TPTP Problem File: SLH0052^1.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/0019_Rewriting_LLRG_LV_Mondaic/prob_00253_010024__14020824_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1530 ( 542 unt; 251 typ; 0 def)
% Number of atoms : 3671 (1022 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 13554 ( 368 ~; 30 |; 235 &;11164 @)
% ( 0 <=>;1757 =>; 0 <=; 0 <~>)
% Maximal formula depth : 22 ( 7 avg)
% Number of types : 25 ( 24 usr)
% Number of type conns : 624 ( 624 >; 0 *; 0 +; 0 <<)
% Number of symbols : 230 ( 227 usr; 26 con; 0-3 aty)
% Number of variables : 3848 ( 161 ^;3550 !; 137 ?;3848 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 14:18:58.135
%------------------------------------------------------------------------------
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thf(sy_c_Term__Context_Oground__ctxt_001tf__a_001tf__b,type,
term_ground_ctxt_a_b: subterm_and_ctxt_a_b > $o ).
thf(sy_c_Terms__Positions_Octxt__well__def__hole__path_001tf__a_001tf__b,type,
terms_8374513854926927137th_a_b: set_Pr4934435412358123699_a_nat > subterm_and_ctxt_a_b > $o ).
thf(sy_c_Terms__Positions_Ofunas__rel_001tf__a_001tf__b_001tf__b,type,
terms_7988297476397195622_a_b_b: set_Pr4386577575007340137rm_a_b > set_Pr4934435412358123699_a_nat ).
thf(sy_c_Terms__Positions_Oinv__const__ctxt_001tf__a_001tf__b,type,
terms_3799363064701517935xt_a_b: set_Pr4934435412358123699_a_nat > b > subterm_and_ctxt_a_b > subterm_and_ctxt_a_b ).
thf(sy_c_Terms__Positions_Olinear__term_001tf__a_001tf__b,type,
terms_5523818207328828897rm_a_b: term_a_b > $o ).
thf(sy_c_Terms__Positions_Oterm__to__sig_001tf__a_001tf__b,type,
terms_8519481630511763164ig_a_b: set_Pr4934435412358123699_a_nat > b > term_a_b > term_a_b ).
thf(sy_c_Transitive__Closure_Oacyclic_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
transi5314701259734769157rm_a_b: set_Pr4386577575007340137rm_a_b > $o ).
thf(sy_c_Transitive__Closure_Ortrancl_001t__Nat__Onat,type,
transi2905341329935302413cl_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).
thf(sy_c_Transitive__Closure_Ortrancl_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,
transi5332651807426350186rm_a_b: set_Pr6848273517546626247rm_a_b > set_Pr6848273517546626247rm_a_b ).
thf(sy_c_Transitive__Closure_Ortrancl_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
transi2615809358984392588rm_a_b: set_Pr2972776593051762503rm_a_b > set_Pr2972776593051762503rm_a_b ).
thf(sy_c_Transitive__Closure_Ortrancl_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
transi7742714808557438673rm_a_b: set_Pr4386577575007340137rm_a_b > set_Pr4386577575007340137rm_a_b ).
thf(sy_c_Transitive__Closure_Ortrancl_001tf__a,type,
transitive_rtrancl_a: set_Product_prod_a_a > set_Product_prod_a_a ).
thf(sy_c_Transitive__Closure_Otrancl_001t__Nat__Onat,type,
transi6264000038957366511cl_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).
thf(sy_c_Transitive__Closure_Otrancl_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,
transi4560671716973800200rm_a_b: set_Pr6848273517546626247rm_a_b > set_Pr6848273517546626247rm_a_b ).
thf(sy_c_Transitive__Closure_Otrancl_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
transi9211502839322181930rm_a_b: set_Pr2972776593051762503rm_a_b > set_Pr2972776593051762503rm_a_b ).
thf(sy_c_Transitive__Closure_Otrancl_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
transi7922773638565587891rm_a_b: set_Pr4386577575007340137rm_a_b > set_Pr4386577575007340137rm_a_b ).
thf(sy_c_Transitive__Closure_Otrancl_001tf__a,type,
transitive_trancl_a: set_Product_prod_a_a > set_Product_prod_a_a ).
thf(sy_c_Wellfounded_Olex__prod_001t__Term__Oterm_Itf__a_Mtf__b_J_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
lex_pr6163557265797435481rm_a_b: set_Pr4386577575007340137rm_a_b > set_Pr4386577575007340137rm_a_b > set_Pr2972776593051762503rm_a_b ).
thf(sy_c_Wellfounded_Omeasure_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
measure_term_a_b: ( term_a_b > nat ) > set_Pr4386577575007340137rm_a_b ).
thf(sy_c_Wellfounded_Omlex__prod_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
mlex_prod_term_a_b: ( term_a_b > nat ) > set_Pr4386577575007340137rm_a_b > set_Pr4386577575007340137rm_a_b ).
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__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
same_f7764617898092871340rm_a_b: ( produc357393685978478089rm_a_b > $o ) > ( produc357393685978478089rm_a_b > set_Pr2972776593051762503rm_a_b ) > set_Pr6848273517546626247rm_a_b ).
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_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__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__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__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J,type,
member5724188588386418708_a_nat: product_prod_a_nat > set_Pr4934435412358123699_a_nat > $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_c_member_001tf__b,type,
member_b: b > set_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_sa____,type,
sa: term_a_b ).
thf(sy_v_t,type,
t: term_a_b ).
thf(sy_v_ta____,type,
ta: term_a_b ).
thf(sy_v_u____,type,
u: term_a_b ).
% Relevant facts (1275)
thf(fact_0_rtrancl__into__rtrancl_Oprems,axiom,
term_ground_a_b @ sa ).
% rtrancl_into_rtrancl.prems
thf(fact_1_assms_I2_J,axiom,
term_ground_a_b @ s ).
% assms(2)
thf(fact_2_rtrancl__into__rtrancl_Ohyps_I2_J,axiom,
( ( term_ground_a_b @ sa )
=> ( term_ground_a_b @ ta ) ) ).
% rtrancl_into_rtrancl.hyps(2)
thf(fact_3_rtrancl__into__rtrancl_Ohyps_I3_J,axiom,
member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ta @ u ) @ ( sig_step_a_b @ f @ ( rstep_a_b @ r ) ) ).
% rtrancl_into_rtrancl.hyps(3)
thf(fact_4_rtrancl__into__rtrancl_Ohyps_I1_J,axiom,
member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ sa @ ta ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ f @ ( rstep_a_b @ r ) ) ) ).
% rtrancl_into_rtrancl.hyps(1)
thf(fact_5_llrg__ground__rhs,axiom,
! [R: set_Pr4386577575007340137rm_a_b,L: term_a_b,R2: term_a_b] :
( ( rewrit5000278123254648670_b_a_b @ R )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ L @ R2 ) @ R )
=> ( term_ground_a_b @ R2 ) ) ) ).
% llrg_ground_rhs
thf(fact_6_ground__adapt__vars,axiom,
! [T: term_a_b] :
( ( term_ground_a_b @ ( term_a2780309480206857766_a_b_b @ T ) )
= ( term_ground_a_b @ T ) ) ).
% ground_adapt_vars
thf(fact_7_adapt__vars__adapt__vars,axiom,
! [T: term_a_b] :
( ( term_ground_a_b @ T )
=> ( ( term_a2780309480206857766_a_b_b @ ( term_a2780309480206857766_a_b_b @ T ) )
= T ) ) ).
% adapt_vars_adapt_vars
thf(fact_8_adapt__vars__inj,axiom,
! [X: term_a_b,Y: term_a_b] :
( ( ( term_a2780309480206857766_a_b_b @ X )
= ( term_a2780309480206857766_a_b_b @ Y ) )
=> ( ( term_ground_a_b @ X )
=> ( ( term_ground_a_b @ Y )
=> ( X = Y ) ) ) ) ).
% adapt_vars_inj
thf(fact_9_adapt__vars__reverse,axiom,
! [T: term_a_b,T2: term_a_b] :
( ( term_ground_a_b @ T )
=> ( ( ( term_a2780309480206857766_a_b_b @ T2 )
= T )
=> ( ( term_a2780309480206857766_a_b_b @ T )
= T2 ) ) ) ).
% adapt_vars_reverse
thf(fact_10_ground__map__term,axiom,
! [F: a > a,H: b > b,T: term_a_b] :
( ( term_ground_a_b @ ( map_term_a_a_b_b @ F @ H @ T ) )
= ( term_ground_a_b @ T ) ) ).
% ground_map_term
thf(fact_11_ground__term__of__gterm,axiom,
! [S: ground_gterm_a] : ( term_ground_a_b @ ( ground5786051259813020157rm_a_b @ S ) ) ).
% ground_term_of_gterm
thf(fact_12_ground__term__to__gtermD,axiom,
! [T: term_a_b] :
( ( term_ground_a_b @ T )
=> ? [T3: ground_gterm_a] :
( T
= ( ground5786051259813020157rm_a_b @ T3 ) ) ) ).
% ground_term_to_gtermD
thf(fact_13_llrg__rrsteps__groundness,axiom,
! [R: set_Pr4386577575007340137rm_a_b,S: term_a_b,T: term_a_b,F2: set_Pr4934435412358123699_a_nat] :
( ( rewrit5000278123254648670_b_a_b @ R )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( sig_step_a_b @ F2 @ ( rrstep_a_b @ R ) ) )
=> ( term_ground_a_b @ T ) ) ) ).
% llrg_rrsteps_groundness
thf(fact_14_gound__linear,axiom,
! [T: term_a_b] :
( ( term_ground_a_b @ T )
=> ( terms_5523818207328828897rm_a_b @ T ) ) ).
% gound_linear
thf(fact_15_ground_Osimps_I1_J,axiom,
! [X: b] :
~ ( term_ground_a_b @ ( var_b_a @ X ) ) ).
% ground.simps(1)
thf(fact_16_ground__subst__apply,axiom,
! [T: term_a_b,Sigma: b > term_a_b] :
( ( term_ground_a_b @ T )
=> ( ( subst_7999470309526761004_a_b_b @ T @ Sigma )
= T ) ) ).
% ground_subst_apply
thf(fact_17_assms_I1_J,axiom,
rewrit5000278123254648670_b_a_b @ r ).
% assms(1)
thf(fact_18_assms_I3_J,axiom,
member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ s @ t ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ f @ ( rstep_a_b @ r ) ) ) ).
% assms(3)
thf(fact_19_linear__term_Osimps_I1_J,axiom,
! [Uu: b] : ( terms_5523818207328828897rm_a_b @ ( var_b_a @ Uu ) ) ).
% linear_term.simps(1)
thf(fact_20_adapt__vars2,axiom,
! [T: term_a_b] :
( ( term_a2780309480206857766_a_b_b @ ( term_a2780309480206857766_a_b_b @ T ) )
= ( term_a2780309480206857766_a_b_b @ T ) ) ).
% adapt_vars2
thf(fact_21_rstep__substI,axiom,
! [S: term_a_b,T: term_a_b,R: set_Pr4386577575007340137rm_a_b,Sigma: b > term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( rstep_a_b @ R ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( subst_7999470309526761004_a_b_b @ S @ Sigma ) @ ( subst_7999470309526761004_a_b_b @ T @ Sigma ) ) @ ( rstep_a_b @ R ) ) ) ).
% rstep_substI
thf(fact_22_rrstep__basicI,axiom,
! [L: term_a_b,R2: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ L @ R2 ) @ R )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ L @ R2 ) @ ( rrstep_a_b @ R ) ) ) ).
% rrstep_basicI
thf(fact_23_subst__apply__term__empty,axiom,
! [T: term_a_b] :
( ( subst_7999470309526761004_a_b_b @ T @ var_b_a )
= T ) ).
% subst_apply_term_empty
thf(fact_24_rstep__ruleI,axiom,
! [L: term_a_b,R2: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ L @ R2 ) @ R )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ L @ R2 ) @ ( rstep_a_b @ R ) ) ) ).
% rstep_ruleI
thf(fact_25_rrstep_Ocases,axiom,
! [A1: term_a_b,A2: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A2 ) @ ( rrstep_a_b @ R ) )
=> ~ ! [L2: term_a_b,R3: term_a_b,Sigma2: b > term_a_b] :
( ( A1
= ( subst_7999470309526761004_a_b_b @ L2 @ Sigma2 ) )
=> ( ( A2
= ( subst_7999470309526761004_a_b_b @ R3 @ Sigma2 ) )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ L2 @ R3 ) @ R ) ) ) ) ).
% rrstep.cases
thf(fact_26_rrstep_Osimps,axiom,
! [A1: term_a_b,A2: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A2 ) @ ( rrstep_a_b @ R ) )
= ( ? [L3: term_a_b,R4: term_a_b,Sigma3: b > term_a_b] :
( ( A1
= ( subst_7999470309526761004_a_b_b @ L3 @ Sigma3 ) )
& ( A2
= ( subst_7999470309526761004_a_b_b @ R4 @ Sigma3 ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ L3 @ R4 ) @ R ) ) ) ) ).
% rrstep.simps
thf(fact_27_rrstep_Ointros,axiom,
! [L: term_a_b,R2: term_a_b,R: set_Pr4386577575007340137rm_a_b,Sigma: b > term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ L @ R2 ) @ R )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( subst_7999470309526761004_a_b_b @ L @ Sigma ) @ ( subst_7999470309526761004_a_b_b @ R2 @ Sigma ) ) @ ( rrstep_a_b @ R ) ) ) ).
% rrstep.intros
thf(fact_28_rrstep__subst,axiom,
! [S: term_a_b,T: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( rrstep_a_b @ R ) )
=> ~ ! [L2: term_a_b,R3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ L2 @ R3 ) @ R )
=> ! [Sigma2: b > term_a_b] :
( ( S
= ( subst_7999470309526761004_a_b_b @ L2 @ Sigma2 ) )
=> ( T
!= ( subst_7999470309526761004_a_b_b @ R3 @ Sigma2 ) ) ) ) ) ).
% rrstep_subst
thf(fact_29_term_Oinject_I1_J,axiom,
! [X1: b,Y1: b] :
( ( ( var_b_a @ X1 )
= ( var_b_a @ Y1 ) )
= ( X1 = Y1 ) ) ).
% term.inject(1)
thf(fact_30_rtrancl__idemp,axiom,
! [R2: set_Pr2972776593051762503rm_a_b] :
( ( transi2615809358984392588rm_a_b @ ( transi2615809358984392588rm_a_b @ R2 ) )
= ( transi2615809358984392588rm_a_b @ R2 ) ) ).
% rtrancl_idemp
thf(fact_31_rtrancl__idemp,axiom,
! [R2: set_Pr4386577575007340137rm_a_b] :
( ( transi7742714808557438673rm_a_b @ ( transi7742714808557438673rm_a_b @ R2 ) )
= ( transi7742714808557438673rm_a_b @ R2 ) ) ).
% rtrancl_idemp
thf(fact_32_r__into__rtrancl,axiom,
! [P: produc4523971788519308903rm_a_b,R2: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ P @ R2 )
=> ( member8417600551952982416rm_a_b @ P @ ( transi2615809358984392588rm_a_b @ R2 ) ) ) ).
% r_into_rtrancl
thf(fact_33_r__into__rtrancl,axiom,
! [P: produc357393685978478089rm_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ P @ R2 )
=> ( member5869715511025134514rm_a_b @ P @ ( transi7742714808557438673rm_a_b @ R2 ) ) ) ).
% r_into_rtrancl
thf(fact_34_converse__rtrancl__induct2,axiom,
! [Ax: produc357393685978478089rm_a_b,Ay: produc357393685978478089rm_a_b,Bx: produc357393685978478089rm_a_b,By: produc357393685978478089rm_a_b,R2: 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 @ R2 ) )
=> ( ( P2 @ Bx @ By )
=> ( ! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,Aa: produc357393685978478089rm_a_b,Ba: produc357393685978478089rm_a_b] :
( ( member5793113912742238992rm_a_b @ ( produc2499998789532207959rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( produc1763473618796451543rm_a_b @ Aa @ Ba ) ) @ R2 )
=> ( ( member5793113912742238992rm_a_b @ ( produc2499998789532207959rm_a_b @ ( produc1763473618796451543rm_a_b @ Aa @ Ba ) @ ( produc1763473618796451543rm_a_b @ Bx @ By ) ) @ ( transi5332651807426350186rm_a_b @ R2 ) )
=> ( ( P2 @ Aa @ Ba )
=> ( P2 @ A @ B ) ) ) )
=> ( P2 @ Ax @ Ay ) ) ) ) ).
% converse_rtrancl_induct2
thf(fact_35_converse__rtrancl__induct2,axiom,
! [Ax: term_a_b,Ay: term_a_b,Bx: term_a_b,By: term_a_b,R2: 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 @ R2 ) )
=> ( ( P2 @ Bx @ By )
=> ( ! [A: term_a_b,B: term_a_b,Aa: term_a_b,Ba: term_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( produc7020197800436672577rm_a_b @ Aa @ Ba ) ) @ R2 )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ Aa @ Ba ) @ ( produc7020197800436672577rm_a_b @ Bx @ By ) ) @ ( transi2615809358984392588rm_a_b @ R2 ) )
=> ( ( P2 @ Aa @ Ba )
=> ( P2 @ A @ B ) ) ) )
=> ( P2 @ Ax @ Ay ) ) ) ) ).
% converse_rtrancl_induct2
thf(fact_36_converse__rtranclE2,axiom,
! [Xa: produc357393685978478089rm_a_b,Xb: produc357393685978478089rm_a_b,Za: produc357393685978478089rm_a_b,Zb: produc357393685978478089rm_a_b,R2: set_Pr6848273517546626247rm_a_b] :
( ( member5793113912742238992rm_a_b @ ( produc2499998789532207959rm_a_b @ ( produc1763473618796451543rm_a_b @ Xa @ Xb ) @ ( produc1763473618796451543rm_a_b @ Za @ Zb ) ) @ ( transi5332651807426350186rm_a_b @ R2 ) )
=> ( ( ( produc1763473618796451543rm_a_b @ Xa @ Xb )
!= ( produc1763473618796451543rm_a_b @ Za @ Zb ) )
=> ~ ! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b] :
( ( member5793113912742238992rm_a_b @ ( produc2499998789532207959rm_a_b @ ( produc1763473618796451543rm_a_b @ Xa @ Xb ) @ ( produc1763473618796451543rm_a_b @ A @ B ) ) @ R2 )
=> ~ ( member5793113912742238992rm_a_b @ ( produc2499998789532207959rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( produc1763473618796451543rm_a_b @ Za @ Zb ) ) @ ( transi5332651807426350186rm_a_b @ R2 ) ) ) ) ) ).
% converse_rtranclE2
thf(fact_37_converse__rtranclE2,axiom,
! [Xa: term_a_b,Xb: term_a_b,Za: term_a_b,Zb: term_a_b,R2: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ Xa @ Xb ) @ ( produc7020197800436672577rm_a_b @ Za @ Zb ) ) @ ( transi2615809358984392588rm_a_b @ R2 ) )
=> ( ( ( produc7020197800436672577rm_a_b @ Xa @ Xb )
!= ( produc7020197800436672577rm_a_b @ Za @ Zb ) )
=> ~ ! [A: term_a_b,B: term_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ Xa @ Xb ) @ ( produc7020197800436672577rm_a_b @ A @ B ) ) @ R2 )
=> ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( produc7020197800436672577rm_a_b @ Za @ Zb ) ) @ ( transi2615809358984392588rm_a_b @ R2 ) ) ) ) ) ).
% converse_rtranclE2
thf(fact_38_rtrancl__induct2,axiom,
! [Ax: produc357393685978478089rm_a_b,Ay: produc357393685978478089rm_a_b,Bx: produc357393685978478089rm_a_b,By: produc357393685978478089rm_a_b,R2: 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 @ R2 ) )
=> ( ( P2 @ Ax @ Ay )
=> ( ! [A: produc357393685978478089rm_a_b,B: 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 @ A @ B ) ) @ ( transi5332651807426350186rm_a_b @ R2 ) )
=> ( ( member5793113912742238992rm_a_b @ ( produc2499998789532207959rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( produc1763473618796451543rm_a_b @ Aa @ Ba ) ) @ R2 )
=> ( ( P2 @ A @ B )
=> ( P2 @ Aa @ Ba ) ) ) )
=> ( P2 @ Bx @ By ) ) ) ) ).
% rtrancl_induct2
thf(fact_39_rtrancl__induct2,axiom,
! [Ax: term_a_b,Ay: term_a_b,Bx: term_a_b,By: term_a_b,R2: 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 @ R2 ) )
=> ( ( P2 @ Ax @ Ay )
=> ( ! [A: term_a_b,B: 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 @ A @ B ) ) @ ( transi2615809358984392588rm_a_b @ R2 ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( produc7020197800436672577rm_a_b @ Aa @ Ba ) ) @ R2 )
=> ( ( P2 @ A @ B )
=> ( P2 @ Aa @ Ba ) ) ) )
=> ( P2 @ Bx @ By ) ) ) ) ).
% rtrancl_induct2
thf(fact_40_subst__term__eqI,axiom,
! [Sigma: b > term_a_b,Tau: b > term_a_b] :
( ! [T4: term_a_b] :
( ( subst_7999470309526761004_a_b_b @ T4 @ Sigma )
= ( subst_7999470309526761004_a_b_b @ T4 @ Tau ) )
=> ( Sigma = Tau ) ) ).
% subst_term_eqI
thf(fact_41_subst__term__eqI,axiom,
! [Sigma: produc357393685978478089rm_a_b > term_a_b,Tau: produc357393685978478089rm_a_b > term_a_b] :
( ! [T4: term_a7872195449889686814rm_a_b] :
( ( subst_4153759000330037794_a_b_b @ T4 @ Sigma )
= ( subst_4153759000330037794_a_b_b @ T4 @ Tau ) )
=> ( Sigma = Tau ) ) ).
% subst_term_eqI
thf(fact_42_converse__rtrancl__into__rtrancl,axiom,
! [A3: produc357393685978478089rm_a_b,B2: produc357393685978478089rm_a_b,R2: set_Pr2972776593051762503rm_a_b,C: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A3 @ B2 ) @ R2 )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B2 @ C ) @ ( transi2615809358984392588rm_a_b @ R2 ) )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A3 @ C ) @ ( transi2615809358984392588rm_a_b @ R2 ) ) ) ) ).
% converse_rtrancl_into_rtrancl
thf(fact_43_converse__rtrancl__into__rtrancl,axiom,
! [A3: term_a_b,B2: term_a_b,R2: set_Pr4386577575007340137rm_a_b,C: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B2 ) @ R2 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ C ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ C ) @ ( transi7742714808557438673rm_a_b @ R2 ) ) ) ) ).
% converse_rtrancl_into_rtrancl
thf(fact_44_converse__rtrancl__induct,axiom,
! [A3: produc357393685978478089rm_a_b,B2: produc357393685978478089rm_a_b,R2: set_Pr2972776593051762503rm_a_b,P2: produc357393685978478089rm_a_b > $o] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A3 @ B2 ) @ ( transi2615809358984392588rm_a_b @ R2 ) )
=> ( ( P2 @ B2 )
=> ( ! [Y2: produc357393685978478089rm_a_b,Z: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y2 @ Z ) @ R2 )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Z @ B2 ) @ ( transi2615809358984392588rm_a_b @ R2 ) )
=> ( ( P2 @ Z )
=> ( P2 @ Y2 ) ) ) )
=> ( P2 @ A3 ) ) ) ) ).
% converse_rtrancl_induct
thf(fact_45_converse__rtrancl__induct,axiom,
! [A3: term_a_b,B2: term_a_b,R2: set_Pr4386577575007340137rm_a_b,P2: term_a_b > $o] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B2 ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
=> ( ( P2 @ B2 )
=> ( ! [Y2: term_a_b,Z: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y2 @ Z ) @ R2 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Z @ B2 ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
=> ( ( P2 @ Z )
=> ( P2 @ Y2 ) ) ) )
=> ( P2 @ A3 ) ) ) ) ).
% converse_rtrancl_induct
thf(fact_46_converse__rtranclE,axiom,
! [X: produc357393685978478089rm_a_b,Z2: produc357393685978478089rm_a_b,R2: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Z2 ) @ ( transi2615809358984392588rm_a_b @ R2 ) )
=> ( ( X != Z2 )
=> ~ ! [Y2: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y2 ) @ R2 )
=> ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y2 @ Z2 ) @ ( transi2615809358984392588rm_a_b @ R2 ) ) ) ) ) ).
% converse_rtranclE
thf(fact_47_converse__rtranclE,axiom,
! [X: term_a_b,Z2: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z2 ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
=> ( ( X != Z2 )
=> ~ ! [Y2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y2 ) @ R2 )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y2 @ Z2 ) @ ( transi7742714808557438673rm_a_b @ R2 ) ) ) ) ) ).
% converse_rtranclE
thf(fact_48_rtrancl__induct,axiom,
! [A3: produc357393685978478089rm_a_b,B2: produc357393685978478089rm_a_b,R2: set_Pr2972776593051762503rm_a_b,P2: produc357393685978478089rm_a_b > $o] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A3 @ B2 ) @ ( transi2615809358984392588rm_a_b @ R2 ) )
=> ( ( P2 @ A3 )
=> ( ! [Y2: produc357393685978478089rm_a_b,Z: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A3 @ Y2 ) @ ( transi2615809358984392588rm_a_b @ R2 ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y2 @ Z ) @ R2 )
=> ( ( P2 @ Y2 )
=> ( P2 @ Z ) ) ) )
=> ( P2 @ B2 ) ) ) ) ).
% rtrancl_induct
thf(fact_49_rtrancl__induct,axiom,
! [A3: term_a_b,B2: term_a_b,R2: set_Pr4386577575007340137rm_a_b,P2: term_a_b > $o] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B2 ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
=> ( ( P2 @ A3 )
=> ( ! [Y2: term_a_b,Z: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ Y2 ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y2 @ Z ) @ R2 )
=> ( ( P2 @ Y2 )
=> ( P2 @ Z ) ) ) )
=> ( P2 @ B2 ) ) ) ) ).
% rtrancl_induct
thf(fact_50_rtrancl__trans,axiom,
! [X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b,R2: set_Pr2972776593051762503rm_a_b,Z2: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ ( transi2615809358984392588rm_a_b @ R2 ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y @ Z2 ) @ ( transi2615809358984392588rm_a_b @ R2 ) )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Z2 ) @ ( transi2615809358984392588rm_a_b @ R2 ) ) ) ) ).
% rtrancl_trans
thf(fact_51_rtrancl__trans,axiom,
! [X: term_a_b,Y: term_a_b,R2: set_Pr4386577575007340137rm_a_b,Z2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ Z2 ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z2 ) @ ( transi7742714808557438673rm_a_b @ R2 ) ) ) ) ).
% rtrancl_trans
thf(fact_52_rtranclE,axiom,
! [A3: produc357393685978478089rm_a_b,B2: produc357393685978478089rm_a_b,R2: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A3 @ B2 ) @ ( transi2615809358984392588rm_a_b @ R2 ) )
=> ( ( A3 != B2 )
=> ~ ! [Y2: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A3 @ Y2 ) @ ( transi2615809358984392588rm_a_b @ R2 ) )
=> ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y2 @ B2 ) @ R2 ) ) ) ) ).
% rtranclE
thf(fact_53_rtranclE,axiom,
! [A3: term_a_b,B2: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B2 ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
=> ( ( A3 != B2 )
=> ~ ! [Y2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ Y2 ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y2 @ B2 ) @ R2 ) ) ) ) ).
% rtranclE
thf(fact_54_rtrancl_Ortrancl__into__rtrancl,axiom,
! [A3: produc357393685978478089rm_a_b,B2: produc357393685978478089rm_a_b,R2: set_Pr2972776593051762503rm_a_b,C: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A3 @ B2 ) @ ( transi2615809358984392588rm_a_b @ R2 ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B2 @ C ) @ R2 )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A3 @ C ) @ ( transi2615809358984392588rm_a_b @ R2 ) ) ) ) ).
% rtrancl.rtrancl_into_rtrancl
thf(fact_55_rtrancl_Ortrancl__into__rtrancl,axiom,
! [A3: term_a_b,B2: term_a_b,R2: set_Pr4386577575007340137rm_a_b,C: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B2 ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ C ) @ R2 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ C ) @ ( transi7742714808557438673rm_a_b @ R2 ) ) ) ) ).
% rtrancl.rtrancl_into_rtrancl
thf(fact_56_rtrancl_Ortrancl__refl,axiom,
! [A3: produc357393685978478089rm_a_b,R2: set_Pr2972776593051762503rm_a_b] : ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A3 @ A3 ) @ ( transi2615809358984392588rm_a_b @ R2 ) ) ).
% rtrancl.rtrancl_refl
thf(fact_57_rtrancl_Ortrancl__refl,axiom,
! [A3: term_a_b,R2: set_Pr4386577575007340137rm_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ A3 ) @ ( transi7742714808557438673rm_a_b @ R2 ) ) ).
% rtrancl.rtrancl_refl
thf(fact_58_mem__Collect__eq,axiom,
! [A3: produc4523971788519308903rm_a_b,P2: produc4523971788519308903rm_a_b > $o] :
( ( member8417600551952982416rm_a_b @ A3 @ ( collec1344989105561874898rm_a_b @ P2 ) )
= ( P2 @ A3 ) ) ).
% mem_Collect_eq
thf(fact_59_mem__Collect__eq,axiom,
! [A3: term_a_b,P2: term_a_b > $o] :
( ( member_term_a_b @ A3 @ ( collect_term_a_b @ P2 ) )
= ( P2 @ A3 ) ) ).
% mem_Collect_eq
thf(fact_60_mem__Collect__eq,axiom,
! [A3: produc357393685978478089rm_a_b,P2: produc357393685978478089rm_a_b > $o] :
( ( member5869715511025134514rm_a_b @ A3 @ ( collec99177395925924084rm_a_b @ P2 ) )
= ( P2 @ A3 ) ) ).
% mem_Collect_eq
thf(fact_61_Collect__mem__eq,axiom,
! [A4: set_Pr2972776593051762503rm_a_b] :
( ( collec1344989105561874898rm_a_b
@ ^ [X2: produc4523971788519308903rm_a_b] : ( member8417600551952982416rm_a_b @ X2 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_62_Collect__mem__eq,axiom,
! [A4: set_term_a_b] :
( ( collect_term_a_b
@ ^ [X2: term_a_b] : ( member_term_a_b @ X2 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_63_Collect__mem__eq,axiom,
! [A4: set_Pr4386577575007340137rm_a_b] :
( ( collec99177395925924084rm_a_b
@ ^ [X2: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ X2 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_64_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_65_rtrancl_Osimps,axiom,
! [A1: produc357393685978478089rm_a_b,A2: produc357393685978478089rm_a_b,R2: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A1 @ A2 ) @ ( transi2615809358984392588rm_a_b @ R2 ) )
= ( ? [A5: produc357393685978478089rm_a_b] :
( ( A1 = A5 )
& ( A2 = A5 ) )
| ? [A5: produc357393685978478089rm_a_b,B3: produc357393685978478089rm_a_b,C2: produc357393685978478089rm_a_b] :
( ( A1 = A5 )
& ( A2 = C2 )
& ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A5 @ B3 ) @ ( transi2615809358984392588rm_a_b @ R2 ) )
& ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B3 @ C2 ) @ R2 ) ) ) ) ).
% rtrancl.simps
thf(fact_66_rtrancl_Osimps,axiom,
! [A1: term_a_b,A2: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A2 ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
= ( ? [A5: term_a_b] :
( ( A1 = A5 )
& ( A2 = A5 ) )
| ? [A5: term_a_b,B3: term_a_b,C2: term_a_b] :
( ( A1 = A5 )
& ( A2 = C2 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A5 @ B3 ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B3 @ C2 ) @ R2 ) ) ) ) ).
% rtrancl.simps
thf(fact_67_rtrancl_Ocases,axiom,
! [A1: produc357393685978478089rm_a_b,A2: produc357393685978478089rm_a_b,R2: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A1 @ A2 ) @ ( transi2615809358984392588rm_a_b @ R2 ) )
=> ( ( A2 != A1 )
=> ~ ! [B: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A1 @ B ) @ ( transi2615809358984392588rm_a_b @ R2 ) )
=> ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B @ A2 ) @ R2 ) ) ) ) ).
% rtrancl.cases
thf(fact_68_rtrancl_Ocases,axiom,
! [A1: term_a_b,A2: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A2 ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
=> ( ( A2 != A1 )
=> ~ ! [B: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ B ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A2 ) @ R2 ) ) ) ) ).
% rtrancl.cases
thf(fact_69_subst__apply__eq__Var,axiom,
! [S: term_a_b,Sigma: b > term_a_b,X: b] :
( ( ( subst_7999470309526761004_a_b_b @ S @ Sigma )
= ( var_b_a @ X ) )
=> ~ ! [Y2: b] :
( ( S
= ( var_b_a @ Y2 ) )
=> ( ( Sigma @ Y2 )
!= ( var_b_a @ X ) ) ) ) ).
% subst_apply_eq_Var
thf(fact_70_subst__apply__eq__Var,axiom,
! [S: term_a7872195449889686814rm_a_b,Sigma: produc357393685978478089rm_a_b > term_a_b,X: b] :
( ( ( subst_4153759000330037794_a_b_b @ S @ Sigma )
= ( var_b_a @ X ) )
=> ~ ! [Y2: produc357393685978478089rm_a_b] :
( ( S
= ( var_Pr9194790515074578440_a_b_a @ Y2 ) )
=> ( ( Sigma @ Y2 )
!= ( var_b_a @ X ) ) ) ) ).
% subst_apply_eq_Var
thf(fact_71_subst__apply__term_Osimps_I1_J,axiom,
! [X: b,Sigma: b > term_a_b] :
( ( subst_7999470309526761004_a_b_b @ ( var_b_a @ X ) @ Sigma )
= ( Sigma @ X ) ) ).
% subst_apply_term.simps(1)
thf(fact_72_subst__apply__term_Osimps_I1_J,axiom,
! [X: produc357393685978478089rm_a_b,Sigma: produc357393685978478089rm_a_b > term_a_b] :
( ( subst_4153759000330037794_a_b_b @ ( var_Pr9194790515074578440_a_b_a @ X ) @ Sigma )
= ( Sigma @ X ) ) ).
% subst_apply_term.simps(1)
thf(fact_73_Term_Oterm_Osimps_I9_J,axiom,
! [F1: a > a,F22: b > b,X1: b] :
( ( map_term_a_a_b_b @ F1 @ F22 @ ( var_b_a @ X1 ) )
= ( var_b_a @ ( F22 @ X1 ) ) ) ).
% Term.term.simps(9)
thf(fact_74_old_Oprod_Oinject,axiom,
! [A3: produc357393685978478089rm_a_b,B2: produc357393685978478089rm_a_b,A6: produc357393685978478089rm_a_b,B4: produc357393685978478089rm_a_b] :
( ( ( produc1763473618796451543rm_a_b @ A3 @ B2 )
= ( produc1763473618796451543rm_a_b @ A6 @ B4 ) )
= ( ( A3 = A6 )
& ( B2 = B4 ) ) ) ).
% old.prod.inject
thf(fact_75_old_Oprod_Oinject,axiom,
! [A3: term_a_b,B2: term_a_b,A6: term_a_b,B4: term_a_b] :
( ( ( produc7020197800436672577rm_a_b @ A3 @ B2 )
= ( produc7020197800436672577rm_a_b @ A6 @ B4 ) )
= ( ( A3 = A6 )
& ( B2 = B4 ) ) ) ).
% old.prod.inject
thf(fact_76_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_77_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_78_gterm__of__term__inv,axiom,
! [T: term_a_b] :
( ( term_ground_a_b @ T )
=> ( ( ground5786051259813020157rm_a_b @ ( ground3594116404241818627rm_a_b @ T ) )
= T ) ) ).
% gterm_of_term_inv
thf(fact_79_rename__subst__domain__range__Var__rhs,axiom,
! [Rho: b > term_a_b] :
( ( rename5388498879213883313_b_a_b @ Rho @ var_b_a )
= var_b_a ) ).
% rename_subst_domain_range_Var_rhs
thf(fact_80_rename__subst__domain__range__Var__lhs,axiom,
! [Sigma: b > term_a_b] :
( ( rename5388498879213883313_b_a_b @ var_b_a @ Sigma )
= Sigma ) ).
% rename_subst_domain_range_Var_lhs
thf(fact_81_restrict__subst__domain__Var,axiom,
! [V: set_b] :
( ( restri22458263168500592in_b_a @ V @ var_b_a )
= var_b_a ) ).
% restrict_subst_domain_Var
thf(fact_82_rstepI,axiom,
! [L: term_a_b,R2: term_a_b,R: set_Pr4386577575007340137rm_a_b,S: term_a_b,C3: subterm_and_ctxt_a_b,Sigma: b > term_a_b,T: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ L @ R2 ) @ R )
=> ( ( S
= ( subter2376574525758040790rm_a_b @ C3 @ ( subst_7999470309526761004_a_b_b @ L @ Sigma ) ) )
=> ( ( T
= ( subter2376574525758040790rm_a_b @ C3 @ ( subst_7999470309526761004_a_b_b @ R2 @ Sigma ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( rstep_a_b @ R ) ) ) ) ) ).
% rstepI
thf(fact_83_map__term__of__gterm,axiom,
! [F: a > a,G: b > b,T: ground_gterm_a] :
( ( map_term_a_a_b_b @ F @ G @ ( ground5786051259813020157rm_a_b @ T ) )
= ( ground5786051259813020157rm_a_b @ ( ground_map_gterm_a_a @ F @ T ) ) ) ).
% map_term_of_gterm
thf(fact_84_ssubst__Pair__rhs,axiom,
! [R2: produc357393685978478089rm_a_b,S: produc357393685978478089rm_a_b,R5: set_Pr2972776593051762503rm_a_b,S2: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ R2 @ S ) @ R5 )
=> ( ( S2 = S )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ R2 @ S2 ) @ R5 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_85_ssubst__Pair__rhs,axiom,
! [R2: term_a_b,S: term_a_b,R5: set_Pr4386577575007340137rm_a_b,S2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ R2 @ S ) @ R5 )
=> ( ( S2 = S )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ R2 @ S2 ) @ R5 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_86_term__of__gterm__inv,axiom,
! [T: ground_gterm_a] :
( ( ground3594116404241818627rm_a_b @ ( ground5786051259813020157rm_a_b @ T ) )
= T ) ).
% term_of_gterm_inv
thf(fact_87_rstep__ctxtI,axiom,
! [S: term_a_b,T: term_a_b,R: set_Pr4386577575007340137rm_a_b,C3: subterm_and_ctxt_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( rstep_a_b @ R ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( subter2376574525758040790rm_a_b @ C3 @ S ) @ ( subter2376574525758040790rm_a_b @ C3 @ T ) ) @ ( rstep_a_b @ R ) ) ) ).
% rstep_ctxtI
thf(fact_88_map__gterm__of__term,axiom,
! [T: term_a_b,F: a > a,G: b > b] :
( ( term_ground_a_b @ T )
=> ( ( ground3594116404241818627rm_a_b @ ( map_term_a_a_b_b @ F @ G @ T ) )
= ( ground_map_gterm_a_a @ F @ ( ground3594116404241818627rm_a_b @ T ) ) ) ) ).
% map_gterm_of_term
thf(fact_89_restrict__subst__domain__def,axiom,
( restri22458263168500592in_b_a
= ( ^ [V2: set_b,Sigma3: b > term_a_b,X2: b] : ( if_term_a_b @ ( member_b @ X2 @ V2 ) @ ( Sigma3 @ X2 ) @ ( var_b_a @ X2 ) ) ) ) ).
% restrict_subst_domain_def
thf(fact_90_rstep__imp__C__s__r,axiom,
! [S: term_a_b,T: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( rstep_a_b @ R ) )
=> ? [C4: subterm_and_ctxt_a_b,Sigma2: b > term_a_b,L2: term_a_b,R3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ L2 @ R3 ) @ R )
& ( S
= ( subter2376574525758040790rm_a_b @ C4 @ ( subst_7999470309526761004_a_b_b @ L2 @ Sigma2 ) ) )
& ( T
= ( subter2376574525758040790rm_a_b @ C4 @ ( subst_7999470309526761004_a_b_b @ R3 @ Sigma2 ) ) ) ) ) ).
% rstep_imp_C_s_r
thf(fact_91_rstep__imp__C__s__r_H,axiom,
! [S: term_a_b,T: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( rstep_a_b @ R ) )
=> ~ ! [C4: subterm_and_ctxt_a_b,L2: term_a_b,R3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ L2 @ R3 ) @ R )
=> ! [Sigma2: b > term_a_b] :
( ( S
= ( subter2376574525758040790rm_a_b @ C4 @ ( subst_7999470309526761004_a_b_b @ L2 @ Sigma2 ) ) )
=> ( T
!= ( subter2376574525758040790rm_a_b @ C4 @ ( subst_7999470309526761004_a_b_b @ R3 @ Sigma2 ) ) ) ) ) ) ).
% rstep_imp_C_s_r'
thf(fact_92_rstep_Ocases,axiom,
! [A1: term_a_b,A2: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A2 ) @ ( rstep_a_b @ R ) )
=> ~ ! [S3: term_a_b,T4: term_a_b,C4: subterm_and_ctxt_a_b] :
( ( A1
= ( subter2376574525758040790rm_a_b @ C4 @ S3 ) )
=> ( ( A2
= ( subter2376574525758040790rm_a_b @ C4 @ T4 ) )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S3 @ T4 ) @ ( rrstep_a_b @ R ) ) ) ) ) ).
% rstep.cases
thf(fact_93_rstep_Osimps,axiom,
! [A1: term_a_b,A2: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A2 ) @ ( rstep_a_b @ R ) )
= ( ? [S4: term_a_b,T5: term_a_b,C5: subterm_and_ctxt_a_b] :
( ( A1
= ( subter2376574525758040790rm_a_b @ C5 @ S4 ) )
& ( A2
= ( subter2376574525758040790rm_a_b @ C5 @ T5 ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S4 @ T5 ) @ ( rrstep_a_b @ R ) ) ) ) ) ).
% rstep.simps
thf(fact_94_rstep_Ointros,axiom,
! [S: term_a_b,T: term_a_b,R: set_Pr4386577575007340137rm_a_b,C3: subterm_and_ctxt_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( rrstep_a_b @ R ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( subter2376574525758040790rm_a_b @ C3 @ S ) @ ( subter2376574525758040790rm_a_b @ C3 @ T ) ) @ ( rstep_a_b @ R ) ) ) ).
% rstep.intros
thf(fact_95_old_Oprod_Oexhaust,axiom,
! [Y: produc4523971788519308903rm_a_b] :
~ ! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b] :
( Y
!= ( produc1763473618796451543rm_a_b @ A @ B ) ) ).
% old.prod.exhaust
thf(fact_96_old_Oprod_Oexhaust,axiom,
! [Y: produc357393685978478089rm_a_b] :
~ ! [A: term_a_b,B: term_a_b] :
( Y
!= ( produc7020197800436672577rm_a_b @ A @ B ) ) ).
% old.prod.exhaust
thf(fact_97_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_98_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_99_prod__cases,axiom,
! [P2: produc4523971788519308903rm_a_b > $o,P: produc4523971788519308903rm_a_b] :
( ! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b] : ( P2 @ ( produc1763473618796451543rm_a_b @ A @ B ) )
=> ( P2 @ P ) ) ).
% prod_cases
thf(fact_100_prod__cases,axiom,
! [P2: produc357393685978478089rm_a_b > $o,P: produc357393685978478089rm_a_b] :
( ! [A: term_a_b,B: term_a_b] : ( P2 @ ( produc7020197800436672577rm_a_b @ A @ B ) )
=> ( P2 @ P ) ) ).
% prod_cases
thf(fact_101_Pair__inject,axiom,
! [A3: produc357393685978478089rm_a_b,B2: produc357393685978478089rm_a_b,A6: produc357393685978478089rm_a_b,B4: produc357393685978478089rm_a_b] :
( ( ( produc1763473618796451543rm_a_b @ A3 @ B2 )
= ( produc1763473618796451543rm_a_b @ A6 @ B4 ) )
=> ~ ( ( A3 = A6 )
=> ( B2 != B4 ) ) ) ).
% Pair_inject
thf(fact_102_Pair__inject,axiom,
! [A3: term_a_b,B2: term_a_b,A6: term_a_b,B4: term_a_b] :
( ( ( produc7020197800436672577rm_a_b @ A3 @ B2 )
= ( produc7020197800436672577rm_a_b @ A6 @ B4 ) )
=> ~ ( ( A3 = A6 )
=> ( B2 != B4 ) ) ) ).
% Pair_inject
thf(fact_103_prod__cases3,axiom,
! [Y: produc4523971788519308903rm_a_b] :
~ ! [A: produc357393685978478089rm_a_b,B: term_a_b,C6: term_a_b] :
( Y
!= ( produc1763473618796451543rm_a_b @ A @ ( produc7020197800436672577rm_a_b @ B @ C6 ) ) ) ).
% prod_cases3
thf(fact_104_prod__induct3,axiom,
! [P2: produc4523971788519308903rm_a_b > $o,X: produc4523971788519308903rm_a_b] :
( ! [A: produc357393685978478089rm_a_b,B: term_a_b,C6: term_a_b] : ( P2 @ ( produc1763473618796451543rm_a_b @ A @ ( produc7020197800436672577rm_a_b @ B @ C6 ) ) )
=> ( P2 @ X ) ) ).
% prod_induct3
thf(fact_105_ctxt__eq,axiom,
! [C3: subterm_and_ctxt_a_b,S: term_a_b,T: term_a_b] :
( ( ( subter2376574525758040790rm_a_b @ C3 @ S )
= ( subter2376574525758040790rm_a_b @ C3 @ T ) )
= ( S = T ) ) ).
% ctxt_eq
thf(fact_106_adapt__vars__ctxt,axiom,
! [C3: subterm_and_ctxt_a_b,T: term_a_b] :
( ( term_a2780309480206857766_a_b_b @ ( subter2376574525758040790rm_a_b @ C3 @ T ) )
= ( subter2376574525758040790rm_a_b @ ( term_a1758706544955394649_a_b_b @ C3 ) @ ( term_a2780309480206857766_a_b_b @ T ) ) ) ).
% adapt_vars_ctxt
thf(fact_107_same__fstI,axiom,
! [P2: produc357393685978478089rm_a_b > $o,X: produc357393685978478089rm_a_b,Y3: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b,R5: produc357393685978478089rm_a_b > set_Pr2972776593051762503rm_a_b] :
( ( P2 @ X )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y3 @ Y ) @ ( R5 @ X ) )
=> ( member5793113912742238992rm_a_b @ ( produc2499998789532207959rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y3 ) @ ( produc1763473618796451543rm_a_b @ X @ Y ) ) @ ( same_f7764617898092871340rm_a_b @ P2 @ R5 ) ) ) ) ).
% same_fstI
thf(fact_108_same__fstI,axiom,
! [P2: term_a_b > $o,X: term_a_b,Y3: term_a_b,Y: term_a_b,R5: term_a_b > set_Pr4386577575007340137rm_a_b] :
( ( P2 @ X )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ Y ) @ ( R5 @ X ) )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y3 ) @ ( produc7020197800436672577rm_a_b @ X @ Y ) ) @ ( same_f843731131676418198rm_a_b @ P2 @ R5 ) ) ) ) ).
% same_fstI
thf(fact_109_ground__ctxt__apply,axiom,
! [C3: subterm_and_ctxt_a_b,T: term_a_b] :
( ( term_ground_a_b @ ( subter2376574525758040790rm_a_b @ C3 @ T ) )
= ( ( term_ground_ctxt_a_b @ C3 )
& ( term_ground_a_b @ T ) ) ) ).
% ground_ctxt_apply
thf(fact_110_restrict__subst__domain__empty,axiom,
! [Sigma: b > term_a_b] :
( ( restri22458263168500592in_b_a @ bot_bot_set_b @ Sigma )
= var_b_a ) ).
% restrict_subst_domain_empty
thf(fact_111_curry__conv,axiom,
( produc6008550485034061667_a_b_o
= ( ^ [F3: produc357393685978478089rm_a_b > $o,A5: term_a_b,B3: term_a_b] : ( F3 @ ( produc7020197800436672577rm_a_b @ A5 @ B3 ) ) ) ) ).
% curry_conv
thf(fact_112_srsteps__with__root__step__def,axiom,
( srstep7844470518422762656ep_a_b
= ( ^ [F4: set_Pr4934435412358123699_a_nat,R6: set_Pr4386577575007340137rm_a_b] : ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F4 @ ( rstep_a_b @ R6 ) ) ) @ ( relcom370159955682700863rm_a_b @ ( sig_step_a_b @ F4 @ ( rrstep_a_b @ R6 ) ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F4 @ ( rstep_a_b @ R6 ) ) ) ) ) ) ) ).
% srsteps_with_root_step_def
thf(fact_113_meetI,axiom,
! [A3: produc357393685978478089rm_a_b,B2: produc357393685978478089rm_a_b,A4: set_Pr2972776593051762503rm_a_b,C: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A3 @ B2 ) @ ( transi2615809358984392588rm_a_b @ A4 ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A3 @ C ) @ ( transi2615809358984392588rm_a_b @ A4 ) )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B2 @ C ) @ ( abstra8296971427885378886rm_a_b @ A4 ) ) ) ) ).
% meetI
thf(fact_114_meetI,axiom,
! [A3: term_a_b,B2: term_a_b,A4: set_Pr4386577575007340137rm_a_b,C: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B2 ) @ ( transi7742714808557438673rm_a_b @ A4 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ C ) @ ( transi7742714808557438673rm_a_b @ A4 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ C ) @ ( abstra1093061187967292567rm_a_b @ A4 ) ) ) ) ).
% meetI
thf(fact_115_curryI,axiom,
! [F: produc4523971788519308903rm_a_b > $o,A3: produc357393685978478089rm_a_b,B2: produc357393685978478089rm_a_b] :
( ( F @ ( produc1763473618796451543rm_a_b @ A3 @ B2 ) )
=> ( produc3713113261896612557_a_b_o @ F @ A3 @ B2 ) ) ).
% curryI
thf(fact_116_curryI,axiom,
! [F: produc357393685978478089rm_a_b > $o,A3: term_a_b,B2: term_a_b] :
( ( F @ ( produc7020197800436672577rm_a_b @ A3 @ B2 ) )
=> ( produc6008550485034061667_a_b_o @ F @ A3 @ B2 ) ) ).
% curryI
thf(fact_117_rtrancl__idemp__self__comp,axiom,
! [R5: set_Pr2972776593051762503rm_a_b] :
( ( relcom6159724393391289352rm_a_b @ ( transi2615809358984392588rm_a_b @ R5 ) @ ( transi2615809358984392588rm_a_b @ R5 ) )
= ( transi2615809358984392588rm_a_b @ R5 ) ) ).
% rtrancl_idemp_self_comp
thf(fact_118_rtrancl__idemp__self__comp,axiom,
! [R5: set_Pr4386577575007340137rm_a_b] :
( ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ R5 ) @ ( transi7742714808557438673rm_a_b @ R5 ) )
= ( transi7742714808557438673rm_a_b @ R5 ) ) ).
% rtrancl_idemp_self_comp
thf(fact_119_relcomp3__I,axiom,
! [T: produc357393685978478089rm_a_b,U: produc357393685978478089rm_a_b,A4: set_Pr2972776593051762503rm_a_b,S: produc357393685978478089rm_a_b,B5: set_Pr2972776593051762503rm_a_b,V3: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ T @ U ) @ A4 )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ S @ T ) @ B5 )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ U @ V3 ) @ B5 )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ S @ V3 ) @ ( relcom6159724393391289352rm_a_b @ B5 @ ( relcom6159724393391289352rm_a_b @ A4 @ B5 ) ) ) ) ) ) ).
% relcomp3_I
thf(fact_120_relcomp3__I,axiom,
! [T: term_a_b,U: term_a_b,A4: set_Pr4386577575007340137rm_a_b,S: term_a_b,B5: set_Pr4386577575007340137rm_a_b,V3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T @ U ) @ A4 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ B5 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ U @ V3 ) @ B5 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ V3 ) @ ( relcom370159955682700863rm_a_b @ B5 @ ( relcom370159955682700863rm_a_b @ A4 @ B5 ) ) ) ) ) ) ).
% relcomp3_I
thf(fact_121_rtrancl__O__shift,axiom,
! [S5: set_Pr4802440762858034464rm_a_b,R5: set_Pr6510905305361987970rm_a_b] :
( ( relcom6590495161205167518rm_a_b @ ( transi7742714808557438673rm_a_b @ ( relcom8422713090239460168rm_a_b @ S5 @ R5 ) ) @ S5 )
= ( relcom4000041261975091669rm_a_b @ S5 @ ( transi2615809358984392588rm_a_b @ ( relcom8086516547131752875rm_a_b @ R5 @ S5 ) ) ) ) ).
% rtrancl_O_shift
thf(fact_122_rtrancl__O__shift,axiom,
! [S5: set_Pr6510905305361987970rm_a_b,R5: set_Pr4802440762858034464rm_a_b] :
( ( relcom695362439311269717rm_a_b @ ( transi2615809358984392588rm_a_b @ ( relcom8086516547131752875rm_a_b @ S5 @ R5 ) ) @ S5 )
= ( relcom3777457926417599474rm_a_b @ S5 @ ( transi7742714808557438673rm_a_b @ ( relcom8422713090239460168rm_a_b @ R5 @ S5 ) ) ) ) ).
% rtrancl_O_shift
thf(fact_123_rtrancl__O__shift,axiom,
! [S5: set_Pr2972776593051762503rm_a_b,R5: set_Pr2972776593051762503rm_a_b] :
( ( relcom6159724393391289352rm_a_b @ ( transi2615809358984392588rm_a_b @ ( relcom6159724393391289352rm_a_b @ S5 @ R5 ) ) @ S5 )
= ( relcom6159724393391289352rm_a_b @ S5 @ ( transi2615809358984392588rm_a_b @ ( relcom6159724393391289352rm_a_b @ R5 @ S5 ) ) ) ) ).
% rtrancl_O_shift
thf(fact_124_rtrancl__O__shift,axiom,
! [S5: set_Pr4386577575007340137rm_a_b,R5: set_Pr4386577575007340137rm_a_b] :
( ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( relcom370159955682700863rm_a_b @ S5 @ R5 ) ) @ S5 )
= ( relcom370159955682700863rm_a_b @ S5 @ ( transi7742714808557438673rm_a_b @ ( relcom370159955682700863rm_a_b @ R5 @ S5 ) ) ) ) ).
% rtrancl_O_shift
thf(fact_125_r__comp__rtrancl__eq,axiom,
! [R2: set_Pr2972776593051762503rm_a_b] :
( ( relcom6159724393391289352rm_a_b @ R2 @ ( transi2615809358984392588rm_a_b @ R2 ) )
= ( relcom6159724393391289352rm_a_b @ ( transi2615809358984392588rm_a_b @ R2 ) @ R2 ) ) ).
% r_comp_rtrancl_eq
thf(fact_126_r__comp__rtrancl__eq,axiom,
! [R2: set_Pr4386577575007340137rm_a_b] :
( ( relcom370159955682700863rm_a_b @ R2 @ ( transi7742714808557438673rm_a_b @ R2 ) )
= ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ R2 ) @ R2 ) ) ).
% r_comp_rtrancl_eq
thf(fact_127_steps__map,axiom,
! [P2: term_a_b > $o,Q: set_Pr4386577575007340137rm_a_b > $o,F: term_a_b > produc357393685978478089rm_a_b,G: set_Pr4386577575007340137rm_a_b > set_Pr2972776593051762503rm_a_b,T: term_a_b,R5: set_Pr4386577575007340137rm_a_b,S5: set_Pr4386577575007340137rm_a_b,U: term_a_b] :
( ! [T4: term_a_b,U2: term_a_b,R7: set_Pr4386577575007340137rm_a_b] :
( ( P2 @ T4 )
=> ( ( Q @ R7 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T4 @ U2 ) @ R7 )
=> ( ( P2 @ U2 )
& ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( F @ T4 ) @ ( F @ U2 ) ) @ ( G @ R7 ) ) ) ) ) )
=> ( ( P2 @ T )
=> ( ( Q @ R5 )
=> ( ( Q @ S5 )
=> ( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T @ U ) @ ( transi7742714808557438673rm_a_b @ R5 ) )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( F @ T ) @ ( F @ U ) ) @ ( transi2615809358984392588rm_a_b @ ( G @ R5 ) ) ) )
& ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T @ U ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ R5 ) @ ( relcom370159955682700863rm_a_b @ S5 @ ( transi7742714808557438673rm_a_b @ R5 ) ) ) )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( F @ T ) @ ( F @ U ) ) @ ( relcom6159724393391289352rm_a_b @ ( transi2615809358984392588rm_a_b @ ( G @ R5 ) ) @ ( relcom6159724393391289352rm_a_b @ ( G @ S5 ) @ ( transi2615809358984392588rm_a_b @ ( G @ R5 ) ) ) ) ) ) ) ) ) ) ) ).
% steps_map
thf(fact_128_steps__map,axiom,
! [P2: produc357393685978478089rm_a_b > $o,Q: set_Pr2972776593051762503rm_a_b > $o,F: produc357393685978478089rm_a_b > term_a_b,G: set_Pr2972776593051762503rm_a_b > set_Pr4386577575007340137rm_a_b,T: produc357393685978478089rm_a_b,R5: set_Pr2972776593051762503rm_a_b,S5: set_Pr2972776593051762503rm_a_b,U: produc357393685978478089rm_a_b] :
( ! [T4: produc357393685978478089rm_a_b,U2: produc357393685978478089rm_a_b,R7: set_Pr2972776593051762503rm_a_b] :
( ( P2 @ T4 )
=> ( ( Q @ R7 )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ T4 @ U2 ) @ R7 )
=> ( ( P2 @ U2 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F @ T4 ) @ ( F @ U2 ) ) @ ( G @ R7 ) ) ) ) ) )
=> ( ( P2 @ T )
=> ( ( Q @ R5 )
=> ( ( Q @ S5 )
=> ( ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ T @ U ) @ ( transi2615809358984392588rm_a_b @ R5 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F @ T ) @ ( F @ U ) ) @ ( transi7742714808557438673rm_a_b @ ( G @ R5 ) ) ) )
& ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ T @ U ) @ ( relcom6159724393391289352rm_a_b @ ( transi2615809358984392588rm_a_b @ R5 ) @ ( relcom6159724393391289352rm_a_b @ S5 @ ( transi2615809358984392588rm_a_b @ R5 ) ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F @ T ) @ ( F @ U ) ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( G @ R5 ) ) @ ( relcom370159955682700863rm_a_b @ ( G @ S5 ) @ ( transi7742714808557438673rm_a_b @ ( G @ R5 ) ) ) ) ) ) ) ) ) ) ) ).
% steps_map
thf(fact_129_steps__map,axiom,
! [P2: produc357393685978478089rm_a_b > $o,Q: set_Pr2972776593051762503rm_a_b > $o,F: produc357393685978478089rm_a_b > produc357393685978478089rm_a_b,G: set_Pr2972776593051762503rm_a_b > set_Pr2972776593051762503rm_a_b,T: produc357393685978478089rm_a_b,R5: set_Pr2972776593051762503rm_a_b,S5: set_Pr2972776593051762503rm_a_b,U: produc357393685978478089rm_a_b] :
( ! [T4: produc357393685978478089rm_a_b,U2: produc357393685978478089rm_a_b,R7: set_Pr2972776593051762503rm_a_b] :
( ( P2 @ T4 )
=> ( ( Q @ R7 )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ T4 @ U2 ) @ R7 )
=> ( ( P2 @ U2 )
& ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( F @ T4 ) @ ( F @ U2 ) ) @ ( G @ R7 ) ) ) ) ) )
=> ( ( P2 @ T )
=> ( ( Q @ R5 )
=> ( ( Q @ S5 )
=> ( ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ T @ U ) @ ( transi2615809358984392588rm_a_b @ R5 ) )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( F @ T ) @ ( F @ U ) ) @ ( transi2615809358984392588rm_a_b @ ( G @ R5 ) ) ) )
& ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ T @ U ) @ ( relcom6159724393391289352rm_a_b @ ( transi2615809358984392588rm_a_b @ R5 ) @ ( relcom6159724393391289352rm_a_b @ S5 @ ( transi2615809358984392588rm_a_b @ R5 ) ) ) )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( F @ T ) @ ( F @ U ) ) @ ( relcom6159724393391289352rm_a_b @ ( transi2615809358984392588rm_a_b @ ( G @ R5 ) ) @ ( relcom6159724393391289352rm_a_b @ ( G @ S5 ) @ ( transi2615809358984392588rm_a_b @ ( G @ R5 ) ) ) ) ) ) ) ) ) ) ) ).
% steps_map
thf(fact_130_steps__map,axiom,
! [P2: term_a_b > $o,Q: set_Pr4386577575007340137rm_a_b > $o,F: term_a_b > term_a_b,G: set_Pr4386577575007340137rm_a_b > set_Pr4386577575007340137rm_a_b,T: term_a_b,R5: set_Pr4386577575007340137rm_a_b,S5: set_Pr4386577575007340137rm_a_b,U: term_a_b] :
( ! [T4: term_a_b,U2: term_a_b,R7: set_Pr4386577575007340137rm_a_b] :
( ( P2 @ T4 )
=> ( ( Q @ R7 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T4 @ U2 ) @ R7 )
=> ( ( P2 @ U2 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F @ T4 ) @ ( F @ U2 ) ) @ ( G @ R7 ) ) ) ) ) )
=> ( ( P2 @ T )
=> ( ( Q @ R5 )
=> ( ( Q @ S5 )
=> ( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T @ U ) @ ( transi7742714808557438673rm_a_b @ R5 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F @ T ) @ ( F @ U ) ) @ ( transi7742714808557438673rm_a_b @ ( G @ R5 ) ) ) )
& ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T @ U ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ R5 ) @ ( relcom370159955682700863rm_a_b @ S5 @ ( transi7742714808557438673rm_a_b @ R5 ) ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F @ T ) @ ( F @ U ) ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( G @ R5 ) ) @ ( relcom370159955682700863rm_a_b @ ( G @ S5 ) @ ( transi7742714808557438673rm_a_b @ ( G @ R5 ) ) ) ) ) ) ) ) ) ) ) ).
% steps_map
thf(fact_131_curryE,axiom,
! [F: produc4523971788519308903rm_a_b > $o,A3: produc357393685978478089rm_a_b,B2: produc357393685978478089rm_a_b] :
( ( produc3713113261896612557_a_b_o @ F @ A3 @ B2 )
=> ( F @ ( produc1763473618796451543rm_a_b @ A3 @ B2 ) ) ) ).
% curryE
thf(fact_132_curryE,axiom,
! [F: produc357393685978478089rm_a_b > $o,A3: term_a_b,B2: term_a_b] :
( ( produc6008550485034061667_a_b_o @ F @ A3 @ B2 )
=> ( F @ ( produc7020197800436672577rm_a_b @ A3 @ B2 ) ) ) ).
% curryE
thf(fact_133_curryD,axiom,
! [F: produc4523971788519308903rm_a_b > $o,A3: produc357393685978478089rm_a_b,B2: produc357393685978478089rm_a_b] :
( ( produc3713113261896612557_a_b_o @ F @ A3 @ B2 )
=> ( F @ ( produc1763473618796451543rm_a_b @ A3 @ B2 ) ) ) ).
% curryD
thf(fact_134_curryD,axiom,
! [F: produc357393685978478089rm_a_b > $o,A3: term_a_b,B2: term_a_b] :
( ( produc6008550485034061667_a_b_o @ F @ A3 @ B2 )
=> ( F @ ( produc7020197800436672577rm_a_b @ A3 @ B2 ) ) ) ).
% curryD
thf(fact_135_meetE,axiom,
! [B2: produc357393685978478089rm_a_b,C: produc357393685978478089rm_a_b,A4: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B2 @ C ) @ ( abstra8296971427885378886rm_a_b @ A4 ) )
=> ~ ! [A: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B2 ) @ ( transi2615809358984392588rm_a_b @ A4 ) )
=> ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ C ) @ ( transi2615809358984392588rm_a_b @ A4 ) ) ) ) ).
