TPTP Problem File: SLH0536^1.p
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%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : ResiduatedTransitionSystem/0000_ResiduatedTransitionSystem/prob_03536_136226__14134370_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1388 ( 318 unt; 103 typ; 0 def)
% Number of atoms : 4420 (1088 equ; 0 cnn)
% Maximal formula atoms : 17 ( 3 avg)
% Number of connectives : 17901 ( 253 ~; 4 |; 463 &;14980 @)
% ( 0 <=>;2201 =>; 0 <=; 0 <~>)
% Maximal formula depth : 25 ( 8 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 1583 (1583 >; 0 *; 0 +; 0 <<)
% Number of symbols : 103 ( 100 usr; 6 con; 0-5 aty)
% Number of variables : 3683 ( 193 ^;3400 !; 90 ?;3683 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 11:47:57.509
%------------------------------------------------------------------------------
% Could-be-implicit typings (3)
thf(ty_n_t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
set_set_a: $tType ).
thf(ty_n_t__Set__Oset_Itf__a_J,type,
set_a: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (100)
thf(sy_c_HOL_OThe_001t__Set__Oset_Itf__a_J,type,
the_set_a: ( set_a > $o ) > set_a ).
thf(sy_c_HOL_OThe_001tf__a,type,
the_a: ( a > $o ) > a ).
thf(sy_c_If_001t__Set__Oset_Itf__a_J,type,
if_set_a: $o > set_a > set_a > set_a ).
thf(sy_c_If_001tf__a,type,
if_a: $o > a > a > a ).
thf(sy_c_Lattices_Oinf__class_Oinf_001_062_It__Set__Oset_Itf__a_J_M_Eo_J,type,
inf_inf_set_a_o: ( set_a > $o ) > ( set_a > $o ) > set_a > $o ).
thf(sy_c_Lattices_Oinf__class_Oinf_001_062_Itf__a_M_Eo_J,type,
inf_inf_a_o: ( a > $o ) > ( a > $o ) > a > $o ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
inf_inf_set_set_a: set_set_a > set_set_a > set_set_a ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_Itf__a_J,type,
inf_inf_set_a: set_a > set_a > set_a ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Set__Oset_Itf__a_J_M_Eo_J,type,
bot_bot_set_a_o: set_a > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_Itf__a_M_Eo_J,type,
bot_bot_a_o: a > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
bot_bot_set_set_a: set_set_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__a_J,type,
bot_bot_set_a: set_a ).
thf(sy_c_Orderings_Oord_OLeast_001t__Set__Oset_Itf__a_J,type,
least_set_a: ( set_a > set_a > $o ) > ( set_a > $o ) > set_a ).
thf(sy_c_Orderings_Oord_OLeast_001tf__a,type,
least_a: ( a > a > $o ) > ( a > $o ) > a ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Set__Oset_Itf__a_J_M_Eo_J,type,
ord_less_eq_set_a_o: ( set_a > $o ) > ( set_a > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_Itf__a_M_Eo_J,type,
ord_less_eq_a_o: ( a > $o ) > ( a > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
ord_le3724670747650509150_set_a: set_set_a > set_set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__a_J,type,
ord_less_eq_set_a: set_a > set_a > $o ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Set__Oset_Itf__a_J,type,
order_Greatest_set_a: ( set_a > $o ) > set_a ).
thf(sy_c_ResiduatedTransitionSystem_Ocoherent__normal__sub__rts_001t__Set__Oset_Itf__a_J,type,
cohere6325062230080414023_set_a: ( set_a > set_a > set_a ) > set_set_a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Ocoherent__normal__sub__rts_001tf__a,type,
cohere6072184133013167079_rts_a: ( a > a > a ) > set_a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Oextensional__rts_001t__Set__Oset_Itf__a_J,type,
extens2802975062453607898_set_a: ( set_a > set_a > set_a ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Oextensional__rts_001tf__a,type,
extensional_rts_a: ( a > a > a ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Oextensional__rts_Ocomp_001t__Set__Oset_Itf__a_J,type,
extens7801945855595804251_set_a: ( set_a > set_a > set_a ) > set_a > set_a > set_a ).
thf(sy_c_ResiduatedTransitionSystem_Oextensional__rts_Ocomp_001tf__a,type,
extensional_comp_a: ( a > a > a ) > a > a > a ).
thf(sy_c_ResiduatedTransitionSystem_Oextensional__rts_Ojoin_001t__Set__Oset_Itf__a_J,type,
extens1973556086528668384_set_a: ( set_a > set_a > set_a ) > set_a > set_a > set_a ).
thf(sy_c_ResiduatedTransitionSystem_Oextensional__rts_Ojoin_001tf__a,type,
extensional_join_a: ( a > a > a ) > a > a > a ).
thf(sy_c_ResiduatedTransitionSystem_Onormal__sub__rts_001tf__a,type,
normal_sub_rts_a: ( a > a > a ) > set_a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Onormal__sub__rts_OCong_001t__Set__Oset_Itf__a_J,type,
normal8977612136997397236_set_a: ( set_a > set_a > set_a ) > set_set_a > set_a > set_a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Onormal__sub__rts_OCong_001tf__a,type,
normal_sub_Cong_a: ( a > a > a ) > set_a > a > a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Onormal__sub__rts_OCong_H_001tf__a,type,
normal_sub_Cong_a2: ( a > a > a ) > set_a > a > a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Onormal__sub__rts_OCong__class_001t__Set__Oset_Itf__a_J,type,
normal2962378890657961070_set_a: ( set_a > set_a > set_a ) > set_set_a > set_a > set_set_a ).
thf(sy_c_ResiduatedTransitionSystem_Onormal__sub__rts_OCong__class_001tf__a,type,
normal7408713899360725774lass_a: ( a > a > a ) > set_a > a > set_a ).
thf(sy_c_ResiduatedTransitionSystem_Onormal__sub__rts_OCong__class__rep_001tf__a,type,
normal3259722184653208495_rep_a: set_a > a ).
thf(sy_c_ResiduatedTransitionSystem_Onormal__sub__rts_Ois__Cong__class_001t__Set__Oset_Itf__a_J,type,
normal4437380936311325560_set_a: ( set_a > set_a > set_a ) > set_set_a > set_set_a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Onormal__sub__rts_Ois__Cong__class_001tf__a,type,
normal8595587647932138008lass_a: ( a > a > a ) > set_a > set_a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Onormal__sub__rts__axioms_001t__Set__Oset_Itf__a_J,type,
normal4776468795420100326_set_a: ( set_a > set_a > set_a ) > set_set_a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Onormal__sub__rts__axioms_001tf__a,type,
normal7698203753654205830ioms_a: ( a > a > a ) > set_a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Opartial__magma_001t__Set__Oset_Itf__a_J,type,
partial_magma_set_a: ( set_a > set_a > set_a ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Opartial__magma_001tf__a,type,
partial_magma_a: ( a > a > a ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Opartial__magma_Onull_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
partia840180994421509092_set_a: ( set_set_a > set_set_a > set_set_a ) > set_set_a ).
thf(sy_c_ResiduatedTransitionSystem_Opartial__magma_Onull_001t__Set__Oset_Itf__a_J,type,
partial_null_set_a: ( set_a > set_a > set_a ) > set_a ).
thf(sy_c_ResiduatedTransitionSystem_Opartial__magma_Onull_001tf__a,type,
partial_null_a: ( a > a > a ) > a ).
thf(sy_c_ResiduatedTransitionSystem_Oquotient__by__coherent__normal_001t__Set__Oset_Itf__a_J,type,
quotie5625257012022141046_set_a: ( set_a > set_a > set_a ) > set_set_a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Oquotient__by__coherent__normal_001tf__a,type,
quotie3282664541148387094rmal_a: ( a > a > a ) > set_a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Oquotient__by__coherent__normal_OResid_001t__Set__Oset_Itf__a_J,type,
quotie3283642546880816345_set_a: ( set_a > set_a > set_a ) > set_set_a > set_set_a > set_set_a > set_set_a ).
thf(sy_c_ResiduatedTransitionSystem_Oquotient__by__coherent__normal_OResid_001tf__a,type,
quotie8165075472272353145esid_a: ( a > a > a ) > set_a > set_a > set_a > set_a ).
thf(sy_c_ResiduatedTransitionSystem_Oresiduation_001t__Set__Oset_Itf__a_J,type,
residuation_set_a: ( set_a > set_a > set_a ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Oresiduation_001tf__a,type,
residuation_a: ( a > a > a ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Oresiduation_Oarr_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
arr_set_set_a: ( set_set_a > set_set_a > set_set_a ) > set_set_a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Oresiduation_Oarr_001t__Set__Oset_Itf__a_J,type,
arr_set_a: ( set_a > set_a > set_a ) > set_a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Oresiduation_Oarr_001tf__a,type,
arr_a: ( a > a > a ) > a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Oresiduation_Ocon_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
con_set_set_a: ( set_set_a > set_set_a > set_set_a ) > set_set_a > set_set_a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Oresiduation_Ocon_001t__Set__Oset_Itf__a_J,type,
con_set_a: ( set_a > set_a > set_a ) > set_a > set_a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Oresiduation_Ocon_001tf__a,type,
con_a: ( a > a > a ) > a > a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Oresiduation_Oide_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
ide_set_set_a: ( set_set_a > set_set_a > set_set_a ) > set_set_a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Oresiduation_Oide_001t__Set__Oset_Itf__a_J,type,
ide_set_a: ( set_a > set_a > set_a ) > set_a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Oresiduation_Oide_001tf__a,type,
ide_a: ( a > a > a ) > a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Oresiduation_Otrg_001t__Set__Oset_Itf__a_J,type,
trg_set_a: ( set_a > set_a > set_a ) > set_a > set_a ).
thf(sy_c_ResiduatedTransitionSystem_Oresiduation_Otrg_001tf__a,type,
trg_a: ( a > a > a ) > a > a ).
thf(sy_c_ResiduatedTransitionSystem_Orts_001t__Set__Oset_Itf__a_J,type,
rts_set_a: ( set_a > set_a > set_a ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Orts_001tf__a,type,
rts_a: ( a > a > a ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Orts_Ocoinitial_001t__Set__Oset_Itf__a_J,type,
coinitial_set_a: ( set_a > set_a > set_a ) > set_a > set_a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Orts_Ocoinitial_001tf__a,type,
coinitial_a: ( a > a > a ) > a > a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Orts_Ocomposable_001t__Set__Oset_Itf__a_J,type,
composable_set_a: ( set_a > set_a > set_a ) > set_a > set_a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Orts_Ocomposable_001tf__a,type,
composable_a: ( a > a > a ) > a > a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Orts_Ocomposite__of_001t__Set__Oset_Itf__a_J,type,
composite_of_set_a: ( set_a > set_a > set_a ) > set_a > set_a > set_a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Orts_Ocomposite__of_001tf__a,type,
composite_of_a: ( a > a > a ) > a > a > a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Orts_Ocoterminal_001t__Set__Oset_Itf__a_J,type,
coterminal_set_a: ( set_a > set_a > set_a ) > set_a > set_a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Orts_Ocoterminal_001tf__a,type,
coterminal_a: ( a > a > a ) > a > a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Orts_Ojoin__of_001t__Set__Oset_Itf__a_J,type,
join_of_set_a: ( set_a > set_a > set_a ) > set_a > set_a > set_a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Orts_Ojoin__of_001tf__a,type,
join_of_a: ( a > a > a ) > a > a > a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Orts_Ojoinable_001t__Set__Oset_Itf__a_J,type,
joinable_set_a: ( set_a > set_a > set_a ) > set_a > set_a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Orts_Ojoinable_001tf__a,type,
joinable_a: ( a > a > a ) > a > a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Orts_Oseq_001t__Set__Oset_Itf__a_J,type,
seq_set_a: ( set_a > set_a > set_a ) > set_a > set_a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Orts_Oseq_001tf__a,type,
seq_a: ( a > a > a ) > a > a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Orts_Osources_001t__Set__Oset_Itf__a_J,type,
sources_set_a: ( set_a > set_a > set_a ) > set_a > set_set_a ).
thf(sy_c_ResiduatedTransitionSystem_Orts_Osources_001tf__a,type,
sources_a: ( a > a > a ) > a > set_a ).
thf(sy_c_ResiduatedTransitionSystem_Orts_Otargets_001t__Set__Oset_Itf__a_J,type,
targets_set_a: ( set_a > set_a > set_a ) > set_a > set_set_a ).
thf(sy_c_ResiduatedTransitionSystem_Orts_Otargets_001tf__a,type,
targets_a: ( a > a > a ) > a > set_a ).
thf(sy_c_ResiduatedTransitionSystem_Orts__axioms_001t__Set__Oset_Itf__a_J,type,
rts_axioms_set_a: ( set_a > set_a > set_a ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Orts__axioms_001tf__a,type,
rts_axioms_a: ( a > a > a ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Osimulation__axioms_001t__Set__Oset_Itf__a_J_001tf__a,type,
simula3408835310535287622et_a_a: ( set_a > set_a > set_a ) > ( a > a > a ) > ( set_a > a ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Osimulation__axioms_001tf__a_001t__Set__Oset_Itf__a_J,type,
simula3192323252075944454_set_a: ( a > a > a ) > ( set_a > set_a > set_a ) > ( a > set_a ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Osimulation__axioms_001tf__a_001tf__a,type,
simula3868467710248865958ms_a_a: ( a > a > a ) > ( a > a > a ) > ( a > a ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Otransformation__axioms_001t__Set__Oset_Itf__a_J_001t__Set__Oset_Itf__a_J,type,
transf2960116383903194536_set_a: ( set_a > set_a > set_a ) > ( set_a > set_a > set_a ) > ( set_a > set_a ) > ( set_a > set_a ) > ( set_a > set_a ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Otransformation__axioms_001t__Set__Oset_Itf__a_J_001tf__a,type,
transf1935308705569152072et_a_a: ( set_a > set_a > set_a ) > ( a > a > a ) > ( set_a > a ) > ( set_a > a ) > ( set_a > a ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Otransformation__axioms_001tf__a_001t__Set__Oset_Itf__a_J,type,
transf1718796647109808904_set_a: ( a > a > a ) > ( set_a > set_a > set_a ) > ( a > set_a ) > ( a > set_a ) > ( a > set_a ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Otransformation__axioms_001tf__a_001tf__a,type,
transf4446446367311712680ms_a_a: ( a > a > a ) > ( a > a > a ) > ( a > a ) > ( a > a ) > ( a > a ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Oweakly__extensional__rts_001t__Set__Oset_Itf__a_J,type,
weakly5936471160286156245_set_a: ( set_a > set_a > set_a ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Oweakly__extensional__rts_001tf__a,type,
weakly1626779504270821493_rts_a: ( a > a > a ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Oweakly__extensional__rts_Osrc_001t__Set__Oset_Itf__a_J,type,
weakly2061155085811118449_set_a: ( set_a > set_a > set_a ) > set_a > set_a ).
thf(sy_c_ResiduatedTransitionSystem_Oweakly__extensional__rts_Osrc_001tf__a,type,
weakly8512939796511659025_src_a: ( a > a > a ) > a > a ).
thf(sy_c_Set_OCollect_001t__Set__Oset_Itf__a_J,type,
collect_set_a: ( set_a > $o ) > set_set_a ).
thf(sy_c_Set_OCollect_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_member_001t__Set__Oset_Itf__a_J,type,
member_set_a: set_a > set_set_a > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v__092_060NN_062,type,
nn: set_a ).
thf(sy_v__092_060T_062,type,
t: set_a ).
thf(sy_v_resid,type,
resid: a > a > a ).
% Relevant facts (1279)
thf(fact_0_R_Ocube,axiom,
! [V: a,T: a,U: a] :
( ( resid @ ( resid @ V @ T ) @ ( resid @ U @ T ) )
= ( resid @ ( resid @ V @ U ) @ ( resid @ T @ U ) ) ) ).
% R.cube
thf(fact_1_R_Oex__un__null,axiom,
? [X: a] :
( ! [T2: a] :
( ( ( resid @ X @ T2 )
= X )
& ( ( resid @ T2 @ X )
= X ) )
& ! [Y: a] :
( ! [T3: a] :
( ( ( resid @ Y @ T3 )
= Y )
& ( ( resid @ T3 @ Y )
= Y ) )
=> ( Y = X ) ) ) ).
% R.ex_un_null
thf(fact_2_sources__char,axiom,
! [T: set_a] :
( ( sources_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T )
= ( collect_set_a
@ ^ [A: set_a] :
( ( arr_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T )
& ( ( weakly2061155085811118449_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T )
= A ) ) ) ) ).
% sources_char
thf(fact_3_N_OCong_092_060_094sub_0620__symmetric,axiom,
! [T: a,T4: a] :
( ( ( member_a @ ( resid @ T @ T4 ) @ nn )
& ( member_a @ ( resid @ T4 @ T ) @ nn ) )
=> ( ( member_a @ ( resid @ T4 @ T ) @ nn )
& ( member_a @ ( resid @ T @ T4 ) @ nn ) ) ) ).
% N.Cong\<^sub>0_symmetric
thf(fact_4_N_OCong_092_060_094sub_0620__transitive,axiom,
! [T: a,T4: a,T5: a] :
( ( ( member_a @ ( resid @ T @ T4 ) @ nn )
& ( member_a @ ( resid @ T4 @ T ) @ nn ) )
=> ( ( ( member_a @ ( resid @ T4 @ T5 ) @ nn )
& ( member_a @ ( resid @ T5 @ T4 ) @ nn ) )
=> ( ( member_a @ ( resid @ T @ T5 ) @ nn )
& ( member_a @ ( resid @ T5 @ T ) @ nn ) ) ) ) ).
% N.Cong\<^sub>0_transitive
thf(fact_5_N_OResid__along__normal__reflects__Cong_092_060_094sub_0620,axiom,
! [T: a,U: a,T4: a] :
( ( ( member_a @ ( resid @ ( resid @ T @ U ) @ ( resid @ T4 @ U ) ) @ nn )
& ( member_a @ ( resid @ ( resid @ T4 @ U ) @ ( resid @ T @ U ) ) @ nn ) )
=> ( ( member_a @ U @ nn )
=> ( ( member_a @ ( resid @ T @ T4 ) @ nn )
& ( member_a @ ( resid @ T4 @ T ) @ nn ) ) ) ) ).
% N.Resid_along_normal_reflects_Cong\<^sub>0
thf(fact_6_N_Obackward__stable,axiom,
! [U: a,T: a] :
( ( member_a @ U @ nn )
=> ( ( member_a @ ( resid @ T @ U ) @ nn )
=> ( member_a @ T @ nn ) ) ) ).
% N.backward_stable
thf(fact_7_ide__char,axiom,
! [U2: set_a] :
( ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ U2 )
= ( ( arr_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ U2 )
& ( ( inf_inf_set_a @ U2 @ nn )
!= bot_bot_set_a ) ) ) ).
% ide_char
thf(fact_8_cube,axiom,
! [V: set_a,T: set_a,U: set_a] :
( ( quotie8165075472272353145esid_a @ resid @ nn @ ( quotie8165075472272353145esid_a @ resid @ nn @ V @ T ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ U @ T ) )
= ( quotie8165075472272353145esid_a @ resid @ nn @ ( quotie8165075472272353145esid_a @ resid @ nn @ V @ U ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ T @ U ) ) ) ).
% cube
thf(fact_9_ex__un__null,axiom,
? [X: set_a] :
( ! [T2: set_a] :
( ( ( quotie8165075472272353145esid_a @ resid @ nn @ X @ T2 )
= X )
& ( ( quotie8165075472272353145esid_a @ resid @ nn @ T2 @ X )
= X ) )
& ! [Y: set_a] :
( ! [T3: set_a] :
( ( ( quotie8165075472272353145esid_a @ resid @ nn @ Y @ T3 )
= Y )
& ( ( quotie8165075472272353145esid_a @ resid @ nn @ T3 @ Y )
= Y ) )
=> ( Y = X ) ) ) ).
% ex_un_null
thf(fact_10_N_OCong_092_060_094sub_0620__imp__coinitial,axiom,
! [T: a,T4: a] :
( ( ( member_a @ ( resid @ T @ T4 ) @ nn )
& ( member_a @ ( resid @ T4 @ T ) @ nn ) )
=> ( ( sources_a @ resid @ T )
= ( sources_a @ resid @ T4 ) ) ) ).
% N.Cong\<^sub>0_imp_coinitial
thf(fact_11_N_OResid__along__normal__preserves__Cong_092_060_094sub_0620,axiom,
! [T: a,T4: a,U: a] :
( ( ( member_a @ ( resid @ T @ T4 ) @ nn )
& ( member_a @ ( resid @ T4 @ T ) @ nn ) )
=> ( ( member_a @ U @ nn )
=> ( ( ( sources_a @ resid @ T )
= ( sources_a @ resid @ U ) )
=> ( ( member_a @ ( resid @ ( resid @ T @ U ) @ ( resid @ T4 @ U ) ) @ nn )
& ( member_a @ ( resid @ ( resid @ T4 @ U ) @ ( resid @ T @ U ) ) @ nn ) ) ) ) ) ).
% N.Resid_along_normal_preserves_Cong\<^sub>0
thf(fact_12_N_OCong__closure__props_I3_J,axiom,
! [T: a,U: a] :
( ( ( member_a @ ( resid @ T @ U ) @ nn )
& ( member_a @ ( resid @ U @ T ) @ nn ) )
=> ( normal_sub_Cong_a @ resid @ nn @ T @ U ) ) ).
% N.Cong_closure_props(3)
thf(fact_13_N_OCong__closure__props_I2_J,axiom,
! [T: a,U: a,V: a] :
( ( normal_sub_Cong_a @ resid @ nn @ T @ U )
=> ( ( normal_sub_Cong_a @ resid @ nn @ U @ V )
=> ( normal_sub_Cong_a @ resid @ nn @ T @ V ) ) ) ).
% N.Cong_closure_props(2)
thf(fact_14_N_OCong__closure__props_I1_J,axiom,
! [T: a,U: a] :
( ( normal_sub_Cong_a @ resid @ nn @ T @ U )
=> ( normal_sub_Cong_a @ resid @ nn @ U @ T ) ) ).
% N.Cong_closure_props(1)
thf(fact_15_N_OCongE,axiom,
! [T: a,T4: a] :
( ( normal_sub_Cong_a @ resid @ nn @ T @ T4 )
=> ~ ! [U3: a] :
( ( member_a @ U3 @ nn )
=> ! [U4: a] :
( ( member_a @ U4 @ nn )
=> ~ ( ( member_a @ ( resid @ ( resid @ T @ U3 ) @ ( resid @ T4 @ U4 ) ) @ nn )
& ( member_a @ ( resid @ ( resid @ T4 @ U4 ) @ ( resid @ T @ U3 ) ) @ nn ) ) ) ) ) ).
% N.CongE
thf(fact_16_N_OCong_092_060_094sub_0620__implies__Cong,axiom,
! [T: a,T4: a] :
( ( ( member_a @ ( resid @ T @ T4 ) @ nn )
& ( member_a @ ( resid @ T4 @ T ) @ nn ) )
=> ( normal_sub_Cong_a @ resid @ nn @ T @ T4 ) ) ).
% N.Cong\<^sub>0_implies_Cong
thf(fact_17_N_OCong__def,axiom,
! [T: a,T4: a] :
( ( normal_sub_Cong_a @ resid @ nn @ T @ T4 )
= ( ? [U5: a,U6: a] :
( ( member_a @ U5 @ nn )
& ( member_a @ U6 @ nn )
& ( member_a @ ( resid @ ( resid @ T @ U5 ) @ ( resid @ T4 @ U6 ) ) @ nn )
& ( member_a @ ( resid @ ( resid @ T4 @ U6 ) @ ( resid @ T @ U5 ) ) @ nn ) ) ) ) ).
% N.Cong_def
thf(fact_18_N_OCong__symmetric,axiom,
! [T: a,T4: a] :
( ( normal_sub_Cong_a @ resid @ nn @ T @ T4 )
=> ( normal_sub_Cong_a @ resid @ nn @ T4 @ T ) ) ).
% N.Cong_symmetric
thf(fact_19_N_OCong__transitive,axiom,
! [T: a,T5: a,T4: a] :
( ( normal_sub_Cong_a @ resid @ nn @ T @ T5 )
=> ( ( normal_sub_Cong_a @ resid @ nn @ T5 @ T4 )
=> ( normal_sub_Cong_a @ resid @ nn @ T @ T4 ) ) ) ).
% N.Cong_transitive
thf(fact_20_N_Onormal__is__Cong__closed,axiom,
! [T: a,T4: a] :
( ( member_a @ T @ nn )
=> ( ( normal_sub_Cong_a @ resid @ nn @ T @ T4 )
=> ( member_a @ T4 @ nn ) ) ) ).
% N.normal_is_Cong_closed
thf(fact_21__092_060open_062arr_A_092_060T_062_A_092_060Longrightarrow_062_A_123a_O_A_092_060exists_062t_Aa_H_O_At_A_092_060in_062_A_092_060T_062_A_092_060and_062_Aa_H_A_092_060in_062_AR_Osources_At_A_092_060and_062_Aa_H_A_092_060approx_062_Aa_125_A_092_060in_062_Asources_A_092_060T_062_092_060close_062,axiom,
( ( arr_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ t )
=> ( member_set_a
@ ( collect_a
@ ^ [A: a] :
? [T6: a,A2: a] :
( ( member_a @ T6 @ t )
& ( member_a @ A2 @ ( sources_a @ resid @ T6 ) )
& ( normal_sub_Cong_a @ resid @ nn @ A2 @ A ) ) )
@ ( sources_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ t ) ) ) ).
% \<open>arr \<T> \<Longrightarrow> {a. \<exists>t a'. t \<in> \<T> \<and> a' \<in> R.sources t \<and> a' \<approx> a} \<in> sources \<T>\<close>
thf(fact_22__C1_C,axiom,
( ( arr_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ t )
=> ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn )
@ ( collect_a
@ ^ [A: a] :
? [T6: a,A2: a] :
( ( member_a @ T6 @ t )
& ( member_a @ A2 @ ( sources_a @ resid @ T6 ) )
& ( normal_sub_Cong_a @ resid @ nn @ A2 @ A ) ) ) ) ) ).
% "1"
thf(fact_23_N_OCong__closure__props_I4_J,axiom,
! [U: a,T: a] :
( ( member_a @ U @ nn )
=> ( ( ( sources_a @ resid @ T )
= ( sources_a @ resid @ U ) )
=> ( normal_sub_Cong_a @ resid @ nn @ T @ ( resid @ T @ U ) ) ) ) ).
% N.Cong_closure_props(4)
thf(fact_24_N_Oin__sources__respects__Cong,axiom,
! [T: a,T4: a,A3: a,A4: a] :
( ( normal_sub_Cong_a @ resid @ nn @ T @ T4 )
=> ( ( member_a @ A3 @ ( sources_a @ resid @ T ) )
=> ( ( member_a @ A4 @ ( sources_a @ resid @ T4 ) )
=> ( normal_sub_Cong_a @ resid @ nn @ A3 @ A4 ) ) ) ) ).
% N.in_sources_respects_Cong
thf(fact_25_N_Osources__are__Cong,axiom,
! [A3: a,T: a,A4: a] :
( ( member_a @ A3 @ ( sources_a @ resid @ T ) )
=> ( ( member_a @ A4 @ ( sources_a @ resid @ T ) )
=> ( normal_sub_Cong_a @ resid @ nn @ A3 @ A4 ) ) ) ).
% N.sources_are_Cong
thf(fact_26_quotient__by__coherent__normal__axioms,axiom,
quotie3282664541148387094rmal_a @ resid @ nn ).
% quotient_by_coherent_normal_axioms
thf(fact_27_arr__has__un__source,axiom,
! [T: set_a] :
( ( arr_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T )
=> ? [X: set_a] :
( ( member_set_a @ X @ ( sources_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T ) )
& ! [Y: set_a] :
( ( member_set_a @ Y @ ( sources_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T ) )
=> ( Y = X ) ) ) ) ).
% arr_has_un_source
thf(fact_28_Arr__Resid,axiom,
! [T7: set_a,U2: set_a] :
( ( ( quotie8165075472272353145esid_a @ resid @ nn @ T7 @ U2 )
!= bot_bot_set_a )
=> ( ( quotie8165075472272353145esid_a @ resid @ nn @ ( quotie8165075472272353145esid_a @ resid @ nn @ T7 @ U2 ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ T7 @ U2 ) )
!= bot_bot_set_a ) ) ).
% Arr_Resid
thf(fact_29_Con__sym,axiom,
! [T7: set_a,U2: set_a] :
( ( ( quotie8165075472272353145esid_a @ resid @ nn @ T7 @ U2 )
!= bot_bot_set_a )
=> ( ( quotie8165075472272353145esid_a @ resid @ nn @ U2 @ T7 )
!= bot_bot_set_a ) ) ).
% Con_sym
thf(fact_30_Cube,axiom,
! [V2: set_a,T7: set_a,U2: set_a] :
( ( ( quotie8165075472272353145esid_a @ resid @ nn @ ( quotie8165075472272353145esid_a @ resid @ nn @ V2 @ T7 ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ U2 @ T7 ) )
!= bot_bot_set_a )
=> ( ( quotie8165075472272353145esid_a @ resid @ nn @ ( quotie8165075472272353145esid_a @ resid @ nn @ V2 @ T7 ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ U2 @ T7 ) )
= ( quotie8165075472272353145esid_a @ resid @ nn @ ( quotie8165075472272353145esid_a @ resid @ nn @ V2 @ U2 ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ T7 @ U2 ) ) ) ) ).
% Cube
thf(fact_31_cong__symmetric,axiom,
! [T: set_a,U: set_a] :
( ( ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ T @ U ) )
& ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ U @ T ) ) )
=> ( ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ U @ T ) )
& ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ T @ U ) ) ) ) ).
% cong_symmetric
thf(fact_32_cong__transitive,axiom,
! [T: set_a,U: set_a,V: set_a] :
( ( ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ T @ U ) )
& ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ U @ T ) ) )
=> ( ( ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ U @ V ) )
& ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ V @ U ) ) )
=> ( ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ T @ V ) )
& ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ V @ T ) ) ) ) ) ).
% cong_transitive
thf(fact_33_extensional,axiom,
! [T: set_a,U: set_a] :
( ( ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ T @ U ) )
& ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ U @ T ) ) )
=> ( T = U ) ) ).
% extensional
thf(fact_34_ide__backward__stable,axiom,
! [A3: set_a,T: set_a] :
( ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ A3 )
=> ( ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ T @ A3 ) )
=> ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T ) ) ) ).
% ide_backward_stable
thf(fact_35_prfx__transitive,axiom,
! [T: set_a,U: set_a,V: set_a] :
( ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ T @ U ) )
=> ( ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ U @ V ) )
=> ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ T @ V ) ) ) ) ).
% prfx_transitive
thf(fact_36_weak__extensionality,axiom,
! [T: set_a,U: set_a] :
( ( ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ T @ U ) )
& ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ U @ T ) ) )
=> ( ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T )
=> ( ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ U )
=> ( T = U ) ) ) ) ).
% weak_extensionality
thf(fact_37_cong__char,axiom,
! [T: set_a,U: set_a] :
( ( ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ T @ U ) )
& ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ U @ T ) ) )
= ( ( arr_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T )
& ( T = U ) ) ) ).
% cong_char
thf(fact_38_cong__reflexive,axiom,
! [T: set_a] :
( ( arr_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T )
=> ( ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ T @ T ) )
& ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ T @ T ) ) ) ) ).
% cong_reflexive
thf(fact_39_ide__implies__arr,axiom,
! [A3: set_a] :
( ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ A3 )
=> ( arr_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ A3 ) ) ).
% ide_implies_arr
thf(fact_40_prfx__reflexive,axiom,
! [T: set_a] :
( ( arr_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T )
=> ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ T @ T ) ) ) ).
% prfx_reflexive
thf(fact_41_source__is__ide,axiom,
! [A3: set_a,T: set_a] :
( ( member_set_a @ A3 @ ( sources_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T ) )
=> ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ A3 ) ) ).
% source_is_ide
thf(fact_42_sources__are__cong,axiom,
! [A3: set_a,T: set_a,A4: set_a] :
( ( member_set_a @ A3 @ ( sources_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T ) )
=> ( ( member_set_a @ A4 @ ( sources_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T ) )
=> ( ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ A3 @ A4 ) )
& ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ A4 @ A3 ) ) ) ) ) ).
% sources_are_cong
thf(fact_43_sources__cong__closed,axiom,
! [A3: set_a,T: set_a,A4: set_a] :
( ( member_set_a @ A3 @ ( sources_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T ) )
=> ( ( ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ A3 @ A4 ) )
& ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ A4 @ A3 ) ) )
=> ( member_set_a @ A4 @ ( sources_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T ) ) ) ) ).
% sources_cong_closed
thf(fact_44_src__in__sources,axiom,
! [T: set_a] :
( ( arr_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T )
=> ( member_set_a @ ( weakly2061155085811118449_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T ) @ ( sources_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T ) ) ) ).
% src_in_sources
thf(fact_45_mem__Collect__eq,axiom,
! [A3: set_a,P: set_a > $o] :
( ( member_set_a @ A3 @ ( collect_set_a @ P ) )
= ( P @ A3 ) ) ).
% mem_Collect_eq
thf(fact_46_mem__Collect__eq,axiom,
! [A3: a,P: a > $o] :
( ( member_a @ A3 @ ( collect_a @ P ) )
= ( P @ A3 ) ) ).
% mem_Collect_eq
thf(fact_47_Collect__mem__eq,axiom,
! [A5: set_set_a] :
( ( collect_set_a
@ ^ [X2: set_a] : ( member_set_a @ X2 @ A5 ) )
= A5 ) ).
% Collect_mem_eq
thf(fact_48_Collect__mem__eq,axiom,
! [A5: set_a] :
( ( collect_a
@ ^ [X2: a] : ( member_a @ X2 @ A5 ) )
= A5 ) ).
% Collect_mem_eq
thf(fact_49_Collect__cong,axiom,
! [P: set_a > $o,Q: set_a > $o] :
( ! [X: set_a] :
( ( P @ X )
= ( Q @ X ) )
=> ( ( collect_set_a @ P )
= ( collect_set_a @ Q ) ) ) ).
% Collect_cong
thf(fact_50_Collect__cong,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X: a] :
( ( P @ X )
= ( Q @ X ) )
=> ( ( collect_a @ P )
= ( collect_a @ Q ) ) ) ).
% Collect_cong
thf(fact_51_R_Osources__eqI,axiom,
! [T: a,T4: a] :
( ( ( inf_inf_set_a @ ( sources_a @ resid @ T ) @ ( sources_a @ resid @ T4 ) )
!= bot_bot_set_a )
=> ( ( sources_a @ resid @ T )
= ( sources_a @ resid @ T4 ) ) ) ).
% R.sources_eqI
thf(fact_52_N_OCongI,axiom,
! [U: a,U7: a,T: a,T4: a] :
( ( member_a @ U @ nn )
=> ( ( member_a @ U7 @ nn )
=> ( ( ( member_a @ ( resid @ ( resid @ T @ U ) @ ( resid @ T4 @ U7 ) ) @ nn )
& ( member_a @ ( resid @ ( resid @ T4 @ U7 ) @ ( resid @ T @ U ) ) @ nn ) )
=> ( normal_sub_Cong_a @ resid @ nn @ T @ T4 ) ) ) ) ).
% N.CongI
thf(fact_53_ide__iff__src__self,axiom,
! [A3: set_a] :
( ( arr_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ A3 )
=> ( ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ A3 )
= ( ( weakly2061155085811118449_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ A3 )
= A3 ) ) ) ).
% ide_iff_src_self
thf(fact_54_ide__src,axiom,
! [T: set_a] :
( ( arr_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T )
=> ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( weakly2061155085811118449_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T ) ) ) ).
% ide_src
thf(fact_55_rts_Osources_Ocong,axiom,
sources_set_a = sources_set_a ).
% rts.sources.cong
thf(fact_56_rts_Osources_Ocong,axiom,
sources_a = sources_a ).
% rts.sources.cong
thf(fact_57_residuation_Oarr_Ocong,axiom,
arr_set_a = arr_set_a ).
% residuation.arr.cong
thf(fact_58_residuation_Oarr_Ocong,axiom,
arr_a = arr_a ).
% residuation.arr.cong
thf(fact_59_normal__sub__rts_OCong_Ocong,axiom,
normal_sub_Cong_a = normal_sub_Cong_a ).
% normal_sub_rts.Cong.cong
thf(fact_60_quotient__by__coherent__normal_OResid_Ocong,axiom,
quotie8165075472272353145esid_a = quotie8165075472272353145esid_a ).
% quotient_by_coherent_normal.Resid.cong
thf(fact_61_partial__magma__axioms,axiom,
partial_magma_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) ).
% partial_magma_axioms
thf(fact_62_R_Opartial__magma__axioms,axiom,
partial_magma_a @ resid ).
% R.partial_magma_axioms
thf(fact_63_arr__src__iff__arr,axiom,
! [T: set_a] :
( ( arr_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( weakly2061155085811118449_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T ) )
= ( arr_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T ) ) ).
% arr_src_iff_arr
thf(fact_64_N_Ocoherent__normal__sub__rts__axioms,axiom,
cohere6072184133013167079_rts_a @ resid @ nn ).
% N.coherent_normal_sub_rts_axioms
thf(fact_65_src__src,axiom,
! [T: set_a] :
( ( weakly2061155085811118449_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( weakly2061155085811118449_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T ) )
= ( weakly2061155085811118449_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T ) ) ).
% src_src
thf(fact_66_N_OCong__class__def,axiom,
! [T: a] :
( ( normal7408713899360725774lass_a @ resid @ nn @ T )
= ( collect_a @ ( normal_sub_Cong_a @ resid @ nn @ T ) ) ) ).
% N.Cong_class_def
thf(fact_67_N_OCong__class__eqI,axiom,
! [T: a,T4: a] :
( ( normal_sub_Cong_a @ resid @ nn @ T @ T4 )
=> ( ( normal7408713899360725774lass_a @ resid @ nn @ T )
= ( normal7408713899360725774lass_a @ resid @ nn @ T4 ) ) ) ).
% N.Cong_class_eqI
thf(fact_68__092_060open_062arr_A_092_060T_062_A_092_060Longrightarrow_062_Acon_A_092_060T_062_A_123a_O_A_092_060exists_062t_Aa_H_O_At_A_092_060in_062_A_092_060T_062_A_092_060and_062_Aa_H_A_092_060in_062_AR_Osources_At_A_092_060and_062_Aa_H_A_092_060approx_062_Aa_125_092_060close_062,axiom,
( ( arr_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ t )
=> ( con_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ t
@ ( collect_a
@ ^ [A: a] :
? [T6: a,A2: a] :
( ( member_a @ T6 @ t )
& ( member_a @ A2 @ ( sources_a @ resid @ T6 ) )
& ( normal_sub_Cong_a @ resid @ nn @ A2 @ A ) ) ) ) ) ).
% \<open>arr \<T> \<Longrightarrow> con \<T> {a. \<exists>t a'. t \<in> \<T> \<and> a' \<in> R.sources t \<and> a' \<approx> a}\<close>
thf(fact_69_coinitialE,axiom,
! [T: set_a,U: set_a] :
( ( coinitial_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U )
=> ~ ( ( arr_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T )
=> ( ( arr_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ U )
=> ( ( sources_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T )
!= ( sources_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ U ) ) ) ) ) ).
% coinitialE
thf(fact_70_coinitial__iff,axiom,
! [T: set_a,T4: set_a] :
( ( coinitial_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ T4 )
= ( ( arr_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T )
& ( arr_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T4 )
& ( ( sources_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T )
= ( sources_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T4 ) ) ) ) ).
% coinitial_iff
thf(fact_71_arr__iff__has__source,axiom,
! [T: set_a] :
( ( arr_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T )
= ( ( sources_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T )
!= bot_bot_set_set_a ) ) ).
% arr_iff_has_source
thf(fact_72_composableD_I2_J,axiom,
! [T: set_a,U: set_a] :
( ( composable_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U )
=> ( arr_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ U ) ) ).
% composableD(2)
thf(fact_73_N_Oelements__are__arr,axiom,
! [T: a] :
( ( member_a @ T @ nn )
=> ( arr_a @ resid @ T ) ) ).
% N.elements_are_arr
thf(fact_74_N_OCong_092_060_094sub_0620__reflexive,axiom,
! [T: a] :
( ( arr_a @ resid @ T )
=> ( ( member_a @ ( resid @ T @ T ) @ nn )
& ( member_a @ ( resid @ T @ T ) @ nn ) ) ) ).
% N.Cong\<^sub>0_reflexive
thf(fact_75_R_Oarr__iff__has__source,axiom,
! [T: a] :
( ( arr_a @ resid @ T )
= ( ( sources_a @ resid @ T )
!= bot_bot_set_a ) ) ).
% R.arr_iff_has_source
thf(fact_76_N_OCong__reflexive,axiom,
! [T: a] :
( ( arr_a @ resid @ T )
=> ( normal_sub_Cong_a @ resid @ nn @ T @ T ) ) ).
% N.Cong_reflexive
thf(fact_77_N_OCong__imp__arr_I1_J,axiom,
! [T: a,T4: a] :
( ( normal_sub_Cong_a @ resid @ nn @ T @ T4 )
=> ( arr_a @ resid @ T ) ) ).
% N.Cong_imp_arr(1)
thf(fact_78_N_OCong__imp__arr_I2_J,axiom,
! [T: a,T4: a] :
( ( normal_sub_Cong_a @ resid @ nn @ T @ T4 )
=> ( arr_a @ resid @ T4 ) ) ).
% N.Cong_imp_arr(2)
thf(fact_79_N_Oarr__in__Cong__class,axiom,
! [T: a] :
( ( arr_a @ resid @ T )
=> ( member_a @ T @ ( normal7408713899360725774lass_a @ resid @ nn @ T ) ) ) ).
% N.arr_in_Cong_class
thf(fact_80_resid__reflects__con,axiom,
! [T: set_a,V: set_a,U: set_a] :
( ( con_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ V )
=> ( ( con_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ U @ V )
=> ( ( con_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ T @ V ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ U @ V ) )
=> ( con_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U ) ) ) ) ).
% resid_reflects_con
thf(fact_81_con__sym,axiom,
! [T: set_a,U: set_a] :
( ( con_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U )
=> ( con_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ U @ T ) ) ).
% con_sym
thf(fact_82_resid__ide__arr,axiom,
! [A3: set_a,T: set_a] :
( ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ A3 )
=> ( ( con_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ A3 @ T )
=> ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ A3 @ T ) ) ) ) ).
% resid_ide_arr
thf(fact_83_resid__arr__ide,axiom,
! [A3: set_a,T: set_a] :
( ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ A3 )
=> ( ( con_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ A3 )
=> ( ( quotie8165075472272353145esid_a @ resid @ nn @ T @ A3 )
= T ) ) ) ).
% resid_arr_ide
thf(fact_84_prfx__implies__con,axiom,
! [T: set_a,U: set_a] :
( ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ T @ U ) )
=> ( con_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U ) ) ).
% prfx_implies_con
thf(fact_85_ide__imp__con__iff__cong,axiom,
! [T: set_a,U: set_a] :
( ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T )
=> ( ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ U )
=> ( ( con_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U )
= ( ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ T @ U ) )
& ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ U @ T ) ) ) ) ) ) ).
% ide_imp_con_iff_cong
thf(fact_86_ide__def,axiom,
! [A3: set_a] :
( ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ A3 )
= ( ( con_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ A3 @ A3 )
& ( ( quotie8165075472272353145esid_a @ resid @ nn @ A3 @ A3 )
= A3 ) ) ) ).
% ide_def
thf(fact_87_ideE,axiom,
! [A3: set_a] :
( ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ A3 )
=> ~ ( ( con_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ A3 @ A3 )
=> ( ( quotie8165075472272353145esid_a @ resid @ nn @ A3 @ A3 )
!= A3 ) ) ) ).
% ideE
thf(fact_88_con__transitive__on__ide,axiom,
! [A3: set_a,B: set_a,C: set_a] :
( ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ A3 )
=> ( ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ B )
=> ( ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ C )
=> ( ( con_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ A3 @ B )
=> ( ( con_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ B @ C )
=> ( con_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ A3 @ C ) ) ) ) ) ) ).
% con_transitive_on_ide
thf(fact_89_con__target,axiom,
! [T: set_a,U: set_a,V: set_a] :
( ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ T @ U ) )
=> ( ( con_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ U @ V )
=> ( con_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ T @ U ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ V @ U ) ) ) ) ).
% con_target
thf(fact_90_con__imp__coinitial__ax,axiom,
! [T: set_a,U: set_a] :
( ( con_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U )
=> ? [A6: set_a] :
( ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ A6 )
& ( con_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ A6 @ T )
& ( con_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ A6 @ U ) ) ) ).
% con_imp_coinitial_ax
thf(fact_91_con__ide__are__eq,axiom,
! [A3: set_a,A4: set_a] :
( ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ A3 )
=> ( ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ A4 )
=> ( ( con_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ A3 @ A4 )
=> ( A3 = A4 ) ) ) ) ).
% con_ide_are_eq
thf(fact_92_cong__subst__right_I1_J,axiom,
! [U: set_a,U7: set_a,T: set_a] :
( ( ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ U @ U7 ) )
& ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ U7 @ U ) ) )
=> ( ( con_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U )
=> ( con_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U7 ) ) ) ).
% cong_subst_right(1)
thf(fact_93_cong__subst__right_I2_J,axiom,
! [U: set_a,U7: set_a,T: set_a] :
( ( ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ U @ U7 ) )
& ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ U7 @ U ) ) )
=> ( ( con_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U )
=> ( ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ ( quotie8165075472272353145esid_a @ resid @ nn @ T @ U ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ T @ U7 ) ) )
& ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ ( quotie8165075472272353145esid_a @ resid @ nn @ T @ U7 ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ T @ U ) ) ) ) ) ) ).
% cong_subst_right(2)
thf(fact_94_cong__subst__left_I1_J,axiom,
! [T: set_a,T4: set_a,U: set_a] :
( ( ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ T @ T4 ) )
& ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ T4 @ T ) ) )
=> ( ( con_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U )
=> ( con_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T4 @ U ) ) ) ).
% cong_subst_left(1)
thf(fact_95_cong__subst__left_I2_J,axiom,
! [T: set_a,T4: set_a,U: set_a] :
( ( ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ T @ T4 ) )
& ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ T4 @ T ) ) )
=> ( ( con_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U )
=> ( ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ ( quotie8165075472272353145esid_a @ resid @ nn @ T @ U ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ T4 @ U ) ) )
& ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ ( quotie8165075472272353145esid_a @ resid @ nn @ T4 @ U ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ T @ U ) ) ) ) ) ) ).
% cong_subst_left(2)
thf(fact_96_arr__resid__iff__con,axiom,
! [T: set_a,U: set_a] :
( ( arr_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ T @ U ) )
= ( con_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U ) ) ).
% arr_resid_iff_con
thf(fact_97_arr__resid,axiom,
! [T: set_a,U: set_a] :
( ( con_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U )
=> ( arr_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ T @ U ) ) ) ).
% arr_resid
thf(fact_98_arr__def,axiom,
! [T: set_a] :
( ( arr_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T )
= ( con_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ T ) ) ).
% arr_def
thf(fact_99_arrE,axiom,
! [T: set_a] :
( ( arr_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T )
=> ( con_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ T ) ) ).
% arrE
thf(fact_100_con__implies__arr_I1_J,axiom,
! [T: set_a,U: set_a] :
( ( con_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U )
=> ( arr_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T ) ) ).
% con_implies_arr(1)
thf(fact_101_con__implies__arr_I2_J,axiom,
! [T: set_a,U: set_a] :
( ( con_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U )
=> ( arr_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ U ) ) ).
% con_implies_arr(2)
thf(fact_102_sources__are__con,axiom,
! [A3: set_a,T: set_a,A4: set_a] :
( ( member_set_a @ A3 @ ( sources_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T ) )
=> ( ( member_set_a @ A4 @ ( sources_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T ) )
=> ( con_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ A3 @ A4 ) ) ) ).
% sources_are_con
thf(fact_103_con__imp__eq__src,axiom,
! [T: set_a,U: set_a] :
( ( con_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U )
=> ( ( weakly2061155085811118449_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T )
= ( weakly2061155085811118449_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ U ) ) ) ).
% con_imp_eq_src
thf(fact_104_cong__implies__coinitial,axiom,
! [U: set_a,U7: set_a] :
( ( ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ U @ U7 ) )
& ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ U7 @ U ) ) )
=> ( coinitial_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ U @ U7 ) ) ).
% cong_implies_coinitial
thf(fact_105_coinitial__ide__are__eq,axiom,
! [A3: set_a,A4: set_a] :
( ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ A3 )
=> ( ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ A4 )
=> ( ( coinitial_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ A3 @ A4 )
=> ( A3 = A4 ) ) ) ) ).
% coinitial_ide_are_eq
thf(fact_106_coinitial__ide__are__cong,axiom,
! [A3: set_a,A4: set_a] :
( ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ A3 )
=> ( ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ A4 )
=> ( ( coinitial_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ A3 @ A4 )
=> ( ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ A3 @ A4 ) )
& ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ A4 @ A3 ) ) ) ) ) ) ).
% coinitial_ide_are_cong
thf(fact_107_resid__ide_I1_J,axiom,
! [A3: set_a,T: set_a] :
( ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ A3 )
=> ( ( coinitial_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ A3 @ T )
=> ( ( quotie8165075472272353145esid_a @ resid @ nn @ T @ A3 )
= T ) ) ) ).
% resid_ide(1)
thf(fact_108_composableD_I1_J,axiom,
! [T: set_a,U: set_a] :
( ( composable_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U )
=> ( arr_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T ) ) ).
% composableD(1)
thf(fact_109_con__imp__coinitial,axiom,
! [T: set_a,U: set_a] :
( ( con_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U )
=> ( coinitial_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U ) ) ).
% con_imp_coinitial
thf(fact_110_sources__con__closed,axiom,
! [A3: set_a,T: set_a,A4: set_a] :
( ( member_set_a @ A3 @ ( sources_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T ) )
=> ( ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ A4 )
=> ( ( con_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ A3 @ A4 )
=> ( member_set_a @ A4 @ ( sources_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T ) ) ) ) ) ).
% sources_con_closed
thf(fact_111_in__sourcesE,axiom,
! [A3: set_a,T: set_a] :
( ( member_set_a @ A3 @ ( sources_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T ) )
=> ~ ( ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ A3 )
=> ~ ( con_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ A3 ) ) ) ).
% in_sourcesE
thf(fact_112_src__eqI,axiom,
! [A3: set_a,T: set_a] :
( ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ A3 )
=> ( ( con_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ A3 @ T )
=> ( ( weakly2061155085811118449_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T )
= A3 ) ) ) ).
% src_eqI
thf(fact_113_coinitial__iff_092_060_094sub_062W_092_060_094sub_062E,axiom,
! [T: set_a,U: set_a] :
( ( coinitial_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U )
= ( ( arr_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T )
& ( arr_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ U )
& ( ( weakly2061155085811118449_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T )
= ( weakly2061155085811118449_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ U ) ) ) ) ).
% coinitial_iff\<^sub>W\<^sub>E
thf(fact_114_coinitialE_092_060_094sub_062W_092_060_094sub_062E,axiom,
! [T: set_a,U: set_a] :
( ( coinitial_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U )
=> ~ ( ( arr_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T )
=> ( ( arr_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ U )
=> ( ( weakly2061155085811118449_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T )
!= ( weakly2061155085811118449_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ U ) ) ) ) ) ).
% coinitialE\<^sub>W\<^sub>E
thf(fact_115_sources__def,axiom,
! [T: set_a] :
( ( sources_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T )
= ( collect_set_a
@ ^ [A: set_a] :
( ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ A )
& ( con_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ A ) ) ) ) ).
% sources_def
thf(fact_116_ideI,axiom,
! [A3: set_a] :
( ( con_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ A3 @ A3 )
=> ( ( ( quotie8165075472272353145esid_a @ resid @ nn @ A3 @ A3 )
= A3 )
=> ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ A3 ) ) ) ).
% ideI
thf(fact_117_arrI,axiom,
! [T: set_a] :
( ( con_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ T )
=> ( arr_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T ) ) ).
% arrI
thf(fact_118_src__ide,axiom,
! [A3: set_a] :
( ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ A3 )
=> ( ( weakly2061155085811118449_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ A3 )
= A3 ) ) ).
% src_ide
thf(fact_119_in__sourcesI,axiom,
! [A3: set_a,T: set_a] :
( ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ A3 )
=> ( ( con_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ A3 )
=> ( member_set_a @ A3 @ ( sources_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T ) ) ) ) ).
% in_sourcesI
thf(fact_120_coinitialI,axiom,
! [T: set_a,U: set_a] :
( ( arr_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T )
=> ( ( ( sources_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T )
= ( sources_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ U ) )
=> ( coinitial_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U ) ) ) ).
% coinitialI
thf(fact_121_coinitialI_092_060_094sub_062W_092_060_094sub_062E,axiom,
! [T: set_a,U: set_a] :
( ( arr_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T )
=> ( ( ( weakly2061155085811118449_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T )
= ( weakly2061155085811118449_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ U ) )
=> ( coinitial_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U ) ) ) ).
% coinitialI\<^sub>W\<^sub>E
thf(fact_122_quotient__by__coherent__normal_Oaxioms_I2_J,axiom,
! [Resid: a > a > a,NN: set_a] :
( ( quotie3282664541148387094rmal_a @ Resid @ NN )
=> ( cohere6072184133013167079_rts_a @ Resid @ NN ) ) ).
% quotient_by_coherent_normal.axioms(2)
thf(fact_123_rts_Ocoinitial_Ocong,axiom,
coinitial_set_a = coinitial_set_a ).
% rts.coinitial.cong
thf(fact_124_rts_Ocoinitial_Ocong,axiom,
coinitial_a = coinitial_a ).
% rts.coinitial.cong
thf(fact_125_rts_Ocomposable_Ocong,axiom,
composable_set_a = composable_set_a ).
% rts.composable.cong
thf(fact_126_rts_Ocomposable_Ocong,axiom,
composable_a = composable_a ).
% rts.composable.cong
thf(fact_127_residuation_Ocon_Ocong,axiom,
con_set_a = con_set_a ).
% residuation.con.cong
thf(fact_128_residuation_Ocon_Ocong,axiom,
con_a = con_a ).
% residuation.con.cong
thf(fact_129_normal__sub__rts_OCong__class_Ocong,axiom,
normal7408713899360725774lass_a = normal7408713899360725774lass_a ).
% normal_sub_rts.Cong_class.cong
thf(fact_130_partial__magma__def,axiom,
( partial_magma_set_a
= ( ^ [OP: set_a > set_a > set_a] :
? [X2: set_a] :
( ! [T6: set_a] :
( ( ( OP @ X2 @ T6 )
= X2 )
& ( ( OP @ T6 @ X2 )
= X2 ) )
& ! [Y2: set_a] :
( ! [T6: set_a] :
( ( ( OP @ Y2 @ T6 )
= Y2 )
& ( ( OP @ T6 @ Y2 )
= Y2 ) )
=> ( Y2 = X2 ) ) ) ) ) ).
% partial_magma_def
thf(fact_131_partial__magma__def,axiom,
( partial_magma_a
= ( ^ [OP: a > a > a] :
? [X2: a] :
( ! [T6: a] :
( ( ( OP @ X2 @ T6 )
= X2 )
& ( ( OP @ T6 @ X2 )
= X2 ) )
& ! [Y2: a] :
( ! [T6: a] :
( ( ( OP @ Y2 @ T6 )
= Y2 )
& ( ( OP @ T6 @ Y2 )
= Y2 ) )
=> ( Y2 = X2 ) ) ) ) ) ).
% partial_magma_def
thf(fact_132_partial__magma_Oex__un__null,axiom,
! [OP2: set_a > set_a > set_a] :
( ( partial_magma_set_a @ OP2 )
=> ? [X: set_a] :
( ! [T2: set_a] :
( ( ( OP2 @ X @ T2 )
= X )
& ( ( OP2 @ T2 @ X )
= X ) )
& ! [Y: set_a] :
( ! [T3: set_a] :
( ( ( OP2 @ Y @ T3 )
= Y )
& ( ( OP2 @ T3 @ Y )
= Y ) )
=> ( Y = X ) ) ) ) ).
% partial_magma.ex_un_null
thf(fact_133_partial__magma_Oex__un__null,axiom,
! [OP2: a > a > a] :
( ( partial_magma_a @ OP2 )
=> ? [X: a] :
( ! [T2: a] :
( ( ( OP2 @ X @ T2 )
= X )
& ( ( OP2 @ T2 @ X )
= X ) )
& ! [Y: a] :
( ! [T3: a] :
( ( ( OP2 @ Y @ T3 )
= Y )
& ( ( OP2 @ T3 @ Y )
= Y ) )
=> ( Y = X ) ) ) ) ).
% partial_magma.ex_un_null
thf(fact_134_partial__magma_Ointro,axiom,
! [OP2: set_a > set_a > set_a] :
( ? [X3: set_a] :
( ! [T3: set_a] :
( ( ( OP2 @ X3 @ T3 )
= X3 )
& ( ( OP2 @ T3 @ X3 )
= X3 ) )
& ! [Y3: set_a] :
( ! [T2: set_a] :
( ( ( OP2 @ Y3 @ T2 )
= Y3 )
& ( ( OP2 @ T2 @ Y3 )
= Y3 ) )
=> ( Y3 = X3 ) ) )
=> ( partial_magma_set_a @ OP2 ) ) ).
% partial_magma.intro
thf(fact_135_partial__magma_Ointro,axiom,
! [OP2: a > a > a] :
( ? [X3: a] :
( ! [T3: a] :
( ( ( OP2 @ X3 @ T3 )
= X3 )
& ( ( OP2 @ T3 @ X3 )
= X3 ) )
& ! [Y3: a] :
( ! [T2: a] :
( ( ( OP2 @ Y3 @ T2 )
= Y3 )
& ( ( OP2 @ T2 @ Y3 )
= Y3 ) )
=> ( Y3 = X3 ) ) )
=> ( partial_magma_a @ OP2 ) ) ).
% partial_magma.intro
thf(fact_136_quotient__by__coherent__normal_OCon__witnesses,axiom,
! [Resid: set_a > set_a > set_a,NN: set_set_a,T7: set_set_a,U2: set_set_a,T: set_a,U: set_a] :
( ( quotie5625257012022141046_set_a @ Resid @ NN )
=> ( ( ( quotie3283642546880816345_set_a @ Resid @ NN @ T7 @ U2 )
!= bot_bot_set_set_a )
=> ( ( member_set_a @ T @ T7 )
=> ( ( member_set_a @ U @ U2 )
=> ? [V3: set_a,W: set_a] :
( ( member_set_a @ V3 @ NN )
& ( member_set_a @ W @ NN )
& ( con_set_a @ Resid @ ( Resid @ T @ V3 ) @ ( Resid @ U @ W ) ) ) ) ) ) ) ).
% quotient_by_coherent_normal.Con_witnesses
thf(fact_137_quotient__by__coherent__normal_OCon__witnesses,axiom,
! [Resid: a > a > a,NN: set_a,T7: set_a,U2: set_a,T: a,U: a] :
( ( quotie3282664541148387094rmal_a @ Resid @ NN )
=> ( ( ( quotie8165075472272353145esid_a @ Resid @ NN @ T7 @ U2 )
!= bot_bot_set_a )
=> ( ( member_a @ T @ T7 )
=> ( ( member_a @ U @ U2 )
=> ? [V3: a,W: a] :
( ( member_a @ V3 @ NN )
& ( member_a @ W @ NN )
& ( con_a @ Resid @ ( Resid @ T @ V3 ) @ ( Resid @ U @ W ) ) ) ) ) ) ) ).
% quotient_by_coherent_normal.Con_witnesses
thf(fact_138_quotient__by__coherent__normal_Ocon__imp__coinitial__members__are__con,axiom,
! [Resid: set_a > set_a > set_a,NN: set_set_a,T7: set_set_a,U2: set_set_a,T: set_a,U: set_a] :
( ( quotie5625257012022141046_set_a @ Resid @ NN )
=> ( ( con_set_set_a @ ( quotie3283642546880816345_set_a @ Resid @ NN ) @ T7 @ U2 )
=> ( ( member_set_a @ T @ T7 )
=> ( ( member_set_a @ U @ U2 )
=> ( ( ( sources_set_a @ Resid @ T )
= ( sources_set_a @ Resid @ U ) )
=> ( con_set_a @ Resid @ T @ U ) ) ) ) ) ) ).
% quotient_by_coherent_normal.con_imp_coinitial_members_are_con
thf(fact_139_quotient__by__coherent__normal_Ocon__imp__coinitial__members__are__con,axiom,
! [Resid: a > a > a,NN: set_a,T7: set_a,U2: set_a,T: a,U: a] :
( ( quotie3282664541148387094rmal_a @ Resid @ NN )
=> ( ( con_set_a @ ( quotie8165075472272353145esid_a @ Resid @ NN ) @ T7 @ U2 )
=> ( ( member_a @ T @ T7 )
=> ( ( member_a @ U @ U2 )
=> ( ( ( sources_a @ Resid @ T )
= ( sources_a @ Resid @ U ) )
=> ( con_a @ Resid @ T @ U ) ) ) ) ) ) ).
% quotient_by_coherent_normal.con_imp_coinitial_members_are_con
thf(fact_140_quotient__by__coherent__normal_Ois__partial__magma,axiom,
! [Resid: a > a > a,NN: set_a] :
( ( quotie3282664541148387094rmal_a @ Resid @ NN )
=> ( partial_magma_set_a @ ( quotie8165075472272353145esid_a @ Resid @ NN ) ) ) ).
% quotient_by_coherent_normal.is_partial_magma
thf(fact_141_quotient__by__coherent__normal_OArr__Resid,axiom,
! [Resid: set_a > set_a > set_a,NN: set_set_a,T7: set_set_a,U2: set_set_a] :
( ( quotie5625257012022141046_set_a @ Resid @ NN )
=> ( ( ( quotie3283642546880816345_set_a @ Resid @ NN @ T7 @ U2 )
!= bot_bot_set_set_a )
=> ( ( quotie3283642546880816345_set_a @ Resid @ NN @ ( quotie3283642546880816345_set_a @ Resid @ NN @ T7 @ U2 ) @ ( quotie3283642546880816345_set_a @ Resid @ NN @ T7 @ U2 ) )
!= bot_bot_set_set_a ) ) ) ).
% quotient_by_coherent_normal.Arr_Resid
thf(fact_142_quotient__by__coherent__normal_OArr__Resid,axiom,
! [Resid: a > a > a,NN: set_a,T7: set_a,U2: set_a] :
( ( quotie3282664541148387094rmal_a @ Resid @ NN )
=> ( ( ( quotie8165075472272353145esid_a @ Resid @ NN @ T7 @ U2 )
!= bot_bot_set_a )
=> ( ( quotie8165075472272353145esid_a @ Resid @ NN @ ( quotie8165075472272353145esid_a @ Resid @ NN @ T7 @ U2 ) @ ( quotie8165075472272353145esid_a @ Resid @ NN @ T7 @ U2 ) )
!= bot_bot_set_a ) ) ) ).
% quotient_by_coherent_normal.Arr_Resid
thf(fact_143_quotient__by__coherent__normal_OCon__sym,axiom,
! [Resid: set_a > set_a > set_a,NN: set_set_a,T7: set_set_a,U2: set_set_a] :
( ( quotie5625257012022141046_set_a @ Resid @ NN )
=> ( ( ( quotie3283642546880816345_set_a @ Resid @ NN @ T7 @ U2 )
!= bot_bot_set_set_a )
=> ( ( quotie3283642546880816345_set_a @ Resid @ NN @ U2 @ T7 )
!= bot_bot_set_set_a ) ) ) ).
