TPTP Problem File: SLH0051^1.p
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%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Rewrite_Properties_Reduction/0020_Rewriting_Properties/prob_00046_001611__14083462_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1542 ( 355 unt; 266 typ; 0 def)
% Number of atoms : 4072 ( 789 equ; 0 cnn)
% Maximal formula atoms : 12 ( 3 avg)
% Number of connectives : 16311 ( 440 ~; 45 |; 262 &;13273 @)
% ( 0 <=>;2291 =>; 0 <=; 0 <~>)
% Maximal formula depth : 22 ( 9 avg)
% Number of types : 33 ( 32 usr)
% Number of type conns : 613 ( 613 >; 0 *; 0 +; 0 <<)
% Number of symbols : 235 ( 234 usr; 17 con; 0-3 aty)
% Number of variables : 4489 ( 230 ^;4096 !; 163 ?;4489 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 14:19:07.583
%------------------------------------------------------------------------------
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thf(sy_c_Terms__Positions_Oinv__const__ctxt_001tf__f_001tf__v,type,
terms_8308829228547689406xt_f_v: set_Pr5245412377734449720_f_nat > v > subterm_and_ctxt_f_v > subterm_and_ctxt_f_v ).
thf(sy_c_Terms__Positions_Oinv__const__ctxt_H_001tf__f_001tf__v,type,
terms_4639549856110724071xt_f_v: set_Pr5245412377734449720_f_nat > v > subterm_and_ctxt_f_v > term_f_v ).
thf(sy_c_Terms__Positions_Oterm__to__sig_001tf__f_001tf__v,type,
terms_3805575757503158827ig_f_v: set_Pr5245412377734449720_f_nat > v > term_f_v > term_f_v ).
thf(sy_c_Transitive__Closure_Oacyclic_001t__Term__Oterm_Itf__f_Mtf__v_J,type,
transi600795472790740820rm_f_v: set_Pr8827868859434726151rm_f_v > $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__f_Mtf__v_J_Mt__Term__Oterm_Itf__f_Mtf__v_J_J_Mt__Product____Type__Oprod_It__Term__Oterm_Itf__f_Mtf__v_J_Mt__Term__Oterm_Itf__f_Mtf__v_J_J_J,type,
transi4215105848801502698rm_f_v: set_Pr4964659734097919175rm_f_v > set_Pr4964659734097919175rm_f_v ).
thf(sy_c_Transitive__Closure_Ortrancl_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__f_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_Itf__f_Mt__Nat__Onat_J_J,type,
transi7343684842217456360_f_nat: set_Pr8434786782240690247_f_nat > set_Pr8434786782240690247_f_nat ).
thf(sy_c_Transitive__Closure_Ortrancl_001t__Product____Type__Oprod_It__Term__Oterm_Itf__f_Mtf__v_J_Mt__Term__Oterm_Itf__f_Mtf__v_J_J,type,
transi1062743709496199978rm_f_v: set_Pr7897041799657265351rm_f_v > set_Pr7897041799657265351rm_f_v ).
thf(sy_c_Transitive__Closure_Ortrancl_001t__Product____Type__Oprod_Itf__f_Mt__Nat__Onat_J,type,
transi8903367562120364223_f_nat: set_Pr516226060960033221_f_nat > set_Pr516226060960033221_f_nat ).
thf(sy_c_Transitive__Closure_Ortrancl_001t__Term__Oterm_Itf__f_Mtf__v_J,type,
transi3028809021613410336rm_f_v: set_Pr8827868859434726151rm_f_v > set_Pr8827868859434726151rm_f_v ).
thf(sy_c_Transitive__Closure_Ortrancl_001tf__f,type,
transitive_rtrancl_f: set_Product_prod_f_f > set_Product_prod_f_f ).
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__f_Mtf__v_J_Mt__Term__Oterm_Itf__f_Mtf__v_J_J_Mt__Product____Type__Oprod_It__Term__Oterm_Itf__f_Mtf__v_J_Mt__Term__Oterm_Itf__f_Mtf__v_J_J_J,type,
transi3443125758348952712rm_f_v: set_Pr4964659734097919175rm_f_v > set_Pr4964659734097919175rm_f_v ).
thf(sy_c_Transitive__Closure_Otrancl_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__f_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_Itf__f_Mt__Nat__Onat_J_J,type,
transi650449548839634566_f_nat: set_Pr8434786782240690247_f_nat > set_Pr8434786782240690247_f_nat ).
thf(sy_c_Transitive__Closure_Otrancl_001t__Product____Type__Oprod_It__Term__Oterm_Itf__f_Mtf__v_J_Mt__Term__Oterm_Itf__f_Mtf__v_J_J,type,
transi7658437189833989320rm_f_v: set_Pr7897041799657265351rm_f_v > set_Pr7897041799657265351rm_f_v ).
thf(sy_c_Transitive__Closure_Otrancl_001t__Product____Type__Oprod_Itf__f_Mt__Nat__Onat_J,type,
transi3759436115690491041_f_nat: set_Pr516226060960033221_f_nat > set_Pr516226060960033221_f_nat ).
thf(sy_c_Transitive__Closure_Otrancl_001t__Term__Oterm_Itf__f_Mtf__v_J,type,
transi3208867851621559554rm_f_v: set_Pr8827868859434726151rm_f_v > set_Pr8827868859434726151rm_f_v ).
thf(sy_c_Transitive__Closure_Otrancl_001tf__f,type,
transitive_trancl_f: set_Product_prod_f_f > set_Product_prod_f_f ).
thf(sy_c_Wellfounded_Olex__prod_001t__Term__Oterm_Itf__f_Mtf__v_J_001t__Term__Oterm_Itf__f_Mtf__v_J,type,
lex_pr1654150031463228279rm_f_v: set_Pr8827868859434726151rm_f_v > set_Pr8827868859434726151rm_f_v > set_Pr7897041799657265351rm_f_v ).
thf(sy_c_Wellfounded_Omeasure_001t__Term__Oterm_Itf__f_Mtf__v_J,type,
measure_term_f_v: ( term_f_v > nat ) > set_Pr8827868859434726151rm_f_v ).
thf(sy_c_Wellfounded_Owf_001t__Product____Type__Oprod_It__Term__Oterm_Itf__f_Mtf__v_J_Mt__Term__Oterm_Itf__f_Mtf__v_J_J,type,
wf_Pro782797968166624016rm_f_v: set_Pr7897041799657265351rm_f_v > $o ).