% meetE
thf(fact_136_meetE,axiom,
! [B2: term_a_b,C: term_a_b,A4: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ C ) @ ( abstra1093061187967292567rm_a_b @ A4 ) )
=> ~ ! [A: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B2 ) @ ( transi7742714808557438673rm_a_b @ A4 ) )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C ) @ ( transi7742714808557438673rm_a_b @ A4 ) ) ) ) ).
% meetE
thf(fact_137_meetD,axiom,
! [B2: produc357393685978478089rm_a_b,C: produc357393685978478089rm_a_b,A4: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B2 @ C ) @ ( abstra8296971427885378886rm_a_b @ A4 ) )
=> ? [A: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B2 ) @ ( transi2615809358984392588rm_a_b @ A4 ) )
& ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ C ) @ ( transi2615809358984392588rm_a_b @ A4 ) ) ) ) ).
% meetD
thf(fact_138_meetD,axiom,
! [B2: term_a_b,C: term_a_b,A4: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ C ) @ ( abstra1093061187967292567rm_a_b @ A4 ) )
=> ? [A: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B2 ) @ ( transi7742714808557438673rm_a_b @ A4 ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C ) @ ( transi7742714808557438673rm_a_b @ A4 ) ) ) ) ).
% meetD
thf(fact_139_relcomp__empty2,axiom,
! [R5: set_Pr4386577575007340137rm_a_b] :
( ( relcom370159955682700863rm_a_b @ R5 @ bot_bo197521221353338581rm_a_b )
= bot_bo197521221353338581rm_a_b ) ).
% relcomp_empty2
thf(fact_140_relcomp__empty1,axiom,
! [R5: set_Pr4386577575007340137rm_a_b] :
( ( relcom370159955682700863rm_a_b @ bot_bo197521221353338581rm_a_b @ R5 )
= bot_bo197521221353338581rm_a_b ) ).
% relcomp_empty1
thf(fact_141_empty__Collect__eq,axiom,
! [P2: produc357393685978478089rm_a_b > $o] :
( ( bot_bo197521221353338581rm_a_b
= ( collec99177395925924084rm_a_b @ P2 ) )
= ( ! [X2: produc357393685978478089rm_a_b] :
~ ( P2 @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_142_empty__Collect__eq,axiom,
! [P2: term_a_b > $o] :
( ( bot_bot_set_term_a_b
= ( collect_term_a_b @ P2 ) )
= ( ! [X2: term_a_b] :
~ ( P2 @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_143_empty__Collect__eq,axiom,
! [P2: b > $o] :
( ( bot_bot_set_b
= ( collect_b @ P2 ) )
= ( ! [X2: b] :
~ ( P2 @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_144_Collect__empty__eq,axiom,
! [P2: produc357393685978478089rm_a_b > $o] :
( ( ( collec99177395925924084rm_a_b @ P2 )
= bot_bo197521221353338581rm_a_b )
= ( ! [X2: produc357393685978478089rm_a_b] :
~ ( P2 @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_145_Collect__empty__eq,axiom,
! [P2: term_a_b > $o] :
( ( ( collect_term_a_b @ P2 )
= bot_bot_set_term_a_b )
= ( ! [X2: term_a_b] :
~ ( P2 @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_146_Collect__empty__eq,axiom,
! [P2: b > $o] :
( ( ( collect_b @ P2 )
= bot_bot_set_b )
= ( ! [X2: b] :
~ ( P2 @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_147_all__not__in__conv,axiom,
! [A4: set_Pr2972776593051762503rm_a_b] :
( ( ! [X2: produc4523971788519308903rm_a_b] :
~ ( member8417600551952982416rm_a_b @ X2 @ A4 ) )
= ( A4 = bot_bo5040873672931234227rm_a_b ) ) ).
% all_not_in_conv
thf(fact_148_all__not__in__conv,axiom,
! [A4: set_term_a_b] :
( ( ! [X2: term_a_b] :
~ ( member_term_a_b @ X2 @ A4 ) )
= ( A4 = bot_bot_set_term_a_b ) ) ).
% all_not_in_conv
thf(fact_149_all__not__in__conv,axiom,
! [A4: set_b] :
( ( ! [X2: b] :
~ ( member_b @ X2 @ A4 ) )
= ( A4 = bot_bot_set_b ) ) ).
% all_not_in_conv
thf(fact_150_all__not__in__conv,axiom,
! [A4: set_Pr4386577575007340137rm_a_b] :
( ( ! [X2: produc357393685978478089rm_a_b] :
~ ( member5869715511025134514rm_a_b @ X2 @ A4 ) )
= ( A4 = bot_bo197521221353338581rm_a_b ) ) ).
% all_not_in_conv
thf(fact_151_empty__iff,axiom,
! [C: produc4523971788519308903rm_a_b] :
~ ( member8417600551952982416rm_a_b @ C @ bot_bo5040873672931234227rm_a_b ) ).
% empty_iff
thf(fact_152_empty__iff,axiom,
! [C: term_a_b] :
~ ( member_term_a_b @ C @ bot_bot_set_term_a_b ) ).
% empty_iff
thf(fact_153_empty__iff,axiom,
! [C: b] :
~ ( member_b @ C @ bot_bot_set_b ) ).
% empty_iff
thf(fact_154_empty__iff,axiom,
! [C: produc357393685978478089rm_a_b] :
~ ( member5869715511025134514rm_a_b @ C @ bot_bo197521221353338581rm_a_b ) ).
% empty_iff
thf(fact_155_bot__apply,axiom,
( bot_bo23760184229683272_a_b_o
= ( ^ [X2: produc357393685978478089rm_a_b] : bot_bot_o ) ) ).
% bot_apply
thf(fact_156_relcompEpair,axiom,
! [A3: produc357393685978478089rm_a_b,C: produc357393685978478089rm_a_b,R2: set_Pr2972776593051762503rm_a_b,S: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A3 @ C ) @ ( relcom6159724393391289352rm_a_b @ R2 @ S ) )
=> ~ ! [B: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A3 @ B ) @ R2 )
=> ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B @ C ) @ S ) ) ) ).
% relcompEpair
thf(fact_157_relcompEpair,axiom,
! [A3: term_a_b,C: term_a_b,R2: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ C ) @ ( relcom370159955682700863rm_a_b @ R2 @ S ) )
=> ~ ! [B: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B ) @ R2 )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C ) @ S ) ) ) ).
% relcompEpair
thf(fact_158_relcompE,axiom,
! [Xz: produc4523971788519308903rm_a_b,R2: set_Pr2972776593051762503rm_a_b,S: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ Xz @ ( relcom6159724393391289352rm_a_b @ R2 @ S ) )
=> ~ ! [X3: produc357393685978478089rm_a_b,Y2: produc357393685978478089rm_a_b,Z: produc357393685978478089rm_a_b] :
( ( Xz
= ( produc1763473618796451543rm_a_b @ X3 @ Z ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X3 @ Y2 ) @ R2 )
=> ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y2 @ Z ) @ S ) ) ) ) ).
% relcompE
thf(fact_159_relcompE,axiom,
! [Xz: produc357393685978478089rm_a_b,R2: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ Xz @ ( relcom370159955682700863rm_a_b @ R2 @ S ) )
=> ~ ! [X3: term_a_b,Y2: term_a_b,Z: term_a_b] :
( ( Xz
= ( produc7020197800436672577rm_a_b @ X3 @ Z ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y2 ) @ R2 )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y2 @ Z ) @ S ) ) ) ) ).
% relcompE
thf(fact_160_relcomp_OrelcompI,axiom,
! [A3: produc357393685978478089rm_a_b,B2: produc357393685978478089rm_a_b,R2: set_Pr2972776593051762503rm_a_b,C: produc357393685978478089rm_a_b,S: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A3 @ B2 ) @ R2 )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B2 @ C ) @ S )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A3 @ C ) @ ( relcom6159724393391289352rm_a_b @ R2 @ S ) ) ) ) ).
% relcomp.relcompI
thf(fact_161_relcomp_OrelcompI,axiom,
! [A3: term_a_b,B2: term_a_b,R2: set_Pr4386577575007340137rm_a_b,C: term_a_b,S: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B2 ) @ R2 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ C ) @ S )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ C ) @ ( relcom370159955682700863rm_a_b @ R2 @ S ) ) ) ) ).
% relcomp.relcompI
thf(fact_162_bot__empty__eq,axiom,
( bot_bo3033454680245804266_a_b_o
= ( ^ [X2: produc4523971788519308903rm_a_b] : ( member8417600551952982416rm_a_b @ X2 @ bot_bo5040873672931234227rm_a_b ) ) ) ).
% bot_empty_eq
thf(fact_163_bot__empty__eq,axiom,
( bot_bot_term_a_b_o
= ( ^ [X2: term_a_b] : ( member_term_a_b @ X2 @ bot_bot_set_term_a_b ) ) ) ).
% bot_empty_eq
thf(fact_164_bot__empty__eq,axiom,
( bot_bot_b_o
= ( ^ [X2: b] : ( member_b @ X2 @ bot_bot_set_b ) ) ) ).
% bot_empty_eq
thf(fact_165_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_166_bot__set__def,axiom,
( bot_bo197521221353338581rm_a_b
= ( collec99177395925924084rm_a_b @ bot_bo23760184229683272_a_b_o ) ) ).
% bot_set_def
thf(fact_167_bot__set__def,axiom,
( bot_bot_set_term_a_b
= ( collect_term_a_b @ bot_bot_term_a_b_o ) ) ).
% bot_set_def
thf(fact_168_bot__set__def,axiom,
( bot_bot_set_b
= ( collect_b @ bot_bot_b_o ) ) ).
% bot_set_def
thf(fact_169_bot__fun__def,axiom,
( bot_bo23760184229683272_a_b_o
= ( ^ [X2: produc357393685978478089rm_a_b] : bot_bot_o ) ) ).
% bot_fun_def
thf(fact_170_emptyE,axiom,
! [A3: produc4523971788519308903rm_a_b] :
~ ( member8417600551952982416rm_a_b @ A3 @ bot_bo5040873672931234227rm_a_b ) ).
% emptyE
thf(fact_171_emptyE,axiom,
! [A3: term_a_b] :
~ ( member_term_a_b @ A3 @ bot_bot_set_term_a_b ) ).
% emptyE
thf(fact_172_emptyE,axiom,
! [A3: b] :
~ ( member_b @ A3 @ bot_bot_set_b ) ).
% emptyE
thf(fact_173_emptyE,axiom,
! [A3: produc357393685978478089rm_a_b] :
~ ( member5869715511025134514rm_a_b @ A3 @ bot_bo197521221353338581rm_a_b ) ).
% emptyE
thf(fact_174_equals0D,axiom,
! [A4: set_Pr2972776593051762503rm_a_b,A3: produc4523971788519308903rm_a_b] :
( ( A4 = bot_bo5040873672931234227rm_a_b )
=> ~ ( member8417600551952982416rm_a_b @ A3 @ A4 ) ) ).
% equals0D
thf(fact_175_equals0D,axiom,
! [A4: set_term_a_b,A3: term_a_b] :
( ( A4 = bot_bot_set_term_a_b )
=> ~ ( member_term_a_b @ A3 @ A4 ) ) ).
% equals0D
thf(fact_176_equals0D,axiom,
! [A4: set_b,A3: b] :
( ( A4 = bot_bot_set_b )
=> ~ ( member_b @ A3 @ A4 ) ) ).
% equals0D
thf(fact_177_equals0D,axiom,
! [A4: set_Pr4386577575007340137rm_a_b,A3: produc357393685978478089rm_a_b] :
( ( A4 = bot_bo197521221353338581rm_a_b )
=> ~ ( member5869715511025134514rm_a_b @ A3 @ A4 ) ) ).
% equals0D
thf(fact_178_equals0I,axiom,
! [A4: set_Pr2972776593051762503rm_a_b] :
( ! [Y2: produc4523971788519308903rm_a_b] :
~ ( member8417600551952982416rm_a_b @ Y2 @ A4 )
=> ( A4 = bot_bo5040873672931234227rm_a_b ) ) ).
% equals0I
thf(fact_179_equals0I,axiom,
! [A4: set_term_a_b] :
( ! [Y2: term_a_b] :
~ ( member_term_a_b @ Y2 @ A4 )
=> ( A4 = bot_bot_set_term_a_b ) ) ).
% equals0I
thf(fact_180_equals0I,axiom,
! [A4: set_b] :
( ! [Y2: b] :
~ ( member_b @ Y2 @ A4 )
=> ( A4 = bot_bot_set_b ) ) ).
% equals0I
thf(fact_181_equals0I,axiom,
! [A4: set_Pr4386577575007340137rm_a_b] :
( ! [Y2: produc357393685978478089rm_a_b] :
~ ( member5869715511025134514rm_a_b @ Y2 @ A4 )
=> ( A4 = bot_bo197521221353338581rm_a_b ) ) ).
% equals0I
thf(fact_182_ex__in__conv,axiom,
! [A4: set_Pr2972776593051762503rm_a_b] :
( ( ? [X2: produc4523971788519308903rm_a_b] : ( member8417600551952982416rm_a_b @ X2 @ A4 ) )
= ( A4 != bot_bo5040873672931234227rm_a_b ) ) ).
% ex_in_conv
thf(fact_183_ex__in__conv,axiom,
! [A4: set_term_a_b] :
( ( ? [X2: term_a_b] : ( member_term_a_b @ X2 @ A4 ) )
= ( A4 != bot_bot_set_term_a_b ) ) ).
% ex_in_conv
thf(fact_184_ex__in__conv,axiom,
! [A4: set_b] :
( ( ? [X2: b] : ( member_b @ X2 @ A4 ) )
= ( A4 != bot_bot_set_b ) ) ).
% ex_in_conv
thf(fact_185_ex__in__conv,axiom,
! [A4: set_Pr4386577575007340137rm_a_b] :
( ( ? [X2: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ X2 @ A4 ) )
= ( A4 != bot_bo197521221353338581rm_a_b ) ) ).
% ex_in_conv
thf(fact_186_O__assoc,axiom,
! [R5: set_Pr4386577575007340137rm_a_b,S5: set_Pr4386577575007340137rm_a_b,T6: set_Pr4386577575007340137rm_a_b] :
( ( relcom370159955682700863rm_a_b @ ( relcom370159955682700863rm_a_b @ R5 @ S5 ) @ T6 )
= ( relcom370159955682700863rm_a_b @ R5 @ ( relcom370159955682700863rm_a_b @ S5 @ T6 ) ) ) ).
% O_assoc
thf(fact_187_relcomp_Ocases,axiom,
! [A1: produc357393685978478089rm_a_b,A2: produc357393685978478089rm_a_b,R2: set_Pr2972776593051762503rm_a_b,S: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A1 @ A2 ) @ ( relcom6159724393391289352rm_a_b @ R2 @ S ) )
=> ~ ! [B: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A1 @ B ) @ R2 )
=> ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B @ A2 ) @ S ) ) ) ).
% relcomp.cases
thf(fact_188_relcomp_Ocases,axiom,
! [A1: term_a_b,A2: term_a_b,R2: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A2 ) @ ( relcom370159955682700863rm_a_b @ R2 @ S ) )
=> ~ ! [B: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ B ) @ R2 )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A2 ) @ S ) ) ) ).
% relcomp.cases
thf(fact_189_relcomp_Osimps,axiom,
! [A1: produc357393685978478089rm_a_b,A2: produc357393685978478089rm_a_b,R2: set_Pr2972776593051762503rm_a_b,S: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A1 @ A2 ) @ ( relcom6159724393391289352rm_a_b @ R2 @ S ) )
= ( ? [A5: produc357393685978478089rm_a_b,B3: produc357393685978478089rm_a_b,C2: produc357393685978478089rm_a_b] :
( ( A1 = A5 )
& ( A2 = C2 )
& ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A5 @ B3 ) @ R2 )
& ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B3 @ C2 ) @ S ) ) ) ) ).
% relcomp.simps
thf(fact_190_relcomp_Osimps,axiom,
! [A1: term_a_b,A2: term_a_b,R2: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A2 ) @ ( relcom370159955682700863rm_a_b @ R2 @ S ) )
= ( ? [A5: term_a_b,B3: term_a_b,C2: term_a_b] :
( ( A1 = A5 )
& ( A2 = C2 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A5 @ B3 ) @ R2 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B3 @ C2 ) @ S ) ) ) ) ).
% relcomp.simps
thf(fact_191_Set_Ois__empty__def,axiom,
( is_emp1328655501402583003rm_a_b
= ( ^ [A7: set_Pr4386577575007340137rm_a_b] : ( A7 = bot_bo197521221353338581rm_a_b ) ) ) ).
% Set.is_empty_def
thf(fact_192_Set_Ois__empty__def,axiom,
( is_empty_term_a_b
= ( ^ [A7: set_term_a_b] : ( A7 = bot_bot_set_term_a_b ) ) ) ).
% Set.is_empty_def
thf(fact_193_Set_Ois__empty__def,axiom,
( is_empty_b
= ( ^ [A7: set_b] : ( A7 = bot_bot_set_b ) ) ) ).
% Set.is_empty_def
thf(fact_194_range__vars__Var,axiom,
( ( range_vars_b_a @ var_b_a )
= bot_bot_set_b ) ).
% range_vars_Var
thf(fact_195_subst__apply__term__ctxt__apply__distrib,axiom,
! [C3: subterm_and_ctxt_a_b,T: term_a_b,Mu: b > term_a_b] :
( ( subst_7999470309526761004_a_b_b @ ( subter2376574525758040790rm_a_b @ C3 @ T ) @ Mu )
= ( subter2376574525758040790rm_a_b @ ( subter4086027020339385547_a_b_b @ C3 @ Mu ) @ ( subst_7999470309526761004_a_b_b @ T @ Mu ) ) ) ).
% subst_apply_term_ctxt_apply_distrib
thf(fact_196_subst__apply__term__ctxt__apply__distrib,axiom,
! [C3: subter6147397892252016653rm_a_b,T: term_a7872195449889686814rm_a_b,Mu: produc357393685978478089rm_a_b > term_a_b] :
( ( subst_4153759000330037794_a_b_b @ ( subter1074907186990661472rm_a_b @ C3 @ T ) @ Mu )
= ( subter2376574525758040790rm_a_b @ ( subter7224274371358213697_a_b_b @ C3 @ Mu ) @ ( subst_4153759000330037794_a_b_b @ T @ Mu ) ) ) ).
% subst_apply_term_ctxt_apply_distrib
thf(fact_197_Id__on__empty,axiom,
( ( id_on_115818524817812774rm_a_b @ bot_bo197521221353338581rm_a_b )
= bot_bo5040873672931234227rm_a_b ) ).
% Id_on_empty
thf(fact_198_Id__on__empty,axiom,
( ( id_on_term_a_b @ bot_bot_set_term_a_b )
= bot_bo197521221353338581rm_a_b ) ).
% Id_on_empty
thf(fact_199_Id__on__empty,axiom,
( ( id_on_b @ bot_bot_set_b )
= bot_bo2792761326896053555od_b_b ) ).
% Id_on_empty
thf(fact_200_normalizability__I_H,axiom,
! [A3: produc357393685978478089rm_a_b,B2: produc357393685978478089rm_a_b,A4: set_Pr2972776593051762503rm_a_b,C: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A3 @ B2 ) @ ( transi2615809358984392588rm_a_b @ A4 ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B2 @ C ) @ ( abstra1496011390211842225rm_a_b @ A4 ) )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A3 @ C ) @ ( abstra1496011390211842225rm_a_b @ A4 ) ) ) ) ).
% normalizability_I'
thf(fact_201_normalizability__I_H,axiom,
! [A3: term_a_b,B2: term_a_b,A4: set_Pr4386577575007340137rm_a_b,C: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B2 ) @ ( transi7742714808557438673rm_a_b @ A4 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ C ) @ ( abstra2972651231392952684rm_a_b @ A4 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ C ) @ ( abstra2972651231392952684rm_a_b @ A4 ) ) ) ) ).
% normalizability_I'
thf(fact_202_meet__def,axiom,
( abstra8296971427885378886rm_a_b
= ( ^ [A7: set_Pr2972776593051762503rm_a_b] : ( relcom6159724393391289352rm_a_b @ ( transi2615809358984392588rm_a_b @ ( conver1575690562132235771rm_a_b @ A7 ) ) @ ( transi2615809358984392588rm_a_b @ A7 ) ) ) ) ).
% meet_def
thf(fact_203_meet__def,axiom,
( abstra1093061187967292567rm_a_b
= ( ^ [A7: set_Pr4386577575007340137rm_a_b] : ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( conver7608559976937845093rm_a_b @ A7 ) ) @ ( transi7742714808557438673rm_a_b @ A7 ) ) ) ) ).
% meet_def
thf(fact_204_Collect__empty__eq__bot,axiom,
! [P2: produc357393685978478089rm_a_b > $o] :
( ( ( collec99177395925924084rm_a_b @ P2 )
= bot_bo197521221353338581rm_a_b )
= ( P2 = bot_bo23760184229683272_a_b_o ) ) ).
% Collect_empty_eq_bot
thf(fact_205_Collect__empty__eq__bot,axiom,
! [P2: term_a_b > $o] :
( ( ( collect_term_a_b @ P2 )
= bot_bot_set_term_a_b )
= ( P2 = bot_bot_term_a_b_o ) ) ).
% Collect_empty_eq_bot
thf(fact_206_Collect__empty__eq__bot,axiom,
! [P2: b > $o] :
( ( ( collect_b @ P2 )
= bot_bot_set_b )
= ( P2 = bot_bot_b_o ) ) ).
% Collect_empty_eq_bot
thf(fact_207_non__strict__into__strict,axiom,
! [NS: set_Pr2972776593051762503rm_a_b,S5: set_Pr2972776593051762503rm_a_b,S: produc357393685978478089rm_a_b,T: produc357393685978478089rm_a_b] :
( ( ord_le2593958535671617255rm_a_b @ ( relcom6159724393391289352rm_a_b @ NS @ S5 ) @ S5 )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ S @ T ) @ ( relcom6159724393391289352rm_a_b @ ( transi2615809358984392588rm_a_b @ NS ) @ S5 ) )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ S @ T ) @ S5 ) ) ) ).
% non_strict_into_strict
thf(fact_208_non__strict__into__strict,axiom,
! [NS: set_Product_prod_a_a,S5: set_Pr4934435412358123699_a_nat,S: a,T: nat] :
( ( ord_le8666007276011122963_a_nat @ ( relcomp_a_a_nat @ NS @ S5 ) @ S5 )
=> ( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ S @ T ) @ ( relcomp_a_a_nat @ ( transitive_rtrancl_a @ NS ) @ S5 ) )
=> ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ S @ T ) @ S5 ) ) ) ).
% non_strict_into_strict
thf(fact_209_non__strict__into__strict,axiom,
! [NS: set_Pr4386577575007340137rm_a_b,S5: set_Pr4386577575007340137rm_a_b,S: term_a_b,T: term_a_b] :
( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ NS @ S5 ) @ S5 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ NS ) @ S5 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ S5 ) ) ) ).
% non_strict_into_strict
thf(fact_210_conversionI_H,axiom,
! [A3: produc357393685978478089rm_a_b,B2: produc357393685978478089rm_a_b,A4: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A3 @ B2 ) @ ( transi2615809358984392588rm_a_b @ A4 ) )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A3 @ B2 ) @ ( abstra6125656530262095991rm_a_b @ A4 ) ) ) ).
% conversionI'
thf(fact_211_conversionI_H,axiom,
! [A3: term_a_b,B2: term_a_b,A4: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B2 ) @ ( transi7742714808557438673rm_a_b @ A4 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B2 ) @ ( abstra1130568886744591910rm_a_b @ A4 ) ) ) ).
% conversionI'
thf(fact_212_order__refl,axiom,
! [X: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ X @ X ) ).
% order_refl
thf(fact_213_order__refl,axiom,
! [X: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ X @ X ) ).
% order_refl
thf(fact_214_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_215_order__refl,axiom,
! [X: set_term_a_b] : ( ord_le2705286416250468010rm_a_b @ X @ X ) ).
% order_refl
thf(fact_216_dual__order_Orefl,axiom,
! [A3: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ A3 @ A3 ) ).
% dual_order.refl
thf(fact_217_dual__order_Orefl,axiom,
! [A3: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ A3 @ A3 ) ).
% dual_order.refl
thf(fact_218_dual__order_Orefl,axiom,
! [A3: nat] : ( ord_less_eq_nat @ A3 @ A3 ) ).
% dual_order.refl
thf(fact_219_dual__order_Orefl,axiom,
! [A3: set_term_a_b] : ( ord_le2705286416250468010rm_a_b @ A3 @ A3 ) ).
% dual_order.refl
thf(fact_220_subsetI,axiom,
! [A4: set_Pr2972776593051762503rm_a_b,B5: set_Pr2972776593051762503rm_a_b] :
( ! [X3: produc4523971788519308903rm_a_b] :
( ( member8417600551952982416rm_a_b @ X3 @ A4 )
=> ( member8417600551952982416rm_a_b @ X3 @ B5 ) )
=> ( ord_le2593958535671617255rm_a_b @ A4 @ B5 ) ) ).
% subsetI
thf(fact_221_subsetI,axiom,
! [A4: set_Pr4934435412358123699_a_nat,B5: set_Pr4934435412358123699_a_nat] :
( ! [X3: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ X3 @ A4 )
=> ( member5724188588386418708_a_nat @ X3 @ B5 ) )
=> ( ord_le8666007276011122963_a_nat @ A4 @ B5 ) ) ).
% subsetI
thf(fact_222_subsetI,axiom,
! [A4: set_term_a_b,B5: set_term_a_b] :
( ! [X3: term_a_b] :
( ( member_term_a_b @ X3 @ A4 )
=> ( member_term_a_b @ X3 @ B5 ) )
=> ( ord_le2705286416250468010rm_a_b @ A4 @ B5 ) ) ).
% subsetI
thf(fact_223_subsetI,axiom,
! [A4: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
( ! [X3: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X3 @ A4 )
=> ( member5869715511025134514rm_a_b @ X3 @ B5 ) )
=> ( ord_le118470702582115849rm_a_b @ A4 @ B5 ) ) ).
% subsetI
thf(fact_224_subset__antisym,axiom,
! [A4: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A4 @ B5 )
=> ( ( ord_le118470702582115849rm_a_b @ B5 @ A4 )
=> ( A4 = B5 ) ) ) ).
% subset_antisym
thf(fact_225_subset__antisym,axiom,
! [A4: set_Pr4934435412358123699_a_nat,B5: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A4 @ B5 )
=> ( ( ord_le8666007276011122963_a_nat @ B5 @ A4 )
=> ( A4 = B5 ) ) ) ).
% subset_antisym
thf(fact_226_subset__antisym,axiom,
! [A4: set_term_a_b,B5: set_term_a_b] :
( ( ord_le2705286416250468010rm_a_b @ A4 @ B5 )
=> ( ( ord_le2705286416250468010rm_a_b @ B5 @ A4 )
=> ( A4 = B5 ) ) ) ).
% subset_antisym
thf(fact_227_converse__inject,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
( ( ( conver7608559976937845093rm_a_b @ R2 )
= ( conver7608559976937845093rm_a_b @ S ) )
= ( R2 = S ) ) ).
% converse_inject
thf(fact_228_converse__inject,axiom,
! [R2: set_Pr2972776593051762503rm_a_b,S: set_Pr2972776593051762503rm_a_b] :
( ( ( conver1575690562132235771rm_a_b @ R2 )
= ( conver1575690562132235771rm_a_b @ S ) )
= ( R2 = S ) ) ).
% converse_inject
thf(fact_229_converse__converse,axiom,
! [R2: set_Pr4386577575007340137rm_a_b] :
( ( conver7608559976937845093rm_a_b @ ( conver7608559976937845093rm_a_b @ R2 ) )
= R2 ) ).
% converse_converse
thf(fact_230_converse__converse,axiom,
! [R2: set_Pr2972776593051762503rm_a_b] :
( ( conver1575690562132235771rm_a_b @ ( conver1575690562132235771rm_a_b @ R2 ) )
= R2 ) ).
% converse_converse
thf(fact_231_conversion__conversion__idemp,axiom,
! [A4: set_Pr4386577575007340137rm_a_b] :
( ( abstra1130568886744591910rm_a_b @ ( abstra1130568886744591910rm_a_b @ A4 ) )
= ( abstra1130568886744591910rm_a_b @ A4 ) ) ).
% conversion_conversion_idemp
thf(fact_232_conversion__conversion__idemp,axiom,
! [A4: set_Pr2972776593051762503rm_a_b] :
( ( abstra6125656530262095991rm_a_b @ ( abstra6125656530262095991rm_a_b @ A4 ) )
= ( abstra6125656530262095991rm_a_b @ A4 ) ) ).
% conversion_conversion_idemp
thf(fact_233_empty__subsetI,axiom,
! [A4: set_b] : ( ord_less_eq_set_b @ bot_bot_set_b @ A4 ) ).
% empty_subsetI
thf(fact_234_empty__subsetI,axiom,
! [A4: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ bot_bo197521221353338581rm_a_b @ A4 ) ).
% empty_subsetI
thf(fact_235_empty__subsetI,axiom,
! [A4: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ bot_bo9049108969261143879_a_nat @ A4 ) ).
% empty_subsetI
thf(fact_236_empty__subsetI,axiom,
! [A4: set_term_a_b] : ( ord_le2705286416250468010rm_a_b @ bot_bot_set_term_a_b @ A4 ) ).
% empty_subsetI
thf(fact_237_subset__empty,axiom,
! [A4: set_b] :
( ( ord_less_eq_set_b @ A4 @ bot_bot_set_b )
= ( A4 = bot_bot_set_b ) ) ).
% subset_empty
thf(fact_238_subset__empty,axiom,
! [A4: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A4 @ bot_bo197521221353338581rm_a_b )
= ( A4 = bot_bo197521221353338581rm_a_b ) ) ).
% subset_empty
thf(fact_239_subset__empty,axiom,
! [A4: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A4 @ bot_bo9049108969261143879_a_nat )
= ( A4 = bot_bo9049108969261143879_a_nat ) ) ).
% subset_empty
thf(fact_240_subset__empty,axiom,
! [A4: set_term_a_b] :
( ( ord_le2705286416250468010rm_a_b @ A4 @ bot_bot_set_term_a_b )
= ( A4 = bot_bot_set_term_a_b ) ) ).
% subset_empty
thf(fact_241_converse__iff,axiom,
! [A3: produc357393685978478089rm_a_b,B2: produc357393685978478089rm_a_b,R2: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A3 @ B2 ) @ ( conver1575690562132235771rm_a_b @ R2 ) )
= ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B2 @ A3 ) @ R2 ) ) ).
% converse_iff
thf(fact_242_converse__iff,axiom,
! [A3: term_a_b,B2: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B2 ) @ ( conver7608559976937845093rm_a_b @ R2 ) )
= ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ A3 ) @ R2 ) ) ).
% converse_iff
thf(fact_243_converse__mono,axiom,
! [R2: set_Pr2972776593051762503rm_a_b,S: set_Pr2972776593051762503rm_a_b] :
( ( ord_le2593958535671617255rm_a_b @ ( conver1575690562132235771rm_a_b @ R2 ) @ ( conver1575690562132235771rm_a_b @ S ) )
= ( ord_le2593958535671617255rm_a_b @ R2 @ S ) ) ).
% converse_mono
thf(fact_244_converse__mono,axiom,
! [R2: set_Pr4934435412358123699_a_nat,S: set_Pr4934435412358123699_a_nat] :
( ( ord_le7924913712489149241_nat_a @ ( converse_a_nat @ R2 ) @ ( converse_a_nat @ S ) )
= ( ord_le8666007276011122963_a_nat @ R2 @ S ) ) ).
% converse_mono
thf(fact_245_converse__mono,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ ( conver7608559976937845093rm_a_b @ R2 ) @ ( conver7608559976937845093rm_a_b @ S ) )
= ( ord_le118470702582115849rm_a_b @ R2 @ S ) ) ).
% converse_mono
thf(fact_246_converse__mono,axiom,
! [R2: set_Pr4193341848836149977_nat_a,S: set_Pr4193341848836149977_nat_a] :
( ( ord_le8666007276011122963_a_nat @ ( converse_nat_a @ R2 ) @ ( converse_nat_a @ S ) )
= ( ord_le7924913712489149241_nat_a @ R2 @ S ) ) ).
% converse_mono
thf(fact_247_Id__onI,axiom,
! [A3: produc4523971788519308903rm_a_b,A4: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ A3 @ A4 )
=> ( member5793113912742238992rm_a_b @ ( produc2499998789532207959rm_a_b @ A3 @ A3 ) @ ( id_on_4065744930832257284rm_a_b @ A4 ) ) ) ).
% Id_onI
thf(fact_248_Id__onI,axiom,
! [A3: produc357393685978478089rm_a_b,A4: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ A3 @ A4 )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A3 @ A3 ) @ ( id_on_115818524817812774rm_a_b @ A4 ) ) ) ).
% Id_onI
thf(fact_249_Id__onI,axiom,
! [A3: term_a_b,A4: set_term_a_b] :
( ( member_term_a_b @ A3 @ A4 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ A3 ) @ ( id_on_term_a_b @ A4 ) ) ) ).
% Id_onI
thf(fact_250_converse__empty,axiom,
( ( conver1575690562132235771rm_a_b @ bot_bo5040873672931234227rm_a_b )
= bot_bo5040873672931234227rm_a_b ) ).
% converse_empty
thf(fact_251_converse__empty,axiom,
( ( conver7608559976937845093rm_a_b @ bot_bo197521221353338581rm_a_b )
= bot_bo197521221353338581rm_a_b ) ).
% converse_empty
thf(fact_252_conversion__rtrancl,axiom,
! [A4: set_Pr2972776593051762503rm_a_b] :
( ( transi2615809358984392588rm_a_b @ ( abstra6125656530262095991rm_a_b @ A4 ) )
= ( abstra6125656530262095991rm_a_b @ A4 ) ) ).
% conversion_rtrancl
thf(fact_253_conversion__rtrancl,axiom,
! [A4: set_Pr4386577575007340137rm_a_b] :
( ( transi7742714808557438673rm_a_b @ ( abstra1130568886744591910rm_a_b @ A4 ) )
= ( abstra1130568886744591910rm_a_b @ A4 ) ) ).
% conversion_rtrancl
thf(fact_254_conversion__O__conversion,axiom,
! [A4: set_Pr4386577575007340137rm_a_b] :
( ( relcom370159955682700863rm_a_b @ ( abstra1130568886744591910rm_a_b @ A4 ) @ ( abstra1130568886744591910rm_a_b @ A4 ) )
= ( abstra1130568886744591910rm_a_b @ A4 ) ) ).
% conversion_O_conversion
thf(fact_255_conversion__O__conversion,axiom,
! [A4: set_Pr2972776593051762503rm_a_b] :
( ( relcom6159724393391289352rm_a_b @ ( abstra6125656530262095991rm_a_b @ A4 ) @ ( abstra6125656530262095991rm_a_b @ A4 ) )
= ( abstra6125656530262095991rm_a_b @ A4 ) ) ).
% conversion_O_conversion
thf(fact_256_conversion__converse,axiom,
! [A4: set_Pr4386577575007340137rm_a_b] :
( ( conver7608559976937845093rm_a_b @ ( abstra1130568886744591910rm_a_b @ A4 ) )
= ( abstra1130568886744591910rm_a_b @ A4 ) ) ).
% conversion_converse
thf(fact_257_conversion__converse,axiom,
! [A4: set_Pr2972776593051762503rm_a_b] :
( ( conver1575690562132235771rm_a_b @ ( abstra6125656530262095991rm_a_b @ A4 ) )
= ( abstra6125656530262095991rm_a_b @ A4 ) ) ).
% conversion_converse
thf(fact_258_converse__Id__on,axiom,
! [A4: set_term_a_b] :
( ( conver7608559976937845093rm_a_b @ ( id_on_term_a_b @ A4 ) )
= ( id_on_term_a_b @ A4 ) ) ).
% converse_Id_on
thf(fact_259_converse__Id__on,axiom,
! [A4: set_Pr4386577575007340137rm_a_b] :
( ( conver1575690562132235771rm_a_b @ ( id_on_115818524817812774rm_a_b @ A4 ) )
= ( id_on_115818524817812774rm_a_b @ A4 ) ) ).
% converse_Id_on
thf(fact_260_nle__le,axiom,
! [A3: nat,B2: nat] :
( ( ~ ( ord_less_eq_nat @ A3 @ B2 ) )
= ( ( ord_less_eq_nat @ B2 @ A3 )
& ( B2 != A3 ) ) ) ).
% nle_le
thf(fact_261_le__cases3,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ( ord_less_eq_nat @ X @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_eq_nat @ X @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ X @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z2 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z2 @ X )
=> ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_262_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_Pr4386577575007340137rm_a_b,Z3: set_Pr4386577575007340137rm_a_b] : ( Y4 = Z3 ) )
= ( ^ [X2: set_Pr4386577575007340137rm_a_b,Y5: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ X2 @ Y5 )
& ( ord_le118470702582115849rm_a_b @ Y5 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_263_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_Pr4934435412358123699_a_nat,Z3: set_Pr4934435412358123699_a_nat] : ( Y4 = Z3 ) )
= ( ^ [X2: set_Pr4934435412358123699_a_nat,Y5: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X2 @ Y5 )
& ( ord_le8666007276011122963_a_nat @ Y5 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_264_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: nat,Z3: nat] : ( Y4 = Z3 ) )
= ( ^ [X2: nat,Y5: nat] :
( ( ord_less_eq_nat @ X2 @ Y5 )
& ( ord_less_eq_nat @ Y5 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_265_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_term_a_b,Z3: set_term_a_b] : ( Y4 = Z3 ) )
= ( ^ [X2: set_term_a_b,Y5: set_term_a_b] :
( ( ord_le2705286416250468010rm_a_b @ X2 @ Y5 )
& ( ord_le2705286416250468010rm_a_b @ Y5 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_266_ord__eq__le__trans,axiom,
! [A3: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b,C: set_Pr4386577575007340137rm_a_b] :
( ( A3 = B2 )
=> ( ( ord_le118470702582115849rm_a_b @ B2 @ C )
=> ( ord_le118470702582115849rm_a_b @ A3 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_267_ord__eq__le__trans,axiom,
! [A3: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat,C: set_Pr4934435412358123699_a_nat] :
( ( A3 = B2 )
=> ( ( ord_le8666007276011122963_a_nat @ B2 @ C )
=> ( ord_le8666007276011122963_a_nat @ A3 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_268_ord__eq__le__trans,axiom,
! [A3: nat,B2: nat,C: nat] :
( ( A3 = B2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_eq_nat @ A3 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_269_ord__eq__le__trans,axiom,
! [A3: set_term_a_b,B2: set_term_a_b,C: set_term_a_b] :
( ( A3 = B2 )
=> ( ( ord_le2705286416250468010rm_a_b @ B2 @ C )
=> ( ord_le2705286416250468010rm_a_b @ A3 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_270_ord__le__eq__trans,axiom,
! [A3: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b,C: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A3 @ B2 )
=> ( ( B2 = C )
=> ( ord_le118470702582115849rm_a_b @ A3 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_271_ord__le__eq__trans,axiom,
! [A3: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat,C: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A3 @ B2 )
=> ( ( B2 = C )
=> ( ord_le8666007276011122963_a_nat @ A3 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_272_ord__le__eq__trans,axiom,
! [A3: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_nat @ A3 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_273_ord__le__eq__trans,axiom,
! [A3: set_term_a_b,B2: set_term_a_b,C: set_term_a_b] :
( ( ord_le2705286416250468010rm_a_b @ A3 @ B2 )
=> ( ( B2 = C )
=> ( ord_le2705286416250468010rm_a_b @ A3 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_274_order__antisym,axiom,
! [X: set_Pr4386577575007340137rm_a_b,Y: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ X @ Y )
=> ( ( ord_le118470702582115849rm_a_b @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_275_order__antisym,axiom,
! [X: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X @ Y )
=> ( ( ord_le8666007276011122963_a_nat @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_276_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_277_order__antisym,axiom,
! [X: set_term_a_b,Y: set_term_a_b] :
( ( ord_le2705286416250468010rm_a_b @ X @ Y )
=> ( ( ord_le2705286416250468010rm_a_b @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_278_order_Otrans,axiom,
! [A3: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b,C: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A3 @ B2 )
=> ( ( ord_le118470702582115849rm_a_b @ B2 @ C )
=> ( ord_le118470702582115849rm_a_b @ A3 @ C ) ) ) ).
% order.trans
thf(fact_279_order_Otrans,axiom,
! [A3: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat,C: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A3 @ B2 )
=> ( ( ord_le8666007276011122963_a_nat @ B2 @ C )
=> ( ord_le8666007276011122963_a_nat @ A3 @ C ) ) ) ).
% order.trans
thf(fact_280_order_Otrans,axiom,
! [A3: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_eq_nat @ A3 @ C ) ) ) ).
% order.trans
thf(fact_281_order_Otrans,axiom,
! [A3: set_term_a_b,B2: set_term_a_b,C: set_term_a_b] :
( ( ord_le2705286416250468010rm_a_b @ A3 @ B2 )
=> ( ( ord_le2705286416250468010rm_a_b @ B2 @ C )
=> ( ord_le2705286416250468010rm_a_b @ A3 @ C ) ) ) ).
% order.trans
thf(fact_282_order__trans,axiom,
! [X: set_Pr4386577575007340137rm_a_b,Y: set_Pr4386577575007340137rm_a_b,Z2: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ X @ Y )
=> ( ( ord_le118470702582115849rm_a_b @ Y @ Z2 )
=> ( ord_le118470702582115849rm_a_b @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_283_order__trans,axiom,
! [X: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat,Z2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X @ Y )
=> ( ( ord_le8666007276011122963_a_nat @ Y @ Z2 )
=> ( ord_le8666007276011122963_a_nat @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_284_order__trans,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z2 )
=> ( ord_less_eq_nat @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_285_order__trans,axiom,
! [X: set_term_a_b,Y: set_term_a_b,Z2: set_term_a_b] :
( ( ord_le2705286416250468010rm_a_b @ X @ Y )
=> ( ( ord_le2705286416250468010rm_a_b @ Y @ Z2 )
=> ( ord_le2705286416250468010rm_a_b @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_286_linorder__wlog,axiom,
! [P2: nat > nat > $o,A3: nat,B2: nat] :
( ! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( P2 @ A @ B ) )
=> ( ! [A: nat,B: nat] :
( ( P2 @ B @ A )
=> ( P2 @ A @ B ) )
=> ( P2 @ A3 @ B2 ) ) ) ).
% linorder_wlog
thf(fact_287_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_Pr4386577575007340137rm_a_b,Z3: set_Pr4386577575007340137rm_a_b] : ( Y4 = Z3 ) )
= ( ^ [A5: set_Pr4386577575007340137rm_a_b,B3: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ B3 @ A5 )
& ( ord_le118470702582115849rm_a_b @ A5 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_288_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_Pr4934435412358123699_a_nat,Z3: set_Pr4934435412358123699_a_nat] : ( Y4 = Z3 ) )
= ( ^ [A5: set_Pr4934435412358123699_a_nat,B3: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ B3 @ A5 )
& ( ord_le8666007276011122963_a_nat @ A5 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_289_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: nat,Z3: nat] : ( Y4 = Z3 ) )
= ( ^ [A5: nat,B3: nat] :
( ( ord_less_eq_nat @ B3 @ A5 )
& ( ord_less_eq_nat @ A5 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_290_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_term_a_b,Z3: set_term_a_b] : ( Y4 = Z3 ) )
= ( ^ [A5: set_term_a_b,B3: set_term_a_b] :
( ( ord_le2705286416250468010rm_a_b @ B3 @ A5 )
& ( ord_le2705286416250468010rm_a_b @ A5 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_291_dual__order_Oantisym,axiom,
! [B2: set_Pr4386577575007340137rm_a_b,A3: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ B2 @ A3 )
=> ( ( ord_le118470702582115849rm_a_b @ A3 @ B2 )
=> ( A3 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_292_dual__order_Oantisym,axiom,
! [B2: set_Pr4934435412358123699_a_nat,A3: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ B2 @ A3 )
=> ( ( ord_le8666007276011122963_a_nat @ A3 @ B2 )
=> ( A3 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_293_dual__order_Oantisym,axiom,
! [B2: nat,A3: nat] :
( ( ord_less_eq_nat @ B2 @ A3 )
=> ( ( ord_less_eq_nat @ A3 @ B2 )
=> ( A3 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_294_dual__order_Oantisym,axiom,
! [B2: set_term_a_b,A3: set_term_a_b] :
( ( ord_le2705286416250468010rm_a_b @ B2 @ A3 )
=> ( ( ord_le2705286416250468010rm_a_b @ A3 @ B2 )
=> ( A3 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_295_dual__order_Otrans,axiom,
! [B2: set_Pr4386577575007340137rm_a_b,A3: set_Pr4386577575007340137rm_a_b,C: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ B2 @ A3 )
=> ( ( ord_le118470702582115849rm_a_b @ C @ B2 )
=> ( ord_le118470702582115849rm_a_b @ C @ A3 ) ) ) ).
% dual_order.trans
thf(fact_296_dual__order_Otrans,axiom,
! [B2: set_Pr4934435412358123699_a_nat,A3: set_Pr4934435412358123699_a_nat,C: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ B2 @ A3 )
=> ( ( ord_le8666007276011122963_a_nat @ C @ B2 )
=> ( ord_le8666007276011122963_a_nat @ C @ A3 ) ) ) ).
% dual_order.trans
thf(fact_297_dual__order_Otrans,axiom,
! [B2: nat,A3: nat,C: nat] :
( ( ord_less_eq_nat @ B2 @ A3 )
=> ( ( ord_less_eq_nat @ C @ B2 )
=> ( ord_less_eq_nat @ C @ A3 ) ) ) ).
% dual_order.trans
thf(fact_298_dual__order_Otrans,axiom,
! [B2: set_term_a_b,A3: set_term_a_b,C: set_term_a_b] :
( ( ord_le2705286416250468010rm_a_b @ B2 @ A3 )
=> ( ( ord_le2705286416250468010rm_a_b @ C @ B2 )
=> ( ord_le2705286416250468010rm_a_b @ C @ A3 ) ) ) ).
% dual_order.trans
thf(fact_299_in__mono,axiom,
! [A4: set_Pr2972776593051762503rm_a_b,B5: set_Pr2972776593051762503rm_a_b,X: produc4523971788519308903rm_a_b] :
( ( ord_le2593958535671617255rm_a_b @ A4 @ B5 )
=> ( ( member8417600551952982416rm_a_b @ X @ A4 )
=> ( member8417600551952982416rm_a_b @ X @ B5 ) ) ) ).
% in_mono
thf(fact_300_in__mono,axiom,
! [A4: set_Pr4934435412358123699_a_nat,B5: set_Pr4934435412358123699_a_nat,X: product_prod_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A4 @ B5 )
=> ( ( member5724188588386418708_a_nat @ X @ A4 )
=> ( member5724188588386418708_a_nat @ X @ B5 ) ) ) ).
% in_mono
thf(fact_301_in__mono,axiom,
! [A4: set_term_a_b,B5: set_term_a_b,X: term_a_b] :
( ( ord_le2705286416250468010rm_a_b @ A4 @ B5 )
=> ( ( member_term_a_b @ X @ A4 )
=> ( member_term_a_b @ X @ B5 ) ) ) ).
% in_mono
thf(fact_302_in__mono,axiom,
! [A4: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b,X: produc357393685978478089rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A4 @ B5 )
=> ( ( member5869715511025134514rm_a_b @ X @ A4 )
=> ( member5869715511025134514rm_a_b @ X @ B5 ) ) ) ).
% in_mono
thf(fact_303_subsetD,axiom,
! [A4: set_Pr2972776593051762503rm_a_b,B5: set_Pr2972776593051762503rm_a_b,C: produc4523971788519308903rm_a_b] :
( ( ord_le2593958535671617255rm_a_b @ A4 @ B5 )
=> ( ( member8417600551952982416rm_a_b @ C @ A4 )
=> ( member8417600551952982416rm_a_b @ C @ B5 ) ) ) ).
% subsetD
thf(fact_304_subsetD,axiom,
! [A4: set_Pr4934435412358123699_a_nat,B5: set_Pr4934435412358123699_a_nat,C: product_prod_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A4 @ B5 )
=> ( ( member5724188588386418708_a_nat @ C @ A4 )
=> ( member5724188588386418708_a_nat @ C @ B5 ) ) ) ).
% subsetD
thf(fact_305_subsetD,axiom,
! [A4: set_term_a_b,B5: set_term_a_b,C: term_a_b] :
( ( ord_le2705286416250468010rm_a_b @ A4 @ B5 )
=> ( ( member_term_a_b @ C @ A4 )
=> ( member_term_a_b @ C @ B5 ) ) ) ).
% subsetD
thf(fact_306_subsetD,axiom,
! [A4: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b,C: produc357393685978478089rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A4 @ B5 )
=> ( ( member5869715511025134514rm_a_b @ C @ A4 )
=> ( member5869715511025134514rm_a_b @ C @ B5 ) ) ) ).
% subsetD
thf(fact_307_equalityE,axiom,
! [A4: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
( ( A4 = B5 )
=> ~ ( ( ord_le118470702582115849rm_a_b @ A4 @ B5 )
=> ~ ( ord_le118470702582115849rm_a_b @ B5 @ A4 ) ) ) ).
% equalityE
thf(fact_308_equalityE,axiom,
! [A4: set_Pr4934435412358123699_a_nat,B5: set_Pr4934435412358123699_a_nat] :
( ( A4 = B5 )
=> ~ ( ( ord_le8666007276011122963_a_nat @ A4 @ B5 )
=> ~ ( ord_le8666007276011122963_a_nat @ B5 @ A4 ) ) ) ).
% equalityE
thf(fact_309_equalityE,axiom,
! [A4: set_term_a_b,B5: set_term_a_b] :
( ( A4 = B5 )
=> ~ ( ( ord_le2705286416250468010rm_a_b @ A4 @ B5 )
=> ~ ( ord_le2705286416250468010rm_a_b @ B5 @ A4 ) ) ) ).
% equalityE
thf(fact_310_subset__eq,axiom,
( ord_le2593958535671617255rm_a_b
= ( ^ [A7: set_Pr2972776593051762503rm_a_b,B6: set_Pr2972776593051762503rm_a_b] :
! [X2: produc4523971788519308903rm_a_b] :
( ( member8417600551952982416rm_a_b @ X2 @ A7 )
=> ( member8417600551952982416rm_a_b @ X2 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_311_subset__eq,axiom,
( ord_le8666007276011122963_a_nat
= ( ^ [A7: set_Pr4934435412358123699_a_nat,B6: set_Pr4934435412358123699_a_nat] :
! [X2: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ X2 @ A7 )
=> ( member5724188588386418708_a_nat @ X2 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_312_subset__eq,axiom,
( ord_le2705286416250468010rm_a_b
= ( ^ [A7: set_term_a_b,B6: set_term_a_b] :
! [X2: term_a_b] :
( ( member_term_a_b @ X2 @ A7 )
=> ( member_term_a_b @ X2 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_313_subset__eq,axiom,
( ord_le118470702582115849rm_a_b
= ( ^ [A7: set_Pr4386577575007340137rm_a_b,B6: set_Pr4386577575007340137rm_a_b] :
! [X2: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X2 @ A7 )
=> ( member5869715511025134514rm_a_b @ X2 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_314_equalityD1,axiom,
! [A4: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
( ( A4 = B5 )
=> ( ord_le118470702582115849rm_a_b @ A4 @ B5 ) ) ).
% equalityD1
thf(fact_315_equalityD1,axiom,
! [A4: set_Pr4934435412358123699_a_nat,B5: set_Pr4934435412358123699_a_nat] :
( ( A4 = B5 )
=> ( ord_le8666007276011122963_a_nat @ A4 @ B5 ) ) ).
% equalityD1
thf(fact_316_equalityD1,axiom,
! [A4: set_term_a_b,B5: set_term_a_b] :
( ( A4 = B5 )
=> ( ord_le2705286416250468010rm_a_b @ A4 @ B5 ) ) ).
% equalityD1
thf(fact_317_equalityD2,axiom,
! [A4: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
( ( A4 = B5 )
=> ( ord_le118470702582115849rm_a_b @ B5 @ A4 ) ) ).
% equalityD2
thf(fact_318_equalityD2,axiom,
! [A4: set_Pr4934435412358123699_a_nat,B5: set_Pr4934435412358123699_a_nat] :
( ( A4 = B5 )
=> ( ord_le8666007276011122963_a_nat @ B5 @ A4 ) ) ).
% equalityD2
thf(fact_319_equalityD2,axiom,
! [A4: set_term_a_b,B5: set_term_a_b] :
( ( A4 = B5 )
=> ( ord_le2705286416250468010rm_a_b @ B5 @ A4 ) ) ).
% equalityD2
thf(fact_320_subset__iff,axiom,
( ord_le2593958535671617255rm_a_b
= ( ^ [A7: set_Pr2972776593051762503rm_a_b,B6: set_Pr2972776593051762503rm_a_b] :
! [T5: produc4523971788519308903rm_a_b] :
( ( member8417600551952982416rm_a_b @ T5 @ A7 )
=> ( member8417600551952982416rm_a_b @ T5 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_321_subset__iff,axiom,
( ord_le8666007276011122963_a_nat
= ( ^ [A7: set_Pr4934435412358123699_a_nat,B6: set_Pr4934435412358123699_a_nat] :
! [T5: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ T5 @ A7 )
=> ( member5724188588386418708_a_nat @ T5 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_322_subset__iff,axiom,
( ord_le2705286416250468010rm_a_b
= ( ^ [A7: set_term_a_b,B6: set_term_a_b] :
! [T5: term_a_b] :
( ( member_term_a_b @ T5 @ A7 )
=> ( member_term_a_b @ T5 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_323_subset__iff,axiom,
( ord_le118470702582115849rm_a_b
= ( ^ [A7: set_Pr4386577575007340137rm_a_b,B6: set_Pr4386577575007340137rm_a_b] :
! [T5: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ T5 @ A7 )
=> ( member5869715511025134514rm_a_b @ T5 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_324_subset__refl,axiom,
! [A4: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ A4 @ A4 ) ).
% subset_refl
thf(fact_325_subset__refl,axiom,
! [A4: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ A4 @ A4 ) ).
% subset_refl
thf(fact_326_subset__refl,axiom,
! [A4: set_term_a_b] : ( ord_le2705286416250468010rm_a_b @ A4 @ A4 ) ).
% subset_refl
thf(fact_327_Collect__mono,axiom,
! [P2: produc357393685978478089rm_a_b > $o,Q: produc357393685978478089rm_a_b > $o] :
( ! [X3: produc357393685978478089rm_a_b] :
( ( P2 @ X3 )
=> ( Q @ X3 ) )
=> ( ord_le118470702582115849rm_a_b @ ( collec99177395925924084rm_a_b @ P2 ) @ ( collec99177395925924084rm_a_b @ Q ) ) ) ).
% Collect_mono
thf(fact_328_Collect__mono,axiom,
! [P2: product_prod_a_nat > $o,Q: product_prod_a_nat > $o] :
( ! [X3: product_prod_a_nat] :
( ( P2 @ X3 )
=> ( Q @ X3 ) )
=> ( ord_le8666007276011122963_a_nat @ ( collec4464134535221767506_a_nat @ P2 ) @ ( collec4464134535221767506_a_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_329_Collect__mono,axiom,
! [P2: term_a_b > $o,Q: term_a_b > $o] :
( ! [X3: term_a_b] :
( ( P2 @ X3 )
=> ( Q @ X3 ) )
=> ( ord_le2705286416250468010rm_a_b @ ( collect_term_a_b @ P2 ) @ ( collect_term_a_b @ Q ) ) ) ).
% Collect_mono
thf(fact_330_subset__trans,axiom,
! [A4: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b,C3: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A4 @ B5 )
=> ( ( ord_le118470702582115849rm_a_b @ B5 @ C3 )
=> ( ord_le118470702582115849rm_a_b @ A4 @ C3 ) ) ) ).
% subset_trans
thf(fact_331_subset__trans,axiom,
! [A4: set_Pr4934435412358123699_a_nat,B5: set_Pr4934435412358123699_a_nat,C3: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A4 @ B5 )
=> ( ( ord_le8666007276011122963_a_nat @ B5 @ C3 )
=> ( ord_le8666007276011122963_a_nat @ A4 @ C3 ) ) ) ).
% subset_trans
thf(fact_332_subset__trans,axiom,
! [A4: set_term_a_b,B5: set_term_a_b,C3: set_term_a_b] :
( ( ord_le2705286416250468010rm_a_b @ A4 @ B5 )
=> ( ( ord_le2705286416250468010rm_a_b @ B5 @ C3 )
=> ( ord_le2705286416250468010rm_a_b @ A4 @ C3 ) ) ) ).
% subset_trans
thf(fact_333_antisym,axiom,
! [A3: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A3 @ B2 )
=> ( ( ord_le118470702582115849rm_a_b @ B2 @ A3 )
=> ( A3 = B2 ) ) ) ).
% antisym
thf(fact_334_antisym,axiom,
! [A3: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A3 @ B2 )
=> ( ( ord_le8666007276011122963_a_nat @ B2 @ A3 )
=> ( A3 = B2 ) ) ) ).
% antisym
thf(fact_335_antisym,axiom,
! [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ A3 )
=> ( A3 = B2 ) ) ) ).
% antisym
thf(fact_336_antisym,axiom,
! [A3: set_term_a_b,B2: set_term_a_b] :
( ( ord_le2705286416250468010rm_a_b @ A3 @ B2 )
=> ( ( ord_le2705286416250468010rm_a_b @ B2 @ A3 )
=> ( A3 = B2 ) ) ) ).
% antisym
thf(fact_337_set__eq__subset,axiom,
( ( ^ [Y4: set_Pr4386577575007340137rm_a_b,Z3: set_Pr4386577575007340137rm_a_b] : ( Y4 = Z3 ) )
= ( ^ [A7: set_Pr4386577575007340137rm_a_b,B6: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A7 @ B6 )
& ( ord_le118470702582115849rm_a_b @ B6 @ A7 ) ) ) ) ).
% set_eq_subset
thf(fact_338_set__eq__subset,axiom,
( ( ^ [Y4: set_Pr4934435412358123699_a_nat,Z3: set_Pr4934435412358123699_a_nat] : ( Y4 = Z3 ) )
= ( ^ [A7: set_Pr4934435412358123699_a_nat,B6: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A7 @ B6 )
& ( ord_le8666007276011122963_a_nat @ B6 @ A7 ) ) ) ) ).
% set_eq_subset
thf(fact_339_set__eq__subset,axiom,
( ( ^ [Y4: set_term_a_b,Z3: set_term_a_b] : ( Y4 = Z3 ) )
= ( ^ [A7: set_term_a_b,B6: set_term_a_b] :
( ( ord_le2705286416250468010rm_a_b @ A7 @ B6 )
& ( ord_le2705286416250468010rm_a_b @ B6 @ A7 ) ) ) ) ).
% set_eq_subset
thf(fact_340_Collect__mono__iff,axiom,
! [P2: produc357393685978478089rm_a_b > $o,Q: produc357393685978478089rm_a_b > $o] :
( ( ord_le118470702582115849rm_a_b @ ( collec99177395925924084rm_a_b @ P2 ) @ ( collec99177395925924084rm_a_b @ Q ) )
= ( ! [X2: produc357393685978478089rm_a_b] :
( ( P2 @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_341_Collect__mono__iff,axiom,
! [P2: product_prod_a_nat > $o,Q: product_prod_a_nat > $o] :
( ( ord_le8666007276011122963_a_nat @ ( collec4464134535221767506_a_nat @ P2 ) @ ( collec4464134535221767506_a_nat @ Q ) )
= ( ! [X2: product_prod_a_nat] :
( ( P2 @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_342_Collect__mono__iff,axiom,
! [P2: term_a_b > $o,Q: term_a_b > $o] :
( ( ord_le2705286416250468010rm_a_b @ ( collect_term_a_b @ P2 ) @ ( collect_term_a_b @ Q ) )
= ( ! [X2: term_a_b] :
( ( P2 @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_343_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_Pr4386577575007340137rm_a_b,Z3: set_Pr4386577575007340137rm_a_b] : ( Y4 = Z3 ) )
= ( ^ [A5: set_Pr4386577575007340137rm_a_b,B3: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A5 @ B3 )
& ( ord_le118470702582115849rm_a_b @ B3 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_344_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_Pr4934435412358123699_a_nat,Z3: set_Pr4934435412358123699_a_nat] : ( Y4 = Z3 ) )
= ( ^ [A5: set_Pr4934435412358123699_a_nat,B3: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A5 @ B3 )
& ( ord_le8666007276011122963_a_nat @ B3 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_345_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: nat,Z3: nat] : ( Y4 = Z3 ) )
= ( ^ [A5: nat,B3: nat] :
( ( ord_less_eq_nat @ A5 @ B3 )
& ( ord_less_eq_nat @ B3 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_346_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_term_a_b,Z3: set_term_a_b] : ( Y4 = Z3 ) )
= ( ^ [A5: set_term_a_b,B3: set_term_a_b] :
( ( ord_le2705286416250468010rm_a_b @ A5 @ B3 )
& ( ord_le2705286416250468010rm_a_b @ B3 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_347_order__subst1,axiom,
! [A3: nat,F: nat > nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_348_order__subst1,axiom,
! [A3: set_Pr4934435412358123699_a_nat,F: nat > set_Pr4934435412358123699_a_nat,B2: nat,C: nat] :
( ( ord_le8666007276011122963_a_nat @ A3 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_le8666007276011122963_a_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le8666007276011122963_a_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_349_order__subst1,axiom,
! [A3: nat,F: set_Pr4934435412358123699_a_nat > nat,B2: set_Pr4934435412358123699_a_nat,C: set_Pr4934435412358123699_a_nat] :
( ( ord_less_eq_nat @ A3 @ ( F @ B2 ) )
=> ( ( ord_le8666007276011122963_a_nat @ B2 @ C )
=> ( ! [X3: set_Pr4934435412358123699_a_nat,Y2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_350_order__subst1,axiom,
! [A3: nat,F: set_term_a_b > nat,B2: set_term_a_b,C: set_term_a_b] :
( ( ord_less_eq_nat @ A3 @ ( F @ B2 ) )
=> ( ( ord_le2705286416250468010rm_a_b @ B2 @ C )
=> ( ! [X3: set_term_a_b,Y2: set_term_a_b] :
( ( ord_le2705286416250468010rm_a_b @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_351_order__subst1,axiom,
! [A3: set_term_a_b,F: nat > set_term_a_b,B2: nat,C: nat] :
( ( ord_le2705286416250468010rm_a_b @ A3 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_le2705286416250468010rm_a_b @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le2705286416250468010rm_a_b @ A3 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_352_order__subst1,axiom,
! [A3: set_Pr4934435412358123699_a_nat,F: set_Pr4934435412358123699_a_nat > set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat,C: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A3 @ ( F @ B2 ) )
=> ( ( ord_le8666007276011122963_a_nat @ B2 @ C )
=> ( ! [X3: set_Pr4934435412358123699_a_nat,Y2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X3 @ Y2 )
=> ( ord_le8666007276011122963_a_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le8666007276011122963_a_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_353_order__subst1,axiom,
! [A3: set_Pr4934435412358123699_a_nat,F: set_term_a_b > set_Pr4934435412358123699_a_nat,B2: set_term_a_b,C: set_term_a_b] :
( ( ord_le8666007276011122963_a_nat @ A3 @ ( F @ B2 ) )
=> ( ( ord_le2705286416250468010rm_a_b @ B2 @ C )
=> ( ! [X3: set_term_a_b,Y2: set_term_a_b] :
( ( ord_le2705286416250468010rm_a_b @ X3 @ Y2 )
=> ( ord_le8666007276011122963_a_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le8666007276011122963_a_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_354_order__subst1,axiom,
! [A3: set_term_a_b,F: set_Pr4934435412358123699_a_nat > set_term_a_b,B2: set_Pr4934435412358123699_a_nat,C: set_Pr4934435412358123699_a_nat] :
( ( ord_le2705286416250468010rm_a_b @ A3 @ ( F @ B2 ) )
=> ( ( ord_le8666007276011122963_a_nat @ B2 @ C )
=> ( ! [X3: set_Pr4934435412358123699_a_nat,Y2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X3 @ Y2 )
=> ( ord_le2705286416250468010rm_a_b @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le2705286416250468010rm_a_b @ A3 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_355_order__subst1,axiom,
! [A3: set_term_a_b,F: set_term_a_b > set_term_a_b,B2: set_term_a_b,C: set_term_a_b] :
( ( ord_le2705286416250468010rm_a_b @ A3 @ ( F @ B2 ) )
=> ( ( ord_le2705286416250468010rm_a_b @ B2 @ C )
=> ( ! [X3: set_term_a_b,Y2: set_term_a_b] :
( ( ord_le2705286416250468010rm_a_b @ X3 @ Y2 )
=> ( ord_le2705286416250468010rm_a_b @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le2705286416250468010rm_a_b @ A3 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_356_order__subst1,axiom,
! [A3: set_Pr4386577575007340137rm_a_b,F: nat > set_Pr4386577575007340137rm_a_b,B2: nat,C: nat] :
( ( ord_le118470702582115849rm_a_b @ A3 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_le118470702582115849rm_a_b @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le118470702582115849rm_a_b @ A3 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_357_order__subst2,axiom,
! [A3: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_358_order__subst2,axiom,
! [A3: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat,F: set_Pr4934435412358123699_a_nat > nat,C: nat] :
( ( ord_le8666007276011122963_a_nat @ A3 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: set_Pr4934435412358123699_a_nat,Y2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_359_order__subst2,axiom,
! [A3: nat,B2: nat,F: nat > set_Pr4934435412358123699_a_nat,C: set_Pr4934435412358123699_a_nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ( ( ord_le8666007276011122963_a_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_le8666007276011122963_a_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le8666007276011122963_a_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_360_order__subst2,axiom,
! [A3: nat,B2: nat,F: nat > set_term_a_b,C: set_term_a_b] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ( ( ord_le2705286416250468010rm_a_b @ ( F @ B2 ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_le2705286416250468010rm_a_b @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le2705286416250468010rm_a_b @ ( F @ A3 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_361_order__subst2,axiom,
! [A3: set_term_a_b,B2: set_term_a_b,F: set_term_a_b > nat,C: nat] :
( ( ord_le2705286416250468010rm_a_b @ A3 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: set_term_a_b,Y2: set_term_a_b] :
( ( ord_le2705286416250468010rm_a_b @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_362_order__subst2,axiom,
! [A3: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat,F: set_Pr4934435412358123699_a_nat > set_Pr4934435412358123699_a_nat,C: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A3 @ B2 )
=> ( ( ord_le8666007276011122963_a_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: set_Pr4934435412358123699_a_nat,Y2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X3 @ Y2 )
=> ( ord_le8666007276011122963_a_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le8666007276011122963_a_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_363_order__subst2,axiom,
! [A3: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat,F: set_Pr4934435412358123699_a_nat > set_term_a_b,C: set_term_a_b] :
( ( ord_le8666007276011122963_a_nat @ A3 @ B2 )
=> ( ( ord_le2705286416250468010rm_a_b @ ( F @ B2 ) @ C )
=> ( ! [X3: set_Pr4934435412358123699_a_nat,Y2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X3 @ Y2 )
=> ( ord_le2705286416250468010rm_a_b @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le2705286416250468010rm_a_b @ ( F @ A3 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_364_order__subst2,axiom,
! [A3: set_term_a_b,B2: set_term_a_b,F: set_term_a_b > set_Pr4934435412358123699_a_nat,C: set_Pr4934435412358123699_a_nat] :
( ( ord_le2705286416250468010rm_a_b @ A3 @ B2 )
=> ( ( ord_le8666007276011122963_a_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: set_term_a_b,Y2: set_term_a_b] :
( ( ord_le2705286416250468010rm_a_b @ X3 @ Y2 )
=> ( ord_le8666007276011122963_a_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le8666007276011122963_a_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_365_order__subst2,axiom,
! [A3: set_term_a_b,B2: set_term_a_b,F: set_term_a_b > set_term_a_b,C: set_term_a_b] :
( ( ord_le2705286416250468010rm_a_b @ A3 @ B2 )
=> ( ( ord_le2705286416250468010rm_a_b @ ( F @ B2 ) @ C )
=> ( ! [X3: set_term_a_b,Y2: set_term_a_b] :
( ( ord_le2705286416250468010rm_a_b @ X3 @ Y2 )
=> ( ord_le2705286416250468010rm_a_b @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le2705286416250468010rm_a_b @ ( F @ A3 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_366_order__subst2,axiom,
! [A3: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b,F: set_Pr4386577575007340137rm_a_b > nat,C: nat] :
( ( ord_le118470702582115849rm_a_b @ A3 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: set_Pr4386577575007340137rm_a_b,Y2: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_367_order__eq__refl,axiom,
! [X: set_Pr4386577575007340137rm_a_b,Y: set_Pr4386577575007340137rm_a_b] :
( ( X = Y )
=> ( ord_le118470702582115849rm_a_b @ X @ Y ) ) ).
