TPTP Problem File: ITP089^1.p

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%------------------------------------------------------------------------------
% File     : ITP089^1 : TPTP v9.0.0. Released v7.5.0.
% Domain   : Interactive Theorem Proving
% Problem  : Sledgehammer KnowledgeKeysSecrets problem prob_396__3295470_1
% Version  : Especial.
% English  :

% Refs     : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
%          : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% Source   : [Des21]
% Names    : KnowledgeKeysSecrets/prob_396__3295470_1 [Des21]

% Status   : Theorem
% Rating   : 0.25 v9.0.0, 0.40 v8.2.0, 0.23 v8.1.0, 0.36 v7.5.0
% Syntax   : Number of formulae    :  447 ( 176 unt;  91 typ;   0 def)
%            Number of atoms       : 1273 ( 451 equ;   0 cnn)
%            Maximal formula atoms :   13 (   3 avg)
%            Number of connectives : 3885 ( 296   ~;  51   |; 125   &;2857   @)
%                                         (   0 <=>; 556  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   19 (   7 avg)
%            Number of types       :   14 (  13 usr)
%            Number of type conns  :  127 ( 127   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :   79 (  78 usr;  10 con; 0-3 aty)
%            Number of variables   :  994 (  56   ^; 893   !;  45   ?; 994   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This file was generated by Sledgehammer 2021-02-23 15:35:26.675
%------------------------------------------------------------------------------
% Could-be-implicit typings (13)
thf(ty_n_t__List__Olist_It__Secrecy____types__OExpression_J,type,
    list_S529945141ession: $tType ).

thf(ty_n_t__Set__Oset_It__Secrecy____types__OSecrets_J,type,
    set_Secrecy_Secrets: $tType ).

thf(ty_n_t__Set__Oset_It__Secrecy____types__OspecID_J,type,
    set_Secrecy_specID: $tType ).

thf(ty_n_t__Set__Oset_It__Secrecy____types__OchanID_J,type,
    set_Secrecy_chanID: $tType ).

thf(ty_n_t__Set__Oset_It__Secrecy____types__OKeys_J,type,
    set_Secrecy_Keys: $tType ).

thf(ty_n_t__Set__Oset_It__Secrecy____types__OKS_J,type,
    set_Secrecy_KS: $tType ).

thf(ty_n_t__Secrecy____types__OExpression,type,
    secrecy_Expression: $tType ).

thf(ty_n_t__Secrecy____types__OSecrets,type,
    secrecy_Secrets: $tType ).

thf(ty_n_t__Secrecy____types__OspecID,type,
    secrecy_specID: $tType ).

thf(ty_n_t__Secrecy____types__OchanID,type,
    secrecy_chanID: $tType ).

thf(ty_n_t__Secrecy____types__OKeys,type,
    secrecy_Keys: $tType ).

thf(ty_n_t__Secrecy____types__OKS,type,
    secrecy_KS: $tType ).

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

% Explicit typings (78)
thf(sy_c_CompLocalSecrets_OLocalSecrets,type,
    localSecrets: secrecy_specID > set_Secrecy_KS ).

thf(sy_c_KnowledgeKeysSecrets__Mirabelle__vytwmzrumu_OeoutKnowCorrect,type,
    knowle233900195orrect: secrecy_specID > secrecy_KS > $o ).

thf(sy_c_KnowledgeKeysSecrets__Mirabelle__vytwmzrumu_Oknow,type,
    knowle945212990e_know: secrecy_specID > secrecy_KS > $o ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Secrecy____types__OKS_J,type,
    sup_su1301156759ecy_KS: set_Secrecy_KS > set_Secrecy_KS > set_Secrecy_KS ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Secrecy____types__OKeys_J,type,
    sup_su552545507y_Keys: set_Secrecy_Keys > set_Secrecy_Keys > set_Secrecy_Keys ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Secrecy____types__OSecrets_J,type,
    sup_su1029080330ecrets: set_Secrecy_Secrets > set_Secrecy_Secrets > set_Secrecy_Secrets ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Secrecy____types__OchanID_J,type,
    sup_su370684444chanID: set_Secrecy_chanID > set_Secrecy_chanID > set_Secrecy_chanID ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Secrecy____types__OspecID_J,type,
    sup_su1712599557specID: set_Secrecy_specID > set_Secrecy_specID > set_Secrecy_specID ).

thf(sy_c_ListExtras_Omem_001t__Secrecy____types__OExpression,type,
    mem_Se1967555118ession: secrecy_Expression > list_S529945141ession > $o ).

thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Secrecy____types__OKS_M_Eo_J,type,
    bot_bot_Secrecy_KS_o: secrecy_KS > $o ).

thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Secrecy____types__OKeys_M_Eo_J,type,
    bot_bo2072920570Keys_o: secrecy_Keys > $o ).

thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Secrecy____types__OSecrets_M_Eo_J,type,
    bot_bo1909946299rets_o: secrecy_Secrets > $o ).

thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Secrecy____types__OchanID_M_Eo_J,type,
    bot_bo1012329985anID_o: secrecy_chanID > $o ).

thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Secrecy____types__OspecID_M_Eo_J,type,
    bot_bo1767780696ecID_o: secrecy_specID > $o ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Secrecy____types__OKS_J,type,
    bot_bo1943466239ecy_KS: set_Secrecy_KS ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Secrecy____types__OKeys_J,type,
    bot_bo701486667y_Keys: set_Secrecy_Keys ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Secrecy____types__OSecrets_J,type,
    bot_bo1124690338ecrets: set_Secrecy_Secrets ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Secrecy____types__OchanID_J,type,
    bot_bo443197316chanID: set_Secrecy_chanID ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Secrecy____types__OspecID_J,type,
    bot_bo1785112429specID: set_Secrecy_specID ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Secrecy____types__OchanID_J,type,
    ord_le1802272080chanID: set_Secrecy_chanID > set_Secrecy_chanID > $o ).

thf(sy_c_Secrecy_OcorrectCompositionKS,type,
    correctCompositionKS: secrecy_specID > $o ).

thf(sy_c_Secrecy_Oeout,type,
    eout: secrecy_specID > secrecy_Expression > $o ).

thf(sy_c_Secrecy_OeoutM,type,
    eoutM: secrecy_specID > set_Secrecy_chanID > secrecy_Expression > $o ).

thf(sy_c_Secrecy_OexprChannel,type,
    exprChannel: secrecy_chanID > secrecy_Expression > $o ).

thf(sy_c_Secrecy_Oine,type,
    ine: secrecy_specID > secrecy_Expression > $o ).

thf(sy_c_Secrecy_OineM,type,
    ineM: secrecy_specID > set_Secrecy_chanID > secrecy_Expression > $o ).

thf(sy_c_Secrecy_Oine__exprChannelSet,type,
    ine_exprChannelSet: secrecy_specID > set_Secrecy_chanID > secrecy_Expression > $o ).

thf(sy_c_Secrecy_Oine__exprChannelSingle,type,
    ine_ex1303305700Single: secrecy_specID > secrecy_chanID > secrecy_Expression > $o ).

thf(sy_c_Secrecy_OnotSpecKeysSecretsExpr,type,
    notSpe1586871396tsExpr: secrecy_specID > list_S529945141ession > $o ).

thf(sy_c_Secrecy_Oout__exprChannelSet,type,
    out_exprChannelSet: secrecy_specID > set_Secrecy_chanID > secrecy_Expression > $o ).

thf(sy_c_Secrecy_Oout__exprChannelSingle,type,
    out_ex985543062Single: secrecy_specID > secrecy_chanID > secrecy_Expression > $o ).

thf(sy_c_Secrecy_OspecKeys,type,
    specKeys: secrecy_specID > set_Secrecy_Keys ).

thf(sy_c_Secrecy_OspecKeysSecrets,type,
    specKeysSecrets: secrecy_specID > set_Secrecy_KS ).

thf(sy_c_Secrecy_OspecSecrets,type,
    specSecrets: secrecy_specID > set_Secrecy_Secrets ).

thf(sy_c_Secrecy__types_OExpression_OkE,type,
    secrecy_kE: secrecy_Keys > secrecy_Expression ).

thf(sy_c_Secrecy__types_OExpression_OsE,type,
    secrecy_sE: secrecy_Secrets > secrecy_Expression ).

thf(sy_c_Secrecy__types_OKS2Expression,type,
    secrec1016742679ession: secrecy_KS > secrecy_Expression ).

thf(sy_c_Secrecy__types_OKS_OkKS,type,
    secrecy_kKS: secrecy_Keys > secrecy_KS ).

thf(sy_c_Secrecy__types_OKS_OsKS,type,
    secrecy_sKS: secrecy_Secrets > secrecy_KS ).

thf(sy_c_Set_OCollect_001t__Secrecy____types__OKS,type,
    collect_Secrecy_KS: ( secrecy_KS > $o ) > set_Secrecy_KS ).

thf(sy_c_Set_OCollect_001t__Secrecy____types__OKeys,type,
    collect_Secrecy_Keys: ( secrecy_Keys > $o ) > set_Secrecy_Keys ).

thf(sy_c_Set_OCollect_001t__Secrecy____types__OSecrets,type,
    collec836099393ecrets: ( secrecy_Secrets > $o ) > set_Secrecy_Secrets ).

thf(sy_c_Set_OCollect_001t__Secrecy____types__OchanID,type,
    collec625564175chanID: ( secrecy_chanID > $o ) > set_Secrecy_chanID ).

thf(sy_c_Set_OCollect_001t__Secrecy____types__OspecID,type,
    collec1545797240specID: ( secrecy_specID > $o ) > set_Secrecy_specID ).

thf(sy_c_Set_Oinsert_001t__Secrecy____types__OKS,type,
    insert_Secrecy_KS: secrecy_KS > set_Secrecy_KS > set_Secrecy_KS ).

thf(sy_c_Set_Oinsert_001t__Secrecy____types__OKeys,type,
    insert_Secrecy_Keys: secrecy_Keys > set_Secrecy_Keys > set_Secrecy_Keys ).

thf(sy_c_Set_Oinsert_001t__Secrecy____types__OSecrets,type,
    insert565337510ecrets: secrecy_Secrets > set_Secrecy_Secrets > set_Secrecy_Secrets ).

thf(sy_c_Set_Oinsert_001t__Secrecy____types__OchanID,type,
    insert751661162chanID: secrecy_chanID > set_Secrecy_chanID > set_Secrecy_chanID ).

thf(sy_c_Set_Oinsert_001t__Secrecy____types__OspecID,type,
    insert1671894227specID: secrecy_specID > set_Secrecy_specID > set_Secrecy_specID ).

thf(sy_c_Set_Ois__empty_001t__Secrecy____types__OchanID,type,
    is_emp928984296chanID: set_Secrecy_chanID > $o ).

thf(sy_c_Set_Ois__empty_001t__Secrecy____types__OspecID,type,
    is_emp1849217361specID: set_Secrecy_specID > $o ).

thf(sy_c_Set_Ois__singleton_001t__Secrecy____types__OKS,type,
    is_sin1442426433ecy_KS: set_Secrecy_KS > $o ).

thf(sy_c_Set_Ois__singleton_001t__Secrecy____types__OKeys,type,
    is_sin1096471309y_Keys: set_Secrecy_Keys > $o ).

thf(sy_c_Set_Ois__singleton_001t__Secrecy____types__OSecrets,type,
    is_sin1002224330ecrets: set_Secrecy_Secrets > $o ).

thf(sy_c_Set_Ois__singleton_001t__Secrecy____types__OchanID,type,
    is_sin581274182chanID: set_Secrecy_chanID > $o ).

thf(sy_c_Set_Ois__singleton_001t__Secrecy____types__OspecID,type,
    is_sin1501507247specID: set_Secrecy_specID > $o ).

thf(sy_c_Set_Othe__elem_001t__Secrecy____types__OchanID,type,
    the_el2054381829chanID: set_Secrecy_chanID > secrecy_chanID ).

thf(sy_c_Set_Othe__elem_001t__Secrecy____types__OspecID,type,
    the_el827131246specID: set_Secrecy_specID > secrecy_specID ).

thf(sy_c_inout_OcorrectCompositionIn,type,
    correctCompositionIn: secrecy_specID > $o ).

thf(sy_c_inout_OcorrectCompositionLoc,type,
    correc965049635ionLoc: secrecy_specID > $o ).

thf(sy_c_inout_OcorrectCompositionOut,type,
    correc990227761ionOut: secrecy_specID > $o ).

thf(sy_c_inout_OinStream,type,
    inStream: secrecy_specID > set_Secrecy_chanID > $o ).

thf(sy_c_inout_Oins,type,
    ins: secrecy_specID > set_Secrecy_chanID ).

thf(sy_c_inout_Oloc,type,
    loc: secrecy_specID > set_Secrecy_chanID ).

thf(sy_c_inout_OlocStream,type,
    locStream: secrecy_specID > set_Secrecy_chanID > $o ).

thf(sy_c_inout_Oout,type,
    out: secrecy_specID > set_Secrecy_chanID ).

thf(sy_c_inout_OoutStream,type,
    outStream: secrecy_specID > set_Secrecy_chanID > $o ).

thf(sy_c_inout_Osubcomponents,type,
    subcomponents: secrecy_specID > set_Secrecy_specID ).

thf(sy_c_member_001t__Secrecy____types__OKS,type,
    member_Secrecy_KS: secrecy_KS > set_Secrecy_KS > $o ).

thf(sy_c_member_001t__Secrecy____types__OKeys,type,
    member_Secrecy_Keys: secrecy_Keys > set_Secrecy_Keys > $o ).

thf(sy_c_member_001t__Secrecy____types__OSecrets,type,
    member2138796287ecrets: secrecy_Secrets > set_Secrecy_Secrets > $o ).

thf(sy_c_member_001t__Secrecy____types__OchanID,type,
    member1967978193chanID: secrecy_chanID > set_Secrecy_chanID > $o ).

thf(sy_c_member_001t__Secrecy____types__OspecID,type,
    member740727610specID: secrecy_specID > set_Secrecy_specID > $o ).

thf(sy_v_P,type,
    p: secrecy_specID ).

thf(sy_v_PQ,type,
    pq: secrecy_specID ).

thf(sy_v_Q,type,
    q: secrecy_specID ).

thf(sy_v_m,type,
    m: secrecy_KS ).

thf(sy_v_secret____,type,
    secret: secrecy_Secrets ).

% Relevant facts (355)
thf(fact_0_cCompLoc,axiom,
    correc965049635ionLoc @ pq ).

% cCompLoc
thf(fact_1_sg1a,axiom,
    ~ ( ine @ p @ ( secrecy_sE @ secret ) ) ).

% sg1a
thf(fact_2_sg2a,axiom,
    ~ ( ine @ q @ ( secrecy_sE @ secret ) ) ).

% sg2a
thf(fact_3_sg1b,axiom,
    ~ ( member_Secrecy_KS @ ( secrecy_sKS @ secret ) @ ( localSecrets @ p ) ) ).

% sg1b
thf(fact_4_sg2b,axiom,
    ~ ( member_Secrecy_KS @ ( secrecy_sKS @ secret ) @ ( localSecrets @ q ) ) ).

% sg2b
thf(fact_5_subPQ,axiom,
    ( ( subcomponents @ pq )
    = ( insert1671894227specID @ p @ ( insert1671894227specID @ q @ bot_bo1785112429specID ) ) ) ).

% subPQ
thf(fact_6_a2,axiom,
    ( m
    = ( secrecy_sKS @ secret ) ) ).

% a2
thf(fact_7_cCompI,axiom,
    correctCompositionIn @ pq ).

% cCompI
thf(fact_8_sg3,axiom,
    ~ ( ine @ pq @ ( secrecy_sE @ secret ) ) ).

% sg3
thf(fact_9_KS_Oinject_I2_J,axiom,
    ! [X2: secrecy_Secrets,Y2: secrecy_Secrets] :
      ( ( ( secrecy_sKS @ X2 )
        = ( secrecy_sKS @ Y2 ) )
      = ( X2 = Y2 ) ) ).

% KS.inject(2)
thf(fact_10_sg1,axiom,
    ~ ( knowle945212990e_know @ p @ ( secrecy_sKS @ secret ) ) ).

% sg1
thf(fact_11_sg2,axiom,
    ~ ( knowle945212990e_know @ q @ ( secrecy_sKS @ secret ) ) ).

% sg2
thf(fact_12_know_Osimps_I2_J,axiom,
    ! [A: secrecy_specID,M: secrecy_Secrets] :
      ( ( knowle945212990e_know @ A @ ( secrecy_sKS @ M ) )
      = ( ( ine @ A @ ( secrecy_sE @ M ) )
        | ( member_Secrecy_KS @ ( secrecy_sKS @ M ) @ ( localSecrets @ A ) ) ) ) ).

% know.simps(2)
thf(fact_13_LocalSecretsComposition__ine1__s,axiom,
    ! [S: secrecy_Secrets,PQ: secrecy_specID,P: secrecy_specID,Q: secrecy_specID] :
      ( ( member_Secrecy_KS @ ( secrecy_sKS @ S ) @ ( localSecrets @ PQ ) )
     => ( ( ( subcomponents @ PQ )
          = ( insert1671894227specID @ P @ ( insert1671894227specID @ Q @ bot_bo1785112429specID ) ) )
       => ( ( correc965049635ionLoc @ PQ )
         => ( ~ ( ine @ Q @ ( secrecy_sE @ S ) )
           => ( ~ ( member_Secrecy_KS @ ( secrecy_sKS @ S ) @ ( localSecrets @ P ) )
             => ( ~ ( member_Secrecy_KS @ ( secrecy_sKS @ S ) @ ( localSecrets @ Q ) )
               => ( ine @ P @ ( secrecy_sE @ S ) ) ) ) ) ) ) ) ).

% LocalSecretsComposition_ine1_s
thf(fact_14_LocalSecretsComposition__ine2__s,axiom,
    ! [S: secrecy_Secrets,PQ: secrecy_specID,P: secrecy_specID,Q: secrecy_specID] :
      ( ( member_Secrecy_KS @ ( secrecy_sKS @ S ) @ ( localSecrets @ PQ ) )
     => ( ( ( subcomponents @ PQ )
          = ( insert1671894227specID @ P @ ( insert1671894227specID @ Q @ bot_bo1785112429specID ) ) )
       => ( ( correc965049635ionLoc @ PQ )
         => ( ~ ( ine @ P @ ( secrecy_sE @ S ) )
           => ( ~ ( member_Secrecy_KS @ ( secrecy_sKS @ S ) @ ( localSecrets @ P ) )
             => ( ~ ( member_Secrecy_KS @ ( secrecy_sKS @ S ) @ ( localSecrets @ Q ) )
               => ( ine @ Q @ ( secrecy_sE @ S ) ) ) ) ) ) ) ) ).

% LocalSecretsComposition_ine2_s
thf(fact_15_LocalSecretsComposition__neg1__s,axiom,
    ! [PQ: secrecy_specID,P: secrecy_specID,Q: secrecy_specID,S: secrecy_Secrets] :
      ( ( ( subcomponents @ PQ )
        = ( insert1671894227specID @ P @ ( insert1671894227specID @ Q @ bot_bo1785112429specID ) ) )
     => ( ( correc965049635ionLoc @ PQ )
       => ( ~ ( ine @ P @ ( secrecy_sE @ S ) )
         => ( ~ ( ine @ Q @ ( secrecy_sE @ S ) )
           => ( ~ ( member_Secrecy_KS @ ( secrecy_sKS @ S ) @ ( localSecrets @ P ) )
             => ( ~ ( member_Secrecy_KS @ ( secrecy_sKS @ S ) @ ( localSecrets @ Q ) )
               => ~ ( member_Secrecy_KS @ ( secrecy_sKS @ S ) @ ( localSecrets @ PQ ) ) ) ) ) ) ) ) ).

% LocalSecretsComposition_neg1_s
thf(fact_16_KS2Expression_Osimps_I2_J,axiom,
    ! [M: secrecy_Secrets] :
      ( ( secrec1016742679ession @ ( secrecy_sKS @ M ) )
      = ( secrecy_sE @ M ) ) ).

% KS2Expression.simps(2)
thf(fact_17_LocalSecretsComposition1,axiom,
    ! [Ls: secrecy_KS,P: secrecy_specID,PQ: secrecy_specID,Q: secrecy_specID] :
      ( ( member_Secrecy_KS @ Ls @ ( localSecrets @ P ) )
     => ( ( ( subcomponents @ PQ )
          = ( insert1671894227specID @ P @ ( insert1671894227specID @ Q @ bot_bo1785112429specID ) ) )
       => ( member_Secrecy_KS @ Ls @ ( localSecrets @ PQ ) ) ) ) ).

% LocalSecretsComposition1
thf(fact_18_not__know__s__not__ine,axiom,
    ! [A: secrecy_specID,M: secrecy_Secrets] :
      ( ~ ( knowle945212990e_know @ A @ ( secrecy_sKS @ M ) )
     => ~ ( ine @ A @ ( secrecy_sE @ M ) ) ) ).

% not_know_s_not_ine
thf(fact_19_notknowQ,axiom,
    ~ ( knowle945212990e_know @ q @ m ) ).

% notknowQ
thf(fact_20_notknowP,axiom,
    ~ ( knowle945212990e_know @ p @ m ) ).

% notknowP
thf(fact_21_Expression_Oinject_I2_J,axiom,
    ! [X2: secrecy_Secrets,Y2: secrecy_Secrets] :
      ( ( ( secrecy_sE @ X2 )
        = ( secrecy_sE @ Y2 ) )
      = ( X2 = Y2 ) ) ).

% Expression.inject(2)
thf(fact_22_know__composition__neg__ine__s,axiom,
    ! [P: secrecy_specID,Secret: secrecy_Secrets,Q: secrecy_specID,PQ: secrecy_specID] :
      ( ~ ( knowle945212990e_know @ P @ ( secrecy_sKS @ Secret ) )
     => ( ~ ( knowle945212990e_know @ Q @ ( secrecy_sKS @ Secret ) )
       => ( ( ( subcomponents @ PQ )
            = ( insert1671894227specID @ P @ ( insert1671894227specID @ Q @ bot_bo1785112429specID ) ) )
         => ( ( correctCompositionIn @ PQ )
           => ~ ( ine @ PQ @ ( secrecy_sE @ Secret ) ) ) ) ) ) ).

% know_composition_neg_ine_s
thf(fact_23_TBtheorem1a,axiom,
    ! [PQ: secrecy_specID,E: secrecy_Expression,P: secrecy_specID,Q: secrecy_specID] :
      ( ( ine @ PQ @ E )
     => ( ( ( subcomponents @ PQ )
          = ( insert1671894227specID @ P @ ( insert1671894227specID @ Q @ bot_bo1785112429specID ) ) )
       => ( ( correctCompositionIn @ PQ )
         => ( ( ine @ P @ E )
            | ( ine @ Q @ E ) ) ) ) ) ).

