TPTP Problem File: SLH0473^1.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 : Rewrite_Properties_Reduction/0020_Rewriting_Properties/prob_00365_016515__14156224_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1487 ( 512 unt; 212 typ; 0 def)
% Number of atoms : 3602 ( 908 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 16022 ( 388 ~; 54 |; 250 &;13453 @)
% ( 0 <=>;1877 =>; 0 <=; 0 <~>)
% Maximal formula depth : 22 ( 8 avg)
% Number of types : 24 ( 23 usr)
% Number of type conns : 491 ( 491 >; 0 *; 0 +; 0 <<)
% Number of symbols : 190 ( 189 usr; 14 con; 0-5 aty)
% Number of variables : 4239 ( 157 ^;3922 !; 160 ?;4239 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 14:19:15.658
%------------------------------------------------------------------------------
% Could-be-implicit typings (23)
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thf(sy_c_Wellfounded_Owf_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
wf_Pro2335863617654816626rm_a_b: set_Pr2972776593051762503rm_a_b > $o ).
thf(sy_c_Wellfounded_Owf_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
wf_term_a_b: set_Pr4386577575007340137rm_a_b > $o ).
thf(sy_c_Wfrec_Osame__fst_001t__Term__Oterm_Itf__a_Mtf__b_J_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
same_f843731131676418198rm_a_b: ( term_a_b > $o ) > ( term_a_b > set_Pr4386577575007340137rm_a_b ) > set_Pr2972776593051762503rm_a_b ).
thf(sy_c_Zorn_OChains_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
chains_term_a_b: set_Pr4386577575007340137rm_a_b > set_set_term_a_b ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_Mt__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_Mt__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J_J,type,
member5793113912742238992rm_a_b: produc2957090337049255143rm_a_b > set_Pr6848273517546626247rm_a_b > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J_J_Mt__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J_J_J,type,
member2018982355343301392rm_a_b: produc8791377122869980391rm_a_b > set_Pr4284312964791606471rm_a_b > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Set__Oset_It__Term__Oterm_Itf__a_Mtf__b_J_J_Mt__Set__Oset_It__Term__Oterm_Itf__a_Mtf__b_J_J_J_Mt__Product____Type__Oprod_It__Set__Oset_It__Term__Oterm_Itf__a_Mtf__b_J_J_Mt__Set__Oset_It__Term__Oterm_Itf__a_Mtf__b_J_J_J_J,type,
member1016977296685850768rm_a_b: produc3601453998762642791rm_a_b > set_Pr7061646551094443591rm_a_b > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_Mt__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
member8417600551952982416rm_a_b: produc4523971788519308903rm_a_b > set_Pr2972776593051762503rm_a_b > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J_J,type,
member1375104729371992464rm_a_b: produc3421996462152255079rm_a_b > set_Pr647378613824986951rm_a_b > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Term__Oterm_Itf__a_Mtf__b_J_J_Mt__Set__Oset_It__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
member8888773126115863070rm_a_b: produc9001763001631046773rm_a_b > set_Pr7151917219565317845rm_a_b > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
member5869715511025134514rm_a_b: produc357393685978478089rm_a_b > set_Pr4386577575007340137rm_a_b > $o ).
thf(sy_c_member_001t__Set__Oset_It__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
member8823642877475723282rm_a_b: set_Pr4386577575007340137rm_a_b > set_se1467346398288745161rm_a_b > $o ).
thf(sy_c_member_001t__Set__Oset_It__Term__Oterm_Itf__a_Mtf__b_J_J,type,
member_set_term_a_b: set_term_a_b > set_set_term_a_b > $o ).
thf(sy_c_member_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
member_term_a_b: term_a_b > set_term_a_b > $o ).
thf(sy_v__092_060F_062,type,
f: set_Pr4934435412358123699_a_nat ).
thf(sy_v__092_060R_062,type,
r: set_Pr4386577575007340137rm_a_b ).
thf(sy_v_s____,type,
s: term_a_b ).
thf(sy_v_t____,type,
t: term_a_b ).
thf(sy_v_u____,type,
u: term_a_b ).
% Relevant facts (1274)
thf(fact_0_False,axiom,
s != t ).
% False
thf(fact_1_a_I2_J,axiom,
member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ t @ u ) @ ( srstep7844470518422762656ep_a_b @ f @ r ) ).
% a(2)
thf(fact_2__092_060open_062_Is_M_At_J_A_092_060in_062_A_Isrstep_A_092_060F_062_A_I_092_060R_062_092_060inverse_062_J_J_092_060_094sup_062_L_092_060close_062,axiom,
member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ s @ t ) @ ( transi7922773638565587891rm_a_b @ ( sig_step_a_b @ f @ ( rstep_a_b @ ( conver7608559976937845093rm_a_b @ r ) ) ) ) ).
% \<open>(s, t) \<in> (srstep \<F> (\<R>\<inverse>))\<^sup>+\<close>
thf(fact_3_assms,axiom,
! [S: term_a_b,T: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( relcom370159955682700863rm_a_b @ ( transi7922773638565587891rm_a_b @ ( sig_step_a_b @ f @ ( rstep_a_b @ ( conver7608559976937845093rm_a_b @ r ) ) ) ) @ ( srstep7844470518422762656ep_a_b @ f @ r ) ) )
=> ( rewrit8865997592796547114dp_a_b @ f @ r @ r @ S @ T ) ) ).
% assms
thf(fact_4__092_060open_062_Is_M_Au_J_A_092_060in_062_A_Isrstep_A_092_060F_062_A_I_092_060R_062_092_060inverse_062_J_J_092_060_094sup_062_K_AO_Asrsteps__with__root__step_A_092_060F_062_A_092_060R_062_092_060close_062,axiom,
member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ s @ u ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ f @ ( rstep_a_b @ ( conver7608559976937845093rm_a_b @ r ) ) ) ) @ ( srstep7844470518422762656ep_a_b @ f @ r ) ) ).
% \<open>(s, u) \<in> (srstep \<F> (\<R>\<inverse>))\<^sup>* O srsteps_with_root_step \<F> \<R>\<close>
thf(fact_5_a_I1_J,axiom,
member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ s @ t ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ f @ ( rstep_a_b @ ( conver7608559976937845093rm_a_b @ r ) ) ) ) ).
% a(1)
thf(fact_6__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062t_O_A_092_060lbrakk_062_Is_M_At_J_A_092_060in_062_A_Isrstep_A_092_060F_062_A_I_092_060R_062_092_060inverse_062_J_J_092_060_094sup_062_K_059_A_It_M_Au_J_A_092_060in_062_Asrsteps__with__root__step_A_092_060F_062_A_092_060R_062_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [T2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ s @ T2 ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ f @ ( rstep_a_b @ ( conver7608559976937845093rm_a_b @ r ) ) ) ) )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T2 @ u ) @ ( srstep7844470518422762656ep_a_b @ f @ r ) ) ) ).
% \<open>\<And>thesis. (\<And>t. \<lbrakk>(s, t) \<in> (srstep \<F> (\<R>\<inverse>))\<^sup>*; (t, u) \<in> srsteps_with_root_step \<F> \<R>\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_7_commute__redp__def,axiom,
( rewrit8865997592796547114dp_a_b
= ( ^ [F: set_Pr4934435412358123699_a_nat,R: set_Pr4386577575007340137rm_a_b,S2: set_Pr4386577575007340137rm_a_b,S3: term_a_b,T3: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S3 @ T3 ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ S2 ) ) ) @ ( transi7742714808557438673rm_a_b @ ( conver7608559976937845093rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R ) ) ) ) ) ) ) ) ).
% commute_redp_def
thf(fact_8_commute__rrstep__intro,axiom,
! [F2: set_Pr4934435412358123699_a_nat,R2: set_Pr4386577575007340137rm_a_b,S4: set_Pr4386577575007340137rm_a_b] :
( ! [S5: term_a_b,T2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S5 @ T2 ) @ ( sup_su6776935440552674877rm_a_b @ ( relcom370159955682700863rm_a_b @ ( srstep7844470518422762656ep_a_b @ F2 @ ( conver7608559976937845093rm_a_b @ R2 ) ) @ ( transi7922773638565587891rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ S4 ) ) ) ) @ ( relcom370159955682700863rm_a_b @ ( transi7922773638565587891rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ ( conver7608559976937845093rm_a_b @ R2 ) ) ) ) @ ( srstep7844470518422762656ep_a_b @ F2 @ S4 ) ) ) )
=> ( rewrit8865997592796547114dp_a_b @ F2 @ R2 @ S4 @ S5 @ T2 ) )
=> ( abstra455195320138238452rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) ) @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ S4 ) ) ) ) ).
% commute_rrstep_intro
thf(fact_9_CR__Aux,axiom,
! [F2: set_Pr4934435412358123699_a_nat,R2: set_Pr4386577575007340137rm_a_b,S: term_a_b,T: term_a_b] :
( ! [S5: term_a_b,T2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S5 @ T2 ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ ( conver7608559976937845093rm_a_b @ R2 ) ) ) ) @ ( srstep7844470518422762656ep_a_b @ F2 @ R2 ) ) )
=> ( rewrit8865997592796547114dp_a_b @ F2 @ R2 @ R2 @ S5 @ T2 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( sup_su6776935440552674877rm_a_b @ ( relcom370159955682700863rm_a_b @ ( srstep7844470518422762656ep_a_b @ F2 @ ( conver7608559976937845093rm_a_b @ R2 ) ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) ) ) ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ ( conver7608559976937845093rm_a_b @ R2 ) ) ) ) @ ( srstep7844470518422762656ep_a_b @ F2 @ R2 ) ) ) )
=> ( rewrit8865997592796547114dp_a_b @ F2 @ R2 @ R2 @ S @ T ) ) ) ).
% CR_Aux
thf(fact_10_commute__to__rrstep,axiom,
! [F2: set_Pr4934435412358123699_a_nat,R2: set_Pr4386577575007340137rm_a_b,S4: set_Pr4386577575007340137rm_a_b,S: term_a_b,T: term_a_b] :
( ( abstra455195320138238452rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) ) @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ S4 ) ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( sup_su6776935440552674877rm_a_b @ ( relcom370159955682700863rm_a_b @ ( srstep7844470518422762656ep_a_b @ F2 @ ( conver7608559976937845093rm_a_b @ R2 ) ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ S4 ) ) ) ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ ( conver7608559976937845093rm_a_b @ R2 ) ) ) ) @ ( srstep7844470518422762656ep_a_b @ F2 @ S4 ) ) ) )
=> ( rewrit8865997592796547114dp_a_b @ F2 @ R2 @ S4 @ S @ T ) ) ) ).
% commute_to_rrstep
thf(fact_11_srsteps__with__root__step__srstepsD,axiom,
! [S: term_a_b,T: term_a_b,F2: set_Pr4934435412358123699_a_nat,R2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( srstep7844470518422762656ep_a_b @ F2 @ R2 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( transi7922773638565587891rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) ) ) ) ) ).
% srsteps_with_root_step_srstepsD
thf(fact_12_srsteps__with__root__step__sresteps__eqD,axiom,
! [S: term_a_b,T: term_a_b,F2: set_Pr4934435412358123699_a_nat,R2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( srstep7844470518422762656ep_a_b @ F2 @ R2 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) ) ) ) ) ).
% srsteps_with_root_step_sresteps_eqD
thf(fact_13_symcl__idemp,axiom,
! [R3: set_Pr2972776593051762503rm_a_b] :
( ( sup_su1845815533690797339rm_a_b @ ( sup_su1845815533690797339rm_a_b @ R3 @ ( conver1575690562132235771rm_a_b @ R3 ) ) @ ( conver1575690562132235771rm_a_b @ ( sup_su1845815533690797339rm_a_b @ R3 @ ( conver1575690562132235771rm_a_b @ R3 ) ) ) )
= ( sup_su1845815533690797339rm_a_b @ R3 @ ( conver1575690562132235771rm_a_b @ R3 ) ) ) ).
% symcl_idemp
thf(fact_14_symcl__idemp,axiom,
! [R3: set_Pr7151917219565317845rm_a_b] :
( ( sup_su1246761269818691241rm_a_b @ ( sup_su1246761269818691241rm_a_b @ R3 @ ( conver1904685531124697809rm_a_b @ R3 ) ) @ ( conver1904685531124697809rm_a_b @ ( sup_su1246761269818691241rm_a_b @ R3 @ ( conver1904685531124697809rm_a_b @ R3 ) ) ) )
= ( sup_su1246761269818691241rm_a_b @ R3 @ ( conver1904685531124697809rm_a_b @ R3 ) ) ) ).
% symcl_idemp
thf(fact_15_symcl__idemp,axiom,
! [R3: set_Pr4386577575007340137rm_a_b] :
( ( sup_su6776935440552674877rm_a_b @ ( sup_su6776935440552674877rm_a_b @ R3 @ ( conver7608559976937845093rm_a_b @ R3 ) ) @ ( conver7608559976937845093rm_a_b @ ( sup_su6776935440552674877rm_a_b @ R3 @ ( conver7608559976937845093rm_a_b @ R3 ) ) ) )
= ( sup_su6776935440552674877rm_a_b @ R3 @ ( conver7608559976937845093rm_a_b @ R3 ) ) ) ).
% symcl_idemp
thf(fact_16_relcomp__distrib,axiom,
! [R4: set_Pr2972776593051762503rm_a_b,S6: set_Pr2972776593051762503rm_a_b,T4: set_Pr2972776593051762503rm_a_b] :
( ( relcom6159724393391289352rm_a_b @ R4 @ ( sup_su1845815533690797339rm_a_b @ S6 @ T4 ) )
= ( sup_su1845815533690797339rm_a_b @ ( relcom6159724393391289352rm_a_b @ R4 @ S6 ) @ ( relcom6159724393391289352rm_a_b @ R4 @ T4 ) ) ) ).
% relcomp_distrib
thf(fact_17_relcomp__distrib,axiom,
! [R4: set_Pr7151917219565317845rm_a_b,S6: set_Pr7151917219565317845rm_a_b,T4: set_Pr7151917219565317845rm_a_b] :
( ( relcom3320975603336820193rm_a_b @ R4 @ ( sup_su1246761269818691241rm_a_b @ S6 @ T4 ) )
= ( sup_su1246761269818691241rm_a_b @ ( relcom3320975603336820193rm_a_b @ R4 @ S6 ) @ ( relcom3320975603336820193rm_a_b @ R4 @ T4 ) ) ) ).
% relcomp_distrib
thf(fact_18_relcomp__distrib,axiom,
! [R4: set_Pr4386577575007340137rm_a_b,S6: set_Pr4386577575007340137rm_a_b,T4: set_Pr4386577575007340137rm_a_b] :
( ( relcom370159955682700863rm_a_b @ R4 @ ( sup_su6776935440552674877rm_a_b @ S6 @ T4 ) )
= ( sup_su6776935440552674877rm_a_b @ ( relcom370159955682700863rm_a_b @ R4 @ S6 ) @ ( relcom370159955682700863rm_a_b @ R4 @ T4 ) ) ) ).
% relcomp_distrib
thf(fact_19_relcomp__distrib2,axiom,
! [S6: set_Pr2972776593051762503rm_a_b,T4: set_Pr2972776593051762503rm_a_b,R4: set_Pr2972776593051762503rm_a_b] :
( ( relcom6159724393391289352rm_a_b @ ( sup_su1845815533690797339rm_a_b @ S6 @ T4 ) @ R4 )
= ( sup_su1845815533690797339rm_a_b @ ( relcom6159724393391289352rm_a_b @ S6 @ R4 ) @ ( relcom6159724393391289352rm_a_b @ T4 @ R4 ) ) ) ).
% relcomp_distrib2
thf(fact_20_relcomp__distrib2,axiom,
! [S6: set_Pr7151917219565317845rm_a_b,T4: set_Pr7151917219565317845rm_a_b,R4: set_Pr7151917219565317845rm_a_b] :
( ( relcom3320975603336820193rm_a_b @ ( sup_su1246761269818691241rm_a_b @ S6 @ T4 ) @ R4 )
= ( sup_su1246761269818691241rm_a_b @ ( relcom3320975603336820193rm_a_b @ S6 @ R4 ) @ ( relcom3320975603336820193rm_a_b @ T4 @ R4 ) ) ) ).
% relcomp_distrib2
thf(fact_21_relcomp__distrib2,axiom,
! [S6: set_Pr4386577575007340137rm_a_b,T4: set_Pr4386577575007340137rm_a_b,R4: set_Pr4386577575007340137rm_a_b] :
( ( relcom370159955682700863rm_a_b @ ( sup_su6776935440552674877rm_a_b @ S6 @ T4 ) @ R4 )
= ( sup_su6776935440552674877rm_a_b @ ( relcom370159955682700863rm_a_b @ S6 @ R4 ) @ ( relcom370159955682700863rm_a_b @ T4 @ R4 ) ) ) ).
% relcomp_distrib2
thf(fact_22_rtrancl__trancl__absorb,axiom,
! [R4: set_Pr2972776593051762503rm_a_b] :
( ( transi9211502839322181930rm_a_b @ ( transi2615809358984392588rm_a_b @ R4 ) )
= ( transi2615809358984392588rm_a_b @ R4 ) ) ).
% rtrancl_trancl_absorb
thf(fact_23_rtrancl__trancl__absorb,axiom,
! [R4: set_Pr4386577575007340137rm_a_b] :
( ( transi7922773638565587891rm_a_b @ ( transi7742714808557438673rm_a_b @ R4 ) )
= ( transi7742714808557438673rm_a_b @ R4 ) ) ).
% rtrancl_trancl_absorb
thf(fact_24_trancl__rtrancl__absorb,axiom,
! [R4: set_Pr2972776593051762503rm_a_b] :
( ( transi2615809358984392588rm_a_b @ ( transi9211502839322181930rm_a_b @ R4 ) )
= ( transi2615809358984392588rm_a_b @ R4 ) ) ).
% trancl_rtrancl_absorb
thf(fact_25_trancl__rtrancl__absorb,axiom,
! [R4: set_Pr4386577575007340137rm_a_b] :
( ( transi7742714808557438673rm_a_b @ ( transi7922773638565587891rm_a_b @ R4 ) )
= ( transi7742714808557438673rm_a_b @ R4 ) ) ).
% trancl_rtrancl_absorb
thf(fact_26_rstep__ruleI,axiom,
! [L: term_a_b,R3: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ L @ R3 ) @ R2 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ L @ R3 ) @ ( rstep_a_b @ R2 ) ) ) ).
% rstep_ruleI
thf(fact_27_rtrancl__idemp__self__comp,axiom,
! [R4: set_Pr7151917219565317845rm_a_b] :
( ( relcom3320975603336820193rm_a_b @ ( transi6904761420441344775rm_a_b @ R4 ) @ ( transi6904761420441344775rm_a_b @ R4 ) )
= ( transi6904761420441344775rm_a_b @ R4 ) ) ).
% rtrancl_idemp_self_comp
thf(fact_28_rtrancl__idemp__self__comp,axiom,
! [R4: set_Pr2972776593051762503rm_a_b] :
( ( relcom6159724393391289352rm_a_b @ ( transi2615809358984392588rm_a_b @ R4 ) @ ( transi2615809358984392588rm_a_b @ R4 ) )
= ( transi2615809358984392588rm_a_b @ R4 ) ) ).
% rtrancl_idemp_self_comp
thf(fact_29_rtrancl__idemp__self__comp,axiom,
! [R4: set_Pr4386577575007340137rm_a_b] :
( ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ R4 ) @ ( transi7742714808557438673rm_a_b @ R4 ) )
= ( transi7742714808557438673rm_a_b @ R4 ) ) ).
% rtrancl_idemp_self_comp
thf(fact_30_converse__iff,axiom,
! [A: set_term_a_b,B: set_term_a_b,R3: set_Pr7151917219565317845rm_a_b] :
( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A @ B ) @ ( conver1904685531124697809rm_a_b @ R3 ) )
= ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ B @ A ) @ R3 ) ) ).
% converse_iff
thf(fact_31_converse__iff,axiom,
! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b,R3: set_Pr647378613824986951rm_a_b] :
( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A @ B ) @ ( conver7531182934069937659rm_a_b @ R3 ) )
= ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ B @ A ) @ R3 ) ) ).
% converse_iff
thf(fact_32_converse__iff,axiom,
! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,R3: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( conver1575690562132235771rm_a_b @ R3 ) )
= ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B @ A ) @ R3 ) ) ).
% converse_iff
thf(fact_33_converse__iff,axiom,
! [A: term_a_b,B: term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( conver7608559976937845093rm_a_b @ R3 ) )
= ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ R3 ) ) ).
% converse_iff
thf(fact_34_srstep__symcl__dist,axiom,
! [F2: set_Pr4934435412358123699_a_nat,R2: set_Pr4386577575007340137rm_a_b] :
( ( sup_su6776935440552674877rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) ) @ ( conver7608559976937845093rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) ) ) )
= ( sig_step_a_b @ F2 @ ( rstep_a_b @ ( sup_su6776935440552674877rm_a_b @ R2 @ ( conver7608559976937845093rm_a_b @ R2 ) ) ) ) ) ).
% srstep_symcl_dist
thf(fact_35_firstStep,axiom,
! [L2: set_Pr647378613824986951rm_a_b,S6: set_Pr647378613824986951rm_a_b,R4: set_Pr647378613824986951rm_a_b,X: set_Pr4386577575007340137rm_a_b,Y: set_Pr4386577575007340137rm_a_b] :
( ( L2
= ( sup_su8433770403025779995rm_a_b @ S6 @ R4 ) )
=> ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ X @ Y ) @ ( transi8812752159401800684rm_a_b @ L2 ) )
=> ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ X @ Y ) @ ( transi8812752159401800684rm_a_b @ R4 ) )
| ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ X @ Y ) @ ( relcom8520403917366345832rm_a_b @ ( transi8812752159401800684rm_a_b @ R4 ) @ ( relcom8520403917366345832rm_a_b @ S6 @ ( transi8812752159401800684rm_a_b @ L2 ) ) ) ) ) ) ) ).
% firstStep
thf(fact_36_firstStep,axiom,
! [L2: set_Pr7151917219565317845rm_a_b,S6: set_Pr7151917219565317845rm_a_b,R4: set_Pr7151917219565317845rm_a_b,X: set_term_a_b,Y: set_term_a_b] :
( ( L2
= ( sup_su1246761269818691241rm_a_b @ S6 @ R4 ) )
=> ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ X @ Y ) @ ( transi6904761420441344775rm_a_b @ L2 ) )
=> ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ X @ Y ) @ ( transi6904761420441344775rm_a_b @ R4 ) )
| ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ X @ Y ) @ ( relcom3320975603336820193rm_a_b @ ( transi6904761420441344775rm_a_b @ R4 ) @ ( relcom3320975603336820193rm_a_b @ S6 @ ( transi6904761420441344775rm_a_b @ L2 ) ) ) ) ) ) ) ).
% firstStep
thf(fact_37_firstStep,axiom,
! [L2: set_Pr2972776593051762503rm_a_b,S6: set_Pr2972776593051762503rm_a_b,R4: set_Pr2972776593051762503rm_a_b,X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b] :
( ( L2
= ( sup_su1845815533690797339rm_a_b @ S6 @ R4 ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ ( transi2615809358984392588rm_a_b @ L2 ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ ( transi2615809358984392588rm_a_b @ R4 ) )
| ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ ( relcom6159724393391289352rm_a_b @ ( transi2615809358984392588rm_a_b @ R4 ) @ ( relcom6159724393391289352rm_a_b @ S6 @ ( transi2615809358984392588rm_a_b @ L2 ) ) ) ) ) ) ) ).
% firstStep
thf(fact_38_firstStep,axiom,
! [L2: set_Pr4386577575007340137rm_a_b,S6: set_Pr4386577575007340137rm_a_b,R4: set_Pr4386577575007340137rm_a_b,X: term_a_b,Y: term_a_b] :
( ( L2
= ( sup_su6776935440552674877rm_a_b @ S6 @ R4 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ L2 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ R4 ) )
| ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ R4 ) @ ( relcom370159955682700863rm_a_b @ S6 @ ( transi7742714808557438673rm_a_b @ L2 ) ) ) ) ) ) ) ).
% firstStep
thf(fact_39_first__step,axiom,
! [C: set_Pr647378613824986951rm_a_b,A2: set_Pr647378613824986951rm_a_b,B2: set_Pr647378613824986951rm_a_b,X: set_Pr4386577575007340137rm_a_b,Y: set_Pr4386577575007340137rm_a_b,Z: set_Pr4386577575007340137rm_a_b] :
( ( C
= ( sup_su8433770403025779995rm_a_b @ A2 @ B2 ) )
=> ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ X @ Y ) @ ( transi8812752159401800684rm_a_b @ C ) )
=> ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ Y @ Z ) @ B2 )
=> ? [Y2: set_Pr4386577575007340137rm_a_b] : ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ X @ Y2 ) @ ( relcom8520403917366345832rm_a_b @ ( transi8812752159401800684rm_a_b @ A2 ) @ B2 ) ) ) ) ) ).
% first_step
thf(fact_40_first__step,axiom,
! [C: set_Pr7151917219565317845rm_a_b,A2: set_Pr7151917219565317845rm_a_b,B2: set_Pr7151917219565317845rm_a_b,X: set_term_a_b,Y: set_term_a_b,Z: set_term_a_b] :
( ( C
= ( sup_su1246761269818691241rm_a_b @ A2 @ B2 ) )
=> ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ X @ Y ) @ ( transi6904761420441344775rm_a_b @ C ) )
=> ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ Y @ Z ) @ B2 )
=> ? [Y2: set_term_a_b] : ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ X @ Y2 ) @ ( relcom3320975603336820193rm_a_b @ ( transi6904761420441344775rm_a_b @ A2 ) @ B2 ) ) ) ) ) ).
% first_step
thf(fact_41_first__step,axiom,
! [C: set_Pr2972776593051762503rm_a_b,A2: set_Pr2972776593051762503rm_a_b,B2: set_Pr2972776593051762503rm_a_b,X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b,Z: produc357393685978478089rm_a_b] :
( ( C
= ( sup_su1845815533690797339rm_a_b @ A2 @ B2 ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ ( transi2615809358984392588rm_a_b @ C ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y @ Z ) @ B2 )
=> ? [Y2: produc357393685978478089rm_a_b] : ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y2 ) @ ( relcom6159724393391289352rm_a_b @ ( transi2615809358984392588rm_a_b @ A2 ) @ B2 ) ) ) ) ) ).
% first_step
thf(fact_42_first__step,axiom,
! [C: set_Pr4386577575007340137rm_a_b,A2: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b,X: term_a_b,Y: term_a_b,Z: term_a_b] :
( ( C
= ( sup_su6776935440552674877rm_a_b @ A2 @ B2 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ C ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ Z ) @ B2 )
=> ? [Y2: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y2 ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ A2 ) @ B2 ) ) ) ) ) ).
% first_step
thf(fact_43_first__step__O,axiom,
! [C: set_Pr647378613824986951rm_a_b,A2: set_Pr647378613824986951rm_a_b,B2: set_Pr647378613824986951rm_a_b,X: set_Pr4386577575007340137rm_a_b,Y: set_Pr4386577575007340137rm_a_b] :
( ( C
= ( sup_su8433770403025779995rm_a_b @ A2 @ B2 ) )
=> ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ X @ Y ) @ ( relcom8520403917366345832rm_a_b @ ( transi8812752159401800684rm_a_b @ C ) @ B2 ) )
=> ? [Y2: set_Pr4386577575007340137rm_a_b] : ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ X @ Y2 ) @ ( relcom8520403917366345832rm_a_b @ ( transi8812752159401800684rm_a_b @ A2 ) @ B2 ) ) ) ) ).
% first_step_O
thf(fact_44_first__step__O,axiom,
! [C: set_Pr7151917219565317845rm_a_b,A2: set_Pr7151917219565317845rm_a_b,B2: set_Pr7151917219565317845rm_a_b,X: set_term_a_b,Y: set_term_a_b] :
( ( C
= ( sup_su1246761269818691241rm_a_b @ A2 @ B2 ) )
=> ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ X @ Y ) @ ( relcom3320975603336820193rm_a_b @ ( transi6904761420441344775rm_a_b @ C ) @ B2 ) )
=> ? [Y2: set_term_a_b] : ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ X @ Y2 ) @ ( relcom3320975603336820193rm_a_b @ ( transi6904761420441344775rm_a_b @ A2 ) @ B2 ) ) ) ) ).
% first_step_O
thf(fact_45_first__step__O,axiom,
! [C: set_Pr2972776593051762503rm_a_b,A2: set_Pr2972776593051762503rm_a_b,B2: set_Pr2972776593051762503rm_a_b,X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b] :
( ( C
= ( sup_su1845815533690797339rm_a_b @ A2 @ B2 ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ ( relcom6159724393391289352rm_a_b @ ( transi2615809358984392588rm_a_b @ C ) @ B2 ) )
=> ? [Y2: produc357393685978478089rm_a_b] : ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y2 ) @ ( relcom6159724393391289352rm_a_b @ ( transi2615809358984392588rm_a_b @ A2 ) @ B2 ) ) ) ) ).
% first_step_O
thf(fact_46_first__step__O,axiom,
! [C: set_Pr4386577575007340137rm_a_b,A2: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b,X: term_a_b,Y: term_a_b] :
( ( C
= ( sup_su6776935440552674877rm_a_b @ A2 @ B2 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ C ) @ B2 ) )
=> ? [Y2: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y2 ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ A2 ) @ B2 ) ) ) ) ).
% first_step_O
thf(fact_47_r__into__rtrancl,axiom,
! [P: produc3421996462152255079rm_a_b,R3: set_Pr647378613824986951rm_a_b] :
( ( member1375104729371992464rm_a_b @ P @ R3 )
=> ( member1375104729371992464rm_a_b @ P @ ( transi8812752159401800684rm_a_b @ R3 ) ) ) ).
% r_into_rtrancl
thf(fact_48_r__into__rtrancl,axiom,
! [P: produc4523971788519308903rm_a_b,R3: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ P @ R3 )
=> ( member8417600551952982416rm_a_b @ P @ ( transi2615809358984392588rm_a_b @ R3 ) ) ) ).
% r_into_rtrancl
thf(fact_49_r__into__rtrancl,axiom,
! [P: produc357393685978478089rm_a_b,R3: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ P @ R3 )
=> ( member5869715511025134514rm_a_b @ P @ ( transi7742714808557438673rm_a_b @ R3 ) ) ) ).
% r_into_rtrancl
thf(fact_50_rtrancl__idemp,axiom,
! [R3: set_Pr2972776593051762503rm_a_b] :
( ( transi2615809358984392588rm_a_b @ ( transi2615809358984392588rm_a_b @ R3 ) )
= ( transi2615809358984392588rm_a_b @ R3 ) ) ).
% rtrancl_idemp
thf(fact_51_rtrancl__idemp,axiom,
! [R3: set_Pr4386577575007340137rm_a_b] :
( ( transi7742714808557438673rm_a_b @ ( transi7742714808557438673rm_a_b @ R3 ) )
= ( transi7742714808557438673rm_a_b @ R3 ) ) ).
% rtrancl_idemp
thf(fact_52_converse__converse,axiom,
! [R3: set_Pr2972776593051762503rm_a_b] :
( ( conver1575690562132235771rm_a_b @ ( conver1575690562132235771rm_a_b @ R3 ) )
= R3 ) ).
% converse_converse
thf(fact_53_converse__converse,axiom,
! [R3: set_Pr4386577575007340137rm_a_b] :
( ( conver7608559976937845093rm_a_b @ ( conver7608559976937845093rm_a_b @ R3 ) )
= R3 ) ).
% converse_converse
thf(fact_54_converse__inject,axiom,
! [R3: set_Pr2972776593051762503rm_a_b,S: set_Pr2972776593051762503rm_a_b] :
( ( ( conver1575690562132235771rm_a_b @ R3 )
= ( conver1575690562132235771rm_a_b @ S ) )
= ( R3 = S ) ) ).
% converse_inject
thf(fact_55_converse__inject,axiom,
! [R3: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
( ( ( conver7608559976937845093rm_a_b @ R3 )
= ( conver7608559976937845093rm_a_b @ S ) )
= ( R3 = S ) ) ).
% converse_inject
thf(fact_56_trancl__induct2,axiom,
! [Ax: produc357393685978478089rm_a_b,Ay: produc357393685978478089rm_a_b,Bx: produc357393685978478089rm_a_b,By: produc357393685978478089rm_a_b,R3: set_Pr6848273517546626247rm_a_b,P2: produc357393685978478089rm_a_b > produc357393685978478089rm_a_b > $o] :
( ( member5793113912742238992rm_a_b @ ( produc2499998789532207959rm_a_b @ ( produc1763473618796451543rm_a_b @ Ax @ Ay ) @ ( produc1763473618796451543rm_a_b @ Bx @ By ) ) @ ( transi4560671716973800200rm_a_b @ R3 ) )
=> ( ! [A3: produc357393685978478089rm_a_b,B3: produc357393685978478089rm_a_b] :
( ( member5793113912742238992rm_a_b @ ( produc2499998789532207959rm_a_b @ ( produc1763473618796451543rm_a_b @ Ax @ Ay ) @ ( produc1763473618796451543rm_a_b @ A3 @ B3 ) ) @ R3 )
=> ( P2 @ A3 @ B3 ) )
=> ( ! [A3: produc357393685978478089rm_a_b,B3: produc357393685978478089rm_a_b,Aa: produc357393685978478089rm_a_b,Ba: produc357393685978478089rm_a_b] :
( ( member5793113912742238992rm_a_b @ ( produc2499998789532207959rm_a_b @ ( produc1763473618796451543rm_a_b @ Ax @ Ay ) @ ( produc1763473618796451543rm_a_b @ A3 @ B3 ) ) @ ( transi4560671716973800200rm_a_b @ R3 ) )
=> ( ( member5793113912742238992rm_a_b @ ( produc2499998789532207959rm_a_b @ ( produc1763473618796451543rm_a_b @ A3 @ B3 ) @ ( produc1763473618796451543rm_a_b @ Aa @ Ba ) ) @ R3 )
=> ( ( P2 @ A3 @ B3 )
=> ( P2 @ Aa @ Ba ) ) ) )
=> ( P2 @ Bx @ By ) ) ) ) ).
% trancl_induct2
thf(fact_57_trancl__induct2,axiom,
! [Ax: set_term_a_b,Ay: set_term_a_b,Bx: set_term_a_b,By: set_term_a_b,R3: set_Pr7061646551094443591rm_a_b,P2: set_term_a_b > set_term_a_b > $o] :
( ( member1016977296685850768rm_a_b @ ( produc1448243995722353623rm_a_b @ ( produc8517751122659631533rm_a_b @ Ax @ Ay ) @ ( produc8517751122659631533rm_a_b @ Bx @ By ) ) @ ( transi6900661278278483862rm_a_b @ R3 ) )
=> ( ! [A3: set_term_a_b,B3: set_term_a_b] :
( ( member1016977296685850768rm_a_b @ ( produc1448243995722353623rm_a_b @ ( produc8517751122659631533rm_a_b @ Ax @ Ay ) @ ( produc8517751122659631533rm_a_b @ A3 @ B3 ) ) @ R3 )
=> ( P2 @ A3 @ B3 ) )
=> ( ! [A3: set_term_a_b,B3: set_term_a_b,Aa: set_term_a_b,Ba: set_term_a_b] :
( ( member1016977296685850768rm_a_b @ ( produc1448243995722353623rm_a_b @ ( produc8517751122659631533rm_a_b @ Ax @ Ay ) @ ( produc8517751122659631533rm_a_b @ A3 @ B3 ) ) @ ( transi6900661278278483862rm_a_b @ R3 ) )
=> ( ( member1016977296685850768rm_a_b @ ( produc1448243995722353623rm_a_b @ ( produc8517751122659631533rm_a_b @ A3 @ B3 ) @ ( produc8517751122659631533rm_a_b @ Aa @ Ba ) ) @ R3 )
=> ( ( P2 @ A3 @ B3 )
=> ( P2 @ Aa @ Ba ) ) ) )
=> ( P2 @ Bx @ By ) ) ) ) ).
% trancl_induct2
thf(fact_58_trancl__induct2,axiom,
! [Ax: set_Pr4386577575007340137rm_a_b,Ay: set_Pr4386577575007340137rm_a_b,Bx: set_Pr4386577575007340137rm_a_b,By: set_Pr4386577575007340137rm_a_b,R3: set_Pr4284312964791606471rm_a_b,P2: set_Pr4386577575007340137rm_a_b > set_Pr4386577575007340137rm_a_b > $o] :
( ( member2018982355343301392rm_a_b @ ( produc765457364750699351rm_a_b @ ( produc3977138704197412567rm_a_b @ Ax @ Ay ) @ ( produc3977138704197412567rm_a_b @ Bx @ By ) ) @ ( transi7714379717876673288rm_a_b @ R3 ) )
=> ( ! [A3: set_Pr4386577575007340137rm_a_b,B3: set_Pr4386577575007340137rm_a_b] :
( ( member2018982355343301392rm_a_b @ ( produc765457364750699351rm_a_b @ ( produc3977138704197412567rm_a_b @ Ax @ Ay ) @ ( produc3977138704197412567rm_a_b @ A3 @ B3 ) ) @ R3 )
=> ( P2 @ A3 @ B3 ) )
=> ( ! [A3: set_Pr4386577575007340137rm_a_b,B3: set_Pr4386577575007340137rm_a_b,Aa: set_Pr4386577575007340137rm_a_b,Ba: set_Pr4386577575007340137rm_a_b] :
( ( member2018982355343301392rm_a_b @ ( produc765457364750699351rm_a_b @ ( produc3977138704197412567rm_a_b @ Ax @ Ay ) @ ( produc3977138704197412567rm_a_b @ A3 @ B3 ) ) @ ( transi7714379717876673288rm_a_b @ R3 ) )
=> ( ( member2018982355343301392rm_a_b @ ( produc765457364750699351rm_a_b @ ( produc3977138704197412567rm_a_b @ A3 @ B3 ) @ ( produc3977138704197412567rm_a_b @ Aa @ Ba ) ) @ R3 )
=> ( ( P2 @ A3 @ B3 )
=> ( P2 @ Aa @ Ba ) ) ) )
=> ( P2 @ Bx @ By ) ) ) ) ).
% trancl_induct2
thf(fact_59_trancl__induct2,axiom,
! [Ax: term_a_b,Ay: term_a_b,Bx: term_a_b,By: term_a_b,R3: set_Pr2972776593051762503rm_a_b,P2: term_a_b > term_a_b > $o] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ Ax @ Ay ) @ ( produc7020197800436672577rm_a_b @ Bx @ By ) ) @ ( transi9211502839322181930rm_a_b @ R3 ) )
=> ( ! [A3: term_a_b,B3: term_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ Ax @ Ay ) @ ( produc7020197800436672577rm_a_b @ A3 @ B3 ) ) @ R3 )
=> ( P2 @ A3 @ B3 ) )
=> ( ! [A3: term_a_b,B3: term_a_b,Aa: term_a_b,Ba: term_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ Ax @ Ay ) @ ( produc7020197800436672577rm_a_b @ A3 @ B3 ) ) @ ( transi9211502839322181930rm_a_b @ R3 ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B3 ) @ ( produc7020197800436672577rm_a_b @ Aa @ Ba ) ) @ R3 )
=> ( ( P2 @ A3 @ B3 )
=> ( P2 @ Aa @ Ba ) ) ) )
=> ( P2 @ Bx @ By ) ) ) ) ).
% trancl_induct2
thf(fact_60_converse__rtrancl__induct2,axiom,
! [Ax: produc357393685978478089rm_a_b,Ay: produc357393685978478089rm_a_b,Bx: produc357393685978478089rm_a_b,By: produc357393685978478089rm_a_b,R3: set_Pr6848273517546626247rm_a_b,P2: produc357393685978478089rm_a_b > produc357393685978478089rm_a_b > $o] :
( ( member5793113912742238992rm_a_b @ ( produc2499998789532207959rm_a_b @ ( produc1763473618796451543rm_a_b @ Ax @ Ay ) @ ( produc1763473618796451543rm_a_b @ Bx @ By ) ) @ ( transi5332651807426350186rm_a_b @ R3 ) )
=> ( ( P2 @ Bx @ By )
=> ( ! [A3: produc357393685978478089rm_a_b,B3: produc357393685978478089rm_a_b,Aa: produc357393685978478089rm_a_b,Ba: produc357393685978478089rm_a_b] :
( ( member5793113912742238992rm_a_b @ ( produc2499998789532207959rm_a_b @ ( produc1763473618796451543rm_a_b @ A3 @ B3 ) @ ( produc1763473618796451543rm_a_b @ Aa @ Ba ) ) @ R3 )
=> ( ( member5793113912742238992rm_a_b @ ( produc2499998789532207959rm_a_b @ ( produc1763473618796451543rm_a_b @ Aa @ Ba ) @ ( produc1763473618796451543rm_a_b @ Bx @ By ) ) @ ( transi5332651807426350186rm_a_b @ R3 ) )
=> ( ( P2 @ Aa @ Ba )
=> ( P2 @ A3 @ B3 ) ) ) )
=> ( P2 @ Ax @ Ay ) ) ) ) ).
% converse_rtrancl_induct2
thf(fact_61_converse__rtrancl__induct2,axiom,
! [Ax: set_term_a_b,Ay: set_term_a_b,Bx: set_term_a_b,By: set_term_a_b,R3: set_Pr7061646551094443591rm_a_b,P2: set_term_a_b > set_term_a_b > $o] :
( ( member1016977296685850768rm_a_b @ ( produc1448243995722353623rm_a_b @ ( produc8517751122659631533rm_a_b @ Ax @ Ay ) @ ( produc8517751122659631533rm_a_b @ Bx @ By ) ) @ ( transi2165682467528973816rm_a_b @ R3 ) )
=> ( ( P2 @ Bx @ By )
=> ( ! [A3: set_term_a_b,B3: set_term_a_b,Aa: set_term_a_b,Ba: set_term_a_b] :
( ( member1016977296685850768rm_a_b @ ( produc1448243995722353623rm_a_b @ ( produc8517751122659631533rm_a_b @ A3 @ B3 ) @ ( produc8517751122659631533rm_a_b @ Aa @ Ba ) ) @ R3 )
=> ( ( member1016977296685850768rm_a_b @ ( produc1448243995722353623rm_a_b @ ( produc8517751122659631533rm_a_b @ Aa @ Ba ) @ ( produc8517751122659631533rm_a_b @ Bx @ By ) ) @ ( transi2165682467528973816rm_a_b @ R3 ) )
=> ( ( P2 @ Aa @ Ba )
=> ( P2 @ A3 @ B3 ) ) ) )
=> ( P2 @ Ax @ Ay ) ) ) ) ).
% converse_rtrancl_induct2
thf(fact_62_converse__rtrancl__induct2,axiom,
! [Ax: set_Pr4386577575007340137rm_a_b,Ay: set_Pr4386577575007340137rm_a_b,Bx: set_Pr4386577575007340137rm_a_b,By: set_Pr4386577575007340137rm_a_b,R3: set_Pr4284312964791606471rm_a_b,P2: set_Pr4386577575007340137rm_a_b > set_Pr4386577575007340137rm_a_b > $o] :
( ( member2018982355343301392rm_a_b @ ( produc765457364750699351rm_a_b @ ( produc3977138704197412567rm_a_b @ Ax @ Ay ) @ ( produc3977138704197412567rm_a_b @ Bx @ By ) ) @ ( transi5504100148209286250rm_a_b @ R3 ) )
=> ( ( P2 @ Bx @ By )
=> ( ! [A3: set_Pr4386577575007340137rm_a_b,B3: set_Pr4386577575007340137rm_a_b,Aa: set_Pr4386577575007340137rm_a_b,Ba: set_Pr4386577575007340137rm_a_b] :
( ( member2018982355343301392rm_a_b @ ( produc765457364750699351rm_a_b @ ( produc3977138704197412567rm_a_b @ A3 @ B3 ) @ ( produc3977138704197412567rm_a_b @ Aa @ Ba ) ) @ R3 )
=> ( ( member2018982355343301392rm_a_b @ ( produc765457364750699351rm_a_b @ ( produc3977138704197412567rm_a_b @ Aa @ Ba ) @ ( produc3977138704197412567rm_a_b @ Bx @ By ) ) @ ( transi5504100148209286250rm_a_b @ R3 ) )
=> ( ( P2 @ Aa @ Ba )
=> ( P2 @ A3 @ B3 ) ) ) )
=> ( P2 @ Ax @ Ay ) ) ) ) ).
% converse_rtrancl_induct2
thf(fact_63_converse__rtrancl__induct2,axiom,
! [Ax: term_a_b,Ay: term_a_b,Bx: term_a_b,By: term_a_b,R3: set_Pr2972776593051762503rm_a_b,P2: term_a_b > term_a_b > $o] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ Ax @ Ay ) @ ( produc7020197800436672577rm_a_b @ Bx @ By ) ) @ ( transi2615809358984392588rm_a_b @ R3 ) )
=> ( ( P2 @ Bx @ By )
=> ( ! [A3: term_a_b,B3: term_a_b,Aa: term_a_b,Ba: term_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B3 ) @ ( produc7020197800436672577rm_a_b @ Aa @ Ba ) ) @ R3 )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ Aa @ Ba ) @ ( produc7020197800436672577rm_a_b @ Bx @ By ) ) @ ( transi2615809358984392588rm_a_b @ R3 ) )
=> ( ( P2 @ Aa @ Ba )
=> ( P2 @ A3 @ B3 ) ) ) )
=> ( P2 @ Ax @ Ay ) ) ) ) ).
% converse_rtrancl_induct2
thf(fact_64_converse__rtranclE2,axiom,
! [Xa: produc357393685978478089rm_a_b,Xb: produc357393685978478089rm_a_b,Za: produc357393685978478089rm_a_b,Zb: produc357393685978478089rm_a_b,R3: set_Pr6848273517546626247rm_a_b] :
( ( member5793113912742238992rm_a_b @ ( produc2499998789532207959rm_a_b @ ( produc1763473618796451543rm_a_b @ Xa @ Xb ) @ ( produc1763473618796451543rm_a_b @ Za @ Zb ) ) @ ( transi5332651807426350186rm_a_b @ R3 ) )
=> ( ( ( produc1763473618796451543rm_a_b @ Xa @ Xb )
!= ( produc1763473618796451543rm_a_b @ Za @ Zb ) )
=> ~ ! [A3: produc357393685978478089rm_a_b,B3: produc357393685978478089rm_a_b] :
( ( member5793113912742238992rm_a_b @ ( produc2499998789532207959rm_a_b @ ( produc1763473618796451543rm_a_b @ Xa @ Xb ) @ ( produc1763473618796451543rm_a_b @ A3 @ B3 ) ) @ R3 )
=> ~ ( member5793113912742238992rm_a_b @ ( produc2499998789532207959rm_a_b @ ( produc1763473618796451543rm_a_b @ A3 @ B3 ) @ ( produc1763473618796451543rm_a_b @ Za @ Zb ) ) @ ( transi5332651807426350186rm_a_b @ R3 ) ) ) ) ) ).
% converse_rtranclE2
thf(fact_65_converse__rtranclE2,axiom,
! [Xa: set_term_a_b,Xb: set_term_a_b,Za: set_term_a_b,Zb: set_term_a_b,R3: set_Pr7061646551094443591rm_a_b] :
( ( member1016977296685850768rm_a_b @ ( produc1448243995722353623rm_a_b @ ( produc8517751122659631533rm_a_b @ Xa @ Xb ) @ ( produc8517751122659631533rm_a_b @ Za @ Zb ) ) @ ( transi2165682467528973816rm_a_b @ R3 ) )
=> ( ( ( produc8517751122659631533rm_a_b @ Xa @ Xb )
!= ( produc8517751122659631533rm_a_b @ Za @ Zb ) )
=> ~ ! [A3: set_term_a_b,B3: set_term_a_b] :
( ( member1016977296685850768rm_a_b @ ( produc1448243995722353623rm_a_b @ ( produc8517751122659631533rm_a_b @ Xa @ Xb ) @ ( produc8517751122659631533rm_a_b @ A3 @ B3 ) ) @ R3 )
=> ~ ( member1016977296685850768rm_a_b @ ( produc1448243995722353623rm_a_b @ ( produc8517751122659631533rm_a_b @ A3 @ B3 ) @ ( produc8517751122659631533rm_a_b @ Za @ Zb ) ) @ ( transi2165682467528973816rm_a_b @ R3 ) ) ) ) ) ).
% converse_rtranclE2
thf(fact_66_converse__rtranclE2,axiom,
! [Xa: set_Pr4386577575007340137rm_a_b,Xb: set_Pr4386577575007340137rm_a_b,Za: set_Pr4386577575007340137rm_a_b,Zb: set_Pr4386577575007340137rm_a_b,R3: set_Pr4284312964791606471rm_a_b] :
( ( member2018982355343301392rm_a_b @ ( produc765457364750699351rm_a_b @ ( produc3977138704197412567rm_a_b @ Xa @ Xb ) @ ( produc3977138704197412567rm_a_b @ Za @ Zb ) ) @ ( transi5504100148209286250rm_a_b @ R3 ) )
=> ( ( ( produc3977138704197412567rm_a_b @ Xa @ Xb )
!= ( produc3977138704197412567rm_a_b @ Za @ Zb ) )
=> ~ ! [A3: set_Pr4386577575007340137rm_a_b,B3: set_Pr4386577575007340137rm_a_b] :
( ( member2018982355343301392rm_a_b @ ( produc765457364750699351rm_a_b @ ( produc3977138704197412567rm_a_b @ Xa @ Xb ) @ ( produc3977138704197412567rm_a_b @ A3 @ B3 ) ) @ R3 )
=> ~ ( member2018982355343301392rm_a_b @ ( produc765457364750699351rm_a_b @ ( produc3977138704197412567rm_a_b @ A3 @ B3 ) @ ( produc3977138704197412567rm_a_b @ Za @ Zb ) ) @ ( transi5504100148209286250rm_a_b @ R3 ) ) ) ) ) ).
% converse_rtranclE2
thf(fact_67_converse__rtranclE2,axiom,
! [Xa: term_a_b,Xb: term_a_b,Za: term_a_b,Zb: term_a_b,R3: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ Xa @ Xb ) @ ( produc7020197800436672577rm_a_b @ Za @ Zb ) ) @ ( transi2615809358984392588rm_a_b @ R3 ) )
=> ( ( ( produc7020197800436672577rm_a_b @ Xa @ Xb )
!= ( produc7020197800436672577rm_a_b @ Za @ Zb ) )
=> ~ ! [A3: term_a_b,B3: term_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ Xa @ Xb ) @ ( produc7020197800436672577rm_a_b @ A3 @ B3 ) ) @ R3 )
=> ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B3 ) @ ( produc7020197800436672577rm_a_b @ Za @ Zb ) ) @ ( transi2615809358984392588rm_a_b @ R3 ) ) ) ) ) ).
% converse_rtranclE2
thf(fact_68_rtrancl__induct2,axiom,
! [Ax: produc357393685978478089rm_a_b,Ay: produc357393685978478089rm_a_b,Bx: produc357393685978478089rm_a_b,By: produc357393685978478089rm_a_b,R3: set_Pr6848273517546626247rm_a_b,P2: produc357393685978478089rm_a_b > produc357393685978478089rm_a_b > $o] :
( ( member5793113912742238992rm_a_b @ ( produc2499998789532207959rm_a_b @ ( produc1763473618796451543rm_a_b @ Ax @ Ay ) @ ( produc1763473618796451543rm_a_b @ Bx @ By ) ) @ ( transi5332651807426350186rm_a_b @ R3 ) )
=> ( ( P2 @ Ax @ Ay )
=> ( ! [A3: produc357393685978478089rm_a_b,B3: produc357393685978478089rm_a_b,Aa: produc357393685978478089rm_a_b,Ba: produc357393685978478089rm_a_b] :
( ( member5793113912742238992rm_a_b @ ( produc2499998789532207959rm_a_b @ ( produc1763473618796451543rm_a_b @ Ax @ Ay ) @ ( produc1763473618796451543rm_a_b @ A3 @ B3 ) ) @ ( transi5332651807426350186rm_a_b @ R3 ) )
=> ( ( member5793113912742238992rm_a_b @ ( produc2499998789532207959rm_a_b @ ( produc1763473618796451543rm_a_b @ A3 @ B3 ) @ ( produc1763473618796451543rm_a_b @ Aa @ Ba ) ) @ R3 )
=> ( ( P2 @ A3 @ B3 )
=> ( P2 @ Aa @ Ba ) ) ) )
=> ( P2 @ Bx @ By ) ) ) ) ).
% rtrancl_induct2
thf(fact_69_rtrancl__induct2,axiom,
! [Ax: set_term_a_b,Ay: set_term_a_b,Bx: set_term_a_b,By: set_term_a_b,R3: set_Pr7061646551094443591rm_a_b,P2: set_term_a_b > set_term_a_b > $o] :
( ( member1016977296685850768rm_a_b @ ( produc1448243995722353623rm_a_b @ ( produc8517751122659631533rm_a_b @ Ax @ Ay ) @ ( produc8517751122659631533rm_a_b @ Bx @ By ) ) @ ( transi2165682467528973816rm_a_b @ R3 ) )
=> ( ( P2 @ Ax @ Ay )
=> ( ! [A3: set_term_a_b,B3: set_term_a_b,Aa: set_term_a_b,Ba: set_term_a_b] :
( ( member1016977296685850768rm_a_b @ ( produc1448243995722353623rm_a_b @ ( produc8517751122659631533rm_a_b @ Ax @ Ay ) @ ( produc8517751122659631533rm_a_b @ A3 @ B3 ) ) @ ( transi2165682467528973816rm_a_b @ R3 ) )
=> ( ( member1016977296685850768rm_a_b @ ( produc1448243995722353623rm_a_b @ ( produc8517751122659631533rm_a_b @ A3 @ B3 ) @ ( produc8517751122659631533rm_a_b @ Aa @ Ba ) ) @ R3 )
=> ( ( P2 @ A3 @ B3 )
=> ( P2 @ Aa @ Ba ) ) ) )
=> ( P2 @ Bx @ By ) ) ) ) ).
% rtrancl_induct2
thf(fact_70_rtrancl__induct2,axiom,
! [Ax: set_Pr4386577575007340137rm_a_b,Ay: set_Pr4386577575007340137rm_a_b,Bx: set_Pr4386577575007340137rm_a_b,By: set_Pr4386577575007340137rm_a_b,R3: set_Pr4284312964791606471rm_a_b,P2: set_Pr4386577575007340137rm_a_b > set_Pr4386577575007340137rm_a_b > $o] :
( ( member2018982355343301392rm_a_b @ ( produc765457364750699351rm_a_b @ ( produc3977138704197412567rm_a_b @ Ax @ Ay ) @ ( produc3977138704197412567rm_a_b @ Bx @ By ) ) @ ( transi5504100148209286250rm_a_b @ R3 ) )
=> ( ( P2 @ Ax @ Ay )
=> ( ! [A3: set_Pr4386577575007340137rm_a_b,B3: set_Pr4386577575007340137rm_a_b,Aa: set_Pr4386577575007340137rm_a_b,Ba: set_Pr4386577575007340137rm_a_b] :
( ( member2018982355343301392rm_a_b @ ( produc765457364750699351rm_a_b @ ( produc3977138704197412567rm_a_b @ Ax @ Ay ) @ ( produc3977138704197412567rm_a_b @ A3 @ B3 ) ) @ ( transi5504100148209286250rm_a_b @ R3 ) )
=> ( ( member2018982355343301392rm_a_b @ ( produc765457364750699351rm_a_b @ ( produc3977138704197412567rm_a_b @ A3 @ B3 ) @ ( produc3977138704197412567rm_a_b @ Aa @ Ba ) ) @ R3 )
=> ( ( P2 @ A3 @ B3 )
=> ( P2 @ Aa @ Ba ) ) ) )
=> ( P2 @ Bx @ By ) ) ) ) ).
% rtrancl_induct2
thf(fact_71_rtrancl__induct2,axiom,
! [Ax: term_a_b,Ay: term_a_b,Bx: term_a_b,By: term_a_b,R3: set_Pr2972776593051762503rm_a_b,P2: term_a_b > term_a_b > $o] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ Ax @ Ay ) @ ( produc7020197800436672577rm_a_b @ Bx @ By ) ) @ ( transi2615809358984392588rm_a_b @ R3 ) )
=> ( ( P2 @ Ax @ Ay )
=> ( ! [A3: term_a_b,B3: term_a_b,Aa: term_a_b,Ba: term_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ Ax @ Ay ) @ ( produc7020197800436672577rm_a_b @ A3 @ B3 ) ) @ ( transi2615809358984392588rm_a_b @ R3 ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B3 ) @ ( produc7020197800436672577rm_a_b @ Aa @ Ba ) ) @ R3 )
=> ( ( P2 @ A3 @ B3 )
=> ( P2 @ Aa @ Ba ) ) ) )
=> ( P2 @ Bx @ By ) ) ) ) ).
% rtrancl_induct2
thf(fact_72_O__assoc,axiom,
! [R4: set_Pr7151917219565317845rm_a_b,S6: set_Pr7151917219565317845rm_a_b,T4: set_Pr7151917219565317845rm_a_b] :
( ( relcom3320975603336820193rm_a_b @ ( relcom3320975603336820193rm_a_b @ R4 @ S6 ) @ T4 )
= ( relcom3320975603336820193rm_a_b @ R4 @ ( relcom3320975603336820193rm_a_b @ S6 @ T4 ) ) ) ).
% O_assoc
thf(fact_73_O__assoc,axiom,
! [R4: set_Pr4386577575007340137rm_a_b,S6: set_Pr4386577575007340137rm_a_b,T4: set_Pr4386577575007340137rm_a_b] :
( ( relcom370159955682700863rm_a_b @ ( relcom370159955682700863rm_a_b @ R4 @ S6 ) @ T4 )
= ( relcom370159955682700863rm_a_b @ R4 @ ( relcom370159955682700863rm_a_b @ S6 @ T4 ) ) ) ).
% O_assoc
thf(fact_74_r__into__trancl_H,axiom,
! [P: produc3421996462152255079rm_a_b,R3: set_Pr647378613824986951rm_a_b] :
( ( member1375104729371992464rm_a_b @ P @ R3 )
=> ( member1375104729371992464rm_a_b @ P @ ( transi6883191930764840330rm_a_b @ R3 ) ) ) ).
% r_into_trancl'
thf(fact_75_r__into__trancl_H,axiom,
! [P: produc4523971788519308903rm_a_b,R3: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ P @ R3 )
=> ( member8417600551952982416rm_a_b @ P @ ( transi9211502839322181930rm_a_b @ R3 ) ) ) ).
% r_into_trancl'
thf(fact_76_r__into__trancl_H,axiom,
! [P: produc357393685978478089rm_a_b,R3: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ P @ R3 )
=> ( member5869715511025134514rm_a_b @ P @ ( transi7922773638565587891rm_a_b @ R3 ) ) ) ).
% r_into_trancl'
thf(fact_77_converse__rtrancl__into__rtrancl,axiom,
! [A: set_term_a_b,B: set_term_a_b,R3: set_Pr7151917219565317845rm_a_b,C2: set_term_a_b] :
( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A @ B ) @ R3 )
=> ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ B @ C2 ) @ ( transi6904761420441344775rm_a_b @ R3 ) )
=> ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A @ C2 ) @ ( transi6904761420441344775rm_a_b @ R3 ) ) ) ) ).
% converse_rtrancl_into_rtrancl
thf(fact_78_converse__rtrancl__into__rtrancl,axiom,
! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b,R3: set_Pr647378613824986951rm_a_b,C2: set_Pr4386577575007340137rm_a_b] :
( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A @ B ) @ R3 )
=> ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ B @ C2 ) @ ( transi8812752159401800684rm_a_b @ R3 ) )
=> ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A @ C2 ) @ ( transi8812752159401800684rm_a_b @ R3 ) ) ) ) ).
% converse_rtrancl_into_rtrancl
thf(fact_79_converse__rtrancl__into__rtrancl,axiom,
! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,R3: set_Pr2972776593051762503rm_a_b,C2: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ R3 )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B @ C2 ) @ ( transi2615809358984392588rm_a_b @ R3 ) )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ C2 ) @ ( transi2615809358984392588rm_a_b @ R3 ) ) ) ) ).
% converse_rtrancl_into_rtrancl
thf(fact_80_converse__rtrancl__into__rtrancl,axiom,
! [A: term_a_b,B: term_a_b,R3: set_Pr4386577575007340137rm_a_b,C2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R3 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C2 ) @ ( transi7742714808557438673rm_a_b @ R3 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C2 ) @ ( transi7742714808557438673rm_a_b @ R3 ) ) ) ) ).
% converse_rtrancl_into_rtrancl
thf(fact_81_converse__rtrancl__induct,axiom,
! [A: set_term_a_b,B: set_term_a_b,R3: set_Pr7151917219565317845rm_a_b,P2: set_term_a_b > $o] :
( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A @ B ) @ ( transi6904761420441344775rm_a_b @ R3 ) )
=> ( ( P2 @ B )
=> ( ! [Y2: set_term_a_b,Z2: set_term_a_b] :
( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ Y2 @ Z2 ) @ R3 )
=> ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ Z2 @ B ) @ ( transi6904761420441344775rm_a_b @ R3 ) )
=> ( ( P2 @ Z2 )
=> ( P2 @ Y2 ) ) ) )
=> ( P2 @ A ) ) ) ) ).
% converse_rtrancl_induct
thf(fact_82_converse__rtrancl__induct,axiom,
! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b,R3: set_Pr647378613824986951rm_a_b,P2: set_Pr4386577575007340137rm_a_b > $o] :
( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A @ B ) @ ( transi8812752159401800684rm_a_b @ R3 ) )
=> ( ( P2 @ B )
=> ( ! [Y2: set_Pr4386577575007340137rm_a_b,Z2: set_Pr4386577575007340137rm_a_b] :
( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ Y2 @ Z2 ) @ R3 )
=> ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ Z2 @ B ) @ ( transi8812752159401800684rm_a_b @ R3 ) )
=> ( ( P2 @ Z2 )
=> ( P2 @ Y2 ) ) ) )
=> ( P2 @ A ) ) ) ) ).
% converse_rtrancl_induct
thf(fact_83_converse__rtrancl__induct,axiom,
! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,R3: set_Pr2972776593051762503rm_a_b,P2: produc357393685978478089rm_a_b > $o] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( transi2615809358984392588rm_a_b @ R3 ) )
=> ( ( P2 @ B )
=> ( ! [Y2: produc357393685978478089rm_a_b,Z2: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y2 @ Z2 ) @ R3 )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Z2 @ B ) @ ( transi2615809358984392588rm_a_b @ R3 ) )
=> ( ( P2 @ Z2 )
=> ( P2 @ Y2 ) ) ) )
=> ( P2 @ A ) ) ) ) ).
% converse_rtrancl_induct
thf(fact_84_converse__rtrancl__induct,axiom,
! [A: term_a_b,B: term_a_b,R3: set_Pr4386577575007340137rm_a_b,P2: term_a_b > $o] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ R3 ) )
=> ( ( P2 @ B )
=> ( ! [Y2: term_a_b,Z2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y2 @ Z2 ) @ R3 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Z2 @ B ) @ ( transi7742714808557438673rm_a_b @ R3 ) )
=> ( ( P2 @ Z2 )
=> ( P2 @ Y2 ) ) ) )
=> ( P2 @ A ) ) ) ) ).
% converse_rtrancl_induct
thf(fact_85_converse__rtranclE,axiom,
! [X: set_term_a_b,Z: set_term_a_b,R3: set_Pr7151917219565317845rm_a_b] :
( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ X @ Z ) @ ( transi6904761420441344775rm_a_b @ R3 ) )
=> ( ( X != Z )
=> ~ ! [Y2: set_term_a_b] :
( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ X @ Y2 ) @ R3 )
=> ~ ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ Y2 @ Z ) @ ( transi6904761420441344775rm_a_b @ R3 ) ) ) ) ) ).
% converse_rtranclE
thf(fact_86_converse__rtranclE,axiom,
! [X: set_Pr4386577575007340137rm_a_b,Z: set_Pr4386577575007340137rm_a_b,R3: set_Pr647378613824986951rm_a_b] :
( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ X @ Z ) @ ( transi8812752159401800684rm_a_b @ R3 ) )
=> ( ( X != Z )
=> ~ ! [Y2: set_Pr4386577575007340137rm_a_b] :
( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ X @ Y2 ) @ R3 )
=> ~ ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ Y2 @ Z ) @ ( transi8812752159401800684rm_a_b @ R3 ) ) ) ) ) ).
% converse_rtranclE
thf(fact_87_converse__rtranclE,axiom,
! [X: produc357393685978478089rm_a_b,Z: produc357393685978478089rm_a_b,R3: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Z ) @ ( transi2615809358984392588rm_a_b @ R3 ) )
=> ( ( X != Z )
=> ~ ! [Y2: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y2 ) @ R3 )
=> ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y2 @ Z ) @ ( transi2615809358984392588rm_a_b @ R3 ) ) ) ) ) ).
% converse_rtranclE
thf(fact_88_converse__rtranclE,axiom,
! [X: term_a_b,Z: term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z ) @ ( transi7742714808557438673rm_a_b @ R3 ) )
=> ( ( X != Z )
=> ~ ! [Y2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y2 ) @ R3 )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y2 @ Z ) @ ( transi7742714808557438673rm_a_b @ R3 ) ) ) ) ) ).
% converse_rtranclE
thf(fact_89_rtrancl__induct,axiom,
! [A: set_term_a_b,B: set_term_a_b,R3: set_Pr7151917219565317845rm_a_b,P2: set_term_a_b > $o] :
( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A @ B ) @ ( transi6904761420441344775rm_a_b @ R3 ) )
=> ( ( P2 @ A )
=> ( ! [Y2: set_term_a_b,Z2: set_term_a_b] :
( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A @ Y2 ) @ ( transi6904761420441344775rm_a_b @ R3 ) )
=> ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ Y2 @ Z2 ) @ R3 )
=> ( ( P2 @ Y2 )
=> ( P2 @ Z2 ) ) ) )
=> ( P2 @ B ) ) ) ) ).
% rtrancl_induct
thf(fact_90_rtrancl__induct,axiom,
! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b,R3: set_Pr647378613824986951rm_a_b,P2: set_Pr4386577575007340137rm_a_b > $o] :
( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A @ B ) @ ( transi8812752159401800684rm_a_b @ R3 ) )
=> ( ( P2 @ A )
=> ( ! [Y2: set_Pr4386577575007340137rm_a_b,Z2: set_Pr4386577575007340137rm_a_b] :
( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A @ Y2 ) @ ( transi8812752159401800684rm_a_b @ R3 ) )
=> ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ Y2 @ Z2 ) @ R3 )
=> ( ( P2 @ Y2 )
=> ( P2 @ Z2 ) ) ) )
=> ( P2 @ B ) ) ) ) ).
% rtrancl_induct
thf(fact_91_rtrancl__induct,axiom,
! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,R3: set_Pr2972776593051762503rm_a_b,P2: produc357393685978478089rm_a_b > $o] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( transi2615809358984392588rm_a_b @ R3 ) )
=> ( ( P2 @ A )
=> ( ! [Y2: produc357393685978478089rm_a_b,Z2: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ Y2 ) @ ( transi2615809358984392588rm_a_b @ R3 ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y2 @ Z2 ) @ R3 )
=> ( ( P2 @ Y2 )
=> ( P2 @ Z2 ) ) ) )
=> ( P2 @ B ) ) ) ) ).
% rtrancl_induct
thf(fact_92_rtrancl__induct,axiom,
! [A: term_a_b,B: term_a_b,R3: set_Pr4386577575007340137rm_a_b,P2: term_a_b > $o] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ R3 ) )
=> ( ( P2 @ A )
=> ( ! [Y2: term_a_b,Z2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ Y2 ) @ ( transi7742714808557438673rm_a_b @ R3 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y2 @ Z2 ) @ R3 )
=> ( ( P2 @ Y2 )
=> ( P2 @ Z2 ) ) ) )
=> ( P2 @ B ) ) ) ) ).
% rtrancl_induct
thf(fact_93_rtrancl__trans,axiom,
! [X: set_term_a_b,Y: set_term_a_b,R3: set_Pr7151917219565317845rm_a_b,Z: set_term_a_b] :
( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ X @ Y ) @ ( transi6904761420441344775rm_a_b @ R3 ) )
=> ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ Y @ Z ) @ ( transi6904761420441344775rm_a_b @ R3 ) )
=> ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ X @ Z ) @ ( transi6904761420441344775rm_a_b @ R3 ) ) ) ) ).
% rtrancl_trans
thf(fact_94_rtrancl__trans,axiom,
! [X: set_Pr4386577575007340137rm_a_b,Y: set_Pr4386577575007340137rm_a_b,R3: set_Pr647378613824986951rm_a_b,Z: set_Pr4386577575007340137rm_a_b] :
( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ X @ Y ) @ ( transi8812752159401800684rm_a_b @ R3 ) )
=> ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ Y @ Z ) @ ( transi8812752159401800684rm_a_b @ R3 ) )
=> ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ X @ Z ) @ ( transi8812752159401800684rm_a_b @ R3 ) ) ) ) ).
% rtrancl_trans
thf(fact_95_rtrancl__trans,axiom,
! [X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b,R3: set_Pr2972776593051762503rm_a_b,Z: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ ( transi2615809358984392588rm_a_b @ R3 ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y @ Z ) @ ( transi2615809358984392588rm_a_b @ R3 ) )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Z ) @ ( transi2615809358984392588rm_a_b @ R3 ) ) ) ) ).
% rtrancl_trans
thf(fact_96_rtrancl__trans,axiom,
! [X: term_a_b,Y: term_a_b,R3: set_Pr4386577575007340137rm_a_b,Z: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ R3 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ Z ) @ ( transi7742714808557438673rm_a_b @ R3 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z ) @ ( transi7742714808557438673rm_a_b @ R3 ) ) ) ) ).
% rtrancl_trans
thf(fact_97_rtranclE,axiom,
! [A: set_term_a_b,B: set_term_a_b,R3: set_Pr7151917219565317845rm_a_b] :
( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A @ B ) @ ( transi6904761420441344775rm_a_b @ R3 ) )
=> ( ( A != B )
=> ~ ! [Y2: set_term_a_b] :
( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A @ Y2 ) @ ( transi6904761420441344775rm_a_b @ R3 ) )
=> ~ ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ Y2 @ B ) @ R3 ) ) ) ) ).
% rtranclE
thf(fact_98_rtranclE,axiom,
! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b,R3: set_Pr647378613824986951rm_a_b] :
( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A @ B ) @ ( transi8812752159401800684rm_a_b @ R3 ) )
=> ( ( A != B )
=> ~ ! [Y2: set_Pr4386577575007340137rm_a_b] :
( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A @ Y2 ) @ ( transi8812752159401800684rm_a_b @ R3 ) )
=> ~ ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ Y2 @ B ) @ R3 ) ) ) ) ).
% rtranclE
thf(fact_99_rtranclE,axiom,
! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,R3: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( transi2615809358984392588rm_a_b @ R3 ) )
=> ( ( A != B )
=> ~ ! [Y2: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ Y2 ) @ ( transi2615809358984392588rm_a_b @ R3 ) )
=> ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y2 @ B ) @ R3 ) ) ) ) ).
% rtranclE
thf(fact_100_rtranclE,axiom,
! [A: term_a_b,B: term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ R3 ) )
=> ( ( A != B )
=> ~ ! [Y2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ Y2 ) @ ( transi7742714808557438673rm_a_b @ R3 ) )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y2 @ B ) @ R3 ) ) ) ) ).
% rtranclE
thf(fact_101_rtrancl_Ortrancl__into__rtrancl,axiom,
! [A: set_term_a_b,B: set_term_a_b,R3: set_Pr7151917219565317845rm_a_b,C2: set_term_a_b] :
( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A @ B ) @ ( transi6904761420441344775rm_a_b @ R3 ) )
=> ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ B @ C2 ) @ R3 )
=> ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A @ C2 ) @ ( transi6904761420441344775rm_a_b @ R3 ) ) ) ) ).
% rtrancl.rtrancl_into_rtrancl
thf(fact_102_rtrancl_Ortrancl__into__rtrancl,axiom,
! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b,R3: set_Pr647378613824986951rm_a_b,C2: set_Pr4386577575007340137rm_a_b] :
( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A @ B ) @ ( transi8812752159401800684rm_a_b @ R3 ) )
=> ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ B @ C2 ) @ R3 )
=> ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A @ C2 ) @ ( transi8812752159401800684rm_a_b @ R3 ) ) ) ) ).