% order_eq_refl
thf(fact_368_order__eq__refl,axiom,
! [X: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat] :
( ( X = Y )
=> ( ord_le8666007276011122963_a_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_369_order__eq__refl,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_370_order__eq__refl,axiom,
! [X: set_term_a_b,Y: set_term_a_b] :
( ( X = Y )
=> ( ord_le2705286416250468010rm_a_b @ X @ Y ) ) ).
% order_eq_refl
thf(fact_371_linorder__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_linear
thf(fact_372_ord__eq__le__subst,axiom,
! [A3: nat,F: nat > nat,B2: nat,C: nat] :
( ( A3
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_373_ord__eq__le__subst,axiom,
! [A3: nat,F: set_Pr4934435412358123699_a_nat > nat,B2: set_Pr4934435412358123699_a_nat,C: set_Pr4934435412358123699_a_nat] :
( ( A3
= ( F @ B2 ) )
=> ( ( ord_le8666007276011122963_a_nat @ B2 @ C )
=> ( ! [X3: set_Pr4934435412358123699_a_nat,Y2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_374_ord__eq__le__subst,axiom,
! [A3: set_Pr4934435412358123699_a_nat,F: nat > set_Pr4934435412358123699_a_nat,B2: nat,C: nat] :
( ( A3
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_le8666007276011122963_a_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le8666007276011122963_a_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_375_ord__eq__le__subst,axiom,
! [A3: set_term_a_b,F: nat > set_term_a_b,B2: nat,C: nat] :
( ( A3
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_le2705286416250468010rm_a_b @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le2705286416250468010rm_a_b @ A3 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_376_ord__eq__le__subst,axiom,
! [A3: nat,F: set_term_a_b > nat,B2: set_term_a_b,C: set_term_a_b] :
( ( A3
= ( F @ B2 ) )
=> ( ( ord_le2705286416250468010rm_a_b @ B2 @ C )
=> ( ! [X3: set_term_a_b,Y2: set_term_a_b] :
( ( ord_le2705286416250468010rm_a_b @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_377_ord__eq__le__subst,axiom,
! [A3: set_Pr4934435412358123699_a_nat,F: set_Pr4934435412358123699_a_nat > set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat,C: set_Pr4934435412358123699_a_nat] :
( ( A3
= ( F @ B2 ) )
=> ( ( ord_le8666007276011122963_a_nat @ B2 @ C )
=> ( ! [X3: set_Pr4934435412358123699_a_nat,Y2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X3 @ Y2 )
=> ( ord_le8666007276011122963_a_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le8666007276011122963_a_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_378_ord__eq__le__subst,axiom,
! [A3: set_term_a_b,F: set_Pr4934435412358123699_a_nat > set_term_a_b,B2: set_Pr4934435412358123699_a_nat,C: set_Pr4934435412358123699_a_nat] :
( ( A3
= ( F @ B2 ) )
=> ( ( ord_le8666007276011122963_a_nat @ B2 @ C )
=> ( ! [X3: set_Pr4934435412358123699_a_nat,Y2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X3 @ Y2 )
=> ( ord_le2705286416250468010rm_a_b @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le2705286416250468010rm_a_b @ A3 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_379_ord__eq__le__subst,axiom,
! [A3: set_Pr4934435412358123699_a_nat,F: set_term_a_b > set_Pr4934435412358123699_a_nat,B2: set_term_a_b,C: set_term_a_b] :
( ( A3
= ( F @ B2 ) )
=> ( ( ord_le2705286416250468010rm_a_b @ B2 @ C )
=> ( ! [X3: set_term_a_b,Y2: set_term_a_b] :
( ( ord_le2705286416250468010rm_a_b @ X3 @ Y2 )
=> ( ord_le8666007276011122963_a_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le8666007276011122963_a_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_380_ord__eq__le__subst,axiom,
! [A3: set_term_a_b,F: set_term_a_b > set_term_a_b,B2: set_term_a_b,C: set_term_a_b] :
( ( A3
= ( F @ B2 ) )
=> ( ( ord_le2705286416250468010rm_a_b @ B2 @ C )
=> ( ! [X3: set_term_a_b,Y2: set_term_a_b] :
( ( ord_le2705286416250468010rm_a_b @ X3 @ Y2 )
=> ( ord_le2705286416250468010rm_a_b @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le2705286416250468010rm_a_b @ A3 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_381_ord__eq__le__subst,axiom,
! [A3: nat,F: set_Pr4386577575007340137rm_a_b > nat,B2: set_Pr4386577575007340137rm_a_b,C: set_Pr4386577575007340137rm_a_b] :
( ( A3
= ( F @ B2 ) )
=> ( ( ord_le118470702582115849rm_a_b @ B2 @ C )
=> ( ! [X3: set_Pr4386577575007340137rm_a_b,Y2: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_382_ord__le__eq__subst,axiom,
! [A3: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_383_ord__le__eq__subst,axiom,
! [A3: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat,F: set_Pr4934435412358123699_a_nat > nat,C: nat] :
( ( ord_le8666007276011122963_a_nat @ A3 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: set_Pr4934435412358123699_a_nat,Y2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_384_ord__le__eq__subst,axiom,
! [A3: nat,B2: nat,F: nat > set_Pr4934435412358123699_a_nat,C: set_Pr4934435412358123699_a_nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_le8666007276011122963_a_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le8666007276011122963_a_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_385_ord__le__eq__subst,axiom,
! [A3: nat,B2: nat,F: nat > set_term_a_b,C: set_term_a_b] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_le2705286416250468010rm_a_b @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le2705286416250468010rm_a_b @ ( F @ A3 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_386_ord__le__eq__subst,axiom,
! [A3: set_term_a_b,B2: set_term_a_b,F: set_term_a_b > nat,C: nat] :
( ( ord_le2705286416250468010rm_a_b @ A3 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: set_term_a_b,Y2: set_term_a_b] :
( ( ord_le2705286416250468010rm_a_b @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_387_ord__le__eq__subst,axiom,
! [A3: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat,F: set_Pr4934435412358123699_a_nat > set_Pr4934435412358123699_a_nat,C: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A3 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: set_Pr4934435412358123699_a_nat,Y2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X3 @ Y2 )
=> ( ord_le8666007276011122963_a_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le8666007276011122963_a_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_388_ord__le__eq__subst,axiom,
! [A3: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat,F: set_Pr4934435412358123699_a_nat > set_term_a_b,C: set_term_a_b] :
( ( ord_le8666007276011122963_a_nat @ A3 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: set_Pr4934435412358123699_a_nat,Y2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X3 @ Y2 )
=> ( ord_le2705286416250468010rm_a_b @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le2705286416250468010rm_a_b @ ( F @ A3 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_389_ord__le__eq__subst,axiom,
! [A3: set_term_a_b,B2: set_term_a_b,F: set_term_a_b > set_Pr4934435412358123699_a_nat,C: set_Pr4934435412358123699_a_nat] :
( ( ord_le2705286416250468010rm_a_b @ A3 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: set_term_a_b,Y2: set_term_a_b] :
( ( ord_le2705286416250468010rm_a_b @ X3 @ Y2 )
=> ( ord_le8666007276011122963_a_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le8666007276011122963_a_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_390_ord__le__eq__subst,axiom,
! [A3: set_term_a_b,B2: set_term_a_b,F: set_term_a_b > set_term_a_b,C: set_term_a_b] :
( ( ord_le2705286416250468010rm_a_b @ A3 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: set_term_a_b,Y2: set_term_a_b] :
( ( ord_le2705286416250468010rm_a_b @ X3 @ Y2 )
=> ( ord_le2705286416250468010rm_a_b @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le2705286416250468010rm_a_b @ ( F @ A3 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_391_ord__le__eq__subst,axiom,
! [A3: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b,F: set_Pr4386577575007340137rm_a_b > nat,C: nat] :
( ( ord_le118470702582115849rm_a_b @ A3 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: set_Pr4386577575007340137rm_a_b,Y2: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_392_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_393_order__antisym__conv,axiom,
! [Y: set_Pr4386577575007340137rm_a_b,X: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ Y @ X )
=> ( ( ord_le118470702582115849rm_a_b @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_394_order__antisym__conv,axiom,
! [Y: set_Pr4934435412358123699_a_nat,X: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ Y @ X )
=> ( ( ord_le8666007276011122963_a_nat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_395_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_396_order__antisym__conv,axiom,
! [Y: set_term_a_b,X: set_term_a_b] :
( ( ord_le2705286416250468010rm_a_b @ Y @ X )
=> ( ( ord_le2705286416250468010rm_a_b @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_397_converse__subset__swap,axiom,
! [R2: set_Pr2972776593051762503rm_a_b,S: set_Pr2972776593051762503rm_a_b] :
( ( ord_le2593958535671617255rm_a_b @ R2 @ ( conver1575690562132235771rm_a_b @ S ) )
= ( ord_le2593958535671617255rm_a_b @ ( conver1575690562132235771rm_a_b @ R2 ) @ S ) ) ).
% converse_subset_swap
thf(fact_398_converse__subset__swap,axiom,
! [R2: set_Pr4193341848836149977_nat_a,S: set_Pr4934435412358123699_a_nat] :
( ( ord_le7924913712489149241_nat_a @ R2 @ ( converse_a_nat @ S ) )
= ( ord_le8666007276011122963_a_nat @ ( converse_nat_a @ R2 ) @ S ) ) ).
% converse_subset_swap
thf(fact_399_converse__subset__swap,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ R2 @ ( conver7608559976937845093rm_a_b @ S ) )
= ( ord_le118470702582115849rm_a_b @ ( conver7608559976937845093rm_a_b @ R2 ) @ S ) ) ).
% converse_subset_swap
thf(fact_400_converse__subset__swap,axiom,
! [R2: set_Pr4934435412358123699_a_nat,S: set_Pr4193341848836149977_nat_a] :
( ( ord_le8666007276011122963_a_nat @ R2 @ ( converse_nat_a @ S ) )
= ( ord_le7924913712489149241_nat_a @ ( converse_a_nat @ R2 ) @ S ) ) ).
% converse_subset_swap
thf(fact_401_conversion__mono,axiom,
! [A4: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A4 @ B5 )
=> ( ord_le118470702582115849rm_a_b @ ( abstra1130568886744591910rm_a_b @ A4 ) @ ( abstra1130568886744591910rm_a_b @ B5 ) ) ) ).
% conversion_mono
thf(fact_402_conversion__mono,axiom,
! [A4: set_Pr2972776593051762503rm_a_b,B5: set_Pr2972776593051762503rm_a_b] :
( ( ord_le2593958535671617255rm_a_b @ A4 @ B5 )
=> ( ord_le2593958535671617255rm_a_b @ ( abstra6125656530262095991rm_a_b @ A4 ) @ ( abstra6125656530262095991rm_a_b @ B5 ) ) ) ).
% conversion_mono
thf(fact_403_sig__step__mono,axiom,
! [F2: set_Pr4934435412358123699_a_nat,G2: set_Pr4934435412358123699_a_nat,R: set_Pr4386577575007340137rm_a_b] :
( ( ord_le8666007276011122963_a_nat @ F2 @ G2 )
=> ( ord_le118470702582115849rm_a_b @ ( sig_step_a_b @ F2 @ R ) @ ( sig_step_a_b @ G2 @ R ) ) ) ).
% sig_step_mono
thf(fact_404_meet__imp__conversion,axiom,
! [R2: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ ( abstra1093061187967292567rm_a_b @ R2 ) @ ( abstra1130568886744591910rm_a_b @ R2 ) ) ).
% meet_imp_conversion
thf(fact_405_meet__imp__conversion,axiom,
! [R2: set_Pr2972776593051762503rm_a_b] : ( ord_le2593958535671617255rm_a_b @ ( abstra8296971427885378886rm_a_b @ R2 ) @ ( abstra6125656530262095991rm_a_b @ R2 ) ) ).
% meet_imp_conversion
thf(fact_406_srsteps__with__root__step__sig__mono,axiom,
! [F2: set_Pr4934435412358123699_a_nat,G2: set_Pr4934435412358123699_a_nat,R: set_Pr4386577575007340137rm_a_b] :
( ( ord_le8666007276011122963_a_nat @ F2 @ G2 )
=> ( ord_le118470702582115849rm_a_b @ ( srstep7844470518422762656ep_a_b @ F2 @ R ) @ ( srstep7844470518422762656ep_a_b @ G2 @ R ) ) ) ).
% srsteps_with_root_step_sig_mono
thf(fact_407_converse_Ocases,axiom,
! [A1: produc357393685978478089rm_a_b,A2: produc357393685978478089rm_a_b,R2: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A1 @ A2 ) @ ( conver1575690562132235771rm_a_b @ R2 ) )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A2 @ A1 ) @ R2 ) ) ).
% converse.cases
thf(fact_408_converse_Ocases,axiom,
! [A1: term_a_b,A2: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A2 ) @ ( conver7608559976937845093rm_a_b @ R2 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ A1 ) @ R2 ) ) ).
% converse.cases
thf(fact_409_converse_Osimps,axiom,
! [A1: produc357393685978478089rm_a_b,A2: produc357393685978478089rm_a_b,R2: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A1 @ A2 ) @ ( conver1575690562132235771rm_a_b @ R2 ) )
= ( ? [A5: produc357393685978478089rm_a_b,B3: produc357393685978478089rm_a_b] :
( ( A1 = B3 )
& ( A2 = A5 )
& ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A5 @ B3 ) @ R2 ) ) ) ) ).
% converse.simps
thf(fact_410_converse_Osimps,axiom,
! [A1: term_a_b,A2: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A2 ) @ ( conver7608559976937845093rm_a_b @ R2 ) )
= ( ? [A5: term_a_b,B3: term_a_b] :
( ( A1 = B3 )
& ( A2 = A5 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A5 @ B3 ) @ R2 ) ) ) ) ).
% converse.simps
thf(fact_411_converseD,axiom,
! [A3: produc357393685978478089rm_a_b,B2: produc357393685978478089rm_a_b,R2: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A3 @ B2 ) @ ( conver1575690562132235771rm_a_b @ R2 ) )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B2 @ A3 ) @ R2 ) ) ).
% converseD
thf(fact_412_converseD,axiom,
! [A3: term_a_b,B2: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B2 ) @ ( conver7608559976937845093rm_a_b @ R2 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ A3 ) @ R2 ) ) ).
% converseD
thf(fact_413_converseE,axiom,
! [Yx: produc4523971788519308903rm_a_b,R2: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ Yx @ ( conver1575690562132235771rm_a_b @ R2 ) )
=> ~ ! [X3: produc357393685978478089rm_a_b,Y2: produc357393685978478089rm_a_b] :
( ( Yx
= ( produc1763473618796451543rm_a_b @ Y2 @ X3 ) )
=> ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X3 @ Y2 ) @ R2 ) ) ) ).
% converseE
thf(fact_414_converseE,axiom,
! [Yx: produc357393685978478089rm_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ Yx @ ( conver7608559976937845093rm_a_b @ R2 ) )
=> ~ ! [X3: term_a_b,Y2: term_a_b] :
( ( Yx
= ( produc7020197800436672577rm_a_b @ Y2 @ X3 ) )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y2 ) @ R2 ) ) ) ).
% converseE
thf(fact_415_converseI,axiom,
! [A3: produc357393685978478089rm_a_b,B2: produc357393685978478089rm_a_b,R2: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A3 @ B2 ) @ R2 )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B2 @ A3 ) @ ( conver1575690562132235771rm_a_b @ R2 ) ) ) ).
% converseI
thf(fact_416_converseI,axiom,
! [A3: term_a_b,B2: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B2 ) @ R2 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ A3 ) @ ( conver7608559976937845093rm_a_b @ R2 ) ) ) ).
% converseI
thf(fact_417_rtrancl__converse,axiom,
! [R2: set_Pr2972776593051762503rm_a_b] :
( ( transi2615809358984392588rm_a_b @ ( conver1575690562132235771rm_a_b @ R2 ) )
= ( conver1575690562132235771rm_a_b @ ( transi2615809358984392588rm_a_b @ R2 ) ) ) ).
% rtrancl_converse
thf(fact_418_rtrancl__converse,axiom,
! [R2: set_Pr4386577575007340137rm_a_b] :
( ( transi7742714808557438673rm_a_b @ ( conver7608559976937845093rm_a_b @ R2 ) )
= ( conver7608559976937845093rm_a_b @ ( transi7742714808557438673rm_a_b @ R2 ) ) ) ).
% rtrancl_converse
thf(fact_419_converse__relcomp,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
( ( conver7608559976937845093rm_a_b @ ( relcom370159955682700863rm_a_b @ R2 @ S ) )
= ( relcom370159955682700863rm_a_b @ ( conver7608559976937845093rm_a_b @ S ) @ ( conver7608559976937845093rm_a_b @ R2 ) ) ) ).
% converse_relcomp
thf(fact_420_converse__relcomp,axiom,
! [R2: set_Pr2972776593051762503rm_a_b,S: set_Pr2972776593051762503rm_a_b] :
( ( conver1575690562132235771rm_a_b @ ( relcom6159724393391289352rm_a_b @ R2 @ S ) )
= ( relcom6159724393391289352rm_a_b @ ( conver1575690562132235771rm_a_b @ S ) @ ( conver1575690562132235771rm_a_b @ R2 ) ) ) ).
% converse_relcomp
thf(fact_421_srstep__monp,axiom,
! [F2: set_Pr4934435412358123699_a_nat,G2: set_Pr4934435412358123699_a_nat,R: set_Pr4386577575007340137rm_a_b] :
( ( ord_le8666007276011122963_a_nat @ F2 @ G2 )
=> ( ord_le118470702582115849rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R ) ) @ ( sig_step_a_b @ G2 @ ( rstep_a_b @ R ) ) ) ) ).
% srstep_monp
thf(fact_422_rstep__converse__dist,axiom,
! [R: set_Pr4386577575007340137rm_a_b] :
( ( conver7608559976937845093rm_a_b @ ( rstep_a_b @ R ) )
= ( rstep_a_b @ ( conver7608559976937845093rm_a_b @ R ) ) ) ).
% rstep_converse_dist
thf(fact_423_subrelI,axiom,
! [R2: set_Pr2972776593051762503rm_a_b,S: set_Pr2972776593051762503rm_a_b] :
( ! [X3: produc357393685978478089rm_a_b,Y2: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X3 @ Y2 ) @ R2 )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X3 @ Y2 ) @ S ) )
=> ( ord_le2593958535671617255rm_a_b @ R2 @ S ) ) ).
% subrelI
thf(fact_424_subrelI,axiom,
! [R2: set_Pr4934435412358123699_a_nat,S: set_Pr4934435412358123699_a_nat] :
( ! [X3: a,Y2: nat] :
( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ X3 @ Y2 ) @ R2 )
=> ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ X3 @ Y2 ) @ S ) )
=> ( ord_le8666007276011122963_a_nat @ R2 @ S ) ) ).
% subrelI
thf(fact_425_subrelI,axiom,
! [R2: 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 ) @ R2 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y2 ) @ S ) )
=> ( ord_le118470702582115849rm_a_b @ R2 @ S ) ) ).
% subrelI
thf(fact_426_bot_Oextremum__uniqueI,axiom,
! [A3: produc357393685978478089rm_a_b > $o] :
( ( ord_le6645574462093738516_a_b_o @ A3 @ bot_bo23760184229683272_a_b_o )
=> ( A3 = bot_bo23760184229683272_a_b_o ) ) ).
% bot.extremum_uniqueI
thf(fact_427_bot_Oextremum__uniqueI,axiom,
! [A3: set_b] :
( ( ord_less_eq_set_b @ A3 @ bot_bot_set_b )
=> ( A3 = bot_bot_set_b ) ) ).
% bot.extremum_uniqueI
thf(fact_428_bot_Oextremum__uniqueI,axiom,
! [A3: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A3 @ bot_bo197521221353338581rm_a_b )
=> ( A3 = bot_bo197521221353338581rm_a_b ) ) ).
% bot.extremum_uniqueI
thf(fact_429_bot_Oextremum__uniqueI,axiom,
! [A3: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A3 @ bot_bo9049108969261143879_a_nat )
=> ( A3 = bot_bo9049108969261143879_a_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_430_bot_Oextremum__uniqueI,axiom,
! [A3: nat] :
( ( ord_less_eq_nat @ A3 @ bot_bot_nat )
=> ( A3 = bot_bot_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_431_bot_Oextremum__uniqueI,axiom,
! [A3: set_term_a_b] :
( ( ord_le2705286416250468010rm_a_b @ A3 @ bot_bot_set_term_a_b )
=> ( A3 = bot_bot_set_term_a_b ) ) ).
% bot.extremum_uniqueI
thf(fact_432_bot_Oextremum__unique,axiom,
! [A3: produc357393685978478089rm_a_b > $o] :
( ( ord_le6645574462093738516_a_b_o @ A3 @ bot_bo23760184229683272_a_b_o )
= ( A3 = bot_bo23760184229683272_a_b_o ) ) ).
% bot.extremum_unique
thf(fact_433_bot_Oextremum__unique,axiom,
! [A3: set_b] :
( ( ord_less_eq_set_b @ A3 @ bot_bot_set_b )
= ( A3 = bot_bot_set_b ) ) ).
% bot.extremum_unique
thf(fact_434_bot_Oextremum__unique,axiom,
! [A3: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A3 @ bot_bo197521221353338581rm_a_b )
= ( A3 = bot_bo197521221353338581rm_a_b ) ) ).
% bot.extremum_unique
thf(fact_435_bot_Oextremum__unique,axiom,
! [A3: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A3 @ bot_bo9049108969261143879_a_nat )
= ( A3 = bot_bo9049108969261143879_a_nat ) ) ).
% bot.extremum_unique
thf(fact_436_bot_Oextremum__unique,axiom,
! [A3: nat] :
( ( ord_less_eq_nat @ A3 @ bot_bot_nat )
= ( A3 = bot_bot_nat ) ) ).
% bot.extremum_unique
thf(fact_437_bot_Oextremum__unique,axiom,
! [A3: set_term_a_b] :
( ( ord_le2705286416250468010rm_a_b @ A3 @ bot_bot_set_term_a_b )
= ( A3 = bot_bot_set_term_a_b ) ) ).
% bot.extremum_unique
thf(fact_438_bot_Oextremum,axiom,
! [A3: produc357393685978478089rm_a_b > $o] : ( ord_le6645574462093738516_a_b_o @ bot_bo23760184229683272_a_b_o @ A3 ) ).
% bot.extremum
thf(fact_439_bot_Oextremum,axiom,
! [A3: set_b] : ( ord_less_eq_set_b @ bot_bot_set_b @ A3 ) ).
% bot.extremum
thf(fact_440_bot_Oextremum,axiom,
! [A3: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ bot_bo197521221353338581rm_a_b @ A3 ) ).
% bot.extremum
thf(fact_441_bot_Oextremum,axiom,
! [A3: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ bot_bo9049108969261143879_a_nat @ A3 ) ).
% bot.extremum
thf(fact_442_bot_Oextremum,axiom,
! [A3: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A3 ) ).
% bot.extremum
thf(fact_443_bot_Oextremum,axiom,
! [A3: set_term_a_b] : ( ord_le2705286416250468010rm_a_b @ bot_bot_set_term_a_b @ A3 ) ).
% bot.extremum
thf(fact_444_subset__emptyI,axiom,
! [A4: set_Pr2972776593051762503rm_a_b] :
( ! [X3: produc4523971788519308903rm_a_b] :
~ ( member8417600551952982416rm_a_b @ X3 @ A4 )
=> ( ord_le2593958535671617255rm_a_b @ A4 @ bot_bo5040873672931234227rm_a_b ) ) ).
% subset_emptyI
thf(fact_445_subset__emptyI,axiom,
! [A4: set_b] :
( ! [X3: b] :
~ ( member_b @ X3 @ A4 )
=> ( ord_less_eq_set_b @ A4 @ bot_bot_set_b ) ) ).
% subset_emptyI
thf(fact_446_subset__emptyI,axiom,
! [A4: set_Pr4934435412358123699_a_nat] :
( ! [X3: product_prod_a_nat] :
~ ( member5724188588386418708_a_nat @ X3 @ A4 )
=> ( ord_le8666007276011122963_a_nat @ A4 @ bot_bo9049108969261143879_a_nat ) ) ).
% subset_emptyI
thf(fact_447_subset__emptyI,axiom,
! [A4: set_term_a_b] :
( ! [X3: term_a_b] :
~ ( member_term_a_b @ X3 @ A4 )
=> ( ord_le2705286416250468010rm_a_b @ A4 @ bot_bot_set_term_a_b ) ) ).
% subset_emptyI
thf(fact_448_subset__emptyI,axiom,
! [A4: set_Pr4386577575007340137rm_a_b] :
( ! [X3: produc357393685978478089rm_a_b] :
~ ( member5869715511025134514rm_a_b @ X3 @ A4 )
=> ( ord_le118470702582115849rm_a_b @ A4 @ bot_bo197521221353338581rm_a_b ) ) ).
% subset_emptyI
thf(fact_449_srrstep__monp,axiom,
! [F2: set_Pr4934435412358123699_a_nat,G2: set_Pr4934435412358123699_a_nat,R: set_Pr4386577575007340137rm_a_b] :
( ( ord_le8666007276011122963_a_nat @ F2 @ G2 )
=> ( ord_le118470702582115849rm_a_b @ ( sig_step_a_b @ F2 @ ( rrstep_a_b @ R ) ) @ ( sig_step_a_b @ G2 @ ( rrstep_a_b @ R ) ) ) ) ).
% srrstep_monp
thf(fact_450_rtrancl__subset__rtrancl,axiom,
! [R2: set_Pr2972776593051762503rm_a_b,S: set_Pr2972776593051762503rm_a_b] :
( ( ord_le2593958535671617255rm_a_b @ R2 @ ( transi2615809358984392588rm_a_b @ S ) )
=> ( ord_le2593958535671617255rm_a_b @ ( transi2615809358984392588rm_a_b @ R2 ) @ ( transi2615809358984392588rm_a_b @ S ) ) ) ).
% rtrancl_subset_rtrancl
thf(fact_451_rtrancl__subset__rtrancl,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ R2 @ ( transi7742714808557438673rm_a_b @ S ) )
=> ( ord_le118470702582115849rm_a_b @ ( transi7742714808557438673rm_a_b @ R2 ) @ ( transi7742714808557438673rm_a_b @ S ) ) ) ).
% rtrancl_subset_rtrancl
thf(fact_452_rtrancl__subset,axiom,
! [R5: set_Pr2972776593051762503rm_a_b,S5: set_Pr2972776593051762503rm_a_b] :
( ( ord_le2593958535671617255rm_a_b @ R5 @ S5 )
=> ( ( ord_le2593958535671617255rm_a_b @ S5 @ ( transi2615809358984392588rm_a_b @ R5 ) )
=> ( ( transi2615809358984392588rm_a_b @ S5 )
= ( transi2615809358984392588rm_a_b @ R5 ) ) ) ) ).
% rtrancl_subset
thf(fact_453_rtrancl__subset,axiom,
! [R5: set_Pr4386577575007340137rm_a_b,S5: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ R5 @ S5 )
=> ( ( ord_le118470702582115849rm_a_b @ S5 @ ( transi7742714808557438673rm_a_b @ R5 ) )
=> ( ( transi7742714808557438673rm_a_b @ S5 )
= ( transi7742714808557438673rm_a_b @ R5 ) ) ) ) ).
% rtrancl_subset
thf(fact_454_rtrancl__mono,axiom,
! [R2: set_Pr2972776593051762503rm_a_b,S: set_Pr2972776593051762503rm_a_b] :
( ( ord_le2593958535671617255rm_a_b @ R2 @ S )
=> ( ord_le2593958535671617255rm_a_b @ ( transi2615809358984392588rm_a_b @ R2 ) @ ( transi2615809358984392588rm_a_b @ S ) ) ) ).
% rtrancl_mono
thf(fact_455_rtrancl__mono,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ R2 @ S )
=> ( ord_le118470702582115849rm_a_b @ ( transi7742714808557438673rm_a_b @ R2 ) @ ( transi7742714808557438673rm_a_b @ S ) ) ) ).
% rtrancl_mono
thf(fact_456_O__mono2,axiom,
! [R5: set_Pr4386577575007340137rm_a_b,R8: set_Pr4386577575007340137rm_a_b,T6: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ R5 @ R8 )
=> ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ R5 @ T6 ) @ ( relcom370159955682700863rm_a_b @ R8 @ T6 ) ) ) ).
% O_mono2
thf(fact_457_O__mono2,axiom,
! [R5: set_Pr4934435412358123699_a_nat,R8: set_Pr4934435412358123699_a_nat,T6: set_Pr1261947904930325089at_nat] :
( ( ord_le8666007276011122963_a_nat @ R5 @ R8 )
=> ( ord_le8666007276011122963_a_nat @ ( relcomp_a_nat_nat @ R5 @ T6 ) @ ( relcomp_a_nat_nat @ R8 @ T6 ) ) ) ).
% O_mono2
thf(fact_458_O__mono1,axiom,
! [R5: set_Pr4386577575007340137rm_a_b,R8: set_Pr4386577575007340137rm_a_b,S5: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ R5 @ R8 )
=> ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ S5 @ R5 ) @ ( relcom370159955682700863rm_a_b @ S5 @ R8 ) ) ) ).
% O_mono1
thf(fact_459_O__mono1,axiom,
! [R5: set_Pr4934435412358123699_a_nat,R8: set_Pr4934435412358123699_a_nat,S5: set_Product_prod_a_a] :
( ( ord_le8666007276011122963_a_nat @ R5 @ R8 )
=> ( ord_le8666007276011122963_a_nat @ ( relcomp_a_a_nat @ S5 @ R5 ) @ ( relcomp_a_a_nat @ S5 @ R8 ) ) ) ).
% O_mono1
thf(fact_460_relcomp__mono,axiom,
! [R9: set_Product_prod_a_a,R2: set_Product_prod_a_a,S2: set_Pr4934435412358123699_a_nat,S: set_Pr4934435412358123699_a_nat] :
( ( ord_le746702958409616551od_a_a @ R9 @ R2 )
=> ( ( ord_le8666007276011122963_a_nat @ S2 @ S )
=> ( ord_le8666007276011122963_a_nat @ ( relcomp_a_a_nat @ R9 @ S2 ) @ ( relcomp_a_a_nat @ R2 @ S ) ) ) ) ).
% relcomp_mono
thf(fact_461_relcomp__mono,axiom,
! [R9: set_Pr4386577575007340137rm_a_b,R2: set_Pr4386577575007340137rm_a_b,S2: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ R9 @ R2 )
=> ( ( ord_le118470702582115849rm_a_b @ S2 @ S )
=> ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ R9 @ S2 ) @ ( relcom370159955682700863rm_a_b @ R2 @ S ) ) ) ) ).
% relcomp_mono
thf(fact_462_relcomp__mono,axiom,
! [R9: set_Pr4934435412358123699_a_nat,R2: set_Pr4934435412358123699_a_nat,S2: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
( ( ord_le8666007276011122963_a_nat @ R9 @ R2 )
=> ( ( ord_le3146513528884898305at_nat @ S2 @ S )
=> ( ord_le8666007276011122963_a_nat @ ( relcomp_a_nat_nat @ R9 @ S2 ) @ ( relcomp_a_nat_nat @ R2 @ S ) ) ) ) ).
% relcomp_mono
thf(fact_463_sig__step__mono2,axiom,
! [R: set_Pr4386577575007340137rm_a_b,L4: set_Pr4386577575007340137rm_a_b,F2: set_Pr4934435412358123699_a_nat] :
( ( ord_le118470702582115849rm_a_b @ R @ L4 )
=> ( ord_le118470702582115849rm_a_b @ ( sig_step_a_b @ F2 @ R ) @ ( sig_step_a_b @ F2 @ L4 ) ) ) ).
% sig_step_mono2
thf(fact_464_conversion__refl,axiom,
! [A3: produc357393685978478089rm_a_b,A4: set_Pr2972776593051762503rm_a_b] : ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A3 @ A3 ) @ ( abstra6125656530262095991rm_a_b @ A4 ) ) ).
% conversion_refl
thf(fact_465_conversion__refl,axiom,
! [A3: term_a_b,A4: set_Pr4386577575007340137rm_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ A3 ) @ ( abstra1130568886744591910rm_a_b @ A4 ) ) ).
% conversion_refl
thf(fact_466_conversion__inv,axiom,
! [X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b,R5: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ ( abstra6125656530262095991rm_a_b @ R5 ) )
= ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y @ X ) @ ( abstra6125656530262095991rm_a_b @ R5 ) ) ) ).
% conversion_inv
thf(fact_467_conversion__inv,axiom,
! [X: term_a_b,Y: term_a_b,R5: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( abstra1130568886744591910rm_a_b @ R5 ) )
= ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X ) @ ( abstra1130568886744591910rm_a_b @ R5 ) ) ) ).
% conversion_inv
thf(fact_468_Id__on__iff,axiom,
! [X: produc4523971788519308903rm_a_b,Y: produc4523971788519308903rm_a_b,A4: set_Pr2972776593051762503rm_a_b] :
( ( member5793113912742238992rm_a_b @ ( produc2499998789532207959rm_a_b @ X @ Y ) @ ( id_on_4065744930832257284rm_a_b @ A4 ) )
= ( ( X = Y )
& ( member8417600551952982416rm_a_b @ X @ A4 ) ) ) ).
% Id_on_iff
thf(fact_469_Id__on__iff,axiom,
! [X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b,A4: set_Pr4386577575007340137rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ ( id_on_115818524817812774rm_a_b @ A4 ) )
= ( ( X = Y )
& ( member5869715511025134514rm_a_b @ X @ A4 ) ) ) ).
% Id_on_iff
thf(fact_470_Id__on__iff,axiom,
! [X: term_a_b,Y: term_a_b,A4: set_term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( id_on_term_a_b @ A4 ) )
= ( ( X = Y )
& ( member_term_a_b @ X @ A4 ) ) ) ).
% Id_on_iff
thf(fact_471_Id__on__eqI,axiom,
! [A3: produc4523971788519308903rm_a_b,B2: produc4523971788519308903rm_a_b,A4: set_Pr2972776593051762503rm_a_b] :
( ( A3 = B2 )
=> ( ( member8417600551952982416rm_a_b @ A3 @ A4 )
=> ( member5793113912742238992rm_a_b @ ( produc2499998789532207959rm_a_b @ A3 @ B2 ) @ ( id_on_4065744930832257284rm_a_b @ A4 ) ) ) ) ).
% Id_on_eqI
thf(fact_472_Id__on__eqI,axiom,
! [A3: produc357393685978478089rm_a_b,B2: produc357393685978478089rm_a_b,A4: set_Pr4386577575007340137rm_a_b] :
( ( A3 = B2 )
=> ( ( member5869715511025134514rm_a_b @ A3 @ A4 )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A3 @ B2 ) @ ( id_on_115818524817812774rm_a_b @ A4 ) ) ) ) ).
% Id_on_eqI
thf(fact_473_Id__on__eqI,axiom,
! [A3: term_a_b,B2: term_a_b,A4: set_term_a_b] :
( ( A3 = B2 )
=> ( ( member_term_a_b @ A3 @ A4 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B2 ) @ ( id_on_term_a_b @ A4 ) ) ) ) ).
% Id_on_eqI
thf(fact_474_Id__onE,axiom,
! [C: produc2957090337049255143rm_a_b,A4: set_Pr2972776593051762503rm_a_b] :
( ( member5793113912742238992rm_a_b @ C @ ( id_on_4065744930832257284rm_a_b @ A4 ) )
=> ~ ! [X3: produc4523971788519308903rm_a_b] :
( ( member8417600551952982416rm_a_b @ X3 @ A4 )
=> ( C
!= ( produc2499998789532207959rm_a_b @ X3 @ X3 ) ) ) ) ).
% Id_onE
thf(fact_475_Id__onE,axiom,
! [C: produc4523971788519308903rm_a_b,A4: set_Pr4386577575007340137rm_a_b] :
( ( member8417600551952982416rm_a_b @ C @ ( id_on_115818524817812774rm_a_b @ A4 ) )
=> ~ ! [X3: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X3 @ A4 )
=> ( C
!= ( produc1763473618796451543rm_a_b @ X3 @ X3 ) ) ) ) ).
% Id_onE
thf(fact_476_Id__onE,axiom,
! [C: produc357393685978478089rm_a_b,A4: set_term_a_b] :
( ( member5869715511025134514rm_a_b @ C @ ( id_on_term_a_b @ A4 ) )
=> ~ ! [X3: term_a_b] :
( ( member_term_a_b @ X3 @ A4 )
=> ( C
!= ( produc7020197800436672577rm_a_b @ X3 @ X3 ) ) ) ) ).
% Id_onE
thf(fact_477_srsteps__eq__monp,axiom,
! [F2: set_Pr4934435412358123699_a_nat,G2: set_Pr4934435412358123699_a_nat,R: 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 @ R ) ) ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ G2 @ ( rstep_a_b @ R ) ) ) ) ) ).
% srsteps_eq_monp
thf(fact_478_mem__range__varsI,axiom,
! [Sigma: b > term_a_b,X: b,Y: b] :
( ( ( Sigma @ X )
= ( var_b_a @ Y ) )
=> ( ( X != Y )
=> ( member_b @ Y @ ( range_vars_b_a @ Sigma ) ) ) ) ).
% mem_range_varsI
thf(fact_479_rtrancl__converseI,axiom,
! [Y: produc357393685978478089rm_a_b,X: produc357393685978478089rm_a_b,R2: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y @ X ) @ ( transi2615809358984392588rm_a_b @ R2 ) )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ ( transi2615809358984392588rm_a_b @ ( conver1575690562132235771rm_a_b @ R2 ) ) ) ) ).
% rtrancl_converseI
thf(fact_480_rtrancl__converseI,axiom,
! [Y: term_a_b,X: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ ( conver7608559976937845093rm_a_b @ R2 ) ) ) ) ).
% rtrancl_converseI
thf(fact_481_rtrancl__converseD,axiom,
! [X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b,R2: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ ( transi2615809358984392588rm_a_b @ ( conver1575690562132235771rm_a_b @ R2 ) ) )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y @ X ) @ ( transi2615809358984392588rm_a_b @ R2 ) ) ) ).
% rtrancl_converseD
thf(fact_482_rtrancl__converseD,axiom,
! [X: term_a_b,Y: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ ( conver7608559976937845093rm_a_b @ R2 ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X ) @ ( transi7742714808557438673rm_a_b @ R2 ) ) ) ).
% rtrancl_converseD
thf(fact_483_peak__iff,axiom,
! [X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b,A4: set_Pr2972776593051762503rm_a_b,B5: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ ( relcom6159724393391289352rm_a_b @ ( conver1575690562132235771rm_a_b @ A4 ) @ B5 ) )
= ( ? [U3: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ U3 @ X ) @ A4 )
& ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ U3 @ Y ) @ B5 ) ) ) ) ).
% peak_iff
thf(fact_484_peak__iff,axiom,
! [X: term_a_b,Y: term_a_b,A4: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( relcom370159955682700863rm_a_b @ ( conver7608559976937845093rm_a_b @ A4 ) @ B5 ) )
= ( ? [U3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ U3 @ X ) @ A4 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ U3 @ Y ) @ B5 ) ) ) ) ).
% peak_iff
thf(fact_485_sig__step__converse__rstep,axiom,
! [F2: set_Pr4934435412358123699_a_nat,R: set_Pr4386577575007340137rm_a_b] :
( ( conver7608559976937845093rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R ) ) )
= ( sig_step_a_b @ F2 @ ( conver7608559976937845093rm_a_b @ ( rstep_a_b @ R ) ) ) ) ).
% sig_step_converse_rstep
thf(fact_486_srstep__converse__dist,axiom,
! [F2: set_Pr4934435412358123699_a_nat,R: set_Pr4386577575007340137rm_a_b] :
( ( conver7608559976937845093rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R ) ) )
= ( sig_step_a_b @ F2 @ ( rstep_a_b @ ( conver7608559976937845093rm_a_b @ R ) ) ) ) ).
% srstep_converse_dist
thf(fact_487_srrstep__converse__dist,axiom,
! [F2: set_Pr4934435412358123699_a_nat,R: set_Pr4386577575007340137rm_a_b] :
( ( conver7608559976937845093rm_a_b @ ( sig_step_a_b @ F2 @ ( rrstep_a_b @ R ) ) )
= ( sig_step_a_b @ F2 @ ( rrstep_a_b @ ( conver7608559976937845093rm_a_b @ R ) ) ) ) ).
% srrstep_converse_dist
thf(fact_488_rtrancl__O__push,axiom,
! [S5: set_Pr2972776593051762503rm_a_b,R5: set_Pr2972776593051762503rm_a_b] :
( ( ord_le2593958535671617255rm_a_b @ ( relcom6159724393391289352rm_a_b @ S5 @ R5 ) @ ( relcom6159724393391289352rm_a_b @ R5 @ ( transi2615809358984392588rm_a_b @ S5 ) ) )
=> ( ord_le2593958535671617255rm_a_b @ ( relcom6159724393391289352rm_a_b @ ( transi2615809358984392588rm_a_b @ S5 ) @ R5 ) @ ( relcom6159724393391289352rm_a_b @ R5 @ ( transi2615809358984392588rm_a_b @ S5 ) ) ) ) ).
% rtrancl_O_push
thf(fact_489_rtrancl__O__push,axiom,
! [S5: set_Pr4386577575007340137rm_a_b,R5: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ S5 @ R5 ) @ ( relcom370159955682700863rm_a_b @ R5 @ ( transi7742714808557438673rm_a_b @ S5 ) ) )
=> ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ S5 ) @ R5 ) @ ( relcom370159955682700863rm_a_b @ R5 @ ( transi7742714808557438673rm_a_b @ S5 ) ) ) ) ).
% rtrancl_O_push
thf(fact_490_compat__tr__compat,axiom,
! [NS: set_Pr4386577575007340137rm_a_b,S5: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ NS @ S5 ) @ S5 )
=> ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ NS ) @ S5 ) @ S5 ) ) ).
% compat_tr_compat
thf(fact_491_compat__tr__compat,axiom,
! [NS: set_Product_prod_a_a,S5: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ ( relcomp_a_a_nat @ NS @ S5 ) @ S5 )
=> ( ord_le8666007276011122963_a_nat @ ( relcomp_a_a_nat @ ( transitive_rtrancl_a @ NS ) @ S5 ) @ S5 ) ) ).
% compat_tr_compat
thf(fact_492_rrstep__rstep__mono,axiom,
! [R: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ ( rrstep_a_b @ R ) @ ( rstep_a_b @ R ) ) ).
% rrstep_rstep_mono
thf(fact_493_commute__def,axiom,
( abstra5145658890658669097rm_a_b
= ( ^ [R4: set_Pr2972776593051762503rm_a_b,S4: set_Pr2972776593051762503rm_a_b] : ( ord_le2593958535671617255rm_a_b @ ( relcom6159724393391289352rm_a_b @ ( transi2615809358984392588rm_a_b @ ( conver1575690562132235771rm_a_b @ R4 ) ) @ ( transi2615809358984392588rm_a_b @ S4 ) ) @ ( relcom6159724393391289352rm_a_b @ ( transi2615809358984392588rm_a_b @ S4 ) @ ( transi2615809358984392588rm_a_b @ ( conver1575690562132235771rm_a_b @ R4 ) ) ) ) ) ) ).
% commute_def
thf(fact_494_commute__def,axiom,
( abstra455195320138238452rm_a_b
= ( ^ [R4: set_Pr4386577575007340137rm_a_b,S4: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( conver7608559976937845093rm_a_b @ R4 ) ) @ ( transi7742714808557438673rm_a_b @ S4 ) ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ S4 ) @ ( transi7742714808557438673rm_a_b @ ( conver7608559976937845093rm_a_b @ R4 ) ) ) ) ) ) ).
% commute_def
thf(fact_495_diamond__I,axiom,
! [R2: set_Pr2972776593051762503rm_a_b] :
( ( ord_le2593958535671617255rm_a_b @ ( relcom6159724393391289352rm_a_b @ ( conver1575690562132235771rm_a_b @ R2 ) @ R2 ) @ ( relcom6159724393391289352rm_a_b @ R2 @ ( conver1575690562132235771rm_a_b @ R2 ) ) )
=> ( abstra7467186385360731951rm_a_b @ R2 ) ) ).
% diamond_I
thf(fact_496_diamond__I,axiom,
! [R2: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ ( conver7608559976937845093rm_a_b @ R2 ) @ R2 ) @ ( relcom370159955682700863rm_a_b @ R2 @ ( conver7608559976937845093rm_a_b @ R2 ) ) )
=> ( abstra8151087076625346926rm_a_b @ R2 ) ) ).
% diamond_I
thf(fact_497_kleene__induct,axiom,
! [A4: set_Pr4802440762858034464rm_a_b,X4: set_Pr4802440762858034464rm_a_b,B5: set_Pr4386577575007340137rm_a_b,C3: set_Pr2972776593051762503rm_a_b] :
( ( ord_le7334406955775145344rm_a_b @ A4 @ X4 )
=> ( ( ord_le7334406955775145344rm_a_b @ ( relcom6590495161205167518rm_a_b @ B5 @ X4 ) @ X4 )
=> ( ( ord_le7334406955775145344rm_a_b @ ( relcom4000041261975091669rm_a_b @ X4 @ C3 ) @ X4 )
=> ( ord_le7334406955775145344rm_a_b @ ( relcom6590495161205167518rm_a_b @ ( transi7742714808557438673rm_a_b @ B5 ) @ ( relcom4000041261975091669rm_a_b @ A4 @ ( transi2615809358984392588rm_a_b @ C3 ) ) ) @ X4 ) ) ) ) ).
% kleene_induct
thf(fact_498_kleene__induct,axiom,
! [A4: set_Pr6510905305361987970rm_a_b,X4: set_Pr6510905305361987970rm_a_b,B5: set_Pr2972776593051762503rm_a_b,C3: set_Pr4386577575007340137rm_a_b] :
( ( ord_le9042871498279098850rm_a_b @ A4 @ X4 )
=> ( ( ord_le9042871498279098850rm_a_b @ ( relcom695362439311269717rm_a_b @ B5 @ X4 ) @ X4 )
=> ( ( ord_le9042871498279098850rm_a_b @ ( relcom3777457926417599474rm_a_b @ X4 @ C3 ) @ X4 )
=> ( ord_le9042871498279098850rm_a_b @ ( relcom695362439311269717rm_a_b @ ( transi2615809358984392588rm_a_b @ B5 ) @ ( relcom3777457926417599474rm_a_b @ A4 @ ( transi7742714808557438673rm_a_b @ C3 ) ) ) @ X4 ) ) ) ) ).
% kleene_induct
thf(fact_499_kleene__induct,axiom,
! [A4: set_Pr2972776593051762503rm_a_b,X4: set_Pr2972776593051762503rm_a_b,B5: set_Pr2972776593051762503rm_a_b,C3: set_Pr2972776593051762503rm_a_b] :
( ( ord_le2593958535671617255rm_a_b @ A4 @ X4 )
=> ( ( ord_le2593958535671617255rm_a_b @ ( relcom6159724393391289352rm_a_b @ B5 @ X4 ) @ X4 )
=> ( ( ord_le2593958535671617255rm_a_b @ ( relcom6159724393391289352rm_a_b @ X4 @ C3 ) @ X4 )
=> ( ord_le2593958535671617255rm_a_b @ ( relcom6159724393391289352rm_a_b @ ( transi2615809358984392588rm_a_b @ B5 ) @ ( relcom6159724393391289352rm_a_b @ A4 @ ( transi2615809358984392588rm_a_b @ C3 ) ) ) @ X4 ) ) ) ) ).
% kleene_induct
thf(fact_500_kleene__induct,axiom,
! [A4: set_Pr4934435412358123699_a_nat,X4: set_Pr4934435412358123699_a_nat,B5: set_Product_prod_a_a,C3: set_Pr1261947904930325089at_nat] :
( ( ord_le8666007276011122963_a_nat @ A4 @ X4 )
=> ( ( ord_le8666007276011122963_a_nat @ ( relcomp_a_a_nat @ B5 @ X4 ) @ X4 )
=> ( ( ord_le8666007276011122963_a_nat @ ( relcomp_a_nat_nat @ X4 @ C3 ) @ X4 )
=> ( ord_le8666007276011122963_a_nat @ ( relcomp_a_a_nat @ ( transitive_rtrancl_a @ B5 ) @ ( relcomp_a_nat_nat @ A4 @ ( transi2905341329935302413cl_nat @ C3 ) ) ) @ X4 ) ) ) ) ).
% kleene_induct
thf(fact_501_kleene__induct,axiom,
! [A4: set_Pr4386577575007340137rm_a_b,X4: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b,C3: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A4 @ X4 )
=> ( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ B5 @ X4 ) @ X4 )
=> ( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ X4 @ C3 ) @ X4 )
=> ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ B5 ) @ ( relcom370159955682700863rm_a_b @ A4 @ ( transi7742714808557438673rm_a_b @ C3 ) ) ) @ X4 ) ) ) ) ).
% kleene_induct
thf(fact_502_less__by__empty,axiom,
! [A4: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
( ( A4 = bot_bo197521221353338581rm_a_b )
=> ( ord_le118470702582115849rm_a_b @ A4 @ B5 ) ) ).
% less_by_empty
thf(fact_503_diamond__def,axiom,
( abstra7467186385360731951rm_a_b
= ( ^ [R4: set_Pr2972776593051762503rm_a_b] : ( ord_le2593958535671617255rm_a_b @ ( relcom6159724393391289352rm_a_b @ ( conver1575690562132235771rm_a_b @ R4 ) @ R4 ) @ ( relcom6159724393391289352rm_a_b @ R4 @ ( conver1575690562132235771rm_a_b @ R4 ) ) ) ) ) ).
% diamond_def
thf(fact_504_diamond__def,axiom,
( abstra8151087076625346926rm_a_b
= ( ^ [R4: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ ( conver7608559976937845093rm_a_b @ R4 ) @ R4 ) @ ( relcom370159955682700863rm_a_b @ R4 @ ( conver7608559976937845093rm_a_b @ R4 ) ) ) ) ) ).
% diamond_def
thf(fact_505_diamond__E,axiom,
! [R2: set_Pr2972776593051762503rm_a_b] :
( ( abstra7467186385360731951rm_a_b @ R2 )
=> ( ord_le2593958535671617255rm_a_b @ ( relcom6159724393391289352rm_a_b @ ( conver1575690562132235771rm_a_b @ R2 ) @ R2 ) @ ( relcom6159724393391289352rm_a_b @ R2 @ ( conver1575690562132235771rm_a_b @ R2 ) ) ) ) ).
% diamond_E
thf(fact_506_diamond__E,axiom,
! [R2: set_Pr4386577575007340137rm_a_b] :
( ( abstra8151087076625346926rm_a_b @ R2 )
=> ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ ( conver7608559976937845093rm_a_b @ R2 ) @ R2 ) @ ( relcom370159955682700863rm_a_b @ R2 @ ( conver7608559976937845093rm_a_b @ R2 ) ) ) ) ).
% diamond_E
thf(fact_507_srsteps__eq__ctxt__closed,axiom,
! [C3: subterm_and_ctxt_a_b,F2: set_Pr4934435412358123699_a_nat,S: term_a_b,T: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( ord_le8666007276011122963_a_nat @ ( term_funas_ctxt_a_b @ C3 ) @ F2 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R ) ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( subter2376574525758040790rm_a_b @ C3 @ S ) @ ( subter2376574525758040790rm_a_b @ C3 @ T ) ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R ) ) ) ) ) ) ).
% srsteps_eq_ctxt_closed
thf(fact_508_restrict__rtrancl,axiom,
! [R2: set_Pr2972776593051762503rm_a_b,S5: set_Pr4386577575007340137rm_a_b] : ( ord_le2593958535671617255rm_a_b @ ( transi2615809358984392588rm_a_b @ ( abstra2354209648719431185rm_a_b @ R2 @ S5 ) ) @ ( transi2615809358984392588rm_a_b @ R2 ) ) ).
% restrict_rtrancl
thf(fact_509_restrict__rtrancl,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,S5: set_term_a_b] : ( ord_le118470702582115849rm_a_b @ ( transi7742714808557438673rm_a_b @ ( abstra6662565036008789004rm_a_b @ R2 @ S5 ) ) @ ( transi7742714808557438673rm_a_b @ R2 ) ) ).
% restrict_rtrancl
thf(fact_510_UNF__onI,axiom,
! [A4: set_Pr2972776593051762503rm_a_b,R2: set_Pr6848273517546626247rm_a_b] :
( ! [A: produc4523971788519308903rm_a_b,B: produc4523971788519308903rm_a_b,C6: produc4523971788519308903rm_a_b] :
( ( member8417600551952982416rm_a_b @ A @ A4 )
=> ( ( member5793113912742238992rm_a_b @ ( produc2499998789532207959rm_a_b @ A @ B ) @ ( abstra7490947582319486223rm_a_b @ R2 ) )
=> ( ( member5793113912742238992rm_a_b @ ( produc2499998789532207959rm_a_b @ A @ C6 ) @ ( abstra7490947582319486223rm_a_b @ R2 ) )
=> ( B = C6 ) ) ) )
=> ( abstra3562400838740635925rm_a_b @ R2 @ A4 ) ) ).
% UNF_onI
thf(fact_511_UNF__onI,axiom,
! [A4: set_Pr4386577575007340137rm_a_b,R2: set_Pr2972776593051762503rm_a_b] :
( ! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,C6: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ A @ A4 )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( abstra1496011390211842225rm_a_b @ R2 ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ C6 ) @ ( abstra1496011390211842225rm_a_b @ R2 ) )
=> ( B = C6 ) ) ) )
=> ( abstra1303798319023342519rm_a_b @ R2 @ A4 ) ) ).
% UNF_onI
thf(fact_512_UNF__onI,axiom,
! [A4: set_term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ! [A: term_a_b,B: term_a_b,C6: term_a_b] :
( ( member_term_a_b @ A @ A4 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( abstra2972651231392952684rm_a_b @ R2 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C6 ) @ ( abstra2972651231392952684rm_a_b @ R2 ) )
=> ( B = C6 ) ) ) )
=> ( abstra3619855735679963878rm_a_b @ R2 @ A4 ) ) ).
% UNF_onI
thf(fact_513_UNF__onD,axiom,
! [R2: set_Pr6848273517546626247rm_a_b,A4: set_Pr2972776593051762503rm_a_b,A3: produc4523971788519308903rm_a_b,B2: produc4523971788519308903rm_a_b,C: produc4523971788519308903rm_a_b] :
( ( abstra3562400838740635925rm_a_b @ R2 @ A4 )
=> ( ( member8417600551952982416rm_a_b @ A3 @ A4 )
=> ( ( member5793113912742238992rm_a_b @ ( produc2499998789532207959rm_a_b @ A3 @ B2 ) @ ( abstra7490947582319486223rm_a_b @ R2 ) )
=> ( ( member5793113912742238992rm_a_b @ ( produc2499998789532207959rm_a_b @ A3 @ C ) @ ( abstra7490947582319486223rm_a_b @ R2 ) )
=> ( B2 = C ) ) ) ) ) ).
% UNF_onD
thf(fact_514_UNF__onD,axiom,
! [R2: set_Pr2972776593051762503rm_a_b,A4: set_Pr4386577575007340137rm_a_b,A3: produc357393685978478089rm_a_b,B2: produc357393685978478089rm_a_b,C: produc357393685978478089rm_a_b] :
( ( abstra1303798319023342519rm_a_b @ R2 @ A4 )
=> ( ( member5869715511025134514rm_a_b @ A3 @ A4 )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A3 @ B2 ) @ ( abstra1496011390211842225rm_a_b @ R2 ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A3 @ C ) @ ( abstra1496011390211842225rm_a_b @ R2 ) )
=> ( B2 = C ) ) ) ) ) ).
% UNF_onD
thf(fact_515_UNF__onD,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,A4: set_term_a_b,A3: term_a_b,B2: term_a_b,C: term_a_b] :
( ( abstra3619855735679963878rm_a_b @ R2 @ A4 )
=> ( ( member_term_a_b @ A3 @ A4 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B2 ) @ ( abstra2972651231392952684rm_a_b @ R2 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ C ) @ ( abstra2972651231392952684rm_a_b @ R2 ) )
=> ( B2 = C ) ) ) ) ) ).
% UNF_onD
thf(fact_516_UNF__onE,axiom,
! [R2: set_Pr6848273517546626247rm_a_b,A4: set_Pr2972776593051762503rm_a_b,A3: produc4523971788519308903rm_a_b,B2: produc4523971788519308903rm_a_b,C: produc4523971788519308903rm_a_b] :
( ( abstra3562400838740635925rm_a_b @ R2 @ A4 )
=> ( ( member8417600551952982416rm_a_b @ A3 @ A4 )
=> ( ( B2 != C )
=> ( ( member5793113912742238992rm_a_b @ ( produc2499998789532207959rm_a_b @ A3 @ B2 ) @ ( abstra7490947582319486223rm_a_b @ R2 ) )
=> ~ ( member5793113912742238992rm_a_b @ ( produc2499998789532207959rm_a_b @ A3 @ C ) @ ( abstra7490947582319486223rm_a_b @ R2 ) ) ) ) ) ) ).
% UNF_onE
thf(fact_517_UNF__onE,axiom,
! [R2: set_Pr2972776593051762503rm_a_b,A4: set_Pr4386577575007340137rm_a_b,A3: produc357393685978478089rm_a_b,B2: produc357393685978478089rm_a_b,C: produc357393685978478089rm_a_b] :
( ( abstra1303798319023342519rm_a_b @ R2 @ A4 )
=> ( ( member5869715511025134514rm_a_b @ A3 @ A4 )
=> ( ( B2 != C )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A3 @ B2 ) @ ( abstra1496011390211842225rm_a_b @ R2 ) )
=> ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A3 @ C ) @ ( abstra1496011390211842225rm_a_b @ R2 ) ) ) ) ) ) ).
% UNF_onE
thf(fact_518_UNF__onE,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,A4: set_term_a_b,A3: term_a_b,B2: term_a_b,C: term_a_b] :
( ( abstra3619855735679963878rm_a_b @ R2 @ A4 )
=> ( ( member_term_a_b @ A3 @ A4 )
=> ( ( B2 != C )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B2 ) @ ( abstra2972651231392952684rm_a_b @ R2 ) )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ C ) @ ( abstra2972651231392952684rm_a_b @ R2 ) ) ) ) ) ) ).
% UNF_onE
thf(fact_519_UNF__on__def,axiom,
( abstra1303798319023342519rm_a_b
= ( ^ [R4: set_Pr2972776593051762503rm_a_b,A7: set_Pr4386577575007340137rm_a_b] :
! [X2: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X2 @ A7 )
=> ! [B3: produc357393685978478089rm_a_b,C2: produc357393685978478089rm_a_b] :
( ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X2 @ B3 ) @ ( abstra1496011390211842225rm_a_b @ R4 ) )
& ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X2 @ C2 ) @ ( abstra1496011390211842225rm_a_b @ R4 ) ) )
=> ( B3 = C2 ) ) ) ) ) ).
% UNF_on_def
thf(fact_520_UNF__on__def,axiom,
( abstra3619855735679963878rm_a_b
= ( ^ [R4: set_Pr4386577575007340137rm_a_b,A7: set_term_a_b] :
! [X2: term_a_b] :
( ( member_term_a_b @ X2 @ A7 )
=> ! [B3: term_a_b,C2: term_a_b] :
( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ B3 ) @ ( abstra2972651231392952684rm_a_b @ R4 ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ C2 ) @ ( abstra2972651231392952684rm_a_b @ R4 ) ) )
=> ( B3 = C2 ) ) ) ) ) ).