% TBtheorem1a
thf(fact_24_TBtheorem3a,axiom,
    ! [P: secrecy_specID,E: secrecy_Expression,Q: secrecy_specID,PQ: secrecy_specID] :
      ( ~ ( ine @ P @ E )
     => ( ~ ( ine @ Q @ E )
       => ( ( ( subcomponents @ PQ )
            = ( insert1671894227specID @ P @ ( insert1671894227specID @ Q @ bot_bo1785112429specID ) ) )
         => ( ( correctCompositionIn @ PQ )
           => ~ ( ine @ PQ @ E ) ) ) ) ) ).

% TBtheorem3a
thf(fact_25_singletonI,axiom,
    ! [A2: secrecy_KS] : ( member_Secrecy_KS @ A2 @ ( insert_Secrecy_KS @ A2 @ bot_bo1943466239ecy_KS ) ) ).

% singletonI
thf(fact_26_singletonI,axiom,
    ! [A2: secrecy_Secrets] : ( member2138796287ecrets @ A2 @ ( insert565337510ecrets @ A2 @ bot_bo1124690338ecrets ) ) ).

% singletonI
thf(fact_27_singletonI,axiom,
    ! [A2: secrecy_Keys] : ( member_Secrecy_Keys @ A2 @ ( insert_Secrecy_Keys @ A2 @ bot_bo701486667y_Keys ) ) ).

% singletonI
thf(fact_28_singletonI,axiom,
    ! [A2: secrecy_specID] : ( member740727610specID @ A2 @ ( insert1671894227specID @ A2 @ bot_bo1785112429specID ) ) ).

% singletonI
thf(fact_29_singletonI,axiom,
    ! [A2: secrecy_chanID] : ( member1967978193chanID @ A2 @ ( insert751661162chanID @ A2 @ bot_bo443197316chanID ) ) ).

% singletonI
thf(fact_30_insertCI,axiom,
    ! [A2: secrecy_KS,B: set_Secrecy_KS,B2: secrecy_KS] :
      ( ( ~ ( member_Secrecy_KS @ A2 @ B )
       => ( A2 = B2 ) )
     => ( member_Secrecy_KS @ A2 @ ( insert_Secrecy_KS @ B2 @ B ) ) ) ).

% insertCI
thf(fact_31_insertCI,axiom,
    ! [A2: secrecy_chanID,B: set_Secrecy_chanID,B2: secrecy_chanID] :
      ( ( ~ ( member1967978193chanID @ A2 @ B )
       => ( A2 = B2 ) )
     => ( member1967978193chanID @ A2 @ ( insert751661162chanID @ B2 @ B ) ) ) ).

% insertCI
thf(fact_32_insertCI,axiom,
    ! [A2: secrecy_specID,B: set_Secrecy_specID,B2: secrecy_specID] :
      ( ( ~ ( member740727610specID @ A2 @ B )
       => ( A2 = B2 ) )
     => ( member740727610specID @ A2 @ ( insert1671894227specID @ B2 @ B ) ) ) ).

% insertCI
thf(fact_33_insertCI,axiom,
    ! [A2: secrecy_Secrets,B: set_Secrecy_Secrets,B2: secrecy_Secrets] :
      ( ( ~ ( member2138796287ecrets @ A2 @ B )
       => ( A2 = B2 ) )
     => ( member2138796287ecrets @ A2 @ ( insert565337510ecrets @ B2 @ B ) ) ) ).

% insertCI
thf(fact_34_insertCI,axiom,
    ! [A2: secrecy_Keys,B: set_Secrecy_Keys,B2: secrecy_Keys] :
      ( ( ~ ( member_Secrecy_Keys @ A2 @ B )
       => ( A2 = B2 ) )
     => ( member_Secrecy_Keys @ A2 @ ( insert_Secrecy_Keys @ B2 @ B ) ) ) ).

% insertCI
thf(fact_35_insert__iff,axiom,
    ! [A2: secrecy_KS,B2: secrecy_KS,A: set_Secrecy_KS] :
      ( ( member_Secrecy_KS @ A2 @ ( insert_Secrecy_KS @ B2 @ A ) )
      = ( ( A2 = B2 )
        | ( member_Secrecy_KS @ A2 @ A ) ) ) ).

% insert_iff
thf(fact_36_insert__iff,axiom,
    ! [A2: secrecy_chanID,B2: secrecy_chanID,A: set_Secrecy_chanID] :
      ( ( member1967978193chanID @ A2 @ ( insert751661162chanID @ B2 @ A ) )
      = ( ( A2 = B2 )
        | ( member1967978193chanID @ A2 @ A ) ) ) ).

% insert_iff
thf(fact_37_insert__iff,axiom,
    ! [A2: secrecy_specID,B2: secrecy_specID,A: set_Secrecy_specID] :
      ( ( member740727610specID @ A2 @ ( insert1671894227specID @ B2 @ A ) )
      = ( ( A2 = B2 )
        | ( member740727610specID @ A2 @ A ) ) ) ).

% insert_iff
thf(fact_38_insert__iff,axiom,
    ! [A2: secrecy_Secrets,B2: secrecy_Secrets,A: set_Secrecy_Secrets] :
      ( ( member2138796287ecrets @ A2 @ ( insert565337510ecrets @ B2 @ A ) )
      = ( ( A2 = B2 )
        | ( member2138796287ecrets @ A2 @ A ) ) ) ).

% insert_iff
thf(fact_39_insert__iff,axiom,
    ! [A2: secrecy_Keys,B2: secrecy_Keys,A: set_Secrecy_Keys] :
      ( ( member_Secrecy_Keys @ A2 @ ( insert_Secrecy_Keys @ B2 @ A ) )
      = ( ( A2 = B2 )
        | ( member_Secrecy_Keys @ A2 @ A ) ) ) ).

% insert_iff
thf(fact_40_insert__absorb2,axiom,
    ! [X: secrecy_specID,A: set_Secrecy_specID] :
      ( ( insert1671894227specID @ X @ ( insert1671894227specID @ X @ A ) )
      = ( insert1671894227specID @ X @ A ) ) ).

% insert_absorb2
thf(fact_41_insert__absorb2,axiom,
    ! [X: secrecy_chanID,A: set_Secrecy_chanID] :
      ( ( insert751661162chanID @ X @ ( insert751661162chanID @ X @ A ) )
      = ( insert751661162chanID @ X @ A ) ) ).

% insert_absorb2
thf(fact_42_empty__iff,axiom,
    ! [C: secrecy_KS] :
      ~ ( member_Secrecy_KS @ C @ bot_bo1943466239ecy_KS ) ).

% empty_iff
thf(fact_43_empty__iff,axiom,
    ! [C: secrecy_Secrets] :
      ~ ( member2138796287ecrets @ C @ bot_bo1124690338ecrets ) ).

% empty_iff
thf(fact_44_empty__iff,axiom,
    ! [C: secrecy_Keys] :
      ~ ( member_Secrecy_Keys @ C @ bot_bo701486667y_Keys ) ).

% empty_iff
thf(fact_45_empty__iff,axiom,
    ! [C: secrecy_specID] :
      ~ ( member740727610specID @ C @ bot_bo1785112429specID ) ).

% empty_iff
thf(fact_46_empty__iff,axiom,
    ! [C: secrecy_chanID] :
      ~ ( member1967978193chanID @ C @ bot_bo443197316chanID ) ).

% empty_iff
thf(fact_47_all__not__in__conv,axiom,
    ! [A: set_Secrecy_KS] :
      ( ( ! [X3: secrecy_KS] :
            ~ ( member_Secrecy_KS @ X3 @ A ) )
      = ( A = bot_bo1943466239ecy_KS ) ) ).

% all_not_in_conv
thf(fact_48_all__not__in__conv,axiom,
    ! [A: set_Secrecy_Secrets] :
      ( ( ! [X3: secrecy_Secrets] :
            ~ ( member2138796287ecrets @ X3 @ A ) )
      = ( A = bot_bo1124690338ecrets ) ) ).

% all_not_in_conv
thf(fact_49_all__not__in__conv,axiom,
    ! [A: set_Secrecy_Keys] :
      ( ( ! [X3: secrecy_Keys] :
            ~ ( member_Secrecy_Keys @ X3 @ A ) )
      = ( A = bot_bo701486667y_Keys ) ) ).

% all_not_in_conv
thf(fact_50_all__not__in__conv,axiom,
    ! [A: set_Secrecy_specID] :
      ( ( ! [X3: secrecy_specID] :
            ~ ( member740727610specID @ X3 @ A ) )
      = ( A = bot_bo1785112429specID ) ) ).

% all_not_in_conv
thf(fact_51_all__not__in__conv,axiom,
    ! [A: set_Secrecy_chanID] :
      ( ( ! [X3: secrecy_chanID] :
            ~ ( member1967978193chanID @ X3 @ A ) )
      = ( A = bot_bo443197316chanID ) ) ).

% all_not_in_conv
thf(fact_52_Collect__empty__eq,axiom,
    ! [P: secrecy_specID > $o] :
      ( ( ( collec1545797240specID @ P )
        = bot_bo1785112429specID )
      = ( ! [X3: secrecy_specID] :
            ~ ( P @ X3 ) ) ) ).

% Collect_empty_eq
thf(fact_53_Collect__empty__eq,axiom,
    ! [P: secrecy_chanID > $o] :
      ( ( ( collec625564175chanID @ P )
        = bot_bo443197316chanID )
      = ( ! [X3: secrecy_chanID] :
            ~ ( P @ X3 ) ) ) ).

% Collect_empty_eq
thf(fact_54_empty__Collect__eq,axiom,
    ! [P: secrecy_specID > $o] :
      ( ( bot_bo1785112429specID
        = ( collec1545797240specID @ P ) )
      = ( ! [X3: secrecy_specID] :
            ~ ( P @ X3 ) ) ) ).

% empty_Collect_eq
thf(fact_55_empty__Collect__eq,axiom,
    ! [P: secrecy_chanID > $o] :
      ( ( bot_bo443197316chanID
        = ( collec625564175chanID @ P ) )
      = ( ! [X3: secrecy_chanID] :
            ~ ( P @ X3 ) ) ) ).

% empty_Collect_eq
thf(fact_56_ex__in__conv,axiom,
    ! [A: set_Secrecy_KS] :
      ( ( ? [X3: secrecy_KS] : ( member_Secrecy_KS @ X3 @ A ) )
      = ( A != bot_bo1943466239ecy_KS ) ) ).

% ex_in_conv
thf(fact_57_ex__in__conv,axiom,
    ! [A: set_Secrecy_Secrets] :
      ( ( ? [X3: secrecy_Secrets] : ( member2138796287ecrets @ X3 @ A ) )
      = ( A != bot_bo1124690338ecrets ) ) ).

% ex_in_conv
thf(fact_58_ex__in__conv,axiom,
    ! [A: set_Secrecy_Keys] :
      ( ( ? [X3: secrecy_Keys] : ( member_Secrecy_Keys @ X3 @ A ) )
      = ( A != bot_bo701486667y_Keys ) ) ).

% ex_in_conv
thf(fact_59_ex__in__conv,axiom,
    ! [A: set_Secrecy_specID] :
      ( ( ? [X3: secrecy_specID] : ( member740727610specID @ X3 @ A ) )
      = ( A != bot_bo1785112429specID ) ) ).

% ex_in_conv
thf(fact_60_ex__in__conv,axiom,
    ! [A: set_Secrecy_chanID] :
      ( ( ? [X3: secrecy_chanID] : ( member1967978193chanID @ X3 @ A ) )
      = ( A != bot_bo443197316chanID ) ) ).

% ex_in_conv
thf(fact_61_equals0I,axiom,
    ! [A: set_Secrecy_KS] :
      ( ! [Y: secrecy_KS] :
          ~ ( member_Secrecy_KS @ Y @ A )
     => ( A = bot_bo1943466239ecy_KS ) ) ).

% equals0I
thf(fact_62_equals0I,axiom,
    ! [A: set_Secrecy_Secrets] :
      ( ! [Y: secrecy_Secrets] :
          ~ ( member2138796287ecrets @ Y @ A )
     => ( A = bot_bo1124690338ecrets ) ) ).

% equals0I
thf(fact_63_equals0I,axiom,
    ! [A: set_Secrecy_Keys] :
      ( ! [Y: secrecy_Keys] :
          ~ ( member_Secrecy_Keys @ Y @ A )
     => ( A = bot_bo701486667y_Keys ) ) ).

% equals0I
thf(fact_64_equals0I,axiom,
    ! [A: set_Secrecy_specID] :
      ( ! [Y: secrecy_specID] :
          ~ ( member740727610specID @ Y @ A )
     => ( A = bot_bo1785112429specID ) ) ).

% equals0I
thf(fact_65_equals0I,axiom,
    ! [A: set_Secrecy_chanID] :
      ( ! [Y: secrecy_chanID] :
          ~ ( member1967978193chanID @ Y @ A )
     => ( A = bot_bo443197316chanID ) ) ).

% equals0I
thf(fact_66_equals0D,axiom,
    ! [A: set_Secrecy_KS,A2: secrecy_KS] :
      ( ( A = bot_bo1943466239ecy_KS )
     => ~ ( member_Secrecy_KS @ A2 @ A ) ) ).

% equals0D
thf(fact_67_equals0D,axiom,
    ! [A: set_Secrecy_Secrets,A2: secrecy_Secrets] :
      ( ( A = bot_bo1124690338ecrets )
     => ~ ( member2138796287ecrets @ A2 @ A ) ) ).

% equals0D
thf(fact_68_equals0D,axiom,
    ! [A: set_Secrecy_Keys,A2: secrecy_Keys] :
      ( ( A = bot_bo701486667y_Keys )
     => ~ ( member_Secrecy_Keys @ A2 @ A ) ) ).

% equals0D
thf(fact_69_equals0D,axiom,
    ! [A: set_Secrecy_specID,A2: secrecy_specID] :
      ( ( A = bot_bo1785112429specID )
     => ~ ( member740727610specID @ A2 @ A ) ) ).

% equals0D
thf(fact_70_equals0D,axiom,
    ! [A: set_Secrecy_chanID,A2: secrecy_chanID] :
      ( ( A = bot_bo443197316chanID )
     => ~ ( member1967978193chanID @ A2 @ A ) ) ).

% equals0D
thf(fact_71_emptyE,axiom,
    ! [A2: secrecy_KS] :
      ~ ( member_Secrecy_KS @ A2 @ bot_bo1943466239ecy_KS ) ).

% emptyE
thf(fact_72_emptyE,axiom,
    ! [A2: secrecy_Secrets] :
      ~ ( member2138796287ecrets @ A2 @ bot_bo1124690338ecrets ) ).

% emptyE
thf(fact_73_emptyE,axiom,
    ! [A2: secrecy_Keys] :
      ~ ( member_Secrecy_Keys @ A2 @ bot_bo701486667y_Keys ) ).

% emptyE
thf(fact_74_emptyE,axiom,
    ! [A2: secrecy_specID] :
      ~ ( member740727610specID @ A2 @ bot_bo1785112429specID ) ).

% emptyE
thf(fact_75_emptyE,axiom,
    ! [A2: secrecy_chanID] :
      ~ ( member1967978193chanID @ A2 @ bot_bo443197316chanID ) ).

% emptyE
thf(fact_76_mk__disjoint__insert,axiom,
    ! [A2: secrecy_KS,A: set_Secrecy_KS] :
      ( ( member_Secrecy_KS @ A2 @ A )
     => ? [B3: set_Secrecy_KS] :
          ( ( A
            = ( insert_Secrecy_KS @ A2 @ B3 ) )
          & ~ ( member_Secrecy_KS @ A2 @ B3 ) ) ) ).

% mk_disjoint_insert
thf(fact_77_mk__disjoint__insert,axiom,
    ! [A2: secrecy_chanID,A: set_Secrecy_chanID] :
      ( ( member1967978193chanID @ A2 @ A )
     => ? [B3: set_Secrecy_chanID] :
          ( ( A
            = ( insert751661162chanID @ A2 @ B3 ) )
          & ~ ( member1967978193chanID @ A2 @ B3 ) ) ) ).

% mk_disjoint_insert
thf(fact_78_mk__disjoint__insert,axiom,
    ! [A2: secrecy_specID,A: set_Secrecy_specID] :
      ( ( member740727610specID @ A2 @ A )
     => ? [B3: set_Secrecy_specID] :
          ( ( A
            = ( insert1671894227specID @ A2 @ B3 ) )
          & ~ ( member740727610specID @ A2 @ B3 ) ) ) ).

% mk_disjoint_insert
thf(fact_79_mk__disjoint__insert,axiom,
    ! [A2: secrecy_Secrets,A: set_Secrecy_Secrets] :
      ( ( member2138796287ecrets @ A2 @ A )
     => ? [B3: set_Secrecy_Secrets] :
          ( ( A
            = ( insert565337510ecrets @ A2 @ B3 ) )
          & ~ ( member2138796287ecrets @ A2 @ B3 ) ) ) ).

% mk_disjoint_insert
thf(fact_80_mk__disjoint__insert,axiom,
    ! [A2: secrecy_Keys,A: set_Secrecy_Keys] :
      ( ( member_Secrecy_Keys @ A2 @ A )
     => ? [B3: set_Secrecy_Keys] :
          ( ( A
            = ( insert_Secrecy_Keys @ A2 @ B3 ) )
          & ~ ( member_Secrecy_Keys @ A2 @ B3 ) ) ) ).

% mk_disjoint_insert
thf(fact_81_insert__commute,axiom,
    ! [X: secrecy_specID,Y3: secrecy_specID,A: set_Secrecy_specID] :
      ( ( insert1671894227specID @ X @ ( insert1671894227specID @ Y3 @ A ) )
      = ( insert1671894227specID @ Y3 @ ( insert1671894227specID @ X @ A ) ) ) ).

% insert_commute
thf(fact_82_insert__commute,axiom,
    ! [X: secrecy_chanID,Y3: secrecy_chanID,A: set_Secrecy_chanID] :
      ( ( insert751661162chanID @ X @ ( insert751661162chanID @ Y3 @ A ) )
      = ( insert751661162chanID @ Y3 @ ( insert751661162chanID @ X @ A ) ) ) ).

% insert_commute
thf(fact_83_insert__eq__iff,axiom,
    ! [A2: secrecy_KS,A: set_Secrecy_KS,B2: secrecy_KS,B: set_Secrecy_KS] :
      ( ~ ( member_Secrecy_KS @ A2 @ A )
     => ( ~ ( member_Secrecy_KS @ B2 @ B )
       => ( ( ( insert_Secrecy_KS @ A2 @ A )
            = ( insert_Secrecy_KS @ B2 @ B ) )
          = ( ( ( A2 = B2 )
             => ( A = B ) )
            & ( ( A2 != B2 )
             => ? [C2: set_Secrecy_KS] :
                  ( ( A
                    = ( insert_Secrecy_KS @ B2 @ C2 ) )
                  & ~ ( member_Secrecy_KS @ B2 @ C2 )
                  & ( B
                    = ( insert_Secrecy_KS @ A2 @ C2 ) )
                  & ~ ( member_Secrecy_KS @ A2 @ C2 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_84_insert__eq__iff,axiom,
    ! [A2: secrecy_chanID,A: set_Secrecy_chanID,B2: secrecy_chanID,B: set_Secrecy_chanID] :
      ( ~ ( member1967978193chanID @ A2 @ A )
     => ( ~ ( member1967978193chanID @ B2 @ B )
       => ( ( ( insert751661162chanID @ A2 @ A )
            = ( insert751661162chanID @ B2 @ B ) )
          = ( ( ( A2 = B2 )
             => ( A = B ) )
            & ( ( A2 != B2 )
             => ? [C2: set_Secrecy_chanID] :
                  ( ( A
                    = ( insert751661162chanID @ B2 @ C2 ) )
                  & ~ ( member1967978193chanID @ B2 @ C2 )
                  & ( B
                    = ( insert751661162chanID @ A2 @ C2 ) )
                  & ~ ( member1967978193chanID @ A2 @ C2 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_85_insert__eq__iff,axiom,
    ! [A2: secrecy_specID,A: set_Secrecy_specID,B2: secrecy_specID,B: set_Secrecy_specID] :
      ( ~ ( member740727610specID @ A2 @ A )
     => ( ~ ( member740727610specID @ B2 @ B )
       => ( ( ( insert1671894227specID @ A2 @ A )
            = ( insert1671894227specID @ B2 @ B ) )
          = ( ( ( A2 = B2 )
             => ( A = B ) )
            & ( ( A2 != B2 )
             => ? [C2: set_Secrecy_specID] :
                  ( ( A
                    = ( insert1671894227specID @ B2 @ C2 ) )
                  & ~ ( member740727610specID @ B2 @ C2 )
                  & ( B
                    = ( insert1671894227specID @ A2 @ C2 ) )
                  & ~ ( member740727610specID @ A2 @ C2 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_86_insert__eq__iff,axiom,
    ! [A2: secrecy_Secrets,A: set_Secrecy_Secrets,B2: secrecy_Secrets,B: set_Secrecy_Secrets] :
      ( ~ ( member2138796287ecrets @ A2 @ A )
     => ( ~ ( member2138796287ecrets @ B2 @ B )
       => ( ( ( insert565337510ecrets @ A2 @ A )
            = ( insert565337510ecrets @ B2 @ B ) )
          = ( ( ( A2 = B2 )
             => ( A = B ) )
            & ( ( A2 != B2 )
             => ? [C2: set_Secrecy_Secrets] :
                  ( ( A
                    = ( insert565337510ecrets @ B2 @ C2 ) )
                  & ~ ( member2138796287ecrets @ B2 @ C2 )
                  & ( B
                    = ( insert565337510ecrets @ A2 @ C2 ) )
                  & ~ ( member2138796287ecrets @ A2 @ C2 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_87_insert__eq__iff,axiom,
    ! [A2: secrecy_Keys,A: set_Secrecy_Keys,B2: secrecy_Keys,B: set_Secrecy_Keys] :
      ( ~ ( member_Secrecy_Keys @ A2 @ A )
     => ( ~ ( member_Secrecy_Keys @ B2 @ B )
       => ( ( ( insert_Secrecy_Keys @ A2 @ A )
            = ( insert_Secrecy_Keys @ B2 @ B ) )
          = ( ( ( A2 = B2 )
             => ( A = B ) )
            & ( ( A2 != B2 )
             => ? [C2: set_Secrecy_Keys] :
                  ( ( A
                    = ( insert_Secrecy_Keys @ B2 @ C2 ) )
                  & ~ ( member_Secrecy_Keys @ B2 @ C2 )
                  & ( B
                    = ( insert_Secrecy_Keys @ A2 @ C2 ) )
                  & ~ ( member_Secrecy_Keys @ A2 @ C2 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_88_insert__absorb,axiom,
    ! [A2: secrecy_KS,A: set_Secrecy_KS] :
      ( ( member_Secrecy_KS @ A2 @ A )
     => ( ( insert_Secrecy_KS @ A2 @ A )
        = A ) ) ).