thf(sy_c_Wellfounded_Owf_001t__Product____Type__Oprod_Itf__f_Mt__Nat__Onat_J,type,
wf_Pro5063733954819197017_f_nat: set_Pr516226060960033221_f_nat > $o ).
thf(sy_c_Wellfounded_Owf_001t__Term__Oterm_Itf__f_Mtf__v_J,type,
wf_term_f_v: set_Pr8827868859434726151rm_f_v > $o ).
thf(sy_c_Wfrec_Osame__fst_001t__Term__Oterm_Itf__f_Mtf__v_J_001t__Term__Oterm_Itf__f_Mtf__v_J,type,
same_f5557695934196986804rm_f_v: ( term_f_v > $o ) > ( term_f_v > set_Pr8827868859434726151rm_f_v ) > set_Pr7897041799657265351rm_f_v ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
member8440522571783428010at_nat: product_prod_nat_nat > set_Pr1261947904930325089at_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mtf__f_J,type,
member8962352073626239679_nat_f: product_prod_nat_f > set_Pr4548512049067879902_nat_f > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Term__Oterm_Itf__f_Mtf__v_J_Mt__Term__Oterm_Itf__f_Mtf__v_J_J_Mt__Product____Type__Oprod_It__Term__Oterm_Itf__f_Mtf__v_J_Mt__Term__Oterm_Itf__f_Mtf__v_J_J_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__Term__Oterm_Itf__f_Mtf__v_J_Mt__Term__Oterm_Itf__f_Mtf__v_J_J_Mt__Product____Type__Oprod_It__Term__Oterm_Itf__f_Mtf__v_J_Mt__Term__Oterm_Itf__f_Mtf__v_J_J_J_J,type,
member1670422607875439376rm_f_v: produc8057771069037231335rm_f_v > set_Pr4964659734097919175rm_f_v > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__f_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_Itf__f_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_Itf__f_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_Itf__f_Mt__Nat__Onat_J_J_J,type,
member3697069702615009936_f_nat: produc1452479431071238503_f_nat > set_Pr8434786782240690247_f_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Term__Oterm_Itf__f_Mtf__v_J_Mt__Term__Oterm_Itf__f_Mtf__v_J_J_Mt__Product____Type__Oprod_It__Term__Oterm_Itf__f_Mtf__v_J_Mt__Term__Oterm_Itf__f_Mtf__v_J_J_J,type,
member7300054593328134928rm_f_v: produc3406425829894461415rm_f_v > set_Pr7897041799657265351rm_f_v > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__f_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_Itf__f_Mt__Nat__Onat_J_J,type,
member7030746907725424910_f_nat: produc2676906022994773093_f_nat > set_Pr516226060960033221_f_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Term__Oterm_Itf__f_Mtf__v_J_Mt__Term__Oterm_Itf__f_Mtf__v_J_J,type,
member4316649861536941904rm_f_v: produc8027700073345061287rm_f_v > set_Pr8827868859434726151rm_f_v > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_Itf__f_Mt__Nat__Onat_J,type,
member2678038196313615385_f_nat: product_prod_f_nat > set_Pr5245412377734449720_f_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_Itf__f_Mtf__f_J,type,
member5935997662887750672od_f_f: product_prod_f_f > set_Product_prod_f_f > $o ).
thf(sy_c_member_001t__Term__Oterm_Itf__f_Mtf__v_J,type,
member_term_f_v: term_f_v > set_term_f_v > $o ).
thf(sy_v_C____,type,
c: subterm_and_ctxt_f_v ).
thf(sy_v__092_060F_062,type,
f2: set_Pr5245412377734449720_f_nat ).
thf(sy_v__092_060R_062,type,
r: set_Pr8827868859434726151rm_f_v ).
thf(sy_v_s____,type,
s: term_f_v ).
thf(sy_v_t____,type,
t: term_f_v ).
thf(sy_v_ta____,type,
ta: term_f_v ).
thf(sy_v_u____,type,
u: term_f_v ).
% Relevant facts (1275)
thf(fact_0_step_Ohyps_I2_J,axiom,
member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ ta @ u ) @ ( sig_step_f_v @ f2 @ ( rstep_f_v @ r ) ) ).
% step.hyps(2)
thf(fact_1_rstep__ruleI,axiom,
! [L: term_f_v,R: term_f_v,R2: set_Pr8827868859434726151rm_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ L @ R ) @ R2 )
=> ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ L @ R ) @ ( rstep_f_v @ R2 ) ) ) ).
% rstep_ruleI
thf(fact_2_step_Ohyps_I1_J,axiom,
member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ s @ ta ) @ ( transi3028809021613410336rm_f_v @ ( sig_step_f_v @ f2 @ ( rstep_f_v @ r ) ) ) ).
% step.hyps(1)
thf(fact_3_prod_Oinject,axiom,
! [X1: produc8027700073345061287rm_f_v,X2: produc8027700073345061287rm_f_v,Y1: produc8027700073345061287rm_f_v,Y2: produc8027700073345061287rm_f_v] :
( ( ( produc3874942763110801239rm_f_v @ X1 @ X2 )
= ( produc3874942763110801239rm_f_v @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_4_prod_Oinject,axiom,
! [X1: f,X2: nat,Y1: f,Y2: nat] :
( ( ( product_Pair_f_nat @ X1 @ X2 )
= ( product_Pair_f_nat @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_5_prod_Oinject,axiom,
! [X1: product_prod_f_nat,X2: product_prod_f_nat,Y1: product_prod_f_nat,Y2: product_prod_f_nat] :
( ( ( produc5861087217798509725_f_nat @ X1 @ X2 )
= ( produc5861087217798509725_f_nat @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_6_prod_Oinject,axiom,
! [X1: term_f_v,X2: term_f_v,Y1: term_f_v,Y2: term_f_v] :
( ( ( produc2510790566102465375rm_f_v @ X1 @ X2 )
= ( produc2510790566102465375rm_f_v @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_7_old_Oprod_Oinject,axiom,
! [A: produc8027700073345061287rm_f_v,B: produc8027700073345061287rm_f_v,A2: produc8027700073345061287rm_f_v,B2: produc8027700073345061287rm_f_v] :
( ( ( produc3874942763110801239rm_f_v @ A @ B )
= ( produc3874942763110801239rm_f_v @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_8_old_Oprod_Oinject,axiom,
! [A: f,B: nat,A2: f,B2: nat] :
( ( ( product_Pair_f_nat @ A @ B )
= ( product_Pair_f_nat @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_9_old_Oprod_Oinject,axiom,
! [A: product_prod_f_nat,B: product_prod_f_nat,A2: product_prod_f_nat,B2: product_prod_f_nat] :
( ( ( produc5861087217798509725_f_nat @ A @ B )
= ( produc5861087217798509725_f_nat @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_10_old_Oprod_Oinject,axiom,
! [A: term_f_v,B: term_f_v,A2: term_f_v,B2: term_f_v] :
( ( ( produc2510790566102465375rm_f_v @ A @ B )
= ( produc2510790566102465375rm_f_v @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_11_steps,axiom,
member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ s @ t ) @ ( transi3028809021613410336rm_f_v @ ( sig_step_f_v @ f2 @ ( rstep_f_v @ r ) ) ) ).