% rtrancl.rtrancl_into_rtrancl
thf(fact_103_rtrancl_Ortrancl__into__rtrancl,axiom,
! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,R3: set_Pr2972776593051762503rm_a_b,C2: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( transi2615809358984392588rm_a_b @ R3 ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B @ C2 ) @ R3 )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ C2 ) @ ( transi2615809358984392588rm_a_b @ R3 ) ) ) ) ).
% rtrancl.rtrancl_into_rtrancl
thf(fact_104_rtrancl_Ortrancl__into__rtrancl,axiom,
! [A: term_a_b,B: term_a_b,R3: set_Pr4386577575007340137rm_a_b,C2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ R3 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C2 ) @ R3 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C2 ) @ ( transi7742714808557438673rm_a_b @ R3 ) ) ) ) ).
% rtrancl.rtrancl_into_rtrancl
thf(fact_105_rtrancl_Ortrancl__refl,axiom,
! [A: set_term_a_b,R3: set_Pr7151917219565317845rm_a_b] : ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A @ A ) @ ( transi6904761420441344775rm_a_b @ R3 ) ) ).
% rtrancl.rtrancl_refl
thf(fact_106_rtrancl_Ortrancl__refl,axiom,
! [A: set_Pr4386577575007340137rm_a_b,R3: set_Pr647378613824986951rm_a_b] : ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A @ A ) @ ( transi8812752159401800684rm_a_b @ R3 ) ) ).
% rtrancl.rtrancl_refl
thf(fact_107_rtrancl_Ortrancl__refl,axiom,
! [A: produc357393685978478089rm_a_b,R3: set_Pr2972776593051762503rm_a_b] : ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ A ) @ ( transi2615809358984392588rm_a_b @ R3 ) ) ).
% rtrancl.rtrancl_refl
thf(fact_108_rtrancl_Ortrancl__refl,axiom,
! [A: term_a_b,R3: set_Pr4386577575007340137rm_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ A ) @ ( transi7742714808557438673rm_a_b @ R3 ) ) ).
% rtrancl.rtrancl_refl
thf(fact_109_rtrancl_Osimps,axiom,
! [A1: set_term_a_b,A22: set_term_a_b,R3: set_Pr7151917219565317845rm_a_b] :
( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A1 @ A22 ) @ ( transi6904761420441344775rm_a_b @ R3 ) )
= ( ? [A4: set_term_a_b] :
( ( A1 = A4 )
& ( A22 = A4 ) )
| ? [A4: set_term_a_b,B4: set_term_a_b,C3: set_term_a_b] :
( ( A1 = A4 )
& ( A22 = C3 )
& ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A4 @ B4 ) @ ( transi6904761420441344775rm_a_b @ R3 ) )
& ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ B4 @ C3 ) @ R3 ) ) ) ) ).
% rtrancl.simps
thf(fact_110_rtrancl_Osimps,axiom,
! [A1: set_Pr4386577575007340137rm_a_b,A22: set_Pr4386577575007340137rm_a_b,R3: set_Pr647378613824986951rm_a_b] :
( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A1 @ A22 ) @ ( transi8812752159401800684rm_a_b @ R3 ) )
= ( ? [A4: set_Pr4386577575007340137rm_a_b] :
( ( A1 = A4 )
& ( A22 = A4 ) )
| ? [A4: set_Pr4386577575007340137rm_a_b,B4: set_Pr4386577575007340137rm_a_b,C3: set_Pr4386577575007340137rm_a_b] :
( ( A1 = A4 )
& ( A22 = C3 )
& ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A4 @ B4 ) @ ( transi8812752159401800684rm_a_b @ R3 ) )
& ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ B4 @ C3 ) @ R3 ) ) ) ) ).
% rtrancl.simps
thf(fact_111_rtrancl_Osimps,axiom,
! [A1: produc357393685978478089rm_a_b,A22: produc357393685978478089rm_a_b,R3: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A1 @ A22 ) @ ( transi2615809358984392588rm_a_b @ R3 ) )
= ( ? [A4: produc357393685978478089rm_a_b] :
( ( A1 = A4 )
& ( A22 = A4 ) )
| ? [A4: produc357393685978478089rm_a_b,B4: produc357393685978478089rm_a_b,C3: produc357393685978478089rm_a_b] :
( ( A1 = A4 )
& ( A22 = C3 )
& ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A4 @ B4 ) @ ( transi2615809358984392588rm_a_b @ R3 ) )
& ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B4 @ C3 ) @ R3 ) ) ) ) ).
% rtrancl.simps
thf(fact_112_rtrancl_Osimps,axiom,
! [A1: term_a_b,A22: term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A22 ) @ ( transi7742714808557438673rm_a_b @ R3 ) )
= ( ? [A4: term_a_b] :
( ( A1 = A4 )
& ( A22 = A4 ) )
| ? [A4: term_a_b,B4: term_a_b,C3: term_a_b] :
( ( A1 = A4 )
& ( A22 = C3 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A4 @ B4 ) @ ( transi7742714808557438673rm_a_b @ R3 ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B4 @ C3 ) @ R3 ) ) ) ) ).
% rtrancl.simps
thf(fact_113_mem__Collect__eq,axiom,
! [A: produc4523971788519308903rm_a_b,P2: produc4523971788519308903rm_a_b > $o] :
( ( member8417600551952982416rm_a_b @ A @ ( collec1344989105561874898rm_a_b @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_114_mem__Collect__eq,axiom,
! [A: term_a_b,P2: term_a_b > $o] :
( ( member_term_a_b @ A @ ( collect_term_a_b @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_115_mem__Collect__eq,axiom,
! [A: produc3421996462152255079rm_a_b,P2: produc3421996462152255079rm_a_b > $o] :
( ( member1375104729371992464rm_a_b @ A @ ( collec1520397187239403986rm_a_b @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_116_mem__Collect__eq,axiom,
! [A: set_term_a_b,P2: set_term_a_b > $o] :
( ( member_set_term_a_b @ A @ ( collect_set_term_a_b @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_117_mem__Collect__eq,axiom,
! [A: set_Pr4386577575007340137rm_a_b,P2: set_Pr4386577575007340137rm_a_b > $o] :
( ( member8823642877475723282rm_a_b @ A @ ( collec4889374776148843860rm_a_b @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_118_mem__Collect__eq,axiom,
! [A: produc357393685978478089rm_a_b,P2: produc357393685978478089rm_a_b > $o] :
( ( member5869715511025134514rm_a_b @ A @ ( collec99177395925924084rm_a_b @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_119_Collect__mem__eq,axiom,
! [A2: set_Pr2972776593051762503rm_a_b] :
( ( collec1344989105561874898rm_a_b
@ ^ [X2: produc4523971788519308903rm_a_b] : ( member8417600551952982416rm_a_b @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_120_Collect__mem__eq,axiom,
! [A2: set_term_a_b] :
( ( collect_term_a_b
@ ^ [X2: term_a_b] : ( member_term_a_b @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_121_Collect__mem__eq,axiom,
! [A2: set_Pr647378613824986951rm_a_b] :
( ( collec1520397187239403986rm_a_b
@ ^ [X2: produc3421996462152255079rm_a_b] : ( member1375104729371992464rm_a_b @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_122_Collect__mem__eq,axiom,
! [A2: set_set_term_a_b] :
( ( collect_set_term_a_b
@ ^ [X2: set_term_a_b] : ( member_set_term_a_b @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_123_Collect__mem__eq,axiom,
! [A2: set_se1467346398288745161rm_a_b] :
( ( collec4889374776148843860rm_a_b
@ ^ [X2: set_Pr4386577575007340137rm_a_b] : ( member8823642877475723282rm_a_b @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_124_Collect__mem__eq,axiom,
! [A2: set_Pr4386577575007340137rm_a_b] :
( ( collec99177395925924084rm_a_b
@ ^ [X2: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_125_Collect__cong,axiom,
! [P2: produc357393685978478089rm_a_b > $o,Q: produc357393685978478089rm_a_b > $o] :
( ! [X3: produc357393685978478089rm_a_b] :
( ( P2 @ X3 )
= ( Q @ X3 ) )
=> ( ( collec99177395925924084rm_a_b @ P2 )
= ( collec99177395925924084rm_a_b @ Q ) ) ) ).
% Collect_cong
thf(fact_126_rtrancl_Ocases,axiom,
! [A1: set_term_a_b,A22: set_term_a_b,R3: set_Pr7151917219565317845rm_a_b] :
( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A1 @ A22 ) @ ( transi6904761420441344775rm_a_b @ R3 ) )
=> ( ( A22 != A1 )
=> ~ ! [B3: set_term_a_b] :
( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A1 @ B3 ) @ ( transi6904761420441344775rm_a_b @ R3 ) )
=> ~ ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ B3 @ A22 ) @ R3 ) ) ) ) ).
% rtrancl.cases
thf(fact_127_rtrancl_Ocases,axiom,
! [A1: set_Pr4386577575007340137rm_a_b,A22: set_Pr4386577575007340137rm_a_b,R3: set_Pr647378613824986951rm_a_b] :
( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A1 @ A22 ) @ ( transi8812752159401800684rm_a_b @ R3 ) )
=> ( ( A22 != A1 )
=> ~ ! [B3: set_Pr4386577575007340137rm_a_b] :
( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A1 @ B3 ) @ ( transi8812752159401800684rm_a_b @ R3 ) )
=> ~ ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ B3 @ A22 ) @ R3 ) ) ) ) ).
% rtrancl.cases
thf(fact_128_rtrancl_Ocases,axiom,
! [A1: produc357393685978478089rm_a_b,A22: produc357393685978478089rm_a_b,R3: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A1 @ A22 ) @ ( transi2615809358984392588rm_a_b @ R3 ) )
=> ( ( A22 != A1 )
=> ~ ! [B3: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A1 @ B3 ) @ ( transi2615809358984392588rm_a_b @ R3 ) )
=> ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B3 @ A22 ) @ R3 ) ) ) ) ).
% rtrancl.cases
thf(fact_129_rtrancl_Ocases,axiom,
! [A1: term_a_b,A22: term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A22 ) @ ( transi7742714808557438673rm_a_b @ R3 ) )
=> ( ( A22 != A1 )
=> ~ ! [B3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ B3 ) @ ( transi7742714808557438673rm_a_b @ R3 ) )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B3 @ A22 ) @ R3 ) ) ) ) ).
% rtrancl.cases
thf(fact_130_relcomp3__I,axiom,
! [T: produc357393685978478089rm_a_b,U: produc357393685978478089rm_a_b,A2: set_Pr2972776593051762503rm_a_b,S: produc357393685978478089rm_a_b,B2: set_Pr2972776593051762503rm_a_b,V: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ T @ U ) @ A2 )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ S @ T ) @ B2 )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ U @ V ) @ B2 )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ S @ V ) @ ( relcom6159724393391289352rm_a_b @ B2 @ ( relcom6159724393391289352rm_a_b @ A2 @ B2 ) ) ) ) ) ) ).
% relcomp3_I
thf(fact_131_relcomp3__I,axiom,
! [T: set_term_a_b,U: set_term_a_b,A2: set_Pr7151917219565317845rm_a_b,S: set_term_a_b,B2: set_Pr7151917219565317845rm_a_b,V: set_term_a_b] :
( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ T @ U ) @ A2 )
=> ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ S @ T ) @ B2 )
=> ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ U @ V ) @ B2 )
=> ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ S @ V ) @ ( relcom3320975603336820193rm_a_b @ B2 @ ( relcom3320975603336820193rm_a_b @ A2 @ B2 ) ) ) ) ) ) ).
% relcomp3_I
thf(fact_132_relcomp3__I,axiom,
! [T: set_Pr4386577575007340137rm_a_b,U: set_Pr4386577575007340137rm_a_b,A2: set_Pr647378613824986951rm_a_b,S: set_Pr4386577575007340137rm_a_b,B2: set_Pr647378613824986951rm_a_b,V: set_Pr4386577575007340137rm_a_b] :
( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ T @ U ) @ A2 )
=> ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ S @ T ) @ B2 )
=> ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ U @ V ) @ B2 )
=> ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ S @ V ) @ ( relcom8520403917366345832rm_a_b @ B2 @ ( relcom8520403917366345832rm_a_b @ A2 @ B2 ) ) ) ) ) ) ).
% relcomp3_I
thf(fact_133_relcomp3__I,axiom,
! [T: term_a_b,U: term_a_b,A2: set_Pr4386577575007340137rm_a_b,S: term_a_b,B2: set_Pr4386577575007340137rm_a_b,V: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T @ U ) @ A2 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ B2 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ U @ V ) @ B2 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ V ) @ ( relcom370159955682700863rm_a_b @ B2 @ ( relcom370159955682700863rm_a_b @ A2 @ B2 ) ) ) ) ) ) ).
% relcomp3_I
thf(fact_134_relcompEpair,axiom,
! [A: produc357393685978478089rm_a_b,C2: produc357393685978478089rm_a_b,R3: set_Pr2972776593051762503rm_a_b,S: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ C2 ) @ ( relcom6159724393391289352rm_a_b @ R3 @ S ) )
=> ~ ! [B3: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B3 ) @ R3 )
=> ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B3 @ C2 ) @ S ) ) ) ).
% relcompEpair
thf(fact_135_relcompEpair,axiom,
! [A: set_term_a_b,C2: set_term_a_b,R3: set_Pr7151917219565317845rm_a_b,S: set_Pr7151917219565317845rm_a_b] :
( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A @ C2 ) @ ( relcom3320975603336820193rm_a_b @ R3 @ S ) )
=> ~ ! [B3: set_term_a_b] :
( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A @ B3 ) @ R3 )
=> ~ ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ B3 @ C2 ) @ S ) ) ) ).
% relcompEpair
thf(fact_136_relcompEpair,axiom,
! [A: set_Pr4386577575007340137rm_a_b,C2: set_Pr4386577575007340137rm_a_b,R3: set_Pr647378613824986951rm_a_b,S: set_Pr647378613824986951rm_a_b] :
( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A @ C2 ) @ ( relcom8520403917366345832rm_a_b @ R3 @ S ) )
=> ~ ! [B3: set_Pr4386577575007340137rm_a_b] :
( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A @ B3 ) @ R3 )
=> ~ ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ B3 @ C2 ) @ S ) ) ) ).
% relcompEpair
thf(fact_137_relcompEpair,axiom,
! [A: term_a_b,C2: term_a_b,R3: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C2 ) @ ( relcom370159955682700863rm_a_b @ R3 @ S ) )
=> ~ ! [B3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B3 ) @ R3 )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B3 @ C2 ) @ S ) ) ) ).
% relcompEpair
thf(fact_138_relcompE,axiom,
! [Xz: produc4523971788519308903rm_a_b,R3: set_Pr2972776593051762503rm_a_b,S: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ Xz @ ( relcom6159724393391289352rm_a_b @ R3 @ S ) )
=> ~ ! [X3: produc357393685978478089rm_a_b,Y2: produc357393685978478089rm_a_b,Z2: produc357393685978478089rm_a_b] :
( ( Xz
= ( produc1763473618796451543rm_a_b @ X3 @ Z2 ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X3 @ Y2 ) @ R3 )
=> ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y2 @ Z2 ) @ S ) ) ) ) ).
% relcompE
thf(fact_139_relcompE,axiom,
! [Xz: produc9001763001631046773rm_a_b,R3: set_Pr7151917219565317845rm_a_b,S: set_Pr7151917219565317845rm_a_b] :
( ( member8888773126115863070rm_a_b @ Xz @ ( relcom3320975603336820193rm_a_b @ R3 @ S ) )
=> ~ ! [X3: set_term_a_b,Y2: set_term_a_b,Z2: set_term_a_b] :
( ( Xz
= ( produc8517751122659631533rm_a_b @ X3 @ Z2 ) )
=> ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ X3 @ Y2 ) @ R3 )
=> ~ ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ Y2 @ Z2 ) @ S ) ) ) ) ).
% relcompE
thf(fact_140_relcompE,axiom,
! [Xz: produc3421996462152255079rm_a_b,R3: set_Pr647378613824986951rm_a_b,S: set_Pr647378613824986951rm_a_b] :
( ( member1375104729371992464rm_a_b @ Xz @ ( relcom8520403917366345832rm_a_b @ R3 @ S ) )
=> ~ ! [X3: set_Pr4386577575007340137rm_a_b,Y2: set_Pr4386577575007340137rm_a_b,Z2: set_Pr4386577575007340137rm_a_b] :
( ( Xz
= ( produc3977138704197412567rm_a_b @ X3 @ Z2 ) )
=> ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ X3 @ Y2 ) @ R3 )
=> ~ ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ Y2 @ Z2 ) @ S ) ) ) ) ).
% relcompE
thf(fact_141_relcompE,axiom,
! [Xz: produc357393685978478089rm_a_b,R3: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ Xz @ ( relcom370159955682700863rm_a_b @ R3 @ S ) )
=> ~ ! [X3: term_a_b,Y2: term_a_b,Z2: term_a_b] :
( ( Xz
= ( produc7020197800436672577rm_a_b @ X3 @ Z2 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y2 ) @ R3 )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y2 @ Z2 ) @ S ) ) ) ) ).
% relcompE
thf(fact_142_relcomp_OrelcompI,axiom,
! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,R3: set_Pr2972776593051762503rm_a_b,C2: produc357393685978478089rm_a_b,S: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ R3 )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B @ C2 ) @ S )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ C2 ) @ ( relcom6159724393391289352rm_a_b @ R3 @ S ) ) ) ) ).
% relcomp.relcompI
thf(fact_143_relcomp_OrelcompI,axiom,
! [A: set_term_a_b,B: set_term_a_b,R3: set_Pr7151917219565317845rm_a_b,C2: set_term_a_b,S: set_Pr7151917219565317845rm_a_b] :
( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A @ B ) @ R3 )
=> ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ B @ C2 ) @ S )
=> ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A @ C2 ) @ ( relcom3320975603336820193rm_a_b @ R3 @ S ) ) ) ) ).
% relcomp.relcompI
thf(fact_144_relcomp_OrelcompI,axiom,
! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b,R3: set_Pr647378613824986951rm_a_b,C2: set_Pr4386577575007340137rm_a_b,S: set_Pr647378613824986951rm_a_b] :
( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A @ B ) @ R3 )
=> ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ B @ C2 ) @ S )
=> ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A @ C2 ) @ ( relcom8520403917366345832rm_a_b @ R3 @ S ) ) ) ) ).
% relcomp.relcompI
thf(fact_145_relcomp_OrelcompI,axiom,
! [A: term_a_b,B: term_a_b,R3: set_Pr4386577575007340137rm_a_b,C2: term_a_b,S: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R3 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C2 ) @ S )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C2 ) @ ( relcom370159955682700863rm_a_b @ R3 @ S ) ) ) ) ).
% relcomp.relcompI
thf(fact_146_relcomp_Osimps,axiom,
! [A1: produc357393685978478089rm_a_b,A22: produc357393685978478089rm_a_b,R3: set_Pr2972776593051762503rm_a_b,S: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A1 @ A22 ) @ ( relcom6159724393391289352rm_a_b @ R3 @ S ) )
= ( ? [A4: produc357393685978478089rm_a_b,B4: produc357393685978478089rm_a_b,C3: produc357393685978478089rm_a_b] :
( ( A1 = A4 )
& ( A22 = C3 )
& ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A4 @ B4 ) @ R3 )
& ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B4 @ C3 ) @ S ) ) ) ) ).
% relcomp.simps
thf(fact_147_relcomp_Osimps,axiom,
! [A1: set_term_a_b,A22: set_term_a_b,R3: set_Pr7151917219565317845rm_a_b,S: set_Pr7151917219565317845rm_a_b] :
( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A1 @ A22 ) @ ( relcom3320975603336820193rm_a_b @ R3 @ S ) )
= ( ? [A4: set_term_a_b,B4: set_term_a_b,C3: set_term_a_b] :
( ( A1 = A4 )
& ( A22 = C3 )
& ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A4 @ B4 ) @ R3 )
& ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ B4 @ C3 ) @ S ) ) ) ) ).
% relcomp.simps
thf(fact_148_relcomp_Osimps,axiom,
! [A1: set_Pr4386577575007340137rm_a_b,A22: set_Pr4386577575007340137rm_a_b,R3: set_Pr647378613824986951rm_a_b,S: set_Pr647378613824986951rm_a_b] :
( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A1 @ A22 ) @ ( relcom8520403917366345832rm_a_b @ R3 @ S ) )
= ( ? [A4: set_Pr4386577575007340137rm_a_b,B4: set_Pr4386577575007340137rm_a_b,C3: set_Pr4386577575007340137rm_a_b] :
( ( A1 = A4 )
& ( A22 = C3 )
& ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A4 @ B4 ) @ R3 )
& ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ B4 @ C3 ) @ S ) ) ) ) ).
% relcomp.simps
thf(fact_149_relcomp_Osimps,axiom,
! [A1: term_a_b,A22: term_a_b,R3: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A22 ) @ ( relcom370159955682700863rm_a_b @ R3 @ S ) )
= ( ? [A4: term_a_b,B4: term_a_b,C3: term_a_b] :
( ( A1 = A4 )
& ( A22 = C3 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A4 @ B4 ) @ R3 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B4 @ C3 ) @ S ) ) ) ) ).
% relcomp.simps
thf(fact_150_relcomp_Ocases,axiom,
! [A1: produc357393685978478089rm_a_b,A22: produc357393685978478089rm_a_b,R3: set_Pr2972776593051762503rm_a_b,S: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A1 @ A22 ) @ ( relcom6159724393391289352rm_a_b @ R3 @ S ) )
=> ~ ! [B3: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A1 @ B3 ) @ R3 )
=> ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B3 @ A22 ) @ S ) ) ) ).
% relcomp.cases
thf(fact_151_relcomp_Ocases,axiom,
! [A1: set_term_a_b,A22: set_term_a_b,R3: set_Pr7151917219565317845rm_a_b,S: set_Pr7151917219565317845rm_a_b] :
( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A1 @ A22 ) @ ( relcom3320975603336820193rm_a_b @ R3 @ S ) )
=> ~ ! [B3: set_term_a_b] :
( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A1 @ B3 ) @ R3 )
=> ~ ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ B3 @ A22 ) @ S ) ) ) ).
% relcomp.cases
thf(fact_152_relcomp_Ocases,axiom,
! [A1: set_Pr4386577575007340137rm_a_b,A22: set_Pr4386577575007340137rm_a_b,R3: set_Pr647378613824986951rm_a_b,S: set_Pr647378613824986951rm_a_b] :
( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A1 @ A22 ) @ ( relcom8520403917366345832rm_a_b @ R3 @ S ) )
=> ~ ! [B3: set_Pr4386577575007340137rm_a_b] :
( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A1 @ B3 ) @ R3 )
=> ~ ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ B3 @ A22 ) @ S ) ) ) ).
% relcomp.cases
thf(fact_153_relcomp_Ocases,axiom,
! [A1: term_a_b,A22: term_a_b,R3: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A22 ) @ ( relcom370159955682700863rm_a_b @ R3 @ S ) )
=> ~ ! [B3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ B3 ) @ R3 )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B3 @ A22 ) @ S ) ) ) ).
% relcomp.cases
thf(fact_154_converseI,axiom,
! [A: set_term_a_b,B: set_term_a_b,R3: set_Pr7151917219565317845rm_a_b] :
( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A @ B ) @ R3 )
=> ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ B @ A ) @ ( conver1904685531124697809rm_a_b @ R3 ) ) ) ).
% converseI
thf(fact_155_converseI,axiom,
! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b,R3: set_Pr647378613824986951rm_a_b] :
( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A @ B ) @ R3 )
=> ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ B @ A ) @ ( conver7531182934069937659rm_a_b @ R3 ) ) ) ).
% converseI
thf(fact_156_converseI,axiom,
! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,R3: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ R3 )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B @ A ) @ ( conver1575690562132235771rm_a_b @ R3 ) ) ) ).
% converseI
thf(fact_157_converseI,axiom,
! [A: term_a_b,B: term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R3 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ ( conver7608559976937845093rm_a_b @ R3 ) ) ) ).
% converseI
thf(fact_158_converseE,axiom,
! [Yx: produc9001763001631046773rm_a_b,R3: set_Pr7151917219565317845rm_a_b] :
( ( member8888773126115863070rm_a_b @ Yx @ ( conver1904685531124697809rm_a_b @ R3 ) )
=> ~ ! [X3: set_term_a_b,Y2: set_term_a_b] :
( ( Yx
= ( produc8517751122659631533rm_a_b @ Y2 @ X3 ) )
=> ~ ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ X3 @ Y2 ) @ R3 ) ) ) ).
% converseE
thf(fact_159_converseE,axiom,
! [Yx: produc3421996462152255079rm_a_b,R3: set_Pr647378613824986951rm_a_b] :
( ( member1375104729371992464rm_a_b @ Yx @ ( conver7531182934069937659rm_a_b @ R3 ) )
=> ~ ! [X3: set_Pr4386577575007340137rm_a_b,Y2: set_Pr4386577575007340137rm_a_b] :
( ( Yx
= ( produc3977138704197412567rm_a_b @ Y2 @ X3 ) )
=> ~ ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ X3 @ Y2 ) @ R3 ) ) ) ).
% converseE
thf(fact_160_converseE,axiom,
! [Yx: produc4523971788519308903rm_a_b,R3: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ Yx @ ( conver1575690562132235771rm_a_b @ R3 ) )
=> ~ ! [X3: produc357393685978478089rm_a_b,Y2: produc357393685978478089rm_a_b] :
( ( Yx
= ( produc1763473618796451543rm_a_b @ Y2 @ X3 ) )
=> ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X3 @ Y2 ) @ R3 ) ) ) ).
% converseE
thf(fact_161_converseE,axiom,
! [Yx: produc357393685978478089rm_a_b,R3: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ Yx @ ( conver7608559976937845093rm_a_b @ R3 ) )
=> ~ ! [X3: term_a_b,Y2: term_a_b] :
( ( Yx
= ( produc7020197800436672577rm_a_b @ Y2 @ X3 ) )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y2 ) @ R3 ) ) ) ).
% converseE
thf(fact_162_converseD,axiom,
! [A: set_term_a_b,B: set_term_a_b,R3: set_Pr7151917219565317845rm_a_b] :
( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A @ B ) @ ( conver1904685531124697809rm_a_b @ R3 ) )
=> ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ B @ A ) @ R3 ) ) ).
% converseD
thf(fact_163_converseD,axiom,
! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b,R3: set_Pr647378613824986951rm_a_b] :
( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A @ B ) @ ( conver7531182934069937659rm_a_b @ R3 ) )
=> ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ B @ A ) @ R3 ) ) ).
% converseD
thf(fact_164_converseD,axiom,
! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,R3: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( conver1575690562132235771rm_a_b @ R3 ) )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B @ A ) @ R3 ) ) ).
% converseD
thf(fact_165_converseD,axiom,
! [A: term_a_b,B: term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( conver7608559976937845093rm_a_b @ R3 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ R3 ) ) ).
% converseD
thf(fact_166_converse_Osimps,axiom,
! [A1: set_term_a_b,A22: set_term_a_b,R3: set_Pr7151917219565317845rm_a_b] :
( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A1 @ A22 ) @ ( conver1904685531124697809rm_a_b @ R3 ) )
= ( ? [A4: set_term_a_b,B4: set_term_a_b] :
( ( A1 = B4 )
& ( A22 = A4 )
& ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A4 @ B4 ) @ R3 ) ) ) ) ).
% converse.simps
thf(fact_167_converse_Osimps,axiom,
! [A1: set_Pr4386577575007340137rm_a_b,A22: set_Pr4386577575007340137rm_a_b,R3: set_Pr647378613824986951rm_a_b] :
( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A1 @ A22 ) @ ( conver7531182934069937659rm_a_b @ R3 ) )
= ( ? [A4: set_Pr4386577575007340137rm_a_b,B4: set_Pr4386577575007340137rm_a_b] :
( ( A1 = B4 )
& ( A22 = A4 )
& ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A4 @ B4 ) @ R3 ) ) ) ) ).
% converse.simps
thf(fact_168_converse_Osimps,axiom,
! [A1: produc357393685978478089rm_a_b,A22: produc357393685978478089rm_a_b,R3: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A1 @ A22 ) @ ( conver1575690562132235771rm_a_b @ R3 ) )
= ( ? [A4: produc357393685978478089rm_a_b,B4: produc357393685978478089rm_a_b] :
( ( A1 = B4 )
& ( A22 = A4 )
& ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A4 @ B4 ) @ R3 ) ) ) ) ).
% converse.simps
thf(fact_169_converse_Osimps,axiom,
! [A1: term_a_b,A22: term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A22 ) @ ( conver7608559976937845093rm_a_b @ R3 ) )
= ( ? [A4: term_a_b,B4: term_a_b] :
( ( A1 = B4 )
& ( A22 = A4 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A4 @ B4 ) @ R3 ) ) ) ) ).
% converse.simps
thf(fact_170_converse_Ocases,axiom,
! [A1: set_term_a_b,A22: set_term_a_b,R3: set_Pr7151917219565317845rm_a_b] :
( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A1 @ A22 ) @ ( conver1904685531124697809rm_a_b @ R3 ) )
=> ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A22 @ A1 ) @ R3 ) ) ).
% converse.cases
thf(fact_171_converse_Ocases,axiom,
! [A1: set_Pr4386577575007340137rm_a_b,A22: set_Pr4386577575007340137rm_a_b,R3: set_Pr647378613824986951rm_a_b] :
( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A1 @ A22 ) @ ( conver7531182934069937659rm_a_b @ R3 ) )
=> ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A22 @ A1 ) @ R3 ) ) ).
% converse.cases
thf(fact_172_converse_Ocases,axiom,
! [A1: produc357393685978478089rm_a_b,A22: produc357393685978478089rm_a_b,R3: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A1 @ A22 ) @ ( conver1575690562132235771rm_a_b @ R3 ) )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A22 @ A1 ) @ R3 ) ) ).
% converse.cases
thf(fact_173_converse_Ocases,axiom,
! [A1: term_a_b,A22: term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A22 ) @ ( conver7608559976937845093rm_a_b @ R3 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A22 @ A1 ) @ R3 ) ) ).
% converse.cases
thf(fact_174_converse__trancl__induct,axiom,
! [A: set_term_a_b,B: set_term_a_b,R3: set_Pr7151917219565317845rm_a_b,P2: set_term_a_b > $o] :
( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A @ B ) @ ( transi2170614581841672169rm_a_b @ R3 ) )
=> ( ! [Y2: set_term_a_b] :
( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ Y2 @ B ) @ R3 )
=> ( P2 @ Y2 ) )
=> ( ! [Y2: set_term_a_b,Z2: set_term_a_b] :
( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ Y2 @ Z2 ) @ R3 )
=> ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ Z2 @ B ) @ ( transi2170614581841672169rm_a_b @ R3 ) )
=> ( ( P2 @ Z2 )
=> ( P2 @ Y2 ) ) ) )
=> ( P2 @ A ) ) ) ) ).
% converse_trancl_induct
thf(fact_175_converse__trancl__induct,axiom,
! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b,R3: set_Pr647378613824986951rm_a_b,P2: set_Pr4386577575007340137rm_a_b > $o] :
( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A @ B ) @ ( transi6883191930764840330rm_a_b @ R3 ) )
=> ( ! [Y2: set_Pr4386577575007340137rm_a_b] :
( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ Y2 @ B ) @ R3 )
=> ( P2 @ Y2 ) )
=> ( ! [Y2: set_Pr4386577575007340137rm_a_b,Z2: set_Pr4386577575007340137rm_a_b] :
( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ Y2 @ Z2 ) @ R3 )
=> ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ Z2 @ B ) @ ( transi6883191930764840330rm_a_b @ R3 ) )
=> ( ( P2 @ Z2 )
=> ( P2 @ Y2 ) ) ) )
=> ( P2 @ A ) ) ) ) ).
% converse_trancl_induct
thf(fact_176_converse__trancl__induct,axiom,
! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,R3: set_Pr2972776593051762503rm_a_b,P2: produc357393685978478089rm_a_b > $o] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( transi9211502839322181930rm_a_b @ R3 ) )
=> ( ! [Y2: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y2 @ B ) @ R3 )
=> ( P2 @ Y2 ) )
=> ( ! [Y2: produc357393685978478089rm_a_b,Z2: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y2 @ Z2 ) @ R3 )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Z2 @ B ) @ ( transi9211502839322181930rm_a_b @ R3 ) )
=> ( ( P2 @ Z2 )
=> ( P2 @ Y2 ) ) ) )
=> ( P2 @ A ) ) ) ) ).
% converse_trancl_induct
thf(fact_177_converse__trancl__induct,axiom,
! [A: term_a_b,B: term_a_b,R3: set_Pr4386577575007340137rm_a_b,P2: term_a_b > $o] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7922773638565587891rm_a_b @ R3 ) )
=> ( ! [Y2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y2 @ B ) @ R3 )
=> ( P2 @ Y2 ) )
=> ( ! [Y2: term_a_b,Z2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y2 @ Z2 ) @ R3 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Z2 @ B ) @ ( transi7922773638565587891rm_a_b @ R3 ) )
=> ( ( P2 @ Z2 )
=> ( P2 @ Y2 ) ) ) )
=> ( P2 @ A ) ) ) ) ).
% converse_trancl_induct
thf(fact_178_trancl__trans__induct,axiom,
! [X: set_term_a_b,Y: set_term_a_b,R3: set_Pr7151917219565317845rm_a_b,P2: set_term_a_b > set_term_a_b > $o] :
( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ X @ Y ) @ ( transi2170614581841672169rm_a_b @ R3 ) )
=> ( ! [X3: set_term_a_b,Y2: set_term_a_b] :
( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ X3 @ Y2 ) @ R3 )
=> ( P2 @ X3 @ Y2 ) )
=> ( ! [X3: set_term_a_b,Y2: set_term_a_b,Z2: set_term_a_b] :
( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ X3 @ Y2 ) @ ( transi2170614581841672169rm_a_b @ R3 ) )
=> ( ( P2 @ X3 @ Y2 )
=> ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ Y2 @ Z2 ) @ ( transi2170614581841672169rm_a_b @ R3 ) )
=> ( ( P2 @ Y2 @ Z2 )
=> ( P2 @ X3 @ Z2 ) ) ) ) )
=> ( P2 @ X @ Y ) ) ) ) ).
% trancl_trans_induct
thf(fact_179_trancl__trans__induct,axiom,
! [X: set_Pr4386577575007340137rm_a_b,Y: set_Pr4386577575007340137rm_a_b,R3: set_Pr647378613824986951rm_a_b,P2: set_Pr4386577575007340137rm_a_b > set_Pr4386577575007340137rm_a_b > $o] :
( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ X @ Y ) @ ( transi6883191930764840330rm_a_b @ R3 ) )
=> ( ! [X3: set_Pr4386577575007340137rm_a_b,Y2: set_Pr4386577575007340137rm_a_b] :
( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ X3 @ Y2 ) @ R3 )
=> ( P2 @ X3 @ Y2 ) )
=> ( ! [X3: set_Pr4386577575007340137rm_a_b,Y2: set_Pr4386577575007340137rm_a_b,Z2: set_Pr4386577575007340137rm_a_b] :
( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ X3 @ Y2 ) @ ( transi6883191930764840330rm_a_b @ R3 ) )
=> ( ( P2 @ X3 @ Y2 )
=> ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ Y2 @ Z2 ) @ ( transi6883191930764840330rm_a_b @ R3 ) )
=> ( ( P2 @ Y2 @ Z2 )
=> ( P2 @ X3 @ Z2 ) ) ) ) )
=> ( P2 @ X @ Y ) ) ) ) ).
% trancl_trans_induct
thf(fact_180_trancl__trans__induct,axiom,
! [X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b,R3: set_Pr2972776593051762503rm_a_b,P2: produc357393685978478089rm_a_b > produc357393685978478089rm_a_b > $o] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ ( transi9211502839322181930rm_a_b @ R3 ) )
=> ( ! [X3: produc357393685978478089rm_a_b,Y2: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X3 @ Y2 ) @ R3 )
=> ( P2 @ X3 @ Y2 ) )
=> ( ! [X3: produc357393685978478089rm_a_b,Y2: produc357393685978478089rm_a_b,Z2: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X3 @ Y2 ) @ ( transi9211502839322181930rm_a_b @ R3 ) )
=> ( ( P2 @ X3 @ Y2 )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y2 @ Z2 ) @ ( transi9211502839322181930rm_a_b @ R3 ) )
=> ( ( P2 @ Y2 @ Z2 )
=> ( P2 @ X3 @ Z2 ) ) ) ) )
=> ( P2 @ X @ Y ) ) ) ) ).
% trancl_trans_induct
thf(fact_181_trancl__trans__induct,axiom,
! [X: term_a_b,Y: term_a_b,R3: set_Pr4386577575007340137rm_a_b,P2: term_a_b > term_a_b > $o] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7922773638565587891rm_a_b @ R3 ) )
=> ( ! [X3: term_a_b,Y2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y2 ) @ R3 )
=> ( P2 @ X3 @ Y2 ) )
=> ( ! [X3: term_a_b,Y2: term_a_b,Z2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y2 ) @ ( transi7922773638565587891rm_a_b @ R3 ) )
=> ( ( P2 @ X3 @ Y2 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y2 @ Z2 ) @ ( transi7922773638565587891rm_a_b @ R3 ) )
=> ( ( P2 @ Y2 @ Z2 )
=> ( P2 @ X3 @ Z2 ) ) ) ) )
=> ( P2 @ X @ Y ) ) ) ) ).
% trancl_trans_induct
thf(fact_182_trancl__into__trancl2,axiom,
! [A: set_term_a_b,B: set_term_a_b,R3: set_Pr7151917219565317845rm_a_b,C2: set_term_a_b] :
( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A @ B ) @ R3 )
=> ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ B @ C2 ) @ ( transi2170614581841672169rm_a_b @ R3 ) )
=> ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A @ C2 ) @ ( transi2170614581841672169rm_a_b @ R3 ) ) ) ) ).
% trancl_into_trancl2
thf(fact_183_trancl__into__trancl2,axiom,
! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b,R3: set_Pr647378613824986951rm_a_b,C2: set_Pr4386577575007340137rm_a_b] :
( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A @ B ) @ R3 )
=> ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ B @ C2 ) @ ( transi6883191930764840330rm_a_b @ R3 ) )
=> ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A @ C2 ) @ ( transi6883191930764840330rm_a_b @ R3 ) ) ) ) ).
% trancl_into_trancl2
thf(fact_184_trancl__into__trancl2,axiom,
! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,R3: set_Pr2972776593051762503rm_a_b,C2: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ R3 )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B @ C2 ) @ ( transi9211502839322181930rm_a_b @ R3 ) )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ C2 ) @ ( transi9211502839322181930rm_a_b @ R3 ) ) ) ) ).
% trancl_into_trancl2
thf(fact_185_trancl__into__trancl2,axiom,
! [A: term_a_b,B: term_a_b,R3: set_Pr4386577575007340137rm_a_b,C2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R3 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C2 ) @ ( transi7922773638565587891rm_a_b @ R3 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C2 ) @ ( transi7922773638565587891rm_a_b @ R3 ) ) ) ) ).
% trancl_into_trancl2
thf(fact_186_Transitive__Closure_Otrancl__into__trancl,axiom,
! [A: set_term_a_b,B: set_term_a_b,R3: set_Pr7151917219565317845rm_a_b,C2: set_term_a_b] :
( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A @ B ) @ ( transi2170614581841672169rm_a_b @ R3 ) )
=> ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ B @ C2 ) @ R3 )
=> ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A @ C2 ) @ ( transi2170614581841672169rm_a_b @ R3 ) ) ) ) ).
% Transitive_Closure.trancl_into_trancl
thf(fact_187_Transitive__Closure_Otrancl__into__trancl,axiom,
! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b,R3: set_Pr647378613824986951rm_a_b,C2: set_Pr4386577575007340137rm_a_b] :
( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A @ B ) @ ( transi6883191930764840330rm_a_b @ R3 ) )
=> ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ B @ C2 ) @ R3 )
=> ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A @ C2 ) @ ( transi6883191930764840330rm_a_b @ R3 ) ) ) ) ).
% Transitive_Closure.trancl_into_trancl
thf(fact_188_Transitive__Closure_Otrancl__into__trancl,axiom,
! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,R3: set_Pr2972776593051762503rm_a_b,C2: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( transi9211502839322181930rm_a_b @ R3 ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B @ C2 ) @ R3 )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ C2 ) @ ( transi9211502839322181930rm_a_b @ R3 ) ) ) ) ).
% Transitive_Closure.trancl_into_trancl
thf(fact_189_Transitive__Closure_Otrancl__into__trancl,axiom,
! [A: term_a_b,B: term_a_b,R3: set_Pr4386577575007340137rm_a_b,C2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7922773638565587891rm_a_b @ R3 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C2 ) @ R3 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C2 ) @ ( transi7922773638565587891rm_a_b @ R3 ) ) ) ) ).
% Transitive_Closure.trancl_into_trancl
thf(fact_190_irrefl__trancl__rD,axiom,
! [R3: set_Pr7151917219565317845rm_a_b,X: set_term_a_b,Y: set_term_a_b] :
( ! [X3: set_term_a_b] :
~ ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ X3 @ X3 ) @ ( transi2170614581841672169rm_a_b @ R3 ) )
=> ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ X @ Y ) @ R3 )
=> ( X != Y ) ) ) ).
% irrefl_trancl_rD
thf(fact_191_irrefl__trancl__rD,axiom,
! [R3: set_Pr647378613824986951rm_a_b,X: set_Pr4386577575007340137rm_a_b,Y: set_Pr4386577575007340137rm_a_b] :
( ! [X3: set_Pr4386577575007340137rm_a_b] :
~ ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ X3 @ X3 ) @ ( transi6883191930764840330rm_a_b @ R3 ) )
=> ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ X @ Y ) @ R3 )
=> ( X != Y ) ) ) ).
% irrefl_trancl_rD
thf(fact_192_irrefl__trancl__rD,axiom,
! [R3: set_Pr2972776593051762503rm_a_b,X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b] :
( ! [X3: produc357393685978478089rm_a_b] :
~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X3 @ X3 ) @ ( transi9211502839322181930rm_a_b @ R3 ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ R3 )
=> ( X != Y ) ) ) ).
% irrefl_trancl_rD
thf(fact_193_irrefl__trancl__rD,axiom,
! [R3: set_Pr4386577575007340137rm_a_b,X: term_a_b,Y: term_a_b] :
( ! [X3: term_a_b] :
~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ X3 ) @ ( transi7922773638565587891rm_a_b @ R3 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ R3 )
=> ( X != Y ) ) ) ).
% irrefl_trancl_rD
thf(fact_194_converse__tranclE,axiom,
! [X: set_term_a_b,Z: set_term_a_b,R3: set_Pr7151917219565317845rm_a_b] :
( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ X @ Z ) @ ( transi2170614581841672169rm_a_b @ R3 ) )
=> ( ~ ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ X @ Z ) @ R3 )
=> ~ ! [Y2: set_term_a_b] :
( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ X @ Y2 ) @ R3 )
=> ~ ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ Y2 @ Z ) @ ( transi2170614581841672169rm_a_b @ R3 ) ) ) ) ) ).
% converse_tranclE
thf(fact_195_converse__tranclE,axiom,
! [X: set_Pr4386577575007340137rm_a_b,Z: set_Pr4386577575007340137rm_a_b,R3: set_Pr647378613824986951rm_a_b] :
( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ X @ Z ) @ ( transi6883191930764840330rm_a_b @ R3 ) )
=> ( ~ ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ X @ Z ) @ R3 )
=> ~ ! [Y2: set_Pr4386577575007340137rm_a_b] :
( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ X @ Y2 ) @ R3 )
=> ~ ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ Y2 @ Z ) @ ( transi6883191930764840330rm_a_b @ R3 ) ) ) ) ) ).
% converse_tranclE
thf(fact_196_converse__tranclE,axiom,
! [X: produc357393685978478089rm_a_b,Z: produc357393685978478089rm_a_b,R3: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Z ) @ ( transi9211502839322181930rm_a_b @ R3 ) )
=> ( ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Z ) @ R3 )
=> ~ ! [Y2: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y2 ) @ R3 )
=> ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y2 @ Z ) @ ( transi9211502839322181930rm_a_b @ R3 ) ) ) ) ) ).
% converse_tranclE
thf(fact_197_converse__tranclE,axiom,
! [X: term_a_b,Z: term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z ) @ ( transi7922773638565587891rm_a_b @ R3 ) )
=> ( ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z ) @ R3 )
=> ~ ! [Y2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y2 ) @ R3 )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y2 @ Z ) @ ( transi7922773638565587891rm_a_b @ R3 ) ) ) ) ) ).
% converse_tranclE
thf(fact_198_r__r__into__trancl,axiom,
! [A: set_term_a_b,B: set_term_a_b,R4: set_Pr7151917219565317845rm_a_b,C2: set_term_a_b] :
( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A @ B ) @ R4 )
=> ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ B @ C2 ) @ R4 )
=> ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A @ C2 ) @ ( transi2170614581841672169rm_a_b @ R4 ) ) ) ) ).
% r_r_into_trancl
thf(fact_199_r__r__into__trancl,axiom,
! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b,R4: set_Pr647378613824986951rm_a_b,C2: set_Pr4386577575007340137rm_a_b] :
( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A @ B ) @ R4 )
=> ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ B @ C2 ) @ R4 )
=> ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A @ C2 ) @ ( transi6883191930764840330rm_a_b @ R4 ) ) ) ) ).
% r_r_into_trancl
thf(fact_200_r__r__into__trancl,axiom,
! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,R4: set_Pr2972776593051762503rm_a_b,C2: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ R4 )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B @ C2 ) @ R4 )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ C2 ) @ ( transi9211502839322181930rm_a_b @ R4 ) ) ) ) ).
% r_r_into_trancl
thf(fact_201_r__r__into__trancl,axiom,
! [A: term_a_b,B: term_a_b,R4: set_Pr4386577575007340137rm_a_b,C2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R4 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C2 ) @ R4 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C2 ) @ ( transi7922773638565587891rm_a_b @ R4 ) ) ) ) ).
% r_r_into_trancl
thf(fact_202_trancl__induct,axiom,
! [A: set_term_a_b,B: set_term_a_b,R3: set_Pr7151917219565317845rm_a_b,P2: set_term_a_b > $o] :
( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A @ B ) @ ( transi2170614581841672169rm_a_b @ R3 ) )
=> ( ! [Y2: set_term_a_b] :
( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A @ Y2 ) @ R3 )
=> ( P2 @ Y2 ) )
=> ( ! [Y2: set_term_a_b,Z2: set_term_a_b] :
( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A @ Y2 ) @ ( transi2170614581841672169rm_a_b @ R3 ) )
=> ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ Y2 @ Z2 ) @ R3 )
=> ( ( P2 @ Y2 )
=> ( P2 @ Z2 ) ) ) )
=> ( P2 @ B ) ) ) ) ).
% trancl_induct
thf(fact_203_trancl__induct,axiom,
! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b,R3: set_Pr647378613824986951rm_a_b,P2: set_Pr4386577575007340137rm_a_b > $o] :
( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A @ B ) @ ( transi6883191930764840330rm_a_b @ R3 ) )
=> ( ! [Y2: set_Pr4386577575007340137rm_a_b] :
( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A @ Y2 ) @ R3 )
=> ( P2 @ Y2 ) )
=> ( ! [Y2: set_Pr4386577575007340137rm_a_b,Z2: set_Pr4386577575007340137rm_a_b] :
( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A @ Y2 ) @ ( transi6883191930764840330rm_a_b @ R3 ) )
=> ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ Y2 @ Z2 ) @ R3 )
=> ( ( P2 @ Y2 )
=> ( P2 @ Z2 ) ) ) )
=> ( P2 @ B ) ) ) ) ).
% trancl_induct
thf(fact_204_trancl__induct,axiom,
! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,R3: set_Pr2972776593051762503rm_a_b,P2: produc357393685978478089rm_a_b > $o] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( transi9211502839322181930rm_a_b @ R3 ) )
=> ( ! [Y2: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ Y2 ) @ R3 )
=> ( P2 @ Y2 ) )
=> ( ! [Y2: produc357393685978478089rm_a_b,Z2: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ Y2 ) @ ( transi9211502839322181930rm_a_b @ R3 ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y2 @ Z2 ) @ R3 )
=> ( ( P2 @ Y2 )
=> ( P2 @ Z2 ) ) ) )
=> ( P2 @ B ) ) ) ) ).
% trancl_induct
thf(fact_205_trancl__induct,axiom,
! [A: term_a_b,B: term_a_b,R3: set_Pr4386577575007340137rm_a_b,P2: term_a_b > $o] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7922773638565587891rm_a_b @ R3 ) )
=> ( ! [Y2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ Y2 ) @ R3 )
=> ( P2 @ Y2 ) )
=> ( ! [Y2: term_a_b,Z2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ Y2 ) @ ( transi7922773638565587891rm_a_b @ R3 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y2 @ Z2 ) @ R3 )
=> ( ( P2 @ Y2 )
=> ( P2 @ Z2 ) ) ) )
=> ( P2 @ B ) ) ) ) ).
% trancl_induct
thf(fact_206_trancl__trans,axiom,
! [X: set_term_a_b,Y: set_term_a_b,R3: set_Pr7151917219565317845rm_a_b,Z: set_term_a_b] :
( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ X @ Y ) @ ( transi2170614581841672169rm_a_b @ R3 ) )
=> ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ Y @ Z ) @ ( transi2170614581841672169rm_a_b @ R3 ) )
=> ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ X @ Z ) @ ( transi2170614581841672169rm_a_b @ R3 ) ) ) ) ).
% trancl_trans
thf(fact_207_trancl__trans,axiom,
! [X: set_Pr4386577575007340137rm_a_b,Y: set_Pr4386577575007340137rm_a_b,R3: set_Pr647378613824986951rm_a_b,Z: set_Pr4386577575007340137rm_a_b] :
( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ X @ Y ) @ ( transi6883191930764840330rm_a_b @ R3 ) )
=> ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ Y @ Z ) @ ( transi6883191930764840330rm_a_b @ R3 ) )
=> ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ X @ Z ) @ ( transi6883191930764840330rm_a_b @ R3 ) ) ) ) ).
% trancl_trans
thf(fact_208_trancl__trans,axiom,
! [X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b,R3: set_Pr2972776593051762503rm_a_b,Z: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ ( transi9211502839322181930rm_a_b @ R3 ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y @ Z ) @ ( transi9211502839322181930rm_a_b @ R3 ) )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Z ) @ ( transi9211502839322181930rm_a_b @ R3 ) ) ) ) ).
% trancl_trans
thf(fact_209_trancl__trans,axiom,
! [X: term_a_b,Y: term_a_b,R3: set_Pr4386577575007340137rm_a_b,Z: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7922773638565587891rm_a_b @ R3 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ Z ) @ ( transi7922773638565587891rm_a_b @ R3 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z ) @ ( transi7922773638565587891rm_a_b @ R3 ) ) ) ) ).
% trancl_trans
thf(fact_210_tranclE,axiom,
! [A: set_term_a_b,B: set_term_a_b,R3: set_Pr7151917219565317845rm_a_b] :
( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A @ B ) @ ( transi2170614581841672169rm_a_b @ R3 ) )
=> ( ~ ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A @ B ) @ R3 )
=> ~ ! [C4: set_term_a_b] :
( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A @ C4 ) @ ( transi2170614581841672169rm_a_b @ R3 ) )
=> ~ ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ C4 @ B ) @ R3 ) ) ) ) ).
% tranclE
thf(fact_211_tranclE,axiom,
! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b,R3: set_Pr647378613824986951rm_a_b] :
( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A @ B ) @ ( transi6883191930764840330rm_a_b @ R3 ) )
=> ( ~ ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A @ B ) @ R3 )
=> ~ ! [C4: set_Pr4386577575007340137rm_a_b] :
( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A @ C4 ) @ ( transi6883191930764840330rm_a_b @ R3 ) )
=> ~ ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ C4 @ B ) @ R3 ) ) ) ) ).
% tranclE
thf(fact_212_tranclE,axiom,
! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,R3: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( transi9211502839322181930rm_a_b @ R3 ) )
=> ( ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ R3 )
=> ~ ! [C4: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ C4 ) @ ( transi9211502839322181930rm_a_b @ R3 ) )
=> ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ C4 @ B ) @ R3 ) ) ) ) ).
% tranclE
thf(fact_213_tranclE,axiom,
! [A: term_a_b,B: term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7922773638565587891rm_a_b @ R3 ) )
=> ( ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R3 )
=> ~ ! [C4: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C4 ) @ ( transi7922773638565587891rm_a_b @ R3 ) )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ C4 @ B ) @ R3 ) ) ) ) ).
% tranclE
thf(fact_214_trancl_Or__into__trancl,axiom,
! [A: set_term_a_b,B: set_term_a_b,R3: set_Pr7151917219565317845rm_a_b] :
( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A @ B ) @ R3 )
=> ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A @ B ) @ ( transi2170614581841672169rm_a_b @ R3 ) ) ) ).
% trancl.r_into_trancl
thf(fact_215_trancl_Or__into__trancl,axiom,
! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b,R3: set_Pr647378613824986951rm_a_b] :
( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A @ B ) @ R3 )
=> ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A @ B ) @ ( transi6883191930764840330rm_a_b @ R3 ) ) ) ).
% trancl.r_into_trancl
thf(fact_216_trancl_Or__into__trancl,axiom,
! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,R3: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ R3 )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( transi9211502839322181930rm_a_b @ R3 ) ) ) ).
% trancl.r_into_trancl
thf(fact_217_trancl_Or__into__trancl,axiom,
! [A: term_a_b,B: term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R3 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7922773638565587891rm_a_b @ R3 ) ) ) ).
% trancl.r_into_trancl
thf(fact_218_trancl_Osimps,axiom,
! [A1: set_term_a_b,A22: set_term_a_b,R3: set_Pr7151917219565317845rm_a_b] :
( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A1 @ A22 ) @ ( transi2170614581841672169rm_a_b @ R3 ) )
= ( ? [A4: set_term_a_b,B4: set_term_a_b] :
( ( A1 = A4 )
& ( A22 = B4 )
& ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A4 @ B4 ) @ R3 ) )
| ? [A4: set_term_a_b,B4: set_term_a_b,C3: set_term_a_b] :
( ( A1 = A4 )
& ( A22 = C3 )
& ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A4 @ B4 ) @ ( transi2170614581841672169rm_a_b @ R3 ) )
& ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ B4 @ C3 ) @ R3 ) ) ) ) ).
% trancl.simps
thf(fact_219_trancl_Osimps,axiom,
! [A1: set_Pr4386577575007340137rm_a_b,A22: set_Pr4386577575007340137rm_a_b,R3: set_Pr647378613824986951rm_a_b] :
( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A1 @ A22 ) @ ( transi6883191930764840330rm_a_b @ R3 ) )
= ( ? [A4: set_Pr4386577575007340137rm_a_b,B4: set_Pr4386577575007340137rm_a_b] :
( ( A1 = A4 )
& ( A22 = B4 )
& ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A4 @ B4 ) @ R3 ) )
| ? [A4: set_Pr4386577575007340137rm_a_b,B4: set_Pr4386577575007340137rm_a_b,C3: set_Pr4386577575007340137rm_a_b] :
( ( A1 = A4 )
& ( A22 = C3 )
& ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A4 @ B4 ) @ ( transi6883191930764840330rm_a_b @ R3 ) )
& ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ B4 @ C3 ) @ R3 ) ) ) ) ).
% trancl.simps
thf(fact_220_trancl_Osimps,axiom,
! [A1: produc357393685978478089rm_a_b,A22: produc357393685978478089rm_a_b,R3: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A1 @ A22 ) @ ( transi9211502839322181930rm_a_b @ R3 ) )
= ( ? [A4: produc357393685978478089rm_a_b,B4: produc357393685978478089rm_a_b] :
( ( A1 = A4 )
& ( A22 = B4 )
& ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A4 @ B4 ) @ R3 ) )
| ? [A4: produc357393685978478089rm_a_b,B4: produc357393685978478089rm_a_b,C3: produc357393685978478089rm_a_b] :
( ( A1 = A4 )
& ( A22 = C3 )
& ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A4 @ B4 ) @ ( transi9211502839322181930rm_a_b @ R3 ) )
& ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B4 @ C3 ) @ R3 ) ) ) ) ).
% trancl.simps
thf(fact_221_trancl_Osimps,axiom,
! [A1: term_a_b,A22: term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A22 ) @ ( transi7922773638565587891rm_a_b @ R3 ) )
= ( ? [A4: term_a_b,B4: term_a_b] :
( ( A1 = A4 )
& ( A22 = B4 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A4 @ B4 ) @ R3 ) )
| ? [A4: term_a_b,B4: term_a_b,C3: term_a_b] :
( ( A1 = A4 )
& ( A22 = C3 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A4 @ B4 ) @ ( transi7922773638565587891rm_a_b @ R3 ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B4 @ C3 ) @ R3 ) ) ) ) ).
% trancl.simps
thf(fact_222_trancl_Ocases,axiom,
! [A1: set_term_a_b,A22: set_term_a_b,R3: set_Pr7151917219565317845rm_a_b] :
( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A1 @ A22 ) @ ( transi2170614581841672169rm_a_b @ R3 ) )
=> ( ~ ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A1 @ A22 ) @ R3 )
=> ~ ! [B3: set_term_a_b] :
( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A1 @ B3 ) @ ( transi2170614581841672169rm_a_b @ R3 ) )
=> ~ ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ B3 @ A22 ) @ R3 ) ) ) ) ).
% trancl.cases
thf(fact_223_trancl_Ocases,axiom,
! [A1: set_Pr4386577575007340137rm_a_b,A22: set_Pr4386577575007340137rm_a_b,R3: set_Pr647378613824986951rm_a_b] :
( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A1 @ A22 ) @ ( transi6883191930764840330rm_a_b @ R3 ) )
=> ( ~ ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A1 @ A22 ) @ R3 )
=> ~ ! [B3: set_Pr4386577575007340137rm_a_b] :
( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A1 @ B3 ) @ ( transi6883191930764840330rm_a_b @ R3 ) )
=> ~ ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ B3 @ A22 ) @ R3 ) ) ) ) ).
% trancl.cases
thf(fact_224_trancl_Ocases,axiom,
! [A1: produc357393685978478089rm_a_b,A22: produc357393685978478089rm_a_b,R3: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A1 @ A22 ) @ ( transi9211502839322181930rm_a_b @ R3 ) )
=> ( ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A1 @ A22 ) @ R3 )
=> ~ ! [B3: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A1 @ B3 ) @ ( transi9211502839322181930rm_a_b @ R3 ) )
=> ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B3 @ A22 ) @ R3 ) ) ) ) ).
% trancl.cases
thf(fact_225_trancl_Ocases,axiom,
! [A1: term_a_b,A22: term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A22 ) @ ( transi7922773638565587891rm_a_b @ R3 ) )
=> ( ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A22 ) @ R3 )
=> ~ ! [B3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ B3 ) @ ( transi7922773638565587891rm_a_b @ R3 ) )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B3 @ A22 ) @ R3 ) ) ) ) ).
% trancl.cases
thf(fact_226_r__comp__rtrancl__eq,axiom,
! [R3: set_Pr7151917219565317845rm_a_b] :
( ( relcom3320975603336820193rm_a_b @ R3 @ ( transi6904761420441344775rm_a_b @ R3 ) )
= ( relcom3320975603336820193rm_a_b @ ( transi6904761420441344775rm_a_b @ R3 ) @ R3 ) ) ).
% r_comp_rtrancl_eq
thf(fact_227_r__comp__rtrancl__eq,axiom,
! [R3: set_Pr2972776593051762503rm_a_b] :
( ( relcom6159724393391289352rm_a_b @ R3 @ ( transi2615809358984392588rm_a_b @ R3 ) )
= ( relcom6159724393391289352rm_a_b @ ( transi2615809358984392588rm_a_b @ R3 ) @ R3 ) ) ).
% r_comp_rtrancl_eq
thf(fact_228_r__comp__rtrancl__eq,axiom,
! [R3: set_Pr4386577575007340137rm_a_b] :
( ( relcom370159955682700863rm_a_b @ R3 @ ( transi7742714808557438673rm_a_b @ R3 ) )
= ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ R3 ) @ R3 ) ) ).
% r_comp_rtrancl_eq
thf(fact_229_rtrancl__O__shift,axiom,
! [S6: set_Pr7151917219565317845rm_a_b,R4: set_Pr7151917219565317845rm_a_b] :
( ( relcom3320975603336820193rm_a_b @ ( transi6904761420441344775rm_a_b @ ( relcom3320975603336820193rm_a_b @ S6 @ R4 ) ) @ S6 )
= ( relcom3320975603336820193rm_a_b @ S6 @ ( transi6904761420441344775rm_a_b @ ( relcom3320975603336820193rm_a_b @ R4 @ S6 ) ) ) ) ).
% rtrancl_O_shift
thf(fact_230_rtrancl__O__shift,axiom,
! [S6: set_Pr4802440762858034464rm_a_b,R4: set_Pr6510905305361987970rm_a_b] :
( ( relcom6590495161205167518rm_a_b @ ( transi7742714808557438673rm_a_b @ ( relcom8422713090239460168rm_a_b @ S6 @ R4 ) ) @ S6 )
= ( relcom4000041261975091669rm_a_b @ S6 @ ( transi2615809358984392588rm_a_b @ ( relcom8086516547131752875rm_a_b @ R4 @ S6 ) ) ) ) ).
% rtrancl_O_shift
thf(fact_231_rtrancl__O__shift,axiom,
! [S6: set_Pr6510905305361987970rm_a_b,R4: set_Pr4802440762858034464rm_a_b] :
( ( relcom695362439311269717rm_a_b @ ( transi2615809358984392588rm_a_b @ ( relcom8086516547131752875rm_a_b @ S6 @ R4 ) ) @ S6 )
= ( relcom3777457926417599474rm_a_b @ S6 @ ( transi7742714808557438673rm_a_b @ ( relcom8422713090239460168rm_a_b @ R4 @ S6 ) ) ) ) ).
% rtrancl_O_shift
thf(fact_232_rtrancl__O__shift,axiom,
! [S6: set_Pr2972776593051762503rm_a_b,R4: set_Pr2972776593051762503rm_a_b] :
( ( relcom6159724393391289352rm_a_b @ ( transi2615809358984392588rm_a_b @ ( relcom6159724393391289352rm_a_b @ S6 @ R4 ) ) @ S6 )
= ( relcom6159724393391289352rm_a_b @ S6 @ ( transi2615809358984392588rm_a_b @ ( relcom6159724393391289352rm_a_b @ R4 @ S6 ) ) ) ) ).
% rtrancl_O_shift
thf(fact_233_rtrancl__O__shift,axiom,
! [S6: set_Pr4386577575007340137rm_a_b,R4: set_Pr4386577575007340137rm_a_b] :
( ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( relcom370159955682700863rm_a_b @ S6 @ R4 ) ) @ S6 )
= ( relcom370159955682700863rm_a_b @ S6 @ ( transi7742714808557438673rm_a_b @ ( relcom370159955682700863rm_a_b @ R4 @ S6 ) ) ) ) ).
% rtrancl_O_shift
thf(fact_234_rtrancl__converse,axiom,
! [R3: set_Pr2972776593051762503rm_a_b] :
( ( transi2615809358984392588rm_a_b @ ( conver1575690562132235771rm_a_b @ R3 ) )
= ( conver1575690562132235771rm_a_b @ ( transi2615809358984392588rm_a_b @ R3 ) ) ) ).
% rtrancl_converse
thf(fact_235_rtrancl__converse,axiom,
! [R3: set_Pr4386577575007340137rm_a_b] :
( ( transi7742714808557438673rm_a_b @ ( conver7608559976937845093rm_a_b @ R3 ) )
= ( conver7608559976937845093rm_a_b @ ( transi7742714808557438673rm_a_b @ R3 ) ) ) ).
% rtrancl_converse
thf(fact_236_rtrancl__Un__rtrancl,axiom,
! [R4: set_Pr7151917219565317845rm_a_b,S6: set_Pr7151917219565317845rm_a_b] :
( ( transi6904761420441344775rm_a_b @ ( sup_su1246761269818691241rm_a_b @ ( transi6904761420441344775rm_a_b @ R4 ) @ ( transi6904761420441344775rm_a_b @ S6 ) ) )
= ( transi6904761420441344775rm_a_b @ ( sup_su1246761269818691241rm_a_b @ R4 @ S6 ) ) ) ).
% rtrancl_Un_rtrancl
thf(fact_237_rtrancl__Un__rtrancl,axiom,
! [R4: set_Pr2972776593051762503rm_a_b,S6: set_Pr2972776593051762503rm_a_b] :
( ( transi2615809358984392588rm_a_b @ ( sup_su1845815533690797339rm_a_b @ ( transi2615809358984392588rm_a_b @ R4 ) @ ( transi2615809358984392588rm_a_b @ S6 ) ) )
= ( transi2615809358984392588rm_a_b @ ( sup_su1845815533690797339rm_a_b @ R4 @ S6 ) ) ) ).
% rtrancl_Un_rtrancl
thf(fact_238_rtrancl__Un__rtrancl,axiom,
! [R4: set_Pr4386577575007340137rm_a_b,S6: set_Pr4386577575007340137rm_a_b] :
( ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ ( transi7742714808557438673rm_a_b @ R4 ) @ ( transi7742714808557438673rm_a_b @ S6 ) ) )
= ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ R4 @ S6 ) ) ) ).
% rtrancl_Un_rtrancl
thf(fact_239_in__rtrancl__UnI,axiom,
! [X: produc3421996462152255079rm_a_b,R4: set_Pr647378613824986951rm_a_b,S6: set_Pr647378613824986951rm_a_b] :
( ( ( member1375104729371992464rm_a_b @ X @ ( transi8812752159401800684rm_a_b @ R4 ) )
| ( member1375104729371992464rm_a_b @ X @ ( transi8812752159401800684rm_a_b @ S6 ) ) )
=> ( member1375104729371992464rm_a_b @ X @ ( transi8812752159401800684rm_a_b @ ( sup_su8433770403025779995rm_a_b @ R4 @ S6 ) ) ) ) ).
% in_rtrancl_UnI
thf(fact_240_in__rtrancl__UnI,axiom,
! [X: produc9001763001631046773rm_a_b,R4: set_Pr7151917219565317845rm_a_b,S6: set_Pr7151917219565317845rm_a_b] :
( ( ( member8888773126115863070rm_a_b @ X @ ( transi6904761420441344775rm_a_b @ R4 ) )
| ( member8888773126115863070rm_a_b @ X @ ( transi6904761420441344775rm_a_b @ S6 ) ) )
=> ( member8888773126115863070rm_a_b @ X @ ( transi6904761420441344775rm_a_b @ ( sup_su1246761269818691241rm_a_b @ R4 @ S6 ) ) ) ) ).
% in_rtrancl_UnI
thf(fact_241_in__rtrancl__UnI,axiom,
! [X: produc4523971788519308903rm_a_b,R4: set_Pr2972776593051762503rm_a_b,S6: set_Pr2972776593051762503rm_a_b] :
( ( ( member8417600551952982416rm_a_b @ X @ ( transi2615809358984392588rm_a_b @ R4 ) )
| ( member8417600551952982416rm_a_b @ X @ ( transi2615809358984392588rm_a_b @ S6 ) ) )
=> ( member8417600551952982416rm_a_b @ X @ ( transi2615809358984392588rm_a_b @ ( sup_su1845815533690797339rm_a_b @ R4 @ S6 ) ) ) ) ).
% in_rtrancl_UnI
thf(fact_242_in__rtrancl__UnI,axiom,
! [X: produc357393685978478089rm_a_b,R4: set_Pr4386577575007340137rm_a_b,S6: set_Pr4386577575007340137rm_a_b] :
( ( ( member5869715511025134514rm_a_b @ X @ ( transi7742714808557438673rm_a_b @ R4 ) )
| ( member5869715511025134514rm_a_b @ X @ ( transi7742714808557438673rm_a_b @ S6 ) ) )
=> ( member5869715511025134514rm_a_b @ X @ ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ R4 @ S6 ) ) ) ) ).
% in_rtrancl_UnI
thf(fact_243_converse__relcomp,axiom,
! [R3: set_Pr7151917219565317845rm_a_b,S: set_Pr7151917219565317845rm_a_b] :
( ( conver1904685531124697809rm_a_b @ ( relcom3320975603336820193rm_a_b @ R3 @ S ) )
= ( relcom3320975603336820193rm_a_b @ ( conver1904685531124697809rm_a_b @ S ) @ ( conver1904685531124697809rm_a_b @ R3 ) ) ) ).
% converse_relcomp
thf(fact_244_converse__relcomp,axiom,
! [R3: set_Pr2972776593051762503rm_a_b,S: set_Pr2972776593051762503rm_a_b] :
( ( conver1575690562132235771rm_a_b @ ( relcom6159724393391289352rm_a_b @ R3 @ S ) )
= ( relcom6159724393391289352rm_a_b @ ( conver1575690562132235771rm_a_b @ S ) @ ( conver1575690562132235771rm_a_b @ R3 ) ) ) ).
% converse_relcomp
thf(fact_245_converse__relcomp,axiom,
! [R3: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
( ( conver7608559976937845093rm_a_b @ ( relcom370159955682700863rm_a_b @ R3 @ S ) )
= ( relcom370159955682700863rm_a_b @ ( conver7608559976937845093rm_a_b @ S ) @ ( conver7608559976937845093rm_a_b @ R3 ) ) ) ).
% converse_relcomp
thf(fact_246_trancl__converse,axiom,
! [R3: set_Pr2972776593051762503rm_a_b] :
( ( transi9211502839322181930rm_a_b @ ( conver1575690562132235771rm_a_b @ R3 ) )
= ( conver1575690562132235771rm_a_b @ ( transi9211502839322181930rm_a_b @ R3 ) ) ) ).
% trancl_converse
thf(fact_247_trancl__converse,axiom,
! [R3: set_Pr4386577575007340137rm_a_b] :
( ( transi7922773638565587891rm_a_b @ ( conver7608559976937845093rm_a_b @ R3 ) )
= ( conver7608559976937845093rm_a_b @ ( transi7922773638565587891rm_a_b @ R3 ) ) ) ).
% trancl_converse
thf(fact_248_symcl__converse,axiom,
! [A2: set_Pr2972776593051762503rm_a_b] :
( ( conver1575690562132235771rm_a_b @ ( sup_su1845815533690797339rm_a_b @ A2 @ ( conver1575690562132235771rm_a_b @ A2 ) ) )
= ( sup_su1845815533690797339rm_a_b @ A2 @ ( conver1575690562132235771rm_a_b @ A2 ) ) ) ).
% symcl_converse
thf(fact_249_symcl__converse,axiom,
! [A2: set_Pr7151917219565317845rm_a_b] :
( ( conver1904685531124697809rm_a_b @ ( sup_su1246761269818691241rm_a_b @ A2 @ ( conver1904685531124697809rm_a_b @ A2 ) ) )
= ( sup_su1246761269818691241rm_a_b @ A2 @ ( conver1904685531124697809rm_a_b @ A2 ) ) ) ).
% symcl_converse
thf(fact_250_symcl__converse,axiom,
! [A2: set_Pr4386577575007340137rm_a_b] :
( ( conver7608559976937845093rm_a_b @ ( sup_su6776935440552674877rm_a_b @ A2 @ ( conver7608559976937845093rm_a_b @ A2 ) ) )
= ( sup_su6776935440552674877rm_a_b @ A2 @ ( conver7608559976937845093rm_a_b @ A2 ) ) ) ).
% symcl_converse
thf(fact_251_symcl__Un,axiom,
! [A2: set_Pr2972776593051762503rm_a_b,B2: set_Pr2972776593051762503rm_a_b] :
( ( sup_su1845815533690797339rm_a_b @ ( sup_su1845815533690797339rm_a_b @ A2 @ B2 ) @ ( conver1575690562132235771rm_a_b @ ( sup_su1845815533690797339rm_a_b @ A2 @ B2 ) ) )
= ( sup_su1845815533690797339rm_a_b @ ( sup_su1845815533690797339rm_a_b @ A2 @ ( conver1575690562132235771rm_a_b @ A2 ) ) @ ( sup_su1845815533690797339rm_a_b @ B2 @ ( conver1575690562132235771rm_a_b @ B2 ) ) ) ) ).
