TPTP Problem File: SLH0580^1.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_03384_129250__14128984_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1463 ( 263 unt; 179 typ; 0 def)
% Number of atoms : 4904 (1135 equ; 0 cnn)
% Maximal formula atoms : 17 ( 3 avg)
% Number of connectives : 17996 ( 302 ~; 14 |; 600 &;14585 @)
% ( 0 <=>;2495 =>; 0 <=; 0 <~>)
% Maximal formula depth : 25 ( 8 avg)
% Number of types : 8 ( 7 usr)
% Number of type conns : 2513 (2513 >; 0 *; 0 +; 0 <<)
% Number of symbols : 174 ( 172 usr; 10 con; 0-5 aty)
% Number of variables : 4157 ( 205 ^;3820 !; 132 ?;4157 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 11:47:56.970
%------------------------------------------------------------------------------
% Could-be-implicit typings (7)
thf(ty_n_t__Product____Type__Oprod_It__Set__Oset_Itf__a_J_Mt__Set__Oset_Itf__a_J_J,type,
produc1703568184450464039_set_a: $tType ).
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__Set__Oset_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J,type,
set_set_set_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_Itf__a_J,type,
set_a: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (172)
thf(sy_c_Finite__Set_OFpow_001tf__a,type,
finite_Fpow_a: set_a > set_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_Lattices_Oinf__class_Oinf_001_062_Itf__a_M_Eo_J,type,
inf_inf_a_o: ( a > $o ) > ( a > $o ) > a > $o ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
inf_inf_set_set_a: set_set_a > set_set_a > set_set_a ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_Itf__a_J,type,
inf_inf_set_a: set_a > set_a > set_a ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_Itf__a_M_Eo_J,type,
bot_bot_a_o: a > $o ).
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_It__Set__Oset_Itf__a_J_J_J,type,
bot_bo3380559777022489994_set_a: set_set_set_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_OLeast_001t__Set__Oset_Itf__a_J,type,
ord_Least_set_a: ( set_a > $o ) > set_a ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_Itf__a_M_Eo_J,type,
ord_less_eq_a_o: ( a > $o ) > ( a > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
ord_le3724670747650509150_set_a: set_set_a > set_set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__a_J,type,
ord_less_eq_set_a: set_a > set_a > $o ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Set__Oset_Itf__a_J,type,
order_Greatest_set_a: ( set_a > $o ) > set_a ).
thf(sy_c_Product__Type_Oprod_Ofst_001t__Set__Oset_Itf__a_J_001t__Set__Oset_Itf__a_J,type,
produc9088895665703139587_set_a: produc1703568184450464039_set_a > set_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_001t__Set__Oset_Itf__a_J_001t__Set__Oset_Itf__a_J,type,
produc1983107199584856133_set_a: produc1703568184450464039_set_a > set_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_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_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,
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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,
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thf(sy_c_ResiduatedTransitionSystem_Oconfluent__rts__axioms_001tf__a,type,
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thf(sy_c_ResiduatedTransitionSystem_Oextensional__rts_001t__Set__Oset_Itf__a_J,type,
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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,
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thf(sy_c_ResiduatedTransitionSystem_Oextensional__rts_Ocomp_001tf__a,type,
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thf(sy_c_ResiduatedTransitionSystem_Oextensional__rts_Ojoin_001t__Set__Oset_Itf__a_J,type,
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thf(sy_c_ResiduatedTransitionSystem_Oextensional__rts_Ojoin_001tf__a,type,
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thf(sy_c_ResiduatedTransitionSystem_Oextensional__rts__axioms_001t__Set__Oset_Itf__a_J,type,
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thf(sy_c_ResiduatedTransitionSystem_Oextensional__rts__axioms_001tf__a,type,
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thf(sy_c_ResiduatedTransitionSystem_Oextensional__rts__with__composites_001t__Set__Oset_Itf__a_J,type,
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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,
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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,
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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_001t__Set__Oset_Itf__a_J,type,
normal8977612136997397236_set_a: ( set_a > set_a > set_a ) > set_set_a > set_a > set_a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Onormal__sub__rts_OCong_001tf__a,type,
normal_sub_Cong_a: ( a > a > a ) > set_a > a > a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Onormal__sub__rts_OCong_H_001t__Set__Oset_Itf__a_J,type,
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thf(sy_c_ResiduatedTransitionSystem_Onormal__sub__rts_OCong_H_001tf__a,type,
normal_sub_Cong_a2: ( a > a > a ) > set_a > a > a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Onormal__sub__rts_OCong__class_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
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thf(sy_c_ResiduatedTransitionSystem_Onormal__sub__rts_OCong__class_001t__Set__Oset_Itf__a_J,type,
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thf(sy_c_ResiduatedTransitionSystem_Onormal__sub__rts_OCong__class_001tf__a,type,
normal7408713899360725774lass_a: ( a > a > a ) > set_a > a > set_a ).
thf(sy_c_ResiduatedTransitionSystem_Onormal__sub__rts_OCong__class__rep_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
normal322541020860755160od_a_a: set_Product_prod_a_a > product_prod_a_a ).
thf(sy_c_ResiduatedTransitionSystem_Onormal__sub__rts_OCong__class__rep_001tf__a,type,
normal3259722184653208495_rep_a: set_a > a ).
thf(sy_c_ResiduatedTransitionSystem_Onormal__sub__rts_Ois__Cong__class_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
normal6700481192199873089od_a_a: ( product_prod_a_a > product_prod_a_a > product_prod_a_a ) > set_Product_prod_a_a > set_Product_prod_a_a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Onormal__sub__rts_Ois__Cong__class_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
normal3204162055925629144_set_a: ( set_set_a > set_set_a > set_set_a ) > set_set_set_a > set_set_set_a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Onormal__sub__rts_Ois__Cong__class_001t__Set__Oset_Itf__a_J,type,
normal4437380936311325560_set_a: ( set_a > set_a > set_a ) > set_set_a > set_set_a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Onormal__sub__rts_Ois__Cong__class_001tf__a,type,
normal8595587647932138008lass_a: ( a > a > a ) > set_a > set_a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Onormal__sub__rts__axioms_001t__Set__Oset_Itf__a_J,type,
normal4776468795420100326_set_a: ( set_a > set_a > set_a ) > set_set_a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Onormal__sub__rts__axioms_001tf__a,type,
normal7698203753654205830ioms_a: ( a > a > a ) > set_a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Opartial__magma_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
partia7170216068986576859_set_a: ( set_set_a > set_set_a > set_set_a ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Opartial__magma_001t__Set__Oset_Itf__a_J,type,
partial_magma_set_a: ( set_a > set_a > set_a ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Opartial__magma_001tf__a,type,
partial_magma_a: ( a > a > a ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Opartial__magma_Onull_001t__Set__Oset_Itf__a_J,type,
partial_null_set_a: ( set_a > set_a > set_a ) > set_a ).
thf(sy_c_ResiduatedTransitionSystem_Opartial__magma_Onull_001tf__a,type,
partial_null_a: ( a > a > a ) > a ).
thf(sy_c_ResiduatedTransitionSystem_Oquotient__by__coherent__normal_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
quotie2905600318227040575od_a_a: ( product_prod_a_a > product_prod_a_a > product_prod_a_a ) > set_Product_prod_a_a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Oquotient__by__coherent__normal_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
quotie4035562349531637206_set_a: ( set_set_a > set_set_a > set_set_a ) > set_set_set_a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Oquotient__by__coherent__normal_001t__Set__Oset_Itf__a_J,type,
quotie5625257012022141046_set_a: ( set_a > set_a > set_a ) > set_set_a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Oquotient__by__coherent__normal_001tf__a,type,
quotie3282664541148387094rmal_a: ( a > a > a ) > set_a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Oquotient__by__coherent__normal_OResid_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
quotie4807335555532242594od_a_a: ( product_prod_a_a > product_prod_a_a > product_prod_a_a ) > set_Product_prod_a_a > set_Product_prod_a_a > set_Product_prod_a_a > set_Product_prod_a_a ).
thf(sy_c_ResiduatedTransitionSystem_Oquotient__by__coherent__normal_OResid_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
quotie4088276940643950137_set_a: ( set_set_a > set_set_a > set_set_a ) > set_set_set_a > set_set_set_a > set_set_set_a > set_set_set_a ).
thf(sy_c_ResiduatedTransitionSystem_Oquotient__by__coherent__normal_OResid_001t__Set__Oset_Itf__a_J,type,
quotie3283642546880816345_set_a: ( set_a > set_a > set_a ) > set_set_a > set_set_a > set_set_a > set_set_a ).
thf(sy_c_ResiduatedTransitionSystem_Oquotient__by__coherent__normal_OResid_001tf__a,type,
quotie8165075472272353145esid_a: ( a > a > a ) > set_a > set_a > set_a > set_a ).
thf(sy_c_ResiduatedTransitionSystem_Oresiduation_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
residu7095500089415647454_set_a: ( set_set_a > set_set_a > set_set_a ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Oresiduation_001t__Set__Oset_Itf__a_J,type,
residuation_set_a: ( set_a > set_a > set_a ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Oresiduation_001tf__a,type,
residuation_a: ( a > a > a ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Oresiduation_Oarr_001t__Set__Oset_Itf__a_J,type,
arr_set_a: ( set_a > set_a > set_a ) > set_a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Oresiduation_Oarr_001tf__a,type,
arr_a: ( a > a > a ) > a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Oresiduation_Ocon_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
con_Product_prod_a_a: ( product_prod_a_a > product_prod_a_a > product_prod_a_a ) > product_prod_a_a > product_prod_a_a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Oresiduation_Ocon_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
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thf(sy_c_ResiduatedTransitionSystem_Oresiduation_Ocon_001t__Set__Oset_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J,type,
con_set_set_set_a: ( set_set_set_a > set_set_set_a > set_set_set_a ) > set_set_set_a > set_set_set_a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Oresiduation_Ocon_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
con_set_set_a: ( set_set_a > set_set_a > set_set_a ) > set_set_a > set_set_a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Oresiduation_Ocon_001t__Set__Oset_Itf__a_J,type,
con_set_a: ( set_a > set_a > set_a ) > set_a > set_a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Oresiduation_Ocon_001tf__a,type,
con_a: ( a > a > a ) > a > a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Oresiduation_Oide_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
ide_set_set_a: ( set_set_a > set_set_a > set_set_a ) > set_set_a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Oresiduation_Oide_001t__Set__Oset_Itf__a_J,type,
ide_set_a: ( set_a > set_a > set_a ) > set_a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Oresiduation_Oide_001tf__a,type,
ide_a: ( a > a > a ) > a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Oresiduation_Otrg_001t__Set__Oset_Itf__a_J,type,
trg_set_a: ( set_a > set_a > set_a ) > set_a > set_a ).
thf(sy_c_ResiduatedTransitionSystem_Oresiduation_Otrg_001tf__a,type,
trg_a: ( a > a > a ) > a > a ).
thf(sy_c_ResiduatedTransitionSystem_Oresiduation__axioms_001t__Set__Oset_Itf__a_J,type,
residu177535419945060507_set_a: ( set_a > set_a > set_a ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Oresiduation__axioms_001tf__a,type,
residuation_axioms_a: ( a > a > a ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Orts_001t__Set__Oset_Itf__a_J,type,
rts_set_a: ( set_a > set_a > set_a ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Orts_001tf__a,type,
rts_a: ( a > a > a ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Orts_Ocoinitial_001t__Set__Oset_Itf__a_J,type,
coinitial_set_a: ( set_a > set_a > set_a ) > set_a > set_a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Orts_Ocoinitial_001tf__a,type,
coinitial_a: ( a > a > a ) > a > a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Orts_Ocomposable_001t__Set__Oset_Itf__a_J,type,
composable_set_a: ( set_a > set_a > set_a ) > set_a > set_a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Orts_Ocomposable_001tf__a,type,
composable_a: ( a > a > a ) > a > a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Orts_Ocomposite__of_001t__Set__Oset_Itf__a_J,type,
composite_of_set_a: ( set_a > set_a > set_a ) > set_a > set_a > set_a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Orts_Ocomposite__of_001tf__a,type,
composite_of_a: ( a > a > a ) > a > a > a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Orts_Ocoterminal_001t__Set__Oset_Itf__a_J,type,
coterminal_set_a: ( set_a > set_a > set_a ) > set_a > set_a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Orts_Ocoterminal_001tf__a,type,
coterminal_a: ( a > a > a ) > a > a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Orts_Ojoin__of_001t__Set__Oset_Itf__a_J,type,
join_of_set_a: ( set_a > set_a > set_a ) > set_a > set_a > set_a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Orts_Ojoin__of_001tf__a,type,
join_of_a: ( a > a > a ) > a > a > a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Orts_Ojoinable_001t__Set__Oset_Itf__a_J,type,
joinable_set_a: ( set_a > set_a > set_a ) > set_a > set_a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Orts_Ojoinable_001tf__a,type,
joinable_a: ( a > a > a ) > a > a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Orts_Oseq_001t__Set__Oset_Itf__a_J,type,
seq_set_a: ( set_a > set_a > set_a ) > set_a > set_a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Orts_Oseq_001tf__a,type,
seq_a: ( a > a > a ) > a > a > $o ).
thf(sy_c_ResiduatedTransitionSystem_Orts_Osources_001t__Set__Oset_Itf__a_J,type,
sources_set_a: ( set_a > set_a > set_a ) > set_a > set_set_a ).
thf(sy_c_ResiduatedTransitionSystem_Orts_Osources_001tf__a,type,
sources_a: ( a > a > a ) > a > set_a ).
thf(sy_c_ResiduatedTransitionSystem_Orts_Otargets_001t__Set__Oset_Itf__a_J,type,
targets_set_a: ( set_a > set_a > set_a ) > set_a > set_set_a ).
thf(sy_c_ResiduatedTransitionSystem_Orts_Otargets_001tf__a,type,
targets_a: ( a > a > a ) > a > set_a ).
thf(sy_c_ResiduatedTransitionSystem_Orts__axioms_001t__Set__Oset_Itf__a_J,type,
rts_axioms_set_a: ( set_a > set_a > set_a ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Orts__axioms_001tf__a,type,
rts_axioms_a: ( a > a > a ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Orts__with__composites_001t__Set__Oset_Itf__a_J,type,
rts_wi6488725688526449878_set_a: ( set_a > set_a > set_a ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Orts__with__composites_001tf__a,type,
rts_wi3777564303360811894ites_a: ( a > a > a ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Orts__with__composites__axioms_001tf__a,type,
rts_wi2614412583573296275ioms_a: ( a > a > a ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Orts__with__joins_001t__Set__Oset_Itf__a_J,type,
rts_with_joins_set_a: ( set_a > set_a > set_a ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Orts__with__joins_001tf__a,type,
rts_with_joins_a: ( a > a > a ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Orts__with__joins__axioms_001t__Set__Oset_Itf__a_J,type,
rts_wi637544758655500588_set_a: ( set_a > set_a > set_a ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Orts__with__joins__axioms_001tf__a,type,
rts_wi560353115624263628ioms_a: ( a > a > a ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Osimulation_001t__Set__Oset_Itf__a_J_001t__Set__Oset_Itf__a_J,type,
simula7336900841036901507_set_a: ( set_a > set_a > set_a ) > ( set_a > set_a > set_a ) > ( set_a > set_a ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Osimulation_001t__Set__Oset_Itf__a_J_001tf__a,type,
simulation_set_a_a: ( set_a > set_a > set_a ) > ( a > a > a ) > ( set_a > a ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Osimulation_001tf__a_001t__Set__Oset_Itf__a_J,type,
simulation_a_set_a: ( a > a > a ) > ( set_a > set_a > set_a ) > ( a > set_a ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Osimulation_001tf__a_001tf__a,type,
simulation_a_a: ( a > a > a ) > ( a > a > a ) > ( a > a ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Osimulation__axioms_001t__Set__Oset_Itf__a_J_001t__Set__Oset_Itf__a_J,type,
simula8704200824037452966_set_a: ( set_a > set_a > set_a ) > ( set_a > set_a > set_a ) > ( set_a > set_a ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Osimulation__axioms_001t__Set__Oset_Itf__a_J_001tf__a,type,
simula3408835310535287622et_a_a: ( set_a > set_a > set_a ) > ( a > a > a ) > ( set_a > a ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Osimulation__axioms_001tf__a_001t__Set__Oset_Itf__a_J,type,
simula3192323252075944454_set_a: ( a > a > a ) > ( set_a > set_a > set_a ) > ( a > set_a ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Osimulation__axioms_001tf__a_001tf__a,type,
simula3868467710248865958ms_a_a: ( a > a > a ) > ( a > a > a ) > ( a > a ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Osimulation__between__extensional__rts_001t__Set__Oset_Itf__a_J_001t__Set__Oset_Itf__a_J,type,
simula5219500460978879148_set_a: ( set_a > set_a > set_a ) > ( set_a > set_a > set_a ) > ( set_a > set_a ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Osimulation__between__extensional__rts_001t__Set__Oset_Itf__a_J_001tf__a,type,
simula4722311157971863884et_a_a: ( set_a > set_a > set_a ) > ( a > a > a ) > ( set_a > a ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Osimulation__between__extensional__rts_001tf__a_001t__Set__Oset_Itf__a_J,type,
simula4505799099512520716_set_a: ( a > a > a ) > ( set_a > set_a > set_a ) > ( a > set_a ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Osimulation__between__extensional__rts_001tf__a_001tf__a,type,
simula722159513454644908ts_a_a: ( a > a > a ) > ( a > a > a ) > ( a > a ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Osimulation__to__weakly__extensional__rts_001tf__a_001t__Set__Oset_Itf__a_J,type,
simula7881043605922138218_set_a: ( a > a > a ) > ( set_a > set_a > set_a ) > ( a > set_a ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Osimulation__to__weakly__extensional__rts_001tf__a_001tf__a,type,
simula2709571904647515914ts_a_a: ( a > a > a ) > ( a > a > a ) > ( a > a ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Otransformation_001t__Set__Oset_Itf__a_J_001t__Set__Oset_Itf__a_J,type,
transf6002003789407478149_set_a: ( set_a > set_a > set_a ) > ( set_a > set_a > set_a ) > ( set_a > set_a ) > ( set_a > set_a ) > ( set_a > set_a ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Otransformation_001t__Set__Oset_Itf__a_J_001tf__a,type,
transf4346523582779806757et_a_a: ( set_a > set_a > set_a ) > ( a > a > a ) > ( set_a > a ) > ( set_a > a ) > ( set_a > a ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Otransformation_001tf__a_001t__Set__Oset_Itf__a_J,type,
transf4130011524320463589_set_a: ( a > a > a ) > ( set_a > set_a > set_a ) > ( a > set_a ) > ( a > set_a ) > ( a > set_a ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Otransformation_001tf__a_001tf__a,type,
transformation_a_a: ( a > a > a ) > ( a > a > a ) > ( a > a ) > ( a > a ) > ( a > a ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Otransformation__axioms_001t__Set__Oset_Itf__a_J_001t__Set__Oset_Itf__a_J,type,
transf2960116383903194536_set_a: ( set_a > set_a > set_a ) > ( set_a > set_a > set_a ) > ( set_a > set_a ) > ( set_a > set_a ) > ( set_a > set_a ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Otransformation__axioms_001t__Set__Oset_Itf__a_J_001tf__a,type,
transf1935308705569152072et_a_a: ( set_a > set_a > set_a ) > ( a > a > a ) > ( set_a > a ) > ( set_a > a ) > ( set_a > a ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Otransformation__axioms_001tf__a_001t__Set__Oset_Itf__a_J,type,
transf1718796647109808904_set_a: ( a > a > a ) > ( set_a > set_a > set_a ) > ( a > set_a ) > ( a > set_a ) > ( a > set_a ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Otransformation__axioms_001tf__a_001tf__a,type,
transf4446446367311712680ms_a_a: ( a > a > a ) > ( a > a > a ) > ( a > a ) > ( a > a ) > ( a > a ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Oweakly__extensional__rts_001t__Set__Oset_Itf__a_J,type,
weakly5936471160286156245_set_a: ( set_a > set_a > set_a ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Oweakly__extensional__rts_001tf__a,type,
weakly1626779504270821493_rts_a: ( a > a > a ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Oweakly__extensional__rts_Osrc_001t__Set__Oset_Itf__a_J,type,
weakly2061155085811118449_set_a: ( set_a > set_a > set_a ) > set_a > set_a ).
thf(sy_c_ResiduatedTransitionSystem_Oweakly__extensional__rts_Osrc_001tf__a,type,
weakly8512939796511659025_src_a: ( a > a > a ) > a > a ).
thf(sy_c_ResiduatedTransitionSystem_Oweakly__extensional__rts__axioms_001t__Set__Oset_Itf__a_J,type,
weakly1639064658423616754_set_a: ( set_a > set_a > set_a ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Oweakly__extensional__rts__axioms_001tf__a,type,
weakly311909585050745746ioms_a: ( a > a > a ) > $o ).
thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
collec3336397797384452498od_a_a: ( product_prod_a_a > $o ) > set_Product_prod_a_a ).
thf(sy_c_Set_OCollect_001t__Set__Oset_Itf__a_J,type,
collect_set_a: ( set_a > $o ) > set_set_a ).
thf(sy_c_Set_OCollect_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_Set_Obind_001tf__a_001tf__a,type,
bind_a_a: set_a > ( a > set_a ) > set_a ).
thf(sy_c_Set_Oimage_001tf__a_001tf__a,type,
image_a_a: ( a > a ) > set_a > set_a ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_Itf__a_J,type,
insert_set_a: set_a > set_set_a > set_set_a ).
thf(sy_c_Set_Oinsert_001tf__a,type,
insert_a: a > set_a > set_a ).
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_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_fChoice_001t__Product____Type__Oprod_It__Set__Oset_Itf__a_J_Mt__Set__Oset_Itf__a_J_J,type,
fChoic7046651920289422459_set_a: ( produc1703568184450464039_set_a > $o ) > produc1703568184450464039_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_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_It__Set__Oset_Itf__a_J_J,type,
member_set_set_a: set_set_a > set_set_set_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_thesis____,type,
thesis: $o ).
% 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_Ocon__sym,axiom,
! [T: a,U: a] :
( ( con_a @ resid @ T @ U )
=> ( con_a @ resid @ U @ T ) ) ).
% R.con_sym
thf(fact_3_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_4_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_5_N_OCong_092_060_094sub_0620__transitive,axiom,
! [T: a,T4: a,T5: a] :
( ( ( member_a @ ( resid @ T @ T4 ) @ nn )
& ( member_a @ ( resid @ T4 @ T ) @ nn ) )
=> ( ( ( member_a @ ( resid @ T4 @ T5 ) @ nn )
& ( member_a @ ( resid @ T5 @ T4 ) @ nn ) )
=> ( ( member_a @ ( resid @ T @ T5 ) @ nn )
& ( member_a @ ( resid @ T5 @ T ) @ nn ) ) ) ) ).
% N.Cong\<^sub>0_transitive
thf(fact_6_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_7_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_8_cube,axiom,
! [V: set_a,T: set_a,U: set_a] :
( ( quotie8165075472272353145esid_a @ resid @ nn @ ( quotie8165075472272353145esid_a @ resid @ nn @ V @ T ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ U @ T ) )
= ( quotie8165075472272353145esid_a @ resid @ nn @ ( quotie8165075472272353145esid_a @ resid @ nn @ V @ U ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ T @ U ) ) ) ).
% cube
thf(fact_9_ex__un__null,axiom,
? [X: set_a] :
( ! [T2: set_a] :
( ( ( quotie8165075472272353145esid_a @ resid @ nn @ X @ T2 )
= X )
& ( ( quotie8165075472272353145esid_a @ resid @ nn @ T2 @ X )
= X ) )
& ! [Y: set_a] :
( ! [T3: set_a] :
( ( ( quotie8165075472272353145esid_a @ resid @ nn @ Y @ T3 )
= Y )
& ( ( quotie8165075472272353145esid_a @ resid @ nn @ T3 @ Y )
= Y ) )
=> ( Y = X ) ) ) ).
% ex_un_null
thf(fact_10_N_OCong_092_060_094sub_0620__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_11_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_12_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_13_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_14_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_15_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_16__092_060T_062_092_060U_062,axiom,
ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ t @ u ) ).
% \<T>\<U>
thf(fact_17_quotient__by__coherent__normal__axioms,axiom,
quotie3282664541148387094rmal_a @ resid @ nn ).
% quotient_by_coherent_normal_axioms
thf(fact_18_residuation_Ocon_Ocong,axiom,
con_set_set_a = con_set_set_a ).
% residuation.con.cong
thf(fact_19_residuation_Ocon_Ocong,axiom,
con_a = con_a ).
% residuation.con.cong
thf(fact_20_residuation_Ocon_Ocong,axiom,
con_set_a = con_set_a ).
% residuation.con.cong
thf(fact_21_normal__sub__rts_OCong__class_Ocong,axiom,
normal2962378890657961070_set_a = normal2962378890657961070_set_a ).
% normal_sub_rts.Cong_class.cong
thf(fact_22_normal__sub__rts_OCong__class_Ocong,axiom,
normal7408713899360725774lass_a = normal7408713899360725774lass_a ).
% normal_sub_rts.Cong_class.cong
thf(fact_23_quotient__by__coherent__normal_OResid_Ocong,axiom,
quotie3283642546880816345_set_a = quotie3283642546880816345_set_a ).
% quotient_by_coherent_normal.Resid.cong
thf(fact_24_quotient__by__coherent__normal_OResid_Ocong,axiom,
quotie8165075472272353145esid_a = quotie8165075472272353145esid_a ).
% quotient_by_coherent_normal.Resid.cong
thf(fact_25_partial__magma__axioms,axiom,
partial_magma_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) ).
% partial_magma_axioms
thf(fact_26_R_Opartial__magma__axioms,axiom,
partial_magma_a @ resid ).
% R.partial_magma_axioms
thf(fact_27__092_060U_062_092_060V_062,axiom,
con_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ u @ v ).
% \<U>\<V>
thf(fact_28_N_Ocoherent__normal__sub__rts__axioms,axiom,
cohere6072184133013167079_rts_a @ resid @ nn ).
% N.coherent_normal_sub_rts_axioms
thf(fact_29_Resid__by__members,axiom,
! [T6: set_a,U3: set_a,T: a,U: a] :
( ( normal8595587647932138008lass_a @ resid @ nn @ T6 )
=> ( ( normal8595587647932138008lass_a @ resid @ nn @ U3 )
=> ( ( member_a @ T @ T6 )
=> ( ( member_a @ U @ U3 )
=> ( ( con_a @ resid @ T @ U )
=> ( ( quotie8165075472272353145esid_a @ resid @ nn @ T6 @ U3 )
= ( normal7408713899360725774lass_a @ resid @ nn @ ( resid @ T @ U ) ) ) ) ) ) ) ) ).
% Resid_by_members
thf(fact_30_residuation__axioms,axiom,
residuation_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) ).
% residuation_axioms
thf(fact_31_Con__witnesses,axiom,
! [T6: set_a,U3: set_a,T: a,U: a] :
( ( ( quotie8165075472272353145esid_a @ resid @ nn @ T6 @ U3 )
!= bot_bot_set_a )
=> ( ( member_a @ T @ T6 )
=> ( ( member_a @ U @ U3 )
=> ? [V2: a,W: a] :
( ( member_a @ V2 @ nn )
& ( member_a @ W @ nn )
& ( con_a @ resid @ ( resid @ T @ V2 ) @ ( resid @ U @ W ) ) ) ) ) ) ).
% Con_witnesses
thf(fact_32_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_33_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_34_Arr__Resid,axiom,
! [T6: set_a,U3: set_a] :
( ( ( quotie8165075472272353145esid_a @ resid @ nn @ T6 @ U3 )
!= bot_bot_set_a )
=> ( ( quotie8165075472272353145esid_a @ resid @ nn @ ( quotie8165075472272353145esid_a @ resid @ nn @ T6 @ U3 ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ T6 @ U3 ) )
!= bot_bot_set_a ) ) ).
% Arr_Resid
thf(fact_35_Con__sym,axiom,
! [T6: set_a,U3: set_a] :
( ( ( quotie8165075472272353145esid_a @ resid @ nn @ T6 @ U3 )
!= bot_bot_set_a )
=> ( ( quotie8165075472272353145esid_a @ resid @ nn @ U3 @ T6 )
!= bot_bot_set_a ) ) ).
% Con_sym
thf(fact_36_Cube,axiom,
! [V3: set_a,T6: set_a,U3: set_a] :
( ( ( quotie8165075472272353145esid_a @ resid @ nn @ ( quotie8165075472272353145esid_a @ resid @ nn @ V3 @ T6 ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ U3 @ T6 ) )
!= bot_bot_set_a )
=> ( ( quotie8165075472272353145esid_a @ resid @ nn @ ( quotie8165075472272353145esid_a @ resid @ nn @ V3 @ T6 ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ U3 @ T6 ) )
= ( quotie8165075472272353145esid_a @ resid @ nn @ ( quotie8165075472272353145esid_a @ resid @ nn @ V3 @ U3 ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ T6 @ U3 ) ) ) ) ).
% Cube
thf(fact_37_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_38_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_39_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_40_con__sym,axiom,
! [T: set_a,U: set_a] :
( ( con_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U )
=> ( con_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ U @ T ) ) ).
% con_sym
thf(fact_41_N_OCong__class__is__nonempty,axiom,
! [T6: set_a] :
( ( normal8595587647932138008lass_a @ resid @ nn @ T6 )
=> ( T6 != bot_bot_set_a ) ) ).
% N.Cong_class_is_nonempty
thf(fact_42_N_Ois__Cong__class__def,axiom,
! [T6: set_a] :
( ( normal8595587647932138008lass_a @ resid @ nn @ T6 )
= ( ? [T7: a] :
( ( member_a @ T7 @ T6 )
& ( T6
= ( normal7408713899360725774lass_a @ resid @ nn @ T7 ) ) ) ) ) ).
% N.is_Cong_class_def
thf(fact_43_is__Cong__class__Resid,axiom,
! [T6: set_a,U3: set_a] :
( ( ( quotie8165075472272353145esid_a @ resid @ nn @ T6 @ U3 )
!= bot_bot_set_a )
=> ( normal8595587647932138008lass_a @ resid @ nn @ ( quotie8165075472272353145esid_a @ resid @ nn @ T6 @ U3 ) ) ) ).
% is_Cong_class_Resid
thf(fact_44_ide__def,axiom,
! [A: set_a] :
( ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ A )
= ( ( con_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ A @ A )
& ( ( quotie8165075472272353145esid_a @ resid @ nn @ A @ A )
= A ) ) ) ).
% ide_def
thf(fact_45_ideE,axiom,
! [A: set_a] :
( ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ A )
=> ~ ( ( con_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ A @ A )
=> ( ( quotie8165075472272353145esid_a @ resid @ nn @ A @ A )
!= A ) ) ) ).
% ideE
thf(fact_46__C1_C,axiom,
! [A2: set_a,T6: set_a] :
( ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ A2 )
=> ( ( con_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T6 @ A2 )
=> ( ( quotie8165075472272353145esid_a @ resid @ nn @ T6 @ A2 )
= T6 ) ) ) ).
% "1"
thf(fact_47__092_060open_062_092_060And_062_092_060U_062_A_092_060T_062_O_Acon_A_092_060T_062_A_092_060U_062_A_092_060Longrightarrow_062_A_092_060exists_062_092_060A_062_O_Aide_A_092_060A_062_A_092_060and_062_Acon_A_092_060A_062_A_092_060T_062_A_092_060and_062_Acon_A_092_060A_062_A_092_060U_062_092_060close_062,axiom,
! [T6: set_a,U3: set_a] :
( ( con_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T6 @ U3 )
=> ? [A3: set_a] :
( ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ A3 )
& ( con_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ A3 @ T6 )
& ( con_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ A3 @ U3 ) ) ) ).
% \<open>\<And>\<U> \<T>. con \<T> \<U> \<Longrightarrow> \<exists>\<A>. ide \<A> \<and> con \<A> \<T> \<and> con \<A> \<U>\<close>
thf(fact_48__092_060open_062_092_060And_062_092_060T_062_A_092_060A_062_O_A_092_060lbrakk_062ide_A_092_060A_062_059_Acon_A_092_060A_062_A_092_060T_062_092_060rbrakk_062_A_092_060Longrightarrow_062_Aide_A_I_092_060A_062_A_092_060lbrace_062_092_092_060rbrace_062_A_092_060T_062_J_092_060close_062,axiom,
! [A2: set_a,T6: set_a] :
( ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ A2 )
=> ( ( con_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ A2 @ T6 )
=> ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ A2 @ T6 ) ) ) ) ).
% \<open>\<And>\<T> \<A>. \<lbrakk>ide \<A>; con \<A> \<T>\<rbrakk> \<Longrightarrow> ide (\<A> \<lbrace>\\<rbrace> \<T>)\<close>
thf(fact_49_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_50_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_51_mem__Collect__eq,axiom,
! [A: a,P: a > $o] :
( ( member_a @ A @ ( collect_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_52_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_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_con__char_092_060_094sub_062Q_092_060_094sub_062C_092_060_094sub_062N,axiom,
! [T6: set_a,U3: set_a] :
( ( con_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T6 @ U3 )
= ( ( normal8595587647932138008lass_a @ resid @ nn @ T6 )
& ( normal8595587647932138008lass_a @ resid @ nn @ U3 )
& ? [T7: a,U4: a] :
( ( member_a @ T7 @ T6 )
& ( member_a @ U4 @ U3 )
& ( con_a @ resid @ T7 @ U4 ) ) ) ) ).
% con_char\<^sub>Q\<^sub>C\<^sub>N
thf(fact_58_Con__char,axiom,
! [T6: set_a,U3: set_a] :
( ( ( quotie8165075472272353145esid_a @ resid @ nn @ T6 @ U3 )
!= bot_bot_set_a )
= ( ( normal8595587647932138008lass_a @ resid @ nn @ T6 )
& ( normal8595587647932138008lass_a @ resid @ nn @ U3 )
& ? [T7: a,U4: a] :
( ( member_a @ T7 @ T6 )
& ( member_a @ U4 @ U3 )
& ( con_a @ resid @ T7 @ U4 ) ) ) ) ).
% Con_char
thf(fact_59_ideI,axiom,
! [A: set_a] :
( ( con_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ A @ A )
=> ( ( ( quotie8165075472272353145esid_a @ resid @ nn @ A @ A )
= A )
=> ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ A ) ) ) ).
% ideI
thf(fact_60_N_Orep__in__Cong__class,axiom,
! [T6: set_a] :
( ( normal8595587647932138008lass_a @ resid @ nn @ T6 )
=> ( member_a @ ( normal3259722184653208495_rep_a @ T6 ) @ T6 ) ) ).
% N.rep_in_Cong_class
thf(fact_61_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_62_residuation_Oaxioms_I1_J,axiom,
! [Resid: a > a > a] :
( ( residuation_a @ Resid )
=> ( partial_magma_a @ Resid ) ) ).
% residuation.axioms(1)
thf(fact_63_quotient__by__coherent__normal_Oaxioms_I2_J,axiom,
! [Resid: set_a > set_a > set_a,NN: set_set_a] :
( ( quotie5625257012022141046_set_a @ Resid @ NN )
=> ( cohere6325062230080414023_set_a @ Resid @ NN ) ) ).
% quotient_by_coherent_normal.axioms(2)
thf(fact_64_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_65_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_66_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_67_rts_Ojoinable_Ocong,axiom,
joinable_set_a = joinable_set_a ).
% rts.joinable.cong
thf(fact_68_rts_Ojoinable_Ocong,axiom,
joinable_a = joinable_a ).
% rts.joinable.cong
thf(fact_69_rts_Ocoinitial_Ocong,axiom,
coinitial_set_a = coinitial_set_a ).
% rts.coinitial.cong
thf(fact_70_rts_Ocoinitial_Ocong,axiom,
coinitial_a = coinitial_a ).
% rts.coinitial.cong
thf(fact_71_normal__sub__rts_Ois__Cong__class_Ocong,axiom,
normal4437380936311325560_set_a = normal4437380936311325560_set_a ).
% normal_sub_rts.is_Cong_class.cong
thf(fact_72_normal__sub__rts_Ois__Cong__class_Ocong,axiom,
normal8595587647932138008lass_a = normal8595587647932138008lass_a ).
% normal_sub_rts.is_Cong_class.cong
thf(fact_73_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,T6: set_Product_prod_a_a,U3: set_Product_prod_a_a] :
( ( quotie2905600318227040575od_a_a @ Resid @ NN )
=> ( ( ( quotie4807335555532242594od_a_a @ Resid @ NN @ T6 @ U3 )
!= bot_bo3357376287454694259od_a_a )
=> ( normal6700481192199873089od_a_a @ Resid @ NN @ ( quotie4807335555532242594od_a_a @ Resid @ NN @ T6 @ U3 ) ) ) ) ).
% quotient_by_coherent_normal.is_Cong_class_Resid
thf(fact_74_quotient__by__coherent__normal_Ois__Cong__class__Resid,axiom,
! [Resid: set_a > set_a > set_a,NN: set_set_a,T6: set_set_a,U3: set_set_a] :
( ( quotie5625257012022141046_set_a @ Resid @ NN )
=> ( ( ( quotie3283642546880816345_set_a @ Resid @ NN @ T6 @ U3 )
!= bot_bot_set_set_a )
=> ( normal4437380936311325560_set_a @ Resid @ NN @ ( quotie3283642546880816345_set_a @ Resid @ NN @ T6 @ U3 ) ) ) ) ).
% quotient_by_coherent_normal.is_Cong_class_Resid
thf(fact_75_quotient__by__coherent__normal_Ois__Cong__class__Resid,axiom,
! [Resid: a > a > a,NN: set_a,T6: set_a,U3: set_a] :
( ( quotie3282664541148387094rmal_a @ Resid @ NN )
=> ( ( ( quotie8165075472272353145esid_a @ Resid @ NN @ T6 @ U3 )
!= bot_bot_set_a )
=> ( normal8595587647932138008lass_a @ Resid @ NN @ ( quotie8165075472272353145esid_a @ Resid @ NN @ T6 @ U3 ) ) ) ) ).
% quotient_by_coherent_normal.is_Cong_class_Resid
thf(fact_76_partial__magma__def,axiom,
( partial_magma_set_a
= ( ^ [OP: set_a > set_a > set_a] :
? [X2: set_a] :
( ! [T7: set_a] :
( ( ( OP @ X2 @ T7 )
= X2 )
& ( ( OP @ T7 @ X2 )
= X2 ) )
& ! [Y2: set_a] :
( ! [T7: set_a] :
( ( ( OP @ Y2 @ T7 )
= Y2 )
& ( ( OP @ T7 @ Y2 )
= Y2 ) )
=> ( Y2 = X2 ) ) ) ) ) ).
% partial_magma_def
thf(fact_77_partial__magma__def,axiom,
( partial_magma_a
= ( ^ [OP: a > a > a] :
? [X2: a] :
( ! [T7: a] :
( ( ( OP @ X2 @ T7 )
= X2 )
& ( ( OP @ T7 @ X2 )
= X2 ) )
& ! [Y2: a] :
( ! [T7: a] :
( ( ( OP @ Y2 @ T7 )
= Y2 )
& ( ( OP @ T7 @ Y2 )
= Y2 ) )
=> ( Y2 = X2 ) ) ) ) ) ).
% partial_magma_def
thf(fact_78_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_79_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_80_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_81_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_82_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,T6: set_Product_prod_a_a,U3: set_Product_prod_a_a] :
( ( quotie2905600318227040575od_a_a @ Resid @ NN )
=> ( ( ( quotie4807335555532242594od_a_a @ Resid @ NN @ T6 @ U3 )
!= bot_bo3357376287454694259od_a_a )
=> ( ( quotie4807335555532242594od_a_a @ Resid @ NN @ ( quotie4807335555532242594od_a_a @ Resid @ NN @ T6 @ U3 ) @ ( quotie4807335555532242594od_a_a @ Resid @ NN @ T6 @ U3 ) )
!= bot_bo3357376287454694259od_a_a ) ) ) ).
% quotient_by_coherent_normal.Arr_Resid
thf(fact_83_quotient__by__coherent__normal_OArr__Resid,axiom,
! [Resid: set_a > set_a > set_a,NN: set_set_a,T6: set_set_a,U3: set_set_a] :
( ( quotie5625257012022141046_set_a @ Resid @ NN )
=> ( ( ( quotie3283642546880816345_set_a @ Resid @ NN @ T6 @ U3 )
!= bot_bot_set_set_a )
=> ( ( quotie3283642546880816345_set_a @ Resid @ NN @ ( quotie3283642546880816345_set_a @ Resid @ NN @ T6 @ U3 ) @ ( quotie3283642546880816345_set_a @ Resid @ NN @ T6 @ U3 ) )
!= bot_bot_set_set_a ) ) ) ).
% quotient_by_coherent_normal.Arr_Resid
thf(fact_84_quotient__by__coherent__normal_OArr__Resid,axiom,
! [Resid: a > a > a,NN: set_a,T6: set_a,U3: set_a] :
( ( quotie3282664541148387094rmal_a @ Resid @ NN )
=> ( ( ( quotie8165075472272353145esid_a @ Resid @ NN @ T6 @ U3 )
!= bot_bot_set_a )
=> ( ( quotie8165075472272353145esid_a @ Resid @ NN @ ( quotie8165075472272353145esid_a @ Resid @ NN @ T6 @ U3 ) @ ( quotie8165075472272353145esid_a @ Resid @ NN @ T6 @ U3 ) )
!= bot_bot_set_a ) ) ) ).