% UNF_on_def
thf(fact_521_srstep__ctxt__closed,axiom,
! [C3: subterm_and_ctxt_a_b,F2: set_Pr4934435412358123699_a_nat,S: term_a_b,T: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( ord_le8666007276011122963_a_nat @ ( term_funas_ctxt_a_b @ C3 ) @ F2 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( subter2376574525758040790rm_a_b @ C3 @ S ) @ ( subter2376574525758040790rm_a_b @ C3 @ T ) ) @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R ) ) ) ) ) ).
% srstep_ctxt_closed
thf(fact_522_sig__steps__join__ctxt__closed,axiom,
! [C3: subterm_and_ctxt_a_b,F2: set_Pr4934435412358123699_a_nat,S: term_a_b,T: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( ord_le8666007276011122963_a_nat @ ( term_funas_ctxt_a_b @ C3 ) @ F2 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( abstra4096080454567261402rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R ) ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( subter2376574525758040790rm_a_b @ C3 @ S ) @ ( subter2376574525758040790rm_a_b @ C3 @ T ) ) @ ( abstra4096080454567261402rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R ) ) ) ) ) ) ).
% sig_steps_join_ctxt_closed
thf(fact_523_srsteps__eq__subst__relcomp__closed,axiom,
! [S: term_a_b,T: term_a_b,F2: set_Pr4934435412358123699_a_nat,R: set_Pr4386577575007340137rm_a_b,S6: 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 @ R ) ) ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ S6 ) ) ) ) )
=> ( ! [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 @ R ) ) ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ S6 ) ) ) ) ) ) ) ).
% srsteps_eq_subst_relcomp_closed
thf(fact_524_funas__ctxt__ctxt__inv__const__ctxt__ind,axiom,
! [C3: subterm_and_ctxt_a_b,F2: set_Pr4934435412358123699_a_nat,V3: b] :
( ( ord_le8666007276011122963_a_nat @ ( term_funas_ctxt_a_b @ C3 ) @ F2 )
=> ( ( terms_3799363064701517935xt_a_b @ F2 @ V3 @ C3 )
= C3 ) ) ).
% funas_ctxt_ctxt_inv_const_ctxt_ind
thf(fact_525_diamond__imp__semi__confluence,axiom,
! [R2: set_Pr2972776593051762503rm_a_b] :
( ( abstra7467186385360731951rm_a_b @ R2 )
=> ( ord_le2593958535671617255rm_a_b @ ( relcom6159724393391289352rm_a_b @ ( conver1575690562132235771rm_a_b @ R2 ) @ ( transi2615809358984392588rm_a_b @ R2 ) ) @ ( abstra4889682940153293507rm_a_b @ R2 ) ) ) ).
% diamond_imp_semi_confluence
thf(fact_526_diamond__imp__semi__confluence,axiom,
! [R2: set_Pr4386577575007340137rm_a_b] :
( ( abstra8151087076625346926rm_a_b @ R2 )
=> ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ ( conver7608559976937845093rm_a_b @ R2 ) @ ( transi7742714808557438673rm_a_b @ R2 ) ) @ ( abstra4096080454567261402rm_a_b @ R2 ) ) ) ).
% diamond_imp_semi_confluence
thf(fact_527_WN__onI,axiom,
! [A4: set_Pr2972776593051762503rm_a_b,R2: set_Pr6848273517546626247rm_a_b] :
( ! [A: produc4523971788519308903rm_a_b] :
( ( member8417600551952982416rm_a_b @ A @ A4 )
=> ? [B7: produc4523971788519308903rm_a_b] : ( member5793113912742238992rm_a_b @ ( produc2499998789532207959rm_a_b @ A @ B7 ) @ ( abstra7490947582319486223rm_a_b @ R2 ) ) )
=> ( abstra1431869762417195701rm_a_b @ R2 @ A4 ) ) ).
% WN_onI
thf(fact_528_WN__onI,axiom,
! [A4: set_Pr4386577575007340137rm_a_b,R2: set_Pr2972776593051762503rm_a_b] :
( ! [A: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ A @ A4 )
=> ? [B7: produc357393685978478089rm_a_b] : ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B7 ) @ ( abstra1496011390211842225rm_a_b @ R2 ) ) )
=> ( abstra2721873542170697047rm_a_b @ R2 @ A4 ) ) ).
% WN_onI
thf(fact_529_WN__onI,axiom,
! [A4: set_term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ! [A: term_a_b] :
( ( member_term_a_b @ A @ A4 )
=> ? [B7: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B7 ) @ ( abstra2972651231392952684rm_a_b @ R2 ) ) )
=> ( abstra4644735066934440262rm_a_b @ R2 @ A4 ) ) ).
% WN_onI
thf(fact_530_srsteps__ctxt__closed,axiom,
! [C3: subterm_and_ctxt_a_b,F2: set_Pr4934435412358123699_a_nat,S: term_a_b,T: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( ord_le8666007276011122963_a_nat @ ( term_funas_ctxt_a_b @ C3 ) @ F2 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( transi7922773638565587891rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R ) ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( subter2376574525758040790rm_a_b @ C3 @ S ) @ ( subter2376574525758040790rm_a_b @ C3 @ T ) ) @ ( transi7922773638565587891rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R ) ) ) ) ) ) ).
% srsteps_ctxt_closed
thf(fact_531_srsteps__eq__subst__closed,axiom,
! [S: term_a_b,T: term_a_b,F2: set_Pr4934435412358123699_a_nat,R: 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 @ R ) ) ) )
=> ( ! [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 @ R ) ) ) ) ) ) ).
% srsteps_eq_subst_closed
thf(fact_532_trancl__rtrancl__absorb,axiom,
! [R5: set_Pr2972776593051762503rm_a_b] :
( ( transi2615809358984392588rm_a_b @ ( transi9211502839322181930rm_a_b @ R5 ) )
= ( transi2615809358984392588rm_a_b @ R5 ) ) ).
% trancl_rtrancl_absorb
thf(fact_533_trancl__rtrancl__absorb,axiom,
! [R5: set_Pr4386577575007340137rm_a_b] :
( ( transi7742714808557438673rm_a_b @ ( transi7922773638565587891rm_a_b @ R5 ) )
= ( transi7742714808557438673rm_a_b @ R5 ) ) ).
% trancl_rtrancl_absorb
thf(fact_534_rtrancl__trancl__absorb,axiom,
! [R5: set_Pr2972776593051762503rm_a_b] :
( ( transi9211502839322181930rm_a_b @ ( transi2615809358984392588rm_a_b @ R5 ) )
= ( transi2615809358984392588rm_a_b @ R5 ) ) ).
% rtrancl_trancl_absorb
thf(fact_535_rtrancl__trancl__absorb,axiom,
! [R5: set_Pr4386577575007340137rm_a_b] :
( ( transi7922773638565587891rm_a_b @ ( transi7742714808557438673rm_a_b @ R5 ) )
= ( transi7742714808557438673rm_a_b @ R5 ) ) ).
% rtrancl_trancl_absorb
thf(fact_536_trancl__empty,axiom,
( ( transi7922773638565587891rm_a_b @ bot_bo197521221353338581rm_a_b )
= bot_bo197521221353338581rm_a_b ) ).
% trancl_empty
thf(fact_537_trancl__empty,axiom,
( ( transi9211502839322181930rm_a_b @ bot_bo5040873672931234227rm_a_b )
= bot_bo5040873672931234227rm_a_b ) ).
% trancl_empty
thf(fact_538_funas__term__adapt__vars,axiom,
! [T: term_a_b] :
( ( term_funas_term_a_b @ ( term_a2780309480206857766_a_b_b @ T ) )
= ( term_funas_term_a_b @ T ) ) ).
% funas_term_adapt_vars
thf(fact_539_joinI__right,axiom,
! [B2: produc357393685978478089rm_a_b,A3: produc357393685978478089rm_a_b,A4: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B2 @ A3 ) @ ( transi2615809358984392588rm_a_b @ A4 ) )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A3 @ B2 ) @ ( abstra4889682940153293507rm_a_b @ A4 ) ) ) ).
% joinI_right
thf(fact_540_joinI__right,axiom,
! [B2: term_a_b,A3: term_a_b,A4: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ A3 ) @ ( transi7742714808557438673rm_a_b @ A4 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B2 ) @ ( abstra4096080454567261402rm_a_b @ A4 ) ) ) ).
% joinI_right
thf(fact_541_joinI__left,axiom,
! [A3: produc357393685978478089rm_a_b,B2: produc357393685978478089rm_a_b,A4: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A3 @ B2 ) @ ( transi2615809358984392588rm_a_b @ A4 ) )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A3 @ B2 ) @ ( abstra4889682940153293507rm_a_b @ A4 ) ) ) ).
% joinI_left
thf(fact_542_joinI__left,axiom,
! [A3: term_a_b,B2: term_a_b,A4: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B2 ) @ ( transi7742714808557438673rm_a_b @ A4 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B2 ) @ ( abstra4096080454567261402rm_a_b @ A4 ) ) ) ).
% joinI_left
thf(fact_543_joinI,axiom,
! [A3: produc357393685978478089rm_a_b,C: produc357393685978478089rm_a_b,A4: set_Pr2972776593051762503rm_a_b,B2: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A3 @ C ) @ ( transi2615809358984392588rm_a_b @ A4 ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B2 @ C ) @ ( transi2615809358984392588rm_a_b @ A4 ) )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A3 @ B2 ) @ ( abstra4889682940153293507rm_a_b @ A4 ) ) ) ) ).
% joinI
thf(fact_544_joinI,axiom,
! [A3: term_a_b,C: term_a_b,A4: set_Pr4386577575007340137rm_a_b,B2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ C ) @ ( transi7742714808557438673rm_a_b @ A4 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ C ) @ ( transi7742714808557438673rm_a_b @ A4 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B2 ) @ ( abstra4096080454567261402rm_a_b @ A4 ) ) ) ) ).
% joinI
thf(fact_545_sig__stepI,axiom,
! [S: term_a_b,F2: set_Pr4934435412358123699_a_nat,T: term_a_b,R: 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 ) @ R )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( sig_step_a_b @ F2 @ R ) ) ) ) ) ).
% sig_stepI
thf(fact_546_r__into__trancl_H,axiom,
! [P: produc4523971788519308903rm_a_b,R2: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ P @ R2 )
=> ( member8417600551952982416rm_a_b @ P @ ( transi9211502839322181930rm_a_b @ R2 ) ) ) ).
% r_into_trancl'
thf(fact_547_r__into__trancl_H,axiom,
! [P: produc357393685978478089rm_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ P @ R2 )
=> ( member5869715511025134514rm_a_b @ P @ ( transi7922773638565587891rm_a_b @ R2 ) ) ) ).
% r_into_trancl'
thf(fact_548_converse__trancl__induct,axiom,
! [A3: produc357393685978478089rm_a_b,B2: produc357393685978478089rm_a_b,R2: set_Pr2972776593051762503rm_a_b,P2: produc357393685978478089rm_a_b > $o] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A3 @ B2 ) @ ( transi9211502839322181930rm_a_b @ R2 ) )
=> ( ! [Y2: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y2 @ B2 ) @ R2 )
=> ( P2 @ Y2 ) )
=> ( ! [Y2: produc357393685978478089rm_a_b,Z: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y2 @ Z ) @ R2 )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Z @ B2 ) @ ( transi9211502839322181930rm_a_b @ R2 ) )
=> ( ( P2 @ Z )
=> ( P2 @ Y2 ) ) ) )
=> ( P2 @ A3 ) ) ) ) ).
% converse_trancl_induct
thf(fact_549_converse__trancl__induct,axiom,
! [A3: term_a_b,B2: term_a_b,R2: set_Pr4386577575007340137rm_a_b,P2: term_a_b > $o] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B2 ) @ ( transi7922773638565587891rm_a_b @ R2 ) )
=> ( ! [Y2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y2 @ B2 ) @ R2 )
=> ( P2 @ Y2 ) )
=> ( ! [Y2: term_a_b,Z: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y2 @ Z ) @ R2 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Z @ B2 ) @ ( transi7922773638565587891rm_a_b @ R2 ) )
=> ( ( P2 @ Z )
=> ( P2 @ Y2 ) ) ) )
=> ( P2 @ A3 ) ) ) ) ).
% converse_trancl_induct
thf(fact_550_trancl__trans__induct,axiom,
! [X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b,R2: set_Pr2972776593051762503rm_a_b,P2: produc357393685978478089rm_a_b > produc357393685978478089rm_a_b > $o] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ ( transi9211502839322181930rm_a_b @ R2 ) )
=> ( ! [X3: produc357393685978478089rm_a_b,Y2: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X3 @ Y2 ) @ R2 )
=> ( P2 @ X3 @ Y2 ) )
=> ( ! [X3: produc357393685978478089rm_a_b,Y2: produc357393685978478089rm_a_b,Z: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X3 @ Y2 ) @ ( transi9211502839322181930rm_a_b @ R2 ) )
=> ( ( P2 @ X3 @ Y2 )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y2 @ Z ) @ ( transi9211502839322181930rm_a_b @ R2 ) )
=> ( ( P2 @ Y2 @ Z )
=> ( P2 @ X3 @ Z ) ) ) ) )
=> ( P2 @ X @ Y ) ) ) ) ).
% trancl_trans_induct
thf(fact_551_trancl__trans__induct,axiom,
! [X: term_a_b,Y: term_a_b,R2: set_Pr4386577575007340137rm_a_b,P2: term_a_b > term_a_b > $o] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7922773638565587891rm_a_b @ R2 ) )
=> ( ! [X3: term_a_b,Y2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y2 ) @ R2 )
=> ( P2 @ X3 @ Y2 ) )
=> ( ! [X3: term_a_b,Y2: term_a_b,Z: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y2 ) @ ( transi7922773638565587891rm_a_b @ R2 ) )
=> ( ( P2 @ X3 @ Y2 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y2 @ Z ) @ ( transi7922773638565587891rm_a_b @ R2 ) )
=> ( ( P2 @ Y2 @ Z )
=> ( P2 @ X3 @ Z ) ) ) ) )
=> ( P2 @ X @ Y ) ) ) ) ).
% trancl_trans_induct
thf(fact_552_trancl__into__trancl2,axiom,
! [A3: produc357393685978478089rm_a_b,B2: produc357393685978478089rm_a_b,R2: set_Pr2972776593051762503rm_a_b,C: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A3 @ B2 ) @ R2 )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B2 @ C ) @ ( transi9211502839322181930rm_a_b @ R2 ) )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A3 @ C ) @ ( transi9211502839322181930rm_a_b @ R2 ) ) ) ) ).
% trancl_into_trancl2
thf(fact_553_trancl__into__trancl2,axiom,
! [A3: term_a_b,B2: term_a_b,R2: set_Pr4386577575007340137rm_a_b,C: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B2 ) @ R2 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ C ) @ ( transi7922773638565587891rm_a_b @ R2 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ C ) @ ( transi7922773638565587891rm_a_b @ R2 ) ) ) ) ).
% trancl_into_trancl2
thf(fact_554_Transitive__Closure_Otrancl__into__trancl,axiom,
! [A3: produc357393685978478089rm_a_b,B2: produc357393685978478089rm_a_b,R2: set_Pr2972776593051762503rm_a_b,C: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A3 @ B2 ) @ ( transi9211502839322181930rm_a_b @ R2 ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B2 @ C ) @ R2 )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A3 @ C ) @ ( transi9211502839322181930rm_a_b @ R2 ) ) ) ) ).
% Transitive_Closure.trancl_into_trancl
thf(fact_555_Transitive__Closure_Otrancl__into__trancl,axiom,
! [A3: term_a_b,B2: term_a_b,R2: set_Pr4386577575007340137rm_a_b,C: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B2 ) @ ( transi7922773638565587891rm_a_b @ R2 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ C ) @ R2 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ C ) @ ( transi7922773638565587891rm_a_b @ R2 ) ) ) ) ).
% Transitive_Closure.trancl_into_trancl
thf(fact_556_irrefl__trancl__rD,axiom,
! [R2: 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 @ R2 ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ R2 )
=> ( X != Y ) ) ) ).
% irrefl_trancl_rD
thf(fact_557_irrefl__trancl__rD,axiom,
! [R2: 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 @ R2 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ R2 )
=> ( X != Y ) ) ) ).
% irrefl_trancl_rD
thf(fact_558_converse__tranclE,axiom,
! [X: produc357393685978478089rm_a_b,Z2: produc357393685978478089rm_a_b,R2: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Z2 ) @ ( transi9211502839322181930rm_a_b @ R2 ) )
=> ( ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Z2 ) @ R2 )
=> ~ ! [Y2: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y2 ) @ R2 )
=> ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y2 @ Z2 ) @ ( transi9211502839322181930rm_a_b @ R2 ) ) ) ) ) ).
% converse_tranclE
thf(fact_559_converse__tranclE,axiom,
! [X: term_a_b,Z2: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z2 ) @ ( transi7922773638565587891rm_a_b @ R2 ) )
=> ( ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z2 ) @ R2 )
=> ~ ! [Y2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y2 ) @ R2 )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y2 @ Z2 ) @ ( transi7922773638565587891rm_a_b @ R2 ) ) ) ) ) ).
% converse_tranclE
thf(fact_560_r__r__into__trancl,axiom,
! [A3: produc357393685978478089rm_a_b,B2: produc357393685978478089rm_a_b,R5: set_Pr2972776593051762503rm_a_b,C: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A3 @ B2 ) @ R5 )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B2 @ C ) @ R5 )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A3 @ C ) @ ( transi9211502839322181930rm_a_b @ R5 ) ) ) ) ).
% r_r_into_trancl
thf(fact_561_r__r__into__trancl,axiom,
! [A3: term_a_b,B2: term_a_b,R5: set_Pr4386577575007340137rm_a_b,C: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B2 ) @ R5 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ C ) @ R5 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ C ) @ ( transi7922773638565587891rm_a_b @ R5 ) ) ) ) ).
% r_r_into_trancl
thf(fact_562_trancl__induct,axiom,
! [A3: produc357393685978478089rm_a_b,B2: produc357393685978478089rm_a_b,R2: set_Pr2972776593051762503rm_a_b,P2: produc357393685978478089rm_a_b > $o] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A3 @ B2 ) @ ( transi9211502839322181930rm_a_b @ R2 ) )
=> ( ! [Y2: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A3 @ Y2 ) @ R2 )
=> ( P2 @ Y2 ) )
=> ( ! [Y2: produc357393685978478089rm_a_b,Z: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A3 @ Y2 ) @ ( transi9211502839322181930rm_a_b @ R2 ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y2 @ Z ) @ R2 )
=> ( ( P2 @ Y2 )
=> ( P2 @ Z ) ) ) )
=> ( P2 @ B2 ) ) ) ) ).
% trancl_induct
thf(fact_563_trancl__induct,axiom,
! [A3: term_a_b,B2: term_a_b,R2: set_Pr4386577575007340137rm_a_b,P2: term_a_b > $o] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B2 ) @ ( transi7922773638565587891rm_a_b @ R2 ) )
=> ( ! [Y2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ Y2 ) @ R2 )
=> ( P2 @ Y2 ) )
=> ( ! [Y2: term_a_b,Z: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ Y2 ) @ ( transi7922773638565587891rm_a_b @ R2 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y2 @ Z ) @ R2 )
=> ( ( P2 @ Y2 )
=> ( P2 @ Z ) ) ) )
=> ( P2 @ B2 ) ) ) ) ).
% trancl_induct
thf(fact_564_trancl__trans,axiom,
! [X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b,R2: set_Pr2972776593051762503rm_a_b,Z2: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ ( transi9211502839322181930rm_a_b @ R2 ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y @ Z2 ) @ ( transi9211502839322181930rm_a_b @ R2 ) )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Z2 ) @ ( transi9211502839322181930rm_a_b @ R2 ) ) ) ) ).
% trancl_trans
thf(fact_565_trancl__trans,axiom,
! [X: term_a_b,Y: term_a_b,R2: set_Pr4386577575007340137rm_a_b,Z2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7922773638565587891rm_a_b @ R2 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ Z2 ) @ ( transi7922773638565587891rm_a_b @ R2 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z2 ) @ ( transi7922773638565587891rm_a_b @ R2 ) ) ) ) ).
% trancl_trans
thf(fact_566_tranclE,axiom,
! [A3: produc357393685978478089rm_a_b,B2: produc357393685978478089rm_a_b,R2: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A3 @ B2 ) @ ( transi9211502839322181930rm_a_b @ R2 ) )
=> ( ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A3 @ B2 ) @ R2 )
=> ~ ! [C6: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A3 @ C6 ) @ ( transi9211502839322181930rm_a_b @ R2 ) )
=> ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ C6 @ B2 ) @ R2 ) ) ) ) ).
% tranclE
thf(fact_567_tranclE,axiom,
! [A3: term_a_b,B2: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B2 ) @ ( transi7922773638565587891rm_a_b @ R2 ) )
=> ( ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B2 ) @ R2 )
=> ~ ! [C6: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ C6 ) @ ( transi7922773638565587891rm_a_b @ R2 ) )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ C6 @ B2 ) @ R2 ) ) ) ) ).
% tranclE
thf(fact_568_trancl_Or__into__trancl,axiom,
! [A3: produc357393685978478089rm_a_b,B2: produc357393685978478089rm_a_b,R2: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A3 @ B2 ) @ R2 )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A3 @ B2 ) @ ( transi9211502839322181930rm_a_b @ R2 ) ) ) ).
% trancl.r_into_trancl
thf(fact_569_trancl_Or__into__trancl,axiom,
! [A3: term_a_b,B2: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B2 ) @ R2 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B2 ) @ ( transi7922773638565587891rm_a_b @ R2 ) ) ) ).
% trancl.r_into_trancl
thf(fact_570_trancl_Osimps,axiom,
! [A1: produc357393685978478089rm_a_b,A2: produc357393685978478089rm_a_b,R2: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A1 @ A2 ) @ ( transi9211502839322181930rm_a_b @ R2 ) )
= ( ? [A5: produc357393685978478089rm_a_b,B3: produc357393685978478089rm_a_b] :
( ( A1 = A5 )
& ( A2 = B3 )
& ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A5 @ B3 ) @ R2 ) )
| ? [A5: produc357393685978478089rm_a_b,B3: produc357393685978478089rm_a_b,C2: produc357393685978478089rm_a_b] :
( ( A1 = A5 )
& ( A2 = C2 )
& ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A5 @ B3 ) @ ( transi9211502839322181930rm_a_b @ R2 ) )
& ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B3 @ C2 ) @ R2 ) ) ) ) ).
% trancl.simps
thf(fact_571_trancl_Osimps,axiom,
! [A1: term_a_b,A2: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A2 ) @ ( transi7922773638565587891rm_a_b @ R2 ) )
= ( ? [A5: term_a_b,B3: term_a_b] :
( ( A1 = A5 )
& ( A2 = B3 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A5 @ B3 ) @ R2 ) )
| ? [A5: term_a_b,B3: term_a_b,C2: term_a_b] :
( ( A1 = A5 )
& ( A2 = C2 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A5 @ B3 ) @ ( transi7922773638565587891rm_a_b @ R2 ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B3 @ C2 ) @ R2 ) ) ) ) ).
% trancl.simps
thf(fact_572_trancl_Ocases,axiom,
! [A1: produc357393685978478089rm_a_b,A2: produc357393685978478089rm_a_b,R2: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A1 @ A2 ) @ ( transi9211502839322181930rm_a_b @ R2 ) )
=> ( ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A1 @ A2 ) @ R2 )
=> ~ ! [B: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A1 @ B ) @ ( transi9211502839322181930rm_a_b @ R2 ) )
=> ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B @ A2 ) @ R2 ) ) ) ) ).
% trancl.cases
thf(fact_573_trancl_Ocases,axiom,
! [A1: term_a_b,A2: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A2 ) @ ( transi7922773638565587891rm_a_b @ R2 ) )
=> ( ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A2 ) @ R2 )
=> ~ ! [B: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ B ) @ ( transi7922773638565587891rm_a_b @ R2 ) )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A2 ) @ R2 ) ) ) ) ).
% trancl.cases
thf(fact_574_trancl__map,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,F: term_a_b > produc357393685978478089rm_a_b,S: set_Pr2972776593051762503rm_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 ) @ R2 )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( F @ X3 ) @ ( F @ Y2 ) ) @ S ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7922773638565587891rm_a_b @ R2 ) )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( F @ X ) @ ( F @ Y ) ) @ ( transi9211502839322181930rm_a_b @ S ) ) ) ) ).
% trancl_map
thf(fact_575_trancl__map,axiom,
! [R2: set_Pr2972776593051762503rm_a_b,F: produc357393685978478089rm_a_b > term_a_b,S: set_Pr4386577575007340137rm_a_b,X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b] :
( ! [X3: produc357393685978478089rm_a_b,Y2: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X3 @ Y2 ) @ R2 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F @ X3 ) @ ( F @ Y2 ) ) @ S ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ ( transi9211502839322181930rm_a_b @ R2 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F @ X ) @ ( F @ Y ) ) @ ( transi7922773638565587891rm_a_b @ S ) ) ) ) ).
% trancl_map
thf(fact_576_trancl__map,axiom,
! [R2: set_Pr2972776593051762503rm_a_b,F: produc357393685978478089rm_a_b > produc357393685978478089rm_a_b,S: set_Pr2972776593051762503rm_a_b,X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b] :
( ! [X3: produc357393685978478089rm_a_b,Y2: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X3 @ Y2 ) @ R2 )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( F @ X3 ) @ ( F @ Y2 ) ) @ S ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ ( transi9211502839322181930rm_a_b @ R2 ) )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( F @ X ) @ ( F @ Y ) ) @ ( transi9211502839322181930rm_a_b @ S ) ) ) ) ).
% trancl_map
thf(fact_577_trancl__map,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,F: 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 ) @ R2 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F @ X3 ) @ ( F @ Y2 ) ) @ S ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7922773638565587891rm_a_b @ R2 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F @ X ) @ ( F @ Y ) ) @ ( transi7922773638565587891rm_a_b @ S ) ) ) ) ).
% trancl_map
thf(fact_578_trancl__mono,axiom,
! [P: produc4523971788519308903rm_a_b,R2: set_Pr2972776593051762503rm_a_b,S: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ P @ ( transi9211502839322181930rm_a_b @ R2 ) )
=> ( ( ord_le2593958535671617255rm_a_b @ R2 @ S )
=> ( member8417600551952982416rm_a_b @ P @ ( transi9211502839322181930rm_a_b @ S ) ) ) ) ).
% trancl_mono
thf(fact_579_trancl__mono,axiom,
! [P: produc357393685978478089rm_a_b,R2: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ P @ ( transi7922773638565587891rm_a_b @ R2 ) )
=> ( ( ord_le118470702582115849rm_a_b @ R2 @ S )
=> ( member5869715511025134514rm_a_b @ P @ ( transi7922773638565587891rm_a_b @ S ) ) ) ) ).
% trancl_mono
thf(fact_580_trancl__mono__set,axiom,
! [R2: set_Pr2972776593051762503rm_a_b,S: set_Pr2972776593051762503rm_a_b] :
( ( ord_le2593958535671617255rm_a_b @ R2 @ S )
=> ( ord_le2593958535671617255rm_a_b @ ( transi9211502839322181930rm_a_b @ R2 ) @ ( transi9211502839322181930rm_a_b @ S ) ) ) ).
% trancl_mono_set
thf(fact_581_trancl__mono__set,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ R2 @ S )
=> ( ord_le118470702582115849rm_a_b @ ( transi7922773638565587891rm_a_b @ R2 ) @ ( transi7922773638565587891rm_a_b @ S ) ) ) ).
% trancl_mono_set
thf(fact_582_funas__term_Osimps_I1_J,axiom,
! [X: b] :
( ( term_funas_term_a_b @ ( var_b_a @ X ) )
= bot_bo9049108969261143879_a_nat ) ).
% funas_term.simps(1)
thf(fact_583_trancl__converse,axiom,
! [R2: set_Pr4386577575007340137rm_a_b] :
( ( transi7922773638565587891rm_a_b @ ( conver7608559976937845093rm_a_b @ R2 ) )
= ( conver7608559976937845093rm_a_b @ ( transi7922773638565587891rm_a_b @ R2 ) ) ) ).
% trancl_converse
thf(fact_584_trancl__converse,axiom,
! [R2: set_Pr2972776593051762503rm_a_b] :
( ( transi9211502839322181930rm_a_b @ ( conver1575690562132235771rm_a_b @ R2 ) )
= ( conver1575690562132235771rm_a_b @ ( transi9211502839322181930rm_a_b @ R2 ) ) ) ).
% trancl_converse
thf(fact_585_join__sym,axiom,
! [S: produc357393685978478089rm_a_b,T: produc357393685978478089rm_a_b,A4: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ S @ T ) @ ( abstra4889682940153293507rm_a_b @ A4 ) )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ T @ S ) @ ( abstra4889682940153293507rm_a_b @ A4 ) ) ) ).
% join_sym
thf(fact_586_join__sym,axiom,
! [S: term_a_b,T: term_a_b,A4: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( abstra4096080454567261402rm_a_b @ A4 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T @ S ) @ ( abstra4096080454567261402rm_a_b @ A4 ) ) ) ).
% join_sym
thf(fact_587_join__mono,axiom,
! [R2: set_Pr2972776593051762503rm_a_b,S: set_Pr2972776593051762503rm_a_b] :
( ( ord_le2593958535671617255rm_a_b @ R2 @ S )
=> ( ord_le2593958535671617255rm_a_b @ ( abstra4889682940153293507rm_a_b @ R2 ) @ ( abstra4889682940153293507rm_a_b @ S ) ) ) ).
% join_mono
thf(fact_588_join__mono,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ R2 @ S )
=> ( ord_le118470702582115849rm_a_b @ ( abstra4096080454567261402rm_a_b @ R2 ) @ ( abstra4096080454567261402rm_a_b @ S ) ) ) ).
% join_mono
thf(fact_589_srstepsD,axiom,
! [S: term_a_b,T: term_a_b,F2: set_Pr4934435412358123699_a_nat,R: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( transi7922773638565587891rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R ) ) ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( transi7922773638565587891rm_a_b @ ( rstep_a_b @ R ) ) )
& ( 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_590_trancl__rtrancl__trancl,axiom,
! [A3: produc357393685978478089rm_a_b,B2: produc357393685978478089rm_a_b,R2: set_Pr2972776593051762503rm_a_b,C: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A3 @ B2 ) @ ( transi9211502839322181930rm_a_b @ R2 ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B2 @ C ) @ ( transi2615809358984392588rm_a_b @ R2 ) )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A3 @ C ) @ ( transi9211502839322181930rm_a_b @ R2 ) ) ) ) ).
% trancl_rtrancl_trancl
thf(fact_591_trancl__rtrancl__trancl,axiom,
! [A3: term_a_b,B2: term_a_b,R2: set_Pr4386577575007340137rm_a_b,C: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B2 ) @ ( transi7922773638565587891rm_a_b @ R2 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ C ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ C ) @ ( transi7922773638565587891rm_a_b @ R2 ) ) ) ) ).
% trancl_rtrancl_trancl
thf(fact_592_rtrancl__trancl__trancl,axiom,
! [X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b,R2: set_Pr2972776593051762503rm_a_b,Z2: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ ( transi2615809358984392588rm_a_b @ R2 ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y @ Z2 ) @ ( transi9211502839322181930rm_a_b @ R2 ) )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Z2 ) @ ( transi9211502839322181930rm_a_b @ R2 ) ) ) ) ).
% rtrancl_trancl_trancl
thf(fact_593_rtrancl__trancl__trancl,axiom,
! [X: term_a_b,Y: term_a_b,R2: set_Pr4386577575007340137rm_a_b,Z2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ Z2 ) @ ( transi7922773638565587891rm_a_b @ R2 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z2 ) @ ( transi7922773638565587891rm_a_b @ R2 ) ) ) ) ).
% rtrancl_trancl_trancl
thf(fact_594_rtrancl__into__trancl2,axiom,
! [A3: produc357393685978478089rm_a_b,B2: produc357393685978478089rm_a_b,R2: set_Pr2972776593051762503rm_a_b,C: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A3 @ B2 ) @ R2 )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B2 @ C ) @ ( transi2615809358984392588rm_a_b @ R2 ) )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A3 @ C ) @ ( transi9211502839322181930rm_a_b @ R2 ) ) ) ) ).
% rtrancl_into_trancl2
thf(fact_595_rtrancl__into__trancl2,axiom,
! [A3: term_a_b,B2: term_a_b,R2: set_Pr4386577575007340137rm_a_b,C: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B2 ) @ R2 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ C ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ C ) @ ( transi7922773638565587891rm_a_b @ R2 ) ) ) ) ).
% rtrancl_into_trancl2
thf(fact_596_rtrancl__into__trancl1,axiom,
! [A3: produc357393685978478089rm_a_b,B2: produc357393685978478089rm_a_b,R2: set_Pr2972776593051762503rm_a_b,C: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A3 @ B2 ) @ ( transi2615809358984392588rm_a_b @ R2 ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B2 @ C ) @ R2 )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A3 @ C ) @ ( transi9211502839322181930rm_a_b @ R2 ) ) ) ) ).
% rtrancl_into_trancl1
thf(fact_597_rtrancl__into__trancl1,axiom,
! [A3: term_a_b,B2: term_a_b,R2: set_Pr4386577575007340137rm_a_b,C: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B2 ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ C ) @ R2 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ C ) @ ( transi7922773638565587891rm_a_b @ R2 ) ) ) ) ).
% rtrancl_into_trancl1
thf(fact_598_rtrancl__eq__or__trancl,axiom,
! [X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b,R5: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ ( transi2615809358984392588rm_a_b @ R5 ) )
= ( ( X = Y )
| ( ( X != Y )
& ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ ( transi9211502839322181930rm_a_b @ R5 ) ) ) ) ) ).
% rtrancl_eq_or_trancl
thf(fact_599_rtrancl__eq__or__trancl,axiom,
! [X: term_a_b,Y: term_a_b,R5: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ R5 ) )
= ( ( X = Y )
| ( ( X != Y )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7922773638565587891rm_a_b @ R5 ) ) ) ) ) ).
% rtrancl_eq_or_trancl
thf(fact_600_trancl__into__rtrancl,axiom,
! [A3: produc357393685978478089rm_a_b,B2: produc357393685978478089rm_a_b,R2: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A3 @ B2 ) @ ( transi9211502839322181930rm_a_b @ R2 ) )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A3 @ B2 ) @ ( transi2615809358984392588rm_a_b @ R2 ) ) ) ).
% trancl_into_rtrancl
thf(fact_601_trancl__into__rtrancl,axiom,
! [A3: term_a_b,B2: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B2 ) @ ( transi7922773638565587891rm_a_b @ R2 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B2 ) @ ( transi7742714808557438673rm_a_b @ R2 ) ) ) ).
% trancl_into_rtrancl
thf(fact_602_tranclD2,axiom,
! [X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b,R5: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ ( transi9211502839322181930rm_a_b @ R5 ) )
=> ? [Z: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Z ) @ ( transi2615809358984392588rm_a_b @ R5 ) )
& ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Z @ Y ) @ R5 ) ) ) ).
% tranclD2
thf(fact_603_tranclD2,axiom,
! [X: term_a_b,Y: term_a_b,R5: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7922773638565587891rm_a_b @ R5 ) )
=> ? [Z: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z ) @ ( transi7742714808557438673rm_a_b @ R5 ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Z @ Y ) @ R5 ) ) ) ).
% tranclD2
thf(fact_604_rtranclD,axiom,
! [A3: produc357393685978478089rm_a_b,B2: produc357393685978478089rm_a_b,R5: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A3 @ B2 ) @ ( transi2615809358984392588rm_a_b @ R5 ) )
=> ( ( A3 = B2 )
| ( ( A3 != B2 )
& ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A3 @ B2 ) @ ( transi9211502839322181930rm_a_b @ R5 ) ) ) ) ) ).
% rtranclD
thf(fact_605_rtranclD,axiom,
! [A3: term_a_b,B2: term_a_b,R5: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B2 ) @ ( transi7742714808557438673rm_a_b @ R5 ) )
=> ( ( A3 = B2 )
| ( ( A3 != B2 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B2 ) @ ( transi7922773638565587891rm_a_b @ R5 ) ) ) ) ) ).
% rtranclD
thf(fact_606_tranclD,axiom,
! [X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b,R5: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ ( transi9211502839322181930rm_a_b @ R5 ) )
=> ? [Z: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Z ) @ R5 )
& ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Z @ Y ) @ ( transi2615809358984392588rm_a_b @ R5 ) ) ) ) ).
% tranclD
thf(fact_607_tranclD,axiom,
! [X: term_a_b,Y: term_a_b,R5: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7922773638565587891rm_a_b @ R5 ) )
=> ? [Z: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z ) @ R5 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Z @ Y ) @ ( transi7742714808557438673rm_a_b @ R5 ) ) ) ) ).
% tranclD
thf(fact_608_trancl__converseI,axiom,
! [X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b,R2: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ ( conver1575690562132235771rm_a_b @ ( transi9211502839322181930rm_a_b @ R2 ) ) )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ ( transi9211502839322181930rm_a_b @ ( conver1575690562132235771rm_a_b @ R2 ) ) ) ) ).
% trancl_converseI
thf(fact_609_trancl__converseI,axiom,
! [X: term_a_b,Y: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( conver7608559976937845093rm_a_b @ ( transi7922773638565587891rm_a_b @ R2 ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7922773638565587891rm_a_b @ ( conver7608559976937845093rm_a_b @ R2 ) ) ) ) ).
% trancl_converseI
thf(fact_610_trancl__converseD,axiom,
! [X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b,R2: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ ( transi9211502839322181930rm_a_b @ ( conver1575690562132235771rm_a_b @ R2 ) ) )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ ( conver1575690562132235771rm_a_b @ ( transi9211502839322181930rm_a_b @ R2 ) ) ) ) ).
% trancl_converseD
thf(fact_611_trancl__converseD,axiom,
! [X: term_a_b,Y: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7922773638565587891rm_a_b @ ( conver7608559976937845093rm_a_b @ R2 ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( conver7608559976937845093rm_a_b @ ( transi7922773638565587891rm_a_b @ R2 ) ) ) ) ).
% trancl_converseD
thf(fact_612_O__rtrancl__O__O,axiom,
! [R5: set_Pr4386577575007340137rm_a_b,S5: set_Pr4386577575007340137rm_a_b] :
( ( relcom370159955682700863rm_a_b @ R5 @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( relcom370159955682700863rm_a_b @ S5 @ R5 ) ) @ S5 ) )
= ( transi7922773638565587891rm_a_b @ ( relcom370159955682700863rm_a_b @ R5 @ S5 ) ) ) ).
% O_rtrancl_O_O
thf(fact_613_O__rtrancl__O__O,axiom,
! [R5: set_Pr6510905305361987970rm_a_b,S5: set_Pr4802440762858034464rm_a_b] :
( ( relcom8086516547131752875rm_a_b @ R5 @ ( relcom6590495161205167518rm_a_b @ ( transi7742714808557438673rm_a_b @ ( relcom8422713090239460168rm_a_b @ S5 @ R5 ) ) @ S5 ) )
= ( transi9211502839322181930rm_a_b @ ( relcom8086516547131752875rm_a_b @ R5 @ S5 ) ) ) ).
% O_rtrancl_O_O
thf(fact_614_O__rtrancl__O__O,axiom,
! [R5: set_Pr4802440762858034464rm_a_b,S5: set_Pr6510905305361987970rm_a_b] :
( ( relcom8422713090239460168rm_a_b @ R5 @ ( relcom695362439311269717rm_a_b @ ( transi2615809358984392588rm_a_b @ ( relcom8086516547131752875rm_a_b @ S5 @ R5 ) ) @ S5 ) )
= ( transi7922773638565587891rm_a_b @ ( relcom8422713090239460168rm_a_b @ R5 @ S5 ) ) ) ).
% O_rtrancl_O_O
thf(fact_615_O__rtrancl__O__O,axiom,
! [R5: set_Pr2972776593051762503rm_a_b,S5: set_Pr2972776593051762503rm_a_b] :
( ( relcom6159724393391289352rm_a_b @ R5 @ ( relcom6159724393391289352rm_a_b @ ( transi2615809358984392588rm_a_b @ ( relcom6159724393391289352rm_a_b @ S5 @ R5 ) ) @ S5 ) )
= ( transi9211502839322181930rm_a_b @ ( relcom6159724393391289352rm_a_b @ R5 @ S5 ) ) ) ).
% O_rtrancl_O_O
thf(fact_616_rtrancl__comp__trancl__conv,axiom,
! [R2: set_Pr2972776593051762503rm_a_b] :
( ( relcom6159724393391289352rm_a_b @ ( transi2615809358984392588rm_a_b @ R2 ) @ R2 )
= ( transi9211502839322181930rm_a_b @ R2 ) ) ).
% rtrancl_comp_trancl_conv
thf(fact_617_rtrancl__comp__trancl__conv,axiom,
! [R2: set_Pr4386577575007340137rm_a_b] :
( ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ R2 ) @ R2 )
= ( transi7922773638565587891rm_a_b @ R2 ) ) ).
% rtrancl_comp_trancl_conv
thf(fact_618_trancl__unfold__left,axiom,
( transi9211502839322181930rm_a_b
= ( ^ [R4: set_Pr2972776593051762503rm_a_b] : ( relcom6159724393391289352rm_a_b @ R4 @ ( transi2615809358984392588rm_a_b @ R4 ) ) ) ) ).
% trancl_unfold_left
thf(fact_619_trancl__unfold__left,axiom,
( transi7922773638565587891rm_a_b
= ( ^ [R4: set_Pr4386577575007340137rm_a_b] : ( relcom370159955682700863rm_a_b @ R4 @ ( transi7742714808557438673rm_a_b @ R4 ) ) ) ) ).
% trancl_unfold_left
thf(fact_620_comp__trancl,axiom,
! [R5: set_Pr2972776593051762503rm_a_b,S5: set_Pr2972776593051762503rm_a_b] :
( ( ord_le2593958535671617255rm_a_b @ ( relcom6159724393391289352rm_a_b @ R5 @ S5 ) @ S5 )
=> ( ord_le2593958535671617255rm_a_b @ ( relcom6159724393391289352rm_a_b @ R5 @ ( transi9211502839322181930rm_a_b @ S5 ) ) @ ( transi9211502839322181930rm_a_b @ S5 ) ) ) ).
% comp_trancl
thf(fact_621_comp__trancl,axiom,
! [R5: set_Pr4386577575007340137rm_a_b,S5: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ R5 @ S5 ) @ S5 )
=> ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ R5 @ ( transi7922773638565587891rm_a_b @ S5 ) ) @ ( transi7922773638565587891rm_a_b @ S5 ) ) ) ).
% comp_trancl
thf(fact_622_kleene__trancl__induct,axiom,
! [A4: set_Pr4802440762858034464rm_a_b,X4: set_Pr4802440762858034464rm_a_b,B5: set_Pr4386577575007340137rm_a_b,C3: set_Pr2972776593051762503rm_a_b] :
( ( ord_le7334406955775145344rm_a_b @ A4 @ X4 )
=> ( ( ord_le7334406955775145344rm_a_b @ ( relcom6590495161205167518rm_a_b @ B5 @ X4 ) @ X4 )
=> ( ( ord_le7334406955775145344rm_a_b @ ( relcom4000041261975091669rm_a_b @ X4 @ C3 ) @ X4 )
=> ( ord_le7334406955775145344rm_a_b @ ( relcom6590495161205167518rm_a_b @ ( transi7922773638565587891rm_a_b @ B5 ) @ ( relcom4000041261975091669rm_a_b @ A4 @ ( transi9211502839322181930rm_a_b @ C3 ) ) ) @ X4 ) ) ) ) ).
% kleene_trancl_induct
thf(fact_623_kleene__trancl__induct,axiom,
! [A4: set_Pr6510905305361987970rm_a_b,X4: set_Pr6510905305361987970rm_a_b,B5: set_Pr2972776593051762503rm_a_b,C3: set_Pr4386577575007340137rm_a_b] :
( ( ord_le9042871498279098850rm_a_b @ A4 @ X4 )
=> ( ( ord_le9042871498279098850rm_a_b @ ( relcom695362439311269717rm_a_b @ B5 @ X4 ) @ X4 )
=> ( ( ord_le9042871498279098850rm_a_b @ ( relcom3777457926417599474rm_a_b @ X4 @ C3 ) @ X4 )
=> ( ord_le9042871498279098850rm_a_b @ ( relcom695362439311269717rm_a_b @ ( transi9211502839322181930rm_a_b @ B5 ) @ ( relcom3777457926417599474rm_a_b @ A4 @ ( transi7922773638565587891rm_a_b @ C3 ) ) ) @ X4 ) ) ) ) ).
% kleene_trancl_induct
thf(fact_624_kleene__trancl__induct,axiom,
! [A4: set_Pr2972776593051762503rm_a_b,X4: set_Pr2972776593051762503rm_a_b,B5: set_Pr2972776593051762503rm_a_b,C3: set_Pr2972776593051762503rm_a_b] :
( ( ord_le2593958535671617255rm_a_b @ A4 @ X4 )
=> ( ( ord_le2593958535671617255rm_a_b @ ( relcom6159724393391289352rm_a_b @ B5 @ X4 ) @ X4 )
=> ( ( ord_le2593958535671617255rm_a_b @ ( relcom6159724393391289352rm_a_b @ X4 @ C3 ) @ X4 )
=> ( ord_le2593958535671617255rm_a_b @ ( relcom6159724393391289352rm_a_b @ ( transi9211502839322181930rm_a_b @ B5 ) @ ( relcom6159724393391289352rm_a_b @ A4 @ ( transi9211502839322181930rm_a_b @ C3 ) ) ) @ X4 ) ) ) ) ).
% kleene_trancl_induct
thf(fact_625_kleene__trancl__induct,axiom,
! [A4: set_Pr4386577575007340137rm_a_b,X4: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b,C3: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A4 @ X4 )
=> ( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ B5 @ X4 ) @ X4 )
=> ( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ X4 @ C3 ) @ X4 )
=> ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ ( transi7922773638565587891rm_a_b @ B5 ) @ ( relcom370159955682700863rm_a_b @ A4 @ ( transi7922773638565587891rm_a_b @ C3 ) ) ) @ X4 ) ) ) ) ).
% kleene_trancl_induct
thf(fact_626_kleene__trancl__induct,axiom,
! [A4: set_Pr4934435412358123699_a_nat,X4: set_Pr4934435412358123699_a_nat,B5: set_Product_prod_a_a,C3: set_Pr1261947904930325089at_nat] :
( ( ord_le8666007276011122963_a_nat @ A4 @ X4 )
=> ( ( ord_le8666007276011122963_a_nat @ ( relcomp_a_a_nat @ B5 @ X4 ) @ X4 )
=> ( ( ord_le8666007276011122963_a_nat @ ( relcomp_a_nat_nat @ X4 @ C3 ) @ X4 )
=> ( ord_le8666007276011122963_a_nat @ ( relcomp_a_a_nat @ ( transitive_trancl_a @ B5 ) @ ( relcomp_a_nat_nat @ A4 @ ( transi6264000038957366511cl_nat @ C3 ) ) ) @ X4 ) ) ) ) ).
% kleene_trancl_induct
thf(fact_627_srsteps__subst__closed,axiom,
! [S: term_a_b,T: term_a_b,F2: set_Pr4934435412358123699_a_nat,R: 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 @ R ) ) ) )
=> ( ! [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 @ R ) ) ) ) ) ) ).
% srsteps_subst_closed
thf(fact_628_rtrancl__join__join,axiom,
! [A3: produc357393685978478089rm_a_b,B2: produc357393685978478089rm_a_b,A4: set_Pr2972776593051762503rm_a_b,C: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A3 @ B2 ) @ ( transi2615809358984392588rm_a_b @ A4 ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B2 @ C ) @ ( abstra4889682940153293507rm_a_b @ A4 ) )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A3 @ C ) @ ( abstra4889682940153293507rm_a_b @ A4 ) ) ) ) ).
% rtrancl_join_join
thf(fact_629_rtrancl__join__join,axiom,
! [A3: term_a_b,B2: term_a_b,A4: set_Pr4386577575007340137rm_a_b,C: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B2 ) @ ( transi7742714808557438673rm_a_b @ A4 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ C ) @ ( abstra4096080454567261402rm_a_b @ A4 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ C ) @ ( abstra4096080454567261402rm_a_b @ A4 ) ) ) ) ).
% rtrancl_join_join
thf(fact_630_join__rtrancl__join,axiom,
! [A3: produc357393685978478089rm_a_b,B2: produc357393685978478089rm_a_b,A4: set_Pr2972776593051762503rm_a_b,C: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A3 @ B2 ) @ ( abstra4889682940153293507rm_a_b @ A4 ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ C @ B2 ) @ ( transi2615809358984392588rm_a_b @ A4 ) )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A3 @ C ) @ ( abstra4889682940153293507rm_a_b @ A4 ) ) ) ) ).
% join_rtrancl_join
thf(fact_631_join__rtrancl__join,axiom,
! [A3: term_a_b,B2: term_a_b,A4: set_Pr4386577575007340137rm_a_b,C: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B2 ) @ ( abstra4096080454567261402rm_a_b @ A4 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ C @ B2 ) @ ( transi7742714808557438673rm_a_b @ A4 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ C ) @ ( abstra4096080454567261402rm_a_b @ A4 ) ) ) ) ).
% join_rtrancl_join
thf(fact_632_joinE,axiom,
! [A3: produc357393685978478089rm_a_b,B2: produc357393685978478089rm_a_b,A4: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A3 @ B2 ) @ ( abstra4889682940153293507rm_a_b @ A4 ) )
=> ~ ! [C6: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A3 @ C6 ) @ ( transi2615809358984392588rm_a_b @ A4 ) )
=> ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B2 @ C6 ) @ ( transi2615809358984392588rm_a_b @ A4 ) ) ) ) ).
% joinE
thf(fact_633_joinE,axiom,
! [A3: term_a_b,B2: term_a_b,A4: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B2 ) @ ( abstra4096080454567261402rm_a_b @ A4 ) )
=> ~ ! [C6: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ C6 ) @ ( transi7742714808557438673rm_a_b @ A4 ) )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ C6 ) @ ( transi7742714808557438673rm_a_b @ A4 ) ) ) ) ).
% joinE
thf(fact_634_joinD,axiom,
! [A3: produc357393685978478089rm_a_b,B2: produc357393685978478089rm_a_b,A4: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A3 @ B2 ) @ ( abstra4889682940153293507rm_a_b @ A4 ) )
=> ? [C6: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A3 @ C6 ) @ ( transi2615809358984392588rm_a_b @ A4 ) )
& ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B2 @ C6 ) @ ( transi2615809358984392588rm_a_b @ A4 ) ) ) ) ).
% joinD
thf(fact_635_joinD,axiom,
! [A3: term_a_b,B2: term_a_b,A4: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B2 ) @ ( abstra4096080454567261402rm_a_b @ A4 ) )
=> ? [C6: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ C6 ) @ ( transi7742714808557438673rm_a_b @ A4 ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ C6 ) @ ( transi7742714808557438673rm_a_b @ A4 ) ) ) ) ).
% joinD
thf(fact_636_join__imp__conversion,axiom,
! [R2: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ ( abstra4096080454567261402rm_a_b @ R2 ) @ ( abstra1130568886744591910rm_a_b @ R2 ) ) ).
% join_imp_conversion
thf(fact_637_join__imp__conversion,axiom,
! [R2: set_Pr2972776593051762503rm_a_b] : ( ord_le2593958535671617255rm_a_b @ ( abstra4889682940153293507rm_a_b @ R2 ) @ ( abstra6125656530262095991rm_a_b @ R2 ) ) ).
% join_imp_conversion
thf(fact_638_sig__stepE,axiom,
! [S: term_a_b,T: term_a_b,F2: set_Pr4934435412358123699_a_nat,R: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( sig_step_a_b @ F2 @ R ) )
=> ~ ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ R )
=> ( ( 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_639_join__def,axiom,
( abstra4889682940153293507rm_a_b
= ( ^ [A7: set_Pr2972776593051762503rm_a_b] : ( relcom6159724393391289352rm_a_b @ ( transi2615809358984392588rm_a_b @ A7 ) @ ( transi2615809358984392588rm_a_b @ ( conver1575690562132235771rm_a_b @ A7 ) ) ) ) ) ).
% join_def
thf(fact_640_join__def,axiom,
( abstra4096080454567261402rm_a_b
= ( ^ [A7: set_Pr4386577575007340137rm_a_b] : ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ A7 ) @ ( transi7742714808557438673rm_a_b @ ( conver7608559976937845093rm_a_b @ A7 ) ) ) ) ) ).
% join_def
thf(fact_641_srstepD,axiom,
! [S: term_a_b,T: term_a_b,F2: set_Pr4934435412358123699_a_nat,R: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R ) ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( rstep_a_b @ R ) )
& ( 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_642_srrstepD,axiom,
! [S: term_a_b,T: term_a_b,F2: set_Pr4934435412358123699_a_nat,R: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( sig_step_a_b @ F2 @ ( rrstep_a_b @ R ) ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( rrstep_a_b @ R ) )
& ( 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_643_WN__on__def,axiom,
( abstra2721873542170697047rm_a_b
= ( ^ [R4: set_Pr2972776593051762503rm_a_b,A7: set_Pr4386577575007340137rm_a_b] :
! [X2: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X2 @ A7 )
=> ? [B3: produc357393685978478089rm_a_b] : ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X2 @ B3 ) @ ( abstra1496011390211842225rm_a_b @ R4 ) ) ) ) ) ).
% WN_on_def
thf(fact_644_WN__on__def,axiom,
( abstra4644735066934440262rm_a_b
= ( ^ [R4: set_Pr4386577575007340137rm_a_b,A7: set_term_a_b] :
! [X2: term_a_b] :
( ( member_term_a_b @ X2 @ A7 )
=> ? [B3: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ B3 ) @ ( abstra2972651231392952684rm_a_b @ R4 ) ) ) ) ) ).
% WN_on_def
thf(fact_645_WN__onE,axiom,
! [R2: set_Pr6848273517546626247rm_a_b,A4: set_Pr2972776593051762503rm_a_b,A3: produc4523971788519308903rm_a_b] :
( ( abstra1431869762417195701rm_a_b @ R2 @ A4 )
=> ( ( member8417600551952982416rm_a_b @ A3 @ A4 )
=> ~ ! [B: produc4523971788519308903rm_a_b] :
~ ( member5793113912742238992rm_a_b @ ( produc2499998789532207959rm_a_b @ A3 @ B ) @ ( abstra7490947582319486223rm_a_b @ R2 ) ) ) ) ).
% WN_onE
thf(fact_646_WN__onE,axiom,
! [R2: set_Pr2972776593051762503rm_a_b,A4: set_Pr4386577575007340137rm_a_b,A3: produc357393685978478089rm_a_b] :
( ( abstra2721873542170697047rm_a_b @ R2 @ A4 )
=> ( ( member5869715511025134514rm_a_b @ A3 @ A4 )
=> ~ ! [B: produc357393685978478089rm_a_b] :
~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A3 @ B ) @ ( abstra1496011390211842225rm_a_b @ R2 ) ) ) ) ).
% WN_onE
thf(fact_647_WN__onE,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,A4: set_term_a_b,A3: term_a_b] :
( ( abstra4644735066934440262rm_a_b @ R2 @ A4 )
=> ( ( member_term_a_b @ A3 @ A4 )
=> ~ ! [B: term_a_b] :
~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B ) @ ( abstra2972651231392952684rm_a_b @ R2 ) ) ) ) ).
% WN_onE
thf(fact_648_WN__onD,axiom,
! [R2: set_Pr6848273517546626247rm_a_b,A4: set_Pr2972776593051762503rm_a_b,A3: produc4523971788519308903rm_a_b] :
( ( abstra1431869762417195701rm_a_b @ R2 @ A4 )
=> ( ( member8417600551952982416rm_a_b @ A3 @ A4 )
=> ? [B: produc4523971788519308903rm_a_b] : ( member5793113912742238992rm_a_b @ ( produc2499998789532207959rm_a_b @ A3 @ B ) @ ( abstra7490947582319486223rm_a_b @ R2 ) ) ) ) ).
% WN_onD
thf(fact_649_WN__onD,axiom,
! [R2: set_Pr2972776593051762503rm_a_b,A4: set_Pr4386577575007340137rm_a_b,A3: produc357393685978478089rm_a_b] :
( ( abstra2721873542170697047rm_a_b @ R2 @ A4 )
=> ( ( member5869715511025134514rm_a_b @ A3 @ A4 )
=> ? [B: produc357393685978478089rm_a_b] : ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A3 @ B ) @ ( abstra1496011390211842225rm_a_b @ R2 ) ) ) ) ).
% WN_onD
thf(fact_650_WN__onD,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,A4: set_term_a_b,A3: term_a_b] :
( ( abstra4644735066934440262rm_a_b @ R2 @ A4 )
=> ( ( member_term_a_b @ A3 @ A4 )
=> ? [B: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B ) @ ( abstra2972651231392952684rm_a_b @ R2 ) ) ) ) ).
% WN_onD
thf(fact_651_srsteps__monp,axiom,
! [F2: set_Pr4934435412358123699_a_nat,G2: set_Pr4934435412358123699_a_nat,R: 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 @ R ) ) ) @ ( transi7922773638565587891rm_a_b @ ( sig_step_a_b @ G2 @ ( rstep_a_b @ R ) ) ) ) ) ).
% srsteps_monp
thf(fact_652_srstep__subst__closed,axiom,
! [S: term_a_b,T: term_a_b,F2: set_Pr4934435412358123699_a_nat,R: 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 @ R ) ) )
=> ( ! [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 @ R ) ) ) ) ) ).
% srstep_subst_closed
thf(fact_653_rsteps__eq__relcomp__srsteps__eq__relcompI,axiom,
! [R: set_Pr4386577575007340137rm_a_b,F2: set_Pr4934435412358123699_a_nat,S6: set_Pr4386577575007340137rm_a_b,S: term_a_b,T: term_a_b] :
( ( ord_le8666007276011122963_a_nat @ ( terms_7988297476397195622_a_b_b @ R ) @ F2 )
=> ( ( ord_le8666007276011122963_a_nat @ ( terms_7988297476397195622_a_b_b @ S6 ) @ 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 @ R ) ) @ ( transi7742714808557438673rm_a_b @ ( rstep_a_b @ S6 ) ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R ) ) ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ S6 ) ) ) ) ) ) ) ) ) ) ).
% rsteps_eq_relcomp_srsteps_eq_relcompI
thf(fact_654_rsteps__srstepsI,axiom,
! [R: 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 @ R ) @ 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 @ R ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( transi7922773638565587891rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R ) ) ) ) ) ) ) ) ).
% rsteps_srstepsI
thf(fact_655_rsteps__eq__srsteps__eqI,axiom,
! [R: 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 @ R ) @ 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 @ R ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R ) ) ) ) ) ) ) ) ).
% rsteps_eq_srsteps_eqI
thf(fact_656_rstep__srstepI,axiom,
! [R: 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 @ R ) @ 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 @ R ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R ) ) ) ) ) ) ) ).
% rstep_srstepI
thf(fact_657_WCR__onI,axiom,
! [A4: set_Pr2972776593051762503rm_a_b,R2: set_Pr6848273517546626247rm_a_b] :
( ! [A: produc4523971788519308903rm_a_b,B: produc4523971788519308903rm_a_b,C6: produc4523971788519308903rm_a_b] :
( ( member8417600551952982416rm_a_b @ A @ A4 )
=> ( ( member5793113912742238992rm_a_b @ ( produc2499998789532207959rm_a_b @ A @ B ) @ R2 )
=> ( ( member5793113912742238992rm_a_b @ ( produc2499998789532207959rm_a_b @ A @ C6 ) @ R2 )
=> ( member5793113912742238992rm_a_b @ ( produc2499998789532207959rm_a_b @ B @ C6 ) @ ( abstra4037915826850753057rm_a_b @ R2 ) ) ) ) )
=> ( abstra4159178081913940476rm_a_b @ R2 @ A4 ) ) ).
% WCR_onI
thf(fact_658_WCR__onI,axiom,
! [A4: set_Pr4386577575007340137rm_a_b,R2: set_Pr2972776593051762503rm_a_b] :
( ! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,C6: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ A @ A4 )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ R2 )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ C6 ) @ R2 )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B @ C6 ) @ ( abstra4889682940153293507rm_a_b @ R2 ) ) ) ) )
=> ( abstra875157153669124638rm_a_b @ R2 @ A4 ) ) ).
% WCR_onI
thf(fact_659_WCR__onI,axiom,
! [A4: set_term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ! [A: term_a_b,B: term_a_b,C6: term_a_b] :
( ( member_term_a_b @ A @ A4 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R2 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C6 ) @ R2 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C6 ) @ ( abstra4096080454567261402rm_a_b @ R2 ) ) ) ) )
=> ( abstra5026849107050860991rm_a_b @ R2 @ A4 ) ) ).
% WCR_onI
thf(fact_660_funas__gterm__gterm__of__term,axiom,
! [T: term_a_b] :
( ( term_ground_a_b @ T )
=> ( ( ground_funas_gterm_a @ ( ground3594116404241818627rm_a_b @ T ) )
= ( term_funas_term_a_b @ T ) ) ) ).
% funas_gterm_gterm_of_term
thf(fact_661_irrefl__tranclI,axiom,
! [R2: set_Pr2972776593051762503rm_a_b,X: produc357393685978478089rm_a_b] :
( ( ( inf_in7724013137991509301rm_a_b @ ( conver1575690562132235771rm_a_b @ R2 ) @ ( transi2615809358984392588rm_a_b @ R2 ) )
= bot_bo5040873672931234227rm_a_b )
=> ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ X ) @ ( transi9211502839322181930rm_a_b @ R2 ) ) ) ).
% irrefl_tranclI
thf(fact_662_irrefl__tranclI,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,X: term_a_b] :
( ( ( inf_in1693821330460682071rm_a_b @ ( conver7608559976937845093rm_a_b @ R2 ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
= bot_bo197521221353338581rm_a_b )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ X ) @ ( transi7922773638565587891rm_a_b @ R2 ) ) ) ).
% irrefl_tranclI
thf(fact_663_Int__iff,axiom,
! [C: produc4523971788519308903rm_a_b,A4: set_Pr2972776593051762503rm_a_b,B5: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ C @ ( inf_in7724013137991509301rm_a_b @ A4 @ B5 ) )
= ( ( member8417600551952982416rm_a_b @ C @ A4 )
& ( member8417600551952982416rm_a_b @ C @ B5 ) ) ) ).
% Int_iff
thf(fact_664_Int__iff,axiom,
! [C: term_a_b,A4: set_term_a_b,B5: set_term_a_b] :
( ( member_term_a_b @ C @ ( inf_inf_set_term_a_b @ A4 @ B5 ) )
= ( ( member_term_a_b @ C @ A4 )
& ( member_term_a_b @ C @ B5 ) ) ) ).
% Int_iff
thf(fact_665_Int__iff,axiom,
! [C: produc357393685978478089rm_a_b,A4: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ C @ ( inf_in1693821330460682071rm_a_b @ A4 @ B5 ) )
= ( ( member5869715511025134514rm_a_b @ C @ A4 )
& ( member5869715511025134514rm_a_b @ C @ B5 ) ) ) ).
% Int_iff
thf(fact_666_IntI,axiom,
! [C: produc4523971788519308903rm_a_b,A4: set_Pr2972776593051762503rm_a_b,B5: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ C @ A4 )
=> ( ( member8417600551952982416rm_a_b @ C @ B5 )
=> ( member8417600551952982416rm_a_b @ C @ ( inf_in7724013137991509301rm_a_b @ A4 @ B5 ) ) ) ) ).
% IntI
thf(fact_667_IntI,axiom,
! [C: term_a_b,A4: set_term_a_b,B5: set_term_a_b] :
( ( member_term_a_b @ C @ A4 )
=> ( ( member_term_a_b @ C @ B5 )
=> ( member_term_a_b @ C @ ( inf_inf_set_term_a_b @ A4 @ B5 ) ) ) ) ).
% IntI
thf(fact_668_IntI,axiom,
! [C: produc357393685978478089rm_a_b,A4: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ C @ A4 )
=> ( ( member5869715511025134514rm_a_b @ C @ B5 )
=> ( member5869715511025134514rm_a_b @ C @ ( inf_in1693821330460682071rm_a_b @ A4 @ B5 ) ) ) ) ).