% symcl_Un
thf(fact_252_symcl__Un,axiom,
! [A2: set_Pr7151917219565317845rm_a_b,B2: set_Pr7151917219565317845rm_a_b] :
( ( sup_su1246761269818691241rm_a_b @ ( sup_su1246761269818691241rm_a_b @ A2 @ B2 ) @ ( conver1904685531124697809rm_a_b @ ( sup_su1246761269818691241rm_a_b @ A2 @ B2 ) ) )
= ( sup_su1246761269818691241rm_a_b @ ( sup_su1246761269818691241rm_a_b @ A2 @ ( conver1904685531124697809rm_a_b @ A2 ) ) @ ( sup_su1246761269818691241rm_a_b @ B2 @ ( conver1904685531124697809rm_a_b @ B2 ) ) ) ) ).
% symcl_Un
thf(fact_253_symcl__Un,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b] :
( ( sup_su6776935440552674877rm_a_b @ ( sup_su6776935440552674877rm_a_b @ A2 @ B2 ) @ ( conver7608559976937845093rm_a_b @ ( sup_su6776935440552674877rm_a_b @ A2 @ B2 ) ) )
= ( sup_su6776935440552674877rm_a_b @ ( sup_su6776935440552674877rm_a_b @ A2 @ ( conver7608559976937845093rm_a_b @ A2 ) ) @ ( sup_su6776935440552674877rm_a_b @ B2 @ ( conver7608559976937845093rm_a_b @ B2 ) ) ) ) ).
% symcl_Un
thf(fact_254_converse__Un,axiom,
! [R3: set_Pr2972776593051762503rm_a_b,S: set_Pr2972776593051762503rm_a_b] :
( ( conver1575690562132235771rm_a_b @ ( sup_su1845815533690797339rm_a_b @ R3 @ S ) )
= ( sup_su1845815533690797339rm_a_b @ ( conver1575690562132235771rm_a_b @ R3 ) @ ( conver1575690562132235771rm_a_b @ S ) ) ) ).
% converse_Un
thf(fact_255_converse__Un,axiom,
! [R3: set_Pr7151917219565317845rm_a_b,S: set_Pr7151917219565317845rm_a_b] :
( ( conver1904685531124697809rm_a_b @ ( sup_su1246761269818691241rm_a_b @ R3 @ S ) )
= ( sup_su1246761269818691241rm_a_b @ ( conver1904685531124697809rm_a_b @ R3 ) @ ( conver1904685531124697809rm_a_b @ S ) ) ) ).
% converse_Un
thf(fact_256_converse__Un,axiom,
! [R3: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
( ( conver7608559976937845093rm_a_b @ ( sup_su6776935440552674877rm_a_b @ R3 @ S ) )
= ( sup_su6776935440552674877rm_a_b @ ( conver7608559976937845093rm_a_b @ R3 ) @ ( conver7608559976937845093rm_a_b @ S ) ) ) ).
% converse_Un
thf(fact_257_rstep__converse__dist,axiom,
! [R2: set_Pr4386577575007340137rm_a_b] :
( ( conver7608559976937845093rm_a_b @ ( rstep_a_b @ R2 ) )
= ( rstep_a_b @ ( conver7608559976937845093rm_a_b @ R2 ) ) ) ).
% rstep_converse_dist
thf(fact_258_steps__map,axiom,
! [P2: term_a_b > $o,Q: set_Pr4386577575007340137rm_a_b > $o,F3: term_a_b > term_a_b,G: set_Pr4386577575007340137rm_a_b > set_Pr4386577575007340137rm_a_b,T: term_a_b,R4: set_Pr4386577575007340137rm_a_b,S6: set_Pr4386577575007340137rm_a_b,U: term_a_b] :
( ! [T2: term_a_b,U2: term_a_b,R5: set_Pr4386577575007340137rm_a_b] :
( ( P2 @ T2 )
=> ( ( Q @ R5 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T2 @ U2 ) @ R5 )
=> ( ( P2 @ U2 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F3 @ T2 ) @ ( F3 @ U2 ) ) @ ( G @ R5 ) ) ) ) ) )
=> ( ( P2 @ T )
=> ( ( Q @ R4 )
=> ( ( Q @ S6 )
=> ( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T @ U ) @ ( transi7742714808557438673rm_a_b @ R4 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F3 @ T ) @ ( F3 @ U ) ) @ ( transi7742714808557438673rm_a_b @ ( G @ R4 ) ) ) )
& ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T @ U ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ R4 ) @ ( relcom370159955682700863rm_a_b @ S6 @ ( transi7742714808557438673rm_a_b @ R4 ) ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F3 @ T ) @ ( F3 @ U ) ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( G @ R4 ) ) @ ( relcom370159955682700863rm_a_b @ ( G @ S6 ) @ ( transi7742714808557438673rm_a_b @ ( G @ R4 ) ) ) ) ) ) ) ) ) ) ) ).
% steps_map
thf(fact_259_steps__map,axiom,
! [P2: set_term_a_b > $o,Q: set_Pr7151917219565317845rm_a_b > $o,F3: set_term_a_b > term_a_b,G: set_Pr7151917219565317845rm_a_b > set_Pr4386577575007340137rm_a_b,T: set_term_a_b,R4: set_Pr7151917219565317845rm_a_b,S6: set_Pr7151917219565317845rm_a_b,U: set_term_a_b] :
( ! [T2: set_term_a_b,U2: set_term_a_b,R5: set_Pr7151917219565317845rm_a_b] :
( ( P2 @ T2 )
=> ( ( Q @ R5 )
=> ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ T2 @ U2 ) @ R5 )
=> ( ( P2 @ U2 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F3 @ T2 ) @ ( F3 @ U2 ) ) @ ( G @ R5 ) ) ) ) ) )
=> ( ( P2 @ T )
=> ( ( Q @ R4 )
=> ( ( Q @ S6 )
=> ( ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ T @ U ) @ ( transi6904761420441344775rm_a_b @ R4 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F3 @ T ) @ ( F3 @ U ) ) @ ( transi7742714808557438673rm_a_b @ ( G @ R4 ) ) ) )
& ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ T @ U ) @ ( relcom3320975603336820193rm_a_b @ ( transi6904761420441344775rm_a_b @ R4 ) @ ( relcom3320975603336820193rm_a_b @ S6 @ ( transi6904761420441344775rm_a_b @ R4 ) ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F3 @ T ) @ ( F3 @ U ) ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( G @ R4 ) ) @ ( relcom370159955682700863rm_a_b @ ( G @ S6 ) @ ( transi7742714808557438673rm_a_b @ ( G @ R4 ) ) ) ) ) ) ) ) ) ) ) ).
% steps_map
thf(fact_260_steps__map,axiom,
! [P2: term_a_b > $o,Q: set_Pr4386577575007340137rm_a_b > $o,F3: term_a_b > set_term_a_b,G: set_Pr4386577575007340137rm_a_b > set_Pr7151917219565317845rm_a_b,T: term_a_b,R4: set_Pr4386577575007340137rm_a_b,S6: set_Pr4386577575007340137rm_a_b,U: term_a_b] :
( ! [T2: term_a_b,U2: term_a_b,R5: set_Pr4386577575007340137rm_a_b] :
( ( P2 @ T2 )
=> ( ( Q @ R5 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T2 @ U2 ) @ R5 )
=> ( ( P2 @ U2 )
& ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ ( F3 @ T2 ) @ ( F3 @ U2 ) ) @ ( G @ R5 ) ) ) ) ) )
=> ( ( P2 @ T )
=> ( ( Q @ R4 )
=> ( ( Q @ S6 )
=> ( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T @ U ) @ ( transi7742714808557438673rm_a_b @ R4 ) )
=> ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ ( F3 @ T ) @ ( F3 @ U ) ) @ ( transi6904761420441344775rm_a_b @ ( G @ R4 ) ) ) )
& ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T @ U ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ R4 ) @ ( relcom370159955682700863rm_a_b @ S6 @ ( transi7742714808557438673rm_a_b @ R4 ) ) ) )
=> ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ ( F3 @ T ) @ ( F3 @ U ) ) @ ( relcom3320975603336820193rm_a_b @ ( transi6904761420441344775rm_a_b @ ( G @ R4 ) ) @ ( relcom3320975603336820193rm_a_b @ ( G @ S6 ) @ ( transi6904761420441344775rm_a_b @ ( G @ R4 ) ) ) ) ) ) ) ) ) ) ) ).
% steps_map
thf(fact_261_steps__map,axiom,
! [P2: set_term_a_b > $o,Q: set_Pr7151917219565317845rm_a_b > $o,F3: set_term_a_b > set_term_a_b,G: set_Pr7151917219565317845rm_a_b > set_Pr7151917219565317845rm_a_b,T: set_term_a_b,R4: set_Pr7151917219565317845rm_a_b,S6: set_Pr7151917219565317845rm_a_b,U: set_term_a_b] :
( ! [T2: set_term_a_b,U2: set_term_a_b,R5: set_Pr7151917219565317845rm_a_b] :
( ( P2 @ T2 )
=> ( ( Q @ R5 )
=> ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ T2 @ U2 ) @ R5 )
=> ( ( P2 @ U2 )
& ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ ( F3 @ T2 ) @ ( F3 @ U2 ) ) @ ( G @ R5 ) ) ) ) ) )
=> ( ( P2 @ T )
=> ( ( Q @ R4 )
=> ( ( Q @ S6 )
=> ( ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ T @ U ) @ ( transi6904761420441344775rm_a_b @ R4 ) )
=> ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ ( F3 @ T ) @ ( F3 @ U ) ) @ ( transi6904761420441344775rm_a_b @ ( G @ R4 ) ) ) )
& ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ T @ U ) @ ( relcom3320975603336820193rm_a_b @ ( transi6904761420441344775rm_a_b @ R4 ) @ ( relcom3320975603336820193rm_a_b @ S6 @ ( transi6904761420441344775rm_a_b @ R4 ) ) ) )
=> ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ ( F3 @ T ) @ ( F3 @ U ) ) @ ( relcom3320975603336820193rm_a_b @ ( transi6904761420441344775rm_a_b @ ( G @ R4 ) ) @ ( relcom3320975603336820193rm_a_b @ ( G @ S6 ) @ ( transi6904761420441344775rm_a_b @ ( G @ R4 ) ) ) ) ) ) ) ) ) ) ) ).
% steps_map
thf(fact_262_steps__map,axiom,
! [P2: term_a_b > $o,Q: set_Pr4386577575007340137rm_a_b > $o,F3: term_a_b > produc357393685978478089rm_a_b,G: set_Pr4386577575007340137rm_a_b > set_Pr2972776593051762503rm_a_b,T: term_a_b,R4: set_Pr4386577575007340137rm_a_b,S6: set_Pr4386577575007340137rm_a_b,U: term_a_b] :
( ! [T2: term_a_b,U2: term_a_b,R5: set_Pr4386577575007340137rm_a_b] :
( ( P2 @ T2 )
=> ( ( Q @ R5 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T2 @ U2 ) @ R5 )
=> ( ( P2 @ U2 )
& ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( F3 @ T2 ) @ ( F3 @ U2 ) ) @ ( G @ R5 ) ) ) ) ) )
=> ( ( P2 @ T )
=> ( ( Q @ R4 )
=> ( ( Q @ S6 )
=> ( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T @ U ) @ ( transi7742714808557438673rm_a_b @ R4 ) )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( F3 @ T ) @ ( F3 @ U ) ) @ ( transi2615809358984392588rm_a_b @ ( G @ R4 ) ) ) )
& ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T @ U ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ R4 ) @ ( relcom370159955682700863rm_a_b @ S6 @ ( transi7742714808557438673rm_a_b @ R4 ) ) ) )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( F3 @ T ) @ ( F3 @ U ) ) @ ( relcom6159724393391289352rm_a_b @ ( transi2615809358984392588rm_a_b @ ( G @ R4 ) ) @ ( relcom6159724393391289352rm_a_b @ ( G @ S6 ) @ ( transi2615809358984392588rm_a_b @ ( G @ R4 ) ) ) ) ) ) ) ) ) ) ) ).
% steps_map
thf(fact_263_steps__map,axiom,
! [P2: produc357393685978478089rm_a_b > $o,Q: set_Pr2972776593051762503rm_a_b > $o,F3: produc357393685978478089rm_a_b > term_a_b,G: set_Pr2972776593051762503rm_a_b > set_Pr4386577575007340137rm_a_b,T: produc357393685978478089rm_a_b,R4: set_Pr2972776593051762503rm_a_b,S6: set_Pr2972776593051762503rm_a_b,U: produc357393685978478089rm_a_b] :
( ! [T2: produc357393685978478089rm_a_b,U2: produc357393685978478089rm_a_b,R5: set_Pr2972776593051762503rm_a_b] :
( ( P2 @ T2 )
=> ( ( Q @ R5 )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ T2 @ U2 ) @ R5 )
=> ( ( P2 @ U2 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F3 @ T2 ) @ ( F3 @ U2 ) ) @ ( G @ R5 ) ) ) ) ) )
=> ( ( P2 @ T )
=> ( ( Q @ R4 )
=> ( ( Q @ S6 )
=> ( ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ T @ U ) @ ( transi2615809358984392588rm_a_b @ R4 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F3 @ T ) @ ( F3 @ U ) ) @ ( transi7742714808557438673rm_a_b @ ( G @ R4 ) ) ) )
& ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ T @ U ) @ ( relcom6159724393391289352rm_a_b @ ( transi2615809358984392588rm_a_b @ R4 ) @ ( relcom6159724393391289352rm_a_b @ S6 @ ( transi2615809358984392588rm_a_b @ R4 ) ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F3 @ T ) @ ( F3 @ U ) ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( G @ R4 ) ) @ ( relcom370159955682700863rm_a_b @ ( G @ S6 ) @ ( transi7742714808557438673rm_a_b @ ( G @ R4 ) ) ) ) ) ) ) ) ) ) ) ).
% steps_map
thf(fact_264_steps__map,axiom,
! [P2: set_Pr4386577575007340137rm_a_b > $o,Q: set_Pr647378613824986951rm_a_b > $o,F3: set_Pr4386577575007340137rm_a_b > term_a_b,G: set_Pr647378613824986951rm_a_b > set_Pr4386577575007340137rm_a_b,T: set_Pr4386577575007340137rm_a_b,R4: set_Pr647378613824986951rm_a_b,S6: set_Pr647378613824986951rm_a_b,U: set_Pr4386577575007340137rm_a_b] :
( ! [T2: set_Pr4386577575007340137rm_a_b,U2: set_Pr4386577575007340137rm_a_b,R5: set_Pr647378613824986951rm_a_b] :
( ( P2 @ T2 )
=> ( ( Q @ R5 )
=> ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ T2 @ U2 ) @ R5 )
=> ( ( P2 @ U2 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F3 @ T2 ) @ ( F3 @ U2 ) ) @ ( G @ R5 ) ) ) ) ) )
=> ( ( P2 @ T )
=> ( ( Q @ R4 )
=> ( ( Q @ S6 )
=> ( ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ T @ U ) @ ( transi8812752159401800684rm_a_b @ R4 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F3 @ T ) @ ( F3 @ U ) ) @ ( transi7742714808557438673rm_a_b @ ( G @ R4 ) ) ) )
& ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ T @ U ) @ ( relcom8520403917366345832rm_a_b @ ( transi8812752159401800684rm_a_b @ R4 ) @ ( relcom8520403917366345832rm_a_b @ S6 @ ( transi8812752159401800684rm_a_b @ R4 ) ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F3 @ T ) @ ( F3 @ U ) ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( G @ R4 ) ) @ ( relcom370159955682700863rm_a_b @ ( G @ S6 ) @ ( transi7742714808557438673rm_a_b @ ( G @ R4 ) ) ) ) ) ) ) ) ) ) ) ).
% steps_map
thf(fact_265_steps__map,axiom,
! [P2: set_term_a_b > $o,Q: set_Pr7151917219565317845rm_a_b > $o,F3: set_term_a_b > produc357393685978478089rm_a_b,G: set_Pr7151917219565317845rm_a_b > set_Pr2972776593051762503rm_a_b,T: set_term_a_b,R4: set_Pr7151917219565317845rm_a_b,S6: set_Pr7151917219565317845rm_a_b,U: set_term_a_b] :
( ! [T2: set_term_a_b,U2: set_term_a_b,R5: set_Pr7151917219565317845rm_a_b] :
( ( P2 @ T2 )
=> ( ( Q @ R5 )
=> ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ T2 @ U2 ) @ R5 )
=> ( ( P2 @ U2 )
& ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( F3 @ T2 ) @ ( F3 @ U2 ) ) @ ( G @ R5 ) ) ) ) ) )
=> ( ( P2 @ T )
=> ( ( Q @ R4 )
=> ( ( Q @ S6 )
=> ( ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ T @ U ) @ ( transi6904761420441344775rm_a_b @ R4 ) )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( F3 @ T ) @ ( F3 @ U ) ) @ ( transi2615809358984392588rm_a_b @ ( G @ R4 ) ) ) )
& ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ T @ U ) @ ( relcom3320975603336820193rm_a_b @ ( transi6904761420441344775rm_a_b @ R4 ) @ ( relcom3320975603336820193rm_a_b @ S6 @ ( transi6904761420441344775rm_a_b @ R4 ) ) ) )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( F3 @ T ) @ ( F3 @ U ) ) @ ( relcom6159724393391289352rm_a_b @ ( transi2615809358984392588rm_a_b @ ( G @ R4 ) ) @ ( relcom6159724393391289352rm_a_b @ ( G @ S6 ) @ ( transi2615809358984392588rm_a_b @ ( G @ R4 ) ) ) ) ) ) ) ) ) ) ) ).
% steps_map
thf(fact_266_steps__map,axiom,
! [P2: term_a_b > $o,Q: set_Pr4386577575007340137rm_a_b > $o,F3: term_a_b > set_Pr4386577575007340137rm_a_b,G: set_Pr4386577575007340137rm_a_b > set_Pr647378613824986951rm_a_b,T: term_a_b,R4: set_Pr4386577575007340137rm_a_b,S6: set_Pr4386577575007340137rm_a_b,U: term_a_b] :
( ! [T2: term_a_b,U2: term_a_b,R5: set_Pr4386577575007340137rm_a_b] :
( ( P2 @ T2 )
=> ( ( Q @ R5 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T2 @ U2 ) @ R5 )
=> ( ( P2 @ U2 )
& ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ ( F3 @ T2 ) @ ( F3 @ U2 ) ) @ ( G @ R5 ) ) ) ) ) )
=> ( ( P2 @ T )
=> ( ( Q @ R4 )
=> ( ( Q @ S6 )
=> ( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T @ U ) @ ( transi7742714808557438673rm_a_b @ R4 ) )
=> ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ ( F3 @ T ) @ ( F3 @ U ) ) @ ( transi8812752159401800684rm_a_b @ ( G @ R4 ) ) ) )
& ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T @ U ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ R4 ) @ ( relcom370159955682700863rm_a_b @ S6 @ ( transi7742714808557438673rm_a_b @ R4 ) ) ) )
=> ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ ( F3 @ T ) @ ( F3 @ U ) ) @ ( relcom8520403917366345832rm_a_b @ ( transi8812752159401800684rm_a_b @ ( G @ R4 ) ) @ ( relcom8520403917366345832rm_a_b @ ( G @ S6 ) @ ( transi8812752159401800684rm_a_b @ ( G @ R4 ) ) ) ) ) ) ) ) ) ) ) ).
% steps_map
thf(fact_267_steps__map,axiom,
! [P2: produc357393685978478089rm_a_b > $o,Q: set_Pr2972776593051762503rm_a_b > $o,F3: produc357393685978478089rm_a_b > set_term_a_b,G: set_Pr2972776593051762503rm_a_b > set_Pr7151917219565317845rm_a_b,T: produc357393685978478089rm_a_b,R4: set_Pr2972776593051762503rm_a_b,S6: set_Pr2972776593051762503rm_a_b,U: produc357393685978478089rm_a_b] :
( ! [T2: produc357393685978478089rm_a_b,U2: produc357393685978478089rm_a_b,R5: set_Pr2972776593051762503rm_a_b] :
( ( P2 @ T2 )
=> ( ( Q @ R5 )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ T2 @ U2 ) @ R5 )
=> ( ( P2 @ U2 )
& ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ ( F3 @ T2 ) @ ( F3 @ U2 ) ) @ ( G @ R5 ) ) ) ) ) )
=> ( ( P2 @ T )
=> ( ( Q @ R4 )
=> ( ( Q @ S6 )
=> ( ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ T @ U ) @ ( transi2615809358984392588rm_a_b @ R4 ) )
=> ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ ( F3 @ T ) @ ( F3 @ U ) ) @ ( transi6904761420441344775rm_a_b @ ( G @ R4 ) ) ) )
& ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ T @ U ) @ ( relcom6159724393391289352rm_a_b @ ( transi2615809358984392588rm_a_b @ R4 ) @ ( relcom6159724393391289352rm_a_b @ S6 @ ( transi2615809358984392588rm_a_b @ R4 ) ) ) )
=> ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ ( F3 @ T ) @ ( F3 @ U ) ) @ ( relcom3320975603336820193rm_a_b @ ( transi6904761420441344775rm_a_b @ ( G @ R4 ) ) @ ( relcom3320975603336820193rm_a_b @ ( G @ S6 ) @ ( transi6904761420441344775rm_a_b @ ( G @ R4 ) ) ) ) ) ) ) ) ) ) ) ).
% steps_map
thf(fact_268_rtrancl__converseI,axiom,
! [Y: set_term_a_b,X: set_term_a_b,R3: set_Pr7151917219565317845rm_a_b] :
( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ Y @ X ) @ ( transi6904761420441344775rm_a_b @ R3 ) )
=> ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ X @ Y ) @ ( transi6904761420441344775rm_a_b @ ( conver1904685531124697809rm_a_b @ R3 ) ) ) ) ).
% rtrancl_converseI
thf(fact_269_rtrancl__converseI,axiom,
! [Y: set_Pr4386577575007340137rm_a_b,X: set_Pr4386577575007340137rm_a_b,R3: set_Pr647378613824986951rm_a_b] :
( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ Y @ X ) @ ( transi8812752159401800684rm_a_b @ R3 ) )
=> ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ X @ Y ) @ ( transi8812752159401800684rm_a_b @ ( conver7531182934069937659rm_a_b @ R3 ) ) ) ) ).
% rtrancl_converseI
thf(fact_270_rtrancl__converseI,axiom,
! [Y: produc357393685978478089rm_a_b,X: produc357393685978478089rm_a_b,R3: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y @ X ) @ ( transi2615809358984392588rm_a_b @ R3 ) )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ ( transi2615809358984392588rm_a_b @ ( conver1575690562132235771rm_a_b @ R3 ) ) ) ) ).
% rtrancl_converseI
thf(fact_271_rtrancl__converseI,axiom,
! [Y: term_a_b,X: term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X ) @ ( transi7742714808557438673rm_a_b @ R3 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ ( conver7608559976937845093rm_a_b @ R3 ) ) ) ) ).
% rtrancl_converseI
thf(fact_272_rtrancl__converseD,axiom,
! [X: set_term_a_b,Y: set_term_a_b,R3: set_Pr7151917219565317845rm_a_b] :
( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ X @ Y ) @ ( transi6904761420441344775rm_a_b @ ( conver1904685531124697809rm_a_b @ R3 ) ) )
=> ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ Y @ X ) @ ( transi6904761420441344775rm_a_b @ R3 ) ) ) ).
% rtrancl_converseD
thf(fact_273_rtrancl__converseD,axiom,
! [X: set_Pr4386577575007340137rm_a_b,Y: set_Pr4386577575007340137rm_a_b,R3: set_Pr647378613824986951rm_a_b] :
( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ X @ Y ) @ ( transi8812752159401800684rm_a_b @ ( conver7531182934069937659rm_a_b @ R3 ) ) )
=> ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ Y @ X ) @ ( transi8812752159401800684rm_a_b @ R3 ) ) ) ).
% rtrancl_converseD
thf(fact_274_rtrancl__converseD,axiom,
! [X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b,R3: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ ( transi2615809358984392588rm_a_b @ ( conver1575690562132235771rm_a_b @ R3 ) ) )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y @ X ) @ ( transi2615809358984392588rm_a_b @ R3 ) ) ) ).
% rtrancl_converseD
thf(fact_275_rtrancl__converseD,axiom,
! [X: term_a_b,Y: term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ ( conver7608559976937845093rm_a_b @ R3 ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X ) @ ( transi7742714808557438673rm_a_b @ R3 ) ) ) ).
% rtrancl_converseD
thf(fact_276_trancl__rtrancl__trancl,axiom,
! [A: set_term_a_b,B: set_term_a_b,R3: set_Pr7151917219565317845rm_a_b,C2: set_term_a_b] :
( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A @ B ) @ ( transi2170614581841672169rm_a_b @ R3 ) )
=> ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ B @ C2 ) @ ( transi6904761420441344775rm_a_b @ R3 ) )
=> ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A @ C2 ) @ ( transi2170614581841672169rm_a_b @ R3 ) ) ) ) ).
% trancl_rtrancl_trancl
thf(fact_277_trancl__rtrancl__trancl,axiom,
! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b,R3: set_Pr647378613824986951rm_a_b,C2: set_Pr4386577575007340137rm_a_b] :
( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A @ B ) @ ( transi6883191930764840330rm_a_b @ R3 ) )
=> ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ B @ C2 ) @ ( transi8812752159401800684rm_a_b @ R3 ) )
=> ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A @ C2 ) @ ( transi6883191930764840330rm_a_b @ R3 ) ) ) ) ).
% trancl_rtrancl_trancl
thf(fact_278_trancl__rtrancl__trancl,axiom,
! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,R3: set_Pr2972776593051762503rm_a_b,C2: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( transi9211502839322181930rm_a_b @ R3 ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B @ C2 ) @ ( transi2615809358984392588rm_a_b @ R3 ) )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ C2 ) @ ( transi9211502839322181930rm_a_b @ R3 ) ) ) ) ).
% trancl_rtrancl_trancl
thf(fact_279_trancl__rtrancl__trancl,axiom,
! [A: term_a_b,B: term_a_b,R3: set_Pr4386577575007340137rm_a_b,C2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7922773638565587891rm_a_b @ R3 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C2 ) @ ( transi7742714808557438673rm_a_b @ R3 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C2 ) @ ( transi7922773638565587891rm_a_b @ R3 ) ) ) ) ).
% trancl_rtrancl_trancl
thf(fact_280_rtrancl__trancl__trancl,axiom,
! [X: set_term_a_b,Y: set_term_a_b,R3: set_Pr7151917219565317845rm_a_b,Z: set_term_a_b] :
( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ X @ Y ) @ ( transi6904761420441344775rm_a_b @ R3 ) )
=> ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ Y @ Z ) @ ( transi2170614581841672169rm_a_b @ R3 ) )
=> ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ X @ Z ) @ ( transi2170614581841672169rm_a_b @ R3 ) ) ) ) ).
% rtrancl_trancl_trancl
thf(fact_281_rtrancl__trancl__trancl,axiom,
! [X: set_Pr4386577575007340137rm_a_b,Y: set_Pr4386577575007340137rm_a_b,R3: set_Pr647378613824986951rm_a_b,Z: set_Pr4386577575007340137rm_a_b] :
( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ X @ Y ) @ ( transi8812752159401800684rm_a_b @ R3 ) )
=> ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ Y @ Z ) @ ( transi6883191930764840330rm_a_b @ R3 ) )
=> ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ X @ Z ) @ ( transi6883191930764840330rm_a_b @ R3 ) ) ) ) ).
% rtrancl_trancl_trancl
thf(fact_282_rtrancl__trancl__trancl,axiom,
! [X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b,R3: set_Pr2972776593051762503rm_a_b,Z: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ ( transi2615809358984392588rm_a_b @ R3 ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y @ Z ) @ ( transi9211502839322181930rm_a_b @ R3 ) )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Z ) @ ( transi9211502839322181930rm_a_b @ R3 ) ) ) ) ).
% rtrancl_trancl_trancl
thf(fact_283_rtrancl__trancl__trancl,axiom,
! [X: term_a_b,Y: term_a_b,R3: set_Pr4386577575007340137rm_a_b,Z: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ R3 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ Z ) @ ( transi7922773638565587891rm_a_b @ R3 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z ) @ ( transi7922773638565587891rm_a_b @ R3 ) ) ) ) ).
% rtrancl_trancl_trancl
thf(fact_284_rtrancl__into__trancl2,axiom,
! [A: set_term_a_b,B: set_term_a_b,R3: set_Pr7151917219565317845rm_a_b,C2: set_term_a_b] :
( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A @ B ) @ R3 )
=> ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ B @ C2 ) @ ( transi6904761420441344775rm_a_b @ R3 ) )
=> ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A @ C2 ) @ ( transi2170614581841672169rm_a_b @ R3 ) ) ) ) ).
% rtrancl_into_trancl2
thf(fact_285_rtrancl__into__trancl2,axiom,
! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b,R3: set_Pr647378613824986951rm_a_b,C2: set_Pr4386577575007340137rm_a_b] :
( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A @ B ) @ R3 )
=> ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ B @ C2 ) @ ( transi8812752159401800684rm_a_b @ R3 ) )
=> ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A @ C2 ) @ ( transi6883191930764840330rm_a_b @ R3 ) ) ) ) ).
% rtrancl_into_trancl2
thf(fact_286_rtrancl__into__trancl2,axiom,
! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,R3: set_Pr2972776593051762503rm_a_b,C2: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ R3 )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B @ C2 ) @ ( transi2615809358984392588rm_a_b @ R3 ) )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ C2 ) @ ( transi9211502839322181930rm_a_b @ R3 ) ) ) ) ).
% rtrancl_into_trancl2
thf(fact_287_rtrancl__into__trancl2,axiom,
! [A: term_a_b,B: term_a_b,R3: set_Pr4386577575007340137rm_a_b,C2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R3 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C2 ) @ ( transi7742714808557438673rm_a_b @ R3 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C2 ) @ ( transi7922773638565587891rm_a_b @ R3 ) ) ) ) ).
% rtrancl_into_trancl2
thf(fact_288_rtrancl__into__trancl1,axiom,
! [A: set_term_a_b,B: set_term_a_b,R3: set_Pr7151917219565317845rm_a_b,C2: set_term_a_b] :
( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A @ B ) @ ( transi6904761420441344775rm_a_b @ R3 ) )
=> ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ B @ C2 ) @ R3 )
=> ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A @ C2 ) @ ( transi2170614581841672169rm_a_b @ R3 ) ) ) ) ).
% rtrancl_into_trancl1
thf(fact_289_rtrancl__into__trancl1,axiom,
! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b,R3: set_Pr647378613824986951rm_a_b,C2: set_Pr4386577575007340137rm_a_b] :
( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A @ B ) @ ( transi8812752159401800684rm_a_b @ R3 ) )
=> ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ B @ C2 ) @ R3 )
=> ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A @ C2 ) @ ( transi6883191930764840330rm_a_b @ R3 ) ) ) ) ).
% rtrancl_into_trancl1
thf(fact_290_rtrancl__into__trancl1,axiom,
! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,R3: set_Pr2972776593051762503rm_a_b,C2: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( transi2615809358984392588rm_a_b @ R3 ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B @ C2 ) @ R3 )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ C2 ) @ ( transi9211502839322181930rm_a_b @ R3 ) ) ) ) ).
% rtrancl_into_trancl1
thf(fact_291_rtrancl__into__trancl1,axiom,
! [A: term_a_b,B: term_a_b,R3: set_Pr4386577575007340137rm_a_b,C2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ R3 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C2 ) @ R3 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C2 ) @ ( transi7922773638565587891rm_a_b @ R3 ) ) ) ) ).
% rtrancl_into_trancl1
thf(fact_292_rtrancl__eq__or__trancl,axiom,
! [X: set_term_a_b,Y: set_term_a_b,R4: set_Pr7151917219565317845rm_a_b] :
( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ X @ Y ) @ ( transi6904761420441344775rm_a_b @ R4 ) )
= ( ( X = Y )
| ( ( X != Y )
& ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ X @ Y ) @ ( transi2170614581841672169rm_a_b @ R4 ) ) ) ) ) ).
% rtrancl_eq_or_trancl
thf(fact_293_rtrancl__eq__or__trancl,axiom,
! [X: set_Pr4386577575007340137rm_a_b,Y: set_Pr4386577575007340137rm_a_b,R4: set_Pr647378613824986951rm_a_b] :
( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ X @ Y ) @ ( transi8812752159401800684rm_a_b @ R4 ) )
= ( ( X = Y )
| ( ( X != Y )
& ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ X @ Y ) @ ( transi6883191930764840330rm_a_b @ R4 ) ) ) ) ) ).
% rtrancl_eq_or_trancl
thf(fact_294_rtrancl__eq__or__trancl,axiom,
! [X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b,R4: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ ( transi2615809358984392588rm_a_b @ R4 ) )
= ( ( X = Y )
| ( ( X != Y )
& ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ ( transi9211502839322181930rm_a_b @ R4 ) ) ) ) ) ).
% rtrancl_eq_or_trancl
thf(fact_295_rtrancl__eq__or__trancl,axiom,
! [X: term_a_b,Y: term_a_b,R4: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ R4 ) )
= ( ( X = Y )
| ( ( X != Y )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7922773638565587891rm_a_b @ R4 ) ) ) ) ) ).
% rtrancl_eq_or_trancl
thf(fact_296_trancl__into__rtrancl,axiom,
! [A: set_term_a_b,B: set_term_a_b,R3: set_Pr7151917219565317845rm_a_b] :
( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A @ B ) @ ( transi2170614581841672169rm_a_b @ R3 ) )
=> ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A @ B ) @ ( transi6904761420441344775rm_a_b @ R3 ) ) ) ).
% trancl_into_rtrancl
thf(fact_297_trancl__into__rtrancl,axiom,
! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b,R3: set_Pr647378613824986951rm_a_b] :
( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A @ B ) @ ( transi6883191930764840330rm_a_b @ R3 ) )
=> ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A @ B ) @ ( transi8812752159401800684rm_a_b @ R3 ) ) ) ).
% trancl_into_rtrancl
thf(fact_298_trancl__into__rtrancl,axiom,
! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,R3: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( transi9211502839322181930rm_a_b @ R3 ) )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( transi2615809358984392588rm_a_b @ R3 ) ) ) ).
% trancl_into_rtrancl
thf(fact_299_trancl__into__rtrancl,axiom,
! [A: term_a_b,B: term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7922773638565587891rm_a_b @ R3 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ R3 ) ) ) ).
% trancl_into_rtrancl
thf(fact_300_tranclD2,axiom,
! [X: set_term_a_b,Y: set_term_a_b,R4: set_Pr7151917219565317845rm_a_b] :
( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ X @ Y ) @ ( transi2170614581841672169rm_a_b @ R4 ) )
=> ? [Z2: set_term_a_b] :
( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ X @ Z2 ) @ ( transi6904761420441344775rm_a_b @ R4 ) )
& ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ Z2 @ Y ) @ R4 ) ) ) ).
% tranclD2
thf(fact_301_tranclD2,axiom,
! [X: set_Pr4386577575007340137rm_a_b,Y: set_Pr4386577575007340137rm_a_b,R4: set_Pr647378613824986951rm_a_b] :
( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ X @ Y ) @ ( transi6883191930764840330rm_a_b @ R4 ) )
=> ? [Z2: set_Pr4386577575007340137rm_a_b] :
( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ X @ Z2 ) @ ( transi8812752159401800684rm_a_b @ R4 ) )
& ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ Z2 @ Y ) @ R4 ) ) ) ).
% tranclD2
thf(fact_302_tranclD2,axiom,
! [X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b,R4: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ ( transi9211502839322181930rm_a_b @ R4 ) )
=> ? [Z2: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Z2 ) @ ( transi2615809358984392588rm_a_b @ R4 ) )
& ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Z2 @ Y ) @ R4 ) ) ) ).
% tranclD2
thf(fact_303_tranclD2,axiom,
! [X: term_a_b,Y: term_a_b,R4: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7922773638565587891rm_a_b @ R4 ) )
=> ? [Z2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z2 ) @ ( transi7742714808557438673rm_a_b @ R4 ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Z2 @ Y ) @ R4 ) ) ) ).
% tranclD2
thf(fact_304_rtranclD,axiom,
! [A: set_term_a_b,B: set_term_a_b,R4: set_Pr7151917219565317845rm_a_b] :
( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A @ B ) @ ( transi6904761420441344775rm_a_b @ R4 ) )
=> ( ( A = B )
| ( ( A != B )
& ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A @ B ) @ ( transi2170614581841672169rm_a_b @ R4 ) ) ) ) ) ).
% rtranclD
thf(fact_305_rtranclD,axiom,
! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b,R4: set_Pr647378613824986951rm_a_b] :
( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A @ B ) @ ( transi8812752159401800684rm_a_b @ R4 ) )
=> ( ( A = B )
| ( ( A != B )
& ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A @ B ) @ ( transi6883191930764840330rm_a_b @ R4 ) ) ) ) ) ).
% rtranclD
thf(fact_306_rtranclD,axiom,
! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,R4: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( transi2615809358984392588rm_a_b @ R4 ) )
=> ( ( A = B )
| ( ( A != B )
& ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( transi9211502839322181930rm_a_b @ R4 ) ) ) ) ) ).
% rtranclD
thf(fact_307_rtranclD,axiom,
! [A: term_a_b,B: term_a_b,R4: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ R4 ) )
=> ( ( A = B )
| ( ( A != B )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7922773638565587891rm_a_b @ R4 ) ) ) ) ) ).
% rtranclD
thf(fact_308_tranclD,axiom,
! [X: set_term_a_b,Y: set_term_a_b,R4: set_Pr7151917219565317845rm_a_b] :
( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ X @ Y ) @ ( transi2170614581841672169rm_a_b @ R4 ) )
=> ? [Z2: set_term_a_b] :
( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ X @ Z2 ) @ R4 )
& ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ Z2 @ Y ) @ ( transi6904761420441344775rm_a_b @ R4 ) ) ) ) ).
% tranclD
thf(fact_309_tranclD,axiom,
! [X: set_Pr4386577575007340137rm_a_b,Y: set_Pr4386577575007340137rm_a_b,R4: set_Pr647378613824986951rm_a_b] :
( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ X @ Y ) @ ( transi6883191930764840330rm_a_b @ R4 ) )
=> ? [Z2: set_Pr4386577575007340137rm_a_b] :
( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ X @ Z2 ) @ R4 )
& ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ Z2 @ Y ) @ ( transi8812752159401800684rm_a_b @ R4 ) ) ) ) ).
% tranclD
thf(fact_310_tranclD,axiom,
! [X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b,R4: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ ( transi9211502839322181930rm_a_b @ R4 ) )
=> ? [Z2: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Z2 ) @ R4 )
& ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Z2 @ Y ) @ ( transi2615809358984392588rm_a_b @ R4 ) ) ) ) ).
% tranclD
thf(fact_311_tranclD,axiom,
! [X: term_a_b,Y: term_a_b,R4: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7922773638565587891rm_a_b @ R4 ) )
=> ? [Z2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z2 ) @ R4 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Z2 @ Y ) @ ( transi7742714808557438673rm_a_b @ R4 ) ) ) ) ).
% tranclD
thf(fact_312_rtrancl__Un__separator__converseE,axiom,
! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b,P2: set_Pr647378613824986951rm_a_b,Q: set_Pr647378613824986951rm_a_b] :
( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A @ B ) @ ( transi8812752159401800684rm_a_b @ ( sup_su8433770403025779995rm_a_b @ P2 @ Q ) ) )
=> ( ! [X3: set_Pr4386577575007340137rm_a_b,Y2: set_Pr4386577575007340137rm_a_b] :
( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ X3 @ B ) @ ( transi8812752159401800684rm_a_b @ P2 ) )
=> ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ Y2 @ X3 ) @ Q )
=> ( Y2 = X3 ) ) )
=> ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A @ B ) @ ( transi8812752159401800684rm_a_b @ P2 ) ) ) ) ).
% rtrancl_Un_separator_converseE
thf(fact_313_rtrancl__Un__separator__converseE,axiom,
! [A: set_term_a_b,B: set_term_a_b,P2: set_Pr7151917219565317845rm_a_b,Q: set_Pr7151917219565317845rm_a_b] :
( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A @ B ) @ ( transi6904761420441344775rm_a_b @ ( sup_su1246761269818691241rm_a_b @ P2 @ Q ) ) )
=> ( ! [X3: set_term_a_b,Y2: set_term_a_b] :
( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ X3 @ B ) @ ( transi6904761420441344775rm_a_b @ P2 ) )
=> ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ Y2 @ X3 ) @ Q )
=> ( Y2 = X3 ) ) )
=> ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A @ B ) @ ( transi6904761420441344775rm_a_b @ P2 ) ) ) ) ).
% rtrancl_Un_separator_converseE
thf(fact_314_rtrancl__Un__separator__converseE,axiom,
! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,P2: set_Pr2972776593051762503rm_a_b,Q: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( transi2615809358984392588rm_a_b @ ( sup_su1845815533690797339rm_a_b @ P2 @ Q ) ) )
=> ( ! [X3: produc357393685978478089rm_a_b,Y2: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X3 @ B ) @ ( transi2615809358984392588rm_a_b @ P2 ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y2 @ X3 ) @ Q )
=> ( Y2 = X3 ) ) )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( transi2615809358984392588rm_a_b @ P2 ) ) ) ) ).
% rtrancl_Un_separator_converseE
thf(fact_315_rtrancl__Un__separator__converseE,axiom,
! [A: term_a_b,B: term_a_b,P2: set_Pr4386577575007340137rm_a_b,Q: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ P2 @ Q ) ) )
=> ( ! [X3: term_a_b,Y2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ B ) @ ( transi7742714808557438673rm_a_b @ P2 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y2 @ X3 ) @ Q )
=> ( Y2 = X3 ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ P2 ) ) ) ) ).
% rtrancl_Un_separator_converseE
thf(fact_316_rtrancl__Un__separatorE,axiom,
! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b,P2: set_Pr647378613824986951rm_a_b,Q: set_Pr647378613824986951rm_a_b] :
( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A @ B ) @ ( transi8812752159401800684rm_a_b @ ( sup_su8433770403025779995rm_a_b @ P2 @ Q ) ) )
=> ( ! [X3: set_Pr4386577575007340137rm_a_b,Y2: set_Pr4386577575007340137rm_a_b] :
( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A @ X3 ) @ ( transi8812752159401800684rm_a_b @ P2 ) )
=> ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ X3 @ Y2 ) @ Q )
=> ( X3 = Y2 ) ) )
=> ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A @ B ) @ ( transi8812752159401800684rm_a_b @ P2 ) ) ) ) ).
% rtrancl_Un_separatorE
thf(fact_317_rtrancl__Un__separatorE,axiom,
! [A: set_term_a_b,B: set_term_a_b,P2: set_Pr7151917219565317845rm_a_b,Q: set_Pr7151917219565317845rm_a_b] :
( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A @ B ) @ ( transi6904761420441344775rm_a_b @ ( sup_su1246761269818691241rm_a_b @ P2 @ Q ) ) )
=> ( ! [X3: set_term_a_b,Y2: set_term_a_b] :
( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A @ X3 ) @ ( transi6904761420441344775rm_a_b @ P2 ) )
=> ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ X3 @ Y2 ) @ Q )
=> ( X3 = Y2 ) ) )
=> ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A @ B ) @ ( transi6904761420441344775rm_a_b @ P2 ) ) ) ) ).
% rtrancl_Un_separatorE
thf(fact_318_rtrancl__Un__separatorE,axiom,
! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,P2: set_Pr2972776593051762503rm_a_b,Q: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( transi2615809358984392588rm_a_b @ ( sup_su1845815533690797339rm_a_b @ P2 @ Q ) ) )
=> ( ! [X3: produc357393685978478089rm_a_b,Y2: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ X3 ) @ ( transi2615809358984392588rm_a_b @ P2 ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X3 @ Y2 ) @ Q )
=> ( X3 = Y2 ) ) )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( transi2615809358984392588rm_a_b @ P2 ) ) ) ) ).
% rtrancl_Un_separatorE
thf(fact_319_rtrancl__Un__separatorE,axiom,
! [A: term_a_b,B: term_a_b,P2: set_Pr4386577575007340137rm_a_b,Q: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ P2 @ Q ) ) )
=> ( ! [X3: term_a_b,Y2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ X3 ) @ ( transi7742714808557438673rm_a_b @ P2 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y2 ) @ Q )
=> ( X3 = Y2 ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ P2 ) ) ) ) ).
% rtrancl_Un_separatorE
thf(fact_320_peak__iff,axiom,
! [X: set_term_a_b,Y: set_term_a_b,A2: set_Pr7151917219565317845rm_a_b,B2: set_Pr7151917219565317845rm_a_b] :
( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ X @ Y ) @ ( relcom3320975603336820193rm_a_b @ ( conver1904685531124697809rm_a_b @ A2 ) @ B2 ) )
= ( ? [U3: set_term_a_b] :
( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ U3 @ X ) @ A2 )
& ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ U3 @ Y ) @ B2 ) ) ) ) ).
% peak_iff
thf(fact_321_peak__iff,axiom,
! [X: set_Pr4386577575007340137rm_a_b,Y: set_Pr4386577575007340137rm_a_b,A2: set_Pr647378613824986951rm_a_b,B2: set_Pr647378613824986951rm_a_b] :
( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ X @ Y ) @ ( relcom8520403917366345832rm_a_b @ ( conver7531182934069937659rm_a_b @ A2 ) @ B2 ) )
= ( ? [U3: set_Pr4386577575007340137rm_a_b] :
( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ U3 @ X ) @ A2 )
& ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ U3 @ Y ) @ B2 ) ) ) ) ).
% peak_iff
thf(fact_322_peak__iff,axiom,
! [X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b,A2: set_Pr2972776593051762503rm_a_b,B2: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ ( relcom6159724393391289352rm_a_b @ ( conver1575690562132235771rm_a_b @ A2 ) @ B2 ) )
= ( ? [U3: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ U3 @ X ) @ A2 )
& ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ U3 @ Y ) @ B2 ) ) ) ) ).
% peak_iff
thf(fact_323_peak__iff,axiom,
! [X: term_a_b,Y: term_a_b,A2: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( relcom370159955682700863rm_a_b @ ( conver7608559976937845093rm_a_b @ A2 ) @ B2 ) )
= ( ? [U3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ U3 @ X ) @ A2 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ U3 @ Y ) @ B2 ) ) ) ) ).
% peak_iff
thf(fact_324_trancl__converseI,axiom,
! [X: set_term_a_b,Y: set_term_a_b,R3: set_Pr7151917219565317845rm_a_b] :
( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ X @ Y ) @ ( conver1904685531124697809rm_a_b @ ( transi2170614581841672169rm_a_b @ R3 ) ) )
=> ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ X @ Y ) @ ( transi2170614581841672169rm_a_b @ ( conver1904685531124697809rm_a_b @ R3 ) ) ) ) ).
% trancl_converseI
thf(fact_325_trancl__converseI,axiom,
! [X: set_Pr4386577575007340137rm_a_b,Y: set_Pr4386577575007340137rm_a_b,R3: set_Pr647378613824986951rm_a_b] :
( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ X @ Y ) @ ( conver7531182934069937659rm_a_b @ ( transi6883191930764840330rm_a_b @ R3 ) ) )
=> ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ X @ Y ) @ ( transi6883191930764840330rm_a_b @ ( conver7531182934069937659rm_a_b @ R3 ) ) ) ) ).
% trancl_converseI
thf(fact_326_trancl__converseI,axiom,
! [X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b,R3: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ ( conver1575690562132235771rm_a_b @ ( transi9211502839322181930rm_a_b @ R3 ) ) )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ ( transi9211502839322181930rm_a_b @ ( conver1575690562132235771rm_a_b @ R3 ) ) ) ) ).
% trancl_converseI
thf(fact_327_trancl__converseI,axiom,
! [X: term_a_b,Y: term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( conver7608559976937845093rm_a_b @ ( transi7922773638565587891rm_a_b @ R3 ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7922773638565587891rm_a_b @ ( conver7608559976937845093rm_a_b @ R3 ) ) ) ) ).
% trancl_converseI
thf(fact_328_trancl__converseD,axiom,
! [X: set_term_a_b,Y: set_term_a_b,R3: set_Pr7151917219565317845rm_a_b] :
( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ X @ Y ) @ ( transi2170614581841672169rm_a_b @ ( conver1904685531124697809rm_a_b @ R3 ) ) )
=> ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ X @ Y ) @ ( conver1904685531124697809rm_a_b @ ( transi2170614581841672169rm_a_b @ R3 ) ) ) ) ).
% trancl_converseD
thf(fact_329_trancl__converseD,axiom,
! [X: set_Pr4386577575007340137rm_a_b,Y: set_Pr4386577575007340137rm_a_b,R3: set_Pr647378613824986951rm_a_b] :
( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ X @ Y ) @ ( transi6883191930764840330rm_a_b @ ( conver7531182934069937659rm_a_b @ R3 ) ) )
=> ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ X @ Y ) @ ( conver7531182934069937659rm_a_b @ ( transi6883191930764840330rm_a_b @ R3 ) ) ) ) ).
% trancl_converseD
thf(fact_330_trancl__converseD,axiom,
! [X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b,R3: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ ( transi9211502839322181930rm_a_b @ ( conver1575690562132235771rm_a_b @ R3 ) ) )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ ( conver1575690562132235771rm_a_b @ ( transi9211502839322181930rm_a_b @ R3 ) ) ) ) ).
% trancl_converseD
thf(fact_331_trancl__converseD,axiom,
! [X: term_a_b,Y: term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7922773638565587891rm_a_b @ ( conver7608559976937845093rm_a_b @ R3 ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( conver7608559976937845093rm_a_b @ ( transi7922773638565587891rm_a_b @ R3 ) ) ) ) ).
% trancl_converseD
thf(fact_332_rtrancl__comp__trancl__conv,axiom,
! [R3: set_Pr7151917219565317845rm_a_b] :
( ( relcom3320975603336820193rm_a_b @ ( transi6904761420441344775rm_a_b @ R3 ) @ R3 )
= ( transi2170614581841672169rm_a_b @ R3 ) ) ).
% rtrancl_comp_trancl_conv
thf(fact_333_rtrancl__comp__trancl__conv,axiom,
! [R3: set_Pr2972776593051762503rm_a_b] :
( ( relcom6159724393391289352rm_a_b @ ( transi2615809358984392588rm_a_b @ R3 ) @ R3 )
= ( transi9211502839322181930rm_a_b @ R3 ) ) ).
% rtrancl_comp_trancl_conv
thf(fact_334_rtrancl__comp__trancl__conv,axiom,
! [R3: set_Pr4386577575007340137rm_a_b] :
( ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ R3 ) @ R3 )
= ( transi7922773638565587891rm_a_b @ R3 ) ) ).
% rtrancl_comp_trancl_conv
thf(fact_335_trancl__unfold__left,axiom,
( transi2170614581841672169rm_a_b
= ( ^ [R6: set_Pr7151917219565317845rm_a_b] : ( relcom3320975603336820193rm_a_b @ R6 @ ( transi6904761420441344775rm_a_b @ R6 ) ) ) ) ).
% trancl_unfold_left
thf(fact_336_trancl__unfold__left,axiom,
( transi9211502839322181930rm_a_b
= ( ^ [R6: set_Pr2972776593051762503rm_a_b] : ( relcom6159724393391289352rm_a_b @ R6 @ ( transi2615809358984392588rm_a_b @ R6 ) ) ) ) ).
% trancl_unfold_left
thf(fact_337_trancl__unfold__left,axiom,
( transi7922773638565587891rm_a_b
= ( ^ [R6: set_Pr4386577575007340137rm_a_b] : ( relcom370159955682700863rm_a_b @ R6 @ ( transi7742714808557438673rm_a_b @ R6 ) ) ) ) ).
% trancl_unfold_left
thf(fact_338_O__rtrancl__O__O,axiom,
! [R4: set_Pr7151917219565317845rm_a_b,S6: set_Pr7151917219565317845rm_a_b] :
( ( relcom3320975603336820193rm_a_b @ R4 @ ( relcom3320975603336820193rm_a_b @ ( transi6904761420441344775rm_a_b @ ( relcom3320975603336820193rm_a_b @ S6 @ R4 ) ) @ S6 ) )
= ( transi2170614581841672169rm_a_b @ ( relcom3320975603336820193rm_a_b @ R4 @ S6 ) ) ) ).
% O_rtrancl_O_O
thf(fact_339_O__rtrancl__O__O,axiom,
! [R4: set_Pr6510905305361987970rm_a_b,S6: set_Pr4802440762858034464rm_a_b] :
( ( relcom8086516547131752875rm_a_b @ R4 @ ( relcom6590495161205167518rm_a_b @ ( transi7742714808557438673rm_a_b @ ( relcom8422713090239460168rm_a_b @ S6 @ R4 ) ) @ S6 ) )
= ( transi9211502839322181930rm_a_b @ ( relcom8086516547131752875rm_a_b @ R4 @ S6 ) ) ) ).
% O_rtrancl_O_O
thf(fact_340_O__rtrancl__O__O,axiom,
! [R4: set_Pr4802440762858034464rm_a_b,S6: set_Pr6510905305361987970rm_a_b] :
( ( relcom8422713090239460168rm_a_b @ R4 @ ( relcom695362439311269717rm_a_b @ ( transi2615809358984392588rm_a_b @ ( relcom8086516547131752875rm_a_b @ S6 @ R4 ) ) @ S6 ) )
= ( transi7922773638565587891rm_a_b @ ( relcom8422713090239460168rm_a_b @ R4 @ S6 ) ) ) ).
% O_rtrancl_O_O
thf(fact_341_O__rtrancl__O__O,axiom,
! [R4: set_Pr2972776593051762503rm_a_b,S6: set_Pr2972776593051762503rm_a_b] :
( ( relcom6159724393391289352rm_a_b @ R4 @ ( relcom6159724393391289352rm_a_b @ ( transi2615809358984392588rm_a_b @ ( relcom6159724393391289352rm_a_b @ S6 @ R4 ) ) @ S6 ) )
= ( transi9211502839322181930rm_a_b @ ( relcom6159724393391289352rm_a_b @ R4 @ S6 ) ) ) ).
% O_rtrancl_O_O
thf(fact_342_O__rtrancl__O__O,axiom,
! [R4: set_Pr4386577575007340137rm_a_b,S6: set_Pr4386577575007340137rm_a_b] :
( ( relcom370159955682700863rm_a_b @ R4 @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( relcom370159955682700863rm_a_b @ S6 @ R4 ) ) @ S6 ) )
= ( transi7922773638565587891rm_a_b @ ( relcom370159955682700863rm_a_b @ R4 @ S6 ) ) ) ).
% O_rtrancl_O_O
thf(fact_343_rtrancl__union__subset__rtrancl__union__trancl,axiom,
! [R3: set_Pr7151917219565317845rm_a_b,S: set_Pr7151917219565317845rm_a_b] :
( ( transi6904761420441344775rm_a_b @ ( sup_su1246761269818691241rm_a_b @ R3 @ ( transi2170614581841672169rm_a_b @ S ) ) )
= ( transi6904761420441344775rm_a_b @ ( sup_su1246761269818691241rm_a_b @ R3 @ S ) ) ) ).
% rtrancl_union_subset_rtrancl_union_trancl
thf(fact_344_rtrancl__union__subset__rtrancl__union__trancl,axiom,
! [R3: set_Pr2972776593051762503rm_a_b,S: set_Pr2972776593051762503rm_a_b] :
( ( transi2615809358984392588rm_a_b @ ( sup_su1845815533690797339rm_a_b @ R3 @ ( transi9211502839322181930rm_a_b @ S ) ) )
= ( transi2615809358984392588rm_a_b @ ( sup_su1845815533690797339rm_a_b @ R3 @ S ) ) ) ).
% rtrancl_union_subset_rtrancl_union_trancl
thf(fact_345_rtrancl__union__subset__rtrancl__union__trancl,axiom,
! [R3: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
( ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ R3 @ ( transi7922773638565587891rm_a_b @ S ) ) )
= ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ R3 @ S ) ) ) ).
% rtrancl_union_subset_rtrancl_union_trancl
thf(fact_346_trancl__unfold,axiom,
( transi2170614581841672169rm_a_b
= ( ^ [R6: set_Pr7151917219565317845rm_a_b] : ( sup_su1246761269818691241rm_a_b @ R6 @ ( relcom3320975603336820193rm_a_b @ ( transi2170614581841672169rm_a_b @ R6 ) @ R6 ) ) ) ) ).
% trancl_unfold
thf(fact_347_trancl__unfold,axiom,
( transi9211502839322181930rm_a_b
= ( ^ [R6: set_Pr2972776593051762503rm_a_b] : ( sup_su1845815533690797339rm_a_b @ R6 @ ( relcom6159724393391289352rm_a_b @ ( transi9211502839322181930rm_a_b @ R6 ) @ R6 ) ) ) ) ).
% trancl_unfold
thf(fact_348_trancl__unfold,axiom,
( transi7922773638565587891rm_a_b
= ( ^ [R6: set_Pr4386577575007340137rm_a_b] : ( sup_su6776935440552674877rm_a_b @ R6 @ ( relcom370159955682700863rm_a_b @ ( transi7922773638565587891rm_a_b @ R6 ) @ R6 ) ) ) ) ).
% trancl_unfold
thf(fact_349_sig__step__converse__rstep,axiom,
! [F2: set_Pr4934435412358123699_a_nat,R2: set_Pr4386577575007340137rm_a_b] :
( ( conver7608559976937845093rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) ) )
= ( sig_step_a_b @ F2 @ ( conver7608559976937845093rm_a_b @ ( rstep_a_b @ R2 ) ) ) ) ).
% sig_step_converse_rstep
thf(fact_350_srstep__converse__dist,axiom,
! [F2: set_Pr4934435412358123699_a_nat,R2: set_Pr4386577575007340137rm_a_b] :
( ( conver7608559976937845093rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) ) )
= ( sig_step_a_b @ F2 @ ( rstep_a_b @ ( conver7608559976937845093rm_a_b @ R2 ) ) ) ) ).
% srstep_converse_dist
thf(fact_351_rstep__smycl__dist,axiom,
! [R2: set_Pr4386577575007340137rm_a_b] :
( ( sup_su6776935440552674877rm_a_b @ ( rstep_a_b @ R2 ) @ ( conver7608559976937845093rm_a_b @ ( rstep_a_b @ R2 ) ) )
= ( rstep_a_b @ ( sup_su6776935440552674877rm_a_b @ R2 @ ( conver7608559976937845093rm_a_b @ R2 ) ) ) ) ).
% rstep_smycl_dist
thf(fact_352_sig__step__symcl__dist,axiom,
! [F2: set_Pr4934435412358123699_a_nat,R2: set_Pr4386577575007340137rm_a_b] :
( ( sup_su6776935440552674877rm_a_b @ ( sig_step_a_b @ F2 @ R2 ) @ ( conver7608559976937845093rm_a_b @ ( sig_step_a_b @ F2 @ R2 ) ) )
= ( sig_step_a_b @ F2 @ ( sup_su6776935440552674877rm_a_b @ R2 @ ( conver7608559976937845093rm_a_b @ R2 ) ) ) ) ).
% sig_step_symcl_dist
thf(fact_353_right__comp__S,axiom,
! [X: set_Pr4386577575007340137rm_a_b,Y: set_Pr4386577575007340137rm_a_b,S6: set_Pr647378613824986951rm_a_b,NS: set_Pr647378613824986951rm_a_b] :
( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ X @ Y ) @ ( relcom8520403917366345832rm_a_b @ S6 @ ( sup_su8433770403025779995rm_a_b @ ( relcom8520403917366345832rm_a_b @ S6 @ ( relcom8520403917366345832rm_a_b @ ( transi8812752159401800684rm_a_b @ S6 ) @ ( transi8812752159401800684rm_a_b @ NS ) ) ) @ ( transi8812752159401800684rm_a_b @ NS ) ) ) )
=> ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ X @ Y ) @ ( relcom8520403917366345832rm_a_b @ S6 @ ( relcom8520403917366345832rm_a_b @ ( transi8812752159401800684rm_a_b @ S6 ) @ ( transi8812752159401800684rm_a_b @ NS ) ) ) ) ) ).
% right_comp_S
thf(fact_354_right__comp__S,axiom,
! [X: set_term_a_b,Y: set_term_a_b,S6: set_Pr7151917219565317845rm_a_b,NS: set_Pr7151917219565317845rm_a_b] :
( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ X @ Y ) @ ( relcom3320975603336820193rm_a_b @ S6 @ ( sup_su1246761269818691241rm_a_b @ ( relcom3320975603336820193rm_a_b @ S6 @ ( relcom3320975603336820193rm_a_b @ ( transi6904761420441344775rm_a_b @ S6 ) @ ( transi6904761420441344775rm_a_b @ NS ) ) ) @ ( transi6904761420441344775rm_a_b @ NS ) ) ) )
=> ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ X @ Y ) @ ( relcom3320975603336820193rm_a_b @ S6 @ ( relcom3320975603336820193rm_a_b @ ( transi6904761420441344775rm_a_b @ S6 ) @ ( transi6904761420441344775rm_a_b @ NS ) ) ) ) ) ).
% right_comp_S
thf(fact_355_right__comp__S,axiom,
! [X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b,S6: set_Pr2972776593051762503rm_a_b,NS: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ ( relcom6159724393391289352rm_a_b @ S6 @ ( sup_su1845815533690797339rm_a_b @ ( relcom6159724393391289352rm_a_b @ S6 @ ( relcom6159724393391289352rm_a_b @ ( transi2615809358984392588rm_a_b @ S6 ) @ ( transi2615809358984392588rm_a_b @ NS ) ) ) @ ( transi2615809358984392588rm_a_b @ NS ) ) ) )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ ( relcom6159724393391289352rm_a_b @ S6 @ ( relcom6159724393391289352rm_a_b @ ( transi2615809358984392588rm_a_b @ S6 ) @ ( transi2615809358984392588rm_a_b @ NS ) ) ) ) ) ).
% right_comp_S
thf(fact_356_right__comp__S,axiom,
! [X: term_a_b,Y: term_a_b,S6: set_Pr4386577575007340137rm_a_b,NS: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( relcom370159955682700863rm_a_b @ S6 @ ( sup_su6776935440552674877rm_a_b @ ( relcom370159955682700863rm_a_b @ S6 @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ S6 ) @ ( transi7742714808557438673rm_a_b @ NS ) ) ) @ ( transi7742714808557438673rm_a_b @ NS ) ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( relcom370159955682700863rm_a_b @ S6 @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ S6 ) @ ( transi7742714808557438673rm_a_b @ NS ) ) ) ) ) ).
% right_comp_S
thf(fact_357_Un__iff,axiom,
! [C2: produc3421996462152255079rm_a_b,A2: set_Pr647378613824986951rm_a_b,B2: set_Pr647378613824986951rm_a_b] :
( ( member1375104729371992464rm_a_b @ C2 @ ( sup_su8433770403025779995rm_a_b @ A2 @ B2 ) )
= ( ( member1375104729371992464rm_a_b @ C2 @ A2 )
| ( member1375104729371992464rm_a_b @ C2 @ B2 ) ) ) ).
% Un_iff
thf(fact_358_Un__iff,axiom,
! [C2: set_term_a_b,A2: set_set_term_a_b,B2: set_set_term_a_b] :
( ( member_set_term_a_b @ C2 @ ( sup_su7127272098975069868rm_a_b @ A2 @ B2 ) )
= ( ( member_set_term_a_b @ C2 @ A2 )
| ( member_set_term_a_b @ C2 @ B2 ) ) ) ).
% Un_iff
thf(fact_359_Un__iff,axiom,
! [C2: set_Pr4386577575007340137rm_a_b,A2: set_se1467346398288745161rm_a_b,B2: set_se1467346398288745161rm_a_b] :
( ( member8823642877475723282rm_a_b @ C2 @ ( sup_su5207735427641754269rm_a_b @ A2 @ B2 ) )
= ( ( member8823642877475723282rm_a_b @ C2 @ A2 )
| ( member8823642877475723282rm_a_b @ C2 @ B2 ) ) ) ).
% Un_iff
thf(fact_360_Un__iff,axiom,
! [C2: produc4523971788519308903rm_a_b,A2: set_Pr2972776593051762503rm_a_b,B2: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ C2 @ ( sup_su1845815533690797339rm_a_b @ A2 @ B2 ) )
= ( ( member8417600551952982416rm_a_b @ C2 @ A2 )
| ( member8417600551952982416rm_a_b @ C2 @ B2 ) ) ) ).
% Un_iff
thf(fact_361_Un__iff,axiom,
! [C2: term_a_b,A2: set_term_a_b,B2: set_term_a_b] :
( ( member_term_a_b @ C2 @ ( sup_sup_set_term_a_b @ A2 @ B2 ) )
= ( ( member_term_a_b @ C2 @ A2 )
| ( member_term_a_b @ C2 @ B2 ) ) ) ).
% Un_iff
thf(fact_362_Un__iff,axiom,
! [C2: produc9001763001631046773rm_a_b,A2: set_Pr7151917219565317845rm_a_b,B2: set_Pr7151917219565317845rm_a_b] :
( ( member8888773126115863070rm_a_b @ C2 @ ( sup_su1246761269818691241rm_a_b @ A2 @ B2 ) )
= ( ( member8888773126115863070rm_a_b @ C2 @ A2 )
| ( member8888773126115863070rm_a_b @ C2 @ B2 ) ) ) ).
% Un_iff
thf(fact_363_Un__iff,axiom,
! [C2: produc357393685978478089rm_a_b,A2: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ C2 @ ( sup_su6776935440552674877rm_a_b @ A2 @ B2 ) )
= ( ( member5869715511025134514rm_a_b @ C2 @ A2 )
| ( member5869715511025134514rm_a_b @ C2 @ B2 ) ) ) ).
% Un_iff
thf(fact_364_UnCI,axiom,
! [C2: produc3421996462152255079rm_a_b,B2: set_Pr647378613824986951rm_a_b,A2: set_Pr647378613824986951rm_a_b] :
( ( ~ ( member1375104729371992464rm_a_b @ C2 @ B2 )
=> ( member1375104729371992464rm_a_b @ C2 @ A2 ) )
=> ( member1375104729371992464rm_a_b @ C2 @ ( sup_su8433770403025779995rm_a_b @ A2 @ B2 ) ) ) ).
% UnCI
thf(fact_365_UnCI,axiom,
! [C2: set_term_a_b,B2: set_set_term_a_b,A2: set_set_term_a_b] :
( ( ~ ( member_set_term_a_b @ C2 @ B2 )
=> ( member_set_term_a_b @ C2 @ A2 ) )
=> ( member_set_term_a_b @ C2 @ ( sup_su7127272098975069868rm_a_b @ A2 @ B2 ) ) ) ).
% UnCI
thf(fact_366_UnCI,axiom,
! [C2: set_Pr4386577575007340137rm_a_b,B2: set_se1467346398288745161rm_a_b,A2: set_se1467346398288745161rm_a_b] :
( ( ~ ( member8823642877475723282rm_a_b @ C2 @ B2 )
=> ( member8823642877475723282rm_a_b @ C2 @ A2 ) )
=> ( member8823642877475723282rm_a_b @ C2 @ ( sup_su5207735427641754269rm_a_b @ A2 @ B2 ) ) ) ).
% UnCI
thf(fact_367_UnCI,axiom,
! [C2: produc4523971788519308903rm_a_b,B2: set_Pr2972776593051762503rm_a_b,A2: set_Pr2972776593051762503rm_a_b] :
( ( ~ ( member8417600551952982416rm_a_b @ C2 @ B2 )
=> ( member8417600551952982416rm_a_b @ C2 @ A2 ) )
=> ( member8417600551952982416rm_a_b @ C2 @ ( sup_su1845815533690797339rm_a_b @ A2 @ B2 ) ) ) ).
% UnCI
thf(fact_368_UnCI,axiom,
! [C2: term_a_b,B2: set_term_a_b,A2: set_term_a_b] :
( ( ~ ( member_term_a_b @ C2 @ B2 )
=> ( member_term_a_b @ C2 @ A2 ) )
=> ( member_term_a_b @ C2 @ ( sup_sup_set_term_a_b @ A2 @ B2 ) ) ) ).
% UnCI
thf(fact_369_UnCI,axiom,
! [C2: produc9001763001631046773rm_a_b,B2: set_Pr7151917219565317845rm_a_b,A2: set_Pr7151917219565317845rm_a_b] :
( ( ~ ( member8888773126115863070rm_a_b @ C2 @ B2 )
=> ( member8888773126115863070rm_a_b @ C2 @ A2 ) )
=> ( member8888773126115863070rm_a_b @ C2 @ ( sup_su1246761269818691241rm_a_b @ A2 @ B2 ) ) ) ).
% UnCI
thf(fact_370_UnCI,axiom,
! [C2: produc357393685978478089rm_a_b,B2: set_Pr4386577575007340137rm_a_b,A2: set_Pr4386577575007340137rm_a_b] :
( ( ~ ( member5869715511025134514rm_a_b @ C2 @ B2 )
=> ( member5869715511025134514rm_a_b @ C2 @ A2 ) )
=> ( member5869715511025134514rm_a_b @ C2 @ ( sup_su6776935440552674877rm_a_b @ A2 @ B2 ) ) ) ).
% UnCI
thf(fact_371_sup_Oright__idem,axiom,
! [A: nat,B: nat] :
( ( sup_sup_nat @ ( sup_sup_nat @ A @ B ) @ B )
= ( sup_sup_nat @ A @ B ) ) ).
% sup.right_idem
thf(fact_372_sup_Oright__idem,axiom,
! [A: set_Pr2972776593051762503rm_a_b,B: set_Pr2972776593051762503rm_a_b] :
( ( sup_su1845815533690797339rm_a_b @ ( sup_su1845815533690797339rm_a_b @ A @ B ) @ B )
= ( sup_su1845815533690797339rm_a_b @ A @ B ) ) ).
% sup.right_idem
thf(fact_373_sup_Oright__idem,axiom,
! [A: set_term_a_b,B: set_term_a_b] :
( ( sup_sup_set_term_a_b @ ( sup_sup_set_term_a_b @ A @ B ) @ B )
= ( sup_sup_set_term_a_b @ A @ B ) ) ).
% sup.right_idem
thf(fact_374_sup_Oright__idem,axiom,
! [A: set_Pr7151917219565317845rm_a_b,B: set_Pr7151917219565317845rm_a_b] :
( ( sup_su1246761269818691241rm_a_b @ ( sup_su1246761269818691241rm_a_b @ A @ B ) @ B )
= ( sup_su1246761269818691241rm_a_b @ A @ B ) ) ).
% sup.right_idem
thf(fact_375_sup_Oright__idem,axiom,
! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ( sup_su6776935440552674877rm_a_b @ ( sup_su6776935440552674877rm_a_b @ A @ B ) @ B )
= ( sup_su6776935440552674877rm_a_b @ A @ B ) ) ).
% sup.right_idem
thf(fact_376_sup__left__idem,axiom,
! [X: nat,Y: nat] :
( ( sup_sup_nat @ X @ ( sup_sup_nat @ X @ Y ) )
= ( sup_sup_nat @ X @ Y ) ) ).
% sup_left_idem
thf(fact_377_sup__left__idem,axiom,
! [X: set_Pr2972776593051762503rm_a_b,Y: set_Pr2972776593051762503rm_a_b] :
( ( sup_su1845815533690797339rm_a_b @ X @ ( sup_su1845815533690797339rm_a_b @ X @ Y ) )
= ( sup_su1845815533690797339rm_a_b @ X @ Y ) ) ).
% sup_left_idem
thf(fact_378_sup__left__idem,axiom,
! [X: set_term_a_b,Y: set_term_a_b] :
( ( sup_sup_set_term_a_b @ X @ ( sup_sup_set_term_a_b @ X @ Y ) )
= ( sup_sup_set_term_a_b @ X @ Y ) ) ).
% sup_left_idem
thf(fact_379_sup__left__idem,axiom,
! [X: set_Pr7151917219565317845rm_a_b,Y: set_Pr7151917219565317845rm_a_b] :
( ( sup_su1246761269818691241rm_a_b @ X @ ( sup_su1246761269818691241rm_a_b @ X @ Y ) )
= ( sup_su1246761269818691241rm_a_b @ X @ Y ) ) ).
% sup_left_idem
thf(fact_380_sup__left__idem,axiom,
! [X: set_Pr4386577575007340137rm_a_b,Y: set_Pr4386577575007340137rm_a_b] :
( ( sup_su6776935440552674877rm_a_b @ X @ ( sup_su6776935440552674877rm_a_b @ X @ Y ) )
= ( sup_su6776935440552674877rm_a_b @ X @ Y ) ) ).
