TPTP Problem File: SLH0302^1.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_03136_117099__14126966_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1535 ( 330 unt; 248 typ; 0 def)
% Number of atoms : 5105 (1192 equ; 0 cnn)
% Maximal formula atoms : 17 ( 3 avg)
% Number of connectives : 18767 ( 369 ~; 26 |; 670 &;15105 @)
% ( 0 <=>;2597 =>; 0 <=; 0 <~>)
% Maximal formula depth : 25 ( 8 avg)
% Number of types : 7 ( 6 usr)
% Number of type conns : 3067 (3067 >; 0 *; 0 +; 0 <<)
% Number of symbols : 245 ( 242 usr; 27 con; 0-5 aty)
% Number of variables : 4167 ( 172 ^;3821 !; 174 ?;4167 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 11:47:56.865
%------------------------------------------------------------------------------
% Could-be-implicit typings (6)
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
set_Product_prod_a_a: $tType ).
thf(ty_n_t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
product_prod_a_a: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
set_set_a: $tType ).
thf(ty_n_t__Set__Oset_I_062_Itf__a_Mtf__a_J_J,type,
set_a_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 (242)
thf(sy_c_BNF__Wellorder__Constructions_OFunc_001tf__a_001tf__a,type,
bNF_We5243062509538606484nc_a_a: set_a > set_a > set_a_a ).
thf(sy_c_Boolean__Algebras_Oabstract__boolean__algebra_001t__Set__Oset_Itf__a_J,type,
boolea6678413348699952596_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_Finite__Set_Ofinite_001t__Set__Oset_Itf__a_J,type,
finite_finite_set_a: set_set_a > $o ).
thf(sy_c_Finite__Set_Ofinite_001tf__a,type,
finite_finite_a: set_a > $o ).
thf(sy_c_Finite__Set_Ofold_001tf__a_001t__Set__Oset_Itf__a_J,type,
finite_fold_a_set_a: ( a > set_a > set_a ) > set_a > set_a > set_a ).
thf(sy_c_Fun_Oinj__on_001t__Set__Oset_Itf__a_J_001tf__a,type,
inj_on_set_a_a: ( set_a > a ) > set_set_a > $o ).
thf(sy_c_Fun_Oinj__on_001tf__a_001t__Set__Oset_Itf__a_J,type,
inj_on_a_set_a: ( a > set_a ) > set_a > $o ).
thf(sy_c_Fun_Oinj__on_001tf__a_001tf__a,type,
inj_on_a_a: ( a > a ) > set_a > $o ).
thf(sy_c_Fun_Othe__inv__into_001tf__a_001tf__a,type,
the_inv_into_a_a: set_a > ( a > a ) > a > a ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
minus_5736297505244876581_set_a: set_set_a > set_set_a > set_set_a ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_Itf__a_J,type,
minus_minus_set_a: set_a > set_a > set_a ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_Itf__a_J,type,
uminus_uminus_set_a: set_a > set_a ).
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_HOL_OUniq_001tf__a,type,
uniq_a: ( a > $o ) > $o ).
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_Inductive_Ocomplete__lattice__class_Olfp_001_062_It__Set__Oset_Itf__a_J_M_Eo_J,type,
comple3243854344658494246et_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_I_062_Itf__a_Mtf__a_J_J,type,
inf_inf_set_a_a: set_a_a > set_a_a > set_a_a ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
inf_in8905007599844390133od_a_a: set_Product_prod_a_a > set_Product_prod_a_a > set_Product_prod_a_a ).
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_Lattices_Osemilattice__neutr__order_001t__Set__Oset_Itf__a_J,type,
semila2496817875450240012_set_a: ( set_a > set_a > set_a ) > set_a > ( set_a > set_a > $o ) > ( set_a > set_a > $o ) > $o ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
sup_sup_set_set_a: set_set_a > set_set_a > set_set_a ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_Itf__a_J,type,
sup_sup_set_a: set_a > set_a > set_a ).
thf(sy_c_Lattices__Big_Osemilattice__inf__class_OInf__fin_001t__Set__Oset_Itf__a_J,type,
lattic8209813465164889211_set_a: set_set_a > set_a ).
thf(sy_c_Lattices__Big_Osemilattice__sup__class_OSup__fin_001t__Set__Oset_Itf__a_J,type,
lattic2918178356826803221_set_a: set_set_a > set_a ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_I_062_Itf__a_Mtf__a_J_M_Eo_J,type,
bot_bot_a_a_o: ( a > a ) > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Product____Type__Oprod_Itf__a_Mtf__a_J_M_Eo_J,type,
bot_bo4160289986317612842_a_a_o: product_prod_a_a > $o ).
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_001_Eo,type,
bot_bot_o: $o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_062_Itf__a_Mtf__a_J_J,type,
bot_bot_set_a_a: set_a_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
bot_bo3357376287454694259od_a_a: set_Product_prod_a_a ).
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__class_Oless_001t__Set__Oset_Itf__a_J,type,
ord_less_set_a: set_a > set_a > $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_Oordering__top_001t__Set__Oset_Itf__a_J,type,
ordering_top_set_a: ( set_a > set_a > $o ) > ( set_a > set_a > $o ) > set_a > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
top_top_set_set_a: set_set_a ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_Itf__a_J,type,
top_top_set_a: set_a ).
thf(sy_c_Partial__Function_Oflat__lub_001t__Set__Oset_Itf__a_J,type,
partia4732287487727000106_set_a: set_a > set_set_a > set_a ).
thf(sy_c_Partial__Function_Oflat__lub_001tf__a,type,
partial_flat_lub_a: a > set_a > a ).
thf(sy_c_Product__Type_Oprod_Ofst_001tf__a_001tf__a,type,
product_fst_a_a: product_prod_a_a > a ).
thf(sy_c_Product__Type_Oprod_Osnd_001tf__a_001tf__a,type,
product_snd_a_a: product_prod_a_a > a ).
thf(sy_c_Relation_Ototalp__on_001tf__a,type,
totalp_on_a: set_a > ( a > a > $o ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Ocoherent__normal__sub__rts_001_062_Itf__a_Mtf__a_J,type,
cohere4631022276737013564ts_a_a: ( ( a > a ) > ( a > a ) > a > a ) > set_a_a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Ocoherent__normal__sub__rts_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
cohere3906135489564710160od_a_a: ( product_prod_a_a > product_prod_a_a > product_prod_a_a ) > set_Product_prod_a_a > $o ).
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_Ocoherent__normal__sub__rts__axioms_001_062_Itf__a_Mtf__a_J,type,
cohere4772091081940722911ms_a_a: ( ( a > a ) > ( a > a ) > a > a ) > set_a_a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Ocoherent__normal__sub__rts__axioms_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
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thf(sy_c_ResiduatedTransitionSystem_Ocoherent__normal__sub__rts__axioms_001t__Set__Oset_Itf__a_J,type,
cohere32089786014956644_set_a: ( set_a > set_a > set_a ) > set_set_a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Ocoherent__normal__sub__rts__axioms_001tf__a,type,
cohere4894532172567702276ioms_a: ( a > a > a ) > set_a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Oconfluent__rts_001t__Set__Oset_Itf__a_J,type,
confluent_rts_set_a: ( set_a > set_a > set_a ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Oconfluent__rts_001tf__a,type,
confluent_rts_a: ( a > a > a ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Oconfluent__rts__axioms_001t__Set__Oset_Itf__a_J,type,
conflu1148668952538903019_set_a: ( set_a > set_a > set_a ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Oconfluent__rts__axioms_001tf__a,type,
conflu3014480972103220363ioms_a: ( a > a > 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_Oextensional__rts__axioms_001tf__a,type,
extens8613361310974791063ioms_a: ( a > a > a ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Oextensional__rts__with__composites_001tf__a,type,
extens4790121754472881640ites_a: ( a > a > a ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Oextensional__rts__with__joins_001t__Set__Oset_Itf__a_J,type,
extens2085910753204196637_set_a: ( set_a > set_a > set_a ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Oextensional__rts__with__joins_001tf__a,type,
extens4936603313648314301oins_a: ( a > a > a ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Oidentity__simulation_001tf__a,type,
identi4709066280192368860tion_a: ( a > a > a ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Onormal__in__extensional__rts__with__composites_001tf__a,type,
normal636964748050715740ites_a: ( a > a > a ) > set_a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Onormal__sub__rts_001_062_Itf__a_Mtf__a_J,type,
normal_sub_rts_a_a: ( ( a > a ) > ( a > a ) > a > a ) > set_a_a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Onormal__sub__rts_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
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thf(sy_c_ResiduatedTransitionSystem_Onormal__sub__rts_001t__Set__Oset_Itf__a_J,type,
normal_sub_rts_set_a: ( set_a > set_a > set_a ) > set_set_a > $o ).
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_001_062_Itf__a_Mtf__a_J,type,
normal_sub_Cong_a_a: ( ( a > a ) > ( a > a ) > a > a ) > set_a_a > ( a > a ) > ( a > a ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Onormal__sub__rts_OCong_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
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thf(sy_c_ResiduatedTransitionSystem_Onormal__sub__rts_OCong_001t__Set__Oset_Itf__a_J,type,
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thf(sy_c_ResiduatedTransitionSystem_Onormal__sub__rts_OCong_001tf__a,type,
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thf(sy_c_ResiduatedTransitionSystem_Onormal__sub__rts_OCong_H_001_062_Itf__a_Mtf__a_J,type,
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thf(sy_c_ResiduatedTransitionSystem_Onormal__sub__rts_OCong__class__rep_001tf__a,type,
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thf(sy_c_ResiduatedTransitionSystem_Onormal__sub__rts_Ois__Cong__class_001_062_Itf__a_Mtf__a_J,type,
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thf(sy_c_ResiduatedTransitionSystem_Opartial__magma_001t__Set__Oset_Itf__a_J,type,
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thf(sy_c_Set_Ois__empty_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
is_emp2937470224744679417od_a_a: set_Product_prod_a_a > $o ).
thf(sy_c_Set_Ois__empty_001t__Set__Oset_Itf__a_J,type,
is_empty_set_a: set_set_a > $o ).
thf(sy_c_Set_Ois__empty_001tf__a,type,
is_empty_a: set_a > $o ).
thf(sy_c_Set_Ois__singleton_001tf__a,type,
is_singleton_a: set_a > $o ).
thf(sy_c_Set_Opairwise_001tf__a,type,
pairwise_a: ( a > a > $o ) > set_a > $o ).
thf(sy_c_Set_Oremove_001tf__a,type,
remove_a: a > set_a > set_a ).
thf(sy_c_Set_Othe__elem_001t__Set__Oset_Itf__a_J,type,
the_elem_set_a: set_set_a > set_a ).
thf(sy_c_Set_Othe__elem_001tf__a,type,
the_elem_a: set_a > a ).
thf(sy_c_Set_Ovimage_001t__Set__Oset_Itf__a_J_001tf__a,type,
vimage_set_a_a: ( set_a > a ) > set_a > set_set_a ).
thf(sy_c_Set_Ovimage_001tf__a_001tf__a,type,
vimage_a_a: ( a > a ) > set_a > set_a ).
thf(sy_c_fChoice_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
fChoic4124218645493772411od_a_a: ( product_prod_a_a > $o ) > product_prod_a_a ).
thf(sy_c_fChoice_001tf__a,type,
fChoice_a: ( a > $o ) > a ).
thf(sy_c_member_001_062_Itf__a_Mtf__a_J,type,
member_a_a: ( a > a ) > set_a_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
member1426531477525435216od_a_a: product_prod_a_a > set_Product_prod_a_a > $o ).
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__092_060U_062,type,
u: set_a ).
thf(sy_v__092_060V_062,type,
v: set_a ).
thf(sy_v_resid,type,
resid: a > a > a ).
thf(sy_v_t_H____,type,
t2: a ).
thf(sy_v_t_Hx_H____,type,
t_x: a ).
thf(sy_v_t____,type,
t3: a ).
thf(sy_v_tx____,type,
tx: a ).
thf(sy_v_u____,type,
u2: a ).
thf(sy_v_v____,type,
v2: a ).
thf(sy_v_w_H____,type,
w: a ).
thf(sy_v_w____,type,
w2: a ).
thf(sy_v_x_H____,type,
x: a ).
thf(sy_v_x____,type,
x2: a ).
thf(sy_v_y_H____,type,
y: a ).
thf(sy_v_y____,type,
y2: a ).
thf(sy_v_z_H____,type,
z: a ).
thf(sy_v_z____,type,
z2: 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_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_3_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_4_R_Oide__backward__stable,axiom,
! [A: a,T: a] :
( ( ide_a @ resid @ A )
=> ( ( ide_a @ resid @ ( resid @ T @ A ) )
=> ( ide_a @ resid @ T ) ) ) ).
% R.ide_backward_stable
thf(fact_5_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_6_residuation_Oide_Ocong,axiom,
ide_a = ide_a ).
% residuation.ide.cong
thf(fact_7_R_Opartial__magma__axioms,axiom,
partial_magma_a @ resid ).
% R.partial_magma_axioms
thf(fact_8_y__coinitial,axiom,
coinitial_a @ resid @ y2 @ ( resid @ u2 @ tx ) ).
% y_coinitial
thf(fact_9_R_Ocong__implies__coterminal,axiom,
! [U: a,U2: a] :
( ( ( ide_a @ resid @ ( resid @ U @ U2 ) )
& ( ide_a @ resid @ ( resid @ U2 @ U ) ) )
=> ( coterminal_a @ resid @ U @ U2 ) ) ).
% R.cong_implies_coterminal
thf(fact_10_y__con,axiom,
con_a @ resid @ y2 @ ( resid @ u2 @ tx ) ).
% y_con
thf(fact_11_Con__z__uw,axiom,
con_a @ resid @ z2 @ ( resid @ u2 @ w2 ) ).
% Con_z_uw
thf(fact_12_R_Ocoinitial__ide__are__cong,axiom,
! [A: a,A2: a] :
( ( ide_a @ resid @ A )
=> ( ( ide_a @ resid @ A2 )
=> ( ( coinitial_a @ resid @ A @ A2 )
=> ( ( ide_a @ resid @ ( resid @ A @ A2 ) )
& ( ide_a @ resid @ ( resid @ A2 @ A ) ) ) ) ) ) ).
% R.coinitial_ide_are_cong
thf(fact_13_R_Ocong__implies__coinitial,axiom,
! [U: a,U2: a] :
( ( ( ide_a @ resid @ ( resid @ U @ U2 ) )
& ( ide_a @ resid @ ( resid @ U2 @ U ) ) )
=> ( coinitial_a @ resid @ U @ U2 ) ) ).
% R.cong_implies_coinitial
thf(fact_14_R_Ocong__respects__seq,axiom,
! [T: a,U: a,T4: a,U2: a] :
( ( seq_a @ resid @ T @ U )
=> ( ( ( ide_a @ resid @ ( resid @ T @ T4 ) )
& ( ide_a @ resid @ ( resid @ T4 @ T ) ) )
=> ( ( ( ide_a @ resid @ ( resid @ U @ U2 ) )
& ( ide_a @ resid @ ( resid @ U2 @ U ) ) )
=> ( seq_a @ resid @ T4 @ U2 ) ) ) ) ).
% R.cong_respects_seq
thf(fact_15_R_Ojoin__of__un__upto__cong,axiom,
! [T: a,U: a,V: a,V2: a] :
( ( join_of_a @ resid @ T @ U @ V )
=> ( ( join_of_a @ resid @ T @ U @ V2 )
=> ( ( ide_a @ resid @ ( resid @ V @ V2 ) )
& ( ide_a @ resid @ ( resid @ V2 @ V ) ) ) ) ) ).
% R.join_of_un_upto_cong
thf(fact_16_R_Oresiduation__axioms,axiom,
residuation_a @ resid ).
% R.residuation_axioms
thf(fact_17__092_060open_062_Iu_A_092_Atx_J_A_092_Ay_A_092_060sim_062_A_I_Iu_A_092_Atx_J_A_092_A_Iw_A_092_Atx_J_J_A_092_A_I_It_Hx_H_A_092_Aw_H_J_A_092_A_Itx_A_092_Aw_J_J_092_060close_062,axiom,
( ( ide_a @ resid @ ( resid @ ( resid @ ( resid @ u2 @ tx ) @ y2 ) @ ( resid @ ( resid @ ( resid @ u2 @ tx ) @ ( resid @ w2 @ tx ) ) @ ( resid @ ( resid @ t_x @ w ) @ ( resid @ tx @ w2 ) ) ) ) )
& ( ide_a @ resid @ ( resid @ ( resid @ ( resid @ ( resid @ u2 @ tx ) @ ( resid @ w2 @ tx ) ) @ ( resid @ ( resid @ t_x @ w ) @ ( resid @ tx @ w2 ) ) ) @ ( resid @ ( resid @ u2 @ tx ) @ y2 ) ) ) ) ).
% \<open>(u \ tx) \ y \<sim> ((u \ tx) \ (w \ tx)) \ ((t'x' \ w') \ (tx \ w))\<close>
thf(fact_18__092_060open_062_I_Iu_A_092_Atx_J_A_092_A_Iw_A_092_Atx_J_J_A_092_A_I_It_Hx_H_A_092_Aw_H_J_A_092_A_Itx_A_092_Aw_J_J_A_092_060sim_062_A_Iu_A_092_Aw_J_A_092_Az_092_060close_062,axiom,
( ( ide_a @ resid @ ( resid @ ( resid @ ( resid @ ( resid @ u2 @ tx ) @ ( resid @ w2 @ tx ) ) @ ( resid @ ( resid @ t_x @ w ) @ ( resid @ tx @ w2 ) ) ) @ ( resid @ ( resid @ u2 @ w2 ) @ z2 ) ) )
& ( ide_a @ resid @ ( resid @ ( resid @ ( resid @ u2 @ w2 ) @ z2 ) @ ( resid @ ( resid @ ( resid @ u2 @ tx ) @ ( resid @ w2 @ tx ) ) @ ( resid @ ( resid @ t_x @ w ) @ ( resid @ tx @ w2 ) ) ) ) ) ) ).
% \<open>((u \ tx) \ (w \ tx)) \ ((t'x' \ w') \ (tx \ w)) \<sim> (u \ w) \ z\<close>
thf(fact_19_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_20_R_Ocon__sym,axiom,
! [T: a,U: a] :
( ( con_a @ resid @ T @ U )
=> ( con_a @ resid @ U @ T ) ) ).
% R.con_sym
thf(fact_21_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_22_R_Oresid__ide__arr,axiom,
! [A: a,T: a] :
( ( ide_a @ resid @ A )
=> ( ( con_a @ resid @ A @ T )
=> ( ide_a @ resid @ ( resid @ A @ T ) ) ) ) ).
% R.resid_ide_arr
thf(fact_23_R_Oresid__arr__ide,axiom,
! [A: a,T: a] :
( ( ide_a @ resid @ A )
=> ( ( con_a @ resid @ T @ A )
=> ( ( resid @ T @ A )
= T ) ) ) ).
% R.resid_arr_ide
thf(fact_24_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_25_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_26_R_Oide__def,axiom,
! [A: a] :
( ( ide_a @ resid @ A )
= ( ( con_a @ resid @ A @ A )
& ( ( resid @ A @ A )
= A ) ) ) ).
% R.ide_def
thf(fact_27_R_OideE,axiom,
! [A: a] :
( ( ide_a @ resid @ A )
=> ~ ( ( con_a @ resid @ A @ A )
=> ( ( resid @ A @ A )
!= A ) ) ) ).
% R.ideE
thf(fact_28_R_Ocon__transitive__on__ide,axiom,
! [A: a,B: a,C: a] :
( ( ide_a @ resid @ A )
=> ( ( ide_a @ resid @ B )
=> ( ( ide_a @ resid @ C )
=> ( ( con_a @ resid @ A @ B )
=> ( ( con_a @ resid @ B @ C )
=> ( con_a @ resid @ A @ C ) ) ) ) ) ) ).
% R.con_transitive_on_ide
thf(fact_29_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_30_R_Ocon__imp__coinitial__ax,axiom,
! [T: a,U: a] :
( ( con_a @ resid @ T @ U )
=> ? [A3: a] :
( ( ide_a @ resid @ A3 )
& ( con_a @ resid @ A3 @ T )
& ( con_a @ resid @ A3 @ U ) ) ) ).
% R.con_imp_coinitial_ax
thf(fact_31_R_Ocong__subst__right_I1_J,axiom,
! [U: a,U2: a,T: a] :
( ( ( ide_a @ resid @ ( resid @ U @ U2 ) )
& ( ide_a @ resid @ ( resid @ U2 @ U ) ) )
=> ( ( con_a @ resid @ T @ U )
=> ( con_a @ resid @ T @ U2 ) ) ) ).
% R.cong_subst_right(1)
thf(fact_32_R_Ocong__subst__right_I2_J,axiom,
! [U: a,U2: a,T: a] :
( ( ( ide_a @ resid @ ( resid @ U @ U2 ) )
& ( ide_a @ resid @ ( resid @ U2 @ U ) ) )
=> ( ( con_a @ resid @ T @ U )
=> ( ( ide_a @ resid @ ( resid @ ( resid @ T @ U ) @ ( resid @ T @ U2 ) ) )
& ( ide_a @ resid @ ( resid @ ( resid @ T @ U2 ) @ ( resid @ T @ U ) ) ) ) ) ) ).
% R.cong_subst_right(2)
thf(fact_33_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_34_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_35_R_Ojoin__of__resid,axiom,
! [T: a,U: a,W: a,V: a] :
( ( join_of_a @ resid @ T @ U @ W )
=> ( ( con_a @ resid @ V @ W )
=> ( join_of_a @ resid @ ( resid @ T @ V ) @ ( resid @ U @ V ) @ ( resid @ W @ V ) ) ) ) ).
% R.join_of_resid
thf(fact_36_R_Ocon__with__join__of__iff_I1_J,axiom,
! [T: a,U: a,W: a,V: a] :
( ( join_of_a @ resid @ T @ U @ W )
=> ( ( ( con_a @ resid @ U @ V )
& ( con_a @ resid @ ( resid @ V @ U ) @ ( resid @ T @ U ) ) )
=> ( con_a @ resid @ W @ V ) ) ) ).
% R.con_with_join_of_iff(1)
thf(fact_37_R_Ocon__with__join__of__iff_I2_J,axiom,
! [T: a,U: a,W: a,V: a] :
( ( join_of_a @ resid @ T @ U @ W )
=> ( ( con_a @ resid @ W @ V )
=> ( ( con_a @ resid @ T @ V )
& ( con_a @ resid @ ( resid @ V @ T ) @ ( resid @ U @ T ) ) ) ) ) ).
% R.con_with_join_of_iff(2)
thf(fact_38_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_39_R_OideI,axiom,
! [A: a] :
( ( con_a @ resid @ A @ A )
=> ( ( ( resid @ A @ A )
= A )
=> ( ide_a @ resid @ A ) ) ) ).
% R.ideI
thf(fact_40_z,axiom,
composite_of_a @ resid @ ( resid @ tx @ w2 ) @ ( resid @ ( resid @ t_x @ w ) @ ( resid @ tx @ w2 ) ) @ z2 ).
% z
thf(fact_41_y__comp,axiom,
composite_of_a @ resid @ ( resid @ w2 @ tx ) @ ( resid @ ( resid @ t_x @ w ) @ ( resid @ tx @ w2 ) ) @ y2 ).
% y_comp
thf(fact_42_residuation_Oaxioms_I1_J,axiom,
! [Resid: a > a > a] :
( ( residuation_a @ Resid )
=> ( partial_magma_a @ Resid ) ) ).
% residuation.axioms(1)
thf(fact_43_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_44_rts_Oseq_Ocong,axiom,
seq_a = seq_a ).
% rts.seq.cong
thf(fact_45_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_46_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_47_mem__Collect__eq,axiom,
! [A: product_prod_a_a,P: product_prod_a_a > $o] :
( ( member1426531477525435216od_a_a @ A @ ( collec3336397797384452498od_a_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_48_mem__Collect__eq,axiom,
! [A: a > a,P: ( a > a ) > $o] :
( ( member_a_a @ A @ ( collect_a_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_49_mem__Collect__eq,axiom,
! [A: set_a,P: set_a > $o] :
( ( member_set_a @ A @ ( collect_set_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_50_mem__Collect__eq,axiom,
! [A: a,P: a > $o] :
( ( member_a @ A @ ( collect_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_51_Collect__mem__eq,axiom,
! [A4: set_Product_prod_a_a] :
( ( collec3336397797384452498od_a_a
@ ^ [X2: product_prod_a_a] : ( member1426531477525435216od_a_a @ X2 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_52_Collect__mem__eq,axiom,
! [A4: set_a_a] :
( ( collect_a_a
@ ^ [X2: a > a] : ( member_a_a @ X2 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_53_Collect__mem__eq,axiom,
! [A4: set_set_a] :
( ( collect_set_a
@ ^ [X2: set_a] : ( member_set_a @ X2 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_54_Collect__mem__eq,axiom,
! [A4: set_a] :
( ( collect_a
@ ^ [X2: a] : ( member_a @ X2 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_55_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_56_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_57_rts_Ojoin__of_Ocong,axiom,
join_of_set_a = join_of_set_a ).
% rts.join_of.cong
thf(fact_58_rts_Ojoin__of_Ocong,axiom,
join_of_a = join_of_a ).
% rts.join_of.cong
thf(fact_59_rts_Ocoinitial_Ocong,axiom,
coinitial_a = coinitial_a ).
% rts.coinitial.cong
thf(fact_60_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_61_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_62_residuation_Ocon_Ocong,axiom,
con_set_a = con_set_a ).
% residuation.con.cong
thf(fact_63_residuation_Ocon_Ocong,axiom,
con_a = con_a ).
% residuation.con.cong
thf(fact_64_rts_Ocoterminal_Ocong,axiom,
coterminal_a = coterminal_a ).
% rts.coterminal.cong
thf(fact_65_partial__magma__def,axiom,
( partial_magma_a
= ( ^ [OP: a > a > a] :
? [X2: a] :
( ! [T5: a] :
( ( ( OP @ X2 @ T5 )
= X2 )
& ( ( OP @ T5 @ X2 )
= X2 ) )
& ! [Y2: a] :
( ! [T5: a] :
( ( ( OP @ Y2 @ T5 )
= Y2 )
& ( ( OP @ T5 @ Y2 )
= Y2 ) )
=> ( Y2 = X2 ) ) ) ) ) ).
% partial_magma_def
thf(fact_66_partial__magma__def,axiom,
( partial_magma_set_a
= ( ^ [OP: set_a > set_a > set_a] :
? [X2: set_a] :
( ! [T5: set_a] :
( ( ( OP @ X2 @ T5 )
= X2 )
& ( ( OP @ T5 @ X2 )
= X2 ) )
& ! [Y2: set_a] :
( ! [T5: set_a] :
( ( ( OP @ Y2 @ T5 )
= Y2 )
& ( ( OP @ T5 @ Y2 )
= Y2 ) )
=> ( Y2 = X2 ) ) ) ) ) ).
% partial_magma_def
thf(fact_67_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_68_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_69_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_70_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_71_residuation_Oide__def,axiom,
! [Resid: set_a > set_a > set_a,A: set_a] :
( ( residuation_set_a @ Resid )
=> ( ( ide_set_a @ Resid @ A )
= ( ( con_set_a @ Resid @ A @ A )
& ( ( Resid @ A @ A )
= A ) ) ) ) ).
% residuation.ide_def
thf(fact_72_residuation_Oide__def,axiom,
! [Resid: a > a > a,A: a] :
( ( residuation_a @ Resid )
=> ( ( ide_a @ Resid @ A )
= ( ( con_a @ Resid @ A @ A )
& ( ( Resid @ A @ A )
= A ) ) ) ) ).
% residuation.ide_def
thf(fact_73_residuation_OideI,axiom,
! [Resid: set_a > set_a > set_a,A: set_a] :
( ( residuation_set_a @ Resid )
=> ( ( con_set_a @ Resid @ A @ A )
=> ( ( ( Resid @ A @ A )
= A )
=> ( ide_set_a @ Resid @ A ) ) ) ) ).
% residuation.ideI
thf(fact_74_residuation_OideI,axiom,
! [Resid: a > a > a,A: a] :
( ( residuation_a @ Resid )
=> ( ( con_a @ Resid @ A @ A )
=> ( ( ( Resid @ A @ A )
= A )
=> ( ide_a @ Resid @ A ) ) ) ) ).
% residuation.ideI
thf(fact_75_residuation_OideE,axiom,
! [Resid: set_a > set_a > set_a,A: set_a] :
( ( residuation_set_a @ Resid )
=> ( ( ide_set_a @ Resid @ A )
=> ~ ( ( con_set_a @ Resid @ A @ A )
=> ( ( Resid @ A @ A )
!= A ) ) ) ) ).
% residuation.ideE
thf(fact_76_residuation_OideE,axiom,
! [Resid: a > a > a,A: a] :
( ( residuation_a @ Resid )
=> ( ( ide_a @ Resid @ A )
=> ~ ( ( con_a @ Resid @ A @ A )
=> ( ( Resid @ A @ A )
!= A ) ) ) ) ).
% residuation.ideE
thf(fact_77_Con__z__vw_H,axiom,
con_a @ resid @ z2 @ ( resid @ v2 @ w ) ).
% Con_z_vw'
thf(fact_78_y_H,axiom,
composite_of_a @ resid @ ( resid @ w @ t_x ) @ ( resid @ ( resid @ tx @ w2 ) @ ( resid @ t_x @ w ) ) @ y ).
% y'
thf(fact_79__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062y_O_AR_Ocomposite__of_A_Iw_A_092_Atx_J_A_I_It_Hx_H_A_092_Aw_H_J_A_092_A_Itx_A_092_Aw_J_J_Ay_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [Y3: a] :
~ ( composite_of_a @ resid @ ( resid @ w2 @ tx ) @ ( resid @ ( resid @ t_x @ w ) @ ( resid @ tx @ w2 ) ) @ Y3 ) ).
% \<open>\<And>thesis. (\<And>y. R.composite_of (w \ tx) ((t'x' \ w') \ (tx \ w)) y \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_80__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062y_H_O_AR_Ocomposite__of_A_Iw_H_A_092_At_Hx_H_J_A_I_Itx_A_092_Aw_J_A_092_A_It_Hx_H_A_092_Aw_H_J_J_Ay_H_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [Y4: a] :
~ ( composite_of_a @ resid @ ( resid @ w @ t_x ) @ ( resid @ ( resid @ tx @ w2 ) @ ( resid @ t_x @ w ) ) @ Y4 ) ).
% \<open>\<And>thesis. (\<And>y'. R.composite_of (w' \ t'x') ((tx \ w) \ (t'x' \ w')) y' \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_81__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062z_O_AR_Ocomposite__of_A_Itx_A_092_Aw_J_A_I_It_Hx_H_A_092_Aw_H_J_A_092_A_Itx_A_092_Aw_J_J_Az_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [Z: a] :
~ ( composite_of_a @ resid @ ( resid @ tx @ w2 ) @ ( resid @ ( resid @ t_x @ w ) @ ( resid @ tx @ w2 ) ) @ Z ) ).
% \<open>\<And>thesis. (\<And>z. R.composite_of (tx \ w) ((t'x' \ w') \ (tx \ w)) z \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_82__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062z_H_O_AR_Ocomposite__of_A_It_Hx_H_A_092_Aw_H_J_A_I_Itx_A_092_Aw_J_A_092_A_It_Hx_H_A_092_Aw_H_J_J_Az_H_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [Z2: a] :
~ ( composite_of_a @ resid @ ( resid @ t_x @ w ) @ ( resid @ ( resid @ tx @ w2 ) @ ( resid @ t_x @ w ) ) @ Z2 ) ).
% \<open>\<And>thesis. (\<And>z'. R.composite_of (t'x' \ w') ((tx \ w) \ (t'x' \ w')) z' \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_83_z_H,axiom,
composite_of_a @ resid @ ( resid @ t_x @ w ) @ ( resid @ ( resid @ tx @ w2 ) @ ( resid @ t_x @ w ) ) @ z ).
% z'
thf(fact_84_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_85_R_Ojoinable__def,axiom,
! [T: a,U: a] :
( ( joinable_a @ resid @ T @ U )
= ( ? [X4: a] : ( join_of_a @ resid @ T @ U @ X4 ) ) ) ).
% R.joinable_def
thf(fact_86_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_87_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_88__C2_C,axiom,
( ( member_a @ ( resid @ ( resid @ tx @ w2 ) @ ( resid @ t_x @ w ) ) @ nn )
& ( member_a @ ( resid @ ( resid @ t_x @ w ) @ ( resid @ tx @ w2 ) ) @ nn ) ) ).
% "2"
thf(fact_89_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_90_N_OCong_092_060_094sub_0620__transitive,axiom,
! [T: a,T4: a,T6: a] :
( ( ( member_a @ ( resid @ T @ T4 ) @ nn )
& ( member_a @ ( resid @ T4 @ T ) @ nn ) )
=> ( ( ( member_a @ ( resid @ T4 @ T6 ) @ nn )
& ( member_a @ ( resid @ T6 @ T4 ) @ nn ) )
=> ( ( member_a @ ( resid @ T @ T6 ) @ nn )
& ( member_a @ ( resid @ T6 @ T ) @ nn ) ) ) ) ).
% N.Cong\<^sub>0_transitive
thf(fact_91_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_92_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_93_R_Ocon__prfx__composite__of_I2_J,axiom,
! [T: a,U: a,W: a,V: a] :
( ( composite_of_a @ resid @ T @ U @ W )
=> ( ( con_a @ resid @ W @ V )
=> ( con_a @ resid @ T @ V ) ) ) ).
% R.con_prfx_composite_of(2)
thf(fact_94_R_Ocon__prfx__composite__of_I1_J,axiom,
! [T: a,U: a,W: a] :
( ( composite_of_a @ resid @ T @ U @ W )
=> ( con_a @ resid @ T @ W ) ) ).
% R.con_prfx_composite_of(1)
thf(fact_95_R_Oresid__composite__of_I4_J,axiom,
! [T: a,U: a,W: a,V: a] :
( ( composite_of_a @ resid @ T @ U @ W )
=> ( ( con_a @ resid @ W @ V )
=> ( composite_of_a @ resid @ ( resid @ T @ V ) @ ( resid @ U @ ( resid @ V @ T ) ) @ ( resid @ W @ V ) ) ) ) ).
% R.resid_composite_of(4)
thf(fact_96_R_Oresid__composite__of_I2_J,axiom,
! [T: a,U: a,W: a,V: a] :
( ( composite_of_a @ resid @ T @ U @ W )
=> ( ( con_a @ resid @ W @ V )
=> ( con_a @ resid @ ( resid @ V @ T ) @ U ) ) ) ).
% R.resid_composite_of(2)
thf(fact_97_R_Oresid__composite__of_I1_J,axiom,
! [T: a,U: a,W: a,V: a] :
( ( composite_of_a @ resid @ T @ U @ W )
=> ( ( con_a @ resid @ W @ V )
=> ( con_a @ resid @ ( resid @ V @ T ) @ ( resid @ W @ T ) ) ) ) ).
% R.resid_composite_of(1)
thf(fact_98_R_Obounded__imp__con,axiom,
! [T: a,U: a,V: a,T4: a,U2: a] :
( ( composite_of_a @ resid @ T @ U @ V )
=> ( ( composite_of_a @ resid @ T4 @ U2 @ V )
=> ( con_a @ resid @ T @ T4 ) ) ) ).
% R.bounded_imp_con
thf(fact_99_R_Ocon__composite__of__iff,axiom,
! [T: a,U: a,V: a,W: a] :
( ( composite_of_a @ resid @ T @ U @ V )
=> ( ( con_a @ resid @ W @ V )
= ( con_a @ resid @ ( resid @ W @ T ) @ U ) ) ) ).
% R.con_composite_of_iff
thf(fact_100_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_101_R_Ocomposite__of__cancel__left,axiom,
! [T: a,U: a,V: a,U2: a] :
( ( composite_of_a @ resid @ T @ U @ V )
=> ( ( composite_of_a @ resid @ T @ U2 @ V )
=> ( ( ide_a @ resid @ ( resid @ U @ U2 ) )
& ( ide_a @ resid @ ( resid @ U2 @ U ) ) ) ) ) ).
% R.composite_of_cancel_left
thf(fact_102_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_103_R_Ocomposite__of__ide__self,axiom,
! [A: a] :
( ( ide_a @ resid @ A )
=> ( composite_of_a @ resid @ A @ A @ A ) ) ).
% R.composite_of_ide_self
thf(fact_104_R_Ocomposite__of__unq__upto__cong,axiom,
! [U: a,T: a,V: a,V2: a] :
( ( composite_of_a @ resid @ U @ T @ V )
=> ( ( composite_of_a @ resid @ U @ T @ V2 )
=> ( ( ide_a @ resid @ ( resid @ V @ V2 ) )
& ( ide_a @ resid @ ( resid @ V2 @ V ) ) ) ) ) ).
% R.composite_of_unq_upto_cong
thf(fact_105_y__in__normal,axiom,
member_a @ y2 @ nn ).
% y_in_normal
thf(fact_106_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_107_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_108_quotient__by__coherent__normal__axioms,axiom,
quotie3282664541148387094rmal_a @ resid @ nn ).
% quotient_by_coherent_normal_axioms
thf(fact_109_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_110_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_111_N_OCong_092_060_094sub_0620__subst__right_I2_J,axiom,
! [U: a,U2: a,T: a] :
( ( ( member_a @ ( resid @ U @ U2 ) @ nn )
& ( member_a @ ( resid @ U2 @ U ) @ nn ) )
=> ( ( con_a @ resid @ T @ U )
=> ( ( member_a @ ( resid @ ( resid @ ( resid @ T @ U ) @ ( resid @ U2 @ U ) ) @ ( resid @ ( resid @ T @ U2 ) @ ( resid @ U @ U2 ) ) ) @ nn )
& ( member_a @ ( resid @ ( resid @ ( resid @ T @ U2 ) @ ( resid @ U @ U2 ) ) @ ( resid @ ( resid @ T @ U ) @ ( resid @ U2 @ U ) ) ) @ nn ) ) ) ) ).
% N.Cong\<^sub>0_subst_right(2)
thf(fact_112_N_OCong_092_060_094sub_0620__subst__right_I1_J,axiom,
! [U: a,U2: a,T: a] :
( ( ( member_a @ ( resid @ U @ U2 ) @ nn )
& ( member_a @ ( resid @ U2 @ U ) @ nn ) )
=> ( ( con_a @ resid @ T @ U )
=> ( con_a @ resid @ T @ U2 ) ) ) ).
% N.Cong\<^sub>0_subst_right(1)
thf(fact_113_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_114_N_OCong_092_060_094sub_0620__subst__Con,axiom,
! [T: a,T4: a,U: a,U2: a] :
( ( ( member_a @ ( resid @ T @ T4 ) @ nn )
& ( member_a @ ( resid @ T4 @ T ) @ nn ) )
=> ( ( ( member_a @ ( resid @ U @ U2 ) @ nn )
& ( member_a @ ( resid @ U2 @ U ) @ nn ) )
=> ( ( con_a @ resid @ T @ U )
= ( con_a @ resid @ T4 @ U2 ) ) ) ) ).
% N.Cong\<^sub>0_subst_Con
thf(fact_115_N_Oide__closed,axiom,
! [A: a] :
( ( ide_a @ resid @ A )
=> ( member_a @ A @ nn ) ) ).
% N.ide_closed
thf(fact_116_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_117_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_118_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_119_N_OCong_092_060_094sub_0620__cancel__left,axiom,
! [T: a,U: a,V: a,U2: a,V2: a] :
( ( composite_of_a @ resid @ T @ U @ V )
=> ( ( composite_of_a @ resid @ T @ U2 @ V2 )
=> ( ( ( member_a @ ( resid @ V @ V2 ) @ nn )
& ( member_a @ ( resid @ V2 @ V ) @ nn ) )
=> ( ( member_a @ ( resid @ U @ U2 ) @ nn )
& ( member_a @ ( resid @ U2 @ U ) @ nn ) ) ) ) ) ).
% N.Cong\<^sub>0_cancel_left
thf(fact_120_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_121_N_OCong_092_060_094sub_0620__iff,axiom,
! [T: a,T4: a] :
( ( ( member_a @ ( resid @ T @ T4 ) @ nn )
& ( member_a @ ( resid @ T4 @ T ) @ nn ) )
= ( ? [U3: a,U4: a,V3: a,V4: a] :
( ( member_a @ U3 @ nn )
& ( member_a @ U4 @ nn )
& ( member_a @ ( resid @ V3 @ V4 ) @ nn )
& ( member_a @ ( resid @ V4 @ V3 ) @ nn )
& ( composite_of_a @ resid @ T @ U3 @ V3 )
& ( composite_of_a @ resid @ T4 @ U4 @ V4 ) ) ) ) ).
% N.Cong\<^sub>0_iff
thf(fact_122_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_123_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_124_R_Ocomposable__def,axiom,
! [T: a,U: a] :
( ( composable_a @ resid @ T @ U )
= ( ? [X4: a] : ( composite_of_a @ resid @ T @ U @ X4 ) ) ) ).
% R.composable_def
thf(fact_125_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_126_R_Oresid__composite__of_I3_J,axiom,
! [T: a,U: a,W: a,V: a] :
( ( composite_of_a @ resid @ T @ U @ W )
=> ( ( con_a @ resid @ W @ V )
=> ( ( ide_a @ resid @ ( resid @ ( resid @ V @ W ) @ ( resid @ ( resid @ V @ T ) @ U ) ) )
& ( ide_a @ resid @ ( resid @ ( resid @ ( resid @ V @ T ) @ U ) @ ( resid @ V @ W ) ) ) ) ) ) ).
% R.resid_composite_of(3)
thf(fact_127_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_128_R_Ocomposite__of__ide__arr,axiom,
! [A: a,T: a] :
( ( ide_a @ resid @ A )
=> ( ( composite_of_a @ resid @ A @ T @ T )
= ( con_a @ resid @ T @ A ) ) ) ).
% R.composite_of_ide_arr
thf(fact_129_N_Odiamond__commutes__upto__Cong_092_060_094sub_0620,axiom,
! [T: a,U: a,V: a,V2: a] :
( ( con_a @ resid @ T @ U )
=> ( ( composite_of_a @ resid @ T @ ( resid @ U @ T ) @ V )
=> ( ( composite_of_a @ resid @ U @ ( resid @ T @ U ) @ V2 )
=> ( ( member_a @ ( resid @ V @ V2 ) @ nn )
& ( member_a @ ( resid @ V2 @ V ) @ nn ) ) ) ) ) ).
% N.diamond_commutes_upto_Cong\<^sub>0
thf(fact_130_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_131_N_Ocomposite__closed__left,axiom,
! [U: a,T: a] :
( ( member_a @ U @ nn )
=> ( ( seq_a @ resid @ U @ T )
=> ? [X_1: a] : ( composite_of_a @ resid @ U @ T @ X_1 ) ) ) ).
% N.composite_closed_left
thf(fact_132_N_Ocomposite__closed__right,axiom,
! [U: a,T: a] :
( ( member_a @ U @ nn )
=> ( ( seq_a @ resid @ T @ U )
=> ? [X_1: a] : ( composite_of_a @ resid @ T @ U @ X_1 ) ) ) ).
% N.composite_closed_right
thf(fact_133__C3_C,axiom,
( ( member_a @ ( resid @ z2 @ z ) @ nn )
& ( member_a @ ( resid @ z @ z2 ) @ nn ) ) ).
% "3"
thf(fact_134_Con__z_H__vw_H,axiom,
con_a @ resid @ z @ ( resid @ v2 @ w ) ).
% Con_z'_vw'
thf(fact_135_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_136_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_137_rts_Ocomposite__of_Ocong,axiom,
composite_of_set_a = composite_of_set_a ).
% rts.composite_of.cong
thf(fact_138_rts_Ocomposite__of_Ocong,axiom,
composite_of_a = composite_of_a ).
% rts.composite_of.cong
thf(fact_139_rts_Ocomposable_Ocong,axiom,
composable_set_a = composable_set_a ).
% rts.composable.cong
thf(fact_140_rts_Ocomposable_Ocong,axiom,
composable_a = composable_a ).
% rts.composable.cong
thf(fact_141_rts_Ojoinable_Ocong,axiom,
joinable_set_a = joinable_set_a ).
% rts.joinable.cong
thf(fact_142_rts_Ojoinable_Ocong,axiom,
joinable_a = joinable_a ).
% rts.joinable.cong
thf(fact_143_N_Ocoherent__normal__sub__rts__axioms,axiom,
cohere6072184133013167079_rts_a @ resid @ nn ).
% N.coherent_normal_sub_rts_axioms
thf(fact_144__092_060open_062R_Otargets_Az_A_061_AR_Otargets_Az_H_092_060close_062,axiom,
( ( targets_a @ resid @ z2 )
= ( targets_a @ resid @ z ) ) ).
% \<open>R.targets z = R.targets z'\<close>
thf(fact_145_t_Hx_H,axiom,
composite_of_a @ resid @ t2 @ x @ t_x ).
% t'x'
thf(fact_146_tx,axiom,
composite_of_a @ resid @ t3 @ x2 @ tx ).
% tx
thf(fact_147_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_148_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_149_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_150_N_OCong_H__if,axiom,
! [U: a,U2: a,T: a,T4: a] :
( ( member_a @ U @ nn )
=> ( ( member_a @ U2 @ nn )
=> ( ( ( member_a @ ( resid @ ( resid @ T @ U ) @ ( resid @ T4 @ U2 ) ) @ nn )
& ( member_a @ ( resid @ ( resid @ T4 @ U2 ) @ ( resid @ T @ U ) ) @ nn ) )
=> ( normal_sub_Cong_a2 @ resid @ nn @ T @ T4 ) ) ) ) ).
% N.Cong'_if
thf(fact_151_ww_H,axiom,
( ( member_a @ w2 @ nn )
& ( member_a @ w @ nn )
& ( member_a @ ( resid @ ( resid @ t3 @ w2 ) @ ( resid @ t2 @ w ) ) @ nn )
& ( member_a @ ( resid @ ( resid @ t2 @ w ) @ ( resid @ t3 @ w2 ) ) @ nn ) ) ).
% ww'
thf(fact_152_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_153_R_Otrg__def,axiom,
! [T: a] :
( ( trg_a @ resid @ T )
= ( resid @ T @ T ) ) ).
% R.trg_def
thf(fact_154_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_155_R_Otargets__are__con,axiom,
! [B: a,T: a,B2: a] :
( ( member_a @ B @ ( targets_a @ resid @ T ) )
=> ( ( member_a @ B2 @ ( targets_a @ resid @ T ) )
=> ( con_a @ resid @ B @ B2 ) ) ) ).
% R.targets_are_con
thf(fact_156_R_Otargets__cong__closed,axiom,
! [B: a,T: a,B2: a] :
( ( member_a @ B @ ( targets_a @ resid @ T ) )
=> ( ( ( ide_a @ resid @ ( resid @ B @ B2 ) )
& ( ide_a @ resid @ ( resid @ B2 @ B ) ) )
=> ( member_a @ B2 @ ( targets_a @ resid @ T ) ) ) ) ).
% R.targets_cong_closed
thf(fact_157_R_Otargets__are__cong,axiom,
! [B: a,T: a,B2: a] :
( ( member_a @ B @ ( targets_a @ resid @ T ) )
=> ( ( member_a @ B2 @ ( targets_a @ resid @ T ) )
=> ( ( ide_a @ resid @ ( resid @ B @ B2 ) )
& ( ide_a @ resid @ ( resid @ B2 @ B ) ) ) ) ) ).
% R.targets_are_cong
thf(fact_158_R_Otarget__is__ide,axiom,
! [A: a,T: a] :
( ( member_a @ A @ ( targets_a @ resid @ T ) )
=> ( ide_a @ resid @ A ) ) ).
% R.target_is_ide
thf(fact_159_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_160_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_161_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_162_R_Otargets__con__closed,axiom,
! [B: a,T: a,B2: a] :
( ( member_a @ B @ ( targets_a @ resid @ T ) )
=> ( ( ide_a @ resid @ B2 )
=> ( ( con_a @ resid @ B @ B2 )
=> ( member_a @ B2 @ ( targets_a @ resid @ T ) ) ) ) ) ).
% R.targets_con_closed
thf(fact_163_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_164__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062tx_O_AR_Ocomposite__of_At_Ax_Atx_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [Tx: a] :
~ ( composite_of_a @ resid @ t3 @ x2 @ Tx ) ).
% \<open>\<And>thesis. (\<And>tx. R.composite_of t x tx \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_165__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062t_Hx_H_O_AR_Ocomposite__of_At_H_Ax_H_At_Hx_H_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [T_x: a] :
~ ( composite_of_a @ resid @ t2 @ x @ T_x ) ).
% \<open>\<And>thesis. (\<And>t'x'. R.composite_of t' x' t'x' \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_166__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062w_Aw_H_O_Aw_A_092_060in_062_A_092_060NN_062_A_092_060and_062_Aw_H_A_092_060in_062_A_092_060NN_062_A_092_060and_062_At_A_092_Aw_A_092_060approx_062_092_060_094sub_0620_At_H_A_092_Aw_H_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [W2: a,W3: a] :
~ ( ( member_a @ W2 @ nn )
& ( member_a @ W3 @ nn )
& ( member_a @ ( resid @ ( resid @ t3 @ W2 ) @ ( resid @ t2 @ W3 ) ) @ nn )
& ( member_a @ ( resid @ ( resid @ t2 @ W3 ) @ ( resid @ t3 @ W2 ) ) @ nn ) ) ).
% \<open>\<And>thesis. (\<And>w w'. w \<in> \<NN> \<and> w' \<in> \<NN> \<and> t \ w \<approx>\<^sub>0 t' \ w' \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_167__092_060open_062R_Oseq_At_Ax_092_060close_062,axiom,
seq_a @ resid @ t3 @ x2 ).
% \<open>R.seq t x\<close>
thf(fact_168__092_060open_062R_Oseq_At_H_Ax_H_092_060close_062,axiom,
seq_a @ resid @ t2 @ x ).
% \<open>R.seq t' x'\<close>
thf(fact_169_xx_H,axiom,
( ( member_a @ x2 @ nn )
& ( member_a @ x @ nn )
& ( con_a @ resid @ ( resid @ ( resid @ u2 @ t3 ) @ x2 ) @ ( resid @ ( resid @ v2 @ t2 ) @ x ) ) ) ).
% xx'
thf(fact_170_w__tx__in___092_060NN_062,axiom,
member_a @ ( resid @ ( resid @ w2 @ t3 ) @ x2 ) @ nn ).
% w_tx_in_\<NN>
thf(fact_171_w_H__t_Hx_H__in___092_060NN_062,axiom,
member_a @ ( resid @ ( resid @ w @ t2 ) @ x ) @ nn ).
% w'_t'x'_in_\<NN>
thf(fact_172__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062x_Ax_H_O_Ax_A_092_060in_062_A_092_060NN_062_A_092_060and_062_Ax_H_A_092_060in_062_A_092_060NN_062_A_092_060and_062_A_Iu_A_092_At_J_A_092_Ax_A_092_060frown_062_A_Iv_A_092_At_H_J_A_092_Ax_H_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [X: a,X5: a] :
~ ( ( member_a @ X @ nn )
& ( member_a @ X5 @ nn )
& ( con_a @ resid @ ( resid @ ( resid @ u2 @ t3 ) @ X ) @ ( resid @ ( resid @ v2 @ t2 ) @ X5 ) ) ) ).
% \<open>\<And>thesis. (\<And>x x'. x \<in> \<NN> \<and> x' \<in> \<NN> \<and> (u \ t) \ x \<frown> (v \ t') \ x' \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_173_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_174_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_175_rts_Otargets_Ocong,axiom,
targets_a = targets_a ).
% rts.targets.cong
thf(fact_176_residuation_Otrg_Ocong,axiom,
trg_set_a = trg_set_a ).
% residuation.trg.cong
thf(fact_177_residuation_Otrg_Ocong,axiom,
trg_a = trg_a ).
% residuation.trg.cong
thf(fact_178_normal__sub__rts_OCong_H_Ocong,axiom,
normal_sub_Cong_a2 = normal_sub_Cong_a2 ).
% normal_sub_rts.Cong'.cong
thf(fact_179_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_180_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_181_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_182_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_183_coherent__normal__sub__rts_OCong_092_060_094sub_0620__composite__of__arr__normal,axiom,
! [Resid: product_prod_a_a > product_prod_a_a > product_prod_a_a,NN: set_Product_prod_a_a,T: product_prod_a_a,U: product_prod_a_a,T4: product_prod_a_a] :
( ( cohere3906135489564710160od_a_a @ Resid @ NN )
=> ( ( compos6970852536677457805od_a_a @ Resid @ T @ U @ T4 )
=> ( ( member1426531477525435216od_a_a @ U @ NN )
=> ( ( member1426531477525435216od_a_a @ ( Resid @ T4 @ T ) @ NN )
& ( member1426531477525435216od_a_a @ ( Resid @ T @ T4 ) @ NN ) ) ) ) ) ).
% coherent_normal_sub_rts.Cong\<^sub>0_composite_of_arr_normal
thf(fact_184_coherent__normal__sub__rts_OCong_092_060_094sub_0620__composite__of__arr__normal,axiom,
! [Resid: ( a > a ) > ( a > a ) > a > a,NN: set_a_a,T: a > a,U: a > a,T4: a > a] :
( ( cohere4631022276737013564ts_a_a @ Resid @ NN )
=> ( ( composite_of_a_a @ Resid @ T @ U @ T4 )
=> ( ( member_a_a @ U @ NN )
=> ( ( member_a_a @ ( Resid @ T4 @ T ) @ NN )
& ( member_a_a @ ( Resid @ T @ T4 ) @ NN ) ) ) ) ) ).
% coherent_normal_sub_rts.Cong\<^sub>0_composite_of_arr_normal
thf(fact_185_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_186_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_187_tt_H,axiom,
normal_sub_Cong_a @ resid @ nn @ t3 @ t2 ).
% tt'
thf(fact_188_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_189_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_190_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_191_N_Ocoherent_H,axiom,
! [V: a,V2: a,W: a,W4: a,T: a,T4: a] :
( ( member_a @ V @ nn )
=> ( ( member_a @ V2 @ nn )
=> ( ( member_a @ W @ nn )
=> ( ( member_a @ W4 @ nn )
=> ( ( ( sources_a @ resid @ V )
= ( sources_a @ resid @ W ) )
=> ( ( ( sources_a @ resid @ V2 )
= ( sources_a @ resid @ W4 ) )
=> ( ( ( targets_a @ resid @ W )
= ( targets_a @ resid @ W4 ) )
=> ( ( ( member_a @ ( resid @ ( resid @ T @ V ) @ ( resid @ T4 @ V2 ) ) @ nn )
& ( member_a @ ( resid @ ( resid @ T4 @ V2 ) @ ( resid @ T @ V ) ) @ nn ) )
=> ( ( member_a @ ( resid @ ( resid @ T @ W ) @ ( resid @ T4 @ W4 ) ) @ nn )
& ( member_a @ ( resid @ ( resid @ T4 @ W4 ) @ ( resid @ T @ W ) ) @ nn ) ) ) ) ) ) ) ) ) ) ).
% N.coherent'
thf(fact_192_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_193_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_194_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_195_R_Oide__trg,axiom,
! [T: a] :
( ( arr_a @ resid @ T )
=> ( ide_a @ resid @ ( trg_a @ resid @ T ) ) ) ).
% R.ide_trg
thf(fact_196_R_Osources__are__con,axiom,
! [A: a,T: a,A2: a] :
( ( member_a @ A @ ( sources_a @ resid @ T ) )
=> ( ( member_a @ A2 @ ( sources_a @ resid @ T ) )
=> ( con_a @ resid @ A @ A2 ) ) ) ).
% R.sources_are_con
thf(fact_197_R_Osource__is__ide,axiom,
! [A: a,T: a] :
( ( member_a @ A @ ( sources_a @ resid @ T ) )
=> ( ide_a @ resid @ A ) ) ).
% R.source_is_ide
thf(fact_198_R_Osources__are__cong,axiom,
! [A: a,T: a,A2: a] :
( ( member_a @ A @ ( sources_a @ resid @ T ) )
=> ( ( member_a @ A2 @ ( sources_a @ resid @ T ) )
=> ( ( ide_a @ resid @ ( resid @ A @ A2 ) )
& ( ide_a @ resid @ ( resid @ A2 @ A ) ) ) ) ) ).
% R.sources_are_cong
thf(fact_199_R_Osources__cong__closed,axiom,
! [A: a,T: a,A2: a] :
( ( member_a @ A @ ( sources_a @ resid @ T ) )
=> ( ( ( ide_a @ resid @ ( resid @ A @ A2 ) )
& ( ide_a @ resid @ ( resid @ A2 @ A ) ) )
=> ( member_a @ A2 @ ( sources_a @ resid @ T ) ) ) ) ).
% R.sources_cong_closed
thf(fact_200_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_201_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_202_R_OarrE,axiom,
! [T: a] :
( ( arr_a @ resid @ T )
=> ( con_a @ resid @ T @ T ) ) ).
% R.arrE
thf(fact_203_R_Oarr__def,axiom,
! [T: a] :
( ( arr_a @ resid @ T )
= ( con_a @ resid @ T @ T ) ) ).
% R.arr_def
thf(fact_204_R_Oarr__resid,axiom,
! [T: a,U: a] :
( ( con_a @ resid @ T @ U )
=> ( arr_a @ resid @ ( resid @ T @ U ) ) ) ).
% R.arr_resid
thf(fact_205_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_206_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_207_R_Oide__implies__arr,axiom,
! [A: a] :
( ( ide_a @ resid @ A )
=> ( arr_a @ resid @ A ) ) ).
% R.ide_implies_arr
thf(fact_208_R_Oprfx__reflexive,axiom,
! [T: a] :
( ( arr_a @ resid @ T )
=> ( ide_a @ resid @ ( resid @ T @ T ) ) ) ).
% R.prfx_reflexive
thf(fact_209_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_210_R_Oresid__source__in__targets,axiom,
! [A: a,T: a] :
( ( member_a @ A @ ( sources_a @ resid @ T ) )
=> ( member_a @ ( resid @ A @ T ) @ ( targets_a @ resid @ T ) ) ) ).
% R.resid_source_in_targets
thf(fact_211_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_212_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_213_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_214_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_215_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_216_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_217_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_218_N_Oelements__are__arr,axiom,
! [T: a] :
( ( member_a @ T @ nn )
=> ( arr_a @ resid @ T ) ) ).
% N.elements_are_arr
thf(fact_219_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_220_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_221_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_222_N_OCongE,axiom,
! [T: a,T4: a] :
( ( normal_sub_Cong_a @ resid @ nn @ T @ T4 )
=> ~ ! [U5: a] :
( ( member_a @ U5 @ nn )
=> ! [U6: a] :
( ( 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.CongE
thf(fact_223_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_224_N_OCong__def,axiom,
! [T: a,T4: a] :
( ( normal_sub_Cong_a @ resid @ nn @ T @ T4 )
= ( ? [U3: a,U4: a] :
( ( member_a @ U3 @ nn )
& ( 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.Cong_def
thf(fact_225_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_226_N_OCong__transitive,axiom,
! [T: a,T6: a,T4: a] :
( ( normal_sub_Cong_a @ resid @ nn @ T @ T6 )
=> ( ( normal_sub_Cong_a @ resid @ nn @ T6 @ T4 )
=> ( normal_sub_Cong_a @ resid @ nn @ T @ T4 ) ) ) ).
% N.Cong_transitive
thf(fact_227_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_228_R_OcomposableD_I2_J,axiom,
! [T: a,U: a] :
( ( composable_a @ resid @ T @ U )
=> ( arr_a @ resid @ U ) ) ).
% R.composableD(2)
thf(fact_229_R_OcomposableD_I1_J,axiom,
! [T: a,U: a] :
( ( composable_a @ resid @ T @ U )
=> ( arr_a @ resid @ T ) ) ).
% R.composableD(1)
thf(fact_230_R_Oin__sourcesE,axiom,
! [A: a,T: a] :
( ( member_a @ A @ ( sources_a @ resid @ T ) )
=> ~ ( ( ide_a @ resid @ A )
=> ~ ( con_a @ resid @ T @ A ) ) ) ).
% R.in_sourcesE
thf(fact_231_R_Osources__con__closed,axiom,
! [A: a,T: a,A2: a] :
( ( member_a @ A @ ( sources_a @ resid @ T ) )
=> ( ( ide_a @ resid @ A2 )
=> ( ( con_a @ resid @ A @ A2 )
=> ( member_a @ A2 @ ( sources_a @ resid @ T ) ) ) ) ) ).
% R.sources_con_closed
thf(fact_232_R_Ocomposite__of__source__arr,axiom,
! [T: a,A: a] :
( ( arr_a @ resid @ T )
=> ( ( member_a @ A @ ( sources_a @ resid @ T ) )
=> ( composite_of_a @ resid @ A @ T @ T ) ) ) ).
% R.composite_of_source_arr
thf(fact_233_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_234_R_Ojoin__of__arr__src_I2_J,axiom,
! [T: a,A: a] :
( ( arr_a @ resid @ T )
=> ( ( member_a @ A @ ( sources_a @ resid @ T ) )
=> ( join_of_a @ resid @ T @ A @ T ) ) ) ).
% R.join_of_arr_src(2)
thf(fact_235_R_Ojoin__of__arr__src_I1_J,axiom,
! [T: a,A: a] :
( ( arr_a @ resid @ T )
=> ( ( member_a @ A @ ( sources_a @ resid @ T ) )
=> ( join_of_a @ resid @ A @ T @ T ) ) ) ).
% R.join_of_arr_src(1)
thf(fact_236_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_237_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_238_N_Osources__are__Cong,axiom,
! [A: a,T: a,A2: a] :
( ( member_a @ A @ ( sources_a @ resid @ T ) )
=> ( ( member_a @ A2 @ ( sources_a @ resid @ T ) )
=> ( normal_sub_Cong_a @ resid @ nn @ A @ A2 ) ) ) ).
% N.sources_are_Cong
thf(fact_239_N_Oin__sources__respects__Cong,axiom,
! [T: a,T4: a,A: a,A2: a] :
( ( normal_sub_Cong_a @ resid @ nn @ T @ T4 )
=> ( ( member_a @ A @ ( sources_a @ resid @ T ) )
=> ( ( member_a @ A2 @ ( sources_a @ resid @ T4 ) )
=> ( normal_sub_Cong_a @ resid @ nn @ A @ A2 ) ) ) ) ).