% insert_absorb
thf(fact_89_insert__absorb,axiom,
    ! [A2: secrecy_chanID,A: set_Secrecy_chanID] :
      ( ( member1967978193chanID @ A2 @ A )
     => ( ( insert751661162chanID @ A2 @ A )
        = A ) ) ).

% insert_absorb
thf(fact_90_insert__absorb,axiom,
    ! [A2: secrecy_specID,A: set_Secrecy_specID] :
      ( ( member740727610specID @ A2 @ A )
     => ( ( insert1671894227specID @ A2 @ A )
        = A ) ) ).

% insert_absorb
thf(fact_91_insert__absorb,axiom,
    ! [A2: secrecy_Secrets,A: set_Secrecy_Secrets] :
      ( ( member2138796287ecrets @ A2 @ A )
     => ( ( insert565337510ecrets @ A2 @ A )
        = A ) ) ).

% insert_absorb
thf(fact_92_insert__absorb,axiom,
    ! [A2: secrecy_Keys,A: set_Secrecy_Keys] :
      ( ( member_Secrecy_Keys @ A2 @ A )
     => ( ( insert_Secrecy_Keys @ A2 @ A )
        = A ) ) ).

% insert_absorb
thf(fact_93_insert__ident,axiom,
    ! [X: secrecy_KS,A: set_Secrecy_KS,B: set_Secrecy_KS] :
      ( ~ ( member_Secrecy_KS @ X @ A )
     => ( ~ ( member_Secrecy_KS @ X @ B )
       => ( ( ( insert_Secrecy_KS @ X @ A )
            = ( insert_Secrecy_KS @ X @ B ) )
          = ( A = B ) ) ) ) ).

% insert_ident
thf(fact_94_insert__ident,axiom,
    ! [X: secrecy_chanID,A: set_Secrecy_chanID,B: set_Secrecy_chanID] :
      ( ~ ( member1967978193chanID @ X @ A )
     => ( ~ ( member1967978193chanID @ X @ B )
       => ( ( ( insert751661162chanID @ X @ A )
            = ( insert751661162chanID @ X @ B ) )
          = ( A = B ) ) ) ) ).

% insert_ident
thf(fact_95_insert__ident,axiom,
    ! [X: secrecy_specID,A: set_Secrecy_specID,B: set_Secrecy_specID] :
      ( ~ ( member740727610specID @ X @ A )
     => ( ~ ( member740727610specID @ X @ B )
       => ( ( ( insert1671894227specID @ X @ A )
            = ( insert1671894227specID @ X @ B ) )
          = ( A = B ) ) ) ) ).

% insert_ident
thf(fact_96_insert__ident,axiom,
    ! [X: secrecy_Secrets,A: set_Secrecy_Secrets,B: set_Secrecy_Secrets] :
      ( ~ ( member2138796287ecrets @ X @ A )
     => ( ~ ( member2138796287ecrets @ X @ B )
       => ( ( ( insert565337510ecrets @ X @ A )
            = ( insert565337510ecrets @ X @ B ) )
          = ( A = B ) ) ) ) ).

% insert_ident
thf(fact_97_insert__ident,axiom,
    ! [X: secrecy_Keys,A: set_Secrecy_Keys,B: set_Secrecy_Keys] :
      ( ~ ( member_Secrecy_Keys @ X @ A )
     => ( ~ ( member_Secrecy_Keys @ X @ B )
       => ( ( ( insert_Secrecy_Keys @ X @ A )
            = ( insert_Secrecy_Keys @ X @ B ) )
          = ( A = B ) ) ) ) ).

% insert_ident
thf(fact_98_Set_Oset__insert,axiom,
    ! [X: secrecy_KS,A: set_Secrecy_KS] :
      ( ( member_Secrecy_KS @ X @ A )
     => ~ ! [B3: set_Secrecy_KS] :
            ( ( A
              = ( insert_Secrecy_KS @ X @ B3 ) )
           => ( member_Secrecy_KS @ X @ B3 ) ) ) ).

% Set.set_insert
thf(fact_99_Set_Oset__insert,axiom,
    ! [X: secrecy_chanID,A: set_Secrecy_chanID] :
      ( ( member1967978193chanID @ X @ A )
     => ~ ! [B3: set_Secrecy_chanID] :
            ( ( A
              = ( insert751661162chanID @ X @ B3 ) )
           => ( member1967978193chanID @ X @ B3 ) ) ) ).

% Set.set_insert
thf(fact_100_Set_Oset__insert,axiom,
    ! [X: secrecy_specID,A: set_Secrecy_specID] :
      ( ( member740727610specID @ X @ A )
     => ~ ! [B3: set_Secrecy_specID] :
            ( ( A
              = ( insert1671894227specID @ X @ B3 ) )
           => ( member740727610specID @ X @ B3 ) ) ) ).

% Set.set_insert
thf(fact_101_Set_Oset__insert,axiom,
    ! [X: secrecy_Secrets,A: set_Secrecy_Secrets] :
      ( ( member2138796287ecrets @ X @ A )
     => ~ ! [B3: set_Secrecy_Secrets] :
            ( ( A
              = ( insert565337510ecrets @ X @ B3 ) )
           => ( member2138796287ecrets @ X @ B3 ) ) ) ).

% Set.set_insert
thf(fact_102_Set_Oset__insert,axiom,
    ! [X: secrecy_Keys,A: set_Secrecy_Keys] :
      ( ( member_Secrecy_Keys @ X @ A )
     => ~ ! [B3: set_Secrecy_Keys] :
            ( ( A
              = ( insert_Secrecy_Keys @ X @ B3 ) )
           => ( member_Secrecy_Keys @ X @ B3 ) ) ) ).

% Set.set_insert
thf(fact_103_mem__Collect__eq,axiom,
    ! [A2: secrecy_KS,P: secrecy_KS > $o] :
      ( ( member_Secrecy_KS @ A2 @ ( collect_Secrecy_KS @ P ) )
      = ( P @ A2 ) ) ).

% mem_Collect_eq
thf(fact_104_mem__Collect__eq,axiom,
    ! [A2: secrecy_chanID,P: secrecy_chanID > $o] :
      ( ( member1967978193chanID @ A2 @ ( collec625564175chanID @ P ) )
      = ( P @ A2 ) ) ).

% mem_Collect_eq
thf(fact_105_mem__Collect__eq,axiom,
    ! [A2: secrecy_specID,P: secrecy_specID > $o] :
      ( ( member740727610specID @ A2 @ ( collec1545797240specID @ P ) )
      = ( P @ A2 ) ) ).

% mem_Collect_eq
thf(fact_106_mem__Collect__eq,axiom,
    ! [A2: secrecy_Secrets,P: secrecy_Secrets > $o] :
      ( ( member2138796287ecrets @ A2 @ ( collec836099393ecrets @ P ) )
      = ( P @ A2 ) ) ).

% mem_Collect_eq
thf(fact_107_mem__Collect__eq,axiom,
    ! [A2: secrecy_Keys,P: secrecy_Keys > $o] :
      ( ( member_Secrecy_Keys @ A2 @ ( collect_Secrecy_Keys @ P ) )
      = ( P @ A2 ) ) ).

% mem_Collect_eq
thf(fact_108_Collect__mem__eq,axiom,
    ! [A: set_Secrecy_KS] :
      ( ( collect_Secrecy_KS
        @ ^ [X3: secrecy_KS] : ( member_Secrecy_KS @ X3 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_109_Collect__mem__eq,axiom,
    ! [A: set_Secrecy_chanID] :
      ( ( collec625564175chanID
        @ ^ [X3: secrecy_chanID] : ( member1967978193chanID @ X3 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_110_Collect__mem__eq,axiom,
    ! [A: set_Secrecy_specID] :
      ( ( collec1545797240specID
        @ ^ [X3: secrecy_specID] : ( member740727610specID @ X3 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_111_Collect__mem__eq,axiom,
    ! [A: set_Secrecy_Secrets] :
      ( ( collec836099393ecrets
        @ ^ [X3: secrecy_Secrets] : ( member2138796287ecrets @ X3 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_112_Collect__mem__eq,axiom,
    ! [A: set_Secrecy_Keys] :
      ( ( collect_Secrecy_Keys
        @ ^ [X3: secrecy_Keys] : ( member_Secrecy_Keys @ X3 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_113_insertI2,axiom,
    ! [A2: secrecy_KS,B: set_Secrecy_KS,B2: secrecy_KS] :
      ( ( member_Secrecy_KS @ A2 @ B )
     => ( member_Secrecy_KS @ A2 @ ( insert_Secrecy_KS @ B2 @ B ) ) ) ).

% insertI2
thf(fact_114_insertI2,axiom,
    ! [A2: secrecy_chanID,B: set_Secrecy_chanID,B2: secrecy_chanID] :
      ( ( member1967978193chanID @ A2 @ B )
     => ( member1967978193chanID @ A2 @ ( insert751661162chanID @ B2 @ B ) ) ) ).

% insertI2
thf(fact_115_insertI2,axiom,
    ! [A2: secrecy_specID,B: set_Secrecy_specID,B2: secrecy_specID] :
      ( ( member740727610specID @ A2 @ B )
     => ( member740727610specID @ A2 @ ( insert1671894227specID @ B2 @ B ) ) ) ).

% insertI2
thf(fact_116_insertI2,axiom,
    ! [A2: secrecy_Secrets,B: set_Secrecy_Secrets,B2: secrecy_Secrets] :
      ( ( member2138796287ecrets @ A2 @ B )
     => ( member2138796287ecrets @ A2 @ ( insert565337510ecrets @ B2 @ B ) ) ) ).

% insertI2
thf(fact_117_insertI2,axiom,
    ! [A2: secrecy_Keys,B: set_Secrecy_Keys,B2: secrecy_Keys] :
      ( ( member_Secrecy_Keys @ A2 @ B )
     => ( member_Secrecy_Keys @ A2 @ ( insert_Secrecy_Keys @ B2 @ B ) ) ) ).

% insertI2
thf(fact_118_insertI1,axiom,
    ! [A2: secrecy_KS,B: set_Secrecy_KS] : ( member_Secrecy_KS @ A2 @ ( insert_Secrecy_KS @ A2 @ B ) ) ).

% insertI1
thf(fact_119_insertI1,axiom,
    ! [A2: secrecy_chanID,B: set_Secrecy_chanID] : ( member1967978193chanID @ A2 @ ( insert751661162chanID @ A2 @ B ) ) ).

% insertI1
thf(fact_120_insertI1,axiom,
    ! [A2: secrecy_specID,B: set_Secrecy_specID] : ( member740727610specID @ A2 @ ( insert1671894227specID @ A2 @ B ) ) ).

% insertI1
thf(fact_121_insertI1,axiom,
    ! [A2: secrecy_Secrets,B: set_Secrecy_Secrets] : ( member2138796287ecrets @ A2 @ ( insert565337510ecrets @ A2 @ B ) ) ).

% insertI1
thf(fact_122_insertI1,axiom,
    ! [A2: secrecy_Keys,B: set_Secrecy_Keys] : ( member_Secrecy_Keys @ A2 @ ( insert_Secrecy_Keys @ A2 @ B ) ) ).

% insertI1
thf(fact_123_insertE,axiom,
    ! [A2: secrecy_KS,B2: secrecy_KS,A: set_Secrecy_KS] :
      ( ( member_Secrecy_KS @ A2 @ ( insert_Secrecy_KS @ B2 @ A ) )
     => ( ( A2 != B2 )
       => ( member_Secrecy_KS @ A2 @ A ) ) ) ).

% insertE
thf(fact_124_insertE,axiom,
    ! [A2: secrecy_chanID,B2: secrecy_chanID,A: set_Secrecy_chanID] :
      ( ( member1967978193chanID @ A2 @ ( insert751661162chanID @ B2 @ A ) )
     => ( ( A2 != B2 )
       => ( member1967978193chanID @ A2 @ A ) ) ) ).

% insertE
thf(fact_125_insertE,axiom,
    ! [A2: secrecy_specID,B2: secrecy_specID,A: set_Secrecy_specID] :
      ( ( member740727610specID @ A2 @ ( insert1671894227specID @ B2 @ A ) )
     => ( ( A2 != B2 )
       => ( member740727610specID @ A2 @ A ) ) ) ).

% insertE
thf(fact_126_insertE,axiom,
    ! [A2: secrecy_Secrets,B2: secrecy_Secrets,A: set_Secrecy_Secrets] :
      ( ( member2138796287ecrets @ A2 @ ( insert565337510ecrets @ B2 @ A ) )
     => ( ( A2 != B2 )
       => ( member2138796287ecrets @ A2 @ A ) ) ) ).

% insertE
thf(fact_127_insertE,axiom,
    ! [A2: secrecy_Keys,B2: secrecy_Keys,A: set_Secrecy_Keys] :
      ( ( member_Secrecy_Keys @ A2 @ ( insert_Secrecy_Keys @ B2 @ A ) )
     => ( ( A2 != B2 )
       => ( member_Secrecy_Keys @ A2 @ A ) ) ) ).

% insertE
thf(fact_128_singleton__inject,axiom,
    ! [A2: secrecy_specID,B2: secrecy_specID] :
      ( ( ( insert1671894227specID @ A2 @ bot_bo1785112429specID )
        = ( insert1671894227specID @ B2 @ bot_bo1785112429specID ) )
     => ( A2 = B2 ) ) ).

% singleton_inject
thf(fact_129_singleton__inject,axiom,
    ! [A2: secrecy_chanID,B2: secrecy_chanID] :
      ( ( ( insert751661162chanID @ A2 @ bot_bo443197316chanID )
        = ( insert751661162chanID @ B2 @ bot_bo443197316chanID ) )
     => ( A2 = B2 ) ) ).

% singleton_inject
thf(fact_130_insert__not__empty,axiom,
    ! [A2: secrecy_specID,A: set_Secrecy_specID] :
      ( ( insert1671894227specID @ A2 @ A )
     != bot_bo1785112429specID ) ).

% insert_not_empty
thf(fact_131_insert__not__empty,axiom,
    ! [A2: secrecy_chanID,A: set_Secrecy_chanID] :
      ( ( insert751661162chanID @ A2 @ A )
     != bot_bo443197316chanID ) ).

% insert_not_empty
thf(fact_132_doubleton__eq__iff,axiom,
    ! [A2: secrecy_specID,B2: secrecy_specID,C: secrecy_specID,D: secrecy_specID] :
      ( ( ( insert1671894227specID @ A2 @ ( insert1671894227specID @ B2 @ bot_bo1785112429specID ) )
        = ( insert1671894227specID @ C @ ( insert1671894227specID @ D @ bot_bo1785112429specID ) ) )
      = ( ( ( A2 = C )
          & ( B2 = D ) )
        | ( ( A2 = D )
          & ( B2 = C ) ) ) ) ).

% doubleton_eq_iff
thf(fact_133_doubleton__eq__iff,axiom,
    ! [A2: secrecy_chanID,B2: secrecy_chanID,C: secrecy_chanID,D: secrecy_chanID] :
      ( ( ( insert751661162chanID @ A2 @ ( insert751661162chanID @ B2 @ bot_bo443197316chanID ) )
        = ( insert751661162chanID @ C @ ( insert751661162chanID @ D @ bot_bo443197316chanID ) ) )
      = ( ( ( A2 = C )
          & ( B2 = D ) )
        | ( ( A2 = D )
          & ( B2 = C ) ) ) ) ).

% doubleton_eq_iff
thf(fact_134_singleton__iff,axiom,
    ! [B2: secrecy_KS,A2: secrecy_KS] :
      ( ( member_Secrecy_KS @ B2 @ ( insert_Secrecy_KS @ A2 @ bot_bo1943466239ecy_KS ) )
      = ( B2 = A2 ) ) ).

% singleton_iff
thf(fact_135_singleton__iff,axiom,
    ! [B2: secrecy_Secrets,A2: secrecy_Secrets] :
      ( ( member2138796287ecrets @ B2 @ ( insert565337510ecrets @ A2 @ bot_bo1124690338ecrets ) )
      = ( B2 = A2 ) ) ).

% singleton_iff
thf(fact_136_singleton__iff,axiom,
    ! [B2: secrecy_Keys,A2: secrecy_Keys] :
      ( ( member_Secrecy_Keys @ B2 @ ( insert_Secrecy_Keys @ A2 @ bot_bo701486667y_Keys ) )
      = ( B2 = A2 ) ) ).

% singleton_iff
thf(fact_137_singleton__iff,axiom,
    ! [B2: secrecy_specID,A2: secrecy_specID] :
      ( ( member740727610specID @ B2 @ ( insert1671894227specID @ A2 @ bot_bo1785112429specID ) )
      = ( B2 = A2 ) ) ).

% singleton_iff
thf(fact_138_singleton__iff,axiom,
    ! [B2: secrecy_chanID,A2: secrecy_chanID] :
      ( ( member1967978193chanID @ B2 @ ( insert751661162chanID @ A2 @ bot_bo443197316chanID ) )
      = ( B2 = A2 ) ) ).

% singleton_iff
thf(fact_139_singletonD,axiom,
    ! [B2: secrecy_KS,A2: secrecy_KS] :
      ( ( member_Secrecy_KS @ B2 @ ( insert_Secrecy_KS @ A2 @ bot_bo1943466239ecy_KS ) )
     => ( B2 = A2 ) ) ).

% singletonD
thf(fact_140_singletonD,axiom,
    ! [B2: secrecy_Secrets,A2: secrecy_Secrets] :
      ( ( member2138796287ecrets @ B2 @ ( insert565337510ecrets @ A2 @ bot_bo1124690338ecrets ) )
     => ( B2 = A2 ) ) ).

% singletonD
thf(fact_141_singletonD,axiom,
    ! [B2: secrecy_Keys,A2: secrecy_Keys] :
      ( ( member_Secrecy_Keys @ B2 @ ( insert_Secrecy_Keys @ A2 @ bot_bo701486667y_Keys ) )
     => ( B2 = A2 ) ) ).

% singletonD
thf(fact_142_singletonD,axiom,
    ! [B2: secrecy_specID,A2: secrecy_specID] :
      ( ( member740727610specID @ B2 @ ( insert1671894227specID @ A2 @ bot_bo1785112429specID ) )
     => ( B2 = A2 ) ) ).

% singletonD
thf(fact_143_singletonD,axiom,
    ! [B2: secrecy_chanID,A2: secrecy_chanID] :
      ( ( member1967978193chanID @ B2 @ ( insert751661162chanID @ A2 @ bot_bo443197316chanID ) )
     => ( B2 = A2 ) ) ).

% singletonD
thf(fact_144_the__elem__eq,axiom,
    ! [X: secrecy_specID] :
      ( ( the_el827131246specID @ ( insert1671894227specID @ X @ bot_bo1785112429specID ) )
      = X ) ).

% the_elem_eq
thf(fact_145_the__elem__eq,axiom,
    ! [X: secrecy_chanID] :
      ( ( the_el2054381829chanID @ ( insert751661162chanID @ X @ bot_bo443197316chanID ) )
      = X ) ).

% the_elem_eq
thf(fact_146_is__singletonI,axiom,
    ! [X: secrecy_specID] : ( is_sin1501507247specID @ ( insert1671894227specID @ X @ bot_bo1785112429specID ) ) ).

% is_singletonI
thf(fact_147_is__singletonI,axiom,
    ! [X: secrecy_chanID] : ( is_sin581274182chanID @ ( insert751661162chanID @ X @ bot_bo443197316chanID ) ) ).

% is_singletonI
thf(fact_148_TBtheorem4a__empty,axiom,
    ! [P: secrecy_specID,E: secrecy_Expression,Q: secrecy_specID,PQ: secrecy_specID] :
      ( ( ( ine @ P @ E )
        | ( ine @ Q @ E ) )
     => ( ( ( subcomponents @ PQ )
          = ( insert1671894227specID @ P @ ( insert1671894227specID @ Q @ bot_bo1785112429specID ) ) )
       => ( ( correctCompositionIn @ PQ )
         => ( ( ( loc @ PQ )
              = bot_bo443197316chanID )
           => ( ine @ PQ @ E ) ) ) ) ) ).

% TBtheorem4a_empty
thf(fact_149_TBtheorem3b,axiom,
    ! [P: secrecy_specID,M2: set_Secrecy_chanID,E: secrecy_Expression,Q: secrecy_specID,PQ: secrecy_specID] :
      ( ~ ( ineM @ P @ M2 @ E )
     => ( ~ ( ineM @ Q @ M2 @ E )
       => ( ( ( subcomponents @ PQ )
            = ( insert1671894227specID @ P @ ( insert1671894227specID @ Q @ bot_bo1785112429specID ) ) )
         => ( ( correctCompositionIn @ PQ )
           => ~ ( ineM @ PQ @ M2 @ E ) ) ) ) ) ).

% TBtheorem3b
thf(fact_150_TBtheorem1b,axiom,
    ! [PQ: secrecy_specID,M2: set_Secrecy_chanID,E: secrecy_Expression,P: secrecy_specID,Q: secrecy_specID] :
      ( ( ineM @ PQ @ M2 @ E )
     => ( ( ( subcomponents @ PQ )
          = ( insert1671894227specID @ P @ ( insert1671894227specID @ Q @ bot_bo1785112429specID ) ) )
       => ( ( correctCompositionIn @ PQ )
         => ( ( ineM @ P @ M2 @ E )
            | ( ineM @ Q @ M2 @ E ) ) ) ) ) ).

% TBtheorem1b
thf(fact_151_Set_Ois__empty__def,axiom,
    ( is_emp1849217361specID
    = ( ^ [A3: set_Secrecy_specID] : ( A3 = bot_bo1785112429specID ) ) ) ).

% Set.is_empty_def
thf(fact_152_Set_Ois__empty__def,axiom,
    ( is_emp928984296chanID
    = ( ^ [A3: set_Secrecy_chanID] : ( A3 = bot_bo443197316chanID ) ) ) ).