% sup_left_idem
thf(fact_381_sup_Oleft__idem,axiom,
! [A: nat,B: nat] :
( ( sup_sup_nat @ A @ ( sup_sup_nat @ A @ B ) )
= ( sup_sup_nat @ A @ B ) ) ).
% sup.left_idem
thf(fact_382_sup_Oleft__idem,axiom,
! [A: set_Pr2972776593051762503rm_a_b,B: set_Pr2972776593051762503rm_a_b] :
( ( sup_su1845815533690797339rm_a_b @ A @ ( sup_su1845815533690797339rm_a_b @ A @ B ) )
= ( sup_su1845815533690797339rm_a_b @ A @ B ) ) ).
% sup.left_idem
thf(fact_383_sup_Oleft__idem,axiom,
! [A: set_term_a_b,B: set_term_a_b] :
( ( sup_sup_set_term_a_b @ A @ ( sup_sup_set_term_a_b @ A @ B ) )
= ( sup_sup_set_term_a_b @ A @ B ) ) ).
% sup.left_idem
thf(fact_384_sup_Oleft__idem,axiom,
! [A: set_Pr7151917219565317845rm_a_b,B: set_Pr7151917219565317845rm_a_b] :
( ( sup_su1246761269818691241rm_a_b @ A @ ( sup_su1246761269818691241rm_a_b @ A @ B ) )
= ( sup_su1246761269818691241rm_a_b @ A @ B ) ) ).
% sup.left_idem
thf(fact_385_sup_Oleft__idem,axiom,
! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ( sup_su6776935440552674877rm_a_b @ A @ ( sup_su6776935440552674877rm_a_b @ A @ B ) )
= ( sup_su6776935440552674877rm_a_b @ A @ B ) ) ).
% sup.left_idem
thf(fact_386_sup__idem,axiom,
! [X: nat] :
( ( sup_sup_nat @ X @ X )
= X ) ).
% sup_idem
thf(fact_387_sup__idem,axiom,
! [X: set_Pr2972776593051762503rm_a_b] :
( ( sup_su1845815533690797339rm_a_b @ X @ X )
= X ) ).
% sup_idem
thf(fact_388_sup__idem,axiom,
! [X: set_term_a_b] :
( ( sup_sup_set_term_a_b @ X @ X )
= X ) ).
% sup_idem
thf(fact_389_sup__idem,axiom,
! [X: set_Pr7151917219565317845rm_a_b] :
( ( sup_su1246761269818691241rm_a_b @ X @ X )
= X ) ).
% sup_idem
thf(fact_390_sup__idem,axiom,
! [X: set_Pr4386577575007340137rm_a_b] :
( ( sup_su6776935440552674877rm_a_b @ X @ X )
= X ) ).
% sup_idem
thf(fact_391_sup_Oidem,axiom,
! [A: nat] :
( ( sup_sup_nat @ A @ A )
= A ) ).
% sup.idem
thf(fact_392_sup_Oidem,axiom,
! [A: set_Pr2972776593051762503rm_a_b] :
( ( sup_su1845815533690797339rm_a_b @ A @ A )
= A ) ).
% sup.idem
thf(fact_393_sup_Oidem,axiom,
! [A: set_term_a_b] :
( ( sup_sup_set_term_a_b @ A @ A )
= A ) ).
% sup.idem
thf(fact_394_sup_Oidem,axiom,
! [A: set_Pr7151917219565317845rm_a_b] :
( ( sup_su1246761269818691241rm_a_b @ A @ A )
= A ) ).
% sup.idem
thf(fact_395_sup_Oidem,axiom,
! [A: set_Pr4386577575007340137rm_a_b] :
( ( sup_su6776935440552674877rm_a_b @ A @ A )
= A ) ).
% sup.idem
thf(fact_396_old_Oprod_Oinject,axiom,
! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,A5: produc357393685978478089rm_a_b,B5: produc357393685978478089rm_a_b] :
( ( ( produc1763473618796451543rm_a_b @ A @ B )
= ( produc1763473618796451543rm_a_b @ A5 @ B5 ) )
= ( ( A = A5 )
& ( B = B5 ) ) ) ).
% old.prod.inject
thf(fact_397_old_Oprod_Oinject,axiom,
! [A: set_term_a_b,B: set_term_a_b,A5: set_term_a_b,B5: set_term_a_b] :
( ( ( produc8517751122659631533rm_a_b @ A @ B )
= ( produc8517751122659631533rm_a_b @ A5 @ B5 ) )
= ( ( A = A5 )
& ( B = B5 ) ) ) ).
% old.prod.inject
thf(fact_398_old_Oprod_Oinject,axiom,
! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b,A5: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
( ( ( produc3977138704197412567rm_a_b @ A @ B )
= ( produc3977138704197412567rm_a_b @ A5 @ B5 ) )
= ( ( A = A5 )
& ( B = B5 ) ) ) ).
% old.prod.inject
thf(fact_399_old_Oprod_Oinject,axiom,
! [A: term_a_b,B: term_a_b,A5: term_a_b,B5: term_a_b] :
( ( ( produc7020197800436672577rm_a_b @ A @ B )
= ( produc7020197800436672577rm_a_b @ A5 @ B5 ) )
= ( ( A = A5 )
& ( B = B5 ) ) ) ).
% old.prod.inject
thf(fact_400_prod_Oinject,axiom,
! [X1: produc357393685978478089rm_a_b,X22: produc357393685978478089rm_a_b,Y1: produc357393685978478089rm_a_b,Y22: produc357393685978478089rm_a_b] :
( ( ( produc1763473618796451543rm_a_b @ X1 @ X22 )
= ( produc1763473618796451543rm_a_b @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X22 = Y22 ) ) ) ).
% prod.inject
thf(fact_401_prod_Oinject,axiom,
! [X1: set_term_a_b,X22: set_term_a_b,Y1: set_term_a_b,Y22: set_term_a_b] :
( ( ( produc8517751122659631533rm_a_b @ X1 @ X22 )
= ( produc8517751122659631533rm_a_b @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X22 = Y22 ) ) ) ).
% prod.inject
thf(fact_402_prod_Oinject,axiom,
! [X1: set_Pr4386577575007340137rm_a_b,X22: set_Pr4386577575007340137rm_a_b,Y1: set_Pr4386577575007340137rm_a_b,Y22: set_Pr4386577575007340137rm_a_b] :
( ( ( produc3977138704197412567rm_a_b @ X1 @ X22 )
= ( produc3977138704197412567rm_a_b @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X22 = Y22 ) ) ) ).
% prod.inject
thf(fact_403_prod_Oinject,axiom,
! [X1: term_a_b,X22: term_a_b,Y1: term_a_b,Y22: term_a_b] :
( ( ( produc7020197800436672577rm_a_b @ X1 @ X22 )
= ( produc7020197800436672577rm_a_b @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X22 = Y22 ) ) ) ).
% prod.inject
thf(fact_404_srsteps__with__root__step__def,axiom,
( srstep7844470518422762656ep_a_b
= ( ^ [F: set_Pr4934435412358123699_a_nat,R: set_Pr4386577575007340137rm_a_b] : ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R ) ) ) @ ( relcom370159955682700863rm_a_b @ ( sig_step_a_b @ F @ ( rrstep_a_b @ R ) ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R ) ) ) ) ) ) ) ).
% srsteps_with_root_step_def
thf(fact_405_rrstep__basicI,axiom,
! [L: term_a_b,R3: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ L @ R3 ) @ R2 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ L @ R3 ) @ ( rrstep_a_b @ R2 ) ) ) ).
% rrstep_basicI
thf(fact_406_old_Oprod_Oexhaust,axiom,
! [Y: produc4523971788519308903rm_a_b] :
~ ! [A3: produc357393685978478089rm_a_b,B3: produc357393685978478089rm_a_b] :
( Y
!= ( produc1763473618796451543rm_a_b @ A3 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_407_old_Oprod_Oexhaust,axiom,
! [Y: produc9001763001631046773rm_a_b] :
~ ! [A3: set_term_a_b,B3: set_term_a_b] :
( Y
!= ( produc8517751122659631533rm_a_b @ A3 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_408_old_Oprod_Oexhaust,axiom,
! [Y: produc3421996462152255079rm_a_b] :
~ ! [A3: set_Pr4386577575007340137rm_a_b,B3: set_Pr4386577575007340137rm_a_b] :
( Y
!= ( produc3977138704197412567rm_a_b @ A3 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_409_old_Oprod_Oexhaust,axiom,
! [Y: produc357393685978478089rm_a_b] :
~ ! [A3: term_a_b,B3: term_a_b] :
( Y
!= ( produc7020197800436672577rm_a_b @ A3 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_410_surj__pair,axiom,
! [P: produc4523971788519308903rm_a_b] :
? [X3: produc357393685978478089rm_a_b,Y2: produc357393685978478089rm_a_b] :
( P
= ( produc1763473618796451543rm_a_b @ X3 @ Y2 ) ) ).
% surj_pair
thf(fact_411_surj__pair,axiom,
! [P: produc9001763001631046773rm_a_b] :
? [X3: set_term_a_b,Y2: set_term_a_b] :
( P
= ( produc8517751122659631533rm_a_b @ X3 @ Y2 ) ) ).
% surj_pair
thf(fact_412_surj__pair,axiom,
! [P: produc3421996462152255079rm_a_b] :
? [X3: set_Pr4386577575007340137rm_a_b,Y2: set_Pr4386577575007340137rm_a_b] :
( P
= ( produc3977138704197412567rm_a_b @ X3 @ Y2 ) ) ).
% surj_pair
thf(fact_413_surj__pair,axiom,
! [P: produc357393685978478089rm_a_b] :
? [X3: term_a_b,Y2: term_a_b] :
( P
= ( produc7020197800436672577rm_a_b @ X3 @ Y2 ) ) ).
% surj_pair
thf(fact_414_prod__cases,axiom,
! [P2: produc4523971788519308903rm_a_b > $o,P: produc4523971788519308903rm_a_b] :
( ! [A3: produc357393685978478089rm_a_b,B3: produc357393685978478089rm_a_b] : ( P2 @ ( produc1763473618796451543rm_a_b @ A3 @ B3 ) )
=> ( P2 @ P ) ) ).
% prod_cases
thf(fact_415_prod__cases,axiom,
! [P2: produc9001763001631046773rm_a_b > $o,P: produc9001763001631046773rm_a_b] :
( ! [A3: set_term_a_b,B3: set_term_a_b] : ( P2 @ ( produc8517751122659631533rm_a_b @ A3 @ B3 ) )
=> ( P2 @ P ) ) ).
% prod_cases
thf(fact_416_prod__cases,axiom,
! [P2: produc3421996462152255079rm_a_b > $o,P: produc3421996462152255079rm_a_b] :
( ! [A3: set_Pr4386577575007340137rm_a_b,B3: set_Pr4386577575007340137rm_a_b] : ( P2 @ ( produc3977138704197412567rm_a_b @ A3 @ B3 ) )
=> ( P2 @ P ) ) ).
% prod_cases
thf(fact_417_prod__cases,axiom,
! [P2: produc357393685978478089rm_a_b > $o,P: produc357393685978478089rm_a_b] :
( ! [A3: term_a_b,B3: term_a_b] : ( P2 @ ( produc7020197800436672577rm_a_b @ A3 @ B3 ) )
=> ( P2 @ P ) ) ).
% prod_cases
thf(fact_418_Pair__inject,axiom,
! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,A5: produc357393685978478089rm_a_b,B5: produc357393685978478089rm_a_b] :
( ( ( produc1763473618796451543rm_a_b @ A @ B )
= ( produc1763473618796451543rm_a_b @ A5 @ B5 ) )
=> ~ ( ( A = A5 )
=> ( B != B5 ) ) ) ).
% Pair_inject
thf(fact_419_Pair__inject,axiom,
! [A: set_term_a_b,B: set_term_a_b,A5: set_term_a_b,B5: set_term_a_b] :
( ( ( produc8517751122659631533rm_a_b @ A @ B )
= ( produc8517751122659631533rm_a_b @ A5 @ B5 ) )
=> ~ ( ( A = A5 )
=> ( B != B5 ) ) ) ).
% Pair_inject
thf(fact_420_Pair__inject,axiom,
! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b,A5: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
( ( ( produc3977138704197412567rm_a_b @ A @ B )
= ( produc3977138704197412567rm_a_b @ A5 @ B5 ) )
=> ~ ( ( A = A5 )
=> ( B != B5 ) ) ) ).
% Pair_inject
thf(fact_421_Pair__inject,axiom,
! [A: term_a_b,B: term_a_b,A5: term_a_b,B5: term_a_b] :
( ( ( produc7020197800436672577rm_a_b @ A @ B )
= ( produc7020197800436672577rm_a_b @ A5 @ B5 ) )
=> ~ ( ( A = A5 )
=> ( B != B5 ) ) ) ).
% Pair_inject
thf(fact_422_prod__cases3,axiom,
! [Y: produc4523971788519308903rm_a_b] :
~ ! [A3: produc357393685978478089rm_a_b,B3: term_a_b,C4: term_a_b] :
( Y
!= ( produc1763473618796451543rm_a_b @ A3 @ ( produc7020197800436672577rm_a_b @ B3 @ C4 ) ) ) ).
% prod_cases3
thf(fact_423_prod__induct3,axiom,
! [P2: produc4523971788519308903rm_a_b > $o,X: produc4523971788519308903rm_a_b] :
( ! [A3: produc357393685978478089rm_a_b,B3: term_a_b,C4: term_a_b] : ( P2 @ ( produc1763473618796451543rm_a_b @ A3 @ ( produc7020197800436672577rm_a_b @ B3 @ C4 ) ) )
=> ( P2 @ X ) ) ).
% prod_induct3
thf(fact_424_inf__sup__aci_I8_J,axiom,
! [X: nat,Y: nat] :
( ( sup_sup_nat @ X @ ( sup_sup_nat @ X @ Y ) )
= ( sup_sup_nat @ X @ Y ) ) ).
% inf_sup_aci(8)
thf(fact_425_inf__sup__aci_I8_J,axiom,
! [X: set_Pr2972776593051762503rm_a_b,Y: set_Pr2972776593051762503rm_a_b] :
( ( sup_su1845815533690797339rm_a_b @ X @ ( sup_su1845815533690797339rm_a_b @ X @ Y ) )
= ( sup_su1845815533690797339rm_a_b @ X @ Y ) ) ).
% inf_sup_aci(8)
thf(fact_426_inf__sup__aci_I8_J,axiom,
! [X: set_term_a_b,Y: set_term_a_b] :
( ( sup_sup_set_term_a_b @ X @ ( sup_sup_set_term_a_b @ X @ Y ) )
= ( sup_sup_set_term_a_b @ X @ Y ) ) ).
% inf_sup_aci(8)
thf(fact_427_inf__sup__aci_I8_J,axiom,
! [X: set_Pr7151917219565317845rm_a_b,Y: set_Pr7151917219565317845rm_a_b] :
( ( sup_su1246761269818691241rm_a_b @ X @ ( sup_su1246761269818691241rm_a_b @ X @ Y ) )
= ( sup_su1246761269818691241rm_a_b @ X @ Y ) ) ).
% inf_sup_aci(8)
thf(fact_428_inf__sup__aci_I8_J,axiom,
! [X: set_Pr4386577575007340137rm_a_b,Y: set_Pr4386577575007340137rm_a_b] :
( ( sup_su6776935440552674877rm_a_b @ X @ ( sup_su6776935440552674877rm_a_b @ X @ Y ) )
= ( sup_su6776935440552674877rm_a_b @ X @ Y ) ) ).
% inf_sup_aci(8)
thf(fact_429_inf__sup__aci_I7_J,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( sup_sup_nat @ X @ ( sup_sup_nat @ Y @ Z ) )
= ( sup_sup_nat @ Y @ ( sup_sup_nat @ X @ Z ) ) ) ).
% inf_sup_aci(7)
thf(fact_430_inf__sup__aci_I7_J,axiom,
! [X: set_Pr2972776593051762503rm_a_b,Y: set_Pr2972776593051762503rm_a_b,Z: set_Pr2972776593051762503rm_a_b] :
( ( sup_su1845815533690797339rm_a_b @ X @ ( sup_su1845815533690797339rm_a_b @ Y @ Z ) )
= ( sup_su1845815533690797339rm_a_b @ Y @ ( sup_su1845815533690797339rm_a_b @ X @ Z ) ) ) ).
% inf_sup_aci(7)
thf(fact_431_inf__sup__aci_I7_J,axiom,
! [X: set_term_a_b,Y: set_term_a_b,Z: set_term_a_b] :
( ( sup_sup_set_term_a_b @ X @ ( sup_sup_set_term_a_b @ Y @ Z ) )
= ( sup_sup_set_term_a_b @ Y @ ( sup_sup_set_term_a_b @ X @ Z ) ) ) ).
% inf_sup_aci(7)
thf(fact_432_inf__sup__aci_I7_J,axiom,
! [X: set_Pr7151917219565317845rm_a_b,Y: set_Pr7151917219565317845rm_a_b,Z: set_Pr7151917219565317845rm_a_b] :
( ( sup_su1246761269818691241rm_a_b @ X @ ( sup_su1246761269818691241rm_a_b @ Y @ Z ) )
= ( sup_su1246761269818691241rm_a_b @ Y @ ( sup_su1246761269818691241rm_a_b @ X @ Z ) ) ) ).
% inf_sup_aci(7)
thf(fact_433_inf__sup__aci_I7_J,axiom,
! [X: set_Pr4386577575007340137rm_a_b,Y: set_Pr4386577575007340137rm_a_b,Z: set_Pr4386577575007340137rm_a_b] :
( ( sup_su6776935440552674877rm_a_b @ X @ ( sup_su6776935440552674877rm_a_b @ Y @ Z ) )
= ( sup_su6776935440552674877rm_a_b @ Y @ ( sup_su6776935440552674877rm_a_b @ X @ Z ) ) ) ).
% inf_sup_aci(7)
thf(fact_434_inf__sup__aci_I6_J,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( sup_sup_nat @ ( sup_sup_nat @ X @ Y ) @ Z )
= ( sup_sup_nat @ X @ ( sup_sup_nat @ Y @ Z ) ) ) ).
% inf_sup_aci(6)
thf(fact_435_inf__sup__aci_I6_J,axiom,
! [X: set_Pr2972776593051762503rm_a_b,Y: set_Pr2972776593051762503rm_a_b,Z: set_Pr2972776593051762503rm_a_b] :
( ( sup_su1845815533690797339rm_a_b @ ( sup_su1845815533690797339rm_a_b @ X @ Y ) @ Z )
= ( sup_su1845815533690797339rm_a_b @ X @ ( sup_su1845815533690797339rm_a_b @ Y @ Z ) ) ) ).
% inf_sup_aci(6)
thf(fact_436_inf__sup__aci_I6_J,axiom,
! [X: set_term_a_b,Y: set_term_a_b,Z: set_term_a_b] :
( ( sup_sup_set_term_a_b @ ( sup_sup_set_term_a_b @ X @ Y ) @ Z )
= ( sup_sup_set_term_a_b @ X @ ( sup_sup_set_term_a_b @ Y @ Z ) ) ) ).
% inf_sup_aci(6)
thf(fact_437_inf__sup__aci_I6_J,axiom,
! [X: set_Pr7151917219565317845rm_a_b,Y: set_Pr7151917219565317845rm_a_b,Z: set_Pr7151917219565317845rm_a_b] :
( ( sup_su1246761269818691241rm_a_b @ ( sup_su1246761269818691241rm_a_b @ X @ Y ) @ Z )
= ( sup_su1246761269818691241rm_a_b @ X @ ( sup_su1246761269818691241rm_a_b @ Y @ Z ) ) ) ).
% inf_sup_aci(6)
thf(fact_438_inf__sup__aci_I6_J,axiom,
! [X: set_Pr4386577575007340137rm_a_b,Y: set_Pr4386577575007340137rm_a_b,Z: set_Pr4386577575007340137rm_a_b] :
( ( sup_su6776935440552674877rm_a_b @ ( sup_su6776935440552674877rm_a_b @ X @ Y ) @ Z )
= ( sup_su6776935440552674877rm_a_b @ X @ ( sup_su6776935440552674877rm_a_b @ Y @ Z ) ) ) ).
% inf_sup_aci(6)
thf(fact_439_inf__sup__aci_I5_J,axiom,
( sup_sup_nat
= ( ^ [X2: nat,Y3: nat] : ( sup_sup_nat @ Y3 @ X2 ) ) ) ).
% inf_sup_aci(5)
thf(fact_440_inf__sup__aci_I5_J,axiom,
( sup_su1845815533690797339rm_a_b
= ( ^ [X2: set_Pr2972776593051762503rm_a_b,Y3: set_Pr2972776593051762503rm_a_b] : ( sup_su1845815533690797339rm_a_b @ Y3 @ X2 ) ) ) ).
% inf_sup_aci(5)
thf(fact_441_inf__sup__aci_I5_J,axiom,
( sup_sup_set_term_a_b
= ( ^ [X2: set_term_a_b,Y3: set_term_a_b] : ( sup_sup_set_term_a_b @ Y3 @ X2 ) ) ) ).
% inf_sup_aci(5)
thf(fact_442_inf__sup__aci_I5_J,axiom,
( sup_su1246761269818691241rm_a_b
= ( ^ [X2: set_Pr7151917219565317845rm_a_b,Y3: set_Pr7151917219565317845rm_a_b] : ( sup_su1246761269818691241rm_a_b @ Y3 @ X2 ) ) ) ).
% inf_sup_aci(5)
thf(fact_443_inf__sup__aci_I5_J,axiom,
( sup_su6776935440552674877rm_a_b
= ( ^ [X2: set_Pr4386577575007340137rm_a_b,Y3: set_Pr4386577575007340137rm_a_b] : ( sup_su6776935440552674877rm_a_b @ Y3 @ X2 ) ) ) ).
% inf_sup_aci(5)
thf(fact_444_sup_Oassoc,axiom,
! [A: nat,B: nat,C2: nat] :
( ( sup_sup_nat @ ( sup_sup_nat @ A @ B ) @ C2 )
= ( sup_sup_nat @ A @ ( sup_sup_nat @ B @ C2 ) ) ) ).
% sup.assoc
thf(fact_445_sup_Oassoc,axiom,
! [A: set_Pr2972776593051762503rm_a_b,B: set_Pr2972776593051762503rm_a_b,C2: set_Pr2972776593051762503rm_a_b] :
( ( sup_su1845815533690797339rm_a_b @ ( sup_su1845815533690797339rm_a_b @ A @ B ) @ C2 )
= ( sup_su1845815533690797339rm_a_b @ A @ ( sup_su1845815533690797339rm_a_b @ B @ C2 ) ) ) ).
% sup.assoc
thf(fact_446_sup_Oassoc,axiom,
! [A: set_term_a_b,B: set_term_a_b,C2: set_term_a_b] :
( ( sup_sup_set_term_a_b @ ( sup_sup_set_term_a_b @ A @ B ) @ C2 )
= ( sup_sup_set_term_a_b @ A @ ( sup_sup_set_term_a_b @ B @ C2 ) ) ) ).
% sup.assoc
thf(fact_447_sup_Oassoc,axiom,
! [A: set_Pr7151917219565317845rm_a_b,B: set_Pr7151917219565317845rm_a_b,C2: set_Pr7151917219565317845rm_a_b] :
( ( sup_su1246761269818691241rm_a_b @ ( sup_su1246761269818691241rm_a_b @ A @ B ) @ C2 )
= ( sup_su1246761269818691241rm_a_b @ A @ ( sup_su1246761269818691241rm_a_b @ B @ C2 ) ) ) ).
% sup.assoc
thf(fact_448_sup_Oassoc,axiom,
! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b,C2: set_Pr4386577575007340137rm_a_b] :
( ( sup_su6776935440552674877rm_a_b @ ( sup_su6776935440552674877rm_a_b @ A @ B ) @ C2 )
= ( sup_su6776935440552674877rm_a_b @ A @ ( sup_su6776935440552674877rm_a_b @ B @ C2 ) ) ) ).
% sup.assoc
thf(fact_449_sup__assoc,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( sup_sup_nat @ ( sup_sup_nat @ X @ Y ) @ Z )
= ( sup_sup_nat @ X @ ( sup_sup_nat @ Y @ Z ) ) ) ).
% sup_assoc
thf(fact_450_sup__assoc,axiom,
! [X: set_Pr2972776593051762503rm_a_b,Y: set_Pr2972776593051762503rm_a_b,Z: set_Pr2972776593051762503rm_a_b] :
( ( sup_su1845815533690797339rm_a_b @ ( sup_su1845815533690797339rm_a_b @ X @ Y ) @ Z )
= ( sup_su1845815533690797339rm_a_b @ X @ ( sup_su1845815533690797339rm_a_b @ Y @ Z ) ) ) ).
% sup_assoc
thf(fact_451_sup__assoc,axiom,
! [X: set_term_a_b,Y: set_term_a_b,Z: set_term_a_b] :
( ( sup_sup_set_term_a_b @ ( sup_sup_set_term_a_b @ X @ Y ) @ Z )
= ( sup_sup_set_term_a_b @ X @ ( sup_sup_set_term_a_b @ Y @ Z ) ) ) ).
% sup_assoc
thf(fact_452_sup__assoc,axiom,
! [X: set_Pr7151917219565317845rm_a_b,Y: set_Pr7151917219565317845rm_a_b,Z: set_Pr7151917219565317845rm_a_b] :
( ( sup_su1246761269818691241rm_a_b @ ( sup_su1246761269818691241rm_a_b @ X @ Y ) @ Z )
= ( sup_su1246761269818691241rm_a_b @ X @ ( sup_su1246761269818691241rm_a_b @ Y @ Z ) ) ) ).
% sup_assoc
thf(fact_453_sup__assoc,axiom,
! [X: set_Pr4386577575007340137rm_a_b,Y: set_Pr4386577575007340137rm_a_b,Z: set_Pr4386577575007340137rm_a_b] :
( ( sup_su6776935440552674877rm_a_b @ ( sup_su6776935440552674877rm_a_b @ X @ Y ) @ Z )
= ( sup_su6776935440552674877rm_a_b @ X @ ( sup_su6776935440552674877rm_a_b @ Y @ Z ) ) ) ).
% sup_assoc
thf(fact_454_sup_Ocommute,axiom,
( sup_sup_nat
= ( ^ [A4: nat,B4: nat] : ( sup_sup_nat @ B4 @ A4 ) ) ) ).
% sup.commute
thf(fact_455_sup_Ocommute,axiom,
( sup_su1845815533690797339rm_a_b
= ( ^ [A4: set_Pr2972776593051762503rm_a_b,B4: set_Pr2972776593051762503rm_a_b] : ( sup_su1845815533690797339rm_a_b @ B4 @ A4 ) ) ) ).
% sup.commute
thf(fact_456_sup_Ocommute,axiom,
( sup_sup_set_term_a_b
= ( ^ [A4: set_term_a_b,B4: set_term_a_b] : ( sup_sup_set_term_a_b @ B4 @ A4 ) ) ) ).
% sup.commute
thf(fact_457_sup_Ocommute,axiom,
( sup_su1246761269818691241rm_a_b
= ( ^ [A4: set_Pr7151917219565317845rm_a_b,B4: set_Pr7151917219565317845rm_a_b] : ( sup_su1246761269818691241rm_a_b @ B4 @ A4 ) ) ) ).
% sup.commute
thf(fact_458_sup_Ocommute,axiom,
( sup_su6776935440552674877rm_a_b
= ( ^ [A4: set_Pr4386577575007340137rm_a_b,B4: set_Pr4386577575007340137rm_a_b] : ( sup_su6776935440552674877rm_a_b @ B4 @ A4 ) ) ) ).
% sup.commute
thf(fact_459_sup__commute,axiom,
( sup_sup_nat
= ( ^ [X2: nat,Y3: nat] : ( sup_sup_nat @ Y3 @ X2 ) ) ) ).
% sup_commute
thf(fact_460_sup__commute,axiom,
( sup_su1845815533690797339rm_a_b
= ( ^ [X2: set_Pr2972776593051762503rm_a_b,Y3: set_Pr2972776593051762503rm_a_b] : ( sup_su1845815533690797339rm_a_b @ Y3 @ X2 ) ) ) ).
% sup_commute
thf(fact_461_sup__commute,axiom,
( sup_sup_set_term_a_b
= ( ^ [X2: set_term_a_b,Y3: set_term_a_b] : ( sup_sup_set_term_a_b @ Y3 @ X2 ) ) ) ).
% sup_commute
thf(fact_462_sup__commute,axiom,
( sup_su1246761269818691241rm_a_b
= ( ^ [X2: set_Pr7151917219565317845rm_a_b,Y3: set_Pr7151917219565317845rm_a_b] : ( sup_su1246761269818691241rm_a_b @ Y3 @ X2 ) ) ) ).
% sup_commute
thf(fact_463_sup__commute,axiom,
( sup_su6776935440552674877rm_a_b
= ( ^ [X2: set_Pr4386577575007340137rm_a_b,Y3: set_Pr4386577575007340137rm_a_b] : ( sup_su6776935440552674877rm_a_b @ Y3 @ X2 ) ) ) ).
% sup_commute
thf(fact_464_sup_Oleft__commute,axiom,
! [B: nat,A: nat,C2: nat] :
( ( sup_sup_nat @ B @ ( sup_sup_nat @ A @ C2 ) )
= ( sup_sup_nat @ A @ ( sup_sup_nat @ B @ C2 ) ) ) ).
% sup.left_commute
thf(fact_465_sup_Oleft__commute,axiom,
! [B: set_Pr2972776593051762503rm_a_b,A: set_Pr2972776593051762503rm_a_b,C2: set_Pr2972776593051762503rm_a_b] :
( ( sup_su1845815533690797339rm_a_b @ B @ ( sup_su1845815533690797339rm_a_b @ A @ C2 ) )
= ( sup_su1845815533690797339rm_a_b @ A @ ( sup_su1845815533690797339rm_a_b @ B @ C2 ) ) ) ).
% sup.left_commute
thf(fact_466_sup_Oleft__commute,axiom,
! [B: set_term_a_b,A: set_term_a_b,C2: set_term_a_b] :
( ( sup_sup_set_term_a_b @ B @ ( sup_sup_set_term_a_b @ A @ C2 ) )
= ( sup_sup_set_term_a_b @ A @ ( sup_sup_set_term_a_b @ B @ C2 ) ) ) ).
% sup.left_commute
thf(fact_467_sup_Oleft__commute,axiom,
! [B: set_Pr7151917219565317845rm_a_b,A: set_Pr7151917219565317845rm_a_b,C2: set_Pr7151917219565317845rm_a_b] :
( ( sup_su1246761269818691241rm_a_b @ B @ ( sup_su1246761269818691241rm_a_b @ A @ C2 ) )
= ( sup_su1246761269818691241rm_a_b @ A @ ( sup_su1246761269818691241rm_a_b @ B @ C2 ) ) ) ).
% sup.left_commute
thf(fact_468_sup_Oleft__commute,axiom,
! [B: set_Pr4386577575007340137rm_a_b,A: set_Pr4386577575007340137rm_a_b,C2: set_Pr4386577575007340137rm_a_b] :
( ( sup_su6776935440552674877rm_a_b @ B @ ( sup_su6776935440552674877rm_a_b @ A @ C2 ) )
= ( sup_su6776935440552674877rm_a_b @ A @ ( sup_su6776935440552674877rm_a_b @ B @ C2 ) ) ) ).
% sup.left_commute
thf(fact_469_sup__left__commute,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( sup_sup_nat @ X @ ( sup_sup_nat @ Y @ Z ) )
= ( sup_sup_nat @ Y @ ( sup_sup_nat @ X @ Z ) ) ) ).
% sup_left_commute
thf(fact_470_sup__left__commute,axiom,
! [X: set_Pr2972776593051762503rm_a_b,Y: set_Pr2972776593051762503rm_a_b,Z: set_Pr2972776593051762503rm_a_b] :
( ( sup_su1845815533690797339rm_a_b @ X @ ( sup_su1845815533690797339rm_a_b @ Y @ Z ) )
= ( sup_su1845815533690797339rm_a_b @ Y @ ( sup_su1845815533690797339rm_a_b @ X @ Z ) ) ) ).
% sup_left_commute
thf(fact_471_sup__left__commute,axiom,
! [X: set_term_a_b,Y: set_term_a_b,Z: set_term_a_b] :
( ( sup_sup_set_term_a_b @ X @ ( sup_sup_set_term_a_b @ Y @ Z ) )
= ( sup_sup_set_term_a_b @ Y @ ( sup_sup_set_term_a_b @ X @ Z ) ) ) ).
% sup_left_commute
thf(fact_472_sup__left__commute,axiom,
! [X: set_Pr7151917219565317845rm_a_b,Y: set_Pr7151917219565317845rm_a_b,Z: set_Pr7151917219565317845rm_a_b] :
( ( sup_su1246761269818691241rm_a_b @ X @ ( sup_su1246761269818691241rm_a_b @ Y @ Z ) )
= ( sup_su1246761269818691241rm_a_b @ Y @ ( sup_su1246761269818691241rm_a_b @ X @ Z ) ) ) ).
% sup_left_commute
thf(fact_473_sup__left__commute,axiom,
! [X: set_Pr4386577575007340137rm_a_b,Y: set_Pr4386577575007340137rm_a_b,Z: set_Pr4386577575007340137rm_a_b] :
( ( sup_su6776935440552674877rm_a_b @ X @ ( sup_su6776935440552674877rm_a_b @ Y @ Z ) )
= ( sup_su6776935440552674877rm_a_b @ Y @ ( sup_su6776935440552674877rm_a_b @ X @ Z ) ) ) ).
% sup_left_commute
thf(fact_474_srrstep__converse__dist,axiom,
! [F2: set_Pr4934435412358123699_a_nat,R2: set_Pr4386577575007340137rm_a_b] :
( ( conver7608559976937845093rm_a_b @ ( sig_step_a_b @ F2 @ ( rrstep_a_b @ R2 ) ) )
= ( sig_step_a_b @ F2 @ ( rrstep_a_b @ ( conver7608559976937845093rm_a_b @ R2 ) ) ) ) ).
% srrstep_converse_dist
thf(fact_475_UnE,axiom,
! [C2: produc3421996462152255079rm_a_b,A2: set_Pr647378613824986951rm_a_b,B2: set_Pr647378613824986951rm_a_b] :
( ( member1375104729371992464rm_a_b @ C2 @ ( sup_su8433770403025779995rm_a_b @ A2 @ B2 ) )
=> ( ~ ( member1375104729371992464rm_a_b @ C2 @ A2 )
=> ( member1375104729371992464rm_a_b @ C2 @ B2 ) ) ) ).
% UnE
thf(fact_476_UnE,axiom,
! [C2: set_term_a_b,A2: set_set_term_a_b,B2: set_set_term_a_b] :
( ( member_set_term_a_b @ C2 @ ( sup_su7127272098975069868rm_a_b @ A2 @ B2 ) )
=> ( ~ ( member_set_term_a_b @ C2 @ A2 )
=> ( member_set_term_a_b @ C2 @ B2 ) ) ) ).
% UnE
thf(fact_477_UnE,axiom,
! [C2: set_Pr4386577575007340137rm_a_b,A2: set_se1467346398288745161rm_a_b,B2: set_se1467346398288745161rm_a_b] :
( ( member8823642877475723282rm_a_b @ C2 @ ( sup_su5207735427641754269rm_a_b @ A2 @ B2 ) )
=> ( ~ ( member8823642877475723282rm_a_b @ C2 @ A2 )
=> ( member8823642877475723282rm_a_b @ C2 @ B2 ) ) ) ).
% UnE
thf(fact_478_UnE,axiom,
! [C2: produc4523971788519308903rm_a_b,A2: set_Pr2972776593051762503rm_a_b,B2: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ C2 @ ( sup_su1845815533690797339rm_a_b @ A2 @ B2 ) )
=> ( ~ ( member8417600551952982416rm_a_b @ C2 @ A2 )
=> ( member8417600551952982416rm_a_b @ C2 @ B2 ) ) ) ).
% UnE
thf(fact_479_UnE,axiom,
! [C2: term_a_b,A2: set_term_a_b,B2: set_term_a_b] :
( ( member_term_a_b @ C2 @ ( sup_sup_set_term_a_b @ A2 @ B2 ) )
=> ( ~ ( member_term_a_b @ C2 @ A2 )
=> ( member_term_a_b @ C2 @ B2 ) ) ) ).
% UnE
thf(fact_480_UnE,axiom,
! [C2: produc9001763001631046773rm_a_b,A2: set_Pr7151917219565317845rm_a_b,B2: set_Pr7151917219565317845rm_a_b] :
( ( member8888773126115863070rm_a_b @ C2 @ ( sup_su1246761269818691241rm_a_b @ A2 @ B2 ) )
=> ( ~ ( member8888773126115863070rm_a_b @ C2 @ A2 )
=> ( member8888773126115863070rm_a_b @ C2 @ B2 ) ) ) ).
% UnE
thf(fact_481_UnE,axiom,
! [C2: produc357393685978478089rm_a_b,A2: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ C2 @ ( sup_su6776935440552674877rm_a_b @ A2 @ B2 ) )
=> ( ~ ( member5869715511025134514rm_a_b @ C2 @ A2 )
=> ( member5869715511025134514rm_a_b @ C2 @ B2 ) ) ) ).
% UnE
thf(fact_482_UnI1,axiom,
! [C2: produc3421996462152255079rm_a_b,A2: set_Pr647378613824986951rm_a_b,B2: set_Pr647378613824986951rm_a_b] :
( ( member1375104729371992464rm_a_b @ C2 @ A2 )
=> ( member1375104729371992464rm_a_b @ C2 @ ( sup_su8433770403025779995rm_a_b @ A2 @ B2 ) ) ) ).
% UnI1
thf(fact_483_UnI1,axiom,
! [C2: set_term_a_b,A2: set_set_term_a_b,B2: set_set_term_a_b] :
( ( member_set_term_a_b @ C2 @ A2 )
=> ( member_set_term_a_b @ C2 @ ( sup_su7127272098975069868rm_a_b @ A2 @ B2 ) ) ) ).
% UnI1
thf(fact_484_UnI1,axiom,
! [C2: set_Pr4386577575007340137rm_a_b,A2: set_se1467346398288745161rm_a_b,B2: set_se1467346398288745161rm_a_b] :
( ( member8823642877475723282rm_a_b @ C2 @ A2 )
=> ( member8823642877475723282rm_a_b @ C2 @ ( sup_su5207735427641754269rm_a_b @ A2 @ B2 ) ) ) ).
% UnI1
thf(fact_485_UnI1,axiom,
! [C2: produc4523971788519308903rm_a_b,A2: set_Pr2972776593051762503rm_a_b,B2: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ C2 @ A2 )
=> ( member8417600551952982416rm_a_b @ C2 @ ( sup_su1845815533690797339rm_a_b @ A2 @ B2 ) ) ) ).
% UnI1
thf(fact_486_UnI1,axiom,
! [C2: term_a_b,A2: set_term_a_b,B2: set_term_a_b] :
( ( member_term_a_b @ C2 @ A2 )
=> ( member_term_a_b @ C2 @ ( sup_sup_set_term_a_b @ A2 @ B2 ) ) ) ).
% UnI1
thf(fact_487_UnI1,axiom,
! [C2: produc9001763001631046773rm_a_b,A2: set_Pr7151917219565317845rm_a_b,B2: set_Pr7151917219565317845rm_a_b] :
( ( member8888773126115863070rm_a_b @ C2 @ A2 )
=> ( member8888773126115863070rm_a_b @ C2 @ ( sup_su1246761269818691241rm_a_b @ A2 @ B2 ) ) ) ).
% UnI1
thf(fact_488_UnI1,axiom,
! [C2: produc357393685978478089rm_a_b,A2: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ C2 @ A2 )
=> ( member5869715511025134514rm_a_b @ C2 @ ( sup_su6776935440552674877rm_a_b @ A2 @ B2 ) ) ) ).
% UnI1
thf(fact_489_UnI2,axiom,
! [C2: produc3421996462152255079rm_a_b,B2: set_Pr647378613824986951rm_a_b,A2: set_Pr647378613824986951rm_a_b] :
( ( member1375104729371992464rm_a_b @ C2 @ B2 )
=> ( member1375104729371992464rm_a_b @ C2 @ ( sup_su8433770403025779995rm_a_b @ A2 @ B2 ) ) ) ).
% UnI2
thf(fact_490_UnI2,axiom,
! [C2: set_term_a_b,B2: set_set_term_a_b,A2: set_set_term_a_b] :
( ( member_set_term_a_b @ C2 @ B2 )
=> ( member_set_term_a_b @ C2 @ ( sup_su7127272098975069868rm_a_b @ A2 @ B2 ) ) ) ).
% UnI2
thf(fact_491_UnI2,axiom,
! [C2: set_Pr4386577575007340137rm_a_b,B2: set_se1467346398288745161rm_a_b,A2: set_se1467346398288745161rm_a_b] :
( ( member8823642877475723282rm_a_b @ C2 @ B2 )
=> ( member8823642877475723282rm_a_b @ C2 @ ( sup_su5207735427641754269rm_a_b @ A2 @ B2 ) ) ) ).
% UnI2
thf(fact_492_UnI2,axiom,
! [C2: produc4523971788519308903rm_a_b,B2: set_Pr2972776593051762503rm_a_b,A2: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ C2 @ B2 )
=> ( member8417600551952982416rm_a_b @ C2 @ ( sup_su1845815533690797339rm_a_b @ A2 @ B2 ) ) ) ).
% UnI2
thf(fact_493_UnI2,axiom,
! [C2: term_a_b,B2: set_term_a_b,A2: set_term_a_b] :
( ( member_term_a_b @ C2 @ B2 )
=> ( member_term_a_b @ C2 @ ( sup_sup_set_term_a_b @ A2 @ B2 ) ) ) ).
% UnI2
thf(fact_494_UnI2,axiom,
! [C2: produc9001763001631046773rm_a_b,B2: set_Pr7151917219565317845rm_a_b,A2: set_Pr7151917219565317845rm_a_b] :
( ( member8888773126115863070rm_a_b @ C2 @ B2 )
=> ( member8888773126115863070rm_a_b @ C2 @ ( sup_su1246761269818691241rm_a_b @ A2 @ B2 ) ) ) ).
% UnI2
thf(fact_495_UnI2,axiom,
! [C2: produc357393685978478089rm_a_b,B2: set_Pr4386577575007340137rm_a_b,A2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ C2 @ B2 )
=> ( member5869715511025134514rm_a_b @ C2 @ ( sup_su6776935440552674877rm_a_b @ A2 @ B2 ) ) ) ).
% UnI2
thf(fact_496_bex__Un,axiom,
! [A2: set_Pr2972776593051762503rm_a_b,B2: set_Pr2972776593051762503rm_a_b,P2: produc4523971788519308903rm_a_b > $o] :
( ( ? [X2: produc4523971788519308903rm_a_b] :
( ( member8417600551952982416rm_a_b @ X2 @ ( sup_su1845815533690797339rm_a_b @ A2 @ B2 ) )
& ( P2 @ X2 ) ) )
= ( ? [X2: produc4523971788519308903rm_a_b] :
( ( member8417600551952982416rm_a_b @ X2 @ A2 )
& ( P2 @ X2 ) )
| ? [X2: produc4523971788519308903rm_a_b] :
( ( member8417600551952982416rm_a_b @ X2 @ B2 )
& ( P2 @ X2 ) ) ) ) ).
% bex_Un
thf(fact_497_bex__Un,axiom,
! [A2: set_term_a_b,B2: set_term_a_b,P2: term_a_b > $o] :
( ( ? [X2: term_a_b] :
( ( member_term_a_b @ X2 @ ( sup_sup_set_term_a_b @ A2 @ B2 ) )
& ( P2 @ X2 ) ) )
= ( ? [X2: term_a_b] :
( ( member_term_a_b @ X2 @ A2 )
& ( P2 @ X2 ) )
| ? [X2: term_a_b] :
( ( member_term_a_b @ X2 @ B2 )
& ( P2 @ X2 ) ) ) ) ).
% bex_Un
thf(fact_498_bex__Un,axiom,
! [A2: set_Pr7151917219565317845rm_a_b,B2: set_Pr7151917219565317845rm_a_b,P2: produc9001763001631046773rm_a_b > $o] :
( ( ? [X2: produc9001763001631046773rm_a_b] :
( ( member8888773126115863070rm_a_b @ X2 @ ( sup_su1246761269818691241rm_a_b @ A2 @ B2 ) )
& ( P2 @ X2 ) ) )
= ( ? [X2: produc9001763001631046773rm_a_b] :
( ( member8888773126115863070rm_a_b @ X2 @ A2 )
& ( P2 @ X2 ) )
| ? [X2: produc9001763001631046773rm_a_b] :
( ( member8888773126115863070rm_a_b @ X2 @ B2 )
& ( P2 @ X2 ) ) ) ) ).
% bex_Un
thf(fact_499_bex__Un,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b,P2: produc357393685978478089rm_a_b > $o] :
( ( ? [X2: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X2 @ ( sup_su6776935440552674877rm_a_b @ A2 @ B2 ) )
& ( P2 @ X2 ) ) )
= ( ? [X2: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X2 @ A2 )
& ( P2 @ X2 ) )
| ? [X2: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X2 @ B2 )
& ( P2 @ X2 ) ) ) ) ).
% bex_Un
thf(fact_500_ball__Un,axiom,
! [A2: set_Pr2972776593051762503rm_a_b,B2: set_Pr2972776593051762503rm_a_b,P2: produc4523971788519308903rm_a_b > $o] :
( ( ! [X2: produc4523971788519308903rm_a_b] :
( ( member8417600551952982416rm_a_b @ X2 @ ( sup_su1845815533690797339rm_a_b @ A2 @ B2 ) )
=> ( P2 @ X2 ) ) )
= ( ! [X2: produc4523971788519308903rm_a_b] :
( ( member8417600551952982416rm_a_b @ X2 @ A2 )
=> ( P2 @ X2 ) )
& ! [X2: produc4523971788519308903rm_a_b] :
( ( member8417600551952982416rm_a_b @ X2 @ B2 )
=> ( P2 @ X2 ) ) ) ) ).
% ball_Un
thf(fact_501_ball__Un,axiom,
! [A2: set_term_a_b,B2: set_term_a_b,P2: term_a_b > $o] :
( ( ! [X2: term_a_b] :
( ( member_term_a_b @ X2 @ ( sup_sup_set_term_a_b @ A2 @ B2 ) )
=> ( P2 @ X2 ) ) )
= ( ! [X2: term_a_b] :
( ( member_term_a_b @ X2 @ A2 )
=> ( P2 @ X2 ) )
& ! [X2: term_a_b] :
( ( member_term_a_b @ X2 @ B2 )
=> ( P2 @ X2 ) ) ) ) ).
% ball_Un
thf(fact_502_ball__Un,axiom,
! [A2: set_Pr7151917219565317845rm_a_b,B2: set_Pr7151917219565317845rm_a_b,P2: produc9001763001631046773rm_a_b > $o] :
( ( ! [X2: produc9001763001631046773rm_a_b] :
( ( member8888773126115863070rm_a_b @ X2 @ ( sup_su1246761269818691241rm_a_b @ A2 @ B2 ) )
=> ( P2 @ X2 ) ) )
= ( ! [X2: produc9001763001631046773rm_a_b] :
( ( member8888773126115863070rm_a_b @ X2 @ A2 )
=> ( P2 @ X2 ) )
& ! [X2: produc9001763001631046773rm_a_b] :
( ( member8888773126115863070rm_a_b @ X2 @ B2 )
=> ( P2 @ X2 ) ) ) ) ).
% ball_Un
thf(fact_503_ball__Un,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b,P2: produc357393685978478089rm_a_b > $o] :
( ( ! [X2: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X2 @ ( sup_su6776935440552674877rm_a_b @ A2 @ B2 ) )
=> ( P2 @ X2 ) ) )
= ( ! [X2: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X2 @ A2 )
=> ( P2 @ X2 ) )
& ! [X2: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X2 @ B2 )
=> ( P2 @ X2 ) ) ) ) ).
% ball_Un
thf(fact_504_Un__assoc,axiom,
! [A2: set_Pr2972776593051762503rm_a_b,B2: set_Pr2972776593051762503rm_a_b,C: set_Pr2972776593051762503rm_a_b] :
( ( sup_su1845815533690797339rm_a_b @ ( sup_su1845815533690797339rm_a_b @ A2 @ B2 ) @ C )
= ( sup_su1845815533690797339rm_a_b @ A2 @ ( sup_su1845815533690797339rm_a_b @ B2 @ C ) ) ) ).
% Un_assoc
thf(fact_505_Un__assoc,axiom,
! [A2: set_term_a_b,B2: set_term_a_b,C: set_term_a_b] :
( ( sup_sup_set_term_a_b @ ( sup_sup_set_term_a_b @ A2 @ B2 ) @ C )
= ( sup_sup_set_term_a_b @ A2 @ ( sup_sup_set_term_a_b @ B2 @ C ) ) ) ).
% Un_assoc
thf(fact_506_Un__assoc,axiom,
! [A2: set_Pr7151917219565317845rm_a_b,B2: set_Pr7151917219565317845rm_a_b,C: set_Pr7151917219565317845rm_a_b] :
( ( sup_su1246761269818691241rm_a_b @ ( sup_su1246761269818691241rm_a_b @ A2 @ B2 ) @ C )
= ( sup_su1246761269818691241rm_a_b @ A2 @ ( sup_su1246761269818691241rm_a_b @ B2 @ C ) ) ) ).
% Un_assoc
thf(fact_507_Un__assoc,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b,C: set_Pr4386577575007340137rm_a_b] :
( ( sup_su6776935440552674877rm_a_b @ ( sup_su6776935440552674877rm_a_b @ A2 @ B2 ) @ C )
= ( sup_su6776935440552674877rm_a_b @ A2 @ ( sup_su6776935440552674877rm_a_b @ B2 @ C ) ) ) ).
% Un_assoc
thf(fact_508_Un__absorb,axiom,
! [A2: set_Pr2972776593051762503rm_a_b] :
( ( sup_su1845815533690797339rm_a_b @ A2 @ A2 )
= A2 ) ).
% Un_absorb
thf(fact_509_Un__absorb,axiom,
! [A2: set_term_a_b] :
( ( sup_sup_set_term_a_b @ A2 @ A2 )
= A2 ) ).
% Un_absorb
thf(fact_510_Un__absorb,axiom,
! [A2: set_Pr7151917219565317845rm_a_b] :
( ( sup_su1246761269818691241rm_a_b @ A2 @ A2 )
= A2 ) ).
% Un_absorb
thf(fact_511_Un__absorb,axiom,
! [A2: set_Pr4386577575007340137rm_a_b] :
( ( sup_su6776935440552674877rm_a_b @ A2 @ A2 )
= A2 ) ).
% Un_absorb
thf(fact_512_Un__commute,axiom,
( sup_su1845815533690797339rm_a_b
= ( ^ [A6: set_Pr2972776593051762503rm_a_b,B6: set_Pr2972776593051762503rm_a_b] : ( sup_su1845815533690797339rm_a_b @ B6 @ A6 ) ) ) ).
% Un_commute
thf(fact_513_Un__commute,axiom,
( sup_sup_set_term_a_b
= ( ^ [A6: set_term_a_b,B6: set_term_a_b] : ( sup_sup_set_term_a_b @ B6 @ A6 ) ) ) ).
% Un_commute
thf(fact_514_Un__commute,axiom,
( sup_su1246761269818691241rm_a_b
= ( ^ [A6: set_Pr7151917219565317845rm_a_b,B6: set_Pr7151917219565317845rm_a_b] : ( sup_su1246761269818691241rm_a_b @ B6 @ A6 ) ) ) ).
% Un_commute
thf(fact_515_Un__commute,axiom,
( sup_su6776935440552674877rm_a_b
= ( ^ [A6: set_Pr4386577575007340137rm_a_b,B6: set_Pr4386577575007340137rm_a_b] : ( sup_su6776935440552674877rm_a_b @ B6 @ A6 ) ) ) ).
% Un_commute
thf(fact_516_Un__left__absorb,axiom,
! [A2: set_term_a_b,B2: set_term_a_b] :
( ( sup_sup_set_term_a_b @ A2 @ ( sup_sup_set_term_a_b @ A2 @ B2 ) )
= ( sup_sup_set_term_a_b @ A2 @ B2 ) ) ).
% Un_left_absorb
thf(fact_517_Un__left__absorb,axiom,
! [A2: set_Pr7151917219565317845rm_a_b,B2: set_Pr7151917219565317845rm_a_b] :
( ( sup_su1246761269818691241rm_a_b @ A2 @ ( sup_su1246761269818691241rm_a_b @ A2 @ B2 ) )
= ( sup_su1246761269818691241rm_a_b @ A2 @ B2 ) ) ).
% Un_left_absorb
thf(fact_518_Un__left__absorb,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b] :
( ( sup_su6776935440552674877rm_a_b @ A2 @ ( sup_su6776935440552674877rm_a_b @ A2 @ B2 ) )
= ( sup_su6776935440552674877rm_a_b @ A2 @ B2 ) ) ).
% Un_left_absorb
thf(fact_519_Un__left__commute,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b,C: set_Pr4386577575007340137rm_a_b] :
( ( sup_su6776935440552674877rm_a_b @ A2 @ ( sup_su6776935440552674877rm_a_b @ B2 @ C ) )
= ( sup_su6776935440552674877rm_a_b @ B2 @ ( sup_su6776935440552674877rm_a_b @ A2 @ C ) ) ) ).
% Un_left_commute
thf(fact_520_srrstep__to__srestep,axiom,
! [S: term_a_b,T: term_a_b,F2: set_Pr4934435412358123699_a_nat,R2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( sig_step_a_b @ F2 @ ( rrstep_a_b @ R2 ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) ) ) ) ).
% srrstep_to_srestep
thf(fact_521_symcl__srsteps__conversion,axiom,
! [S: term_a_b,T: term_a_b,F2: set_Pr4934435412358123699_a_nat,R2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ ( sup_su6776935440552674877rm_a_b @ R2 @ ( conver7608559976937845093rm_a_b @ R2 ) ) ) ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( abstra1130568886744591910rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) ) ) ) ) ).
% symcl_srsteps_conversion
thf(fact_522_meetI,axiom,
! [A: term_a_b,B: term_a_b,A2: set_Pr4386577575007340137rm_a_b,C2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ A2 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C2 ) @ ( transi7742714808557438673rm_a_b @ A2 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C2 ) @ ( abstra1093061187967292567rm_a_b @ A2 ) ) ) ) ).
% meetI
thf(fact_523_meet__def,axiom,
( abstra1093061187967292567rm_a_b
= ( ^ [A6: set_Pr4386577575007340137rm_a_b] : ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( conver7608559976937845093rm_a_b @ A6 ) ) @ ( transi7742714808557438673rm_a_b @ A6 ) ) ) ) ).
% meet_def
thf(fact_524_all__ctxt__closed__sig__rsteps,axiom,
! [F2: set_Pr4934435412358123699_a_nat,R2: set_Pr4386577575007340137rm_a_b] : ( terms_5226143800768910156ed_a_b @ F2 @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) ) ) ) ).
% all_ctxt_closed_sig_rsteps
thf(fact_525_trancl__map,axiom,
! [R3: set_Pr4386577575007340137rm_a_b,F3: term_a_b > term_a_b,S: set_Pr4386577575007340137rm_a_b,X: term_a_b,Y: term_a_b] :
( ! [X3: term_a_b,Y2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y2 ) @ R3 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F3 @ X3 ) @ ( F3 @ Y2 ) ) @ S ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7922773638565587891rm_a_b @ R3 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F3 @ X ) @ ( F3 @ Y ) ) @ ( transi7922773638565587891rm_a_b @ S ) ) ) ) ).
% trancl_map
thf(fact_526_sig__step__conversion__dist,axiom,
! [F2: set_Pr4934435412358123699_a_nat,R2: set_Pr4386577575007340137rm_a_b] :
( ( abstra1130568886744591910rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) ) )
= ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ ( sup_su6776935440552674877rm_a_b @ R2 @ ( conver7608559976937845093rm_a_b @ R2 ) ) ) ) ) ) ).
% sig_step_conversion_dist
thf(fact_527_symcl__srstep__conversion,axiom,
! [S: term_a_b,T: term_a_b,F2: set_Pr4934435412358123699_a_nat,R2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ ( sup_su6776935440552674877rm_a_b @ R2 @ ( conver7608559976937845093rm_a_b @ R2 ) ) ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( abstra1130568886744591910rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) ) ) ) ) ).
% symcl_srstep_conversion
thf(fact_528_conversionI,axiom,
! [A: term_a_b,B: term_a_b,A2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ A2 @ ( conver7608559976937845093rm_a_b @ A2 ) ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( abstra1130568886744591910rm_a_b @ A2 ) ) ) ).
% conversionI
thf(fact_529_conversion__rtrancl,axiom,
! [A2: set_Pr4386577575007340137rm_a_b] :
( ( transi7742714808557438673rm_a_b @ ( abstra1130568886744591910rm_a_b @ A2 ) )
= ( abstra1130568886744591910rm_a_b @ A2 ) ) ).
% conversion_rtrancl
thf(fact_530_conversion__O__conversion,axiom,
! [A2: set_Pr4386577575007340137rm_a_b] :
( ( relcom370159955682700863rm_a_b @ ( abstra1130568886744591910rm_a_b @ A2 ) @ ( abstra1130568886744591910rm_a_b @ A2 ) )
= ( abstra1130568886744591910rm_a_b @ A2 ) ) ).
% conversion_O_conversion
thf(fact_531_conversion__converse,axiom,
! [A2: set_Pr4386577575007340137rm_a_b] :
( ( conver7608559976937845093rm_a_b @ ( abstra1130568886744591910rm_a_b @ A2 ) )
= ( abstra1130568886744591910rm_a_b @ A2 ) ) ).
% conversion_converse
thf(fact_532_conversionI_H,axiom,
! [A: term_a_b,B: term_a_b,A2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ A2 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( abstra1130568886744591910rm_a_b @ A2 ) ) ) ).
% conversionI'
thf(fact_533_conversion__refl,axiom,
! [A: term_a_b,A2: set_Pr4386577575007340137rm_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ A ) @ ( abstra1130568886744591910rm_a_b @ A2 ) ) ).
% conversion_refl
thf(fact_534_conversion__inv,axiom,
! [X: term_a_b,Y: term_a_b,R4: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( abstra1130568886744591910rm_a_b @ R4 ) )
= ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X ) @ ( abstra1130568886744591910rm_a_b @ R4 ) ) ) ).
% conversion_inv
thf(fact_535_all__ctxt__closed__srstep__conversions,axiom,
! [F2: set_Pr4934435412358123699_a_nat,R2: set_Pr4386577575007340137rm_a_b] : ( terms_5226143800768910156ed_a_b @ F2 @ ( abstra1130568886744591910rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) ) ) ) ).
% all_ctxt_closed_srstep_conversions
thf(fact_536_conversion__def,axiom,
( abstra1130568886744591910rm_a_b
= ( ^ [A6: set_Pr4386577575007340137rm_a_b] : ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ A6 @ ( conver7608559976937845093rm_a_b @ A6 ) ) ) ) ) ).
% conversion_def
thf(fact_537_meetE,axiom,
! [B: term_a_b,C2: term_a_b,A2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C2 ) @ ( abstra1093061187967292567rm_a_b @ A2 ) )
=> ~ ! [A3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B ) @ ( transi7742714808557438673rm_a_b @ A2 ) )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ C2 ) @ ( transi7742714808557438673rm_a_b @ A2 ) ) ) ) ).
% meetE
thf(fact_538_meetD,axiom,
! [B: term_a_b,C2: term_a_b,A2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C2 ) @ ( abstra1093061187967292567rm_a_b @ A2 ) )
=> ? [A3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B ) @ ( transi7742714808557438673rm_a_b @ A2 ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ C2 ) @ ( transi7742714808557438673rm_a_b @ A2 ) ) ) ) ).
% meetD
thf(fact_539_conversionE,axiom,
! [A: term_a_b,B: term_a_b,A2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( abstra1130568886744591910rm_a_b @ A2 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ A2 @ ( conver7608559976937845093rm_a_b @ A2 ) ) ) ) ) ).
% conversionE
thf(fact_540_CE__on__def,axiom,
( rewrit660295756789745446rm_a_b
= ( ^ [R6: set_Pr4386577575007340137rm_a_b,S3: set_Pr4386577575007340137rm_a_b,A6: set_term_a_b] :
! [X2: term_a_b] :
( ( member_term_a_b @ X2 @ A6 )
=> ! [B4: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ B4 ) @ ( abstra1130568886744591910rm_a_b @ R6 ) )
= ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ B4 ) @ ( abstra1130568886744591910rm_a_b @ S3 ) ) ) ) ) ) ).
% CE_on_def
thf(fact_541_comp__rtrancl__trancl,axiom,
! [R4: set_Pr4386577575007340137rm_a_b,S6: set_Pr4386577575007340137rm_a_b,S: term_a_b,T: term_a_b] :
( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ R4 @ S6 ) @ S6 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ R4 @ S6 ) ) @ S6 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( transi7922773638565587891rm_a_b @ S6 ) ) ) ) ).
% comp_rtrancl_trancl
thf(fact_542_restrict__SN__trancl__simp,axiom,
! [A2: set_Pr4386577575007340137rm_a_b] :
( ( transi7922773638565587891rm_a_b @ ( abstra4933154510623179783rm_a_b @ A2 @ A2 ) )
= ( abstra4933154510623179783rm_a_b @ ( transi7922773638565587891rm_a_b @ A2 ) @ A2 ) ) ).
% restrict_SN_trancl_simp
thf(fact_543_commute__def,axiom,
( abstra455195320138238452rm_a_b
= ( ^ [R6: set_Pr4386577575007340137rm_a_b,S3: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( conver7608559976937845093rm_a_b @ R6 ) ) @ ( transi7742714808557438673rm_a_b @ S3 ) ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ S3 ) @ ( transi7742714808557438673rm_a_b @ ( conver7608559976937845093rm_a_b @ R6 ) ) ) ) ) ) ).
% commute_def
thf(fact_544_same__fstI,axiom,
! [P2: term_a_b > $o,X: term_a_b,Y4: term_a_b,Y: term_a_b,R4: term_a_b > set_Pr4386577575007340137rm_a_b] :
( ( P2 @ X )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y4 @ Y ) @ ( R4 @ X ) )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y4 ) @ ( produc7020197800436672577rm_a_b @ X @ Y ) ) @ ( same_f843731131676418198rm_a_b @ P2 @ R4 ) ) ) ) ).
% same_fstI
thf(fact_545_compatible__rtrancl__split,axiom,
! [NS: set_Pr4386577575007340137rm_a_b,S6: set_Pr4386577575007340137rm_a_b,X: term_a_b,Y: term_a_b] :
( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ NS @ S6 ) @ S6 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ NS @ S6 ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( sup_su6776935440552674877rm_a_b @ ( relcom370159955682700863rm_a_b @ S6 @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ S6 ) @ ( transi7742714808557438673rm_a_b @ NS ) ) ) @ ( transi7742714808557438673rm_a_b @ NS ) ) ) ) ) ).
% compatible_rtrancl_split
thf(fact_546_trancl__reflcl,axiom,
! [R3: set_Pr4386577575007340137rm_a_b] :
( ( transi7922773638565587891rm_a_b @ ( sup_su6776935440552674877rm_a_b @ R3 @ id_term_a_b ) )
= ( transi7742714808557438673rm_a_b @ R3 ) ) ).
% trancl_reflcl
thf(fact_547_subsetI,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b] :
( ! [X3: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X3 @ A2 )
=> ( member5869715511025134514rm_a_b @ X3 @ B2 ) )
=> ( ord_le118470702582115849rm_a_b @ A2 @ B2 ) ) ).
% subsetI
thf(fact_548_le__sup__iff,axiom,
! [X: set_Pr4386577575007340137rm_a_b,Y: set_Pr4386577575007340137rm_a_b,Z: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ ( sup_su6776935440552674877rm_a_b @ X @ Y ) @ Z )
= ( ( ord_le118470702582115849rm_a_b @ X @ Z )
& ( ord_le118470702582115849rm_a_b @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_549_le__sup__iff,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ X @ Y ) @ Z )
= ( ( ord_less_eq_nat @ X @ Z )
& ( ord_less_eq_nat @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_550_sup_Obounded__iff,axiom,
! [B: set_Pr4386577575007340137rm_a_b,C2: set_Pr4386577575007340137rm_a_b,A: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ ( sup_su6776935440552674877rm_a_b @ B @ C2 ) @ A )
= ( ( ord_le118470702582115849rm_a_b @ B @ A )
& ( ord_le118470702582115849rm_a_b @ C2 @ A ) ) ) ).
% sup.bounded_iff
thf(fact_551_sup_Obounded__iff,axiom,
! [B: nat,C2: nat,A: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C2 ) @ A )
= ( ( ord_less_eq_nat @ B @ A )
& ( ord_less_eq_nat @ C2 @ A ) ) ) ).
% sup.bounded_iff
thf(fact_552_Un__subset__iff,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b,C: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ ( sup_su6776935440552674877rm_a_b @ A2 @ B2 ) @ C )
= ( ( ord_le118470702582115849rm_a_b @ A2 @ C )
& ( ord_le118470702582115849rm_a_b @ B2 @ C ) ) ) ).
% Un_subset_iff
thf(fact_553_IdI,axiom,
! [A: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ A ) @ id_term_a_b ) ).
% IdI
thf(fact_554_pair__in__Id__conv,axiom,
! [A: term_a_b,B: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ id_term_a_b )
= ( A = B ) ) ).
% pair_in_Id_conv
thf(fact_555_converse__mono,axiom,
! [R3: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ ( conver7608559976937845093rm_a_b @ R3 ) @ ( conver7608559976937845093rm_a_b @ S ) )
= ( ord_le118470702582115849rm_a_b @ R3 @ S ) ) ).
% converse_mono
thf(fact_556_Id__O__R,axiom,
! [R4: set_Pr4386577575007340137rm_a_b] :
( ( relcom370159955682700863rm_a_b @ id_term_a_b @ R4 )
= R4 ) ).
% Id_O_R
thf(fact_557_R__O__Id,axiom,
! [R4: set_Pr4386577575007340137rm_a_b] :
( ( relcom370159955682700863rm_a_b @ R4 @ id_term_a_b )
= R4 ) ).
% R_O_Id
thf(fact_558_converse__Id,axiom,
( ( conver7608559976937845093rm_a_b @ id_term_a_b )
= id_term_a_b ) ).
% converse_Id
thf(fact_559_rtrancl__reflcl,axiom,
! [R4: set_Pr4386577575007340137rm_a_b] :
( ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ R4 @ id_term_a_b ) )
= ( transi7742714808557438673rm_a_b @ R4 ) ) ).
% rtrancl_reflcl
thf(fact_560_rtrancl__reflcl__absorb,axiom,
! [R4: set_Pr4386577575007340137rm_a_b] :
( ( sup_su6776935440552674877rm_a_b @ ( transi7742714808557438673rm_a_b @ R4 ) @ id_term_a_b )
= ( transi7742714808557438673rm_a_b @ R4 ) ) ).
% rtrancl_reflcl_absorb
thf(fact_561_srsteps__with__root__step__sig__mono,axiom,
! [F2: set_Pr4934435412358123699_a_nat,G2: set_Pr4934435412358123699_a_nat,R2: set_Pr4386577575007340137rm_a_b] :
( ( ord_le8666007276011122963_a_nat @ F2 @ G2 )
=> ( ord_le118470702582115849rm_a_b @ ( srstep7844470518422762656ep_a_b @ F2 @ R2 ) @ ( srstep7844470518422762656ep_a_b @ G2 @ R2 ) ) ) ).
% srsteps_with_root_step_sig_mono
thf(fact_562_sig__step__mono,axiom,
! [F2: set_Pr4934435412358123699_a_nat,G2: set_Pr4934435412358123699_a_nat,R2: set_Pr4386577575007340137rm_a_b] :
( ( ord_le8666007276011122963_a_nat @ F2 @ G2 )
=> ( ord_le118470702582115849rm_a_b @ ( sig_step_a_b @ F2 @ R2 ) @ ( sig_step_a_b @ G2 @ R2 ) ) ) ).
% sig_step_mono
thf(fact_563_subset__iff,axiom,
( ord_le118470702582115849rm_a_b
= ( ^ [A6: set_Pr4386577575007340137rm_a_b,B6: set_Pr4386577575007340137rm_a_b] :
! [T3: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ T3 @ A6 )
=> ( member5869715511025134514rm_a_b @ T3 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_564_subset__eq,axiom,
( ord_le118470702582115849rm_a_b
= ( ^ [A6: set_Pr4386577575007340137rm_a_b,B6: set_Pr4386577575007340137rm_a_b] :
! [X2: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X2 @ A6 )
=> ( member5869715511025134514rm_a_b @ X2 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_565_subsetD,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b,C2: produc357393685978478089rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A2 @ B2 )
=> ( ( member5869715511025134514rm_a_b @ C2 @ A2 )
=> ( member5869715511025134514rm_a_b @ C2 @ B2 ) ) ) ).
% subsetD
thf(fact_566_in__mono,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b,X: produc357393685978478089rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A2 @ B2 )
=> ( ( member5869715511025134514rm_a_b @ X @ A2 )
=> ( member5869715511025134514rm_a_b @ X @ B2 ) ) ) ).
% in_mono
thf(fact_567_srstep__monp,axiom,
! [F2: set_Pr4934435412358123699_a_nat,G2: set_Pr4934435412358123699_a_nat,R2: set_Pr4386577575007340137rm_a_b] :
( ( ord_le8666007276011122963_a_nat @ F2 @ G2 )
=> ( ord_le118470702582115849rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) ) @ ( sig_step_a_b @ G2 @ ( rstep_a_b @ R2 ) ) ) ) ).
% srstep_monp
thf(fact_568_IdE,axiom,
! [P: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ P @ id_term_a_b )
=> ~ ! [X3: term_a_b] :
( P
!= ( produc7020197800436672577rm_a_b @ X3 @ X3 ) ) ) ).
% IdE
thf(fact_569_subrelI,axiom,
! [R3: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
( ! [X3: term_a_b,Y2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y2 ) @ R3 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y2 ) @ S ) )
=> ( ord_le118470702582115849rm_a_b @ R3 @ S ) ) ).
% subrelI
thf(fact_570_inf__sup__ord_I4_J,axiom,
! [Y: set_Pr4386577575007340137rm_a_b,X: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ Y @ ( sup_su6776935440552674877rm_a_b @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_571_inf__sup__ord_I4_J,axiom,
! [Y: nat,X: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_572_inf__sup__ord_I3_J,axiom,
! [X: set_Pr4386577575007340137rm_a_b,Y: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ X @ ( sup_su6776935440552674877rm_a_b @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_573_inf__sup__ord_I3_J,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ X @ ( sup_sup_nat @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_574_le__supE,axiom,
! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b,X: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ ( sup_su6776935440552674877rm_a_b @ A @ B ) @ X )
=> ~ ( ( ord_le118470702582115849rm_a_b @ A @ X )
=> ~ ( ord_le118470702582115849rm_a_b @ B @ X ) ) ) ).
% le_supE
thf(fact_575_le__supE,axiom,
! [A: nat,B: nat,X: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B ) @ X )
=> ~ ( ( ord_less_eq_nat @ A @ X )
=> ~ ( ord_less_eq_nat @ B @ X ) ) ) ).
% le_supE
thf(fact_576_le__supI,axiom,
! [A: set_Pr4386577575007340137rm_a_b,X: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A @ X )
=> ( ( ord_le118470702582115849rm_a_b @ B @ X )
=> ( ord_le118470702582115849rm_a_b @ ( sup_su6776935440552674877rm_a_b @ A @ B ) @ X ) ) ) ).
% le_supI
thf(fact_577_le__supI,axiom,
! [A: nat,X: nat,B: nat] :
( ( ord_less_eq_nat @ A @ X )
=> ( ( ord_less_eq_nat @ B @ X )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B ) @ X ) ) ) ).
% le_supI
thf(fact_578_sup__ge1,axiom,
! [X: set_Pr4386577575007340137rm_a_b,Y: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ X @ ( sup_su6776935440552674877rm_a_b @ X @ Y ) ) ).
% sup_ge1
thf(fact_579_sup__ge1,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ X @ ( sup_sup_nat @ X @ Y ) ) ).
% sup_ge1
thf(fact_580_sup__ge2,axiom,
! [Y: set_Pr4386577575007340137rm_a_b,X: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ Y @ ( sup_su6776935440552674877rm_a_b @ X @ Y ) ) ).
