616set_fm
          @ ( image_rule_set_fm
            @ ^ [X3: rule] : C2
            @ A3 ) )
        = C2 ) ) ).

% cSUP_const
thf(fact_686_cSUP__const,axiom,
    ! [A3: set_o,C2: set_fm] :
      ( ( A3 != bot_bot_set_o )
     => ( ( comple2134933779557159616set_fm
          @ ( image_o_set_fm
            @ ^ [X3: $o] : C2
            @ A3 ) )
        = C2 ) ) ).

% cSUP_const
thf(fact_687_cSUP__const,axiom,
    ! [A3: set_nat,C2: set_fm] :
      ( ( A3 != bot_bot_set_nat )
     => ( ( comple2134933779557159616set_fm
          @ ( image_nat_set_fm
            @ ^ [X3: nat] : C2
            @ A3 ) )
        = C2 ) ) ).

% cSUP_const
thf(fact_688_cSUP__const,axiom,
    ! [A3: set_rule,C2: $o] :
      ( ( A3 != bot_bot_set_rule )
     => ( ( complete_Sup_Sup_o
          @ ( image_rule_o
            @ ^ [X3: rule] : C2
            @ A3 ) )
        = C2 ) ) ).

% cSUP_const
thf(fact_689_cSUP__const,axiom,
    ! [A3: set_o,C2: $o] :
      ( ( A3 != bot_bot_set_o )
     => ( ( complete_Sup_Sup_o
          @ ( image_o_o
            @ ^ [X3: $o] : C2
            @ A3 ) )
        = C2 ) ) ).

% cSUP_const
thf(fact_690_cSUP__const,axiom,
    ! [A3: set_nat,C2: $o] :
      ( ( A3 != bot_bot_set_nat )
     => ( ( complete_Sup_Sup_o
          @ ( image_nat_o
            @ ^ [X3: nat] : C2
            @ A3 ) )
        = C2 ) ) ).

% cSUP_const
thf(fact_691_cSUP__const,axiom,
    ! [A3: set_rule,C2: nat] :
      ( ( A3 != bot_bot_set_rule )
     => ( ( complete_Sup_Sup_nat
          @ ( image_rule_nat
            @ ^ [X3: rule] : C2
            @ A3 ) )
        = C2 ) ) ).

% cSUP_const
thf(fact_692_cSUP__const,axiom,
    ! [A3: set_o,C2: nat] :
      ( ( A3 != bot_bot_set_o )
     => ( ( complete_Sup_Sup_nat
          @ ( image_o_nat
            @ ^ [X3: $o] : C2
            @ A3 ) )
        = C2 ) ) ).

% cSUP_const
thf(fact_693_cSUP__const,axiom,
    ! [A3: set_nat,C2: nat] :
      ( ( A3 != bot_bot_set_nat )
     => ( ( complete_Sup_Sup_nat
          @ ( image_nat_nat
            @ ^ [X3: nat] : C2
            @ A3 ) )
        = C2 ) ) ).

% cSUP_const
thf(fact_694_ccSUP__const,axiom,
    ! [A3: set_Pr1822751329126368876m_rule,F: set_fm] :
      ( ( A3 != bot_bo3426410157013319936m_rule )
     => ( ( comple2134933779557159616set_fm
          @ ( image_377115208147135190set_fm
            @ ^ [I: produc340336539035504054m_rule] : F
            @ A3 ) )
        = F ) ) ).

% ccSUP_const
thf(fact_695_ccSUP__const,axiom,
    ! [A3: set_rule,F: set_fm] :
      ( ( A3 != bot_bot_set_rule )
     => ( ( comple2134933779557159616set_fm
          @ ( image_rule_set_fm
            @ ^ [I: rule] : F
            @ A3 ) )
        = F ) ) ).

% ccSUP_const
thf(fact_696_ccSUP__const,axiom,
    ! [A3: set_o,F: set_fm] :
      ( ( A3 != bot_bot_set_o )
     => ( ( comple2134933779557159616set_fm
          @ ( image_o_set_fm
            @ ^ [I: $o] : F
            @ A3 ) )
        = F ) ) ).

% ccSUP_const
thf(fact_697_ccSUP__const,axiom,
    ! [A3: set_nat,F: set_fm] :
      ( ( A3 != bot_bot_set_nat )
     => ( ( comple2134933779557159616set_fm
          @ ( image_nat_set_fm
            @ ^ [I: nat] : F
            @ A3 ) )
        = F ) ) ).

% ccSUP_const
thf(fact_698_ccSUP__const,axiom,
    ! [A3: set_rule,F: $o] :
      ( ( A3 != bot_bot_set_rule )
     => ( ( complete_Sup_Sup_o
          @ ( image_rule_o
            @ ^ [I: rule] : F
            @ A3 ) )
        = F ) ) ).

% ccSUP_const
thf(fact_699_ccSUP__const,axiom,
    ! [A3: set_o,F: $o] :
      ( ( A3 != bot_bot_set_o )
     => ( ( complete_Sup_Sup_o
          @ ( image_o_o
            @ ^ [I: $o] : F
            @ A3 ) )
        = F ) ) ).

% ccSUP_const
thf(fact_700_ccSUP__const,axiom,
    ! [A3: set_nat,F: $o] :
      ( ( A3 != bot_bot_set_nat )
     => ( ( complete_Sup_Sup_o
          @ ( image_nat_o
            @ ^ [I: nat] : F
            @ A3 ) )
        = F ) ) ).

% ccSUP_const
thf(fact_701_ccSUP__bot,axiom,
    ! [A3: set_Pr1822751329126368876m_rule] :
      ( ( comple2134933779557159616set_fm
        @ ( image_377115208147135190set_fm
          @ ^ [X3: produc340336539035504054m_rule] : bot_bot_set_fm
          @ A3 ) )
      = bot_bot_set_fm ) ).

% ccSUP_bot
thf(fact_702_ccSup__empty,axiom,
    ( ( comple7665822446944348827t_rule @ bot_bot_set_set_rule )
    = bot_bot_set_rule ) ).

% ccSup_empty
thf(fact_703_ccSup__empty,axiom,
    ( ( comple90263536869209701_set_o @ bot_bot_set_set_o )
    = bot_bot_set_o ) ).

% ccSup_empty
thf(fact_704_ccSup__empty,axiom,
    ( ( comple7399068483239264473et_nat @ bot_bot_set_set_nat )
    = bot_bot_set_nat ) ).

% ccSup_empty
thf(fact_705_ccSup__empty,axiom,
    ( ( comple2134933779557159616set_fm @ bot_bot_set_set_fm )
    = bot_bot_set_fm ) ).

% ccSup_empty
thf(fact_706_ccSup__empty,axiom,
    ( ( complete_Sup_Sup_o @ bot_bot_set_o )
    = bot_bot_o ) ).

% ccSup_empty
thf(fact_707_Union__image__empty,axiom,
    ! [A3: set_nat,F: rule > set_nat] :
      ( ( sup_sup_set_nat @ A3 @ ( comple7399068483239264473et_nat @ ( image_rule_set_nat @ F @ bot_bot_set_rule ) ) )
      = A3 ) ).

% Union_image_empty
thf(fact_708_Union__image__empty,axiom,
    ! [A3: set_nat,F: $o > set_nat] :
      ( ( sup_sup_set_nat @ A3 @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ bot_bot_set_o ) ) )
      = A3 ) ).

% Union_image_empty
thf(fact_709_Union__image__empty,axiom,
    ! [A3: set_nat,F: nat > set_nat] :
      ( ( sup_sup_set_nat @ A3 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ bot_bot_set_nat ) ) )
      = A3 ) ).

% Union_image_empty
thf(fact_710_Union__image__empty,axiom,
    ! [A3: set_fm,F: produc340336539035504054m_rule > set_fm] :
      ( ( sup_sup_set_fm @ A3 @ ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ F @ bot_bo3426410157013319936m_rule ) ) )
      = A3 ) ).

% Union_image_empty
thf(fact_711_Union__image__empty,axiom,
    ! [A3: set_fm,F: rule > set_fm] :
      ( ( sup_sup_set_fm @ A3 @ ( comple2134933779557159616set_fm @ ( image_rule_set_fm @ F @ bot_bot_set_rule ) ) )
      = A3 ) ).

% Union_image_empty
thf(fact_712_Union__image__empty,axiom,
    ! [A3: set_fm,F: $o > set_fm] :
      ( ( sup_sup_set_fm @ A3 @ ( comple2134933779557159616set_fm @ ( image_o_set_fm @ F @ bot_bot_set_o ) ) )
      = A3 ) ).

% Union_image_empty
thf(fact_713_Union__image__empty,axiom,
    ! [A3: set_fm,F: nat > set_fm] :
      ( ( sup_sup_set_fm @ A3 @ ( comple2134933779557159616set_fm @ ( image_nat_set_fm @ F @ bot_bot_set_nat ) ) )
      = A3 ) ).

% Union_image_empty
thf(fact_714_RuleSystem__Defs_ONE__R,axiom,
    ! [Rules: stream2709947120125613254m_rule] :
      ( ( sset_P4484857331586881186m_rule @ Rules )
     != bot_bo3426410157013319936m_rule ) ).

% RuleSystem_Defs.NE_R
thf(fact_715_RuleSystem__Defs_ONE__R,axiom,
    ! [Rules: stream_rule] :
      ( ( sset_rule @ Rules )
     != bot_bot_set_rule ) ).

% RuleSystem_Defs.NE_R
thf(fact_716_RuleSystem__Defs_ONE__R,axiom,
    ! [Rules: stream_o] :
      ( ( sset_o @ Rules )
     != bot_bot_set_o ) ).

% RuleSystem_Defs.NE_R
thf(fact_717_RuleSystem__Defs_ONE__R,axiom,
    ! [Rules: stream_nat] :
      ( ( sset_nat @ Rules )
     != bot_bot_set_nat ) ).

% RuleSystem_Defs.NE_R
thf(fact_718_bot__set__def,axiom,
    ( bot_bot_set_rule
    = ( collect_rule @ bot_bot_rule_o ) ) ).

% bot_set_def
thf(fact_719_bot__set__def,axiom,
    ( bot_bot_set_o
    = ( collect_o @ bot_bot_o_o ) ) ).

% bot_set_def
thf(fact_720_bot__set__def,axiom,
    ( bot_bot_set_nat
    = ( collect_nat @ bot_bot_nat_o ) ) ).

% bot_set_def
thf(fact_721_boolean__algebra__cancel_Osup1,axiom,
    ! [A3: set_nat,K: set_nat,A: set_nat,B2: set_nat] :
      ( ( A3
        = ( sup_sup_set_nat @ K @ A ) )
     => ( ( sup_sup_set_nat @ A3 @ B2 )
        = ( sup_sup_set_nat @ K @ ( sup_sup_set_nat @ A @ B2 ) ) ) ) ).

% boolean_algebra_cancel.sup1
thf(fact_722_boolean__algebra__cancel_Osup2,axiom,
    ! [B: set_nat,K: set_nat,B2: set_nat,A: set_nat] :
      ( ( B
        = ( sup_sup_set_nat @ K @ B2 ) )
     => ( ( sup_sup_set_nat @ A @ B )
        = ( sup_sup_set_nat @ K @ ( sup_sup_set_nat @ A @ B2 ) ) ) ) ).

% boolean_algebra_cancel.sup2
thf(fact_723_RuleSystem__Defs_Oepath_Ocong,axiom,
    abstra6789711989322986974ist_fm = abstra6789711989322986974ist_fm ).

% RuleSystem_Defs.epath.cong
thf(fact_724_RuleSystem__Defs_OSaturated_Ocong,axiom,
    abstra6097777249025082867ist_fm = abstra6097777249025082867ist_fm ).

% RuleSystem_Defs.Saturated.cong
thf(fact_725_boolean__algebra_Odisj__zero__right,axiom,
    ! [X: set_rule] :
      ( ( sup_sup_set_rule @ X @ bot_bot_set_rule )
      = X ) ).

% boolean_algebra.disj_zero_right
thf(fact_726_boolean__algebra_Odisj__zero__right,axiom,
    ! [X: set_o] :
      ( ( sup_sup_set_o @ X @ bot_bot_set_o )
      = X ) ).

% boolean_algebra.disj_zero_right
thf(fact_727_boolean__algebra_Odisj__zero__right,axiom,
    ! [X: set_nat] :
      ( ( sup_sup_set_nat @ X @ bot_bot_set_nat )
      = X ) ).

% boolean_algebra.disj_zero_right
thf(fact_728_i_ONE__R,axiom,
    ! [Rules: stream2709947120125613254m_rule] :
      ( ( sset_P4484857331586881186m_rule @ Rules )
     != bot_bo3426410157013319936m_rule ) ).

% i.NE_R
thf(fact_729_i_ONE__R,axiom,
    ! [Rules: stream_rule] :
      ( ( sset_rule @ Rules )
     != bot_bot_set_rule ) ).

% i.NE_R
thf(fact_730_i_ONE__R,axiom,
    ! [Rules: stream_o] :
      ( ( sset_o @ Rules )
     != bot_bot_set_o ) ).

% i.NE_R
thf(fact_731_i_ONE__R,axiom,
    ! [Rules: stream_nat] :
      ( ( sset_nat @ Rules )
     != bot_bot_set_nat ) ).

% i.NE_R
thf(fact_732_eq__snd__iff,axiom,
    ! [B2: rule,P: produc340336539035504054m_rule] :
      ( ( B2
        = ( produc6514711208036230658m_rule @ P ) )
      = ( ? [A2: produc6018962875968178549ist_fm] :
            ( P
            = ( produc1733806532565653680m_rule @ A2 @ B2 ) ) ) ) ).

% eq_snd_iff
thf(fact_733_snd__conv,axiom,
    ! [X1: produc6018962875968178549ist_fm,X22: rule] :
      ( ( produc6514711208036230658m_rule @ ( produc1733806532565653680m_rule @ X1 @ X22 ) )
      = X22 ) ).

% snd_conv
thf(fact_734_snd__eqD,axiom,
    ! [X: produc6018962875968178549ist_fm,Y: rule,A: rule] :
      ( ( ( produc6514711208036230658m_rule @ ( produc1733806532565653680m_rule @ X @ Y ) )
        = A )
     => ( Y = A ) ) ).

% snd_eqD
thf(fact_735_sndI,axiom,
    ! [X: produc340336539035504054m_rule,Y: produc6018962875968178549ist_fm,Z4: rule] :
      ( ( X
        = ( produc1733806532565653680m_rule @ Y @ Z4 ) )
     => ( ( produc6514711208036230658m_rule @ X )
        = Z4 ) ) ).

% sndI
thf(fact_736_cSUP__UNION,axiom,
    ! [A3: set_fm,B: fm > set_rule,F: rule > $o] :
      ( ( A3 != bot_bot_set_fm )
     => ( ! [X2: fm] :
            ( ( member_fm3 @ X2 @ A3 )
           => ( ( B @ X2 )
             != bot_bot_set_rule ) )
       => ( ( condit5488710616941104124bove_o
            @ ( comple90263536869209701_set_o
              @ ( image_fm_set_o
                @ ^ [X3: fm] : ( image_rule_o @ F @ ( B @ X3 ) )
                @ A3 ) ) )
         => ( ( complete_Sup_Sup_o @ ( image_rule_o @ F @ ( comple7665822446944348827t_rule @ ( image_fm_set_rule @ B @ A3 ) ) ) )
            = ( complete_Sup_Sup_o
              @ ( image_fm_o
                @ ^ [X3: fm] : ( complete_Sup_Sup_o @ ( image_rule_o @ F @ ( B @ X3 ) ) )
                @ A3 ) ) ) ) ) ) ).

% cSUP_UNION
thf(fact_737_cSUP__UNION,axiom,
    ! [A3: set_fm,B: fm > set_o,F: $o > $o] :
      ( ( A3 != bot_bot_set_fm )
     => ( ! [X2: fm] :
            ( ( member_fm3 @ X2 @ A3 )
           => ( ( B @ X2 )
             != bot_bot_set_o ) )
       => ( ( condit5488710616941104124bove_o
            @ ( comple90263536869209701_set_o
              @ ( image_fm_set_o
                @ ^ [X3: fm] : ( image_o_o @ F @ ( B @ X3 ) )
                @ A3 ) ) )
         => ( ( complete_Sup_Sup_o @ ( image_o_o @ F @ ( comple90263536869209701_set_o @ ( image_fm_set_o @ B @ A3 ) ) ) )
            = ( complete_Sup_Sup_o
              @ ( image_fm_o
                @ ^ [X3: fm] : ( complete_Sup_Sup_o @ ( image_o_o @ F @ ( B @ X3 ) ) )
                @ A3 ) ) ) ) ) ) ).

% cSUP_UNION
thf(fact_738_cSUP__UNION,axiom,
    ! [A3: set_fm,B: fm > set_nat,F: nat > $o] :
      ( ( A3 != bot_bot_set_fm )
     => ( ! [X2: fm] :
            ( ( member_fm3 @ X2 @ A3 )
           => ( ( B @ X2 )
             != bot_bot_set_nat ) )
       => ( ( condit5488710616941104124bove_o
            @ ( comple90263536869209701_set_o
              @ ( image_fm_set_o
                @ ^ [X3: fm] : ( image_nat_o @ F @ ( B @ X3 ) )
                @ A3 ) ) )
         => ( ( complete_Sup_Sup_o @ ( image_nat_o @ F @ ( comple7399068483239264473et_nat @ ( image_fm_set_nat @ B @ A3 ) ) ) )
            = ( complete_Sup_Sup_o
              @ ( image_fm_o
                @ ^ [X3: fm] : ( complete_Sup_Sup_o @ ( image_nat_o @ F @ ( B @ X3 ) ) )
                @ A3 ) ) ) ) ) ) ).

% cSUP_UNION
thf(fact_739_cSUP__UNION,axiom,
    ! [A3: set_rule,B: rule > set_rule,F: rule > $o] :
      ( ( A3 != bot_bot_set_rule )
     => ( ! [X2: rule] :
            ( ( member_rule3 @ X2 @ A3 )
           => ( ( B @ X2 )
             != bot_bot_set_rule ) )
       => ( ( condit5488710616941104124bove_o
            @ ( comple90263536869209701_set_o
              @ ( image_rule_set_o
                @ ^ [X3: rule] : ( image_rule_o @ F @ ( B @ X3 ) )
                @ A3 ) ) )
         => ( ( complete_Sup_Sup_o @ ( image_rule_o @ F @ ( comple7665822446944348827t_rule @ ( image_rule_set_rule @ B @ A3 ) ) ) )
            = ( complete_Sup_Sup_o
              @ ( image_rule_o
                @ ^ [X3: rule] : ( complete_Sup_Sup_o @ ( image_rule_o @ F @ ( B @ X3 ) ) )
                @ A3 ) ) ) ) ) ) ).

% cSUP_UNION
thf(fact_740_cSUP__UNION,axiom,
    ! [A3: set_rule,B: rule > set_o,F: $o > $o] :
      ( ( A3 != bot_bot_set_rule )
     => ( ! [X2: rule] :
            ( ( member_rule3 @ X2 @ A3 )
           => ( ( B @ X2 )
             != bot_bot_set_o ) )
       => ( ( condit5488710616941104124bove_o
            @ ( comple90263536869209701_set_o
              @ ( image_rule_set_o
                @ ^ [X3: rule] : ( image_o_o @ F @ ( B @ X3 ) )
                @ A3 ) ) )
         => ( ( complete_Sup_Sup_o @ ( image_o_o @ F @ ( comple90263536869209701_set_o @ ( image_rule_set_o @ B @ A3 ) ) ) )
            = ( complete_Sup_Sup_o
              @ ( image_rule_o
                @ ^ [X3: rule] : ( complete_Sup_Sup_o @ ( image_o_o @ F @ ( B @ X3 ) ) )
                @ A3 ) ) ) ) ) ) ).

% cSUP_UNION
thf(fact_741_cSUP__UNION,axiom,
    ! [A3: set_rule,B: rule > set_nat,F: nat > $o] :
      ( ( A3 != bot_bot_set_rule )
     => ( ! [X2: rule] :
            ( ( member_rule3 @ X2 @ A3 )
           => ( ( B @ X2 )
             != bot_bot_set_nat ) )
       => ( ( condit5488710616941104124bove_o
            @ ( comple90263536869209701_set_o
              @ ( image_rule_set_o
                @ ^ [X3: rule] : ( image_nat_o @ F @ ( B @ X3 ) )
                @ A3 ) ) )
         => ( ( complete_Sup_Sup_o @ ( image_nat_o @ F @ ( comple7399068483239264473et_nat @ ( image_rule_set_nat @ B @ A3 ) ) ) )
            = ( complete_Sup_Sup_o
              @ ( image_rule_o
                @ ^ [X3: rule] : ( complete_Sup_Sup_o @ ( image_nat_o @ F @ ( B @ X3 ) ) )
                @ A3 ) ) ) ) ) ) ).

% cSUP_UNION
thf(fact_742_cSUP__UNION,axiom,
    ! [A3: set_o,B: $o > set_rule,F: rule > $o] :
      ( ( A3 != bot_bot_set_o )
     => ( ! [X2: $o] :
            ( ( member_o3 @ X2 @ A3 )
           => ( ( B @ X2 )
             != bot_bot_set_rule ) )
       => ( ( condit5488710616941104124bove_o
            @ ( comple90263536869209701_set_o
              @ ( image_o_set_o
                @ ^ [X3: $o] : ( image_rule_o @ F @ ( B @ X3 ) )
                @ A3 ) ) )
         => ( ( complete_Sup_Sup_o @ ( image_rule_o @ F @ ( comple7665822446944348827t_rule @ ( image_o_set_rule @ B @ A3 ) ) ) )
            = ( complete_Sup_Sup_o
              @ ( image_o_o
                @ ^ [X3: $o] : ( complete_Sup_Sup_o @ ( image_rule_o @ F @ ( B @ X3 ) ) )
                @ A3 ) ) ) ) ) ) ).

% cSUP_UNION
thf(fact_743_cSUP__UNION,axiom,
    ! [A3: set_o,B: $o > set_o,F: $o > $o] :
      ( ( A3 != bot_bot_set_o )
     => ( ! [X2: $o] :
            ( ( member_o3 @ X2 @ A3 )
           => ( ( B @ X2 )
             != bot_bot_set_o ) )
       => ( ( condit5488710616941104124bove_o
            @ ( comple90263536869209701_set_o
              @ ( image_o_set_o
                @ ^ [X3: $o] : ( image_o_o @ F @ ( B @ X3 ) )
                @ A3 ) ) )
         => ( ( complete_Sup_Sup_o @ ( image_o_o @ F @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B @ A3 ) ) ) )
            = ( complete_Sup_Sup_o
              @ ( image_o_o
                @ ^ [X3: $o] : ( complete_Sup_Sup_o @ ( image_o_o @ F @ ( B @ X3 ) ) )
                @ A3 ) ) ) ) ) ) ).

% cSUP_UNION
thf(fact_744_cSUP__UNION,axiom,
    ! [A3: set_o,B: $o > set_nat,F: nat > $o] :
      ( ( A3 != bot_bot_set_o )
     => ( ! [X2: $o] :
            ( ( member_o3 @ X2 @ A3 )
           => ( ( B @ X2 )
             != bot_bot_set_nat ) )
       => ( ( condit5488710616941104124bove_o
            @ ( comple90263536869209701_set_o
              @ ( image_o_set_o
                @ ^ [X3: $o] : ( image_nat_o @ F @ ( B @ X3 ) )
                @ A3 ) ) )
         => ( ( complete_Sup_Sup_o @ ( image_nat_o @ F @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ B @ A3 ) ) ) )
            = ( complete_Sup_Sup_o
              @ ( image_o_o
                @ ^ [X3: $o] : ( complete_Sup_Sup_o @ ( image_nat_o @ F @ ( B @ X3 ) ) )
                @ A3 ) ) ) ) ) ) ).

% cSUP_UNION
thf(fact_745_cSUP__UNION,axiom,
    ! [A3: set_nat,B: nat > set_rule,F: rule > $o] :
      ( ( A3 != bot_bot_set_nat )
     => ( ! [X2: nat] :
            ( ( member_nat3 @ X2 @ A3 )
           => ( ( B @ X2 )
             != bot_bot_set_rule ) )
       => ( ( condit5488710616941104124bove_o
            @ ( comple90263536869209701_set_o
              @ ( image_nat_set_o
                @ ^ [X3: nat] : ( image_rule_o @ F @ ( B @ X3 ) )
                @ A3 ) ) )
         => ( ( complete_Sup_Sup_o @ ( image_rule_o @ F @ ( comple7665822446944348827t_rule @ ( image_nat_set_rule @ B @ A3 ) ) ) )
            = ( complete_Sup_Sup_o
              @ ( image_nat_o
                @ ^ [X3: nat] : ( complete_Sup_Sup_o @ ( image_rule_o @ F @ ( B @ X3 ) ) )
                @ A3 ) ) ) ) ) ) ).

% cSUP_UNION
thf(fact_746_Set_Ois__empty__def,axiom,
    ( is_empty_rule
    = ( ^ [A4: set_rule] : ( A4 = bot_bot_set_rule ) ) ) ).

% Set.is_empty_def
thf(fact_747_Set_Ois__empty__def,axiom,
    ( is_empty_o
    = ( ^ [A4: set_o] : ( A4 = bot_bot_set_o ) ) ) ).

% Set.is_empty_def
thf(fact_748_Set_Ois__empty__def,axiom,
    ( is_empty_nat
    = ( ^ [A4: set_nat] : ( A4 = bot_bot_set_nat ) ) ) ).

% Set.is_empty_def
thf(fact_749_bdd__above__Un,axiom,
    ! [A3: set_nat,B: set_nat] :
      ( ( condit2214826472909112428ve_nat @ ( sup_sup_set_nat @ A3 @ B ) )
      = ( ( condit2214826472909112428ve_nat @ A3 )
        & ( condit2214826472909112428ve_nat @ B ) ) ) ).

% bdd_above_Un
thf(fact_750_bdd__above__image__sup,axiom,
    ! [F: produc340336539035504054m_rule > set_fm,G: produc340336539035504054m_rule > set_fm,A3: set_Pr1822751329126368876m_rule] :
      ( ( condit7632555770478454199set_fm
        @ ( image_377115208147135190set_fm
          @ ^ [X3: produc340336539035504054m_rule] : ( sup_sup_set_fm @ ( F @ X3 ) @ ( G @ X3 ) )
          @ A3 ) )
      = ( ( condit7632555770478454199set_fm @ ( image_377115208147135190set_fm @ F @ A3 ) )
        & ( condit7632555770478454199set_fm @ ( image_377115208147135190set_fm @ G @ A3 ) ) ) ) ).

% bdd_above_image_sup
thf(fact_751_bdd__above__image__sup,axiom,
    ! [F: nat > nat,G: nat > nat,A3: set_nat] :
      ( ( condit2214826472909112428ve_nat
        @ ( image_nat_nat
          @ ^ [X3: nat] : ( sup_sup_nat @ ( F @ X3 ) @ ( G @ X3 ) )
          @ A3 ) )
      = ( ( condit2214826472909112428ve_nat @ ( image_nat_nat @ F @ A3 ) )
        & ( condit2214826472909112428ve_nat @ ( image_nat_nat @ G @ A3 ) ) ) ) ).

% bdd_above_image_sup
thf(fact_752_cSup__union__distrib,axiom,
    ! [A3: set_set_nat,B: set_set_nat] :
      ( ( A3 != bot_bot_set_set_nat )
     => ( ( condit5477540289124974626et_nat @ A3 )
       => ( ( B != bot_bot_set_set_nat )
         => ( ( condit5477540289124974626et_nat @ B )
           => ( ( comple7399068483239264473et_nat @ ( sup_sup_set_set_nat @ A3 @ B ) )
              = ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ A3 ) @ ( comple7399068483239264473et_nat @ B ) ) ) ) ) ) ) ).

% cSup_union_distrib
thf(fact_753_cSup__union__distrib,axiom,
    ! [A3: set_set_fm,B: set_set_fm] :
      ( ( A3 != bot_bot_set_set_fm )
     => ( ( condit7632555770478454199set_fm @ A3 )
       => ( ( B != bot_bot_set_set_fm )
         => ( ( condit7632555770478454199set_fm @ B )
           => ( ( comple2134933779557159616set_fm @ ( sup_sup_set_set_fm @ A3 @ B ) )
              = ( sup_sup_set_fm @ ( comple2134933779557159616set_fm @ A3 ) @ ( comple2134933779557159616set_fm @ B ) ) ) ) ) ) ) ).

% cSup_union_distrib
thf(fact_754_cSup__union__distrib,axiom,
    ! [A3: set_o,B: set_o] :
      ( ( A3 != bot_bot_set_o )
     => ( ( condit5488710616941104124bove_o @ A3 )
       => ( ( B != bot_bot_set_o )
         => ( ( condit5488710616941104124bove_o @ B )
           => ( ( complete_Sup_Sup_o @ ( sup_sup_set_o @ A3 @ B ) )
              = ( sup_sup_o @ ( complete_Sup_Sup_o @ A3 ) @ ( complete_Sup_Sup_o @ B ) ) ) ) ) ) ) ).

% cSup_union_distrib
thf(fact_755_cSup__union__distrib,axiom,
    ! [A3: set_nat,B: set_nat] :
      ( ( A3 != bot_bot_set_nat )
     => ( ( condit2214826472909112428ve_nat @ A3 )
       => ( ( B != bot_bot_set_nat )
         => ( ( condit2214826472909112428ve_nat @ B )
           => ( ( complete_Sup_Sup_nat @ ( sup_sup_set_nat @ A3 @ B ) )
              = ( sup_sup_nat @ ( complete_Sup_Sup_nat @ A3 ) @ ( complete_Sup_Sup_nat @ B ) ) ) ) ) ) ) ).

% cSup_union_distrib
thf(fact_756_conditionally__complete__lattice__class_OSUP__sup__distrib,axiom,
    ! [A3: set_rule,F: rule > $o,G: rule > $o] :
      ( ( A3 != bot_bot_set_rule )
     => ( ( condit5488710616941104124bove_o @ ( image_rule_o @ F @ A3 ) )
       => ( ( condit5488710616941104124bove_o @ ( image_rule_o @ G @ A3 ) )
         => ( ( sup_sup_o @ ( complete_Sup_Sup_o @ ( image_rule_o @ F @ A3 ) ) @ ( complete_Sup_Sup_o @ ( image_rule_o @ G @ A3 ) ) )
            = ( complete_Sup_Sup_o
              @ ( image_rule_o
                @ ^ [A2: rule] : ( sup_sup_o @ ( F @ A2 ) @ ( G @ A2 ) )
                @ A3 ) ) ) ) ) ) ).

% conditionally_complete_lattice_class.SUP_sup_distrib
thf(fact_757_conditionally__complete__lattice__class_OSUP__sup__distrib,axiom,
    ! [A3: set_o,F: $o > $o,G: $o > $o] :
      ( ( A3 != bot_bot_set_o )
     => ( ( condit5488710616941104124bove_o @ ( image_o_o @ F @ A3 ) )
       => ( ( condit5488710616941104124bove_o @ ( image_o_o @ G @ A3 ) )
         => ( ( sup_sup_o @ ( complete_Sup_Sup_o @ ( image_o_o @ F @ A3 ) ) @ ( complete_Sup_Sup_o @ ( image_o_o @ G @ A3 ) ) )
            = ( complete_Sup_Sup_o
              @ ( image_o_o
                @ ^ [A2: $o] : ( sup_sup_o @ ( F @ A2 ) @ ( G @ A2 ) )
                @ A3 ) ) ) ) ) ) ).

