TPTP Problem File: ITP086^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : ITP086^1 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer KnowledgeKeysSecrets problem prob_194__3293238_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_194__3293238_1 [Des21]
% Status : Theorem
% Rating : 0.40 v8.2.0, 0.23 v8.1.0, 0.27 v7.5.0
% Syntax : Number of formulae : 414 ( 177 unt; 58 typ; 0 def)
% Number of atoms : 1148 ( 411 equ; 0 cnn)
% Maximal formula atoms : 10 ( 3 avg)
% Number of connectives : 3132 ( 131 ~; 36 |; 110 &;2409 @)
% ( 0 <=>; 446 =>; 0 <=; 0 <~>)
% Maximal formula depth : 19 ( 6 avg)
% Number of types : 7 ( 6 usr)
% Number of type conns : 132 ( 132 >; 0 *; 0 +; 0 <<)
% Number of symbols : 53 ( 52 usr; 8 con; 0-3 aty)
% Number of variables : 1013 ( 84 ^; 902 !; 27 ?;1013 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 15:34:07.612
%------------------------------------------------------------------------------
% Could-be-implicit typings (6)
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__Secrecy____types__OExpression,type,
secrecy_Expression: $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__Nat__Onat,type,
nat: $tType ).
% Explicit typings (52)
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Secrecy____types__OchanID_J,type,
uminus1044094777chanID: set_Secrecy_chanID > set_Secrecy_chanID ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Secrecy____types__OspecID_J,type,
uminus238526242specID: set_Secrecy_specID > set_Secrecy_specID ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Secrecy____types__OchanID_J,type,
inf_in1474033538chanID: set_Secrecy_chanID > set_Secrecy_chanID > set_Secrecy_chanID ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Secrecy____types__OspecID_J,type,
inf_in668465003specID: set_Secrecy_specID > set_Secrecy_specID > set_Secrecy_specID ).
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_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_001_Eo,type,
bot_bot_o: $o ).
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_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Secrecy____types__OspecID_J,type,
ord_le996703545specID: set_Secrecy_specID > set_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_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_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__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__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_Opairwise_001t__Secrecy____types__OchanID,type,
pairwi1741995279chanID: ( secrecy_chanID > secrecy_chanID > $o ) > set_Secrecy_chanID > $o ).
thf(sy_c_Set_Opairwise_001t__Secrecy____types__OspecID,type,
pairwi514744696specID: ( secrecy_specID > secrecy_specID > $o ) > 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_OcorrectInOutLoc,type,
correctInOutLoc: 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__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_ch,type,
ch: secrecy_chanID ).
thf(sy_v_m,type,
m: secrecy_Expression ).
% Relevant facts (355)
thf(fact_0_assms_I2_J,axiom,
correctCompositionIn @ pq ).
% assms(2)
thf(fact_1_assms_I4_J,axiom,
exprChannel @ ch @ m ).
% assms(4)
thf(fact_2_assms_I3_J,axiom,
member1967978193chanID @ ch @ ( ins @ p ) ).
% assms(3)
thf(fact_3_assms_I5_J,axiom,
! [X: secrecy_chanID] :
( ( member1967978193chanID @ X @ ( ins @ pq ) )
=> ~ ( exprChannel @ X @ m ) ) ).
% assms(5)
thf(fact_4_assms_I1_J,axiom,
( ( subcomponents @ pq )
= ( insert1671894227specID @ p @ ( insert1671894227specID @ q @ bot_bo1785112429specID ) ) ) ).
% assms(1)
thf(fact_5_locStream__def,axiom,
( locStream
= ( ^ [X2: secrecy_specID] :
( ^ [Y: set_Secrecy_chanID,Z: set_Secrecy_chanID] : Y = Z
@ ( loc @ X2 ) ) ) ) ).
% locStream_def
thf(fact_6_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_7_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_8_correctCompositionIn__prop1,axiom,
! [PQ: secrecy_specID,P: secrecy_specID,Q: secrecy_specID,X3: secrecy_chanID] :
( ( ( subcomponents @ PQ )
= ( insert1671894227specID @ P @ ( insert1671894227specID @ Q @ bot_bo1785112429specID ) ) )
=> ( ( correctCompositionIn @ PQ )
=> ( ( member1967978193chanID @ X3 @ ( ins @ PQ ) )
=> ( ( member1967978193chanID @ X3 @ ( ins @ P ) )
| ( member1967978193chanID @ X3 @ ( ins @ Q ) ) ) ) ) ) ).
% correctCompositionIn_prop1
thf(fact_9_singletonI,axiom,
! [A: secrecy_specID] : ( member740727610specID @ A @ ( insert1671894227specID @ A @ bot_bo1785112429specID ) ) ).
% singletonI
thf(fact_10_singletonI,axiom,
! [A: secrecy_chanID] : ( member1967978193chanID @ A @ ( insert751661162chanID @ A @ bot_bo443197316chanID ) ) ).
% singletonI
thf(fact_11_TBtheorem4b__P,axiom,
! [P: secrecy_specID,M: set_Secrecy_chanID,E: secrecy_Expression,PQ: secrecy_specID,Q: secrecy_specID] :
( ( ineM @ P @ M @ 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 @ M ) )
=> ( ineM @ PQ @ M @ E ) ) ) ) ) ).
% TBtheorem4b_P
thf(fact_12_TBtheorem4b__PQ,axiom,
! [P: secrecy_specID,M: set_Secrecy_chanID,E: secrecy_Expression,Q: secrecy_specID,PQ: secrecy_specID] :
( ( ( ineM @ P @ M @ E )
| ( ineM @ Q @ M @ E ) )
=> ( ( ( subcomponents @ PQ )
= ( insert1671894227specID @ P @ ( insert1671894227specID @ Q @ bot_bo1785112429specID ) ) )
=> ( ( correctCompositionIn @ PQ )
=> ( ? [Ch2: secrecy_chanID] :
( ( ( member1967978193chanID @ Ch2 @ ( ins @ P ) )
| ( member1967978193chanID @ Ch2 @ ( ins @ Q ) ) )
& ( member1967978193chanID @ Ch2 @ M )
& ( exprChannel @ Ch2 @ E )
& ~ ( member1967978193chanID @ Ch2 @ ( loc @ PQ ) ) )
=> ( ineM @ PQ @ M @ E ) ) ) ) ) ).
% TBtheorem4b_PQ
thf(fact_13_TBlemma3b,axiom,
! [P: secrecy_specID,M: set_Secrecy_chanID,E: secrecy_Expression,Q: secrecy_specID,PQ: secrecy_specID,Ch: secrecy_chanID] :
( ~ ( ineM @ P @ M @ E )
=> ( ~ ( ineM @ Q @ M @ E )
=> ( ( ( subcomponents @ PQ )
= ( insert1671894227specID @ P @ ( insert1671894227specID @ Q @ bot_bo1785112429specID ) ) )
=> ( ( correctCompositionIn @ PQ )
=> ( ( member1967978193chanID @ Ch @ M )
=> ( ( member1967978193chanID @ Ch @ ( ins @ PQ ) )
=> ~ ( exprChannel @ Ch @ E ) ) ) ) ) ) ) ).
% TBlemma3b
thf(fact_14_inStream__def,axiom,
( inStream
= ( ^ [X2: secrecy_specID] :
( ^ [Y: set_Secrecy_chanID,Z: set_Secrecy_chanID] : Y = Z
@ ( ins @ X2 ) ) ) ) ).
% inStream_def
thf(fact_15_insertCI,axiom,
! [A: secrecy_chanID,B: set_Secrecy_chanID,B2: secrecy_chanID] :
( ( ~ ( member1967978193chanID @ A @ B )
=> ( A = B2 ) )
=> ( member1967978193chanID @ A @ ( insert751661162chanID @ B2 @ B ) ) ) ).
% insertCI
thf(fact_16_insertCI,axiom,
! [A: secrecy_specID,B: set_Secrecy_specID,B2: secrecy_specID] :
( ( ~ ( member740727610specID @ A @ B )
=> ( A = B2 ) )
=> ( member740727610specID @ A @ ( insert1671894227specID @ B2 @ B ) ) ) ).
% insertCI
thf(fact_17_insert__iff,axiom,
! [A: secrecy_chanID,B2: secrecy_chanID,A2: set_Secrecy_chanID] :
( ( member1967978193chanID @ A @ ( insert751661162chanID @ B2 @ A2 ) )
= ( ( A = B2 )
| ( member1967978193chanID @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_18_insert__iff,axiom,
! [A: secrecy_specID,B2: secrecy_specID,A2: set_Secrecy_specID] :
( ( member740727610specID @ A @ ( insert1671894227specID @ B2 @ A2 ) )
= ( ( A = B2 )
| ( member740727610specID @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_19_insert__absorb2,axiom,
! [X3: secrecy_specID,A2: set_Secrecy_specID] :
( ( insert1671894227specID @ X3 @ ( insert1671894227specID @ X3 @ A2 ) )
= ( insert1671894227specID @ X3 @ A2 ) ) ).
% insert_absorb2
thf(fact_20_insert__absorb2,axiom,
! [X3: secrecy_chanID,A2: set_Secrecy_chanID] :
( ( insert751661162chanID @ X3 @ ( insert751661162chanID @ X3 @ A2 ) )
= ( insert751661162chanID @ X3 @ A2 ) ) ).
% insert_absorb2
thf(fact_21_empty__iff,axiom,
! [C: secrecy_specID] :
~ ( member740727610specID @ C @ bot_bo1785112429specID ) ).
% empty_iff
thf(fact_22_empty__iff,axiom,
! [C: secrecy_chanID] :
~ ( member1967978193chanID @ C @ bot_bo443197316chanID ) ).
% empty_iff
thf(fact_23_all__not__in__conv,axiom,
! [A2: set_Secrecy_specID] :
( ( ! [X2: secrecy_specID] :
~ ( member740727610specID @ X2 @ A2 ) )
= ( A2 = bot_bo1785112429specID ) ) ).
% all_not_in_conv
thf(fact_24_all__not__in__conv,axiom,
! [A2: set_Secrecy_chanID] :
( ( ! [X2: secrecy_chanID] :
~ ( member1967978193chanID @ X2 @ A2 ) )
= ( A2 = bot_bo443197316chanID ) ) ).
% all_not_in_conv
thf(fact_25_Collect__empty__eq,axiom,
! [P: secrecy_specID > $o] :
( ( ( collec1545797240specID @ P )
= bot_bo1785112429specID )
= ( ! [X2: secrecy_specID] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_26_Collect__empty__eq,axiom,
! [P: secrecy_chanID > $o] :
( ( ( collec625564175chanID @ P )
= bot_bo443197316chanID )
= ( ! [X2: secrecy_chanID] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_27_empty__Collect__eq,axiom,
! [P: secrecy_specID > $o] :
( ( bot_bo1785112429specID
= ( collec1545797240specID @ P ) )
= ( ! [X2: secrecy_specID] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_28_empty__Collect__eq,axiom,
! [P: secrecy_chanID > $o] :
( ( bot_bo443197316chanID
= ( collec625564175chanID @ P ) )
= ( ! [X2: secrecy_chanID] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_29_ineM__def,axiom,
( ineM
= ( ^ [SP: secrecy_specID,M2: set_Secrecy_chanID,E2: secrecy_Expression] :
? [Ch3: secrecy_chanID] :
( ( member1967978193chanID @ Ch3 @ ( ins @ SP ) )
& ( member1967978193chanID @ Ch3 @ M2 )
& ( exprChannel @ Ch3 @ E2 ) ) ) ) ).
% ineM_def
thf(fact_30_ineM__L1,axiom,
! [Ch: secrecy_chanID,M: set_Secrecy_chanID,P: secrecy_specID,E: secrecy_Expression] :
( ( member1967978193chanID @ Ch @ M )
=> ( ( member1967978193chanID @ Ch @ ( ins @ P ) )
=> ( ( exprChannel @ Ch @ E )
=> ( ineM @ P @ M @ E ) ) ) ) ).
% ineM_L1
thf(fact_31_ex__in__conv,axiom,
! [A2: set_Secrecy_specID] :
( ( ? [X2: secrecy_specID] : ( member740727610specID @ X2 @ A2 ) )
= ( A2 != bot_bo1785112429specID ) ) ).
% ex_in_conv
thf(fact_32_ex__in__conv,axiom,
! [A2: set_Secrecy_chanID] :
( ( ? [X2: secrecy_chanID] : ( member1967978193chanID @ X2 @ A2 ) )
= ( A2 != bot_bo443197316chanID ) ) ).
% ex_in_conv
thf(fact_33_equals0I,axiom,
! [A2: set_Secrecy_specID] :
( ! [Y2: secrecy_specID] :
~ ( member740727610specID @ Y2 @ A2 )
=> ( A2 = bot_bo1785112429specID ) ) ).
% equals0I
thf(fact_34_equals0I,axiom,
! [A2: set_Secrecy_chanID] :
( ! [Y2: secrecy_chanID] :
~ ( member1967978193chanID @ Y2 @ A2 )
=> ( A2 = bot_bo443197316chanID ) ) ).
% equals0I
thf(fact_35_equals0D,axiom,
! [A2: set_Secrecy_specID,A: secrecy_specID] :
( ( A2 = bot_bo1785112429specID )
=> ~ ( member740727610specID @ A @ A2 ) ) ).
% equals0D
thf(fact_36_equals0D,axiom,
! [A2: set_Secrecy_chanID,A: secrecy_chanID] :
( ( A2 = bot_bo443197316chanID )
=> ~ ( member1967978193chanID @ A @ A2 ) ) ).
% equals0D
thf(fact_37_emptyE,axiom,
! [A: secrecy_specID] :
~ ( member740727610specID @ A @ bot_bo1785112429specID ) ).
% emptyE
thf(fact_38_emptyE,axiom,
! [A: secrecy_chanID] :
~ ( member1967978193chanID @ A @ bot_bo443197316chanID ) ).
% emptyE
thf(fact_39_mk__disjoint__insert,axiom,
! [A: secrecy_chanID,A2: set_Secrecy_chanID] :
( ( member1967978193chanID @ A @ A2 )
=> ? [B3: set_Secrecy_chanID] :
( ( A2
= ( insert751661162chanID @ A @ B3 ) )
& ~ ( member1967978193chanID @ A @ B3 ) ) ) ).
% mk_disjoint_insert
thf(fact_40_mk__disjoint__insert,axiom,
! [A: secrecy_specID,A2: set_Secrecy_specID] :
( ( member740727610specID @ A @ A2 )
=> ? [B3: set_Secrecy_specID] :
( ( A2
= ( insert1671894227specID @ A @ B3 ) )
& ~ ( member740727610specID @ A @ B3 ) ) ) ).
% mk_disjoint_insert
thf(fact_41_insert__commute,axiom,
! [X3: secrecy_specID,Y3: secrecy_specID,A2: set_Secrecy_specID] :
( ( insert1671894227specID @ X3 @ ( insert1671894227specID @ Y3 @ A2 ) )
= ( insert1671894227specID @ Y3 @ ( insert1671894227specID @ X3 @ A2 ) ) ) ).
% insert_commute
thf(fact_42_insert__commute,axiom,
! [X3: secrecy_chanID,Y3: secrecy_chanID,A2: set_Secrecy_chanID] :
( ( insert751661162chanID @ X3 @ ( insert751661162chanID @ Y3 @ A2 ) )
= ( insert751661162chanID @ Y3 @ ( insert751661162chanID @ X3 @ A2 ) ) ) ).