% quotient_by_coherent_normal.Con_sym
thf(fact_144_quotient__by__coherent__normal_OCon__sym,axiom,
! [Resid: a > a > a,NN: set_a,T7: set_a,U2: set_a] :
( ( quotie3282664541148387094rmal_a @ Resid @ NN )
=> ( ( ( quotie8165075472272353145esid_a @ Resid @ NN @ T7 @ U2 )
!= bot_bot_set_a )
=> ( ( quotie8165075472272353145esid_a @ Resid @ NN @ U2 @ T7 )
!= bot_bot_set_a ) ) ) ).
% quotient_by_coherent_normal.Con_sym
thf(fact_145_quotient__by__coherent__normal_OCube,axiom,
! [Resid: set_a > set_a > set_a,NN: set_set_a,V2: set_set_a,T7: set_set_a,U2: set_set_a] :
( ( quotie5625257012022141046_set_a @ Resid @ NN )
=> ( ( ( quotie3283642546880816345_set_a @ Resid @ NN @ ( quotie3283642546880816345_set_a @ Resid @ NN @ V2 @ T7 ) @ ( quotie3283642546880816345_set_a @ Resid @ NN @ U2 @ T7 ) )
!= bot_bot_set_set_a )
=> ( ( quotie3283642546880816345_set_a @ Resid @ NN @ ( quotie3283642546880816345_set_a @ Resid @ NN @ V2 @ T7 ) @ ( quotie3283642546880816345_set_a @ Resid @ NN @ U2 @ T7 ) )
= ( quotie3283642546880816345_set_a @ Resid @ NN @ ( quotie3283642546880816345_set_a @ Resid @ NN @ V2 @ U2 ) @ ( quotie3283642546880816345_set_a @ Resid @ NN @ T7 @ U2 ) ) ) ) ) ).
% quotient_by_coherent_normal.Cube
thf(fact_146_quotient__by__coherent__normal_OCube,axiom,
! [Resid: a > a > a,NN: set_a,V2: set_a,T7: set_a,U2: set_a] :
( ( quotie3282664541148387094rmal_a @ Resid @ NN )
=> ( ( ( quotie8165075472272353145esid_a @ Resid @ NN @ ( quotie8165075472272353145esid_a @ Resid @ NN @ V2 @ T7 ) @ ( quotie8165075472272353145esid_a @ Resid @ NN @ U2 @ T7 ) )
!= bot_bot_set_a )
=> ( ( quotie8165075472272353145esid_a @ Resid @ NN @ ( quotie8165075472272353145esid_a @ Resid @ NN @ V2 @ T7 ) @ ( quotie8165075472272353145esid_a @ Resid @ NN @ U2 @ T7 ) )
= ( quotie8165075472272353145esid_a @ Resid @ NN @ ( quotie8165075472272353145esid_a @ Resid @ NN @ V2 @ U2 ) @ ( quotie8165075472272353145esid_a @ Resid @ NN @ T7 @ U2 ) ) ) ) ) ).
% quotient_by_coherent_normal.Cube
thf(fact_147_residuation_Oide_Ocong,axiom,
ide_set_a = ide_set_a ).
% residuation.ide.cong
thf(fact_148_residuation_Oide_Ocong,axiom,
ide_a = ide_a ).
% residuation.ide.cong
thf(fact_149_coherent__normal__sub__rts_OCong__subst__con,axiom,
! [Resid: set_a > set_a > set_a,NN: set_set_a,T: set_a,U: set_a,T4: set_a,U7: set_a] :
( ( cohere6325062230080414023_set_a @ Resid @ NN )
=> ( ( ( sources_set_a @ Resid @ T )
= ( sources_set_a @ Resid @ U ) )
=> ( ( ( sources_set_a @ Resid @ T4 )
= ( sources_set_a @ Resid @ U7 ) )
=> ( ( normal8977612136997397236_set_a @ Resid @ NN @ T @ T4 )
=> ( ( normal8977612136997397236_set_a @ Resid @ NN @ U @ U7 )
=> ( ( con_set_a @ Resid @ T @ U )
= ( con_set_a @ Resid @ T4 @ U7 ) ) ) ) ) ) ) ).
% coherent_normal_sub_rts.Cong_subst_con
thf(fact_150_coherent__normal__sub__rts_OCong__subst__con,axiom,
! [Resid: a > a > a,NN: set_a,T: a,U: a,T4: a,U7: a] :
( ( cohere6072184133013167079_rts_a @ Resid @ NN )
=> ( ( ( sources_a @ Resid @ T )
= ( sources_a @ Resid @ U ) )
=> ( ( ( sources_a @ Resid @ T4 )
= ( sources_a @ Resid @ U7 ) )
=> ( ( normal_sub_Cong_a @ Resid @ NN @ T @ T4 )
=> ( ( normal_sub_Cong_a @ Resid @ NN @ U @ U7 )
=> ( ( con_a @ Resid @ T @ U )
= ( con_a @ Resid @ T4 @ U7 ) ) ) ) ) ) ) ).
% coherent_normal_sub_rts.Cong_subst_con
thf(fact_151_coherent__normal__sub__rts_OCong__subst_I1_J,axiom,
! [Resid: set_a > set_a > set_a,NN: set_set_a,T: set_a,T4: set_a,U: set_a,U7: set_a] :
( ( cohere6325062230080414023_set_a @ Resid @ NN )
=> ( ( normal8977612136997397236_set_a @ Resid @ NN @ T @ T4 )
=> ( ( normal8977612136997397236_set_a @ Resid @ NN @ U @ U7 )
=> ( ( con_set_a @ Resid @ T @ U )
=> ( ( ( sources_set_a @ Resid @ T4 )
= ( sources_set_a @ Resid @ U7 ) )
=> ( con_set_a @ Resid @ T4 @ U7 ) ) ) ) ) ) ).
% coherent_normal_sub_rts.Cong_subst(1)
thf(fact_152_coherent__normal__sub__rts_OCong__subst_I1_J,axiom,
! [Resid: a > a > a,NN: set_a,T: a,T4: a,U: a,U7: a] :
( ( cohere6072184133013167079_rts_a @ Resid @ NN )
=> ( ( normal_sub_Cong_a @ Resid @ NN @ T @ T4 )
=> ( ( normal_sub_Cong_a @ Resid @ NN @ U @ U7 )
=> ( ( con_a @ Resid @ T @ U )
=> ( ( ( sources_a @ Resid @ T4 )
= ( sources_a @ Resid @ U7 ) )
=> ( con_a @ Resid @ T4 @ U7 ) ) ) ) ) ) ).
% coherent_normal_sub_rts.Cong_subst(1)
thf(fact_153_coherent__normal__sub__rts_OCong__subst_I2_J,axiom,
! [Resid: set_a > set_a > set_a,NN: set_set_a,T: set_a,T4: set_a,U: set_a,U7: set_a] :
( ( cohere6325062230080414023_set_a @ Resid @ NN )
=> ( ( normal8977612136997397236_set_a @ Resid @ NN @ T @ T4 )
=> ( ( normal8977612136997397236_set_a @ Resid @ NN @ U @ U7 )
=> ( ( con_set_a @ Resid @ T @ U )
=> ( ( ( sources_set_a @ Resid @ T4 )
= ( sources_set_a @ Resid @ U7 ) )
=> ( normal8977612136997397236_set_a @ Resid @ NN @ ( Resid @ T @ U ) @ ( Resid @ T4 @ U7 ) ) ) ) ) ) ) ).
% coherent_normal_sub_rts.Cong_subst(2)
thf(fact_154_coherent__normal__sub__rts_OCong__subst_I2_J,axiom,
! [Resid: a > a > a,NN: set_a,T: a,T4: a,U: a,U7: a] :
( ( cohere6072184133013167079_rts_a @ Resid @ NN )
=> ( ( normal_sub_Cong_a @ Resid @ NN @ T @ T4 )
=> ( ( normal_sub_Cong_a @ Resid @ NN @ U @ U7 )
=> ( ( con_a @ Resid @ T @ U )
=> ( ( ( sources_a @ Resid @ T4 )
= ( sources_a @ Resid @ U7 ) )
=> ( normal_sub_Cong_a @ Resid @ NN @ ( Resid @ T @ U ) @ ( Resid @ T4 @ U7 ) ) ) ) ) ) ) ).
% coherent_normal_sub_rts.Cong_subst(2)
thf(fact_155_weakly__extensional__rts_Osrc_Ocong,axiom,
weakly2061155085811118449_set_a = weakly2061155085811118449_set_a ).
% weakly_extensional_rts.src.cong
thf(fact_156_quotient__by__coherent__normal_Oide__char,axiom,
! [Resid: set_a > set_a > set_a,NN: set_set_a,U2: set_set_a] :
( ( quotie5625257012022141046_set_a @ Resid @ NN )
=> ( ( ide_set_set_a @ ( quotie3283642546880816345_set_a @ Resid @ NN ) @ U2 )
= ( ( arr_set_set_a @ ( quotie3283642546880816345_set_a @ Resid @ NN ) @ U2 )
& ( ( inf_inf_set_set_a @ U2 @ NN )
!= bot_bot_set_set_a ) ) ) ) ).
% quotient_by_coherent_normal.ide_char
thf(fact_157_quotient__by__coherent__normal_Oide__char,axiom,
! [Resid: a > a > a,NN: set_a,U2: set_a] :
( ( quotie3282664541148387094rmal_a @ Resid @ NN )
=> ( ( ide_set_a @ ( quotie8165075472272353145esid_a @ Resid @ NN ) @ U2 )
= ( ( arr_set_a @ ( quotie8165075472272353145esid_a @ Resid @ NN ) @ U2 )
& ( ( inf_inf_set_a @ U2 @ NN )
!= bot_bot_set_a ) ) ) ) ).
% quotient_by_coherent_normal.ide_char
thf(fact_158_cong__implies__coterminal,axiom,
! [U: set_a,U7: set_a] :
( ( ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ U @ U7 ) )
& ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ U7 @ U ) ) )
=> ( coterminal_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ U @ U7 ) ) ).
% cong_implies_coterminal
thf(fact_159_coinitial__def,axiom,
! [T: set_a,U: set_a] :
( ( coinitial_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U )
= ( ( inf_inf_set_set_a @ ( sources_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T ) @ ( sources_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ U ) )
!= bot_bot_set_set_a ) ) ).
% coinitial_def
thf(fact_160_R_Ocoinitial__def,axiom,
! [T: a,U: a] :
( ( coinitial_a @ resid @ T @ U )
= ( ( inf_inf_set_a @ ( sources_a @ resid @ T ) @ ( sources_a @ resid @ U ) )
!= bot_bot_set_a ) ) ).
% R.coinitial_def
thf(fact_161_con__imp__common__source,axiom,
! [T: set_a,U: set_a] :
( ( con_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U )
=> ( ( inf_inf_set_set_a @ ( sources_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T ) @ ( sources_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ U ) )
!= bot_bot_set_set_a ) ) ).
% con_imp_common_source
thf(fact_162_joinable__implies__coinitial,axiom,
! [T: set_a,U: set_a] :
( ( joinable_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U )
=> ( coinitial_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U ) ) ).
% joinable_implies_coinitial
thf(fact_163_ide__char_H,axiom,
! [A7: set_a] :
( ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ A7 )
= ( ( arr_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ A7 )
& ( ord_less_eq_set_a @ A7 @ nn ) ) ) ).
% ide_char'
thf(fact_164_inf__bot__left,axiom,
! [X4: set_a] :
( ( inf_inf_set_a @ bot_bot_set_a @ X4 )
= bot_bot_set_a ) ).
% inf_bot_left
thf(fact_165_inf__bot__left,axiom,
! [X4: set_set_a] :
( ( inf_inf_set_set_a @ bot_bot_set_set_a @ X4 )
= bot_bot_set_set_a ) ).
% inf_bot_left
thf(fact_166_inf__bot__right,axiom,
! [X4: set_a] :
( ( inf_inf_set_a @ X4 @ bot_bot_set_a )
= bot_bot_set_a ) ).
% inf_bot_right
thf(fact_167_inf__bot__right,axiom,
! [X4: set_set_a] :
( ( inf_inf_set_set_a @ X4 @ bot_bot_set_set_a )
= bot_bot_set_set_a ) ).
% inf_bot_right
thf(fact_168_boolean__algebra_Oconj__zero__left,axiom,
! [X4: set_a] :
( ( inf_inf_set_a @ bot_bot_set_a @ X4 )
= bot_bot_set_a ) ).
% boolean_algebra.conj_zero_left
thf(fact_169_boolean__algebra_Oconj__zero__left,axiom,
! [X4: set_set_a] :
( ( inf_inf_set_set_a @ bot_bot_set_set_a @ X4 )
= bot_bot_set_set_a ) ).
% boolean_algebra.conj_zero_left
thf(fact_170_boolean__algebra_Oconj__zero__right,axiom,
! [X4: set_a] :
( ( inf_inf_set_a @ X4 @ bot_bot_set_a )
= bot_bot_set_a ) ).
% boolean_algebra.conj_zero_right
thf(fact_171_boolean__algebra_Oconj__zero__right,axiom,
! [X4: set_set_a] :
( ( inf_inf_set_set_a @ X4 @ bot_bot_set_set_a )
= bot_bot_set_set_a ) ).
% boolean_algebra.conj_zero_right
thf(fact_172_R_Ocong__subst__left_I2_J,axiom,
! [T: a,T4: a,U: a] :
( ( ( ide_a @ resid @ ( resid @ T @ T4 ) )
& ( ide_a @ resid @ ( resid @ T4 @ T ) ) )
=> ( ( con_a @ resid @ T @ U )
=> ( ( ide_a @ resid @ ( resid @ ( resid @ T @ U ) @ ( resid @ T4 @ U ) ) )
& ( ide_a @ resid @ ( resid @ ( resid @ T4 @ U ) @ ( resid @ T @ U ) ) ) ) ) ) ).
% R.cong_subst_left(2)
thf(fact_173_R_Ocong__subst__left_I1_J,axiom,
! [T: a,T4: a,U: a] :
( ( ( ide_a @ resid @ ( resid @ T @ T4 ) )
& ( ide_a @ resid @ ( resid @ T4 @ T ) ) )
=> ( ( con_a @ resid @ T @ U )
=> ( con_a @ resid @ T4 @ U ) ) ) ).
% R.cong_subst_left(1)
thf(fact_174_R_Ocong__subst__right_I2_J,axiom,
! [U: a,U7: a,T: a] :
( ( ( ide_a @ resid @ ( resid @ U @ U7 ) )
& ( ide_a @ resid @ ( resid @ U7 @ U ) ) )
=> ( ( con_a @ resid @ T @ U )
=> ( ( ide_a @ resid @ ( resid @ ( resid @ T @ U ) @ ( resid @ T @ U7 ) ) )
& ( ide_a @ resid @ ( resid @ ( resid @ T @ U7 ) @ ( resid @ T @ U ) ) ) ) ) ) ).
% R.cong_subst_right(2)
thf(fact_175_R_Ocong__subst__right_I1_J,axiom,
! [U: a,U7: a,T: a] :
( ( ( ide_a @ resid @ ( resid @ U @ U7 ) )
& ( ide_a @ resid @ ( resid @ U7 @ U ) ) )
=> ( ( con_a @ resid @ T @ U )
=> ( con_a @ resid @ T @ U7 ) ) ) ).
% R.cong_subst_right(1)
thf(fact_176_R_Ocon__imp__coinitial__ax,axiom,
! [T: a,U: a] :
( ( con_a @ resid @ T @ U )
=> ? [A6: a] :
( ( ide_a @ resid @ A6 )
& ( con_a @ resid @ A6 @ T )
& ( con_a @ resid @ A6 @ U ) ) ) ).
% R.con_imp_coinitial_ax
thf(fact_177_R_Ocon__target,axiom,
! [T: a,U: a,V: a] :
( ( ide_a @ resid @ ( resid @ T @ U ) )
=> ( ( con_a @ resid @ U @ V )
=> ( con_a @ resid @ ( resid @ T @ U ) @ ( resid @ V @ U ) ) ) ) ).
% R.con_target
thf(fact_178_R_Ocon__transitive__on__ide,axiom,
! [A3: a,B: a,C: a] :
( ( ide_a @ resid @ A3 )
=> ( ( ide_a @ resid @ B )
=> ( ( ide_a @ resid @ C )
=> ( ( con_a @ resid @ A3 @ B )
=> ( ( con_a @ resid @ B @ C )
=> ( con_a @ resid @ A3 @ C ) ) ) ) ) ) ).
% R.con_transitive_on_ide
thf(fact_179_R_OideE,axiom,
! [A3: a] :
( ( ide_a @ resid @ A3 )
=> ~ ( ( con_a @ resid @ A3 @ A3 )
=> ( ( resid @ A3 @ A3 )
!= A3 ) ) ) ).
% R.ideE
thf(fact_180_R_Oide__def,axiom,
! [A3: a] :
( ( ide_a @ resid @ A3 )
= ( ( con_a @ resid @ A3 @ A3 )
& ( ( resid @ A3 @ A3 )
= A3 ) ) ) ).
% R.ide_def
thf(fact_181_R_Oide__imp__con__iff__cong,axiom,
! [T: a,U: a] :
( ( ide_a @ resid @ T )
=> ( ( ide_a @ resid @ U )
=> ( ( con_a @ resid @ T @ U )
= ( ( ide_a @ resid @ ( resid @ T @ U ) )
& ( ide_a @ resid @ ( resid @ U @ T ) ) ) ) ) ) ).
% R.ide_imp_con_iff_cong
thf(fact_182_R_Oprfx__implies__con,axiom,
! [T: a,U: a] :
( ( ide_a @ resid @ ( resid @ T @ U ) )
=> ( con_a @ resid @ T @ U ) ) ).
% R.prfx_implies_con
thf(fact_183_R_Oresid__arr__ide,axiom,
! [A3: a,T: a] :
( ( ide_a @ resid @ A3 )
=> ( ( con_a @ resid @ T @ A3 )
=> ( ( resid @ T @ A3 )
= T ) ) ) ).
% R.resid_arr_ide
thf(fact_184_R_Oresid__ide__arr,axiom,
! [A3: a,T: a] :
( ( ide_a @ resid @ A3 )
=> ( ( con_a @ resid @ A3 @ T )
=> ( ide_a @ resid @ ( resid @ A3 @ T ) ) ) ) ).
% R.resid_ide_arr
thf(fact_185_R_Ocon__sym,axiom,
! [T: a,U: a] :
( ( con_a @ resid @ T @ U )
=> ( con_a @ resid @ U @ T ) ) ).
% R.con_sym
thf(fact_186_R_Oresid__reflects__con,axiom,
! [T: a,V: a,U: a] :
( ( con_a @ resid @ T @ V )
=> ( ( con_a @ resid @ U @ V )
=> ( ( con_a @ resid @ ( resid @ T @ V ) @ ( resid @ U @ V ) )
=> ( con_a @ resid @ T @ U ) ) ) ) ).
% R.resid_reflects_con
thf(fact_187_R_Ocong__symmetric,axiom,
! [T: a,U: a] :
( ( ( ide_a @ resid @ ( resid @ T @ U ) )
& ( ide_a @ resid @ ( resid @ U @ T ) ) )
=> ( ( ide_a @ resid @ ( resid @ U @ T ) )
& ( ide_a @ resid @ ( resid @ T @ U ) ) ) ) ).
% R.cong_symmetric
thf(fact_188_R_Ocong__transitive,axiom,
! [T: a,U: a,V: a] :
( ( ( ide_a @ resid @ ( resid @ T @ U ) )
& ( ide_a @ resid @ ( resid @ U @ T ) ) )
=> ( ( ( ide_a @ resid @ ( resid @ U @ V ) )
& ( ide_a @ resid @ ( resid @ V @ U ) ) )
=> ( ( ide_a @ resid @ ( resid @ T @ V ) )
& ( ide_a @ resid @ ( resid @ V @ T ) ) ) ) ) ).
% R.cong_transitive
thf(fact_189_R_Oide__backward__stable,axiom,
! [A3: a,T: a] :
( ( ide_a @ resid @ A3 )
=> ( ( ide_a @ resid @ ( resid @ T @ A3 ) )
=> ( ide_a @ resid @ T ) ) ) ).
% R.ide_backward_stable
thf(fact_190_R_Oprfx__transitive,axiom,
! [T: a,U: a,V: a] :
( ( ide_a @ resid @ ( resid @ T @ U ) )
=> ( ( ide_a @ resid @ ( resid @ U @ V ) )
=> ( ide_a @ resid @ ( resid @ T @ V ) ) ) ) ).
% R.prfx_transitive
thf(fact_191_R_Oin__sourcesE,axiom,
! [A3: a,T: a] :
( ( member_a @ A3 @ ( sources_a @ resid @ T ) )
=> ~ ( ( ide_a @ resid @ A3 )
=> ~ ( con_a @ resid @ T @ A3 ) ) ) ).
% R.in_sourcesE
thf(fact_192_R_Osources__con__closed,axiom,
! [A3: a,T: a,A4: a] :
( ( member_a @ A3 @ ( sources_a @ resid @ T ) )
=> ( ( ide_a @ resid @ A4 )
=> ( ( con_a @ resid @ A3 @ A4 )
=> ( member_a @ A4 @ ( sources_a @ resid @ T ) ) ) ) ) ).
% R.sources_con_closed
thf(fact_193_R_Osources__are__con,axiom,
! [A3: a,T: a,A4: a] :
( ( member_a @ A3 @ ( sources_a @ resid @ T ) )
=> ( ( member_a @ A4 @ ( sources_a @ resid @ T ) )
=> ( con_a @ resid @ A3 @ A4 ) ) ) ).
% R.sources_are_con
thf(fact_194_R_Osource__is__ide,axiom,
! [A3: a,T: a] :
( ( member_a @ A3 @ ( sources_a @ resid @ T ) )
=> ( ide_a @ resid @ A3 ) ) ).
% R.source_is_ide
thf(fact_195_R_Osources__are__cong,axiom,
! [A3: a,T: a,A4: a] :
( ( member_a @ A3 @ ( sources_a @ resid @ T ) )
=> ( ( member_a @ A4 @ ( sources_a @ resid @ T ) )
=> ( ( ide_a @ resid @ ( resid @ A3 @ A4 ) )
& ( ide_a @ resid @ ( resid @ A4 @ A3 ) ) ) ) ) ).
% R.sources_are_cong
thf(fact_196_R_Osources__cong__closed,axiom,
! [A3: a,T: a,A4: a] :
( ( member_a @ A3 @ ( sources_a @ resid @ T ) )
=> ( ( ( ide_a @ resid @ ( resid @ A3 @ A4 ) )
& ( ide_a @ resid @ ( resid @ A4 @ A3 ) ) )
=> ( member_a @ A4 @ ( sources_a @ resid @ T ) ) ) ) ).
% R.sources_cong_closed
thf(fact_197_R_Ocon__implies__arr_I2_J,axiom,
! [T: a,U: a] :
( ( con_a @ resid @ T @ U )
=> ( arr_a @ resid @ U ) ) ).
% R.con_implies_arr(2)
thf(fact_198_R_Ocon__implies__arr_I1_J,axiom,
! [T: a,U: a] :
( ( con_a @ resid @ T @ U )
=> ( arr_a @ resid @ T ) ) ).
% R.con_implies_arr(1)
thf(fact_199_R_OarrE,axiom,
! [T: a] :
( ( arr_a @ resid @ T )
=> ( con_a @ resid @ T @ T ) ) ).
% R.arrE
thf(fact_200_R_Oarr__def,axiom,
! [T: a] :
( ( arr_a @ resid @ T )
= ( con_a @ resid @ T @ T ) ) ).
% R.arr_def
thf(fact_201_R_Oarr__resid,axiom,
! [T: a,U: a] :
( ( con_a @ resid @ T @ U )
=> ( arr_a @ resid @ ( resid @ T @ U ) ) ) ).
% R.arr_resid
thf(fact_202_R_Oarr__resid__iff__con,axiom,
! [T: a,U: a] :
( ( arr_a @ resid @ ( resid @ T @ U ) )
= ( con_a @ resid @ T @ U ) ) ).
% R.arr_resid_iff_con
thf(fact_203_R_Ocong__reflexive,axiom,
! [T: a] :
( ( arr_a @ resid @ T )
=> ( ( ide_a @ resid @ ( resid @ T @ T ) )
& ( ide_a @ resid @ ( resid @ T @ T ) ) ) ) ).
% R.cong_reflexive
thf(fact_204_R_Oide__implies__arr,axiom,
! [A3: a] :
( ( ide_a @ resid @ A3 )
=> ( arr_a @ resid @ A3 ) ) ).
% R.ide_implies_arr
thf(fact_205_R_Oprfx__reflexive,axiom,
! [T: a] :
( ( arr_a @ resid @ T )
=> ( ide_a @ resid @ ( resid @ T @ T ) ) ) ).
% R.prfx_reflexive
thf(fact_206_R_OcomposableD_I2_J,axiom,
! [T: a,U: a] :
( ( composable_a @ resid @ T @ U )
=> ( arr_a @ resid @ U ) ) ).
% R.composableD(2)
thf(fact_207_R_OcomposableD_I1_J,axiom,
! [T: a,U: a] :
( ( composable_a @ resid @ T @ U )
=> ( arr_a @ resid @ T ) ) ).
% R.composableD(1)
thf(fact_208_R_Ocon__imp__coinitial,axiom,
! [T: a,U: a] :
( ( con_a @ resid @ T @ U )
=> ( coinitial_a @ resid @ T @ U ) ) ).
% R.con_imp_coinitial
thf(fact_209_R_Ocoinitial__ide__are__cong,axiom,
! [A3: a,A4: a] :
( ( ide_a @ resid @ A3 )
=> ( ( ide_a @ resid @ A4 )
=> ( ( coinitial_a @ resid @ A3 @ A4 )
=> ( ( ide_a @ resid @ ( resid @ A3 @ A4 ) )
& ( ide_a @ resid @ ( resid @ A4 @ A3 ) ) ) ) ) ) ).
% R.coinitial_ide_are_cong
thf(fact_210_R_Ocong__implies__coinitial,axiom,
! [U: a,U7: a] :
( ( ( ide_a @ resid @ ( resid @ U @ U7 ) )
& ( ide_a @ resid @ ( resid @ U7 @ U ) ) )
=> ( coinitial_a @ resid @ U @ U7 ) ) ).
% R.cong_implies_coinitial
thf(fact_211_N_OCong_092_060_094sub_0620__subst__left_I2_J,axiom,
! [T: a,T4: a,U: a] :
( ( ( member_a @ ( resid @ T @ T4 ) @ nn )
& ( member_a @ ( resid @ T4 @ T ) @ nn ) )
=> ( ( con_a @ resid @ T @ U )
=> ( ( member_a @ ( resid @ ( resid @ T @ U ) @ ( resid @ T4 @ U ) ) @ nn )
& ( member_a @ ( resid @ ( resid @ T4 @ U ) @ ( resid @ T @ U ) ) @ nn ) ) ) ) ).
% N.Cong\<^sub>0_subst_left(2)
thf(fact_212_N_OCong_092_060_094sub_0620__subst__left_I1_J,axiom,
! [T: a,T4: a,U: a] :
( ( ( member_a @ ( resid @ T @ T4 ) @ nn )
& ( member_a @ ( resid @ T4 @ T ) @ nn ) )
=> ( ( con_a @ resid @ T @ U )
=> ( con_a @ resid @ T4 @ U ) ) ) ).
% N.Cong\<^sub>0_subst_left(1)
thf(fact_213_N_OCong_092_060_094sub_0620__subst__right_I2_J,axiom,
! [U: a,U7: a,T: a] :
( ( ( member_a @ ( resid @ U @ U7 ) @ nn )
& ( member_a @ ( resid @ U7 @ U ) @ nn ) )
=> ( ( con_a @ resid @ T @ U )
=> ( ( member_a @ ( resid @ ( resid @ ( resid @ T @ U ) @ ( resid @ U7 @ U ) ) @ ( resid @ ( resid @ T @ U7 ) @ ( resid @ U @ U7 ) ) ) @ nn )
& ( member_a @ ( resid @ ( resid @ ( resid @ T @ U7 ) @ ( resid @ U @ U7 ) ) @ ( resid @ ( resid @ T @ U ) @ ( resid @ U7 @ U ) ) ) @ nn ) ) ) ) ).
% N.Cong\<^sub>0_subst_right(2)
thf(fact_214_N_OCong_092_060_094sub_0620__subst__right_I1_J,axiom,
! [U: a,U7: a,T: a] :
( ( ( member_a @ ( resid @ U @ U7 ) @ nn )
& ( member_a @ ( resid @ U7 @ U ) @ nn ) )
=> ( ( con_a @ resid @ T @ U )
=> ( con_a @ resid @ T @ U7 ) ) ) ).
% N.Cong\<^sub>0_subst_right(1)
thf(fact_215_N_OCong_092_060_094sub_0620__imp__con,axiom,
! [T: a,T4: a] :
( ( ( member_a @ ( resid @ T @ T4 ) @ nn )
& ( member_a @ ( resid @ T4 @ T ) @ nn ) )
=> ( con_a @ resid @ T @ T4 ) ) ).
% N.Cong\<^sub>0_imp_con
thf(fact_216_N_OCong_092_060_094sub_0620__subst__Con,axiom,
! [T: a,T4: a,U: a,U7: a] :
( ( ( member_a @ ( resid @ T @ T4 ) @ nn )
& ( member_a @ ( resid @ T4 @ T ) @ nn ) )
=> ( ( ( member_a @ ( resid @ U @ U7 ) @ nn )
& ( member_a @ ( resid @ U7 @ U ) @ nn ) )
=> ( ( con_a @ resid @ T @ U )
= ( con_a @ resid @ T4 @ U7 ) ) ) ) ).
% N.Cong\<^sub>0_subst_Con
thf(fact_217_N_Oide__closed,axiom,
! [A3: a] :
( ( ide_a @ resid @ A3 )
=> ( member_a @ A3 @ nn ) ) ).
% N.ide_closed
thf(fact_218_N_Oprfx__closed,axiom,
! [U: a,T: a] :
( ( member_a @ U @ nn )
=> ( ( ide_a @ resid @ ( resid @ T @ U ) )
=> ( member_a @ T @ nn ) ) ) ).
% N.prfx_closed
thf(fact_219_R_Osources__def,axiom,
! [T: a] :
( ( sources_a @ resid @ T )
= ( collect_a
@ ^ [A: a] :
( ( ide_a @ resid @ A )
& ( con_a @ resid @ T @ A ) ) ) ) ).
% R.sources_def
thf(fact_220_N_OResid__along__normal__preserves__reflects__con,axiom,
! [U: a,T: a,T4: a] :
( ( member_a @ U @ nn )
=> ( ( ( sources_a @ resid @ T )
= ( sources_a @ resid @ U ) )
=> ( ( con_a @ resid @ ( resid @ T @ U ) @ ( resid @ T4 @ U ) )
= ( con_a @ resid @ T @ T4 ) ) ) ) ).
% N.Resid_along_normal_preserves_reflects_con
thf(fact_221_N_Oresid__along__elem__preserves__con,axiom,
! [T: a,T4: a,U: a] :
( ( con_a @ resid @ T @ T4 )
=> ( ( coinitial_a @ resid @ T @ U )
=> ( ( member_a @ U @ nn )
=> ( con_a @ resid @ ( resid @ T @ U ) @ ( resid @ T4 @ U ) ) ) ) ) ).
% N.resid_along_elem_preserves_con
thf(fact_222_N_Oforward__stable,axiom,
! [U: a,T: a] :
( ( member_a @ U @ nn )
=> ( ( coinitial_a @ resid @ T @ U )
=> ( member_a @ ( resid @ U @ T ) @ nn ) ) ) ).
% N.forward_stable
thf(fact_223_inf__right__idem,axiom,
! [X4: set_a,Y4: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ X4 @ Y4 ) @ Y4 )
= ( inf_inf_set_a @ X4 @ Y4 ) ) ).
% inf_right_idem
thf(fact_224_inf__right__idem,axiom,
! [X4: set_set_a,Y4: set_set_a] :
( ( inf_inf_set_set_a @ ( inf_inf_set_set_a @ X4 @ Y4 ) @ Y4 )
= ( inf_inf_set_set_a @ X4 @ Y4 ) ) ).
% inf_right_idem
thf(fact_225_inf_Oright__idem,axiom,
! [A3: set_a,B: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A3 @ B ) @ B )
= ( inf_inf_set_a @ A3 @ B ) ) ).
% inf.right_idem
thf(fact_226_inf_Oright__idem,axiom,
! [A3: set_set_a,B: set_set_a] :
( ( inf_inf_set_set_a @ ( inf_inf_set_set_a @ A3 @ B ) @ B )
= ( inf_inf_set_set_a @ A3 @ B ) ) ).
% inf.right_idem
thf(fact_227_inf__left__idem,axiom,
! [X4: set_a,Y4: set_a] :
( ( inf_inf_set_a @ X4 @ ( inf_inf_set_a @ X4 @ Y4 ) )
= ( inf_inf_set_a @ X4 @ Y4 ) ) ).
% inf_left_idem
thf(fact_228_inf__left__idem,axiom,
! [X4: set_set_a,Y4: set_set_a] :
( ( inf_inf_set_set_a @ X4 @ ( inf_inf_set_set_a @ X4 @ Y4 ) )
= ( inf_inf_set_set_a @ X4 @ Y4 ) ) ).
% inf_left_idem
thf(fact_229_inf_Oleft__idem,axiom,
! [A3: set_a,B: set_a] :
( ( inf_inf_set_a @ A3 @ ( inf_inf_set_a @ A3 @ B ) )
= ( inf_inf_set_a @ A3 @ B ) ) ).
% inf.left_idem
thf(fact_230_inf_Oleft__idem,axiom,
! [A3: set_set_a,B: set_set_a] :
( ( inf_inf_set_set_a @ A3 @ ( inf_inf_set_set_a @ A3 @ B ) )
= ( inf_inf_set_set_a @ A3 @ B ) ) ).
% inf.left_idem
thf(fact_231_inf__idem,axiom,
! [X4: set_a] :
( ( inf_inf_set_a @ X4 @ X4 )
= X4 ) ).
% inf_idem
thf(fact_232_inf__idem,axiom,
! [X4: set_set_a] :
( ( inf_inf_set_set_a @ X4 @ X4 )
= X4 ) ).
% inf_idem
thf(fact_233_inf_Oidem,axiom,
! [A3: set_a] :
( ( inf_inf_set_a @ A3 @ A3 )
= A3 ) ).
% inf.idem
thf(fact_234_inf_Oidem,axiom,
! [A3: set_set_a] :
( ( inf_inf_set_set_a @ A3 @ A3 )
= A3 ) ).
% inf.idem
thf(fact_235_R_OcoinitialE,axiom,
! [T: a,U: a] :
( ( coinitial_a @ resid @ T @ U )
=> ~ ( ( arr_a @ resid @ T )
=> ( ( arr_a @ resid @ U )
=> ( ( sources_a @ resid @ T )
!= ( sources_a @ resid @ U ) ) ) ) ) ).
% R.coinitialE
thf(fact_236_R_Ocoinitial__iff,axiom,
! [T: a,T4: a] :
( ( coinitial_a @ resid @ T @ T4 )
= ( ( arr_a @ resid @ T )
& ( arr_a @ resid @ T4 )
& ( ( sources_a @ resid @ T )
= ( sources_a @ resid @ T4 ) ) ) ) ).
% R.coinitial_iff
thf(fact_237_Con__witnesses,axiom,
! [T7: set_a,U2: set_a,T: a,U: a] :
( ( ( quotie8165075472272353145esid_a @ resid @ nn @ T7 @ U2 )
!= bot_bot_set_a )
=> ( ( member_a @ T @ T7 )
=> ( ( member_a @ U @ U2 )
=> ? [V3: a,W: a] :
( ( member_a @ V3 @ nn )
& ( member_a @ W @ nn )
& ( con_a @ resid @ ( resid @ T @ V3 ) @ ( resid @ U @ W ) ) ) ) ) ) ).
% Con_witnesses
thf(fact_238_N_OCong__subst_I2_J,axiom,
! [T: a,T4: a,U: a,U7: a] :
( ( normal_sub_Cong_a @ resid @ nn @ T @ T4 )
=> ( ( normal_sub_Cong_a @ resid @ nn @ U @ U7 )
=> ( ( con_a @ resid @ T @ U )
=> ( ( ( sources_a @ resid @ T4 )
= ( sources_a @ resid @ U7 ) )
=> ( normal_sub_Cong_a @ resid @ nn @ ( resid @ T @ U ) @ ( resid @ T4 @ U7 ) ) ) ) ) ) ).
% N.Cong_subst(2)
thf(fact_239_N_OCong__subst_I1_J,axiom,
! [T: a,T4: a,U: a,U7: a] :
( ( normal_sub_Cong_a @ resid @ nn @ T @ T4 )
=> ( ( normal_sub_Cong_a @ resid @ nn @ U @ U7 )
=> ( ( con_a @ resid @ T @ U )
=> ( ( ( sources_a @ resid @ T4 )
= ( sources_a @ resid @ U7 ) )
=> ( con_a @ resid @ T4 @ U7 ) ) ) ) ) ).
% N.Cong_subst(1)
thf(fact_240_N_OCong__subst__con,axiom,
! [T: a,U: a,T4: a,U7: a] :
( ( ( sources_a @ resid @ T )
= ( sources_a @ resid @ U ) )
=> ( ( ( sources_a @ resid @ T4 )
= ( sources_a @ resid @ U7 ) )
=> ( ( normal_sub_Cong_a @ resid @ nn @ T @ T4 )
=> ( ( normal_sub_Cong_a @ resid @ nn @ U @ U7 )
=> ( ( con_a @ resid @ T @ U )
= ( con_a @ resid @ T4 @ U7 ) ) ) ) ) ) ).
% N.Cong_subst_con
thf(fact_241_R_Ocon__imp__common__source,axiom,
! [T: a,U: a] :
( ( con_a @ resid @ T @ U )
=> ( ( inf_inf_set_a @ ( sources_a @ resid @ T ) @ ( sources_a @ resid @ U ) )
!= bot_bot_set_a ) ) ).
% R.con_imp_common_source
thf(fact_242_con__imp__coinitial__members__are__con,axiom,
! [T7: set_a,U2: set_a,T: a,U: a] :
( ( con_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T7 @ U2 )
=> ( ( member_a @ T @ T7 )
=> ( ( member_a @ U @ U2 )
=> ( ( ( sources_a @ resid @ T )
= ( sources_a @ resid @ U ) )
=> ( con_a @ resid @ T @ U ) ) ) ) ) ).
% con_imp_coinitial_members_are_con
thf(fact_243_joinable__implies__con,axiom,
! [T: set_a,U: set_a] :
( ( joinable_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U )
=> ( con_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U ) ) ).
% joinable_implies_con
thf(fact_244_R_OideI,axiom,
! [A3: a] :
( ( con_a @ resid @ A3 @ A3 )
=> ( ( ( resid @ A3 @ A3 )
= A3 )
=> ( ide_a @ resid @ A3 ) ) ) ).
% R.ideI
thf(fact_245_inf_Obounded__iff,axiom,
! [A3: set_set_a,B: set_set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A3 @ ( inf_inf_set_set_a @ B @ C ) )
= ( ( ord_le3724670747650509150_set_a @ A3 @ B )
& ( ord_le3724670747650509150_set_a @ A3 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_246_inf_Obounded__iff,axiom,
! [A3: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A3 @ ( inf_inf_set_a @ B @ C ) )
= ( ( ord_less_eq_set_a @ A3 @ B )
& ( ord_less_eq_set_a @ A3 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_247_le__inf__iff,axiom,
! [X4: set_set_a,Y4: set_set_a,Z: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X4 @ ( inf_inf_set_set_a @ Y4 @ Z ) )
= ( ( ord_le3724670747650509150_set_a @ X4 @ Y4 )
& ( ord_le3724670747650509150_set_a @ X4 @ Z ) ) ) ).
% le_inf_iff
thf(fact_248_le__inf__iff,axiom,
! [X4: set_a,Y4: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ X4 @ ( inf_inf_set_a @ Y4 @ Z ) )
= ( ( ord_less_eq_set_a @ X4 @ Y4 )
& ( ord_less_eq_set_a @ X4 @ Z ) ) ) ).
% le_inf_iff
thf(fact_249_sources__eqI,axiom,
! [T: set_a,T4: set_a] :
( ( ( inf_inf_set_set_a @ ( sources_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T ) @ ( sources_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T4 ) )
!= bot_bot_set_set_a )
=> ( ( sources_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T )
= ( sources_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T4 ) ) ) ).
% sources_eqI
thf(fact_250_R_Oin__sourcesI,axiom,
! [A3: a,T: a] :
( ( ide_a @ resid @ A3 )
=> ( ( con_a @ resid @ T @ A3 )
=> ( member_a @ A3 @ ( sources_a @ resid @ T ) ) ) ) ).
% R.in_sourcesI
thf(fact_251_R_OarrI,axiom,
! [T: a] :
( ( con_a @ resid @ T @ T )
=> ( arr_a @ resid @ T ) ) ).
% R.arrI
thf(fact_252_R_OcoinitialI,axiom,
! [T: a,U: a] :
( ( arr_a @ resid @ T )
=> ( ( ( sources_a @ resid @ T )
= ( sources_a @ resid @ U ) )
=> ( coinitial_a @ resid @ T @ U ) ) ) ).
% R.coinitialI
thf(fact_253_rts_Ocoterminal_Ocong,axiom,
coterminal_set_a = coterminal_set_a ).
% rts.coterminal.cong
thf(fact_254_rts_Ocoterminal_Ocong,axiom,
coterminal_a = coterminal_a ).
% rts.coterminal.cong
thf(fact_255_rts_Ojoinable_Ocong,axiom,
joinable_set_a = joinable_set_a ).
% rts.joinable.cong
thf(fact_256_rts_Ojoinable_Ocong,axiom,
joinable_a = joinable_a ).
% rts.joinable.cong
thf(fact_257_inf_OcoboundedI2,axiom,
! [B: set_set_a,C: set_set_a,A3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B @ C )
=> ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A3 @ B ) @ C ) ) ).
% inf.coboundedI2
thf(fact_258_inf_OcoboundedI2,axiom,
! [B: set_a,C: set_a,A3: set_a] :
( ( ord_less_eq_set_a @ B @ C )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A3 @ B ) @ C ) ) ).
% inf.coboundedI2
thf(fact_259_inf_OcoboundedI1,axiom,
! [A3: set_set_a,C: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A3 @ C )
=> ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A3 @ B ) @ C ) ) ).
% inf.coboundedI1
thf(fact_260_inf_OcoboundedI1,axiom,
! [A3: set_a,C: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A3 @ C )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A3 @ B ) @ C ) ) ).
% inf.coboundedI1
thf(fact_261_inf_Oabsorb__iff2,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [B2: set_set_a,A: set_set_a] :
( ( inf_inf_set_set_a @ A @ B2 )
= B2 ) ) ) ).
% inf.absorb_iff2
thf(fact_262_inf_Oabsorb__iff2,axiom,
( ord_less_eq_set_a
= ( ^ [B2: set_a,A: set_a] :
( ( inf_inf_set_a @ A @ B2 )
= B2 ) ) ) ).
% inf.absorb_iff2
thf(fact_263_inf_Oabsorb__iff1,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A: set_set_a,B2: set_set_a] :
( ( inf_inf_set_set_a @ A @ B2 )
= A ) ) ) ).
% inf.absorb_iff1
thf(fact_264_inf_Oabsorb__iff1,axiom,
( ord_less_eq_set_a
= ( ^ [A: set_a,B2: set_a] :
( ( inf_inf_set_a @ A @ B2 )
= A ) ) ) ).
% inf.absorb_iff1
thf(fact_265_inf_Ocobounded2,axiom,
! [A3: set_set_a,B: set_set_a] : ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A3 @ B ) @ B ) ).
% inf.cobounded2
thf(fact_266_inf_Ocobounded2,axiom,
! [A3: set_a,B: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A3 @ B ) @ B ) ).
% inf.cobounded2
thf(fact_267_inf_Ocobounded1,axiom,
! [A3: set_set_a,B: set_set_a] : ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A3 @ B ) @ A3 ) ).
% inf.cobounded1
thf(fact_268_inf_Ocobounded1,axiom,
! [A3: set_a,B: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A3 @ B ) @ A3 ) ).
% inf.cobounded1
thf(fact_269_inf_Oorder__iff,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A: set_set_a,B2: set_set_a] :
( A
= ( inf_inf_set_set_a @ A @ B2 ) ) ) ) ).
% inf.order_iff
thf(fact_270_inf_Oorder__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A: set_a,B2: set_a] :
( A
= ( inf_inf_set_a @ A @ B2 ) ) ) ) ).
% inf.order_iff
thf(fact_271_inf__greatest,axiom,
! [X4: set_set_a,Y4: set_set_a,Z: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X4 @ Y4 )
=> ( ( ord_le3724670747650509150_set_a @ X4 @ Z )
=> ( ord_le3724670747650509150_set_a @ X4 @ ( inf_inf_set_set_a @ Y4 @ Z ) ) ) ) ).
% inf_greatest
thf(fact_272_inf__greatest,axiom,
! [X4: set_a,Y4: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y4 )
=> ( ( ord_less_eq_set_a @ X4 @ Z )
=> ( ord_less_eq_set_a @ X4 @ ( inf_inf_set_a @ Y4 @ Z ) ) ) ) ).
% inf_greatest
thf(fact_273_inf_OboundedI,axiom,
! [A3: set_set_a,B: set_set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A3 @ B )
=> ( ( ord_le3724670747650509150_set_a @ A3 @ C )
=> ( ord_le3724670747650509150_set_a @ A3 @ ( inf_inf_set_set_a @ B @ C ) ) ) ) ).
% inf.boundedI
thf(fact_274_inf_OboundedI,axiom,
! [A3: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A3 @ B )
=> ( ( ord_less_eq_set_a @ A3 @ C )
=> ( ord_less_eq_set_a @ A3 @ ( inf_inf_set_a @ B @ C ) ) ) ) ).
% inf.boundedI
thf(fact_275_inf_OboundedE,axiom,
! [A3: set_set_a,B: set_set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A3 @ ( inf_inf_set_set_a @ B @ C ) )
=> ~ ( ( ord_le3724670747650509150_set_a @ A3 @ B )
=> ~ ( ord_le3724670747650509150_set_a @ A3 @ C ) ) ) ).
% inf.boundedE
thf(fact_276_inf_OboundedE,axiom,
! [A3: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A3 @ ( inf_inf_set_a @ B @ C ) )
=> ~ ( ( ord_less_eq_set_a @ A3 @ B )
=> ~ ( ord_less_eq_set_a @ A3 @ C ) ) ) ).
% inf.boundedE
thf(fact_277_inf__absorb2,axiom,
! [Y4: set_set_a,X4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ Y4 @ X4 )
=> ( ( inf_inf_set_set_a @ X4 @ Y4 )
= Y4 ) ) ).
% inf_absorb2
thf(fact_278_inf__absorb2,axiom,
! [Y4: set_a,X4: set_a] :
( ( ord_less_eq_set_a @ Y4 @ X4 )
=> ( ( inf_inf_set_a @ X4 @ Y4 )
= Y4 ) ) ).
% inf_absorb2
thf(fact_279_inf__absorb1,axiom,
! [X4: set_set_a,Y4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X4 @ Y4 )
=> ( ( inf_inf_set_set_a @ X4 @ Y4 )
= X4 ) ) ).
% inf_absorb1
thf(fact_280_inf__absorb1,axiom,
! [X4: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y4 )
=> ( ( inf_inf_set_a @ X4 @ Y4 )
= X4 ) ) ).
% inf_absorb1
thf(fact_281_inf_Oabsorb2,axiom,
! [B: set_set_a,A3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B @ A3 )
=> ( ( inf_inf_set_set_a @ A3 @ B )
= B ) ) ).
% inf.absorb2
thf(fact_282_inf_Oabsorb2,axiom,
! [B: set_a,A3: set_a] :
( ( ord_less_eq_set_a @ B @ A3 )
=> ( ( inf_inf_set_a @ A3 @ B )
= B ) ) ).
% inf.absorb2
thf(fact_283_inf_Oabsorb1,axiom,
! [A3: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A3 @ B )
=> ( ( inf_inf_set_set_a @ A3 @ B )
= A3 ) ) ).
% inf.absorb1
thf(fact_284_inf_Oabsorb1,axiom,
! [A3: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A3 @ B )
=> ( ( inf_inf_set_a @ A3 @ B )
= A3 ) ) ).
% inf.absorb1
thf(fact_285_le__iff__inf,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [X2: set_set_a,Y2: set_set_a] :
( ( inf_inf_set_set_a @ X2 @ Y2 )
= X2 ) ) ) ).
% le_iff_inf
thf(fact_286_le__iff__inf,axiom,
( ord_less_eq_set_a
= ( ^ [X2: set_a,Y2: set_a] :
( ( inf_inf_set_a @ X2 @ Y2 )
= X2 ) ) ) ).
% le_iff_inf
thf(fact_287_inf__unique,axiom,
! [F: set_set_a > set_set_a > set_set_a,X4: set_set_a,Y4: set_set_a] :
( ! [X: set_set_a,Y3: set_set_a] : ( ord_le3724670747650509150_set_a @ ( F @ X @ Y3 ) @ X )
=> ( ! [X: set_set_a,Y3: set_set_a] : ( ord_le3724670747650509150_set_a @ ( F @ X @ Y3 ) @ Y3 )
=> ( ! [X: set_set_a,Y3: set_set_a,Z2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X @ Y3 )
=> ( ( ord_le3724670747650509150_set_a @ X @ Z2 )
=> ( ord_le3724670747650509150_set_a @ X @ ( F @ Y3 @ Z2 ) ) ) )
=> ( ( inf_inf_set_set_a @ X4 @ Y4 )
= ( F @ X4 @ Y4 ) ) ) ) ) ).
% inf_unique
thf(fact_288_inf__unique,axiom,
! [F: set_a > set_a > set_a,X4: set_a,Y4: set_a] :
( ! [X: set_a,Y3: set_a] : ( ord_less_eq_set_a @ ( F @ X @ Y3 ) @ X )
=> ( ! [X: set_a,Y3: set_a] : ( ord_less_eq_set_a @ ( F @ X @ Y3 ) @ Y3 )
=> ( ! [X: set_a,Y3: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ X @ Y3 )
=> ( ( ord_less_eq_set_a @ X @ Z2 )
=> ( ord_less_eq_set_a @ X @ ( F @ Y3 @ Z2 ) ) ) )
=> ( ( inf_inf_set_a @ X4 @ Y4 )
= ( F @ X4 @ Y4 ) ) ) ) ) ).
% inf_unique
thf(fact_289_inf_OorderI,axiom,
! [A3: set_set_a,B: set_set_a] :
( ( A3
= ( inf_inf_set_set_a @ A3 @ B ) )
=> ( ord_le3724670747650509150_set_a @ A3 @ B ) ) ).
% inf.orderI
thf(fact_290_inf_OorderI,axiom,
! [A3: set_a,B: set_a] :
( ( A3
= ( inf_inf_set_a @ A3 @ B ) )
=> ( ord_less_eq_set_a @ A3 @ B ) ) ).
% inf.orderI
thf(fact_291_inf_OorderE,axiom,
! [A3: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A3 @ B )
=> ( A3
= ( inf_inf_set_set_a @ A3 @ B ) ) ) ).
% inf.orderE
thf(fact_292_inf_OorderE,axiom,
! [A3: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A3 @ B )
=> ( A3
= ( inf_inf_set_a @ A3 @ B ) ) ) ).
% inf.orderE
thf(fact_293_le__infI2,axiom,
! [B: set_set_a,X4: set_set_a,A3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B @ X4 )
=> ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A3 @ B ) @ X4 ) ) ).
% le_infI2
thf(fact_294_le__infI2,axiom,
! [B: set_a,X4: set_a,A3: set_a] :
( ( ord_less_eq_set_a @ B @ X4 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A3 @ B ) @ X4 ) ) ).
% le_infI2
thf(fact_295_le__infI1,axiom,
! [A3: set_set_a,X4: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A3 @ X4 )
=> ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A3 @ B ) @ X4 ) ) ).
% le_infI1
thf(fact_296_le__infI1,axiom,
! [A3: set_a,X4: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A3 @ X4 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A3 @ B ) @ X4 ) ) ).
% le_infI1
thf(fact_297_inf__mono,axiom,
! [A3: set_set_a,C: set_set_a,B: set_set_a,D: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A3 @ C )
=> ( ( ord_le3724670747650509150_set_a @ B @ D )
=> ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A3 @ B ) @ ( inf_inf_set_set_a @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_298_inf__mono,axiom,
! [A3: set_a,C: set_a,B: set_a,D: set_a] :
( ( ord_less_eq_set_a @ A3 @ C )
=> ( ( ord_less_eq_set_a @ B @ D )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A3 @ B ) @ ( inf_inf_set_a @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_299_le__infI,axiom,
! [X4: set_set_a,A3: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X4 @ A3 )
=> ( ( ord_le3724670747650509150_set_a @ X4 @ B )
=> ( ord_le3724670747650509150_set_a @ X4 @ ( inf_inf_set_set_a @ A3 @ B ) ) ) ) ).
% le_infI
thf(fact_300_le__infI,axiom,
! [X4: set_a,A3: set_a,B: set_a] :
( ( ord_less_eq_set_a @ X4 @ A3 )
=> ( ( ord_less_eq_set_a @ X4 @ B )
=> ( ord_less_eq_set_a @ X4 @ ( inf_inf_set_a @ A3 @ B ) ) ) ) ).
% le_infI
thf(fact_301_le__infE,axiom,
! [X4: set_set_a,A3: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X4 @ ( inf_inf_set_set_a @ A3 @ B ) )
=> ~ ( ( ord_le3724670747650509150_set_a @ X4 @ A3 )
=> ~ ( ord_le3724670747650509150_set_a @ X4 @ B ) ) ) ).
% le_infE
thf(fact_302_le__infE,axiom,
! [X4: set_a,A3: set_a,B: set_a] :
( ( ord_less_eq_set_a @ X4 @ ( inf_inf_set_a @ A3 @ B ) )
=> ~ ( ( ord_less_eq_set_a @ X4 @ A3 )
=> ~ ( ord_less_eq_set_a @ X4 @ B ) ) ) ).
% le_infE
thf(fact_303_inf__le2,axiom,
! [X4: set_set_a,Y4: set_set_a] : ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ X4 @ Y4 ) @ Y4 ) ).
% inf_le2
thf(fact_304_inf__le2,axiom,
! [X4: set_a,Y4: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X4 @ Y4 ) @ Y4 ) ).
% inf_le2
thf(fact_305_inf__le1,axiom,
! [X4: set_set_a,Y4: set_set_a] : ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ X4 @ Y4 ) @ X4 ) ).
% inf_le1
thf(fact_306_inf__le1,axiom,
! [X4: set_a,Y4: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X4 @ Y4 ) @ X4 ) ).
% inf_le1
thf(fact_307_inf__sup__ord_I1_J,axiom,
! [X4: set_set_a,Y4: set_set_a] : ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ X4 @ Y4 ) @ X4 ) ).
% inf_sup_ord(1)
thf(fact_308_inf__sup__ord_I1_J,axiom,
! [X4: set_a,Y4: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X4 @ Y4 ) @ X4 ) ).
% inf_sup_ord(1)
thf(fact_309_inf__sup__ord_I2_J,axiom,
! [X4: set_set_a,Y4: set_set_a] : ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ X4 @ Y4 ) @ Y4 ) ).
% inf_sup_ord(2)
thf(fact_310_inf__sup__ord_I2_J,axiom,
! [X4: set_a,Y4: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X4 @ Y4 ) @ Y4 ) ).
% inf_sup_ord(2)
thf(fact_311_inf__left__commute,axiom,
! [X4: set_a,Y4: set_a,Z: set_a] :
( ( inf_inf_set_a @ X4 @ ( inf_inf_set_a @ Y4 @ Z ) )
= ( inf_inf_set_a @ Y4 @ ( inf_inf_set_a @ X4 @ Z ) ) ) ).
% inf_left_commute
thf(fact_312_inf__left__commute,axiom,
! [X4: set_set_a,Y4: set_set_a,Z: set_set_a] :
( ( inf_inf_set_set_a @ X4 @ ( inf_inf_set_set_a @ Y4 @ Z ) )
= ( inf_inf_set_set_a @ Y4 @ ( inf_inf_set_set_a @ X4 @ Z ) ) ) ).
% inf_left_commute
thf(fact_313_inf_Oleft__commute,axiom,
! [B: set_a,A3: set_a,C: set_a] :
( ( inf_inf_set_a @ B @ ( inf_inf_set_a @ A3 @ C ) )
= ( inf_inf_set_a @ A3 @ ( inf_inf_set_a @ B @ C ) ) ) ).
% inf.left_commute
thf(fact_314_inf_Oleft__commute,axiom,
! [B: set_set_a,A3: set_set_a,C: set_set_a] :
( ( inf_inf_set_set_a @ B @ ( inf_inf_set_set_a @ A3 @ C ) )
= ( inf_inf_set_set_a @ A3 @ ( inf_inf_set_set_a @ B @ C ) ) ) ).
% inf.left_commute
thf(fact_315_boolean__algebra__cancel_Oinf2,axiom,
! [B3: set_a,K: set_a,B: set_a,A3: set_a] :
( ( B3
= ( inf_inf_set_a @ K @ B ) )
=> ( ( inf_inf_set_a @ A3 @ B3 )
= ( inf_inf_set_a @ K @ ( inf_inf_set_a @ A3 @ B ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_316_boolean__algebra__cancel_Oinf2,axiom,
! [B3: set_set_a,K: set_set_a,B: set_set_a,A3: set_set_a] :
( ( B3
= ( inf_inf_set_set_a @ K @ B ) )
=> ( ( inf_inf_set_set_a @ A3 @ B3 )
= ( inf_inf_set_set_a @ K @ ( inf_inf_set_set_a @ A3 @ B ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_317_boolean__algebra__cancel_Oinf1,axiom,
! [A5: set_a,K: set_a,A3: set_a,B: set_a] :
( ( A5
= ( inf_inf_set_a @ K @ A3 ) )
=> ( ( inf_inf_set_a @ A5 @ B )
= ( inf_inf_set_a @ K @ ( inf_inf_set_a @ A3 @ B ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_318_boolean__algebra__cancel_Oinf1,axiom,
! [A5: set_set_a,K: set_set_a,A3: set_set_a,B: set_set_a] :
( ( A5
= ( inf_inf_set_set_a @ K @ A3 ) )
=> ( ( inf_inf_set_set_a @ A5 @ B )
= ( inf_inf_set_set_a @ K @ ( inf_inf_set_set_a @ A3 @ B ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_319_inf__commute,axiom,
( inf_inf_set_a
= ( ^ [X2: set_a,Y2: set_a] : ( inf_inf_set_a @ Y2 @ X2 ) ) ) ).
% inf_commute
thf(fact_320_inf__commute,axiom,
( inf_inf_set_set_a
= ( ^ [X2: set_set_a,Y2: set_set_a] : ( inf_inf_set_set_a @ Y2 @ X2 ) ) ) ).
% inf_commute
thf(fact_321_inf_Ocommute,axiom,
( inf_inf_set_a
= ( ^ [A: set_a,B2: set_a] : ( inf_inf_set_a @ B2 @ A ) ) ) ).
% inf.commute
thf(fact_322_inf_Ocommute,axiom,
( inf_inf_set_set_a
= ( ^ [A: set_set_a,B2: set_set_a] : ( inf_inf_set_set_a @ B2 @ A ) ) ) ).
% inf.commute
thf(fact_323_inf__assoc,axiom,
! [X4: set_a,Y4: set_a,Z: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ X4 @ Y4 ) @ Z )
= ( inf_inf_set_a @ X4 @ ( inf_inf_set_a @ Y4 @ Z ) ) ) ).
% inf_assoc
thf(fact_324_inf__assoc,axiom,
! [X4: set_set_a,Y4: set_set_a,Z: set_set_a] :
( ( inf_inf_set_set_a @ ( inf_inf_set_set_a @ X4 @ Y4 ) @ Z )
= ( inf_inf_set_set_a @ X4 @ ( inf_inf_set_set_a @ Y4 @ Z ) ) ) ).
% inf_assoc
thf(fact_325_inf_Oassoc,axiom,
! [A3: set_a,B: set_a,C: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A3 @ B ) @ C )
= ( inf_inf_set_a @ A3 @ ( inf_inf_set_a @ B @ C ) ) ) ).
% inf.assoc
thf(fact_326_inf_Oassoc,axiom,
! [A3: set_set_a,B: set_set_a,C: set_set_a] :
( ( inf_inf_set_set_a @ ( inf_inf_set_set_a @ A3 @ B ) @ C )
= ( inf_inf_set_set_a @ A3 @ ( inf_inf_set_set_a @ B @ C ) ) ) ).
% inf.assoc
thf(fact_327_inf__sup__aci_I1_J,axiom,
( inf_inf_set_a
= ( ^ [X2: set_a,Y2: set_a] : ( inf_inf_set_a @ Y2 @ X2 ) ) ) ).
% inf_sup_aci(1)
thf(fact_328_inf__sup__aci_I1_J,axiom,
( inf_inf_set_set_a
= ( ^ [X2: set_set_a,Y2: set_set_a] : ( inf_inf_set_set_a @ Y2 @ X2 ) ) ) ).
% inf_sup_aci(1)
thf(fact_329_inf__sup__aci_I2_J,axiom,
! [X4: set_a,Y4: set_a,Z: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ X4 @ Y4 ) @ Z )
= ( inf_inf_set_a @ X4 @ ( inf_inf_set_a @ Y4 @ Z ) ) ) ).
% inf_sup_aci(2)
thf(fact_330_inf__sup__aci_I2_J,axiom,
! [X4: set_set_a,Y4: set_set_a,Z: set_set_a] :
( ( inf_inf_set_set_a @ ( inf_inf_set_set_a @ X4 @ Y4 ) @ Z )
= ( inf_inf_set_set_a @ X4 @ ( inf_inf_set_set_a @ Y4 @ Z ) ) ) ).
% inf_sup_aci(2)
thf(fact_331_inf__sup__aci_I3_J,axiom,
! [X4: set_a,Y4: set_a,Z: set_a] :
( ( inf_inf_set_a @ X4 @ ( inf_inf_set_a @ Y4 @ Z ) )
= ( inf_inf_set_a @ Y4 @ ( inf_inf_set_a @ X4 @ Z ) ) ) ).
% inf_sup_aci(3)
thf(fact_332_inf__sup__aci_I3_J,axiom,
! [X4: set_set_a,Y4: set_set_a,Z: set_set_a] :
( ( inf_inf_set_set_a @ X4 @ ( inf_inf_set_set_a @ Y4 @ Z ) )
= ( inf_inf_set_set_a @ Y4 @ ( inf_inf_set_set_a @ X4 @ Z ) ) ) ).
% inf_sup_aci(3)
thf(fact_333_inf__sup__aci_I4_J,axiom,
! [X4: set_a,Y4: set_a] :
( ( inf_inf_set_a @ X4 @ ( inf_inf_set_a @ X4 @ Y4 ) )
= ( inf_inf_set_a @ X4 @ Y4 ) ) ).
% inf_sup_aci(4)
thf(fact_334_inf__sup__aci_I4_J,axiom,
! [X4: set_set_a,Y4: set_set_a] :
( ( inf_inf_set_set_a @ X4 @ ( inf_inf_set_set_a @ X4 @ Y4 ) )
= ( inf_inf_set_set_a @ X4 @ Y4 ) ) ).
% inf_sup_aci(4)
thf(fact_335_quotient__by__coherent__normal_Oide__char_H,axiom,
! [Resid: a > a > a,NN: set_a,A7: set_a] :
( ( quotie3282664541148387094rmal_a @ Resid @ NN )
=> ( ( ide_set_a @ ( quotie8165075472272353145esid_a @ Resid @ NN ) @ A7 )
= ( ( arr_set_a @ ( quotie8165075472272353145esid_a @ Resid @ NN ) @ A7 )
& ( ord_less_eq_set_a @ A7 @ NN ) ) ) ) ).