% steps
thf(fact_12_step_Ohyps_I3_J,axiom,
member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ ( subter6886040689604212261rm_f_v @ c @ s ) @ ( subter6886040689604212261rm_f_v @ c @ ta ) ) @ ( transi3028809021613410336rm_f_v @ ( sig_step_f_v @ f2 @ ( rstep_f_v @ r ) ) ) ).
% step.hyps(3)
thf(fact_13_old_Oprod_Oexhaust,axiom,
! [Y: produc3406425829894461415rm_f_v] :
~ ! [A3: produc8027700073345061287rm_f_v,B3: produc8027700073345061287rm_f_v] :
( Y
!= ( produc3874942763110801239rm_f_v @ A3 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_14_old_Oprod_Oexhaust,axiom,
! [Y: product_prod_f_nat] :
~ ! [A3: f,B3: nat] :
( Y
!= ( product_Pair_f_nat @ A3 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_15_old_Oprod_Oexhaust,axiom,
! [Y: produc2676906022994773093_f_nat] :
~ ! [A3: product_prod_f_nat,B3: product_prod_f_nat] :
( Y
!= ( produc5861087217798509725_f_nat @ A3 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_16_old_Oprod_Oexhaust,axiom,
! [Y: produc8027700073345061287rm_f_v] :
~ ! [A3: term_f_v,B3: term_f_v] :
( Y
!= ( produc2510790566102465375rm_f_v @ A3 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_17_surj__pair,axiom,
! [P: produc3406425829894461415rm_f_v] :
? [X: produc8027700073345061287rm_f_v,Y3: produc8027700073345061287rm_f_v] :
( P
= ( produc3874942763110801239rm_f_v @ X @ Y3 ) ) ).
% surj_pair
thf(fact_18_surj__pair,axiom,
! [P: product_prod_f_nat] :
? [X: f,Y3: nat] :
( P
= ( product_Pair_f_nat @ X @ Y3 ) ) ).
% surj_pair
thf(fact_19_surj__pair,axiom,
! [P: produc2676906022994773093_f_nat] :
? [X: product_prod_f_nat,Y3: product_prod_f_nat] :
( P
= ( produc5861087217798509725_f_nat @ X @ Y3 ) ) ).
% surj_pair
thf(fact_20_surj__pair,axiom,
! [P: produc8027700073345061287rm_f_v] :
? [X: term_f_v,Y3: term_f_v] :
( P
= ( produc2510790566102465375rm_f_v @ X @ Y3 ) ) ).
% surj_pair
thf(fact_21_prod__cases,axiom,
! [P2: produc3406425829894461415rm_f_v > $o,P: produc3406425829894461415rm_f_v] :
( ! [A3: produc8027700073345061287rm_f_v,B3: produc8027700073345061287rm_f_v] : ( P2 @ ( produc3874942763110801239rm_f_v @ A3 @ B3 ) )
=> ( P2 @ P ) ) ).
% prod_cases
thf(fact_22_prod__cases,axiom,
! [P2: product_prod_f_nat > $o,P: product_prod_f_nat] :
( ! [A3: f,B3: nat] : ( P2 @ ( product_Pair_f_nat @ A3 @ B3 ) )
=> ( P2 @ P ) ) ).
% prod_cases
thf(fact_23_prod__cases,axiom,
! [P2: produc2676906022994773093_f_nat > $o,P: produc2676906022994773093_f_nat] :
( ! [A3: product_prod_f_nat,B3: product_prod_f_nat] : ( P2 @ ( produc5861087217798509725_f_nat @ A3 @ B3 ) )
=> ( P2 @ P ) ) ).
% prod_cases
thf(fact_24_prod__cases,axiom,
! [P2: produc8027700073345061287rm_f_v > $o,P: produc8027700073345061287rm_f_v] :
( ! [A3: term_f_v,B3: term_f_v] : ( P2 @ ( produc2510790566102465375rm_f_v @ A3 @ B3 ) )
=> ( P2 @ P ) ) ).
% prod_cases
thf(fact_25_Pair__inject,axiom,
! [A: produc8027700073345061287rm_f_v,B: produc8027700073345061287rm_f_v,A2: produc8027700073345061287rm_f_v,B2: produc8027700073345061287rm_f_v] :
( ( ( produc3874942763110801239rm_f_v @ A @ B )
= ( produc3874942763110801239rm_f_v @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_26_Pair__inject,axiom,
! [A: f,B: nat,A2: f,B2: nat] :
( ( ( product_Pair_f_nat @ A @ B )
= ( product_Pair_f_nat @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_27_Pair__inject,axiom,
! [A: product_prod_f_nat,B: product_prod_f_nat,A2: product_prod_f_nat,B2: product_prod_f_nat] :
( ( ( produc5861087217798509725_f_nat @ A @ B )
= ( produc5861087217798509725_f_nat @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_28_Pair__inject,axiom,
! [A: term_f_v,B: term_f_v,A2: term_f_v,B2: term_f_v] :
( ( ( produc2510790566102465375rm_f_v @ A @ B )
= ( produc2510790566102465375rm_f_v @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_29_prod__cases3,axiom,
! [Y: produc3406425829894461415rm_f_v] :
~ ! [A3: produc8027700073345061287rm_f_v,B3: term_f_v,C: term_f_v] :
( Y
!= ( produc3874942763110801239rm_f_v @ A3 @ ( produc2510790566102465375rm_f_v @ B3 @ C ) ) ) ).
% prod_cases3
thf(fact_30_prod__cases3,axiom,
! [Y: produc2676906022994773093_f_nat] :
~ ! [A3: product_prod_f_nat,B3: f,C: nat] :
( Y
!= ( produc5861087217798509725_f_nat @ A3 @ ( product_Pair_f_nat @ B3 @ C ) ) ) ).