% insert_commute
thf(fact_43_insert__eq__iff,axiom,
! [A: secrecy_chanID,A2: set_Secrecy_chanID,B2: secrecy_chanID,B: set_Secrecy_chanID] :
( ~ ( member1967978193chanID @ A @ A2 )
=> ( ~ ( member1967978193chanID @ B2 @ B )
=> ( ( ( insert751661162chanID @ A @ A2 )
= ( insert751661162chanID @ B2 @ B ) )
= ( ( ( A = B2 )
=> ( A2 = B ) )
& ( ( A != B2 )
=> ? [C2: set_Secrecy_chanID] :
( ( A2
= ( insert751661162chanID @ B2 @ C2 ) )
& ~ ( member1967978193chanID @ B2 @ C2 )
& ( B
= ( insert751661162chanID @ A @ C2 ) )
& ~ ( member1967978193chanID @ A @ C2 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_44_insert__eq__iff,axiom,
! [A: secrecy_specID,A2: set_Secrecy_specID,B2: secrecy_specID,B: set_Secrecy_specID] :
( ~ ( member740727610specID @ A @ A2 )
=> ( ~ ( member740727610specID @ B2 @ B )
=> ( ( ( insert1671894227specID @ A @ A2 )
= ( insert1671894227specID @ B2 @ B ) )
= ( ( ( A = B2 )
=> ( A2 = B ) )
& ( ( A != B2 )
=> ? [C2: set_Secrecy_specID] :
( ( A2
= ( insert1671894227specID @ B2 @ C2 ) )
& ~ ( member740727610specID @ B2 @ C2 )
& ( B
= ( insert1671894227specID @ A @ C2 ) )
& ~ ( member740727610specID @ A @ C2 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_45_insert__absorb,axiom,
! [A: secrecy_chanID,A2: set_Secrecy_chanID] :
( ( member1967978193chanID @ A @ A2 )
=> ( ( insert751661162chanID @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_46_insert__absorb,axiom,
! [A: secrecy_specID,A2: set_Secrecy_specID] :
( ( member740727610specID @ A @ A2 )
=> ( ( insert1671894227specID @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_47_insert__ident,axiom,
! [X3: secrecy_chanID,A2: set_Secrecy_chanID,B: set_Secrecy_chanID] :
( ~ ( member1967978193chanID @ X3 @ A2 )
=> ( ~ ( member1967978193chanID @ X3 @ B )
=> ( ( ( insert751661162chanID @ X3 @ A2 )
= ( insert751661162chanID @ X3 @ B ) )
= ( A2 = B ) ) ) ) ).
% insert_ident
thf(fact_48_insert__ident,axiom,
! [X3: secrecy_specID,A2: set_Secrecy_specID,B: set_Secrecy_specID] :
( ~ ( member740727610specID @ X3 @ A2 )
=> ( ~ ( member740727610specID @ X3 @ B )
=> ( ( ( insert1671894227specID @ X3 @ A2 )
= ( insert1671894227specID @ X3 @ B ) )
= ( A2 = B ) ) ) ) ).
% insert_ident
thf(fact_49_Set_Oset__insert,axiom,
! [X3: secrecy_chanID,A2: set_Secrecy_chanID] :
( ( member1967978193chanID @ X3 @ A2 )
=> ~ ! [B3: set_Secrecy_chanID] :
( ( A2
= ( insert751661162chanID @ X3 @ B3 ) )
=> ( member1967978193chanID @ X3 @ B3 ) ) ) ).
% Set.set_insert
thf(fact_50_Set_Oset__insert,axiom,
! [X3: secrecy_specID,A2: set_Secrecy_specID] :
( ( member740727610specID @ X3 @ A2 )
=> ~ ! [B3: set_Secrecy_specID] :
( ( A2
= ( insert1671894227specID @ X3 @ B3 ) )
=> ( member740727610specID @ X3 @ B3 ) ) ) ).
% Set.set_insert
thf(fact_51_insertI2,axiom,
! [A: secrecy_chanID,B: set_Secrecy_chanID,B2: secrecy_chanID] :
( ( member1967978193chanID @ A @ B )
=> ( member1967978193chanID @ A @ ( insert751661162chanID @ B2 @ B ) ) ) ).
% insertI2
thf(fact_52_insertI2,axiom,
! [A: secrecy_specID,B: set_Secrecy_specID,B2: secrecy_specID] :
( ( member740727610specID @ A @ B )
=> ( member740727610specID @ A @ ( insert1671894227specID @ B2 @ B ) ) ) ).
% insertI2
thf(fact_53_insertI1,axiom,
! [A: secrecy_chanID,B: set_Secrecy_chanID] : ( member1967978193chanID @ A @ ( insert751661162chanID @ A @ B ) ) ).
% insertI1
thf(fact_54_insertI1,axiom,
! [A: secrecy_specID,B: set_Secrecy_specID] : ( member740727610specID @ A @ ( insert1671894227specID @ A @ B ) ) ).
% insertI1
thf(fact_55_insertE,axiom,
! [A: secrecy_chanID,B2: secrecy_chanID,A2: set_Secrecy_chanID] :
( ( member1967978193chanID @ A @ ( insert751661162chanID @ B2 @ A2 ) )
=> ( ( A != B2 )
=> ( member1967978193chanID @ A @ A2 ) ) ) ).
% insertE
thf(fact_56_insertE,axiom,
! [A: secrecy_specID,B2: secrecy_specID,A2: set_Secrecy_specID] :
( ( member740727610specID @ A @ ( insert1671894227specID @ B2 @ A2 ) )
=> ( ( A != B2 )
=> ( member740727610specID @ A @ A2 ) ) ) ).
% insertE
thf(fact_57_TBtheorem3b,axiom,
! [P: secrecy_specID,M: set_Secrecy_chanID,E: secrecy_Expression,Q: secrecy_specID,PQ: secrecy_specID] :
( ~ ( ineM @ P @ M @ E )
=> ( ~ ( ineM @ Q @ M @ E )
=> ( ( ( subcomponents @ PQ )
= ( insert1671894227specID @ P @ ( insert1671894227specID @ Q @ bot_bo1785112429specID ) ) )
=> ( ( correctCompositionIn @ PQ )
=> ~ ( ineM @ PQ @ M @ E ) ) ) ) ) ).
% TBtheorem3b
thf(fact_58_TBtheorem1b,axiom,
! [PQ: secrecy_specID,M: set_Secrecy_chanID,E: secrecy_Expression,P: secrecy_specID,Q: secrecy_specID] :
( ( ineM @ PQ @ M @ E )
=> ( ( ( subcomponents @ PQ )
= ( insert1671894227specID @ P @ ( insert1671894227specID @ Q @ bot_bo1785112429specID ) ) )
=> ( ( correctCompositionIn @ PQ )
=> ( ( ineM @ P @ M @ E )
| ( ineM @ Q @ M @ E ) ) ) ) ) ).
% TBtheorem1b
thf(fact_59_singleton__inject,axiom,
! [A: secrecy_specID,B2: secrecy_specID] :
( ( ( insert1671894227specID @ A @ bot_bo1785112429specID )
= ( insert1671894227specID @ B2 @ bot_bo1785112429specID ) )
=> ( A = B2 ) ) ).
% singleton_inject
thf(fact_60_singleton__inject,axiom,
! [A: secrecy_chanID,B2: secrecy_chanID] :
( ( ( insert751661162chanID @ A @ bot_bo443197316chanID )
= ( insert751661162chanID @ B2 @ bot_bo443197316chanID ) )
=> ( A = B2 ) ) ).
% singleton_inject
thf(fact_61_insert__not__empty,axiom,
! [A: secrecy_specID,A2: set_Secrecy_specID] :
( ( insert1671894227specID @ A @ A2 )
!= bot_bo1785112429specID ) ).
% insert_not_empty
thf(fact_62_insert__not__empty,axiom,
! [A: secrecy_chanID,A2: set_Secrecy_chanID] :
( ( insert751661162chanID @ A @ A2 )
!= bot_bo443197316chanID ) ).
% insert_not_empty
thf(fact_63_doubleton__eq__iff,axiom,
! [A: secrecy_specID,B2: secrecy_specID,C: secrecy_specID,D: secrecy_specID] :
( ( ( insert1671894227specID @ A @ ( insert1671894227specID @ B2 @ bot_bo1785112429specID ) )
= ( insert1671894227specID @ C @ ( insert1671894227specID @ D @ bot_bo1785112429specID ) ) )
= ( ( ( A = C )
& ( B2 = D ) )
| ( ( A = D )
& ( B2 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_64_doubleton__eq__iff,axiom,
! [A: secrecy_chanID,B2: secrecy_chanID,C: secrecy_chanID,D: secrecy_chanID] :
( ( ( insert751661162chanID @ A @ ( insert751661162chanID @ B2 @ bot_bo443197316chanID ) )
= ( insert751661162chanID @ C @ ( insert751661162chanID @ D @ bot_bo443197316chanID ) ) )
= ( ( ( A = C )
& ( B2 = D ) )
| ( ( A = D )
& ( B2 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_65_singleton__iff,axiom,
! [B2: secrecy_specID,A: secrecy_specID] :
( ( member740727610specID @ B2 @ ( insert1671894227specID @ A @ bot_bo1785112429specID ) )
= ( B2 = A ) ) ).
% singleton_iff
thf(fact_66_singleton__iff,axiom,
! [B2: secrecy_chanID,A: secrecy_chanID] :
( ( member1967978193chanID @ B2 @ ( insert751661162chanID @ A @ bot_bo443197316chanID ) )
= ( B2 = A ) ) ).
% singleton_iff
thf(fact_67_singletonD,axiom,
! [B2: secrecy_specID,A: secrecy_specID] :
( ( member740727610specID @ B2 @ ( insert1671894227specID @ A @ bot_bo1785112429specID ) )
=> ( B2 = A ) ) ).
% singletonD
thf(fact_68_singletonD,axiom,
! [B2: secrecy_chanID,A: secrecy_chanID] :
( ( member1967978193chanID @ B2 @ ( insert751661162chanID @ A @ bot_bo443197316chanID ) )
=> ( B2 = A ) ) ).
% singletonD
thf(fact_69_TBtheorem4b__notP1,axiom,
! [P: secrecy_specID,M: set_Secrecy_chanID,E: secrecy_Expression,Q: secrecy_specID,PQ: secrecy_specID] :
( ( ineM @ P @ M @ E )
=> ( ~ ( ineM @ Q @ M @ E )
=> ( ( ( subcomponents @ PQ )
= ( insert1671894227specID @ P @ ( insert1671894227specID @ Q @ bot_bo1785112429specID ) ) )
=> ( ( correctCompositionIn @ PQ )
=> ( ? [Ch2: secrecy_chanID] :
( ( ine_ex1303305700Single @ P @ Ch2 @ E )
& ( member1967978193chanID @ Ch2 @ M )
& ( member1967978193chanID @ Ch2 @ ( loc @ PQ ) ) )
=> ~ ( ineM @ PQ @ M @ E ) ) ) ) ) ) ).
% TBtheorem4b_notP1
thf(fact_70_TBtheorem4b__notPQ,axiom,
! [PQ: secrecy_specID,P: secrecy_specID,Q: secrecy_specID,ChSetP: set_Secrecy_chanID,E: secrecy_Expression,ChSetQ: set_Secrecy_chanID,M: 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 @ M @ E ) ) ) ) ) ) ) ).
% TBtheorem4b_notPQ
thf(fact_71_mem__Collect__eq,axiom,
! [A: secrecy_chanID,P: secrecy_chanID > $o] :
( ( member1967978193chanID @ A @ ( collec625564175chanID @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_72_mem__Collect__eq,axiom,
! [A: secrecy_specID,P: secrecy_specID > $o] :
( ( member740727610specID @ A @ ( collec1545797240specID @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_73_Collect__mem__eq,axiom,
! [A2: set_Secrecy_chanID] :
( ( collec625564175chanID
@ ^ [X2: secrecy_chanID] : ( member1967978193chanID @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_74_Collect__mem__eq,axiom,
! [A2: set_Secrecy_specID] :
( ( collec1545797240specID
@ ^ [X2: secrecy_specID] : ( member740727610specID @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_75_Collect__cong,axiom,
! [P: secrecy_chanID > $o,Q: secrecy_chanID > $o] :
( ! [X4: secrecy_chanID] :
( ( P @ X4 )
= ( Q @ X4 ) )
=> ( ( collec625564175chanID @ P )
= ( collec625564175chanID @ Q ) ) ) ).
% Collect_cong
thf(fact_76_Collect__cong,axiom,
! [P: secrecy_specID > $o,Q: secrecy_specID > $o] :
( ! [X4: secrecy_specID] :
( ( P @ X4 )
= ( Q @ X4 ) )
=> ( ( collec1545797240specID @ P )
= ( collec1545797240specID @ Q ) ) ) ).
% Collect_cong
thf(fact_77_TBtheorem4b__notP2,axiom,
! [Q: secrecy_specID,M: set_Secrecy_chanID,E: secrecy_Expression,PQ: secrecy_specID,P: secrecy_specID,ChSet: set_Secrecy_chanID] :
( ~ ( ineM @ Q @ M @ 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 @ M @ E ) ) ) ) ) ) ).
% TBtheorem4b_notP2
thf(fact_78_the__elem__eq,axiom,
! [X3: secrecy_specID] :
( ( the_el827131246specID @ ( insert1671894227specID @ X3 @ bot_bo1785112429specID ) )
= X3 ) ).
% the_elem_eq
thf(fact_79_the__elem__eq,axiom,
! [X3: secrecy_chanID] :
( ( the_el2054381829chanID @ ( insert751661162chanID @ X3 @ bot_bo443197316chanID ) )
= X3 ) ).
% the_elem_eq
thf(fact_80_bot__apply,axiom,
( bot_bo1012329985anID_o
= ( ^ [X2: secrecy_chanID] : bot_bot_o ) ) ).
% bot_apply
thf(fact_81_bot__apply,axiom,
( bot_bo1767780696ecID_o
= ( ^ [X2: secrecy_specID] : bot_bot_o ) ) ).
% bot_apply
thf(fact_82_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_83_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_84_is__singletonI,axiom,
! [X3: secrecy_specID] : ( is_sin1501507247specID @ ( insert1671894227specID @ X3 @ bot_bo1785112429specID ) ) ).
% is_singletonI
thf(fact_85_is__singletonI,axiom,
! [X3: secrecy_chanID] : ( is_sin581274182chanID @ ( insert751661162chanID @ X3 @ bot_bo443197316chanID ) ) ).
% is_singletonI
thf(fact_86_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_87_ine__exprChannelSingle__def,axiom,
( ine_ex1303305700Single
= ( ^ [SP: secrecy_specID,Ch3: secrecy_chanID,E2: secrecy_Expression] :
( ( member1967978193chanID @ Ch3 @ ( ins @ SP ) )
& ( exprChannel @ Ch3 @ E2 )
& ! [X2: secrecy_chanID,T: nat] :
( ( ( member1967978193chanID @ X2 @ ( ins @ SP ) )
& ( X2 != Ch3 ) )
=> ~ ( exprChannel @ X2 @ E2 ) ) ) ) ) ).
% ine_exprChannelSingle_def
thf(fact_88_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_89_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_90_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_91_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_92_bot__set__def,axiom,
( bot_bo1785112429specID
= ( collec1545797240specID @ bot_bo1767780696ecID_o ) ) ).
% bot_set_def
thf(fact_93_bot__set__def,axiom,
( bot_bo443197316chanID
= ( collec625564175chanID @ bot_bo1012329985anID_o ) ) ).