% quotient_by_coherent_normal.ide_char'
thf(fact_336_Int__subset__iff,axiom,
! [C2: set_set_a,A5: set_set_a,B3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ C2 @ ( inf_inf_set_set_a @ A5 @ B3 ) )
= ( ( ord_le3724670747650509150_set_a @ C2 @ A5 )
& ( ord_le3724670747650509150_set_a @ C2 @ B3 ) ) ) ).
% Int_subset_iff
thf(fact_337_Int__subset__iff,axiom,
! [C2: set_a,A5: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ C2 @ ( inf_inf_set_a @ A5 @ B3 ) )
= ( ( ord_less_eq_set_a @ C2 @ A5 )
& ( ord_less_eq_set_a @ C2 @ B3 ) ) ) ).
% Int_subset_iff
thf(fact_338_empty__subsetI,axiom,
! [A5: set_set_a] : ( ord_le3724670747650509150_set_a @ bot_bot_set_set_a @ A5 ) ).
% empty_subsetI
thf(fact_339_empty__subsetI,axiom,
! [A5: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A5 ) ).
% empty_subsetI
thf(fact_340_subset__empty,axiom,
! [A5: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A5 @ bot_bot_set_set_a )
= ( A5 = bot_bot_set_set_a ) ) ).
% subset_empty
thf(fact_341_subset__empty,axiom,
! [A5: set_a] :
( ( ord_less_eq_set_a @ A5 @ bot_bot_set_a )
= ( A5 = bot_bot_set_a ) ) ).
% subset_empty
thf(fact_342_composable__imp__seq,axiom,
! [T: set_a,U: set_a] :
( ( composable_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U )
=> ( seq_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U ) ) ).
% composable_imp_seq
thf(fact_343_N_OCong__class__eqI_H,axiom,
! [T7: set_a,U2: set_a] :
( ( normal8595587647932138008lass_a @ resid @ nn @ T7 )
=> ( ( normal8595587647932138008lass_a @ resid @ nn @ U2 )
=> ( ( ( inf_inf_set_a @ T7 @ U2 )
!= bot_bot_set_a )
=> ( T7 = U2 ) ) ) ) ).
% N.Cong_class_eqI'
thf(fact_344_cong__respects__seq,axiom,
! [T: set_a,U: set_a,T4: set_a,U7: set_a] :
( ( seq_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U )
=> ( ( ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ T @ T4 ) )
& ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ T4 @ T ) ) )
=> ( ( ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ U @ U7 ) )
& ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ U7 @ U ) ) )
=> ( seq_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T4 @ U7 ) ) ) ) ).
% cong_respects_seq
thf(fact_345_N_Ois__Cong__classI,axiom,
! [T: a] :
( ( arr_a @ resid @ T )
=> ( normal8595587647932138008lass_a @ resid @ nn @ ( normal7408713899360725774lass_a @ resid @ nn @ T ) ) ) ).
% N.is_Cong_classI
thf(fact_346_residuation__axioms,axiom,
residuation_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) ).
% residuation_axioms
thf(fact_347_coterminal__def,axiom,
! [T: set_a,U: set_a] :
( ( coterminal_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U )
= ( ( inf_inf_set_set_a @ ( targets_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T ) @ ( targets_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ U ) )
!= bot_bot_set_set_a ) ) ).
% coterminal_def
thf(fact_348_R_Ocong__implies__coterminal,axiom,
! [U: a,U7: a] :
( ( ( ide_a @ resid @ ( resid @ U @ U7 ) )
& ( ide_a @ resid @ ( resid @ U7 @ U ) ) )
=> ( coterminal_a @ resid @ U @ U7 ) ) ).
% R.cong_implies_coterminal
thf(fact_349_R_Ojoinable__implies__con,axiom,
! [T: a,U: a] :
( ( joinable_a @ resid @ T @ U )
=> ( con_a @ resid @ T @ U ) ) ).
% R.joinable_implies_con
thf(fact_350_R_Ojoinable__implies__coinitial,axiom,
! [T: a,U: a] :
( ( joinable_a @ resid @ T @ U )
=> ( coinitial_a @ resid @ T @ U ) ) ).
% R.joinable_implies_coinitial
thf(fact_351_empty__iff,axiom,
! [C: a] :
~ ( member_a @ C @ bot_bot_set_a ) ).
% empty_iff
thf(fact_352_empty__iff,axiom,
! [C: set_a] :
~ ( member_set_a @ C @ bot_bot_set_set_a ) ).
% empty_iff
thf(fact_353_all__not__in__conv,axiom,
! [A5: set_a] :
( ( ! [X2: a] :
~ ( member_a @ X2 @ A5 ) )
= ( A5 = bot_bot_set_a ) ) ).
% all_not_in_conv
thf(fact_354_all__not__in__conv,axiom,
! [A5: set_set_a] :
( ( ! [X2: set_a] :
~ ( member_set_a @ X2 @ A5 ) )
= ( A5 = bot_bot_set_set_a ) ) ).
% all_not_in_conv
thf(fact_355_Collect__empty__eq,axiom,
! [P: a > $o] :
( ( ( collect_a @ P )
= bot_bot_set_a )
= ( ! [X2: a] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_356_Collect__empty__eq,axiom,
! [P: set_a > $o] :
( ( ( collect_set_a @ P )
= bot_bot_set_set_a )
= ( ! [X2: set_a] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_357_empty__Collect__eq,axiom,
! [P: a > $o] :
( ( bot_bot_set_a
= ( collect_a @ P ) )
= ( ! [X2: a] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_358_empty__Collect__eq,axiom,
! [P: set_a > $o] :
( ( bot_bot_set_set_a
= ( collect_set_a @ P ) )
= ( ! [X2: set_a] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_359_subsetI,axiom,
! [A5: set_set_a,B3: set_set_a] :
( ! [X: set_a] :
( ( member_set_a @ X @ A5 )
=> ( member_set_a @ X @ B3 ) )
=> ( ord_le3724670747650509150_set_a @ A5 @ B3 ) ) ).
% subsetI
thf(fact_360_subsetI,axiom,
! [A5: set_a,B3: set_a] :
( ! [X: a] :
( ( member_a @ X @ A5 )
=> ( member_a @ X @ B3 ) )
=> ( ord_less_eq_set_a @ A5 @ B3 ) ) ).
% subsetI
thf(fact_361_subset__antisym,axiom,
! [A5: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A5 @ B3 )
=> ( ( ord_less_eq_set_a @ B3 @ A5 )
=> ( A5 = B3 ) ) ) ).
% subset_antisym
thf(fact_362_IntI,axiom,
! [C: a,A5: set_a,B3: set_a] :
( ( member_a @ C @ A5 )
=> ( ( member_a @ C @ B3 )
=> ( member_a @ C @ ( inf_inf_set_a @ A5 @ B3 ) ) ) ) ).
% IntI
thf(fact_363_IntI,axiom,
! [C: set_a,A5: set_set_a,B3: set_set_a] :
( ( member_set_a @ C @ A5 )
=> ( ( member_set_a @ C @ B3 )
=> ( member_set_a @ C @ ( inf_inf_set_set_a @ A5 @ B3 ) ) ) ) ).
% IntI
thf(fact_364_Int__iff,axiom,
! [C: a,A5: set_a,B3: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A5 @ B3 ) )
= ( ( member_a @ C @ A5 )
& ( member_a @ C @ B3 ) ) ) ).
% Int_iff
thf(fact_365_Int__iff,axiom,
! [C: set_a,A5: set_set_a,B3: set_set_a] :
( ( member_set_a @ C @ ( inf_inf_set_set_a @ A5 @ B3 ) )
= ( ( member_set_a @ C @ A5 )
& ( member_set_a @ C @ B3 ) ) ) ).
% Int_iff
thf(fact_366_N_OCong__class__memb__is__arr,axiom,
! [T7: set_a,T: a] :
( ( normal8595587647932138008lass_a @ resid @ nn @ T7 )
=> ( ( member_a @ T @ T7 )
=> ( arr_a @ resid @ T ) ) ) ).
% N.Cong_class_memb_is_arr
thf(fact_367_N_OCong__class__membs__are__Cong,axiom,
! [T7: set_a,T: a,T4: a] :
( ( normal8595587647932138008lass_a @ resid @ nn @ T7 )
=> ( ( member_a @ T @ T7 )
=> ( ( member_a @ T4 @ T7 )
=> ( normal_sub_Cong_a @ resid @ nn @ T @ T4 ) ) ) ) ).
% N.Cong_class_membs_are_Cong
thf(fact_368_N_OCong__class__is__nonempty,axiom,
! [T7: set_a] :
( ( normal8595587647932138008lass_a @ resid @ nn @ T7 )
=> ( T7 != bot_bot_set_a ) ) ).
% N.Cong_class_is_nonempty
thf(fact_369_N_Ois__Cong__class__def,axiom,
! [T7: set_a] :
( ( normal8595587647932138008lass_a @ resid @ nn @ T7 )
= ( ? [T6: a] :
( ( member_a @ T6 @ T7 )
& ( T7
= ( normal7408713899360725774lass_a @ resid @ nn @ T6 ) ) ) ) ) ).
% N.is_Cong_class_def
thf(fact_370_targets__cong__closed,axiom,
! [B: set_a,T: set_a,B4: set_a] :
( ( member_set_a @ B @ ( targets_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T ) )
=> ( ( ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ B @ B4 ) )
& ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ B4 @ B ) ) )
=> ( member_set_a @ B4 @ ( targets_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T ) ) ) ) ).
% targets_cong_closed
thf(fact_371_targets__are__cong,axiom,
! [B: set_a,T: set_a,B4: set_a] :
( ( member_set_a @ B @ ( targets_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T ) )
=> ( ( member_set_a @ B4 @ ( targets_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T ) )
=> ( ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ B @ B4 ) )
& ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ B4 @ B ) ) ) ) ) ).
% targets_are_cong
thf(fact_372_target__is__ide,axiom,
! [A3: set_a,T: set_a] :
( ( member_set_a @ A3 @ ( targets_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T ) )
=> ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ A3 ) ) ).
% target_is_ide
thf(fact_373_targets__resid__sym,axiom,
! [T: set_a,U: set_a] :
( ( con_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U )
=> ( ( targets_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ T @ U ) )
= ( targets_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ U @ T ) ) ) ) ).
% targets_resid_sym
thf(fact_374_targets__are__con,axiom,
! [B: set_a,T: set_a,B4: set_a] :
( ( member_set_a @ B @ ( targets_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T ) )
=> ( ( member_set_a @ B4 @ ( targets_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T ) )
=> ( con_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ B @ B4 ) ) ) ).
% targets_are_con
thf(fact_375_arr__has__un__target,axiom,
! [T: set_a] :
( ( arr_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T )
=> ? [X: set_a] :
( ( member_set_a @ X @ ( targets_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T ) )
& ! [Y: set_a] :
( ( member_set_a @ Y @ ( targets_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T ) )
=> ( Y = X ) ) ) ) ).
% arr_has_un_target
thf(fact_376_resid__source__in__targets,axiom,
! [A3: set_a,T: set_a] :
( ( member_set_a @ A3 @ ( sources_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T ) )
=> ( member_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn @ A3 @ T ) @ ( targets_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T ) ) ) ).
% resid_source_in_targets
thf(fact_377_arr__char,axiom,
! [T7: set_a] :
( ( arr_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T7 )
= ( normal8595587647932138008lass_a @ resid @ nn @ T7 ) ) ).
% arr_char
thf(fact_378_is__Cong__class__Resid,axiom,
! [T7: set_a,U2: set_a] :
( ( ( quotie8165075472272353145esid_a @ resid @ nn @ T7 @ U2 )
!= bot_bot_set_a )
=> ( normal8595587647932138008lass_a @ resid @ nn @ ( quotie8165075472272353145esid_a @ resid @ nn @ T7 @ U2 ) ) ) ).
% is_Cong_class_Resid
thf(fact_379_N_Ois__Cong__classE,axiom,
! [T7: set_a] :
( ( normal8595587647932138008lass_a @ resid @ nn @ T7 )
=> ~ ( ( T7 != bot_bot_set_a )
=> ( ! [T2: a] :
( ( member_a @ T2 @ T7 )
=> ! [T8: a] :
( ( member_a @ T8 @ T7 )
=> ( normal_sub_Cong_a @ resid @ nn @ T2 @ T8 ) ) )
=> ~ ! [T2: a] :
( ( member_a @ T2 @ T7 )
=> ! [T8: a] :
( ( normal_sub_Cong_a @ resid @ nn @ T8 @ T2 )
=> ( member_a @ T8 @ T7 ) ) ) ) ) ) ).
% N.is_Cong_classE
thf(fact_380_targets__con__closed,axiom,
! [B: set_a,T: set_a,B4: set_a] :
( ( member_set_a @ B @ ( targets_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T ) )
=> ( ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ B4 )
=> ( ( con_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ B @ B4 )
=> ( member_set_a @ B4 @ ( targets_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T ) ) ) ) ) ).
% targets_con_closed
thf(fact_381_con__char_092_060_094sub_062Q_092_060_094sub_062C_092_060_094sub_062N,axiom,
! [T7: set_a,U2: set_a] :
( ( con_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T7 @ U2 )
= ( ( normal8595587647932138008lass_a @ resid @ nn @ T7 )
& ( normal8595587647932138008lass_a @ resid @ nn @ U2 )
& ? [T6: a,U5: a] :
( ( member_a @ T6 @ T7 )
& ( member_a @ U5 @ U2 )
& ( con_a @ resid @ T6 @ U5 ) ) ) ) ).
% con_char\<^sub>Q\<^sub>C\<^sub>N
thf(fact_382_Con__char,axiom,
! [T7: set_a,U2: set_a] :
( ( ( quotie8165075472272353145esid_a @ resid @ nn @ T7 @ U2 )
!= bot_bot_set_a )
= ( ( normal8595587647932138008lass_a @ resid @ nn @ T7 )
& ( normal8595587647932138008lass_a @ resid @ nn @ U2 )
& ? [T6: a,U5: a] :
( ( member_a @ T6 @ T7 )
& ( member_a @ U5 @ U2 )
& ( con_a @ resid @ T6 @ U5 ) ) ) ) ).
% Con_char
thf(fact_383_composableD_I3_J,axiom,
! [T: set_a,U: set_a] :
( ( composable_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U )
=> ( ( targets_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T )
= ( sources_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ U ) ) ) ).
% composableD(3)
thf(fact_384_coterminal__iff,axiom,
! [T: set_a,T4: set_a] :
( ( coterminal_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ T4 )
= ( ( arr_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T )
& ( arr_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T4 )
& ( ( targets_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T )
= ( targets_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T4 ) ) ) ) ).
% coterminal_iff
thf(fact_385_coterminalE,axiom,
! [T: set_a,U: set_a] :
( ( coterminal_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U )
=> ~ ( ( arr_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T )
=> ( ( arr_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ U )
=> ( ( targets_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T )
!= ( targets_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ U ) ) ) ) ) ).
% coterminalE
thf(fact_386_targets__char,axiom,
! [T: set_a] :
( ( targets_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T )
= ( collect_set_a
@ ^ [B2: set_a] :
( ( arr_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T )
& ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ ( quotie8165075472272353145esid_a @ resid @ nn @ T @ T ) @ B2 ) )
& ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ B2 @ ( quotie8165075472272353145esid_a @ resid @ nn @ T @ T ) ) ) ) ) ) ).
% targets_char
thf(fact_387_arr__iff__has__target,axiom,
! [T: set_a] :
( ( arr_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T )
= ( ( targets_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T )
!= bot_bot_set_set_a ) ) ).
% arr_iff_has_target
thf(fact_388_Resid__by__members,axiom,
! [T7: set_a,U2: set_a,T: a,U: a] :
( ( normal8595587647932138008lass_a @ resid @ nn @ T7 )
=> ( ( normal8595587647932138008lass_a @ resid @ nn @ U2 )
=> ( ( member_a @ T @ T7 )
=> ( ( member_a @ U @ U2 )
=> ( ( con_a @ resid @ T @ U )
=> ( ( quotie8165075472272353145esid_a @ resid @ nn @ T7 @ U2 )
= ( normal7408713899360725774lass_a @ resid @ nn @ ( resid @ T @ U ) ) ) ) ) ) ) ) ).
% Resid_by_members
thf(fact_389_targets__eqI,axiom,
! [T: set_a,T4: set_a] :
( ( ( inf_inf_set_set_a @ ( targets_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T ) @ ( targets_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T4 ) )
!= bot_bot_set_set_a )
=> ( ( targets_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T )
= ( targets_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T4 ) ) ) ).
% targets_eqI
thf(fact_390_seq__def,axiom,
! [T: set_a,U: set_a] :
( ( seq_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U )
= ( ( arr_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T )
& ( arr_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ U )
& ( ( targets_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T )
= ( sources_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ U ) ) ) ) ).
% seq_def
thf(fact_391_seqE,axiom,
! [T: set_a,U: set_a] :
( ( seq_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U )
=> ~ ( ( arr_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T )
=> ( ( arr_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ U )
=> ( ( targets_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T )
!= ( sources_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ U ) ) ) ) ) ).
% seqE
thf(fact_392_N_Ois__Cong__classI_H,axiom,
! [T7: set_a] :
( ( T7 != bot_bot_set_a )
=> ( ! [T3: a,T9: a] :
( ( member_a @ T3 @ T7 )
=> ( ( member_a @ T9 @ T7 )
=> ( normal_sub_Cong_a @ resid @ nn @ T3 @ T9 ) ) )
=> ( ! [T3: a,T9: a] :
( ( member_a @ T3 @ T7 )
=> ( ( normal_sub_Cong_a @ resid @ nn @ T9 @ T3 )
=> ( member_a @ T9 @ T7 ) ) )
=> ( normal8595587647932138008lass_a @ resid @ nn @ T7 ) ) ) ) ).
% N.is_Cong_classI'
thf(fact_393_sources__resid,axiom,
! [T: set_a,U: set_a] :
( ( con_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U )
=> ( ( sources_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ T @ U ) )
= ( targets_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ U ) ) ) ).
% sources_resid
thf(fact_394_coterminalI,axiom,
! [T: set_a,U: set_a] :
( ( arr_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T )
=> ( ( ( targets_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T )
= ( targets_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ U ) )
=> ( coterminal_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U ) ) ) ).
% coterminalI
thf(fact_395_seqI,axiom,
! [T: set_a,U: set_a] :
( ( arr_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T )
=> ( ( arr_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ U )
=> ( ( ( targets_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T )
= ( sources_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ U ) )
=> ( seq_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U ) ) ) ) ).
% seqI
thf(fact_396_N_Orep__in__Cong__class,axiom,
! [T7: set_a] :
( ( normal8595587647932138008lass_a @ resid @ nn @ T7 )
=> ( member_a @ ( normal3259722184653208495_rep_a @ T7 ) @ T7 ) ) ).
% N.rep_in_Cong_class
thf(fact_397_N_OCong__class__memb__Cong__rep,axiom,
! [T7: set_a,T: a] :
( ( normal8595587647932138008lass_a @ resid @ nn @ T7 )
=> ( ( member_a @ T @ T7 )
=> ( normal_sub_Cong_a @ resid @ nn @ T @ ( normal3259722184653208495_rep_a @ T7 ) ) ) ) ).
% N.Cong_class_memb_Cong_rep
thf(fact_398_rts_Oseq_Ocong,axiom,
seq_set_a = seq_set_a ).
% rts.seq.cong
thf(fact_399_rts_Oseq_Ocong,axiom,
seq_a = seq_a ).
% rts.seq.cong
thf(fact_400_residuation_Ocube,axiom,
! [Resid: set_a > set_a > set_a,V: set_a,T: set_a,U: set_a] :
( ( residuation_set_a @ Resid )
=> ( ( Resid @ ( Resid @ V @ T ) @ ( Resid @ U @ T ) )
= ( Resid @ ( Resid @ V @ U ) @ ( Resid @ T @ U ) ) ) ) ).
% residuation.cube
thf(fact_401_residuation_Ocube,axiom,
! [Resid: a > a > a,V: a,T: a,U: a] :
( ( residuation_a @ Resid )
=> ( ( Resid @ ( Resid @ V @ T ) @ ( Resid @ U @ T ) )
= ( Resid @ ( Resid @ V @ U ) @ ( Resid @ T @ U ) ) ) ) ).
% residuation.cube
thf(fact_402_rts_Otargets_Ocong,axiom,
targets_set_a = targets_set_a ).
% rts.targets.cong
thf(fact_403_rts_Otargets_Ocong,axiom,
targets_a = targets_a ).
% rts.targets.cong
thf(fact_404_normal__sub__rts_Ois__Cong__class_Ocong,axiom,
normal8595587647932138008lass_a = normal8595587647932138008lass_a ).
% normal_sub_rts.is_Cong_class.cong
thf(fact_405_less__eq__set__def,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A8: set_set_a,B5: set_set_a] :
( ord_less_eq_set_a_o
@ ^ [X2: set_a] : ( member_set_a @ X2 @ A8 )
@ ^ [X2: set_a] : ( member_set_a @ X2 @ B5 ) ) ) ) ).
% less_eq_set_def
thf(fact_406_less__eq__set__def,axiom,
( ord_less_eq_set_a
= ( ^ [A8: set_a,B5: set_a] :
( ord_less_eq_a_o
@ ^ [X2: a] : ( member_a @ X2 @ A8 )
@ ^ [X2: a] : ( member_a @ X2 @ B5 ) ) ) ) ).
% less_eq_set_def
thf(fact_407_residuation_Ocon__sym,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a] :
( ( residuation_set_a @ Resid )
=> ( ( con_set_a @ Resid @ T @ U )
=> ( con_set_a @ Resid @ U @ T ) ) ) ).
% residuation.con_sym
thf(fact_408_residuation_Ocon__sym,axiom,
! [Resid: a > a > a,T: a,U: a] :
( ( residuation_a @ Resid )
=> ( ( con_a @ Resid @ T @ U )
=> ( con_a @ Resid @ U @ T ) ) ) ).
% residuation.con_sym
thf(fact_409_residuation_Oaxioms_I1_J,axiom,
! [Resid: set_a > set_a > set_a] :
( ( residuation_set_a @ Resid )
=> ( partial_magma_set_a @ Resid ) ) ).
% residuation.axioms(1)
thf(fact_410_residuation_Oaxioms_I1_J,axiom,
! [Resid: a > a > a] :
( ( residuation_a @ Resid )
=> ( partial_magma_a @ Resid ) ) ).
% residuation.axioms(1)
thf(fact_411_coherent__normal__sub__rts_Ocoherent_H,axiom,
! [Resid: set_a > set_a > set_a,NN: set_set_a,V: set_a,V4: set_a,W2: set_a,W3: set_a,T: set_a,T4: set_a] :
( ( cohere6325062230080414023_set_a @ Resid @ NN )
=> ( ( member_set_a @ V @ NN )
=> ( ( member_set_a @ V4 @ NN )
=> ( ( member_set_a @ W2 @ NN )
=> ( ( member_set_a @ W3 @ NN )
=> ( ( ( sources_set_a @ Resid @ V )
= ( sources_set_a @ Resid @ W2 ) )
=> ( ( ( sources_set_a @ Resid @ V4 )
= ( sources_set_a @ Resid @ W3 ) )
=> ( ( ( targets_set_a @ Resid @ W2 )
= ( targets_set_a @ Resid @ W3 ) )
=> ( ( ( member_set_a @ ( Resid @ ( Resid @ T @ V ) @ ( Resid @ T4 @ V4 ) ) @ NN )
& ( member_set_a @ ( Resid @ ( Resid @ T4 @ V4 ) @ ( Resid @ T @ V ) ) @ NN ) )
=> ( ( member_set_a @ ( Resid @ ( Resid @ T @ W2 ) @ ( Resid @ T4 @ W3 ) ) @ NN )
& ( member_set_a @ ( Resid @ ( Resid @ T4 @ W3 ) @ ( Resid @ T @ W2 ) ) @ NN ) ) ) ) ) ) ) ) ) ) ) ).
% coherent_normal_sub_rts.coherent'
thf(fact_412_coherent__normal__sub__rts_Ocoherent_H,axiom,
! [Resid: a > a > a,NN: set_a,V: a,V4: a,W2: a,W3: a,T: a,T4: a] :
( ( cohere6072184133013167079_rts_a @ Resid @ NN )
=> ( ( member_a @ V @ NN )
=> ( ( member_a @ V4 @ NN )
=> ( ( member_a @ W2 @ NN )
=> ( ( member_a @ W3 @ NN )
=> ( ( ( sources_a @ Resid @ V )
= ( sources_a @ Resid @ W2 ) )
=> ( ( ( sources_a @ Resid @ V4 )
= ( sources_a @ Resid @ W3 ) )
=> ( ( ( targets_a @ Resid @ W2 )
= ( targets_a @ Resid @ W3 ) )
=> ( ( ( member_a @ ( Resid @ ( Resid @ T @ V ) @ ( Resid @ T4 @ V4 ) ) @ NN )
& ( member_a @ ( Resid @ ( Resid @ T4 @ V4 ) @ ( Resid @ T @ V ) ) @ NN ) )
=> ( ( member_a @ ( Resid @ ( Resid @ T @ W2 ) @ ( Resid @ T4 @ W3 ) ) @ NN )
& ( member_a @ ( Resid @ ( Resid @ T4 @ W3 ) @ ( Resid @ T @ W2 ) ) @ NN ) ) ) ) ) ) ) ) ) ) ) ).
% coherent_normal_sub_rts.coherent'
thf(fact_413_residuation_OideE,axiom,
! [Resid: set_a > set_a > set_a,A3: set_a] :
( ( residuation_set_a @ Resid )
=> ( ( ide_set_a @ Resid @ A3 )
=> ~ ( ( con_set_a @ Resid @ A3 @ A3 )
=> ( ( Resid @ A3 @ A3 )
!= A3 ) ) ) ) ).
% residuation.ideE
thf(fact_414_residuation_OideE,axiom,
! [Resid: a > a > a,A3: a] :
( ( residuation_a @ Resid )
=> ( ( ide_a @ Resid @ A3 )
=> ~ ( ( con_a @ Resid @ A3 @ A3 )
=> ( ( Resid @ A3 @ A3 )
!= A3 ) ) ) ) ).
% residuation.ideE
thf(fact_415_residuation_OideI,axiom,
! [Resid: set_a > set_a > set_a,A3: set_a] :
( ( residuation_set_a @ Resid )
=> ( ( con_set_a @ Resid @ A3 @ A3 )
=> ( ( ( Resid @ A3 @ A3 )
= A3 )
=> ( ide_set_a @ Resid @ A3 ) ) ) ) ).
% residuation.ideI
thf(fact_416_residuation_OideI,axiom,
! [Resid: a > a > a,A3: a] :
( ( residuation_a @ Resid )
=> ( ( con_a @ Resid @ A3 @ A3 )
=> ( ( ( Resid @ A3 @ A3 )
= A3 )
=> ( ide_a @ Resid @ A3 ) ) ) ) ).
% residuation.ideI
thf(fact_417_residuation_Oide__def,axiom,
! [Resid: set_a > set_a > set_a,A3: set_a] :
( ( residuation_set_a @ Resid )
=> ( ( ide_set_a @ Resid @ A3 )
= ( ( con_set_a @ Resid @ A3 @ A3 )
& ( ( Resid @ A3 @ A3 )
= A3 ) ) ) ) ).
% residuation.ide_def
thf(fact_418_residuation_Oide__def,axiom,
! [Resid: a > a > a,A3: a] :
( ( residuation_a @ Resid )
=> ( ( ide_a @ Resid @ A3 )
= ( ( con_a @ Resid @ A3 @ A3 )
& ( ( Resid @ A3 @ A3 )
= A3 ) ) ) ) ).
% residuation.ide_def
thf(fact_419_residuation_Oide__implies__arr,axiom,
! [Resid: set_a > set_a > set_a,A3: set_a] :
( ( residuation_set_a @ Resid )
=> ( ( ide_set_a @ Resid @ A3 )
=> ( arr_set_a @ Resid @ A3 ) ) ) ).
% residuation.ide_implies_arr
thf(fact_420_residuation_Oide__implies__arr,axiom,
! [Resid: a > a > a,A3: a] :
( ( residuation_a @ Resid )
=> ( ( ide_a @ Resid @ A3 )
=> ( arr_a @ Resid @ A3 ) ) ) ).
% residuation.ide_implies_arr
thf(fact_421_residuation_Oarr__resid__iff__con,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a] :
( ( residuation_set_a @ Resid )
=> ( ( arr_set_a @ Resid @ ( Resid @ T @ U ) )
= ( con_set_a @ Resid @ T @ U ) ) ) ).
% residuation.arr_resid_iff_con
thf(fact_422_residuation_Oarr__resid__iff__con,axiom,
! [Resid: a > a > a,T: a,U: a] :
( ( residuation_a @ Resid )
=> ( ( arr_a @ Resid @ ( Resid @ T @ U ) )
= ( con_a @ Resid @ T @ U ) ) ) ).
% residuation.arr_resid_iff_con
thf(fact_423_residuation_Oarr__resid,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a] :
( ( residuation_set_a @ Resid )
=> ( ( con_set_a @ Resid @ T @ U )
=> ( arr_set_a @ Resid @ ( Resid @ T @ U ) ) ) ) ).
% residuation.arr_resid
thf(fact_424_residuation_Oarr__resid,axiom,
! [Resid: a > a > a,T: a,U: a] :
( ( residuation_a @ Resid )
=> ( ( con_a @ Resid @ T @ U )
=> ( arr_a @ Resid @ ( Resid @ T @ U ) ) ) ) ).
% residuation.arr_resid
thf(fact_425_residuation_Oarr__def,axiom,
! [Resid: set_a > set_a > set_a,T: set_a] :
( ( residuation_set_a @ Resid )
=> ( ( arr_set_a @ Resid @ T )
= ( con_set_a @ Resid @ T @ T ) ) ) ).
% residuation.arr_def
thf(fact_426_residuation_Oarr__def,axiom,
! [Resid: a > a > a,T: a] :
( ( residuation_a @ Resid )
=> ( ( arr_a @ Resid @ T )
= ( con_a @ Resid @ T @ T ) ) ) ).
% residuation.arr_def
thf(fact_427_residuation_OarrI,axiom,
! [Resid: set_a > set_a > set_a,T: set_a] :
( ( residuation_set_a @ Resid )
=> ( ( con_set_a @ Resid @ T @ T )
=> ( arr_set_a @ Resid @ T ) ) ) ).
% residuation.arrI
thf(fact_428_residuation_OarrI,axiom,
! [Resid: a > a > a,T: a] :
( ( residuation_a @ Resid )
=> ( ( con_a @ Resid @ T @ T )
=> ( arr_a @ Resid @ T ) ) ) ).
% residuation.arrI
thf(fact_429_residuation_OarrE,axiom,
! [Resid: set_a > set_a > set_a,T: set_a] :
( ( residuation_set_a @ Resid )
=> ( ( arr_set_a @ Resid @ T )
=> ( con_set_a @ Resid @ T @ T ) ) ) ).
% residuation.arrE
thf(fact_430_residuation_OarrE,axiom,
! [Resid: a > a > a,T: a] :
( ( residuation_a @ Resid )
=> ( ( arr_a @ Resid @ T )
=> ( con_a @ Resid @ T @ T ) ) ) ).
% residuation.arrE
thf(fact_431_residuation_Ocon__implies__arr_I1_J,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a] :
( ( residuation_set_a @ Resid )
=> ( ( con_set_a @ Resid @ T @ U )
=> ( arr_set_a @ Resid @ T ) ) ) ).
% residuation.con_implies_arr(1)
thf(fact_432_residuation_Ocon__implies__arr_I1_J,axiom,
! [Resid: a > a > a,T: a,U: a] :
( ( residuation_a @ Resid )
=> ( ( con_a @ Resid @ T @ U )
=> ( arr_a @ Resid @ T ) ) ) ).
% residuation.con_implies_arr(1)
thf(fact_433_residuation_Ocon__implies__arr_I2_J,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a] :
( ( residuation_set_a @ Resid )
=> ( ( con_set_a @ Resid @ T @ U )
=> ( arr_set_a @ Resid @ U ) ) ) ).
% residuation.con_implies_arr(2)
thf(fact_434_residuation_Ocon__implies__arr_I2_J,axiom,
! [Resid: a > a > a,T: a,U: a] :
( ( residuation_a @ Resid )
=> ( ( con_a @ Resid @ T @ U )
=> ( arr_a @ Resid @ U ) ) ) ).
% residuation.con_implies_arr(2)
thf(fact_435_quotient__by__coherent__normal_Ois__residuation,axiom,
! [Resid: a > a > a,NN: set_a] :
( ( quotie3282664541148387094rmal_a @ Resid @ NN )
=> ( residuation_set_a @ ( quotie8165075472272353145esid_a @ Resid @ NN ) ) ) ).
% quotient_by_coherent_normal.is_residuation
thf(fact_436_emptyE,axiom,
! [A3: a] :
~ ( member_a @ A3 @ bot_bot_set_a ) ).
% emptyE
thf(fact_437_emptyE,axiom,
! [A3: set_a] :
~ ( member_set_a @ A3 @ bot_bot_set_set_a ) ).
% emptyE
thf(fact_438_equals0D,axiom,
! [A5: set_a,A3: a] :
( ( A5 = bot_bot_set_a )
=> ~ ( member_a @ A3 @ A5 ) ) ).
% equals0D
thf(fact_439_equals0D,axiom,
! [A5: set_set_a,A3: set_a] :
( ( A5 = bot_bot_set_set_a )
=> ~ ( member_set_a @ A3 @ A5 ) ) ).
% equals0D
thf(fact_440_equals0I,axiom,
! [A5: set_a] :
( ! [Y3: a] :
~ ( member_a @ Y3 @ A5 )
=> ( A5 = bot_bot_set_a ) ) ).
% equals0I
thf(fact_441_equals0I,axiom,
! [A5: set_set_a] :
( ! [Y3: set_a] :
~ ( member_set_a @ Y3 @ A5 )
=> ( A5 = bot_bot_set_set_a ) ) ).
% equals0I
thf(fact_442_ex__in__conv,axiom,
! [A5: set_a] :
( ( ? [X2: a] : ( member_a @ X2 @ A5 ) )
= ( A5 != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_443_ex__in__conv,axiom,
! [A5: set_set_a] :
( ( ? [X2: set_a] : ( member_set_a @ X2 @ A5 ) )
= ( A5 != bot_bot_set_set_a ) ) ).
% ex_in_conv
thf(fact_444_bot__set__def,axiom,
( bot_bot_set_a
= ( collect_a @ bot_bot_a_o ) ) ).
% bot_set_def
thf(fact_445_bot__set__def,axiom,
( bot_bot_set_set_a
= ( collect_set_a @ bot_bot_set_a_o ) ) ).
% bot_set_def
thf(fact_446_in__mono,axiom,
! [A5: set_set_a,B3: set_set_a,X4: set_a] :
( ( ord_le3724670747650509150_set_a @ A5 @ B3 )
=> ( ( member_set_a @ X4 @ A5 )
=> ( member_set_a @ X4 @ B3 ) ) ) ).
% in_mono
thf(fact_447_in__mono,axiom,
! [A5: set_a,B3: set_a,X4: a] :
( ( ord_less_eq_set_a @ A5 @ B3 )
=> ( ( member_a @ X4 @ A5 )
=> ( member_a @ X4 @ B3 ) ) ) ).
% in_mono
thf(fact_448_subsetD,axiom,
! [A5: set_set_a,B3: set_set_a,C: set_a] :
( ( ord_le3724670747650509150_set_a @ A5 @ B3 )
=> ( ( member_set_a @ C @ A5 )
=> ( member_set_a @ C @ B3 ) ) ) ).
% subsetD
thf(fact_449_subsetD,axiom,
! [A5: set_a,B3: set_a,C: a] :
( ( ord_less_eq_set_a @ A5 @ B3 )
=> ( ( member_a @ C @ A5 )
=> ( member_a @ C @ B3 ) ) ) ).
% subsetD
thf(fact_450_equalityE,axiom,
! [A5: set_a,B3: set_a] :
( ( A5 = B3 )
=> ~ ( ( ord_less_eq_set_a @ A5 @ B3 )
=> ~ ( ord_less_eq_set_a @ B3 @ A5 ) ) ) ).
% equalityE
thf(fact_451_subset__eq,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A8: set_set_a,B5: set_set_a] :
! [X2: set_a] :
( ( member_set_a @ X2 @ A8 )
=> ( member_set_a @ X2 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_452_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A8: set_a,B5: set_a] :
! [X2: a] :
( ( member_a @ X2 @ A8 )
=> ( member_a @ X2 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_453_equalityD1,axiom,
! [A5: set_a,B3: set_a] :
( ( A5 = B3 )
=> ( ord_less_eq_set_a @ A5 @ B3 ) ) ).
% equalityD1
thf(fact_454_equalityD2,axiom,
! [A5: set_a,B3: set_a] :
( ( A5 = B3 )
=> ( ord_less_eq_set_a @ B3 @ A5 ) ) ).
% equalityD2
thf(fact_455_subset__iff,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A8: set_set_a,B5: set_set_a] :
! [T6: set_a] :
( ( member_set_a @ T6 @ A8 )
=> ( member_set_a @ T6 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_456_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A8: set_a,B5: set_a] :
! [T6: a] :
( ( member_a @ T6 @ A8 )
=> ( member_a @ T6 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_457_subset__refl,axiom,
! [A5: set_a] : ( ord_less_eq_set_a @ A5 @ A5 ) ).
% subset_refl
thf(fact_458_Collect__mono,axiom,
! [P: set_a > $o,Q: set_a > $o] :
( ! [X: set_a] :
( ( P @ X )
=> ( Q @ X ) )
=> ( ord_le3724670747650509150_set_a @ ( collect_set_a @ P ) @ ( collect_set_a @ Q ) ) ) ).
% Collect_mono
thf(fact_459_Collect__mono,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X: a] :
( ( P @ X )
=> ( Q @ X ) )
=> ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% Collect_mono
thf(fact_460_subset__trans,axiom,
! [A5: set_a,B3: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A5 @ B3 )
=> ( ( ord_less_eq_set_a @ B3 @ C2 )
=> ( ord_less_eq_set_a @ A5 @ C2 ) ) ) ).
% subset_trans
thf(fact_461_set__eq__subset,axiom,
( ( ^ [Y5: set_a,Z3: set_a] : ( Y5 = Z3 ) )
= ( ^ [A8: set_a,B5: set_a] :
( ( ord_less_eq_set_a @ A8 @ B5 )
& ( ord_less_eq_set_a @ B5 @ A8 ) ) ) ) ).
% set_eq_subset
thf(fact_462_Collect__mono__iff,axiom,
! [P: set_a > $o,Q: set_a > $o] :
( ( ord_le3724670747650509150_set_a @ ( collect_set_a @ P ) @ ( collect_set_a @ Q ) )
= ( ! [X2: set_a] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_463_Collect__mono__iff,axiom,
! [P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) )
= ( ! [X2: a] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_464_coherent__normal__sub__rts_Ocoherent,axiom,
! [Resid: set_a > set_a > set_a,NN: set_set_a,T: set_a,U: set_a,U7: set_a] :
( ( cohere6325062230080414023_set_a @ Resid @ NN )
=> ( ( arr_set_a @ Resid @ T )
=> ( ( member_set_a @ U @ NN )
=> ( ( member_set_a @ U7 @ NN )
=> ( ( ( sources_set_a @ Resid @ U )
= ( sources_set_a @ Resid @ U7 ) )
=> ( ( ( targets_set_a @ Resid @ U )
= ( targets_set_a @ Resid @ U7 ) )
=> ( ( ( sources_set_a @ Resid @ T )
= ( sources_set_a @ Resid @ U ) )
=> ( ( member_set_a @ ( Resid @ ( Resid @ T @ U ) @ ( Resid @ T @ U7 ) ) @ NN )
& ( member_set_a @ ( Resid @ ( Resid @ T @ U7 ) @ ( Resid @ T @ U ) ) @ NN ) ) ) ) ) ) ) ) ) ).
% coherent_normal_sub_rts.coherent
thf(fact_465_coherent__normal__sub__rts_Ocoherent,axiom,
! [Resid: a > a > a,NN: set_a,T: a,U: a,U7: a] :
( ( cohere6072184133013167079_rts_a @ Resid @ NN )
=> ( ( arr_a @ Resid @ T )
=> ( ( member_a @ U @ NN )
=> ( ( member_a @ U7 @ NN )
=> ( ( ( sources_a @ Resid @ U )
= ( sources_a @ Resid @ U7 ) )
=> ( ( ( targets_a @ Resid @ U )
= ( targets_a @ Resid @ U7 ) )
=> ( ( ( sources_a @ Resid @ T )
= ( sources_a @ Resid @ U ) )
=> ( ( member_a @ ( Resid @ ( Resid @ T @ U ) @ ( Resid @ T @ U7 ) ) @ NN )
& ( member_a @ ( Resid @ ( Resid @ T @ U7 ) @ ( Resid @ T @ U ) ) @ NN ) ) ) ) ) ) ) ) ) ).
% coherent_normal_sub_rts.coherent
thf(fact_466_IntE,axiom,
! [C: a,A5: set_a,B3: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A5 @ B3 ) )
=> ~ ( ( member_a @ C @ A5 )
=> ~ ( member_a @ C @ B3 ) ) ) ).
% IntE
thf(fact_467_IntE,axiom,
! [C: set_a,A5: set_set_a,B3: set_set_a] :
( ( member_set_a @ C @ ( inf_inf_set_set_a @ A5 @ B3 ) )
=> ~ ( ( member_set_a @ C @ A5 )
=> ~ ( member_set_a @ C @ B3 ) ) ) ).
% IntE
thf(fact_468_IntD1,axiom,
! [C: a,A5: set_a,B3: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A5 @ B3 ) )
=> ( member_a @ C @ A5 ) ) ).
% IntD1
thf(fact_469_IntD1,axiom,
! [C: set_a,A5: set_set_a,B3: set_set_a] :
( ( member_set_a @ C @ ( inf_inf_set_set_a @ A5 @ B3 ) )
=> ( member_set_a @ C @ A5 ) ) ).
% IntD1
thf(fact_470_IntD2,axiom,
! [C: a,A5: set_a,B3: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A5 @ B3 ) )
=> ( member_a @ C @ B3 ) ) ).
% IntD2
thf(fact_471_IntD2,axiom,
! [C: set_a,A5: set_set_a,B3: set_set_a] :
( ( member_set_a @ C @ ( inf_inf_set_set_a @ A5 @ B3 ) )
=> ( member_set_a @ C @ B3 ) ) ).
% IntD2
thf(fact_472_Int__assoc,axiom,
! [A5: set_a,B3: set_a,C2: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A5 @ B3 ) @ C2 )
= ( inf_inf_set_a @ A5 @ ( inf_inf_set_a @ B3 @ C2 ) ) ) ).
% Int_assoc
thf(fact_473_Int__assoc,axiom,
! [A5: set_set_a,B3: set_set_a,C2: set_set_a] :
( ( inf_inf_set_set_a @ ( inf_inf_set_set_a @ A5 @ B3 ) @ C2 )
= ( inf_inf_set_set_a @ A5 @ ( inf_inf_set_set_a @ B3 @ C2 ) ) ) ).
% Int_assoc
thf(fact_474_Int__absorb,axiom,
! [A5: set_a] :
( ( inf_inf_set_a @ A5 @ A5 )
= A5 ) ).
% Int_absorb
thf(fact_475_Int__absorb,axiom,
! [A5: set_set_a] :
( ( inf_inf_set_set_a @ A5 @ A5 )
= A5 ) ).
% Int_absorb
thf(fact_476_Int__commute,axiom,
( inf_inf_set_a
= ( ^ [A8: set_a,B5: set_a] : ( inf_inf_set_a @ B5 @ A8 ) ) ) ).
% Int_commute
thf(fact_477_Int__commute,axiom,
( inf_inf_set_set_a
= ( ^ [A8: set_set_a,B5: set_set_a] : ( inf_inf_set_set_a @ B5 @ A8 ) ) ) ).
% Int_commute
thf(fact_478_Int__left__absorb,axiom,
! [A5: set_a,B3: set_a] :
( ( inf_inf_set_a @ A5 @ ( inf_inf_set_a @ A5 @ B3 ) )
= ( inf_inf_set_a @ A5 @ B3 ) ) ).
% Int_left_absorb
thf(fact_479_Int__left__absorb,axiom,
! [A5: set_set_a,B3: set_set_a] :
( ( inf_inf_set_set_a @ A5 @ ( inf_inf_set_set_a @ A5 @ B3 ) )
= ( inf_inf_set_set_a @ A5 @ B3 ) ) ).
% Int_left_absorb
thf(fact_480_Int__left__commute,axiom,
! [A5: set_a,B3: set_a,C2: set_a] :
( ( inf_inf_set_a @ A5 @ ( inf_inf_set_a @ B3 @ C2 ) )
= ( inf_inf_set_a @ B3 @ ( inf_inf_set_a @ A5 @ C2 ) ) ) ).
% Int_left_commute
thf(fact_481_Int__left__commute,axiom,
! [A5: set_set_a,B3: set_set_a,C2: set_set_a] :
( ( inf_inf_set_set_a @ A5 @ ( inf_inf_set_set_a @ B3 @ C2 ) )
= ( inf_inf_set_set_a @ B3 @ ( inf_inf_set_set_a @ A5 @ C2 ) ) ) ).
% Int_left_commute
thf(fact_482_quotient__by__coherent__normal_Ois__Cong__class__Resid,axiom,
! [Resid: set_a > set_a > set_a,NN: set_set_a,T7: set_set_a,U2: set_set_a] :
( ( quotie5625257012022141046_set_a @ Resid @ NN )
=> ( ( ( quotie3283642546880816345_set_a @ Resid @ NN @ T7 @ U2 )
!= bot_bot_set_set_a )
=> ( normal4437380936311325560_set_a @ Resid @ NN @ ( quotie3283642546880816345_set_a @ Resid @ NN @ T7 @ U2 ) ) ) ) ).
% quotient_by_coherent_normal.is_Cong_class_Resid
thf(fact_483_quotient__by__coherent__normal_Ois__Cong__class__Resid,axiom,
! [Resid: a > a > a,NN: set_a,T7: set_a,U2: set_a] :
( ( quotie3282664541148387094rmal_a @ Resid @ NN )
=> ( ( ( quotie8165075472272353145esid_a @ Resid @ NN @ T7 @ U2 )
!= bot_bot_set_a )
=> ( normal8595587647932138008lass_a @ Resid @ NN @ ( quotie8165075472272353145esid_a @ Resid @ NN @ T7 @ U2 ) ) ) ) ).
% quotient_by_coherent_normal.is_Cong_class_Resid
thf(fact_484_quotient__by__coherent__normal_Oarr__char,axiom,
! [Resid: a > a > a,NN: set_a,T7: set_a] :
( ( quotie3282664541148387094rmal_a @ Resid @ NN )
=> ( ( arr_set_a @ ( quotie8165075472272353145esid_a @ Resid @ NN ) @ T7 )
= ( normal8595587647932138008lass_a @ Resid @ NN @ T7 ) ) ) ).
% quotient_by_coherent_normal.arr_char
thf(fact_485_empty__def,axiom,
( bot_bot_set_a
= ( collect_a
@ ^ [X2: a] : $false ) ) ).
% empty_def
thf(fact_486_empty__def,axiom,
( bot_bot_set_set_a
= ( collect_set_a
@ ^ [X2: set_a] : $false ) ) ).
% empty_def
thf(fact_487_Collect__subset,axiom,
! [A5: set_set_a,P: set_a > $o] :
( ord_le3724670747650509150_set_a
@ ( collect_set_a
@ ^ [X2: set_a] :
( ( member_set_a @ X2 @ A5 )
& ( P @ X2 ) ) )
@ A5 ) ).
% Collect_subset
thf(fact_488_Collect__subset,axiom,
! [A5: set_a,P: a > $o] :
( ord_less_eq_set_a
@ ( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ A5 )
& ( P @ X2 ) ) )
@ A5 ) ).
% Collect_subset
thf(fact_489_Collect__conj__eq,axiom,
! [P: a > $o,Q: a > $o] :
( ( collect_a
@ ^ [X2: a] :
( ( P @ X2 )
& ( Q @ X2 ) ) )
= ( inf_inf_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_490_Collect__conj__eq,axiom,
! [P: set_a > $o,Q: set_a > $o] :
( ( collect_set_a
@ ^ [X2: set_a] :
( ( P @ X2 )
& ( Q @ X2 ) ) )
= ( inf_inf_set_set_a @ ( collect_set_a @ P ) @ ( collect_set_a @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_491_Int__Collect,axiom,
! [X4: a,A5: set_a,P: a > $o] :
( ( member_a @ X4 @ ( inf_inf_set_a @ A5 @ ( collect_a @ P ) ) )
= ( ( member_a @ X4 @ A5 )
& ( P @ X4 ) ) ) ).
% Int_Collect
thf(fact_492_Int__Collect,axiom,
! [X4: set_a,A5: set_set_a,P: set_a > $o] :
( ( member_set_a @ X4 @ ( inf_inf_set_set_a @ A5 @ ( collect_set_a @ P ) ) )
= ( ( member_set_a @ X4 @ A5 )
& ( P @ X4 ) ) ) ).
% Int_Collect
thf(fact_493_Int__def,axiom,
( inf_inf_set_a
= ( ^ [A8: set_a,B5: set_a] :
( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ A8 )
& ( member_a @ X2 @ B5 ) ) ) ) ) ).
% Int_def
thf(fact_494_Int__def,axiom,
( inf_inf_set_set_a
= ( ^ [A8: set_set_a,B5: set_set_a] :
( collect_set_a
@ ^ [X2: set_a] :
( ( member_set_a @ X2 @ A8 )
& ( member_set_a @ X2 @ B5 ) ) ) ) ) ).
% Int_def
thf(fact_495_inf__set__def,axiom,
( inf_inf_set_a
= ( ^ [A8: set_a,B5: set_a] :
( collect_a
@ ( inf_inf_a_o
@ ^ [X2: a] : ( member_a @ X2 @ A8 )
@ ^ [X2: a] : ( member_a @ X2 @ B5 ) ) ) ) ) ).
% inf_set_def
thf(fact_496_inf__set__def,axiom,
( inf_inf_set_set_a
= ( ^ [A8: set_set_a,B5: set_set_a] :
( collect_set_a
@ ( inf_inf_set_a_o
@ ^ [X2: set_a] : ( member_set_a @ X2 @ A8 )
@ ^ [X2: set_a] : ( member_set_a @ X2 @ B5 ) ) ) ) ) ).
% inf_set_def
thf(fact_497_quotient__by__coherent__normal_OCon__char,axiom,
! [Resid: set_a > set_a > set_a,NN: set_set_a,T7: set_set_a,U2: set_set_a] :
( ( quotie5625257012022141046_set_a @ Resid @ NN )
=> ( ( ( quotie3283642546880816345_set_a @ Resid @ NN @ T7 @ U2 )
!= bot_bot_set_set_a )
= ( ( normal4437380936311325560_set_a @ Resid @ NN @ T7 )
& ( normal4437380936311325560_set_a @ Resid @ NN @ U2 )
& ? [T6: set_a,U5: set_a] :
( ( member_set_a @ T6 @ T7 )
& ( member_set_a @ U5 @ U2 )
& ( con_set_a @ Resid @ T6 @ U5 ) ) ) ) ) ).
% quotient_by_coherent_normal.Con_char
thf(fact_498_quotient__by__coherent__normal_OCon__char,axiom,
! [Resid: a > a > a,NN: set_a,T7: set_a,U2: set_a] :
( ( quotie3282664541148387094rmal_a @ Resid @ NN )
=> ( ( ( quotie8165075472272353145esid_a @ Resid @ NN @ T7 @ U2 )
!= bot_bot_set_a )
= ( ( normal8595587647932138008lass_a @ Resid @ NN @ T7 )
& ( normal8595587647932138008lass_a @ Resid @ NN @ U2 )
& ? [T6: a,U5: a] :
( ( member_a @ T6 @ T7 )
& ( member_a @ U5 @ U2 )
& ( con_a @ Resid @ T6 @ U5 ) ) ) ) ) ).
% quotient_by_coherent_normal.Con_char
thf(fact_499_quotient__by__coherent__normal_Ocon__char_092_060_094sub_062Q_092_060_094sub_062C_092_060_094sub_062N,axiom,
! [Resid: set_a > set_a > set_a,NN: set_set_a,T7: set_set_a,U2: set_set_a] :
( ( quotie5625257012022141046_set_a @ Resid @ NN )
=> ( ( con_set_set_a @ ( quotie3283642546880816345_set_a @ Resid @ NN ) @ T7 @ U2 )
= ( ( normal4437380936311325560_set_a @ Resid @ NN @ T7 )
& ( normal4437380936311325560_set_a @ Resid @ NN @ U2 )
& ? [T6: set_a,U5: set_a] :
( ( member_set_a @ T6 @ T7 )
& ( member_set_a @ U5 @ U2 )
& ( con_set_a @ Resid @ T6 @ U5 ) ) ) ) ) ).
% quotient_by_coherent_normal.con_char\<^sub>Q\<^sub>C\<^sub>N
thf(fact_500_quotient__by__coherent__normal_Ocon__char_092_060_094sub_062Q_092_060_094sub_062C_092_060_094sub_062N,axiom,
! [Resid: a > a > a,NN: set_a,T7: set_a,U2: set_a] :
( ( quotie3282664541148387094rmal_a @ Resid @ NN )
=> ( ( con_set_a @ ( quotie8165075472272353145esid_a @ Resid @ NN ) @ T7 @ U2 )
= ( ( normal8595587647932138008lass_a @ Resid @ NN @ T7 )
& ( normal8595587647932138008lass_a @ Resid @ NN @ U2 )
& ? [T6: a,U5: a] :
( ( member_a @ T6 @ T7 )
& ( member_a @ U5 @ U2 )
& ( con_a @ Resid @ T6 @ U5 ) ) ) ) ) ).
% quotient_by_coherent_normal.con_char\<^sub>Q\<^sub>C\<^sub>N
thf(fact_501_quotient__by__coherent__normal_OResid__by__members,axiom,
! [Resid: set_a > set_a > set_a,NN: set_set_a,T7: set_set_a,U2: set_set_a,T: set_a,U: set_a] :
( ( quotie5625257012022141046_set_a @ Resid @ NN )
=> ( ( normal4437380936311325560_set_a @ Resid @ NN @ T7 )
=> ( ( normal4437380936311325560_set_a @ Resid @ NN @ U2 )
=> ( ( member_set_a @ T @ T7 )
=> ( ( member_set_a @ U @ U2 )
=> ( ( con_set_a @ Resid @ T @ U )
=> ( ( quotie3283642546880816345_set_a @ Resid @ NN @ T7 @ U2 )
= ( normal2962378890657961070_set_a @ Resid @ NN @ ( Resid @ T @ U ) ) ) ) ) ) ) ) ) ).
% quotient_by_coherent_normal.Resid_by_members
thf(fact_502_quotient__by__coherent__normal_OResid__by__members,axiom,
! [Resid: a > a > a,NN: set_a,T7: set_a,U2: set_a,T: a,U: a] :
( ( quotie3282664541148387094rmal_a @ Resid @ NN )
=> ( ( normal8595587647932138008lass_a @ Resid @ NN @ T7 )
=> ( ( normal8595587647932138008lass_a @ Resid @ NN @ U2 )
=> ( ( member_a @ T @ T7 )
=> ( ( member_a @ U @ U2 )
=> ( ( con_a @ Resid @ T @ U )
=> ( ( quotie8165075472272353145esid_a @ Resid @ NN @ T7 @ U2 )
= ( normal7408713899360725774lass_a @ Resid @ NN @ ( Resid @ T @ U ) ) ) ) ) ) ) ) ) ).
% quotient_by_coherent_normal.Resid_by_members
thf(fact_503_Int__emptyI,axiom,
! [A5: set_a,B3: set_a] :
( ! [X: a] :
( ( member_a @ X @ A5 )
=> ~ ( member_a @ X @ B3 ) )
=> ( ( inf_inf_set_a @ A5 @ B3 )
= bot_bot_set_a ) ) ).
% Int_emptyI
thf(fact_504_Int__emptyI,axiom,
! [A5: set_set_a,B3: set_set_a] :
( ! [X: set_a] :
( ( member_set_a @ X @ A5 )
=> ~ ( member_set_a @ X @ B3 ) )
=> ( ( inf_inf_set_set_a @ A5 @ B3 )
= bot_bot_set_set_a ) ) ).
% Int_emptyI
thf(fact_505_disjoint__iff,axiom,
! [A5: set_a,B3: set_a] :
( ( ( inf_inf_set_a @ A5 @ B3 )
= bot_bot_set_a )
= ( ! [X2: a] :
( ( member_a @ X2 @ A5 )
=> ~ ( member_a @ X2 @ B3 ) ) ) ) ).
% disjoint_iff
thf(fact_506_disjoint__iff,axiom,
! [A5: set_set_a,B3: set_set_a] :
( ( ( inf_inf_set_set_a @ A5 @ B3 )
= bot_bot_set_set_a )
= ( ! [X2: set_a] :
( ( member_set_a @ X2 @ A5 )
=> ~ ( member_set_a @ X2 @ B3 ) ) ) ) ).
% disjoint_iff
thf(fact_507_Int__empty__left,axiom,
! [B3: set_a] :
( ( inf_inf_set_a @ bot_bot_set_a @ B3 )
= bot_bot_set_a ) ).
% Int_empty_left
thf(fact_508_Int__empty__left,axiom,
! [B3: set_set_a] :
( ( inf_inf_set_set_a @ bot_bot_set_set_a @ B3 )
= bot_bot_set_set_a ) ).
% Int_empty_left
thf(fact_509_Int__empty__right,axiom,
! [A5: set_a] :
( ( inf_inf_set_a @ A5 @ bot_bot_set_a )
= bot_bot_set_a ) ).
% Int_empty_right
thf(fact_510_Int__empty__right,axiom,
! [A5: set_set_a] :
( ( inf_inf_set_set_a @ A5 @ bot_bot_set_set_a )
= bot_bot_set_set_a ) ).
% Int_empty_right
thf(fact_511_disjoint__iff__not__equal,axiom,
! [A5: set_a,B3: set_a] :
( ( ( inf_inf_set_a @ A5 @ B3 )
= bot_bot_set_a )
= ( ! [X2: a] :
( ( member_a @ X2 @ A5 )
=> ! [Y2: a] :
( ( member_a @ Y2 @ B3 )
=> ( X2 != Y2 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_512_disjoint__iff__not__equal,axiom,
! [A5: set_set_a,B3: set_set_a] :
( ( ( inf_inf_set_set_a @ A5 @ B3 )
= bot_bot_set_set_a )
= ( ! [X2: set_a] :
( ( member_set_a @ X2 @ A5 )
=> ! [Y2: set_a] :
( ( member_set_a @ Y2 @ B3 )
=> ( X2 != Y2 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_513_Int__mono,axiom,
! [A5: set_set_a,C2: set_set_a,B3: set_set_a,D2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A5 @ C2 )
=> ( ( ord_le3724670747650509150_set_a @ B3 @ D2 )
=> ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A5 @ B3 ) @ ( inf_inf_set_set_a @ C2 @ D2 ) ) ) ) ).
% Int_mono
thf(fact_514_Int__mono,axiom,
! [A5: set_a,C2: set_a,B3: set_a,D2: set_a] :
( ( ord_less_eq_set_a @ A5 @ C2 )
=> ( ( ord_less_eq_set_a @ B3 @ D2 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A5 @ B3 ) @ ( inf_inf_set_a @ C2 @ D2 ) ) ) ) ).
% Int_mono
thf(fact_515_Int__lower1,axiom,
! [A5: set_set_a,B3: set_set_a] : ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A5 @ B3 ) @ A5 ) ).
% Int_lower1
thf(fact_516_Int__lower1,axiom,
! [A5: set_a,B3: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A5 @ B3 ) @ A5 ) ).
% Int_lower1
thf(fact_517_Int__lower2,axiom,
! [A5: set_set_a,B3: set_set_a] : ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A5 @ B3 ) @ B3 ) ).
% Int_lower2
thf(fact_518_Int__lower2,axiom,
! [A5: set_a,B3: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A5 @ B3 ) @ B3 ) ).
% Int_lower2
thf(fact_519_Int__absorb1,axiom,
! [B3: set_set_a,A5: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B3 @ A5 )
=> ( ( inf_inf_set_set_a @ A5 @ B3 )
= B3 ) ) ).
% Int_absorb1
thf(fact_520_Int__absorb1,axiom,
! [B3: set_a,A5: set_a] :
( ( ord_less_eq_set_a @ B3 @ A5 )
=> ( ( inf_inf_set_a @ A5 @ B3 )
= B3 ) ) ).
% Int_absorb1
thf(fact_521_Int__absorb2,axiom,
! [A5: set_set_a,B3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A5 @ B3 )
=> ( ( inf_inf_set_set_a @ A5 @ B3 )
= A5 ) ) ).
% Int_absorb2
thf(fact_522_Int__absorb2,axiom,
! [A5: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A5 @ B3 )
=> ( ( inf_inf_set_a @ A5 @ B3 )
= A5 ) ) ).
% Int_absorb2
thf(fact_523_Int__greatest,axiom,
! [C2: set_set_a,A5: set_set_a,B3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ C2 @ A5 )
=> ( ( ord_le3724670747650509150_set_a @ C2 @ B3 )
=> ( ord_le3724670747650509150_set_a @ C2 @ ( inf_inf_set_set_a @ A5 @ B3 ) ) ) ) ).
% Int_greatest
thf(fact_524_Int__greatest,axiom,
! [C2: set_a,A5: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ C2 @ A5 )
=> ( ( ord_less_eq_set_a @ C2 @ B3 )
=> ( ord_less_eq_set_a @ C2 @ ( inf_inf_set_a @ A5 @ B3 ) ) ) ) ).
% Int_greatest
thf(fact_525_Int__Collect__mono,axiom,
! [A5: set_set_a,B3: set_set_a,P: set_a > $o,Q: set_a > $o] :
( ( ord_le3724670747650509150_set_a @ A5 @ B3 )
=> ( ! [X: set_a] :
( ( member_set_a @ X @ A5 )
=> ( ( P @ X )
=> ( Q @ X ) ) )
=> ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A5 @ ( collect_set_a @ P ) ) @ ( inf_inf_set_set_a @ B3 @ ( collect_set_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_526_Int__Collect__mono,axiom,
! [A5: set_a,B3: set_a,P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ A5 @ B3 )
=> ( ! [X: a] :
( ( member_a @ X @ A5 )
=> ( ( P @ X )
=> ( Q @ X ) ) )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A5 @ ( collect_a @ P ) ) @ ( inf_inf_set_a @ B3 @ ( collect_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_527_N_OCong__class__rep,axiom,
! [T7: set_a] :
( ( normal8595587647932138008lass_a @ resid @ nn @ T7 )
=> ( ( normal7408713899360725774lass_a @ resid @ nn @ ( normal3259722184653208495_rep_a @ T7 ) )
= T7 ) ) ).
% N.Cong_class_rep
thf(fact_528_N_OCong_H_Osimps,axiom,
! [A1: a,A22: a] :
( ( normal_sub_Cong_a2 @ resid @ nn @ A1 @ A22 )
= ( ? [T6: a,U5: a] :
( ( A1 = U5 )
& ( A22 = T6 )
& ( normal_sub_Cong_a2 @ resid @ nn @ T6 @ U5 ) )
| ? [T6: a,U5: a,V5: a] :
( ( A1 = T6 )
& ( A22 = V5 )
& ( normal_sub_Cong_a2 @ resid @ nn @ T6 @ U5 )
& ( normal_sub_Cong_a2 @ resid @ nn @ U5 @ V5 ) )
| ? [T6: a,U5: a] :
( ( A1 = T6 )
& ( A22 = U5 )
& ( member_a @ ( resid @ T6 @ U5 ) @ nn )
& ( member_a @ ( resid @ U5 @ T6 ) @ nn ) )
| ? [T6: a,U5: a] :
( ( A1 = T6 )
& ( A22
= ( resid @ T6 @ U5 ) )
& ( arr_a @ resid @ T6 )
& ( member_a @ U5 @ nn )
& ( ( sources_a @ resid @ T6 )
= ( sources_a @ resid @ U5 ) ) ) ) ) ).