% Set.is_empty_def
thf(fact_153_correctCompositionIn__prop1,axiom,
    ! [PQ: secrecy_specID,P: secrecy_specID,Q: secrecy_specID,X: secrecy_chanID] :
      ( ( ( subcomponents @ PQ )
        = ( insert1671894227specID @ P @ ( insert1671894227specID @ Q @ bot_bo1785112429specID ) ) )
     => ( ( correctCompositionIn @ PQ )
       => ( ( member1967978193chanID @ X @ ( ins @ PQ ) )
         => ( ( member1967978193chanID @ X @ ( ins @ P ) )
            | ( member1967978193chanID @ X @ ( ins @ Q ) ) ) ) ) ) ).

% correctCompositionIn_prop1
thf(fact_154_correctCompositionIn__L2,axiom,
    ! [PQ: secrecy_specID,P: secrecy_specID,Q: secrecy_specID,Ch: secrecy_chanID] :
      ( ( ( subcomponents @ PQ )
        = ( insert1671894227specID @ P @ ( insert1671894227specID @ Q @ bot_bo1785112429specID ) ) )
     => ( ( correctCompositionIn @ PQ )
       => ( ( member1967978193chanID @ Ch @ ( ins @ PQ ) )
         => ( ( member1967978193chanID @ Ch @ ( ins @ P ) )
            | ( member1967978193chanID @ Ch @ ( ins @ Q ) ) ) ) ) ) ).

% correctCompositionIn_L2
thf(fact_155_bot__set__def,axiom,
    ( bot_bo1785112429specID
    = ( collec1545797240specID @ bot_bo1767780696ecID_o ) ) ).

% bot_set_def
thf(fact_156_bot__set__def,axiom,
    ( bot_bo443197316chanID
    = ( collec625564175chanID @ bot_bo1012329985anID_o ) ) ).

% bot_set_def
thf(fact_157_ineM__ine,axiom,
    ! [P: secrecy_specID,M2: set_Secrecy_chanID,E: secrecy_Expression] :
      ( ( ineM @ P @ M2 @ E )
     => ( ine @ P @ E ) ) ).

% ineM_ine
thf(fact_158_not__ine__ineM,axiom,
    ! [P: secrecy_specID,E: secrecy_Expression,M2: set_Secrecy_chanID] :
      ( ~ ( ine @ P @ E )
     => ~ ( ineM @ P @ M2 @ E ) ) ).

% not_ine_ineM
thf(fact_159_is__singleton__the__elem,axiom,
    ( is_sin1501507247specID
    = ( ^ [A3: set_Secrecy_specID] :
          ( A3
          = ( insert1671894227specID @ ( the_el827131246specID @ A3 ) @ bot_bo1785112429specID ) ) ) ) ).

% is_singleton_the_elem
thf(fact_160_is__singleton__the__elem,axiom,
    ( is_sin581274182chanID
    = ( ^ [A3: set_Secrecy_chanID] :
          ( A3
          = ( insert751661162chanID @ ( the_el2054381829chanID @ A3 ) @ bot_bo443197316chanID ) ) ) ) ).

% is_singleton_the_elem
thf(fact_161_correctCompositionIn__L1,axiom,
    ! [PQ: secrecy_specID,P: secrecy_specID,Q: secrecy_specID,Ch: secrecy_chanID] :
      ( ( ( subcomponents @ PQ )
        = ( insert1671894227specID @ P @ ( insert1671894227specID @ Q @ bot_bo1785112429specID ) ) )
     => ( ( correctCompositionIn @ PQ )
       => ( ~ ( member1967978193chanID @ Ch @ ( loc @ PQ ) )
         => ( ( member1967978193chanID @ Ch @ ( ins @ P ) )
           => ( member1967978193chanID @ Ch @ ( ins @ PQ ) ) ) ) ) ) ).

% correctCompositionIn_L1
thf(fact_162_is__singletonI_H,axiom,
    ! [A: set_Secrecy_KS] :
      ( ( A != bot_bo1943466239ecy_KS )
     => ( ! [X4: secrecy_KS,Y: secrecy_KS] :
            ( ( member_Secrecy_KS @ X4 @ A )
           => ( ( member_Secrecy_KS @ Y @ A )
             => ( X4 = Y ) ) )
       => ( is_sin1442426433ecy_KS @ A ) ) ) ).

% is_singletonI'
thf(fact_163_is__singletonI_H,axiom,
    ! [A: set_Secrecy_Secrets] :
      ( ( A != bot_bo1124690338ecrets )
     => ( ! [X4: secrecy_Secrets,Y: secrecy_Secrets] :
            ( ( member2138796287ecrets @ X4 @ A )
           => ( ( member2138796287ecrets @ Y @ A )
             => ( X4 = Y ) ) )
       => ( is_sin1002224330ecrets @ A ) ) ) ).

% is_singletonI'
thf(fact_164_is__singletonI_H,axiom,
    ! [A: set_Secrecy_Keys] :
      ( ( A != bot_bo701486667y_Keys )
     => ( ! [X4: secrecy_Keys,Y: secrecy_Keys] :
            ( ( member_Secrecy_Keys @ X4 @ A )
           => ( ( member_Secrecy_Keys @ Y @ A )
             => ( X4 = Y ) ) )
       => ( is_sin1096471309y_Keys @ A ) ) ) ).

% is_singletonI'
thf(fact_165_is__singletonI_H,axiom,
    ! [A: set_Secrecy_specID] :
      ( ( A != bot_bo1785112429specID )
     => ( ! [X4: secrecy_specID,Y: secrecy_specID] :
            ( ( member740727610specID @ X4 @ A )
           => ( ( member740727610specID @ Y @ A )
             => ( X4 = Y ) ) )
       => ( is_sin1501507247specID @ A ) ) ) ).

% is_singletonI'
thf(fact_166_is__singletonI_H,axiom,
    ! [A: set_Secrecy_chanID] :
      ( ( A != bot_bo443197316chanID )
     => ( ! [X4: secrecy_chanID,Y: secrecy_chanID] :
            ( ( member1967978193chanID @ X4 @ A )
           => ( ( member1967978193chanID @ Y @ A )
             => ( X4 = Y ) ) )
       => ( is_sin581274182chanID @ A ) ) ) ).

% is_singletonI'
thf(fact_167_is__singletonE,axiom,
    ! [A: set_Secrecy_specID] :
      ( ( is_sin1501507247specID @ A )
     => ~ ! [X4: secrecy_specID] :
            ( A
           != ( insert1671894227specID @ X4 @ bot_bo1785112429specID ) ) ) ).

% is_singletonE
thf(fact_168_is__singletonE,axiom,
    ! [A: set_Secrecy_chanID] :
      ( ( is_sin581274182chanID @ A )
     => ~ ! [X4: secrecy_chanID] :
            ( A
           != ( insert751661162chanID @ X4 @ bot_bo443197316chanID ) ) ) ).

% is_singletonE
thf(fact_169_is__singleton__def,axiom,
    ( is_sin1501507247specID
    = ( ^ [A3: set_Secrecy_specID] :
        ? [X3: secrecy_specID] :
          ( A3
          = ( insert1671894227specID @ X3 @ bot_bo1785112429specID ) ) ) ) ).

% is_singleton_def
thf(fact_170_is__singleton__def,axiom,
    ( is_sin581274182chanID
    = ( ^ [A3: set_Secrecy_chanID] :
        ? [X3: secrecy_chanID] :
          ( A3
          = ( insert751661162chanID @ X3 @ bot_bo443197316chanID ) ) ) ) ).

% is_singleton_def
thf(fact_171_subcomponents__loc,axiom,
    ! [X: secrecy_specID] :
      ( ( correc965049635ionLoc @ X )
     => ( ( ( subcomponents @ X )
          = bot_bo1785112429specID )
       => ( ( loc @ X )
          = bot_bo443197316chanID ) ) ) ).

% subcomponents_loc
thf(fact_172_TBtheorem4b__notP1,axiom,
    ! [P: secrecy_specID,M2: set_Secrecy_chanID,E: secrecy_Expression,Q: secrecy_specID,PQ: secrecy_specID] :
      ( ( ineM @ P @ M2 @ E )
     => ( ~ ( ineM @ Q @ M2 @ E )
       => ( ( ( subcomponents @ PQ )
            = ( insert1671894227specID @ P @ ( insert1671894227specID @ Q @ bot_bo1785112429specID ) ) )
         => ( ( correctCompositionIn @ PQ )
           => ( ? [Ch2: secrecy_chanID] :
                  ( ( ine_ex1303305700Single @ P @ Ch2 @ E )
                  & ( member1967978193chanID @ Ch2 @ M2 )
                  & ( member1967978193chanID @ Ch2 @ ( loc @ PQ ) ) )
             => ~ ( ineM @ PQ @ M2 @ E ) ) ) ) ) ) ).

% TBtheorem4b_notP1
thf(fact_173_TBtheorem4a__notP1,axiom,
    ! [P: secrecy_specID,E: secrecy_Expression,Q: secrecy_specID,PQ: secrecy_specID] :
      ( ( ine @ P @ E )
     => ( ~ ( ine @ Q @ E )
       => ( ( ( subcomponents @ PQ )
            = ( insert1671894227specID @ P @ ( insert1671894227specID @ Q @ bot_bo1785112429specID ) ) )
         => ( ( correctCompositionIn @ PQ )
           => ( ? [Ch2: secrecy_chanID] :
                  ( ( ine_ex1303305700Single @ P @ Ch2 @ E )
                  & ( member1967978193chanID @ Ch2 @ ( loc @ PQ ) ) )
             => ~ ( ine @ PQ @ E ) ) ) ) ) ) ).

% TBtheorem4a_notP1
thf(fact_174_TBtheorem4b__notP2,axiom,
    ! [Q: secrecy_specID,M2: set_Secrecy_chanID,E: secrecy_Expression,PQ: secrecy_specID,P: secrecy_specID,ChSet: set_Secrecy_chanID] :
      ( ~ ( ineM @ Q @ M2 @ E )
     => ( ( ( subcomponents @ PQ )
          = ( insert1671894227specID @ P @ ( insert1671894227specID @ Q @ bot_bo1785112429specID ) ) )
       => ( ( correctCompositionIn @ PQ )
         => ( ( ine_exprChannelSet @ P @ ChSet @ E )
           => ( ! [X4: secrecy_chanID] :
                  ( ( member1967978193chanID @ X4 @ ChSet )
                 => ( member1967978193chanID @ X4 @ ( loc @ PQ ) ) )
             => ~ ( ineM @ PQ @ M2 @ E ) ) ) ) ) ) ).

% TBtheorem4b_notP2
thf(fact_175_TBtheorem4b__notPQ,axiom,
    ! [PQ: secrecy_specID,P: secrecy_specID,Q: secrecy_specID,ChSetP: set_Secrecy_chanID,E: secrecy_Expression,ChSetQ: set_Secrecy_chanID,M2: set_Secrecy_chanID] :
      ( ( ( subcomponents @ PQ )
        = ( insert1671894227specID @ P @ ( insert1671894227specID @ Q @ bot_bo1785112429specID ) ) )
     => ( ( correctCompositionIn @ PQ )
       => ( ( ine_exprChannelSet @ P @ ChSetP @ E )
         => ( ( ine_exprChannelSet @ Q @ ChSetQ @ E )
           => ( ! [X4: secrecy_chanID] :
                  ( ( member1967978193chanID @ X4 @ ChSetP )
                 => ( member1967978193chanID @ X4 @ ( loc @ PQ ) ) )
             => ( ! [X4: secrecy_chanID] :
                    ( ( member1967978193chanID @ X4 @ ChSetQ )
                   => ( member1967978193chanID @ X4 @ ( loc @ PQ ) ) )
               => ~ ( ineM @ PQ @ M2 @ E ) ) ) ) ) ) ) ).

% TBtheorem4b_notPQ
thf(fact_176_TBtheorem4a__notP2,axiom,
    ! [Q: secrecy_specID,E: secrecy_Expression,PQ: secrecy_specID,P: secrecy_specID,ChSet: set_Secrecy_chanID] :
      ( ~ ( ine @ Q @ E )
     => ( ( ( subcomponents @ PQ )
          = ( insert1671894227specID @ P @ ( insert1671894227specID @ Q @ bot_bo1785112429specID ) ) )
       => ( ( correctCompositionIn @ PQ )
         => ( ( ine_exprChannelSet @ P @ ChSet @ E )
           => ( ! [X4: secrecy_chanID] :
                  ( ( member1967978193chanID @ X4 @ ChSet )
                 => ( member1967978193chanID @ X4 @ ( loc @ PQ ) ) )
             => ~ ( ine @ PQ @ E ) ) ) ) ) ) ).

% TBtheorem4a_notP2
thf(fact_177_TBtheorem4a__notPQ,axiom,
    ! [PQ: secrecy_specID,P: secrecy_specID,Q: secrecy_specID,ChSetP: set_Secrecy_chanID,E: secrecy_Expression,ChSetQ: set_Secrecy_chanID] :
      ( ( ( subcomponents @ PQ )
        = ( insert1671894227specID @ P @ ( insert1671894227specID @ Q @ bot_bo1785112429specID ) ) )
     => ( ( correctCompositionIn @ PQ )
       => ( ( ine_exprChannelSet @ P @ ChSetP @ E )
         => ( ( ine_exprChannelSet @ Q @ ChSetQ @ E )
           => ( ! [X4: secrecy_chanID] :
                  ( ( member1967978193chanID @ X4 @ ChSetP )
                 => ( member1967978193chanID @ X4 @ ( loc @ PQ ) ) )
             => ( ! [X4: secrecy_chanID] :
                    ( ( member1967978193chanID @ X4 @ ChSetQ )
                   => ( member1967978193chanID @ X4 @ ( loc @ PQ ) ) )
               => ~ ( ine @ PQ @ E ) ) ) ) ) ) ) ).

% TBtheorem4a_notPQ
thf(fact_178_ine__exprChannelSingle__Set,axiom,
    ! [P: secrecy_specID,Ch: secrecy_chanID,E: secrecy_Expression] :
      ( ( ine_ex1303305700Single @ P @ Ch @ E )
     => ( ine_exprChannelSet @ P @ ( insert751661162chanID @ Ch @ bot_bo443197316chanID ) @ E ) ) ).

% ine_exprChannelSingle_Set
thf(fact_179_ine__exprChannelSet__Single,axiom,
    ! [P: secrecy_specID,Ch: secrecy_chanID,E: secrecy_Expression] :
      ( ( ine_exprChannelSet @ P @ ( insert751661162chanID @ Ch @ bot_bo443197316chanID ) @ E )
     => ( ine_ex1303305700Single @ P @ Ch @ E ) ) ).

% ine_exprChannelSet_Single
thf(fact_180_ine__empty__exprChannelSet,axiom,
    ! [P: secrecy_specID,ChSet: set_Secrecy_chanID,E: secrecy_Expression] :
      ( ( ine_exprChannelSet @ P @ ChSet @ E )
     => ( ( ChSet = bot_bo443197316chanID )
       => ~ ( ine @ P @ E ) ) ) ).

% ine_empty_exprChannelSet
thf(fact_181_ine__nonempty__exprChannelSet,axiom,
    ! [P: secrecy_specID,ChSet: set_Secrecy_chanID,E: secrecy_Expression] :
      ( ( ine_exprChannelSet @ P @ ChSet @ E )
     => ( ( ChSet != bot_bo443197316chanID )
       => ( ine @ P @ E ) ) ) ).

% ine_nonempty_exprChannelSet
thf(fact_182_locStream__def,axiom,
    ( locStream
    = ( ^ [X3: secrecy_specID] :
          ( ^ [Y4: set_Secrecy_chanID,Z: set_Secrecy_chanID] : ( Y4 = Z )
          @ ( loc @ X3 ) ) ) ) ).

% locStream_def
thf(fact_183_inStream__def,axiom,
    ( inStream
    = ( ^ [X3: secrecy_specID] :
          ( ^ [Y4: set_Secrecy_chanID,Z: set_Secrecy_chanID] : ( Y4 = Z )
          @ ( ins @ X3 ) ) ) ) ).

% inStream_def
thf(fact_184_TBtheorem4b__PQ,axiom,
    ! [P: secrecy_specID,M2: set_Secrecy_chanID,E: secrecy_Expression,Q: secrecy_specID,PQ: secrecy_specID] :
      ( ( ( ineM @ P @ M2 @ E )
        | ( ineM @ Q @ M2 @ E ) )
     => ( ( ( subcomponents @ PQ )
          = ( insert1671894227specID @ P @ ( insert1671894227specID @ Q @ bot_bo1785112429specID ) ) )
       => ( ( correctCompositionIn @ PQ )
         => ( ? [Ch2: secrecy_chanID] :
                ( ( ( member1967978193chanID @ Ch2 @ ( ins @ P ) )
                  | ( member1967978193chanID @ Ch2 @ ( ins @ Q ) ) )
                & ( member1967978193chanID @ Ch2 @ M2 )
                & ( exprChannel @ Ch2 @ E )
                & ~ ( member1967978193chanID @ Ch2 @ ( loc @ PQ ) ) )
           => ( ineM @ PQ @ M2 @ E ) ) ) ) ) ).

% TBtheorem4b_PQ
thf(fact_185_TBtheorem4b__P,axiom,
    ! [P: secrecy_specID,M2: set_Secrecy_chanID,E: secrecy_Expression,PQ: secrecy_specID,Q: secrecy_specID] :
      ( ( ineM @ P @ M2 @ E )
     => ( ( ( subcomponents @ PQ )
          = ( insert1671894227specID @ P @ ( insert1671894227specID @ Q @ bot_bo1785112429specID ) ) )
       => ( ( correctCompositionIn @ PQ )
         => ( ? [Ch2: secrecy_chanID] :
                ( ( member1967978193chanID @ Ch2 @ ( ins @ Q ) )
                & ( exprChannel @ Ch2 @ E )
                & ~ ( member1967978193chanID @ Ch2 @ ( loc @ PQ ) )
                & ( member1967978193chanID @ Ch2 @ M2 ) )
           => ( ineM @ PQ @ M2 @ E ) ) ) ) ) ).

% TBtheorem4b_P
thf(fact_186_TBtheorem4a__PQ,axiom,
    ! [P: secrecy_specID,E: secrecy_Expression,Q: secrecy_specID,PQ: secrecy_specID] :
      ( ( ( ine @ P @ E )
        | ( ine @ Q @ E ) )
     => ( ( ( subcomponents @ PQ )
          = ( insert1671894227specID @ P @ ( insert1671894227specID @ Q @ bot_bo1785112429specID ) ) )
       => ( ( correctCompositionIn @ PQ )
         => ( ? [Ch2: secrecy_chanID] :
                ( ( ( member1967978193chanID @ Ch2 @ ( ins @ P ) )
                  | ( member1967978193chanID @ Ch2 @ ( ins @ Q ) ) )
                & ( exprChannel @ Ch2 @ E )
                & ~ ( member1967978193chanID @ Ch2 @ ( loc @ PQ ) ) )
           => ( ine @ PQ @ E ) ) ) ) ) ).

% TBtheorem4a_PQ
thf(fact_187_TBtheorem4a__P,axiom,
    ! [P: secrecy_specID,E: secrecy_Expression,PQ: secrecy_specID,Q: secrecy_specID] :
      ( ( ine @ P @ E )
     => ( ( ( subcomponents @ PQ )
          = ( insert1671894227specID @ P @ ( insert1671894227specID @ Q @ bot_bo1785112429specID ) ) )
       => ( ( correctCompositionIn @ PQ )
         => ( ? [Ch2: secrecy_chanID] :
                ( ( member1967978193chanID @ Ch2 @ ( ins @ P ) )
                & ( exprChannel @ Ch2 @ E )
                & ~ ( member1967978193chanID @ Ch2 @ ( loc @ PQ ) ) )
           => ( ine @ PQ @ E ) ) ) ) ) ).

% TBtheorem4a_P
thf(fact_188_ine__def,axiom,
    ( ine
    = ( ^ [SP: secrecy_specID,E2: secrecy_Expression] :
        ? [Ch3: secrecy_chanID] :
          ( ( member1967978193chanID @ Ch3 @ ( ins @ SP ) )
          & ( exprChannel @ Ch3 @ E2 ) ) ) ) ).

% ine_def
thf(fact_189_ine__ins__neg1,axiom,
    ! [P: secrecy_specID,M: secrecy_Expression,X: secrecy_chanID] :
      ( ~ ( ine @ P @ M )
     => ( ( exprChannel @ X @ M )
       => ~ ( member1967978193chanID @ X @ ( ins @ P ) ) ) ) ).

% ine_ins_neg1
thf(fact_190_ineM__def,axiom,
    ( ineM
    = ( ^ [SP: secrecy_specID,M3: set_Secrecy_chanID,E2: secrecy_Expression] :
        ? [Ch3: secrecy_chanID] :
          ( ( member1967978193chanID @ Ch3 @ ( ins @ SP ) )
          & ( member1967978193chanID @ Ch3 @ M3 )
          & ( exprChannel @ Ch3 @ E2 ) ) ) ) ).

% ineM_def
thf(fact_191_ineM__L1,axiom,
    ! [Ch: secrecy_chanID,M2: set_Secrecy_chanID,P: secrecy_specID,E: secrecy_Expression] :
      ( ( member1967978193chanID @ Ch @ M2 )
     => ( ( member1967978193chanID @ Ch @ ( ins @ P ) )
       => ( ( exprChannel @ Ch @ E )
         => ( ineM @ P @ M2 @ E ) ) ) ) ).

% ineM_L1
thf(fact_192_ine__exprChannelSet__def,axiom,
    ( ine_exprChannelSet
    = ( ^ [SP: secrecy_specID,ChSet2: set_Secrecy_chanID,E2: secrecy_Expression] :
          ( ! [X3: secrecy_chanID] :
              ( ( member1967978193chanID @ X3 @ ChSet2 )
             => ( ( member1967978193chanID @ X3 @ ( ins @ SP ) )
                & ( exprChannel @ X3 @ E2 ) ) )
          & ! [X3: secrecy_chanID] :
              ( ( ~ ( member1967978193chanID @ X3 @ ChSet2 )
                & ( member1967978193chanID @ X3 @ ( ins @ SP ) ) )
             => ~ ( exprChannel @ X3 @ E2 ) ) ) ) ) ).

% ine_exprChannelSet_def
thf(fact_193_ine__exprChannelSingle__def,axiom,
    ( ine_ex1303305700Single
    = ( ^ [SP: secrecy_specID,Ch3: secrecy_chanID,E2: secrecy_Expression] :
          ( ( member1967978193chanID @ Ch3 @ ( ins @ SP ) )
          & ( exprChannel @ Ch3 @ E2 )
          & ! [X3: secrecy_chanID,T: nat] :
              ( ( ( member1967978193chanID @ X3 @ ( ins @ SP ) )
                & ( X3 != Ch3 ) )
             => ~ ( exprChannel @ X3 @ E2 ) ) ) ) ) ).