% quotient_by_coherent_normal.Arr_Resid
thf(fact_85_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,T6: set_Product_prod_a_a,U3: set_Product_prod_a_a] :
( ( quotie2905600318227040575od_a_a @ Resid @ NN )
=> ( ( ( quotie4807335555532242594od_a_a @ Resid @ NN @ T6 @ U3 )
!= bot_bo3357376287454694259od_a_a )
=> ( ( quotie4807335555532242594od_a_a @ Resid @ NN @ U3 @ T6 )
!= bot_bo3357376287454694259od_a_a ) ) ) ).
% quotient_by_coherent_normal.Con_sym
thf(fact_86_quotient__by__coherent__normal_OCon__sym,axiom,
! [Resid: set_a > set_a > set_a,NN: set_set_a,T6: set_set_a,U3: set_set_a] :
( ( quotie5625257012022141046_set_a @ Resid @ NN )
=> ( ( ( quotie3283642546880816345_set_a @ Resid @ NN @ T6 @ U3 )
!= bot_bot_set_set_a )
=> ( ( quotie3283642546880816345_set_a @ Resid @ NN @ U3 @ T6 )
!= bot_bot_set_set_a ) ) ) ).
% quotient_by_coherent_normal.Con_sym
thf(fact_87_quotient__by__coherent__normal_OCon__sym,axiom,
! [Resid: a > a > a,NN: set_a,T6: set_a,U3: set_a] :
( ( quotie3282664541148387094rmal_a @ Resid @ NN )
=> ( ( ( quotie8165075472272353145esid_a @ Resid @ NN @ T6 @ U3 )
!= bot_bot_set_a )
=> ( ( quotie8165075472272353145esid_a @ Resid @ NN @ U3 @ T6 )
!= bot_bot_set_a ) ) ) ).
% quotient_by_coherent_normal.Con_sym
thf(fact_88_quotient__by__coherent__normal_OCube,axiom,
! [Resid: product_prod_a_a > product_prod_a_a > product_prod_a_a,NN: set_Product_prod_a_a,V3: set_Product_prod_a_a,T6: set_Product_prod_a_a,U3: set_Product_prod_a_a] :
( ( quotie2905600318227040575od_a_a @ Resid @ NN )
=> ( ( ( quotie4807335555532242594od_a_a @ Resid @ NN @ ( quotie4807335555532242594od_a_a @ Resid @ NN @ V3 @ T6 ) @ ( quotie4807335555532242594od_a_a @ Resid @ NN @ U3 @ T6 ) )
!= bot_bo3357376287454694259od_a_a )
=> ( ( quotie4807335555532242594od_a_a @ Resid @ NN @ ( quotie4807335555532242594od_a_a @ Resid @ NN @ V3 @ T6 ) @ ( quotie4807335555532242594od_a_a @ Resid @ NN @ U3 @ T6 ) )
= ( quotie4807335555532242594od_a_a @ Resid @ NN @ ( quotie4807335555532242594od_a_a @ Resid @ NN @ V3 @ U3 ) @ ( quotie4807335555532242594od_a_a @ Resid @ NN @ T6 @ U3 ) ) ) ) ) ).
% quotient_by_coherent_normal.Cube
thf(fact_89_quotient__by__coherent__normal_OCube,axiom,
! [Resid: set_a > set_a > set_a,NN: set_set_a,V3: set_set_a,T6: set_set_a,U3: set_set_a] :
( ( quotie5625257012022141046_set_a @ Resid @ NN )
=> ( ( ( quotie3283642546880816345_set_a @ Resid @ NN @ ( quotie3283642546880816345_set_a @ Resid @ NN @ V3 @ T6 ) @ ( quotie3283642546880816345_set_a @ Resid @ NN @ U3 @ T6 ) )
!= bot_bot_set_set_a )
=> ( ( quotie3283642546880816345_set_a @ Resid @ NN @ ( quotie3283642546880816345_set_a @ Resid @ NN @ V3 @ T6 ) @ ( quotie3283642546880816345_set_a @ Resid @ NN @ U3 @ T6 ) )
= ( quotie3283642546880816345_set_a @ Resid @ NN @ ( quotie3283642546880816345_set_a @ Resid @ NN @ V3 @ U3 ) @ ( quotie3283642546880816345_set_a @ Resid @ NN @ T6 @ U3 ) ) ) ) ) ).
% quotient_by_coherent_normal.Cube
thf(fact_90_quotient__by__coherent__normal_OCube,axiom,
! [Resid: a > a > a,NN: set_a,V3: set_a,T6: set_a,U3: set_a] :
( ( quotie3282664541148387094rmal_a @ Resid @ NN )
=> ( ( ( quotie8165075472272353145esid_a @ Resid @ NN @ ( quotie8165075472272353145esid_a @ Resid @ NN @ V3 @ T6 ) @ ( quotie8165075472272353145esid_a @ Resid @ NN @ U3 @ T6 ) )
!= bot_bot_set_a )
=> ( ( quotie8165075472272353145esid_a @ Resid @ NN @ ( quotie8165075472272353145esid_a @ Resid @ NN @ V3 @ T6 ) @ ( quotie8165075472272353145esid_a @ Resid @ NN @ U3 @ T6 ) )
= ( quotie8165075472272353145esid_a @ Resid @ NN @ ( quotie8165075472272353145esid_a @ Resid @ NN @ V3 @ U3 ) @ ( quotie8165075472272353145esid_a @ Resid @ NN @ T6 @ U3 ) ) ) ) ) ).
% quotient_by_coherent_normal.Cube
thf(fact_91_quotient__by__coherent__normal_Ois__residuation,axiom,
! [Resid: set_a > set_a > set_a,NN: set_set_a] :
( ( quotie5625257012022141046_set_a @ Resid @ NN )
=> ( residu7095500089415647454_set_a @ ( quotie3283642546880816345_set_a @ Resid @ NN ) ) ) ).
% quotient_by_coherent_normal.is_residuation
thf(fact_92_quotient__by__coherent__normal_Ois__residuation,axiom,
! [Resid: a > a > a,NN: set_a] :
( ( quotie3282664541148387094rmal_a @ Resid @ NN )
=> ( residuation_set_a @ ( quotie8165075472272353145esid_a @ Resid @ NN ) ) ) ).
% quotient_by_coherent_normal.is_residuation
thf(fact_93_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,T6: set_Product_prod_a_a,U3: set_Product_prod_a_a] :
( ( quotie2905600318227040575od_a_a @ Resid @ NN )
=> ( ( ( quotie4807335555532242594od_a_a @ Resid @ NN @ T6 @ U3 )
!= bot_bo3357376287454694259od_a_a )
= ( ( normal6700481192199873089od_a_a @ Resid @ NN @ T6 )
& ( normal6700481192199873089od_a_a @ Resid @ NN @ U3 )
& ? [T7: product_prod_a_a,U4: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ T7 @ T6 )
& ( member1426531477525435216od_a_a @ U4 @ U3 )
& ( con_Product_prod_a_a @ Resid @ T7 @ U4 ) ) ) ) ) ).
% quotient_by_coherent_normal.Con_char
thf(fact_94_quotient__by__coherent__normal_OCon__char,axiom,
! [Resid: set_set_a > set_set_a > set_set_a,NN: set_set_set_a,T6: set_set_set_a,U3: set_set_set_a] :
( ( quotie4035562349531637206_set_a @ Resid @ NN )
=> ( ( ( quotie4088276940643950137_set_a @ Resid @ NN @ T6 @ U3 )
!= bot_bo3380559777022489994_set_a )
= ( ( normal3204162055925629144_set_a @ Resid @ NN @ T6 )
& ( normal3204162055925629144_set_a @ Resid @ NN @ U3 )
& ? [T7: set_set_a,U4: set_set_a] :
( ( member_set_set_a @ T7 @ T6 )
& ( member_set_set_a @ U4 @ U3 )
& ( con_set_set_a @ Resid @ T7 @ U4 ) ) ) ) ) ).
% quotient_by_coherent_normal.Con_char
thf(fact_95_quotient__by__coherent__normal_OCon__char,axiom,
! [Resid: set_a > set_a > set_a,NN: set_set_a,T6: set_set_a,U3: set_set_a] :
( ( quotie5625257012022141046_set_a @ Resid @ NN )
=> ( ( ( quotie3283642546880816345_set_a @ Resid @ NN @ T6 @ U3 )
!= bot_bot_set_set_a )
= ( ( normal4437380936311325560_set_a @ Resid @ NN @ T6 )
& ( normal4437380936311325560_set_a @ Resid @ NN @ U3 )
& ? [T7: set_a,U4: set_a] :
( ( member_set_a @ T7 @ T6 )
& ( member_set_a @ U4 @ U3 )
& ( con_set_a @ Resid @ T7 @ U4 ) ) ) ) ) ).
% quotient_by_coherent_normal.Con_char
thf(fact_96_quotient__by__coherent__normal_OCon__char,axiom,
! [Resid: a > a > a,NN: set_a,T6: set_a,U3: set_a] :
( ( quotie3282664541148387094rmal_a @ Resid @ NN )
=> ( ( ( quotie8165075472272353145esid_a @ Resid @ NN @ T6 @ U3 )
!= bot_bot_set_a )
= ( ( normal8595587647932138008lass_a @ Resid @ NN @ T6 )
& ( normal8595587647932138008lass_a @ Resid @ NN @ U3 )
& ? [T7: a,U4: a] :
( ( member_a @ T7 @ T6 )
& ( member_a @ U4 @ U3 )
& ( con_a @ Resid @ T7 @ U4 ) ) ) ) ) ).
% quotient_by_coherent_normal.Con_char
thf(fact_97_quotient__by__coherent__normal_Ois__partial__magma,axiom,
! [Resid: set_a > set_a > set_a,NN: set_set_a] :
( ( quotie5625257012022141046_set_a @ Resid @ NN )
=> ( partia7170216068986576859_set_a @ ( quotie3283642546880816345_set_a @ Resid @ NN ) ) ) ).
% quotient_by_coherent_normal.is_partial_magma
thf(fact_98_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_99_residuation_Oide__def,axiom,
! [Resid: set_set_a > set_set_a > set_set_a,A: set_set_a] :
( ( residu7095500089415647454_set_a @ Resid )
=> ( ( ide_set_set_a @ Resid @ A )
= ( ( con_set_set_a @ Resid @ A @ A )
& ( ( Resid @ A @ A )
= A ) ) ) ) ).
% residuation.ide_def
thf(fact_100_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_101_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_102_residuation_OideI,axiom,
! [Resid: set_set_a > set_set_a > set_set_a,A: set_set_a] :
( ( residu7095500089415647454_set_a @ Resid )
=> ( ( con_set_set_a @ Resid @ A @ A )
=> ( ( ( Resid @ A @ A )
= A )
=> ( ide_set_set_a @ Resid @ A ) ) ) ) ).
% residuation.ideI
thf(fact_103_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_104_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_105_residuation_OideE,axiom,
! [Resid: set_set_a > set_set_a > set_set_a,A: set_set_a] :
( ( residu7095500089415647454_set_a @ Resid )
=> ( ( ide_set_set_a @ Resid @ A )
=> ~ ( ( con_set_set_a @ Resid @ A @ A )
=> ( ( Resid @ A @ A )
!= A ) ) ) ) ).
% residuation.ideE
thf(fact_106_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_107_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_108_residuation_Oide_Ocong,axiom,
ide_set_a = ide_set_a ).
% residuation.ide.cong
thf(fact_109_residuation_Oide_Ocong,axiom,
ide_a = ide_a ).
% residuation.ide.cong
thf(fact_110_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,T6: set_Product_prod_a_a,U3: 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 @ T6 @ U3 )
!= bot_bo3357376287454694259od_a_a )
=> ( ( member1426531477525435216od_a_a @ T @ T6 )
=> ( ( member1426531477525435216od_a_a @ U @ U3 )
=> ? [V2: product_prod_a_a,W: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ V2 @ NN )
& ( member1426531477525435216od_a_a @ W @ NN )
& ( con_Product_prod_a_a @ Resid @ ( Resid @ T @ V2 ) @ ( Resid @ U @ W ) ) ) ) ) ) ) ).
% quotient_by_coherent_normal.Con_witnesses
thf(fact_111_quotient__by__coherent__normal_OCon__witnesses,axiom,
! [Resid: set_set_a > set_set_a > set_set_a,NN: set_set_set_a,T6: set_set_set_a,U3: set_set_set_a,T: set_set_a,U: set_set_a] :
( ( quotie4035562349531637206_set_a @ Resid @ NN )
=> ( ( ( quotie4088276940643950137_set_a @ Resid @ NN @ T6 @ U3 )
!= bot_bo3380559777022489994_set_a )
=> ( ( member_set_set_a @ T @ T6 )
=> ( ( member_set_set_a @ U @ U3 )
=> ? [V2: set_set_a,W: set_set_a] :
( ( member_set_set_a @ V2 @ NN )
& ( member_set_set_a @ W @ NN )
& ( con_set_set_a @ Resid @ ( Resid @ T @ V2 ) @ ( Resid @ U @ W ) ) ) ) ) ) ) ).
% quotient_by_coherent_normal.Con_witnesses
thf(fact_112_quotient__by__coherent__normal_OCon__witnesses,axiom,
! [Resid: set_a > set_a > set_a,NN: set_set_a,T6: set_set_a,U3: set_set_a,T: set_a,U: set_a] :
( ( quotie5625257012022141046_set_a @ Resid @ NN )
=> ( ( ( quotie3283642546880816345_set_a @ Resid @ NN @ T6 @ U3 )
!= bot_bot_set_set_a )
=> ( ( member_set_a @ T @ T6 )
=> ( ( member_set_a @ U @ U3 )
=> ? [V2: set_a,W: set_a] :
( ( member_set_a @ V2 @ NN )
& ( member_set_a @ W @ NN )
& ( con_set_a @ Resid @ ( Resid @ T @ V2 ) @ ( Resid @ U @ W ) ) ) ) ) ) ) ).
% quotient_by_coherent_normal.Con_witnesses
thf(fact_113_quotient__by__coherent__normal_OCon__witnesses,axiom,
! [Resid: a > a > a,NN: set_a,T6: set_a,U3: set_a,T: a,U: a] :
( ( quotie3282664541148387094rmal_a @ Resid @ NN )
=> ( ( ( quotie8165075472272353145esid_a @ Resid @ NN @ T6 @ U3 )
!= bot_bot_set_a )
=> ( ( member_a @ T @ T6 )
=> ( ( member_a @ U @ U3 )
=> ? [V2: a,W: a] :
( ( member_a @ V2 @ NN )
& ( member_a @ W @ NN )
& ( con_a @ Resid @ ( Resid @ T @ V2 ) @ ( Resid @ U @ W ) ) ) ) ) ) ) ).
% quotient_by_coherent_normal.Con_witnesses
thf(fact_114_quotient__by__coherent__normal_Ocon__char_092_060_094sub_062Q_092_060_094sub_062C_092_060_094sub_062N,axiom,
! [Resid: product_prod_a_a > product_prod_a_a > product_prod_a_a,NN: set_Product_prod_a_a,T6: set_Product_prod_a_a,U3: set_Product_prod_a_a] :
( ( quotie2905600318227040575od_a_a @ Resid @ NN )
=> ( ( con_se2857841251380680691od_a_a @ ( quotie4807335555532242594od_a_a @ Resid @ NN ) @ T6 @ U3 )
= ( ( normal6700481192199873089od_a_a @ Resid @ NN @ T6 )
& ( normal6700481192199873089od_a_a @ Resid @ NN @ U3 )
& ? [T7: product_prod_a_a,U4: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ T7 @ T6 )
& ( member1426531477525435216od_a_a @ U4 @ U3 )
& ( con_Product_prod_a_a @ Resid @ T7 @ U4 ) ) ) ) ) ).
% quotient_by_coherent_normal.con_char\<^sub>Q\<^sub>C\<^sub>N
thf(fact_115_quotient__by__coherent__normal_Ocon__char_092_060_094sub_062Q_092_060_094sub_062C_092_060_094sub_062N,axiom,
! [Resid: set_set_a > set_set_a > set_set_a,NN: set_set_set_a,T6: set_set_set_a,U3: set_set_set_a] :
( ( quotie4035562349531637206_set_a @ Resid @ NN )
=> ( ( con_set_set_set_a @ ( quotie4088276940643950137_set_a @ Resid @ NN ) @ T6 @ U3 )
= ( ( normal3204162055925629144_set_a @ Resid @ NN @ T6 )
& ( normal3204162055925629144_set_a @ Resid @ NN @ U3 )
& ? [T7: set_set_a,U4: set_set_a] :
( ( member_set_set_a @ T7 @ T6 )
& ( member_set_set_a @ U4 @ U3 )
& ( con_set_set_a @ Resid @ T7 @ U4 ) ) ) ) ) ).
% quotient_by_coherent_normal.con_char\<^sub>Q\<^sub>C\<^sub>N
thf(fact_116_quotient__by__coherent__normal_Ocon__char_092_060_094sub_062Q_092_060_094sub_062C_092_060_094sub_062N,axiom,
! [Resid: set_a > set_a > set_a,NN: set_set_a,T6: set_set_a,U3: set_set_a] :
( ( quotie5625257012022141046_set_a @ Resid @ NN )
=> ( ( con_set_set_a @ ( quotie3283642546880816345_set_a @ Resid @ NN ) @ T6 @ U3 )
= ( ( normal4437380936311325560_set_a @ Resid @ NN @ T6 )
& ( normal4437380936311325560_set_a @ Resid @ NN @ U3 )
& ? [T7: set_a,U4: set_a] :
( ( member_set_a @ T7 @ T6 )
& ( member_set_a @ U4 @ U3 )
& ( con_set_a @ Resid @ T7 @ U4 ) ) ) ) ) ).
% quotient_by_coherent_normal.con_char\<^sub>Q\<^sub>C\<^sub>N
thf(fact_117_quotient__by__coherent__normal_Ocon__char_092_060_094sub_062Q_092_060_094sub_062C_092_060_094sub_062N,axiom,
! [Resid: a > a > a,NN: set_a,T6: set_a,U3: set_a] :
( ( quotie3282664541148387094rmal_a @ Resid @ NN )
=> ( ( con_set_a @ ( quotie8165075472272353145esid_a @ Resid @ NN ) @ T6 @ U3 )
= ( ( normal8595587647932138008lass_a @ Resid @ NN @ T6 )
& ( normal8595587647932138008lass_a @ Resid @ NN @ U3 )
& ? [T7: a,U4: a] :
( ( member_a @ T7 @ T6 )
& ( member_a @ U4 @ U3 )
& ( con_a @ Resid @ T7 @ U4 ) ) ) ) ) ).
% quotient_by_coherent_normal.con_char\<^sub>Q\<^sub>C\<^sub>N
thf(fact_118_residuation_Ocon__sym,axiom,
! [Resid: set_set_a > set_set_a > set_set_a,T: set_set_a,U: set_set_a] :
( ( residu7095500089415647454_set_a @ Resid )
=> ( ( con_set_set_a @ Resid @ T @ U )
=> ( con_set_set_a @ Resid @ U @ T ) ) ) ).
% residuation.con_sym
thf(fact_119_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_120_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_121_quotient__by__coherent__normal_OResid__by__members,axiom,
! [Resid: set_set_a > set_set_a > set_set_a,NN: set_set_set_a,T6: set_set_set_a,U3: set_set_set_a,T: set_set_a,U: set_set_a] :
( ( quotie4035562349531637206_set_a @ Resid @ NN )
=> ( ( normal3204162055925629144_set_a @ Resid @ NN @ T6 )
=> ( ( normal3204162055925629144_set_a @ Resid @ NN @ U3 )
=> ( ( member_set_set_a @ T @ T6 )
=> ( ( member_set_set_a @ U @ U3 )
=> ( ( con_set_set_a @ Resid @ T @ U )
=> ( ( quotie4088276940643950137_set_a @ Resid @ NN @ T6 @ U3 )
= ( normal2632836576369441230_set_a @ Resid @ NN @ ( Resid @ T @ U ) ) ) ) ) ) ) ) ) ).
% quotient_by_coherent_normal.Resid_by_members
thf(fact_122_quotient__by__coherent__normal_OResid__by__members,axiom,
! [Resid: set_a > set_a > set_a,NN: set_set_a,T6: set_set_a,U3: set_set_a,T: set_a,U: set_a] :
( ( quotie5625257012022141046_set_a @ Resid @ NN )
=> ( ( normal4437380936311325560_set_a @ Resid @ NN @ T6 )
=> ( ( normal4437380936311325560_set_a @ Resid @ NN @ U3 )
=> ( ( member_set_a @ T @ T6 )
=> ( ( member_set_a @ U @ U3 )
=> ( ( con_set_a @ Resid @ T @ U )
=> ( ( quotie3283642546880816345_set_a @ Resid @ NN @ T6 @ U3 )
= ( normal2962378890657961070_set_a @ Resid @ NN @ ( Resid @ T @ U ) ) ) ) ) ) ) ) ) ).
% quotient_by_coherent_normal.Resid_by_members
thf(fact_123_quotient__by__coherent__normal_OResid__by__members,axiom,
! [Resid: a > a > a,NN: set_a,T6: set_a,U3: set_a,T: a,U: a] :
( ( quotie3282664541148387094rmal_a @ Resid @ NN )
=> ( ( normal8595587647932138008lass_a @ Resid @ NN @ T6 )
=> ( ( normal8595587647932138008lass_a @ Resid @ NN @ U3 )
=> ( ( member_a @ T @ T6 )
=> ( ( member_a @ U @ U3 )
=> ( ( con_a @ Resid @ T @ U )
=> ( ( quotie8165075472272353145esid_a @ Resid @ NN @ T6 @ U3 )
= ( normal7408713899360725774lass_a @ Resid @ NN @ ( Resid @ T @ U ) ) ) ) ) ) ) ) ) ).
% quotient_by_coherent_normal.Resid_by_members
thf(fact_124_N_OCong__class__rep,axiom,
! [T6: set_a] :
( ( normal8595587647932138008lass_a @ resid @ nn @ T6 )
=> ( ( normal7408713899360725774lass_a @ resid @ nn @ ( normal3259722184653208495_rep_a @ T6 ) )
= T6 ) ) ).
% N.Cong_class_rep
thf(fact_125_N_OCong__class__eqI_H,axiom,
! [T6: set_a,U3: set_a] :
( ( normal8595587647932138008lass_a @ resid @ nn @ T6 )
=> ( ( normal8595587647932138008lass_a @ resid @ nn @ U3 )
=> ( ( ( inf_inf_set_a @ T6 @ U3 )
!= bot_bot_set_a )
=> ( T6 = U3 ) ) ) ) ).
% N.Cong_class_eqI'
thf(fact_126_null__char,axiom,
( ( partial_null_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) )
= bot_bot_set_a ) ).
% null_char
thf(fact_127_con__def,axiom,
! [T: set_a,U: set_a] :
( ( con_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U )
= ( ( quotie8165075472272353145esid_a @ resid @ nn @ T @ U )
!= ( partial_null_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) ) ) ) ).
% con_def
thf(fact_128_conE,axiom,
! [T: set_a,U: set_a] :
( ( con_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U )
=> ( ( quotie8165075472272353145esid_a @ resid @ nn @ T @ U )
!= ( partial_null_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) ) ) ) ).
% conE
thf(fact_129_con__imp__coinitial__members__are__con,axiom,
! [T6: set_a,U3: set_a,T: a,U: a] :
( ( con_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T6 @ U3 )
=> ( ( member_a @ T @ T6 )
=> ( ( member_a @ U @ U3 )
=> ( ( ( sources_a @ resid @ T )
= ( sources_a @ resid @ U ) )
=> ( con_a @ resid @ T @ U ) ) ) ) ) ).
% con_imp_coinitial_members_are_con
thf(fact_130_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_131_N_Ois__Cong__classE,axiom,
! [T6: set_a] :
( ( normal8595587647932138008lass_a @ resid @ nn @ T6 )
=> ~ ( ( T6 != bot_bot_set_a )
=> ( ! [T2: a] :
( ( member_a @ T2 @ T6 )
=> ! [T8: a] :
( ( member_a @ T8 @ T6 )
=> ( normal_sub_Cong_a @ resid @ nn @ T2 @ T8 ) ) )
=> ~ ! [T2: a] :
( ( member_a @ T2 @ T6 )
=> ! [T8: a] :
( ( normal_sub_Cong_a @ resid @ nn @ T8 @ T2 )
=> ( member_a @ T8 @ T6 ) ) ) ) ) ) ).
% N.is_Cong_classE
thf(fact_132_ide__implies__arr,axiom,
! [A: set_a] :
( ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ A )
=> ( arr_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ A ) ) ).
% ide_implies_arr
thf(fact_133_arr__char,axiom,
! [T6: set_a] :
( ( arr_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T6 )
= ( normal8595587647932138008lass_a @ resid @ nn @ T6 ) ) ).
% arr_char
thf(fact_134_arr__resid__iff__con,axiom,
! [T: set_a,U: set_a] :
( ( arr_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ T @ U ) )
= ( con_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U ) ) ).
% arr_resid_iff_con
thf(fact_135_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_136_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_137_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_138_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_139_R_Oresiduation__axioms,axiom,
residuation_a @ resid ).
% R.residuation_axioms
thf(fact_140_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_141_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_142_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_143_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_144_R_Ocon__imp__coinitial__ax,axiom,
! [T: a,U: a] :
( ( con_a @ resid @ T @ U )
=> ? [A5: a] :
( ( ide_a @ resid @ A5 )
& ( con_a @ resid @ A5 @ T )
& ( con_a @ resid @ A5 @ U ) ) ) ).
% R.con_imp_coinitial_ax
thf(fact_145_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_146_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_147_R_OideE,axiom,
! [A: a] :
( ( ide_a @ resid @ A )
=> ~ ( ( con_a @ resid @ A @ A )
=> ( ( resid @ A @ A )
!= A ) ) ) ).
% R.ideE
thf(fact_148_R_Oide__def,axiom,
! [A: a] :
( ( ide_a @ resid @ A )
= ( ( con_a @ resid @ A @ A )
& ( ( resid @ A @ A )
= A ) ) ) ).
% R.ide_def
thf(fact_149_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_150_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_151_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_152_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_153_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_154_R_Osources__are__cong,axiom,
! [A: a,T: a,A6: a] :
( ( member_a @ A @ ( sources_a @ resid @ T ) )
=> ( ( member_a @ A6 @ ( sources_a @ resid @ T ) )
=> ( ( ide_a @ resid @ ( resid @ A @ A6 ) )
& ( ide_a @ resid @ ( resid @ A6 @ A ) ) ) ) ) ).
% R.sources_are_cong
thf(fact_155_R_Osources__cong__closed,axiom,
! [A: a,T: a,A6: a] :
( ( member_a @ A @ ( sources_a @ resid @ T ) )
=> ( ( ( ide_a @ resid @ ( resid @ A @ A6 ) )
& ( ide_a @ resid @ ( resid @ A6 @ A ) ) )
=> ( member_a @ A6 @ ( sources_a @ resid @ T ) ) ) ) ).
% R.sources_cong_closed
thf(fact_156_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_157_R_Oide__implies__arr,axiom,
! [A: a] :
( ( ide_a @ resid @ A )
=> ( arr_a @ resid @ A ) ) ).
% R.ide_implies_arr
thf(fact_158_R_Oprfx__reflexive,axiom,
! [T: a] :
( ( arr_a @ resid @ T )
=> ( ide_a @ resid @ ( resid @ T @ T ) ) ) ).
% R.prfx_reflexive
thf(fact_159_R_Ocoinitial__ide__are__cong,axiom,
! [A: a,A6: a] :
( ( ide_a @ resid @ A )
=> ( ( ide_a @ resid @ A6 )
=> ( ( coinitial_a @ resid @ A @ A6 )
=> ( ( ide_a @ resid @ ( resid @ A @ A6 ) )
& ( ide_a @ resid @ ( resid @ A6 @ A ) ) ) ) ) ) ).
% R.coinitial_ide_are_cong
thf(fact_160_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_161_N_Oide__closed,axiom,
! [A: a] :
( ( ide_a @ resid @ A )
=> ( member_a @ A @ nn ) ) ).
% N.ide_closed
thf(fact_162_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_163_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_164_R_Osources__con__closed,axiom,
! [A: a,T: a,A6: a] :
( ( member_a @ A @ ( sources_a @ resid @ T ) )
=> ( ( ide_a @ resid @ A6 )
=> ( ( con_a @ resid @ A @ A6 )
=> ( member_a @ A6 @ ( sources_a @ resid @ T ) ) ) ) ) ).
% R.sources_con_closed
thf(fact_165_R_Osources__are__con,axiom,
! [A: a,T: a,A6: a] :
( ( member_a @ A @ ( sources_a @ resid @ T ) )
=> ( ( member_a @ A6 @ ( sources_a @ resid @ T ) )
=> ( con_a @ resid @ A @ A6 ) ) ) ).
% R.sources_are_con
thf(fact_166_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_167_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_168_R_OarrE,axiom,
! [T: a] :
( ( arr_a @ resid @ T )
=> ( con_a @ resid @ T @ T ) ) ).
% R.arrE
thf(fact_169_R_Oarr__def,axiom,
! [T: a] :
( ( arr_a @ resid @ T )
= ( con_a @ resid @ T @ T ) ) ).
% R.arr_def
thf(fact_170_R_Oarr__resid,axiom,
! [T: a,U: a] :
( ( con_a @ resid @ T @ U )
=> ( arr_a @ resid @ ( resid @ T @ U ) ) ) ).
% R.arr_resid
thf(fact_171_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_172_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_173_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_174_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_175_N_Oelements__are__arr,axiom,
! [T: a] :
( ( member_a @ T @ nn )
=> ( arr_a @ resid @ T ) ) ).
% N.elements_are_arr
thf(fact_176_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_177_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_178_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_179_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_180_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_181_N_OCong__def,axiom,
! [T: a,T4: a] :
( ( normal_sub_Cong_a @ resid @ nn @ T @ T4 )
= ( ? [U4: a,U7: a] :
( ( member_a @ U4 @ nn )
& ( member_a @ U7 @ nn )
& ( member_a @ ( resid @ ( resid @ T @ U4 ) @ ( resid @ T4 @ U7 ) ) @ nn )
& ( member_a @ ( resid @ ( resid @ T4 @ U7 ) @ ( resid @ T @ U4 ) ) @ nn ) ) ) ) ).
% N.Cong_def
thf(fact_182_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_183_N_OCong__transitive,axiom,
! [T: a,T5: a,T4: a] :
( ( normal_sub_Cong_a @ resid @ nn @ T @ T5 )
=> ( ( normal_sub_Cong_a @ resid @ nn @ T5 @ T4 )
=> ( normal_sub_Cong_a @ resid @ nn @ T @ T4 ) ) ) ).
% N.Cong_transitive
thf(fact_184_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_185_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_186_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_187_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_188_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_189_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_190_N_Oin__sources__respects__Cong,axiom,
! [T: a,T4: a,A: a,A6: a] :
( ( normal_sub_Cong_a @ resid @ nn @ T @ T4 )
=> ( ( member_a @ A @ ( sources_a @ resid @ T ) )
=> ( ( member_a @ A6 @ ( sources_a @ resid @ T4 ) )
=> ( normal_sub_Cong_a @ resid @ nn @ A @ A6 ) ) ) ) ).
% N.in_sources_respects_Cong
thf(fact_191_N_Osources__are__Cong,axiom,
! [A: a,T: a,A6: a] :
( ( member_a @ A @ ( sources_a @ resid @ T ) )
=> ( ( member_a @ A6 @ ( sources_a @ resid @ T ) )
=> ( normal_sub_Cong_a @ resid @ nn @ A @ A6 ) ) ) ).
% N.sources_are_Cong
thf(fact_192_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_193_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_194_N_OCong__reflexive,axiom,
! [T: a] :
( ( arr_a @ resid @ T )
=> ( normal_sub_Cong_a @ resid @ nn @ T @ T ) ) ).
% N.Cong_reflexive
thf(fact_195_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_196_con__imp__arr__resid,axiom,
! [T: set_a,U: set_a] :
( ( ( quotie8165075472272353145esid_a @ resid @ nn @ T @ U )
!= ( partial_null_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) ) )
=> ( ( quotie8165075472272353145esid_a @ resid @ nn @ ( quotie8165075472272353145esid_a @ resid @ nn @ T @ U ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ T @ U ) )
!= ( partial_null_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) ) ) ) ).
% con_imp_arr_resid
thf(fact_197_con__sym__ax,axiom,
! [T: set_a,U: set_a] :
( ( ( quotie8165075472272353145esid_a @ resid @ nn @ T @ U )
!= ( partial_null_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) ) )
=> ( ( quotie8165075472272353145esid_a @ resid @ nn @ U @ T )
!= ( partial_null_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) ) ) ) ).
% con_sym_ax
thf(fact_198_cube__ax,axiom,
! [V: set_a,T: set_a,U: set_a] :
( ( ( quotie8165075472272353145esid_a @ resid @ nn @ ( quotie8165075472272353145esid_a @ resid @ nn @ V @ T ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ U @ T ) )
!= ( partial_null_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) ) )
=> ( ( quotie8165075472272353145esid_a @ resid @ nn @ ( quotie8165075472272353145esid_a @ resid @ nn @ V @ T ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ U @ T ) )
= ( quotie8165075472272353145esid_a @ resid @ nn @ ( quotie8165075472272353145esid_a @ resid @ nn @ V @ U ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ T @ U ) ) ) ) ).
% cube_ax
thf(fact_199_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_200_N_OCong__class__memb__is__arr,axiom,
! [T6: set_a,T: a] :
( ( normal8595587647932138008lass_a @ resid @ nn @ T6 )
=> ( ( member_a @ T @ T6 )
=> ( arr_a @ resid @ T ) ) ) ).
% N.Cong_class_memb_is_arr
thf(fact_201_N_OCong__class__membs__are__Cong,axiom,
! [T6: set_a,T: a,T4: a] :
( ( normal8595587647932138008lass_a @ resid @ nn @ T6 )
=> ( ( member_a @ T @ T6 )
=> ( ( member_a @ T4 @ T6 )
=> ( normal_sub_Cong_a @ resid @ nn @ T @ T4 ) ) ) ) ).
% N.Cong_class_membs_are_Cong
thf(fact_202_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_203_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_204_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_205_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_206_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_207_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_208_con__implies__arr_I2_J,axiom,
! [T: set_a,U: set_a] :
( ( con_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U )
=> ( arr_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ U ) ) ).
% con_implies_arr(2)
thf(fact_209_con__implies__arr_I1_J,axiom,
! [T: set_a,U: set_a] :
( ( con_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U )
=> ( arr_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T ) ) ).
% con_implies_arr(1)
thf(fact_210_arrE,axiom,
! [T: set_a] :
( ( arr_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T )
=> ( con_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ T ) ) ).
% arrE
thf(fact_211_arr__def,axiom,
! [T: set_a] :
( ( arr_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T )
= ( con_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ T ) ) ).
% arr_def
thf(fact_212_arr__resid,axiom,
! [T: set_a,U: set_a] :
( ( con_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U )
=> ( arr_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( quotie8165075472272353145esid_a @ resid @ nn @ T @ U ) ) ) ).
% arr_resid
thf(fact_213_not__arr__null,axiom,
~ ( arr_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( partial_null_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) ) ) ).
% not_arr_null
thf(fact_214_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_215_N_OCong__class__memb__Cong__rep,axiom,
! [T6: set_a,T: a] :
( ( normal8595587647932138008lass_a @ resid @ nn @ T6 )
=> ( ( member_a @ T @ T6 )
=> ( normal_sub_Cong_a @ resid @ nn @ T @ ( normal3259722184653208495_rep_a @ T6 ) ) ) ) ).
% N.Cong_class_memb_Cong_rep
thf(fact_216_ide__char,axiom,
! [U3: set_a] :
( ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ U3 )
= ( ( arr_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ U3 )
& ( ( inf_inf_set_a @ U3 @ nn )
!= bot_bot_set_a ) ) ) ).
% ide_char
thf(fact_217_R_OideI,axiom,
! [A: a] :
( ( con_a @ resid @ A @ A )
=> ( ( ( resid @ A @ A )
= A )
=> ( ide_a @ resid @ A ) ) ) ).
% R.ideI
thf(fact_218_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_219_R_OarrI,axiom,
! [T: a] :
( ( con_a @ resid @ T @ T )
=> ( arr_a @ resid @ T ) ) ).
% R.arrI
thf(fact_220_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_221_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_222_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_223_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_224_conI,axiom,
! [T: set_a,U: set_a] :
( ( ( quotie8165075472272353145esid_a @ resid @ nn @ T @ U )
!= ( partial_null_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) ) )
=> ( con_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ U ) ) ).
% conI
thf(fact_225_arrI,axiom,
! [T: set_a] :
( ( con_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T @ T )
=> ( arr_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T ) ) ).
% arrI
thf(fact_226_N_Ois__Cong__classI_H,axiom,
! [T6: set_a] :
( ( T6 != bot_bot_set_a )
=> ( ! [T3: a,T9: a] :
( ( member_a @ T3 @ T6 )
=> ( ( member_a @ T9 @ T6 )
=> ( normal_sub_Cong_a @ resid @ nn @ T3 @ T9 ) ) )
=> ( ! [T3: a,T9: a] :
( ( member_a @ T3 @ T6 )
=> ( ( normal_sub_Cong_a @ resid @ nn @ T9 @ T3 )
=> ( member_a @ T9 @ T6 ) ) )
=> ( normal8595587647932138008lass_a @ resid @ nn @ T6 ) ) ) ) ).
% N.is_Cong_classI'
thf(fact_227_normal__sub__rts_OCong_Ocong,axiom,
normal_sub_Cong_a = normal_sub_Cong_a ).
% normal_sub_rts.Cong.cong
thf(fact_228_partial__magma_Onull_Ocong,axiom,
partial_null_set_a = partial_null_set_a ).
% partial_magma.null.cong
thf(fact_229_partial__magma_Onull_Ocong,axiom,
partial_null_a = partial_null_a ).
% partial_magma.null.cong
thf(fact_230_residuation_Oarr_Ocong,axiom,
arr_a = arr_a ).
% residuation.arr.cong
thf(fact_231_residuation_Oarr_Ocong,axiom,
arr_set_a = arr_set_a ).
% residuation.arr.cong
thf(fact_232_rts_Osources_Ocong,axiom,
sources_a = sources_a ).
% rts.sources.cong
thf(fact_233_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_234_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_235_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_236_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_237_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_238_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_239_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_240_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_241_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_242_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_243_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_244_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_245_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_246_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_247_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_248_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_249_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_250_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_251_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_252_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_253_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_254_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_255_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_256_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_257_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_258_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_259_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_260_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_261_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_262_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_263_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_264_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_265_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_266_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_267_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_268_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_269_quotient__by__coherent__normal_Oide__char,axiom,
! [Resid: a > a > a,NN: set_a,U3: set_a] :
( ( quotie3282664541148387094rmal_a @ Resid @ NN )
=> ( ( ide_set_a @ ( quotie8165075472272353145esid_a @ Resid @ NN ) @ U3 )
= ( ( arr_set_a @ ( quotie8165075472272353145esid_a @ Resid @ NN ) @ U3 )
& ( ( inf_inf_set_a @ U3 @ NN )
!= bot_bot_set_a ) ) ) ) ).
% quotient_by_coherent_normal.ide_char
thf(fact_270_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_271_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_272_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_273_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_274_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_275_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_276_quotient__by__coherent__normal_Oarr__char,axiom,
! [Resid: a > a > a,NN: set_a,T6: set_a] :
( ( quotie3282664541148387094rmal_a @ Resid @ NN )
=> ( ( arr_set_a @ ( quotie8165075472272353145esid_a @ Resid @ NN ) @ T6 )
= ( normal8595587647932138008lass_a @ Resid @ NN @ T6 ) ) ) ).
% quotient_by_coherent_normal.arr_char
thf(fact_277_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_278_quotient__by__coherent__normal_Ocon__imp__coinitial__members__are__con,axiom,
! [Resid: set_a > set_a > set_a,NN: set_set_a,T6: set_set_a,U3: set_set_a,T: set_a,U: set_a] :
( ( quotie5625257012022141046_set_a @ Resid @ NN )
=> ( ( con_set_set_a @ ( quotie3283642546880816345_set_a @ Resid @ NN ) @ T6 @ U3 )
=> ( ( member_set_a @ T @ T6 )
=> ( ( member_set_a @ U @ U3 )
=> ( ( ( sources_set_a @ Resid @ T )
= ( sources_set_a @ Resid @ U ) )
=> ( con_set_a @ Resid @ T @ U ) ) ) ) ) ) ).
% quotient_by_coherent_normal.con_imp_coinitial_members_are_con
thf(fact_279_quotient__by__coherent__normal_Ocon__imp__coinitial__members__are__con,axiom,
! [Resid: a > a > a,NN: set_a,T6: set_a,U3: set_a,T: a,U: a] :
( ( quotie3282664541148387094rmal_a @ Resid @ NN )
=> ( ( con_set_a @ ( quotie8165075472272353145esid_a @ Resid @ NN ) @ T6 @ U3 )
=> ( ( member_a @ T @ T6 )
=> ( ( member_a @ U @ U3 )
=> ( ( ( sources_a @ Resid @ T )
= ( sources_a @ Resid @ U ) )
=> ( con_a @ Resid @ T @ U ) ) ) ) ) ) ).
% quotient_by_coherent_normal.con_imp_coinitial_members_are_con
thf(fact_280_ide__char_H,axiom,
! [A2: set_a] :
( ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ A2 )
= ( ( arr_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ A2 )
& ( ord_less_eq_set_a @ A2 @ nn ) ) ) ).
% ide_char'
thf(fact_281_boolean__algebra_Oconj__zero__right,axiom,
! [X4: set_a] :
( ( inf_inf_set_a @ X4 @ bot_bot_set_a )
= bot_bot_set_a ) ).