% bot_set_def
thf(fact_94_not__ine__ineM,axiom,
! [P: secrecy_specID,E: secrecy_Expression,M: set_Secrecy_chanID] :
( ~ ( ine @ P @ E )
=> ~ ( ineM @ P @ M @ E ) ) ).
% not_ine_ineM
thf(fact_95_ineM__ine,axiom,
! [P: secrecy_specID,M: set_Secrecy_chanID,E: secrecy_Expression] :
( ( ineM @ P @ M @ E )
=> ( ine @ P @ E ) ) ).
% ineM_ine
thf(fact_96_is__singleton__the__elem,axiom,
( is_sin1501507247specID
= ( ^ [A3: set_Secrecy_specID] :
( A3
= ( insert1671894227specID @ ( the_el827131246specID @ A3 ) @ bot_bo1785112429specID ) ) ) ) ).
% is_singleton_the_elem
thf(fact_97_is__singleton__the__elem,axiom,
( is_sin581274182chanID
= ( ^ [A3: set_Secrecy_chanID] :
( A3
= ( insert751661162chanID @ ( the_el2054381829chanID @ A3 ) @ bot_bo443197316chanID ) ) ) ) ).
% is_singleton_the_elem
thf(fact_98_ine__def,axiom,
( ine
= ( ^ [SP: secrecy_specID,E2: secrecy_Expression] :
? [Ch3: secrecy_chanID] :
( ( member1967978193chanID @ Ch3 @ ( ins @ SP ) )
& ( exprChannel @ Ch3 @ E2 ) ) ) ) ).
% ine_def
thf(fact_99_ine__ins__neg1,axiom,
! [P: secrecy_specID,M3: secrecy_Expression,X3: secrecy_chanID] :
( ~ ( ine @ P @ M3 )
=> ( ( exprChannel @ X3 @ M3 )
=> ~ ( member1967978193chanID @ X3 @ ( ins @ P ) ) ) ) ).
% ine_ins_neg1
thf(fact_100_is__singletonI_H,axiom,
! [A2: set_Secrecy_specID] :
( ( A2 != bot_bo1785112429specID )
=> ( ! [X4: secrecy_specID,Y2: secrecy_specID] :
( ( member740727610specID @ X4 @ A2 )
=> ( ( member740727610specID @ Y2 @ A2 )
=> ( X4 = Y2 ) ) )
=> ( is_sin1501507247specID @ A2 ) ) ) ).
% is_singletonI'
thf(fact_101_is__singletonI_H,axiom,
! [A2: set_Secrecy_chanID] :
( ( A2 != bot_bo443197316chanID )
=> ( ! [X4: secrecy_chanID,Y2: secrecy_chanID] :
( ( member1967978193chanID @ X4 @ A2 )
=> ( ( member1967978193chanID @ Y2 @ A2 )
=> ( X4 = Y2 ) ) )
=> ( is_sin581274182chanID @ A2 ) ) ) ).
% is_singletonI'
thf(fact_102_bot__fun__def,axiom,
( bot_bo1012329985anID_o
= ( ^ [X2: secrecy_chanID] : bot_bot_o ) ) ).
% bot_fun_def
thf(fact_103_bot__fun__def,axiom,
( bot_bo1767780696ecID_o
= ( ^ [X2: secrecy_specID] : bot_bot_o ) ) ).
% bot_fun_def
thf(fact_104_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_105_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_106_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_107_ine__exprChannelSet__def,axiom,
( ine_exprChannelSet
= ( ^ [SP: secrecy_specID,ChSet2: set_Secrecy_chanID,E2: secrecy_Expression] :
( ! [X2: secrecy_chanID] :
( ( member1967978193chanID @ X2 @ ChSet2 )
=> ( ( member1967978193chanID @ X2 @ ( ins @ SP ) )
& ( exprChannel @ X2 @ E2 ) ) )
& ! [X2: secrecy_chanID] :
( ( ~ ( member1967978193chanID @ X2 @ ChSet2 )
& ( member1967978193chanID @ X2 @ ( ins @ SP ) ) )
=> ~ ( exprChannel @ X2 @ E2 ) ) ) ) ) ).
% ine_exprChannelSet_def
thf(fact_108_is__singletonE,axiom,
! [A2: set_Secrecy_specID] :
( ( is_sin1501507247specID @ A2 )
=> ~ ! [X4: secrecy_specID] :
( A2
!= ( insert1671894227specID @ X4 @ bot_bo1785112429specID ) ) ) ).
% is_singletonE
thf(fact_109_is__singletonE,axiom,
! [A2: set_Secrecy_chanID] :
( ( is_sin581274182chanID @ A2 )
=> ~ ! [X4: secrecy_chanID] :
( A2
!= ( insert751661162chanID @ X4 @ bot_bo443197316chanID ) ) ) ).
% is_singletonE
thf(fact_110_is__singleton__def,axiom,
( is_sin1501507247specID
= ( ^ [A3: set_Secrecy_specID] :
? [X2: secrecy_specID] :
( A3
= ( insert1671894227specID @ X2 @ bot_bo1785112429specID ) ) ) ) ).
% is_singleton_def
thf(fact_111_is__singleton__def,axiom,
( is_sin581274182chanID
= ( ^ [A3: set_Secrecy_chanID] :
? [X2: secrecy_chanID] :
( A3
= ( insert751661162chanID @ X2 @ bot_bo443197316chanID ) ) ) ) ).
% is_singleton_def
thf(fact_112_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_113_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_114_subcomponents__loc,axiom,
! [X3: secrecy_specID] :
( ( correc965049635ionLoc @ X3 )
=> ( ( ( subcomponents @ X3 )
= bot_bo1785112429specID )
=> ( ( loc @ X3 )
= bot_bo443197316chanID ) ) ) ).
% subcomponents_loc
thf(fact_115_Set_Ois__empty__def,axiom,
( is_emp1849217361specID
= ( ^ [A3: set_Secrecy_specID] : A3 = bot_bo1785112429specID ) ) ).
% Set.is_empty_def
thf(fact_116_Set_Ois__empty__def,axiom,
( is_emp928984296chanID
= ( ^ [A3: set_Secrecy_chanID] : A3 = bot_bo443197316chanID ) ) ).
% Set.is_empty_def
thf(fact_117_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_118_bot__empty__eq,axiom,
( bot_bo1767780696ecID_o
= ( ^ [X2: secrecy_specID] : ( member740727610specID @ X2 @ bot_bo1785112429specID ) ) ) ).
% bot_empty_eq
thf(fact_119_bot__empty__eq,axiom,
( bot_bo1012329985anID_o
= ( ^ [X2: secrecy_chanID] : ( member1967978193chanID @ X2 @ bot_bo443197316chanID ) ) ) ).
% bot_empty_eq
thf(fact_120_Collect__empty__eq__bot,axiom,
! [P: secrecy_specID > $o] :
( ( ( collec1545797240specID @ P )
= bot_bo1785112429specID )
= ( P = bot_bo1767780696ecID_o ) ) ).
% Collect_empty_eq_bot
thf(fact_121_Collect__empty__eq__bot,axiom,
! [P: secrecy_chanID > $o] :
( ( ( collec625564175chanID @ P )
= bot_bo443197316chanID )
= ( P = bot_bo1012329985anID_o ) ) ).
% Collect_empty_eq_bot
thf(fact_122_pairwise__singleton,axiom,
! [P: secrecy_specID > secrecy_specID > $o,A2: secrecy_specID] : ( pairwi514744696specID @ P @ ( insert1671894227specID @ A2 @ bot_bo1785112429specID ) ) ).
% pairwise_singleton
thf(fact_123_pairwise__singleton,axiom,
! [P: secrecy_chanID > secrecy_chanID > $o,A2: secrecy_chanID] : ( pairwi1741995279chanID @ P @ ( insert751661162chanID @ A2 @ bot_bo443197316chanID ) ) ).
% pairwise_singleton
thf(fact_124_singleton__insert__inj__eq,axiom,
! [B2: secrecy_specID,A: secrecy_specID,A2: set_Secrecy_specID] :
( ( ( insert1671894227specID @ B2 @ bot_bo1785112429specID )
= ( insert1671894227specID @ A @ A2 ) )
= ( ( A = B2 )
& ( ord_le996703545specID @ A2 @ ( insert1671894227specID @ B2 @ bot_bo1785112429specID ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_125_singleton__insert__inj__eq,axiom,
! [B2: secrecy_chanID,A: secrecy_chanID,A2: set_Secrecy_chanID] :
( ( ( insert751661162chanID @ B2 @ bot_bo443197316chanID )
= ( insert751661162chanID @ A @ A2 ) )
= ( ( A = B2 )
& ( ord_le1802272080chanID @ A2 @ ( insert751661162chanID @ B2 @ bot_bo443197316chanID ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_126_order__refl,axiom,
! [X3: set_Secrecy_chanID] : ( ord_le1802272080chanID @ X3 @ X3 ) ).
% order_refl
thf(fact_127_order__refl,axiom,
! [X3: set_Secrecy_specID] : ( ord_le996703545specID @ X3 @ X3 ) ).
% order_refl
thf(fact_128_subsetI,axiom,
! [A2: set_Secrecy_chanID,B: set_Secrecy_chanID] :
( ! [X4: secrecy_chanID] :
( ( member1967978193chanID @ X4 @ A2 )
=> ( member1967978193chanID @ X4 @ B ) )
=> ( ord_le1802272080chanID @ A2 @ B ) ) ).
% subsetI
thf(fact_129_subsetI,axiom,
! [A2: set_Secrecy_specID,B: set_Secrecy_specID] :
( ! [X4: secrecy_specID] :
( ( member740727610specID @ X4 @ A2 )
=> ( member740727610specID @ X4 @ B ) )
=> ( ord_le996703545specID @ A2 @ B ) ) ).
% subsetI
thf(fact_130_subset__antisym,axiom,
! [A2: set_Secrecy_chanID,B: set_Secrecy_chanID] :
( ( ord_le1802272080chanID @ A2 @ B )
=> ( ( ord_le1802272080chanID @ B @ A2 )
=> ( A2 = B ) ) ) ).
% subset_antisym
thf(fact_131_subset__antisym,axiom,
! [A2: set_Secrecy_specID,B: set_Secrecy_specID] :
( ( ord_le996703545specID @ A2 @ B )
=> ( ( ord_le996703545specID @ B @ A2 )
=> ( A2 = B ) ) ) ).
% subset_antisym
thf(fact_132_empty__subsetI,axiom,
! [A2: set_Secrecy_specID] : ( ord_le996703545specID @ bot_bo1785112429specID @ A2 ) ).
% empty_subsetI
thf(fact_133_empty__subsetI,axiom,
! [A2: set_Secrecy_chanID] : ( ord_le1802272080chanID @ bot_bo443197316chanID @ A2 ) ).
% empty_subsetI
thf(fact_134_subset__empty,axiom,
! [A2: set_Secrecy_specID] :
( ( ord_le996703545specID @ A2 @ bot_bo1785112429specID )
= ( A2 = bot_bo1785112429specID ) ) ).
% subset_empty
thf(fact_135_subset__empty,axiom,
! [A2: set_Secrecy_chanID] :
( ( ord_le1802272080chanID @ A2 @ bot_bo443197316chanID )
= ( A2 = bot_bo443197316chanID ) ) ).
% subset_empty
thf(fact_136_insert__subset,axiom,
! [X3: secrecy_chanID,A2: set_Secrecy_chanID,B: set_Secrecy_chanID] :
( ( ord_le1802272080chanID @ ( insert751661162chanID @ X3 @ A2 ) @ B )
= ( ( member1967978193chanID @ X3 @ B )
& ( ord_le1802272080chanID @ A2 @ B ) ) ) ).
% insert_subset
thf(fact_137_insert__subset,axiom,
! [X3: secrecy_specID,A2: set_Secrecy_specID,B: set_Secrecy_specID] :
( ( ord_le996703545specID @ ( insert1671894227specID @ X3 @ A2 ) @ B )
= ( ( member740727610specID @ X3 @ B )
& ( ord_le996703545specID @ A2 @ B ) ) ) ).