% ine_exprChannelSingle_def
thf(fact_194_LocalSecretsComposition__exprChannel__s,axiom,
    ! [X: secrecy_chanID,Secrets: secrecy_Secrets,P: secrecy_specID,Q: secrecy_specID] :
      ( ( exprChannel @ X @ ( secrecy_sE @ Secrets ) )
     => ( ~ ( ine @ P @ ( secrecy_sE @ Secrets ) )
       => ( ~ ( ine @ Q @ ( secrecy_sE @ Secrets ) )
         => ( ~ ( member1967978193chanID @ X @ ( ins @ P ) )
            & ~ ( member1967978193chanID @ X @ ( ins @ Q ) ) ) ) ) ) ).

% LocalSecretsComposition_exprChannel_s
thf(fact_195_correctCompositionInLoc__exprChannel,axiom,
    ! [PQ: secrecy_specID,P: secrecy_specID,Q: secrecy_specID,Ch: secrecy_chanID,M: secrecy_Expression] :
      ( ( ( subcomponents @ PQ )
        = ( insert1671894227specID @ P @ ( insert1671894227specID @ Q @ bot_bo1785112429specID ) ) )
     => ( ( correctCompositionIn @ PQ )
       => ( ( member1967978193chanID @ Ch @ ( ins @ P ) )
         => ( ( exprChannel @ Ch @ M )
           => ( ! [X4: secrecy_chanID] :
                  ( ( member1967978193chanID @ X4 @ ( ins @ PQ ) )
                 => ~ ( exprChannel @ X4 @ M ) )
             => ( member1967978193chanID @ Ch @ ( loc @ PQ ) ) ) ) ) ) ) ).

% correctCompositionInLoc_exprChannel
thf(fact_196_TBlemma3b,axiom,
    ! [P: secrecy_specID,M2: set_Secrecy_chanID,E: secrecy_Expression,Q: secrecy_specID,PQ: secrecy_specID,Ch: secrecy_chanID] :
      ( ~ ( ineM @ P @ M2 @ E )
     => ( ~ ( ineM @ Q @ M2 @ E )
       => ( ( ( subcomponents @ PQ )
            = ( insert1671894227specID @ P @ ( insert1671894227specID @ Q @ bot_bo1785112429specID ) ) )
         => ( ( correctCompositionIn @ PQ )
           => ( ( member1967978193chanID @ Ch @ M2 )
             => ( ( member1967978193chanID @ Ch @ ( ins @ PQ ) )
               => ~ ( exprChannel @ Ch @ E ) ) ) ) ) ) ) ).

% TBlemma3b
thf(fact_197_Collect__empty__eq__bot,axiom,
    ! [P: secrecy_specID > $o] :
      ( ( ( collec1545797240specID @ P )
        = bot_bo1785112429specID )
      = ( P = bot_bo1767780696ecID_o ) ) ).

% Collect_empty_eq_bot
thf(fact_198_Collect__empty__eq__bot,axiom,
    ! [P: secrecy_chanID > $o] :
      ( ( ( collec625564175chanID @ P )
        = bot_bo443197316chanID )
      = ( P = bot_bo1012329985anID_o ) ) ).

% Collect_empty_eq_bot
thf(fact_199_bot__empty__eq,axiom,
    ( bot_bot_Secrecy_KS_o
    = ( ^ [X3: secrecy_KS] : ( member_Secrecy_KS @ X3 @ bot_bo1943466239ecy_KS ) ) ) ).

% bot_empty_eq
thf(fact_200_bot__empty__eq,axiom,
    ( bot_bo1909946299rets_o
    = ( ^ [X3: secrecy_Secrets] : ( member2138796287ecrets @ X3 @ bot_bo1124690338ecrets ) ) ) ).

% bot_empty_eq
thf(fact_201_bot__empty__eq,axiom,
    ( bot_bo2072920570Keys_o
    = ( ^ [X3: secrecy_Keys] : ( member_Secrecy_Keys @ X3 @ bot_bo701486667y_Keys ) ) ) ).

% bot_empty_eq
thf(fact_202_bot__empty__eq,axiom,
    ( bot_bo1767780696ecID_o
    = ( ^ [X3: secrecy_specID] : ( member740727610specID @ X3 @ bot_bo1785112429specID ) ) ) ).

% bot_empty_eq
thf(fact_203_bot__empty__eq,axiom,
    ( bot_bo1012329985anID_o
    = ( ^ [X3: secrecy_chanID] : ( member1967978193chanID @ X3 @ bot_bo443197316chanID ) ) ) ).

% bot_empty_eq
thf(fact_204_LocalSecretsComposition__neg__loc__s,axiom,
    ! [Secret: secrecy_Secrets,P: secrecy_specID,Ch: secrecy_chanID] :
      ( ~ ( member_Secrecy_KS @ ( secrecy_sKS @ Secret ) @ ( localSecrets @ P ) )
     => ( ( exprChannel @ Ch @ ( secrecy_sE @ Secret ) )
       => ( ~ ( member_Secrecy_KS @ ( secrecy_sKS @ Secret ) @ ( specKeysSecrets @ P ) )
         => ~ ( member1967978193chanID @ Ch @ ( loc @ P ) ) ) ) ) ).

% LocalSecretsComposition_neg_loc_s
thf(fact_205_correctCompositionKS__exprChannel__s__P,axiom,
    ! [PQ: secrecy_specID,P: secrecy_specID,Q: secrecy_specID,Secret: secrecy_Secrets,Ch: secrecy_chanID] :
      ( ( ( subcomponents @ PQ )
        = ( insert1671894227specID @ P @ ( insert1671894227specID @ Q @ bot_bo1785112429specID ) ) )
     => ( ( correctCompositionKS @ PQ )
       => ( ~ ( member_Secrecy_KS @ ( secrecy_sKS @ Secret ) @ ( localSecrets @ PQ ) )
         => ( ( member1967978193chanID @ Ch @ ( ins @ P ) )
           => ( ( exprChannel @ Ch @ ( secrecy_sE @ Secret ) )
             => ( ~ ( member_Secrecy_KS @ ( secrecy_sKS @ Secret ) @ ( specKeysSecrets @ PQ ) )
               => ( ( correctCompositionIn @ PQ )
                 => ( ( member1967978193chanID @ Ch @ ( ins @ PQ ) )
                    & ( exprChannel @ Ch @ ( secrecy_sE @ Secret ) ) ) ) ) ) ) ) ) ) ).

% correctCompositionKS_exprChannel_s_P
thf(fact_206_correctCompositionKS__exprChannel__s__Q,axiom,
    ! [PQ: secrecy_specID,P: secrecy_specID,Q: secrecy_specID,Secret: secrecy_Secrets,Ch: secrecy_chanID] :
      ( ( ( subcomponents @ PQ )
        = ( insert1671894227specID @ P @ ( insert1671894227specID @ Q @ bot_bo1785112429specID ) ) )
     => ( ( correctCompositionKS @ PQ )
       => ( ~ ( member_Secrecy_KS @ ( secrecy_sKS @ Secret ) @ ( localSecrets @ PQ ) )
         => ( ( member1967978193chanID @ Ch @ ( ins @ Q ) )
           => ( ( exprChannel @ Ch @ ( secrecy_sE @ Secret ) )
             => ( ~ ( member_Secrecy_KS @ ( secrecy_sKS @ Secret ) @ ( specKeysSecrets @ PQ ) )
               => ( ( correctCompositionIn @ PQ )
                 => ( ( member1967978193chanID @ Ch @ ( ins @ PQ ) )
                    & ( exprChannel @ Ch @ ( secrecy_sE @ Secret ) ) ) ) ) ) ) ) ) ) ).

% correctCompositionKS_exprChannel_s_Q
thf(fact_207_correctCompositionKS__negQ,axiom,
    ! [PQ: secrecy_specID,P: secrecy_specID,Q: secrecy_specID,Ks: secrecy_KS] :
      ( ( ( subcomponents @ PQ )
        = ( insert1671894227specID @ P @ ( insert1671894227specID @ Q @ bot_bo1785112429specID ) ) )
     => ( ( correctCompositionKS @ PQ )
       => ( ~ ( member_Secrecy_KS @ Ks @ ( specKeysSecrets @ PQ ) )
         => ~ ( member_Secrecy_KS @ Ks @ ( specKeysSecrets @ Q ) ) ) ) ) ).

% correctCompositionKS_negQ
thf(fact_208_correctCompositionKS__negP,axiom,
    ! [PQ: secrecy_specID,P: secrecy_specID,Q: secrecy_specID,Ks: secrecy_KS] :
      ( ( ( subcomponents @ PQ )
        = ( insert1671894227specID @ P @ ( insert1671894227specID @ Q @ bot_bo1785112429specID ) ) )
     => ( ( correctCompositionKS @ PQ )
       => ( ~ ( member_Secrecy_KS @ Ks @ ( specKeysSecrets @ PQ ) )
         => ~ ( member_Secrecy_KS @ Ks @ ( specKeysSecrets @ P ) ) ) ) ) ).

% correctCompositionKS_negP
thf(fact_209_correctCompositionKS__neg1,axiom,
    ! [PQ: secrecy_specID,P: secrecy_specID,Q: secrecy_specID,Ks: secrecy_KS] :
      ( ( ( subcomponents @ PQ )
        = ( insert1671894227specID @ P @ ( insert1671894227specID @ Q @ bot_bo1785112429specID ) ) )
     => ( ( correctCompositionKS @ PQ )
       => ( ~ ( member_Secrecy_KS @ Ks @ ( specKeysSecrets @ P ) )
         => ( ~ ( member_Secrecy_KS @ Ks @ ( specKeysSecrets @ Q ) )
           => ~ ( member_Secrecy_KS @ Ks @ ( specKeysSecrets @ PQ ) ) ) ) ) ) ).

% correctCompositionKS_neg1
thf(fact_210_correctCompositionKS__PQ,axiom,
    ! [PQ: secrecy_specID,P: secrecy_specID,Q: secrecy_specID,Ks: secrecy_KS] :
      ( ( ( subcomponents @ PQ )
        = ( insert1671894227specID @ P @ ( insert1671894227specID @ Q @ bot_bo1785112429specID ) ) )
     => ( ( correctCompositionKS @ PQ )
       => ( ( member_Secrecy_KS @ Ks @ ( specKeysSecrets @ PQ ) )
         => ( ( member_Secrecy_KS @ Ks @ ( specKeysSecrets @ P ) )
            | ( member_Secrecy_KS @ Ks @ ( specKeysSecrets @ Q ) ) ) ) ) ) ).

% correctCompositionKS_PQ
thf(fact_211_know__composition2,axiom,
    ! [M: secrecy_KS,P: secrecy_specID,Q: secrecy_specID,PQ: secrecy_specID] :
      ( ~ ( member_Secrecy_KS @ M @ ( specKeysSecrets @ P ) )
     => ( ~ ( member_Secrecy_KS @ M @ ( specKeysSecrets @ Q ) )
       => ( ( knowle945212990e_know @ Q @ M )
         => ( ( ( subcomponents @ PQ )
              = ( insert1671894227specID @ P @ ( insert1671894227specID @ Q @ bot_bo1785112429specID ) ) )
           => ( ( correctCompositionIn @ PQ )
             => ( ( correctCompositionKS @ PQ )
               => ( knowle945212990e_know @ PQ @ M ) ) ) ) ) ) ) ).

% know_composition2
thf(fact_212_know__composition1,axiom,
    ! [M: secrecy_KS,P: secrecy_specID,Q: secrecy_specID,PQ: secrecy_specID] :
      ( ~ ( member_Secrecy_KS @ M @ ( specKeysSecrets @ P ) )
     => ( ~ ( member_Secrecy_KS @ M @ ( specKeysSecrets @ Q ) )
       => ( ( knowle945212990e_know @ P @ M )
         => ( ( ( subcomponents @ PQ )
              = ( insert1671894227specID @ P @ ( insert1671894227specID @ Q @ bot_bo1785112429specID ) ) )
           => ( ( correctCompositionIn @ PQ )
             => ( ( correctCompositionKS @ PQ )
               => ( knowle945212990e_know @ PQ @ M ) ) ) ) ) ) ) ).

% know_composition1
thf(fact_213_know__composition,axiom,
    ! [M: secrecy_KS,P: secrecy_specID,Q: secrecy_specID,PQ: secrecy_specID] :
      ( ~ ( member_Secrecy_KS @ M @ ( specKeysSecrets @ P ) )
     => ( ~ ( member_Secrecy_KS @ M @ ( specKeysSecrets @ Q ) )
       => ( ( ( knowle945212990e_know @ P @ M )
            | ( knowle945212990e_know @ Q @ M ) )
         => ( ( ( subcomponents @ PQ )
              = ( insert1671894227specID @ P @ ( insert1671894227specID @ Q @ bot_bo1785112429specID ) ) )
           => ( ( correctCompositionIn @ PQ )
             => ( ( correctCompositionKS @ PQ )
               => ( knowle945212990e_know @ PQ @ M ) ) ) ) ) ) ) ).

% know_composition
thf(fact_214_correctCompositionKS__exprChannel__s__Qex,axiom,
    ! [PQ: secrecy_specID,P: secrecy_specID,Q: secrecy_specID,Secret: secrecy_Secrets,Ch: secrecy_chanID] :
      ( ( ( subcomponents @ PQ )
        = ( insert1671894227specID @ P @ ( insert1671894227specID @ Q @ bot_bo1785112429specID ) ) )
     => ( ( correctCompositionKS @ PQ )
       => ( ~ ( member_Secrecy_KS @ ( secrecy_sKS @ Secret ) @ ( localSecrets @ PQ ) )
         => ( ( member1967978193chanID @ Ch @ ( ins @ Q ) )
           => ( ( exprChannel @ Ch @ ( secrecy_sE @ Secret ) )
             => ( ~ ( member_Secrecy_KS @ ( secrecy_sKS @ Secret ) @ ( specKeysSecrets @ PQ ) )
               => ( ( correctCompositionIn @ PQ )
                 => ? [Ch4: secrecy_chanID] :
                      ( ( member1967978193chanID @ Ch4 @ ( ins @ PQ ) )
                      & ( exprChannel @ Ch4 @ ( secrecy_sE @ Secret ) ) ) ) ) ) ) ) ) ) ).

% correctCompositionKS_exprChannel_s_Qex
thf(fact_215_correctCompositionKS__exprChannel__s__Pex,axiom,
    ! [PQ: secrecy_specID,P: secrecy_specID,Q: secrecy_specID,Secret: secrecy_Secrets,Ch: secrecy_chanID] :
      ( ( ( subcomponents @ PQ )
        = ( insert1671894227specID @ P @ ( insert1671894227specID @ Q @ bot_bo1785112429specID ) ) )
     => ( ( correctCompositionKS @ PQ )
       => ( ~ ( member_Secrecy_KS @ ( secrecy_sKS @ Secret ) @ ( localSecrets @ PQ ) )
         => ( ( member1967978193chanID @ Ch @ ( ins @ P ) )
           => ( ( exprChannel @ Ch @ ( secrecy_sE @ Secret ) )
             => ( ~ ( member_Secrecy_KS @ ( secrecy_sKS @ Secret ) @ ( specKeysSecrets @ PQ ) )
               => ( ( correctCompositionIn @ PQ )
                 => ? [Ch4: secrecy_chanID] :
                      ( ( member1967978193chanID @ Ch4 @ ( ins @ PQ ) )
                      & ( exprChannel @ Ch4 @ ( secrecy_sE @ Secret ) ) ) ) ) ) ) ) ) ) ).

% correctCompositionKS_exprChannel_s_Pex
thf(fact_216_LocalSecretsComposition__neg__s,axiom,
    ! [PQ: secrecy_specID,P: secrecy_specID,Q: secrecy_specID,M: secrecy_Secrets] :
      ( ( ( subcomponents @ PQ )
        = ( insert1671894227specID @ P @ ( insert1671894227specID @ Q @ bot_bo1785112429specID ) ) )
     => ( ( correc965049635ionLoc @ PQ )
       => ( ( correctCompositionKS @ PQ )
         => ( ~ ( member_Secrecy_KS @ ( secrecy_sKS @ M ) @ ( specKeysSecrets @ P ) )
           => ( ~ ( member_Secrecy_KS @ ( secrecy_sKS @ M ) @ ( specKeysSecrets @ Q ) )
             => ( ~ ( ine @ P @ ( secrecy_sE @ M ) )
               => ( ~ ( ine @ Q @ ( secrecy_sE @ M ) )
                 => ( ~ ( member_Secrecy_KS @ ( secrecy_sKS @ M ) @ ( sup_su1301156759ecy_KS @ ( localSecrets @ P ) @ ( localSecrets @ Q ) ) )
                   => ~ ( member_Secrecy_KS @ ( secrecy_sKS @ M ) @ ( localSecrets @ PQ ) ) ) ) ) ) ) ) ) ) ).

% LocalSecretsComposition_neg_s
thf(fact_217_correctCompositionKS__exprChannel__k__Qex,axiom,
    ! [PQ: secrecy_specID,P: secrecy_specID,Q: secrecy_specID,Key: secrecy_Keys,Ch: secrecy_chanID] :
      ( ( ( subcomponents @ PQ )
        = ( insert1671894227specID @ P @ ( insert1671894227specID @ Q @ bot_bo1785112429specID ) ) )
     => ( ( correctCompositionKS @ PQ )
       => ( ~ ( member_Secrecy_KS @ ( secrecy_kKS @ Key ) @ ( localSecrets @ PQ ) )
         => ( ( member1967978193chanID @ Ch @ ( ins @ Q ) )
           => ( ( exprChannel @ Ch @ ( secrecy_kE @ Key ) )
             => ( ~ ( member_Secrecy_KS @ ( secrecy_kKS @ Key ) @ ( specKeysSecrets @ PQ ) )
               => ( ( correctCompositionIn @ PQ )
                 => ? [Ch4: secrecy_chanID] :
                      ( ( member1967978193chanID @ Ch4 @ ( ins @ PQ ) )
                      & ( exprChannel @ Ch4 @ ( secrecy_kE @ Key ) ) ) ) ) ) ) ) ) ) ).

% correctCompositionKS_exprChannel_k_Qex
thf(fact_218_correctCompositionKS__exprChannel__k__Pex,axiom,
    ! [PQ: secrecy_specID,P: secrecy_specID,Q: secrecy_specID,Key: secrecy_Keys,Ch: secrecy_chanID] :
      ( ( ( subcomponents @ PQ )
        = ( insert1671894227specID @ P @ ( insert1671894227specID @ Q @ bot_bo1785112429specID ) ) )
     => ( ( correctCompositionKS @ PQ )
       => ( ~ ( member_Secrecy_KS @ ( secrecy_kKS @ Key ) @ ( localSecrets @ PQ ) )
         => ( ( member1967978193chanID @ Ch @ ( ins @ P ) )
           => ( ( exprChannel @ Ch @ ( secrecy_kE @ Key ) )
             => ( ~ ( member_Secrecy_KS @ ( secrecy_kKS @ Key ) @ ( specKeysSecrets @ PQ ) )
               => ( ( correctCompositionIn @ PQ )
                 => ? [Ch4: secrecy_chanID] :
                      ( ( member1967978193chanID @ Ch4 @ ( ins @ PQ ) )
                      & ( exprChannel @ Ch4 @ ( secrecy_kE @ Key ) ) ) ) ) ) ) ) ) ) ).

% correctCompositionKS_exprChannel_k_Pex
thf(fact_219_correctCompositionKS__exprChannel__k__Q,axiom,
    ! [PQ: secrecy_specID,P: secrecy_specID,Q: secrecy_specID,Key: secrecy_Keys,Ch: secrecy_chanID] :
      ( ( ( subcomponents @ PQ )
        = ( insert1671894227specID @ P @ ( insert1671894227specID @ Q @ bot_bo1785112429specID ) ) )
     => ( ( correctCompositionKS @ PQ )
       => ( ~ ( member_Secrecy_KS @ ( secrecy_kKS @ Key ) @ ( localSecrets @ PQ ) )
         => ( ( member1967978193chanID @ Ch @ ( ins @ Q ) )
           => ( ( exprChannel @ Ch @ ( secrecy_kE @ Key ) )
             => ( ~ ( member_Secrecy_KS @ ( secrecy_kKS @ Key ) @ ( specKeysSecrets @ PQ ) )
               => ( ( correctCompositionIn @ PQ )
                 => ( ( member1967978193chanID @ Ch @ ( ins @ PQ ) )
                    & ( exprChannel @ Ch @ ( secrecy_kE @ Key ) ) ) ) ) ) ) ) ) ) ).

% correctCompositionKS_exprChannel_k_Q
thf(fact_220_correctCompositionKS__exprChannel__k__P,axiom,
    ! [PQ: secrecy_specID,P: secrecy_specID,Q: secrecy_specID,Key: secrecy_Keys,Ch: secrecy_chanID] :
      ( ( ( subcomponents @ PQ )
        = ( insert1671894227specID @ P @ ( insert1671894227specID @ Q @ bot_bo1785112429specID ) ) )
     => ( ( correctCompositionKS @ PQ )
       => ( ~ ( member_Secrecy_KS @ ( secrecy_kKS @ Key ) @ ( localSecrets @ PQ ) )
         => ( ( member1967978193chanID @ Ch @ ( ins @ P ) )
           => ( ( exprChannel @ Ch @ ( secrecy_kE @ Key ) )
             => ( ~ ( member_Secrecy_KS @ ( secrecy_kKS @ Key ) @ ( specKeysSecrets @ PQ ) )
               => ( ( correctCompositionIn @ PQ )
                 => ( ( member1967978193chanID @ Ch @ ( ins @ PQ ) )
                    & ( exprChannel @ Ch @ ( secrecy_kE @ Key ) ) ) ) ) ) ) ) ) ) ).

% correctCompositionKS_exprChannel_k_P
thf(fact_221_Un__iff,axiom,
    ! [C: secrecy_specID,A: set_Secrecy_specID,B: set_Secrecy_specID] :
      ( ( member740727610specID @ C @ ( sup_su1712599557specID @ A @ B ) )
      = ( ( member740727610specID @ C @ A )
        | ( member740727610specID @ C @ B ) ) ) ).

% Un_iff
thf(fact_222_Un__iff,axiom,
    ! [C: secrecy_Secrets,A: set_Secrecy_Secrets,B: set_Secrecy_Secrets] :
      ( ( member2138796287ecrets @ C @ ( sup_su1029080330ecrets @ A @ B ) )
      = ( ( member2138796287ecrets @ C @ A )
        | ( member2138796287ecrets @ C @ B ) ) ) ).