% boolean_algebra.conj_zero_right
thf(fact_282_boolean__algebra_Oconj__zero__left,axiom,
! [X4: set_a] :
( ( inf_inf_set_a @ bot_bot_set_a @ X4 )
= bot_bot_set_a ) ).
% boolean_algebra.conj_zero_left
thf(fact_283_inf__bot__right,axiom,
! [X4: set_a] :
( ( inf_inf_set_a @ X4 @ bot_bot_set_a )
= bot_bot_set_a ) ).
% inf_bot_right
thf(fact_284_inf__bot__left,axiom,
! [X4: set_a] :
( ( inf_inf_set_a @ bot_bot_set_a @ X4 )
= bot_bot_set_a ) ).
% inf_bot_left
thf(fact_285_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_286_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_287_N_OCong_H_Osimps,axiom,
! [A1: a,A22: a] :
( ( normal_sub_Cong_a2 @ resid @ nn @ A1 @ A22 )
= ( ? [T7: a,U4: a] :
( ( A1 = U4 )
& ( A22 = T7 )
& ( normal_sub_Cong_a2 @ resid @ nn @ T7 @ U4 ) )
| ? [T7: a,U4: a,V4: a] :
( ( A1 = T7 )
& ( A22 = V4 )
& ( normal_sub_Cong_a2 @ resid @ nn @ T7 @ U4 )
& ( normal_sub_Cong_a2 @ resid @ nn @ U4 @ V4 ) )
| ? [T7: a,U4: a] :
( ( A1 = T7 )
& ( A22 = U4 )
& ( member_a @ ( resid @ T7 @ U4 ) @ nn )
& ( member_a @ ( resid @ U4 @ T7 ) @ nn ) )
| ? [T7: a,U4: a] :
( ( A1 = T7 )
& ( A22
= ( resid @ T7 @ U4 ) )
& ( arr_a @ resid @ T7 )
& ( member_a @ U4 @ nn )
& ( ( sources_a @ resid @ T7 )
= ( sources_a @ resid @ U4 ) ) ) ) ) ).
% N.Cong'.simps
thf(fact_288_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_289_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_290_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_291_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_292_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_293_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_294_R_OconE,axiom,
! [T: a,U: a] :
( ( con_a @ resid @ T @ U )
=> ( ( resid @ T @ U )
!= ( partial_null_a @ resid ) ) ) ).
% R.conE
thf(fact_295_R_Onot__arr__null,axiom,
~ ( arr_a @ resid @ ( partial_null_a @ resid ) ) ).
% R.not_arr_null
thf(fact_296_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_297_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_298_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_299_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_300_inf_Oidem,axiom,
! [A: set_a] :
( ( inf_inf_set_a @ A @ A )
= A ) ).
% inf.idem
thf(fact_301_inf__idem,axiom,
! [X4: set_a] :
( ( inf_inf_set_a @ X4 @ X4 )
= X4 ) ).
% inf_idem
thf(fact_302_inf_Oleft__idem,axiom,
! [A: set_a,B: set_a] :
( ( inf_inf_set_a @ A @ ( inf_inf_set_a @ A @ B ) )
= ( inf_inf_set_a @ A @ B ) ) ).
% inf.left_idem
thf(fact_303_inf__left__idem,axiom,
! [X4: set_a,Y4: set_a] :
( ( inf_inf_set_a @ X4 @ ( inf_inf_set_a @ X4 @ Y4 ) )
= ( inf_inf_set_a @ X4 @ Y4 ) ) ).
% inf_left_idem
thf(fact_304_inf_Oright__idem,axiom,
! [A: set_a,B: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A @ B ) @ B )
= ( inf_inf_set_a @ A @ B ) ) ).
% inf.right_idem
thf(fact_305_inf__right__idem,axiom,
! [X4: set_a,Y4: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ X4 @ Y4 ) @ Y4 )
= ( inf_inf_set_a @ X4 @ Y4 ) ) ).
% inf_right_idem
thf(fact_306_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_307_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_308_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_309_le__inf__iff,axiom,
! [X4: set_a,Y4: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ X4 @ ( inf_inf_set_a @ Y4 @ Z ) )
= ( ( ord_less_eq_set_a @ X4 @ Y4 )
& ( ord_less_eq_set_a @ X4 @ Z ) ) ) ).
% le_inf_iff
thf(fact_310_inf_Obounded__iff,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ ( inf_inf_set_a @ B @ C ) )
= ( ( ord_less_eq_set_a @ A @ B )
& ( ord_less_eq_set_a @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_311_R_OconI,axiom,
! [T: a,U: a] :
( ( ( resid @ T @ U )
!= ( partial_null_a @ resid ) )
=> ( con_a @ resid @ T @ U ) ) ).
% R.conI
thf(fact_312_normal__sub__rts_OCong_H_Ocong,axiom,
normal_sub_Cong_a2 = normal_sub_Cong_a2 ).
% normal_sub_rts.Cong'.cong
thf(fact_313_rts_Ocoterminal_Ocong,axiom,
coterminal_a = coterminal_a ).
% rts.coterminal.cong
thf(fact_314_inf__sup__ord_I2_J,axiom,
! [X4: set_a,Y4: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X4 @ Y4 ) @ Y4 ) ).
% inf_sup_ord(2)
thf(fact_315_inf__sup__ord_I1_J,axiom,
! [X4: set_a,Y4: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X4 @ Y4 ) @ X4 ) ).
% inf_sup_ord(1)
thf(fact_316_inf__le1,axiom,
! [X4: set_a,Y4: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X4 @ Y4 ) @ X4 ) ).
% inf_le1
thf(fact_317_inf__le2,axiom,
! [X4: set_a,Y4: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X4 @ Y4 ) @ Y4 ) ).
% inf_le2
thf(fact_318_le__infE,axiom,
! [X4: set_a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ X4 @ ( inf_inf_set_a @ A @ B ) )
=> ~ ( ( ord_less_eq_set_a @ X4 @ A )
=> ~ ( ord_less_eq_set_a @ X4 @ B ) ) ) ).
% le_infE
thf(fact_319_le__infI,axiom,
! [X4: set_a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ X4 @ A )
=> ( ( ord_less_eq_set_a @ X4 @ B )
=> ( ord_less_eq_set_a @ X4 @ ( inf_inf_set_a @ A @ B ) ) ) ) ).
% le_infI
thf(fact_320_inf__mono,axiom,
! [A: set_a,C: set_a,B: set_a,D: set_a] :
( ( ord_less_eq_set_a @ A @ C )
=> ( ( ord_less_eq_set_a @ B @ D )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ ( inf_inf_set_a @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_321_le__infI1,axiom,
! [A: set_a,X4: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ X4 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ X4 ) ) ).
% le_infI1
thf(fact_322_le__infI2,axiom,
! [B: set_a,X4: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B @ X4 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ X4 ) ) ).
% le_infI2
thf(fact_323_inf_OorderE,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( A
= ( inf_inf_set_a @ A @ B ) ) ) ).
% inf.orderE
thf(fact_324_inf_OorderI,axiom,
! [A: set_a,B: set_a] :
( ( A
= ( inf_inf_set_a @ A @ B ) )
=> ( ord_less_eq_set_a @ A @ B ) ) ).
% inf.orderI
thf(fact_325_inf__unique,axiom,
! [F: set_a > set_a > set_a,X4: set_a,Y4: set_a] :
( ! [X: set_a,Y3: set_a] : ( ord_less_eq_set_a @ ( F @ X @ Y3 ) @ X )
=> ( ! [X: set_a,Y3: set_a] : ( ord_less_eq_set_a @ ( F @ X @ Y3 ) @ Y3 )
=> ( ! [X: set_a,Y3: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ X @ Y3 )
=> ( ( ord_less_eq_set_a @ X @ Z2 )
=> ( ord_less_eq_set_a @ X @ ( F @ Y3 @ Z2 ) ) ) )
=> ( ( inf_inf_set_a @ X4 @ Y4 )
= ( F @ X4 @ Y4 ) ) ) ) ) ).
% inf_unique
thf(fact_326_le__iff__inf,axiom,
( ord_less_eq_set_a
= ( ^ [X2: set_a,Y2: set_a] :
( ( inf_inf_set_a @ X2 @ Y2 )
= X2 ) ) ) ).
% le_iff_inf
thf(fact_327_inf_Oabsorb1,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( inf_inf_set_a @ A @ B )
= A ) ) ).
% inf.absorb1
thf(fact_328_inf_Oabsorb2,axiom,
! [B: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( inf_inf_set_a @ A @ B )
= B ) ) ).
% inf.absorb2
thf(fact_329_inf__absorb1,axiom,
! [X4: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y4 )
=> ( ( inf_inf_set_a @ X4 @ Y4 )
= X4 ) ) ).
% inf_absorb1
thf(fact_330_inf__absorb2,axiom,
! [Y4: set_a,X4: set_a] :
( ( ord_less_eq_set_a @ Y4 @ X4 )
=> ( ( inf_inf_set_a @ X4 @ Y4 )
= Y4 ) ) ).
% inf_absorb2
thf(fact_331_inf_OboundedE,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ ( inf_inf_set_a @ B @ C ) )
=> ~ ( ( ord_less_eq_set_a @ A @ B )
=> ~ ( ord_less_eq_set_a @ A @ C ) ) ) ).
% inf.boundedE
thf(fact_332_inf_OboundedI,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ A @ C )
=> ( ord_less_eq_set_a @ A @ ( inf_inf_set_a @ B @ C ) ) ) ) ).
% inf.boundedI
thf(fact_333_inf__greatest,axiom,
! [X4: set_a,Y4: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y4 )
=> ( ( ord_less_eq_set_a @ X4 @ Z )
=> ( ord_less_eq_set_a @ X4 @ ( inf_inf_set_a @ Y4 @ Z ) ) ) ) ).
% inf_greatest
thf(fact_334_inf_Oorder__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A7: set_a,B2: set_a] :
( A7
= ( inf_inf_set_a @ A7 @ B2 ) ) ) ) ).
% inf.order_iff
thf(fact_335_inf_Ocobounded1,axiom,
! [A: set_a,B: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ A ) ).
% inf.cobounded1
thf(fact_336_inf_Ocobounded2,axiom,
! [A: set_a,B: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ B ) ).
% inf.cobounded2
thf(fact_337_inf_Oabsorb__iff1,axiom,
( ord_less_eq_set_a
= ( ^ [A7: set_a,B2: set_a] :
( ( inf_inf_set_a @ A7 @ B2 )
= A7 ) ) ) ).
% inf.absorb_iff1
thf(fact_338_inf_Oabsorb__iff2,axiom,
( ord_less_eq_set_a
= ( ^ [B2: set_a,A7: set_a] :
( ( inf_inf_set_a @ A7 @ B2 )
= B2 ) ) ) ).
% inf.absorb_iff2
thf(fact_339_inf_OcoboundedI1,axiom,
! [A: set_a,C: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ C )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ C ) ) ).
% inf.coboundedI1
thf(fact_340_inf_OcoboundedI2,axiom,
! [B: set_a,C: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B @ C )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ C ) ) ).
% inf.coboundedI2
thf(fact_341_inf__sup__aci_I4_J,axiom,
! [X4: set_a,Y4: set_a] :
( ( inf_inf_set_a @ X4 @ ( inf_inf_set_a @ X4 @ Y4 ) )
= ( inf_inf_set_a @ X4 @ Y4 ) ) ).
% inf_sup_aci(4)
thf(fact_342_inf__sup__aci_I3_J,axiom,
! [X4: set_a,Y4: set_a,Z: set_a] :
( ( inf_inf_set_a @ X4 @ ( inf_inf_set_a @ Y4 @ Z ) )
= ( inf_inf_set_a @ Y4 @ ( inf_inf_set_a @ X4 @ Z ) ) ) ).
% inf_sup_aci(3)
thf(fact_343_inf__sup__aci_I2_J,axiom,
! [X4: set_a,Y4: set_a,Z: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ X4 @ Y4 ) @ Z )
= ( inf_inf_set_a @ X4 @ ( inf_inf_set_a @ Y4 @ Z ) ) ) ).
% inf_sup_aci(2)
thf(fact_344_inf__sup__aci_I1_J,axiom,
( inf_inf_set_a
= ( ^ [X2: set_a,Y2: set_a] : ( inf_inf_set_a @ Y2 @ X2 ) ) ) ).
% inf_sup_aci(1)
thf(fact_345_inf_Oassoc,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A @ B ) @ C )
= ( inf_inf_set_a @ A @ ( inf_inf_set_a @ B @ C ) ) ) ).
% inf.assoc
thf(fact_346_inf__assoc,axiom,
! [X4: set_a,Y4: set_a,Z: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ X4 @ Y4 ) @ Z )
= ( inf_inf_set_a @ X4 @ ( inf_inf_set_a @ Y4 @ Z ) ) ) ).
% inf_assoc
thf(fact_347_inf_Ocommute,axiom,
( inf_inf_set_a
= ( ^ [A7: set_a,B2: set_a] : ( inf_inf_set_a @ B2 @ A7 ) ) ) ).
% inf.commute
thf(fact_348_inf__commute,axiom,
( inf_inf_set_a
= ( ^ [X2: set_a,Y2: set_a] : ( inf_inf_set_a @ Y2 @ X2 ) ) ) ).
% inf_commute
thf(fact_349_boolean__algebra__cancel_Oinf1,axiom,
! [A4: set_a,K: set_a,A: set_a,B: set_a] :
( ( A4
= ( inf_inf_set_a @ K @ A ) )
=> ( ( inf_inf_set_a @ A4 @ B )
= ( inf_inf_set_a @ K @ ( inf_inf_set_a @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_350_boolean__algebra__cancel_Oinf2,axiom,
! [B3: set_a,K: set_a,B: set_a,A: set_a] :
( ( B3
= ( inf_inf_set_a @ K @ B ) )
=> ( ( inf_inf_set_a @ A @ B3 )
= ( inf_inf_set_a @ K @ ( inf_inf_set_a @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_351_inf_Oleft__commute,axiom,
! [B: set_a,A: set_a,C: set_a] :
( ( inf_inf_set_a @ B @ ( inf_inf_set_a @ A @ C ) )
= ( inf_inf_set_a @ A @ ( inf_inf_set_a @ B @ C ) ) ) ).
% inf.left_commute
thf(fact_352_inf__left__commute,axiom,
! [X4: set_a,Y4: set_a,Z: set_a] :
( ( inf_inf_set_a @ X4 @ ( inf_inf_set_a @ Y4 @ Z ) )
= ( inf_inf_set_a @ Y4 @ ( inf_inf_set_a @ X4 @ Z ) ) ) ).
% inf_left_commute
thf(fact_353_quotient__by__coherent__normal_Oide__char_H,axiom,
! [Resid: a > a > a,NN: set_a,A2: set_a] :
( ( quotie3282664541148387094rmal_a @ Resid @ NN )
=> ( ( ide_set_a @ ( quotie8165075472272353145esid_a @ Resid @ NN ) @ A2 )
= ( ( arr_set_a @ ( quotie8165075472272353145esid_a @ Resid @ NN ) @ A2 )
& ( ord_less_eq_set_a @ A2 @ NN ) ) ) ) ).
% quotient_by_coherent_normal.ide_char'
thf(fact_354_Int__subset__iff,axiom,
! [C2: set_a,A4: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ C2 @ ( inf_inf_set_a @ A4 @ B3 ) )
= ( ( ord_less_eq_set_a @ C2 @ A4 )
& ( ord_less_eq_set_a @ C2 @ B3 ) ) ) ).
% Int_subset_iff
thf(fact_355_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_356_empty__subsetI,axiom,
! [A4: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A4 ) ).
% empty_subsetI
thf(fact_357__092_060open_062_092_060And_062_092_060T_062_O_Aarr_A_092_060T_062_A_092_060Longrightarrow_062_Aide_A_Itrg_A_092_060T_062_J_092_060close_062,axiom,
! [T6: set_a] :
( ( arr_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T6 )
=> ( ide_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ ( trg_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T6 ) ) ) ).
% \<open>\<And>\<T>. arr \<T> \<Longrightarrow> ide (trg \<T>)\<close>
thf(fact_358_R_OcomposableD_I1_J,axiom,
! [T: a,U: a] :
( ( composable_a @ resid @ T @ U )
=> ( arr_a @ resid @ T ) ) ).
% R.composableD(1)
thf(fact_359_R_OcomposableD_I2_J,axiom,
! [T: a,U: a] :
( ( composable_a @ resid @ T @ U )
=> ( arr_a @ resid @ U ) ) ).
% R.composableD(2)
thf(fact_360_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_361_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_362_R_Otargets__are__cong,axiom,
! [B: a,T: a,B4: a] :
( ( member_a @ B @ ( targets_a @ resid @ T ) )
=> ( ( member_a @ B4 @ ( targets_a @ resid @ T ) )
=> ( ( ide_a @ resid @ ( resid @ B @ B4 ) )
& ( ide_a @ resid @ ( resid @ B4 @ B ) ) ) ) ) ).
% R.targets_are_cong
thf(fact_363_R_Otargets__cong__closed,axiom,
! [B: a,T: a,B4: a] :
( ( member_a @ B @ ( targets_a @ resid @ T ) )
=> ( ( ( ide_a @ resid @ ( resid @ B @ B4 ) )
& ( ide_a @ resid @ ( resid @ B4 @ B ) ) )
=> ( member_a @ B4 @ ( targets_a @ resid @ T ) ) ) ) ).
% R.targets_cong_closed
thf(fact_364_R_Otargets__are__con,axiom,
! [B: a,T: a,B4: a] :
( ( member_a @ B @ ( targets_a @ resid @ T ) )
=> ( ( member_a @ B4 @ ( targets_a @ resid @ T ) )
=> ( con_a @ resid @ B @ B4 ) ) ) ).
% R.targets_are_con
thf(fact_365_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_366_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_367_empty__Collect__eq,axiom,
! [P: a > $o] :
( ( bot_bot_set_a
= ( collect_a @ P ) )
= ( ! [X2: a] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_368_Collect__empty__eq,axiom,
! [P: a > $o] :
( ( ( collect_a @ P )
= bot_bot_set_a )
= ( ! [X2: a] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_369_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_370_empty__iff,axiom,
! [C: a] :
~ ( member_a @ C @ bot_bot_set_a ) ).
% empty_iff
thf(fact_371_subset__antisym,axiom,
! [A4: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A4 @ B3 )
=> ( ( ord_less_eq_set_a @ B3 @ A4 )
=> ( A4 = B3 ) ) ) ).
% subset_antisym
thf(fact_372_subsetI,axiom,
! [A4: set_a,B3: set_a] :
( ! [X: a] :
( ( member_a @ X @ A4 )
=> ( member_a @ X @ B3 ) )
=> ( ord_less_eq_set_a @ A4 @ B3 ) ) ).
% subsetI
thf(fact_373_Int__iff,axiom,
! [C: a,A4: set_a,B3: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A4 @ B3 ) )
= ( ( member_a @ C @ A4 )
& ( member_a @ C @ B3 ) ) ) ).
% Int_iff
thf(fact_374_IntI,axiom,
! [C: a,A4: set_a,B3: set_a] :
( ( member_a @ C @ A4 )
=> ( ( member_a @ C @ B3 )
=> ( member_a @ C @ ( inf_inf_set_a @ A4 @ B3 ) ) ) ) ).
% IntI
thf(fact_375_R_Otargets__con__closed,axiom,
! [B: a,T: a,B4: a] :
( ( member_a @ B @ ( targets_a @ resid @ T ) )
=> ( ( ide_a @ resid @ B4 )
=> ( ( con_a @ resid @ B @ B4 )
=> ( member_a @ B4 @ ( targets_a @ resid @ T ) ) ) ) ) ).
% R.targets_con_closed
thf(fact_376_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_377_N_Ocoherent_H,axiom,
! [V: a,V5: a,W2: a,W3: a,T: a,T4: a] :
( ( member_a @ V @ nn )
=> ( ( member_a @ V5 @ nn )
=> ( ( member_a @ W2 @ nn )
=> ( ( member_a @ W3 @ nn )
=> ( ( ( sources_a @ resid @ V )
= ( sources_a @ resid @ W2 ) )
=> ( ( ( sources_a @ resid @ V5 )
= ( sources_a @ resid @ W3 ) )
=> ( ( ( targets_a @ resid @ W2 )
= ( targets_a @ resid @ W3 ) )
=> ( ( ( member_a @ ( resid @ ( resid @ T @ V ) @ ( resid @ T4 @ V5 ) ) @ nn )
& ( member_a @ ( resid @ ( resid @ T4 @ V5 ) @ ( resid @ T @ V ) ) @ nn ) )
=> ( ( member_a @ ( resid @ ( resid @ T @ W2 ) @ ( resid @ T4 @ W3 ) ) @ nn )
& ( member_a @ ( resid @ ( resid @ T4 @ W3 ) @ ( resid @ T @ W2 ) ) @ nn ) ) ) ) ) ) ) ) ) ) ).
% N.coherent'
thf(fact_378_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_379_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_380_N_Oin__targets__respects__Cong,axiom,
! [T: a,T4: a,B: a,B4: a] :
( ( normal_sub_Cong_a @ resid @ nn @ T @ T4 )
=> ( ( member_a @ B @ ( targets_a @ resid @ T ) )
=> ( ( member_a @ B4 @ ( targets_a @ resid @ T4 ) )
=> ( normal_sub_Cong_a @ resid @ nn @ B @ B4 ) ) ) ) ).
% N.in_targets_respects_Cong
thf(fact_381_N_Otargets__are__Cong,axiom,
! [B: a,T: a,B4: a] :
( ( member_a @ B @ ( targets_a @ resid @ T ) )
=> ( ( member_a @ B4 @ ( targets_a @ resid @ T ) )
=> ( normal_sub_Cong_a @ resid @ nn @ B @ B4 ) ) ) ).
% N.targets_are_Cong
thf(fact_382_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_383_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_384_trg__def,axiom,
! [T: set_a] :
( ( trg_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) @ T )
= ( quotie8165075472272353145esid_a @ resid @ nn @ T @ T ) ) ).
% trg_def
thf(fact_385_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_386_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_387_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_388_residuation_Otrg_Ocong,axiom,
trg_set_a = trg_set_a ).
% residuation.trg.cong
thf(fact_389_residuation_Otrg_Ocong,axiom,
trg_a = trg_a ).
% residuation.trg.cong
thf(fact_390_rts_Ocomposable_Ocong,axiom,
composable_a = composable_a ).
% rts.composable.cong
thf(fact_391_rts_Otargets_Ocong,axiom,
targets_a = targets_a ).
% rts.targets.cong
thf(fact_392_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_393_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_394_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_395_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_396_coherent__normal__sub__rts_Ocoherent_H,axiom,
! [Resid: a > a > a,NN: set_a,V: a,V5: a,W2: a,W3: a,T: a,T4: a] :
( ( cohere6072184133013167079_rts_a @ Resid @ NN )
=> ( ( member_a @ V @ NN )
=> ( ( member_a @ V5 @ NN )
=> ( ( member_a @ W2 @ NN )
=> ( ( member_a @ W3 @ NN )
=> ( ( ( sources_a @ Resid @ V )
= ( sources_a @ Resid @ W2 ) )
=> ( ( ( sources_a @ Resid @ V5 )
= ( sources_a @ Resid @ W3 ) )
=> ( ( ( targets_a @ Resid @ W2 )
= ( targets_a @ Resid @ W3 ) )
=> ( ( ( member_a @ ( Resid @ ( Resid @ T @ V ) @ ( Resid @ T4 @ V5 ) ) @ NN )
& ( member_a @ ( Resid @ ( Resid @ T4 @ V5 ) @ ( Resid @ T @ V ) ) @ NN ) )
=> ( ( member_a @ ( Resid @ ( Resid @ T @ W2 ) @ ( Resid @ T4 @ W3 ) ) @ NN )
& ( member_a @ ( Resid @ ( Resid @ T4 @ W3 ) @ ( Resid @ T @ W2 ) ) @ NN ) ) ) ) ) ) ) ) ) ) ) ).
% coherent_normal_sub_rts.coherent'
thf(fact_397_ex__in__conv,axiom,
! [A4: set_a] :
( ( ? [X2: a] : ( member_a @ X2 @ A4 ) )
= ( A4 != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_398_equals0I,axiom,
! [A4: set_a] :
( ! [Y3: a] :
~ ( member_a @ Y3 @ A4 )
=> ( A4 = bot_bot_set_a ) ) ).
% equals0I
thf(fact_399_equals0D,axiom,
! [A4: set_a,A: a] :
( ( A4 = bot_bot_set_a )
=> ~ ( member_a @ A @ A4 ) ) ).
% equals0D
thf(fact_400_emptyE,axiom,
! [A: a] :
~ ( member_a @ A @ bot_bot_set_a ) ).
% emptyE
thf(fact_401_bot__set__def,axiom,
( bot_bot_set_a
= ( collect_a @ bot_bot_a_o ) ) ).
% bot_set_def
thf(fact_402_Collect__mono__iff,axiom,
! [P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) )
= ( ! [X2: a] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_403_set__eq__subset,axiom,
( ( ^ [Y5: set_a,Z3: set_a] : ( Y5 = Z3 ) )
= ( ^ [A8: set_a,B5: set_a] :
( ( ord_less_eq_set_a @ A8 @ B5 )
& ( ord_less_eq_set_a @ B5 @ A8 ) ) ) ) ).
% set_eq_subset
thf(fact_404_subset__trans,axiom,
! [A4: set_a,B3: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A4 @ B3 )
=> ( ( ord_less_eq_set_a @ B3 @ C2 )
=> ( ord_less_eq_set_a @ A4 @ C2 ) ) ) ).
% subset_trans
thf(fact_405_Collect__mono,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X: a] :
( ( P @ X )
=> ( Q @ X ) )
=> ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% Collect_mono
thf(fact_406_subset__refl,axiom,
! [A4: set_a] : ( ord_less_eq_set_a @ A4 @ A4 ) ).
% subset_refl
thf(fact_407_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A8: set_a,B5: set_a] :
! [T7: a] :
( ( member_a @ T7 @ A8 )
=> ( member_a @ T7 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_408_equalityD2,axiom,
! [A4: set_a,B3: set_a] :
( ( A4 = B3 )
=> ( ord_less_eq_set_a @ B3 @ A4 ) ) ).
% equalityD2
thf(fact_409_equalityD1,axiom,
! [A4: set_a,B3: set_a] :
( ( A4 = B3 )
=> ( ord_less_eq_set_a @ A4 @ B3 ) ) ).
% equalityD1
thf(fact_410_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A8: set_a,B5: set_a] :
! [X2: a] :
( ( member_a @ X2 @ A8 )
=> ( member_a @ X2 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_411_equalityE,axiom,
! [A4: set_a,B3: set_a] :
( ( A4 = B3 )
=> ~ ( ( ord_less_eq_set_a @ A4 @ B3 )
=> ~ ( ord_less_eq_set_a @ B3 @ A4 ) ) ) ).
% equalityE
thf(fact_412_subsetD,axiom,
! [A4: set_a,B3: set_a,C: a] :
( ( ord_less_eq_set_a @ A4 @ B3 )
=> ( ( member_a @ C @ A4 )
=> ( member_a @ C @ B3 ) ) ) ).
% subsetD
thf(fact_413_in__mono,axiom,
! [A4: set_a,B3: set_a,X4: a] :
( ( ord_less_eq_set_a @ A4 @ B3 )
=> ( ( member_a @ X4 @ A4 )
=> ( member_a @ X4 @ B3 ) ) ) ).
% in_mono
thf(fact_414_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_415_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_416_Int__left__commute,axiom,
! [A4: set_a,B3: set_a,C2: set_a] :
( ( inf_inf_set_a @ A4 @ ( inf_inf_set_a @ B3 @ C2 ) )
= ( inf_inf_set_a @ B3 @ ( inf_inf_set_a @ A4 @ C2 ) ) ) ).
% Int_left_commute
thf(fact_417_Int__left__absorb,axiom,
! [A4: set_a,B3: set_a] :
( ( inf_inf_set_a @ A4 @ ( inf_inf_set_a @ A4 @ B3 ) )
= ( inf_inf_set_a @ A4 @ B3 ) ) ).
% Int_left_absorb
thf(fact_418_Int__commute,axiom,
( inf_inf_set_a
= ( ^ [A8: set_a,B5: set_a] : ( inf_inf_set_a @ B5 @ A8 ) ) ) ).
% Int_commute
thf(fact_419_Int__absorb,axiom,
! [A4: set_a] :
( ( inf_inf_set_a @ A4 @ A4 )
= A4 ) ).
% Int_absorb
thf(fact_420_Int__assoc,axiom,
! [A4: set_a,B3: set_a,C2: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A4 @ B3 ) @ C2 )
= ( inf_inf_set_a @ A4 @ ( inf_inf_set_a @ B3 @ C2 ) ) ) ).
% Int_assoc
thf(fact_421_IntD2,axiom,
! [C: a,A4: set_a,B3: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A4 @ B3 ) )
=> ( member_a @ C @ B3 ) ) ).
% IntD2
thf(fact_422_IntD1,axiom,
! [C: a,A4: set_a,B3: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A4 @ B3 ) )
=> ( member_a @ C @ A4 ) ) ).
% IntD1
thf(fact_423_IntE,axiom,
! [C: a,A4: set_a,B3: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A4 @ B3 ) )
=> ~ ( ( member_a @ C @ A4 )
=> ~ ( member_a @ C @ B3 ) ) ) ).
% IntE
thf(fact_424_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_425_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_426_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_427_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_428_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_429_Int__Collect__mono,axiom,
! [A4: set_a,B3: set_a,P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ A4 @ B3 )
=> ( ! [X: a] :
( ( member_a @ X @ A4 )
=> ( ( P @ X )
=> ( Q @ X ) ) )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A4 @ ( collect_a @ P ) ) @ ( inf_inf_set_a @ B3 @ ( collect_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_430_Int__greatest,axiom,
! [C2: set_a,A4: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ C2 @ A4 )
=> ( ( ord_less_eq_set_a @ C2 @ B3 )
=> ( ord_less_eq_set_a @ C2 @ ( inf_inf_set_a @ A4 @ B3 ) ) ) ) ).
% Int_greatest
thf(fact_431_Int__absorb2,axiom,
! [A4: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A4 @ B3 )
=> ( ( inf_inf_set_a @ A4 @ B3 )
= A4 ) ) ).
% Int_absorb2
thf(fact_432_Int__absorb1,axiom,
! [B3: set_a,A4: set_a] :
( ( ord_less_eq_set_a @ B3 @ A4 )
=> ( ( inf_inf_set_a @ A4 @ B3 )
= B3 ) ) ).
% Int_absorb1
thf(fact_433_Int__lower2,axiom,
! [A4: set_a,B3: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A4 @ B3 ) @ B3 ) ).
% Int_lower2
thf(fact_434_Int__lower1,axiom,
! [A4: set_a,B3: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A4 @ B3 ) @ A4 ) ).
% Int_lower1
thf(fact_435_Int__mono,axiom,
! [A4: set_a,C2: set_a,B3: set_a,D2: set_a] :
( ( ord_less_eq_set_a @ A4 @ C2 )
=> ( ( ord_less_eq_set_a @ B3 @ D2 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A4 @ B3 ) @ ( inf_inf_set_a @ C2 @ D2 ) ) ) ) ).
% Int_mono
thf(fact_436_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_437_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_438_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_439_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_440_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_441_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_442_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_443_R_Otrg__def,axiom,
! [T: a] :
( ( trg_a @ resid @ T )
= ( resid @ T @ T ) ) ).
% R.trg_def
thf(fact_444_R_Ocomposite__of__unq__upto__cong,axiom,
! [U: a,T: a,V: a,V5: a] :
( ( composite_of_a @ resid @ U @ T @ V )
=> ( ( composite_of_a @ resid @ U @ T @ V5 )
=> ( ( ide_a @ resid @ ( resid @ V @ V5 ) )
& ( ide_a @ resid @ ( resid @ V5 @ V ) ) ) ) ) ).
% R.composite_of_unq_upto_cong
thf(fact_445_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_446_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_447_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_448_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_449_R_Ocon__composite__of__iff,axiom,
! [T: a,U: a,V: a,W2: a] :
( ( composite_of_a @ resid @ T @ U @ V )
=> ( ( con_a @ resid @ W2 @ V )
= ( con_a @ resid @ ( resid @ W2 @ T ) @ U ) ) ) ).
% R.con_composite_of_iff
thf(fact_450_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_451_R_Oresid__composite__of_I1_J,axiom,
! [T: a,U: a,W2: a,V: a] :
( ( composite_of_a @ resid @ T @ U @ W2 )
=> ( ( con_a @ resid @ W2 @ V )
=> ( con_a @ resid @ ( resid @ V @ T ) @ ( resid @ W2 @ T ) ) ) ) ).
% R.resid_composite_of(1)
thf(fact_452_R_Oresid__composite__of_I2_J,axiom,
! [T: a,U: a,W2: a,V: a] :
( ( composite_of_a @ resid @ T @ U @ W2 )
=> ( ( con_a @ resid @ W2 @ V )
=> ( con_a @ resid @ ( resid @ V @ T ) @ U ) ) ) ).
% R.resid_composite_of(2)
thf(fact_453_R_Oresid__composite__of_I4_J,axiom,
! [T: a,U: a,W2: a,V: a] :
( ( composite_of_a @ resid @ T @ U @ W2 )
=> ( ( con_a @ resid @ W2 @ V )
=> ( composite_of_a @ resid @ ( resid @ T @ V ) @ ( resid @ U @ ( resid @ V @ T ) ) @ ( resid @ W2 @ V ) ) ) ) ).
% R.resid_composite_of(4)
thf(fact_454_R_Ocon__prfx__composite__of_I1_J,axiom,
! [T: a,U: a,W2: a] :
( ( composite_of_a @ resid @ T @ U @ W2 )
=> ( con_a @ resid @ T @ W2 ) ) ).
% R.con_prfx_composite_of(1)
thf(fact_455_R_Ocon__prfx__composite__of_I2_J,axiom,
! [T: a,U: a,W2: a,V: a] :
( ( composite_of_a @ resid @ T @ U @ W2 )
=> ( ( con_a @ resid @ W2 @ V )
=> ( con_a @ resid @ T @ V ) ) ) ).
% R.con_prfx_composite_of(2)
thf(fact_456_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_457_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_458_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_459_R_Ojoin__of__un__upto__cong,axiom,
! [T: a,U: a,V: a,V5: a] :
( ( join_of_a @ resid @ T @ U @ V )
=> ( ( join_of_a @ resid @ T @ U @ V5 )
=> ( ( ide_a @ resid @ ( resid @ V @ V5 ) )
& ( ide_a @ resid @ ( resid @ V5 @ V ) ) ) ) ) ).
% R.join_of_un_upto_cong
thf(fact_460_R_Ojoin__of__resid,axiom,
! [T: a,U: a,W2: a,V: a] :
( ( join_of_a @ resid @ T @ U @ W2 )
=> ( ( con_a @ resid @ V @ W2 )
=> ( join_of_a @ resid @ ( resid @ T @ V ) @ ( resid @ U @ V ) @ ( resid @ W2 @ V ) ) ) ) ).
% R.join_of_resid
thf(fact_461_R_Ocon__with__join__of__iff_I1_J,axiom,
! [T: a,U: a,W2: a,V: a] :
( ( join_of_a @ resid @ T @ U @ W2 )
=> ( ( ( con_a @ resid @ U @ V )
& ( con_a @ resid @ ( resid @ V @ U ) @ ( resid @ T @ U ) ) )
=> ( con_a @ resid @ W2 @ V ) ) ) ).
% R.con_with_join_of_iff(1)
thf(fact_462_R_Ocon__with__join__of__iff_I2_J,axiom,
! [T: a,U: a,W2: a,V: a] :
( ( join_of_a @ resid @ T @ U @ W2 )
=> ( ( con_a @ resid @ W2 @ V )
=> ( ( con_a @ resid @ T @ V )
& ( con_a @ resid @ ( resid @ V @ T ) @ ( resid @ U @ T ) ) ) ) ) ).
% R.con_with_join_of_iff(2)
thf(fact_463_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_464_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_465_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_466_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_467_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_468_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_469_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_470_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_471_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_472_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_473_N_OCong_092_060_094sub_0620__iff,axiom,
! [T: a,T4: a] :
( ( ( member_a @ ( resid @ T @ T4 ) @ nn )
& ( member_a @ ( resid @ T4 @ T ) @ nn ) )
= ( ? [U4: a,U7: a,V4: a,V6: a] :
( ( member_a @ U4 @ nn )
& ( member_a @ U7 @ nn )
& ( member_a @ ( resid @ V4 @ V6 ) @ nn )
& ( member_a @ ( resid @ V6 @ V4 ) @ nn )
& ( composite_of_a @ resid @ T @ U4 @ V4 )
& ( composite_of_a @ resid @ T4 @ U7 @ V6 ) ) ) ) ).
% N.Cong\<^sub>0_iff
thf(fact_474_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_475_N_OCong_092_060_094sub_0620__cancel__left,axiom,
! [T: a,U: a,V: a,U2: a,V5: a] :
( ( composite_of_a @ resid @ T @ U @ V )
=> ( ( composite_of_a @ resid @ T @ U2 @ V5 )
=> ( ( ( member_a @ ( resid @ V @ V5 ) @ nn )
& ( member_a @ ( resid @ V5 @ V ) @ nn ) )
=> ( ( member_a @ ( resid @ U @ U2 ) @ nn )
& ( member_a @ ( resid @ U2 @ U ) @ nn ) ) ) ) ) ).
% N.Cong\<^sub>0_cancel_left
thf(fact_476_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_477_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_478_R_Ocomposable__def,axiom,
! [T: a,U: a] :
( ( composable_a @ resid @ T @ U )
= ( ? [X5: a] : ( composite_of_a @ resid @ T @ U @ X5 ) ) ) ).
% R.composable_def
thf(fact_479_R_Ojoinable__def,axiom,
! [T: a,U: a] :
( ( joinable_a @ resid @ T @ U )
= ( ? [X5: a] : ( join_of_a @ resid @ T @ U @ X5 ) ) ) ).
% R.joinable_def
thf(fact_480_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_481_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_482_R_Oresid__composite__of_I3_J,axiom,
! [T: a,U: a,W2: a,V: a] :
( ( composite_of_a @ resid @ T @ U @ W2 )
=> ( ( con_a @ resid @ W2 @ V )
=> ( ( ide_a @ resid @ ( resid @ ( resid @ V @ W2 ) @ ( resid @ ( resid @ V @ T ) @ U ) ) )
& ( ide_a @ resid @ ( resid @ ( resid @ ( resid @ V @ T ) @ U ) @ ( resid @ V @ W2 ) ) ) ) ) ) ).
% R.resid_composite_of(3)
thf(fact_483_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_484_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_485_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_486_R_Oide__trg,axiom,
! [T: a] :
( ( arr_a @ resid @ T )
=> ( ide_a @ resid @ ( trg_a @ resid @ T ) ) ) ).
% R.ide_trg
thf(fact_487_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_488_N_Odiamond__commutes__upto__Cong_092_060_094sub_0620,axiom,
! [T: a,U: a,V: a,V5: a] :
( ( con_a @ resid @ T @ U )
=> ( ( composite_of_a @ resid @ T @ ( resid @ U @ T ) @ V )
=> ( ( composite_of_a @ resid @ U @ ( resid @ T @ U ) @ V5 )
=> ( ( member_a @ ( resid @ V @ V5 ) @ nn )
& ( member_a @ ( resid @ V5 @ V ) @ nn ) ) ) ) ) ).
% N.diamond_commutes_upto_Cong\<^sub>0
thf(fact_489_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_490_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_491_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_492_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_493_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_494_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_495_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_496_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_497_rts_Oseq_Ocong,axiom,
seq_a = seq_a ).
% rts.seq.cong
thf(fact_498_rts_Ojoin__of_Ocong,axiom,
join_of_a = join_of_a ).
% rts.join_of.cong
thf(fact_499_rts_Ocomposite__of_Ocong,axiom,
composite_of_a = composite_of_a ).
% rts.composite_of.cong
thf(fact_500_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_501_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_502_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 ) )
=> ( ! [A5: set_a] :
( ( ide_set_a @ Resid @ A5 )
=> ( member_set_a @ A5 @ 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_503_normal__sub__rts__axioms_Ointro,axiom,
! [NN: set_a,Resid: a > a > a] :
( ! [T3: a] :
( ( member_a @ T3 @ NN )
=> ( arr_a @ Resid @ T3 ) )
=> ( ! [A5: a] :
( ( ide_a @ Resid @ A5 )
=> ( member_a @ A5 @ 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_504_normal__sub__rts__axioms__def,axiom,
( normal4776468795420100326_set_a
= ( ^ [Resid2: set_a > set_a > set_a,NN2: set_set_a] :
( ! [T7: set_a] :
( ( member_set_a @ T7 @ NN2 )
=> ( arr_set_a @ Resid2 @ T7 ) )
& ! [A7: set_a] :
( ( ide_set_a @ Resid2 @ A7 )
=> ( member_set_a @ A7 @ NN2 ) )
& ! [U4: set_a,T7: set_a] :
( ( member_set_a @ U4 @ NN2 )
=> ( ( coinitial_set_a @ Resid2 @ T7 @ U4 )
=> ( member_set_a @ ( Resid2 @ U4 @ T7 ) @ NN2 ) ) )
& ! [U4: set_a,T7: set_a] :
( ( member_set_a @ U4 @ NN2 )
=> ( ( member_set_a @ ( Resid2 @ T7 @ U4 ) @ NN2 )
=> ( member_set_a @ T7 @ NN2 ) ) )
& ! [U4: set_a,T7: set_a] :
( ( member_set_a @ U4 @ NN2 )
=> ( ( seq_set_a @ Resid2 @ U4 @ T7 )
=> ? [X5: set_a] : ( composite_of_set_a @ Resid2 @ U4 @ T7 @ X5 ) ) )
& ! [U4: set_a,T7: set_a] :
( ( member_set_a @ U4 @ NN2 )
=> ( ( seq_set_a @ Resid2 @ T7 @ U4 )
=> ? [X5: set_a] : ( composite_of_set_a @ Resid2 @ T7 @ U4 @ X5 ) ) ) ) ) ) ).