% N.Cong'.simps
thf(fact_529_N_OCong_H_Ocases,axiom,
! [A1: a,A22: a] :
( ( normal_sub_Cong_a2 @ resid @ nn @ A1 @ A22 )
=> ( ~ ( normal_sub_Cong_a2 @ resid @ nn @ A22 @ A1 )
=> ( ! [U3: a] :
( ( normal_sub_Cong_a2 @ resid @ nn @ A1 @ U3 )
=> ~ ( normal_sub_Cong_a2 @ resid @ nn @ U3 @ A22 ) )
=> ( ~ ( ( member_a @ ( resid @ A1 @ A22 ) @ nn )
& ( member_a @ ( resid @ A22 @ A1 ) @ nn ) )
=> ~ ! [U3: a] :
( ( A22
= ( resid @ A1 @ U3 ) )
=> ( ( arr_a @ resid @ A1 )
=> ( ( member_a @ U3 @ nn )
=> ( ( sources_a @ resid @ A1 )
!= ( sources_a @ resid @ U3 ) ) ) ) ) ) ) ) ) ).
% N.Cong'.cases
thf(fact_530_N_OCong_H_Ointros_I4_J,axiom,
! [T: a,U: a] :
( ( arr_a @ resid @ T )
=> ( ( member_a @ U @ nn )
=> ( ( ( sources_a @ resid @ T )
= ( sources_a @ resid @ U ) )
=> ( normal_sub_Cong_a2 @ resid @ nn @ T @ ( resid @ T @ U ) ) ) ) ) ).
% N.Cong'.intros(4)
thf(fact_531_join__of__arr__src_I1_J,axiom,
! [T: set_a,A3: set_a] :
( ( arr_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T )
=> ( ( member_set_a @ A3 @ ( sources_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T ) )
=> ( join_of_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ A3 @ T @ T ) ) ) ).
% join_of_arr_src(1)
thf(fact_532_join__of__arr__src_I2_J,axiom,
! [T: set_a,A3: set_a] :
( ( arr_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T )
=> ( ( member_set_a @ A3 @ ( sources_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T ) )
=> ( join_of_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ A3 @ T ) ) ) ).
% join_of_arr_src(2)
thf(fact_533_R_Oresiduation__axioms,axiom,
residuation_a @ resid ).
% R.residuation_axioms
thf(fact_534_R_Otarget__is__ide,axiom,
! [A3: a,T: a] :
( ( member_a @ A3 @ ( targets_a @ resid @ T ) )
=> ( ide_a @ resid @ A3 ) ) ).
% R.target_is_ide
thf(fact_535_R_Otargets__are__cong,axiom,
! [B: a,T: a,B4: a] :
( ( member_a @ B @ ( targets_a @ resid @ T ) )
=> ( ( member_a @ B4 @ ( targets_a @ resid @ T ) )
=> ( ( ide_a @ resid @ ( resid @ B @ B4 ) )
& ( ide_a @ resid @ ( resid @ B4 @ B ) ) ) ) ) ).
% R.targets_are_cong
thf(fact_536_R_Otargets__cong__closed,axiom,
! [B: a,T: a,B4: a] :
( ( member_a @ B @ ( targets_a @ resid @ T ) )
=> ( ( ( ide_a @ resid @ ( resid @ B @ B4 ) )
& ( ide_a @ resid @ ( resid @ B4 @ B ) ) )
=> ( member_a @ B4 @ ( targets_a @ resid @ T ) ) ) ) ).
% R.targets_cong_closed
thf(fact_537_R_Ocong__respects__seq,axiom,
! [T: a,U: a,T4: a,U7: a] :
( ( seq_a @ resid @ T @ U )
=> ( ( ( ide_a @ resid @ ( resid @ T @ T4 ) )
& ( ide_a @ resid @ ( resid @ T4 @ T ) ) )
=> ( ( ( ide_a @ resid @ ( resid @ U @ U7 ) )
& ( ide_a @ resid @ ( resid @ U7 @ U ) ) )
=> ( seq_a @ resid @ T4 @ U7 ) ) ) ) ).
% R.cong_respects_seq
thf(fact_538_R_Otargets__are__con,axiom,
! [B: a,T: a,B4: a] :
( ( member_a @ B @ ( targets_a @ resid @ T ) )
=> ( ( member_a @ B4 @ ( targets_a @ resid @ T ) )
=> ( con_a @ resid @ B @ B4 ) ) ) ).
% R.targets_are_con
thf(fact_539_R_Otargets__resid__sym,axiom,
! [T: a,U: a] :
( ( con_a @ resid @ T @ U )
=> ( ( targets_a @ resid @ ( resid @ T @ U ) )
= ( targets_a @ resid @ ( resid @ U @ T ) ) ) ) ).
% R.targets_resid_sym
thf(fact_540_R_Oresid__source__in__targets,axiom,
! [A3: a,T: a] :
( ( member_a @ A3 @ ( sources_a @ resid @ T ) )
=> ( member_a @ ( resid @ A3 @ T ) @ ( targets_a @ resid @ T ) ) ) ).
% R.resid_source_in_targets
thf(fact_541_R_Ocomposable__imp__seq,axiom,
! [T: a,U: a] :
( ( composable_a @ resid @ T @ U )
=> ( seq_a @ resid @ T @ U ) ) ).
% R.composable_imp_seq
thf(fact_542_R_Otargets__con__closed,axiom,
! [B: a,T: a,B4: a] :
( ( member_a @ B @ ( targets_a @ resid @ T ) )
=> ( ( ide_a @ resid @ B4 )
=> ( ( con_a @ resid @ B @ B4 )
=> ( member_a @ B4 @ ( targets_a @ resid @ T ) ) ) ) ) ).
% R.targets_con_closed
thf(fact_543_R_OseqE,axiom,
! [T: a,U: a] :
( ( seq_a @ resid @ T @ U )
=> ~ ( ( arr_a @ resid @ T )
=> ( ( arr_a @ resid @ U )
=> ( ( targets_a @ resid @ T )
!= ( sources_a @ resid @ U ) ) ) ) ) ).
% R.seqE
thf(fact_544_R_Oseq__def,axiom,
! [T: a,U: a] :
( ( seq_a @ resid @ T @ U )
= ( ( arr_a @ resid @ T )
& ( arr_a @ resid @ U )
& ( ( targets_a @ resid @ T )
= ( sources_a @ resid @ U ) ) ) ) ).
% R.seq_def
thf(fact_545_R_Oarr__iff__has__target,axiom,
! [T: a] :
( ( arr_a @ resid @ T )
= ( ( targets_a @ resid @ T )
!= bot_bot_set_a ) ) ).
% R.arr_iff_has_target
thf(fact_546_N_Ocoherent_H,axiom,
! [V: a,V4: a,W2: a,W3: a,T: a,T4: a] :
( ( member_a @ V @ nn )
=> ( ( member_a @ V4 @ nn )
=> ( ( member_a @ W2 @ nn )
=> ( ( member_a @ W3 @ nn )
=> ( ( ( sources_a @ resid @ V )
= ( sources_a @ resid @ W2 ) )
=> ( ( ( sources_a @ resid @ V4 )
= ( sources_a @ resid @ W3 ) )
=> ( ( ( targets_a @ resid @ W2 )
= ( targets_a @ resid @ W3 ) )
=> ( ( ( member_a @ ( resid @ ( resid @ T @ V ) @ ( resid @ T4 @ V4 ) ) @ nn )
& ( member_a @ ( resid @ ( resid @ T4 @ V4 ) @ ( resid @ T @ V ) ) @ nn ) )
=> ( ( member_a @ ( resid @ ( resid @ T @ W2 ) @ ( resid @ T4 @ W3 ) ) @ nn )
& ( member_a @ ( resid @ ( resid @ T4 @ W3 ) @ ( resid @ T @ W2 ) ) @ nn ) ) ) ) ) ) ) ) ) ) ).
% N.coherent'
thf(fact_547_R_Otargets__char,axiom,
! [T: a] :
( ( targets_a @ resid @ T )
= ( collect_a
@ ^ [B2: a] :
( ( arr_a @ resid @ T )
& ( ide_a @ resid @ ( resid @ ( resid @ T @ T ) @ B2 ) )
& ( ide_a @ resid @ ( resid @ B2 @ ( resid @ T @ T ) ) ) ) ) ) ).
% R.targets_char
thf(fact_548_N_Oin__targets__respects__Cong,axiom,
! [T: a,T4: a,B: a,B4: a] :
( ( normal_sub_Cong_a @ resid @ nn @ T @ T4 )
=> ( ( member_a @ B @ ( targets_a @ resid @ T ) )
=> ( ( member_a @ B4 @ ( targets_a @ resid @ T4 ) )
=> ( normal_sub_Cong_a @ resid @ nn @ B @ B4 ) ) ) ) ).
% N.in_targets_respects_Cong
thf(fact_549_N_Otargets__are__Cong,axiom,
! [B: a,T: a,B4: a] :
( ( member_a @ B @ ( targets_a @ resid @ T ) )
=> ( ( member_a @ B4 @ ( targets_a @ resid @ T ) )
=> ( normal_sub_Cong_a @ resid @ nn @ B @ B4 ) ) ) ).
% N.targets_are_Cong
thf(fact_550_R_OcoterminalE,axiom,
! [T: a,U: a] :
( ( coterminal_a @ resid @ T @ U )
=> ~ ( ( arr_a @ resid @ T )
=> ( ( arr_a @ resid @ U )
=> ( ( targets_a @ resid @ T )
!= ( targets_a @ resid @ U ) ) ) ) ) ).
% R.coterminalE
thf(fact_551_R_Ocoterminal__iff,axiom,
! [T: a,T4: a] :
( ( coterminal_a @ resid @ T @ T4 )
= ( ( arr_a @ resid @ T )
& ( arr_a @ resid @ T4 )
& ( ( targets_a @ resid @ T )
= ( targets_a @ resid @ T4 ) ) ) ) ).
% R.coterminal_iff
thf(fact_552_R_OcomposableD_I3_J,axiom,
! [T: a,U: a] :
( ( composable_a @ resid @ T @ U )
=> ( ( targets_a @ resid @ T )
= ( sources_a @ resid @ U ) ) ) ).
% R.composableD(3)
thf(fact_553_N_OCong_H_Ointros_I3_J,axiom,
! [T: a,U: a] :
( ( ( member_a @ ( resid @ T @ U ) @ nn )
& ( member_a @ ( resid @ U @ T ) @ nn ) )
=> ( normal_sub_Cong_a2 @ resid @ nn @ T @ U ) ) ).
% N.Cong'.intros(3)
thf(fact_554_N_OCong_H_Ointros_I2_J,axiom,
! [T: a,U: a,V: a] :
( ( normal_sub_Cong_a2 @ resid @ nn @ T @ U )
=> ( ( normal_sub_Cong_a2 @ resid @ nn @ U @ V )
=> ( normal_sub_Cong_a2 @ resid @ nn @ T @ V ) ) ) ).
% N.Cong'.intros(2)
thf(fact_555_N_OCong_H_Ointros_I1_J,axiom,
! [T: a,U: a] :
( ( normal_sub_Cong_a2 @ resid @ nn @ T @ U )
=> ( normal_sub_Cong_a2 @ resid @ nn @ U @ T ) ) ).
% N.Cong'.intros(1)
thf(fact_556_N_OCong_H__if,axiom,
! [U: a,U7: a,T: a,T4: a] :
( ( member_a @ U @ nn )
=> ( ( member_a @ U7 @ nn )
=> ( ( ( member_a @ ( resid @ ( resid @ T @ U ) @ ( resid @ T4 @ U7 ) ) @ nn )
& ( member_a @ ( resid @ ( resid @ T4 @ U7 ) @ ( resid @ T @ U ) ) @ nn ) )
=> ( normal_sub_Cong_a2 @ resid @ nn @ T @ T4 ) ) ) ) ).
% N.Cong'_if
thf(fact_557_R_Otargets__eqI,axiom,
! [T: a,T4: a] :
( ( ( inf_inf_set_a @ ( targets_a @ resid @ T ) @ ( targets_a @ resid @ T4 ) )
!= bot_bot_set_a )
=> ( ( targets_a @ resid @ T )
= ( targets_a @ resid @ T4 ) ) ) ).
% R.targets_eqI
thf(fact_558_join__of__symmetric,axiom,
! [T: set_a,U: set_a,V: set_a] :
( ( join_of_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U @ V )
=> ( join_of_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ U @ T @ V ) ) ).
% join_of_symmetric
thf(fact_559_join__of__unique,axiom,
! [T: set_a,U: set_a,V: set_a,V4: set_a] :
( ( join_of_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U @ V )
=> ( ( join_of_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U @ V4 )
=> ( V = V4 ) ) ) ).
% join_of_unique
thf(fact_560_N_Ocoherent,axiom,
! [T: a,U: a,U7: a] :
( ( arr_a @ resid @ T )
=> ( ( member_a @ U @ nn )
=> ( ( member_a @ U7 @ nn )
=> ( ( ( sources_a @ resid @ U )
= ( sources_a @ resid @ U7 ) )
=> ( ( ( targets_a @ resid @ U )
= ( targets_a @ resid @ U7 ) )
=> ( ( ( sources_a @ resid @ T )
= ( sources_a @ resid @ U ) )
=> ( ( member_a @ ( resid @ ( resid @ T @ U ) @ ( resid @ T @ U7 ) ) @ nn )
& ( member_a @ ( resid @ ( resid @ T @ U7 ) @ ( resid @ T @ U ) ) @ nn ) ) ) ) ) ) ) ) ).
% N.coherent
thf(fact_561_N_OCong__char,axiom,
! [T: a,T4: a] :
( ( normal_sub_Cong_a @ resid @ nn @ T @ T4 )
= ( normal_sub_Cong_a2 @ resid @ nn @ T @ T4 ) ) ).
% N.Cong_char
thf(fact_562_R_Ocoterminal__def,axiom,
! [T: a,U: a] :
( ( coterminal_a @ resid @ T @ U )
= ( ( inf_inf_set_a @ ( targets_a @ resid @ T ) @ ( targets_a @ resid @ U ) )
!= bot_bot_set_a ) ) ).
% R.coterminal_def
thf(fact_563_join__of__un__upto__cong,axiom,
! [T: set_a,U: set_a,V: set_a,V4: set_a] :
( ( join_of_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U @ V )
=> ( ( join_of_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U @ V4 )
=> ( ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ V @ V4 ) )
& ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ V4 @ V ) ) ) ) ) ).
% join_of_un_upto_cong
thf(fact_564_con__with__join__of__iff_I2_J,axiom,
! [T: set_a,U: set_a,W2: set_a,V: set_a] :
( ( join_of_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U @ W2 )
=> ( ( con_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ W2 @ V )
=> ( ( con_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ V )
& ( con_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ V @ T ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ U @ T ) ) ) ) ) ).
% con_with_join_of_iff(2)
thf(fact_565_con__with__join__of__iff_I1_J,axiom,
! [T: set_a,U: set_a,W2: set_a,V: set_a] :
( ( join_of_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U @ W2 )
=> ( ( ( con_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ U @ V )
& ( con_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ V @ U ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ T @ U ) ) )
=> ( con_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ W2 @ V ) ) ) ).
% con_with_join_of_iff(1)
thf(fact_566_join__of__resid,axiom,
! [T: set_a,U: set_a,W2: set_a,V: set_a] :
( ( join_of_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U @ W2 )
=> ( ( con_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ V @ W2 )
=> ( join_of_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ T @ V ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ U @ V ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ W2 @ V ) ) ) ) ).
% join_of_resid
thf(fact_567_join__of__arr__self,axiom,
! [T: set_a] :
( ( arr_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T )
=> ( join_of_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ T @ T ) ) ).
% join_of_arr_self
thf(fact_568_sources__join__of_I2_J,axiom,
! [T: set_a,U: set_a,V: set_a] :
( ( join_of_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U @ V )
=> ( ( sources_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ U )
= ( sources_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ V ) ) ) ).
% sources_join_of(2)
thf(fact_569_sources__join__of_I1_J,axiom,
! [T: set_a,U: set_a,V: set_a] :
( ( join_of_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U @ V )
=> ( ( sources_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T )
= ( sources_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ V ) ) ) ).
% sources_join_of(1)
thf(fact_570_targets__join__of_I2_J,axiom,
! [T: set_a,U: set_a,V: set_a] :
( ( join_of_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U @ V )
=> ( ( targets_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ U @ T ) )
= ( targets_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ V ) ) ) ).
% targets_join_of(2)
thf(fact_571_targets__join__of_I1_J,axiom,
! [T: set_a,U: set_a,V: set_a] :
( ( join_of_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U @ V )
=> ( ( targets_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ T @ U ) )
= ( targets_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ V ) ) ) ).
% targets_join_of(1)
thf(fact_572_src__join__of_I2_J,axiom,
! [T: set_a,U: set_a,V: set_a] :
( ( join_of_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U @ V )
=> ( ( weakly2061155085811118449_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ U )
= ( weakly2061155085811118449_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ V ) ) ) ).
% src_join_of(2)
thf(fact_573_src__join__of_I1_J,axiom,
! [T: set_a,U: set_a,V: set_a] :
( ( join_of_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U @ V )
=> ( ( weakly2061155085811118449_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T )
= ( weakly2061155085811118449_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ V ) ) ) ).
% src_join_of(1)
thf(fact_574_joinable__def,axiom,
! [T: set_a,U: set_a] :
( ( joinable_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U )
= ( ? [X5: set_a] : ( join_of_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U @ X5 ) ) ) ).
% joinable_def
thf(fact_575_R_Osources__resid,axiom,
! [T: a,U: a] :
( ( con_a @ resid @ T @ U )
=> ( ( sources_a @ resid @ ( resid @ T @ U ) )
= ( targets_a @ resid @ U ) ) ) ).
% R.sources_resid
thf(fact_576_R_OseqI,axiom,
! [T: a,U: a] :
( ( arr_a @ resid @ T )
=> ( ( arr_a @ resid @ U )
=> ( ( ( targets_a @ resid @ T )
= ( sources_a @ resid @ U ) )
=> ( seq_a @ resid @ T @ U ) ) ) ) ).
% R.seqI
thf(fact_577_R_OcoterminalI,axiom,
! [T: a,U: a] :
( ( arr_a @ resid @ T )
=> ( ( ( targets_a @ resid @ T )
= ( targets_a @ resid @ U ) )
=> ( coterminal_a @ resid @ T @ U ) ) ) ).
% R.coterminalI
thf(fact_578_normal__sub__rts_OCong_H_Ocong,axiom,
normal_sub_Cong_a2 = normal_sub_Cong_a2 ).
% normal_sub_rts.Cong'.cong
thf(fact_579_rts_Ojoin__of_Ocong,axiom,
join_of_set_a = join_of_set_a ).
% rts.join_of.cong
thf(fact_580_rts_Ojoin__of_Ocong,axiom,
join_of_a = join_of_a ).
% rts.join_of.cong
thf(fact_581_join__is__join__of,axiom,
! [T: set_a,U: set_a] :
( ( joinable_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U )
=> ( join_of_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U @ ( extens1973556086528668384_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U ) ) ) ).
% join_is_join_of
thf(fact_582_src__join,axiom,
! [T: set_a,U: set_a] :
( ( joinable_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U )
=> ( ( weakly2061155085811118449_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( extens1973556086528668384_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U ) )
= ( weakly2061155085811118449_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T ) ) ) ).
% src_join
thf(fact_583_joinable__iff__arr__join,axiom,
! [T: set_a,U: set_a] :
( ( joinable_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U )
= ( arr_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( extens1973556086528668384_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U ) ) ) ).
% joinable_iff_arr_join
thf(fact_584_arr__prfx__join__self,axiom,
! [T: set_a,U: set_a] :
( ( joinable_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U )
=> ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ T @ ( extens1973556086528668384_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U ) ) ) ) ).
% arr_prfx_join_self
thf(fact_585_R_Ojoin__of__symmetric,axiom,
! [T: a,U: a,V: a] :
( ( join_of_a @ resid @ T @ U @ V )
=> ( join_of_a @ resid @ U @ T @ V ) ) ).
% R.join_of_symmetric
thf(fact_586_R_Ojoin__of__un__upto__cong,axiom,
! [T: a,U: a,V: a,V4: a] :
( ( join_of_a @ resid @ T @ U @ V )
=> ( ( join_of_a @ resid @ T @ U @ V4 )
=> ( ( ide_a @ resid @ ( resid @ V @ V4 ) )
& ( ide_a @ resid @ ( resid @ V4 @ V ) ) ) ) ) ).
% R.join_of_un_upto_cong
thf(fact_587_R_Ojoin__of__resid,axiom,
! [T: a,U: a,W2: a,V: a] :
( ( join_of_a @ resid @ T @ U @ W2 )
=> ( ( con_a @ resid @ V @ W2 )
=> ( join_of_a @ resid @ ( resid @ T @ V ) @ ( resid @ U @ V ) @ ( resid @ W2 @ V ) ) ) ) ).
% R.join_of_resid
thf(fact_588_R_Ocon__with__join__of__iff_I1_J,axiom,
! [T: a,U: a,W2: a,V: a] :
( ( join_of_a @ resid @ T @ U @ W2 )
=> ( ( ( con_a @ resid @ U @ V )
& ( con_a @ resid @ ( resid @ V @ U ) @ ( resid @ T @ U ) ) )
=> ( con_a @ resid @ W2 @ V ) ) ) ).
% R.con_with_join_of_iff(1)
thf(fact_589_R_Ocon__with__join__of__iff_I2_J,axiom,
! [T: a,U: a,W2: a,V: a] :
( ( join_of_a @ resid @ T @ U @ W2 )
=> ( ( con_a @ resid @ W2 @ V )
=> ( ( con_a @ resid @ T @ V )
& ( con_a @ resid @ ( resid @ V @ T ) @ ( resid @ U @ T ) ) ) ) ) ).
% R.con_with_join_of_iff(2)
thf(fact_590_R_Osources__join__of_I1_J,axiom,
! [T: a,U: a,V: a] :
( ( join_of_a @ resid @ T @ U @ V )
=> ( ( sources_a @ resid @ T )
= ( sources_a @ resid @ V ) ) ) ).
% R.sources_join_of(1)
thf(fact_591_R_Osources__join__of_I2_J,axiom,
! [T: a,U: a,V: a] :
( ( join_of_a @ resid @ T @ U @ V )
=> ( ( sources_a @ resid @ U )
= ( sources_a @ resid @ V ) ) ) ).
% R.sources_join_of(2)
thf(fact_592_R_Ojoin__of__arr__self,axiom,
! [T: a] :
( ( arr_a @ resid @ T )
=> ( join_of_a @ resid @ T @ T @ T ) ) ).
% R.join_of_arr_self
thf(fact_593_R_Otargets__join__of_I1_J,axiom,
! [T: a,U: a,V: a] :
( ( join_of_a @ resid @ T @ U @ V )
=> ( ( targets_a @ resid @ ( resid @ T @ U ) )
= ( targets_a @ resid @ V ) ) ) ).
% R.targets_join_of(1)
thf(fact_594_R_Otargets__join__of_I2_J,axiom,
! [T: a,U: a,V: a] :
( ( join_of_a @ resid @ T @ U @ V )
=> ( ( targets_a @ resid @ ( resid @ U @ T ) )
= ( targets_a @ resid @ V ) ) ) ).
% R.targets_join_of(2)
thf(fact_595_R_Ojoinable__def,axiom,
! [T: a,U: a] :
( ( joinable_a @ resid @ T @ U )
= ( ? [X5: a] : ( join_of_a @ resid @ T @ U @ X5 ) ) ) ).
% R.joinable_def
thf(fact_596_R_Ojoin__of__arr__src_I1_J,axiom,
! [T: a,A3: a] :
( ( arr_a @ resid @ T )
=> ( ( member_a @ A3 @ ( sources_a @ resid @ T ) )
=> ( join_of_a @ resid @ A3 @ T @ T ) ) ) ).
% R.join_of_arr_src(1)
thf(fact_597_R_Ojoin__of__arr__src_I2_J,axiom,
! [T: a,A3: a] :
( ( arr_a @ resid @ T )
=> ( ( member_a @ A3 @ ( sources_a @ resid @ T ) )
=> ( join_of_a @ resid @ T @ A3 @ T ) ) ) ).
% R.join_of_arr_src(2)
thf(fact_598_order__refl,axiom,
! [X4: set_a] : ( ord_less_eq_set_a @ X4 @ X4 ) ).
% order_refl
thf(fact_599_dual__order_Orefl,axiom,
! [A3: set_a] : ( ord_less_eq_set_a @ A3 @ A3 ) ).
% dual_order.refl
thf(fact_600_join__eqI,axiom,
! [T: set_a,V: set_a,U: set_a] :
( ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ T @ V ) )
=> ( ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ U @ V ) )
=> ( ( ( quotie8165075472272353145esid_a @ resid @ nn @ V @ U )
= ( quotie8165075472272353145esid_a @ resid @ nn @ T @ U ) )
=> ( ( ( quotie8165075472272353145esid_a @ resid @ nn @ V @ T )
= ( quotie8165075472272353145esid_a @ resid @ nn @ U @ T ) )
=> ( ( extens1973556086528668384_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U )
= V ) ) ) ) ) ).
% join_eqI
thf(fact_601_join__self,axiom,
! [T: set_a] :
( ( arr_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T )
=> ( ( extens1973556086528668384_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ T )
= T ) ) ).
% join_self
thf(fact_602_join__src,axiom,
! [T: set_a] :
( ( arr_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T )
=> ( ( extens1973556086528668384_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( weakly2061155085811118449_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T ) @ T )
= T ) ) ).
% join_src
thf(fact_603_resid__join_092_060_094sub_062E_I1_J,axiom,
! [T: set_a,U: set_a,V: set_a] :
( ( joinable_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U )
=> ( ( con_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ V @ ( extens1973556086528668384_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U ) )
=> ( ( quotie8165075472272353145esid_a @ resid @ nn @ V @ ( extens1973556086528668384_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U ) )
= ( quotie8165075472272353145esid_a @ resid @ nn @ ( quotie8165075472272353145esid_a @ resid @ nn @ V @ U ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ T @ U ) ) ) ) ) ).
% resid_join\<^sub>E(1)
thf(fact_604_resid__join_092_060_094sub_062E_I2_J,axiom,
! [T: set_a,U: set_a,V: set_a] :
( ( joinable_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U )
=> ( ( con_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ V @ ( extens1973556086528668384_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U ) )
=> ( ( quotie8165075472272353145esid_a @ resid @ nn @ V @ ( extens1973556086528668384_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U ) )
= ( quotie8165075472272353145esid_a @ resid @ nn @ ( quotie8165075472272353145esid_a @ resid @ nn @ V @ T ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ U @ T ) ) ) ) ) ).
% resid_join\<^sub>E(2)
thf(fact_605_resid__join_092_060_094sub_062E_I3_J,axiom,
! [T: set_a,U: set_a,V: set_a] :
( ( joinable_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U )
=> ( ( con_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ V @ ( extens1973556086528668384_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U ) )
=> ( ( quotie8165075472272353145esid_a @ resid @ nn @ ( extens1973556086528668384_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U ) @ V )
= ( extens1973556086528668384_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ T @ V ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ U @ V ) ) ) ) ) ).
% resid_join\<^sub>E(3)
thf(fact_606_extensional__rts_Ojoin_Ocong,axiom,
extens1973556086528668384_set_a = extens1973556086528668384_set_a ).
% extensional_rts.join.cong
thf(fact_607_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_a,Z3: set_a] : ( Y5 = Z3 ) )
= ( ^ [X2: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y2 )
& ( ord_less_eq_set_a @ Y2 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_608_ord__eq__le__trans,axiom,
! [A3: set_a,B: set_a,C: set_a] :
( ( A3 = B )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ord_less_eq_set_a @ A3 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_609_ord__le__eq__trans,axiom,
! [A3: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A3 @ B )
=> ( ( B = C )
=> ( ord_less_eq_set_a @ A3 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_610_order__antisym,axiom,
! [X4: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y4 )
=> ( ( ord_less_eq_set_a @ Y4 @ X4 )
=> ( X4 = Y4 ) ) ) ).
% order_antisym
thf(fact_611_order_Otrans,axiom,
! [A3: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A3 @ B )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ord_less_eq_set_a @ A3 @ C ) ) ) ).
% order.trans
thf(fact_612_order__trans,axiom,
! [X4: set_a,Y4: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y4 )
=> ( ( ord_less_eq_set_a @ Y4 @ Z )
=> ( ord_less_eq_set_a @ X4 @ Z ) ) ) ).
% order_trans
thf(fact_613_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: set_a,Z3: set_a] : ( Y5 = Z3 ) )
= ( ^ [A: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ B2 @ A )
& ( ord_less_eq_set_a @ A @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_614_dual__order_Oantisym,axiom,
! [B: set_a,A3: set_a] :
( ( ord_less_eq_set_a @ B @ A3 )
=> ( ( ord_less_eq_set_a @ A3 @ B )
=> ( A3 = B ) ) ) ).
% dual_order.antisym
thf(fact_615_dual__order_Otrans,axiom,
! [B: set_a,A3: set_a,C: set_a] :
( ( ord_less_eq_set_a @ B @ A3 )
=> ( ( ord_less_eq_set_a @ C @ B )
=> ( ord_less_eq_set_a @ C @ A3 ) ) ) ).
% dual_order.trans
thf(fact_616_antisym,axiom,
! [A3: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A3 @ B )
=> ( ( ord_less_eq_set_a @ B @ A3 )
=> ( A3 = B ) ) ) ).
% antisym
thf(fact_617_order__eq__iff,axiom,
( ( ^ [Y5: set_a,Z3: set_a] : ( Y5 = Z3 ) )
= ( ^ [A: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A @ B2 )
& ( ord_less_eq_set_a @ B2 @ A ) ) ) ) ).
% order_eq_iff
thf(fact_618_order__subst1,axiom,
! [A3: set_a,F: set_a > set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A3 @ ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ A3 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_619_order__subst2,axiom,
! [A3: set_a,B: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_eq_set_a @ A3 @ B )
=> ( ( ord_less_eq_set_a @ ( F @ B ) @ C )
=> ( ! [X: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A3 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_620_order__eq__refl,axiom,
! [X4: set_a,Y4: set_a] :
( ( X4 = Y4 )
=> ( ord_less_eq_set_a @ X4 @ Y4 ) ) ).
% order_eq_refl
thf(fact_621_ord__eq__le__subst,axiom,
! [A3: set_a,F: set_a > set_a,B: set_a,C: set_a] :
( ( A3
= ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ A3 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_622_ord__le__eq__subst,axiom,
! [A3: set_a,B: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_eq_set_a @ A3 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A3 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_623_order__antisym__conv,axiom,
! [Y4: set_a,X4: set_a] :
( ( ord_less_eq_set_a @ Y4 @ X4 )
=> ( ( ord_less_eq_set_a @ X4 @ Y4 )
= ( X4 = Y4 ) ) ) ).
% order_antisym_conv
thf(fact_624_bot_Oextremum,axiom,
! [A3: set_set_a] : ( ord_le3724670747650509150_set_a @ bot_bot_set_set_a @ A3 ) ).
% bot.extremum
thf(fact_625_bot_Oextremum,axiom,
! [A3: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A3 ) ).
% bot.extremum
thf(fact_626_bot_Oextremum__unique,axiom,
! [A3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A3 @ bot_bot_set_set_a )
= ( A3 = bot_bot_set_set_a ) ) ).
% bot.extremum_unique
thf(fact_627_bot_Oextremum__unique,axiom,
! [A3: set_a] :
( ( ord_less_eq_set_a @ A3 @ bot_bot_set_a )
= ( A3 = bot_bot_set_a ) ) ).
% bot.extremum_unique
thf(fact_628_bot_Oextremum__uniqueI,axiom,
! [A3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A3 @ bot_bot_set_set_a )
=> ( A3 = bot_bot_set_set_a ) ) ).
% bot.extremum_uniqueI
thf(fact_629_bot_Oextremum__uniqueI,axiom,
! [A3: set_a] :
( ( ord_less_eq_set_a @ A3 @ bot_bot_set_a )
=> ( A3 = bot_bot_set_a ) ) ).
% bot.extremum_uniqueI
thf(fact_630_comp__join_I1_J,axiom,
! [T: set_a,U: set_a,U7: set_a] :
( ( joinable_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( extens7801945855595804251_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U ) @ ( extens7801945855595804251_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U7 ) )
=> ( composable_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ ( extens1973556086528668384_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ U @ U7 ) ) ) ).
% comp_join(1)
thf(fact_631_seqE_092_060_094sub_062W_092_060_094sub_062E,axiom,
! [T: set_a,U: set_a] :
( ( seq_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U )
=> ~ ( ( arr_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ U )
=> ( ( arr_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T )
=> ( ( trg_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T )
!= ( weakly2061155085811118449_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ U ) ) ) ) ) ).
% seqE\<^sub>W\<^sub>E
thf(fact_632_targets__def,axiom,
! [T: set_a] :
( ( targets_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T )
= ( collect_set_a
@ ^ [B2: set_a] :
( ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ B2 )
& ( con_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( trg_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T ) @ B2 ) ) ) ) ).
% targets_def
thf(fact_633_joinable__iff__join__not__null,axiom,
! [T: set_a,U: set_a] :
( ( joinable_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U )
= ( ( extens1973556086528668384_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U )
!= ( partial_null_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) ) ) ) ).
% joinable_iff_join_not_null
thf(fact_634_in__targetsE,axiom,
! [B: set_a,T: set_a] :
( ( member_set_a @ B @ ( targets_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T ) )
=> ~ ( ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ B )
=> ~ ( con_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( trg_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T ) @ B ) ) ) ).
% in_targetsE
thf(fact_635_apex__sym,axiom,
! [T: set_a,U: set_a] :
( ( trg_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ T @ U ) )
= ( trg_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ U @ T ) ) ) ).
% apex_sym
thf(fact_636_trg__def,axiom,
! [T: set_a] :
( ( trg_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T )
= ( quotie8165075472272353145esid_a @ resid @ nn @ T @ T ) ) ).
% trg_def
thf(fact_637_con__imp__arr__resid,axiom,
! [T: set_a,U: set_a] :
( ( ( quotie8165075472272353145esid_a @ resid @ nn @ T @ U )
!= ( partial_null_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) ) )
=> ( ( quotie8165075472272353145esid_a @ resid @ nn @ ( quotie8165075472272353145esid_a @ resid @ nn @ T @ U ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ T @ U ) )
!= ( partial_null_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) ) ) ) ).
% con_imp_arr_resid
thf(fact_638_con__sym__ax,axiom,
! [T: set_a,U: set_a] :
( ( ( quotie8165075472272353145esid_a @ resid @ nn @ T @ U )
!= ( partial_null_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) ) )
=> ( ( quotie8165075472272353145esid_a @ resid @ nn @ U @ T )
!= ( partial_null_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) ) ) ) ).
% con_sym_ax
thf(fact_639_cube__ax,axiom,
! [V: set_a,T: set_a,U: set_a] :
( ( ( quotie8165075472272353145esid_a @ resid @ nn @ ( quotie8165075472272353145esid_a @ resid @ nn @ V @ T ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ U @ T ) )
!= ( partial_null_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) ) )
=> ( ( quotie8165075472272353145esid_a @ resid @ nn @ ( quotie8165075472272353145esid_a @ resid @ nn @ V @ T ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ U @ T ) )
= ( quotie8165075472272353145esid_a @ resid @ nn @ ( quotie8165075472272353145esid_a @ resid @ nn @ V @ U ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ T @ U ) ) ) ) ).
% cube_ax
thf(fact_640_null__eqI,axiom,
! [N: set_a] :
( ! [T3: set_a] :
( ( ( quotie8165075472272353145esid_a @ resid @ nn @ N @ T3 )
= N )
& ( ( quotie8165075472272353145esid_a @ resid @ nn @ T3 @ N )
= N ) )
=> ( N
= ( partial_null_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) ) ) ) ).
% null_eqI
thf(fact_641_comp__eqI,axiom,
! [T: set_a,V: set_a,U: set_a] :
( ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ T @ V ) )
=> ( ( U
= ( quotie8165075472272353145esid_a @ resid @ nn @ V @ T ) )
=> ( ( extens7801945855595804251_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U )
= V ) ) ) ).
% comp_eqI
thf(fact_642_comp__ide__self,axiom,
! [A3: set_a] :
( ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ A3 )
=> ( ( extens7801945855595804251_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ A3 @ A3 )
= A3 ) ) ).
% comp_ide_self
thf(fact_643_prfx__decomp,axiom,
! [T: set_a,U: set_a] :
( ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ T @ U ) )
=> ( ( extens7801945855595804251_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ ( quotie8165075472272353145esid_a @ resid @ nn @ U @ T ) )
= U ) ) ).
% prfx_decomp
thf(fact_644_resid__comp_I2_J,axiom,
! [T: set_a,U: set_a,W2: set_a] :
( ( con_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( extens7801945855595804251_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U ) @ W2 )
=> ( ( quotie8165075472272353145esid_a @ resid @ nn @ ( extens7801945855595804251_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U ) @ W2 )
= ( extens7801945855595804251_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ T @ W2 ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ U @ ( quotie8165075472272353145esid_a @ resid @ nn @ W2 @ T ) ) ) ) ) ).
% resid_comp(2)
thf(fact_645_resid__comp_I1_J,axiom,
! [T: set_a,U: set_a,W2: set_a] :
( ( con_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( extens7801945855595804251_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U ) @ W2 )
=> ( ( quotie8165075472272353145esid_a @ resid @ nn @ W2 @ ( extens7801945855595804251_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U ) )
= ( quotie8165075472272353145esid_a @ resid @ nn @ ( quotie8165075472272353145esid_a @ resid @ nn @ W2 @ T ) @ U ) ) ) ).
% resid_comp(1)
thf(fact_646_trg__resid__sym,axiom,
! [T: set_a,U: set_a] :
( ( con_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U )
=> ( ( trg_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ T @ U ) )
= ( trg_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ U @ T ) ) ) ) ).
% trg_resid_sym
thf(fact_647_comp__cancel__left,axiom,
! [T: set_a,U: set_a,V: set_a] :
( ( arr_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( extens7801945855595804251_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U ) )
=> ( ( ( extens7801945855595804251_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U )
= ( extens7801945855595804251_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ V ) )
=> ( U = V ) ) ) ).
% comp_cancel_left
thf(fact_648_conE,axiom,
! [T: set_a,U: set_a] :
( ( con_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U )
=> ( ( quotie8165075472272353145esid_a @ resid @ nn @ T @ U )
!= ( partial_null_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) ) ) ) ).
% conE
thf(fact_649_con__def,axiom,
! [T: set_a,U: set_a] :
( ( con_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U )
= ( ( quotie8165075472272353145esid_a @ resid @ nn @ T @ U )
!= ( partial_null_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) ) ) ) ).
% con_def
thf(fact_650_not__arr__null,axiom,
~ ( arr_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( partial_null_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) ) ) ).
% not_arr_null
thf(fact_651_trg__join__of_I2_J,axiom,
! [T: set_a,U: set_a,V: set_a] :
( ( join_of_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U @ V )
=> ( ( trg_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ U @ T ) )
= ( trg_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ V ) ) ) ).
% trg_join_of(2)
thf(fact_652_trg__join__of_I1_J,axiom,
! [T: set_a,U: set_a,V: set_a] :
( ( join_of_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U @ V )
=> ( ( trg_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ T @ U ) )
= ( trg_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ V ) ) ) ).
% trg_join_of(1)
thf(fact_653_comp__assoc,axiom,
! [T: set_a,U: set_a,V: set_a] :
( ( composable_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( extens7801945855595804251_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U ) @ V )
=> ( ( extens7801945855595804251_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ ( extens7801945855595804251_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ U @ V ) )
= ( extens7801945855595804251_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( extens7801945855595804251_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U ) @ V ) ) ) ).
% comp_assoc
thf(fact_654_join__sym,axiom,
! [T: set_a,U: set_a] :
( ( ( extens1973556086528668384_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U )
!= ( partial_null_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) ) )
=> ( ( extens1973556086528668384_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U )
= ( extens1973556086528668384_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ U @ T ) ) ) ).
% join_sym
thf(fact_655_null__char,axiom,
( ( partial_null_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) )
= bot_bot_set_a ) ).
% null_char
thf(fact_656_bounded__imp__con_092_060_094sub_062E,axiom,
! [T: set_a,U: set_a,T4: set_a,U7: set_a] :
( ( ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ ( extens7801945855595804251_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U ) @ ( extens7801945855595804251_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T4 @ U7 ) ) )
& ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ ( extens7801945855595804251_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T4 @ U7 ) @ ( extens7801945855595804251_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U ) ) ) )
=> ( con_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ T4 ) ) ).
% bounded_imp_con\<^sub>E
thf(fact_657_prfx__comp,axiom,
! [U: set_a,T: set_a,V: set_a] :
( ( arr_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ U )
=> ( ( ( extens7801945855595804251_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ V )
= U )
=> ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ T @ U ) ) ) ) ).
% prfx_comp
thf(fact_658_ide__iff__trg__self,axiom,
! [A3: set_a] :
( ( arr_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ A3 )
=> ( ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ A3 )
= ( ( trg_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ A3 )
= A3 ) ) ) ).
% ide_iff_trg_self
thf(fact_659_ide__trg,axiom,
! [T: set_a] :
( ( arr_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T )
=> ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( trg_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T ) ) ) ).
% ide_trg
thf(fact_660_trg__in__targets,axiom,
! [T: set_a] :
( ( arr_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T )
=> ( member_set_a @ ( trg_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T ) @ ( targets_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T ) ) ) ).
% trg_in_targets
thf(fact_661_con__comp__iff,axiom,
! [W2: set_a,T: set_a,U: set_a] :
( ( con_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ W2 @ ( extens7801945855595804251_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U ) )
= ( ( composable_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U )
& ( con_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ W2 @ T ) @ U ) ) ) ).
% con_comp_iff
thf(fact_662_resid__ide_I2_J,axiom,
! [A3: set_a,T: set_a] :
( ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ A3 )
=> ( ( coinitial_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ A3 @ T )
=> ( ( quotie8165075472272353145esid_a @ resid @ nn @ A3 @ T )
= ( trg_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T ) ) ) ) ).
% resid_ide(2)
thf(fact_663_composable__iff__arr__comp,axiom,
! [T: set_a,U: set_a] :
( ( composable_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U )
= ( arr_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( extens7801945855595804251_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U ) ) ) ).
% composable_iff_arr_comp
thf(fact_664_composable__iff__comp__not__null,axiom,
! [T: set_a,U: set_a] :
( ( composable_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U )
= ( ( extens7801945855595804251_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U )
!= ( partial_null_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) ) ) ) ).
% composable_iff_comp_not_null
thf(fact_665_coterminal__iff__con__trg,axiom,
! [T: set_a,U: set_a] :
( ( coterminal_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U )
= ( con_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( trg_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T ) @ ( trg_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ U ) ) ) ).
% coterminal_iff_con_trg
thf(fact_666_coterminalE_092_060_094sub_062W_092_060_094sub_062E,axiom,
! [T: set_a,U: set_a] :
( ( coterminal_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U )
=> ~ ( ( arr_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T )
=> ( ( arr_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ U )
=> ( ( trg_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T )
!= ( trg_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ U ) ) ) ) ) ).
% coterminalE\<^sub>W\<^sub>E
thf(fact_667_coterminal__iff_092_060_094sub_062W_092_060_094sub_062E,axiom,
! [T: set_a,U: set_a] :
( ( coterminal_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U )
= ( ( arr_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T )
& ( arr_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ U )
& ( ( trg_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T )
= ( trg_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ U ) ) ) ) ).
% coterminal_iff\<^sub>W\<^sub>E
thf(fact_668_comp__join_I2_J,axiom,
! [T: set_a,U: set_a,U7: set_a] :
( ( joinable_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( extens7801945855595804251_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U ) @ ( extens7801945855595804251_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U7 ) )
=> ( ( extens7801945855595804251_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ ( extens1973556086528668384_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ U @ U7 ) )
= ( extens1973556086528668384_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( extens7801945855595804251_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U ) @ ( extens7801945855595804251_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U7 ) ) ) ) ).
% comp_join(2)
thf(fact_669_trg__join,axiom,
! [T: set_a,U: set_a] :
( ( joinable_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U )
=> ( ( trg_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( extens1973556086528668384_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U ) )
= ( trg_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ T @ U ) ) ) ) ).
% trg_join
thf(fact_670_targets__char_092_060_094sub_062W_092_060_094sub_062E,axiom,
! [T: set_a] :
( ( targets_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T )
= ( collect_set_a
@ ^ [B2: set_a] :
( ( arr_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T )
& ( ( trg_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T )
= B2 ) ) ) ) ).
% targets_char\<^sub>W\<^sub>E
thf(fact_671_trg__trg,axiom,
! [T: set_a] :
( ( trg_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( trg_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T ) )
= ( trg_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T ) ) ).
% trg_trg
thf(fact_672_null__is__zero_I2_J,axiom,
! [T: set_a] :
( ( quotie8165075472272353145esid_a @ resid @ nn @ T @ ( partial_null_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) ) )
= ( partial_null_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) ) ) ).
% null_is_zero(2)
thf(fact_673_null__is__zero_I1_J,axiom,
! [T: set_a] :
( ( quotie8165075472272353145esid_a @ resid @ nn @ ( partial_null_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) ) @ T )
= ( partial_null_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) ) ) ).
% null_is_zero(1)
thf(fact_674_trg__ide,axiom,
! [A3: set_a] :
( ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ A3 )
=> ( ( trg_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ A3 )
= A3 ) ) ).
% trg_ide
thf(fact_675_comp__resid__prfx,axiom,
! [T: set_a,U: set_a] :
( ( arr_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( extens7801945855595804251_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U ) )
=> ( ( quotie8165075472272353145esid_a @ resid @ nn @ ( extens7801945855595804251_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U ) @ T )
= U ) ) ).
% comp_resid_prfx
thf(fact_676_arr__trg__iff__arr,axiom,
! [T: set_a] :
( ( arr_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( trg_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T ) )
= ( arr_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T ) ) ).
% arr_trg_iff_arr
thf(fact_677_conI,axiom,
! [T: set_a,U: set_a] :
( ( ( quotie8165075472272353145esid_a @ resid @ nn @ T @ U )
!= ( partial_null_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) ) )
=> ( con_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U ) ) ).
% conI
thf(fact_678_src__trg,axiom,
! [T: set_a] :
( ( weakly2061155085811118449_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( trg_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T ) )
= ( trg_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T ) ) ).
% src_trg
thf(fact_679_trg__src,axiom,
! [T: set_a] :
( ( trg_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( weakly2061155085811118449_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T ) )
= ( weakly2061155085811118449_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T ) ) ).
% trg_src
thf(fact_680_comp__null_I2_J,axiom,
! [T: set_a] :
( ( extens7801945855595804251_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ ( partial_null_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) ) )
= ( partial_null_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) ) ) ).
% comp_null(2)
thf(fact_681_comp__null_I1_J,axiom,
! [T: set_a] :
( ( extens7801945855595804251_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( partial_null_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) ) @ T )
= ( partial_null_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) ) ) ).
% comp_null(1)
thf(fact_682_comp__arr__trg,axiom,
! [T: set_a,B: set_a] :
( ( arr_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T )
=> ( ( ( trg_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T )
= B )
=> ( ( extens7801945855595804251_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ B )
= T ) ) ) ).
% comp_arr_trg
thf(fact_683_src__resid,axiom,
! [T: set_a,U: set_a] :
( ( con_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U )
=> ( ( weakly2061155085811118449_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ T @ U ) )
= ( trg_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ U ) ) ) ).
% src_resid
thf(fact_684_comp__src__arr,axiom,
! [T: set_a,A3: set_a] :
( ( arr_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T )
=> ( ( ( weakly2061155085811118449_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T )
= A3 )
=> ( ( extens7801945855595804251_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ A3 @ T )
= T ) ) ) ).
% comp_src_arr
thf(fact_685_con__compI_I2_J,axiom,
! [T: set_a,U: set_a,W2: set_a] :
( ( composable_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U )
=> ( ( con_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ W2 @ T ) @ U )
=> ( con_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( extens7801945855595804251_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U ) @ W2 ) ) ) ).
% con_compI(2)
thf(fact_686_con__compI_I1_J,axiom,
! [T: set_a,U: set_a,W2: set_a] :
( ( composable_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U )
=> ( ( con_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ W2 @ T ) @ U )
=> ( con_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ W2 @ ( extens7801945855595804251_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U ) ) ) ) ).
% con_compI(1)
thf(fact_687_arr__comp,axiom,
! [T: set_a,U: set_a] :
( ( composable_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U )
=> ( arr_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( extens7801945855595804251_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U ) ) ) ).
% arr_comp
thf(fact_688_trg__comp,axiom,
! [T: set_a,U: set_a] :
( ( composable_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U )
=> ( ( trg_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( extens7801945855595804251_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U ) )
= ( trg_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ U ) ) ) ).
% trg_comp
thf(fact_689_src__comp,axiom,
! [T: set_a,U: set_a] :
( ( composable_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U )
=> ( ( weakly2061155085811118449_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( extens7801945855595804251_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U ) )
= ( weakly2061155085811118449_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T ) ) ) ).
% src_comp
thf(fact_690_coterminalI_092_060_094sub_062W_092_060_094sub_062E,axiom,
! [T: set_a,U: set_a] :
( ( arr_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T )
=> ( ( ( trg_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T )
= ( trg_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ U ) )
=> ( coterminal_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U ) ) ) ).
% coterminalI\<^sub>W\<^sub>E
thf(fact_691_in__targetsI,axiom,
! [B: set_a,T: set_a] :
( ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ B )
=> ( ( con_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( trg_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T ) @ B )
=> ( member_set_a @ B @ ( targets_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T ) ) ) ) ).
% in_targetsI
thf(fact_692_seqI_092_060_094sub_062W_092_060_094sub_062E,axiom,
! [U: set_a,T: set_a] :
( ( arr_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ U )
=> ( ( arr_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T )
=> ( ( ( trg_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T )
= ( weakly2061155085811118449_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ U ) )
=> ( seq_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U ) ) ) ) ).
% seqI\<^sub>W\<^sub>E
thf(fact_693_residuation_Otrg__def,axiom,
! [Resid: set_a > set_a > set_a,T: set_a] :
( ( residuation_set_a @ Resid )
=> ( ( trg_set_a @ Resid @ T )
= ( Resid @ T @ T ) ) ) ).
% residuation.trg_def
thf(fact_694_residuation_Otrg__def,axiom,
! [Resid: a > a > a,T: a] :
( ( residuation_a @ Resid )
=> ( ( trg_a @ Resid @ T )
= ( Resid @ T @ T ) ) ) ).
% residuation.trg_def
thf(fact_695_residuation_Oresid__arr__self,axiom,
! [Resid: set_a > set_a > set_a,T: set_a] :
( ( residuation_set_a @ Resid )
=> ( ( Resid @ T @ T )
= ( trg_set_a @ Resid @ T ) ) ) ).
% residuation.resid_arr_self
thf(fact_696_residuation_Oresid__arr__self,axiom,
! [Resid: a > a > a,T: a] :
( ( residuation_a @ Resid )
=> ( ( Resid @ T @ T )
= ( trg_a @ Resid @ T ) ) ) ).
% residuation.resid_arr_self
thf(fact_697_extensional__rts_Ocomp_Ocong,axiom,
extens7801945855595804251_set_a = extens7801945855595804251_set_a ).
% extensional_rts.comp.cong
thf(fact_698_partial__magma_Onull_Ocong,axiom,
partial_null_set_a = partial_null_set_a ).
% partial_magma.null.cong
thf(fact_699_partial__magma_Onull_Ocong,axiom,
partial_null_a = partial_null_a ).
% partial_magma.null.cong
thf(fact_700_residuation_Otrg_Ocong,axiom,
trg_set_a = trg_set_a ).
% residuation.trg.cong
thf(fact_701_residuation_Otrg_Ocong,axiom,
trg_a = trg_a ).
% residuation.trg.cong
thf(fact_702_residuation_Ocon__imp__arr__resid,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a] :
( ( residuation_set_a @ Resid )
=> ( ( ( Resid @ T @ U )
!= ( partial_null_set_a @ Resid ) )
=> ( ( Resid @ ( Resid @ T @ U ) @ ( Resid @ T @ U ) )
!= ( partial_null_set_a @ Resid ) ) ) ) ).
% residuation.con_imp_arr_resid
thf(fact_703_residuation_Ocon__imp__arr__resid,axiom,
! [Resid: a > a > a,T: a,U: a] :
( ( residuation_a @ Resid )
=> ( ( ( Resid @ T @ U )
!= ( partial_null_a @ Resid ) )
=> ( ( Resid @ ( Resid @ T @ U ) @ ( Resid @ T @ U ) )
!= ( partial_null_a @ Resid ) ) ) ) ).
% residuation.con_imp_arr_resid
thf(fact_704_residuation_Ocon__sym__ax,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a] :
( ( residuation_set_a @ Resid )
=> ( ( ( Resid @ T @ U )
!= ( partial_null_set_a @ Resid ) )
=> ( ( Resid @ U @ T )
!= ( partial_null_set_a @ Resid ) ) ) ) ).
% residuation.con_sym_ax
thf(fact_705_residuation_Ocon__sym__ax,axiom,
! [Resid: a > a > a,T: a,U: a] :
( ( residuation_a @ Resid )
=> ( ( ( Resid @ T @ U )
!= ( partial_null_a @ Resid ) )
=> ( ( Resid @ U @ T )
!= ( partial_null_a @ Resid ) ) ) ) ).
% residuation.con_sym_ax
thf(fact_706_residuation_Ocube__ax,axiom,
! [Resid: set_a > set_a > set_a,V: set_a,T: set_a,U: set_a] :
( ( residuation_set_a @ Resid )
=> ( ( ( Resid @ ( Resid @ V @ T ) @ ( Resid @ U @ T ) )
!= ( partial_null_set_a @ Resid ) )
=> ( ( Resid @ ( Resid @ V @ T ) @ ( Resid @ U @ T ) )
= ( Resid @ ( Resid @ V @ U ) @ ( Resid @ T @ U ) ) ) ) ) ).
% residuation.cube_ax
thf(fact_707_residuation_Ocube__ax,axiom,
! [Resid: a > a > a,V: a,T: a,U: a] :
( ( residuation_a @ Resid )
=> ( ( ( Resid @ ( Resid @ V @ T ) @ ( Resid @ U @ T ) )
!= ( partial_null_a @ Resid ) )
=> ( ( Resid @ ( Resid @ V @ T ) @ ( Resid @ U @ T ) )
= ( Resid @ ( Resid @ V @ U ) @ ( Resid @ T @ U ) ) ) ) ) ).
% residuation.cube_ax
thf(fact_708_partial__magma_Onull__eqI,axiom,
! [OP2: set_a > set_a > set_a,N: set_a] :
( ( partial_magma_set_a @ OP2 )
=> ( ! [T3: set_a] :
( ( ( OP2 @ N @ T3 )
= N )
& ( ( OP2 @ T3 @ N )
= N ) )
=> ( N
= ( partial_null_set_a @ OP2 ) ) ) ) ).
% partial_magma.null_eqI
thf(fact_709_partial__magma_Onull__eqI,axiom,
! [OP2: a > a > a,N: a] :
( ( partial_magma_a @ OP2 )
=> ( ! [T3: a] :
( ( ( OP2 @ N @ T3 )
= N )
& ( ( OP2 @ T3 @ N )
= N ) )
=> ( N
= ( partial_null_a @ OP2 ) ) ) ) ).
% partial_magma.null_eqI
thf(fact_710_partial__magma_Onull__is__zero_I1_J,axiom,
! [OP2: set_a > set_a > set_a,T: set_a] :
( ( partial_magma_set_a @ OP2 )
=> ( ( OP2 @ ( partial_null_set_a @ OP2 ) @ T )
= ( partial_null_set_a @ OP2 ) ) ) ).
% partial_magma.null_is_zero(1)
thf(fact_711_partial__magma_Onull__is__zero_I1_J,axiom,
! [OP2: a > a > a,T: a] :
( ( partial_magma_a @ OP2 )
=> ( ( OP2 @ ( partial_null_a @ OP2 ) @ T )
= ( partial_null_a @ OP2 ) ) ) ).
% partial_magma.null_is_zero(1)
thf(fact_712_partial__magma_Onull__is__zero_I2_J,axiom,
! [OP2: set_a > set_a > set_a,T: set_a] :
( ( partial_magma_set_a @ OP2 )
=> ( ( OP2 @ T @ ( partial_null_set_a @ OP2 ) )
= ( partial_null_set_a @ OP2 ) ) ) ).
% partial_magma.null_is_zero(2)
thf(fact_713_partial__magma_Onull__is__zero_I2_J,axiom,
! [OP2: a > a > a,T: a] :
( ( partial_magma_a @ OP2 )
=> ( ( OP2 @ T @ ( partial_null_a @ OP2 ) )
= ( partial_null_a @ OP2 ) ) ) ).
% partial_magma.null_is_zero(2)
thf(fact_714_residuation_OconE,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a] :
( ( residuation_set_a @ Resid )
=> ( ( con_set_a @ Resid @ T @ U )
=> ( ( Resid @ T @ U )
!= ( partial_null_set_a @ Resid ) ) ) ) ).
% residuation.conE
thf(fact_715_residuation_OconE,axiom,
! [Resid: a > a > a,T: a,U: a] :
( ( residuation_a @ Resid )
=> ( ( con_a @ Resid @ T @ U )
=> ( ( Resid @ T @ U )
!= ( partial_null_a @ Resid ) ) ) ) ).
% residuation.conE
thf(fact_716_residuation_OconI,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a] :
( ( residuation_set_a @ Resid )
=> ( ( ( Resid @ T @ U )
!= ( partial_null_set_a @ Resid ) )
=> ( con_set_a @ Resid @ T @ U ) ) ) ).
% residuation.conI
thf(fact_717_residuation_OconI,axiom,
! [Resid: a > a > a,T: a,U: a] :
( ( residuation_a @ Resid )
=> ( ( ( Resid @ T @ U )
!= ( partial_null_a @ Resid ) )
=> ( con_a @ Resid @ T @ U ) ) ) ).
% residuation.conI
thf(fact_718_residuation_Ocon__def,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a] :
( ( residuation_set_a @ Resid )
=> ( ( con_set_a @ Resid @ T @ U )
= ( ( Resid @ T @ U )
!= ( partial_null_set_a @ Resid ) ) ) ) ).
% residuation.con_def
thf(fact_719_residuation_Ocon__def,axiom,
! [Resid: a > a > a,T: a,U: a] :
( ( residuation_a @ Resid )
=> ( ( con_a @ Resid @ T @ U )
= ( ( Resid @ T @ U )
!= ( partial_null_a @ Resid ) ) ) ) ).
% residuation.con_def
thf(fact_720_residuation_Onot__arr__null,axiom,
! [Resid: set_a > set_a > set_a] :
( ( residuation_set_a @ Resid )
=> ~ ( arr_set_a @ Resid @ ( partial_null_set_a @ Resid ) ) ) ).
% residuation.not_arr_null
thf(fact_721_residuation_Onot__arr__null,axiom,
! [Resid: a > a > a] :
( ( residuation_a @ Resid )
=> ~ ( arr_a @ Resid @ ( partial_null_a @ Resid ) ) ) ).
% residuation.not_arr_null
thf(fact_722_quotient__by__coherent__normal_Onull__char,axiom,
! [Resid: set_a > set_a > set_a,NN: set_set_a] :
( ( quotie5625257012022141046_set_a @ Resid @ NN )
=> ( ( partia840180994421509092_set_a @ ( quotie3283642546880816345_set_a @ Resid @ NN ) )
= bot_bot_set_set_a ) ) ).
% quotient_by_coherent_normal.null_char
thf(fact_723_quotient__by__coherent__normal_Onull__char,axiom,
! [Resid: a > a > a,NN: set_a] :
( ( quotie3282664541148387094rmal_a @ Resid @ NN )
=> ( ( partial_null_set_a @ ( quotie8165075472272353145esid_a @ Resid @ NN ) )
= bot_bot_set_a ) ) ).
% quotient_by_coherent_normal.null_char
thf(fact_724_join__def,axiom,
! [T: set_a,U: set_a] :
( ( ( joinable_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U )
=> ( ( extens1973556086528668384_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U )
= ( the_set_a @ ( join_of_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U ) ) ) )
& ( ~ ( joinable_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U )
=> ( ( extens1973556086528668384_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U )
= ( partial_null_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) ) ) ) ) ).
% join_def
thf(fact_725_src__def,axiom,
! [T: set_a] :
( ( ( arr_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T )
=> ( ( weakly2061155085811118449_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T )
= ( the_set_a
@ ^ [A: set_a] : ( member_set_a @ A @ ( sources_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T ) ) ) ) )
& ( ~ ( arr_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T )
=> ( ( weakly2061155085811118449_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T )
= ( partial_null_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) ) ) ) ) ).
% src_def
thf(fact_726_transformation__axioms_Ointro,axiom,
! [A5: a > a > a,Tau: a > a,B3: a > a > a,F2: a > a,G: a > a] :
( ! [F3: a] :
( ~ ( arr_a @ A5 @ F3 )
=> ( ( Tau @ F3 )
= ( partial_null_a @ B3 ) ) )
=> ( ! [F3: a] :
( ( ide_a @ A5 @ F3 )
=> ( ( weakly8512939796511659025_src_a @ B3 @ ( Tau @ F3 ) )
= ( F2 @ ( weakly8512939796511659025_src_a @ A5 @ F3 ) ) ) )
=> ( ! [F3: a] :
( ( ide_a @ A5 @ F3 )
=> ( ( trg_a @ B3 @ ( Tau @ F3 ) )
= ( G @ ( trg_a @ A5 @ F3 ) ) ) )
=> ( ! [F3: a] :
( ( arr_a @ A5 @ F3 )
=> ( ( B3 @ ( Tau @ ( weakly8512939796511659025_src_a @ A5 @ F3 ) ) @ ( F2 @ F3 ) )
= ( Tau @ ( trg_a @ A5 @ F3 ) ) ) )
=> ( ! [F3: a] :
( ( arr_a @ A5 @ F3 )
=> ( ( B3 @ ( F2 @ F3 ) @ ( Tau @ ( weakly8512939796511659025_src_a @ A5 @ F3 ) ) )
= ( G @ F3 ) ) )
=> ( ! [F3: a] :
( ( arr_a @ A5 @ F3 )
=> ( join_of_a @ B3 @ ( Tau @ ( weakly8512939796511659025_src_a @ A5 @ F3 ) ) @ ( F2 @ F3 ) @ ( Tau @ F3 ) ) )
=> ( transf4446446367311712680ms_a_a @ A5 @ B3 @ F2 @ G @ Tau ) ) ) ) ) ) ) ).
% transformation_axioms.intro
thf(fact_727_transformation__axioms_Ointro,axiom,
! [A5: set_a > set_a > set_a,Tau: set_a > a,B3: a > a > a,F2: set_a > a,G: set_a > a] :
( ! [F3: set_a] :
( ~ ( arr_set_a @ A5 @ F3 )
=> ( ( Tau @ F3 )
= ( partial_null_a @ B3 ) ) )
=> ( ! [F3: set_a] :
( ( ide_set_a @ A5 @ F3 )
=> ( ( weakly8512939796511659025_src_a @ B3 @ ( Tau @ F3 ) )
= ( F2 @ ( weakly2061155085811118449_set_a @ A5 @ F3 ) ) ) )
=> ( ! [F3: set_a] :
( ( ide_set_a @ A5 @ F3 )
=> ( ( trg_a @ B3 @ ( Tau @ F3 ) )
= ( G @ ( trg_set_a @ A5 @ F3 ) ) ) )
=> ( ! [F3: set_a] :
( ( arr_set_a @ A5 @ F3 )
=> ( ( B3 @ ( Tau @ ( weakly2061155085811118449_set_a @ A5 @ F3 ) ) @ ( F2 @ F3 ) )
= ( Tau @ ( trg_set_a @ A5 @ F3 ) ) ) )
=> ( ! [F3: set_a] :
( ( arr_set_a @ A5 @ F3 )
=> ( ( B3 @ ( F2 @ F3 ) @ ( Tau @ ( weakly2061155085811118449_set_a @ A5 @ F3 ) ) )
= ( G @ F3 ) ) )
=> ( ! [F3: set_a] :
( ( arr_set_a @ A5 @ F3 )
=> ( join_of_a @ B3 @ ( Tau @ ( weakly2061155085811118449_set_a @ A5 @ F3 ) ) @ ( F2 @ F3 ) @ ( Tau @ F3 ) ) )
=> ( transf1935308705569152072et_a_a @ A5 @ B3 @ F2 @ G @ Tau ) ) ) ) ) ) ) ).