% conditionally_complete_lattice_class.SUP_sup_distrib
thf(fact_758_conditionally__complete__lattice__class_OSUP__sup__distrib,axiom,
    ! [A3: set_nat,F: nat > $o,G: nat > $o] :
      ( ( A3 != bot_bot_set_nat )
     => ( ( condit5488710616941104124bove_o @ ( image_nat_o @ F @ A3 ) )
       => ( ( condit5488710616941104124bove_o @ ( image_nat_o @ G @ A3 ) )
         => ( ( sup_sup_o @ ( complete_Sup_Sup_o @ ( image_nat_o @ F @ A3 ) ) @ ( complete_Sup_Sup_o @ ( image_nat_o @ G @ A3 ) ) )
            = ( complete_Sup_Sup_o
              @ ( image_nat_o
                @ ^ [A2: nat] : ( sup_sup_o @ ( F @ A2 ) @ ( G @ A2 ) )
                @ A3 ) ) ) ) ) ) ).

% conditionally_complete_lattice_class.SUP_sup_distrib
thf(fact_759_conditionally__complete__lattice__class_OSUP__sup__distrib,axiom,
    ! [A3: set_rule,F: rule > nat,G: rule > nat] :
      ( ( A3 != bot_bot_set_rule )
     => ( ( condit2214826472909112428ve_nat @ ( image_rule_nat @ F @ A3 ) )
       => ( ( condit2214826472909112428ve_nat @ ( image_rule_nat @ G @ A3 ) )
         => ( ( sup_sup_nat @ ( complete_Sup_Sup_nat @ ( image_rule_nat @ F @ A3 ) ) @ ( complete_Sup_Sup_nat @ ( image_rule_nat @ G @ A3 ) ) )
            = ( complete_Sup_Sup_nat
              @ ( image_rule_nat
                @ ^ [A2: rule] : ( sup_sup_nat @ ( F @ A2 ) @ ( G @ A2 ) )
                @ A3 ) ) ) ) ) ) ).

% conditionally_complete_lattice_class.SUP_sup_distrib
thf(fact_760_conditionally__complete__lattice__class_OSUP__sup__distrib,axiom,
    ! [A3: set_o,F: $o > nat,G: $o > nat] :
      ( ( A3 != bot_bot_set_o )
     => ( ( condit2214826472909112428ve_nat @ ( image_o_nat @ F @ A3 ) )
       => ( ( condit2214826472909112428ve_nat @ ( image_o_nat @ G @ A3 ) )
         => ( ( sup_sup_nat @ ( complete_Sup_Sup_nat @ ( image_o_nat @ F @ A3 ) ) @ ( complete_Sup_Sup_nat @ ( image_o_nat @ G @ A3 ) ) )
            = ( complete_Sup_Sup_nat
              @ ( image_o_nat
                @ ^ [A2: $o] : ( sup_sup_nat @ ( F @ A2 ) @ ( G @ A2 ) )
                @ A3 ) ) ) ) ) ) ).

% conditionally_complete_lattice_class.SUP_sup_distrib
thf(fact_761_conditionally__complete__lattice__class_OSUP__sup__distrib,axiom,
    ! [A3: set_nat,F: nat > nat,G: nat > nat] :
      ( ( A3 != bot_bot_set_nat )
     => ( ( condit2214826472909112428ve_nat @ ( image_nat_nat @ F @ A3 ) )
       => ( ( condit2214826472909112428ve_nat @ ( image_nat_nat @ G @ A3 ) )
         => ( ( sup_sup_nat @ ( complete_Sup_Sup_nat @ ( image_nat_nat @ F @ A3 ) ) @ ( complete_Sup_Sup_nat @ ( image_nat_nat @ G @ A3 ) ) )
            = ( complete_Sup_Sup_nat
              @ ( image_nat_nat
                @ ^ [A2: nat] : ( sup_sup_nat @ ( F @ A2 ) @ ( G @ A2 ) )
                @ A3 ) ) ) ) ) ) ).

% conditionally_complete_lattice_class.SUP_sup_distrib
thf(fact_762_conditionally__complete__lattice__class_OSUP__sup__distrib,axiom,
    ! [A3: set_rule,F: rule > set_nat,G: rule > set_nat] :
      ( ( A3 != bot_bot_set_rule )
     => ( ( condit5477540289124974626et_nat @ ( image_rule_set_nat @ F @ A3 ) )
       => ( ( condit5477540289124974626et_nat @ ( image_rule_set_nat @ G @ A3 ) )
         => ( ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ ( image_rule_set_nat @ F @ A3 ) ) @ ( comple7399068483239264473et_nat @ ( image_rule_set_nat @ G @ A3 ) ) )
            = ( comple7399068483239264473et_nat
              @ ( image_rule_set_nat
                @ ^ [A2: rule] : ( sup_sup_set_nat @ ( F @ A2 ) @ ( G @ A2 ) )
                @ A3 ) ) ) ) ) ) ).

% conditionally_complete_lattice_class.SUP_sup_distrib
thf(fact_763_conditionally__complete__lattice__class_OSUP__sup__distrib,axiom,
    ! [A3: set_o,F: $o > set_nat,G: $o > set_nat] :
      ( ( A3 != bot_bot_set_o )
     => ( ( condit5477540289124974626et_nat @ ( image_o_set_nat @ F @ A3 ) )
       => ( ( condit5477540289124974626et_nat @ ( image_o_set_nat @ G @ A3 ) )
         => ( ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ A3 ) ) @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ G @ A3 ) ) )
            = ( comple7399068483239264473et_nat
              @ ( image_o_set_nat
                @ ^ [A2: $o] : ( sup_sup_set_nat @ ( F @ A2 ) @ ( G @ A2 ) )
                @ A3 ) ) ) ) ) ) ).

% conditionally_complete_lattice_class.SUP_sup_distrib
thf(fact_764_conditionally__complete__lattice__class_OSUP__sup__distrib,axiom,
    ! [A3: set_nat,F: nat > set_nat,G: nat > set_nat] :
      ( ( A3 != bot_bot_set_nat )
     => ( ( condit5477540289124974626et_nat @ ( image_nat_set_nat @ F @ A3 ) )
       => ( ( condit5477540289124974626et_nat @ ( image_nat_set_nat @ G @ A3 ) )
         => ( ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A3 ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ A3 ) ) )
            = ( comple7399068483239264473et_nat
              @ ( image_nat_set_nat
                @ ^ [A2: nat] : ( sup_sup_set_nat @ ( F @ A2 ) @ ( G @ A2 ) )
                @ A3 ) ) ) ) ) ) ).

% conditionally_complete_lattice_class.SUP_sup_distrib
thf(fact_765_conditionally__complete__lattice__class_OSUP__sup__distrib,axiom,
    ! [A3: set_rule,F: rule > set_fm,G: rule > set_fm] :
      ( ( A3 != bot_bot_set_rule )
     => ( ( condit7632555770478454199set_fm @ ( image_rule_set_fm @ F @ A3 ) )
       => ( ( condit7632555770478454199set_fm @ ( image_rule_set_fm @ G @ A3 ) )
         => ( ( sup_sup_set_fm @ ( comple2134933779557159616set_fm @ ( image_rule_set_fm @ F @ A3 ) ) @ ( comple2134933779557159616set_fm @ ( image_rule_set_fm @ G @ A3 ) ) )
            = ( comple2134933779557159616set_fm
              @ ( image_rule_set_fm
                @ ^ [A2: rule] : ( sup_sup_set_fm @ ( F @ A2 ) @ ( G @ A2 ) )
                @ A3 ) ) ) ) ) ) ).

% conditionally_complete_lattice_class.SUP_sup_distrib
thf(fact_766_cSUP__union,axiom,
    ! [A3: set_rule,F: rule > $o,B: set_rule] :
      ( ( A3 != bot_bot_set_rule )
     => ( ( condit5488710616941104124bove_o @ ( image_rule_o @ F @ A3 ) )
       => ( ( B != bot_bot_set_rule )
         => ( ( condit5488710616941104124bove_o @ ( image_rule_o @ F @ B ) )
           => ( ( complete_Sup_Sup_o @ ( image_rule_o @ F @ ( sup_sup_set_rule @ A3 @ B ) ) )
              = ( sup_sup_o @ ( complete_Sup_Sup_o @ ( image_rule_o @ F @ A3 ) ) @ ( complete_Sup_Sup_o @ ( image_rule_o @ F @ B ) ) ) ) ) ) ) ) ).

% cSUP_union
thf(fact_767_cSUP__union,axiom,
    ! [A3: set_o,F: $o > $o,B: set_o] :
      ( ( A3 != bot_bot_set_o )
     => ( ( condit5488710616941104124bove_o @ ( image_o_o @ F @ A3 ) )
       => ( ( B != bot_bot_set_o )
         => ( ( condit5488710616941104124bove_o @ ( image_o_o @ F @ B ) )
           => ( ( complete_Sup_Sup_o @ ( image_o_o @ F @ ( sup_sup_set_o @ A3 @ B ) ) )
              = ( sup_sup_o @ ( complete_Sup_Sup_o @ ( image_o_o @ F @ A3 ) ) @ ( complete_Sup_Sup_o @ ( image_o_o @ F @ B ) ) ) ) ) ) ) ) ).

% cSUP_union
thf(fact_768_cSUP__union,axiom,
    ! [A3: set_nat,F: nat > $o,B: set_nat] :
      ( ( A3 != bot_bot_set_nat )
     => ( ( condit5488710616941104124bove_o @ ( image_nat_o @ F @ A3 ) )
       => ( ( B != bot_bot_set_nat )
         => ( ( condit5488710616941104124bove_o @ ( image_nat_o @ F @ B ) )
           => ( ( complete_Sup_Sup_o @ ( image_nat_o @ F @ ( sup_sup_set_nat @ A3 @ B ) ) )
              = ( sup_sup_o @ ( complete_Sup_Sup_o @ ( image_nat_o @ F @ A3 ) ) @ ( complete_Sup_Sup_o @ ( image_nat_o @ F @ B ) ) ) ) ) ) ) ) ).

% cSUP_union
thf(fact_769_cSUP__union,axiom,
    ! [A3: set_rule,F: rule > nat,B: set_rule] :
      ( ( A3 != bot_bot_set_rule )
     => ( ( condit2214826472909112428ve_nat @ ( image_rule_nat @ F @ A3 ) )
       => ( ( B != bot_bot_set_rule )
         => ( ( condit2214826472909112428ve_nat @ ( image_rule_nat @ F @ B ) )
           => ( ( complete_Sup_Sup_nat @ ( image_rule_nat @ F @ ( sup_sup_set_rule @ A3 @ B ) ) )
              = ( sup_sup_nat @ ( complete_Sup_Sup_nat @ ( image_rule_nat @ F @ A3 ) ) @ ( complete_Sup_Sup_nat @ ( image_rule_nat @ F @ B ) ) ) ) ) ) ) ) ).

% cSUP_union
thf(fact_770_cSUP__union,axiom,
    ! [A3: set_o,F: $o > nat,B: set_o] :
      ( ( A3 != bot_bot_set_o )
     => ( ( condit2214826472909112428ve_nat @ ( image_o_nat @ F @ A3 ) )
       => ( ( B != bot_bot_set_o )
         => ( ( condit2214826472909112428ve_nat @ ( image_o_nat @ F @ B ) )
           => ( ( complete_Sup_Sup_nat @ ( image_o_nat @ F @ ( sup_sup_set_o @ A3 @ B ) ) )
              = ( sup_sup_nat @ ( complete_Sup_Sup_nat @ ( image_o_nat @ F @ A3 ) ) @ ( complete_Sup_Sup_nat @ ( image_o_nat @ F @ B ) ) ) ) ) ) ) ) ).

% cSUP_union
thf(fact_771_cSUP__union,axiom,
    ! [A3: set_nat,F: nat > nat,B: set_nat] :
      ( ( A3 != bot_bot_set_nat )
     => ( ( condit2214826472909112428ve_nat @ ( image_nat_nat @ F @ A3 ) )
       => ( ( B != bot_bot_set_nat )
         => ( ( condit2214826472909112428ve_nat @ ( image_nat_nat @ F @ B ) )
           => ( ( complete_Sup_Sup_nat @ ( image_nat_nat @ F @ ( sup_sup_set_nat @ A3 @ B ) ) )
              = ( sup_sup_nat @ ( complete_Sup_Sup_nat @ ( image_nat_nat @ F @ A3 ) ) @ ( complete_Sup_Sup_nat @ ( image_nat_nat @ F @ B ) ) ) ) ) ) ) ) ).

% cSUP_union
thf(fact_772_cSUP__union,axiom,
    ! [A3: set_rule,F: rule > set_nat,B: set_rule] :
      ( ( A3 != bot_bot_set_rule )
     => ( ( condit5477540289124974626et_nat @ ( image_rule_set_nat @ F @ A3 ) )
       => ( ( B != bot_bot_set_rule )
         => ( ( condit5477540289124974626et_nat @ ( image_rule_set_nat @ F @ B ) )
           => ( ( comple7399068483239264473et_nat @ ( image_rule_set_nat @ F @ ( sup_sup_set_rule @ A3 @ B ) ) )
              = ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ ( image_rule_set_nat @ F @ A3 ) ) @ ( comple7399068483239264473et_nat @ ( image_rule_set_nat @ F @ B ) ) ) ) ) ) ) ) ).

% cSUP_union
thf(fact_773_cSUP__union,axiom,
    ! [A3: set_o,F: $o > set_nat,B: set_o] :
      ( ( A3 != bot_bot_set_o )
     => ( ( condit5477540289124974626et_nat @ ( image_o_set_nat @ F @ A3 ) )
       => ( ( B != bot_bot_set_o )
         => ( ( condit5477540289124974626et_nat @ ( image_o_set_nat @ F @ B ) )
           => ( ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ ( sup_sup_set_o @ A3 @ B ) ) )
              = ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ A3 ) ) @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ B ) ) ) ) ) ) ) ) ).

% cSUP_union
thf(fact_774_cSUP__union,axiom,
    ! [A3: set_nat,F: nat > set_nat,B: set_nat] :
      ( ( A3 != bot_bot_set_nat )
     => ( ( condit5477540289124974626et_nat @ ( image_nat_set_nat @ F @ A3 ) )
       => ( ( B != bot_bot_set_nat )
         => ( ( condit5477540289124974626et_nat @ ( image_nat_set_nat @ F @ B ) )
           => ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ ( sup_sup_set_nat @ A3 @ B ) ) )
              = ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A3 ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ B ) ) ) ) ) ) ) ) ).

% cSUP_union
thf(fact_775_cSUP__union,axiom,
    ! [A3: set_rule,F: rule > set_fm,B: set_rule] :
      ( ( A3 != bot_bot_set_rule )
     => ( ( condit7632555770478454199set_fm @ ( image_rule_set_fm @ F @ A3 ) )
       => ( ( B != bot_bot_set_rule )
         => ( ( condit7632555770478454199set_fm @ ( image_rule_set_fm @ F @ B ) )
           => ( ( comple2134933779557159616set_fm @ ( image_rule_set_fm @ F @ ( sup_sup_set_rule @ A3 @ B ) ) )
              = ( sup_sup_set_fm @ ( comple2134933779557159616set_fm @ ( image_rule_set_fm @ F @ A3 ) ) @ ( comple2134933779557159616set_fm @ ( image_rule_set_fm @ F @ B ) ) ) ) ) ) ) ) ).

% cSUP_union
thf(fact_776_is__empty__set,axiom,
    ! [Xs2: list_fm] :
      ( ( is_empty_fm @ ( set_fm2 @ Xs2 ) )
      = ( null_fm @ Xs2 ) ) ).

% is_empty_set
thf(fact_777_is__empty__set,axiom,
    ! [Xs2: list_list_fm] :
      ( ( is_empty_list_fm @ ( set_list_fm2 @ Xs2 ) )
      = ( null_list_fm @ Xs2 ) ) ).

% is_empty_set
thf(fact_778_is__empty__set,axiom,
    ! [Xs2: list_tm] :
      ( ( is_empty_tm @ ( set_tm2 @ Xs2 ) )
      = ( null_tm @ Xs2 ) ) ).

% is_empty_set
thf(fact_779_cSUP__insert,axiom,
    ! [A3: set_rule,F: rule > $o,A: rule] :
      ( ( A3 != bot_bot_set_rule )
     => ( ( condit5488710616941104124bove_o @ ( image_rule_o @ F @ A3 ) )
       => ( ( complete_Sup_Sup_o @ ( image_rule_o @ F @ ( insert_rule @ A @ A3 ) ) )
          = ( sup_sup_o @ ( F @ A ) @ ( complete_Sup_Sup_o @ ( image_rule_o @ F @ A3 ) ) ) ) ) ) ).

% cSUP_insert
thf(fact_780_cSUP__insert,axiom,
    ! [A3: set_o,F: $o > $o,A: $o] :
      ( ( A3 != bot_bot_set_o )
     => ( ( condit5488710616941104124bove_o @ ( image_o_o @ F @ A3 ) )
       => ( ( complete_Sup_Sup_o @ ( image_o_o @ F @ ( insert_o @ A @ A3 ) ) )
          = ( sup_sup_o @ ( F @ A ) @ ( complete_Sup_Sup_o @ ( image_o_o @ F @ A3 ) ) ) ) ) ) ).

% cSUP_insert
thf(fact_781_cSUP__insert,axiom,
    ! [A3: set_nat,F: nat > $o,A: nat] :
      ( ( A3 != bot_bot_set_nat )
     => ( ( condit5488710616941104124bove_o @ ( image_nat_o @ F @ A3 ) )
       => ( ( complete_Sup_Sup_o @ ( image_nat_o @ F @ ( insert_nat @ A @ A3 ) ) )
          = ( sup_sup_o @ ( F @ A ) @ ( complete_Sup_Sup_o @ ( image_nat_o @ F @ A3 ) ) ) ) ) ) ).

% cSUP_insert
thf(fact_782_cSUP__insert,axiom,
    ! [A3: set_rule,F: rule > nat,A: rule] :
      ( ( A3 != bot_bot_set_rule )
     => ( ( condit2214826472909112428ve_nat @ ( image_rule_nat @ F @ A3 ) )
       => ( ( complete_Sup_Sup_nat @ ( image_rule_nat @ F @ ( insert_rule @ A @ A3 ) ) )
          = ( sup_sup_nat @ ( F @ A ) @ ( complete_Sup_Sup_nat @ ( image_rule_nat @ F @ A3 ) ) ) ) ) ) ).

% cSUP_insert
thf(fact_783_cSUP__insert,axiom,
    ! [A3: set_o,F: $o > nat,A: $o] :
      ( ( A3 != bot_bot_set_o )
     => ( ( condit2214826472909112428ve_nat @ ( image_o_nat @ F @ A3 ) )
       => ( ( complete_Sup_Sup_nat @ ( image_o_nat @ F @ ( insert_o @ A @ A3 ) ) )
          = ( sup_sup_nat @ ( F @ A ) @ ( complete_Sup_Sup_nat @ ( image_o_nat @ F @ A3 ) ) ) ) ) ) ).

% cSUP_insert
thf(fact_784_cSUP__insert,axiom,
    ! [A3: set_nat,F: nat > nat,A: nat] :
      ( ( A3 != bot_bot_set_nat )
     => ( ( condit2214826472909112428ve_nat @ ( image_nat_nat @ F @ A3 ) )
       => ( ( complete_Sup_Sup_nat @ ( image_nat_nat @ F @ ( insert_nat @ A @ A3 ) ) )
          = ( sup_sup_nat @ ( F @ A ) @ ( complete_Sup_Sup_nat @ ( image_nat_nat @ F @ A3 ) ) ) ) ) ) ).

% cSUP_insert
thf(fact_785_cSUP__insert,axiom,
    ! [A3: set_rule,F: rule > set_nat,A: rule] :
      ( ( A3 != bot_bot_set_rule )
     => ( ( condit5477540289124974626et_nat @ ( image_rule_set_nat @ F @ A3 ) )
       => ( ( comple7399068483239264473et_nat @ ( image_rule_set_nat @ F @ ( insert_rule @ A @ A3 ) ) )
          = ( sup_sup_set_nat @ ( F @ A ) @ ( comple7399068483239264473et_nat @ ( image_rule_set_nat @ F @ A3 ) ) ) ) ) ) ).

% cSUP_insert
thf(fact_786_cSUP__insert,axiom,
    ! [A3: set_o,F: $o > set_nat,A: $o] :
      ( ( A3 != bot_bot_set_o )
     => ( ( condit5477540289124974626et_nat @ ( image_o_set_nat @ F @ A3 ) )
       => ( ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ ( insert_o @ A @ A3 ) ) )
          = ( sup_sup_set_nat @ ( F @ A ) @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ A3 ) ) ) ) ) ) ).

% cSUP_insert
thf(fact_787_cSUP__insert,axiom,
    ! [A3: set_nat,F: nat > set_nat,A: nat] :
      ( ( A3 != bot_bot_set_nat )
     => ( ( condit5477540289124974626et_nat @ ( image_nat_set_nat @ F @ A3 ) )
       => ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ ( insert_nat @ A @ A3 ) ) )
          = ( sup_sup_set_nat @ ( F @ A ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A3 ) ) ) ) ) ) ).

% cSUP_insert
thf(fact_788_cSUP__insert,axiom,
    ! [A3: set_rule,F: rule > set_fm,A: rule] :
      ( ( A3 != bot_bot_set_rule )
     => ( ( condit7632555770478454199set_fm @ ( image_rule_set_fm @ F @ A3 ) )
       => ( ( comple2134933779557159616set_fm @ ( image_rule_set_fm @ F @ ( insert_rule @ A @ A3 ) ) )
          = ( sup_sup_set_fm @ ( F @ A ) @ ( comple2134933779557159616set_fm @ ( image_rule_set_fm @ F @ A3 ) ) ) ) ) ) ).

% cSUP_insert
thf(fact_789_refl__on__UNION,axiom,
    ! [S4: set_Pr1822751329126368876m_rule,A3: produc340336539035504054m_rule > set_fm,R: produc340336539035504054m_rule > set_Pr4706815898642364871_fm_fm] :
      ( ! [X2: produc340336539035504054m_rule] :
          ( ( member7231649785386036813m_rule @ X2 @ S4 )
         => ( refl_on_fm @ ( A3 @ X2 ) @ ( R @ X2 ) ) )
     => ( refl_on_fm @ ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ A3 @ S4 ) ) @ ( comple3052221160759688756_fm_fm @ ( image_9204174774680167498_fm_fm @ R @ S4 ) ) ) ) ).

% refl_on_UNION
thf(fact_790_RuleSystem__Defs_OSaturated__def,axiom,
    ( abstra6097777249025082867ist_fm
    = ( ^ [Eff: rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rules2: stream_rule,Steps2: stream2709947120125613254m_rule] :
        ! [X3: rule] :
          ( ( member_rule3 @ X3 @ ( sset_rule @ Rules2 ) )
         => ( abstra2533313685312581075ist_fm @ Eff @ X3 @ Steps2 ) ) ) ) ).

% RuleSystem_Defs.Saturated_def
thf(fact_791_less__cSUP__iff,axiom,
    ! [A3: set_rule,F: rule > nat,A: nat] :
      ( ( A3 != bot_bot_set_rule )
     => ( ( condit2214826472909112428ve_nat @ ( image_rule_nat @ F @ A3 ) )
       => ( ( ord_less_nat @ A @ ( complete_Sup_Sup_nat @ ( image_rule_nat @ F @ A3 ) ) )
          = ( ? [X3: rule] :
                ( ( member_rule3 @ X3 @ A3 )
                & ( ord_less_nat @ A @ ( F @ X3 ) ) ) ) ) ) ) ).

% less_cSUP_iff
thf(fact_792_less__cSUP__iff,axiom,
    ! [A3: set_o,F: $o > nat,A: nat] :
      ( ( A3 != bot_bot_set_o )
     => ( ( condit2214826472909112428ve_nat @ ( image_o_nat @ F @ A3 ) )
       => ( ( ord_less_nat @ A @ ( complete_Sup_Sup_nat @ ( image_o_nat @ F @ A3 ) ) )
          = ( ? [X3: $o] :
                ( ( member_o3 @ X3 @ A3 )
                & ( ord_less_nat @ A @ ( F @ X3 ) ) ) ) ) ) ) ).

% less_cSUP_iff
thf(fact_793_less__cSUP__iff,axiom,
    ! [A3: set_nat,F: nat > nat,A: nat] :
      ( ( A3 != bot_bot_set_nat )
     => ( ( condit2214826472909112428ve_nat @ ( image_nat_nat @ F @ A3 ) )
       => ( ( ord_less_nat @ A @ ( complete_Sup_Sup_nat @ ( image_nat_nat @ F @ A3 ) ) )
          = ( ? [X3: nat] :
                ( ( member_nat3 @ X3 @ A3 )
                & ( ord_less_nat @ A @ ( F @ X3 ) ) ) ) ) ) ) ).

% less_cSUP_iff
thf(fact_794_insert__absorb2,axiom,
    ! [X: $o,A3: set_o] :
      ( ( insert_o @ X @ ( insert_o @ X @ A3 ) )
      = ( insert_o @ X @ A3 ) ) ).

% insert_absorb2
thf(fact_795_insert__absorb2,axiom,
    ! [X: nat,A3: set_nat] :
      ( ( insert_nat @ X @ ( insert_nat @ X @ A3 ) )
      = ( insert_nat @ X @ A3 ) ) ).

% insert_absorb2
thf(fact_796_insert__iff,axiom,
    ! [A: rule,B2: rule,A3: set_rule] :
      ( ( member_rule3 @ A @ ( insert_rule @ B2 @ A3 ) )
      = ( ( A = B2 )
        | ( member_rule3 @ A @ A3 ) ) ) ).

% insert_iff
thf(fact_797_insert__iff,axiom,
    ! [A: list_fm,B2: list_fm,A3: set_list_fm] :
      ( ( member_list_fm3 @ A @ ( insert_list_fm @ B2 @ A3 ) )
      = ( ( A = B2 )
        | ( member_list_fm3 @ A @ A3 ) ) ) ).

% insert_iff
thf(fact_798_insert__iff,axiom,
    ! [A: fm,B2: fm,A3: set_fm] :
      ( ( member_fm3 @ A @ ( insert_fm @ B2 @ A3 ) )
      = ( ( A = B2 )
        | ( member_fm3 @ A @ A3 ) ) ) ).

% insert_iff
thf(fact_799_insert__iff,axiom,
    ! [A: $o,B2: $o,A3: set_o] :
      ( ( member_o3 @ A @ ( insert_o @ B2 @ A3 ) )
      = ( ( A = B2 )
        | ( member_o3 @ A @ A3 ) ) ) ).

% insert_iff
thf(fact_800_insert__iff,axiom,
    ! [A: nat,B2: nat,A3: set_nat] :
      ( ( member_nat3 @ A @ ( insert_nat @ B2 @ A3 ) )
      = ( ( A = B2 )
        | ( member_nat3 @ A @ A3 ) ) ) ).

% insert_iff
thf(fact_801_insertCI,axiom,
    ! [A: rule,B: set_rule,B2: rule] :
      ( ( ~ ( member_rule3 @ A @ B )
       => ( A = B2 ) )
     => ( member_rule3 @ A @ ( insert_rule @ B2 @ B ) ) ) ).

% insertCI
thf(fact_802_insertCI,axiom,
    ! [A: list_fm,B: set_list_fm,B2: list_fm] :
      ( ( ~ ( member_list_fm3 @ A @ B )
       => ( A = B2 ) )
     => ( member_list_fm3 @ A @ ( insert_list_fm @ B2 @ B ) ) ) ).

% insertCI
thf(fact_803_insertCI,axiom,
    ! [A: fm,B: set_fm,B2: fm] :
      ( ( ~ ( member_fm3 @ A @ B )
       => ( A = B2 ) )
     => ( member_fm3 @ A @ ( insert_fm @ B2 @ B ) ) ) ).

% insertCI
thf(fact_804_insertCI,axiom,
    ! [A: $o,B: set_o,B2: $o] :
      ( ( ~ ( member_o3 @ A @ B )
       => ( A = B2 ) )
     => ( member_o3 @ A @ ( insert_o @ B2 @ B ) ) ) ).

% insertCI
thf(fact_805_insertCI,axiom,
    ! [A: nat,B: set_nat,B2: nat] :
      ( ( ~ ( member_nat3 @ A @ B )
       => ( A = B2 ) )
     => ( member_nat3 @ A @ ( insert_nat @ B2 @ B ) ) ) ).

% insertCI
thf(fact_806_sdrop__add,axiom,
    ! [N2: nat,M2: nat,S: stream2709947120125613254m_rule] :
      ( ( sdrop_8169176516188972301m_rule @ N2 @ ( sdrop_8169176516188972301m_rule @ M2 @ S ) )
      = ( sdrop_8169176516188972301m_rule @ ( plus_plus_nat @ M2 @ N2 ) @ S ) ) ).

% sdrop_add
thf(fact_807_sdrop__add,axiom,
    ! [N2: nat,M2: nat,S: stream_rule] :
      ( ( sdrop_rule @ N2 @ ( sdrop_rule @ M2 @ S ) )
      = ( sdrop_rule @ ( plus_plus_nat @ M2 @ N2 ) @ S ) ) ).

% sdrop_add
thf(fact_808_image__insert,axiom,
    ! [F: produc340336539035504054m_rule > set_fm,A: produc340336539035504054m_rule,B: set_Pr1822751329126368876m_rule] :
      ( ( image_377115208147135190set_fm @ F @ ( insert775845832097999846m_rule @ A @ B ) )
      = ( insert_set_fm @ ( F @ A ) @ ( image_377115208147135190set_fm @ F @ B ) ) ) ).

% image_insert
thf(fact_809_image__insert,axiom,
    ! [F: $o > $o,A: $o,B: set_o] :
      ( ( image_o_o @ F @ ( insert_o @ A @ B ) )
      = ( insert_o @ ( F @ A ) @ ( image_o_o @ F @ B ) ) ) ).

% image_insert
thf(fact_810_image__insert,axiom,
    ! [F: $o > nat,A: $o,B: set_o] :
      ( ( image_o_nat @ F @ ( insert_o @ A @ B ) )
      = ( insert_nat @ ( F @ A ) @ ( image_o_nat @ F @ B ) ) ) ).

% image_insert
thf(fact_811_image__insert,axiom,
    ! [F: nat > $o,A: nat,B: set_nat] :
      ( ( image_nat_o @ F @ ( insert_nat @ A @ B ) )
      = ( insert_o @ ( F @ A ) @ ( image_nat_o @ F @ B ) ) ) ).

% image_insert
thf(fact_812_image__insert,axiom,
    ! [F: nat > nat,A: nat,B: set_nat] :
      ( ( image_nat_nat @ F @ ( insert_nat @ A @ B ) )
      = ( insert_nat @ ( F @ A ) @ ( image_nat_nat @ F @ B ) ) ) ).

% image_insert
thf(fact_813_insert__image,axiom,
    ! [X: rule,A3: set_rule,F: rule > $o] :
      ( ( member_rule3 @ X @ A3 )
     => ( ( insert_o @ ( F @ X ) @ ( image_rule_o @ F @ A3 ) )
        = ( image_rule_o @ F @ A3 ) ) ) ).

% insert_image
thf(fact_814_insert__image,axiom,
    ! [X: rule,A3: set_rule,F: rule > nat] :
      ( ( member_rule3 @ X @ A3 )
     => ( ( insert_nat @ ( F @ X ) @ ( image_rule_nat @ F @ A3 ) )
        = ( image_rule_nat @ F @ A3 ) ) ) ).

% insert_image
thf(fact_815_insert__image,axiom,
    ! [X: fm,A3: set_fm,F: fm > $o] :
      ( ( member_fm3 @ X @ A3 )
     => ( ( insert_o @ ( F @ X ) @ ( image_fm_o @ F @ A3 ) )
        = ( image_fm_o @ F @ A3 ) ) ) ).