% sup_ge2
thf(fact_581_sup__ge2,axiom,
! [Y: nat,X: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X @ Y ) ) ).
% sup_ge2
thf(fact_582_le__supI1,axiom,
! [X: set_Pr4386577575007340137rm_a_b,A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ X @ A )
=> ( ord_le118470702582115849rm_a_b @ X @ ( sup_su6776935440552674877rm_a_b @ A @ B ) ) ) ).
% le_supI1
thf(fact_583_le__supI1,axiom,
! [X: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ X @ A )
=> ( ord_less_eq_nat @ X @ ( sup_sup_nat @ A @ B ) ) ) ).
% le_supI1
thf(fact_584_le__supI2,axiom,
! [X: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b,A: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ X @ B )
=> ( ord_le118470702582115849rm_a_b @ X @ ( sup_su6776935440552674877rm_a_b @ A @ B ) ) ) ).
% le_supI2
thf(fact_585_le__supI2,axiom,
! [X: nat,B: nat,A: nat] :
( ( ord_less_eq_nat @ X @ B )
=> ( ord_less_eq_nat @ X @ ( sup_sup_nat @ A @ B ) ) ) ).
% le_supI2
thf(fact_586_sup_Omono,axiom,
! [C2: set_Pr4386577575007340137rm_a_b,A: set_Pr4386577575007340137rm_a_b,D: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ C2 @ A )
=> ( ( ord_le118470702582115849rm_a_b @ D @ B )
=> ( ord_le118470702582115849rm_a_b @ ( sup_su6776935440552674877rm_a_b @ C2 @ D ) @ ( sup_su6776935440552674877rm_a_b @ A @ B ) ) ) ) ).
% sup.mono
thf(fact_587_sup_Omono,axiom,
! [C2: nat,A: nat,D: nat,B: nat] :
( ( ord_less_eq_nat @ C2 @ A )
=> ( ( ord_less_eq_nat @ D @ B )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ C2 @ D ) @ ( sup_sup_nat @ A @ B ) ) ) ) ).
% sup.mono
thf(fact_588_sup__mono,axiom,
! [A: set_Pr4386577575007340137rm_a_b,C2: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b,D: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A @ C2 )
=> ( ( ord_le118470702582115849rm_a_b @ B @ D )
=> ( ord_le118470702582115849rm_a_b @ ( sup_su6776935440552674877rm_a_b @ A @ B ) @ ( sup_su6776935440552674877rm_a_b @ C2 @ D ) ) ) ) ).
% sup_mono
thf(fact_589_sup__mono,axiom,
! [A: nat,C2: nat,B: nat,D: nat] :
( ( ord_less_eq_nat @ A @ C2 )
=> ( ( ord_less_eq_nat @ B @ D )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B ) @ ( sup_sup_nat @ C2 @ D ) ) ) ) ).
% sup_mono
thf(fact_590_sup__least,axiom,
! [Y: set_Pr4386577575007340137rm_a_b,X: set_Pr4386577575007340137rm_a_b,Z: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ Y @ X )
=> ( ( ord_le118470702582115849rm_a_b @ Z @ X )
=> ( ord_le118470702582115849rm_a_b @ ( sup_su6776935440552674877rm_a_b @ Y @ Z ) @ X ) ) ) ).
% sup_least
thf(fact_591_sup__least,axiom,
! [Y: nat,X: nat,Z: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_less_eq_nat @ Z @ X )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ Y @ Z ) @ X ) ) ) ).
% sup_least
thf(fact_592_le__iff__sup,axiom,
( ord_le118470702582115849rm_a_b
= ( ^ [X2: set_Pr4386577575007340137rm_a_b,Y3: set_Pr4386577575007340137rm_a_b] :
( ( sup_su6776935440552674877rm_a_b @ X2 @ Y3 )
= Y3 ) ) ) ).
% le_iff_sup
thf(fact_593_le__iff__sup,axiom,
( ord_less_eq_nat
= ( ^ [X2: nat,Y3: nat] :
( ( sup_sup_nat @ X2 @ Y3 )
= Y3 ) ) ) ).
% le_iff_sup
thf(fact_594_sup_OorderE,axiom,
! [B: set_Pr4386577575007340137rm_a_b,A: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ B @ A )
=> ( A
= ( sup_su6776935440552674877rm_a_b @ A @ B ) ) ) ).
% sup.orderE
thf(fact_595_sup_OorderE,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( A
= ( sup_sup_nat @ A @ B ) ) ) ).
% sup.orderE
thf(fact_596_sup_OorderI,axiom,
! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ( A
= ( sup_su6776935440552674877rm_a_b @ A @ B ) )
=> ( ord_le118470702582115849rm_a_b @ B @ A ) ) ).
% sup.orderI
thf(fact_597_sup_OorderI,axiom,
! [A: nat,B: nat] :
( ( A
= ( sup_sup_nat @ A @ B ) )
=> ( ord_less_eq_nat @ B @ A ) ) ).
% sup.orderI
thf(fact_598_sup__unique,axiom,
! [F3: set_Pr4386577575007340137rm_a_b > set_Pr4386577575007340137rm_a_b > set_Pr4386577575007340137rm_a_b,X: set_Pr4386577575007340137rm_a_b,Y: set_Pr4386577575007340137rm_a_b] :
( ! [X3: set_Pr4386577575007340137rm_a_b,Y2: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ X3 @ ( F3 @ X3 @ Y2 ) )
=> ( ! [X3: set_Pr4386577575007340137rm_a_b,Y2: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ Y2 @ ( F3 @ X3 @ Y2 ) )
=> ( ! [X3: set_Pr4386577575007340137rm_a_b,Y2: set_Pr4386577575007340137rm_a_b,Z2: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ Y2 @ X3 )
=> ( ( ord_le118470702582115849rm_a_b @ Z2 @ X3 )
=> ( ord_le118470702582115849rm_a_b @ ( F3 @ Y2 @ Z2 ) @ X3 ) ) )
=> ( ( sup_su6776935440552674877rm_a_b @ X @ Y )
= ( F3 @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_599_sup__unique,axiom,
! [F3: nat > nat > nat,X: nat,Y: nat] :
( ! [X3: nat,Y2: nat] : ( ord_less_eq_nat @ X3 @ ( F3 @ X3 @ Y2 ) )
=> ( ! [X3: nat,Y2: nat] : ( ord_less_eq_nat @ Y2 @ ( F3 @ X3 @ Y2 ) )
=> ( ! [X3: nat,Y2: nat,Z2: nat] :
( ( ord_less_eq_nat @ Y2 @ X3 )
=> ( ( ord_less_eq_nat @ Z2 @ X3 )
=> ( ord_less_eq_nat @ ( F3 @ Y2 @ Z2 ) @ X3 ) ) )
=> ( ( sup_sup_nat @ X @ Y )
= ( F3 @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_600_sup_Oabsorb1,axiom,
! [B: set_Pr4386577575007340137rm_a_b,A: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ B @ A )
=> ( ( sup_su6776935440552674877rm_a_b @ A @ B )
= A ) ) ).
% sup.absorb1
thf(fact_601_sup_Oabsorb1,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( sup_sup_nat @ A @ B )
= A ) ) ).
% sup.absorb1
thf(fact_602_sup_Oabsorb2,axiom,
! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A @ B )
=> ( ( sup_su6776935440552674877rm_a_b @ A @ B )
= B ) ) ).
% sup.absorb2
thf(fact_603_sup_Oabsorb2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( sup_sup_nat @ A @ B )
= B ) ) ).
% sup.absorb2
thf(fact_604_sup__absorb1,axiom,
! [Y: set_Pr4386577575007340137rm_a_b,X: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ Y @ X )
=> ( ( sup_su6776935440552674877rm_a_b @ X @ Y )
= X ) ) ).
% sup_absorb1
thf(fact_605_sup__absorb1,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( sup_sup_nat @ X @ Y )
= X ) ) ).
% sup_absorb1
thf(fact_606_sup__absorb2,axiom,
! [X: set_Pr4386577575007340137rm_a_b,Y: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ X @ Y )
=> ( ( sup_su6776935440552674877rm_a_b @ X @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_607_sup__absorb2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( sup_sup_nat @ X @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_608_sup_OboundedE,axiom,
! [B: set_Pr4386577575007340137rm_a_b,C2: set_Pr4386577575007340137rm_a_b,A: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ ( sup_su6776935440552674877rm_a_b @ B @ C2 ) @ A )
=> ~ ( ( ord_le118470702582115849rm_a_b @ B @ A )
=> ~ ( ord_le118470702582115849rm_a_b @ C2 @ A ) ) ) ).
% sup.boundedE
thf(fact_609_sup_OboundedE,axiom,
! [B: nat,C2: nat,A: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C2 ) @ A )
=> ~ ( ( ord_less_eq_nat @ B @ A )
=> ~ ( ord_less_eq_nat @ C2 @ A ) ) ) ).
% sup.boundedE
thf(fact_610_sup_OboundedI,axiom,
! [B: set_Pr4386577575007340137rm_a_b,A: set_Pr4386577575007340137rm_a_b,C2: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ B @ A )
=> ( ( ord_le118470702582115849rm_a_b @ C2 @ A )
=> ( ord_le118470702582115849rm_a_b @ ( sup_su6776935440552674877rm_a_b @ B @ C2 ) @ A ) ) ) ).
% sup.boundedI
thf(fact_611_sup_OboundedI,axiom,
! [B: nat,A: nat,C2: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C2 @ A )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C2 ) @ A ) ) ) ).
% sup.boundedI
thf(fact_612_sup_Oorder__iff,axiom,
( ord_le118470702582115849rm_a_b
= ( ^ [B4: set_Pr4386577575007340137rm_a_b,A4: set_Pr4386577575007340137rm_a_b] :
( A4
= ( sup_su6776935440552674877rm_a_b @ A4 @ B4 ) ) ) ) ).
% sup.order_iff
thf(fact_613_sup_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [B4: nat,A4: nat] :
( A4
= ( sup_sup_nat @ A4 @ B4 ) ) ) ) ).
% sup.order_iff
thf(fact_614_sup_Ocobounded1,axiom,
! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ A @ ( sup_su6776935440552674877rm_a_b @ A @ B ) ) ).
% sup.cobounded1
thf(fact_615_sup_Ocobounded1,axiom,
! [A: nat,B: nat] : ( ord_less_eq_nat @ A @ ( sup_sup_nat @ A @ B ) ) ).
% sup.cobounded1
thf(fact_616_sup_Ocobounded2,axiom,
! [B: set_Pr4386577575007340137rm_a_b,A: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ B @ ( sup_su6776935440552674877rm_a_b @ A @ B ) ) ).
% sup.cobounded2
thf(fact_617_sup_Ocobounded2,axiom,
! [B: nat,A: nat] : ( ord_less_eq_nat @ B @ ( sup_sup_nat @ A @ B ) ) ).
% sup.cobounded2
thf(fact_618_sup_Oabsorb__iff1,axiom,
( ord_le118470702582115849rm_a_b
= ( ^ [B4: set_Pr4386577575007340137rm_a_b,A4: set_Pr4386577575007340137rm_a_b] :
( ( sup_su6776935440552674877rm_a_b @ A4 @ B4 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_619_sup_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [B4: nat,A4: nat] :
( ( sup_sup_nat @ A4 @ B4 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_620_sup_Oabsorb__iff2,axiom,
( ord_le118470702582115849rm_a_b
= ( ^ [A4: set_Pr4386577575007340137rm_a_b,B4: set_Pr4386577575007340137rm_a_b] :
( ( sup_su6776935440552674877rm_a_b @ A4 @ B4 )
= B4 ) ) ) ).
% sup.absorb_iff2
thf(fact_621_sup_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B4: nat] :
( ( sup_sup_nat @ A4 @ B4 )
= B4 ) ) ) ).
% sup.absorb_iff2
thf(fact_622_sup_OcoboundedI1,axiom,
! [C2: set_Pr4386577575007340137rm_a_b,A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ C2 @ A )
=> ( ord_le118470702582115849rm_a_b @ C2 @ ( sup_su6776935440552674877rm_a_b @ A @ B ) ) ) ).
% sup.coboundedI1
thf(fact_623_sup_OcoboundedI1,axiom,
! [C2: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ C2 @ A )
=> ( ord_less_eq_nat @ C2 @ ( sup_sup_nat @ A @ B ) ) ) ).
% sup.coboundedI1
thf(fact_624_sup_OcoboundedI2,axiom,
! [C2: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b,A: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ C2 @ B )
=> ( ord_le118470702582115849rm_a_b @ C2 @ ( sup_su6776935440552674877rm_a_b @ A @ B ) ) ) ).
% sup.coboundedI2
thf(fact_625_sup_OcoboundedI2,axiom,
! [C2: nat,B: nat,A: nat] :
( ( ord_less_eq_nat @ C2 @ B )
=> ( ord_less_eq_nat @ C2 @ ( sup_sup_nat @ A @ B ) ) ) ).
% sup.coboundedI2
thf(fact_626_srrstep__monp,axiom,
! [F2: set_Pr4934435412358123699_a_nat,G2: set_Pr4934435412358123699_a_nat,R2: set_Pr4386577575007340137rm_a_b] :
( ( ord_le8666007276011122963_a_nat @ F2 @ G2 )
=> ( ord_le118470702582115849rm_a_b @ ( sig_step_a_b @ F2 @ ( rrstep_a_b @ R2 ) ) @ ( sig_step_a_b @ G2 @ ( rrstep_a_b @ R2 ) ) ) ) ).
% srrstep_monp
thf(fact_627_Un__mono,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,C: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b,D2: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A2 @ C )
=> ( ( ord_le118470702582115849rm_a_b @ B2 @ D2 )
=> ( ord_le118470702582115849rm_a_b @ ( sup_su6776935440552674877rm_a_b @ A2 @ B2 ) @ ( sup_su6776935440552674877rm_a_b @ C @ D2 ) ) ) ) ).
% Un_mono
thf(fact_628_Un__least,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,C: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A2 @ C )
=> ( ( ord_le118470702582115849rm_a_b @ B2 @ C )
=> ( ord_le118470702582115849rm_a_b @ ( sup_su6776935440552674877rm_a_b @ A2 @ B2 ) @ C ) ) ) ).
% Un_least
thf(fact_629_Un__upper1,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ A2 @ ( sup_su6776935440552674877rm_a_b @ A2 @ B2 ) ) ).
% Un_upper1
thf(fact_630_Un__upper2,axiom,
! [B2: set_Pr4386577575007340137rm_a_b,A2: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ B2 @ ( sup_su6776935440552674877rm_a_b @ A2 @ B2 ) ) ).
% Un_upper2
thf(fact_631_Un__absorb1,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A2 @ B2 )
=> ( ( sup_su6776935440552674877rm_a_b @ A2 @ B2 )
= B2 ) ) ).
% Un_absorb1
thf(fact_632_Un__absorb2,axiom,
! [B2: set_Pr4386577575007340137rm_a_b,A2: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ B2 @ A2 )
=> ( ( sup_su6776935440552674877rm_a_b @ A2 @ B2 )
= A2 ) ) ).
% Un_absorb2
thf(fact_633_subset__UnE,axiom,
! [C: set_Pr4386577575007340137rm_a_b,A2: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ C @ ( sup_su6776935440552674877rm_a_b @ A2 @ B2 ) )
=> ~ ! [A7: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A7 @ A2 )
=> ! [B7: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ B7 @ B2 )
=> ( C
!= ( sup_su6776935440552674877rm_a_b @ A7 @ B7 ) ) ) ) ) ).
% subset_UnE
thf(fact_634_subset__Un__eq,axiom,
( ord_le118470702582115849rm_a_b
= ( ^ [A6: set_Pr4386577575007340137rm_a_b,B6: set_Pr4386577575007340137rm_a_b] :
( ( sup_su6776935440552674877rm_a_b @ A6 @ B6 )
= B6 ) ) ) ).
% subset_Un_eq
thf(fact_635_rtrancl__mono,axiom,
! [R3: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ R3 @ S )
=> ( ord_le118470702582115849rm_a_b @ ( transi7742714808557438673rm_a_b @ R3 ) @ ( transi7742714808557438673rm_a_b @ S ) ) ) ).
% rtrancl_mono
thf(fact_636_rtrancl__subset,axiom,
! [R4: set_Pr4386577575007340137rm_a_b,S6: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ R4 @ S6 )
=> ( ( ord_le118470702582115849rm_a_b @ S6 @ ( transi7742714808557438673rm_a_b @ R4 ) )
=> ( ( transi7742714808557438673rm_a_b @ S6 )
= ( transi7742714808557438673rm_a_b @ R4 ) ) ) ) ).
% rtrancl_subset
thf(fact_637_rtrancl__subset__rtrancl,axiom,
! [R3: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ R3 @ ( transi7742714808557438673rm_a_b @ S ) )
=> ( ord_le118470702582115849rm_a_b @ ( transi7742714808557438673rm_a_b @ R3 ) @ ( transi7742714808557438673rm_a_b @ S ) ) ) ).
% rtrancl_subset_rtrancl
thf(fact_638_relcomp__mono,axiom,
! [R7: set_Pr4386577575007340137rm_a_b,R3: set_Pr4386577575007340137rm_a_b,S7: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ R7 @ R3 )
=> ( ( ord_le118470702582115849rm_a_b @ S7 @ S )
=> ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ R7 @ S7 ) @ ( relcom370159955682700863rm_a_b @ R3 @ S ) ) ) ) ).
% relcomp_mono
thf(fact_639_O__mono1,axiom,
! [R4: set_Pr4386577575007340137rm_a_b,R8: set_Pr4386577575007340137rm_a_b,S6: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ R4 @ R8 )
=> ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ S6 @ R4 ) @ ( relcom370159955682700863rm_a_b @ S6 @ R8 ) ) ) ).
% O_mono1
thf(fact_640_O__mono2,axiom,
! [R4: set_Pr4386577575007340137rm_a_b,R8: set_Pr4386577575007340137rm_a_b,T4: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ R4 @ R8 )
=> ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ R4 @ T4 ) @ ( relcom370159955682700863rm_a_b @ R8 @ T4 ) ) ) ).
% O_mono2
thf(fact_641_converse__subset__swap,axiom,
! [R3: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ R3 @ ( conver7608559976937845093rm_a_b @ S ) )
= ( ord_le118470702582115849rm_a_b @ ( conver7608559976937845093rm_a_b @ R3 ) @ S ) ) ).
% converse_subset_swap
thf(fact_642_trancl__mono,axiom,
! [P: produc357393685978478089rm_a_b,R3: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ P @ ( transi7922773638565587891rm_a_b @ R3 ) )
=> ( ( ord_le118470702582115849rm_a_b @ R3 @ S )
=> ( member5869715511025134514rm_a_b @ P @ ( transi7922773638565587891rm_a_b @ S ) ) ) ) ).
% trancl_mono
thf(fact_643_trancl__mono__set,axiom,
! [R3: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ R3 @ S )
=> ( ord_le118470702582115849rm_a_b @ ( transi7922773638565587891rm_a_b @ R3 ) @ ( transi7922773638565587891rm_a_b @ S ) ) ) ).
% trancl_mono_set
thf(fact_644_sig__step__mono2,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,L3: set_Pr4386577575007340137rm_a_b,F2: set_Pr4934435412358123699_a_nat] :
( ( ord_le118470702582115849rm_a_b @ R2 @ L3 )
=> ( ord_le118470702582115849rm_a_b @ ( sig_step_a_b @ F2 @ R2 ) @ ( sig_step_a_b @ F2 @ L3 ) ) ) ).
% sig_step_mono2
thf(fact_645_srsteps__eq__monp,axiom,
! [F2: set_Pr4934435412358123699_a_nat,G2: set_Pr4934435412358123699_a_nat,R2: set_Pr4386577575007340137rm_a_b] :
( ( ord_le8666007276011122963_a_nat @ F2 @ G2 )
=> ( ord_le118470702582115849rm_a_b @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) ) ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ G2 @ ( rstep_a_b @ R2 ) ) ) ) ) ).
% srsteps_eq_monp
thf(fact_646_srsteps__monp,axiom,
! [F2: set_Pr4934435412358123699_a_nat,G2: set_Pr4934435412358123699_a_nat,R2: set_Pr4386577575007340137rm_a_b] :
( ( ord_le8666007276011122963_a_nat @ F2 @ G2 )
=> ( ord_le118470702582115849rm_a_b @ ( transi7922773638565587891rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) ) ) @ ( transi7922773638565587891rm_a_b @ ( sig_step_a_b @ G2 @ ( rstep_a_b @ R2 ) ) ) ) ) ).
% srsteps_monp
thf(fact_647_rtrancl__O__push,axiom,
! [S6: set_Pr4386577575007340137rm_a_b,R4: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ S6 @ R4 ) @ ( relcom370159955682700863rm_a_b @ R4 @ ( transi7742714808557438673rm_a_b @ S6 ) ) )
=> ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ S6 ) @ R4 ) @ ( relcom370159955682700863rm_a_b @ R4 @ ( transi7742714808557438673rm_a_b @ S6 ) ) ) ) ).
% rtrancl_O_push
thf(fact_648_compat__tr__compat,axiom,
! [NS: set_Pr4386577575007340137rm_a_b,S6: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ NS @ S6 ) @ S6 )
=> ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ NS ) @ S6 ) @ S6 ) ) ).
% compat_tr_compat
thf(fact_649_kleene__induct,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,X4: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b,C: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A2 @ X4 )
=> ( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ B2 @ X4 ) @ X4 )
=> ( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ X4 @ C ) @ X4 )
=> ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ B2 ) @ ( relcom370159955682700863rm_a_b @ A2 @ ( transi7742714808557438673rm_a_b @ C ) ) ) @ X4 ) ) ) ) ).
% kleene_induct
thf(fact_650_rtrancl__Un__subset,axiom,
! [R4: set_Pr4386577575007340137rm_a_b,S6: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ ( sup_su6776935440552674877rm_a_b @ ( transi7742714808557438673rm_a_b @ R4 ) @ ( transi7742714808557438673rm_a_b @ S6 ) ) @ ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ R4 @ S6 ) ) ) ).
% rtrancl_Un_subset
thf(fact_651_comp__trancl,axiom,
! [R4: set_Pr4386577575007340137rm_a_b,S6: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ R4 @ S6 ) @ S6 )
=> ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ R4 @ ( transi7922773638565587891rm_a_b @ S6 ) ) @ ( transi7922773638565587891rm_a_b @ S6 ) ) ) ).
% comp_trancl
thf(fact_652_kleene__trancl__induct,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,X4: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b,C: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A2 @ X4 )
=> ( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ B2 @ X4 ) @ X4 )
=> ( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ X4 @ C ) @ X4 )
=> ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ ( transi7922773638565587891rm_a_b @ B2 ) @ ( relcom370159955682700863rm_a_b @ A2 @ ( transi7922773638565587891rm_a_b @ C ) ) ) @ X4 ) ) ) ) ).
% kleene_trancl_induct
thf(fact_653_trancl__union__right,axiom,
! [R3: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ ( transi7922773638565587891rm_a_b @ R3 ) @ ( transi7922773638565587891rm_a_b @ ( sup_su6776935440552674877rm_a_b @ S @ R3 ) ) ) ).
% trancl_union_right
thf(fact_654_rrstep__rstep__mono,axiom,
! [R2: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ ( rrstep_a_b @ R2 ) @ ( rstep_a_b @ R2 ) ) ).
% rrstep_rstep_mono
thf(fact_655_non__strict__into__strict,axiom,
! [NS: set_Pr4386577575007340137rm_a_b,S6: set_Pr4386577575007340137rm_a_b,S: term_a_b,T: term_a_b] :
( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ NS @ S6 ) @ S6 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ NS ) @ S6 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ S6 ) ) ) ).
% non_strict_into_strict
thf(fact_656_rtrancl__unfold,axiom,
( transi7742714808557438673rm_a_b
= ( ^ [R6: set_Pr4386577575007340137rm_a_b] : ( sup_su6776935440552674877rm_a_b @ id_term_a_b @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ R6 ) @ R6 ) ) ) ) ).
% rtrancl_unfold
thf(fact_657_rtrancl__U__push,axiom,
! [S6: set_Pr4386577575007340137rm_a_b,R4: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ S6 @ R4 ) @ ( relcom370159955682700863rm_a_b @ R4 @ ( transi7742714808557438673rm_a_b @ S6 ) ) )
=> ( ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ S6 @ R4 ) )
= ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ R4 ) @ ( transi7742714808557438673rm_a_b @ S6 ) ) ) ) ).
% rtrancl_U_push
thf(fact_658_compatible__conv,axiom,
! [NS: set_Pr4386577575007340137rm_a_b,S6: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ NS @ S6 ) @ S6 )
=> ( ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ NS @ S6 ) ) @ ( relcom370159955682700863rm_a_b @ S6 @ ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ NS @ S6 ) ) ) )
= ( relcom370159955682700863rm_a_b @ S6 @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ S6 ) @ ( transi7742714808557438673rm_a_b @ NS ) ) ) ) ) ).
% compatible_conv
thf(fact_659_rtrancl__trancl__reflcl,axiom,
( transi7742714808557438673rm_a_b
= ( ^ [R6: set_Pr4386577575007340137rm_a_b] : ( sup_su6776935440552674877rm_a_b @ ( transi7922773638565587891rm_a_b @ R6 ) @ id_term_a_b ) ) ) ).
% rtrancl_trancl_reflcl
thf(fact_660_trancl__o__refl__is__trancl,axiom,
! [R3: set_Pr4386577575007340137rm_a_b] :
( ( relcom370159955682700863rm_a_b @ ( transi7922773638565587891rm_a_b @ R3 ) @ ( sup_su6776935440552674877rm_a_b @ R3 @ id_term_a_b ) )
= ( transi7922773638565587891rm_a_b @ R3 ) ) ).
% trancl_o_refl_is_trancl
thf(fact_661_SCR__on__def,axiom,
( rewrit4966517460798004134rm_a_b
= ( ^ [R6: set_Pr4386577575007340137rm_a_b,A6: set_term_a_b] :
! [X2: term_a_b] :
( ( member_term_a_b @ X2 @ A6 )
=> ! [B4: term_a_b,C3: term_a_b] :
( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ B4 ) @ R6 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ C3 ) @ R6 ) )
=> ? [D3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B4 @ D3 ) @ ( sup_su6776935440552674877rm_a_b @ R6 @ id_term_a_b ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ C3 @ D3 ) @ ( transi7742714808557438673rm_a_b @ R6 ) ) ) ) ) ) ) ).
% SCR_on_def
thf(fact_662_quasi__commute__def,axiom,
( abstra4317384144805385751rm_a_b
= ( ^ [R6: set_Pr4386577575007340137rm_a_b,S3: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ S3 @ R6 ) @ ( relcom370159955682700863rm_a_b @ R6 @ ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ R6 @ S3 ) ) ) ) ) ) ).
% quasi_commute_def
thf(fact_663_diamond__I,axiom,
! [R3: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ ( conver7608559976937845093rm_a_b @ R3 ) @ R3 ) @ ( relcom370159955682700863rm_a_b @ R3 @ ( conver7608559976937845093rm_a_b @ R3 ) ) )
=> ( abstra8151087076625346926rm_a_b @ R3 ) ) ).
% diamond_I
thf(fact_664_strongly__confluent__on__E11,axiom,
! [R3: set_Pr2972776593051762503rm_a_b,A2: set_Pr4386577575007340137rm_a_b,X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b,Z: produc357393685978478089rm_a_b] :
( ( abstra837640964651051114rm_a_b @ R3 @ A2 )
=> ( ( member5869715511025134514rm_a_b @ X @ A2 )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ R3 )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Z ) @ R3 )
=> ? [U2: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y @ U2 ) @ ( transi2615809358984392588rm_a_b @ R3 ) )
& ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Z @ U2 ) @ ( sup_su1845815533690797339rm_a_b @ R3 @ id_Pro3488709451141449061rm_a_b ) ) ) ) ) ) ) ).
% strongly_confluent_on_E11
thf(fact_665_strongly__confluent__on__E11,axiom,
! [R3: set_Pr4386577575007340137rm_a_b,A2: set_term_a_b,X: term_a_b,Y: term_a_b,Z: term_a_b] :
( ( abstra5664096641628173427rm_a_b @ R3 @ A2 )
=> ( ( member_term_a_b @ X @ A2 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ R3 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z ) @ R3 )
=> ? [U2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ U2 ) @ ( transi7742714808557438673rm_a_b @ R3 ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Z @ U2 ) @ ( sup_su6776935440552674877rm_a_b @ R3 @ id_term_a_b ) ) ) ) ) ) ) ).
% strongly_confluent_on_E11
thf(fact_666_strongly__confluent__on__def,axiom,
( abstra5664096641628173427rm_a_b
= ( ^ [R6: set_Pr4386577575007340137rm_a_b,A6: set_term_a_b] :
! [X2: term_a_b] :
( ( member_term_a_b @ X2 @ A6 )
=> ! [Y3: term_a_b,Z3: term_a_b] :
( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y3 ) @ R6 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Z3 ) @ R6 ) )
=> ? [U3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ U3 ) @ ( transi7742714808557438673rm_a_b @ R6 ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Z3 @ U3 ) @ ( sup_su6776935440552674877rm_a_b @ R6 @ id_term_a_b ) ) ) ) ) ) ) ).
% strongly_confluent_on_def
thf(fact_667_sig__step__rsteps__eq__dist,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,F2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ ( terms_7988297476397195622_a_b_b @ R2 ) @ F2 )
=> ( ( sup_su6776935440552674877rm_a_b @ ( sig_step_a_b @ F2 @ ( transi7922773638565587891rm_a_b @ ( rstep_a_b @ R2 ) ) ) @ id_term_a_b )
= ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) ) ) ) ) ).
% sig_step_rsteps_eq_dist
thf(fact_668_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_669_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_670_funas__rel__converse,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,F2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ ( terms_7988297476397195622_a_b_b @ R2 ) @ F2 )
=> ( ord_le8666007276011122963_a_nat @ ( terms_7988297476397195622_a_b_b @ ( conver7608559976937845093rm_a_b @ R2 ) ) @ F2 ) ) ).
% funas_rel_converse
thf(fact_671_order__antisym__conv,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_672_linorder__le__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_673_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F3: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F3 @ B )
= C2 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F3 @ X3 ) @ ( F3 @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F3 @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_674_ord__eq__le__subst,axiom,
! [A: nat,F3: nat > nat,B: nat,C2: nat] :
( ( A
= ( F3 @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F3 @ X3 ) @ ( F3 @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F3 @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_675_linorder__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_linear
thf(fact_676_order__eq__refl,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_677_order__subst2,axiom,
! [A: nat,B: nat,F3: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F3 @ B ) @ C2 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F3 @ X3 ) @ ( F3 @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F3 @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_678_order__subst1,axiom,
! [A: nat,F3: nat > nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ ( F3 @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F3 @ X3 ) @ ( F3 @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F3 @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_679_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 ) )
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ A4 @ B4 )
& ( ord_less_eq_nat @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_680_antisym,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_681_dual__order_Otrans,axiom,
! [B: nat,A: nat,C2: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C2 @ B )
=> ( ord_less_eq_nat @ C2 @ A ) ) ) ).
% dual_order.trans
thf(fact_682_dual__order_Oantisym,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_683_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 ) )
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ B4 @ A4 )
& ( ord_less_eq_nat @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_684_linorder__wlog,axiom,
! [P2: nat > nat > $o,A: nat,B: nat] :
( ! [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( P2 @ A3 @ B3 ) )
=> ( ! [A3: nat,B3: nat] :
( ( P2 @ B3 @ A3 )
=> ( P2 @ A3 @ B3 ) )
=> ( P2 @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_685_order__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z )
=> ( ord_less_eq_nat @ X @ Z ) ) ) ).
% order_trans
thf(fact_686_order_Otrans,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ord_less_eq_nat @ A @ C2 ) ) ) ).
% order.trans
thf(fact_687_order__antisym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_688_ord__le__eq__trans,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( B = C2 )
=> ( ord_less_eq_nat @ A @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_689_ord__eq__le__trans,axiom,
! [A: nat,B: nat,C2: nat] :
( ( A = B )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ord_less_eq_nat @ A @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_690_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 ) )
= ( ^ [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
& ( ord_less_eq_nat @ Y3 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_691_le__cases3,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ( ord_less_eq_nat @ X @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z ) )
=> ( ( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_eq_nat @ X @ Z ) )
=> ( ( ( ord_less_eq_nat @ X @ Z )
=> ~ ( ord_less_eq_nat @ Z @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z )
=> ~ ( ord_less_eq_nat @ Z @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z @ X )
=> ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_692_nle__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_eq_nat @ A @ B ) )
= ( ( ord_less_eq_nat @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_693_qc__imp__qc__trancl,axiom,
! [R3: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
( ( abstra4317384144805385751rm_a_b @ R3 @ S )
=> ( abstra4317384144805385751rm_a_b @ R3 @ ( transi7922773638565587891rm_a_b @ S ) ) ) ).
% qc_imp_qc_trancl
thf(fact_694_sig__step__rsteps__dist,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,F2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ ( terms_7988297476397195622_a_b_b @ R2 ) @ F2 )
=> ( ( sig_step_a_b @ F2 @ ( transi7922773638565587891rm_a_b @ ( rstep_a_b @ R2 ) ) )
= ( transi7922773638565587891rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) ) ) ) ) ).
% sig_step_rsteps_dist
thf(fact_695_diamond__def,axiom,
( abstra8151087076625346926rm_a_b
= ( ^ [R6: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ ( conver7608559976937845093rm_a_b @ R6 ) @ R6 ) @ ( relcom370159955682700863rm_a_b @ R6 @ ( conver7608559976937845093rm_a_b @ R6 ) ) ) ) ) ).
% diamond_def
thf(fact_696_diamond__E,axiom,
! [R3: set_Pr4386577575007340137rm_a_b] :
( ( abstra8151087076625346926rm_a_b @ R3 )
=> ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ ( conver7608559976937845093rm_a_b @ R3 ) @ R3 ) @ ( relcom370159955682700863rm_a_b @ R3 @ ( conver7608559976937845093rm_a_b @ R3 ) ) ) ) ).
% diamond_E
thf(fact_697_rsteps__eq__relcomp__srsteps__eq__relcompI,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,F2: set_Pr4934435412358123699_a_nat,S4: set_Pr4386577575007340137rm_a_b,S: term_a_b,T: term_a_b] :
( ( ord_le8666007276011122963_a_nat @ ( terms_7988297476397195622_a_b_b @ R2 ) @ F2 )
=> ( ( ord_le8666007276011122963_a_nat @ ( terms_7988297476397195622_a_b_b @ S4 ) @ F2 )
=> ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ S ) @ F2 )
=> ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ T ) @ F2 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( rstep_a_b @ R2 ) ) @ ( transi7742714808557438673rm_a_b @ ( rstep_a_b @ S4 ) ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) ) ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ S4 ) ) ) ) ) ) ) ) ) ) ).
% rsteps_eq_relcomp_srsteps_eq_relcompI
thf(fact_698_strongly__confluentI,axiom,
! [R3: set_Pr4386577575007340137rm_a_b] :
( ! [X3: term_a_b,Y2: term_a_b,Z2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y2 ) @ R3 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Z2 ) @ R3 )
=> ? [U4: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y2 @ U4 ) @ ( transi7742714808557438673rm_a_b @ R3 ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Z2 @ U4 ) @ ( sup_su6776935440552674877rm_a_b @ R3 @ id_term_a_b ) ) ) ) )
=> ( abstra5664096641628173427rm_a_b @ R3 @ top_top_set_term_a_b ) ) ).
% strongly_confluentI
thf(fact_699_rsteps__srstepsI,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,F2: set_Pr4934435412358123699_a_nat,S: term_a_b,T: term_a_b] :
( ( ord_le8666007276011122963_a_nat @ ( terms_7988297476397195622_a_b_b @ R2 ) @ F2 )
=> ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ S ) @ F2 )
=> ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ T ) @ F2 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( transi7922773638565587891rm_a_b @ ( rstep_a_b @ R2 ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( transi7922773638565587891rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) ) ) ) ) ) ) ) ).
% rsteps_srstepsI
thf(fact_700_rsteps__eq__srsteps__eqI,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,F2: set_Pr4934435412358123699_a_nat,S: term_a_b,T: term_a_b] :
( ( ord_le8666007276011122963_a_nat @ ( terms_7988297476397195622_a_b_b @ R2 ) @ F2 )
=> ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ S ) @ F2 )
=> ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ T ) @ F2 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( transi7742714808557438673rm_a_b @ ( rstep_a_b @ R2 ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) ) ) ) ) ) ) ) ).
% rsteps_eq_srsteps_eqI
thf(fact_701_IdD,axiom,
! [A: term_a_b,B: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ id_term_a_b )
=> ( A = B ) ) ).
% IdD
thf(fact_702_restrict__rtrancl,axiom,
! [R3: set_Pr4386577575007340137rm_a_b,S6: set_term_a_b] : ( ord_le118470702582115849rm_a_b @ ( transi7742714808557438673rm_a_b @ ( abstra6662565036008789004rm_a_b @ R3 @ S6 ) ) @ ( transi7742714808557438673rm_a_b @ R3 ) ) ).
% restrict_rtrancl
thf(fact_703_UNIV__I,axiom,
! [X: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ X @ top_to1314267278846557113rm_a_b ) ).
% UNIV_I
thf(fact_704_converse__UNIV,axiom,
( ( conver7608559976937845093rm_a_b @ top_to1314267278846557113rm_a_b )
= top_to1314267278846557113rm_a_b ) ).
% converse_UNIV
thf(fact_705_sup__top__left,axiom,
! [X: set_Pr4386577575007340137rm_a_b] :
( ( sup_su6776935440552674877rm_a_b @ top_to1314267278846557113rm_a_b @ X )
= top_to1314267278846557113rm_a_b ) ).
% sup_top_left
thf(fact_706_sup__top__right,axiom,
! [X: set_Pr4386577575007340137rm_a_b] :
( ( sup_su6776935440552674877rm_a_b @ X @ top_to1314267278846557113rm_a_b )
= top_to1314267278846557113rm_a_b ) ).
% sup_top_right
thf(fact_707_sig__stepI,axiom,
! [S: term_a_b,F2: set_Pr4934435412358123699_a_nat,T: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ S ) @ F2 )
=> ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ T ) @ F2 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ R2 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( sig_step_a_b @ F2 @ R2 ) ) ) ) ) ).
% sig_stepI
thf(fact_708_all__ctxt__closed__relcomp,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,F2: set_Pr4934435412358123699_a_nat,S4: set_Pr4386577575007340137rm_a_b] :
( ! [S5: term_a_b,T2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S5 @ T2 ) @ R2 )
=> ( ( S5 != T2 )
=> ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ S5 ) @ F2 )
& ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ T2 ) @ F2 ) ) ) )
=> ( ! [S5: term_a_b,T2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S5 @ T2 ) @ S4 )
=> ( ( S5 != T2 )
=> ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ S5 ) @ F2 )
& ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ T2 ) @ F2 ) ) ) )
=> ( ( terms_5226143800768910156ed_a_b @ F2 @ R2 )
=> ( ( terms_5226143800768910156ed_a_b @ F2 @ S4 )
=> ( terms_5226143800768910156ed_a_b @ F2 @ ( relcom370159955682700863rm_a_b @ R2 @ S4 ) ) ) ) ) ) ).
% all_ctxt_closed_relcomp
thf(fact_709_rstep__srstepI,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,F2: set_Pr4934435412358123699_a_nat,S: term_a_b,T: term_a_b] :
( ( ord_le8666007276011122963_a_nat @ ( terms_7988297476397195622_a_b_b @ R2 ) @ F2 )
=> ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ S ) @ F2 )
=> ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ T ) @ F2 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( rstep_a_b @ R2 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) ) ) ) ) ) ) ).
% rstep_srstepI
thf(fact_710_UNIV__witness,axiom,
? [X3: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ X3 @ top_to1314267278846557113rm_a_b ) ).
% UNIV_witness
thf(fact_711_UNIV__eq__I,axiom,
! [A2: set_Pr4386577575007340137rm_a_b] :
( ! [X3: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ X3 @ A2 )
=> ( top_to1314267278846557113rm_a_b = A2 ) ) ).
% UNIV_eq_I
thf(fact_712_Un__UNIV__left,axiom,
! [B2: set_Pr4386577575007340137rm_a_b] :
( ( sup_su6776935440552674877rm_a_b @ top_to1314267278846557113rm_a_b @ B2 )
= top_to1314267278846557113rm_a_b ) ).
% Un_UNIV_left
thf(fact_713_Un__UNIV__right,axiom,
! [A2: set_Pr4386577575007340137rm_a_b] :
( ( sup_su6776935440552674877rm_a_b @ A2 @ top_to1314267278846557113rm_a_b )
= top_to1314267278846557113rm_a_b ) ).
% Un_UNIV_right
thf(fact_714_sig__stepE,axiom,
! [S: term_a_b,T: term_a_b,F2: set_Pr4934435412358123699_a_nat,R2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( sig_step_a_b @ F2 @ R2 ) )
=> ~ ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ R2 )
=> ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ S ) @ F2 )
=> ~ ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ T ) @ F2 ) ) ) ) ).
% sig_stepE
thf(fact_715_all__ctxt__closed__sig__reflE,axiom,
! [F2: set_Pr4934435412358123699_a_nat,R2: set_Pr4386577575007340137rm_a_b,T: term_a_b] :
( ( terms_5226143800768910156ed_a_b @ F2 @ R2 )
=> ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ T ) @ F2 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T @ T ) @ R2 ) ) ) ).
% all_ctxt_closed_sig_reflE
thf(fact_716_srstepD,axiom,
! [S: term_a_b,T: term_a_b,F2: set_Pr4934435412358123699_a_nat,R2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( rstep_a_b @ R2 ) )
& ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ S ) @ F2 )
& ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ T ) @ F2 ) ) ) ).
% srstepD
thf(fact_717_srrstepD,axiom,
! [S: term_a_b,T: term_a_b,F2: set_Pr4934435412358123699_a_nat,R2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( sig_step_a_b @ F2 @ ( rrstep_a_b @ R2 ) ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( rrstep_a_b @ R2 ) )
& ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ S ) @ F2 )
& ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ T ) @ F2 ) ) ) ).
% srrstepD
thf(fact_718_sigstep__trancl__funas_I2_J,axiom,
! [S: term_a_b,T: term_a_b,F2: set_Pr4934435412358123699_a_nat,S4: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ S4 ) ) ) )
=> ( ( S != T )
=> ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ T ) @ F2 ) ) ) ).
% sigstep_trancl_funas(2)
thf(fact_719_sigstep__trancl__funas_I1_J,axiom,
! [S: term_a_b,T: term_a_b,F2: set_Pr4934435412358123699_a_nat,S4: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ S4 ) ) ) )
=> ( ( S != T )
=> ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ S ) @ F2 ) ) ) ).
% sigstep_trancl_funas(1)
thf(fact_720_srstepsD,axiom,
! [S: term_a_b,T: term_a_b,F2: set_Pr4934435412358123699_a_nat,R2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( transi7922773638565587891rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) ) ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( transi7922773638565587891rm_a_b @ ( rstep_a_b @ R2 ) ) )
& ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ S ) @ F2 )
& ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ T ) @ F2 ) ) ) ).
% srstepsD
thf(fact_721_boolean__algebra_Odisj__one__right,axiom,
! [X: set_Pr4386577575007340137rm_a_b] :
( ( sup_su6776935440552674877rm_a_b @ X @ top_to1314267278846557113rm_a_b )
= top_to1314267278846557113rm_a_b ) ).
% boolean_algebra.disj_one_right
thf(fact_722_boolean__algebra_Odisj__one__left,axiom,
! [X: set_Pr4386577575007340137rm_a_b] :
( ( sup_su6776935440552674877rm_a_b @ top_to1314267278846557113rm_a_b @ X )
= top_to1314267278846557113rm_a_b ) ).
% boolean_algebra.disj_one_left
thf(fact_723_rstep__trancl__sig__step__l,axiom,
! [S: term_a_b,T: term_a_b,R2: set_Pr4386577575007340137rm_a_b,F2: set_Pr4934435412358123699_a_nat,V: b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( transi7922773638565587891rm_a_b @ ( rstep_a_b @ R2 ) ) )
=> ( ( ord_le8666007276011122963_a_nat @ ( terms_7988297476397195622_a_b_b @ R2 ) @ F2 )
=> ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ T ) @ F2 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( terms_8519481630511763164ig_a_b @ F2 @ V @ S ) @ T ) @ ( transi7922773638565587891rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) ) ) ) ) ) ) ).
% rstep_trancl_sig_step_l
thf(fact_724_rstep__trancl__sig__step__r,axiom,
! [S: term_a_b,T: term_a_b,R2: set_Pr4386577575007340137rm_a_b,F2: set_Pr4934435412358123699_a_nat,V: b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( transi7922773638565587891rm_a_b @ ( rstep_a_b @ R2 ) ) )
=> ( ( ord_le8666007276011122963_a_nat @ ( terms_7988297476397195622_a_b_b @ R2 ) @ F2 )
=> ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ S ) @ F2 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ ( terms_8519481630511763164ig_a_b @ F2 @ V @ T ) ) @ ( transi7922773638565587891rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) ) ) ) ) ) ) ).
% rstep_trancl_sig_step_r
thf(fact_725_rstep__term__to__sig__r,axiom,
! [S: term_a_b,T: term_a_b,R2: set_Pr4386577575007340137rm_a_b,F2: set_Pr4934435412358123699_a_nat,V: b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( rstep_a_b @ R2 ) )
=> ( ( ord_le8666007276011122963_a_nat @ ( terms_7988297476397195622_a_b_b @ R2 ) @ F2 )
=> ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ S ) @ F2 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ ( terms_8519481630511763164ig_a_b @ F2 @ V @ T ) ) @ ( rstep_a_b @ R2 ) ) ) ) ) ).
% rstep_term_to_sig_r
thf(fact_726_rstep__term__to__sig__l,axiom,
! [S: term_a_b,T: term_a_b,R2: set_Pr4386577575007340137rm_a_b,F2: set_Pr4934435412358123699_a_nat,V: b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( rstep_a_b @ R2 ) )
=> ( ( ord_le8666007276011122963_a_nat @ ( terms_7988297476397195622_a_b_b @ R2 ) @ F2 )
=> ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ T ) @ F2 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( terms_8519481630511763164ig_a_b @ F2 @ V @ S ) @ T ) @ ( rstep_a_b @ R2 ) ) ) ) ) ).
% rstep_term_to_sig_l
thf(fact_727_top__empty__eq,axiom,
( top_to1262317115727556708_a_b_o
= ( ^ [X2: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ X2 @ top_to1314267278846557113rm_a_b ) ) ) ).
% top_empty_eq
thf(fact_728_boolean__algebra__cancel_Osup2,axiom,
! [B2: set_Pr4386577575007340137rm_a_b,K: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b,A: set_Pr4386577575007340137rm_a_b] :
( ( B2
= ( sup_su6776935440552674877rm_a_b @ K @ B ) )
=> ( ( sup_su6776935440552674877rm_a_b @ A @ B2 )
= ( sup_su6776935440552674877rm_a_b @ K @ ( sup_su6776935440552674877rm_a_b @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_729_boolean__algebra__cancel_Osup1,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,K: set_Pr4386577575007340137rm_a_b,A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ( A2
= ( sup_su6776935440552674877rm_a_b @ K @ A ) )
=> ( ( sup_su6776935440552674877rm_a_b @ A2 @ B )
= ( sup_su6776935440552674877rm_a_b @ K @ ( sup_su6776935440552674877rm_a_b @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_730_srsteps__eq__subst__relcomp__closed,axiom,
! [S: term_a_b,T: term_a_b,F2: set_Pr4934435412358123699_a_nat,R2: set_Pr4386577575007340137rm_a_b,S4: set_Pr4386577575007340137rm_a_b,Sigma: b > term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) ) ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ S4 ) ) ) ) )
=> ( ! [X3: b] : ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ ( Sigma @ X3 ) ) @ F2 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( subst_7999470309526761004_a_b_b @ S @ Sigma ) @ ( subst_7999470309526761004_a_b_b @ T @ Sigma ) ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) ) ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ S4 ) ) ) ) ) ) ) ).
% srsteps_eq_subst_relcomp_closed
thf(fact_731_strongly__confluent__E1n,axiom,
! [R3: set_Pr4386577575007340137rm_a_b,X: term_a_b,Y: term_a_b,Z: term_a_b,N: nat] :
( ( abstra5664096641628173427rm_a_b @ R3 @ top_top_set_term_a_b )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( sup_su6776935440552674877rm_a_b @ R3 @ id_term_a_b ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z ) @ ( compow4057154403645558940rm_a_b @ N @ R3 ) )
=> ? [U2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ U2 ) @ ( transi7742714808557438673rm_a_b @ R3 ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Z @ U2 ) @ ( sup_su6776935440552674877rm_a_b @ R3 @ id_term_a_b ) ) ) ) ) ) ).
% strongly_confluent_E1n
thf(fact_732_diamond__imp__semi__confluence,axiom,
! [R3: set_Pr4386577575007340137rm_a_b] :
( ( abstra8151087076625346926rm_a_b @ R3 )
=> ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ ( conver7608559976937845093rm_a_b @ R3 ) @ ( transi7742714808557438673rm_a_b @ R3 ) ) @ ( abstra4096080454567261402rm_a_b @ R3 ) ) ) ).
% diamond_imp_semi_confluence
thf(fact_733_joinI,axiom,
! [A: term_a_b,C2: term_a_b,A2: set_Pr4386577575007340137rm_a_b,B: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C2 ) @ ( transi7742714808557438673rm_a_b @ A2 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C2 ) @ ( transi7742714808557438673rm_a_b @ A2 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( abstra4096080454567261402rm_a_b @ A2 ) ) ) ) ).
% joinI
thf(fact_734_joinI__left,axiom,
! [A: term_a_b,B: term_a_b,A2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ A2 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( abstra4096080454567261402rm_a_b @ A2 ) ) ) ).
% joinI_left
thf(fact_735_joinI__right,axiom,
! [B: term_a_b,A: term_a_b,A2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ ( transi7742714808557438673rm_a_b @ A2 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( abstra4096080454567261402rm_a_b @ A2 ) ) ) ).
% joinI_right
thf(fact_736_rstep__substI,axiom,
! [S: term_a_b,T: term_a_b,R2: set_Pr4386577575007340137rm_a_b,Sigma: b > term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( rstep_a_b @ R2 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( subst_7999470309526761004_a_b_b @ S @ Sigma ) @ ( subst_7999470309526761004_a_b_b @ T @ Sigma ) ) @ ( rstep_a_b @ R2 ) ) ) ).
% rstep_substI
thf(fact_737_relpow__image,axiom,
! [R3: set_Pr4386577575007340137rm_a_b,F3: term_a_b > term_a_b,R7: set_Pr4386577575007340137rm_a_b,S: term_a_b,T: term_a_b,N: nat] :
( ! [S5: term_a_b,T2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S5 @ T2 ) @ R3 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F3 @ S5 ) @ ( F3 @ T2 ) ) @ R7 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( compow4057154403645558940rm_a_b @ N @ R3 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F3 @ S ) @ ( F3 @ T ) ) @ ( compow4057154403645558940rm_a_b @ N @ R7 ) ) ) ) ).
% relpow_image
thf(fact_738_relpow__Suc__D2_H,axiom,
! [N: nat,R4: set_Pr4386577575007340137rm_a_b,X5: term_a_b,Y6: term_a_b,Z5: term_a_b] :
( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X5 @ Y6 ) @ ( compow4057154403645558940rm_a_b @ N @ R4 ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y6 @ Z5 ) @ R4 ) )
=> ? [W: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X5 @ W ) @ R4 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ W @ Z5 ) @ ( compow4057154403645558940rm_a_b @ N @ R4 ) ) ) ) ).
% relpow_Suc_D2'
thf(fact_739_relpow__refl__mono,axiom,
! [Rel: set_Pr4386577575007340137rm_a_b,M: nat,N: nat,A: term_a_b,B: term_a_b] :
( ! [X3: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ X3 ) @ Rel )
=> ( ( ord_less_eq_nat @ M @ N )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( compow4057154403645558940rm_a_b @ M @ Rel ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( compow4057154403645558940rm_a_b @ N @ Rel ) ) ) ) ) ).
% relpow_refl_mono
thf(fact_740_rtrancl__power,axiom,
! [P: produc357393685978478089rm_a_b,R4: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ P @ ( transi7742714808557438673rm_a_b @ R4 ) )
= ( ? [N2: nat] : ( member5869715511025134514rm_a_b @ P @ ( compow4057154403645558940rm_a_b @ N2 @ R4 ) ) ) ) ).
% rtrancl_power
thf(fact_741_relpow__imp__rtrancl,axiom,
! [P: produc357393685978478089rm_a_b,N: nat,R4: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ P @ ( compow4057154403645558940rm_a_b @ N @ R4 ) )
=> ( member5869715511025134514rm_a_b @ P @ ( transi7742714808557438673rm_a_b @ R4 ) ) ) ).
% relpow_imp_rtrancl
thf(fact_742_rtrancl__imp__relpow,axiom,
! [P: produc357393685978478089rm_a_b,R4: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ P @ ( transi7742714808557438673rm_a_b @ R4 ) )
=> ? [N3: nat] : ( member5869715511025134514rm_a_b @ P @ ( compow4057154403645558940rm_a_b @ N3 @ R4 ) ) ) ).
% rtrancl_imp_relpow
thf(fact_743_relpow__commute,axiom,
! [R4: set_Pr4386577575007340137rm_a_b,N: nat] :
( ( relcom370159955682700863rm_a_b @ R4 @ ( compow4057154403645558940rm_a_b @ N @ R4 ) )
= ( relcom370159955682700863rm_a_b @ ( compow4057154403645558940rm_a_b @ N @ R4 ) @ R4 ) ) ).
% relpow_commute
thf(fact_744_converse__power,axiom,
! [N: nat,R3: set_Pr4386577575007340137rm_a_b] :
( ( compow4057154403645558940rm_a_b @ N @ ( conver7608559976937845093rm_a_b @ R3 ) )
= ( conver7608559976937845093rm_a_b @ ( compow4057154403645558940rm_a_b @ N @ R3 ) ) ) ).
% converse_power
thf(fact_745_join__sym,axiom,
! [S: term_a_b,T: term_a_b,A2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( abstra4096080454567261402rm_a_b @ A2 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T @ S ) @ ( abstra4096080454567261402rm_a_b @ A2 ) ) ) ).
% join_sym
thf(fact_746_rtrancl__len__E,axiom,
! [X: term_a_b,Y: term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ R3 ) )
=> ~ ! [N3: nat] :
~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( compow4057154403645558940rm_a_b @ N3 @ R3 ) ) ) ).
% rtrancl_len_E
thf(fact_747_rrstep__subst,axiom,
! [S: term_a_b,T: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( rrstep_a_b @ R2 ) )
=> ~ ! [L4: term_a_b,R9: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ L4 @ R9 ) @ R2 )
=> ! [Sigma2: b > term_a_b] :
( ( S
= ( subst_7999470309526761004_a_b_b @ L4 @ Sigma2 ) )
=> ( T
!= ( subst_7999470309526761004_a_b_b @ R9 @ Sigma2 ) ) ) ) ) ).
% rrstep_subst
thf(fact_748_rrstep_Ointros,axiom,
! [L: term_a_b,R3: term_a_b,R2: set_Pr4386577575007340137rm_a_b,Sigma: b > term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ L @ R3 ) @ R2 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( subst_7999470309526761004_a_b_b @ L @ Sigma ) @ ( subst_7999470309526761004_a_b_b @ R3 @ Sigma ) ) @ ( rrstep_a_b @ R2 ) ) ) ).
% rrstep.intros
thf(fact_749_rrstep_Osimps,axiom,
! [A1: term_a_b,A22: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A22 ) @ ( rrstep_a_b @ R2 ) )
= ( ? [L5: term_a_b,R6: term_a_b,Sigma3: b > term_a_b] :
( ( A1
= ( subst_7999470309526761004_a_b_b @ L5 @ Sigma3 ) )
& ( A22
= ( subst_7999470309526761004_a_b_b @ R6 @ Sigma3 ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ L5 @ R6 ) @ R2 ) ) ) ) ).
% rrstep.simps
thf(fact_750_rrstep_Ocases,axiom,
! [A1: term_a_b,A22: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A22 ) @ ( rrstep_a_b @ R2 ) )
=> ~ ! [L4: term_a_b,R9: term_a_b,Sigma2: b > term_a_b] :
( ( A1
= ( subst_7999470309526761004_a_b_b @ L4 @ Sigma2 ) )
=> ( ( A22
= ( subst_7999470309526761004_a_b_b @ R9 @ Sigma2 ) )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ L4 @ R9 ) @ R2 ) ) ) ) ).
% rrstep.cases
thf(fact_751_joinD,axiom,
! [A: term_a_b,B: term_a_b,A2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( abstra4096080454567261402rm_a_b @ A2 ) )
=> ? [C4: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C4 ) @ ( transi7742714808557438673rm_a_b @ A2 ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C4 ) @ ( transi7742714808557438673rm_a_b @ A2 ) ) ) ) ).
% joinD
thf(fact_752_joinE,axiom,
! [A: term_a_b,B: term_a_b,A2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( abstra4096080454567261402rm_a_b @ A2 ) )
=> ~ ! [C4: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C4 ) @ ( transi7742714808557438673rm_a_b @ A2 ) )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C4 ) @ ( transi7742714808557438673rm_a_b @ A2 ) ) ) ) ).
% joinE
thf(fact_753_join__rtrancl__join,axiom,
! [A: term_a_b,B: term_a_b,A2: set_Pr4386577575007340137rm_a_b,C2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( abstra4096080454567261402rm_a_b @ A2 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ C2 @ B ) @ ( transi7742714808557438673rm_a_b @ A2 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C2 ) @ ( abstra4096080454567261402rm_a_b @ A2 ) ) ) ) ).
% join_rtrancl_join
thf(fact_754_rtrancl__join__join,axiom,
! [A: term_a_b,B: term_a_b,A2: set_Pr4386577575007340137rm_a_b,C2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ A2 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C2 ) @ ( abstra4096080454567261402rm_a_b @ A2 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C2 ) @ ( abstra4096080454567261402rm_a_b @ A2 ) ) ) ) ).
% rtrancl_join_join
thf(fact_755_trancl__steps__relpow,axiom,
! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b,X: term_a_b,Y: term_a_b,N: nat] :
( ( ord_le118470702582115849rm_a_b @ A @ ( transi7922773638565587891rm_a_b @ B ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( compow4057154403645558940rm_a_b @ N @ A ) )
=> ? [M2: nat] :
( ( ord_less_eq_nat @ N @ M2 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( compow4057154403645558940rm_a_b @ M2 @ B ) ) ) ) ) ).
% trancl_steps_relpow
thf(fact_756_join__def,axiom,
( abstra4096080454567261402rm_a_b
= ( ^ [A6: set_Pr4386577575007340137rm_a_b] : ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ A6 ) @ ( transi7742714808557438673rm_a_b @ ( conver7608559976937845093rm_a_b @ A6 ) ) ) ) ) ).
% join_def
thf(fact_757_srstep__subst__closed,axiom,
! [S: term_a_b,T: term_a_b,F2: set_Pr4934435412358123699_a_nat,R2: set_Pr4386577575007340137rm_a_b,Sigma: b > term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) ) )
=> ( ! [X3: b] : ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ ( Sigma @ X3 ) ) @ F2 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( subst_7999470309526761004_a_b_b @ S @ Sigma ) @ ( subst_7999470309526761004_a_b_b @ T @ Sigma ) ) @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) ) ) ) ) ).
% srstep_subst_closed
thf(fact_758_srsteps__eq__subst__closed,axiom,
! [S: term_a_b,T: term_a_b,F2: set_Pr4934435412358123699_a_nat,R2: set_Pr4386577575007340137rm_a_b,Sigma: b > term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) ) ) )
=> ( ! [X3: b] : ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ ( Sigma @ X3 ) ) @ F2 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( subst_7999470309526761004_a_b_b @ S @ Sigma ) @ ( subst_7999470309526761004_a_b_b @ T @ Sigma ) ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) ) ) ) ) ) ).
% srsteps_eq_subst_closed
thf(fact_759_srsteps__subst__closed,axiom,
! [S: term_a_b,T: term_a_b,F2: set_Pr4934435412358123699_a_nat,R2: set_Pr4386577575007340137rm_a_b,Sigma: b > term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( transi7922773638565587891rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) ) ) )
=> ( ! [X3: b] : ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ ( Sigma @ X3 ) ) @ F2 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( subst_7999470309526761004_a_b_b @ S @ Sigma ) @ ( subst_7999470309526761004_a_b_b @ T @ Sigma ) ) @ ( transi7922773638565587891rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) ) ) ) ) ) ).
% srsteps_subst_closed
thf(fact_760_WCR__alt__def,axiom,
! [A2: set_Pr4386577575007340137rm_a_b] :
( ( abstra5026849107050860991rm_a_b @ A2 @ top_top_set_term_a_b )
= ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ ( conver7608559976937845093rm_a_b @ A2 ) @ A2 ) @ ( abstra4096080454567261402rm_a_b @ A2 ) ) ) ).
% WCR_alt_def
thf(fact_761_semi__confluence__imp__CR,axiom,
! [R3: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ ( conver7608559976937845093rm_a_b @ R3 ) @ ( transi7742714808557438673rm_a_b @ R3 ) ) @ ( abstra4096080454567261402rm_a_b @ R3 ) )
=> ( abstra8448919418672941150rm_a_b @ R3 @ top_top_set_term_a_b ) ) ).
% semi_confluence_imp_CR
thf(fact_762_WCR__onI,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,R3: set_Pr2972776593051762503rm_a_b] :
( ! [A3: produc357393685978478089rm_a_b,B3: produc357393685978478089rm_a_b,C4: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ A3 @ A2 )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A3 @ B3 ) @ R3 )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A3 @ C4 ) @ R3 )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B3 @ C4 ) @ ( abstra4889682940153293507rm_a_b @ R3 ) ) ) ) )
=> ( abstra875157153669124638rm_a_b @ R3 @ A2 ) ) ).
% WCR_onI
thf(fact_763_WCR__onI,axiom,
! [A2: set_term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
( ! [A3: term_a_b,B3: term_a_b,C4: term_a_b] :
( ( member_term_a_b @ A3 @ A2 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B3 ) @ R3 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ C4 ) @ R3 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B3 @ C4 ) @ ( abstra4096080454567261402rm_a_b @ R3 ) ) ) ) )
=> ( abstra5026849107050860991rm_a_b @ R3 @ A2 ) ) ).
% WCR_onI
thf(fact_764_CR__onI,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,R3: set_Pr2972776593051762503rm_a_b] :
( ! [A3: produc357393685978478089rm_a_b,B3: produc357393685978478089rm_a_b,C4: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ A3 @ A2 )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A3 @ B3 ) @ ( transi2615809358984392588rm_a_b @ R3 ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A3 @ C4 ) @ ( transi2615809358984392588rm_a_b @ R3 ) )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B3 @ C4 ) @ ( abstra4889682940153293507rm_a_b @ R3 ) ) ) ) )
=> ( abstra7359694574113289279rm_a_b @ R3 @ A2 ) ) ).
% CR_onI
thf(fact_765_CR__onI,axiom,
! [A2: set_term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
( ! [A3: term_a_b,B3: term_a_b,C4: term_a_b] :
( ( member_term_a_b @ A3 @ A2 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B3 ) @ ( transi7742714808557438673rm_a_b @ R3 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ C4 ) @ ( transi7742714808557438673rm_a_b @ R3 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B3 @ C4 ) @ ( abstra4096080454567261402rm_a_b @ R3 ) ) ) ) )
=> ( abstra8448919418672941150rm_a_b @ R3 @ A2 ) ) ).
% CR_onI
thf(fact_766_CR__iff__self__commute,axiom,
! [R3: set_Pr4386577575007340137rm_a_b] :
( ( abstra8448919418672941150rm_a_b @ R3 @ top_top_set_term_a_b )
= ( abstra455195320138238452rm_a_b @ R3 @ R3 ) ) ).
% CR_iff_self_commute
thf(fact_767_CR__on__def,axiom,
( abstra8448919418672941150rm_a_b
= ( ^ [R6: set_Pr4386577575007340137rm_a_b,A6: set_term_a_b] :
! [X2: term_a_b] :
( ( member_term_a_b @ X2 @ A6 )
=> ! [B4: term_a_b,C3: term_a_b] :
( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ B4 ) @ ( transi7742714808557438673rm_a_b @ R6 ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ C3 ) @ ( transi7742714808557438673rm_a_b @ R6 ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B4 @ C3 ) @ ( abstra4096080454567261402rm_a_b @ R6 ) ) ) ) ) ) ).
% CR_on_def
thf(fact_768_CR__onE,axiom,
! [R3: set_Pr2972776593051762503rm_a_b,A2: set_Pr4386577575007340137rm_a_b,A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,C2: produc357393685978478089rm_a_b] :
( ( abstra7359694574113289279rm_a_b @ R3 @ A2 )
=> ( ( member5869715511025134514rm_a_b @ A @ A2 )
=> ( ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B @ C2 ) @ ( abstra4889682940153293507rm_a_b @ R3 ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( transi2615809358984392588rm_a_b @ R3 ) )
=> ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ C2 ) @ ( transi2615809358984392588rm_a_b @ R3 ) ) ) ) ) ) ).
% CR_onE
thf(fact_769_CR__onE,axiom,
! [R3: set_Pr4386577575007340137rm_a_b,A2: set_term_a_b,A: term_a_b,B: term_a_b,C2: term_a_b] :
( ( abstra8448919418672941150rm_a_b @ R3 @ A2 )
=> ( ( member_term_a_b @ A @ A2 )
=> ( ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C2 ) @ ( abstra4096080454567261402rm_a_b @ R3 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ R3 ) )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C2 ) @ ( transi7742714808557438673rm_a_b @ R3 ) ) ) ) ) ) ).
% CR_onE
thf(fact_770_CR__onD,axiom,
! [R3: set_Pr2972776593051762503rm_a_b,A2: set_Pr4386577575007340137rm_a_b,A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,C2: produc357393685978478089rm_a_b] :
( ( abstra7359694574113289279rm_a_b @ R3 @ A2 )
=> ( ( member5869715511025134514rm_a_b @ A @ A2 )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( transi2615809358984392588rm_a_b @ R3 ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ C2 ) @ ( transi2615809358984392588rm_a_b @ R3 ) )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B @ C2 ) @ ( abstra4889682940153293507rm_a_b @ R3 ) ) ) ) ) ) ).
% CR_onD
thf(fact_771_CR__onD,axiom,
! [R3: set_Pr4386577575007340137rm_a_b,A2: set_term_a_b,A: term_a_b,B: term_a_b,C2: term_a_b] :
( ( abstra8448919418672941150rm_a_b @ R3 @ A2 )
=> ( ( member_term_a_b @ A @ A2 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ R3 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C2 ) @ ( transi7742714808557438673rm_a_b @ R3 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C2 ) @ ( abstra4096080454567261402rm_a_b @ R3 ) ) ) ) ) ) ).
% CR_onD
thf(fact_772_WCR__on__def,axiom,
( abstra5026849107050860991rm_a_b
= ( ^ [R6: set_Pr4386577575007340137rm_a_b,A6: set_term_a_b] :
! [X2: term_a_b] :
( ( member_term_a_b @ X2 @ A6 )
=> ! [B4: term_a_b,C3: term_a_b] :
( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ B4 ) @ R6 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ C3 ) @ R6 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B4 @ C3 ) @ ( abstra4096080454567261402rm_a_b @ R6 ) ) ) ) ) ) ).
% WCR_on_def
thf(fact_773_WCR__onE,axiom,
! [R3: set_Pr2972776593051762503rm_a_b,A2: set_Pr4386577575007340137rm_a_b,A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,C2: produc357393685978478089rm_a_b] :
( ( abstra875157153669124638rm_a_b @ R3 @ A2 )
=> ( ( member5869715511025134514rm_a_b @ A @ A2 )
=> ( ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B @ C2 ) @ ( abstra4889682940153293507rm_a_b @ R3 ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ R3 )
=> ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ C2 ) @ R3 ) ) ) ) ) ).
% WCR_onE
thf(fact_774_WCR__onE,axiom,
! [R3: set_Pr4386577575007340137rm_a_b,A2: set_term_a_b,A: term_a_b,B: term_a_b,C2: term_a_b] :
( ( abstra5026849107050860991rm_a_b @ R3 @ A2 )
=> ( ( member_term_a_b @ A @ A2 )
=> ( ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C2 ) @ ( abstra4096080454567261402rm_a_b @ R3 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R3 )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C2 ) @ R3 ) ) ) ) ) ).
% WCR_onE
thf(fact_775_WCR__onD,axiom,
! [R3: set_Pr2972776593051762503rm_a_b,A2: set_Pr4386577575007340137rm_a_b,A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,C2: produc357393685978478089rm_a_b] :
( ( abstra875157153669124638rm_a_b @ R3 @ A2 )
=> ( ( member5869715511025134514rm_a_b @ A @ A2 )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ R3 )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ C2 ) @ R3 )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B @ C2 ) @ ( abstra4889682940153293507rm_a_b @ R3 ) ) ) ) ) ) ).
% WCR_onD
thf(fact_776_WCR__onD,axiom,
! [R3: set_Pr4386577575007340137rm_a_b,A2: set_term_a_b,A: term_a_b,B: term_a_b,C2: term_a_b] :
( ( abstra5026849107050860991rm_a_b @ R3 @ A2 )
=> ( ( member_term_a_b @ A @ A2 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R3 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C2 ) @ R3 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C2 ) @ ( abstra4096080454567261402rm_a_b @ R3 ) ) ) ) ) ) ).
% WCR_onD
thf(fact_777_CR__join__left__I,axiom,
! [R3: set_Pr4386577575007340137rm_a_b,X: term_a_b,Y: term_a_b,Z: term_a_b] :
( ( abstra8448919418672941150rm_a_b @ R3 @ top_top_set_term_a_b )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ R3 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z ) @ ( abstra4096080454567261402rm_a_b @ R3 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ Z ) @ ( abstra4096080454567261402rm_a_b @ R3 ) ) ) ) ) ).
% CR_join_left_I
thf(fact_778_CR__join__right__I,axiom,
! [R3: set_Pr4386577575007340137rm_a_b,X: term_a_b,Y: term_a_b,Z: term_a_b] :
( ( abstra8448919418672941150rm_a_b @ R3 @ top_top_set_term_a_b )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( abstra4096080454567261402rm_a_b @ R3 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ Z ) @ ( transi7742714808557438673rm_a_b @ R3 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z ) @ ( abstra4096080454567261402rm_a_b @ R3 ) ) ) ) ) ).
% CR_join_right_I
thf(fact_779_partially__localize__CR,axiom,
! [R3: set_Pr4386577575007340137rm_a_b] :
( ( abstra8448919418672941150rm_a_b @ R3 @ top_top_set_term_a_b )
= ( ! [X2: term_a_b,Y3: term_a_b,Z3: term_a_b] :
( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y3 ) @ R3 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Z3 ) @ ( transi7742714808557438673rm_a_b @ R3 ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ Z3 ) @ ( abstra4096080454567261402rm_a_b @ R3 ) ) ) ) ) ).
% partially_localize_CR
thf(fact_780_CR__divergence__imp__join,axiom,
! [R3: set_Pr4386577575007340137rm_a_b,X: term_a_b,Y: term_a_b,Z: term_a_b] :
( ( abstra8448919418672941150rm_a_b @ R3 @ top_top_set_term_a_b )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ R3 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z ) @ ( transi7742714808557438673rm_a_b @ R3 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ Z ) @ ( abstra4096080454567261402rm_a_b @ R3 ) ) ) ) ) ).
% CR_divergence_imp_join
thf(fact_781_diamond__imp__CR_H,axiom,
! [S: set_Pr4386577575007340137rm_a_b,R3: set_Pr4386577575007340137rm_a_b] :
( ( abstra8151087076625346926rm_a_b @ S )
=> ( ( ord_le118470702582115849rm_a_b @ R3 @ S )
=> ( ( ord_le118470702582115849rm_a_b @ S @ ( transi7742714808557438673rm_a_b @ R3 ) )
=> ( abstra8448919418672941150rm_a_b @ R3 @ top_top_set_term_a_b ) ) ) ) ).