% transformation_axioms.intro
thf(fact_728_transformation__axioms_Ointro,axiom,
! [A5: a > a > a,Tau: a > set_a,B3: set_a > set_a > set_a,F2: a > set_a,G: a > set_a] :
( ! [F3: a] :
( ~ ( arr_a @ A5 @ F3 )
=> ( ( Tau @ F3 )
= ( partial_null_set_a @ B3 ) ) )
=> ( ! [F3: a] :
( ( ide_a @ A5 @ F3 )
=> ( ( weakly2061155085811118449_set_a @ B3 @ ( Tau @ F3 ) )
= ( F2 @ ( weakly8512939796511659025_src_a @ A5 @ F3 ) ) ) )
=> ( ! [F3: a] :
( ( ide_a @ A5 @ F3 )
=> ( ( trg_set_a @ B3 @ ( Tau @ F3 ) )
= ( G @ ( trg_a @ A5 @ F3 ) ) ) )
=> ( ! [F3: a] :
( ( arr_a @ A5 @ F3 )
=> ( ( B3 @ ( Tau @ ( weakly8512939796511659025_src_a @ A5 @ F3 ) ) @ ( F2 @ F3 ) )
= ( Tau @ ( trg_a @ A5 @ F3 ) ) ) )
=> ( ! [F3: a] :
( ( arr_a @ A5 @ F3 )
=> ( ( B3 @ ( F2 @ F3 ) @ ( Tau @ ( weakly8512939796511659025_src_a @ A5 @ F3 ) ) )
= ( G @ F3 ) ) )
=> ( ! [F3: a] :
( ( arr_a @ A5 @ F3 )
=> ( join_of_set_a @ B3 @ ( Tau @ ( weakly8512939796511659025_src_a @ A5 @ F3 ) ) @ ( F2 @ F3 ) @ ( Tau @ F3 ) ) )
=> ( transf1718796647109808904_set_a @ A5 @ B3 @ F2 @ G @ Tau ) ) ) ) ) ) ) ).
% transformation_axioms.intro
thf(fact_729_transformation__axioms_Ointro,axiom,
! [A5: set_a > set_a > set_a,Tau: set_a > set_a,B3: set_a > set_a > set_a,F2: set_a > set_a,G: set_a > set_a] :
( ! [F3: set_a] :
( ~ ( arr_set_a @ A5 @ F3 )
=> ( ( Tau @ F3 )
= ( partial_null_set_a @ B3 ) ) )
=> ( ! [F3: set_a] :
( ( ide_set_a @ A5 @ F3 )
=> ( ( weakly2061155085811118449_set_a @ B3 @ ( Tau @ F3 ) )
= ( F2 @ ( weakly2061155085811118449_set_a @ A5 @ F3 ) ) ) )
=> ( ! [F3: set_a] :
( ( ide_set_a @ A5 @ F3 )
=> ( ( trg_set_a @ B3 @ ( Tau @ F3 ) )
= ( G @ ( trg_set_a @ A5 @ F3 ) ) ) )
=> ( ! [F3: set_a] :
( ( arr_set_a @ A5 @ F3 )
=> ( ( B3 @ ( Tau @ ( weakly2061155085811118449_set_a @ A5 @ F3 ) ) @ ( F2 @ F3 ) )
= ( Tau @ ( trg_set_a @ A5 @ F3 ) ) ) )
=> ( ! [F3: set_a] :
( ( arr_set_a @ A5 @ F3 )
=> ( ( B3 @ ( F2 @ F3 ) @ ( Tau @ ( weakly2061155085811118449_set_a @ A5 @ F3 ) ) )
= ( G @ F3 ) ) )
=> ( ! [F3: set_a] :
( ( arr_set_a @ A5 @ F3 )
=> ( join_of_set_a @ B3 @ ( Tau @ ( weakly2061155085811118449_set_a @ A5 @ F3 ) ) @ ( F2 @ F3 ) @ ( Tau @ F3 ) ) )
=> ( transf2960116383903194536_set_a @ A5 @ B3 @ F2 @ G @ Tau ) ) ) ) ) ) ) ).
% transformation_axioms.intro
thf(fact_730_transformation__axioms__def,axiom,
( transf4446446367311712680ms_a_a
= ( ^ [A8: a > a > a,B5: a > a > a,F4: a > a,G2: a > a,Tau2: a > a] :
( ! [F5: a] :
( ~ ( arr_a @ A8 @ F5 )
=> ( ( Tau2 @ F5 )
= ( partial_null_a @ B5 ) ) )
& ! [F5: a] :
( ( ide_a @ A8 @ F5 )
=> ( ( weakly8512939796511659025_src_a @ B5 @ ( Tau2 @ F5 ) )
= ( F4 @ ( weakly8512939796511659025_src_a @ A8 @ F5 ) ) ) )
& ! [F5: a] :
( ( ide_a @ A8 @ F5 )
=> ( ( trg_a @ B5 @ ( Tau2 @ F5 ) )
= ( G2 @ ( trg_a @ A8 @ F5 ) ) ) )
& ! [F5: a] :
( ( arr_a @ A8 @ F5 )
=> ( ( B5 @ ( Tau2 @ ( weakly8512939796511659025_src_a @ A8 @ F5 ) ) @ ( F4 @ F5 ) )
= ( Tau2 @ ( trg_a @ A8 @ F5 ) ) ) )
& ! [F5: a] :
( ( arr_a @ A8 @ F5 )
=> ( ( B5 @ ( F4 @ F5 ) @ ( Tau2 @ ( weakly8512939796511659025_src_a @ A8 @ F5 ) ) )
= ( G2 @ F5 ) ) )
& ! [F5: a] :
( ( arr_a @ A8 @ F5 )
=> ( join_of_a @ B5 @ ( Tau2 @ ( weakly8512939796511659025_src_a @ A8 @ F5 ) ) @ ( F4 @ F5 ) @ ( Tau2 @ F5 ) ) ) ) ) ) ).
% transformation_axioms_def
thf(fact_731_transformation__axioms__def,axiom,
( transf1935308705569152072et_a_a
= ( ^ [A8: set_a > set_a > set_a,B5: a > a > a,F4: set_a > a,G2: set_a > a,Tau2: set_a > a] :
( ! [F5: set_a] :
( ~ ( arr_set_a @ A8 @ F5 )
=> ( ( Tau2 @ F5 )
= ( partial_null_a @ B5 ) ) )
& ! [F5: set_a] :
( ( ide_set_a @ A8 @ F5 )
=> ( ( weakly8512939796511659025_src_a @ B5 @ ( Tau2 @ F5 ) )
= ( F4 @ ( weakly2061155085811118449_set_a @ A8 @ F5 ) ) ) )
& ! [F5: set_a] :
( ( ide_set_a @ A8 @ F5 )
=> ( ( trg_a @ B5 @ ( Tau2 @ F5 ) )
= ( G2 @ ( trg_set_a @ A8 @ F5 ) ) ) )
& ! [F5: set_a] :
( ( arr_set_a @ A8 @ F5 )
=> ( ( B5 @ ( Tau2 @ ( weakly2061155085811118449_set_a @ A8 @ F5 ) ) @ ( F4 @ F5 ) )
= ( Tau2 @ ( trg_set_a @ A8 @ F5 ) ) ) )
& ! [F5: set_a] :
( ( arr_set_a @ A8 @ F5 )
=> ( ( B5 @ ( F4 @ F5 ) @ ( Tau2 @ ( weakly2061155085811118449_set_a @ A8 @ F5 ) ) )
= ( G2 @ F5 ) ) )
& ! [F5: set_a] :
( ( arr_set_a @ A8 @ F5 )
=> ( join_of_a @ B5 @ ( Tau2 @ ( weakly2061155085811118449_set_a @ A8 @ F5 ) ) @ ( F4 @ F5 ) @ ( Tau2 @ F5 ) ) ) ) ) ) ).
% transformation_axioms_def
thf(fact_732_transformation__axioms__def,axiom,
( transf1718796647109808904_set_a
= ( ^ [A8: a > a > a,B5: set_a > set_a > set_a,F4: a > set_a,G2: a > set_a,Tau2: a > set_a] :
( ! [F5: a] :
( ~ ( arr_a @ A8 @ F5 )
=> ( ( Tau2 @ F5 )
= ( partial_null_set_a @ B5 ) ) )
& ! [F5: a] :
( ( ide_a @ A8 @ F5 )
=> ( ( weakly2061155085811118449_set_a @ B5 @ ( Tau2 @ F5 ) )
= ( F4 @ ( weakly8512939796511659025_src_a @ A8 @ F5 ) ) ) )
& ! [F5: a] :
( ( ide_a @ A8 @ F5 )
=> ( ( trg_set_a @ B5 @ ( Tau2 @ F5 ) )
= ( G2 @ ( trg_a @ A8 @ F5 ) ) ) )
& ! [F5: a] :
( ( arr_a @ A8 @ F5 )
=> ( ( B5 @ ( Tau2 @ ( weakly8512939796511659025_src_a @ A8 @ F5 ) ) @ ( F4 @ F5 ) )
= ( Tau2 @ ( trg_a @ A8 @ F5 ) ) ) )
& ! [F5: a] :
( ( arr_a @ A8 @ F5 )
=> ( ( B5 @ ( F4 @ F5 ) @ ( Tau2 @ ( weakly8512939796511659025_src_a @ A8 @ F5 ) ) )
= ( G2 @ F5 ) ) )
& ! [F5: a] :
( ( arr_a @ A8 @ F5 )
=> ( join_of_set_a @ B5 @ ( Tau2 @ ( weakly8512939796511659025_src_a @ A8 @ F5 ) ) @ ( F4 @ F5 ) @ ( Tau2 @ F5 ) ) ) ) ) ) ).
% transformation_axioms_def
thf(fact_733_transformation__axioms__def,axiom,
( transf2960116383903194536_set_a
= ( ^ [A8: set_a > set_a > set_a,B5: set_a > set_a > set_a,F4: set_a > set_a,G2: set_a > set_a,Tau2: set_a > set_a] :
( ! [F5: set_a] :
( ~ ( arr_set_a @ A8 @ F5 )
=> ( ( Tau2 @ F5 )
= ( partial_null_set_a @ B5 ) ) )
& ! [F5: set_a] :
( ( ide_set_a @ A8 @ F5 )
=> ( ( weakly2061155085811118449_set_a @ B5 @ ( Tau2 @ F5 ) )
= ( F4 @ ( weakly2061155085811118449_set_a @ A8 @ F5 ) ) ) )
& ! [F5: set_a] :
( ( ide_set_a @ A8 @ F5 )
=> ( ( trg_set_a @ B5 @ ( Tau2 @ F5 ) )
= ( G2 @ ( trg_set_a @ A8 @ F5 ) ) ) )
& ! [F5: set_a] :
( ( arr_set_a @ A8 @ F5 )
=> ( ( B5 @ ( Tau2 @ ( weakly2061155085811118449_set_a @ A8 @ F5 ) ) @ ( F4 @ F5 ) )
= ( Tau2 @ ( trg_set_a @ A8 @ F5 ) ) ) )
& ! [F5: set_a] :
( ( arr_set_a @ A8 @ F5 )
=> ( ( B5 @ ( F4 @ F5 ) @ ( Tau2 @ ( weakly2061155085811118449_set_a @ A8 @ F5 ) ) )
= ( G2 @ F5 ) ) )
& ! [F5: set_a] :
( ( arr_set_a @ A8 @ F5 )
=> ( join_of_set_a @ B5 @ ( Tau2 @ ( weakly2061155085811118449_set_a @ A8 @ F5 ) ) @ ( F4 @ F5 ) @ ( Tau2 @ F5 ) ) ) ) ) ) ).
% transformation_axioms_def
thf(fact_734_R_Otrg__def,axiom,
! [T: a] :
( ( trg_a @ resid @ T )
= ( resid @ T @ T ) ) ).
% R.trg_def
thf(fact_735_R_Onull__eqI,axiom,
! [N: a] :
( ! [T3: a] :
( ( ( resid @ N @ T3 )
= N )
& ( ( resid @ T3 @ N )
= N ) )
=> ( N
= ( partial_null_a @ resid ) ) ) ).
% R.null_eqI
thf(fact_736_R_Ocube__ax,axiom,
! [V: a,T: a,U: a] :
( ( ( resid @ ( resid @ V @ T ) @ ( resid @ U @ T ) )
!= ( partial_null_a @ resid ) )
=> ( ( resid @ ( resid @ V @ T ) @ ( resid @ U @ T ) )
= ( resid @ ( resid @ V @ U ) @ ( resid @ T @ U ) ) ) ) ).
% R.cube_ax
thf(fact_737_R_Ocon__sym__ax,axiom,
! [T: a,U: a] :
( ( ( resid @ T @ U )
!= ( partial_null_a @ resid ) )
=> ( ( resid @ U @ T )
!= ( partial_null_a @ resid ) ) ) ).
% R.con_sym_ax
thf(fact_738_R_Ocon__imp__arr__resid,axiom,
! [T: a,U: a] :
( ( ( resid @ T @ U )
!= ( partial_null_a @ resid ) )
=> ( ( resid @ ( resid @ T @ U ) @ ( resid @ T @ U ) )
!= ( partial_null_a @ resid ) ) ) ).
% R.con_imp_arr_resid
thf(fact_739_R_Ocon__def,axiom,
! [T: a,U: a] :
( ( con_a @ resid @ T @ U )
= ( ( resid @ T @ U )
!= ( partial_null_a @ resid ) ) ) ).
% R.con_def
thf(fact_740_R_OconE,axiom,
! [T: a,U: a] :
( ( con_a @ resid @ T @ U )
=> ( ( resid @ T @ U )
!= ( partial_null_a @ resid ) ) ) ).
% R.conE
thf(fact_741_R_Onot__arr__null,axiom,
~ ( arr_a @ resid @ ( partial_null_a @ resid ) ) ).
% R.not_arr_null
thf(fact_742_R_Oide__trg,axiom,
! [T: a] :
( ( arr_a @ resid @ T )
=> ( ide_a @ resid @ ( trg_a @ resid @ T ) ) ) ).
% R.ide_trg
thf(fact_743_R_Otrg__in__targets,axiom,
! [T: a] :
( ( arr_a @ resid @ T )
=> ( member_a @ ( trg_a @ resid @ T ) @ ( targets_a @ resid @ T ) ) ) ).
% R.trg_in_targets
thf(fact_744_R_Ocoterminal__iff__con__trg,axiom,
! [T: a,U: a] :
( ( coterminal_a @ resid @ T @ U )
= ( con_a @ resid @ ( trg_a @ resid @ T ) @ ( trg_a @ resid @ U ) ) ) ).
% R.coterminal_iff_con_trg
thf(fact_745_R_Oin__targetsE,axiom,
! [B: a,T: a] :
( ( member_a @ B @ ( targets_a @ resid @ T ) )
=> ~ ( ( ide_a @ resid @ B )
=> ~ ( con_a @ resid @ ( trg_a @ resid @ T ) @ B ) ) ) ).
% R.in_targetsE
thf(fact_746_R_Otargets__def,axiom,
! [T: a] :
( ( targets_a @ resid @ T )
= ( collect_a
@ ^ [B2: a] :
( ( ide_a @ resid @ B2 )
& ( con_a @ resid @ ( trg_a @ resid @ T ) @ B2 ) ) ) ) ).
% R.targets_def
thf(fact_747_R_Onull__is__zero_I1_J,axiom,
! [T: a] :
( ( resid @ ( partial_null_a @ resid ) @ T )
= ( partial_null_a @ resid ) ) ).
% R.null_is_zero(1)
thf(fact_748_R_Onull__is__zero_I2_J,axiom,
! [T: a] :
( ( resid @ T @ ( partial_null_a @ resid ) )
= ( partial_null_a @ resid ) ) ).
% R.null_is_zero(2)
thf(fact_749_R_OconI,axiom,
! [T: a,U: a] :
( ( ( resid @ T @ U )
!= ( partial_null_a @ resid ) )
=> ( con_a @ resid @ T @ U ) ) ).
% R.conI
thf(fact_750_R_Oin__targetsI,axiom,
! [B: a,T: a] :
( ( ide_a @ resid @ B )
=> ( ( con_a @ resid @ ( trg_a @ resid @ T ) @ B )
=> ( member_a @ B @ ( targets_a @ resid @ T ) ) ) ) ).
% R.in_targetsI
thf(fact_751_comp__def,axiom,
! [T: set_a,U: set_a] :
( ( ( composable_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U )
=> ( ( extens7801945855595804251_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U )
= ( the_set_a @ ( composite_of_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U ) ) ) )
& ( ~ ( composable_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U )
=> ( ( extens7801945855595804251_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U )
= ( partial_null_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) ) ) ) ) ).
% comp_def
thf(fact_752_null__def,axiom,
( ( partial_null_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) )
= ( the_set_a
@ ^ [N2: set_a] :
! [T6: set_a] :
( ( ( quotie8165075472272353145esid_a @ resid @ nn @ N2 @ T6 )
= N2 )
& ( ( quotie8165075472272353145esid_a @ resid @ nn @ T6 @ N2 )
= N2 ) ) ) ) ).
% null_def
thf(fact_753_the__equality,axiom,
! [P: set_a > $o,A3: set_a] :
( ( P @ A3 )
=> ( ! [X: set_a] :
( ( P @ X )
=> ( X = A3 ) )
=> ( ( the_set_a @ P )
= A3 ) ) ) ).
% the_equality
thf(fact_754_the__equality,axiom,
! [P: a > $o,A3: a] :
( ( P @ A3 )
=> ( ! [X: a] :
( ( P @ X )
=> ( X = A3 ) )
=> ( ( the_a @ P )
= A3 ) ) ) ).
% the_equality
thf(fact_755_composite__of__unique,axiom,
! [T: set_a,U: set_a,V: set_a,V4: set_a] :
( ( composite_of_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U @ V )
=> ( ( composite_of_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U @ V4 )
=> ( V = V4 ) ) ) ).
% composite_of_unique
thf(fact_756_composite__ofE,axiom,
! [U: set_a,T: set_a,V: set_a] :
( ( composite_of_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ U @ T @ V )
=> ~ ( ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ U @ V ) )
=> ~ ( ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ ( quotie8165075472272353145esid_a @ resid @ nn @ V @ U ) @ T ) )
& ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ T @ ( quotie8165075472272353145esid_a @ resid @ nn @ V @ U ) ) ) ) ) ) ).
% composite_ofE
thf(fact_757_composite__of__cancel__left,axiom,
! [T: set_a,U: set_a,V: set_a,U7: set_a] :
( ( composite_of_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U @ V )
=> ( ( composite_of_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U7 @ V )
=> ( ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ U @ U7 ) )
& ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ U7 @ U ) ) ) ) ) ).
% composite_of_cancel_left
thf(fact_758_composite__of__def,axiom,
! [U: set_a,T: set_a,V: set_a] :
( ( composite_of_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ U @ T @ V )
= ( ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ U @ V ) )
& ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ ( quotie8165075472272353145esid_a @ resid @ nn @ V @ U ) @ T ) )
& ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ T @ ( quotie8165075472272353145esid_a @ resid @ nn @ V @ U ) ) ) ) ) ).
% composite_of_def
thf(fact_759_composite__of__ide__self,axiom,
! [A3: set_a] :
( ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ A3 )
=> ( composite_of_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ A3 @ A3 @ A3 ) ) ).
% composite_of_ide_self
thf(fact_760_composite__of__unq__upto__cong,axiom,
! [U: set_a,T: set_a,V: set_a,V4: set_a] :
( ( composite_of_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ U @ T @ V )
=> ( ( composite_of_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ U @ T @ V4 )
=> ( ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ V @ V4 ) )
& ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ V4 @ V ) ) ) ) ) ).
% composite_of_unq_upto_cong
thf(fact_761_con__prfx__composite__of_I2_J,axiom,
! [T: set_a,U: set_a,W2: set_a,V: set_a] :
( ( composite_of_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U @ W2 )
=> ( ( con_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ W2 @ V )
=> ( con_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ V ) ) ) ).
% con_prfx_composite_of(2)
thf(fact_762_con__prfx__composite__of_I1_J,axiom,
! [T: set_a,U: set_a,W2: set_a] :
( ( composite_of_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U @ W2 )
=> ( con_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ W2 ) ) ).
% con_prfx_composite_of(1)
thf(fact_763_resid__composite__of_I4_J,axiom,
! [T: set_a,U: set_a,W2: set_a,V: set_a] :
( ( composite_of_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U @ W2 )
=> ( ( con_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ W2 @ V )
=> ( composite_of_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ T @ V ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ U @ ( quotie8165075472272353145esid_a @ resid @ nn @ V @ T ) ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ W2 @ V ) ) ) ) ).
% resid_composite_of(4)
thf(fact_764_resid__composite__of_I2_J,axiom,
! [T: set_a,U: set_a,W2: set_a,V: set_a] :
( ( composite_of_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U @ W2 )
=> ( ( con_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ W2 @ V )
=> ( con_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ V @ T ) @ U ) ) ) ).
% resid_composite_of(2)
thf(fact_765_resid__composite__of_I1_J,axiom,
! [T: set_a,U: set_a,W2: set_a,V: set_a] :
( ( composite_of_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U @ W2 )
=> ( ( con_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ W2 @ V )
=> ( con_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ V @ T ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ W2 @ T ) ) ) ) ).
% resid_composite_of(1)
thf(fact_766_bounded__imp__con,axiom,
! [T: set_a,U: set_a,V: set_a,T4: set_a,U7: set_a] :
( ( composite_of_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U @ V )
=> ( ( composite_of_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T4 @ U7 @ V )
=> ( con_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ T4 ) ) ) ).
% bounded_imp_con
thf(fact_767_con__composite__of__iff,axiom,
! [T: set_a,U: set_a,V: set_a,W2: set_a] :
( ( composite_of_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U @ V )
=> ( ( con_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ W2 @ V )
= ( con_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ W2 @ T ) @ U ) ) ) ).
% con_composite_of_iff
thf(fact_768_arr__composite__of,axiom,
! [U: set_a,T: set_a,V: set_a] :
( ( composite_of_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ U @ T @ V )
=> ( arr_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ V ) ) ).
% arr_composite_of
thf(fact_769_sources__composite__of,axiom,
! [U: set_a,T: set_a,V: set_a] :
( ( composite_of_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ U @ T @ V )
=> ( ( sources_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ V )
= ( sources_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ U ) ) ) ).
% sources_composite_of
thf(fact_770_comp__is__composite__of_I2_J,axiom,
! [T: set_a,U: set_a,V: set_a] :
( ( composite_of_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U @ V )
=> ( ( extens7801945855595804251_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U )
= V ) ) ).
% comp_is_composite_of(2)
thf(fact_771_comp__is__composite__of_I1_J,axiom,
! [T: set_a,U: set_a,V: set_a] :
( ( composite_of_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U @ V )
=> ( composite_of_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U @ ( extens7801945855595804251_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U ) ) ) ).
% comp_is_composite_of(1)
thf(fact_772_targets__composite__of,axiom,
! [U: set_a,T: set_a,V: set_a] :
( ( composite_of_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ U @ T @ V )
=> ( ( targets_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ V )
= ( targets_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T ) ) ) ).
% targets_composite_of
thf(fact_773_trg__composite__of,axiom,
! [U: set_a,T: set_a,V: set_a] :
( ( composite_of_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ U @ T @ V )
=> ( ( trg_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ V )
= ( trg_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T ) ) ) ).
% trg_composite_of
thf(fact_774_src__composite__of,axiom,
! [U: set_a,T: set_a,V: set_a] :
( ( composite_of_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ U @ T @ V )
=> ( ( weakly2061155085811118449_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ V )
= ( weakly2061155085811118449_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ U ) ) ) ).
% src_composite_of
thf(fact_775_join__ofE,axiom,
! [T: set_a,U: set_a,V: set_a] :
( ( join_of_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U @ V )
=> ~ ( ( composite_of_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ ( quotie8165075472272353145esid_a @ resid @ nn @ U @ T ) @ V )
=> ~ ( composite_of_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ U @ ( quotie8165075472272353145esid_a @ resid @ nn @ T @ U ) @ V ) ) ) ).
% join_ofE
thf(fact_776_join__of__def,axiom,
! [T: set_a,U: set_a,V: set_a] :
( ( join_of_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U @ V )
= ( ( composite_of_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ ( quotie8165075472272353145esid_a @ resid @ nn @ U @ T ) @ V )
& ( composite_of_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ U @ ( quotie8165075472272353145esid_a @ resid @ nn @ T @ U ) @ V ) ) ) ).
% join_of_def
thf(fact_777_composable__def,axiom,
! [T: set_a,U: set_a] :
( ( composable_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U )
= ( ? [X5: set_a] : ( composite_of_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U @ X5 ) ) ) ).
% composable_def
thf(fact_778_the__sym__eq__trivial,axiom,
! [X4: set_a] :
( ( the_set_a
@ ( ^ [Y5: set_a,Z3: set_a] : ( Y5 = Z3 )
@ X4 ) )
= X4 ) ).
% the_sym_eq_trivial
thf(fact_779_the__sym__eq__trivial,axiom,
! [X4: a] :
( ( the_a
@ ( ^ [Y5: a,Z3: a] : ( Y5 = Z3 )
@ X4 ) )
= X4 ) ).
% the_sym_eq_trivial
thf(fact_780_the__eq__trivial,axiom,
! [A3: set_a] :
( ( the_set_a
@ ^ [X2: set_a] : ( X2 = A3 ) )
= A3 ) ).
% the_eq_trivial
thf(fact_781_the__eq__trivial,axiom,
! [A3: a] :
( ( the_a
@ ^ [X2: a] : ( X2 = A3 ) )
= A3 ) ).
% the_eq_trivial
thf(fact_782_resid__composite__of_I3_J,axiom,
! [T: set_a,U: set_a,W2: set_a,V: set_a] :
( ( composite_of_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U @ W2 )
=> ( ( con_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ W2 @ V )
=> ( ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ ( quotie8165075472272353145esid_a @ resid @ nn @ V @ W2 ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ ( quotie8165075472272353145esid_a @ resid @ nn @ V @ T ) @ U ) ) )
& ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ ( quotie8165075472272353145esid_a @ resid @ nn @ ( quotie8165075472272353145esid_a @ resid @ nn @ V @ T ) @ U ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ V @ W2 ) ) ) ) ) ) ).
% resid_composite_of(3)
thf(fact_783_composite__of__arr__ide,axiom,
! [B: set_a,T: set_a] :
( ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ B )
=> ( ( composite_of_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ B @ T )
= ( con_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ T @ T ) @ B ) ) ) ).
% composite_of_arr_ide
thf(fact_784_composite__of__ide__arr,axiom,
! [A3: set_a,T: set_a] :
( ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ A3 )
=> ( ( composite_of_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ A3 @ T @ T )
= ( con_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ A3 ) ) ) ).
% composite_of_ide_arr
thf(fact_785_composite__of__source__arr,axiom,
! [T: set_a,A3: set_a] :
( ( arr_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T )
=> ( ( member_set_a @ A3 @ ( sources_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T ) )
=> ( composite_of_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ A3 @ T @ T ) ) ) ).
% composite_of_source_arr
thf(fact_786_composite__of__arr__target,axiom,
! [T: set_a,B: set_a] :
( ( arr_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T )
=> ( ( member_set_a @ B @ ( targets_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T ) )
=> ( composite_of_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ B @ T ) ) ) ).
% composite_of_arr_target
thf(fact_787_composite__ofI,axiom,
! [U: set_a,V: set_a,T: set_a] :
( ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ U @ V ) )
=> ( ( ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ ( quotie8165075472272353145esid_a @ resid @ nn @ V @ U ) @ T ) )
& ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ T @ ( quotie8165075472272353145esid_a @ resid @ nn @ V @ U ) ) ) )
=> ( composite_of_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ U @ T @ V ) ) ) ).
% composite_ofI
thf(fact_788_join__ofI,axiom,
! [T: set_a,U: set_a,V: set_a] :
( ( composite_of_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ ( quotie8165075472272353145esid_a @ resid @ nn @ U @ T ) @ V )
=> ( ( composite_of_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ U @ ( quotie8165075472272353145esid_a @ resid @ nn @ T @ U ) @ V )
=> ( join_of_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U @ V ) ) ) ).
% join_ofI
thf(fact_789_rts_Ocomposite__of_Ocong,axiom,
composite_of_set_a = composite_of_set_a ).
% rts.composite_of.cong
thf(fact_790_rts_Ocomposite__of_Ocong,axiom,
composite_of_a = composite_of_a ).
% rts.composite_of.cong
thf(fact_791_coherent__normal__sub__rts_OCong_092_060_094sub_0620__composite__of__arr__normal,axiom,
! [Resid: set_a > set_a > set_a,NN: set_set_a,T: set_a,U: set_a,T4: set_a] :
( ( cohere6325062230080414023_set_a @ Resid @ NN )
=> ( ( composite_of_set_a @ Resid @ T @ U @ T4 )
=> ( ( member_set_a @ U @ NN )
=> ( ( member_set_a @ ( Resid @ T4 @ T ) @ NN )
& ( member_set_a @ ( Resid @ T @ T4 ) @ NN ) ) ) ) ) ).
% coherent_normal_sub_rts.Cong\<^sub>0_composite_of_arr_normal
thf(fact_792_coherent__normal__sub__rts_OCong_092_060_094sub_0620__composite__of__arr__normal,axiom,
! [Resid: a > a > a,NN: set_a,T: a,U: a,T4: a] :
( ( cohere6072184133013167079_rts_a @ Resid @ NN )
=> ( ( composite_of_a @ Resid @ T @ U @ T4 )
=> ( ( member_a @ U @ NN )
=> ( ( member_a @ ( Resid @ T4 @ T ) @ NN )
& ( member_a @ ( Resid @ T @ T4 ) @ NN ) ) ) ) ) ).
% coherent_normal_sub_rts.Cong\<^sub>0_composite_of_arr_normal
thf(fact_793_coherent__normal__sub__rts_OCong__composite__of__normal__arr,axiom,
! [Resid: set_a > set_a > set_a,NN: set_set_a,U: set_a,T: set_a,T4: set_a] :
( ( cohere6325062230080414023_set_a @ Resid @ NN )
=> ( ( composite_of_set_a @ Resid @ U @ T @ T4 )
=> ( ( member_set_a @ U @ NN )
=> ( normal8977612136997397236_set_a @ Resid @ NN @ T4 @ T ) ) ) ) ).
% coherent_normal_sub_rts.Cong_composite_of_normal_arr
thf(fact_794_coherent__normal__sub__rts_OCong__composite__of__normal__arr,axiom,
! [Resid: a > a > a,NN: set_a,U: a,T: a,T4: a] :
( ( cohere6072184133013167079_rts_a @ Resid @ NN )
=> ( ( composite_of_a @ Resid @ U @ T @ T4 )
=> ( ( member_a @ U @ NN )
=> ( normal_sub_Cong_a @ Resid @ NN @ T4 @ T ) ) ) ) ).
% coherent_normal_sub_rts.Cong_composite_of_normal_arr
thf(fact_795_partial__magma_Onull__def,axiom,
! [OP2: set_a > set_a > set_a] :
( ( partial_magma_set_a @ OP2 )
=> ( ( partial_null_set_a @ OP2 )
= ( the_set_a
@ ^ [N2: set_a] :
! [T6: set_a] :
( ( ( OP2 @ N2 @ T6 )
= N2 )
& ( ( OP2 @ T6 @ N2 )
= N2 ) ) ) ) ) ).
% partial_magma.null_def
thf(fact_796_partial__magma_Onull__def,axiom,
! [OP2: a > a > a] :
( ( partial_magma_a @ OP2 )
=> ( ( partial_null_a @ OP2 )
= ( the_a
@ ^ [N2: a] :
! [T6: a] :
( ( ( OP2 @ N2 @ T6 )
= N2 )
& ( ( OP2 @ T6 @ N2 )
= N2 ) ) ) ) ) ).
% partial_magma.null_def
thf(fact_797_the1__equality,axiom,
! [P: set_a > $o,A3: set_a] :
( ? [X3: set_a] :
( ( P @ X3 )
& ! [Y3: set_a] :
( ( P @ Y3 )
=> ( Y3 = X3 ) ) )
=> ( ( P @ A3 )
=> ( ( the_set_a @ P )
= A3 ) ) ) ).
% the1_equality
thf(fact_798_the1__equality,axiom,
! [P: a > $o,A3: a] :
( ? [X3: a] :
( ( P @ X3 )
& ! [Y3: a] :
( ( P @ Y3 )
=> ( Y3 = X3 ) ) )
=> ( ( P @ A3 )
=> ( ( the_a @ P )
= A3 ) ) ) ).
% the1_equality
thf(fact_799_the1I2,axiom,
! [P: set_a > $o,Q: set_a > $o] :
( ? [X3: set_a] :
( ( P @ X3 )
& ! [Y3: set_a] :
( ( P @ Y3 )
=> ( Y3 = X3 ) ) )
=> ( ! [X: set_a] :
( ( P @ X )
=> ( Q @ X ) )
=> ( Q @ ( the_set_a @ P ) ) ) ) ).
% the1I2
thf(fact_800_the1I2,axiom,
! [P: a > $o,Q: a > $o] :
( ? [X3: a] :
( ( P @ X3 )
& ! [Y3: a] :
( ( P @ Y3 )
=> ( Y3 = X3 ) ) )
=> ( ! [X: a] :
( ( P @ X )
=> ( Q @ X ) )
=> ( Q @ ( the_a @ P ) ) ) ) ).
% the1I2
thf(fact_801_If__def,axiom,
( if_set_a
= ( ^ [P2: $o,X2: set_a,Y2: set_a] :
( the_set_a
@ ^ [Z4: set_a] :
( ( P2
=> ( Z4 = X2 ) )
& ( ~ P2
=> ( Z4 = Y2 ) ) ) ) ) ) ).
% If_def
thf(fact_802_If__def,axiom,
( if_a
= ( ^ [P2: $o,X2: a,Y2: a] :
( the_a
@ ^ [Z4: a] :
( ( P2
=> ( Z4 = X2 ) )
& ( ~ P2
=> ( Z4 = Y2 ) ) ) ) ) ) ).
% If_def
thf(fact_803_theI2,axiom,
! [P: set_a > $o,A3: set_a,Q: set_a > $o] :
( ( P @ A3 )
=> ( ! [X: set_a] :
( ( P @ X )
=> ( X = A3 ) )
=> ( ! [X: set_a] :
( ( P @ X )
=> ( Q @ X ) )
=> ( Q @ ( the_set_a @ P ) ) ) ) ) ).
% theI2
thf(fact_804_theI2,axiom,
! [P: a > $o,A3: a,Q: a > $o] :
( ( P @ A3 )
=> ( ! [X: a] :
( ( P @ X )
=> ( X = A3 ) )
=> ( ! [X: a] :
( ( P @ X )
=> ( Q @ X ) )
=> ( Q @ ( the_a @ P ) ) ) ) ) ).
% theI2
thf(fact_805_theI_H,axiom,
! [P: set_a > $o] :
( ? [X3: set_a] :
( ( P @ X3 )
& ! [Y3: set_a] :
( ( P @ Y3 )
=> ( Y3 = X3 ) ) )
=> ( P @ ( the_set_a @ P ) ) ) ).
% theI'
thf(fact_806_theI_H,axiom,
! [P: a > $o] :
( ? [X3: a] :
( ( P @ X3 )
& ! [Y3: a] :
( ( P @ Y3 )
=> ( Y3 = X3 ) ) )
=> ( P @ ( the_a @ P ) ) ) ).
% theI'
thf(fact_807_theI,axiom,
! [P: set_a > $o,A3: set_a] :
( ( P @ A3 )
=> ( ! [X: set_a] :
( ( P @ X )
=> ( X = A3 ) )
=> ( P @ ( the_set_a @ P ) ) ) ) ).
% theI
thf(fact_808_theI,axiom,
! [P: a > $o,A3: a] :
( ( P @ A3 )
=> ( ! [X: a] :
( ( P @ X )
=> ( X = A3 ) )
=> ( P @ ( the_a @ P ) ) ) ) ).
% theI
thf(fact_809_normal__sub__rts__axioms__def,axiom,
( normal4776468795420100326_set_a
= ( ^ [Resid2: set_a > set_a > set_a,NN2: set_set_a] :
( ! [T6: set_a] :
( ( member_set_a @ T6 @ NN2 )
=> ( arr_set_a @ Resid2 @ T6 ) )
& ! [A: set_a] :
( ( ide_set_a @ Resid2 @ A )
=> ( member_set_a @ A @ NN2 ) )
& ! [U5: set_a,T6: set_a] :
( ( member_set_a @ U5 @ NN2 )
=> ( ( coinitial_set_a @ Resid2 @ T6 @ U5 )
=> ( member_set_a @ ( Resid2 @ U5 @ T6 ) @ NN2 ) ) )
& ! [U5: set_a,T6: set_a] :
( ( member_set_a @ U5 @ NN2 )
=> ( ( member_set_a @ ( Resid2 @ T6 @ U5 ) @ NN2 )
=> ( member_set_a @ T6 @ NN2 ) ) )
& ! [U5: set_a,T6: set_a] :
( ( member_set_a @ U5 @ NN2 )
=> ( ( seq_set_a @ Resid2 @ U5 @ T6 )
=> ? [X5: set_a] : ( composite_of_set_a @ Resid2 @ U5 @ T6 @ X5 ) ) )
& ! [U5: set_a,T6: set_a] :
( ( member_set_a @ U5 @ NN2 )
=> ( ( seq_set_a @ Resid2 @ T6 @ U5 )
=> ? [X5: set_a] : ( composite_of_set_a @ Resid2 @ T6 @ U5 @ X5 ) ) ) ) ) ) ).
% normal_sub_rts_axioms_def
thf(fact_810_normal__sub__rts__axioms__def,axiom,
( normal7698203753654205830ioms_a
= ( ^ [Resid2: a > a > a,NN2: set_a] :
( ! [T6: a] :
( ( member_a @ T6 @ NN2 )
=> ( arr_a @ Resid2 @ T6 ) )
& ! [A: a] :
( ( ide_a @ Resid2 @ A )
=> ( member_a @ A @ NN2 ) )
& ! [U5: a,T6: a] :
( ( member_a @ U5 @ NN2 )
=> ( ( coinitial_a @ Resid2 @ T6 @ U5 )
=> ( member_a @ ( Resid2 @ U5 @ T6 ) @ NN2 ) ) )
& ! [U5: a,T6: a] :
( ( member_a @ U5 @ NN2 )
=> ( ( member_a @ ( Resid2 @ T6 @ U5 ) @ NN2 )
=> ( member_a @ T6 @ NN2 ) ) )
& ! [U5: a,T6: a] :
( ( member_a @ U5 @ NN2 )
=> ( ( seq_a @ Resid2 @ U5 @ T6 )
=> ? [X5: a] : ( composite_of_a @ Resid2 @ U5 @ T6 @ X5 ) ) )
& ! [U5: a,T6: a] :
( ( member_a @ U5 @ NN2 )
=> ( ( seq_a @ Resid2 @ T6 @ U5 )
=> ? [X5: a] : ( composite_of_a @ Resid2 @ T6 @ U5 @ X5 ) ) ) ) ) ) ).
% normal_sub_rts_axioms_def
thf(fact_811_normal__sub__rts__axioms_Ointro,axiom,
! [NN: set_set_a,Resid: set_a > set_a > set_a] :
( ! [T3: set_a] :
( ( member_set_a @ T3 @ NN )
=> ( arr_set_a @ Resid @ T3 ) )
=> ( ! [A6: set_a] :
( ( ide_set_a @ Resid @ A6 )
=> ( member_set_a @ A6 @ NN ) )
=> ( ! [U3: set_a,T3: set_a] :
( ( member_set_a @ U3 @ NN )
=> ( ( coinitial_set_a @ Resid @ T3 @ U3 )
=> ( member_set_a @ ( Resid @ U3 @ T3 ) @ NN ) ) )
=> ( ! [U3: set_a,T3: set_a] :
( ( member_set_a @ U3 @ NN )
=> ( ( member_set_a @ ( Resid @ T3 @ U3 ) @ NN )
=> ( member_set_a @ T3 @ NN ) ) )
=> ( ! [U3: set_a,T3: set_a] :
( ( member_set_a @ U3 @ NN )
=> ( ( seq_set_a @ Resid @ U3 @ T3 )
=> ? [X_1: set_a] : ( composite_of_set_a @ Resid @ U3 @ T3 @ X_1 ) ) )
=> ( ! [U3: set_a,T3: set_a] :
( ( member_set_a @ U3 @ NN )
=> ( ( seq_set_a @ Resid @ T3 @ U3 )
=> ? [X_1: set_a] : ( composite_of_set_a @ Resid @ T3 @ U3 @ X_1 ) ) )
=> ( normal4776468795420100326_set_a @ Resid @ NN ) ) ) ) ) ) ) ).
% normal_sub_rts_axioms.intro
thf(fact_812_normal__sub__rts__axioms_Ointro,axiom,
! [NN: set_a,Resid: a > a > a] :
( ! [T3: a] :
( ( member_a @ T3 @ NN )
=> ( arr_a @ Resid @ T3 ) )
=> ( ! [A6: a] :
( ( ide_a @ Resid @ A6 )
=> ( member_a @ A6 @ NN ) )
=> ( ! [U3: a,T3: a] :
( ( member_a @ U3 @ NN )
=> ( ( coinitial_a @ Resid @ T3 @ U3 )
=> ( member_a @ ( Resid @ U3 @ T3 ) @ NN ) ) )
=> ( ! [U3: a,T3: a] :
( ( member_a @ U3 @ NN )
=> ( ( member_a @ ( Resid @ T3 @ U3 ) @ NN )
=> ( member_a @ T3 @ NN ) ) )
=> ( ! [U3: a,T3: a] :
( ( member_a @ U3 @ NN )
=> ( ( seq_a @ Resid @ U3 @ T3 )
=> ? [X_1: a] : ( composite_of_a @ Resid @ U3 @ T3 @ X_1 ) ) )
=> ( ! [U3: a,T3: a] :
( ( member_a @ U3 @ NN )
=> ( ( seq_a @ Resid @ T3 @ U3 )
=> ? [X_1: a] : ( composite_of_a @ Resid @ T3 @ U3 @ X_1 ) ) )
=> ( normal7698203753654205830ioms_a @ Resid @ NN ) ) ) ) ) ) ) ).
% normal_sub_rts_axioms.intro
thf(fact_813_Greatest__def,axiom,
( order_Greatest_set_a
= ( ^ [P2: set_a > $o] :
( the_set_a
@ ^ [X2: set_a] :
( ( P2 @ X2 )
& ! [Y2: set_a] :
( ( P2 @ Y2 )
=> ( ord_less_eq_set_a @ Y2 @ X2 ) ) ) ) ) ) ).
% Greatest_def
thf(fact_814_R_Ocomposite__of__unq__upto__cong,axiom,
! [U: a,T: a,V: a,V4: a] :
( ( composite_of_a @ resid @ U @ T @ V )
=> ( ( composite_of_a @ resid @ U @ T @ V4 )
=> ( ( ide_a @ resid @ ( resid @ V @ V4 ) )
& ( ide_a @ resid @ ( resid @ V4 @ V ) ) ) ) ) ).
% R.composite_of_unq_upto_cong
thf(fact_815_R_Ocomposite__of__ide__self,axiom,
! [A3: a] :
( ( ide_a @ resid @ A3 )
=> ( composite_of_a @ resid @ A3 @ A3 @ A3 ) ) ).
% R.composite_of_ide_self
thf(fact_816_R_Ocomposite__of__def,axiom,
! [U: a,T: a,V: a] :
( ( composite_of_a @ resid @ U @ T @ V )
= ( ( ide_a @ resid @ ( resid @ U @ V ) )
& ( ide_a @ resid @ ( resid @ ( resid @ V @ U ) @ T ) )
& ( ide_a @ resid @ ( resid @ T @ ( resid @ V @ U ) ) ) ) ) ).
% R.composite_of_def
thf(fact_817_R_Ocomposite__of__cancel__left,axiom,
! [T: a,U: a,V: a,U7: a] :
( ( composite_of_a @ resid @ T @ U @ V )
=> ( ( composite_of_a @ resid @ T @ U7 @ V )
=> ( ( ide_a @ resid @ ( resid @ U @ U7 ) )
& ( ide_a @ resid @ ( resid @ U7 @ U ) ) ) ) ) ).
% R.composite_of_cancel_left
thf(fact_818_R_Ocomposite__ofE,axiom,
! [U: a,T: a,V: a] :
( ( composite_of_a @ resid @ U @ T @ V )
=> ~ ( ( ide_a @ resid @ ( resid @ U @ V ) )
=> ~ ( ( ide_a @ resid @ ( resid @ ( resid @ V @ U ) @ T ) )
& ( ide_a @ resid @ ( resid @ T @ ( resid @ V @ U ) ) ) ) ) ) ).
% R.composite_ofE
thf(fact_819_R_Ocon__composite__of__iff,axiom,
! [T: a,U: a,V: a,W2: a] :
( ( composite_of_a @ resid @ T @ U @ V )
=> ( ( con_a @ resid @ W2 @ V )
= ( con_a @ resid @ ( resid @ W2 @ T ) @ U ) ) ) ).
% R.con_composite_of_iff
thf(fact_820_R_Obounded__imp__con,axiom,
! [T: a,U: a,V: a,T4: a,U7: a] :
( ( composite_of_a @ resid @ T @ U @ V )
=> ( ( composite_of_a @ resid @ T4 @ U7 @ V )
=> ( con_a @ resid @ T @ T4 ) ) ) ).
% R.bounded_imp_con
thf(fact_821_R_Oresid__composite__of_I1_J,axiom,
! [T: a,U: a,W2: a,V: a] :
( ( composite_of_a @ resid @ T @ U @ W2 )
=> ( ( con_a @ resid @ W2 @ V )
=> ( con_a @ resid @ ( resid @ V @ T ) @ ( resid @ W2 @ T ) ) ) ) ).
% R.resid_composite_of(1)
thf(fact_822_R_Oresid__composite__of_I2_J,axiom,
! [T: a,U: a,W2: a,V: a] :
( ( composite_of_a @ resid @ T @ U @ W2 )
=> ( ( con_a @ resid @ W2 @ V )
=> ( con_a @ resid @ ( resid @ V @ T ) @ U ) ) ) ).
% R.resid_composite_of(2)
thf(fact_823_R_Oresid__composite__of_I4_J,axiom,
! [T: a,U: a,W2: a,V: a] :
( ( composite_of_a @ resid @ T @ U @ W2 )
=> ( ( con_a @ resid @ W2 @ V )
=> ( composite_of_a @ resid @ ( resid @ T @ V ) @ ( resid @ U @ ( resid @ V @ T ) ) @ ( resid @ W2 @ V ) ) ) ) ).
% R.resid_composite_of(4)
thf(fact_824_R_Ocon__prfx__composite__of_I1_J,axiom,
! [T: a,U: a,W2: a] :
( ( composite_of_a @ resid @ T @ U @ W2 )
=> ( con_a @ resid @ T @ W2 ) ) ).
% R.con_prfx_composite_of(1)
thf(fact_825_R_Ocon__prfx__composite__of_I2_J,axiom,
! [T: a,U: a,W2: a,V: a] :
( ( composite_of_a @ resid @ T @ U @ W2 )
=> ( ( con_a @ resid @ W2 @ V )
=> ( con_a @ resid @ T @ V ) ) ) ).
% R.con_prfx_composite_of(2)
thf(fact_826_R_Osources__composite__of,axiom,
! [U: a,T: a,V: a] :
( ( composite_of_a @ resid @ U @ T @ V )
=> ( ( sources_a @ resid @ V )
= ( sources_a @ resid @ U ) ) ) ).
% R.sources_composite_of
thf(fact_827_R_Oarr__composite__of,axiom,
! [U: a,T: a,V: a] :
( ( composite_of_a @ resid @ U @ T @ V )
=> ( arr_a @ resid @ V ) ) ).
% R.arr_composite_of
thf(fact_828_R_Otargets__composite__of,axiom,
! [U: a,T: a,V: a] :
( ( composite_of_a @ resid @ U @ T @ V )
=> ( ( targets_a @ resid @ V )
= ( targets_a @ resid @ T ) ) ) ).
% R.targets_composite_of
thf(fact_829_R_Ojoin__ofE,axiom,
! [T: a,U: a,V: a] :
( ( join_of_a @ resid @ T @ U @ V )
=> ~ ( ( composite_of_a @ resid @ T @ ( resid @ U @ T ) @ V )
=> ~ ( composite_of_a @ resid @ U @ ( resid @ T @ U ) @ V ) ) ) ).
% R.join_ofE
thf(fact_830_R_Ojoin__of__def,axiom,
! [T: a,U: a,V: a] :
( ( join_of_a @ resid @ T @ U @ V )
= ( ( composite_of_a @ resid @ T @ ( resid @ U @ T ) @ V )
& ( composite_of_a @ resid @ U @ ( resid @ T @ U ) @ V ) ) ) ).
% R.join_of_def
thf(fact_831_N_Ocomposite__of__arr__normal,axiom,
! [Arr: a > $o,T: a,U: a,T4: a] :
( ( Arr @ T )
=> ( ( member_a @ U @ nn )
=> ( ( composite_of_a @ resid @ T @ U @ T4 )
=> ( ( member_a @ ( resid @ T4 @ T ) @ nn )
& ( member_a @ ( resid @ T @ T4 ) @ nn ) ) ) ) ) ).
% N.composite_of_arr_normal
thf(fact_832_N_Ocomposite__closed,axiom,
! [T: a,U: a,V: a] :
( ( member_a @ T @ nn )
=> ( ( member_a @ U @ nn )
=> ( ( composite_of_a @ resid @ T @ U @ V )
=> ( member_a @ V @ nn ) ) ) ) ).
% N.composite_closed
thf(fact_833_N_OCong_092_060_094sub_0620__iff,axiom,
! [T: a,T4: a] :
( ( ( member_a @ ( resid @ T @ T4 ) @ nn )
& ( member_a @ ( resid @ T4 @ T ) @ nn ) )
= ( ? [U5: a,U6: a,V5: a,V6: a] :
( ( member_a @ U5 @ nn )
& ( member_a @ U6 @ nn )
& ( member_a @ ( resid @ V5 @ V6 ) @ nn )
& ( member_a @ ( resid @ V6 @ V5 ) @ nn )
& ( composite_of_a @ resid @ T @ U5 @ V5 )
& ( composite_of_a @ resid @ T4 @ U6 @ V6 ) ) ) ) ).
% N.Cong\<^sub>0_iff
thf(fact_834_N_OCong_092_060_094sub_0620__composite__of__arr__normal,axiom,
! [T: a,U: a,T4: a] :
( ( composite_of_a @ resid @ T @ U @ T4 )
=> ( ( member_a @ U @ nn )
=> ( ( member_a @ ( resid @ T4 @ T ) @ nn )
& ( member_a @ ( resid @ T @ T4 ) @ nn ) ) ) ) ).
% N.Cong\<^sub>0_composite_of_arr_normal
thf(fact_835_N_OCong_092_060_094sub_0620__cancel__left,axiom,
! [T: a,U: a,V: a,U7: a,V4: a] :
( ( composite_of_a @ resid @ T @ U @ V )
=> ( ( composite_of_a @ resid @ T @ U7 @ V4 )
=> ( ( ( member_a @ ( resid @ V @ V4 ) @ nn )
& ( member_a @ ( resid @ V4 @ V ) @ nn ) )
=> ( ( member_a @ ( resid @ U @ U7 ) @ nn )
& ( member_a @ ( resid @ U7 @ U ) @ nn ) ) ) ) ) ).
% N.Cong\<^sub>0_cancel_left
thf(fact_836_N_Ofactor__closed_I1_J,axiom,
! [T: a,U: a,V: a] :
( ( composite_of_a @ resid @ T @ U @ V )
=> ( ( member_a @ V @ nn )
=> ( member_a @ T @ nn ) ) ) ).
% N.factor_closed(1)
thf(fact_837_N_Ofactor__closed_I2_J,axiom,
! [T: a,U: a,V: a] :
( ( composite_of_a @ resid @ T @ U @ V )
=> ( ( member_a @ V @ nn )
=> ( member_a @ U @ nn ) ) ) ).
% N.factor_closed(2)
thf(fact_838_R_Ocomposable__def,axiom,
! [T: a,U: a] :
( ( composable_a @ resid @ T @ U )
= ( ? [X5: a] : ( composite_of_a @ resid @ T @ U @ X5 ) ) ) ).
% R.composable_def
thf(fact_839_R_Onull__def,axiom,
( ( partial_null_a @ resid )
= ( the_a
@ ^ [N2: a] :
! [T6: a] :
( ( ( resid @ N2 @ T6 )
= N2 )
& ( ( resid @ T6 @ N2 )
= N2 ) ) ) ) ).
% R.null_def
thf(fact_840_R_Oresid__composite__of_I3_J,axiom,
! [T: a,U: a,W2: a,V: a] :
( ( composite_of_a @ resid @ T @ U @ W2 )
=> ( ( con_a @ resid @ W2 @ V )
=> ( ( ide_a @ resid @ ( resid @ ( resid @ V @ W2 ) @ ( resid @ ( resid @ V @ T ) @ U ) ) )
& ( ide_a @ resid @ ( resid @ ( resid @ ( resid @ V @ T ) @ U ) @ ( resid @ V @ W2 ) ) ) ) ) ) ).
% R.resid_composite_of(3)
thf(fact_841_R_Ocomposite__of__arr__ide,axiom,
! [B: a,T: a] :
( ( ide_a @ resid @ B )
=> ( ( composite_of_a @ resid @ T @ B @ T )
= ( con_a @ resid @ ( resid @ T @ T ) @ B ) ) ) ).
% R.composite_of_arr_ide
thf(fact_842_R_Ocomposite__of__ide__arr,axiom,
! [A3: a,T: a] :
( ( ide_a @ resid @ A3 )
=> ( ( composite_of_a @ resid @ A3 @ T @ T )
= ( con_a @ resid @ T @ A3 ) ) ) ).
% R.composite_of_ide_arr
thf(fact_843_R_Ocomposite__of__source__arr,axiom,
! [T: a,A3: a] :
( ( arr_a @ resid @ T )
=> ( ( member_a @ A3 @ ( sources_a @ resid @ T ) )
=> ( composite_of_a @ resid @ A3 @ T @ T ) ) ) ).
% R.composite_of_source_arr
thf(fact_844_R_Ocomposite__of__arr__target,axiom,
! [T: a,B: a] :
( ( arr_a @ resid @ T )
=> ( ( member_a @ B @ ( targets_a @ resid @ T ) )
=> ( composite_of_a @ resid @ T @ B @ T ) ) ) ).
% R.composite_of_arr_target
thf(fact_845_N_Odiamond__commutes__upto__Cong_092_060_094sub_0620,axiom,
! [T: a,U: a,V: a,V4: a] :
( ( con_a @ resid @ T @ U )
=> ( ( composite_of_a @ resid @ T @ ( resid @ U @ T ) @ V )
=> ( ( composite_of_a @ resid @ U @ ( resid @ T @ U ) @ V4 )
=> ( ( member_a @ ( resid @ V @ V4 ) @ nn )
& ( member_a @ ( resid @ V4 @ V ) @ nn ) ) ) ) ) ).
% N.diamond_commutes_upto_Cong\<^sub>0
thf(fact_846_N_OCong__composite__of__normal__arr,axiom,
! [U: a,T: a,T4: a] :
( ( composite_of_a @ resid @ U @ T @ T4 )
=> ( ( member_a @ U @ nn )
=> ( normal_sub_Cong_a @ resid @ nn @ T4 @ T ) ) ) ).
% N.Cong_composite_of_normal_arr
thf(fact_847_N_Ocomposite__closed__right,axiom,
! [U: a,T: a] :
( ( member_a @ U @ nn )
=> ( ( seq_a @ resid @ T @ U )
=> ? [X_12: a] : ( composite_of_a @ resid @ T @ U @ X_12 ) ) ) ).
% N.composite_closed_right
thf(fact_848_N_Ocomposite__closed__left,axiom,
! [U: a,T: a] :
( ( member_a @ U @ nn )
=> ( ( seq_a @ resid @ U @ T )
=> ? [X_12: a] : ( composite_of_a @ resid @ U @ T @ X_12 ) ) ) ).
% N.composite_closed_left
thf(fact_849_N_Ocomposite__of__normal__arr,axiom,
! [T: a,U: a,T4: a] :
( ( arr_a @ resid @ T )
=> ( ( member_a @ U @ nn )
=> ( ( composite_of_a @ resid @ U @ T @ T4 )
=> ( normal_sub_Cong_a @ resid @ nn @ T4 @ T ) ) ) ) ).
% N.composite_of_normal_arr
thf(fact_850_R_Ocomposite__ofI,axiom,
! [U: a,V: a,T: a] :
( ( ide_a @ resid @ ( resid @ U @ V ) )
=> ( ( ( ide_a @ resid @ ( resid @ ( resid @ V @ U ) @ T ) )
& ( ide_a @ resid @ ( resid @ T @ ( resid @ V @ U ) ) ) )
=> ( composite_of_a @ resid @ U @ T @ V ) ) ) ).
% R.composite_ofI
thf(fact_851_R_Ojoin__ofI,axiom,
! [T: a,U: a,V: a] :
( ( composite_of_a @ resid @ T @ ( resid @ U @ T ) @ V )
=> ( ( composite_of_a @ resid @ U @ ( resid @ T @ U ) @ V )
=> ( join_of_a @ resid @ T @ U @ V ) ) ) ).
% R.join_ofI
thf(fact_852_GreatestI2__order,axiom,
! [P: set_a > $o,X4: set_a,Q: set_a > $o] :
( ( P @ X4 )
=> ( ! [Y3: set_a] :
( ( P @ Y3 )
=> ( ord_less_eq_set_a @ Y3 @ X4 ) )
=> ( ! [X: set_a] :
( ( P @ X )
=> ( ! [Y: set_a] :
( ( P @ Y )
=> ( ord_less_eq_set_a @ Y @ X ) )
=> ( Q @ X ) ) )
=> ( Q @ ( order_Greatest_set_a @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_853_Greatest__equality,axiom,
! [P: set_a > $o,X4: set_a] :
( ( P @ X4 )
=> ( ! [Y3: set_a] :
( ( P @ Y3 )
=> ( ord_less_eq_set_a @ Y3 @ X4 ) )
=> ( ( order_Greatest_set_a @ P )
= X4 ) ) ) ).
% Greatest_equality
thf(fact_854_weakly__extensional__rts__axioms,axiom,
weakly5936471160286156245_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) ).
% weakly_extensional_rts_axioms
thf(fact_855_rts__axioms_Ointro,axiom,
! [Resid: set_a > set_a > set_a] :
( ! [T3: set_a] :
( ( arr_set_a @ Resid @ T3 )
=> ( ide_set_a @ Resid @ ( trg_set_a @ Resid @ T3 ) ) )
=> ( ! [A6: set_a,T3: set_a] :
( ( ide_set_a @ Resid @ A6 )
=> ( ( con_set_a @ Resid @ T3 @ A6 )
=> ( ( Resid @ T3 @ A6 )
= T3 ) ) )
=> ( ! [A6: set_a,T3: set_a] :
( ( ide_set_a @ Resid @ A6 )
=> ( ( con_set_a @ Resid @ A6 @ T3 )
=> ( ide_set_a @ Resid @ ( Resid @ A6 @ T3 ) ) ) )
=> ( ! [T3: set_a,U3: set_a] :
( ( con_set_a @ Resid @ T3 @ U3 )
=> ? [A9: set_a] :
( ( ide_set_a @ Resid @ A9 )
& ( con_set_a @ Resid @ A9 @ T3 )
& ( con_set_a @ Resid @ A9 @ U3 ) ) )
=> ( ! [T3: set_a,U3: set_a,V3: set_a] :
( ( ide_set_a @ Resid @ ( Resid @ T3 @ U3 ) )
=> ( ( con_set_a @ Resid @ U3 @ V3 )
=> ( con_set_a @ Resid @ ( Resid @ T3 @ U3 ) @ ( Resid @ V3 @ U3 ) ) ) )
=> ( rts_axioms_set_a @ Resid ) ) ) ) ) ) ).
% rts_axioms.intro
thf(fact_856_rts__axioms_Ointro,axiom,
! [Resid: a > a > a] :
( ! [T3: a] :
( ( arr_a @ Resid @ T3 )
=> ( ide_a @ Resid @ ( trg_a @ Resid @ T3 ) ) )
=> ( ! [A6: a,T3: a] :
( ( ide_a @ Resid @ A6 )
=> ( ( con_a @ Resid @ T3 @ A6 )
=> ( ( Resid @ T3 @ A6 )
= T3 ) ) )
=> ( ! [A6: a,T3: a] :
( ( ide_a @ Resid @ A6 )
=> ( ( con_a @ Resid @ A6 @ T3 )
=> ( ide_a @ Resid @ ( Resid @ A6 @ T3 ) ) ) )
=> ( ! [T3: a,U3: a] :
( ( con_a @ Resid @ T3 @ U3 )
=> ? [A9: a] :
( ( ide_a @ Resid @ A9 )
& ( con_a @ Resid @ A9 @ T3 )
& ( con_a @ Resid @ A9 @ U3 ) ) )
=> ( ! [T3: a,U3: a,V3: a] :
( ( ide_a @ Resid @ ( Resid @ T3 @ U3 ) )
=> ( ( con_a @ Resid @ U3 @ V3 )
=> ( con_a @ Resid @ ( Resid @ T3 @ U3 ) @ ( Resid @ V3 @ U3 ) ) ) )
=> ( rts_axioms_a @ Resid ) ) ) ) ) ) ).
% rts_axioms.intro
thf(fact_857_rts__axioms__def,axiom,
( rts_axioms_set_a
= ( ^ [Resid2: set_a > set_a > set_a] :
( ! [T6: set_a] :
( ( arr_set_a @ Resid2 @ T6 )
=> ( ide_set_a @ Resid2 @ ( trg_set_a @ Resid2 @ T6 ) ) )
& ! [A: set_a,T6: set_a] :
( ( ide_set_a @ Resid2 @ A )
=> ( ( con_set_a @ Resid2 @ T6 @ A )
=> ( ( Resid2 @ T6 @ A )
= T6 ) ) )
& ! [A: set_a,T6: set_a] :
( ( ide_set_a @ Resid2 @ A )
=> ( ( con_set_a @ Resid2 @ A @ T6 )
=> ( ide_set_a @ Resid2 @ ( Resid2 @ A @ T6 ) ) ) )
& ! [T6: set_a,U5: set_a] :
( ( con_set_a @ Resid2 @ T6 @ U5 )
=> ? [A: set_a] :
( ( ide_set_a @ Resid2 @ A )
& ( con_set_a @ Resid2 @ A @ T6 )
& ( con_set_a @ Resid2 @ A @ U5 ) ) )
& ! [T6: set_a,U5: set_a,V5: set_a] :
( ( ide_set_a @ Resid2 @ ( Resid2 @ T6 @ U5 ) )
=> ( ( con_set_a @ Resid2 @ U5 @ V5 )
=> ( con_set_a @ Resid2 @ ( Resid2 @ T6 @ U5 ) @ ( Resid2 @ V5 @ U5 ) ) ) ) ) ) ) ).