% insert_image
thf(fact_816_insert__image,axiom,
    ! [X: fm,A3: set_fm,F: fm > nat] :
      ( ( member_fm3 @ X @ A3 )
     => ( ( insert_nat @ ( F @ X ) @ ( image_fm_nat @ F @ A3 ) )
        = ( image_fm_nat @ F @ A3 ) ) ) ).

% insert_image
thf(fact_817_insert__image,axiom,
    ! [X: $o,A3: set_o,F: $o > $o] :
      ( ( member_o3 @ X @ A3 )
     => ( ( insert_o @ ( F @ X ) @ ( image_o_o @ F @ A3 ) )
        = ( image_o_o @ F @ A3 ) ) ) ).

% insert_image
thf(fact_818_insert__image,axiom,
    ! [X: $o,A3: set_o,F: $o > nat] :
      ( ( member_o3 @ X @ A3 )
     => ( ( insert_nat @ ( F @ X ) @ ( image_o_nat @ F @ A3 ) )
        = ( image_o_nat @ F @ A3 ) ) ) ).

% insert_image
thf(fact_819_insert__image,axiom,
    ! [X: nat,A3: set_nat,F: nat > $o] :
      ( ( member_nat3 @ X @ A3 )
     => ( ( insert_o @ ( F @ X ) @ ( image_nat_o @ F @ A3 ) )
        = ( image_nat_o @ F @ A3 ) ) ) ).

% insert_image
thf(fact_820_insert__image,axiom,
    ! [X: nat,A3: set_nat,F: nat > nat] :
      ( ( member_nat3 @ X @ A3 )
     => ( ( insert_nat @ ( F @ X ) @ ( image_nat_nat @ F @ A3 ) )
        = ( image_nat_nat @ F @ A3 ) ) ) ).

% insert_image
thf(fact_821_insert__image,axiom,
    ! [X: list_fm,A3: set_list_fm,F: list_fm > $o] :
      ( ( member_list_fm3 @ X @ A3 )
     => ( ( insert_o @ ( F @ X ) @ ( image_list_fm_o @ F @ A3 ) )
        = ( image_list_fm_o @ F @ A3 ) ) ) ).

% insert_image
thf(fact_822_insert__image,axiom,
    ! [X: list_fm,A3: set_list_fm,F: list_fm > nat] :
      ( ( member_list_fm3 @ X @ A3 )
     => ( ( insert_nat @ ( F @ X ) @ ( image_list_fm_nat @ F @ A3 ) )
        = ( image_list_fm_nat @ F @ A3 ) ) ) ).

% insert_image
thf(fact_823_singletonI,axiom,
    ! [A: list_fm] : ( member_list_fm3 @ A @ ( insert_list_fm @ A @ bot_bot_set_list_fm ) ) ).

% singletonI
thf(fact_824_singletonI,axiom,
    ! [A: fm] : ( member_fm3 @ A @ ( insert_fm @ A @ bot_bot_set_fm ) ) ).

% singletonI
thf(fact_825_singletonI,axiom,
    ! [A: rule] : ( member_rule3 @ A @ ( insert_rule @ A @ bot_bot_set_rule ) ) ).

% singletonI
thf(fact_826_singletonI,axiom,
    ! [A: $o] : ( member_o3 @ A @ ( insert_o @ A @ bot_bot_set_o ) ) ).

% singletonI
thf(fact_827_singletonI,axiom,
    ! [A: nat] : ( member_nat3 @ A @ ( insert_nat @ A @ bot_bot_set_nat ) ) ).

% singletonI
thf(fact_828_List_Ofinite__set,axiom,
    ! [Xs2: list_fm] : ( finite_finite_fm @ ( set_fm2 @ Xs2 ) ) ).

% List.finite_set
thf(fact_829_List_Ofinite__set,axiom,
    ! [Xs2: list_list_fm] : ( finite834038518558952994ist_fm @ ( set_list_fm2 @ Xs2 ) ) ).

% List.finite_set
thf(fact_830_List_Ofinite__set,axiom,
    ! [Xs2: list_tm] : ( finite_finite_tm @ ( set_tm2 @ Xs2 ) ) ).

% List.finite_set
thf(fact_831_List_Ofinite__set,axiom,
    ! [Xs2: list_nat] : ( finite_finite_nat @ ( set_nat2 @ Xs2 ) ) ).

% List.finite_set
thf(fact_832_Un__insert__left,axiom,
    ! [A: $o,B: set_o,C: set_o] :
      ( ( sup_sup_set_o @ ( insert_o @ A @ B ) @ C )
      = ( insert_o @ A @ ( sup_sup_set_o @ B @ C ) ) ) ).

% Un_insert_left
thf(fact_833_Un__insert__left,axiom,
    ! [A: nat,B: set_nat,C: set_nat] :
      ( ( sup_sup_set_nat @ ( insert_nat @ A @ B ) @ C )
      = ( insert_nat @ A @ ( sup_sup_set_nat @ B @ C ) ) ) ).

% Un_insert_left
thf(fact_834_Un__insert__right,axiom,
    ! [A3: set_o,A: $o,B: set_o] :
      ( ( sup_sup_set_o @ A3 @ ( insert_o @ A @ B ) )
      = ( insert_o @ A @ ( sup_sup_set_o @ A3 @ B ) ) ) ).

% Un_insert_right
thf(fact_835_Un__insert__right,axiom,
    ! [A3: set_nat,A: nat,B: set_nat] :
      ( ( sup_sup_set_nat @ A3 @ ( insert_nat @ A @ B ) )
      = ( insert_nat @ A @ ( sup_sup_set_nat @ A3 @ B ) ) ) ).

% Un_insert_right
thf(fact_836_singleton__conv2,axiom,
    ! [A: rule] :
      ( ( collect_rule
        @ ( ^ [Y5: rule,Z5: rule] : ( Y5 = Z5 )
          @ A ) )
      = ( insert_rule @ A @ bot_bot_set_rule ) ) ).

% singleton_conv2
thf(fact_837_singleton__conv2,axiom,
    ! [A: $o] :
      ( ( collect_o
        @ ( ^ [Y5: $o,Z5: $o] : ( Y5 = Z5 )
          @ A ) )
      = ( insert_o @ A @ bot_bot_set_o ) ) ).

% singleton_conv2
thf(fact_838_singleton__conv2,axiom,
    ! [A: nat] :
      ( ( collect_nat
        @ ( ^ [Y5: nat,Z5: nat] : ( Y5 = Z5 )
          @ A ) )
      = ( insert_nat @ A @ bot_bot_set_nat ) ) ).

% singleton_conv2
thf(fact_839_singleton__conv,axiom,
    ! [A: rule] :
      ( ( collect_rule
        @ ^ [X3: rule] : ( X3 = A ) )
      = ( insert_rule @ A @ bot_bot_set_rule ) ) ).

% singleton_conv
thf(fact_840_singleton__conv,axiom,
    ! [A: $o] :
      ( ( collect_o
        @ ^ [X3: $o] : ( X3 = A ) )
      = ( insert_o @ A @ bot_bot_set_o ) ) ).

% singleton_conv
thf(fact_841_singleton__conv,axiom,
    ! [A: nat] :
      ( ( collect_nat
        @ ^ [X3: nat] : ( X3 = A ) )
      = ( insert_nat @ A @ bot_bot_set_nat ) ) ).

% singleton_conv
thf(fact_842_cSup__singleton,axiom,
    ! [X: set_fm] :
      ( ( comple2134933779557159616set_fm @ ( insert_set_fm @ X @ bot_bot_set_set_fm ) )
      = X ) ).

% cSup_singleton
thf(fact_843_cSup__singleton,axiom,
    ! [X: $o] :
      ( ( complete_Sup_Sup_o @ ( insert_o @ X @ bot_bot_set_o ) )
      = X ) ).

% cSup_singleton
thf(fact_844_cSup__singleton,axiom,
    ! [X: nat] :
      ( ( complete_Sup_Sup_nat @ ( insert_nat @ X @ bot_bot_set_nat ) )
      = X ) ).

% cSup_singleton
thf(fact_845_Sup__insert,axiom,
    ! [A: set_nat,A3: set_set_nat] :
      ( ( comple7399068483239264473et_nat @ ( insert_set_nat @ A @ A3 ) )
      = ( sup_sup_set_nat @ A @ ( comple7399068483239264473et_nat @ A3 ) ) ) ).

% Sup_insert
thf(fact_846_Sup__insert,axiom,
    ! [A: set_fm,A3: set_set_fm] :
      ( ( comple2134933779557159616set_fm @ ( insert_set_fm @ A @ A3 ) )
      = ( sup_sup_set_fm @ A @ ( comple2134933779557159616set_fm @ A3 ) ) ) ).

% Sup_insert
thf(fact_847_Sup__insert,axiom,
    ! [A: $o,A3: set_o] :
      ( ( complete_Sup_Sup_o @ ( insert_o @ A @ A3 ) )
      = ( sup_sup_o @ A @ ( complete_Sup_Sup_o @ A3 ) ) ) ).

% Sup_insert
thf(fact_848_UN__simps_I1_J,axiom,
    ! [C: set_rule,A: rule,B: rule > set_rule] :
      ( ( ( C = bot_bot_set_rule )
       => ( ( comple7665822446944348827t_rule
            @ ( image_rule_set_rule
              @ ^ [X3: rule] : ( insert_rule @ A @ ( B @ X3 ) )
              @ C ) )
          = bot_bot_set_rule ) )
      & ( ( C != bot_bot_set_rule )
       => ( ( comple7665822446944348827t_rule
            @ ( image_rule_set_rule
              @ ^ [X3: rule] : ( insert_rule @ A @ ( B @ X3 ) )
              @ C ) )
          = ( insert_rule @ A @ ( comple7665822446944348827t_rule @ ( image_rule_set_rule @ B @ C ) ) ) ) ) ) ).

% UN_simps(1)
thf(fact_849_UN__simps_I1_J,axiom,
    ! [C: set_rule,A: $o,B: rule > set_o] :
      ( ( ( C = bot_bot_set_rule )
       => ( ( comple90263536869209701_set_o
            @ ( image_rule_set_o
              @ ^ [X3: rule] : ( insert_o @ A @ ( B @ X3 ) )
              @ C ) )
          = bot_bot_set_o ) )
      & ( ( C != bot_bot_set_rule )
       => ( ( comple90263536869209701_set_o
            @ ( image_rule_set_o
              @ ^ [X3: rule] : ( insert_o @ A @ ( B @ X3 ) )
              @ C ) )
          = ( insert_o @ A @ ( comple90263536869209701_set_o @ ( image_rule_set_o @ B @ C ) ) ) ) ) ) ).

% UN_simps(1)
thf(fact_850_UN__simps_I1_J,axiom,
    ! [C: set_rule,A: nat,B: rule > set_nat] :
      ( ( ( C = bot_bot_set_rule )
       => ( ( comple7399068483239264473et_nat
            @ ( image_rule_set_nat
              @ ^ [X3: rule] : ( insert_nat @ A @ ( B @ X3 ) )
              @ C ) )
          = bot_bot_set_nat ) )
      & ( ( C != bot_bot_set_rule )
       => ( ( comple7399068483239264473et_nat
            @ ( image_rule_set_nat
              @ ^ [X3: rule] : ( insert_nat @ A @ ( B @ X3 ) )
              @ C ) )
          = ( insert_nat @ A @ ( comple7399068483239264473et_nat @ ( image_rule_set_nat @ B @ C ) ) ) ) ) ) ).

% UN_simps(1)
thf(fact_851_UN__simps_I1_J,axiom,
    ! [C: set_o,A: rule,B: $o > set_rule] :
      ( ( ( C = bot_bot_set_o )
       => ( ( comple7665822446944348827t_rule
            @ ( image_o_set_rule
              @ ^ [X3: $o] : ( insert_rule @ A @ ( B @ X3 ) )
              @ C ) )
          = bot_bot_set_rule ) )
      & ( ( C != bot_bot_set_o )
       => ( ( comple7665822446944348827t_rule
            @ ( image_o_set_rule
              @ ^ [X3: $o] : ( insert_rule @ A @ ( B @ X3 ) )
              @ C ) )
          = ( insert_rule @ A @ ( comple7665822446944348827t_rule @ ( image_o_set_rule @ B @ C ) ) ) ) ) ) ).

% UN_simps(1)
thf(fact_852_UN__simps_I1_J,axiom,
    ! [C: set_o,A: $o,B: $o > set_o] :
      ( ( ( C = bot_bot_set_o )
       => ( ( comple90263536869209701_set_o
            @ ( image_o_set_o
              @ ^ [X3: $o] : ( insert_o @ A @ ( B @ X3 ) )
              @ C ) )
          = bot_bot_set_o ) )
      & ( ( C != bot_bot_set_o )
       => ( ( comple90263536869209701_set_o
            @ ( image_o_set_o
              @ ^ [X3: $o] : ( insert_o @ A @ ( B @ X3 ) )
              @ C ) )
          = ( insert_o @ A @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B @ C ) ) ) ) ) ) ).

% UN_simps(1)
thf(fact_853_UN__simps_I1_J,axiom,
    ! [C: set_o,A: nat,B: $o > set_nat] :
      ( ( ( C = bot_bot_set_o )
       => ( ( comple7399068483239264473et_nat
            @ ( image_o_set_nat
              @ ^ [X3: $o] : ( insert_nat @ A @ ( B @ X3 ) )
              @ C ) )
          = bot_bot_set_nat ) )
      & ( ( C != bot_bot_set_o )
       => ( ( comple7399068483239264473et_nat
            @ ( image_o_set_nat
              @ ^ [X3: $o] : ( insert_nat @ A @ ( B @ X3 ) )
              @ C ) )
          = ( insert_nat @ A @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ B @ C ) ) ) ) ) ) ).

% UN_simps(1)
thf(fact_854_UN__simps_I1_J,axiom,
    ! [C: set_nat,A: rule,B: nat > set_rule] :
      ( ( ( C = bot_bot_set_nat )
       => ( ( comple7665822446944348827t_rule
            @ ( image_nat_set_rule
              @ ^ [X3: nat] : ( insert_rule @ A @ ( B @ X3 ) )
              @ C ) )
          = bot_bot_set_rule ) )
      & ( ( C != bot_bot_set_nat )
       => ( ( comple7665822446944348827t_rule
            @ ( image_nat_set_rule
              @ ^ [X3: nat] : ( insert_rule @ A @ ( B @ X3 ) )
              @ C ) )
          = ( insert_rule @ A @ ( comple7665822446944348827t_rule @ ( image_nat_set_rule @ B @ C ) ) ) ) ) ) ).

% UN_simps(1)
thf(fact_855_UN__simps_I1_J,axiom,
    ! [C: set_nat,A: $o,B: nat > set_o] :
      ( ( ( C = bot_bot_set_nat )
       => ( ( comple90263536869209701_set_o
            @ ( image_nat_set_o
              @ ^ [X3: nat] : ( insert_o @ A @ ( B @ X3 ) )
              @ C ) )
          = bot_bot_set_o ) )
      & ( ( C != bot_bot_set_nat )
       => ( ( comple90263536869209701_set_o
            @ ( image_nat_set_o
              @ ^ [X3: nat] : ( insert_o @ A @ ( B @ X3 ) )
              @ C ) )
          = ( insert_o @ A @ ( comple90263536869209701_set_o @ ( image_nat_set_o @ B @ C ) ) ) ) ) ) ).

% UN_simps(1)
thf(fact_856_UN__simps_I1_J,axiom,
    ! [C: set_nat,A: nat,B: nat > set_nat] :
      ( ( ( C = bot_bot_set_nat )
       => ( ( comple7399068483239264473et_nat
            @ ( image_nat_set_nat
              @ ^ [X3: nat] : ( insert_nat @ A @ ( B @ X3 ) )
              @ C ) )
          = bot_bot_set_nat ) )
      & ( ( C != bot_bot_set_nat )
       => ( ( comple7399068483239264473et_nat
            @ ( image_nat_set_nat
              @ ^ [X3: nat] : ( insert_nat @ A @ ( B @ X3 ) )
              @ C ) )
          = ( insert_nat @ A @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B @ C ) ) ) ) ) ) ).

% UN_simps(1)
thf(fact_857_UN__simps_I1_J,axiom,
    ! [C: set_rule,A: fm,B: rule > set_fm] :
      ( ( ( C = bot_bot_set_rule )
       => ( ( comple2134933779557159616set_fm
            @ ( image_rule_set_fm
              @ ^ [X3: rule] : ( insert_fm @ A @ ( B @ X3 ) )
              @ C ) )
          = bot_bot_set_fm ) )
      & ( ( C != bot_bot_set_rule )
       => ( ( comple2134933779557159616set_fm
            @ ( image_rule_set_fm
              @ ^ [X3: rule] : ( insert_fm @ A @ ( B @ X3 ) )
              @ C ) )
          = ( insert_fm @ A @ ( comple2134933779557159616set_fm @ ( image_rule_set_fm @ B @ C ) ) ) ) ) ) ).

% UN_simps(1)
thf(fact_858_UN__singleton,axiom,
    ! [A3: set_rule] :
      ( ( comple7665822446944348827t_rule
        @ ( image_rule_set_rule
          @ ^ [X3: rule] : ( insert_rule @ X3 @ bot_bot_set_rule )
          @ A3 ) )
      = A3 ) ).

% UN_singleton
thf(fact_859_UN__singleton,axiom,
    ! [A3: set_o] :
      ( ( comple90263536869209701_set_o
        @ ( image_o_set_o
          @ ^ [X3: $o] : ( insert_o @ X3 @ bot_bot_set_o )
          @ A3 ) )
      = A3 ) ).

% UN_singleton
thf(fact_860_UN__singleton,axiom,
    ! [A3: set_nat] :
      ( ( comple7399068483239264473et_nat
        @ ( image_nat_set_nat
          @ ^ [X3: nat] : ( insert_nat @ X3 @ bot_bot_set_nat )
          @ A3 ) )
      = A3 ) ).

% UN_singleton
thf(fact_861_UN__singleton,axiom,
    ! [A3: set_fm] :
      ( ( comple2134933779557159616set_fm
        @ ( image_fm_set_fm
          @ ^ [X3: fm] : ( insert_fm @ X3 @ bot_bot_set_fm )
          @ A3 ) )
      = A3 ) ).

% UN_singleton
thf(fact_862_UN__insert,axiom,
    ! [B: $o > set_nat,A: $o,A3: set_o] :
      ( ( comple7399068483239264473et_nat @ ( image_o_set_nat @ B @ ( insert_o @ A @ A3 ) ) )
      = ( sup_sup_set_nat @ ( B @ A ) @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ B @ A3 ) ) ) ) ).

% UN_insert
thf(fact_863_UN__insert,axiom,
    ! [B: nat > set_nat,A: nat,A3: set_nat] :
      ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B @ ( insert_nat @ A @ A3 ) ) )
      = ( sup_sup_set_nat @ ( B @ A ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B @ A3 ) ) ) ) ).

% UN_insert
thf(fact_864_UN__insert,axiom,
    ! [B: produc340336539035504054m_rule > set_fm,A: produc340336539035504054m_rule,A3: set_Pr1822751329126368876m_rule] :
      ( ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ B @ ( insert775845832097999846m_rule @ A @ A3 ) ) )
      = ( sup_sup_set_fm @ ( B @ A ) @ ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ B @ A3 ) ) ) ) ).

% UN_insert
thf(fact_865_UN__insert,axiom,
    ! [B: $o > set_fm,A: $o,A3: set_o] :
      ( ( comple2134933779557159616set_fm @ ( image_o_set_fm @ B @ ( insert_o @ A @ A3 ) ) )
      = ( sup_sup_set_fm @ ( B @ A ) @ ( comple2134933779557159616set_fm @ ( image_o_set_fm @ B @ A3 ) ) ) ) ).

% UN_insert
thf(fact_866_UN__insert,axiom,
    ! [B: nat > set_fm,A: nat,A3: set_nat] :
      ( ( comple2134933779557159616set_fm @ ( image_nat_set_fm @ B @ ( insert_nat @ A @ A3 ) ) )
      = ( sup_sup_set_fm @ ( B @ A ) @ ( comple2134933779557159616set_fm @ ( image_nat_set_fm @ B @ A3 ) ) ) ) ).

% UN_insert
thf(fact_867_mk__disjoint__insert,axiom,
    ! [A: rule,A3: set_rule] :
      ( ( member_rule3 @ A @ A3 )
     => ? [B5: set_rule] :
          ( ( A3
            = ( insert_rule @ A @ B5 ) )
          & ~ ( member_rule3 @ A @ B5 ) ) ) ).

% mk_disjoint_insert
thf(fact_868_mk__disjoint__insert,axiom,
    ! [A: list_fm,A3: set_list_fm] :
      ( ( member_list_fm3 @ A @ A3 )
     => ? [B5: set_list_fm] :
          ( ( A3
            = ( insert_list_fm @ A @ B5 ) )
          & ~ ( member_list_fm3 @ A @ B5 ) ) ) ).

% mk_disjoint_insert
thf(fact_869_mk__disjoint__insert,axiom,
    ! [A: fm,A3: set_fm] :
      ( ( member_fm3 @ A @ A3 )
     => ? [B5: set_fm] :
          ( ( A3
            = ( insert_fm @ A @ B5 ) )
          & ~ ( member_fm3 @ A @ B5 ) ) ) ).

% mk_disjoint_insert
thf(fact_870_mk__disjoint__insert,axiom,
    ! [A: $o,A3: set_o] :
      ( ( member_o3 @ A @ A3 )
     => ? [B5: set_o] :
          ( ( A3
            = ( insert_o @ A @ B5 ) )
          & ~ ( member_o3 @ A @ B5 ) ) ) ).

% mk_disjoint_insert
thf(fact_871_mk__disjoint__insert,axiom,
    ! [A: nat,A3: set_nat] :
      ( ( member_nat3 @ A @ A3 )
     => ? [B5: set_nat] :
          ( ( A3
            = ( insert_nat @ A @ B5 ) )
          & ~ ( member_nat3 @ A @ B5 ) ) ) ).

% mk_disjoint_insert
thf(fact_872_insert__commute,axiom,
    ! [X: $o,Y: $o,A3: set_o] :
      ( ( insert_o @ X @ ( insert_o @ Y @ A3 ) )
      = ( insert_o @ Y @ ( insert_o @ X @ A3 ) ) ) ).

% insert_commute
thf(fact_873_insert__commute,axiom,
    ! [X: nat,Y: nat,A3: set_nat] :
      ( ( insert_nat @ X @ ( insert_nat @ Y @ A3 ) )
      = ( insert_nat @ Y @ ( insert_nat @ X @ A3 ) ) ) ).

% insert_commute
thf(fact_874_insert__Collect,axiom,
    ! [A: $o,P3: $o > $o] :
      ( ( insert_o @ A @ ( collect_o @ P3 ) )
      = ( collect_o
        @ ^ [U: $o] :
            ( ( U != A )
           => ( P3 @ U ) ) ) ) ).

% insert_Collect
thf(fact_875_insert__Collect,axiom,
    ! [A: nat,P3: nat > $o] :
      ( ( insert_nat @ A @ ( collect_nat @ P3 ) )
      = ( collect_nat
        @ ^ [U: nat] :
            ( ( U != A )
           => ( P3 @ U ) ) ) ) ).

% insert_Collect
thf(fact_876_insert__eq__iff,axiom,
    ! [A: rule,A3: set_rule,B2: rule,B: set_rule] :
      ( ~ ( member_rule3 @ A @ A3 )
     => ( ~ ( member_rule3 @ B2 @ B )
       => ( ( ( insert_rule @ A @ A3 )
            = ( insert_rule @ B2 @ B ) )
          = ( ( ( A = B2 )
             => ( A3 = B ) )
            & ( ( A != B2 )
             => ? [C3: set_rule] :
                  ( ( A3
                    = ( insert_rule @ B2 @ C3 ) )
                  & ~ ( member_rule3 @ B2 @ C3 )
                  & ( B
                    = ( insert_rule @ A @ C3 ) )
                  & ~ ( member_rule3 @ A @ C3 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_877_insert__eq__iff,axiom,
    ! [A: list_fm,A3: set_list_fm,B2: list_fm,B: set_list_fm] :
      ( ~ ( member_list_fm3 @ A @ A3 )
     => ( ~ ( member_list_fm3 @ B2 @ B )
       => ( ( ( insert_list_fm @ A @ A3 )
            = ( insert_list_fm @ B2 @ B ) )
          = ( ( ( A = B2 )
             => ( A3 = B ) )
            & ( ( A != B2 )
             => ? [C3: set_list_fm] :
                  ( ( A3
                    = ( insert_list_fm @ B2 @ C3 ) )
                  & ~ ( member_list_fm3 @ B2 @ C3 )
                  & ( B
                    = ( insert_list_fm @ A @ C3 ) )
                  & ~ ( member_list_fm3 @ A @ C3 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_878_insert__eq__iff,axiom,
    ! [A: fm,A3: set_fm,B2: fm,B: set_fm] :
      ( ~ ( member_fm3 @ A @ A3 )
     => ( ~ ( member_fm3 @ B2 @ B )
       => ( ( ( insert_fm @ A @ A3 )
            = ( insert_fm @ B2 @ B ) )
          = ( ( ( A = B2 )
             => ( A3 = B ) )
            & ( ( A != B2 )
             => ? [C3: set_fm] :
                  ( ( A3
                    = ( insert_fm @ B2 @ C3 ) )
                  & ~ ( member_fm3 @ B2 @ C3 )
                  & ( B
                    = ( insert_fm @ A @ C3 ) )
                  & ~ ( member_fm3 @ A @ C3 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_879_insert__eq__iff,axiom,
    ! [A: $o,A3: set_o,B2: $o,B: set_o] :
      ( ~ ( member_o3 @ A @ A3 )
     => ( ~ ( member_o3 @ B2 @ B )
       => ( ( ( insert_o @ A @ A3 )
            = ( insert_o @ B2 @ B ) )
          = ( ( ( A = B2 )
             => ( A3 = B ) )
            & ( ( A = ~ B2 )
             => ? [C3: set_o] :
                  ( ( A3
                    = ( insert_o @ B2 @ C3 ) )
                  & ~ ( member_o3 @ B2 @ C3 )
                  & ( B
                    = ( insert_o @ A @ C3 ) )
                  & ~ ( member_o3 @ A @ C3 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_880_insert__eq__iff,axiom,
    ! [A: nat,A3: set_nat,B2: nat,B: set_nat] :
      ( ~ ( member_nat3 @ A @ A3 )
     => ( ~ ( member_nat3 @ B2 @ B )
       => ( ( ( insert_nat @ A @ A3 )
            = ( insert_nat @ B2 @ B ) )
          = ( ( ( A = B2 )
             => ( A3 = B ) )
            & ( ( A != B2 )
             => ? [C3: set_nat] :
                  ( ( A3
                    = ( insert_nat @ B2 @ C3 ) )
                  & ~ ( member_nat3 @ B2 @ C3 )
                  & ( B
                    = ( insert_nat @ A @ C3 ) )
                  & ~ ( member_nat3 @ A @ C3 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_881_insert__absorb,axiom,
    ! [A: rule,A3: set_rule] :
      ( ( member_rule3 @ A @ A3 )
     => ( ( insert_rule @ A @ A3 )
        = A3 ) ) ).

% insert_absorb
thf(fact_882_insert__absorb,axiom,
    ! [A: list_fm,A3: set_list_fm] :
      ( ( member_list_fm3 @ A @ A3 )
     => ( ( insert_list_fm @ A @ A3 )
        = A3 ) ) ).

% insert_absorb
thf(fact_883_insert__absorb,axiom,
    ! [A: fm,A3: set_fm] :
      ( ( member_fm3 @ A @ A3 )
     => ( ( insert_fm @ A @ A3 )
        = A3 ) ) ).

% insert_absorb
thf(fact_884_insert__absorb,axiom,
    ! [A: $o,A3: set_o] :
      ( ( member_o3 @ A @ A3 )
     => ( ( insert_o @ A @ A3 )
        = A3 ) ) ).

% insert_absorb
thf(fact_885_insert__absorb,axiom,
    ! [A: nat,A3: set_nat] :
      ( ( member_nat3 @ A @ A3 )
     => ( ( insert_nat @ A @ A3 )
        = A3 ) ) ).

% insert_absorb
thf(fact_886_insert__ident,axiom,
    ! [X: rule,A3: set_rule,B: set_rule] :
      ( ~ ( member_rule3 @ X @ A3 )
     => ( ~ ( member_rule3 @ X @ B )
       => ( ( ( insert_rule @ X @ A3 )
            = ( insert_rule @ X @ B ) )
          = ( A3 = B ) ) ) ) ).

% insert_ident
thf(fact_887_insert__ident,axiom,
    ! [X: list_fm,A3: set_list_fm,B: set_list_fm] :
      ( ~ ( member_list_fm3 @ X @ A3 )
     => ( ~ ( member_list_fm3 @ X @ B )
       => ( ( ( insert_list_fm @ X @ A3 )
            = ( insert_list_fm @ X @ B ) )
          = ( A3 = B ) ) ) ) ).

% insert_ident
thf(fact_888_insert__ident,axiom,
    ! [X: fm,A3: set_fm,B: set_fm] :
      ( ~ ( member_fm3 @ X @ A3 )
     => ( ~ ( member_fm3 @ X @ B )
       => ( ( ( insert_fm @ X @ A3 )
            = ( insert_fm @ X @ B ) )
          = ( A3 = B ) ) ) ) ).

% insert_ident
thf(fact_889_insert__ident,axiom,
    ! [X: $o,A3: set_o,B: set_o] :
      ( ~ ( member_o3 @ X @ A3 )
     => ( ~ ( member_o3 @ X @ B )
       => ( ( ( insert_o @ X @ A3 )
            = ( insert_o @ X @ B ) )
          = ( A3 = B ) ) ) ) ).

% insert_ident
thf(fact_890_insert__ident,axiom,
    ! [X: nat,A3: set_nat,B: set_nat] :
      ( ~ ( member_nat3 @ X @ A3 )
     => ( ~ ( member_nat3 @ X @ B )
       => ( ( ( insert_nat @ X @ A3 )
            = ( insert_nat @ X @ B ) )
          = ( A3 = B ) ) ) ) ).

% insert_ident
thf(fact_891_insert__compr,axiom,
    ( insert_rule
    = ( ^ [A2: rule,B3: set_rule] :
          ( collect_rule
          @ ^ [X3: rule] :
              ( ( X3 = A2 )
              | ( member_rule3 @ X3 @ B3 ) ) ) ) ) ).

% insert_compr
thf(fact_892_insert__compr,axiom,
    ( insert_list_fm
    = ( ^ [A2: list_fm,B3: set_list_fm] :
          ( collect_list_fm
          @ ^ [X3: list_fm] :
              ( ( X3 = A2 )
              | ( member_list_fm3 @ X3 @ B3 ) ) ) ) ) ).

% insert_compr
thf(fact_893_insert__compr,axiom,
    ( insert_fm
    = ( ^ [A2: fm,B3: set_fm] :
          ( collect_fm
          @ ^ [X3: fm] :
              ( ( X3 = A2 )
              | ( member_fm3 @ X3 @ B3 ) ) ) ) ) ).

% insert_compr
thf(fact_894_insert__compr,axiom,
    ( insert_o
    = ( ^ [A2: $o,B3: set_o] :
          ( collect_o
          @ ^ [X3: $o] :
              ( ( X3 = A2 )
              | ( member_o3 @ X3 @ B3 ) ) ) ) ) ).

% insert_compr
thf(fact_895_insert__compr,axiom,
    ( insert_nat
    = ( ^ [A2: nat,B3: set_nat] :
          ( collect_nat
          @ ^ [X3: nat] :
              ( ( X3 = A2 )
              | ( member_nat3 @ X3 @ B3 ) ) ) ) ) ).