% diamond_imp_CR'
thf(fact_782_qc__SN__relto__iff,axiom,
! [R3: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ R3 @ S ) @ ( relcom370159955682700863rm_a_b @ S @ ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ S @ R3 ) ) ) )
=> ( ( abstra4720023341729745482rm_a_b @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ R3 ) @ ( relcom370159955682700863rm_a_b @ S @ ( transi7742714808557438673rm_a_b @ R3 ) ) ) @ top_top_set_term_a_b )
= ( abstra4720023341729745482rm_a_b @ S @ top_top_set_term_a_b ) ) ) ).
% qc_SN_relto_iff
thf(fact_783_compatible__SN_H,axiom,
! [NS: set_Pr4386577575007340137rm_a_b,S6: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ NS @ S6 ) @ S6 )
=> ( ( abstra4720023341729745482rm_a_b @ S6 @ top_top_set_term_a_b )
=> ( abstra4720023341729745482rm_a_b @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ NS @ S6 ) ) @ ( relcom370159955682700863rm_a_b @ S6 @ ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ NS @ S6 ) ) ) ) @ top_top_set_term_a_b ) ) ) ).
% compatible_SN'
thf(fact_784_SN__on__trancl__SN__on__conv,axiom,
! [R4: set_Pr4386577575007340137rm_a_b,T4: set_term_a_b] :
( ( abstra4720023341729745482rm_a_b @ ( transi7922773638565587891rm_a_b @ R4 ) @ T4 )
= ( abstra4720023341729745482rm_a_b @ R4 @ T4 ) ) ).
% SN_on_trancl_SN_on_conv
thf(fact_785_SN__on__trancl__imp__SN__on,axiom,
! [R4: set_Pr4386577575007340137rm_a_b,T4: set_term_a_b] :
( ( abstra4720023341729745482rm_a_b @ ( transi7922773638565587891rm_a_b @ R4 ) @ T4 )
=> ( abstra4720023341729745482rm_a_b @ R4 @ T4 ) ) ).
% SN_on_trancl_imp_SN_on
thf(fact_786_SN__on__trancl,axiom,
! [R3: set_Pr4386577575007340137rm_a_b,A2: set_term_a_b] :
( ( abstra4720023341729745482rm_a_b @ R3 @ A2 )
=> ( abstra4720023341729745482rm_a_b @ ( transi7922773638565587891rm_a_b @ R3 ) @ A2 ) ) ).
% SN_on_trancl
thf(fact_787_SN__on__irrefl,axiom,
! [R3: set_Pr4386577575007340137rm_a_b,A2: set_term_a_b] :
( ( abstra4720023341729745482rm_a_b @ R3 @ A2 )
=> ! [X5: term_a_b] :
( ( member_term_a_b @ X5 @ A2 )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X5 @ X5 ) @ R3 ) ) ) ).
% SN_on_irrefl
thf(fact_788_SN__on__induct,axiom,
! [R4: set_Pr2972776593051762503rm_a_b,A2: set_Pr4386577575007340137rm_a_b,S: produc357393685978478089rm_a_b,P2: produc357393685978478089rm_a_b > $o] :
( ( abstra2398554102055911763rm_a_b @ R4 @ A2 )
=> ( ( member5869715511025134514rm_a_b @ S @ A2 )
=> ( ! [T2: produc357393685978478089rm_a_b] :
( ! [U4: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ T2 @ U4 ) @ R4 )
=> ( P2 @ U4 ) )
=> ( P2 @ T2 ) )
=> ( P2 @ S ) ) ) ) ).
% SN_on_induct
thf(fact_789_SN__on__induct,axiom,
! [R4: set_Pr4386577575007340137rm_a_b,A2: set_term_a_b,S: term_a_b,P2: term_a_b > $o] :
( ( abstra4720023341729745482rm_a_b @ R4 @ A2 )
=> ( ( member_term_a_b @ S @ A2 )
=> ( ! [T2: term_a_b] :
( ! [U4: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T2 @ U4 ) @ R4 )
=> ( P2 @ U4 ) )
=> ( P2 @ T2 ) )
=> ( P2 @ S ) ) ) ) ).
% SN_on_induct
thf(fact_790_SN__on__Un2,axiom,
! [R4: set_Pr2972776593051762503rm_a_b,X4: set_Pr4386577575007340137rm_a_b,Y7: set_Pr4386577575007340137rm_a_b] :
( ( abstra2398554102055911763rm_a_b @ R4 @ X4 )
=> ( ( abstra2398554102055911763rm_a_b @ R4 @ Y7 )
=> ( abstra2398554102055911763rm_a_b @ R4 @ ( sup_su6776935440552674877rm_a_b @ X4 @ Y7 ) ) ) ) ).
% SN_on_Un2
thf(fact_791_SN__induct__rule,axiom,
! [R3: set_Pr4386577575007340137rm_a_b,P2: term_a_b > $o,A: term_a_b] :
( ( abstra4720023341729745482rm_a_b @ R3 @ top_top_set_term_a_b )
=> ( ! [A3: term_a_b] :
( ! [B8: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B8 ) @ R3 )
=> ( P2 @ B8 ) )
=> ( P2 @ A3 ) )
=> ( P2 @ A ) ) ) ).
% SN_induct_rule
thf(fact_792_SN__imp__minimal,axiom,
! [A2: set_Pr2972776593051762503rm_a_b] :
( ( abstra2398554102055911763rm_a_b @ A2 @ top_to1314267278846557113rm_a_b )
=> ! [Q2: set_Pr4386577575007340137rm_a_b] :
( ? [X5: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ X5 @ Q2 )
=> ? [X3: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X3 @ Q2 )
& ! [Y6: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X3 @ Y6 ) @ A2 )
=> ~ ( member5869715511025134514rm_a_b @ Y6 @ Q2 ) ) ) ) ) ).
% SN_imp_minimal
thf(fact_793_SN__imp__minimal,axiom,
! [A2: set_Pr4386577575007340137rm_a_b] :
( ( abstra4720023341729745482rm_a_b @ A2 @ top_top_set_term_a_b )
=> ! [Q2: set_term_a_b] :
( ? [X5: term_a_b] : ( member_term_a_b @ X5 @ Q2 )
=> ? [X3: term_a_b] :
( ( member_term_a_b @ X3 @ Q2 )
& ! [Y6: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y6 ) @ A2 )
=> ~ ( member_term_a_b @ Y6 @ Q2 ) ) ) ) ) ).
% SN_imp_minimal
thf(fact_794_refl__not__SN,axiom,
! [X: term_a_b,R4: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ X ) @ R4 )
=> ~ ( abstra4720023341729745482rm_a_b @ R4 @ top_top_set_term_a_b ) ) ).
% refl_not_SN
thf(fact_795_SN__O__comm,axiom,
! [R4: set_Pr4386577575007340137rm_a_b,S6: set_Pr4386577575007340137rm_a_b] :
( ( abstra4720023341729745482rm_a_b @ ( relcom370159955682700863rm_a_b @ R4 @ S6 ) @ top_top_set_term_a_b )
= ( abstra4720023341729745482rm_a_b @ ( relcom370159955682700863rm_a_b @ S6 @ R4 ) @ top_top_set_term_a_b ) ) ).
% SN_O_comm
thf(fact_796_SN__imp__SN__trancl,axiom,
! [R4: set_Pr4386577575007340137rm_a_b] :
( ( abstra4720023341729745482rm_a_b @ R4 @ top_top_set_term_a_b )
=> ( abstra4720023341729745482rm_a_b @ ( transi7922773638565587891rm_a_b @ R4 ) @ top_top_set_term_a_b ) ) ).
% SN_imp_SN_trancl
thf(fact_797_SN__trancl__imp__SN,axiom,
! [R4: set_Pr4386577575007340137rm_a_b] :
( ( abstra4720023341729745482rm_a_b @ ( transi7922773638565587891rm_a_b @ R4 ) @ top_top_set_term_a_b )
=> ( abstra4720023341729745482rm_a_b @ R4 @ top_top_set_term_a_b ) ) ).
% SN_trancl_imp_SN
thf(fact_798_SN__trancl__SN__conv,axiom,
! [R4: set_Pr4386577575007340137rm_a_b] :
( ( abstra4720023341729745482rm_a_b @ ( transi7922773638565587891rm_a_b @ R4 ) @ top_top_set_term_a_b )
= ( abstra4720023341729745482rm_a_b @ R4 @ top_top_set_term_a_b ) ) ).
% SN_trancl_SN_conv
thf(fact_799_SN__on__O__push,axiom,
! [S6: set_Pr4386577575007340137rm_a_b,R4: set_Pr4386577575007340137rm_a_b,X4: set_term_a_b] :
( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ S6 @ R4 ) @ ( relcom370159955682700863rm_a_b @ R4 @ ( transi7742714808557438673rm_a_b @ S6 ) ) )
=> ( ( abstra4720023341729745482rm_a_b @ R4 @ X4 )
=> ( abstra4720023341729745482rm_a_b @ ( relcom370159955682700863rm_a_b @ R4 @ ( transi7742714808557438673rm_a_b @ S6 ) ) @ X4 ) ) ) ).
% SN_on_O_push
thf(fact_800_quasi__commute__imp__SN,axiom,
! [R3: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
( ( abstra4720023341729745482rm_a_b @ R3 @ top_top_set_term_a_b )
=> ( ( abstra4720023341729745482rm_a_b @ S @ top_top_set_term_a_b )
=> ( ( abstra4317384144805385751rm_a_b @ R3 @ S )
=> ( abstra4720023341729745482rm_a_b @ ( sup_su6776935440552674877rm_a_b @ R3 @ S ) @ top_top_set_term_a_b ) ) ) ) ).
% quasi_commute_imp_SN
thf(fact_801_compatible__SN,axiom,
! [S6: set_Pr4386577575007340137rm_a_b,NS: set_Pr4386577575007340137rm_a_b] :
( ( abstra4720023341729745482rm_a_b @ S6 @ top_top_set_term_a_b )
=> ( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ NS @ S6 ) @ S6 )
=> ( abstra4720023341729745482rm_a_b @ ( relcom370159955682700863rm_a_b @ S6 @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ S6 ) @ ( transi7742714808557438673rm_a_b @ NS ) ) ) @ top_top_set_term_a_b ) ) ) ).
% compatible_SN
thf(fact_802_the__NF__conv,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,A: term_a_b,B: term_a_b] :
( ( abstra4720023341729745482rm_a_b @ A2 @ top_top_set_term_a_b )
=> ( ( abstra8448919418672941150rm_a_b @ A2 @ top_top_set_term_a_b )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( abstra1130568886744591910rm_a_b @ A2 ) )
=> ( ( abstra2242782261322123075rm_a_b @ A2 @ A )
= ( abstra2242782261322123075rm_a_b @ A2 @ B ) ) ) ) ) ).
% the_NF_conv
thf(fact_803_the__NF__steps,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,A: term_a_b,B: term_a_b] :
( ( abstra4720023341729745482rm_a_b @ A2 @ top_top_set_term_a_b )
=> ( ( abstra8448919418672941150rm_a_b @ A2 @ top_top_set_term_a_b )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ A2 ) )
=> ( ( abstra2242782261322123075rm_a_b @ A2 @ A )
= ( abstra2242782261322123075rm_a_b @ A2 @ B ) ) ) ) ) ).
% the_NF_steps
thf(fact_804_the__NF__step,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,A: term_a_b,B: term_a_b] :
( ( abstra4720023341729745482rm_a_b @ A2 @ top_top_set_term_a_b )
=> ( ( abstra8448919418672941150rm_a_b @ A2 @ top_top_set_term_a_b )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ A2 )
=> ( ( abstra2242782261322123075rm_a_b @ A2 @ A )
= ( abstra2242782261322123075rm_a_b @ A2 @ B ) ) ) ) ) ).
% the_NF_step
thf(fact_805_the__NF,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,A: term_a_b] :
( ( abstra4720023341729745482rm_a_b @ A2 @ top_top_set_term_a_b )
=> ( ( abstra8448919418672941150rm_a_b @ A2 @ top_top_set_term_a_b )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ ( abstra2242782261322123075rm_a_b @ A2 @ A ) ) @ ( abstra2972651231392952684rm_a_b @ A2 ) ) ) ) ).
% the_NF
thf(fact_806_weak__diamond__def,axiom,
( abstra5190899124789103926rm_a_b
= ( ^ [R6: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ ( minus_5192120951422937424rm_a_b @ ( relcom370159955682700863rm_a_b @ ( conver7608559976937845093rm_a_b @ R6 ) @ R6 ) @ id_term_a_b ) @ ( relcom370159955682700863rm_a_b @ R6 @ ( conver7608559976937845093rm_a_b @ R6 ) ) ) ) ) ).
% weak_diamond_def
thf(fact_807_the__NF__NF,axiom,
! [A2: set_Pr2972776593051762503rm_a_b,A: produc357393685978478089rm_a_b] :
( ( abstra2398554102055911763rm_a_b @ A2 @ top_to1314267278846557113rm_a_b )
=> ( ( abstra7359694574113289279rm_a_b @ A2 @ top_to1314267278846557113rm_a_b )
=> ( member5869715511025134514rm_a_b @ ( abstra8972549633426078106rm_a_b @ A2 @ A ) @ ( abstra1902783451776770197rm_a_b @ A2 ) ) ) ) ).
% the_NF_NF
thf(fact_808_Diff__iff,axiom,
! [C2: produc357393685978478089rm_a_b,A2: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ C2 @ ( minus_5192120951422937424rm_a_b @ A2 @ B2 ) )
= ( ( member5869715511025134514rm_a_b @ C2 @ A2 )
& ~ ( member5869715511025134514rm_a_b @ C2 @ B2 ) ) ) ).
% Diff_iff
thf(fact_809_DiffI,axiom,
! [C2: produc357393685978478089rm_a_b,A2: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ C2 @ A2 )
=> ( ~ ( member5869715511025134514rm_a_b @ C2 @ B2 )
=> ( member5869715511025134514rm_a_b @ C2 @ ( minus_5192120951422937424rm_a_b @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_810_Un__Diff__cancel,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b] :
( ( sup_su6776935440552674877rm_a_b @ A2 @ ( minus_5192120951422937424rm_a_b @ B2 @ A2 ) )
= ( sup_su6776935440552674877rm_a_b @ A2 @ B2 ) ) ).
% Un_Diff_cancel
thf(fact_811_Un__Diff__cancel2,axiom,
! [B2: set_Pr4386577575007340137rm_a_b,A2: set_Pr4386577575007340137rm_a_b] :
( ( sup_su6776935440552674877rm_a_b @ ( minus_5192120951422937424rm_a_b @ B2 @ A2 ) @ A2 )
= ( sup_su6776935440552674877rm_a_b @ B2 @ A2 ) ) ).
% Un_Diff_cancel2
thf(fact_812_NF__I,axiom,
! [A: produc357393685978478089rm_a_b,A2: set_Pr2972776593051762503rm_a_b] :
( ! [B3: produc357393685978478089rm_a_b] :
~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B3 ) @ A2 )
=> ( member5869715511025134514rm_a_b @ A @ ( abstra1902783451776770197rm_a_b @ A2 ) ) ) ).
% NF_I
thf(fact_813_NF__I,axiom,
! [A: term_a_b,A2: set_Pr4386577575007340137rm_a_b] :
( ! [B3: term_a_b] :
~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B3 ) @ A2 )
=> ( member_term_a_b @ A @ ( abstract_NF_term_a_b @ A2 ) ) ) ).
% NF_I
thf(fact_814_normalizability__I_H,axiom,
! [A: term_a_b,B: term_a_b,A2: set_Pr4386577575007340137rm_a_b,C2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ A2 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C2 ) @ ( abstra2972651231392952684rm_a_b @ A2 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C2 ) @ ( abstra2972651231392952684rm_a_b @ A2 ) ) ) ) ).
% normalizability_I'
thf(fact_815_WN__onI,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,R3: set_Pr2972776593051762503rm_a_b] :
( ! [A3: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ A3 @ A2 )
=> ? [B8: produc357393685978478089rm_a_b] : ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A3 @ B8 ) @ ( abstra1496011390211842225rm_a_b @ R3 ) ) )
=> ( abstra2721873542170697047rm_a_b @ R3 @ A2 ) ) ).
% WN_onI
thf(fact_816_WN__onI,axiom,
! [A2: set_term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
( ! [A3: term_a_b] :
( ( member_term_a_b @ A3 @ A2 )
=> ? [B8: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B8 ) @ ( abstra2972651231392952684rm_a_b @ R3 ) ) )
=> ( abstra4644735066934440262rm_a_b @ R3 @ A2 ) ) ).
% WN_onI
thf(fact_817_UNF__onI,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,R3: set_Pr2972776593051762503rm_a_b] :
( ! [A3: produc357393685978478089rm_a_b,B3: produc357393685978478089rm_a_b,C4: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ A3 @ A2 )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A3 @ B3 ) @ ( abstra1496011390211842225rm_a_b @ R3 ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A3 @ C4 ) @ ( abstra1496011390211842225rm_a_b @ R3 ) )
=> ( B3 = C4 ) ) ) )
=> ( abstra1303798319023342519rm_a_b @ R3 @ A2 ) ) ).
% UNF_onI
thf(fact_818_UNF__onI,axiom,
! [A2: set_term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
( ! [A3: term_a_b,B3: term_a_b,C4: term_a_b] :
( ( member_term_a_b @ A3 @ A2 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B3 ) @ ( abstra2972651231392952684rm_a_b @ R3 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ C4 ) @ ( abstra2972651231392952684rm_a_b @ R3 ) )
=> ( B3 = C4 ) ) ) )
=> ( abstra3619855735679963878rm_a_b @ R3 @ A2 ) ) ).
% UNF_onI
thf(fact_819_normalizability__I,axiom,
! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,A2: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( transi2615809358984392588rm_a_b @ A2 ) )
=> ( ( member5869715511025134514rm_a_b @ B @ ( abstra1902783451776770197rm_a_b @ A2 ) )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( abstra1496011390211842225rm_a_b @ A2 ) ) ) ) ).
% normalizability_I
thf(fact_820_normalizability__I,axiom,
! [A: term_a_b,B: term_a_b,A2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ A2 ) )
=> ( ( member_term_a_b @ B @ ( abstract_NF_term_a_b @ A2 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( abstra2972651231392952684rm_a_b @ A2 ) ) ) ) ).
% normalizability_I
thf(fact_821_NF__iff__no__step,axiom,
! [A: produc357393685978478089rm_a_b,A2: set_Pr2972776593051762503rm_a_b] :
( ( member5869715511025134514rm_a_b @ A @ ( abstra1902783451776770197rm_a_b @ A2 ) )
= ( ! [B4: produc357393685978478089rm_a_b] :
~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B4 ) @ A2 ) ) ) ).
% NF_iff_no_step
thf(fact_822_NF__iff__no__step,axiom,
! [A: term_a_b,A2: set_Pr4386577575007340137rm_a_b] :
( ( member_term_a_b @ A @ ( abstract_NF_term_a_b @ A2 ) )
= ( ! [B4: term_a_b] :
~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B4 ) @ A2 ) ) ) ).
% NF_iff_no_step
thf(fact_823_NF__no__step,axiom,
! [A: produc357393685978478089rm_a_b,A2: set_Pr2972776593051762503rm_a_b] :
( ( member5869715511025134514rm_a_b @ A @ ( abstra1902783451776770197rm_a_b @ A2 ) )
=> ! [B8: produc357393685978478089rm_a_b] :
~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B8 ) @ A2 ) ) ).
% NF_no_step
thf(fact_824_NF__no__step,axiom,
! [A: term_a_b,A2: set_Pr4386577575007340137rm_a_b] :
( ( member_term_a_b @ A @ ( abstract_NF_term_a_b @ A2 ) )
=> ! [B8: term_a_b] :
~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B8 ) @ A2 ) ) ).
% NF_no_step
thf(fact_825_NF__E,axiom,
! [A: produc357393685978478089rm_a_b,A2: set_Pr2972776593051762503rm_a_b,B: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ A @ ( abstra1902783451776770197rm_a_b @ A2 ) )
=> ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ A2 ) ) ).
% NF_E
thf(fact_826_NF__E,axiom,
! [A: term_a_b,A2: set_Pr4386577575007340137rm_a_b,B: term_a_b] :
( ( member_term_a_b @ A @ ( abstract_NF_term_a_b @ A2 ) )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ A2 ) ) ).
% NF_E
thf(fact_827_Un__Diff,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b,C: set_Pr4386577575007340137rm_a_b] :
( ( minus_5192120951422937424rm_a_b @ ( sup_su6776935440552674877rm_a_b @ A2 @ B2 ) @ C )
= ( sup_su6776935440552674877rm_a_b @ ( minus_5192120951422937424rm_a_b @ A2 @ C ) @ ( minus_5192120951422937424rm_a_b @ B2 @ C ) ) ) ).
% Un_Diff
thf(fact_828_WN__on__def,axiom,
( abstra4644735066934440262rm_a_b
= ( ^ [R6: set_Pr4386577575007340137rm_a_b,A6: set_term_a_b] :
! [X2: term_a_b] :
( ( member_term_a_b @ X2 @ A6 )
=> ? [B4: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ B4 ) @ ( abstra2972651231392952684rm_a_b @ R6 ) ) ) ) ) ).
% WN_on_def
thf(fact_829_WN__onE,axiom,
! [R3: set_Pr2972776593051762503rm_a_b,A2: set_Pr4386577575007340137rm_a_b,A: produc357393685978478089rm_a_b] :
( ( abstra2721873542170697047rm_a_b @ R3 @ A2 )
=> ( ( member5869715511025134514rm_a_b @ A @ A2 )
=> ~ ! [B3: produc357393685978478089rm_a_b] :
~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B3 ) @ ( abstra1496011390211842225rm_a_b @ R3 ) ) ) ) ).
% WN_onE
thf(fact_830_WN__onE,axiom,
! [R3: set_Pr4386577575007340137rm_a_b,A2: set_term_a_b,A: term_a_b] :
( ( abstra4644735066934440262rm_a_b @ R3 @ A2 )
=> ( ( member_term_a_b @ A @ A2 )
=> ~ ! [B3: term_a_b] :
~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B3 ) @ ( abstra2972651231392952684rm_a_b @ R3 ) ) ) ) ).
% WN_onE
thf(fact_831_WN__onD,axiom,
! [R3: set_Pr2972776593051762503rm_a_b,A2: set_Pr4386577575007340137rm_a_b,A: produc357393685978478089rm_a_b] :
( ( abstra2721873542170697047rm_a_b @ R3 @ A2 )
=> ( ( member5869715511025134514rm_a_b @ A @ A2 )
=> ? [B3: produc357393685978478089rm_a_b] : ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B3 ) @ ( abstra1496011390211842225rm_a_b @ R3 ) ) ) ) ).
% WN_onD
thf(fact_832_WN__onD,axiom,
! [R3: set_Pr4386577575007340137rm_a_b,A2: set_term_a_b,A: term_a_b] :
( ( abstra4644735066934440262rm_a_b @ R3 @ A2 )
=> ( ( member_term_a_b @ A @ A2 )
=> ? [B3: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B3 ) @ ( abstra2972651231392952684rm_a_b @ R3 ) ) ) ) ).
% WN_onD
thf(fact_833_DiffD2,axiom,
! [C2: produc357393685978478089rm_a_b,A2: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ C2 @ ( minus_5192120951422937424rm_a_b @ A2 @ B2 ) )
=> ~ ( member5869715511025134514rm_a_b @ C2 @ B2 ) ) ).
% DiffD2
thf(fact_834_DiffD1,axiom,
! [C2: produc357393685978478089rm_a_b,A2: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ C2 @ ( minus_5192120951422937424rm_a_b @ A2 @ B2 ) )
=> ( member5869715511025134514rm_a_b @ C2 @ A2 ) ) ).
% DiffD1
thf(fact_835_DiffE,axiom,
! [C2: produc357393685978478089rm_a_b,A2: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ C2 @ ( minus_5192120951422937424rm_a_b @ A2 @ B2 ) )
=> ~ ( ( member5869715511025134514rm_a_b @ C2 @ A2 )
=> ( member5869715511025134514rm_a_b @ C2 @ B2 ) ) ) ).
% DiffE
thf(fact_836_normalizability__E,axiom,
! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,A2: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( abstra1496011390211842225rm_a_b @ A2 ) )
=> ~ ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( transi2615809358984392588rm_a_b @ A2 ) )
=> ~ ( member5869715511025134514rm_a_b @ B @ ( abstra1902783451776770197rm_a_b @ A2 ) ) ) ) ).
% normalizability_E
thf(fact_837_normalizability__E,axiom,
! [A: term_a_b,B: term_a_b,A2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( abstra2972651231392952684rm_a_b @ A2 ) )
=> ~ ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ A2 ) )
=> ~ ( member_term_a_b @ B @ ( abstract_NF_term_a_b @ A2 ) ) ) ) ).
% normalizability_E
thf(fact_838_Diff__subset__conv,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b,C: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ ( minus_5192120951422937424rm_a_b @ A2 @ B2 ) @ C )
= ( ord_le118470702582115849rm_a_b @ A2 @ ( sup_su6776935440552674877rm_a_b @ B2 @ C ) ) ) ).
% Diff_subset_conv
thf(fact_839_Diff__partition,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A2 @ B2 )
=> ( ( sup_su6776935440552674877rm_a_b @ A2 @ ( minus_5192120951422937424rm_a_b @ B2 @ A2 ) )
= B2 ) ) ).
% Diff_partition
thf(fact_840_rtrancl__r__diff__Id,axiom,
! [R3: set_Pr4386577575007340137rm_a_b] :
( ( transi7742714808557438673rm_a_b @ ( minus_5192120951422937424rm_a_b @ R3 @ id_term_a_b ) )
= ( transi7742714808557438673rm_a_b @ R3 ) ) ).
% rtrancl_r_diff_Id
thf(fact_841_NF__not__suc,axiom,
! [X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b,R3: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ ( transi2615809358984392588rm_a_b @ R3 ) )
=> ( ( member5869715511025134514rm_a_b @ X @ ( abstra1902783451776770197rm_a_b @ R3 ) )
=> ( X = Y ) ) ) ).
% NF_not_suc
thf(fact_842_NF__not__suc,axiom,
! [X: term_a_b,Y: term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ R3 ) )
=> ( ( member_term_a_b @ X @ ( abstract_NF_term_a_b @ R3 ) )
=> ( X = Y ) ) ) ).
% NF_not_suc
thf(fact_843_NF__no__trancl__step,axiom,
! [A: produc357393685978478089rm_a_b,A2: set_Pr2972776593051762503rm_a_b] :
( ( member5869715511025134514rm_a_b @ A @ ( abstra1902783451776770197rm_a_b @ A2 ) )
=> ! [B8: produc357393685978478089rm_a_b] :
~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B8 ) @ ( transi9211502839322181930rm_a_b @ A2 ) ) ) ).
% NF_no_trancl_step
thf(fact_844_NF__no__trancl__step,axiom,
! [A: term_a_b,A2: set_Pr4386577575007340137rm_a_b] :
( ( member_term_a_b @ A @ ( abstract_NF_term_a_b @ A2 ) )
=> ! [B8: term_a_b] :
~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B8 ) @ ( transi7922773638565587891rm_a_b @ A2 ) ) ) ).
% NF_no_trancl_step
thf(fact_845_CR__NF__conv,axiom,
! [R3: set_Pr2972776593051762503rm_a_b,T: produc357393685978478089rm_a_b,U: produc357393685978478089rm_a_b] :
( ( abstra7359694574113289279rm_a_b @ R3 @ top_to1314267278846557113rm_a_b )
=> ( ( member5869715511025134514rm_a_b @ T @ ( abstra1902783451776770197rm_a_b @ R3 ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ U @ T ) @ ( abstra6125656530262095991rm_a_b @ R3 ) )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ U @ T ) @ ( abstra1496011390211842225rm_a_b @ R3 ) ) ) ) ) ).
% CR_NF_conv
thf(fact_846_CR__NF__conv,axiom,
! [R3: set_Pr4386577575007340137rm_a_b,T: term_a_b,U: term_a_b] :
( ( abstra8448919418672941150rm_a_b @ R3 @ top_top_set_term_a_b )
=> ( ( member_term_a_b @ T @ ( abstract_NF_term_a_b @ R3 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ U @ T ) @ ( abstra1130568886744591910rm_a_b @ R3 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ U @ T ) @ ( abstra2972651231392952684rm_a_b @ R3 ) ) ) ) ) ).
% CR_NF_conv
thf(fact_847_join__NF__imp__eq,axiom,
! [X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b,R3: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ ( abstra4889682940153293507rm_a_b @ R3 ) )
=> ( ( member5869715511025134514rm_a_b @ X @ ( abstra1902783451776770197rm_a_b @ R3 ) )
=> ( ( member5869715511025134514rm_a_b @ Y @ ( abstra1902783451776770197rm_a_b @ R3 ) )
=> ( X = Y ) ) ) ) ).
% join_NF_imp_eq
thf(fact_848_join__NF__imp__eq,axiom,
! [X: term_a_b,Y: term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( abstra4096080454567261402rm_a_b @ R3 ) )
=> ( ( member_term_a_b @ X @ ( abstract_NF_term_a_b @ R3 ) )
=> ( ( member_term_a_b @ Y @ ( abstract_NF_term_a_b @ R3 ) )
=> ( X = Y ) ) ) ) ).
% join_NF_imp_eq
thf(fact_849_rtrancl__diff__decomp,axiom,
! [X: term_a_b,Y: term_a_b,A2: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( minus_5192120951422937424rm_a_b @ ( transi7742714808557438673rm_a_b @ A2 ) @ ( transi7742714808557438673rm_a_b @ B2 ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ A2 ) @ ( relcom370159955682700863rm_a_b @ ( minus_5192120951422937424rm_a_b @ A2 @ B2 ) @ ( transi7742714808557438673rm_a_b @ A2 ) ) ) ) ) ).
% rtrancl_diff_decomp
thf(fact_850_UNF__on__def,axiom,
( abstra3619855735679963878rm_a_b
= ( ^ [R6: set_Pr4386577575007340137rm_a_b,A6: set_term_a_b] :
! [X2: term_a_b] :
( ( member_term_a_b @ X2 @ A6 )
=> ! [B4: term_a_b,C3: term_a_b] :
( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ B4 ) @ ( abstra2972651231392952684rm_a_b @ R6 ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ C3 ) @ ( abstra2972651231392952684rm_a_b @ R6 ) ) )
=> ( B4 = C3 ) ) ) ) ) ).
% UNF_on_def
thf(fact_851_UNF__onE,axiom,
! [R3: set_Pr2972776593051762503rm_a_b,A2: set_Pr4386577575007340137rm_a_b,A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,C2: produc357393685978478089rm_a_b] :
( ( abstra1303798319023342519rm_a_b @ R3 @ A2 )
=> ( ( member5869715511025134514rm_a_b @ A @ A2 )
=> ( ( B != C2 )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( abstra1496011390211842225rm_a_b @ R3 ) )
=> ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ C2 ) @ ( abstra1496011390211842225rm_a_b @ R3 ) ) ) ) ) ) ).
% UNF_onE
thf(fact_852_UNF__onE,axiom,
! [R3: set_Pr4386577575007340137rm_a_b,A2: set_term_a_b,A: term_a_b,B: term_a_b,C2: term_a_b] :
( ( abstra3619855735679963878rm_a_b @ R3 @ A2 )
=> ( ( member_term_a_b @ A @ A2 )
=> ( ( B != C2 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( abstra2972651231392952684rm_a_b @ R3 ) )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C2 ) @ ( abstra2972651231392952684rm_a_b @ R3 ) ) ) ) ) ) ).
% UNF_onE
thf(fact_853_UNF__onD,axiom,
! [R3: set_Pr2972776593051762503rm_a_b,A2: set_Pr4386577575007340137rm_a_b,A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,C2: produc357393685978478089rm_a_b] :
( ( abstra1303798319023342519rm_a_b @ R3 @ A2 )
=> ( ( member5869715511025134514rm_a_b @ A @ A2 )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( abstra1496011390211842225rm_a_b @ R3 ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ C2 ) @ ( abstra1496011390211842225rm_a_b @ R3 ) )
=> ( B = C2 ) ) ) ) ) ).
% UNF_onD
thf(fact_854_UNF__onD,axiom,
! [R3: set_Pr4386577575007340137rm_a_b,A2: set_term_a_b,A: term_a_b,B: term_a_b,C2: term_a_b] :
( ( abstra3619855735679963878rm_a_b @ R3 @ A2 )
=> ( ( member_term_a_b @ A @ A2 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( abstra2972651231392952684rm_a_b @ R3 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C2 ) @ ( abstra2972651231392952684rm_a_b @ R3 ) )
=> ( B = C2 ) ) ) ) ) ).
% UNF_onD
thf(fact_855_NF__join__imp__reach,axiom,
! [Y: produc357393685978478089rm_a_b,A2: set_Pr2972776593051762503rm_a_b,X: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ Y @ ( abstra1902783451776770197rm_a_b @ A2 ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ ( abstra4889682940153293507rm_a_b @ A2 ) )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ ( transi2615809358984392588rm_a_b @ A2 ) ) ) ) ).
% NF_join_imp_reach
thf(fact_856_NF__join__imp__reach,axiom,
! [Y: term_a_b,A2: set_Pr4386577575007340137rm_a_b,X: term_a_b] :
( ( member_term_a_b @ Y @ ( abstract_NF_term_a_b @ A2 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( abstra4096080454567261402rm_a_b @ A2 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ A2 ) ) ) ) ).
% NF_join_imp_reach
thf(fact_857_UNC__E,axiom,
! [A2: set_Pr2972776593051762503rm_a_b,A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b] :
( ( abstra8480497335821202873rm_a_b @ A2 )
=> ( ( A != B )
=> ( ( member5869715511025134514rm_a_b @ A @ ( abstra1902783451776770197rm_a_b @ A2 ) )
=> ( ( member5869715511025134514rm_a_b @ B @ ( abstra1902783451776770197rm_a_b @ A2 ) )
=> ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( abstra6125656530262095991rm_a_b @ A2 ) ) ) ) ) ) ).
% UNC_E
thf(fact_858_UNC__E,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,A: term_a_b,B: term_a_b] :
( ( abstra4649771476585718372rm_a_b @ A2 )
=> ( ( A != B )
=> ( ( member_term_a_b @ A @ ( abstract_NF_term_a_b @ A2 ) )
=> ( ( member_term_a_b @ B @ ( abstract_NF_term_a_b @ A2 ) )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( abstra1130568886744591910rm_a_b @ A2 ) ) ) ) ) ) ).
% UNC_E
thf(fact_859_UNC__I,axiom,
! [A2: set_Pr2972776593051762503rm_a_b] :
( ! [A3: produc357393685978478089rm_a_b,B3: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ A3 @ ( abstra1902783451776770197rm_a_b @ A2 ) )
=> ( ( member5869715511025134514rm_a_b @ B3 @ ( abstra1902783451776770197rm_a_b @ A2 ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A3 @ B3 ) @ ( abstra6125656530262095991rm_a_b @ A2 ) )
=> ( A3 = B3 ) ) ) )
=> ( abstra8480497335821202873rm_a_b @ A2 ) ) ).
% UNC_I
thf(fact_860_UNC__I,axiom,
! [A2: set_Pr4386577575007340137rm_a_b] :
( ! [A3: term_a_b,B3: term_a_b] :
( ( member_term_a_b @ A3 @ ( abstract_NF_term_a_b @ A2 ) )
=> ( ( member_term_a_b @ B3 @ ( abstract_NF_term_a_b @ A2 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B3 ) @ ( abstra1130568886744591910rm_a_b @ A2 ) )
=> ( A3 = B3 ) ) ) )
=> ( abstra4649771476585718372rm_a_b @ A2 ) ) ).
% UNC_I
thf(fact_861_UNC__def,axiom,
( abstra8480497335821202873rm_a_b
= ( ^ [A6: set_Pr2972776593051762503rm_a_b] :
! [A4: produc357393685978478089rm_a_b,B4: produc357393685978478089rm_a_b] :
( ( ( member5869715511025134514rm_a_b @ A4 @ ( abstra1902783451776770197rm_a_b @ A6 ) )
& ( member5869715511025134514rm_a_b @ B4 @ ( abstra1902783451776770197rm_a_b @ A6 ) )
& ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A4 @ B4 ) @ ( abstra6125656530262095991rm_a_b @ A6 ) ) )
=> ( A4 = B4 ) ) ) ) ).
% UNC_def
thf(fact_862_UNC__def,axiom,
( abstra4649771476585718372rm_a_b
= ( ^ [A6: set_Pr4386577575007340137rm_a_b] :
! [A4: term_a_b,B4: term_a_b] :
( ( ( member_term_a_b @ A4 @ ( abstract_NF_term_a_b @ A6 ) )
& ( member_term_a_b @ B4 @ ( abstract_NF_term_a_b @ A6 ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A4 @ B4 ) @ ( abstra1130568886744591910rm_a_b @ A6 ) ) )
=> ( A4 = B4 ) ) ) ) ).
% UNC_def
thf(fact_863_NFP__stepD,axiom,
! [R3: set_Pr2972776593051762503rm_a_b,A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,C2: produc357393685978478089rm_a_b] :
( ( rewrit2138790387785322561rm_a_b @ R3 @ top_to1314267278846557113rm_a_b )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( transi2615809358984392588rm_a_b @ R3 ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ C2 ) @ ( transi2615809358984392588rm_a_b @ R3 ) )
=> ( ( member5869715511025134514rm_a_b @ C2 @ ( abstra1902783451776770197rm_a_b @ R3 ) )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B @ C2 ) @ ( transi2615809358984392588rm_a_b @ R3 ) ) ) ) ) ) ).
% NFP_stepD
thf(fact_864_NFP__stepD,axiom,
! [R3: set_Pr4386577575007340137rm_a_b,A: term_a_b,B: term_a_b,C2: term_a_b] :
( ( rewrit6304709981509430748rm_a_b @ R3 @ top_top_set_term_a_b )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ R3 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C2 ) @ ( transi7742714808557438673rm_a_b @ R3 ) )
=> ( ( member_term_a_b @ C2 @ ( abstract_NF_term_a_b @ R3 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C2 ) @ ( transi7742714808557438673rm_a_b @ R3 ) ) ) ) ) ) ).
% NFP_stepD
thf(fact_865_some__NF__UNF,axiom,
! [R3: set_Pr2972776593051762503rm_a_b,X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b] :
( ( abstra1303798319023342519rm_a_b @ R3 @ top_to1314267278846557113rm_a_b )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ ( transi2615809358984392588rm_a_b @ R3 ) )
=> ( ( member5869715511025134514rm_a_b @ Y @ ( abstra1902783451776770197rm_a_b @ R3 ) )
=> ( Y
= ( abstra8203652949708803681rm_a_b @ R3 @ X ) ) ) ) ) ).
% some_NF_UNF
thf(fact_866_some__NF__UNF,axiom,
! [R3: set_Pr4386577575007340137rm_a_b,X: term_a_b,Y: term_a_b] :
( ( abstra3619855735679963878rm_a_b @ R3 @ top_top_set_term_a_b )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ R3 ) )
=> ( ( member_term_a_b @ Y @ ( abstract_NF_term_a_b @ R3 ) )
=> ( Y
= ( abstra2589370493620291516rm_a_b @ R3 @ X ) ) ) ) ) ).
% some_NF_UNF
thf(fact_867_NFP__on__def,axiom,
( rewrit6304709981509430748rm_a_b
= ( ^ [R6: set_Pr4386577575007340137rm_a_b,A6: set_term_a_b] :
! [X2: term_a_b] :
( ( member_term_a_b @ X2 @ A6 )
=> ! [B4: term_a_b,C3: term_a_b] :
( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ B4 ) @ ( transi7742714808557438673rm_a_b @ R6 ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ C3 ) @ ( abstra2972651231392952684rm_a_b @ R6 ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B4 @ C3 ) @ ( transi7742714808557438673rm_a_b @ R6 ) ) ) ) ) ) ).
% NFP_on_def
thf(fact_868_semi__complete__imp__conversionIff__same__NF,axiom,
! [R3: set_Pr4386577575007340137rm_a_b,X: term_a_b,Y: term_a_b] :
( ( abstra7162307272211697846rm_a_b @ R3 @ top_top_set_term_a_b )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( abstra1130568886744591910rm_a_b @ R3 ) )
= ( ! [U3: term_a_b,V2: term_a_b] :
( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ U3 ) @ ( abstra2972651231392952684rm_a_b @ R3 ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ V2 ) @ ( abstra2972651231392952684rm_a_b @ R3 ) ) )
=> ( U3 = V2 ) ) ) ) ) ).
% semi_complete_imp_conversionIff_same_NF
thf(fact_869_diff__diff__cancel,axiom,
! [I: nat,N: nat] :
( ( ord_less_eq_nat @ I @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_870_NE__on__def,axiom,
( rewrit7370782028743737947rm_a_b
= ( ^ [R6: set_Pr4386577575007340137rm_a_b,S3: set_Pr4386577575007340137rm_a_b,A6: set_term_a_b] :
! [X2: term_a_b] :
( ( member_term_a_b @ X2 @ A6 )
=> ! [B4: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ B4 ) @ ( abstra2972651231392952684rm_a_b @ R6 ) )
= ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ B4 ) @ ( abstra2972651231392952684rm_a_b @ S3 ) ) ) ) ) ) ).
% NE_on_def
thf(fact_871_Nat_Oex__has__greatest__nat,axiom,
! [P2: nat > $o,K: nat,B: nat] :
( ( P2 @ K )
=> ( ! [Y2: nat] :
( ( P2 @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ B ) )
=> ? [X3: nat] :
( ( P2 @ X3 )
& ! [Y6: nat] :
( ( P2 @ Y6 )
=> ( ord_less_eq_nat @ Y6 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_872_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
| ( ord_less_eq_nat @ N @ M ) ) ).
% nat_le_linear
thf(fact_873_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( M = N ) ) ) ).
% le_antisym
thf(fact_874_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_875_le__trans,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I @ K ) ) ) ).
% le_trans
thf(fact_876_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_877_diff__le__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).
% diff_le_mono2
thf(fact_878_le__diff__iff_H,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ord_less_eq_nat @ A @ C2 )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C2 @ A ) @ ( minus_minus_nat @ C2 @ B ) )
= ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% le_diff_iff'
thf(fact_879_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_880_diff__le__mono,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% diff_le_mono
thf(fact_881_Nat_Odiff__diff__eq,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_882_le__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% le_diff_iff
thf(fact_883_eq__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ( minus_minus_nat @ M @ K )
= ( minus_minus_nat @ N @ K ) )
= ( M = N ) ) ) ) ).
% eq_diff_iff
thf(fact_884_rtrancl__Int__subset,axiom,
! [S: set_Pr4386577575007340137rm_a_b,R3: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ id_term_a_b @ S )
=> ( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ ( inf_in1693821330460682071rm_a_b @ ( transi7742714808557438673rm_a_b @ R3 ) @ S ) @ R3 ) @ S )
=> ( ord_le118470702582115849rm_a_b @ ( transi7742714808557438673rm_a_b @ R3 ) @ S ) ) ) ).
% rtrancl_Int_subset
thf(fact_885_ideriv__SN,axiom,
! [S6: set_Pr4386577575007340137rm_a_b,NS: set_Pr4386577575007340137rm_a_b,R4: set_Pr4386577575007340137rm_a_b,As: nat > term_a_b] :
( ( abstra4720023341729745482rm_a_b @ S6 @ top_top_set_term_a_b )
=> ( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ NS @ S6 ) @ S6 )
=> ( ( ord_le118470702582115849rm_a_b @ R4 @ ( sup_su6776935440552674877rm_a_b @ NS @ S6 ) )
=> ~ ( abstra5386884192082944997rm_a_b @ ( inf_in1693821330460682071rm_a_b @ S6 @ R4 ) @ ( minus_5192120951422937424rm_a_b @ R4 @ S6 ) @ As ) ) ) ) ).
% ideriv_SN
thf(fact_886_relChain__def,axiom,
( bNF_Ca2413601583040309898_b_nat
= ( ^ [R6: set_Pr4386577575007340137rm_a_b,As2: term_a_b > nat] :
! [I2: term_a_b,J2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ I2 @ J2 ) @ R6 )
=> ( ord_less_eq_nat @ ( As2 @ I2 ) @ ( As2 @ J2 ) ) ) ) ) ).
% relChain_def
thf(fact_887_IntI,axiom,
! [C2: produc357393685978478089rm_a_b,A2: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ C2 @ A2 )
=> ( ( member5869715511025134514rm_a_b @ C2 @ B2 )
=> ( member5869715511025134514rm_a_b @ C2 @ ( inf_in1693821330460682071rm_a_b @ A2 @ B2 ) ) ) ) ).
% IntI
thf(fact_888_Int__iff,axiom,
! [C2: produc357393685978478089rm_a_b,A2: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ C2 @ ( inf_in1693821330460682071rm_a_b @ A2 @ B2 ) )
= ( ( member5869715511025134514rm_a_b @ C2 @ A2 )
& ( member5869715511025134514rm_a_b @ C2 @ B2 ) ) ) ).
% Int_iff
thf(fact_889_inf_Obounded__iff,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B @ C2 ) )
= ( ( ord_less_eq_nat @ A @ B )
& ( ord_less_eq_nat @ A @ C2 ) ) ) ).
% inf.bounded_iff
thf(fact_890_le__inf__iff,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ ( inf_inf_nat @ Y @ Z ) )
= ( ( ord_less_eq_nat @ X @ Y )
& ( ord_less_eq_nat @ X @ Z ) ) ) ).
% le_inf_iff
thf(fact_891_inf__sup__absorb,axiom,
! [X: set_Pr4386577575007340137rm_a_b,Y: set_Pr4386577575007340137rm_a_b] :
( ( inf_in1693821330460682071rm_a_b @ X @ ( sup_su6776935440552674877rm_a_b @ X @ Y ) )
= X ) ).
% inf_sup_absorb
thf(fact_892_sup__inf__absorb,axiom,
! [X: set_Pr4386577575007340137rm_a_b,Y: set_Pr4386577575007340137rm_a_b] :
( ( sup_su6776935440552674877rm_a_b @ X @ ( inf_in1693821330460682071rm_a_b @ X @ Y ) )
= X ) ).
% sup_inf_absorb
thf(fact_893_Int__Un__eq_I4_J,axiom,
! [T4: set_Pr4386577575007340137rm_a_b,S6: set_Pr4386577575007340137rm_a_b] :
( ( sup_su6776935440552674877rm_a_b @ T4 @ ( inf_in1693821330460682071rm_a_b @ S6 @ T4 ) )
= T4 ) ).
% Int_Un_eq(4)
thf(fact_894_Int__Un__eq_I3_J,axiom,
! [S6: set_Pr4386577575007340137rm_a_b,T4: set_Pr4386577575007340137rm_a_b] :
( ( sup_su6776935440552674877rm_a_b @ S6 @ ( inf_in1693821330460682071rm_a_b @ S6 @ T4 ) )
= S6 ) ).
% Int_Un_eq(3)
thf(fact_895_Int__Un__eq_I2_J,axiom,
! [S6: set_Pr4386577575007340137rm_a_b,T4: set_Pr4386577575007340137rm_a_b] :
( ( sup_su6776935440552674877rm_a_b @ ( inf_in1693821330460682071rm_a_b @ S6 @ T4 ) @ T4 )
= T4 ) ).
% Int_Un_eq(2)
thf(fact_896_Int__Un__eq_I1_J,axiom,
! [S6: set_Pr4386577575007340137rm_a_b,T4: set_Pr4386577575007340137rm_a_b] :
( ( sup_su6776935440552674877rm_a_b @ ( inf_in1693821330460682071rm_a_b @ S6 @ T4 ) @ S6 )
= S6 ) ).
% Int_Un_eq(1)
thf(fact_897_Un__Int__eq_I4_J,axiom,
! [T4: set_Pr4386577575007340137rm_a_b,S6: set_Pr4386577575007340137rm_a_b] :
( ( inf_in1693821330460682071rm_a_b @ T4 @ ( sup_su6776935440552674877rm_a_b @ S6 @ T4 ) )
= T4 ) ).
% Un_Int_eq(4)
thf(fact_898_Un__Int__eq_I3_J,axiom,
! [S6: set_Pr4386577575007340137rm_a_b,T4: set_Pr4386577575007340137rm_a_b] :
( ( inf_in1693821330460682071rm_a_b @ S6 @ ( sup_su6776935440552674877rm_a_b @ S6 @ T4 ) )
= S6 ) ).
% Un_Int_eq(3)
thf(fact_899_Un__Int__eq_I2_J,axiom,
! [S6: set_Pr4386577575007340137rm_a_b,T4: set_Pr4386577575007340137rm_a_b] :
( ( inf_in1693821330460682071rm_a_b @ ( sup_su6776935440552674877rm_a_b @ S6 @ T4 ) @ T4 )
= T4 ) ).
% Un_Int_eq(2)
thf(fact_900_Un__Int__eq_I1_J,axiom,
! [S6: set_Pr4386577575007340137rm_a_b,T4: set_Pr4386577575007340137rm_a_b] :
( ( inf_in1693821330460682071rm_a_b @ ( sup_su6776935440552674877rm_a_b @ S6 @ T4 ) @ S6 )
= S6 ) ).
% Un_Int_eq(1)
thf(fact_901_irrefl__on__converse,axiom,
! [A2: set_term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
( ( irrefl_on_term_a_b @ A2 @ ( conver7608559976937845093rm_a_b @ R3 ) )
= ( irrefl_on_term_a_b @ A2 @ R3 ) ) ).
% irrefl_on_converse
thf(fact_902_converse__Int,axiom,
! [R3: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
( ( conver7608559976937845093rm_a_b @ ( inf_in1693821330460682071rm_a_b @ R3 @ S ) )
= ( inf_in1693821330460682071rm_a_b @ ( conver7608559976937845093rm_a_b @ R3 ) @ ( conver7608559976937845093rm_a_b @ S ) ) ) ).
% converse_Int
thf(fact_903_IntE,axiom,
! [C2: produc357393685978478089rm_a_b,A2: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ C2 @ ( inf_in1693821330460682071rm_a_b @ A2 @ B2 ) )
=> ~ ( ( member5869715511025134514rm_a_b @ C2 @ A2 )
=> ~ ( member5869715511025134514rm_a_b @ C2 @ B2 ) ) ) ).
% IntE
thf(fact_904_IntD1,axiom,
! [C2: produc357393685978478089rm_a_b,A2: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ C2 @ ( inf_in1693821330460682071rm_a_b @ A2 @ B2 ) )
=> ( member5869715511025134514rm_a_b @ C2 @ A2 ) ) ).
% IntD1
thf(fact_905_IntD2,axiom,
! [C2: produc357393685978478089rm_a_b,A2: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ C2 @ ( inf_in1693821330460682071rm_a_b @ A2 @ B2 ) )
=> ( member5869715511025134514rm_a_b @ C2 @ B2 ) ) ).
% IntD2
thf(fact_906_Un__Int__crazy,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b,C: set_Pr4386577575007340137rm_a_b] :
( ( sup_su6776935440552674877rm_a_b @ ( sup_su6776935440552674877rm_a_b @ ( inf_in1693821330460682071rm_a_b @ A2 @ B2 ) @ ( inf_in1693821330460682071rm_a_b @ B2 @ C ) ) @ ( inf_in1693821330460682071rm_a_b @ C @ A2 ) )
= ( inf_in1693821330460682071rm_a_b @ ( inf_in1693821330460682071rm_a_b @ ( sup_su6776935440552674877rm_a_b @ A2 @ B2 ) @ ( sup_su6776935440552674877rm_a_b @ B2 @ C ) ) @ ( sup_su6776935440552674877rm_a_b @ C @ A2 ) ) ) ).
% Un_Int_crazy
thf(fact_907_Int__Un__distrib,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b,C: set_Pr4386577575007340137rm_a_b] :
( ( inf_in1693821330460682071rm_a_b @ A2 @ ( sup_su6776935440552674877rm_a_b @ B2 @ C ) )
= ( sup_su6776935440552674877rm_a_b @ ( inf_in1693821330460682071rm_a_b @ A2 @ B2 ) @ ( inf_in1693821330460682071rm_a_b @ A2 @ C ) ) ) ).
% Int_Un_distrib
thf(fact_908_Un__Int__distrib,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b,C: set_Pr4386577575007340137rm_a_b] :
( ( sup_su6776935440552674877rm_a_b @ A2 @ ( inf_in1693821330460682071rm_a_b @ B2 @ C ) )
= ( inf_in1693821330460682071rm_a_b @ ( sup_su6776935440552674877rm_a_b @ A2 @ B2 ) @ ( sup_su6776935440552674877rm_a_b @ A2 @ C ) ) ) ).
% Un_Int_distrib
thf(fact_909_Int__Un__distrib2,axiom,
! [B2: set_Pr4386577575007340137rm_a_b,C: set_Pr4386577575007340137rm_a_b,A2: set_Pr4386577575007340137rm_a_b] :
( ( inf_in1693821330460682071rm_a_b @ ( sup_su6776935440552674877rm_a_b @ B2 @ C ) @ A2 )
= ( sup_su6776935440552674877rm_a_b @ ( inf_in1693821330460682071rm_a_b @ B2 @ A2 ) @ ( inf_in1693821330460682071rm_a_b @ C @ A2 ) ) ) ).
% Int_Un_distrib2
thf(fact_910_Un__Int__distrib2,axiom,
! [B2: set_Pr4386577575007340137rm_a_b,C: set_Pr4386577575007340137rm_a_b,A2: set_Pr4386577575007340137rm_a_b] :
( ( sup_su6776935440552674877rm_a_b @ ( inf_in1693821330460682071rm_a_b @ B2 @ C ) @ A2 )
= ( inf_in1693821330460682071rm_a_b @ ( sup_su6776935440552674877rm_a_b @ B2 @ A2 ) @ ( sup_su6776935440552674877rm_a_b @ C @ A2 ) ) ) ).
% Un_Int_distrib2
thf(fact_911_Int__Collect__mono,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b,P2: produc357393685978478089rm_a_b > $o,Q: produc357393685978478089rm_a_b > $o] :
( ( ord_le118470702582115849rm_a_b @ A2 @ B2 )
=> ( ! [X3: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X3 @ A2 )
=> ( ( P2 @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_le118470702582115849rm_a_b @ ( inf_in1693821330460682071rm_a_b @ A2 @ ( collec99177395925924084rm_a_b @ P2 ) ) @ ( inf_in1693821330460682071rm_a_b @ B2 @ ( collec99177395925924084rm_a_b @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_912_distrib__imp1,axiom,
! [X: set_Pr4386577575007340137rm_a_b,Y: set_Pr4386577575007340137rm_a_b,Z: set_Pr4386577575007340137rm_a_b] :
( ! [X3: set_Pr4386577575007340137rm_a_b,Y2: set_Pr4386577575007340137rm_a_b,Z2: set_Pr4386577575007340137rm_a_b] :
( ( inf_in1693821330460682071rm_a_b @ X3 @ ( sup_su6776935440552674877rm_a_b @ Y2 @ Z2 ) )
= ( sup_su6776935440552674877rm_a_b @ ( inf_in1693821330460682071rm_a_b @ X3 @ Y2 ) @ ( inf_in1693821330460682071rm_a_b @ X3 @ Z2 ) ) )
=> ( ( sup_su6776935440552674877rm_a_b @ X @ ( inf_in1693821330460682071rm_a_b @ Y @ Z ) )
= ( inf_in1693821330460682071rm_a_b @ ( sup_su6776935440552674877rm_a_b @ X @ Y ) @ ( sup_su6776935440552674877rm_a_b @ X @ Z ) ) ) ) ).
% distrib_imp1
thf(fact_913_distrib__imp2,axiom,
! [X: set_Pr4386577575007340137rm_a_b,Y: set_Pr4386577575007340137rm_a_b,Z: set_Pr4386577575007340137rm_a_b] :
( ! [X3: set_Pr4386577575007340137rm_a_b,Y2: set_Pr4386577575007340137rm_a_b,Z2: set_Pr4386577575007340137rm_a_b] :
( ( sup_su6776935440552674877rm_a_b @ X3 @ ( inf_in1693821330460682071rm_a_b @ Y2 @ Z2 ) )
= ( inf_in1693821330460682071rm_a_b @ ( sup_su6776935440552674877rm_a_b @ X3 @ Y2 ) @ ( sup_su6776935440552674877rm_a_b @ X3 @ Z2 ) ) )
=> ( ( inf_in1693821330460682071rm_a_b @ X @ ( sup_su6776935440552674877rm_a_b @ Y @ Z ) )
= ( sup_su6776935440552674877rm_a_b @ ( inf_in1693821330460682071rm_a_b @ X @ Y ) @ ( inf_in1693821330460682071rm_a_b @ X @ Z ) ) ) ) ).
% distrib_imp2
thf(fact_914_inf__sup__distrib1,axiom,
! [X: set_Pr4386577575007340137rm_a_b,Y: set_Pr4386577575007340137rm_a_b,Z: set_Pr4386577575007340137rm_a_b] :
( ( inf_in1693821330460682071rm_a_b @ X @ ( sup_su6776935440552674877rm_a_b @ Y @ Z ) )
= ( sup_su6776935440552674877rm_a_b @ ( inf_in1693821330460682071rm_a_b @ X @ Y ) @ ( inf_in1693821330460682071rm_a_b @ X @ Z ) ) ) ).
% inf_sup_distrib1
thf(fact_915_inf__sup__distrib2,axiom,
! [Y: set_Pr4386577575007340137rm_a_b,Z: set_Pr4386577575007340137rm_a_b,X: set_Pr4386577575007340137rm_a_b] :
( ( inf_in1693821330460682071rm_a_b @ ( sup_su6776935440552674877rm_a_b @ Y @ Z ) @ X )
= ( sup_su6776935440552674877rm_a_b @ ( inf_in1693821330460682071rm_a_b @ Y @ X ) @ ( inf_in1693821330460682071rm_a_b @ Z @ X ) ) ) ).
% inf_sup_distrib2
thf(fact_916_sup__inf__distrib1,axiom,
! [X: set_Pr4386577575007340137rm_a_b,Y: set_Pr4386577575007340137rm_a_b,Z: set_Pr4386577575007340137rm_a_b] :
( ( sup_su6776935440552674877rm_a_b @ X @ ( inf_in1693821330460682071rm_a_b @ Y @ Z ) )
= ( inf_in1693821330460682071rm_a_b @ ( sup_su6776935440552674877rm_a_b @ X @ Y ) @ ( sup_su6776935440552674877rm_a_b @ X @ Z ) ) ) ).
% sup_inf_distrib1
thf(fact_917_sup__inf__distrib2,axiom,
! [Y: set_Pr4386577575007340137rm_a_b,Z: set_Pr4386577575007340137rm_a_b,X: set_Pr4386577575007340137rm_a_b] :
( ( sup_su6776935440552674877rm_a_b @ ( inf_in1693821330460682071rm_a_b @ Y @ Z ) @ X )
= ( inf_in1693821330460682071rm_a_b @ ( sup_su6776935440552674877rm_a_b @ Y @ X ) @ ( sup_su6776935440552674877rm_a_b @ Z @ X ) ) ) ).
% sup_inf_distrib2
thf(fact_918_boolean__algebra_Odisj__conj__distrib2,axiom,
! [Y: set_Pr4386577575007340137rm_a_b,Z: set_Pr4386577575007340137rm_a_b,X: set_Pr4386577575007340137rm_a_b] :
( ( sup_su6776935440552674877rm_a_b @ ( inf_in1693821330460682071rm_a_b @ Y @ Z ) @ X )
= ( inf_in1693821330460682071rm_a_b @ ( sup_su6776935440552674877rm_a_b @ Y @ X ) @ ( sup_su6776935440552674877rm_a_b @ Z @ X ) ) ) ).
% boolean_algebra.disj_conj_distrib2
thf(fact_919_boolean__algebra_Oconj__disj__distrib2,axiom,
! [Y: set_Pr4386577575007340137rm_a_b,Z: set_Pr4386577575007340137rm_a_b,X: set_Pr4386577575007340137rm_a_b] :
( ( inf_in1693821330460682071rm_a_b @ ( sup_su6776935440552674877rm_a_b @ Y @ Z ) @ X )
= ( sup_su6776935440552674877rm_a_b @ ( inf_in1693821330460682071rm_a_b @ Y @ X ) @ ( inf_in1693821330460682071rm_a_b @ Z @ X ) ) ) ).
% boolean_algebra.conj_disj_distrib2
thf(fact_920_boolean__algebra_Odisj__conj__distrib,axiom,
! [X: set_Pr4386577575007340137rm_a_b,Y: set_Pr4386577575007340137rm_a_b,Z: set_Pr4386577575007340137rm_a_b] :
( ( sup_su6776935440552674877rm_a_b @ X @ ( inf_in1693821330460682071rm_a_b @ Y @ Z ) )
= ( inf_in1693821330460682071rm_a_b @ ( sup_su6776935440552674877rm_a_b @ X @ Y ) @ ( sup_su6776935440552674877rm_a_b @ X @ Z ) ) ) ).
% boolean_algebra.disj_conj_distrib
thf(fact_921_boolean__algebra_Oconj__disj__distrib,axiom,
! [X: set_Pr4386577575007340137rm_a_b,Y: set_Pr4386577575007340137rm_a_b,Z: set_Pr4386577575007340137rm_a_b] :
( ( inf_in1693821330460682071rm_a_b @ X @ ( sup_su6776935440552674877rm_a_b @ Y @ Z ) )
= ( sup_su6776935440552674877rm_a_b @ ( inf_in1693821330460682071rm_a_b @ X @ Y ) @ ( inf_in1693821330460682071rm_a_b @ X @ Z ) ) ) ).
% boolean_algebra.conj_disj_distrib
thf(fact_922_inf__sup__ord_I2_J,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_923_inf__sup__ord_I1_J,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_924_inf__le1,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_925_inf__le2,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_926_le__infE,axiom,
! [X: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ X @ ( inf_inf_nat @ A @ B ) )
=> ~ ( ( ord_less_eq_nat @ X @ A )
=> ~ ( ord_less_eq_nat @ X @ B ) ) ) ).
% le_infE
thf(fact_927_le__infI,axiom,
! [X: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ X @ A )
=> ( ( ord_less_eq_nat @ X @ B )
=> ( ord_less_eq_nat @ X @ ( inf_inf_nat @ A @ B ) ) ) ) ).
% le_infI
thf(fact_928_inf__mono,axiom,
! [A: nat,C2: nat,B: nat,D: nat] :
( ( ord_less_eq_nat @ A @ C2 )
=> ( ( ord_less_eq_nat @ B @ D )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ ( inf_inf_nat @ C2 @ D ) ) ) ) ).
% inf_mono
thf(fact_929_le__infI1,axiom,
! [A: nat,X: nat,B: nat] :
( ( ord_less_eq_nat @ A @ X )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ X ) ) ).
% le_infI1
thf(fact_930_le__infI2,axiom,
! [B: nat,X: nat,A: nat] :
( ( ord_less_eq_nat @ B @ X )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ X ) ) ).
% le_infI2
thf(fact_931_inf_OorderE,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( A
= ( inf_inf_nat @ A @ B ) ) ) ).
% inf.orderE
thf(fact_932_inf_OorderI,axiom,
! [A: nat,B: nat] :
( ( A
= ( inf_inf_nat @ A @ B ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% inf.orderI
thf(fact_933_inf__unique,axiom,
! [F3: nat > nat > nat,X: nat,Y: nat] :
( ! [X3: nat,Y2: nat] : ( ord_less_eq_nat @ ( F3 @ X3 @ Y2 ) @ X3 )
=> ( ! [X3: nat,Y2: nat] : ( ord_less_eq_nat @ ( F3 @ X3 @ Y2 ) @ Y2 )
=> ( ! [X3: nat,Y2: nat,Z2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ( ord_less_eq_nat @ X3 @ Z2 )
=> ( ord_less_eq_nat @ X3 @ ( F3 @ Y2 @ Z2 ) ) ) )
=> ( ( inf_inf_nat @ X @ Y )
= ( F3 @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_934_le__iff__inf,axiom,
( ord_less_eq_nat
= ( ^ [X2: nat,Y3: nat] :
( ( inf_inf_nat @ X2 @ Y3 )
= X2 ) ) ) ).
% le_iff_inf
thf(fact_935_inf_Oabsorb1,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( inf_inf_nat @ A @ B )
= A ) ) ).
% inf.absorb1
thf(fact_936_inf_Oabsorb2,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( inf_inf_nat @ A @ B )
= B ) ) ).
% inf.absorb2
thf(fact_937_inf__absorb1,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( inf_inf_nat @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_938_inf__absorb2,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( inf_inf_nat @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_939_inf_OboundedE,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B @ C2 ) )
=> ~ ( ( ord_less_eq_nat @ A @ B )
=> ~ ( ord_less_eq_nat @ A @ C2 ) ) ) ).
% inf.boundedE
thf(fact_940_inf_OboundedI,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ A @ C2 )
=> ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B @ C2 ) ) ) ) ).
% inf.boundedI
thf(fact_941_inf__greatest,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Z )
=> ( ord_less_eq_nat @ X @ ( inf_inf_nat @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_942_inf_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B4: nat] :
( A4
= ( inf_inf_nat @ A4 @ B4 ) ) ) ) ).
% inf.order_iff
thf(fact_943_inf_Ocobounded1,axiom,
! [A: nat,B: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ A ) ).
% inf.cobounded1
thf(fact_944_inf_Ocobounded2,axiom,
! [A: nat,B: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ B ) ).
% inf.cobounded2
thf(fact_945_inf_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B4: nat] :
( ( inf_inf_nat @ A4 @ B4 )
= A4 ) ) ) ).
% inf.absorb_iff1
thf(fact_946_inf_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [B4: nat,A4: nat] :
( ( inf_inf_nat @ A4 @ B4 )
= B4 ) ) ) ).
% inf.absorb_iff2
thf(fact_947_inf_OcoboundedI1,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ord_less_eq_nat @ A @ C2 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ C2 ) ) ).
% inf.coboundedI1
thf(fact_948_inf_OcoboundedI2,axiom,
! [B: nat,C2: nat,A: nat] :
( ( ord_less_eq_nat @ B @ C2 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ C2 ) ) ).
% inf.coboundedI2
thf(fact_949_irrefl__onD,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,R3: set_Pr2972776593051762503rm_a_b,A: produc357393685978478089rm_a_b] :
( ( irrefl6620156600162922239rm_a_b @ A2 @ R3 )
=> ( ( member5869715511025134514rm_a_b @ A @ A2 )
=> ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ A ) @ R3 ) ) ) ).
% irrefl_onD
thf(fact_950_irrefl__onD,axiom,
! [A2: set_term_a_b,R3: set_Pr4386577575007340137rm_a_b,A: term_a_b] :
( ( irrefl_on_term_a_b @ A2 @ R3 )
=> ( ( member_term_a_b @ A @ A2 )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ A ) @ R3 ) ) ) ).
% irrefl_onD
thf(fact_951_irrefl__onI,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,R3: set_Pr2972776593051762503rm_a_b] :
( ! [A3: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ A3 @ A2 )
=> ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A3 @ A3 ) @ R3 ) )
=> ( irrefl6620156600162922239rm_a_b @ A2 @ R3 ) ) ).
% irrefl_onI
thf(fact_952_irrefl__onI,axiom,
! [A2: set_term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
( ! [A3: term_a_b] :
( ( member_term_a_b @ A3 @ A2 )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ A3 ) @ R3 ) )
=> ( irrefl_on_term_a_b @ A2 @ R3 ) ) ).
% irrefl_onI
thf(fact_953_irrefl__on__def,axiom,
( irrefl_on_term_a_b
= ( ^ [A6: set_term_a_b,R6: set_Pr4386577575007340137rm_a_b] :
! [X2: term_a_b] :
( ( member_term_a_b @ X2 @ A6 )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ X2 ) @ R6 ) ) ) ) ).
% irrefl_on_def
thf(fact_954_distrib__inf__le,axiom,
! [X: set_Pr4386577575007340137rm_a_b,Y: set_Pr4386577575007340137rm_a_b,Z: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ ( sup_su6776935440552674877rm_a_b @ ( inf_in1693821330460682071rm_a_b @ X @ Y ) @ ( inf_in1693821330460682071rm_a_b @ X @ Z ) ) @ ( inf_in1693821330460682071rm_a_b @ X @ ( sup_su6776935440552674877rm_a_b @ Y @ Z ) ) ) ).
% distrib_inf_le
thf(fact_955_distrib__inf__le,axiom,
! [X: nat,Y: nat,Z: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ ( inf_inf_nat @ X @ Y ) @ ( inf_inf_nat @ X @ Z ) ) @ ( inf_inf_nat @ X @ ( sup_sup_nat @ Y @ Z ) ) ) ).
% distrib_inf_le
thf(fact_956_distrib__sup__le,axiom,
! [X: set_Pr4386577575007340137rm_a_b,Y: set_Pr4386577575007340137rm_a_b,Z: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ ( sup_su6776935440552674877rm_a_b @ X @ ( inf_in1693821330460682071rm_a_b @ Y @ Z ) ) @ ( inf_in1693821330460682071rm_a_b @ ( sup_su6776935440552674877rm_a_b @ X @ Y ) @ ( sup_su6776935440552674877rm_a_b @ X @ Z ) ) ) ).
% distrib_sup_le
thf(fact_957_distrib__sup__le,axiom,
! [X: nat,Y: nat,Z: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ X @ ( inf_inf_nat @ Y @ Z ) ) @ ( inf_inf_nat @ ( sup_sup_nat @ X @ Y ) @ ( sup_sup_nat @ X @ Z ) ) ) ).
% distrib_sup_le
thf(fact_958_Un__Int__assoc__eq,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b,C: set_Pr4386577575007340137rm_a_b] :
( ( ( sup_su6776935440552674877rm_a_b @ ( inf_in1693821330460682071rm_a_b @ A2 @ B2 ) @ C )
= ( inf_in1693821330460682071rm_a_b @ A2 @ ( sup_su6776935440552674877rm_a_b @ B2 @ C ) ) )
= ( ord_le118470702582115849rm_a_b @ C @ A2 ) ) ).
% Un_Int_assoc_eq
thf(fact_959_Diff__Un,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b,C: set_Pr4386577575007340137rm_a_b] :
( ( minus_5192120951422937424rm_a_b @ A2 @ ( sup_su6776935440552674877rm_a_b @ B2 @ C ) )
= ( inf_in1693821330460682071rm_a_b @ ( minus_5192120951422937424rm_a_b @ A2 @ B2 ) @ ( minus_5192120951422937424rm_a_b @ A2 @ C ) ) ) ).
% Diff_Un
thf(fact_960_Diff__Int,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b,C: set_Pr4386577575007340137rm_a_b] :
( ( minus_5192120951422937424rm_a_b @ A2 @ ( inf_in1693821330460682071rm_a_b @ B2 @ C ) )
= ( sup_su6776935440552674877rm_a_b @ ( minus_5192120951422937424rm_a_b @ A2 @ B2 ) @ ( minus_5192120951422937424rm_a_b @ A2 @ C ) ) ) ).
% Diff_Int
thf(fact_961_Int__Diff__Un,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b] :
( ( sup_su6776935440552674877rm_a_b @ ( inf_in1693821330460682071rm_a_b @ A2 @ B2 ) @ ( minus_5192120951422937424rm_a_b @ A2 @ B2 ) )
= A2 ) ).
% Int_Diff_Un
thf(fact_962_Un__Diff__Int,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b] :
( ( sup_su6776935440552674877rm_a_b @ ( minus_5192120951422937424rm_a_b @ A2 @ B2 ) @ ( inf_in1693821330460682071rm_a_b @ A2 @ B2 ) )
= A2 ) ).
% Un_Diff_Int
thf(fact_963_irreflD,axiom,
! [R3: set_Pr4386577575007340137rm_a_b,X: term_a_b] :
( ( irrefl_on_term_a_b @ top_top_set_term_a_b @ R3 )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ X ) @ R3 ) ) ).
% irreflD
thf(fact_964_irreflI,axiom,
! [R3: set_Pr4386577575007340137rm_a_b] :
( ! [A3: term_a_b] :
~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ A3 ) @ R3 )
=> ( irrefl_on_term_a_b @ top_top_set_term_a_b @ R3 ) ) ).
% irreflI
thf(fact_965_trancl__Int__subset,axiom,
! [R3: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ R3 @ S )
=> ( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ ( inf_in1693821330460682071rm_a_b @ ( transi7922773638565587891rm_a_b @ R3 ) @ S ) @ R3 ) @ S )
=> ( ord_le118470702582115849rm_a_b @ ( transi7922773638565587891rm_a_b @ R3 ) @ S ) ) ) ).