% Un_iff
thf(fact_223_Un__iff,axiom,
    ! [C: secrecy_Keys,A: set_Secrecy_Keys,B: set_Secrecy_Keys] :
      ( ( member_Secrecy_Keys @ C @ ( sup_su552545507y_Keys @ A @ B ) )
      = ( ( member_Secrecy_Keys @ C @ A )
        | ( member_Secrecy_Keys @ C @ B ) ) ) ).

% Un_iff
thf(fact_224_Un__iff,axiom,
    ! [C: secrecy_KS,A: set_Secrecy_KS,B: set_Secrecy_KS] :
      ( ( member_Secrecy_KS @ C @ ( sup_su1301156759ecy_KS @ A @ B ) )
      = ( ( member_Secrecy_KS @ C @ A )
        | ( member_Secrecy_KS @ C @ B ) ) ) ).

% Un_iff
thf(fact_225_Un__iff,axiom,
    ! [C: secrecy_chanID,A: set_Secrecy_chanID,B: set_Secrecy_chanID] :
      ( ( member1967978193chanID @ C @ ( sup_su370684444chanID @ A @ B ) )
      = ( ( member1967978193chanID @ C @ A )
        | ( member1967978193chanID @ C @ B ) ) ) ).

% Un_iff
thf(fact_226_UnCI,axiom,
    ! [C: secrecy_specID,B: set_Secrecy_specID,A: set_Secrecy_specID] :
      ( ( ~ ( member740727610specID @ C @ B )
       => ( member740727610specID @ C @ A ) )
     => ( member740727610specID @ C @ ( sup_su1712599557specID @ A @ B ) ) ) ).

% UnCI
thf(fact_227_UnCI,axiom,
    ! [C: secrecy_Secrets,B: set_Secrecy_Secrets,A: set_Secrecy_Secrets] :
      ( ( ~ ( member2138796287ecrets @ C @ B )
       => ( member2138796287ecrets @ C @ A ) )
     => ( member2138796287ecrets @ C @ ( sup_su1029080330ecrets @ A @ B ) ) ) ).

% UnCI
thf(fact_228_UnCI,axiom,
    ! [C: secrecy_Keys,B: set_Secrecy_Keys,A: set_Secrecy_Keys] :
      ( ( ~ ( member_Secrecy_Keys @ C @ B )
       => ( member_Secrecy_Keys @ C @ A ) )
     => ( member_Secrecy_Keys @ C @ ( sup_su552545507y_Keys @ A @ B ) ) ) ).

% UnCI
thf(fact_229_UnCI,axiom,
    ! [C: secrecy_KS,B: set_Secrecy_KS,A: set_Secrecy_KS] :
      ( ( ~ ( member_Secrecy_KS @ C @ B )
       => ( member_Secrecy_KS @ C @ A ) )
     => ( member_Secrecy_KS @ C @ ( sup_su1301156759ecy_KS @ A @ B ) ) ) ).

% UnCI
thf(fact_230_UnCI,axiom,
    ! [C: secrecy_chanID,B: set_Secrecy_chanID,A: set_Secrecy_chanID] :
      ( ( ~ ( member1967978193chanID @ C @ B )
       => ( member1967978193chanID @ C @ A ) )
     => ( member1967978193chanID @ C @ ( sup_su370684444chanID @ A @ B ) ) ) ).

% UnCI
thf(fact_231_KS_Oinject_I1_J,axiom,
    ! [X1: secrecy_Keys,Y1: secrecy_Keys] :
      ( ( ( secrecy_kKS @ X1 )
        = ( secrecy_kKS @ Y1 ) )
      = ( X1 = Y1 ) ) ).

% KS.inject(1)
thf(fact_232_Expression_Oinject_I1_J,axiom,
    ! [X1: secrecy_Keys,Y1: secrecy_Keys] :
      ( ( ( secrecy_kE @ X1 )
        = ( secrecy_kE @ Y1 ) )
      = ( X1 = Y1 ) ) ).

% Expression.inject(1)
thf(fact_233_Un__empty,axiom,
    ! [A: set_Secrecy_KS,B: set_Secrecy_KS] :
      ( ( ( sup_su1301156759ecy_KS @ A @ B )
        = bot_bo1943466239ecy_KS )
      = ( ( A = bot_bo1943466239ecy_KS )
        & ( B = bot_bo1943466239ecy_KS ) ) ) ).

% Un_empty
thf(fact_234_Un__empty,axiom,
    ! [A: set_Secrecy_specID,B: set_Secrecy_specID] :
      ( ( ( sup_su1712599557specID @ A @ B )
        = bot_bo1785112429specID )
      = ( ( A = bot_bo1785112429specID )
        & ( B = bot_bo1785112429specID ) ) ) ).

% Un_empty
thf(fact_235_Un__empty,axiom,
    ! [A: set_Secrecy_chanID,B: set_Secrecy_chanID] :
      ( ( ( sup_su370684444chanID @ A @ B )
        = bot_bo443197316chanID )
      = ( ( A = bot_bo443197316chanID )
        & ( B = bot_bo443197316chanID ) ) ) ).

% Un_empty
thf(fact_236_Un__insert__left,axiom,
    ! [A2: secrecy_specID,B: set_Secrecy_specID,C3: set_Secrecy_specID] :
      ( ( sup_su1712599557specID @ ( insert1671894227specID @ A2 @ B ) @ C3 )
      = ( insert1671894227specID @ A2 @ ( sup_su1712599557specID @ B @ C3 ) ) ) ).

% Un_insert_left
thf(fact_237_Un__insert__left,axiom,
    ! [A2: secrecy_KS,B: set_Secrecy_KS,C3: set_Secrecy_KS] :
      ( ( sup_su1301156759ecy_KS @ ( insert_Secrecy_KS @ A2 @ B ) @ C3 )
      = ( insert_Secrecy_KS @ A2 @ ( sup_su1301156759ecy_KS @ B @ C3 ) ) ) ).

% Un_insert_left
thf(fact_238_Un__insert__left,axiom,
    ! [A2: secrecy_chanID,B: set_Secrecy_chanID,C3: set_Secrecy_chanID] :
      ( ( sup_su370684444chanID @ ( insert751661162chanID @ A2 @ B ) @ C3 )
      = ( insert751661162chanID @ A2 @ ( sup_su370684444chanID @ B @ C3 ) ) ) ).

% Un_insert_left
thf(fact_239_Un__insert__right,axiom,
    ! [A: set_Secrecy_specID,A2: secrecy_specID,B: set_Secrecy_specID] :
      ( ( sup_su1712599557specID @ A @ ( insert1671894227specID @ A2 @ B ) )
      = ( insert1671894227specID @ A2 @ ( sup_su1712599557specID @ A @ B ) ) ) ).

% Un_insert_right
thf(fact_240_Un__insert__right,axiom,
    ! [A: set_Secrecy_KS,A2: secrecy_KS,B: set_Secrecy_KS] :
      ( ( sup_su1301156759ecy_KS @ A @ ( insert_Secrecy_KS @ A2 @ B ) )
      = ( insert_Secrecy_KS @ A2 @ ( sup_su1301156759ecy_KS @ A @ B ) ) ) ).

% Un_insert_right
thf(fact_241_Un__insert__right,axiom,
    ! [A: set_Secrecy_chanID,A2: secrecy_chanID,B: set_Secrecy_chanID] :
      ( ( sup_su370684444chanID @ A @ ( insert751661162chanID @ A2 @ B ) )
      = ( insert751661162chanID @ A2 @ ( sup_su370684444chanID @ A @ B ) ) ) ).

% Un_insert_right
thf(fact_242_not__know__k__not__ine,axiom,
    ! [A: secrecy_specID,M: secrecy_Keys] :
      ( ~ ( knowle945212990e_know @ A @ ( secrecy_kKS @ M ) )
     => ~ ( ine @ A @ ( secrecy_kE @ M ) ) ) ).

% not_know_k_not_ine
thf(fact_243_Un__empty__right,axiom,
    ! [A: set_Secrecy_KS] :
      ( ( sup_su1301156759ecy_KS @ A @ bot_bo1943466239ecy_KS )
      = A ) ).

% Un_empty_right
thf(fact_244_Un__empty__right,axiom,
    ! [A: set_Secrecy_specID] :
      ( ( sup_su1712599557specID @ A @ bot_bo1785112429specID )
      = A ) ).

% Un_empty_right
thf(fact_245_Un__empty__right,axiom,
    ! [A: set_Secrecy_chanID] :
      ( ( sup_su370684444chanID @ A @ bot_bo443197316chanID )
      = A ) ).

% Un_empty_right
thf(fact_246_Un__empty__left,axiom,
    ! [B: set_Secrecy_KS] :
      ( ( sup_su1301156759ecy_KS @ bot_bo1943466239ecy_KS @ B )
      = B ) ).

% Un_empty_left
thf(fact_247_Un__empty__left,axiom,
    ! [B: set_Secrecy_specID] :
      ( ( sup_su1712599557specID @ bot_bo1785112429specID @ B )
      = B ) ).

% Un_empty_left
thf(fact_248_Un__empty__left,axiom,
    ! [B: set_Secrecy_chanID] :
      ( ( sup_su370684444chanID @ bot_bo443197316chanID @ B )
      = B ) ).

% Un_empty_left
thf(fact_249_KS2Expression_Osimps_I1_J,axiom,
    ! [M: secrecy_Keys] :
      ( ( secrec1016742679ession @ ( secrecy_kKS @ M ) )
      = ( secrecy_kE @ M ) ) ).

% KS2Expression.simps(1)
thf(fact_250_Un__left__commute,axiom,
    ! [A: set_Secrecy_KS,B: set_Secrecy_KS,C3: set_Secrecy_KS] :
      ( ( sup_su1301156759ecy_KS @ A @ ( sup_su1301156759ecy_KS @ B @ C3 ) )
      = ( sup_su1301156759ecy_KS @ B @ ( sup_su1301156759ecy_KS @ A @ C3 ) ) ) ).

% Un_left_commute
thf(fact_251_Un__left__commute,axiom,
    ! [A: set_Secrecy_chanID,B: set_Secrecy_chanID,C3: set_Secrecy_chanID] :
      ( ( sup_su370684444chanID @ A @ ( sup_su370684444chanID @ B @ C3 ) )
      = ( sup_su370684444chanID @ B @ ( sup_su370684444chanID @ A @ C3 ) ) ) ).

% Un_left_commute
thf(fact_252_Un__left__absorb,axiom,
    ! [A: set_Secrecy_KS,B: set_Secrecy_KS] :
      ( ( sup_su1301156759ecy_KS @ A @ ( sup_su1301156759ecy_KS @ A @ B ) )
      = ( sup_su1301156759ecy_KS @ A @ B ) ) ).

% Un_left_absorb
thf(fact_253_Un__left__absorb,axiom,
    ! [A: set_Secrecy_chanID,B: set_Secrecy_chanID] :
      ( ( sup_su370684444chanID @ A @ ( sup_su370684444chanID @ A @ B ) )
      = ( sup_su370684444chanID @ A @ B ) ) ).

% Un_left_absorb
thf(fact_254_Un__commute,axiom,
    ( sup_su1301156759ecy_KS
    = ( ^ [A3: set_Secrecy_KS,B4: set_Secrecy_KS] : ( sup_su1301156759ecy_KS @ B4 @ A3 ) ) ) ).

% Un_commute
thf(fact_255_Un__commute,axiom,
    ( sup_su370684444chanID
    = ( ^ [A3: set_Secrecy_chanID,B4: set_Secrecy_chanID] : ( sup_su370684444chanID @ B4 @ A3 ) ) ) ).

% Un_commute
thf(fact_256_Un__absorb,axiom,
    ! [A: set_Secrecy_KS] :
      ( ( sup_su1301156759ecy_KS @ A @ A )
      = A ) ).

% Un_absorb
thf(fact_257_Un__absorb,axiom,
    ! [A: set_Secrecy_chanID] :
      ( ( sup_su370684444chanID @ A @ A )
      = A ) ).

% Un_absorb
thf(fact_258_Un__assoc,axiom,
    ! [A: set_Secrecy_KS,B: set_Secrecy_KS,C3: set_Secrecy_KS] :
      ( ( sup_su1301156759ecy_KS @ ( sup_su1301156759ecy_KS @ A @ B ) @ C3 )
      = ( sup_su1301156759ecy_KS @ A @ ( sup_su1301156759ecy_KS @ B @ C3 ) ) ) ).

% Un_assoc
thf(fact_259_Un__assoc,axiom,
    ! [A: set_Secrecy_chanID,B: set_Secrecy_chanID,C3: set_Secrecy_chanID] :
      ( ( sup_su370684444chanID @ ( sup_su370684444chanID @ A @ B ) @ C3 )
      = ( sup_su370684444chanID @ A @ ( sup_su370684444chanID @ B @ C3 ) ) ) ).

% Un_assoc
thf(fact_260_ball__Un,axiom,
    ! [A: set_Secrecy_KS,B: set_Secrecy_KS,P: secrecy_KS > $o] :
      ( ( ! [X3: secrecy_KS] :
            ( ( member_Secrecy_KS @ X3 @ ( sup_su1301156759ecy_KS @ A @ B ) )
           => ( P @ X3 ) ) )
      = ( ! [X3: secrecy_KS] :
            ( ( member_Secrecy_KS @ X3 @ A )
           => ( P @ X3 ) )
        & ! [X3: secrecy_KS] :
            ( ( member_Secrecy_KS @ X3 @ B )
           => ( P @ X3 ) ) ) ) ).

% ball_Un
thf(fact_261_ball__Un,axiom,
    ! [A: set_Secrecy_chanID,B: set_Secrecy_chanID,P: secrecy_chanID > $o] :
      ( ( ! [X3: secrecy_chanID] :
            ( ( member1967978193chanID @ X3 @ ( sup_su370684444chanID @ A @ B ) )
           => ( P @ X3 ) ) )
      = ( ! [X3: secrecy_chanID] :
            ( ( member1967978193chanID @ X3 @ A )
           => ( P @ X3 ) )
        & ! [X3: secrecy_chanID] :
            ( ( member1967978193chanID @ X3 @ B )
           => ( P @ X3 ) ) ) ) ).

% ball_Un
thf(fact_262_bex__Un,axiom,
    ! [A: set_Secrecy_KS,B: set_Secrecy_KS,P: secrecy_KS > $o] :
      ( ( ? [X3: secrecy_KS] :
            ( ( member_Secrecy_KS @ X3 @ ( sup_su1301156759ecy_KS @ A @ B ) )
            & ( P @ X3 ) ) )
      = ( ? [X3: secrecy_KS] :
            ( ( member_Secrecy_KS @ X3 @ A )
            & ( P @ X3 ) )
        | ? [X3: secrecy_KS] :
            ( ( member_Secrecy_KS @ X3 @ B )
            & ( P @ X3 ) ) ) ) ).

% bex_Un
thf(fact_263_bex__Un,axiom,
    ! [A: set_Secrecy_chanID,B: set_Secrecy_chanID,P: secrecy_chanID > $o] :
      ( ( ? [X3: secrecy_chanID] :
            ( ( member1967978193chanID @ X3 @ ( sup_su370684444chanID @ A @ B ) )
            & ( P @ X3 ) ) )
      = ( ? [X3: secrecy_chanID] :
            ( ( member1967978193chanID @ X3 @ A )
            & ( P @ X3 ) )
        | ? [X3: secrecy_chanID] :
            ( ( member1967978193chanID @ X3 @ B )
            & ( P @ X3 ) ) ) ) ).

% bex_Un
thf(fact_264_UnI2,axiom,
    ! [C: secrecy_specID,B: set_Secrecy_specID,A: set_Secrecy_specID] :
      ( ( member740727610specID @ C @ B )
     => ( member740727610specID @ C @ ( sup_su1712599557specID @ A @ B ) ) ) ).

% UnI2
thf(fact_265_UnI2,axiom,
    ! [C: secrecy_Secrets,B: set_Secrecy_Secrets,A: set_Secrecy_Secrets] :
      ( ( member2138796287ecrets @ C @ B )
     => ( member2138796287ecrets @ C @ ( sup_su1029080330ecrets @ A @ B ) ) ) ).

% UnI2
thf(fact_266_UnI2,axiom,
    ! [C: secrecy_Keys,B: set_Secrecy_Keys,A: set_Secrecy_Keys] :
      ( ( member_Secrecy_Keys @ C @ B )
     => ( member_Secrecy_Keys @ C @ ( sup_su552545507y_Keys @ A @ B ) ) ) ).

% UnI2
thf(fact_267_UnI2,axiom,
    ! [C: secrecy_KS,B: set_Secrecy_KS,A: set_Secrecy_KS] :
      ( ( member_Secrecy_KS @ C @ B )
     => ( member_Secrecy_KS @ C @ ( sup_su1301156759ecy_KS @ A @ B ) ) ) ).

% UnI2
thf(fact_268_UnI2,axiom,
    ! [C: secrecy_chanID,B: set_Secrecy_chanID,A: set_Secrecy_chanID] :
      ( ( member1967978193chanID @ C @ B )
     => ( member1967978193chanID @ C @ ( sup_su370684444chanID @ A @ B ) ) ) ).

% UnI2
thf(fact_269_UnI1,axiom,
    ! [C: secrecy_specID,A: set_Secrecy_specID,B: set_Secrecy_specID] :
      ( ( member740727610specID @ C @ A )
     => ( member740727610specID @ C @ ( sup_su1712599557specID @ A @ B ) ) ) ).

% UnI1
thf(fact_270_UnI1,axiom,
    ! [C: secrecy_Secrets,A: set_Secrecy_Secrets,B: set_Secrecy_Secrets] :
      ( ( member2138796287ecrets @ C @ A )
     => ( member2138796287ecrets @ C @ ( sup_su1029080330ecrets @ A @ B ) ) ) ).

% UnI1
thf(fact_271_UnI1,axiom,
    ! [C: secrecy_Keys,A: set_Secrecy_Keys,B: set_Secrecy_Keys] :
      ( ( member_Secrecy_Keys @ C @ A )
     => ( member_Secrecy_Keys @ C @ ( sup_su552545507y_Keys @ A @ B ) ) ) ).

% UnI1
thf(fact_272_UnI1,axiom,
    ! [C: secrecy_KS,A: set_Secrecy_KS,B: set_Secrecy_KS] :
      ( ( member_Secrecy_KS @ C @ A )
     => ( member_Secrecy_KS @ C @ ( sup_su1301156759ecy_KS @ A @ B ) ) ) ).

% UnI1
thf(fact_273_UnI1,axiom,
    ! [C: secrecy_chanID,A: set_Secrecy_chanID,B: set_Secrecy_chanID] :
      ( ( member1967978193chanID @ C @ A )
     => ( member1967978193chanID @ C @ ( sup_su370684444chanID @ A @ B ) ) ) ).

% UnI1
thf(fact_274_UnE,axiom,
    ! [C: secrecy_specID,A: set_Secrecy_specID,B: set_Secrecy_specID] :
      ( ( member740727610specID @ C @ ( sup_su1712599557specID @ A @ B ) )
     => ( ~ ( member740727610specID @ C @ A )
       => ( member740727610specID @ C @ B ) ) ) ).

% UnE
thf(fact_275_UnE,axiom,
    ! [C: secrecy_Secrets,A: set_Secrecy_Secrets,B: set_Secrecy_Secrets] :
      ( ( member2138796287ecrets @ C @ ( sup_su1029080330ecrets @ A @ B ) )
     => ( ~ ( member2138796287ecrets @ C @ A )
       => ( member2138796287ecrets @ C @ B ) ) ) ).

% UnE
thf(fact_276_UnE,axiom,
    ! [C: secrecy_Keys,A: set_Secrecy_Keys,B: set_Secrecy_Keys] :
      ( ( member_Secrecy_Keys @ C @ ( sup_su552545507y_Keys @ A @ B ) )
     => ( ~ ( member_Secrecy_Keys @ C @ A )
       => ( member_Secrecy_Keys @ C @ B ) ) ) ).

% UnE
thf(fact_277_UnE,axiom,
    ! [C: secrecy_KS,A: set_Secrecy_KS,B: set_Secrecy_KS] :
      ( ( member_Secrecy_KS @ C @ ( sup_su1301156759ecy_KS @ A @ B ) )
     => ( ~ ( member_Secrecy_KS @ C @ A )
       => ( member_Secrecy_KS @ C @ B ) ) ) ).

% UnE
thf(fact_278_UnE,axiom,
    ! [C: secrecy_chanID,A: set_Secrecy_chanID,B: set_Secrecy_chanID] :
      ( ( member1967978193chanID @ C @ ( sup_su370684444chanID @ A @ B ) )
     => ( ~ ( member1967978193chanID @ C @ A )
       => ( member1967978193chanID @ C @ B ) ) ) ).

% UnE
thf(fact_279_KS_Oexhaust,axiom,
    ! [Y3: secrecy_KS] :
      ( ! [X12: secrecy_Keys] :
          ( Y3
         != ( secrecy_kKS @ X12 ) )
     => ~ ! [X22: secrecy_Secrets] :
            ( Y3
           != ( secrecy_sKS @ X22 ) ) ) ).

% KS.exhaust
thf(fact_280_KS_Oinduct,axiom,
    ! [P: secrecy_KS > $o,KS: secrecy_KS] :
      ( ! [X4: secrecy_Keys] : ( P @ ( secrecy_kKS @ X4 ) )
     => ( ! [X4: secrecy_Secrets] : ( P @ ( secrecy_sKS @ X4 ) )
       => ( P @ KS ) ) ) ).

% KS.induct
thf(fact_281_KS_Odistinct_I1_J,axiom,
    ! [X1: secrecy_Keys,X2: secrecy_Secrets] :
      ( ( secrecy_kKS @ X1 )
     != ( secrecy_sKS @ X2 ) ) ).

% KS.distinct(1)
thf(fact_282_Expression_Odistinct_I1_J,axiom,
    ! [X1: secrecy_Keys,X2: secrecy_Secrets] :
      ( ( secrecy_kE @ X1 )
     != ( secrecy_sE @ X2 ) ) ).

% Expression.distinct(1)
thf(fact_283_know_Osimps_I1_J,axiom,
    ! [A: secrecy_specID,M: secrecy_Keys] :
      ( ( knowle945212990e_know @ A @ ( secrecy_kKS @ M ) )
      = ( ( ine @ A @ ( secrecy_kE @ M ) )
        | ( member_Secrecy_KS @ ( secrecy_kKS @ M ) @ ( localSecrets @ A ) ) ) ) ).

% know.simps(1)
thf(fact_284_insert__is__Un,axiom,
    ( insert_Secrecy_KS
    = ( ^ [A4: secrecy_KS] : ( sup_su1301156759ecy_KS @ ( insert_Secrecy_KS @ A4 @ bot_bo1943466239ecy_KS ) ) ) ) ).