% insert_compr
thf(fact_896_Set_Oset__insert,axiom,
    ! [X: rule,A3: set_rule] :
      ( ( member_rule3 @ X @ A3 )
     => ~ ! [B5: set_rule] :
            ( ( A3
              = ( insert_rule @ X @ B5 ) )
           => ( member_rule3 @ X @ B5 ) ) ) ).

% Set.set_insert
thf(fact_897_Set_Oset__insert,axiom,
    ! [X: list_fm,A3: set_list_fm] :
      ( ( member_list_fm3 @ X @ A3 )
     => ~ ! [B5: set_list_fm] :
            ( ( A3
              = ( insert_list_fm @ X @ B5 ) )
           => ( member_list_fm3 @ X @ B5 ) ) ) ).

% Set.set_insert
thf(fact_898_Set_Oset__insert,axiom,
    ! [X: fm,A3: set_fm] :
      ( ( member_fm3 @ X @ A3 )
     => ~ ! [B5: set_fm] :
            ( ( A3
              = ( insert_fm @ X @ B5 ) )
           => ( member_fm3 @ X @ B5 ) ) ) ).

% Set.set_insert
thf(fact_899_Set_Oset__insert,axiom,
    ! [X: $o,A3: set_o] :
      ( ( member_o3 @ X @ A3 )
     => ~ ! [B5: set_o] :
            ( ( A3
              = ( insert_o @ X @ B5 ) )
           => ( member_o3 @ X @ B5 ) ) ) ).

% Set.set_insert
thf(fact_900_Set_Oset__insert,axiom,
    ! [X: nat,A3: set_nat] :
      ( ( member_nat3 @ X @ A3 )
     => ~ ! [B5: set_nat] :
            ( ( A3
              = ( insert_nat @ X @ B5 ) )
           => ( member_nat3 @ X @ B5 ) ) ) ).

% Set.set_insert
thf(fact_901_insertI2,axiom,
    ! [A: rule,B: set_rule,B2: rule] :
      ( ( member_rule3 @ A @ B )
     => ( member_rule3 @ A @ ( insert_rule @ B2 @ B ) ) ) ).

% insertI2
thf(fact_902_insertI2,axiom,
    ! [A: list_fm,B: set_list_fm,B2: list_fm] :
      ( ( member_list_fm3 @ A @ B )
     => ( member_list_fm3 @ A @ ( insert_list_fm @ B2 @ B ) ) ) ).

% insertI2
thf(fact_903_insertI2,axiom,
    ! [A: fm,B: set_fm,B2: fm] :
      ( ( member_fm3 @ A @ B )
     => ( member_fm3 @ A @ ( insert_fm @ B2 @ B ) ) ) ).

% insertI2
thf(fact_904_insertI2,axiom,
    ! [A: $o,B: set_o,B2: $o] :
      ( ( member_o3 @ A @ B )
     => ( member_o3 @ A @ ( insert_o @ B2 @ B ) ) ) ).

% insertI2
thf(fact_905_insertI2,axiom,
    ! [A: nat,B: set_nat,B2: nat] :
      ( ( member_nat3 @ A @ B )
     => ( member_nat3 @ A @ ( insert_nat @ B2 @ B ) ) ) ).

% insertI2
thf(fact_906_insertI1,axiom,
    ! [A: rule,B: set_rule] : ( member_rule3 @ A @ ( insert_rule @ A @ B ) ) ).

% insertI1
thf(fact_907_insertI1,axiom,
    ! [A: list_fm,B: set_list_fm] : ( member_list_fm3 @ A @ ( insert_list_fm @ A @ B ) ) ).

% insertI1
thf(fact_908_insertI1,axiom,
    ! [A: fm,B: set_fm] : ( member_fm3 @ A @ ( insert_fm @ A @ B ) ) ).

% insertI1
thf(fact_909_insertI1,axiom,
    ! [A: $o,B: set_o] : ( member_o3 @ A @ ( insert_o @ A @ B ) ) ).

% insertI1
thf(fact_910_insertI1,axiom,
    ! [A: nat,B: set_nat] : ( member_nat3 @ A @ ( insert_nat @ A @ B ) ) ).

% insertI1
thf(fact_911_insertE,axiom,
    ! [A: rule,B2: rule,A3: set_rule] :
      ( ( member_rule3 @ A @ ( insert_rule @ B2 @ A3 ) )
     => ( ( A != B2 )
       => ( member_rule3 @ A @ A3 ) ) ) ).

% insertE
thf(fact_912_insertE,axiom,
    ! [A: list_fm,B2: list_fm,A3: set_list_fm] :
      ( ( member_list_fm3 @ A @ ( insert_list_fm @ B2 @ A3 ) )
     => ( ( A != B2 )
       => ( member_list_fm3 @ A @ A3 ) ) ) ).

% insertE
thf(fact_913_insertE,axiom,
    ! [A: fm,B2: fm,A3: set_fm] :
      ( ( member_fm3 @ A @ ( insert_fm @ B2 @ A3 ) )
     => ( ( A != B2 )
       => ( member_fm3 @ A @ A3 ) ) ) ).

% insertE
thf(fact_914_insertE,axiom,
    ! [A: $o,B2: $o,A3: set_o] :
      ( ( member_o3 @ A @ ( insert_o @ B2 @ A3 ) )
     => ( ( A = ~ B2 )
       => ( member_o3 @ A @ A3 ) ) ) ).

% insertE
thf(fact_915_insertE,axiom,
    ! [A: nat,B2: nat,A3: set_nat] :
      ( ( member_nat3 @ A @ ( insert_nat @ B2 @ A3 ) )
     => ( ( A != B2 )
       => ( member_nat3 @ A @ A3 ) ) ) ).

% insertE
thf(fact_916_finite__imp__Sup__less,axiom,
    ! [X4: set_nat,X: nat,A: nat] :
      ( ( finite_finite_nat @ X4 )
     => ( ( member_nat3 @ X @ X4 )
       => ( ! [X2: nat] :
              ( ( member_nat3 @ X2 @ X4 )
             => ( ord_less_nat @ X2 @ A ) )
         => ( ord_less_nat @ ( complete_Sup_Sup_nat @ X4 ) @ A ) ) ) ) ).

% finite_imp_Sup_less
thf(fact_917_finite__Sup__less__iff,axiom,
    ! [X4: set_nat,A: nat] :
      ( ( finite_finite_nat @ X4 )
     => ( ( X4 != bot_bot_set_nat )
       => ( ( ord_less_nat @ ( complete_Sup_Sup_nat @ X4 ) @ A )
          = ( ! [X3: nat] :
                ( ( member_nat3 @ X3 @ X4 )
               => ( ord_less_nat @ X3 @ A ) ) ) ) ) ) ).

% finite_Sup_less_iff
thf(fact_918_singleton__inject,axiom,
    ! [A: rule,B2: rule] :
      ( ( ( insert_rule @ A @ bot_bot_set_rule )
        = ( insert_rule @ B2 @ bot_bot_set_rule ) )
     => ( A = B2 ) ) ).

% singleton_inject
thf(fact_919_singleton__inject,axiom,
    ! [A: $o,B2: $o] :
      ( ( ( insert_o @ A @ bot_bot_set_o )
        = ( insert_o @ B2 @ bot_bot_set_o ) )
     => ( A = B2 ) ) ).

% singleton_inject
thf(fact_920_singleton__inject,axiom,
    ! [A: nat,B2: nat] :
      ( ( ( insert_nat @ A @ bot_bot_set_nat )
        = ( insert_nat @ B2 @ bot_bot_set_nat ) )
     => ( A = B2 ) ) ).

% singleton_inject
thf(fact_921_insert__not__empty,axiom,
    ! [A: rule,A3: set_rule] :
      ( ( insert_rule @ A @ A3 )
     != bot_bot_set_rule ) ).

% insert_not_empty
thf(fact_922_insert__not__empty,axiom,
    ! [A: $o,A3: set_o] :
      ( ( insert_o @ A @ A3 )
     != bot_bot_set_o ) ).

% insert_not_empty
thf(fact_923_insert__not__empty,axiom,
    ! [A: nat,A3: set_nat] :
      ( ( insert_nat @ A @ A3 )
     != bot_bot_set_nat ) ).

% insert_not_empty
thf(fact_924_doubleton__eq__iff,axiom,
    ! [A: rule,B2: rule,C2: rule,D2: rule] :
      ( ( ( insert_rule @ A @ ( insert_rule @ B2 @ bot_bot_set_rule ) )
        = ( insert_rule @ C2 @ ( insert_rule @ D2 @ bot_bot_set_rule ) ) )
      = ( ( ( A = C2 )
          & ( B2 = D2 ) )
        | ( ( A = D2 )
          & ( B2 = C2 ) ) ) ) ).

% doubleton_eq_iff
thf(fact_925_doubleton__eq__iff,axiom,
    ! [A: $o,B2: $o,C2: $o,D2: $o] :
      ( ( ( insert_o @ A @ ( insert_o @ B2 @ bot_bot_set_o ) )
        = ( insert_o @ C2 @ ( insert_o @ D2 @ bot_bot_set_o ) ) )
      = ( ( ( A = C2 )
          & ( B2 = D2 ) )
        | ( ( A = D2 )
          & ( B2 = C2 ) ) ) ) ).

% doubleton_eq_iff
thf(fact_926_doubleton__eq__iff,axiom,
    ! [A: nat,B2: nat,C2: nat,D2: nat] :
      ( ( ( insert_nat @ A @ ( insert_nat @ B2 @ bot_bot_set_nat ) )
        = ( insert_nat @ C2 @ ( insert_nat @ D2 @ bot_bot_set_nat ) ) )
      = ( ( ( A = C2 )
          & ( B2 = D2 ) )
        | ( ( A = D2 )
          & ( B2 = C2 ) ) ) ) ).

% doubleton_eq_iff
thf(fact_927_singleton__iff,axiom,
    ! [B2: list_fm,A: list_fm] :
      ( ( member_list_fm3 @ B2 @ ( insert_list_fm @ A @ bot_bot_set_list_fm ) )
      = ( B2 = A ) ) ).

% singleton_iff
thf(fact_928_singleton__iff,axiom,
    ! [B2: fm,A: fm] :
      ( ( member_fm3 @ B2 @ ( insert_fm @ A @ bot_bot_set_fm ) )
      = ( B2 = A ) ) ).

% singleton_iff
thf(fact_929_singleton__iff,axiom,
    ! [B2: rule,A: rule] :
      ( ( member_rule3 @ B2 @ ( insert_rule @ A @ bot_bot_set_rule ) )
      = ( B2 = A ) ) ).

% singleton_iff
thf(fact_930_singleton__iff,axiom,
    ! [B2: $o,A: $o] :
      ( ( member_o3 @ B2 @ ( insert_o @ A @ bot_bot_set_o ) )
      = ( B2 = A ) ) ).

% singleton_iff
thf(fact_931_singleton__iff,axiom,
    ! [B2: nat,A: nat] :
      ( ( member_nat3 @ B2 @ ( insert_nat @ A @ bot_bot_set_nat ) )
      = ( B2 = A ) ) ).

% singleton_iff
thf(fact_932_singletonD,axiom,
    ! [B2: list_fm,A: list_fm] :
      ( ( member_list_fm3 @ B2 @ ( insert_list_fm @ A @ bot_bot_set_list_fm ) )
     => ( B2 = A ) ) ).

% singletonD
thf(fact_933_singletonD,axiom,
    ! [B2: fm,A: fm] :
      ( ( member_fm3 @ B2 @ ( insert_fm @ A @ bot_bot_set_fm ) )
     => ( B2 = A ) ) ).

% singletonD
thf(fact_934_singletonD,axiom,
    ! [B2: rule,A: rule] :
      ( ( member_rule3 @ B2 @ ( insert_rule @ A @ bot_bot_set_rule ) )
     => ( B2 = A ) ) ).

% singletonD
thf(fact_935_singletonD,axiom,
    ! [B2: $o,A: $o] :
      ( ( member_o3 @ B2 @ ( insert_o @ A @ bot_bot_set_o ) )
     => ( B2 = A ) ) ).

% singletonD
thf(fact_936_singletonD,axiom,
    ! [B2: nat,A: nat] :
      ( ( member_nat3 @ B2 @ ( insert_nat @ A @ bot_bot_set_nat ) )
     => ( B2 = A ) ) ).

% singletonD
thf(fact_937_not__psubset__empty,axiom,
    ! [A3: set_rule] :
      ~ ( ord_less_set_rule @ A3 @ bot_bot_set_rule ) ).

% not_psubset_empty
thf(fact_938_not__psubset__empty,axiom,
    ! [A3: set_o] :
      ~ ( ord_less_set_o @ A3 @ bot_bot_set_o ) ).

% not_psubset_empty
thf(fact_939_not__psubset__empty,axiom,
    ! [A3: set_nat] :
      ~ ( ord_less_set_nat @ A3 @ bot_bot_set_nat ) ).

% not_psubset_empty
thf(fact_940_finite__list,axiom,
    ! [A3: set_fm] :
      ( ( finite_finite_fm @ A3 )
     => ? [Xs3: list_fm] :
          ( ( set_fm2 @ Xs3 )
          = A3 ) ) ).

% finite_list
thf(fact_941_finite__list,axiom,
    ! [A3: set_list_fm] :
      ( ( finite834038518558952994ist_fm @ A3 )
     => ? [Xs3: list_list_fm] :
          ( ( set_list_fm2 @ Xs3 )
          = A3 ) ) ).

% finite_list
thf(fact_942_finite__list,axiom,
    ! [A3: set_tm] :
      ( ( finite_finite_tm @ A3 )
     => ? [Xs3: list_tm] :
          ( ( set_tm2 @ Xs3 )
          = A3 ) ) ).

% finite_list
thf(fact_943_finite__list,axiom,
    ! [A3: set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ? [Xs3: list_nat] :
          ( ( set_nat2 @ Xs3 )
          = A3 ) ) ).

% finite_list
thf(fact_944_less__supI1,axiom,
    ! [X: set_nat,A: set_nat,B2: set_nat] :
      ( ( ord_less_set_nat @ X @ A )
     => ( ord_less_set_nat @ X @ ( sup_sup_set_nat @ A @ B2 ) ) ) ).

% less_supI1
thf(fact_945_less__supI1,axiom,
    ! [X: nat,A: nat,B2: nat] :
      ( ( ord_less_nat @ X @ A )
     => ( ord_less_nat @ X @ ( sup_sup_nat @ A @ B2 ) ) ) ).

% less_supI1
thf(fact_946_less__supI2,axiom,
    ! [X: set_nat,B2: set_nat,A: set_nat] :
      ( ( ord_less_set_nat @ X @ B2 )
     => ( ord_less_set_nat @ X @ ( sup_sup_set_nat @ A @ B2 ) ) ) ).

% less_supI2
thf(fact_947_less__supI2,axiom,
    ! [X: nat,B2: nat,A: nat] :
      ( ( ord_less_nat @ X @ B2 )
     => ( ord_less_nat @ X @ ( sup_sup_nat @ A @ B2 ) ) ) ).

% less_supI2
thf(fact_948_sup_Oabsorb3,axiom,
    ! [B2: set_nat,A: set_nat] :
      ( ( ord_less_set_nat @ B2 @ A )
     => ( ( sup_sup_set_nat @ A @ B2 )
        = A ) ) ).

% sup.absorb3
thf(fact_949_sup_Oabsorb3,axiom,
    ! [B2: nat,A: nat] :
      ( ( ord_less_nat @ B2 @ A )
     => ( ( sup_sup_nat @ A @ B2 )
        = A ) ) ).

% sup.absorb3
thf(fact_950_sup_Oabsorb4,axiom,
    ! [A: set_nat,B2: set_nat] :
      ( ( ord_less_set_nat @ A @ B2 )
     => ( ( sup_sup_set_nat @ A @ B2 )
        = B2 ) ) ).

% sup.absorb4
thf(fact_951_sup_Oabsorb4,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_nat @ A @ B2 )
     => ( ( sup_sup_nat @ A @ B2 )
        = B2 ) ) ).

% sup.absorb4
thf(fact_952_sup_Ostrict__boundedE,axiom,
    ! [B2: set_nat,C2: set_nat,A: set_nat] :
      ( ( ord_less_set_nat @ ( sup_sup_set_nat @ B2 @ C2 ) @ A )
     => ~ ( ( ord_less_set_nat @ B2 @ A )
         => ~ ( ord_less_set_nat @ C2 @ A ) ) ) ).

% sup.strict_boundedE
thf(fact_953_sup_Ostrict__boundedE,axiom,
    ! [B2: nat,C2: nat,A: nat] :
      ( ( ord_less_nat @ ( sup_sup_nat @ B2 @ C2 ) @ A )
     => ~ ( ( ord_less_nat @ B2 @ A )
         => ~ ( ord_less_nat @ C2 @ A ) ) ) ).

% sup.strict_boundedE
thf(fact_954_sup_Ostrict__order__iff,axiom,
    ( ord_less_set_nat
    = ( ^ [B4: set_nat,A2: set_nat] :
          ( ( A2
            = ( sup_sup_set_nat @ A2 @ B4 ) )
          & ( A2 != B4 ) ) ) ) ).

% sup.strict_order_iff
thf(fact_955_sup_Ostrict__order__iff,axiom,
    ( ord_less_nat
    = ( ^ [B4: nat,A2: nat] :
          ( ( A2
            = ( sup_sup_nat @ A2 @ B4 ) )
          & ( A2 != B4 ) ) ) ) ).

% sup.strict_order_iff
thf(fact_956_sup_Ostrict__coboundedI1,axiom,
    ! [C2: set_nat,A: set_nat,B2: set_nat] :
      ( ( ord_less_set_nat @ C2 @ A )
     => ( ord_less_set_nat @ C2 @ ( sup_sup_set_nat @ A @ B2 ) ) ) ).

% sup.strict_coboundedI1
thf(fact_957_sup_Ostrict__coboundedI1,axiom,
    ! [C2: nat,A: nat,B2: nat] :
      ( ( ord_less_nat @ C2 @ A )
     => ( ord_less_nat @ C2 @ ( sup_sup_nat @ A @ B2 ) ) ) ).

% sup.strict_coboundedI1
thf(fact_958_sup_Ostrict__coboundedI2,axiom,
    ! [C2: set_nat,B2: set_nat,A: set_nat] :
      ( ( ord_less_set_nat @ C2 @ B2 )
     => ( ord_less_set_nat @ C2 @ ( sup_sup_set_nat @ A @ B2 ) ) ) ).

% sup.strict_coboundedI2
thf(fact_959_sup_Ostrict__coboundedI2,axiom,
    ! [C2: nat,B2: nat,A: nat] :
      ( ( ord_less_nat @ C2 @ B2 )
     => ( ord_less_nat @ C2 @ ( sup_sup_nat @ A @ B2 ) ) ) ).

% sup.strict_coboundedI2
thf(fact_960_Collect__conv__if2,axiom,
    ! [P3: rule > $o,A: rule] :
      ( ( ( P3 @ A )
       => ( ( collect_rule
            @ ^ [X3: rule] :
                ( ( A = X3 )
                & ( P3 @ X3 ) ) )
          = ( insert_rule @ A @ bot_bot_set_rule ) ) )
      & ( ~ ( P3 @ A )
       => ( ( collect_rule
            @ ^ [X3: rule] :
                ( ( A = X3 )
                & ( P3 @ X3 ) ) )
          = bot_bot_set_rule ) ) ) ).

% Collect_conv_if2
thf(fact_961_Collect__conv__if2,axiom,
    ! [P3: $o > $o,A: $o] :
      ( ( ( P3 @ A )
       => ( ( collect_o
            @ ^ [X3: $o] :
                ( ( A = X3 )
                & ( P3 @ X3 ) ) )
          = ( insert_o @ A @ bot_bot_set_o ) ) )
      & ( ~ ( P3 @ A )
       => ( ( collect_o
            @ ^ [X3: $o] :
                ( ( A = X3 )
                & ( P3 @ X3 ) ) )
          = bot_bot_set_o ) ) ) ).

% Collect_conv_if2
thf(fact_962_Collect__conv__if2,axiom,
    ! [P3: nat > $o,A: nat] :
      ( ( ( P3 @ A )
       => ( ( collect_nat
            @ ^ [X3: nat] :
                ( ( A = X3 )
                & ( P3 @ X3 ) ) )
          = ( insert_nat @ A @ bot_bot_set_nat ) ) )
      & ( ~ ( P3 @ A )
       => ( ( collect_nat
            @ ^ [X3: nat] :
                ( ( A = X3 )
                & ( P3 @ X3 ) ) )
          = bot_bot_set_nat ) ) ) ).

% Collect_conv_if2
thf(fact_963_Collect__conv__if,axiom,
    ! [P3: rule > $o,A: rule] :
      ( ( ( P3 @ A )
       => ( ( collect_rule
            @ ^ [X3: rule] :
                ( ( X3 = A )
                & ( P3 @ X3 ) ) )
          = ( insert_rule @ A @ bot_bot_set_rule ) ) )
      & ( ~ ( P3 @ A )
       => ( ( collect_rule
            @ ^ [X3: rule] :
                ( ( X3 = A )
                & ( P3 @ X3 ) ) )
          = bot_bot_set_rule ) ) ) ).

% Collect_conv_if
thf(fact_964_Collect__conv__if,axiom,
    ! [P3: $o > $o,A: $o] :
      ( ( ( P3 @ A )
       => ( ( collect_o
            @ ^ [X3: $o] :
                ( ( X3 = A )
                & ( P3 @ X3 ) ) )
          = ( insert_o @ A @ bot_bot_set_o ) ) )
      & ( ~ ( P3 @ A )
       => ( ( collect_o
            @ ^ [X3: $o] :
                ( ( X3 = A )
                & ( P3 @ X3 ) ) )
          = bot_bot_set_o ) ) ) ).

% Collect_conv_if
thf(fact_965_Collect__conv__if,axiom,
    ! [P3: nat > $o,A: nat] :
      ( ( ( P3 @ A )
       => ( ( collect_nat
            @ ^ [X3: nat] :
                ( ( X3 = A )
                & ( P3 @ X3 ) ) )
          = ( insert_nat @ A @ bot_bot_set_nat ) ) )
      & ( ~ ( P3 @ A )
       => ( ( collect_nat
            @ ^ [X3: nat] :
                ( ( X3 = A )
                & ( P3 @ X3 ) ) )
          = bot_bot_set_nat ) ) ) ).

% Collect_conv_if
thf(fact_966_insert__def,axiom,
    ( insert_o
    = ( ^ [A2: $o] :
          ( sup_sup_set_o
          @ ( collect_o
            @ ^ [X3: $o] : ( X3 = A2 ) ) ) ) ) ).

% insert_def
thf(fact_967_insert__def,axiom,
    ( insert_nat
    = ( ^ [A2: nat] :
          ( sup_sup_set_nat
          @ ( collect_nat
            @ ^ [X3: nat] : ( X3 = A2 ) ) ) ) ) ).

% insert_def
thf(fact_968_singleton__Un__iff,axiom,
    ! [X: rule,A3: set_rule,B: set_rule] :
      ( ( ( insert_rule @ X @ bot_bot_set_rule )
        = ( sup_sup_set_rule @ A3 @ B ) )
      = ( ( ( A3 = bot_bot_set_rule )
          & ( B
            = ( insert_rule @ X @ bot_bot_set_rule ) ) )
        | ( ( A3
            = ( insert_rule @ X @ bot_bot_set_rule ) )
          & ( B = bot_bot_set_rule ) )
        | ( ( A3
            = ( insert_rule @ X @ bot_bot_set_rule ) )
          & ( B
            = ( insert_rule @ X @ bot_bot_set_rule ) ) ) ) ) ).

% singleton_Un_iff
thf(fact_969_singleton__Un__iff,axiom,
    ! [X: $o,A3: set_o,B: set_o] :
      ( ( ( insert_o @ X @ bot_bot_set_o )
        = ( sup_sup_set_o @ A3 @ B ) )
      = ( ( ( A3 = bot_bot_set_o )
          & ( B
            = ( insert_o @ X @ bot_bot_set_o ) ) )
        | ( ( A3
            = ( insert_o @ X @ bot_bot_set_o ) )
          & ( B = bot_bot_set_o ) )
        | ( ( A3
            = ( insert_o @ X @ bot_bot_set_o ) )
          & ( B
            = ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ).

% singleton_Un_iff
thf(fact_970_singleton__Un__iff,axiom,
    ! [X: nat,A3: set_nat,B: set_nat] :
      ( ( ( insert_nat @ X @ bot_bot_set_nat )
        = ( sup_sup_set_nat @ A3 @ B ) )
      = ( ( ( A3 = bot_bot_set_nat )
          & ( B
            = ( insert_nat @ X @ bot_bot_set_nat ) ) )
        | ( ( A3
            = ( insert_nat @ X @ bot_bot_set_nat ) )
          & ( B = bot_bot_set_nat ) )
        | ( ( A3
            = ( insert_nat @ X @ bot_bot_set_nat ) )
          & ( B
            = ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ).

% singleton_Un_iff
thf(fact_971_Un__singleton__iff,axiom,
    ! [A3: set_rule,B: set_rule,X: rule] :
      ( ( ( sup_sup_set_rule @ A3 @ B )
        = ( insert_rule @ X @ bot_bot_set_rule ) )
      = ( ( ( A3 = bot_bot_set_rule )
          & ( B
            = ( insert_rule @ X @ bot_bot_set_rule ) ) )
        | ( ( A3
            = ( insert_rule @ X @ bot_bot_set_rule ) )
          & ( B = bot_bot_set_rule ) )
        | ( ( A3
            = ( insert_rule @ X @ bot_bot_set_rule ) )
          & ( B
            = ( insert_rule @ X @ bot_bot_set_rule ) ) ) ) ) ).

% Un_singleton_iff
thf(fact_972_Un__singleton__iff,axiom,
    ! [A3: set_o,B: set_o,X: $o] :
      ( ( ( sup_sup_set_o @ A3 @ B )
        = ( insert_o @ X @ bot_bot_set_o ) )
      = ( ( ( A3 = bot_bot_set_o )
          & ( B
            = ( insert_o @ X @ bot_bot_set_o ) ) )
        | ( ( A3
            = ( insert_o @ X @ bot_bot_set_o ) )
          & ( B = bot_bot_set_o ) )
        | ( ( A3
            = ( insert_o @ X @ bot_bot_set_o ) )
          & ( B
            = ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ).

% Un_singleton_iff
thf(fact_973_Un__singleton__iff,axiom,
    ! [A3: set_nat,B: set_nat,X: nat] :
      ( ( ( sup_sup_set_nat @ A3 @ B )
        = ( insert_nat @ X @ bot_bot_set_nat ) )
      = ( ( ( A3 = bot_bot_set_nat )
          & ( B
            = ( insert_nat @ X @ bot_bot_set_nat ) ) )
        | ( ( A3
            = ( insert_nat @ X @ bot_bot_set_nat ) )
          & ( B = bot_bot_set_nat ) )
        | ( ( A3
            = ( insert_nat @ X @ bot_bot_set_nat ) )
          & ( B
            = ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ).

% Un_singleton_iff
thf(fact_974_insert__is__Un,axiom,
    ( insert_rule
    = ( ^ [A2: rule] : ( sup_sup_set_rule @ ( insert_rule @ A2 @ bot_bot_set_rule ) ) ) ) ).

% insert_is_Un
thf(fact_975_insert__is__Un,axiom,
    ( insert_o
    = ( ^ [A2: $o] : ( sup_sup_set_o @ ( insert_o @ A2 @ bot_bot_set_o ) ) ) ) ).

% insert_is_Un
thf(fact_976_insert__is__Un,axiom,
    ( insert_nat
    = ( ^ [A2: nat] : ( sup_sup_set_nat @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) ) ) ).

% insert_is_Un
thf(fact_977_less__cSupE,axiom,
    ! [Y: nat,X4: set_nat] :
      ( ( ord_less_nat @ Y @ ( complete_Sup_Sup_nat @ X4 ) )
     => ( ( X4 != bot_bot_set_nat )
       => ~ ! [X2: nat] :
              ( ( member_nat3 @ X2 @ X4 )
             => ~ ( ord_less_nat @ Y @ X2 ) ) ) ) ).

% less_cSupE
thf(fact_978_less__cSupD,axiom,
    ! [X4: set_nat,Z4: nat] :
      ( ( X4 != bot_bot_set_nat )
     => ( ( ord_less_nat @ Z4 @ ( complete_Sup_Sup_nat @ X4 ) )
       => ? [X2: nat] :
            ( ( member_nat3 @ X2 @ X4 )
            & ( ord_less_nat @ Z4 @ X2 ) ) ) ) ).

% less_cSupD
thf(fact_979_Union__insert,axiom,
    ! [A: set_nat,B: set_set_nat] :
      ( ( comple7399068483239264473et_nat @ ( insert_set_nat @ A @ B ) )
      = ( sup_sup_set_nat @ A @ ( comple7399068483239264473et_nat @ B ) ) ) ).

% Union_insert
thf(fact_980_Union__insert,axiom,
    ! [A: set_fm,B: set_set_fm] :
      ( ( comple2134933779557159616set_fm @ ( insert_set_fm @ A @ B ) )
      = ( sup_sup_set_fm @ A @ ( comple2134933779557159616set_fm @ B ) ) ) ).

% Union_insert
thf(fact_981_refl__on__Un,axiom,
    ! [A3: set_nat,R: set_Pr1261947904930325089at_nat,B: set_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( refl_on_nat @ A3 @ R )
     => ( ( refl_on_nat @ B @ S )
       => ( refl_on_nat @ ( sup_sup_set_nat @ A3 @ B ) @ ( sup_su6327502436637775413at_nat @ R @ S ) ) ) ) ).

% refl_on_Un
thf(fact_982_image__constant,axiom,
    ! [X: rule,A3: set_rule,C2: rule] :
      ( ( member_rule3 @ X @ A3 )
     => ( ( image_rule_rule
          @ ^ [X3: rule] : C2
          @ A3 )
        = ( insert_rule @ C2 @ bot_bot_set_rule ) ) ) ).

% image_constant
thf(fact_983_image__constant,axiom,
    ! [X: fm,A3: set_fm,C2: rule] :
      ( ( member_fm3 @ X @ A3 )
     => ( ( image_fm_rule
          @ ^ [X3: fm] : C2
          @ A3 )
        = ( insert_rule @ C2 @ bot_bot_set_rule ) ) ) ).

% image_constant
thf(fact_984_image__constant,axiom,
    ! [X: $o,A3: set_o,C2: rule] :
      ( ( member_o3 @ X @ A3 )
     => ( ( image_o_rule
          @ ^ [X3: $o] : C2
          @ A3 )
        = ( insert_rule @ C2 @ bot_bot_set_rule ) ) ) ).

% image_constant
thf(fact_985_image__constant,axiom,
    ! [X: nat,A3: set_nat,C2: rule] :
      ( ( member_nat3 @ X @ A3 )
     => ( ( image_nat_rule
          @ ^ [X3: nat] : C2
          @ A3 )
        = ( insert_rule @ C2 @ bot_bot_set_rule ) ) ) ).

% image_constant
thf(fact_986_image__constant,axiom,
    ! [X: rule,A3: set_rule,C2: $o] :
      ( ( member_rule3 @ X @ A3 )
     => ( ( image_rule_o
          @ ^ [X3: rule] : C2
          @ A3 )
        = ( insert_o @ C2 @ bot_bot_set_o ) ) ) ).

% image_constant
thf(fact_987_image__constant,axiom,
    ! [X: fm,A3: set_fm,C2: $o] :
      ( ( member_fm3 @ X @ A3 )
     => ( ( image_fm_o
          @ ^ [X3: fm] : C2
          @ A3 )
        = ( insert_o @ C2 @ bot_bot_set_o ) ) ) ).

% image_constant
thf(fact_988_image__constant,axiom,
    ! [X: $o,A3: set_o,C2: $o] :
      ( ( member_o3 @ X @ A3 )
     => ( ( image_o_o
          @ ^ [X3: $o] : C2
          @ A3 )
        = ( insert_o @ C2 @ bot_bot_set_o ) ) ) ).