% normal_sub_rts_axioms_def
thf(fact_505_normal__sub__rts__axioms__def,axiom,
( normal7698203753654205830ioms_a
= ( ^ [Resid2: a > a > a,NN2: set_a] :
( ! [T7: a] :
( ( member_a @ T7 @ NN2 )
=> ( arr_a @ Resid2 @ T7 ) )
& ! [A7: a] :
( ( ide_a @ Resid2 @ A7 )
=> ( member_a @ A7 @ NN2 ) )
& ! [U4: a,T7: a] :
( ( member_a @ U4 @ NN2 )
=> ( ( coinitial_a @ Resid2 @ T7 @ U4 )
=> ( member_a @ ( Resid2 @ U4 @ T7 ) @ NN2 ) ) )
& ! [U4: a,T7: a] :
( ( member_a @ U4 @ NN2 )
=> ( ( member_a @ ( Resid2 @ T7 @ U4 ) @ NN2 )
=> ( member_a @ T7 @ NN2 ) ) )
& ! [U4: a,T7: a] :
( ( member_a @ U4 @ NN2 )
=> ( ( seq_a @ Resid2 @ U4 @ T7 )
=> ? [X5: a] : ( composite_of_a @ Resid2 @ U4 @ T7 @ X5 ) ) )
& ! [U4: a,T7: a] :
( ( member_a @ U4 @ NN2 )
=> ( ( seq_a @ Resid2 @ T7 @ U4 )
=> ? [X5: a] : ( composite_of_a @ Resid2 @ T7 @ U4 @ X5 ) ) ) ) ) ) ).
% normal_sub_rts_axioms_def
thf(fact_506_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 ) ) )
=> ( ! [A5: set_a,T3: set_a] :
( ( ide_set_a @ Resid @ A5 )
=> ( ( con_set_a @ Resid @ T3 @ A5 )
=> ( ( Resid @ T3 @ A5 )
= T3 ) ) )
=> ( ! [A5: set_a,T3: set_a] :
( ( ide_set_a @ Resid @ A5 )
=> ( ( con_set_a @ Resid @ A5 @ T3 )
=> ( ide_set_a @ Resid @ ( Resid @ A5 @ T3 ) ) ) )
=> ( ! [T3: set_a,U5: set_a] :
( ( con_set_a @ Resid @ T3 @ U5 )
=> ? [A9: set_a] :
( ( ide_set_a @ Resid @ A9 )
& ( con_set_a @ Resid @ A9 @ T3 )
& ( con_set_a @ Resid @ A9 @ U5 ) ) )
=> ( ! [T3: set_a,U5: set_a,V2: set_a] :
( ( ide_set_a @ Resid @ ( Resid @ T3 @ U5 ) )
=> ( ( con_set_a @ Resid @ U5 @ V2 )
=> ( con_set_a @ Resid @ ( Resid @ T3 @ U5 ) @ ( Resid @ V2 @ U5 ) ) ) )
=> ( rts_axioms_set_a @ Resid ) ) ) ) ) ) ).
% rts_axioms.intro
thf(fact_507_rts__axioms_Ointro,axiom,
! [Resid: a > a > a] :
( ! [T3: a] :
( ( arr_a @ Resid @ T3 )
=> ( ide_a @ Resid @ ( trg_a @ Resid @ T3 ) ) )
=> ( ! [A5: a,T3: a] :
( ( ide_a @ Resid @ A5 )
=> ( ( con_a @ Resid @ T3 @ A5 )
=> ( ( Resid @ T3 @ A5 )
= T3 ) ) )
=> ( ! [A5: a,T3: a] :
( ( ide_a @ Resid @ A5 )
=> ( ( con_a @ Resid @ A5 @ T3 )
=> ( ide_a @ Resid @ ( Resid @ A5 @ T3 ) ) ) )
=> ( ! [T3: a,U5: a] :
( ( con_a @ Resid @ T3 @ U5 )
=> ? [A9: a] :
( ( ide_a @ Resid @ A9 )
& ( con_a @ Resid @ A9 @ T3 )
& ( con_a @ Resid @ A9 @ U5 ) ) )
=> ( ! [T3: a,U5: a,V2: a] :
( ( ide_a @ Resid @ ( Resid @ T3 @ U5 ) )
=> ( ( con_a @ Resid @ U5 @ V2 )
=> ( con_a @ Resid @ ( Resid @ T3 @ U5 ) @ ( Resid @ V2 @ U5 ) ) ) )
=> ( rts_axioms_a @ Resid ) ) ) ) ) ) ).
% rts_axioms.intro
thf(fact_508_rts__axioms__def,axiom,
( rts_axioms_set_a
= ( ^ [Resid2: set_a > set_a > set_a] :
( ! [T7: set_a] :
( ( arr_set_a @ Resid2 @ T7 )
=> ( ide_set_a @ Resid2 @ ( trg_set_a @ Resid2 @ T7 ) ) )
& ! [A7: set_a,T7: set_a] :
( ( ide_set_a @ Resid2 @ A7 )
=> ( ( con_set_a @ Resid2 @ T7 @ A7 )
=> ( ( Resid2 @ T7 @ A7 )
= T7 ) ) )
& ! [A7: set_a,T7: set_a] :
( ( ide_set_a @ Resid2 @ A7 )
=> ( ( con_set_a @ Resid2 @ A7 @ T7 )
=> ( ide_set_a @ Resid2 @ ( Resid2 @ A7 @ T7 ) ) ) )
& ! [T7: set_a,U4: set_a] :
( ( con_set_a @ Resid2 @ T7 @ U4 )
=> ? [A7: set_a] :
( ( ide_set_a @ Resid2 @ A7 )
& ( con_set_a @ Resid2 @ A7 @ T7 )
& ( con_set_a @ Resid2 @ A7 @ U4 ) ) )
& ! [T7: set_a,U4: set_a,V4: set_a] :
( ( ide_set_a @ Resid2 @ ( Resid2 @ T7 @ U4 ) )
=> ( ( con_set_a @ Resid2 @ U4 @ V4 )
=> ( con_set_a @ Resid2 @ ( Resid2 @ T7 @ U4 ) @ ( Resid2 @ V4 @ U4 ) ) ) ) ) ) ) ).
% rts_axioms_def
thf(fact_509_rts__axioms__def,axiom,
( rts_axioms_a
= ( ^ [Resid2: a > a > a] :
( ! [T7: a] :
( ( arr_a @ Resid2 @ T7 )
=> ( ide_a @ Resid2 @ ( trg_a @ Resid2 @ T7 ) ) )
& ! [A7: a,T7: a] :
( ( ide_a @ Resid2 @ A7 )
=> ( ( con_a @ Resid2 @ T7 @ A7 )
=> ( ( Resid2 @ T7 @ A7 )
= T7 ) ) )
& ! [A7: a,T7: a] :
( ( ide_a @ Resid2 @ A7 )
=> ( ( con_a @ Resid2 @ A7 @ T7 )
=> ( ide_a @ Resid2 @ ( Resid2 @ A7 @ T7 ) ) ) )
& ! [T7: a,U4: a] :
( ( con_a @ Resid2 @ T7 @ U4 )
=> ? [A7: a] :
( ( ide_a @ Resid2 @ A7 )
& ( con_a @ Resid2 @ A7 @ T7 )
& ( con_a @ Resid2 @ A7 @ U4 ) ) )
& ! [T7: a,U4: a,V4: a] :
( ( ide_a @ Resid2 @ ( Resid2 @ T7 @ U4 ) )
=> ( ( con_a @ Resid2 @ U4 @ V4 )
=> ( con_a @ Resid2 @ ( Resid2 @ T7 @ U4 ) @ ( Resid2 @ V4 @ U4 ) ) ) ) ) ) ) ).
% rts_axioms_def
thf(fact_510_dual__order_Orefl,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% dual_order.refl
thf(fact_511_order__refl,axiom,
! [X4: set_a] : ( ord_less_eq_set_a @ X4 @ X4 ) ).
% order_refl
thf(fact_512_order__antisym__conv,axiom,
! [Y4: set_a,X4: set_a] :
( ( ord_less_eq_set_a @ Y4 @ X4 )
=> ( ( ord_less_eq_set_a @ X4 @ Y4 )
= ( X4 = Y4 ) ) ) ).
% order_antisym_conv
thf(fact_513_ord__le__eq__subst,axiom,
! [A: set_a,B: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_514_ord__eq__le__subst,axiom,
! [A: set_a,F: set_a > set_a,B: set_a,C: set_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_515_order__eq__refl,axiom,
! [X4: set_a,Y4: set_a] :
( ( X4 = Y4 )
=> ( ord_less_eq_set_a @ X4 @ Y4 ) ) ).
% order_eq_refl
thf(fact_516_order__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ ( F @ B ) @ C )
=> ( ! [X: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_517_order__subst1,axiom,
! [A: set_a,F: set_a > set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_518_order__eq__iff,axiom,
( ( ^ [Y5: set_a,Z3: set_a] : ( Y5 = Z3 ) )
= ( ^ [A7: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A7 @ B2 )
& ( ord_less_eq_set_a @ B2 @ A7 ) ) ) ) ).
% order_eq_iff
thf(fact_519_antisym,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_520_dual__order_Otrans,axiom,
! [B: set_a,A: set_a,C: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( ord_less_eq_set_a @ C @ B )
=> ( ord_less_eq_set_a @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_521_dual__order_Oantisym,axiom,
! [B: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( ord_less_eq_set_a @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_522_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: set_a,Z3: set_a] : ( Y5 = Z3 ) )
= ( ^ [A7: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ B2 @ A7 )
& ( ord_less_eq_set_a @ A7 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_523_order__trans,axiom,
! [X4: set_a,Y4: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y4 )
=> ( ( ord_less_eq_set_a @ Y4 @ Z )
=> ( ord_less_eq_set_a @ X4 @ Z ) ) ) ).
% order_trans
thf(fact_524_order_Otrans,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ord_less_eq_set_a @ A @ C ) ) ) ).
% order.trans
thf(fact_525_order__antisym,axiom,
! [X4: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y4 )
=> ( ( ord_less_eq_set_a @ Y4 @ X4 )
=> ( X4 = Y4 ) ) ) ).
% order_antisym
thf(fact_526_ord__le__eq__trans,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_set_a @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_527_ord__eq__le__trans,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( A = B )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ord_less_eq_set_a @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_528_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_a,Z3: set_a] : ( Y5 = Z3 ) )
= ( ^ [X2: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y2 )
& ( ord_less_eq_set_a @ Y2 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_529_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_530_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_531_bot_Oextremum,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A ) ).
% bot.extremum
thf(fact_532_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_533_bot__empty__eq,axiom,
( bot_bot_a_o
= ( ^ [X2: a] : ( member_a @ X2 @ bot_bot_set_a ) ) ) ).
% bot_empty_eq
thf(fact_534_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_535_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_536_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_537_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_538_simulation__axioms__def,axiom,
( simula3192323252075944454_set_a
= ( ^ [A8: a > a > a,B5: set_a > set_a > set_a,F3: a > set_a] :
( ! [T7: a] :
( ~ ( arr_a @ A8 @ T7 )
=> ( ( F3 @ T7 )
= ( partial_null_set_a @ B5 ) ) )
& ! [T7: a,U4: a] :
( ( con_a @ A8 @ T7 @ U4 )
=> ( con_set_a @ B5 @ ( F3 @ T7 ) @ ( F3 @ U4 ) ) )
& ! [T7: a,U4: a] :
( ( con_a @ A8 @ T7 @ U4 )
=> ( ( F3 @ ( A8 @ T7 @ U4 ) )
= ( B5 @ ( F3 @ T7 ) @ ( F3 @ U4 ) ) ) ) ) ) ) ).
% simulation_axioms_def
thf(fact_539_simulation__axioms__def,axiom,
( simula8704200824037452966_set_a
= ( ^ [A8: set_a > set_a > set_a,B5: set_a > set_a > set_a,F3: set_a > set_a] :
( ! [T7: set_a] :
( ~ ( arr_set_a @ A8 @ T7 )
=> ( ( F3 @ T7 )
= ( partial_null_set_a @ B5 ) ) )
& ! [T7: set_a,U4: set_a] :
( ( con_set_a @ A8 @ T7 @ U4 )
=> ( con_set_a @ B5 @ ( F3 @ T7 ) @ ( F3 @ U4 ) ) )
& ! [T7: set_a,U4: set_a] :
( ( con_set_a @ A8 @ T7 @ U4 )
=> ( ( F3 @ ( A8 @ T7 @ U4 ) )
= ( B5 @ ( F3 @ T7 ) @ ( F3 @ U4 ) ) ) ) ) ) ) ).
% simulation_axioms_def
thf(fact_540_simulation__axioms__def,axiom,
( simula3868467710248865958ms_a_a
= ( ^ [A8: a > a > a,B5: a > a > a,F3: a > a] :
( ! [T7: a] :
( ~ ( arr_a @ A8 @ T7 )
=> ( ( F3 @ T7 )
= ( partial_null_a @ B5 ) ) )
& ! [T7: a,U4: a] :
( ( con_a @ A8 @ T7 @ U4 )
=> ( con_a @ B5 @ ( F3 @ T7 ) @ ( F3 @ U4 ) ) )
& ! [T7: a,U4: a] :
( ( con_a @ A8 @ T7 @ U4 )
=> ( ( F3 @ ( A8 @ T7 @ U4 ) )
= ( B5 @ ( F3 @ T7 ) @ ( F3 @ U4 ) ) ) ) ) ) ) ).
% simulation_axioms_def
thf(fact_541_simulation__axioms__def,axiom,
( simula3408835310535287622et_a_a
= ( ^ [A8: set_a > set_a > set_a,B5: a > a > a,F3: set_a > a] :
( ! [T7: set_a] :
( ~ ( arr_set_a @ A8 @ T7 )
=> ( ( F3 @ T7 )
= ( partial_null_a @ B5 ) ) )
& ! [T7: set_a,U4: set_a] :
( ( con_set_a @ A8 @ T7 @ U4 )
=> ( con_a @ B5 @ ( F3 @ T7 ) @ ( F3 @ U4 ) ) )
& ! [T7: set_a,U4: set_a] :
( ( con_set_a @ A8 @ T7 @ U4 )
=> ( ( F3 @ ( A8 @ T7 @ U4 ) )
= ( B5 @ ( F3 @ T7 ) @ ( F3 @ U4 ) ) ) ) ) ) ) ).
% simulation_axioms_def
thf(fact_542_coherent__normal__sub__rts__axioms__def,axiom,
( cohere4894532172567702276ioms_a
= ( ^ [Resid2: a > a > a,NN2: set_a] :
! [T7: a,U4: a,U7: a] :
( ( arr_a @ Resid2 @ T7 )
=> ( ( member_a @ U4 @ NN2 )
=> ( ( member_a @ U7 @ NN2 )
=> ( ( ( sources_a @ Resid2 @ U4 )
= ( sources_a @ Resid2 @ U7 ) )
=> ( ( ( targets_a @ Resid2 @ U4 )
= ( targets_a @ Resid2 @ U7 ) )
=> ( ( ( sources_a @ Resid2 @ T7 )
= ( sources_a @ Resid2 @ U4 ) )
=> ( ( member_a @ ( Resid2 @ ( Resid2 @ T7 @ U4 ) @ ( Resid2 @ T7 @ U7 ) ) @ NN2 )
& ( member_a @ ( Resid2 @ ( Resid2 @ T7 @ U7 ) @ ( Resid2 @ T7 @ U4 ) ) @ NN2 ) ) ) ) ) ) ) ) ) ) ).
% coherent_normal_sub_rts_axioms_def
thf(fact_543_coherent__normal__sub__rts__axioms__def,axiom,
( cohere32089786014956644_set_a
= ( ^ [Resid2: set_a > set_a > set_a,NN2: set_set_a] :
! [T7: set_a,U4: set_a,U7: set_a] :
( ( arr_set_a @ Resid2 @ T7 )
=> ( ( member_set_a @ U4 @ NN2 )
=> ( ( member_set_a @ U7 @ NN2 )
=> ( ( ( sources_set_a @ Resid2 @ U4 )
= ( sources_set_a @ Resid2 @ U7 ) )
=> ( ( ( targets_set_a @ Resid2 @ U4 )
= ( targets_set_a @ Resid2 @ U7 ) )
=> ( ( ( sources_set_a @ Resid2 @ T7 )
= ( sources_set_a @ Resid2 @ U4 ) )
=> ( ( member_set_a @ ( Resid2 @ ( Resid2 @ T7 @ U4 ) @ ( Resid2 @ T7 @ U7 ) ) @ NN2 )
& ( member_set_a @ ( Resid2 @ ( Resid2 @ T7 @ U7 ) @ ( Resid2 @ T7 @ U4 ) ) @ NN2 ) ) ) ) ) ) ) ) ) ) ).
% coherent_normal_sub_rts_axioms_def
thf(fact_544_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_545_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_546_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_547_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_548_N_Onormal__sub__rts__axioms,axiom,
normal_sub_rts_a @ resid @ nn ).
% N.normal_sub_rts_axioms
thf(fact_549_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_550_residuation_Ointro,axiom,
! [Resid: a > a > a] :
( ( partial_magma_a @ Resid )
=> ( ( residuation_axioms_a @ Resid )
=> ( residuation_a @ Resid ) ) ) ).
% residuation.intro
thf(fact_551_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_552_residuation__def,axiom,
( residuation_a
= ( ^ [Resid2: a > a > a] :
( ( partial_magma_a @ Resid2 )
& ( residuation_axioms_a @ Resid2 ) ) ) ) ).
% residuation_def
thf(fact_553_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_554_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_555_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_556_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_557_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_558_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_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__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_561_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_562_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_563_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_564_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_565_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_566_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_567_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_568_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_569_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_570_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_571_normal__sub__rts_OCong_092_060_094sub_0620__transitive,axiom,
! [Resid: a > a > a,NN: set_a,T: a,T4: a,T5: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( ( member_a @ ( Resid @ T @ T4 ) @ NN )
& ( member_a @ ( Resid @ T4 @ T ) @ NN ) )
=> ( ( ( member_a @ ( Resid @ T4 @ T5 ) @ NN )
& ( member_a @ ( Resid @ T5 @ T4 ) @ NN ) )
=> ( ( member_a @ ( Resid @ T @ T5 ) @ NN )
& ( member_a @ ( Resid @ T5 @ T ) @ NN ) ) ) ) ) ).
% normal_sub_rts.Cong\<^sub>0_transitive
thf(fact_572_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_573_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_574_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_575_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_576_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_577_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_578_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_579_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_580_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_581_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_582_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_583_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_584_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 )
= ( ? [U4: a,U7: a] :
( ( member_a @ U4 @ NN )
& ( member_a @ U7 @ NN )
& ( member_a @ ( Resid @ ( Resid @ T @ U4 ) @ ( Resid @ T4 @ U7 ) ) @ NN )
& ( member_a @ ( Resid @ ( Resid @ T4 @ U7 ) @ ( Resid @ T @ U4 ) ) @ NN ) ) ) ) ) ).
% normal_sub_rts.Cong_def
thf(fact_585_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_586_normal__sub__rts_OCong__transitive,axiom,
! [Resid: a > a > a,NN: set_a,T: a,T5: a,T4: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( normal_sub_Cong_a @ Resid @ NN @ T @ T5 )
=> ( ( normal_sub_Cong_a @ Resid @ NN @ T5 @ T4 )
=> ( normal_sub_Cong_a @ Resid @ NN @ T @ T4 ) ) ) ) ).
% normal_sub_rts.Cong_transitive
thf(fact_587_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_588_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_589_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_590_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_591_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 ) )
= ( ? [U4: a,U7: a,V4: a,V6: a] :
( ( member_a @ U4 @ NN )
& ( member_a @ U7 @ NN )
& ( member_a @ ( Resid @ V4 @ V6 ) @ NN )
& ( member_a @ ( Resid @ V6 @ V4 ) @ NN )
& ( composite_of_a @ Resid @ T @ U4 @ V4 )
& ( composite_of_a @ Resid @ T4 @ U7 @ V6 ) ) ) ) ) ).
% normal_sub_rts.Cong\<^sub>0_iff
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: 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_594_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,V5: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( composite_of_a @ Resid @ T @ U @ V )
=> ( ( composite_of_a @ Resid @ T @ U2 @ V5 )
=> ( ( ( member_a @ ( Resid @ V @ V5 ) @ NN )
& ( member_a @ ( Resid @ V5 @ V ) @ NN ) )
=> ( ( member_a @ ( Resid @ U @ U2 ) @ NN )
& ( member_a @ ( Resid @ U2 @ U ) @ NN ) ) ) ) ) ) ).
% normal_sub_rts.Cong\<^sub>0_cancel_left
thf(fact_595_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_596_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_597_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_598_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_599_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_600_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_601_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_602_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_603_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_604_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,V5: 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 ) @ V5 )
=> ( ( member_set_a @ ( Resid @ V @ V5 ) @ NN )
& ( member_set_a @ ( Resid @ V5 @ V ) @ NN ) ) ) ) ) ) ).
% normal_sub_rts.diamond_commutes_upto_Cong\<^sub>0
thf(fact_605_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,V5: 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 ) @ V5 )
=> ( ( member_a @ ( Resid @ V @ V5 ) @ NN )
& ( member_a @ ( Resid @ V5 @ V ) @ NN ) ) ) ) ) ) ).
% normal_sub_rts.diamond_commutes_upto_Cong\<^sub>0
thf(fact_606_normal__sub__rts_OCong__class__is__nonempty,axiom,
! [Resid: a > a > a,NN: set_a,T6: set_a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( normal8595587647932138008lass_a @ Resid @ NN @ T6 )
=> ( T6 != bot_bot_set_a ) ) ) ).
% normal_sub_rts.Cong_class_is_nonempty
thf(fact_607_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_608_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_609_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_610_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_611_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_612_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_613_normal__sub__rts_Oin__sources__respects__Cong,axiom,
! [Resid: a > a > a,NN: set_a,T: a,T4: a,A: a,A6: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( normal_sub_Cong_a @ Resid @ NN @ T @ T4 )
=> ( ( member_a @ A @ ( sources_a @ Resid @ T ) )
=> ( ( member_a @ A6 @ ( sources_a @ Resid @ T4 ) )
=> ( normal_sub_Cong_a @ Resid @ NN @ A @ A6 ) ) ) ) ) ).
% normal_sub_rts.in_sources_respects_Cong
thf(fact_614_normal__sub__rts_Osources__are__Cong,axiom,
! [Resid: a > a > a,NN: set_a,A: a,T: a,A6: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( member_a @ A @ ( sources_a @ Resid @ T ) )
=> ( ( member_a @ A6 @ ( sources_a @ Resid @ T ) )
=> ( normal_sub_Cong_a @ Resid @ NN @ A @ A6 ) ) ) ) ).
% normal_sub_rts.sources_are_Cong
thf(fact_615_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_616_normal__sub__rts_Oin__targets__respects__Cong,axiom,
! [Resid: a > a > a,NN: set_a,T: a,T4: a,B: a,B4: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( normal_sub_Cong_a @ Resid @ NN @ T @ T4 )
=> ( ( member_a @ B @ ( targets_a @ Resid @ T ) )
=> ( ( member_a @ B4 @ ( targets_a @ Resid @ T4 ) )
=> ( normal_sub_Cong_a @ Resid @ NN @ B @ B4 ) ) ) ) ) ).
% normal_sub_rts.in_targets_respects_Cong
thf(fact_617_normal__sub__rts_Otargets__are__Cong,axiom,
! [Resid: a > a > a,NN: set_a,B: a,T: a,B4: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( member_a @ B @ ( targets_a @ Resid @ T ) )
=> ( ( member_a @ B4 @ ( targets_a @ Resid @ T ) )
=> ( normal_sub_Cong_a @ Resid @ NN @ B @ B4 ) ) ) ) ).
% normal_sub_rts.targets_are_Cong
thf(fact_618_normal__sub__rts_OCong__class__memb__is__arr,axiom,
! [Resid: set_a > set_a > set_a,NN: set_set_a,T6: set_set_a,T: set_a] :
( ( normal_sub_rts_set_a @ Resid @ NN )
=> ( ( normal4437380936311325560_set_a @ Resid @ NN @ T6 )
=> ( ( member_set_a @ T @ T6 )
=> ( arr_set_a @ Resid @ T ) ) ) ) ).
% normal_sub_rts.Cong_class_memb_is_arr
thf(fact_619_normal__sub__rts_OCong__class__memb__is__arr,axiom,
! [Resid: a > a > a,NN: set_a,T6: set_a,T: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( normal8595587647932138008lass_a @ Resid @ NN @ T6 )
=> ( ( member_a @ T @ T6 )
=> ( arr_a @ Resid @ T ) ) ) ) ).
% normal_sub_rts.Cong_class_memb_is_arr
thf(fact_620_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_621_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_622_normal__sub__rts_OCong__class__membs__are__Cong,axiom,
! [Resid: a > a > a,NN: set_a,T6: set_a,T: a,T4: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( normal8595587647932138008lass_a @ Resid @ NN @ T6 )
=> ( ( member_a @ T @ T6 )
=> ( ( member_a @ T4 @ T6 )
=> ( normal_sub_Cong_a @ Resid @ NN @ T @ T4 ) ) ) ) ) ).
% normal_sub_rts.Cong_class_membs_are_Cong
thf(fact_623_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_624_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_625_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_626_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_627_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_628_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_629_normal__sub__rts_Ois__Cong__class__def,axiom,
! [Resid: a > a > a,NN: set_a,T6: set_a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( normal8595587647932138008lass_a @ Resid @ NN @ T6 )
= ( ? [T7: a] :
( ( member_a @ T7 @ T6 )
& ( T6
= ( normal7408713899360725774lass_a @ Resid @ NN @ T7 ) ) ) ) ) ) ).
% normal_sub_rts.is_Cong_class_def
thf(fact_630_normal__sub__rts_Orep__in__Cong__class,axiom,
! [Resid: a > a > a,NN: set_a,T6: set_a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( normal8595587647932138008lass_a @ Resid @ NN @ T6 )
=> ( member_a @ ( normal3259722184653208495_rep_a @ T6 ) @ T6 ) ) ) ).
% normal_sub_rts.rep_in_Cong_class
thf(fact_631_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_632_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_633_normal__sub__rts_OCong__class__eqI_H,axiom,
! [Resid: a > a > a,NN: set_a,T6: set_a,U3: set_a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( normal8595587647932138008lass_a @ Resid @ NN @ T6 )
=> ( ( normal8595587647932138008lass_a @ Resid @ NN @ U3 )
=> ( ( ( inf_inf_set_a @ T6 @ U3 )
!= bot_bot_set_a )
=> ( T6 = U3 ) ) ) ) ) ).
% normal_sub_rts.Cong_class_eqI'
thf(fact_634_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_635_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_636_normal__sub__rts_Ois__Cong__classI_H,axiom,
! [Resid: a > a > a,NN: set_a,T6: set_a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( T6 != bot_bot_set_a )
=> ( ! [T3: a,T9: a] :
( ( member_a @ T3 @ T6 )
=> ( ( member_a @ T9 @ T6 )
=> ( normal_sub_Cong_a @ Resid @ NN @ T3 @ T9 ) ) )
=> ( ! [T3: a,T9: a] :
( ( member_a @ T3 @ T6 )
=> ( ( normal_sub_Cong_a @ Resid @ NN @ T9 @ T3 )
=> ( member_a @ T9 @ T6 ) ) )
=> ( normal8595587647932138008lass_a @ Resid @ NN @ T6 ) ) ) ) ) ).
% normal_sub_rts.is_Cong_classI'
thf(fact_637_normal__sub__rts_Ois__Cong__classE,axiom,
! [Resid: a > a > a,NN: set_a,T6: set_a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( normal8595587647932138008lass_a @ Resid @ NN @ T6 )
=> ~ ( ( T6 != bot_bot_set_a )
=> ( ! [T2: a] :
( ( member_a @ T2 @ T6 )
=> ! [T8: a] :
( ( member_a @ T8 @ T6 )
=> ( normal_sub_Cong_a @ Resid @ NN @ T2 @ T8 ) ) )
=> ~ ! [T2: a] :
( ( member_a @ T2 @ T6 )
=> ! [T8: a] :
( ( normal_sub_Cong_a @ Resid @ NN @ T8 @ T2 )
=> ( member_a @ T8 @ T6 ) ) ) ) ) ) ) ).
% normal_sub_rts.is_Cong_classE
thf(fact_638_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 )
= ( ? [T7: set_a,U4: set_a] :
( ( A1 = U4 )
& ( A22 = T7 )
& ( normal8837514132249976843_set_a @ Resid @ NN @ T7 @ U4 ) )
| ? [T7: set_a,U4: set_a,V4: set_a] :
( ( A1 = T7 )
& ( A22 = V4 )
& ( normal8837514132249976843_set_a @ Resid @ NN @ T7 @ U4 )
& ( normal8837514132249976843_set_a @ Resid @ NN @ U4 @ V4 ) )
| ? [T7: set_a,U4: set_a] :
( ( A1 = T7 )
& ( A22 = U4 )
& ( member_set_a @ ( Resid @ T7 @ U4 ) @ NN )
& ( member_set_a @ ( Resid @ U4 @ T7 ) @ NN ) )
| ? [T7: set_a,U4: set_a] :
( ( A1 = T7 )
& ( A22
= ( Resid @ T7 @ U4 ) )
& ( arr_set_a @ Resid @ T7 )
& ( member_set_a @ U4 @ NN )
& ( ( sources_set_a @ Resid @ T7 )
= ( sources_set_a @ Resid @ U4 ) ) ) ) ) ) ).
% normal_sub_rts.Cong'.simps
thf(fact_639_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 )
= ( ? [T7: a,U4: a] :
( ( A1 = U4 )
& ( A22 = T7 )
& ( normal_sub_Cong_a2 @ Resid @ NN @ T7 @ U4 ) )
| ? [T7: a,U4: a,V4: a] :
( ( A1 = T7 )
& ( A22 = V4 )
& ( normal_sub_Cong_a2 @ Resid @ NN @ T7 @ U4 )
& ( normal_sub_Cong_a2 @ Resid @ NN @ U4 @ V4 ) )
| ? [T7: a,U4: a] :
( ( A1 = T7 )
& ( A22 = U4 )
& ( member_a @ ( Resid @ T7 @ U4 ) @ NN )
& ( member_a @ ( Resid @ U4 @ T7 ) @ NN ) )
| ? [T7: a,U4: a] :
( ( A1 = T7 )
& ( A22
= ( Resid @ T7 @ U4 ) )
& ( arr_a @ Resid @ T7 )
& ( member_a @ U4 @ NN )
& ( ( sources_a @ Resid @ T7 )
= ( sources_a @ Resid @ U4 ) ) ) ) ) ) ).
% normal_sub_rts.Cong'.simps
thf(fact_640_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_641_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_642_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_643_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_644_residuation__axioms__def,axiom,
( residu177535419945060507_set_a
= ( ^ [Resid2: set_a > set_a > set_a] :
( ! [T7: set_a,U4: set_a] :
( ( ( Resid2 @ T7 @ U4 )
!= ( partial_null_set_a @ Resid2 ) )
=> ( ( Resid2 @ U4 @ T7 )
!= ( partial_null_set_a @ Resid2 ) ) )
& ! [T7: set_a,U4: set_a] :
( ( ( Resid2 @ T7 @ U4 )
!= ( partial_null_set_a @ Resid2 ) )
=> ( ( Resid2 @ ( Resid2 @ T7 @ U4 ) @ ( Resid2 @ T7 @ U4 ) )
!= ( partial_null_set_a @ Resid2 ) ) )
& ! [V4: set_a,T7: set_a,U4: set_a] :
( ( ( Resid2 @ ( Resid2 @ V4 @ T7 ) @ ( Resid2 @ U4 @ T7 ) )
!= ( partial_null_set_a @ Resid2 ) )
=> ( ( Resid2 @ ( Resid2 @ V4 @ T7 ) @ ( Resid2 @ U4 @ T7 ) )
= ( Resid2 @ ( Resid2 @ V4 @ U4 ) @ ( Resid2 @ T7 @ U4 ) ) ) ) ) ) ) ).
% residuation_axioms_def
thf(fact_645_residuation__axioms__def,axiom,
( residuation_axioms_a
= ( ^ [Resid2: a > a > a] :
( ! [T7: a,U4: a] :
( ( ( Resid2 @ T7 @ U4 )
!= ( partial_null_a @ Resid2 ) )
=> ( ( Resid2 @ U4 @ T7 )
!= ( partial_null_a @ Resid2 ) ) )
& ! [T7: a,U4: a] :
( ( ( Resid2 @ T7 @ U4 )
!= ( partial_null_a @ Resid2 ) )
=> ( ( Resid2 @ ( Resid2 @ T7 @ U4 ) @ ( Resid2 @ T7 @ U4 ) )
!= ( partial_null_a @ Resid2 ) ) )
& ! [V4: a,T7: a,U4: a] :
( ( ( Resid2 @ ( Resid2 @ V4 @ T7 ) @ ( Resid2 @ U4 @ T7 ) )
!= ( partial_null_a @ Resid2 ) )
=> ( ( Resid2 @ ( Resid2 @ V4 @ T7 ) @ ( Resid2 @ U4 @ T7 ) )
= ( Resid2 @ ( Resid2 @ V4 @ U4 ) @ ( Resid2 @ T7 @ U4 ) ) ) ) ) ) ) ).
% residuation_axioms_def
thf(fact_646_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 ) ) )
=> ( ! [V2: set_a,T3: set_a,U5: set_a] :
( ( ( Resid @ ( Resid @ V2 @ T3 ) @ ( Resid @ U5 @ T3 ) )
!= ( partial_null_set_a @ Resid ) )
=> ( ( Resid @ ( Resid @ V2 @ T3 ) @ ( Resid @ U5 @ T3 ) )
= ( Resid @ ( Resid @ V2 @ U5 ) @ ( Resid @ T3 @ U5 ) ) ) )
=> ( residu177535419945060507_set_a @ Resid ) ) ) ) ).
% residuation_axioms.intro
thf(fact_647_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 ) ) )
=> ( ! [V2: a,T3: a,U5: a] :
( ( ( Resid @ ( Resid @ V2 @ T3 ) @ ( Resid @ U5 @ T3 ) )
!= ( partial_null_a @ Resid ) )
=> ( ( Resid @ ( Resid @ V2 @ T3 ) @ ( Resid @ U5 @ T3 ) )
= ( Resid @ ( Resid @ V2 @ U5 ) @ ( Resid @ T3 @ U5 ) ) ) )
=> ( residuation_axioms_a @ Resid ) ) ) ) ).
% residuation_axioms.intro
thf(fact_648_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_649_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_650_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_651_residuation_Oaxioms_I2_J,axiom,
! [Resid: a > a > a] :
( ( residuation_a @ Resid )
=> ( residuation_axioms_a @ Resid ) ) ).
% residuation.axioms(2)
thf(fact_652_normal__sub__rts_OCong__class__memb__Cong__rep,axiom,
! [Resid: a > a > a,NN: set_a,T6: set_a,T: a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( normal8595587647932138008lass_a @ Resid @ NN @ T6 )
=> ( ( member_a @ T @ T6 )
=> ( normal_sub_Cong_a @ Resid @ NN @ T @ ( normal3259722184653208495_rep_a @ T6 ) ) ) ) ) ).
% normal_sub_rts.Cong_class_memb_Cong_rep
thf(fact_653_normal__sub__rts_OCong__class__rep,axiom,
! [Resid: a > a > a,NN: set_a,T6: set_a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( normal8595587647932138008lass_a @ Resid @ NN @ T6 )
=> ( ( normal7408713899360725774lass_a @ Resid @ NN @ ( normal3259722184653208495_rep_a @ T6 ) )
= T6 ) ) ) ).
% normal_sub_rts.Cong_class_rep
thf(fact_654_rts__with__composites__axioms__def,axiom,
( rts_wi2614412583573296275ioms_a
= ( ^ [Resid2: a > a > a] :
! [T7: a,U4: a] :
( ( seq_a @ Resid2 @ T7 @ U4 )
=> ( composable_a @ Resid2 @ T7 @ U4 ) ) ) ) ).
% rts_with_composites_axioms_def
thf(fact_655_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_656_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_657_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_658_rts__with__joins__axioms__def,axiom,
( rts_wi560353115624263628ioms_a
= ( ^ [Resid2: a > a > a] :
! [T7: a,U4: a] :
( ( con_a @ Resid2 @ T7 @ U4 )
=> ( joinable_a @ Resid2 @ T7 @ U4 ) ) ) ) ).
% rts_with_joins_axioms_def
thf(fact_659_rts__with__joins__axioms__def,axiom,
( rts_wi637544758655500588_set_a
= ( ^ [Resid2: set_a > set_a > set_a] :
! [T7: set_a,U4: set_a] :
( ( con_set_a @ Resid2 @ T7 @ U4 )
=> ( joinable_set_a @ Resid2 @ T7 @ U4 ) ) ) ) ).
% rts_with_joins_axioms_def
thf(fact_660_confluent__rts__axioms__def,axiom,
( conflu3014480972103220363ioms_a
= ( ^ [Resid2: a > a > a] :
! [T7: a,U4: a] :
( ( coinitial_a @ Resid2 @ T7 @ U4 )
=> ( con_a @ Resid2 @ T7 @ U4 ) ) ) ) ).
% confluent_rts_axioms_def
thf(fact_661_confluent__rts__axioms__def,axiom,
( conflu1148668952538903019_set_a
= ( ^ [Resid2: set_a > set_a > set_a] :
! [T7: set_a,U4: set_a] :
( ( coinitial_set_a @ Resid2 @ T7 @ U4 )
=> ( con_set_a @ Resid2 @ T7 @ U4 ) ) ) ) ).
% confluent_rts_axioms_def
thf(fact_662_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_663_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_664_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_665_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_666_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_667_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_668_Set_Ois__empty__def,axiom,
( is_empty_a
= ( ^ [A8: set_a] : ( A8 = bot_bot_set_a ) ) ) ).
% Set.is_empty_def
thf(fact_669_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_670_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_671_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_672_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_673_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_674_rts_OseqE,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a] :
( ( rts_set_a @ Resid )
=> ( ( seq_set_a @ Resid @ T @ U )
=> ~ ( ( arr_set_a @ Resid @ T )
=> ( ( arr_set_a @ Resid @ U )
=> ( ( targets_set_a @ Resid @ T )
!= ( sources_set_a @ Resid @ U ) ) ) ) ) ) ).
% rts.seqE
thf(fact_675_R_Orts__axioms,axiom,
rts_a @ resid ).
% R.rts_axioms
thf(fact_676_confluent__rts_Oaxioms_I1_J,axiom,
! [Resid: a > a > a] :
( ( confluent_rts_a @ Resid )
=> ( rts_a @ Resid ) ) ).
% confluent_rts.axioms(1)
thf(fact_677_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_678_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_679_rts_Oaxioms_I1_J,axiom,
! [Resid: a > a > a] :
( ( rts_a @ Resid )
=> ( residuation_a @ Resid ) ) ).
% rts.axioms(1)
thf(fact_680_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_681_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_682_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_683_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_684_rts_Oide__backward__stable,axiom,
! [Resid: set_a > set_a > set_a,A: set_a,T: set_a] :
( ( rts_set_a @ Resid )
=> ( ( ide_set_a @ Resid @ A )
=> ( ( ide_set_a @ Resid @ ( Resid @ T @ A ) )
=> ( ide_set_a @ Resid @ T ) ) ) ) ).
% rts.ide_backward_stable
thf(fact_685_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_686_rts_Oprfx__transitive,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a,V: set_a] :
( ( rts_set_a @ Resid )
=> ( ( ide_set_a @ Resid @ ( Resid @ T @ U ) )
=> ( ( ide_set_a @ Resid @ ( Resid @ U @ V ) )
=> ( ide_set_a @ Resid @ ( Resid @ T @ V ) ) ) ) ) ).
% rts.prfx_transitive
thf(fact_687_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_688_rts_Ocong__transitive,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a,V: set_a] :
( ( rts_set_a @ Resid )
=> ( ( ( ide_set_a @ Resid @ ( Resid @ T @ U ) )
& ( ide_set_a @ Resid @ ( Resid @ U @ T ) ) )
=> ( ( ( ide_set_a @ Resid @ ( Resid @ U @ V ) )
& ( ide_set_a @ Resid @ ( Resid @ V @ U ) ) )
=> ( ( ide_set_a @ Resid @ ( Resid @ T @ V ) )
& ( ide_set_a @ Resid @ ( Resid @ V @ T ) ) ) ) ) ) ).
% rts.cong_transitive
thf(fact_689_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_690_rts_Ocong__symmetric,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a] :
( ( rts_set_a @ Resid )
=> ( ( ( ide_set_a @ Resid @ ( Resid @ T @ U ) )
& ( ide_set_a @ Resid @ ( Resid @ U @ T ) ) )
=> ( ( ide_set_a @ Resid @ ( Resid @ U @ T ) )
& ( ide_set_a @ Resid @ ( Resid @ T @ U ) ) ) ) ) ).
% rts.cong_symmetric
thf(fact_691_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_692_rts_Oaxioms_I2_J,axiom,
! [Resid: a > a > a] :
( ( rts_a @ Resid )
=> ( rts_axioms_a @ Resid ) ) ).