% N.in_sources_respects_Cong
thf(fact_240_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_241_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_242_N_OCong__reflexive,axiom,
! [T: a] :
( ( arr_a @ resid @ T )
=> ( normal_sub_Cong_a @ resid @ nn @ T @ T ) ) ).
% N.Cong_reflexive
thf(fact_243_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_244_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_245_N_Otargets__are__Cong,axiom,
! [B: a,T: a,B2: a] :
( ( member_a @ B @ ( targets_a @ resid @ T ) )
=> ( ( member_a @ B2 @ ( targets_a @ resid @ T ) )
=> ( normal_sub_Cong_a @ resid @ nn @ B @ B2 ) ) ) ).
% N.targets_are_Cong
thf(fact_246_N_Oin__targets__respects__Cong,axiom,
! [T: a,T4: a,B: a,B2: a] :
( ( normal_sub_Cong_a @ resid @ nn @ T @ T4 )
=> ( ( member_a @ B @ ( targets_a @ resid @ T ) )
=> ( ( member_a @ B2 @ ( targets_a @ resid @ T4 ) )
=> ( normal_sub_Cong_a @ resid @ nn @ B @ B2 ) ) ) ) ).
% N.in_targets_respects_Cong
thf(fact_247_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_248_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_249_N_OCong__subst__con,axiom,
! [T: a,U: a,T4: a,U2: a] :
( ( ( sources_a @ resid @ T )
= ( sources_a @ resid @ U ) )
=> ( ( ( sources_a @ resid @ T4 )
= ( sources_a @ resid @ U2 ) )
=> ( ( normal_sub_Cong_a @ resid @ nn @ T @ T4 )
=> ( ( normal_sub_Cong_a @ resid @ nn @ U @ U2 )
=> ( ( con_a @ resid @ T @ U )
= ( con_a @ resid @ T4 @ U2 ) ) ) ) ) ) ).
% N.Cong_subst_con
thf(fact_250_N_OCong__subst_I1_J,axiom,
! [T: a,T4: a,U: a,U2: a] :
( ( normal_sub_Cong_a @ resid @ nn @ T @ T4 )
=> ( ( normal_sub_Cong_a @ resid @ nn @ U @ U2 )
=> ( ( con_a @ resid @ T @ U )
=> ( ( ( sources_a @ resid @ T4 )
= ( sources_a @ resid @ U2 ) )
=> ( con_a @ resid @ T4 @ U2 ) ) ) ) ) ).
% N.Cong_subst(1)
thf(fact_251_N_OCong__subst_I2_J,axiom,
! [T: a,T4: a,U: a,U2: a] :
( ( normal_sub_Cong_a @ resid @ nn @ T @ T4 )
=> ( ( normal_sub_Cong_a @ resid @ nn @ U @ U2 )
=> ( ( con_a @ resid @ T @ U )
=> ( ( ( sources_a @ resid @ T4 )
= ( sources_a @ resid @ U2 ) )
=> ( normal_sub_Cong_a @ resid @ nn @ ( resid @ T @ U ) @ ( resid @ T4 @ U2 ) ) ) ) ) ) ).
% N.Cong_subst(2)
thf(fact_252_N_Ocoherent,axiom,
! [T: a,U: a,U2: a] :
( ( arr_a @ resid @ T )
=> ( ( member_a @ U @ nn )
=> ( ( member_a @ U2 @ nn )
=> ( ( ( sources_a @ resid @ U )
= ( sources_a @ resid @ U2 ) )
=> ( ( ( targets_a @ resid @ U )
= ( targets_a @ resid @ U2 ) )
=> ( ( ( sources_a @ resid @ T )
= ( sources_a @ resid @ U ) )
=> ( ( member_a @ ( resid @ ( resid @ T @ U ) @ ( resid @ T @ U2 ) ) @ nn )
& ( member_a @ ( resid @ ( resid @ T @ U2 ) @ ( resid @ T @ U ) ) @ nn ) ) ) ) ) ) ) ) ).
% N.coherent
thf(fact_253_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_254_N_OCong_H_Osimps,axiom,
! [A1: a,A22: a] :
( ( normal_sub_Cong_a2 @ resid @ nn @ A1 @ A22 )
= ( ? [T5: a,U3: a] :
( ( A1 = U3 )
& ( A22 = T5 )
& ( normal_sub_Cong_a2 @ resid @ nn @ T5 @ U3 ) )
| ? [T5: a,U3: a,V3: a] :
( ( A1 = T5 )
& ( A22 = V3 )
& ( normal_sub_Cong_a2 @ resid @ nn @ T5 @ U3 )
& ( normal_sub_Cong_a2 @ resid @ nn @ U3 @ V3 ) )
| ? [T5: a,U3: a] :
( ( A1 = T5 )
& ( A22 = U3 )
& ( member_a @ ( resid @ T5 @ U3 ) @ nn )
& ( member_a @ ( resid @ U3 @ T5 ) @ nn ) )
| ? [T5: a,U3: a] :
( ( A1 = T5 )
& ( A22
= ( resid @ T5 @ U3 ) )
& ( arr_a @ resid @ T5 )
& ( member_a @ U3 @ nn )
& ( ( sources_a @ resid @ T5 )
= ( sources_a @ resid @ U3 ) ) ) ) ) ).
% N.Cong'.simps
thf(fact_255_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 )
=> ( ! [U5: a] :
( ( normal_sub_Cong_a2 @ resid @ nn @ A1 @ U5 )
=> ~ ( normal_sub_Cong_a2 @ resid @ nn @ U5 @ A22 ) )
=> ( ~ ( ( member_a @ ( resid @ A1 @ A22 ) @ nn )
& ( member_a @ ( resid @ A22 @ A1 ) @ nn ) )
=> ~ ! [U5: a] :
( ( A22
= ( resid @ A1 @ U5 ) )
=> ( ( arr_a @ resid @ A1 )
=> ( ( member_a @ U5 @ nn )
=> ( ( sources_a @ resid @ A1 )
!= ( sources_a @ resid @ U5 ) ) ) ) ) ) ) ) ) ).
% N.Cong'.cases
thf(fact_256_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_257_R_OarrI,axiom,
! [T: a] :
( ( con_a @ resid @ T @ T )
=> ( arr_a @ resid @ T ) ) ).
% R.arrI
thf(fact_258_N_OCongI,axiom,
! [U: a,U2: a,T: a,T4: a] :
( ( member_a @ U @ nn )
=> ( ( member_a @ U2 @ nn )
=> ( ( ( member_a @ ( resid @ ( resid @ T @ U ) @ ( resid @ T4 @ U2 ) ) @ nn )
& ( member_a @ ( resid @ ( resid @ T4 @ U2 ) @ ( resid @ T @ U ) ) @ nn ) )
=> ( normal_sub_Cong_a @ resid @ nn @ T @ T4 ) ) ) ) ).
% N.CongI
thf(fact_259_R_Oin__sourcesI,axiom,
! [A: a,T: a] :
( ( ide_a @ resid @ A )
=> ( ( con_a @ resid @ T @ A )
=> ( member_a @ A @ ( sources_a @ resid @ T ) ) ) ) ).
% R.in_sourcesI
thf(fact_260_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_261_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_262_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_263_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_264_normal__sub__rts_OCong_Ocong,axiom,
normal_sub_Cong_a = normal_sub_Cong_a ).
% normal_sub_rts.Cong.cong
thf(fact_265_residuation_Oarr_Ocong,axiom,
arr_set_a = arr_set_a ).
% residuation.arr.cong
thf(fact_266_residuation_Oarr_Ocong,axiom,
arr_a = arr_a ).
% residuation.arr.cong
thf(fact_267_rts_Osources_Ocong,axiom,
sources_set_a = sources_set_a ).
% rts.sources.cong
thf(fact_268_rts_Osources_Ocong,axiom,
sources_a = sources_a ).
% rts.sources.cong
thf(fact_269_coherent__normal__sub__rts_Ocoherent,axiom,
! [Resid: product_prod_a_a > product_prod_a_a > product_prod_a_a,NN: set_Product_prod_a_a,T: product_prod_a_a,U: product_prod_a_a,U2: product_prod_a_a] :
( ( cohere3906135489564710160od_a_a @ Resid @ NN )
=> ( ( arr_Product_prod_a_a @ Resid @ T )
=> ( ( member1426531477525435216od_a_a @ U @ NN )
=> ( ( member1426531477525435216od_a_a @ U2 @ NN )
=> ( ( ( source6950040787684646355od_a_a @ Resid @ U )
= ( source6950040787684646355od_a_a @ Resid @ U2 ) )
=> ( ( ( target5293506191220573129od_a_a @ Resid @ U )
= ( target5293506191220573129od_a_a @ Resid @ U2 ) )
=> ( ( ( source6950040787684646355od_a_a @ Resid @ T )
= ( source6950040787684646355od_a_a @ Resid @ U ) )
=> ( ( member1426531477525435216od_a_a @ ( Resid @ ( Resid @ T @ U ) @ ( Resid @ T @ U2 ) ) @ NN )
& ( member1426531477525435216od_a_a @ ( Resid @ ( Resid @ T @ U2 ) @ ( Resid @ T @ U ) ) @ NN ) ) ) ) ) ) ) ) ) ).
% coherent_normal_sub_rts.coherent
thf(fact_270_coherent__normal__sub__rts_Ocoherent,axiom,
! [Resid: ( a > a ) > ( a > a ) > a > a,NN: set_a_a,T: a > a,U: a > a,U2: a > a] :
( ( cohere4631022276737013564ts_a_a @ Resid @ NN )
=> ( ( arr_a_a @ Resid @ T )
=> ( ( member_a_a @ U @ NN )
=> ( ( member_a_a @ U2 @ NN )
=> ( ( ( sources_a_a @ Resid @ U )
= ( sources_a_a @ Resid @ U2 ) )
=> ( ( ( targets_a_a @ Resid @ U )
= ( targets_a_a @ Resid @ U2 ) )
=> ( ( ( sources_a_a @ Resid @ T )
= ( sources_a_a @ Resid @ U ) )
=> ( ( member_a_a @ ( Resid @ ( Resid @ T @ U ) @ ( Resid @ T @ U2 ) ) @ NN )
& ( member_a_a @ ( Resid @ ( Resid @ T @ U2 ) @ ( Resid @ T @ U ) ) @ NN ) ) ) ) ) ) ) ) ) ).
% coherent_normal_sub_rts.coherent
thf(fact_271_coherent__normal__sub__rts_Ocoherent,axiom,
! [Resid: set_a > set_a > set_a,NN: set_set_a,T: set_a,U: set_a,U2: set_a] :
( ( cohere6325062230080414023_set_a @ Resid @ NN )
=> ( ( arr_set_a @ Resid @ T )
=> ( ( member_set_a @ U @ NN )
=> ( ( member_set_a @ U2 @ NN )
=> ( ( ( sources_set_a @ Resid @ U )
= ( sources_set_a @ Resid @ U2 ) )
=> ( ( ( targets_set_a @ Resid @ U )
= ( targets_set_a @ Resid @ U2 ) )
=> ( ( ( sources_set_a @ Resid @ T )
= ( sources_set_a @ Resid @ U ) )
=> ( ( member_set_a @ ( Resid @ ( Resid @ T @ U ) @ ( Resid @ T @ U2 ) ) @ NN )
& ( member_set_a @ ( Resid @ ( Resid @ T @ U2 ) @ ( Resid @ T @ U ) ) @ NN ) ) ) ) ) ) ) ) ) ).
% coherent_normal_sub_rts.coherent
thf(fact_272_coherent__normal__sub__rts_Ocoherent,axiom,
! [Resid: a > a > a,NN: set_a,T: a,U: a,U2: a] :
( ( cohere6072184133013167079_rts_a @ Resid @ NN )
=> ( ( arr_a @ Resid @ T )
=> ( ( member_a @ U @ NN )
=> ( ( member_a @ U2 @ NN )
=> ( ( ( sources_a @ Resid @ U )
= ( sources_a @ Resid @ U2 ) )
=> ( ( ( targets_a @ Resid @ U )
= ( targets_a @ Resid @ U2 ) )
=> ( ( ( sources_a @ Resid @ T )
= ( sources_a @ Resid @ U ) )
=> ( ( member_a @ ( Resid @ ( Resid @ T @ U ) @ ( Resid @ T @ U2 ) ) @ NN )
& ( member_a @ ( Resid @ ( Resid @ T @ U2 ) @ ( Resid @ T @ U ) ) @ NN ) ) ) ) ) ) ) ) ) ).
% coherent_normal_sub_rts.coherent
thf(fact_273_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,U2: 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 @ U2 ) )
=> ( ( normal8977612136997397236_set_a @ Resid @ NN @ T @ T4 )
=> ( ( normal8977612136997397236_set_a @ Resid @ NN @ U @ U2 )
=> ( ( con_set_a @ Resid @ T @ U )
= ( con_set_a @ Resid @ T4 @ U2 ) ) ) ) ) ) ) ).
% coherent_normal_sub_rts.Cong_subst_con
thf(fact_274_coherent__normal__sub__rts_OCong__subst__con,axiom,
! [Resid: a > a > a,NN: set_a,T: a,U: a,T4: a,U2: a] :
( ( cohere6072184133013167079_rts_a @ Resid @ NN )
=> ( ( ( sources_a @ Resid @ T )
= ( sources_a @ Resid @ U ) )
=> ( ( ( sources_a @ Resid @ T4 )
= ( sources_a @ Resid @ U2 ) )
=> ( ( normal_sub_Cong_a @ Resid @ NN @ T @ T4 )
=> ( ( normal_sub_Cong_a @ Resid @ NN @ U @ U2 )
=> ( ( con_a @ Resid @ T @ U )
= ( con_a @ Resid @ T4 @ U2 ) ) ) ) ) ) ) ).
% coherent_normal_sub_rts.Cong_subst_con
thf(fact_275_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,U2: set_a] :
( ( cohere6325062230080414023_set_a @ Resid @ NN )
=> ( ( normal8977612136997397236_set_a @ Resid @ NN @ T @ T4 )
=> ( ( normal8977612136997397236_set_a @ Resid @ NN @ U @ U2 )
=> ( ( con_set_a @ Resid @ T @ U )
=> ( ( ( sources_set_a @ Resid @ T4 )
= ( sources_set_a @ Resid @ U2 ) )
=> ( con_set_a @ Resid @ T4 @ U2 ) ) ) ) ) ) ).
% coherent_normal_sub_rts.Cong_subst(1)
thf(fact_276_coherent__normal__sub__rts_OCong__subst_I1_J,axiom,
! [Resid: a > a > a,NN: set_a,T: a,T4: a,U: a,U2: a] :
( ( cohere6072184133013167079_rts_a @ Resid @ NN )
=> ( ( normal_sub_Cong_a @ Resid @ NN @ T @ T4 )
=> ( ( normal_sub_Cong_a @ Resid @ NN @ U @ U2 )
=> ( ( con_a @ Resid @ T @ U )
=> ( ( ( sources_a @ Resid @ T4 )
= ( sources_a @ Resid @ U2 ) )
=> ( con_a @ Resid @ T4 @ U2 ) ) ) ) ) ) ).
% coherent_normal_sub_rts.Cong_subst(1)
thf(fact_277_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,U2: set_a] :
( ( cohere6325062230080414023_set_a @ Resid @ NN )
=> ( ( normal8977612136997397236_set_a @ Resid @ NN @ T @ T4 )
=> ( ( normal8977612136997397236_set_a @ Resid @ NN @ U @ U2 )
=> ( ( con_set_a @ Resid @ T @ U )
=> ( ( ( sources_set_a @ Resid @ T4 )
= ( sources_set_a @ Resid @ U2 ) )
=> ( normal8977612136997397236_set_a @ Resid @ NN @ ( Resid @ T @ U ) @ ( Resid @ T4 @ U2 ) ) ) ) ) ) ) ).
% coherent_normal_sub_rts.Cong_subst(2)
thf(fact_278_coherent__normal__sub__rts_OCong__subst_I2_J,axiom,
! [Resid: a > a > a,NN: set_a,T: a,T4: a,U: a,U2: a] :
( ( cohere6072184133013167079_rts_a @ Resid @ NN )
=> ( ( normal_sub_Cong_a @ Resid @ NN @ T @ T4 )
=> ( ( normal_sub_Cong_a @ Resid @ NN @ U @ U2 )
=> ( ( con_a @ Resid @ T @ U )
=> ( ( ( sources_a @ Resid @ T4 )
= ( sources_a @ Resid @ U2 ) )
=> ( normal_sub_Cong_a @ Resid @ NN @ ( Resid @ T @ U ) @ ( Resid @ T4 @ U2 ) ) ) ) ) ) ) ).
% coherent_normal_sub_rts.Cong_subst(2)
thf(fact_279_residuation_Oide__implies__arr,axiom,
! [Resid: set_a > set_a > set_a,A: set_a] :
( ( residuation_set_a @ Resid )
=> ( ( ide_set_a @ Resid @ A )
=> ( arr_set_a @ Resid @ A ) ) ) ).
% residuation.ide_implies_arr
thf(fact_280_residuation_Oide__implies__arr,axiom,
! [Resid: a > a > a,A: a] :
( ( residuation_a @ Resid )
=> ( ( ide_a @ Resid @ A )
=> ( arr_a @ Resid @ A ) ) ) ).
% residuation.ide_implies_arr
thf(fact_281_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_282_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_283_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_284_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_285_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_286_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_287_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_288_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_289_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_290_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_291_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_292_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_293_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_294_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_295_coherent__normal__sub__rts_Ocoherent_H,axiom,
! [Resid: product_prod_a_a > product_prod_a_a > product_prod_a_a,NN: set_Product_prod_a_a,V: product_prod_a_a,V2: product_prod_a_a,W: product_prod_a_a,W4: product_prod_a_a,T: product_prod_a_a,T4: product_prod_a_a] :
( ( cohere3906135489564710160od_a_a @ Resid @ NN )
=> ( ( member1426531477525435216od_a_a @ V @ NN )
=> ( ( member1426531477525435216od_a_a @ V2 @ NN )
=> ( ( member1426531477525435216od_a_a @ W @ NN )
=> ( ( member1426531477525435216od_a_a @ W4 @ NN )
=> ( ( ( source6950040787684646355od_a_a @ Resid @ V )
= ( source6950040787684646355od_a_a @ Resid @ W ) )
=> ( ( ( source6950040787684646355od_a_a @ Resid @ V2 )
= ( source6950040787684646355od_a_a @ Resid @ W4 ) )
=> ( ( ( target5293506191220573129od_a_a @ Resid @ W )
= ( target5293506191220573129od_a_a @ Resid @ W4 ) )
=> ( ( ( member1426531477525435216od_a_a @ ( Resid @ ( Resid @ T @ V ) @ ( Resid @ T4 @ V2 ) ) @ NN )
& ( member1426531477525435216od_a_a @ ( Resid @ ( Resid @ T4 @ V2 ) @ ( Resid @ T @ V ) ) @ NN ) )
=> ( ( member1426531477525435216od_a_a @ ( Resid @ ( Resid @ T @ W ) @ ( Resid @ T4 @ W4 ) ) @ NN )
& ( member1426531477525435216od_a_a @ ( Resid @ ( Resid @ T4 @ W4 ) @ ( Resid @ T @ W ) ) @ NN ) ) ) ) ) ) ) ) ) ) ) ).
% coherent_normal_sub_rts.coherent'
thf(fact_296_coherent__normal__sub__rts_Ocoherent_H,axiom,
! [Resid: ( a > a ) > ( a > a ) > a > a,NN: set_a_a,V: a > a,V2: a > a,W: a > a,W4: a > a,T: a > a,T4: a > a] :
( ( cohere4631022276737013564ts_a_a @ Resid @ NN )
=> ( ( member_a_a @ V @ NN )
=> ( ( member_a_a @ V2 @ NN )
=> ( ( member_a_a @ W @ NN )
=> ( ( member_a_a @ W4 @ NN )
=> ( ( ( sources_a_a @ Resid @ V )
= ( sources_a_a @ Resid @ W ) )
=> ( ( ( sources_a_a @ Resid @ V2 )
= ( sources_a_a @ Resid @ W4 ) )
=> ( ( ( targets_a_a @ Resid @ W )
= ( targets_a_a @ Resid @ W4 ) )
=> ( ( ( member_a_a @ ( Resid @ ( Resid @ T @ V ) @ ( Resid @ T4 @ V2 ) ) @ NN )
& ( member_a_a @ ( Resid @ ( Resid @ T4 @ V2 ) @ ( Resid @ T @ V ) ) @ NN ) )
=> ( ( member_a_a @ ( Resid @ ( Resid @ T @ W ) @ ( Resid @ T4 @ W4 ) ) @ NN )
& ( member_a_a @ ( Resid @ ( Resid @ T4 @ W4 ) @ ( Resid @ T @ W ) ) @ NN ) ) ) ) ) ) ) ) ) ) ) ).
% coherent_normal_sub_rts.coherent'
thf(fact_297_coherent__normal__sub__rts_Ocoherent_H,axiom,
! [Resid: set_a > set_a > set_a,NN: set_set_a,V: set_a,V2: set_a,W: set_a,W4: set_a,T: set_a,T4: set_a] :
( ( cohere6325062230080414023_set_a @ Resid @ NN )
=> ( ( member_set_a @ V @ NN )
=> ( ( member_set_a @ V2 @ NN )
=> ( ( member_set_a @ W @ NN )
=> ( ( member_set_a @ W4 @ NN )
=> ( ( ( sources_set_a @ Resid @ V )
= ( sources_set_a @ Resid @ W ) )
=> ( ( ( sources_set_a @ Resid @ V2 )
= ( sources_set_a @ Resid @ W4 ) )
=> ( ( ( targets_set_a @ Resid @ W )
= ( targets_set_a @ Resid @ W4 ) )
=> ( ( ( member_set_a @ ( Resid @ ( Resid @ T @ V ) @ ( Resid @ T4 @ V2 ) ) @ NN )
& ( member_set_a @ ( Resid @ ( Resid @ T4 @ V2 ) @ ( Resid @ T @ V ) ) @ NN ) )
=> ( ( member_set_a @ ( Resid @ ( Resid @ T @ W ) @ ( Resid @ T4 @ W4 ) ) @ NN )
& ( member_set_a @ ( Resid @ ( Resid @ T4 @ W4 ) @ ( Resid @ T @ W ) ) @ NN ) ) ) ) ) ) ) ) ) ) ) ).
% coherent_normal_sub_rts.coherent'
thf(fact_298_coherent__normal__sub__rts_Ocoherent_H,axiom,
! [Resid: a > a > a,NN: set_a,V: a,V2: a,W: a,W4: a,T: a,T4: a] :
( ( cohere6072184133013167079_rts_a @ Resid @ NN )
=> ( ( member_a @ V @ NN )
=> ( ( member_a @ V2 @ NN )
=> ( ( member_a @ W @ NN )
=> ( ( member_a @ W4 @ NN )
=> ( ( ( sources_a @ Resid @ V )
= ( sources_a @ Resid @ W ) )
=> ( ( ( sources_a @ Resid @ V2 )
= ( sources_a @ Resid @ W4 ) )
=> ( ( ( targets_a @ Resid @ W )
= ( targets_a @ Resid @ W4 ) )
=> ( ( ( member_a @ ( Resid @ ( Resid @ T @ V ) @ ( Resid @ T4 @ V2 ) ) @ NN )
& ( member_a @ ( Resid @ ( Resid @ T4 @ V2 ) @ ( Resid @ T @ V ) ) @ NN ) )
=> ( ( member_a @ ( Resid @ ( Resid @ T @ W ) @ ( Resid @ T4 @ W4 ) ) @ NN )
& ( member_a @ ( Resid @ ( Resid @ T4 @ W4 ) @ ( Resid @ T @ W ) ) @ NN ) ) ) ) ) ) ) ) ) ) ) ).
% coherent_normal_sub_rts.coherent'
thf(fact_299_coherent__normal__sub__rts_OCong__composite__of__normal__arr,axiom,
! [Resid: product_prod_a_a > product_prod_a_a > product_prod_a_a,NN: set_Product_prod_a_a,U: product_prod_a_a,T: product_prod_a_a,T4: product_prod_a_a] :
( ( cohere3906135489564710160od_a_a @ Resid @ NN )
=> ( ( compos6970852536677457805od_a_a @ Resid @ U @ T @ T4 )
=> ( ( member1426531477525435216od_a_a @ U @ NN )
=> ( normal7727773393199519165od_a_a @ Resid @ NN @ T4 @ T ) ) ) ) ).
% coherent_normal_sub_rts.Cong_composite_of_normal_arr
thf(fact_300_coherent__normal__sub__rts_OCong__composite__of__normal__arr,axiom,
! [Resid: ( a > a ) > ( a > a ) > a > a,NN: set_a_a,U: a > a,T: a > a,T4: a > a] :
( ( cohere4631022276737013564ts_a_a @ Resid @ NN )
=> ( ( composite_of_a_a @ Resid @ U @ T @ T4 )
=> ( ( member_a_a @ U @ NN )
=> ( normal_sub_Cong_a_a @ Resid @ NN @ T4 @ T ) ) ) ) ).
% coherent_normal_sub_rts.Cong_composite_of_normal_arr
thf(fact_301_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_302_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_303_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_304_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_305_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_306_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_307_normal__sub__rts__axioms__def,axiom,
( normal7992730464706559919od_a_a
= ( ^ [Resid2: product_prod_a_a > product_prod_a_a > product_prod_a_a,NN2: set_Product_prod_a_a] :
( ! [T5: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ T5 @ NN2 )
=> ( arr_Product_prod_a_a @ Resid2 @ T5 ) )
& ! [A5: product_prod_a_a] :
( ( ide_Product_prod_a_a @ Resid2 @ A5 )
=> ( member1426531477525435216od_a_a @ A5 @ NN2 ) )
& ! [U3: product_prod_a_a,T5: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ U3 @ NN2 )
=> ( ( coinit880904215945527283od_a_a @ Resid2 @ T5 @ U3 )
=> ( member1426531477525435216od_a_a @ ( Resid2 @ U3 @ T5 ) @ NN2 ) ) )
& ! [U3: product_prod_a_a,T5: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ U3 @ NN2 )
=> ( ( member1426531477525435216od_a_a @ ( Resid2 @ T5 @ U3 ) @ NN2 )
=> ( member1426531477525435216od_a_a @ T5 @ NN2 ) ) )
& ! [U3: product_prod_a_a,T5: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ U3 @ NN2 )
=> ( ( seq_Product_prod_a_a @ Resid2 @ U3 @ T5 )
=> ? [X4: product_prod_a_a] : ( compos6970852536677457805od_a_a @ Resid2 @ U3 @ T5 @ X4 ) ) )
& ! [U3: product_prod_a_a,T5: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ U3 @ NN2 )
=> ( ( seq_Product_prod_a_a @ Resid2 @ T5 @ U3 )
=> ? [X4: product_prod_a_a] : ( compos6970852536677457805od_a_a @ Resid2 @ T5 @ U3 @ X4 ) ) ) ) ) ) ).
% normal_sub_rts_axioms_def
thf(fact_308_normal__sub__rts__axioms__def,axiom,
( normal4896315260473203165ms_a_a
= ( ^ [Resid2: ( a > a ) > ( a > a ) > a > a,NN2: set_a_a] :
( ! [T5: a > a] :
( ( member_a_a @ T5 @ NN2 )
=> ( arr_a_a @ Resid2 @ T5 ) )
& ! [A5: a > a] :
( ( ide_a_a @ Resid2 @ A5 )
=> ( member_a_a @ A5 @ NN2 ) )
& ! [U3: a > a,T5: a > a] :
( ( member_a_a @ U3 @ NN2 )
=> ( ( coinitial_a_a @ Resid2 @ T5 @ U3 )
=> ( member_a_a @ ( Resid2 @ U3 @ T5 ) @ NN2 ) ) )
& ! [U3: a > a,T5: a > a] :
( ( member_a_a @ U3 @ NN2 )
=> ( ( member_a_a @ ( Resid2 @ T5 @ U3 ) @ NN2 )
=> ( member_a_a @ T5 @ NN2 ) ) )
& ! [U3: a > a,T5: a > a] :
( ( member_a_a @ U3 @ NN2 )
=> ( ( seq_a_a @ Resid2 @ U3 @ T5 )
=> ? [X4: a > a] : ( composite_of_a_a @ Resid2 @ U3 @ T5 @ X4 ) ) )
& ! [U3: a > a,T5: a > a] :
( ( member_a_a @ U3 @ NN2 )
=> ( ( seq_a_a @ Resid2 @ T5 @ U3 )
=> ? [X4: a > a] : ( composite_of_a_a @ Resid2 @ T5 @ U3 @ X4 ) ) ) ) ) ) ).
% normal_sub_rts_axioms_def
thf(fact_309_normal__sub__rts__axioms__def,axiom,
( normal4776468795420100326_set_a
= ( ^ [Resid2: set_a > set_a > set_a,NN2: set_set_a] :
( ! [T5: set_a] :
( ( member_set_a @ T5 @ NN2 )
=> ( arr_set_a @ Resid2 @ T5 ) )
& ! [A5: set_a] :
( ( ide_set_a @ Resid2 @ A5 )
=> ( member_set_a @ A5 @ NN2 ) )
& ! [U3: set_a,T5: set_a] :
( ( member_set_a @ U3 @ NN2 )
=> ( ( coinitial_set_a @ Resid2 @ T5 @ U3 )
=> ( member_set_a @ ( Resid2 @ U3 @ T5 ) @ NN2 ) ) )
& ! [U3: set_a,T5: set_a] :
( ( member_set_a @ U3 @ NN2 )
=> ( ( member_set_a @ ( Resid2 @ T5 @ U3 ) @ NN2 )
=> ( member_set_a @ T5 @ NN2 ) ) )
& ! [U3: set_a,T5: set_a] :
( ( member_set_a @ U3 @ NN2 )
=> ( ( seq_set_a @ Resid2 @ U3 @ T5 )
=> ? [X4: set_a] : ( composite_of_set_a @ Resid2 @ U3 @ T5 @ X4 ) ) )
& ! [U3: set_a,T5: set_a] :
( ( member_set_a @ U3 @ NN2 )
=> ( ( seq_set_a @ Resid2 @ T5 @ U3 )
=> ? [X4: set_a] : ( composite_of_set_a @ Resid2 @ T5 @ U3 @ X4 ) ) ) ) ) ) ).
% normal_sub_rts_axioms_def
thf(fact_310_normal__sub__rts__axioms__def,axiom,
( normal7698203753654205830ioms_a
= ( ^ [Resid2: a > a > a,NN2: set_a] :
( ! [T5: a] :
( ( member_a @ T5 @ NN2 )
=> ( arr_a @ Resid2 @ T5 ) )
& ! [A5: a] :
( ( ide_a @ Resid2 @ A5 )
=> ( member_a @ A5 @ NN2 ) )
& ! [U3: a,T5: a] :
( ( member_a @ U3 @ NN2 )
=> ( ( coinitial_a @ Resid2 @ T5 @ U3 )
=> ( member_a @ ( Resid2 @ U3 @ T5 ) @ NN2 ) ) )
& ! [U3: a,T5: a] :
( ( member_a @ U3 @ NN2 )
=> ( ( member_a @ ( Resid2 @ T5 @ U3 ) @ NN2 )
=> ( member_a @ T5 @ NN2 ) ) )
& ! [U3: a,T5: a] :
( ( member_a @ U3 @ NN2 )
=> ( ( seq_a @ Resid2 @ U3 @ T5 )
=> ? [X4: a] : ( composite_of_a @ Resid2 @ U3 @ T5 @ X4 ) ) )
& ! [U3: a,T5: a] :
( ( member_a @ U3 @ NN2 )
=> ( ( seq_a @ Resid2 @ T5 @ U3 )
=> ? [X4: a] : ( composite_of_a @ Resid2 @ T5 @ U3 @ X4 ) ) ) ) ) ) ).
% normal_sub_rts_axioms_def
thf(fact_311_normal__sub__rts__axioms_Ointro,axiom,
! [NN: set_Product_prod_a_a,Resid: product_prod_a_a > product_prod_a_a > product_prod_a_a] :
( ! [T3: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ T3 @ NN )
=> ( arr_Product_prod_a_a @ Resid @ T3 ) )
=> ( ! [A3: product_prod_a_a] :
( ( ide_Product_prod_a_a @ Resid @ A3 )
=> ( member1426531477525435216od_a_a @ A3 @ NN ) )
=> ( ! [U5: product_prod_a_a,T3: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ U5 @ NN )
=> ( ( coinit880904215945527283od_a_a @ Resid @ T3 @ U5 )
=> ( member1426531477525435216od_a_a @ ( Resid @ U5 @ T3 ) @ NN ) ) )
=> ( ! [U5: product_prod_a_a,T3: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ U5 @ NN )
=> ( ( member1426531477525435216od_a_a @ ( Resid @ T3 @ U5 ) @ NN )
=> ( member1426531477525435216od_a_a @ T3 @ NN ) ) )
=> ( ! [U5: product_prod_a_a,T3: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ U5 @ NN )
=> ( ( seq_Product_prod_a_a @ Resid @ U5 @ T3 )
=> ? [X_12: product_prod_a_a] : ( compos6970852536677457805od_a_a @ Resid @ U5 @ T3 @ X_12 ) ) )
=> ( ! [U5: product_prod_a_a,T3: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ U5 @ NN )
=> ( ( seq_Product_prod_a_a @ Resid @ T3 @ U5 )
=> ? [X_12: product_prod_a_a] : ( compos6970852536677457805od_a_a @ Resid @ T3 @ U5 @ X_12 ) ) )
=> ( normal7992730464706559919od_a_a @ Resid @ NN ) ) ) ) ) ) ) ).
% normal_sub_rts_axioms.intro
thf(fact_312_normal__sub__rts__axioms_Ointro,axiom,
! [NN: set_a_a,Resid: ( a > a ) > ( a > a ) > a > a] :
( ! [T3: a > a] :
( ( member_a_a @ T3 @ NN )
=> ( arr_a_a @ Resid @ T3 ) )
=> ( ! [A3: a > a] :
( ( ide_a_a @ Resid @ A3 )
=> ( member_a_a @ A3 @ NN ) )
=> ( ! [U5: a > a,T3: a > a] :
( ( member_a_a @ U5 @ NN )
=> ( ( coinitial_a_a @ Resid @ T3 @ U5 )
=> ( member_a_a @ ( Resid @ U5 @ T3 ) @ NN ) ) )
=> ( ! [U5: a > a,T3: a > a] :
( ( member_a_a @ U5 @ NN )
=> ( ( member_a_a @ ( Resid @ T3 @ U5 ) @ NN )
=> ( member_a_a @ T3 @ NN ) ) )
=> ( ! [U5: a > a,T3: a > a] :
( ( member_a_a @ U5 @ NN )
=> ( ( seq_a_a @ Resid @ U5 @ T3 )
=> ? [X_12: a > a] : ( composite_of_a_a @ Resid @ U5 @ T3 @ X_12 ) ) )
=> ( ! [U5: a > a,T3: a > a] :
( ( member_a_a @ U5 @ NN )
=> ( ( seq_a_a @ Resid @ T3 @ U5 )
=> ? [X_12: a > a] : ( composite_of_a_a @ Resid @ T3 @ U5 @ X_12 ) ) )
=> ( normal4896315260473203165ms_a_a @ Resid @ NN ) ) ) ) ) ) ) ).
% normal_sub_rts_axioms.intro
thf(fact_313_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 ) )
=> ( ! [A3: set_a] :
( ( ide_set_a @ Resid @ A3 )
=> ( member_set_a @ A3 @ NN ) )
=> ( ! [U5: set_a,T3: set_a] :
( ( member_set_a @ U5 @ NN )
=> ( ( coinitial_set_a @ Resid @ T3 @ U5 )
=> ( member_set_a @ ( Resid @ U5 @ T3 ) @ NN ) ) )
=> ( ! [U5: set_a,T3: set_a] :
( ( member_set_a @ U5 @ NN )
=> ( ( member_set_a @ ( Resid @ T3 @ U5 ) @ NN )
=> ( member_set_a @ T3 @ NN ) ) )
=> ( ! [U5: set_a,T3: set_a] :
( ( member_set_a @ U5 @ NN )
=> ( ( seq_set_a @ Resid @ U5 @ T3 )
=> ? [X_12: set_a] : ( composite_of_set_a @ Resid @ U5 @ T3 @ X_12 ) ) )
=> ( ! [U5: set_a,T3: set_a] :
( ( member_set_a @ U5 @ NN )
=> ( ( seq_set_a @ Resid @ T3 @ U5 )
=> ? [X_12: set_a] : ( composite_of_set_a @ Resid @ T3 @ U5 @ X_12 ) ) )
=> ( normal4776468795420100326_set_a @ Resid @ NN ) ) ) ) ) ) ) ).
% normal_sub_rts_axioms.intro
thf(fact_314_normal__sub__rts__axioms_Ointro,axiom,
! [NN: set_a,Resid: a > a > a] :
( ! [T3: a] :
( ( member_a @ T3 @ NN )
=> ( arr_a @ Resid @ T3 ) )
=> ( ! [A3: a] :
( ( ide_a @ Resid @ A3 )
=> ( member_a @ A3 @ NN ) )
=> ( ! [U5: a,T3: a] :
( ( member_a @ U5 @ NN )
=> ( ( coinitial_a @ Resid @ T3 @ U5 )
=> ( member_a @ ( Resid @ U5 @ T3 ) @ NN ) ) )
=> ( ! [U5: a,T3: a] :
( ( member_a @ U5 @ NN )
=> ( ( member_a @ ( Resid @ T3 @ U5 ) @ NN )
=> ( member_a @ T3 @ NN ) ) )
=> ( ! [U5: a,T3: a] :
( ( member_a @ U5 @ NN )
=> ( ( seq_a @ Resid @ U5 @ T3 )
=> ? [X_12: a] : ( composite_of_a @ Resid @ U5 @ T3 @ X_12 ) ) )
=> ( ! [U5: a,T3: a] :
( ( member_a @ U5 @ NN )
=> ( ( seq_a @ Resid @ T3 @ U5 )
=> ? [X_12: a] : ( composite_of_a @ Resid @ T3 @ U5 @ X_12 ) ) )
=> ( normal7698203753654205830ioms_a @ Resid @ NN ) ) ) ) ) ) ) ).
% normal_sub_rts_axioms.intro
thf(fact_315_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_316_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_317_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_318_N_Ois__Cong__class__def,axiom,
! [T7: set_a] :
( ( normal8595587647932138008lass_a @ resid @ nn @ T7 )
= ( ? [T5: a] :
( ( member_a @ T5 @ T7 )
& ( T7
= ( normal7408713899360725774lass_a @ resid @ nn @ T5 ) ) ) ) ) ).
% N.is_Cong_class_def
thf(fact_319_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_320_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_321_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_322_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_323_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_324_normal__sub__rts_OCong__class_Ocong,axiom,
normal7408713899360725774lass_a = normal7408713899360725774lass_a ).
% normal_sub_rts.Cong_class.cong
thf(fact_325_normal__sub__rts_Ois__Cong__class_Ocong,axiom,
normal8595587647932138008lass_a = normal8595587647932138008lass_a ).
% normal_sub_rts.is_Cong_class.cong
thf(fact_326_Resid__by__members,axiom,
! [T7: set_a,U7: set_a,T: a,U: a] :
( ( normal8595587647932138008lass_a @ resid @ nn @ T7 )
=> ( ( normal8595587647932138008lass_a @ resid @ nn @ U7 )
=> ( ( member_a @ T @ T7 )
=> ( ( member_a @ U @ U7 )
=> ( ( con_a @ resid @ T @ U )
=> ( ( quotie8165075472272353145esid_a @ resid @ nn @ T7 @ U7 )
= ( normal7408713899360725774lass_a @ resid @ nn @ ( resid @ T @ U ) ) ) ) ) ) ) ) ).
% Resid_by_members
thf(fact_327_Con__char,axiom,
! [T7: set_a,U7: set_a] :
( ( ( quotie8165075472272353145esid_a @ resid @ nn @ T7 @ U7 )
!= bot_bot_set_a )
= ( ( normal8595587647932138008lass_a @ resid @ nn @ T7 )
& ( normal8595587647932138008lass_a @ resid @ nn @ U7 )
& ? [T5: a,U3: a] :
( ( member_a @ T5 @ T7 )
& ( member_a @ U3 @ U7 )
& ( con_a @ resid @ T5 @ U3 ) ) ) ) ).
% Con_char
thf(fact_328_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_329_is__Cong__class__Resid,axiom,
! [T7: set_a,U7: set_a] :
( ( ( quotie8165075472272353145esid_a @ resid @ nn @ T7 @ U7 )
!= bot_bot_set_a )
=> ( normal8595587647932138008lass_a @ resid @ nn @ ( quotie8165075472272353145esid_a @ resid @ nn @ T7 @ U7 ) ) ) ).
% is_Cong_class_Resid
thf(fact_330_N_OCong__class__eqI_H,axiom,
! [T7: set_a,U7: set_a] :
( ( normal8595587647932138008lass_a @ resid @ nn @ T7 )
=> ( ( normal8595587647932138008lass_a @ resid @ nn @ U7 )
=> ( ( ( inf_inf_set_a @ T7 @ U7 )
!= bot_bot_set_a )
=> ( T7 = U7 ) ) ) ) ).
% N.Cong_class_eqI'
thf(fact_331_Con__witnesses,axiom,
! [T7: set_a,U7: set_a,T: a,U: a] :
( ( ( quotie8165075472272353145esid_a @ resid @ nn @ T7 @ U7 )
!= bot_bot_set_a )
=> ( ( member_a @ T @ T7 )
=> ( ( member_a @ U @ U7 )
=> ? [V5: a,W2: a] :
( ( member_a @ V5 @ nn )
& ( member_a @ W2 @ nn )
& ( con_a @ resid @ ( resid @ T @ V5 ) @ ( resid @ U @ W2 ) ) ) ) ) ) ).
% Con_witnesses
thf(fact_332_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_333_partial__magma__axioms,axiom,
partial_magma_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) ).
% partial_magma_axioms
thf(fact_334_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_335_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_336_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_337_Arr__Resid,axiom,
! [T7: set_a,U7: set_a] :
( ( ( quotie8165075472272353145esid_a @ resid @ nn @ T7 @ U7 )
!= bot_bot_set_a )
=> ( ( quotie8165075472272353145esid_a @ resid @ nn @ ( quotie8165075472272353145esid_a @ resid @ nn @ T7 @ U7 ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ T7 @ U7 ) )
!= bot_bot_set_a ) ) ).
% Arr_Resid
thf(fact_338_Con__sym,axiom,
! [T7: set_a,U7: set_a] :
( ( ( quotie8165075472272353145esid_a @ resid @ nn @ T7 @ U7 )
!= bot_bot_set_a )
=> ( ( quotie8165075472272353145esid_a @ resid @ nn @ U7 @ T7 )
!= bot_bot_set_a ) ) ).
% Con_sym
thf(fact_339_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_340_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_341_quotient__by__coherent__normal_OResid_Ocong,axiom,
quotie8165075472272353145esid_a = quotie8165075472272353145esid_a ).
% quotient_by_coherent_normal.Resid.cong
thf(fact_342_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_343_quotient__by__coherent__normal_OArr__Resid,axiom,
! [Resid: product_prod_a_a > product_prod_a_a > product_prod_a_a,NN: set_Product_prod_a_a,T7: set_Product_prod_a_a,U7: set_Product_prod_a_a] :
( ( quotie2905600318227040575od_a_a @ Resid @ NN )
=> ( ( ( quotie4807335555532242594od_a_a @ Resid @ NN @ T7 @ U7 )
!= bot_bo3357376287454694259od_a_a )
=> ( ( quotie4807335555532242594od_a_a @ Resid @ NN @ ( quotie4807335555532242594od_a_a @ Resid @ NN @ T7 @ U7 ) @ ( quotie4807335555532242594od_a_a @ Resid @ NN @ T7 @ U7 ) )
!= bot_bo3357376287454694259od_a_a ) ) ) ).
% quotient_by_coherent_normal.Arr_Resid
thf(fact_344_quotient__by__coherent__normal_OArr__Resid,axiom,
! [Resid: set_a > set_a > set_a,NN: set_set_a,T7: set_set_a,U7: set_set_a] :
( ( quotie5625257012022141046_set_a @ Resid @ NN )
=> ( ( ( quotie3283642546880816345_set_a @ Resid @ NN @ T7 @ U7 )
!= bot_bot_set_set_a )
=> ( ( quotie3283642546880816345_set_a @ Resid @ NN @ ( quotie3283642546880816345_set_a @ Resid @ NN @ T7 @ U7 ) @ ( quotie3283642546880816345_set_a @ Resid @ NN @ T7 @ U7 ) )
!= bot_bot_set_set_a ) ) ) ).
% quotient_by_coherent_normal.Arr_Resid
thf(fact_345_quotient__by__coherent__normal_OArr__Resid,axiom,
! [Resid: ( a > a ) > ( a > a ) > a > a,NN: set_a_a,T7: set_a_a,U7: set_a_a] :
( ( quotie8968171584129522765al_a_a @ Resid @ NN )
=> ( ( ( quotie4957313913467096874id_a_a @ Resid @ NN @ T7 @ U7 )
!= bot_bot_set_a_a )
=> ( ( quotie4957313913467096874id_a_a @ Resid @ NN @ ( quotie4957313913467096874id_a_a @ Resid @ NN @ T7 @ U7 ) @ ( quotie4957313913467096874id_a_a @ Resid @ NN @ T7 @ U7 ) )
!= bot_bot_set_a_a ) ) ) ).
% quotient_by_coherent_normal.Arr_Resid
thf(fact_346_quotient__by__coherent__normal_OArr__Resid,axiom,
! [Resid: a > a > a,NN: set_a,T7: set_a,U7: set_a] :
( ( quotie3282664541148387094rmal_a @ Resid @ NN )
=> ( ( ( quotie8165075472272353145esid_a @ Resid @ NN @ T7 @ U7 )
!= bot_bot_set_a )
=> ( ( quotie8165075472272353145esid_a @ Resid @ NN @ ( quotie8165075472272353145esid_a @ Resid @ NN @ T7 @ U7 ) @ ( quotie8165075472272353145esid_a @ Resid @ NN @ T7 @ U7 ) )
!= bot_bot_set_a ) ) ) ).
% quotient_by_coherent_normal.Arr_Resid
thf(fact_347_quotient__by__coherent__normal_OCon__sym,axiom,
! [Resid: product_prod_a_a > product_prod_a_a > product_prod_a_a,NN: set_Product_prod_a_a,T7: set_Product_prod_a_a,U7: set_Product_prod_a_a] :
( ( quotie2905600318227040575od_a_a @ Resid @ NN )
=> ( ( ( quotie4807335555532242594od_a_a @ Resid @ NN @ T7 @ U7 )
!= bot_bo3357376287454694259od_a_a )
=> ( ( quotie4807335555532242594od_a_a @ Resid @ NN @ U7 @ T7 )
!= bot_bo3357376287454694259od_a_a ) ) ) ).
% quotient_by_coherent_normal.Con_sym
thf(fact_348_quotient__by__coherent__normal_OCon__sym,axiom,
! [Resid: set_a > set_a > set_a,NN: set_set_a,T7: set_set_a,U7: set_set_a] :
( ( quotie5625257012022141046_set_a @ Resid @ NN )
=> ( ( ( quotie3283642546880816345_set_a @ Resid @ NN @ T7 @ U7 )
!= bot_bot_set_set_a )
=> ( ( quotie3283642546880816345_set_a @ Resid @ NN @ U7 @ T7 )
!= bot_bot_set_set_a ) ) ) ).
% quotient_by_coherent_normal.Con_sym
thf(fact_349_quotient__by__coherent__normal_OCon__sym,axiom,
! [Resid: ( a > a ) > ( a > a ) > a > a,NN: set_a_a,T7: set_a_a,U7: set_a_a] :
( ( quotie8968171584129522765al_a_a @ Resid @ NN )
=> ( ( ( quotie4957313913467096874id_a_a @ Resid @ NN @ T7 @ U7 )
!= bot_bot_set_a_a )
=> ( ( quotie4957313913467096874id_a_a @ Resid @ NN @ U7 @ T7 )
!= bot_bot_set_a_a ) ) ) ).
% quotient_by_coherent_normal.Con_sym
thf(fact_350_quotient__by__coherent__normal_OCon__sym,axiom,
! [Resid: a > a > a,NN: set_a,T7: set_a,U7: set_a] :
( ( quotie3282664541148387094rmal_a @ Resid @ NN )
=> ( ( ( quotie8165075472272353145esid_a @ Resid @ NN @ T7 @ U7 )
!= bot_bot_set_a )
=> ( ( quotie8165075472272353145esid_a @ Resid @ NN @ U7 @ T7 )
!= bot_bot_set_a ) ) ) ).
% quotient_by_coherent_normal.Con_sym
thf(fact_351_quotient__by__coherent__normal_OCon__witnesses,axiom,
! [Resid: product_prod_a_a > product_prod_a_a > product_prod_a_a,NN: set_Product_prod_a_a,T7: set_Product_prod_a_a,U7: set_Product_prod_a_a,T: product_prod_a_a,U: product_prod_a_a] :
( ( quotie2905600318227040575od_a_a @ Resid @ NN )
=> ( ( ( quotie4807335555532242594od_a_a @ Resid @ NN @ T7 @ U7 )
!= bot_bo3357376287454694259od_a_a )
=> ( ( member1426531477525435216od_a_a @ T @ T7 )
=> ( ( member1426531477525435216od_a_a @ U @ U7 )
=> ? [V5: product_prod_a_a,W2: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ V5 @ NN )
& ( member1426531477525435216od_a_a @ W2 @ NN )
& ( con_Product_prod_a_a @ Resid @ ( Resid @ T @ V5 ) @ ( Resid @ U @ W2 ) ) ) ) ) ) ) ).
% quotient_by_coherent_normal.Con_witnesses
thf(fact_352_quotient__by__coherent__normal_OCon__witnesses,axiom,
! [Resid: ( a > a ) > ( a > a ) > a > a,NN: set_a_a,T7: set_a_a,U7: set_a_a,T: a > a,U: a > a] :
( ( quotie8968171584129522765al_a_a @ Resid @ NN )
=> ( ( ( quotie4957313913467096874id_a_a @ Resid @ NN @ T7 @ U7 )
!= bot_bot_set_a_a )
=> ( ( member_a_a @ T @ T7 )
=> ( ( member_a_a @ U @ U7 )
=> ? [V5: a > a,W2: a > a] :
( ( member_a_a @ V5 @ NN )
& ( member_a_a @ W2 @ NN )
& ( con_a_a @ Resid @ ( Resid @ T @ V5 ) @ ( Resid @ U @ W2 ) ) ) ) ) ) ) ).
% quotient_by_coherent_normal.Con_witnesses
thf(fact_353_quotient__by__coherent__normal_OCon__witnesses,axiom,
! [Resid: set_a > set_a > set_a,NN: set_set_a,T7: set_set_a,U7: set_set_a,T: set_a,U: set_a] :
( ( quotie5625257012022141046_set_a @ Resid @ NN )
=> ( ( ( quotie3283642546880816345_set_a @ Resid @ NN @ T7 @ U7 )
!= bot_bot_set_set_a )
=> ( ( member_set_a @ T @ T7 )
=> ( ( member_set_a @ U @ U7 )
=> ? [V5: set_a,W2: set_a] :
( ( member_set_a @ V5 @ NN )
& ( member_set_a @ W2 @ NN )
& ( con_set_a @ Resid @ ( Resid @ T @ V5 ) @ ( Resid @ U @ W2 ) ) ) ) ) ) ) ).
% quotient_by_coherent_normal.Con_witnesses
thf(fact_354_quotient__by__coherent__normal_OCon__witnesses,axiom,
! [Resid: a > a > a,NN: set_a,T7: set_a,U7: set_a,T: a,U: a] :
( ( quotie3282664541148387094rmal_a @ Resid @ NN )
=> ( ( ( quotie8165075472272353145esid_a @ Resid @ NN @ T7 @ U7 )
!= bot_bot_set_a )
=> ( ( member_a @ T @ T7 )
=> ( ( member_a @ U @ U7 )
=> ? [V5: a,W2: a] :
( ( member_a @ V5 @ NN )
& ( member_a @ W2 @ NN )
& ( con_a @ Resid @ ( Resid @ T @ V5 ) @ ( Resid @ U @ W2 ) ) ) ) ) ) ) ).
% quotient_by_coherent_normal.Con_witnesses
thf(fact_355_quotient__by__coherent__normal_Ois__Cong__class__Resid,axiom,
! [Resid: product_prod_a_a > product_prod_a_a > product_prod_a_a,NN: set_Product_prod_a_a,T7: set_Product_prod_a_a,U7: set_Product_prod_a_a] :
( ( quotie2905600318227040575od_a_a @ Resid @ NN )
=> ( ( ( quotie4807335555532242594od_a_a @ Resid @ NN @ T7 @ U7 )
!= bot_bo3357376287454694259od_a_a )
=> ( normal6700481192199873089od_a_a @ Resid @ NN @ ( quotie4807335555532242594od_a_a @ Resid @ NN @ T7 @ U7 ) ) ) ) ).
% quotient_by_coherent_normal.is_Cong_class_Resid
thf(fact_356_quotient__by__coherent__normal_Ois__Cong__class__Resid,axiom,
! [Resid: set_a > set_a > set_a,NN: set_set_a,T7: set_set_a,U7: set_set_a] :
( ( quotie5625257012022141046_set_a @ Resid @ NN )
=> ( ( ( quotie3283642546880816345_set_a @ Resid @ NN @ T7 @ U7 )
!= bot_bot_set_set_a )
=> ( normal4437380936311325560_set_a @ Resid @ NN @ ( quotie3283642546880816345_set_a @ Resid @ NN @ T7 @ U7 ) ) ) ) ).
% quotient_by_coherent_normal.is_Cong_class_Resid
thf(fact_357_quotient__by__coherent__normal_Ois__Cong__class__Resid,axiom,
! [Resid: ( a > a ) > ( a > a ) > a > a,NN: set_a_a,T7: set_a_a,U7: set_a_a] :
( ( quotie8968171584129522765al_a_a @ Resid @ NN )
=> ( ( ( quotie4957313913467096874id_a_a @ Resid @ NN @ T7 @ U7 )
!= bot_bot_set_a_a )
=> ( normal6067668076082862283ss_a_a @ Resid @ NN @ ( quotie4957313913467096874id_a_a @ Resid @ NN @ T7 @ U7 ) ) ) ) ).
% quotient_by_coherent_normal.is_Cong_class_Resid
thf(fact_358_quotient__by__coherent__normal_Ois__Cong__class__Resid,axiom,
! [Resid: a > a > a,NN: set_a,T7: set_a,U7: set_a] :
( ( quotie3282664541148387094rmal_a @ Resid @ NN )
=> ( ( ( quotie8165075472272353145esid_a @ Resid @ NN @ T7 @ U7 )
!= bot_bot_set_a )
=> ( normal8595587647932138008lass_a @ Resid @ NN @ ( quotie8165075472272353145esid_a @ Resid @ NN @ T7 @ U7 ) ) ) ) ).
% quotient_by_coherent_normal.is_Cong_class_Resid
thf(fact_359_quotient__by__coherent__normal_OCon__char,axiom,
! [Resid: product_prod_a_a > product_prod_a_a > product_prod_a_a,NN: set_Product_prod_a_a,T7: set_Product_prod_a_a,U7: set_Product_prod_a_a] :
( ( quotie2905600318227040575od_a_a @ Resid @ NN )
=> ( ( ( quotie4807335555532242594od_a_a @ Resid @ NN @ T7 @ U7 )
!= bot_bo3357376287454694259od_a_a )
= ( ( normal6700481192199873089od_a_a @ Resid @ NN @ T7 )
& ( normal6700481192199873089od_a_a @ Resid @ NN @ U7 )
& ? [T5: product_prod_a_a,U3: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ T5 @ T7 )
& ( member1426531477525435216od_a_a @ U3 @ U7 )
& ( con_Product_prod_a_a @ Resid @ T5 @ U3 ) ) ) ) ) ).
% quotient_by_coherent_normal.Con_char
thf(fact_360_quotient__by__coherent__normal_OCon__char,axiom,
! [Resid: ( a > a ) > ( a > a ) > a > a,NN: set_a_a,T7: set_a_a,U7: set_a_a] :
( ( quotie8968171584129522765al_a_a @ Resid @ NN )
=> ( ( ( quotie4957313913467096874id_a_a @ Resid @ NN @ T7 @ U7 )
!= bot_bot_set_a_a )
= ( ( normal6067668076082862283ss_a_a @ Resid @ NN @ T7 )
& ( normal6067668076082862283ss_a_a @ Resid @ NN @ U7 )
& ? [T5: a > a,U3: a > a] :
( ( member_a_a @ T5 @ T7 )
& ( member_a_a @ U3 @ U7 )
& ( con_a_a @ Resid @ T5 @ U3 ) ) ) ) ) ).
% quotient_by_coherent_normal.Con_char
thf(fact_361_quotient__by__coherent__normal_OCon__char,axiom,
! [Resid: set_a > set_a > set_a,NN: set_set_a,T7: set_set_a,U7: set_set_a] :
( ( quotie5625257012022141046_set_a @ Resid @ NN )
=> ( ( ( quotie3283642546880816345_set_a @ Resid @ NN @ T7 @ U7 )
!= bot_bot_set_set_a )
= ( ( normal4437380936311325560_set_a @ Resid @ NN @ T7 )
& ( normal4437380936311325560_set_a @ Resid @ NN @ U7 )
& ? [T5: set_a,U3: set_a] :
( ( member_set_a @ T5 @ T7 )
& ( member_set_a @ U3 @ U7 )
& ( con_set_a @ Resid @ T5 @ U3 ) ) ) ) ) ).
% quotient_by_coherent_normal.Con_char
thf(fact_362_quotient__by__coherent__normal_OCon__char,axiom,
! [Resid: a > a > a,NN: set_a,T7: set_a,U7: set_a] :
( ( quotie3282664541148387094rmal_a @ Resid @ NN )
=> ( ( ( quotie8165075472272353145esid_a @ Resid @ NN @ T7 @ U7 )
!= bot_bot_set_a )
= ( ( normal8595587647932138008lass_a @ Resid @ NN @ T7 )
& ( normal8595587647932138008lass_a @ Resid @ NN @ U7 )
& ? [T5: a,U3: a] :
( ( member_a @ T5 @ T7 )
& ( member_a @ U3 @ U7 )
& ( con_a @ Resid @ T5 @ U3 ) ) ) ) ) ).
% quotient_by_coherent_normal.Con_char
thf(fact_363_quotient__by__coherent__normal_OResid__by__members,axiom,
! [Resid: product_prod_a_a > product_prod_a_a > product_prod_a_a,NN: set_Product_prod_a_a,T7: set_Product_prod_a_a,U7: set_Product_prod_a_a,T: product_prod_a_a,U: product_prod_a_a] :
( ( quotie2905600318227040575od_a_a @ Resid @ NN )
=> ( ( normal6700481192199873089od_a_a @ Resid @ NN @ T7 )
=> ( ( normal6700481192199873089od_a_a @ Resid @ NN @ U7 )
=> ( ( member1426531477525435216od_a_a @ T @ T7 )
=> ( ( member1426531477525435216od_a_a @ U @ U7 )
=> ( ( con_Product_prod_a_a @ Resid @ T @ U )
=> ( ( quotie4807335555532242594od_a_a @ Resid @ NN @ T7 @ U7 )
= ( normal8582136959803729207od_a_a @ Resid @ NN @ ( Resid @ T @ U ) ) ) ) ) ) ) ) ) ).
% quotient_by_coherent_normal.Resid_by_members
thf(fact_364_quotient__by__coherent__normal_OResid__by__members,axiom,
! [Resid: ( a > a ) > ( a > a ) > a > a,NN: set_a_a,T7: set_a_a,U7: set_a_a,T: a > a,U: a > a] :
( ( quotie8968171584129522765al_a_a @ Resid @ NN )
=> ( ( normal6067668076082862283ss_a_a @ Resid @ NN @ T7 )
=> ( ( normal6067668076082862283ss_a_a @ Resid @ NN @ U7 )
=> ( ( member_a_a @ T @ T7 )
=> ( ( member_a_a @ U @ U7 )
=> ( ( con_a_a @ Resid @ T @ U )
=> ( ( quotie4957313913467096874id_a_a @ Resid @ NN @ T7 @ U7 )
= ( normal3779061517355887189ss_a_a @ Resid @ NN @ ( Resid @ T @ U ) ) ) ) ) ) ) ) ) ).
% quotient_by_coherent_normal.Resid_by_members
thf(fact_365_quotient__by__coherent__normal_OResid__by__members,axiom,
! [Resid: set_a > set_a > set_a,NN: set_set_a,T7: set_set_a,U7: 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 @ U7 )
=> ( ( member_set_a @ T @ T7 )
=> ( ( member_set_a @ U @ U7 )
=> ( ( con_set_a @ Resid @ T @ U )
=> ( ( quotie3283642546880816345_set_a @ Resid @ NN @ T7 @ U7 )
= ( normal2962378890657961070_set_a @ Resid @ NN @ ( Resid @ T @ U ) ) ) ) ) ) ) ) ) ).
% quotient_by_coherent_normal.Resid_by_members
thf(fact_366_quotient__by__coherent__normal_OResid__by__members,axiom,
! [Resid: a > a > a,NN: set_a,T7: set_a,U7: set_a,T: a,U: a] :
( ( quotie3282664541148387094rmal_a @ Resid @ NN )
=> ( ( normal8595587647932138008lass_a @ Resid @ NN @ T7 )
=> ( ( normal8595587647932138008lass_a @ Resid @ NN @ U7 )
=> ( ( member_a @ T @ T7 )
=> ( ( member_a @ U @ U7 )
=> ( ( con_a @ Resid @ T @ U )
=> ( ( quotie8165075472272353145esid_a @ Resid @ NN @ T7 @ U7 )
= ( normal7408713899360725774lass_a @ Resid @ NN @ ( Resid @ T @ U ) ) ) ) ) ) ) ) ) ).
% quotient_by_coherent_normal.Resid_by_members
thf(fact_367_t_Hv,axiom,
( ( member_a @ t2 @ t )
& ( member_a @ v2 @ v )
& ( con_a @ resid @ t2 @ v2 )
& ( ( quotie8165075472272353145esid_a @ resid @ nn @ t @ v )
= ( normal7408713899360725774lass_a @ resid @ nn @ ( resid @ t2 @ v2 ) ) ) ) ).