% image_constant
thf(fact_989_image__constant,axiom,
    ! [X: nat,A3: set_nat,C2: $o] :
      ( ( member_nat3 @ X @ A3 )
     => ( ( image_nat_o
          @ ^ [X3: nat] : C2
          @ A3 )
        = ( insert_o @ C2 @ bot_bot_set_o ) ) ) ).

% image_constant
thf(fact_990_image__constant,axiom,
    ! [X: rule,A3: set_rule,C2: nat] :
      ( ( member_rule3 @ X @ A3 )
     => ( ( image_rule_nat
          @ ^ [X3: rule] : C2
          @ A3 )
        = ( insert_nat @ C2 @ bot_bot_set_nat ) ) ) ).

% image_constant
thf(fact_991_image__constant,axiom,
    ! [X: fm,A3: set_fm,C2: nat] :
      ( ( member_fm3 @ X @ A3 )
     => ( ( image_fm_nat
          @ ^ [X3: fm] : C2
          @ A3 )
        = ( insert_nat @ C2 @ bot_bot_set_nat ) ) ) ).

% image_constant
thf(fact_992_image__constant__conv,axiom,
    ! [A3: set_Pr1822751329126368876m_rule,C2: set_fm] :
      ( ( ( A3 = bot_bo3426410157013319936m_rule )
       => ( ( image_377115208147135190set_fm
            @ ^ [X3: produc340336539035504054m_rule] : C2
            @ A3 )
          = bot_bot_set_set_fm ) )
      & ( ( A3 != bot_bo3426410157013319936m_rule )
       => ( ( image_377115208147135190set_fm
            @ ^ [X3: produc340336539035504054m_rule] : C2
            @ A3 )
          = ( insert_set_fm @ C2 @ bot_bot_set_set_fm ) ) ) ) ).

% image_constant_conv
thf(fact_993_image__constant__conv,axiom,
    ! [A3: set_rule,C2: rule] :
      ( ( ( A3 = bot_bot_set_rule )
       => ( ( image_rule_rule
            @ ^ [X3: rule] : C2
            @ A3 )
          = bot_bot_set_rule ) )
      & ( ( A3 != bot_bot_set_rule )
       => ( ( image_rule_rule
            @ ^ [X3: rule] : C2
            @ A3 )
          = ( insert_rule @ C2 @ bot_bot_set_rule ) ) ) ) ).

% image_constant_conv
thf(fact_994_image__constant__conv,axiom,
    ! [A3: set_rule,C2: $o] :
      ( ( ( A3 = bot_bot_set_rule )
       => ( ( image_rule_o
            @ ^ [X3: rule] : C2
            @ A3 )
          = bot_bot_set_o ) )
      & ( ( A3 != bot_bot_set_rule )
       => ( ( image_rule_o
            @ ^ [X3: rule] : C2
            @ A3 )
          = ( insert_o @ C2 @ bot_bot_set_o ) ) ) ) ).

% image_constant_conv
thf(fact_995_image__constant__conv,axiom,
    ! [A3: set_rule,C2: nat] :
      ( ( ( A3 = bot_bot_set_rule )
       => ( ( image_rule_nat
            @ ^ [X3: rule] : C2
            @ A3 )
          = bot_bot_set_nat ) )
      & ( ( A3 != bot_bot_set_rule )
       => ( ( image_rule_nat
            @ ^ [X3: rule] : C2
            @ A3 )
          = ( insert_nat @ C2 @ bot_bot_set_nat ) ) ) ) ).

% image_constant_conv
thf(fact_996_image__constant__conv,axiom,
    ! [A3: set_o,C2: rule] :
      ( ( ( A3 = bot_bot_set_o )
       => ( ( image_o_rule
            @ ^ [X3: $o] : C2
            @ A3 )
          = bot_bot_set_rule ) )
      & ( ( A3 != bot_bot_set_o )
       => ( ( image_o_rule
            @ ^ [X3: $o] : C2
            @ A3 )
          = ( insert_rule @ C2 @ bot_bot_set_rule ) ) ) ) ).

% image_constant_conv
thf(fact_997_image__constant__conv,axiom,
    ! [A3: set_o,C2: $o] :
      ( ( ( A3 = bot_bot_set_o )
       => ( ( image_o_o
            @ ^ [X3: $o] : C2
            @ A3 )
          = bot_bot_set_o ) )
      & ( ( A3 != bot_bot_set_o )
       => ( ( image_o_o
            @ ^ [X3: $o] : C2
            @ A3 )
          = ( insert_o @ C2 @ bot_bot_set_o ) ) ) ) ).

% image_constant_conv
thf(fact_998_image__constant__conv,axiom,
    ! [A3: set_o,C2: nat] :
      ( ( ( A3 = bot_bot_set_o )
       => ( ( image_o_nat
            @ ^ [X3: $o] : C2
            @ A3 )
          = bot_bot_set_nat ) )
      & ( ( A3 != bot_bot_set_o )
       => ( ( image_o_nat
            @ ^ [X3: $o] : C2
            @ A3 )
          = ( insert_nat @ C2 @ bot_bot_set_nat ) ) ) ) ).

% image_constant_conv
thf(fact_999_image__constant__conv,axiom,
    ! [A3: set_nat,C2: rule] :
      ( ( ( A3 = bot_bot_set_nat )
       => ( ( image_nat_rule
            @ ^ [X3: nat] : C2
            @ A3 )
          = bot_bot_set_rule ) )
      & ( ( A3 != bot_bot_set_nat )
       => ( ( image_nat_rule
            @ ^ [X3: nat] : C2
            @ A3 )
          = ( insert_rule @ C2 @ bot_bot_set_rule ) ) ) ) ).

% image_constant_conv
thf(fact_1000_image__constant__conv,axiom,
    ! [A3: set_nat,C2: $o] :
      ( ( ( A3 = bot_bot_set_nat )
       => ( ( image_nat_o
            @ ^ [X3: nat] : C2
            @ A3 )
          = bot_bot_set_o ) )
      & ( ( A3 != bot_bot_set_nat )
       => ( ( image_nat_o
            @ ^ [X3: nat] : C2
            @ A3 )
          = ( insert_o @ C2 @ bot_bot_set_o ) ) ) ) ).

% image_constant_conv
thf(fact_1001_image__constant__conv,axiom,
    ! [A3: set_nat,C2: nat] :
      ( ( ( A3 = bot_bot_set_nat )
       => ( ( image_nat_nat
            @ ^ [X3: nat] : C2
            @ A3 )
          = bot_bot_set_nat ) )
      & ( ( A3 != bot_bot_set_nat )
       => ( ( image_nat_nat
            @ ^ [X3: nat] : C2
            @ A3 )
          = ( insert_nat @ C2 @ bot_bot_set_nat ) ) ) ) ).

% image_constant_conv
thf(fact_1002_sdrop__snth,axiom,
    ! [N2: nat,S: stream2709947120125613254m_rule,M2: nat] :
      ( ( snth_P6679518042731451922m_rule @ ( sdrop_8169176516188972301m_rule @ N2 @ S ) @ M2 )
      = ( snth_P6679518042731451922m_rule @ S @ ( plus_plus_nat @ N2 @ M2 ) ) ) ).

% sdrop_snth
thf(fact_1003_sdrop__snth,axiom,
    ! [N2: nat,S: stream_rule,M2: nat] :
      ( ( snth_rule @ ( sdrop_rule @ N2 @ S ) @ M2 )
      = ( snth_rule @ S @ ( plus_plus_nat @ N2 @ M2 ) ) ) ).

% sdrop_snth
thf(fact_1004_SUP__lessD,axiom,
    ! [F: rule > $o,A3: set_rule,Y: $o,I4: rule] :
      ( ( ord_less_o @ ( complete_Sup_Sup_o @ ( image_rule_o @ F @ A3 ) ) @ Y )
     => ( ( member_rule3 @ I4 @ A3 )
       => ( ord_less_o @ ( F @ I4 ) @ Y ) ) ) ).

% SUP_lessD
thf(fact_1005_SUP__lessD,axiom,
    ! [F: fm > $o,A3: set_fm,Y: $o,I4: fm] :
      ( ( ord_less_o @ ( complete_Sup_Sup_o @ ( image_fm_o @ F @ A3 ) ) @ Y )
     => ( ( member_fm3 @ I4 @ A3 )
       => ( ord_less_o @ ( F @ I4 ) @ Y ) ) ) ).

% SUP_lessD
thf(fact_1006_SUP__lessD,axiom,
    ! [F: $o > $o,A3: set_o,Y: $o,I4: $o] :
      ( ( ord_less_o @ ( complete_Sup_Sup_o @ ( image_o_o @ F @ A3 ) ) @ Y )
     => ( ( member_o3 @ I4 @ A3 )
       => ( ord_less_o @ ( F @ I4 ) @ Y ) ) ) ).

% SUP_lessD
thf(fact_1007_SUP__lessD,axiom,
    ! [F: nat > $o,A3: set_nat,Y: $o,I4: nat] :
      ( ( ord_less_o @ ( complete_Sup_Sup_o @ ( image_nat_o @ F @ A3 ) ) @ Y )
     => ( ( member_nat3 @ I4 @ A3 )
       => ( ord_less_o @ ( F @ I4 ) @ Y ) ) ) ).

% SUP_lessD
thf(fact_1008_SUP__lessD,axiom,
    ! [F: rule > set_fm,A3: set_rule,Y: set_fm,I4: rule] :
      ( ( ord_less_set_fm @ ( comple2134933779557159616set_fm @ ( image_rule_set_fm @ F @ A3 ) ) @ Y )
     => ( ( member_rule3 @ I4 @ A3 )
       => ( ord_less_set_fm @ ( F @ I4 ) @ Y ) ) ) ).

% SUP_lessD
thf(fact_1009_SUP__lessD,axiom,
    ! [F: fm > set_fm,A3: set_fm,Y: set_fm,I4: fm] :
      ( ( ord_less_set_fm @ ( comple2134933779557159616set_fm @ ( image_fm_set_fm @ F @ A3 ) ) @ Y )
     => ( ( member_fm3 @ I4 @ A3 )
       => ( ord_less_set_fm @ ( F @ I4 ) @ Y ) ) ) ).

% SUP_lessD
thf(fact_1010_SUP__lessD,axiom,
    ! [F: $o > set_fm,A3: set_o,Y: set_fm,I4: $o] :
      ( ( ord_less_set_fm @ ( comple2134933779557159616set_fm @ ( image_o_set_fm @ F @ A3 ) ) @ Y )
     => ( ( member_o3 @ I4 @ A3 )
       => ( ord_less_set_fm @ ( F @ I4 ) @ Y ) ) ) ).

% SUP_lessD
thf(fact_1011_SUP__lessD,axiom,
    ! [F: nat > set_fm,A3: set_nat,Y: set_fm,I4: nat] :
      ( ( ord_less_set_fm @ ( comple2134933779557159616set_fm @ ( image_nat_set_fm @ F @ A3 ) ) @ Y )
     => ( ( member_nat3 @ I4 @ A3 )
       => ( ord_less_set_fm @ ( F @ I4 ) @ Y ) ) ) ).

% SUP_lessD
thf(fact_1012_SUP__lessD,axiom,
    ! [F: list_fm > $o,A3: set_list_fm,Y: $o,I4: list_fm] :
      ( ( ord_less_o @ ( complete_Sup_Sup_o @ ( image_list_fm_o @ F @ A3 ) ) @ Y )
     => ( ( member_list_fm3 @ I4 @ A3 )
       => ( ord_less_o @ ( F @ I4 ) @ Y ) ) ) ).

% SUP_lessD
thf(fact_1013_SUP__lessD,axiom,
    ! [F: list_fm > set_fm,A3: set_list_fm,Y: set_fm,I4: list_fm] :
      ( ( ord_less_set_fm @ ( comple2134933779557159616set_fm @ ( image_list_fm_set_fm @ F @ A3 ) ) @ Y )
     => ( ( member_list_fm3 @ I4 @ A3 )
       => ( ord_less_set_fm @ ( F @ I4 ) @ Y ) ) ) ).

% SUP_lessD
thf(fact_1014_UN__insert__distrib,axiom,
    ! [U2: rule,A3: set_rule,A: $o,B: rule > set_o] :
      ( ( member_rule3 @ U2 @ A3 )
     => ( ( comple90263536869209701_set_o
          @ ( image_rule_set_o
            @ ^ [X3: rule] : ( insert_o @ A @ ( B @ X3 ) )
            @ A3 ) )
        = ( insert_o @ A @ ( comple90263536869209701_set_o @ ( image_rule_set_o @ B @ A3 ) ) ) ) ) ).

% UN_insert_distrib
thf(fact_1015_UN__insert__distrib,axiom,
    ! [U2: rule,A3: set_rule,A: nat,B: rule > set_nat] :
      ( ( member_rule3 @ U2 @ A3 )
     => ( ( comple7399068483239264473et_nat
          @ ( image_rule_set_nat
            @ ^ [X3: rule] : ( insert_nat @ A @ ( B @ X3 ) )
            @ A3 ) )
        = ( insert_nat @ A @ ( comple7399068483239264473et_nat @ ( image_rule_set_nat @ B @ A3 ) ) ) ) ) ).

% UN_insert_distrib
thf(fact_1016_UN__insert__distrib,axiom,
    ! [U2: fm,A3: set_fm,A: $o,B: fm > set_o] :
      ( ( member_fm3 @ U2 @ A3 )
     => ( ( comple90263536869209701_set_o
          @ ( image_fm_set_o
            @ ^ [X3: fm] : ( insert_o @ A @ ( B @ X3 ) )
            @ A3 ) )
        = ( insert_o @ A @ ( comple90263536869209701_set_o @ ( image_fm_set_o @ B @ A3 ) ) ) ) ) ).

% UN_insert_distrib
thf(fact_1017_UN__insert__distrib,axiom,
    ! [U2: fm,A3: set_fm,A: nat,B: fm > set_nat] :
      ( ( member_fm3 @ U2 @ A3 )
     => ( ( comple7399068483239264473et_nat
          @ ( image_fm_set_nat
            @ ^ [X3: fm] : ( insert_nat @ A @ ( B @ X3 ) )
            @ A3 ) )
        = ( insert_nat @ A @ ( comple7399068483239264473et_nat @ ( image_fm_set_nat @ B @ A3 ) ) ) ) ) ).

% UN_insert_distrib
thf(fact_1018_UN__insert__distrib,axiom,
    ! [U2: $o,A3: set_o,A: $o,B: $o > set_o] :
      ( ( member_o3 @ U2 @ A3 )
     => ( ( comple90263536869209701_set_o
          @ ( image_o_set_o
            @ ^ [X3: $o] : ( insert_o @ A @ ( B @ X3 ) )
            @ A3 ) )
        = ( insert_o @ A @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B @ A3 ) ) ) ) ) ).

% UN_insert_distrib
thf(fact_1019_UN__insert__distrib,axiom,
    ! [U2: $o,A3: set_o,A: nat,B: $o > set_nat] :
      ( ( member_o3 @ U2 @ A3 )
     => ( ( comple7399068483239264473et_nat
          @ ( image_o_set_nat
            @ ^ [X3: $o] : ( insert_nat @ A @ ( B @ X3 ) )
            @ A3 ) )
        = ( insert_nat @ A @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ B @ A3 ) ) ) ) ) ).

% UN_insert_distrib
thf(fact_1020_UN__insert__distrib,axiom,
    ! [U2: nat,A3: set_nat,A: $o,B: nat > set_o] :
      ( ( member_nat3 @ U2 @ A3 )
     => ( ( comple90263536869209701_set_o
          @ ( image_nat_set_o
            @ ^ [X3: nat] : ( insert_o @ A @ ( B @ X3 ) )
            @ A3 ) )
        = ( insert_o @ A @ ( comple90263536869209701_set_o @ ( image_nat_set_o @ B @ A3 ) ) ) ) ) ).

% UN_insert_distrib
thf(fact_1021_UN__insert__distrib,axiom,
    ! [U2: nat,A3: set_nat,A: nat,B: nat > set_nat] :
      ( ( member_nat3 @ U2 @ A3 )
     => ( ( comple7399068483239264473et_nat
          @ ( image_nat_set_nat
            @ ^ [X3: nat] : ( insert_nat @ A @ ( B @ X3 ) )
            @ A3 ) )
        = ( insert_nat @ A @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B @ A3 ) ) ) ) ) ).

% UN_insert_distrib
thf(fact_1022_UN__insert__distrib,axiom,
    ! [U2: rule,A3: set_rule,A: fm,B: rule > set_fm] :
      ( ( member_rule3 @ U2 @ A3 )
     => ( ( comple2134933779557159616set_fm
          @ ( image_rule_set_fm
            @ ^ [X3: rule] : ( insert_fm @ A @ ( B @ X3 ) )
            @ A3 ) )
        = ( insert_fm @ A @ ( comple2134933779557159616set_fm @ ( image_rule_set_fm @ B @ A3 ) ) ) ) ) ).

% UN_insert_distrib
thf(fact_1023_UN__insert__distrib,axiom,
    ! [U2: fm,A3: set_fm,A: fm,B: fm > set_fm] :
      ( ( member_fm3 @ U2 @ A3 )
     => ( ( comple2134933779557159616set_fm
          @ ( image_fm_set_fm
            @ ^ [X3: fm] : ( insert_fm @ A @ ( B @ X3 ) )
            @ A3 ) )
        = ( insert_fm @ A @ ( comple2134933779557159616set_fm @ ( image_fm_set_fm @ B @ A3 ) ) ) ) ) ).

% UN_insert_distrib
thf(fact_1024_Union__image__insert,axiom,
    ! [F: $o > set_nat,A: $o,B: set_o] :
      ( ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ ( insert_o @ A @ B ) ) )
      = ( sup_sup_set_nat @ ( F @ A ) @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ B ) ) ) ) ).

% Union_image_insert
thf(fact_1025_Union__image__insert,axiom,
    ! [F: nat > set_nat,A: nat,B: set_nat] :
      ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ ( insert_nat @ A @ B ) ) )
      = ( sup_sup_set_nat @ ( F @ A ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ B ) ) ) ) ).

% Union_image_insert
thf(fact_1026_Union__image__insert,axiom,
    ! [F: produc340336539035504054m_rule > set_fm,A: produc340336539035504054m_rule,B: set_Pr1822751329126368876m_rule] :
      ( ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ F @ ( insert775845832097999846m_rule @ A @ B ) ) )
      = ( sup_sup_set_fm @ ( F @ A ) @ ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ F @ B ) ) ) ) ).

% Union_image_insert
thf(fact_1027_Union__image__insert,axiom,
    ! [F: $o > set_fm,A: $o,B: set_o] :
      ( ( comple2134933779557159616set_fm @ ( image_o_set_fm @ F @ ( insert_o @ A @ B ) ) )
      = ( sup_sup_set_fm @ ( F @ A ) @ ( comple2134933779557159616set_fm @ ( image_o_set_fm @ F @ B ) ) ) ) ).

% Union_image_insert
thf(fact_1028_Union__image__insert,axiom,
    ! [F: nat > set_fm,A: nat,B: set_nat] :
      ( ( comple2134933779557159616set_fm @ ( image_nat_set_fm @ F @ ( insert_nat @ A @ B ) ) )
      = ( sup_sup_set_fm @ ( F @ A ) @ ( comple2134933779557159616set_fm @ ( image_nat_set_fm @ F @ B ) ) ) ) ).

% Union_image_insert
thf(fact_1029_rules__repeat,axiom,
    ! [M2: nat,R: rule] :
    ? [N: nat] :
      ( ( ord_less_nat @ M2 @ N )
      & ( ( snth_rule @ rules @ N )
        = R ) ) ).

% rules_repeat
thf(fact_1030_less__cSup__iff,axiom,
    ! [X4: set_nat,Y: nat] :
      ( ( X4 != bot_bot_set_nat )
     => ( ( condit2214826472909112428ve_nat @ X4 )
       => ( ( ord_less_nat @ Y @ ( complete_Sup_Sup_nat @ X4 ) )
          = ( ? [X3: nat] :
                ( ( member_nat3 @ X3 @ X4 )
                & ( ord_less_nat @ Y @ X3 ) ) ) ) ) ) ).

% less_cSup_iff
thf(fact_1031_SUP__insert,axiom,
    ! [F: $o > set_nat,A: $o,A3: set_o] :
      ( ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ ( insert_o @ A @ A3 ) ) )
      = ( sup_sup_set_nat @ ( F @ A ) @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ A3 ) ) ) ) ).

% SUP_insert
thf(fact_1032_SUP__insert,axiom,
    ! [F: nat > set_nat,A: nat,A3: set_nat] :
      ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ ( insert_nat @ A @ A3 ) ) )
      = ( sup_sup_set_nat @ ( F @ A ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A3 ) ) ) ) ).

% SUP_insert
thf(fact_1033_SUP__insert,axiom,
    ! [F: produc340336539035504054m_rule > set_fm,A: produc340336539035504054m_rule,A3: set_Pr1822751329126368876m_rule] :
      ( ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ F @ ( insert775845832097999846m_rule @ A @ A3 ) ) )
      = ( sup_sup_set_fm @ ( F @ A ) @ ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ F @ A3 ) ) ) ) ).

% SUP_insert
thf(fact_1034_SUP__insert,axiom,
    ! [F: $o > set_fm,A: $o,A3: set_o] :
      ( ( comple2134933779557159616set_fm @ ( image_o_set_fm @ F @ ( insert_o @ A @ A3 ) ) )
      = ( sup_sup_set_fm @ ( F @ A ) @ ( comple2134933779557159616set_fm @ ( image_o_set_fm @ F @ A3 ) ) ) ) ).

% SUP_insert
thf(fact_1035_SUP__insert,axiom,
    ! [F: nat > set_fm,A: nat,A3: set_nat] :
      ( ( comple2134933779557159616set_fm @ ( image_nat_set_fm @ F @ ( insert_nat @ A @ A3 ) ) )
      = ( sup_sup_set_fm @ ( F @ A ) @ ( comple2134933779557159616set_fm @ ( image_nat_set_fm @ F @ A3 ) ) ) ) ).

% SUP_insert
thf(fact_1036_SUP__insert,axiom,
    ! [F: $o > $o,A: $o,A3: set_o] :
      ( ( complete_Sup_Sup_o @ ( image_o_o @ F @ ( insert_o @ A @ A3 ) ) )
      = ( sup_sup_o @ ( F @ A ) @ ( complete_Sup_Sup_o @ ( image_o_o @ F @ A3 ) ) ) ) ).

% SUP_insert
thf(fact_1037_SUP__insert,axiom,
    ! [F: nat > $o,A: nat,A3: set_nat] :
      ( ( complete_Sup_Sup_o @ ( image_nat_o @ F @ ( insert_nat @ A @ A3 ) ) )
      = ( sup_sup_o @ ( F @ A ) @ ( complete_Sup_Sup_o @ ( image_nat_o @ F @ A3 ) ) ) ) ).

% SUP_insert
thf(fact_1038_UN__extend__simps_I1_J,axiom,
    ! [C: set_rule,A: rule,B: rule > set_rule] :
      ( ( ( C = bot_bot_set_rule )
       => ( ( insert_rule @ A @ ( comple7665822446944348827t_rule @ ( image_rule_set_rule @ B @ C ) ) )
          = ( insert_rule @ A @ bot_bot_set_rule ) ) )
      & ( ( C != bot_bot_set_rule )
       => ( ( insert_rule @ A @ ( comple7665822446944348827t_rule @ ( image_rule_set_rule @ B @ C ) ) )
          = ( comple7665822446944348827t_rule
            @ ( image_rule_set_rule
              @ ^ [X3: rule] : ( insert_rule @ A @ ( B @ X3 ) )
              @ C ) ) ) ) ) ).

% UN_extend_simps(1)
thf(fact_1039_UN__extend__simps_I1_J,axiom,
    ! [C: set_rule,A: $o,B: rule > set_o] :
      ( ( ( C = bot_bot_set_rule )
       => ( ( insert_o @ A @ ( comple90263536869209701_set_o @ ( image_rule_set_o @ B @ C ) ) )
          = ( insert_o @ A @ bot_bot_set_o ) ) )
      & ( ( C != bot_bot_set_rule )
       => ( ( insert_o @ A @ ( comple90263536869209701_set_o @ ( image_rule_set_o @ B @ C ) ) )
          = ( comple90263536869209701_set_o
            @ ( image_rule_set_o
              @ ^ [X3: rule] : ( insert_o @ A @ ( B @ X3 ) )
              @ C ) ) ) ) ) ).

% UN_extend_simps(1)
thf(fact_1040_UN__extend__simps_I1_J,axiom,
    ! [C: set_rule,A: nat,B: rule > set_nat] :
      ( ( ( C = bot_bot_set_rule )
       => ( ( insert_nat @ A @ ( comple7399068483239264473et_nat @ ( image_rule_set_nat @ B @ C ) ) )
          = ( insert_nat @ A @ bot_bot_set_nat ) ) )
      & ( ( C != bot_bot_set_rule )
       => ( ( insert_nat @ A @ ( comple7399068483239264473et_nat @ ( image_rule_set_nat @ B @ C ) ) )
          = ( comple7399068483239264473et_nat
            @ ( image_rule_set_nat
              @ ^ [X3: rule] : ( insert_nat @ A @ ( B @ X3 ) )
              @ C ) ) ) ) ) ).

% UN_extend_simps(1)
thf(fact_1041_UN__extend__simps_I1_J,axiom,
    ! [C: set_o,A: rule,B: $o > set_rule] :
      ( ( ( C = bot_bot_set_o )
       => ( ( insert_rule @ A @ ( comple7665822446944348827t_rule @ ( image_o_set_rule @ B @ C ) ) )
          = ( insert_rule @ A @ bot_bot_set_rule ) ) )
      & ( ( C != bot_bot_set_o )
       => ( ( insert_rule @ A @ ( comple7665822446944348827t_rule @ ( image_o_set_rule @ B @ C ) ) )
          = ( comple7665822446944348827t_rule
            @ ( image_o_set_rule
              @ ^ [X3: $o] : ( insert_rule @ A @ ( B @ X3 ) )
              @ C ) ) ) ) ) ).

% UN_extend_simps(1)
thf(fact_1042_UN__extend__simps_I1_J,axiom,
    ! [C: set_o,A: $o,B: $o > set_o] :
      ( ( ( C = bot_bot_set_o )
       => ( ( insert_o @ A @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B @ C ) ) )
          = ( insert_o @ A @ bot_bot_set_o ) ) )
      & ( ( C != bot_bot_set_o )
       => ( ( insert_o @ A @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B @ C ) ) )
          = ( comple90263536869209701_set_o
            @ ( image_o_set_o
              @ ^ [X3: $o] : ( insert_o @ A @ ( B @ X3 ) )
              @ C ) ) ) ) ) ).

% UN_extend_simps(1)
thf(fact_1043_UN__extend__simps_I1_J,axiom,
    ! [C: set_o,A: nat,B: $o > set_nat] :
      ( ( ( C = bot_bot_set_o )
       => ( ( insert_nat @ A @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ B @ C ) ) )
          = ( insert_nat @ A @ bot_bot_set_nat ) ) )
      & ( ( C != bot_bot_set_o )
       => ( ( insert_nat @ A @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ B @ C ) ) )
          = ( comple7399068483239264473et_nat
            @ ( image_o_set_nat
              @ ^ [X3: $o] : ( insert_nat @ A @ ( B @ X3 ) )
              @ C ) ) ) ) ) ).

% UN_extend_simps(1)
thf(fact_1044_UN__extend__simps_I1_J,axiom,
    ! [C: set_nat,A: rule,B: nat > set_rule] :
      ( ( ( C = bot_bot_set_nat )
       => ( ( insert_rule @ A @ ( comple7665822446944348827t_rule @ ( image_nat_set_rule @ B @ C ) ) )
          = ( insert_rule @ A @ bot_bot_set_rule ) ) )
      & ( ( C != bot_bot_set_nat )
       => ( ( insert_rule @ A @ ( comple7665822446944348827t_rule @ ( image_nat_set_rule @ B @ C ) ) )
          = ( comple7665822446944348827t_rule
            @ ( image_nat_set_rule
              @ ^ [X3: nat] : ( insert_rule @ A @ ( B @ X3 ) )
              @ C ) ) ) ) ) ).

% UN_extend_simps(1)
thf(fact_1045_UN__extend__simps_I1_J,axiom,
    ! [C: set_nat,A: $o,B: nat > set_o] :
      ( ( ( C = bot_bot_set_nat )
       => ( ( insert_o @ A @ ( comple90263536869209701_set_o @ ( image_nat_set_o @ B @ C ) ) )
          = ( insert_o @ A @ bot_bot_set_o ) ) )
      & ( ( C != bot_bot_set_nat )
       => ( ( insert_o @ A @ ( comple90263536869209701_set_o @ ( image_nat_set_o @ B @ C ) ) )
          = ( comple90263536869209701_set_o
            @ ( image_nat_set_o
              @ ^ [X3: nat] : ( insert_o @ A @ ( B @ X3 ) )
              @ C ) ) ) ) ) ).

% UN_extend_simps(1)
thf(fact_1046_UN__extend__simps_I1_J,axiom,
    ! [C: set_nat,A: nat,B: nat > set_nat] :
      ( ( ( C = bot_bot_set_nat )
       => ( ( insert_nat @ A @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B @ C ) ) )
          = ( insert_nat @ A @ bot_bot_set_nat ) ) )
      & ( ( C != bot_bot_set_nat )
       => ( ( insert_nat @ A @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B @ C ) ) )
          = ( comple7399068483239264473et_nat
            @ ( image_nat_set_nat
              @ ^ [X3: nat] : ( insert_nat @ A @ ( B @ X3 ) )
              @ C ) ) ) ) ) ).

% UN_extend_simps(1)
thf(fact_1047_UN__extend__simps_I1_J,axiom,
    ! [C: set_rule,A: fm,B: rule > set_fm] :
      ( ( ( C = bot_bot_set_rule )
       => ( ( insert_fm @ A @ ( comple2134933779557159616set_fm @ ( image_rule_set_fm @ B @ C ) ) )
          = ( insert_fm @ A @ bot_bot_set_fm ) ) )
      & ( ( C != bot_bot_set_rule )
       => ( ( insert_fm @ A @ ( comple2134933779557159616set_fm @ ( image_rule_set_fm @ B @ C ) ) )
          = ( comple2134933779557159616set_fm
            @ ( image_rule_set_fm
              @ ^ [X3: rule] : ( insert_fm @ A @ ( B @ X3 ) )
              @ C ) ) ) ) ) ).

% UN_extend_simps(1)
thf(fact_1048_cSUP__lessD,axiom,
    ! [F: rule > $o,A3: set_rule,Y: $o,I4: rule] :
      ( ( condit5488710616941104124bove_o @ ( image_rule_o @ F @ A3 ) )
     => ( ( ord_less_o @ ( complete_Sup_Sup_o @ ( image_rule_o @ F @ A3 ) ) @ Y )
       => ( ( member_rule3 @ I4 @ A3 )
         => ( ord_less_o @ ( F @ I4 ) @ Y ) ) ) ) ).

% cSUP_lessD
thf(fact_1049_cSUP__lessD,axiom,
    ! [F: fm > $o,A3: set_fm,Y: $o,I4: fm] :
      ( ( condit5488710616941104124bove_o @ ( image_fm_o @ F @ A3 ) )
     => ( ( ord_less_o @ ( complete_Sup_Sup_o @ ( image_fm_o @ F @ A3 ) ) @ Y )
       => ( ( member_fm3 @ I4 @ A3 )
         => ( ord_less_o @ ( F @ I4 ) @ Y ) ) ) ) ).