% rts_axioms_def
thf(fact_858_rts__axioms__def,axiom,
( rts_axioms_a
= ( ^ [Resid2: a > a > a] :
( ! [T6: a] :
( ( arr_a @ Resid2 @ T6 )
=> ( ide_a @ Resid2 @ ( trg_a @ Resid2 @ T6 ) ) )
& ! [A: a,T6: a] :
( ( ide_a @ Resid2 @ A )
=> ( ( con_a @ Resid2 @ T6 @ A )
=> ( ( Resid2 @ T6 @ A )
= T6 ) ) )
& ! [A: a,T6: a] :
( ( ide_a @ Resid2 @ A )
=> ( ( con_a @ Resid2 @ A @ T6 )
=> ( ide_a @ Resid2 @ ( Resid2 @ A @ T6 ) ) ) )
& ! [T6: a,U5: a] :
( ( con_a @ Resid2 @ T6 @ U5 )
=> ? [A: a] :
( ( ide_a @ Resid2 @ A )
& ( con_a @ Resid2 @ A @ T6 )
& ( con_a @ Resid2 @ A @ U5 ) ) )
& ! [T6: a,U5: a,V5: a] :
( ( ide_a @ Resid2 @ ( Resid2 @ T6 @ U5 ) )
=> ( ( con_a @ Resid2 @ U5 @ V5 )
=> ( con_a @ Resid2 @ ( Resid2 @ T6 @ U5 ) @ ( Resid2 @ V5 @ U5 ) ) ) ) ) ) ) ).
% rts_axioms_def
thf(fact_859_weakly__extensional__rts_Oweak__extensionality,axiom,
! [Resid: a > a > a,T: a,U: a] :
( ( weakly1626779504270821493_rts_a @ Resid )
=> ( ( ( ide_a @ Resid @ ( Resid @ T @ U ) )
& ( ide_a @ Resid @ ( Resid @ U @ T ) ) )
=> ( ( ide_a @ Resid @ T )
=> ( ( ide_a @ Resid @ U )
=> ( T = U ) ) ) ) ) ).
% weakly_extensional_rts.weak_extensionality
thf(fact_860_weakly__extensional__rts_Oweak__extensionality,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a] :
( ( weakly5936471160286156245_set_a @ Resid )
=> ( ( ( ide_set_a @ Resid @ ( Resid @ T @ U ) )
& ( ide_set_a @ Resid @ ( Resid @ U @ T ) ) )
=> ( ( ide_set_a @ Resid @ T )
=> ( ( ide_set_a @ Resid @ U )
=> ( T = U ) ) ) ) ) ).
% weakly_extensional_rts.weak_extensionality
thf(fact_861_weakly__extensional__rts_Osrc__src,axiom,
! [Resid: set_a > set_a > set_a,T: set_a] :
( ( weakly5936471160286156245_set_a @ Resid )
=> ( ( weakly2061155085811118449_set_a @ Resid @ ( weakly2061155085811118449_set_a @ Resid @ T ) )
= ( weakly2061155085811118449_set_a @ Resid @ T ) ) ) ).
% weakly_extensional_rts.src_src
thf(fact_862_weakly__extensional__rts_Otrg__trg,axiom,
! [Resid: a > a > a,T: a] :
( ( weakly1626779504270821493_rts_a @ Resid )
=> ( ( trg_a @ Resid @ ( trg_a @ Resid @ T ) )
= ( trg_a @ Resid @ T ) ) ) ).
% weakly_extensional_rts.trg_trg
thf(fact_863_weakly__extensional__rts_Otrg__trg,axiom,
! [Resid: set_a > set_a > set_a,T: set_a] :
( ( weakly5936471160286156245_set_a @ Resid )
=> ( ( trg_set_a @ Resid @ ( trg_set_a @ Resid @ T ) )
= ( trg_set_a @ Resid @ T ) ) ) ).
% weakly_extensional_rts.trg_trg
thf(fact_864_weakly__extensional__rts_Oapex__sym,axiom,
! [Resid: a > a > a,T: a,U: a] :
( ( weakly1626779504270821493_rts_a @ Resid )
=> ( ( trg_a @ Resid @ ( Resid @ T @ U ) )
= ( trg_a @ Resid @ ( Resid @ U @ T ) ) ) ) ).
% weakly_extensional_rts.apex_sym
thf(fact_865_weakly__extensional__rts_Oapex__sym,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a] :
( ( weakly5936471160286156245_set_a @ Resid )
=> ( ( trg_set_a @ Resid @ ( Resid @ T @ U ) )
= ( trg_set_a @ Resid @ ( Resid @ U @ T ) ) ) ) ).
% weakly_extensional_rts.apex_sym
thf(fact_866_weakly__extensional__rts_Ocon__ide__are__eq,axiom,
! [Resid: a > a > a,A3: a,A4: a] :
( ( weakly1626779504270821493_rts_a @ Resid )
=> ( ( ide_a @ Resid @ A3 )
=> ( ( ide_a @ Resid @ A4 )
=> ( ( con_a @ Resid @ A3 @ A4 )
=> ( A3 = A4 ) ) ) ) ) ).
% weakly_extensional_rts.con_ide_are_eq
thf(fact_867_weakly__extensional__rts_Ocon__ide__are__eq,axiom,
! [Resid: set_a > set_a > set_a,A3: set_a,A4: set_a] :
( ( weakly5936471160286156245_set_a @ Resid )
=> ( ( ide_set_a @ Resid @ A3 )
=> ( ( ide_set_a @ Resid @ A4 )
=> ( ( con_set_a @ Resid @ A3 @ A4 )
=> ( A3 = A4 ) ) ) ) ) ).
% weakly_extensional_rts.con_ide_are_eq
thf(fact_868_weakly__extensional__rts_Oarr__has__un__source,axiom,
! [Resid: a > a > a,T: a] :
( ( weakly1626779504270821493_rts_a @ Resid )
=> ( ( arr_a @ Resid @ T )
=> ? [X: a] :
( ( member_a @ X @ ( sources_a @ Resid @ T ) )
& ! [Y: a] :
( ( member_a @ Y @ ( sources_a @ Resid @ T ) )
=> ( Y = X ) ) ) ) ) ).
% weakly_extensional_rts.arr_has_un_source
thf(fact_869_weakly__extensional__rts_Oarr__has__un__source,axiom,
! [Resid: set_a > set_a > set_a,T: set_a] :
( ( weakly5936471160286156245_set_a @ Resid )
=> ( ( arr_set_a @ Resid @ T )
=> ? [X: set_a] :
( ( member_set_a @ X @ ( sources_set_a @ Resid @ T ) )
& ! [Y: set_a] :
( ( member_set_a @ Y @ ( sources_set_a @ Resid @ T ) )
=> ( Y = X ) ) ) ) ) ).
% weakly_extensional_rts.arr_has_un_source
thf(fact_870_weakly__extensional__rts_Otrg__ide,axiom,
! [Resid: a > a > a,A3: a] :
( ( weakly1626779504270821493_rts_a @ Resid )
=> ( ( ide_a @ Resid @ A3 )
=> ( ( trg_a @ Resid @ A3 )
= A3 ) ) ) ).
% weakly_extensional_rts.trg_ide
thf(fact_871_weakly__extensional__rts_Otrg__ide,axiom,
! [Resid: set_a > set_a > set_a,A3: set_a] :
( ( weakly5936471160286156245_set_a @ Resid )
=> ( ( ide_set_a @ Resid @ A3 )
=> ( ( trg_set_a @ Resid @ A3 )
= A3 ) ) ) ).
% weakly_extensional_rts.trg_ide
thf(fact_872_weakly__extensional__rts_Oarr__has__un__target,axiom,
! [Resid: a > a > a,T: a] :
( ( weakly1626779504270821493_rts_a @ Resid )
=> ( ( arr_a @ Resid @ T )
=> ? [X: a] :
( ( member_a @ X @ ( targets_a @ Resid @ T ) )
& ! [Y: a] :
( ( member_a @ Y @ ( targets_a @ Resid @ T ) )
=> ( Y = X ) ) ) ) ) ).
% weakly_extensional_rts.arr_has_un_target
thf(fact_873_weakly__extensional__rts_Oarr__has__un__target,axiom,
! [Resid: set_a > set_a > set_a,T: set_a] :
( ( weakly5936471160286156245_set_a @ Resid )
=> ( ( arr_set_a @ Resid @ T )
=> ? [X: set_a] :
( ( member_set_a @ X @ ( targets_set_a @ Resid @ T ) )
& ! [Y: set_a] :
( ( member_set_a @ Y @ ( targets_set_a @ Resid @ T ) )
=> ( Y = X ) ) ) ) ) ).
% weakly_extensional_rts.arr_has_un_target
thf(fact_874_weakly__extensional__rts_Otrg__resid__sym,axiom,
! [Resid: a > a > a,T: a,U: a] :
( ( weakly1626779504270821493_rts_a @ Resid )
=> ( ( con_a @ Resid @ T @ U )
=> ( ( trg_a @ Resid @ ( Resid @ T @ U ) )
= ( trg_a @ Resid @ ( Resid @ U @ T ) ) ) ) ) ).
% weakly_extensional_rts.trg_resid_sym
thf(fact_875_weakly__extensional__rts_Otrg__resid__sym,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a] :
( ( weakly5936471160286156245_set_a @ Resid )
=> ( ( con_set_a @ Resid @ T @ U )
=> ( ( trg_set_a @ Resid @ ( Resid @ T @ U ) )
= ( trg_set_a @ Resid @ ( Resid @ U @ T ) ) ) ) ) ).
% weakly_extensional_rts.trg_resid_sym
thf(fact_876_weakly__extensional__rts_Osrc__ide,axiom,
! [Resid: a > a > a,A3: a] :
( ( weakly1626779504270821493_rts_a @ Resid )
=> ( ( ide_a @ Resid @ A3 )
=> ( ( weakly8512939796511659025_src_a @ Resid @ A3 )
= A3 ) ) ) ).
% weakly_extensional_rts.src_ide
thf(fact_877_weakly__extensional__rts_Osrc__ide,axiom,
! [Resid: set_a > set_a > set_a,A3: set_a] :
( ( weakly5936471160286156245_set_a @ Resid )
=> ( ( ide_set_a @ Resid @ A3 )
=> ( ( weakly2061155085811118449_set_a @ Resid @ A3 )
= A3 ) ) ) ).
% weakly_extensional_rts.src_ide
thf(fact_878_weakly__extensional__rts_Ocon__imp__eq__src,axiom,
! [Resid: a > a > a,T: a,U: a] :
( ( weakly1626779504270821493_rts_a @ Resid )
=> ( ( con_a @ Resid @ T @ U )
=> ( ( weakly8512939796511659025_src_a @ Resid @ T )
= ( weakly8512939796511659025_src_a @ Resid @ U ) ) ) ) ).
% weakly_extensional_rts.con_imp_eq_src
thf(fact_879_weakly__extensional__rts_Ocon__imp__eq__src,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a] :
( ( weakly5936471160286156245_set_a @ Resid )
=> ( ( con_set_a @ Resid @ T @ U )
=> ( ( weakly2061155085811118449_set_a @ Resid @ T )
= ( weakly2061155085811118449_set_a @ Resid @ U ) ) ) ) ).
% weakly_extensional_rts.con_imp_eq_src
thf(fact_880_weakly__extensional__rts_Oarr__trg__iff__arr,axiom,
! [Resid: a > a > a,T: a] :
( ( weakly1626779504270821493_rts_a @ Resid )
=> ( ( arr_a @ Resid @ ( trg_a @ Resid @ T ) )
= ( arr_a @ Resid @ T ) ) ) ).
% weakly_extensional_rts.arr_trg_iff_arr
thf(fact_881_weakly__extensional__rts_Oarr__trg__iff__arr,axiom,
! [Resid: set_a > set_a > set_a,T: set_a] :
( ( weakly5936471160286156245_set_a @ Resid )
=> ( ( arr_set_a @ Resid @ ( trg_set_a @ Resid @ T ) )
= ( arr_set_a @ Resid @ T ) ) ) ).
% weakly_extensional_rts.arr_trg_iff_arr
thf(fact_882_weakly__extensional__rts_Oarr__src__iff__arr,axiom,
! [Resid: a > a > a,T: a] :
( ( weakly1626779504270821493_rts_a @ Resid )
=> ( ( arr_a @ Resid @ ( weakly8512939796511659025_src_a @ Resid @ T ) )
= ( arr_a @ Resid @ T ) ) ) ).
% weakly_extensional_rts.arr_src_iff_arr
thf(fact_883_weakly__extensional__rts_Oarr__src__iff__arr,axiom,
! [Resid: set_a > set_a > set_a,T: set_a] :
( ( weakly5936471160286156245_set_a @ Resid )
=> ( ( arr_set_a @ Resid @ ( weakly2061155085811118449_set_a @ Resid @ T ) )
= ( arr_set_a @ Resid @ T ) ) ) ).
% weakly_extensional_rts.arr_src_iff_arr
thf(fact_884_weakly__extensional__rts_Otrg__composite__of,axiom,
! [Resid: a > a > a,U: a,T: a,V: a] :
( ( weakly1626779504270821493_rts_a @ Resid )
=> ( ( composite_of_a @ Resid @ U @ T @ V )
=> ( ( trg_a @ Resid @ V )
= ( trg_a @ Resid @ T ) ) ) ) ).
% weakly_extensional_rts.trg_composite_of
thf(fact_885_weakly__extensional__rts_Otrg__composite__of,axiom,
! [Resid: set_a > set_a > set_a,U: set_a,T: set_a,V: set_a] :
( ( weakly5936471160286156245_set_a @ Resid )
=> ( ( composite_of_set_a @ Resid @ U @ T @ V )
=> ( ( trg_set_a @ Resid @ V )
= ( trg_set_a @ Resid @ T ) ) ) ) ).
% weakly_extensional_rts.trg_composite_of
thf(fact_886_weakly__extensional__rts_Osrc__composite__of,axiom,
! [Resid: a > a > a,U: a,T: a,V: a] :
( ( weakly1626779504270821493_rts_a @ Resid )
=> ( ( composite_of_a @ Resid @ U @ T @ V )
=> ( ( weakly8512939796511659025_src_a @ Resid @ V )
= ( weakly8512939796511659025_src_a @ Resid @ U ) ) ) ) ).
% weakly_extensional_rts.src_composite_of
thf(fact_887_weakly__extensional__rts_Osrc__composite__of,axiom,
! [Resid: set_a > set_a > set_a,U: set_a,T: set_a,V: set_a] :
( ( weakly5936471160286156245_set_a @ Resid )
=> ( ( composite_of_set_a @ Resid @ U @ T @ V )
=> ( ( weakly2061155085811118449_set_a @ Resid @ V )
= ( weakly2061155085811118449_set_a @ Resid @ U ) ) ) ) ).
% weakly_extensional_rts.src_composite_of
thf(fact_888_weakly__extensional__rts_Ocoinitial__ide__are__eq,axiom,
! [Resid: a > a > a,A3: a,A4: a] :
( ( weakly1626779504270821493_rts_a @ Resid )
=> ( ( ide_a @ Resid @ A3 )
=> ( ( ide_a @ Resid @ A4 )
=> ( ( coinitial_a @ Resid @ A3 @ A4 )
=> ( A3 = A4 ) ) ) ) ) ).
% weakly_extensional_rts.coinitial_ide_are_eq
thf(fact_889_weakly__extensional__rts_Ocoinitial__ide__are__eq,axiom,
! [Resid: set_a > set_a > set_a,A3: set_a,A4: set_a] :
( ( weakly5936471160286156245_set_a @ Resid )
=> ( ( ide_set_a @ Resid @ A3 )
=> ( ( ide_set_a @ Resid @ A4 )
=> ( ( coinitial_set_a @ Resid @ A3 @ A4 )
=> ( A3 = A4 ) ) ) ) ) ).
% weakly_extensional_rts.coinitial_ide_are_eq
thf(fact_890_weakly__extensional__rts_Oresid__ide_I1_J,axiom,
! [Resid: a > a > a,A3: a,T: a] :
( ( weakly1626779504270821493_rts_a @ Resid )
=> ( ( ide_a @ Resid @ A3 )
=> ( ( coinitial_a @ Resid @ A3 @ T )
=> ( ( Resid @ T @ A3 )
= T ) ) ) ) ).
% weakly_extensional_rts.resid_ide(1)
thf(fact_891_weakly__extensional__rts_Oresid__ide_I1_J,axiom,
! [Resid: set_a > set_a > set_a,A3: set_a,T: set_a] :
( ( weakly5936471160286156245_set_a @ Resid )
=> ( ( ide_set_a @ Resid @ A3 )
=> ( ( coinitial_set_a @ Resid @ A3 @ T )
=> ( ( Resid @ T @ A3 )
= T ) ) ) ) ).
% weakly_extensional_rts.resid_ide(1)
thf(fact_892_weakly__extensional__rts_Osrc__trg,axiom,
! [Resid: a > a > a,T: a] :
( ( weakly1626779504270821493_rts_a @ Resid )
=> ( ( weakly8512939796511659025_src_a @ Resid @ ( trg_a @ Resid @ T ) )
= ( trg_a @ Resid @ T ) ) ) ).
% weakly_extensional_rts.src_trg
thf(fact_893_weakly__extensional__rts_Osrc__trg,axiom,
! [Resid: set_a > set_a > set_a,T: set_a] :
( ( weakly5936471160286156245_set_a @ Resid )
=> ( ( weakly2061155085811118449_set_a @ Resid @ ( trg_set_a @ Resid @ T ) )
= ( trg_set_a @ Resid @ T ) ) ) ).
% weakly_extensional_rts.src_trg
thf(fact_894_weakly__extensional__rts_Otrg__src,axiom,
! [Resid: a > a > a,T: a] :
( ( weakly1626779504270821493_rts_a @ Resid )
=> ( ( trg_a @ Resid @ ( weakly8512939796511659025_src_a @ Resid @ T ) )
= ( weakly8512939796511659025_src_a @ Resid @ T ) ) ) ).
% weakly_extensional_rts.trg_src
thf(fact_895_weakly__extensional__rts_Otrg__src,axiom,
! [Resid: set_a > set_a > set_a,T: set_a] :
( ( weakly5936471160286156245_set_a @ Resid )
=> ( ( trg_set_a @ Resid @ ( weakly2061155085811118449_set_a @ Resid @ T ) )
= ( weakly2061155085811118449_set_a @ Resid @ T ) ) ) ).
% weakly_extensional_rts.trg_src
thf(fact_896_weakly__extensional__rts_Otrg__join__of_I1_J,axiom,
! [Resid: a > a > a,T: a,U: a,V: a] :
( ( weakly1626779504270821493_rts_a @ Resid )
=> ( ( join_of_a @ Resid @ T @ U @ V )
=> ( ( trg_a @ Resid @ ( Resid @ T @ U ) )
= ( trg_a @ Resid @ V ) ) ) ) ).
% weakly_extensional_rts.trg_join_of(1)
thf(fact_897_weakly__extensional__rts_Otrg__join__of_I1_J,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a,V: set_a] :
( ( weakly5936471160286156245_set_a @ Resid )
=> ( ( join_of_set_a @ Resid @ T @ U @ V )
=> ( ( trg_set_a @ Resid @ ( Resid @ T @ U ) )
= ( trg_set_a @ Resid @ V ) ) ) ) ).
% weakly_extensional_rts.trg_join_of(1)
thf(fact_898_weakly__extensional__rts_Otrg__join__of_I2_J,axiom,
! [Resid: a > a > a,T: a,U: a,V: a] :
( ( weakly1626779504270821493_rts_a @ Resid )
=> ( ( join_of_a @ Resid @ T @ U @ V )
=> ( ( trg_a @ Resid @ ( Resid @ U @ T ) )
= ( trg_a @ Resid @ V ) ) ) ) ).
% weakly_extensional_rts.trg_join_of(2)
thf(fact_899_weakly__extensional__rts_Otrg__join__of_I2_J,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a,V: set_a] :
( ( weakly5936471160286156245_set_a @ Resid )
=> ( ( join_of_set_a @ Resid @ T @ U @ V )
=> ( ( trg_set_a @ Resid @ ( Resid @ U @ T ) )
= ( trg_set_a @ Resid @ V ) ) ) ) ).
% weakly_extensional_rts.trg_join_of(2)
thf(fact_900_weakly__extensional__rts_Osrc__join__of_I2_J,axiom,
! [Resid: a > a > a,T: a,U: a,V: a] :
( ( weakly1626779504270821493_rts_a @ Resid )
=> ( ( join_of_a @ Resid @ T @ U @ V )
=> ( ( weakly8512939796511659025_src_a @ Resid @ U )
= ( weakly8512939796511659025_src_a @ Resid @ V ) ) ) ) ).
% weakly_extensional_rts.src_join_of(2)
thf(fact_901_weakly__extensional__rts_Osrc__join__of_I2_J,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a,V: set_a] :
( ( weakly5936471160286156245_set_a @ Resid )
=> ( ( join_of_set_a @ Resid @ T @ U @ V )
=> ( ( weakly2061155085811118449_set_a @ Resid @ U )
= ( weakly2061155085811118449_set_a @ Resid @ V ) ) ) ) ).
% weakly_extensional_rts.src_join_of(2)
thf(fact_902_weakly__extensional__rts_Osrc__join__of_I1_J,axiom,
! [Resid: a > a > a,T: a,U: a,V: a] :
( ( weakly1626779504270821493_rts_a @ Resid )
=> ( ( join_of_a @ Resid @ T @ U @ V )
=> ( ( weakly8512939796511659025_src_a @ Resid @ T )
= ( weakly8512939796511659025_src_a @ Resid @ V ) ) ) ) ).
% weakly_extensional_rts.src_join_of(1)
thf(fact_903_weakly__extensional__rts_Osrc__join__of_I1_J,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a,V: set_a] :
( ( weakly5936471160286156245_set_a @ Resid )
=> ( ( join_of_set_a @ Resid @ T @ U @ V )
=> ( ( weakly2061155085811118449_set_a @ Resid @ T )
= ( weakly2061155085811118449_set_a @ Resid @ V ) ) ) ) ).
% weakly_extensional_rts.src_join_of(1)
thf(fact_904_weakly__extensional__rts_Osrc__eqI,axiom,
! [Resid: a > a > a,A3: a,T: a] :
( ( weakly1626779504270821493_rts_a @ Resid )
=> ( ( ide_a @ Resid @ A3 )
=> ( ( con_a @ Resid @ A3 @ T )
=> ( ( weakly8512939796511659025_src_a @ Resid @ T )
= A3 ) ) ) ) ).
% weakly_extensional_rts.src_eqI
thf(fact_905_weakly__extensional__rts_Osrc__eqI,axiom,
! [Resid: set_a > set_a > set_a,A3: set_a,T: set_a] :
( ( weakly5936471160286156245_set_a @ Resid )
=> ( ( ide_set_a @ Resid @ A3 )
=> ( ( con_set_a @ Resid @ A3 @ T )
=> ( ( weakly2061155085811118449_set_a @ Resid @ T )
= A3 ) ) ) ) ).
% weakly_extensional_rts.src_eqI
thf(fact_906_weakly__extensional__rts_Oide__iff__trg__self,axiom,
! [Resid: a > a > a,A3: a] :
( ( weakly1626779504270821493_rts_a @ Resid )
=> ( ( arr_a @ Resid @ A3 )
=> ( ( ide_a @ Resid @ A3 )
= ( ( trg_a @ Resid @ A3 )
= A3 ) ) ) ) ).
% weakly_extensional_rts.ide_iff_trg_self
thf(fact_907_weakly__extensional__rts_Oide__iff__trg__self,axiom,
! [Resid: set_a > set_a > set_a,A3: set_a] :
( ( weakly5936471160286156245_set_a @ Resid )
=> ( ( arr_set_a @ Resid @ A3 )
=> ( ( ide_set_a @ Resid @ A3 )
= ( ( trg_set_a @ Resid @ A3 )
= A3 ) ) ) ) ).
% weakly_extensional_rts.ide_iff_trg_self
thf(fact_908_weakly__extensional__rts_Oide__src,axiom,
! [Resid: a > a > a,T: a] :
( ( weakly1626779504270821493_rts_a @ Resid )
=> ( ( arr_a @ Resid @ T )
=> ( ide_a @ Resid @ ( weakly8512939796511659025_src_a @ Resid @ T ) ) ) ) ).
% weakly_extensional_rts.ide_src
thf(fact_909_weakly__extensional__rts_Oide__src,axiom,
! [Resid: set_a > set_a > set_a,T: set_a] :
( ( weakly5936471160286156245_set_a @ Resid )
=> ( ( arr_set_a @ Resid @ T )
=> ( ide_set_a @ Resid @ ( weakly2061155085811118449_set_a @ Resid @ T ) ) ) ) ).
% weakly_extensional_rts.ide_src
thf(fact_910_weakly__extensional__rts_Oide__iff__src__self,axiom,
! [Resid: a > a > a,A3: a] :
( ( weakly1626779504270821493_rts_a @ Resid )
=> ( ( arr_a @ Resid @ A3 )
=> ( ( ide_a @ Resid @ A3 )
= ( ( weakly8512939796511659025_src_a @ Resid @ A3 )
= A3 ) ) ) ) ).
% weakly_extensional_rts.ide_iff_src_self
thf(fact_911_weakly__extensional__rts_Oide__iff__src__self,axiom,
! [Resid: set_a > set_a > set_a,A3: set_a] :
( ( weakly5936471160286156245_set_a @ Resid )
=> ( ( arr_set_a @ Resid @ A3 )
=> ( ( ide_set_a @ Resid @ A3 )
= ( ( weakly2061155085811118449_set_a @ Resid @ A3 )
= A3 ) ) ) ) ).
% weakly_extensional_rts.ide_iff_src_self
thf(fact_912_weakly__extensional__rts_Osrc__in__sources,axiom,
! [Resid: a > a > a,T: a] :
( ( weakly1626779504270821493_rts_a @ Resid )
=> ( ( arr_a @ Resid @ T )
=> ( member_a @ ( weakly8512939796511659025_src_a @ Resid @ T ) @ ( sources_a @ Resid @ T ) ) ) ) ).
% weakly_extensional_rts.src_in_sources
thf(fact_913_weakly__extensional__rts_Osrc__in__sources,axiom,
! [Resid: set_a > set_a > set_a,T: set_a] :
( ( weakly5936471160286156245_set_a @ Resid )
=> ( ( arr_set_a @ Resid @ T )
=> ( member_set_a @ ( weakly2061155085811118449_set_a @ Resid @ T ) @ ( sources_set_a @ Resid @ T ) ) ) ) ).
% weakly_extensional_rts.src_in_sources
thf(fact_914_weakly__extensional__rts_Osrc__resid,axiom,
! [Resid: a > a > a,T: a,U: a] :
( ( weakly1626779504270821493_rts_a @ Resid )
=> ( ( con_a @ Resid @ T @ U )
=> ( ( weakly8512939796511659025_src_a @ Resid @ ( Resid @ T @ U ) )
= ( trg_a @ Resid @ U ) ) ) ) ).
% weakly_extensional_rts.src_resid
thf(fact_915_weakly__extensional__rts_Osrc__resid,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a] :
( ( weakly5936471160286156245_set_a @ Resid )
=> ( ( con_set_a @ Resid @ T @ U )
=> ( ( weakly2061155085811118449_set_a @ Resid @ ( Resid @ T @ U ) )
= ( trg_set_a @ Resid @ U ) ) ) ) ).
% weakly_extensional_rts.src_resid
thf(fact_916_weakly__extensional__rts_Oresid__ide_I2_J,axiom,
! [Resid: a > a > a,A3: a,T: a] :
( ( weakly1626779504270821493_rts_a @ Resid )
=> ( ( ide_a @ Resid @ A3 )
=> ( ( coinitial_a @ Resid @ A3 @ T )
=> ( ( Resid @ A3 @ T )
= ( trg_a @ Resid @ T ) ) ) ) ) ).
% weakly_extensional_rts.resid_ide(2)
thf(fact_917_weakly__extensional__rts_Oresid__ide_I2_J,axiom,
! [Resid: set_a > set_a > set_a,A3: set_a,T: set_a] :
( ( weakly5936471160286156245_set_a @ Resid )
=> ( ( ide_set_a @ Resid @ A3 )
=> ( ( coinitial_set_a @ Resid @ A3 @ T )
=> ( ( Resid @ A3 @ T )
= ( trg_set_a @ Resid @ T ) ) ) ) ) ).
% weakly_extensional_rts.resid_ide(2)
thf(fact_918_weakly__extensional__rts_OcoinitialE_092_060_094sub_062W_092_060_094sub_062E,axiom,
! [Resid: a > a > a,T: a,U: a] :
( ( weakly1626779504270821493_rts_a @ Resid )
=> ( ( coinitial_a @ Resid @ T @ U )
=> ~ ( ( arr_a @ Resid @ T )
=> ( ( arr_a @ Resid @ U )
=> ( ( weakly8512939796511659025_src_a @ Resid @ T )
!= ( weakly8512939796511659025_src_a @ Resid @ U ) ) ) ) ) ) ).
% weakly_extensional_rts.coinitialE\<^sub>W\<^sub>E
thf(fact_919_weakly__extensional__rts_OcoinitialE_092_060_094sub_062W_092_060_094sub_062E,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a] :
( ( weakly5936471160286156245_set_a @ Resid )
=> ( ( coinitial_set_a @ Resid @ T @ U )
=> ~ ( ( arr_set_a @ Resid @ T )
=> ( ( arr_set_a @ Resid @ U )
=> ( ( weakly2061155085811118449_set_a @ Resid @ T )
!= ( weakly2061155085811118449_set_a @ Resid @ U ) ) ) ) ) ) ).
% weakly_extensional_rts.coinitialE\<^sub>W\<^sub>E
thf(fact_920_weakly__extensional__rts_OcoinitialI_092_060_094sub_062W_092_060_094sub_062E,axiom,
! [Resid: a > a > a,T: a,U: a] :
( ( weakly1626779504270821493_rts_a @ Resid )
=> ( ( arr_a @ Resid @ T )
=> ( ( ( weakly8512939796511659025_src_a @ Resid @ T )
= ( weakly8512939796511659025_src_a @ Resid @ U ) )
=> ( coinitial_a @ Resid @ T @ U ) ) ) ) ).
% weakly_extensional_rts.coinitialI\<^sub>W\<^sub>E
thf(fact_921_weakly__extensional__rts_OcoinitialI_092_060_094sub_062W_092_060_094sub_062E,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a] :
( ( weakly5936471160286156245_set_a @ Resid )
=> ( ( arr_set_a @ Resid @ T )
=> ( ( ( weakly2061155085811118449_set_a @ Resid @ T )
= ( weakly2061155085811118449_set_a @ Resid @ U ) )
=> ( coinitial_set_a @ Resid @ T @ U ) ) ) ) ).
% weakly_extensional_rts.coinitialI\<^sub>W\<^sub>E
thf(fact_922_weakly__extensional__rts_Ocoinitial__iff_092_060_094sub_062W_092_060_094sub_062E,axiom,
! [Resid: a > a > a,T: a,U: a] :
( ( weakly1626779504270821493_rts_a @ Resid )
=> ( ( coinitial_a @ Resid @ T @ U )
= ( ( arr_a @ Resid @ T )
& ( arr_a @ Resid @ U )
& ( ( weakly8512939796511659025_src_a @ Resid @ T )
= ( weakly8512939796511659025_src_a @ Resid @ U ) ) ) ) ) ).
% weakly_extensional_rts.coinitial_iff\<^sub>W\<^sub>E
thf(fact_923_weakly__extensional__rts_Ocoinitial__iff_092_060_094sub_062W_092_060_094sub_062E,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a] :
( ( weakly5936471160286156245_set_a @ Resid )
=> ( ( coinitial_set_a @ Resid @ T @ U )
= ( ( arr_set_a @ Resid @ T )
& ( arr_set_a @ Resid @ U )
& ( ( weakly2061155085811118449_set_a @ Resid @ T )
= ( weakly2061155085811118449_set_a @ Resid @ U ) ) ) ) ) ).
% weakly_extensional_rts.coinitial_iff\<^sub>W\<^sub>E
thf(fact_924_weakly__extensional__rts_Ocoterminal__iff_092_060_094sub_062W_092_060_094sub_062E,axiom,
! [Resid: a > a > a,T: a,U: a] :
( ( weakly1626779504270821493_rts_a @ Resid )
=> ( ( coterminal_a @ Resid @ T @ U )
= ( ( arr_a @ Resid @ T )
& ( arr_a @ Resid @ U )
& ( ( trg_a @ Resid @ T )
= ( trg_a @ Resid @ U ) ) ) ) ) ).
% weakly_extensional_rts.coterminal_iff\<^sub>W\<^sub>E
thf(fact_925_weakly__extensional__rts_Ocoterminal__iff_092_060_094sub_062W_092_060_094sub_062E,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a] :
( ( weakly5936471160286156245_set_a @ Resid )
=> ( ( coterminal_set_a @ Resid @ T @ U )
= ( ( arr_set_a @ Resid @ T )
& ( arr_set_a @ Resid @ U )
& ( ( trg_set_a @ Resid @ T )
= ( trg_set_a @ Resid @ U ) ) ) ) ) ).
% weakly_extensional_rts.coterminal_iff\<^sub>W\<^sub>E
thf(fact_926_weakly__extensional__rts_OcoterminalI_092_060_094sub_062W_092_060_094sub_062E,axiom,
! [Resid: a > a > a,T: a,U: a] :
( ( weakly1626779504270821493_rts_a @ Resid )
=> ( ( arr_a @ Resid @ T )
=> ( ( ( trg_a @ Resid @ T )
= ( trg_a @ Resid @ U ) )
=> ( coterminal_a @ Resid @ T @ U ) ) ) ) ).
% weakly_extensional_rts.coterminalI\<^sub>W\<^sub>E
thf(fact_927_weakly__extensional__rts_OcoterminalI_092_060_094sub_062W_092_060_094sub_062E,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a] :
( ( weakly5936471160286156245_set_a @ Resid )
=> ( ( arr_set_a @ Resid @ T )
=> ( ( ( trg_set_a @ Resid @ T )
= ( trg_set_a @ Resid @ U ) )
=> ( coterminal_set_a @ Resid @ T @ U ) ) ) ) ).
% weakly_extensional_rts.coterminalI\<^sub>W\<^sub>E
thf(fact_928_weakly__extensional__rts_OcoterminalE_092_060_094sub_062W_092_060_094sub_062E,axiom,
! [Resid: a > a > a,T: a,U: a] :
( ( weakly1626779504270821493_rts_a @ Resid )
=> ( ( coterminal_a @ Resid @ T @ U )
=> ~ ( ( arr_a @ Resid @ T )
=> ( ( arr_a @ Resid @ U )
=> ( ( trg_a @ Resid @ T )
!= ( trg_a @ Resid @ U ) ) ) ) ) ) ).
% weakly_extensional_rts.coterminalE\<^sub>W\<^sub>E
thf(fact_929_weakly__extensional__rts_OcoterminalE_092_060_094sub_062W_092_060_094sub_062E,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a] :
( ( weakly5936471160286156245_set_a @ Resid )
=> ( ( coterminal_set_a @ Resid @ T @ U )
=> ~ ( ( arr_set_a @ Resid @ T )
=> ( ( arr_set_a @ Resid @ U )
=> ( ( trg_set_a @ Resid @ T )
!= ( trg_set_a @ Resid @ U ) ) ) ) ) ) ).
% weakly_extensional_rts.coterminalE\<^sub>W\<^sub>E
thf(fact_930_weakly__extensional__rts_Osources__char,axiom,
! [Resid: a > a > a,T: a] :
( ( weakly1626779504270821493_rts_a @ Resid )
=> ( ( sources_a @ Resid @ T )
= ( collect_a
@ ^ [A: a] :
( ( arr_a @ Resid @ T )
& ( ( weakly8512939796511659025_src_a @ Resid @ T )
= A ) ) ) ) ) ).
% weakly_extensional_rts.sources_char
thf(fact_931_weakly__extensional__rts_Osources__char,axiom,
! [Resid: set_a > set_a > set_a,T: set_a] :
( ( weakly5936471160286156245_set_a @ Resid )
=> ( ( sources_set_a @ Resid @ T )
= ( collect_set_a
@ ^ [A: set_a] :
( ( arr_set_a @ Resid @ T )
& ( ( weakly2061155085811118449_set_a @ Resid @ T )
= A ) ) ) ) ) ).
% weakly_extensional_rts.sources_char
thf(fact_932_weakly__extensional__rts_Otargets__char_092_060_094sub_062W_092_060_094sub_062E,axiom,
! [Resid: a > a > a,T: a] :
( ( weakly1626779504270821493_rts_a @ Resid )
=> ( ( targets_a @ Resid @ T )
= ( collect_a
@ ^ [B2: a] :
( ( arr_a @ Resid @ T )
& ( ( trg_a @ Resid @ T )
= B2 ) ) ) ) ) ).
% weakly_extensional_rts.targets_char\<^sub>W\<^sub>E
thf(fact_933_weakly__extensional__rts_Otargets__char_092_060_094sub_062W_092_060_094sub_062E,axiom,
! [Resid: set_a > set_a > set_a,T: set_a] :
( ( weakly5936471160286156245_set_a @ Resid )
=> ( ( targets_set_a @ Resid @ T )
= ( collect_set_a
@ ^ [B2: set_a] :
( ( arr_set_a @ Resid @ T )
& ( ( trg_set_a @ Resid @ T )
= B2 ) ) ) ) ) ).
% weakly_extensional_rts.targets_char\<^sub>W\<^sub>E
thf(fact_934_weakly__extensional__rts_OseqI_092_060_094sub_062W_092_060_094sub_062E,axiom,
! [Resid: a > a > a,U: a,T: a] :
( ( weakly1626779504270821493_rts_a @ Resid )
=> ( ( arr_a @ Resid @ U )
=> ( ( arr_a @ Resid @ T )
=> ( ( ( trg_a @ Resid @ T )
= ( weakly8512939796511659025_src_a @ Resid @ U ) )
=> ( seq_a @ Resid @ T @ U ) ) ) ) ) ).
% weakly_extensional_rts.seqI\<^sub>W\<^sub>E
thf(fact_935_weakly__extensional__rts_OseqI_092_060_094sub_062W_092_060_094sub_062E,axiom,
! [Resid: set_a > set_a > set_a,U: set_a,T: set_a] :
( ( weakly5936471160286156245_set_a @ Resid )
=> ( ( arr_set_a @ Resid @ U )
=> ( ( arr_set_a @ Resid @ T )
=> ( ( ( trg_set_a @ Resid @ T )
= ( weakly2061155085811118449_set_a @ Resid @ U ) )
=> ( seq_set_a @ Resid @ T @ U ) ) ) ) ) ).
% weakly_extensional_rts.seqI\<^sub>W\<^sub>E
thf(fact_936_weakly__extensional__rts_OseqE_092_060_094sub_062W_092_060_094sub_062E,axiom,
! [Resid: a > a > a,T: a,U: a] :
( ( weakly1626779504270821493_rts_a @ Resid )
=> ( ( seq_a @ Resid @ T @ U )
=> ~ ( ( arr_a @ Resid @ U )
=> ( ( arr_a @ Resid @ T )
=> ( ( trg_a @ Resid @ T )
!= ( weakly8512939796511659025_src_a @ Resid @ U ) ) ) ) ) ) ).
% weakly_extensional_rts.seqE\<^sub>W\<^sub>E
thf(fact_937_weakly__extensional__rts_OseqE_092_060_094sub_062W_092_060_094sub_062E,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a] :
( ( weakly5936471160286156245_set_a @ Resid )
=> ( ( seq_set_a @ Resid @ T @ U )
=> ~ ( ( arr_set_a @ Resid @ U )
=> ( ( arr_set_a @ Resid @ T )
=> ( ( trg_set_a @ Resid @ T )
!= ( weakly2061155085811118449_set_a @ Resid @ U ) ) ) ) ) ) ).
% weakly_extensional_rts.seqE\<^sub>W\<^sub>E
thf(fact_938_weakly__extensional__rts_Osrc__def,axiom,
! [Resid: a > a > a,T: a] :
( ( weakly1626779504270821493_rts_a @ Resid )
=> ( ( ( arr_a @ Resid @ T )
=> ( ( weakly8512939796511659025_src_a @ Resid @ T )
= ( the_a
@ ^ [A: a] : ( member_a @ A @ ( sources_a @ Resid @ T ) ) ) ) )
& ( ~ ( arr_a @ Resid @ T )
=> ( ( weakly8512939796511659025_src_a @ Resid @ T )
= ( partial_null_a @ Resid ) ) ) ) ) ).
% weakly_extensional_rts.src_def
thf(fact_939_weakly__extensional__rts_Osrc__def,axiom,
! [Resid: set_a > set_a > set_a,T: set_a] :
( ( weakly5936471160286156245_set_a @ Resid )
=> ( ( ( arr_set_a @ Resid @ T )
=> ( ( weakly2061155085811118449_set_a @ Resid @ T )
= ( the_set_a
@ ^ [A: set_a] : ( member_set_a @ A @ ( sources_set_a @ Resid @ T ) ) ) ) )
& ( ~ ( arr_set_a @ Resid @ T )
=> ( ( weakly2061155085811118449_set_a @ Resid @ T )
= ( partial_null_set_a @ Resid ) ) ) ) ) ).
% weakly_extensional_rts.src_def
thf(fact_940_inf__Int__eq,axiom,
! [R: set_a,S: set_a] :
( ( inf_inf_a_o
@ ^ [X2: a] : ( member_a @ X2 @ R )
@ ^ [X2: a] : ( member_a @ X2 @ S ) )
= ( ^ [X2: a] : ( member_a @ X2 @ ( inf_inf_set_a @ R @ S ) ) ) ) ).
% inf_Int_eq
thf(fact_941_inf__Int__eq,axiom,
! [R: set_set_a,S: set_set_a] :
( ( inf_inf_set_a_o
@ ^ [X2: set_a] : ( member_set_a @ X2 @ R )
@ ^ [X2: set_a] : ( member_set_a @ X2 @ S ) )
= ( ^ [X2: set_a] : ( member_set_a @ X2 @ ( inf_inf_set_set_a @ R @ S ) ) ) ) ).
% inf_Int_eq
thf(fact_942_is__extensional__rts,axiom,
extens2802975062453607898_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) ).
% is_extensional_rts
thf(fact_943_pred__subset__eq,axiom,
! [R: set_set_a,S: set_set_a] :
( ( ord_less_eq_set_a_o
@ ^ [X2: set_a] : ( member_set_a @ X2 @ R )
@ ^ [X2: set_a] : ( member_set_a @ X2 @ S ) )
= ( ord_le3724670747650509150_set_a @ R @ S ) ) ).
% pred_subset_eq
thf(fact_944_pred__subset__eq,axiom,
! [R: set_a,S: set_a] :
( ( ord_less_eq_a_o
@ ^ [X2: a] : ( member_a @ X2 @ R )
@ ^ [X2: a] : ( member_a @ X2 @ S ) )
= ( ord_less_eq_set_a @ R @ S ) ) ).
% pred_subset_eq
thf(fact_945_extensional__rts_Ocomposite__of__unique,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a,V: set_a,V4: set_a] :
( ( extens2802975062453607898_set_a @ Resid )
=> ( ( composite_of_set_a @ Resid @ T @ U @ V )
=> ( ( composite_of_set_a @ Resid @ T @ U @ V4 )
=> ( V = V4 ) ) ) ) ).
% extensional_rts.composite_of_unique
thf(fact_946_extensional__rts_Ocomposite__of__unique,axiom,
! [Resid: a > a > a,T: a,U: a,V: a,V4: a] :
( ( extensional_rts_a @ Resid )
=> ( ( composite_of_a @ Resid @ T @ U @ V )
=> ( ( composite_of_a @ Resid @ T @ U @ V4 )
=> ( V = V4 ) ) ) ) ).
% extensional_rts.composite_of_unique
thf(fact_947_extensional__rts_Ojoin__of__unique,axiom,
! [Resid: a > a > a,T: a,U: a,V: a,V4: a] :
( ( extensional_rts_a @ Resid )
=> ( ( join_of_a @ Resid @ T @ U @ V )
=> ( ( join_of_a @ Resid @ T @ U @ V4 )
=> ( V = V4 ) ) ) ) ).
% extensional_rts.join_of_unique
thf(fact_948_extensional__rts_Ojoin__of__unique,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a,V: set_a,V4: set_a] :
( ( extens2802975062453607898_set_a @ Resid )
=> ( ( join_of_set_a @ Resid @ T @ U @ V )
=> ( ( join_of_set_a @ Resid @ T @ U @ V4 )
=> ( V = V4 ) ) ) ) ).
% extensional_rts.join_of_unique
thf(fact_949_extensional__rts_Oextensional,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a] :
( ( extens2802975062453607898_set_a @ Resid )
=> ( ( ( ide_set_a @ Resid @ ( Resid @ T @ U ) )
& ( ide_set_a @ Resid @ ( Resid @ U @ T ) ) )
=> ( T = U ) ) ) ).
% extensional_rts.extensional
thf(fact_950_extensional__rts_Oextensional,axiom,
! [Resid: a > a > a,T: a,U: a] :
( ( extensional_rts_a @ Resid )
=> ( ( ( ide_a @ Resid @ ( Resid @ T @ U ) )
& ( ide_a @ Resid @ ( Resid @ U @ T ) ) )
=> ( T = U ) ) ) ).
% extensional_rts.extensional
thf(fact_951_extensional__rts_Ocong__char,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a] :
( ( extens2802975062453607898_set_a @ Resid )
=> ( ( ( ide_set_a @ Resid @ ( Resid @ T @ U ) )
& ( ide_set_a @ Resid @ ( Resid @ U @ T ) ) )
= ( ( arr_set_a @ Resid @ T )
& ( T = U ) ) ) ) ).
% extensional_rts.cong_char
thf(fact_952_extensional__rts_Ocong__char,axiom,
! [Resid: a > a > a,T: a,U: a] :
( ( extensional_rts_a @ Resid )
=> ( ( ( ide_a @ Resid @ ( Resid @ T @ U ) )
& ( ide_a @ Resid @ ( Resid @ U @ T ) ) )
= ( ( arr_a @ Resid @ T )
& ( T = U ) ) ) ) ).
% extensional_rts.cong_char
thf(fact_953_extensional__rts_Ocomp__ide__self,axiom,
! [Resid: a > a > a,A3: a] :
( ( extensional_rts_a @ Resid )
=> ( ( ide_a @ Resid @ A3 )
=> ( ( extensional_comp_a @ Resid @ A3 @ A3 )
= A3 ) ) ) ).
% extensional_rts.comp_ide_self
thf(fact_954_extensional__rts_Ocomp__ide__self,axiom,
! [Resid: set_a > set_a > set_a,A3: set_a] :
( ( extens2802975062453607898_set_a @ Resid )
=> ( ( ide_set_a @ Resid @ A3 )
=> ( ( extens7801945855595804251_set_a @ Resid @ A3 @ A3 )
= A3 ) ) ) ).
% extensional_rts.comp_ide_self
thf(fact_955_extensional__rts_Oprfx__decomp,axiom,
! [Resid: a > a > a,T: a,U: a] :
( ( extensional_rts_a @ Resid )
=> ( ( ide_a @ Resid @ ( Resid @ T @ U ) )
=> ( ( extensional_comp_a @ Resid @ T @ ( Resid @ U @ T ) )
= U ) ) ) ).
% extensional_rts.prfx_decomp
thf(fact_956_extensional__rts_Oprfx__decomp,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a] :
( ( extens2802975062453607898_set_a @ Resid )
=> ( ( ide_set_a @ Resid @ ( Resid @ T @ U ) )
=> ( ( extens7801945855595804251_set_a @ Resid @ T @ ( Resid @ U @ T ) )
= U ) ) ) ).
% extensional_rts.prfx_decomp
thf(fact_957_extensional__rts_Ocomp__eqI,axiom,
! [Resid: a > a > a,T: a,V: a,U: a] :
( ( extensional_rts_a @ Resid )
=> ( ( ide_a @ Resid @ ( Resid @ T @ V ) )
=> ( ( U
= ( Resid @ V @ T ) )
=> ( ( extensional_comp_a @ Resid @ T @ U )
= V ) ) ) ) ).
% extensional_rts.comp_eqI
thf(fact_958_extensional__rts_Ocomp__eqI,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,V: set_a,U: set_a] :
( ( extens2802975062453607898_set_a @ Resid )
=> ( ( ide_set_a @ Resid @ ( Resid @ T @ V ) )
=> ( ( U
= ( Resid @ V @ T ) )
=> ( ( extens7801945855595804251_set_a @ Resid @ T @ U )
= V ) ) ) ) ).
% extensional_rts.comp_eqI
thf(fact_959_extensional__rts_Oresid__comp_I1_J,axiom,
! [Resid: a > a > a,T: a,U: a,W2: a] :
( ( extensional_rts_a @ Resid )
=> ( ( con_a @ Resid @ ( extensional_comp_a @ Resid @ T @ U ) @ W2 )
=> ( ( Resid @ W2 @ ( extensional_comp_a @ Resid @ T @ U ) )
= ( Resid @ ( Resid @ W2 @ T ) @ U ) ) ) ) ).
% extensional_rts.resid_comp(1)
thf(fact_960_extensional__rts_Oresid__comp_I1_J,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a,W2: set_a] :
( ( extens2802975062453607898_set_a @ Resid )
=> ( ( con_set_a @ Resid @ ( extens7801945855595804251_set_a @ Resid @ T @ U ) @ W2 )
=> ( ( Resid @ W2 @ ( extens7801945855595804251_set_a @ Resid @ T @ U ) )
= ( Resid @ ( Resid @ W2 @ T ) @ U ) ) ) ) ).
% extensional_rts.resid_comp(1)
thf(fact_961_extensional__rts_Oresid__comp_I2_J,axiom,
! [Resid: a > a > a,T: a,U: a,W2: a] :
( ( extensional_rts_a @ Resid )
=> ( ( con_a @ Resid @ ( extensional_comp_a @ Resid @ T @ U ) @ W2 )
=> ( ( Resid @ ( extensional_comp_a @ Resid @ T @ U ) @ W2 )
= ( extensional_comp_a @ Resid @ ( Resid @ T @ W2 ) @ ( Resid @ U @ ( Resid @ W2 @ T ) ) ) ) ) ) ).
% extensional_rts.resid_comp(2)
thf(fact_962_extensional__rts_Oresid__comp_I2_J,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a,W2: set_a] :
( ( extens2802975062453607898_set_a @ Resid )
=> ( ( con_set_a @ Resid @ ( extens7801945855595804251_set_a @ Resid @ T @ U ) @ W2 )
=> ( ( Resid @ ( extens7801945855595804251_set_a @ Resid @ T @ U ) @ W2 )
= ( extens7801945855595804251_set_a @ Resid @ ( Resid @ T @ W2 ) @ ( Resid @ U @ ( Resid @ W2 @ T ) ) ) ) ) ) ).
% extensional_rts.resid_comp(2)
thf(fact_963_extensional__rts_Ocomp__cancel__left,axiom,
! [Resid: a > a > a,T: a,U: a,V: a] :
( ( extensional_rts_a @ Resid )
=> ( ( arr_a @ Resid @ ( extensional_comp_a @ Resid @ T @ U ) )
=> ( ( ( extensional_comp_a @ Resid @ T @ U )
= ( extensional_comp_a @ Resid @ T @ V ) )
=> ( U = V ) ) ) ) ).
% extensional_rts.comp_cancel_left
thf(fact_964_extensional__rts_Ocomp__cancel__left,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a,V: set_a] :
( ( extens2802975062453607898_set_a @ Resid )
=> ( ( arr_set_a @ Resid @ ( extens7801945855595804251_set_a @ Resid @ T @ U ) )
=> ( ( ( extens7801945855595804251_set_a @ Resid @ T @ U )
= ( extens7801945855595804251_set_a @ Resid @ T @ V ) )
=> ( U = V ) ) ) ) ).
% extensional_rts.comp_cancel_left
thf(fact_965_extensional__rts_Ocomp__resid__prfx,axiom,
! [Resid: a > a > a,T: a,U: a] :
( ( extensional_rts_a @ Resid )
=> ( ( arr_a @ Resid @ ( extensional_comp_a @ Resid @ T @ U ) )
=> ( ( Resid @ ( extensional_comp_a @ Resid @ T @ U ) @ T )
= U ) ) ) ).
% extensional_rts.comp_resid_prfx
thf(fact_966_extensional__rts_Ocomp__resid__prfx,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a] :
( ( extens2802975062453607898_set_a @ Resid )
=> ( ( arr_set_a @ Resid @ ( extens7801945855595804251_set_a @ Resid @ T @ U ) )
=> ( ( Resid @ ( extens7801945855595804251_set_a @ Resid @ T @ U ) @ T )
= U ) ) ) ).
% extensional_rts.comp_resid_prfx
thf(fact_967_extensional__rts_Ojoin__eqI,axiom,
! [Resid: a > a > a,T: a,V: a,U: a] :
( ( extensional_rts_a @ Resid )
=> ( ( ide_a @ Resid @ ( Resid @ T @ V ) )
=> ( ( ide_a @ Resid @ ( Resid @ U @ V ) )
=> ( ( ( Resid @ V @ U )
= ( Resid @ T @ U ) )
=> ( ( ( Resid @ V @ T )
= ( Resid @ U @ T ) )
=> ( ( extensional_join_a @ Resid @ T @ U )
= V ) ) ) ) ) ) ).
% extensional_rts.join_eqI
thf(fact_968_extensional__rts_Ojoin__eqI,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,V: set_a,U: set_a] :
( ( extens2802975062453607898_set_a @ Resid )
=> ( ( ide_set_a @ Resid @ ( Resid @ T @ V ) )
=> ( ( ide_set_a @ Resid @ ( Resid @ U @ V ) )
=> ( ( ( Resid @ V @ U )
= ( Resid @ T @ U ) )
=> ( ( ( Resid @ V @ T )
= ( Resid @ U @ T ) )
=> ( ( extens1973556086528668384_set_a @ Resid @ T @ U )
= V ) ) ) ) ) ) ).
% extensional_rts.join_eqI
thf(fact_969_extensional__rts_Ocomp__is__composite__of_I2_J,axiom,
! [Resid: a > a > a,T: a,U: a,V: a] :
( ( extensional_rts_a @ Resid )
=> ( ( composite_of_a @ Resid @ T @ U @ V )
=> ( ( extensional_comp_a @ Resid @ T @ U )
= V ) ) ) ).
% extensional_rts.comp_is_composite_of(2)
thf(fact_970_extensional__rts_Ocomp__is__composite__of_I2_J,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a,V: set_a] :
( ( extens2802975062453607898_set_a @ Resid )
=> ( ( composite_of_set_a @ Resid @ T @ U @ V )
=> ( ( extens7801945855595804251_set_a @ Resid @ T @ U )
= V ) ) ) ).
% extensional_rts.comp_is_composite_of(2)
thf(fact_971_extensional__rts_Ocomp__is__composite__of_I1_J,axiom,
! [Resid: a > a > a,T: a,U: a,V: a] :
( ( extensional_rts_a @ Resid )
=> ( ( composite_of_a @ Resid @ T @ U @ V )
=> ( composite_of_a @ Resid @ T @ U @ ( extensional_comp_a @ Resid @ T @ U ) ) ) ) ).
% extensional_rts.comp_is_composite_of(1)
thf(fact_972_extensional__rts_Ocomp__is__composite__of_I1_J,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a,V: set_a] :
( ( extens2802975062453607898_set_a @ Resid )
=> ( ( composite_of_set_a @ Resid @ T @ U @ V )
=> ( composite_of_set_a @ Resid @ T @ U @ ( extens7801945855595804251_set_a @ Resid @ T @ U ) ) ) ) ).
% extensional_rts.comp_is_composite_of(1)
thf(fact_973_extensional__rts_Ojoin__self,axiom,
! [Resid: a > a > a,T: a] :
( ( extensional_rts_a @ Resid )
=> ( ( arr_a @ Resid @ T )
=> ( ( extensional_join_a @ Resid @ T @ T )
= T ) ) ) ).
% extensional_rts.join_self
thf(fact_974_extensional__rts_Ojoin__self,axiom,
! [Resid: set_a > set_a > set_a,T: set_a] :
( ( extens2802975062453607898_set_a @ Resid )
=> ( ( arr_set_a @ Resid @ T )
=> ( ( extens1973556086528668384_set_a @ Resid @ T @ T )
= T ) ) ) ).
% extensional_rts.join_self
thf(fact_975_extensional__rts_Ocomp__null_I2_J,axiom,
! [Resid: a > a > a,T: a] :
( ( extensional_rts_a @ Resid )
=> ( ( extensional_comp_a @ Resid @ T @ ( partial_null_a @ Resid ) )
= ( partial_null_a @ Resid ) ) ) ).
% extensional_rts.comp_null(2)
thf(fact_976_extensional__rts_Ocomp__null_I2_J,axiom,
! [Resid: set_a > set_a > set_a,T: set_a] :
( ( extens2802975062453607898_set_a @ Resid )
=> ( ( extens7801945855595804251_set_a @ Resid @ T @ ( partial_null_set_a @ Resid ) )
= ( partial_null_set_a @ Resid ) ) ) ).
% extensional_rts.comp_null(2)
thf(fact_977_extensional__rts_Ocomp__null_I1_J,axiom,
! [Resid: a > a > a,T: a] :
( ( extensional_rts_a @ Resid )
=> ( ( extensional_comp_a @ Resid @ ( partial_null_a @ Resid ) @ T )
= ( partial_null_a @ Resid ) ) ) ).
% extensional_rts.comp_null(1)
thf(fact_978_extensional__rts_Ocomp__null_I1_J,axiom,
! [Resid: set_a > set_a > set_a,T: set_a] :
( ( extens2802975062453607898_set_a @ Resid )
=> ( ( extens7801945855595804251_set_a @ Resid @ ( partial_null_set_a @ Resid ) @ T )
= ( partial_null_set_a @ Resid ) ) ) ).
% extensional_rts.comp_null(1)
thf(fact_979_quotient__by__coherent__normal_Ois__extensional__rts,axiom,
! [Resid: a > a > a,NN: set_a] :
( ( quotie3282664541148387094rmal_a @ Resid @ NN )
=> ( extens2802975062453607898_set_a @ ( quotie8165075472272353145esid_a @ Resid @ NN ) ) ) ).
% quotient_by_coherent_normal.is_extensional_rts
thf(fact_980_extensional__rts_Ojoin__sym,axiom,
! [Resid: a > a > a,T: a,U: a] :
( ( extensional_rts_a @ Resid )
=> ( ( ( extensional_join_a @ Resid @ T @ U )
!= ( partial_null_a @ Resid ) )
=> ( ( extensional_join_a @ Resid @ T @ U )
= ( extensional_join_a @ Resid @ U @ T ) ) ) ) ).
% extensional_rts.join_sym
thf(fact_981_extensional__rts_Ojoin__sym,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a] :
( ( extens2802975062453607898_set_a @ Resid )
=> ( ( ( extens1973556086528668384_set_a @ Resid @ T @ U )
!= ( partial_null_set_a @ Resid ) )
=> ( ( extens1973556086528668384_set_a @ Resid @ T @ U )
= ( extens1973556086528668384_set_a @ Resid @ U @ T ) ) ) ) ).
% extensional_rts.join_sym
thf(fact_982_extensional__rts_Ocomp__assoc,axiom,
! [Resid: a > a > a,T: a,U: a,V: a] :
( ( extensional_rts_a @ Resid )
=> ( ( composable_a @ Resid @ ( extensional_comp_a @ Resid @ T @ U ) @ V )
=> ( ( extensional_comp_a @ Resid @ T @ ( extensional_comp_a @ Resid @ U @ V ) )
= ( extensional_comp_a @ Resid @ ( extensional_comp_a @ Resid @ T @ U ) @ V ) ) ) ) ).
% extensional_rts.comp_assoc
thf(fact_983_extensional__rts_Ocomp__assoc,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a,V: set_a] :
( ( extens2802975062453607898_set_a @ Resid )
=> ( ( composable_set_a @ Resid @ ( extens7801945855595804251_set_a @ Resid @ T @ U ) @ V )
=> ( ( extens7801945855595804251_set_a @ Resid @ T @ ( extens7801945855595804251_set_a @ Resid @ U @ V ) )
= ( extens7801945855595804251_set_a @ Resid @ ( extens7801945855595804251_set_a @ Resid @ T @ U ) @ V ) ) ) ) ).
% extensional_rts.comp_assoc
thf(fact_984_extensional__rts_Obounded__imp__con_092_060_094sub_062E,axiom,
! [Resid: a > a > a,T: a,U: a,T4: a,U7: a] :
( ( extensional_rts_a @ Resid )
=> ( ( ( ide_a @ Resid @ ( Resid @ ( extensional_comp_a @ Resid @ T @ U ) @ ( extensional_comp_a @ Resid @ T4 @ U7 ) ) )
& ( ide_a @ Resid @ ( Resid @ ( extensional_comp_a @ Resid @ T4 @ U7 ) @ ( extensional_comp_a @ Resid @ T @ U ) ) ) )
=> ( con_a @ Resid @ T @ T4 ) ) ) ).
% extensional_rts.bounded_imp_con\<^sub>E
thf(fact_985_extensional__rts_Obounded__imp__con_092_060_094sub_062E,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a,T4: set_a,U7: set_a] :
( ( extens2802975062453607898_set_a @ Resid )
=> ( ( ( ide_set_a @ Resid @ ( Resid @ ( extens7801945855595804251_set_a @ Resid @ T @ U ) @ ( extens7801945855595804251_set_a @ Resid @ T4 @ U7 ) ) )
& ( ide_set_a @ Resid @ ( Resid @ ( extens7801945855595804251_set_a @ Resid @ T4 @ U7 ) @ ( extens7801945855595804251_set_a @ Resid @ T @ U ) ) ) )
=> ( con_set_a @ Resid @ T @ T4 ) ) ) ).
% extensional_rts.bounded_imp_con\<^sub>E
thf(fact_986_extensional__rts_Oprfx__comp,axiom,
! [Resid: a > a > a,U: a,T: a,V: a] :
( ( extensional_rts_a @ Resid )
=> ( ( arr_a @ Resid @ U )
=> ( ( ( extensional_comp_a @ Resid @ T @ V )
= U )
=> ( ide_a @ Resid @ ( Resid @ T @ U ) ) ) ) ) ).
% extensional_rts.prfx_comp
thf(fact_987_extensional__rts_Oprfx__comp,axiom,
! [Resid: set_a > set_a > set_a,U: set_a,T: set_a,V: set_a] :
( ( extens2802975062453607898_set_a @ Resid )
=> ( ( arr_set_a @ Resid @ U )
=> ( ( ( extens7801945855595804251_set_a @ Resid @ T @ V )
= U )
=> ( ide_set_a @ Resid @ ( Resid @ T @ U ) ) ) ) ) ).
% extensional_rts.prfx_comp
thf(fact_988_extensional__rts_Ocomp__arr__trg,axiom,
! [Resid: a > a > a,T: a,B: a] :
( ( extensional_rts_a @ Resid )
=> ( ( arr_a @ Resid @ T )
=> ( ( ( trg_a @ Resid @ T )
= B )
=> ( ( extensional_comp_a @ Resid @ T @ B )
= T ) ) ) ) ).
% extensional_rts.comp_arr_trg
thf(fact_989_extensional__rts_Ocomp__arr__trg,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,B: set_a] :
( ( extens2802975062453607898_set_a @ Resid )
=> ( ( arr_set_a @ Resid @ T )
=> ( ( ( trg_set_a @ Resid @ T )
= B )
=> ( ( extens7801945855595804251_set_a @ Resid @ T @ B )
= T ) ) ) ) ).