% t'v
thf(fact_368_tu,axiom,
( ( member_a @ t3 @ t )
& ( member_a @ u2 @ u )
& ( con_a @ resid @ t3 @ u2 )
& ( ( quotie8165075472272353145esid_a @ resid @ nn @ t @ u )
= ( normal7408713899360725774lass_a @ resid @ nn @ ( resid @ t3 @ u2 ) ) ) ) ).
% tu
thf(fact_369_inf__bot__left,axiom,
! [X6: a > $o] :
( ( inf_inf_a_o @ bot_bot_a_o @ X6 )
= bot_bot_a_o ) ).
% inf_bot_left
thf(fact_370_inf__bot__left,axiom,
! [X6: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ bot_bo3357376287454694259od_a_a @ X6 )
= bot_bo3357376287454694259od_a_a ) ).
% inf_bot_left
thf(fact_371_inf__bot__left,axiom,
! [X6: set_set_a] :
( ( inf_inf_set_set_a @ bot_bot_set_set_a @ X6 )
= bot_bot_set_set_a ) ).
% inf_bot_left
thf(fact_372_inf__bot__left,axiom,
! [X6: set_a_a] :
( ( inf_inf_set_a_a @ bot_bot_set_a_a @ X6 )
= bot_bot_set_a_a ) ).
% inf_bot_left
thf(fact_373_inf__bot__left,axiom,
! [X6: set_a] :
( ( inf_inf_set_a @ bot_bot_set_a @ X6 )
= bot_bot_set_a ) ).
% inf_bot_left
thf(fact_374_inf__bot__right,axiom,
! [X6: a > $o] :
( ( inf_inf_a_o @ X6 @ bot_bot_a_o )
= bot_bot_a_o ) ).
% inf_bot_right
thf(fact_375_inf__bot__right,axiom,
! [X6: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ X6 @ bot_bo3357376287454694259od_a_a )
= bot_bo3357376287454694259od_a_a ) ).
% inf_bot_right
thf(fact_376_inf__bot__right,axiom,
! [X6: set_set_a] :
( ( inf_inf_set_set_a @ X6 @ bot_bot_set_set_a )
= bot_bot_set_set_a ) ).
% inf_bot_right
thf(fact_377_inf__bot__right,axiom,
! [X6: set_a_a] :
( ( inf_inf_set_a_a @ X6 @ bot_bot_set_a_a )
= bot_bot_set_a_a ) ).
% inf_bot_right
thf(fact_378_inf__bot__right,axiom,
! [X6: set_a] :
( ( inf_inf_set_a @ X6 @ bot_bot_set_a )
= bot_bot_set_a ) ).
% inf_bot_right
thf(fact_379_boolean__algebra_Oconj__zero__left,axiom,
! [X6: a > $o] :
( ( inf_inf_a_o @ bot_bot_a_o @ X6 )
= bot_bot_a_o ) ).
% boolean_algebra.conj_zero_left
thf(fact_380_boolean__algebra_Oconj__zero__left,axiom,
! [X6: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ bot_bo3357376287454694259od_a_a @ X6 )
= bot_bo3357376287454694259od_a_a ) ).
% boolean_algebra.conj_zero_left
thf(fact_381_boolean__algebra_Oconj__zero__left,axiom,
! [X6: set_set_a] :
( ( inf_inf_set_set_a @ bot_bot_set_set_a @ X6 )
= bot_bot_set_set_a ) ).
% boolean_algebra.conj_zero_left
thf(fact_382_boolean__algebra_Oconj__zero__left,axiom,
! [X6: set_a_a] :
( ( inf_inf_set_a_a @ bot_bot_set_a_a @ X6 )
= bot_bot_set_a_a ) ).
% boolean_algebra.conj_zero_left
thf(fact_383_boolean__algebra_Oconj__zero__left,axiom,
! [X6: set_a] :
( ( inf_inf_set_a @ bot_bot_set_a @ X6 )
= bot_bot_set_a ) ).
% boolean_algebra.conj_zero_left
thf(fact_384_boolean__algebra_Oconj__zero__right,axiom,
! [X6: a > $o] :
( ( inf_inf_a_o @ X6 @ bot_bot_a_o )
= bot_bot_a_o ) ).
% boolean_algebra.conj_zero_right
thf(fact_385_boolean__algebra_Oconj__zero__right,axiom,
! [X6: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ X6 @ bot_bo3357376287454694259od_a_a )
= bot_bo3357376287454694259od_a_a ) ).
% boolean_algebra.conj_zero_right
thf(fact_386_boolean__algebra_Oconj__zero__right,axiom,
! [X6: set_set_a] :
( ( inf_inf_set_set_a @ X6 @ bot_bot_set_set_a )
= bot_bot_set_set_a ) ).
% boolean_algebra.conj_zero_right
thf(fact_387_boolean__algebra_Oconj__zero__right,axiom,
! [X6: set_a_a] :
( ( inf_inf_set_a_a @ X6 @ bot_bot_set_a_a )
= bot_bot_set_a_a ) ).
% boolean_algebra.conj_zero_right
thf(fact_388_boolean__algebra_Oconj__zero__right,axiom,
! [X6: set_a] :
( ( inf_inf_set_a @ X6 @ bot_bot_set_a )
= bot_bot_set_a ) ).
% boolean_algebra.conj_zero_right
thf(fact_389_null__char,axiom,
( ( partial_null_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) )
= bot_bot_set_a ) ).
% null_char
thf(fact_390_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_391_assms,axiom,
( ( quotie8165075472272353145esid_a @ resid @ nn @ ( quotie8165075472272353145esid_a @ resid @ nn @ v @ t ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ u @ t ) )
!= bot_bot_set_a ) ).
% assms
thf(fact_392__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062t_Au_O_At_A_092_060in_062_A_092_060T_062_A_092_060and_062_Au_A_092_060in_062_A_092_060U_062_A_092_060and_062_At_A_092_060frown_062_Au_A_092_060and_062_A_092_060T_062_A_092_060lbrace_062_092_092_060rbrace_062_A_092_060U_062_A_061_A_092_060lbrace_062t_A_092_Au_092_060rbrace_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [T3: a,U5: a] :
~ ( ( member_a @ T3 @ t )
& ( member_a @ U5 @ u )
& ( con_a @ resid @ T3 @ U5 )
& ( ( quotie8165075472272353145esid_a @ resid @ nn @ t @ u )
= ( normal7408713899360725774lass_a @ resid @ nn @ ( resid @ T3 @ U5 ) ) ) ) ).
% \<open>\<And>thesis. (\<And>t u. t \<in> \<T> \<and> u \<in> \<U> \<and> t \<frown> u \<and> \<T> \<lbrace>\\<rbrace> \<U> = \<lbrace>t \ u\<rbrace> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_393__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062t_H_Av_O_At_H_A_092_060in_062_A_092_060T_062_A_092_060and_062_Av_A_092_060in_062_A_092_060V_062_A_092_060and_062_At_H_A_092_060frown_062_Av_A_092_060and_062_A_092_060T_062_A_092_060lbrace_062_092_092_060rbrace_062_A_092_060V_062_A_061_A_092_060lbrace_062t_H_A_092_Av_092_060rbrace_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [T9: a,V5: a] :
~ ( ( member_a @ T9 @ t )
& ( member_a @ V5 @ v )
& ( con_a @ resid @ T9 @ V5 )
& ( ( quotie8165075472272353145esid_a @ resid @ nn @ t @ v )
= ( normal7408713899360725774lass_a @ resid @ nn @ ( resid @ T9 @ V5 ) ) ) ) ).
% \<open>\<And>thesis. (\<And>t' v. t' \<in> \<T> \<and> v \<in> \<V> \<and> t' \<frown> v \<and> \<T> \<lbrace>\\<rbrace> \<V> = \<lbrace>t' \ v\<rbrace> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_394__C1_C,axiom,
( ( ( quotie8165075472272353145esid_a @ resid @ nn @ u @ t )
= ( normal7408713899360725774lass_a @ resid @ nn @ ( resid @ u2 @ t3 ) ) )
& ( ( quotie8165075472272353145esid_a @ resid @ nn @ v @ t )
= ( normal7408713899360725774lass_a @ resid @ nn @ ( resid @ v2 @ t2 ) ) ) ) ).
% "1"
thf(fact_395_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_396_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_397_partial__magma_Onull_Ocong,axiom,
partial_null_set_a = partial_null_set_a ).
% partial_magma.null.cong
thf(fact_398_partial__magma_Onull_Ocong,axiom,
partial_null_a = partial_null_a ).
% partial_magma.null.cong
thf(fact_399_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_400_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_401_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_402_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_403_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_404_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_405_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_406_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_407_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_408_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_409_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_410_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_411_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_412_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_413_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_414_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_415_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_416_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_417_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_418_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_419_quotient__by__coherent__normal_Onull__char,axiom,
! [Resid: product_prod_a_a > product_prod_a_a > product_prod_a_a,NN: set_Product_prod_a_a] :
( ( quotie2905600318227040575od_a_a @ Resid @ NN )
=> ( ( partia768789332685730093od_a_a @ ( quotie4807335555532242594od_a_a @ Resid @ NN ) )
= bot_bo3357376287454694259od_a_a ) ) ).
% quotient_by_coherent_normal.null_char
thf(fact_420_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_421_quotient__by__coherent__normal_Onull__char,axiom,
! [Resid: ( a > a ) > ( a > a ) > a > a,NN: set_a_a] :
( ( quotie8968171584129522765al_a_a @ Resid @ NN )
=> ( ( partial_null_set_a_a @ ( quotie4957313913467096874id_a_a @ Resid @ NN ) )
= bot_bot_set_a_a ) ) ).
% quotient_by_coherent_normal.null_char
thf(fact_422_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_423_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 ) ) )
=> ( ! [A3: set_a,T3: set_a] :
( ( ide_set_a @ Resid @ A3 )
=> ( ( con_set_a @ Resid @ T3 @ A3 )
=> ( ( Resid @ T3 @ A3 )
= T3 ) ) )
=> ( ! [A3: set_a,T3: set_a] :
( ( ide_set_a @ Resid @ A3 )
=> ( ( con_set_a @ Resid @ A3 @ T3 )
=> ( ide_set_a @ Resid @ ( Resid @ A3 @ T3 ) ) ) )
=> ( ! [T3: set_a,U5: set_a] :
( ( con_set_a @ Resid @ T3 @ U5 )
=> ? [A6: set_a] :
( ( ide_set_a @ Resid @ A6 )
& ( con_set_a @ Resid @ A6 @ T3 )
& ( con_set_a @ Resid @ A6 @ U5 ) ) )
=> ( ! [T3: set_a,U5: set_a,V5: set_a] :
( ( ide_set_a @ Resid @ ( Resid @ T3 @ U5 ) )
=> ( ( con_set_a @ Resid @ U5 @ V5 )
=> ( con_set_a @ Resid @ ( Resid @ T3 @ U5 ) @ ( Resid @ V5 @ U5 ) ) ) )
=> ( rts_axioms_set_a @ Resid ) ) ) ) ) ) ).
% rts_axioms.intro
thf(fact_424_rts__axioms_Ointro,axiom,
! [Resid: a > a > a] :
( ! [T3: a] :
( ( arr_a @ Resid @ T3 )
=> ( ide_a @ Resid @ ( trg_a @ Resid @ T3 ) ) )
=> ( ! [A3: a,T3: a] :
( ( ide_a @ Resid @ A3 )
=> ( ( con_a @ Resid @ T3 @ A3 )
=> ( ( Resid @ T3 @ A3 )
= T3 ) ) )
=> ( ! [A3: a,T3: a] :
( ( ide_a @ Resid @ A3 )
=> ( ( con_a @ Resid @ A3 @ T3 )
=> ( ide_a @ Resid @ ( Resid @ A3 @ T3 ) ) ) )
=> ( ! [T3: a,U5: a] :
( ( con_a @ Resid @ T3 @ U5 )
=> ? [A6: a] :
( ( ide_a @ Resid @ A6 )
& ( con_a @ Resid @ A6 @ T3 )
& ( con_a @ Resid @ A6 @ U5 ) ) )
=> ( ! [T3: a,U5: a,V5: a] :
( ( ide_a @ Resid @ ( Resid @ T3 @ U5 ) )
=> ( ( con_a @ Resid @ U5 @ V5 )
=> ( con_a @ Resid @ ( Resid @ T3 @ U5 ) @ ( Resid @ V5 @ U5 ) ) ) )
=> ( rts_axioms_a @ Resid ) ) ) ) ) ) ).
% rts_axioms.intro
thf(fact_425_rts__axioms__def,axiom,
( rts_axioms_set_a
= ( ^ [Resid2: set_a > set_a > set_a] :
( ! [T5: set_a] :
( ( arr_set_a @ Resid2 @ T5 )
=> ( ide_set_a @ Resid2 @ ( trg_set_a @ Resid2 @ T5 ) ) )
& ! [A5: set_a,T5: set_a] :
( ( ide_set_a @ Resid2 @ A5 )
=> ( ( con_set_a @ Resid2 @ T5 @ A5 )
=> ( ( Resid2 @ T5 @ A5 )
= T5 ) ) )
& ! [A5: set_a,T5: set_a] :
( ( ide_set_a @ Resid2 @ A5 )
=> ( ( con_set_a @ Resid2 @ A5 @ T5 )
=> ( ide_set_a @ Resid2 @ ( Resid2 @ A5 @ T5 ) ) ) )
& ! [T5: set_a,U3: set_a] :
( ( con_set_a @ Resid2 @ T5 @ U3 )
=> ? [A5: set_a] :
( ( ide_set_a @ Resid2 @ A5 )
& ( con_set_a @ Resid2 @ A5 @ T5 )
& ( con_set_a @ Resid2 @ A5 @ U3 ) ) )
& ! [T5: set_a,U3: set_a,V3: set_a] :
( ( ide_set_a @ Resid2 @ ( Resid2 @ T5 @ U3 ) )
=> ( ( con_set_a @ Resid2 @ U3 @ V3 )
=> ( con_set_a @ Resid2 @ ( Resid2 @ T5 @ U3 ) @ ( Resid2 @ V3 @ U3 ) ) ) ) ) ) ) ).
% rts_axioms_def
thf(fact_426_rts__axioms__def,axiom,
( rts_axioms_a
= ( ^ [Resid2: a > a > a] :
( ! [T5: a] :
( ( arr_a @ Resid2 @ T5 )
=> ( ide_a @ Resid2 @ ( trg_a @ Resid2 @ T5 ) ) )
& ! [A5: a,T5: a] :
( ( ide_a @ Resid2 @ A5 )
=> ( ( con_a @ Resid2 @ T5 @ A5 )
=> ( ( Resid2 @ T5 @ A5 )
= T5 ) ) )
& ! [A5: a,T5: a] :
( ( ide_a @ Resid2 @ A5 )
=> ( ( con_a @ Resid2 @ A5 @ T5 )
=> ( ide_a @ Resid2 @ ( Resid2 @ A5 @ T5 ) ) ) )
& ! [T5: a,U3: a] :
( ( con_a @ Resid2 @ T5 @ U3 )
=> ? [A5: a] :
( ( ide_a @ Resid2 @ A5 )
& ( con_a @ Resid2 @ A5 @ T5 )
& ( con_a @ Resid2 @ A5 @ U3 ) ) )
& ! [T5: a,U3: a,V3: a] :
( ( ide_a @ Resid2 @ ( Resid2 @ T5 @ U3 ) )
=> ( ( con_a @ Resid2 @ U3 @ V3 )
=> ( con_a @ Resid2 @ ( Resid2 @ T5 @ U3 ) @ ( Resid2 @ V3 @ U3 ) ) ) ) ) ) ) ).
% rts_axioms_def
thf(fact_427_empty__iff,axiom,
! [C: product_prod_a_a] :
~ ( member1426531477525435216od_a_a @ C @ bot_bo3357376287454694259od_a_a ) ).
% empty_iff
thf(fact_428_empty__iff,axiom,
! [C: set_a] :
~ ( member_set_a @ C @ bot_bot_set_set_a ) ).
% empty_iff
thf(fact_429_empty__iff,axiom,
! [C: a > a] :
~ ( member_a_a @ C @ bot_bot_set_a_a ) ).
% empty_iff
thf(fact_430_empty__iff,axiom,
! [C: a] :
~ ( member_a @ C @ bot_bot_set_a ) ).
% empty_iff
thf(fact_431_all__not__in__conv,axiom,
! [A4: set_Product_prod_a_a] :
( ( ! [X2: product_prod_a_a] :
~ ( member1426531477525435216od_a_a @ X2 @ A4 ) )
= ( A4 = bot_bo3357376287454694259od_a_a ) ) ).
% all_not_in_conv
thf(fact_432_all__not__in__conv,axiom,
! [A4: set_set_a] :
( ( ! [X2: set_a] :
~ ( member_set_a @ X2 @ A4 ) )
= ( A4 = bot_bot_set_set_a ) ) ).
% all_not_in_conv
thf(fact_433_all__not__in__conv,axiom,
! [A4: set_a_a] :
( ( ! [X2: a > a] :
~ ( member_a_a @ X2 @ A4 ) )
= ( A4 = bot_bot_set_a_a ) ) ).
% all_not_in_conv
thf(fact_434_all__not__in__conv,axiom,
! [A4: set_a] :
( ( ! [X2: a] :
~ ( member_a @ X2 @ A4 ) )
= ( A4 = bot_bot_set_a ) ) ).
% all_not_in_conv
thf(fact_435_Collect__empty__eq,axiom,
! [P: product_prod_a_a > $o] :
( ( ( collec3336397797384452498od_a_a @ P )
= bot_bo3357376287454694259od_a_a )
= ( ! [X2: product_prod_a_a] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_436_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_437_Collect__empty__eq,axiom,
! [P: ( a > a ) > $o] :
( ( ( collect_a_a @ P )
= bot_bot_set_a_a )
= ( ! [X2: a > a] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_438_Collect__empty__eq,axiom,
! [P: a > $o] :
( ( ( collect_a @ P )
= bot_bot_set_a )
= ( ! [X2: a] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_439_empty__Collect__eq,axiom,
! [P: product_prod_a_a > $o] :
( ( bot_bo3357376287454694259od_a_a
= ( collec3336397797384452498od_a_a @ P ) )
= ( ! [X2: product_prod_a_a] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_440_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_441_empty__Collect__eq,axiom,
! [P: ( a > a ) > $o] :
( ( bot_bot_set_a_a
= ( collect_a_a @ P ) )
= ( ! [X2: a > a] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_442_empty__Collect__eq,axiom,
! [P: a > $o] :
( ( bot_bot_set_a
= ( collect_a @ P ) )
= ( ! [X2: a] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_443_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_444_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_445_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_446_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_447_R_OconE,axiom,
! [T: a,U: a] :
( ( con_a @ resid @ T @ U )
=> ( ( resid @ T @ U )
!= ( partial_null_a @ resid ) ) ) ).
% R.conE
thf(fact_448_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_449_R_Onot__arr__null,axiom,
~ ( arr_a @ resid @ ( partial_null_a @ resid ) ) ).
% R.not_arr_null
thf(fact_450_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_451_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_452_R_OconI,axiom,
! [T: a,U: a] :
( ( ( resid @ T @ U )
!= ( partial_null_a @ resid ) )
=> ( con_a @ resid @ T @ U ) ) ).
% R.conI
thf(fact_453_ex__in__conv,axiom,
! [A4: set_Product_prod_a_a] :
( ( ? [X2: product_prod_a_a] : ( member1426531477525435216od_a_a @ X2 @ A4 ) )
= ( A4 != bot_bo3357376287454694259od_a_a ) ) ).
% ex_in_conv
thf(fact_454_ex__in__conv,axiom,
! [A4: set_set_a] :
( ( ? [X2: set_a] : ( member_set_a @ X2 @ A4 ) )
= ( A4 != bot_bot_set_set_a ) ) ).
% ex_in_conv
thf(fact_455_ex__in__conv,axiom,
! [A4: set_a_a] :
( ( ? [X2: a > a] : ( member_a_a @ X2 @ A4 ) )
= ( A4 != bot_bot_set_a_a ) ) ).
% ex_in_conv
thf(fact_456_ex__in__conv,axiom,
! [A4: set_a] :
( ( ? [X2: a] : ( member_a @ X2 @ A4 ) )
= ( A4 != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_457_equals0I,axiom,
! [A4: set_Product_prod_a_a] :
( ! [Y3: product_prod_a_a] :
~ ( member1426531477525435216od_a_a @ Y3 @ A4 )
=> ( A4 = bot_bo3357376287454694259od_a_a ) ) ).
% equals0I
thf(fact_458_equals0I,axiom,
! [A4: set_set_a] :
( ! [Y3: set_a] :
~ ( member_set_a @ Y3 @ A4 )
=> ( A4 = bot_bot_set_set_a ) ) ).
% equals0I
thf(fact_459_equals0I,axiom,
! [A4: set_a_a] :
( ! [Y3: a > a] :
~ ( member_a_a @ Y3 @ A4 )
=> ( A4 = bot_bot_set_a_a ) ) ).
% equals0I
thf(fact_460_equals0I,axiom,
! [A4: set_a] :
( ! [Y3: a] :
~ ( member_a @ Y3 @ A4 )
=> ( A4 = bot_bot_set_a ) ) ).
% equals0I
thf(fact_461_equals0D,axiom,
! [A4: set_Product_prod_a_a,A: product_prod_a_a] :
( ( A4 = bot_bo3357376287454694259od_a_a )
=> ~ ( member1426531477525435216od_a_a @ A @ A4 ) ) ).
% equals0D
thf(fact_462_equals0D,axiom,
! [A4: set_set_a,A: set_a] :
( ( A4 = bot_bot_set_set_a )
=> ~ ( member_set_a @ A @ A4 ) ) ).
% equals0D
thf(fact_463_equals0D,axiom,
! [A4: set_a_a,A: a > a] :
( ( A4 = bot_bot_set_a_a )
=> ~ ( member_a_a @ A @ A4 ) ) ).
% equals0D
thf(fact_464_equals0D,axiom,
! [A4: set_a,A: a] :
( ( A4 = bot_bot_set_a )
=> ~ ( member_a @ A @ A4 ) ) ).
% equals0D
thf(fact_465_emptyE,axiom,
! [A: product_prod_a_a] :
~ ( member1426531477525435216od_a_a @ A @ bot_bo3357376287454694259od_a_a ) ).
% emptyE
thf(fact_466_emptyE,axiom,
! [A: set_a] :
~ ( member_set_a @ A @ bot_bot_set_set_a ) ).
% emptyE
thf(fact_467_emptyE,axiom,
! [A: a > a] :
~ ( member_a_a @ A @ bot_bot_set_a_a ) ).
% emptyE
thf(fact_468_emptyE,axiom,
! [A: a] :
~ ( member_a @ A @ bot_bot_set_a ) ).
% emptyE
thf(fact_469_bot__set__def,axiom,
( bot_bo3357376287454694259od_a_a
= ( collec3336397797384452498od_a_a @ bot_bo4160289986317612842_a_a_o ) ) ).
% bot_set_def
thf(fact_470_bot__set__def,axiom,
( bot_bot_set_set_a
= ( collect_set_a @ bot_bot_set_a_o ) ) ).
% bot_set_def
thf(fact_471_bot__set__def,axiom,
( bot_bot_set_a_a
= ( collect_a_a @ bot_bot_a_a_o ) ) ).
% bot_set_def
thf(fact_472_bot__set__def,axiom,
( bot_bot_set_a
= ( collect_a @ bot_bot_a_o ) ) ).
% bot_set_def
thf(fact_473_Int__emptyI,axiom,
! [A4: set_Product_prod_a_a,B3: set_Product_prod_a_a] :
( ! [X: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X @ A4 )
=> ~ ( member1426531477525435216od_a_a @ X @ B3 ) )
=> ( ( inf_in8905007599844390133od_a_a @ A4 @ B3 )
= bot_bo3357376287454694259od_a_a ) ) ).
% Int_emptyI
thf(fact_474_Int__emptyI,axiom,
! [A4: set_set_a,B3: set_set_a] :
( ! [X: set_a] :
( ( member_set_a @ X @ A4 )
=> ~ ( member_set_a @ X @ B3 ) )
=> ( ( inf_inf_set_set_a @ A4 @ B3 )
= bot_bot_set_set_a ) ) ).
% Int_emptyI
thf(fact_475_Int__emptyI,axiom,
! [A4: set_a_a,B3: set_a_a] :
( ! [X: a > a] :
( ( member_a_a @ X @ A4 )
=> ~ ( member_a_a @ X @ B3 ) )
=> ( ( inf_inf_set_a_a @ A4 @ B3 )
= bot_bot_set_a_a ) ) ).
% Int_emptyI
thf(fact_476_Int__emptyI,axiom,
! [A4: set_a,B3: set_a] :
( ! [X: a] :
( ( member_a @ X @ A4 )
=> ~ ( member_a @ X @ B3 ) )
=> ( ( inf_inf_set_a @ A4 @ B3 )
= bot_bot_set_a ) ) ).
% Int_emptyI
thf(fact_477_disjoint__iff,axiom,
! [A4: set_Product_prod_a_a,B3: set_Product_prod_a_a] :
( ( ( inf_in8905007599844390133od_a_a @ A4 @ B3 )
= bot_bo3357376287454694259od_a_a )
= ( ! [X2: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X2 @ A4 )
=> ~ ( member1426531477525435216od_a_a @ X2 @ B3 ) ) ) ) ).
% disjoint_iff
thf(fact_478_disjoint__iff,axiom,
! [A4: set_set_a,B3: set_set_a] :
( ( ( inf_inf_set_set_a @ A4 @ B3 )
= bot_bot_set_set_a )
= ( ! [X2: set_a] :
( ( member_set_a @ X2 @ A4 )
=> ~ ( member_set_a @ X2 @ B3 ) ) ) ) ).
% disjoint_iff
thf(fact_479_disjoint__iff,axiom,
! [A4: set_a_a,B3: set_a_a] :
( ( ( inf_inf_set_a_a @ A4 @ B3 )
= bot_bot_set_a_a )
= ( ! [X2: a > a] :
( ( member_a_a @ X2 @ A4 )
=> ~ ( member_a_a @ X2 @ B3 ) ) ) ) ).
% disjoint_iff
thf(fact_480_disjoint__iff,axiom,
! [A4: set_a,B3: set_a] :
( ( ( inf_inf_set_a @ A4 @ B3 )
= bot_bot_set_a )
= ( ! [X2: a] :
( ( member_a @ X2 @ A4 )
=> ~ ( member_a @ X2 @ B3 ) ) ) ) ).
% disjoint_iff
thf(fact_481_Int__empty__left,axiom,
! [B3: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ bot_bo3357376287454694259od_a_a @ B3 )
= bot_bo3357376287454694259od_a_a ) ).
% Int_empty_left
thf(fact_482_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_483_Int__empty__left,axiom,
! [B3: set_a_a] :
( ( inf_inf_set_a_a @ bot_bot_set_a_a @ B3 )
= bot_bot_set_a_a ) ).
% Int_empty_left
thf(fact_484_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_485_Int__empty__right,axiom,
! [A4: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ A4 @ bot_bo3357376287454694259od_a_a )
= bot_bo3357376287454694259od_a_a ) ).
% Int_empty_right
thf(fact_486_Int__empty__right,axiom,
! [A4: set_set_a] :
( ( inf_inf_set_set_a @ A4 @ bot_bot_set_set_a )
= bot_bot_set_set_a ) ).
% Int_empty_right
thf(fact_487_Int__empty__right,axiom,
! [A4: set_a_a] :
( ( inf_inf_set_a_a @ A4 @ bot_bot_set_a_a )
= bot_bot_set_a_a ) ).
% Int_empty_right
thf(fact_488_Int__empty__right,axiom,
! [A4: set_a] :
( ( inf_inf_set_a @ A4 @ bot_bot_set_a )
= bot_bot_set_a ) ).
% Int_empty_right
thf(fact_489_disjoint__iff__not__equal,axiom,
! [A4: set_Product_prod_a_a,B3: set_Product_prod_a_a] :
( ( ( inf_in8905007599844390133od_a_a @ A4 @ B3 )
= bot_bo3357376287454694259od_a_a )
= ( ! [X2: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X2 @ A4 )
=> ! [Y2: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Y2 @ B3 )
=> ( X2 != Y2 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_490_disjoint__iff__not__equal,axiom,
! [A4: set_set_a,B3: set_set_a] :
( ( ( inf_inf_set_set_a @ A4 @ B3 )
= bot_bot_set_set_a )
= ( ! [X2: set_a] :
( ( member_set_a @ X2 @ A4 )
=> ! [Y2: set_a] :
( ( member_set_a @ Y2 @ B3 )
=> ( X2 != Y2 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_491_disjoint__iff__not__equal,axiom,
! [A4: set_a_a,B3: set_a_a] :
( ( ( inf_inf_set_a_a @ A4 @ B3 )
= bot_bot_set_a_a )
= ( ! [X2: a > a] :
( ( member_a_a @ X2 @ A4 )
=> ! [Y2: a > a] :
( ( member_a_a @ Y2 @ B3 )
=> ( X2 != Y2 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_492_disjoint__iff__not__equal,axiom,
! [A4: set_a,B3: set_a] :
( ( ( inf_inf_set_a @ A4 @ B3 )
= bot_bot_set_a )
= ( ! [X2: a] :
( ( member_a @ X2 @ A4 )
=> ! [Y2: a] :
( ( member_a @ Y2 @ B3 )
=> ( X2 != Y2 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_493_bot__apply,axiom,
( bot_bot_a_o
= ( ^ [X2: a] : bot_bot_o ) ) ).
% bot_apply
thf(fact_494_simulation__axioms__def,axiom,
( simula8704200824037452966_set_a
= ( ^ [A7: set_a > set_a > set_a,B4: set_a > set_a > set_a,F: set_a > set_a] :
( ! [T5: set_a] :
( ~ ( arr_set_a @ A7 @ T5 )
=> ( ( F @ T5 )
= ( partial_null_set_a @ B4 ) ) )
& ! [T5: set_a,U3: set_a] :
( ( con_set_a @ A7 @ T5 @ U3 )
=> ( con_set_a @ B4 @ ( F @ T5 ) @ ( F @ U3 ) ) )
& ! [T5: set_a,U3: set_a] :
( ( con_set_a @ A7 @ T5 @ U3 )
=> ( ( F @ ( A7 @ T5 @ U3 ) )
= ( B4 @ ( F @ T5 ) @ ( F @ U3 ) ) ) ) ) ) ) ).
% simulation_axioms_def
thf(fact_495_simulation__axioms__def,axiom,
( simula3408835310535287622et_a_a
= ( ^ [A7: set_a > set_a > set_a,B4: a > a > a,F: set_a > a] :
( ! [T5: set_a] :
( ~ ( arr_set_a @ A7 @ T5 )
=> ( ( F @ T5 )
= ( partial_null_a @ B4 ) ) )
& ! [T5: set_a,U3: set_a] :
( ( con_set_a @ A7 @ T5 @ U3 )
=> ( con_a @ B4 @ ( F @ T5 ) @ ( F @ U3 ) ) )
& ! [T5: set_a,U3: set_a] :
( ( con_set_a @ A7 @ T5 @ U3 )
=> ( ( F @ ( A7 @ T5 @ U3 ) )
= ( B4 @ ( F @ T5 ) @ ( F @ U3 ) ) ) ) ) ) ) ).
% simulation_axioms_def
thf(fact_496_simulation__axioms__def,axiom,
( simula3192323252075944454_set_a
= ( ^ [A7: a > a > a,B4: set_a > set_a > set_a,F: a > set_a] :
( ! [T5: a] :
( ~ ( arr_a @ A7 @ T5 )
=> ( ( F @ T5 )
= ( partial_null_set_a @ B4 ) ) )
& ! [T5: a,U3: a] :
( ( con_a @ A7 @ T5 @ U3 )
=> ( con_set_a @ B4 @ ( F @ T5 ) @ ( F @ U3 ) ) )
& ! [T5: a,U3: a] :
( ( con_a @ A7 @ T5 @ U3 )
=> ( ( F @ ( A7 @ T5 @ U3 ) )
= ( B4 @ ( F @ T5 ) @ ( F @ U3 ) ) ) ) ) ) ) ).
% simulation_axioms_def
thf(fact_497_simulation__axioms__def,axiom,
( simula3868467710248865958ms_a_a
= ( ^ [A7: a > a > a,B4: a > a > a,F: a > a] :
( ! [T5: a] :
( ~ ( arr_a @ A7 @ T5 )
=> ( ( F @ T5 )
= ( partial_null_a @ B4 ) ) )
& ! [T5: a,U3: a] :
( ( con_a @ A7 @ T5 @ U3 )
=> ( con_a @ B4 @ ( F @ T5 ) @ ( F @ U3 ) ) )
& ! [T5: a,U3: a] :
( ( con_a @ A7 @ T5 @ U3 )
=> ( ( F @ ( A7 @ T5 @ U3 ) )
= ( B4 @ ( F @ T5 ) @ ( F @ U3 ) ) ) ) ) ) ) ).
% simulation_axioms_def
thf(fact_498_simulation__axioms_Ointro,axiom,
! [A4: set_a > set_a > set_a,F2: set_a > set_a,B3: set_a > set_a > set_a] :
( ! [T3: set_a] :
( ~ ( arr_set_a @ A4 @ T3 )
=> ( ( F2 @ T3 )
= ( partial_null_set_a @ B3 ) ) )
=> ( ! [T3: set_a,U5: set_a] :
( ( con_set_a @ A4 @ T3 @ U5 )
=> ( con_set_a @ B3 @ ( F2 @ T3 ) @ ( F2 @ U5 ) ) )
=> ( ! [T3: set_a,U5: set_a] :
( ( con_set_a @ A4 @ T3 @ U5 )
=> ( ( F2 @ ( A4 @ T3 @ U5 ) )
= ( B3 @ ( F2 @ T3 ) @ ( F2 @ U5 ) ) ) )
=> ( simula8704200824037452966_set_a @ A4 @ B3 @ F2 ) ) ) ) ).
% simulation_axioms.intro
thf(fact_499_simulation__axioms_Ointro,axiom,
! [A4: set_a > set_a > set_a,F2: set_a > a,B3: a > a > a] :
( ! [T3: set_a] :
( ~ ( arr_set_a @ A4 @ T3 )
=> ( ( F2 @ T3 )
= ( partial_null_a @ B3 ) ) )
=> ( ! [T3: set_a,U5: set_a] :
( ( con_set_a @ A4 @ T3 @ U5 )
=> ( con_a @ B3 @ ( F2 @ T3 ) @ ( F2 @ U5 ) ) )
=> ( ! [T3: set_a,U5: set_a] :
( ( con_set_a @ A4 @ T3 @ U5 )
=> ( ( F2 @ ( A4 @ T3 @ U5 ) )
= ( B3 @ ( F2 @ T3 ) @ ( F2 @ U5 ) ) ) )
=> ( simula3408835310535287622et_a_a @ A4 @ B3 @ F2 ) ) ) ) ).
% simulation_axioms.intro
thf(fact_500_simulation__axioms_Ointro,axiom,
! [A4: a > a > a,F2: a > set_a,B3: set_a > set_a > set_a] :
( ! [T3: a] :
( ~ ( arr_a @ A4 @ T3 )
=> ( ( F2 @ T3 )
= ( partial_null_set_a @ B3 ) ) )
=> ( ! [T3: a,U5: a] :
( ( con_a @ A4 @ T3 @ U5 )
=> ( con_set_a @ B3 @ ( F2 @ T3 ) @ ( F2 @ U5 ) ) )
=> ( ! [T3: a,U5: a] :
( ( con_a @ A4 @ T3 @ U5 )
=> ( ( F2 @ ( A4 @ T3 @ U5 ) )
= ( B3 @ ( F2 @ T3 ) @ ( F2 @ U5 ) ) ) )
=> ( simula3192323252075944454_set_a @ A4 @ B3 @ F2 ) ) ) ) ).
% simulation_axioms.intro
thf(fact_501_simulation__axioms_Ointro,axiom,
! [A4: a > a > a,F2: a > a,B3: a > a > a] :
( ! [T3: a] :
( ~ ( arr_a @ A4 @ T3 )
=> ( ( F2 @ T3 )
= ( partial_null_a @ B3 ) ) )
=> ( ! [T3: a,U5: a] :
( ( con_a @ A4 @ T3 @ U5 )
=> ( con_a @ B3 @ ( F2 @ T3 ) @ ( F2 @ U5 ) ) )
=> ( ! [T3: a,U5: a] :
( ( con_a @ A4 @ T3 @ U5 )
=> ( ( F2 @ ( A4 @ T3 @ U5 ) )
= ( B3 @ ( F2 @ T3 ) @ ( F2 @ U5 ) ) ) )
=> ( simula3868467710248865958ms_a_a @ A4 @ B3 @ F2 ) ) ) ) ).
% simulation_axioms.intro
thf(fact_502_coherent__normal__sub__rts__axioms_Ointro,axiom,
! [Resid: product_prod_a_a > product_prod_a_a > product_prod_a_a,NN: set_Product_prod_a_a] :
( ! [T3: product_prod_a_a,U5: product_prod_a_a,U6: product_prod_a_a] :
( ( arr_Product_prod_a_a @ Resid @ T3 )
=> ( ( member1426531477525435216od_a_a @ U5 @ NN )
=> ( ( member1426531477525435216od_a_a @ U6 @ NN )
=> ( ( ( source6950040787684646355od_a_a @ Resid @ U5 )
= ( source6950040787684646355od_a_a @ Resid @ U6 ) )
=> ( ( ( target5293506191220573129od_a_a @ Resid @ U5 )
= ( target5293506191220573129od_a_a @ Resid @ U6 ) )
=> ( ( ( source6950040787684646355od_a_a @ Resid @ T3 )
= ( source6950040787684646355od_a_a @ Resid @ U5 ) )
=> ( ( member1426531477525435216od_a_a @ ( Resid @ ( Resid @ T3 @ U5 ) @ ( Resid @ T3 @ U6 ) ) @ NN )
& ( member1426531477525435216od_a_a @ ( Resid @ ( Resid @ T3 @ U6 ) @ ( Resid @ T3 @ U5 ) ) @ NN ) ) ) ) ) ) ) )
=> ( cohere5271700870576340781od_a_a @ Resid @ NN ) ) ).
% coherent_normal_sub_rts_axioms.intro
thf(fact_503_coherent__normal__sub__rts__axioms_Ointro,axiom,
! [Resid: ( a > a ) > ( a > a ) > a > a,NN: set_a_a] :
( ! [T3: a > a,U5: a > a,U6: a > a] :
( ( arr_a_a @ Resid @ T3 )
=> ( ( member_a_a @ U5 @ NN )
=> ( ( member_a_a @ U6 @ NN )
=> ( ( ( sources_a_a @ Resid @ U5 )
= ( sources_a_a @ Resid @ U6 ) )
=> ( ( ( targets_a_a @ Resid @ U5 )
= ( targets_a_a @ Resid @ U6 ) )
=> ( ( ( sources_a_a @ Resid @ T3 )
= ( sources_a_a @ Resid @ U5 ) )
=> ( ( member_a_a @ ( Resid @ ( Resid @ T3 @ U5 ) @ ( Resid @ T3 @ U6 ) ) @ NN )
& ( member_a_a @ ( Resid @ ( Resid @ T3 @ U6 ) @ ( Resid @ T3 @ U5 ) ) @ NN ) ) ) ) ) ) ) )
=> ( cohere4772091081940722911ms_a_a @ Resid @ NN ) ) ).
% coherent_normal_sub_rts_axioms.intro
thf(fact_504_coherent__normal__sub__rts__axioms_Ointro,axiom,
! [Resid: set_a > set_a > set_a,NN: set_set_a] :
( ! [T3: set_a,U5: set_a,U6: set_a] :
( ( arr_set_a @ Resid @ T3 )
=> ( ( member_set_a @ U5 @ NN )
=> ( ( member_set_a @ U6 @ NN )
=> ( ( ( sources_set_a @ Resid @ U5 )
= ( sources_set_a @ Resid @ U6 ) )
=> ( ( ( targets_set_a @ Resid @ U5 )
= ( targets_set_a @ Resid @ U6 ) )
=> ( ( ( sources_set_a @ Resid @ T3 )
= ( sources_set_a @ Resid @ U5 ) )
=> ( ( member_set_a @ ( Resid @ ( Resid @ T3 @ U5 ) @ ( Resid @ T3 @ U6 ) ) @ NN )
& ( member_set_a @ ( Resid @ ( Resid @ T3 @ U6 ) @ ( Resid @ T3 @ U5 ) ) @ NN ) ) ) ) ) ) ) )
=> ( cohere32089786014956644_set_a @ Resid @ NN ) ) ).
% coherent_normal_sub_rts_axioms.intro
thf(fact_505_coherent__normal__sub__rts__axioms_Ointro,axiom,
! [Resid: a > a > a,NN: set_a] :
( ! [T3: a,U5: a,U6: a] :
( ( arr_a @ Resid @ T3 )
=> ( ( member_a @ U5 @ NN )
=> ( ( member_a @ U6 @ NN )
=> ( ( ( sources_a @ Resid @ U5 )
= ( sources_a @ Resid @ U6 ) )
=> ( ( ( targets_a @ Resid @ U5 )
= ( targets_a @ Resid @ U6 ) )
=> ( ( ( sources_a @ Resid @ T3 )
= ( sources_a @ Resid @ U5 ) )
=> ( ( member_a @ ( Resid @ ( Resid @ T3 @ U5 ) @ ( Resid @ T3 @ U6 ) ) @ NN )
& ( member_a @ ( Resid @ ( Resid @ T3 @ U6 ) @ ( Resid @ T3 @ U5 ) ) @ NN ) ) ) ) ) ) ) )
=> ( cohere4894532172567702276ioms_a @ Resid @ NN ) ) ).
% coherent_normal_sub_rts_axioms.intro
thf(fact_506_coherent__normal__sub__rts__axioms__def,axiom,
( cohere5271700870576340781od_a_a
= ( ^ [Resid2: product_prod_a_a > product_prod_a_a > product_prod_a_a,NN2: set_Product_prod_a_a] :
! [T5: product_prod_a_a,U3: product_prod_a_a,U4: product_prod_a_a] :
( ( arr_Product_prod_a_a @ Resid2 @ T5 )
=> ( ( member1426531477525435216od_a_a @ U3 @ NN2 )
=> ( ( member1426531477525435216od_a_a @ U4 @ NN2 )
=> ( ( ( source6950040787684646355od_a_a @ Resid2 @ U3 )
= ( source6950040787684646355od_a_a @ Resid2 @ U4 ) )
=> ( ( ( target5293506191220573129od_a_a @ Resid2 @ U3 )
= ( target5293506191220573129od_a_a @ Resid2 @ U4 ) )
=> ( ( ( source6950040787684646355od_a_a @ Resid2 @ T5 )
= ( source6950040787684646355od_a_a @ Resid2 @ U3 ) )
=> ( ( member1426531477525435216od_a_a @ ( Resid2 @ ( Resid2 @ T5 @ U3 ) @ ( Resid2 @ T5 @ U4 ) ) @ NN2 )
& ( member1426531477525435216od_a_a @ ( Resid2 @ ( Resid2 @ T5 @ U4 ) @ ( Resid2 @ T5 @ U3 ) ) @ NN2 ) ) ) ) ) ) ) ) ) ) ).
% coherent_normal_sub_rts_axioms_def
thf(fact_507_coherent__normal__sub__rts__axioms__def,axiom,
( cohere4772091081940722911ms_a_a
= ( ^ [Resid2: ( a > a ) > ( a > a ) > a > a,NN2: set_a_a] :
! [T5: a > a,U3: a > a,U4: a > a] :
( ( arr_a_a @ Resid2 @ T5 )
=> ( ( member_a_a @ U3 @ NN2 )
=> ( ( member_a_a @ U4 @ NN2 )
=> ( ( ( sources_a_a @ Resid2 @ U3 )
= ( sources_a_a @ Resid2 @ U4 ) )
=> ( ( ( targets_a_a @ Resid2 @ U3 )
= ( targets_a_a @ Resid2 @ U4 ) )
=> ( ( ( sources_a_a @ Resid2 @ T5 )
= ( sources_a_a @ Resid2 @ U3 ) )
=> ( ( member_a_a @ ( Resid2 @ ( Resid2 @ T5 @ U3 ) @ ( Resid2 @ T5 @ U4 ) ) @ NN2 )
& ( member_a_a @ ( Resid2 @ ( Resid2 @ T5 @ U4 ) @ ( Resid2 @ T5 @ U3 ) ) @ NN2 ) ) ) ) ) ) ) ) ) ) ).
% coherent_normal_sub_rts_axioms_def
thf(fact_508_coherent__normal__sub__rts__axioms__def,axiom,
( cohere32089786014956644_set_a
= ( ^ [Resid2: set_a > set_a > set_a,NN2: set_set_a] :
! [T5: set_a,U3: set_a,U4: set_a] :
( ( arr_set_a @ Resid2 @ T5 )
=> ( ( member_set_a @ U3 @ NN2 )
=> ( ( member_set_a @ U4 @ NN2 )
=> ( ( ( sources_set_a @ Resid2 @ U3 )
= ( sources_set_a @ Resid2 @ U4 ) )
=> ( ( ( targets_set_a @ Resid2 @ U3 )
= ( targets_set_a @ Resid2 @ U4 ) )
=> ( ( ( sources_set_a @ Resid2 @ T5 )
= ( sources_set_a @ Resid2 @ U3 ) )
=> ( ( member_set_a @ ( Resid2 @ ( Resid2 @ T5 @ U3 ) @ ( Resid2 @ T5 @ U4 ) ) @ NN2 )
& ( member_set_a @ ( Resid2 @ ( Resid2 @ T5 @ U4 ) @ ( Resid2 @ T5 @ U3 ) ) @ NN2 ) ) ) ) ) ) ) ) ) ) ).
% coherent_normal_sub_rts_axioms_def
thf(fact_509_coherent__normal__sub__rts__axioms__def,axiom,
( cohere4894532172567702276ioms_a
= ( ^ [Resid2: a > a > a,NN2: set_a] :
! [T5: a,U3: a,U4: a] :
( ( arr_a @ Resid2 @ T5 )
=> ( ( member_a @ U3 @ NN2 )
=> ( ( member_a @ U4 @ NN2 )
=> ( ( ( sources_a @ Resid2 @ U3 )
= ( sources_a @ Resid2 @ U4 ) )
=> ( ( ( targets_a @ Resid2 @ U3 )
= ( targets_a @ Resid2 @ U4 ) )
=> ( ( ( sources_a @ Resid2 @ T5 )
= ( sources_a @ Resid2 @ U3 ) )
=> ( ( member_a @ ( Resid2 @ ( Resid2 @ T5 @ U3 ) @ ( Resid2 @ T5 @ U4 ) ) @ NN2 )
& ( member_a @ ( Resid2 @ ( Resid2 @ T5 @ U4 ) @ ( Resid2 @ T5 @ U3 ) ) @ NN2 ) ) ) ) ) ) ) ) ) ) ).
% coherent_normal_sub_rts_axioms_def
thf(fact_510_N_Onormal__sub__rts__axioms,axiom,
normal_sub_rts_a @ resid @ nn ).
% N.normal_sub_rts_axioms
thf(fact_511_normal__sub__rts_Obackward__stable,axiom,
! [Resid: product_prod_a_a > product_prod_a_a > product_prod_a_a,NN: set_Product_prod_a_a,U: product_prod_a_a,T: product_prod_a_a] :
( ( normal2251803270671997330od_a_a @ Resid @ NN )
=> ( ( member1426531477525435216od_a_a @ U @ NN )
=> ( ( member1426531477525435216od_a_a @ ( Resid @ T @ U ) @ NN )
=> ( member1426531477525435216od_a_a @ T @ NN ) ) ) ) ).
% normal_sub_rts.backward_stable
thf(fact_512_normal__sub__rts_Obackward__stable,axiom,
! [Resid: set_a > set_a > set_a,NN: set_set_a,U: set_a,T: set_a] :
( ( normal_sub_rts_set_a @ Resid @ NN )
=> ( ( member_set_a @ U @ NN )
=> ( ( member_set_a @ ( Resid @ T @ U ) @ NN )
=> ( member_set_a @ T @ NN ) ) ) ) ).
% normal_sub_rts.backward_stable
thf(fact_513_normal__sub__rts_Obackward__stable,axiom,
! [Resid: ( a > a ) > ( a > a ) > a > a,NN: set_a_a,U: a > a,T: a > a] :
( ( normal_sub_rts_a_a @ Resid @ NN )
=> ( ( member_a_a @ U @ NN )
=> ( ( member_a_a @ ( Resid @ T @ U ) @ NN )
=> ( member_a_a @ T @ NN ) ) ) ) ).
% normal_sub_rts.backward_stable
thf(fact_514_normal__sub__rts_Obackward__stable,axiom,
! [Resid: a > a > a,NN: set_a,U: a,T: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( member_a @ U @ NN )
=> ( ( member_a @ ( Resid @ T @ U ) @ NN )
=> ( member_a @ T @ NN ) ) ) ) ).
% normal_sub_rts.backward_stable
thf(fact_515_normal__sub__rts_OCong_092_060_094sub_0620__symmetric,axiom,
! [Resid: product_prod_a_a > product_prod_a_a > product_prod_a_a,NN: set_Product_prod_a_a,T: product_prod_a_a,T4: product_prod_a_a] :
( ( normal2251803270671997330od_a_a @ Resid @ NN )
=> ( ( ( member1426531477525435216od_a_a @ ( Resid @ T @ T4 ) @ NN )
& ( member1426531477525435216od_a_a @ ( Resid @ T4 @ T ) @ NN ) )
=> ( ( member1426531477525435216od_a_a @ ( Resid @ T4 @ T ) @ NN )
& ( member1426531477525435216od_a_a @ ( Resid @ T @ T4 ) @ NN ) ) ) ) ).
% normal_sub_rts.Cong\<^sub>0_symmetric
thf(fact_516_normal__sub__rts_OCong_092_060_094sub_0620__symmetric,axiom,
! [Resid: set_a > set_a > set_a,NN: set_set_a,T: set_a,T4: set_a] :
( ( normal_sub_rts_set_a @ Resid @ NN )
=> ( ( ( member_set_a @ ( Resid @ T @ T4 ) @ NN )
& ( member_set_a @ ( Resid @ T4 @ T ) @ NN ) )
=> ( ( member_set_a @ ( Resid @ T4 @ T ) @ NN )
& ( member_set_a @ ( Resid @ T @ T4 ) @ NN ) ) ) ) ).
% normal_sub_rts.Cong\<^sub>0_symmetric
thf(fact_517_normal__sub__rts_OCong_092_060_094sub_0620__symmetric,axiom,
! [Resid: ( a > a ) > ( a > a ) > a > a,NN: set_a_a,T: a > a,T4: a > a] :
( ( normal_sub_rts_a_a @ Resid @ NN )
=> ( ( ( member_a_a @ ( Resid @ T @ T4 ) @ NN )
& ( member_a_a @ ( Resid @ T4 @ T ) @ NN ) )
=> ( ( member_a_a @ ( Resid @ T4 @ T ) @ NN )
& ( member_a_a @ ( Resid @ T @ T4 ) @ NN ) ) ) ) ).
% normal_sub_rts.Cong\<^sub>0_symmetric
thf(fact_518_normal__sub__rts_OCong_092_060_094sub_0620__symmetric,axiom,
! [Resid: a > a > a,NN: set_a,T: a,T4: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( ( member_a @ ( Resid @ T @ T4 ) @ NN )
& ( member_a @ ( Resid @ T4 @ T ) @ NN ) )
=> ( ( member_a @ ( Resid @ T4 @ T ) @ NN )
& ( member_a @ ( Resid @ T @ T4 ) @ NN ) ) ) ) ).
% normal_sub_rts.Cong\<^sub>0_symmetric
thf(fact_519_normal__sub__rts_OCong_092_060_094sub_0620__transitive,axiom,
! [Resid: product_prod_a_a > product_prod_a_a > product_prod_a_a,NN: set_Product_prod_a_a,T: product_prod_a_a,T4: product_prod_a_a,T6: product_prod_a_a] :
( ( normal2251803270671997330od_a_a @ Resid @ NN )
=> ( ( ( member1426531477525435216od_a_a @ ( Resid @ T @ T4 ) @ NN )
& ( member1426531477525435216od_a_a @ ( Resid @ T4 @ T ) @ NN ) )
=> ( ( ( member1426531477525435216od_a_a @ ( Resid @ T4 @ T6 ) @ NN )
& ( member1426531477525435216od_a_a @ ( Resid @ T6 @ T4 ) @ NN ) )
=> ( ( member1426531477525435216od_a_a @ ( Resid @ T @ T6 ) @ NN )
& ( member1426531477525435216od_a_a @ ( Resid @ T6 @ T ) @ NN ) ) ) ) ) ).
% normal_sub_rts.Cong\<^sub>0_transitive
thf(fact_520_normal__sub__rts_OCong_092_060_094sub_0620__transitive,axiom,
! [Resid: set_a > set_a > set_a,NN: set_set_a,T: set_a,T4: set_a,T6: set_a] :
( ( normal_sub_rts_set_a @ Resid @ NN )
=> ( ( ( member_set_a @ ( Resid @ T @ T4 ) @ NN )
& ( member_set_a @ ( Resid @ T4 @ T ) @ NN ) )
=> ( ( ( member_set_a @ ( Resid @ T4 @ T6 ) @ NN )
& ( member_set_a @ ( Resid @ T6 @ T4 ) @ NN ) )
=> ( ( member_set_a @ ( Resid @ T @ T6 ) @ NN )
& ( member_set_a @ ( Resid @ T6 @ T ) @ NN ) ) ) ) ) ).
% normal_sub_rts.Cong\<^sub>0_transitive
thf(fact_521_normal__sub__rts_OCong_092_060_094sub_0620__transitive,axiom,
! [Resid: ( a > a ) > ( a > a ) > a > a,NN: set_a_a,T: a > a,T4: a > a,T6: a > a] :
( ( normal_sub_rts_a_a @ Resid @ NN )
=> ( ( ( member_a_a @ ( Resid @ T @ T4 ) @ NN )
& ( member_a_a @ ( Resid @ T4 @ T ) @ NN ) )
=> ( ( ( member_a_a @ ( Resid @ T4 @ T6 ) @ NN )
& ( member_a_a @ ( Resid @ T6 @ T4 ) @ NN ) )
=> ( ( member_a_a @ ( Resid @ T @ T6 ) @ NN )
& ( member_a_a @ ( Resid @ T6 @ T ) @ NN ) ) ) ) ) ).
% normal_sub_rts.Cong\<^sub>0_transitive
thf(fact_522_normal__sub__rts_OCong_092_060_094sub_0620__transitive,axiom,
! [Resid: a > a > a,NN: set_a,T: a,T4: a,T6: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( ( member_a @ ( Resid @ T @ T4 ) @ NN )
& ( member_a @ ( Resid @ T4 @ T ) @ NN ) )
=> ( ( ( member_a @ ( Resid @ T4 @ T6 ) @ NN )
& ( member_a @ ( Resid @ T6 @ T4 ) @ NN ) )
=> ( ( member_a @ ( Resid @ T @ T6 ) @ NN )
& ( member_a @ ( Resid @ T6 @ T ) @ NN ) ) ) ) ) ).
% normal_sub_rts.Cong\<^sub>0_transitive
thf(fact_523_normal__sub__rts_OResid__along__normal__reflects__Cong_092_060_094sub_0620,axiom,
! [Resid: product_prod_a_a > product_prod_a_a > product_prod_a_a,NN: set_Product_prod_a_a,T: product_prod_a_a,U: product_prod_a_a,T4: product_prod_a_a] :
( ( normal2251803270671997330od_a_a @ Resid @ NN )
=> ( ( ( member1426531477525435216od_a_a @ ( Resid @ ( Resid @ T @ U ) @ ( Resid @ T4 @ U ) ) @ NN )
& ( member1426531477525435216od_a_a @ ( Resid @ ( Resid @ T4 @ U ) @ ( Resid @ T @ U ) ) @ NN ) )
=> ( ( member1426531477525435216od_a_a @ U @ NN )
=> ( ( member1426531477525435216od_a_a @ ( Resid @ T @ T4 ) @ NN )
& ( member1426531477525435216od_a_a @ ( Resid @ T4 @ T ) @ NN ) ) ) ) ) ).
% normal_sub_rts.Resid_along_normal_reflects_Cong\<^sub>0
thf(fact_524_normal__sub__rts_OResid__along__normal__reflects__Cong_092_060_094sub_0620,axiom,
! [Resid: set_a > set_a > set_a,NN: set_set_a,T: set_a,U: set_a,T4: set_a] :
( ( normal_sub_rts_set_a @ Resid @ NN )
=> ( ( ( member_set_a @ ( Resid @ ( Resid @ T @ U ) @ ( Resid @ T4 @ U ) ) @ NN )
& ( member_set_a @ ( Resid @ ( Resid @ T4 @ U ) @ ( Resid @ T @ U ) ) @ NN ) )
=> ( ( member_set_a @ U @ NN )
=> ( ( member_set_a @ ( Resid @ T @ T4 ) @ NN )
& ( member_set_a @ ( Resid @ T4 @ T ) @ NN ) ) ) ) ) ).
% normal_sub_rts.Resid_along_normal_reflects_Cong\<^sub>0
thf(fact_525_normal__sub__rts_OResid__along__normal__reflects__Cong_092_060_094sub_0620,axiom,
! [Resid: ( a > a ) > ( a > a ) > a > a,NN: set_a_a,T: a > a,U: a > a,T4: a > a] :
( ( normal_sub_rts_a_a @ Resid @ NN )
=> ( ( ( member_a_a @ ( Resid @ ( Resid @ T @ U ) @ ( Resid @ T4 @ U ) ) @ NN )
& ( member_a_a @ ( Resid @ ( Resid @ T4 @ U ) @ ( Resid @ T @ U ) ) @ NN ) )
=> ( ( member_a_a @ U @ NN )
=> ( ( member_a_a @ ( Resid @ T @ T4 ) @ NN )
& ( member_a_a @ ( Resid @ T4 @ T ) @ NN ) ) ) ) ) ).
% normal_sub_rts.Resid_along_normal_reflects_Cong\<^sub>0
thf(fact_526_normal__sub__rts_OResid__along__normal__reflects__Cong_092_060_094sub_0620,axiom,
! [Resid: a > a > a,NN: set_a,T: a,U: a,T4: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( ( 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 ) ) ) ) ) ).
% normal_sub_rts.Resid_along_normal_reflects_Cong\<^sub>0
thf(fact_527_coherent__normal__sub__rts__def,axiom,
( cohere6072184133013167079_rts_a
= ( ^ [Resid2: a > a > a,NN2: set_a] :
( ( normal_sub_rts_a @ Resid2 @ NN2 )
& ( cohere4894532172567702276ioms_a @ Resid2 @ NN2 ) ) ) ) ).
% coherent_normal_sub_rts_def
thf(fact_528_coherent__normal__sub__rts_Ointro,axiom,
! [Resid: a > a > a,NN: set_a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( cohere4894532172567702276ioms_a @ Resid @ NN )
=> ( cohere6072184133013167079_rts_a @ Resid @ NN ) ) ) ).
% coherent_normal_sub_rts.intro
thf(fact_529_normal__sub__rts_Oprfx__closed,axiom,
! [Resid: product_prod_a_a > product_prod_a_a > product_prod_a_a,NN: set_Product_prod_a_a,U: product_prod_a_a,T: product_prod_a_a] :
( ( normal2251803270671997330od_a_a @ Resid @ NN )
=> ( ( member1426531477525435216od_a_a @ U @ NN )
=> ( ( ide_Product_prod_a_a @ Resid @ ( Resid @ T @ U ) )
=> ( member1426531477525435216od_a_a @ T @ NN ) ) ) ) ).
% normal_sub_rts.prfx_closed
thf(fact_530_normal__sub__rts_Oprfx__closed,axiom,
! [Resid: set_a > set_a > set_a,NN: set_set_a,U: set_a,T: set_a] :
( ( normal_sub_rts_set_a @ Resid @ NN )
=> ( ( member_set_a @ U @ NN )
=> ( ( ide_set_a @ Resid @ ( Resid @ T @ U ) )
=> ( member_set_a @ T @ NN ) ) ) ) ).
% normal_sub_rts.prfx_closed
thf(fact_531_normal__sub__rts_Oprfx__closed,axiom,
! [Resid: ( a > a ) > ( a > a ) > a > a,NN: set_a_a,U: a > a,T: a > a] :
( ( normal_sub_rts_a_a @ Resid @ NN )
=> ( ( member_a_a @ U @ NN )
=> ( ( ide_a_a @ Resid @ ( Resid @ T @ U ) )
=> ( member_a_a @ T @ NN ) ) ) ) ).
% normal_sub_rts.prfx_closed
thf(fact_532_normal__sub__rts_Oprfx__closed,axiom,
! [Resid: a > a > a,NN: set_a,U: a,T: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( member_a @ U @ NN )
=> ( ( ide_a @ Resid @ ( Resid @ T @ U ) )
=> ( member_a @ T @ NN ) ) ) ) ).
% normal_sub_rts.prfx_closed
thf(fact_533_normal__sub__rts_Oide__closed,axiom,
! [Resid: product_prod_a_a > product_prod_a_a > product_prod_a_a,NN: set_Product_prod_a_a,A: product_prod_a_a] :
( ( normal2251803270671997330od_a_a @ Resid @ NN )
=> ( ( ide_Product_prod_a_a @ Resid @ A )
=> ( member1426531477525435216od_a_a @ A @ NN ) ) ) ).
% normal_sub_rts.ide_closed
thf(fact_534_normal__sub__rts_Oide__closed,axiom,
! [Resid: set_a > set_a > set_a,NN: set_set_a,A: set_a] :
( ( normal_sub_rts_set_a @ Resid @ NN )
=> ( ( ide_set_a @ Resid @ A )
=> ( member_set_a @ A @ NN ) ) ) ).
% normal_sub_rts.ide_closed
thf(fact_535_normal__sub__rts_Oide__closed,axiom,
! [Resid: ( a > a ) > ( a > a ) > a > a,NN: set_a_a,A: a > a] :
( ( normal_sub_rts_a_a @ Resid @ NN )
=> ( ( ide_a_a @ Resid @ A )
=> ( member_a_a @ A @ NN ) ) ) ).
% normal_sub_rts.ide_closed
thf(fact_536_normal__sub__rts_Oide__closed,axiom,
! [Resid: a > a > a,NN: set_a,A: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( ide_a @ Resid @ A )
=> ( member_a @ A @ NN ) ) ) ).