% insert_subset
thf(fact_138_singleton__insert__inj__eq_H,axiom,
! [A: secrecy_specID,A2: set_Secrecy_specID,B2: secrecy_specID] :
( ( ( insert1671894227specID @ A @ A2 )
= ( insert1671894227specID @ B2 @ bot_bo1785112429specID ) )
= ( ( A = B2 )
& ( ord_le996703545specID @ A2 @ ( insert1671894227specID @ B2 @ bot_bo1785112429specID ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_139_singleton__insert__inj__eq_H,axiom,
! [A: secrecy_chanID,A2: set_Secrecy_chanID,B2: secrecy_chanID] :
( ( ( insert751661162chanID @ A @ A2 )
= ( insert751661162chanID @ B2 @ bot_bo443197316chanID ) )
= ( ( A = B2 )
& ( ord_le1802272080chanID @ A2 @ ( insert751661162chanID @ B2 @ bot_bo443197316chanID ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_140_order__subst1,axiom,
! [A: set_Secrecy_chanID,F: set_Secrecy_chanID > set_Secrecy_chanID,B2: set_Secrecy_chanID,C: set_Secrecy_chanID] :
( ( ord_le1802272080chanID @ A @ ( F @ B2 ) )
=> ( ( ord_le1802272080chanID @ B2 @ C )
=> ( ! [X4: set_Secrecy_chanID,Y2: set_Secrecy_chanID] :
( ( ord_le1802272080chanID @ X4 @ Y2 )
=> ( ord_le1802272080chanID @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_le1802272080chanID @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_141_order__subst1,axiom,
! [A: set_Secrecy_chanID,F: set_Secrecy_specID > set_Secrecy_chanID,B2: set_Secrecy_specID,C: set_Secrecy_specID] :
( ( ord_le1802272080chanID @ A @ ( F @ B2 ) )
=> ( ( ord_le996703545specID @ B2 @ C )
=> ( ! [X4: set_Secrecy_specID,Y2: set_Secrecy_specID] :
( ( ord_le996703545specID @ X4 @ Y2 )
=> ( ord_le1802272080chanID @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_le1802272080chanID @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_142_order__subst1,axiom,
! [A: set_Secrecy_specID,F: set_Secrecy_chanID > set_Secrecy_specID,B2: set_Secrecy_chanID,C: set_Secrecy_chanID] :
( ( ord_le996703545specID @ A @ ( F @ B2 ) )
=> ( ( ord_le1802272080chanID @ B2 @ C )
=> ( ! [X4: set_Secrecy_chanID,Y2: set_Secrecy_chanID] :
( ( ord_le1802272080chanID @ X4 @ Y2 )
=> ( ord_le996703545specID @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_le996703545specID @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_143_order__subst1,axiom,
! [A: set_Secrecy_specID,F: set_Secrecy_specID > set_Secrecy_specID,B2: set_Secrecy_specID,C: set_Secrecy_specID] :
( ( ord_le996703545specID @ A @ ( F @ B2 ) )
=> ( ( ord_le996703545specID @ B2 @ C )
=> ( ! [X4: set_Secrecy_specID,Y2: set_Secrecy_specID] :
( ( ord_le996703545specID @ X4 @ Y2 )
=> ( ord_le996703545specID @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_le996703545specID @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_144_order__subst2,axiom,
! [A: set_Secrecy_chanID,B2: set_Secrecy_chanID,F: set_Secrecy_chanID > set_Secrecy_chanID,C: set_Secrecy_chanID] :
( ( ord_le1802272080chanID @ A @ B2 )
=> ( ( ord_le1802272080chanID @ ( F @ B2 ) @ C )
=> ( ! [X4: set_Secrecy_chanID,Y2: set_Secrecy_chanID] :
( ( ord_le1802272080chanID @ X4 @ Y2 )
=> ( ord_le1802272080chanID @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_le1802272080chanID @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_145_order__subst2,axiom,
! [A: set_Secrecy_chanID,B2: set_Secrecy_chanID,F: set_Secrecy_chanID > set_Secrecy_specID,C: set_Secrecy_specID] :
( ( ord_le1802272080chanID @ A @ B2 )
=> ( ( ord_le996703545specID @ ( F @ B2 ) @ C )
=> ( ! [X4: set_Secrecy_chanID,Y2: set_Secrecy_chanID] :
( ( ord_le1802272080chanID @ X4 @ Y2 )
=> ( ord_le996703545specID @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_le996703545specID @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_146_order__subst2,axiom,
! [A: set_Secrecy_specID,B2: set_Secrecy_specID,F: set_Secrecy_specID > set_Secrecy_chanID,C: set_Secrecy_chanID] :
( ( ord_le996703545specID @ A @ B2 )
=> ( ( ord_le1802272080chanID @ ( F @ B2 ) @ C )
=> ( ! [X4: set_Secrecy_specID,Y2: set_Secrecy_specID] :
( ( ord_le996703545specID @ X4 @ Y2 )
=> ( ord_le1802272080chanID @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_le1802272080chanID @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_147_order__subst2,axiom,
! [A: set_Secrecy_specID,B2: set_Secrecy_specID,F: set_Secrecy_specID > set_Secrecy_specID,C: set_Secrecy_specID] :
( ( ord_le996703545specID @ A @ B2 )
=> ( ( ord_le996703545specID @ ( F @ B2 ) @ C )
=> ( ! [X4: set_Secrecy_specID,Y2: set_Secrecy_specID] :
( ( ord_le996703545specID @ X4 @ Y2 )
=> ( ord_le996703545specID @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_le996703545specID @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_148_ord__eq__le__subst,axiom,
! [A: set_Secrecy_chanID,F: set_Secrecy_chanID > set_Secrecy_chanID,B2: set_Secrecy_chanID,C: set_Secrecy_chanID] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_le1802272080chanID @ B2 @ C )
=> ( ! [X4: set_Secrecy_chanID,Y2: set_Secrecy_chanID] :
( ( ord_le1802272080chanID @ X4 @ Y2 )
=> ( ord_le1802272080chanID @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_le1802272080chanID @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_149_ord__eq__le__subst,axiom,
! [A: set_Secrecy_specID,F: set_Secrecy_chanID > set_Secrecy_specID,B2: set_Secrecy_chanID,C: set_Secrecy_chanID] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_le1802272080chanID @ B2 @ C )
=> ( ! [X4: set_Secrecy_chanID,Y2: set_Secrecy_chanID] :
( ( ord_le1802272080chanID @ X4 @ Y2 )
=> ( ord_le996703545specID @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_le996703545specID @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_150_ord__eq__le__subst,axiom,
! [A: set_Secrecy_chanID,F: set_Secrecy_specID > set_Secrecy_chanID,B2: set_Secrecy_specID,C: set_Secrecy_specID] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_le996703545specID @ B2 @ C )
=> ( ! [X4: set_Secrecy_specID,Y2: set_Secrecy_specID] :
( ( ord_le996703545specID @ X4 @ Y2 )
=> ( ord_le1802272080chanID @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_le1802272080chanID @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_151_ord__eq__le__subst,axiom,
! [A: set_Secrecy_specID,F: set_Secrecy_specID > set_Secrecy_specID,B2: set_Secrecy_specID,C: set_Secrecy_specID] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_le996703545specID @ B2 @ C )
=> ( ! [X4: set_Secrecy_specID,Y2: set_Secrecy_specID] :
( ( ord_le996703545specID @ X4 @ Y2 )
=> ( ord_le996703545specID @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_le996703545specID @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_152_ord__le__eq__subst,axiom,
! [A: set_Secrecy_chanID,B2: set_Secrecy_chanID,F: set_Secrecy_chanID > set_Secrecy_chanID,C: set_Secrecy_chanID] :
( ( ord_le1802272080chanID @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: set_Secrecy_chanID,Y2: set_Secrecy_chanID] :
( ( ord_le1802272080chanID @ X4 @ Y2 )
=> ( ord_le1802272080chanID @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_le1802272080chanID @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_153_ord__le__eq__subst,axiom,
! [A: set_Secrecy_chanID,B2: set_Secrecy_chanID,F: set_Secrecy_chanID > set_Secrecy_specID,C: set_Secrecy_specID] :
( ( ord_le1802272080chanID @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: set_Secrecy_chanID,Y2: set_Secrecy_chanID] :
( ( ord_le1802272080chanID @ X4 @ Y2 )
=> ( ord_le996703545specID @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_le996703545specID @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_154_ord__le__eq__subst,axiom,
! [A: set_Secrecy_specID,B2: set_Secrecy_specID,F: set_Secrecy_specID > set_Secrecy_chanID,C: set_Secrecy_chanID] :
( ( ord_le996703545specID @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: set_Secrecy_specID,Y2: set_Secrecy_specID] :
( ( ord_le996703545specID @ X4 @ Y2 )
=> ( ord_le1802272080chanID @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_le1802272080chanID @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_155_ord__le__eq__subst,axiom,
! [A: set_Secrecy_specID,B2: set_Secrecy_specID,F: set_Secrecy_specID > set_Secrecy_specID,C: set_Secrecy_specID] :
( ( ord_le996703545specID @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: set_Secrecy_specID,Y2: set_Secrecy_specID] :
( ( ord_le996703545specID @ X4 @ Y2 )
=> ( ord_le996703545specID @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_le996703545specID @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_156_eq__iff,axiom,
( ( ^ [Y: set_Secrecy_chanID,Z: set_Secrecy_chanID] : Y = Z )
= ( ^ [X2: set_Secrecy_chanID,Y4: set_Secrecy_chanID] :
( ( ord_le1802272080chanID @ X2 @ Y4 )
& ( ord_le1802272080chanID @ Y4 @ X2 ) ) ) ) ).
% eq_iff
thf(fact_157_eq__iff,axiom,
( ( ^ [Y: set_Secrecy_specID,Z: set_Secrecy_specID] : Y = Z )
= ( ^ [X2: set_Secrecy_specID,Y4: set_Secrecy_specID] :
( ( ord_le996703545specID @ X2 @ Y4 )
& ( ord_le996703545specID @ Y4 @ X2 ) ) ) ) ).
% eq_iff
thf(fact_158_antisym,axiom,
! [X3: set_Secrecy_chanID,Y3: set_Secrecy_chanID] :
( ( ord_le1802272080chanID @ X3 @ Y3 )
=> ( ( ord_le1802272080chanID @ Y3 @ X3 )
=> ( X3 = Y3 ) ) ) ).
% antisym
thf(fact_159_antisym,axiom,
! [X3: set_Secrecy_specID,Y3: set_Secrecy_specID] :
( ( ord_le996703545specID @ X3 @ Y3 )
=> ( ( ord_le996703545specID @ Y3 @ X3 )
=> ( X3 = Y3 ) ) ) ).
% antisym
thf(fact_160_eq__refl,axiom,
! [X3: set_Secrecy_chanID,Y3: set_Secrecy_chanID] :
( ( X3 = Y3 )
=> ( ord_le1802272080chanID @ X3 @ Y3 ) ) ).
% eq_refl
thf(fact_161_eq__refl,axiom,
! [X3: set_Secrecy_specID,Y3: set_Secrecy_specID] :
( ( X3 = Y3 )
=> ( ord_le996703545specID @ X3 @ Y3 ) ) ).
% eq_refl
thf(fact_162_order_Otrans,axiom,
! [A: set_Secrecy_chanID,B2: set_Secrecy_chanID,C: set_Secrecy_chanID] :
( ( ord_le1802272080chanID @ A @ B2 )
=> ( ( ord_le1802272080chanID @ B2 @ C )
=> ( ord_le1802272080chanID @ A @ C ) ) ) ).
% order.trans
thf(fact_163_order_Otrans,axiom,
! [A: set_Secrecy_specID,B2: set_Secrecy_specID,C: set_Secrecy_specID] :
( ( ord_le996703545specID @ A @ B2 )
=> ( ( ord_le996703545specID @ B2 @ C )
=> ( ord_le996703545specID @ A @ C ) ) ) ).
% order.trans
thf(fact_164_antisym__conv,axiom,
! [Y3: set_Secrecy_chanID,X3: set_Secrecy_chanID] :
( ( ord_le1802272080chanID @ Y3 @ X3 )
=> ( ( ord_le1802272080chanID @ X3 @ Y3 )
= ( X3 = Y3 ) ) ) ).
% antisym_conv
thf(fact_165_antisym__conv,axiom,
! [Y3: set_Secrecy_specID,X3: set_Secrecy_specID] :
( ( ord_le996703545specID @ Y3 @ X3 )
=> ( ( ord_le996703545specID @ X3 @ Y3 )
= ( X3 = Y3 ) ) ) ).
% antisym_conv
thf(fact_166_order__class_Oorder_Oeq__iff,axiom,
( ( ^ [Y: set_Secrecy_chanID,Z: set_Secrecy_chanID] : Y = Z )
= ( ^ [A4: set_Secrecy_chanID,B4: set_Secrecy_chanID] :
( ( ord_le1802272080chanID @ A4 @ B4 )
& ( ord_le1802272080chanID @ B4 @ A4 ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_167_order__class_Oorder_Oeq__iff,axiom,
( ( ^ [Y: set_Secrecy_specID,Z: set_Secrecy_specID] : Y = Z )
= ( ^ [A4: set_Secrecy_specID,B4: set_Secrecy_specID] :
( ( ord_le996703545specID @ A4 @ B4 )
& ( ord_le996703545specID @ B4 @ A4 ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_168_ord__eq__le__trans,axiom,
! [A: set_Secrecy_chanID,B2: set_Secrecy_chanID,C: set_Secrecy_chanID] :
( ( A = B2 )
=> ( ( ord_le1802272080chanID @ B2 @ C )
=> ( ord_le1802272080chanID @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_169_ord__eq__le__trans,axiom,
! [A: set_Secrecy_specID,B2: set_Secrecy_specID,C: set_Secrecy_specID] :
( ( A = B2 )
=> ( ( ord_le996703545specID @ B2 @ C )
=> ( ord_le996703545specID @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_170_ord__le__eq__trans,axiom,
! [A: set_Secrecy_chanID,B2: set_Secrecy_chanID,C: set_Secrecy_chanID] :
( ( ord_le1802272080chanID @ A @ B2 )
=> ( ( B2 = C )
=> ( ord_le1802272080chanID @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_171_ord__le__eq__trans,axiom,
! [A: set_Secrecy_specID,B2: set_Secrecy_specID,C: set_Secrecy_specID] :
( ( ord_le996703545specID @ A @ B2 )
=> ( ( B2 = C )
=> ( ord_le996703545specID @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_172_order__class_Oorder_Oantisym,axiom,
! [A: set_Secrecy_chanID,B2: set_Secrecy_chanID] :
( ( ord_le1802272080chanID @ A @ B2 )
=> ( ( ord_le1802272080chanID @ B2 @ A )
=> ( A = B2 ) ) ) ).
% order_class.order.antisym
thf(fact_173_order__class_Oorder_Oantisym,axiom,
! [A: set_Secrecy_specID,B2: set_Secrecy_specID] :
( ( ord_le996703545specID @ A @ B2 )
=> ( ( ord_le996703545specID @ B2 @ A )
=> ( A = B2 ) ) ) ).
% order_class.order.antisym
thf(fact_174_order__trans,axiom,
! [X3: set_Secrecy_chanID,Y3: set_Secrecy_chanID,Z2: set_Secrecy_chanID] :
( ( ord_le1802272080chanID @ X3 @ Y3 )
=> ( ( ord_le1802272080chanID @ Y3 @ Z2 )
=> ( ord_le1802272080chanID @ X3 @ Z2 ) ) ) ).
% order_trans
thf(fact_175_order__trans,axiom,
! [X3: set_Secrecy_specID,Y3: set_Secrecy_specID,Z2: set_Secrecy_specID] :
( ( ord_le996703545specID @ X3 @ Y3 )
=> ( ( ord_le996703545specID @ Y3 @ Z2 )
=> ( ord_le996703545specID @ X3 @ Z2 ) ) ) ).
% order_trans
thf(fact_176_dual__order_Orefl,axiom,
! [A: set_Secrecy_specID] : ( ord_le996703545specID @ A @ A ) ).
% dual_order.refl
thf(fact_177_in__mono,axiom,
! [A2: set_Secrecy_chanID,B: set_Secrecy_chanID,X3: secrecy_chanID] :
( ( ord_le1802272080chanID @ A2 @ B )
=> ( ( member1967978193chanID @ X3 @ A2 )
=> ( member1967978193chanID @ X3 @ B ) ) ) ).
% in_mono
thf(fact_178_in__mono,axiom,
! [A2: set_Secrecy_specID,B: set_Secrecy_specID,X3: secrecy_specID] :
( ( ord_le996703545specID @ A2 @ B )
=> ( ( member740727610specID @ X3 @ A2 )
=> ( member740727610specID @ X3 @ B ) ) ) ).
% in_mono
thf(fact_179_subsetD,axiom,
! [A2: set_Secrecy_chanID,B: set_Secrecy_chanID,C: secrecy_chanID] :
( ( ord_le1802272080chanID @ A2 @ B )
=> ( ( member1967978193chanID @ C @ A2 )
=> ( member1967978193chanID @ C @ B ) ) ) ).
% subsetD
thf(fact_180_subsetD,axiom,
! [A2: set_Secrecy_specID,B: set_Secrecy_specID,C: secrecy_specID] :
( ( ord_le996703545specID @ A2 @ B )
=> ( ( member740727610specID @ C @ A2 )
=> ( member740727610specID @ C @ B ) ) ) ).
% subsetD
thf(fact_181_pairwiseD,axiom,
! [R: secrecy_chanID > secrecy_chanID > $o,S: set_Secrecy_chanID,X3: secrecy_chanID,Y3: secrecy_chanID] :
( ( pairwi1741995279chanID @ R @ S )
=> ( ( member1967978193chanID @ X3 @ S )
=> ( ( member1967978193chanID @ Y3 @ S )
=> ( ( X3 != Y3 )
=> ( R @ X3 @ Y3 ) ) ) ) ) ).
% pairwiseD
thf(fact_182_pairwiseD,axiom,
! [R: secrecy_specID > secrecy_specID > $o,S: set_Secrecy_specID,X3: secrecy_specID,Y3: secrecy_specID] :
( ( pairwi514744696specID @ R @ S )
=> ( ( member740727610specID @ X3 @ S )
=> ( ( member740727610specID @ Y3 @ S )
=> ( ( X3 != Y3 )
=> ( R @ X3 @ Y3 ) ) ) ) ) ).
% pairwiseD
thf(fact_183_pairwiseI,axiom,
! [S: set_Secrecy_chanID,R: secrecy_chanID > secrecy_chanID > $o] :
( ! [X4: secrecy_chanID,Y2: secrecy_chanID] :
( ( member1967978193chanID @ X4 @ S )
=> ( ( member1967978193chanID @ Y2 @ S )
=> ( ( X4 != Y2 )
=> ( R @ X4 @ Y2 ) ) ) )
=> ( pairwi1741995279chanID @ R @ S ) ) ).