% cSUP_lessD
thf(fact_1050_cSUP__lessD,axiom,
    ! [F: $o > $o,A3: set_o,Y: $o,I4: $o] :
      ( ( condit5488710616941104124bove_o @ ( image_o_o @ F @ A3 ) )
     => ( ( ord_less_o @ ( complete_Sup_Sup_o @ ( image_o_o @ F @ A3 ) ) @ Y )
       => ( ( member_o3 @ I4 @ A3 )
         => ( ord_less_o @ ( F @ I4 ) @ Y ) ) ) ) ).

% cSUP_lessD
thf(fact_1051_cSUP__lessD,axiom,
    ! [F: nat > $o,A3: set_nat,Y: $o,I4: nat] :
      ( ( condit5488710616941104124bove_o @ ( image_nat_o @ F @ A3 ) )
     => ( ( ord_less_o @ ( complete_Sup_Sup_o @ ( image_nat_o @ F @ A3 ) ) @ Y )
       => ( ( member_nat3 @ I4 @ A3 )
         => ( ord_less_o @ ( F @ I4 ) @ Y ) ) ) ) ).

% cSUP_lessD
thf(fact_1052_cSUP__lessD,axiom,
    ! [F: rule > nat,A3: set_rule,Y: nat,I4: rule] :
      ( ( condit2214826472909112428ve_nat @ ( image_rule_nat @ F @ A3 ) )
     => ( ( ord_less_nat @ ( complete_Sup_Sup_nat @ ( image_rule_nat @ F @ A3 ) ) @ Y )
       => ( ( member_rule3 @ I4 @ A3 )
         => ( ord_less_nat @ ( F @ I4 ) @ Y ) ) ) ) ).

% cSUP_lessD
thf(fact_1053_cSUP__lessD,axiom,
    ! [F: fm > nat,A3: set_fm,Y: nat,I4: fm] :
      ( ( condit2214826472909112428ve_nat @ ( image_fm_nat @ F @ A3 ) )
     => ( ( ord_less_nat @ ( complete_Sup_Sup_nat @ ( image_fm_nat @ F @ A3 ) ) @ Y )
       => ( ( member_fm3 @ I4 @ A3 )
         => ( ord_less_nat @ ( F @ I4 ) @ Y ) ) ) ) ).

% cSUP_lessD
thf(fact_1054_cSUP__lessD,axiom,
    ! [F: $o > nat,A3: set_o,Y: nat,I4: $o] :
      ( ( condit2214826472909112428ve_nat @ ( image_o_nat @ F @ A3 ) )
     => ( ( ord_less_nat @ ( complete_Sup_Sup_nat @ ( image_o_nat @ F @ A3 ) ) @ Y )
       => ( ( member_o3 @ I4 @ A3 )
         => ( ord_less_nat @ ( F @ I4 ) @ Y ) ) ) ) ).

% cSUP_lessD
thf(fact_1055_cSUP__lessD,axiom,
    ! [F: nat > nat,A3: set_nat,Y: nat,I4: nat] :
      ( ( condit2214826472909112428ve_nat @ ( image_nat_nat @ F @ A3 ) )
     => ( ( ord_less_nat @ ( complete_Sup_Sup_nat @ ( image_nat_nat @ F @ A3 ) ) @ Y )
       => ( ( member_nat3 @ I4 @ A3 )
         => ( ord_less_nat @ ( F @ I4 ) @ Y ) ) ) ) ).

% cSUP_lessD
thf(fact_1056_cSUP__lessD,axiom,
    ! [F: rule > set_fm,A3: set_rule,Y: set_fm,I4: rule] :
      ( ( condit7632555770478454199set_fm @ ( image_rule_set_fm @ F @ A3 ) )
     => ( ( ord_less_set_fm @ ( comple2134933779557159616set_fm @ ( image_rule_set_fm @ F @ A3 ) ) @ Y )
       => ( ( member_rule3 @ I4 @ A3 )
         => ( ord_less_set_fm @ ( F @ I4 ) @ Y ) ) ) ) ).

% cSUP_lessD
thf(fact_1057_cSUP__lessD,axiom,
    ! [F: fm > set_fm,A3: set_fm,Y: set_fm,I4: fm] :
      ( ( condit7632555770478454199set_fm @ ( image_fm_set_fm @ F @ A3 ) )
     => ( ( ord_less_set_fm @ ( comple2134933779557159616set_fm @ ( image_fm_set_fm @ F @ A3 ) ) @ Y )
       => ( ( member_fm3 @ I4 @ A3 )
         => ( ord_less_set_fm @ ( F @ I4 ) @ Y ) ) ) ) ).

% cSUP_lessD
thf(fact_1058_cSup__insert__If,axiom,
    ! [X4: set_set_nat,A: set_nat] :
      ( ( condit5477540289124974626et_nat @ X4 )
     => ( ( ( X4 = bot_bot_set_set_nat )
         => ( ( comple7399068483239264473et_nat @ ( insert_set_nat @ A @ X4 ) )
            = A ) )
        & ( ( X4 != bot_bot_set_set_nat )
         => ( ( comple7399068483239264473et_nat @ ( insert_set_nat @ A @ X4 ) )
            = ( sup_sup_set_nat @ A @ ( comple7399068483239264473et_nat @ X4 ) ) ) ) ) ) ).

% cSup_insert_If
thf(fact_1059_cSup__insert__If,axiom,
    ! [X4: set_set_fm,A: set_fm] :
      ( ( condit7632555770478454199set_fm @ X4 )
     => ( ( ( X4 = bot_bot_set_set_fm )
         => ( ( comple2134933779557159616set_fm @ ( insert_set_fm @ A @ X4 ) )
            = A ) )
        & ( ( X4 != bot_bot_set_set_fm )
         => ( ( comple2134933779557159616set_fm @ ( insert_set_fm @ A @ X4 ) )
            = ( sup_sup_set_fm @ A @ ( comple2134933779557159616set_fm @ X4 ) ) ) ) ) ) ).

% cSup_insert_If
thf(fact_1060_cSup__insert__If,axiom,
    ! [X4: set_o,A: $o] :
      ( ( condit5488710616941104124bove_o @ X4 )
     => ( ( complete_Sup_Sup_o @ ( insert_o @ A @ X4 ) )
        = ( ( ( X4 = bot_bot_set_o )
           => A )
          & ( ( X4 != bot_bot_set_o )
           => ( sup_sup_o @ A @ ( complete_Sup_Sup_o @ X4 ) ) ) ) ) ) ).

% cSup_insert_If
thf(fact_1061_cSup__insert__If,axiom,
    ! [X4: set_nat,A: nat] :
      ( ( condit2214826472909112428ve_nat @ X4 )
     => ( ( ( X4 = bot_bot_set_nat )
         => ( ( complete_Sup_Sup_nat @ ( insert_nat @ A @ X4 ) )
            = A ) )
        & ( ( X4 != bot_bot_set_nat )
         => ( ( complete_Sup_Sup_nat @ ( insert_nat @ A @ X4 ) )
            = ( sup_sup_nat @ A @ ( complete_Sup_Sup_nat @ X4 ) ) ) ) ) ) ).

% cSup_insert_If
thf(fact_1062_cSup__insert,axiom,
    ! [X4: set_set_nat,A: set_nat] :
      ( ( X4 != bot_bot_set_set_nat )
     => ( ( condit5477540289124974626et_nat @ X4 )
       => ( ( comple7399068483239264473et_nat @ ( insert_set_nat @ A @ X4 ) )
          = ( sup_sup_set_nat @ A @ ( comple7399068483239264473et_nat @ X4 ) ) ) ) ) ).

% cSup_insert
thf(fact_1063_cSup__insert,axiom,
    ! [X4: set_set_fm,A: set_fm] :
      ( ( X4 != bot_bot_set_set_fm )
     => ( ( condit7632555770478454199set_fm @ X4 )
       => ( ( comple2134933779557159616set_fm @ ( insert_set_fm @ A @ X4 ) )
          = ( sup_sup_set_fm @ A @ ( comple2134933779557159616set_fm @ X4 ) ) ) ) ) ).

% cSup_insert
thf(fact_1064_cSup__insert,axiom,
    ! [X4: set_o,A: $o] :
      ( ( X4 != bot_bot_set_o )
     => ( ( condit5488710616941104124bove_o @ X4 )
       => ( ( complete_Sup_Sup_o @ ( insert_o @ A @ X4 ) )
          = ( sup_sup_o @ A @ ( complete_Sup_Sup_o @ X4 ) ) ) ) ) ).

% cSup_insert
thf(fact_1065_cSup__insert,axiom,
    ! [X4: set_nat,A: nat] :
      ( ( X4 != bot_bot_set_nat )
     => ( ( condit2214826472909112428ve_nat @ X4 )
       => ( ( complete_Sup_Sup_nat @ ( insert_nat @ A @ X4 ) )
          = ( sup_sup_nat @ A @ ( complete_Sup_Sup_nat @ X4 ) ) ) ) ) ).

% cSup_insert
thf(fact_1066_UNION__singleton__eq__range,axiom,
    ! [F: produc340336539035504054m_rule > set_fm,A3: set_Pr1822751329126368876m_rule] :
      ( ( comple4326961365146087328set_fm
        @ ( image_1174956300053327798set_fm
          @ ^ [X3: produc340336539035504054m_rule] : ( insert_set_fm @ ( F @ X3 ) @ bot_bot_set_set_fm )
          @ A3 ) )
      = ( image_377115208147135190set_fm @ F @ A3 ) ) ).

% UNION_singleton_eq_range
thf(fact_1067_UNION__singleton__eq__range,axiom,
    ! [F: nat > nat,A3: set_nat] :
      ( ( comple7399068483239264473et_nat
        @ ( image_nat_set_nat
          @ ^ [X3: nat] : ( insert_nat @ ( F @ X3 ) @ bot_bot_set_nat )
          @ A3 ) )
      = ( image_nat_nat @ F @ A3 ) ) ).

% UNION_singleton_eq_range
thf(fact_1068_UNION__singleton__eq__range,axiom,
    ! [F: produc340336539035504054m_rule > fm,A3: set_Pr1822751329126368876m_rule] :
      ( ( comple2134933779557159616set_fm
        @ ( image_377115208147135190set_fm
          @ ^ [X3: produc340336539035504054m_rule] : ( insert_fm @ ( F @ X3 ) @ bot_bot_set_fm )
          @ A3 ) )
      = ( image_2303374963288105462ule_fm @ F @ A3 ) ) ).

% UNION_singleton_eq_range
thf(fact_1069_finite__UN__I,axiom,
    ! [A3: set_rule,B: rule > set_nat] :
      ( ( finite_finite_rule @ A3 )
     => ( ! [A5: rule] :
            ( ( member_rule3 @ A5 @ A3 )
           => ( finite_finite_nat @ ( B @ A5 ) ) )
       => ( finite_finite_nat @ ( comple7399068483239264473et_nat @ ( image_rule_set_nat @ B @ A3 ) ) ) ) ) ).

% finite_UN_I
thf(fact_1070_finite__UN__I,axiom,
    ! [A3: set_fm,B: fm > set_nat] :
      ( ( finite_finite_fm @ A3 )
     => ( ! [A5: fm] :
            ( ( member_fm3 @ A5 @ A3 )
           => ( finite_finite_nat @ ( B @ A5 ) ) )
       => ( finite_finite_nat @ ( comple7399068483239264473et_nat @ ( image_fm_set_nat @ B @ A3 ) ) ) ) ) ).

% finite_UN_I
thf(fact_1071_finite__UN__I,axiom,
    ! [A3: set_o,B: $o > set_nat] :
      ( ( finite_finite_o @ A3 )
     => ( ! [A5: $o] :
            ( ( member_o3 @ A5 @ A3 )
           => ( finite_finite_nat @ ( B @ A5 ) ) )
       => ( finite_finite_nat @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ B @ A3 ) ) ) ) ) ).

% finite_UN_I
thf(fact_1072_finite__UN__I,axiom,
    ! [A3: set_nat,B: nat > set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ! [A5: nat] :
            ( ( member_nat3 @ A5 @ A3 )
           => ( finite_finite_nat @ ( B @ A5 ) ) )
       => ( finite_finite_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B @ A3 ) ) ) ) ) ).

% finite_UN_I
thf(fact_1073_finite__UN__I,axiom,
    ! [A3: set_rule,B: rule > set_fm] :
      ( ( finite_finite_rule @ A3 )
     => ( ! [A5: rule] :
            ( ( member_rule3 @ A5 @ A3 )
           => ( finite_finite_fm @ ( B @ A5 ) ) )
       => ( finite_finite_fm @ ( comple2134933779557159616set_fm @ ( image_rule_set_fm @ B @ A3 ) ) ) ) ) ).

% finite_UN_I
thf(fact_1074_finite__UN__I,axiom,
    ! [A3: set_fm,B: fm > set_fm] :
      ( ( finite_finite_fm @ A3 )
     => ( ! [A5: fm] :
            ( ( member_fm3 @ A5 @ A3 )
           => ( finite_finite_fm @ ( B @ A5 ) ) )
       => ( finite_finite_fm @ ( comple2134933779557159616set_fm @ ( image_fm_set_fm @ B @ A3 ) ) ) ) ) ).

% finite_UN_I
thf(fact_1075_finite__UN__I,axiom,
    ! [A3: set_o,B: $o > set_fm] :
      ( ( finite_finite_o @ A3 )
     => ( ! [A5: $o] :
            ( ( member_o3 @ A5 @ A3 )
           => ( finite_finite_fm @ ( B @ A5 ) ) )
       => ( finite_finite_fm @ ( comple2134933779557159616set_fm @ ( image_o_set_fm @ B @ A3 ) ) ) ) ) ).

% finite_UN_I
thf(fact_1076_finite__UN__I,axiom,
    ! [A3: set_nat,B: nat > set_fm] :
      ( ( finite_finite_nat @ A3 )
     => ( ! [A5: nat] :
            ( ( member_nat3 @ A5 @ A3 )
           => ( finite_finite_fm @ ( B @ A5 ) ) )
       => ( finite_finite_fm @ ( comple2134933779557159616set_fm @ ( image_nat_set_fm @ B @ A3 ) ) ) ) ) ).

% finite_UN_I
thf(fact_1077_finite__UN__I,axiom,
    ! [A3: set_list_fm,B: list_fm > set_nat] :
      ( ( finite834038518558952994ist_fm @ A3 )
     => ( ! [A5: list_fm] :
            ( ( member_list_fm3 @ A5 @ A3 )
           => ( finite_finite_nat @ ( B @ A5 ) ) )
       => ( finite_finite_nat @ ( comple7399068483239264473et_nat @ ( image_3154564810619571642et_nat @ B @ A3 ) ) ) ) ) ).

% finite_UN_I
thf(fact_1078_finite__UN__I,axiom,
    ! [A3: set_list_fm,B: list_fm > set_fm] :
      ( ( finite834038518558952994ist_fm @ A3 )
     => ( ! [A5: list_fm] :
            ( ( member_list_fm3 @ A5 @ A3 )
           => ( finite_finite_fm @ ( B @ A5 ) ) )
       => ( finite_finite_fm @ ( comple2134933779557159616set_fm @ ( image_list_fm_set_fm @ B @ A3 ) ) ) ) ) ).

% finite_UN_I
thf(fact_1079_finite__UN,axiom,
    ! [A3: set_nat,B: nat > set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( finite_finite_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B @ A3 ) ) )
        = ( ! [X3: nat] :
              ( ( member_nat3 @ X3 @ A3 )
             => ( finite_finite_nat @ ( B @ X3 ) ) ) ) ) ) ).

% finite_UN
thf(fact_1080_finite__UN,axiom,
    ! [A3: set_Pr1822751329126368876m_rule,B: produc340336539035504054m_rule > set_fm] :
      ( ( finite3934748277521269581m_rule @ A3 )
     => ( ( finite_finite_fm @ ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ B @ A3 ) ) )
        = ( ! [X3: produc340336539035504054m_rule] :
              ( ( member7231649785386036813m_rule @ X3 @ A3 )
             => ( finite_finite_fm @ ( B @ X3 ) ) ) ) ) ) ).

% finite_UN
thf(fact_1081_finite__UN,axiom,
    ! [A3: set_nat,B: nat > set_fm] :
      ( ( finite_finite_nat @ A3 )
     => ( ( finite_finite_fm @ ( comple2134933779557159616set_fm @ ( image_nat_set_fm @ B @ A3 ) ) )
        = ( ! [X3: nat] :
              ( ( member_nat3 @ X3 @ A3 )
             => ( finite_finite_fm @ ( B @ X3 ) ) ) ) ) ) ).

% finite_UN
thf(fact_1082_ccpo__Sup__singleton,axiom,
    ! [X: set_fm] :
      ( ( comple2134933779557159616set_fm @ ( insert_set_fm @ X @ bot_bot_set_set_fm ) )
      = X ) ).

% ccpo_Sup_singleton
thf(fact_1083_ccpo__Sup__singleton,axiom,
    ! [X: $o] :
      ( ( complete_Sup_Sup_o @ ( insert_o @ X @ bot_bot_set_o ) )
      = X ) ).

% ccpo_Sup_singleton
thf(fact_1084_finite__Un,axiom,
    ! [F2: set_nat,G2: set_nat] :
      ( ( finite_finite_nat @ ( sup_sup_set_nat @ F2 @ G2 ) )
      = ( ( finite_finite_nat @ F2 )
        & ( finite_finite_nat @ G2 ) ) ) ).

% finite_Un
thf(fact_1085_finite__Union,axiom,
    ! [A3: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ A3 )
     => ( ! [M4: set_nat] :
            ( ( member_set_nat @ M4 @ A3 )
           => ( finite_finite_nat @ M4 ) )
       => ( finite_finite_nat @ ( comple7399068483239264473et_nat @ A3 ) ) ) ) ).

% finite_Union
thf(fact_1086_finite__Union,axiom,
    ! [A3: set_set_fm] :
      ( ( finite_finite_set_fm @ A3 )
     => ( ! [M4: set_fm] :
            ( ( member_set_fm @ M4 @ A3 )
           => ( finite_finite_fm @ M4 ) )
       => ( finite_finite_fm @ ( comple2134933779557159616set_fm @ A3 ) ) ) ) ).

% finite_Union
thf(fact_1087_finite__imageI,axiom,
    ! [F2: set_Pr1822751329126368876m_rule,H2: produc340336539035504054m_rule > set_fm] :
      ( ( finite3934748277521269581m_rule @ F2 )
     => ( finite_finite_set_fm @ ( image_377115208147135190set_fm @ H2 @ F2 ) ) ) ).

% finite_imageI
thf(fact_1088_finite__imageI,axiom,
    ! [F2: set_nat,H2: nat > nat] :
      ( ( finite_finite_nat @ F2 )
     => ( finite_finite_nat @ ( image_nat_nat @ H2 @ F2 ) ) ) ).

% finite_imageI
thf(fact_1089_finite__Sup__in,axiom,
    ! [A3: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ A3 )
     => ( ( A3 != bot_bot_set_set_nat )
       => ( ! [X2: set_nat,Y4: set_nat] :
              ( ( member_set_nat @ X2 @ A3 )
             => ( ( member_set_nat @ Y4 @ A3 )
               => ( member_set_nat @ ( sup_sup_set_nat @ X2 @ Y4 ) @ A3 ) ) )
         => ( member_set_nat @ ( comple7399068483239264473et_nat @ A3 ) @ A3 ) ) ) ) ).

% finite_Sup_in
thf(fact_1090_finite__Sup__in,axiom,
    ! [A3: set_set_fm] :
      ( ( finite_finite_set_fm @ A3 )
     => ( ( A3 != bot_bot_set_set_fm )
       => ( ! [X2: set_fm,Y4: set_fm] :
              ( ( member_set_fm @ X2 @ A3 )
             => ( ( member_set_fm @ Y4 @ A3 )
               => ( member_set_fm @ ( sup_sup_set_fm @ X2 @ Y4 ) @ A3 ) ) )
         => ( member_set_fm @ ( comple2134933779557159616set_fm @ A3 ) @ A3 ) ) ) ) ).

% finite_Sup_in
thf(fact_1091_finite__Sup__in,axiom,
    ! [A3: set_o] :
      ( ( finite_finite_o @ A3 )
     => ( ( A3 != bot_bot_set_o )
       => ( ! [X2: $o,Y4: $o] :
              ( ( member_o3 @ X2 @ A3 )
             => ( ( member_o3 @ Y4 @ A3 )
               => ( member_o3 @ ( sup_sup_o @ X2 @ Y4 ) @ A3 ) ) )
         => ( member_o3 @ ( complete_Sup_Sup_o @ A3 ) @ A3 ) ) ) ) ).

% finite_Sup_in
thf(fact_1092_finite__Collect__conjI,axiom,
    ! [P3: nat > $o,Q2: nat > $o] :
      ( ( ( finite_finite_nat @ ( collect_nat @ P3 ) )
        | ( finite_finite_nat @ ( collect_nat @ Q2 ) ) )
     => ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X3: nat] :
              ( ( P3 @ X3 )
              & ( Q2 @ X3 ) ) ) ) ) ).

% finite_Collect_conjI
thf(fact_1093_finite__Collect__disjI,axiom,
    ! [P3: nat > $o,Q2: nat > $o] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X3: nat] :
              ( ( P3 @ X3 )
              | ( Q2 @ X3 ) ) ) )
      = ( ( finite_finite_nat @ ( collect_nat @ P3 ) )
        & ( finite_finite_nat @ ( collect_nat @ Q2 ) ) ) ) ).

% finite_Collect_disjI
thf(fact_1094_finite__Collect__less__nat,axiom,
    ! [K: nat] :
      ( finite_finite_nat
      @ ( collect_nat
        @ ^ [N4: nat] : ( ord_less_nat @ N4 @ K ) ) ) ).

% finite_Collect_less_nat
thf(fact_1095_less__set__def,axiom,
    ( ord_less_set_rule
    = ( ^ [A4: set_rule,B3: set_rule] :
          ( ord_less_rule_o
          @ ^ [X3: rule] : ( member_rule3 @ X3 @ A4 )
          @ ^ [X3: rule] : ( member_rule3 @ X3 @ B3 ) ) ) ) ).

% less_set_def
thf(fact_1096_less__set__def,axiom,
    ( ord_less_set_list_fm
    = ( ^ [A4: set_list_fm,B3: set_list_fm] :
          ( ord_less_list_fm_o
          @ ^ [X3: list_fm] : ( member_list_fm3 @ X3 @ A4 )
          @ ^ [X3: list_fm] : ( member_list_fm3 @ X3 @ B3 ) ) ) ) ).

% less_set_def
thf(fact_1097_less__set__def,axiom,
    ( ord_less_set_fm
    = ( ^ [A4: set_fm,B3: set_fm] :
          ( ord_less_fm_o
          @ ^ [X3: fm] : ( member_fm3 @ X3 @ A4 )
          @ ^ [X3: fm] : ( member_fm3 @ X3 @ B3 ) ) ) ) ).

% less_set_def
thf(fact_1098_less__set__def,axiom,
    ( ord_less_set_o
    = ( ^ [A4: set_o,B3: set_o] :
          ( ord_less_o_o
          @ ^ [X3: $o] : ( member_o3 @ X3 @ A4 )
          @ ^ [X3: $o] : ( member_o3 @ X3 @ B3 ) ) ) ) ).

% less_set_def
thf(fact_1099_less__set__def,axiom,
    ( ord_less_set_nat
    = ( ^ [A4: set_nat,B3: set_nat] :
          ( ord_less_nat_o
          @ ^ [X3: nat] : ( member_nat3 @ X3 @ A4 )
          @ ^ [X3: nat] : ( member_nat3 @ X3 @ B3 ) ) ) ) ).

% less_set_def
thf(fact_1100_psubsetD,axiom,
    ! [A3: set_rule,B: set_rule,C2: rule] :
      ( ( ord_less_set_rule @ A3 @ B )
     => ( ( member_rule3 @ C2 @ A3 )
       => ( member_rule3 @ C2 @ B ) ) ) ).

% psubsetD
thf(fact_1101_psubsetD,axiom,
    ! [A3: set_list_fm,B: set_list_fm,C2: list_fm] :
      ( ( ord_less_set_list_fm @ A3 @ B )
     => ( ( member_list_fm3 @ C2 @ A3 )
       => ( member_list_fm3 @ C2 @ B ) ) ) ).

% psubsetD
thf(fact_1102_psubsetD,axiom,
    ! [A3: set_fm,B: set_fm,C2: fm] :
      ( ( ord_less_set_fm @ A3 @ B )
     => ( ( member_fm3 @ C2 @ A3 )
       => ( member_fm3 @ C2 @ B ) ) ) ).

% psubsetD
thf(fact_1103_psubsetD,axiom,
    ! [A3: set_o,B: set_o,C2: $o] :
      ( ( ord_less_set_o @ A3 @ B )
     => ( ( member_o3 @ C2 @ A3 )
       => ( member_o3 @ C2 @ B ) ) ) ).

% psubsetD
thf(fact_1104_psubsetD,axiom,
    ! [A3: set_nat,B: set_nat,C2: nat] :
      ( ( ord_less_set_nat @ A3 @ B )
     => ( ( member_nat3 @ C2 @ A3 )
       => ( member_nat3 @ C2 @ B ) ) ) ).

% psubsetD
thf(fact_1105_nat__seg__image__imp__finite,axiom,
    ! [A3: set_nat,F: nat > nat,N2: nat] :
      ( ( A3
        = ( image_nat_nat @ F
          @ ( collect_nat
            @ ^ [I: nat] : ( ord_less_nat @ I @ N2 ) ) ) )
     => ( finite_finite_nat @ A3 ) ) ).

% nat_seg_image_imp_finite
thf(fact_1106_finite__conv__nat__seg__image,axiom,
    ( finite_finite_nat
    = ( ^ [A4: set_nat] :
        ? [N4: nat,F3: nat > nat] :
          ( A4
          = ( image_nat_nat @ F3
            @ ( collect_nat
              @ ^ [I: nat] : ( ord_less_nat @ I @ N4 ) ) ) ) ) ) ).

% finite_conv_nat_seg_image
thf(fact_1107_not__finite__existsD,axiom,
    ! [P3: nat > $o] :
      ( ~ ( finite_finite_nat @ ( collect_nat @ P3 ) )
     => ? [X_1: nat] : ( P3 @ X_1 ) ) ).

% not_finite_existsD
thf(fact_1108_pigeonhole__infinite__rel,axiom,
    ! [A3: set_rule,B: set_nat,R2: rule > nat > $o] :
      ( ~ ( finite_finite_rule @ A3 )
     => ( ( finite_finite_nat @ B )
       => ( ! [X2: rule] :
              ( ( member_rule3 @ X2 @ A3 )
             => ? [Xa2: nat] :
                  ( ( member_nat3 @ Xa2 @ B )
                  & ( R2 @ X2 @ Xa2 ) ) )
         => ? [X2: nat] :
              ( ( member_nat3 @ X2 @ B )
              & ~ ( finite_finite_rule
                  @ ( collect_rule
                    @ ^ [A2: rule] :
                        ( ( member_rule3 @ A2 @ A3 )
                        & ( R2 @ A2 @ X2 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_1109_pigeonhole__infinite__rel,axiom,
    ! [A3: set_list_fm,B: set_nat,R2: list_fm > nat > $o] :
      ( ~ ( finite834038518558952994ist_fm @ A3 )
     => ( ( finite_finite_nat @ B )
       => ( ! [X2: list_fm] :
              ( ( member_list_fm3 @ X2 @ A3 )
             => ? [Xa2: nat] :
                  ( ( member_nat3 @ Xa2 @ B )
                  & ( R2 @ X2 @ Xa2 ) ) )
         => ? [X2: nat] :
              ( ( member_nat3 @ X2 @ B )
              & ~ ( finite834038518558952994ist_fm
                  @ ( collect_list_fm
                    @ ^ [A2: list_fm] :
                        ( ( member_list_fm3 @ A2 @ A3 )
                        & ( R2 @ A2 @ X2 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_1110_pigeonhole__infinite__rel,axiom,
    ! [A3: set_fm,B: set_nat,R2: fm > nat > $o] :
      ( ~ ( finite_finite_fm @ A3 )
     => ( ( finite_finite_nat @ B )
       => ( ! [X2: fm] :
              ( ( member_fm3 @ X2 @ A3 )
             => ? [Xa2: nat] :
                  ( ( member_nat3 @ Xa2 @ B )
                  & ( R2 @ X2 @ Xa2 ) ) )
         => ? [X2: nat] :
              ( ( member_nat3 @ X2 @ B )
              & ~ ( finite_finite_fm
                  @ ( collect_fm
                    @ ^ [A2: fm] :
                        ( ( member_fm3 @ A2 @ A3 )
                        & ( R2 @ A2 @ X2 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_1111_pigeonhole__infinite__rel,axiom,
    ! [A3: set_o,B: set_nat,R2: $o > nat > $o] :
      ( ~ ( finite_finite_o @ A3 )
     => ( ( finite_finite_nat @ B )
       => ( ! [X2: $o] :
              ( ( member_o3 @ X2 @ A3 )
             => ? [Xa2: nat] :
                  ( ( member_nat3 @ Xa2 @ B )
                  & ( R2 @ X2 @ Xa2 ) ) )
         => ? [X2: nat] :
              ( ( member_nat3 @ X2 @ B )
              & ~ ( finite_finite_o
                  @ ( collect_o
                    @ ^ [A2: $o] :
                        ( ( member_o3 @ A2 @ A3 )
                        & ( R2 @ A2 @ X2 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_1112_pigeonhole__infinite__rel,axiom,
    ! [A3: set_nat,B: set_nat,R2: nat > nat > $o] :
      ( ~ ( finite_finite_nat @ A3 )
     => ( ( finite_finite_nat @ B )
       => ( ! [X2: nat] :
              ( ( member_nat3 @ X2 @ A3 )
             => ? [Xa2: nat] :
                  ( ( member_nat3 @ Xa2 @ B )
                  & ( R2 @ X2 @ Xa2 ) ) )
         => ? [X2: nat] :
              ( ( member_nat3 @ X2 @ B )
              & ~ ( finite_finite_nat
                  @ ( collect_nat
                    @ ^ [A2: nat] :
                        ( ( member_nat3 @ A2 @ A3 )
                        & ( R2 @ A2 @ X2 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_1113_finite__UnionD,axiom,
    ! [A3: set_set_nat] :
      ( ( finite_finite_nat @ ( comple7399068483239264473et_nat @ A3 ) )
     => ( finite1152437895449049373et_nat @ A3 ) ) ).

% finite_UnionD
thf(fact_1114_finite__UnionD,axiom,
    ! [A3: set_set_fm] :
      ( ( finite_finite_fm @ ( comple2134933779557159616set_fm @ A3 ) )
     => ( finite_finite_set_fm @ A3 ) ) ).

% finite_UnionD
thf(fact_1115_infinite__Un,axiom,
    ! [S4: set_nat,T: set_nat] :
      ( ( ~ ( finite_finite_nat @ ( sup_sup_set_nat @ S4 @ T ) ) )
      = ( ~ ( finite_finite_nat @ S4 )
        | ~ ( finite_finite_nat @ T ) ) ) ).

% infinite_Un
thf(fact_1116_Un__infinite,axiom,
    ! [S4: set_nat,T: set_nat] :
      ( ~ ( finite_finite_nat @ S4 )
     => ~ ( finite_finite_nat @ ( sup_sup_set_nat @ S4 @ T ) ) ) ).

% Un_infinite
thf(fact_1117_finite__UnI,axiom,
    ! [F2: set_nat,G2: set_nat] :
      ( ( finite_finite_nat @ F2 )
     => ( ( finite_finite_nat @ G2 )
       => ( finite_finite_nat @ ( sup_sup_set_nat @ F2 @ G2 ) ) ) ) ).