% prod_cases3
thf(fact_31_rstep__ctxtI,axiom,
! [S: term_f_v,T: term_f_v,R2: set_Pr8827868859434726151rm_f_v,C2: subterm_and_ctxt_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ ( rstep_f_v @ R2 ) )
=> ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ ( subter6886040689604212261rm_f_v @ C2 @ S ) @ ( subter6886040689604212261rm_f_v @ C2 @ T ) ) @ ( rstep_f_v @ R2 ) ) ) ).
% rstep_ctxtI
thf(fact_32_prod__induct3,axiom,
! [P2: produc3406425829894461415rm_f_v > $o,X3: produc3406425829894461415rm_f_v] :
( ! [A3: produc8027700073345061287rm_f_v,B3: term_f_v,C: term_f_v] : ( P2 @ ( produc3874942763110801239rm_f_v @ A3 @ ( produc2510790566102465375rm_f_v @ B3 @ C ) ) )
=> ( P2 @ X3 ) ) ).
% prod_induct3
thf(fact_33_prod__induct3,axiom,
! [P2: produc2676906022994773093_f_nat > $o,X3: produc2676906022994773093_f_nat] :
( ! [A3: product_prod_f_nat,B3: f,C: nat] : ( P2 @ ( produc5861087217798509725_f_nat @ A3 @ ( product_Pair_f_nat @ B3 @ C ) ) )
=> ( P2 @ X3 ) ) ).
% prod_induct3
thf(fact_34_ctxt__eq,axiom,
! [C2: subterm_and_ctxt_f_v,S: term_f_v,T: term_f_v] :
( ( ( subter6886040689604212261rm_f_v @ C2 @ S )
= ( subter6886040689604212261rm_f_v @ C2 @ T ) )
= ( S = T ) ) ).
% ctxt_eq
thf(fact_35_rtrancl__idemp,axiom,
! [R: set_Pr7897041799657265351rm_f_v] :
( ( transi1062743709496199978rm_f_v @ ( transi1062743709496199978rm_f_v @ R ) )
= ( transi1062743709496199978rm_f_v @ R ) ) ).
% rtrancl_idemp
thf(fact_36_rtrancl__idemp,axiom,
! [R: set_Product_prod_f_f] :
( ( transitive_rtrancl_f @ ( transitive_rtrancl_f @ R ) )
= ( transitive_rtrancl_f @ R ) ) ).
% rtrancl_idemp
thf(fact_37_rtrancl__idemp,axiom,
! [R: set_Pr1261947904930325089at_nat] :
( ( transi2905341329935302413cl_nat @ ( transi2905341329935302413cl_nat @ R ) )
= ( transi2905341329935302413cl_nat @ R ) ) ).
% rtrancl_idemp
thf(fact_38_rtrancl__idemp,axiom,
! [R: set_Pr8827868859434726151rm_f_v] :
( ( transi3028809021613410336rm_f_v @ ( transi3028809021613410336rm_f_v @ R ) )
= ( transi3028809021613410336rm_f_v @ R ) ) ).
% rtrancl_idemp
thf(fact_39_r__into__rtrancl,axiom,
! [P: produc2676906022994773093_f_nat,R: set_Pr516226060960033221_f_nat] :
( ( member7030746907725424910_f_nat @ P @ R )
=> ( member7030746907725424910_f_nat @ P @ ( transi8903367562120364223_f_nat @ R ) ) ) ).
% r_into_rtrancl
thf(fact_40_r__into__rtrancl,axiom,
! [P: produc3406425829894461415rm_f_v,R: set_Pr7897041799657265351rm_f_v] :
( ( member7300054593328134928rm_f_v @ P @ R )
=> ( member7300054593328134928rm_f_v @ P @ ( transi1062743709496199978rm_f_v @ R ) ) ) ).
% r_into_rtrancl
thf(fact_41_r__into__rtrancl,axiom,
! [P: product_prod_f_f,R: set_Product_prod_f_f] :
( ( member5935997662887750672od_f_f @ P @ R )
=> ( member5935997662887750672od_f_f @ P @ ( transitive_rtrancl_f @ R ) ) ) ).
% r_into_rtrancl
thf(fact_42_r__into__rtrancl,axiom,
! [P: product_prod_nat_nat,R: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ P @ R )
=> ( member8440522571783428010at_nat @ P @ ( transi2905341329935302413cl_nat @ R ) ) ) ).
% r_into_rtrancl
thf(fact_43_r__into__rtrancl,axiom,
! [P: produc8027700073345061287rm_f_v,R: set_Pr8827868859434726151rm_f_v] :
( ( member4316649861536941904rm_f_v @ P @ R )
=> ( member4316649861536941904rm_f_v @ P @ ( transi3028809021613410336rm_f_v @ R ) ) ) ).
% r_into_rtrancl
thf(fact_44_C,axiom,
ord_le8976984241387448984_f_nat @ ( term_funas_ctxt_f_v @ c ) @ f2 ).
% C
thf(fact_45_converse__rtrancl__into__rtrancl,axiom,
! [A: product_prod_f_nat,B: product_prod_f_nat,R: set_Pr516226060960033221_f_nat,C3: product_prod_f_nat] :
( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ A @ B ) @ R )
=> ( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ B @ C3 ) @ ( transi8903367562120364223_f_nat @ R ) )
=> ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ A @ C3 ) @ ( transi8903367562120364223_f_nat @ R ) ) ) ) ).
% converse_rtrancl_into_rtrancl
thf(fact_46_converse__rtrancl__into__rtrancl,axiom,
! [A: produc8027700073345061287rm_f_v,B: produc8027700073345061287rm_f_v,R: set_Pr7897041799657265351rm_f_v,C3: produc8027700073345061287rm_f_v] :
( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A @ B ) @ R )
=> ( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ B @ C3 ) @ ( transi1062743709496199978rm_f_v @ R ) )
=> ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A @ C3 ) @ ( transi1062743709496199978rm_f_v @ R ) ) ) ) ).
% converse_rtrancl_into_rtrancl
thf(fact_47_converse__rtrancl__into__rtrancl,axiom,
! [A: f,B: f,R: set_Product_prod_f_f,C3: f] :
( ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ A @ B ) @ R )
=> ( ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ B @ C3 ) @ ( transitive_rtrancl_f @ R ) )
=> ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ A @ C3 ) @ ( transitive_rtrancl_f @ R ) ) ) ) ).