% pairwiseI
thf(fact_184_pairwiseI,axiom,
! [S: set_Secrecy_specID,R: secrecy_specID > secrecy_specID > $o] :
( ! [X4: secrecy_specID,Y2: secrecy_specID] :
( ( member740727610specID @ X4 @ S )
=> ( ( member740727610specID @ Y2 @ S )
=> ( ( X4 != Y2 )
=> ( R @ X4 @ Y2 ) ) ) )
=> ( pairwi514744696specID @ R @ S ) ) ).
% pairwiseI
thf(fact_185_subset__eq,axiom,
( ord_le1802272080chanID
= ( ^ [A3: set_Secrecy_chanID,B5: set_Secrecy_chanID] :
! [X2: secrecy_chanID] :
( ( member1967978193chanID @ X2 @ A3 )
=> ( member1967978193chanID @ X2 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_186_subset__eq,axiom,
( ord_le996703545specID
= ( ^ [A3: set_Secrecy_specID,B5: set_Secrecy_specID] :
! [X2: secrecy_specID] :
( ( member740727610specID @ X2 @ A3 )
=> ( member740727610specID @ X2 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_187_subset__iff,axiom,
( ord_le1802272080chanID
= ( ^ [A3: set_Secrecy_chanID,B5: set_Secrecy_chanID] :
! [T: secrecy_chanID] :
( ( member1967978193chanID @ T @ A3 )
=> ( member1967978193chanID @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_188_subset__iff,axiom,
( ord_le996703545specID
= ( ^ [A3: set_Secrecy_specID,B5: set_Secrecy_specID] :
! [T: secrecy_specID] :
( ( member740727610specID @ T @ A3 )
=> ( member740727610specID @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_189_bot_Oextremum__uniqueI,axiom,
! [A: set_Secrecy_specID] :
( ( ord_le996703545specID @ A @ bot_bo1785112429specID )
=> ( A = bot_bo1785112429specID ) ) ).
% bot.extremum_uniqueI
thf(fact_190_bot_Oextremum__uniqueI,axiom,
! [A: set_Secrecy_chanID] :
( ( ord_le1802272080chanID @ A @ bot_bo443197316chanID )
=> ( A = bot_bo443197316chanID ) ) ).
% bot.extremum_uniqueI
thf(fact_191_bot_Oextremum__unique,axiom,
! [A: set_Secrecy_specID] :
( ( ord_le996703545specID @ A @ bot_bo1785112429specID )
= ( A = bot_bo1785112429specID ) ) ).
% bot.extremum_unique
thf(fact_192_bot_Oextremum__unique,axiom,
! [A: set_Secrecy_chanID] :
( ( ord_le1802272080chanID @ A @ bot_bo443197316chanID )
= ( A = bot_bo443197316chanID ) ) ).
% bot.extremum_unique
thf(fact_193_bot_Oextremum,axiom,
! [A: set_Secrecy_specID] : ( ord_le996703545specID @ bot_bo1785112429specID @ A ) ).
% bot.extremum
thf(fact_194_bot_Oextremum,axiom,
! [A: set_Secrecy_chanID] : ( ord_le1802272080chanID @ bot_bo443197316chanID @ A ) ).
% bot.extremum
thf(fact_195_subset__insertI2,axiom,
! [A2: set_Secrecy_specID,B: set_Secrecy_specID,B2: secrecy_specID] :
( ( ord_le996703545specID @ A2 @ B )
=> ( ord_le996703545specID @ A2 @ ( insert1671894227specID @ B2 @ B ) ) ) ).
% subset_insertI2
thf(fact_196_subset__insertI2,axiom,
! [A2: set_Secrecy_chanID,B: set_Secrecy_chanID,B2: secrecy_chanID] :
( ( ord_le1802272080chanID @ A2 @ B )
=> ( ord_le1802272080chanID @ A2 @ ( insert751661162chanID @ B2 @ B ) ) ) ).
% subset_insertI2
thf(fact_197_subset__insertI,axiom,
! [B: set_Secrecy_specID,A: secrecy_specID] : ( ord_le996703545specID @ B @ ( insert1671894227specID @ A @ B ) ) ).
% subset_insertI
thf(fact_198_subset__insertI,axiom,
! [B: set_Secrecy_chanID,A: secrecy_chanID] : ( ord_le1802272080chanID @ B @ ( insert751661162chanID @ A @ B ) ) ).
% subset_insertI
thf(fact_199_subset__insert,axiom,
! [X3: secrecy_chanID,A2: set_Secrecy_chanID,B: set_Secrecy_chanID] :
( ~ ( member1967978193chanID @ X3 @ A2 )
=> ( ( ord_le1802272080chanID @ A2 @ ( insert751661162chanID @ X3 @ B ) )
= ( ord_le1802272080chanID @ A2 @ B ) ) ) ).
% subset_insert
thf(fact_200_subset__insert,axiom,
! [X3: secrecy_specID,A2: set_Secrecy_specID,B: set_Secrecy_specID] :
( ~ ( member740727610specID @ X3 @ A2 )
=> ( ( ord_le996703545specID @ A2 @ ( insert1671894227specID @ X3 @ B ) )
= ( ord_le996703545specID @ A2 @ B ) ) ) ).
% subset_insert
thf(fact_201_insert__mono,axiom,
! [C3: set_Secrecy_specID,D2: set_Secrecy_specID,A: secrecy_specID] :
( ( ord_le996703545specID @ C3 @ D2 )
=> ( ord_le996703545specID @ ( insert1671894227specID @ A @ C3 ) @ ( insert1671894227specID @ A @ D2 ) ) ) ).
% insert_mono
thf(fact_202_insert__mono,axiom,
! [C3: set_Secrecy_chanID,D2: set_Secrecy_chanID,A: secrecy_chanID] :
( ( ord_le1802272080chanID @ C3 @ D2 )
=> ( ord_le1802272080chanID @ ( insert751661162chanID @ A @ C3 ) @ ( insert751661162chanID @ A @ D2 ) ) ) ).
% insert_mono
thf(fact_203_pairwise__empty,axiom,
! [P: secrecy_specID > secrecy_specID > $o] : ( pairwi514744696specID @ P @ bot_bo1785112429specID ) ).
% pairwise_empty
thf(fact_204_pairwise__empty,axiom,
! [P: secrecy_chanID > secrecy_chanID > $o] : ( pairwi1741995279chanID @ P @ bot_bo443197316chanID ) ).
% pairwise_empty
thf(fact_205_pairwise__insert,axiom,
! [R2: secrecy_chanID > secrecy_chanID > $o,X3: secrecy_chanID,S2: set_Secrecy_chanID] :
( ( pairwi1741995279chanID @ R2 @ ( insert751661162chanID @ X3 @ S2 ) )
= ( ! [Y4: secrecy_chanID] :
( ( ( member1967978193chanID @ Y4 @ S2 )
& ( Y4 != X3 ) )
=> ( ( R2 @ X3 @ Y4 )
& ( R2 @ Y4 @ X3 ) ) )
& ( pairwi1741995279chanID @ R2 @ S2 ) ) ) ).
% pairwise_insert
thf(fact_206_pairwise__insert,axiom,
! [R2: secrecy_specID > secrecy_specID > $o,X3: secrecy_specID,S2: set_Secrecy_specID] :
( ( pairwi514744696specID @ R2 @ ( insert1671894227specID @ X3 @ S2 ) )
= ( ! [Y4: secrecy_specID] :
( ( ( member740727610specID @ Y4 @ S2 )
& ( Y4 != X3 ) )
=> ( ( R2 @ X3 @ Y4 )
& ( R2 @ Y4 @ X3 ) ) )
& ( pairwi514744696specID @ R2 @ S2 ) ) ) ).
% pairwise_insert
thf(fact_207_subset__singleton__iff,axiom,
! [X5: set_Secrecy_specID,A: secrecy_specID] :
( ( ord_le996703545specID @ X5 @ ( insert1671894227specID @ A @ bot_bo1785112429specID ) )
= ( ( X5 = bot_bo1785112429specID )
| ( X5
= ( insert1671894227specID @ A @ bot_bo1785112429specID ) ) ) ) ).
% subset_singleton_iff
thf(fact_208_subset__singleton__iff,axiom,
! [X5: set_Secrecy_chanID,A: secrecy_chanID] :
( ( ord_le1802272080chanID @ X5 @ ( insert751661162chanID @ A @ bot_bo443197316chanID ) )
= ( ( X5 = bot_bo443197316chanID )
| ( X5
= ( insert751661162chanID @ A @ bot_bo443197316chanID ) ) ) ) ).
% subset_singleton_iff
thf(fact_209_subset__singletonD,axiom,
! [A2: set_Secrecy_specID,X3: secrecy_specID] :
( ( ord_le996703545specID @ A2 @ ( insert1671894227specID @ X3 @ bot_bo1785112429specID ) )
=> ( ( A2 = bot_bo1785112429specID )
| ( A2
= ( insert1671894227specID @ X3 @ bot_bo1785112429specID ) ) ) ) ).
% subset_singletonD
thf(fact_210_subset__singletonD,axiom,
! [A2: set_Secrecy_chanID,X3: secrecy_chanID] :
( ( ord_le1802272080chanID @ A2 @ ( insert751661162chanID @ X3 @ bot_bo443197316chanID ) )
=> ( ( A2 = bot_bo443197316chanID )
| ( A2
= ( insert751661162chanID @ X3 @ bot_bo443197316chanID ) ) ) ) ).
% subset_singletonD
thf(fact_211_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_212_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_213_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_214_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_215_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_216_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_217_TBtheorem2b,axiom,
! [PQ: secrecy_specID,M: set_Secrecy_chanID,E: secrecy_Expression,P: secrecy_specID,Q: secrecy_specID] :
( ( eoutM @ PQ @ M @ E )
=> ( ( ( subcomponents @ PQ )
= ( insert1671894227specID @ P @ ( insert1671894227specID @ Q @ bot_bo1785112429specID ) ) )
=> ( ( correc990227761ionOut @ PQ )
=> ( ( eoutM @ P @ M @ E )
| ( eoutM @ Q @ M @ E ) ) ) ) ) ).
% TBtheorem2b
thf(fact_218_eoutM__def,axiom,
( eoutM
= ( ^ [SP: secrecy_specID,M2: set_Secrecy_chanID,E2: secrecy_Expression] :
? [Ch3: secrecy_chanID] :
( ( member1967978193chanID @ Ch3 @ ( out @ SP ) )
& ( member1967978193chanID @ Ch3 @ M2 )
& ( exprChannel @ Ch3 @ E2 ) ) ) ) ).
% eoutM_def
thf(fact_219_out__exprChannelSet__def,axiom,
( out_exprChannelSet
= ( ^ [SP: secrecy_specID,ChSet2: set_Secrecy_chanID,E2: secrecy_Expression] :
( ! [X2: secrecy_chanID] :
( ( member1967978193chanID @ X2 @ ChSet2 )
=> ( ( member1967978193chanID @ X2 @ ( out @ SP ) )
& ( exprChannel @ X2 @ E2 ) ) )
& ! [X2: secrecy_chanID] :
( ( ~ ( member1967978193chanID @ X2 @ ChSet2 )
& ( member1967978193chanID @ X2 @ ( out @ SP ) ) )
=> ~ ( exprChannel @ X2 @ E2 ) ) ) ) ) ).
% out_exprChannelSet_def
thf(fact_220_out__exprChannelSingle__def,axiom,
( out_ex985543062Single
= ( ^ [SP: secrecy_specID,Ch3: secrecy_chanID,E2: secrecy_Expression] :
( ( member1967978193chanID @ Ch3 @ ( out @ SP ) )
& ( exprChannel @ Ch3 @ E2 )
& ! [X2: secrecy_chanID,T: nat] :
( ( ( member1967978193chanID @ X2 @ ( out @ SP ) )
& ( X2 != Ch3 ) )
=> ~ ( exprChannel @ X2 @ E2 ) ) ) ) ) ).
% out_exprChannelSingle_def
thf(fact_221_not__eout__eoutM,axiom,
! [P: secrecy_specID,E: secrecy_Expression,M: set_Secrecy_chanID] :
( ~ ( eout @ P @ E )
=> ~ ( eoutM @ P @ M @ E ) ) ).
% not_eout_eoutM
thf(fact_222_eoutM__eout,axiom,
! [P: secrecy_specID,M: set_Secrecy_chanID,E: secrecy_Expression] :
( ( eoutM @ P @ M @ E )
=> ( eout @ P @ E ) ) ).
% eoutM_eout
thf(fact_223_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_224_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_225_eout__def,axiom,
( eout
= ( ^ [SP: secrecy_specID,E2: secrecy_Expression] :
? [Ch3: secrecy_chanID] :
( ( member1967978193chanID @ Ch3 @ ( out @ SP ) )
& ( exprChannel @ Ch3 @ E2 ) ) ) ) ).
% eout_def
thf(fact_226_TBtheorem5b__notP2,axiom,
! [Q: secrecy_specID,M: set_Secrecy_chanID,E: secrecy_Expression,PQ: secrecy_specID,P: secrecy_specID,ChSet: set_Secrecy_chanID] :
( ~ ( eoutM @ Q @ M @ 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 @ M @ E ) ) ) ) ) ) ).
% TBtheorem5b_notP2
thf(fact_227_TBtheore54b__P,axiom,
! [P: secrecy_specID,M: set_Secrecy_chanID,E: secrecy_Expression,PQ: secrecy_specID,Q: secrecy_specID] :
( ( eoutM @ P @ M @ 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 @ M ) )
=> ( eoutM @ PQ @ M @ E ) ) ) ) ) ).
% TBtheore54b_P
thf(fact_228_TBtheorem5b__PQ,axiom,
! [P: secrecy_specID,M: set_Secrecy_chanID,E: secrecy_Expression,Q: secrecy_specID,PQ: secrecy_specID] :
( ( ( eoutM @ P @ M @ E )
| ( eoutM @ Q @ M @ E ) )
=> ( ( ( subcomponents @ PQ )
= ( insert1671894227specID @ P @ ( insert1671894227specID @ Q @ bot_bo1785112429specID ) ) )
=> ( ( correc990227761ionOut @ PQ )
=> ( ? [Ch2: secrecy_chanID] :
( ( ( member1967978193chanID @ Ch2 @ ( out @ P ) )
| ( member1967978193chanID @ Ch2 @ ( out @ Q ) ) )
& ( member1967978193chanID @ Ch2 @ M )
& ( exprChannel @ Ch2 @ E )
& ~ ( member1967978193chanID @ Ch2 @ ( loc @ PQ ) ) )
=> ( eoutM @ PQ @ M @ E ) ) ) ) ) ).
% TBtheorem5b_PQ
thf(fact_229_TBtheorem5b__notP1,axiom,
! [P: secrecy_specID,M: set_Secrecy_chanID,E: secrecy_Expression,Q: secrecy_specID,PQ: secrecy_specID] :
( ( eoutM @ P @ M @ E )
=> ( ~ ( eoutM @ Q @ M @ E )
=> ( ( ( subcomponents @ PQ )
= ( insert1671894227specID @ P @ ( insert1671894227specID @ Q @ bot_bo1785112429specID ) ) )
=> ( ( correc990227761ionOut @ PQ )
=> ( ? [Ch2: secrecy_chanID] :
( ( out_ex985543062Single @ P @ Ch2 @ E )
& ( member1967978193chanID @ Ch2 @ M )
& ( member1967978193chanID @ Ch2 @ ( loc @ PQ ) ) )
=> ~ ( eoutM @ PQ @ M @ E ) ) ) ) ) ) ).