% finite_UnI
thf(fact_1118_pigeonhole__infinite,axiom,
    ! [A3: set_Pr1822751329126368876m_rule,F: produc340336539035504054m_rule > set_fm] :
      ( ~ ( finite3934748277521269581m_rule @ A3 )
     => ( ( finite_finite_set_fm @ ( image_377115208147135190set_fm @ F @ A3 ) )
       => ? [X2: produc340336539035504054m_rule] :
            ( ( member7231649785386036813m_rule @ X2 @ A3 )
            & ~ ( finite3934748277521269581m_rule
                @ ( collec1009276759030335883m_rule
                  @ ^ [A2: produc340336539035504054m_rule] :
                      ( ( member7231649785386036813m_rule @ A2 @ A3 )
                      & ( ( F @ A2 )
                        = ( F @ X2 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite
thf(fact_1119_pigeonhole__infinite,axiom,
    ! [A3: set_rule,F: rule > nat] :
      ( ~ ( finite_finite_rule @ A3 )
     => ( ( finite_finite_nat @ ( image_rule_nat @ F @ A3 ) )
       => ? [X2: rule] :
            ( ( member_rule3 @ X2 @ A3 )
            & ~ ( finite_finite_rule
                @ ( collect_rule
                  @ ^ [A2: rule] :
                      ( ( member_rule3 @ A2 @ A3 )
                      & ( ( F @ A2 )
                        = ( F @ X2 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite
thf(fact_1120_pigeonhole__infinite,axiom,
    ! [A3: set_list_fm,F: list_fm > nat] :
      ( ~ ( finite834038518558952994ist_fm @ A3 )
     => ( ( finite_finite_nat @ ( image_list_fm_nat @ F @ A3 ) )
       => ? [X2: list_fm] :
            ( ( member_list_fm3 @ X2 @ A3 )
            & ~ ( finite834038518558952994ist_fm
                @ ( collect_list_fm
                  @ ^ [A2: list_fm] :
                      ( ( member_list_fm3 @ A2 @ A3 )
                      & ( ( F @ A2 )
                        = ( F @ X2 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite
thf(fact_1121_pigeonhole__infinite,axiom,
    ! [A3: set_fm,F: fm > nat] :
      ( ~ ( finite_finite_fm @ A3 )
     => ( ( finite_finite_nat @ ( image_fm_nat @ F @ A3 ) )
       => ? [X2: fm] :
            ( ( member_fm3 @ X2 @ A3 )
            & ~ ( finite_finite_fm
                @ ( collect_fm
                  @ ^ [A2: fm] :
                      ( ( member_fm3 @ A2 @ A3 )
                      & ( ( F @ A2 )
                        = ( F @ X2 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite
thf(fact_1122_pigeonhole__infinite,axiom,
    ! [A3: set_o,F: $o > nat] :
      ( ~ ( finite_finite_o @ A3 )
     => ( ( finite_finite_nat @ ( image_o_nat @ F @ A3 ) )
       => ? [X2: $o] :
            ( ( member_o3 @ X2 @ A3 )
            & ~ ( finite_finite_o
                @ ( collect_o
                  @ ^ [A2: $o] :
                      ( ( member_o3 @ A2 @ A3 )
                      & ( ( F @ A2 )
                        = ( F @ X2 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite
thf(fact_1123_pigeonhole__infinite,axiom,
    ! [A3: set_nat,F: nat > nat] :
      ( ~ ( finite_finite_nat @ A3 )
     => ( ( finite_finite_nat @ ( image_nat_nat @ F @ A3 ) )
       => ? [X2: nat] :
            ( ( member_nat3 @ X2 @ A3 )
            & ~ ( finite_finite_nat
                @ ( collect_nat
                  @ ^ [A2: nat] :
                      ( ( member_nat3 @ A2 @ A3 )
                      & ( ( F @ A2 )
                        = ( F @ X2 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite
thf(fact_1124_Saturated__def,axiom,
    ! [Steps: stream2709947120125613254m_rule] :
      ( ( abstra6097777249025082867ist_fm @ eff @ rules @ Steps )
      = ( ! [X3: rule] :
            ( ( member_rule3 @ X3 @ ( sset_rule @ rules ) )
           => ( abstra2533313685312581075ist_fm @ eff @ X3 @ Steps ) ) ) ) ).

% Saturated_def
thf(fact_1125_Union__maximal__sets,axiom,
    ! [F4: set_set_fm] :
      ( ( finite_finite_set_fm @ F4 )
     => ( ( comple2134933779557159616set_fm
          @ ( collect_set_fm
            @ ^ [T2: set_fm] :
                ( ( member_set_fm @ T2 @ F4 )
                & ! [X3: set_fm] :
                    ( ( member_set_fm @ X3 @ F4 )
                   => ~ ( ord_less_set_fm @ T2 @ X3 ) ) ) ) )
        = ( comple2134933779557159616set_fm @ F4 ) ) ) ).

% Union_maximal_sets
thf(fact_1126_is__singletonI,axiom,
    ! [X: rule] : ( is_singleton_rule @ ( insert_rule @ X @ bot_bot_set_rule ) ) ).

% is_singletonI
thf(fact_1127_is__singletonI,axiom,
    ! [X: $o] : ( is_singleton_o @ ( insert_o @ X @ bot_bot_set_o ) ) ).

% is_singletonI
thf(fact_1128_is__singletonI,axiom,
    ! [X: nat] : ( is_singleton_nat @ ( insert_nat @ X @ bot_bot_set_nat ) ) ).

% is_singletonI
thf(fact_1129_Id__on__fold,axiom,
    ! [A3: set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( id_on_nat @ A3 )
        = ( finite3745491028973389255at_nat
          @ ^ [X3: nat] : ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ X3 @ X3 ) )
          @ bot_bo2099793752762293965at_nat
          @ A3 ) ) ) ).

% Id_on_fold
thf(fact_1130_is__singletonI_H,axiom,
    ! [A3: set_list_fm] :
      ( ( A3 != bot_bot_set_list_fm )
     => ( ! [X2: list_fm,Y4: list_fm] :
            ( ( member_list_fm3 @ X2 @ A3 )
           => ( ( member_list_fm3 @ Y4 @ A3 )
             => ( X2 = Y4 ) ) )
       => ( is_singleton_list_fm @ A3 ) ) ) ).

% is_singletonI'
thf(fact_1131_is__singletonI_H,axiom,
    ! [A3: set_fm] :
      ( ( A3 != bot_bot_set_fm )
     => ( ! [X2: fm,Y4: fm] :
            ( ( member_fm3 @ X2 @ A3 )
           => ( ( member_fm3 @ Y4 @ A3 )
             => ( X2 = Y4 ) ) )
       => ( is_singleton_fm @ A3 ) ) ) ).

% is_singletonI'
thf(fact_1132_is__singletonI_H,axiom,
    ! [A3: set_rule] :
      ( ( A3 != bot_bot_set_rule )
     => ( ! [X2: rule,Y4: rule] :
            ( ( member_rule3 @ X2 @ A3 )
           => ( ( member_rule3 @ Y4 @ A3 )
             => ( X2 = Y4 ) ) )
       => ( is_singleton_rule @ A3 ) ) ) ).

% is_singletonI'
thf(fact_1133_is__singletonI_H,axiom,
    ! [A3: set_o] :
      ( ( A3 != bot_bot_set_o )
     => ( ! [X2: $o,Y4: $o] :
            ( ( member_o3 @ X2 @ A3 )
           => ( ( member_o3 @ Y4 @ A3 )
             => ( X2 = Y4 ) ) )
       => ( is_singleton_o @ A3 ) ) ) ).

% is_singletonI'
thf(fact_1134_is__singletonI_H,axiom,
    ! [A3: set_nat] :
      ( ( A3 != bot_bot_set_nat )
     => ( ! [X2: nat,Y4: nat] :
            ( ( member_nat3 @ X2 @ A3 )
           => ( ( member_nat3 @ Y4 @ A3 )
             => ( X2 = Y4 ) ) )
       => ( is_singleton_nat @ A3 ) ) ) ).

% is_singletonI'
thf(fact_1135_union__fold__insert,axiom,
    ! [A3: set_o,B: set_o] :
      ( ( finite_finite_o @ A3 )
     => ( ( sup_sup_set_o @ A3 @ B )
        = ( finite_fold_o_set_o @ insert_o @ B @ A3 ) ) ) ).

% union_fold_insert
thf(fact_1136_union__fold__insert,axiom,
    ! [A3: set_nat,B: set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( sup_sup_set_nat @ A3 @ B )
        = ( finite5529483035118572448et_nat @ insert_nat @ B @ A3 ) ) ) ).

% union_fold_insert
thf(fact_1137_sup__Sup__fold__sup,axiom,
    ! [A3: set_set_nat,B: set_nat] :
      ( ( finite1152437895449049373et_nat @ A3 )
     => ( ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ A3 ) @ B )
        = ( finite677925301803182934et_nat @ sup_sup_set_nat @ B @ A3 ) ) ) ).

% sup_Sup_fold_sup
thf(fact_1138_sup__Sup__fold__sup,axiom,
    ! [A3: set_set_fm,B: set_fm] :
      ( ( finite_finite_set_fm @ A3 )
     => ( ( sup_sup_set_fm @ ( comple2134933779557159616set_fm @ A3 ) @ B )
        = ( finite3569889474304629640set_fm @ sup_sup_set_fm @ B @ A3 ) ) ) ).

% sup_Sup_fold_sup
thf(fact_1139_sup__Sup__fold__sup,axiom,
    ! [A3: set_o,B: $o] :
      ( ( finite_finite_o @ A3 )
     => ( ( sup_sup_o @ ( complete_Sup_Sup_o @ A3 ) @ B )
        = ( finite_fold_o_o @ sup_sup_o @ B @ A3 ) ) ) ).

% sup_Sup_fold_sup
thf(fact_1140_NE__R,axiom,
    ( ( sset_rule @ rules )
   != bot_bot_set_rule ) ).

% NE_R
thf(fact_1141_Sup__fold__sup,axiom,
    ! [A3: set_set_rule] :
      ( ( finite6045212024966962911t_rule @ A3 )
     => ( ( comple7665822446944348827t_rule @ A3 )
        = ( finite9137686000838441434t_rule @ sup_sup_set_rule @ bot_bot_set_rule @ A3 ) ) ) ).

% Sup_fold_sup
thf(fact_1142_Sup__fold__sup,axiom,
    ! [A3: set_set_o] :
      ( ( finite_finite_set_o @ A3 )
     => ( ( comple90263536869209701_set_o @ A3 )
        = ( finite4337638375924247368_set_o @ sup_sup_set_o @ bot_bot_set_o @ A3 ) ) ) ).

% Sup_fold_sup
thf(fact_1143_Sup__fold__sup,axiom,
    ! [A3: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ A3 )
     => ( ( comple7399068483239264473et_nat @ A3 )
        = ( finite677925301803182934et_nat @ sup_sup_set_nat @ bot_bot_set_nat @ A3 ) ) ) ).

% Sup_fold_sup
thf(fact_1144_Sup__fold__sup,axiom,
    ! [A3: set_set_fm] :
      ( ( finite_finite_set_fm @ A3 )
     => ( ( comple2134933779557159616set_fm @ A3 )
        = ( finite3569889474304629640set_fm @ sup_sup_set_fm @ bot_bot_set_fm @ A3 ) ) ) ).

% Sup_fold_sup
thf(fact_1145_Sup__fold__sup,axiom,
    ! [A3: set_o] :
      ( ( finite_finite_o @ A3 )
     => ( ( complete_Sup_Sup_o @ A3 )
        = ( finite_fold_o_o @ sup_sup_o @ bot_bot_o @ A3 ) ) ) ).

% Sup_fold_sup
thf(fact_1146_image__fold__insert,axiom,
    ! [A3: set_Pr1822751329126368876m_rule,F: produc340336539035504054m_rule > set_fm] :
      ( ( finite3934748277521269581m_rule @ A3 )
     => ( ( image_377115208147135190set_fm @ F @ A3 )
        = ( finite760053451692937779set_fm
          @ ^ [K2: produc340336539035504054m_rule] : ( insert_set_fm @ ( F @ K2 ) )
          @ bot_bot_set_set_fm
          @ A3 ) ) ) ).

% image_fold_insert
thf(fact_1147_image__fold__insert,axiom,
    ! [A3: set_nat,F: nat > rule] :
      ( ( finite_finite_nat @ A3 )
     => ( ( image_nat_rule @ F @ A3 )
        = ( finite2403555171470153314t_rule
          @ ^ [K2: nat] : ( insert_rule @ ( F @ K2 ) )
          @ bot_bot_set_rule
          @ A3 ) ) ) ).

% image_fold_insert
thf(fact_1148_image__fold__insert,axiom,
    ! [A3: set_nat,F: nat > $o] :
      ( ( finite_finite_nat @ A3 )
     => ( ( image_nat_o @ F @ A3 )
        = ( finite3217087857726763998_set_o
          @ ^ [K2: nat] : ( insert_o @ ( F @ K2 ) )
          @ bot_bot_set_o
          @ A3 ) ) ) ).

% image_fold_insert
thf(fact_1149_image__fold__insert,axiom,
    ! [A3: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( image_nat_nat @ F @ A3 )
        = ( finite5529483035118572448et_nat
          @ ^ [K2: nat] : ( insert_nat @ ( F @ K2 ) )
          @ bot_bot_set_nat
          @ A3 ) ) ) ).

% image_fold_insert
thf(fact_1150_is__singleton__def,axiom,
    ( is_singleton_rule
    = ( ^ [A4: set_rule] :
        ? [X3: rule] :
          ( A4
          = ( insert_rule @ X3 @ bot_bot_set_rule ) ) ) ) ).

% is_singleton_def
thf(fact_1151_is__singleton__def,axiom,
    ( is_singleton_o
    = ( ^ [A4: set_o] :
        ? [X3: $o] :
          ( A4
          = ( insert_o @ X3 @ bot_bot_set_o ) ) ) ) ).

% is_singleton_def
thf(fact_1152_is__singleton__def,axiom,
    ( is_singleton_nat
    = ( ^ [A4: set_nat] :
        ? [X3: nat] :
          ( A4
          = ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ) ).

% is_singleton_def
thf(fact_1153_is__singletonE,axiom,
    ! [A3: set_rule] :
      ( ( is_singleton_rule @ A3 )
     => ~ ! [X2: rule] :
            ( A3
           != ( insert_rule @ X2 @ bot_bot_set_rule ) ) ) ).

% is_singletonE
thf(fact_1154_is__singletonE,axiom,
    ! [A3: set_o] :
      ( ( is_singleton_o @ A3 )
     => ~ ! [X2: $o] :
            ( A3
           != ( insert_o @ X2 @ bot_bot_set_o ) ) ) ).

% is_singletonE
thf(fact_1155_is__singletonE,axiom,
    ! [A3: set_nat] :
      ( ( is_singleton_nat @ A3 )
     => ~ ! [X2: nat] :
            ( A3
           != ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ).

% is_singletonE
thf(fact_1156_finite__M__bounded__by__nat,axiom,
    ! [P3: nat > $o,I4: nat] :
      ( finite_finite_nat
      @ ( collect_nat
        @ ^ [K2: nat] :
            ( ( P3 @ K2 )
            & ( ord_less_nat @ K2 @ I4 ) ) ) ) ).

% finite_M_bounded_by_nat
thf(fact_1157_Set__filter__fold,axiom,
    ! [A3: set_rule,P3: rule > $o] :
      ( ( finite_finite_rule @ A3 )
     => ( ( filter_rule @ P3 @ A3 )
        = ( finite8771924891862877220t_rule
          @ ^ [X3: rule,A6: set_rule] : ( if_set_rule @ ( P3 @ X3 ) @ ( insert_rule @ X3 @ A6 ) @ A6 )
          @ bot_bot_set_rule
          @ A3 ) ) ) ).

% Set_filter_fold
thf(fact_1158_Set__filter__fold,axiom,
    ! [A3: set_o,P3: $o > $o] :
      ( ( finite_finite_o @ A3 )
     => ( ( filter_o @ P3 @ A3 )
        = ( finite_fold_o_set_o
          @ ^ [X3: $o,A6: set_o] : ( if_set_o @ ( P3 @ X3 ) @ ( insert_o @ X3 @ A6 ) @ A6 )
          @ bot_bot_set_o
          @ A3 ) ) ) ).

% Set_filter_fold
thf(fact_1159_Set__filter__fold,axiom,
    ! [A3: set_nat,P3: nat > $o] :
      ( ( finite_finite_nat @ A3 )
     => ( ( filter_nat @ P3 @ A3 )
        = ( finite5529483035118572448et_nat
          @ ^ [X3: nat,A6: set_nat] : ( if_set_nat @ ( P3 @ X3 ) @ ( insert_nat @ X3 @ A6 ) @ A6 )
          @ bot_bot_set_nat
          @ A3 ) ) ) ).

% Set_filter_fold
thf(fact_1160_holds__eq2,axiom,
    ! [X: rule] :
      ( ( linear7354821883049305950s_rule
        @ ^ [Y2: rule] : ( Y2 = X ) )
      = ( linear5032321834864961386D_rule @ ( insert_rule @ X @ bot_bot_set_rule ) ) ) ).

% holds_eq2
thf(fact_1161_holds__eq2,axiom,
    ! [X: $o] :
      ( ( linear7433082607169829324olds_o
        @ ^ [Y2: $o] : ( Y2 = X ) )
      = ( linear3076992658408248000_HLD_o @ ( insert_o @ X @ bot_bot_set_o ) ) ) ).

% holds_eq2
thf(fact_1162_holds__eq2,axiom,
    ! [X: nat] :
      ( ( linear3087052677352696476ds_nat
        @ ^ [Y2: nat] : ( Y2 = X ) )
      = ( linear5551378545506465448LD_nat @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ).

% holds_eq2
thf(fact_1163_holds__eq2,axiom,
    ! [X: produc340336539035504054m_rule] :
      ( ( linear7925859179493631106m_rule
        @ ^ [Y2: produc340336539035504054m_rule] : ( Y2 = X ) )
      = ( linear2885739836753817742m_rule @ ( insert775845832097999846m_rule @ X @ bot_bo3426410157013319936m_rule ) ) ) ).

% holds_eq2
thf(fact_1164_is__singleton__the__elem,axiom,
    ( is_singleton_rule
    = ( ^ [A4: set_rule] :
          ( A4
          = ( insert_rule @ ( the_elem_rule @ A4 ) @ bot_bot_set_rule ) ) ) ) ).

% is_singleton_the_elem
thf(fact_1165_is__singleton__the__elem,axiom,
    ( is_singleton_o
    = ( ^ [A4: set_o] :
          ( A4
          = ( insert_o @ ( the_elem_o @ A4 ) @ bot_bot_set_o ) ) ) ) ).

% is_singleton_the_elem
thf(fact_1166_is__singleton__the__elem,axiom,
    ( is_singleton_nat
    = ( ^ [A4: set_nat] :
          ( A4
          = ( insert_nat @ ( the_elem_nat @ A4 ) @ bot_bot_set_nat ) ) ) ) ).

% is_singleton_the_elem
thf(fact_1167_member__filter,axiom,
    ! [X: rule,P3: rule > $o,A3: set_rule] :
      ( ( member_rule3 @ X @ ( filter_rule @ P3 @ A3 ) )
      = ( ( member_rule3 @ X @ A3 )
        & ( P3 @ X ) ) ) ).

% member_filter
thf(fact_1168_member__filter,axiom,
    ! [X: list_fm,P3: list_fm > $o,A3: set_list_fm] :
      ( ( member_list_fm3 @ X @ ( filter_list_fm @ P3 @ A3 ) )
      = ( ( member_list_fm3 @ X @ A3 )
        & ( P3 @ X ) ) ) ).

% member_filter
thf(fact_1169_member__filter,axiom,
    ! [X: fm,P3: fm > $o,A3: set_fm] :
      ( ( member_fm3 @ X @ ( filter_fm @ P3 @ A3 ) )
      = ( ( member_fm3 @ X @ A3 )
        & ( P3 @ X ) ) ) ).

% member_filter
thf(fact_1170_member__filter,axiom,
    ! [X: $o,P3: $o > $o,A3: set_o] :
      ( ( member_o3 @ X @ ( filter_o @ P3 @ A3 ) )
      = ( ( member_o3 @ X @ A3 )
        & ( P3 @ X ) ) ) ).

% member_filter
thf(fact_1171_member__filter,axiom,
    ! [X: nat,P3: nat > $o,A3: set_nat] :
      ( ( member_nat3 @ X @ ( filter_nat @ P3 @ A3 ) )
      = ( ( member_nat3 @ X @ A3 )
        & ( P3 @ X ) ) ) ).

% member_filter
thf(fact_1172_the__elem__eq,axiom,
    ! [X: rule] :
      ( ( the_elem_rule @ ( insert_rule @ X @ bot_bot_set_rule ) )
      = X ) ).

% the_elem_eq
thf(fact_1173_the__elem__eq,axiom,
    ! [X: $o] :
      ( ( the_elem_o @ ( insert_o @ X @ bot_bot_set_o ) )
      = X ) ).

% the_elem_eq
thf(fact_1174_the__elem__eq,axiom,
    ! [X: nat] :
      ( ( the_elem_nat @ ( insert_nat @ X @ bot_bot_set_nat ) )
      = X ) ).

% the_elem_eq
thf(fact_1175_holds__eq1,axiom,
    ! [X: rule] :
      ( ( linear7354821883049305950s_rule
        @ ( ^ [Y5: rule,Z5: rule] : ( Y5 = Z5 )
          @ X ) )
      = ( linear5032321834864961386D_rule @ ( insert_rule @ X @ bot_bot_set_rule ) ) ) ).

% holds_eq1
thf(fact_1176_holds__eq1,axiom,
    ! [X: $o] :
      ( ( linear7433082607169829324olds_o
        @ ( ^ [Y5: $o,Z5: $o] : ( Y5 = Z5 )
          @ X ) )
      = ( linear3076992658408248000_HLD_o @ ( insert_o @ X @ bot_bot_set_o ) ) ) ).

% holds_eq1
thf(fact_1177_holds__eq1,axiom,
    ! [X: nat] :
      ( ( linear3087052677352696476ds_nat
        @ ( ^ [Y5: nat,Z5: nat] : ( Y5 = Z5 )
          @ X ) )
      = ( linear5551378545506465448LD_nat @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ).

% holds_eq1
thf(fact_1178_holds__eq1,axiom,
    ! [X: produc340336539035504054m_rule] :
      ( ( linear7925859179493631106m_rule
        @ ( ^ [Y5: produc340336539035504054m_rule,Z5: produc340336539035504054m_rule] : ( Y5 = Z5 )
          @ X ) )
      = ( linear2885739836753817742m_rule @ ( insert775845832097999846m_rule @ X @ bot_bo3426410157013319936m_rule ) ) ) ).

% holds_eq1
thf(fact_1179_Set_Ofilter__def,axiom,
    ( filter_rule
    = ( ^ [P4: rule > $o,A4: set_rule] :
          ( collect_rule
          @ ^ [A2: rule] :
              ( ( member_rule3 @ A2 @ A4 )
              & ( P4 @ A2 ) ) ) ) ) ).

% Set.filter_def
thf(fact_1180_Set_Ofilter__def,axiom,
    ( filter_list_fm
    = ( ^ [P4: list_fm > $o,A4: set_list_fm] :
          ( collect_list_fm
          @ ^ [A2: list_fm] :
              ( ( member_list_fm3 @ A2 @ A4 )
              & ( P4 @ A2 ) ) ) ) ) ).

% Set.filter_def
thf(fact_1181_Set_Ofilter__def,axiom,
    ( filter_fm
    = ( ^ [P4: fm > $o,A4: set_fm] :
          ( collect_fm
          @ ^ [A2: fm] :
              ( ( member_fm3 @ A2 @ A4 )
              & ( P4 @ A2 ) ) ) ) ) ).

% Set.filter_def
thf(fact_1182_Set_Ofilter__def,axiom,
    ( filter_o
    = ( ^ [P4: $o > $o,A4: set_o] :
          ( collect_o
          @ ^ [A2: $o] :
              ( ( member_o3 @ A2 @ A4 )
              & ( P4 @ A2 ) ) ) ) ) ).

% Set.filter_def
thf(fact_1183_Set_Ofilter__def,axiom,
    ( filter_nat
    = ( ^ [P4: nat > $o,A4: set_nat] :
          ( collect_nat
          @ ^ [A2: nat] :
              ( ( member_nat3 @ A2 @ A4 )
              & ( P4 @ A2 ) ) ) ) ) ).

% Set.filter_def
thf(fact_1184_HLD__iff,axiom,
    ( linear4005088938741646753ist_fm
    = ( ^ [S3: set_list_fm,Omega2: stream_list_fm] : ( member_list_fm3 @ ( shd_list_fm @ Omega2 ) @ S3 ) ) ) ).

% HLD_iff
thf(fact_1185_HLD__iff,axiom,
    ( linear2973384439519985051HLD_fm
    = ( ^ [S3: set_fm,Omega2: stream_fm] : ( member_fm3 @ ( shd_fm @ Omega2 ) @ S3 ) ) ) ).

% HLD_iff
thf(fact_1186_HLD__iff,axiom,
    ( linear3076992658408248000_HLD_o
    = ( ^ [S3: set_o,Omega2: stream_o] : ( member_o3 @ ( shd_o @ Omega2 ) @ S3 ) ) ) ).

% HLD_iff
thf(fact_1187_HLD__iff,axiom,
    ( linear5551378545506465448LD_nat
    = ( ^ [S3: set_nat,Omega2: stream_nat] : ( member_nat3 @ ( shd_nat @ Omega2 ) @ S3 ) ) ) ).

% HLD_iff
thf(fact_1188_HLD__iff,axiom,
    ( linear2885739836753817742m_rule
    = ( ^ [S3: set_Pr1822751329126368876m_rule,Omega2: stream2709947120125613254m_rule] : ( member7231649785386036813m_rule @ ( shd_Pr4562317740776619530m_rule @ Omega2 ) @ S3 ) ) ) ).

% HLD_iff
thf(fact_1189_HLD__iff,axiom,
    ( linear5032321834864961386D_rule
    = ( ^ [S3: set_rule,Omega2: stream_rule] : ( member_rule3 @ ( shd_rule @ Omega2 ) @ S3 ) ) ) ).

% HLD_iff
thf(fact_1190_comp__fun__commute__filter__fold,axiom,
    ! [P3: $o > $o] :
      ( finite7927022897302369457_set_o
      @ ^ [X3: $o,A6: set_o] : ( if_set_o @ ( P3 @ X3 ) @ ( insert_o @ X3 @ A6 ) @ A6 ) ) ).

% comp_fun_commute_filter_fold
thf(fact_1191_comp__fun__commute__filter__fold,axiom,
    ! [P3: nat > $o] :
      ( finite4743618823921050025et_nat
      @ ^ [X3: nat,A6: set_nat] : ( if_set_nat @ ( P3 @ X3 ) @ ( insert_nat @ X3 @ A6 ) @ A6 ) ) ).

% comp_fun_commute_filter_fold
thf(fact_1192_HLD__def,axiom,
    ( linear5032321834864961386D_rule
    = ( ^ [S3: set_rule] :
          ( linear7354821883049305950s_rule
          @ ^ [X3: rule] : ( member_rule3 @ X3 @ S3 ) ) ) ) ).

% HLD_def
thf(fact_1193_HLD__def,axiom,
    ( linear4005088938741646753ist_fm
    = ( ^ [S3: set_list_fm] :
          ( linear6347412853745146029ist_fm
          @ ^ [X3: list_fm] : ( member_list_fm3 @ X3 @ S3 ) ) ) ) ).

% HLD_def
thf(fact_1194_HLD__def,axiom,
    ( linear2973384439519985051HLD_fm
    = ( ^ [S3: set_fm] :
          ( linear3702449374084777639lds_fm
          @ ^ [X3: fm] : ( member_fm3 @ X3 @ S3 ) ) ) ) ).

% HLD_def
thf(fact_1195_HLD__def,axiom,
    ( linear3076992658408248000_HLD_o
    = ( ^ [S3: set_o] :
          ( linear7433082607169829324olds_o
          @ ^ [X3: $o] : ( member_o3 @ X3 @ S3 ) ) ) ) ).

% HLD_def
thf(fact_1196_HLD__def,axiom,
    ( linear5551378545506465448LD_nat
    = ( ^ [S3: set_nat] :
          ( linear3087052677352696476ds_nat
          @ ^ [X3: nat] : ( member_nat3 @ X3 @ S3 ) ) ) ) ).

% HLD_def
thf(fact_1197_HLD__def,axiom,
    ( linear2885739836753817742m_rule
    = ( ^ [S3: set_Pr1822751329126368876m_rule] :
          ( linear7925859179493631106m_rule
          @ ^ [X3: produc340336539035504054m_rule] : ( member7231649785386036813m_rule @ X3 @ S3 ) ) ) ) ).

% HLD_def
thf(fact_1198_Ball__fold,axiom,
    ! [A3: set_nat,P3: nat > $o] :
      ( ( finite_finite_nat @ A3 )
     => ( ( ! [X3: nat] :
              ( ( member_nat3 @ X3 @ A3 )
             => ( P3 @ X3 ) ) )
        = ( finite_fold_nat_o
          @ ^ [K2: nat,S3: $o] :
              ( S3
              & ( P3 @ K2 ) )
          @ $true
          @ A3 ) ) ) ).

% Ball_fold
thf(fact_1199_the__elem__image__unique,axiom,
    ! [A3: set_Pr1822751329126368876m_rule,F: produc340336539035504054m_rule > set_fm,X: produc340336539035504054m_rule] :
      ( ( A3 != bot_bo3426410157013319936m_rule )
     => ( ! [Y4: produc340336539035504054m_rule] :
            ( ( member7231649785386036813m_rule @ Y4 @ A3 )
           => ( ( F @ Y4 )
              = ( F @ X ) ) )
       => ( ( the_elem_set_fm @ ( image_377115208147135190set_fm @ F @ A3 ) )
          = ( F @ X ) ) ) ) ).

% the_elem_image_unique
thf(fact_1200_the__elem__image__unique,axiom,
    ! [A3: set_nat,F: nat > nat,X: nat] :
      ( ( A3 != bot_bot_set_nat )
     => ( ! [Y4: nat] :
            ( ( member_nat3 @ Y4 @ A3 )
           => ( ( F @ Y4 )
              = ( F @ X ) ) )
       => ( ( the_elem_nat @ ( image_nat_nat @ F @ A3 ) )
          = ( F @ X ) ) ) ) ).

% the_elem_image_unique
thf(fact_1201_Pow__fold,axiom,
    ! [A3: set_rule] :
      ( ( finite_finite_rule @ A3 )
     => ( ( pow_rule @ A3 )
        = ( finite3753076602029976794t_rule
          @ ^ [X3: rule,A4: set_set_rule] : ( sup_sup_set_set_rule @ A4 @ ( image_6923031405177057687t_rule @ ( insert_rule @ X3 ) @ A4 ) )
          @ ( insert_set_rule @ bot_bot_set_rule @ bot_bot_set_set_rule )
          @ A3 ) ) ) ).

% Pow_fold
thf(fact_1202_Pow__fold,axiom,
    ! [A3: set_o] :
      ( ( finite_finite_o @ A3 )
     => ( ( pow_o @ A3 )
        = ( finite2775045464208429320_set_o
          @ ^ [X3: $o,A4: set_set_o] : ( sup_sup_set_set_o @ A4 @ ( image_set_o_set_o @ ( insert_o @ X3 ) @ A4 ) )
          @ ( insert_set_o @ bot_bot_set_o @ bot_bot_set_set_o )
          @ A3 ) ) ) ).