% converse_rtrancl_into_rtrancl
thf(fact_48_converse__rtrancl__into__rtrancl,axiom,
! [A: nat,B: nat,R: set_Pr1261947904930325089at_nat,C3: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ C3 ) @ ( transi2905341329935302413cl_nat @ R ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ C3 ) @ ( transi2905341329935302413cl_nat @ R ) ) ) ) ).
% converse_rtrancl_into_rtrancl
thf(fact_49_converse__rtrancl__into__rtrancl,axiom,
! [A: term_f_v,B: term_f_v,R: set_Pr8827868859434726151rm_f_v,C3: term_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ R )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B @ C3 ) @ ( transi3028809021613410336rm_f_v @ R ) )
=> ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ C3 ) @ ( transi3028809021613410336rm_f_v @ R ) ) ) ) ).
% converse_rtrancl_into_rtrancl
thf(fact_50_converse__rtrancl__induct,axiom,
! [A: product_prod_f_nat,B: product_prod_f_nat,R: set_Pr516226060960033221_f_nat,P2: product_prod_f_nat > $o] :
( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ A @ B ) @ ( transi8903367562120364223_f_nat @ R ) )
=> ( ( P2 @ B )
=> ( ! [Y3: product_prod_f_nat,Z: product_prod_f_nat] :
( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ Y3 @ Z ) @ R )
=> ( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ Z @ B ) @ ( transi8903367562120364223_f_nat @ R ) )
=> ( ( P2 @ Z )
=> ( P2 @ Y3 ) ) ) )
=> ( P2 @ A ) ) ) ) ).
% converse_rtrancl_induct
thf(fact_51_converse__rtrancl__induct,axiom,
! [A: produc8027700073345061287rm_f_v,B: produc8027700073345061287rm_f_v,R: set_Pr7897041799657265351rm_f_v,P2: produc8027700073345061287rm_f_v > $o] :
( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A @ B ) @ ( transi1062743709496199978rm_f_v @ R ) )
=> ( ( P2 @ B )
=> ( ! [Y3: produc8027700073345061287rm_f_v,Z: produc8027700073345061287rm_f_v] :
( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ Y3 @ Z ) @ R )
=> ( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ Z @ B ) @ ( transi1062743709496199978rm_f_v @ R ) )
=> ( ( P2 @ Z )
=> ( P2 @ Y3 ) ) ) )
=> ( P2 @ A ) ) ) ) ).
% converse_rtrancl_induct
thf(fact_52_converse__rtrancl__induct,axiom,
! [A: f,B: f,R: set_Product_prod_f_f,P2: f > $o] :
( ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ A @ B ) @ ( transitive_rtrancl_f @ R ) )
=> ( ( P2 @ B )
=> ( ! [Y3: f,Z: f] :
( ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ Y3 @ Z ) @ R )
=> ( ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ Z @ B ) @ ( transitive_rtrancl_f @ R ) )
=> ( ( P2 @ Z )
=> ( P2 @ Y3 ) ) ) )
=> ( P2 @ A ) ) ) ) ).
% converse_rtrancl_induct
thf(fact_53_converse__rtrancl__induct,axiom,
! [A: nat,B: nat,R: set_Pr1261947904930325089at_nat,P2: nat > $o] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( transi2905341329935302413cl_nat @ R ) )
=> ( ( P2 @ B )
=> ( ! [Y3: nat,Z: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ Z ) @ R )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Z @ B ) @ ( transi2905341329935302413cl_nat @ R ) )
=> ( ( P2 @ Z )
=> ( P2 @ Y3 ) ) ) )
=> ( P2 @ A ) ) ) ) ).
% converse_rtrancl_induct
thf(fact_54_converse__rtrancl__induct,axiom,
! [A: term_f_v,B: term_f_v,R: set_Pr8827868859434726151rm_f_v,P2: term_f_v > $o] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( transi3028809021613410336rm_f_v @ R ) )
=> ( ( P2 @ B )
=> ( ! [Y3: term_f_v,Z: term_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y3 @ Z ) @ R )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Z @ B ) @ ( transi3028809021613410336rm_f_v @ R ) )
=> ( ( P2 @ Z )
=> ( P2 @ Y3 ) ) ) )
=> ( P2 @ A ) ) ) ) ).
% converse_rtrancl_induct
thf(fact_55_converse__rtranclE,axiom,
! [X3: product_prod_f_nat,Z2: product_prod_f_nat,R: set_Pr516226060960033221_f_nat] :
( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ X3 @ Z2 ) @ ( transi8903367562120364223_f_nat @ R ) )
=> ( ( X3 != Z2 )
=> ~ ! [Y3: product_prod_f_nat] :
( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ X3 @ Y3 ) @ R )
=> ~ ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ Y3 @ Z2 ) @ ( transi8903367562120364223_f_nat @ R ) ) ) ) ) ).
% converse_rtranclE
thf(fact_56_converse__rtranclE,axiom,
! [X3: produc8027700073345061287rm_f_v,Z2: produc8027700073345061287rm_f_v,R: set_Pr7897041799657265351rm_f_v] :
( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ X3 @ Z2 ) @ ( transi1062743709496199978rm_f_v @ R ) )
=> ( ( X3 != Z2 )
=> ~ ! [Y3: produc8027700073345061287rm_f_v] :
( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ X3 @ Y3 ) @ R )
=> ~ ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ Y3 @ Z2 ) @ ( transi1062743709496199978rm_f_v @ R ) ) ) ) ) ).
% converse_rtranclE
thf(fact_57_converse__rtranclE,axiom,
! [X3: f,Z2: f,R: set_Product_prod_f_f] :
( ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ X3 @ Z2 ) @ ( transitive_rtrancl_f @ R ) )
=> ( ( X3 != Z2 )
=> ~ ! [Y3: f] :
( ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ X3 @ Y3 ) @ R )
=> ~ ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ Y3 @ Z2 ) @ ( transitive_rtrancl_f @ R ) ) ) ) ) ).
% converse_rtranclE
thf(fact_58_converse__rtranclE,axiom,
! [X3: nat,Z2: nat,R: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Z2 ) @ ( transi2905341329935302413cl_nat @ R ) )
=> ( ( X3 != Z2 )
=> ~ ! [Y3: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ Z2 ) @ ( transi2905341329935302413cl_nat @ R ) ) ) ) ) ).