% IntI
thf(fact_669_Int__subset__iff,axiom,
! [C3: set_Pr4386577575007340137rm_a_b,A4: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ C3 @ ( inf_in1693821330460682071rm_a_b @ A4 @ B5 ) )
= ( ( ord_le118470702582115849rm_a_b @ C3 @ A4 )
& ( ord_le118470702582115849rm_a_b @ C3 @ B5 ) ) ) ).
% Int_subset_iff
thf(fact_670_Int__subset__iff,axiom,
! [C3: set_Pr4934435412358123699_a_nat,A4: set_Pr4934435412358123699_a_nat,B5: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ C3 @ ( inf_in3813033988066259269_a_nat @ A4 @ B5 ) )
= ( ( ord_le8666007276011122963_a_nat @ C3 @ A4 )
& ( ord_le8666007276011122963_a_nat @ C3 @ B5 ) ) ) ).
% Int_subset_iff
thf(fact_671_Int__subset__iff,axiom,
! [C3: set_term_a_b,A4: set_term_a_b,B5: set_term_a_b] :
( ( ord_le2705286416250468010rm_a_b @ C3 @ ( inf_inf_set_term_a_b @ A4 @ B5 ) )
= ( ( ord_le2705286416250468010rm_a_b @ C3 @ A4 )
& ( ord_le2705286416250468010rm_a_b @ C3 @ B5 ) ) ) ).
% Int_subset_iff
thf(fact_672_Int__left__commute,axiom,
! [A4: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b,C3: set_Pr4386577575007340137rm_a_b] :
( ( inf_in1693821330460682071rm_a_b @ A4 @ ( inf_in1693821330460682071rm_a_b @ B5 @ C3 ) )
= ( inf_in1693821330460682071rm_a_b @ B5 @ ( inf_in1693821330460682071rm_a_b @ A4 @ C3 ) ) ) ).
% Int_left_commute
thf(fact_673_Int__left__absorb,axiom,
! [A4: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
( ( inf_in1693821330460682071rm_a_b @ A4 @ ( inf_in1693821330460682071rm_a_b @ A4 @ B5 ) )
= ( inf_in1693821330460682071rm_a_b @ A4 @ B5 ) ) ).
% Int_left_absorb
thf(fact_674_Int__commute,axiom,
( inf_in1693821330460682071rm_a_b
= ( ^ [A7: set_Pr4386577575007340137rm_a_b,B6: set_Pr4386577575007340137rm_a_b] : ( inf_in1693821330460682071rm_a_b @ B6 @ A7 ) ) ) ).
% Int_commute
thf(fact_675_Int__absorb,axiom,
! [A4: set_Pr4386577575007340137rm_a_b] :
( ( inf_in1693821330460682071rm_a_b @ A4 @ A4 )
= A4 ) ).
% Int_absorb
thf(fact_676_Int__assoc,axiom,
! [A4: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b,C3: set_Pr4386577575007340137rm_a_b] :
( ( inf_in1693821330460682071rm_a_b @ ( inf_in1693821330460682071rm_a_b @ A4 @ B5 ) @ C3 )
= ( inf_in1693821330460682071rm_a_b @ A4 @ ( inf_in1693821330460682071rm_a_b @ B5 @ C3 ) ) ) ).
% Int_assoc
thf(fact_677_IntD2,axiom,
! [C: produc4523971788519308903rm_a_b,A4: set_Pr2972776593051762503rm_a_b,B5: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ C @ ( inf_in7724013137991509301rm_a_b @ A4 @ B5 ) )
=> ( member8417600551952982416rm_a_b @ C @ B5 ) ) ).
% IntD2
thf(fact_678_IntD2,axiom,
! [C: term_a_b,A4: set_term_a_b,B5: set_term_a_b] :
( ( member_term_a_b @ C @ ( inf_inf_set_term_a_b @ A4 @ B5 ) )
=> ( member_term_a_b @ C @ B5 ) ) ).
% IntD2
thf(fact_679_IntD2,axiom,
! [C: produc357393685978478089rm_a_b,A4: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ C @ ( inf_in1693821330460682071rm_a_b @ A4 @ B5 ) )
=> ( member5869715511025134514rm_a_b @ C @ B5 ) ) ).
% IntD2
thf(fact_680_IntD1,axiom,
! [C: produc4523971788519308903rm_a_b,A4: set_Pr2972776593051762503rm_a_b,B5: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ C @ ( inf_in7724013137991509301rm_a_b @ A4 @ B5 ) )
=> ( member8417600551952982416rm_a_b @ C @ A4 ) ) ).
% IntD1
thf(fact_681_IntD1,axiom,
! [C: term_a_b,A4: set_term_a_b,B5: set_term_a_b] :
( ( member_term_a_b @ C @ ( inf_inf_set_term_a_b @ A4 @ B5 ) )
=> ( member_term_a_b @ C @ A4 ) ) ).
% IntD1
thf(fact_682_IntD1,axiom,
! [C: produc357393685978478089rm_a_b,A4: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ C @ ( inf_in1693821330460682071rm_a_b @ A4 @ B5 ) )
=> ( member5869715511025134514rm_a_b @ C @ A4 ) ) ).
% IntD1
thf(fact_683_IntE,axiom,
! [C: produc4523971788519308903rm_a_b,A4: set_Pr2972776593051762503rm_a_b,B5: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ C @ ( inf_in7724013137991509301rm_a_b @ A4 @ B5 ) )
=> ~ ( ( member8417600551952982416rm_a_b @ C @ A4 )
=> ~ ( member8417600551952982416rm_a_b @ C @ B5 ) ) ) ).
% IntE
thf(fact_684_IntE,axiom,
! [C: term_a_b,A4: set_term_a_b,B5: set_term_a_b] :
( ( member_term_a_b @ C @ ( inf_inf_set_term_a_b @ A4 @ B5 ) )
=> ~ ( ( member_term_a_b @ C @ A4 )
=> ~ ( member_term_a_b @ C @ B5 ) ) ) ).
% IntE
thf(fact_685_IntE,axiom,
! [C: produc357393685978478089rm_a_b,A4: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ C @ ( inf_in1693821330460682071rm_a_b @ A4 @ B5 ) )
=> ~ ( ( member5869715511025134514rm_a_b @ C @ A4 )
=> ~ ( member5869715511025134514rm_a_b @ C @ B5 ) ) ) ).
% IntE
thf(fact_686_Int__emptyI,axiom,
! [A4: set_Pr2972776593051762503rm_a_b,B5: set_Pr2972776593051762503rm_a_b] :
( ! [X3: produc4523971788519308903rm_a_b] :
( ( member8417600551952982416rm_a_b @ X3 @ A4 )
=> ~ ( member8417600551952982416rm_a_b @ X3 @ B5 ) )
=> ( ( inf_in7724013137991509301rm_a_b @ A4 @ B5 )
= bot_bo5040873672931234227rm_a_b ) ) ).
% Int_emptyI
thf(fact_687_Int__emptyI,axiom,
! [A4: set_term_a_b,B5: set_term_a_b] :
( ! [X3: term_a_b] :
( ( member_term_a_b @ X3 @ A4 )
=> ~ ( member_term_a_b @ X3 @ B5 ) )
=> ( ( inf_inf_set_term_a_b @ A4 @ B5 )
= bot_bot_set_term_a_b ) ) ).
% Int_emptyI
thf(fact_688_Int__emptyI,axiom,
! [A4: set_b,B5: set_b] :
( ! [X3: b] :
( ( member_b @ X3 @ A4 )
=> ~ ( member_b @ X3 @ B5 ) )
=> ( ( inf_inf_set_b @ A4 @ B5 )
= bot_bot_set_b ) ) ).
% Int_emptyI
thf(fact_689_Int__emptyI,axiom,
! [A4: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
( ! [X3: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X3 @ A4 )
=> ~ ( member5869715511025134514rm_a_b @ X3 @ B5 ) )
=> ( ( inf_in1693821330460682071rm_a_b @ A4 @ B5 )
= bot_bo197521221353338581rm_a_b ) ) ).
% Int_emptyI
thf(fact_690_disjoint__iff,axiom,
! [A4: set_Pr2972776593051762503rm_a_b,B5: set_Pr2972776593051762503rm_a_b] :
( ( ( inf_in7724013137991509301rm_a_b @ A4 @ B5 )
= bot_bo5040873672931234227rm_a_b )
= ( ! [X2: produc4523971788519308903rm_a_b] :
( ( member8417600551952982416rm_a_b @ X2 @ A4 )
=> ~ ( member8417600551952982416rm_a_b @ X2 @ B5 ) ) ) ) ).
% disjoint_iff
thf(fact_691_disjoint__iff,axiom,
! [A4: set_term_a_b,B5: set_term_a_b] :
( ( ( inf_inf_set_term_a_b @ A4 @ B5 )
= bot_bot_set_term_a_b )
= ( ! [X2: term_a_b] :
( ( member_term_a_b @ X2 @ A4 )
=> ~ ( member_term_a_b @ X2 @ B5 ) ) ) ) ).
% disjoint_iff
thf(fact_692_disjoint__iff,axiom,
! [A4: set_b,B5: set_b] :
( ( ( inf_inf_set_b @ A4 @ B5 )
= bot_bot_set_b )
= ( ! [X2: b] :
( ( member_b @ X2 @ A4 )
=> ~ ( member_b @ X2 @ B5 ) ) ) ) ).
% disjoint_iff
thf(fact_693_disjoint__iff,axiom,
! [A4: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
( ( ( inf_in1693821330460682071rm_a_b @ A4 @ B5 )
= bot_bo197521221353338581rm_a_b )
= ( ! [X2: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X2 @ A4 )
=> ~ ( member5869715511025134514rm_a_b @ X2 @ B5 ) ) ) ) ).
% disjoint_iff
thf(fact_694_Int__empty__left,axiom,
! [B5: set_Pr4386577575007340137rm_a_b] :
( ( inf_in1693821330460682071rm_a_b @ bot_bo197521221353338581rm_a_b @ B5 )
= bot_bo197521221353338581rm_a_b ) ).
% Int_empty_left
thf(fact_695_Int__empty__left,axiom,
! [B5: set_term_a_b] :
( ( inf_inf_set_term_a_b @ bot_bot_set_term_a_b @ B5 )
= bot_bot_set_term_a_b ) ).
% Int_empty_left
thf(fact_696_Int__empty__left,axiom,
! [B5: set_b] :
( ( inf_inf_set_b @ bot_bot_set_b @ B5 )
= bot_bot_set_b ) ).
% Int_empty_left
thf(fact_697_Int__empty__right,axiom,
! [A4: set_Pr4386577575007340137rm_a_b] :
( ( inf_in1693821330460682071rm_a_b @ A4 @ bot_bo197521221353338581rm_a_b )
= bot_bo197521221353338581rm_a_b ) ).
% Int_empty_right
thf(fact_698_Int__empty__right,axiom,
! [A4: set_term_a_b] :
( ( inf_inf_set_term_a_b @ A4 @ bot_bot_set_term_a_b )
= bot_bot_set_term_a_b ) ).
% Int_empty_right
thf(fact_699_Int__empty__right,axiom,
! [A4: set_b] :
( ( inf_inf_set_b @ A4 @ bot_bot_set_b )
= bot_bot_set_b ) ).
% Int_empty_right
thf(fact_700_disjoint__iff__not__equal,axiom,
! [A4: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
( ( ( inf_in1693821330460682071rm_a_b @ A4 @ B5 )
= bot_bo197521221353338581rm_a_b )
= ( ! [X2: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X2 @ A4 )
=> ! [Y5: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ Y5 @ B5 )
=> ( X2 != Y5 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_701_disjoint__iff__not__equal,axiom,
! [A4: set_term_a_b,B5: set_term_a_b] :
( ( ( inf_inf_set_term_a_b @ A4 @ B5 )
= bot_bot_set_term_a_b )
= ( ! [X2: term_a_b] :
( ( member_term_a_b @ X2 @ A4 )
=> ! [Y5: term_a_b] :
( ( member_term_a_b @ Y5 @ B5 )
=> ( X2 != Y5 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_702_disjoint__iff__not__equal,axiom,
! [A4: set_b,B5: set_b] :
( ( ( inf_inf_set_b @ A4 @ B5 )
= bot_bot_set_b )
= ( ! [X2: b] :
( ( member_b @ X2 @ A4 )
=> ! [Y5: b] :
( ( member_b @ Y5 @ B5 )
=> ( X2 != Y5 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_703_Int__Collect__mono,axiom,
! [A4: set_Pr2972776593051762503rm_a_b,B5: set_Pr2972776593051762503rm_a_b,P2: produc4523971788519308903rm_a_b > $o,Q: produc4523971788519308903rm_a_b > $o] :
( ( ord_le2593958535671617255rm_a_b @ A4 @ B5 )
=> ( ! [X3: produc4523971788519308903rm_a_b] :
( ( member8417600551952982416rm_a_b @ X3 @ A4 )
=> ( ( P2 @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_le2593958535671617255rm_a_b @ ( inf_in7724013137991509301rm_a_b @ A4 @ ( collec1344989105561874898rm_a_b @ P2 ) ) @ ( inf_in7724013137991509301rm_a_b @ B5 @ ( collec1344989105561874898rm_a_b @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_704_Int__Collect__mono,axiom,
! [A4: set_Pr4934435412358123699_a_nat,B5: set_Pr4934435412358123699_a_nat,P2: product_prod_a_nat > $o,Q: product_prod_a_nat > $o] :
( ( ord_le8666007276011122963_a_nat @ A4 @ B5 )
=> ( ! [X3: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ X3 @ A4 )
=> ( ( P2 @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_le8666007276011122963_a_nat @ ( inf_in3813033988066259269_a_nat @ A4 @ ( collec4464134535221767506_a_nat @ P2 ) ) @ ( inf_in3813033988066259269_a_nat @ B5 @ ( collec4464134535221767506_a_nat @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_705_Int__Collect__mono,axiom,
! [A4: set_term_a_b,B5: set_term_a_b,P2: term_a_b > $o,Q: term_a_b > $o] :
( ( ord_le2705286416250468010rm_a_b @ A4 @ B5 )
=> ( ! [X3: term_a_b] :
( ( member_term_a_b @ X3 @ A4 )
=> ( ( P2 @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_le2705286416250468010rm_a_b @ ( inf_inf_set_term_a_b @ A4 @ ( collect_term_a_b @ P2 ) ) @ ( inf_inf_set_term_a_b @ B5 @ ( collect_term_a_b @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_706_Int__Collect__mono,axiom,
! [A4: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b,P2: produc357393685978478089rm_a_b > $o,Q: produc357393685978478089rm_a_b > $o] :
( ( ord_le118470702582115849rm_a_b @ A4 @ B5 )
=> ( ! [X3: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X3 @ A4 )
=> ( ( P2 @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_le118470702582115849rm_a_b @ ( inf_in1693821330460682071rm_a_b @ A4 @ ( collec99177395925924084rm_a_b @ P2 ) ) @ ( inf_in1693821330460682071rm_a_b @ B5 @ ( collec99177395925924084rm_a_b @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_707_Int__greatest,axiom,
! [C3: set_Pr4386577575007340137rm_a_b,A4: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ C3 @ A4 )
=> ( ( ord_le118470702582115849rm_a_b @ C3 @ B5 )
=> ( ord_le118470702582115849rm_a_b @ C3 @ ( inf_in1693821330460682071rm_a_b @ A4 @ B5 ) ) ) ) ).
% Int_greatest
thf(fact_708_Int__greatest,axiom,
! [C3: set_Pr4934435412358123699_a_nat,A4: set_Pr4934435412358123699_a_nat,B5: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ C3 @ A4 )
=> ( ( ord_le8666007276011122963_a_nat @ C3 @ B5 )
=> ( ord_le8666007276011122963_a_nat @ C3 @ ( inf_in3813033988066259269_a_nat @ A4 @ B5 ) ) ) ) ).
% Int_greatest
thf(fact_709_Int__greatest,axiom,
! [C3: set_term_a_b,A4: set_term_a_b,B5: set_term_a_b] :
( ( ord_le2705286416250468010rm_a_b @ C3 @ A4 )
=> ( ( ord_le2705286416250468010rm_a_b @ C3 @ B5 )
=> ( ord_le2705286416250468010rm_a_b @ C3 @ ( inf_inf_set_term_a_b @ A4 @ B5 ) ) ) ) ).
% Int_greatest
thf(fact_710_Int__absorb2,axiom,
! [A4: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A4 @ B5 )
=> ( ( inf_in1693821330460682071rm_a_b @ A4 @ B5 )
= A4 ) ) ).
% Int_absorb2
thf(fact_711_Int__absorb2,axiom,
! [A4: set_Pr4934435412358123699_a_nat,B5: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A4 @ B5 )
=> ( ( inf_in3813033988066259269_a_nat @ A4 @ B5 )
= A4 ) ) ).
% Int_absorb2
thf(fact_712_Int__absorb2,axiom,
! [A4: set_term_a_b,B5: set_term_a_b] :
( ( ord_le2705286416250468010rm_a_b @ A4 @ B5 )
=> ( ( inf_inf_set_term_a_b @ A4 @ B5 )
= A4 ) ) ).
% Int_absorb2
thf(fact_713_Int__absorb1,axiom,
! [B5: set_Pr4386577575007340137rm_a_b,A4: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ B5 @ A4 )
=> ( ( inf_in1693821330460682071rm_a_b @ A4 @ B5 )
= B5 ) ) ).
% Int_absorb1
thf(fact_714_Int__absorb1,axiom,
! [B5: set_Pr4934435412358123699_a_nat,A4: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ B5 @ A4 )
=> ( ( inf_in3813033988066259269_a_nat @ A4 @ B5 )
= B5 ) ) ).
% Int_absorb1
thf(fact_715_Int__absorb1,axiom,
! [B5: set_term_a_b,A4: set_term_a_b] :
( ( ord_le2705286416250468010rm_a_b @ B5 @ A4 )
=> ( ( inf_inf_set_term_a_b @ A4 @ B5 )
= B5 ) ) ).
% Int_absorb1
thf(fact_716_Int__lower2,axiom,
! [A4: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ ( inf_in1693821330460682071rm_a_b @ A4 @ B5 ) @ B5 ) ).
% Int_lower2
thf(fact_717_Int__lower2,axiom,
! [A4: set_Pr4934435412358123699_a_nat,B5: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ ( inf_in3813033988066259269_a_nat @ A4 @ B5 ) @ B5 ) ).
% Int_lower2
thf(fact_718_Int__lower2,axiom,
! [A4: set_term_a_b,B5: set_term_a_b] : ( ord_le2705286416250468010rm_a_b @ ( inf_inf_set_term_a_b @ A4 @ B5 ) @ B5 ) ).
% Int_lower2
thf(fact_719_Int__lower1,axiom,
! [A4: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ ( inf_in1693821330460682071rm_a_b @ A4 @ B5 ) @ A4 ) ).
% Int_lower1
thf(fact_720_Int__lower1,axiom,
! [A4: set_Pr4934435412358123699_a_nat,B5: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ ( inf_in3813033988066259269_a_nat @ A4 @ B5 ) @ A4 ) ).
% Int_lower1
thf(fact_721_Int__lower1,axiom,
! [A4: set_term_a_b,B5: set_term_a_b] : ( ord_le2705286416250468010rm_a_b @ ( inf_inf_set_term_a_b @ A4 @ B5 ) @ A4 ) ).
% Int_lower1
thf(fact_722_Int__mono,axiom,
! [A4: set_Pr4386577575007340137rm_a_b,C3: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b,D: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A4 @ C3 )
=> ( ( ord_le118470702582115849rm_a_b @ B5 @ D )
=> ( ord_le118470702582115849rm_a_b @ ( inf_in1693821330460682071rm_a_b @ A4 @ B5 ) @ ( inf_in1693821330460682071rm_a_b @ C3 @ D ) ) ) ) ).
% Int_mono
thf(fact_723_Int__mono,axiom,
! [A4: set_Pr4934435412358123699_a_nat,C3: set_Pr4934435412358123699_a_nat,B5: set_Pr4934435412358123699_a_nat,D: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A4 @ C3 )
=> ( ( ord_le8666007276011122963_a_nat @ B5 @ D )
=> ( ord_le8666007276011122963_a_nat @ ( inf_in3813033988066259269_a_nat @ A4 @ B5 ) @ ( inf_in3813033988066259269_a_nat @ C3 @ D ) ) ) ) ).
% Int_mono
thf(fact_724_Int__mono,axiom,
! [A4: set_term_a_b,C3: set_term_a_b,B5: set_term_a_b,D: set_term_a_b] :
( ( ord_le2705286416250468010rm_a_b @ A4 @ C3 )
=> ( ( ord_le2705286416250468010rm_a_b @ B5 @ D )
=> ( ord_le2705286416250468010rm_a_b @ ( inf_inf_set_term_a_b @ A4 @ B5 ) @ ( inf_inf_set_term_a_b @ C3 @ D ) ) ) ) ).
% Int_mono
thf(fact_725_converse__Int,axiom,
! [R2: set_Pr2972776593051762503rm_a_b,S: set_Pr2972776593051762503rm_a_b] :
( ( conver1575690562132235771rm_a_b @ ( inf_in7724013137991509301rm_a_b @ R2 @ S ) )
= ( inf_in7724013137991509301rm_a_b @ ( conver1575690562132235771rm_a_b @ R2 ) @ ( conver1575690562132235771rm_a_b @ S ) ) ) ).
% converse_Int
thf(fact_726_converse__Int,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
( ( conver7608559976937845093rm_a_b @ ( inf_in1693821330460682071rm_a_b @ R2 @ S ) )
= ( inf_in1693821330460682071rm_a_b @ ( conver7608559976937845093rm_a_b @ R2 ) @ ( conver7608559976937845093rm_a_b @ S ) ) ) ).
% converse_Int
thf(fact_727_trancl__induct2,axiom,
! [Ax: produc357393685978478089rm_a_b,Ay: produc357393685978478089rm_a_b,Bx: produc357393685978478089rm_a_b,By: produc357393685978478089rm_a_b,R2: 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 @ R2 ) )
=> ( ! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b] :
( ( member5793113912742238992rm_a_b @ ( produc2499998789532207959rm_a_b @ ( produc1763473618796451543rm_a_b @ Ax @ Ay ) @ ( produc1763473618796451543rm_a_b @ A @ B ) ) @ R2 )
=> ( P2 @ A @ B ) )
=> ( ! [A: produc357393685978478089rm_a_b,B: 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 @ A @ B ) ) @ ( transi4560671716973800200rm_a_b @ R2 ) )
=> ( ( member5793113912742238992rm_a_b @ ( produc2499998789532207959rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( produc1763473618796451543rm_a_b @ Aa @ Ba ) ) @ R2 )
=> ( ( P2 @ A @ B )
=> ( P2 @ Aa @ Ba ) ) ) )
=> ( P2 @ Bx @ By ) ) ) ) ).
% trancl_induct2
thf(fact_728_trancl__induct2,axiom,
! [Ax: term_a_b,Ay: term_a_b,Bx: term_a_b,By: term_a_b,R2: 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 @ R2 ) )
=> ( ! [A: term_a_b,B: term_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ Ax @ Ay ) @ ( produc7020197800436672577rm_a_b @ A @ B ) ) @ R2 )
=> ( P2 @ A @ B ) )
=> ( ! [A: term_a_b,B: 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 @ A @ B ) ) @ ( transi9211502839322181930rm_a_b @ R2 ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( produc7020197800436672577rm_a_b @ Aa @ Ba ) ) @ R2 )
=> ( ( P2 @ A @ B )
=> ( P2 @ Aa @ Ba ) ) ) )
=> ( P2 @ Bx @ By ) ) ) ) ).
% trancl_induct2
thf(fact_729_funas__rel__converse,axiom,
! [R: set_Pr4386577575007340137rm_a_b,F2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ ( terms_7988297476397195622_a_b_b @ R ) @ F2 )
=> ( ord_le8666007276011122963_a_nat @ ( terms_7988297476397195622_a_b_b @ ( conver7608559976937845093rm_a_b @ R ) ) @ F2 ) ) ).
% funas_rel_converse
thf(fact_730_trancl__Int__subset,axiom,
! [R2: set_Pr2972776593051762503rm_a_b,S: set_Pr2972776593051762503rm_a_b] :
( ( ord_le2593958535671617255rm_a_b @ R2 @ S )
=> ( ( ord_le2593958535671617255rm_a_b @ ( relcom6159724393391289352rm_a_b @ ( inf_in7724013137991509301rm_a_b @ ( transi9211502839322181930rm_a_b @ R2 ) @ S ) @ R2 ) @ S )
=> ( ord_le2593958535671617255rm_a_b @ ( transi9211502839322181930rm_a_b @ R2 ) @ S ) ) ) ).
% trancl_Int_subset
thf(fact_731_trancl__Int__subset,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ R2 @ S )
=> ( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ ( inf_in1693821330460682071rm_a_b @ ( transi7922773638565587891rm_a_b @ R2 ) @ S ) @ R2 ) @ S )
=> ( ord_le118470702582115849rm_a_b @ ( transi7922773638565587891rm_a_b @ R2 ) @ S ) ) ) ).
% trancl_Int_subset
thf(fact_732_funas__term__of__gterm__conv,axiom,
! [T: ground_gterm_a] :
( ( term_funas_term_a_b @ ( ground5786051259813020157rm_a_b @ T ) )
= ( ground_funas_gterm_a @ T ) ) ).
% funas_term_of_gterm_conv
thf(fact_733_WCR__on__def,axiom,
( abstra875157153669124638rm_a_b
= ( ^ [R4: set_Pr2972776593051762503rm_a_b,A7: set_Pr4386577575007340137rm_a_b] :
! [X2: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X2 @ A7 )
=> ! [B3: produc357393685978478089rm_a_b,C2: produc357393685978478089rm_a_b] :
( ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X2 @ B3 ) @ R4 )
& ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X2 @ C2 ) @ R4 ) )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B3 @ C2 ) @ ( abstra4889682940153293507rm_a_b @ R4 ) ) ) ) ) ) ).
% WCR_on_def
thf(fact_734_WCR__on__def,axiom,
( abstra5026849107050860991rm_a_b
= ( ^ [R4: set_Pr4386577575007340137rm_a_b,A7: set_term_a_b] :
! [X2: term_a_b] :
( ( member_term_a_b @ X2 @ A7 )
=> ! [B3: term_a_b,C2: term_a_b] :
( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ B3 ) @ R4 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ C2 ) @ R4 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B3 @ C2 ) @ ( abstra4096080454567261402rm_a_b @ R4 ) ) ) ) ) ) ).
% WCR_on_def
thf(fact_735_WCR__onE,axiom,
! [R2: set_Pr6848273517546626247rm_a_b,A4: set_Pr2972776593051762503rm_a_b,A3: produc4523971788519308903rm_a_b,B2: produc4523971788519308903rm_a_b,C: produc4523971788519308903rm_a_b] :
( ( abstra4159178081913940476rm_a_b @ R2 @ A4 )
=> ( ( member8417600551952982416rm_a_b @ A3 @ A4 )
=> ( ~ ( member5793113912742238992rm_a_b @ ( produc2499998789532207959rm_a_b @ B2 @ C ) @ ( abstra4037915826850753057rm_a_b @ R2 ) )
=> ( ( member5793113912742238992rm_a_b @ ( produc2499998789532207959rm_a_b @ A3 @ B2 ) @ R2 )
=> ~ ( member5793113912742238992rm_a_b @ ( produc2499998789532207959rm_a_b @ A3 @ C ) @ R2 ) ) ) ) ) ).
% WCR_onE
thf(fact_736_WCR__onE,axiom,
! [R2: set_Pr2972776593051762503rm_a_b,A4: set_Pr4386577575007340137rm_a_b,A3: produc357393685978478089rm_a_b,B2: produc357393685978478089rm_a_b,C: produc357393685978478089rm_a_b] :
( ( abstra875157153669124638rm_a_b @ R2 @ A4 )
=> ( ( member5869715511025134514rm_a_b @ A3 @ A4 )
=> ( ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B2 @ C ) @ ( abstra4889682940153293507rm_a_b @ R2 ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A3 @ B2 ) @ R2 )
=> ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A3 @ C ) @ R2 ) ) ) ) ) ).
% WCR_onE
thf(fact_737_WCR__onE,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,A4: set_term_a_b,A3: term_a_b,B2: term_a_b,C: term_a_b] :
( ( abstra5026849107050860991rm_a_b @ R2 @ A4 )
=> ( ( member_term_a_b @ A3 @ A4 )
=> ( ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ C ) @ ( abstra4096080454567261402rm_a_b @ R2 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B2 ) @ R2 )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ C ) @ R2 ) ) ) ) ) ).
% WCR_onE
thf(fact_738_WCR__onD,axiom,
! [R2: set_Pr6848273517546626247rm_a_b,A4: set_Pr2972776593051762503rm_a_b,A3: produc4523971788519308903rm_a_b,B2: produc4523971788519308903rm_a_b,C: produc4523971788519308903rm_a_b] :
( ( abstra4159178081913940476rm_a_b @ R2 @ A4 )
=> ( ( member8417600551952982416rm_a_b @ A3 @ A4 )
=> ( ( member5793113912742238992rm_a_b @ ( produc2499998789532207959rm_a_b @ A3 @ B2 ) @ R2 )
=> ( ( member5793113912742238992rm_a_b @ ( produc2499998789532207959rm_a_b @ A3 @ C ) @ R2 )
=> ( member5793113912742238992rm_a_b @ ( produc2499998789532207959rm_a_b @ B2 @ C ) @ ( abstra4037915826850753057rm_a_b @ R2 ) ) ) ) ) ) ).
% WCR_onD
thf(fact_739_WCR__onD,axiom,
! [R2: set_Pr2972776593051762503rm_a_b,A4: set_Pr4386577575007340137rm_a_b,A3: produc357393685978478089rm_a_b,B2: produc357393685978478089rm_a_b,C: produc357393685978478089rm_a_b] :
( ( abstra875157153669124638rm_a_b @ R2 @ A4 )
=> ( ( member5869715511025134514rm_a_b @ A3 @ A4 )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A3 @ B2 ) @ R2 )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A3 @ C ) @ R2 )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B2 @ C ) @ ( abstra4889682940153293507rm_a_b @ R2 ) ) ) ) ) ) ).
% WCR_onD
thf(fact_740_WCR__onD,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,A4: set_term_a_b,A3: term_a_b,B2: term_a_b,C: term_a_b] :
( ( abstra5026849107050860991rm_a_b @ R2 @ A4 )
=> ( ( member_term_a_b @ A3 @ A4 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B2 ) @ R2 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ C ) @ R2 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ C ) @ ( abstra4096080454567261402rm_a_b @ R2 ) ) ) ) ) ) ).
% WCR_onD
thf(fact_741_sig__step__rsteps__dist,axiom,
! [R: set_Pr4386577575007340137rm_a_b,F2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ ( terms_7988297476397195622_a_b_b @ R ) @ F2 )
=> ( ( sig_step_a_b @ F2 @ ( transi7922773638565587891rm_a_b @ ( rstep_a_b @ R ) ) )
= ( transi7922773638565587891rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R ) ) ) ) ) ).
% sig_step_rsteps_dist
thf(fact_742_boolean__algebra_Oconj__zero__right,axiom,
! [X: set_Pr4386577575007340137rm_a_b] :
( ( inf_in1693821330460682071rm_a_b @ X @ bot_bo197521221353338581rm_a_b )
= bot_bo197521221353338581rm_a_b ) ).
% boolean_algebra.conj_zero_right
thf(fact_743_boolean__algebra_Oconj__zero__right,axiom,
! [X: produc357393685978478089rm_a_b > $o] :
( ( inf_in7523581892642934342_a_b_o @ X @ bot_bo23760184229683272_a_b_o )
= bot_bo23760184229683272_a_b_o ) ).
% boolean_algebra.conj_zero_right
thf(fact_744_boolean__algebra_Oconj__zero__right,axiom,
! [X: set_term_a_b] :
( ( inf_inf_set_term_a_b @ X @ bot_bot_set_term_a_b )
= bot_bot_set_term_a_b ) ).
% boolean_algebra.conj_zero_right
thf(fact_745_boolean__algebra_Oconj__zero__right,axiom,
! [X: set_b] :
( ( inf_inf_set_b @ X @ bot_bot_set_b )
= bot_bot_set_b ) ).
% boolean_algebra.conj_zero_right
thf(fact_746_boolean__algebra_Oconj__zero__left,axiom,
! [X: set_Pr4386577575007340137rm_a_b] :
( ( inf_in1693821330460682071rm_a_b @ bot_bo197521221353338581rm_a_b @ X )
= bot_bo197521221353338581rm_a_b ) ).
% boolean_algebra.conj_zero_left
thf(fact_747_boolean__algebra_Oconj__zero__left,axiom,
! [X: produc357393685978478089rm_a_b > $o] :
( ( inf_in7523581892642934342_a_b_o @ bot_bo23760184229683272_a_b_o @ X )
= bot_bo23760184229683272_a_b_o ) ).
% boolean_algebra.conj_zero_left
thf(fact_748_boolean__algebra_Oconj__zero__left,axiom,
! [X: set_term_a_b] :
( ( inf_inf_set_term_a_b @ bot_bot_set_term_a_b @ X )
= bot_bot_set_term_a_b ) ).
% boolean_algebra.conj_zero_left
thf(fact_749_boolean__algebra_Oconj__zero__left,axiom,
! [X: set_b] :
( ( inf_inf_set_b @ bot_bot_set_b @ X )
= bot_bot_set_b ) ).
% boolean_algebra.conj_zero_left
thf(fact_750_inf__bot__right,axiom,
! [X: set_Pr4386577575007340137rm_a_b] :
( ( inf_in1693821330460682071rm_a_b @ X @ bot_bo197521221353338581rm_a_b )
= bot_bo197521221353338581rm_a_b ) ).
% inf_bot_right
thf(fact_751_inf__bot__right,axiom,
! [X: produc357393685978478089rm_a_b > $o] :
( ( inf_in7523581892642934342_a_b_o @ X @ bot_bo23760184229683272_a_b_o )
= bot_bo23760184229683272_a_b_o ) ).
% inf_bot_right
thf(fact_752_inf__bot__right,axiom,
! [X: set_term_a_b] :
( ( inf_inf_set_term_a_b @ X @ bot_bot_set_term_a_b )
= bot_bot_set_term_a_b ) ).
% inf_bot_right
thf(fact_753_inf__bot__right,axiom,
! [X: set_b] :
( ( inf_inf_set_b @ X @ bot_bot_set_b )
= bot_bot_set_b ) ).
% inf_bot_right
thf(fact_754_inf__bot__left,axiom,
! [X: set_Pr4386577575007340137rm_a_b] :
( ( inf_in1693821330460682071rm_a_b @ bot_bo197521221353338581rm_a_b @ X )
= bot_bo197521221353338581rm_a_b ) ).
% inf_bot_left
thf(fact_755_inf__bot__left,axiom,
! [X: produc357393685978478089rm_a_b > $o] :
( ( inf_in7523581892642934342_a_b_o @ bot_bo23760184229683272_a_b_o @ X )
= bot_bo23760184229683272_a_b_o ) ).
% inf_bot_left
thf(fact_756_inf__bot__left,axiom,
! [X: set_term_a_b] :
( ( inf_inf_set_term_a_b @ bot_bot_set_term_a_b @ X )
= bot_bot_set_term_a_b ) ).
% inf_bot_left
thf(fact_757_inf__bot__left,axiom,
! [X: set_b] :
( ( inf_inf_set_b @ bot_bot_set_b @ X )
= bot_bot_set_b ) ).
% inf_bot_left
thf(fact_758_rstep__trancl__sig__step__r,axiom,
! [S: term_a_b,T: term_a_b,R: set_Pr4386577575007340137rm_a_b,F2: set_Pr4934435412358123699_a_nat,V3: b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( transi7922773638565587891rm_a_b @ ( rstep_a_b @ R ) ) )
=> ( ( ord_le8666007276011122963_a_nat @ ( terms_7988297476397195622_a_b_b @ R ) @ F2 )
=> ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ S ) @ F2 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ ( terms_8519481630511763164ig_a_b @ F2 @ V3 @ T ) ) @ ( transi7922773638565587891rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R ) ) ) ) ) ) ) ).
% rstep_trancl_sig_step_r
thf(fact_759_rstep__trancl__sig__step__l,axiom,
! [S: term_a_b,T: term_a_b,R: set_Pr4386577575007340137rm_a_b,F2: set_Pr4934435412358123699_a_nat,V3: b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( transi7922773638565587891rm_a_b @ ( rstep_a_b @ R ) ) )
=> ( ( ord_le8666007276011122963_a_nat @ ( terms_7988297476397195622_a_b_b @ R ) @ F2 )
=> ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ T ) @ F2 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( terms_8519481630511763164ig_a_b @ F2 @ V3 @ S ) @ T ) @ ( transi7922773638565587891rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R ) ) ) ) ) ) ) ).
% rstep_trancl_sig_step_l
thf(fact_760_term__to__sig__id,axiom,
! [T: term_a_b,F2: set_Pr4934435412358123699_a_nat,V3: b] :
( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ T ) @ F2 )
=> ( ( terms_8519481630511763164ig_a_b @ F2 @ V3 @ T )
= T ) ) ).
% term_to_sig_id
thf(fact_761_term__to__sig_Osimps_I1_J,axiom,
! [F2: set_Pr4934435412358123699_a_nat,V3: b,X: b] :
( ( terms_8519481630511763164ig_a_b @ F2 @ V3 @ ( var_b_a @ X ) )
= ( var_b_a @ X ) ) ).
% term_to_sig.simps(1)
thf(fact_762_fuans__term__term__to__sig,axiom,
! [F2: set_Pr4934435412358123699_a_nat,V3: b,T: term_a_b] : ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ ( terms_8519481630511763164ig_a_b @ F2 @ V3 @ T ) ) @ F2 ) ).
% fuans_term_term_to_sig
thf(fact_763_rstep__term__to__sig__l,axiom,
! [S: term_a_b,T: term_a_b,R: set_Pr4386577575007340137rm_a_b,F2: set_Pr4934435412358123699_a_nat,V3: b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( rstep_a_b @ R ) )
=> ( ( ord_le8666007276011122963_a_nat @ ( terms_7988297476397195622_a_b_b @ R ) @ F2 )
=> ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ T ) @ F2 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( terms_8519481630511763164ig_a_b @ F2 @ V3 @ S ) @ T ) @ ( rstep_a_b @ R ) ) ) ) ) ).
% rstep_term_to_sig_l
thf(fact_764_rstep__term__to__sig__r,axiom,
! [S: term_a_b,T: term_a_b,R: set_Pr4386577575007340137rm_a_b,F2: set_Pr4934435412358123699_a_nat,V3: b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( rstep_a_b @ R ) )
=> ( ( ord_le8666007276011122963_a_nat @ ( terms_7988297476397195622_a_b_b @ R ) @ F2 )
=> ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ S ) @ F2 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ ( terms_8519481630511763164ig_a_b @ F2 @ V3 @ T ) ) @ ( rstep_a_b @ R ) ) ) ) ) ).
% rstep_term_to_sig_r
thf(fact_765_term__to__sig__ctxt__apply,axiom,
! [F2: set_Pr4934435412358123699_a_nat,C3: subterm_and_ctxt_a_b,V3: b,S: term_a_b] :
( ( terms_8374513854926927137th_a_b @ F2 @ C3 )
=> ( ( terms_8519481630511763164ig_a_b @ F2 @ V3 @ ( subter2376574525758040790rm_a_b @ C3 @ S ) )
= ( subter2376574525758040790rm_a_b @ ( terms_3799363064701517935xt_a_b @ F2 @ V3 @ C3 ) @ ( terms_8519481630511763164ig_a_b @ F2 @ V3 @ S ) ) ) ) ).
% term_to_sig_ctxt_apply
thf(fact_766_WCR__alt__def,axiom,
! [A4: set_Pr2972776593051762503rm_a_b] :
( ( abstra875157153669124638rm_a_b @ A4 @ top_to1314267278846557113rm_a_b )
= ( ord_le2593958535671617255rm_a_b @ ( relcom6159724393391289352rm_a_b @ ( conver1575690562132235771rm_a_b @ A4 ) @ A4 ) @ ( abstra4889682940153293507rm_a_b @ A4 ) ) ) ).
% WCR_alt_def
thf(fact_767_WCR__alt__def,axiom,
! [A4: set_Pr4386577575007340137rm_a_b] :
( ( abstra5026849107050860991rm_a_b @ A4 @ top_top_set_term_a_b )
= ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ ( conver7608559976937845093rm_a_b @ A4 ) @ A4 ) @ ( abstra4096080454567261402rm_a_b @ A4 ) ) ) ).
% WCR_alt_def
thf(fact_768_conversionI,axiom,
! [A3: produc357393685978478089rm_a_b,B2: produc357393685978478089rm_a_b,A4: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A3 @ B2 ) @ ( transi2615809358984392588rm_a_b @ ( sup_su1845815533690797339rm_a_b @ A4 @ ( conver1575690562132235771rm_a_b @ A4 ) ) ) )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A3 @ B2 ) @ ( abstra6125656530262095991rm_a_b @ A4 ) ) ) ).
% conversionI
thf(fact_769_conversionI,axiom,
! [A3: term_a_b,B2: term_a_b,A4: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B2 ) @ ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ A4 @ ( conver7608559976937845093rm_a_b @ A4 ) ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B2 ) @ ( abstra1130568886744591910rm_a_b @ A4 ) ) ) ).
% conversionI
thf(fact_770_restrict__SN__trancl__simp,axiom,
! [A4: set_Pr4386577575007340137rm_a_b] :
( ( transi7922773638565587891rm_a_b @ ( abstra4933154510623179783rm_a_b @ A4 @ A4 ) )
= ( abstra4933154510623179783rm_a_b @ ( transi7922773638565587891rm_a_b @ A4 ) @ A4 ) ) ).
% restrict_SN_trancl_simp
thf(fact_771_restrict__SN__trancl__simp,axiom,
! [A4: set_Pr2972776593051762503rm_a_b] :
( ( transi9211502839322181930rm_a_b @ ( abstra5272529552662653398rm_a_b @ A4 @ A4 ) )
= ( abstra5272529552662653398rm_a_b @ ( transi9211502839322181930rm_a_b @ A4 ) @ A4 ) ) ).
% restrict_SN_trancl_simp
thf(fact_772_comp__rtrancl__trancl,axiom,
! [R5: set_Pr2972776593051762503rm_a_b,S5: set_Pr2972776593051762503rm_a_b,S: produc357393685978478089rm_a_b,T: produc357393685978478089rm_a_b] :
( ( ord_le2593958535671617255rm_a_b @ ( relcom6159724393391289352rm_a_b @ R5 @ S5 ) @ S5 )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ S @ T ) @ ( relcom6159724393391289352rm_a_b @ ( transi2615809358984392588rm_a_b @ ( sup_su1845815533690797339rm_a_b @ R5 @ S5 ) ) @ S5 ) )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ S @ T ) @ ( transi9211502839322181930rm_a_b @ S5 ) ) ) ) ).
% comp_rtrancl_trancl
thf(fact_773_comp__rtrancl__trancl,axiom,
! [R5: set_Pr4386577575007340137rm_a_b,S5: set_Pr4386577575007340137rm_a_b,S: term_a_b,T: term_a_b] :
( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ R5 @ S5 ) @ S5 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ R5 @ S5 ) ) @ S5 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( transi7922773638565587891rm_a_b @ S5 ) ) ) ) ).
% comp_rtrancl_trancl
thf(fact_774_top__apply,axiom,
( top_to1262317115727556708_a_b_o
= ( ^ [X2: produc357393685978478089rm_a_b] : top_top_o ) ) ).
% top_apply
thf(fact_775_UNIV__I,axiom,
! [X: produc4523971788519308903rm_a_b] : ( member8417600551952982416rm_a_b @ X @ top_to1302229308806327447rm_a_b ) ).
% UNIV_I
thf(fact_776_UNIV__I,axiom,
! [X: term_a_b] : ( member_term_a_b @ X @ top_top_set_term_a_b ) ).
% UNIV_I
thf(fact_777_UNIV__I,axiom,
! [X: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ X @ top_to1314267278846557113rm_a_b ) ).
% UNIV_I
thf(fact_778_Un__iff,axiom,
! [C: produc4523971788519308903rm_a_b,A4: set_Pr2972776593051762503rm_a_b,B5: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ C @ ( sup_su1845815533690797339rm_a_b @ A4 @ B5 ) )
= ( ( member8417600551952982416rm_a_b @ C @ A4 )
| ( member8417600551952982416rm_a_b @ C @ B5 ) ) ) ).
% Un_iff
thf(fact_779_Un__iff,axiom,
! [C: term_a_b,A4: set_term_a_b,B5: set_term_a_b] :
( ( member_term_a_b @ C @ ( sup_sup_set_term_a_b @ A4 @ B5 ) )
= ( ( member_term_a_b @ C @ A4 )
| ( member_term_a_b @ C @ B5 ) ) ) ).
% Un_iff
thf(fact_780_Un__iff,axiom,
! [C: produc357393685978478089rm_a_b,A4: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ C @ ( sup_su6776935440552674877rm_a_b @ A4 @ B5 ) )
= ( ( member5869715511025134514rm_a_b @ C @ A4 )
| ( member5869715511025134514rm_a_b @ C @ B5 ) ) ) ).
% Un_iff
thf(fact_781_UnCI,axiom,
! [C: produc4523971788519308903rm_a_b,B5: set_Pr2972776593051762503rm_a_b,A4: set_Pr2972776593051762503rm_a_b] :
( ( ~ ( member8417600551952982416rm_a_b @ C @ B5 )
=> ( member8417600551952982416rm_a_b @ C @ A4 ) )
=> ( member8417600551952982416rm_a_b @ C @ ( sup_su1845815533690797339rm_a_b @ A4 @ B5 ) ) ) ).
% UnCI
thf(fact_782_UnCI,axiom,
! [C: term_a_b,B5: set_term_a_b,A4: set_term_a_b] :
( ( ~ ( member_term_a_b @ C @ B5 )
=> ( member_term_a_b @ C @ A4 ) )
=> ( member_term_a_b @ C @ ( sup_sup_set_term_a_b @ A4 @ B5 ) ) ) ).
% UnCI
thf(fact_783_UnCI,axiom,
! [C: produc357393685978478089rm_a_b,B5: set_Pr4386577575007340137rm_a_b,A4: set_Pr4386577575007340137rm_a_b] :
( ( ~ ( member5869715511025134514rm_a_b @ C @ B5 )
=> ( member5869715511025134514rm_a_b @ C @ A4 ) )
=> ( member5869715511025134514rm_a_b @ C @ ( sup_su6776935440552674877rm_a_b @ A4 @ B5 ) ) ) ).
% UnCI
thf(fact_784_converse__UNIV,axiom,
( ( conver1575690562132235771rm_a_b @ top_to1302229308806327447rm_a_b )
= top_to1302229308806327447rm_a_b ) ).
% converse_UNIV
thf(fact_785_converse__UNIV,axiom,
( ( conver7608559976937845093rm_a_b @ top_to1314267278846557113rm_a_b )
= top_to1314267278846557113rm_a_b ) ).
% converse_UNIV
thf(fact_786_sup__bot__left,axiom,
! [X: set_Pr2972776593051762503rm_a_b] :
( ( sup_su1845815533690797339rm_a_b @ bot_bo5040873672931234227rm_a_b @ X )
= X ) ).
% sup_bot_left
thf(fact_787_sup__bot__left,axiom,
! [X: set_Pr4386577575007340137rm_a_b] :
( ( sup_su6776935440552674877rm_a_b @ bot_bo197521221353338581rm_a_b @ X )
= X ) ).
% sup_bot_left
thf(fact_788_sup__bot__left,axiom,
! [X: produc357393685978478089rm_a_b > $o] :
( ( sup_su2803897115920335584_a_b_o @ bot_bo23760184229683272_a_b_o @ X )
= X ) ).
% sup_bot_left
thf(fact_789_sup__bot__left,axiom,
! [X: set_term_a_b] :
( ( sup_sup_set_term_a_b @ bot_bot_set_term_a_b @ X )
= X ) ).
% sup_bot_left
thf(fact_790_sup__bot__left,axiom,
! [X: set_b] :
( ( sup_sup_set_b @ bot_bot_set_b @ X )
= X ) ).
% sup_bot_left
thf(fact_791_sup__bot__right,axiom,
! [X: set_Pr2972776593051762503rm_a_b] :
( ( sup_su1845815533690797339rm_a_b @ X @ bot_bo5040873672931234227rm_a_b )
= X ) ).
% sup_bot_right
thf(fact_792_sup__bot__right,axiom,
! [X: set_Pr4386577575007340137rm_a_b] :
( ( sup_su6776935440552674877rm_a_b @ X @ bot_bo197521221353338581rm_a_b )
= X ) ).
% sup_bot_right
thf(fact_793_sup__bot__right,axiom,
! [X: produc357393685978478089rm_a_b > $o] :
( ( sup_su2803897115920335584_a_b_o @ X @ bot_bo23760184229683272_a_b_o )
= X ) ).
% sup_bot_right
thf(fact_794_sup__bot__right,axiom,
! [X: set_term_a_b] :
( ( sup_sup_set_term_a_b @ X @ bot_bot_set_term_a_b )
= X ) ).
% sup_bot_right
thf(fact_795_sup__bot__right,axiom,
! [X: set_b] :
( ( sup_sup_set_b @ X @ bot_bot_set_b )
= X ) ).
% sup_bot_right
thf(fact_796_bot__eq__sup__iff,axiom,
! [X: set_Pr2972776593051762503rm_a_b,Y: set_Pr2972776593051762503rm_a_b] :
( ( bot_bo5040873672931234227rm_a_b
= ( sup_su1845815533690797339rm_a_b @ X @ Y ) )
= ( ( X = bot_bo5040873672931234227rm_a_b )
& ( Y = bot_bo5040873672931234227rm_a_b ) ) ) ).
% bot_eq_sup_iff
thf(fact_797_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_798_bot__eq__sup__iff,axiom,
! [X: produc357393685978478089rm_a_b > $o,Y: produc357393685978478089rm_a_b > $o] :
( ( bot_bo23760184229683272_a_b_o
= ( sup_su2803897115920335584_a_b_o @ X @ Y ) )
= ( ( X = bot_bo23760184229683272_a_b_o )
& ( Y = bot_bo23760184229683272_a_b_o ) ) ) ).
% bot_eq_sup_iff
thf(fact_799_bot__eq__sup__iff,axiom,
! [X: set_term_a_b,Y: set_term_a_b] :
( ( bot_bot_set_term_a_b
= ( sup_sup_set_term_a_b @ X @ Y ) )
= ( ( X = bot_bot_set_term_a_b )
& ( Y = bot_bot_set_term_a_b ) ) ) ).
% bot_eq_sup_iff
thf(fact_800_bot__eq__sup__iff,axiom,
! [X: set_b,Y: set_b] :
( ( bot_bot_set_b
= ( sup_sup_set_b @ X @ Y ) )
= ( ( X = bot_bot_set_b )
& ( Y = bot_bot_set_b ) ) ) ).
% bot_eq_sup_iff
thf(fact_801_sup__eq__bot__iff,axiom,
! [X: set_Pr2972776593051762503rm_a_b,Y: set_Pr2972776593051762503rm_a_b] :
( ( ( sup_su1845815533690797339rm_a_b @ X @ Y )
= bot_bo5040873672931234227rm_a_b )
= ( ( X = bot_bo5040873672931234227rm_a_b )
& ( Y = bot_bo5040873672931234227rm_a_b ) ) ) ).
% sup_eq_bot_iff
thf(fact_802_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_803_sup__eq__bot__iff,axiom,
! [X: produc357393685978478089rm_a_b > $o,Y: produc357393685978478089rm_a_b > $o] :
( ( ( sup_su2803897115920335584_a_b_o @ X @ Y )
= bot_bo23760184229683272_a_b_o )
= ( ( X = bot_bo23760184229683272_a_b_o )
& ( Y = bot_bo23760184229683272_a_b_o ) ) ) ).
% sup_eq_bot_iff
thf(fact_804_sup__eq__bot__iff,axiom,
! [X: set_term_a_b,Y: set_term_a_b] :
( ( ( sup_sup_set_term_a_b @ X @ Y )
= bot_bot_set_term_a_b )
= ( ( X = bot_bot_set_term_a_b )
& ( Y = bot_bot_set_term_a_b ) ) ) ).
% sup_eq_bot_iff
thf(fact_805_sup__eq__bot__iff,axiom,
! [X: set_b,Y: set_b] :
( ( ( sup_sup_set_b @ X @ Y )
= bot_bot_set_b )
= ( ( X = bot_bot_set_b )
& ( Y = bot_bot_set_b ) ) ) ).
% sup_eq_bot_iff
thf(fact_806_sup__bot_Oeq__neutr__iff,axiom,
! [A3: set_Pr2972776593051762503rm_a_b,B2: set_Pr2972776593051762503rm_a_b] :
( ( ( sup_su1845815533690797339rm_a_b @ A3 @ B2 )
= bot_bo5040873672931234227rm_a_b )
= ( ( A3 = bot_bo5040873672931234227rm_a_b )
& ( B2 = bot_bo5040873672931234227rm_a_b ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_807_sup__bot_Oeq__neutr__iff,axiom,
! [A3: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b] :
( ( ( sup_su6776935440552674877rm_a_b @ A3 @ B2 )
= bot_bo197521221353338581rm_a_b )
= ( ( A3 = bot_bo197521221353338581rm_a_b )
& ( B2 = bot_bo197521221353338581rm_a_b ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_808_sup__bot_Oeq__neutr__iff,axiom,
! [A3: produc357393685978478089rm_a_b > $o,B2: produc357393685978478089rm_a_b > $o] :
( ( ( sup_su2803897115920335584_a_b_o @ A3 @ B2 )
= bot_bo23760184229683272_a_b_o )
= ( ( A3 = bot_bo23760184229683272_a_b_o )
& ( B2 = bot_bo23760184229683272_a_b_o ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_809_sup__bot_Oeq__neutr__iff,axiom,
! [A3: set_term_a_b,B2: set_term_a_b] :
( ( ( sup_sup_set_term_a_b @ A3 @ B2 )
= bot_bot_set_term_a_b )
= ( ( A3 = bot_bot_set_term_a_b )
& ( B2 = bot_bot_set_term_a_b ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_810_sup__bot_Oeq__neutr__iff,axiom,
! [A3: set_b,B2: set_b] :
( ( ( sup_sup_set_b @ A3 @ B2 )
= bot_bot_set_b )
= ( ( A3 = bot_bot_set_b )
& ( B2 = bot_bot_set_b ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_811_sup__bot_Oleft__neutral,axiom,
! [A3: set_Pr2972776593051762503rm_a_b] :
( ( sup_su1845815533690797339rm_a_b @ bot_bo5040873672931234227rm_a_b @ A3 )
= A3 ) ).
% sup_bot.left_neutral
thf(fact_812_sup__bot_Oleft__neutral,axiom,
! [A3: set_Pr4386577575007340137rm_a_b] :
( ( sup_su6776935440552674877rm_a_b @ bot_bo197521221353338581rm_a_b @ A3 )
= A3 ) ).
% sup_bot.left_neutral
thf(fact_813_sup__bot_Oleft__neutral,axiom,
! [A3: produc357393685978478089rm_a_b > $o] :
( ( sup_su2803897115920335584_a_b_o @ bot_bo23760184229683272_a_b_o @ A3 )
= A3 ) ).
% sup_bot.left_neutral
thf(fact_814_sup__bot_Oleft__neutral,axiom,
! [A3: set_term_a_b] :
( ( sup_sup_set_term_a_b @ bot_bot_set_term_a_b @ A3 )
= A3 ) ).
% sup_bot.left_neutral
thf(fact_815_sup__bot_Oleft__neutral,axiom,
! [A3: set_b] :
( ( sup_sup_set_b @ bot_bot_set_b @ A3 )
= A3 ) ).
% sup_bot.left_neutral
thf(fact_816_sup__bot_Oneutr__eq__iff,axiom,
! [A3: set_Pr2972776593051762503rm_a_b,B2: set_Pr2972776593051762503rm_a_b] :
( ( bot_bo5040873672931234227rm_a_b
= ( sup_su1845815533690797339rm_a_b @ A3 @ B2 ) )
= ( ( A3 = bot_bo5040873672931234227rm_a_b )
& ( B2 = bot_bo5040873672931234227rm_a_b ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_817_sup__bot_Oneutr__eq__iff,axiom,
! [A3: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b] :
( ( bot_bo197521221353338581rm_a_b
= ( sup_su6776935440552674877rm_a_b @ A3 @ B2 ) )
= ( ( A3 = bot_bo197521221353338581rm_a_b )
& ( B2 = bot_bo197521221353338581rm_a_b ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_818_sup__bot_Oneutr__eq__iff,axiom,
! [A3: produc357393685978478089rm_a_b > $o,B2: produc357393685978478089rm_a_b > $o] :
( ( bot_bo23760184229683272_a_b_o
= ( sup_su2803897115920335584_a_b_o @ A3 @ B2 ) )
= ( ( A3 = bot_bo23760184229683272_a_b_o )
& ( B2 = bot_bo23760184229683272_a_b_o ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_819_sup__bot_Oneutr__eq__iff,axiom,
! [A3: set_term_a_b,B2: set_term_a_b] :
( ( bot_bot_set_term_a_b
= ( sup_sup_set_term_a_b @ A3 @ B2 ) )
= ( ( A3 = bot_bot_set_term_a_b )
& ( B2 = bot_bot_set_term_a_b ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_820_sup__bot_Oneutr__eq__iff,axiom,
! [A3: set_b,B2: set_b] :
( ( bot_bot_set_b
= ( sup_sup_set_b @ A3 @ B2 ) )
= ( ( A3 = bot_bot_set_b )
& ( B2 = bot_bot_set_b ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_821_sup__bot_Oright__neutral,axiom,
! [A3: set_Pr2972776593051762503rm_a_b] :
( ( sup_su1845815533690797339rm_a_b @ A3 @ bot_bo5040873672931234227rm_a_b )
= A3 ) ).
% sup_bot.right_neutral
thf(fact_822_sup__bot_Oright__neutral,axiom,
! [A3: set_Pr4386577575007340137rm_a_b] :
( ( sup_su6776935440552674877rm_a_b @ A3 @ bot_bo197521221353338581rm_a_b )
= A3 ) ).
% sup_bot.right_neutral
thf(fact_823_sup__bot_Oright__neutral,axiom,
! [A3: produc357393685978478089rm_a_b > $o] :
( ( sup_su2803897115920335584_a_b_o @ A3 @ bot_bo23760184229683272_a_b_o )
= A3 ) ).
% sup_bot.right_neutral
thf(fact_824_sup__bot_Oright__neutral,axiom,
! [A3: set_term_a_b] :
( ( sup_sup_set_term_a_b @ A3 @ bot_bot_set_term_a_b )
= A3 ) ).
% sup_bot.right_neutral
thf(fact_825_sup__bot_Oright__neutral,axiom,
! [A3: set_b] :
( ( sup_sup_set_b @ A3 @ bot_bot_set_b )
= A3 ) ).
% sup_bot.right_neutral
thf(fact_826_Int__UNIV,axiom,
! [A4: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
( ( ( inf_in1693821330460682071rm_a_b @ A4 @ B5 )
= top_to1314267278846557113rm_a_b )
= ( ( A4 = top_to1314267278846557113rm_a_b )
& ( B5 = top_to1314267278846557113rm_a_b ) ) ) ).
% Int_UNIV
thf(fact_827_Int__UNIV,axiom,
! [A4: set_term_a_b,B5: set_term_a_b] :
( ( ( inf_inf_set_term_a_b @ A4 @ B5 )
= top_top_set_term_a_b )
= ( ( A4 = top_top_set_term_a_b )
& ( B5 = top_top_set_term_a_b ) ) ) ).
% Int_UNIV
thf(fact_828_Un__empty,axiom,
! [A4: set_Pr2972776593051762503rm_a_b,B5: set_Pr2972776593051762503rm_a_b] :
( ( ( sup_su1845815533690797339rm_a_b @ A4 @ B5 )
= bot_bo5040873672931234227rm_a_b )
= ( ( A4 = bot_bo5040873672931234227rm_a_b )
& ( B5 = bot_bo5040873672931234227rm_a_b ) ) ) ).
% Un_empty
thf(fact_829_Un__empty,axiom,
! [A4: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
( ( ( sup_su6776935440552674877rm_a_b @ A4 @ B5 )
= bot_bo197521221353338581rm_a_b )
= ( ( A4 = bot_bo197521221353338581rm_a_b )
& ( B5 = bot_bo197521221353338581rm_a_b ) ) ) ).
% Un_empty
thf(fact_830_Un__empty,axiom,
! [A4: set_term_a_b,B5: set_term_a_b] :
( ( ( sup_sup_set_term_a_b @ A4 @ B5 )
= bot_bot_set_term_a_b )
= ( ( A4 = bot_bot_set_term_a_b )
& ( B5 = bot_bot_set_term_a_b ) ) ) ).
% Un_empty
thf(fact_831_Un__empty,axiom,
! [A4: set_b,B5: set_b] :
( ( ( sup_sup_set_b @ A4 @ B5 )
= bot_bot_set_b )
= ( ( A4 = bot_bot_set_b )
& ( B5 = bot_bot_set_b ) ) ) ).
% Un_empty
thf(fact_832_Un__subset__iff,axiom,
! [A4: set_Pr2972776593051762503rm_a_b,B5: set_Pr2972776593051762503rm_a_b,C3: set_Pr2972776593051762503rm_a_b] :
( ( ord_le2593958535671617255rm_a_b @ ( sup_su1845815533690797339rm_a_b @ A4 @ B5 ) @ C3 )
= ( ( ord_le2593958535671617255rm_a_b @ A4 @ C3 )
& ( ord_le2593958535671617255rm_a_b @ B5 @ C3 ) ) ) ).
% Un_subset_iff
thf(fact_833_Un__subset__iff,axiom,
! [A4: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b,C3: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ ( sup_su6776935440552674877rm_a_b @ A4 @ B5 ) @ C3 )
= ( ( ord_le118470702582115849rm_a_b @ A4 @ C3 )
& ( ord_le118470702582115849rm_a_b @ B5 @ C3 ) ) ) ).
% Un_subset_iff
thf(fact_834_Un__subset__iff,axiom,
! [A4: set_Pr4934435412358123699_a_nat,B5: set_Pr4934435412358123699_a_nat,C3: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ ( sup_su459911885395995103_a_nat @ A4 @ B5 ) @ C3 )
= ( ( ord_le8666007276011122963_a_nat @ A4 @ C3 )
& ( ord_le8666007276011122963_a_nat @ B5 @ C3 ) ) ) ).
% Un_subset_iff
thf(fact_835_Un__subset__iff,axiom,
! [A4: set_term_a_b,B5: set_term_a_b,C3: set_term_a_b] :
( ( ord_le2705286416250468010rm_a_b @ ( sup_sup_set_term_a_b @ A4 @ B5 ) @ C3 )
= ( ( ord_le2705286416250468010rm_a_b @ A4 @ C3 )
& ( ord_le2705286416250468010rm_a_b @ B5 @ C3 ) ) ) ).
% Un_subset_iff
thf(fact_836_Un__Int__eq_I1_J,axiom,
! [S5: set_Pr4386577575007340137rm_a_b,T6: set_Pr4386577575007340137rm_a_b] :
( ( inf_in1693821330460682071rm_a_b @ ( sup_su6776935440552674877rm_a_b @ S5 @ T6 ) @ S5 )
= S5 ) ).
% Un_Int_eq(1)
thf(fact_837_Un__Int__eq_I1_J,axiom,
! [S5: set_Pr2972776593051762503rm_a_b,T6: set_Pr2972776593051762503rm_a_b] :
( ( inf_in7724013137991509301rm_a_b @ ( sup_su1845815533690797339rm_a_b @ S5 @ T6 ) @ S5 )
= S5 ) ).
% Un_Int_eq(1)
thf(fact_838_Un__Int__eq_I1_J,axiom,
! [S5: set_term_a_b,T6: set_term_a_b] :
( ( inf_inf_set_term_a_b @ ( sup_sup_set_term_a_b @ S5 @ T6 ) @ S5 )
= S5 ) ).