% insert_is_Un
thf(fact_285_insert__is__Un,axiom,
    ( insert1671894227specID
    = ( ^ [A4: secrecy_specID] : ( sup_su1712599557specID @ ( insert1671894227specID @ A4 @ bot_bo1785112429specID ) ) ) ) ).

% insert_is_Un
thf(fact_286_insert__is__Un,axiom,
    ( insert751661162chanID
    = ( ^ [A4: secrecy_chanID] : ( sup_su370684444chanID @ ( insert751661162chanID @ A4 @ bot_bo443197316chanID ) ) ) ) ).

% insert_is_Un
thf(fact_287_Un__singleton__iff,axiom,
    ! [A: set_Secrecy_KS,B: set_Secrecy_KS,X: secrecy_KS] :
      ( ( ( sup_su1301156759ecy_KS @ A @ B )
        = ( insert_Secrecy_KS @ X @ bot_bo1943466239ecy_KS ) )
      = ( ( ( A = bot_bo1943466239ecy_KS )
          & ( B
            = ( insert_Secrecy_KS @ X @ bot_bo1943466239ecy_KS ) ) )
        | ( ( A
            = ( insert_Secrecy_KS @ X @ bot_bo1943466239ecy_KS ) )
          & ( B = bot_bo1943466239ecy_KS ) )
        | ( ( A
            = ( insert_Secrecy_KS @ X @ bot_bo1943466239ecy_KS ) )
          & ( B
            = ( insert_Secrecy_KS @ X @ bot_bo1943466239ecy_KS ) ) ) ) ) ).

% Un_singleton_iff
thf(fact_288_Un__singleton__iff,axiom,
    ! [A: set_Secrecy_specID,B: set_Secrecy_specID,X: secrecy_specID] :
      ( ( ( sup_su1712599557specID @ A @ B )
        = ( insert1671894227specID @ X @ bot_bo1785112429specID ) )
      = ( ( ( A = bot_bo1785112429specID )
          & ( B
            = ( insert1671894227specID @ X @ bot_bo1785112429specID ) ) )
        | ( ( A
            = ( insert1671894227specID @ X @ bot_bo1785112429specID ) )
          & ( B = bot_bo1785112429specID ) )
        | ( ( A
            = ( insert1671894227specID @ X @ bot_bo1785112429specID ) )
          & ( B
            = ( insert1671894227specID @ X @ bot_bo1785112429specID ) ) ) ) ) ).

% Un_singleton_iff
thf(fact_289_Un__singleton__iff,axiom,
    ! [A: set_Secrecy_chanID,B: set_Secrecy_chanID,X: secrecy_chanID] :
      ( ( ( sup_su370684444chanID @ A @ B )
        = ( insert751661162chanID @ X @ bot_bo443197316chanID ) )
      = ( ( ( A = bot_bo443197316chanID )
          & ( B
            = ( insert751661162chanID @ X @ bot_bo443197316chanID ) ) )
        | ( ( A
            = ( insert751661162chanID @ X @ bot_bo443197316chanID ) )
          & ( B = bot_bo443197316chanID ) )
        | ( ( A
            = ( insert751661162chanID @ X @ bot_bo443197316chanID ) )
          & ( B
            = ( insert751661162chanID @ X @ bot_bo443197316chanID ) ) ) ) ) ).

% Un_singleton_iff
thf(fact_290_singleton__Un__iff,axiom,
    ! [X: secrecy_KS,A: set_Secrecy_KS,B: set_Secrecy_KS] :
      ( ( ( insert_Secrecy_KS @ X @ bot_bo1943466239ecy_KS )
        = ( sup_su1301156759ecy_KS @ A @ B ) )
      = ( ( ( A = bot_bo1943466239ecy_KS )
          & ( B
            = ( insert_Secrecy_KS @ X @ bot_bo1943466239ecy_KS ) ) )
        | ( ( A
            = ( insert_Secrecy_KS @ X @ bot_bo1943466239ecy_KS ) )
          & ( B = bot_bo1943466239ecy_KS ) )
        | ( ( A
            = ( insert_Secrecy_KS @ X @ bot_bo1943466239ecy_KS ) )
          & ( B
            = ( insert_Secrecy_KS @ X @ bot_bo1943466239ecy_KS ) ) ) ) ) ).

% singleton_Un_iff
thf(fact_291_singleton__Un__iff,axiom,
    ! [X: secrecy_specID,A: set_Secrecy_specID,B: set_Secrecy_specID] :
      ( ( ( insert1671894227specID @ X @ bot_bo1785112429specID )
        = ( sup_su1712599557specID @ A @ B ) )
      = ( ( ( A = bot_bo1785112429specID )
          & ( B
            = ( insert1671894227specID @ X @ bot_bo1785112429specID ) ) )
        | ( ( A
            = ( insert1671894227specID @ X @ bot_bo1785112429specID ) )
          & ( B = bot_bo1785112429specID ) )
        | ( ( A
            = ( insert1671894227specID @ X @ bot_bo1785112429specID ) )
          & ( B
            = ( insert1671894227specID @ X @ bot_bo1785112429specID ) ) ) ) ) ).

% singleton_Un_iff
thf(fact_292_singleton__Un__iff,axiom,
    ! [X: secrecy_chanID,A: set_Secrecy_chanID,B: set_Secrecy_chanID] :
      ( ( ( insert751661162chanID @ X @ bot_bo443197316chanID )
        = ( sup_su370684444chanID @ A @ B ) )
      = ( ( ( A = bot_bo443197316chanID )
          & ( B
            = ( insert751661162chanID @ X @ bot_bo443197316chanID ) ) )
        | ( ( A
            = ( insert751661162chanID @ X @ bot_bo443197316chanID ) )
          & ( B = bot_bo443197316chanID ) )
        | ( ( A
            = ( insert751661162chanID @ X @ bot_bo443197316chanID ) )
          & ( B
            = ( insert751661162chanID @ X @ bot_bo443197316chanID ) ) ) ) ) ).

% singleton_Un_iff
thf(fact_293_LocalSecretsComposition__neg__loc__k,axiom,
    ! [Key: secrecy_Keys,P: secrecy_specID,Ch: secrecy_chanID] :
      ( ~ ( member_Secrecy_KS @ ( secrecy_kKS @ Key ) @ ( localSecrets @ P ) )
     => ( ( exprChannel @ Ch @ ( secrecy_kE @ Key ) )
       => ( ~ ( member_Secrecy_KS @ ( secrecy_kKS @ Key ) @ ( specKeysSecrets @ P ) )
         => ~ ( member1967978193chanID @ Ch @ ( loc @ P ) ) ) ) ) ).

% LocalSecretsComposition_neg_loc_k
thf(fact_294_LocalSecretsComposition__neg__k,axiom,
    ! [PQ: secrecy_specID,P: secrecy_specID,Q: secrecy_specID,M: secrecy_Keys] :
      ( ( ( subcomponents @ PQ )
        = ( insert1671894227specID @ P @ ( insert1671894227specID @ Q @ bot_bo1785112429specID ) ) )
     => ( ( correc965049635ionLoc @ PQ )
       => ( ( correctCompositionKS @ PQ )
         => ( ~ ( member_Secrecy_KS @ ( secrecy_kKS @ M ) @ ( specKeysSecrets @ P ) )
           => ( ~ ( member_Secrecy_KS @ ( secrecy_kKS @ M ) @ ( specKeysSecrets @ Q ) )
             => ( ~ ( ine @ P @ ( secrecy_kE @ M ) )
               => ( ~ ( ine @ Q @ ( secrecy_kE @ M ) )
                 => ( ~ ( member_Secrecy_KS @ ( secrecy_kKS @ M ) @ ( sup_su1301156759ecy_KS @ ( localSecrets @ P ) @ ( localSecrets @ Q ) ) )
                   => ~ ( member_Secrecy_KS @ ( secrecy_kKS @ M ) @ ( localSecrets @ PQ ) ) ) ) ) ) ) ) ) ) ).

% LocalSecretsComposition_neg_k
thf(fact_295_know__composition__neg__ine__k,axiom,
    ! [P: secrecy_specID,Key: secrecy_Keys,Q: secrecy_specID,PQ: secrecy_specID] :
      ( ~ ( knowle945212990e_know @ P @ ( secrecy_kKS @ Key ) )
     => ( ~ ( knowle945212990e_know @ Q @ ( secrecy_kKS @ Key ) )
       => ( ( ( subcomponents @ PQ )
            = ( insert1671894227specID @ P @ ( insert1671894227specID @ Q @ bot_bo1785112429specID ) ) )
         => ( ( correctCompositionIn @ PQ )
           => ~ ( ine @ PQ @ ( secrecy_kE @ Key ) ) ) ) ) ) ).

% know_composition_neg_ine_k
thf(fact_296_LocalSecretsComposition__ine1__k,axiom,
    ! [K: secrecy_Keys,PQ: secrecy_specID,P: secrecy_specID,Q: secrecy_specID] :
      ( ( member_Secrecy_KS @ ( secrecy_kKS @ K ) @ ( localSecrets @ PQ ) )
     => ( ( ( subcomponents @ PQ )
          = ( insert1671894227specID @ P @ ( insert1671894227specID @ Q @ bot_bo1785112429specID ) ) )
       => ( ( correc965049635ionLoc @ PQ )
         => ( ~ ( ine @ Q @ ( secrecy_kE @ K ) )
           => ( ~ ( member_Secrecy_KS @ ( secrecy_kKS @ K ) @ ( localSecrets @ P ) )
             => ( ~ ( member_Secrecy_KS @ ( secrecy_kKS @ K ) @ ( localSecrets @ Q ) )
               => ( ine @ P @ ( secrecy_kE @ K ) ) ) ) ) ) ) ) ).

% LocalSecretsComposition_ine1_k
thf(fact_297_LocalSecretsComposition__ine2__k,axiom,
    ! [K: secrecy_Keys,PQ: secrecy_specID,P: secrecy_specID,Q: secrecy_specID] :
      ( ( member_Secrecy_KS @ ( secrecy_kKS @ K ) @ ( localSecrets @ PQ ) )
     => ( ( ( subcomponents @ PQ )
          = ( insert1671894227specID @ P @ ( insert1671894227specID @ Q @ bot_bo1785112429specID ) ) )
       => ( ( correc965049635ionLoc @ PQ )
         => ( ~ ( ine @ P @ ( secrecy_kE @ K ) )
           => ( ~ ( member_Secrecy_KS @ ( secrecy_kKS @ K ) @ ( localSecrets @ P ) )
             => ( ~ ( member_Secrecy_KS @ ( secrecy_kKS @ K ) @ ( localSecrets @ Q ) )
               => ( ine @ Q @ ( secrecy_kE @ K ) ) ) ) ) ) ) ) ).

% LocalSecretsComposition_ine2_k
thf(fact_298_LocalSecretsComposition__neg1__k,axiom,
    ! [PQ: secrecy_specID,P: secrecy_specID,Q: secrecy_specID,Keys: secrecy_Keys] :
      ( ( ( subcomponents @ PQ )
        = ( insert1671894227specID @ P @ ( insert1671894227specID @ Q @ bot_bo1785112429specID ) ) )
     => ( ( correc965049635ionLoc @ PQ )
       => ( ~ ( ine @ P @ ( secrecy_kE @ Keys ) )
         => ( ~ ( ine @ Q @ ( secrecy_kE @ Keys ) )
           => ( ~ ( member_Secrecy_KS @ ( secrecy_kKS @ Keys ) @ ( localSecrets @ P ) )
             => ( ~ ( member_Secrecy_KS @ ( secrecy_kKS @ Keys ) @ ( localSecrets @ Q ) )
               => ~ ( member_Secrecy_KS @ ( secrecy_kKS @ Keys ) @ ( localSecrets @ PQ ) ) ) ) ) ) ) ) ).

% LocalSecretsComposition_neg1_k
thf(fact_299_LocalSecretsComposition__exprChannel__k,axiom,
    ! [X: secrecy_chanID,Keys: secrecy_Keys,P: secrecy_specID,Q: secrecy_specID] :
      ( ( exprChannel @ X @ ( secrecy_kE @ Keys ) )
     => ( ~ ( ine @ P @ ( secrecy_kE @ Keys ) )
       => ( ~ ( ine @ Q @ ( secrecy_kE @ Keys ) )
         => ( ~ ( member1967978193chanID @ X @ ( ins @ P ) )
            & ~ ( member1967978193chanID @ X @ ( ins @ Q ) ) ) ) ) ) ).

% LocalSecretsComposition_exprChannel_k
thf(fact_300_LocalSecretsComposition__neg,axiom,
    ! [PQ: secrecy_specID,P: secrecy_specID,Q: secrecy_specID,Ks: secrecy_KS] :
      ( ( ( subcomponents @ PQ )
        = ( insert1671894227specID @ P @ ( insert1671894227specID @ Q @ bot_bo1785112429specID ) ) )
     => ( ( correc965049635ionLoc @ PQ )
       => ( ( correctCompositionKS @ PQ )
         => ( ~ ( member_Secrecy_KS @ Ks @ ( specKeysSecrets @ P ) )
           => ( ~ ( member_Secrecy_KS @ Ks @ ( specKeysSecrets @ Q ) )
             => ( ! [M4: secrecy_Keys] :
                    ( ( Ks
                      = ( secrecy_kKS @ M4 ) )
                   => ( ~ ( ine @ P @ ( secrecy_kE @ M4 ) )
                      & ~ ( ine @ Q @ ( secrecy_kE @ M4 ) ) ) )
               => ( ! [M4: secrecy_Secrets] :
                      ( ( Ks
                        = ( secrecy_sKS @ M4 ) )
                     => ( ~ ( ine @ P @ ( secrecy_sE @ M4 ) )
                        & ~ ( ine @ Q @ ( secrecy_sE @ M4 ) ) ) )
                 => ( ~ ( member_Secrecy_KS @ Ks @ ( sup_su1301156759ecy_KS @ ( localSecrets @ P ) @ ( localSecrets @ Q ) ) )
                   => ~ ( member_Secrecy_KS @ Ks @ ( localSecrets @ PQ ) ) ) ) ) ) ) ) ) ) ).

% LocalSecretsComposition_neg
thf(fact_301_LocalSecretsComposition__neg1,axiom,
    ! [PQ: secrecy_specID,P: secrecy_specID,Q: secrecy_specID,Ks: secrecy_KS] :
      ( ( ( subcomponents @ PQ )
        = ( insert1671894227specID @ P @ ( insert1671894227specID @ Q @ bot_bo1785112429specID ) ) )
     => ( ( correc965049635ionLoc @ PQ )
       => ( ! [M4: secrecy_Keys] :
              ( ( Ks
                = ( secrecy_kKS @ M4 ) )
             => ( ~ ( ine @ P @ ( secrecy_kE @ M4 ) )
                & ~ ( ine @ Q @ ( secrecy_kE @ M4 ) ) ) )
         => ( ! [M4: secrecy_Secrets] :
                ( ( Ks
                  = ( secrecy_sKS @ M4 ) )
               => ( ~ ( ine @ P @ ( secrecy_sE @ M4 ) )
                  & ~ ( ine @ Q @ ( secrecy_sE @ M4 ) ) ) )
           => ( ~ ( member_Secrecy_KS @ Ks @ ( localSecrets @ P ) )
             => ( ~ ( member_Secrecy_KS @ Ks @ ( localSecrets @ Q ) )
               => ~ ( member_Secrecy_KS @ Ks @ ( localSecrets @ PQ ) ) ) ) ) ) ) ) ).

% LocalSecretsComposition_neg1
thf(fact_302_sup__bot__left,axiom,
    ! [X: set_Secrecy_KS] :
      ( ( sup_su1301156759ecy_KS @ bot_bo1943466239ecy_KS @ X )
      = X ) ).

% sup_bot_left
thf(fact_303_sup__bot__left,axiom,
    ! [X: set_Secrecy_specID] :
      ( ( sup_su1712599557specID @ bot_bo1785112429specID @ X )
      = X ) ).

% sup_bot_left
thf(fact_304_sup__bot__left,axiom,
    ! [X: set_Secrecy_chanID] :
      ( ( sup_su370684444chanID @ bot_bo443197316chanID @ X )
      = X ) ).

% sup_bot_left
thf(fact_305_sup__bot__right,axiom,
    ! [X: set_Secrecy_KS] :
      ( ( sup_su1301156759ecy_KS @ X @ bot_bo1943466239ecy_KS )
      = X ) ).

% sup_bot_right
thf(fact_306_sup__bot__right,axiom,
    ! [X: set_Secrecy_specID] :
      ( ( sup_su1712599557specID @ X @ bot_bo1785112429specID )
      = X ) ).

% sup_bot_right
thf(fact_307_sup__bot__right,axiom,
    ! [X: set_Secrecy_chanID] :
      ( ( sup_su370684444chanID @ X @ bot_bo443197316chanID )
      = X ) ).

% sup_bot_right
thf(fact_308_bot__eq__sup__iff,axiom,
    ! [X: set_Secrecy_KS,Y3: set_Secrecy_KS] :
      ( ( bot_bo1943466239ecy_KS
        = ( sup_su1301156759ecy_KS @ X @ Y3 ) )
      = ( ( X = bot_bo1943466239ecy_KS )
        & ( Y3 = bot_bo1943466239ecy_KS ) ) ) ).

% bot_eq_sup_iff
thf(fact_309_bot__eq__sup__iff,axiom,
    ! [X: set_Secrecy_specID,Y3: set_Secrecy_specID] :
      ( ( bot_bo1785112429specID
        = ( sup_su1712599557specID @ X @ Y3 ) )
      = ( ( X = bot_bo1785112429specID )
        & ( Y3 = bot_bo1785112429specID ) ) ) ).

% bot_eq_sup_iff
thf(fact_310_bot__eq__sup__iff,axiom,
    ! [X: set_Secrecy_chanID,Y3: set_Secrecy_chanID] :
      ( ( bot_bo443197316chanID
        = ( sup_su370684444chanID @ X @ Y3 ) )
      = ( ( X = bot_bo443197316chanID )
        & ( Y3 = bot_bo443197316chanID ) ) ) ).

% bot_eq_sup_iff
thf(fact_311_sup__eq__bot__iff,axiom,
    ! [X: set_Secrecy_KS,Y3: set_Secrecy_KS] :
      ( ( ( sup_su1301156759ecy_KS @ X @ Y3 )
        = bot_bo1943466239ecy_KS )
      = ( ( X = bot_bo1943466239ecy_KS )
        & ( Y3 = bot_bo1943466239ecy_KS ) ) ) ).

% sup_eq_bot_iff
thf(fact_312_sup__eq__bot__iff,axiom,
    ! [X: set_Secrecy_specID,Y3: set_Secrecy_specID] :
      ( ( ( sup_su1712599557specID @ X @ Y3 )
        = bot_bo1785112429specID )
      = ( ( X = bot_bo1785112429specID )
        & ( Y3 = bot_bo1785112429specID ) ) ) ).

% sup_eq_bot_iff
thf(fact_313_sup__eq__bot__iff,axiom,
    ! [X: set_Secrecy_chanID,Y3: set_Secrecy_chanID] :
      ( ( ( sup_su370684444chanID @ X @ Y3 )
        = bot_bo443197316chanID )
      = ( ( X = bot_bo443197316chanID )
        & ( Y3 = bot_bo443197316chanID ) ) ) ).

% sup_eq_bot_iff
thf(fact_314_sup__bot_Oright__neutral,axiom,
    ! [A2: set_Secrecy_specID] :
      ( ( sup_su1712599557specID @ A2 @ bot_bo1785112429specID )
      = A2 ) ).

% sup_bot.right_neutral
thf(fact_315_sup__bot_Oright__neutral,axiom,
    ! [A2: set_Secrecy_chanID] :
      ( ( sup_su370684444chanID @ A2 @ bot_bo443197316chanID )
      = A2 ) ).

% sup_bot.right_neutral
thf(fact_316_ineM__Un1,axiom,
    ! [P: secrecy_specID,A: set_Secrecy_chanID,E: secrecy_Expression,B: set_Secrecy_chanID] :
      ( ( ineM @ P @ A @ E )
     => ( ineM @ P @ ( sup_su370684444chanID @ A @ B ) @ E ) ) ).

% ineM_Un1
thf(fact_317_notSpecKeysSecretsExpr__def,axiom,
    ( notSpe1586871396tsExpr
    = ( ^ [P2: secrecy_specID,E3: list_S529945141ession] :
          ( ! [X3: secrecy_Keys] :
              ( ( mem_Se1967555118ession @ ( secrecy_kE @ X3 ) @ E3 )
             => ~ ( member_Secrecy_KS @ ( secrecy_kKS @ X3 ) @ ( specKeysSecrets @ P2 ) ) )
          & ! [Y5: secrecy_Secrets] :
              ( ( mem_Se1967555118ession @ ( secrecy_sE @ Y5 ) @ E3 )
             => ~ ( member_Secrecy_KS @ ( secrecy_sKS @ Y5 ) @ ( specKeysSecrets @ P2 ) ) ) ) ) ) ).

% notSpecKeysSecretsExpr_def
thf(fact_318_TBtheorem5a__empty,axiom,
    ! [P: secrecy_specID,E: secrecy_Expression,Q: secrecy_specID,PQ: secrecy_specID] :
      ( ( ( eout @ P @ E )
        | ( eout @ Q @ E ) )
     => ( ( ( subcomponents @ PQ )
          = ( insert1671894227specID @ P @ ( insert1671894227specID @ Q @ bot_bo1785112429specID ) ) )
       => ( ( correc990227761ionOut @ PQ )
         => ( ( ( loc @ PQ )
              = bot_bo443197316chanID )
           => ( eout @ PQ @ E ) ) ) ) ) ).

% TBtheorem5a_empty
thf(fact_319_correctCompositionKS__subcomp4,axiom,
    ! [C3: secrecy_specID,X: secrecy_specID,Xa: secrecy_Secrets] :
      ( ( correctCompositionKS @ C3 )
     => ( ( member740727610specID @ X @ ( subcomponents @ C3 ) )
       => ( ( member2138796287ecrets @ Xa @ ( specSecrets @ X ) )
         => ( member2138796287ecrets @ Xa @ ( specSecrets @ C3 ) ) ) ) ) ).

% correctCompositionKS_subcomp4
thf(fact_320_TBtheorem2a,axiom,
    ! [PQ: secrecy_specID,E: secrecy_Expression,P: secrecy_specID,Q: secrecy_specID] :
      ( ( eout @ PQ @ E )
     => ( ( ( subcomponents @ PQ )
          = ( insert1671894227specID @ P @ ( insert1671894227specID @ Q @ bot_bo1785112429specID ) ) )
       => ( ( correc990227761ionOut @ PQ )
         => ( ( eout @ P @ E )
            | ( eout @ Q @ E ) ) ) ) ) ).