% trancl_Int_subset
thf(fact_966_ideriv__split,axiom,
! [R4: set_Pr4386577575007340137rm_a_b,S6: set_Pr4386577575007340137rm_a_b,As: nat > term_a_b,D2: set_Pr4386577575007340137rm_a_b] :
( ( abstra5386884192082944997rm_a_b @ R4 @ S6 @ As )
=> ( ~ ( abstra5386884192082944997rm_a_b @ ( inf_in1693821330460682071rm_a_b @ D2 @ ( sup_su6776935440552674877rm_a_b @ R4 @ S6 ) ) @ ( minus_5192120951422937424rm_a_b @ ( sup_su6776935440552674877rm_a_b @ R4 @ S6 ) @ D2 ) @ As )
=> ? [I3: nat] : ( abstra5386884192082944997rm_a_b @ ( minus_5192120951422937424rm_a_b @ R4 @ D2 ) @ ( minus_5192120951422937424rm_a_b @ S6 @ D2 ) @ ( abstra6393647268163345180rm_a_b @ As @ I3 ) ) ) ) ).
% ideriv_split
thf(fact_967_irrefl__tranclI,axiom,
! [R3: set_Pr4386577575007340137rm_a_b,X: term_a_b] :
( ( ( inf_in1693821330460682071rm_a_b @ ( conver7608559976937845093rm_a_b @ R3 ) @ ( transi7742714808557438673rm_a_b @ R3 ) )
= bot_bo197521221353338581rm_a_b )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ X ) @ ( transi7922773638565587891rm_a_b @ R3 ) ) ) ).
% irrefl_tranclI
thf(fact_968_acyclic__irrefl,axiom,
( transi5314701259734769157rm_a_b
= ( ^ [R6: set_Pr4386577575007340137rm_a_b] : ( irrefl_on_term_a_b @ top_top_set_term_a_b @ ( transi7922773638565587891rm_a_b @ R6 ) ) ) ) ).
% acyclic_irrefl
thf(fact_969_SN__on__Image__push,axiom,
! [S6: set_Pr4386577575007340137rm_a_b,R4: set_Pr4386577575007340137rm_a_b,X4: set_term_a_b] :
( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ S6 @ R4 ) @ ( relcom370159955682700863rm_a_b @ R4 @ ( transi7742714808557438673rm_a_b @ S6 ) ) )
=> ( ( abstra4720023341729745482rm_a_b @ R4 @ X4 )
=> ( abstra4720023341729745482rm_a_b @ R4 @ ( image_6923787618150785661rm_a_b @ ( transi7742714808557438673rm_a_b @ S6 ) @ X4 ) ) ) ) ).
% SN_on_Image_push
thf(fact_970_all__not__in__conv,axiom,
! [A2: set_Pr4386577575007340137rm_a_b] :
( ( ! [X2: produc357393685978478089rm_a_b] :
~ ( member5869715511025134514rm_a_b @ X2 @ A2 ) )
= ( A2 = bot_bo197521221353338581rm_a_b ) ) ).
% all_not_in_conv
thf(fact_971_empty__iff,axiom,
! [C2: produc357393685978478089rm_a_b] :
~ ( member5869715511025134514rm_a_b @ C2 @ bot_bo197521221353338581rm_a_b ) ).
% empty_iff
thf(fact_972_sup__bot_Oright__neutral,axiom,
! [A: set_Pr4386577575007340137rm_a_b] :
( ( sup_su6776935440552674877rm_a_b @ A @ bot_bo197521221353338581rm_a_b )
= A ) ).
% sup_bot.right_neutral
thf(fact_973_sup__bot_Oneutr__eq__iff,axiom,
! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ( bot_bo197521221353338581rm_a_b
= ( sup_su6776935440552674877rm_a_b @ A @ B ) )
= ( ( A = bot_bo197521221353338581rm_a_b )
& ( B = bot_bo197521221353338581rm_a_b ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_974_sup__bot_Oleft__neutral,axiom,
! [A: set_Pr4386577575007340137rm_a_b] :
( ( sup_su6776935440552674877rm_a_b @ bot_bo197521221353338581rm_a_b @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_975_sup__bot_Oeq__neutr__iff,axiom,
! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ( ( sup_su6776935440552674877rm_a_b @ A @ B )
= bot_bo197521221353338581rm_a_b )
= ( ( A = bot_bo197521221353338581rm_a_b )
& ( B = bot_bo197521221353338581rm_a_b ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_976_sup__eq__bot__iff,axiom,
! [X: set_Pr4386577575007340137rm_a_b,Y: set_Pr4386577575007340137rm_a_b] :
( ( ( sup_su6776935440552674877rm_a_b @ X @ Y )
= bot_bo197521221353338581rm_a_b )
= ( ( X = bot_bo197521221353338581rm_a_b )
& ( Y = bot_bo197521221353338581rm_a_b ) ) ) ).
% sup_eq_bot_iff
thf(fact_977_bot__eq__sup__iff,axiom,
! [X: set_Pr4386577575007340137rm_a_b,Y: set_Pr4386577575007340137rm_a_b] :
( ( bot_bo197521221353338581rm_a_b
= ( sup_su6776935440552674877rm_a_b @ X @ Y ) )
= ( ( X = bot_bo197521221353338581rm_a_b )
& ( Y = bot_bo197521221353338581rm_a_b ) ) ) ).
% bot_eq_sup_iff
thf(fact_978_sup__bot__right,axiom,
! [X: set_Pr4386577575007340137rm_a_b] :
( ( sup_su6776935440552674877rm_a_b @ X @ bot_bo197521221353338581rm_a_b )
= X ) ).
% sup_bot_right
thf(fact_979_sup__bot__left,axiom,
! [X: set_Pr4386577575007340137rm_a_b] :
( ( sup_su6776935440552674877rm_a_b @ bot_bo197521221353338581rm_a_b @ X )
= X ) ).
% sup_bot_left
thf(fact_980_Un__empty,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b] :
( ( ( sup_su6776935440552674877rm_a_b @ A2 @ B2 )
= bot_bo197521221353338581rm_a_b )
= ( ( A2 = bot_bo197521221353338581rm_a_b )
& ( B2 = bot_bo197521221353338581rm_a_b ) ) ) ).
% Un_empty
thf(fact_981_ImageI,axiom,
! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,R3: set_Pr2972776593051762503rm_a_b,A2: set_Pr4386577575007340137rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ R3 )
=> ( ( member5869715511025134514rm_a_b @ A @ A2 )
=> ( member5869715511025134514rm_a_b @ B @ ( image_871640889055226643rm_a_b @ R3 @ A2 ) ) ) ) ).
% ImageI
thf(fact_982_ImageI,axiom,
! [A: term_a_b,B: term_a_b,R3: set_Pr4386577575007340137rm_a_b,A2: set_term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R3 )
=> ( ( member_term_a_b @ A @ A2 )
=> ( member_term_a_b @ B @ ( image_6923787618150785661rm_a_b @ R3 @ A2 ) ) ) ) ).
% ImageI
thf(fact_983_relcomp__empty1,axiom,
! [R4: set_Pr4386577575007340137rm_a_b] :
( ( relcom370159955682700863rm_a_b @ bot_bo197521221353338581rm_a_b @ R4 )
= bot_bo197521221353338581rm_a_b ) ).
% relcomp_empty1
thf(fact_984_relcomp__empty2,axiom,
! [R4: set_Pr4386577575007340137rm_a_b] :
( ( relcom370159955682700863rm_a_b @ R4 @ bot_bo197521221353338581rm_a_b )
= bot_bo197521221353338581rm_a_b ) ).
% relcomp_empty2
thf(fact_985_converse__empty,axiom,
( ( conver7608559976937845093rm_a_b @ bot_bo197521221353338581rm_a_b )
= bot_bo197521221353338581rm_a_b ) ).
% converse_empty
thf(fact_986_trancl__empty,axiom,
( ( transi7922773638565587891rm_a_b @ bot_bo197521221353338581rm_a_b )
= bot_bo197521221353338581rm_a_b ) ).
% trancl_empty
thf(fact_987_acyclic__converse,axiom,
! [R3: set_Pr4386577575007340137rm_a_b] :
( ( transi5314701259734769157rm_a_b @ ( conver7608559976937845093rm_a_b @ R3 ) )
= ( transi5314701259734769157rm_a_b @ R3 ) ) ).
% acyclic_converse
thf(fact_988_SN__on__Image__rtrancl__iff,axiom,
! [R4: set_Pr4386577575007340137rm_a_b,X4: set_term_a_b] :
( ( abstra4720023341729745482rm_a_b @ R4 @ ( image_6923787618150785661rm_a_b @ ( transi7742714808557438673rm_a_b @ R4 ) @ X4 ) )
= ( abstra4720023341729745482rm_a_b @ R4 @ X4 ) ) ).
% SN_on_Image_rtrancl_iff
thf(fact_989_rtrancl__empty,axiom,
( ( transi7742714808557438673rm_a_b @ bot_bo197521221353338581rm_a_b )
= id_term_a_b ) ).
% rtrancl_empty
thf(fact_990_Int__emptyI,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b] :
( ! [X3: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X3 @ A2 )
=> ~ ( member5869715511025134514rm_a_b @ X3 @ B2 ) )
=> ( ( inf_in1693821330460682071rm_a_b @ A2 @ B2 )
= bot_bo197521221353338581rm_a_b ) ) ).
% Int_emptyI
thf(fact_991_disjoint__iff,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b] :
( ( ( inf_in1693821330460682071rm_a_b @ A2 @ B2 )
= bot_bo197521221353338581rm_a_b )
= ( ! [X2: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X2 @ A2 )
=> ~ ( member5869715511025134514rm_a_b @ X2 @ B2 ) ) ) ) ).
% disjoint_iff
thf(fact_992_Un__empty__right,axiom,
! [A2: set_Pr4386577575007340137rm_a_b] :
( ( sup_su6776935440552674877rm_a_b @ A2 @ bot_bo197521221353338581rm_a_b )
= A2 ) ).
% Un_empty_right
thf(fact_993_Un__empty__left,axiom,
! [B2: set_Pr4386577575007340137rm_a_b] :
( ( sup_su6776935440552674877rm_a_b @ bot_bo197521221353338581rm_a_b @ B2 )
= B2 ) ).
% Un_empty_left
thf(fact_994_bot_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
=> ( A = bot_bot_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_995_bot_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
= ( A = bot_bot_nat ) ) ).
% bot.extremum_unique
thf(fact_996_bot_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A ) ).
% bot.extremum
thf(fact_997_boolean__algebra_Odisj__zero__right,axiom,
! [X: set_Pr4386577575007340137rm_a_b] :
( ( sup_su6776935440552674877rm_a_b @ X @ bot_bo197521221353338581rm_a_b )
= X ) ).
% boolean_algebra.disj_zero_right
thf(fact_998_relcomp__Image,axiom,
! [X4: set_Pr4386577575007340137rm_a_b,Y7: set_Pr4386577575007340137rm_a_b,Z6: set_term_a_b] :
( ( image_6923787618150785661rm_a_b @ ( relcom370159955682700863rm_a_b @ X4 @ Y7 ) @ Z6 )
= ( image_6923787618150785661rm_a_b @ Y7 @ ( image_6923787618150785661rm_a_b @ X4 @ Z6 ) ) ) ).
% relcomp_Image
thf(fact_999_rev__ImageI,axiom,
! [A: produc357393685978478089rm_a_b,A2: set_Pr4386577575007340137rm_a_b,B: produc357393685978478089rm_a_b,R3: set_Pr2972776593051762503rm_a_b] :
( ( member5869715511025134514rm_a_b @ A @ A2 )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ R3 )
=> ( member5869715511025134514rm_a_b @ B @ ( image_871640889055226643rm_a_b @ R3 @ A2 ) ) ) ) ).
% rev_ImageI
thf(fact_1000_rev__ImageI,axiom,
! [A: term_a_b,A2: set_term_a_b,B: term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
( ( member_term_a_b @ A @ A2 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R3 )
=> ( member_term_a_b @ B @ ( image_6923787618150785661rm_a_b @ R3 @ A2 ) ) ) ) ).
% rev_ImageI
thf(fact_1001_Image__iff,axiom,
! [B: term_a_b,R3: set_Pr4386577575007340137rm_a_b,A2: set_term_a_b] :
( ( member_term_a_b @ B @ ( image_6923787618150785661rm_a_b @ R3 @ A2 ) )
= ( ? [X2: term_a_b] :
( ( member_term_a_b @ X2 @ A2 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ B ) @ R3 ) ) ) ) ).
% Image_iff
thf(fact_1002_ImageE,axiom,
! [B: produc357393685978478089rm_a_b,R3: set_Pr2972776593051762503rm_a_b,A2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ B @ ( image_871640889055226643rm_a_b @ R3 @ A2 ) )
=> ~ ! [X3: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X3 @ B ) @ R3 )
=> ~ ( member5869715511025134514rm_a_b @ X3 @ A2 ) ) ) ).
% ImageE
thf(fact_1003_ImageE,axiom,
! [B: term_a_b,R3: set_Pr4386577575007340137rm_a_b,A2: set_term_a_b] :
( ( member_term_a_b @ B @ ( image_6923787618150785661rm_a_b @ R3 @ A2 ) )
=> ~ ! [X3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ B ) @ R3 )
=> ~ ( member_term_a_b @ X3 @ A2 ) ) ) ).
% ImageE
thf(fact_1004_ex__in__conv,axiom,
! [A2: set_Pr4386577575007340137rm_a_b] :
( ( ? [X2: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ X2 @ A2 ) )
= ( A2 != bot_bo197521221353338581rm_a_b ) ) ).
% ex_in_conv
thf(fact_1005_equals0I,axiom,
! [A2: set_Pr4386577575007340137rm_a_b] :
( ! [Y2: produc357393685978478089rm_a_b] :
~ ( member5869715511025134514rm_a_b @ Y2 @ A2 )
=> ( A2 = bot_bo197521221353338581rm_a_b ) ) ).
% equals0I
thf(fact_1006_equals0D,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,A: produc357393685978478089rm_a_b] :
( ( A2 = bot_bo197521221353338581rm_a_b )
=> ~ ( member5869715511025134514rm_a_b @ A @ A2 ) ) ).
% equals0D
thf(fact_1007_emptyE,axiom,
! [A: produc357393685978478089rm_a_b] :
~ ( member5869715511025134514rm_a_b @ A @ bot_bo197521221353338581rm_a_b ) ).
% emptyE
thf(fact_1008_Image__Un,axiom,
! [R4: set_Pr2972776593051762503rm_a_b,A2: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b] :
( ( image_871640889055226643rm_a_b @ R4 @ ( sup_su6776935440552674877rm_a_b @ A2 @ B2 ) )
= ( sup_su6776935440552674877rm_a_b @ ( image_871640889055226643rm_a_b @ R4 @ A2 ) @ ( image_871640889055226643rm_a_b @ R4 @ B2 ) ) ) ).
% Image_Un
thf(fact_1009_rtrancl__Image__step,axiom,
! [A: produc357393685978478089rm_a_b,R3: set_Pr2972776593051762503rm_a_b,A2: set_Pr4386577575007340137rm_a_b,B: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ A @ ( image_871640889055226643rm_a_b @ ( transi2615809358984392588rm_a_b @ R3 ) @ A2 ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( transi2615809358984392588rm_a_b @ R3 ) )
=> ( member5869715511025134514rm_a_b @ B @ ( image_871640889055226643rm_a_b @ ( transi2615809358984392588rm_a_b @ R3 ) @ A2 ) ) ) ) ).
% rtrancl_Image_step
thf(fact_1010_rtrancl__Image__step,axiom,
! [A: term_a_b,R3: set_Pr4386577575007340137rm_a_b,A2: set_term_a_b,B: term_a_b] :
( ( member_term_a_b @ A @ ( image_6923787618150785661rm_a_b @ ( transi7742714808557438673rm_a_b @ R3 ) @ A2 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ R3 ) )
=> ( member_term_a_b @ B @ ( image_6923787618150785661rm_a_b @ ( transi7742714808557438673rm_a_b @ R3 ) @ A2 ) ) ) ) ).
% rtrancl_Image_step
thf(fact_1011_Image__closed__trancl,axiom,
! [R3: set_Pr4386577575007340137rm_a_b,X4: set_term_a_b] :
( ( ord_le2705286416250468010rm_a_b @ ( image_6923787618150785661rm_a_b @ R3 @ X4 ) @ X4 )
=> ( ( image_6923787618150785661rm_a_b @ ( transi7742714808557438673rm_a_b @ R3 ) @ X4 )
= X4 ) ) ).
% Image_closed_trancl
thf(fact_1012_Un__Image,axiom,
! [R4: set_Pr4386577575007340137rm_a_b,S6: set_Pr4386577575007340137rm_a_b,A2: set_term_a_b] :
( ( image_6923787618150785661rm_a_b @ ( sup_su6776935440552674877rm_a_b @ R4 @ S6 ) @ A2 )
= ( sup_sup_set_term_a_b @ ( image_6923787618150785661rm_a_b @ R4 @ A2 ) @ ( image_6923787618150785661rm_a_b @ S6 @ A2 ) ) ) ).
% Un_Image
thf(fact_1013_SN__on__Image__rtrancl,axiom,
! [R3: set_Pr4386577575007340137rm_a_b,A2: set_term_a_b] :
( ( abstra4720023341729745482rm_a_b @ R3 @ A2 )
=> ( abstra4720023341729745482rm_a_b @ R3 @ ( image_6923787618150785661rm_a_b @ ( transi7742714808557438673rm_a_b @ R3 ) @ A2 ) ) ) ).
% SN_on_Image_rtrancl
thf(fact_1014_SN__on__O__comm,axiom,
! [R4: set_Pr4386577575007340137rm_a_b,S6: set_Pr4386577575007340137rm_a_b,X4: set_term_a_b] :
( ( abstra4720023341729745482rm_a_b @ ( relcom370159955682700863rm_a_b @ R4 @ S6 ) @ ( image_6923787618150785661rm_a_b @ S6 @ X4 ) )
=> ( abstra4720023341729745482rm_a_b @ ( relcom370159955682700863rm_a_b @ S6 @ R4 ) @ X4 ) ) ).
% SN_on_O_comm
thf(fact_1015_SN__on__Image__normalizable,axiom,
! [R3: set_Pr2972776593051762503rm_a_b,A2: set_Pr4386577575007340137rm_a_b] :
( ( abstra2398554102055911763rm_a_b @ R3 @ A2 )
=> ( ? [X5: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ X5 @ A2 )
=> ? [B3: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ B3 @ ( image_871640889055226643rm_a_b @ ( abstra1496011390211842225rm_a_b @ R3 ) @ A2 ) ) ) ) ).
% SN_on_Image_normalizable
thf(fact_1016_acyclicI,axiom,
! [R3: set_Pr4386577575007340137rm_a_b] :
( ! [X3: term_a_b] :
~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ X3 ) @ ( transi7922773638565587891rm_a_b @ R3 ) )
=> ( transi5314701259734769157rm_a_b @ R3 ) ) ).
% acyclicI
thf(fact_1017_acyclic__def,axiom,
( transi5314701259734769157rm_a_b
= ( ^ [R6: set_Pr4386577575007340137rm_a_b] :
! [X2: term_a_b] :
~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ X2 ) @ ( transi7922773638565587891rm_a_b @ R6 ) ) ) ) ).
% acyclic_def
thf(fact_1018_boolean__algebra_Ocomplement__unique,axiom,
! [A: set_Pr4386577575007340137rm_a_b,X: set_Pr4386577575007340137rm_a_b,Y: set_Pr4386577575007340137rm_a_b] :
( ( ( inf_in1693821330460682071rm_a_b @ A @ X )
= bot_bo197521221353338581rm_a_b )
=> ( ( ( sup_su6776935440552674877rm_a_b @ A @ X )
= top_to1314267278846557113rm_a_b )
=> ( ( ( inf_in1693821330460682071rm_a_b @ A @ Y )
= bot_bo197521221353338581rm_a_b )
=> ( ( ( sup_su6776935440552674877rm_a_b @ A @ Y )
= top_to1314267278846557113rm_a_b )
=> ( X = Y ) ) ) ) ) ).
% boolean_algebra.complement_unique
thf(fact_1019_rtrancl__Un2__separatorE,axiom,
! [B2: set_Pr4386577575007340137rm_a_b,A2: set_Pr4386577575007340137rm_a_b] :
( ( ( relcom370159955682700863rm_a_b @ B2 @ A2 )
= bot_bo197521221353338581rm_a_b )
=> ( ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ A2 @ B2 ) )
= ( sup_su6776935440552674877rm_a_b @ ( transi7742714808557438673rm_a_b @ A2 ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ A2 ) @ ( transi7742714808557438673rm_a_b @ B2 ) ) ) ) ) ).
% rtrancl_Un2_separatorE
thf(fact_1020_trancl__Un2__separatorE,axiom,
! [B2: set_Pr4386577575007340137rm_a_b,A2: set_Pr4386577575007340137rm_a_b] :
( ( ( relcom370159955682700863rm_a_b @ B2 @ A2 )
= bot_bo197521221353338581rm_a_b )
=> ( ( transi7922773638565587891rm_a_b @ ( sup_su6776935440552674877rm_a_b @ A2 @ B2 ) )
= ( sup_su6776935440552674877rm_a_b @ ( sup_su6776935440552674877rm_a_b @ ( transi7922773638565587891rm_a_b @ A2 ) @ ( relcom370159955682700863rm_a_b @ ( transi7922773638565587891rm_a_b @ A2 ) @ ( transi7922773638565587891rm_a_b @ B2 ) ) ) @ ( transi7922773638565587891rm_a_b @ B2 ) ) ) ) ).
% trancl_Un2_separatorE
thf(fact_1021_union__comp__emptyR,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b,C: set_Pr4386577575007340137rm_a_b] :
( ( ( relcom370159955682700863rm_a_b @ A2 @ B2 )
= bot_bo197521221353338581rm_a_b )
=> ( ( ( relcom370159955682700863rm_a_b @ A2 @ C )
= bot_bo197521221353338581rm_a_b )
=> ( ( relcom370159955682700863rm_a_b @ A2 @ ( sup_su6776935440552674877rm_a_b @ B2 @ C ) )
= bot_bo197521221353338581rm_a_b ) ) ) ).
% union_comp_emptyR
thf(fact_1022_union__comp__emptyL,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,C: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b] :
( ( ( relcom370159955682700863rm_a_b @ A2 @ C )
= bot_bo197521221353338581rm_a_b )
=> ( ( ( relcom370159955682700863rm_a_b @ B2 @ C )
= bot_bo197521221353338581rm_a_b )
=> ( ( relcom370159955682700863rm_a_b @ ( sup_su6776935440552674877rm_a_b @ A2 @ B2 ) @ C )
= bot_bo197521221353338581rm_a_b ) ) ) ).
% union_comp_emptyL
thf(fact_1023_bot__empty__eq,axiom,
( bot_bo23760184229683272_a_b_o
= ( ^ [X2: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ X2 @ bot_bo197521221353338581rm_a_b ) ) ) ).
% bot_empty_eq
thf(fact_1024_some__NF,axiom,
! [R3: set_Pr2972776593051762503rm_a_b,X: produc357393685978478089rm_a_b] :
( ( abstra2398554102055911763rm_a_b @ R3 @ ( insert7009541432154983385rm_a_b @ X @ bot_bo197521221353338581rm_a_b ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ ( abstra8203652949708803681rm_a_b @ R3 @ X ) ) @ ( transi2615809358984392588rm_a_b @ R3 ) )
& ( member5869715511025134514rm_a_b @ ( abstra8203652949708803681rm_a_b @ R3 @ X ) @ ( abstra1902783451776770197rm_a_b @ R3 ) ) ) ) ).
% some_NF
thf(fact_1025_some__NF,axiom,
! [R3: set_Pr4386577575007340137rm_a_b,X: term_a_b] :
( ( abstra4720023341729745482rm_a_b @ R3 @ ( insert_term_a_b @ X @ bot_bot_set_term_a_b ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ ( abstra2589370493620291516rm_a_b @ R3 @ X ) ) @ ( transi7742714808557438673rm_a_b @ R3 ) )
& ( member_term_a_b @ ( abstra2589370493620291516rm_a_b @ R3 @ X ) @ ( abstract_NF_term_a_b @ R3 ) ) ) ) ).
% some_NF
thf(fact_1026_subset__emptyI,axiom,
! [A2: set_Pr4386577575007340137rm_a_b] :
( ! [X3: produc357393685978478089rm_a_b] :
~ ( member5869715511025134514rm_a_b @ X3 @ A2 )
=> ( ord_le118470702582115849rm_a_b @ A2 @ bot_bo197521221353338581rm_a_b ) ) ).
% subset_emptyI
thf(fact_1027_CR__on__singletonI,axiom,
! [A: term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
( ! [B3: term_a_b,C4: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B3 ) @ ( transi7742714808557438673rm_a_b @ R3 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C4 ) @ ( transi7742714808557438673rm_a_b @ R3 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B3 @ C4 ) @ ( abstra4096080454567261402rm_a_b @ R3 ) ) ) )
=> ( abstra8448919418672941150rm_a_b @ R3 @ ( insert_term_a_b @ A @ bot_bot_set_term_a_b ) ) ) ).
% CR_on_singletonI
thf(fact_1028_insert__iff,axiom,
! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,A2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ A @ ( insert7009541432154983385rm_a_b @ B @ A2 ) )
= ( ( A = B )
| ( member5869715511025134514rm_a_b @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_1029_insertCI,axiom,
! [A: produc357393685978478089rm_a_b,B2: set_Pr4386577575007340137rm_a_b,B: produc357393685978478089rm_a_b] :
( ( ~ ( member5869715511025134514rm_a_b @ A @ B2 )
=> ( A = B ) )
=> ( member5869715511025134514rm_a_b @ A @ ( insert7009541432154983385rm_a_b @ B @ B2 ) ) ) ).
% insertCI
thf(fact_1030_singletonI,axiom,
! [A: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ A @ ( insert7009541432154983385rm_a_b @ A @ bot_bo197521221353338581rm_a_b ) ) ).
% singletonI
thf(fact_1031_insert__subset,axiom,
! [X: produc357393685978478089rm_a_b,A2: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ ( insert7009541432154983385rm_a_b @ X @ A2 ) @ B2 )
= ( ( member5869715511025134514rm_a_b @ X @ B2 )
& ( ord_le118470702582115849rm_a_b @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_1032_Un__insert__right,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,A: produc357393685978478089rm_a_b,B2: set_Pr4386577575007340137rm_a_b] :
( ( sup_su6776935440552674877rm_a_b @ A2 @ ( insert7009541432154983385rm_a_b @ A @ B2 ) )
= ( insert7009541432154983385rm_a_b @ A @ ( sup_su6776935440552674877rm_a_b @ A2 @ B2 ) ) ) ).
% Un_insert_right
thf(fact_1033_Un__insert__left,axiom,
! [A: produc357393685978478089rm_a_b,B2: set_Pr4386577575007340137rm_a_b,C: set_Pr4386577575007340137rm_a_b] :
( ( sup_su6776935440552674877rm_a_b @ ( insert7009541432154983385rm_a_b @ A @ B2 ) @ C )
= ( insert7009541432154983385rm_a_b @ A @ ( sup_su6776935440552674877rm_a_b @ B2 @ C ) ) ) ).
% Un_insert_left
thf(fact_1034_Int__insert__right__if1,axiom,
! [A: produc357393685978478089rm_a_b,A2: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ A @ A2 )
=> ( ( inf_in1693821330460682071rm_a_b @ A2 @ ( insert7009541432154983385rm_a_b @ A @ B2 ) )
= ( insert7009541432154983385rm_a_b @ A @ ( inf_in1693821330460682071rm_a_b @ A2 @ B2 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_1035_Int__insert__right__if0,axiom,
! [A: produc357393685978478089rm_a_b,A2: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b] :
( ~ ( member5869715511025134514rm_a_b @ A @ A2 )
=> ( ( inf_in1693821330460682071rm_a_b @ A2 @ ( insert7009541432154983385rm_a_b @ A @ B2 ) )
= ( inf_in1693821330460682071rm_a_b @ A2 @ B2 ) ) ) ).
% Int_insert_right_if0
thf(fact_1036_Int__insert__left__if1,axiom,
! [A: produc357393685978478089rm_a_b,C: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ A @ C )
=> ( ( inf_in1693821330460682071rm_a_b @ ( insert7009541432154983385rm_a_b @ A @ B2 ) @ C )
= ( insert7009541432154983385rm_a_b @ A @ ( inf_in1693821330460682071rm_a_b @ B2 @ C ) ) ) ) ).
% Int_insert_left_if1
thf(fact_1037_Int__insert__left__if0,axiom,
! [A: produc357393685978478089rm_a_b,C: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b] :
( ~ ( member5869715511025134514rm_a_b @ A @ C )
=> ( ( inf_in1693821330460682071rm_a_b @ ( insert7009541432154983385rm_a_b @ A @ B2 ) @ C )
= ( inf_in1693821330460682071rm_a_b @ B2 @ C ) ) ) ).
% Int_insert_left_if0
thf(fact_1038_insert__Diff1,axiom,
! [X: produc357393685978478089rm_a_b,B2: set_Pr4386577575007340137rm_a_b,A2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ X @ B2 )
=> ( ( minus_5192120951422937424rm_a_b @ ( insert7009541432154983385rm_a_b @ X @ A2 ) @ B2 )
= ( minus_5192120951422937424rm_a_b @ A2 @ B2 ) ) ) ).
% insert_Diff1
thf(fact_1039_Diff__insert0,axiom,
! [X: produc357393685978478089rm_a_b,A2: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b] :
( ~ ( member5869715511025134514rm_a_b @ X @ A2 )
=> ( ( minus_5192120951422937424rm_a_b @ A2 @ ( insert7009541432154983385rm_a_b @ X @ B2 ) )
= ( minus_5192120951422937424rm_a_b @ A2 @ B2 ) ) ) ).
% Diff_insert0
thf(fact_1040_disjoint__insert_I2_J,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,B: produc357393685978478089rm_a_b,B2: set_Pr4386577575007340137rm_a_b] :
( ( bot_bo197521221353338581rm_a_b
= ( inf_in1693821330460682071rm_a_b @ A2 @ ( insert7009541432154983385rm_a_b @ B @ B2 ) ) )
= ( ~ ( member5869715511025134514rm_a_b @ B @ A2 )
& ( bot_bo197521221353338581rm_a_b
= ( inf_in1693821330460682071rm_a_b @ A2 @ B2 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_1041_disjoint__insert_I1_J,axiom,
! [B2: set_Pr4386577575007340137rm_a_b,A: produc357393685978478089rm_a_b,A2: set_Pr4386577575007340137rm_a_b] :
( ( ( inf_in1693821330460682071rm_a_b @ B2 @ ( insert7009541432154983385rm_a_b @ A @ A2 ) )
= bot_bo197521221353338581rm_a_b )
= ( ~ ( member5869715511025134514rm_a_b @ A @ B2 )
& ( ( inf_in1693821330460682071rm_a_b @ B2 @ A2 )
= bot_bo197521221353338581rm_a_b ) ) ) ).
% disjoint_insert(1)
thf(fact_1042_insert__disjoint_I2_J,axiom,
! [A: produc357393685978478089rm_a_b,A2: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b] :
( ( bot_bo197521221353338581rm_a_b
= ( inf_in1693821330460682071rm_a_b @ ( insert7009541432154983385rm_a_b @ A @ A2 ) @ B2 ) )
= ( ~ ( member5869715511025134514rm_a_b @ A @ B2 )
& ( bot_bo197521221353338581rm_a_b
= ( inf_in1693821330460682071rm_a_b @ A2 @ B2 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_1043_insert__disjoint_I1_J,axiom,
! [A: produc357393685978478089rm_a_b,A2: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b] :
( ( ( inf_in1693821330460682071rm_a_b @ ( insert7009541432154983385rm_a_b @ A @ A2 ) @ B2 )
= bot_bo197521221353338581rm_a_b )
= ( ~ ( member5869715511025134514rm_a_b @ A @ B2 )
& ( ( inf_in1693821330460682071rm_a_b @ A2 @ B2 )
= bot_bo197521221353338581rm_a_b ) ) ) ).
% insert_disjoint(1)
thf(fact_1044_acyclic__insert,axiom,
! [Y: term_a_b,X: term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
( ( transi5314701259734769157rm_a_b @ ( insert7009541432154983385rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X ) @ R3 ) )
= ( ( transi5314701259734769157rm_a_b @ R3 )
& ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ R3 ) ) ) ) ).
% acyclic_insert
thf(fact_1045_Image__singleton__iff,axiom,
! [B: term_a_b,R3: set_Pr4386577575007340137rm_a_b,A: term_a_b] :
( ( member_term_a_b @ B @ ( image_6923787618150785661rm_a_b @ R3 @ ( insert_term_a_b @ A @ bot_bot_set_term_a_b ) ) )
= ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R3 ) ) ).
% Image_singleton_iff
thf(fact_1046_singleton__iff,axiom,
! [B: produc357393685978478089rm_a_b,A: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ B @ ( insert7009541432154983385rm_a_b @ A @ bot_bo197521221353338581rm_a_b ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_1047_singletonD,axiom,
! [B: produc357393685978478089rm_a_b,A: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ B @ ( insert7009541432154983385rm_a_b @ A @ bot_bo197521221353338581rm_a_b ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_1048_insert__is__Un,axiom,
( insert7009541432154983385rm_a_b
= ( ^ [A4: produc357393685978478089rm_a_b] : ( sup_su6776935440552674877rm_a_b @ ( insert7009541432154983385rm_a_b @ A4 @ bot_bo197521221353338581rm_a_b ) ) ) ) ).
% insert_is_Un
thf(fact_1049_Un__singleton__iff,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b,X: produc357393685978478089rm_a_b] :
( ( ( sup_su6776935440552674877rm_a_b @ A2 @ B2 )
= ( insert7009541432154983385rm_a_b @ X @ bot_bo197521221353338581rm_a_b ) )
= ( ( ( A2 = bot_bo197521221353338581rm_a_b )
& ( B2
= ( insert7009541432154983385rm_a_b @ X @ bot_bo197521221353338581rm_a_b ) ) )
| ( ( A2
= ( insert7009541432154983385rm_a_b @ X @ bot_bo197521221353338581rm_a_b ) )
& ( B2 = bot_bo197521221353338581rm_a_b ) )
| ( ( A2
= ( insert7009541432154983385rm_a_b @ X @ bot_bo197521221353338581rm_a_b ) )
& ( B2
= ( insert7009541432154983385rm_a_b @ X @ bot_bo197521221353338581rm_a_b ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_1050_singleton__Un__iff,axiom,
! [X: produc357393685978478089rm_a_b,A2: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b] :
( ( ( insert7009541432154983385rm_a_b @ X @ bot_bo197521221353338581rm_a_b )
= ( sup_su6776935440552674877rm_a_b @ A2 @ B2 ) )
= ( ( ( A2 = bot_bo197521221353338581rm_a_b )
& ( B2
= ( insert7009541432154983385rm_a_b @ X @ bot_bo197521221353338581rm_a_b ) ) )
| ( ( A2
= ( insert7009541432154983385rm_a_b @ X @ bot_bo197521221353338581rm_a_b ) )
& ( B2 = bot_bo197521221353338581rm_a_b ) )
| ( ( A2
= ( insert7009541432154983385rm_a_b @ X @ bot_bo197521221353338581rm_a_b ) )
& ( B2
= ( insert7009541432154983385rm_a_b @ X @ bot_bo197521221353338581rm_a_b ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_1051_Diff__insert__absorb,axiom,
! [X: produc357393685978478089rm_a_b,A2: set_Pr4386577575007340137rm_a_b] :
( ~ ( member5869715511025134514rm_a_b @ X @ A2 )
=> ( ( minus_5192120951422937424rm_a_b @ ( insert7009541432154983385rm_a_b @ X @ A2 ) @ ( insert7009541432154983385rm_a_b @ X @ bot_bo197521221353338581rm_a_b ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_1052_insert__Diff,axiom,
! [A: produc357393685978478089rm_a_b,A2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ A @ A2 )
=> ( ( insert7009541432154983385rm_a_b @ A @ ( minus_5192120951422937424rm_a_b @ A2 @ ( insert7009541432154983385rm_a_b @ A @ bot_bo197521221353338581rm_a_b ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_1053_subset__Diff__insert,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b,X: produc357393685978478089rm_a_b,C: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A2 @ ( minus_5192120951422937424rm_a_b @ B2 @ ( insert7009541432154983385rm_a_b @ X @ C ) ) )
= ( ( ord_le118470702582115849rm_a_b @ A2 @ ( minus_5192120951422937424rm_a_b @ B2 @ C ) )
& ~ ( member5869715511025134514rm_a_b @ X @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_1054_insert__subsetI,axiom,
! [X: produc357393685978478089rm_a_b,A2: set_Pr4386577575007340137rm_a_b,X4: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ X @ A2 )
=> ( ( ord_le118470702582115849rm_a_b @ X4 @ A2 )
=> ( ord_le118470702582115849rm_a_b @ ( insert7009541432154983385rm_a_b @ X @ X4 ) @ A2 ) ) ) ).
% insert_subsetI
thf(fact_1055_subset__insert,axiom,
! [X: produc357393685978478089rm_a_b,A2: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b] :
( ~ ( member5869715511025134514rm_a_b @ X @ A2 )
=> ( ( ord_le118470702582115849rm_a_b @ A2 @ ( insert7009541432154983385rm_a_b @ X @ B2 ) )
= ( ord_le118470702582115849rm_a_b @ A2 @ B2 ) ) ) ).
% subset_insert
thf(fact_1056_mk__disjoint__insert,axiom,
! [A: produc357393685978478089rm_a_b,A2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ A @ A2 )
=> ? [B9: set_Pr4386577575007340137rm_a_b] :
( ( A2
= ( insert7009541432154983385rm_a_b @ A @ B9 ) )
& ~ ( member5869715511025134514rm_a_b @ A @ B9 ) ) ) ).
% mk_disjoint_insert
thf(fact_1057_insert__eq__iff,axiom,
! [A: produc357393685978478089rm_a_b,A2: set_Pr4386577575007340137rm_a_b,B: produc357393685978478089rm_a_b,B2: set_Pr4386577575007340137rm_a_b] :
( ~ ( member5869715511025134514rm_a_b @ A @ A2 )
=> ( ~ ( member5869715511025134514rm_a_b @ B @ B2 )
=> ( ( ( insert7009541432154983385rm_a_b @ A @ A2 )
= ( insert7009541432154983385rm_a_b @ B @ B2 ) )
= ( ( ( A = B )
=> ( A2 = B2 ) )
& ( ( A != B )
=> ? [C5: set_Pr4386577575007340137rm_a_b] :
( ( A2
= ( insert7009541432154983385rm_a_b @ B @ C5 ) )
& ~ ( member5869715511025134514rm_a_b @ B @ C5 )
& ( B2
= ( insert7009541432154983385rm_a_b @ A @ C5 ) )
& ~ ( member5869715511025134514rm_a_b @ A @ C5 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_1058_insert__absorb,axiom,
! [A: produc357393685978478089rm_a_b,A2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ A @ A2 )
=> ( ( insert7009541432154983385rm_a_b @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_1059_insert__ident,axiom,
! [X: produc357393685978478089rm_a_b,A2: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b] :
( ~ ( member5869715511025134514rm_a_b @ X @ A2 )
=> ( ~ ( member5869715511025134514rm_a_b @ X @ B2 )
=> ( ( ( insert7009541432154983385rm_a_b @ X @ A2 )
= ( insert7009541432154983385rm_a_b @ X @ B2 ) )
= ( A2 = B2 ) ) ) ) ).
% insert_ident
thf(fact_1060_Set_Oset__insert,axiom,
! [X: produc357393685978478089rm_a_b,A2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ X @ A2 )
=> ~ ! [B9: set_Pr4386577575007340137rm_a_b] :
( ( A2
= ( insert7009541432154983385rm_a_b @ X @ B9 ) )
=> ( member5869715511025134514rm_a_b @ X @ B9 ) ) ) ).
% Set.set_insert
thf(fact_1061_insertI2,axiom,
! [A: produc357393685978478089rm_a_b,B2: set_Pr4386577575007340137rm_a_b,B: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ A @ B2 )
=> ( member5869715511025134514rm_a_b @ A @ ( insert7009541432154983385rm_a_b @ B @ B2 ) ) ) ).
% insertI2
thf(fact_1062_insertI1,axiom,
! [A: produc357393685978478089rm_a_b,B2: set_Pr4386577575007340137rm_a_b] : ( member5869715511025134514rm_a_b @ A @ ( insert7009541432154983385rm_a_b @ A @ B2 ) ) ).
% insertI1
thf(fact_1063_insertE,axiom,
! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,A2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ A @ ( insert7009541432154983385rm_a_b @ B @ A2 ) )
=> ( ( A != B )
=> ( member5869715511025134514rm_a_b @ A @ A2 ) ) ) ).
% insertE
thf(fact_1064_insert__Diff__if,axiom,
! [X: produc357393685978478089rm_a_b,B2: set_Pr4386577575007340137rm_a_b,A2: set_Pr4386577575007340137rm_a_b] :
( ( ( member5869715511025134514rm_a_b @ X @ B2 )
=> ( ( minus_5192120951422937424rm_a_b @ ( insert7009541432154983385rm_a_b @ X @ A2 ) @ B2 )
= ( minus_5192120951422937424rm_a_b @ A2 @ B2 ) ) )
& ( ~ ( member5869715511025134514rm_a_b @ X @ B2 )
=> ( ( minus_5192120951422937424rm_a_b @ ( insert7009541432154983385rm_a_b @ X @ A2 ) @ B2 )
= ( insert7009541432154983385rm_a_b @ X @ ( minus_5192120951422937424rm_a_b @ A2 @ B2 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_1065_Int__insert__right,axiom,
! [A: produc357393685978478089rm_a_b,A2: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b] :
( ( ( member5869715511025134514rm_a_b @ A @ A2 )
=> ( ( inf_in1693821330460682071rm_a_b @ A2 @ ( insert7009541432154983385rm_a_b @ A @ B2 ) )
= ( insert7009541432154983385rm_a_b @ A @ ( inf_in1693821330460682071rm_a_b @ A2 @ B2 ) ) ) )
& ( ~ ( member5869715511025134514rm_a_b @ A @ A2 )
=> ( ( inf_in1693821330460682071rm_a_b @ A2 @ ( insert7009541432154983385rm_a_b @ A @ B2 ) )
= ( inf_in1693821330460682071rm_a_b @ A2 @ B2 ) ) ) ) ).
% Int_insert_right
thf(fact_1066_Int__insert__left,axiom,
! [A: produc357393685978478089rm_a_b,C: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b] :
( ( ( member5869715511025134514rm_a_b @ A @ C )
=> ( ( inf_in1693821330460682071rm_a_b @ ( insert7009541432154983385rm_a_b @ A @ B2 ) @ C )
= ( insert7009541432154983385rm_a_b @ A @ ( inf_in1693821330460682071rm_a_b @ B2 @ C ) ) ) )
& ( ~ ( member5869715511025134514rm_a_b @ A @ C )
=> ( ( inf_in1693821330460682071rm_a_b @ ( insert7009541432154983385rm_a_b @ A @ B2 ) @ C )
= ( inf_in1693821330460682071rm_a_b @ B2 @ C ) ) ) ) ).
% Int_insert_left
thf(fact_1067_subset__insert__iff,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,X: produc357393685978478089rm_a_b,B2: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A2 @ ( insert7009541432154983385rm_a_b @ X @ B2 ) )
= ( ( ( member5869715511025134514rm_a_b @ X @ A2 )
=> ( ord_le118470702582115849rm_a_b @ ( minus_5192120951422937424rm_a_b @ A2 @ ( insert7009541432154983385rm_a_b @ X @ bot_bo197521221353338581rm_a_b ) ) @ B2 ) )
& ( ~ ( member5869715511025134514rm_a_b @ X @ A2 )
=> ( ord_le118470702582115849rm_a_b @ A2 @ B2 ) ) ) ) ).
% subset_insert_iff
thf(fact_1068_SN__on__all__reducts__SN__on__conv,axiom,
! [R3: set_Pr4386577575007340137rm_a_b,A: term_a_b] :
( ( abstra4720023341729745482rm_a_b @ R3 @ ( insert_term_a_b @ A @ bot_bot_set_term_a_b ) )
= ( ! [B4: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B4 ) @ R3 )
=> ( abstra4720023341729745482rm_a_b @ R3 @ ( insert_term_a_b @ B4 @ bot_bot_set_term_a_b ) ) ) ) ) ).
% SN_on_all_reducts_SN_on_conv
thf(fact_1069_lower__set__imp__not__SN__on,axiom,
! [S: produc357393685978478089rm_a_b,X4: set_Pr4386577575007340137rm_a_b,R4: set_Pr2972776593051762503rm_a_b] :
( ( member5869715511025134514rm_a_b @ S @ X4 )
=> ( ! [X3: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X3 @ X4 )
=> ? [Xa2: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ Xa2 @ X4 )
& ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X3 @ Xa2 ) @ R4 ) ) )
=> ~ ( abstra2398554102055911763rm_a_b @ R4 @ ( insert7009541432154983385rm_a_b @ S @ bot_bo197521221353338581rm_a_b ) ) ) ) ).
% lower_set_imp_not_SN_on
thf(fact_1070_lower__set__imp__not__SN__on,axiom,
! [S: term_a_b,X4: set_term_a_b,R4: set_Pr4386577575007340137rm_a_b] :
( ( member_term_a_b @ S @ X4 )
=> ( ! [X3: term_a_b] :
( ( member_term_a_b @ X3 @ X4 )
=> ? [Xa2: term_a_b] :
( ( member_term_a_b @ Xa2 @ X4 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Xa2 ) @ R4 ) ) )
=> ~ ( abstra4720023341729745482rm_a_b @ R4 @ ( insert_term_a_b @ S @ bot_bot_set_term_a_b ) ) ) ) ).
% lower_set_imp_not_SN_on
thf(fact_1071_SN__on__induct__acc__style,axiom,
! [R4: set_Pr4386577575007340137rm_a_b,A: term_a_b,P2: term_a_b > $o] :
( ( abstra4720023341729745482rm_a_b @ R4 @ ( insert_term_a_b @ A @ bot_bot_set_term_a_b ) )
=> ( ! [X3: term_a_b] :
( ( abstra4720023341729745482rm_a_b @ R4 @ ( insert_term_a_b @ X3 @ bot_bot_set_term_a_b ) )
=> ( ! [Y6: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y6 ) @ R4 )
=> ( P2 @ Y6 ) )
=> ( P2 @ X3 ) ) )
=> ( P2 @ A ) ) ) ).
% SN_on_induct_acc_style
thf(fact_1072_step__preserves__SN__on,axiom,
! [A: term_a_b,B: term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R3 )
=> ( ( abstra4720023341729745482rm_a_b @ R3 @ ( insert_term_a_b @ A @ bot_bot_set_term_a_b ) )
=> ( abstra4720023341729745482rm_a_b @ R3 @ ( insert_term_a_b @ B @ bot_bot_set_term_a_b ) ) ) ) ).
% step_preserves_SN_on
thf(fact_1073_SN__on__imp__on__minimal,axiom,
! [R3: set_Pr2972776593051762503rm_a_b,X: produc357393685978478089rm_a_b] :
( ( abstra2398554102055911763rm_a_b @ R3 @ ( insert7009541432154983385rm_a_b @ X @ bot_bo197521221353338581rm_a_b ) )
=> ! [Q2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ X @ Q2 )
=> ? [X3: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X3 @ Q2 )
& ! [Y6: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X3 @ Y6 ) @ R3 )
=> ~ ( member5869715511025134514rm_a_b @ Y6 @ Q2 ) ) ) ) ) ).
% SN_on_imp_on_minimal
thf(fact_1074_SN__on__imp__on__minimal,axiom,
! [R3: set_Pr4386577575007340137rm_a_b,X: term_a_b] :
( ( abstra4720023341729745482rm_a_b @ R3 @ ( insert_term_a_b @ X @ bot_bot_set_term_a_b ) )
=> ! [Q2: set_term_a_b] :
( ( member_term_a_b @ X @ Q2 )
=> ? [X3: term_a_b] :
( ( member_term_a_b @ X3 @ Q2 )
& ! [Y6: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y6 ) @ R3 )
=> ~ ( member_term_a_b @ Y6 @ Q2 ) ) ) ) ) ).
% SN_on_imp_on_minimal
thf(fact_1075_step__reflects__SN__on,axiom,
! [A: term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
( ! [B3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B3 ) @ R3 )
=> ( abstra4720023341729745482rm_a_b @ R3 @ ( insert_term_a_b @ B3 @ bot_bot_set_term_a_b ) ) )
=> ( abstra4720023341729745482rm_a_b @ R3 @ ( insert_term_a_b @ A @ bot_bot_set_term_a_b ) ) ) ).
% step_reflects_SN_on
thf(fact_1076_SN__on__induct_H,axiom,
! [R4: set_Pr4386577575007340137rm_a_b,S: term_a_b,P2: term_a_b > $o] :
( ( abstra4720023341729745482rm_a_b @ R4 @ ( insert_term_a_b @ S @ bot_bot_set_term_a_b ) )
=> ( ! [T2: term_a_b] :
( ! [U4: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T2 @ U4 ) @ R4 )
=> ( P2 @ U4 ) )
=> ( P2 @ T2 ) )
=> ( P2 @ S ) ) ) ).
% SN_on_induct'
thf(fact_1077_ssubst__Pair__rhs,axiom,
! [R3: term_a_b,S: term_a_b,R4: set_Pr4386577575007340137rm_a_b,S7: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ R3 @ S ) @ R4 )
=> ( ( S7 = S )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ R3 @ S7 ) @ R4 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_1078_NF__imp__SN__on,axiom,
! [A: produc357393685978478089rm_a_b,R4: set_Pr2972776593051762503rm_a_b] :
( ( member5869715511025134514rm_a_b @ A @ ( abstra1902783451776770197rm_a_b @ R4 ) )
=> ( abstra2398554102055911763rm_a_b @ R4 @ ( insert7009541432154983385rm_a_b @ A @ bot_bo197521221353338581rm_a_b ) ) ) ).
% NF_imp_SN_on
thf(fact_1079_no__step,axiom,
! [A2: set_Pr2972776593051762503rm_a_b,A: produc357393685978478089rm_a_b] :
( ( ( image_871640889055226643rm_a_b @ A2 @ ( insert7009541432154983385rm_a_b @ A @ bot_bo197521221353338581rm_a_b ) )
= bot_bo197521221353338581rm_a_b )
=> ( member5869715511025134514rm_a_b @ A @ ( abstra1902783451776770197rm_a_b @ A2 ) ) ) ).
% no_step
thf(fact_1080_steps__preserve__SN__on,axiom,
! [A: term_a_b,B: term_a_b,A2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ A2 ) )
=> ( ( abstra4720023341729745482rm_a_b @ A2 @ ( insert_term_a_b @ A @ bot_bot_set_term_a_b ) )
=> ( abstra4720023341729745482rm_a_b @ A2 @ ( insert_term_a_b @ B @ bot_bot_set_term_a_b ) ) ) ) ).
% steps_preserve_SN_on
thf(fact_1081_steps__reflect__SN__on,axiom,
! [R3: set_Pr4386577575007340137rm_a_b,B: term_a_b,A: term_a_b] :
( ~ ( abstra4720023341729745482rm_a_b @ R3 @ ( insert_term_a_b @ B @ bot_bot_set_term_a_b ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ R3 ) )
=> ~ ( abstra4720023341729745482rm_a_b @ R3 @ ( insert_term_a_b @ A @ bot_bot_set_term_a_b ) ) ) ) ).
% steps_reflect_SN_on
thf(fact_1082_SN__on__imp__normalizability,axiom,
! [R3: set_Pr4386577575007340137rm_a_b,A: term_a_b] :
( ( abstra4720023341729745482rm_a_b @ R3 @ ( insert_term_a_b @ A @ bot_bot_set_term_a_b ) )
=> ? [B3: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B3 ) @ ( abstra2972651231392952684rm_a_b @ R3 ) ) ) ).
% SN_on_imp_normalizability
thf(fact_1083_SN__reaches__NF,axiom,
! [R3: set_Pr2972776593051762503rm_a_b,X: produc357393685978478089rm_a_b] :
( ( abstra2398554102055911763rm_a_b @ R3 @ ( insert7009541432154983385rm_a_b @ X @ bot_bo197521221353338581rm_a_b ) )
=> ? [Y2: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y2 ) @ ( transi2615809358984392588rm_a_b @ R3 ) )
& ( member5869715511025134514rm_a_b @ Y2 @ ( abstra1902783451776770197rm_a_b @ R3 ) ) ) ) ).
% SN_reaches_NF
thf(fact_1084_SN__reaches__NF,axiom,
! [R3: set_Pr4386577575007340137rm_a_b,X: term_a_b] :
( ( abstra4720023341729745482rm_a_b @ R3 @ ( insert_term_a_b @ X @ bot_bot_set_term_a_b ) )
=> ? [Y2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y2 ) @ ( transi7742714808557438673rm_a_b @ R3 ) )
& ( member_term_a_b @ Y2 @ ( abstract_NF_term_a_b @ R3 ) ) ) ) ).
% SN_reaches_NF
thf(fact_1085_min__ext__compat,axiom,
! [R4: set_Pr4386577575007340137rm_a_b,S6: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ R4 @ S6 ) @ R4 )
=> ( ord_le2663425029446080117rm_a_b @ ( relcom3320975603336820193rm_a_b @ ( min_ext_term_a_b @ R4 ) @ ( sup_su1246761269818691241rm_a_b @ ( min_ext_term_a_b @ S6 ) @ ( insert807748097412076101rm_a_b @ ( produc8517751122659631533rm_a_b @ bot_bot_set_term_a_b @ bot_bot_set_term_a_b ) @ bot_bo6544111711586975553rm_a_b ) ) ) @ ( min_ext_term_a_b @ R4 ) ) ) ).
% min_ext_compat
thf(fact_1086_max__ext__compat,axiom,
! [R4: set_Pr4386577575007340137rm_a_b,S6: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ R4 @ S6 ) @ R4 )
=> ( ord_le2663425029446080117rm_a_b @ ( relcom3320975603336820193rm_a_b @ ( max_ext_term_a_b @ R4 ) @ ( sup_su1246761269818691241rm_a_b @ ( max_ext_term_a_b @ S6 ) @ ( insert807748097412076101rm_a_b @ ( produc8517751122659631533rm_a_b @ bot_bot_set_term_a_b @ bot_bot_set_term_a_b ) @ bot_bo6544111711586975553rm_a_b ) ) ) @ ( max_ext_term_a_b @ R4 ) ) ) ).
% max_ext_compat
thf(fact_1087_is__singletonI_H,axiom,
! [A2: set_Pr4386577575007340137rm_a_b] :
( ( A2 != bot_bo197521221353338581rm_a_b )
=> ( ! [X3: produc357393685978478089rm_a_b,Y2: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X3 @ A2 )
=> ( ( member5869715511025134514rm_a_b @ Y2 @ A2 )
=> ( X3 = Y2 ) ) )
=> ( is_sin332545968047684349rm_a_b @ A2 ) ) ) ).
% is_singletonI'
thf(fact_1088_max__ext__additive,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b,R4: set_Pr2972776593051762503rm_a_b,C: set_Pr4386577575007340137rm_a_b,D2: set_Pr4386577575007340137rm_a_b] :
( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ A2 @ B2 ) @ ( max_ex2405036269911770436rm_a_b @ R4 ) )
=> ( ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ C @ D2 ) @ ( max_ex2405036269911770436rm_a_b @ R4 ) )
=> ( member1375104729371992464rm_a_b @ ( produc3977138704197412567rm_a_b @ ( sup_su6776935440552674877rm_a_b @ A2 @ C ) @ ( sup_su6776935440552674877rm_a_b @ B2 @ D2 ) ) @ ( max_ex2405036269911770436rm_a_b @ R4 ) ) ) ) ).
% max_ext_additive
thf(fact_1089_Field__insert,axiom,
! [A: term_a_b,B: term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
( ( field_term_a_b @ ( insert7009541432154983385rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R3 ) )
= ( sup_sup_set_term_a_b @ ( insert_term_a_b @ A @ ( insert_term_a_b @ B @ bot_bot_set_term_a_b ) ) @ ( field_term_a_b @ R3 ) ) ) ).
% Field_insert
thf(fact_1090_Field__insert,axiom,
! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,R3: set_Pr2972776593051762503rm_a_b] :
( ( field_6884932134483023318rm_a_b @ ( insert3577861631138384951rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ R3 ) )
= ( sup_su6776935440552674877rm_a_b @ ( insert7009541432154983385rm_a_b @ A @ ( insert7009541432154983385rm_a_b @ B @ bot_bo197521221353338581rm_a_b ) ) @ ( field_6884932134483023318rm_a_b @ R3 ) ) ) ).
% Field_insert
thf(fact_1091_Field__converse,axiom,
! [R3: set_Pr4386577575007340137rm_a_b] :
( ( field_term_a_b @ ( conver7608559976937845093rm_a_b @ R3 ) )
= ( field_term_a_b @ R3 ) ) ).
% Field_converse
thf(fact_1092_Field__Un,axiom,
! [R3: set_Pr2972776593051762503rm_a_b,S: set_Pr2972776593051762503rm_a_b] :
( ( field_6884932134483023318rm_a_b @ ( sup_su1845815533690797339rm_a_b @ R3 @ S ) )
= ( sup_su6776935440552674877rm_a_b @ ( field_6884932134483023318rm_a_b @ R3 ) @ ( field_6884932134483023318rm_a_b @ S ) ) ) ).
% Field_Un
thf(fact_1093_Field__Un,axiom,
! [R3: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
( ( field_term_a_b @ ( sup_su6776935440552674877rm_a_b @ R3 @ S ) )
= ( sup_sup_set_term_a_b @ ( field_term_a_b @ R3 ) @ ( field_term_a_b @ S ) ) ) ).
% Field_Un
thf(fact_1094_FieldI2,axiom,
! [I: produc357393685978478089rm_a_b,J: produc357393685978478089rm_a_b,R4: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ I @ J ) @ R4 )
=> ( member5869715511025134514rm_a_b @ J @ ( field_6884932134483023318rm_a_b @ R4 ) ) ) ).
% FieldI2
thf(fact_1095_FieldI2,axiom,
! [I: term_a_b,J: term_a_b,R4: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ I @ J ) @ R4 )
=> ( member_term_a_b @ J @ ( field_term_a_b @ R4 ) ) ) ).
% FieldI2
thf(fact_1096_FieldI1,axiom,
! [I: produc357393685978478089rm_a_b,J: produc357393685978478089rm_a_b,R4: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ I @ J ) @ R4 )
=> ( member5869715511025134514rm_a_b @ I @ ( field_6884932134483023318rm_a_b @ R4 ) ) ) ).
% FieldI1
thf(fact_1097_FieldI1,axiom,
! [I: term_a_b,J: term_a_b,R4: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ I @ J ) @ R4 )
=> ( member_term_a_b @ I @ ( field_term_a_b @ R4 ) ) ) ).
% FieldI1
thf(fact_1098_cofinal__def,axiom,
( bNF_Ca4789820777293777539rm_a_b
= ( ^ [A6: set_term_a_b,R6: set_Pr4386577575007340137rm_a_b] :
! [X2: term_a_b] :
( ( member_term_a_b @ X2 @ ( field_term_a_b @ R6 ) )
=> ? [Y3: term_a_b] :
( ( member_term_a_b @ Y3 @ A6 )
& ( X2 != Y3 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y3 ) @ R6 ) ) ) ) ) ).
% cofinal_def
thf(fact_1099_subset__Image1__Image1__iff,axiom,
! [R3: set_Pr2972776593051762503rm_a_b,A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b] :
( ( order_4024118566430455712rm_a_b @ ( field_6884932134483023318rm_a_b @ R3 ) @ R3 )
=> ( ( member5869715511025134514rm_a_b @ A @ ( field_6884932134483023318rm_a_b @ R3 ) )
=> ( ( member5869715511025134514rm_a_b @ B @ ( field_6884932134483023318rm_a_b @ R3 ) )
=> ( ( ord_le118470702582115849rm_a_b @ ( image_871640889055226643rm_a_b @ R3 @ ( insert7009541432154983385rm_a_b @ A @ bot_bo197521221353338581rm_a_b ) ) @ ( image_871640889055226643rm_a_b @ R3 @ ( insert7009541432154983385rm_a_b @ B @ bot_bo197521221353338581rm_a_b ) ) )
= ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B @ A ) @ R3 ) ) ) ) ) ).
% subset_Image1_Image1_iff
thf(fact_1100_subset__Image1__Image1__iff,axiom,
! [R3: set_Pr4386577575007340137rm_a_b,A: term_a_b,B: term_a_b] :
( ( order_7053473298986962365rm_a_b @ ( field_term_a_b @ R3 ) @ R3 )
=> ( ( member_term_a_b @ A @ ( field_term_a_b @ R3 ) )
=> ( ( member_term_a_b @ B @ ( field_term_a_b @ R3 ) )
=> ( ( ord_le2705286416250468010rm_a_b @ ( image_6923787618150785661rm_a_b @ R3 @ ( insert_term_a_b @ A @ bot_bot_set_term_a_b ) ) @ ( image_6923787618150785661rm_a_b @ R3 @ ( insert_term_a_b @ B @ bot_bot_set_term_a_b ) ) )
= ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ R3 ) ) ) ) ) ).
% subset_Image1_Image1_iff
thf(fact_1101_preorder__on__converse,axiom,
! [A2: set_term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
( ( order_7053473298986962365rm_a_b @ A2 @ ( conver7608559976937845093rm_a_b @ R3 ) )
= ( order_7053473298986962365rm_a_b @ A2 @ R3 ) ) ).
% preorder_on_converse
thf(fact_1102_subset__Image__Image__iff,axiom,
! [R3: set_Pr4386577575007340137rm_a_b,A2: set_term_a_b,B2: set_term_a_b] :
( ( order_7053473298986962365rm_a_b @ ( field_term_a_b @ R3 ) @ R3 )
=> ( ( ord_le2705286416250468010rm_a_b @ A2 @ ( field_term_a_b @ R3 ) )
=> ( ( ord_le2705286416250468010rm_a_b @ B2 @ ( field_term_a_b @ R3 ) )
=> ( ( ord_le2705286416250468010rm_a_b @ ( image_6923787618150785661rm_a_b @ R3 @ A2 ) @ ( image_6923787618150785661rm_a_b @ R3 @ B2 ) )
= ( ! [X2: term_a_b] :
( ( member_term_a_b @ X2 @ A2 )
=> ? [Y3: term_a_b] :
( ( member_term_a_b @ Y3 @ B2 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ X2 ) @ R3 ) ) ) ) ) ) ) ) ).
% subset_Image_Image_iff
thf(fact_1103_Total__subset__Id,axiom,
! [R3: set_Pr4386577575007340137rm_a_b] :
( ( total_on_term_a_b @ ( field_term_a_b @ R3 ) @ R3 )
=> ( ( ord_le118470702582115849rm_a_b @ R3 @ id_term_a_b )
=> ( ( R3 = bot_bo197521221353338581rm_a_b )
| ? [A3: term_a_b] :
( R3
= ( insert7009541432154983385rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ A3 ) @ bot_bo197521221353338581rm_a_b ) ) ) ) ) ).
% Total_subset_Id
thf(fact_1104_linear__order__on__singleton,axiom,
! [X: term_a_b] : ( order_5388802246213473311rm_a_b @ ( insert_term_a_b @ X @ bot_bot_set_term_a_b ) @ ( insert7009541432154983385rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ X ) @ bot_bo197521221353338581rm_a_b ) ) ).
% linear_order_on_singleton
thf(fact_1105_total__on__converse,axiom,
! [A2: set_term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
( ( total_on_term_a_b @ A2 @ ( conver7608559976937845093rm_a_b @ R3 ) )
= ( total_on_term_a_b @ A2 @ R3 ) ) ).
% total_on_converse
thf(fact_1106_linear__order__on__converse,axiom,
! [A2: set_term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
( ( order_5388802246213473311rm_a_b @ A2 @ ( conver7608559976937845093rm_a_b @ R3 ) )
= ( order_5388802246213473311rm_a_b @ A2 @ R3 ) ) ).
% linear_order_on_converse
thf(fact_1107_total__on__trancl,axiom,
! [A2: set_term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
( ( total_on_term_a_b @ A2 @ R3 )
=> ( total_on_term_a_b @ A2 @ ( transi7922773638565587891rm_a_b @ R3 ) ) ) ).
% total_on_trancl
thf(fact_1108_totalI,axiom,
! [R3: set_Pr4386577575007340137rm_a_b] :
( ! [X3: term_a_b,Y2: term_a_b] :
( ( X3 != Y2 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y2 ) @ R3 )
| ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y2 @ X3 ) @ R3 ) ) )
=> ( total_on_term_a_b @ top_top_set_term_a_b @ R3 ) ) ).
% totalI
thf(fact_1109_total__on__def,axiom,
( total_on_term_a_b
= ( ^ [A6: set_term_a_b,R6: set_Pr4386577575007340137rm_a_b] :
! [X2: term_a_b] :
( ( member_term_a_b @ X2 @ A6 )
=> ! [Y3: term_a_b] :
( ( member_term_a_b @ Y3 @ A6 )
=> ( ( X2 != Y3 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y3 ) @ R6 )
| ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ X2 ) @ R6 ) ) ) ) ) ) ) ).
% total_on_def
thf(fact_1110_total__onI,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,R3: set_Pr2972776593051762503rm_a_b] :
( ! [X3: produc357393685978478089rm_a_b,Y2: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X3 @ A2 )
=> ( ( member5869715511025134514rm_a_b @ Y2 @ A2 )
=> ( ( X3 != Y2 )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X3 @ Y2 ) @ R3 )
| ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y2 @ X3 ) @ R3 ) ) ) ) )
=> ( total_4010809703584253837rm_a_b @ A2 @ R3 ) ) ).
% total_onI
thf(fact_1111_total__onI,axiom,
! [A2: set_term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
( ! [X3: term_a_b,Y2: term_a_b] :
( ( member_term_a_b @ X3 @ A2 )
=> ( ( member_term_a_b @ Y2 @ A2 )
=> ( ( X3 != Y2 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y2 ) @ R3 )
| ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y2 @ X3 ) @ R3 ) ) ) ) )
=> ( total_on_term_a_b @ A2 @ R3 ) ) ).
% total_onI
thf(fact_1112_Linear__order__in__diff__Id,axiom,
! [R3: set_Pr2972776593051762503rm_a_b,A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b] :
( ( order_681589137112254398rm_a_b @ ( field_6884932134483023318rm_a_b @ R3 ) @ R3 )
=> ( ( member5869715511025134514rm_a_b @ A @ ( field_6884932134483023318rm_a_b @ R3 ) )
=> ( ( member5869715511025134514rm_a_b @ B @ ( field_6884932134483023318rm_a_b @ R3 ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ R3 )
= ( ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B @ A ) @ ( minus_45395518312058030rm_a_b @ R3 @ id_Pro3488709451141449061rm_a_b ) ) ) ) ) ) ) ).
% Linear_order_in_diff_Id
thf(fact_1113_Linear__order__in__diff__Id,axiom,
! [R3: set_Pr4386577575007340137rm_a_b,A: term_a_b,B: term_a_b] :
( ( order_5388802246213473311rm_a_b @ ( field_term_a_b @ R3 ) @ R3 )
=> ( ( member_term_a_b @ A @ ( field_term_a_b @ R3 ) )
=> ( ( member_term_a_b @ B @ ( field_term_a_b @ R3 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R3 )
= ( ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ ( minus_5192120951422937424rm_a_b @ R3 @ id_term_a_b ) ) ) ) ) ) ) ).
% Linear_order_in_diff_Id
thf(fact_1114_Linear__order__wf__diff__Id,axiom,
! [R3: set_Pr4386577575007340137rm_a_b] :
( ( order_5388802246213473311rm_a_b @ ( field_term_a_b @ R3 ) @ R3 )
=> ( ( wf_term_a_b @ ( minus_5192120951422937424rm_a_b @ R3 @ id_term_a_b ) )
= ( ! [A6: set_term_a_b] :
( ( ord_le2705286416250468010rm_a_b @ A6 @ ( field_term_a_b @ R3 ) )
=> ( ( A6 != bot_bot_set_term_a_b )
=> ? [X2: term_a_b] :
( ( member_term_a_b @ X2 @ A6 )
& ! [Y3: term_a_b] :
( ( member_term_a_b @ Y3 @ A6 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y3 ) @ R3 ) ) ) ) ) ) ) ) ).
% Linear_order_wf_diff_Id
thf(fact_1115_wf__insert,axiom,
! [Y: term_a_b,X: term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
( ( wf_term_a_b @ ( insert7009541432154983385rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X ) @ R3 ) )
= ( ( wf_term_a_b @ R3 )
& ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ R3 ) ) ) ) ).
% wf_insert
thf(fact_1116_wf__trancl,axiom,
! [R3: set_Pr4386577575007340137rm_a_b] :
( ( wf_term_a_b @ R3 )
=> ( wf_term_a_b @ ( transi7922773638565587891rm_a_b @ R3 ) ) ) ).
% wf_trancl
thf(fact_1117_wf__converse__trancl,axiom,
! [R3: set_Pr4386577575007340137rm_a_b] :
( ( wf_term_a_b @ ( conver7608559976937845093rm_a_b @ R3 ) )
=> ( wf_term_a_b @ ( conver7608559976937845093rm_a_b @ ( transi7922773638565587891rm_a_b @ R3 ) ) ) ) ).
% wf_converse_trancl
thf(fact_1118_wf__union__merge,axiom,
! [R4: set_Pr4386577575007340137rm_a_b,S6: set_Pr4386577575007340137rm_a_b] :
( ( wf_term_a_b @ ( sup_su6776935440552674877rm_a_b @ R4 @ S6 ) )
= ( wf_term_a_b @ ( sup_su6776935440552674877rm_a_b @ ( sup_su6776935440552674877rm_a_b @ ( relcom370159955682700863rm_a_b @ R4 @ R4 ) @ ( relcom370159955682700863rm_a_b @ S6 @ R4 ) ) @ S6 ) ) ) ).
% wf_union_merge
thf(fact_1119_wf__relcomp__compatible,axiom,
! [R4: set_Pr4386577575007340137rm_a_b,S6: set_Pr4386577575007340137rm_a_b] :
( ( wf_term_a_b @ R4 )
=> ( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ R4 @ S6 ) @ ( relcom370159955682700863rm_a_b @ S6 @ R4 ) )
=> ( wf_term_a_b @ ( relcom370159955682700863rm_a_b @ S6 @ R4 ) ) ) ) ).
% wf_relcomp_compatible
thf(fact_1120_wf__def,axiom,
( wf_term_a_b
= ( ^ [R6: set_Pr4386577575007340137rm_a_b] :
! [P3: term_a_b > $o] :
( ! [X2: term_a_b] :
( ! [Y3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ X2 ) @ R6 )
=> ( P3 @ Y3 ) )
=> ( P3 @ X2 ) )
=> ! [X6: term_a_b] : ( P3 @ X6 ) ) ) ) ).
% wf_def
thf(fact_1121_wfE__min,axiom,
! [R4: set_Pr2972776593051762503rm_a_b,X: produc357393685978478089rm_a_b,Q: set_Pr4386577575007340137rm_a_b] :
( ( wf_Pro2335863617654816626rm_a_b @ R4 )
=> ( ( member5869715511025134514rm_a_b @ X @ Q )
=> ~ ! [Z2: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ Z2 @ Q )
=> ~ ! [Y6: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y6 @ Z2 ) @ R4 )
=> ~ ( member5869715511025134514rm_a_b @ Y6 @ Q ) ) ) ) ) ).
% wfE_min
thf(fact_1122_wfE__min,axiom,
! [R4: set_Pr4386577575007340137rm_a_b,X: term_a_b,Q: set_term_a_b] :
( ( wf_term_a_b @ R4 )
=> ( ( member_term_a_b @ X @ Q )
=> ~ ! [Z2: term_a_b] :
( ( member_term_a_b @ Z2 @ Q )
=> ~ ! [Y6: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y6 @ Z2 ) @ R4 )
=> ~ ( member_term_a_b @ Y6 @ Q ) ) ) ) ) ).
% wfE_min
thf(fact_1123_wfI__min,axiom,
! [R4: set_Pr2972776593051762503rm_a_b] :
( ! [X3: produc357393685978478089rm_a_b,Q3: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ X3 @ Q3 )
=> ? [Xa2: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ Xa2 @ Q3 )
& ! [Y2: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y2 @ Xa2 ) @ R4 )
=> ~ ( member5869715511025134514rm_a_b @ Y2 @ Q3 ) ) ) )
=> ( wf_Pro2335863617654816626rm_a_b @ R4 ) ) ).