% rts.axioms(2)
thf(fact_693_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_694_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_695_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_696_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_697_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_698_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_699_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_700_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 )
=> ? [A5: set_a] :
( ( ide_set_a @ Resid @ A5 )
& ( con_set_a @ Resid @ A5 @ T )
& ( con_set_a @ Resid @ A5 @ U ) ) ) ) ).
% rts.con_imp_coinitial_ax
thf(fact_701_rts_Ocon__imp__coinitial__ax,axiom,
! [Resid: a > a > a,T: a,U: a] :
( ( rts_a @ Resid )
=> ( ( con_a @ Resid @ T @ U )
=> ? [A5: a] :
( ( ide_a @ Resid @ A5 )
& ( con_a @ Resid @ A5 @ T )
& ( con_a @ Resid @ A5 @ U ) ) ) ) ).
% rts.con_imp_coinitial_ax
thf(fact_702_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_703_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_704_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_705_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_706_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_707_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_708_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_709_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_710_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_711_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_712_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_713_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_714_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_715_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_716_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_717_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_718_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_719_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_720_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_721_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_722_rts_Osources__cong__closed,axiom,
! [Resid: set_a > set_a > set_a,A: set_a,T: set_a,A6: set_a] :
( ( rts_set_a @ Resid )
=> ( ( member_set_a @ A @ ( sources_set_a @ Resid @ T ) )
=> ( ( ( ide_set_a @ Resid @ ( Resid @ A @ A6 ) )
& ( ide_set_a @ Resid @ ( Resid @ A6 @ A ) ) )
=> ( member_set_a @ A6 @ ( sources_set_a @ Resid @ T ) ) ) ) ) ).
% rts.sources_cong_closed
thf(fact_723_rts_Osources__cong__closed,axiom,
! [Resid: a > a > a,A: a,T: a,A6: a] :
( ( rts_a @ Resid )
=> ( ( member_a @ A @ ( sources_a @ Resid @ T ) )
=> ( ( ( ide_a @ Resid @ ( Resid @ A @ A6 ) )
& ( ide_a @ Resid @ ( Resid @ A6 @ A ) ) )
=> ( member_a @ A6 @ ( sources_a @ Resid @ T ) ) ) ) ) ).
% rts.sources_cong_closed
thf(fact_724_rts_Osources__are__cong,axiom,
! [Resid: set_a > set_a > set_a,A: set_a,T: set_a,A6: set_a] :
( ( rts_set_a @ Resid )
=> ( ( member_set_a @ A @ ( sources_set_a @ Resid @ T ) )
=> ( ( member_set_a @ A6 @ ( sources_set_a @ Resid @ T ) )
=> ( ( ide_set_a @ Resid @ ( Resid @ A @ A6 ) )
& ( ide_set_a @ Resid @ ( Resid @ A6 @ A ) ) ) ) ) ) ).
% rts.sources_are_cong
thf(fact_725_rts_Osources__are__cong,axiom,
! [Resid: a > a > a,A: a,T: a,A6: a] :
( ( rts_a @ Resid )
=> ( ( member_a @ A @ ( sources_a @ Resid @ T ) )
=> ( ( member_a @ A6 @ ( sources_a @ Resid @ T ) )
=> ( ( ide_a @ Resid @ ( Resid @ A @ A6 ) )
& ( ide_a @ Resid @ ( Resid @ A6 @ A ) ) ) ) ) ) ).
% rts.sources_are_cong
thf(fact_726_rts_Osource__is__ide,axiom,
! [Resid: set_a > set_a > set_a,A: set_a,T: set_a] :
( ( rts_set_a @ Resid )
=> ( ( member_set_a @ A @ ( sources_set_a @ Resid @ T ) )
=> ( ide_set_a @ Resid @ A ) ) ) ).
% rts.source_is_ide
thf(fact_727_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_728_rts_Osources__are__con,axiom,
! [Resid: a > a > a,A: a,T: a,A6: a] :
( ( rts_a @ Resid )
=> ( ( member_a @ A @ ( sources_a @ Resid @ T ) )
=> ( ( member_a @ A6 @ ( sources_a @ Resid @ T ) )
=> ( con_a @ Resid @ A @ A6 ) ) ) ) ).
% rts.sources_are_con
thf(fact_729_rts_Osources__are__con,axiom,
! [Resid: set_a > set_a > set_a,A: set_a,T: set_a,A6: set_a] :
( ( rts_set_a @ Resid )
=> ( ( member_set_a @ A @ ( sources_set_a @ Resid @ T ) )
=> ( ( member_set_a @ A6 @ ( sources_set_a @ Resid @ T ) )
=> ( con_set_a @ Resid @ A @ A6 ) ) ) ) ).
% rts.sources_are_con
thf(fact_730_rts_Ocomposite__ofE,axiom,
! [Resid: set_a > set_a > set_a,U: set_a,T: set_a,V: set_a] :
( ( rts_set_a @ Resid )
=> ( ( composite_of_set_a @ Resid @ U @ T @ V )
=> ~ ( ( ide_set_a @ Resid @ ( Resid @ U @ V ) )
=> ~ ( ( ide_set_a @ Resid @ ( Resid @ ( Resid @ V @ U ) @ T ) )
& ( ide_set_a @ Resid @ ( Resid @ T @ ( Resid @ V @ U ) ) ) ) ) ) ) ).
% rts.composite_ofE
thf(fact_731_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_732_rts_Ocomposite__ofI,axiom,
! [Resid: set_a > set_a > set_a,U: set_a,V: set_a,T: set_a] :
( ( rts_set_a @ Resid )
=> ( ( ide_set_a @ Resid @ ( Resid @ U @ V ) )
=> ( ( ( ide_set_a @ Resid @ ( Resid @ ( Resid @ V @ U ) @ T ) )
& ( ide_set_a @ Resid @ ( Resid @ T @ ( Resid @ V @ U ) ) ) )
=> ( composite_of_set_a @ Resid @ U @ T @ V ) ) ) ) ).
% rts.composite_ofI
thf(fact_733_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_734_rts_Ocomposite__of__def,axiom,
! [Resid: set_a > set_a > set_a,U: set_a,T: set_a,V: set_a] :
( ( rts_set_a @ Resid )
=> ( ( composite_of_set_a @ Resid @ U @ T @ V )
= ( ( ide_set_a @ Resid @ ( Resid @ U @ V ) )
& ( ide_set_a @ Resid @ ( Resid @ ( Resid @ V @ U ) @ T ) )
& ( ide_set_a @ Resid @ ( Resid @ T @ ( Resid @ V @ U ) ) ) ) ) ) ).
% rts.composite_of_def
thf(fact_735_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_736_rts_Ocomposite__of__ide__self,axiom,
! [Resid: set_a > set_a > set_a,A: set_a] :
( ( rts_set_a @ Resid )
=> ( ( ide_set_a @ Resid @ A )
=> ( composite_of_set_a @ Resid @ A @ A @ A ) ) ) ).
% rts.composite_of_ide_self
thf(fact_737_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_738_rts_Ocomposite__of__cancel__left,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a,V: set_a,U2: set_a] :
( ( rts_set_a @ Resid )
=> ( ( composite_of_set_a @ Resid @ T @ U @ V )
=> ( ( composite_of_set_a @ Resid @ T @ U2 @ V )
=> ( ( ide_set_a @ Resid @ ( Resid @ U @ U2 ) )
& ( ide_set_a @ Resid @ ( Resid @ U2 @ U ) ) ) ) ) ) ).
% rts.composite_of_cancel_left
thf(fact_739_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_740_rts_Ocomposite__of__unq__upto__cong,axiom,
! [Resid: set_a > set_a > set_a,U: set_a,T: set_a,V: set_a,V5: set_a] :
( ( rts_set_a @ Resid )
=> ( ( composite_of_set_a @ Resid @ U @ T @ V )
=> ( ( composite_of_set_a @ Resid @ U @ T @ V5 )
=> ( ( ide_set_a @ Resid @ ( Resid @ V @ V5 ) )
& ( ide_set_a @ Resid @ ( Resid @ V5 @ V ) ) ) ) ) ) ).
% rts.composite_of_unq_upto_cong
thf(fact_741_rts_Ocomposite__of__unq__upto__cong,axiom,
! [Resid: a > a > a,U: a,T: a,V: a,V5: a] :
( ( rts_a @ Resid )
=> ( ( composite_of_a @ Resid @ U @ T @ V )
=> ( ( composite_of_a @ Resid @ U @ T @ V5 )
=> ( ( ide_a @ Resid @ ( Resid @ V @ V5 ) )
& ( ide_a @ Resid @ ( Resid @ V5 @ V ) ) ) ) ) ) ).
% rts.composite_of_unq_upto_cong
thf(fact_742_rts_Oresid__composite__of_I4_J,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a,W2: set_a,V: set_a] :
( ( rts_set_a @ Resid )
=> ( ( composite_of_set_a @ Resid @ T @ U @ W2 )
=> ( ( con_set_a @ Resid @ W2 @ V )
=> ( composite_of_set_a @ Resid @ ( Resid @ T @ V ) @ ( Resid @ U @ ( Resid @ V @ T ) ) @ ( Resid @ W2 @ V ) ) ) ) ) ).
% rts.resid_composite_of(4)
thf(fact_743_rts_Oresid__composite__of_I4_J,axiom,
! [Resid: a > a > a,T: a,U: a,W2: a,V: a] :
( ( rts_a @ Resid )
=> ( ( composite_of_a @ Resid @ T @ U @ W2 )
=> ( ( con_a @ Resid @ W2 @ V )
=> ( composite_of_a @ Resid @ ( Resid @ T @ V ) @ ( Resid @ U @ ( Resid @ V @ T ) ) @ ( Resid @ W2 @ V ) ) ) ) ) ).
% rts.resid_composite_of(4)
thf(fact_744_rts_Oresid__composite__of_I2_J,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a,W2: set_a,V: set_a] :
( ( rts_set_a @ Resid )
=> ( ( composite_of_set_a @ Resid @ T @ U @ W2 )
=> ( ( con_set_a @ Resid @ W2 @ V )
=> ( con_set_a @ Resid @ ( Resid @ V @ T ) @ U ) ) ) ) ).
% rts.resid_composite_of(2)
thf(fact_745_rts_Oresid__composite__of_I2_J,axiom,
! [Resid: a > a > a,T: a,U: a,W2: a,V: a] :
( ( rts_a @ Resid )
=> ( ( composite_of_a @ Resid @ T @ U @ W2 )
=> ( ( con_a @ Resid @ W2 @ V )
=> ( con_a @ Resid @ ( Resid @ V @ T ) @ U ) ) ) ) ).
% rts.resid_composite_of(2)
thf(fact_746_rts_Oresid__composite__of_I1_J,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a,W2: set_a,V: set_a] :
( ( rts_set_a @ Resid )
=> ( ( composite_of_set_a @ Resid @ T @ U @ W2 )
=> ( ( con_set_a @ Resid @ W2 @ V )
=> ( con_set_a @ Resid @ ( Resid @ V @ T ) @ ( Resid @ W2 @ T ) ) ) ) ) ).
% rts.resid_composite_of(1)
thf(fact_747_rts_Oresid__composite__of_I1_J,axiom,
! [Resid: a > a > a,T: a,U: a,W2: a,V: a] :
( ( rts_a @ Resid )
=> ( ( composite_of_a @ Resid @ T @ U @ W2 )
=> ( ( con_a @ Resid @ W2 @ V )
=> ( con_a @ Resid @ ( Resid @ V @ T ) @ ( Resid @ W2 @ T ) ) ) ) ) ).
% rts.resid_composite_of(1)
thf(fact_748_rts_Ocon__prfx__composite__of_I2_J,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a,W2: set_a,V: set_a] :
( ( rts_set_a @ Resid )
=> ( ( composite_of_set_a @ Resid @ T @ U @ W2 )
=> ( ( con_set_a @ Resid @ W2 @ V )
=> ( con_set_a @ Resid @ T @ V ) ) ) ) ).
% rts.con_prfx_composite_of(2)
thf(fact_749_rts_Ocon__prfx__composite__of_I2_J,axiom,
! [Resid: a > a > a,T: a,U: a,W2: a,V: a] :
( ( rts_a @ Resid )
=> ( ( composite_of_a @ Resid @ T @ U @ W2 )
=> ( ( con_a @ Resid @ W2 @ V )
=> ( con_a @ Resid @ T @ V ) ) ) ) ).
% rts.con_prfx_composite_of(2)
thf(fact_750_rts_Ocon__prfx__composite__of_I1_J,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a,W2: set_a] :
( ( rts_set_a @ Resid )
=> ( ( composite_of_set_a @ Resid @ T @ U @ W2 )
=> ( con_set_a @ Resid @ T @ W2 ) ) ) ).
% rts.con_prfx_composite_of(1)
thf(fact_751_rts_Ocon__prfx__composite__of_I1_J,axiom,
! [Resid: a > a > a,T: a,U: a,W2: a] :
( ( rts_a @ Resid )
=> ( ( composite_of_a @ Resid @ T @ U @ W2 )
=> ( con_a @ Resid @ T @ W2 ) ) ) ).
% rts.con_prfx_composite_of(1)
thf(fact_752_rts_Obounded__imp__con,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a,V: set_a,T4: set_a,U2: set_a] :
( ( rts_set_a @ Resid )
=> ( ( composite_of_set_a @ Resid @ T @ U @ V )
=> ( ( composite_of_set_a @ Resid @ T4 @ U2 @ V )
=> ( con_set_a @ Resid @ T @ T4 ) ) ) ) ).
% rts.bounded_imp_con
thf(fact_753_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_754_rts_Ocon__composite__of__iff,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a,V: set_a,W2: set_a] :
( ( rts_set_a @ Resid )
=> ( ( composite_of_set_a @ Resid @ T @ U @ V )
=> ( ( con_set_a @ Resid @ W2 @ V )
= ( con_set_a @ Resid @ ( Resid @ W2 @ T ) @ U ) ) ) ) ).
% rts.con_composite_of_iff
thf(fact_755_rts_Ocon__composite__of__iff,axiom,
! [Resid: a > a > a,T: a,U: a,V: a,W2: a] :
( ( rts_a @ Resid )
=> ( ( composite_of_a @ Resid @ T @ U @ V )
=> ( ( con_a @ Resid @ W2 @ V )
= ( con_a @ Resid @ ( Resid @ W2 @ T ) @ U ) ) ) ) ).
% rts.con_composite_of_iff
thf(fact_756_rts_Otarget__is__ide,axiom,
! [Resid: set_a > set_a > set_a,A: set_a,T: set_a] :
( ( rts_set_a @ Resid )
=> ( ( member_set_a @ A @ ( targets_set_a @ Resid @ T ) )
=> ( ide_set_a @ Resid @ A ) ) ) ).
% rts.target_is_ide
thf(fact_757_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_758_rts_Otargets__are__cong,axiom,
! [Resid: set_a > set_a > set_a,B: set_a,T: set_a,B4: set_a] :
( ( rts_set_a @ Resid )
=> ( ( member_set_a @ B @ ( targets_set_a @ Resid @ T ) )
=> ( ( member_set_a @ B4 @ ( targets_set_a @ Resid @ T ) )
=> ( ( ide_set_a @ Resid @ ( Resid @ B @ B4 ) )
& ( ide_set_a @ Resid @ ( Resid @ B4 @ B ) ) ) ) ) ) ).
% rts.targets_are_cong
thf(fact_759_rts_Otargets__are__cong,axiom,
! [Resid: a > a > a,B: a,T: a,B4: a] :
( ( rts_a @ Resid )
=> ( ( member_a @ B @ ( targets_a @ Resid @ T ) )
=> ( ( member_a @ B4 @ ( targets_a @ Resid @ T ) )
=> ( ( ide_a @ Resid @ ( Resid @ B @ B4 ) )
& ( ide_a @ Resid @ ( Resid @ B4 @ B ) ) ) ) ) ) ).
% rts.targets_are_cong
thf(fact_760_rts_Otargets__cong__closed,axiom,
! [Resid: set_a > set_a > set_a,B: set_a,T: set_a,B4: set_a] :
( ( rts_set_a @ Resid )
=> ( ( member_set_a @ B @ ( targets_set_a @ Resid @ T ) )
=> ( ( ( ide_set_a @ Resid @ ( Resid @ B @ B4 ) )
& ( ide_set_a @ Resid @ ( Resid @ B4 @ B ) ) )
=> ( member_set_a @ B4 @ ( targets_set_a @ Resid @ T ) ) ) ) ) ).
% rts.targets_cong_closed
thf(fact_761_rts_Otargets__cong__closed,axiom,
! [Resid: a > a > a,B: a,T: a,B4: a] :
( ( rts_a @ Resid )
=> ( ( member_a @ B @ ( targets_a @ Resid @ T ) )
=> ( ( ( ide_a @ Resid @ ( Resid @ B @ B4 ) )
& ( ide_a @ Resid @ ( Resid @ B4 @ B ) ) )
=> ( member_a @ B4 @ ( targets_a @ Resid @ T ) ) ) ) ) ).
% rts.targets_cong_closed
thf(fact_762_rts_Otargets__are__con,axiom,
! [Resid: a > a > a,B: a,T: a,B4: a] :
( ( rts_a @ Resid )
=> ( ( member_a @ B @ ( targets_a @ Resid @ T ) )
=> ( ( member_a @ B4 @ ( targets_a @ Resid @ T ) )
=> ( con_a @ Resid @ B @ B4 ) ) ) ) ).
% rts.targets_are_con
thf(fact_763_rts_Otargets__are__con,axiom,
! [Resid: set_a > set_a > set_a,B: set_a,T: set_a,B4: set_a] :
( ( rts_set_a @ Resid )
=> ( ( member_set_a @ B @ ( targets_set_a @ Resid @ T ) )
=> ( ( member_set_a @ B4 @ ( targets_set_a @ Resid @ T ) )
=> ( con_set_a @ Resid @ B @ B4 ) ) ) ) ).
% rts.targets_are_con
thf(fact_764_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_765_rts_Otargets__resid__sym,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a] :
( ( rts_set_a @ Resid )
=> ( ( con_set_a @ Resid @ T @ U )
=> ( ( targets_set_a @ Resid @ ( Resid @ T @ U ) )
= ( targets_set_a @ Resid @ ( Resid @ U @ T ) ) ) ) ) ).
% rts.targets_resid_sym
thf(fact_766_rts_Oarr__composite__of,axiom,
! [Resid: set_a > set_a > set_a,U: set_a,T: set_a,V: set_a] :
( ( rts_set_a @ Resid )
=> ( ( composite_of_set_a @ Resid @ U @ T @ V )
=> ( arr_set_a @ Resid @ V ) ) ) ).
% rts.arr_composite_of
thf(fact_767_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_768_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_769_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_770_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_771_rts_Ojoin__of__un__upto__cong,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a,V: set_a,V5: set_a] :
( ( rts_set_a @ Resid )
=> ( ( join_of_set_a @ Resid @ T @ U @ V )
=> ( ( join_of_set_a @ Resid @ T @ U @ V5 )
=> ( ( ide_set_a @ Resid @ ( Resid @ V @ V5 ) )
& ( ide_set_a @ Resid @ ( Resid @ V5 @ V ) ) ) ) ) ) ).
% rts.join_of_un_upto_cong
thf(fact_772_rts_Ojoin__of__un__upto__cong,axiom,
! [Resid: a > a > a,T: a,U: a,V: a,V5: a] :
( ( rts_a @ Resid )
=> ( ( join_of_a @ Resid @ T @ U @ V )
=> ( ( join_of_a @ Resid @ T @ U @ V5 )
=> ( ( ide_a @ Resid @ ( Resid @ V @ V5 ) )
& ( ide_a @ Resid @ ( Resid @ V5 @ V ) ) ) ) ) ) ).
% rts.join_of_un_upto_cong
thf(fact_773_rts_Ocon__with__join__of__iff_I2_J,axiom,
! [Resid: a > a > a,T: a,U: a,W2: a,V: a] :
( ( rts_a @ Resid )
=> ( ( join_of_a @ Resid @ T @ U @ W2 )
=> ( ( con_a @ Resid @ W2 @ V )
=> ( ( con_a @ Resid @ T @ V )
& ( con_a @ Resid @ ( Resid @ V @ T ) @ ( Resid @ U @ T ) ) ) ) ) ) ).
% rts.con_with_join_of_iff(2)
thf(fact_774_rts_Ocon__with__join__of__iff_I2_J,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a,W2: set_a,V: set_a] :
( ( rts_set_a @ Resid )
=> ( ( join_of_set_a @ Resid @ T @ U @ W2 )
=> ( ( con_set_a @ Resid @ W2 @ V )
=> ( ( con_set_a @ Resid @ T @ V )
& ( con_set_a @ Resid @ ( Resid @ V @ T ) @ ( Resid @ U @ T ) ) ) ) ) ) ).
% rts.con_with_join_of_iff(2)
thf(fact_775_rts_Ocon__with__join__of__iff_I1_J,axiom,
! [Resid: a > a > a,T: a,U: a,W2: a,V: a] :
( ( rts_a @ Resid )
=> ( ( join_of_a @ Resid @ T @ U @ W2 )
=> ( ( ( con_a @ Resid @ U @ V )
& ( con_a @ Resid @ ( Resid @ V @ U ) @ ( Resid @ T @ U ) ) )
=> ( con_a @ Resid @ W2 @ V ) ) ) ) ).
% rts.con_with_join_of_iff(1)
thf(fact_776_rts_Ocon__with__join__of__iff_I1_J,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a,W2: set_a,V: set_a] :
( ( rts_set_a @ Resid )
=> ( ( join_of_set_a @ Resid @ T @ U @ W2 )
=> ( ( ( con_set_a @ Resid @ U @ V )
& ( con_set_a @ Resid @ ( Resid @ V @ U ) @ ( Resid @ T @ U ) ) )
=> ( con_set_a @ Resid @ W2 @ V ) ) ) ) ).
% rts.con_with_join_of_iff(1)
thf(fact_777_rts_Ojoin__of__resid,axiom,
! [Resid: a > a > a,T: a,U: a,W2: a,V: a] :
( ( rts_a @ Resid )
=> ( ( join_of_a @ Resid @ T @ U @ W2 )
=> ( ( con_a @ Resid @ V @ W2 )
=> ( join_of_a @ Resid @ ( Resid @ T @ V ) @ ( Resid @ U @ V ) @ ( Resid @ W2 @ V ) ) ) ) ) ).
% rts.join_of_resid
thf(fact_778_rts_Ojoin__of__resid,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a,W2: set_a,V: set_a] :
( ( rts_set_a @ Resid )
=> ( ( join_of_set_a @ Resid @ T @ U @ W2 )
=> ( ( con_set_a @ Resid @ V @ W2 )
=> ( join_of_set_a @ Resid @ ( Resid @ T @ V ) @ ( Resid @ U @ V ) @ ( Resid @ W2 @ V ) ) ) ) ) ).
% rts.join_of_resid
thf(fact_779_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_780_rts_Ojoin__of__arr__self,axiom,
! [Resid: set_a > set_a > set_a,T: set_a] :
( ( rts_set_a @ Resid )
=> ( ( arr_set_a @ Resid @ T )
=> ( join_of_set_a @ Resid @ T @ T @ T ) ) ) ).
% rts.join_of_arr_self
thf(fact_781_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_782_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_783_rts_Ocong__respects__seq,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a,T4: set_a,U2: set_a] :
( ( rts_set_a @ Resid )
=> ( ( seq_set_a @ Resid @ T @ U )
=> ( ( ( ide_set_a @ Resid @ ( Resid @ T @ T4 ) )
& ( ide_set_a @ Resid @ ( Resid @ T4 @ T ) ) )
=> ( ( ( ide_set_a @ Resid @ ( Resid @ U @ U2 ) )
& ( ide_set_a @ Resid @ ( Resid @ U2 @ U ) ) )
=> ( seq_set_a @ Resid @ T4 @ U2 ) ) ) ) ) ).
% rts.cong_respects_seq
thf(fact_784_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_785_rts_Ocoinitial__ide__are__cong,axiom,
! [Resid: set_a > set_a > set_a,A: set_a,A6: set_a] :
( ( rts_set_a @ Resid )
=> ( ( ide_set_a @ Resid @ A )
=> ( ( ide_set_a @ Resid @ A6 )
=> ( ( coinitial_set_a @ Resid @ A @ A6 )
=> ( ( ide_set_a @ Resid @ ( Resid @ A @ A6 ) )
& ( ide_set_a @ Resid @ ( Resid @ A6 @ A ) ) ) ) ) ) ) ).
% rts.coinitial_ide_are_cong
thf(fact_786_rts_Ocoinitial__ide__are__cong,axiom,
! [Resid: a > a > a,A: a,A6: a] :
( ( rts_a @ Resid )
=> ( ( ide_a @ Resid @ A )
=> ( ( ide_a @ Resid @ A6 )
=> ( ( coinitial_a @ Resid @ A @ A6 )
=> ( ( ide_a @ Resid @ ( Resid @ A @ A6 ) )
& ( ide_a @ Resid @ ( Resid @ A6 @ A ) ) ) ) ) ) ) ).
% rts.coinitial_ide_are_cong
thf(fact_787_rts_Ocong__implies__coinitial,axiom,
! [Resid: set_a > set_a > set_a,U: set_a,U2: set_a] :
( ( rts_set_a @ Resid )
=> ( ( ( ide_set_a @ Resid @ ( Resid @ U @ U2 ) )
& ( ide_set_a @ Resid @ ( Resid @ U2 @ U ) ) )
=> ( coinitial_set_a @ Resid @ U @ U2 ) ) ) ).
% rts.cong_implies_coinitial
thf(fact_788_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_789_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_790_rts_Ocon__imp__coinitial,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a] :
( ( rts_set_a @ Resid )
=> ( ( con_set_a @ Resid @ T @ U )
=> ( coinitial_set_a @ Resid @ T @ U ) ) ) ).
% rts.con_imp_coinitial
thf(fact_791_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_792_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_793_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_794_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_795_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_796_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_797_rts_Ojoinable__implies__con,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a] :
( ( rts_set_a @ Resid )
=> ( ( joinable_set_a @ Resid @ T @ U )
=> ( con_set_a @ Resid @ T @ U ) ) ) ).
% rts.joinable_implies_con
thf(fact_798_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_799_rts_OcomposableD_I1_J,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a] :
( ( rts_set_a @ Resid )
=> ( ( composable_set_a @ Resid @ T @ U )
=> ( arr_set_a @ Resid @ T ) ) ) ).
% rts.composableD(1)
thf(fact_800_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_801_rts_OcomposableD_I2_J,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a] :
( ( rts_set_a @ Resid )
=> ( ( composable_set_a @ Resid @ T @ U )
=> ( arr_set_a @ Resid @ U ) ) ) ).
% rts.composableD(2)
thf(fact_802_rts_Ocomposable__def,axiom,
! [Resid: a > a > a,T: a,U: a] :
( ( rts_a @ Resid )
=> ( ( composable_a @ Resid @ T @ U )
= ( ? [X5: a] : ( composite_of_a @ Resid @ T @ U @ X5 ) ) ) ) ).
% rts.composable_def
thf(fact_803_rts_Ocong__implies__coterminal,axiom,
! [Resid: set_a > set_a > set_a,U: set_a,U2: set_a] :
( ( rts_set_a @ Resid )
=> ( ( ( ide_set_a @ Resid @ ( Resid @ U @ U2 ) )
& ( ide_set_a @ Resid @ ( Resid @ U2 @ U ) ) )
=> ( coterminal_set_a @ Resid @ U @ U2 ) ) ) ).
% rts.cong_implies_coterminal
thf(fact_804_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_805_rts_Ojoinable__def,axiom,
! [Resid: a > a > a,T: a,U: a] :
( ( rts_a @ Resid )
=> ( ( joinable_a @ Resid @ T @ U )
= ( ? [X5: a] : ( join_of_a @ Resid @ T @ U @ X5 ) ) ) ) ).
% rts.joinable_def
thf(fact_806_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_807_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_808_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_809_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_810_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_811_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_812_rts__def,axiom,
( rts_set_a
= ( ^ [Resid2: set_a > set_a > set_a] :
( ( residuation_set_a @ Resid2 )
& ( rts_axioms_set_a @ Resid2 ) ) ) ) ).
% rts_def
thf(fact_813_rts__def,axiom,
( rts_a
= ( ^ [Resid2: a > a > a] :
( ( residuation_a @ Resid2 )
& ( rts_axioms_a @ Resid2 ) ) ) ) ).
% rts_def
thf(fact_814_rts_Ointro,axiom,
! [Resid: set_a > set_a > set_a] :
( ( residuation_set_a @ Resid )
=> ( ( rts_axioms_set_a @ Resid )
=> ( rts_set_a @ Resid ) ) ) ).
% rts.intro
thf(fact_815_rts_Ointro,axiom,
! [Resid: a > a > a] :
( ( residuation_a @ Resid )
=> ( ( rts_axioms_a @ Resid )
=> ( rts_a @ Resid ) ) ) ).
% rts.intro
thf(fact_816_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_817_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_818_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_819_rts_Oarr__iff__has__source,axiom,
! [Resid: set_a > set_a > set_a,T: set_a] :
( ( rts_set_a @ Resid )
=> ( ( arr_set_a @ Resid @ T )
= ( ( sources_set_a @ Resid @ T )
!= bot_bot_set_set_a ) ) ) ).
% rts.arr_iff_has_source
thf(fact_820_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_821_rts_Oarr__iff__has__target,axiom,
! [Resid: set_a > set_a > set_a,T: set_a] :
( ( rts_set_a @ Resid )
=> ( ( arr_set_a @ Resid @ T )
= ( ( targets_set_a @ Resid @ T )
!= bot_bot_set_set_a ) ) ) ).
% rts.arr_iff_has_target
thf(fact_822_rts_Oin__sourcesE,axiom,
! [Resid: set_a > set_a > set_a,A: set_a,T: set_a] :
( ( rts_set_a @ Resid )
=> ( ( member_set_a @ A @ ( sources_set_a @ Resid @ T ) )
=> ~ ( ( ide_set_a @ Resid @ A )
=> ~ ( con_set_a @ Resid @ T @ A ) ) ) ) ).
% rts.in_sourcesE
thf(fact_823_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_824_rts_Oin__sourcesI,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 )
=> ( member_set_a @ A @ ( sources_set_a @ Resid @ T ) ) ) ) ) ).
% rts.in_sourcesI
thf(fact_825_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_826_rts_Osources__con__closed,axiom,
! [Resid: set_a > set_a > set_a,A: set_a,T: set_a,A6: set_a] :
( ( rts_set_a @ Resid )
=> ( ( member_set_a @ A @ ( sources_set_a @ Resid @ T ) )
=> ( ( ide_set_a @ Resid @ A6 )
=> ( ( con_set_a @ Resid @ A @ A6 )
=> ( member_set_a @ A6 @ ( sources_set_a @ Resid @ T ) ) ) ) ) ) ).
% rts.sources_con_closed
thf(fact_827_rts_Osources__con__closed,axiom,
! [Resid: a > a > a,A: a,T: a,A6: a] :
( ( rts_a @ Resid )
=> ( ( member_a @ A @ ( sources_a @ Resid @ T ) )
=> ( ( ide_a @ Resid @ A6 )
=> ( ( con_a @ Resid @ A @ A6 )
=> ( member_a @ A6 @ ( sources_a @ Resid @ T ) ) ) ) ) ) ).
% rts.sources_con_closed
thf(fact_828_rts_Ocomposite__of__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 )
=> ( ( composite_of_set_a @ Resid @ A @ T @ T )
= ( con_set_a @ Resid @ T @ A ) ) ) ) ).
% rts.composite_of_ide_arr
thf(fact_829_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_830_rts_Ocomposite__of__arr__ide,axiom,
! [Resid: set_a > set_a > set_a,B: set_a,T: set_a] :
( ( rts_set_a @ Resid )
=> ( ( ide_set_a @ Resid @ B )
=> ( ( composite_of_set_a @ Resid @ T @ B @ T )
= ( con_set_a @ Resid @ ( Resid @ T @ T ) @ B ) ) ) ) ).
% rts.composite_of_arr_ide
thf(fact_831_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_832_rts_Oresid__composite__of_I3_J,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a,W2: set_a,V: set_a] :
( ( rts_set_a @ Resid )
=> ( ( composite_of_set_a @ Resid @ T @ U @ W2 )
=> ( ( con_set_a @ Resid @ W2 @ V )
=> ( ( ide_set_a @ Resid @ ( Resid @ ( Resid @ V @ W2 ) @ ( Resid @ ( Resid @ V @ T ) @ U ) ) )
& ( ide_set_a @ Resid @ ( Resid @ ( Resid @ ( Resid @ V @ T ) @ U ) @ ( Resid @ V @ W2 ) ) ) ) ) ) ) ).
% rts.resid_composite_of(3)
thf(fact_833_rts_Oresid__composite__of_I3_J,axiom,
! [Resid: a > a > a,T: a,U: a,W2: a,V: a] :
( ( rts_a @ Resid )
=> ( ( composite_of_a @ Resid @ T @ U @ W2 )
=> ( ( con_a @ Resid @ W2 @ V )
=> ( ( ide_a @ Resid @ ( Resid @ ( Resid @ V @ W2 ) @ ( Resid @ ( Resid @ V @ T ) @ U ) ) )
& ( ide_a @ Resid @ ( Resid @ ( Resid @ ( Resid @ V @ T ) @ U ) @ ( Resid @ V @ W2 ) ) ) ) ) ) ) ).
% rts.resid_composite_of(3)
thf(fact_834_rts_Otargets__con__closed,axiom,
! [Resid: set_a > set_a > set_a,B: set_a,T: set_a,B4: set_a] :
( ( rts_set_a @ Resid )
=> ( ( member_set_a @ B @ ( targets_set_a @ Resid @ T ) )
=> ( ( ide_set_a @ Resid @ B4 )
=> ( ( con_set_a @ Resid @ B @ B4 )
=> ( member_set_a @ B4 @ ( targets_set_a @ Resid @ T ) ) ) ) ) ) ).
% rts.targets_con_closed
thf(fact_835_rts_Otargets__con__closed,axiom,
! [Resid: a > a > a,B: a,T: a,B4: a] :
( ( rts_a @ Resid )
=> ( ( member_a @ B @ ( targets_a @ Resid @ T ) )
=> ( ( ide_a @ Resid @ B4 )
=> ( ( con_a @ Resid @ B @ B4 )
=> ( member_a @ B4 @ ( targets_a @ Resid @ T ) ) ) ) ) ) ).
% rts.targets_con_closed
thf(fact_836_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_837_rts_Osources__resid,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a] :
( ( rts_set_a @ Resid )
=> ( ( con_set_a @ Resid @ T @ U )
=> ( ( sources_set_a @ Resid @ ( Resid @ T @ U ) )
= ( targets_set_a @ Resid @ U ) ) ) ) ).
% rts.sources_resid
thf(fact_838_rts_Ocomposite__of__source__arr,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,A: set_a] :
( ( rts_set_a @ Resid )
=> ( ( arr_set_a @ Resid @ T )
=> ( ( member_set_a @ A @ ( sources_set_a @ Resid @ T ) )
=> ( composite_of_set_a @ Resid @ A @ T @ T ) ) ) ) ).
% rts.composite_of_source_arr
thf(fact_839_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_840_rts_Oide__trg,axiom,
! [Resid: set_a > set_a > set_a,T: set_a] :
( ( rts_set_a @ Resid )
=> ( ( arr_set_a @ Resid @ T )
=> ( ide_set_a @ Resid @ ( trg_set_a @ Resid @ T ) ) ) ) ).
% rts.ide_trg
thf(fact_841_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_842_rts_Ocomposite__of__arr__target,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,B: set_a] :
( ( rts_set_a @ Resid )
=> ( ( arr_set_a @ Resid @ T )
=> ( ( member_set_a @ B @ ( targets_set_a @ Resid @ T ) )
=> ( composite_of_set_a @ Resid @ T @ B @ T ) ) ) ) ).
% rts.composite_of_arr_target
thf(fact_843_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_844_confluent__rts_Ointro,axiom,
! [Resid: a > a > a] :
( ( rts_a @ Resid )
=> ( ( conflu3014480972103220363ioms_a @ Resid )
=> ( confluent_rts_a @ Resid ) ) ) ).
% confluent_rts.intro
thf(fact_845_confluent__rts__def,axiom,
( confluent_rts_a
= ( ^ [Resid2: a > a > a] :
( ( rts_a @ Resid2 )
& ( conflu3014480972103220363ioms_a @ Resid2 ) ) ) ) ).
% confluent_rts_def
thf(fact_846_rts_Otrg__in__targets,axiom,
! [Resid: set_a > set_a > set_a,T: set_a] :
( ( rts_set_a @ Resid )
=> ( ( arr_set_a @ Resid @ T )
=> ( member_set_a @ ( trg_set_a @ Resid @ T ) @ ( targets_set_a @ Resid @ T ) ) ) ) ).
% rts.trg_in_targets
thf(fact_847_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_848_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_849_rts_Ojoin__of__arr__src_I2_J,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,A: set_a] :
( ( rts_set_a @ Resid )
=> ( ( arr_set_a @ Resid @ T )
=> ( ( member_set_a @ A @ ( sources_set_a @ Resid @ T ) )
=> ( join_of_set_a @ Resid @ T @ A @ T ) ) ) ) ).
% rts.join_of_arr_src(2)
thf(fact_850_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_851_rts_Ojoin__of__arr__src_I1_J,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,A: set_a] :
( ( rts_set_a @ Resid )
=> ( ( arr_set_a @ Resid @ T )
=> ( ( member_set_a @ A @ ( sources_set_a @ Resid @ T ) )
=> ( join_of_set_a @ Resid @ A @ T @ T ) ) ) ) ).
% rts.join_of_arr_src(1)
thf(fact_852_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_853_rts_Ocoinitial__iff,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,T4: set_a] :
( ( rts_set_a @ Resid )
=> ( ( coinitial_set_a @ Resid @ T @ T4 )
= ( ( arr_set_a @ Resid @ T )
& ( arr_set_a @ Resid @ T4 )
& ( ( sources_set_a @ Resid @ T )
= ( sources_set_a @ Resid @ T4 ) ) ) ) ) ).
% rts.coinitial_iff
thf(fact_854_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_855_rts_OcoinitialI,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a] :
( ( rts_set_a @ Resid )
=> ( ( arr_set_a @ Resid @ T )
=> ( ( ( sources_set_a @ Resid @ T )
= ( sources_set_a @ Resid @ U ) )
=> ( coinitial_set_a @ Resid @ T @ U ) ) ) ) ).
% rts.coinitialI
thf(fact_856_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_857_rts_OcoinitialE,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a] :
( ( rts_set_a @ Resid )
=> ( ( coinitial_set_a @ Resid @ T @ U )
=> ~ ( ( arr_set_a @ Resid @ T )
=> ( ( arr_set_a @ Resid @ U )
=> ( ( sources_set_a @ Resid @ T )
!= ( sources_set_a @ Resid @ U ) ) ) ) ) ) ).
% rts.coinitialE
thf(fact_858_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_859_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_860_rts_Ocoterminal__iff,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,T4: set_a] :
( ( rts_set_a @ Resid )
=> ( ( coterminal_set_a @ Resid @ T @ T4 )
= ( ( arr_set_a @ Resid @ T )
& ( arr_set_a @ Resid @ T4 )
& ( ( targets_set_a @ Resid @ T )
= ( targets_set_a @ Resid @ T4 ) ) ) ) ) ).
% rts.coterminal_iff
thf(fact_861_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_862_rts_OcoterminalI,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a] :
( ( rts_set_a @ Resid )
=> ( ( arr_set_a @ Resid @ T )
=> ( ( ( targets_set_a @ Resid @ T )
= ( targets_set_a @ Resid @ U ) )
=> ( coterminal_set_a @ Resid @ T @ U ) ) ) ) ).
% rts.coterminalI
thf(fact_863_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_864_rts_OcoterminalE,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a] :
( ( rts_set_a @ Resid )
=> ( ( coterminal_set_a @ Resid @ T @ U )
=> ~ ( ( arr_set_a @ Resid @ T )
=> ( ( arr_set_a @ Resid @ U )
=> ( ( targets_set_a @ Resid @ T )
!= ( targets_set_a @ Resid @ U ) ) ) ) ) ) ).
% rts.coterminalE
thf(fact_865_rts_Ocoterminal__iff__con__trg,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a] :
( ( rts_set_a @ Resid )
=> ( ( coterminal_set_a @ Resid @ T @ U )
= ( con_set_a @ Resid @ ( trg_set_a @ Resid @ T ) @ ( trg_set_a @ Resid @ U ) ) ) ) ).
% rts.coterminal_iff_con_trg
thf(fact_866_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_867_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_868_rts_Ocon__imp__common__source,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a] :
( ( rts_set_a @ Resid )
=> ( ( con_set_a @ Resid @ T @ U )
=> ( ( inf_inf_set_set_a @ ( sources_set_a @ Resid @ T ) @ ( sources_set_a @ Resid @ U ) )
!= bot_bot_set_set_a ) ) ) ).
% rts.con_imp_common_source
thf(fact_869_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_870_rts_Oin__targetsI,axiom,
! [Resid: set_a > set_a > set_a,B: set_a,T: set_a] :
( ( rts_set_a @ Resid )
=> ( ( ide_set_a @ Resid @ B )
=> ( ( con_set_a @ Resid @ ( trg_set_a @ Resid @ T ) @ B )
=> ( member_set_a @ B @ ( targets_set_a @ Resid @ T ) ) ) ) ) ).
% rts.in_targetsI
thf(fact_871_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_872_rts_Oin__targetsE,axiom,
! [Resid: set_a > set_a > set_a,B: set_a,T: set_a] :
( ( rts_set_a @ Resid )
=> ( ( member_set_a @ B @ ( targets_set_a @ Resid @ T ) )
=> ~ ( ( ide_set_a @ Resid @ B )
=> ~ ( con_set_a @ Resid @ ( trg_set_a @ Resid @ T ) @ B ) ) ) ) ).