% normal_sub_rts.ide_closed
thf(fact_537_normal__sub__rts_OCong_092_060_094sub_0620__subst__left_I2_J,axiom,
! [Resid: product_prod_a_a > product_prod_a_a > product_prod_a_a,NN: set_Product_prod_a_a,T: product_prod_a_a,T4: product_prod_a_a,U: product_prod_a_a] :
( ( normal2251803270671997330od_a_a @ Resid @ NN )
=> ( ( ( member1426531477525435216od_a_a @ ( Resid @ T @ T4 ) @ NN )
& ( member1426531477525435216od_a_a @ ( Resid @ T4 @ T ) @ NN ) )
=> ( ( con_Product_prod_a_a @ Resid @ T @ U )
=> ( ( member1426531477525435216od_a_a @ ( Resid @ ( Resid @ T @ U ) @ ( Resid @ T4 @ U ) ) @ NN )
& ( member1426531477525435216od_a_a @ ( Resid @ ( Resid @ T4 @ U ) @ ( Resid @ T @ U ) ) @ NN ) ) ) ) ) ).
% normal_sub_rts.Cong\<^sub>0_subst_left(2)
thf(fact_538_normal__sub__rts_OCong_092_060_094sub_0620__subst__left_I2_J,axiom,
! [Resid: ( a > a ) > ( a > a ) > a > a,NN: set_a_a,T: a > a,T4: a > a,U: a > a] :
( ( normal_sub_rts_a_a @ Resid @ NN )
=> ( ( ( member_a_a @ ( Resid @ T @ T4 ) @ NN )
& ( member_a_a @ ( Resid @ T4 @ T ) @ NN ) )
=> ( ( con_a_a @ Resid @ T @ U )
=> ( ( member_a_a @ ( Resid @ ( Resid @ T @ U ) @ ( Resid @ T4 @ U ) ) @ NN )
& ( member_a_a @ ( Resid @ ( Resid @ T4 @ U ) @ ( Resid @ T @ U ) ) @ NN ) ) ) ) ) ).
% normal_sub_rts.Cong\<^sub>0_subst_left(2)
thf(fact_539_normal__sub__rts_OCong_092_060_094sub_0620__subst__left_I2_J,axiom,
! [Resid: set_a > set_a > set_a,NN: set_set_a,T: set_a,T4: set_a,U: set_a] :
( ( normal_sub_rts_set_a @ Resid @ NN )
=> ( ( ( member_set_a @ ( Resid @ T @ T4 ) @ NN )
& ( member_set_a @ ( Resid @ T4 @ T ) @ NN ) )
=> ( ( con_set_a @ Resid @ T @ U )
=> ( ( member_set_a @ ( Resid @ ( Resid @ T @ U ) @ ( Resid @ T4 @ U ) ) @ NN )
& ( member_set_a @ ( Resid @ ( Resid @ T4 @ U ) @ ( Resid @ T @ U ) ) @ NN ) ) ) ) ) ).
% normal_sub_rts.Cong\<^sub>0_subst_left(2)
thf(fact_540_normal__sub__rts_OCong_092_060_094sub_0620__subst__left_I2_J,axiom,
! [Resid: a > a > a,NN: set_a,T: a,T4: a,U: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( ( 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 ) ) ) ) ) ).
% normal_sub_rts.Cong\<^sub>0_subst_left(2)
thf(fact_541_normal__sub__rts_OCong_092_060_094sub_0620__subst__left_I1_J,axiom,
! [Resid: product_prod_a_a > product_prod_a_a > product_prod_a_a,NN: set_Product_prod_a_a,T: product_prod_a_a,T4: product_prod_a_a,U: product_prod_a_a] :
( ( normal2251803270671997330od_a_a @ Resid @ NN )
=> ( ( ( member1426531477525435216od_a_a @ ( Resid @ T @ T4 ) @ NN )
& ( member1426531477525435216od_a_a @ ( Resid @ T4 @ T ) @ NN ) )
=> ( ( con_Product_prod_a_a @ Resid @ T @ U )
=> ( con_Product_prod_a_a @ Resid @ T4 @ U ) ) ) ) ).
% normal_sub_rts.Cong\<^sub>0_subst_left(1)
thf(fact_542_normal__sub__rts_OCong_092_060_094sub_0620__subst__left_I1_J,axiom,
! [Resid: ( a > a ) > ( a > a ) > a > a,NN: set_a_a,T: a > a,T4: a > a,U: a > a] :
( ( normal_sub_rts_a_a @ Resid @ NN )
=> ( ( ( member_a_a @ ( Resid @ T @ T4 ) @ NN )
& ( member_a_a @ ( Resid @ T4 @ T ) @ NN ) )
=> ( ( con_a_a @ Resid @ T @ U )
=> ( con_a_a @ Resid @ T4 @ U ) ) ) ) ).
% normal_sub_rts.Cong\<^sub>0_subst_left(1)
thf(fact_543_normal__sub__rts_OCong_092_060_094sub_0620__subst__left_I1_J,axiom,
! [Resid: set_a > set_a > set_a,NN: set_set_a,T: set_a,T4: set_a,U: set_a] :
( ( normal_sub_rts_set_a @ Resid @ NN )
=> ( ( ( member_set_a @ ( Resid @ T @ T4 ) @ NN )
& ( member_set_a @ ( Resid @ T4 @ T ) @ NN ) )
=> ( ( con_set_a @ Resid @ T @ U )
=> ( con_set_a @ Resid @ T4 @ U ) ) ) ) ).
% normal_sub_rts.Cong\<^sub>0_subst_left(1)
thf(fact_544_normal__sub__rts_OCong_092_060_094sub_0620__subst__left_I1_J,axiom,
! [Resid: a > a > a,NN: set_a,T: a,T4: a,U: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( ( member_a @ ( Resid @ T @ T4 ) @ NN )
& ( member_a @ ( Resid @ T4 @ T ) @ NN ) )
=> ( ( con_a @ Resid @ T @ U )
=> ( con_a @ Resid @ T4 @ U ) ) ) ) ).
% normal_sub_rts.Cong\<^sub>0_subst_left(1)
thf(fact_545_normal__sub__rts_OCong_092_060_094sub_0620__subst__right_I2_J,axiom,
! [Resid: product_prod_a_a > product_prod_a_a > product_prod_a_a,NN: set_Product_prod_a_a,U: product_prod_a_a,U2: product_prod_a_a,T: product_prod_a_a] :
( ( normal2251803270671997330od_a_a @ Resid @ NN )
=> ( ( ( member1426531477525435216od_a_a @ ( Resid @ U @ U2 ) @ NN )
& ( member1426531477525435216od_a_a @ ( Resid @ U2 @ U ) @ NN ) )
=> ( ( con_Product_prod_a_a @ Resid @ T @ U )
=> ( ( member1426531477525435216od_a_a @ ( Resid @ ( Resid @ ( Resid @ T @ U ) @ ( Resid @ U2 @ U ) ) @ ( Resid @ ( Resid @ T @ U2 ) @ ( Resid @ U @ U2 ) ) ) @ NN )
& ( member1426531477525435216od_a_a @ ( Resid @ ( Resid @ ( Resid @ T @ U2 ) @ ( Resid @ U @ U2 ) ) @ ( Resid @ ( Resid @ T @ U ) @ ( Resid @ U2 @ U ) ) ) @ NN ) ) ) ) ) ).
% normal_sub_rts.Cong\<^sub>0_subst_right(2)
thf(fact_546_normal__sub__rts_OCong_092_060_094sub_0620__subst__right_I2_J,axiom,
! [Resid: ( a > a ) > ( a > a ) > a > a,NN: set_a_a,U: a > a,U2: a > a,T: a > a] :
( ( normal_sub_rts_a_a @ Resid @ NN )
=> ( ( ( member_a_a @ ( Resid @ U @ U2 ) @ NN )
& ( member_a_a @ ( Resid @ U2 @ U ) @ NN ) )
=> ( ( con_a_a @ Resid @ T @ U )
=> ( ( member_a_a @ ( Resid @ ( Resid @ ( Resid @ T @ U ) @ ( Resid @ U2 @ U ) ) @ ( Resid @ ( Resid @ T @ U2 ) @ ( Resid @ U @ U2 ) ) ) @ NN )
& ( member_a_a @ ( Resid @ ( Resid @ ( Resid @ T @ U2 ) @ ( Resid @ U @ U2 ) ) @ ( Resid @ ( Resid @ T @ U ) @ ( Resid @ U2 @ U ) ) ) @ NN ) ) ) ) ) ).
% normal_sub_rts.Cong\<^sub>0_subst_right(2)
thf(fact_547_normal__sub__rts_OCong_092_060_094sub_0620__subst__right_I2_J,axiom,
! [Resid: set_a > set_a > set_a,NN: set_set_a,U: set_a,U2: set_a,T: set_a] :
( ( normal_sub_rts_set_a @ Resid @ NN )
=> ( ( ( member_set_a @ ( Resid @ U @ U2 ) @ NN )
& ( member_set_a @ ( Resid @ U2 @ U ) @ NN ) )
=> ( ( con_set_a @ Resid @ T @ U )
=> ( ( member_set_a @ ( Resid @ ( Resid @ ( Resid @ T @ U ) @ ( Resid @ U2 @ U ) ) @ ( Resid @ ( Resid @ T @ U2 ) @ ( Resid @ U @ U2 ) ) ) @ NN )
& ( member_set_a @ ( Resid @ ( Resid @ ( Resid @ T @ U2 ) @ ( Resid @ U @ U2 ) ) @ ( Resid @ ( Resid @ T @ U ) @ ( Resid @ U2 @ U ) ) ) @ NN ) ) ) ) ) ).
% normal_sub_rts.Cong\<^sub>0_subst_right(2)
thf(fact_548_normal__sub__rts_OCong_092_060_094sub_0620__subst__right_I2_J,axiom,
! [Resid: a > a > a,NN: set_a,U: a,U2: a,T: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( ( member_a @ ( Resid @ U @ U2 ) @ NN )
& ( member_a @ ( Resid @ U2 @ U ) @ NN ) )
=> ( ( con_a @ Resid @ T @ U )
=> ( ( member_a @ ( Resid @ ( Resid @ ( Resid @ T @ U ) @ ( Resid @ U2 @ U ) ) @ ( Resid @ ( Resid @ T @ U2 ) @ ( Resid @ U @ U2 ) ) ) @ NN )
& ( member_a @ ( Resid @ ( Resid @ ( Resid @ T @ U2 ) @ ( Resid @ U @ U2 ) ) @ ( Resid @ ( Resid @ T @ U ) @ ( Resid @ U2 @ U ) ) ) @ NN ) ) ) ) ) ).
% normal_sub_rts.Cong\<^sub>0_subst_right(2)
thf(fact_549_normal__sub__rts_OCong_092_060_094sub_0620__subst__right_I1_J,axiom,
! [Resid: product_prod_a_a > product_prod_a_a > product_prod_a_a,NN: set_Product_prod_a_a,U: product_prod_a_a,U2: product_prod_a_a,T: product_prod_a_a] :
( ( normal2251803270671997330od_a_a @ Resid @ NN )
=> ( ( ( member1426531477525435216od_a_a @ ( Resid @ U @ U2 ) @ NN )
& ( member1426531477525435216od_a_a @ ( Resid @ U2 @ U ) @ NN ) )
=> ( ( con_Product_prod_a_a @ Resid @ T @ U )
=> ( con_Product_prod_a_a @ Resid @ T @ U2 ) ) ) ) ).
% normal_sub_rts.Cong\<^sub>0_subst_right(1)
thf(fact_550_normal__sub__rts_OCong_092_060_094sub_0620__subst__right_I1_J,axiom,
! [Resid: ( a > a ) > ( a > a ) > a > a,NN: set_a_a,U: a > a,U2: a > a,T: a > a] :
( ( normal_sub_rts_a_a @ Resid @ NN )
=> ( ( ( member_a_a @ ( Resid @ U @ U2 ) @ NN )
& ( member_a_a @ ( Resid @ U2 @ U ) @ NN ) )
=> ( ( con_a_a @ Resid @ T @ U )
=> ( con_a_a @ Resid @ T @ U2 ) ) ) ) ).
% normal_sub_rts.Cong\<^sub>0_subst_right(1)
thf(fact_551_normal__sub__rts_OCong_092_060_094sub_0620__subst__right_I1_J,axiom,
! [Resid: set_a > set_a > set_a,NN: set_set_a,U: set_a,U2: set_a,T: set_a] :
( ( normal_sub_rts_set_a @ Resid @ NN )
=> ( ( ( member_set_a @ ( Resid @ U @ U2 ) @ NN )
& ( member_set_a @ ( Resid @ U2 @ U ) @ NN ) )
=> ( ( con_set_a @ Resid @ T @ U )
=> ( con_set_a @ Resid @ T @ U2 ) ) ) ) ).
% normal_sub_rts.Cong\<^sub>0_subst_right(1)
thf(fact_552_normal__sub__rts_OCong_092_060_094sub_0620__subst__right_I1_J,axiom,
! [Resid: a > a > a,NN: set_a,U: a,U2: a,T: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( ( member_a @ ( Resid @ U @ U2 ) @ NN )
& ( member_a @ ( Resid @ U2 @ U ) @ NN ) )
=> ( ( con_a @ Resid @ T @ U )
=> ( con_a @ Resid @ T @ U2 ) ) ) ) ).
% normal_sub_rts.Cong\<^sub>0_subst_right(1)
thf(fact_553_normal__sub__rts_OCong_092_060_094sub_0620__imp__con,axiom,
! [Resid: product_prod_a_a > product_prod_a_a > product_prod_a_a,NN: set_Product_prod_a_a,T: product_prod_a_a,T4: product_prod_a_a] :
( ( normal2251803270671997330od_a_a @ Resid @ NN )
=> ( ( ( member1426531477525435216od_a_a @ ( Resid @ T @ T4 ) @ NN )
& ( member1426531477525435216od_a_a @ ( Resid @ T4 @ T ) @ NN ) )
=> ( con_Product_prod_a_a @ Resid @ T @ T4 ) ) ) ).
% normal_sub_rts.Cong\<^sub>0_imp_con
thf(fact_554_normal__sub__rts_OCong_092_060_094sub_0620__imp__con,axiom,
! [Resid: ( a > a ) > ( a > a ) > a > a,NN: set_a_a,T: a > a,T4: a > a] :
( ( normal_sub_rts_a_a @ Resid @ NN )
=> ( ( ( member_a_a @ ( Resid @ T @ T4 ) @ NN )
& ( member_a_a @ ( Resid @ T4 @ T ) @ NN ) )
=> ( con_a_a @ Resid @ T @ T4 ) ) ) ).
% normal_sub_rts.Cong\<^sub>0_imp_con
thf(fact_555_normal__sub__rts_OCong_092_060_094sub_0620__imp__con,axiom,
! [Resid: set_a > set_a > set_a,NN: set_set_a,T: set_a,T4: set_a] :
( ( normal_sub_rts_set_a @ Resid @ NN )
=> ( ( ( member_set_a @ ( Resid @ T @ T4 ) @ NN )
& ( member_set_a @ ( Resid @ T4 @ T ) @ NN ) )
=> ( con_set_a @ Resid @ T @ T4 ) ) ) ).
% normal_sub_rts.Cong\<^sub>0_imp_con
thf(fact_556_normal__sub__rts_OCong_092_060_094sub_0620__imp__con,axiom,
! [Resid: a > a > a,NN: set_a,T: a,T4: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( ( member_a @ ( Resid @ T @ T4 ) @ NN )
& ( member_a @ ( Resid @ T4 @ T ) @ NN ) )
=> ( con_a @ Resid @ T @ T4 ) ) ) ).
% normal_sub_rts.Cong\<^sub>0_imp_con
thf(fact_557_normal__sub__rts_OCong_092_060_094sub_0620__subst__Con,axiom,
! [Resid: product_prod_a_a > product_prod_a_a > product_prod_a_a,NN: set_Product_prod_a_a,T: product_prod_a_a,T4: product_prod_a_a,U: product_prod_a_a,U2: product_prod_a_a] :
( ( normal2251803270671997330od_a_a @ Resid @ NN )
=> ( ( ( member1426531477525435216od_a_a @ ( Resid @ T @ T4 ) @ NN )
& ( member1426531477525435216od_a_a @ ( Resid @ T4 @ T ) @ NN ) )
=> ( ( ( member1426531477525435216od_a_a @ ( Resid @ U @ U2 ) @ NN )
& ( member1426531477525435216od_a_a @ ( Resid @ U2 @ U ) @ NN ) )
=> ( ( con_Product_prod_a_a @ Resid @ T @ U )
= ( con_Product_prod_a_a @ Resid @ T4 @ U2 ) ) ) ) ) ).
% normal_sub_rts.Cong\<^sub>0_subst_Con
thf(fact_558_normal__sub__rts_OCong_092_060_094sub_0620__subst__Con,axiom,
! [Resid: ( a > a ) > ( a > a ) > a > a,NN: set_a_a,T: a > a,T4: a > a,U: a > a,U2: a > a] :
( ( normal_sub_rts_a_a @ Resid @ NN )
=> ( ( ( member_a_a @ ( Resid @ T @ T4 ) @ NN )
& ( member_a_a @ ( Resid @ T4 @ T ) @ NN ) )
=> ( ( ( member_a_a @ ( Resid @ U @ U2 ) @ NN )
& ( member_a_a @ ( Resid @ U2 @ U ) @ NN ) )
=> ( ( con_a_a @ Resid @ T @ U )
= ( con_a_a @ Resid @ T4 @ U2 ) ) ) ) ) ).
% normal_sub_rts.Cong\<^sub>0_subst_Con
thf(fact_559_normal__sub__rts_OCong_092_060_094sub_0620__subst__Con,axiom,
! [Resid: set_a > set_a > set_a,NN: set_set_a,T: set_a,T4: set_a,U: set_a,U2: set_a] :
( ( normal_sub_rts_set_a @ Resid @ NN )
=> ( ( ( member_set_a @ ( Resid @ T @ T4 ) @ NN )
& ( member_set_a @ ( Resid @ T4 @ T ) @ NN ) )
=> ( ( ( member_set_a @ ( Resid @ U @ U2 ) @ NN )
& ( member_set_a @ ( Resid @ U2 @ U ) @ NN ) )
=> ( ( con_set_a @ Resid @ T @ U )
= ( con_set_a @ Resid @ T4 @ U2 ) ) ) ) ) ).
% normal_sub_rts.Cong\<^sub>0_subst_Con
thf(fact_560_normal__sub__rts_OCong_092_060_094sub_0620__subst__Con,axiom,
! [Resid: a > a > a,NN: set_a,T: a,T4: a,U: a,U2: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( ( member_a @ ( Resid @ T @ T4 ) @ NN )
& ( member_a @ ( Resid @ T4 @ T ) @ NN ) )
=> ( ( ( member_a @ ( Resid @ U @ U2 ) @ NN )
& ( member_a @ ( Resid @ U2 @ U ) @ NN ) )
=> ( ( con_a @ Resid @ T @ U )
= ( con_a @ Resid @ T4 @ U2 ) ) ) ) ) ).
% normal_sub_rts.Cong\<^sub>0_subst_Con
thf(fact_561_normal__sub__rts_Oelements__are__arr,axiom,
! [Resid: product_prod_a_a > product_prod_a_a > product_prod_a_a,NN: set_Product_prod_a_a,T: product_prod_a_a] :
( ( normal2251803270671997330od_a_a @ Resid @ NN )
=> ( ( member1426531477525435216od_a_a @ T @ NN )
=> ( arr_Product_prod_a_a @ Resid @ T ) ) ) ).
% normal_sub_rts.elements_are_arr
thf(fact_562_normal__sub__rts_Oelements__are__arr,axiom,
! [Resid: ( a > a ) > ( a > a ) > a > a,NN: set_a_a,T: a > a] :
( ( normal_sub_rts_a_a @ Resid @ NN )
=> ( ( member_a_a @ T @ NN )
=> ( arr_a_a @ Resid @ T ) ) ) ).
% normal_sub_rts.elements_are_arr
thf(fact_563_normal__sub__rts_Oelements__are__arr,axiom,
! [Resid: set_a > set_a > set_a,NN: set_set_a,T: set_a] :
( ( normal_sub_rts_set_a @ Resid @ NN )
=> ( ( member_set_a @ T @ NN )
=> ( arr_set_a @ Resid @ T ) ) ) ).
% normal_sub_rts.elements_are_arr
thf(fact_564_normal__sub__rts_Oelements__are__arr,axiom,
! [Resid: a > a > a,NN: set_a,T: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( member_a @ T @ NN )
=> ( arr_a @ Resid @ T ) ) ) ).
% normal_sub_rts.elements_are_arr
thf(fact_565_normal__sub__rts_OCong_092_060_094sub_0620__reflexive,axiom,
! [Resid: product_prod_a_a > product_prod_a_a > product_prod_a_a,NN: set_Product_prod_a_a,T: product_prod_a_a] :
( ( normal2251803270671997330od_a_a @ Resid @ NN )
=> ( ( arr_Product_prod_a_a @ Resid @ T )
=> ( ( member1426531477525435216od_a_a @ ( Resid @ T @ T ) @ NN )
& ( member1426531477525435216od_a_a @ ( Resid @ T @ T ) @ NN ) ) ) ) ).
% normal_sub_rts.Cong\<^sub>0_reflexive
thf(fact_566_normal__sub__rts_OCong_092_060_094sub_0620__reflexive,axiom,
! [Resid: ( a > a ) > ( a > a ) > a > a,NN: set_a_a,T: a > a] :
( ( normal_sub_rts_a_a @ Resid @ NN )
=> ( ( arr_a_a @ Resid @ T )
=> ( ( member_a_a @ ( Resid @ T @ T ) @ NN )
& ( member_a_a @ ( Resid @ T @ T ) @ NN ) ) ) ) ).
% normal_sub_rts.Cong\<^sub>0_reflexive
thf(fact_567_normal__sub__rts_OCong_092_060_094sub_0620__reflexive,axiom,
! [Resid: set_a > set_a > set_a,NN: set_set_a,T: set_a] :
( ( normal_sub_rts_set_a @ Resid @ NN )
=> ( ( arr_set_a @ Resid @ T )
=> ( ( member_set_a @ ( Resid @ T @ T ) @ NN )
& ( member_set_a @ ( Resid @ T @ T ) @ NN ) ) ) ) ).
% normal_sub_rts.Cong\<^sub>0_reflexive
thf(fact_568_normal__sub__rts_OCong_092_060_094sub_0620__reflexive,axiom,
! [Resid: a > a > a,NN: set_a,T: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( arr_a @ Resid @ T )
=> ( ( member_a @ ( Resid @ T @ T ) @ NN )
& ( member_a @ ( Resid @ T @ T ) @ NN ) ) ) ) ).
% normal_sub_rts.Cong\<^sub>0_reflexive
thf(fact_569_normal__sub__rts_OCong_092_060_094sub_0620__imp__coinitial,axiom,
! [Resid: product_prod_a_a > product_prod_a_a > product_prod_a_a,NN: set_Product_prod_a_a,T: product_prod_a_a,T4: product_prod_a_a] :
( ( normal2251803270671997330od_a_a @ Resid @ NN )
=> ( ( ( member1426531477525435216od_a_a @ ( Resid @ T @ T4 ) @ NN )
& ( member1426531477525435216od_a_a @ ( Resid @ T4 @ T ) @ NN ) )
=> ( ( source6950040787684646355od_a_a @ Resid @ T )
= ( source6950040787684646355od_a_a @ Resid @ T4 ) ) ) ) ).
% normal_sub_rts.Cong\<^sub>0_imp_coinitial
thf(fact_570_normal__sub__rts_OCong_092_060_094sub_0620__imp__coinitial,axiom,
! [Resid: ( a > a ) > ( a > a ) > a > a,NN: set_a_a,T: a > a,T4: a > a] :
( ( normal_sub_rts_a_a @ Resid @ NN )
=> ( ( ( member_a_a @ ( Resid @ T @ T4 ) @ NN )
& ( member_a_a @ ( Resid @ T4 @ T ) @ NN ) )
=> ( ( sources_a_a @ Resid @ T )
= ( sources_a_a @ Resid @ T4 ) ) ) ) ).
% normal_sub_rts.Cong\<^sub>0_imp_coinitial
thf(fact_571_normal__sub__rts_OCong_092_060_094sub_0620__imp__coinitial,axiom,
! [Resid: set_a > set_a > set_a,NN: set_set_a,T: set_a,T4: set_a] :
( ( normal_sub_rts_set_a @ Resid @ NN )
=> ( ( ( member_set_a @ ( Resid @ T @ T4 ) @ NN )
& ( member_set_a @ ( Resid @ T4 @ T ) @ NN ) )
=> ( ( sources_set_a @ Resid @ T )
= ( sources_set_a @ Resid @ T4 ) ) ) ) ).
% normal_sub_rts.Cong\<^sub>0_imp_coinitial
thf(fact_572_normal__sub__rts_OCong_092_060_094sub_0620__imp__coinitial,axiom,
! [Resid: a > a > a,NN: set_a,T: a,T4: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( ( member_a @ ( Resid @ T @ T4 ) @ NN )
& ( member_a @ ( Resid @ T4 @ T ) @ NN ) )
=> ( ( sources_a @ Resid @ T )
= ( sources_a @ Resid @ T4 ) ) ) ) ).
% normal_sub_rts.Cong\<^sub>0_imp_coinitial
thf(fact_573_normal__sub__rts_OResid__along__normal__preserves__Cong_092_060_094sub_0620,axiom,
! [Resid: product_prod_a_a > product_prod_a_a > product_prod_a_a,NN: set_Product_prod_a_a,T: product_prod_a_a,T4: product_prod_a_a,U: product_prod_a_a] :
( ( normal2251803270671997330od_a_a @ Resid @ NN )
=> ( ( ( member1426531477525435216od_a_a @ ( Resid @ T @ T4 ) @ NN )
& ( member1426531477525435216od_a_a @ ( Resid @ T4 @ T ) @ NN ) )
=> ( ( member1426531477525435216od_a_a @ U @ NN )
=> ( ( ( source6950040787684646355od_a_a @ Resid @ T )
= ( source6950040787684646355od_a_a @ Resid @ U ) )
=> ( ( member1426531477525435216od_a_a @ ( Resid @ ( Resid @ T @ U ) @ ( Resid @ T4 @ U ) ) @ NN )
& ( member1426531477525435216od_a_a @ ( Resid @ ( Resid @ T4 @ U ) @ ( Resid @ T @ U ) ) @ NN ) ) ) ) ) ) ).
% normal_sub_rts.Resid_along_normal_preserves_Cong\<^sub>0
thf(fact_574_normal__sub__rts_OResid__along__normal__preserves__Cong_092_060_094sub_0620,axiom,
! [Resid: ( a > a ) > ( a > a ) > a > a,NN: set_a_a,T: a > a,T4: a > a,U: a > a] :
( ( normal_sub_rts_a_a @ Resid @ NN )
=> ( ( ( member_a_a @ ( Resid @ T @ T4 ) @ NN )
& ( member_a_a @ ( Resid @ T4 @ T ) @ NN ) )
=> ( ( member_a_a @ U @ NN )
=> ( ( ( sources_a_a @ Resid @ T )
= ( sources_a_a @ Resid @ U ) )
=> ( ( member_a_a @ ( Resid @ ( Resid @ T @ U ) @ ( Resid @ T4 @ U ) ) @ NN )
& ( member_a_a @ ( Resid @ ( Resid @ T4 @ U ) @ ( Resid @ T @ U ) ) @ NN ) ) ) ) ) ) ).
% normal_sub_rts.Resid_along_normal_preserves_Cong\<^sub>0
thf(fact_575_normal__sub__rts_OResid__along__normal__preserves__Cong_092_060_094sub_0620,axiom,
! [Resid: set_a > set_a > set_a,NN: set_set_a,T: set_a,T4: set_a,U: set_a] :
( ( normal_sub_rts_set_a @ Resid @ NN )
=> ( ( ( member_set_a @ ( Resid @ T @ T4 ) @ NN )
& ( member_set_a @ ( Resid @ T4 @ T ) @ NN ) )
=> ( ( member_set_a @ U @ NN )
=> ( ( ( sources_set_a @ Resid @ T )
= ( sources_set_a @ Resid @ U ) )
=> ( ( member_set_a @ ( Resid @ ( Resid @ T @ U ) @ ( Resid @ T4 @ U ) ) @ NN )
& ( member_set_a @ ( Resid @ ( Resid @ T4 @ U ) @ ( Resid @ T @ U ) ) @ NN ) ) ) ) ) ) ).
% normal_sub_rts.Resid_along_normal_preserves_Cong\<^sub>0
thf(fact_576_normal__sub__rts_OResid__along__normal__preserves__Cong_092_060_094sub_0620,axiom,
! [Resid: a > a > a,NN: set_a,T: a,T4: a,U: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( ( 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 ) ) ) ) ) ) ).
% normal_sub_rts.Resid_along_normal_preserves_Cong\<^sub>0
thf(fact_577_normal__sub__rts_Ofactor__closed_I2_J,axiom,
! [Resid: product_prod_a_a > product_prod_a_a > product_prod_a_a,NN: set_Product_prod_a_a,T: product_prod_a_a,U: product_prod_a_a,V: product_prod_a_a] :
( ( normal2251803270671997330od_a_a @ Resid @ NN )
=> ( ( compos6970852536677457805od_a_a @ Resid @ T @ U @ V )
=> ( ( member1426531477525435216od_a_a @ V @ NN )
=> ( member1426531477525435216od_a_a @ U @ NN ) ) ) ) ).
% normal_sub_rts.factor_closed(2)
thf(fact_578_normal__sub__rts_Ofactor__closed_I2_J,axiom,
! [Resid: ( a > a ) > ( a > a ) > a > a,NN: set_a_a,T: a > a,U: a > a,V: a > a] :
( ( normal_sub_rts_a_a @ Resid @ NN )
=> ( ( composite_of_a_a @ Resid @ T @ U @ V )
=> ( ( member_a_a @ V @ NN )
=> ( member_a_a @ U @ NN ) ) ) ) ).
% normal_sub_rts.factor_closed(2)
thf(fact_579_normal__sub__rts_Ofactor__closed_I2_J,axiom,
! [Resid: set_a > set_a > set_a,NN: set_set_a,T: set_a,U: set_a,V: set_a] :
( ( normal_sub_rts_set_a @ Resid @ NN )
=> ( ( composite_of_set_a @ Resid @ T @ U @ V )
=> ( ( member_set_a @ V @ NN )
=> ( member_set_a @ U @ NN ) ) ) ) ).
% normal_sub_rts.factor_closed(2)
thf(fact_580_normal__sub__rts_Ofactor__closed_I2_J,axiom,
! [Resid: a > a > a,NN: set_a,T: a,U: a,V: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( composite_of_a @ Resid @ T @ U @ V )
=> ( ( member_a @ V @ NN )
=> ( member_a @ U @ NN ) ) ) ) ).
% normal_sub_rts.factor_closed(2)
thf(fact_581_normal__sub__rts_Ofactor__closed_I1_J,axiom,
! [Resid: product_prod_a_a > product_prod_a_a > product_prod_a_a,NN: set_Product_prod_a_a,T: product_prod_a_a,U: product_prod_a_a,V: product_prod_a_a] :
( ( normal2251803270671997330od_a_a @ Resid @ NN )
=> ( ( compos6970852536677457805od_a_a @ Resid @ T @ U @ V )
=> ( ( member1426531477525435216od_a_a @ V @ NN )
=> ( member1426531477525435216od_a_a @ T @ NN ) ) ) ) ).
% normal_sub_rts.factor_closed(1)
thf(fact_582_normal__sub__rts_Ofactor__closed_I1_J,axiom,
! [Resid: ( a > a ) > ( a > a ) > a > a,NN: set_a_a,T: a > a,U: a > a,V: a > a] :
( ( normal_sub_rts_a_a @ Resid @ NN )
=> ( ( composite_of_a_a @ Resid @ T @ U @ V )
=> ( ( member_a_a @ V @ NN )
=> ( member_a_a @ T @ NN ) ) ) ) ).
% normal_sub_rts.factor_closed(1)
thf(fact_583_normal__sub__rts_Ofactor__closed_I1_J,axiom,
! [Resid: set_a > set_a > set_a,NN: set_set_a,T: set_a,U: set_a,V: set_a] :
( ( normal_sub_rts_set_a @ Resid @ NN )
=> ( ( composite_of_set_a @ Resid @ T @ U @ V )
=> ( ( member_set_a @ V @ NN )
=> ( member_set_a @ T @ NN ) ) ) ) ).
% normal_sub_rts.factor_closed(1)
thf(fact_584_normal__sub__rts_Ofactor__closed_I1_J,axiom,
! [Resid: a > a > a,NN: set_a,T: a,U: a,V: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( composite_of_a @ Resid @ T @ U @ V )
=> ( ( member_a @ V @ NN )
=> ( member_a @ T @ NN ) ) ) ) ).
% normal_sub_rts.factor_closed(1)
thf(fact_585_normal__sub__rts_OCong_092_060_094sub_0620__iff,axiom,
! [Resid: product_prod_a_a > product_prod_a_a > product_prod_a_a,NN: set_Product_prod_a_a,T: product_prod_a_a,T4: product_prod_a_a] :
( ( normal2251803270671997330od_a_a @ Resid @ NN )
=> ( ( ( member1426531477525435216od_a_a @ ( Resid @ T @ T4 ) @ NN )
& ( member1426531477525435216od_a_a @ ( Resid @ T4 @ T ) @ NN ) )
= ( ? [U3: product_prod_a_a,U4: product_prod_a_a,V3: product_prod_a_a,V4: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ U3 @ NN )
& ( member1426531477525435216od_a_a @ U4 @ NN )
& ( member1426531477525435216od_a_a @ ( Resid @ V3 @ V4 ) @ NN )
& ( member1426531477525435216od_a_a @ ( Resid @ V4 @ V3 ) @ NN )
& ( compos6970852536677457805od_a_a @ Resid @ T @ U3 @ V3 )
& ( compos6970852536677457805od_a_a @ Resid @ T4 @ U4 @ V4 ) ) ) ) ) ).
% normal_sub_rts.Cong\<^sub>0_iff
thf(fact_586_normal__sub__rts_OCong_092_060_094sub_0620__iff,axiom,
! [Resid: ( a > a ) > ( a > a ) > a > a,NN: set_a_a,T: a > a,T4: a > a] :
( ( normal_sub_rts_a_a @ Resid @ NN )
=> ( ( ( member_a_a @ ( Resid @ T @ T4 ) @ NN )
& ( member_a_a @ ( Resid @ T4 @ T ) @ NN ) )
= ( ? [U3: a > a,U4: a > a,V3: a > a,V4: a > a] :
( ( member_a_a @ U3 @ NN )
& ( member_a_a @ U4 @ NN )
& ( member_a_a @ ( Resid @ V3 @ V4 ) @ NN )
& ( member_a_a @ ( Resid @ V4 @ V3 ) @ NN )
& ( composite_of_a_a @ Resid @ T @ U3 @ V3 )
& ( composite_of_a_a @ Resid @ T4 @ U4 @ V4 ) ) ) ) ) ).
% normal_sub_rts.Cong\<^sub>0_iff
thf(fact_587_normal__sub__rts_OCong_092_060_094sub_0620__iff,axiom,
! [Resid: set_a > set_a > set_a,NN: set_set_a,T: set_a,T4: set_a] :
( ( normal_sub_rts_set_a @ Resid @ NN )
=> ( ( ( member_set_a @ ( Resid @ T @ T4 ) @ NN )
& ( member_set_a @ ( Resid @ T4 @ T ) @ NN ) )
= ( ? [U3: set_a,U4: set_a,V3: set_a,V4: set_a] :
( ( member_set_a @ U3 @ NN )
& ( member_set_a @ U4 @ NN )
& ( member_set_a @ ( Resid @ V3 @ V4 ) @ NN )
& ( member_set_a @ ( Resid @ V4 @ V3 ) @ NN )
& ( composite_of_set_a @ Resid @ T @ U3 @ V3 )
& ( composite_of_set_a @ Resid @ T4 @ U4 @ V4 ) ) ) ) ) ).
% normal_sub_rts.Cong\<^sub>0_iff
thf(fact_588_normal__sub__rts_OCong_092_060_094sub_0620__iff,axiom,
! [Resid: a > a > a,NN: set_a,T: a,T4: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( ( member_a @ ( Resid @ T @ T4 ) @ NN )
& ( member_a @ ( Resid @ T4 @ T ) @ NN ) )
= ( ? [U3: a,U4: a,V3: a,V4: a] :
( ( member_a @ U3 @ NN )
& ( member_a @ U4 @ NN )
& ( member_a @ ( Resid @ V3 @ V4 ) @ NN )
& ( member_a @ ( Resid @ V4 @ V3 ) @ NN )
& ( composite_of_a @ Resid @ T @ U3 @ V3 )
& ( composite_of_a @ Resid @ T4 @ U4 @ V4 ) ) ) ) ) ).
% normal_sub_rts.Cong\<^sub>0_iff
thf(fact_589_normal__sub__rts_Ocomposite__closed,axiom,
! [Resid: product_prod_a_a > product_prod_a_a > product_prod_a_a,NN: set_Product_prod_a_a,T: product_prod_a_a,U: product_prod_a_a,V: product_prod_a_a] :
( ( normal2251803270671997330od_a_a @ Resid @ NN )
=> ( ( member1426531477525435216od_a_a @ T @ NN )
=> ( ( member1426531477525435216od_a_a @ U @ NN )
=> ( ( compos6970852536677457805od_a_a @ Resid @ T @ U @ V )
=> ( member1426531477525435216od_a_a @ V @ NN ) ) ) ) ) ).
% normal_sub_rts.composite_closed
thf(fact_590_normal__sub__rts_Ocomposite__closed,axiom,
! [Resid: ( a > a ) > ( a > a ) > a > a,NN: set_a_a,T: a > a,U: a > a,V: a > a] :
( ( normal_sub_rts_a_a @ Resid @ NN )
=> ( ( member_a_a @ T @ NN )
=> ( ( member_a_a @ U @ NN )
=> ( ( composite_of_a_a @ Resid @ T @ U @ V )
=> ( member_a_a @ V @ NN ) ) ) ) ) ).
% normal_sub_rts.composite_closed
thf(fact_591_normal__sub__rts_Ocomposite__closed,axiom,
! [Resid: set_a > set_a > set_a,NN: set_set_a,T: set_a,U: set_a,V: set_a] :
( ( normal_sub_rts_set_a @ Resid @ NN )
=> ( ( member_set_a @ T @ NN )
=> ( ( member_set_a @ U @ NN )
=> ( ( composite_of_set_a @ Resid @ T @ U @ V )
=> ( member_set_a @ V @ NN ) ) ) ) ) ).
% normal_sub_rts.composite_closed
thf(fact_592_normal__sub__rts_Ocomposite__closed,axiom,
! [Resid: a > a > a,NN: set_a,T: a,U: a,V: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( member_a @ T @ NN )
=> ( ( member_a @ U @ NN )
=> ( ( composite_of_a @ Resid @ T @ U @ V )
=> ( member_a @ V @ NN ) ) ) ) ) ).
% normal_sub_rts.composite_closed
thf(fact_593_normal__sub__rts_Ocomposite__of__arr__normal,axiom,
! [Resid: product_prod_a_a > product_prod_a_a > product_prod_a_a,NN: set_Product_prod_a_a,Arr: product_prod_a_a > $o,T: product_prod_a_a,U: product_prod_a_a,T4: product_prod_a_a] :
( ( normal2251803270671997330od_a_a @ Resid @ NN )
=> ( ( Arr @ T )
=> ( ( member1426531477525435216od_a_a @ U @ NN )
=> ( ( compos6970852536677457805od_a_a @ Resid @ T @ U @ T4 )
=> ( ( member1426531477525435216od_a_a @ ( Resid @ T4 @ T ) @ NN )
& ( member1426531477525435216od_a_a @ ( Resid @ T @ T4 ) @ NN ) ) ) ) ) ) ).
% normal_sub_rts.composite_of_arr_normal
thf(fact_594_normal__sub__rts_Ocomposite__of__arr__normal,axiom,
! [Resid: ( a > a ) > ( a > a ) > a > a,NN: set_a_a,Arr: ( a > a ) > $o,T: a > a,U: a > a,T4: a > a] :
( ( normal_sub_rts_a_a @ Resid @ NN )
=> ( ( Arr @ T )
=> ( ( member_a_a @ U @ NN )
=> ( ( composite_of_a_a @ Resid @ T @ U @ T4 )
=> ( ( member_a_a @ ( Resid @ T4 @ T ) @ NN )
& ( member_a_a @ ( Resid @ T @ T4 ) @ NN ) ) ) ) ) ) ).
% normal_sub_rts.composite_of_arr_normal
thf(fact_595_normal__sub__rts_Ocomposite__of__arr__normal,axiom,
! [Resid: set_a > set_a > set_a,NN: set_set_a,Arr: set_a > $o,T: set_a,U: set_a,T4: set_a] :
( ( normal_sub_rts_set_a @ Resid @ NN )
=> ( ( Arr @ T )
=> ( ( member_set_a @ U @ NN )
=> ( ( composite_of_set_a @ Resid @ T @ U @ T4 )
=> ( ( member_set_a @ ( Resid @ T4 @ T ) @ NN )
& ( member_set_a @ ( Resid @ T @ T4 ) @ NN ) ) ) ) ) ) ).
% normal_sub_rts.composite_of_arr_normal
thf(fact_596_normal__sub__rts_Ocomposite__of__arr__normal,axiom,
! [Resid: a > a > a,NN: set_a,Arr: a > $o,T: a,U: a,T4: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( 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 ) ) ) ) ) ) ).
% normal_sub_rts.composite_of_arr_normal
thf(fact_597_normal__sub__rts_OCong_092_060_094sub_0620__cancel__left,axiom,
! [Resid: product_prod_a_a > product_prod_a_a > product_prod_a_a,NN: set_Product_prod_a_a,T: product_prod_a_a,U: product_prod_a_a,V: product_prod_a_a,U2: product_prod_a_a,V2: product_prod_a_a] :
( ( normal2251803270671997330od_a_a @ Resid @ NN )
=> ( ( compos6970852536677457805od_a_a @ Resid @ T @ U @ V )
=> ( ( compos6970852536677457805od_a_a @ Resid @ T @ U2 @ V2 )
=> ( ( ( member1426531477525435216od_a_a @ ( Resid @ V @ V2 ) @ NN )
& ( member1426531477525435216od_a_a @ ( Resid @ V2 @ V ) @ NN ) )
=> ( ( member1426531477525435216od_a_a @ ( Resid @ U @ U2 ) @ NN )
& ( member1426531477525435216od_a_a @ ( Resid @ U2 @ U ) @ NN ) ) ) ) ) ) ).
% normal_sub_rts.Cong\<^sub>0_cancel_left
thf(fact_598_normal__sub__rts_OCong_092_060_094sub_0620__cancel__left,axiom,
! [Resid: ( a > a ) > ( a > a ) > a > a,NN: set_a_a,T: a > a,U: a > a,V: a > a,U2: a > a,V2: a > a] :
( ( normal_sub_rts_a_a @ Resid @ NN )
=> ( ( composite_of_a_a @ Resid @ T @ U @ V )
=> ( ( composite_of_a_a @ Resid @ T @ U2 @ V2 )
=> ( ( ( member_a_a @ ( Resid @ V @ V2 ) @ NN )
& ( member_a_a @ ( Resid @ V2 @ V ) @ NN ) )
=> ( ( member_a_a @ ( Resid @ U @ U2 ) @ NN )
& ( member_a_a @ ( Resid @ U2 @ U ) @ NN ) ) ) ) ) ) ).
% normal_sub_rts.Cong\<^sub>0_cancel_left
thf(fact_599_normal__sub__rts_OCong_092_060_094sub_0620__cancel__left,axiom,
! [Resid: set_a > set_a > set_a,NN: set_set_a,T: set_a,U: set_a,V: set_a,U2: set_a,V2: set_a] :
( ( normal_sub_rts_set_a @ Resid @ NN )
=> ( ( composite_of_set_a @ Resid @ T @ U @ V )
=> ( ( composite_of_set_a @ Resid @ T @ U2 @ V2 )
=> ( ( ( member_set_a @ ( Resid @ V @ V2 ) @ NN )
& ( member_set_a @ ( Resid @ V2 @ V ) @ NN ) )
=> ( ( member_set_a @ ( Resid @ U @ U2 ) @ NN )
& ( member_set_a @ ( Resid @ U2 @ U ) @ NN ) ) ) ) ) ) ).
% normal_sub_rts.Cong\<^sub>0_cancel_left
thf(fact_600_normal__sub__rts_OCong_092_060_094sub_0620__cancel__left,axiom,
! [Resid: a > a > a,NN: set_a,T: a,U: a,V: a,U2: a,V2: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( composite_of_a @ Resid @ T @ U @ V )
=> ( ( composite_of_a @ Resid @ T @ U2 @ V2 )
=> ( ( ( member_a @ ( Resid @ V @ V2 ) @ NN )
& ( member_a @ ( Resid @ V2 @ V ) @ NN ) )
=> ( ( member_a @ ( Resid @ U @ U2 ) @ NN )
& ( member_a @ ( Resid @ U2 @ U ) @ NN ) ) ) ) ) ) ).
% normal_sub_rts.Cong\<^sub>0_cancel_left
thf(fact_601_normal__sub__rts_OCong__closure__props_I3_J,axiom,
! [Resid: product_prod_a_a > product_prod_a_a > product_prod_a_a,NN: set_Product_prod_a_a,T: product_prod_a_a,U: product_prod_a_a] :
( ( normal2251803270671997330od_a_a @ Resid @ NN )
=> ( ( ( member1426531477525435216od_a_a @ ( Resid @ T @ U ) @ NN )
& ( member1426531477525435216od_a_a @ ( Resid @ U @ T ) @ NN ) )
=> ( normal7727773393199519165od_a_a @ Resid @ NN @ T @ U ) ) ) ).
% normal_sub_rts.Cong_closure_props(3)
thf(fact_602_normal__sub__rts_OCong__closure__props_I3_J,axiom,
! [Resid: set_a > set_a > set_a,NN: set_set_a,T: set_a,U: set_a] :
( ( normal_sub_rts_set_a @ Resid @ NN )
=> ( ( ( member_set_a @ ( Resid @ T @ U ) @ NN )
& ( member_set_a @ ( Resid @ U @ T ) @ NN ) )
=> ( normal8977612136997397236_set_a @ Resid @ NN @ T @ U ) ) ) ).
% normal_sub_rts.Cong_closure_props(3)
thf(fact_603_normal__sub__rts_OCong__closure__props_I3_J,axiom,
! [Resid: ( a > a ) > ( a > a ) > a > a,NN: set_a_a,T: a > a,U: a > a] :
( ( normal_sub_rts_a_a @ Resid @ NN )
=> ( ( ( member_a_a @ ( Resid @ T @ U ) @ NN )
& ( member_a_a @ ( Resid @ U @ T ) @ NN ) )
=> ( normal_sub_Cong_a_a @ Resid @ NN @ T @ U ) ) ) ).
% normal_sub_rts.Cong_closure_props(3)
thf(fact_604_normal__sub__rts_OCong__closure__props_I3_J,axiom,
! [Resid: a > a > a,NN: set_a,T: a,U: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( ( member_a @ ( Resid @ T @ U ) @ NN )
& ( member_a @ ( Resid @ U @ T ) @ NN ) )
=> ( normal_sub_Cong_a @ Resid @ NN @ T @ U ) ) ) ).
% normal_sub_rts.Cong_closure_props(3)
thf(fact_605_normal__sub__rts_OCong__closure__props_I2_J,axiom,
! [Resid: a > a > a,NN: set_a,T: a,U: a,V: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( normal_sub_Cong_a @ Resid @ NN @ T @ U )
=> ( ( normal_sub_Cong_a @ Resid @ NN @ U @ V )
=> ( normal_sub_Cong_a @ Resid @ NN @ T @ V ) ) ) ) ).
% normal_sub_rts.Cong_closure_props(2)
thf(fact_606_normal__sub__rts_OCong__closure__props_I1_J,axiom,
! [Resid: a > a > a,NN: set_a,T: a,U: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( normal_sub_Cong_a @ Resid @ NN @ T @ U )
=> ( normal_sub_Cong_a @ Resid @ NN @ U @ T ) ) ) ).
% normal_sub_rts.Cong_closure_props(1)
thf(fact_607_normal__sub__rts_OCongE,axiom,
! [Resid: product_prod_a_a > product_prod_a_a > product_prod_a_a,NN: set_Product_prod_a_a,T: product_prod_a_a,T4: product_prod_a_a] :
( ( normal2251803270671997330od_a_a @ Resid @ NN )
=> ( ( normal7727773393199519165od_a_a @ Resid @ NN @ T @ T4 )
=> ~ ! [U5: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ U5 @ NN )
=> ! [U6: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ U6 @ NN )
=> ~ ( ( member1426531477525435216od_a_a @ ( Resid @ ( Resid @ T @ U5 ) @ ( Resid @ T4 @ U6 ) ) @ NN )
& ( member1426531477525435216od_a_a @ ( Resid @ ( Resid @ T4 @ U6 ) @ ( Resid @ T @ U5 ) ) @ NN ) ) ) ) ) ) ).
% normal_sub_rts.CongE
thf(fact_608_normal__sub__rts_OCongE,axiom,
! [Resid: set_a > set_a > set_a,NN: set_set_a,T: set_a,T4: set_a] :
( ( normal_sub_rts_set_a @ Resid @ NN )
=> ( ( normal8977612136997397236_set_a @ Resid @ NN @ T @ T4 )
=> ~ ! [U5: set_a] :
( ( member_set_a @ U5 @ NN )
=> ! [U6: set_a] :
( ( member_set_a @ U6 @ NN )
=> ~ ( ( member_set_a @ ( Resid @ ( Resid @ T @ U5 ) @ ( Resid @ T4 @ U6 ) ) @ NN )
& ( member_set_a @ ( Resid @ ( Resid @ T4 @ U6 ) @ ( Resid @ T @ U5 ) ) @ NN ) ) ) ) ) ) ).
% normal_sub_rts.CongE
thf(fact_609_normal__sub__rts_OCongE,axiom,
! [Resid: ( a > a ) > ( a > a ) > a > a,NN: set_a_a,T: a > a,T4: a > a] :
( ( normal_sub_rts_a_a @ Resid @ NN )
=> ( ( normal_sub_Cong_a_a @ Resid @ NN @ T @ T4 )
=> ~ ! [U5: a > a] :
( ( member_a_a @ U5 @ NN )
=> ! [U6: a > a] :
( ( member_a_a @ U6 @ NN )
=> ~ ( ( member_a_a @ ( Resid @ ( Resid @ T @ U5 ) @ ( Resid @ T4 @ U6 ) ) @ NN )
& ( member_a_a @ ( Resid @ ( Resid @ T4 @ U6 ) @ ( Resid @ T @ U5 ) ) @ NN ) ) ) ) ) ) ).
% normal_sub_rts.CongE
thf(fact_610_normal__sub__rts_OCongE,axiom,
! [Resid: a > a > a,NN: set_a,T: a,T4: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( normal_sub_Cong_a @ Resid @ NN @ T @ T4 )
=> ~ ! [U5: a] :
( ( member_a @ U5 @ NN )
=> ! [U6: a] :
( ( member_a @ U6 @ NN )
=> ~ ( ( member_a @ ( Resid @ ( Resid @ T @ U5 ) @ ( Resid @ T4 @ U6 ) ) @ NN )
& ( member_a @ ( Resid @ ( Resid @ T4 @ U6 ) @ ( Resid @ T @ U5 ) ) @ NN ) ) ) ) ) ) ).
% normal_sub_rts.CongE
thf(fact_611_normal__sub__rts_OCongI,axiom,
! [Resid: product_prod_a_a > product_prod_a_a > product_prod_a_a,NN: set_Product_prod_a_a,U: product_prod_a_a,U2: product_prod_a_a,T: product_prod_a_a,T4: product_prod_a_a] :
( ( normal2251803270671997330od_a_a @ Resid @ NN )
=> ( ( member1426531477525435216od_a_a @ U @ NN )
=> ( ( member1426531477525435216od_a_a @ U2 @ NN )
=> ( ( ( member1426531477525435216od_a_a @ ( Resid @ ( Resid @ T @ U ) @ ( Resid @ T4 @ U2 ) ) @ NN )
& ( member1426531477525435216od_a_a @ ( Resid @ ( Resid @ T4 @ U2 ) @ ( Resid @ T @ U ) ) @ NN ) )
=> ( normal7727773393199519165od_a_a @ Resid @ NN @ T @ T4 ) ) ) ) ) ).
% normal_sub_rts.CongI
thf(fact_612_normal__sub__rts_OCongI,axiom,
! [Resid: set_a > set_a > set_a,NN: set_set_a,U: set_a,U2: set_a,T: set_a,T4: set_a] :
( ( normal_sub_rts_set_a @ Resid @ NN )
=> ( ( member_set_a @ U @ NN )
=> ( ( member_set_a @ U2 @ NN )
=> ( ( ( member_set_a @ ( Resid @ ( Resid @ T @ U ) @ ( Resid @ T4 @ U2 ) ) @ NN )
& ( member_set_a @ ( Resid @ ( Resid @ T4 @ U2 ) @ ( Resid @ T @ U ) ) @ NN ) )
=> ( normal8977612136997397236_set_a @ Resid @ NN @ T @ T4 ) ) ) ) ) ).
% normal_sub_rts.CongI
thf(fact_613_normal__sub__rts_OCongI,axiom,
! [Resid: ( a > a ) > ( a > a ) > a > a,NN: set_a_a,U: a > a,U2: a > a,T: a > a,T4: a > a] :
( ( normal_sub_rts_a_a @ Resid @ NN )
=> ( ( member_a_a @ U @ NN )
=> ( ( member_a_a @ U2 @ NN )
=> ( ( ( member_a_a @ ( Resid @ ( Resid @ T @ U ) @ ( Resid @ T4 @ U2 ) ) @ NN )
& ( member_a_a @ ( Resid @ ( Resid @ T4 @ U2 ) @ ( Resid @ T @ U ) ) @ NN ) )
=> ( normal_sub_Cong_a_a @ Resid @ NN @ T @ T4 ) ) ) ) ) ).
% normal_sub_rts.CongI
thf(fact_614_normal__sub__rts_OCongI,axiom,
! [Resid: a > a > a,NN: set_a,U: a,U2: a,T: a,T4: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( member_a @ U @ NN )
=> ( ( member_a @ U2 @ NN )
=> ( ( ( member_a @ ( Resid @ ( Resid @ T @ U ) @ ( Resid @ T4 @ U2 ) ) @ NN )
& ( member_a @ ( Resid @ ( Resid @ T4 @ U2 ) @ ( Resid @ T @ U ) ) @ NN ) )
=> ( normal_sub_Cong_a @ Resid @ NN @ T @ T4 ) ) ) ) ) ).
% normal_sub_rts.CongI
thf(fact_615_normal__sub__rts_OCong__def,axiom,
! [Resid: product_prod_a_a > product_prod_a_a > product_prod_a_a,NN: set_Product_prod_a_a,T: product_prod_a_a,T4: product_prod_a_a] :
( ( normal2251803270671997330od_a_a @ Resid @ NN )
=> ( ( normal7727773393199519165od_a_a @ Resid @ NN @ T @ T4 )
= ( ? [U3: product_prod_a_a,U4: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ U3 @ NN )
& ( member1426531477525435216od_a_a @ U4 @ NN )
& ( member1426531477525435216od_a_a @ ( Resid @ ( Resid @ T @ U3 ) @ ( Resid @ T4 @ U4 ) ) @ NN )
& ( member1426531477525435216od_a_a @ ( Resid @ ( Resid @ T4 @ U4 ) @ ( Resid @ T @ U3 ) ) @ NN ) ) ) ) ) ).
% normal_sub_rts.Cong_def
thf(fact_616_normal__sub__rts_OCong__def,axiom,
! [Resid: set_a > set_a > set_a,NN: set_set_a,T: set_a,T4: set_a] :
( ( normal_sub_rts_set_a @ Resid @ NN )
=> ( ( normal8977612136997397236_set_a @ Resid @ NN @ T @ T4 )
= ( ? [U3: set_a,U4: set_a] :
( ( member_set_a @ U3 @ NN )
& ( member_set_a @ U4 @ NN )
& ( member_set_a @ ( Resid @ ( Resid @ T @ U3 ) @ ( Resid @ T4 @ U4 ) ) @ NN )
& ( member_set_a @ ( Resid @ ( Resid @ T4 @ U4 ) @ ( Resid @ T @ U3 ) ) @ NN ) ) ) ) ) ).
% normal_sub_rts.Cong_def
thf(fact_617_normal__sub__rts_OCong__def,axiom,
! [Resid: ( a > a ) > ( a > a ) > a > a,NN: set_a_a,T: a > a,T4: a > a] :
( ( normal_sub_rts_a_a @ Resid @ NN )
=> ( ( normal_sub_Cong_a_a @ Resid @ NN @ T @ T4 )
= ( ? [U3: a > a,U4: a > a] :
( ( member_a_a @ U3 @ NN )
& ( member_a_a @ U4 @ NN )
& ( member_a_a @ ( Resid @ ( Resid @ T @ U3 ) @ ( Resid @ T4 @ U4 ) ) @ NN )
& ( member_a_a @ ( Resid @ ( Resid @ T4 @ U4 ) @ ( Resid @ T @ U3 ) ) @ NN ) ) ) ) ) ).
% normal_sub_rts.Cong_def
thf(fact_618_normal__sub__rts_OCong__def,axiom,
! [Resid: a > a > a,NN: set_a,T: a,T4: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( normal_sub_Cong_a @ Resid @ NN @ T @ T4 )
= ( ? [U3: a,U4: a] :
( ( member_a @ U3 @ NN )
& ( member_a @ U4 @ NN )
& ( member_a @ ( Resid @ ( Resid @ T @ U3 ) @ ( Resid @ T4 @ U4 ) ) @ NN )
& ( member_a @ ( Resid @ ( Resid @ T4 @ U4 ) @ ( Resid @ T @ U3 ) ) @ NN ) ) ) ) ) ).
% normal_sub_rts.Cong_def
thf(fact_619_normal__sub__rts_OCong__symmetric,axiom,
! [Resid: a > a > a,NN: set_a,T: a,T4: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( normal_sub_Cong_a @ Resid @ NN @ T @ T4 )
=> ( normal_sub_Cong_a @ Resid @ NN @ T4 @ T ) ) ) ).
% normal_sub_rts.Cong_symmetric
thf(fact_620_normal__sub__rts_OCong__transitive,axiom,
! [Resid: a > a > a,NN: set_a,T: a,T6: a,T4: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( normal_sub_Cong_a @ Resid @ NN @ T @ T6 )
=> ( ( normal_sub_Cong_a @ Resid @ NN @ T6 @ T4 )
=> ( normal_sub_Cong_a @ Resid @ NN @ T @ T4 ) ) ) ) ).
% normal_sub_rts.Cong_transitive
thf(fact_621_normal__sub__rts_Onormal__is__Cong__closed,axiom,
! [Resid: product_prod_a_a > product_prod_a_a > product_prod_a_a,NN: set_Product_prod_a_a,T: product_prod_a_a,T4: product_prod_a_a] :
( ( normal2251803270671997330od_a_a @ Resid @ NN )
=> ( ( member1426531477525435216od_a_a @ T @ NN )
=> ( ( normal7727773393199519165od_a_a @ Resid @ NN @ T @ T4 )
=> ( member1426531477525435216od_a_a @ T4 @ NN ) ) ) ) ).
% normal_sub_rts.normal_is_Cong_closed
thf(fact_622_normal__sub__rts_Onormal__is__Cong__closed,axiom,
! [Resid: set_a > set_a > set_a,NN: set_set_a,T: set_a,T4: set_a] :
( ( normal_sub_rts_set_a @ Resid @ NN )
=> ( ( member_set_a @ T @ NN )
=> ( ( normal8977612136997397236_set_a @ Resid @ NN @ T @ T4 )
=> ( member_set_a @ T4 @ NN ) ) ) ) ).
% normal_sub_rts.normal_is_Cong_closed
thf(fact_623_normal__sub__rts_Onormal__is__Cong__closed,axiom,
! [Resid: ( a > a ) > ( a > a ) > a > a,NN: set_a_a,T: a > a,T4: a > a] :
( ( normal_sub_rts_a_a @ Resid @ NN )
=> ( ( member_a_a @ T @ NN )
=> ( ( normal_sub_Cong_a_a @ Resid @ NN @ T @ T4 )
=> ( member_a_a @ T4 @ NN ) ) ) ) ).
% normal_sub_rts.normal_is_Cong_closed
thf(fact_624_normal__sub__rts_Onormal__is__Cong__closed,axiom,
! [Resid: a > a > a,NN: set_a,T: a,T4: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( member_a @ T @ NN )
=> ( ( normal_sub_Cong_a @ Resid @ NN @ T @ T4 )
=> ( member_a @ T4 @ NN ) ) ) ) ).
% normal_sub_rts.normal_is_Cong_closed
thf(fact_625_normal__sub__rts_OCong_092_060_094sub_0620__implies__Cong,axiom,
! [Resid: product_prod_a_a > product_prod_a_a > product_prod_a_a,NN: set_Product_prod_a_a,T: product_prod_a_a,T4: product_prod_a_a] :
( ( normal2251803270671997330od_a_a @ Resid @ NN )
=> ( ( ( member1426531477525435216od_a_a @ ( Resid @ T @ T4 ) @ NN )
& ( member1426531477525435216od_a_a @ ( Resid @ T4 @ T ) @ NN ) )
=> ( normal7727773393199519165od_a_a @ Resid @ NN @ T @ T4 ) ) ) ).
% normal_sub_rts.Cong\<^sub>0_implies_Cong
thf(fact_626_normal__sub__rts_OCong_092_060_094sub_0620__implies__Cong,axiom,
! [Resid: set_a > set_a > set_a,NN: set_set_a,T: set_a,T4: set_a] :
( ( normal_sub_rts_set_a @ Resid @ NN )
=> ( ( ( member_set_a @ ( Resid @ T @ T4 ) @ NN )
& ( member_set_a @ ( Resid @ T4 @ T ) @ NN ) )
=> ( normal8977612136997397236_set_a @ Resid @ NN @ T @ T4 ) ) ) ).