% converse_rtranclE
thf(fact_59_converse__rtranclE,axiom,
! [X3: term_f_v,Z2: term_f_v,R: set_Pr8827868859434726151rm_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Z2 ) @ ( transi3028809021613410336rm_f_v @ R ) )
=> ( ( X3 != Z2 )
=> ~ ! [Y3: term_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Y3 ) @ R )
=> ~ ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y3 @ Z2 ) @ ( transi3028809021613410336rm_f_v @ R ) ) ) ) ) ).
% converse_rtranclE
thf(fact_60_rtrancl__induct,axiom,
! [A: product_prod_f_nat,B: product_prod_f_nat,R: set_Pr516226060960033221_f_nat,P2: product_prod_f_nat > $o] :
( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ A @ B ) @ ( transi8903367562120364223_f_nat @ R ) )
=> ( ( P2 @ A )
=> ( ! [Y3: product_prod_f_nat,Z: product_prod_f_nat] :
( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ A @ Y3 ) @ ( transi8903367562120364223_f_nat @ R ) )
=> ( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ Y3 @ Z ) @ R )
=> ( ( P2 @ Y3 )
=> ( P2 @ Z ) ) ) )
=> ( P2 @ B ) ) ) ) ).
% rtrancl_induct
thf(fact_61_rtrancl__induct,axiom,
! [A: produc8027700073345061287rm_f_v,B: produc8027700073345061287rm_f_v,R: set_Pr7897041799657265351rm_f_v,P2: produc8027700073345061287rm_f_v > $o] :
( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A @ B ) @ ( transi1062743709496199978rm_f_v @ R ) )
=> ( ( P2 @ A )
=> ( ! [Y3: produc8027700073345061287rm_f_v,Z: produc8027700073345061287rm_f_v] :
( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A @ Y3 ) @ ( transi1062743709496199978rm_f_v @ R ) )
=> ( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ Y3 @ Z ) @ R )
=> ( ( P2 @ Y3 )
=> ( P2 @ Z ) ) ) )
=> ( P2 @ B ) ) ) ) ).
% rtrancl_induct
thf(fact_62_rtrancl__induct,axiom,
! [A: f,B: f,R: set_Product_prod_f_f,P2: f > $o] :
( ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ A @ B ) @ ( transitive_rtrancl_f @ R ) )
=> ( ( P2 @ A )
=> ( ! [Y3: f,Z: f] :
( ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ A @ Y3 ) @ ( transitive_rtrancl_f @ R ) )
=> ( ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ Y3 @ Z ) @ R )
=> ( ( P2 @ Y3 )
=> ( P2 @ Z ) ) ) )
=> ( P2 @ B ) ) ) ) ).
% rtrancl_induct
thf(fact_63_rtrancl__induct,axiom,
! [A: nat,B: nat,R: set_Pr1261947904930325089at_nat,P2: nat > $o] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( transi2905341329935302413cl_nat @ R ) )
=> ( ( P2 @ A )
=> ( ! [Y3: nat,Z: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ Y3 ) @ ( transi2905341329935302413cl_nat @ R ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ Z ) @ R )
=> ( ( P2 @ Y3 )
=> ( P2 @ Z ) ) ) )
=> ( P2 @ B ) ) ) ) ).
% rtrancl_induct
thf(fact_64_rtrancl__induct,axiom,
! [A: term_f_v,B: term_f_v,R: set_Pr8827868859434726151rm_f_v,P2: term_f_v > $o] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( transi3028809021613410336rm_f_v @ R ) )
=> ( ( P2 @ A )
=> ( ! [Y3: term_f_v,Z: term_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ Y3 ) @ ( transi3028809021613410336rm_f_v @ R ) )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y3 @ Z ) @ R )
=> ( ( P2 @ Y3 )
=> ( P2 @ Z ) ) ) )
=> ( P2 @ B ) ) ) ) ).
% rtrancl_induct
thf(fact_65_rtrancl__trans,axiom,
! [X3: product_prod_f_nat,Y: product_prod_f_nat,R: set_Pr516226060960033221_f_nat,Z2: product_prod_f_nat] :
( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ X3 @ Y ) @ ( transi8903367562120364223_f_nat @ R ) )
=> ( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ Y @ Z2 ) @ ( transi8903367562120364223_f_nat @ R ) )
=> ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ X3 @ Z2 ) @ ( transi8903367562120364223_f_nat @ R ) ) ) ) ).
% rtrancl_trans
thf(fact_66_rtrancl__trans,axiom,
! [X3: produc8027700073345061287rm_f_v,Y: produc8027700073345061287rm_f_v,R: set_Pr7897041799657265351rm_f_v,Z2: produc8027700073345061287rm_f_v] :
( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ X3 @ Y ) @ ( transi1062743709496199978rm_f_v @ R ) )
=> ( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ Y @ Z2 ) @ ( transi1062743709496199978rm_f_v @ R ) )
=> ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ X3 @ Z2 ) @ ( transi1062743709496199978rm_f_v @ R ) ) ) ) ).
% rtrancl_trans
thf(fact_67_rtrancl__trans,axiom,
! [X3: f,Y: f,R: set_Product_prod_f_f,Z2: f] :
( ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ X3 @ Y ) @ ( transitive_rtrancl_f @ R ) )
=> ( ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ Y @ Z2 ) @ ( transitive_rtrancl_f @ R ) )
=> ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ X3 @ Z2 ) @ ( transitive_rtrancl_f @ R ) ) ) ) ).
% rtrancl_trans
thf(fact_68_rtrancl__trans,axiom,
! [X3: nat,Y: nat,R: set_Pr1261947904930325089at_nat,Z2: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y ) @ ( transi2905341329935302413cl_nat @ R ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ Z2 ) @ ( transi2905341329935302413cl_nat @ R ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Z2 ) @ ( transi2905341329935302413cl_nat @ R ) ) ) ) ).
% rtrancl_trans
thf(fact_69_rtrancl__trans,axiom,
! [X3: term_f_v,Y: term_f_v,R: set_Pr8827868859434726151rm_f_v,Z2: term_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Y ) @ ( transi3028809021613410336rm_f_v @ R ) )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y @ Z2 ) @ ( transi3028809021613410336rm_f_v @ R ) )
=> ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Z2 ) @ ( transi3028809021613410336rm_f_v @ R ) ) ) ) ).
% rtrancl_trans
thf(fact_70_rtranclE,axiom,
! [A: product_prod_f_nat,B: product_prod_f_nat,R: set_Pr516226060960033221_f_nat] :
( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ A @ B ) @ ( transi8903367562120364223_f_nat @ R ) )
=> ( ( A != B )
=> ~ ! [Y3: product_prod_f_nat] :
( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ A @ Y3 ) @ ( transi8903367562120364223_f_nat @ R ) )
=> ~ ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ Y3 @ B ) @ R ) ) ) ) ).