% rts.in_targetsE
thf(fact_873_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_874_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_875_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_876_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_877_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_878_rts_OseqI,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a] :
( ( rts_set_a @ Resid )
=> ( ( arr_set_a @ Resid @ T )
=> ( ( arr_set_a @ Resid @ U )
=> ( ( ( targets_set_a @ Resid @ T )
= ( sources_set_a @ Resid @ U ) )
=> ( seq_set_a @ Resid @ T @ U ) ) ) ) ) ).
% rts.seqI
thf(fact_879_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_880_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_881_identity__simulation_Ointro,axiom,
! [Resid: a > a > a] :
( ( rts_a @ Resid )
=> ( identi4709066280192368860tion_a @ Resid ) ) ).
% identity_simulation.intro
thf(fact_882_identity__simulation_Oaxioms,axiom,
! [Resid: a > a > a] :
( ( identi4709066280192368860tion_a @ Resid )
=> ( rts_a @ Resid ) ) ).
% identity_simulation.axioms
thf(fact_883_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_884_rts__with__composites_Odiamond__commutes__upto__cong,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a,V: set_a,V5: set_a] :
( ( rts_wi6488725688526449878_set_a @ Resid )
=> ( ( composite_of_set_a @ Resid @ T @ ( Resid @ U @ T ) @ V )
=> ( ( composite_of_set_a @ Resid @ U @ ( Resid @ T @ U ) @ V5 )
=> ( ( ide_set_a @ Resid @ ( Resid @ V @ V5 ) )
& ( ide_set_a @ Resid @ ( Resid @ V5 @ V ) ) ) ) ) ) ).
% rts_with_composites.diamond_commutes_upto_cong
thf(fact_885_rts__with__composites_Odiamond__commutes__upto__cong,axiom,
! [Resid: a > a > a,T: a,U: a,V: a,V5: a] :
( ( rts_wi3777564303360811894ites_a @ Resid )
=> ( ( composite_of_a @ Resid @ T @ ( Resid @ U @ T ) @ V )
=> ( ( composite_of_a @ Resid @ U @ ( Resid @ T @ U ) @ V5 )
=> ( ( ide_a @ Resid @ ( Resid @ V @ V5 ) )
& ( ide_a @ Resid @ ( Resid @ V5 @ V ) ) ) ) ) ) ).
% rts_with_composites.diamond_commutes_upto_cong
thf(fact_886_rts__with__composites_Oobtains__composite__of,axiom,
! [Resid: a > a > a,G: a,F: a] :
( ( rts_wi3777564303360811894ites_a @ Resid )
=> ( ( seq_a @ Resid @ G @ F )
=> ~ ! [H: a] :
~ ( composite_of_a @ Resid @ G @ F @ H ) ) ) ).
% rts_with_composites.obtains_composite_of
thf(fact_887_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_888_rts__with__composites_Ocomposable__iff__seq,axiom,
! [Resid: a > a > a,G: a,F: a] :
( ( rts_wi3777564303360811894ites_a @ Resid )
=> ( ( composable_a @ Resid @ G @ F )
= ( seq_a @ Resid @ G @ F ) ) ) ).
% rts_with_composites.composable_iff_seq
thf(fact_889_identity__simulation__def,axiom,
identi4709066280192368860tion_a = rts_a ).
% identity_simulation_def
thf(fact_890_R_Otargets__def,axiom,
! [T: a] :
( ( targets_a @ resid @ T )
= ( collect_a
@ ^ [B2: a] :
( ( ide_a @ resid @ B2 )
& ( con_a @ resid @ ( trg_a @ resid @ T ) @ B2 ) ) ) ) ).
% R.targets_def
thf(fact_891_extensional__rts__with__joins_Otrg__join_092_060_094sub_062E_092_060_094sub_062J,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a] :
( ( extens2085910753204196637_set_a @ Resid )
=> ( ( con_set_a @ Resid @ T @ U )
=> ( ( trg_set_a @ Resid @ ( extens1973556086528668384_set_a @ Resid @ T @ U ) )
= ( trg_set_a @ Resid @ ( Resid @ T @ U ) ) ) ) ) ).
% extensional_rts_with_joins.trg_join\<^sub>E\<^sub>J
thf(fact_892_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_893_transformation__axioms_Ointro,axiom,
! [A4: set_a > set_a > set_a,Tau: set_a > set_a,B3: set_a > set_a > set_a,F2: set_a > set_a,G2: set_a > set_a] :
( ! [F4: set_a] :
( ~ ( arr_set_a @ A4 @ F4 )
=> ( ( Tau @ F4 )
= ( partial_null_set_a @ B3 ) ) )
=> ( ! [F4: set_a] :
( ( ide_set_a @ A4 @ F4 )
=> ( ( weakly2061155085811118449_set_a @ B3 @ ( Tau @ F4 ) )
= ( F2 @ ( weakly2061155085811118449_set_a @ A4 @ F4 ) ) ) )
=> ( ! [F4: set_a] :
( ( ide_set_a @ A4 @ F4 )
=> ( ( trg_set_a @ B3 @ ( Tau @ F4 ) )
= ( G2 @ ( trg_set_a @ A4 @ F4 ) ) ) )
=> ( ! [F4: set_a] :
( ( arr_set_a @ A4 @ F4 )
=> ( ( B3 @ ( Tau @ ( weakly2061155085811118449_set_a @ A4 @ F4 ) ) @ ( F2 @ F4 ) )
= ( Tau @ ( trg_set_a @ A4 @ F4 ) ) ) )
=> ( ! [F4: set_a] :
( ( arr_set_a @ A4 @ F4 )
=> ( ( B3 @ ( F2 @ F4 ) @ ( Tau @ ( weakly2061155085811118449_set_a @ A4 @ F4 ) ) )
= ( G2 @ F4 ) ) )
=> ( ! [F4: set_a] :
( ( arr_set_a @ A4 @ F4 )
=> ( join_of_set_a @ B3 @ ( Tau @ ( weakly2061155085811118449_set_a @ A4 @ F4 ) ) @ ( F2 @ F4 ) @ ( Tau @ F4 ) ) )
=> ( transf2960116383903194536_set_a @ A4 @ B3 @ F2 @ G2 @ Tau ) ) ) ) ) ) ) ).
% transformation_axioms.intro
thf(fact_894_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_895_transformation__axioms_Ointro,axiom,
! [A4: set_a > set_a > set_a,Tau: set_a > a,B3: a > a > a,F2: set_a > a,G2: set_a > a] :
( ! [F4: set_a] :
( ~ ( arr_set_a @ A4 @ F4 )
=> ( ( Tau @ F4 )
= ( partial_null_a @ B3 ) ) )
=> ( ! [F4: set_a] :
( ( ide_set_a @ A4 @ F4 )
=> ( ( weakly8512939796511659025_src_a @ B3 @ ( Tau @ F4 ) )
= ( F2 @ ( weakly2061155085811118449_set_a @ A4 @ F4 ) ) ) )
=> ( ! [F4: set_a] :
( ( ide_set_a @ A4 @ F4 )
=> ( ( trg_a @ B3 @ ( Tau @ F4 ) )
= ( G2 @ ( trg_set_a @ A4 @ F4 ) ) ) )
=> ( ! [F4: set_a] :
( ( arr_set_a @ A4 @ F4 )
=> ( ( B3 @ ( Tau @ ( weakly2061155085811118449_set_a @ A4 @ F4 ) ) @ ( F2 @ F4 ) )
= ( Tau @ ( trg_set_a @ A4 @ F4 ) ) ) )
=> ( ! [F4: set_a] :
( ( arr_set_a @ A4 @ F4 )
=> ( ( B3 @ ( F2 @ F4 ) @ ( Tau @ ( weakly2061155085811118449_set_a @ A4 @ F4 ) ) )
= ( G2 @ F4 ) ) )
=> ( ! [F4: set_a] :
( ( arr_set_a @ A4 @ F4 )
=> ( join_of_a @ B3 @ ( Tau @ ( weakly2061155085811118449_set_a @ A4 @ F4 ) ) @ ( F2 @ F4 ) @ ( Tau @ F4 ) ) )
=> ( transf1935308705569152072et_a_a @ A4 @ B3 @ F2 @ G2 @ Tau ) ) ) ) ) ) ) ).
% transformation_axioms.intro
thf(fact_896_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_897_R_Osources__def,axiom,
! [T: a] :
( ( sources_a @ resid @ T )
= ( collect_a
@ ^ [A7: a] :
( ( ide_a @ resid @ A7 )
& ( con_a @ resid @ T @ A7 ) ) ) ) ).
% R.sources_def
thf(fact_898_R_Otargets__char,axiom,
! [T: a] :
( ( targets_a @ resid @ T )
= ( collect_a
@ ^ [B2: a] :
( ( arr_a @ resid @ T )
& ( ide_a @ resid @ ( resid @ ( resid @ T @ T ) @ B2 ) )
& ( ide_a @ resid @ ( resid @ B2 @ ( resid @ T @ T ) ) ) ) ) ) ).
% R.targets_char
thf(fact_899_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_900_rts_Osources__def,axiom,
! [Resid: set_a > set_a > set_a,T: set_a] :
( ( rts_set_a @ Resid )
=> ( ( sources_set_a @ Resid @ T )
= ( collect_set_a
@ ^ [A7: set_a] :
( ( ide_set_a @ Resid @ A7 )
& ( con_set_a @ Resid @ T @ A7 ) ) ) ) ) ).
% rts.sources_def
thf(fact_901_rts_Osources__def,axiom,
! [Resid: a > a > a,T: a] :
( ( rts_a @ Resid )
=> ( ( sources_a @ Resid @ T )
= ( collect_a
@ ^ [A7: a] :
( ( ide_a @ Resid @ A7 )
& ( con_a @ Resid @ T @ A7 ) ) ) ) ) ).
% rts.sources_def
thf(fact_902_rts_Otargets__char,axiom,
! [Resid: set_a > set_a > set_a,T: set_a] :
( ( rts_set_a @ Resid )
=> ( ( targets_set_a @ Resid @ T )
= ( collect_set_a
@ ^ [B2: set_a] :
( ( arr_set_a @ Resid @ T )
& ( ide_set_a @ Resid @ ( Resid @ ( Resid @ T @ T ) @ B2 ) )
& ( ide_set_a @ Resid @ ( Resid @ B2 @ ( Resid @ T @ T ) ) ) ) ) ) ) ).
% rts.targets_char
thf(fact_903_rts_Otargets__char,axiom,
! [Resid: a > a > a,T: a] :
( ( rts_a @ Resid )
=> ( ( targets_a @ Resid @ T )
= ( collect_a
@ ^ [B2: a] :
( ( arr_a @ Resid @ T )
& ( ide_a @ Resid @ ( Resid @ ( Resid @ T @ T ) @ B2 ) )
& ( ide_a @ Resid @ ( Resid @ B2 @ ( Resid @ T @ T ) ) ) ) ) ) ) ).
% rts.targets_char
thf(fact_904_prop__restrict,axiom,
! [X4: a,Z4: set_a,X6: set_a,P: a > $o] :
( ( member_a @ X4 @ Z4 )
=> ( ( ord_less_eq_set_a @ Z4
@ ( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ X6 )
& ( P @ X2 ) ) ) )
=> ( P @ X4 ) ) ) ).
% prop_restrict
thf(fact_905_Collect__restrict,axiom,
! [X6: set_a,P: a > $o] :
( ord_less_eq_set_a
@ ( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ X6 )
& ( P @ X2 ) ) )
@ X6 ) ).
% Collect_restrict
thf(fact_906_empty__def,axiom,
( bot_bot_set_a
= ( collect_a
@ ^ [X2: a] : $false ) ) ).
% empty_def
thf(fact_907_Collect__conj__eq,axiom,
! [P: a > $o,Q: a > $o] :
( ( collect_a
@ ^ [X2: a] :
( ( P @ X2 )
& ( Q @ X2 ) ) )
= ( inf_inf_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_908_inf__set__def,axiom,
( inf_inf_set_a
= ( ^ [A8: set_a,B5: set_a] :
( collect_a
@ ( inf_inf_a_o
@ ^ [X2: a] : ( member_a @ X2 @ A8 )
@ ^ [X2: a] : ( member_a @ X2 @ B5 ) ) ) ) ) ).
% inf_set_def
thf(fact_909_Int__Collect,axiom,
! [X4: a,A4: set_a,P: a > $o] :
( ( member_a @ X4 @ ( inf_inf_set_a @ A4 @ ( collect_a @ P ) ) )
= ( ( member_a @ X4 @ A4 )
& ( P @ X4 ) ) ) ).
% Int_Collect
thf(fact_910_Int__def,axiom,
( inf_inf_set_a
= ( ^ [A8: set_a,B5: set_a] :
( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ A8 )
& ( member_a @ X2 @ B5 ) ) ) ) ) ).
% Int_def
thf(fact_911_pred__subset__eq,axiom,
! [R: set_a,S: set_a] :
( ( ord_less_eq_a_o
@ ^ [X2: a] : ( member_a @ X2 @ R )
@ ^ [X2: a] : ( member_a @ X2 @ S ) )
= ( ord_less_eq_set_a @ R @ S ) ) ).
% pred_subset_eq
thf(fact_912_less__eq__set__def,axiom,
( ord_less_eq_set_a
= ( ^ [A8: set_a,B5: set_a] :
( ord_less_eq_a_o
@ ^ [X2: a] : ( member_a @ X2 @ A8 )
@ ^ [X2: a] : ( member_a @ X2 @ B5 ) ) ) ) ).
% less_eq_set_def
thf(fact_913_Collect__subset,axiom,
! [A4: set_a,P: a > $o] :
( ord_less_eq_set_a
@ ( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ A4 )
& ( P @ X2 ) ) )
@ A4 ) ).
% Collect_subset
thf(fact_914_inf__Int__eq,axiom,
! [R: set_a,S: set_a] :
( ( inf_inf_a_o
@ ^ [X2: a] : ( member_a @ X2 @ R )
@ ^ [X2: a] : ( member_a @ X2 @ S ) )
= ( ^ [X2: a] : ( member_a @ X2 @ ( inf_inf_set_a @ R @ S ) ) ) ) ).
% inf_Int_eq
thf(fact_915_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_916_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_917_extensional__rts__with__joins_Osrc__join_092_060_094sub_062E_092_060_094sub_062J,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a] :
( ( extens2085910753204196637_set_a @ Resid )
=> ( ( con_set_a @ Resid @ T @ U )
=> ( ( weakly2061155085811118449_set_a @ Resid @ ( extens1973556086528668384_set_a @ Resid @ T @ U ) )
= ( weakly2061155085811118449_set_a @ Resid @ T ) ) ) ) ).
% extensional_rts_with_joins.src_join\<^sub>E\<^sub>J
thf(fact_918_rts_Otargets__def,axiom,
! [Resid: set_a > set_a > set_a,T: set_a] :
( ( rts_set_a @ Resid )
=> ( ( targets_set_a @ Resid @ T )
= ( collect_set_a
@ ^ [B2: set_a] :
( ( ide_set_a @ Resid @ B2 )
& ( con_set_a @ Resid @ ( trg_set_a @ Resid @ T ) @ B2 ) ) ) ) ) ).
% rts.targets_def
thf(fact_919_rts_Otargets__def,axiom,
! [Resid: a > a > a,T: a] :
( ( rts_a @ Resid )
=> ( ( targets_a @ Resid @ T )
= ( collect_a
@ ^ [B2: a] :
( ( ide_a @ Resid @ B2 )
& ( con_a @ Resid @ ( trg_a @ Resid @ T ) @ B2 ) ) ) ) ) ).
% rts.targets_def
thf(fact_920_extensional__rts__with__joins_Ojoin__is__lub,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,V: set_a,U: set_a] :
( ( extens2085910753204196637_set_a @ Resid )
=> ( ( ide_set_a @ Resid @ ( Resid @ T @ V ) )
=> ( ( ide_set_a @ Resid @ ( Resid @ U @ V ) )
=> ( ide_set_a @ Resid @ ( Resid @ ( extens1973556086528668384_set_a @ Resid @ T @ U ) @ V ) ) ) ) ) ).
% extensional_rts_with_joins.join_is_lub
thf(fact_921_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_922_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_923_extensional__rts__with__joins_Oresid__join_092_060_094sub_062E_092_060_094sub_062J_I2_J,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a,V: set_a] :
( ( extens2085910753204196637_set_a @ Resid )
=> ( ( con_set_a @ Resid @ T @ U )
=> ( ( con_set_a @ Resid @ V @ ( extens1973556086528668384_set_a @ Resid @ T @ U ) )
=> ( ( Resid @ ( extens1973556086528668384_set_a @ Resid @ T @ U ) @ V )
= ( extens1973556086528668384_set_a @ Resid @ ( Resid @ T @ V ) @ ( Resid @ U @ V ) ) ) ) ) ) ).
% extensional_rts_with_joins.resid_join\<^sub>E\<^sub>J(2)
thf(fact_924_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_925_extensional__rts__with__joins_Oresid__join_092_060_094sub_062E_092_060_094sub_062J_I1_J,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a,V: set_a] :
( ( extens2085910753204196637_set_a @ Resid )
=> ( ( con_set_a @ Resid @ T @ U )
=> ( ( con_set_a @ Resid @ V @ ( extens1973556086528668384_set_a @ Resid @ T @ U ) )
=> ( ( Resid @ V @ ( extens1973556086528668384_set_a @ Resid @ T @ U ) )
= ( Resid @ ( Resid @ V @ T ) @ ( Resid @ U @ T ) ) ) ) ) ) ).
% extensional_rts_with_joins.resid_join\<^sub>E\<^sub>J(1)
thf(fact_926_transformation__axioms__def,axiom,
( transf2960116383903194536_set_a
= ( ^ [A8: set_a > set_a > set_a,B5: set_a > set_a > set_a,F3: set_a > set_a,G3: set_a > set_a,Tau2: set_a > set_a] :
( ! [F5: set_a] :
( ~ ( arr_set_a @ A8 @ F5 )
=> ( ( Tau2 @ F5 )
= ( partial_null_set_a @ B5 ) ) )
& ! [F5: set_a] :
( ( ide_set_a @ A8 @ F5 )
=> ( ( weakly2061155085811118449_set_a @ B5 @ ( Tau2 @ F5 ) )
= ( F3 @ ( weakly2061155085811118449_set_a @ A8 @ F5 ) ) ) )
& ! [F5: set_a] :
( ( ide_set_a @ A8 @ F5 )
=> ( ( trg_set_a @ B5 @ ( Tau2 @ F5 ) )
= ( G3 @ ( trg_set_a @ A8 @ F5 ) ) ) )
& ! [F5: set_a] :
( ( arr_set_a @ A8 @ F5 )
=> ( ( B5 @ ( Tau2 @ ( weakly2061155085811118449_set_a @ A8 @ F5 ) ) @ ( F3 @ F5 ) )
= ( Tau2 @ ( trg_set_a @ A8 @ F5 ) ) ) )
& ! [F5: set_a] :
( ( arr_set_a @ A8 @ F5 )
=> ( ( B5 @ ( F3 @ F5 ) @ ( Tau2 @ ( weakly2061155085811118449_set_a @ A8 @ F5 ) ) )
= ( G3 @ F5 ) ) )
& ! [F5: set_a] :
( ( arr_set_a @ A8 @ F5 )
=> ( join_of_set_a @ B5 @ ( Tau2 @ ( weakly2061155085811118449_set_a @ A8 @ F5 ) ) @ ( F3 @ F5 ) @ ( Tau2 @ F5 ) ) ) ) ) ) ).
% transformation_axioms_def
thf(fact_927_transformation__axioms__def,axiom,
( transf1718796647109808904_set_a
= ( ^ [A8: a > a > a,B5: set_a > set_a > set_a,F3: a > set_a,G3: a > set_a,Tau2: a > set_a] :
( ! [F5: a] :
( ~ ( arr_a @ A8 @ F5 )
=> ( ( Tau2 @ F5 )
= ( partial_null_set_a @ B5 ) ) )
& ! [F5: a] :
( ( ide_a @ A8 @ F5 )
=> ( ( weakly2061155085811118449_set_a @ B5 @ ( Tau2 @ F5 ) )
= ( F3 @ ( weakly8512939796511659025_src_a @ A8 @ F5 ) ) ) )
& ! [F5: a] :
( ( ide_a @ A8 @ F5 )
=> ( ( trg_set_a @ B5 @ ( Tau2 @ F5 ) )
= ( G3 @ ( trg_a @ A8 @ F5 ) ) ) )
& ! [F5: a] :
( ( arr_a @ A8 @ F5 )
=> ( ( B5 @ ( Tau2 @ ( weakly8512939796511659025_src_a @ A8 @ F5 ) ) @ ( F3 @ F5 ) )
= ( Tau2 @ ( trg_a @ A8 @ F5 ) ) ) )
& ! [F5: a] :
( ( arr_a @ A8 @ F5 )
=> ( ( B5 @ ( F3 @ F5 ) @ ( Tau2 @ ( weakly8512939796511659025_src_a @ A8 @ F5 ) ) )
= ( G3 @ F5 ) ) )
& ! [F5: a] :
( ( arr_a @ A8 @ F5 )
=> ( join_of_set_a @ B5 @ ( Tau2 @ ( weakly8512939796511659025_src_a @ A8 @ F5 ) ) @ ( F3 @ F5 ) @ ( Tau2 @ F5 ) ) ) ) ) ) ).
% transformation_axioms_def
thf(fact_928_transformation__axioms__def,axiom,
( transf1935308705569152072et_a_a
= ( ^ [A8: set_a > set_a > set_a,B5: a > a > a,F3: set_a > a,G3: set_a > a,Tau2: set_a > a] :
( ! [F5: set_a] :
( ~ ( arr_set_a @ A8 @ F5 )
=> ( ( Tau2 @ F5 )
= ( partial_null_a @ B5 ) ) )
& ! [F5: set_a] :
( ( ide_set_a @ A8 @ F5 )
=> ( ( weakly8512939796511659025_src_a @ B5 @ ( Tau2 @ F5 ) )
= ( F3 @ ( weakly2061155085811118449_set_a @ A8 @ F5 ) ) ) )
& ! [F5: set_a] :
( ( ide_set_a @ A8 @ F5 )
=> ( ( trg_a @ B5 @ ( Tau2 @ F5 ) )
= ( G3 @ ( trg_set_a @ A8 @ F5 ) ) ) )
& ! [F5: set_a] :
( ( arr_set_a @ A8 @ F5 )
=> ( ( B5 @ ( Tau2 @ ( weakly2061155085811118449_set_a @ A8 @ F5 ) ) @ ( F3 @ F5 ) )
= ( Tau2 @ ( trg_set_a @ A8 @ F5 ) ) ) )
& ! [F5: set_a] :
( ( arr_set_a @ A8 @ F5 )
=> ( ( B5 @ ( F3 @ F5 ) @ ( Tau2 @ ( weakly2061155085811118449_set_a @ A8 @ F5 ) ) )
= ( G3 @ F5 ) ) )
& ! [F5: set_a] :
( ( arr_set_a @ A8 @ F5 )
=> ( join_of_a @ B5 @ ( Tau2 @ ( weakly2061155085811118449_set_a @ A8 @ F5 ) ) @ ( F3 @ F5 ) @ ( Tau2 @ F5 ) ) ) ) ) ) ).
% transformation_axioms_def
thf(fact_929_transformation__axioms__def,axiom,
( transf4446446367311712680ms_a_a
= ( ^ [A8: a > a > a,B5: a > a > a,F3: a > a,G3: a > a,Tau2: a > a] :
( ! [F5: a] :
( ~ ( arr_a @ A8 @ F5 )
=> ( ( Tau2 @ F5 )
= ( partial_null_a @ B5 ) ) )
& ! [F5: a] :
( ( ide_a @ A8 @ F5 )
=> ( ( weakly8512939796511659025_src_a @ B5 @ ( Tau2 @ F5 ) )
= ( F3 @ ( weakly8512939796511659025_src_a @ A8 @ F5 ) ) ) )
& ! [F5: a] :
( ( ide_a @ A8 @ F5 )
=> ( ( trg_a @ B5 @ ( Tau2 @ F5 ) )
= ( G3 @ ( trg_a @ A8 @ F5 ) ) ) )
& ! [F5: a] :
( ( arr_a @ A8 @ F5 )
=> ( ( B5 @ ( Tau2 @ ( weakly8512939796511659025_src_a @ A8 @ F5 ) ) @ ( F3 @ F5 ) )
= ( Tau2 @ ( trg_a @ A8 @ F5 ) ) ) )
& ! [F5: a] :
( ( arr_a @ A8 @ F5 )
=> ( ( B5 @ ( F3 @ F5 ) @ ( Tau2 @ ( weakly8512939796511659025_src_a @ A8 @ F5 ) ) )
= ( G3 @ F5 ) ) )
& ! [F5: a] :
( ( arr_a @ A8 @ F5 )
=> ( join_of_a @ B5 @ ( Tau2 @ ( weakly8512939796511659025_src_a @ A8 @ F5 ) ) @ ( F3 @ F5 ) @ ( Tau2 @ F5 ) ) ) ) ) ) ).
% transformation_axioms_def
thf(fact_930_N_OCong__class__rep__def,axiom,
( normal322541020860755160od_a_a
= ( ^ [T10: set_Product_prod_a_a] :
( fChoic4124218645493772411od_a_a
@ ^ [T7: product_prod_a_a] : ( member1426531477525435216od_a_a @ T7 @ T10 ) ) ) ) ).
% N.Cong_class_rep_def
thf(fact_931_N_OCong__class__rep__def,axiom,
( normal3259722184653208495_rep_a
= ( ^ [T10: set_a] :
( fChoice_a
@ ^ [T7: a] : ( member_a @ T7 @ T10 ) ) ) ) ).
% N.Cong_class_rep_def
thf(fact_932_transformation_Onaturality3,axiom,
! [A4: a > a > a,B3: a > a > a,F2: a > a,G2: a > a,Tau: a > a,F: a] :
( ( transformation_a_a @ A4 @ B3 @ F2 @ G2 @ Tau )
=> ( ( arr_a @ A4 @ F )
=> ( join_of_a @ B3 @ ( Tau @ ( weakly8512939796511659025_src_a @ A4 @ F ) ) @ ( F2 @ F ) @ ( Tau @ F ) ) ) ) ).
% transformation.naturality3
thf(fact_933_transformation_Onaturality3,axiom,
! [A4: set_a > set_a > set_a,B3: a > a > a,F2: set_a > a,G2: set_a > a,Tau: set_a > a,F: set_a] :
( ( transf4346523582779806757et_a_a @ A4 @ B3 @ F2 @ G2 @ Tau )
=> ( ( arr_set_a @ A4 @ F )
=> ( join_of_a @ B3 @ ( Tau @ ( weakly2061155085811118449_set_a @ A4 @ F ) ) @ ( F2 @ F ) @ ( Tau @ F ) ) ) ) ).
% transformation.naturality3
thf(fact_934_normal__sub__rts_OCong__class__rep__def,axiom,
! [Resid: a > a > a,NN: set_a,T6: set_Product_prod_a_a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( normal322541020860755160od_a_a @ T6 )
= ( fChoic4124218645493772411od_a_a
@ ^ [T7: product_prod_a_a] : ( member1426531477525435216od_a_a @ T7 @ T6 ) ) ) ) ).
% normal_sub_rts.Cong_class_rep_def
thf(fact_935_normal__sub__rts_OCong__class__rep__def,axiom,
! [Resid: a > a > a,NN: set_a,T6: set_a] :
( ( normal_sub_rts_a @ Resid @ NN )
=> ( ( normal3259722184653208495_rep_a @ T6 )
= ( fChoice_a
@ ^ [T7: a] : ( member_a @ T7 @ T6 ) ) ) ) ).
% normal_sub_rts.Cong_class_rep_def
thf(fact_936_transformation_Opreserves__trg,axiom,
! [A4: set_a > set_a > set_a,B3: set_a > set_a > set_a,F2: set_a > set_a,G2: set_a > set_a,Tau: set_a > set_a,F: set_a] :
( ( transf6002003789407478149_set_a @ A4 @ B3 @ F2 @ G2 @ Tau )
=> ( ( ide_set_a @ A4 @ F )
=> ( ( trg_set_a @ B3 @ ( Tau @ F ) )
= ( G2 @ ( trg_set_a @ A4 @ F ) ) ) ) ) ).
% transformation.preserves_trg
thf(fact_937_transformation_Opreserves__trg,axiom,
! [A4: a > a > a,B3: set_a > set_a > set_a,F2: a > set_a,G2: a > set_a,Tau: a > set_a,F: a] :
( ( transf4130011524320463589_set_a @ A4 @ B3 @ F2 @ G2 @ Tau )
=> ( ( ide_a @ A4 @ F )
=> ( ( trg_set_a @ B3 @ ( Tau @ F ) )
= ( G2 @ ( trg_a @ A4 @ F ) ) ) ) ) ).
% transformation.preserves_trg
thf(fact_938_transformation_Opreserves__trg,axiom,
! [A4: set_a > set_a > set_a,B3: a > a > a,F2: set_a > a,G2: set_a > a,Tau: set_a > a,F: set_a] :
( ( transf4346523582779806757et_a_a @ A4 @ B3 @ F2 @ G2 @ Tau )
=> ( ( ide_set_a @ A4 @ F )
=> ( ( trg_a @ B3 @ ( Tau @ F ) )
= ( G2 @ ( trg_set_a @ A4 @ F ) ) ) ) ) ).
% transformation.preserves_trg
thf(fact_939_transformation_Opreserves__trg,axiom,
! [A4: a > a > a,B3: a > a > a,F2: a > a,G2: a > a,Tau: a > a,F: a] :
( ( transformation_a_a @ A4 @ B3 @ F2 @ G2 @ Tau )
=> ( ( ide_a @ A4 @ F )
=> ( ( trg_a @ B3 @ ( Tau @ F ) )
= ( G2 @ ( trg_a @ A4 @ F ) ) ) ) ) ).
% transformation.preserves_trg
thf(fact_940_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,F: a] :
( ( transf4130011524320463589_set_a @ A4 @ B3 @ F2 @ G2 @ Tau )
=> ( ~ ( arr_a @ A4 @ F )
=> ( ( Tau @ F )
= ( partial_null_set_a @ B3 ) ) ) ) ).
% transformation.extensional
thf(fact_941_transformation_Oextensional,axiom,
! [A4: set_a > set_a > set_a,B3: set_a > set_a > set_a,F2: set_a > set_a,G2: set_a > set_a,Tau: set_a > set_a,F: set_a] :
( ( transf6002003789407478149_set_a @ A4 @ B3 @ F2 @ G2 @ Tau )
=> ( ~ ( arr_set_a @ A4 @ F )
=> ( ( Tau @ F )
= ( partial_null_set_a @ B3 ) ) ) ) ).
% transformation.extensional
thf(fact_942_transformation_Oextensional,axiom,
! [A4: a > a > a,B3: a > a > a,F2: a > a,G2: a > a,Tau: a > a,F: a] :
( ( transformation_a_a @ A4 @ B3 @ F2 @ G2 @ Tau )
=> ( ~ ( arr_a @ A4 @ F )
=> ( ( Tau @ F )
= ( partial_null_a @ B3 ) ) ) ) ).
% transformation.extensional
thf(fact_943_transformation_Oextensional,axiom,
! [A4: set_a > set_a > set_a,B3: a > a > a,F2: set_a > a,G2: set_a > a,Tau: set_a > a,F: set_a] :
( ( transf4346523582779806757et_a_a @ A4 @ B3 @ F2 @ G2 @ Tau )
=> ( ~ ( arr_set_a @ A4 @ F )
=> ( ( Tau @ F )
= ( partial_null_a @ B3 ) ) ) ) ).
% transformation.extensional
thf(fact_944_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_945_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_946_conj__subset__def,axiom,
! [A4: set_a,P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ A4
@ ( collect_a
@ ^ [X2: a] :
( ( P @ X2 )
& ( Q @ X2 ) ) ) )
= ( ( ord_less_eq_set_a @ A4 @ ( collect_a @ P ) )
& ( ord_less_eq_set_a @ A4 @ ( collect_a @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_947_simulation__between__extensional__rts_Opreserves__trg,axiom,
! [A4: set_a > set_a > set_a,B3: set_a > set_a > set_a,F2: set_a > set_a,T: set_a] :
( ( simula5219500460978879148_set_a @ A4 @ B3 @ F2 )
=> ( ( trg_set_a @ B3 @ ( F2 @ T ) )
= ( F2 @ ( trg_set_a @ A4 @ T ) ) ) ) ).
% simulation_between_extensional_rts.preserves_trg
thf(fact_948_simulation__between__extensional__rts_Opreserves__trg,axiom,
! [A4: a > a > a,B3: set_a > set_a > set_a,F2: a > set_a,T: a] :
( ( simula4505799099512520716_set_a @ A4 @ B3 @ F2 )
=> ( ( trg_set_a @ B3 @ ( F2 @ T ) )
= ( F2 @ ( trg_a @ A4 @ T ) ) ) ) ).
% simulation_between_extensional_rts.preserves_trg
thf(fact_949_simulation__between__extensional__rts_Opreserves__trg,axiom,
! [A4: set_a > set_a > set_a,B3: a > a > a,F2: set_a > a,T: set_a] :
( ( simula4722311157971863884et_a_a @ A4 @ B3 @ F2 )
=> ( ( trg_a @ B3 @ ( F2 @ T ) )
= ( F2 @ ( trg_set_a @ A4 @ T ) ) ) ) ).
% simulation_between_extensional_rts.preserves_trg
thf(fact_950_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_951_subset__Collect__iff,axiom,
! [B3: set_a,A4: set_a,P: a > $o] :
( ( ord_less_eq_set_a @ B3 @ A4 )
=> ( ( ord_less_eq_set_a @ B3
@ ( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ A4 )
& ( P @ X2 ) ) ) )
= ( ! [X2: a] :
( ( member_a @ X2 @ B3 )
=> ( P @ X2 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_952_subset__CollectI,axiom,
! [B3: set_a,A4: set_a,Q: a > $o,P: a > $o] :
( ( ord_less_eq_set_a @ B3 @ A4 )
=> ( ! [X: a] :
( ( member_a @ X @ B3 )
=> ( ( Q @ X )
=> ( P @ X ) ) )
=> ( ord_less_eq_set_a
@ ( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ B3 )
& ( Q @ X2 ) ) )
@ ( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ A4 )
& ( P @ X2 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_953_weakly__extensional__rts_OseqE_092_060_094sub_062W_092_060_094sub_062E,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a] :
( ( weakly5936471160286156245_set_a @ Resid )
=> ( ( seq_set_a @ Resid @ T @ U )
=> ~ ( ( arr_set_a @ Resid @ U )
=> ( ( arr_set_a @ Resid @ T )
=> ( ( trg_set_a @ Resid @ T )
!= ( weakly2061155085811118449_set_a @ Resid @ U ) ) ) ) ) ) ).
% weakly_extensional_rts.seqE\<^sub>W\<^sub>E
thf(fact_954_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_955_weakly__extensional__rts_Ocon__ide__are__eq,axiom,
! [Resid: set_a > set_a > set_a,A: set_a,A6: set_a] :
( ( weakly5936471160286156245_set_a @ Resid )
=> ( ( ide_set_a @ Resid @ A )
=> ( ( ide_set_a @ Resid @ A6 )
=> ( ( con_set_a @ Resid @ A @ A6 )
=> ( A = A6 ) ) ) ) ) ).
% weakly_extensional_rts.con_ide_are_eq
thf(fact_956_weakly__extensional__rts_Ocon__ide__are__eq,axiom,
! [Resid: a > a > a,A: a,A6: a] :
( ( weakly1626779504270821493_rts_a @ Resid )
=> ( ( ide_a @ Resid @ A )
=> ( ( ide_a @ Resid @ A6 )
=> ( ( con_a @ Resid @ A @ A6 )
=> ( A = A6 ) ) ) ) ) ).
% weakly_extensional_rts.con_ide_are_eq
thf(fact_957_weakly__extensional__rts_Otrg__trg,axiom,
! [Resid: set_a > set_a > set_a,T: set_a] :
( ( weakly5936471160286156245_set_a @ Resid )
=> ( ( trg_set_a @ Resid @ ( trg_set_a @ Resid @ T ) )
= ( trg_set_a @ Resid @ T ) ) ) ).
% weakly_extensional_rts.trg_trg
thf(fact_958_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_959_weakly__extensional__rts_Oapex__sym,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a] :
( ( weakly5936471160286156245_set_a @ Resid )
=> ( ( trg_set_a @ Resid @ ( Resid @ T @ U ) )
= ( trg_set_a @ Resid @ ( Resid @ U @ T ) ) ) ) ).
% weakly_extensional_rts.apex_sym
thf(fact_960_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_961_weakly__extensional__rts_Oweak__extensionality,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a] :
( ( weakly5936471160286156245_set_a @ Resid )
=> ( ( ( ide_set_a @ Resid @ ( Resid @ T @ U ) )
& ( ide_set_a @ Resid @ ( Resid @ U @ T ) ) )
=> ( ( ide_set_a @ Resid @ T )
=> ( ( ide_set_a @ Resid @ U )
=> ( T = U ) ) ) ) ) ).
% weakly_extensional_rts.weak_extensionality
thf(fact_962_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_963_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_964_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_965_weakly__extensional__rts_Oarr__has__un__source,axiom,
! [Resid: set_a > set_a > set_a,T: set_a] :
( ( weakly5936471160286156245_set_a @ Resid )
=> ( ( arr_set_a @ Resid @ T )
=> ? [X: set_a] :
( ( member_set_a @ X @ ( sources_set_a @ Resid @ T ) )
& ! [Y: set_a] :
( ( member_set_a @ Y @ ( sources_set_a @ Resid @ T ) )
=> ( Y = X ) ) ) ) ) ).
% weakly_extensional_rts.arr_has_un_source
thf(fact_966_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_967_weakly__extensional__rts_Oarr__has__un__target,axiom,
! [Resid: set_a > set_a > set_a,T: set_a] :
( ( weakly5936471160286156245_set_a @ Resid )
=> ( ( arr_set_a @ Resid @ T )
=> ? [X: set_a] :
( ( member_set_a @ X @ ( targets_set_a @ Resid @ T ) )
& ! [Y: set_a] :
( ( member_set_a @ Y @ ( targets_set_a @ Resid @ T ) )
=> ( Y = X ) ) ) ) ) ).
% weakly_extensional_rts.arr_has_un_target
thf(fact_968_weakly__extensional__rts_Otrg__ide,axiom,
! [Resid: set_a > set_a > set_a,A: set_a] :
( ( weakly5936471160286156245_set_a @ Resid )
=> ( ( ide_set_a @ Resid @ A )
=> ( ( trg_set_a @ Resid @ A )
= A ) ) ) ).
% weakly_extensional_rts.trg_ide
thf(fact_969_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_970_weakly__extensional__rts_Otrg__resid__sym,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a] :
( ( weakly5936471160286156245_set_a @ Resid )
=> ( ( con_set_a @ Resid @ T @ U )
=> ( ( trg_set_a @ Resid @ ( Resid @ T @ U ) )
= ( trg_set_a @ Resid @ ( Resid @ U @ T ) ) ) ) ) ).
% weakly_extensional_rts.trg_resid_sym
thf(fact_971_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_972_weakly__extensional__rts_Osrc__ide,axiom,
! [Resid: set_a > set_a > set_a,A: set_a] :
( ( weakly5936471160286156245_set_a @ Resid )
=> ( ( ide_set_a @ Resid @ A )
=> ( ( weakly2061155085811118449_set_a @ Resid @ A )
= A ) ) ) ).
% weakly_extensional_rts.src_ide
thf(fact_973_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_974_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_975_weakly__extensional__rts_Ocon__imp__eq__src,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a] :
( ( weakly5936471160286156245_set_a @ Resid )
=> ( ( con_set_a @ Resid @ T @ U )
=> ( ( weakly2061155085811118449_set_a @ Resid @ T )
= ( weakly2061155085811118449_set_a @ Resid @ U ) ) ) ) ).
% weakly_extensional_rts.con_imp_eq_src
thf(fact_976_weakly__extensional__rts_Oarr__trg__iff__arr,axiom,
! [Resid: set_a > set_a > set_a,T: set_a] :
( ( weakly5936471160286156245_set_a @ Resid )
=> ( ( arr_set_a @ Resid @ ( trg_set_a @ Resid @ T ) )
= ( arr_set_a @ Resid @ T ) ) ) ).
% weakly_extensional_rts.arr_trg_iff_arr
thf(fact_977_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_978_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_979_weakly__extensional__rts_Oarr__src__iff__arr,axiom,
! [Resid: set_a > set_a > set_a,T: set_a] :
( ( weakly5936471160286156245_set_a @ Resid )
=> ( ( arr_set_a @ Resid @ ( weakly2061155085811118449_set_a @ Resid @ T ) )
= ( arr_set_a @ Resid @ T ) ) ) ).
% weakly_extensional_rts.arr_src_iff_arr
thf(fact_980_weakly__extensional__rts_Otrg__composite__of,axiom,
! [Resid: set_a > set_a > set_a,U: set_a,T: set_a,V: set_a] :
( ( weakly5936471160286156245_set_a @ Resid )
=> ( ( composite_of_set_a @ Resid @ U @ T @ V )
=> ( ( trg_set_a @ Resid @ V )
= ( trg_set_a @ Resid @ T ) ) ) ) ).
% weakly_extensional_rts.trg_composite_of
thf(fact_981_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_982_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_983_weakly__extensional__rts_Ocoinitial__ide__are__eq,axiom,
! [Resid: set_a > set_a > set_a,A: set_a,A6: set_a] :
( ( weakly5936471160286156245_set_a @ Resid )
=> ( ( ide_set_a @ Resid @ A )
=> ( ( ide_set_a @ Resid @ A6 )
=> ( ( coinitial_set_a @ Resid @ A @ A6 )
=> ( A = A6 ) ) ) ) ) ).