% extensional_rts.comp_arr_trg
thf(fact_990_extensional__rts_Ocomp__src__arr,axiom,
! [Resid: a > a > a,T: a,A3: a] :
( ( extensional_rts_a @ Resid )
=> ( ( arr_a @ Resid @ T )
=> ( ( ( weakly8512939796511659025_src_a @ Resid @ T )
= A3 )
=> ( ( extensional_comp_a @ Resid @ A3 @ T )
= T ) ) ) ) ).
% extensional_rts.comp_src_arr
thf(fact_991_extensional__rts_Ocomp__src__arr,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,A3: set_a] :
( ( extens2802975062453607898_set_a @ Resid )
=> ( ( arr_set_a @ Resid @ T )
=> ( ( ( weakly2061155085811118449_set_a @ Resid @ T )
= A3 )
=> ( ( extens7801945855595804251_set_a @ Resid @ A3 @ T )
= T ) ) ) ) ).
% extensional_rts.comp_src_arr
thf(fact_992_extensional__rts_Ojoin__src,axiom,
! [Resid: a > a > a,T: a] :
( ( extensional_rts_a @ Resid )
=> ( ( arr_a @ Resid @ T )
=> ( ( extensional_join_a @ Resid @ ( weakly8512939796511659025_src_a @ Resid @ T ) @ T )
= T ) ) ) ).
% extensional_rts.join_src
thf(fact_993_extensional__rts_Ojoin__src,axiom,
! [Resid: set_a > set_a > set_a,T: set_a] :
( ( extens2802975062453607898_set_a @ Resid )
=> ( ( arr_set_a @ Resid @ T )
=> ( ( extens1973556086528668384_set_a @ Resid @ ( weakly2061155085811118449_set_a @ Resid @ T ) @ T )
= T ) ) ) ).
% extensional_rts.join_src
thf(fact_994_extensional__rts_Ocon__comp__iff,axiom,
! [Resid: a > a > a,W2: a,T: a,U: a] :
( ( extensional_rts_a @ Resid )
=> ( ( con_a @ Resid @ W2 @ ( extensional_comp_a @ Resid @ T @ U ) )
= ( ( composable_a @ Resid @ T @ U )
& ( con_a @ Resid @ ( Resid @ W2 @ T ) @ U ) ) ) ) ).
% extensional_rts.con_comp_iff
thf(fact_995_extensional__rts_Ocon__comp__iff,axiom,
! [Resid: set_a > set_a > set_a,W2: set_a,T: set_a,U: set_a] :
( ( extens2802975062453607898_set_a @ Resid )
=> ( ( con_set_a @ Resid @ W2 @ ( extens7801945855595804251_set_a @ Resid @ T @ U ) )
= ( ( composable_set_a @ Resid @ T @ U )
& ( con_set_a @ Resid @ ( Resid @ W2 @ T ) @ U ) ) ) ) ).
% extensional_rts.con_comp_iff
thf(fact_996_extensional__rts_Ocon__compI_I1_J,axiom,
! [Resid: a > a > a,T: a,U: a,W2: a] :
( ( extensional_rts_a @ Resid )
=> ( ( composable_a @ Resid @ T @ U )
=> ( ( con_a @ Resid @ ( Resid @ W2 @ T ) @ U )
=> ( con_a @ Resid @ W2 @ ( extensional_comp_a @ Resid @ T @ U ) ) ) ) ) ).
% extensional_rts.con_compI(1)
thf(fact_997_extensional__rts_Ocon__compI_I1_J,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a,W2: set_a] :
( ( extens2802975062453607898_set_a @ Resid )
=> ( ( composable_set_a @ Resid @ T @ U )
=> ( ( con_set_a @ Resid @ ( Resid @ W2 @ T ) @ U )
=> ( con_set_a @ Resid @ W2 @ ( extens7801945855595804251_set_a @ Resid @ T @ U ) ) ) ) ) ).
% extensional_rts.con_compI(1)
thf(fact_998_extensional__rts_Ocon__compI_I2_J,axiom,
! [Resid: a > a > a,T: a,U: a,W2: a] :
( ( extensional_rts_a @ Resid )
=> ( ( composable_a @ Resid @ T @ U )
=> ( ( con_a @ Resid @ ( Resid @ W2 @ T ) @ U )
=> ( con_a @ Resid @ ( extensional_comp_a @ Resid @ T @ U ) @ W2 ) ) ) ) ).
% extensional_rts.con_compI(2)
thf(fact_999_extensional__rts_Ocon__compI_I2_J,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a,W2: set_a] :
( ( extens2802975062453607898_set_a @ Resid )
=> ( ( composable_set_a @ Resid @ T @ U )
=> ( ( con_set_a @ Resid @ ( Resid @ W2 @ T ) @ U )
=> ( con_set_a @ Resid @ ( extens7801945855595804251_set_a @ Resid @ T @ U ) @ W2 ) ) ) ) ).
% extensional_rts.con_compI(2)
thf(fact_1000_extensional__rts_Ocomposable__iff__arr__comp,axiom,
! [Resid: a > a > a,T: a,U: a] :
( ( extensional_rts_a @ Resid )
=> ( ( composable_a @ Resid @ T @ U )
= ( arr_a @ Resid @ ( extensional_comp_a @ Resid @ T @ U ) ) ) ) ).
% extensional_rts.composable_iff_arr_comp
thf(fact_1001_extensional__rts_Ocomposable__iff__arr__comp,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a] :
( ( extens2802975062453607898_set_a @ Resid )
=> ( ( composable_set_a @ Resid @ T @ U )
= ( arr_set_a @ Resid @ ( extens7801945855595804251_set_a @ Resid @ T @ U ) ) ) ) ).
% extensional_rts.composable_iff_arr_comp
thf(fact_1002_extensional__rts_Oarr__comp,axiom,
! [Resid: a > a > a,T: a,U: a] :
( ( extensional_rts_a @ Resid )
=> ( ( composable_a @ Resid @ T @ U )
=> ( arr_a @ Resid @ ( extensional_comp_a @ Resid @ T @ U ) ) ) ) ).
% extensional_rts.arr_comp
thf(fact_1003_extensional__rts_Oarr__comp,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a] :
( ( extens2802975062453607898_set_a @ Resid )
=> ( ( composable_set_a @ Resid @ T @ U )
=> ( arr_set_a @ Resid @ ( extens7801945855595804251_set_a @ Resid @ T @ U ) ) ) ) ).
% extensional_rts.arr_comp
thf(fact_1004_extensional__rts_Oarr__prfx__join__self,axiom,
! [Resid: a > a > a,T: a,U: a] :
( ( extensional_rts_a @ Resid )
=> ( ( joinable_a @ Resid @ T @ U )
=> ( ide_a @ Resid @ ( Resid @ T @ ( extensional_join_a @ Resid @ T @ U ) ) ) ) ) ).
% extensional_rts.arr_prfx_join_self
thf(fact_1005_extensional__rts_Oarr__prfx__join__self,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a] :
( ( extens2802975062453607898_set_a @ Resid )
=> ( ( joinable_set_a @ Resid @ T @ U )
=> ( ide_set_a @ Resid @ ( Resid @ T @ ( extens1973556086528668384_set_a @ Resid @ T @ U ) ) ) ) ) ).
% extensional_rts.arr_prfx_join_self
thf(fact_1006_extensional__rts_Oresid__join_092_060_094sub_062E_I3_J,axiom,
! [Resid: a > a > a,T: a,U: a,V: a] :
( ( extensional_rts_a @ Resid )
=> ( ( joinable_a @ Resid @ T @ U )
=> ( ( con_a @ Resid @ V @ ( extensional_join_a @ Resid @ T @ U ) )
=> ( ( Resid @ ( extensional_join_a @ Resid @ T @ U ) @ V )
= ( extensional_join_a @ Resid @ ( Resid @ T @ V ) @ ( Resid @ U @ V ) ) ) ) ) ) ).
% extensional_rts.resid_join\<^sub>E(3)
thf(fact_1007_extensional__rts_Oresid__join_092_060_094sub_062E_I3_J,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a,V: set_a] :
( ( extens2802975062453607898_set_a @ Resid )
=> ( ( joinable_set_a @ Resid @ T @ U )
=> ( ( con_set_a @ Resid @ V @ ( extens1973556086528668384_set_a @ Resid @ T @ U ) )
=> ( ( Resid @ ( extens1973556086528668384_set_a @ Resid @ T @ U ) @ V )
= ( extens1973556086528668384_set_a @ Resid @ ( Resid @ T @ V ) @ ( Resid @ U @ V ) ) ) ) ) ) ).
% extensional_rts.resid_join\<^sub>E(3)
thf(fact_1008_extensional__rts_Oresid__join_092_060_094sub_062E_I2_J,axiom,
! [Resid: a > a > a,T: a,U: a,V: a] :
( ( extensional_rts_a @ Resid )
=> ( ( joinable_a @ Resid @ T @ U )
=> ( ( con_a @ Resid @ V @ ( extensional_join_a @ Resid @ T @ U ) )
=> ( ( Resid @ V @ ( extensional_join_a @ Resid @ T @ U ) )
= ( Resid @ ( Resid @ V @ T ) @ ( Resid @ U @ T ) ) ) ) ) ) ).
% extensional_rts.resid_join\<^sub>E(2)
thf(fact_1009_extensional__rts_Oresid__join_092_060_094sub_062E_I2_J,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a,V: set_a] :
( ( extens2802975062453607898_set_a @ Resid )
=> ( ( joinable_set_a @ Resid @ T @ U )
=> ( ( con_set_a @ Resid @ V @ ( extens1973556086528668384_set_a @ Resid @ T @ U ) )
=> ( ( Resid @ V @ ( extens1973556086528668384_set_a @ Resid @ T @ U ) )
= ( Resid @ ( Resid @ V @ T ) @ ( Resid @ U @ T ) ) ) ) ) ) ).
% extensional_rts.resid_join\<^sub>E(2)
thf(fact_1010_extensional__rts_Oresid__join_092_060_094sub_062E_I1_J,axiom,
! [Resid: a > a > a,T: a,U: a,V: a] :
( ( extensional_rts_a @ Resid )
=> ( ( joinable_a @ Resid @ T @ U )
=> ( ( con_a @ Resid @ V @ ( extensional_join_a @ Resid @ T @ U ) )
=> ( ( Resid @ V @ ( extensional_join_a @ Resid @ T @ U ) )
= ( Resid @ ( Resid @ V @ U ) @ ( Resid @ T @ U ) ) ) ) ) ) ).
% extensional_rts.resid_join\<^sub>E(1)
thf(fact_1011_extensional__rts_Oresid__join_092_060_094sub_062E_I1_J,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a,V: set_a] :
( ( extens2802975062453607898_set_a @ Resid )
=> ( ( joinable_set_a @ Resid @ T @ U )
=> ( ( con_set_a @ Resid @ V @ ( extens1973556086528668384_set_a @ Resid @ T @ U ) )
=> ( ( Resid @ V @ ( extens1973556086528668384_set_a @ Resid @ T @ U ) )
= ( Resid @ ( Resid @ V @ U ) @ ( Resid @ T @ U ) ) ) ) ) ) ).
% extensional_rts.resid_join\<^sub>E(1)
thf(fact_1012_extensional__rts_Otrg__comp,axiom,
! [Resid: a > a > a,T: a,U: a] :
( ( extensional_rts_a @ Resid )
=> ( ( composable_a @ Resid @ T @ U )
=> ( ( trg_a @ Resid @ ( extensional_comp_a @ Resid @ T @ U ) )
= ( trg_a @ Resid @ U ) ) ) ) ).
% extensional_rts.trg_comp
thf(fact_1013_extensional__rts_Otrg__comp,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a] :
( ( extens2802975062453607898_set_a @ Resid )
=> ( ( composable_set_a @ Resid @ T @ U )
=> ( ( trg_set_a @ Resid @ ( extens7801945855595804251_set_a @ Resid @ T @ U ) )
= ( trg_set_a @ Resid @ U ) ) ) ) ).
% extensional_rts.trg_comp
thf(fact_1014_extensional__rts_Osrc__comp,axiom,
! [Resid: a > a > a,T: a,U: a] :
( ( extensional_rts_a @ Resid )
=> ( ( composable_a @ Resid @ T @ U )
=> ( ( weakly8512939796511659025_src_a @ Resid @ ( extensional_comp_a @ Resid @ T @ U ) )
= ( weakly8512939796511659025_src_a @ Resid @ T ) ) ) ) ).
% extensional_rts.src_comp
thf(fact_1015_extensional__rts_Osrc__comp,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a] :
( ( extens2802975062453607898_set_a @ Resid )
=> ( ( composable_set_a @ Resid @ T @ U )
=> ( ( weakly2061155085811118449_set_a @ Resid @ ( extens7801945855595804251_set_a @ Resid @ T @ U ) )
= ( weakly2061155085811118449_set_a @ Resid @ T ) ) ) ) ).
% extensional_rts.src_comp
thf(fact_1016_extensional__rts_Ojoinable__iff__arr__join,axiom,
! [Resid: a > a > a,T: a,U: a] :
( ( extensional_rts_a @ Resid )
=> ( ( joinable_a @ Resid @ T @ U )
= ( arr_a @ Resid @ ( extensional_join_a @ Resid @ T @ U ) ) ) ) ).
% extensional_rts.joinable_iff_arr_join
thf(fact_1017_extensional__rts_Ojoinable__iff__arr__join,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a] :
( ( extens2802975062453607898_set_a @ Resid )
=> ( ( joinable_set_a @ Resid @ T @ U )
= ( arr_set_a @ Resid @ ( extens1973556086528668384_set_a @ Resid @ T @ U ) ) ) ) ).
% extensional_rts.joinable_iff_arr_join
thf(fact_1018_extensional__rts_Ocomposable__iff__comp__not__null,axiom,
! [Resid: a > a > a,T: a,U: a] :
( ( extensional_rts_a @ Resid )
=> ( ( composable_a @ Resid @ T @ U )
= ( ( extensional_comp_a @ Resid @ T @ U )
!= ( partial_null_a @ Resid ) ) ) ) ).
% extensional_rts.composable_iff_comp_not_null
thf(fact_1019_extensional__rts_Ocomposable__iff__comp__not__null,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a] :
( ( extens2802975062453607898_set_a @ Resid )
=> ( ( composable_set_a @ Resid @ T @ U )
= ( ( extens7801945855595804251_set_a @ Resid @ T @ U )
!= ( partial_null_set_a @ Resid ) ) ) ) ).
% extensional_rts.composable_iff_comp_not_null
thf(fact_1020_extensional__rts_Ocomp__join_I2_J,axiom,
! [Resid: a > a > a,T: a,U: a,U7: a] :
( ( extensional_rts_a @ Resid )
=> ( ( joinable_a @ Resid @ ( extensional_comp_a @ Resid @ T @ U ) @ ( extensional_comp_a @ Resid @ T @ U7 ) )
=> ( ( extensional_comp_a @ Resid @ T @ ( extensional_join_a @ Resid @ U @ U7 ) )
= ( extensional_join_a @ Resid @ ( extensional_comp_a @ Resid @ T @ U ) @ ( extensional_comp_a @ Resid @ T @ U7 ) ) ) ) ) ).
% extensional_rts.comp_join(2)
thf(fact_1021_extensional__rts_Ocomp__join_I2_J,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a,U7: set_a] :
( ( extens2802975062453607898_set_a @ Resid )
=> ( ( joinable_set_a @ Resid @ ( extens7801945855595804251_set_a @ Resid @ T @ U ) @ ( extens7801945855595804251_set_a @ Resid @ T @ U7 ) )
=> ( ( extens7801945855595804251_set_a @ Resid @ T @ ( extens1973556086528668384_set_a @ Resid @ U @ U7 ) )
= ( extens1973556086528668384_set_a @ Resid @ ( extens7801945855595804251_set_a @ Resid @ T @ U ) @ ( extens7801945855595804251_set_a @ Resid @ T @ U7 ) ) ) ) ) ).
% extensional_rts.comp_join(2)
thf(fact_1022_extensional__rts_Otrg__join,axiom,
! [Resid: a > a > a,T: a,U: a] :
( ( extensional_rts_a @ Resid )
=> ( ( joinable_a @ Resid @ T @ U )
=> ( ( trg_a @ Resid @ ( extensional_join_a @ Resid @ T @ U ) )
= ( trg_a @ Resid @ ( Resid @ T @ U ) ) ) ) ) ).
% extensional_rts.trg_join
thf(fact_1023_extensional__rts_Otrg__join,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a] :
( ( extens2802975062453607898_set_a @ Resid )
=> ( ( joinable_set_a @ Resid @ T @ U )
=> ( ( trg_set_a @ Resid @ ( extens1973556086528668384_set_a @ Resid @ T @ U ) )
= ( trg_set_a @ Resid @ ( Resid @ T @ U ) ) ) ) ) ).
% extensional_rts.trg_join
thf(fact_1024_extensional__rts_Osrc__join,axiom,
! [Resid: a > a > a,T: a,U: a] :
( ( extensional_rts_a @ Resid )
=> ( ( joinable_a @ Resid @ T @ U )
=> ( ( weakly8512939796511659025_src_a @ Resid @ ( extensional_join_a @ Resid @ T @ U ) )
= ( weakly8512939796511659025_src_a @ Resid @ T ) ) ) ) ).
% extensional_rts.src_join
thf(fact_1025_extensional__rts_Osrc__join,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a] :
( ( extens2802975062453607898_set_a @ Resid )
=> ( ( joinable_set_a @ Resid @ T @ U )
=> ( ( weakly2061155085811118449_set_a @ Resid @ ( extens1973556086528668384_set_a @ Resid @ T @ U ) )
= ( weakly2061155085811118449_set_a @ Resid @ T ) ) ) ) ).
% extensional_rts.src_join
thf(fact_1026_extensional__rts_Ojoinable__iff__join__not__null,axiom,
! [Resid: a > a > a,T: a,U: a] :
( ( extensional_rts_a @ Resid )
=> ( ( joinable_a @ Resid @ T @ U )
= ( ( extensional_join_a @ Resid @ T @ U )
!= ( partial_null_a @ Resid ) ) ) ) ).
% extensional_rts.joinable_iff_join_not_null
thf(fact_1027_extensional__rts_Ojoinable__iff__join__not__null,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a] :
( ( extens2802975062453607898_set_a @ Resid )
=> ( ( joinable_set_a @ Resid @ T @ U )
= ( ( extens1973556086528668384_set_a @ Resid @ T @ U )
!= ( partial_null_set_a @ Resid ) ) ) ) ).
% extensional_rts.joinable_iff_join_not_null
thf(fact_1028_extensional__rts_Ojoin__is__join__of,axiom,
! [Resid: a > a > a,T: a,U: a] :
( ( extensional_rts_a @ Resid )
=> ( ( joinable_a @ Resid @ T @ U )
=> ( join_of_a @ Resid @ T @ U @ ( extensional_join_a @ Resid @ T @ U ) ) ) ) ).
% extensional_rts.join_is_join_of
thf(fact_1029_extensional__rts_Ojoin__is__join__of,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a] :
( ( extens2802975062453607898_set_a @ Resid )
=> ( ( joinable_set_a @ Resid @ T @ U )
=> ( join_of_set_a @ Resid @ T @ U @ ( extens1973556086528668384_set_a @ Resid @ T @ U ) ) ) ) ).
% extensional_rts.join_is_join_of
thf(fact_1030_extensional__rts_Ocomp__join_I1_J,axiom,
! [Resid: a > a > a,T: a,U: a,U7: a] :
( ( extensional_rts_a @ Resid )
=> ( ( joinable_a @ Resid @ ( extensional_comp_a @ Resid @ T @ U ) @ ( extensional_comp_a @ Resid @ T @ U7 ) )
=> ( composable_a @ Resid @ T @ ( extensional_join_a @ Resid @ U @ U7 ) ) ) ) ).
% extensional_rts.comp_join(1)
thf(fact_1031_extensional__rts_Ocomp__join_I1_J,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a,U7: set_a] :
( ( extens2802975062453607898_set_a @ Resid )
=> ( ( joinable_set_a @ Resid @ ( extens7801945855595804251_set_a @ Resid @ T @ U ) @ ( extens7801945855595804251_set_a @ Resid @ T @ U7 ) )
=> ( composable_set_a @ Resid @ T @ ( extens1973556086528668384_set_a @ Resid @ U @ U7 ) ) ) ) ).
% extensional_rts.comp_join(1)
thf(fact_1032_extensional__rts_Ocomp__def,axiom,
! [Resid: a > a > a,T: a,U: a] :
( ( extensional_rts_a @ Resid )
=> ( ( ( composable_a @ Resid @ T @ U )
=> ( ( extensional_comp_a @ Resid @ T @ U )
= ( the_a @ ( composite_of_a @ Resid @ T @ U ) ) ) )
& ( ~ ( composable_a @ Resid @ T @ U )
=> ( ( extensional_comp_a @ Resid @ T @ U )
= ( partial_null_a @ Resid ) ) ) ) ) ).
% extensional_rts.comp_def
thf(fact_1033_extensional__rts_Ocomp__def,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a] :
( ( extens2802975062453607898_set_a @ Resid )
=> ( ( ( composable_set_a @ Resid @ T @ U )
=> ( ( extens7801945855595804251_set_a @ Resid @ T @ U )
= ( the_set_a @ ( composite_of_set_a @ Resid @ T @ U ) ) ) )
& ( ~ ( composable_set_a @ Resid @ T @ U )
=> ( ( extens7801945855595804251_set_a @ Resid @ T @ U )
= ( partial_null_set_a @ Resid ) ) ) ) ) ).
% extensional_rts.comp_def
thf(fact_1034_extensional__rts_Ojoin__def,axiom,
! [Resid: a > a > a,T: a,U: a] :
( ( extensional_rts_a @ Resid )
=> ( ( ( joinable_a @ Resid @ T @ U )
=> ( ( extensional_join_a @ Resid @ T @ U )
= ( the_a @ ( join_of_a @ Resid @ T @ U ) ) ) )
& ( ~ ( joinable_a @ Resid @ T @ U )
=> ( ( extensional_join_a @ Resid @ T @ U )
= ( partial_null_a @ Resid ) ) ) ) ) ).
% extensional_rts.join_def
thf(fact_1035_extensional__rts_Ojoin__def,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a] :
( ( extens2802975062453607898_set_a @ Resid )
=> ( ( ( joinable_set_a @ Resid @ T @ U )
=> ( ( extens1973556086528668384_set_a @ Resid @ T @ U )
= ( the_set_a @ ( join_of_set_a @ Resid @ T @ U ) ) ) )
& ( ~ ( joinable_set_a @ Resid @ T @ U )
=> ( ( extens1973556086528668384_set_a @ Resid @ T @ U )
= ( partial_null_set_a @ Resid ) ) ) ) ) ).
% extensional_rts.join_def
thf(fact_1036_bot__empty__eq,axiom,
( bot_bot_a_o
= ( ^ [X2: a] : ( member_a @ X2 @ bot_bot_set_a ) ) ) ).
% bot_empty_eq
thf(fact_1037_bot__empty__eq,axiom,
( bot_bot_set_a_o
= ( ^ [X2: set_a] : ( member_set_a @ X2 @ bot_bot_set_set_a ) ) ) ).
% bot_empty_eq
thf(fact_1038_Collect__empty__eq__bot,axiom,
! [P: a > $o] :
( ( ( collect_a @ P )
= bot_bot_set_a )
= ( P = bot_bot_a_o ) ) ).
% Collect_empty_eq_bot
thf(fact_1039_Collect__empty__eq__bot,axiom,
! [P: set_a > $o] :
( ( ( collect_set_a @ P )
= bot_bot_set_set_a )
= ( P = bot_bot_set_a_o ) ) ).
% Collect_empty_eq_bot
thf(fact_1040_ord_OLeast__def,axiom,
( least_set_a
= ( ^ [Less_eq: set_a > set_a > $o,P2: set_a > $o] :
( the_set_a
@ ^ [X2: set_a] :
( ( P2 @ X2 )
& ! [Y2: set_a] :
( ( P2 @ Y2 )
=> ( Less_eq @ X2 @ Y2 ) ) ) ) ) ) ).
% ord.Least_def
thf(fact_1041_ord_OLeast__def,axiom,
( least_a
= ( ^ [Less_eq: a > a > $o,P2: a > $o] :
( the_a
@ ^ [X2: a] :
( ( P2 @ X2 )
& ! [Y2: a] :
( ( P2 @ Y2 )
=> ( Less_eq @ X2 @ Y2 ) ) ) ) ) ) ).
% ord.Least_def
thf(fact_1042_rts__axioms,axiom,
rts_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) ).
% rts_axioms
thf(fact_1043_weakly__extensional__rts_Oaxioms_I1_J,axiom,
! [Resid: a > a > a] :
( ( weakly1626779504270821493_rts_a @ Resid )
=> ( rts_a @ Resid ) ) ).
% weakly_extensional_rts.axioms(1)
thf(fact_1044_weakly__extensional__rts_Oaxioms_I1_J,axiom,
! [Resid: set_a > set_a > set_a] :
( ( weakly5936471160286156245_set_a @ Resid )
=> ( rts_set_a @ Resid ) ) ).
% weakly_extensional_rts.axioms(1)
thf(fact_1045_extensional__rts_Oaxioms_I1_J,axiom,
! [Resid: a > a > a] :
( ( extensional_rts_a @ Resid )
=> ( rts_a @ Resid ) ) ).
% extensional_rts.axioms(1)
thf(fact_1046_extensional__rts_Oaxioms_I1_J,axiom,
! [Resid: set_a > set_a > set_a] :
( ( extens2802975062453607898_set_a @ Resid )
=> ( rts_set_a @ Resid ) ) ).
% extensional_rts.axioms(1)
thf(fact_1047_quotient__by__coherent__normal_Ois__rts,axiom,
! [Resid: a > a > a,NN: set_a] :
( ( quotie3282664541148387094rmal_a @ Resid @ NN )
=> ( rts_set_a @ ( quotie8165075472272353145esid_a @ Resid @ NN ) ) ) ).
% quotient_by_coherent_normal.is_rts
thf(fact_1048_rts_Ocomposable__def,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a] :
( ( rts_set_a @ Resid )
=> ( ( composable_set_a @ Resid @ T @ U )
= ( ? [X5: set_a] : ( composite_of_set_a @ Resid @ T @ U @ X5 ) ) ) ) ).
% rts.composable_def
thf(fact_1049_rts_Ocomposable__def,axiom,
! [Resid: a > a > a,T: a,U: a] :
( ( rts_a @ Resid )
=> ( ( composable_a @ Resid @ T @ U )
= ( ? [X5: a] : ( composite_of_a @ Resid @ T @ U @ X5 ) ) ) ) ).
% rts.composable_def
thf(fact_1050_rts_Ocong__implies__coterminal,axiom,
! [Resid: set_a > set_a > set_a,U: set_a,U7: set_a] :
( ( rts_set_a @ Resid )
=> ( ( ( ide_set_a @ Resid @ ( Resid @ U @ U7 ) )
& ( ide_set_a @ Resid @ ( Resid @ U7 @ U ) ) )
=> ( coterminal_set_a @ Resid @ U @ U7 ) ) ) ).
% rts.cong_implies_coterminal
thf(fact_1051_rts_Ocong__implies__coterminal,axiom,
! [Resid: a > a > a,U: a,U7: a] :
( ( rts_a @ Resid )
=> ( ( ( ide_a @ Resid @ ( Resid @ U @ U7 ) )
& ( ide_a @ Resid @ ( Resid @ U7 @ U ) ) )
=> ( coterminal_a @ Resid @ U @ U7 ) ) ) ).
% rts.cong_implies_coterminal
thf(fact_1052_rts_Ojoinable__def,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a] :
( ( rts_set_a @ Resid )
=> ( ( joinable_set_a @ Resid @ T @ U )
= ( ? [X5: set_a] : ( join_of_set_a @ Resid @ T @ U @ X5 ) ) ) ) ).
% rts.joinable_def
thf(fact_1053_rts_Ojoinable__def,axiom,
! [Resid: a > a > a,T: a,U: a] :
( ( rts_a @ Resid )
=> ( ( joinable_a @ Resid @ T @ U )
= ( ? [X5: a] : ( join_of_a @ Resid @ T @ U @ X5 ) ) ) ) ).
% rts.joinable_def
thf(fact_1054_rts_Ocomposable__imp__seq,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a] :
( ( rts_set_a @ Resid )
=> ( ( composable_set_a @ Resid @ T @ U )
=> ( seq_set_a @ Resid @ T @ U ) ) ) ).
% rts.composable_imp_seq
thf(fact_1055_rts_Ocomposable__imp__seq,axiom,
! [Resid: a > a > a,T: a,U: a] :
( ( rts_a @ Resid )
=> ( ( composable_a @ Resid @ T @ U )
=> ( seq_a @ Resid @ T @ U ) ) ) ).
% rts.composable_imp_seq
thf(fact_1056_rts_Ojoinable__implies__coinitial,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a] :
( ( rts_set_a @ Resid )
=> ( ( joinable_set_a @ Resid @ T @ U )
=> ( coinitial_set_a @ Resid @ T @ U ) ) ) ).
% rts.joinable_implies_coinitial
thf(fact_1057_rts_Ojoinable__implies__coinitial,axiom,
! [Resid: a > a > a,T: a,U: a] :
( ( rts_a @ Resid )
=> ( ( joinable_a @ Resid @ T @ U )
=> ( coinitial_a @ Resid @ T @ U ) ) ) ).
% rts.joinable_implies_coinitial
thf(fact_1058_rts_Oaxioms_I1_J,axiom,
! [Resid: set_a > set_a > set_a] :
( ( rts_set_a @ Resid )
=> ( residuation_set_a @ Resid ) ) ).
% rts.axioms(1)
thf(fact_1059_rts_Oaxioms_I1_J,axiom,
! [Resid: a > a > a] :
( ( rts_a @ Resid )
=> ( residuation_a @ Resid ) ) ).
% rts.axioms(1)
thf(fact_1060_quotient__by__coherent__normal_Oaxioms_I1_J,axiom,
! [Resid: set_a > set_a > set_a,NN: set_set_a] :
( ( quotie5625257012022141046_set_a @ Resid @ NN )
=> ( rts_set_a @ Resid ) ) ).
% quotient_by_coherent_normal.axioms(1)
thf(fact_1061_quotient__by__coherent__normal_Oaxioms_I1_J,axiom,
! [Resid: a > a > a,NN: set_a] :
( ( quotie3282664541148387094rmal_a @ Resid @ NN )
=> ( rts_a @ Resid ) ) ).
% quotient_by_coherent_normal.axioms(1)
thf(fact_1062_rts_Oresid__reflects__con,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,V: set_a,U: set_a] :
( ( rts_set_a @ Resid )
=> ( ( con_set_a @ Resid @ T @ V )
=> ( ( con_set_a @ Resid @ U @ V )
=> ( ( con_set_a @ Resid @ ( Resid @ T @ V ) @ ( Resid @ U @ V ) )
=> ( con_set_a @ Resid @ T @ U ) ) ) ) ) ).
% rts.resid_reflects_con
thf(fact_1063_rts_Oresid__reflects__con,axiom,
! [Resid: a > a > a,T: a,V: a,U: a] :
( ( rts_a @ Resid )
=> ( ( con_a @ Resid @ T @ V )
=> ( ( con_a @ Resid @ U @ V )
=> ( ( con_a @ Resid @ ( Resid @ T @ V ) @ ( Resid @ U @ V ) )
=> ( con_a @ Resid @ T @ U ) ) ) ) ) ).
% rts.resid_reflects_con
thf(fact_1064_rts_Ocong__symmetric,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a] :
( ( rts_set_a @ Resid )
=> ( ( ( ide_set_a @ Resid @ ( Resid @ T @ U ) )
& ( ide_set_a @ Resid @ ( Resid @ U @ T ) ) )
=> ( ( ide_set_a @ Resid @ ( Resid @ U @ T ) )
& ( ide_set_a @ Resid @ ( Resid @ T @ U ) ) ) ) ) ).
% rts.cong_symmetric
thf(fact_1065_rts_Ocong__symmetric,axiom,
! [Resid: a > a > a,T: a,U: a] :
( ( rts_a @ Resid )
=> ( ( ( ide_a @ Resid @ ( Resid @ T @ U ) )
& ( ide_a @ Resid @ ( Resid @ U @ T ) ) )
=> ( ( ide_a @ Resid @ ( Resid @ U @ T ) )
& ( ide_a @ Resid @ ( Resid @ T @ U ) ) ) ) ) ).
% rts.cong_symmetric
thf(fact_1066_rts_Ocong__transitive,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a,V: set_a] :
( ( rts_set_a @ Resid )
=> ( ( ( ide_set_a @ Resid @ ( Resid @ T @ U ) )
& ( ide_set_a @ Resid @ ( Resid @ U @ T ) ) )
=> ( ( ( ide_set_a @ Resid @ ( Resid @ U @ V ) )
& ( ide_set_a @ Resid @ ( Resid @ V @ U ) ) )
=> ( ( ide_set_a @ Resid @ ( Resid @ T @ V ) )
& ( ide_set_a @ Resid @ ( Resid @ V @ T ) ) ) ) ) ) ).
% rts.cong_transitive
thf(fact_1067_rts_Ocong__transitive,axiom,
! [Resid: a > a > a,T: a,U: a,V: a] :
( ( rts_a @ Resid )
=> ( ( ( ide_a @ Resid @ ( Resid @ T @ U ) )
& ( ide_a @ Resid @ ( Resid @ U @ T ) ) )
=> ( ( ( ide_a @ Resid @ ( Resid @ U @ V ) )
& ( ide_a @ Resid @ ( Resid @ V @ U ) ) )
=> ( ( ide_a @ Resid @ ( Resid @ T @ V ) )
& ( ide_a @ Resid @ ( Resid @ V @ T ) ) ) ) ) ) ).
% rts.cong_transitive
thf(fact_1068_rts_Oprfx__transitive,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a,V: set_a] :
( ( rts_set_a @ Resid )
=> ( ( ide_set_a @ Resid @ ( Resid @ T @ U ) )
=> ( ( ide_set_a @ Resid @ ( Resid @ U @ V ) )
=> ( ide_set_a @ Resid @ ( Resid @ T @ V ) ) ) ) ) ).
% rts.prfx_transitive
thf(fact_1069_rts_Oprfx__transitive,axiom,
! [Resid: a > a > a,T: a,U: a,V: a] :
( ( rts_a @ Resid )
=> ( ( ide_a @ Resid @ ( Resid @ T @ U ) )
=> ( ( ide_a @ Resid @ ( Resid @ U @ V ) )
=> ( ide_a @ Resid @ ( Resid @ T @ V ) ) ) ) ) ).
% rts.prfx_transitive
thf(fact_1070_rts_Oide__backward__stable,axiom,
! [Resid: set_a > set_a > set_a,A3: set_a,T: set_a] :
( ( rts_set_a @ Resid )
=> ( ( ide_set_a @ Resid @ A3 )
=> ( ( ide_set_a @ Resid @ ( Resid @ T @ A3 ) )
=> ( ide_set_a @ Resid @ T ) ) ) ) ).
% rts.ide_backward_stable
thf(fact_1071_rts_Oide__backward__stable,axiom,
! [Resid: a > a > a,A3: a,T: a] :
( ( rts_a @ Resid )
=> ( ( ide_a @ Resid @ A3 )
=> ( ( ide_a @ Resid @ ( Resid @ T @ A3 ) )
=> ( ide_a @ Resid @ T ) ) ) ) ).
% rts.ide_backward_stable
thf(fact_1072_rts_Ojoin__of__symmetric,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a,V: set_a] :
( ( rts_set_a @ Resid )
=> ( ( join_of_set_a @ Resid @ T @ U @ V )
=> ( join_of_set_a @ Resid @ U @ T @ V ) ) ) ).
% rts.join_of_symmetric
thf(fact_1073_rts_Ojoin__of__symmetric,axiom,
! [Resid: a > a > a,T: a,U: a,V: a] :
( ( rts_a @ Resid )
=> ( ( join_of_a @ Resid @ T @ U @ V )
=> ( join_of_a @ Resid @ U @ T @ V ) ) ) ).
% rts.join_of_symmetric
thf(fact_1074_rts_OcomposableD_I1_J,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a] :
( ( rts_set_a @ Resid )
=> ( ( composable_set_a @ Resid @ T @ U )
=> ( arr_set_a @ Resid @ T ) ) ) ).
% rts.composableD(1)
thf(fact_1075_rts_OcomposableD_I1_J,axiom,
! [Resid: a > a > a,T: a,U: a] :
( ( rts_a @ Resid )
=> ( ( composable_a @ Resid @ T @ U )
=> ( arr_a @ Resid @ T ) ) ) ).
% rts.composableD(1)
thf(fact_1076_rts_OcomposableD_I2_J,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a] :
( ( rts_set_a @ Resid )
=> ( ( composable_set_a @ Resid @ T @ U )
=> ( arr_set_a @ Resid @ U ) ) ) ).
% rts.composableD(2)
thf(fact_1077_rts_OcomposableD_I2_J,axiom,
! [Resid: a > a > a,T: a,U: a] :
( ( rts_a @ Resid )
=> ( ( composable_a @ Resid @ T @ U )
=> ( arr_a @ Resid @ U ) ) ) ).
% rts.composableD(2)
thf(fact_1078_rts_Ojoinable__implies__con,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a] :
( ( rts_set_a @ Resid )
=> ( ( joinable_set_a @ Resid @ T @ U )
=> ( con_set_a @ Resid @ T @ U ) ) ) ).
% rts.joinable_implies_con
thf(fact_1079_rts_Ojoinable__implies__con,axiom,
! [Resid: a > a > a,T: a,U: a] :
( ( rts_a @ Resid )
=> ( ( joinable_a @ Resid @ T @ U )
=> ( con_a @ Resid @ T @ U ) ) ) ).
% rts.joinable_implies_con
thf(fact_1080_rts_Ocon__imp__coinitial,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a] :
( ( rts_set_a @ Resid )
=> ( ( con_set_a @ Resid @ T @ U )
=> ( coinitial_set_a @ Resid @ T @ U ) ) ) ).
% rts.con_imp_coinitial
thf(fact_1081_rts_Ocon__imp__coinitial,axiom,
! [Resid: a > a > a,T: a,U: a] :
( ( rts_a @ Resid )
=> ( ( con_a @ Resid @ T @ U )
=> ( coinitial_a @ Resid @ T @ U ) ) ) ).
% rts.con_imp_coinitial
thf(fact_1082_rts_Ojoin__of__def,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a,V: set_a] :
( ( rts_set_a @ Resid )
=> ( ( join_of_set_a @ Resid @ T @ U @ V )
= ( ( composite_of_set_a @ Resid @ T @ ( Resid @ U @ T ) @ V )
& ( composite_of_set_a @ Resid @ U @ ( Resid @ T @ U ) @ V ) ) ) ) ).
% rts.join_of_def
thf(fact_1083_rts_Ojoin__of__def,axiom,
! [Resid: a > a > a,T: a,U: a,V: a] :
( ( rts_a @ Resid )
=> ( ( join_of_a @ Resid @ T @ U @ V )
= ( ( composite_of_a @ Resid @ T @ ( Resid @ U @ T ) @ V )
& ( composite_of_a @ Resid @ U @ ( Resid @ T @ U ) @ V ) ) ) ) ).
% rts.join_of_def
thf(fact_1084_rts_Ojoin__ofI,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a,V: set_a] :
( ( rts_set_a @ Resid )
=> ( ( composite_of_set_a @ Resid @ T @ ( Resid @ U @ T ) @ V )
=> ( ( composite_of_set_a @ Resid @ U @ ( Resid @ T @ U ) @ V )
=> ( join_of_set_a @ Resid @ T @ U @ V ) ) ) ) ).
% rts.join_ofI
thf(fact_1085_rts_Ojoin__ofI,axiom,
! [Resid: a > a > a,T: a,U: a,V: a] :
( ( rts_a @ Resid )
=> ( ( composite_of_a @ Resid @ T @ ( Resid @ U @ T ) @ V )
=> ( ( composite_of_a @ Resid @ U @ ( Resid @ T @ U ) @ V )
=> ( join_of_a @ Resid @ T @ U @ V ) ) ) ) ).
% rts.join_ofI
thf(fact_1086_rts_Ojoin__ofE,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a,V: set_a] :
( ( rts_set_a @ Resid )
=> ( ( join_of_set_a @ Resid @ T @ U @ V )
=> ~ ( ( composite_of_set_a @ Resid @ T @ ( Resid @ U @ T ) @ V )
=> ~ ( composite_of_set_a @ Resid @ U @ ( Resid @ T @ U ) @ V ) ) ) ) ).
% rts.join_ofE
thf(fact_1087_rts_Ojoin__ofE,axiom,
! [Resid: a > a > a,T: a,U: a,V: a] :
( ( rts_a @ Resid )
=> ( ( join_of_a @ Resid @ T @ U @ V )
=> ~ ( ( composite_of_a @ Resid @ T @ ( Resid @ U @ T ) @ V )
=> ~ ( composite_of_a @ Resid @ U @ ( Resid @ T @ U ) @ V ) ) ) ) ).
% rts.join_ofE
thf(fact_1088_rts_Otargets__join__of_I1_J,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a,V: set_a] :
( ( rts_set_a @ Resid )
=> ( ( join_of_set_a @ Resid @ T @ U @ V )
=> ( ( targets_set_a @ Resid @ ( Resid @ T @ U ) )
= ( targets_set_a @ Resid @ V ) ) ) ) ).
% rts.targets_join_of(1)
thf(fact_1089_rts_Otargets__join__of_I1_J,axiom,
! [Resid: a > a > a,T: a,U: a,V: a] :
( ( rts_a @ Resid )
=> ( ( join_of_a @ Resid @ T @ U @ V )
=> ( ( targets_a @ Resid @ ( Resid @ T @ U ) )
= ( targets_a @ Resid @ V ) ) ) ) ).
% rts.targets_join_of(1)
thf(fact_1090_rts_Otargets__join__of_I2_J,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a,V: set_a] :
( ( rts_set_a @ Resid )
=> ( ( join_of_set_a @ Resid @ T @ U @ V )
=> ( ( targets_set_a @ Resid @ ( Resid @ U @ T ) )
= ( targets_set_a @ Resid @ V ) ) ) ) ).
% rts.targets_join_of(2)
thf(fact_1091_rts_Otargets__join__of_I2_J,axiom,
! [Resid: a > a > a,T: a,U: a,V: a] :
( ( rts_a @ Resid )
=> ( ( join_of_a @ Resid @ T @ U @ V )
=> ( ( targets_a @ Resid @ ( Resid @ U @ T ) )
= ( targets_a @ Resid @ V ) ) ) ) ).
% rts.targets_join_of(2)
thf(fact_1092_rts_Ocong__respects__seq,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a,T4: set_a,U7: set_a] :
( ( rts_set_a @ Resid )
=> ( ( seq_set_a @ Resid @ T @ U )
=> ( ( ( ide_set_a @ Resid @ ( Resid @ T @ T4 ) )
& ( ide_set_a @ Resid @ ( Resid @ T4 @ T ) ) )
=> ( ( ( ide_set_a @ Resid @ ( Resid @ U @ U7 ) )
& ( ide_set_a @ Resid @ ( Resid @ U7 @ U ) ) )
=> ( seq_set_a @ Resid @ T4 @ U7 ) ) ) ) ) ).
% rts.cong_respects_seq
thf(fact_1093_rts_Ocong__respects__seq,axiom,
! [Resid: a > a > a,T: a,U: a,T4: a,U7: a] :
( ( rts_a @ Resid )
=> ( ( seq_a @ Resid @ T @ U )
=> ( ( ( ide_a @ Resid @ ( Resid @ T @ T4 ) )
& ( ide_a @ Resid @ ( Resid @ T4 @ T ) ) )
=> ( ( ( ide_a @ Resid @ ( Resid @ U @ U7 ) )
& ( ide_a @ Resid @ ( Resid @ U7 @ U ) ) )
=> ( seq_a @ Resid @ T4 @ U7 ) ) ) ) ) ).
% rts.cong_respects_seq
thf(fact_1094_rts_Ocoinitial__ide__are__cong,axiom,
! [Resid: set_a > set_a > set_a,A3: set_a,A4: set_a] :
( ( rts_set_a @ Resid )
=> ( ( ide_set_a @ Resid @ A3 )
=> ( ( ide_set_a @ Resid @ A4 )
=> ( ( coinitial_set_a @ Resid @ A3 @ A4 )
=> ( ( ide_set_a @ Resid @ ( Resid @ A3 @ A4 ) )
& ( ide_set_a @ Resid @ ( Resid @ A4 @ A3 ) ) ) ) ) ) ) ).
% rts.coinitial_ide_are_cong
thf(fact_1095_rts_Ocoinitial__ide__are__cong,axiom,
! [Resid: a > a > a,A3: a,A4: a] :
( ( rts_a @ Resid )
=> ( ( ide_a @ Resid @ A3 )
=> ( ( ide_a @ Resid @ A4 )
=> ( ( coinitial_a @ Resid @ A3 @ A4 )
=> ( ( ide_a @ Resid @ ( Resid @ A3 @ A4 ) )
& ( ide_a @ Resid @ ( Resid @ A4 @ A3 ) ) ) ) ) ) ) ).
% rts.coinitial_ide_are_cong
thf(fact_1096_rts_Ocong__implies__coinitial,axiom,
! [Resid: set_a > set_a > set_a,U: set_a,U7: set_a] :
( ( rts_set_a @ Resid )
=> ( ( ( ide_set_a @ Resid @ ( Resid @ U @ U7 ) )
& ( ide_set_a @ Resid @ ( Resid @ U7 @ U ) ) )
=> ( coinitial_set_a @ Resid @ U @ U7 ) ) ) ).
% rts.cong_implies_coinitial
thf(fact_1097_rts_Ocong__implies__coinitial,axiom,
! [Resid: a > a > a,U: a,U7: a] :
( ( rts_a @ Resid )
=> ( ( ( ide_a @ Resid @ ( Resid @ U @ U7 ) )
& ( ide_a @ Resid @ ( Resid @ U7 @ U ) ) )
=> ( coinitial_a @ Resid @ U @ U7 ) ) ) ).
% rts.cong_implies_coinitial
thf(fact_1098_rts_Osources__join__of_I2_J,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a,V: set_a] :
( ( rts_set_a @ Resid )
=> ( ( join_of_set_a @ Resid @ T @ U @ V )
=> ( ( sources_set_a @ Resid @ U )
= ( sources_set_a @ Resid @ V ) ) ) ) ).
% rts.sources_join_of(2)
thf(fact_1099_rts_Osources__join__of_I2_J,axiom,
! [Resid: a > a > a,T: a,U: a,V: a] :
( ( rts_a @ Resid )
=> ( ( join_of_a @ Resid @ T @ U @ V )
=> ( ( sources_a @ Resid @ U )
= ( sources_a @ Resid @ V ) ) ) ) ).
% rts.sources_join_of(2)
thf(fact_1100_rts_Osources__join__of_I1_J,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a,V: set_a] :
( ( rts_set_a @ Resid )
=> ( ( join_of_set_a @ Resid @ T @ U @ V )
=> ( ( sources_set_a @ Resid @ T )
= ( sources_set_a @ Resid @ V ) ) ) ) ).
% rts.sources_join_of(1)
thf(fact_1101_rts_Osources__join__of_I1_J,axiom,
! [Resid: a > a > a,T: a,U: a,V: a] :
( ( rts_a @ Resid )
=> ( ( join_of_a @ Resid @ T @ U @ V )
=> ( ( sources_a @ Resid @ T )
= ( sources_a @ Resid @ V ) ) ) ) ).
% rts.sources_join_of(1)
thf(fact_1102_rts_Ojoin__of__arr__self,axiom,
! [Resid: set_a > set_a > set_a,T: set_a] :
( ( rts_set_a @ Resid )
=> ( ( arr_set_a @ Resid @ T )
=> ( join_of_set_a @ Resid @ T @ T @ T ) ) ) ).
% rts.join_of_arr_self
thf(fact_1103_rts_Ojoin__of__arr__self,axiom,
! [Resid: a > a > a,T: a] :
( ( rts_a @ Resid )
=> ( ( arr_a @ Resid @ T )
=> ( join_of_a @ Resid @ T @ T @ T ) ) ) ).
% rts.join_of_arr_self
thf(fact_1104_rts_Ocon__with__join__of__iff_I2_J,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a,W2: set_a,V: set_a] :
( ( rts_set_a @ Resid )
=> ( ( join_of_set_a @ Resid @ T @ U @ W2 )
=> ( ( con_set_a @ Resid @ W2 @ V )
=> ( ( con_set_a @ Resid @ T @ V )
& ( con_set_a @ Resid @ ( Resid @ V @ T ) @ ( Resid @ U @ T ) ) ) ) ) ) ).
% rts.con_with_join_of_iff(2)
thf(fact_1105_rts_Ocon__with__join__of__iff_I2_J,axiom,
! [Resid: a > a > a,T: a,U: a,W2: a,V: a] :
( ( rts_a @ Resid )
=> ( ( join_of_a @ Resid @ T @ U @ W2 )
=> ( ( con_a @ Resid @ W2 @ V )
=> ( ( con_a @ Resid @ T @ V )
& ( con_a @ Resid @ ( Resid @ V @ T ) @ ( Resid @ U @ T ) ) ) ) ) ) ).
% rts.con_with_join_of_iff(2)
thf(fact_1106_rts_Ocon__with__join__of__iff_I1_J,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a,W2: set_a,V: set_a] :
( ( rts_set_a @ Resid )
=> ( ( join_of_set_a @ Resid @ T @ U @ W2 )
=> ( ( ( con_set_a @ Resid @ U @ V )
& ( con_set_a @ Resid @ ( Resid @ V @ U ) @ ( Resid @ T @ U ) ) )
=> ( con_set_a @ Resid @ W2 @ V ) ) ) ) ).
% rts.con_with_join_of_iff(1)
thf(fact_1107_rts_Ocon__with__join__of__iff_I1_J,axiom,
! [Resid: a > a > a,T: a,U: a,W2: a,V: a] :
( ( rts_a @ Resid )
=> ( ( join_of_a @ Resid @ T @ U @ W2 )
=> ( ( ( con_a @ Resid @ U @ V )
& ( con_a @ Resid @ ( Resid @ V @ U ) @ ( Resid @ T @ U ) ) )
=> ( con_a @ Resid @ W2 @ V ) ) ) ) ).
% rts.con_with_join_of_iff(1)
thf(fact_1108_rts_Ojoin__of__resid,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a,W2: set_a,V: set_a] :
( ( rts_set_a @ Resid )
=> ( ( join_of_set_a @ Resid @ T @ U @ W2 )
=> ( ( con_set_a @ Resid @ V @ W2 )
=> ( join_of_set_a @ Resid @ ( Resid @ T @ V ) @ ( Resid @ U @ V ) @ ( Resid @ W2 @ V ) ) ) ) ) ).
% rts.join_of_resid
thf(fact_1109_rts_Ojoin__of__resid,axiom,
! [Resid: a > a > a,T: a,U: a,W2: a,V: a] :
( ( rts_a @ Resid )
=> ( ( join_of_a @ Resid @ T @ U @ W2 )
=> ( ( con_a @ Resid @ V @ W2 )
=> ( join_of_a @ Resid @ ( Resid @ T @ V ) @ ( Resid @ U @ V ) @ ( Resid @ W2 @ V ) ) ) ) ) ).
% rts.join_of_resid
thf(fact_1110_rts_Otargets__composite__of,axiom,
! [Resid: set_a > set_a > set_a,U: set_a,T: set_a,V: set_a] :
( ( rts_set_a @ Resid )
=> ( ( composite_of_set_a @ Resid @ U @ T @ V )
=> ( ( targets_set_a @ Resid @ V )
= ( targets_set_a @ Resid @ T ) ) ) ) ).
% rts.targets_composite_of
thf(fact_1111_rts_Otargets__composite__of,axiom,
! [Resid: a > a > a,U: a,T: a,V: a] :
( ( rts_a @ Resid )
=> ( ( composite_of_a @ Resid @ U @ T @ V )
=> ( ( targets_a @ Resid @ V )
= ( targets_a @ Resid @ T ) ) ) ) ).
% rts.targets_composite_of
thf(fact_1112_rts_Ojoin__of__un__upto__cong,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a,V: set_a,V4: set_a] :
( ( rts_set_a @ Resid )
=> ( ( join_of_set_a @ Resid @ T @ U @ V )
=> ( ( join_of_set_a @ Resid @ T @ U @ V4 )
=> ( ( ide_set_a @ Resid @ ( Resid @ V @ V4 ) )
& ( ide_set_a @ Resid @ ( Resid @ V4 @ V ) ) ) ) ) ) ).
% rts.join_of_un_upto_cong
thf(fact_1113_rts_Ojoin__of__un__upto__cong,axiom,
! [Resid: a > a > a,T: a,U: a,V: a,V4: a] :
( ( rts_a @ Resid )
=> ( ( join_of_a @ Resid @ T @ U @ V )
=> ( ( join_of_a @ Resid @ T @ U @ V4 )
=> ( ( ide_a @ Resid @ ( Resid @ V @ V4 ) )
& ( ide_a @ Resid @ ( Resid @ V4 @ V ) ) ) ) ) ) ).
% rts.join_of_un_upto_cong
thf(fact_1114_rts_Osources__composite__of,axiom,
! [Resid: set_a > set_a > set_a,U: set_a,T: set_a,V: set_a] :
( ( rts_set_a @ Resid )
=> ( ( composite_of_set_a @ Resid @ U @ T @ V )
=> ( ( sources_set_a @ Resid @ V )
= ( sources_set_a @ Resid @ U ) ) ) ) ).
% rts.sources_composite_of
thf(fact_1115_rts_Osources__composite__of,axiom,
! [Resid: a > a > a,U: a,T: a,V: a] :
( ( rts_a @ Resid )
=> ( ( composite_of_a @ Resid @ U @ T @ V )
=> ( ( sources_a @ Resid @ V )
= ( sources_a @ Resid @ U ) ) ) ) ).
% rts.sources_composite_of
thf(fact_1116_rts_Oresid__source__in__targets,axiom,
! [Resid: set_a > set_a > set_a,A3: set_a,T: set_a] :
( ( rts_set_a @ Resid )
=> ( ( member_set_a @ A3 @ ( sources_set_a @ Resid @ T ) )
=> ( member_set_a @ ( Resid @ A3 @ T ) @ ( targets_set_a @ Resid @ T ) ) ) ) ).
% rts.resid_source_in_targets
thf(fact_1117_rts_Oresid__source__in__targets,axiom,
! [Resid: a > a > a,A3: a,T: a] :
( ( rts_a @ Resid )
=> ( ( member_a @ A3 @ ( sources_a @ Resid @ T ) )
=> ( member_a @ ( Resid @ A3 @ T ) @ ( targets_a @ Resid @ T ) ) ) ) ).
% rts.resid_source_in_targets
thf(fact_1118_rts_Oarr__composite__of,axiom,
! [Resid: set_a > set_a > set_a,U: set_a,T: set_a,V: set_a] :
( ( rts_set_a @ Resid )
=> ( ( composite_of_set_a @ Resid @ U @ T @ V )
=> ( arr_set_a @ Resid @ V ) ) ) ).
% rts.arr_composite_of
thf(fact_1119_rts_Oarr__composite__of,axiom,
! [Resid: a > a > a,U: a,T: a,V: a] :
( ( rts_a @ Resid )
=> ( ( composite_of_a @ Resid @ U @ T @ V )
=> ( arr_a @ Resid @ V ) ) ) ).
% rts.arr_composite_of
thf(fact_1120_rts_Ocon__composite__of__iff,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a,V: set_a,W2: set_a] :
( ( rts_set_a @ Resid )
=> ( ( composite_of_set_a @ Resid @ T @ U @ V )
=> ( ( con_set_a @ Resid @ W2 @ V )
= ( con_set_a @ Resid @ ( Resid @ W2 @ T ) @ U ) ) ) ) ).
% rts.con_composite_of_iff
thf(fact_1121_rts_Ocon__composite__of__iff,axiom,
! [Resid: a > a > a,T: a,U: a,V: a,W2: a] :
( ( rts_a @ Resid )
=> ( ( composite_of_a @ Resid @ T @ U @ V )
=> ( ( con_a @ Resid @ W2 @ V )
= ( con_a @ Resid @ ( Resid @ W2 @ T ) @ U ) ) ) ) ).
% rts.con_composite_of_iff
thf(fact_1122_rts_Obounded__imp__con,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a,V: set_a,T4: set_a,U7: set_a] :
( ( rts_set_a @ Resid )
=> ( ( composite_of_set_a @ Resid @ T @ U @ V )
=> ( ( composite_of_set_a @ Resid @ T4 @ U7 @ V )
=> ( con_set_a @ Resid @ T @ T4 ) ) ) ) ).
% rts.bounded_imp_con
thf(fact_1123_rts_Obounded__imp__con,axiom,
! [Resid: a > a > a,T: a,U: a,V: a,T4: a,U7: a] :
( ( rts_a @ Resid )
=> ( ( composite_of_a @ Resid @ T @ U @ V )
=> ( ( composite_of_a @ Resid @ T4 @ U7 @ V )
=> ( con_a @ Resid @ T @ T4 ) ) ) ) ).
% rts.bounded_imp_con
thf(fact_1124_rts_Ocon__prfx__composite__of_I1_J,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a,W2: set_a] :
( ( rts_set_a @ Resid )
=> ( ( composite_of_set_a @ Resid @ T @ U @ W2 )
=> ( con_set_a @ Resid @ T @ W2 ) ) ) ).
% rts.con_prfx_composite_of(1)
thf(fact_1125_rts_Ocon__prfx__composite__of_I1_J,axiom,
! [Resid: a > a > a,T: a,U: a,W2: a] :
( ( rts_a @ Resid )
=> ( ( composite_of_a @ Resid @ T @ U @ W2 )
=> ( con_a @ Resid @ T @ W2 ) ) ) ).
% rts.con_prfx_composite_of(1)
thf(fact_1126_rts_Ocon__prfx__composite__of_I2_J,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a,W2: set_a,V: set_a] :
( ( rts_set_a @ Resid )
=> ( ( composite_of_set_a @ Resid @ T @ U @ W2 )
=> ( ( con_set_a @ Resid @ W2 @ V )
=> ( con_set_a @ Resid @ T @ V ) ) ) ) ).
% rts.con_prfx_composite_of(2)
thf(fact_1127_rts_Ocon__prfx__composite__of_I2_J,axiom,
! [Resid: a > a > a,T: a,U: a,W2: a,V: a] :
( ( rts_a @ Resid )
=> ( ( composite_of_a @ Resid @ T @ U @ W2 )
=> ( ( con_a @ Resid @ W2 @ V )
=> ( con_a @ Resid @ T @ V ) ) ) ) ).
% rts.con_prfx_composite_of(2)
thf(fact_1128_rts_Oresid__composite__of_I1_J,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a,W2: set_a,V: set_a] :
( ( rts_set_a @ Resid )
=> ( ( composite_of_set_a @ Resid @ T @ U @ W2 )
=> ( ( con_set_a @ Resid @ W2 @ V )
=> ( con_set_a @ Resid @ ( Resid @ V @ T ) @ ( Resid @ W2 @ T ) ) ) ) ) ).
% rts.resid_composite_of(1)
thf(fact_1129_rts_Oresid__composite__of_I1_J,axiom,
! [Resid: a > a > a,T: a,U: a,W2: a,V: a] :
( ( rts_a @ Resid )
=> ( ( composite_of_a @ Resid @ T @ U @ W2 )
=> ( ( con_a @ Resid @ W2 @ V )
=> ( con_a @ Resid @ ( Resid @ V @ T ) @ ( Resid @ W2 @ T ) ) ) ) ) ).
% rts.resid_composite_of(1)
thf(fact_1130_rts_Oresid__composite__of_I2_J,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a,W2: set_a,V: set_a] :
( ( rts_set_a @ Resid )
=> ( ( composite_of_set_a @ Resid @ T @ U @ W2 )
=> ( ( con_set_a @ Resid @ W2 @ V )
=> ( con_set_a @ Resid @ ( Resid @ V @ T ) @ U ) ) ) ) ).
% rts.resid_composite_of(2)
thf(fact_1131_rts_Oresid__composite__of_I2_J,axiom,
! [Resid: a > a > a,T: a,U: a,W2: a,V: a] :
( ( rts_a @ Resid )
=> ( ( composite_of_a @ Resid @ T @ U @ W2 )
=> ( ( con_a @ Resid @ W2 @ V )
=> ( con_a @ Resid @ ( Resid @ V @ T ) @ U ) ) ) ) ).
% rts.resid_composite_of(2)
thf(fact_1132_rts_Oresid__composite__of_I4_J,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a,W2: set_a,V: set_a] :
( ( rts_set_a @ Resid )
=> ( ( composite_of_set_a @ Resid @ T @ U @ W2 )
=> ( ( con_set_a @ Resid @ W2 @ V )
=> ( composite_of_set_a @ Resid @ ( Resid @ T @ V ) @ ( Resid @ U @ ( Resid @ V @ T ) ) @ ( Resid @ W2 @ V ) ) ) ) ) ).
% rts.resid_composite_of(4)
thf(fact_1133_rts_Oresid__composite__of_I4_J,axiom,
! [Resid: a > a > a,T: a,U: a,W2: a,V: a] :
( ( rts_a @ Resid )
=> ( ( composite_of_a @ Resid @ T @ U @ W2 )
=> ( ( con_a @ Resid @ W2 @ V )
=> ( composite_of_a @ Resid @ ( Resid @ T @ V ) @ ( Resid @ U @ ( Resid @ V @ T ) ) @ ( Resid @ W2 @ V ) ) ) ) ) ).
% rts.resid_composite_of(4)
thf(fact_1134_rts_Otargets__are__con,axiom,
! [Resid: set_a > set_a > set_a,B: set_a,T: set_a,B4: set_a] :
( ( rts_set_a @ Resid )
=> ( ( member_set_a @ B @ ( targets_set_a @ Resid @ T ) )
=> ( ( member_set_a @ B4 @ ( targets_set_a @ Resid @ T ) )
=> ( con_set_a @ Resid @ B @ B4 ) ) ) ) ).
% rts.targets_are_con
thf(fact_1135_rts_Otargets__are__con,axiom,
! [Resid: a > a > a,B: a,T: a,B4: a] :
( ( rts_a @ Resid )
=> ( ( member_a @ B @ ( targets_a @ Resid @ T ) )
=> ( ( member_a @ B4 @ ( targets_a @ Resid @ T ) )
=> ( con_a @ Resid @ B @ B4 ) ) ) ) ).
% rts.targets_are_con
thf(fact_1136_rts_Otargets__resid__sym,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a] :
( ( rts_set_a @ Resid )
=> ( ( con_set_a @ Resid @ T @ U )
=> ( ( targets_set_a @ Resid @ ( Resid @ T @ U ) )
= ( targets_set_a @ Resid @ ( Resid @ U @ T ) ) ) ) ) ).
% rts.targets_resid_sym
thf(fact_1137_rts_Otargets__resid__sym,axiom,
! [Resid: a > a > a,T: a,U: a] :
( ( rts_a @ Resid )
=> ( ( con_a @ Resid @ T @ U )
=> ( ( targets_a @ Resid @ ( Resid @ T @ U ) )
= ( targets_a @ Resid @ ( Resid @ U @ T ) ) ) ) ) ).
% rts.targets_resid_sym
thf(fact_1138_rts_Ocomposite__of__unq__upto__cong,axiom,
! [Resid: set_a > set_a > set_a,U: set_a,T: set_a,V: set_a,V4: set_a] :
( ( rts_set_a @ Resid )
=> ( ( composite_of_set_a @ Resid @ U @ T @ V )
=> ( ( composite_of_set_a @ Resid @ U @ T @ V4 )
=> ( ( ide_set_a @ Resid @ ( Resid @ V @ V4 ) )
& ( ide_set_a @ Resid @ ( Resid @ V4 @ V ) ) ) ) ) ) ).