% rtranclE
thf(fact_71_rtranclE,axiom,
! [A: produc8027700073345061287rm_f_v,B: produc8027700073345061287rm_f_v,R: set_Pr7897041799657265351rm_f_v] :
( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A @ B ) @ ( transi1062743709496199978rm_f_v @ R ) )
=> ( ( A != B )
=> ~ ! [Y3: produc8027700073345061287rm_f_v] :
( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A @ Y3 ) @ ( transi1062743709496199978rm_f_v @ R ) )
=> ~ ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ Y3 @ B ) @ R ) ) ) ) ).
% rtranclE
thf(fact_72_rtranclE,axiom,
! [A: f,B: f,R: set_Product_prod_f_f] :
( ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ A @ B ) @ ( transitive_rtrancl_f @ R ) )
=> ( ( A != B )
=> ~ ! [Y3: f] :
( ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ A @ Y3 ) @ ( transitive_rtrancl_f @ R ) )
=> ~ ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ Y3 @ B ) @ R ) ) ) ) ).
% rtranclE
thf(fact_73_rtranclE,axiom,
! [A: nat,B: nat,R: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( transi2905341329935302413cl_nat @ R ) )
=> ( ( A != B )
=> ~ ! [Y3: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ Y3 ) @ ( transi2905341329935302413cl_nat @ R ) )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ B ) @ R ) ) ) ) ).
% rtranclE
thf(fact_74_rtranclE,axiom,
! [A: term_f_v,B: term_f_v,R: set_Pr8827868859434726151rm_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( transi3028809021613410336rm_f_v @ R ) )
=> ( ( A != B )
=> ~ ! [Y3: term_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ Y3 ) @ ( transi3028809021613410336rm_f_v @ R ) )
=> ~ ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y3 @ B ) @ R ) ) ) ) ).
% rtranclE
thf(fact_75_rtrancl_Ortrancl__into__rtrancl,axiom,
! [A: product_prod_f_nat,B: product_prod_f_nat,R: set_Pr516226060960033221_f_nat,C3: product_prod_f_nat] :
( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ A @ B ) @ ( transi8903367562120364223_f_nat @ R ) )
=> ( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ B @ C3 ) @ R )
=> ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ A @ C3 ) @ ( transi8903367562120364223_f_nat @ R ) ) ) ) ).
% rtrancl.rtrancl_into_rtrancl
thf(fact_76_rtrancl_Ortrancl__into__rtrancl,axiom,
! [A: produc8027700073345061287rm_f_v,B: produc8027700073345061287rm_f_v,R: set_Pr7897041799657265351rm_f_v,C3: produc8027700073345061287rm_f_v] :
( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A @ B ) @ ( transi1062743709496199978rm_f_v @ R ) )
=> ( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ B @ C3 ) @ R )
=> ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A @ C3 ) @ ( transi1062743709496199978rm_f_v @ R ) ) ) ) ).
% rtrancl.rtrancl_into_rtrancl
thf(fact_77_rtrancl_Ortrancl__into__rtrancl,axiom,
! [A: f,B: f,R: set_Product_prod_f_f,C3: f] :
( ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ A @ B ) @ ( transitive_rtrancl_f @ R ) )
=> ( ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ B @ C3 ) @ R )
=> ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ A @ C3 ) @ ( transitive_rtrancl_f @ R ) ) ) ) ).
% rtrancl.rtrancl_into_rtrancl
thf(fact_78_rtrancl_Ortrancl__into__rtrancl,axiom,
! [A: nat,B: nat,R: set_Pr1261947904930325089at_nat,C3: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( transi2905341329935302413cl_nat @ R ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ C3 ) @ R )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ C3 ) @ ( transi2905341329935302413cl_nat @ R ) ) ) ) ).
% rtrancl.rtrancl_into_rtrancl
thf(fact_79_rtrancl_Ortrancl__into__rtrancl,axiom,
! [A: term_f_v,B: term_f_v,R: set_Pr8827868859434726151rm_f_v,C3: term_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( transi3028809021613410336rm_f_v @ R ) )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B @ C3 ) @ R )
=> ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ C3 ) @ ( transi3028809021613410336rm_f_v @ R ) ) ) ) ).
% rtrancl.rtrancl_into_rtrancl
thf(fact_80_s,axiom,
ord_le8976984241387448984_f_nat @ ( term_funas_term_f_v @ s ) @ f2 ).
% s
thf(fact_81_t,axiom,
ord_le8976984241387448984_f_nat @ ( term_funas_term_f_v @ t ) @ f2 ).
% t
thf(fact_82_rtrancl__induct2,axiom,
! [Ax: produc8027700073345061287rm_f_v,Ay: produc8027700073345061287rm_f_v,Bx: produc8027700073345061287rm_f_v,By: produc8027700073345061287rm_f_v,R: set_Pr4964659734097919175rm_f_v,P2: produc8027700073345061287rm_f_v > produc8027700073345061287rm_f_v > $o] :
( ( member1670422607875439376rm_f_v @ ( produc1558868356289833815rm_f_v @ ( produc3874942763110801239rm_f_v @ Ax @ Ay ) @ ( produc3874942763110801239rm_f_v @ Bx @ By ) ) @ ( transi4215105848801502698rm_f_v @ R ) )
=> ( ( P2 @ Ax @ Ay )
=> ( ! [A3: produc8027700073345061287rm_f_v,B3: produc8027700073345061287rm_f_v,Aa: produc8027700073345061287rm_f_v,Ba: produc8027700073345061287rm_f_v] :
( ( member1670422607875439376rm_f_v @ ( produc1558868356289833815rm_f_v @ ( produc3874942763110801239rm_f_v @ Ax @ Ay ) @ ( produc3874942763110801239rm_f_v @ A3 @ B3 ) ) @ ( transi4215105848801502698rm_f_v @ R ) )
=> ( ( member1670422607875439376rm_f_v @ ( produc1558868356289833815rm_f_v @ ( produc3874942763110801239rm_f_v @ A3 @ B3 ) @ ( produc3874942763110801239rm_f_v @ Aa @ Ba ) ) @ R )
=> ( ( P2 @ A3 @ B3 )
=> ( P2 @ Aa @ Ba ) ) ) )
=> ( P2 @ Bx @ By ) ) ) ) ).