% TBtheorem5b_notP1
thf(fact_230_correctCompositionOut__prop1,axiom,
! [PQ: secrecy_specID,P: secrecy_specID,Q: secrecy_specID,X3: secrecy_chanID] :
( ( ( subcomponents @ PQ )
= ( insert1671894227specID @ P @ ( insert1671894227specID @ Q @ bot_bo1785112429specID ) ) )
=> ( ( correc990227761ionOut @ PQ )
=> ( ( member1967978193chanID @ X3 @ ( out @ PQ ) )
=> ( ( member1967978193chanID @ X3 @ ( out @ P ) )
| ( member1967978193chanID @ X3 @ ( out @ Q ) ) ) ) ) ) ).
% correctCompositionOut_prop1
thf(fact_231_outStream__def,axiom,
( outStream
= ( ^ [X2: secrecy_specID] :
( ^ [Y: set_Secrecy_chanID,Z: set_Secrecy_chanID] : Y = Z
@ ( out @ X2 ) ) ) ) ).
% outStream_def
thf(fact_232_TBtheorem5b__notPQ,axiom,
! [PQ: secrecy_specID,P: secrecy_specID,Q: secrecy_specID,ChSetP: set_Secrecy_chanID,E: secrecy_Expression,ChSetQ: set_Secrecy_chanID,M: set_Secrecy_chanID] :
( ( ( subcomponents @ PQ )
= ( insert1671894227specID @ P @ ( insert1671894227specID @ Q @ bot_bo1785112429specID ) ) )
=> ( ( correc990227761ionOut @ PQ )
=> ( ( out_exprChannelSet @ P @ ChSetP @ E )
=> ( ( out_exprChannelSet @ Q @ ChSetQ @ E )
=> ( ( M
= ( 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 @ M @ E ) ) ) ) ) ) ) ) ).
% TBtheorem5b_notPQ
thf(fact_233_insert__subsetI,axiom,
! [X3: secrecy_chanID,A2: set_Secrecy_chanID,X5: set_Secrecy_chanID] :
( ( member1967978193chanID @ X3 @ A2 )
=> ( ( ord_le1802272080chanID @ X5 @ A2 )
=> ( ord_le1802272080chanID @ ( insert751661162chanID @ X3 @ X5 ) @ A2 ) ) ) ).
% insert_subsetI
thf(fact_234_insert__subsetI,axiom,
! [X3: secrecy_specID,A2: set_Secrecy_specID,X5: set_Secrecy_specID] :
( ( member740727610specID @ X3 @ A2 )
=> ( ( ord_le996703545specID @ X5 @ A2 )
=> ( ord_le996703545specID @ ( insert1671894227specID @ X3 @ X5 ) @ A2 ) ) ) ).
% insert_subsetI
thf(fact_235_UnCI,axiom,
! [C: secrecy_specID,B: set_Secrecy_specID,A2: set_Secrecy_specID] :
( ( ~ ( member740727610specID @ C @ B )
=> ( member740727610specID @ C @ A2 ) )
=> ( member740727610specID @ C @ ( sup_su1712599557specID @ A2 @ B ) ) ) ).
% UnCI
thf(fact_236_UnCI,axiom,
! [C: secrecy_chanID,B: set_Secrecy_chanID,A2: set_Secrecy_chanID] :
( ( ~ ( member1967978193chanID @ C @ B )
=> ( member1967978193chanID @ C @ A2 ) )
=> ( member1967978193chanID @ C @ ( sup_su370684444chanID @ A2 @ B ) ) ) ).
% UnCI
thf(fact_237_Un__iff,axiom,
! [C: secrecy_specID,A2: set_Secrecy_specID,B: set_Secrecy_specID] :
( ( member740727610specID @ C @ ( sup_su1712599557specID @ A2 @ B ) )
= ( ( member740727610specID @ C @ A2 )
| ( member740727610specID @ C @ B ) ) ) ).
% Un_iff
thf(fact_238_Un__iff,axiom,
! [C: secrecy_chanID,A2: set_Secrecy_chanID,B: set_Secrecy_chanID] :
( ( member1967978193chanID @ C @ ( sup_su370684444chanID @ A2 @ B ) )
= ( ( member1967978193chanID @ C @ A2 )
| ( member1967978193chanID @ C @ B ) ) ) ).
% Un_iff
thf(fact_239_Un__empty,axiom,
! [A2: set_Secrecy_specID,B: set_Secrecy_specID] :
( ( ( sup_su1712599557specID @ A2 @ B )
= bot_bo1785112429specID )
= ( ( A2 = bot_bo1785112429specID )
& ( B = bot_bo1785112429specID ) ) ) ).
% Un_empty
thf(fact_240_Un__empty,axiom,
! [A2: set_Secrecy_chanID,B: set_Secrecy_chanID] :
( ( ( sup_su370684444chanID @ A2 @ B )
= bot_bo443197316chanID )
= ( ( A2 = bot_bo443197316chanID )
& ( B = bot_bo443197316chanID ) ) ) ).
% Un_empty
thf(fact_241_Un__insert__left,axiom,
! [A: secrecy_specID,B: set_Secrecy_specID,C3: set_Secrecy_specID] :
( ( sup_su1712599557specID @ ( insert1671894227specID @ A @ B ) @ C3 )
= ( insert1671894227specID @ A @ ( sup_su1712599557specID @ B @ C3 ) ) ) ).
% Un_insert_left
thf(fact_242_Un__insert__left,axiom,
! [A: secrecy_chanID,B: set_Secrecy_chanID,C3: set_Secrecy_chanID] :
( ( sup_su370684444chanID @ ( insert751661162chanID @ A @ B ) @ C3 )
= ( insert751661162chanID @ A @ ( sup_su370684444chanID @ B @ C3 ) ) ) ).
% Un_insert_left
thf(fact_243_Un__insert__right,axiom,
! [A2: set_Secrecy_specID,A: secrecy_specID,B: set_Secrecy_specID] :
( ( sup_su1712599557specID @ A2 @ ( insert1671894227specID @ A @ B ) )
= ( insert1671894227specID @ A @ ( sup_su1712599557specID @ A2 @ B ) ) ) ).
% Un_insert_right
thf(fact_244_Un__insert__right,axiom,
! [A2: set_Secrecy_chanID,A: secrecy_chanID,B: set_Secrecy_chanID] :
( ( sup_su370684444chanID @ A2 @ ( insert751661162chanID @ A @ B ) )
= ( insert751661162chanID @ A @ ( sup_su370684444chanID @ A2 @ B ) ) ) ).
% Un_insert_right
thf(fact_245_Un__subset__iff,axiom,
! [A2: set_Secrecy_chanID,B: set_Secrecy_chanID,C3: set_Secrecy_chanID] :
( ( ord_le1802272080chanID @ ( sup_su370684444chanID @ A2 @ B ) @ C3 )
= ( ( ord_le1802272080chanID @ A2 @ C3 )
& ( ord_le1802272080chanID @ B @ C3 ) ) ) ).
% Un_subset_iff
thf(fact_246_Un__empty__left,axiom,
! [B: set_Secrecy_specID] :
( ( sup_su1712599557specID @ bot_bo1785112429specID @ B )
= B ) ).
% Un_empty_left
thf(fact_247_Un__empty__left,axiom,
! [B: set_Secrecy_chanID] :
( ( sup_su370684444chanID @ bot_bo443197316chanID @ B )
= B ) ).
% Un_empty_left
thf(fact_248_Un__empty__right,axiom,
! [A2: set_Secrecy_specID] :
( ( sup_su1712599557specID @ A2 @ bot_bo1785112429specID )
= A2 ) ).
% Un_empty_right
thf(fact_249_Un__empty__right,axiom,
! [A2: set_Secrecy_chanID] :
( ( sup_su370684444chanID @ A2 @ bot_bo443197316chanID )
= A2 ) ).
% Un_empty_right
thf(fact_250_subset__Un__eq,axiom,
( ord_le1802272080chanID
= ( ^ [A3: set_Secrecy_chanID,B5: set_Secrecy_chanID] :
( ( sup_su370684444chanID @ A3 @ B5 )
= B5 ) ) ) ).
% subset_Un_eq
thf(fact_251_subset__UnE,axiom,
! [C3: set_Secrecy_chanID,A2: set_Secrecy_chanID,B: set_Secrecy_chanID] :
( ( ord_le1802272080chanID @ C3 @ ( sup_su370684444chanID @ A2 @ B ) )
=> ~ ! [A5: set_Secrecy_chanID] :
( ( ord_le1802272080chanID @ A5 @ A2 )
=> ! [B6: set_Secrecy_chanID] :
( ( ord_le1802272080chanID @ B6 @ B )
=> ( C3
!= ( sup_su370684444chanID @ A5 @ B6 ) ) ) ) ) ).
% subset_UnE
thf(fact_252_Un__absorb2,axiom,
! [B: set_Secrecy_chanID,A2: set_Secrecy_chanID] :
( ( ord_le1802272080chanID @ B @ A2 )
=> ( ( sup_su370684444chanID @ A2 @ B )
= A2 ) ) ).
% Un_absorb2
thf(fact_253_Un__absorb1,axiom,
! [A2: set_Secrecy_chanID,B: set_Secrecy_chanID] :
( ( ord_le1802272080chanID @ A2 @ B )
=> ( ( sup_su370684444chanID @ A2 @ B )
= B ) ) ).
% Un_absorb1
thf(fact_254_Un__upper2,axiom,
! [B: set_Secrecy_chanID,A2: set_Secrecy_chanID] : ( ord_le1802272080chanID @ B @ ( sup_su370684444chanID @ A2 @ B ) ) ).
% Un_upper2
thf(fact_255_Un__upper1,axiom,
! [A2: set_Secrecy_chanID,B: set_Secrecy_chanID] : ( ord_le1802272080chanID @ A2 @ ( sup_su370684444chanID @ A2 @ B ) ) ).
% Un_upper1
thf(fact_256_Un__least,axiom,
! [A2: set_Secrecy_chanID,C3: set_Secrecy_chanID,B: set_Secrecy_chanID] :
( ( ord_le1802272080chanID @ A2 @ C3 )
=> ( ( ord_le1802272080chanID @ B @ C3 )
=> ( ord_le1802272080chanID @ ( sup_su370684444chanID @ A2 @ B ) @ C3 ) ) ) ).
% Un_least
thf(fact_257_Un__mono,axiom,
! [A2: set_Secrecy_chanID,C3: set_Secrecy_chanID,B: set_Secrecy_chanID,D2: set_Secrecy_chanID] :
( ( ord_le1802272080chanID @ A2 @ C3 )
=> ( ( ord_le1802272080chanID @ B @ D2 )
=> ( ord_le1802272080chanID @ ( sup_su370684444chanID @ A2 @ B ) @ ( sup_su370684444chanID @ C3 @ D2 ) ) ) ) ).
% Un_mono
thf(fact_258_ineM__Un1,axiom,
! [P: secrecy_specID,A2: set_Secrecy_chanID,E: secrecy_Expression,B: set_Secrecy_chanID] :
( ( ineM @ P @ A2 @ E )
=> ( ineM @ P @ ( sup_su370684444chanID @ A2 @ B ) @ E ) ) ).
% ineM_Un1
thf(fact_259_UnE,axiom,
! [C: secrecy_specID,A2: set_Secrecy_specID,B: set_Secrecy_specID] :
( ( member740727610specID @ C @ ( sup_su1712599557specID @ A2 @ B ) )
=> ( ~ ( member740727610specID @ C @ A2 )
=> ( member740727610specID @ C @ B ) ) ) ).
% UnE
thf(fact_260_UnE,axiom,
! [C: secrecy_chanID,A2: set_Secrecy_chanID,B: set_Secrecy_chanID] :
( ( member1967978193chanID @ C @ ( sup_su370684444chanID @ A2 @ B ) )
=> ( ~ ( member1967978193chanID @ C @ A2 )
=> ( member1967978193chanID @ C @ B ) ) ) ).
% UnE
thf(fact_261_UnI1,axiom,
! [C: secrecy_specID,A2: set_Secrecy_specID,B: set_Secrecy_specID] :
( ( member740727610specID @ C @ A2 )
=> ( member740727610specID @ C @ ( sup_su1712599557specID @ A2 @ B ) ) ) ).
% UnI1
thf(fact_262_UnI1,axiom,
! [C: secrecy_chanID,A2: set_Secrecy_chanID,B: set_Secrecy_chanID] :
( ( member1967978193chanID @ C @ A2 )
=> ( member1967978193chanID @ C @ ( sup_su370684444chanID @ A2 @ B ) ) ) ).
% UnI1
thf(fact_263_UnI2,axiom,
! [C: secrecy_specID,B: set_Secrecy_specID,A2: set_Secrecy_specID] :
( ( member740727610specID @ C @ B )
=> ( member740727610specID @ C @ ( sup_su1712599557specID @ A2 @ B ) ) ) ).
% UnI2
thf(fact_264_UnI2,axiom,
! [C: secrecy_chanID,B: set_Secrecy_chanID,A2: set_Secrecy_chanID] :
( ( member1967978193chanID @ C @ B )
=> ( member1967978193chanID @ C @ ( sup_su370684444chanID @ A2 @ B ) ) ) ).
% UnI2
thf(fact_265_bex__Un,axiom,
! [A2: set_Secrecy_chanID,B: set_Secrecy_chanID,P: secrecy_chanID > $o] :
( ( ? [X2: secrecy_chanID] :
( ( member1967978193chanID @ X2 @ ( sup_su370684444chanID @ A2 @ B ) )
& ( P @ X2 ) ) )
= ( ? [X2: secrecy_chanID] :
( ( member1967978193chanID @ X2 @ A2 )
& ( P @ X2 ) )
| ? [X2: secrecy_chanID] :
( ( member1967978193chanID @ X2 @ B )
& ( P @ X2 ) ) ) ) ).
% bex_Un
thf(fact_266_ball__Un,axiom,
! [A2: set_Secrecy_chanID,B: set_Secrecy_chanID,P: secrecy_chanID > $o] :
( ( ! [X2: secrecy_chanID] :
( ( member1967978193chanID @ X2 @ ( sup_su370684444chanID @ A2 @ B ) )
=> ( P @ X2 ) ) )
= ( ! [X2: secrecy_chanID] :
( ( member1967978193chanID @ X2 @ A2 )
=> ( P @ X2 ) )
& ! [X2: secrecy_chanID] :
( ( member1967978193chanID @ X2 @ B )
=> ( P @ X2 ) ) ) ) ).
% ball_Un
thf(fact_267_Un__assoc,axiom,
! [A2: set_Secrecy_chanID,B: set_Secrecy_chanID,C3: set_Secrecy_chanID] :
( ( sup_su370684444chanID @ ( sup_su370684444chanID @ A2 @ B ) @ C3 )
= ( sup_su370684444chanID @ A2 @ ( sup_su370684444chanID @ B @ C3 ) ) ) ).
% Un_assoc
thf(fact_268_Un__absorb,axiom,
! [A2: set_Secrecy_chanID] :
( ( sup_su370684444chanID @ A2 @ A2 )
= A2 ) ).
% Un_absorb
thf(fact_269_Un__commute,axiom,
( sup_su370684444chanID
= ( ^ [A3: set_Secrecy_chanID,B5: set_Secrecy_chanID] : ( sup_su370684444chanID @ B5 @ A3 ) ) ) ).
% Un_commute
thf(fact_270_Un__left__absorb,axiom,
! [A2: set_Secrecy_chanID,B: set_Secrecy_chanID] :
( ( sup_su370684444chanID @ A2 @ ( sup_su370684444chanID @ A2 @ B ) )
= ( sup_su370684444chanID @ A2 @ B ) ) ).