% normal_sub_rts.Cong\<^sub>0_implies_Cong
thf(fact_627_normal__sub__rts_OCong_092_060_094sub_0620__implies__Cong,axiom,
! [Resid: ( a > a ) > ( a > a ) > a > a,NN: set_a_a,T: a > a,T4: a > a] :
( ( normal_sub_rts_a_a @ Resid @ NN )
=> ( ( ( member_a_a @ ( Resid @ T @ T4 ) @ NN )
& ( member_a_a @ ( Resid @ T4 @ T ) @ NN ) )
=> ( normal_sub_Cong_a_a @ Resid @ NN @ T @ T4 ) ) ) ).
% normal_sub_rts.Cong\<^sub>0_implies_Cong
thf(fact_628_normal__sub__rts_OCong_092_060_094sub_0620__implies__Cong,axiom,
! [Resid: a > a > a,NN: set_a,T: a,T4: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( ( member_a @ ( Resid @ T @ T4 ) @ NN )
& ( member_a @ ( Resid @ T4 @ T ) @ NN ) )
=> ( normal_sub_Cong_a @ Resid @ NN @ T @ T4 ) ) ) ).
% normal_sub_rts.Cong\<^sub>0_implies_Cong
thf(fact_629_normal__sub__rts_OCong_H_Ointros_I3_J,axiom,
! [Resid: product_prod_a_a > product_prod_a_a > product_prod_a_a,NN: set_Product_prod_a_a,T: product_prod_a_a,U: product_prod_a_a] :
( ( normal2251803270671997330od_a_a @ Resid @ NN )
=> ( ( ( member1426531477525435216od_a_a @ ( Resid @ T @ U ) @ NN )
& ( member1426531477525435216od_a_a @ ( Resid @ U @ T ) @ NN ) )
=> ( normal2813552597983363028od_a_a @ Resid @ NN @ T @ U ) ) ) ).
% normal_sub_rts.Cong'.intros(3)
thf(fact_630_normal__sub__rts_OCong_H_Ointros_I3_J,axiom,
! [Resid: set_a > set_a > set_a,NN: set_set_a,T: set_a,U: set_a] :
( ( normal_sub_rts_set_a @ Resid @ NN )
=> ( ( ( member_set_a @ ( Resid @ T @ U ) @ NN )
& ( member_set_a @ ( Resid @ U @ T ) @ NN ) )
=> ( normal8837514132249976843_set_a @ Resid @ NN @ T @ U ) ) ) ).
% normal_sub_rts.Cong'.intros(3)
thf(fact_631_normal__sub__rts_OCong_H_Ointros_I3_J,axiom,
! [Resid: ( a > a ) > ( a > a ) > a > a,NN: set_a_a,T: a > a,U: a > a] :
( ( normal_sub_rts_a_a @ Resid @ NN )
=> ( ( ( member_a_a @ ( Resid @ T @ U ) @ NN )
& ( member_a_a @ ( Resid @ U @ T ) @ NN ) )
=> ( normal_sub_Cong_a_a2 @ Resid @ NN @ T @ U ) ) ) ).
% normal_sub_rts.Cong'.intros(3)
thf(fact_632_normal__sub__rts_OCong_H_Ointros_I3_J,axiom,
! [Resid: a > a > a,NN: set_a,T: a,U: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( ( member_a @ ( Resid @ T @ U ) @ NN )
& ( member_a @ ( Resid @ U @ T ) @ NN ) )
=> ( normal_sub_Cong_a2 @ Resid @ NN @ T @ U ) ) ) ).
% normal_sub_rts.Cong'.intros(3)
thf(fact_633_normal__sub__rts_OCong_H_Ointros_I2_J,axiom,
! [Resid: a > a > a,NN: set_a,T: a,U: a,V: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( normal_sub_Cong_a2 @ Resid @ NN @ T @ U )
=> ( ( normal_sub_Cong_a2 @ Resid @ NN @ U @ V )
=> ( normal_sub_Cong_a2 @ Resid @ NN @ T @ V ) ) ) ) ).
% normal_sub_rts.Cong'.intros(2)
thf(fact_634_normal__sub__rts_OCong_H_Ointros_I1_J,axiom,
! [Resid: a > a > a,NN: set_a,T: a,U: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( normal_sub_Cong_a2 @ Resid @ NN @ T @ U )
=> ( normal_sub_Cong_a2 @ Resid @ NN @ U @ T ) ) ) ).
% normal_sub_rts.Cong'.intros(1)
thf(fact_635_normal__sub__rts_OCong_H__if,axiom,
! [Resid: product_prod_a_a > product_prod_a_a > product_prod_a_a,NN: set_Product_prod_a_a,U: product_prod_a_a,U2: product_prod_a_a,T: product_prod_a_a,T4: product_prod_a_a] :
( ( normal2251803270671997330od_a_a @ Resid @ NN )
=> ( ( member1426531477525435216od_a_a @ U @ NN )
=> ( ( member1426531477525435216od_a_a @ U2 @ NN )
=> ( ( ( member1426531477525435216od_a_a @ ( Resid @ ( Resid @ T @ U ) @ ( Resid @ T4 @ U2 ) ) @ NN )
& ( member1426531477525435216od_a_a @ ( Resid @ ( Resid @ T4 @ U2 ) @ ( Resid @ T @ U ) ) @ NN ) )
=> ( normal2813552597983363028od_a_a @ Resid @ NN @ T @ T4 ) ) ) ) ) ).
% normal_sub_rts.Cong'_if
thf(fact_636_normal__sub__rts_OCong_H__if,axiom,
! [Resid: set_a > set_a > set_a,NN: set_set_a,U: set_a,U2: set_a,T: set_a,T4: set_a] :
( ( normal_sub_rts_set_a @ Resid @ NN )
=> ( ( member_set_a @ U @ NN )
=> ( ( member_set_a @ U2 @ NN )
=> ( ( ( member_set_a @ ( Resid @ ( Resid @ T @ U ) @ ( Resid @ T4 @ U2 ) ) @ NN )
& ( member_set_a @ ( Resid @ ( Resid @ T4 @ U2 ) @ ( Resid @ T @ U ) ) @ NN ) )
=> ( normal8837514132249976843_set_a @ Resid @ NN @ T @ T4 ) ) ) ) ) ).
% normal_sub_rts.Cong'_if
thf(fact_637_normal__sub__rts_OCong_H__if,axiom,
! [Resid: ( a > a ) > ( a > a ) > a > a,NN: set_a_a,U: a > a,U2: a > a,T: a > a,T4: a > a] :
( ( normal_sub_rts_a_a @ Resid @ NN )
=> ( ( member_a_a @ U @ NN )
=> ( ( member_a_a @ U2 @ NN )
=> ( ( ( member_a_a @ ( Resid @ ( Resid @ T @ U ) @ ( Resid @ T4 @ U2 ) ) @ NN )
& ( member_a_a @ ( Resid @ ( Resid @ T4 @ U2 ) @ ( Resid @ T @ U ) ) @ NN ) )
=> ( normal_sub_Cong_a_a2 @ Resid @ NN @ T @ T4 ) ) ) ) ) ).
% normal_sub_rts.Cong'_if
thf(fact_638_normal__sub__rts_OCong_H__if,axiom,
! [Resid: a > a > a,NN: set_a,U: a,U2: a,T: a,T4: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( member_a @ U @ NN )
=> ( ( member_a @ U2 @ NN )
=> ( ( ( member_a @ ( Resid @ ( Resid @ T @ U ) @ ( Resid @ T4 @ U2 ) ) @ NN )
& ( member_a @ ( Resid @ ( Resid @ T4 @ U2 ) @ ( Resid @ T @ U ) ) @ NN ) )
=> ( normal_sub_Cong_a2 @ Resid @ NN @ T @ T4 ) ) ) ) ) ).
% normal_sub_rts.Cong'_if
thf(fact_639_normal__sub__rts_Oforward__stable,axiom,
! [Resid: product_prod_a_a > product_prod_a_a > product_prod_a_a,NN: set_Product_prod_a_a,U: product_prod_a_a,T: product_prod_a_a] :
( ( normal2251803270671997330od_a_a @ Resid @ NN )
=> ( ( member1426531477525435216od_a_a @ U @ NN )
=> ( ( coinit880904215945527283od_a_a @ Resid @ T @ U )
=> ( member1426531477525435216od_a_a @ ( Resid @ U @ T ) @ NN ) ) ) ) ).
% normal_sub_rts.forward_stable
thf(fact_640_normal__sub__rts_Oforward__stable,axiom,
! [Resid: set_a > set_a > set_a,NN: set_set_a,U: set_a,T: set_a] :
( ( normal_sub_rts_set_a @ Resid @ NN )
=> ( ( member_set_a @ U @ NN )
=> ( ( coinitial_set_a @ Resid @ T @ U )
=> ( member_set_a @ ( Resid @ U @ T ) @ NN ) ) ) ) ).
% normal_sub_rts.forward_stable
thf(fact_641_normal__sub__rts_Oforward__stable,axiom,
! [Resid: ( a > a ) > ( a > a ) > a > a,NN: set_a_a,U: a > a,T: a > a] :
( ( normal_sub_rts_a_a @ Resid @ NN )
=> ( ( member_a_a @ U @ NN )
=> ( ( coinitial_a_a @ Resid @ T @ U )
=> ( member_a_a @ ( Resid @ U @ T ) @ NN ) ) ) ) ).
% normal_sub_rts.forward_stable
thf(fact_642_normal__sub__rts_Oforward__stable,axiom,
! [Resid: a > a > a,NN: set_a,U: a,T: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( member_a @ U @ NN )
=> ( ( coinitial_a @ Resid @ T @ U )
=> ( member_a @ ( Resid @ U @ T ) @ NN ) ) ) ) ).
% normal_sub_rts.forward_stable
thf(fact_643_coherent__normal__sub__rts_Oaxioms_I1_J,axiom,
! [Resid: a > a > a,NN: set_a] :
( ( cohere6072184133013167079_rts_a @ Resid @ NN )
=> ( normal_sub_rts_a @ Resid @ NN ) ) ).
% coherent_normal_sub_rts.axioms(1)
thf(fact_644_normal__sub__rts_Oaxioms_I2_J,axiom,
! [Resid: a > a > a,NN: set_a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( normal7698203753654205830ioms_a @ Resid @ NN ) ) ).
% normal_sub_rts.axioms(2)
thf(fact_645_normal__sub__rts_OResid__along__normal__preserves__reflects__con,axiom,
! [Resid: product_prod_a_a > product_prod_a_a > product_prod_a_a,NN: set_Product_prod_a_a,U: product_prod_a_a,T: product_prod_a_a,T4: product_prod_a_a] :
( ( normal2251803270671997330od_a_a @ Resid @ NN )
=> ( ( member1426531477525435216od_a_a @ U @ NN )
=> ( ( ( source6950040787684646355od_a_a @ Resid @ T )
= ( source6950040787684646355od_a_a @ Resid @ U ) )
=> ( ( con_Product_prod_a_a @ Resid @ ( Resid @ T @ U ) @ ( Resid @ T4 @ U ) )
= ( con_Product_prod_a_a @ Resid @ T @ T4 ) ) ) ) ) ).
% normal_sub_rts.Resid_along_normal_preserves_reflects_con
thf(fact_646_normal__sub__rts_OResid__along__normal__preserves__reflects__con,axiom,
! [Resid: ( a > a ) > ( a > a ) > a > a,NN: set_a_a,U: a > a,T: a > a,T4: a > a] :
( ( normal_sub_rts_a_a @ Resid @ NN )
=> ( ( member_a_a @ U @ NN )
=> ( ( ( sources_a_a @ Resid @ T )
= ( sources_a_a @ Resid @ U ) )
=> ( ( con_a_a @ Resid @ ( Resid @ T @ U ) @ ( Resid @ T4 @ U ) )
= ( con_a_a @ Resid @ T @ T4 ) ) ) ) ) ).
% normal_sub_rts.Resid_along_normal_preserves_reflects_con
thf(fact_647_normal__sub__rts_OResid__along__normal__preserves__reflects__con,axiom,
! [Resid: set_a > set_a > set_a,NN: set_set_a,U: set_a,T: set_a,T4: set_a] :
( ( normal_sub_rts_set_a @ Resid @ NN )
=> ( ( member_set_a @ U @ NN )
=> ( ( ( sources_set_a @ Resid @ T )
= ( sources_set_a @ Resid @ U ) )
=> ( ( con_set_a @ Resid @ ( Resid @ T @ U ) @ ( Resid @ T4 @ U ) )
= ( con_set_a @ Resid @ T @ T4 ) ) ) ) ) ).
% normal_sub_rts.Resid_along_normal_preserves_reflects_con
thf(fact_648_normal__sub__rts_OResid__along__normal__preserves__reflects__con,axiom,
! [Resid: a > a > a,NN: set_a,U: a,T: a,T4: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( 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 ) ) ) ) ) ).
% normal_sub_rts.Resid_along_normal_preserves_reflects_con
thf(fact_649_normal__sub__rts_Odiamond__commutes__upto__Cong_092_060_094sub_0620,axiom,
! [Resid: product_prod_a_a > product_prod_a_a > product_prod_a_a,NN: set_Product_prod_a_a,T: product_prod_a_a,U: product_prod_a_a,V: product_prod_a_a,V2: product_prod_a_a] :
( ( normal2251803270671997330od_a_a @ Resid @ NN )
=> ( ( con_Product_prod_a_a @ Resid @ T @ U )
=> ( ( compos6970852536677457805od_a_a @ Resid @ T @ ( Resid @ U @ T ) @ V )
=> ( ( compos6970852536677457805od_a_a @ Resid @ U @ ( Resid @ T @ U ) @ V2 )
=> ( ( member1426531477525435216od_a_a @ ( Resid @ V @ V2 ) @ NN )
& ( member1426531477525435216od_a_a @ ( Resid @ V2 @ V ) @ NN ) ) ) ) ) ) ).
% normal_sub_rts.diamond_commutes_upto_Cong\<^sub>0
thf(fact_650_normal__sub__rts_Odiamond__commutes__upto__Cong_092_060_094sub_0620,axiom,
! [Resid: ( a > a ) > ( a > a ) > a > a,NN: set_a_a,T: a > a,U: a > a,V: a > a,V2: a > a] :
( ( normal_sub_rts_a_a @ Resid @ NN )
=> ( ( con_a_a @ Resid @ T @ U )
=> ( ( composite_of_a_a @ Resid @ T @ ( Resid @ U @ T ) @ V )
=> ( ( composite_of_a_a @ Resid @ U @ ( Resid @ T @ U ) @ V2 )
=> ( ( member_a_a @ ( Resid @ V @ V2 ) @ NN )
& ( member_a_a @ ( Resid @ V2 @ V ) @ NN ) ) ) ) ) ) ).
% normal_sub_rts.diamond_commutes_upto_Cong\<^sub>0
thf(fact_651_normal__sub__rts_Odiamond__commutes__upto__Cong_092_060_094sub_0620,axiom,
! [Resid: set_a > set_a > set_a,NN: set_set_a,T: set_a,U: set_a,V: set_a,V2: set_a] :
( ( normal_sub_rts_set_a @ Resid @ NN )
=> ( ( con_set_a @ Resid @ T @ U )
=> ( ( composite_of_set_a @ Resid @ T @ ( Resid @ U @ T ) @ V )
=> ( ( composite_of_set_a @ Resid @ U @ ( Resid @ T @ U ) @ V2 )
=> ( ( member_set_a @ ( Resid @ V @ V2 ) @ NN )
& ( member_set_a @ ( Resid @ V2 @ V ) @ NN ) ) ) ) ) ) ).
% normal_sub_rts.diamond_commutes_upto_Cong\<^sub>0
thf(fact_652_normal__sub__rts_Odiamond__commutes__upto__Cong_092_060_094sub_0620,axiom,
! [Resid: a > a > a,NN: set_a,T: a,U: a,V: a,V2: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( con_a @ Resid @ T @ U )
=> ( ( composite_of_a @ Resid @ T @ ( Resid @ U @ T ) @ V )
=> ( ( composite_of_a @ Resid @ U @ ( Resid @ T @ U ) @ V2 )
=> ( ( member_a @ ( Resid @ V @ V2 ) @ NN )
& ( member_a @ ( Resid @ V2 @ V ) @ NN ) ) ) ) ) ) ).
% normal_sub_rts.diamond_commutes_upto_Cong\<^sub>0
thf(fact_653_normal__sub__rts_OCong__class__is__nonempty,axiom,
! [Resid: product_prod_a_a > product_prod_a_a > product_prod_a_a,NN: set_Product_prod_a_a,T7: set_Product_prod_a_a] :
( ( normal2251803270671997330od_a_a @ Resid @ NN )
=> ( ( normal6700481192199873089od_a_a @ Resid @ NN @ T7 )
=> ( T7 != bot_bo3357376287454694259od_a_a ) ) ) ).
% normal_sub_rts.Cong_class_is_nonempty
thf(fact_654_normal__sub__rts_OCong__class__is__nonempty,axiom,
! [Resid: set_a > set_a > set_a,NN: set_set_a,T7: set_set_a] :
( ( normal_sub_rts_set_a @ Resid @ NN )
=> ( ( normal4437380936311325560_set_a @ Resid @ NN @ T7 )
=> ( T7 != bot_bot_set_set_a ) ) ) ).
% normal_sub_rts.Cong_class_is_nonempty
thf(fact_655_normal__sub__rts_OCong__class__is__nonempty,axiom,
! [Resid: ( a > a ) > ( a > a ) > a > a,NN: set_a_a,T7: set_a_a] :
( ( normal_sub_rts_a_a @ Resid @ NN )
=> ( ( normal6067668076082862283ss_a_a @ Resid @ NN @ T7 )
=> ( T7 != bot_bot_set_a_a ) ) ) ).
% normal_sub_rts.Cong_class_is_nonempty
thf(fact_656_normal__sub__rts_OCong__class__is__nonempty,axiom,
! [Resid: a > a > a,NN: set_a,T7: set_a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( normal8595587647932138008lass_a @ Resid @ NN @ T7 )
=> ( T7 != bot_bot_set_a ) ) ) ).
% normal_sub_rts.Cong_class_is_nonempty
thf(fact_657_normal__sub__rts_OCong__imp__arr_I2_J,axiom,
! [Resid: set_a > set_a > set_a,NN: set_set_a,T: set_a,T4: set_a] :
( ( normal_sub_rts_set_a @ Resid @ NN )
=> ( ( normal8977612136997397236_set_a @ Resid @ NN @ T @ T4 )
=> ( arr_set_a @ Resid @ T4 ) ) ) ).
% normal_sub_rts.Cong_imp_arr(2)
thf(fact_658_normal__sub__rts_OCong__imp__arr_I2_J,axiom,
! [Resid: a > a > a,NN: set_a,T: a,T4: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( normal_sub_Cong_a @ Resid @ NN @ T @ T4 )
=> ( arr_a @ Resid @ T4 ) ) ) ).
% normal_sub_rts.Cong_imp_arr(2)
thf(fact_659_normal__sub__rts_OCong__imp__arr_I1_J,axiom,
! [Resid: set_a > set_a > set_a,NN: set_set_a,T: set_a,T4: set_a] :
( ( normal_sub_rts_set_a @ Resid @ NN )
=> ( ( normal8977612136997397236_set_a @ Resid @ NN @ T @ T4 )
=> ( arr_set_a @ Resid @ T ) ) ) ).
% normal_sub_rts.Cong_imp_arr(1)
thf(fact_660_normal__sub__rts_OCong__imp__arr_I1_J,axiom,
! [Resid: a > a > a,NN: set_a,T: a,T4: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( normal_sub_Cong_a @ Resid @ NN @ T @ T4 )
=> ( arr_a @ Resid @ T ) ) ) ).
% normal_sub_rts.Cong_imp_arr(1)
thf(fact_661_normal__sub__rts_OCong__reflexive,axiom,
! [Resid: set_a > set_a > set_a,NN: set_set_a,T: set_a] :
( ( normal_sub_rts_set_a @ Resid @ NN )
=> ( ( arr_set_a @ Resid @ T )
=> ( normal8977612136997397236_set_a @ Resid @ NN @ T @ T ) ) ) ).
% normal_sub_rts.Cong_reflexive
thf(fact_662_normal__sub__rts_OCong__reflexive,axiom,
! [Resid: a > a > a,NN: set_a,T: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( arr_a @ Resid @ T )
=> ( normal_sub_Cong_a @ Resid @ NN @ T @ T ) ) ) ).
% normal_sub_rts.Cong_reflexive
thf(fact_663_normal__sub__rts_OCong__closure__props_I4_J,axiom,
! [Resid: product_prod_a_a > product_prod_a_a > product_prod_a_a,NN: set_Product_prod_a_a,U: product_prod_a_a,T: product_prod_a_a] :
( ( normal2251803270671997330od_a_a @ Resid @ NN )
=> ( ( member1426531477525435216od_a_a @ U @ NN )
=> ( ( ( source6950040787684646355od_a_a @ Resid @ T )
= ( source6950040787684646355od_a_a @ Resid @ U ) )
=> ( normal7727773393199519165od_a_a @ Resid @ NN @ T @ ( Resid @ T @ U ) ) ) ) ) ).
% normal_sub_rts.Cong_closure_props(4)
thf(fact_664_normal__sub__rts_OCong__closure__props_I4_J,axiom,
! [Resid: ( a > a ) > ( a > a ) > a > a,NN: set_a_a,U: a > a,T: a > a] :
( ( normal_sub_rts_a_a @ Resid @ NN )
=> ( ( member_a_a @ U @ NN )
=> ( ( ( sources_a_a @ Resid @ T )
= ( sources_a_a @ Resid @ U ) )
=> ( normal_sub_Cong_a_a @ Resid @ NN @ T @ ( Resid @ T @ U ) ) ) ) ) ).
% normal_sub_rts.Cong_closure_props(4)
thf(fact_665_normal__sub__rts_OCong__closure__props_I4_J,axiom,
! [Resid: set_a > set_a > set_a,NN: set_set_a,U: set_a,T: set_a] :
( ( normal_sub_rts_set_a @ Resid @ NN )
=> ( ( member_set_a @ U @ NN )
=> ( ( ( sources_set_a @ Resid @ T )
= ( sources_set_a @ Resid @ U ) )
=> ( normal8977612136997397236_set_a @ Resid @ NN @ T @ ( Resid @ T @ U ) ) ) ) ) ).
% normal_sub_rts.Cong_closure_props(4)
thf(fact_666_normal__sub__rts_OCong__closure__props_I4_J,axiom,
! [Resid: a > a > a,NN: set_a,U: a,T: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( member_a @ U @ NN )
=> ( ( ( sources_a @ Resid @ T )
= ( sources_a @ Resid @ U ) )
=> ( normal_sub_Cong_a @ Resid @ NN @ T @ ( Resid @ T @ U ) ) ) ) ) ).
% normal_sub_rts.Cong_closure_props(4)
thf(fact_667_normal__sub__rts_Osources__are__Cong,axiom,
! [Resid: product_prod_a_a > product_prod_a_a > product_prod_a_a,NN: set_Product_prod_a_a,A: product_prod_a_a,T: product_prod_a_a,A2: product_prod_a_a] :
( ( normal2251803270671997330od_a_a @ Resid @ NN )
=> ( ( member1426531477525435216od_a_a @ A @ ( source6950040787684646355od_a_a @ Resid @ T ) )
=> ( ( member1426531477525435216od_a_a @ A2 @ ( source6950040787684646355od_a_a @ Resid @ T ) )
=> ( normal7727773393199519165od_a_a @ Resid @ NN @ A @ A2 ) ) ) ) ).
% normal_sub_rts.sources_are_Cong
thf(fact_668_normal__sub__rts_Osources__are__Cong,axiom,
! [Resid: ( a > a ) > ( a > a ) > a > a,NN: set_a_a,A: a > a,T: a > a,A2: a > a] :
( ( normal_sub_rts_a_a @ Resid @ NN )
=> ( ( member_a_a @ A @ ( sources_a_a @ Resid @ T ) )
=> ( ( member_a_a @ A2 @ ( sources_a_a @ Resid @ T ) )
=> ( normal_sub_Cong_a_a @ Resid @ NN @ A @ A2 ) ) ) ) ).
% normal_sub_rts.sources_are_Cong
thf(fact_669_normal__sub__rts_Osources__are__Cong,axiom,
! [Resid: set_a > set_a > set_a,NN: set_set_a,A: set_a,T: set_a,A2: set_a] :
( ( normal_sub_rts_set_a @ Resid @ NN )
=> ( ( member_set_a @ A @ ( sources_set_a @ Resid @ T ) )
=> ( ( member_set_a @ A2 @ ( sources_set_a @ Resid @ T ) )
=> ( normal8977612136997397236_set_a @ Resid @ NN @ A @ A2 ) ) ) ) ).
% normal_sub_rts.sources_are_Cong
thf(fact_670_normal__sub__rts_Osources__are__Cong,axiom,
! [Resid: a > a > a,NN: set_a,A: a,T: a,A2: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( member_a @ A @ ( sources_a @ Resid @ T ) )
=> ( ( member_a @ A2 @ ( sources_a @ Resid @ T ) )
=> ( normal_sub_Cong_a @ Resid @ NN @ A @ A2 ) ) ) ) ).
% normal_sub_rts.sources_are_Cong
thf(fact_671_normal__sub__rts_Oin__sources__respects__Cong,axiom,
! [Resid: product_prod_a_a > product_prod_a_a > product_prod_a_a,NN: set_Product_prod_a_a,T: product_prod_a_a,T4: product_prod_a_a,A: product_prod_a_a,A2: product_prod_a_a] :
( ( normal2251803270671997330od_a_a @ Resid @ NN )
=> ( ( normal7727773393199519165od_a_a @ Resid @ NN @ T @ T4 )
=> ( ( member1426531477525435216od_a_a @ A @ ( source6950040787684646355od_a_a @ Resid @ T ) )
=> ( ( member1426531477525435216od_a_a @ A2 @ ( source6950040787684646355od_a_a @ Resid @ T4 ) )
=> ( normal7727773393199519165od_a_a @ Resid @ NN @ A @ A2 ) ) ) ) ) ).
% normal_sub_rts.in_sources_respects_Cong
thf(fact_672_normal__sub__rts_Oin__sources__respects__Cong,axiom,
! [Resid: ( a > a ) > ( a > a ) > a > a,NN: set_a_a,T: a > a,T4: a > a,A: a > a,A2: a > a] :
( ( normal_sub_rts_a_a @ Resid @ NN )
=> ( ( normal_sub_Cong_a_a @ Resid @ NN @ T @ T4 )
=> ( ( member_a_a @ A @ ( sources_a_a @ Resid @ T ) )
=> ( ( member_a_a @ A2 @ ( sources_a_a @ Resid @ T4 ) )
=> ( normal_sub_Cong_a_a @ Resid @ NN @ A @ A2 ) ) ) ) ) ).
% normal_sub_rts.in_sources_respects_Cong
thf(fact_673_normal__sub__rts_Oin__sources__respects__Cong,axiom,
! [Resid: set_a > set_a > set_a,NN: set_set_a,T: set_a,T4: set_a,A: set_a,A2: set_a] :
( ( normal_sub_rts_set_a @ Resid @ NN )
=> ( ( normal8977612136997397236_set_a @ Resid @ NN @ T @ T4 )
=> ( ( member_set_a @ A @ ( sources_set_a @ Resid @ T ) )
=> ( ( member_set_a @ A2 @ ( sources_set_a @ Resid @ T4 ) )
=> ( normal8977612136997397236_set_a @ Resid @ NN @ A @ A2 ) ) ) ) ) ).
% normal_sub_rts.in_sources_respects_Cong
thf(fact_674_normal__sub__rts_Oin__sources__respects__Cong,axiom,
! [Resid: a > a > a,NN: set_a,T: a,T4: a,A: a,A2: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( normal_sub_Cong_a @ Resid @ NN @ T @ T4 )
=> ( ( member_a @ A @ ( sources_a @ Resid @ T ) )
=> ( ( member_a @ A2 @ ( sources_a @ Resid @ T4 ) )
=> ( normal_sub_Cong_a @ Resid @ NN @ A @ A2 ) ) ) ) ) ).
% normal_sub_rts.in_sources_respects_Cong
thf(fact_675_normal__sub__rts_Oin__targets__respects__Cong,axiom,
! [Resid: product_prod_a_a > product_prod_a_a > product_prod_a_a,NN: set_Product_prod_a_a,T: product_prod_a_a,T4: product_prod_a_a,B: product_prod_a_a,B2: product_prod_a_a] :
( ( normal2251803270671997330od_a_a @ Resid @ NN )
=> ( ( normal7727773393199519165od_a_a @ Resid @ NN @ T @ T4 )
=> ( ( member1426531477525435216od_a_a @ B @ ( target5293506191220573129od_a_a @ Resid @ T ) )
=> ( ( member1426531477525435216od_a_a @ B2 @ ( target5293506191220573129od_a_a @ Resid @ T4 ) )
=> ( normal7727773393199519165od_a_a @ Resid @ NN @ B @ B2 ) ) ) ) ) ).
% normal_sub_rts.in_targets_respects_Cong
thf(fact_676_normal__sub__rts_Oin__targets__respects__Cong,axiom,
! [Resid: set_a > set_a > set_a,NN: set_set_a,T: set_a,T4: set_a,B: set_a,B2: set_a] :
( ( normal_sub_rts_set_a @ Resid @ NN )
=> ( ( normal8977612136997397236_set_a @ Resid @ NN @ T @ T4 )
=> ( ( member_set_a @ B @ ( targets_set_a @ Resid @ T ) )
=> ( ( member_set_a @ B2 @ ( targets_set_a @ Resid @ T4 ) )
=> ( normal8977612136997397236_set_a @ Resid @ NN @ B @ B2 ) ) ) ) ) ).
% normal_sub_rts.in_targets_respects_Cong
thf(fact_677_normal__sub__rts_Oin__targets__respects__Cong,axiom,
! [Resid: ( a > a ) > ( a > a ) > a > a,NN: set_a_a,T: a > a,T4: a > a,B: a > a,B2: a > a] :
( ( normal_sub_rts_a_a @ Resid @ NN )
=> ( ( normal_sub_Cong_a_a @ Resid @ NN @ T @ T4 )
=> ( ( member_a_a @ B @ ( targets_a_a @ Resid @ T ) )
=> ( ( member_a_a @ B2 @ ( targets_a_a @ Resid @ T4 ) )
=> ( normal_sub_Cong_a_a @ Resid @ NN @ B @ B2 ) ) ) ) ) ).
% normal_sub_rts.in_targets_respects_Cong
thf(fact_678_normal__sub__rts_Oin__targets__respects__Cong,axiom,
! [Resid: a > a > a,NN: set_a,T: a,T4: a,B: a,B2: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( normal_sub_Cong_a @ Resid @ NN @ T @ T4 )
=> ( ( member_a @ B @ ( targets_a @ Resid @ T ) )
=> ( ( member_a @ B2 @ ( targets_a @ Resid @ T4 ) )
=> ( normal_sub_Cong_a @ Resid @ NN @ B @ B2 ) ) ) ) ) ).
% normal_sub_rts.in_targets_respects_Cong
thf(fact_679_normal__sub__rts_Otargets__are__Cong,axiom,
! [Resid: product_prod_a_a > product_prod_a_a > product_prod_a_a,NN: set_Product_prod_a_a,B: product_prod_a_a,T: product_prod_a_a,B2: product_prod_a_a] :
( ( normal2251803270671997330od_a_a @ Resid @ NN )
=> ( ( member1426531477525435216od_a_a @ B @ ( target5293506191220573129od_a_a @ Resid @ T ) )
=> ( ( member1426531477525435216od_a_a @ B2 @ ( target5293506191220573129od_a_a @ Resid @ T ) )
=> ( normal7727773393199519165od_a_a @ Resid @ NN @ B @ B2 ) ) ) ) ).
% normal_sub_rts.targets_are_Cong
thf(fact_680_normal__sub__rts_Otargets__are__Cong,axiom,
! [Resid: set_a > set_a > set_a,NN: set_set_a,B: set_a,T: set_a,B2: set_a] :
( ( normal_sub_rts_set_a @ Resid @ NN )
=> ( ( member_set_a @ B @ ( targets_set_a @ Resid @ T ) )
=> ( ( member_set_a @ B2 @ ( targets_set_a @ Resid @ T ) )
=> ( normal8977612136997397236_set_a @ Resid @ NN @ B @ B2 ) ) ) ) ).
% normal_sub_rts.targets_are_Cong
thf(fact_681_normal__sub__rts_Otargets__are__Cong,axiom,
! [Resid: ( a > a ) > ( a > a ) > a > a,NN: set_a_a,B: a > a,T: a > a,B2: a > a] :
( ( normal_sub_rts_a_a @ Resid @ NN )
=> ( ( member_a_a @ B @ ( targets_a_a @ Resid @ T ) )
=> ( ( member_a_a @ B2 @ ( targets_a_a @ Resid @ T ) )
=> ( normal_sub_Cong_a_a @ Resid @ NN @ B @ B2 ) ) ) ) ).
% normal_sub_rts.targets_are_Cong
thf(fact_682_normal__sub__rts_Otargets__are__Cong,axiom,
! [Resid: a > a > a,NN: set_a,B: a,T: a,B2: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( member_a @ B @ ( targets_a @ Resid @ T ) )
=> ( ( member_a @ B2 @ ( targets_a @ Resid @ T ) )
=> ( normal_sub_Cong_a @ Resid @ NN @ B @ B2 ) ) ) ) ).
% normal_sub_rts.targets_are_Cong
thf(fact_683_normal__sub__rts_Oresid__along__elem__preserves__con,axiom,
! [Resid: product_prod_a_a > product_prod_a_a > product_prod_a_a,NN: set_Product_prod_a_a,T: product_prod_a_a,T4: product_prod_a_a,U: product_prod_a_a] :
( ( normal2251803270671997330od_a_a @ Resid @ NN )
=> ( ( con_Product_prod_a_a @ Resid @ T @ T4 )
=> ( ( coinit880904215945527283od_a_a @ Resid @ T @ U )
=> ( ( member1426531477525435216od_a_a @ U @ NN )
=> ( con_Product_prod_a_a @ Resid @ ( Resid @ T @ U ) @ ( Resid @ T4 @ U ) ) ) ) ) ) ).
% normal_sub_rts.resid_along_elem_preserves_con
thf(fact_684_normal__sub__rts_Oresid__along__elem__preserves__con,axiom,
! [Resid: ( a > a ) > ( a > a ) > a > a,NN: set_a_a,T: a > a,T4: a > a,U: a > a] :
( ( normal_sub_rts_a_a @ Resid @ NN )
=> ( ( con_a_a @ Resid @ T @ T4 )
=> ( ( coinitial_a_a @ Resid @ T @ U )
=> ( ( member_a_a @ U @ NN )
=> ( con_a_a @ Resid @ ( Resid @ T @ U ) @ ( Resid @ T4 @ U ) ) ) ) ) ) ).
% normal_sub_rts.resid_along_elem_preserves_con
thf(fact_685_normal__sub__rts_Oresid__along__elem__preserves__con,axiom,
! [Resid: set_a > set_a > set_a,NN: set_set_a,T: set_a,T4: set_a,U: set_a] :
( ( normal_sub_rts_set_a @ Resid @ NN )
=> ( ( con_set_a @ Resid @ T @ T4 )
=> ( ( coinitial_set_a @ Resid @ T @ U )
=> ( ( member_set_a @ U @ NN )
=> ( con_set_a @ Resid @ ( Resid @ T @ U ) @ ( Resid @ T4 @ U ) ) ) ) ) ) ).
% normal_sub_rts.resid_along_elem_preserves_con
thf(fact_686_normal__sub__rts_Oresid__along__elem__preserves__con,axiom,
! [Resid: a > a > a,NN: set_a,T: a,T4: a,U: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( con_a @ Resid @ T @ T4 )
=> ( ( coinitial_a @ Resid @ T @ U )
=> ( ( member_a @ U @ NN )
=> ( con_a @ Resid @ ( Resid @ T @ U ) @ ( Resid @ T4 @ U ) ) ) ) ) ) ).
% normal_sub_rts.resid_along_elem_preserves_con
thf(fact_687_normal__sub__rts_OCong__class__memb__is__arr,axiom,
! [Resid: product_prod_a_a > product_prod_a_a > product_prod_a_a,NN: set_Product_prod_a_a,T7: set_Product_prod_a_a,T: product_prod_a_a] :
( ( normal2251803270671997330od_a_a @ Resid @ NN )
=> ( ( normal6700481192199873089od_a_a @ Resid @ NN @ T7 )
=> ( ( member1426531477525435216od_a_a @ T @ T7 )
=> ( arr_Product_prod_a_a @ Resid @ T ) ) ) ) ).
% normal_sub_rts.Cong_class_memb_is_arr
thf(fact_688_normal__sub__rts_OCong__class__memb__is__arr,axiom,
! [Resid: ( a > a ) > ( a > a ) > a > a,NN: set_a_a,T7: set_a_a,T: a > a] :
( ( normal_sub_rts_a_a @ Resid @ NN )
=> ( ( normal6067668076082862283ss_a_a @ Resid @ NN @ T7 )
=> ( ( member_a_a @ T @ T7 )
=> ( arr_a_a @ Resid @ T ) ) ) ) ).
% normal_sub_rts.Cong_class_memb_is_arr
thf(fact_689_normal__sub__rts_OCong__class__memb__is__arr,axiom,
! [Resid: set_a > set_a > set_a,NN: set_set_a,T7: set_set_a,T: set_a] :
( ( normal_sub_rts_set_a @ Resid @ NN )
=> ( ( normal4437380936311325560_set_a @ Resid @ NN @ T7 )
=> ( ( member_set_a @ T @ T7 )
=> ( arr_set_a @ Resid @ T ) ) ) ) ).
% normal_sub_rts.Cong_class_memb_is_arr
thf(fact_690_normal__sub__rts_OCong__class__memb__is__arr,axiom,
! [Resid: a > a > a,NN: set_a,T7: set_a,T: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( normal8595587647932138008lass_a @ Resid @ NN @ T7 )
=> ( ( member_a @ T @ T7 )
=> ( arr_a @ Resid @ T ) ) ) ) ).
% normal_sub_rts.Cong_class_memb_is_arr
thf(fact_691_normal__sub__rts_Ocomposite__closed__left,axiom,
! [Resid: product_prod_a_a > product_prod_a_a > product_prod_a_a,NN: set_Product_prod_a_a,U: product_prod_a_a,T: product_prod_a_a] :
( ( normal2251803270671997330od_a_a @ Resid @ NN )
=> ( ( member1426531477525435216od_a_a @ U @ NN )
=> ( ( seq_Product_prod_a_a @ Resid @ U @ T )
=> ? [X_1: product_prod_a_a] : ( compos6970852536677457805od_a_a @ Resid @ U @ T @ X_1 ) ) ) ) ).
% normal_sub_rts.composite_closed_left
thf(fact_692_normal__sub__rts_Ocomposite__closed__left,axiom,
! [Resid: ( a > a ) > ( a > a ) > a > a,NN: set_a_a,U: a > a,T: a > a] :
( ( normal_sub_rts_a_a @ Resid @ NN )
=> ( ( member_a_a @ U @ NN )
=> ( ( seq_a_a @ Resid @ U @ T )
=> ? [X_1: a > a] : ( composite_of_a_a @ Resid @ U @ T @ X_1 ) ) ) ) ).
% normal_sub_rts.composite_closed_left
thf(fact_693_normal__sub__rts_Ocomposite__closed__left,axiom,
! [Resid: set_a > set_a > set_a,NN: set_set_a,U: set_a,T: set_a] :
( ( normal_sub_rts_set_a @ Resid @ NN )
=> ( ( member_set_a @ U @ NN )
=> ( ( seq_set_a @ Resid @ U @ T )
=> ? [X_1: set_a] : ( composite_of_set_a @ Resid @ U @ T @ X_1 ) ) ) ) ).
% normal_sub_rts.composite_closed_left
thf(fact_694_normal__sub__rts_Ocomposite__closed__left,axiom,
! [Resid: a > a > a,NN: set_a,U: a,T: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( member_a @ U @ NN )
=> ( ( seq_a @ Resid @ U @ T )
=> ? [X_1: a] : ( composite_of_a @ Resid @ U @ T @ X_1 ) ) ) ) ).
% normal_sub_rts.composite_closed_left
thf(fact_695_normal__sub__rts_Ocomposite__closed__right,axiom,
! [Resid: product_prod_a_a > product_prod_a_a > product_prod_a_a,NN: set_Product_prod_a_a,U: product_prod_a_a,T: product_prod_a_a] :
( ( normal2251803270671997330od_a_a @ Resid @ NN )
=> ( ( member1426531477525435216od_a_a @ U @ NN )
=> ( ( seq_Product_prod_a_a @ Resid @ T @ U )
=> ? [X_1: product_prod_a_a] : ( compos6970852536677457805od_a_a @ Resid @ T @ U @ X_1 ) ) ) ) ).
% normal_sub_rts.composite_closed_right
thf(fact_696_normal__sub__rts_Ocomposite__closed__right,axiom,
! [Resid: ( a > a ) > ( a > a ) > a > a,NN: set_a_a,U: a > a,T: a > a] :
( ( normal_sub_rts_a_a @ Resid @ NN )
=> ( ( member_a_a @ U @ NN )
=> ( ( seq_a_a @ Resid @ T @ U )
=> ? [X_1: a > a] : ( composite_of_a_a @ Resid @ T @ U @ X_1 ) ) ) ) ).
% normal_sub_rts.composite_closed_right
thf(fact_697_normal__sub__rts_Ocomposite__closed__right,axiom,
! [Resid: set_a > set_a > set_a,NN: set_set_a,U: set_a,T: set_a] :
( ( normal_sub_rts_set_a @ Resid @ NN )
=> ( ( member_set_a @ U @ NN )
=> ( ( seq_set_a @ Resid @ T @ U )
=> ? [X_1: set_a] : ( composite_of_set_a @ Resid @ T @ U @ X_1 ) ) ) ) ).
% normal_sub_rts.composite_closed_right
thf(fact_698_normal__sub__rts_Ocomposite__closed__right,axiom,
! [Resid: a > a > a,NN: set_a,U: a,T: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( member_a @ U @ NN )
=> ( ( seq_a @ Resid @ T @ U )
=> ? [X_1: a] : ( composite_of_a @ Resid @ T @ U @ X_1 ) ) ) ) ).
% normal_sub_rts.composite_closed_right
thf(fact_699_normal__sub__rts_OCong__char,axiom,
! [Resid: a > a > a,NN: set_a,T: a,T4: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( normal_sub_Cong_a @ Resid @ NN @ T @ T4 )
= ( normal_sub_Cong_a2 @ Resid @ NN @ T @ T4 ) ) ) ).
% normal_sub_rts.Cong_char
thf(fact_700_normal__sub__rts_OCong__class__membs__are__Cong,axiom,
! [Resid: product_prod_a_a > product_prod_a_a > product_prod_a_a,NN: set_Product_prod_a_a,T7: set_Product_prod_a_a,T: product_prod_a_a,T4: product_prod_a_a] :
( ( normal2251803270671997330od_a_a @ Resid @ NN )
=> ( ( normal6700481192199873089od_a_a @ Resid @ NN @ T7 )
=> ( ( member1426531477525435216od_a_a @ T @ T7 )
=> ( ( member1426531477525435216od_a_a @ T4 @ T7 )
=> ( normal7727773393199519165od_a_a @ Resid @ NN @ T @ T4 ) ) ) ) ) ).
% normal_sub_rts.Cong_class_membs_are_Cong
thf(fact_701_normal__sub__rts_OCong__class__membs__are__Cong,axiom,
! [Resid: set_a > set_a > set_a,NN: set_set_a,T7: set_set_a,T: set_a,T4: set_a] :
( ( normal_sub_rts_set_a @ Resid @ NN )
=> ( ( normal4437380936311325560_set_a @ Resid @ NN @ T7 )
=> ( ( member_set_a @ T @ T7 )
=> ( ( member_set_a @ T4 @ T7 )
=> ( normal8977612136997397236_set_a @ Resid @ NN @ T @ T4 ) ) ) ) ) ).
% normal_sub_rts.Cong_class_membs_are_Cong
thf(fact_702_normal__sub__rts_OCong__class__membs__are__Cong,axiom,
! [Resid: ( a > a ) > ( a > a ) > a > a,NN: set_a_a,T7: set_a_a,T: a > a,T4: a > a] :
( ( normal_sub_rts_a_a @ Resid @ NN )
=> ( ( normal6067668076082862283ss_a_a @ Resid @ NN @ T7 )
=> ( ( member_a_a @ T @ T7 )
=> ( ( member_a_a @ T4 @ T7 )
=> ( normal_sub_Cong_a_a @ Resid @ NN @ T @ T4 ) ) ) ) ) ).
% normal_sub_rts.Cong_class_membs_are_Cong
thf(fact_703_normal__sub__rts_OCong__class__membs__are__Cong,axiom,
! [Resid: a > a > a,NN: set_a,T7: set_a,T: a,T4: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( normal8595587647932138008lass_a @ Resid @ NN @ T7 )
=> ( ( member_a @ T @ T7 )
=> ( ( member_a @ T4 @ T7 )
=> ( normal_sub_Cong_a @ Resid @ NN @ T @ T4 ) ) ) ) ) ).
% normal_sub_rts.Cong_class_membs_are_Cong
thf(fact_704_normal__sub__rts_Oarr__in__Cong__class,axiom,
! [Resid: product_prod_a_a > product_prod_a_a > product_prod_a_a,NN: set_Product_prod_a_a,T: product_prod_a_a] :
( ( normal2251803270671997330od_a_a @ Resid @ NN )
=> ( ( arr_Product_prod_a_a @ Resid @ T )
=> ( member1426531477525435216od_a_a @ T @ ( normal8582136959803729207od_a_a @ Resid @ NN @ T ) ) ) ) ).
% normal_sub_rts.arr_in_Cong_class
thf(fact_705_normal__sub__rts_Oarr__in__Cong__class,axiom,
! [Resid: ( a > a ) > ( a > a ) > a > a,NN: set_a_a,T: a > a] :
( ( normal_sub_rts_a_a @ Resid @ NN )
=> ( ( arr_a_a @ Resid @ T )
=> ( member_a_a @ T @ ( normal3779061517355887189ss_a_a @ Resid @ NN @ T ) ) ) ) ).
% normal_sub_rts.arr_in_Cong_class
thf(fact_706_normal__sub__rts_Oarr__in__Cong__class,axiom,
! [Resid: set_a > set_a > set_a,NN: set_set_a,T: set_a] :
( ( normal_sub_rts_set_a @ Resid @ NN )
=> ( ( arr_set_a @ Resid @ T )
=> ( member_set_a @ T @ ( normal2962378890657961070_set_a @ Resid @ NN @ T ) ) ) ) ).
% normal_sub_rts.arr_in_Cong_class
thf(fact_707_normal__sub__rts_Oarr__in__Cong__class,axiom,
! [Resid: a > a > a,NN: set_a,T: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( arr_a @ Resid @ T )
=> ( member_a @ T @ ( normal7408713899360725774lass_a @ Resid @ NN @ T ) ) ) ) ).
% normal_sub_rts.arr_in_Cong_class
thf(fact_708_normal__sub__rts_OCong__class__eqI,axiom,
! [Resid: a > a > a,NN: set_a,T: a,T4: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( normal_sub_Cong_a @ Resid @ NN @ T @ T4 )
=> ( ( normal7408713899360725774lass_a @ Resid @ NN @ T )
= ( normal7408713899360725774lass_a @ Resid @ NN @ T4 ) ) ) ) ).
% normal_sub_rts.Cong_class_eqI
thf(fact_709_normal__sub__rts_Ois__Cong__class__def,axiom,
! [Resid: product_prod_a_a > product_prod_a_a > product_prod_a_a,NN: set_Product_prod_a_a,T7: set_Product_prod_a_a] :
( ( normal2251803270671997330od_a_a @ Resid @ NN )
=> ( ( normal6700481192199873089od_a_a @ Resid @ NN @ T7 )
= ( ? [T5: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ T5 @ T7 )
& ( T7
= ( normal8582136959803729207od_a_a @ Resid @ NN @ T5 ) ) ) ) ) ) ).
% normal_sub_rts.is_Cong_class_def
thf(fact_710_normal__sub__rts_Ois__Cong__class__def,axiom,
! [Resid: set_a > set_a > set_a,NN: set_set_a,T7: set_set_a] :
( ( normal_sub_rts_set_a @ Resid @ NN )
=> ( ( normal4437380936311325560_set_a @ Resid @ NN @ T7 )
= ( ? [T5: set_a] :
( ( member_set_a @ T5 @ T7 )
& ( T7
= ( normal2962378890657961070_set_a @ Resid @ NN @ T5 ) ) ) ) ) ) ).
% normal_sub_rts.is_Cong_class_def
thf(fact_711_normal__sub__rts_Ois__Cong__class__def,axiom,
! [Resid: ( a > a ) > ( a > a ) > a > a,NN: set_a_a,T7: set_a_a] :
( ( normal_sub_rts_a_a @ Resid @ NN )
=> ( ( normal6067668076082862283ss_a_a @ Resid @ NN @ T7 )
= ( ? [T5: a > a] :
( ( member_a_a @ T5 @ T7 )
& ( T7
= ( normal3779061517355887189ss_a_a @ Resid @ NN @ T5 ) ) ) ) ) ) ).
% normal_sub_rts.is_Cong_class_def
thf(fact_712_normal__sub__rts_Ois__Cong__class__def,axiom,
! [Resid: a > a > a,NN: set_a,T7: set_a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( normal8595587647932138008lass_a @ Resid @ NN @ T7 )
= ( ? [T5: a] :
( ( member_a @ T5 @ T7 )
& ( T7
= ( normal7408713899360725774lass_a @ Resid @ NN @ T5 ) ) ) ) ) ) ).
% normal_sub_rts.is_Cong_class_def
thf(fact_713_normal__sub__rts_Orep__in__Cong__class,axiom,
! [Resid: set_a > set_a > set_a,NN: set_set_a,T7: set_set_a] :
( ( normal_sub_rts_set_a @ Resid @ NN )
=> ( ( normal4437380936311325560_set_a @ Resid @ NN @ T7 )
=> ( member_set_a @ ( normal2464625585956020495_set_a @ T7 ) @ T7 ) ) ) ).
% normal_sub_rts.rep_in_Cong_class
thf(fact_714_normal__sub__rts_Orep__in__Cong__class,axiom,
! [Resid: ( a > a ) > ( a > a ) > a > a,NN: set_a_a,T7: set_a_a] :
( ( normal_sub_rts_a_a @ Resid @ NN )
=> ( ( normal6067668076082862283ss_a_a @ Resid @ NN @ T7 )
=> ( member_a_a @ ( normal1637961696771286388ep_a_a @ T7 ) @ T7 ) ) ) ).
% normal_sub_rts.rep_in_Cong_class
thf(fact_715_normal__sub__rts_Orep__in__Cong__class,axiom,
! [Resid: product_prod_a_a > product_prod_a_a > product_prod_a_a,NN: set_Product_prod_a_a,T7: set_Product_prod_a_a] :
( ( normal2251803270671997330od_a_a @ Resid @ NN )
=> ( ( normal6700481192199873089od_a_a @ Resid @ NN @ T7 )
=> ( member1426531477525435216od_a_a @ ( normal322541020860755160od_a_a @ T7 ) @ T7 ) ) ) ).
% normal_sub_rts.rep_in_Cong_class
thf(fact_716_normal__sub__rts_Orep__in__Cong__class,axiom,
! [Resid: a > a > a,NN: set_a,T7: set_a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( normal8595587647932138008lass_a @ Resid @ NN @ T7 )
=> ( member_a @ ( normal3259722184653208495_rep_a @ T7 ) @ T7 ) ) ) ).
% normal_sub_rts.rep_in_Cong_class
thf(fact_717_normal__sub__rts_OCong__class__eqI_H,axiom,
! [Resid: product_prod_a_a > product_prod_a_a > product_prod_a_a,NN: set_Product_prod_a_a,T7: set_Product_prod_a_a,U7: set_Product_prod_a_a] :
( ( normal2251803270671997330od_a_a @ Resid @ NN )
=> ( ( normal6700481192199873089od_a_a @ Resid @ NN @ T7 )
=> ( ( normal6700481192199873089od_a_a @ Resid @ NN @ U7 )
=> ( ( ( inf_in8905007599844390133od_a_a @ T7 @ U7 )
!= bot_bo3357376287454694259od_a_a )
=> ( T7 = U7 ) ) ) ) ) ).
% normal_sub_rts.Cong_class_eqI'
thf(fact_718_normal__sub__rts_OCong__class__eqI_H,axiom,
! [Resid: set_a > set_a > set_a,NN: set_set_a,T7: set_set_a,U7: set_set_a] :
( ( normal_sub_rts_set_a @ Resid @ NN )
=> ( ( normal4437380936311325560_set_a @ Resid @ NN @ T7 )
=> ( ( normal4437380936311325560_set_a @ Resid @ NN @ U7 )
=> ( ( ( inf_inf_set_set_a @ T7 @ U7 )
!= bot_bot_set_set_a )
=> ( T7 = U7 ) ) ) ) ) ).
% normal_sub_rts.Cong_class_eqI'
thf(fact_719_normal__sub__rts_OCong__class__eqI_H,axiom,
! [Resid: ( a > a ) > ( a > a ) > a > a,NN: set_a_a,T7: set_a_a,U7: set_a_a] :
( ( normal_sub_rts_a_a @ Resid @ NN )
=> ( ( normal6067668076082862283ss_a_a @ Resid @ NN @ T7 )
=> ( ( normal6067668076082862283ss_a_a @ Resid @ NN @ U7 )
=> ( ( ( inf_inf_set_a_a @ T7 @ U7 )
!= bot_bot_set_a_a )
=> ( T7 = U7 ) ) ) ) ) ).
% normal_sub_rts.Cong_class_eqI'
thf(fact_720_normal__sub__rts_OCong__class__eqI_H,axiom,
! [Resid: a > a > a,NN: set_a,T7: set_a,U7: set_a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( normal8595587647932138008lass_a @ Resid @ NN @ T7 )
=> ( ( normal8595587647932138008lass_a @ Resid @ NN @ U7 )
=> ( ( ( inf_inf_set_a @ T7 @ U7 )
!= bot_bot_set_a )
=> ( T7 = U7 ) ) ) ) ) ).
% normal_sub_rts.Cong_class_eqI'
thf(fact_721_normal__sub__rts_Ocomposite__of__normal__arr,axiom,
! [Resid: product_prod_a_a > product_prod_a_a > product_prod_a_a,NN: set_Product_prod_a_a,T: product_prod_a_a,U: product_prod_a_a,T4: product_prod_a_a] :
( ( normal2251803270671997330od_a_a @ Resid @ NN )
=> ( ( arr_Product_prod_a_a @ Resid @ T )
=> ( ( member1426531477525435216od_a_a @ U @ NN )
=> ( ( compos6970852536677457805od_a_a @ Resid @ U @ T @ T4 )
=> ( normal7727773393199519165od_a_a @ Resid @ NN @ T4 @ T ) ) ) ) ) ).
% normal_sub_rts.composite_of_normal_arr
thf(fact_722_normal__sub__rts_Ocomposite__of__normal__arr,axiom,
! [Resid: ( a > a ) > ( a > a ) > a > a,NN: set_a_a,T: a > a,U: a > a,T4: a > a] :
( ( normal_sub_rts_a_a @ Resid @ NN )
=> ( ( arr_a_a @ Resid @ T )
=> ( ( member_a_a @ U @ NN )
=> ( ( composite_of_a_a @ Resid @ U @ T @ T4 )
=> ( normal_sub_Cong_a_a @ Resid @ NN @ T4 @ T ) ) ) ) ) ).
% normal_sub_rts.composite_of_normal_arr
thf(fact_723_normal__sub__rts_Ocomposite__of__normal__arr,axiom,
! [Resid: set_a > set_a > set_a,NN: set_set_a,T: set_a,U: set_a,T4: set_a] :
( ( normal_sub_rts_set_a @ Resid @ NN )
=> ( ( arr_set_a @ Resid @ T )
=> ( ( member_set_a @ U @ NN )
=> ( ( composite_of_set_a @ Resid @ U @ T @ T4 )
=> ( normal8977612136997397236_set_a @ Resid @ NN @ T4 @ T ) ) ) ) ) ).
% normal_sub_rts.composite_of_normal_arr
thf(fact_724_normal__sub__rts_Ocomposite__of__normal__arr,axiom,
! [Resid: a > a > a,NN: set_a,T: a,U: a,T4: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( arr_a @ Resid @ T )
=> ( ( member_a @ U @ NN )
=> ( ( composite_of_a @ Resid @ U @ T @ T4 )
=> ( normal_sub_Cong_a @ Resid @ NN @ T4 @ T ) ) ) ) ) ).
% normal_sub_rts.composite_of_normal_arr
thf(fact_725_normal__sub__rts_Ois__Cong__classI_H,axiom,
! [Resid: product_prod_a_a > product_prod_a_a > product_prod_a_a,NN: set_Product_prod_a_a,T7: set_Product_prod_a_a] :
( ( normal2251803270671997330od_a_a @ Resid @ NN )
=> ( ( T7 != bot_bo3357376287454694259od_a_a )
=> ( ! [T3: product_prod_a_a,T9: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ T3 @ T7 )
=> ( ( member1426531477525435216od_a_a @ T9 @ T7 )
=> ( normal7727773393199519165od_a_a @ Resid @ NN @ T3 @ T9 ) ) )
=> ( ! [T3: product_prod_a_a,T9: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ T3 @ T7 )
=> ( ( normal7727773393199519165od_a_a @ Resid @ NN @ T9 @ T3 )
=> ( member1426531477525435216od_a_a @ T9 @ T7 ) ) )
=> ( normal6700481192199873089od_a_a @ Resid @ NN @ T7 ) ) ) ) ) ).
% normal_sub_rts.is_Cong_classI'
thf(fact_726_normal__sub__rts_Ois__Cong__classI_H,axiom,
! [Resid: set_a > set_a > set_a,NN: set_set_a,T7: set_set_a] :
( ( normal_sub_rts_set_a @ Resid @ NN )
=> ( ( T7 != bot_bot_set_set_a )
=> ( ! [T3: set_a,T9: set_a] :
( ( member_set_a @ T3 @ T7 )
=> ( ( member_set_a @ T9 @ T7 )
=> ( normal8977612136997397236_set_a @ Resid @ NN @ T3 @ T9 ) ) )
=> ( ! [T3: set_a,T9: set_a] :
( ( member_set_a @ T3 @ T7 )
=> ( ( normal8977612136997397236_set_a @ Resid @ NN @ T9 @ T3 )
=> ( member_set_a @ T9 @ T7 ) ) )
=> ( normal4437380936311325560_set_a @ Resid @ NN @ T7 ) ) ) ) ) ).
% normal_sub_rts.is_Cong_classI'
thf(fact_727_normal__sub__rts_Ois__Cong__classI_H,axiom,
! [Resid: ( a > a ) > ( a > a ) > a > a,NN: set_a_a,T7: set_a_a] :
( ( normal_sub_rts_a_a @ Resid @ NN )
=> ( ( T7 != bot_bot_set_a_a )
=> ( ! [T3: a > a,T9: a > a] :
( ( member_a_a @ T3 @ T7 )
=> ( ( member_a_a @ T9 @ T7 )
=> ( normal_sub_Cong_a_a @ Resid @ NN @ T3 @ T9 ) ) )
=> ( ! [T3: a > a,T9: a > a] :
( ( member_a_a @ T3 @ T7 )
=> ( ( normal_sub_Cong_a_a @ Resid @ NN @ T9 @ T3 )
=> ( member_a_a @ T9 @ T7 ) ) )
=> ( normal6067668076082862283ss_a_a @ Resid @ NN @ T7 ) ) ) ) ) ).
% normal_sub_rts.is_Cong_classI'
thf(fact_728_normal__sub__rts_Ois__Cong__classI_H,axiom,
! [Resid: a > a > a,NN: set_a,T7: set_a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( 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 ) ) ) ) ) ).
% normal_sub_rts.is_Cong_classI'
thf(fact_729_normal__sub__rts_Ois__Cong__classE,axiom,
! [Resid: product_prod_a_a > product_prod_a_a > product_prod_a_a,NN: set_Product_prod_a_a,T7: set_Product_prod_a_a] :
( ( normal2251803270671997330od_a_a @ Resid @ NN )
=> ( ( normal6700481192199873089od_a_a @ Resid @ NN @ T7 )
=> ~ ( ( T7 != bot_bo3357376287454694259od_a_a )
=> ( ! [T2: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ T2 @ T7 )
=> ! [T8: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ T8 @ T7 )
=> ( normal7727773393199519165od_a_a @ Resid @ NN @ T2 @ T8 ) ) )
=> ~ ! [T2: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ T2 @ T7 )
=> ! [T8: product_prod_a_a] :
( ( normal7727773393199519165od_a_a @ Resid @ NN @ T8 @ T2 )
=> ( member1426531477525435216od_a_a @ T8 @ T7 ) ) ) ) ) ) ) ).
% normal_sub_rts.is_Cong_classE
thf(fact_730_normal__sub__rts_Ois__Cong__classE,axiom,
! [Resid: set_a > set_a > set_a,NN: set_set_a,T7: set_set_a] :
( ( normal_sub_rts_set_a @ Resid @ NN )
=> ( ( normal4437380936311325560_set_a @ Resid @ NN @ T7 )
=> ~ ( ( T7 != bot_bot_set_set_a )
=> ( ! [T2: set_a] :
( ( member_set_a @ T2 @ T7 )
=> ! [T8: set_a] :
( ( member_set_a @ T8 @ T7 )
=> ( normal8977612136997397236_set_a @ Resid @ NN @ T2 @ T8 ) ) )
=> ~ ! [T2: set_a] :
( ( member_set_a @ T2 @ T7 )
=> ! [T8: set_a] :
( ( normal8977612136997397236_set_a @ Resid @ NN @ T8 @ T2 )
=> ( member_set_a @ T8 @ T7 ) ) ) ) ) ) ) ).
% normal_sub_rts.is_Cong_classE
thf(fact_731_normal__sub__rts_Ois__Cong__classE,axiom,
! [Resid: ( a > a ) > ( a > a ) > a > a,NN: set_a_a,T7: set_a_a] :
( ( normal_sub_rts_a_a @ Resid @ NN )
=> ( ( normal6067668076082862283ss_a_a @ Resid @ NN @ T7 )
=> ~ ( ( T7 != bot_bot_set_a_a )
=> ( ! [T2: a > a] :
( ( member_a_a @ T2 @ T7 )
=> ! [T8: a > a] :
( ( member_a_a @ T8 @ T7 )
=> ( normal_sub_Cong_a_a @ Resid @ NN @ T2 @ T8 ) ) )
=> ~ ! [T2: a > a] :
( ( member_a_a @ T2 @ T7 )
=> ! [T8: a > a] :
( ( normal_sub_Cong_a_a @ Resid @ NN @ T8 @ T2 )
=> ( member_a_a @ T8 @ T7 ) ) ) ) ) ) ) ).