% Pow_fold
thf(fact_1203_Pow__fold,axiom,
    ! [A3: set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( pow_nat @ A3 )
        = ( finite4178521680790401110et_nat
          @ ^ [X3: nat,A4: set_set_nat] : ( sup_sup_set_set_nat @ A4 @ ( image_7916887816326733075et_nat @ ( insert_nat @ X3 ) @ A4 ) )
          @ ( insert_set_nat @ bot_bot_set_nat @ bot_bot_set_set_nat )
          @ A3 ) ) ) ).

% Pow_fold
thf(fact_1204_Field__insert,axiom,
    ! [A: rule,B2: rule,R: set_Pr7340090144965549541e_rule] :
      ( ( field_rule @ ( insert1446918821690244693e_rule @ ( produc5849431337705160893e_rule @ A @ B2 ) @ R ) )
      = ( sup_sup_set_rule @ ( insert_rule @ A @ ( insert_rule @ B2 @ bot_bot_set_rule ) ) @ ( field_rule @ R ) ) ) ).

% Field_insert
thf(fact_1205_Field__insert,axiom,
    ! [A: $o,B2: $o,R: set_Product_prod_o_o] :
      ( ( field_o @ ( insert6201435330877294327od_o_o @ ( product_Pair_o_o @ A @ B2 ) @ R ) )
      = ( sup_sup_set_o @ ( insert_o @ A @ ( insert_o @ B2 @ bot_bot_set_o ) ) @ ( field_o @ R ) ) ) ).

% Field_insert
thf(fact_1206_Field__insert,axiom,
    ! [A: nat,B2: nat,R: set_Pr1261947904930325089at_nat] :
      ( ( field_nat @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ A @ B2 ) @ R ) )
      = ( sup_sup_set_nat @ ( insert_nat @ A @ ( insert_nat @ B2 @ bot_bot_set_nat ) ) @ ( field_nat @ R ) ) ) ).

% Field_insert
thf(fact_1207_set__rec,axiom,
    ( set_fm2
    = ( rec_list_set_fm_fm @ bot_bot_set_fm
      @ ^ [X3: fm,Uu: list_fm] : ( insert_fm @ X3 ) ) ) ).

% set_rec
thf(fact_1208_set__rec,axiom,
    ( set_list_fm2
    = ( rec_li3562514433299826070ist_fm @ bot_bot_set_list_fm
      @ ^ [X3: list_fm,Uu: list_list_fm] : ( insert_list_fm @ X3 ) ) ) ).

% set_rec
thf(fact_1209_set__rec,axiom,
    ( set_tm2
    = ( rec_list_set_tm_tm @ bot_bot_set_tm
      @ ^ [X3: tm,Uu: list_tm] : ( insert_tm @ X3 ) ) ) ).

% set_rec
thf(fact_1210_set__rec,axiom,
    ( set_rule2
    = ( rec_li2005092410820253458e_rule @ bot_bot_set_rule
      @ ^ [X3: rule,Uu: list_rule] : ( insert_rule @ X3 ) ) ) ).

% set_rec
thf(fact_1211_set__rec,axiom,
    ( set_o2
    = ( rec_list_set_o_o @ bot_bot_set_o
      @ ^ [X3: $o,Uu: list_o] : ( insert_o @ X3 ) ) ) ).

% set_rec
thf(fact_1212_set__rec,axiom,
    ( set_nat2
    = ( rec_list_set_nat_nat @ bot_bot_set_nat
      @ ^ [X3: nat,Uu: list_nat] : ( insert_nat @ X3 ) ) ) ).

% set_rec
thf(fact_1213_ev__at__HLD__single__imp__snth,axiom,
    ! [X: produc340336539035504054m_rule,N2: nat,Omega: stream2709947120125613254m_rule] :
      ( ( linear1910601085820404996m_rule @ ( linear2885739836753817742m_rule @ ( insert775845832097999846m_rule @ X @ bot_bo3426410157013319936m_rule ) ) @ N2 @ Omega )
     => ( ( snth_P6679518042731451922m_rule @ Omega @ N2 )
        = X ) ) ).

% ev_at_HLD_single_imp_snth
thf(fact_1214_ev__at__HLD__single__imp__snth,axiom,
    ! [X: rule,N2: nat,Omega: stream_rule] :
      ( ( linear214229915476510176t_rule @ ( linear5032321834864961386D_rule @ ( insert_rule @ X @ bot_bot_set_rule ) ) @ N2 @ Omega )
     => ( ( snth_rule @ Omega @ N2 )
        = X ) ) ).

% ev_at_HLD_single_imp_snth
thf(fact_1215_ev__at__HLD__single__imp__snth,axiom,
    ! [X: $o,N2: nat,Omega: stream_o] :
      ( ( linear8449623917756454090v_at_o @ ( linear3076992658408248000_HLD_o @ ( insert_o @ X @ bot_bot_set_o ) ) @ N2 @ Omega )
     => ( ( snth_o @ Omega @ N2 )
        = X ) ) ).

% ev_at_HLD_single_imp_snth
thf(fact_1216_ev__at__HLD__single__imp__snth,axiom,
    ! [X: nat,N2: nat,Omega: stream_nat] :
      ( ( linear7589697474662451998at_nat @ ( linear5551378545506465448LD_nat @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ N2 @ Omega )
     => ( ( snth_nat @ Omega @ N2 )
        = X ) ) ).

% ev_at_HLD_single_imp_snth
thf(fact_1217_SUP__fold__sup,axiom,
    ! [A3: set_nat,F: nat > set_rule] :
      ( ( finite_finite_nat @ A3 )
     => ( ( comple7665822446944348827t_rule @ ( image_nat_set_rule @ F @ A3 ) )
        = ( finite2403555171470153314t_rule @ ( comp_s9031224967322063916le_nat @ sup_sup_set_rule @ F ) @ bot_bot_set_rule @ A3 ) ) ) ).

% SUP_fold_sup
thf(fact_1218_SUP__fold__sup,axiom,
    ! [A3: set_nat,F: nat > set_o] :
      ( ( finite_finite_nat @ A3 )
     => ( ( comple90263536869209701_set_o @ ( image_nat_set_o @ F @ A3 ) )
        = ( finite3217087857726763998_set_o @ ( comp_s4424444925706540684_o_nat @ sup_sup_set_o @ F ) @ bot_bot_set_o @ A3 ) ) ) ).

% SUP_fold_sup
thf(fact_1219_SUP__fold__sup,axiom,
    ! [A3: set_nat,F: nat > set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A3 ) )
        = ( finite5529483035118572448et_nat @ ( comp_s2351873599094224870at_nat @ sup_sup_set_nat @ F ) @ bot_bot_set_nat @ A3 ) ) ) ).

% SUP_fold_sup
thf(fact_1220_SUP__fold__sup,axiom,
    ! [A3: set_Pr1822751329126368876m_rule,F: produc340336539035504054m_rule > set_fm] :
      ( ( finite3934748277521269581m_rule @ A3 )
     => ( ( comple2134933779557159616set_fm @ ( image_377115208147135190set_fm @ F @ A3 ) )
        = ( finite7900053181138256211set_fm @ ( comp_s4147295670828738445m_rule @ sup_sup_set_fm @ F ) @ bot_bot_set_fm @ A3 ) ) ) ).

% SUP_fold_sup
thf(fact_1221_SUP__fold__sup,axiom,
    ! [A3: set_nat,F: nat > set_fm] :
      ( ( finite_finite_nat @ A3 )
     => ( ( comple2134933779557159616set_fm @ ( image_nat_set_fm @ F @ A3 ) )
        = ( finite2377096568234930617set_fm @ ( comp_s7470357424480357799fm_nat @ sup_sup_set_fm @ F ) @ bot_bot_set_fm @ A3 ) ) ) ).

% SUP_fold_sup
thf(fact_1222_SUP__fold__sup,axiom,
    ! [A3: set_nat,F: nat > $o] :
      ( ( finite_finite_nat @ A3 )
     => ( ( complete_Sup_Sup_o @ ( image_nat_o @ F @ A3 ) )
        = ( finite_fold_nat_o @ ( comp_o_o_o_nat @ sup_sup_o @ F ) @ bot_bot_o @ A3 ) ) ) ).

% SUP_fold_sup
thf(fact_1223_Union__Pow__eq,axiom,
    ! [A3: set_fm] :
      ( ( comple2134933779557159616set_fm @ ( pow_fm @ A3 ) )
      = A3 ) ).

% Union_Pow_eq
thf(fact_1224_Field__Un,axiom,
    ! [R: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( field_nat @ ( sup_su6327502436637775413at_nat @ R @ S ) )
      = ( sup_sup_set_nat @ ( field_nat @ R ) @ ( field_nat @ S ) ) ) ).

% Field_Un
thf(fact_1225_Field__Union,axiom,
    ! [R2: set_se7864838648521419175_fm_fm] :
      ( ( field_fm @ ( comple3052221160759688756_fm_fm @ R2 ) )
      = ( comple2134933779557159616set_fm @ ( image_1125575614967635793set_fm @ field_fm @ R2 ) ) ) ).

% Field_Union
thf(fact_1226_Pow__empty,axiom,
    ( ( pow_rule @ bot_bot_set_rule )
    = ( insert_set_rule @ bot_bot_set_rule @ bot_bot_set_set_rule ) ) ).

% Pow_empty
thf(fact_1227_Pow__empty,axiom,
    ( ( pow_o @ bot_bot_set_o )
    = ( insert_set_o @ bot_bot_set_o @ bot_bot_set_set_o ) ) ).

% Pow_empty
thf(fact_1228_Pow__empty,axiom,
    ( ( pow_nat @ bot_bot_set_nat )
    = ( insert_set_nat @ bot_bot_set_nat @ bot_bot_set_set_nat ) ) ).

% Pow_empty
thf(fact_1229_Pow__singleton__iff,axiom,
    ! [X4: set_rule,Y3: set_rule] :
      ( ( ( pow_rule @ X4 )
        = ( insert_set_rule @ Y3 @ bot_bot_set_set_rule ) )
      = ( ( X4 = bot_bot_set_rule )
        & ( Y3 = bot_bot_set_rule ) ) ) ).

% Pow_singleton_iff
thf(fact_1230_Pow__singleton__iff,axiom,
    ! [X4: set_o,Y3: set_o] :
      ( ( ( pow_o @ X4 )
        = ( insert_set_o @ Y3 @ bot_bot_set_set_o ) )
      = ( ( X4 = bot_bot_set_o )
        & ( Y3 = bot_bot_set_o ) ) ) ).

% Pow_singleton_iff
thf(fact_1231_Pow__singleton__iff,axiom,
    ! [X4: set_nat,Y3: set_nat] :
      ( ( ( pow_nat @ X4 )
        = ( insert_set_nat @ Y3 @ bot_bot_set_set_nat ) )
      = ( ( X4 = bot_bot_set_nat )
        & ( Y3 = bot_bot_set_nat ) ) ) ).

% Pow_singleton_iff
thf(fact_1232_Sup_OSUP__image,axiom,
    ! [Sup: set_set_fm > set_fm,G: set_fm > set_fm,F: produc340336539035504054m_rule > set_fm,A3: set_Pr1822751329126368876m_rule] :
      ( ( Sup @ ( image_set_fm_set_fm @ G @ ( image_377115208147135190set_fm @ F @ A3 ) ) )
      = ( Sup @ ( image_377115208147135190set_fm @ ( comp_s4523611521753001433m_rule @ G @ F ) @ A3 ) ) ) ).

% Sup.SUP_image
thf(fact_1233_Sup_OSUP__image,axiom,
    ! [Sup: set_set_fm > set_fm,G: produc340336539035504054m_rule > set_fm,F: produc340336539035504054m_rule > produc340336539035504054m_rule,A3: set_Pr1822751329126368876m_rule] :
      ( ( Sup @ ( image_377115208147135190set_fm @ G @ ( image_2498997284783760755m_rule @ F @ A3 ) ) )
      = ( Sup @ ( image_377115208147135190set_fm @ ( comp_P1105420087566651696m_rule @ G @ F ) @ A3 ) ) ) ).

% Sup.SUP_image
thf(fact_1234_Sup_OSUP__image,axiom,
    ! [Sup: set_nat > nat,G: nat > nat,F: nat > nat,A3: set_nat] :
      ( ( Sup @ ( image_nat_nat @ G @ ( image_nat_nat @ F @ A3 ) ) )
      = ( Sup @ ( image_nat_nat @ ( comp_nat_nat_nat @ G @ F ) @ A3 ) ) ) ).

% Sup.SUP_image
thf(fact_1235_Inf_OINF__image,axiom,
    ! [Inf: set_nat > nat,G: nat > nat,F: nat > nat,A3: set_nat] :
      ( ( Inf @ ( image_nat_nat @ G @ ( image_nat_nat @ F @ A3 ) ) )
      = ( Inf @ ( image_nat_nat @ ( comp_nat_nat_nat @ G @ F ) @ A3 ) ) ) ).

% Inf.INF_image
thf(fact_1236_parts__in__children,axiom,
    ! [P: fm,Z4: list_fm,Z3: list_fm,A3: list_tm,R: rule] :
      ( ( member_fm3 @ P @ ( set_fm2 @ Z4 ) )
     => ( ( member_list_fm3 @ Z3 @ ( set_list_fm2 @ ( children @ A3 @ R @ Z4 ) ) )
       => ? [B5: list_tm,Xs3: list_fm] :
            ( ( ord_less_eq_set_tm @ ( set_tm2 @ A3 ) @ ( set_tm2 @ B5 ) )
            & ( member_list_fm3 @ Xs3 @ ( set_list_fm2 @ ( parts @ B5 @ R @ P ) ) )
            & ( ord_less_eq_set_fm @ ( set_fm2 @ Xs3 ) @ ( set_fm2 @ Z3 ) ) ) ) ) ).

% parts_in_children
thf(fact_1237_epath__sdrop__ptms,axiom,
    ! [Steps: stream2709947120125613254m_rule,N2: nat] :
      ( ( abstra6789711989322986974ist_fm @ eff @ rules @ Steps )
     => ( ord_less_eq_set_tm @ ( set_tm2 @ ( ptms @ ( shd_Pr4562317740776619530m_rule @ Steps ) ) ) @ ( set_tm2 @ ( ptms @ ( shd_Pr4562317740776619530m_rule @ ( sdrop_8169176516188972301m_rule @ N2 @ Steps ) ) ) ) ) ) ).

% epath_sdrop_ptms
thf(fact_1238_finite__Collect__le__nat,axiom,
    ! [K: nat] :
      ( finite_finite_nat
      @ ( collect_nat
        @ ^ [N4: nat] : ( ord_less_eq_nat @ N4 @ K ) ) ) ).

% finite_Collect_le_nat
thf(fact_1239_finite__less__ub,axiom,
    ! [F: nat > nat,U2: nat] :
      ( ! [N: nat] : ( ord_less_eq_nat @ N @ ( F @ N ) )
     => ( finite_finite_nat
        @ ( collect_nat
          @ ^ [N4: nat] : ( ord_less_eq_nat @ ( F @ N4 ) @ U2 ) ) ) ) ).

% finite_less_ub
thf(fact_1240_fair__rules,axiom,
    abstra3799686578551160190r_rule @ rules @ rules ).

% fair_rules
thf(fact_1241_sdrop__fair,axiom,
    ! [Rs: stream_rule,M2: nat] :
      ( ( abstra3799686578551160190r_rule @ rules @ Rs )
     => ( abstra3799686578551160190r_rule @ rules @ ( sdrop_rule @ M2 @ Rs ) ) ) ).

% sdrop_fair
thf(fact_1242_pos,axiom,
    ! [Rs: stream_rule,R: rule] :
      ( ( abstra3799686578551160190r_rule @ rules @ Rs )
     => ( ( member_rule3 @ R @ ( sset_rule @ rules ) )
       => ( ( shd_rule @ ( sdrop_rule @ ( abstract_pos_rule @ Rs @ R ) @ Rs ) )
          = R ) ) ) ).

% pos
thf(fact_1243_enabled__def,axiom,
    ! [R: rule,S: produc6018962875968178549ist_fm] :
      ( ( abstra1707737187183579335ist_fm @ eff @ R @ S )
      = ( ? [X7: fset_P8989946509869081563ist_fm] : ( eff @ R @ S @ X7 ) ) ) ).

% enabled_def
thf(fact_1244_minWait__least,axiom,
    ! [N2: nat,Rs: stream_rule,S: produc6018962875968178549ist_fm] :
      ( ( abstra1707737187183579335ist_fm @ eff @ ( shd_rule @ ( sdrop_rule @ N2 @ Rs ) ) @ S )
     => ( ord_less_eq_nat @ ( abstra1963046427795717555ist_fm @ eff @ Rs @ S ) @ N2 ) ) ).

% minWait_least
thf(fact_1245_minWait__def,axiom,
    ! [Rs: stream_rule,S: produc6018962875968178549ist_fm] :
      ( ( abstra1963046427795717555ist_fm @ eff @ Rs @ S )
      = ( ord_Least_nat
        @ ^ [N4: nat] : ( abstra1707737187183579335ist_fm @ eff @ ( shd_rule @ ( sdrop_rule @ N4 @ Rs ) ) @ S ) ) ) ).

% minWait_def
thf(fact_1246_minWait__le__pos,axiom,
    ! [Rs: stream_rule,R: rule,S: produc6018962875968178549ist_fm] :
      ( ( abstra3799686578551160190r_rule @ rules @ Rs )
     => ( ( member_rule3 @ R @ ( sset_rule @ rules ) )
       => ( ( abstra1707737187183579335ist_fm @ eff @ R @ S )
         => ( ord_less_eq_nat @ ( abstra1963046427795717555ist_fm @ eff @ Rs @ S ) @ ( abstract_pos_rule @ Rs @ R ) ) ) ) ) ).

% minWait_le_pos
thf(fact_1247_minWait__ex,axiom,
    ! [S: produc6018962875968178549ist_fm,Rs: stream_rule] :
      ( ( member4699826688122452638ist_fm @ S @ top_to1730629564055774885ist_fm )
     => ( ( abstra3799686578551160190r_rule @ rules @ Rs )
       => ? [N: nat] : ( abstra1707737187183579335ist_fm @ eff @ ( shd_rule @ ( sdrop_rule @ N @ Rs ) ) @ S ) ) ) ).

% minWait_ex
thf(fact_1248_trim__fair,axiom,
    ! [S: produc6018962875968178549ist_fm,Rs: stream_rule] :
      ( ( member4699826688122452638ist_fm @ S @ top_to1730629564055774885ist_fm )
     => ( ( abstra3799686578551160190r_rule @ rules @ Rs )
       => ( abstra3799686578551160190r_rule @ rules @ ( abstra7806236140005899928ist_fm @ eff @ Rs @ S ) ) ) ) ).

% trim_fair
thf(fact_1249_RuleSystem__axioms,axiom,
    abstra5221733350967904376ist_fm @ eff @ rules @ top_to1730629564055774885ist_fm ).

% RuleSystem_axioms
thf(fact_1250_trim__alt,axiom,
    ! [S: produc6018962875968178549ist_fm,Rs: stream_rule] :
      ( ( member4699826688122452638ist_fm @ S @ top_to1730629564055774885ist_fm )
     => ( ( abstra3799686578551160190r_rule @ rules @ Rs )
       => ( ( abstra7806236140005899928ist_fm @ eff @ Rs @ S )
          = ( sdrop_rule @ ( abstra1963046427795717555ist_fm @ eff @ Rs @ S ) @ Rs ) ) ) ) ).

% trim_alt
thf(fact_1251_trim__def,axiom,
    ! [Rs: stream_rule,S: produc6018962875968178549ist_fm] :
      ( ( abstra7806236140005899928ist_fm @ eff @ Rs @ S )
      = ( sdrop_while_rule
        @ ^ [R3: rule] :
            ~ ( abstra1707737187183579335ist_fm @ eff @ R3 @ S )
        @ Rs ) ) ).

% trim_def
thf(fact_1252_trim__in__R,axiom,
    ! [S: produc6018962875968178549ist_fm,Rs: stream_rule] :
      ( ( member4699826688122452638ist_fm @ S @ top_to1730629564055774885ist_fm )
     => ( ( abstra3799686578551160190r_rule @ rules @ Rs )
       => ( member_rule3 @ ( shd_rule @ ( abstra7806236140005899928ist_fm @ eff @ Rs @ S ) ) @ ( sset_rule @ rules ) ) ) ) ).

% trim_in_R
thf(fact_1253_trim__enabled,axiom,
    ! [S: produc6018962875968178549ist_fm,Rs: stream_rule] :
      ( ( member4699826688122452638ist_fm @ S @ top_to1730629564055774885ist_fm )
     => ( ( abstra3799686578551160190r_rule @ rules @ Rs )
       => ( abstra1707737187183579335ist_fm @ eff @ ( shd_rule @ ( abstra7806236140005899928ist_fm @ eff @ Rs @ S ) ) @ S ) ) ) ).

% trim_enabled
thf(fact_1254_enabled__R,axiom,
    ! [S: produc6018962875968178549ist_fm] :
      ( ( member4699826688122452638ist_fm @ S @ top_to1730629564055774885ist_fm )
     => ? [X2: rule] :
          ( ( member_rule3 @ X2 @ ( sset_rule @ rules ) )
          & ? [X_1: fset_P8989946509869081563ist_fm] : ( eff @ X2 @ S @ X_1 ) ) ) ).

% enabled_R
thf(fact_1255_all__rules__persistent,axiom,
    ! [R4: rule] :
      ( ( member_rule3 @ R4 @ ( sset_rule @ rules ) )
     => ( abstra5255361903751151037ist_fm @ eff @ rules @ top_to1730629564055774885ist_fm @ R4 ) ) ).

% all_rules_persistent
thf(fact_1256_per,axiom,
    ! [R: rule] :
      ( ( member_rule3 @ R @ ( sset_rule @ rules ) )
     => ( abstra5255361903751151037ist_fm @ eff @ rules @ top_to1730629564055774885ist_fm @ R ) ) ).

% per
thf(fact_1257_PersistentRuleSystem__axioms,axiom,
    abstra3967921542344661089ist_fm @ eff @ rules @ top_to1730629564055774885ist_fm ).

% PersistentRuleSystem_axioms
thf(fact_1258_card__Collect__less__nat,axiom,
    ! [N2: nat] :
      ( ( finite_card_nat
        @ ( collect_nat
          @ ^ [I: nat] : ( ord_less_nat @ I @ N2 ) ) )
      = N2 ) ).

% card_Collect_less_nat
thf(fact_1259_UNIV__bool,axiom,
    ( top_top_set_o
    = ( insert_o @ $false @ ( insert_o @ $true @ bot_bot_set_o ) ) ) ).

% UNIV_bool
thf(fact_1260_Sup__nat__empty,axiom,
    ( ( complete_Sup_Sup_nat @ bot_bot_set_nat )
    = zero_zero_nat ) ).

% Sup_nat_empty
thf(fact_1261_ex__nat__less__eq,axiom,
    ! [N2: nat,P3: nat > $o] :
      ( ( ? [M: nat] :
            ( ( ord_less_nat @ M @ N2 )
            & ( P3 @ M ) ) )
      = ( ? [X3: nat] :
            ( ( member_nat3 @ X3 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) )
            & ( P3 @ X3 ) ) ) ) ).

% ex_nat_less_eq
thf(fact_1262_all__nat__less__eq,axiom,
    ! [N2: nat,P3: nat > $o] :
      ( ( ! [M: nat] :
            ( ( ord_less_nat @ M @ N2 )
           => ( P3 @ M ) ) )
      = ( ! [X3: nat] :
            ( ( member_nat3 @ X3 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) )
           => ( P3 @ X3 ) ) ) ) ).

% all_nat_less_eq
thf(fact_1263_atLeastLessThan__add__Un,axiom,
    ! [I4: nat,J2: nat,K: nat] :
      ( ( ord_less_eq_nat @ I4 @ J2 )
     => ( ( set_or4665077453230672383an_nat @ I4 @ ( plus_plus_nat @ J2 @ K ) )
        = ( sup_sup_set_nat @ ( set_or4665077453230672383an_nat @ I4 @ J2 ) @ ( set_or4665077453230672383an_nat @ J2 @ ( plus_plus_nat @ J2 @ K ) ) ) ) ) ).

% atLeastLessThan_add_Un
thf(fact_1264_image__Suc__atLeastLessThan,axiom,
    ! [I4: nat,J2: nat] :
      ( ( image_nat_nat @ suc @ ( set_or4665077453230672383an_nat @ I4 @ J2 ) )
      = ( set_or4665077453230672383an_nat @ ( suc @ I4 ) @ ( suc @ J2 ) ) ) ).

% image_Suc_atLeastLessThan
thf(fact_1265_card__Collect__le__nat,axiom,
    ! [N2: nat] :
      ( ( finite_card_nat
        @ ( collect_nat
          @ ^ [I: nat] : ( ord_less_eq_nat @ I @ N2 ) ) )
      = ( suc @ N2 ) ) ).

% card_Collect_le_nat
thf(fact_1266_UNIV__nat__eq,axiom,
    ( top_top_set_nat
    = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ top_top_set_nat ) ) ) ).

% UNIV_nat_eq
thf(fact_1267_zero__notin__Suc__image,axiom,
    ! [A3: set_nat] :
      ~ ( member_nat3 @ zero_zero_nat @ ( image_nat_nat @ suc @ A3 ) ) ).

% zero_notin_Suc_image
thf(fact_1268_atLeast0__lessThan__Suc__eq__insert__0,axiom,
    ! [N2: nat] :
      ( ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N2 ) )
      = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) ) ) ).

% atLeast0_lessThan_Suc_eq_insert_0
thf(fact_1269_card__less,axiom,
    ! [M3: set_nat,I4: nat] :
      ( ( member_nat3 @ zero_zero_nat @ M3 )
     => ( ( finite_card_nat
          @ ( collect_nat
            @ ^ [K2: nat] :
                ( ( member_nat3 @ K2 @ M3 )
                & ( ord_less_nat @ K2 @ ( suc @ I4 ) ) ) ) )
       != zero_zero_nat ) ) ).

% card_less
thf(fact_1270_card__less__Suc,axiom,
    ! [M3: set_nat,I4: nat] :
      ( ( member_nat3 @ zero_zero_nat @ M3 )
     => ( ( suc
          @ ( finite_card_nat
            @ ( collect_nat
              @ ^ [K2: nat] :
                  ( ( member_nat3 @ ( suc @ K2 ) @ M3 )
                  & ( ord_less_nat @ K2 @ I4 ) ) ) ) )
        = ( finite_card_nat
          @ ( collect_nat
            @ ^ [K2: nat] :
                ( ( member_nat3 @ K2 @ M3 )
                & ( ord_less_nat @ K2 @ ( suc @ I4 ) ) ) ) ) ) ) ).

% card_less_Suc
thf(fact_1271_card__less__Suc2,axiom,
    ! [M3: set_nat,I4: nat] :
      ( ~ ( member_nat3 @ zero_zero_nat @ M3 )
     => ( ( finite_card_nat
          @ ( collect_nat
            @ ^ [K2: nat] :
                ( ( member_nat3 @ ( suc @ K2 ) @ M3 )
                & ( ord_less_nat @ K2 @ I4 ) ) ) )
        = ( finite_card_nat
          @ ( collect_nat
            @ ^ [K2: nat] :
                ( ( member_nat3 @ K2 @ M3 )
                & ( ord_less_nat @ K2 @ ( suc @ I4 ) ) ) ) ) ) ) ).

% card_less_Suc2
thf(fact_1272_Least__Suc,axiom,
    ! [P3: nat > $o,N2: nat] :
      ( ( P3 @ N2 )
     => ( ~ ( P3 @ zero_zero_nat )
       => ( ( ord_Least_nat @ P3 )
          = ( suc
            @ ( ord_Least_nat
              @ ^ [M: nat] : ( P3 @ ( suc @ M ) ) ) ) ) ) ) ).

% Least_Suc
thf(fact_1273_range__enumerate,axiom,
    ! [S4: set_nat] :
      ( ~ ( finite_finite_nat @ S4 )
     => ( ( image_nat_nat @ ( infini8530281810654367211te_nat @ S4 ) @ top_top_set_nat )
        = S4 ) ) ).

% range_enumerate
thf(fact_1274_image__minus__const__atLeastLessThan__nat,axiom,
    ! [C2: nat,Y: nat,X: nat] :
      ( ( ( ord_less_nat @ C2 @ Y )
       => ( ( image_nat_nat
            @ ^ [I: nat] : ( minus_minus_nat @ I @ C2 )
            @ ( set_or4665077453230672383an_nat @ X @ Y ) )
          = ( set_or4665077453230672383an_nat @ ( minus_minus_nat @ X @ C2 ) @ ( minus_minus_nat @ Y @ C2 ) ) ) )
      & ( ~ ( ord_less_nat @ C2 @ Y )
       => ( ( ( ord_less_nat @ X @ Y )
           => ( ( image_nat_nat
                @ ^ [I: nat] : ( minus_minus_nat @ I @ C2 )
                @ ( set_or4665077453230672383an_nat @ X @ Y ) )
              = ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) )
          & ( ~ ( ord_less_nat @ X @ Y )
           => ( ( image_nat_nat
                @ ^ [I: nat] : ( minus_minus_nat @ I @ C2 )
                @ ( set_or4665077453230672383an_nat @ X @ Y ) )
              = bot_bot_set_nat ) ) ) ) ) ).

% image_minus_const_atLeastLessThan_nat
thf(fact_1275_stake__pos__minWait,axiom,
    ! [Rs: stream_rule,S: produc6018962875968178549ist_fm,R: rule] :
      ( ( abstra3799686578551160190r_rule @ rules @ Rs )
     => ( ( ord_less_nat @ ( abstra1963046427795717555ist_fm @ eff @ Rs @ S ) @ ( abstract_pos_rule @ Rs @ R ) )
       => ( ( member_rule3 @ R @ ( sset_rule @ rules ) )
         => ( ( member4699826688122452638ist_fm @ S @ top_to1730629564055774885ist_fm )
           => ( ( abstract_pos_rule @ ( stl_rule @ ( abstra7806236140005899928ist_fm @ eff @ Rs @ S ) ) @ R )
              = ( minus_minus_nat @ ( abstract_pos_rule @ Rs @ R ) @ ( suc @ ( abstra1963046427795717555ist_fm @ eff @ Rs @ S ) ) ) ) ) ) ) ) ).

% stake_pos_minWait
thf(fact_1276_fair__stl,axiom,
    ! [Rs: stream_rule] :
      ( ( abstra3799686578551160190r_rule @ rules @ Rs )
     => ( abstra3799686578551160190r_rule @ rules @ ( stl_rule @ Rs ) ) ) ).

% fair_stl

% Helper facts (7)
thf(help_If_2_1_If_001t__Set__Oset_I_Eo_J_T,axiom,
    ! [X: set_o,Y: set_o] :
      ( ( if_set_o @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Set__Oset_I_Eo_J_T,axiom,
    ! [X: set_o,Y: set_o] :
      ( ( if_set_o @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( if_set_nat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( if_set_nat @ $true @ X @ Y )
      = X ) ).

thf(help_If_3_1_If_001t__Set__Oset_It__Prover__Orule_J_T,axiom,
    ! [P3: $o] :
      ( ( P3 = $true )
      | ( P3 = $false ) ) ).

thf(help_If_2_1_If_001t__Set__Oset_It__Prover__Orule_J_T,axiom,
    ! [X: set_rule,Y: set_rule] :
      ( ( if_set_rule @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Set__Oset_It__Prover__Orule_J_T,axiom,
    ! [X: set_rule,Y: set_rule] :
      ( ( if_set_rule @ $true @ X @ Y )
      = X ) ).

% Conjectures (1)
thf(conj_0,conjecture,
    ( list_a9038620553111681026m_rule
    @ ^ [Xs: produc340336539035504054m_rule] :
        ~ ( affects @ ( produc6514711208036230658m_rule @ Xs ) @ ( dis @ p @ q ) )
    @ pre ) ).

%------------------------------------------------------------------------------