% TBtheorem2a
thf(fact_321_correctCompositionKS__subcomp2,axiom,
    ! [C3: secrecy_specID,X: secrecy_specID,Xa: secrecy_Secrets] :
      ( ( correctCompositionKS @ C3 )
     => ( ( member740727610specID @ X @ ( subcomponents @ C3 ) )
       => ( ( member2138796287ecrets @ Xa @ ( specSecrets @ C3 ) )
         => ? [X4: secrecy_specID] :
              ( ( member740727610specID @ X4 @ ( subcomponents @ C3 ) )
              & ( member2138796287ecrets @ Xa @ ( specSecrets @ X4 ) ) ) ) ) ) ).

% correctCompositionKS_subcomp2
thf(fact_322_TBtheorem5a__notP1,axiom,
    ! [P: secrecy_specID,E: secrecy_Expression,Q: secrecy_specID,PQ: secrecy_specID] :
      ( ( eout @ P @ E )
     => ( ~ ( eout @ Q @ E )
       => ( ( ( subcomponents @ PQ )
            = ( insert1671894227specID @ P @ ( insert1671894227specID @ Q @ bot_bo1785112429specID ) ) )
         => ( ( correc990227761ionOut @ PQ )
           => ( ? [Ch2: secrecy_chanID] :
                  ( ( out_ex985543062Single @ P @ Ch2 @ E )
                  & ( member1967978193chanID @ Ch2 @ ( loc @ PQ ) ) )
             => ~ ( eout @ PQ @ E ) ) ) ) ) ) ).

% TBtheorem5a_notP1
thf(fact_323_TBtheorem5a__notP2,axiom,
    ! [Q: secrecy_specID,E: secrecy_Expression,PQ: secrecy_specID,P: secrecy_specID,ChSet: set_Secrecy_chanID] :
      ( ~ ( eout @ Q @ E )
     => ( ( ( subcomponents @ PQ )
          = ( insert1671894227specID @ P @ ( insert1671894227specID @ Q @ bot_bo1785112429specID ) ) )
       => ( ( correc990227761ionOut @ PQ )
         => ( ( out_exprChannelSet @ P @ ChSet @ E )
           => ( ! [X4: secrecy_chanID] :
                  ( ( member1967978193chanID @ X4 @ ChSet )
                 => ( member1967978193chanID @ X4 @ ( loc @ PQ ) ) )
             => ~ ( eout @ PQ @ E ) ) ) ) ) ) ).

% TBtheorem5a_notP2
thf(fact_324_TBtheorem5a__notPQ,axiom,
    ! [PQ: secrecy_specID,P: secrecy_specID,Q: secrecy_specID,ChSetP: set_Secrecy_chanID,E: secrecy_Expression,ChSetQ: set_Secrecy_chanID] :
      ( ( ( subcomponents @ PQ )
        = ( insert1671894227specID @ P @ ( insert1671894227specID @ Q @ bot_bo1785112429specID ) ) )
     => ( ( correc990227761ionOut @ PQ )
       => ( ( out_exprChannelSet @ P @ ChSetP @ E )
         => ( ( out_exprChannelSet @ Q @ ChSetQ @ E )
           => ( ! [X4: secrecy_chanID] :
                  ( ( member1967978193chanID @ X4 @ ChSetP )
                 => ( member1967978193chanID @ X4 @ ( loc @ PQ ) ) )
             => ( ! [X4: secrecy_chanID] :
                    ( ( member1967978193chanID @ X4 @ ChSetQ )
                   => ( member1967978193chanID @ X4 @ ( loc @ PQ ) ) )
               => ~ ( eout @ PQ @ E ) ) ) ) ) ) ) ).

% TBtheorem5a_notPQ
thf(fact_325_out__exprChannelSingle__Set,axiom,
    ! [P: secrecy_specID,Ch: secrecy_chanID,E: secrecy_Expression] :
      ( ( out_ex985543062Single @ P @ Ch @ E )
     => ( out_exprChannelSet @ P @ ( insert751661162chanID @ Ch @ bot_bo443197316chanID ) @ E ) ) ).

% out_exprChannelSingle_Set
thf(fact_326_out__exprChannelSet__Single,axiom,
    ! [P: secrecy_specID,Ch: secrecy_chanID,E: secrecy_Expression] :
      ( ( out_exprChannelSet @ P @ ( insert751661162chanID @ Ch @ bot_bo443197316chanID ) @ E )
     => ( out_ex985543062Single @ P @ Ch @ E ) ) ).

% out_exprChannelSet_Single
thf(fact_327_TBtheorem5b__notPQ,axiom,
    ! [PQ: secrecy_specID,P: secrecy_specID,Q: secrecy_specID,ChSetP: set_Secrecy_chanID,E: secrecy_Expression,ChSetQ: set_Secrecy_chanID,M2: set_Secrecy_chanID] :
      ( ( ( subcomponents @ PQ )
        = ( insert1671894227specID @ P @ ( insert1671894227specID @ Q @ bot_bo1785112429specID ) ) )
     => ( ( correc990227761ionOut @ PQ )
       => ( ( out_exprChannelSet @ P @ ChSetP @ E )
         => ( ( out_exprChannelSet @ Q @ ChSetQ @ E )
           => ( ( M2
                = ( sup_su370684444chanID @ ChSetP @ ChSetQ ) )
             => ( ! [X4: secrecy_chanID] :
                    ( ( member1967978193chanID @ X4 @ ChSetP )
                   => ( member1967978193chanID @ X4 @ ( loc @ PQ ) ) )
               => ( ! [X4: secrecy_chanID] :
                      ( ( member1967978193chanID @ X4 @ ChSetQ )
                     => ( member1967978193chanID @ X4 @ ( loc @ PQ ) ) )
                 => ~ ( eoutM @ PQ @ M2 @ E ) ) ) ) ) ) ) ) ).

% TBtheorem5b_notPQ
thf(fact_328_eout__know__nonKS__s,axiom,
    ! [M: secrecy_Secrets,A: secrecy_specID] :
      ( ~ ( member2138796287ecrets @ M @ ( specSecrets @ A ) )
     => ( ( eout @ A @ ( secrecy_sE @ M ) )
       => ( ( knowle233900195orrect @ A @ ( secrecy_sKS @ M ) )
         => ( knowle945212990e_know @ A @ ( secrecy_sKS @ M ) ) ) ) ) ).

% eout_know_nonKS_s
thf(fact_329_eoutKnowCorrect__L1s,axiom,
    ! [C3: secrecy_specID,M: secrecy_Secrets] :
      ( ( knowle233900195orrect @ C3 @ ( secrecy_sKS @ M ) )
     => ( ( eout @ C3 @ ( secrecy_sE @ M ) )
       => ( ( member2138796287ecrets @ M @ ( specSecrets @ C3 ) )
          | ( knowle945212990e_know @ C3 @ ( secrecy_sKS @ M ) ) ) ) ) ).

% eoutKnowCorrect_L1s
thf(fact_330_eoutM__eout,axiom,
    ! [P: secrecy_specID,M2: set_Secrecy_chanID,E: secrecy_Expression] :
      ( ( eoutM @ P @ M2 @ E )
     => ( eout @ P @ E ) ) ).

% eoutM_eout
thf(fact_331_not__eout__eoutM,axiom,
    ! [P: secrecy_specID,E: secrecy_Expression,M2: set_Secrecy_chanID] :
      ( ~ ( eout @ P @ E )
     => ~ ( eoutM @ P @ M2 @ E ) ) ).

% not_eout_eoutM
thf(fact_332_TBtheorem2b,axiom,
    ! [PQ: secrecy_specID,M2: set_Secrecy_chanID,E: secrecy_Expression,P: secrecy_specID,Q: secrecy_specID] :
      ( ( eoutM @ PQ @ M2 @ E )
     => ( ( ( subcomponents @ PQ )
          = ( insert1671894227specID @ P @ ( insert1671894227specID @ Q @ bot_bo1785112429specID ) ) )
       => ( ( correc990227761ionOut @ PQ )
         => ( ( eoutM @ P @ M2 @ E )
            | ( eoutM @ Q @ M2 @ E ) ) ) ) ) ).

% TBtheorem2b
thf(fact_333_TBtheorem5b__notP2,axiom,
    ! [Q: secrecy_specID,M2: set_Secrecy_chanID,E: secrecy_Expression,PQ: secrecy_specID,P: secrecy_specID,ChSet: set_Secrecy_chanID] :
      ( ~ ( eoutM @ Q @ M2 @ E )
     => ( ( ( subcomponents @ PQ )
          = ( insert1671894227specID @ P @ ( insert1671894227specID @ Q @ bot_bo1785112429specID ) ) )
       => ( ( correc990227761ionOut @ PQ )
         => ( ( out_exprChannelSet @ P @ ChSet @ E )
           => ( ! [X4: secrecy_chanID] :
                  ( ( member1967978193chanID @ X4 @ ChSet )
                 => ( member1967978193chanID @ X4 @ ( loc @ PQ ) ) )
             => ~ ( eoutM @ PQ @ M2 @ E ) ) ) ) ) ) ).

% TBtheorem5b_notP2
thf(fact_334_TBtheorem5b__notP1,axiom,
    ! [P: secrecy_specID,M2: set_Secrecy_chanID,E: secrecy_Expression,Q: secrecy_specID,PQ: secrecy_specID] :
      ( ( eoutM @ P @ M2 @ E )
     => ( ~ ( eoutM @ Q @ M2 @ E )
       => ( ( ( subcomponents @ PQ )
            = ( insert1671894227specID @ P @ ( insert1671894227specID @ Q @ bot_bo1785112429specID ) ) )
         => ( ( correc990227761ionOut @ PQ )
           => ( ? [Ch2: secrecy_chanID] :
                  ( ( out_ex985543062Single @ P @ Ch2 @ E )
                  & ( member1967978193chanID @ Ch2 @ M2 )
                  & ( member1967978193chanID @ Ch2 @ ( loc @ PQ ) ) )
             => ~ ( eoutM @ PQ @ M2 @ E ) ) ) ) ) ) ).

% TBtheorem5b_notP1
thf(fact_335_eout__know__s,axiom,
    ! [C3: secrecy_specID,M: secrecy_Secrets] :
      ( ( knowle233900195orrect @ C3 @ ( secrecy_sKS @ M ) )
      = ( ( eout @ C3 @ ( secrecy_sE @ M ) )
        = ( ( member2138796287ecrets @ M @ ( specSecrets @ C3 ) )
          | ( knowle945212990e_know @ C3 @ ( secrecy_sKS @ M ) ) ) ) ) ).

% eout_know_s
thf(fact_336_not__know__s__not__eout,axiom,
    ! [M: secrecy_Secrets,A: secrecy_specID] :
      ( ~ ( member2138796287ecrets @ M @ ( specSecrets @ A ) )
     => ( ~ ( knowle945212990e_know @ A @ ( secrecy_sKS @ M ) )
       => ( ( knowle233900195orrect @ A @ ( secrecy_sKS @ M ) )
         => ~ ( eout @ A @ ( secrecy_sE @ M ) ) ) ) ) ).

% not_know_s_not_eout
thf(fact_337_TBtheore54b__P,axiom,
    ! [P: secrecy_specID,M2: set_Secrecy_chanID,E: secrecy_Expression,PQ: secrecy_specID,Q: secrecy_specID] :
      ( ( eoutM @ P @ M2 @ E )
     => ( ( ( subcomponents @ PQ )
          = ( insert1671894227specID @ P @ ( insert1671894227specID @ Q @ bot_bo1785112429specID ) ) )
       => ( ( correc990227761ionOut @ PQ )
         => ( ? [Ch2: secrecy_chanID] :
                ( ( member1967978193chanID @ Ch2 @ ( out @ Q ) )
                & ( exprChannel @ Ch2 @ E )
                & ~ ( member1967978193chanID @ Ch2 @ ( loc @ PQ ) )
                & ( member1967978193chanID @ Ch2 @ M2 ) )
           => ( eoutM @ PQ @ M2 @ E ) ) ) ) ) ).

% TBtheore54b_P
thf(fact_338_TBtheorem5b__PQ,axiom,
    ! [P: secrecy_specID,M2: set_Secrecy_chanID,E: secrecy_Expression,Q: secrecy_specID,PQ: secrecy_specID] :
      ( ( ( eoutM @ P @ M2 @ E )
        | ( eoutM @ Q @ M2 @ E ) )
     => ( ( ( subcomponents @ PQ )
          = ( insert1671894227specID @ P @ ( insert1671894227specID @ Q @ bot_bo1785112429specID ) ) )
       => ( ( correc990227761ionOut @ PQ )
         => ( ? [Ch2: secrecy_chanID] :
                ( ( ( member1967978193chanID @ Ch2 @ ( out @ P ) )
                  | ( member1967978193chanID @ Ch2 @ ( out @ Q ) ) )
                & ( member1967978193chanID @ Ch2 @ M2 )
                & ( exprChannel @ Ch2 @ E )
                & ~ ( member1967978193chanID @ Ch2 @ ( loc @ PQ ) ) )
           => ( eoutM @ PQ @ M2 @ E ) ) ) ) ) ).

% TBtheorem5b_PQ
thf(fact_339_TBtheorem45a__P,axiom,
    ! [P: secrecy_specID,E: secrecy_Expression,PQ: secrecy_specID,Q: secrecy_specID] :
      ( ( eout @ P @ E )
     => ( ( ( subcomponents @ PQ )
          = ( insert1671894227specID @ P @ ( insert1671894227specID @ Q @ bot_bo1785112429specID ) ) )
       => ( ( correc990227761ionOut @ PQ )
         => ( ? [Ch2: secrecy_chanID] :
                ( ( member1967978193chanID @ Ch2 @ ( out @ P ) )
                & ( exprChannel @ Ch2 @ E )
                & ~ ( member1967978193chanID @ Ch2 @ ( loc @ PQ ) ) )
           => ( eout @ PQ @ E ) ) ) ) ) ).

% TBtheorem45a_P
thf(fact_340_eout__def,axiom,
    ( eout
    = ( ^ [SP: secrecy_specID,E2: secrecy_Expression] :
        ? [Ch3: secrecy_chanID] :
          ( ( member1967978193chanID @ Ch3 @ ( out @ SP ) )
          & ( exprChannel @ Ch3 @ E2 ) ) ) ) ).

% eout_def
thf(fact_341_eoutM__def,axiom,
    ( eoutM
    = ( ^ [SP: secrecy_specID,M3: set_Secrecy_chanID,E2: secrecy_Expression] :
        ? [Ch3: secrecy_chanID] :
          ( ( member1967978193chanID @ Ch3 @ ( out @ SP ) )
          & ( member1967978193chanID @ Ch3 @ M3 )
          & ( exprChannel @ Ch3 @ E2 ) ) ) ) ).

% eoutM_def
thf(fact_342_out__exprChannelSet__def,axiom,
    ( out_exprChannelSet
    = ( ^ [SP: secrecy_specID,ChSet2: set_Secrecy_chanID,E2: secrecy_Expression] :
          ( ! [X3: secrecy_chanID] :
              ( ( member1967978193chanID @ X3 @ ChSet2 )
             => ( ( member1967978193chanID @ X3 @ ( out @ SP ) )
                & ( exprChannel @ X3 @ E2 ) ) )
          & ! [X3: secrecy_chanID] :
              ( ( ~ ( member1967978193chanID @ X3 @ ChSet2 )
                & ( member1967978193chanID @ X3 @ ( out @ SP ) ) )
             => ~ ( exprChannel @ X3 @ E2 ) ) ) ) ) ).

% out_exprChannelSet_def
thf(fact_343_out__exprChannelSingle__def,axiom,
    ( out_ex985543062Single
    = ( ^ [SP: secrecy_specID,Ch3: secrecy_chanID,E2: secrecy_Expression] :
          ( ( member1967978193chanID @ Ch3 @ ( out @ SP ) )
          & ( exprChannel @ Ch3 @ E2 )
          & ! [X3: secrecy_chanID,T: nat] :
              ( ( ( member1967978193chanID @ X3 @ ( out @ SP ) )
                & ( X3 != Ch3 ) )
             => ~ ( exprChannel @ X3 @ E2 ) ) ) ) ) ).

% out_exprChannelSingle_def
thf(fact_344_correctCompositionOut__prop1,axiom,
    ! [PQ: secrecy_specID,P: secrecy_specID,Q: secrecy_specID,X: secrecy_chanID] :
      ( ( ( subcomponents @ PQ )
        = ( insert1671894227specID @ P @ ( insert1671894227specID @ Q @ bot_bo1785112429specID ) ) )
     => ( ( correc990227761ionOut @ PQ )
       => ( ( member1967978193chanID @ X @ ( out @ PQ ) )
         => ( ( member1967978193chanID @ X @ ( out @ P ) )
            | ( member1967978193chanID @ X @ ( out @ Q ) ) ) ) ) ) ).

% correctCompositionOut_prop1
thf(fact_345_TBtheorem5a__PQ,axiom,
    ! [P: secrecy_specID,E: secrecy_Expression,Q: secrecy_specID,PQ: secrecy_specID] :
      ( ( ( eout @ P @ E )
        | ( eout @ Q @ E ) )
     => ( ( ( subcomponents @ PQ )
          = ( insert1671894227specID @ P @ ( insert1671894227specID @ Q @ bot_bo1785112429specID ) ) )
       => ( ( correc990227761ionOut @ PQ )
         => ( ? [Ch2: secrecy_chanID] :
                ( ( ( member1967978193chanID @ Ch2 @ ( out @ P ) )
                  | ( member1967978193chanID @ Ch2 @ ( out @ Q ) ) )
                & ( exprChannel @ Ch2 @ E )
                & ~ ( member1967978193chanID @ Ch2 @ ( loc @ PQ ) ) )
           => ( eout @ PQ @ E ) ) ) ) ) ).

% TBtheorem5a_PQ
thf(fact_346_outStream__def,axiom,
    ( outStream
    = ( ^ [X3: secrecy_specID] :
          ( ^ [Y4: set_Secrecy_chanID,Z: set_Secrecy_chanID] : ( Y4 = Z )
          @ ( out @ X3 ) ) ) ) ).

% outStream_def
thf(fact_347_eout__know__k,axiom,
    ! [C3: secrecy_specID,M: secrecy_Keys] :
      ( ( knowle233900195orrect @ C3 @ ( secrecy_kKS @ M ) )
      = ( ( eout @ C3 @ ( secrecy_kE @ M ) )
        = ( ( member_Secrecy_Keys @ M @ ( specKeys @ C3 ) )
          | ( knowle945212990e_know @ C3 @ ( secrecy_kKS @ M ) ) ) ) ) ).

% eout_know_k
thf(fact_348_correctCompositionKS__subcomp1,axiom,
    ! [C3: secrecy_specID,X: secrecy_specID,Xa: secrecy_Keys] :
      ( ( correctCompositionKS @ C3 )
     => ( ( member740727610specID @ X @ ( subcomponents @ C3 ) )
       => ( ( member_Secrecy_Keys @ Xa @ ( specKeys @ C3 ) )
         => ? [X4: secrecy_specID] :
              ( ( member740727610specID @ X4 @ ( subcomponents @ C3 ) )
              & ( member_Secrecy_Keys @ Xa @ ( specKeys @ X4 ) ) ) ) ) ) ).

% correctCompositionKS_subcomp1
thf(fact_349_correctCompositionKS__subcomp3,axiom,
    ! [C3: secrecy_specID,X: secrecy_specID,Xa: secrecy_Keys] :
      ( ( correctCompositionKS @ C3 )
     => ( ( member740727610specID @ X @ ( subcomponents @ C3 ) )
       => ( ( member_Secrecy_Keys @ Xa @ ( specKeys @ X ) )
         => ( member_Secrecy_Keys @ Xa @ ( specKeys @ C3 ) ) ) ) ) ).

% correctCompositionKS_subcomp3
thf(fact_350_eout__know__nonKS__k,axiom,
    ! [M: secrecy_Keys,A: secrecy_specID] :
      ( ~ ( member_Secrecy_Keys @ M @ ( specKeys @ A ) )
     => ( ( eout @ A @ ( secrecy_kE @ M ) )
       => ( ( knowle233900195orrect @ A @ ( secrecy_kKS @ M ) )
         => ( knowle945212990e_know @ A @ ( secrecy_kKS @ M ) ) ) ) ) ).

% eout_know_nonKS_k
thf(fact_351_eoutKnowCorrect__L1k,axiom,
    ! [C3: secrecy_specID,M: secrecy_Keys] :
      ( ( knowle233900195orrect @ C3 @ ( secrecy_kKS @ M ) )
     => ( ( eout @ C3 @ ( secrecy_kE @ M ) )
       => ( ( member_Secrecy_Keys @ M @ ( specKeys @ C3 ) )
          | ( knowle945212990e_know @ C3 @ ( secrecy_kKS @ M ) ) ) ) ) ).

% eoutKnowCorrect_L1k
thf(fact_352_not__know__k__not__eout,axiom,
    ! [M: secrecy_Keys,A: secrecy_specID] :
      ( ~ ( member_Secrecy_Keys @ M @ ( specKeys @ A ) )
     => ( ~ ( knowle945212990e_know @ A @ ( secrecy_kKS @ M ) )
       => ( ( knowle233900195orrect @ A @ ( secrecy_kKS @ M ) )
         => ~ ( eout @ A @ ( secrecy_kE @ M ) ) ) ) ) ).

% not_know_k_not_eout
thf(fact_353_adv__not__know2,axiom,
    ! [P: secrecy_specID,A: secrecy_specID,M: secrecy_Secrets] :
      ( ( ord_le1802272080chanID @ ( out @ P ) @ ( ins @ A ) )
     => ( ~ ( knowle945212990e_know @ A @ ( secrecy_sKS @ M ) )
       => ~ ( eout @ P @ ( secrecy_sE @ M ) ) ) ) ).

% adv_not_know2
thf(fact_354_adv__not__know1,axiom,
    ! [P: secrecy_specID,A: secrecy_specID,M: secrecy_Keys] :
      ( ( ord_le1802272080chanID @ ( out @ P ) @ ( ins @ A ) )
     => ( ~ ( knowle945212990e_know @ A @ ( secrecy_kKS @ M ) )
       => ~ ( eout @ P @ ( secrecy_kE @ M ) ) ) ) ).

% adv_not_know1

% Conjectures (1)
thf(conj_0,conjecture,
    ~ ( member_Secrecy_KS @ ( secrecy_sKS @ secret ) @ ( localSecrets @ pq ) ) ).

%------------------------------------------------------------------------------