% Un_Int_eq(1)
thf(fact_839_Un__Int__eq_I2_J,axiom,
! [S5: set_Pr4386577575007340137rm_a_b,T6: set_Pr4386577575007340137rm_a_b] :
( ( inf_in1693821330460682071rm_a_b @ ( sup_su6776935440552674877rm_a_b @ S5 @ T6 ) @ T6 )
= T6 ) ).
% Un_Int_eq(2)
thf(fact_840_Un__Int__eq_I2_J,axiom,
! [S5: set_Pr2972776593051762503rm_a_b,T6: set_Pr2972776593051762503rm_a_b] :
( ( inf_in7724013137991509301rm_a_b @ ( sup_su1845815533690797339rm_a_b @ S5 @ T6 ) @ T6 )
= T6 ) ).
% Un_Int_eq(2)
thf(fact_841_Un__Int__eq_I2_J,axiom,
! [S5: set_term_a_b,T6: set_term_a_b] :
( ( inf_inf_set_term_a_b @ ( sup_sup_set_term_a_b @ S5 @ T6 ) @ T6 )
= T6 ) ).
% Un_Int_eq(2)
thf(fact_842_Un__Int__eq_I3_J,axiom,
! [S5: set_Pr4386577575007340137rm_a_b,T6: set_Pr4386577575007340137rm_a_b] :
( ( inf_in1693821330460682071rm_a_b @ S5 @ ( sup_su6776935440552674877rm_a_b @ S5 @ T6 ) )
= S5 ) ).
% Un_Int_eq(3)
thf(fact_843_Un__Int__eq_I3_J,axiom,
! [S5: set_Pr2972776593051762503rm_a_b,T6: set_Pr2972776593051762503rm_a_b] :
( ( inf_in7724013137991509301rm_a_b @ S5 @ ( sup_su1845815533690797339rm_a_b @ S5 @ T6 ) )
= S5 ) ).
% Un_Int_eq(3)
thf(fact_844_Un__Int__eq_I3_J,axiom,
! [S5: set_term_a_b,T6: set_term_a_b] :
( ( inf_inf_set_term_a_b @ S5 @ ( sup_sup_set_term_a_b @ S5 @ T6 ) )
= S5 ) ).
% Un_Int_eq(3)
thf(fact_845_Un__Int__eq_I4_J,axiom,
! [T6: set_Pr4386577575007340137rm_a_b,S5: set_Pr4386577575007340137rm_a_b] :
( ( inf_in1693821330460682071rm_a_b @ T6 @ ( sup_su6776935440552674877rm_a_b @ S5 @ T6 ) )
= T6 ) ).
% Un_Int_eq(4)
thf(fact_846_Un__Int__eq_I4_J,axiom,
! [T6: set_Pr2972776593051762503rm_a_b,S5: set_Pr2972776593051762503rm_a_b] :
( ( inf_in7724013137991509301rm_a_b @ T6 @ ( sup_su1845815533690797339rm_a_b @ S5 @ T6 ) )
= T6 ) ).
% Un_Int_eq(4)
thf(fact_847_Un__Int__eq_I4_J,axiom,
! [T6: set_term_a_b,S5: set_term_a_b] :
( ( inf_inf_set_term_a_b @ T6 @ ( sup_sup_set_term_a_b @ S5 @ T6 ) )
= T6 ) ).
% Un_Int_eq(4)
thf(fact_848_Int__Un__eq_I1_J,axiom,
! [S5: set_Pr4386577575007340137rm_a_b,T6: set_Pr4386577575007340137rm_a_b] :
( ( sup_su6776935440552674877rm_a_b @ ( inf_in1693821330460682071rm_a_b @ S5 @ T6 ) @ S5 )
= S5 ) ).
% Int_Un_eq(1)
thf(fact_849_Int__Un__eq_I1_J,axiom,
! [S5: set_Pr2972776593051762503rm_a_b,T6: set_Pr2972776593051762503rm_a_b] :
( ( sup_su1845815533690797339rm_a_b @ ( inf_in7724013137991509301rm_a_b @ S5 @ T6 ) @ S5 )
= S5 ) ).
% Int_Un_eq(1)
thf(fact_850_Int__Un__eq_I1_J,axiom,
! [S5: set_term_a_b,T6: set_term_a_b] :
( ( sup_sup_set_term_a_b @ ( inf_inf_set_term_a_b @ S5 @ T6 ) @ S5 )
= S5 ) ).
% Int_Un_eq(1)
thf(fact_851_Int__Un__eq_I2_J,axiom,
! [S5: set_Pr4386577575007340137rm_a_b,T6: set_Pr4386577575007340137rm_a_b] :
( ( sup_su6776935440552674877rm_a_b @ ( inf_in1693821330460682071rm_a_b @ S5 @ T6 ) @ T6 )
= T6 ) ).
% Int_Un_eq(2)
thf(fact_852_Int__Un__eq_I2_J,axiom,
! [S5: set_Pr2972776593051762503rm_a_b,T6: set_Pr2972776593051762503rm_a_b] :
( ( sup_su1845815533690797339rm_a_b @ ( inf_in7724013137991509301rm_a_b @ S5 @ T6 ) @ T6 )
= T6 ) ).
% Int_Un_eq(2)
thf(fact_853_Int__Un__eq_I2_J,axiom,
! [S5: set_term_a_b,T6: set_term_a_b] :
( ( sup_sup_set_term_a_b @ ( inf_inf_set_term_a_b @ S5 @ T6 ) @ T6 )
= T6 ) ).
% Int_Un_eq(2)
thf(fact_854_Int__Un__eq_I3_J,axiom,
! [S5: set_Pr4386577575007340137rm_a_b,T6: set_Pr4386577575007340137rm_a_b] :
( ( sup_su6776935440552674877rm_a_b @ S5 @ ( inf_in1693821330460682071rm_a_b @ S5 @ T6 ) )
= S5 ) ).
% Int_Un_eq(3)
thf(fact_855_Int__Un__eq_I3_J,axiom,
! [S5: set_Pr2972776593051762503rm_a_b,T6: set_Pr2972776593051762503rm_a_b] :
( ( sup_su1845815533690797339rm_a_b @ S5 @ ( inf_in7724013137991509301rm_a_b @ S5 @ T6 ) )
= S5 ) ).
% Int_Un_eq(3)
thf(fact_856_Int__Un__eq_I3_J,axiom,
! [S5: set_term_a_b,T6: set_term_a_b] :
( ( sup_sup_set_term_a_b @ S5 @ ( inf_inf_set_term_a_b @ S5 @ T6 ) )
= S5 ) ).
% Int_Un_eq(3)
thf(fact_857_Int__Un__eq_I4_J,axiom,
! [T6: set_Pr4386577575007340137rm_a_b,S5: set_Pr4386577575007340137rm_a_b] :
( ( sup_su6776935440552674877rm_a_b @ T6 @ ( inf_in1693821330460682071rm_a_b @ S5 @ T6 ) )
= T6 ) ).
% Int_Un_eq(4)
thf(fact_858_Int__Un__eq_I4_J,axiom,
! [T6: set_Pr2972776593051762503rm_a_b,S5: set_Pr2972776593051762503rm_a_b] :
( ( sup_su1845815533690797339rm_a_b @ T6 @ ( inf_in7724013137991509301rm_a_b @ S5 @ T6 ) )
= T6 ) ).
% Int_Un_eq(4)
thf(fact_859_Int__Un__eq_I4_J,axiom,
! [T6: set_term_a_b,S5: set_term_a_b] :
( ( sup_sup_set_term_a_b @ T6 @ ( inf_inf_set_term_a_b @ S5 @ T6 ) )
= T6 ) ).
% Int_Un_eq(4)
thf(fact_860_relcomp__distrib,axiom,
! [R5: set_Pr4386577575007340137rm_a_b,S5: set_Pr4386577575007340137rm_a_b,T6: set_Pr4386577575007340137rm_a_b] :
( ( relcom370159955682700863rm_a_b @ R5 @ ( sup_su6776935440552674877rm_a_b @ S5 @ T6 ) )
= ( sup_su6776935440552674877rm_a_b @ ( relcom370159955682700863rm_a_b @ R5 @ S5 ) @ ( relcom370159955682700863rm_a_b @ R5 @ T6 ) ) ) ).
% relcomp_distrib
thf(fact_861_relcomp__distrib,axiom,
! [R5: set_Pr2972776593051762503rm_a_b,S5: set_Pr2972776593051762503rm_a_b,T6: set_Pr2972776593051762503rm_a_b] :
( ( relcom6159724393391289352rm_a_b @ R5 @ ( sup_su1845815533690797339rm_a_b @ S5 @ T6 ) )
= ( sup_su1845815533690797339rm_a_b @ ( relcom6159724393391289352rm_a_b @ R5 @ S5 ) @ ( relcom6159724393391289352rm_a_b @ R5 @ T6 ) ) ) ).
% relcomp_distrib
thf(fact_862_relcomp__distrib2,axiom,
! [S5: set_Pr4386577575007340137rm_a_b,T6: set_Pr4386577575007340137rm_a_b,R5: set_Pr4386577575007340137rm_a_b] :
( ( relcom370159955682700863rm_a_b @ ( sup_su6776935440552674877rm_a_b @ S5 @ T6 ) @ R5 )
= ( sup_su6776935440552674877rm_a_b @ ( relcom370159955682700863rm_a_b @ S5 @ R5 ) @ ( relcom370159955682700863rm_a_b @ T6 @ R5 ) ) ) ).
% relcomp_distrib2
thf(fact_863_relcomp__distrib2,axiom,
! [S5: set_Pr2972776593051762503rm_a_b,T6: set_Pr2972776593051762503rm_a_b,R5: set_Pr2972776593051762503rm_a_b] :
( ( relcom6159724393391289352rm_a_b @ ( sup_su1845815533690797339rm_a_b @ S5 @ T6 ) @ R5 )
= ( sup_su1845815533690797339rm_a_b @ ( relcom6159724393391289352rm_a_b @ S5 @ R5 ) @ ( relcom6159724393391289352rm_a_b @ T6 @ R5 ) ) ) ).
% relcomp_distrib2
thf(fact_864_symcl__idemp,axiom,
! [R2: set_Pr4386577575007340137rm_a_b] :
( ( sup_su6776935440552674877rm_a_b @ ( sup_su6776935440552674877rm_a_b @ R2 @ ( conver7608559976937845093rm_a_b @ R2 ) ) @ ( conver7608559976937845093rm_a_b @ ( sup_su6776935440552674877rm_a_b @ R2 @ ( conver7608559976937845093rm_a_b @ R2 ) ) ) )
= ( sup_su6776935440552674877rm_a_b @ R2 @ ( conver7608559976937845093rm_a_b @ R2 ) ) ) ).
% symcl_idemp
thf(fact_865_symcl__idemp,axiom,
! [R2: set_Pr2972776593051762503rm_a_b] :
( ( sup_su1845815533690797339rm_a_b @ ( sup_su1845815533690797339rm_a_b @ R2 @ ( conver1575690562132235771rm_a_b @ R2 ) ) @ ( conver1575690562132235771rm_a_b @ ( sup_su1845815533690797339rm_a_b @ R2 @ ( conver1575690562132235771rm_a_b @ R2 ) ) ) )
= ( sup_su1845815533690797339rm_a_b @ R2 @ ( conver1575690562132235771rm_a_b @ R2 ) ) ) ).
% symcl_idemp
thf(fact_866_funas__ctxt__apply,axiom,
! [C3: subterm_and_ctxt_a_b,T: term_a_b] :
( ( term_funas_term_a_b @ ( subter2376574525758040790rm_a_b @ C3 @ T ) )
= ( sup_su459911885395995103_a_nat @ ( term_funas_ctxt_a_b @ C3 ) @ ( term_funas_term_a_b @ T ) ) ) ).
% funas_ctxt_apply
thf(fact_867_boolean__algebra_Ocomplement__unique,axiom,
! [A3: set_Pr2972776593051762503rm_a_b,X: set_Pr2972776593051762503rm_a_b,Y: set_Pr2972776593051762503rm_a_b] :
( ( ( inf_in7724013137991509301rm_a_b @ A3 @ X )
= bot_bo5040873672931234227rm_a_b )
=> ( ( ( sup_su1845815533690797339rm_a_b @ A3 @ X )
= top_to1302229308806327447rm_a_b )
=> ( ( ( inf_in7724013137991509301rm_a_b @ A3 @ Y )
= bot_bo5040873672931234227rm_a_b )
=> ( ( ( sup_su1845815533690797339rm_a_b @ A3 @ Y )
= top_to1302229308806327447rm_a_b )
=> ( X = Y ) ) ) ) ) ).
% boolean_algebra.complement_unique
thf(fact_868_boolean__algebra_Ocomplement__unique,axiom,
! [A3: set_b,X: set_b,Y: set_b] :
( ( ( inf_inf_set_b @ A3 @ X )
= bot_bot_set_b )
=> ( ( ( sup_sup_set_b @ A3 @ X )
= top_top_set_b )
=> ( ( ( inf_inf_set_b @ A3 @ Y )
= bot_bot_set_b )
=> ( ( ( sup_sup_set_b @ A3 @ Y )
= top_top_set_b )
=> ( X = Y ) ) ) ) ) ).
% boolean_algebra.complement_unique
thf(fact_869_boolean__algebra_Ocomplement__unique,axiom,
! [A3: set_Pr4386577575007340137rm_a_b,X: set_Pr4386577575007340137rm_a_b,Y: set_Pr4386577575007340137rm_a_b] :
( ( ( inf_in1693821330460682071rm_a_b @ A3 @ X )
= bot_bo197521221353338581rm_a_b )
=> ( ( ( sup_su6776935440552674877rm_a_b @ A3 @ X )
= top_to1314267278846557113rm_a_b )
=> ( ( ( inf_in1693821330460682071rm_a_b @ A3 @ Y )
= bot_bo197521221353338581rm_a_b )
=> ( ( ( sup_su6776935440552674877rm_a_b @ A3 @ Y )
= top_to1314267278846557113rm_a_b )
=> ( X = Y ) ) ) ) ) ).
% boolean_algebra.complement_unique
thf(fact_870_boolean__algebra_Ocomplement__unique,axiom,
! [A3: set_term_a_b,X: set_term_a_b,Y: set_term_a_b] :
( ( ( inf_inf_set_term_a_b @ A3 @ X )
= bot_bot_set_term_a_b )
=> ( ( ( sup_sup_set_term_a_b @ A3 @ X )
= top_top_set_term_a_b )
=> ( ( ( inf_inf_set_term_a_b @ A3 @ Y )
= bot_bot_set_term_a_b )
=> ( ( ( sup_sup_set_term_a_b @ A3 @ Y )
= top_top_set_term_a_b )
=> ( X = Y ) ) ) ) ) ).
% boolean_algebra.complement_unique
thf(fact_871_boolean__algebra_Ocomplement__unique,axiom,
! [A3: produc357393685978478089rm_a_b > $o,X: produc357393685978478089rm_a_b > $o,Y: produc357393685978478089rm_a_b > $o] :
( ( ( inf_in7523581892642934342_a_b_o @ A3 @ X )
= bot_bo23760184229683272_a_b_o )
=> ( ( ( sup_su2803897115920335584_a_b_o @ A3 @ X )
= top_to1262317115727556708_a_b_o )
=> ( ( ( inf_in7523581892642934342_a_b_o @ A3 @ Y )
= bot_bo23760184229683272_a_b_o )
=> ( ( ( sup_su2803897115920335584_a_b_o @ A3 @ Y )
= top_to1262317115727556708_a_b_o )
=> ( X = Y ) ) ) ) ) ).
% boolean_algebra.complement_unique
thf(fact_872_top__greatest,axiom,
! [A3: produc357393685978478089rm_a_b > $o] : ( ord_le6645574462093738516_a_b_o @ A3 @ top_to1262317115727556708_a_b_o ) ).
% top_greatest
thf(fact_873_top__greatest,axiom,
! [A3: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ A3 @ top_to1314267278846557113rm_a_b ) ).
% top_greatest
thf(fact_874_top__greatest,axiom,
! [A3: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ A3 @ top_to3353692345378799459_a_nat ) ).
% top_greatest
thf(fact_875_top__greatest,axiom,
! [A3: set_term_a_b] : ( ord_le2705286416250468010rm_a_b @ A3 @ top_top_set_term_a_b ) ).
% top_greatest
thf(fact_876_top_Oextremum__unique,axiom,
! [A3: produc357393685978478089rm_a_b > $o] :
( ( ord_le6645574462093738516_a_b_o @ top_to1262317115727556708_a_b_o @ A3 )
= ( A3 = top_to1262317115727556708_a_b_o ) ) ).
% top.extremum_unique
thf(fact_877_top_Oextremum__unique,axiom,
! [A3: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ top_to1314267278846557113rm_a_b @ A3 )
= ( A3 = top_to1314267278846557113rm_a_b ) ) ).
% top.extremum_unique
thf(fact_878_top_Oextremum__unique,axiom,
! [A3: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ top_to3353692345378799459_a_nat @ A3 )
= ( A3 = top_to3353692345378799459_a_nat ) ) ).
% top.extremum_unique
thf(fact_879_top_Oextremum__unique,axiom,
! [A3: set_term_a_b] :
( ( ord_le2705286416250468010rm_a_b @ top_top_set_term_a_b @ A3 )
= ( A3 = top_top_set_term_a_b ) ) ).
% top.extremum_unique
thf(fact_880_top_Oextremum__uniqueI,axiom,
! [A3: produc357393685978478089rm_a_b > $o] :
( ( ord_le6645574462093738516_a_b_o @ top_to1262317115727556708_a_b_o @ A3 )
=> ( A3 = top_to1262317115727556708_a_b_o ) ) ).
% top.extremum_uniqueI
thf(fact_881_top_Oextremum__uniqueI,axiom,
! [A3: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ top_to1314267278846557113rm_a_b @ A3 )
=> ( A3 = top_to1314267278846557113rm_a_b ) ) ).
% top.extremum_uniqueI
thf(fact_882_top_Oextremum__uniqueI,axiom,
! [A3: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ top_to3353692345378799459_a_nat @ A3 )
=> ( A3 = top_to3353692345378799459_a_nat ) ) ).
% top.extremum_uniqueI
thf(fact_883_top_Oextremum__uniqueI,axiom,
! [A3: set_term_a_b] :
( ( ord_le2705286416250468010rm_a_b @ top_top_set_term_a_b @ A3 )
=> ( A3 = top_top_set_term_a_b ) ) ).
% top.extremum_uniqueI
thf(fact_884_Id__on__union,axiom,
! [A4: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
( ( id_on_115818524817812774rm_a_b @ ( sup_su6776935440552674877rm_a_b @ A4 @ B5 ) )
= ( sup_su1845815533690797339rm_a_b @ ( id_on_115818524817812774rm_a_b @ A4 ) @ ( id_on_115818524817812774rm_a_b @ B5 ) ) ) ).
% Id_on_union
thf(fact_885_Id__on__union,axiom,
! [A4: set_Pr2972776593051762503rm_a_b,B5: set_Pr2972776593051762503rm_a_b] :
( ( id_on_4065744930832257284rm_a_b @ ( sup_su1845815533690797339rm_a_b @ A4 @ B5 ) )
= ( sup_su657794527144859291rm_a_b @ ( id_on_4065744930832257284rm_a_b @ A4 ) @ ( id_on_4065744930832257284rm_a_b @ B5 ) ) ) ).
% Id_on_union
thf(fact_886_Id__on__union,axiom,
! [A4: set_term_a_b,B5: set_term_a_b] :
( ( id_on_term_a_b @ ( sup_sup_set_term_a_b @ A4 @ B5 ) )
= ( sup_su6776935440552674877rm_a_b @ ( id_on_term_a_b @ A4 ) @ ( id_on_term_a_b @ B5 ) ) ) ).
% Id_on_union
thf(fact_887_Un__left__commute,axiom,
! [A4: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b,C3: set_Pr4386577575007340137rm_a_b] :
( ( sup_su6776935440552674877rm_a_b @ A4 @ ( sup_su6776935440552674877rm_a_b @ B5 @ C3 ) )
= ( sup_su6776935440552674877rm_a_b @ B5 @ ( sup_su6776935440552674877rm_a_b @ A4 @ C3 ) ) ) ).
% Un_left_commute
thf(fact_888_Un__left__commute,axiom,
! [A4: set_Pr2972776593051762503rm_a_b,B5: set_Pr2972776593051762503rm_a_b,C3: set_Pr2972776593051762503rm_a_b] :
( ( sup_su1845815533690797339rm_a_b @ A4 @ ( sup_su1845815533690797339rm_a_b @ B5 @ C3 ) )
= ( sup_su1845815533690797339rm_a_b @ B5 @ ( sup_su1845815533690797339rm_a_b @ A4 @ C3 ) ) ) ).
% Un_left_commute
thf(fact_889_Un__left__commute,axiom,
! [A4: set_term_a_b,B5: set_term_a_b,C3: set_term_a_b] :
( ( sup_sup_set_term_a_b @ A4 @ ( sup_sup_set_term_a_b @ B5 @ C3 ) )
= ( sup_sup_set_term_a_b @ B5 @ ( sup_sup_set_term_a_b @ A4 @ C3 ) ) ) ).
% Un_left_commute
thf(fact_890_Un__left__absorb,axiom,
! [A4: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
( ( sup_su6776935440552674877rm_a_b @ A4 @ ( sup_su6776935440552674877rm_a_b @ A4 @ B5 ) )
= ( sup_su6776935440552674877rm_a_b @ A4 @ B5 ) ) ).
% Un_left_absorb
thf(fact_891_Un__left__absorb,axiom,
! [A4: set_Pr2972776593051762503rm_a_b,B5: set_Pr2972776593051762503rm_a_b] :
( ( sup_su1845815533690797339rm_a_b @ A4 @ ( sup_su1845815533690797339rm_a_b @ A4 @ B5 ) )
= ( sup_su1845815533690797339rm_a_b @ A4 @ B5 ) ) ).
% Un_left_absorb
thf(fact_892_Un__left__absorb,axiom,
! [A4: set_term_a_b,B5: set_term_a_b] :
( ( sup_sup_set_term_a_b @ A4 @ ( sup_sup_set_term_a_b @ A4 @ B5 ) )
= ( sup_sup_set_term_a_b @ A4 @ B5 ) ) ).
% Un_left_absorb
thf(fact_893_Un__UNIV__right,axiom,
! [A4: set_Pr2972776593051762503rm_a_b] :
( ( sup_su1845815533690797339rm_a_b @ A4 @ top_to1302229308806327447rm_a_b )
= top_to1302229308806327447rm_a_b ) ).
% Un_UNIV_right
thf(fact_894_Un__UNIV__right,axiom,
! [A4: set_Pr4386577575007340137rm_a_b] :
( ( sup_su6776935440552674877rm_a_b @ A4 @ top_to1314267278846557113rm_a_b )
= top_to1314267278846557113rm_a_b ) ).
% Un_UNIV_right
thf(fact_895_Un__UNIV__right,axiom,
! [A4: set_term_a_b] :
( ( sup_sup_set_term_a_b @ A4 @ top_top_set_term_a_b )
= top_top_set_term_a_b ) ).
% Un_UNIV_right
thf(fact_896_Un__UNIV__left,axiom,
! [B5: set_Pr2972776593051762503rm_a_b] :
( ( sup_su1845815533690797339rm_a_b @ top_to1302229308806327447rm_a_b @ B5 )
= top_to1302229308806327447rm_a_b ) ).
% Un_UNIV_left
thf(fact_897_Un__UNIV__left,axiom,
! [B5: set_Pr4386577575007340137rm_a_b] :
( ( sup_su6776935440552674877rm_a_b @ top_to1314267278846557113rm_a_b @ B5 )
= top_to1314267278846557113rm_a_b ) ).
% Un_UNIV_left
thf(fact_898_Un__UNIV__left,axiom,
! [B5: set_term_a_b] :
( ( sup_sup_set_term_a_b @ top_top_set_term_a_b @ B5 )
= top_top_set_term_a_b ) ).
% Un_UNIV_left
thf(fact_899_UNIV__witness,axiom,
? [X3: produc4523971788519308903rm_a_b] : ( member8417600551952982416rm_a_b @ X3 @ top_to1302229308806327447rm_a_b ) ).
% UNIV_witness
thf(fact_900_UNIV__witness,axiom,
? [X3: term_a_b] : ( member_term_a_b @ X3 @ top_top_set_term_a_b ) ).
% UNIV_witness
thf(fact_901_UNIV__witness,axiom,
? [X3: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ X3 @ top_to1314267278846557113rm_a_b ) ).
% UNIV_witness
thf(fact_902_Un__commute,axiom,
( sup_su6776935440552674877rm_a_b
= ( ^ [A7: set_Pr4386577575007340137rm_a_b,B6: set_Pr4386577575007340137rm_a_b] : ( sup_su6776935440552674877rm_a_b @ B6 @ A7 ) ) ) ).
% Un_commute
thf(fact_903_Un__commute,axiom,
( sup_su1845815533690797339rm_a_b
= ( ^ [A7: set_Pr2972776593051762503rm_a_b,B6: set_Pr2972776593051762503rm_a_b] : ( sup_su1845815533690797339rm_a_b @ B6 @ A7 ) ) ) ).
% Un_commute
thf(fact_904_Un__commute,axiom,
( sup_sup_set_term_a_b
= ( ^ [A7: set_term_a_b,B6: set_term_a_b] : ( sup_sup_set_term_a_b @ B6 @ A7 ) ) ) ).
% Un_commute
thf(fact_905_Un__absorb,axiom,
! [A4: set_Pr4386577575007340137rm_a_b] :
( ( sup_su6776935440552674877rm_a_b @ A4 @ A4 )
= A4 ) ).
% Un_absorb
thf(fact_906_Un__absorb,axiom,
! [A4: set_Pr2972776593051762503rm_a_b] :
( ( sup_su1845815533690797339rm_a_b @ A4 @ A4 )
= A4 ) ).
% Un_absorb
thf(fact_907_Un__absorb,axiom,
! [A4: set_term_a_b] :
( ( sup_sup_set_term_a_b @ A4 @ A4 )
= A4 ) ).
% Un_absorb
thf(fact_908_UNIV__eq__I,axiom,
! [A4: set_Pr2972776593051762503rm_a_b] :
( ! [X3: produc4523971788519308903rm_a_b] : ( member8417600551952982416rm_a_b @ X3 @ A4 )
=> ( top_to1302229308806327447rm_a_b = A4 ) ) ).
% UNIV_eq_I
thf(fact_909_UNIV__eq__I,axiom,
! [A4: set_term_a_b] :
( ! [X3: term_a_b] : ( member_term_a_b @ X3 @ A4 )
=> ( top_top_set_term_a_b = A4 ) ) ).
% UNIV_eq_I
thf(fact_910_UNIV__eq__I,axiom,
! [A4: set_Pr4386577575007340137rm_a_b] :
( ! [X3: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ X3 @ A4 )
=> ( top_to1314267278846557113rm_a_b = A4 ) ) ).
% UNIV_eq_I
thf(fact_911_Un__assoc,axiom,
! [A4: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b,C3: set_Pr4386577575007340137rm_a_b] :
( ( sup_su6776935440552674877rm_a_b @ ( sup_su6776935440552674877rm_a_b @ A4 @ B5 ) @ C3 )
= ( sup_su6776935440552674877rm_a_b @ A4 @ ( sup_su6776935440552674877rm_a_b @ B5 @ C3 ) ) ) ).
% Un_assoc
thf(fact_912_Un__assoc,axiom,
! [A4: set_Pr2972776593051762503rm_a_b,B5: set_Pr2972776593051762503rm_a_b,C3: set_Pr2972776593051762503rm_a_b] :
( ( sup_su1845815533690797339rm_a_b @ ( sup_su1845815533690797339rm_a_b @ A4 @ B5 ) @ C3 )
= ( sup_su1845815533690797339rm_a_b @ A4 @ ( sup_su1845815533690797339rm_a_b @ B5 @ C3 ) ) ) ).
% Un_assoc
thf(fact_913_Un__assoc,axiom,
! [A4: set_term_a_b,B5: set_term_a_b,C3: set_term_a_b] :
( ( sup_sup_set_term_a_b @ ( sup_sup_set_term_a_b @ A4 @ B5 ) @ C3 )
= ( sup_sup_set_term_a_b @ A4 @ ( sup_sup_set_term_a_b @ B5 @ C3 ) ) ) ).
% Un_assoc
thf(fact_914_ball__Un,axiom,
! [A4: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b,P2: produc357393685978478089rm_a_b > $o] :
( ( ! [X2: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X2 @ ( sup_su6776935440552674877rm_a_b @ A4 @ B5 ) )
=> ( P2 @ X2 ) ) )
= ( ! [X2: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X2 @ A4 )
=> ( P2 @ X2 ) )
& ! [X2: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X2 @ B5 )
=> ( P2 @ X2 ) ) ) ) ).
% ball_Un
thf(fact_915_ball__Un,axiom,
! [A4: set_Pr2972776593051762503rm_a_b,B5: set_Pr2972776593051762503rm_a_b,P2: produc4523971788519308903rm_a_b > $o] :
( ( ! [X2: produc4523971788519308903rm_a_b] :
( ( member8417600551952982416rm_a_b @ X2 @ ( sup_su1845815533690797339rm_a_b @ A4 @ B5 ) )
=> ( P2 @ X2 ) ) )
= ( ! [X2: produc4523971788519308903rm_a_b] :
( ( member8417600551952982416rm_a_b @ X2 @ A4 )
=> ( P2 @ X2 ) )
& ! [X2: produc4523971788519308903rm_a_b] :
( ( member8417600551952982416rm_a_b @ X2 @ B5 )
=> ( P2 @ X2 ) ) ) ) ).
% ball_Un
thf(fact_916_ball__Un,axiom,
! [A4: set_term_a_b,B5: set_term_a_b,P2: term_a_b > $o] :
( ( ! [X2: term_a_b] :
( ( member_term_a_b @ X2 @ ( sup_sup_set_term_a_b @ A4 @ B5 ) )
=> ( P2 @ X2 ) ) )
= ( ! [X2: term_a_b] :
( ( member_term_a_b @ X2 @ A4 )
=> ( P2 @ X2 ) )
& ! [X2: term_a_b] :
( ( member_term_a_b @ X2 @ B5 )
=> ( P2 @ X2 ) ) ) ) ).
% ball_Un
thf(fact_917_bex__Un,axiom,
! [A4: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b,P2: produc357393685978478089rm_a_b > $o] :
( ( ? [X2: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X2 @ ( sup_su6776935440552674877rm_a_b @ A4 @ B5 ) )
& ( P2 @ X2 ) ) )
= ( ? [X2: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X2 @ A4 )
& ( P2 @ X2 ) )
| ? [X2: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X2 @ B5 )
& ( P2 @ X2 ) ) ) ) ).
% bex_Un
thf(fact_918_bex__Un,axiom,
! [A4: set_Pr2972776593051762503rm_a_b,B5: set_Pr2972776593051762503rm_a_b,P2: produc4523971788519308903rm_a_b > $o] :
( ( ? [X2: produc4523971788519308903rm_a_b] :
( ( member8417600551952982416rm_a_b @ X2 @ ( sup_su1845815533690797339rm_a_b @ A4 @ B5 ) )
& ( P2 @ X2 ) ) )
= ( ? [X2: produc4523971788519308903rm_a_b] :
( ( member8417600551952982416rm_a_b @ X2 @ A4 )
& ( P2 @ X2 ) )
| ? [X2: produc4523971788519308903rm_a_b] :
( ( member8417600551952982416rm_a_b @ X2 @ B5 )
& ( P2 @ X2 ) ) ) ) ).
% bex_Un
thf(fact_919_bex__Un,axiom,
! [A4: set_term_a_b,B5: set_term_a_b,P2: term_a_b > $o] :
( ( ? [X2: term_a_b] :
( ( member_term_a_b @ X2 @ ( sup_sup_set_term_a_b @ A4 @ B5 ) )
& ( P2 @ X2 ) ) )
= ( ? [X2: term_a_b] :
( ( member_term_a_b @ X2 @ A4 )
& ( P2 @ X2 ) )
| ? [X2: term_a_b] :
( ( member_term_a_b @ X2 @ B5 )
& ( P2 @ X2 ) ) ) ) ).
% bex_Un
thf(fact_920_UnI2,axiom,
! [C: produc4523971788519308903rm_a_b,B5: set_Pr2972776593051762503rm_a_b,A4: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ C @ B5 )
=> ( member8417600551952982416rm_a_b @ C @ ( sup_su1845815533690797339rm_a_b @ A4 @ B5 ) ) ) ).
% UnI2
thf(fact_921_UnI2,axiom,
! [C: term_a_b,B5: set_term_a_b,A4: set_term_a_b] :
( ( member_term_a_b @ C @ B5 )
=> ( member_term_a_b @ C @ ( sup_sup_set_term_a_b @ A4 @ B5 ) ) ) ).
% UnI2
thf(fact_922_UnI2,axiom,
! [C: produc357393685978478089rm_a_b,B5: set_Pr4386577575007340137rm_a_b,A4: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ C @ B5 )
=> ( member5869715511025134514rm_a_b @ C @ ( sup_su6776935440552674877rm_a_b @ A4 @ B5 ) ) ) ).
% UnI2
thf(fact_923_UnI1,axiom,
! [C: produc4523971788519308903rm_a_b,A4: set_Pr2972776593051762503rm_a_b,B5: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ C @ A4 )
=> ( member8417600551952982416rm_a_b @ C @ ( sup_su1845815533690797339rm_a_b @ A4 @ B5 ) ) ) ).
% UnI1
thf(fact_924_UnI1,axiom,
! [C: term_a_b,A4: set_term_a_b,B5: set_term_a_b] :
( ( member_term_a_b @ C @ A4 )
=> ( member_term_a_b @ C @ ( sup_sup_set_term_a_b @ A4 @ B5 ) ) ) ).
% UnI1
thf(fact_925_UnI1,axiom,
! [C: produc357393685978478089rm_a_b,A4: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ C @ A4 )
=> ( member5869715511025134514rm_a_b @ C @ ( sup_su6776935440552674877rm_a_b @ A4 @ B5 ) ) ) ).
% UnI1
thf(fact_926_UnE,axiom,
! [C: produc4523971788519308903rm_a_b,A4: set_Pr2972776593051762503rm_a_b,B5: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ C @ ( sup_su1845815533690797339rm_a_b @ A4 @ B5 ) )
=> ( ~ ( member8417600551952982416rm_a_b @ C @ A4 )
=> ( member8417600551952982416rm_a_b @ C @ B5 ) ) ) ).
% UnE
thf(fact_927_UnE,axiom,
! [C: term_a_b,A4: set_term_a_b,B5: set_term_a_b] :
( ( member_term_a_b @ C @ ( sup_sup_set_term_a_b @ A4 @ B5 ) )
=> ( ~ ( member_term_a_b @ C @ A4 )
=> ( member_term_a_b @ C @ B5 ) ) ) ).
% UnE
thf(fact_928_UnE,axiom,
! [C: produc357393685978478089rm_a_b,A4: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ C @ ( sup_su6776935440552674877rm_a_b @ A4 @ B5 ) )
=> ( ~ ( member5869715511025134514rm_a_b @ C @ A4 )
=> ( member5869715511025134514rm_a_b @ C @ B5 ) ) ) ).
% UnE
thf(fact_929_boolean__algebra_Odisj__zero__right,axiom,
! [X: set_Pr2972776593051762503rm_a_b] :
( ( sup_su1845815533690797339rm_a_b @ X @ bot_bo5040873672931234227rm_a_b )
= X ) ).
% boolean_algebra.disj_zero_right
thf(fact_930_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_931_boolean__algebra_Odisj__zero__right,axiom,
! [X: produc357393685978478089rm_a_b > $o] :
( ( sup_su2803897115920335584_a_b_o @ X @ bot_bo23760184229683272_a_b_o )
= X ) ).
% boolean_algebra.disj_zero_right
thf(fact_932_boolean__algebra_Odisj__zero__right,axiom,
! [X: set_term_a_b] :
( ( sup_sup_set_term_a_b @ X @ bot_bot_set_term_a_b )
= X ) ).
% boolean_algebra.disj_zero_right
thf(fact_933_boolean__algebra_Odisj__zero__right,axiom,
! [X: set_b] :
( ( sup_sup_set_b @ X @ bot_bot_set_b )
= X ) ).
% boolean_algebra.disj_zero_right
thf(fact_934_Un__empty__left,axiom,
! [B5: set_Pr2972776593051762503rm_a_b] :
( ( sup_su1845815533690797339rm_a_b @ bot_bo5040873672931234227rm_a_b @ B5 )
= B5 ) ).
% Un_empty_left
thf(fact_935_Un__empty__left,axiom,
! [B5: set_Pr4386577575007340137rm_a_b] :
( ( sup_su6776935440552674877rm_a_b @ bot_bo197521221353338581rm_a_b @ B5 )
= B5 ) ).
% Un_empty_left
thf(fact_936_Un__empty__left,axiom,
! [B5: set_term_a_b] :
( ( sup_sup_set_term_a_b @ bot_bot_set_term_a_b @ B5 )
= B5 ) ).
% Un_empty_left
thf(fact_937_Un__empty__left,axiom,
! [B5: set_b] :
( ( sup_sup_set_b @ bot_bot_set_b @ B5 )
= B5 ) ).
% Un_empty_left
thf(fact_938_Un__empty__right,axiom,
! [A4: set_Pr2972776593051762503rm_a_b] :
( ( sup_su1845815533690797339rm_a_b @ A4 @ bot_bo5040873672931234227rm_a_b )
= A4 ) ).
% Un_empty_right
thf(fact_939_Un__empty__right,axiom,
! [A4: set_Pr4386577575007340137rm_a_b] :
( ( sup_su6776935440552674877rm_a_b @ A4 @ bot_bo197521221353338581rm_a_b )
= A4 ) ).
% Un_empty_right
thf(fact_940_Un__empty__right,axiom,
! [A4: set_term_a_b] :
( ( sup_sup_set_term_a_b @ A4 @ bot_bot_set_term_a_b )
= A4 ) ).
% Un_empty_right
thf(fact_941_Un__empty__right,axiom,
! [A4: set_b] :
( ( sup_sup_set_b @ A4 @ bot_bot_set_b )
= A4 ) ).
% Un_empty_right
thf(fact_942_subset__Un__eq,axiom,
( ord_le2593958535671617255rm_a_b
= ( ^ [A7: set_Pr2972776593051762503rm_a_b,B6: set_Pr2972776593051762503rm_a_b] :
( ( sup_su1845815533690797339rm_a_b @ A7 @ B6 )
= B6 ) ) ) ).
% subset_Un_eq
thf(fact_943_subset__Un__eq,axiom,
( ord_le118470702582115849rm_a_b
= ( ^ [A7: set_Pr4386577575007340137rm_a_b,B6: set_Pr4386577575007340137rm_a_b] :
( ( sup_su6776935440552674877rm_a_b @ A7 @ B6 )
= B6 ) ) ) ).
% subset_Un_eq
thf(fact_944_subset__Un__eq,axiom,
( ord_le8666007276011122963_a_nat
= ( ^ [A7: set_Pr4934435412358123699_a_nat,B6: set_Pr4934435412358123699_a_nat] :
( ( sup_su459911885395995103_a_nat @ A7 @ B6 )
= B6 ) ) ) ).
% subset_Un_eq
thf(fact_945_subset__Un__eq,axiom,
( ord_le2705286416250468010rm_a_b
= ( ^ [A7: set_term_a_b,B6: set_term_a_b] :
( ( sup_sup_set_term_a_b @ A7 @ B6 )
= B6 ) ) ) ).
% subset_Un_eq
thf(fact_946_subset__UnE,axiom,
! [C3: set_Pr2972776593051762503rm_a_b,A4: set_Pr2972776593051762503rm_a_b,B5: set_Pr2972776593051762503rm_a_b] :
( ( ord_le2593958535671617255rm_a_b @ C3 @ ( sup_su1845815533690797339rm_a_b @ A4 @ B5 ) )
=> ~ ! [A8: set_Pr2972776593051762503rm_a_b] :
( ( ord_le2593958535671617255rm_a_b @ A8 @ A4 )
=> ! [B8: set_Pr2972776593051762503rm_a_b] :
( ( ord_le2593958535671617255rm_a_b @ B8 @ B5 )
=> ( C3
!= ( sup_su1845815533690797339rm_a_b @ A8 @ B8 ) ) ) ) ) ).
% subset_UnE
thf(fact_947_subset__UnE,axiom,
! [C3: set_Pr4386577575007340137rm_a_b,A4: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ C3 @ ( sup_su6776935440552674877rm_a_b @ A4 @ B5 ) )
=> ~ ! [A8: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A8 @ A4 )
=> ! [B8: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ B8 @ B5 )
=> ( C3
!= ( sup_su6776935440552674877rm_a_b @ A8 @ B8 ) ) ) ) ) ).
% subset_UnE
thf(fact_948_subset__UnE,axiom,
! [C3: set_Pr4934435412358123699_a_nat,A4: set_Pr4934435412358123699_a_nat,B5: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ C3 @ ( sup_su459911885395995103_a_nat @ A4 @ B5 ) )
=> ~ ! [A8: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A8 @ A4 )
=> ! [B8: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ B8 @ B5 )
=> ( C3
!= ( sup_su459911885395995103_a_nat @ A8 @ B8 ) ) ) ) ) ).
% subset_UnE
thf(fact_949_subset__UnE,axiom,
! [C3: set_term_a_b,A4: set_term_a_b,B5: set_term_a_b] :
( ( ord_le2705286416250468010rm_a_b @ C3 @ ( sup_sup_set_term_a_b @ A4 @ B5 ) )
=> ~ ! [A8: set_term_a_b] :
( ( ord_le2705286416250468010rm_a_b @ A8 @ A4 )
=> ! [B8: set_term_a_b] :
( ( ord_le2705286416250468010rm_a_b @ B8 @ B5 )
=> ( C3
!= ( sup_sup_set_term_a_b @ A8 @ B8 ) ) ) ) ) ).
% subset_UnE
thf(fact_950_Un__absorb2,axiom,
! [B5: set_Pr2972776593051762503rm_a_b,A4: set_Pr2972776593051762503rm_a_b] :
( ( ord_le2593958535671617255rm_a_b @ B5 @ A4 )
=> ( ( sup_su1845815533690797339rm_a_b @ A4 @ B5 )
= A4 ) ) ).
% Un_absorb2
thf(fact_951_Un__absorb2,axiom,
! [B5: set_Pr4386577575007340137rm_a_b,A4: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ B5 @ A4 )
=> ( ( sup_su6776935440552674877rm_a_b @ A4 @ B5 )
= A4 ) ) ).
% Un_absorb2
thf(fact_952_Un__absorb2,axiom,
! [B5: set_Pr4934435412358123699_a_nat,A4: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ B5 @ A4 )
=> ( ( sup_su459911885395995103_a_nat @ A4 @ B5 )
= A4 ) ) ).
% Un_absorb2
thf(fact_953_Un__absorb2,axiom,
! [B5: set_term_a_b,A4: set_term_a_b] :
( ( ord_le2705286416250468010rm_a_b @ B5 @ A4 )
=> ( ( sup_sup_set_term_a_b @ A4 @ B5 )
= A4 ) ) ).
% Un_absorb2
thf(fact_954_in__rtrancl__UnI,axiom,
! [X: produc357393685978478089rm_a_b,R5: set_Pr4386577575007340137rm_a_b,S5: set_Pr4386577575007340137rm_a_b] :
( ( ( member5869715511025134514rm_a_b @ X @ ( transi7742714808557438673rm_a_b @ R5 ) )
| ( member5869715511025134514rm_a_b @ X @ ( transi7742714808557438673rm_a_b @ S5 ) ) )
=> ( member5869715511025134514rm_a_b @ X @ ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ R5 @ S5 ) ) ) ) ).
% in_rtrancl_UnI
thf(fact_955_rtrancl__Un__rtrancl,axiom,
! [R5: set_Pr4386577575007340137rm_a_b,S5: set_Pr4386577575007340137rm_a_b] :
( ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ ( transi7742714808557438673rm_a_b @ R5 ) @ ( transi7742714808557438673rm_a_b @ S5 ) ) )
= ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ R5 @ S5 ) ) ) ).
% rtrancl_Un_rtrancl
thf(fact_956_rtrancl__Un__separatorE,axiom,
! [A3: term_a_b,B2: term_a_b,P2: set_Pr4386577575007340137rm_a_b,Q: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B2 ) @ ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ P2 @ Q ) ) )
=> ( ! [X3: term_a_b,Y2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ X3 ) @ ( transi7742714808557438673rm_a_b @ P2 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y2 ) @ Q )
=> ( X3 = Y2 ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B2 ) @ ( transi7742714808557438673rm_a_b @ P2 ) ) ) ) ).
% rtrancl_Un_separatorE
thf(fact_957_rtrancl__Un__separator__converseE,axiom,
! [A3: term_a_b,B2: term_a_b,P2: set_Pr4386577575007340137rm_a_b,Q: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B2 ) @ ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ P2 @ Q ) ) )
=> ( ! [X3: term_a_b,Y2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ B2 ) @ ( transi7742714808557438673rm_a_b @ P2 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y2 @ X3 ) @ Q )
=> ( Y2 = X3 ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B2 ) @ ( transi7742714808557438673rm_a_b @ P2 ) ) ) ) ).
% rtrancl_Un_separator_converseE
thf(fact_958_rtrancl__Un__subset,axiom,
! [R5: set_Pr4386577575007340137rm_a_b,S5: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ ( sup_su6776935440552674877rm_a_b @ ( transi7742714808557438673rm_a_b @ R5 ) @ ( transi7742714808557438673rm_a_b @ S5 ) ) @ ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ R5 @ S5 ) ) ) ).
% rtrancl_Un_subset
thf(fact_959_rtrancl__union__subset__rtrancl__union__trancl,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
( ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ R2 @ ( transi7922773638565587891rm_a_b @ S ) ) )
= ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ R2 @ S ) ) ) ).
% rtrancl_union_subset_rtrancl_union_trancl
thf(fact_960_rstep__smycl__dist,axiom,
! [R: set_Pr4386577575007340137rm_a_b] :
( ( sup_su6776935440552674877rm_a_b @ ( rstep_a_b @ R ) @ ( conver7608559976937845093rm_a_b @ ( rstep_a_b @ R ) ) )
= ( rstep_a_b @ ( sup_su6776935440552674877rm_a_b @ R @ ( conver7608559976937845093rm_a_b @ R ) ) ) ) ).
% rstep_smycl_dist
thf(fact_961_sig__step__symcl__dist,axiom,
! [F2: set_Pr4934435412358123699_a_nat,R: set_Pr4386577575007340137rm_a_b] :
( ( sup_su6776935440552674877rm_a_b @ ( sig_step_a_b @ F2 @ R ) @ ( conver7608559976937845093rm_a_b @ ( sig_step_a_b @ F2 @ R ) ) )
= ( sig_step_a_b @ F2 @ ( sup_su6776935440552674877rm_a_b @ R @ ( conver7608559976937845093rm_a_b @ R ) ) ) ) ).
% sig_step_symcl_dist
thf(fact_962_firstStep,axiom,
! [L5: set_Pr4386577575007340137rm_a_b,S5: set_Pr4386577575007340137rm_a_b,R5: set_Pr4386577575007340137rm_a_b,X: term_a_b,Y: term_a_b] :
( ( L5
= ( sup_su6776935440552674877rm_a_b @ S5 @ R5 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ L5 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ R5 ) )
| ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ R5 ) @ ( relcom370159955682700863rm_a_b @ S5 @ ( transi7742714808557438673rm_a_b @ L5 ) ) ) ) ) ) ) ).
% firstStep
thf(fact_963_first__step,axiom,
! [C3: set_Pr4386577575007340137rm_a_b,A4: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b,X: term_a_b,Y: term_a_b,Z2: term_a_b] :
( ( C3
= ( sup_su6776935440552674877rm_a_b @ A4 @ B5 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ C3 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ Z2 ) @ B5 )
=> ? [Y2: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y2 ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ A4 ) @ B5 ) ) ) ) ) ).
% first_step
thf(fact_964_first__step__O,axiom,
! [C3: set_Pr4386577575007340137rm_a_b,A4: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b,X: term_a_b,Y: term_a_b] :
( ( C3
= ( sup_su6776935440552674877rm_a_b @ A4 @ B5 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ C3 ) @ B5 ) )
=> ? [Y2: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y2 ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ A4 ) @ B5 ) ) ) ) ).
% first_step_O
thf(fact_965_right__comp__S,axiom,
! [X: term_a_b,Y: term_a_b,S5: set_Pr4386577575007340137rm_a_b,NS: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( relcom370159955682700863rm_a_b @ S5 @ ( sup_su6776935440552674877rm_a_b @ ( relcom370159955682700863rm_a_b @ S5 @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ S5 ) @ ( transi7742714808557438673rm_a_b @ NS ) ) ) @ ( transi7742714808557438673rm_a_b @ NS ) ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( relcom370159955682700863rm_a_b @ S5 @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ S5 ) @ ( transi7742714808557438673rm_a_b @ NS ) ) ) ) ) ).
% right_comp_S
thf(fact_966_compatible__conv,axiom,
! [NS: set_Pr4386577575007340137rm_a_b,S5: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ NS @ S5 ) @ S5 )
=> ( ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ NS @ S5 ) ) @ ( relcom370159955682700863rm_a_b @ S5 @ ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ NS @ S5 ) ) ) )
= ( relcom370159955682700863rm_a_b @ S5 @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ S5 ) @ ( transi7742714808557438673rm_a_b @ NS ) ) ) ) ) ).
% compatible_conv
thf(fact_967_rtrancl__U__push,axiom,
! [S5: set_Pr4386577575007340137rm_a_b,R5: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ S5 @ R5 ) @ ( relcom370159955682700863rm_a_b @ R5 @ ( transi7742714808557438673rm_a_b @ S5 ) ) )
=> ( ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ S5 @ R5 ) )
= ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ R5 ) @ ( transi7742714808557438673rm_a_b @ S5 ) ) ) ) ).
% rtrancl_U_push
thf(fact_968_rtrancl__Un2__separatorE,axiom,
! [B5: set_Pr4386577575007340137rm_a_b,A4: set_Pr4386577575007340137rm_a_b] :
( ( ( relcom370159955682700863rm_a_b @ B5 @ A4 )
= bot_bo197521221353338581rm_a_b )
=> ( ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ A4 @ B5 ) )
= ( sup_su6776935440552674877rm_a_b @ ( transi7742714808557438673rm_a_b @ A4 ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ A4 ) @ ( transi7742714808557438673rm_a_b @ B5 ) ) ) ) ) ).
% rtrancl_Un2_separatorE
thf(fact_969_conversion__def,axiom,
( abstra1130568886744591910rm_a_b
= ( ^ [A7: set_Pr4386577575007340137rm_a_b] : ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ A7 @ ( conver7608559976937845093rm_a_b @ A7 ) ) ) ) ) ).
% conversion_def
thf(fact_970_srstep__symcl__dist,axiom,
! [F2: set_Pr4934435412358123699_a_nat,R: set_Pr4386577575007340137rm_a_b] :
( ( sup_su6776935440552674877rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R ) ) @ ( conver7608559976937845093rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R ) ) ) )
= ( sig_step_a_b @ F2 @ ( rstep_a_b @ ( sup_su6776935440552674877rm_a_b @ R @ ( conver7608559976937845093rm_a_b @ R ) ) ) ) ) ).
% srstep_symcl_dist
thf(fact_971_compatible__rtrancl__split,axiom,
! [NS: set_Pr4386577575007340137rm_a_b,S5: set_Pr4386577575007340137rm_a_b,X: term_a_b,Y: term_a_b] :
( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ NS @ S5 ) @ S5 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ NS @ S5 ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( sup_su6776935440552674877rm_a_b @ ( relcom370159955682700863rm_a_b @ S5 @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ S5 ) @ ( transi7742714808557438673rm_a_b @ NS ) ) ) @ ( transi7742714808557438673rm_a_b @ NS ) ) ) ) ) ).
% compatible_rtrancl_split
thf(fact_972_conversionE,axiom,
! [A3: term_a_b,B2: term_a_b,A4: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B2 ) @ ( abstra1130568886744591910rm_a_b @ A4 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B2 ) @ ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ A4 @ ( conver7608559976937845093rm_a_b @ A4 ) ) ) ) ) ).
% conversionE
thf(fact_973_sig__step__conversion__dist,axiom,
! [F2: set_Pr4934435412358123699_a_nat,R: set_Pr4386577575007340137rm_a_b] :
( ( abstra1130568886744591910rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R ) ) )
= ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ ( sup_su6776935440552674877rm_a_b @ R @ ( conver7608559976937845093rm_a_b @ R ) ) ) ) ) ) ).
% sig_step_conversion_dist
thf(fact_974_semi__complete__imp__conversionIff__same__NF,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,X: term_a_b,Y: term_a_b] :
( ( abstra7162307272211697846rm_a_b @ R2 @ top_top_set_term_a_b )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( abstra1130568886744591910rm_a_b @ R2 ) )
= ( ! [U3: term_a_b,V4: term_a_b] :
( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ U3 ) @ ( abstra2972651231392952684rm_a_b @ R2 ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ V4 ) @ ( abstra2972651231392952684rm_a_b @ R2 ) ) )
=> ( U3 = V4 ) ) ) ) ) ).
% semi_complete_imp_conversionIff_same_NF
thf(fact_975_quasi__commute__def,axiom,
( abstra4317384144805385751rm_a_b
= ( ^ [R4: set_Pr4386577575007340137rm_a_b,S4: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ S4 @ R4 ) @ ( relcom370159955682700863rm_a_b @ R4 @ ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ R4 @ S4 ) ) ) ) ) ) ).
% quasi_commute_def
thf(fact_976_semi__confluence__imp__CR,axiom,
! [R2: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ ( conver7608559976937845093rm_a_b @ R2 ) @ ( transi7742714808557438673rm_a_b @ R2 ) ) @ ( abstra4096080454567261402rm_a_b @ R2 ) )
=> ( abstra8448919418672941150rm_a_b @ R2 @ top_top_set_term_a_b ) ) ).
% semi_confluence_imp_CR
thf(fact_977_sig__step__rsteps__eq__dist,axiom,
! [R: set_Pr4386577575007340137rm_a_b,F2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ ( terms_7988297476397195622_a_b_b @ R ) @ F2 )
=> ( ( sup_su6776935440552674877rm_a_b @ ( sig_step_a_b @ F2 @ ( transi7922773638565587891rm_a_b @ ( rstep_a_b @ R ) ) ) @ id_term_a_b )
= ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R ) ) ) ) ) ).
% sig_step_rsteps_eq_dist
thf(fact_978_pair__in__Id__conv,axiom,
! [A3: term_a_b,B2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B2 ) @ id_term_a_b )
= ( A3 = B2 ) ) ).
% pair_in_Id_conv
thf(fact_979_IdI,axiom,
! [A3: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ A3 ) @ id_term_a_b ) ).
% IdI
thf(fact_980_rtrancl__reflcl__absorb,axiom,
! [R5: set_Pr4386577575007340137rm_a_b] :
( ( sup_su6776935440552674877rm_a_b @ ( transi7742714808557438673rm_a_b @ R5 ) @ id_term_a_b )
= ( transi7742714808557438673rm_a_b @ R5 ) ) ).
% rtrancl_reflcl_absorb
thf(fact_981_rtrancl__reflcl,axiom,
! [R5: set_Pr4386577575007340137rm_a_b] :
( ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ R5 @ id_term_a_b ) )
= ( transi7742714808557438673rm_a_b @ R5 ) ) ).
% rtrancl_reflcl
thf(fact_982_rtrancl__empty,axiom,
( ( transi7742714808557438673rm_a_b @ bot_bo197521221353338581rm_a_b )
= id_term_a_b ) ).
% rtrancl_empty
thf(fact_983_trancl__reflcl,axiom,
! [R2: set_Pr4386577575007340137rm_a_b] :
( ( transi7922773638565587891rm_a_b @ ( sup_su6776935440552674877rm_a_b @ R2 @ id_term_a_b ) )
= ( transi7742714808557438673rm_a_b @ R2 ) ) ).
% trancl_reflcl
thf(fact_984_CR__onI,axiom,
! [A4: set_Pr4386577575007340137rm_a_b,R2: set_Pr2972776593051762503rm_a_b] :
( ! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,C6: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ A @ A4 )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( transi2615809358984392588rm_a_b @ R2 ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ C6 ) @ ( transi2615809358984392588rm_a_b @ R2 ) )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B @ C6 ) @ ( abstra4889682940153293507rm_a_b @ R2 ) ) ) ) )
=> ( abstra7359694574113289279rm_a_b @ R2 @ A4 ) ) ).
% CR_onI
thf(fact_985_CR__onI,axiom,
! [A4: set_term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ! [A: term_a_b,B: term_a_b,C6: term_a_b] :
( ( member_term_a_b @ A @ A4 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C6 ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C6 ) @ ( abstra4096080454567261402rm_a_b @ R2 ) ) ) ) )
=> ( abstra8448919418672941150rm_a_b @ R2 @ A4 ) ) ).
% CR_onI
thf(fact_986_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_987_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_988_rtrancl__unfold,axiom,
( transi7742714808557438673rm_a_b
= ( ^ [R4: set_Pr4386577575007340137rm_a_b] : ( sup_su6776935440552674877rm_a_b @ id_term_a_b @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ R4 ) @ R4 ) ) ) ) ).
% rtrancl_unfold
thf(fact_989_rtrancl__trancl__reflcl,axiom,
( transi7742714808557438673rm_a_b
= ( ^ [R4: set_Pr4386577575007340137rm_a_b] : ( sup_su6776935440552674877rm_a_b @ ( transi7922773638565587891rm_a_b @ R4 ) @ id_term_a_b ) ) ) ).
% rtrancl_trancl_reflcl
thf(fact_990_CR__on__def,axiom,
( abstra8448919418672941150rm_a_b
= ( ^ [R4: set_Pr4386577575007340137rm_a_b,A7: set_term_a_b] :
! [X2: term_a_b] :
( ( member_term_a_b @ X2 @ A7 )
=> ! [B3: term_a_b,C2: term_a_b] :
( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ B3 ) @ ( transi7742714808557438673rm_a_b @ R4 ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ C2 ) @ ( transi7742714808557438673rm_a_b @ R4 ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B3 @ C2 ) @ ( abstra4096080454567261402rm_a_b @ R4 ) ) ) ) ) ) ).
% CR_on_def
thf(fact_991_CR__onE,axiom,
! [R2: set_Pr2972776593051762503rm_a_b,A4: set_Pr4386577575007340137rm_a_b,A3: produc357393685978478089rm_a_b,B2: produc357393685978478089rm_a_b,C: produc357393685978478089rm_a_b] :
( ( abstra7359694574113289279rm_a_b @ R2 @ A4 )
=> ( ( member5869715511025134514rm_a_b @ A3 @ A4 )
=> ( ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B2 @ C ) @ ( abstra4889682940153293507rm_a_b @ R2 ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A3 @ B2 ) @ ( transi2615809358984392588rm_a_b @ R2 ) )
=> ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A3 @ C ) @ ( transi2615809358984392588rm_a_b @ R2 ) ) ) ) ) ) ).
% CR_onE
thf(fact_992_CR__onE,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,A4: set_term_a_b,A3: term_a_b,B2: term_a_b,C: term_a_b] :
( ( abstra8448919418672941150rm_a_b @ R2 @ A4 )
=> ( ( member_term_a_b @ A3 @ A4 )
=> ( ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ C ) @ ( abstra4096080454567261402rm_a_b @ R2 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B2 ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ C ) @ ( transi7742714808557438673rm_a_b @ R2 ) ) ) ) ) ) ).
% CR_onE
thf(fact_993_CR__onD,axiom,
! [R2: set_Pr2972776593051762503rm_a_b,A4: set_Pr4386577575007340137rm_a_b,A3: produc357393685978478089rm_a_b,B2: produc357393685978478089rm_a_b,C: produc357393685978478089rm_a_b] :
( ( abstra7359694574113289279rm_a_b @ R2 @ A4 )
=> ( ( member5869715511025134514rm_a_b @ A3 @ A4 )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A3 @ B2 ) @ ( transi2615809358984392588rm_a_b @ R2 ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A3 @ C ) @ ( transi2615809358984392588rm_a_b @ R2 ) )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B2 @ C ) @ ( abstra4889682940153293507rm_a_b @ R2 ) ) ) ) ) ) ).
% CR_onD
thf(fact_994_CR__onD,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,A4: set_term_a_b,A3: term_a_b,B2: term_a_b,C: term_a_b] :
( ( abstra8448919418672941150rm_a_b @ R2 @ A4 )
=> ( ( member_term_a_b @ A3 @ A4 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B2 ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ C ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ C ) @ ( abstra4096080454567261402rm_a_b @ R2 ) ) ) ) ) ) ).
% CR_onD
thf(fact_995_rtrancl__Int__subset,axiom,
! [S: set_Pr4386577575007340137rm_a_b,R2: 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 @ R2 ) @ S ) @ R2 ) @ S )
=> ( ord_le118470702582115849rm_a_b @ ( transi7742714808557438673rm_a_b @ R2 ) @ S ) ) ) ).
% rtrancl_Int_subset
thf(fact_996_CR__divergence__imp__join,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,X: term_a_b,Y: term_a_b,Z2: term_a_b] :
( ( abstra8448919418672941150rm_a_b @ R2 @ top_top_set_term_a_b )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z2 ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ Z2 ) @ ( abstra4096080454567261402rm_a_b @ R2 ) ) ) ) ) ).
% CR_divergence_imp_join
thf(fact_997_partially__localize__CR,axiom,
! [R2: set_Pr4386577575007340137rm_a_b] :
( ( abstra8448919418672941150rm_a_b @ R2 @ top_top_set_term_a_b )
= ( ! [X2: term_a_b,Y5: term_a_b,Z4: term_a_b] :
( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y5 ) @ R2 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Z4 ) @ ( transi7742714808557438673rm_a_b @ R2 ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y5 @ Z4 ) @ ( abstra4096080454567261402rm_a_b @ R2 ) ) ) ) ) ).
% partially_localize_CR
thf(fact_998_CR__join__right__I,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,X: term_a_b,Y: term_a_b,Z2: term_a_b] :
( ( abstra8448919418672941150rm_a_b @ R2 @ top_top_set_term_a_b )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( abstra4096080454567261402rm_a_b @ R2 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ Z2 ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z2 ) @ ( abstra4096080454567261402rm_a_b @ R2 ) ) ) ) ) ).
% CR_join_right_I
thf(fact_999_CR__join__left__I,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,X: term_a_b,Y: term_a_b,Z2: term_a_b] :
( ( abstra8448919418672941150rm_a_b @ R2 @ top_top_set_term_a_b )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z2 ) @ ( abstra4096080454567261402rm_a_b @ R2 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ Z2 ) @ ( abstra4096080454567261402rm_a_b @ R2 ) ) ) ) ) ).
% CR_join_left_I
thf(fact_1000_diamond__imp__CR_H,axiom,
! [S: set_Pr4386577575007340137rm_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ( abstra8151087076625346926rm_a_b @ S )
=> ( ( ord_le118470702582115849rm_a_b @ R2 @ S )
=> ( ( ord_le118470702582115849rm_a_b @ S @ ( transi7742714808557438673rm_a_b @ R2 ) )
=> ( abstra8448919418672941150rm_a_b @ R2 @ top_top_set_term_a_b ) ) ) ) ).
% diamond_imp_CR'
thf(fact_1001_strongly__confluentI,axiom,
! [R2: set_Pr4386577575007340137rm_a_b] :
( ! [X3: term_a_b,Y2: term_a_b,Z: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y2 ) @ R2 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Z ) @ R2 )
=> ? [U4: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y2 @ U4 ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Z @ U4 ) @ ( sup_su6776935440552674877rm_a_b @ R2 @ id_term_a_b ) ) ) ) )
=> ( abstra5664096641628173427rm_a_b @ R2 @ top_top_set_term_a_b ) ) ).