% weakly_extensional_rts.coinitial_ide_are_eq
thf(fact_984_weakly__extensional__rts_Ocoinitial__ide__are__eq,axiom,
! [Resid: a > a > a,A: a,A6: a] :
( ( weakly1626779504270821493_rts_a @ Resid )
=> ( ( ide_a @ Resid @ A )
=> ( ( ide_a @ Resid @ A6 )
=> ( ( coinitial_a @ Resid @ A @ A6 )
=> ( A = A6 ) ) ) ) ) ).
% weakly_extensional_rts.coinitial_ide_are_eq
thf(fact_985_weakly__extensional__rts_Oresid__ide_I1_J,axiom,
! [Resid: set_a > set_a > set_a,A: set_a,T: set_a] :
( ( weakly5936471160286156245_set_a @ Resid )
=> ( ( ide_set_a @ Resid @ A )
=> ( ( coinitial_set_a @ Resid @ A @ T )
=> ( ( Resid @ T @ A )
= T ) ) ) ) ).
% weakly_extensional_rts.resid_ide(1)
thf(fact_986_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_987_weakly__extensional__rts_Otrg__src,axiom,
! [Resid: set_a > set_a > set_a,T: set_a] :
( ( weakly5936471160286156245_set_a @ Resid )
=> ( ( trg_set_a @ Resid @ ( weakly2061155085811118449_set_a @ Resid @ T ) )
= ( weakly2061155085811118449_set_a @ Resid @ T ) ) ) ).
% weakly_extensional_rts.trg_src
thf(fact_988_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_989_weakly__extensional__rts_Osrc__trg,axiom,
! [Resid: set_a > set_a > set_a,T: set_a] :
( ( weakly5936471160286156245_set_a @ Resid )
=> ( ( weakly2061155085811118449_set_a @ Resid @ ( trg_set_a @ Resid @ T ) )
= ( trg_set_a @ Resid @ T ) ) ) ).
% weakly_extensional_rts.src_trg
thf(fact_990_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_991_weakly__extensional__rts_Otrg__join__of_I2_J,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a,V: set_a] :
( ( weakly5936471160286156245_set_a @ Resid )
=> ( ( join_of_set_a @ Resid @ T @ U @ V )
=> ( ( trg_set_a @ Resid @ ( Resid @ U @ T ) )
= ( trg_set_a @ Resid @ V ) ) ) ) ).
% weakly_extensional_rts.trg_join_of(2)
thf(fact_992_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_993_weakly__extensional__rts_Otrg__join__of_I1_J,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a,V: set_a] :
( ( weakly5936471160286156245_set_a @ Resid )
=> ( ( join_of_set_a @ Resid @ T @ U @ V )
=> ( ( trg_set_a @ Resid @ ( Resid @ T @ U ) )
= ( trg_set_a @ Resid @ V ) ) ) ) ).
% weakly_extensional_rts.trg_join_of(1)
thf(fact_994_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_995_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_996_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_997_weakly__extensional__rts_Osrc__eqI,axiom,
! [Resid: set_a > set_a > set_a,A: set_a,T: set_a] :
( ( weakly5936471160286156245_set_a @ Resid )
=> ( ( ide_set_a @ Resid @ A )
=> ( ( con_set_a @ Resid @ A @ T )
=> ( ( weakly2061155085811118449_set_a @ Resid @ T )
= A ) ) ) ) ).
% weakly_extensional_rts.src_eqI
thf(fact_998_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_999_weakly__extensional__rts_Oide__iff__trg__self,axiom,
! [Resid: set_a > set_a > set_a,A: set_a] :
( ( weakly5936471160286156245_set_a @ Resid )
=> ( ( arr_set_a @ Resid @ A )
=> ( ( ide_set_a @ Resid @ A )
= ( ( trg_set_a @ Resid @ A )
= A ) ) ) ) ).
% weakly_extensional_rts.ide_iff_trg_self
thf(fact_1000_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_1001_weakly__extensional__rts_Oide__iff__src__self,axiom,
! [Resid: set_a > set_a > set_a,A: set_a] :
( ( weakly5936471160286156245_set_a @ Resid )
=> ( ( arr_set_a @ Resid @ A )
=> ( ( ide_set_a @ Resid @ A )
= ( ( weakly2061155085811118449_set_a @ Resid @ A )
= A ) ) ) ) ).
% weakly_extensional_rts.ide_iff_src_self
thf(fact_1002_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_1003_weakly__extensional__rts_Oide__src,axiom,
! [Resid: set_a > set_a > set_a,T: set_a] :
( ( weakly5936471160286156245_set_a @ Resid )
=> ( ( arr_set_a @ Resid @ T )
=> ( ide_set_a @ Resid @ ( weakly2061155085811118449_set_a @ Resid @ T ) ) ) ) ).
% weakly_extensional_rts.ide_src
thf(fact_1004_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_1005_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_1006_weakly__extensional__rts_Osrc__in__sources,axiom,
! [Resid: set_a > set_a > set_a,T: set_a] :
( ( weakly5936471160286156245_set_a @ Resid )
=> ( ( arr_set_a @ Resid @ T )
=> ( member_set_a @ ( weakly2061155085811118449_set_a @ Resid @ T ) @ ( sources_set_a @ Resid @ T ) ) ) ) ).
% weakly_extensional_rts.src_in_sources
thf(fact_1007_weakly__extensional__rts_Osrc__resid,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a] :
( ( weakly5936471160286156245_set_a @ Resid )
=> ( ( con_set_a @ Resid @ T @ U )
=> ( ( weakly2061155085811118449_set_a @ Resid @ ( Resid @ T @ U ) )
= ( trg_set_a @ Resid @ U ) ) ) ) ).
% weakly_extensional_rts.src_resid
thf(fact_1008_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_1009_weakly__extensional__rts_Oresid__ide_I2_J,axiom,
! [Resid: set_a > set_a > set_a,A: set_a,T: set_a] :
( ( weakly5936471160286156245_set_a @ Resid )
=> ( ( ide_set_a @ Resid @ A )
=> ( ( coinitial_set_a @ Resid @ A @ T )
=> ( ( Resid @ A @ T )
= ( trg_set_a @ Resid @ T ) ) ) ) ) ).
% weakly_extensional_rts.resid_ide(2)
thf(fact_1010_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_1011_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_1012_weakly__extensional__rts_Ocoinitial__iff_092_060_094sub_062W_092_060_094sub_062E,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a] :
( ( weakly5936471160286156245_set_a @ Resid )
=> ( ( coinitial_set_a @ Resid @ T @ U )
= ( ( arr_set_a @ Resid @ T )
& ( arr_set_a @ Resid @ U )
& ( ( weakly2061155085811118449_set_a @ Resid @ T )
= ( weakly2061155085811118449_set_a @ Resid @ U ) ) ) ) ) ).
% weakly_extensional_rts.coinitial_iff\<^sub>W\<^sub>E
thf(fact_1013_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_1014_weakly__extensional__rts_OcoinitialI_092_060_094sub_062W_092_060_094sub_062E,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a] :
( ( weakly5936471160286156245_set_a @ Resid )
=> ( ( arr_set_a @ Resid @ T )
=> ( ( ( weakly2061155085811118449_set_a @ Resid @ T )
= ( weakly2061155085811118449_set_a @ Resid @ U ) )
=> ( coinitial_set_a @ Resid @ T @ U ) ) ) ) ).
% weakly_extensional_rts.coinitialI\<^sub>W\<^sub>E
thf(fact_1015_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_1016_weakly__extensional__rts_OcoinitialE_092_060_094sub_062W_092_060_094sub_062E,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a] :
( ( weakly5936471160286156245_set_a @ Resid )
=> ( ( coinitial_set_a @ Resid @ T @ U )
=> ~ ( ( arr_set_a @ Resid @ T )
=> ( ( arr_set_a @ Resid @ U )
=> ( ( weakly2061155085811118449_set_a @ Resid @ T )
!= ( weakly2061155085811118449_set_a @ Resid @ U ) ) ) ) ) ) ).
% weakly_extensional_rts.coinitialE\<^sub>W\<^sub>E
thf(fact_1017_weakly__extensional__rts_OcoterminalE_092_060_094sub_062W_092_060_094sub_062E,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a] :
( ( weakly5936471160286156245_set_a @ Resid )
=> ( ( coterminal_set_a @ Resid @ T @ U )
=> ~ ( ( arr_set_a @ Resid @ T )
=> ( ( arr_set_a @ Resid @ U )
=> ( ( trg_set_a @ Resid @ T )
!= ( trg_set_a @ Resid @ U ) ) ) ) ) ) ).
% weakly_extensional_rts.coterminalE\<^sub>W\<^sub>E
thf(fact_1018_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_1019_weakly__extensional__rts_OcoterminalI_092_060_094sub_062W_092_060_094sub_062E,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a] :
( ( weakly5936471160286156245_set_a @ Resid )
=> ( ( arr_set_a @ Resid @ T )
=> ( ( ( trg_set_a @ Resid @ T )
= ( trg_set_a @ Resid @ U ) )
=> ( coterminal_set_a @ Resid @ T @ U ) ) ) ) ).
% weakly_extensional_rts.coterminalI\<^sub>W\<^sub>E
thf(fact_1020_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_1021_weakly__extensional__rts_Ocoterminal__iff_092_060_094sub_062W_092_060_094sub_062E,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a] :
( ( weakly5936471160286156245_set_a @ Resid )
=> ( ( coterminal_set_a @ Resid @ T @ U )
= ( ( arr_set_a @ Resid @ T )
& ( arr_set_a @ Resid @ U )
& ( ( trg_set_a @ Resid @ T )
= ( trg_set_a @ Resid @ U ) ) ) ) ) ).
% weakly_extensional_rts.coterminal_iff\<^sub>W\<^sub>E
thf(fact_1022_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_1023_weakly__extensional__rts_Osources__char,axiom,
! [Resid: a > a > a,T: a] :
( ( weakly1626779504270821493_rts_a @ Resid )
=> ( ( sources_a @ Resid @ T )
= ( collect_a
@ ^ [A7: a] :
( ( arr_a @ Resid @ T )
& ( ( weakly8512939796511659025_src_a @ Resid @ T )
= A7 ) ) ) ) ) ).
% weakly_extensional_rts.sources_char
thf(fact_1024_weakly__extensional__rts_Osources__char,axiom,
! [Resid: set_a > set_a > set_a,T: set_a] :
( ( weakly5936471160286156245_set_a @ Resid )
=> ( ( sources_set_a @ Resid @ T )
= ( collect_set_a
@ ^ [A7: set_a] :
( ( arr_set_a @ Resid @ T )
& ( ( weakly2061155085811118449_set_a @ Resid @ T )
= A7 ) ) ) ) ) ).
% weakly_extensional_rts.sources_char
thf(fact_1025_weakly__extensional__rts_Otargets__char_092_060_094sub_062W_092_060_094sub_062E,axiom,
! [Resid: set_a > set_a > set_a,T: set_a] :
( ( weakly5936471160286156245_set_a @ Resid )
=> ( ( targets_set_a @ Resid @ T )
= ( collect_set_a
@ ^ [B2: set_a] :
( ( arr_set_a @ Resid @ T )
& ( ( trg_set_a @ Resid @ T )
= B2 ) ) ) ) ) ).
% weakly_extensional_rts.targets_char\<^sub>W\<^sub>E
thf(fact_1026_weakly__extensional__rts_Otargets__char_092_060_094sub_062W_092_060_094sub_062E,axiom,
! [Resid: a > a > a,T: a] :
( ( weakly1626779504270821493_rts_a @ Resid )
=> ( ( targets_a @ Resid @ T )
= ( collect_a
@ ^ [B2: a] :
( ( arr_a @ Resid @ T )
& ( ( trg_a @ Resid @ T )
= B2 ) ) ) ) ) ).
% weakly_extensional_rts.targets_char\<^sub>W\<^sub>E
thf(fact_1027_weakly__extensional__rts_OseqI_092_060_094sub_062W_092_060_094sub_062E,axiom,
! [Resid: set_a > set_a > set_a,U: set_a,T: set_a] :
( ( weakly5936471160286156245_set_a @ Resid )
=> ( ( arr_set_a @ Resid @ U )
=> ( ( arr_set_a @ Resid @ T )
=> ( ( ( trg_set_a @ Resid @ T )
= ( weakly2061155085811118449_set_a @ Resid @ U ) )
=> ( seq_set_a @ Resid @ T @ U ) ) ) ) ) ).
% weakly_extensional_rts.seqI\<^sub>W\<^sub>E
thf(fact_1028_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_1029_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
@ ^ [A7: set_a] : ( member_set_a @ A7 @ ( 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_1030_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
@ ^ [A7: a] : ( member_a @ A7 @ ( sources_a @ Resid @ T ) ) ) ) )
& ( ~ ( arr_a @ Resid @ T )
=> ( ( weakly8512939796511659025_src_a @ Resid @ T )
= ( partial_null_a @ Resid ) ) ) ) ) ).
% weakly_extensional_rts.src_def
thf(fact_1031_weakly__extensional__rts__def,axiom,
( weakly1626779504270821493_rts_a
= ( ^ [Resid2: a > a > a] :
( ( rts_a @ Resid2 )
& ( weakly311909585050745746ioms_a @ Resid2 ) ) ) ) ).
% weakly_extensional_rts_def
thf(fact_1032_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_1033_weakly__extensional__rts__axioms_Ointro,axiom,
! [Resid: set_a > set_a > set_a] :
( ! [T3: set_a,U5: set_a] :
( ( ( ide_set_a @ Resid @ ( Resid @ T3 @ U5 ) )
& ( ide_set_a @ Resid @ ( Resid @ U5 @ T3 ) ) )
=> ( ( ide_set_a @ Resid @ T3 )
=> ( ( ide_set_a @ Resid @ U5 )
=> ( T3 = U5 ) ) ) )
=> ( weakly1639064658423616754_set_a @ Resid ) ) ).
% weakly_extensional_rts_axioms.intro
thf(fact_1034_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_1035_weakly__extensional__rts__axioms__def,axiom,
( weakly1639064658423616754_set_a
= ( ^ [Resid2: set_a > set_a > set_a] :
! [T7: set_a,U4: set_a] :
( ( ( ide_set_a @ Resid2 @ ( Resid2 @ T7 @ U4 ) )
& ( ide_set_a @ Resid2 @ ( Resid2 @ U4 @ T7 ) ) )
=> ( ( ide_set_a @ Resid2 @ T7 )
=> ( ( ide_set_a @ Resid2 @ U4 )
=> ( T7 = U4 ) ) ) ) ) ) ).
% weakly_extensional_rts_axioms_def
thf(fact_1036_weakly__extensional__rts__axioms__def,axiom,
( weakly311909585050745746ioms_a
= ( ^ [Resid2: a > a > a] :
! [T7: a,U4: a] :
( ( ( ide_a @ Resid2 @ ( Resid2 @ T7 @ U4 ) )
& ( ide_a @ Resid2 @ ( Resid2 @ U4 @ T7 ) ) )
=> ( ( ide_a @ Resid2 @ T7 )
=> ( ( ide_a @ Resid2 @ U4 )
=> ( T7 = U4 ) ) ) ) ) ) ).
% weakly_extensional_rts_axioms_def
thf(fact_1037_null__def,axiom,
( ( partial_null_set_a @ ( quotie8165075472272353145esid_a @ resid @ nn ) )
= ( the_set_a
@ ^ [N2: set_a] :
! [T7: set_a] :
( ( ( quotie8165075472272353145esid_a @ resid @ nn @ N2 @ T7 )
= N2 )
& ( ( quotie8165075472272353145esid_a @ resid @ nn @ T7 @ N2 )
= N2 ) ) ) ) ).
% null_def
thf(fact_1038_R_Onull__def,axiom,
( ( partial_null_a @ resid )
= ( the_a
@ ^ [N2: a] :
! [T7: a] :
( ( ( resid @ N2 @ T7 )
= N2 )
& ( ( resid @ T7 @ N2 )
= N2 ) ) ) ) ).
% R.null_def
thf(fact_1039_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] :
! [T7: set_a] :
( ( ( OP2 @ N2 @ T7 )
= N2 )
& ( ( OP2 @ T7 @ N2 )
= N2 ) ) ) ) ) ).
% partial_magma.null_def
thf(fact_1040_partial__magma_Onull__def,axiom,
! [OP2: a > a > a] :
( ( partial_magma_a @ OP2 )
=> ( ( partial_null_a @ OP2 )
= ( the_a
@ ^ [N2: a] :
! [T7: a] :
( ( ( OP2 @ N2 @ T7 )
= N2 )
& ( ( OP2 @ T7 @ N2 )
= N2 ) ) ) ) ) ).
% partial_magma.null_def
thf(fact_1041_Greatest__def,axiom,
( order_Greatest_set_a
= ( ^ [P2: set_a > $o] :
( the_set_a
@ ^ [X2: set_a] :
( ( P2 @ X2 )
& ! [Y2: set_a] :
( ( P2 @ Y2 )
=> ( ord_less_eq_set_a @ Y2 @ X2 ) ) ) ) ) ) ).
% Greatest_def
thf(fact_1042_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_1043_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_1044_extensional__rts_Ojoin__eqI,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,V: set_a,U: set_a] :
( ( extens2802975062453607898_set_a @ Resid )
=> ( ( ide_set_a @ Resid @ ( Resid @ T @ V ) )
=> ( ( ide_set_a @ Resid @ ( Resid @ U @ V ) )
=> ( ( ( Resid @ V @ U )
= ( Resid @ T @ U ) )
=> ( ( ( Resid @ V @ T )
= ( Resid @ U @ T ) )
=> ( ( extens1973556086528668384_set_a @ Resid @ T @ U )
= V ) ) ) ) ) ) ).
% extensional_rts.join_eqI
thf(fact_1045_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_1046_extensional__rts_Ocong__char,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a] :
( ( extens2802975062453607898_set_a @ Resid )
=> ( ( ( ide_set_a @ Resid @ ( Resid @ T @ U ) )
& ( ide_set_a @ Resid @ ( Resid @ U @ T ) ) )
= ( ( arr_set_a @ Resid @ T )
& ( T = U ) ) ) ) ).
% extensional_rts.cong_char
thf(fact_1047_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_1048_extensional__rts_Oaxioms_I1_J,axiom,
! [Resid: a > a > a] :
( ( extensional_rts_a @ Resid )
=> ( rts_a @ Resid ) ) ).
% extensional_rts.axioms(1)
thf(fact_1049_extensional__rts_Oextensional,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a] :
( ( extens2802975062453607898_set_a @ Resid )
=> ( ( ( ide_set_a @ Resid @ ( Resid @ T @ U ) )
& ( ide_set_a @ Resid @ ( Resid @ U @ T ) ) )
=> ( T = U ) ) ) ).
% extensional_rts.extensional
thf(fact_1050_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_1051_extensional__rts_Ocomposite__of__unique,axiom,
! [Resid: a > a > a,T: a,U: a,V: a,V5: a] :
( ( extensional_rts_a @ Resid )
=> ( ( composite_of_a @ Resid @ T @ U @ V )
=> ( ( composite_of_a @ Resid @ T @ U @ V5 )
=> ( V = V5 ) ) ) ) ).
% extensional_rts.composite_of_unique
thf(fact_1052_extensional__rts_Ojoin__of__unique,axiom,
! [Resid: a > a > a,T: a,U: a,V: a,V5: a] :
( ( extensional_rts_a @ Resid )
=> ( ( join_of_a @ Resid @ T @ U @ V )
=> ( ( join_of_a @ Resid @ T @ U @ V5 )
=> ( V = V5 ) ) ) ) ).
% extensional_rts.join_of_unique
thf(fact_1053_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_1054_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_1055_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_1056_extensional__rts_Ojoin__self,axiom,
! [Resid: set_a > set_a > set_a,T: set_a] :
( ( extens2802975062453607898_set_a @ Resid )
=> ( ( arr_set_a @ Resid @ T )
=> ( ( extens1973556086528668384_set_a @ Resid @ T @ T )
= T ) ) ) ).
% extensional_rts.join_self
thf(fact_1057_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_1058_extensional__rts_Ojoin__src,axiom,
! [Resid: set_a > set_a > set_a,T: set_a] :
( ( extens2802975062453607898_set_a @ Resid )
=> ( ( arr_set_a @ Resid @ T )
=> ( ( extens1973556086528668384_set_a @ Resid @ ( weakly2061155085811118449_set_a @ Resid @ T ) @ T )
= T ) ) ) ).
% extensional_rts.join_src
thf(fact_1059_extensional__rts_Oarr__prfx__join__self,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a] :
( ( extens2802975062453607898_set_a @ Resid )
=> ( ( joinable_set_a @ Resid @ T @ U )
=> ( ide_set_a @ Resid @ ( Resid @ T @ ( extens1973556086528668384_set_a @ Resid @ T @ U ) ) ) ) ) ).
% extensional_rts.arr_prfx_join_self
thf(fact_1060_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_1061_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_1062_extensional__rts_Oresid__join_092_060_094sub_062E_I1_J,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a,V: set_a] :
( ( extens2802975062453607898_set_a @ Resid )
=> ( ( joinable_set_a @ Resid @ T @ U )
=> ( ( con_set_a @ Resid @ V @ ( extens1973556086528668384_set_a @ Resid @ T @ U ) )
=> ( ( Resid @ V @ ( extens1973556086528668384_set_a @ Resid @ T @ U ) )
= ( Resid @ ( Resid @ V @ U ) @ ( Resid @ T @ U ) ) ) ) ) ) ).
% extensional_rts.resid_join\<^sub>E(1)
thf(fact_1063_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_1064_extensional__rts_Oresid__join_092_060_094sub_062E_I2_J,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a,V: set_a] :
( ( extens2802975062453607898_set_a @ Resid )
=> ( ( joinable_set_a @ Resid @ T @ U )
=> ( ( con_set_a @ Resid @ V @ ( extens1973556086528668384_set_a @ Resid @ T @ U ) )
=> ( ( Resid @ V @ ( extens1973556086528668384_set_a @ Resid @ T @ U ) )
= ( Resid @ ( Resid @ V @ T ) @ ( Resid @ U @ T ) ) ) ) ) ) ).
% extensional_rts.resid_join\<^sub>E(2)
thf(fact_1065_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_1066_extensional__rts_Oresid__join_092_060_094sub_062E_I3_J,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a,V: set_a] :
( ( extens2802975062453607898_set_a @ Resid )
=> ( ( joinable_set_a @ Resid @ T @ U )
=> ( ( con_set_a @ Resid @ V @ ( extens1973556086528668384_set_a @ Resid @ T @ U ) )
=> ( ( Resid @ ( extens1973556086528668384_set_a @ Resid @ T @ U ) @ V )
= ( extens1973556086528668384_set_a @ Resid @ ( Resid @ T @ V ) @ ( Resid @ U @ V ) ) ) ) ) ) ).
% extensional_rts.resid_join\<^sub>E(3)
thf(fact_1067_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_1068_extensional__rts_Ojoinable__iff__arr__join,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a] :
( ( extens2802975062453607898_set_a @ Resid )
=> ( ( joinable_set_a @ Resid @ T @ U )
= ( arr_set_a @ Resid @ ( extens1973556086528668384_set_a @ Resid @ T @ U ) ) ) ) ).
% extensional_rts.joinable_iff_arr_join
thf(fact_1069_extensional__rts_Otrg__join,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a] :
( ( extens2802975062453607898_set_a @ Resid )
=> ( ( joinable_set_a @ Resid @ T @ U )
=> ( ( trg_set_a @ Resid @ ( extens1973556086528668384_set_a @ Resid @ T @ U ) )
= ( trg_set_a @ Resid @ ( Resid @ T @ U ) ) ) ) ) ).
% extensional_rts.trg_join
thf(fact_1070_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_1071_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_1072_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_1073_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_1074_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_1075_GreatestI2__order,axiom,
! [P: set_a > $o,X4: set_a,Q: set_a > $o] :
( ( P @ X4 )
=> ( ! [Y3: set_a] :
( ( P @ Y3 )
=> ( ord_less_eq_set_a @ Y3 @ X4 ) )
=> ( ! [X: set_a] :
( ( P @ X )
=> ( ! [Y: set_a] :
( ( P @ Y )
=> ( ord_less_eq_set_a @ Y @ X ) )
=> ( Q @ X ) ) )
=> ( Q @ ( order_Greatest_set_a @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_1076_Greatest__equality,axiom,
! [P: set_a > $o,X4: set_a] :
( ( P @ X4 )
=> ( ! [Y3: set_a] :
( ( P @ Y3 )
=> ( ord_less_eq_set_a @ Y3 @ X4 ) )
=> ( ( order_Greatest_set_a @ P )
= X4 ) ) ) ).
% Greatest_equality
thf(fact_1077_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_1078_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_1079_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_1080_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_1081_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_1082_extensional__rts__def,axiom,
( extensional_rts_a
= ( ^ [Resid2: a > a > a] :
( ( rts_a @ Resid2 )
& ( extens8613361310974791063ioms_a @ Resid2 ) ) ) ) ).
% extensional_rts_def
thf(fact_1083_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_1084_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_1085_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_1086_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_1087_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_1088_extensional__rts_Ocomp__cancel__left,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a,V: set_a] :
( ( extens2802975062453607898_set_a @ Resid )
=> ( ( arr_set_a @ Resid @ ( extens7801945855595804251_set_a @ Resid @ T @ U ) )
=> ( ( ( extens7801945855595804251_set_a @ Resid @ T @ U )
= ( extens7801945855595804251_set_a @ Resid @ T @ V ) )
=> ( U = V ) ) ) ) ).
% extensional_rts.comp_cancel_left
thf(fact_1089_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_1090_extensional__rts_Ocomp__resid__prfx,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a] :
( ( extens2802975062453607898_set_a @ Resid )
=> ( ( arr_set_a @ Resid @ ( extens7801945855595804251_set_a @ Resid @ T @ U ) )
=> ( ( Resid @ ( extens7801945855595804251_set_a @ Resid @ T @ U ) @ T )
= U ) ) ) ).
% extensional_rts.comp_resid_prfx
thf(fact_1091_extensional__rts_Ocomp__ide__self,axiom,
! [Resid: set_a > set_a > set_a,A: set_a] :
( ( extens2802975062453607898_set_a @ Resid )
=> ( ( ide_set_a @ Resid @ A )
=> ( ( extens7801945855595804251_set_a @ Resid @ A @ A )
= A ) ) ) ).
% extensional_rts.comp_ide_self
thf(fact_1092_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_1093_extensional__rts_Oprfx__decomp,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a] :
( ( extens2802975062453607898_set_a @ Resid )
=> ( ( ide_set_a @ Resid @ ( Resid @ T @ U ) )
=> ( ( extens7801945855595804251_set_a @ Resid @ T @ ( Resid @ U @ T ) )
= U ) ) ) ).
% extensional_rts.prfx_decomp
thf(fact_1094_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_1095_extensional__rts_Ocomp__eqI,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,V: set_a,U: set_a] :
( ( extens2802975062453607898_set_a @ Resid )
=> ( ( ide_set_a @ Resid @ ( Resid @ T @ V ) )
=> ( ( U
= ( Resid @ V @ T ) )
=> ( ( extens7801945855595804251_set_a @ Resid @ T @ U )
= V ) ) ) ) ).
% extensional_rts.comp_eqI
thf(fact_1096_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_1097_extensional__rts_Oresid__comp_I1_J,axiom,
! [Resid: a > a > a,T: a,U: a,W2: a] :
( ( extensional_rts_a @ Resid )
=> ( ( con_a @ Resid @ ( extensional_comp_a @ Resid @ T @ U ) @ W2 )
=> ( ( Resid @ W2 @ ( extensional_comp_a @ Resid @ T @ U ) )
= ( Resid @ ( Resid @ W2 @ T ) @ U ) ) ) ) ).
% extensional_rts.resid_comp(1)
thf(fact_1098_extensional__rts_Oresid__comp_I1_J,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a,W2: set_a] :
( ( extens2802975062453607898_set_a @ Resid )
=> ( ( con_set_a @ Resid @ ( extens7801945855595804251_set_a @ Resid @ T @ U ) @ W2 )
=> ( ( Resid @ W2 @ ( extens7801945855595804251_set_a @ Resid @ T @ U ) )
= ( Resid @ ( Resid @ W2 @ T ) @ U ) ) ) ) ).
% extensional_rts.resid_comp(1)
thf(fact_1099_extensional__rts_Oresid__comp_I2_J,axiom,
! [Resid: a > a > a,T: a,U: a,W2: a] :
( ( extensional_rts_a @ Resid )
=> ( ( con_a @ Resid @ ( extensional_comp_a @ Resid @ T @ U ) @ W2 )
=> ( ( Resid @ ( extensional_comp_a @ Resid @ T @ U ) @ W2 )
= ( extensional_comp_a @ Resid @ ( Resid @ T @ W2 ) @ ( Resid @ U @ ( Resid @ W2 @ T ) ) ) ) ) ) ).
% extensional_rts.resid_comp(2)
thf(fact_1100_extensional__rts_Oresid__comp_I2_J,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a,W2: set_a] :
( ( extens2802975062453607898_set_a @ Resid )
=> ( ( con_set_a @ Resid @ ( extens7801945855595804251_set_a @ Resid @ T @ U ) @ W2 )
=> ( ( Resid @ ( extens7801945855595804251_set_a @ Resid @ T @ U ) @ W2 )
= ( extens7801945855595804251_set_a @ Resid @ ( Resid @ T @ W2 ) @ ( Resid @ U @ ( Resid @ W2 @ T ) ) ) ) ) ) ).
% extensional_rts.resid_comp(2)
thf(fact_1101_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_1102_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_1103_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_1104_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_1105_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_1106_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_1107_extensional__rts__axioms_Ointro,axiom,
! [Resid: set_a > set_a > set_a] :
( ! [T3: set_a,U5: set_a] :
( ( ( ide_set_a @ Resid @ ( Resid @ T3 @ U5 ) )
& ( ide_set_a @ Resid @ ( Resid @ U5 @ T3 ) ) )
=> ( T3 = U5 ) )
=> ( extens4895702437644178167_set_a @ Resid ) ) ).
% extensional_rts_axioms.intro
thf(fact_1108_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_1109_extensional__rts__axioms__def,axiom,
( extens4895702437644178167_set_a
= ( ^ [Resid2: set_a > set_a > set_a] :
! [T7: set_a,U4: set_a] :
( ( ( ide_set_a @ Resid2 @ ( Resid2 @ T7 @ U4 ) )
& ( ide_set_a @ Resid2 @ ( Resid2 @ U4 @ T7 ) ) )
=> ( T7 = U4 ) ) ) ) ).
% extensional_rts_axioms_def
thf(fact_1110_extensional__rts__axioms__def,axiom,
( extens8613361310974791063ioms_a
= ( ^ [Resid2: a > a > a] :
! [T7: a,U4: a] :
( ( ( ide_a @ Resid2 @ ( Resid2 @ T7 @ U4 ) )
& ( ide_a @ Resid2 @ ( Resid2 @ U4 @ T7 ) ) )
=> ( T7 = U4 ) ) ) ) ).
% extensional_rts_axioms_def
thf(fact_1111_extensional__rts_Obounded__imp__con_092_060_094sub_062E,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a,T4: set_a,U2: set_a] :
( ( extens2802975062453607898_set_a @ Resid )
=> ( ( ( ide_set_a @ Resid @ ( Resid @ ( extens7801945855595804251_set_a @ Resid @ T @ U ) @ ( extens7801945855595804251_set_a @ Resid @ T4 @ U2 ) ) )
& ( ide_set_a @ Resid @ ( Resid @ ( extens7801945855595804251_set_a @ Resid @ T4 @ U2 ) @ ( extens7801945855595804251_set_a @ Resid @ T @ U ) ) ) )
=> ( con_set_a @ Resid @ T @ T4 ) ) ) ).
% extensional_rts.bounded_imp_con\<^sub>E
thf(fact_1112_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_1113_extensional__rts_Oprfx__comp,axiom,
! [Resid: set_a > set_a > set_a,U: set_a,T: set_a,V: set_a] :
( ( extens2802975062453607898_set_a @ Resid )
=> ( ( arr_set_a @ Resid @ U )
=> ( ( ( extens7801945855595804251_set_a @ Resid @ T @ V )
= U )
=> ( ide_set_a @ Resid @ ( Resid @ T @ U ) ) ) ) ) ).
% extensional_rts.prfx_comp
thf(fact_1114_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_1115_extensional__rts_Ocomp__arr__trg,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,B: set_a] :
( ( extens2802975062453607898_set_a @ Resid )
=> ( ( arr_set_a @ Resid @ T )
=> ( ( ( trg_set_a @ Resid @ T )
= B )
=> ( ( extens7801945855595804251_set_a @ Resid @ T @ B )
= T ) ) ) ) ).
% extensional_rts.comp_arr_trg
thf(fact_1116_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_1117_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_1118_extensional__rts_Ocomp__src__arr,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,A: set_a] :
( ( extens2802975062453607898_set_a @ Resid )
=> ( ( arr_set_a @ Resid @ T )
=> ( ( ( weakly2061155085811118449_set_a @ Resid @ T )
= A )
=> ( ( extens7801945855595804251_set_a @ Resid @ A @ T )
= T ) ) ) ) ).
% extensional_rts.comp_src_arr
thf(fact_1119_extensional__rts_Ocon__comp__iff,axiom,
! [Resid: a > a > a,W2: a,T: a,U: a] :
( ( extensional_rts_a @ Resid )
=> ( ( con_a @ Resid @ W2 @ ( extensional_comp_a @ Resid @ T @ U ) )
= ( ( composable_a @ Resid @ T @ U )
& ( con_a @ Resid @ ( Resid @ W2 @ T ) @ U ) ) ) ) ).
% extensional_rts.con_comp_iff
thf(fact_1120_extensional__rts_Ocon__comp__iff,axiom,
! [Resid: set_a > set_a > set_a,W2: set_a,T: set_a,U: set_a] :
( ( extens2802975062453607898_set_a @ Resid )
=> ( ( con_set_a @ Resid @ W2 @ ( extens7801945855595804251_set_a @ Resid @ T @ U ) )
= ( ( composable_set_a @ Resid @ T @ U )
& ( con_set_a @ Resid @ ( Resid @ W2 @ T ) @ U ) ) ) ) ).
% extensional_rts.con_comp_iff
thf(fact_1121_extensional__rts_Ocon__compI_I1_J,axiom,
! [Resid: a > a > a,T: a,U: a,W2: a] :
( ( extensional_rts_a @ Resid )
=> ( ( composable_a @ Resid @ T @ U )
=> ( ( con_a @ Resid @ ( Resid @ W2 @ T ) @ U )
=> ( con_a @ Resid @ W2 @ ( extensional_comp_a @ Resid @ T @ U ) ) ) ) ) ).
% extensional_rts.con_compI(1)
thf(fact_1122_extensional__rts_Ocon__compI_I1_J,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a,W2: set_a] :
( ( extens2802975062453607898_set_a @ Resid )
=> ( ( composable_set_a @ Resid @ T @ U )
=> ( ( con_set_a @ Resid @ ( Resid @ W2 @ T ) @ U )
=> ( con_set_a @ Resid @ W2 @ ( extens7801945855595804251_set_a @ Resid @ T @ U ) ) ) ) ) ).
% extensional_rts.con_compI(1)
thf(fact_1123_extensional__rts_Ocon__compI_I2_J,axiom,
! [Resid: a > a > a,T: a,U: a,W2: a] :
( ( extensional_rts_a @ Resid )
=> ( ( composable_a @ Resid @ T @ U )
=> ( ( con_a @ Resid @ ( Resid @ W2 @ T ) @ U )
=> ( con_a @ Resid @ ( extensional_comp_a @ Resid @ T @ U ) @ W2 ) ) ) ) ).
% extensional_rts.con_compI(2)
thf(fact_1124_extensional__rts_Ocon__compI_I2_J,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a,W2: set_a] :
( ( extens2802975062453607898_set_a @ Resid )
=> ( ( composable_set_a @ Resid @ T @ U )
=> ( ( con_set_a @ Resid @ ( Resid @ W2 @ T ) @ U )
=> ( con_set_a @ Resid @ ( extens7801945855595804251_set_a @ Resid @ T @ U ) @ W2 ) ) ) ) ).
% extensional_rts.con_compI(2)
thf(fact_1125_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_1126_extensional__rts_Ocomposable__iff__arr__comp,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a] :
( ( extens2802975062453607898_set_a @ Resid )
=> ( ( composable_set_a @ Resid @ T @ U )
= ( arr_set_a @ Resid @ ( extens7801945855595804251_set_a @ Resid @ T @ U ) ) ) ) ).
% extensional_rts.composable_iff_arr_comp
thf(fact_1127_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_1128_extensional__rts_Oarr__comp,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a] :
( ( extens2802975062453607898_set_a @ Resid )
=> ( ( composable_set_a @ Resid @ T @ U )
=> ( arr_set_a @ Resid @ ( extens7801945855595804251_set_a @ Resid @ T @ U ) ) ) ) ).
% extensional_rts.arr_comp
thf(fact_1129_extensional__rts_Otrg__comp,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a] :
( ( extens2802975062453607898_set_a @ Resid )
=> ( ( composable_set_a @ Resid @ T @ U )
=> ( ( trg_set_a @ Resid @ ( extens7801945855595804251_set_a @ Resid @ T @ U ) )
= ( trg_set_a @ Resid @ U ) ) ) ) ).
% extensional_rts.trg_comp
thf(fact_1130_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_1131_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_1132_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_1133_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_1134_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_1135_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_1136_extensional__rts_Ointro,axiom,
! [Resid: a > a > a] :
( ( rts_a @ Resid )
=> ( ( extens8613361310974791063ioms_a @ Resid )
=> ( extensional_rts_a @ Resid ) ) ) ).
% extensional_rts.intro
thf(fact_1137_extensional__rts__with__composites_Oarr__compE_092_060_094sub_062E_092_060_094sub_062C,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a] :
( ( extens4585908139652882248_set_a @ Resid )
=> ( ( arr_set_a @ Resid @ ( extens7801945855595804251_set_a @ Resid @ T @ U ) )
=> ~ ( ( arr_set_a @ Resid @ T )
=> ( ( arr_set_a @ Resid @ U )
=> ( ( trg_set_a @ Resid @ T )
!= ( weakly2061155085811118449_set_a @ Resid @ U ) ) ) ) ) ) ).
% extensional_rts_with_composites.arr_compE\<^sub>E\<^sub>C
thf(fact_1138_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_1139_extensional__rts__with__composites_Oarr__comp_092_060_094sub_062E_092_060_094sub_062C,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a] :
( ( extens4585908139652882248_set_a @ Resid )
=> ( ( arr_set_a @ Resid @ T )
=> ( ( arr_set_a @ Resid @ U )
=> ( ( ( trg_set_a @ Resid @ T )
= ( weakly2061155085811118449_set_a @ Resid @ U ) )
=> ( arr_set_a @ Resid @ ( extens7801945855595804251_set_a @ Resid @ T @ U ) ) ) ) ) ) ).
% extensional_rts_with_composites.arr_comp\<^sub>E\<^sub>C
thf(fact_1140_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_1141_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_1142_extensional__rts__with__composites_Oresid__common__prefix,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a,V: set_a] :
( ( extens4585908139652882248_set_a @ Resid )
=> ( ( con_set_a @ Resid @ ( extens7801945855595804251_set_a @ Resid @ T @ U ) @ ( extens7801945855595804251_set_a @ Resid @ T @ V ) )
=> ( ( Resid @ ( extens7801945855595804251_set_a @ Resid @ T @ U ) @ ( extens7801945855595804251_set_a @ Resid @ T @ V ) )
= ( Resid @ U @ V ) ) ) ) ).
% extensional_rts_with_composites.resid_common_prefix
thf(fact_1143_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_1144_extensional__rts__with__composites_Ojoin__expansion_I2_J,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a] :
( ( extens4585908139652882248_set_a @ Resid )
=> ( ( con_set_a @ Resid @ T @ U )
=> ( seq_set_a @ Resid @ T @ ( Resid @ U @ T ) ) ) ) ).
% extensional_rts_with_composites.join_expansion(2)
thf(fact_1145_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_1146_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_1147_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_1148_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_1149_extensional__rts__with__composites_Ojoin__expansion_I1_J,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a] :
( ( extens4585908139652882248_set_a @ Resid )
=> ( ( con_set_a @ Resid @ T @ U )
=> ( ( extens1973556086528668384_set_a @ Resid @ T @ U )
= ( extens7801945855595804251_set_a @ Resid @ T @ ( Resid @ U @ T ) ) ) ) ) ).
% extensional_rts_with_composites.join_expansion(1)
thf(fact_1150_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_1151_extensional__rts__with__composites_Ojoin3__expansion,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a,V: set_a] :
( ( extens4585908139652882248_set_a @ Resid )
=> ( ( con_set_a @ Resid @ T @ U )
=> ( ( con_set_a @ Resid @ T @ V )
=> ( ( con_set_a @ Resid @ U @ V )
=> ( ( extens1973556086528668384_set_a @ Resid @ ( extens1973556086528668384_set_a @ Resid @ T @ U ) @ V )
= ( extens7801945855595804251_set_a @ Resid @ ( extens7801945855595804251_set_a @ Resid @ T @ ( Resid @ U @ T ) ) @ ( Resid @ ( Resid @ V @ T ) @ ( Resid @ U @ T ) ) ) ) ) ) ) ) ).