% Un_left_absorb
thf(fact_271_Un__left__commute,axiom,
! [A2: set_Secrecy_chanID,B: set_Secrecy_chanID,C3: set_Secrecy_chanID] :
( ( sup_su370684444chanID @ A2 @ ( sup_su370684444chanID @ B @ C3 ) )
= ( sup_su370684444chanID @ B @ ( sup_su370684444chanID @ A2 @ C3 ) ) ) ).
% Un_left_commute
thf(fact_272_singleton__Un__iff,axiom,
! [X3: secrecy_specID,A2: set_Secrecy_specID,B: set_Secrecy_specID] :
( ( ( insert1671894227specID @ X3 @ bot_bo1785112429specID )
= ( sup_su1712599557specID @ A2 @ B ) )
= ( ( ( A2 = bot_bo1785112429specID )
& ( B
= ( insert1671894227specID @ X3 @ bot_bo1785112429specID ) ) )
| ( ( A2
= ( insert1671894227specID @ X3 @ bot_bo1785112429specID ) )
& ( B = bot_bo1785112429specID ) )
| ( ( A2
= ( insert1671894227specID @ X3 @ bot_bo1785112429specID ) )
& ( B
= ( insert1671894227specID @ X3 @ bot_bo1785112429specID ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_273_singleton__Un__iff,axiom,
! [X3: secrecy_chanID,A2: set_Secrecy_chanID,B: set_Secrecy_chanID] :
( ( ( insert751661162chanID @ X3 @ bot_bo443197316chanID )
= ( sup_su370684444chanID @ A2 @ B ) )
= ( ( ( A2 = bot_bo443197316chanID )
& ( B
= ( insert751661162chanID @ X3 @ bot_bo443197316chanID ) ) )
| ( ( A2
= ( insert751661162chanID @ X3 @ bot_bo443197316chanID ) )
& ( B = bot_bo443197316chanID ) )
| ( ( A2
= ( insert751661162chanID @ X3 @ bot_bo443197316chanID ) )
& ( B
= ( insert751661162chanID @ X3 @ bot_bo443197316chanID ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_274_Un__singleton__iff,axiom,
! [A2: set_Secrecy_specID,B: set_Secrecy_specID,X3: secrecy_specID] :
( ( ( sup_su1712599557specID @ A2 @ B )
= ( insert1671894227specID @ X3 @ bot_bo1785112429specID ) )
= ( ( ( A2 = bot_bo1785112429specID )
& ( B
= ( insert1671894227specID @ X3 @ bot_bo1785112429specID ) ) )
| ( ( A2
= ( insert1671894227specID @ X3 @ bot_bo1785112429specID ) )
& ( B = bot_bo1785112429specID ) )
| ( ( A2
= ( insert1671894227specID @ X3 @ bot_bo1785112429specID ) )
& ( B
= ( insert1671894227specID @ X3 @ bot_bo1785112429specID ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_275_Un__singleton__iff,axiom,
! [A2: set_Secrecy_chanID,B: set_Secrecy_chanID,X3: secrecy_chanID] :
( ( ( sup_su370684444chanID @ A2 @ B )
= ( insert751661162chanID @ X3 @ bot_bo443197316chanID ) )
= ( ( ( A2 = bot_bo443197316chanID )
& ( B
= ( insert751661162chanID @ X3 @ bot_bo443197316chanID ) ) )
| ( ( A2
= ( insert751661162chanID @ X3 @ bot_bo443197316chanID ) )
& ( B = bot_bo443197316chanID ) )
| ( ( A2
= ( insert751661162chanID @ X3 @ bot_bo443197316chanID ) )
& ( B
= ( insert751661162chanID @ X3 @ bot_bo443197316chanID ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_276_insert__is__Un,axiom,
( insert1671894227specID
= ( ^ [A4: secrecy_specID] : ( sup_su1712599557specID @ ( insert1671894227specID @ A4 @ bot_bo1785112429specID ) ) ) ) ).
% insert_is_Un
thf(fact_277_insert__is__Un,axiom,
( insert751661162chanID
= ( ^ [A4: secrecy_chanID] : ( sup_su370684444chanID @ ( insert751661162chanID @ A4 @ bot_bo443197316chanID ) ) ) ) ).
% insert_is_Un
thf(fact_278_subset__emptyI,axiom,
! [A2: set_Secrecy_specID] :
( ! [X4: secrecy_specID] :
~ ( member740727610specID @ X4 @ A2 )
=> ( ord_le996703545specID @ A2 @ bot_bo1785112429specID ) ) ).
% subset_emptyI
thf(fact_279_subset__emptyI,axiom,
! [A2: set_Secrecy_chanID] :
( ! [X4: secrecy_chanID] :
~ ( member1967978193chanID @ X4 @ A2 )
=> ( ord_le1802272080chanID @ A2 @ bot_bo443197316chanID ) ) ).
% subset_emptyI
thf(fact_280_sup__bot_Oright__neutral,axiom,
! [A: set_Secrecy_specID] :
( ( sup_su1712599557specID @ A @ bot_bo1785112429specID )
= A ) ).
% sup_bot.right_neutral
thf(fact_281_sup__bot_Oright__neutral,axiom,
! [A: set_Secrecy_chanID] :
( ( sup_su370684444chanID @ A @ bot_bo443197316chanID )
= A ) ).
% sup_bot.right_neutral
thf(fact_282_sup__bot_Oneutr__eq__iff,axiom,
! [A: set_Secrecy_specID,B2: set_Secrecy_specID] :
( ( bot_bo1785112429specID
= ( sup_su1712599557specID @ A @ B2 ) )
= ( ( A = bot_bo1785112429specID )
& ( B2 = bot_bo1785112429specID ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_283_sup__bot_Oneutr__eq__iff,axiom,
! [A: set_Secrecy_chanID,B2: set_Secrecy_chanID] :
( ( bot_bo443197316chanID
= ( sup_su370684444chanID @ A @ B2 ) )
= ( ( A = bot_bo443197316chanID )
& ( B2 = bot_bo443197316chanID ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_284_sup__bot_Oleft__neutral,axiom,
! [A: set_Secrecy_specID] :
( ( sup_su1712599557specID @ bot_bo1785112429specID @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_285_sup__bot_Oleft__neutral,axiom,
! [A: set_Secrecy_chanID] :
( ( sup_su370684444chanID @ bot_bo443197316chanID @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_286_le__sup__iff,axiom,
! [X3: set_Secrecy_chanID,Y3: set_Secrecy_chanID,Z2: set_Secrecy_chanID] :
( ( ord_le1802272080chanID @ ( sup_su370684444chanID @ X3 @ Y3 ) @ Z2 )
= ( ( ord_le1802272080chanID @ X3 @ Z2 )
& ( ord_le1802272080chanID @ Y3 @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_287_sup_Obounded__iff,axiom,
! [B2: set_Secrecy_chanID,C: set_Secrecy_chanID,A: set_Secrecy_chanID] :
( ( ord_le1802272080chanID @ ( sup_su370684444chanID @ B2 @ C ) @ A )
= ( ( ord_le1802272080chanID @ B2 @ A )
& ( ord_le1802272080chanID @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_288_sup__bot__left,axiom,
! [X3: set_Secrecy_specID] :
( ( sup_su1712599557specID @ bot_bo1785112429specID @ X3 )
= X3 ) ).
% sup_bot_left
thf(fact_289_sup__bot__left,axiom,
! [X3: set_Secrecy_chanID] :
( ( sup_su370684444chanID @ bot_bo443197316chanID @ X3 )
= X3 ) ).
% sup_bot_left
thf(fact_290_sup__bot__right,axiom,
! [X3: set_Secrecy_specID] :
( ( sup_su1712599557specID @ X3 @ bot_bo1785112429specID )
= X3 ) ).
% sup_bot_right
thf(fact_291_sup__bot__right,axiom,
! [X3: set_Secrecy_chanID] :
( ( sup_su370684444chanID @ X3 @ bot_bo443197316chanID )
= X3 ) ).
% sup_bot_right
thf(fact_292_bot__eq__sup__iff,axiom,
! [X3: set_Secrecy_specID,Y3: set_Secrecy_specID] :
( ( bot_bo1785112429specID
= ( sup_su1712599557specID @ X3 @ Y3 ) )
= ( ( X3 = bot_bo1785112429specID )
& ( Y3 = bot_bo1785112429specID ) ) ) ).
% bot_eq_sup_iff
thf(fact_293_bot__eq__sup__iff,axiom,
! [X3: set_Secrecy_chanID,Y3: set_Secrecy_chanID] :
( ( bot_bo443197316chanID
= ( sup_su370684444chanID @ X3 @ Y3 ) )
= ( ( X3 = bot_bo443197316chanID )
& ( Y3 = bot_bo443197316chanID ) ) ) ).
% bot_eq_sup_iff
thf(fact_294_sup__eq__bot__iff,axiom,
! [X3: set_Secrecy_specID,Y3: set_Secrecy_specID] :
( ( ( sup_su1712599557specID @ X3 @ Y3 )
= bot_bo1785112429specID )
= ( ( X3 = bot_bo1785112429specID )
& ( Y3 = bot_bo1785112429specID ) ) ) ).
% sup_eq_bot_iff
thf(fact_295_sup__eq__bot__iff,axiom,
! [X3: set_Secrecy_chanID,Y3: set_Secrecy_chanID] :
( ( ( sup_su370684444chanID @ X3 @ Y3 )
= bot_bo443197316chanID )
= ( ( X3 = bot_bo443197316chanID )
& ( Y3 = bot_bo443197316chanID ) ) ) ).
% sup_eq_bot_iff
thf(fact_296_sup__bot_Oeq__neutr__iff,axiom,
! [A: set_Secrecy_specID,B2: set_Secrecy_specID] :
( ( ( sup_su1712599557specID @ A @ B2 )
= bot_bo1785112429specID )
= ( ( A = bot_bo1785112429specID )
& ( B2 = bot_bo1785112429specID ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_297_sup__bot_Oeq__neutr__iff,axiom,
! [A: set_Secrecy_chanID,B2: set_Secrecy_chanID] :
( ( ( sup_su370684444chanID @ A @ B2 )
= bot_bo443197316chanID )
= ( ( A = bot_bo443197316chanID )
& ( B2 = bot_bo443197316chanID ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_298_inf__sup__ord_I4_J,axiom,
! [Y3: set_Secrecy_chanID,X3: set_Secrecy_chanID] : ( ord_le1802272080chanID @ Y3 @ ( sup_su370684444chanID @ X3 @ Y3 ) ) ).
% inf_sup_ord(4)
thf(fact_299_inf__sup__ord_I3_J,axiom,
! [X3: set_Secrecy_chanID,Y3: set_Secrecy_chanID] : ( ord_le1802272080chanID @ X3 @ ( sup_su370684444chanID @ X3 @ Y3 ) ) ).
% inf_sup_ord(3)
thf(fact_300_le__supE,axiom,
! [A: set_Secrecy_chanID,B2: set_Secrecy_chanID,X3: set_Secrecy_chanID] :
( ( ord_le1802272080chanID @ ( sup_su370684444chanID @ A @ B2 ) @ X3 )
=> ~ ( ( ord_le1802272080chanID @ A @ X3 )
=> ~ ( ord_le1802272080chanID @ B2 @ X3 ) ) ) ).
% le_supE
thf(fact_301_le__supI,axiom,
! [A: set_Secrecy_chanID,X3: set_Secrecy_chanID,B2: set_Secrecy_chanID] :
( ( ord_le1802272080chanID @ A @ X3 )
=> ( ( ord_le1802272080chanID @ B2 @ X3 )
=> ( ord_le1802272080chanID @ ( sup_su370684444chanID @ A @ B2 ) @ X3 ) ) ) ).
% le_supI
thf(fact_302_sup__ge1,axiom,
! [X3: set_Secrecy_chanID,Y3: set_Secrecy_chanID] : ( ord_le1802272080chanID @ X3 @ ( sup_su370684444chanID @ X3 @ Y3 ) ) ).
% sup_ge1
thf(fact_303_sup__ge2,axiom,
! [Y3: set_Secrecy_chanID,X3: set_Secrecy_chanID] : ( ord_le1802272080chanID @ Y3 @ ( sup_su370684444chanID @ X3 @ Y3 ) ) ).
% sup_ge2
thf(fact_304_le__supI1,axiom,
! [X3: set_Secrecy_chanID,A: set_Secrecy_chanID,B2: set_Secrecy_chanID] :
( ( ord_le1802272080chanID @ X3 @ A )
=> ( ord_le1802272080chanID @ X3 @ ( sup_su370684444chanID @ A @ B2 ) ) ) ).
% le_supI1
thf(fact_305_le__supI2,axiom,
! [X3: set_Secrecy_chanID,B2: set_Secrecy_chanID,A: set_Secrecy_chanID] :
( ( ord_le1802272080chanID @ X3 @ B2 )
=> ( ord_le1802272080chanID @ X3 @ ( sup_su370684444chanID @ A @ B2 ) ) ) ).
% le_supI2
thf(fact_306_sup_Omono,axiom,
! [C: set_Secrecy_chanID,A: set_Secrecy_chanID,D: set_Secrecy_chanID,B2: set_Secrecy_chanID] :
( ( ord_le1802272080chanID @ C @ A )
=> ( ( ord_le1802272080chanID @ D @ B2 )
=> ( ord_le1802272080chanID @ ( sup_su370684444chanID @ C @ D ) @ ( sup_su370684444chanID @ A @ B2 ) ) ) ) ).
% sup.mono
thf(fact_307_sup__mono,axiom,
! [A: set_Secrecy_chanID,C: set_Secrecy_chanID,B2: set_Secrecy_chanID,D: set_Secrecy_chanID] :
( ( ord_le1802272080chanID @ A @ C )
=> ( ( ord_le1802272080chanID @ B2 @ D )
=> ( ord_le1802272080chanID @ ( sup_su370684444chanID @ A @ B2 ) @ ( sup_su370684444chanID @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_308_sup__least,axiom,
! [Y3: set_Secrecy_chanID,X3: set_Secrecy_chanID,Z2: set_Secrecy_chanID] :
( ( ord_le1802272080chanID @ Y3 @ X3 )
=> ( ( ord_le1802272080chanID @ Z2 @ X3 )
=> ( ord_le1802272080chanID @ ( sup_su370684444chanID @ Y3 @ Z2 ) @ X3 ) ) ) ).
% sup_least
thf(fact_309_le__iff__sup,axiom,
( ord_le1802272080chanID
= ( ^ [X2: set_Secrecy_chanID,Y4: set_Secrecy_chanID] :
( ( sup_su370684444chanID @ X2 @ Y4 )
= Y4 ) ) ) ).
% le_iff_sup
thf(fact_310_sup_OorderE,axiom,
! [B2: set_Secrecy_chanID,A: set_Secrecy_chanID] :
( ( ord_le1802272080chanID @ B2 @ A )
=> ( A
= ( sup_su370684444chanID @ A @ B2 ) ) ) ).
% sup.orderE
thf(fact_311_sup_OorderI,axiom,
! [A: set_Secrecy_chanID,B2: set_Secrecy_chanID] :
( ( A
= ( sup_su370684444chanID @ A @ B2 ) )
=> ( ord_le1802272080chanID @ B2 @ A ) ) ).