% strongly_confluentI
thf(fact_1002_strongly__confluent__on__E11,axiom,
! [R2: set_Pr2972776593051762503rm_a_b,A4: set_Pr4386577575007340137rm_a_b,X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b,Z2: produc357393685978478089rm_a_b] :
( ( abstra837640964651051114rm_a_b @ R2 @ A4 )
=> ( ( member5869715511025134514rm_a_b @ X @ A4 )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ R2 )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Z2 ) @ R2 )
=> ? [U2: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y @ U2 ) @ ( transi2615809358984392588rm_a_b @ R2 ) )
& ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Z2 @ U2 ) @ ( sup_su1845815533690797339rm_a_b @ R2 @ id_Pro3488709451141449061rm_a_b ) ) ) ) ) ) ) ).
% strongly_confluent_on_E11
thf(fact_1003_strongly__confluent__on__E11,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,A4: set_term_a_b,X: term_a_b,Y: term_a_b,Z2: term_a_b] :
( ( abstra5664096641628173427rm_a_b @ R2 @ A4 )
=> ( ( member_term_a_b @ X @ A4 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ R2 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z2 ) @ R2 )
=> ? [U2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ U2 ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Z2 @ U2 ) @ ( sup_su6776935440552674877rm_a_b @ R2 @ id_term_a_b ) ) ) ) ) ) ) ).
% strongly_confluent_on_E11
thf(fact_1004_strongly__confluent__on__def,axiom,
( abstra5664096641628173427rm_a_b
= ( ^ [R4: set_Pr4386577575007340137rm_a_b,A7: set_term_a_b] :
! [X2: term_a_b] :
( ( member_term_a_b @ X2 @ A7 )
=> ! [Y5: term_a_b,Z4: term_a_b] :
( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y5 ) @ R4 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Z4 ) @ R4 ) )
=> ? [U3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y5 @ U3 ) @ ( transi7742714808557438673rm_a_b @ R4 ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Z4 @ U3 ) @ ( sup_su6776935440552674877rm_a_b @ R4 @ id_term_a_b ) ) ) ) ) ) ) ).
% strongly_confluent_on_def
thf(fact_1005_IdD,axiom,
! [A3: term_a_b,B2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B2 ) @ id_term_a_b )
=> ( A3 = B2 ) ) ).
% IdD
thf(fact_1006_strongly__confluent__E1n,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,X: term_a_b,Y: term_a_b,Z2: term_a_b,N: nat] :
( ( abstra5664096641628173427rm_a_b @ R2 @ top_top_set_term_a_b )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( sup_su6776935440552674877rm_a_b @ R2 @ id_term_a_b ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z2 ) @ ( compow4057154403645558940rm_a_b @ N @ R2 ) )
=> ? [U2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ U2 ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Z2 @ U2 ) @ ( sup_su6776935440552674877rm_a_b @ R2 @ id_term_a_b ) ) ) ) ) ) ).
% strongly_confluent_E1n
thf(fact_1007_CR__NF__conv,axiom,
! [R2: set_Pr2972776593051762503rm_a_b,T: produc357393685978478089rm_a_b,U: produc357393685978478089rm_a_b] :
( ( abstra7359694574113289279rm_a_b @ R2 @ top_to1314267278846557113rm_a_b )
=> ( ( member5869715511025134514rm_a_b @ T @ ( abstra1902783451776770197rm_a_b @ R2 ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ U @ T ) @ ( abstra6125656530262095991rm_a_b @ R2 ) )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ U @ T ) @ ( abstra1496011390211842225rm_a_b @ R2 ) ) ) ) ) ).
% CR_NF_conv
thf(fact_1008_CR__NF__conv,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,T: term_a_b,U: term_a_b] :
( ( abstra8448919418672941150rm_a_b @ R2 @ top_top_set_term_a_b )
=> ( ( member_term_a_b @ T @ ( abstract_NF_term_a_b @ R2 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ U @ T ) @ ( abstra1130568886744591910rm_a_b @ R2 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ U @ T ) @ ( abstra2972651231392952684rm_a_b @ R2 ) ) ) ) ) ).
% CR_NF_conv
thf(fact_1009_DiffI,axiom,
! [C: produc357393685978478089rm_a_b,A4: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ C @ A4 )
=> ( ~ ( member5869715511025134514rm_a_b @ C @ B5 )
=> ( member5869715511025134514rm_a_b @ C @ ( minus_5192120951422937424rm_a_b @ A4 @ B5 ) ) ) ) ).
% DiffI
thf(fact_1010_Diff__iff,axiom,
! [C: produc357393685978478089rm_a_b,A4: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ C @ ( minus_5192120951422937424rm_a_b @ A4 @ B5 ) )
= ( ( member5869715511025134514rm_a_b @ C @ A4 )
& ~ ( member5869715511025134514rm_a_b @ C @ B5 ) ) ) ).
% Diff_iff
thf(fact_1011_NF__I,axiom,
! [A3: produc357393685978478089rm_a_b,A4: set_Pr2972776593051762503rm_a_b] :
( ! [B: produc357393685978478089rm_a_b] :
~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A3 @ B ) @ A4 )
=> ( member5869715511025134514rm_a_b @ A3 @ ( abstra1902783451776770197rm_a_b @ A4 ) ) ) ).
% NF_I
thf(fact_1012_NF__I,axiom,
! [A3: term_a_b,A4: set_Pr4386577575007340137rm_a_b] :
( ! [B: term_a_b] :
~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B ) @ A4 )
=> ( member_term_a_b @ A3 @ ( abstract_NF_term_a_b @ A4 ) ) ) ).
% NF_I
thf(fact_1013_normalizability__I,axiom,
! [A3: produc357393685978478089rm_a_b,B2: produc357393685978478089rm_a_b,A4: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A3 @ B2 ) @ ( transi2615809358984392588rm_a_b @ A4 ) )
=> ( ( member5869715511025134514rm_a_b @ B2 @ ( abstra1902783451776770197rm_a_b @ A4 ) )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A3 @ B2 ) @ ( abstra1496011390211842225rm_a_b @ A4 ) ) ) ) ).
% normalizability_I
thf(fact_1014_normalizability__I,axiom,
! [A3: term_a_b,B2: term_a_b,A4: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B2 ) @ ( transi7742714808557438673rm_a_b @ A4 ) )
=> ( ( member_term_a_b @ B2 @ ( abstract_NF_term_a_b @ A4 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B2 ) @ ( abstra2972651231392952684rm_a_b @ A4 ) ) ) ) ).
% normalizability_I
thf(fact_1015_relpow__refl__mono,axiom,
! [Rel: set_Pr4386577575007340137rm_a_b,M: nat,N: nat,A3: term_a_b,B2: 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 @ A3 @ B2 ) @ ( compow4057154403645558940rm_a_b @ M @ Rel ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B2 ) @ ( compow4057154403645558940rm_a_b @ N @ Rel ) ) ) ) ) ).
% relpow_refl_mono
thf(fact_1016_relpow__Suc__D2_H,axiom,
! [N: nat,R5: 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 @ R5 ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y6 @ Z5 ) @ R5 ) )
=> ? [W: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X5 @ W ) @ R5 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ W @ Z5 ) @ ( compow4057154403645558940rm_a_b @ N @ R5 ) ) ) ) ).
% relpow_Suc_D2'
thf(fact_1017_relpow__image,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,F: term_a_b > term_a_b,R9: set_Pr4386577575007340137rm_a_b,S: term_a_b,T: term_a_b,N: nat] :
( ! [S3: term_a_b,T4: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S3 @ T4 ) @ R2 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F @ S3 ) @ ( F @ T4 ) ) @ R9 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( compow4057154403645558940rm_a_b @ N @ R2 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F @ S ) @ ( F @ T ) ) @ ( compow4057154403645558940rm_a_b @ N @ R9 ) ) ) ) ).
% relpow_image
thf(fact_1018_DiffE,axiom,
! [C: produc357393685978478089rm_a_b,A4: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ C @ ( minus_5192120951422937424rm_a_b @ A4 @ B5 ) )
=> ~ ( ( member5869715511025134514rm_a_b @ C @ A4 )
=> ( member5869715511025134514rm_a_b @ C @ B5 ) ) ) ).
% DiffE
thf(fact_1019_DiffD1,axiom,
! [C: produc357393685978478089rm_a_b,A4: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ C @ ( minus_5192120951422937424rm_a_b @ A4 @ B5 ) )
=> ( member5869715511025134514rm_a_b @ C @ A4 ) ) ).
% DiffD1
thf(fact_1020_DiffD2,axiom,
! [C: produc357393685978478089rm_a_b,A4: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ C @ ( minus_5192120951422937424rm_a_b @ A4 @ B5 ) )
=> ~ ( member5869715511025134514rm_a_b @ C @ B5 ) ) ).
% DiffD2
thf(fact_1021_NF__E,axiom,
! [A3: produc357393685978478089rm_a_b,A4: set_Pr2972776593051762503rm_a_b,B2: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ A3 @ ( abstra1902783451776770197rm_a_b @ A4 ) )
=> ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A3 @ B2 ) @ A4 ) ) ).
% NF_E
thf(fact_1022_NF__E,axiom,
! [A3: term_a_b,A4: set_Pr4386577575007340137rm_a_b,B2: term_a_b] :
( ( member_term_a_b @ A3 @ ( abstract_NF_term_a_b @ A4 ) )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B2 ) @ A4 ) ) ).
% NF_E
thf(fact_1023_NF__no__step,axiom,
! [A3: produc357393685978478089rm_a_b,A4: set_Pr2972776593051762503rm_a_b] :
( ( member5869715511025134514rm_a_b @ A3 @ ( abstra1902783451776770197rm_a_b @ A4 ) )
=> ! [B7: produc357393685978478089rm_a_b] :
~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A3 @ B7 ) @ A4 ) ) ).
% NF_no_step
thf(fact_1024_NF__no__step,axiom,
! [A3: term_a_b,A4: set_Pr4386577575007340137rm_a_b] :
( ( member_term_a_b @ A3 @ ( abstract_NF_term_a_b @ A4 ) )
=> ! [B7: term_a_b] :
~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B7 ) @ A4 ) ) ).
% NF_no_step
thf(fact_1025_NF__iff__no__step,axiom,
! [A3: produc357393685978478089rm_a_b,A4: set_Pr2972776593051762503rm_a_b] :
( ( member5869715511025134514rm_a_b @ A3 @ ( abstra1902783451776770197rm_a_b @ A4 ) )
= ( ! [B3: produc357393685978478089rm_a_b] :
~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A3 @ B3 ) @ A4 ) ) ) ).
% NF_iff_no_step
thf(fact_1026_NF__iff__no__step,axiom,
! [A3: term_a_b,A4: set_Pr4386577575007340137rm_a_b] :
( ( member_term_a_b @ A3 @ ( abstract_NF_term_a_b @ A4 ) )
= ( ! [B3: term_a_b] :
~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B3 ) @ A4 ) ) ) ).
% NF_iff_no_step
thf(fact_1027_rtrancl__power,axiom,
! [P: produc357393685978478089rm_a_b,R5: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ P @ ( transi7742714808557438673rm_a_b @ R5 ) )
= ( ? [N2: nat] : ( member5869715511025134514rm_a_b @ P @ ( compow4057154403645558940rm_a_b @ N2 @ R5 ) ) ) ) ).
% rtrancl_power
thf(fact_1028_relpow__imp__rtrancl,axiom,
! [P: produc357393685978478089rm_a_b,N: nat,R5: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ P @ ( compow4057154403645558940rm_a_b @ N @ R5 ) )
=> ( member5869715511025134514rm_a_b @ P @ ( transi7742714808557438673rm_a_b @ R5 ) ) ) ).
% relpow_imp_rtrancl
thf(fact_1029_rtrancl__imp__relpow,axiom,
! [P: produc357393685978478089rm_a_b,R5: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ P @ ( transi7742714808557438673rm_a_b @ R5 ) )
=> ? [N3: nat] : ( member5869715511025134514rm_a_b @ P @ ( compow4057154403645558940rm_a_b @ N3 @ R5 ) ) ) ).
% rtrancl_imp_relpow
thf(fact_1030_rtrancl__len__E,axiom,
! [X: term_a_b,Y: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
=> ~ ! [N3: nat] :
~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( compow4057154403645558940rm_a_b @ N3 @ R2 ) ) ) ).
% rtrancl_len_E
thf(fact_1031_rtrancl__r__diff__Id,axiom,
! [R2: set_Pr4386577575007340137rm_a_b] :
( ( transi7742714808557438673rm_a_b @ ( minus_5192120951422937424rm_a_b @ R2 @ id_term_a_b ) )
= ( transi7742714808557438673rm_a_b @ R2 ) ) ).
% rtrancl_r_diff_Id
thf(fact_1032_NF__not__suc,axiom,
! [X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b,R2: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ ( transi2615809358984392588rm_a_b @ R2 ) )
=> ( ( member5869715511025134514rm_a_b @ X @ ( abstra1902783451776770197rm_a_b @ R2 ) )
=> ( X = Y ) ) ) ).
% NF_not_suc
thf(fact_1033_NF__not__suc,axiom,
! [X: term_a_b,Y: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
=> ( ( member_term_a_b @ X @ ( abstract_NF_term_a_b @ R2 ) )
=> ( X = Y ) ) ) ).
% NF_not_suc
thf(fact_1034_NF__no__trancl__step,axiom,
! [A3: produc357393685978478089rm_a_b,A4: set_Pr2972776593051762503rm_a_b] :
( ( member5869715511025134514rm_a_b @ A3 @ ( abstra1902783451776770197rm_a_b @ A4 ) )
=> ! [B7: produc357393685978478089rm_a_b] :
~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A3 @ B7 ) @ ( transi9211502839322181930rm_a_b @ A4 ) ) ) ).
% NF_no_trancl_step
thf(fact_1035_NF__no__trancl__step,axiom,
! [A3: term_a_b,A4: set_Pr4386577575007340137rm_a_b] :
( ( member_term_a_b @ A3 @ ( abstract_NF_term_a_b @ A4 ) )
=> ! [B7: term_a_b] :
~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B7 ) @ ( transi7922773638565587891rm_a_b @ A4 ) ) ) ).
% NF_no_trancl_step
thf(fact_1036_join__NF__imp__eq,axiom,
! [X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b,R2: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ ( abstra4889682940153293507rm_a_b @ R2 ) )
=> ( ( member5869715511025134514rm_a_b @ X @ ( abstra1902783451776770197rm_a_b @ R2 ) )
=> ( ( member5869715511025134514rm_a_b @ Y @ ( abstra1902783451776770197rm_a_b @ R2 ) )
=> ( X = Y ) ) ) ) ).
% join_NF_imp_eq
thf(fact_1037_join__NF__imp__eq,axiom,
! [X: term_a_b,Y: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( abstra4096080454567261402rm_a_b @ R2 ) )
=> ( ( member_term_a_b @ X @ ( abstract_NF_term_a_b @ R2 ) )
=> ( ( member_term_a_b @ Y @ ( abstract_NF_term_a_b @ R2 ) )
=> ( X = Y ) ) ) ) ).
% join_NF_imp_eq
thf(fact_1038_trancl__steps__relpow,axiom,
! [A3: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b,X: term_a_b,Y: term_a_b,N: nat] :
( ( ord_le118470702582115849rm_a_b @ A3 @ ( transi7922773638565587891rm_a_b @ B2 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( compow4057154403645558940rm_a_b @ N @ A3 ) )
=> ? [M2: nat] :
( ( ord_less_eq_nat @ N @ M2 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( compow4057154403645558940rm_a_b @ M2 @ B2 ) ) ) ) ) ).
% trancl_steps_relpow
thf(fact_1039_rtrancl__diff__decomp,axiom,
! [X: term_a_b,Y: term_a_b,A4: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( minus_5192120951422937424rm_a_b @ ( transi7742714808557438673rm_a_b @ A4 ) @ ( transi7742714808557438673rm_a_b @ B5 ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ A4 ) @ ( relcom370159955682700863rm_a_b @ ( minus_5192120951422937424rm_a_b @ A4 @ B5 ) @ ( transi7742714808557438673rm_a_b @ A4 ) ) ) ) ) ).
% rtrancl_diff_decomp
thf(fact_1040_NF__join__imp__reach,axiom,
! [Y: produc357393685978478089rm_a_b,A4: set_Pr2972776593051762503rm_a_b,X: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ Y @ ( abstra1902783451776770197rm_a_b @ A4 ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ ( abstra4889682940153293507rm_a_b @ A4 ) )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ ( transi2615809358984392588rm_a_b @ A4 ) ) ) ) ).
% NF_join_imp_reach
thf(fact_1041_NF__join__imp__reach,axiom,
! [Y: term_a_b,A4: set_Pr4386577575007340137rm_a_b,X: term_a_b] :
( ( member_term_a_b @ Y @ ( abstract_NF_term_a_b @ A4 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( abstra4096080454567261402rm_a_b @ A4 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ A4 ) ) ) ) ).
% NF_join_imp_reach
thf(fact_1042_normalizability__E,axiom,
! [A3: produc357393685978478089rm_a_b,B2: produc357393685978478089rm_a_b,A4: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A3 @ B2 ) @ ( abstra1496011390211842225rm_a_b @ A4 ) )
=> ~ ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A3 @ B2 ) @ ( transi2615809358984392588rm_a_b @ A4 ) )
=> ~ ( member5869715511025134514rm_a_b @ B2 @ ( abstra1902783451776770197rm_a_b @ A4 ) ) ) ) ).
% normalizability_E
thf(fact_1043_normalizability__E,axiom,
! [A3: term_a_b,B2: term_a_b,A4: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B2 ) @ ( abstra2972651231392952684rm_a_b @ A4 ) )
=> ~ ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B2 ) @ ( transi7742714808557438673rm_a_b @ A4 ) )
=> ~ ( member_term_a_b @ B2 @ ( abstract_NF_term_a_b @ A4 ) ) ) ) ).
% normalizability_E
thf(fact_1044_some__NF__UNF,axiom,
! [R2: set_Pr2972776593051762503rm_a_b,X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b] :
( ( abstra1303798319023342519rm_a_b @ R2 @ top_to1314267278846557113rm_a_b )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ ( transi2615809358984392588rm_a_b @ R2 ) )
=> ( ( member5869715511025134514rm_a_b @ Y @ ( abstra1902783451776770197rm_a_b @ R2 ) )
=> ( Y
= ( abstra8203652949708803681rm_a_b @ R2 @ X ) ) ) ) ) ).
% some_NF_UNF
thf(fact_1045_some__NF__UNF,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,X: term_a_b,Y: term_a_b] :
( ( abstra3619855735679963878rm_a_b @ R2 @ top_top_set_term_a_b )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
=> ( ( member_term_a_b @ Y @ ( abstract_NF_term_a_b @ R2 ) )
=> ( Y
= ( abstra2589370493620291516rm_a_b @ R2 @ X ) ) ) ) ) ).
% some_NF_UNF
thf(fact_1046_UNC__E,axiom,
! [A4: set_Pr2972776593051762503rm_a_b,A3: produc357393685978478089rm_a_b,B2: produc357393685978478089rm_a_b] :
( ( abstra8480497335821202873rm_a_b @ A4 )
=> ( ( A3 != B2 )
=> ( ( member5869715511025134514rm_a_b @ A3 @ ( abstra1902783451776770197rm_a_b @ A4 ) )
=> ( ( member5869715511025134514rm_a_b @ B2 @ ( abstra1902783451776770197rm_a_b @ A4 ) )
=> ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A3 @ B2 ) @ ( abstra6125656530262095991rm_a_b @ A4 ) ) ) ) ) ) ).
% UNC_E
thf(fact_1047_UNC__E,axiom,
! [A4: set_Pr4386577575007340137rm_a_b,A3: term_a_b,B2: term_a_b] :
( ( abstra4649771476585718372rm_a_b @ A4 )
=> ( ( A3 != B2 )
=> ( ( member_term_a_b @ A3 @ ( abstract_NF_term_a_b @ A4 ) )
=> ( ( member_term_a_b @ B2 @ ( abstract_NF_term_a_b @ A4 ) )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B2 ) @ ( abstra1130568886744591910rm_a_b @ A4 ) ) ) ) ) ) ).
% UNC_E
thf(fact_1048_UNC__I,axiom,
! [A4: set_Pr2972776593051762503rm_a_b] :
( ! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ A @ ( abstra1902783451776770197rm_a_b @ A4 ) )
=> ( ( member5869715511025134514rm_a_b @ B @ ( abstra1902783451776770197rm_a_b @ A4 ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( abstra6125656530262095991rm_a_b @ A4 ) )
=> ( A = B ) ) ) )
=> ( abstra8480497335821202873rm_a_b @ A4 ) ) ).
% UNC_I
thf(fact_1049_UNC__I,axiom,
! [A4: set_Pr4386577575007340137rm_a_b] :
( ! [A: term_a_b,B: term_a_b] :
( ( member_term_a_b @ A @ ( abstract_NF_term_a_b @ A4 ) )
=> ( ( member_term_a_b @ B @ ( abstract_NF_term_a_b @ A4 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( abstra1130568886744591910rm_a_b @ A4 ) )
=> ( A = B ) ) ) )
=> ( abstra4649771476585718372rm_a_b @ A4 ) ) ).
% UNC_I
thf(fact_1050_UNC__def,axiom,
( abstra8480497335821202873rm_a_b
= ( ^ [A7: set_Pr2972776593051762503rm_a_b] :
! [A5: produc357393685978478089rm_a_b,B3: produc357393685978478089rm_a_b] :
( ( ( member5869715511025134514rm_a_b @ A5 @ ( abstra1902783451776770197rm_a_b @ A7 ) )
& ( member5869715511025134514rm_a_b @ B3 @ ( abstra1902783451776770197rm_a_b @ A7 ) )
& ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A5 @ B3 ) @ ( abstra6125656530262095991rm_a_b @ A7 ) ) )
=> ( A5 = B3 ) ) ) ) ).
% UNC_def
thf(fact_1051_UNC__def,axiom,
( abstra4649771476585718372rm_a_b
= ( ^ [A7: set_Pr4386577575007340137rm_a_b] :
! [A5: term_a_b,B3: term_a_b] :
( ( ( member_term_a_b @ A5 @ ( abstract_NF_term_a_b @ A7 ) )
& ( member_term_a_b @ B3 @ ( abstract_NF_term_a_b @ A7 ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A5 @ B3 ) @ ( abstra1130568886744591910rm_a_b @ A7 ) ) )
=> ( A5 = B3 ) ) ) ) ).
% UNC_def
thf(fact_1052_CR__on__singletonI,axiom,
! [A3: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ! [B: term_a_b,C6: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ C6 ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C6 ) @ ( abstra4096080454567261402rm_a_b @ R2 ) ) ) )
=> ( abstra8448919418672941150rm_a_b @ R2 @ ( insert_term_a_b @ A3 @ bot_bot_set_term_a_b ) ) ) ).
% CR_on_singletonI
thf(fact_1053_compatible__SN_H,axiom,
! [NS: set_Pr4386577575007340137rm_a_b,S5: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ NS @ S5 ) @ S5 )
=> ( ( abstra4720023341729745482rm_a_b @ S5 @ top_top_set_term_a_b )
=> ( abstra4720023341729745482rm_a_b @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ NS @ S5 ) ) @ ( relcom370159955682700863rm_a_b @ S5 @ ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ NS @ S5 ) ) ) ) @ top_top_set_term_a_b ) ) ) ).
% compatible_SN'
thf(fact_1054_qc__SN__relto__iff,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ R2 @ S ) @ ( relcom370159955682700863rm_a_b @ S @ ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ S @ R2 ) ) ) )
=> ( ( abstra4720023341729745482rm_a_b @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ R2 ) @ ( relcom370159955682700863rm_a_b @ S @ ( transi7742714808557438673rm_a_b @ R2 ) ) ) @ top_top_set_term_a_b )
= ( abstra4720023341729745482rm_a_b @ S @ top_top_set_term_a_b ) ) ) ).
% qc_SN_relto_iff
thf(fact_1055_insertCI,axiom,
! [A3: produc357393685978478089rm_a_b,B5: set_Pr4386577575007340137rm_a_b,B2: produc357393685978478089rm_a_b] :
( ( ~ ( member5869715511025134514rm_a_b @ A3 @ B5 )
=> ( A3 = B2 ) )
=> ( member5869715511025134514rm_a_b @ A3 @ ( insert7009541432154983385rm_a_b @ B2 @ B5 ) ) ) ).
% insertCI
thf(fact_1056_insert__iff,axiom,
! [A3: produc357393685978478089rm_a_b,B2: produc357393685978478089rm_a_b,A4: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ A3 @ ( insert7009541432154983385rm_a_b @ B2 @ A4 ) )
= ( ( A3 = B2 )
| ( member5869715511025134514rm_a_b @ A3 @ A4 ) ) ) ).
% insert_iff
thf(fact_1057_singletonI,axiom,
! [A3: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ A3 @ ( insert7009541432154983385rm_a_b @ A3 @ bot_bo197521221353338581rm_a_b ) ) ).
% singletonI
thf(fact_1058_insert__subset,axiom,
! [X: produc357393685978478089rm_a_b,A4: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ ( insert7009541432154983385rm_a_b @ X @ A4 ) @ B5 )
= ( ( member5869715511025134514rm_a_b @ X @ B5 )
& ( ord_le118470702582115849rm_a_b @ A4 @ B5 ) ) ) ).
% insert_subset
thf(fact_1059_Int__insert__right__if1,axiom,
! [A3: produc357393685978478089rm_a_b,A4: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ A3 @ A4 )
=> ( ( inf_in1693821330460682071rm_a_b @ A4 @ ( insert7009541432154983385rm_a_b @ A3 @ B5 ) )
= ( insert7009541432154983385rm_a_b @ A3 @ ( inf_in1693821330460682071rm_a_b @ A4 @ B5 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_1060_Int__insert__right__if0,axiom,
! [A3: produc357393685978478089rm_a_b,A4: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
( ~ ( member5869715511025134514rm_a_b @ A3 @ A4 )
=> ( ( inf_in1693821330460682071rm_a_b @ A4 @ ( insert7009541432154983385rm_a_b @ A3 @ B5 ) )
= ( inf_in1693821330460682071rm_a_b @ A4 @ B5 ) ) ) ).
% Int_insert_right_if0
thf(fact_1061_Int__insert__left__if1,axiom,
! [A3: produc357393685978478089rm_a_b,C3: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ A3 @ C3 )
=> ( ( inf_in1693821330460682071rm_a_b @ ( insert7009541432154983385rm_a_b @ A3 @ B5 ) @ C3 )
= ( insert7009541432154983385rm_a_b @ A3 @ ( inf_in1693821330460682071rm_a_b @ B5 @ C3 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_1062_Int__insert__left__if0,axiom,
! [A3: produc357393685978478089rm_a_b,C3: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
( ~ ( member5869715511025134514rm_a_b @ A3 @ C3 )
=> ( ( inf_in1693821330460682071rm_a_b @ ( insert7009541432154983385rm_a_b @ A3 @ B5 ) @ C3 )
= ( inf_in1693821330460682071rm_a_b @ B5 @ C3 ) ) ) ).
% Int_insert_left_if0
thf(fact_1063_Diff__insert0,axiom,
! [X: produc357393685978478089rm_a_b,A4: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
( ~ ( member5869715511025134514rm_a_b @ X @ A4 )
=> ( ( minus_5192120951422937424rm_a_b @ A4 @ ( insert7009541432154983385rm_a_b @ X @ B5 ) )
= ( minus_5192120951422937424rm_a_b @ A4 @ B5 ) ) ) ).
% Diff_insert0
thf(fact_1064_insert__Diff1,axiom,
! [X: produc357393685978478089rm_a_b,B5: set_Pr4386577575007340137rm_a_b,A4: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ X @ B5 )
=> ( ( minus_5192120951422937424rm_a_b @ ( insert7009541432154983385rm_a_b @ X @ A4 ) @ B5 )
= ( minus_5192120951422937424rm_a_b @ A4 @ B5 ) ) ) ).
% insert_Diff1
thf(fact_1065_disjoint__insert_I2_J,axiom,
! [A4: set_Pr4386577575007340137rm_a_b,B2: produc357393685978478089rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
( ( bot_bo197521221353338581rm_a_b
= ( inf_in1693821330460682071rm_a_b @ A4 @ ( insert7009541432154983385rm_a_b @ B2 @ B5 ) ) )
= ( ~ ( member5869715511025134514rm_a_b @ B2 @ A4 )
& ( bot_bo197521221353338581rm_a_b
= ( inf_in1693821330460682071rm_a_b @ A4 @ B5 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_1066_disjoint__insert_I1_J,axiom,
! [B5: set_Pr4386577575007340137rm_a_b,A3: produc357393685978478089rm_a_b,A4: set_Pr4386577575007340137rm_a_b] :
( ( ( inf_in1693821330460682071rm_a_b @ B5 @ ( insert7009541432154983385rm_a_b @ A3 @ A4 ) )
= bot_bo197521221353338581rm_a_b )
= ( ~ ( member5869715511025134514rm_a_b @ A3 @ B5 )
& ( ( inf_in1693821330460682071rm_a_b @ B5 @ A4 )
= bot_bo197521221353338581rm_a_b ) ) ) ).
% disjoint_insert(1)
thf(fact_1067_insert__disjoint_I2_J,axiom,
! [A3: produc357393685978478089rm_a_b,A4: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
( ( bot_bo197521221353338581rm_a_b
= ( inf_in1693821330460682071rm_a_b @ ( insert7009541432154983385rm_a_b @ A3 @ A4 ) @ B5 ) )
= ( ~ ( member5869715511025134514rm_a_b @ A3 @ B5 )
& ( bot_bo197521221353338581rm_a_b
= ( inf_in1693821330460682071rm_a_b @ A4 @ B5 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_1068_insert__disjoint_I1_J,axiom,
! [A3: produc357393685978478089rm_a_b,A4: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
( ( ( inf_in1693821330460682071rm_a_b @ ( insert7009541432154983385rm_a_b @ A3 @ A4 ) @ B5 )
= bot_bo197521221353338581rm_a_b )
= ( ~ ( member5869715511025134514rm_a_b @ A3 @ B5 )
& ( ( inf_in1693821330460682071rm_a_b @ A4 @ B5 )
= bot_bo197521221353338581rm_a_b ) ) ) ).
% insert_disjoint(1)
thf(fact_1069_insert__Diff__if,axiom,
! [X: produc357393685978478089rm_a_b,B5: set_Pr4386577575007340137rm_a_b,A4: set_Pr4386577575007340137rm_a_b] :
( ( ( member5869715511025134514rm_a_b @ X @ B5 )
=> ( ( minus_5192120951422937424rm_a_b @ ( insert7009541432154983385rm_a_b @ X @ A4 ) @ B5 )
= ( minus_5192120951422937424rm_a_b @ A4 @ B5 ) ) )
& ( ~ ( member5869715511025134514rm_a_b @ X @ B5 )
=> ( ( minus_5192120951422937424rm_a_b @ ( insert7009541432154983385rm_a_b @ X @ A4 ) @ B5 )
= ( insert7009541432154983385rm_a_b @ X @ ( minus_5192120951422937424rm_a_b @ A4 @ B5 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_1070_subset__insert,axiom,
! [X: produc357393685978478089rm_a_b,A4: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
( ~ ( member5869715511025134514rm_a_b @ X @ A4 )
=> ( ( ord_le118470702582115849rm_a_b @ A4 @ ( insert7009541432154983385rm_a_b @ X @ B5 ) )
= ( ord_le118470702582115849rm_a_b @ A4 @ B5 ) ) ) ).
% subset_insert
thf(fact_1071_insert__subsetI,axiom,
! [X: produc357393685978478089rm_a_b,A4: set_Pr4386577575007340137rm_a_b,X4: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ X @ A4 )
=> ( ( ord_le118470702582115849rm_a_b @ X4 @ A4 )
=> ( ord_le118470702582115849rm_a_b @ ( insert7009541432154983385rm_a_b @ X @ X4 ) @ A4 ) ) ) ).
% insert_subsetI
thf(fact_1072_steps__reflect__SN__on,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,B2: term_a_b,A3: term_a_b] :
( ~ ( abstra4720023341729745482rm_a_b @ R2 @ ( insert_term_a_b @ B2 @ bot_bot_set_term_a_b ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B2 ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
=> ~ ( abstra4720023341729745482rm_a_b @ R2 @ ( insert_term_a_b @ A3 @ bot_bot_set_term_a_b ) ) ) ) ).
% steps_reflect_SN_on
thf(fact_1073_steps__preserve__SN__on,axiom,
! [A3: term_a_b,B2: term_a_b,A4: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B2 ) @ ( transi7742714808557438673rm_a_b @ A4 ) )
=> ( ( abstra4720023341729745482rm_a_b @ A4 @ ( insert_term_a_b @ A3 @ bot_bot_set_term_a_b ) )
=> ( abstra4720023341729745482rm_a_b @ A4 @ ( insert_term_a_b @ B2 @ bot_bot_set_term_a_b ) ) ) ) ).
% steps_preserve_SN_on
thf(fact_1074_SN__on__imp__normalizability,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,A3: term_a_b] :
( ( abstra4720023341729745482rm_a_b @ R2 @ ( insert_term_a_b @ A3 @ bot_bot_set_term_a_b ) )
=> ? [B: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B ) @ ( abstra2972651231392952684rm_a_b @ R2 ) ) ) ).
% SN_on_imp_normalizability
thf(fact_1075_SN__on__induct_H,axiom,
! [R5: set_Pr4386577575007340137rm_a_b,S: term_a_b,P2: term_a_b > $o] :
( ( abstra4720023341729745482rm_a_b @ R5 @ ( insert_term_a_b @ S @ bot_bot_set_term_a_b ) )
=> ( ! [T4: term_a_b] :
( ! [U4: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T4 @ U4 ) @ R5 )
=> ( P2 @ U4 ) )
=> ( P2 @ T4 ) )
=> ( P2 @ S ) ) ) ).
% SN_on_induct'
thf(fact_1076_step__reflects__SN__on,axiom,
! [A3: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ! [B: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B ) @ R2 )
=> ( abstra4720023341729745482rm_a_b @ R2 @ ( insert_term_a_b @ B @ bot_bot_set_term_a_b ) ) )
=> ( abstra4720023341729745482rm_a_b @ R2 @ ( insert_term_a_b @ A3 @ bot_bot_set_term_a_b ) ) ) ).
% step_reflects_SN_on
thf(fact_1077_SN__on__imp__on__minimal,axiom,
! [R2: set_Pr2972776593051762503rm_a_b,X: produc357393685978478089rm_a_b] :
( ( abstra2398554102055911763rm_a_b @ R2 @ ( 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 ) @ R2 )
=> ~ ( member5869715511025134514rm_a_b @ Y6 @ Q2 ) ) ) ) ) ).
% SN_on_imp_on_minimal
thf(fact_1078_SN__on__imp__on__minimal,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,X: term_a_b] :
( ( abstra4720023341729745482rm_a_b @ R2 @ ( 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 ) @ R2 )
=> ~ ( member_term_a_b @ Y6 @ Q2 ) ) ) ) ) ).
% SN_on_imp_on_minimal
thf(fact_1079_step__preserves__SN__on,axiom,
! [A3: term_a_b,B2: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B2 ) @ R2 )
=> ( ( abstra4720023341729745482rm_a_b @ R2 @ ( insert_term_a_b @ A3 @ bot_bot_set_term_a_b ) )
=> ( abstra4720023341729745482rm_a_b @ R2 @ ( insert_term_a_b @ B2 @ bot_bot_set_term_a_b ) ) ) ) ).
% step_preserves_SN_on
thf(fact_1080_SN__on__induct__acc__style,axiom,
! [R5: set_Pr4386577575007340137rm_a_b,A3: term_a_b,P2: term_a_b > $o] :
( ( abstra4720023341729745482rm_a_b @ R5 @ ( insert_term_a_b @ A3 @ bot_bot_set_term_a_b ) )
=> ( ! [X3: term_a_b] :
( ( abstra4720023341729745482rm_a_b @ R5 @ ( insert_term_a_b @ X3 @ bot_bot_set_term_a_b ) )
=> ( ! [Y6: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y6 ) @ R5 )
=> ( P2 @ Y6 ) )
=> ( P2 @ X3 ) ) )
=> ( P2 @ A3 ) ) ) ).
% SN_on_induct_acc_style
thf(fact_1081_lower__set__imp__not__SN__on,axiom,
! [S: produc357393685978478089rm_a_b,X4: set_Pr4386577575007340137rm_a_b,R5: 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 ) @ R5 ) ) )
=> ~ ( abstra2398554102055911763rm_a_b @ R5 @ ( insert7009541432154983385rm_a_b @ S @ bot_bo197521221353338581rm_a_b ) ) ) ) ).
% lower_set_imp_not_SN_on
thf(fact_1082_lower__set__imp__not__SN__on,axiom,
! [S: term_a_b,X4: set_term_a_b,R5: 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 ) @ R5 ) ) )
=> ~ ( abstra4720023341729745482rm_a_b @ R5 @ ( insert_term_a_b @ S @ bot_bot_set_term_a_b ) ) ) ) ).
% lower_set_imp_not_SN_on
thf(fact_1083_SN__on__all__reducts__SN__on__conv,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,A3: term_a_b] :
( ( abstra4720023341729745482rm_a_b @ R2 @ ( insert_term_a_b @ A3 @ bot_bot_set_term_a_b ) )
= ( ! [B3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B3 ) @ R2 )
=> ( abstra4720023341729745482rm_a_b @ R2 @ ( insert_term_a_b @ B3 @ bot_bot_set_term_a_b ) ) ) ) ) ).
% SN_on_all_reducts_SN_on_conv
thf(fact_1084_insertE,axiom,
! [A3: produc357393685978478089rm_a_b,B2: produc357393685978478089rm_a_b,A4: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ A3 @ ( insert7009541432154983385rm_a_b @ B2 @ A4 ) )
=> ( ( A3 != B2 )
=> ( member5869715511025134514rm_a_b @ A3 @ A4 ) ) ) ).
% insertE
thf(fact_1085_insertI1,axiom,
! [A3: produc357393685978478089rm_a_b,B5: set_Pr4386577575007340137rm_a_b] : ( member5869715511025134514rm_a_b @ A3 @ ( insert7009541432154983385rm_a_b @ A3 @ B5 ) ) ).
% insertI1
thf(fact_1086_insertI2,axiom,
! [A3: produc357393685978478089rm_a_b,B5: set_Pr4386577575007340137rm_a_b,B2: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ A3 @ B5 )
=> ( member5869715511025134514rm_a_b @ A3 @ ( insert7009541432154983385rm_a_b @ B2 @ B5 ) ) ) ).
% insertI2
thf(fact_1087_Set_Oset__insert,axiom,
! [X: produc357393685978478089rm_a_b,A4: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ X @ A4 )
=> ~ ! [B9: set_Pr4386577575007340137rm_a_b] :
( ( A4
= ( insert7009541432154983385rm_a_b @ X @ B9 ) )
=> ( member5869715511025134514rm_a_b @ X @ B9 ) ) ) ).
% Set.set_insert
thf(fact_1088_insert__ident,axiom,
! [X: produc357393685978478089rm_a_b,A4: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
( ~ ( member5869715511025134514rm_a_b @ X @ A4 )
=> ( ~ ( member5869715511025134514rm_a_b @ X @ B5 )
=> ( ( ( insert7009541432154983385rm_a_b @ X @ A4 )
= ( insert7009541432154983385rm_a_b @ X @ B5 ) )
= ( A4 = B5 ) ) ) ) ).
% insert_ident
thf(fact_1089_insert__absorb,axiom,
! [A3: produc357393685978478089rm_a_b,A4: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ A3 @ A4 )
=> ( ( insert7009541432154983385rm_a_b @ A3 @ A4 )
= A4 ) ) ).
% insert_absorb
thf(fact_1090_insert__eq__iff,axiom,
! [A3: produc357393685978478089rm_a_b,A4: set_Pr4386577575007340137rm_a_b,B2: produc357393685978478089rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
( ~ ( member5869715511025134514rm_a_b @ A3 @ A4 )
=> ( ~ ( member5869715511025134514rm_a_b @ B2 @ B5 )
=> ( ( ( insert7009541432154983385rm_a_b @ A3 @ A4 )
= ( insert7009541432154983385rm_a_b @ B2 @ B5 ) )
= ( ( ( A3 = B2 )
=> ( A4 = B5 ) )
& ( ( A3 != B2 )
=> ? [C5: set_Pr4386577575007340137rm_a_b] :
( ( A4
= ( insert7009541432154983385rm_a_b @ B2 @ C5 ) )
& ~ ( member5869715511025134514rm_a_b @ B2 @ C5 )
& ( B5
= ( insert7009541432154983385rm_a_b @ A3 @ C5 ) )
& ~ ( member5869715511025134514rm_a_b @ A3 @ C5 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_1091_mk__disjoint__insert,axiom,
! [A3: produc357393685978478089rm_a_b,A4: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ A3 @ A4 )
=> ? [B9: set_Pr4386577575007340137rm_a_b] :
( ( A4
= ( insert7009541432154983385rm_a_b @ A3 @ B9 ) )
& ~ ( member5869715511025134514rm_a_b @ A3 @ B9 ) ) ) ).
% mk_disjoint_insert
thf(fact_1092_SN__on__induct,axiom,
! [R5: set_Pr2972776593051762503rm_a_b,A4: set_Pr4386577575007340137rm_a_b,S: produc357393685978478089rm_a_b,P2: produc357393685978478089rm_a_b > $o] :
( ( abstra2398554102055911763rm_a_b @ R5 @ A4 )
=> ( ( member5869715511025134514rm_a_b @ S @ A4 )
=> ( ! [T4: produc357393685978478089rm_a_b] :
( ! [U4: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ T4 @ U4 ) @ R5 )
=> ( P2 @ U4 ) )
=> ( P2 @ T4 ) )
=> ( P2 @ S ) ) ) ) ).
% SN_on_induct
thf(fact_1093_SN__on__induct,axiom,
! [R5: set_Pr4386577575007340137rm_a_b,A4: set_term_a_b,S: term_a_b,P2: term_a_b > $o] :
( ( abstra4720023341729745482rm_a_b @ R5 @ A4 )
=> ( ( member_term_a_b @ S @ A4 )
=> ( ! [T4: term_a_b] :
( ! [U4: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T4 @ U4 ) @ R5 )
=> ( P2 @ U4 ) )
=> ( P2 @ T4 ) )
=> ( P2 @ S ) ) ) ) ).
% SN_on_induct
thf(fact_1094_SN__on__irrefl,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,A4: set_term_a_b] :
( ( abstra4720023341729745482rm_a_b @ R2 @ A4 )
=> ! [X5: term_a_b] :
( ( member_term_a_b @ X5 @ A4 )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X5 @ X5 ) @ R2 ) ) ) ).
% SN_on_irrefl
thf(fact_1095_Int__insert__right,axiom,
! [A3: produc357393685978478089rm_a_b,A4: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
( ( ( member5869715511025134514rm_a_b @ A3 @ A4 )
=> ( ( inf_in1693821330460682071rm_a_b @ A4 @ ( insert7009541432154983385rm_a_b @ A3 @ B5 ) )
= ( insert7009541432154983385rm_a_b @ A3 @ ( inf_in1693821330460682071rm_a_b @ A4 @ B5 ) ) ) )
& ( ~ ( member5869715511025134514rm_a_b @ A3 @ A4 )
=> ( ( inf_in1693821330460682071rm_a_b @ A4 @ ( insert7009541432154983385rm_a_b @ A3 @ B5 ) )
= ( inf_in1693821330460682071rm_a_b @ A4 @ B5 ) ) ) ) ).
% Int_insert_right
thf(fact_1096_Int__insert__left,axiom,
! [A3: produc357393685978478089rm_a_b,C3: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
( ( ( member5869715511025134514rm_a_b @ A3 @ C3 )
=> ( ( inf_in1693821330460682071rm_a_b @ ( insert7009541432154983385rm_a_b @ A3 @ B5 ) @ C3 )
= ( insert7009541432154983385rm_a_b @ A3 @ ( inf_in1693821330460682071rm_a_b @ B5 @ C3 ) ) ) )
& ( ~ ( member5869715511025134514rm_a_b @ A3 @ C3 )
=> ( ( inf_in1693821330460682071rm_a_b @ ( insert7009541432154983385rm_a_b @ A3 @ B5 ) @ C3 )
= ( inf_in1693821330460682071rm_a_b @ B5 @ C3 ) ) ) ) ).
% Int_insert_left
thf(fact_1097_singletonD,axiom,
! [B2: produc357393685978478089rm_a_b,A3: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ B2 @ ( insert7009541432154983385rm_a_b @ A3 @ bot_bo197521221353338581rm_a_b ) )
=> ( B2 = A3 ) ) ).
% singletonD
thf(fact_1098_singleton__iff,axiom,
! [B2: produc357393685978478089rm_a_b,A3: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ B2 @ ( insert7009541432154983385rm_a_b @ A3 @ bot_bo197521221353338581rm_a_b ) )
= ( B2 = A3 ) ) ).
% singleton_iff
thf(fact_1099_NF__imp__SN__on,axiom,
! [A3: produc357393685978478089rm_a_b,R5: set_Pr2972776593051762503rm_a_b] :
( ( member5869715511025134514rm_a_b @ A3 @ ( abstra1902783451776770197rm_a_b @ R5 ) )
=> ( abstra2398554102055911763rm_a_b @ R5 @ ( insert7009541432154983385rm_a_b @ A3 @ bot_bo197521221353338581rm_a_b ) ) ) ).
% NF_imp_SN_on
thf(fact_1100_refl__not__SN,axiom,
! [X: term_a_b,R5: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ X ) @ R5 )
=> ~ ( abstra4720023341729745482rm_a_b @ R5 @ top_top_set_term_a_b ) ) ).
% refl_not_SN
thf(fact_1101_SN__imp__minimal,axiom,
! [A4: set_Pr2972776593051762503rm_a_b] :
( ( abstra2398554102055911763rm_a_b @ A4 @ 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 ) @ A4 )
=> ~ ( member5869715511025134514rm_a_b @ Y6 @ Q2 ) ) ) ) ) ).
% SN_imp_minimal
thf(fact_1102_SN__imp__minimal,axiom,
! [A4: set_Pr4386577575007340137rm_a_b] :
( ( abstra4720023341729745482rm_a_b @ A4 @ 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 ) @ A4 )
=> ~ ( member_term_a_b @ Y6 @ Q2 ) ) ) ) ) ).
% SN_imp_minimal
thf(fact_1103_SN__induct__rule,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,P2: term_a_b > $o,A3: term_a_b] :
( ( abstra4720023341729745482rm_a_b @ R2 @ top_top_set_term_a_b )
=> ( ! [A: term_a_b] :
( ! [B7: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B7 ) @ R2 )
=> ( P2 @ B7 ) )
=> ( P2 @ A ) )
=> ( P2 @ A3 ) ) ) ).
% SN_induct_rule
thf(fact_1104_SN__reaches__NF,axiom,
! [R2: set_Pr2972776593051762503rm_a_b,X: produc357393685978478089rm_a_b] :
( ( abstra2398554102055911763rm_a_b @ R2 @ ( insert7009541432154983385rm_a_b @ X @ bot_bo197521221353338581rm_a_b ) )
=> ? [Y2: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y2 ) @ ( transi2615809358984392588rm_a_b @ R2 ) )
& ( member5869715511025134514rm_a_b @ Y2 @ ( abstra1902783451776770197rm_a_b @ R2 ) ) ) ) ).
% SN_reaches_NF
thf(fact_1105_SN__reaches__NF,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,X: term_a_b] :
( ( abstra4720023341729745482rm_a_b @ R2 @ ( 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 @ R2 ) )
& ( member_term_a_b @ Y2 @ ( abstract_NF_term_a_b @ R2 ) ) ) ) ).
% SN_reaches_NF
thf(fact_1106_Diff__insert__absorb,axiom,
! [X: produc357393685978478089rm_a_b,A4: set_Pr4386577575007340137rm_a_b] :
( ~ ( member5869715511025134514rm_a_b @ X @ A4 )
=> ( ( minus_5192120951422937424rm_a_b @ ( insert7009541432154983385rm_a_b @ X @ A4 ) @ ( insert7009541432154983385rm_a_b @ X @ bot_bo197521221353338581rm_a_b ) )
= A4 ) ) ).
% Diff_insert_absorb
thf(fact_1107_insert__Diff,axiom,
! [A3: produc357393685978478089rm_a_b,A4: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ A3 @ A4 )
=> ( ( insert7009541432154983385rm_a_b @ A3 @ ( minus_5192120951422937424rm_a_b @ A4 @ ( insert7009541432154983385rm_a_b @ A3 @ bot_bo197521221353338581rm_a_b ) ) )
= A4 ) ) ).
% insert_Diff
thf(fact_1108_subset__Diff__insert,axiom,
! [A4: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b,X: produc357393685978478089rm_a_b,C3: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A4 @ ( minus_5192120951422937424rm_a_b @ B5 @ ( insert7009541432154983385rm_a_b @ X @ C3 ) ) )
= ( ( ord_le118470702582115849rm_a_b @ A4 @ ( minus_5192120951422937424rm_a_b @ B5 @ C3 ) )
& ~ ( member5869715511025134514rm_a_b @ X @ A4 ) ) ) ).
% subset_Diff_insert
thf(fact_1109_some__NF,axiom,
! [R2: set_Pr2972776593051762503rm_a_b,X: produc357393685978478089rm_a_b] :
( ( abstra2398554102055911763rm_a_b @ R2 @ ( insert7009541432154983385rm_a_b @ X @ bot_bo197521221353338581rm_a_b ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ ( abstra8203652949708803681rm_a_b @ R2 @ X ) ) @ ( transi2615809358984392588rm_a_b @ R2 ) )
& ( member5869715511025134514rm_a_b @ ( abstra8203652949708803681rm_a_b @ R2 @ X ) @ ( abstra1902783451776770197rm_a_b @ R2 ) ) ) ) ).
% some_NF
thf(fact_1110_some__NF,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,X: term_a_b] :
( ( abstra4720023341729745482rm_a_b @ R2 @ ( insert_term_a_b @ X @ bot_bot_set_term_a_b ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ ( abstra2589370493620291516rm_a_b @ R2 @ X ) ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
& ( member_term_a_b @ ( abstra2589370493620291516rm_a_b @ R2 @ X ) @ ( abstract_NF_term_a_b @ R2 ) ) ) ) ).
% some_NF
thf(fact_1111_SN__on__O__push,axiom,
! [S5: set_Pr4386577575007340137rm_a_b,R5: set_Pr4386577575007340137rm_a_b,X4: set_term_a_b] :
( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ S5 @ R5 ) @ ( relcom370159955682700863rm_a_b @ R5 @ ( transi7742714808557438673rm_a_b @ S5 ) ) )
=> ( ( abstra4720023341729745482rm_a_b @ R5 @ X4 )
=> ( abstra4720023341729745482rm_a_b @ ( relcom370159955682700863rm_a_b @ R5 @ ( transi7742714808557438673rm_a_b @ S5 ) ) @ X4 ) ) ) ).
% SN_on_O_push
thf(fact_1112_subset__insert__iff,axiom,
! [A4: set_Pr4386577575007340137rm_a_b,X: produc357393685978478089rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A4 @ ( insert7009541432154983385rm_a_b @ X @ B5 ) )
= ( ( ( member5869715511025134514rm_a_b @ X @ A4 )
=> ( ord_le118470702582115849rm_a_b @ ( minus_5192120951422937424rm_a_b @ A4 @ ( insert7009541432154983385rm_a_b @ X @ bot_bo197521221353338581rm_a_b ) ) @ B5 ) )
& ( ~ ( member5869715511025134514rm_a_b @ X @ A4 )
=> ( ord_le118470702582115849rm_a_b @ A4 @ B5 ) ) ) ) ).
% subset_insert_iff
thf(fact_1113_compatible__SN,axiom,
! [S5: set_Pr4386577575007340137rm_a_b,NS: set_Pr4386577575007340137rm_a_b] :
( ( abstra4720023341729745482rm_a_b @ S5 @ top_top_set_term_a_b )
=> ( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ NS @ S5 ) @ S5 )
=> ( abstra4720023341729745482rm_a_b @ ( relcom370159955682700863rm_a_b @ S5 @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ S5 ) @ ( transi7742714808557438673rm_a_b @ NS ) ) ) @ top_top_set_term_a_b ) ) ) ).
% compatible_SN
thf(fact_1114_the__NF,axiom,
! [A4: set_Pr4386577575007340137rm_a_b,A3: term_a_b] :
( ( abstra4720023341729745482rm_a_b @ A4 @ top_top_set_term_a_b )
=> ( ( abstra8448919418672941150rm_a_b @ A4 @ top_top_set_term_a_b )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ ( abstra2242782261322123075rm_a_b @ A4 @ A3 ) ) @ ( abstra2972651231392952684rm_a_b @ A4 ) ) ) ) ).
% the_NF
thf(fact_1115_the__NF__step,axiom,
! [A4: set_Pr4386577575007340137rm_a_b,A3: term_a_b,B2: term_a_b] :
( ( abstra4720023341729745482rm_a_b @ A4 @ top_top_set_term_a_b )
=> ( ( abstra8448919418672941150rm_a_b @ A4 @ top_top_set_term_a_b )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B2 ) @ A4 )
=> ( ( abstra2242782261322123075rm_a_b @ A4 @ A3 )
= ( abstra2242782261322123075rm_a_b @ A4 @ B2 ) ) ) ) ) ).
% the_NF_step
thf(fact_1116_the__NF__NF,axiom,
! [A4: set_Pr2972776593051762503rm_a_b,A3: produc357393685978478089rm_a_b] :
( ( abstra2398554102055911763rm_a_b @ A4 @ top_to1314267278846557113rm_a_b )
=> ( ( abstra7359694574113289279rm_a_b @ A4 @ top_to1314267278846557113rm_a_b )
=> ( member5869715511025134514rm_a_b @ ( abstra8972549633426078106rm_a_b @ A4 @ A3 ) @ ( abstra1902783451776770197rm_a_b @ A4 ) ) ) ) ).
% the_NF_NF
thf(fact_1117_the__NF__steps,axiom,
! [A4: set_Pr4386577575007340137rm_a_b,A3: term_a_b,B2: term_a_b] :
( ( abstra4720023341729745482rm_a_b @ A4 @ top_top_set_term_a_b )
=> ( ( abstra8448919418672941150rm_a_b @ A4 @ top_top_set_term_a_b )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B2 ) @ ( transi7742714808557438673rm_a_b @ A4 ) )
=> ( ( abstra2242782261322123075rm_a_b @ A4 @ A3 )
= ( abstra2242782261322123075rm_a_b @ A4 @ B2 ) ) ) ) ) ).
% the_NF_steps
thf(fact_1118_the__NF__conv,axiom,
! [A4: set_Pr4386577575007340137rm_a_b,A3: term_a_b,B2: term_a_b] :
( ( abstra4720023341729745482rm_a_b @ A4 @ top_top_set_term_a_b )
=> ( ( abstra8448919418672941150rm_a_b @ A4 @ top_top_set_term_a_b )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B2 ) @ ( abstra1130568886744591910rm_a_b @ A4 ) )
=> ( ( abstra2242782261322123075rm_a_b @ A4 @ A3 )
= ( abstra2242782261322123075rm_a_b @ A4 @ B2 ) ) ) ) ) ).
% the_NF_conv
thf(fact_1119_is__singletonI_H,axiom,
! [A4: set_Pr4386577575007340137rm_a_b] :
( ( A4 != bot_bo197521221353338581rm_a_b )
=> ( ! [X3: produc357393685978478089rm_a_b,Y2: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X3 @ A4 )
=> ( ( member5869715511025134514rm_a_b @ Y2 @ A4 )
=> ( X3 = Y2 ) ) )
=> ( is_sin332545968047684349rm_a_b @ A4 ) ) ) ).
% is_singletonI'
thf(fact_1120_Field__insert,axiom,
! [A3: term_a_b,B2: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ( field_term_a_b @ ( insert7009541432154983385rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B2 ) @ R2 ) )
= ( sup_sup_set_term_a_b @ ( insert_term_a_b @ A3 @ ( insert_term_a_b @ B2 @ bot_bot_set_term_a_b ) ) @ ( field_term_a_b @ R2 ) ) ) ).
% Field_insert
thf(fact_1121_acyclic__insert,axiom,
! [Y: term_a_b,X: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ( transi5314701259734769157rm_a_b @ ( insert7009541432154983385rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X ) @ R2 ) )
= ( ( transi5314701259734769157rm_a_b @ R2 )
& ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ R2 ) ) ) ) ).
% acyclic_insert
thf(fact_1122_member__remove,axiom,
! [X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b,A4: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ X @ ( remove8523906646656265582rm_a_b @ Y @ A4 ) )
= ( ( member5869715511025134514rm_a_b @ X @ A4 )
& ( X != Y ) ) ) ).
% member_remove
thf(fact_1123_FieldI2,axiom,
! [I: produc357393685978478089rm_a_b,J: produc357393685978478089rm_a_b,R5: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ I @ J ) @ R5 )
=> ( member5869715511025134514rm_a_b @ J @ ( field_6884932134483023318rm_a_b @ R5 ) ) ) ).
% FieldI2
thf(fact_1124_FieldI2,axiom,
! [I: term_a_b,J: term_a_b,R5: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ I @ J ) @ R5 )
=> ( member_term_a_b @ J @ ( field_term_a_b @ R5 ) ) ) ).
% FieldI2
thf(fact_1125_FieldI1,axiom,
! [I: produc357393685978478089rm_a_b,J: produc357393685978478089rm_a_b,R5: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ I @ J ) @ R5 )
=> ( member5869715511025134514rm_a_b @ I @ ( field_6884932134483023318rm_a_b @ R5 ) ) ) ).
% FieldI1
thf(fact_1126_FieldI1,axiom,
! [I: term_a_b,J: term_a_b,R5: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ I @ J ) @ R5 )
=> ( member_term_a_b @ I @ ( field_term_a_b @ R5 ) ) ) ).
% FieldI1
thf(fact_1127_acyclic__def,axiom,
( transi5314701259734769157rm_a_b
= ( ^ [R4: set_Pr4386577575007340137rm_a_b] :
! [X2: term_a_b] :
~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ X2 ) @ ( transi7922773638565587891rm_a_b @ R4 ) ) ) ) ).
% acyclic_def
thf(fact_1128_acyclicI,axiom,
! [R2: set_Pr4386577575007340137rm_a_b] :
( ! [X3: term_a_b] :
~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ X3 ) @ ( transi7922773638565587891rm_a_b @ R2 ) )
=> ( transi5314701259734769157rm_a_b @ R2 ) ) ).
% acyclicI
thf(fact_1129_wo__rel_Ocases__Total3,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,A3: term_a_b,B2: term_a_b,Phi: term_a_b > term_a_b > $o] :
( ( bNF_We251836928773027112rm_a_b @ R2 )
=> ( ( ord_le2705286416250468010rm_a_b @ ( insert_term_a_b @ A3 @ ( insert_term_a_b @ B2 @ bot_bot_set_term_a_b ) ) @ ( field_term_a_b @ R2 ) )
=> ( ( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B2 ) @ ( minus_5192120951422937424rm_a_b @ R2 @ id_term_a_b ) )
| ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ A3 ) @ ( minus_5192120951422937424rm_a_b @ R2 @ id_term_a_b ) ) )
=> ( Phi @ A3 @ B2 ) )
=> ( ( ( A3 = B2 )
=> ( Phi @ A3 @ B2 ) )
=> ( Phi @ A3 @ B2 ) ) ) ) ) ).
% wo_rel.cases_Total3
thf(fact_1130_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_1131_Total__subset__Id,axiom,
! [R2: set_Pr4386577575007340137rm_a_b] :
( ( total_on_term_a_b @ ( field_term_a_b @ R2 ) @ R2 )
=> ( ( ord_le118470702582115849rm_a_b @ R2 @ id_term_a_b )
=> ( ( R2 = bot_bo197521221353338581rm_a_b )
| ? [A: term_a_b] :
( R2
= ( insert7009541432154983385rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ A ) @ bot_bo197521221353338581rm_a_b ) ) ) ) ) ).
% Total_subset_Id
thf(fact_1132_wo__rel_Oin__notinI,axiom,
! [R2: set_Pr2972776593051762503rm_a_b,J: produc357393685978478089rm_a_b,I: produc357393685978478089rm_a_b] :
( ( bNF_We1688910813892497397rm_a_b @ R2 )
=> ( ( ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ J @ I ) @ R2 )
| ( J = I ) )
=> ( ( member5869715511025134514rm_a_b @ I @ ( field_6884932134483023318rm_a_b @ R2 ) )
=> ( ( member5869715511025134514rm_a_b @ J @ ( field_6884932134483023318rm_a_b @ R2 ) )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ I @ J ) @ R2 ) ) ) ) ) ).
% wo_rel.in_notinI
thf(fact_1133_wo__rel_Oin__notinI,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,J: term_a_b,I: term_a_b] :
( ( bNF_We251836928773027112rm_a_b @ R2 )
=> ( ( ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ J @ I ) @ R2 )
| ( J = I ) )
=> ( ( member_term_a_b @ I @ ( field_term_a_b @ R2 ) )
=> ( ( member_term_a_b @ J @ ( field_term_a_b @ R2 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ I @ J ) @ R2 ) ) ) ) ) ).
% wo_rel.in_notinI
thf(fact_1134_wo__rel_OTOTALS,axiom,
! [R2: set_Pr4386577575007340137rm_a_b] :
( ( bNF_We251836928773027112rm_a_b @ R2 )
=> ! [X5: term_a_b] :
( ( member_term_a_b @ X5 @ ( field_term_a_b @ R2 ) )
=> ! [Xa2: term_a_b] :
( ( member_term_a_b @ Xa2 @ ( field_term_a_b @ R2 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X5 @ Xa2 ) @ R2 )
| ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Xa2 @ X5 ) @ R2 ) ) ) ) ) ).
% wo_rel.TOTALS
thf(fact_1135_wo__rel_Owell__order__induct,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,P2: term_a_b > $o,A3: term_a_b] :
( ( bNF_We251836928773027112rm_a_b @ R2 )
=> ( ! [X3: term_a_b] :
( ! [Y6: term_a_b] :
( ( ( Y6 != X3 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y6 @ X3 ) @ R2 ) )
=> ( P2 @ Y6 ) )
=> ( P2 @ X3 ) )
=> ( P2 @ A3 ) ) ) ).
% wo_rel.well_order_induct
thf(fact_1136_refl__on__domain,axiom,
! [A4: set_Pr4386577575007340137rm_a_b,R2: set_Pr2972776593051762503rm_a_b,A3: produc357393685978478089rm_a_b,B2: produc357393685978478089rm_a_b] :
( ( refl_o5344404488616963464rm_a_b @ A4 @ R2 )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A3 @ B2 ) @ R2 )
=> ( ( member5869715511025134514rm_a_b @ A3 @ A4 )
& ( member5869715511025134514rm_a_b @ B2 @ A4 ) ) ) ) ).
% refl_on_domain
thf(fact_1137_refl__on__domain,axiom,
! [A4: set_term_a_b,R2: set_Pr4386577575007340137rm_a_b,A3: term_a_b,B2: term_a_b] :
( ( refl_on_term_a_b @ A4 @ R2 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B2 ) @ R2 )
=> ( ( member_term_a_b @ A3 @ A4 )
& ( member_term_a_b @ B2 @ A4 ) ) ) ) ).
% refl_on_domain
thf(fact_1138_total__onI,axiom,
! [A4: set_Pr4386577575007340137rm_a_b,R2: set_Pr2972776593051762503rm_a_b] :
( ! [X3: produc357393685978478089rm_a_b,Y2: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X3 @ A4 )
=> ( ( member5869715511025134514rm_a_b @ Y2 @ A4 )
=> ( ( X3 != Y2 )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X3 @ Y2 ) @ R2 )
| ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y2 @ X3 ) @ R2 ) ) ) ) )
=> ( total_4010809703584253837rm_a_b @ A4 @ R2 ) ) ).
% total_onI
thf(fact_1139_total__onI,axiom,
! [A4: set_term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ! [X3: term_a_b,Y2: term_a_b] :
( ( member_term_a_b @ X3 @ A4 )
=> ( ( member_term_a_b @ Y2 @ A4 )
=> ( ( X3 != Y2 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y2 ) @ R2 )
| ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y2 @ X3 ) @ R2 ) ) ) ) )
=> ( total_on_term_a_b @ A4 @ R2 ) ) ).