% extensional_rts_with_composites.join3_expansion
thf(fact_1152_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,W2: a] :
( ( extens4790121754472881640ites_a @ Resid )
=> ( ( con_a @ Resid @ ( extensional_comp_a @ Resid @ T @ U ) @ W2 )
= ( ( seq_a @ Resid @ T @ U )
& ( con_a @ Resid @ U @ ( Resid @ W2 @ T ) ) ) ) ) ).
% extensional_rts_with_composites.con_comp_iff\<^sub>E\<^sub>C(2)
thf(fact_1153_extensional__rts__with__composites_Ocon__comp__iff_092_060_094sub_062E_092_060_094sub_062C_I2_J,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a,W2: set_a] :
( ( extens4585908139652882248_set_a @ Resid )
=> ( ( con_set_a @ Resid @ ( extens7801945855595804251_set_a @ Resid @ T @ U ) @ W2 )
= ( ( seq_set_a @ Resid @ T @ U )
& ( con_set_a @ Resid @ U @ ( Resid @ W2 @ T ) ) ) ) ) ).
% extensional_rts_with_composites.con_comp_iff\<^sub>E\<^sub>C(2)
thf(fact_1154_extensional__rts__with__composites_Ocon__comp__iff_092_060_094sub_062E_092_060_094sub_062C_I1_J,axiom,
! [Resid: a > a > a,W2: a,T: a,U: a] :
( ( extens4790121754472881640ites_a @ Resid )
=> ( ( con_a @ Resid @ W2 @ ( extensional_comp_a @ Resid @ T @ U ) )
= ( ( seq_a @ Resid @ T @ U )
& ( con_a @ Resid @ U @ ( Resid @ W2 @ T ) ) ) ) ) ).
% extensional_rts_with_composites.con_comp_iff\<^sub>E\<^sub>C(1)
thf(fact_1155_extensional__rts__with__composites_Ocon__comp__iff_092_060_094sub_062E_092_060_094sub_062C_I1_J,axiom,
! [Resid: set_a > set_a > set_a,W2: set_a,T: set_a,U: set_a] :
( ( extens4585908139652882248_set_a @ Resid )
=> ( ( con_set_a @ Resid @ W2 @ ( extens7801945855595804251_set_a @ Resid @ T @ U ) )
= ( ( seq_set_a @ Resid @ T @ U )
& ( con_set_a @ Resid @ U @ ( Resid @ W2 @ T ) ) ) ) ) ).
% extensional_rts_with_composites.con_comp_iff\<^sub>E\<^sub>C(1)
thf(fact_1156_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_1157_extensional__rts__with__composites_Oseq__implies__arr__comp,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a] :
( ( extens4585908139652882248_set_a @ Resid )
=> ( ( seq_set_a @ Resid @ T @ U )
=> ( arr_set_a @ Resid @ ( extens7801945855595804251_set_a @ Resid @ T @ U ) ) ) ) ).
% extensional_rts_with_composites.seq_implies_arr_comp
thf(fact_1158_extensional__rts__with__composites_Otrg__comp_092_060_094sub_062E_092_060_094sub_062C,axiom,
! [Resid: set_a > set_a > set_a,T: set_a,U: set_a] :
( ( extens4585908139652882248_set_a @ Resid )
=> ( ( seq_set_a @ Resid @ T @ U )
=> ( ( trg_set_a @ Resid @ ( extens7801945855595804251_set_a @ Resid @ T @ U ) )
= ( trg_set_a @ Resid @ U ) ) ) ) ).
% extensional_rts_with_composites.trg_comp\<^sub>E\<^sub>C
thf(fact_1159_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_1160_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_1161_Least__def,axiom,
( ord_Least_set_a
= ( ^ [P2: set_a > $o] :
( the_set_a
@ ^ [X2: set_a] :
( ( P2 @ X2 )
& ! [Y2: set_a] :
( ( P2 @ Y2 )
=> ( ord_less_eq_set_a @ X2 @ Y2 ) ) ) ) ) ) ).
% Least_def
thf(fact_1162_Least1I,axiom,
! [P: set_a > $o] :
( ? [X3: set_a] :
( ( P @ X3 )
& ! [Y3: set_a] :
( ( P @ Y3 )
=> ( ord_less_eq_set_a @ X3 @ Y3 ) )
& ! [Y3: set_a] :
( ( ( P @ Y3 )
& ! [Ya: set_a] :
( ( P @ Ya )
=> ( ord_less_eq_set_a @ Y3 @ Ya ) ) )
=> ( Y3 = X3 ) ) )
=> ( P @ ( ord_Least_set_a @ P ) ) ) ).
% Least1I
thf(fact_1163_Least1__le,axiom,
! [P: set_a > $o,Z: set_a] :
( ? [X3: set_a] :
( ( P @ X3 )
& ! [Y3: set_a] :
( ( P @ Y3 )
=> ( ord_less_eq_set_a @ X3 @ Y3 ) )
& ! [Y3: set_a] :
( ( ( P @ Y3 )
& ! [Ya: set_a] :
( ( P @ Ya )
=> ( ord_less_eq_set_a @ Y3 @ Ya ) ) )
=> ( Y3 = X3 ) ) )
=> ( ( P @ Z )
=> ( ord_less_eq_set_a @ ( ord_Least_set_a @ P ) @ Z ) ) ) ).
% Least1_le
thf(fact_1164_LeastI2__order,axiom,
! [P: set_a > $o,X4: set_a,Q: set_a > $o] :
( ( P @ X4 )
=> ( ! [Y3: set_a] :
( ( P @ Y3 )
=> ( ord_less_eq_set_a @ X4 @ Y3 ) )
=> ( ! [X: set_a] :
( ( P @ X )
=> ( ! [Y: set_a] :
( ( P @ Y )
=> ( ord_less_eq_set_a @ X @ Y ) )
=> ( Q @ X ) ) )
=> ( Q @ ( ord_Least_set_a @ P ) ) ) ) ) ).
% LeastI2_order
thf(fact_1165_Least__equality,axiom,
! [P: set_a > $o,X4: set_a] :
( ( P @ X4 )
=> ( ! [Y3: set_a] :
( ( P @ Y3 )
=> ( ord_less_eq_set_a @ X4 @ Y3 ) )
=> ( ( ord_Least_set_a @ P )
= X4 ) ) ) ).
% Least_equality
thf(fact_1166_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_1167_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_1168_simulation_Opreserves__ide,axiom,
! [A4: set_a > set_a > set_a,B3: set_a > set_a > set_a,F2: set_a > set_a,A: set_a] :
( ( simula7336900841036901507_set_a @ A4 @ B3 @ F2 )
=> ( ( ide_set_a @ A4 @ A )
=> ( ide_set_a @ B3 @ ( F2 @ A ) ) ) ) ).
% simulation.preserves_ide
thf(fact_1169_simulation_Opreserves__ide,axiom,
! [A4: set_a > set_a > set_a,B3: a > a > a,F2: set_a > a,A: set_a] :
( ( simulation_set_a_a @ A4 @ B3 @ F2 )
=> ( ( ide_set_a @ A4 @ A )
=> ( ide_a @ B3 @ ( F2 @ A ) ) ) ) ).
% simulation.preserves_ide
thf(fact_1170_simulation_Opreserves__ide,axiom,
! [A4: a > a > a,B3: set_a > set_a > set_a,F2: a > set_a,A: a] :
( ( simulation_a_set_a @ A4 @ B3 @ F2 )
=> ( ( ide_a @ A4 @ A )
=> ( ide_set_a @ B3 @ ( F2 @ A ) ) ) ) ).
% simulation.preserves_ide
thf(fact_1171_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_1172_simulation_Opreserves__cong,axiom,
! [A4: set_a > set_a > set_a,B3: set_a > set_a > set_a,F2: set_a > set_a,T: set_a,U: set_a] :
( ( simula7336900841036901507_set_a @ A4 @ B3 @ F2 )
=> ( ( ( ide_set_a @ A4 @ ( A4 @ T @ U ) )
& ( ide_set_a @ A4 @ ( A4 @ U @ T ) ) )
=> ( ( ide_set_a @ B3 @ ( B3 @ ( F2 @ T ) @ ( F2 @ U ) ) )
& ( ide_set_a @ B3 @ ( B3 @ ( F2 @ U ) @ ( F2 @ T ) ) ) ) ) ) ).
% simulation.preserves_cong
thf(fact_1173_simulation_Opreserves__cong,axiom,
! [A4: set_a > set_a > set_a,B3: a > a > a,F2: set_a > a,T: set_a,U: set_a] :
( ( simulation_set_a_a @ A4 @ B3 @ F2 )
=> ( ( ( ide_set_a @ A4 @ ( A4 @ T @ U ) )
& ( ide_set_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_1174_simulation_Opreserves__cong,axiom,
! [A4: a > a > a,B3: set_a > set_a > set_a,F2: a > set_a,T: a,U: a] :
( ( simulation_a_set_a @ A4 @ B3 @ F2 )
=> ( ( ( ide_a @ A4 @ ( A4 @ T @ U ) )
& ( ide_a @ A4 @ ( A4 @ U @ T ) ) )
=> ( ( ide_set_a @ B3 @ ( B3 @ ( F2 @ T ) @ ( F2 @ U ) ) )
& ( ide_set_a @ B3 @ ( B3 @ ( F2 @ U ) @ ( F2 @ T ) ) ) ) ) ) ).
% simulation.preserves_cong
thf(fact_1175_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_1176_simulation_Opreserves__prfx,axiom,
! [A4: set_a > set_a > set_a,B3: set_a > set_a > set_a,F2: set_a > set_a,T: set_a,U: set_a] :
( ( simula7336900841036901507_set_a @ A4 @ B3 @ F2 )
=> ( ( ide_set_a @ A4 @ ( A4 @ T @ U ) )
=> ( ide_set_a @ B3 @ ( B3 @ ( F2 @ T ) @ ( F2 @ U ) ) ) ) ) ).
% simulation.preserves_prfx
thf(fact_1177_simulation_Opreserves__prfx,axiom,
! [A4: set_a > set_a > set_a,B3: a > a > a,F2: set_a > a,T: set_a,U: set_a] :
( ( simulation_set_a_a @ A4 @ B3 @ F2 )
=> ( ( ide_set_a @ A4 @ ( A4 @ T @ U ) )
=> ( ide_a @ B3 @ ( B3 @ ( F2 @ T ) @ ( F2 @ U ) ) ) ) ) ).
% simulation.preserves_prfx
thf(fact_1178_simulation_Opreserves__prfx,axiom,
! [A4: a > a > a,B3: set_a > set_a > set_a,F2: a > set_a,T: a,U: a] :
( ( simulation_a_set_a @ A4 @ B3 @ F2 )
=> ( ( ide_a @ A4 @ ( A4 @ T @ U ) )
=> ( ide_set_a @ B3 @ ( B3 @ ( F2 @ T ) @ ( F2 @ U ) ) ) ) ) ).
% simulation.preserves_prfx
thf(fact_1179_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_1180_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_1181_simulation_Opreserves__con,axiom,
! [A4: a > a > a,B3: set_a > set_a > set_a,F2: a > set_a,T: a,U: a] :
( ( simulation_a_set_a @ A4 @ B3 @ F2 )
=> ( ( con_a @ A4 @ T @ U )
=> ( con_set_a @ B3 @ ( F2 @ T ) @ ( F2 @ U ) ) ) ) ).
% simulation.preserves_con
thf(fact_1182_simulation_Opreserves__con,axiom,
! [A4: set_a > set_a > set_a,B3: a > a > a,F2: set_a > a,T: set_a,U: set_a] :
( ( simulation_set_a_a @ A4 @ B3 @ F2 )
=> ( ( con_set_a @ A4 @ T @ U )
=> ( con_a @ B3 @ ( F2 @ T ) @ ( F2 @ U ) ) ) ) ).
% simulation.preserves_con
thf(fact_1183_simulation_Opreserves__con,axiom,
! [A4: set_a > set_a > set_a,B3: set_a > set_a > set_a,F2: set_a > set_a,T: set_a,U: set_a] :
( ( simula7336900841036901507_set_a @ A4 @ B3 @ F2 )
=> ( ( con_set_a @ A4 @ T @ U )
=> ( con_set_a @ B3 @ ( F2 @ T ) @ ( F2 @ U ) ) ) ) ).
% simulation.preserves_con
thf(fact_1184_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_1185_simulation_Opreserves__reflects__arr,axiom,
! [A4: set_a > set_a > set_a,B3: a > a > a,F2: set_a > a,T: set_a] :
( ( simulation_set_a_a @ A4 @ B3 @ F2 )
=> ( ( arr_a @ B3 @ ( F2 @ T ) )
= ( arr_set_a @ A4 @ T ) ) ) ).
% simulation.preserves_reflects_arr
thf(fact_1186_simulation_Opreserves__reflects__arr,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_set_a @ B3 @ ( F2 @ T ) )
= ( arr_a @ A4 @ T ) ) ) ).
% simulation.preserves_reflects_arr
thf(fact_1187_simulation_Opreserves__reflects__arr,axiom,
! [A4: set_a > set_a > set_a,B3: set_a > set_a > set_a,F2: set_a > set_a,T: set_a] :
( ( simula7336900841036901507_set_a @ A4 @ B3 @ F2 )
=> ( ( arr_set_a @ B3 @ ( F2 @ T ) )
= ( arr_set_a @ A4 @ T ) ) ) ).
% simulation.preserves_reflects_arr
thf(fact_1188_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_1189_simulation_Oextensional,axiom,
! [A4: set_a > set_a > set_a,B3: set_a > set_a > set_a,F2: set_a > set_a,T: set_a] :
( ( simula7336900841036901507_set_a @ A4 @ B3 @ F2 )
=> ( ~ ( arr_set_a @ A4 @ T )
=> ( ( F2 @ T )
= ( partial_null_set_a @ B3 ) ) ) ) ).
% simulation.extensional
thf(fact_1190_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_1191_simulation_Oextensional,axiom,
! [A4: set_a > set_a > set_a,B3: a > a > a,F2: set_a > a,T: set_a] :
( ( simulation_set_a_a @ A4 @ B3 @ F2 )
=> ( ~ ( arr_set_a @ A4 @ T )
=> ( ( F2 @ T )
= ( partial_null_a @ B3 ) ) ) ) ).
% simulation.extensional
thf(fact_1192_simulation_Opreserves__trg,axiom,
! [A4: set_a > set_a > set_a,B3: set_a > set_a > set_a,F2: set_a > set_a,T: set_a] :
( ( simula7336900841036901507_set_a @ A4 @ B3 @ F2 )
=> ( ( arr_set_a @ A4 @ T )
=> ( ( F2 @ ( trg_set_a @ A4 @ T ) )
= ( trg_set_a @ B3 @ ( F2 @ T ) ) ) ) ) ).
% simulation.preserves_trg
thf(fact_1193_simulation_Opreserves__trg,axiom,
! [A4: set_a > set_a > set_a,B3: a > a > a,F2: set_a > a,T: set_a] :
( ( simulation_set_a_a @ A4 @ B3 @ F2 )
=> ( ( arr_set_a @ A4 @ T )
=> ( ( F2 @ ( trg_set_a @ A4 @ T ) )
= ( trg_a @ B3 @ ( F2 @ T ) ) ) ) ) ).
% simulation.preserves_trg
thf(fact_1194_simulation_Opreserves__trg,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 @ ( trg_a @ A4 @ T ) )
= ( trg_set_a @ B3 @ ( F2 @ T ) ) ) ) ) ).
% simulation.preserves_trg
thf(fact_1195_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_1196_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_1197_simulation__def,axiom,
( simulation_a_a
= ( ^ [A8: a > a > a,B5: a > a > a,F3: a > a] :
( ( rts_a @ A8 )
& ( rts_a @ B5 )
& ( simula3868467710248865958ms_a_a @ A8 @ B5 @ F3 ) ) ) ) ).
% simulation_def
thf(fact_1198_Resid__def,axiom,
! [T6: set_a,U3: set_a] :
( ( ( ( normal8595587647932138008lass_a @ resid @ nn @ T6 )
& ( normal8595587647932138008lass_a @ resid @ nn @ U3 )
& ? [T2: a] :
( ( member_a @ T2 @ T6 )
& ? [U8: a] :
( ( member_a @ U8 @ U3 )
& ( con_a @ resid @ T2 @ U8 ) ) ) )
=> ( ( quotie8165075472272353145esid_a @ resid @ nn @ T6 @ U3 )
= ( normal7408713899360725774lass_a @ resid @ nn
@ ( resid
@ ( product_fst_a_a
@ ( fChoic4124218645493772411od_a_a
@ ^ [Tu: product_prod_a_a] :
( ( member_a @ ( product_fst_a_a @ Tu ) @ T6 )
& ( member_a @ ( product_snd_a_a @ Tu ) @ U3 )
& ( 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 ) @ T6 )
& ( member_a @ ( product_snd_a_a @ Tu ) @ U3 )
& ( con_a @ resid @ ( product_fst_a_a @ Tu ) @ ( product_snd_a_a @ Tu ) ) ) ) ) ) ) ) )
& ( ~ ( ( normal8595587647932138008lass_a @ resid @ nn @ T6 )
& ( normal8595587647932138008lass_a @ resid @ nn @ U3 )
& ? [T3: a] :
( ( member_a @ T3 @ T6 )
& ? [U5: a] :
( ( member_a @ U5 @ U3 )
& ( con_a @ resid @ T3 @ U5 ) ) ) )
=> ( ( quotie8165075472272353145esid_a @ resid @ nn @ T6 @ U3 )
= bot_bot_set_a ) ) ) ).
% Resid_def
thf(fact_1199_quotient__by__coherent__normal_OResid__def,axiom,
! [Resid: set_a > set_a > set_a,NN: set_set_a,T6: set_set_a,U3: set_set_a] :
( ( quotie5625257012022141046_set_a @ Resid @ NN )
=> ( ( ( ( normal4437380936311325560_set_a @ Resid @ NN @ T6 )
& ( normal4437380936311325560_set_a @ Resid @ NN @ U3 )
& ? [T2: set_a] :
( ( member_set_a @ T2 @ T6 )
& ? [U8: set_a] :
( ( member_set_a @ U8 @ U3 )
& ( con_set_a @ Resid @ T2 @ U8 ) ) ) )
=> ( ( quotie3283642546880816345_set_a @ Resid @ NN @ T6 @ U3 )
= ( normal2962378890657961070_set_a @ Resid @ NN
@ ( Resid
@ ( produc9088895665703139587_set_a
@ ( fChoic7046651920289422459_set_a
@ ^ [Tu: produc1703568184450464039_set_a] :
( ( member_set_a @ ( produc9088895665703139587_set_a @ Tu ) @ T6 )
& ( member_set_a @ ( produc1983107199584856133_set_a @ Tu ) @ U3 )
& ( con_set_a @ Resid @ ( produc9088895665703139587_set_a @ Tu ) @ ( produc1983107199584856133_set_a @ Tu ) ) ) ) )
@ ( produc1983107199584856133_set_a
@ ( fChoic7046651920289422459_set_a
@ ^ [Tu: produc1703568184450464039_set_a] :
( ( member_set_a @ ( produc9088895665703139587_set_a @ Tu ) @ T6 )
& ( member_set_a @ ( produc1983107199584856133_set_a @ Tu ) @ U3 )
& ( con_set_a @ Resid @ ( produc9088895665703139587_set_a @ Tu ) @ ( produc1983107199584856133_set_a @ Tu ) ) ) ) ) ) ) ) )
& ( ~ ( ( normal4437380936311325560_set_a @ Resid @ NN @ T6 )
& ( normal4437380936311325560_set_a @ Resid @ NN @ U3 )
& ? [T3: set_a] :
( ( member_set_a @ T3 @ T6 )
& ? [U5: set_a] :
( ( member_set_a @ U5 @ U3 )
& ( con_set_a @ Resid @ T3 @ U5 ) ) ) )
=> ( ( quotie3283642546880816345_set_a @ Resid @ NN @ T6 @ U3 )
= bot_bot_set_set_a ) ) ) ) ).
% quotient_by_coherent_normal.Resid_def
thf(fact_1200_quotient__by__coherent__normal_OResid__def,axiom,
! [Resid: a > a > a,NN: set_a,T6: set_a,U3: set_a] :
( ( quotie3282664541148387094rmal_a @ Resid @ NN )
=> ( ( ( ( normal8595587647932138008lass_a @ Resid @ NN @ T6 )
& ( normal8595587647932138008lass_a @ Resid @ NN @ U3 )
& ? [T2: a] :
( ( member_a @ T2 @ T6 )
& ? [U8: a] :
( ( member_a @ U8 @ U3 )
& ( con_a @ Resid @ T2 @ U8 ) ) ) )
=> ( ( quotie8165075472272353145esid_a @ Resid @ NN @ T6 @ U3 )
= ( normal7408713899360725774lass_a @ Resid @ NN
@ ( Resid
@ ( product_fst_a_a
@ ( fChoic4124218645493772411od_a_a
@ ^ [Tu: product_prod_a_a] :
( ( member_a @ ( product_fst_a_a @ Tu ) @ T6 )
& ( member_a @ ( product_snd_a_a @ Tu ) @ U3 )
& ( 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 ) @ T6 )
& ( member_a @ ( product_snd_a_a @ Tu ) @ U3 )
& ( con_a @ Resid @ ( product_fst_a_a @ Tu ) @ ( product_snd_a_a @ Tu ) ) ) ) ) ) ) ) )
& ( ~ ( ( normal8595587647932138008lass_a @ Resid @ NN @ T6 )
& ( normal8595587647932138008lass_a @ Resid @ NN @ U3 )
& ? [T3: a] :
( ( member_a @ T3 @ T6 )
& ? [U5: a] :
( ( member_a @ U5 @ U3 )
& ( con_a @ Resid @ T3 @ U5 ) ) ) )
=> ( ( quotie8165075472272353145esid_a @ Resid @ NN @ T6 @ U3 )
= bot_bot_set_a ) ) ) ) ).
% quotient_by_coherent_normal.Resid_def
thf(fact_1201_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_1202_verit__comp__simplify1_I2_J,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_1203_image__eqI,axiom,
! [B: a,F: a > a,X4: a,A4: set_a] :
( ( B
= ( F @ X4 ) )
=> ( ( member_a @ X4 @ A4 )
=> ( member_a @ B @ ( image_a_a @ F @ A4 ) ) ) ) ).
% image_eqI
thf(fact_1204_image__is__empty,axiom,
! [F: a > a,A4: set_a] :
( ( ( image_a_a @ F @ A4 )
= bot_bot_set_a )
= ( A4 = bot_bot_set_a ) ) ).
% image_is_empty
thf(fact_1205_empty__is__image,axiom,
! [F: a > a,A4: set_a] :
( ( bot_bot_set_a
= ( image_a_a @ F @ A4 ) )
= ( A4 = bot_bot_set_a ) ) ).
% empty_is_image
thf(fact_1206_image__empty,axiom,
! [F: a > a] :
( ( image_a_a @ F @ bot_bot_set_a )
= bot_bot_set_a ) ).
% image_empty
thf(fact_1207_Setcompr__eq__image,axiom,
! [F: a > a,A4: set_a] :
( ( collect_a
@ ^ [Uu: a] :
? [X2: a] :
( ( Uu
= ( F @ X2 ) )
& ( member_a @ X2 @ A4 ) ) )
= ( image_a_a @ F @ A4 ) ) ).
% Setcompr_eq_image
thf(fact_1208_setcompr__eq__image,axiom,
! [F: a > a,P: a > $o] :
( ( collect_a
@ ^ [Uu: a] :
? [X2: a] :
( ( Uu
= ( F @ X2 ) )
& ( P @ X2 ) ) )
= ( image_a_a @ F @ ( collect_a @ P ) ) ) ).
% setcompr_eq_image
thf(fact_1209_Compr__image__eq,axiom,
! [F: a > a,A4: set_a,P: a > $o] :
( ( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ ( image_a_a @ F @ A4 ) )
& ( P @ X2 ) ) )
= ( image_a_a @ F
@ ( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ A4 )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1210_imageE,axiom,
! [B: a,F: a > a,A4: set_a] :
( ( member_a @ B @ ( image_a_a @ F @ A4 ) )
=> ~ ! [X: a] :
( ( B
= ( F @ X ) )
=> ~ ( member_a @ X @ A4 ) ) ) ).
% imageE
thf(fact_1211_imageI,axiom,
! [X4: a,A4: set_a,F: a > a] :
( ( member_a @ X4 @ A4 )
=> ( member_a @ ( F @ X4 ) @ ( image_a_a @ F @ A4 ) ) ) ).
% imageI
thf(fact_1212_rev__image__eqI,axiom,
! [X4: a,A4: set_a,B: a,F: a > a] :
( ( member_a @ X4 @ A4 )
=> ( ( B
= ( F @ X4 ) )
=> ( member_a @ B @ ( image_a_a @ F @ A4 ) ) ) ) ).
% rev_image_eqI
thf(fact_1213_subset__image__iff,axiom,
! [B3: set_a,F: a > a,A4: set_a] :
( ( ord_less_eq_set_a @ B3 @ ( image_a_a @ F @ A4 ) )
= ( ? [AA: set_a] :
( ( ord_less_eq_set_a @ AA @ A4 )
& ( B3
= ( image_a_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_1214_subset__imageE,axiom,
! [B3: set_a,F: a > a,A4: set_a] :
( ( ord_less_eq_set_a @ B3 @ ( image_a_a @ F @ A4 ) )
=> ~ ! [C3: set_a] :
( ( ord_less_eq_set_a @ C3 @ A4 )
=> ( B3
!= ( image_a_a @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_1215_image__subsetI,axiom,
! [A4: set_a,F: a > a,B3: set_a] :
( ! [X: a] :
( ( member_a @ X @ A4 )
=> ( member_a @ ( F @ X ) @ B3 ) )
=> ( ord_less_eq_set_a @ ( image_a_a @ F @ A4 ) @ B3 ) ) ).
% image_subsetI
thf(fact_1216_image__mono,axiom,
! [A4: set_a,B3: set_a,F: a > a] :
( ( ord_less_eq_set_a @ A4 @ B3 )
=> ( ord_less_eq_set_a @ ( image_a_a @ F @ A4 ) @ ( image_a_a @ F @ B3 ) ) ) ).
% image_mono
thf(fact_1217_image__Int__subset,axiom,
! [F: a > a,A4: set_a,B3: set_a] : ( ord_less_eq_set_a @ ( image_a_a @ F @ ( inf_inf_set_a @ A4 @ B3 ) ) @ ( inf_inf_set_a @ ( image_a_a @ F @ A4 ) @ ( image_a_a @ F @ B3 ) ) ) ).
% image_Int_subset
thf(fact_1218_image__Collect__subsetI,axiom,
! [P: a > $o,F: a > a,B3: set_a] :
( ! [X: a] :
( ( P @ X )
=> ( member_a @ ( F @ X ) @ B3 ) )
=> ( ord_less_eq_set_a @ ( image_a_a @ F @ ( collect_a @ P ) ) @ B3 ) ) ).
% image_Collect_subsetI
thf(fact_1219_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_1220_all__subset__image,axiom,
! [F: a > a,A4: set_a,P: set_a > $o] :
( ( ! [B5: set_a] :
( ( ord_less_eq_set_a @ B5 @ ( image_a_a @ F @ A4 ) )
=> ( P @ B5 ) ) )
= ( ! [B5: set_a] :
( ( ord_less_eq_set_a @ B5 @ A4 )
=> ( P @ ( image_a_a @ F @ B5 ) ) ) ) ) ).
% all_subset_image
thf(fact_1221_the__elem__def,axiom,
( the_elem_set_a
= ( ^ [X5: set_set_a] :
( the_set_a
@ ^ [X2: set_a] :
( X5
= ( insert_set_a @ X2 @ bot_bot_set_set_a ) ) ) ) ) ).
% the_elem_def
thf(fact_1222_the__elem__def,axiom,
( the_elem_a
= ( ^ [X5: set_a] :
( the_a
@ ^ [X2: a] :
( X5
= ( insert_a @ X2 @ bot_bot_set_a ) ) ) ) ) ).
% the_elem_def
thf(fact_1223_insertCI,axiom,
! [A: a,B3: set_a,B: a] :
( ( ~ ( member_a @ A @ B3 )
=> ( A = B ) )
=> ( member_a @ A @ ( insert_a @ B @ B3 ) ) ) ).
% insertCI
thf(fact_1224_insert__iff,axiom,
! [A: a,B: a,A4: set_a] :
( ( member_a @ A @ ( insert_a @ B @ A4 ) )
= ( ( A = B )
| ( member_a @ A @ A4 ) ) ) ).
% insert_iff
thf(fact_1225_singletonI,axiom,
! [A: a] : ( member_a @ A @ ( insert_a @ A @ bot_bot_set_a ) ) ).
% singletonI
thf(fact_1226_insert__subset,axiom,
! [X4: a,A4: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ ( insert_a @ X4 @ A4 ) @ B3 )
= ( ( member_a @ X4 @ B3 )
& ( ord_less_eq_set_a @ A4 @ B3 ) ) ) ).
% insert_subset
thf(fact_1227_Int__insert__left__if0,axiom,
! [A: a,C2: set_a,B3: set_a] :
( ~ ( member_a @ A @ C2 )
=> ( ( inf_inf_set_a @ ( insert_a @ A @ B3 ) @ C2 )
= ( inf_inf_set_a @ B3 @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_1228_Int__insert__left__if1,axiom,
! [A: a,C2: set_a,B3: set_a] :
( ( member_a @ A @ C2 )
=> ( ( inf_inf_set_a @ ( insert_a @ A @ B3 ) @ C2 )
= ( insert_a @ A @ ( inf_inf_set_a @ B3 @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_1229_insert__inter__insert,axiom,
! [A: a,A4: set_a,B3: set_a] :
( ( inf_inf_set_a @ ( insert_a @ A @ A4 ) @ ( insert_a @ A @ B3 ) )
= ( insert_a @ A @ ( inf_inf_set_a @ A4 @ B3 ) ) ) ).
% insert_inter_insert
thf(fact_1230_Int__insert__right__if0,axiom,
! [A: a,A4: set_a,B3: set_a] :
( ~ ( member_a @ A @ A4 )
=> ( ( inf_inf_set_a @ A4 @ ( insert_a @ A @ B3 ) )
= ( inf_inf_set_a @ A4 @ B3 ) ) ) ).
% Int_insert_right_if0
thf(fact_1231_Int__insert__right__if1,axiom,
! [A: a,A4: set_a,B3: set_a] :
( ( member_a @ A @ A4 )
=> ( ( inf_inf_set_a @ A4 @ ( insert_a @ A @ B3 ) )
= ( insert_a @ A @ ( inf_inf_set_a @ A4 @ B3 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_1232_singleton__conv,axiom,
! [A: a] :
( ( collect_a
@ ^ [X2: a] : ( X2 = A ) )
= ( insert_a @ A @ bot_bot_set_a ) ) ).
% singleton_conv
thf(fact_1233_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_1234_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_1235_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_1236_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_1237_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_1238_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_1239_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_1240_the__elem__eq,axiom,
! [X4: a] :
( ( the_elem_a @ ( insert_a @ X4 @ bot_bot_set_a ) )
= X4 ) ).
% the_elem_eq
thf(fact_1241_Fpow__mono,axiom,
! [A4: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A4 @ B3 )
=> ( ord_le3724670747650509150_set_a @ ( finite_Fpow_a @ A4 ) @ ( finite_Fpow_a @ B3 ) ) ) ).
% Fpow_mono
thf(fact_1242_empty__in__Fpow,axiom,
! [A4: set_a] : ( member_set_a @ bot_bot_set_a @ ( finite_Fpow_a @ A4 ) ) ).
% empty_in_Fpow
thf(fact_1243_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_1244_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_1245_insert__Collect,axiom,
! [A: a,P: a > $o] :
( ( insert_a @ A @ ( collect_a @ P ) )
= ( collect_a
@ ^ [U4: a] :
( ( U4 != A )
=> ( P @ U4 ) ) ) ) ).
% insert_Collect
thf(fact_1246_insert__compr,axiom,
( insert_a
= ( ^ [A7: a,B5: set_a] :
( collect_a
@ ^ [X2: a] :
( ( X2 = A7 )
| ( member_a @ X2 @ B5 ) ) ) ) ) ).
% insert_compr
thf(fact_1247_insertE,axiom,
! [A: a,B: a,A4: set_a] :
( ( member_a @ A @ ( insert_a @ B @ A4 ) )
=> ( ( A != B )
=> ( member_a @ A @ A4 ) ) ) ).
% insertE
thf(fact_1248_insertI1,axiom,
! [A: a,B3: set_a] : ( member_a @ A @ ( insert_a @ A @ B3 ) ) ).
% insertI1
thf(fact_1249_insertI2,axiom,
! [A: a,B3: set_a,B: a] :
( ( member_a @ A @ B3 )
=> ( member_a @ A @ ( insert_a @ B @ B3 ) ) ) ).
% insertI2
thf(fact_1250_Set_Oset__insert,axiom,
! [X4: a,A4: set_a] :
( ( member_a @ X4 @ A4 )
=> ~ ! [B6: set_a] :
( ( A4
= ( insert_a @ X4 @ B6 ) )
=> ( member_a @ X4 @ B6 ) ) ) ).
% Set.set_insert
thf(fact_1251_insert__ident,axiom,
! [X4: a,A4: set_a,B3: set_a] :
( ~ ( member_a @ X4 @ A4 )
=> ( ~ ( member_a @ X4 @ B3 )
=> ( ( ( insert_a @ X4 @ A4 )
= ( insert_a @ X4 @ B3 ) )
= ( A4 = B3 ) ) ) ) ).
% insert_ident
thf(fact_1252_insert__absorb,axiom,
! [A: a,A4: set_a] :
( ( member_a @ A @ A4 )
=> ( ( insert_a @ A @ A4 )
= A4 ) ) ).
% insert_absorb
thf(fact_1253_insert__eq__iff,axiom,
! [A: a,A4: set_a,B: a,B3: set_a] :
( ~ ( member_a @ A @ A4 )
=> ( ~ ( member_a @ B @ B3 )
=> ( ( ( insert_a @ A @ A4 )
= ( insert_a @ B @ B3 ) )
= ( ( ( A = B )
=> ( A4 = B3 ) )
& ( ( A != B )
=> ? [C4: set_a] :
( ( A4
= ( insert_a @ B @ C4 ) )
& ~ ( member_a @ B @ C4 )
& ( B3
= ( insert_a @ A @ C4 ) )
& ~ ( member_a @ A @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_1254_mk__disjoint__insert,axiom,
! [A: a,A4: set_a] :
( ( member_a @ A @ A4 )
=> ? [B6: set_a] :
( ( A4
= ( insert_a @ A @ B6 ) )
& ~ ( member_a @ A @ B6 ) ) ) ).
% mk_disjoint_insert
thf(fact_1255_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_1256_insert__not__empty,axiom,
! [A: a,A4: set_a] :
( ( insert_a @ A @ A4 )
!= bot_bot_set_a ) ).
% insert_not_empty
thf(fact_1257_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_1258_singleton__iff,axiom,
! [B: a,A: a] :
( ( member_a @ B @ ( insert_a @ A @ bot_bot_set_a ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_1259_singletonD,axiom,
! [B: a,A: a] :
( ( member_a @ B @ ( insert_a @ A @ bot_bot_set_a ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_1260_subset__singletonD,axiom,
! [A4: set_a,X4: a] :
( ( ord_less_eq_set_a @ A4 @ ( insert_a @ X4 @ bot_bot_set_a ) )
=> ( ( A4 = bot_bot_set_a )
| ( A4
= ( insert_a @ X4 @ bot_bot_set_a ) ) ) ) ).
% subset_singletonD
thf(fact_1261_subset__singleton__iff,axiom,
! [X6: set_a,A: a] :
( ( ord_less_eq_set_a @ X6 @ ( insert_a @ A @ bot_bot_set_a ) )
= ( ( X6 = bot_bot_set_a )
| ( X6
= ( insert_a @ A @ bot_bot_set_a ) ) ) ) ).
% subset_singleton_iff
thf(fact_1262_insert__subsetI,axiom,
! [X4: a,A4: set_a,X6: set_a] :
( ( member_a @ X4 @ A4 )
=> ( ( ord_less_eq_set_a @ X6 @ A4 )
=> ( ord_less_eq_set_a @ ( insert_a @ X4 @ X6 ) @ A4 ) ) ) ).
% insert_subsetI
thf(fact_1263_Int__insert__left,axiom,
! [A: a,C2: set_a,B3: set_a] :
( ( ( member_a @ A @ C2 )
=> ( ( inf_inf_set_a @ ( insert_a @ A @ B3 ) @ C2 )
= ( insert_a @ A @ ( inf_inf_set_a @ B3 @ C2 ) ) ) )
& ( ~ ( member_a @ A @ C2 )
=> ( ( inf_inf_set_a @ ( insert_a @ A @ B3 ) @ C2 )
= ( inf_inf_set_a @ B3 @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_1264_Int__insert__right,axiom,
! [A: a,A4: set_a,B3: set_a] :
( ( ( member_a @ A @ A4 )
=> ( ( inf_inf_set_a @ A4 @ ( insert_a @ A @ B3 ) )
= ( insert_a @ A @ ( inf_inf_set_a @ A4 @ B3 ) ) ) )
& ( ~ ( member_a @ A @ A4 )
=> ( ( inf_inf_set_a @ A4 @ ( insert_a @ A @ B3 ) )
= ( inf_inf_set_a @ A4 @ B3 ) ) ) ) ).
% Int_insert_right
thf(fact_1265_subset__insertI2,axiom,
! [A4: set_a,B3: set_a,B: a] :
( ( ord_less_eq_set_a @ A4 @ B3 )
=> ( ord_less_eq_set_a @ A4 @ ( insert_a @ B @ B3 ) ) ) ).
% subset_insertI2
thf(fact_1266_subset__insertI,axiom,
! [B3: set_a,A: a] : ( ord_less_eq_set_a @ B3 @ ( insert_a @ A @ B3 ) ) ).
% subset_insertI
thf(fact_1267_subset__insert,axiom,
! [X4: a,A4: set_a,B3: set_a] :
( ~ ( member_a @ X4 @ A4 )
=> ( ( ord_less_eq_set_a @ A4 @ ( insert_a @ X4 @ B3 ) )
= ( ord_less_eq_set_a @ A4 @ B3 ) ) ) ).
% subset_insert
thf(fact_1268_insert__mono,axiom,
! [C2: set_a,D2: set_a,A: a] :
( ( ord_less_eq_set_a @ C2 @ D2 )
=> ( ord_less_eq_set_a @ ( insert_a @ A @ C2 ) @ ( insert_a @ A @ D2 ) ) ) ).
% insert_mono
thf(fact_1269_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_1270_image__constant,axiom,
! [X4: a,A4: set_a,C: a] :
( ( member_a @ X4 @ A4 )
=> ( ( image_a_a
@ ^ [X2: a] : C
@ A4 )
= ( insert_a @ C @ bot_bot_set_a ) ) ) ).
% image_constant
thf(fact_1271_is__singleton__the__elem,axiom,
( is_singleton_a
= ( ^ [A8: set_a] :
( A8
= ( insert_a @ ( the_elem_a @ A8 ) @ bot_bot_set_a ) ) ) ) ).
% is_singleton_the_elem
thf(fact_1272_is__singletonI,axiom,
! [X4: a] : ( is_singleton_a @ ( insert_a @ X4 @ bot_bot_set_a ) ) ).
% is_singletonI
thf(fact_1273_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_1274_is__singletonE,axiom,
! [A4: set_a] :
( ( is_singleton_a @ A4 )
=> ~ ! [X: a] :
( A4
!= ( insert_a @ X @ bot_bot_set_a ) ) ) ).
% is_singletonE
thf(fact_1275_is__singleton__def,axiom,
( is_singleton_a
= ( ^ [A8: set_a] :
? [X2: a] :
( A8
= ( insert_a @ X2 @ bot_bot_set_a ) ) ) ) ).
% is_singleton_def
thf(fact_1276_subset__singleton__iff__Uniq,axiom,
! [A4: set_a] :
( ( ? [A7: a] : ( ord_less_eq_set_a @ A4 @ ( insert_a @ A7 @ bot_bot_set_a ) ) )
= ( uniq_a
@ ^ [X2: a] : ( member_a @ X2 @ A4 ) ) ) ).
% subset_singleton_iff_Uniq
thf(fact_1277_empty__bind,axiom,
! [F: a > set_a] :
( ( bind_a_a @ bot_bot_set_a @ F )
= bot_bot_set_a ) ).
% empty_bind
thf(fact_1278_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
% Helper facts (3)
thf(help_fChoice_1_1_fChoice_001tf__a_T,axiom,
! [P: a > $o] :
( ( P @ ( fChoice_a @ P ) )
= ( ? [X5: a] : ( P @ X5 ) ) ) ).
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 ) )
= ( ? [X5: product_prod_a_a] : ( P @ X5 ) ) ) ).
thf(help_fChoice_1_1_fChoice_001t__Product____Type__Oprod_It__Set__Oset_Itf__a_J_Mt__Set__Oset_Itf__a_J_J_T,axiom,
! [P: produc1703568184450464039_set_a > $o] :
( ( P @ ( fChoic7046651920289422459_set_a @ P ) )
= ( ? [X5: produc1703568184450464039_set_a] : ( P @ X5 ) ) ) ).
% Conjectures (2)
thf(conj_0,hypothesis,
! [T2: a,U8: a] :
( ( ( member_a @ T2 @ t )
& ( member_a @ U8 @ u )
& ( con_a @ resid @ T2 @ U8 )
& ( ( quotie8165075472272353145esid_a @ resid @ nn @ t @ u )
= ( normal7408713899360725774lass_a @ resid @ nn @ ( resid @ T2 @ U8 ) ) ) )
=> thesis ) ).
thf(conj_1,conjecture,
thesis ).
%------------------------------------------------------------------------------