% sup.orderI
thf(fact_312_sup__unique,axiom,
! [F: set_Secrecy_chanID > set_Secrecy_chanID > set_Secrecy_chanID,X3: set_Secrecy_chanID,Y3: set_Secrecy_chanID] :
( ! [X4: set_Secrecy_chanID,Y2: set_Secrecy_chanID] : ( ord_le1802272080chanID @ X4 @ ( F @ X4 @ Y2 ) )
=> ( ! [X4: set_Secrecy_chanID,Y2: set_Secrecy_chanID] : ( ord_le1802272080chanID @ Y2 @ ( F @ X4 @ Y2 ) )
=> ( ! [X4: set_Secrecy_chanID,Y2: set_Secrecy_chanID,Z3: set_Secrecy_chanID] :
( ( ord_le1802272080chanID @ Y2 @ X4 )
=> ( ( ord_le1802272080chanID @ Z3 @ X4 )
=> ( ord_le1802272080chanID @ ( F @ Y2 @ Z3 ) @ X4 ) ) )
=> ( ( sup_su370684444chanID @ X3 @ Y3 )
= ( F @ X3 @ Y3 ) ) ) ) ) ).
% sup_unique
thf(fact_313_sup_Oabsorb1,axiom,
! [B2: set_Secrecy_chanID,A: set_Secrecy_chanID] :
( ( ord_le1802272080chanID @ B2 @ A )
=> ( ( sup_su370684444chanID @ A @ B2 )
= A ) ) ).
% sup.absorb1
thf(fact_314_sup_Oabsorb2,axiom,
! [A: set_Secrecy_chanID,B2: set_Secrecy_chanID] :
( ( ord_le1802272080chanID @ A @ B2 )
=> ( ( sup_su370684444chanID @ A @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_315_sup__absorb1,axiom,
! [Y3: set_Secrecy_chanID,X3: set_Secrecy_chanID] :
( ( ord_le1802272080chanID @ Y3 @ X3 )
=> ( ( sup_su370684444chanID @ X3 @ Y3 )
= X3 ) ) ).
% sup_absorb1
thf(fact_316_sup__absorb2,axiom,
! [X3: set_Secrecy_chanID,Y3: set_Secrecy_chanID] :
( ( ord_le1802272080chanID @ X3 @ Y3 )
=> ( ( sup_su370684444chanID @ X3 @ Y3 )
= Y3 ) ) ).
% sup_absorb2
thf(fact_317_sup_OboundedE,axiom,
! [B2: set_Secrecy_chanID,C: set_Secrecy_chanID,A: set_Secrecy_chanID] :
( ( ord_le1802272080chanID @ ( sup_su370684444chanID @ B2 @ C ) @ A )
=> ~ ( ( ord_le1802272080chanID @ B2 @ A )
=> ~ ( ord_le1802272080chanID @ C @ A ) ) ) ).
% sup.boundedE
thf(fact_318_sup_OboundedI,axiom,
! [B2: set_Secrecy_chanID,A: set_Secrecy_chanID,C: set_Secrecy_chanID] :
( ( ord_le1802272080chanID @ B2 @ A )
=> ( ( ord_le1802272080chanID @ C @ A )
=> ( ord_le1802272080chanID @ ( sup_su370684444chanID @ B2 @ C ) @ A ) ) ) ).
% sup.boundedI
thf(fact_319_sup_Oorder__iff,axiom,
( ord_le1802272080chanID
= ( ^ [B4: set_Secrecy_chanID,A4: set_Secrecy_chanID] :
( A4
= ( sup_su370684444chanID @ A4 @ B4 ) ) ) ) ).
% sup.order_iff
thf(fact_320_sup_Ocobounded1,axiom,
! [A: set_Secrecy_chanID,B2: set_Secrecy_chanID] : ( ord_le1802272080chanID @ A @ ( sup_su370684444chanID @ A @ B2 ) ) ).
% sup.cobounded1
thf(fact_321_sup_Ocobounded2,axiom,
! [B2: set_Secrecy_chanID,A: set_Secrecy_chanID] : ( ord_le1802272080chanID @ B2 @ ( sup_su370684444chanID @ A @ B2 ) ) ).
% sup.cobounded2
thf(fact_322_sup_Oabsorb__iff1,axiom,
( ord_le1802272080chanID
= ( ^ [B4: set_Secrecy_chanID,A4: set_Secrecy_chanID] :
( ( sup_su370684444chanID @ A4 @ B4 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_323_sup_Oabsorb__iff2,axiom,
( ord_le1802272080chanID
= ( ^ [A4: set_Secrecy_chanID,B4: set_Secrecy_chanID] :
( ( sup_su370684444chanID @ A4 @ B4 )
= B4 ) ) ) ).
% sup.absorb_iff2
thf(fact_324_sup_OcoboundedI1,axiom,
! [C: set_Secrecy_chanID,A: set_Secrecy_chanID,B2: set_Secrecy_chanID] :
( ( ord_le1802272080chanID @ C @ A )
=> ( ord_le1802272080chanID @ C @ ( sup_su370684444chanID @ A @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_325_sup_OcoboundedI2,axiom,
! [C: set_Secrecy_chanID,B2: set_Secrecy_chanID,A: set_Secrecy_chanID] :
( ( ord_le1802272080chanID @ C @ B2 )
=> ( ord_le1802272080chanID @ C @ ( sup_su370684444chanID @ A @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_326_correctInOutLoc__def,axiom,
( correctInOutLoc
= ( ^ [X2: secrecy_specID] :
( ( ( inf_in1474033538chanID @ ( ins @ X2 ) @ ( out @ X2 ) )
= bot_bo443197316chanID )
& ( ( inf_in1474033538chanID @ ( ins @ X2 ) @ ( loc @ X2 ) )
= bot_bo443197316chanID )
& ( ( inf_in1474033538chanID @ ( loc @ X2 ) @ ( out @ X2 ) )
= bot_bo443197316chanID ) ) ) ) ).
% correctInOutLoc_def
thf(fact_327_subset__Compl__singleton,axiom,
! [A2: set_Secrecy_specID,B2: secrecy_specID] :
( ( ord_le996703545specID @ A2 @ ( uminus238526242specID @ ( insert1671894227specID @ B2 @ bot_bo1785112429specID ) ) )
= ( ~ ( member740727610specID @ B2 @ A2 ) ) ) ).
% subset_Compl_singleton
thf(fact_328_subset__Compl__singleton,axiom,
! [A2: set_Secrecy_chanID,B2: secrecy_chanID] :
( ( ord_le1802272080chanID @ A2 @ ( uminus1044094777chanID @ ( insert751661162chanID @ B2 @ bot_bo443197316chanID ) ) )
= ( ~ ( member1967978193chanID @ B2 @ A2 ) ) ) ).
% subset_Compl_singleton
thf(fact_329_IntI,axiom,
! [C: secrecy_specID,A2: set_Secrecy_specID,B: set_Secrecy_specID] :
( ( member740727610specID @ C @ A2 )
=> ( ( member740727610specID @ C @ B )
=> ( member740727610specID @ C @ ( inf_in668465003specID @ A2 @ B ) ) ) ) ).
% IntI
thf(fact_330_IntI,axiom,
! [C: secrecy_chanID,A2: set_Secrecy_chanID,B: set_Secrecy_chanID] :
( ( member1967978193chanID @ C @ A2 )
=> ( ( member1967978193chanID @ C @ B )
=> ( member1967978193chanID @ C @ ( inf_in1474033538chanID @ A2 @ B ) ) ) ) ).
% IntI
thf(fact_331_Int__iff,axiom,
! [C: secrecy_specID,A2: set_Secrecy_specID,B: set_Secrecy_specID] :
( ( member740727610specID @ C @ ( inf_in668465003specID @ A2 @ B ) )
= ( ( member740727610specID @ C @ A2 )
& ( member740727610specID @ C @ B ) ) ) ).
% Int_iff
thf(fact_332_Int__iff,axiom,
! [C: secrecy_chanID,A2: set_Secrecy_chanID,B: set_Secrecy_chanID] :
( ( member1967978193chanID @ C @ ( inf_in1474033538chanID @ A2 @ B ) )
= ( ( member1967978193chanID @ C @ A2 )
& ( member1967978193chanID @ C @ B ) ) ) ).
% Int_iff
thf(fact_333_ComplI,axiom,
! [C: secrecy_chanID,A2: set_Secrecy_chanID] :
( ~ ( member1967978193chanID @ C @ A2 )
=> ( member1967978193chanID @ C @ ( uminus1044094777chanID @ A2 ) ) ) ).
% ComplI
thf(fact_334_ComplI,axiom,
! [C: secrecy_specID,A2: set_Secrecy_specID] :
( ~ ( member740727610specID @ C @ A2 )
=> ( member740727610specID @ C @ ( uminus238526242specID @ A2 ) ) ) ).
% ComplI
thf(fact_335_Compl__iff,axiom,
! [C: secrecy_chanID,A2: set_Secrecy_chanID] :
( ( member1967978193chanID @ C @ ( uminus1044094777chanID @ A2 ) )
= ( ~ ( member1967978193chanID @ C @ A2 ) ) ) ).
% Compl_iff
thf(fact_336_Compl__iff,axiom,
! [C: secrecy_specID,A2: set_Secrecy_specID] :
( ( member740727610specID @ C @ ( uminus238526242specID @ A2 ) )
= ( ~ ( member740727610specID @ C @ A2 ) ) ) ).
% Compl_iff
thf(fact_337_le__inf__iff,axiom,
! [X3: set_Secrecy_chanID,Y3: set_Secrecy_chanID,Z2: set_Secrecy_chanID] :
( ( ord_le1802272080chanID @ X3 @ ( inf_in1474033538chanID @ Y3 @ Z2 ) )
= ( ( ord_le1802272080chanID @ X3 @ Y3 )
& ( ord_le1802272080chanID @ X3 @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_338_inf_Obounded__iff,axiom,
! [A: set_Secrecy_chanID,B2: set_Secrecy_chanID,C: set_Secrecy_chanID] :
( ( ord_le1802272080chanID @ A @ ( inf_in1474033538chanID @ B2 @ C ) )
= ( ( ord_le1802272080chanID @ A @ B2 )
& ( ord_le1802272080chanID @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_339_inf__bot__left,axiom,
! [X3: set_Secrecy_specID] :
( ( inf_in668465003specID @ bot_bo1785112429specID @ X3 )
= bot_bo1785112429specID ) ).
% inf_bot_left
thf(fact_340_inf__bot__left,axiom,
! [X3: set_Secrecy_chanID] :
( ( inf_in1474033538chanID @ bot_bo443197316chanID @ X3 )
= bot_bo443197316chanID ) ).
% inf_bot_left
thf(fact_341_inf__bot__right,axiom,
! [X3: set_Secrecy_specID] :
( ( inf_in668465003specID @ X3 @ bot_bo1785112429specID )
= bot_bo1785112429specID ) ).
% inf_bot_right
thf(fact_342_inf__bot__right,axiom,
! [X3: set_Secrecy_chanID] :
( ( inf_in1474033538chanID @ X3 @ bot_bo443197316chanID )
= bot_bo443197316chanID ) ).
% inf_bot_right
thf(fact_343_Int__insert__left__if0,axiom,
! [A: secrecy_specID,C3: set_Secrecy_specID,B: set_Secrecy_specID] :
( ~ ( member740727610specID @ A @ C3 )
=> ( ( inf_in668465003specID @ ( insert1671894227specID @ A @ B ) @ C3 )
= ( inf_in668465003specID @ B @ C3 ) ) ) ).
% Int_insert_left_if0
thf(fact_344_Int__insert__left__if0,axiom,
! [A: secrecy_chanID,C3: set_Secrecy_chanID,B: set_Secrecy_chanID] :
( ~ ( member1967978193chanID @ A @ C3 )
=> ( ( inf_in1474033538chanID @ ( insert751661162chanID @ A @ B ) @ C3 )
= ( inf_in1474033538chanID @ B @ C3 ) ) ) ).
% Int_insert_left_if0
thf(fact_345_Int__insert__left__if1,axiom,
! [A: secrecy_specID,C3: set_Secrecy_specID,B: set_Secrecy_specID] :
( ( member740727610specID @ A @ C3 )
=> ( ( inf_in668465003specID @ ( insert1671894227specID @ A @ B ) @ C3 )
= ( insert1671894227specID @ A @ ( inf_in668465003specID @ B @ C3 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_346_Int__insert__left__if1,axiom,
! [A: secrecy_chanID,C3: set_Secrecy_chanID,B: set_Secrecy_chanID] :
( ( member1967978193chanID @ A @ C3 )
=> ( ( inf_in1474033538chanID @ ( insert751661162chanID @ A @ B ) @ C3 )
= ( insert751661162chanID @ A @ ( inf_in1474033538chanID @ B @ C3 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_347_insert__inter__insert,axiom,
! [A: secrecy_specID,A2: set_Secrecy_specID,B: set_Secrecy_specID] :
( ( inf_in668465003specID @ ( insert1671894227specID @ A @ A2 ) @ ( insert1671894227specID @ A @ B ) )
= ( insert1671894227specID @ A @ ( inf_in668465003specID @ A2 @ B ) ) ) ).
% insert_inter_insert
thf(fact_348_insert__inter__insert,axiom,
! [A: secrecy_chanID,A2: set_Secrecy_chanID,B: set_Secrecy_chanID] :
( ( inf_in1474033538chanID @ ( insert751661162chanID @ A @ A2 ) @ ( insert751661162chanID @ A @ B ) )
= ( insert751661162chanID @ A @ ( inf_in1474033538chanID @ A2 @ B ) ) ) ).
% insert_inter_insert
thf(fact_349_Int__insert__right__if0,axiom,
! [A: secrecy_specID,A2: set_Secrecy_specID,B: set_Secrecy_specID] :
( ~ ( member740727610specID @ A @ A2 )
=> ( ( inf_in668465003specID @ A2 @ ( insert1671894227specID @ A @ B ) )
= ( inf_in668465003specID @ A2 @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_350_Int__insert__right__if0,axiom,
! [A: secrecy_chanID,A2: set_Secrecy_chanID,B: set_Secrecy_chanID] :
( ~ ( member1967978193chanID @ A @ A2 )
=> ( ( inf_in1474033538chanID @ A2 @ ( insert751661162chanID @ A @ B ) )
= ( inf_in1474033538chanID @ A2 @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_351_Int__insert__right__if1,axiom,
! [A: secrecy_specID,A2: set_Secrecy_specID,B: set_Secrecy_specID] :
( ( member740727610specID @ A @ A2 )
=> ( ( inf_in668465003specID @ A2 @ ( insert1671894227specID @ A @ B ) )
= ( insert1671894227specID @ A @ ( inf_in668465003specID @ A2 @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_352_Int__insert__right__if1,axiom,
! [A: secrecy_chanID,A2: set_Secrecy_chanID,B: set_Secrecy_chanID] :
( ( member1967978193chanID @ A @ A2 )
=> ( ( inf_in1474033538chanID @ A2 @ ( insert751661162chanID @ A @ B ) )
= ( insert751661162chanID @ A @ ( inf_in1474033538chanID @ A2 @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_353_Int__subset__iff,axiom,
! [C3: set_Secrecy_chanID,A2: set_Secrecy_chanID,B: set_Secrecy_chanID] :
( ( ord_le1802272080chanID @ C3 @ ( inf_in1474033538chanID @ A2 @ B ) )
= ( ( ord_le1802272080chanID @ C3 @ A2 )
& ( ord_le1802272080chanID @ C3 @ B ) ) ) ).
% Int_subset_iff
thf(fact_354_Int__Un__eq_I4_J,axiom,
! [T2: set_Secrecy_chanID,S: set_Secrecy_chanID] :
( ( sup_su370684444chanID @ T2 @ ( inf_in1474033538chanID @ S @ T2 ) )
= T2 ) ).
% Int_Un_eq(4)
% Conjectures (1)
thf(conj_0,conjecture,
member1967978193chanID @ ch @ ( loc @ pq ) ).
%------------------------------------------------------------------------------