% total_onI
thf(fact_1140_total__on__def,axiom,
( total_on_term_a_b
= ( ^ [A7: set_term_a_b,R4: set_Pr4386577575007340137rm_a_b] :
! [X2: term_a_b] :
( ( member_term_a_b @ X2 @ A7 )
=> ! [Y5: term_a_b] :
( ( member_term_a_b @ Y5 @ A7 )
=> ( ( X2 != Y5 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y5 ) @ R4 )
| ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y5 @ X2 ) @ R4 ) ) ) ) ) ) ) ).
% total_on_def
thf(fact_1141_refl__onD,axiom,
! [A4: set_Pr4386577575007340137rm_a_b,R2: set_Pr2972776593051762503rm_a_b,A3: produc357393685978478089rm_a_b] :
( ( refl_o5344404488616963464rm_a_b @ A4 @ R2 )
=> ( ( member5869715511025134514rm_a_b @ A3 @ A4 )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A3 @ A3 ) @ R2 ) ) ) ).
% refl_onD
thf(fact_1142_refl__onD,axiom,
! [A4: set_term_a_b,R2: set_Pr4386577575007340137rm_a_b,A3: term_a_b] :
( ( refl_on_term_a_b @ A4 @ R2 )
=> ( ( member_term_a_b @ A3 @ A4 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ A3 ) @ R2 ) ) ) ).
% refl_onD
thf(fact_1143_refl__onD1,axiom,
! [A4: set_Pr4386577575007340137rm_a_b,R2: set_Pr2972776593051762503rm_a_b,X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b] :
( ( refl_o5344404488616963464rm_a_b @ A4 @ R2 )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ R2 )
=> ( member5869715511025134514rm_a_b @ X @ A4 ) ) ) ).
% refl_onD1
thf(fact_1144_refl__onD1,axiom,
! [A4: set_term_a_b,R2: set_Pr4386577575007340137rm_a_b,X: term_a_b,Y: term_a_b] :
( ( refl_on_term_a_b @ A4 @ R2 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ R2 )
=> ( member_term_a_b @ X @ A4 ) ) ) ).
% refl_onD1
thf(fact_1145_refl__onD2,axiom,
! [A4: set_Pr4386577575007340137rm_a_b,R2: set_Pr2972776593051762503rm_a_b,X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b] :
( ( refl_o5344404488616963464rm_a_b @ A4 @ R2 )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ R2 )
=> ( member5869715511025134514rm_a_b @ Y @ A4 ) ) ) ).
% refl_onD2
thf(fact_1146_refl__onD2,axiom,
! [A4: set_term_a_b,R2: set_Pr4386577575007340137rm_a_b,X: term_a_b,Y: term_a_b] :
( ( refl_on_term_a_b @ A4 @ R2 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ R2 )
=> ( member_term_a_b @ Y @ A4 ) ) ) ).
% refl_onD2
thf(fact_1147_reflD,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,A3: term_a_b] :
( ( refl_on_term_a_b @ top_top_set_term_a_b @ R2 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ A3 ) @ R2 ) ) ).
% reflD
thf(fact_1148_reflI,axiom,
! [R2: set_Pr4386577575007340137rm_a_b] :
( ! [X3: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ X3 ) @ R2 )
=> ( refl_on_term_a_b @ top_top_set_term_a_b @ R2 ) ) ).
% reflI
thf(fact_1149_well__order__induct__imp,axiom,
! [R2: set_Pr2972776593051762503rm_a_b,P2: produc357393685978478089rm_a_b > $o,A3: produc357393685978478089rm_a_b] :
( ( bNF_We1688910813892497397rm_a_b @ R2 )
=> ( ! [X3: produc357393685978478089rm_a_b] :
( ! [Y6: produc357393685978478089rm_a_b] :
( ( ( Y6 != X3 )
& ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y6 @ X3 ) @ R2 ) )
=> ( ( member5869715511025134514rm_a_b @ Y6 @ ( field_6884932134483023318rm_a_b @ R2 ) )
=> ( P2 @ Y6 ) ) )
=> ( ( member5869715511025134514rm_a_b @ X3 @ ( field_6884932134483023318rm_a_b @ R2 ) )
=> ( P2 @ X3 ) ) )
=> ( ( member5869715511025134514rm_a_b @ A3 @ ( field_6884932134483023318rm_a_b @ R2 ) )
=> ( P2 @ A3 ) ) ) ) ).
% well_order_induct_imp
thf(fact_1150_well__order__induct__imp,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,P2: term_a_b > $o,A3: term_a_b] :
( ( bNF_We251836928773027112rm_a_b @ R2 )
=> ( ! [X3: term_a_b] :
( ! [Y6: term_a_b] :
( ( ( Y6 != X3 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y6 @ X3 ) @ R2 ) )
=> ( ( member_term_a_b @ Y6 @ ( field_term_a_b @ R2 ) )
=> ( P2 @ Y6 ) ) )
=> ( ( member_term_a_b @ X3 @ ( field_term_a_b @ R2 ) )
=> ( P2 @ X3 ) ) )
=> ( ( member_term_a_b @ A3 @ ( field_term_a_b @ R2 ) )
=> ( P2 @ A3 ) ) ) ) ).
% well_order_induct_imp
thf(fact_1151_refl__rtrancl,axiom,
! [R2: set_Pr4386577575007340137rm_a_b] : ( refl_on_term_a_b @ top_top_set_term_a_b @ ( transi7742714808557438673rm_a_b @ R2 ) ) ).
% refl_rtrancl
thf(fact_1152_totalI,axiom,
! [R2: set_Pr4386577575007340137rm_a_b] :
( ! [X3: term_a_b,Y2: term_a_b] :
( ( X3 != Y2 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y2 ) @ R2 )
| ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y2 @ X3 ) @ R2 ) ) )
=> ( total_on_term_a_b @ top_top_set_term_a_b @ R2 ) ) ).
% totalI
thf(fact_1153_wo__rel_Ocases__Total,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,A3: term_a_b,B2: term_a_b,Phi: term_a_b > term_a_b > $o] :
( ( bNF_We251836928773027112rm_a_b @ R2 )
=> ( ( ord_le2705286416250468010rm_a_b @ ( insert_term_a_b @ A3 @ ( insert_term_a_b @ B2 @ bot_bot_set_term_a_b ) ) @ ( field_term_a_b @ R2 ) )
=> ( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B2 ) @ R2 )
=> ( Phi @ A3 @ B2 ) )
=> ( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ A3 ) @ R2 )
=> ( Phi @ A3 @ B2 ) )
=> ( Phi @ A3 @ B2 ) ) ) ) ) ).
% wo_rel.cases_Total
thf(fact_1154_wo__rel_Ominim__in,axiom,
! [R2: set_Pr2972776593051762503rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
( ( bNF_We1688910813892497397rm_a_b @ R2 )
=> ( ( ord_le118470702582115849rm_a_b @ B5 @ ( field_6884932134483023318rm_a_b @ R2 ) )
=> ( ( B5 != bot_bo197521221353338581rm_a_b )
=> ( member5869715511025134514rm_a_b @ ( bNF_We1338740440619677141rm_a_b @ R2 @ B5 ) @ B5 ) ) ) ) ).
% wo_rel.minim_in
thf(fact_1155_wo__rel_OisMinim__def,axiom,
! [R2: set_Pr2972776593051762503rm_a_b,A4: set_Pr4386577575007340137rm_a_b,B2: produc357393685978478089rm_a_b] :
( ( bNF_We1688910813892497397rm_a_b @ R2 )
=> ( ( bNF_We8430659725849320639rm_a_b @ R2 @ A4 @ B2 )
= ( ( member5869715511025134514rm_a_b @ B2 @ A4 )
& ! [X2: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X2 @ A4 )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B2 @ X2 ) @ R2 ) ) ) ) ) ).
% wo_rel.isMinim_def
thf(fact_1156_wo__rel_OisMinim__def,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,A4: set_term_a_b,B2: term_a_b] :
( ( bNF_We251836928773027112rm_a_b @ R2 )
=> ( ( bNF_We695392074589568990rm_a_b @ R2 @ A4 @ B2 )
= ( ( member_term_a_b @ B2 @ A4 )
& ! [X2: term_a_b] :
( ( member_term_a_b @ X2 @ A4 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ X2 ) @ R2 ) ) ) ) ) ).
% wo_rel.isMinim_def
thf(fact_1157_wo__rel_Oequals__minim,axiom,
! [R2: set_Pr2972776593051762503rm_a_b,B5: set_Pr4386577575007340137rm_a_b,A3: produc357393685978478089rm_a_b] :
( ( bNF_We1688910813892497397rm_a_b @ R2 )
=> ( ( ord_le118470702582115849rm_a_b @ B5 @ ( field_6884932134483023318rm_a_b @ R2 ) )
=> ( ( member5869715511025134514rm_a_b @ A3 @ B5 )
=> ( ! [B: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ B @ B5 )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A3 @ B ) @ R2 ) )
=> ( A3
= ( bNF_We1338740440619677141rm_a_b @ R2 @ B5 ) ) ) ) ) ) ).
% wo_rel.equals_minim
thf(fact_1158_wo__rel_Oequals__minim,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,B5: set_term_a_b,A3: term_a_b] :
( ( bNF_We251836928773027112rm_a_b @ R2 )
=> ( ( ord_le2705286416250468010rm_a_b @ B5 @ ( field_term_a_b @ R2 ) )
=> ( ( member_term_a_b @ A3 @ B5 )
=> ( ! [B: term_a_b] :
( ( member_term_a_b @ B @ B5 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B ) @ R2 ) )
=> ( A3
= ( bNF_We6258903063523145544rm_a_b @ R2 @ B5 ) ) ) ) ) ) ).
% wo_rel.equals_minim
thf(fact_1159_wo__rel_Ominim__least,axiom,
! [R2: set_Pr2972776593051762503rm_a_b,B5: set_Pr4386577575007340137rm_a_b,B2: produc357393685978478089rm_a_b] :
( ( bNF_We1688910813892497397rm_a_b @ R2 )
=> ( ( ord_le118470702582115849rm_a_b @ B5 @ ( field_6884932134483023318rm_a_b @ R2 ) )
=> ( ( member5869715511025134514rm_a_b @ B2 @ B5 )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( bNF_We1338740440619677141rm_a_b @ R2 @ B5 ) @ B2 ) @ R2 ) ) ) ) ).
% wo_rel.minim_least
thf(fact_1160_wo__rel_Ominim__least,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,B5: set_term_a_b,B2: term_a_b] :
( ( bNF_We251836928773027112rm_a_b @ R2 )
=> ( ( ord_le2705286416250468010rm_a_b @ B5 @ ( field_term_a_b @ R2 ) )
=> ( ( member_term_a_b @ B2 @ B5 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( bNF_We6258903063523145544rm_a_b @ R2 @ B5 ) @ B2 ) @ R2 ) ) ) ) ).
% wo_rel.minim_least
thf(fact_1161_wo__rel_Ominim__inField,axiom,
! [R2: set_Pr2972776593051762503rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
( ( bNF_We1688910813892497397rm_a_b @ R2 )
=> ( ( ord_le118470702582115849rm_a_b @ B5 @ ( field_6884932134483023318rm_a_b @ R2 ) )
=> ( ( B5 != bot_bo197521221353338581rm_a_b )
=> ( member5869715511025134514rm_a_b @ ( bNF_We1338740440619677141rm_a_b @ R2 @ B5 ) @ ( field_6884932134483023318rm_a_b @ R2 ) ) ) ) ) ).
% wo_rel.minim_inField
thf(fact_1162_wo__rel_Omax2__greater__among,axiom,
! [R2: set_Pr2972776593051762503rm_a_b,A3: produc357393685978478089rm_a_b,B2: produc357393685978478089rm_a_b] :
( ( bNF_We1688910813892497397rm_a_b @ R2 )
=> ( ( member5869715511025134514rm_a_b @ A3 @ ( field_6884932134483023318rm_a_b @ R2 ) )
=> ( ( member5869715511025134514rm_a_b @ B2 @ ( field_6884932134483023318rm_a_b @ R2 ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A3 @ ( bNF_We1369222903935127127rm_a_b @ R2 @ A3 @ B2 ) ) @ R2 )
& ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B2 @ ( bNF_We1369222903935127127rm_a_b @ R2 @ A3 @ B2 ) ) @ R2 )
& ( member5869715511025134514rm_a_b @ ( bNF_We1369222903935127127rm_a_b @ R2 @ A3 @ B2 ) @ ( insert7009541432154983385rm_a_b @ A3 @ ( insert7009541432154983385rm_a_b @ B2 @ bot_bo197521221353338581rm_a_b ) ) ) ) ) ) ) ).
% wo_rel.max2_greater_among
thf(fact_1163_wo__rel_Omax2__greater__among,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,A3: term_a_b,B2: term_a_b] :
( ( bNF_We251836928773027112rm_a_b @ R2 )
=> ( ( member_term_a_b @ A3 @ ( field_term_a_b @ R2 ) )
=> ( ( member_term_a_b @ B2 @ ( field_term_a_b @ R2 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ ( bNF_We4618759378340593734rm_a_b @ R2 @ A3 @ B2 ) ) @ R2 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ ( bNF_We4618759378340593734rm_a_b @ R2 @ A3 @ B2 ) ) @ R2 )
& ( member_term_a_b @ ( bNF_We4618759378340593734rm_a_b @ R2 @ A3 @ B2 ) @ ( insert_term_a_b @ A3 @ ( insert_term_a_b @ B2 @ bot_bot_set_term_a_b ) ) ) ) ) ) ) ).
% wo_rel.max2_greater_among
thf(fact_1164_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_1165_wo__rel_Omax2__def,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,A3: term_a_b,B2: term_a_b] :
( ( bNF_We251836928773027112rm_a_b @ R2 )
=> ( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B2 ) @ R2 )
=> ( ( bNF_We4618759378340593734rm_a_b @ R2 @ A3 @ B2 )
= B2 ) )
& ( ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B2 ) @ R2 )
=> ( ( bNF_We4618759378340593734rm_a_b @ R2 @ A3 @ B2 )
= A3 ) ) ) ) ).
% wo_rel.max2_def
thf(fact_1166_wo__rel_Omax2__greater,axiom,
! [R2: set_Pr2972776593051762503rm_a_b,A3: produc357393685978478089rm_a_b,B2: produc357393685978478089rm_a_b] :
( ( bNF_We1688910813892497397rm_a_b @ R2 )
=> ( ( member5869715511025134514rm_a_b @ A3 @ ( field_6884932134483023318rm_a_b @ R2 ) )
=> ( ( member5869715511025134514rm_a_b @ B2 @ ( field_6884932134483023318rm_a_b @ R2 ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A3 @ ( bNF_We1369222903935127127rm_a_b @ R2 @ A3 @ B2 ) ) @ R2 )
& ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B2 @ ( bNF_We1369222903935127127rm_a_b @ R2 @ A3 @ B2 ) ) @ R2 ) ) ) ) ) ).
% wo_rel.max2_greater
thf(fact_1167_wo__rel_Omax2__greater,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,A3: term_a_b,B2: term_a_b] :
( ( bNF_We251836928773027112rm_a_b @ R2 )
=> ( ( member_term_a_b @ A3 @ ( field_term_a_b @ R2 ) )
=> ( ( member_term_a_b @ B2 @ ( field_term_a_b @ R2 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ ( bNF_We4618759378340593734rm_a_b @ R2 @ A3 @ B2 ) ) @ R2 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ ( bNF_We4618759378340593734rm_a_b @ R2 @ A3 @ B2 ) ) @ R2 ) ) ) ) ) ).
% wo_rel.max2_greater
thf(fact_1168_wo__rel_Omax2__equals2,axiom,
! [R2: set_Pr2972776593051762503rm_a_b,A3: produc357393685978478089rm_a_b,B2: produc357393685978478089rm_a_b] :
( ( bNF_We1688910813892497397rm_a_b @ R2 )
=> ( ( member5869715511025134514rm_a_b @ A3 @ ( field_6884932134483023318rm_a_b @ R2 ) )
=> ( ( member5869715511025134514rm_a_b @ B2 @ ( field_6884932134483023318rm_a_b @ R2 ) )
=> ( ( ( bNF_We1369222903935127127rm_a_b @ R2 @ A3 @ B2 )
= B2 )
= ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A3 @ B2 ) @ R2 ) ) ) ) ) ).
% wo_rel.max2_equals2
thf(fact_1169_wo__rel_Omax2__equals2,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,A3: term_a_b,B2: term_a_b] :
( ( bNF_We251836928773027112rm_a_b @ R2 )
=> ( ( member_term_a_b @ A3 @ ( field_term_a_b @ R2 ) )
=> ( ( member_term_a_b @ B2 @ ( field_term_a_b @ R2 ) )
=> ( ( ( bNF_We4618759378340593734rm_a_b @ R2 @ A3 @ B2 )
= B2 )
= ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B2 ) @ R2 ) ) ) ) ) ).
% wo_rel.max2_equals2
thf(fact_1170_wo__rel_Omax2__equals1,axiom,
! [R2: set_Pr2972776593051762503rm_a_b,A3: produc357393685978478089rm_a_b,B2: produc357393685978478089rm_a_b] :
( ( bNF_We1688910813892497397rm_a_b @ R2 )
=> ( ( member5869715511025134514rm_a_b @ A3 @ ( field_6884932134483023318rm_a_b @ R2 ) )
=> ( ( member5869715511025134514rm_a_b @ B2 @ ( field_6884932134483023318rm_a_b @ R2 ) )
=> ( ( ( bNF_We1369222903935127127rm_a_b @ R2 @ A3 @ B2 )
= A3 )
= ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B2 @ A3 ) @ R2 ) ) ) ) ) ).
% wo_rel.max2_equals1
thf(fact_1171_wo__rel_Omax2__equals1,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,A3: term_a_b,B2: term_a_b] :
( ( bNF_We251836928773027112rm_a_b @ R2 )
=> ( ( member_term_a_b @ A3 @ ( field_term_a_b @ R2 ) )
=> ( ( member_term_a_b @ B2 @ ( field_term_a_b @ R2 ) )
=> ( ( ( bNF_We4618759378340593734rm_a_b @ R2 @ A3 @ B2 )
= A3 )
= ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ A3 ) @ R2 ) ) ) ) ) ).
% wo_rel.max2_equals1
thf(fact_1172_Linear__order__in__diff__Id,axiom,
! [R2: set_Pr2972776593051762503rm_a_b,A3: produc357393685978478089rm_a_b,B2: produc357393685978478089rm_a_b] :
( ( order_681589137112254398rm_a_b @ ( field_6884932134483023318rm_a_b @ R2 ) @ R2 )
=> ( ( member5869715511025134514rm_a_b @ A3 @ ( field_6884932134483023318rm_a_b @ R2 ) )
=> ( ( member5869715511025134514rm_a_b @ B2 @ ( field_6884932134483023318rm_a_b @ R2 ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A3 @ B2 ) @ R2 )
= ( ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B2 @ A3 ) @ ( minus_45395518312058030rm_a_b @ R2 @ id_Pro3488709451141449061rm_a_b ) ) ) ) ) ) ) ).
% Linear_order_in_diff_Id
thf(fact_1173_Linear__order__in__diff__Id,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,A3: term_a_b,B2: term_a_b] :
( ( order_5388802246213473311rm_a_b @ ( field_term_a_b @ R2 ) @ R2 )
=> ( ( member_term_a_b @ A3 @ ( field_term_a_b @ R2 ) )
=> ( ( member_term_a_b @ B2 @ ( field_term_a_b @ R2 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B2 ) @ R2 )
= ( ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ A3 ) @ ( minus_5192120951422937424rm_a_b @ R2 @ id_term_a_b ) ) ) ) ) ) ) ).
% Linear_order_in_diff_Id
thf(fact_1174_wo__rel_Omax2__among,axiom,
! [R2: set_Pr2972776593051762503rm_a_b,A3: produc357393685978478089rm_a_b,B2: produc357393685978478089rm_a_b] :
( ( bNF_We1688910813892497397rm_a_b @ R2 )
=> ( ( member5869715511025134514rm_a_b @ A3 @ ( field_6884932134483023318rm_a_b @ R2 ) )
=> ( ( member5869715511025134514rm_a_b @ B2 @ ( field_6884932134483023318rm_a_b @ R2 ) )
=> ( member5869715511025134514rm_a_b @ ( bNF_We1369222903935127127rm_a_b @ R2 @ A3 @ B2 ) @ ( insert7009541432154983385rm_a_b @ A3 @ ( insert7009541432154983385rm_a_b @ B2 @ bot_bo197521221353338581rm_a_b ) ) ) ) ) ) ).
% wo_rel.max2_among
thf(fact_1175_Linear__order__wf__diff__Id,axiom,
! [R2: set_Pr4386577575007340137rm_a_b] :
( ( order_5388802246213473311rm_a_b @ ( field_term_a_b @ R2 ) @ R2 )
=> ( ( wf_term_a_b @ ( minus_5192120951422937424rm_a_b @ R2 @ id_term_a_b ) )
= ( ! [A7: set_term_a_b] :
( ( ord_le2705286416250468010rm_a_b @ A7 @ ( field_term_a_b @ R2 ) )
=> ( ( A7 != bot_bot_set_term_a_b )
=> ? [X2: term_a_b] :
( ( member_term_a_b @ X2 @ A7 )
& ! [Y5: term_a_b] :
( ( member_term_a_b @ Y5 @ A7 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y5 ) @ R2 ) ) ) ) ) ) ) ) ).
% Linear_order_wf_diff_Id
thf(fact_1176_Linear__order__Well__order__iff,axiom,
! [R2: set_Pr4386577575007340137rm_a_b] :
( ( order_5388802246213473311rm_a_b @ ( field_term_a_b @ R2 ) @ R2 )
=> ( ( order_5719896216891381494rm_a_b @ ( field_term_a_b @ R2 ) @ R2 )
= ( ! [A7: set_term_a_b] :
( ( ord_le2705286416250468010rm_a_b @ A7 @ ( field_term_a_b @ R2 ) )
=> ( ( A7 != bot_bot_set_term_a_b )
=> ? [X2: term_a_b] :
( ( member_term_a_b @ X2 @ A7 )
& ! [Y5: term_a_b] :
( ( member_term_a_b @ Y5 @ A7 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y5 ) @ R2 ) ) ) ) ) ) ) ) ).
% Linear_order_Well_order_iff
thf(fact_1177_well__order__on__domain,axiom,
! [A4: set_Pr4386577575007340137rm_a_b,R2: set_Pr2972776593051762503rm_a_b,A3: produc357393685978478089rm_a_b,B2: produc357393685978478089rm_a_b] :
( ( order_8637870607158948775rm_a_b @ A4 @ R2 )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A3 @ B2 ) @ R2 )
=> ( ( member5869715511025134514rm_a_b @ A3 @ A4 )
& ( member5869715511025134514rm_a_b @ B2 @ A4 ) ) ) ) ).
% well_order_on_domain
thf(fact_1178_well__order__on__domain,axiom,
! [A4: set_term_a_b,R2: set_Pr4386577575007340137rm_a_b,A3: term_a_b,B2: term_a_b] :
( ( order_5719896216891381494rm_a_b @ A4 @ R2 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B2 ) @ R2 )
=> ( ( member_term_a_b @ A3 @ A4 )
& ( member_term_a_b @ B2 @ A4 ) ) ) ) ).
% well_order_on_domain
thf(fact_1179_minimal__imp__wf,axiom,
! [R2: 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 ) @ R2 )
=> ~ ( member5869715511025134514rm_a_b @ Y2 @ Q3 ) ) ) )
=> ( wf_Pro2335863617654816626rm_a_b @ ( conver1575690562132235771rm_a_b @ R2 ) ) ) ).
% minimal_imp_wf
thf(fact_1180_minimal__imp__wf,axiom,
! [R2: 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 ) @ R2 )
=> ~ ( member_term_a_b @ Y2 @ Q3 ) ) ) )
=> ( wf_term_a_b @ ( conver7608559976937845093rm_a_b @ R2 ) ) ) ).
% minimal_imp_wf
thf(fact_1181_wf__eq__minimal2,axiom,
( wf_term_a_b
= ( ^ [R4: set_Pr4386577575007340137rm_a_b] :
! [A7: set_term_a_b] :
( ( ( ord_le2705286416250468010rm_a_b @ A7 @ ( field_term_a_b @ R4 ) )
& ( A7 != bot_bot_set_term_a_b ) )
=> ? [X2: term_a_b] :
( ( member_term_a_b @ X2 @ A7 )
& ! [Y5: term_a_b] :
( ( member_term_a_b @ Y5 @ A7 )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y5 @ X2 ) @ R4 ) ) ) ) ) ) ).
% wf_eq_minimal2
thf(fact_1182_wf__insert,axiom,
! [Y: term_a_b,X: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ( wf_term_a_b @ ( insert7009541432154983385rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X ) @ R2 ) )
= ( ( wf_term_a_b @ R2 )
& ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ R2 ) ) ) ) ).
% wf_insert
thf(fact_1183_wf__def,axiom,
( wf_term_a_b
= ( ^ [R4: set_Pr4386577575007340137rm_a_b] :
! [P3: term_a_b > $o] :
( ! [X2: term_a_b] :
( ! [Y5: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y5 @ X2 ) @ R4 )
=> ( P3 @ Y5 ) )
=> ( P3 @ X2 ) )
=> ! [X6: term_a_b] : ( P3 @ X6 ) ) ) ) ).
% wf_def
thf(fact_1184_wfE__min,axiom,
! [R5: set_Pr2972776593051762503rm_a_b,X: produc357393685978478089rm_a_b,Q: set_Pr4386577575007340137rm_a_b] :
( ( wf_Pro2335863617654816626rm_a_b @ R5 )
=> ( ( member5869715511025134514rm_a_b @ X @ Q )
=> ~ ! [Z: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ Z @ Q )
=> ~ ! [Y6: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y6 @ Z ) @ R5 )
=> ~ ( member5869715511025134514rm_a_b @ Y6 @ Q ) ) ) ) ) ).
% wfE_min
thf(fact_1185_wfE__min,axiom,
! [R5: set_Pr4386577575007340137rm_a_b,X: term_a_b,Q: set_term_a_b] :
( ( wf_term_a_b @ R5 )
=> ( ( member_term_a_b @ X @ Q )
=> ~ ! [Z: term_a_b] :
( ( member_term_a_b @ Z @ Q )
=> ~ ! [Y6: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y6 @ Z ) @ R5 )
=> ~ ( member_term_a_b @ Y6 @ Q ) ) ) ) ) ).
% wfE_min
thf(fact_1186_wfI__min,axiom,
! [R5: 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 ) @ R5 )
=> ~ ( member5869715511025134514rm_a_b @ Y2 @ Q3 ) ) ) )
=> ( wf_Pro2335863617654816626rm_a_b @ R5 ) ) ).
% wfI_min
thf(fact_1187_wfI__min,axiom,
! [R5: 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 ) @ R5 )
=> ~ ( member_term_a_b @ Y2 @ Q3 ) ) ) )
=> ( wf_term_a_b @ R5 ) ) ).
% wfI_min
thf(fact_1188_wfUNIVI,axiom,
! [R2: 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 ) @ R2 )
=> ( P4 @ Y2 ) )
=> ( P4 @ Xa2 ) )
=> ( P4 @ X3 ) )
=> ( wf_term_a_b @ R2 ) ) ).
% wfUNIVI
thf(fact_1189_wf__asym,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,A3: term_a_b,X: term_a_b] :
( ( wf_term_a_b @ R2 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ X ) @ R2 )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ A3 ) @ R2 ) ) ) ).
% wf_asym
thf(fact_1190_wf__induct,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,P2: term_a_b > $o,A3: term_a_b] :
( ( wf_term_a_b @ R2 )
=> ( ! [X3: term_a_b] :
( ! [Y6: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y6 @ X3 ) @ R2 )
=> ( P2 @ Y6 ) )
=> ( P2 @ X3 ) )
=> ( P2 @ A3 ) ) ) ).
% wf_induct
thf(fact_1191_wf__irrefl,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,A3: term_a_b] :
( ( wf_term_a_b @ R2 )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ A3 ) @ R2 ) ) ).
% wf_irrefl
thf(fact_1192_wf__not__sym,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,A3: term_a_b,X: term_a_b] :
( ( wf_term_a_b @ R2 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ X ) @ R2 )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ A3 ) @ R2 ) ) ) ).
% wf_not_sym
thf(fact_1193_wf__not__refl,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,A3: term_a_b] :
( ( wf_term_a_b @ R2 )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ A3 ) @ R2 ) ) ).
% wf_not_refl
thf(fact_1194_wf__eq__minimal,axiom,
( wf_Pro2335863617654816626rm_a_b
= ( ^ [R4: 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 )
& ! [Y5: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y5 @ X2 ) @ R4 )
=> ~ ( member5869715511025134514rm_a_b @ Y5 @ Q4 ) ) ) ) ) ) ).
% wf_eq_minimal
thf(fact_1195_wf__eq__minimal,axiom,
( wf_term_a_b
= ( ^ [R4: 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 )
& ! [Y5: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y5 @ X2 ) @ R4 )
=> ~ ( member_term_a_b @ Y5 @ Q4 ) ) ) ) ) ) ).
% wf_eq_minimal
thf(fact_1196_wf__induct__rule,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,P2: term_a_b > $o,A3: term_a_b] :
( ( wf_term_a_b @ R2 )
=> ( ! [X3: term_a_b] :
( ! [Y6: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y6 @ X3 ) @ R2 )
=> ( P2 @ Y6 ) )
=> ( P2 @ X3 ) )
=> ( P2 @ A3 ) ) ) ).
% wf_induct_rule
thf(fact_1197_wf__if__convertible__to__wf,axiom,
! [S: set_Pr4386577575007340137rm_a_b,R2: set_Pr4386577575007340137rm_a_b,F: 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 ) @ R2 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F @ X3 ) @ ( F @ Y2 ) ) @ S ) )
=> ( wf_term_a_b @ R2 ) ) ) ).
% wf_if_convertible_to_wf
thf(fact_1198_wfE__min_H,axiom,
! [R5: set_Pr2972776593051762503rm_a_b,Q: set_Pr4386577575007340137rm_a_b] :
( ( wf_Pro2335863617654816626rm_a_b @ R5 )
=> ( ( Q != bot_bo197521221353338581rm_a_b )
=> ~ ! [Z: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ Z @ Q )
=> ~ ! [Y6: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y6 @ Z ) @ R5 )
=> ~ ( member5869715511025134514rm_a_b @ Y6 @ Q ) ) ) ) ) ).
% wfE_min'
thf(fact_1199_wfE__min_H,axiom,
! [R5: set_Pr4386577575007340137rm_a_b,Q: set_term_a_b] :
( ( wf_term_a_b @ R5 )
=> ( ( Q != bot_bot_set_term_a_b )
=> ~ ! [Z: term_a_b] :
( ( member_term_a_b @ Z @ Q )
=> ~ ! [Y6: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y6 @ Z ) @ R5 )
=> ~ ( member_term_a_b @ Y6 @ Q ) ) ) ) ) ).
% wfE_min'
thf(fact_1200_qc__wf__relto__iff,axiom,
! [R5: set_Pr4386577575007340137rm_a_b,S5: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ R5 @ S5 ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ R5 @ S5 ) ) @ R5 ) )
=> ( ( wf_term_a_b @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ S5 ) @ ( relcom370159955682700863rm_a_b @ R5 @ ( transi7742714808557438673rm_a_b @ S5 ) ) ) )
= ( wf_term_a_b @ R5 ) ) ) ).
% qc_wf_relto_iff
thf(fact_1201_bsqr__max2,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,A1: term_a_b,A2: term_a_b,B1: term_a_b,B22: term_a_b] :
( ( order_5719896216891381494rm_a_b @ ( field_term_a_b @ R2 ) @ R2 )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A2 ) @ ( produc7020197800436672577rm_a_b @ B1 @ B22 ) ) @ ( bNF_We8522098035112346167rm_a_b @ R2 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( bNF_We4618759378340593734rm_a_b @ R2 @ A1 @ A2 ) @ ( bNF_We4618759378340593734rm_a_b @ R2 @ B1 @ B22 ) ) @ R2 ) ) ) ).
% bsqr_max2
thf(fact_1202_in__lex__prod,axiom,
! [A3: term_a_b,B2: term_a_b,A6: term_a_b,B4: term_a_b,R2: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B2 ) @ ( produc7020197800436672577rm_a_b @ A6 @ B4 ) ) @ ( lex_pr6163557265797435481rm_a_b @ R2 @ S ) )
= ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ A6 ) @ R2 )
| ( ( A3 = A6 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ B4 ) @ S ) ) ) ) ).
% in_lex_prod
thf(fact_1203_non__strict__ending,axiom,
! [T: nat > term_a_b,R5: set_Pr4386577575007340137rm_a_b,S5: set_Pr4386577575007340137rm_a_b] :
( ! [I2: nat] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( T @ I2 ) @ ( T @ ( suc @ I2 ) ) ) @ ( sup_su6776935440552674877rm_a_b @ R5 @ S5 ) )
=> ( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ R5 @ S5 ) @ S5 )
=> ( ( abstra4720023341729745482rm_a_b @ S5 @ ( insert_term_a_b @ ( T @ zero_zero_nat ) @ bot_bot_set_term_a_b ) )
=> ? [J2: nat] :
! [I3: nat] :
( ( ord_less_eq_nat @ J2 @ I3 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( T @ I3 ) @ ( T @ ( suc @ I3 ) ) ) @ ( minus_5192120951422937424rm_a_b @ R5 @ S5 ) ) ) ) ) ) ).
% non_strict_ending
thf(fact_1204_not__SN__onI,axiom,
! [F: nat > produc357393685978478089rm_a_b,X4: set_Pr4386577575007340137rm_a_b,R5: set_Pr2972776593051762503rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( F @ zero_zero_nat ) @ X4 )
=> ( ! [I2: nat] : ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( F @ I2 ) @ ( F @ ( suc @ I2 ) ) ) @ R5 )
=> ~ ( abstra2398554102055911763rm_a_b @ R5 @ X4 ) ) ) ).
% not_SN_onI
thf(fact_1205_not__SN__onI,axiom,
! [F: nat > term_a_b,X4: set_term_a_b,R5: set_Pr4386577575007340137rm_a_b] :
( ( member_term_a_b @ ( F @ zero_zero_nat ) @ X4 )
=> ( ! [I2: nat] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F @ I2 ) @ ( F @ ( suc @ I2 ) ) ) @ R5 )
=> ~ ( abstra4720023341729745482rm_a_b @ R5 @ X4 ) ) ) ).
% not_SN_onI
thf(fact_1206_SN__onI,axiom,
! [A4: set_Pr4386577575007340137rm_a_b,R2: set_Pr2972776593051762503rm_a_b] :
( ! [F5: nat > produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( F5 @ zero_zero_nat ) @ A4 )
=> ~ ! [I3: nat] : ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( F5 @ I3 ) @ ( F5 @ ( suc @ I3 ) ) ) @ R2 ) )
=> ( abstra2398554102055911763rm_a_b @ R2 @ A4 ) ) ).
% SN_onI
thf(fact_1207_SN__onI,axiom,
! [A4: set_term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ! [F5: nat > term_a_b] :
( ( member_term_a_b @ ( F5 @ zero_zero_nat ) @ A4 )
=> ~ ! [I3: nat] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F5 @ I3 ) @ ( F5 @ ( suc @ I3 ) ) ) @ R2 ) )
=> ( abstra4720023341729745482rm_a_b @ R2 @ A4 ) ) ).
% SN_onI
thf(fact_1208_relpow__0__E,axiom,
! [X: term_a_b,Y: term_a_b,R5: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( compow4057154403645558940rm_a_b @ zero_zero_nat @ R5 ) )
=> ( X = Y ) ) ).
% relpow_0_E
thf(fact_1209_relpow__0__I,axiom,
! [X: term_a_b,R5: set_Pr4386577575007340137rm_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ X ) @ ( compow4057154403645558940rm_a_b @ zero_zero_nat @ R5 ) ) ).
% relpow_0_I
thf(fact_1210_relpow__Suc__E,axiom,
! [X: term_a_b,Z2: term_a_b,N: nat,R5: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z2 ) @ ( compow4057154403645558940rm_a_b @ ( suc @ N ) @ R5 ) )
=> ~ ! [Y2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y2 ) @ ( compow4057154403645558940rm_a_b @ N @ R5 ) )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y2 @ Z2 ) @ R5 ) ) ) ).
% relpow_Suc_E
thf(fact_1211_relpow__Suc__I,axiom,
! [X: term_a_b,Y: term_a_b,N: nat,R5: set_Pr4386577575007340137rm_a_b,Z2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( compow4057154403645558940rm_a_b @ N @ R5 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ Z2 ) @ R5 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z2 ) @ ( compow4057154403645558940rm_a_b @ ( suc @ N ) @ R5 ) ) ) ) ).
% relpow_Suc_I
thf(fact_1212_relpow__Suc__D2,axiom,
! [X: term_a_b,Z2: term_a_b,N: nat,R5: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z2 ) @ ( compow4057154403645558940rm_a_b @ ( suc @ N ) @ R5 ) )
=> ? [Y2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y2 ) @ R5 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y2 @ Z2 ) @ ( compow4057154403645558940rm_a_b @ N @ R5 ) ) ) ) ).
% relpow_Suc_D2
thf(fact_1213_relpow__Suc__E2,axiom,
! [X: term_a_b,Z2: term_a_b,N: nat,R5: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z2 ) @ ( compow4057154403645558940rm_a_b @ ( suc @ N ) @ R5 ) )
=> ~ ! [Y2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y2 ) @ R5 )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y2 @ Z2 ) @ ( compow4057154403645558940rm_a_b @ N @ R5 ) ) ) ) ).
% relpow_Suc_E2
thf(fact_1214_relpow__Suc__I2,axiom,
! [X: term_a_b,Y: term_a_b,R5: set_Pr4386577575007340137rm_a_b,Z2: term_a_b,N: nat] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ R5 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ Z2 ) @ ( compow4057154403645558940rm_a_b @ N @ R5 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z2 ) @ ( compow4057154403645558940rm_a_b @ ( suc @ N ) @ R5 ) ) ) ) ).
% relpow_Suc_I2
thf(fact_1215_relpow__E,axiom,
! [X: term_a_b,Z2: term_a_b,N: nat,R5: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z2 ) @ ( compow4057154403645558940rm_a_b @ N @ R5 ) )
=> ( ( ( N = zero_zero_nat )
=> ( X != Z2 ) )
=> ~ ! [Y2: term_a_b,M2: nat] :
( ( N
= ( suc @ M2 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y2 ) @ ( compow4057154403645558940rm_a_b @ M2 @ R5 ) )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y2 @ Z2 ) @ R5 ) ) ) ) ) ).
% relpow_E
thf(fact_1216_relpow__E2,axiom,
! [X: term_a_b,Z2: term_a_b,N: nat,R5: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z2 ) @ ( compow4057154403645558940rm_a_b @ N @ R5 ) )
=> ( ( ( N = zero_zero_nat )
=> ( X != Z2 ) )
=> ~ ! [Y2: term_a_b,M2: nat] :
( ( N
= ( suc @ M2 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y2 ) @ R5 )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y2 @ Z2 ) @ ( compow4057154403645558940rm_a_b @ M2 @ R5 ) ) ) ) ) ) ).
% relpow_E2
thf(fact_1217_not__SN__onE,axiom,
! [R2: set_Pr2972776593051762503rm_a_b,A4: set_Pr4386577575007340137rm_a_b] :
( ~ ( abstra2398554102055911763rm_a_b @ R2 @ A4 )
=> ~ ! [F5: nat > produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( F5 @ zero_zero_nat ) @ A4 )
=> ~ ! [I3: nat] : ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( F5 @ I3 ) @ ( F5 @ ( suc @ I3 ) ) ) @ R2 ) ) ) ).
% not_SN_onE
thf(fact_1218_not__SN__onE,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,A4: set_term_a_b] :
( ~ ( abstra4720023341729745482rm_a_b @ R2 @ A4 )
=> ~ ! [F5: nat > term_a_b] :
( ( member_term_a_b @ ( F5 @ zero_zero_nat ) @ A4 )
=> ~ ! [I3: nat] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F5 @ I3 ) @ ( F5 @ ( suc @ I3 ) ) ) @ R2 ) ) ) ).
% not_SN_onE
thf(fact_1219_SN__on__def,axiom,
( abstra2398554102055911763rm_a_b
= ( ^ [R4: set_Pr2972776593051762503rm_a_b,A7: set_Pr4386577575007340137rm_a_b] :
~ ? [F3: nat > produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( F3 @ zero_zero_nat ) @ A7 )
& ! [I4: nat] : ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( F3 @ I4 ) @ ( F3 @ ( suc @ I4 ) ) ) @ R4 ) ) ) ) ).
% SN_on_def
thf(fact_1220_SN__on__def,axiom,
( abstra4720023341729745482rm_a_b
= ( ^ [R4: set_Pr4386577575007340137rm_a_b,A7: set_term_a_b] :
~ ? [F3: nat > term_a_b] :
( ( member_term_a_b @ ( F3 @ zero_zero_nat ) @ A7 )
& ! [I4: nat] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F3 @ I4 ) @ ( F3 @ ( suc @ I4 ) ) ) @ R4 ) ) ) ) ).
% SN_on_def
thf(fact_1221_SN__onE,axiom,
! [R2: set_Pr2972776593051762503rm_a_b,A4: set_Pr4386577575007340137rm_a_b] :
( ( abstra2398554102055911763rm_a_b @ R2 @ A4 )
=> ~ ? [F6: nat > produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( F6 @ zero_zero_nat ) @ A4 )
& ! [I2: nat] : ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( F6 @ I2 ) @ ( F6 @ ( suc @ I2 ) ) ) @ R2 ) ) ) ).
% SN_onE
thf(fact_1222_SN__onE,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,A4: set_term_a_b] :
( ( abstra4720023341729745482rm_a_b @ R2 @ A4 )
=> ~ ? [F6: nat > term_a_b] :
( ( member_term_a_b @ ( F6 @ zero_zero_nat ) @ A4 )
& ! [I2: nat] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F6 @ I2 ) @ ( F6 @ ( suc @ I2 ) ) ) @ R2 ) ) ) ).
% SN_onE
thf(fact_1223_Abstract__Rewriting_Ochain__mono,axiom,
! [R8: set_Pr4386577575007340137rm_a_b,R5: set_Pr4386577575007340137rm_a_b,Seq: nat > term_a_b] :
( ( ord_le118470702582115849rm_a_b @ R8 @ R5 )
=> ( ! [I2: nat] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( Seq @ I2 ) @ ( Seq @ ( suc @ I2 ) ) ) @ R8 )
=> ! [I3: nat] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( Seq @ I3 ) @ ( Seq @ ( suc @ I3 ) ) ) @ R5 ) ) ) ).
% Abstract_Rewriting.chain_mono
thf(fact_1224_relpow__Suc__E2_H,axiom,
! [X: term_a_b,Z2: term_a_b,N: nat,A4: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z2 ) @ ( compow4057154403645558940rm_a_b @ ( suc @ N ) @ A4 ) )
=> ~ ! [Y2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y2 ) @ A4 )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y2 @ Z2 ) @ ( transi7742714808557438673rm_a_b @ A4 ) ) ) ) ).
% relpow_Suc_E2'
thf(fact_1225_chain__imp__not__SN__on,axiom,
! [F: nat > term_a_b,R2: set_Pr4386577575007340137rm_a_b,I: nat] :
( ! [I2: nat] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F @ I2 ) @ ( F @ ( suc @ I2 ) ) ) @ R2 )
=> ~ ( abstra4720023341729745482rm_a_b @ R2 @ ( insert_term_a_b @ ( F @ I ) @ bot_bot_set_term_a_b ) ) ) ).
% chain_imp_not_SN_on
thf(fact_1226_chain__imp__rtrancl,axiom,
! [S5: nat > term_a_b,R2: set_Pr4386577575007340137rm_a_b,I: nat,J: nat] :
( ! [I2: nat] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( S5 @ I2 ) @ ( S5 @ ( suc @ I2 ) ) ) @ R2 )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( S5 @ I ) @ ( S5 @ J ) ) @ ( transi7742714808557438673rm_a_b @ R2 ) ) ) ) ).
% chain_imp_rtrancl
thf(fact_1227_wf__iff__no__infinite__down__chain,axiom,
( wf_term_a_b
= ( ^ [R4: set_Pr4386577575007340137rm_a_b] :
~ ? [F3: nat > term_a_b] :
! [I4: nat] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F3 @ ( suc @ I4 ) ) @ ( F3 @ I4 ) ) @ R4 ) ) ) ).
% wf_iff_no_infinite_down_chain
thf(fact_1228_wf__no__infinite__down__chainE,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,F: nat > term_a_b] :
( ( wf_term_a_b @ R2 )
=> ~ ! [K: nat] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F @ ( suc @ K ) ) @ ( F @ K ) ) @ R2 ) ) ).
% wf_no_infinite_down_chainE
thf(fact_1229_chain__Un__SN__on__imp__first__step,axiom,
! [T: nat > term_a_b,R5: set_Pr4386577575007340137rm_a_b,S5: set_Pr4386577575007340137rm_a_b] :
( ! [I2: nat] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( T @ I2 ) @ ( T @ ( suc @ I2 ) ) ) @ ( sup_su6776935440552674877rm_a_b @ R5 @ S5 ) )
=> ( ( abstra4720023341729745482rm_a_b @ S5 @ ( insert_term_a_b @ ( T @ zero_zero_nat ) @ bot_bot_set_term_a_b ) )
=> ? [I2: nat] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( T @ I2 ) @ ( T @ ( suc @ I2 ) ) ) @ R5 )
& ! [J3: nat] :
( ( ord_less_nat @ J3 @ I2 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( T @ J3 ) @ ( T @ ( suc @ J3 ) ) ) @ S5 )
& ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( T @ J3 ) @ ( T @ ( suc @ J3 ) ) ) @ R5 ) ) ) ) ) ) ).
% chain_Un_SN_on_imp_first_step
thf(fact_1230_underS__incl__iff,axiom,
! [R2: set_Pr2972776593051762503rm_a_b,A3: produc357393685978478089rm_a_b,B2: produc357393685978478089rm_a_b] :
( ( order_681589137112254398rm_a_b @ ( field_6884932134483023318rm_a_b @ R2 ) @ R2 )
=> ( ( member5869715511025134514rm_a_b @ A3 @ ( field_6884932134483023318rm_a_b @ R2 ) )
=> ( ( member5869715511025134514rm_a_b @ B2 @ ( field_6884932134483023318rm_a_b @ R2 ) )
=> ( ( ord_le118470702582115849rm_a_b @ ( order_8163520290499871439rm_a_b @ R2 @ A3 ) @ ( order_8163520290499871439rm_a_b @ R2 @ B2 ) )
= ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A3 @ B2 ) @ R2 ) ) ) ) ) ).
% underS_incl_iff
thf(fact_1231_underS__incl__iff,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,A3: term_a_b,B2: term_a_b] :
( ( order_5388802246213473311rm_a_b @ ( field_term_a_b @ R2 ) @ R2 )
=> ( ( member_term_a_b @ A3 @ ( field_term_a_b @ R2 ) )
=> ( ( member_term_a_b @ B2 @ ( field_term_a_b @ R2 ) )
=> ( ( ord_le2705286416250468010rm_a_b @ ( order_1619770106104096206rm_a_b @ R2 @ A3 ) @ ( order_1619770106104096206rm_a_b @ R2 @ B2 ) )
= ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B2 ) @ R2 ) ) ) ) ) ).
% underS_incl_iff
thf(fact_1232_underS__I,axiom,
! [I: produc357393685978478089rm_a_b,J: produc357393685978478089rm_a_b,R5: set_Pr2972776593051762503rm_a_b] :
( ( I != J )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ I @ J ) @ R5 )
=> ( member5869715511025134514rm_a_b @ I @ ( order_8163520290499871439rm_a_b @ R5 @ J ) ) ) ) ).
% underS_I
thf(fact_1233_underS__I,axiom,
! [I: term_a_b,J: term_a_b,R5: set_Pr4386577575007340137rm_a_b] :
( ( I != J )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ I @ J ) @ R5 )
=> ( member_term_a_b @ I @ ( order_1619770106104096206rm_a_b @ R5 @ J ) ) ) ) ).
% underS_I
thf(fact_1234_underS__E,axiom,
! [I: produc357393685978478089rm_a_b,R5: set_Pr2972776593051762503rm_a_b,J: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ I @ ( order_8163520290499871439rm_a_b @ R5 @ J ) )
=> ( ( I != J )
& ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ I @ J ) @ R5 ) ) ) ).
% underS_E
thf(fact_1235_underS__E,axiom,
! [I: term_a_b,R5: set_Pr4386577575007340137rm_a_b,J: term_a_b] :
( ( member_term_a_b @ I @ ( order_1619770106104096206rm_a_b @ R5 @ J ) )
=> ( ( I != J )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ I @ J ) @ R5 ) ) ) ).
% underS_E
thf(fact_1236_BNF__Least__Fixpoint_OunderS__Field,axiom,
! [I: produc357393685978478089rm_a_b,R5: set_Pr2972776593051762503rm_a_b,J: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ I @ ( order_8163520290499871439rm_a_b @ R5 @ J ) )
=> ( member5869715511025134514rm_a_b @ I @ ( field_6884932134483023318rm_a_b @ R5 ) ) ) ).
% BNF_Least_Fixpoint.underS_Field
thf(fact_1237_underS__empty,axiom,
! [A3: produc357393685978478089rm_a_b,R2: set_Pr2972776593051762503rm_a_b] :
( ~ ( member5869715511025134514rm_a_b @ A3 @ ( field_6884932134483023318rm_a_b @ R2 ) )
=> ( ( order_8163520290499871439rm_a_b @ R2 @ A3 )
= bot_bo197521221353338581rm_a_b ) ) ).
% underS_empty
thf(fact_1238_chain__imp__trancl,axiom,
! [S5: nat > term_a_b,R2: set_Pr4386577575007340137rm_a_b,I: nat,J: nat] :
( ! [I2: nat] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( S5 @ I2 ) @ ( S5 @ ( suc @ I2 ) ) ) @ R2 )
=> ( ( ord_less_nat @ I @ J )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( S5 @ I ) @ ( S5 @ J ) ) @ ( transi7922773638565587891rm_a_b @ R2 ) ) ) ) ).
% chain_imp_trancl
thf(fact_1239_wf__bounded__measure,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,Ub: term_a_b > nat,F: term_a_b > nat] :
( ! [A: term_a_b,B: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ R2 )
=> ( ( ord_less_eq_nat @ ( Ub @ B ) @ ( Ub @ A ) )
& ( ord_less_eq_nat @ ( F @ B ) @ ( Ub @ A ) )
& ( ord_less_nat @ ( F @ A ) @ ( F @ B ) ) ) )
=> ( wf_term_a_b @ R2 ) ) ).
% wf_bounded_measure
thf(fact_1240_trancl__power,axiom,
! [P: produc357393685978478089rm_a_b,R5: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ P @ ( transi7922773638565587891rm_a_b @ R5 ) )
= ( ? [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
& ( member5869715511025134514rm_a_b @ P @ ( compow4057154403645558940rm_a_b @ N2 @ R5 ) ) ) ) ) ).
% trancl_power
thf(fact_1241_rtrancl__imp__seq,axiom,
! [X: term_a_b,Y: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
=> ? [F5: nat > term_a_b,N3: nat] :
( ( ( F5 @ zero_zero_nat )
= X )
& ( ( F5 @ N3 )
= Y )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ N3 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F5 @ I3 ) @ ( F5 @ ( suc @ I3 ) ) ) @ R2 ) ) ) ) ).
% rtrancl_imp_seq
thf(fact_1242_rtrancl__fun__conv,axiom,
! [S: term_a_b,T: term_a_b,R5: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( transi7742714808557438673rm_a_b @ R5 ) )
= ( ? [F3: nat > term_a_b,N2: nat] :
( ( ( F3 @ zero_zero_nat )
= S )
& ( ( F3 @ N2 )
= T )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ N2 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F3 @ I4 ) @ ( F3 @ ( suc @ I4 ) ) ) @ R5 ) ) ) ) ) ).
% rtrancl_fun_conv
thf(fact_1243_relpow__seq,axiom,
! [X: term_a_b,Y: term_a_b,N: nat,R2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( compow4057154403645558940rm_a_b @ N @ R2 ) )
=> ? [F5: nat > term_a_b] :
( ( ( F5 @ zero_zero_nat )
= X )
& ( ( F5 @ N )
= Y )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F5 @ I3 ) @ ( F5 @ ( suc @ I3 ) ) ) @ R2 ) ) ) ) ).
% relpow_seq
thf(fact_1244_relpow__fun__conv,axiom,
! [A3: term_a_b,B2: term_a_b,N: nat,R5: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B2 ) @ ( compow4057154403645558940rm_a_b @ N @ R5 ) )
= ( ? [F3: nat > term_a_b] :
( ( ( F3 @ zero_zero_nat )
= A3 )
& ( ( F3 @ N )
= B2 )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ N )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F3 @ I4 ) @ ( F3 @ ( suc @ I4 ) ) ) @ R5 ) ) ) ) ) ).
% relpow_fun_conv
thf(fact_1245_in__measure,axiom,
! [X: term_a_b,Y: term_a_b,F: term_a_b > nat] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( measure_term_a_b @ F ) )
= ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) ) ) ).
% in_measure
thf(fact_1246_Refl__under__underS,axiom,
! [R2: set_Pr2972776593051762503rm_a_b,A3: produc357393685978478089rm_a_b] :
( ( refl_o5344404488616963464rm_a_b @ ( field_6884932134483023318rm_a_b @ R2 ) @ R2 )
=> ( ( member5869715511025134514rm_a_b @ A3 @ ( field_6884932134483023318rm_a_b @ R2 ) )
=> ( ( order_346605463002971118rm_a_b @ R2 @ A3 )
= ( sup_su6776935440552674877rm_a_b @ ( order_8163520290499871439rm_a_b @ R2 @ A3 ) @ ( insert7009541432154983385rm_a_b @ A3 @ bot_bo197521221353338581rm_a_b ) ) ) ) ) ).
% Refl_under_underS
thf(fact_1247_psubset__imp__ex__mem,axiom,
! [A4: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
( ( ord_le4883060154188259093rm_a_b @ A4 @ B5 )
=> ? [B: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ B @ ( minus_5192120951422937424rm_a_b @ B5 @ A4 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_1248_psubset__insert__iff,axiom,
! [A4: set_Pr4386577575007340137rm_a_b,X: produc357393685978478089rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
( ( ord_le4883060154188259093rm_a_b @ A4 @ ( insert7009541432154983385rm_a_b @ X @ B5 ) )
= ( ( ( member5869715511025134514rm_a_b @ X @ B5 )
=> ( ord_le4883060154188259093rm_a_b @ A4 @ B5 ) )
& ( ~ ( member5869715511025134514rm_a_b @ X @ B5 )
=> ( ( ( member5869715511025134514rm_a_b @ X @ A4 )
=> ( ord_le4883060154188259093rm_a_b @ ( minus_5192120951422937424rm_a_b @ A4 @ ( insert7009541432154983385rm_a_b @ X @ bot_bo197521221353338581rm_a_b ) ) @ B5 ) )
& ( ~ ( member5869715511025134514rm_a_b @ X @ A4 )
=> ( ord_le118470702582115849rm_a_b @ A4 @ B5 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_1249_ground__ctxt__substI,axiom,
! [C3: subter6147397892252016653rm_a_b,Sigma: produc357393685978478089rm_a_b > term_a_b] :
( ! [X3: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X3 @ ( subter7674038538283327737rm_a_b @ C3 ) )
=> ( term_ground_a_b @ ( Sigma @ X3 ) ) )
=> ( term_ground_ctxt_a_b @ ( subter7224274371358213697_a_b_b @ C3 @ Sigma ) ) ) ).
% ground_ctxt_substI
thf(fact_1250_psubsetD,axiom,
! [A4: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b,C: produc357393685978478089rm_a_b] :
( ( ord_le4883060154188259093rm_a_b @ A4 @ B5 )
=> ( ( member5869715511025134514rm_a_b @ C @ A4 )
=> ( member5869715511025134514rm_a_b @ C @ B5 ) ) ) ).
% psubsetD
thf(fact_1251_Range__insert,axiom,
! [A3: term_a_b,B2: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ( range_8688746797885257759rm_a_b @ ( insert7009541432154983385rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B2 ) @ R2 ) )
= ( insert_term_a_b @ B2 @ ( range_8688746797885257759rm_a_b @ R2 ) ) ) ).
% Range_insert
thf(fact_1252_mlex__leq,axiom,
! [F: term_a_b > nat,X: term_a_b,Y: term_a_b,R5: set_Pr4386577575007340137rm_a_b] :
( ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ R5 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( mlex_prod_term_a_b @ F @ R5 ) ) ) ) ).
% mlex_leq
thf(fact_1253_Range__rtrancl,axiom,
! [R5: set_Pr4386577575007340137rm_a_b] :
( ( range_8688746797885257759rm_a_b @ ( transi7742714808557438673rm_a_b @ R5 ) )
= top_top_set_term_a_b ) ).
% Range_rtrancl
thf(fact_1254_Range_Ocases,axiom,
! [A3: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ( member_term_a_b @ A3 @ ( range_8688746797885257759rm_a_b @ R2 ) )
=> ~ ! [A: term_a_b] :
~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ A3 ) @ R2 ) ) ).
% Range.cases
thf(fact_1255_Range_Osimps,axiom,
! [A3: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ( member_term_a_b @ A3 @ ( range_8688746797885257759rm_a_b @ R2 ) )
= ( ? [A5: term_a_b,B3: term_a_b] :
( ( A3 = B3 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A5 @ B3 ) @ R2 ) ) ) ) ).
% Range.simps
thf(fact_1256_Range_Ointros,axiom,
! [A3: term_a_b,B2: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B2 ) @ R2 )
=> ( member_term_a_b @ B2 @ ( range_8688746797885257759rm_a_b @ R2 ) ) ) ).
% Range.intros
thf(fact_1257_RangeE,axiom,
! [B2: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ( member_term_a_b @ B2 @ ( range_8688746797885257759rm_a_b @ R2 ) )
=> ~ ! [A: term_a_b] :
~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B2 ) @ R2 ) ) ).
% RangeE
thf(fact_1258_Range__iff,axiom,
! [A3: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ( member_term_a_b @ A3 @ ( range_8688746797885257759rm_a_b @ R2 ) )
= ( ? [Y5: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y5 @ A3 ) @ R2 ) ) ) ).
% Range_iff
thf(fact_1259_mlex__less,axiom,
! [F: term_a_b > nat,X: term_a_b,Y: term_a_b,R5: set_Pr4386577575007340137rm_a_b] :
( ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( mlex_prod_term_a_b @ F @ R5 ) ) ) ).
% mlex_less
thf(fact_1260_mlex__iff,axiom,
! [X: term_a_b,Y: term_a_b,F: term_a_b > nat,R5: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( mlex_prod_term_a_b @ F @ R5 ) )
= ( ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) )
| ( ( ( F @ X )
= ( F @ Y ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ R5 ) ) ) ) ).
% mlex_iff
thf(fact_1261_Domain__rtrancl,axiom,
! [R5: set_Pr4386577575007340137rm_a_b] :
( ( domain6518119340328324150rm_a_b @ ( transi7742714808557438673rm_a_b @ R5 ) )
= top_top_set_term_a_b ) ).
% Domain_rtrancl
thf(fact_1262_Domain__insert,axiom,
! [A3: term_a_b,B2: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ( domain6518119340328324150rm_a_b @ ( insert7009541432154983385rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B2 ) @ R2 ) )
= ( insert_term_a_b @ A3 @ ( domain6518119340328324150rm_a_b @ R2 ) ) ) ).
% Domain_insert
thf(fact_1263_Domain__iff,axiom,
! [A3: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ( member_term_a_b @ A3 @ ( domain6518119340328324150rm_a_b @ R2 ) )
= ( ? [Y5: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ Y5 ) @ R2 ) ) ) ).
% Domain_iff
thf(fact_1264_DomainE,axiom,
! [A3: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ( member_term_a_b @ A3 @ ( domain6518119340328324150rm_a_b @ R2 ) )
=> ~ ! [B: term_a_b] :
~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B ) @ R2 ) ) ).
% DomainE
thf(fact_1265_Domain_ODomainI,axiom,
! [A3: term_a_b,B2: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B2 ) @ R2 )
=> ( member_term_a_b @ A3 @ ( domain6518119340328324150rm_a_b @ R2 ) ) ) ).
% Domain.DomainI
thf(fact_1266_Domain_Osimps,axiom,
! [A3: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ( member_term_a_b @ A3 @ ( domain6518119340328324150rm_a_b @ R2 ) )
= ( ? [A5: term_a_b,B3: term_a_b] :
( ( A3 = A5 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A5 @ B3 ) @ R2 ) ) ) ) ).
% Domain.simps
thf(fact_1267_Domain_Ocases,axiom,
! [A3: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ( member_term_a_b @ A3 @ ( domain6518119340328324150rm_a_b @ R2 ) )
=> ~ ! [B: term_a_b] :
~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B ) @ R2 ) ) ).
% Domain.cases
thf(fact_1268_Not__Domain__rtrancl,axiom,
! [X: produc357393685978478089rm_a_b,R5: set_Pr2972776593051762503rm_a_b,Y: produc357393685978478089rm_a_b] :
( ~ ( member5869715511025134514rm_a_b @ X @ ( domain7769946957726046796rm_a_b @ R5 ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ ( transi2615809358984392588rm_a_b @ R5 ) )
= ( X = Y ) ) ) ).
% Not_Domain_rtrancl
thf(fact_1269_Not__Domain__rtrancl,axiom,
! [X: term_a_b,R5: set_Pr4386577575007340137rm_a_b,Y: term_a_b] :
( ~ ( member_term_a_b @ X @ ( domain6518119340328324150rm_a_b @ R5 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ R5 ) )
= ( X = Y ) ) ) ).
% Not_Domain_rtrancl
thf(fact_1270_gterm__of__term__inj,axiom,
! [S5: set_term_a_b] :
( ! [T4: term_a_b] :
( ( member_term_a_b @ T4 @ S5 )
=> ( term_ground_a_b @ T4 ) )
=> ( inj_on1763302225069724520term_a @ ground3594116404241818627rm_a_b @ S5 ) ) ).
% gterm_of_term_inj
thf(fact_1271_apsnd__conv,axiom,
! [F: term_a_b > term_a_b,X: term_a_b,Y: term_a_b] :
( ( produc6963505556121627836rm_a_b @ F @ ( produc7020197800436672577rm_a_b @ X @ Y ) )
= ( produc7020197800436672577rm_a_b @ X @ ( F @ Y ) ) ) ).
% apsnd_conv
thf(fact_1272_apfst__conv,axiom,
! [F: term_a_b > term_a_b,X: term_a_b,Y: term_a_b] :
( ( produc4217944709291808634rm_a_b @ F @ ( produc7020197800436672577rm_a_b @ X @ Y ) )
= ( produc7020197800436672577rm_a_b @ ( F @ X ) @ Y ) ) ).
% apfst_conv
thf(fact_1273_ground__vars__term__empty,axiom,
! [T: term_a_b] :
( ( term_ground_a_b @ T )
=> ( ( vars_term_a_b @ T )
= bot_bot_set_b ) ) ).
% ground_vars_term_empty
thf(fact_1274_ground__substI,axiom,
! [S: term_a7872195449889686814rm_a_b,Sigma: produc357393685978478089rm_a_b > term_a_b] :
( ! [X3: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X3 @ ( vars_t1977812157029556855rm_a_b @ S ) )
=> ( term_ground_a_b @ ( Sigma @ X3 ) ) )
=> ( term_ground_a_b @ ( subst_4153759000330037794_a_b_b @ S @ Sigma ) ) ) ).
% ground_substI
% Helper facts (3)
thf(help_If_3_1_If_001t__Term__Oterm_Itf__a_Mtf__b_J_T,axiom,
! [P2: $o] :
( ( P2 = $true )
| ( P2 = $false ) ) ).
thf(help_If_2_1_If_001t__Term__Oterm_Itf__a_Mtf__b_J_T,axiom,
! [X: term_a_b,Y: term_a_b] :
( ( if_term_a_b @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Term__Oterm_Itf__a_Mtf__b_J_T,axiom,
! [X: term_a_b,Y: term_a_b] :
( ( if_term_a_b @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
term_ground_a_b @ ta ).
%------------------------------------------------------------------------------