% normal_sub_rts.is_Cong_classE
thf(fact_732_normal__sub__rts_Ois__Cong__classE,axiom,
! [Resid: a > a > a,NN: set_a,T7: set_a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( 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 ) ) ) ) ) ) ) ).
% normal_sub_rts.is_Cong_classE
thf(fact_733_normal__sub__rts_OCong_H_Osimps,axiom,
! [Resid: product_prod_a_a > product_prod_a_a > product_prod_a_a,NN: set_Product_prod_a_a,A1: product_prod_a_a,A22: product_prod_a_a] :
( ( normal2251803270671997330od_a_a @ Resid @ NN )
=> ( ( normal2813552597983363028od_a_a @ Resid @ NN @ A1 @ A22 )
= ( ? [T5: product_prod_a_a,U3: product_prod_a_a] :
( ( A1 = U3 )
& ( A22 = T5 )
& ( normal2813552597983363028od_a_a @ Resid @ NN @ T5 @ U3 ) )
| ? [T5: product_prod_a_a,U3: product_prod_a_a,V3: product_prod_a_a] :
( ( A1 = T5 )
& ( A22 = V3 )
& ( normal2813552597983363028od_a_a @ Resid @ NN @ T5 @ U3 )
& ( normal2813552597983363028od_a_a @ Resid @ NN @ U3 @ V3 ) )
| ? [T5: product_prod_a_a,U3: product_prod_a_a] :
( ( A1 = T5 )
& ( A22 = U3 )
& ( member1426531477525435216od_a_a @ ( Resid @ T5 @ U3 ) @ NN )
& ( member1426531477525435216od_a_a @ ( Resid @ U3 @ T5 ) @ NN ) )
| ? [T5: product_prod_a_a,U3: product_prod_a_a] :
( ( A1 = T5 )
& ( A22
= ( Resid @ T5 @ U3 ) )
& ( arr_Product_prod_a_a @ Resid @ T5 )
& ( member1426531477525435216od_a_a @ U3 @ NN )
& ( ( source6950040787684646355od_a_a @ Resid @ T5 )
= ( source6950040787684646355od_a_a @ Resid @ U3 ) ) ) ) ) ) ).
% normal_sub_rts.Cong'.simps
thf(fact_734_normal__sub__rts_OCong_H_Osimps,axiom,
! [Resid: ( a > a ) > ( a > a ) > a > a,NN: set_a_a,A1: a > a,A22: a > a] :
( ( normal_sub_rts_a_a @ Resid @ NN )
=> ( ( normal_sub_Cong_a_a2 @ Resid @ NN @ A1 @ A22 )
= ( ? [T5: a > a,U3: a > a] :
( ( A1 = U3 )
& ( A22 = T5 )
& ( normal_sub_Cong_a_a2 @ Resid @ NN @ T5 @ U3 ) )
| ? [T5: a > a,U3: a > a,V3: a > a] :
( ( A1 = T5 )
& ( A22 = V3 )
& ( normal_sub_Cong_a_a2 @ Resid @ NN @ T5 @ U3 )
& ( normal_sub_Cong_a_a2 @ Resid @ NN @ U3 @ V3 ) )
| ? [T5: a > a,U3: a > a] :
( ( A1 = T5 )
& ( A22 = U3 )
& ( member_a_a @ ( Resid @ T5 @ U3 ) @ NN )
& ( member_a_a @ ( Resid @ U3 @ T5 ) @ NN ) )
| ? [T5: a > a,U3: a > a] :
( ( A1 = T5 )
& ( A22
= ( Resid @ T5 @ U3 ) )
& ( arr_a_a @ Resid @ T5 )
& ( member_a_a @ U3 @ NN )
& ( ( sources_a_a @ Resid @ T5 )
= ( sources_a_a @ Resid @ U3 ) ) ) ) ) ) ).
% normal_sub_rts.Cong'.simps
thf(fact_735_normal__sub__rts_OCong_H_Osimps,axiom,
! [Resid: set_a > set_a > set_a,NN: set_set_a,A1: set_a,A22: set_a] :
( ( normal_sub_rts_set_a @ Resid @ NN )
=> ( ( normal8837514132249976843_set_a @ Resid @ NN @ A1 @ A22 )
= ( ? [T5: set_a,U3: set_a] :
( ( A1 = U3 )
& ( A22 = T5 )
& ( normal8837514132249976843_set_a @ Resid @ NN @ T5 @ U3 ) )
| ? [T5: set_a,U3: set_a,V3: set_a] :
( ( A1 = T5 )
& ( A22 = V3 )
& ( normal8837514132249976843_set_a @ Resid @ NN @ T5 @ U3 )
& ( normal8837514132249976843_set_a @ Resid @ NN @ U3 @ V3 ) )
| ? [T5: set_a,U3: set_a] :
( ( A1 = T5 )
& ( A22 = U3 )
& ( member_set_a @ ( Resid @ T5 @ U3 ) @ NN )
& ( member_set_a @ ( Resid @ U3 @ T5 ) @ NN ) )
| ? [T5: set_a,U3: set_a] :
( ( A1 = T5 )
& ( A22
= ( Resid @ T5 @ U3 ) )
& ( arr_set_a @ Resid @ T5 )
& ( member_set_a @ U3 @ NN )
& ( ( sources_set_a @ Resid @ T5 )
= ( sources_set_a @ Resid @ U3 ) ) ) ) ) ) ).
% normal_sub_rts.Cong'.simps
thf(fact_736_normal__sub__rts_OCong_H_Osimps,axiom,
! [Resid: a > a > a,NN: set_a,A1: a,A22: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( normal_sub_Cong_a2 @ Resid @ NN @ A1 @ A22 )
= ( ? [T5: a,U3: a] :
( ( A1 = U3 )
& ( A22 = T5 )
& ( normal_sub_Cong_a2 @ Resid @ NN @ T5 @ U3 ) )
| ? [T5: a,U3: a,V3: a] :
( ( A1 = T5 )
& ( A22 = V3 )
& ( normal_sub_Cong_a2 @ Resid @ NN @ T5 @ U3 )
& ( normal_sub_Cong_a2 @ Resid @ NN @ U3 @ V3 ) )
| ? [T5: a,U3: a] :
( ( A1 = T5 )
& ( A22 = U3 )
& ( member_a @ ( Resid @ T5 @ U3 ) @ NN )
& ( member_a @ ( Resid @ U3 @ T5 ) @ NN ) )
| ? [T5: a,U3: a] :
( ( A1 = T5 )
& ( A22
= ( Resid @ T5 @ U3 ) )
& ( arr_a @ Resid @ T5 )
& ( member_a @ U3 @ NN )
& ( ( sources_a @ Resid @ T5 )
= ( sources_a @ Resid @ U3 ) ) ) ) ) ) ).
% normal_sub_rts.Cong'.simps
thf(fact_737_normal__sub__rts_OCong_H_Ocases,axiom,
! [Resid: product_prod_a_a > product_prod_a_a > product_prod_a_a,NN: set_Product_prod_a_a,A1: product_prod_a_a,A22: product_prod_a_a] :
( ( normal2251803270671997330od_a_a @ Resid @ NN )
=> ( ( normal2813552597983363028od_a_a @ Resid @ NN @ A1 @ A22 )
=> ( ~ ( normal2813552597983363028od_a_a @ Resid @ NN @ A22 @ A1 )
=> ( ! [U5: product_prod_a_a] :
( ( normal2813552597983363028od_a_a @ Resid @ NN @ A1 @ U5 )
=> ~ ( normal2813552597983363028od_a_a @ Resid @ NN @ U5 @ A22 ) )
=> ( ~ ( ( member1426531477525435216od_a_a @ ( Resid @ A1 @ A22 ) @ NN )
& ( member1426531477525435216od_a_a @ ( Resid @ A22 @ A1 ) @ NN ) )
=> ~ ! [U5: product_prod_a_a] :
( ( A22
= ( Resid @ A1 @ U5 ) )
=> ( ( arr_Product_prod_a_a @ Resid @ A1 )
=> ( ( member1426531477525435216od_a_a @ U5 @ NN )
=> ( ( source6950040787684646355od_a_a @ Resid @ A1 )
!= ( source6950040787684646355od_a_a @ Resid @ U5 ) ) ) ) ) ) ) ) ) ) ).
% normal_sub_rts.Cong'.cases
thf(fact_738_normal__sub__rts_OCong_H_Ocases,axiom,
! [Resid: ( a > a ) > ( a > a ) > a > a,NN: set_a_a,A1: a > a,A22: a > a] :
( ( normal_sub_rts_a_a @ Resid @ NN )
=> ( ( normal_sub_Cong_a_a2 @ Resid @ NN @ A1 @ A22 )
=> ( ~ ( normal_sub_Cong_a_a2 @ Resid @ NN @ A22 @ A1 )
=> ( ! [U5: a > a] :
( ( normal_sub_Cong_a_a2 @ Resid @ NN @ A1 @ U5 )
=> ~ ( normal_sub_Cong_a_a2 @ Resid @ NN @ U5 @ A22 ) )
=> ( ~ ( ( member_a_a @ ( Resid @ A1 @ A22 ) @ NN )
& ( member_a_a @ ( Resid @ A22 @ A1 ) @ NN ) )
=> ~ ! [U5: a > a] :
( ( A22
= ( Resid @ A1 @ U5 ) )
=> ( ( arr_a_a @ Resid @ A1 )
=> ( ( member_a_a @ U5 @ NN )
=> ( ( sources_a_a @ Resid @ A1 )
!= ( sources_a_a @ Resid @ U5 ) ) ) ) ) ) ) ) ) ) ).
% normal_sub_rts.Cong'.cases
thf(fact_739_normal__sub__rts_OCong_H_Ocases,axiom,
! [Resid: set_a > set_a > set_a,NN: set_set_a,A1: set_a,A22: set_a] :
( ( normal_sub_rts_set_a @ Resid @ NN )
=> ( ( normal8837514132249976843_set_a @ Resid @ NN @ A1 @ A22 )
=> ( ~ ( normal8837514132249976843_set_a @ Resid @ NN @ A22 @ A1 )
=> ( ! [U5: set_a] :
( ( normal8837514132249976843_set_a @ Resid @ NN @ A1 @ U5 )
=> ~ ( normal8837514132249976843_set_a @ Resid @ NN @ U5 @ A22 ) )
=> ( ~ ( ( member_set_a @ ( Resid @ A1 @ A22 ) @ NN )
& ( member_set_a @ ( Resid @ A22 @ A1 ) @ NN ) )
=> ~ ! [U5: set_a] :
( ( A22
= ( Resid @ A1 @ U5 ) )
=> ( ( arr_set_a @ Resid @ A1 )
=> ( ( member_set_a @ U5 @ NN )
=> ( ( sources_set_a @ Resid @ A1 )
!= ( sources_set_a @ Resid @ U5 ) ) ) ) ) ) ) ) ) ) ).
% normal_sub_rts.Cong'.cases
thf(fact_740_normal__sub__rts_OCong_H_Ocases,axiom,
! [Resid: a > a > a,NN: set_a,A1: a,A22: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( normal_sub_Cong_a2 @ Resid @ NN @ A1 @ A22 )
=> ( ~ ( normal_sub_Cong_a2 @ Resid @ NN @ A22 @ A1 )
=> ( ! [U5: a] :
( ( normal_sub_Cong_a2 @ Resid @ NN @ A1 @ U5 )
=> ~ ( normal_sub_Cong_a2 @ Resid @ NN @ U5 @ A22 ) )
=> ( ~ ( ( member_a @ ( Resid @ A1 @ A22 ) @ NN )
& ( member_a @ ( Resid @ A22 @ A1 ) @ NN ) )
=> ~ ! [U5: a] :
( ( A22
= ( Resid @ A1 @ U5 ) )
=> ( ( arr_a @ Resid @ A1 )
=> ( ( member_a @ U5 @ NN )
=> ( ( sources_a @ Resid @ A1 )
!= ( sources_a @ Resid @ U5 ) ) ) ) ) ) ) ) ) ) ).
% normal_sub_rts.Cong'.cases
thf(fact_741_normal__sub__rts_OCong_H_Ointros_I4_J,axiom,
! [Resid: product_prod_a_a > product_prod_a_a > product_prod_a_a,NN: set_Product_prod_a_a,T: product_prod_a_a,U: product_prod_a_a] :
( ( normal2251803270671997330od_a_a @ Resid @ NN )
=> ( ( arr_Product_prod_a_a @ Resid @ T )
=> ( ( member1426531477525435216od_a_a @ U @ NN )
=> ( ( ( source6950040787684646355od_a_a @ Resid @ T )
= ( source6950040787684646355od_a_a @ Resid @ U ) )
=> ( normal2813552597983363028od_a_a @ Resid @ NN @ T @ ( Resid @ T @ U ) ) ) ) ) ) ).
% normal_sub_rts.Cong'.intros(4)
thf(fact_742_normal__sub__rts_OCong_H_Ointros_I4_J,axiom,
! [Resid: ( a > a ) > ( a > a ) > a > a,NN: set_a_a,T: a > a,U: a > a] :
( ( normal_sub_rts_a_a @ Resid @ NN )
=> ( ( arr_a_a @ Resid @ T )
=> ( ( member_a_a @ U @ NN )
=> ( ( ( sources_a_a @ Resid @ T )
= ( sources_a_a @ Resid @ U ) )
=> ( normal_sub_Cong_a_a2 @ Resid @ NN @ T @ ( Resid @ T @ U ) ) ) ) ) ) ).
% normal_sub_rts.Cong'.intros(4)
thf(fact_743_normal__sub__rts_OCong_H_Ointros_I4_J,axiom,
! [Resid: set_a > set_a > set_a,NN: set_set_a,T: set_a,U: set_a] :
( ( normal_sub_rts_set_a @ Resid @ NN )
=> ( ( arr_set_a @ Resid @ T )
=> ( ( member_set_a @ U @ NN )
=> ( ( ( sources_set_a @ Resid @ T )
= ( sources_set_a @ Resid @ U ) )
=> ( normal8837514132249976843_set_a @ Resid @ NN @ T @ ( Resid @ T @ U ) ) ) ) ) ) ).
% normal_sub_rts.Cong'.intros(4)
thf(fact_744_normal__sub__rts_OCong_H_Ointros_I4_J,axiom,
! [Resid: a > a > a,NN: set_a,T: a,U: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( 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 ) ) ) ) ) ) ).
% normal_sub_rts.Cong'.intros(4)
thf(fact_745_normal__sub__rts_Ois__Cong__classI,axiom,
! [Resid: set_a > set_a > set_a,NN: set_set_a,T: set_a] :
( ( normal_sub_rts_set_a @ Resid @ NN )
=> ( ( arr_set_a @ Resid @ T )
=> ( normal4437380936311325560_set_a @ Resid @ NN @ ( normal2962378890657961070_set_a @ Resid @ NN @ T ) ) ) ) ).
% normal_sub_rts.is_Cong_classI
thf(fact_746_normal__sub__rts_Ois__Cong__classI,axiom,
! [Resid: a > a > a,NN: set_a,T: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( arr_a @ Resid @ T )
=> ( normal8595587647932138008lass_a @ Resid @ NN @ ( normal7408713899360725774lass_a @ Resid @ NN @ T ) ) ) ) ).
% normal_sub_rts.is_Cong_classI
thf(fact_747_bot__fun__def,axiom,
( bot_bot_a_o
= ( ^ [X2: a] : bot_bot_o ) ) ).
% bot_fun_def
thf(fact_748_normal__sub__rts_OCong__class__memb__Cong__rep,axiom,
! [Resid: set_a > set_a > set_a,NN: set_set_a,T7: set_set_a,T: set_a] :
( ( normal_sub_rts_set_a @ Resid @ NN )
=> ( ( normal4437380936311325560_set_a @ Resid @ NN @ T7 )
=> ( ( member_set_a @ T @ T7 )
=> ( normal8977612136997397236_set_a @ Resid @ NN @ T @ ( normal2464625585956020495_set_a @ T7 ) ) ) ) ) ).
% normal_sub_rts.Cong_class_memb_Cong_rep
thf(fact_749_normal__sub__rts_OCong__class__memb__Cong__rep,axiom,
! [Resid: ( a > a ) > ( a > a ) > a > a,NN: set_a_a,T7: set_a_a,T: a > a] :
( ( normal_sub_rts_a_a @ Resid @ NN )
=> ( ( normal6067668076082862283ss_a_a @ Resid @ NN @ T7 )
=> ( ( member_a_a @ T @ T7 )
=> ( normal_sub_Cong_a_a @ Resid @ NN @ T @ ( normal1637961696771286388ep_a_a @ T7 ) ) ) ) ) ).
% normal_sub_rts.Cong_class_memb_Cong_rep
thf(fact_750_normal__sub__rts_OCong__class__memb__Cong__rep,axiom,
! [Resid: product_prod_a_a > product_prod_a_a > product_prod_a_a,NN: set_Product_prod_a_a,T7: set_Product_prod_a_a,T: product_prod_a_a] :
( ( normal2251803270671997330od_a_a @ Resid @ NN )
=> ( ( normal6700481192199873089od_a_a @ Resid @ NN @ T7 )
=> ( ( member1426531477525435216od_a_a @ T @ T7 )
=> ( normal7727773393199519165od_a_a @ Resid @ NN @ T @ ( normal322541020860755160od_a_a @ T7 ) ) ) ) ) ).
% normal_sub_rts.Cong_class_memb_Cong_rep
thf(fact_751_normal__sub__rts_OCong__class__memb__Cong__rep,axiom,
! [Resid: a > a > a,NN: set_a,T7: set_a,T: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( normal8595587647932138008lass_a @ Resid @ NN @ T7 )
=> ( ( member_a @ T @ T7 )
=> ( normal_sub_Cong_a @ Resid @ NN @ T @ ( normal3259722184653208495_rep_a @ T7 ) ) ) ) ) ).
% normal_sub_rts.Cong_class_memb_Cong_rep
thf(fact_752_coherent__normal__sub__rts_Oaxioms_I2_J,axiom,
! [Resid: a > a > a,NN: set_a] :
( ( cohere6072184133013167079_rts_a @ Resid @ NN )
=> ( cohere4894532172567702276ioms_a @ Resid @ NN ) ) ).
% coherent_normal_sub_rts.axioms(2)
thf(fact_753_normal__sub__rts_OCong__class__rep,axiom,
! [Resid: product_prod_a_a > product_prod_a_a > product_prod_a_a,NN: set_Product_prod_a_a,T7: set_Product_prod_a_a] :
( ( normal2251803270671997330od_a_a @ Resid @ NN )
=> ( ( normal6700481192199873089od_a_a @ Resid @ NN @ T7 )
=> ( ( normal8582136959803729207od_a_a @ Resid @ NN @ ( normal322541020860755160od_a_a @ T7 ) )
= T7 ) ) ) ).
% normal_sub_rts.Cong_class_rep
thf(fact_754_normal__sub__rts_OCong__class__rep,axiom,
! [Resid: a > a > a,NN: set_a,T7: set_a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( normal8595587647932138008lass_a @ Resid @ NN @ T7 )
=> ( ( normal7408713899360725774lass_a @ Resid @ NN @ ( normal3259722184653208495_rep_a @ T7 ) )
= T7 ) ) ) ).
% normal_sub_rts.Cong_class_rep
thf(fact_755_Collect__empty__eq__bot,axiom,
! [P: product_prod_a_a > $o] :
( ( ( collec3336397797384452498od_a_a @ P )
= bot_bo3357376287454694259od_a_a )
= ( P = bot_bo4160289986317612842_a_a_o ) ) ).
% Collect_empty_eq_bot
thf(fact_756_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_757_Collect__empty__eq__bot,axiom,
! [P: ( a > a ) > $o] :
( ( ( collect_a_a @ P )
= bot_bot_set_a_a )
= ( P = bot_bot_a_a_o ) ) ).
% Collect_empty_eq_bot
thf(fact_758_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_759_bot__empty__eq,axiom,
( bot_bo4160289986317612842_a_a_o
= ( ^ [X2: product_prod_a_a] : ( member1426531477525435216od_a_a @ X2 @ bot_bo3357376287454694259od_a_a ) ) ) ).
% bot_empty_eq
thf(fact_760_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_761_bot__empty__eq,axiom,
( bot_bot_a_a_o
= ( ^ [X2: a > a] : ( member_a_a @ X2 @ bot_bot_set_a_a ) ) ) ).
% bot_empty_eq
thf(fact_762_bot__empty__eq,axiom,
( bot_bot_a_o
= ( ^ [X2: a] : ( member_a @ X2 @ bot_bot_set_a ) ) ) ).
% bot_empty_eq
thf(fact_763_residuation_Ointro,axiom,
! [Resid: a > a > a] :
( ( partial_magma_a @ Resid )
=> ( ( residuation_axioms_a @ Resid )
=> ( residuation_a @ Resid ) ) ) ).
% residuation.intro
thf(fact_764_residuation_Ointro,axiom,
! [Resid: set_a > set_a > set_a] :
( ( partial_magma_set_a @ Resid )
=> ( ( residu177535419945060507_set_a @ Resid )
=> ( residuation_set_a @ Resid ) ) ) ).
% residuation.intro
thf(fact_765_residuation__def,axiom,
( residuation_a
= ( ^ [Resid2: a > a > a] :
( ( partial_magma_a @ Resid2 )
& ( residuation_axioms_a @ Resid2 ) ) ) ) ).
% residuation_def
thf(fact_766_residuation__def,axiom,
( residuation_set_a
= ( ^ [Resid2: set_a > set_a > set_a] :
( ( partial_magma_set_a @ Resid2 )
& ( residu177535419945060507_set_a @ Resid2 ) ) ) ) ).
% residuation_def
thf(fact_767_rts__with__composites__axioms_Ointro,axiom,
! [Resid: set_a > set_a > set_a] :
( ! [T3: set_a,U5: set_a] :
( ( seq_set_a @ Resid @ T3 @ U5 )
=> ( composable_set_a @ Resid @ T3 @ U5 ) )
=> ( rts_wi3761204822955878899_set_a @ Resid ) ) ).
% rts_with_composites_axioms.intro
thf(fact_768_rts__with__composites__axioms_Ointro,axiom,
! [Resid: a > a > a] :
( ! [T3: a,U5: a] :
( ( seq_a @ Resid @ T3 @ U5 )
=> ( composable_a @ Resid @ T3 @ U5 ) )
=> ( rts_wi2614412583573296275ioms_a @ Resid ) ) ).
% rts_with_composites_axioms.intro
thf(fact_769_rts__with__composites__axioms__def,axiom,
( rts_wi3761204822955878899_set_a
= ( ^ [Resid2: set_a > set_a > set_a] :
! [T5: set_a,U3: set_a] :
( ( seq_set_a @ Resid2 @ T5 @ U3 )
=> ( composable_set_a @ Resid2 @ T5 @ U3 ) ) ) ) ).
% rts_with_composites_axioms_def
thf(fact_770_rts__with__composites__axioms__def,axiom,
( rts_wi2614412583573296275ioms_a
= ( ^ [Resid2: a > a > a] :
! [T5: a,U3: a] :
( ( seq_a @ Resid2 @ T5 @ U3 )
=> ( composable_a @ Resid2 @ T5 @ U3 ) ) ) ) ).
% rts_with_composites_axioms_def
thf(fact_771_residuation__axioms__def,axiom,
( residu177535419945060507_set_a
= ( ^ [Resid2: set_a > set_a > set_a] :
( ! [T5: set_a,U3: set_a] :
( ( ( Resid2 @ T5 @ U3 )
!= ( partial_null_set_a @ Resid2 ) )
=> ( ( Resid2 @ U3 @ T5 )
!= ( partial_null_set_a @ Resid2 ) ) )
& ! [T5: set_a,U3: set_a] :
( ( ( Resid2 @ T5 @ U3 )
!= ( partial_null_set_a @ Resid2 ) )
=> ( ( Resid2 @ ( Resid2 @ T5 @ U3 ) @ ( Resid2 @ T5 @ U3 ) )
!= ( partial_null_set_a @ Resid2 ) ) )
& ! [V3: set_a,T5: set_a,U3: set_a] :
( ( ( Resid2 @ ( Resid2 @ V3 @ T5 ) @ ( Resid2 @ U3 @ T5 ) )
!= ( partial_null_set_a @ Resid2 ) )
=> ( ( Resid2 @ ( Resid2 @ V3 @ T5 ) @ ( Resid2 @ U3 @ T5 ) )
= ( Resid2 @ ( Resid2 @ V3 @ U3 ) @ ( Resid2 @ T5 @ U3 ) ) ) ) ) ) ) ).
% residuation_axioms_def
thf(fact_772_residuation__axioms__def,axiom,
( residuation_axioms_a
= ( ^ [Resid2: a > a > a] :
( ! [T5: a,U3: a] :
( ( ( Resid2 @ T5 @ U3 )
!= ( partial_null_a @ Resid2 ) )
=> ( ( Resid2 @ U3 @ T5 )
!= ( partial_null_a @ Resid2 ) ) )
& ! [T5: a,U3: a] :
( ( ( Resid2 @ T5 @ U3 )
!= ( partial_null_a @ Resid2 ) )
=> ( ( Resid2 @ ( Resid2 @ T5 @ U3 ) @ ( Resid2 @ T5 @ U3 ) )
!= ( partial_null_a @ Resid2 ) ) )
& ! [V3: a,T5: a,U3: a] :
( ( ( Resid2 @ ( Resid2 @ V3 @ T5 ) @ ( Resid2 @ U3 @ T5 ) )
!= ( partial_null_a @ Resid2 ) )
=> ( ( Resid2 @ ( Resid2 @ V3 @ T5 ) @ ( Resid2 @ U3 @ T5 ) )
= ( Resid2 @ ( Resid2 @ V3 @ U3 ) @ ( Resid2 @ T5 @ U3 ) ) ) ) ) ) ) ).
% residuation_axioms_def
thf(fact_773_residuation__axioms_Ointro,axiom,
! [Resid: set_a > set_a > set_a] :
( ! [T3: set_a,U5: set_a] :
( ( ( Resid @ T3 @ U5 )
!= ( partial_null_set_a @ Resid ) )
=> ( ( Resid @ U5 @ T3 )
!= ( partial_null_set_a @ Resid ) ) )
=> ( ! [T3: set_a,U5: set_a] :
( ( ( Resid @ T3 @ U5 )
!= ( partial_null_set_a @ Resid ) )
=> ( ( Resid @ ( Resid @ T3 @ U5 ) @ ( Resid @ T3 @ U5 ) )
!= ( partial_null_set_a @ Resid ) ) )
=> ( ! [V5: set_a,T3: set_a,U5: set_a] :
( ( ( Resid @ ( Resid @ V5 @ T3 ) @ ( Resid @ U5 @ T3 ) )
!= ( partial_null_set_a @ Resid ) )
=> ( ( Resid @ ( Resid @ V5 @ T3 ) @ ( Resid @ U5 @ T3 ) )
= ( Resid @ ( Resid @ V5 @ U5 ) @ ( Resid @ T3 @ U5 ) ) ) )
=> ( residu177535419945060507_set_a @ Resid ) ) ) ) ).
% residuation_axioms.intro
thf(fact_774_residuation__axioms_Ointro,axiom,
! [Resid: a > a > a] :
( ! [T3: a,U5: a] :
( ( ( Resid @ T3 @ U5 )
!= ( partial_null_a @ Resid ) )
=> ( ( Resid @ U5 @ T3 )
!= ( partial_null_a @ Resid ) ) )
=> ( ! [T3: a,U5: a] :
( ( ( Resid @ T3 @ U5 )
!= ( partial_null_a @ Resid ) )
=> ( ( Resid @ ( Resid @ T3 @ U5 ) @ ( Resid @ T3 @ U5 ) )
!= ( partial_null_a @ Resid ) ) )
=> ( ! [V5: a,T3: a,U5: a] :
( ( ( Resid @ ( Resid @ V5 @ T3 ) @ ( Resid @ U5 @ T3 ) )
!= ( partial_null_a @ Resid ) )
=> ( ( Resid @ ( Resid @ V5 @ T3 ) @ ( Resid @ U5 @ T3 ) )
= ( Resid @ ( Resid @ V5 @ U5 ) @ ( Resid @ T3 @ U5 ) ) ) )
=> ( residuation_axioms_a @ Resid ) ) ) ) ).
% residuation_axioms.intro
thf(fact_775_residuation_Oaxioms_I2_J,axiom,
! [Resid: set_a > set_a > set_a] :
( ( residuation_set_a @ Resid )
=> ( residu177535419945060507_set_a @ Resid ) ) ).
% residuation.axioms(2)
thf(fact_776_residuation_Oaxioms_I2_J,axiom,
! [Resid: a > a > a] :
( ( residuation_a @ Resid )
=> ( residuation_axioms_a @ Resid ) ) ).
% residuation.axioms(2)
thf(fact_777_rts__with__joins__axioms__def,axiom,
( rts_wi637544758655500588_set_a
= ( ^ [Resid2: set_a > set_a > set_a] :
! [T5: set_a,U3: set_a] :
( ( con_set_a @ Resid2 @ T5 @ U3 )
=> ( joinable_set_a @ Resid2 @ T5 @ U3 ) ) ) ) ).
% rts_with_joins_axioms_def
thf(fact_778_rts__with__joins__axioms__def,axiom,
( rts_wi560353115624263628ioms_a
= ( ^ [Resid2: a > a > a] :
! [T5: a,U3: a] :
( ( con_a @ Resid2 @ T5 @ U3 )
=> ( joinable_a @ Resid2 @ T5 @ U3 ) ) ) ) ).
% rts_with_joins_axioms_def
thf(fact_779_rts__with__joins__axioms_Ointro,axiom,
! [Resid: set_a > set_a > set_a] :
( ! [T3: set_a,U5: set_a] :
( ( con_set_a @ Resid @ T3 @ U5 )
=> ( joinable_set_a @ Resid @ T3 @ U5 ) )
=> ( rts_wi637544758655500588_set_a @ Resid ) ) ).
% rts_with_joins_axioms.intro
thf(fact_780_rts__with__joins__axioms_Ointro,axiom,
! [Resid: a > a > a] :
( ! [T3: a,U5: a] :
( ( con_a @ Resid @ T3 @ U5 )
=> ( joinable_a @ Resid @ T3 @ U5 ) )
=> ( rts_wi560353115624263628ioms_a @ Resid ) ) ).
% rts_with_joins_axioms.intro
thf(fact_781_confluent__rts__axioms__def,axiom,
( conflu1148668952538903019_set_a
= ( ^ [Resid2: set_a > set_a > set_a] :
! [T5: set_a,U3: set_a] :
( ( coinitial_set_a @ Resid2 @ T5 @ U3 )
=> ( con_set_a @ Resid2 @ T5 @ U3 ) ) ) ) ).
% confluent_rts_axioms_def
thf(fact_782_confluent__rts__axioms__def,axiom,
( conflu3014480972103220363ioms_a
= ( ^ [Resid2: a > a > a] :
! [T5: a,U3: a] :
( ( coinitial_a @ Resid2 @ T5 @ U3 )
=> ( con_a @ Resid2 @ T5 @ U3 ) ) ) ) ).
% confluent_rts_axioms_def
thf(fact_783_confluent__rts__axioms_Ointro,axiom,
! [Resid: set_a > set_a > set_a] :
( ! [T3: set_a,U5: set_a] :
( ( coinitial_set_a @ Resid @ T3 @ U5 )
=> ( con_set_a @ Resid @ T3 @ U5 ) )
=> ( conflu1148668952538903019_set_a @ Resid ) ) ).
% confluent_rts_axioms.intro
thf(fact_784_confluent__rts__axioms_Ointro,axiom,
! [Resid: a > a > a] :
( ! [T3: a,U5: a] :
( ( coinitial_a @ Resid @ T3 @ U5 )
=> ( con_a @ Resid @ T3 @ U5 ) )
=> ( conflu3014480972103220363ioms_a @ Resid ) ) ).
% confluent_rts_axioms.intro
thf(fact_785_confluent__rts_Oconfluence,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a] :
( ( confluent_rts_set_a @ Resid )
=> ( ( coinitial_set_a @ Resid @ T @ U )
=> ( con_set_a @ Resid @ T @ U ) ) ) ).
% confluent_rts.confluence
thf(fact_786_confluent__rts_Oconfluence,axiom,
! [Resid: a > a > a,T: a,U: a] :
( ( confluent_rts_a @ Resid )
=> ( ( coinitial_a @ Resid @ T @ U )
=> ( con_a @ Resid @ T @ U ) ) ) ).
% confluent_rts.confluence
thf(fact_787_confluent__rts_Oaxioms_I2_J,axiom,
! [Resid: a > a > a] :
( ( confluent_rts_a @ Resid )
=> ( conflu3014480972103220363ioms_a @ Resid ) ) ).
% confluent_rts.axioms(2)
thf(fact_788_simulation__to__weakly__extensional__rts_Opreserves__trg,axiom,
! [A4: product_prod_a_a > product_prod_a_a > product_prod_a_a,B3: a > a > a,F2: product_prod_a_a > a,B: product_prod_a_a,T: product_prod_a_a] :
( ( simula3723153512272018977_a_a_a @ A4 @ B3 @ F2 )
=> ( ( member1426531477525435216od_a_a @ B @ ( target5293506191220573129od_a_a @ A4 @ T ) )
=> ( ( trg_a @ B3 @ ( F2 @ T ) )
= ( F2 @ B ) ) ) ) ).
% simulation_to_weakly_extensional_rts.preserves_trg
thf(fact_789_simulation__to__weakly__extensional__rts_Opreserves__trg,axiom,
! [A4: set_a > set_a > set_a,B3: a > a > a,F2: set_a > a,B: set_a,T: set_a] :
( ( simula8097555664381481386et_a_a @ A4 @ B3 @ F2 )
=> ( ( member_set_a @ B @ ( targets_set_a @ A4 @ T ) )
=> ( ( trg_a @ B3 @ ( F2 @ T ) )
= ( F2 @ B ) ) ) ) ).
% simulation_to_weakly_extensional_rts.preserves_trg
thf(fact_790_simulation__to__weakly__extensional__rts_Opreserves__trg,axiom,
! [A4: ( a > a ) > ( a > a ) > a > a,B3: a > a > a,F2: ( a > a ) > a,B: a > a,T: a > a] :
( ( simula2983153686137172173_a_a_a @ A4 @ B3 @ F2 )
=> ( ( member_a_a @ B @ ( targets_a_a @ A4 @ T ) )
=> ( ( trg_a @ B3 @ ( F2 @ T ) )
= ( F2 @ B ) ) ) ) ).
% simulation_to_weakly_extensional_rts.preserves_trg
thf(fact_791_simulation__to__weakly__extensional__rts_Opreserves__trg,axiom,
! [A4: product_prod_a_a > product_prod_a_a > product_prod_a_a,B3: set_a > set_a > set_a,F2: product_prod_a_a > set_a,B: product_prod_a_a,T: product_prod_a_a] :
( ( simula5313398963451091841_set_a @ A4 @ B3 @ F2 )
=> ( ( member1426531477525435216od_a_a @ B @ ( target5293506191220573129od_a_a @ A4 @ T ) )
=> ( ( trg_set_a @ B3 @ ( F2 @ T ) )
= ( F2 @ B ) ) ) ) ).
% simulation_to_weakly_extensional_rts.preserves_trg
thf(fact_792_simulation__to__weakly__extensional__rts_Opreserves__trg,axiom,
! [A4: set_a > set_a > set_a,B3: set_a > set_a > set_a,F2: set_a > set_a,B: set_a,T: set_a] :
( ( simula348638060201904906_set_a @ A4 @ B3 @ F2 )
=> ( ( member_set_a @ B @ ( targets_set_a @ A4 @ T ) )
=> ( ( trg_set_a @ B3 @ ( F2 @ T ) )
= ( F2 @ B ) ) ) ) ).
% simulation_to_weakly_extensional_rts.preserves_trg
thf(fact_793_simulation__to__weakly__extensional__rts_Opreserves__trg,axiom,
! [A4: ( a > a ) > ( a > a ) > a > a,B3: set_a > set_a > set_a,F2: ( a > a ) > set_a,B: a > a,T: a > a] :
( ( simula2357104975539035693_set_a @ A4 @ B3 @ F2 )
=> ( ( member_a_a @ B @ ( targets_a_a @ A4 @ T ) )
=> ( ( trg_set_a @ B3 @ ( F2 @ T ) )
= ( F2 @ B ) ) ) ) ).
% simulation_to_weakly_extensional_rts.preserves_trg
thf(fact_794_simulation__to__weakly__extensional__rts_Opreserves__trg,axiom,
! [A4: a > a > a,B3: set_a > set_a > set_a,F2: a > set_a,B: a,T: a] :
( ( simula7881043605922138218_set_a @ A4 @ B3 @ F2 )
=> ( ( member_a @ B @ ( targets_a @ A4 @ T ) )
=> ( ( trg_set_a @ B3 @ ( F2 @ T ) )
= ( F2 @ B ) ) ) ) ).
% simulation_to_weakly_extensional_rts.preserves_trg
thf(fact_795_simulation__to__weakly__extensional__rts_Opreserves__trg,axiom,
! [A4: a > a > a,B3: a > a > a,F2: a > a,B: a,T: a] :
( ( simula2709571904647515914ts_a_a @ A4 @ B3 @ F2 )
=> ( ( member_a @ B @ ( targets_a @ A4 @ T ) )
=> ( ( trg_a @ B3 @ ( F2 @ T ) )
= ( F2 @ B ) ) ) ) ).
% simulation_to_weakly_extensional_rts.preserves_trg
thf(fact_796_Set_Ois__empty__def,axiom,
( is_emp2937470224744679417od_a_a
= ( ^ [A7: set_Product_prod_a_a] : ( A7 = bot_bo3357376287454694259od_a_a ) ) ) ).
% Set.is_empty_def
thf(fact_797_Set_Ois__empty__def,axiom,
( is_empty_set_a
= ( ^ [A7: set_set_a] : ( A7 = bot_bot_set_set_a ) ) ) ).
% Set.is_empty_def
thf(fact_798_Set_Ois__empty__def,axiom,
( is_empty_a_a
= ( ^ [A7: set_a_a] : ( A7 = bot_bot_set_a_a ) ) ) ).
% Set.is_empty_def
thf(fact_799_Set_Ois__empty__def,axiom,
( is_empty_a
= ( ^ [A7: set_a] : ( A7 = bot_bot_set_a ) ) ) ).
% Set.is_empty_def
thf(fact_800_rts__with__joins_Ohas__joins,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a] :
( ( rts_with_joins_set_a @ Resid )
=> ( ( con_set_a @ Resid @ T @ U )
=> ( joinable_set_a @ Resid @ T @ U ) ) ) ).
% rts_with_joins.has_joins
thf(fact_801_rts__with__joins_Ohas__joins,axiom,
! [Resid: a > a > a,T: a,U: a] :
( ( rts_with_joins_a @ Resid )
=> ( ( con_a @ Resid @ T @ U )
=> ( joinable_a @ Resid @ T @ U ) ) ) ).
% rts_with_joins.has_joins
thf(fact_802_extensional__rts__with__joins_Ojoinable__iff__con,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a] :
( ( extens2085910753204196637_set_a @ Resid )
=> ( ( joinable_set_a @ Resid @ T @ U )
= ( con_set_a @ Resid @ T @ U ) ) ) ).
% extensional_rts_with_joins.joinable_iff_con
thf(fact_803_extensional__rts__with__joins_Ojoinable__iff__con,axiom,
! [Resid: a > a > a,T: a,U: a] :
( ( extens4936603313648314301oins_a @ Resid )
=> ( ( joinable_a @ Resid @ T @ U )
= ( con_a @ Resid @ T @ U ) ) ) ).
% extensional_rts_with_joins.joinable_iff_con
thf(fact_804_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_805_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_806_R_Orts__axioms,axiom,
rts_a @ resid ).
% R.rts_axioms
thf(fact_807_rts__with__joins_Ointro,axiom,
! [Resid: a > a > a] :
( ( rts_a @ Resid )
=> ( ( rts_wi560353115624263628ioms_a @ Resid )
=> ( rts_with_joins_a @ Resid ) ) ) ).
% rts_with_joins.intro
thf(fact_808_rts__with__joins__def,axiom,
( rts_with_joins_a
= ( ^ [Resid2: a > a > a] :
( ( rts_a @ Resid2 )
& ( rts_wi560353115624263628ioms_a @ Resid2 ) ) ) ) ).
% rts_with_joins_def
thf(fact_809_rts_Oide__backward__stable,axiom,
! [Resid: a > a > a,A: a,T: a] :
( ( rts_a @ Resid )
=> ( ( ide_a @ Resid @ A )
=> ( ( ide_a @ Resid @ ( Resid @ T @ A ) )
=> ( ide_a @ Resid @ T ) ) ) ) ).
% rts.ide_backward_stable
thf(fact_810_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_811_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_812_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_813_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_814_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_815_extensional__rts__with__joins_Oaxioms_I1_J,axiom,
! [Resid: a > a > a] :
( ( extens4936603313648314301oins_a @ Resid )
=> ( rts_with_joins_a @ Resid ) ) ).
% extensional_rts_with_joins.axioms(1)
thf(fact_816_rts__with__joins_Oaxioms_I1_J,axiom,
! [Resid: a > a > a] :
( ( rts_with_joins_a @ Resid )
=> ( rts_a @ Resid ) ) ).
% rts_with_joins.axioms(1)
thf(fact_817_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_818_rts_Oaxioms_I1_J,axiom,
! [Resid: a > a > a] :
( ( rts_a @ Resid )
=> ( residuation_a @ Resid ) ) ).
% rts.axioms(1)
thf(fact_819_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_820_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_821_normal__sub__rts_Oaxioms_I1_J,axiom,
! [Resid: a > a > a,NN: set_a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( rts_a @ Resid ) ) ).
% normal_sub_rts.axioms(1)
thf(fact_822_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_823_rts_Oaxioms_I2_J,axiom,
! [Resid: a > a > a] :
( ( rts_a @ Resid )
=> ( rts_axioms_a @ Resid ) ) ).
% rts.axioms(2)
thf(fact_824_confluent__rts_Oaxioms_I1_J,axiom,
! [Resid: a > a > a] :
( ( confluent_rts_a @ Resid )
=> ( rts_a @ Resid ) ) ).
% confluent_rts.axioms(1)
thf(fact_825_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_826_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_827_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_828_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_829_rts_Ocong__subst__right_I2_J,axiom,
! [Resid: set_a > set_a > set_a,U: set_a,U2: set_a,T: set_a] :
( ( rts_set_a @ Resid )
=> ( ( ( ide_set_a @ Resid @ ( Resid @ U @ U2 ) )
& ( ide_set_a @ Resid @ ( Resid @ U2 @ U ) ) )
=> ( ( con_set_a @ Resid @ T @ U )
=> ( ( ide_set_a @ Resid @ ( Resid @ ( Resid @ T @ U ) @ ( Resid @ T @ U2 ) ) )
& ( ide_set_a @ Resid @ ( Resid @ ( Resid @ T @ U2 ) @ ( Resid @ T @ U ) ) ) ) ) ) ) ).
% rts.cong_subst_right(2)
thf(fact_830_rts_Ocong__subst__right_I2_J,axiom,
! [Resid: a > a > a,U: a,U2: a,T: a] :
( ( rts_a @ Resid )
=> ( ( ( ide_a @ Resid @ ( Resid @ U @ U2 ) )
& ( ide_a @ Resid @ ( Resid @ U2 @ U ) ) )
=> ( ( con_a @ Resid @ T @ U )
=> ( ( ide_a @ Resid @ ( Resid @ ( Resid @ T @ U ) @ ( Resid @ T @ U2 ) ) )
& ( ide_a @ Resid @ ( Resid @ ( Resid @ T @ U2 ) @ ( Resid @ T @ U ) ) ) ) ) ) ) ).
% rts.cong_subst_right(2)
thf(fact_831_rts_Ocong__subst__right_I1_J,axiom,
! [Resid: set_a > set_a > set_a,U: set_a,U2: set_a,T: set_a] :
( ( rts_set_a @ Resid )
=> ( ( ( ide_set_a @ Resid @ ( Resid @ U @ U2 ) )
& ( ide_set_a @ Resid @ ( Resid @ U2 @ U ) ) )
=> ( ( con_set_a @ Resid @ T @ U )
=> ( con_set_a @ Resid @ T @ U2 ) ) ) ) ).
% rts.cong_subst_right(1)
thf(fact_832_rts_Ocong__subst__right_I1_J,axiom,
! [Resid: a > a > a,U: a,U2: a,T: a] :
( ( rts_a @ Resid )
=> ( ( ( ide_a @ Resid @ ( Resid @ U @ U2 ) )
& ( ide_a @ Resid @ ( Resid @ U2 @ U ) ) )
=> ( ( con_a @ Resid @ T @ U )
=> ( con_a @ Resid @ T @ U2 ) ) ) ) ).
% rts.cong_subst_right(1)
thf(fact_833_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_834_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_835_rts_Oresid__arr__ide,axiom,
! [Resid: set_a > set_a > set_a,A: set_a,T: set_a] :
( ( rts_set_a @ Resid )
=> ( ( ide_set_a @ Resid @ A )
=> ( ( con_set_a @ Resid @ T @ A )
=> ( ( Resid @ T @ A )
= T ) ) ) ) ).
% rts.resid_arr_ide
thf(fact_836_rts_Oresid__arr__ide,axiom,
! [Resid: a > a > a,A: a,T: a] :
( ( rts_a @ Resid )
=> ( ( ide_a @ Resid @ A )
=> ( ( con_a @ Resid @ T @ A )
=> ( ( Resid @ T @ A )
= T ) ) ) ) ).
% rts.resid_arr_ide
thf(fact_837_rts_Oresid__ide__arr,axiom,
! [Resid: set_a > set_a > set_a,A: set_a,T: set_a] :
( ( rts_set_a @ Resid )
=> ( ( ide_set_a @ Resid @ A )
=> ( ( con_set_a @ Resid @ A @ T )
=> ( ide_set_a @ Resid @ ( Resid @ A @ T ) ) ) ) ) ).
% rts.resid_ide_arr
thf(fact_838_rts_Oresid__ide__arr,axiom,
! [Resid: a > a > a,A: a,T: a] :
( ( rts_a @ Resid )
=> ( ( ide_a @ Resid @ A )
=> ( ( con_a @ Resid @ A @ T )
=> ( ide_a @ Resid @ ( Resid @ A @ T ) ) ) ) ) ).
% rts.resid_ide_arr
thf(fact_839_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_840_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_841_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 )
=> ? [A3: set_a] :
( ( ide_set_a @ Resid @ A3 )
& ( con_set_a @ Resid @ A3 @ T )
& ( con_set_a @ Resid @ A3 @ U ) ) ) ) ).
% rts.con_imp_coinitial_ax
thf(fact_842_rts_Ocon__imp__coinitial__ax,axiom,
! [Resid: a > a > a,T: a,U: a] :
( ( rts_a @ Resid )
=> ( ( con_a @ Resid @ T @ U )
=> ? [A3: a] :
( ( ide_a @ Resid @ A3 )
& ( con_a @ Resid @ A3 @ T )
& ( con_a @ Resid @ A3 @ U ) ) ) ) ).
% rts.con_imp_coinitial_ax
thf(fact_843_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_844_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_845_rts_Ocon__transitive__on__ide,axiom,
! [Resid: set_a > set_a > set_a,A: set_a,B: set_a,C: set_a] :
( ( rts_set_a @ Resid )
=> ( ( ide_set_a @ Resid @ A )
=> ( ( ide_set_a @ Resid @ B )
=> ( ( ide_set_a @ Resid @ C )
=> ( ( con_set_a @ Resid @ A @ B )
=> ( ( con_set_a @ Resid @ B @ C )
=> ( con_set_a @ Resid @ A @ C ) ) ) ) ) ) ) ).
% rts.con_transitive_on_ide
thf(fact_846_rts_Ocon__transitive__on__ide,axiom,
! [Resid: a > a > a,A: a,B: a,C: a] :
( ( rts_a @ Resid )
=> ( ( ide_a @ Resid @ A )
=> ( ( ide_a @ Resid @ B )
=> ( ( ide_a @ Resid @ C )
=> ( ( con_a @ Resid @ A @ B )
=> ( ( con_a @ Resid @ B @ C )
=> ( con_a @ Resid @ A @ C ) ) ) ) ) ) ) ).
% rts.con_transitive_on_ide
thf(fact_847_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_848_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_849_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_850_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_851_rts_Osources__cong__closed,axiom,
! [Resid: ( a > a ) > ( a > a ) > a > a,A: a > a,T: a > a,A2: a > a] :
( ( rts_a_a @ Resid )
=> ( ( member_a_a @ A @ ( sources_a_a @ Resid @ T ) )
=> ( ( ( ide_a_a @ Resid @ ( Resid @ A @ A2 ) )
& ( ide_a_a @ Resid @ ( Resid @ A2 @ A ) ) )
=> ( member_a_a @ A2 @ ( sources_a_a @ Resid @ T ) ) ) ) ) ).
% rts.sources_cong_closed
thf(fact_852_rts_Osources__cong__closed,axiom,
! [Resid: set_a > set_a > set_a,A: set_a,T: set_a,A2: set_a] :
( ( rts_set_a @ Resid )
=> ( ( member_set_a @ A @ ( sources_set_a @ Resid @ T ) )
=> ( ( ( ide_set_a @ Resid @ ( Resid @ A @ A2 ) )
& ( ide_set_a @ Resid @ ( Resid @ A2 @ A ) ) )
=> ( member_set_a @ A2 @ ( sources_set_a @ Resid @ T ) ) ) ) ) ).
% rts.sources_cong_closed
thf(fact_853_rts_Osources__cong__closed,axiom,
! [Resid: a > a > a,A: a,T: a,A2: a] :
( ( rts_a @ Resid )
=> ( ( member_a @ A @ ( sources_a @ Resid @ T ) )
=> ( ( ( ide_a @ Resid @ ( Resid @ A @ A2 ) )
& ( ide_a @ Resid @ ( Resid @ A2 @ A ) ) )
=> ( member_a @ A2 @ ( sources_a @ Resid @ T ) ) ) ) ) ).
% rts.sources_cong_closed
thf(fact_854_rts_Osources__are__cong,axiom,
! [Resid: a > a > a,A: a,T: a,A2: a] :
( ( rts_a @ Resid )
=> ( ( member_a @ A @ ( sources_a @ Resid @ T ) )
=> ( ( member_a @ A2 @ ( sources_a @ Resid @ T ) )
=> ( ( ide_a @ Resid @ ( Resid @ A @ A2 ) )
& ( ide_a @ Resid @ ( Resid @ A2 @ A ) ) ) ) ) ) ).
% rts.sources_are_cong
thf(fact_855_rts_Osource__is__ide,axiom,
! [Resid: a > a > a,A: a,T: a] :
( ( rts_a @ Resid )
=> ( ( member_a @ A @ ( sources_a @ Resid @ T ) )
=> ( ide_a @ Resid @ A ) ) ) ).
% rts.source_is_ide
thf(fact_856_rts_Osources__are__con,axiom,
! [Resid: a > a > a,A: a,T: a,A2: a] :
( ( rts_a @ Resid )
=> ( ( member_a @ A @ ( sources_a @ Resid @ T ) )
=> ( ( member_a @ A2 @ ( sources_a @ Resid @ T ) )
=> ( con_a @ Resid @ A @ A2 ) ) ) ) ).
% rts.sources_are_con
thf(fact_857_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_858_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_859_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_860_rts_Ocomposite__of__ide__self,axiom,
! [Resid: a > a > a,A: a] :
( ( rts_a @ Resid )
=> ( ( ide_a @ Resid @ A )
=> ( composite_of_a @ Resid @ A @ A @ A ) ) ) ).
% rts.composite_of_ide_self
thf(fact_861_rts_Ocomposite__of__cancel__left,axiom,
! [Resid: a > a > a,T: a,U: a,V: a,U2: a] :
( ( rts_a @ Resid )
=> ( ( composite_of_a @ Resid @ T @ U @ V )
=> ( ( composite_of_a @ Resid @ T @ U2 @ V )
=> ( ( ide_a @ Resid @ ( Resid @ U @ U2 ) )
& ( ide_a @ Resid @ ( Resid @ U2 @ U ) ) ) ) ) ) ).
% rts.composite_of_cancel_left
thf(fact_862_rts_Ocomposite__of__unq__upto__cong,axiom,
! [Resid: a > a > a,U: a,T: a,V: a,V2: a] :
( ( rts_a @ Resid )
=> ( ( composite_of_a @ Resid @ U @ T @ V )
=> ( ( composite_of_a @ Resid @ U @ T @ V2 )
=> ( ( ide_a @ Resid @ ( Resid @ V @ V2 ) )
& ( ide_a @ Resid @ ( Resid @ V2 @ V ) ) ) ) ) ) ).
% rts.composite_of_unq_upto_cong
thf(fact_863_rts_Ocon__composite__of__iff,axiom,
! [Resid: a > a > a,T: a,U: a,V: a,W: a] :
( ( rts_a @ Resid )
=> ( ( composite_of_a @ Resid @ T @ U @ V )
=> ( ( con_a @ Resid @ W @ V )
= ( con_a @ Resid @ ( Resid @ W @ T ) @ U ) ) ) ) ).
% rts.con_composite_of_iff
thf(fact_864_rts_Obounded__imp__con,axiom,
! [Resid: a > a > a,T: a,U: a,V: a,T4: a,U2: a] :
( ( rts_a @ Resid )
=> ( ( composite_of_a @ Resid @ T @ U @ V )
=> ( ( composite_of_a @ Resid @ T4 @ U2 @ V )
=> ( con_a @ Resid @ T @ T4 ) ) ) ) ).
% rts.bounded_imp_con
thf(fact_865_rts_Ocon__prfx__composite__of_I1_J,axiom,
! [Resid: a > a > a,T: a,U: a,W: a] :
( ( rts_a @ Resid )
=> ( ( composite_of_a @ Resid @ T @ U @ W )
=> ( con_a @ Resid @ T @ W ) ) ) ).
% rts.con_prfx_composite_of(1)
thf(fact_866_rts_Ocon__prfx__composite__of_I2_J,axiom,
! [Resid: a > a > a,T: a,U: a,W: a,V: a] :
( ( rts_a @ Resid )
=> ( ( composite_of_a @ Resid @ T @ U @ W )
=> ( ( con_a @ Resid @ W @ V )
=> ( con_a @ Resid @ T @ V ) ) ) ) ).
% rts.con_prfx_composite_of(2)
thf(fact_867_rts_Oresid__composite__of_I1_J,axiom,
! [Resid: a > a > a,T: a,U: a,W: a,V: a] :
( ( rts_a @ Resid )
=> ( ( composite_of_a @ Resid @ T @ U @ W )
=> ( ( con_a @ Resid @ W @ V )
=> ( con_a @ Resid @ ( Resid @ V @ T ) @ ( Resid @ W @ T ) ) ) ) ) ).
% rts.resid_composite_of(1)
thf(fact_868_rts_Oresid__composite__of_I2_J,axiom,
! [Resid: a > a > a,T: a,U: a,W: a,V: a] :
( ( rts_a @ Resid )
=> ( ( composite_of_a @ Resid @ T @ U @ W )
=> ( ( con_a @ Resid @ W @ V )
=> ( con_a @ Resid @ ( Resid @ V @ T ) @ U ) ) ) ) ).
% rts.resid_composite_of(2)
thf(fact_869_rts_Oresid__composite__of_I4_J,axiom,
! [Resid: a > a > a,T: a,U: a,W: a,V: a] :
( ( rts_a @ Resid )
=> ( ( composite_of_a @ Resid @ T @ U @ W )
=> ( ( con_a @ Resid @ W @ V )
=> ( composite_of_a @ Resid @ ( Resid @ T @ V ) @ ( Resid @ U @ ( Resid @ V @ T ) ) @ ( Resid @ W @ V ) ) ) ) ) ).
% rts.resid_composite_of(4)
thf(fact_870_rts_Otargets__cong__closed,axiom,
! [Resid: a > a > a,B: a,T: a,B2: a] :
( ( rts_a @ Resid )
=> ( ( member_a @ B @ ( targets_a @ Resid @ T ) )
=> ( ( ( ide_a @ Resid @ ( Resid @ B @ B2 ) )
& ( ide_a @ Resid @ ( Resid @ B2 @ B ) ) )
=> ( member_a @ B2 @ ( targets_a @ Resid @ T ) ) ) ) ) ).
% rts.targets_cong_closed
thf(fact_871_rts_Otargets__are__cong,axiom,
! [Resid: a > a > a,B: a,T: a,B2: a] :
( ( rts_a @ Resid )
=> ( ( member_a @ B @ ( targets_a @ Resid @ T ) )
=> ( ( member_a @ B2 @ ( targets_a @ Resid @ T ) )
=> ( ( ide_a @ Resid @ ( Resid @ B @ B2 ) )
& ( ide_a @ Resid @ ( Resid @ B2 @ B ) ) ) ) ) ) ).
% rts.targets_are_cong
thf(fact_872_rts_Otarget__is__ide,axiom,
! [Resid: a > a > a,A: a,T: a] :
( ( rts_a @ Resid )
=> ( ( member_a @ A @ ( targets_a @ Resid @ T ) )
=> ( ide_a @ Resid @ A ) ) ) ).
% rts.target_is_ide
thf(fact_873_rts_Otargets__are__con,axiom,
! [Resid: a > a > a,B: a,T: a,B2: a] :
( ( rts_a @ Resid )
=> ( ( member_a @ B @ ( targets_a @ Resid @ T ) )
=> ( ( member_a @ B2 @ ( targets_a @ Resid @ T ) )
=> ( con_a @ Resid @ B @ B2 ) ) ) ) ).
% rts.targets_are_con
thf(fact_874_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_875_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_876_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_877_rts_Oresid__source__in__targets,axiom,
! [Resid: a > a > a,A: a,T: a] :
( ( rts_a @ Resid )
=> ( ( member_a @ A @ ( sources_a @ Resid @ T ) )
=> ( member_a @ ( Resid @ A @ T ) @ ( targets_a @ Resid @ T ) ) ) ) ).
% rts.resid_source_in_targets
thf(fact_878_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_879_rts_Ojoin__of__un__upto__cong,axiom,
! [Resid: a > a > a,T: a,U: a,V: a,V2: a] :
( ( rts_a @ Resid )
=> ( ( join_of_a @ Resid @ T @ U @ V )
=> ( ( join_of_a @ Resid @ T @ U @ V2 )
=> ( ( ide_a @ Resid @ ( Resid @ V @ V2 ) )
& ( ide_a @ Resid @ ( Resid @ V2 @ V ) ) ) ) ) ) ).
% rts.join_of_un_upto_cong
thf(fact_880_rts_Ojoin__of__resid,axiom,
! [Resid: a > a > a,T: a,U: a,W: a,V: a] :
( ( rts_a @ Resid )
=> ( ( join_of_a @ Resid @ T @ U @ W )
=> ( ( con_a @ Resid @ V @ W )
=> ( join_of_a @ Resid @ ( Resid @ T @ V ) @ ( Resid @ U @ V ) @ ( Resid @ W @ V ) ) ) ) ) ).
% rts.join_of_resid
thf(fact_881_rts_Ocon__with__join__of__iff_I1_J,axiom,
! [Resid: a > a > a,T: a,U: a,W: a,V: a] :
( ( rts_a @ Resid )
=> ( ( join_of_a @ Resid @ T @ U @ W )
=> ( ( ( con_a @ Resid @ U @ V )
& ( con_a @ Resid @ ( Resid @ V @ U ) @ ( Resid @ T @ U ) ) )
=> ( con_a @ Resid @ W @ V ) ) ) ) ).
% rts.con_with_join_of_iff(1)
thf(fact_882_rts_Ocon__with__join__of__iff_I2_J,axiom,
! [Resid: a > a > a,T: a,U: a,W: a,V: a] :
( ( rts_a @ Resid )
=> ( ( join_of_a @ Resid @ T @ U @ W )
=> ( ( con_a @ Resid @ W @ V )
=> ( ( con_a @ Resid @ T @ V )
& ( con_a @ Resid @ ( Resid @ V @ T ) @ ( Resid @ U @ T ) ) ) ) ) ) ).
% rts.con_with_join_of_iff(2)
thf(fact_883_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_884_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_885_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_886_rts_Ocong__respects__seq,axiom,
! [Resid: a > a > a,T: a,U: a,T4: a,U2: 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 @ U2 ) )
& ( ide_a @ Resid @ ( Resid @ U2 @ U ) ) )
=> ( seq_a @ Resid @ T4 @ U2 ) ) ) ) ) ).
% rts.cong_respects_seq
thf(fact_887_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_888_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_889_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_890_rts_Ocoinitial__ide__are__cong,axiom,
! [Resid: a > a > a,A: a,A2: a] :
( ( rts_a @ Resid )
=> ( ( ide_a @ Resid @ A )
=> ( ( ide_a @ Resid @ A2 )
=> ( ( coinitial_a @ Resid @ A @ A2 )
=> ( ( ide_a @ Resid @ ( Resid @ A @ A2 ) )
& ( ide_a @ Resid @ ( Resid @ A2 @ A ) ) ) ) ) ) ) ).
% rts.coinitial_ide_are_cong
thf(fact_891_rts_Ocong__implies__coinitial,axiom,
! [Resid: a > a > a,U: a,U2: a] :
( ( rts_a @ Resid )
=> ( ( ( ide_a @ Resid @ ( Resid @ U @ U2 ) )
& ( ide_a @ Resid @ ( Resid @ U2 @ U ) ) )
=> ( coinitial_a @ Resid @ U @ U2 ) ) ) ).
% rts.cong_implies_coinitial
thf(fact_892_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_893_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_894_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_895_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_896_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_897_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_898_rts_Ocomposable__def,axiom,
! [Resid: a > a > a,T: a,U: a] :
( ( rts_a @ Resid )
=> ( ( composable_a @ Resid @ T @ U )
= ( ? [X4: a] : ( composite_of_a @ Resid @ T @ U @ X4 ) ) ) ) ).