% wfI_min
thf(fact_1124_wfI__min,axiom,
! [R4: set_Pr4386577575007340137rm_a_b] :
( ! [X3: term_a_b,Q3: set_term_a_b] :
( ( member_term_a_b @ X3 @ Q3 )
=> ? [Xa2: term_a_b] :
( ( member_term_a_b @ Xa2 @ Q3 )
& ! [Y2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y2 @ Xa2 ) @ R4 )
=> ~ ( member_term_a_b @ Y2 @ Q3 ) ) ) )
=> ( wf_term_a_b @ R4 ) ) ).
% wfI_min
thf(fact_1125_wfUNIVI,axiom,
! [R3: set_Pr4386577575007340137rm_a_b] :
( ! [P4: term_a_b > $o,X3: term_a_b] :
( ! [Xa2: term_a_b] :
( ! [Y2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y2 @ Xa2 ) @ R3 )
=> ( P4 @ Y2 ) )
=> ( P4 @ Xa2 ) )
=> ( P4 @ X3 ) )
=> ( wf_term_a_b @ R3 ) ) ).
% wfUNIVI
thf(fact_1126_wf__asym,axiom,
! [R3: set_Pr4386577575007340137rm_a_b,A: term_a_b,X: term_a_b] :
( ( wf_term_a_b @ R3 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ X ) @ R3 )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ A ) @ R3 ) ) ) ).
% wf_asym
thf(fact_1127_wf__induct,axiom,
! [R3: set_Pr4386577575007340137rm_a_b,P2: term_a_b > $o,A: term_a_b] :
( ( wf_term_a_b @ R3 )
=> ( ! [X3: term_a_b] :
( ! [Y6: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y6 @ X3 ) @ R3 )
=> ( P2 @ Y6 ) )
=> ( P2 @ X3 ) )
=> ( P2 @ A ) ) ) ).
% wf_induct
thf(fact_1128_wf__irrefl,axiom,
! [R3: set_Pr4386577575007340137rm_a_b,A: term_a_b] :
( ( wf_term_a_b @ R3 )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ A ) @ R3 ) ) ).
% wf_irrefl
thf(fact_1129_wf__not__sym,axiom,
! [R3: set_Pr4386577575007340137rm_a_b,A: term_a_b,X: term_a_b] :
( ( wf_term_a_b @ R3 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ X ) @ R3 )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ A ) @ R3 ) ) ) ).
% wf_not_sym
thf(fact_1130_wf__not__refl,axiom,
! [R3: set_Pr4386577575007340137rm_a_b,A: term_a_b] :
( ( wf_term_a_b @ R3 )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ A ) @ R3 ) ) ).
% wf_not_refl
thf(fact_1131_wf__eq__minimal,axiom,
( wf_Pro2335863617654816626rm_a_b
= ( ^ [R6: set_Pr2972776593051762503rm_a_b] :
! [Q4: set_Pr4386577575007340137rm_a_b] :
( ? [X2: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ X2 @ Q4 )
=> ? [X2: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X2 @ Q4 )
& ! [Y3: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y3 @ X2 ) @ R6 )
=> ~ ( member5869715511025134514rm_a_b @ Y3 @ Q4 ) ) ) ) ) ) ).
% wf_eq_minimal
thf(fact_1132_wf__eq__minimal,axiom,
( wf_term_a_b
= ( ^ [R6: set_Pr4386577575007340137rm_a_b] :
! [Q4: set_term_a_b] :
( ? [X2: term_a_b] : ( member_term_a_b @ X2 @ Q4 )
=> ? [X2: term_a_b] :
( ( member_term_a_b @ X2 @ Q4 )
& ! [Y3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ X2 ) @ R6 )
=> ~ ( member_term_a_b @ Y3 @ Q4 ) ) ) ) ) ) ).
% wf_eq_minimal
thf(fact_1133_wf__induct__rule,axiom,
! [R3: set_Pr4386577575007340137rm_a_b,P2: term_a_b > $o,A: term_a_b] :
( ( wf_term_a_b @ R3 )
=> ( ! [X3: term_a_b] :
( ! [Y6: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y6 @ X3 ) @ R3 )
=> ( P2 @ Y6 ) )
=> ( P2 @ X3 ) )
=> ( P2 @ A ) ) ) ).
% wf_induct_rule
thf(fact_1134_wf__if__convertible__to__wf,axiom,
! [S: set_Pr4386577575007340137rm_a_b,R3: set_Pr4386577575007340137rm_a_b,F3: term_a_b > term_a_b] :
( ( wf_term_a_b @ S )
=> ( ! [X3: term_a_b,Y2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y2 ) @ R3 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F3 @ X3 ) @ ( F3 @ Y2 ) ) @ S ) )
=> ( wf_term_a_b @ R3 ) ) ) ).
% wf_if_convertible_to_wf
thf(fact_1135_wf__comp__self,axiom,
( wf_term_a_b
= ( ^ [R10: set_Pr4386577575007340137rm_a_b] : ( wf_term_a_b @ ( relcom370159955682700863rm_a_b @ R10 @ R10 ) ) ) ) ).
% wf_comp_self
thf(fact_1136_wfE__min_H,axiom,
! [R4: set_Pr2972776593051762503rm_a_b,Q: set_Pr4386577575007340137rm_a_b] :
( ( wf_Pro2335863617654816626rm_a_b @ R4 )
=> ( ( Q != bot_bo197521221353338581rm_a_b )
=> ~ ! [Z2: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ Z2 @ Q )
=> ~ ! [Y6: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y6 @ Z2 ) @ R4 )
=> ~ ( member5869715511025134514rm_a_b @ Y6 @ Q ) ) ) ) ) ).
% wfE_min'
thf(fact_1137_wfE__min_H,axiom,
! [R4: set_Pr4386577575007340137rm_a_b,Q: set_term_a_b] :
( ( wf_term_a_b @ R4 )
=> ( ( Q != bot_bot_set_term_a_b )
=> ~ ! [Z2: term_a_b] :
( ( member_term_a_b @ Z2 @ Q )
=> ~ ! [Y6: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y6 @ Z2 ) @ R4 )
=> ~ ( member_term_a_b @ Y6 @ Q ) ) ) ) ) ).
% wfE_min'
thf(fact_1138_wf__no__loop,axiom,
! [R4: set_Pr4386577575007340137rm_a_b] :
( ( ( relcom370159955682700863rm_a_b @ R4 @ R4 )
= bot_bo197521221353338581rm_a_b )
=> ( wf_term_a_b @ R4 ) ) ).
% wf_no_loop
thf(fact_1139_minimal__imp__wf,axiom,
! [R3: set_Pr2972776593051762503rm_a_b] :
( ! [Q3: set_Pr4386577575007340137rm_a_b] :
( ? [X3: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ X3 @ Q3 )
=> ? [X5: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X5 @ Q3 )
& ! [Y2: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X5 @ Y2 ) @ R3 )
=> ~ ( member5869715511025134514rm_a_b @ Y2 @ Q3 ) ) ) )
=> ( wf_Pro2335863617654816626rm_a_b @ ( conver1575690562132235771rm_a_b @ R3 ) ) ) ).
% minimal_imp_wf
thf(fact_1140_minimal__imp__wf,axiom,
! [R3: set_Pr4386577575007340137rm_a_b] :
( ! [Q3: set_term_a_b] :
( ? [X3: term_a_b] : ( member_term_a_b @ X3 @ Q3 )
=> ? [X5: term_a_b] :
( ( member_term_a_b @ X5 @ Q3 )
& ! [Y2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X5 @ Y2 ) @ R3 )
=> ~ ( member_term_a_b @ Y2 @ Q3 ) ) ) )
=> ( wf_term_a_b @ ( conver7608559976937845093rm_a_b @ R3 ) ) ) ).
% minimal_imp_wf
thf(fact_1141_wf__imp__SN,axiom,
! [A2: set_Pr4386577575007340137rm_a_b] :
( ( wf_term_a_b @ ( conver7608559976937845093rm_a_b @ A2 ) )
=> ( abstra4720023341729745482rm_a_b @ A2 @ top_top_set_term_a_b ) ) ).
% wf_imp_SN
thf(fact_1142_SN__imp__wf,axiom,
! [R3: set_Pr4386577575007340137rm_a_b] :
( ( abstra4720023341729745482rm_a_b @ R3 @ top_top_set_term_a_b )
=> ( wf_term_a_b @ ( conver7608559976937845093rm_a_b @ R3 ) ) ) ).
% SN_imp_wf
thf(fact_1143_SN__iff__wf,axiom,
! [A2: set_Pr4386577575007340137rm_a_b] :
( ( abstra4720023341729745482rm_a_b @ A2 @ top_top_set_term_a_b )
= ( wf_term_a_b @ ( conver7608559976937845093rm_a_b @ A2 ) ) ) ).
% SN_iff_wf
thf(fact_1144_wf__union__compatible,axiom,
! [R4: set_Pr4386577575007340137rm_a_b,S6: set_Pr4386577575007340137rm_a_b] :
( ( wf_term_a_b @ R4 )
=> ( ( wf_term_a_b @ S6 )
=> ( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ R4 @ S6 ) @ R4 )
=> ( wf_term_a_b @ ( sup_su6776935440552674877rm_a_b @ R4 @ S6 ) ) ) ) ) ).
% wf_union_compatible
thf(fact_1145_wf__eq__minimal2,axiom,
( wf_term_a_b
= ( ^ [R6: set_Pr4386577575007340137rm_a_b] :
! [A6: set_term_a_b] :
( ( ( ord_le2705286416250468010rm_a_b @ A6 @ ( field_term_a_b @ R6 ) )
& ( A6 != bot_bot_set_term_a_b ) )
=> ? [X2: term_a_b] :
( ( member_term_a_b @ X2 @ A6 )
& ! [Y3: term_a_b] :
( ( member_term_a_b @ Y3 @ A6 )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ X2 ) @ R6 ) ) ) ) ) ) ).
% wf_eq_minimal2
thf(fact_1146_qc__wf__relto__iff,axiom,
! [R4: set_Pr4386577575007340137rm_a_b,S6: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ R4 @ S6 ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ R4 @ S6 ) ) @ R4 ) )
=> ( ( wf_term_a_b @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ S6 ) @ ( relcom370159955682700863rm_a_b @ R4 @ ( transi7742714808557438673rm_a_b @ S6 ) ) ) )
= ( wf_term_a_b @ R4 ) ) ) ).
% qc_wf_relto_iff
thf(fact_1147_reduction__pairI,axiom,
! [R4: set_Pr4386577575007340137rm_a_b,S6: set_Pr4386577575007340137rm_a_b] :
( ( wf_term_a_b @ R4 )
=> ( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ R4 @ S6 ) @ R4 )
=> ( fun_re2623890006011866704rm_a_b @ ( produc3977138704197412567rm_a_b @ R4 @ S6 ) ) ) ) ).
% reduction_pairI
thf(fact_1148_member__remove,axiom,
! [X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b,A2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ X @ ( remove8523906646656265582rm_a_b @ Y @ A2 ) )
= ( ( member5869715511025134514rm_a_b @ X @ A2 )
& ( X != Y ) ) ) ).
% member_remove
thf(fact_1149_wo__rel_Ocases__Total3,axiom,
! [R3: set_Pr4386577575007340137rm_a_b,A: term_a_b,B: term_a_b,Phi: term_a_b > term_a_b > $o] :
( ( bNF_We251836928773027112rm_a_b @ R3 )
=> ( ( ord_le2705286416250468010rm_a_b @ ( insert_term_a_b @ A @ ( insert_term_a_b @ B @ bot_bot_set_term_a_b ) ) @ ( field_term_a_b @ R3 ) )
=> ( ( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( minus_5192120951422937424rm_a_b @ R3 @ id_term_a_b ) )
| ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ ( minus_5192120951422937424rm_a_b @ R3 @ id_term_a_b ) ) )
=> ( Phi @ A @ B ) )
=> ( ( ( A = B )
=> ( Phi @ A @ B ) )
=> ( Phi @ A @ B ) ) ) ) ) ).
% wo_rel.cases_Total3
thf(fact_1150_refl__on__singleton,axiom,
! [X: term_a_b] : ( refl_on_term_a_b @ ( insert_term_a_b @ X @ bot_bot_set_term_a_b ) @ ( insert7009541432154983385rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ X ) @ bot_bo197521221353338581rm_a_b ) ) ).
% refl_on_singleton
thf(fact_1151_refl__on__converse,axiom,
! [A2: set_term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
( ( refl_on_term_a_b @ A2 @ ( conver7608559976937845093rm_a_b @ R3 ) )
= ( refl_on_term_a_b @ A2 @ R3 ) ) ).
% refl_on_converse
thf(fact_1152_refl__rtrancl,axiom,
! [R3: set_Pr4386577575007340137rm_a_b] : ( refl_on_term_a_b @ top_top_set_term_a_b @ ( transi7742714808557438673rm_a_b @ R3 ) ) ).
% refl_rtrancl
thf(fact_1153_refl__on__Un,axiom,
! [A2: set_term_a_b,R3: set_Pr4386577575007340137rm_a_b,B2: set_term_a_b,S: set_Pr4386577575007340137rm_a_b] :
( ( refl_on_term_a_b @ A2 @ R3 )
=> ( ( refl_on_term_a_b @ B2 @ S )
=> ( refl_on_term_a_b @ ( sup_sup_set_term_a_b @ A2 @ B2 ) @ ( sup_su6776935440552674877rm_a_b @ R3 @ S ) ) ) ) ).
% refl_on_Un
thf(fact_1154_refl__on__Un,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,R3: set_Pr2972776593051762503rm_a_b,B2: set_Pr4386577575007340137rm_a_b,S: set_Pr2972776593051762503rm_a_b] :
( ( refl_o5344404488616963464rm_a_b @ A2 @ R3 )
=> ( ( refl_o5344404488616963464rm_a_b @ B2 @ S )
=> ( refl_o5344404488616963464rm_a_b @ ( sup_su6776935440552674877rm_a_b @ A2 @ B2 ) @ ( sup_su1845815533690797339rm_a_b @ R3 @ S ) ) ) ) ).
% refl_on_Un
thf(fact_1155_wo__rel_OTOTALS,axiom,
! [R3: set_Pr4386577575007340137rm_a_b] :
( ( bNF_We251836928773027112rm_a_b @ R3 )
=> ! [X5: term_a_b] :
( ( member_term_a_b @ X5 @ ( field_term_a_b @ R3 ) )
=> ! [Xa2: term_a_b] :
( ( member_term_a_b @ Xa2 @ ( field_term_a_b @ R3 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X5 @ Xa2 ) @ R3 )
| ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Xa2 @ X5 ) @ R3 ) ) ) ) ) ).
% wo_rel.TOTALS
thf(fact_1156_wo__rel_Oin__notinI,axiom,
! [R3: set_Pr2972776593051762503rm_a_b,J: produc357393685978478089rm_a_b,I: produc357393685978478089rm_a_b] :
( ( bNF_We1688910813892497397rm_a_b @ R3 )
=> ( ( ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ J @ I ) @ R3 )
| ( J = I ) )
=> ( ( member5869715511025134514rm_a_b @ I @ ( field_6884932134483023318rm_a_b @ R3 ) )
=> ( ( member5869715511025134514rm_a_b @ J @ ( field_6884932134483023318rm_a_b @ R3 ) )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ I @ J ) @ R3 ) ) ) ) ) ).
% wo_rel.in_notinI
thf(fact_1157_wo__rel_Oin__notinI,axiom,
! [R3: set_Pr4386577575007340137rm_a_b,J: term_a_b,I: term_a_b] :
( ( bNF_We251836928773027112rm_a_b @ R3 )
=> ( ( ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ J @ I ) @ R3 )
| ( J = I ) )
=> ( ( member_term_a_b @ I @ ( field_term_a_b @ R3 ) )
=> ( ( member_term_a_b @ J @ ( field_term_a_b @ R3 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ I @ J ) @ R3 ) ) ) ) ) ).
% wo_rel.in_notinI
thf(fact_1158_well__order__induct__imp,axiom,
! [R3: set_Pr2972776593051762503rm_a_b,P2: produc357393685978478089rm_a_b > $o,A: produc357393685978478089rm_a_b] :
( ( bNF_We1688910813892497397rm_a_b @ R3 )
=> ( ! [X3: produc357393685978478089rm_a_b] :
( ! [Y6: produc357393685978478089rm_a_b] :
( ( ( Y6 != X3 )
& ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y6 @ X3 ) @ R3 ) )
=> ( ( member5869715511025134514rm_a_b @ Y6 @ ( field_6884932134483023318rm_a_b @ R3 ) )
=> ( P2 @ Y6 ) ) )
=> ( ( member5869715511025134514rm_a_b @ X3 @ ( field_6884932134483023318rm_a_b @ R3 ) )
=> ( P2 @ X3 ) ) )
=> ( ( member5869715511025134514rm_a_b @ A @ ( field_6884932134483023318rm_a_b @ R3 ) )
=> ( P2 @ A ) ) ) ) ).
% well_order_induct_imp
thf(fact_1159_well__order__induct__imp,axiom,
! [R3: set_Pr4386577575007340137rm_a_b,P2: term_a_b > $o,A: term_a_b] :
( ( bNF_We251836928773027112rm_a_b @ R3 )
=> ( ! [X3: term_a_b] :
( ! [Y6: term_a_b] :
( ( ( Y6 != X3 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y6 @ X3 ) @ R3 ) )
=> ( ( member_term_a_b @ Y6 @ ( field_term_a_b @ R3 ) )
=> ( P2 @ Y6 ) ) )
=> ( ( member_term_a_b @ X3 @ ( field_term_a_b @ R3 ) )
=> ( P2 @ X3 ) ) )
=> ( ( member_term_a_b @ A @ ( field_term_a_b @ R3 ) )
=> ( P2 @ A ) ) ) ) ).
% well_order_induct_imp
thf(fact_1160_wo__rel_Owell__order__induct,axiom,
! [R3: set_Pr4386577575007340137rm_a_b,P2: term_a_b > $o,A: term_a_b] :
( ( bNF_We251836928773027112rm_a_b @ R3 )
=> ( ! [X3: term_a_b] :
( ! [Y6: term_a_b] :
( ( ( Y6 != X3 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y6 @ X3 ) @ R3 ) )
=> ( P2 @ Y6 ) )
=> ( P2 @ X3 ) )
=> ( P2 @ A ) ) ) ).
% wo_rel.well_order_induct
thf(fact_1161_refl__onD,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,R3: set_Pr2972776593051762503rm_a_b,A: produc357393685978478089rm_a_b] :
( ( refl_o5344404488616963464rm_a_b @ A2 @ R3 )
=> ( ( member5869715511025134514rm_a_b @ A @ A2 )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ A ) @ R3 ) ) ) ).
% refl_onD
thf(fact_1162_refl__onD,axiom,
! [A2: set_term_a_b,R3: set_Pr4386577575007340137rm_a_b,A: term_a_b] :
( ( refl_on_term_a_b @ A2 @ R3 )
=> ( ( member_term_a_b @ A @ A2 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ A ) @ R3 ) ) ) ).
% refl_onD
thf(fact_1163_refl__onD1,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,R3: set_Pr2972776593051762503rm_a_b,X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b] :
( ( refl_o5344404488616963464rm_a_b @ A2 @ R3 )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ R3 )
=> ( member5869715511025134514rm_a_b @ X @ A2 ) ) ) ).
% refl_onD1
thf(fact_1164_refl__onD1,axiom,
! [A2: set_term_a_b,R3: set_Pr4386577575007340137rm_a_b,X: term_a_b,Y: term_a_b] :
( ( refl_on_term_a_b @ A2 @ R3 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ R3 )
=> ( member_term_a_b @ X @ A2 ) ) ) ).
% refl_onD1
thf(fact_1165_refl__onD2,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,R3: set_Pr2972776593051762503rm_a_b,X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b] :
( ( refl_o5344404488616963464rm_a_b @ A2 @ R3 )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ R3 )
=> ( member5869715511025134514rm_a_b @ Y @ A2 ) ) ) ).
% refl_onD2
thf(fact_1166_refl__onD2,axiom,
! [A2: set_term_a_b,R3: set_Pr4386577575007340137rm_a_b,X: term_a_b,Y: term_a_b] :
( ( refl_on_term_a_b @ A2 @ R3 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ R3 )
=> ( member_term_a_b @ Y @ A2 ) ) ) ).
% refl_onD2
thf(fact_1167_reflI,axiom,
! [R3: set_Pr4386577575007340137rm_a_b] :
( ! [X3: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ X3 ) @ R3 )
=> ( refl_on_term_a_b @ top_top_set_term_a_b @ R3 ) ) ).
% reflI
thf(fact_1168_reflD,axiom,
! [R3: set_Pr4386577575007340137rm_a_b,A: term_a_b] :
( ( refl_on_term_a_b @ top_top_set_term_a_b @ R3 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ A ) @ R3 ) ) ).
% reflD
thf(fact_1169_refl__on__domain,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,R3: set_Pr2972776593051762503rm_a_b,A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b] :
( ( refl_o5344404488616963464rm_a_b @ A2 @ R3 )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ R3 )
=> ( ( member5869715511025134514rm_a_b @ A @ A2 )
& ( member5869715511025134514rm_a_b @ B @ A2 ) ) ) ) ).
% refl_on_domain
thf(fact_1170_refl__on__domain,axiom,
! [A2: set_term_a_b,R3: set_Pr4386577575007340137rm_a_b,A: term_a_b,B: term_a_b] :
( ( refl_on_term_a_b @ A2 @ R3 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R3 )
=> ( ( member_term_a_b @ A @ A2 )
& ( member_term_a_b @ B @ A2 ) ) ) ) ).
% refl_on_domain
thf(fact_1171_refl__reflcl,axiom,
! [R3: set_Pr4386577575007340137rm_a_b] : ( refl_on_term_a_b @ top_top_set_term_a_b @ ( sup_su6776935440552674877rm_a_b @ R3 @ id_term_a_b ) ) ).
% refl_reflcl
thf(fact_1172_wo__rel_Ocases__Total,axiom,
! [R3: set_Pr4386577575007340137rm_a_b,A: term_a_b,B: term_a_b,Phi: term_a_b > term_a_b > $o] :
( ( bNF_We251836928773027112rm_a_b @ R3 )
=> ( ( ord_le2705286416250468010rm_a_b @ ( insert_term_a_b @ A @ ( insert_term_a_b @ B @ bot_bot_set_term_a_b ) ) @ ( field_term_a_b @ R3 ) )
=> ( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R3 )
=> ( Phi @ A @ B ) )
=> ( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ R3 )
=> ( Phi @ A @ B ) )
=> ( Phi @ A @ B ) ) ) ) ) ).
% wo_rel.cases_Total
thf(fact_1173_refl__on__comp__subset,axiom,
! [A2: set_term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
( ( refl_on_term_a_b @ A2 @ R3 )
=> ( ord_le118470702582115849rm_a_b @ R3 @ ( relcom370159955682700863rm_a_b @ ( conver7608559976937845093rm_a_b @ R3 ) @ R3 ) ) ) ).
% refl_on_comp_subset
thf(fact_1174_wo__rel_OisMinim__def,axiom,
! [R3: set_Pr2972776593051762503rm_a_b,A2: set_Pr4386577575007340137rm_a_b,B: produc357393685978478089rm_a_b] :
( ( bNF_We1688910813892497397rm_a_b @ R3 )
=> ( ( bNF_We8430659725849320639rm_a_b @ R3 @ A2 @ B )
= ( ( member5869715511025134514rm_a_b @ B @ A2 )
& ! [X2: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X2 @ A2 )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B @ X2 ) @ R3 ) ) ) ) ) ).
% wo_rel.isMinim_def
thf(fact_1175_wo__rel_OisMinim__def,axiom,
! [R3: set_Pr4386577575007340137rm_a_b,A2: set_term_a_b,B: term_a_b] :
( ( bNF_We251836928773027112rm_a_b @ R3 )
=> ( ( bNF_We695392074589568990rm_a_b @ R3 @ A2 @ B )
= ( ( member_term_a_b @ B @ A2 )
& ! [X2: term_a_b] :
( ( member_term_a_b @ X2 @ A2 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ X2 ) @ R3 ) ) ) ) ) ).
% wo_rel.isMinim_def
thf(fact_1176_wo__rel_Ominim__inField,axiom,
! [R3: set_Pr2972776593051762503rm_a_b,B2: set_Pr4386577575007340137rm_a_b] :
( ( bNF_We1688910813892497397rm_a_b @ R3 )
=> ( ( ord_le118470702582115849rm_a_b @ B2 @ ( field_6884932134483023318rm_a_b @ R3 ) )
=> ( ( B2 != bot_bo197521221353338581rm_a_b )
=> ( member5869715511025134514rm_a_b @ ( bNF_We1338740440619677141rm_a_b @ R3 @ B2 ) @ ( field_6884932134483023318rm_a_b @ R3 ) ) ) ) ) ).
% wo_rel.minim_inField
thf(fact_1177_wo__rel_Ominim__least,axiom,
! [R3: set_Pr2972776593051762503rm_a_b,B2: set_Pr4386577575007340137rm_a_b,B: produc357393685978478089rm_a_b] :
( ( bNF_We1688910813892497397rm_a_b @ R3 )
=> ( ( ord_le118470702582115849rm_a_b @ B2 @ ( field_6884932134483023318rm_a_b @ R3 ) )
=> ( ( member5869715511025134514rm_a_b @ B @ B2 )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( bNF_We1338740440619677141rm_a_b @ R3 @ B2 ) @ B ) @ R3 ) ) ) ) ).
% wo_rel.minim_least
thf(fact_1178_wo__rel_Ominim__least,axiom,
! [R3: set_Pr4386577575007340137rm_a_b,B2: set_term_a_b,B: term_a_b] :
( ( bNF_We251836928773027112rm_a_b @ R3 )
=> ( ( ord_le2705286416250468010rm_a_b @ B2 @ ( field_term_a_b @ R3 ) )
=> ( ( member_term_a_b @ B @ B2 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( bNF_We6258903063523145544rm_a_b @ R3 @ B2 ) @ B ) @ R3 ) ) ) ) ).
% wo_rel.minim_least
thf(fact_1179_wo__rel_Oequals__minim,axiom,
! [R3: set_Pr2972776593051762503rm_a_b,B2: set_Pr4386577575007340137rm_a_b,A: produc357393685978478089rm_a_b] :
( ( bNF_We1688910813892497397rm_a_b @ R3 )
=> ( ( ord_le118470702582115849rm_a_b @ B2 @ ( field_6884932134483023318rm_a_b @ R3 ) )
=> ( ( member5869715511025134514rm_a_b @ A @ B2 )
=> ( ! [B3: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ B3 @ B2 )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B3 ) @ R3 ) )
=> ( A
= ( bNF_We1338740440619677141rm_a_b @ R3 @ B2 ) ) ) ) ) ) ).
% wo_rel.equals_minim
thf(fact_1180_wo__rel_Oequals__minim,axiom,
! [R3: set_Pr4386577575007340137rm_a_b,B2: set_term_a_b,A: term_a_b] :
( ( bNF_We251836928773027112rm_a_b @ R3 )
=> ( ( ord_le2705286416250468010rm_a_b @ B2 @ ( field_term_a_b @ R3 ) )
=> ( ( member_term_a_b @ A @ B2 )
=> ( ! [B3: term_a_b] :
( ( member_term_a_b @ B3 @ B2 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B3 ) @ R3 ) )
=> ( A
= ( bNF_We6258903063523145544rm_a_b @ R3 @ B2 ) ) ) ) ) ) ).
% wo_rel.equals_minim
thf(fact_1181_wo__rel_Ominim__in,axiom,
! [R3: set_Pr2972776593051762503rm_a_b,B2: set_Pr4386577575007340137rm_a_b] :
( ( bNF_We1688910813892497397rm_a_b @ R3 )
=> ( ( ord_le118470702582115849rm_a_b @ B2 @ ( field_6884932134483023318rm_a_b @ R3 ) )
=> ( ( B2 != bot_bo197521221353338581rm_a_b )
=> ( member5869715511025134514rm_a_b @ ( bNF_We1338740440619677141rm_a_b @ R3 @ B2 ) @ B2 ) ) ) ) ).
% wo_rel.minim_in
thf(fact_1182_wo__rel_Omax2__greater__among,axiom,
! [R3: set_Pr2972776593051762503rm_a_b,A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b] :
( ( bNF_We1688910813892497397rm_a_b @ R3 )
=> ( ( member5869715511025134514rm_a_b @ A @ ( field_6884932134483023318rm_a_b @ R3 ) )
=> ( ( member5869715511025134514rm_a_b @ B @ ( field_6884932134483023318rm_a_b @ R3 ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ ( bNF_We1369222903935127127rm_a_b @ R3 @ A @ B ) ) @ R3 )
& ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B @ ( bNF_We1369222903935127127rm_a_b @ R3 @ A @ B ) ) @ R3 )
& ( member5869715511025134514rm_a_b @ ( bNF_We1369222903935127127rm_a_b @ R3 @ A @ B ) @ ( insert7009541432154983385rm_a_b @ A @ ( insert7009541432154983385rm_a_b @ B @ bot_bo197521221353338581rm_a_b ) ) ) ) ) ) ) ).
% wo_rel.max2_greater_among
thf(fact_1183_wo__rel_Omax2__greater__among,axiom,
! [R3: set_Pr4386577575007340137rm_a_b,A: term_a_b,B: term_a_b] :
( ( bNF_We251836928773027112rm_a_b @ R3 )
=> ( ( member_term_a_b @ A @ ( field_term_a_b @ R3 ) )
=> ( ( member_term_a_b @ B @ ( field_term_a_b @ R3 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ ( bNF_We4618759378340593734rm_a_b @ R3 @ A @ B ) ) @ R3 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ ( bNF_We4618759378340593734rm_a_b @ R3 @ A @ B ) ) @ R3 )
& ( member_term_a_b @ ( bNF_We4618759378340593734rm_a_b @ R3 @ A @ B ) @ ( insert_term_a_b @ A @ ( insert_term_a_b @ B @ bot_bot_set_term_a_b ) ) ) ) ) ) ) ).
% wo_rel.max2_greater_among
thf(fact_1184_equiv__class__nondisjoint,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,R3: set_Pr2972776593051762503rm_a_b,X: produc357393685978478089rm_a_b,A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b] :
( ( equiv_19968616203005119rm_a_b @ A2 @ R3 )
=> ( ( member5869715511025134514rm_a_b @ X @ ( inf_in1693821330460682071rm_a_b @ ( image_871640889055226643rm_a_b @ R3 @ ( insert7009541432154983385rm_a_b @ A @ bot_bo197521221353338581rm_a_b ) ) @ ( image_871640889055226643rm_a_b @ R3 @ ( insert7009541432154983385rm_a_b @ B @ bot_bo197521221353338581rm_a_b ) ) ) )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ R3 ) ) ) ).
% equiv_class_nondisjoint
thf(fact_1185_equiv__class__nondisjoint,axiom,
! [A2: set_term_a_b,R3: set_Pr4386577575007340137rm_a_b,X: term_a_b,A: term_a_b,B: term_a_b] :
( ( equiv_equiv_term_a_b @ A2 @ R3 )
=> ( ( member_term_a_b @ X @ ( inf_inf_set_term_a_b @ ( image_6923787618150785661rm_a_b @ R3 @ ( insert_term_a_b @ A @ bot_bot_set_term_a_b ) ) @ ( image_6923787618150785661rm_a_b @ R3 @ ( insert_term_a_b @ B @ bot_bot_set_term_a_b ) ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R3 ) ) ) ).
% equiv_class_nondisjoint
thf(fact_1186_equiv__comp__eq,axiom,
! [A2: set_term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
( ( equiv_equiv_term_a_b @ A2 @ R3 )
=> ( ( relcom370159955682700863rm_a_b @ ( conver7608559976937845093rm_a_b @ R3 ) @ R3 )
= R3 ) ) ).
% equiv_comp_eq
thf(fact_1187_wo__rel_Omax2__def,axiom,
! [R3: set_Pr4386577575007340137rm_a_b,A: term_a_b,B: term_a_b] :
( ( bNF_We251836928773027112rm_a_b @ R3 )
=> ( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R3 )
=> ( ( bNF_We4618759378340593734rm_a_b @ R3 @ A @ B )
= B ) )
& ( ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R3 )
=> ( ( bNF_We4618759378340593734rm_a_b @ R3 @ A @ B )
= A ) ) ) ) ).
% wo_rel.max2_def
thf(fact_1188_eq__equiv__class,axiom,
! [R3: set_Pr2972776593051762503rm_a_b,A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,A2: set_Pr4386577575007340137rm_a_b] :
( ( ( image_871640889055226643rm_a_b @ R3 @ ( insert7009541432154983385rm_a_b @ A @ bot_bo197521221353338581rm_a_b ) )
= ( image_871640889055226643rm_a_b @ R3 @ ( insert7009541432154983385rm_a_b @ B @ bot_bo197521221353338581rm_a_b ) ) )
=> ( ( equiv_19968616203005119rm_a_b @ A2 @ R3 )
=> ( ( member5869715511025134514rm_a_b @ B @ A2 )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ R3 ) ) ) ) ).
% eq_equiv_class
thf(fact_1189_eq__equiv__class,axiom,
! [R3: set_Pr4386577575007340137rm_a_b,A: term_a_b,B: term_a_b,A2: set_term_a_b] :
( ( ( image_6923787618150785661rm_a_b @ R3 @ ( insert_term_a_b @ A @ bot_bot_set_term_a_b ) )
= ( image_6923787618150785661rm_a_b @ R3 @ ( insert_term_a_b @ B @ bot_bot_set_term_a_b ) ) )
=> ( ( equiv_equiv_term_a_b @ A2 @ R3 )
=> ( ( member_term_a_b @ B @ A2 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R3 ) ) ) ) ).
% eq_equiv_class
thf(fact_1190_equiv__class__eq,axiom,
! [A2: set_term_a_b,R3: set_Pr4386577575007340137rm_a_b,A: term_a_b,B: term_a_b] :
( ( equiv_equiv_term_a_b @ A2 @ R3 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R3 )
=> ( ( image_6923787618150785661rm_a_b @ R3 @ ( insert_term_a_b @ A @ bot_bot_set_term_a_b ) )
= ( image_6923787618150785661rm_a_b @ R3 @ ( insert_term_a_b @ B @ bot_bot_set_term_a_b ) ) ) ) ) ).
% equiv_class_eq
thf(fact_1191_eq__equiv__class__iff,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,R3: set_Pr2972776593051762503rm_a_b,X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b] :
( ( equiv_19968616203005119rm_a_b @ A2 @ R3 )
=> ( ( member5869715511025134514rm_a_b @ X @ A2 )
=> ( ( member5869715511025134514rm_a_b @ Y @ A2 )
=> ( ( ( image_871640889055226643rm_a_b @ R3 @ ( insert7009541432154983385rm_a_b @ X @ bot_bo197521221353338581rm_a_b ) )
= ( image_871640889055226643rm_a_b @ R3 @ ( insert7009541432154983385rm_a_b @ Y @ bot_bo197521221353338581rm_a_b ) ) )
= ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ R3 ) ) ) ) ) ).
% eq_equiv_class_iff
thf(fact_1192_eq__equiv__class__iff,axiom,
! [A2: set_term_a_b,R3: set_Pr4386577575007340137rm_a_b,X: term_a_b,Y: term_a_b] :
( ( equiv_equiv_term_a_b @ A2 @ R3 )
=> ( ( member_term_a_b @ X @ A2 )
=> ( ( member_term_a_b @ Y @ A2 )
=> ( ( ( image_6923787618150785661rm_a_b @ R3 @ ( insert_term_a_b @ X @ bot_bot_set_term_a_b ) )
= ( image_6923787618150785661rm_a_b @ R3 @ ( insert_term_a_b @ Y @ bot_bot_set_term_a_b ) ) )
= ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ R3 ) ) ) ) ) ).
% eq_equiv_class_iff
thf(fact_1193_equiv__class__eq__iff,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,R3: set_Pr2972776593051762503rm_a_b,X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b] :
( ( equiv_19968616203005119rm_a_b @ A2 @ R3 )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ R3 )
= ( ( ( image_871640889055226643rm_a_b @ R3 @ ( insert7009541432154983385rm_a_b @ X @ bot_bo197521221353338581rm_a_b ) )
= ( image_871640889055226643rm_a_b @ R3 @ ( insert7009541432154983385rm_a_b @ Y @ bot_bo197521221353338581rm_a_b ) ) )
& ( member5869715511025134514rm_a_b @ X @ A2 )
& ( member5869715511025134514rm_a_b @ Y @ A2 ) ) ) ) ).
% equiv_class_eq_iff
thf(fact_1194_equiv__class__eq__iff,axiom,
! [A2: set_term_a_b,R3: set_Pr4386577575007340137rm_a_b,X: term_a_b,Y: term_a_b] :
( ( equiv_equiv_term_a_b @ A2 @ R3 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ R3 )
= ( ( ( image_6923787618150785661rm_a_b @ R3 @ ( insert_term_a_b @ X @ bot_bot_set_term_a_b ) )
= ( image_6923787618150785661rm_a_b @ R3 @ ( insert_term_a_b @ Y @ bot_bot_set_term_a_b ) ) )
& ( member_term_a_b @ X @ A2 )
& ( member_term_a_b @ Y @ A2 ) ) ) ) ).
% equiv_class_eq_iff
thf(fact_1195_wo__rel_Omax2__greater,axiom,
! [R3: set_Pr2972776593051762503rm_a_b,A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b] :
( ( bNF_We1688910813892497397rm_a_b @ R3 )
=> ( ( member5869715511025134514rm_a_b @ A @ ( field_6884932134483023318rm_a_b @ R3 ) )
=> ( ( member5869715511025134514rm_a_b @ B @ ( field_6884932134483023318rm_a_b @ R3 ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ ( bNF_We1369222903935127127rm_a_b @ R3 @ A @ B ) ) @ R3 )
& ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B @ ( bNF_We1369222903935127127rm_a_b @ R3 @ A @ B ) ) @ R3 ) ) ) ) ) ).
% wo_rel.max2_greater
thf(fact_1196_wo__rel_Omax2__greater,axiom,
! [R3: set_Pr4386577575007340137rm_a_b,A: term_a_b,B: term_a_b] :
( ( bNF_We251836928773027112rm_a_b @ R3 )
=> ( ( member_term_a_b @ A @ ( field_term_a_b @ R3 ) )
=> ( ( member_term_a_b @ B @ ( field_term_a_b @ R3 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ ( bNF_We4618759378340593734rm_a_b @ R3 @ A @ B ) ) @ R3 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ ( bNF_We4618759378340593734rm_a_b @ R3 @ A @ B ) ) @ R3 ) ) ) ) ) ).
% wo_rel.max2_greater
thf(fact_1197_wo__rel_Omax2__equals2,axiom,
! [R3: set_Pr2972776593051762503rm_a_b,A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b] :
( ( bNF_We1688910813892497397rm_a_b @ R3 )
=> ( ( member5869715511025134514rm_a_b @ A @ ( field_6884932134483023318rm_a_b @ R3 ) )
=> ( ( member5869715511025134514rm_a_b @ B @ ( field_6884932134483023318rm_a_b @ R3 ) )
=> ( ( ( bNF_We1369222903935127127rm_a_b @ R3 @ A @ B )
= B )
= ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ R3 ) ) ) ) ) ).
% wo_rel.max2_equals2
thf(fact_1198_wo__rel_Omax2__equals2,axiom,
! [R3: set_Pr4386577575007340137rm_a_b,A: term_a_b,B: term_a_b] :
( ( bNF_We251836928773027112rm_a_b @ R3 )
=> ( ( member_term_a_b @ A @ ( field_term_a_b @ R3 ) )
=> ( ( member_term_a_b @ B @ ( field_term_a_b @ R3 ) )
=> ( ( ( bNF_We4618759378340593734rm_a_b @ R3 @ A @ B )
= B )
= ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R3 ) ) ) ) ) ).
% wo_rel.max2_equals2
thf(fact_1199_wo__rel_Omax2__equals1,axiom,
! [R3: set_Pr2972776593051762503rm_a_b,A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b] :
( ( bNF_We1688910813892497397rm_a_b @ R3 )
=> ( ( member5869715511025134514rm_a_b @ A @ ( field_6884932134483023318rm_a_b @ R3 ) )
=> ( ( member5869715511025134514rm_a_b @ B @ ( field_6884932134483023318rm_a_b @ R3 ) )
=> ( ( ( bNF_We1369222903935127127rm_a_b @ R3 @ A @ B )
= A )
= ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B @ A ) @ R3 ) ) ) ) ) ).
% wo_rel.max2_equals1
thf(fact_1200_wo__rel_Omax2__equals1,axiom,
! [R3: set_Pr4386577575007340137rm_a_b,A: term_a_b,B: term_a_b] :
( ( bNF_We251836928773027112rm_a_b @ R3 )
=> ( ( member_term_a_b @ A @ ( field_term_a_b @ R3 ) )
=> ( ( member_term_a_b @ B @ ( field_term_a_b @ R3 ) )
=> ( ( ( bNF_We4618759378340593734rm_a_b @ R3 @ A @ B )
= A )
= ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ R3 ) ) ) ) ) ).
% wo_rel.max2_equals1
thf(fact_1201_equiv__class__subset,axiom,
! [A2: set_term_a_b,R3: set_Pr4386577575007340137rm_a_b,A: term_a_b,B: term_a_b] :
( ( equiv_equiv_term_a_b @ A2 @ R3 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R3 )
=> ( ord_le2705286416250468010rm_a_b @ ( image_6923787618150785661rm_a_b @ R3 @ ( insert_term_a_b @ A @ bot_bot_set_term_a_b ) ) @ ( image_6923787618150785661rm_a_b @ R3 @ ( insert_term_a_b @ B @ bot_bot_set_term_a_b ) ) ) ) ) ).
% equiv_class_subset
thf(fact_1202_subset__equiv__class,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,R3: set_Pr2972776593051762503rm_a_b,B: produc357393685978478089rm_a_b,A: produc357393685978478089rm_a_b] :
( ( equiv_19968616203005119rm_a_b @ A2 @ R3 )
=> ( ( ord_le118470702582115849rm_a_b @ ( image_871640889055226643rm_a_b @ R3 @ ( insert7009541432154983385rm_a_b @ B @ bot_bo197521221353338581rm_a_b ) ) @ ( image_871640889055226643rm_a_b @ R3 @ ( insert7009541432154983385rm_a_b @ A @ bot_bo197521221353338581rm_a_b ) ) )
=> ( ( member5869715511025134514rm_a_b @ B @ A2 )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ R3 ) ) ) ) ).
% subset_equiv_class
thf(fact_1203_subset__equiv__class,axiom,
! [A2: set_term_a_b,R3: set_Pr4386577575007340137rm_a_b,B: term_a_b,A: term_a_b] :
( ( equiv_equiv_term_a_b @ A2 @ R3 )
=> ( ( ord_le2705286416250468010rm_a_b @ ( image_6923787618150785661rm_a_b @ R3 @ ( insert_term_a_b @ B @ bot_bot_set_term_a_b ) ) @ ( image_6923787618150785661rm_a_b @ R3 @ ( insert_term_a_b @ A @ bot_bot_set_term_a_b ) ) )
=> ( ( member_term_a_b @ B @ A2 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R3 ) ) ) ) ).
% subset_equiv_class
thf(fact_1204_proj__iff,axiom,
! [A2: set_term_a_b,R3: set_Pr4386577575007340137rm_a_b,X: term_a_b,Y: term_a_b] :
( ( equiv_equiv_term_a_b @ A2 @ R3 )
=> ( ( ord_le2705286416250468010rm_a_b @ ( insert_term_a_b @ X @ ( insert_term_a_b @ Y @ bot_bot_set_term_a_b ) ) @ A2 )
=> ( ( ( equiv_8008468056096195508rm_a_b @ R3 @ X )
= ( equiv_8008468056096195508rm_a_b @ R3 @ Y ) )
= ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ R3 ) ) ) ) ).
% proj_iff
thf(fact_1205_disjnt__equiv__class,axiom,
! [A2: set_term_a_b,R3: set_Pr4386577575007340137rm_a_b,A: term_a_b,B: term_a_b] :
( ( equiv_equiv_term_a_b @ A2 @ R3 )
=> ( ( disjnt_term_a_b @ ( image_6923787618150785661rm_a_b @ R3 @ ( insert_term_a_b @ A @ bot_bot_set_term_a_b ) ) @ ( image_6923787618150785661rm_a_b @ R3 @ ( insert_term_a_b @ B @ bot_bot_set_term_a_b ) ) )
= ( ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R3 ) ) ) ) ).
% disjnt_equiv_class
thf(fact_1206_disjnt__insert1,axiom,
! [A: produc357393685978478089rm_a_b,X4: set_Pr4386577575007340137rm_a_b,Y7: set_Pr4386577575007340137rm_a_b] :
( ( disjnt884005487775839824rm_a_b @ ( insert7009541432154983385rm_a_b @ A @ X4 ) @ Y7 )
= ( ~ ( member5869715511025134514rm_a_b @ A @ Y7 )
& ( disjnt884005487775839824rm_a_b @ X4 @ Y7 ) ) ) ).
% disjnt_insert1
thf(fact_1207_disjnt__insert2,axiom,
! [Y7: set_Pr4386577575007340137rm_a_b,A: produc357393685978478089rm_a_b,X4: set_Pr4386577575007340137rm_a_b] :
( ( disjnt884005487775839824rm_a_b @ Y7 @ ( insert7009541432154983385rm_a_b @ A @ X4 ) )
= ( ~ ( member5869715511025134514rm_a_b @ A @ Y7 )
& ( disjnt884005487775839824rm_a_b @ Y7 @ X4 ) ) ) ).
% disjnt_insert2
thf(fact_1208_disjnt__Un2,axiom,
! [C: set_Pr4386577575007340137rm_a_b,A2: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b] :
( ( disjnt884005487775839824rm_a_b @ C @ ( sup_su6776935440552674877rm_a_b @ A2 @ B2 ) )
= ( ( disjnt884005487775839824rm_a_b @ C @ A2 )
& ( disjnt884005487775839824rm_a_b @ C @ B2 ) ) ) ).
% disjnt_Un2
thf(fact_1209_disjnt__Un1,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b,C: set_Pr4386577575007340137rm_a_b] :
( ( disjnt884005487775839824rm_a_b @ ( sup_su6776935440552674877rm_a_b @ A2 @ B2 ) @ C )
= ( ( disjnt884005487775839824rm_a_b @ A2 @ C )
& ( disjnt884005487775839824rm_a_b @ B2 @ C ) ) ) ).
% disjnt_Un1
thf(fact_1210_disjnt__iff,axiom,
( disjnt884005487775839824rm_a_b
= ( ^ [A6: set_Pr4386577575007340137rm_a_b,B6: set_Pr4386577575007340137rm_a_b] :
! [X2: produc357393685978478089rm_a_b] :
~ ( ( member5869715511025134514rm_a_b @ X2 @ A6 )
& ( member5869715511025134514rm_a_b @ X2 @ B6 ) ) ) ) ).
% disjnt_iff
thf(fact_1211_disjnt__insert,axiom,
! [X: produc357393685978478089rm_a_b,N4: set_Pr4386577575007340137rm_a_b,M3: set_Pr4386577575007340137rm_a_b] :
( ~ ( member5869715511025134514rm_a_b @ X @ N4 )
=> ( ( disjnt884005487775839824rm_a_b @ M3 @ N4 )
=> ( disjnt884005487775839824rm_a_b @ ( insert7009541432154983385rm_a_b @ X @ M3 ) @ N4 ) ) ) ).
% disjnt_insert
thf(fact_1212_in__quotient__imp__in__rel,axiom,
! [A2: set_term_a_b,R3: set_Pr4386577575007340137rm_a_b,X4: set_term_a_b,X: term_a_b,Y: term_a_b] :
( ( equiv_equiv_term_a_b @ A2 @ R3 )
=> ( ( member_set_term_a_b @ X4 @ ( equiv_4975866614301036301rm_a_b @ A2 @ R3 ) )
=> ( ( ord_le2705286416250468010rm_a_b @ ( insert_term_a_b @ X @ ( insert_term_a_b @ Y @ bot_bot_set_term_a_b ) ) @ X4 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ R3 ) ) ) ) ).
% in_quotient_imp_in_rel
thf(fact_1213_partial__order__on__converse,axiom,
! [A2: set_term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
( ( order_4104115877010626143rm_a_b @ A2 @ ( conver7608559976937845093rm_a_b @ R3 ) )
= ( order_4104115877010626143rm_a_b @ A2 @ R3 ) ) ).
% partial_order_on_converse
thf(fact_1214_quotient__eqI,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,R3: set_Pr2972776593051762503rm_a_b,X4: set_Pr4386577575007340137rm_a_b,Y7: set_Pr4386577575007340137rm_a_b,X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b] :
( ( equiv_19968616203005119rm_a_b @ A2 @ R3 )
=> ( ( member8823642877475723282rm_a_b @ X4 @ ( equiv_8528125694393760336rm_a_b @ A2 @ R3 ) )
=> ( ( member8823642877475723282rm_a_b @ Y7 @ ( equiv_8528125694393760336rm_a_b @ A2 @ R3 ) )
=> ( ( member5869715511025134514rm_a_b @ X @ X4 )
=> ( ( member5869715511025134514rm_a_b @ Y @ Y7 )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ R3 )
=> ( X4 = Y7 ) ) ) ) ) ) ) ).
% quotient_eqI
thf(fact_1215_quotient__eqI,axiom,
! [A2: set_term_a_b,R3: set_Pr4386577575007340137rm_a_b,X4: set_term_a_b,Y7: set_term_a_b,X: term_a_b,Y: term_a_b] :
( ( equiv_equiv_term_a_b @ A2 @ R3 )
=> ( ( member_set_term_a_b @ X4 @ ( equiv_4975866614301036301rm_a_b @ A2 @ R3 ) )
=> ( ( member_set_term_a_b @ Y7 @ ( equiv_4975866614301036301rm_a_b @ A2 @ R3 ) )
=> ( ( member_term_a_b @ X @ X4 )
=> ( ( member_term_a_b @ Y @ Y7 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ R3 )
=> ( X4 = Y7 ) ) ) ) ) ) ) ).
% quotient_eqI
thf(fact_1216_quotient__eq__iff,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,R3: set_Pr2972776593051762503rm_a_b,X4: set_Pr4386577575007340137rm_a_b,Y7: set_Pr4386577575007340137rm_a_b,X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b] :
( ( equiv_19968616203005119rm_a_b @ A2 @ R3 )
=> ( ( member8823642877475723282rm_a_b @ X4 @ ( equiv_8528125694393760336rm_a_b @ A2 @ R3 ) )
=> ( ( member8823642877475723282rm_a_b @ Y7 @ ( equiv_8528125694393760336rm_a_b @ A2 @ R3 ) )
=> ( ( member5869715511025134514rm_a_b @ X @ X4 )
=> ( ( member5869715511025134514rm_a_b @ Y @ Y7 )
=> ( ( X4 = Y7 )
= ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ R3 ) ) ) ) ) ) ) ).
% quotient_eq_iff
thf(fact_1217_quotient__eq__iff,axiom,
! [A2: set_term_a_b,R3: set_Pr4386577575007340137rm_a_b,X4: set_term_a_b,Y7: set_term_a_b,X: term_a_b,Y: term_a_b] :
( ( equiv_equiv_term_a_b @ A2 @ R3 )
=> ( ( member_set_term_a_b @ X4 @ ( equiv_4975866614301036301rm_a_b @ A2 @ R3 ) )
=> ( ( member_set_term_a_b @ Y7 @ ( equiv_4975866614301036301rm_a_b @ A2 @ R3 ) )
=> ( ( member_term_a_b @ X @ X4 )
=> ( ( member_term_a_b @ Y @ Y7 )
=> ( ( X4 = Y7 )
= ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ R3 ) ) ) ) ) ) ) ).
% quotient_eq_iff
thf(fact_1218_in__quotient__imp__closed,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,R3: set_Pr2972776593051762503rm_a_b,X4: set_Pr4386577575007340137rm_a_b,X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b] :
( ( equiv_19968616203005119rm_a_b @ A2 @ R3 )
=> ( ( member8823642877475723282rm_a_b @ X4 @ ( equiv_8528125694393760336rm_a_b @ A2 @ R3 ) )
=> ( ( member5869715511025134514rm_a_b @ X @ X4 )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ R3 )
=> ( member5869715511025134514rm_a_b @ Y @ X4 ) ) ) ) ) ).
% in_quotient_imp_closed
thf(fact_1219_in__quotient__imp__closed,axiom,
! [A2: set_term_a_b,R3: set_Pr4386577575007340137rm_a_b,X4: set_term_a_b,X: term_a_b,Y: term_a_b] :
( ( equiv_equiv_term_a_b @ A2 @ R3 )
=> ( ( member_set_term_a_b @ X4 @ ( equiv_4975866614301036301rm_a_b @ A2 @ R3 ) )
=> ( ( member_term_a_b @ X @ X4 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ R3 )
=> ( member_term_a_b @ Y @ X4 ) ) ) ) ) ).
% in_quotient_imp_closed
thf(fact_1220_eq__equiv__class__iff2,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,R3: set_Pr2972776593051762503rm_a_b,X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b] :
( ( equiv_19968616203005119rm_a_b @ A2 @ R3 )
=> ( ( member5869715511025134514rm_a_b @ X @ A2 )
=> ( ( member5869715511025134514rm_a_b @ Y @ A2 )
=> ( ( ( equiv_8528125694393760336rm_a_b @ ( insert7009541432154983385rm_a_b @ X @ bot_bo197521221353338581rm_a_b ) @ R3 )
= ( equiv_8528125694393760336rm_a_b @ ( insert7009541432154983385rm_a_b @ Y @ bot_bo197521221353338581rm_a_b ) @ R3 ) )
= ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ R3 ) ) ) ) ) ).
% eq_equiv_class_iff2
thf(fact_1221_eq__equiv__class__iff2,axiom,
! [A2: set_term_a_b,R3: set_Pr4386577575007340137rm_a_b,X: term_a_b,Y: term_a_b] :
( ( equiv_equiv_term_a_b @ A2 @ R3 )
=> ( ( member_term_a_b @ X @ A2 )
=> ( ( member_term_a_b @ Y @ A2 )
=> ( ( ( equiv_4975866614301036301rm_a_b @ ( insert_term_a_b @ X @ bot_bot_set_term_a_b ) @ R3 )
= ( equiv_4975866614301036301rm_a_b @ ( insert_term_a_b @ Y @ bot_bot_set_term_a_b ) @ R3 ) )
= ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ R3 ) ) ) ) ) ).
% eq_equiv_class_iff2
thf(fact_1222_Zorns__po__lemma,axiom,
! [R3: set_Pr4386577575007340137rm_a_b] :
( ( order_4104115877010626143rm_a_b @ ( field_term_a_b @ R3 ) @ R3 )
=> ( ! [C6: set_term_a_b] :
( ( member_set_term_a_b @ C6 @ ( chains_term_a_b @ R3 ) )
=> ? [X5: term_a_b] :
( ( member_term_a_b @ X5 @ ( field_term_a_b @ R3 ) )
& ! [Xa3: term_a_b] :
( ( member_term_a_b @ Xa3 @ C6 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Xa3 @ X5 ) @ R3 ) ) ) )
=> ? [X3: term_a_b] :
( ( member_term_a_b @ X3 @ ( field_term_a_b @ R3 ) )
& ! [Xa2: term_a_b] :
( ( member_term_a_b @ Xa2 @ ( field_term_a_b @ R3 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Xa2 ) @ R3 )
=> ( Xa2 = X3 ) ) ) ) ) ) ).
% Zorns_po_lemma
thf(fact_1223_in__lex__prod,axiom,
! [A: term_a_b,B: term_a_b,A5: term_a_b,B5: term_a_b,R3: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( produc7020197800436672577rm_a_b @ A5 @ B5 ) ) @ ( lex_pr6163557265797435481rm_a_b @ R3 @ S ) )
= ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ A5 ) @ R3 )
| ( ( A = A5 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ B5 ) @ S ) ) ) ) ).
% in_lex_prod
thf(fact_1224_Image__Int__eq,axiom,
! [R4: set_Pr4386577575007340137rm_a_b,A2: set_term_a_b,B2: set_term_a_b] :
( ( single1584833695948963373rm_a_b @ ( conver7608559976937845093rm_a_b @ R4 ) )
=> ( ( image_6923787618150785661rm_a_b @ R4 @ ( inf_inf_set_term_a_b @ A2 @ B2 ) )
= ( inf_inf_set_term_a_b @ ( image_6923787618150785661rm_a_b @ R4 @ A2 ) @ ( image_6923787618150785661rm_a_b @ R4 @ B2 ) ) ) ) ).
% Image_Int_eq
thf(fact_1225_single__valued__def,axiom,
( single1584833695948963373rm_a_b
= ( ^ [R6: set_Pr4386577575007340137rm_a_b] :
! [X2: term_a_b,Y3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y3 ) @ R6 )
=> ! [Z3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Z3 ) @ R6 )
=> ( Y3 = Z3 ) ) ) ) ) ).
% single_valued_def
thf(fact_1226_single__valuedI,axiom,
! [R3: set_Pr4386577575007340137rm_a_b] :
( ! [X3: term_a_b,Y2: term_a_b,Z2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y2 ) @ R3 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Z2 ) @ R3 )
=> ( Y2 = Z2 ) ) )
=> ( single1584833695948963373rm_a_b @ R3 ) ) ).
% single_valuedI
thf(fact_1227_single__valuedD,axiom,
! [R3: set_Pr4386577575007340137rm_a_b,X: term_a_b,Y: term_a_b,Z: term_a_b] :
( ( single1584833695948963373rm_a_b @ R3 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ R3 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z ) @ R3 )
=> ( Y = Z ) ) ) ) ).
% single_valuedD
thf(fact_1228_single__valued__relcomp,axiom,
! [R3: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
( ( single1584833695948963373rm_a_b @ R3 )
=> ( ( single1584833695948963373rm_a_b @ S )
=> ( single1584833695948963373rm_a_b @ ( relcom370159955682700863rm_a_b @ R3 @ S ) ) ) ) ).
% single_valued_relcomp
thf(fact_1229_single__valued__confluent,axiom,
! [R3: set_Pr4386577575007340137rm_a_b,X: term_a_b,Y: term_a_b,Z: term_a_b] :
( ( single1584833695948963373rm_a_b @ R3 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ R3 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z ) @ ( transi7742714808557438673rm_a_b @ R3 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ Z ) @ ( transi7742714808557438673rm_a_b @ R3 ) )
| ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Z @ Y ) @ ( transi7742714808557438673rm_a_b @ R3 ) ) ) ) ) ) ).
% single_valued_confluent
thf(fact_1230_Linear__order__Well__order__iff,axiom,
! [R3: set_Pr4386577575007340137rm_a_b] :
( ( order_5388802246213473311rm_a_b @ ( field_term_a_b @ R3 ) @ R3 )
=> ( ( order_5719896216891381494rm_a_b @ ( field_term_a_b @ R3 ) @ R3 )
= ( ! [A6: set_term_a_b] :
( ( ord_le2705286416250468010rm_a_b @ A6 @ ( field_term_a_b @ R3 ) )
=> ( ( A6 != bot_bot_set_term_a_b )
=> ? [X2: term_a_b] :
( ( member_term_a_b @ X2 @ A6 )
& ! [Y3: term_a_b] :
( ( member_term_a_b @ Y3 @ A6 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y3 ) @ R3 ) ) ) ) ) ) ) ) ).
% Linear_order_Well_order_iff
thf(fact_1231_non__strict__ending,axiom,
! [T: nat > term_a_b,R4: set_Pr4386577575007340137rm_a_b,S6: set_Pr4386577575007340137rm_a_b] :
( ! [I3: nat] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( T @ I3 ) @ ( T @ ( suc @ I3 ) ) ) @ ( sup_su6776935440552674877rm_a_b @ R4 @ S6 ) )
=> ( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ R4 @ S6 ) @ S6 )
=> ( ( abstra4720023341729745482rm_a_b @ S6 @ ( insert_term_a_b @ ( T @ zero_zero_nat ) @ bot_bot_set_term_a_b ) )
=> ? [J3: nat] :
! [I4: nat] :
( ( ord_less_eq_nat @ J3 @ I4 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( T @ I4 ) @ ( T @ ( suc @ I4 ) ) ) @ ( minus_5192120951422937424rm_a_b @ R4 @ S6 ) ) ) ) ) ) ).
% non_strict_ending
thf(fact_1232_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_1233_Suc__le__mono,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M ) )
= ( ord_less_eq_nat @ N @ M ) ) ).
% Suc_le_mono
thf(fact_1234_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_1235_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_1236_diff__is__0__eq_H,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( minus_minus_nat @ M @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_1237_diff__is__0__eq,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M @ N ) ) ).
% diff_is_0_eq
thf(fact_1238_SN__onI,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,R3: set_Pr2972776593051762503rm_a_b] :
( ! [F4: nat > produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( F4 @ zero_zero_nat ) @ A2 )
=> ~ ! [I4: nat] : ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( F4 @ I4 ) @ ( F4 @ ( suc @ I4 ) ) ) @ R3 ) )
=> ( abstra2398554102055911763rm_a_b @ R3 @ A2 ) ) ).
% SN_onI
thf(fact_1239_SN__onI,axiom,
! [A2: set_term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
( ! [F4: nat > term_a_b] :
( ( member_term_a_b @ ( F4 @ zero_zero_nat ) @ A2 )
=> ~ ! [I4: nat] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F4 @ I4 ) @ ( F4 @ ( suc @ I4 ) ) ) @ R3 ) )
=> ( abstra4720023341729745482rm_a_b @ R3 @ A2 ) ) ).
% SN_onI
thf(fact_1240_not__SN__onI,axiom,
! [F3: nat > produc357393685978478089rm_a_b,X4: set_Pr4386577575007340137rm_a_b,R4: set_Pr2972776593051762503rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( F3 @ zero_zero_nat ) @ X4 )
=> ( ! [I3: nat] : ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( F3 @ I3 ) @ ( F3 @ ( suc @ I3 ) ) ) @ R4 )
=> ~ ( abstra2398554102055911763rm_a_b @ R4 @ X4 ) ) ) ).
% not_SN_onI
thf(fact_1241_not__SN__onI,axiom,
! [F3: nat > term_a_b,X4: set_term_a_b,R4: set_Pr4386577575007340137rm_a_b] :
( ( member_term_a_b @ ( F3 @ zero_zero_nat ) @ X4 )
=> ( ! [I3: nat] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F3 @ I3 ) @ ( F3 @ ( suc @ I3 ) ) ) @ R4 )
=> ~ ( abstra4720023341729745482rm_a_b @ R4 @ X4 ) ) ) ).
% not_SN_onI
thf(fact_1242_Abstract__Rewriting_Ochain__mono,axiom,
! [R8: set_Pr4386577575007340137rm_a_b,R4: set_Pr4386577575007340137rm_a_b,Seq: nat > term_a_b] :
( ( ord_le118470702582115849rm_a_b @ R8 @ R4 )
=> ( ! [I3: nat] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( Seq @ I3 ) @ ( Seq @ ( suc @ I3 ) ) ) @ R8 )
=> ! [I4: nat] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( Seq @ I4 ) @ ( Seq @ ( suc @ I4 ) ) ) @ R4 ) ) ) ).
% Abstract_Rewriting.chain_mono
thf(fact_1243_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_1244_bot__nat__0_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
= ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_1245_bot__nat__0_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_1246_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_1247_Suc__leD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_leD
thf(fact_1248_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_1249_le__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_eq_nat @ M @ N )
=> ( M
= ( suc @ N ) ) ) ) ).
% le_SucE
thf(fact_1250_le__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_1251_Suc__le__D,axiom,
! [N: nat,M4: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M4 )
=> ? [M2: nat] :
( M4
= ( suc @ M2 ) ) ) ).
% Suc_le_D
thf(fact_1252_le__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
= ( ( ord_less_eq_nat @ M @ N )
| ( M
= ( suc @ N ) ) ) ) ).
% le_Suc_eq
thf(fact_1253_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_1254_not__less__eq__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_eq_nat @ M @ N ) )
= ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).
% not_less_eq_eq
thf(fact_1255_full__nat__induct,axiom,
! [P2: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M5: nat] :
( ( ord_less_eq_nat @ ( suc @ M5 ) @ N3 )
=> ( P2 @ M5 ) )
=> ( P2 @ N3 ) )
=> ( P2 @ N ) ) ).
% full_nat_induct
thf(fact_1256_nat__induct__at__least,axiom,
! [M: nat,N: nat,P2: nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( P2 @ M )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ M @ N3 )
=> ( ( P2 @ N3 )
=> ( P2 @ ( suc @ N3 ) ) ) )
=> ( P2 @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_1257_transitive__stepwise__le,axiom,
! [M: nat,N: nat,R4: nat > nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ! [X3: nat] : ( R4 @ X3 @ X3 )
=> ( ! [X3: nat,Y2: nat,Z2: nat] :
( ( R4 @ X3 @ Y2 )
=> ( ( R4 @ Y2 @ Z2 )
=> ( R4 @ X3 @ Z2 ) ) )
=> ( ! [N3: nat] : ( R4 @ N3 @ ( suc @ N3 ) )
=> ( R4 @ M @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_1258_Suc__diff__le,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( minus_minus_nat @ ( suc @ M ) @ N )
= ( suc @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% Suc_diff_le
thf(fact_1259_lift__Suc__mono__le,axiom,
! [F3: nat > nat,N: nat,N5: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ ( F3 @ N3 ) @ ( F3 @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N @ N5 )
=> ( ord_less_eq_nat @ ( F3 @ N ) @ ( F3 @ N5 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_1260_lift__Suc__antimono__le,axiom,
! [F3: nat > nat,N: nat,N5: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ ( F3 @ ( suc @ N3 ) ) @ ( F3 @ N3 ) )
=> ( ( ord_less_eq_nat @ N @ N5 )
=> ( ord_less_eq_nat @ ( F3 @ N5 ) @ ( F3 @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_1261_relpow__Suc__E2_H,axiom,
! [X: term_a_b,Z: term_a_b,N: nat,A2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z ) @ ( compow4057154403645558940rm_a_b @ ( suc @ N ) @ A2 ) )
=> ~ ! [Y2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y2 ) @ A2 )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y2 @ Z ) @ ( transi7742714808557438673rm_a_b @ A2 ) ) ) ) ).
% relpow_Suc_E2'
thf(fact_1262_relpow_Osimps_I2_J,axiom,
! [N: nat,R4: set_Pr4386577575007340137rm_a_b] :
( ( compow4057154403645558940rm_a_b @ ( suc @ N ) @ R4 )
= ( relcom370159955682700863rm_a_b @ ( compow4057154403645558940rm_a_b @ N @ R4 ) @ R4 ) ) ).
% relpow.simps(2)
thf(fact_1263_relpow__Suc,axiom,
! [N: nat,R3: set_Pr4386577575007340137rm_a_b] :
( ( compow4057154403645558940rm_a_b @ ( suc @ N ) @ R3 )
= ( relcom370159955682700863rm_a_b @ R3 @ ( compow4057154403645558940rm_a_b @ N @ R3 ) ) ) ).
% relpow_Suc
thf(fact_1264_SN__onE,axiom,
! [R3: set_Pr2972776593051762503rm_a_b,A2: set_Pr4386577575007340137rm_a_b] :
( ( abstra2398554102055911763rm_a_b @ R3 @ A2 )
=> ~ ? [F5: nat > produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( F5 @ zero_zero_nat ) @ A2 )
& ! [I3: nat] : ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( F5 @ I3 ) @ ( F5 @ ( suc @ I3 ) ) ) @ R3 ) ) ) ).
% SN_onE
thf(fact_1265_SN__onE,axiom,
! [R3: set_Pr4386577575007340137rm_a_b,A2: set_term_a_b] :
( ( abstra4720023341729745482rm_a_b @ R3 @ A2 )
=> ~ ? [F5: nat > term_a_b] :
( ( member_term_a_b @ ( F5 @ zero_zero_nat ) @ A2 )
& ! [I3: nat] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F5 @ I3 ) @ ( F5 @ ( suc @ I3 ) ) ) @ R3 ) ) ) ).
% SN_onE
thf(fact_1266_SN__on__def,axiom,
( abstra2398554102055911763rm_a_b
= ( ^ [R6: set_Pr2972776593051762503rm_a_b,A6: set_Pr4386577575007340137rm_a_b] :
~ ? [F6: nat > produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( F6 @ zero_zero_nat ) @ A6 )
& ! [I2: nat] : ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( F6 @ I2 ) @ ( F6 @ ( suc @ I2 ) ) ) @ R6 ) ) ) ) ).
% SN_on_def
thf(fact_1267_SN__on__def,axiom,
( abstra4720023341729745482rm_a_b
= ( ^ [R6: set_Pr4386577575007340137rm_a_b,A6: set_term_a_b] :
~ ? [F6: nat > term_a_b] :
( ( member_term_a_b @ ( F6 @ zero_zero_nat ) @ A6 )
& ! [I2: nat] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F6 @ I2 ) @ ( F6 @ ( suc @ I2 ) ) ) @ R6 ) ) ) ) ).
% SN_on_def
thf(fact_1268_not__SN__onE,axiom,
! [R3: set_Pr2972776593051762503rm_a_b,A2: set_Pr4386577575007340137rm_a_b] :
( ~ ( abstra2398554102055911763rm_a_b @ R3 @ A2 )
=> ~ ! [F4: nat > produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( F4 @ zero_zero_nat ) @ A2 )
=> ~ ! [I4: nat] : ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( F4 @ I4 ) @ ( F4 @ ( suc @ I4 ) ) ) @ R3 ) ) ) ).
% not_SN_onE
thf(fact_1269_not__SN__onE,axiom,
! [R3: set_Pr4386577575007340137rm_a_b,A2: set_term_a_b] :
( ~ ( abstra4720023341729745482rm_a_b @ R3 @ A2 )
=> ~ ! [F4: nat > term_a_b] :
( ( member_term_a_b @ ( F4 @ zero_zero_nat ) @ A2 )
=> ~ ! [I4: nat] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F4 @ I4 ) @ ( F4 @ ( suc @ I4 ) ) ) @ R3 ) ) ) ).
% not_SN_onE
thf(fact_1270_relpow__E,axiom,
! [X: term_a_b,Z: term_a_b,N: nat,R4: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z ) @ ( compow4057154403645558940rm_a_b @ N @ R4 ) )
=> ( ( ( N = zero_zero_nat )
=> ( X != Z ) )
=> ~ ! [Y2: term_a_b,M2: nat] :
( ( N
= ( suc @ M2 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y2 ) @ ( compow4057154403645558940rm_a_b @ M2 @ R4 ) )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y2 @ Z ) @ R4 ) ) ) ) ) ).
% relpow_E
thf(fact_1271_relpow__E2,axiom,
! [X: term_a_b,Z: term_a_b,N: nat,R4: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z ) @ ( compow4057154403645558940rm_a_b @ N @ R4 ) )
=> ( ( ( N = zero_zero_nat )
=> ( X != Z ) )
=> ~ ! [Y2: term_a_b,M2: nat] :
( ( N
= ( suc @ M2 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y2 ) @ R4 )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y2 @ Z ) @ ( compow4057154403645558940rm_a_b @ M2 @ R4 ) ) ) ) ) ) ).
% relpow_E2
thf(fact_1272_relpow__Suc__I2,axiom,
! [X: term_a_b,Y: term_a_b,R4: set_Pr4386577575007340137rm_a_b,Z: term_a_b,N: nat] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ R4 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ Z ) @ ( compow4057154403645558940rm_a_b @ N @ R4 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z ) @ ( compow4057154403645558940rm_a_b @ ( suc @ N ) @ R4 ) ) ) ) ).
% relpow_Suc_I2
thf(fact_1273_relpow__Suc__E2,axiom,
! [X: term_a_b,Z: term_a_b,N: nat,R4: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z ) @ ( compow4057154403645558940rm_a_b @ ( suc @ N ) @ R4 ) )
=> ~ ! [Y2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y2 ) @ R4 )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y2 @ Z ) @ ( compow4057154403645558940rm_a_b @ N @ R4 ) ) ) ) ).
% relpow_Suc_E2
% Conjectures (1)
thf(conj_0,conjecture,
rewrit8865997592796547114dp_a_b @ f @ r @ r @ s @ u ).
%------------------------------------------------------------------------------