% rts.composite_of_unq_upto_cong
thf(fact_1139_rts_Ocomposite__of__unq__upto__cong,axiom,
! [Resid: a > a > a,U: a,T: a,V: a,V4: a] :
( ( rts_a @ Resid )
=> ( ( composite_of_a @ Resid @ U @ T @ V )
=> ( ( composite_of_a @ Resid @ U @ T @ V4 )
=> ( ( ide_a @ Resid @ ( Resid @ V @ V4 ) )
& ( ide_a @ Resid @ ( Resid @ V4 @ V ) ) ) ) ) ) ).
% rts.composite_of_unq_upto_cong
thf(fact_1140_rts_Ocomposite__of__cancel__left,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a,V: set_a,U7: set_a] :
( ( rts_set_a @ Resid )
=> ( ( composite_of_set_a @ Resid @ T @ U @ V )
=> ( ( composite_of_set_a @ Resid @ T @ U7 @ V )
=> ( ( ide_set_a @ Resid @ ( Resid @ U @ U7 ) )
& ( ide_set_a @ Resid @ ( Resid @ U7 @ U ) ) ) ) ) ) ).
% rts.composite_of_cancel_left
thf(fact_1141_rts_Ocomposite__of__cancel__left,axiom,
! [Resid: a > a > a,T: a,U: a,V: a,U7: a] :
( ( rts_a @ Resid )
=> ( ( composite_of_a @ Resid @ T @ U @ V )
=> ( ( composite_of_a @ Resid @ T @ U7 @ V )
=> ( ( ide_a @ Resid @ ( Resid @ U @ U7 ) )
& ( ide_a @ Resid @ ( Resid @ U7 @ U ) ) ) ) ) ) ).
% rts.composite_of_cancel_left
thf(fact_1142_rts_Ocomposite__of__ide__self,axiom,
! [Resid: set_a > set_a > set_a,A3: set_a] :
( ( rts_set_a @ Resid )
=> ( ( ide_set_a @ Resid @ A3 )
=> ( composite_of_set_a @ Resid @ A3 @ A3 @ A3 ) ) ) ).
% rts.composite_of_ide_self
thf(fact_1143_rts_Ocomposite__of__ide__self,axiom,
! [Resid: a > a > a,A3: a] :
( ( rts_a @ Resid )
=> ( ( ide_a @ Resid @ A3 )
=> ( composite_of_a @ Resid @ A3 @ A3 @ A3 ) ) ) ).
% rts.composite_of_ide_self
thf(fact_1144_rts_Ocomposite__of__def,axiom,
! [Resid: set_a > set_a > set_a,U: set_a,T: set_a,V: set_a] :
( ( rts_set_a @ Resid )
=> ( ( composite_of_set_a @ Resid @ U @ T @ V )
= ( ( ide_set_a @ Resid @ ( Resid @ U @ V ) )
& ( ide_set_a @ Resid @ ( Resid @ ( Resid @ V @ U ) @ T ) )
& ( ide_set_a @ Resid @ ( Resid @ T @ ( Resid @ V @ U ) ) ) ) ) ) ).
% rts.composite_of_def
thf(fact_1145_rts_Ocomposite__of__def,axiom,
! [Resid: a > a > a,U: a,T: a,V: a] :
( ( rts_a @ Resid )
=> ( ( composite_of_a @ Resid @ U @ T @ V )
= ( ( ide_a @ Resid @ ( Resid @ U @ V ) )
& ( ide_a @ Resid @ ( Resid @ ( Resid @ V @ U ) @ T ) )
& ( ide_a @ Resid @ ( Resid @ T @ ( Resid @ V @ U ) ) ) ) ) ) ).
% rts.composite_of_def
thf(fact_1146_rts_Ocomposite__ofI,axiom,
! [Resid: set_a > set_a > set_a,U: set_a,V: set_a,T: set_a] :
( ( rts_set_a @ Resid )
=> ( ( ide_set_a @ Resid @ ( Resid @ U @ V ) )
=> ( ( ( ide_set_a @ Resid @ ( Resid @ ( Resid @ V @ U ) @ T ) )
& ( ide_set_a @ Resid @ ( Resid @ T @ ( Resid @ V @ U ) ) ) )
=> ( composite_of_set_a @ Resid @ U @ T @ V ) ) ) ) ).
% rts.composite_ofI
thf(fact_1147_rts_Ocomposite__ofI,axiom,
! [Resid: a > a > a,U: a,V: a,T: a] :
( ( rts_a @ Resid )
=> ( ( ide_a @ Resid @ ( Resid @ U @ V ) )
=> ( ( ( ide_a @ Resid @ ( Resid @ ( Resid @ V @ U ) @ T ) )
& ( ide_a @ Resid @ ( Resid @ T @ ( Resid @ V @ U ) ) ) )
=> ( composite_of_a @ Resid @ U @ T @ V ) ) ) ) ).
% rts.composite_ofI
thf(fact_1148_rts_Ocomposite__ofE,axiom,
! [Resid: set_a > set_a > set_a,U: set_a,T: set_a,V: set_a] :
( ( rts_set_a @ Resid )
=> ( ( composite_of_set_a @ Resid @ U @ T @ V )
=> ~ ( ( ide_set_a @ Resid @ ( Resid @ U @ V ) )
=> ~ ( ( ide_set_a @ Resid @ ( Resid @ ( Resid @ V @ U ) @ T ) )
& ( ide_set_a @ Resid @ ( Resid @ T @ ( Resid @ V @ U ) ) ) ) ) ) ) ).
% rts.composite_ofE
thf(fact_1149_rts_Ocomposite__ofE,axiom,
! [Resid: a > a > a,U: a,T: a,V: a] :
( ( rts_a @ Resid )
=> ( ( composite_of_a @ Resid @ U @ T @ V )
=> ~ ( ( ide_a @ Resid @ ( Resid @ U @ V ) )
=> ~ ( ( ide_a @ Resid @ ( Resid @ ( Resid @ V @ U ) @ T ) )
& ( ide_a @ Resid @ ( Resid @ T @ ( Resid @ V @ U ) ) ) ) ) ) ) ).
% rts.composite_ofE
thf(fact_1150_rts_Otarget__is__ide,axiom,
! [Resid: set_a > set_a > set_a,A3: set_a,T: set_a] :
( ( rts_set_a @ Resid )
=> ( ( member_set_a @ A3 @ ( targets_set_a @ Resid @ T ) )
=> ( ide_set_a @ Resid @ A3 ) ) ) ).
% rts.target_is_ide
thf(fact_1151_rts_Otarget__is__ide,axiom,
! [Resid: a > a > a,A3: a,T: a] :
( ( rts_a @ Resid )
=> ( ( member_a @ A3 @ ( targets_a @ Resid @ T ) )
=> ( ide_a @ Resid @ A3 ) ) ) ).
% rts.target_is_ide
thf(fact_1152_rts_Otargets__are__cong,axiom,
! [Resid: set_a > set_a > set_a,B: set_a,T: set_a,B4: set_a] :
( ( rts_set_a @ Resid )
=> ( ( member_set_a @ B @ ( targets_set_a @ Resid @ T ) )
=> ( ( member_set_a @ B4 @ ( targets_set_a @ Resid @ T ) )
=> ( ( ide_set_a @ Resid @ ( Resid @ B @ B4 ) )
& ( ide_set_a @ Resid @ ( Resid @ B4 @ B ) ) ) ) ) ) ).
% rts.targets_are_cong
thf(fact_1153_rts_Otargets__are__cong,axiom,
! [Resid: a > a > a,B: a,T: a,B4: a] :
( ( rts_a @ Resid )
=> ( ( member_a @ B @ ( targets_a @ Resid @ T ) )
=> ( ( member_a @ B4 @ ( targets_a @ Resid @ T ) )
=> ( ( ide_a @ Resid @ ( Resid @ B @ B4 ) )
& ( ide_a @ Resid @ ( Resid @ B4 @ B ) ) ) ) ) ) ).
% rts.targets_are_cong
thf(fact_1154_rts_Otargets__cong__closed,axiom,
! [Resid: set_a > set_a > set_a,B: set_a,T: set_a,B4: set_a] :
( ( rts_set_a @ Resid )
=> ( ( member_set_a @ B @ ( targets_set_a @ Resid @ T ) )
=> ( ( ( ide_set_a @ Resid @ ( Resid @ B @ B4 ) )
& ( ide_set_a @ Resid @ ( Resid @ B4 @ B ) ) )
=> ( member_set_a @ B4 @ ( targets_set_a @ Resid @ T ) ) ) ) ) ).
% rts.targets_cong_closed
thf(fact_1155_rts_Otargets__cong__closed,axiom,
! [Resid: a > a > a,B: a,T: a,B4: a] :
( ( rts_a @ Resid )
=> ( ( member_a @ B @ ( targets_a @ Resid @ T ) )
=> ( ( ( ide_a @ Resid @ ( Resid @ B @ B4 ) )
& ( ide_a @ Resid @ ( Resid @ B4 @ B ) ) )
=> ( member_a @ B4 @ ( targets_a @ Resid @ T ) ) ) ) ) ).
% rts.targets_cong_closed
thf(fact_1156_rts_Osources__are__con,axiom,
! [Resid: set_a > set_a > set_a,A3: set_a,T: set_a,A4: set_a] :
( ( rts_set_a @ Resid )
=> ( ( member_set_a @ A3 @ ( sources_set_a @ Resid @ T ) )
=> ( ( member_set_a @ A4 @ ( sources_set_a @ Resid @ T ) )
=> ( con_set_a @ Resid @ A3 @ A4 ) ) ) ) ).
% rts.sources_are_con
thf(fact_1157_rts_Osources__are__con,axiom,
! [Resid: a > a > a,A3: a,T: a,A4: a] :
( ( rts_a @ Resid )
=> ( ( member_a @ A3 @ ( sources_a @ Resid @ T ) )
=> ( ( member_a @ A4 @ ( sources_a @ Resid @ T ) )
=> ( con_a @ Resid @ A3 @ A4 ) ) ) ) ).
% rts.sources_are_con
thf(fact_1158_rts_Osources__cong__closed,axiom,
! [Resid: set_a > set_a > set_a,A3: set_a,T: set_a,A4: set_a] :
( ( rts_set_a @ Resid )
=> ( ( member_set_a @ A3 @ ( sources_set_a @ Resid @ T ) )
=> ( ( ( ide_set_a @ Resid @ ( Resid @ A3 @ A4 ) )
& ( ide_set_a @ Resid @ ( Resid @ A4 @ A3 ) ) )
=> ( member_set_a @ A4 @ ( sources_set_a @ Resid @ T ) ) ) ) ) ).
% rts.sources_cong_closed
thf(fact_1159_rts_Osources__cong__closed,axiom,
! [Resid: a > a > a,A3: a,T: a,A4: a] :
( ( rts_a @ Resid )
=> ( ( member_a @ A3 @ ( sources_a @ Resid @ T ) )
=> ( ( ( ide_a @ Resid @ ( Resid @ A3 @ A4 ) )
& ( ide_a @ Resid @ ( Resid @ A4 @ A3 ) ) )
=> ( member_a @ A4 @ ( sources_a @ Resid @ T ) ) ) ) ) ).
% rts.sources_cong_closed
thf(fact_1160_rts_Osources__are__cong,axiom,
! [Resid: set_a > set_a > set_a,A3: set_a,T: set_a,A4: set_a] :
( ( rts_set_a @ Resid )
=> ( ( member_set_a @ A3 @ ( sources_set_a @ Resid @ T ) )
=> ( ( member_set_a @ A4 @ ( sources_set_a @ Resid @ T ) )
=> ( ( ide_set_a @ Resid @ ( Resid @ A3 @ A4 ) )
& ( ide_set_a @ Resid @ ( Resid @ A4 @ A3 ) ) ) ) ) ) ).
% rts.sources_are_cong
thf(fact_1161_rts_Osources__are__cong,axiom,
! [Resid: a > a > a,A3: a,T: a,A4: a] :
( ( rts_a @ Resid )
=> ( ( member_a @ A3 @ ( sources_a @ Resid @ T ) )
=> ( ( member_a @ A4 @ ( sources_a @ Resid @ T ) )
=> ( ( ide_a @ Resid @ ( Resid @ A3 @ A4 ) )
& ( ide_a @ Resid @ ( Resid @ A4 @ A3 ) ) ) ) ) ) ).
% rts.sources_are_cong
thf(fact_1162_rts_Osource__is__ide,axiom,
! [Resid: set_a > set_a > set_a,A3: set_a,T: set_a] :
( ( rts_set_a @ Resid )
=> ( ( member_set_a @ A3 @ ( sources_set_a @ Resid @ T ) )
=> ( ide_set_a @ Resid @ A3 ) ) ) ).
% rts.source_is_ide
thf(fact_1163_rts_Osource__is__ide,axiom,
! [Resid: a > a > a,A3: a,T: a] :
( ( rts_a @ Resid )
=> ( ( member_a @ A3 @ ( sources_a @ Resid @ T ) )
=> ( ide_a @ Resid @ A3 ) ) ) ).
% rts.source_is_ide
thf(fact_1164_rts_Oprfx__reflexive,axiom,
! [Resid: set_a > set_a > set_a,T: set_a] :
( ( rts_set_a @ Resid )
=> ( ( arr_set_a @ Resid @ T )
=> ( ide_set_a @ Resid @ ( Resid @ T @ T ) ) ) ) ).
% rts.prfx_reflexive
thf(fact_1165_rts_Oprfx__reflexive,axiom,
! [Resid: a > a > a,T: a] :
( ( rts_a @ Resid )
=> ( ( arr_a @ Resid @ T )
=> ( ide_a @ Resid @ ( Resid @ T @ T ) ) ) ) ).
% rts.prfx_reflexive
thf(fact_1166_rts_Ocong__reflexive,axiom,
! [Resid: set_a > set_a > set_a,T: set_a] :
( ( rts_set_a @ Resid )
=> ( ( arr_set_a @ Resid @ T )
=> ( ( ide_set_a @ Resid @ ( Resid @ T @ T ) )
& ( ide_set_a @ Resid @ ( Resid @ T @ T ) ) ) ) ) ).
% rts.cong_reflexive
thf(fact_1167_rts_Ocong__reflexive,axiom,
! [Resid: a > a > a,T: a] :
( ( rts_a @ Resid )
=> ( ( arr_a @ Resid @ T )
=> ( ( ide_a @ Resid @ ( Resid @ T @ T ) )
& ( ide_a @ Resid @ ( Resid @ T @ T ) ) ) ) ) ).
% rts.cong_reflexive
thf(fact_1168_rts_Ocong__subst__left_I2_J,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,T4: set_a,U: set_a] :
( ( rts_set_a @ Resid )
=> ( ( ( ide_set_a @ Resid @ ( Resid @ T @ T4 ) )
& ( ide_set_a @ Resid @ ( Resid @ T4 @ T ) ) )
=> ( ( con_set_a @ Resid @ T @ U )
=> ( ( ide_set_a @ Resid @ ( Resid @ ( Resid @ T @ U ) @ ( Resid @ T4 @ U ) ) )
& ( ide_set_a @ Resid @ ( Resid @ ( Resid @ T4 @ U ) @ ( Resid @ T @ U ) ) ) ) ) ) ) ).
% rts.cong_subst_left(2)
thf(fact_1169_rts_Ocong__subst__left_I2_J,axiom,
! [Resid: a > a > a,T: a,T4: a,U: a] :
( ( rts_a @ Resid )
=> ( ( ( ide_a @ Resid @ ( Resid @ T @ T4 ) )
& ( ide_a @ Resid @ ( Resid @ T4 @ T ) ) )
=> ( ( con_a @ Resid @ T @ U )
=> ( ( ide_a @ Resid @ ( Resid @ ( Resid @ T @ U ) @ ( Resid @ T4 @ U ) ) )
& ( ide_a @ Resid @ ( Resid @ ( Resid @ T4 @ U ) @ ( Resid @ T @ U ) ) ) ) ) ) ) ).
% rts.cong_subst_left(2)
thf(fact_1170_rts_Ocong__subst__left_I1_J,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,T4: set_a,U: set_a] :
( ( rts_set_a @ Resid )
=> ( ( ( ide_set_a @ Resid @ ( Resid @ T @ T4 ) )
& ( ide_set_a @ Resid @ ( Resid @ T4 @ T ) ) )
=> ( ( con_set_a @ Resid @ T @ U )
=> ( con_set_a @ Resid @ T4 @ U ) ) ) ) ).
% rts.cong_subst_left(1)
thf(fact_1171_rts_Ocong__subst__left_I1_J,axiom,
! [Resid: a > a > a,T: a,T4: a,U: a] :
( ( rts_a @ Resid )
=> ( ( ( ide_a @ Resid @ ( Resid @ T @ T4 ) )
& ( ide_a @ Resid @ ( Resid @ T4 @ T ) ) )
=> ( ( con_a @ Resid @ T @ U )
=> ( con_a @ Resid @ T4 @ U ) ) ) ) ).
% rts.cong_subst_left(1)
thf(fact_1172_rts_Ocong__subst__right_I2_J,axiom,
! [Resid: set_a > set_a > set_a,U: set_a,U7: set_a,T: set_a] :
( ( rts_set_a @ Resid )
=> ( ( ( ide_set_a @ Resid @ ( Resid @ U @ U7 ) )
& ( ide_set_a @ Resid @ ( Resid @ U7 @ U ) ) )
=> ( ( con_set_a @ Resid @ T @ U )
=> ( ( ide_set_a @ Resid @ ( Resid @ ( Resid @ T @ U ) @ ( Resid @ T @ U7 ) ) )
& ( ide_set_a @ Resid @ ( Resid @ ( Resid @ T @ U7 ) @ ( Resid @ T @ U ) ) ) ) ) ) ) ).
% rts.cong_subst_right(2)
thf(fact_1173_rts_Ocong__subst__right_I2_J,axiom,
! [Resid: a > a > a,U: a,U7: a,T: a] :
( ( rts_a @ Resid )
=> ( ( ( ide_a @ Resid @ ( Resid @ U @ U7 ) )
& ( ide_a @ Resid @ ( Resid @ U7 @ U ) ) )
=> ( ( con_a @ Resid @ T @ U )
=> ( ( ide_a @ Resid @ ( Resid @ ( Resid @ T @ U ) @ ( Resid @ T @ U7 ) ) )
& ( ide_a @ Resid @ ( Resid @ ( Resid @ T @ U7 ) @ ( Resid @ T @ U ) ) ) ) ) ) ) ).
% rts.cong_subst_right(2)
thf(fact_1174_rts_Ocong__subst__right_I1_J,axiom,
! [Resid: set_a > set_a > set_a,U: set_a,U7: set_a,T: set_a] :
( ( rts_set_a @ Resid )
=> ( ( ( ide_set_a @ Resid @ ( Resid @ U @ U7 ) )
& ( ide_set_a @ Resid @ ( Resid @ U7 @ U ) ) )
=> ( ( con_set_a @ Resid @ T @ U )
=> ( con_set_a @ Resid @ T @ U7 ) ) ) ) ).
% rts.cong_subst_right(1)
thf(fact_1175_rts_Ocong__subst__right_I1_J,axiom,
! [Resid: a > a > a,U: a,U7: a,T: a] :
( ( rts_a @ Resid )
=> ( ( ( ide_a @ Resid @ ( Resid @ U @ U7 ) )
& ( ide_a @ Resid @ ( Resid @ U7 @ U ) ) )
=> ( ( con_a @ Resid @ T @ U )
=> ( con_a @ Resid @ T @ U7 ) ) ) ) ).
% rts.cong_subst_right(1)
thf(fact_1176_rts_Ocon__target,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a,V: set_a] :
( ( rts_set_a @ Resid )
=> ( ( ide_set_a @ Resid @ ( Resid @ T @ U ) )
=> ( ( con_set_a @ Resid @ U @ V )
=> ( con_set_a @ Resid @ ( Resid @ T @ U ) @ ( Resid @ V @ U ) ) ) ) ) ).
% rts.con_target
thf(fact_1177_rts_Ocon__target,axiom,
! [Resid: a > a > a,T: a,U: a,V: a] :
( ( rts_a @ Resid )
=> ( ( ide_a @ Resid @ ( Resid @ T @ U ) )
=> ( ( con_a @ Resid @ U @ V )
=> ( con_a @ Resid @ ( Resid @ T @ U ) @ ( Resid @ V @ U ) ) ) ) ) ).
% rts.con_target
thf(fact_1178_rts_Oresid__arr__ide,axiom,
! [Resid: set_a > set_a > set_a,A3: set_a,T: set_a] :
( ( rts_set_a @ Resid )
=> ( ( ide_set_a @ Resid @ A3 )
=> ( ( con_set_a @ Resid @ T @ A3 )
=> ( ( Resid @ T @ A3 )
= T ) ) ) ) ).
% rts.resid_arr_ide
thf(fact_1179_rts_Oresid__arr__ide,axiom,
! [Resid: a > a > a,A3: a,T: a] :
( ( rts_a @ Resid )
=> ( ( ide_a @ Resid @ A3 )
=> ( ( con_a @ Resid @ T @ A3 )
=> ( ( Resid @ T @ A3 )
= T ) ) ) ) ).
% rts.resid_arr_ide
thf(fact_1180_rts_Oresid__ide__arr,axiom,
! [Resid: set_a > set_a > set_a,A3: set_a,T: set_a] :
( ( rts_set_a @ Resid )
=> ( ( ide_set_a @ Resid @ A3 )
=> ( ( con_set_a @ Resid @ A3 @ T )
=> ( ide_set_a @ Resid @ ( Resid @ A3 @ T ) ) ) ) ) ).
% rts.resid_ide_arr
thf(fact_1181_rts_Oresid__ide__arr,axiom,
! [Resid: a > a > a,A3: a,T: a] :
( ( rts_a @ Resid )
=> ( ( ide_a @ Resid @ A3 )
=> ( ( con_a @ Resid @ A3 @ T )
=> ( ide_a @ Resid @ ( Resid @ A3 @ T ) ) ) ) ) ).
% rts.resid_ide_arr
thf(fact_1182_rts_Oprfx__implies__con,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a] :
( ( rts_set_a @ Resid )
=> ( ( ide_set_a @ Resid @ ( Resid @ T @ U ) )
=> ( con_set_a @ Resid @ T @ U ) ) ) ).
% rts.prfx_implies_con
thf(fact_1183_rts_Oprfx__implies__con,axiom,
! [Resid: a > a > a,T: a,U: a] :
( ( rts_a @ Resid )
=> ( ( ide_a @ Resid @ ( Resid @ T @ U ) )
=> ( con_a @ Resid @ T @ U ) ) ) ).
% rts.prfx_implies_con
thf(fact_1184_rts_Ocon__imp__coinitial__ax,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a] :
( ( rts_set_a @ Resid )
=> ( ( con_set_a @ Resid @ T @ U )
=> ? [A6: set_a] :
( ( ide_set_a @ Resid @ A6 )
& ( con_set_a @ Resid @ A6 @ T )
& ( con_set_a @ Resid @ A6 @ U ) ) ) ) ).
% rts.con_imp_coinitial_ax
thf(fact_1185_rts_Ocon__imp__coinitial__ax,axiom,
! [Resid: a > a > a,T: a,U: a] :
( ( rts_a @ Resid )
=> ( ( con_a @ Resid @ T @ U )
=> ? [A6: a] :
( ( ide_a @ Resid @ A6 )
& ( con_a @ Resid @ A6 @ T )
& ( con_a @ Resid @ A6 @ U ) ) ) ) ).
% rts.con_imp_coinitial_ax
thf(fact_1186_rts_Oide__imp__con__iff__cong,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a] :
( ( rts_set_a @ Resid )
=> ( ( ide_set_a @ Resid @ T )
=> ( ( ide_set_a @ Resid @ U )
=> ( ( con_set_a @ Resid @ T @ U )
= ( ( ide_set_a @ Resid @ ( Resid @ T @ U ) )
& ( ide_set_a @ Resid @ ( Resid @ U @ T ) ) ) ) ) ) ) ).
% rts.ide_imp_con_iff_cong
thf(fact_1187_rts_Oide__imp__con__iff__cong,axiom,
! [Resid: a > a > a,T: a,U: a] :
( ( rts_a @ Resid )
=> ( ( ide_a @ Resid @ T )
=> ( ( ide_a @ Resid @ U )
=> ( ( con_a @ Resid @ T @ U )
= ( ( ide_a @ Resid @ ( Resid @ T @ U ) )
& ( ide_a @ Resid @ ( Resid @ U @ T ) ) ) ) ) ) ) ).
% rts.ide_imp_con_iff_cong
thf(fact_1188_rts_Ocon__transitive__on__ide,axiom,
! [Resid: set_a > set_a > set_a,A3: set_a,B: set_a,C: set_a] :
( ( rts_set_a @ Resid )
=> ( ( ide_set_a @ Resid @ A3 )
=> ( ( ide_set_a @ Resid @ B )
=> ( ( ide_set_a @ Resid @ C )
=> ( ( con_set_a @ Resid @ A3 @ B )
=> ( ( con_set_a @ Resid @ B @ C )
=> ( con_set_a @ Resid @ A3 @ C ) ) ) ) ) ) ) ).
% rts.con_transitive_on_ide
thf(fact_1189_rts_Ocon__transitive__on__ide,axiom,
! [Resid: a > a > a,A3: a,B: a,C: a] :
( ( rts_a @ Resid )
=> ( ( ide_a @ Resid @ A3 )
=> ( ( ide_a @ Resid @ B )
=> ( ( ide_a @ Resid @ C )
=> ( ( con_a @ Resid @ A3 @ B )
=> ( ( con_a @ Resid @ B @ C )
=> ( con_a @ Resid @ A3 @ C ) ) ) ) ) ) ) ).
% rts.con_transitive_on_ide
thf(fact_1190_rts_Oaxioms_I2_J,axiom,
! [Resid: set_a > set_a > set_a] :
( ( rts_set_a @ Resid )
=> ( rts_axioms_set_a @ Resid ) ) ).
% rts.axioms(2)
thf(fact_1191_rts_Oaxioms_I2_J,axiom,
! [Resid: a > a > a] :
( ( rts_a @ Resid )
=> ( rts_axioms_a @ Resid ) ) ).
% rts.axioms(2)
thf(fact_1192_quotient__by__coherent__normal__def,axiom,
( quotie5625257012022141046_set_a
= ( ^ [Resid2: set_a > set_a > set_a,NN2: set_set_a] :
( ( rts_set_a @ Resid2 )
& ( cohere6325062230080414023_set_a @ Resid2 @ NN2 ) ) ) ) ).
% quotient_by_coherent_normal_def
thf(fact_1193_quotient__by__coherent__normal__def,axiom,
( quotie3282664541148387094rmal_a
= ( ^ [Resid2: a > a > a,NN2: set_a] :
( ( rts_a @ Resid2 )
& ( cohere6072184133013167079_rts_a @ Resid2 @ NN2 ) ) ) ) ).
% quotient_by_coherent_normal_def
thf(fact_1194_quotient__by__coherent__normal_Ointro,axiom,
! [Resid: set_a > set_a > set_a,NN: set_set_a] :
( ( rts_set_a @ Resid )
=> ( ( cohere6325062230080414023_set_a @ Resid @ NN )
=> ( quotie5625257012022141046_set_a @ Resid @ NN ) ) ) ).
% quotient_by_coherent_normal.intro
thf(fact_1195_quotient__by__coherent__normal_Ointro,axiom,
! [Resid: a > a > a,NN: set_a] :
( ( rts_a @ Resid )
=> ( ( cohere6072184133013167079_rts_a @ Resid @ NN )
=> ( quotie3282664541148387094rmal_a @ Resid @ NN ) ) ) ).
% quotient_by_coherent_normal.intro
thf(fact_1196_rts_Ointro,axiom,
! [Resid: set_a > set_a > set_a] :
( ( residuation_set_a @ Resid )
=> ( ( rts_axioms_set_a @ Resid )
=> ( rts_set_a @ Resid ) ) ) ).
% rts.intro
thf(fact_1197_rts_Ointro,axiom,
! [Resid: a > a > a] :
( ( residuation_a @ Resid )
=> ( ( rts_axioms_a @ Resid )
=> ( rts_a @ Resid ) ) ) ).
% rts.intro
thf(fact_1198_rts__def,axiom,
( rts_set_a
= ( ^ [Resid2: set_a > set_a > set_a] :
( ( residuation_set_a @ Resid2 )
& ( rts_axioms_set_a @ Resid2 ) ) ) ) ).
% rts_def
thf(fact_1199_rts__def,axiom,
( rts_a
= ( ^ [Resid2: a > a > a] :
( ( residuation_a @ Resid2 )
& ( rts_axioms_a @ Resid2 ) ) ) ) ).
% rts_def
thf(fact_1200_rts_Osources__eqI,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,T4: set_a] :
( ( rts_set_a @ Resid )
=> ( ( ( inf_inf_set_set_a @ ( sources_set_a @ Resid @ T ) @ ( sources_set_a @ Resid @ T4 ) )
!= bot_bot_set_set_a )
=> ( ( sources_set_a @ Resid @ T )
= ( sources_set_a @ Resid @ T4 ) ) ) ) ).
% rts.sources_eqI
thf(fact_1201_rts_Osources__eqI,axiom,
! [Resid: a > a > a,T: a,T4: a] :
( ( rts_a @ Resid )
=> ( ( ( inf_inf_set_a @ ( sources_a @ Resid @ T ) @ ( sources_a @ Resid @ T4 ) )
!= bot_bot_set_a )
=> ( ( sources_a @ Resid @ T )
= ( sources_a @ Resid @ T4 ) ) ) ) ).
% rts.sources_eqI
thf(fact_1202_rts_Otargets__eqI,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,T4: set_a] :
( ( rts_set_a @ Resid )
=> ( ( ( inf_inf_set_set_a @ ( targets_set_a @ Resid @ T ) @ ( targets_set_a @ Resid @ T4 ) )
!= bot_bot_set_set_a )
=> ( ( targets_set_a @ Resid @ T )
= ( targets_set_a @ Resid @ T4 ) ) ) ) ).
% rts.targets_eqI
thf(fact_1203_rts_Otargets__eqI,axiom,
! [Resid: a > a > a,T: a,T4: a] :
( ( rts_a @ Resid )
=> ( ( ( inf_inf_set_a @ ( targets_a @ Resid @ T ) @ ( targets_a @ Resid @ T4 ) )
!= bot_bot_set_a )
=> ( ( targets_a @ Resid @ T )
= ( targets_a @ Resid @ T4 ) ) ) ) ).
% rts.targets_eqI
thf(fact_1204_rts_Oarr__iff__has__source,axiom,
! [Resid: set_a > set_a > set_a,T: set_a] :
( ( rts_set_a @ Resid )
=> ( ( arr_set_a @ Resid @ T )
= ( ( sources_set_a @ Resid @ T )
!= bot_bot_set_set_a ) ) ) ).
% rts.arr_iff_has_source
thf(fact_1205_rts_Oarr__iff__has__source,axiom,
! [Resid: a > a > a,T: a] :
( ( rts_a @ Resid )
=> ( ( arr_a @ Resid @ T )
= ( ( sources_a @ Resid @ T )
!= bot_bot_set_a ) ) ) ).
% rts.arr_iff_has_source
thf(fact_1206_rts_Oarr__iff__has__target,axiom,
! [Resid: set_a > set_a > set_a,T: set_a] :
( ( rts_set_a @ Resid )
=> ( ( arr_set_a @ Resid @ T )
= ( ( targets_set_a @ Resid @ T )
!= bot_bot_set_set_a ) ) ) ).
% rts.arr_iff_has_target
thf(fact_1207_rts_Oarr__iff__has__target,axiom,
! [Resid: a > a > a,T: a] :
( ( rts_a @ Resid )
=> ( ( arr_a @ Resid @ T )
= ( ( targets_a @ Resid @ T )
!= bot_bot_set_a ) ) ) ).
% rts.arr_iff_has_target
thf(fact_1208_rts_Osources__con__closed,axiom,
! [Resid: set_a > set_a > set_a,A3: set_a,T: set_a,A4: set_a] :
( ( rts_set_a @ Resid )
=> ( ( member_set_a @ A3 @ ( sources_set_a @ Resid @ T ) )
=> ( ( ide_set_a @ Resid @ A4 )
=> ( ( con_set_a @ Resid @ A3 @ A4 )
=> ( member_set_a @ A4 @ ( sources_set_a @ Resid @ T ) ) ) ) ) ) ).
% rts.sources_con_closed
thf(fact_1209_rts_Osources__con__closed,axiom,
! [Resid: a > a > a,A3: a,T: a,A4: a] :
( ( rts_a @ Resid )
=> ( ( member_a @ A3 @ ( sources_a @ Resid @ T ) )
=> ( ( ide_a @ Resid @ A4 )
=> ( ( con_a @ Resid @ A3 @ A4 )
=> ( member_a @ A4 @ ( sources_a @ Resid @ T ) ) ) ) ) ) ).
% rts.sources_con_closed
thf(fact_1210_rts_Oin__sourcesI,axiom,
! [Resid: set_a > set_a > set_a,A3: set_a,T: set_a] :
( ( rts_set_a @ Resid )
=> ( ( ide_set_a @ Resid @ A3 )
=> ( ( con_set_a @ Resid @ T @ A3 )
=> ( member_set_a @ A3 @ ( sources_set_a @ Resid @ T ) ) ) ) ) ).
% rts.in_sourcesI
thf(fact_1211_rts_Oin__sourcesI,axiom,
! [Resid: a > a > a,A3: a,T: a] :
( ( rts_a @ Resid )
=> ( ( ide_a @ Resid @ A3 )
=> ( ( con_a @ Resid @ T @ A3 )
=> ( member_a @ A3 @ ( sources_a @ Resid @ T ) ) ) ) ) ).
% rts.in_sourcesI
thf(fact_1212_rts_Oin__sourcesE,axiom,
! [Resid: set_a > set_a > set_a,A3: set_a,T: set_a] :
( ( rts_set_a @ Resid )
=> ( ( member_set_a @ A3 @ ( sources_set_a @ Resid @ T ) )
=> ~ ( ( ide_set_a @ Resid @ A3 )
=> ~ ( con_set_a @ Resid @ T @ A3 ) ) ) ) ).
% rts.in_sourcesE
thf(fact_1213_rts_Oin__sourcesE,axiom,
! [Resid: a > a > a,A3: a,T: a] :
( ( rts_a @ Resid )
=> ( ( member_a @ A3 @ ( sources_a @ Resid @ T ) )
=> ~ ( ( ide_a @ Resid @ A3 )
=> ~ ( con_a @ Resid @ T @ A3 ) ) ) ) ).
% rts.in_sourcesE
thf(fact_1214_rts_Otargets__con__closed,axiom,
! [Resid: set_a > set_a > set_a,B: set_a,T: set_a,B4: set_a] :
( ( rts_set_a @ Resid )
=> ( ( member_set_a @ B @ ( targets_set_a @ Resid @ T ) )
=> ( ( ide_set_a @ Resid @ B4 )
=> ( ( con_set_a @ Resid @ B @ B4 )
=> ( member_set_a @ B4 @ ( targets_set_a @ Resid @ T ) ) ) ) ) ) ).
% rts.targets_con_closed
thf(fact_1215_rts_Otargets__con__closed,axiom,
! [Resid: a > a > a,B: a,T: a,B4: a] :
( ( rts_a @ Resid )
=> ( ( member_a @ B @ ( targets_a @ Resid @ T ) )
=> ( ( ide_a @ Resid @ B4 )
=> ( ( con_a @ Resid @ B @ B4 )
=> ( member_a @ B4 @ ( targets_a @ Resid @ T ) ) ) ) ) ) ).
% rts.targets_con_closed
thf(fact_1216_rts_Oresid__composite__of_I3_J,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a,W2: set_a,V: set_a] :
( ( rts_set_a @ Resid )
=> ( ( composite_of_set_a @ Resid @ T @ U @ W2 )
=> ( ( con_set_a @ Resid @ W2 @ V )
=> ( ( ide_set_a @ Resid @ ( Resid @ ( Resid @ V @ W2 ) @ ( Resid @ ( Resid @ V @ T ) @ U ) ) )
& ( ide_set_a @ Resid @ ( Resid @ ( Resid @ ( Resid @ V @ T ) @ U ) @ ( Resid @ V @ W2 ) ) ) ) ) ) ) ).
% rts.resid_composite_of(3)
thf(fact_1217_rts_Oresid__composite__of_I3_J,axiom,
! [Resid: a > a > a,T: a,U: a,W2: a,V: a] :
( ( rts_a @ Resid )
=> ( ( composite_of_a @ Resid @ T @ U @ W2 )
=> ( ( con_a @ Resid @ W2 @ V )
=> ( ( ide_a @ Resid @ ( Resid @ ( Resid @ V @ W2 ) @ ( Resid @ ( Resid @ V @ T ) @ U ) ) )
& ( ide_a @ Resid @ ( Resid @ ( Resid @ ( Resid @ V @ T ) @ U ) @ ( Resid @ V @ W2 ) ) ) ) ) ) ) ).
% rts.resid_composite_of(3)
thf(fact_1218_rts_Ocomposite__of__arr__ide,axiom,
! [Resid: set_a > set_a > set_a,B: set_a,T: set_a] :
( ( rts_set_a @ Resid )
=> ( ( ide_set_a @ Resid @ B )
=> ( ( composite_of_set_a @ Resid @ T @ B @ T )
= ( con_set_a @ Resid @ ( Resid @ T @ T ) @ B ) ) ) ) ).
% rts.composite_of_arr_ide
thf(fact_1219_rts_Ocomposite__of__arr__ide,axiom,
! [Resid: a > a > a,B: a,T: a] :
( ( rts_a @ Resid )
=> ( ( ide_a @ Resid @ B )
=> ( ( composite_of_a @ Resid @ T @ B @ T )
= ( con_a @ Resid @ ( Resid @ T @ T ) @ B ) ) ) ) ).
% rts.composite_of_arr_ide
thf(fact_1220_rts_Ocomposite__of__ide__arr,axiom,
! [Resid: set_a > set_a > set_a,A3: set_a,T: set_a] :
( ( rts_set_a @ Resid )
=> ( ( ide_set_a @ Resid @ A3 )
=> ( ( composite_of_set_a @ Resid @ A3 @ T @ T )
= ( con_set_a @ Resid @ T @ A3 ) ) ) ) ).
% rts.composite_of_ide_arr
thf(fact_1221_rts_Ocomposite__of__ide__arr,axiom,
! [Resid: a > a > a,A3: a,T: a] :
( ( rts_a @ Resid )
=> ( ( ide_a @ Resid @ A3 )
=> ( ( composite_of_a @ Resid @ A3 @ T @ T )
= ( con_a @ Resid @ T @ A3 ) ) ) ) ).
% rts.composite_of_ide_arr
thf(fact_1222_rts_Osources__resid,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a] :
( ( rts_set_a @ Resid )
=> ( ( con_set_a @ Resid @ T @ U )
=> ( ( sources_set_a @ Resid @ ( Resid @ T @ U ) )
= ( targets_set_a @ Resid @ U ) ) ) ) ).
% rts.sources_resid
thf(fact_1223_rts_Osources__resid,axiom,
! [Resid: a > a > a,T: a,U: a] :
( ( rts_a @ Resid )
=> ( ( con_a @ Resid @ T @ U )
=> ( ( sources_a @ Resid @ ( Resid @ T @ U ) )
= ( targets_a @ Resid @ U ) ) ) ) ).
% rts.sources_resid
thf(fact_1224_rts_Oide__trg,axiom,
! [Resid: set_a > set_a > set_a,T: set_a] :
( ( rts_set_a @ Resid )
=> ( ( arr_set_a @ Resid @ T )
=> ( ide_set_a @ Resid @ ( trg_set_a @ Resid @ T ) ) ) ) ).
% rts.ide_trg
thf(fact_1225_rts_Oide__trg,axiom,
! [Resid: a > a > a,T: a] :
( ( rts_a @ Resid )
=> ( ( arr_a @ Resid @ T )
=> ( ide_a @ Resid @ ( trg_a @ Resid @ T ) ) ) ) ).
% rts.ide_trg
thf(fact_1226_rts_Ocomposite__of__source__arr,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,A3: set_a] :
( ( rts_set_a @ Resid )
=> ( ( arr_set_a @ Resid @ T )
=> ( ( member_set_a @ A3 @ ( sources_set_a @ Resid @ T ) )
=> ( composite_of_set_a @ Resid @ A3 @ T @ T ) ) ) ) ).
% rts.composite_of_source_arr
thf(fact_1227_rts_Ocomposite__of__source__arr,axiom,
! [Resid: a > a > a,T: a,A3: a] :
( ( rts_a @ Resid )
=> ( ( arr_a @ Resid @ T )
=> ( ( member_a @ A3 @ ( sources_a @ Resid @ T ) )
=> ( composite_of_a @ Resid @ A3 @ T @ T ) ) ) ) ).
% rts.composite_of_source_arr
thf(fact_1228_rts_Ocomposite__of__arr__target,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,B: set_a] :
( ( rts_set_a @ Resid )
=> ( ( arr_set_a @ Resid @ T )
=> ( ( member_set_a @ B @ ( targets_set_a @ Resid @ T ) )
=> ( composite_of_set_a @ Resid @ T @ B @ T ) ) ) ) ).
% rts.composite_of_arr_target
thf(fact_1229_rts_Ocomposite__of__arr__target,axiom,
! [Resid: a > a > a,T: a,B: a] :
( ( rts_a @ Resid )
=> ( ( arr_a @ Resid @ T )
=> ( ( member_a @ B @ ( targets_a @ Resid @ T ) )
=> ( composite_of_a @ Resid @ T @ B @ T ) ) ) ) ).
% rts.composite_of_arr_target
thf(fact_1230_rts_Otrg__in__targets,axiom,
! [Resid: set_a > set_a > set_a,T: set_a] :
( ( rts_set_a @ Resid )
=> ( ( arr_set_a @ Resid @ T )
=> ( member_set_a @ ( trg_set_a @ Resid @ T ) @ ( targets_set_a @ Resid @ T ) ) ) ) ).
% rts.trg_in_targets
thf(fact_1231_rts_Otrg__in__targets,axiom,
! [Resid: a > a > a,T: a] :
( ( rts_a @ Resid )
=> ( ( arr_a @ Resid @ T )
=> ( member_a @ ( trg_a @ Resid @ T ) @ ( targets_a @ Resid @ T ) ) ) ) ).
% rts.trg_in_targets
thf(fact_1232_rts_Ojoin__of__arr__src_I2_J,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,A3: set_a] :
( ( rts_set_a @ Resid )
=> ( ( arr_set_a @ Resid @ T )
=> ( ( member_set_a @ A3 @ ( sources_set_a @ Resid @ T ) )
=> ( join_of_set_a @ Resid @ T @ A3 @ T ) ) ) ) ).
% rts.join_of_arr_src(2)
thf(fact_1233_rts_Ojoin__of__arr__src_I2_J,axiom,
! [Resid: a > a > a,T: a,A3: a] :
( ( rts_a @ Resid )
=> ( ( arr_a @ Resid @ T )
=> ( ( member_a @ A3 @ ( sources_a @ Resid @ T ) )
=> ( join_of_a @ Resid @ T @ A3 @ T ) ) ) ) ).
% rts.join_of_arr_src(2)
thf(fact_1234_rts_Ojoin__of__arr__src_I1_J,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,A3: set_a] :
( ( rts_set_a @ Resid )
=> ( ( arr_set_a @ Resid @ T )
=> ( ( member_set_a @ A3 @ ( sources_set_a @ Resid @ T ) )
=> ( join_of_set_a @ Resid @ A3 @ T @ T ) ) ) ) ).
% rts.join_of_arr_src(1)
thf(fact_1235_rts_Ojoin__of__arr__src_I1_J,axiom,
! [Resid: a > a > a,T: a,A3: a] :
( ( rts_a @ Resid )
=> ( ( arr_a @ Resid @ T )
=> ( ( member_a @ A3 @ ( sources_a @ Resid @ T ) )
=> ( join_of_a @ Resid @ A3 @ T @ T ) ) ) ) ).
% rts.join_of_arr_src(1)
thf(fact_1236_rts_OcoinitialE,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a] :
( ( rts_set_a @ Resid )
=> ( ( coinitial_set_a @ Resid @ T @ U )
=> ~ ( ( arr_set_a @ Resid @ T )
=> ( ( arr_set_a @ Resid @ U )
=> ( ( sources_set_a @ Resid @ T )
!= ( sources_set_a @ Resid @ U ) ) ) ) ) ) ).
% rts.coinitialE
thf(fact_1237_rts_OcoinitialE,axiom,
! [Resid: a > a > a,T: a,U: a] :
( ( rts_a @ Resid )
=> ( ( coinitial_a @ Resid @ T @ U )
=> ~ ( ( arr_a @ Resid @ T )
=> ( ( arr_a @ Resid @ U )
=> ( ( sources_a @ Resid @ T )
!= ( sources_a @ Resid @ U ) ) ) ) ) ) ).
% rts.coinitialE
thf(fact_1238_rts_OcoinitialI,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a] :
( ( rts_set_a @ Resid )
=> ( ( arr_set_a @ Resid @ T )
=> ( ( ( sources_set_a @ Resid @ T )
= ( sources_set_a @ Resid @ U ) )
=> ( coinitial_set_a @ Resid @ T @ U ) ) ) ) ).
% rts.coinitialI
thf(fact_1239_rts_OcoinitialI,axiom,
! [Resid: a > a > a,T: a,U: a] :
( ( rts_a @ Resid )
=> ( ( arr_a @ Resid @ T )
=> ( ( ( sources_a @ Resid @ T )
= ( sources_a @ Resid @ U ) )
=> ( coinitial_a @ Resid @ T @ U ) ) ) ) ).
% rts.coinitialI
thf(fact_1240_rts_Ocoinitial__iff,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,T4: set_a] :
( ( rts_set_a @ Resid )
=> ( ( coinitial_set_a @ Resid @ T @ T4 )
= ( ( arr_set_a @ Resid @ T )
& ( arr_set_a @ Resid @ T4 )
& ( ( sources_set_a @ Resid @ T )
= ( sources_set_a @ Resid @ T4 ) ) ) ) ) ).
% rts.coinitial_iff
thf(fact_1241_rts_Ocoinitial__iff,axiom,
! [Resid: a > a > a,T: a,T4: a] :
( ( rts_a @ Resid )
=> ( ( coinitial_a @ Resid @ T @ T4 )
= ( ( arr_a @ Resid @ T )
& ( arr_a @ Resid @ T4 )
& ( ( sources_a @ Resid @ T )
= ( sources_a @ Resid @ T4 ) ) ) ) ) ).
% rts.coinitial_iff
thf(fact_1242_rts_OcomposableD_I3_J,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a] :
( ( rts_set_a @ Resid )
=> ( ( composable_set_a @ Resid @ T @ U )
=> ( ( targets_set_a @ Resid @ T )
= ( sources_set_a @ Resid @ U ) ) ) ) ).
% rts.composableD(3)
thf(fact_1243_rts_OcomposableD_I3_J,axiom,
! [Resid: a > a > a,T: a,U: a] :
( ( rts_a @ Resid )
=> ( ( composable_a @ Resid @ T @ U )
=> ( ( targets_a @ Resid @ T )
= ( sources_a @ Resid @ U ) ) ) ) ).
% rts.composableD(3)
thf(fact_1244_rts_OcoterminalE,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a] :
( ( rts_set_a @ Resid )
=> ( ( coterminal_set_a @ Resid @ T @ U )
=> ~ ( ( arr_set_a @ Resid @ T )
=> ( ( arr_set_a @ Resid @ U )
=> ( ( targets_set_a @ Resid @ T )
!= ( targets_set_a @ Resid @ U ) ) ) ) ) ) ).
% rts.coterminalE
thf(fact_1245_rts_OcoterminalE,axiom,
! [Resid: a > a > a,T: a,U: a] :
( ( rts_a @ Resid )
=> ( ( coterminal_a @ Resid @ T @ U )
=> ~ ( ( arr_a @ Resid @ T )
=> ( ( arr_a @ Resid @ U )
=> ( ( targets_a @ Resid @ T )
!= ( targets_a @ Resid @ U ) ) ) ) ) ) ).
% rts.coterminalE
thf(fact_1246_rts_OcoterminalI,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a] :
( ( rts_set_a @ Resid )
=> ( ( arr_set_a @ Resid @ T )
=> ( ( ( targets_set_a @ Resid @ T )
= ( targets_set_a @ Resid @ U ) )
=> ( coterminal_set_a @ Resid @ T @ U ) ) ) ) ).
% rts.coterminalI
thf(fact_1247_rts_OcoterminalI,axiom,
! [Resid: a > a > a,T: a,U: a] :
( ( rts_a @ Resid )
=> ( ( arr_a @ Resid @ T )
=> ( ( ( targets_a @ Resid @ T )
= ( targets_a @ Resid @ U ) )
=> ( coterminal_a @ Resid @ T @ U ) ) ) ) ).
% rts.coterminalI
thf(fact_1248_rts_Ocoterminal__iff,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,T4: set_a] :
( ( rts_set_a @ Resid )
=> ( ( coterminal_set_a @ Resid @ T @ T4 )
= ( ( arr_set_a @ Resid @ T )
& ( arr_set_a @ Resid @ T4 )
& ( ( targets_set_a @ Resid @ T )
= ( targets_set_a @ Resid @ T4 ) ) ) ) ) ).
% rts.coterminal_iff
thf(fact_1249_rts_Ocoterminal__iff,axiom,
! [Resid: a > a > a,T: a,T4: a] :
( ( rts_a @ Resid )
=> ( ( coterminal_a @ Resid @ T @ T4 )
= ( ( arr_a @ Resid @ T )
& ( arr_a @ Resid @ T4 )
& ( ( targets_a @ Resid @ T )
= ( targets_a @ Resid @ T4 ) ) ) ) ) ).
% rts.coterminal_iff
thf(fact_1250_rts_Ocoterminal__iff__con__trg,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a] :
( ( rts_set_a @ Resid )
=> ( ( coterminal_set_a @ Resid @ T @ U )
= ( con_set_a @ Resid @ ( trg_set_a @ Resid @ T ) @ ( trg_set_a @ Resid @ U ) ) ) ) ).
% rts.coterminal_iff_con_trg
thf(fact_1251_rts_Ocoterminal__iff__con__trg,axiom,
! [Resid: a > a > a,T: a,U: a] :
( ( rts_a @ Resid )
=> ( ( coterminal_a @ Resid @ T @ U )
= ( con_a @ Resid @ ( trg_a @ Resid @ T ) @ ( trg_a @ Resid @ U ) ) ) ) ).
% rts.coterminal_iff_con_trg
thf(fact_1252_rts_Osources__def,axiom,
! [Resid: set_a > set_a > set_a,T: set_a] :
( ( rts_set_a @ Resid )
=> ( ( sources_set_a @ Resid @ T )
= ( collect_set_a
@ ^ [A: set_a] :
( ( ide_set_a @ Resid @ A )
& ( con_set_a @ Resid @ T @ A ) ) ) ) ) ).
% rts.sources_def
thf(fact_1253_rts_Osources__def,axiom,
! [Resid: a > a > a,T: a] :
( ( rts_a @ Resid )
=> ( ( sources_a @ Resid @ T )
= ( collect_a
@ ^ [A: a] :
( ( ide_a @ Resid @ A )
& ( con_a @ Resid @ T @ A ) ) ) ) ) ).
% rts.sources_def
thf(fact_1254_rts_Otargets__char,axiom,
! [Resid: set_a > set_a > set_a,T: set_a] :
( ( rts_set_a @ Resid )
=> ( ( targets_set_a @ Resid @ T )
= ( collect_set_a
@ ^ [B2: set_a] :
( ( arr_set_a @ Resid @ T )
& ( ide_set_a @ Resid @ ( Resid @ ( Resid @ T @ T ) @ B2 ) )
& ( ide_set_a @ Resid @ ( Resid @ B2 @ ( Resid @ T @ T ) ) ) ) ) ) ) ).
% rts.targets_char
thf(fact_1255_rts_Otargets__char,axiom,
! [Resid: a > a > a,T: a] :
( ( rts_a @ Resid )
=> ( ( targets_a @ Resid @ T )
= ( collect_a
@ ^ [B2: a] :
( ( arr_a @ Resid @ T )
& ( ide_a @ Resid @ ( Resid @ ( Resid @ T @ T ) @ B2 ) )
& ( ide_a @ Resid @ ( Resid @ B2 @ ( Resid @ T @ T ) ) ) ) ) ) ) ).
% rts.targets_char
thf(fact_1256_rts_Ocon__imp__common__source,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a] :
( ( rts_set_a @ Resid )
=> ( ( con_set_a @ Resid @ T @ U )
=> ( ( inf_inf_set_set_a @ ( sources_set_a @ Resid @ T ) @ ( sources_set_a @ Resid @ U ) )
!= bot_bot_set_set_a ) ) ) ).
% rts.con_imp_common_source
thf(fact_1257_rts_Ocon__imp__common__source,axiom,
! [Resid: a > a > a,T: a,U: a] :
( ( rts_a @ Resid )
=> ( ( con_a @ Resid @ T @ U )
=> ( ( inf_inf_set_a @ ( sources_a @ Resid @ T ) @ ( sources_a @ Resid @ U ) )
!= bot_bot_set_a ) ) ) ).
% rts.con_imp_common_source
thf(fact_1258_rts_Oin__targetsE,axiom,
! [Resid: set_a > set_a > set_a,B: set_a,T: set_a] :
( ( rts_set_a @ Resid )
=> ( ( member_set_a @ B @ ( targets_set_a @ Resid @ T ) )
=> ~ ( ( ide_set_a @ Resid @ B )
=> ~ ( con_set_a @ Resid @ ( trg_set_a @ Resid @ T ) @ B ) ) ) ) ).
% rts.in_targetsE
thf(fact_1259_rts_Oin__targetsE,axiom,
! [Resid: a > a > a,B: a,T: a] :
( ( rts_a @ Resid )
=> ( ( member_a @ B @ ( targets_a @ Resid @ T ) )
=> ~ ( ( ide_a @ Resid @ B )
=> ~ ( con_a @ Resid @ ( trg_a @ Resid @ T ) @ B ) ) ) ) ).
% rts.in_targetsE
thf(fact_1260_rts_Oin__targetsI,axiom,
! [Resid: set_a > set_a > set_a,B: set_a,T: set_a] :
( ( rts_set_a @ Resid )
=> ( ( ide_set_a @ Resid @ B )
=> ( ( con_set_a @ Resid @ ( trg_set_a @ Resid @ T ) @ B )
=> ( member_set_a @ B @ ( targets_set_a @ Resid @ T ) ) ) ) ) ).
% rts.in_targetsI
thf(fact_1261_rts_Oin__targetsI,axiom,
! [Resid: a > a > a,B: a,T: a] :
( ( rts_a @ Resid )
=> ( ( ide_a @ Resid @ B )
=> ( ( con_a @ Resid @ ( trg_a @ Resid @ T ) @ B )
=> ( member_a @ B @ ( targets_a @ Resid @ T ) ) ) ) ) ).
% rts.in_targetsI
thf(fact_1262_rts_Ocoinitial__def,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a] :
( ( rts_set_a @ Resid )
=> ( ( coinitial_set_a @ Resid @ T @ U )
= ( ( inf_inf_set_set_a @ ( sources_set_a @ Resid @ T ) @ ( sources_set_a @ Resid @ U ) )
!= bot_bot_set_set_a ) ) ) ).
% rts.coinitial_def
thf(fact_1263_rts_Ocoinitial__def,axiom,
! [Resid: a > a > a,T: a,U: a] :
( ( rts_a @ Resid )
=> ( ( coinitial_a @ Resid @ T @ U )
= ( ( inf_inf_set_a @ ( sources_a @ Resid @ T ) @ ( sources_a @ Resid @ U ) )
!= bot_bot_set_a ) ) ) ).
% rts.coinitial_def
thf(fact_1264_rts_Ocoterminal__def,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a] :
( ( rts_set_a @ Resid )
=> ( ( coterminal_set_a @ Resid @ T @ U )
= ( ( inf_inf_set_set_a @ ( targets_set_a @ Resid @ T ) @ ( targets_set_a @ Resid @ U ) )
!= bot_bot_set_set_a ) ) ) ).
% rts.coterminal_def
thf(fact_1265_rts_Ocoterminal__def,axiom,
! [Resid: a > a > a,T: a,U: a] :
( ( rts_a @ Resid )
=> ( ( coterminal_a @ Resid @ T @ U )
= ( ( inf_inf_set_a @ ( targets_a @ Resid @ T ) @ ( targets_a @ Resid @ U ) )
!= bot_bot_set_a ) ) ) ).
% rts.coterminal_def
thf(fact_1266_rts_Oseq__def,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a] :
( ( rts_set_a @ Resid )
=> ( ( seq_set_a @ Resid @ T @ U )
= ( ( arr_set_a @ Resid @ T )
& ( arr_set_a @ Resid @ U )
& ( ( targets_set_a @ Resid @ T )
= ( sources_set_a @ Resid @ U ) ) ) ) ) ).
% rts.seq_def
thf(fact_1267_rts_Oseq__def,axiom,
! [Resid: a > a > a,T: a,U: a] :
( ( rts_a @ Resid )
=> ( ( seq_a @ Resid @ T @ U )
= ( ( arr_a @ Resid @ T )
& ( arr_a @ Resid @ U )
& ( ( targets_a @ Resid @ T )
= ( sources_a @ Resid @ U ) ) ) ) ) ).
% rts.seq_def
thf(fact_1268_rts_OseqI,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a] :
( ( rts_set_a @ Resid )
=> ( ( arr_set_a @ Resid @ T )
=> ( ( arr_set_a @ Resid @ U )
=> ( ( ( targets_set_a @ Resid @ T )
= ( sources_set_a @ Resid @ U ) )
=> ( seq_set_a @ Resid @ T @ U ) ) ) ) ) ).
% rts.seqI
thf(fact_1269_rts_OseqI,axiom,
! [Resid: a > a > a,T: a,U: a] :
( ( rts_a @ Resid )
=> ( ( arr_a @ Resid @ T )
=> ( ( arr_a @ Resid @ U )
=> ( ( ( targets_a @ Resid @ T )
= ( sources_a @ Resid @ U ) )
=> ( seq_a @ Resid @ T @ U ) ) ) ) ) ).
% rts.seqI
thf(fact_1270_rts_OseqE,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a] :
( ( rts_set_a @ Resid )
=> ( ( seq_set_a @ Resid @ T @ U )
=> ~ ( ( arr_set_a @ Resid @ T )
=> ( ( arr_set_a @ Resid @ U )
=> ( ( targets_set_a @ Resid @ T )
!= ( sources_set_a @ Resid @ U ) ) ) ) ) ) ).
% rts.seqE
thf(fact_1271_rts_OseqE,axiom,
! [Resid: a > a > a,T: a,U: a] :
( ( rts_a @ Resid )
=> ( ( seq_a @ Resid @ T @ U )
=> ~ ( ( arr_a @ Resid @ T )
=> ( ( arr_a @ Resid @ U )
=> ( ( targets_a @ Resid @ T )
!= ( sources_a @ Resid @ U ) ) ) ) ) ) ).
% rts.seqE
thf(fact_1272_rts_Otargets__def,axiom,
! [Resid: set_a > set_a > set_a,T: set_a] :
( ( rts_set_a @ Resid )
=> ( ( targets_set_a @ Resid @ T )
= ( collect_set_a
@ ^ [B2: set_a] :
( ( ide_set_a @ Resid @ B2 )
& ( con_set_a @ Resid @ ( trg_set_a @ Resid @ T ) @ B2 ) ) ) ) ) ).
% rts.targets_def
thf(fact_1273_rts_Otargets__def,axiom,
! [Resid: a > a > a,T: a] :
( ( rts_a @ Resid )
=> ( ( targets_a @ Resid @ T )
= ( collect_a
@ ^ [B2: a] :
( ( ide_a @ Resid @ B2 )
& ( con_a @ Resid @ ( trg_a @ Resid @ T ) @ B2 ) ) ) ) ) ).
% rts.targets_def
thf(fact_1274_simulation__axioms__def,axiom,
( simula3192323252075944454_set_a
= ( ^ [A8: a > a > a,B5: set_a > set_a > set_a,F4: a > set_a] :
( ! [T6: a] :
( ~ ( arr_a @ A8 @ T6 )
=> ( ( F4 @ T6 )
= ( partial_null_set_a @ B5 ) ) )
& ! [T6: a,U5: a] :
( ( con_a @ A8 @ T6 @ U5 )
=> ( con_set_a @ B5 @ ( F4 @ T6 ) @ ( F4 @ U5 ) ) )
& ! [T6: a,U5: a] :
( ( con_a @ A8 @ T6 @ U5 )
=> ( ( F4 @ ( A8 @ T6 @ U5 ) )
= ( B5 @ ( F4 @ T6 ) @ ( F4 @ U5 ) ) ) ) ) ) ) ).
% simulation_axioms_def
thf(fact_1275_simulation__axioms__def,axiom,
( simula3408835310535287622et_a_a
= ( ^ [A8: set_a > set_a > set_a,B5: a > a > a,F4: set_a > a] :
( ! [T6: set_a] :
( ~ ( arr_set_a @ A8 @ T6 )
=> ( ( F4 @ T6 )
= ( partial_null_a @ B5 ) ) )
& ! [T6: set_a,U5: set_a] :
( ( con_set_a @ A8 @ T6 @ U5 )
=> ( con_a @ B5 @ ( F4 @ T6 ) @ ( F4 @ U5 ) ) )
& ! [T6: set_a,U5: set_a] :
( ( con_set_a @ A8 @ T6 @ U5 )
=> ( ( F4 @ ( A8 @ T6 @ U5 ) )
= ( B5 @ ( F4 @ T6 ) @ ( F4 @ U5 ) ) ) ) ) ) ) ).
% simulation_axioms_def
thf(fact_1276_simulation__axioms__def,axiom,
( simula3868467710248865958ms_a_a
= ( ^ [A8: a > a > a,B5: a > a > a,F4: a > a] :
( ! [T6: a] :
( ~ ( arr_a @ A8 @ T6 )
=> ( ( F4 @ T6 )
= ( partial_null_a @ B5 ) ) )
& ! [T6: a,U5: a] :
( ( con_a @ A8 @ T6 @ U5 )
=> ( con_a @ B5 @ ( F4 @ T6 ) @ ( F4 @ U5 ) ) )
& ! [T6: a,U5: a] :
( ( con_a @ A8 @ T6 @ U5 )
=> ( ( F4 @ ( A8 @ T6 @ U5 ) )
= ( B5 @ ( F4 @ T6 ) @ ( F4 @ U5 ) ) ) ) ) ) ) ).
% simulation_axioms_def
thf(fact_1277_R_Orts__axioms,axiom,
rts_a @ resid ).
% R.rts_axioms
thf(fact_1278_N_Onormal__sub__rts__axioms,axiom,
normal_sub_rts_a @ resid @ nn ).
% N.normal_sub_rts_axioms
% Helper facts (5)
thf(help_If_2_1_If_001tf__a_T,axiom,
! [X4: a,Y4: a] :
( ( if_a @ $false @ X4 @ Y4 )
= Y4 ) ).
thf(help_If_1_1_If_001tf__a_T,axiom,
! [X4: a,Y4: a] :
( ( if_a @ $true @ X4 @ Y4 )
= X4 ) ).
thf(help_If_3_1_If_001t__Set__Oset_Itf__a_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Set__Oset_Itf__a_J_T,axiom,
! [X4: set_a,Y4: set_a] :
( ( if_set_a @ $false @ X4 @ Y4 )
= Y4 ) ).
thf(help_If_1_1_If_001t__Set__Oset_Itf__a_J_T,axiom,
! [X4: set_a,Y4: set_a] :
( ( if_set_a @ $true @ X4 @ Y4 )
= X4 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( sources_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ t )
= ( collect_set_a
@ ^ [A10: set_a] :
( ( arr_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ t )
& ( A10
= ( collect_a
@ ^ [A: a] :
? [T6: a,A2: a] :
( ( member_a @ T6 @ t )
& ( member_a @ A2 @ ( sources_a @ resid @ T6 ) )
& ( normal_sub_Cong_a @ resid @ nn @ A2 @ A ) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------