% rts.composable_def
thf(fact_899_rts_Ocong__implies__coterminal,axiom,
! [Resid: a > a > a,U: a,U2: a] :
( ( rts_a @ Resid )
=> ( ( ( ide_a @ Resid @ ( Resid @ U @ U2 ) )
& ( ide_a @ Resid @ ( Resid @ U2 @ U ) ) )
=> ( coterminal_a @ Resid @ U @ U2 ) ) ) ).
% rts.cong_implies_coterminal
thf(fact_900_rts_Ojoinable__def,axiom,
! [Resid: a > a > a,T: a,U: a] :
( ( rts_a @ Resid )
=> ( ( joinable_a @ Resid @ T @ U )
= ( ? [X4: a] : ( join_of_a @ Resid @ T @ U @ X4 ) ) ) ) ).
% rts.joinable_def
thf(fact_901_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_902_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_903_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_904_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_905_normal__sub__rts_Ointro,axiom,
! [Resid: a > a > a,NN: set_a] :
( ( rts_a @ Resid )
=> ( ( normal7698203753654205830ioms_a @ Resid @ NN )
=> ( normal_sub_rts_a @ Resid @ NN ) ) ) ).
% normal_sub_rts.intro
thf(fact_906_normal__sub__rts__def,axiom,
( normal_sub_rts_a
= ( ^ [Resid2: a > a > a,NN2: set_a] :
( ( rts_a @ Resid2 )
& ( normal7698203753654205830ioms_a @ Resid2 @ NN2 ) ) ) ) ).
% normal_sub_rts_def
thf(fact_907_rts_Ointro,axiom,
! [Resid: a > a > a] :
( ( residuation_a @ Resid )
=> ( ( rts_axioms_a @ Resid )
=> ( rts_a @ Resid ) ) ) ).
% rts.intro
thf(fact_908_rts__def,axiom,
( rts_a
= ( ^ [Resid2: a > a > a] :
( ( residuation_a @ Resid2 )
& ( rts_axioms_a @ Resid2 ) ) ) ) ).
% rts_def
thf(fact_909_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_910_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_911_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_912_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_913_rts_Osources__con__closed,axiom,
! [Resid: a > a > a,A: a,T: a,A2: a] :
( ( rts_a @ Resid )
=> ( ( member_a @ A @ ( sources_a @ Resid @ T ) )
=> ( ( ide_a @ Resid @ A2 )
=> ( ( con_a @ Resid @ A @ A2 )
=> ( member_a @ A2 @ ( sources_a @ Resid @ T ) ) ) ) ) ) ).
% rts.sources_con_closed
thf(fact_914_rts_Oin__sourcesI,axiom,
! [Resid: a > a > a,A: a,T: a] :
( ( rts_a @ Resid )
=> ( ( ide_a @ Resid @ A )
=> ( ( con_a @ Resid @ T @ A )
=> ( member_a @ A @ ( sources_a @ Resid @ T ) ) ) ) ) ).
% rts.in_sourcesI
thf(fact_915_rts_Oin__sourcesE,axiom,
! [Resid: a > a > a,A: a,T: a] :
( ( rts_a @ Resid )
=> ( ( member_a @ A @ ( sources_a @ Resid @ T ) )
=> ~ ( ( ide_a @ Resid @ A )
=> ~ ( con_a @ Resid @ T @ A ) ) ) ) ).
% rts.in_sourcesE
thf(fact_916_rts_Ocomposite__of__ide__arr,axiom,
! [Resid: a > a > a,A: a,T: a] :
( ( rts_a @ Resid )
=> ( ( ide_a @ Resid @ A )
=> ( ( composite_of_a @ Resid @ A @ T @ T )
= ( con_a @ Resid @ T @ A ) ) ) ) ).
% rts.composite_of_ide_arr
thf(fact_917_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_918_rts_Oresid__composite__of_I3_J,axiom,
! [Resid: a > a > a,T: a,U: a,W: a,V: a] :
( ( rts_a @ Resid )
=> ( ( composite_of_a @ Resid @ T @ U @ W )
=> ( ( con_a @ Resid @ W @ V )
=> ( ( ide_a @ Resid @ ( Resid @ ( Resid @ V @ W ) @ ( Resid @ ( Resid @ V @ T ) @ U ) ) )
& ( ide_a @ Resid @ ( Resid @ ( Resid @ ( Resid @ V @ T ) @ U ) @ ( Resid @ V @ W ) ) ) ) ) ) ) ).
% rts.resid_composite_of(3)
thf(fact_919_rts_Otargets__con__closed,axiom,
! [Resid: a > a > a,B: a,T: a,B2: a] :
( ( rts_a @ Resid )
=> ( ( member_a @ B @ ( targets_a @ Resid @ T ) )
=> ( ( ide_a @ Resid @ B2 )
=> ( ( con_a @ Resid @ B @ B2 )
=> ( member_a @ B2 @ ( targets_a @ Resid @ T ) ) ) ) ) ) ).
% rts.targets_con_closed
thf(fact_920_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_921_rts_Ocomposite__of__source__arr,axiom,
! [Resid: a > a > a,T: a,A: a] :
( ( rts_a @ Resid )
=> ( ( arr_a @ Resid @ T )
=> ( ( member_a @ A @ ( sources_a @ Resid @ T ) )
=> ( composite_of_a @ Resid @ A @ T @ T ) ) ) ) ).
% rts.composite_of_source_arr
thf(fact_922_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_923_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_924_confluent__rts__def,axiom,
( confluent_rts_a
= ( ^ [Resid2: a > a > a] :
( ( rts_a @ Resid2 )
& ( conflu3014480972103220363ioms_a @ Resid2 ) ) ) ) ).
% confluent_rts_def
thf(fact_925_confluent__rts_Ointro,axiom,
! [Resid: a > a > a] :
( ( rts_a @ Resid )
=> ( ( conflu3014480972103220363ioms_a @ Resid )
=> ( confluent_rts_a @ Resid ) ) ) ).
% confluent_rts.intro
thf(fact_926_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_927_rts_Ojoin__of__arr__src_I1_J,axiom,
! [Resid: a > a > a,T: a,A: a] :
( ( rts_a @ Resid )
=> ( ( arr_a @ Resid @ T )
=> ( ( member_a @ A @ ( sources_a @ Resid @ T ) )
=> ( join_of_a @ Resid @ A @ T @ T ) ) ) ) ).
% rts.join_of_arr_src(1)
thf(fact_928_rts_Ojoin__of__arr__src_I2_J,axiom,
! [Resid: a > a > a,T: a,A: a] :
( ( rts_a @ Resid )
=> ( ( arr_a @ Resid @ T )
=> ( ( member_a @ A @ ( sources_a @ Resid @ T ) )
=> ( join_of_a @ Resid @ T @ A @ T ) ) ) ) ).
% rts.join_of_arr_src(2)
thf(fact_929_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_930_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_931_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_932_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_933_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_934_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_935_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_936_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_937_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_938_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_939_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_940_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_941_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_942_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_943_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_944_rts__with__composites_Ointro,axiom,
! [Resid: a > a > a] :
( ( rts_a @ Resid )
=> ( ( rts_wi2614412583573296275ioms_a @ Resid )
=> ( rts_wi3777564303360811894ites_a @ Resid ) ) ) ).
% rts_with_composites.intro
thf(fact_945_rts__with__composites__def,axiom,
( rts_wi3777564303360811894ites_a
= ( ^ [Resid2: a > a > a] :
( ( rts_a @ Resid2 )
& ( rts_wi2614412583573296275ioms_a @ Resid2 ) ) ) ) ).
% rts_with_composites_def
thf(fact_946_identity__simulation__def,axiom,
identi4709066280192368860tion_a = rts_a ).
% identity_simulation_def
thf(fact_947_identity__simulation_Oaxioms,axiom,
! [Resid: a > a > a] :
( ( identi4709066280192368860tion_a @ Resid )
=> ( rts_a @ Resid ) ) ).
% identity_simulation.axioms
thf(fact_948_identity__simulation_Ointro,axiom,
! [Resid: a > a > a] :
( ( rts_a @ Resid )
=> ( identi4709066280192368860tion_a @ Resid ) ) ).
% identity_simulation.intro
thf(fact_949_rts__with__composites_Oaxioms_I1_J,axiom,
! [Resid: a > a > a] :
( ( rts_wi3777564303360811894ites_a @ Resid )
=> ( rts_a @ Resid ) ) ).
% rts_with_composites.axioms(1)
thf(fact_950_rts__with__composites_Odiamond__commutes__upto__cong,axiom,
! [Resid: a > a > a,T: a,U: a,V: a,V2: a] :
( ( rts_wi3777564303360811894ites_a @ Resid )
=> ( ( composite_of_a @ Resid @ T @ ( Resid @ U @ T ) @ V )
=> ( ( composite_of_a @ Resid @ U @ ( Resid @ T @ U ) @ V2 )
=> ( ( ide_a @ Resid @ ( Resid @ V @ V2 ) )
& ( ide_a @ Resid @ ( Resid @ V2 @ V ) ) ) ) ) ) ).
% rts_with_composites.diamond_commutes_upto_cong
thf(fact_951_rts__with__composites_Oobtains__composite__of,axiom,
! [Resid: a > a > a,G: a,F3: a] :
( ( rts_wi3777564303360811894ites_a @ Resid )
=> ( ( seq_a @ Resid @ G @ F3 )
=> ~ ! [H: a] :
~ ( composite_of_a @ Resid @ G @ F3 @ H ) ) ) ).
% rts_with_composites.obtains_composite_of
thf(fact_952_rts__with__composites_Ohas__composites,axiom,
! [Resid: a > a > a,T: a,U: a] :
( ( rts_wi3777564303360811894ites_a @ Resid )
=> ( ( seq_a @ Resid @ T @ U )
=> ( composable_a @ Resid @ T @ U ) ) ) ).
% rts_with_composites.has_composites
thf(fact_953_rts__with__composites_Ocomposable__iff__seq,axiom,
! [Resid: a > a > a,G: a,F3: a] :
( ( rts_wi3777564303360811894ites_a @ Resid )
=> ( ( composable_a @ Resid @ G @ F3 )
= ( seq_a @ Resid @ G @ F3 ) ) ) ).
% rts_with_composites.composable_iff_seq
thf(fact_954_R_Otargets__def,axiom,
! [T: a] :
( ( targets_a @ resid @ T )
= ( collect_a
@ ^ [B5: a] :
( ( ide_a @ resid @ B5 )
& ( con_a @ resid @ ( trg_a @ resid @ T ) @ B5 ) ) ) ) ).
% R.targets_def
thf(fact_955_extensional__rts__with__joins_Otrg__join_092_060_094sub_062E_092_060_094sub_062J,axiom,
! [Resid: a > a > a,T: a,U: a] :
( ( extens4936603313648314301oins_a @ Resid )
=> ( ( con_a @ Resid @ T @ U )
=> ( ( trg_a @ Resid @ ( extensional_join_a @ Resid @ T @ U ) )
= ( trg_a @ Resid @ ( Resid @ T @ U ) ) ) ) ) ).
% extensional_rts_with_joins.trg_join\<^sub>E\<^sub>J
thf(fact_956_transformation__axioms_Ointro,axiom,
! [A4: a > a > a,Tau: a > set_a,B3: set_a > set_a > set_a,F2: a > set_a,G2: a > set_a] :
( ! [F4: a] :
( ~ ( arr_a @ A4 @ F4 )
=> ( ( Tau @ F4 )
= ( partial_null_set_a @ B3 ) ) )
=> ( ! [F4: a] :
( ( ide_a @ A4 @ F4 )
=> ( ( weakly2061155085811118449_set_a @ B3 @ ( Tau @ F4 ) )
= ( F2 @ ( weakly8512939796511659025_src_a @ A4 @ F4 ) ) ) )
=> ( ! [F4: a] :
( ( ide_a @ A4 @ F4 )
=> ( ( trg_set_a @ B3 @ ( Tau @ F4 ) )
= ( G2 @ ( trg_a @ A4 @ F4 ) ) ) )
=> ( ! [F4: a] :
( ( arr_a @ A4 @ F4 )
=> ( ( B3 @ ( Tau @ ( weakly8512939796511659025_src_a @ A4 @ F4 ) ) @ ( F2 @ F4 ) )
= ( Tau @ ( trg_a @ A4 @ F4 ) ) ) )
=> ( ! [F4: a] :
( ( arr_a @ A4 @ F4 )
=> ( ( B3 @ ( F2 @ F4 ) @ ( Tau @ ( weakly8512939796511659025_src_a @ A4 @ F4 ) ) )
= ( G2 @ F4 ) ) )
=> ( ! [F4: a] :
( ( arr_a @ A4 @ F4 )
=> ( join_of_set_a @ B3 @ ( Tau @ ( weakly8512939796511659025_src_a @ A4 @ F4 ) ) @ ( F2 @ F4 ) @ ( Tau @ F4 ) ) )
=> ( transf1718796647109808904_set_a @ A4 @ B3 @ F2 @ G2 @ Tau ) ) ) ) ) ) ) ).
% transformation_axioms.intro
thf(fact_957_transformation__axioms_Ointro,axiom,
! [A4: a > a > a,Tau: a > a,B3: a > a > a,F2: a > a,G2: a > a] :
( ! [F4: a] :
( ~ ( arr_a @ A4 @ F4 )
=> ( ( Tau @ F4 )
= ( partial_null_a @ B3 ) ) )
=> ( ! [F4: a] :
( ( ide_a @ A4 @ F4 )
=> ( ( weakly8512939796511659025_src_a @ B3 @ ( Tau @ F4 ) )
= ( F2 @ ( weakly8512939796511659025_src_a @ A4 @ F4 ) ) ) )
=> ( ! [F4: a] :
( ( ide_a @ A4 @ F4 )
=> ( ( trg_a @ B3 @ ( Tau @ F4 ) )
= ( G2 @ ( trg_a @ A4 @ F4 ) ) ) )
=> ( ! [F4: a] :
( ( arr_a @ A4 @ F4 )
=> ( ( B3 @ ( Tau @ ( weakly8512939796511659025_src_a @ A4 @ F4 ) ) @ ( F2 @ F4 ) )
= ( Tau @ ( trg_a @ A4 @ F4 ) ) ) )
=> ( ! [F4: a] :
( ( arr_a @ A4 @ F4 )
=> ( ( B3 @ ( F2 @ F4 ) @ ( Tau @ ( weakly8512939796511659025_src_a @ A4 @ F4 ) ) )
= ( G2 @ F4 ) ) )
=> ( ! [F4: a] :
( ( arr_a @ A4 @ F4 )
=> ( join_of_a @ B3 @ ( Tau @ ( weakly8512939796511659025_src_a @ A4 @ F4 ) ) @ ( F2 @ F4 ) @ ( Tau @ F4 ) ) )
=> ( transf4446446367311712680ms_a_a @ A4 @ B3 @ F2 @ G2 @ Tau ) ) ) ) ) ) ) ).
% transformation_axioms.intro
thf(fact_958_transformation__axioms__def,axiom,
( transf1718796647109808904_set_a
= ( ^ [A7: a > a > a,B4: set_a > set_a > set_a,F: a > set_a,G3: a > set_a,Tau2: a > set_a] :
( ! [F5: a] :
( ~ ( arr_a @ A7 @ F5 )
=> ( ( Tau2 @ F5 )
= ( partial_null_set_a @ B4 ) ) )
& ! [F5: a] :
( ( ide_a @ A7 @ F5 )
=> ( ( weakly2061155085811118449_set_a @ B4 @ ( Tau2 @ F5 ) )
= ( F @ ( weakly8512939796511659025_src_a @ A7 @ F5 ) ) ) )
& ! [F5: a] :
( ( ide_a @ A7 @ F5 )
=> ( ( trg_set_a @ B4 @ ( Tau2 @ F5 ) )
= ( G3 @ ( trg_a @ A7 @ F5 ) ) ) )
& ! [F5: a] :
( ( arr_a @ A7 @ F5 )
=> ( ( B4 @ ( Tau2 @ ( weakly8512939796511659025_src_a @ A7 @ F5 ) ) @ ( F @ F5 ) )
= ( Tau2 @ ( trg_a @ A7 @ F5 ) ) ) )
& ! [F5: a] :
( ( arr_a @ A7 @ F5 )
=> ( ( B4 @ ( F @ F5 ) @ ( Tau2 @ ( weakly8512939796511659025_src_a @ A7 @ F5 ) ) )
= ( G3 @ F5 ) ) )
& ! [F5: a] :
( ( arr_a @ A7 @ F5 )
=> ( join_of_set_a @ B4 @ ( Tau2 @ ( weakly8512939796511659025_src_a @ A7 @ F5 ) ) @ ( F @ F5 ) @ ( Tau2 @ F5 ) ) ) ) ) ) ).
% transformation_axioms_def
thf(fact_959_transformation__axioms__def,axiom,
( transf4446446367311712680ms_a_a
= ( ^ [A7: a > a > a,B4: a > a > a,F: a > a,G3: a > a,Tau2: a > a] :
( ! [F5: a] :
( ~ ( arr_a @ A7 @ F5 )
=> ( ( Tau2 @ F5 )
= ( partial_null_a @ B4 ) ) )
& ! [F5: a] :
( ( ide_a @ A7 @ F5 )
=> ( ( weakly8512939796511659025_src_a @ B4 @ ( Tau2 @ F5 ) )
= ( F @ ( weakly8512939796511659025_src_a @ A7 @ F5 ) ) ) )
& ! [F5: a] :
( ( ide_a @ A7 @ F5 )
=> ( ( trg_a @ B4 @ ( Tau2 @ F5 ) )
= ( G3 @ ( trg_a @ A7 @ F5 ) ) ) )
& ! [F5: a] :
( ( arr_a @ A7 @ F5 )
=> ( ( B4 @ ( Tau2 @ ( weakly8512939796511659025_src_a @ A7 @ F5 ) ) @ ( F @ F5 ) )
= ( Tau2 @ ( trg_a @ A7 @ F5 ) ) ) )
& ! [F5: a] :
( ( arr_a @ A7 @ F5 )
=> ( ( B4 @ ( F @ F5 ) @ ( Tau2 @ ( weakly8512939796511659025_src_a @ A7 @ F5 ) ) )
= ( G3 @ F5 ) ) )
& ! [F5: a] :
( ( arr_a @ A7 @ F5 )
=> ( join_of_a @ B4 @ ( Tau2 @ ( weakly8512939796511659025_src_a @ A7 @ F5 ) ) @ ( F @ F5 ) @ ( Tau2 @ F5 ) ) ) ) ) ) ).
% transformation_axioms_def
thf(fact_960_R_Osources__def,axiom,
! [T: a] :
( ( sources_a @ resid @ T )
= ( collect_a
@ ^ [A5: a] :
( ( ide_a @ resid @ A5 )
& ( con_a @ resid @ T @ A5 ) ) ) ) ).
% R.sources_def
thf(fact_961_R_Otargets__char,axiom,
! [T: a] :
( ( targets_a @ resid @ T )
= ( collect_a
@ ^ [B5: a] :
( ( arr_a @ resid @ T )
& ( ide_a @ resid @ ( resid @ ( resid @ T @ T ) @ B5 ) )
& ( ide_a @ resid @ ( resid @ B5 @ ( resid @ T @ T ) ) ) ) ) ) ).
% R.targets_char
thf(fact_962_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_963_normal__sub__rts_OCong__class__def,axiom,
! [Resid: a > a > a,NN: set_a,T: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( normal7408713899360725774lass_a @ Resid @ NN @ T )
= ( collect_a @ ( normal_sub_Cong_a @ Resid @ NN @ T ) ) ) ) ).
% normal_sub_rts.Cong_class_def
thf(fact_964_empty__def,axiom,
( bot_bot_set_a
= ( collect_a
@ ^ [X2: a] : $false ) ) ).
% empty_def
thf(fact_965_extensional__rts__with__joins_Osrc__join_092_060_094sub_062E_092_060_094sub_062J,axiom,
! [Resid: a > a > a,T: a,U: a] :
( ( extens4936603313648314301oins_a @ Resid )
=> ( ( con_a @ Resid @ T @ U )
=> ( ( weakly8512939796511659025_src_a @ Resid @ ( extensional_join_a @ Resid @ T @ U ) )
= ( weakly8512939796511659025_src_a @ Resid @ T ) ) ) ) ).
% extensional_rts_with_joins.src_join\<^sub>E\<^sub>J
thf(fact_966_rts_Osources__def,axiom,
! [Resid: a > a > a,T: a] :
( ( rts_a @ Resid )
=> ( ( sources_a @ Resid @ T )
= ( collect_a
@ ^ [A5: a] :
( ( ide_a @ Resid @ A5 )
& ( con_a @ Resid @ T @ A5 ) ) ) ) ) ).
% rts.sources_def
thf(fact_967_rts_Otargets__char,axiom,
! [Resid: a > a > a,T: a] :
( ( rts_a @ Resid )
=> ( ( targets_a @ Resid @ T )
= ( collect_a
@ ^ [B5: a] :
( ( arr_a @ Resid @ T )
& ( ide_a @ Resid @ ( Resid @ ( Resid @ T @ T ) @ B5 ) )
& ( ide_a @ Resid @ ( Resid @ B5 @ ( Resid @ T @ T ) ) ) ) ) ) ) ).
% rts.targets_char
thf(fact_968_rts_Otargets__def,axiom,
! [Resid: a > a > a,T: a] :
( ( rts_a @ Resid )
=> ( ( targets_a @ Resid @ T )
= ( collect_a
@ ^ [B5: a] :
( ( ide_a @ Resid @ B5 )
& ( con_a @ Resid @ ( trg_a @ Resid @ T ) @ B5 ) ) ) ) ) ).
% rts.targets_def
thf(fact_969_extensional__rts__with__joins_Ojoin__is__lub,axiom,
! [Resid: a > a > a,T: a,V: a,U: a] :
( ( extens4936603313648314301oins_a @ Resid )
=> ( ( ide_a @ Resid @ ( Resid @ T @ V ) )
=> ( ( ide_a @ Resid @ ( Resid @ U @ V ) )
=> ( ide_a @ Resid @ ( Resid @ ( extensional_join_a @ Resid @ T @ U ) @ V ) ) ) ) ) ).
% extensional_rts_with_joins.join_is_lub
thf(fact_970_extensional__rts__with__joins_Oresid__join_092_060_094sub_062E_092_060_094sub_062J_I2_J,axiom,
! [Resid: a > a > a,T: a,U: a,V: a] :
( ( extens4936603313648314301oins_a @ Resid )
=> ( ( con_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_with_joins.resid_join\<^sub>E\<^sub>J(2)
thf(fact_971_extensional__rts__with__joins_Oresid__join_092_060_094sub_062E_092_060_094sub_062J_I1_J,axiom,
! [Resid: a > a > a,T: a,U: a,V: a] :
( ( extens4936603313648314301oins_a @ Resid )
=> ( ( con_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_with_joins.resid_join\<^sub>E\<^sub>J(1)
thf(fact_972_N_OCong__class__rep__def,axiom,
( normal322541020860755160od_a_a
= ( ^ [T10: set_Product_prod_a_a] :
( fChoic4124218645493772411od_a_a
@ ^ [T5: product_prod_a_a] : ( member1426531477525435216od_a_a @ T5 @ T10 ) ) ) ) ).
% N.Cong_class_rep_def
thf(fact_973_N_OCong__class__rep__def,axiom,
( normal3259722184653208495_rep_a
= ( ^ [T10: set_a] :
( fChoice_a
@ ^ [T5: a] : ( member_a @ T5 @ T10 ) ) ) ) ).
% N.Cong_class_rep_def
thf(fact_974_transformation_Onaturality3,axiom,
! [A4: a > a > a,B3: a > a > a,F2: a > a,G2: a > a,Tau: a > a,F3: a] :
( ( transformation_a_a @ A4 @ B3 @ F2 @ G2 @ Tau )
=> ( ( arr_a @ A4 @ F3 )
=> ( join_of_a @ B3 @ ( Tau @ ( weakly8512939796511659025_src_a @ A4 @ F3 ) ) @ ( F2 @ F3 ) @ ( Tau @ F3 ) ) ) ) ).
% transformation.naturality3
thf(fact_975_normal__sub__rts_OCong__class__rep__def,axiom,
! [Resid: a > a > a,NN: set_a,T7: set_Product_prod_a_a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( normal322541020860755160od_a_a @ T7 )
= ( fChoic4124218645493772411od_a_a
@ ^ [T5: product_prod_a_a] : ( member1426531477525435216od_a_a @ T5 @ T7 ) ) ) ) ).
% normal_sub_rts.Cong_class_rep_def
thf(fact_976_normal__sub__rts_OCong__class__rep__def,axiom,
! [Resid: a > a > a,NN: set_a,T7: set_a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( normal3259722184653208495_rep_a @ T7 )
= ( fChoice_a
@ ^ [T5: a] : ( member_a @ T5 @ T7 ) ) ) ) ).
% normal_sub_rts.Cong_class_rep_def
thf(fact_977_transformation_Opreserves__trg,axiom,
! [A4: a > a > a,B3: a > a > a,F2: a > a,G2: a > a,Tau: a > a,F3: a] :
( ( transformation_a_a @ A4 @ B3 @ F2 @ G2 @ Tau )
=> ( ( ide_a @ A4 @ F3 )
=> ( ( trg_a @ B3 @ ( Tau @ F3 ) )
= ( G2 @ ( trg_a @ A4 @ F3 ) ) ) ) ) ).
% transformation.preserves_trg
thf(fact_978_transformation_Oextensional,axiom,
! [A4: a > a > a,B3: set_a > set_a > set_a,F2: a > set_a,G2: a > set_a,Tau: a > set_a,F3: a] :
( ( transf4130011524320463589_set_a @ A4 @ B3 @ F2 @ G2 @ Tau )
=> ( ~ ( arr_a @ A4 @ F3 )
=> ( ( Tau @ F3 )
= ( partial_null_set_a @ B3 ) ) ) ) ).
% transformation.extensional
thf(fact_979_transformation_Oextensional,axiom,
! [A4: a > a > a,B3: a > a > a,F2: a > a,G2: a > a,Tau: a > a,F3: a] :
( ( transformation_a_a @ A4 @ B3 @ F2 @ G2 @ Tau )
=> ( ~ ( arr_a @ A4 @ F3 )
=> ( ( Tau @ F3 )
= ( partial_null_a @ B3 ) ) ) ) ).
% transformation.extensional
thf(fact_980_some__in__eq,axiom,
! [A4: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a
@ ( fChoic4124218645493772411od_a_a
@ ^ [X2: product_prod_a_a] : ( member1426531477525435216od_a_a @ X2 @ A4 ) )
@ A4 )
= ( A4 != bot_bo3357376287454694259od_a_a ) ) ).
% some_in_eq
thf(fact_981_some__in__eq,axiom,
! [A4: set_a] :
( ( member_a
@ ( fChoice_a
@ ^ [X2: a] : ( member_a @ X2 @ A4 ) )
@ A4 )
= ( A4 != bot_bot_set_a ) ) ).
% some_in_eq
thf(fact_982_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_983_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_984_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_985_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_986_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_987_weakly__extensional__rts_Ocon__ide__are__eq,axiom,
! [Resid: a > a > a,A: a,A2: a] :
( ( weakly1626779504270821493_rts_a @ Resid )
=> ( ( ide_a @ Resid @ A )
=> ( ( ide_a @ Resid @ A2 )
=> ( ( con_a @ Resid @ A @ A2 )
=> ( A = A2 ) ) ) ) ) ).
% weakly_extensional_rts.con_ide_are_eq
thf(fact_988_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_989_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_990_weakly__extensional__rts_Otrg__ide,axiom,
! [Resid: a > a > a,A: a] :
( ( weakly1626779504270821493_rts_a @ Resid )
=> ( ( ide_a @ Resid @ A )
=> ( ( trg_a @ Resid @ A )
= A ) ) ) ).
% weakly_extensional_rts.trg_ide
thf(fact_991_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_992_weakly__extensional__rts_Osrc__ide,axiom,
! [Resid: a > a > a,A: a] :
( ( weakly1626779504270821493_rts_a @ Resid )
=> ( ( ide_a @ Resid @ A )
=> ( ( weakly8512939796511659025_src_a @ Resid @ A )
= A ) ) ) ).
% weakly_extensional_rts.src_ide
thf(fact_993_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_994_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_995_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_996_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_997_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_998_weakly__extensional__rts_Oresid__ide_I1_J,axiom,
! [Resid: a > a > a,A: a,T: a] :
( ( weakly1626779504270821493_rts_a @ Resid )
=> ( ( ide_a @ Resid @ A )
=> ( ( coinitial_a @ Resid @ A @ T )
=> ( ( Resid @ T @ A )
= T ) ) ) ) ).
% weakly_extensional_rts.resid_ide(1)
thf(fact_999_weakly__extensional__rts_Ocoinitial__ide__are__eq,axiom,
! [Resid: a > a > a,A: a,A2: a] :
( ( weakly1626779504270821493_rts_a @ Resid )
=> ( ( ide_a @ Resid @ A )
=> ( ( ide_a @ Resid @ A2 )
=> ( ( coinitial_a @ Resid @ A @ A2 )
=> ( A = A2 ) ) ) ) ) ).
% weakly_extensional_rts.coinitial_ide_are_eq
thf(fact_1000_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_1001_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_1002_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_1003_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_1004_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_1005_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_1006_simulation__between__extensional__rts_Opreserves__trg,axiom,
! [A4: a > a > a,B3: a > a > a,F2: a > a,T: a] :
( ( simula722159513454644908ts_a_a @ A4 @ B3 @ F2 )
=> ( ( trg_a @ B3 @ ( F2 @ T ) )
= ( F2 @ ( trg_a @ A4 @ T ) ) ) ) ).
% simulation_between_extensional_rts.preserves_trg
thf(fact_1007_weakly__extensional__rts_Osrc__eqI,axiom,
! [Resid: a > a > a,A: a,T: a] :
( ( weakly1626779504270821493_rts_a @ Resid )
=> ( ( ide_a @ Resid @ A )
=> ( ( con_a @ Resid @ A @ T )
=> ( ( weakly8512939796511659025_src_a @ Resid @ T )
= A ) ) ) ) ).
% weakly_extensional_rts.src_eqI
thf(fact_1008_weakly__extensional__rts_Oide__iff__trg__self,axiom,
! [Resid: a > a > a,A: a] :
( ( weakly1626779504270821493_rts_a @ Resid )
=> ( ( arr_a @ Resid @ A )
=> ( ( ide_a @ Resid @ A )
= ( ( trg_a @ Resid @ A )
= A ) ) ) ) ).
% weakly_extensional_rts.ide_iff_trg_self
thf(fact_1009_weakly__extensional__rts_Oide__iff__src__self,axiom,
! [Resid: a > a > a,A: a] :
( ( weakly1626779504270821493_rts_a @ Resid )
=> ( ( arr_a @ Resid @ A )
=> ( ( ide_a @ Resid @ A )
= ( ( weakly8512939796511659025_src_a @ Resid @ A )
= A ) ) ) ) ).
% weakly_extensional_rts.ide_iff_src_self
thf(fact_1010_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_1011_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_1012_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_1013_weakly__extensional__rts_Oresid__ide_I2_J,axiom,
! [Resid: a > a > a,A: a,T: a] :
( ( weakly1626779504270821493_rts_a @ Resid )
=> ( ( ide_a @ Resid @ A )
=> ( ( coinitial_a @ Resid @ A @ T )
=> ( ( Resid @ A @ T )
= ( trg_a @ Resid @ T ) ) ) ) ) ).
% weakly_extensional_rts.resid_ide(2)
thf(fact_1014_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_1015_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_1016_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_1017_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_1018_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_1019_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_1020_weakly__extensional__rts_Osources__char,axiom,
! [Resid: a > a > a,T: a] :
( ( weakly1626779504270821493_rts_a @ Resid )
=> ( ( sources_a @ Resid @ T )
= ( collect_a
@ ^ [A5: a] :
( ( arr_a @ Resid @ T )
& ( ( weakly8512939796511659025_src_a @ Resid @ T )
= A5 ) ) ) ) ) ).
% weakly_extensional_rts.sources_char
thf(fact_1021_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
@ ^ [B5: a] :
( ( arr_a @ Resid @ T )
& ( ( trg_a @ Resid @ T )
= B5 ) ) ) ) ) ).
% weakly_extensional_rts.targets_char\<^sub>W\<^sub>E
thf(fact_1022_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_1023_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
@ ^ [A5: set_a] : ( member_set_a @ A5 @ ( 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_1024_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
@ ^ [A5: a] : ( member_a @ A5 @ ( sources_a @ Resid @ T ) ) ) ) )
& ( ~ ( arr_a @ Resid @ T )
=> ( ( weakly8512939796511659025_src_a @ Resid @ T )
= ( partial_null_a @ Resid ) ) ) ) ) ).
% weakly_extensional_rts.src_def
thf(fact_1025_weakly__extensional__rts__def,axiom,
( weakly1626779504270821493_rts_a
= ( ^ [Resid2: a > a > a] :
( ( rts_a @ Resid2 )
& ( weakly311909585050745746ioms_a @ Resid2 ) ) ) ) ).
% weakly_extensional_rts_def
thf(fact_1026_weakly__extensional__rts_Ointro,axiom,
! [Resid: a > a > a] :
( ( rts_a @ Resid )
=> ( ( weakly311909585050745746ioms_a @ Resid )
=> ( weakly1626779504270821493_rts_a @ Resid ) ) ) ).
% weakly_extensional_rts.intro
thf(fact_1027_weakly__extensional__rts__axioms__def,axiom,
( weakly311909585050745746ioms_a
= ( ^ [Resid2: a > a > a] :
! [T5: a,U3: a] :
( ( ( ide_a @ Resid2 @ ( Resid2 @ T5 @ U3 ) )
& ( ide_a @ Resid2 @ ( Resid2 @ U3 @ T5 ) ) )
=> ( ( ide_a @ Resid2 @ T5 )
=> ( ( ide_a @ Resid2 @ U3 )
=> ( T5 = U3 ) ) ) ) ) ) ).
% weakly_extensional_rts_axioms_def
thf(fact_1028_weakly__extensional__rts__axioms_Ointro,axiom,
! [Resid: a > a > a] :
( ! [T3: a,U5: a] :
( ( ( ide_a @ Resid @ ( Resid @ T3 @ U5 ) )
& ( ide_a @ Resid @ ( Resid @ U5 @ T3 ) ) )
=> ( ( ide_a @ Resid @ T3 )
=> ( ( ide_a @ Resid @ U5 )
=> ( T3 = U5 ) ) ) )
=> ( weakly311909585050745746ioms_a @ Resid ) ) ).
% weakly_extensional_rts_axioms.intro
thf(fact_1029_null__def,axiom,
( ( partial_null_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) )
= ( the_set_a
@ ^ [N2: set_a] :
! [T5: set_a] :
( ( ( quotie8165075472272353145esid_a @ resid @ nn @ N2 @ T5 )
= N2 )
& ( ( quotie8165075472272353145esid_a @ resid @ nn @ T5 @ N2 )
= N2 ) ) ) ) ).
% null_def
thf(fact_1030_R_Onull__def,axiom,
( ( partial_null_a @ resid )
= ( the_a
@ ^ [N2: a] :
! [T5: a] :
( ( ( resid @ N2 @ T5 )
= N2 )
& ( ( resid @ T5 @ N2 )
= N2 ) ) ) ) ).
% R.null_def
thf(fact_1031_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_1032_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_1033_extensional__rts_Oaxioms_I1_J,axiom,
! [Resid: a > a > a] :
( ( extensional_rts_a @ Resid )
=> ( rts_a @ Resid ) ) ).
% extensional_rts.axioms(1)
thf(fact_1034_extensional__rts_Ojoin__of__unique,axiom,
! [Resid: a > a > a,T: a,U: a,V: a,V2: a] :
( ( extensional_rts_a @ Resid )
=> ( ( join_of_a @ Resid @ T @ U @ V )
=> ( ( join_of_a @ Resid @ T @ U @ V2 )
=> ( V = V2 ) ) ) ) ).
% extensional_rts.join_of_unique
thf(fact_1035_extensional__rts_Ocomposite__of__unique,axiom,
! [Resid: a > a > a,T: a,U: a,V: a,V2: a] :
( ( extensional_rts_a @ Resid )
=> ( ( composite_of_a @ Resid @ T @ U @ V )
=> ( ( composite_of_a @ Resid @ T @ U @ V2 )
=> ( V = V2 ) ) ) ) ).
% extensional_rts.composite_of_unique
thf(fact_1036_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_1037_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_1038_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_1039_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_1040_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_1041_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_1042_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_1043_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_1044_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_1045_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_1046_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_1047_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_1048_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_1049_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_1050_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_1051_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_1052_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_1053_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] :
! [T5: set_a] :
( ( ( OP2 @ N2 @ T5 )
= N2 )
& ( ( OP2 @ T5 @ N2 )
= N2 ) ) ) ) ) ).
% partial_magma.null_def
thf(fact_1054_partial__magma_Onull__def,axiom,
! [OP2: a > a > a] :
( ( partial_magma_a @ OP2 )
=> ( ( partial_null_a @ OP2 )
= ( the_a
@ ^ [N2: a] :
! [T5: a] :
( ( ( OP2 @ N2 @ T5 )
= N2 )
& ( ( OP2 @ T5 @ N2 )
= N2 ) ) ) ) ) ).
% partial_magma.null_def
thf(fact_1055_normal__in__extensional__rts__with__composites_Ointro,axiom,
! [Resid: a > a > a,NN: set_a] :
( ( extensional_rts_a @ Resid )
=> ( ( rts_wi3777564303360811894ites_a @ Resid )
=> ( ( normal_sub_rts_a @ Resid @ NN )
=> ( normal636964748050715740ites_a @ Resid @ NN ) ) ) ) ).
% normal_in_extensional_rts_with_composites.intro
thf(fact_1056_normal__in__extensional__rts__with__composites__def,axiom,
( normal636964748050715740ites_a
= ( ^ [Resid2: a > a > a,NN2: set_a] :
( ( extensional_rts_a @ Resid2 )
& ( rts_wi3777564303360811894ites_a @ Resid2 )
& ( normal_sub_rts_a @ Resid2 @ NN2 ) ) ) ) ).
% normal_in_extensional_rts_with_composites_def
thf(fact_1057_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_1058_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_1059_normal__in__extensional__rts__with__composites_Ofactor__closed_092_060_094sub_062E_092_060_094sub_062C_I2_J,axiom,
! [Resid: a > a > a,NN: set_a,T: a,U: a] :
( ( normal636964748050715740ites_a @ Resid @ NN )
=> ( ( member_a @ ( extensional_comp_a @ Resid @ T @ U ) @ NN )
=> ( member_a @ U @ NN ) ) ) ).
% normal_in_extensional_rts_with_composites.factor_closed\<^sub>E\<^sub>C(2)
thf(fact_1060_normal__in__extensional__rts__with__composites_Ofactor__closed_092_060_094sub_062E_092_060_094sub_062C_I1_J,axiom,
! [Resid: a > a > a,NN: set_a,T: a,U: a] :
( ( normal636964748050715740ites_a @ Resid @ NN )
=> ( ( member_a @ ( extensional_comp_a @ Resid @ T @ U ) @ NN )
=> ( member_a @ T @ NN ) ) ) ).
% normal_in_extensional_rts_with_composites.factor_closed\<^sub>E\<^sub>C(1)
thf(fact_1061_normal__in__extensional__rts__with__composites_Ocomp__in__normal__iff,axiom,
! [Resid: a > a > a,NN: set_a,T: a,U: a] :
( ( normal636964748050715740ites_a @ Resid @ NN )
=> ( ( member_a @ ( extensional_comp_a @ Resid @ T @ U ) @ NN )
= ( ( member_a @ T @ NN )
& ( member_a @ U @ NN )
& ( seq_a @ Resid @ T @ U ) ) ) ) ).
% normal_in_extensional_rts_with_composites.comp_in_normal_iff
thf(fact_1062_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_1063_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_1064_extensional__rts_Ocomp__ide__self,axiom,
! [Resid: a > a > a,A: a] :
( ( extensional_rts_a @ Resid )
=> ( ( ide_a @ Resid @ A )
=> ( ( extensional_comp_a @ Resid @ A @ A )
= A ) ) ) ).
% extensional_rts.comp_ide_self
thf(fact_1065_extensional__rts_Oresid__comp_I1_J,axiom,
! [Resid: a > a > a,T: a,U: a,W: a] :
( ( extensional_rts_a @ Resid )
=> ( ( con_a @ Resid @ ( extensional_comp_a @ Resid @ T @ U ) @ W )
=> ( ( Resid @ W @ ( extensional_comp_a @ Resid @ T @ U ) )
= ( Resid @ ( Resid @ W @ T ) @ U ) ) ) ) ).
% extensional_rts.resid_comp(1)
thf(fact_1066_extensional__rts_Oresid__comp_I2_J,axiom,
! [Resid: a > a > a,T: a,U: a,W: a] :
( ( extensional_rts_a @ Resid )
=> ( ( con_a @ Resid @ ( extensional_comp_a @ Resid @ T @ U ) @ W )
=> ( ( Resid @ ( extensional_comp_a @ Resid @ T @ U ) @ W )
= ( extensional_comp_a @ Resid @ ( Resid @ T @ W ) @ ( Resid @ U @ ( Resid @ W @ T ) ) ) ) ) ) ).
% extensional_rts.resid_comp(2)
thf(fact_1067_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_1068_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_1069_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_1070_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_1071_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_1072_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_1073_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_1074_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_1075_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_1076_normal__in__extensional__rts__with__composites_Oaxioms_I3_J,axiom,
! [Resid: a > a > a,NN: set_a] :
( ( normal636964748050715740ites_a @ Resid @ NN )
=> ( normal_sub_rts_a @ Resid @ NN ) ) ).
% normal_in_extensional_rts_with_composites.axioms(3)
thf(fact_1077_extensional__rts_Obounded__imp__con_092_060_094sub_062E,axiom,
! [Resid: a > a > a,T: a,U: a,T4: a,U2: a] :
( ( extensional_rts_a @ Resid )
=> ( ( ( ide_a @ Resid @ ( Resid @ ( extensional_comp_a @ Resid @ T @ U ) @ ( extensional_comp_a @ Resid @ T4 @ U2 ) ) )
& ( ide_a @ Resid @ ( Resid @ ( extensional_comp_a @ Resid @ T4 @ U2 ) @ ( extensional_comp_a @ Resid @ T @ U ) ) ) )
=> ( con_a @ Resid @ T @ T4 ) ) ) ).
% extensional_rts.bounded_imp_con\<^sub>E
thf(fact_1078_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_1079_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_1080_extensional__rts_Ocomp__src__arr,axiom,
! [Resid: a > a > a,T: a,A: a] :
( ( extensional_rts_a @ Resid )
=> ( ( arr_a @ Resid @ T )
=> ( ( ( weakly8512939796511659025_src_a @ Resid @ T )
= A )
=> ( ( extensional_comp_a @ Resid @ A @ T )
= T ) ) ) ) ).
% extensional_rts.comp_src_arr
thf(fact_1081_extensional__rts_Ocon__compI_I2_J,axiom,
! [Resid: a > a > a,T: a,U: a,W: a] :
( ( extensional_rts_a @ Resid )
=> ( ( composable_a @ Resid @ T @ U )
=> ( ( con_a @ Resid @ ( Resid @ W @ T ) @ U )
=> ( con_a @ Resid @ ( extensional_comp_a @ Resid @ T @ U ) @ W ) ) ) ) ).
% extensional_rts.con_compI(2)
thf(fact_1082_extensional__rts_Ocon__compI_I1_J,axiom,
! [Resid: a > a > a,T: a,U: a,W: a] :
( ( extensional_rts_a @ Resid )
=> ( ( composable_a @ Resid @ T @ U )
=> ( ( con_a @ Resid @ ( Resid @ W @ T ) @ U )
=> ( con_a @ Resid @ W @ ( extensional_comp_a @ Resid @ T @ U ) ) ) ) ) ).
% extensional_rts.con_compI(1)
thf(fact_1083_extensional__rts_Ocon__comp__iff,axiom,
! [Resid: a > a > a,W: a,T: a,U: a] :
( ( extensional_rts_a @ Resid )
=> ( ( con_a @ Resid @ W @ ( extensional_comp_a @ Resid @ T @ U ) )
= ( ( composable_a @ Resid @ T @ U )
& ( con_a @ Resid @ ( Resid @ W @ T ) @ U ) ) ) ) ).
% extensional_rts.con_comp_iff
thf(fact_1084_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_1085_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_1086_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_1087_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_1088_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_1089_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_1090_extensional__rts_Ocomp__join_I2_J,axiom,
! [Resid: a > a > a,T: a,U: a,U2: a] :
( ( extensional_rts_a @ Resid )
=> ( ( joinable_a @ Resid @ ( extensional_comp_a @ Resid @ T @ U ) @ ( extensional_comp_a @ Resid @ T @ U2 ) )
=> ( ( extensional_comp_a @ Resid @ T @ ( extensional_join_a @ Resid @ U @ U2 ) )
= ( extensional_join_a @ Resid @ ( extensional_comp_a @ Resid @ T @ U ) @ ( extensional_comp_a @ Resid @ T @ U2 ) ) ) ) ) ).
% extensional_rts.comp_join(2)
thf(fact_1091_extensional__rts_Ocomp__join_I1_J,axiom,
! [Resid: a > a > a,T: a,U: a,U2: a] :
( ( extensional_rts_a @ Resid )
=> ( ( joinable_a @ Resid @ ( extensional_comp_a @ Resid @ T @ U ) @ ( extensional_comp_a @ Resid @ T @ U2 ) )
=> ( composable_a @ Resid @ T @ ( extensional_join_a @ Resid @ U @ U2 ) ) ) ) ).
% extensional_rts.comp_join(1)
thf(fact_1092_extensional__rts__with__composites_Oarr__compE_092_060_094sub_062E_092_060_094sub_062C,axiom,
! [Resid: a > a > a,T: a,U: a] :
( ( extens4790121754472881640ites_a @ Resid )
=> ( ( arr_a @ Resid @ ( extensional_comp_a @ Resid @ T @ U ) )
=> ~ ( ( arr_a @ Resid @ T )
=> ( ( arr_a @ Resid @ U )
=> ( ( trg_a @ Resid @ T )
!= ( weakly8512939796511659025_src_a @ Resid @ U ) ) ) ) ) ) ).
% extensional_rts_with_composites.arr_compE\<^sub>E\<^sub>C
thf(fact_1093_extensional__rts__with__composites_Oarr__comp_092_060_094sub_062E_092_060_094sub_062C,axiom,
! [Resid: a > a > a,T: a,U: a] :
( ( extens4790121754472881640ites_a @ Resid )
=> ( ( arr_a @ Resid @ T )
=> ( ( arr_a @ Resid @ U )
=> ( ( ( trg_a @ Resid @ T )
= ( weakly8512939796511659025_src_a @ Resid @ U ) )
=> ( arr_a @ Resid @ ( extensional_comp_a @ Resid @ T @ U ) ) ) ) ) ) ).
% extensional_rts_with_composites.arr_comp\<^sub>E\<^sub>C
thf(fact_1094_extensional__rts_Ointro,axiom,
! [Resid: a > a > a] :
( ( rts_a @ Resid )
=> ( ( extens8613361310974791063ioms_a @ Resid )
=> ( extensional_rts_a @ Resid ) ) ) ).
% extensional_rts.intro
thf(fact_1095_extensional__rts__axioms_Ointro,axiom,
! [Resid: a > a > a] :
( ! [T3: a,U5: a] :
( ( ( ide_a @ Resid @ ( Resid @ T3 @ U5 ) )
& ( ide_a @ Resid @ ( Resid @ U5 @ T3 ) ) )
=> ( T3 = U5 ) )
=> ( extens8613361310974791063ioms_a @ Resid ) ) ).
% extensional_rts_axioms.intro
thf(fact_1096_extensional__rts__axioms__def,axiom,
( extens8613361310974791063ioms_a
= ( ^ [Resid2: a > a > a] :
! [T5: a,U3: a] :
( ( ( ide_a @ Resid2 @ ( Resid2 @ T5 @ U3 ) )
& ( ide_a @ Resid2 @ ( Resid2 @ U3 @ T5 ) ) )
=> ( T5 = U3 ) ) ) ) ).
% extensional_rts_axioms_def
thf(fact_1097_extensional__rts__with__composites_Oresid__common__prefix,axiom,
! [Resid: a > a > a,T: a,U: a,V: a] :
( ( extens4790121754472881640ites_a @ Resid )
=> ( ( con_a @ Resid @ ( extensional_comp_a @ Resid @ T @ U ) @ ( extensional_comp_a @ Resid @ T @ V ) )
=> ( ( Resid @ ( extensional_comp_a @ Resid @ T @ U ) @ ( extensional_comp_a @ Resid @ T @ V ) )
= ( Resid @ U @ V ) ) ) ) ).
% extensional_rts_with_composites.resid_common_prefix
thf(fact_1098_extensional__rts__with__composites_Ojoin__expansion_I2_J,axiom,
! [Resid: a > a > a,T: a,U: a] :
( ( extens4790121754472881640ites_a @ Resid )
=> ( ( con_a @ Resid @ T @ U )
=> ( seq_a @ Resid @ T @ ( Resid @ U @ T ) ) ) ) ).
% extensional_rts_with_composites.join_expansion(2)
thf(fact_1099_extensional__rts__with__composites_Oinduced__arrow_I3_J,axiom,
! [Resid: a > a > a,T: a,U: a,T4: a,U2: a,V: a] :
( ( extens4790121754472881640ites_a @ Resid )
=> ( ( seq_a @ Resid @ T @ U )
=> ( ( ( extensional_comp_a @ Resid @ T @ U )
= ( extensional_comp_a @ Resid @ T4 @ U2 ) )
=> ( ( ( extensional_comp_a @ Resid @ ( Resid @ T4 @ T ) @ V )
= U )
=> ( V
= ( Resid @ U @ ( Resid @ T4 @ T ) ) ) ) ) ) ) ).
% extensional_rts_with_composites.induced_arrow(3)
thf(fact_1100_extensional__rts__with__composites_Oinduced__arrow_I2_J,axiom,
! [Resid: a > a > a,T: a,U: a,T4: a,U2: a] :
( ( extens4790121754472881640ites_a @ Resid )
=> ( ( seq_a @ Resid @ T @ U )
=> ( ( ( extensional_comp_a @ Resid @ T @ U )
= ( extensional_comp_a @ Resid @ T4 @ U2 ) )
=> ( ( extensional_comp_a @ Resid @ ( Resid @ T @ T4 ) @ ( Resid @ U @ ( Resid @ T4 @ T ) ) )
= U2 ) ) ) ) ).
% extensional_rts_with_composites.induced_arrow(2)
thf(fact_1101_extensional__rts__with__composites_Oinduced__arrow_I1_J,axiom,
! [Resid: a > a > a,T: a,U: a,T4: a,U2: a] :
( ( extens4790121754472881640ites_a @ Resid )
=> ( ( seq_a @ Resid @ T @ U )
=> ( ( ( extensional_comp_a @ Resid @ T @ U )
= ( extensional_comp_a @ Resid @ T4 @ U2 ) )
=> ( ( extensional_comp_a @ Resid @ ( Resid @ T4 @ T ) @ ( Resid @ U @ ( Resid @ T4 @ T ) ) )
= U ) ) ) ) ).
% extensional_rts_with_composites.induced_arrow(1)
thf(fact_1102_extensional__rts__with__composites_Ojoin__expansion_I1_J,axiom,
! [Resid: a > a > a,T: a,U: a] :
( ( extens4790121754472881640ites_a @ Resid )
=> ( ( con_a @ Resid @ T @ U )
=> ( ( extensional_join_a @ Resid @ T @ U )
= ( extensional_comp_a @ Resid @ T @ ( Resid @ U @ T ) ) ) ) ) ).
% extensional_rts_with_composites.join_expansion(1)
thf(fact_1103_extensional__rts__with__composites_Ojoin3__expansion,axiom,
! [Resid: a > a > a,T: a,U: a,V: a] :
( ( extens4790121754472881640ites_a @ Resid )
=> ( ( con_a @ Resid @ T @ U )
=> ( ( con_a @ Resid @ T @ V )
=> ( ( con_a @ Resid @ U @ V )
=> ( ( extensional_join_a @ Resid @ ( extensional_join_a @ Resid @ T @ U ) @ V )
= ( extensional_comp_a @ Resid @ ( extensional_comp_a @ Resid @ T @ ( Resid @ U @ T ) ) @ ( Resid @ ( Resid @ V @ T ) @ ( Resid @ U @ T ) ) ) ) ) ) ) ) ).
% extensional_rts_with_composites.join3_expansion
thf(fact_1104_extensional__rts__with__composites_Ocon__comp__iff_092_060_094sub_062E_092_060_094sub_062C_I2_J,axiom,
! [Resid: a > a > a,T: a,U: a,W: a] :
( ( extens4790121754472881640ites_a @ Resid )
=> ( ( con_a @ Resid @ ( extensional_comp_a @ Resid @ T @ U ) @ W )
= ( ( seq_a @ Resid @ T @ U )
& ( con_a @ Resid @ U @ ( Resid @ W @ T ) ) ) ) ) ).
% extensional_rts_with_composites.con_comp_iff\<^sub>E\<^sub>C(2)
thf(fact_1105_extensional__rts__with__composites_Ocon__comp__iff_092_060_094sub_062E_092_060_094sub_062C_I1_J,axiom,
! [Resid: a > a > a,W: a,T: a,U: a] :
( ( extens4790121754472881640ites_a @ Resid )
=> ( ( con_a @ Resid @ W @ ( extensional_comp_a @ Resid @ T @ U ) )
= ( ( seq_a @ Resid @ T @ U )
& ( con_a @ Resid @ U @ ( Resid @ W @ T ) ) ) ) ) ).
% extensional_rts_with_composites.con_comp_iff\<^sub>E\<^sub>C(1)
thf(fact_1106_extensional__rts__with__composites_Oseq__implies__arr__comp,axiom,
! [Resid: a > a > a,T: a,U: a] :
( ( extens4790121754472881640ites_a @ Resid )
=> ( ( seq_a @ Resid @ T @ U )
=> ( arr_a @ Resid @ ( extensional_comp_a @ Resid @ T @ U ) ) ) ) ).
% extensional_rts_with_composites.seq_implies_arr_comp
thf(fact_1107_extensional__rts__with__composites_Otrg__comp_092_060_094sub_062E_092_060_094sub_062C,axiom,
! [Resid: a > a > a,T: a,U: a] :
( ( extens4790121754472881640ites_a @ Resid )
=> ( ( seq_a @ Resid @ T @ U )
=> ( ( trg_a @ Resid @ ( extensional_comp_a @ Resid @ T @ U ) )
= ( trg_a @ Resid @ U ) ) ) ) ).
% extensional_rts_with_composites.trg_comp\<^sub>E\<^sub>C
thf(fact_1108_extensional__rts__with__composites_Osrc__comp_092_060_094sub_062E_092_060_094sub_062C,axiom,
! [Resid: a > a > a,T: a,U: a] :
( ( extens4790121754472881640ites_a @ Resid )
=> ( ( seq_a @ Resid @ T @ U )
=> ( ( weakly8512939796511659025_src_a @ Resid @ ( extensional_comp_a @ Resid @ T @ U ) )
= ( weakly8512939796511659025_src_a @ Resid @ T ) ) ) ) ).
% extensional_rts_with_composites.src_comp\<^sub>E\<^sub>C
thf(fact_1109_extensional__rts__def,axiom,
( extensional_rts_a
= ( ^ [Resid2: a > a > a] :
( ( rts_a @ Resid2 )
& ( extens8613361310974791063ioms_a @ Resid2 ) ) ) ) ).
% extensional_rts_def
thf(fact_1110_simulation_Opreserves__joins,axiom,
! [A4: a > a > a,B3: a > a > a,F2: a > a,T: a,U: a,V: a] :
( ( simulation_a_a @ A4 @ B3 @ F2 )
=> ( ( join_of_a @ A4 @ T @ U @ V )
=> ( join_of_a @ B3 @ ( F2 @ T ) @ ( F2 @ U ) @ ( F2 @ V ) ) ) ) ).
% simulation.preserves_joins
thf(fact_1111_simulation_Opreserves__composites,axiom,
! [A4: a > a > a,B3: a > a > a,F2: a > a,T: a,U: a,V: a] :
( ( simulation_a_a @ A4 @ B3 @ F2 )
=> ( ( composite_of_a @ A4 @ T @ U @ V )
=> ( composite_of_a @ B3 @ ( F2 @ T ) @ ( F2 @ U ) @ ( F2 @ V ) ) ) ) ).
% simulation.preserves_composites
thf(fact_1112_simulation_Opreserves__reflects__arr,axiom,
! [A4: a > a > a,B3: a > a > a,F2: a > a,T: a] :
( ( simulation_a_a @ A4 @ B3 @ F2 )
=> ( ( arr_a @ B3 @ ( F2 @ T ) )
= ( arr_a @ A4 @ T ) ) ) ).
% simulation.preserves_reflects_arr
thf(fact_1113_simulation_Opreserves__con,axiom,
! [A4: a > a > a,B3: a > a > a,F2: a > a,T: a,U: a] :
( ( simulation_a_a @ A4 @ B3 @ F2 )
=> ( ( con_a @ A4 @ T @ U )
=> ( con_a @ B3 @ ( F2 @ T ) @ ( F2 @ U ) ) ) ) ).
% simulation.preserves_con
thf(fact_1114_simulation_Opreserves__ide,axiom,
! [A4: a > a > a,B3: a > a > a,F2: a > a,A: a] :
( ( simulation_a_a @ A4 @ B3 @ F2 )
=> ( ( ide_a @ A4 @ A )
=> ( ide_a @ B3 @ ( F2 @ A ) ) ) ) ).
% simulation.preserves_ide
thf(fact_1115_simulation_Opreserves__cong,axiom,
! [A4: a > a > a,B3: a > a > a,F2: a > a,T: a,U: a] :
( ( simulation_a_a @ A4 @ B3 @ F2 )
=> ( ( ( ide_a @ A4 @ ( A4 @ T @ U ) )
& ( ide_a @ A4 @ ( A4 @ U @ T ) ) )
=> ( ( ide_a @ B3 @ ( B3 @ ( F2 @ T ) @ ( F2 @ U ) ) )
& ( ide_a @ B3 @ ( B3 @ ( F2 @ U ) @ ( F2 @ T ) ) ) ) ) ) ).
% simulation.preserves_cong
thf(fact_1116_simulation_Opreserves__prfx,axiom,
! [A4: a > a > a,B3: a > a > a,F2: a > a,T: a,U: a] :
( ( simulation_a_a @ A4 @ B3 @ F2 )
=> ( ( ide_a @ A4 @ ( A4 @ T @ U ) )
=> ( ide_a @ B3 @ ( B3 @ ( F2 @ T ) @ ( F2 @ U ) ) ) ) ) ).
% simulation.preserves_prfx
thf(fact_1117_simulation_Opreserves__trg,axiom,
! [A4: a > a > a,B3: a > a > a,F2: a > a,T: a] :
( ( simulation_a_a @ A4 @ B3 @ F2 )
=> ( ( arr_a @ A4 @ T )
=> ( ( F2 @ ( trg_a @ A4 @ T ) )
= ( trg_a @ B3 @ ( F2 @ T ) ) ) ) ) ).
% simulation.preserves_trg
thf(fact_1118_simulation_Oextensional,axiom,
! [A4: a > a > a,B3: set_a > set_a > set_a,F2: a > set_a,T: a] :
( ( simulation_a_set_a @ A4 @ B3 @ F2 )
=> ( ~ ( arr_a @ A4 @ T )
=> ( ( F2 @ T )
= ( partial_null_set_a @ B3 ) ) ) ) ).
% simulation.extensional
thf(fact_1119_simulation_Oextensional,axiom,
! [A4: a > a > a,B3: a > a > a,F2: a > a,T: a] :
( ( simulation_a_a @ A4 @ B3 @ F2 )
=> ( ~ ( arr_a @ A4 @ T )
=> ( ( F2 @ T )
= ( partial_null_a @ B3 ) ) ) ) ).
% simulation.extensional
thf(fact_1120_simulation__def,axiom,
( simulation_a_a
= ( ^ [A7: a > a > a,B4: a > a > a,F: a > a] :
( ( rts_a @ A7 )
& ( rts_a @ B4 )
& ( simula3868467710248865958ms_a_a @ A7 @ B4 @ F ) ) ) ) ).
% simulation_def
thf(fact_1121_simulation_Ointro,axiom,
! [A4: a > a > a,B3: a > a > a,F2: a > a] :
( ( rts_a @ A4 )
=> ( ( rts_a @ B3 )
=> ( ( simula3868467710248865958ms_a_a @ A4 @ B3 @ F2 )
=> ( simulation_a_a @ A4 @ B3 @ F2 ) ) ) ) ).
% simulation.intro
thf(fact_1122_Resid__def,axiom,
! [T7: set_a,U7: set_a] :
( ( ( ( normal8595587647932138008lass_a @ resid @ nn @ T7 )
& ( normal8595587647932138008lass_a @ resid @ nn @ U7 )
& ? [T2: a] :
( ( member_a @ T2 @ T7 )
& ? [U8: a] :
( ( member_a @ U8 @ U7 )
& ( con_a @ resid @ T2 @ U8 ) ) ) )
=> ( ( quotie8165075472272353145esid_a @ resid @ nn @ T7 @ U7 )
= ( normal7408713899360725774lass_a @ resid @ nn
@ ( resid
@ ( product_fst_a_a
@ ( fChoic4124218645493772411od_a_a
@ ^ [Tu: product_prod_a_a] :
( ( member_a @ ( product_fst_a_a @ Tu ) @ T7 )
& ( member_a @ ( product_snd_a_a @ Tu ) @ U7 )
& ( con_a @ resid @ ( product_fst_a_a @ Tu ) @ ( product_snd_a_a @ Tu ) ) ) ) )
@ ( product_snd_a_a
@ ( fChoic4124218645493772411od_a_a
@ ^ [Tu: product_prod_a_a] :
( ( member_a @ ( product_fst_a_a @ Tu ) @ T7 )
& ( member_a @ ( product_snd_a_a @ Tu ) @ U7 )
& ( con_a @ resid @ ( product_fst_a_a @ Tu ) @ ( product_snd_a_a @ Tu ) ) ) ) ) ) ) ) )
& ( ~ ( ( normal8595587647932138008lass_a @ resid @ nn @ T7 )
& ( normal8595587647932138008lass_a @ resid @ nn @ U7 )
& ? [T3: a] :
( ( member_a @ T3 @ T7 )
& ? [U5: a] :
( ( member_a @ U5 @ U7 )
& ( con_a @ resid @ T3 @ U5 ) ) ) )
=> ( ( quotie8165075472272353145esid_a @ resid @ nn @ T7 @ U7 )
= bot_bot_set_a ) ) ) ).
% Resid_def
thf(fact_1123_quotient__by__coherent__normal_OResid__def,axiom,
! [Resid: a > a > a,NN: set_a,T7: set_a,U7: set_a] :
( ( quotie3282664541148387094rmal_a @ Resid @ NN )
=> ( ( ( ( normal8595587647932138008lass_a @ Resid @ NN @ T7 )
& ( normal8595587647932138008lass_a @ Resid @ NN @ U7 )
& ? [T2: a] :
( ( member_a @ T2 @ T7 )
& ? [U8: a] :
( ( member_a @ U8 @ U7 )
& ( con_a @ Resid @ T2 @ U8 ) ) ) )
=> ( ( quotie8165075472272353145esid_a @ Resid @ NN @ T7 @ U7 )
= ( normal7408713899360725774lass_a @ Resid @ NN
@ ( Resid
@ ( product_fst_a_a
@ ( fChoic4124218645493772411od_a_a
@ ^ [Tu: product_prod_a_a] :
( ( member_a @ ( product_fst_a_a @ Tu ) @ T7 )
& ( member_a @ ( product_snd_a_a @ Tu ) @ U7 )
& ( con_a @ Resid @ ( product_fst_a_a @ Tu ) @ ( product_snd_a_a @ Tu ) ) ) ) )
@ ( product_snd_a_a
@ ( fChoic4124218645493772411od_a_a
@ ^ [Tu: product_prod_a_a] :
( ( member_a @ ( product_fst_a_a @ Tu ) @ T7 )
& ( member_a @ ( product_snd_a_a @ Tu ) @ U7 )
& ( con_a @ Resid @ ( product_fst_a_a @ Tu ) @ ( product_snd_a_a @ Tu ) ) ) ) ) ) ) ) )
& ( ~ ( ( normal8595587647932138008lass_a @ Resid @ NN @ T7 )
& ( normal8595587647932138008lass_a @ Resid @ NN @ U7 )
& ? [T3: a] :
( ( member_a @ T3 @ T7 )
& ? [U5: a] :
( ( member_a @ U5 @ U7 )
& ( con_a @ Resid @ T3 @ U5 ) ) ) )
=> ( ( quotie8165075472272353145esid_a @ Resid @ NN @ T7 @ U7 )
= bot_bot_set_a ) ) ) ) ).
% quotient_by_coherent_normal.Resid_def
thf(fact_1124_is__singletonI_H,axiom,
! [A4: set_a] :
( ( A4 != bot_bot_set_a )
=> ( ! [X: a,Y3: a] :
( ( member_a @ X @ A4 )
=> ( ( member_a @ Y3 @ A4 )
=> ( X = Y3 ) ) )
=> ( is_singleton_a @ A4 ) ) ) ).
% is_singletonI'
thf(fact_1125_is__singletonI,axiom,
! [X6: a] : ( is_singleton_a @ ( insert_a @ X6 @ bot_bot_set_a ) ) ).
% is_singletonI
thf(fact_1126_is__singletonE,axiom,
! [A4: set_a] :
( ( is_singleton_a @ A4 )
=> ~ ! [X: a] :
( A4
!= ( insert_a @ X @ bot_bot_set_a ) ) ) ).
% is_singletonE
thf(fact_1127_singletonI,axiom,
! [A: a] : ( member_a @ A @ ( insert_a @ A @ bot_bot_set_a ) ) ).
% singletonI
thf(fact_1128_singleton__conv,axiom,
! [A: a] :
( ( collect_a
@ ^ [X2: a] : ( X2 = A ) )
= ( insert_a @ A @ bot_bot_set_a ) ) ).
% singleton_conv
thf(fact_1129_singleton__conv2,axiom,
! [A: a] :
( ( collect_a
@ ( ^ [Y5: a,Z3: a] : ( Y5 = Z3 )
@ A ) )
= ( insert_a @ A @ bot_bot_set_a ) ) ).
% singleton_conv2
thf(fact_1130_disjoint__insert_I2_J,axiom,
! [A4: set_a,B: a,B3: set_a] :
( ( bot_bot_set_a
= ( inf_inf_set_a @ A4 @ ( insert_a @ B @ B3 ) ) )
= ( ~ ( member_a @ B @ A4 )
& ( bot_bot_set_a
= ( inf_inf_set_a @ A4 @ B3 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_1131_disjoint__insert_I1_J,axiom,
! [B3: set_a,A: a,A4: set_a] :
( ( ( inf_inf_set_a @ B3 @ ( insert_a @ A @ A4 ) )
= bot_bot_set_a )
= ( ~ ( member_a @ A @ B3 )
& ( ( inf_inf_set_a @ B3 @ A4 )
= bot_bot_set_a ) ) ) ).
% disjoint_insert(1)
thf(fact_1132_insert__disjoint_I2_J,axiom,
! [A: a,A4: set_a,B3: set_a] :
( ( bot_bot_set_a
= ( inf_inf_set_a @ ( insert_a @ A @ A4 ) @ B3 ) )
= ( ~ ( member_a @ A @ B3 )
& ( bot_bot_set_a
= ( inf_inf_set_a @ A4 @ B3 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_1133_insert__disjoint_I1_J,axiom,
! [A: a,A4: set_a,B3: set_a] :
( ( ( inf_inf_set_a @ ( insert_a @ A @ A4 ) @ B3 )
= bot_bot_set_a )
= ( ~ ( member_a @ A @ B3 )
& ( ( inf_inf_set_a @ A4 @ B3 )
= bot_bot_set_a ) ) ) ).
% insert_disjoint(1)
thf(fact_1134_singletonD,axiom,
! [B: a,A: a] :
( ( member_a @ B @ ( insert_a @ A @ bot_bot_set_a ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_1135_singleton__iff,axiom,
! [B: a,A: a] :
( ( member_a @ B @ ( insert_a @ A @ bot_bot_set_a ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_1136_doubleton__eq__iff,axiom,
! [A: a,B: a,C: a,D: a] :
( ( ( insert_a @ A @ ( insert_a @ B @ bot_bot_set_a ) )
= ( insert_a @ C @ ( insert_a @ D @ bot_bot_set_a ) ) )
= ( ( ( A = C )
& ( B = D ) )
| ( ( A = D )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_1137_insert__not__empty,axiom,
! [A: a,A4: set_a] :
( ( insert_a @ A @ A4 )
!= bot_bot_set_a ) ).
% insert_not_empty
thf(fact_1138_singleton__inject,axiom,
! [A: a,B: a] :
( ( ( insert_a @ A @ bot_bot_set_a )
= ( insert_a @ B @ bot_bot_set_a ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_1139_Collect__conv__if,axiom,
! [P: a > $o,A: a] :
( ( ( P @ A )
=> ( ( collect_a
@ ^ [X2: a] :
( ( X2 = A )
& ( P @ X2 ) ) )
= ( insert_a @ A @ bot_bot_set_a ) ) )
& ( ~ ( P @ A )
=> ( ( collect_a
@ ^ [X2: a] :
( ( X2 = A )
& ( P @ X2 ) ) )
= bot_bot_set_a ) ) ) ).
% Collect_conv_if
thf(fact_1140_Collect__conv__if2,axiom,
! [P: a > $o,A: a] :
( ( ( P @ A )
=> ( ( collect_a
@ ^ [X2: a] :
( ( A = X2 )
& ( P @ X2 ) ) )
= ( insert_a @ A @ bot_bot_set_a ) ) )
& ( ~ ( P @ A )
=> ( ( collect_a
@ ^ [X2: a] :
( ( A = X2 )
& ( P @ X2 ) ) )
= bot_bot_set_a ) ) ) ).
% Collect_conv_if2
thf(fact_1141_is__singleton__def,axiom,
( is_singleton_a
= ( ^ [A7: set_a] :
? [X2: a] :
( A7
= ( insert_a @ X2 @ bot_bot_set_a ) ) ) ) ).
% is_singleton_def
thf(fact_1142_is__singleton__the__elem,axiom,
( is_singleton_a
= ( ^ [A7: set_a] :
( A7
= ( insert_a @ ( the_elem_a @ A7 ) @ bot_bot_set_a ) ) ) ) ).
% is_singleton_the_elem
thf(fact_1143_the__elem__def,axiom,
( the_elem_set_a
= ( ^ [X4: set_set_a] :
( the_set_a
@ ^ [X2: set_a] :
( X4
= ( insert_set_a @ X2 @ bot_bot_set_set_a ) ) ) ) ) ).
% the_elem_def
thf(fact_1144_the__elem__def,axiom,
( the_elem_a
= ( ^ [X4: set_a] :
( the_a
@ ^ [X2: a] :
( X4
= ( insert_a @ X2 @ bot_bot_set_a ) ) ) ) ) ).
% the_elem_def
thf(fact_1145_the__elem__eq,axiom,
! [X6: a] :
( ( the_elem_a @ ( insert_a @ X6 @ bot_bot_set_a ) )
= X6 ) ).
% the_elem_eq
thf(fact_1146_totalp__on__singleton,axiom,
! [X6: a,R: a > a > $o] : ( totalp_on_a @ ( insert_a @ X6 @ bot_bot_set_a ) @ R ) ).
% totalp_on_singleton
thf(fact_1147_pairwise__singleton,axiom,
! [P: a > a > $o,A4: a] : ( pairwise_a @ P @ ( insert_a @ A4 @ bot_bot_set_a ) ) ).
% pairwise_singleton
thf(fact_1148_pairwise__empty,axiom,
! [P: a > a > $o] : ( pairwise_a @ P @ bot_bot_set_a ) ).
% pairwise_empty
thf(fact_1149_totalp__on__empty,axiom,
! [R: a > a > $o] : ( totalp_on_a @ bot_bot_set_a @ R ) ).
% totalp_on_empty
thf(fact_1150_pairwise__alt,axiom,
( pairwise_a
= ( ^ [R2: a > a > $o,S: set_a] :
! [X2: a] :
( ( member_a @ X2 @ S )
=> ! [Y2: a] :
( ( member_a @ Y2 @ ( minus_minus_set_a @ S @ ( insert_a @ X2 @ bot_bot_set_a ) ) )
=> ( R2 @ X2 @ Y2 ) ) ) ) ) ).
% pairwise_alt
thf(fact_1151_singleton__insert__inj__eq_H,axiom,
! [A: a,A4: set_a,B: a] :
( ( ( insert_a @ A @ A4 )
= ( insert_a @ B @ bot_bot_set_a ) )
= ( ( A = B )
& ( ord_less_eq_set_a @ A4 @ ( insert_a @ B @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_1152_empty__subsetI,axiom,
! [A4: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A4 ) ).
% empty_subsetI
thf(fact_1153_subset__empty,axiom,
! [A4: set_a] :
( ( ord_less_eq_set_a @ A4 @ bot_bot_set_a )
= ( A4 = bot_bot_set_a ) ) ).
% subset_empty
thf(fact_1154_Diff__cancel,axiom,
! [A4: set_a] :
( ( minus_minus_set_a @ A4 @ A4 )
= bot_bot_set_a ) ).
% Diff_cancel
thf(fact_1155_empty__Diff,axiom,
! [A4: set_a] :
( ( minus_minus_set_a @ bot_bot_set_a @ A4 )
= bot_bot_set_a ) ).
% empty_Diff
thf(fact_1156_Diff__empty,axiom,
! [A4: set_a] :
( ( minus_minus_set_a @ A4 @ bot_bot_set_a )
= A4 ) ).
% Diff_empty
thf(fact_1157_singleton__insert__inj__eq,axiom,
! [B: a,A: a,A4: set_a] :
( ( ( insert_a @ B @ bot_bot_set_a )
= ( insert_a @ A @ A4 ) )
= ( ( A = B )
& ( ord_less_eq_set_a @ A4 @ ( insert_a @ B @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_1158_Diff__eq__empty__iff,axiom,
! [A4: set_a,B3: set_a] :
( ( ( minus_minus_set_a @ A4 @ B3 )
= bot_bot_set_a )
= ( ord_less_eq_set_a @ A4 @ B3 ) ) ).
% Diff_eq_empty_iff
thf(fact_1159_insert__Diff__single,axiom,
! [A: a,A4: set_a] :
( ( insert_a @ A @ ( minus_minus_set_a @ A4 @ ( insert_a @ A @ bot_bot_set_a ) ) )
= ( insert_a @ A @ A4 ) ) ).
% insert_Diff_single
thf(fact_1160_Diff__disjoint,axiom,
! [A4: set_a,B3: set_a] :
( ( inf_inf_set_a @ A4 @ ( minus_minus_set_a @ B3 @ A4 ) )
= bot_bot_set_a ) ).
% Diff_disjoint
thf(fact_1161_subset__singleton__iff,axiom,
! [X7: set_a,A: a] :
( ( ord_less_eq_set_a @ X7 @ ( insert_a @ A @ bot_bot_set_a ) )
= ( ( X7 = bot_bot_set_a )
| ( X7
= ( insert_a @ A @ bot_bot_set_a ) ) ) ) ).
% subset_singleton_iff
thf(fact_1162_Diff__single__insert,axiom,
! [A4: set_a,X6: a,B3: set_a] :
( ( ord_less_eq_set_a @ ( minus_minus_set_a @ A4 @ ( insert_a @ X6 @ bot_bot_set_a ) ) @ B3 )
=> ( ord_less_eq_set_a @ A4 @ ( insert_a @ X6 @ B3 ) ) ) ).
% Diff_single_insert
thf(fact_1163_Diff__insert__absorb,axiom,
! [X6: a,A4: set_a] :
( ~ ( member_a @ X6 @ A4 )
=> ( ( minus_minus_set_a @ ( insert_a @ X6 @ A4 ) @ ( insert_a @ X6 @ bot_bot_set_a ) )
= A4 ) ) ).
% Diff_insert_absorb
thf(fact_1164_subset__singletonD,axiom,
! [A4: set_a,X6: a] :
( ( ord_less_eq_set_a @ A4 @ ( insert_a @ X6 @ bot_bot_set_a ) )
=> ( ( A4 = bot_bot_set_a )
| ( A4
= ( insert_a @ X6 @ bot_bot_set_a ) ) ) ) ).
% subset_singletonD
thf(fact_1165_subset__insert__iff,axiom,
! [A4: set_a,X6: a,B3: set_a] :
( ( ord_less_eq_set_a @ A4 @ ( insert_a @ X6 @ B3 ) )
= ( ( ( member_a @ X6 @ A4 )
=> ( ord_less_eq_set_a @ ( minus_minus_set_a @ A4 @ ( insert_a @ X6 @ bot_bot_set_a ) ) @ B3 ) )
& ( ~ ( member_a @ X6 @ A4 )
=> ( ord_less_eq_set_a @ A4 @ B3 ) ) ) ) ).
% subset_insert_iff
thf(fact_1166_Diff__insert2,axiom,
! [A4: set_a,A: a,B3: set_a] :
( ( minus_minus_set_a @ A4 @ ( insert_a @ A @ B3 ) )
= ( minus_minus_set_a @ ( minus_minus_set_a @ A4 @ ( insert_a @ A @ bot_bot_set_a ) ) @ B3 ) ) ).
% Diff_insert2
thf(fact_1167_insert__Diff,axiom,
! [A: a,A4: set_a] :
( ( member_a @ A @ A4 )
=> ( ( insert_a @ A @ ( minus_minus_set_a @ A4 @ ( insert_a @ A @ bot_bot_set_a ) ) )
= A4 ) ) ).
% insert_Diff
thf(fact_1168_Diff__insert,axiom,
! [A4: set_a,A: a,B3: set_a] :
( ( minus_minus_set_a @ A4 @ ( insert_a @ A @ B3 ) )
= ( minus_minus_set_a @ ( minus_minus_set_a @ A4 @ B3 ) @ ( insert_a @ A @ bot_bot_set_a ) ) ) ).
% Diff_insert
thf(fact_1169_Diff__triv,axiom,
! [A4: set_a,B3: set_a] :
( ( ( inf_inf_set_a @ A4 @ B3 )
= bot_bot_set_a )
=> ( ( minus_minus_set_a @ A4 @ B3 )
= A4 ) ) ).
% Diff_triv
thf(fact_1170_Int__Diff__disjoint,axiom,
! [A4: set_a,B3: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A4 @ B3 ) @ ( minus_minus_set_a @ A4 @ B3 ) )
= bot_bot_set_a ) ).
% Int_Diff_disjoint
thf(fact_1171_bot_Oextremum__uniqueI,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ A @ bot_bot_set_a )
=> ( A = bot_bot_set_a ) ) ).
% bot.extremum_uniqueI
thf(fact_1172_bot_Oextremum__unique,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ A @ bot_bot_set_a )
= ( A = bot_bot_set_a ) ) ).
% bot.extremum_unique
thf(fact_1173_bot_Oextremum,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A ) ).
% bot.extremum
thf(fact_1174_diff__shunt__var,axiom,
! [X6: set_a,Y6: set_a] :
( ( ( minus_minus_set_a @ X6 @ Y6 )
= bot_bot_set_a )
= ( ord_less_eq_set_a @ X6 @ Y6 ) ) ).
% diff_shunt_var
thf(fact_1175_subset__emptyI,axiom,
! [A4: set_a] :
( ! [X: a] :
~ ( member_a @ X @ A4 )
=> ( ord_less_eq_set_a @ A4 @ bot_bot_set_a ) ) ).
% subset_emptyI
thf(fact_1176_remove__def,axiom,
( remove_a
= ( ^ [X2: a,A7: set_a] : ( minus_minus_set_a @ A7 @ ( insert_a @ X2 @ bot_bot_set_a ) ) ) ) ).
% remove_def
thf(fact_1177_flat__lub__def,axiom,
( partia4732287487727000106_set_a
= ( ^ [B5: set_a,A7: set_set_a] :
( if_set_a @ ( ord_le3724670747650509150_set_a @ A7 @ ( insert_set_a @ B5 @ bot_bot_set_set_a ) ) @ B5
@ ( the_set_a
@ ^ [X2: set_a] : ( member_set_a @ X2 @ ( minus_5736297505244876581_set_a @ A7 @ ( insert_set_a @ B5 @ bot_bot_set_set_a ) ) ) ) ) ) ) ).
% flat_lub_def
thf(fact_1178_flat__lub__def,axiom,
( partial_flat_lub_a
= ( ^ [B5: a,A7: set_a] :
( if_a @ ( ord_less_eq_set_a @ A7 @ ( insert_a @ B5 @ bot_bot_set_a ) ) @ B5
@ ( the_a
@ ^ [X2: a] : ( member_a @ X2 @ ( minus_minus_set_a @ A7 @ ( insert_a @ B5 @ bot_bot_set_a ) ) ) ) ) ) ) ).
% flat_lub_def
thf(fact_1179_subset__singleton__iff__Uniq,axiom,
! [A4: set_a] :
( ( ? [A5: a] : ( ord_less_eq_set_a @ A4 @ ( insert_a @ A5 @ bot_bot_set_a ) ) )
= ( uniq_a
@ ^ [X2: a] : ( member_a @ X2 @ A4 ) ) ) ).
% subset_singleton_iff_Uniq
thf(fact_1180_psubset__insert__iff,axiom,
! [A4: set_a,X6: a,B3: set_a] :
( ( ord_less_set_a @ A4 @ ( insert_a @ X6 @ B3 ) )
= ( ( ( member_a @ X6 @ B3 )
=> ( ord_less_set_a @ A4 @ B3 ) )
& ( ~ ( member_a @ X6 @ B3 )
=> ( ( ( member_a @ X6 @ A4 )
=> ( ord_less_set_a @ ( minus_minus_set_a @ A4 @ ( insert_a @ X6 @ bot_bot_set_a ) ) @ B3 ) )
& ( ~ ( member_a @ X6 @ A4 )
=> ( ord_less_eq_set_a @ A4 @ B3 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_1181_subset__Compl__singleton,axiom,
! [A4: set_a,B: a] :
( ( ord_less_eq_set_a @ A4 @ ( uminus_uminus_set_a @ ( insert_a @ B @ bot_bot_set_a ) ) )
= ( ~ ( member_a @ B @ A4 ) ) ) ).
% subset_Compl_singleton
thf(fact_1182_boolean__algebra_Oconj__cancel__right,axiom,
! [X6: set_a] :
( ( inf_inf_set_a @ X6 @ ( uminus_uminus_set_a @ X6 ) )
= bot_bot_set_a ) ).
% boolean_algebra.conj_cancel_right
thf(fact_1183_boolean__algebra_Oconj__cancel__left,axiom,
! [X6: set_a] :
( ( inf_inf_set_a @ ( uminus_uminus_set_a @ X6 ) @ X6 )
= bot_bot_set_a ) ).
% boolean_algebra.conj_cancel_left
thf(fact_1184_inf__compl__bot__right,axiom,
! [X6: set_a,Y6: set_a] :
( ( inf_inf_set_a @ X6 @ ( inf_inf_set_a @ Y6 @ ( uminus_uminus_set_a @ X6 ) ) )
= bot_bot_set_a ) ).
% inf_compl_bot_right
thf(fact_1185_inf__compl__bot__left2,axiom,
! [X6: set_a,Y6: set_a] :
( ( inf_inf_set_a @ X6 @ ( inf_inf_set_a @ ( uminus_uminus_set_a @ X6 ) @ Y6 ) )
= bot_bot_set_a ) ).
% inf_compl_bot_left2
thf(fact_1186_inf__compl__bot__left1,axiom,
! [X6: set_a,Y6: set_a] :
( ( inf_inf_set_a @ ( uminus_uminus_set_a @ X6 ) @ ( inf_inf_set_a @ X6 @ Y6 ) )
= bot_bot_set_a ) ).
% inf_compl_bot_left1
thf(fact_1187_Compl__disjoint2,axiom,
! [A4: set_a] :
( ( inf_inf_set_a @ ( uminus_uminus_set_a @ A4 ) @ A4 )
= bot_bot_set_a ) ).
% Compl_disjoint2
thf(fact_1188_Compl__disjoint,axiom,
! [A4: set_a] :
( ( inf_inf_set_a @ A4 @ ( uminus_uminus_set_a @ A4 ) )
= bot_bot_set_a ) ).
% Compl_disjoint
thf(fact_1189_not__psubset__empty,axiom,
! [A4: set_a] :
~ ( ord_less_set_a @ A4 @ bot_bot_set_a ) ).
% not_psubset_empty
thf(fact_1190_bot_Oextremum__strict,axiom,
! [A: set_a] :
~ ( ord_less_set_a @ A @ bot_bot_set_a ) ).
% bot.extremum_strict
thf(fact_1191_bot_Onot__eq__extremum,axiom,
! [A: set_a] :
( ( A != bot_bot_set_a )
= ( ord_less_set_a @ bot_bot_set_a @ A ) ) ).
% bot.not_eq_extremum
thf(fact_1192_inf__cancel__left2,axiom,
! [X6: set_a,A: set_a,B: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ ( uminus_uminus_set_a @ X6 ) @ A ) @ ( inf_inf_set_a @ X6 @ B ) )
= bot_bot_set_a ) ).
% inf_cancel_left2
thf(fact_1193_inf__cancel__left1,axiom,
! [X6: set_a,A: set_a,B: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ X6 @ A ) @ ( inf_inf_set_a @ ( uminus_uminus_set_a @ X6 ) @ B ) )
= bot_bot_set_a ) ).
% inf_cancel_left1
thf(fact_1194_subset__Compl__self__eq,axiom,
! [A4: set_a] :
( ( ord_less_eq_set_a @ A4 @ ( uminus_uminus_set_a @ A4 ) )
= ( A4 = bot_bot_set_a ) ) ).
% subset_Compl_self_eq
thf(fact_1195_inf__shunt,axiom,
! [X6: set_a,Y6: set_a] :
( ( ( inf_inf_set_a @ X6 @ Y6 )
= bot_bot_set_a )
= ( ord_less_eq_set_a @ X6 @ ( uminus_uminus_set_a @ Y6 ) ) ) ).
% inf_shunt
thf(fact_1196_disjoint__eq__subset__Compl,axiom,
! [A4: set_a,B3: set_a] :
( ( ( inf_inf_set_a @ A4 @ B3 )
= bot_bot_set_a )
= ( ord_less_eq_set_a @ A4 @ ( uminus_uminus_set_a @ B3 ) ) ) ).
% disjoint_eq_subset_Compl
thf(fact_1197_Compl__insert,axiom,
! [X6: a,A4: set_a] :
( ( uminus_uminus_set_a @ ( insert_a @ X6 @ A4 ) )
= ( minus_minus_set_a @ ( uminus_uminus_set_a @ A4 ) @ ( insert_a @ X6 @ bot_bot_set_a ) ) ) ).
% Compl_insert
thf(fact_1198_bot_Oordering__top__axioms,axiom,
( ordering_top_set_a
@ ^ [X2: set_a,Y2: set_a] : ( ord_less_eq_set_a @ Y2 @ X2 )
@ ^ [X2: set_a,Y2: set_a] : ( ord_less_set_a @ Y2 @ X2 )
@ bot_bot_set_a ) ).
% bot.ordering_top_axioms
thf(fact_1199_simulation_Opreserves__targets,axiom,
! [A4: a > a > a,B3: a > a > a,F2: a > a,T: a] :
( ( simulation_a_a @ A4 @ B3 @ F2 )
=> ( ord_less_eq_set_a @ ( image_a_a @ F2 @ ( targets_a @ A4 @ T ) ) @ ( targets_a @ B3 @ ( F2 @ T ) ) ) ) ).
% simulation.preserves_targets
thf(fact_1200_image__is__empty,axiom,
! [F3: a > a,A4: set_a] :
( ( ( image_a_a @ F3 @ A4 )
= bot_bot_set_a )
= ( A4 = bot_bot_set_a ) ) ).
% image_is_empty
thf(fact_1201_empty__is__image,axiom,
! [F3: a > a,A4: set_a] :
( ( bot_bot_set_a
= ( image_a_a @ F3 @ A4 ) )
= ( A4 = bot_bot_set_a ) ) ).
% empty_is_image
thf(fact_1202_image__empty,axiom,
! [F3: a > a] :
( ( image_a_a @ F3 @ bot_bot_set_a )
= bot_bot_set_a ) ).
% image_empty
thf(fact_1203_image__constant,axiom,
! [X6: a,A4: set_a,C: a] :
( ( member_a @ X6 @ A4 )
=> ( ( image_a_a
@ ^ [X2: a] : C
@ A4 )
= ( insert_a @ C @ bot_bot_set_a ) ) ) ).
% image_constant
thf(fact_1204_image__constant__conv,axiom,
! [A4: set_a,C: a] :
( ( ( A4 = bot_bot_set_a )
=> ( ( image_a_a
@ ^ [X2: a] : C
@ A4 )
= bot_bot_set_a ) )
& ( ( A4 != bot_bot_set_a )
=> ( ( image_a_a
@ ^ [X2: a] : C
@ A4 )
= ( insert_a @ C @ bot_bot_set_a ) ) ) ) ).
% image_constant_conv
thf(fact_1205_in__image__insert__iff,axiom,
! [B3: set_set_a,X6: a,A4: set_a] :
( ! [C2: set_a] :
( ( member_set_a @ C2 @ B3 )
=> ~ ( member_a @ X6 @ C2 ) )
=> ( ( member_set_a @ A4 @ ( image_set_a_set_a @ ( insert_a @ X6 ) @ B3 ) )
= ( ( member_a @ X6 @ A4 )
& ( member_set_a @ ( minus_minus_set_a @ A4 @ ( insert_a @ X6 @ bot_bot_set_a ) ) @ B3 ) ) ) ) ).
% in_image_insert_iff
thf(fact_1206_simulation_Opreserves__sources,axiom,
! [A4: a > a > a,B3: a > a > a,F2: a > a,T: a] :
( ( simulation_a_a @ A4 @ B3 @ F2 )
=> ( ord_less_eq_set_a @ ( image_a_a @ F2 @ ( sources_a @ A4 @ T ) ) @ ( sources_a @ B3 @ ( F2 @ T ) ) ) ) ).
% simulation.preserves_sources
thf(fact_1207_empty__bind,axiom,
! [F3: a > set_a] :
( ( bind_a_a @ bot_bot_set_a @ F3 )
= bot_bot_set_a ) ).
% empty_bind
thf(fact_1208_bind__const,axiom,
! [A4: set_a,B3: set_a] :
( ( ( A4 = bot_bot_set_a )
=> ( ( bind_a_a @ A4
@ ^ [Uu: a] : B3 )
= bot_bot_set_a ) )
& ( ( A4 != bot_bot_set_a )
=> ( ( bind_a_a @ A4
@ ^ [Uu: a] : B3 )
= B3 ) ) ) ).
% bind_const
thf(fact_1209_Collect__const,axiom,
! [P: $o] :
( ( P
=> ( ( collect_a
@ ^ [S2: a] : P )
= top_top_set_a ) )
& ( ~ P
=> ( ( collect_a
@ ^ [S2: a] : P )
= bot_bot_set_a ) ) ) ).
% Collect_const
thf(fact_1210_boolean__algebra_Ocompl__zero,axiom,
( ( uminus_uminus_set_a @ bot_bot_set_a )
= top_top_set_a ) ).
% boolean_algebra.compl_zero
thf(fact_1211_boolean__algebra_Ocompl__one,axiom,
( ( uminus_uminus_set_a @ top_top_set_a )
= bot_bot_set_a ) ).
% boolean_algebra.compl_one
thf(fact_1212_Diff__UNIV,axiom,
! [A4: set_a] :
( ( minus_minus_set_a @ A4 @ top_top_set_a )
= bot_bot_set_a ) ).
% Diff_UNIV
thf(fact_1213_empty__not__UNIV,axiom,
bot_bot_set_a != top_top_set_a ).
% empty_not_UNIV
thf(fact_1214_Compl__UNIV__eq,axiom,
( ( uminus_uminus_set_a @ top_top_set_a )
= bot_bot_set_a ) ).
% Compl_UNIV_eq
thf(fact_1215_Compl__empty__eq,axiom,
( ( uminus_uminus_set_a @ bot_bot_set_a )
= top_top_set_a ) ).
% Compl_empty_eq
thf(fact_1216_inj__on__insert,axiom,
! [F3: a > a,A: a,A4: set_a] :
( ( inj_on_a_a @ F3 @ ( insert_a @ A @ A4 ) )
= ( ( inj_on_a_a @ F3 @ A4 )
& ~ ( member_a @ ( F3 @ A ) @ ( image_a_a @ F3 @ ( minus_minus_set_a @ A4 @ ( insert_a @ A @ bot_bot_set_a ) ) ) ) ) ) ).
% inj_on_insert
thf(fact_1217_boolean__algebra__class_Oboolean__algebra_Ocompl__unique,axiom,
! [X6: set_a,Y6: set_a] :
( ( ( inf_inf_set_a @ X6 @ Y6 )
= bot_bot_set_a )
=> ( ( ( sup_sup_set_a @ X6 @ Y6 )
= top_top_set_a )
=> ( ( uminus_uminus_set_a @ X6 )
= Y6 ) ) ) ).
% boolean_algebra_class.boolean_algebra.compl_unique
thf(fact_1218_Un__empty,axiom,
! [A4: set_a,B3: set_a] :
( ( ( sup_sup_set_a @ A4 @ B3 )
= bot_bot_set_a )
= ( ( A4 = bot_bot_set_a )
& ( B3 = bot_bot_set_a ) ) ) ).
% Un_empty
thf(fact_1219_sup__bot_Oright__neutral,axiom,
! [A: set_a] :
( ( sup_sup_set_a @ A @ bot_bot_set_a )
= A ) ).
% sup_bot.right_neutral
thf(fact_1220_sup__bot_Oneutr__eq__iff,axiom,
! [A: set_a,B: set_a] :
( ( bot_bot_set_a
= ( sup_sup_set_a @ A @ B ) )
= ( ( A = bot_bot_set_a )
& ( B = bot_bot_set_a ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_1221_sup__bot_Oleft__neutral,axiom,
! [A: set_a] :
( ( sup_sup_set_a @ bot_bot_set_a @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_1222_sup__bot_Oeq__neutr__iff,axiom,
! [A: set_a,B: set_a] :
( ( ( sup_sup_set_a @ A @ B )
= bot_bot_set_a )
= ( ( A = bot_bot_set_a )
& ( B = bot_bot_set_a ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_1223_sup__eq__bot__iff,axiom,
! [X6: set_a,Y6: set_a] :
( ( ( sup_sup_set_a @ X6 @ Y6 )
= bot_bot_set_a )
= ( ( X6 = bot_bot_set_a )
& ( Y6 = bot_bot_set_a ) ) ) ).
% sup_eq_bot_iff
thf(fact_1224_bot__eq__sup__iff,axiom,
! [X6: set_a,Y6: set_a] :
( ( bot_bot_set_a
= ( sup_sup_set_a @ X6 @ Y6 ) )
= ( ( X6 = bot_bot_set_a )
& ( Y6 = bot_bot_set_a ) ) ) ).
% bot_eq_sup_iff
thf(fact_1225_sup__bot__right,axiom,
! [X6: set_a] :
( ( sup_sup_set_a @ X6 @ bot_bot_set_a )
= X6 ) ).
% sup_bot_right
thf(fact_1226_sup__bot__left,axiom,
! [X6: set_a] :
( ( sup_sup_set_a @ bot_bot_set_a @ X6 )
= X6 ) ).
% sup_bot_left
thf(fact_1227_inj__on__disjoint__Un,axiom,
! [F3: a > a,A4: set_a,G: a > a,B3: set_a] :
( ( inj_on_a_a @ F3 @ A4 )
=> ( ( inj_on_a_a @ G @ B3 )
=> ( ( ( inf_inf_set_a @ ( image_a_a @ F3 @ A4 ) @ ( image_a_a @ G @ B3 ) )
= bot_bot_set_a )
=> ( inj_on_a_a
@ ^ [X2: a] : ( if_a @ ( member_a @ X2 @ A4 ) @ ( F3 @ X2 ) @ ( G @ X2 ) )
@ ( sup_sup_set_a @ A4 @ B3 ) ) ) ) ) ).
% inj_on_disjoint_Un
thf(fact_1228_inj__singleton,axiom,
! [A4: set_a] :
( inj_on_a_set_a
@ ^ [X2: a] : ( insert_a @ X2 @ bot_bot_set_a )
@ A4 ) ).
% inj_singleton
thf(fact_1229_Un__empty__left,axiom,
! [B3: set_a] :
( ( sup_sup_set_a @ bot_bot_set_a @ B3 )
= B3 ) ).
% Un_empty_left
thf(fact_1230_Un__empty__right,axiom,
! [A4: set_a] :
( ( sup_sup_set_a @ A4 @ bot_bot_set_a )
= A4 ) ).
% Un_empty_right
thf(fact_1231_boolean__algebra_Odisj__zero__right,axiom,
! [X6: set_a] :
( ( sup_sup_set_a @ X6 @ bot_bot_set_a )
= X6 ) ).
% boolean_algebra.disj_zero_right
thf(fact_1232_insert__is__Un,axiom,
( insert_a
= ( ^ [A5: a] : ( sup_sup_set_a @ ( insert_a @ A5 @ bot_bot_set_a ) ) ) ) ).
% insert_is_Un
thf(fact_1233_Un__singleton__iff,axiom,
! [A4: set_a,B3: set_a,X6: a] :
( ( ( sup_sup_set_a @ A4 @ B3 )
= ( insert_a @ X6 @ bot_bot_set_a ) )
= ( ( ( A4 = bot_bot_set_a )
& ( B3
= ( insert_a @ X6 @ bot_bot_set_a ) ) )
| ( ( A4
= ( insert_a @ X6 @ bot_bot_set_a ) )
& ( B3 = bot_bot_set_a ) )
| ( ( A4
= ( insert_a @ X6 @ bot_bot_set_a ) )
& ( B3
= ( insert_a @ X6 @ bot_bot_set_a ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_1234_singleton__Un__iff,axiom,
! [X6: a,A4: set_a,B3: set_a] :
( ( ( insert_a @ X6 @ bot_bot_set_a )
= ( sup_sup_set_a @ A4 @ B3 ) )
= ( ( ( A4 = bot_bot_set_a )
& ( B3
= ( insert_a @ X6 @ bot_bot_set_a ) ) )
| ( ( A4
= ( insert_a @ X6 @ bot_bot_set_a ) )
& ( B3 = bot_bot_set_a ) )
| ( ( A4
= ( insert_a @ X6 @ bot_bot_set_a ) )
& ( B3
= ( insert_a @ X6 @ bot_bot_set_a ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_1235_boolean__algebra_Ocomplement__unique,axiom,
! [A: set_a,X6: set_a,Y6: set_a] :
( ( ( inf_inf_set_a @ A @ X6 )
= bot_bot_set_a )
=> ( ( ( sup_sup_set_a @ A @ X6 )
= top_top_set_a )
=> ( ( ( inf_inf_set_a @ A @ Y6 )
= bot_bot_set_a )
=> ( ( ( sup_sup_set_a @ A @ Y6 )
= top_top_set_a )
=> ( X6 = Y6 ) ) ) ) ) ).
% boolean_algebra.complement_unique
thf(fact_1236_sup__bot_Osemilattice__neutr__order__axioms,axiom,
( semila2496817875450240012_set_a @ sup_sup_set_a @ bot_bot_set_a
@ ^ [X2: set_a,Y2: set_a] : ( ord_less_eq_set_a @ Y2 @ X2 )
@ ^ [X2: set_a,Y2: set_a] : ( ord_less_set_a @ Y2 @ X2 ) ) ).
% sup_bot.semilattice_neutr_order_axioms
thf(fact_1237_inj__on__vimage__singleton,axiom,
! [F3: set_a > a,A4: set_set_a,A: a] :
( ( inj_on_set_a_a @ F3 @ A4 )
=> ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ ( vimage_set_a_a @ F3 @ ( insert_a @ A @ bot_bot_set_a ) ) @ A4 )
@ ( insert_set_a
@ ( the_set_a
@ ^ [X2: set_a] :
( ( member_set_a @ X2 @ A4 )
& ( ( F3 @ X2 )
= A ) ) )
@ bot_bot_set_set_a ) ) ) ).
% inj_on_vimage_singleton
thf(fact_1238_inj__on__vimage__singleton,axiom,
! [F3: a > a,A4: set_a,A: a] :
( ( inj_on_a_a @ F3 @ A4 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ ( vimage_a_a @ F3 @ ( insert_a @ A @ bot_bot_set_a ) ) @ A4 )
@ ( insert_a
@ ( the_a
@ ^ [X2: a] :
( ( member_a @ X2 @ A4 )
& ( ( F3 @ X2 )
= A ) ) )
@ bot_bot_set_a ) ) ) ).
% inj_on_vimage_singleton
thf(fact_1239_vimage__empty,axiom,
! [F3: a > a] :
( ( vimage_a_a @ F3 @ bot_bot_set_a )
= bot_bot_set_a ) ).
% vimage_empty
thf(fact_1240_vimage__const,axiom,
! [C: a,A4: set_a] :
( ( ( member_a @ C @ A4 )
=> ( ( vimage_a_a
@ ^ [X2: a] : C
@ A4 )
= top_top_set_a ) )
& ( ~ ( member_a @ C @ A4 )
=> ( ( vimage_a_a
@ ^ [X2: a] : C
@ A4 )
= bot_bot_set_a ) ) ) ).
% vimage_const
thf(fact_1241_vimage__if,axiom,
! [C: a,A4: set_a,D: a,B3: set_a] :
( ( ( member_a @ C @ A4 )
=> ( ( ( member_a @ D @ A4 )
=> ( ( vimage_a_a
@ ^ [X2: a] : ( if_a @ ( member_a @ X2 @ B3 ) @ C @ D )
@ A4 )
= top_top_set_a ) )
& ( ~ ( member_a @ D @ A4 )
=> ( ( vimage_a_a
@ ^ [X2: a] : ( if_a @ ( member_a @ X2 @ B3 ) @ C @ D )
@ A4 )
= B3 ) ) ) )
& ( ~ ( member_a @ C @ A4 )
=> ( ( ( member_a @ D @ A4 )
=> ( ( vimage_a_a
@ ^ [X2: a] : ( if_a @ ( member_a @ X2 @ B3 ) @ C @ D )
@ A4 )
= ( uminus_uminus_set_a @ B3 ) ) )
& ( ~ ( member_a @ D @ A4 )
=> ( ( vimage_a_a
@ ^ [X2: a] : ( if_a @ ( member_a @ X2 @ B3 ) @ C @ D )
@ A4 )
= bot_bot_set_a ) ) ) ) ) ).
% vimage_if
thf(fact_1242_surj__vimage__empty,axiom,
! [F3: a > a,A4: set_a] :
( ( ( image_a_a @ F3 @ top_top_set_a )
= top_top_set_a )
=> ( ( ( vimage_a_a @ F3 @ A4 )
= bot_bot_set_a )
= ( A4 = bot_bot_set_a ) ) ) ).
% surj_vimage_empty
thf(fact_1243_vimage__singleton__eq,axiom,
! [A: a,F3: a > a,B: a] :
( ( member_a @ A @ ( vimage_a_a @ F3 @ ( insert_a @ B @ bot_bot_set_a ) ) )
= ( ( F3 @ A )
= B ) ) ).
% vimage_singleton_eq
thf(fact_1244_inj__vimage__singleton,axiom,
! [F3: set_a > a,A: a] :
( ( inj_on_set_a_a @ F3 @ top_top_set_set_a )
=> ( ord_le3724670747650509150_set_a @ ( vimage_set_a_a @ F3 @ ( insert_a @ A @ bot_bot_set_a ) )
@ ( insert_set_a
@ ( the_set_a
@ ^ [X2: set_a] :
( ( F3 @ X2 )
= A ) )
@ bot_bot_set_set_a ) ) ) ).
% inj_vimage_singleton
thf(fact_1245_inj__vimage__singleton,axiom,
! [F3: a > a,A: a] :
( ( inj_on_a_a @ F3 @ top_top_set_a )
=> ( ord_less_eq_set_a @ ( vimage_a_a @ F3 @ ( insert_a @ A @ bot_bot_set_a ) )
@ ( insert_a
@ ( the_a
@ ^ [X2: a] :
( ( F3 @ X2 )
= A ) )
@ bot_bot_set_a ) ) ) ).
% inj_vimage_singleton
thf(fact_1246_boolean__algebra_Oabstract__boolean__algebra__axioms,axiom,
boolea6678413348699952596_set_a @ inf_inf_set_a @ sup_sup_set_a @ uminus_uminus_set_a @ bot_bot_set_a @ top_top_set_a ).
% boolean_algebra.abstract_boolean_algebra_axioms
thf(fact_1247_If__the__inv__into__in__Func,axiom,
! [G: a > a,C3: set_a,B3: set_a,X6: a] :
( ( inj_on_a_a @ G @ C3 )
=> ( ( ord_less_eq_set_a @ C3 @ ( sup_sup_set_a @ B3 @ ( insert_a @ X6 @ bot_bot_set_a ) ) )
=> ( member_a_a
@ ^ [I: a] : ( if_a @ ( member_a @ I @ ( image_a_a @ G @ C3 ) ) @ ( the_inv_into_a_a @ C3 @ G @ I ) @ X6 )
@ ( bNF_We5243062509538606484nc_a_a @ top_top_set_a @ ( sup_sup_set_a @ B3 @ ( insert_a @ X6 @ bot_bot_set_a ) ) ) ) ) ) ).
% If_the_inv_into_in_Func
thf(fact_1248_Func__is__emp,axiom,
! [A4: set_a,B3: set_a] :
( ( ( bNF_We5243062509538606484nc_a_a @ A4 @ B3 )
= bot_bot_set_a_a )
= ( ( A4 != bot_bot_set_a )
& ( B3 = bot_bot_set_a ) ) ) ).
% Func_is_emp
thf(fact_1249_inf__img__fin__domE_H,axiom,
! [F3: a > a,A4: set_a] :
( ( finite_finite_a @ ( image_a_a @ F3 @ A4 ) )
=> ( ~ ( finite_finite_a @ A4 )
=> ~ ! [Y3: a] :
( ( member_a @ Y3 @ ( image_a_a @ F3 @ A4 ) )
=> ( finite_finite_a @ ( inf_inf_set_a @ ( vimage_a_a @ F3 @ ( insert_a @ Y3 @ bot_bot_set_a ) ) @ A4 ) ) ) ) ) ).
% inf_img_fin_domE'
thf(fact_1250_finite__finite__vimage__IntI,axiom,
! [F2: set_a,H2: a > a,A4: set_a] :
( ( finite_finite_a @ F2 )
=> ( ! [Y3: a] :
( ( member_a @ Y3 @ F2 )
=> ( finite_finite_a @ ( inf_inf_set_a @ ( vimage_a_a @ H2 @ ( insert_a @ Y3 @ bot_bot_set_a ) ) @ A4 ) ) )
=> ( finite_finite_a @ ( inf_inf_set_a @ ( vimage_a_a @ H2 @ F2 ) @ A4 ) ) ) ) ).
% finite_finite_vimage_IntI
thf(fact_1251_infinite__remove,axiom,
! [S3: set_a,A: a] :
( ~ ( finite_finite_a @ S3 )
=> ~ ( finite_finite_a @ ( minus_minus_set_a @ S3 @ ( insert_a @ A @ bot_bot_set_a ) ) ) ) ).
% infinite_remove
thf(fact_1252_infinite__coinduct,axiom,
! [X7: set_a > $o,A4: set_a] :
( ( X7 @ A4 )
=> ( ! [A8: set_a] :
( ( X7 @ A8 )
=> ? [X3: a] :
( ( member_a @ X3 @ A8 )
& ( ( X7 @ ( minus_minus_set_a @ A8 @ ( insert_a @ X3 @ bot_bot_set_a ) ) )
| ~ ( finite_finite_a @ ( minus_minus_set_a @ A8 @ ( insert_a @ X3 @ bot_bot_set_a ) ) ) ) ) )
=> ~ ( finite_finite_a @ A4 ) ) ) ).
% infinite_coinduct
thf(fact_1253_finite__empty__induct,axiom,
! [A4: set_a,P: set_a > $o] :
( ( finite_finite_a @ A4 )
=> ( ( P @ A4 )
=> ( ! [A3: a,A8: set_a] :
( ( finite_finite_a @ A8 )
=> ( ( member_a @ A3 @ A8 )
=> ( ( P @ A8 )
=> ( P @ ( minus_minus_set_a @ A8 @ ( insert_a @ A3 @ bot_bot_set_a ) ) ) ) ) )
=> ( P @ bot_bot_set_a ) ) ) ) ).
% finite_empty_induct
thf(fact_1254_finite__subset__induct,axiom,
! [F2: set_a,A4: set_a,P: set_a > $o] :
( ( finite_finite_a @ F2 )
=> ( ( ord_less_eq_set_a @ F2 @ A4 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A3: a,F6: set_a] :
( ( finite_finite_a @ F6 )
=> ( ( member_a @ A3 @ A4 )
=> ( ~ ( member_a @ A3 @ F6 )
=> ( ( P @ F6 )
=> ( P @ ( insert_a @ A3 @ F6 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_1255_finite__subset__induct_H,axiom,
! [F2: set_a,A4: set_a,P: set_a > $o] :
( ( finite_finite_a @ F2 )
=> ( ( ord_less_eq_set_a @ F2 @ A4 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A3: a,F6: set_a] :
( ( finite_finite_a @ F6 )
=> ( ( member_a @ A3 @ A4 )
=> ( ( ord_less_eq_set_a @ F6 @ A4 )
=> ( ~ ( member_a @ A3 @ F6 )
=> ( ( P @ F6 )
=> ( P @ ( insert_a @ A3 @ F6 ) ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_1256_finite_Ocases,axiom,
! [A: set_a] :
( ( finite_finite_a @ A )
=> ( ( A != bot_bot_set_a )
=> ~ ! [A8: set_a] :
( ? [A3: a] :
( A
= ( insert_a @ A3 @ A8 ) )
=> ~ ( finite_finite_a @ A8 ) ) ) ) ).
% finite.cases
thf(fact_1257_finite_Osimps,axiom,
( finite_finite_a
= ( ^ [A5: set_a] :
( ( A5 = bot_bot_set_a )
| ? [A7: set_a,B5: a] :
( ( A5
= ( insert_a @ B5 @ A7 ) )
& ( finite_finite_a @ A7 ) ) ) ) ) ).
% finite.simps
thf(fact_1258_finite__induct,axiom,
! [F2: set_a,P: set_a > $o] :
( ( finite_finite_a @ F2 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X: a,F6: set_a] :
( ( finite_finite_a @ F6 )
=> ( ~ ( member_a @ X @ F6 )
=> ( ( P @ F6 )
=> ( P @ ( insert_a @ X @ F6 ) ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% finite_induct
thf(fact_1259_finite__ne__induct,axiom,
! [F2: set_a,P: set_a > $o] :
( ( finite_finite_a @ F2 )
=> ( ( F2 != bot_bot_set_a )
=> ( ! [X: a] : ( P @ ( insert_a @ X @ bot_bot_set_a ) )
=> ( ! [X: a,F6: set_a] :
( ( finite_finite_a @ F6 )
=> ( ( F6 != bot_bot_set_a )
=> ( ~ ( member_a @ X @ F6 )
=> ( ( P @ F6 )
=> ( P @ ( insert_a @ X @ F6 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_1260_infinite__finite__induct,axiom,
! [P: set_a > $o,A4: set_a] :
( ! [A8: set_a] :
( ~ ( finite_finite_a @ A8 )
=> ( P @ A8 ) )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X: a,F6: set_a] :
( ( finite_finite_a @ F6 )
=> ( ~ ( member_a @ X @ F6 )
=> ( ( P @ F6 )
=> ( P @ ( insert_a @ X @ F6 ) ) ) ) )
=> ( P @ A4 ) ) ) ) ).
% infinite_finite_induct
thf(fact_1261_infinite__imp__nonempty,axiom,
! [S3: set_a] :
( ~ ( finite_finite_a @ S3 )
=> ( S3 != bot_bot_set_a ) ) ).
% infinite_imp_nonempty
thf(fact_1262_finite_OemptyI,axiom,
finite_finite_a @ bot_bot_set_a ).
% finite.emptyI
thf(fact_1263_finite__induct__select,axiom,
! [S3: set_a,P: set_a > $o] :
( ( finite_finite_a @ S3 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [T11: set_a] :
( ( ord_less_set_a @ T11 @ S3 )
=> ( ( P @ T11 )
=> ? [X3: a] :
( ( member_a @ X3 @ ( minus_minus_set_a @ S3 @ T11 ) )
& ( P @ ( insert_a @ X3 @ T11 ) ) ) ) )
=> ( P @ S3 ) ) ) ) ).
% finite_induct_select
thf(fact_1264_finite__remove__induct,axiom,
! [B3: set_a,P: set_a > $o] :
( ( finite_finite_a @ B3 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A8: set_a] :
( ( finite_finite_a @ A8 )
=> ( ( A8 != bot_bot_set_a )
=> ( ( ord_less_eq_set_a @ A8 @ B3 )
=> ( ! [X3: a] :
( ( member_a @ X3 @ A8 )
=> ( P @ ( minus_minus_set_a @ A8 @ ( insert_a @ X3 @ bot_bot_set_a ) ) ) )
=> ( P @ A8 ) ) ) ) )
=> ( P @ B3 ) ) ) ) ).
% finite_remove_induct
thf(fact_1265_remove__induct,axiom,
! [P: set_a > $o,B3: set_a] :
( ( P @ bot_bot_set_a )
=> ( ( ~ ( finite_finite_a @ B3 )
=> ( P @ B3 ) )
=> ( ! [A8: set_a] :
( ( finite_finite_a @ A8 )
=> ( ( A8 != bot_bot_set_a )
=> ( ( ord_less_eq_set_a @ A8 @ B3 )
=> ( ! [X3: a] :
( ( member_a @ X3 @ A8 )
=> ( P @ ( minus_minus_set_a @ A8 @ ( insert_a @ X3 @ bot_bot_set_a ) ) ) )
=> ( P @ A8 ) ) ) ) )
=> ( P @ B3 ) ) ) ) ).
% remove_induct
thf(fact_1266_inf__img__fin__dom_H,axiom,
! [F3: a > a,A4: set_a] :
( ( finite_finite_a @ ( image_a_a @ F3 @ A4 ) )
=> ( ~ ( finite_finite_a @ A4 )
=> ? [X: a] :
( ( member_a @ X @ ( image_a_a @ F3 @ A4 ) )
& ~ ( finite_finite_a @ ( inf_inf_set_a @ ( vimage_a_a @ F3 @ ( insert_a @ X @ bot_bot_set_a ) ) @ A4 ) ) ) ) ) ).
% inf_img_fin_dom'
thf(fact_1267_inf__Sup1__distrib,axiom,
! [A4: set_set_a,X6: set_a] :
( ( finite_finite_set_a @ A4 )
=> ( ( A4 != bot_bot_set_set_a )
=> ( ( inf_inf_set_a @ X6 @ ( lattic2918178356826803221_set_a @ A4 ) )
= ( lattic2918178356826803221_set_a
@ ( collect_set_a
@ ^ [Uu: set_a] :
? [A5: set_a] :
( ( Uu
= ( inf_inf_set_a @ X6 @ A5 ) )
& ( member_set_a @ A5 @ A4 ) ) ) ) ) ) ) ).
% inf_Sup1_distrib
thf(fact_1268_inf__Sup2__distrib,axiom,
! [A4: set_set_a,B3: set_set_a] :
( ( finite_finite_set_a @ A4 )
=> ( ( A4 != bot_bot_set_set_a )
=> ( ( finite_finite_set_a @ B3 )
=> ( ( B3 != bot_bot_set_set_a )
=> ( ( inf_inf_set_a @ ( lattic2918178356826803221_set_a @ A4 ) @ ( lattic2918178356826803221_set_a @ B3 ) )
= ( lattic2918178356826803221_set_a
@ ( collect_set_a
@ ^ [Uu: set_a] :
? [A5: set_a,B5: set_a] :
( ( Uu
= ( inf_inf_set_a @ A5 @ B5 ) )
& ( member_set_a @ A5 @ A4 )
& ( member_set_a @ B5 @ B3 ) ) ) ) ) ) ) ) ) ).
% inf_Sup2_distrib
thf(fact_1269_Inf__fin_Oremove,axiom,
! [A4: set_set_a,X6: set_a] :
( ( finite_finite_set_a @ A4 )
=> ( ( member_set_a @ X6 @ A4 )
=> ( ( ( ( minus_5736297505244876581_set_a @ A4 @ ( insert_set_a @ X6 @ bot_bot_set_set_a ) )
= bot_bot_set_set_a )
=> ( ( lattic8209813465164889211_set_a @ A4 )
= X6 ) )
& ( ( ( minus_5736297505244876581_set_a @ A4 @ ( insert_set_a @ X6 @ bot_bot_set_set_a ) )
!= bot_bot_set_set_a )
=> ( ( lattic8209813465164889211_set_a @ A4 )
= ( inf_inf_set_a @ X6 @ ( lattic8209813465164889211_set_a @ ( minus_5736297505244876581_set_a @ A4 @ ( insert_set_a @ X6 @ bot_bot_set_set_a ) ) ) ) ) ) ) ) ) ).
% Inf_fin.remove
thf(fact_1270_Inf__fin_Oinsert__remove,axiom,
! [A4: set_set_a,X6: set_a] :
( ( finite_finite_set_a @ A4 )
=> ( ( ( ( minus_5736297505244876581_set_a @ A4 @ ( insert_set_a @ X6 @ bot_bot_set_set_a ) )
= bot_bot_set_set_a )
=> ( ( lattic8209813465164889211_set_a @ ( insert_set_a @ X6 @ A4 ) )
= X6 ) )
& ( ( ( minus_5736297505244876581_set_a @ A4 @ ( insert_set_a @ X6 @ bot_bot_set_set_a ) )
!= bot_bot_set_set_a )
=> ( ( lattic8209813465164889211_set_a @ ( insert_set_a @ X6 @ A4 ) )
= ( inf_inf_set_a @ X6 @ ( lattic8209813465164889211_set_a @ ( minus_5736297505244876581_set_a @ A4 @ ( insert_set_a @ X6 @ bot_bot_set_set_a ) ) ) ) ) ) ) ) ).
% Inf_fin.insert_remove
thf(fact_1271_Inf__fin_Oinsert,axiom,
! [A4: set_set_a,X6: set_a] :
( ( finite_finite_set_a @ A4 )
=> ( ( A4 != bot_bot_set_set_a )
=> ( ( lattic8209813465164889211_set_a @ ( insert_set_a @ X6 @ A4 ) )
= ( inf_inf_set_a @ X6 @ ( lattic8209813465164889211_set_a @ A4 ) ) ) ) ) ).
% Inf_fin.insert
thf(fact_1272_Inf__fin_Ohom__commute,axiom,
! [H2: set_a > set_a,N3: set_set_a] :
( ! [X: set_a,Y3: set_a] :
( ( H2 @ ( inf_inf_set_a @ X @ Y3 ) )
= ( inf_inf_set_a @ ( H2 @ X ) @ ( H2 @ Y3 ) ) )
=> ( ( finite_finite_set_a @ N3 )
=> ( ( N3 != bot_bot_set_set_a )
=> ( ( H2 @ ( lattic8209813465164889211_set_a @ N3 ) )
= ( lattic8209813465164889211_set_a @ ( image_set_a_set_a @ H2 @ N3 ) ) ) ) ) ) ).
% Inf_fin.hom_commute
thf(fact_1273_Inf__fin_Osubset,axiom,
! [A4: set_set_a,B3: set_set_a] :
( ( finite_finite_set_a @ A4 )
=> ( ( B3 != bot_bot_set_set_a )
=> ( ( ord_le3724670747650509150_set_a @ B3 @ A4 )
=> ( ( inf_inf_set_a @ ( lattic8209813465164889211_set_a @ B3 ) @ ( lattic8209813465164889211_set_a @ A4 ) )
= ( lattic8209813465164889211_set_a @ A4 ) ) ) ) ) ).
% Inf_fin.subset
thf(fact_1274_Inf__fin_Oinsert__not__elem,axiom,
! [A4: set_set_a,X6: set_a] :
( ( finite_finite_set_a @ A4 )
=> ( ~ ( member_set_a @ X6 @ A4 )
=> ( ( A4 != bot_bot_set_set_a )
=> ( ( lattic8209813465164889211_set_a @ ( insert_set_a @ X6 @ A4 ) )
= ( inf_inf_set_a @ X6 @ ( lattic8209813465164889211_set_a @ A4 ) ) ) ) ) ) ).
% Inf_fin.insert_not_elem
thf(fact_1275_Inf__fin_Oclosed,axiom,
! [A4: set_set_a] :
( ( finite_finite_set_a @ A4 )
=> ( ( A4 != bot_bot_set_set_a )
=> ( ! [X: set_a,Y3: set_a] : ( member_set_a @ ( inf_inf_set_a @ X @ Y3 ) @ ( insert_set_a @ X @ ( insert_set_a @ Y3 @ bot_bot_set_set_a ) ) )
=> ( member_set_a @ ( lattic8209813465164889211_set_a @ A4 ) @ A4 ) ) ) ) ).
% Inf_fin.closed
thf(fact_1276_Inf__fin_Ounion,axiom,
! [A4: set_set_a,B3: set_set_a] :
( ( finite_finite_set_a @ A4 )
=> ( ( A4 != bot_bot_set_set_a )
=> ( ( finite_finite_set_a @ B3 )
=> ( ( B3 != bot_bot_set_set_a )
=> ( ( lattic8209813465164889211_set_a @ ( sup_sup_set_set_a @ A4 @ B3 ) )
= ( inf_inf_set_a @ ( lattic8209813465164889211_set_a @ A4 ) @ ( lattic8209813465164889211_set_a @ B3 ) ) ) ) ) ) ) ).
% Inf_fin.union
thf(fact_1277_finite__def,axiom,
( finite_finite_a
= ( comple3243854344658494246et_a_o
@ ^ [P2: set_a > $o,X2: set_a] :
( ( X2 = bot_bot_set_a )
| ? [A7: set_a,A5: a] :
( ( X2
= ( insert_a @ A5 @ A7 ) )
& ( P2 @ A7 ) ) ) ) ) ).
% finite_def
thf(fact_1278_Set__filter__fold,axiom,
! [A4: set_a,P: a > $o] :
( ( finite_finite_a @ A4 )
=> ( ( filter_a @ P @ A4 )
= ( finite_fold_a_set_a
@ ^ [X2: a,A9: set_a] : ( if_set_a @ ( P @ X2 ) @ ( insert_a @ X2 @ A9 ) @ A9 )
@ bot_bot_set_a
@ A4 ) ) ) ).
% Set_filter_fold
% Helper facts (7)
thf(help_If_2_1_If_001tf__a_T,axiom,
! [X6: a,Y6: a] :
( ( if_a @ $false @ X6 @ Y6 )
= Y6 ) ).
thf(help_If_1_1_If_001tf__a_T,axiom,
! [X6: a,Y6: a] :
( ( if_a @ $true @ X6 @ Y6 )
= X6 ) ).
thf(help_fChoice_1_1_fChoice_001tf__a_T,axiom,
! [P: a > $o] :
( ( P @ ( fChoice_a @ P ) )
= ( ? [X4: a] : ( P @ 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,
! [X6: set_a,Y6: set_a] :
( ( if_set_a @ $false @ X6 @ Y6 )
= Y6 ) ).
thf(help_If_1_1_If_001t__Set__Oset_Itf__a_J_T,axiom,
! [X6: set_a,Y6: set_a] :
( ( if_set_a @ $true @ X6 @ Y6 )
= X6 ) ).
thf(help_fChoice_1_1_fChoice_001t__Product____Type__Oprod_Itf__a_Mtf__a_J_T,axiom,
! [P: product_prod_a_a > $o] :
( ( P @ ( fChoic4124218645493772411od_a_a @ P ) )
= ( ? [X4: product_prod_a_a] : ( P @ X4 ) ) ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( ide_a @ resid @ ( resid @ ( resid @ ( resid @ u2 @ w2 ) @ z2 ) @ ( resid @ ( resid @ u2 @ tx ) @ y2 ) ) )
& ( ide_a @ resid @ ( resid @ ( resid @ ( resid @ u2 @ tx ) @ y2 ) @ ( resid @ ( resid @ u2 @ w2 ) @ z2 ) ) ) ) ).
%------------------------------------------------------------------------------