% rtrancl_induct2
thf(fact_83_rtrancl__induct2,axiom,
! [Ax: product_prod_f_nat,Ay: product_prod_f_nat,Bx: product_prod_f_nat,By: product_prod_f_nat,R: set_Pr8434786782240690247_f_nat,P2: product_prod_f_nat > product_prod_f_nat > $o] :
( ( member3697069702615009936_f_nat @ ( produc425345625175287255_f_nat @ ( produc5861087217798509725_f_nat @ Ax @ Ay ) @ ( produc5861087217798509725_f_nat @ Bx @ By ) ) @ ( transi7343684842217456360_f_nat @ R ) )
=> ( ( P2 @ Ax @ Ay )
=> ( ! [A3: product_prod_f_nat,B3: product_prod_f_nat,Aa: product_prod_f_nat,Ba: product_prod_f_nat] :
( ( member3697069702615009936_f_nat @ ( produc425345625175287255_f_nat @ ( produc5861087217798509725_f_nat @ Ax @ Ay ) @ ( produc5861087217798509725_f_nat @ A3 @ B3 ) ) @ ( transi7343684842217456360_f_nat @ R ) )
=> ( ( member3697069702615009936_f_nat @ ( produc425345625175287255_f_nat @ ( produc5861087217798509725_f_nat @ A3 @ B3 ) @ ( produc5861087217798509725_f_nat @ Aa @ Ba ) ) @ R )
=> ( ( P2 @ A3 @ B3 )
=> ( P2 @ Aa @ Ba ) ) ) )
=> ( P2 @ Bx @ By ) ) ) ) ).
% rtrancl_induct2
thf(fact_84_rtrancl__induct2,axiom,
! [Ax: f,Ay: nat,Bx: f,By: nat,R: set_Pr516226060960033221_f_nat,P2: f > nat > $o] :
( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ ( product_Pair_f_nat @ Ax @ Ay ) @ ( product_Pair_f_nat @ Bx @ By ) ) @ ( transi8903367562120364223_f_nat @ R ) )
=> ( ( P2 @ Ax @ Ay )
=> ( ! [A3: f,B3: nat,Aa: f,Ba: nat] :
( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ ( product_Pair_f_nat @ Ax @ Ay ) @ ( product_Pair_f_nat @ A3 @ B3 ) ) @ ( transi8903367562120364223_f_nat @ R ) )
=> ( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ ( product_Pair_f_nat @ A3 @ B3 ) @ ( product_Pair_f_nat @ Aa @ Ba ) ) @ R )
=> ( ( P2 @ A3 @ B3 )
=> ( P2 @ Aa @ Ba ) ) ) )
=> ( P2 @ Bx @ By ) ) ) ) ).
% rtrancl_induct2
thf(fact_85_rtrancl__induct2,axiom,
! [Ax: term_f_v,Ay: term_f_v,Bx: term_f_v,By: term_f_v,R: set_Pr7897041799657265351rm_f_v,P2: term_f_v > term_f_v > $o] :
( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ ( produc2510790566102465375rm_f_v @ Ax @ Ay ) @ ( produc2510790566102465375rm_f_v @ Bx @ By ) ) @ ( transi1062743709496199978rm_f_v @ R ) )
=> ( ( P2 @ Ax @ Ay )
=> ( ! [A3: term_f_v,B3: term_f_v,Aa: term_f_v,Ba: term_f_v] :
( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ ( produc2510790566102465375rm_f_v @ Ax @ Ay ) @ ( produc2510790566102465375rm_f_v @ A3 @ B3 ) ) @ ( transi1062743709496199978rm_f_v @ R ) )
=> ( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ ( produc2510790566102465375rm_f_v @ A3 @ B3 ) @ ( produc2510790566102465375rm_f_v @ Aa @ Ba ) ) @ R )
=> ( ( P2 @ A3 @ B3 )
=> ( P2 @ Aa @ Ba ) ) ) )
=> ( P2 @ Bx @ By ) ) ) ) ).
% rtrancl_induct2
thf(fact_86_converse__rtranclE2,axiom,
! [Xa: produc8027700073345061287rm_f_v,Xb: produc8027700073345061287rm_f_v,Za: produc8027700073345061287rm_f_v,Zb: produc8027700073345061287rm_f_v,R: set_Pr4964659734097919175rm_f_v] :
( ( member1670422607875439376rm_f_v @ ( produc1558868356289833815rm_f_v @ ( produc3874942763110801239rm_f_v @ Xa @ Xb ) @ ( produc3874942763110801239rm_f_v @ Za @ Zb ) ) @ ( transi4215105848801502698rm_f_v @ R ) )
=> ( ( ( produc3874942763110801239rm_f_v @ Xa @ Xb )
!= ( produc3874942763110801239rm_f_v @ Za @ Zb ) )
=> ~ ! [A3: produc8027700073345061287rm_f_v,B3: produc8027700073345061287rm_f_v] :
( ( member1670422607875439376rm_f_v @ ( produc1558868356289833815rm_f_v @ ( produc3874942763110801239rm_f_v @ Xa @ Xb ) @ ( produc3874942763110801239rm_f_v @ A3 @ B3 ) ) @ R )
=> ~ ( member1670422607875439376rm_f_v @ ( produc1558868356289833815rm_f_v @ ( produc3874942763110801239rm_f_v @ A3 @ B3 ) @ ( produc3874942763110801239rm_f_v @ Za @ Zb ) ) @ ( transi4215105848801502698rm_f_v @ R ) ) ) ) ) ).
% converse_rtranclE2
thf(fact_87_converse__rtranclE2,axiom,
! [Xa: product_prod_f_nat,Xb: product_prod_f_nat,Za: product_prod_f_nat,Zb: product_prod_f_nat,R: set_Pr8434786782240690247_f_nat] :
( ( member3697069702615009936_f_nat @ ( produc425345625175287255_f_nat @ ( produc5861087217798509725_f_nat @ Xa @ Xb ) @ ( produc5861087217798509725_f_nat @ Za @ Zb ) ) @ ( transi7343684842217456360_f_nat @ R ) )
=> ( ( ( produc5861087217798509725_f_nat @ Xa @ Xb )
!= ( produc5861087217798509725_f_nat @ Za @ Zb ) )
=> ~ ! [A3: product_prod_f_nat,B3: product_prod_f_nat] :
( ( member3697069702615009936_f_nat @ ( produc425345625175287255_f_nat @ ( produc5861087217798509725_f_nat @ Xa @ Xb ) @ ( produc5861087217798509725_f_nat @ A3 @ B3 ) ) @ R )
=> ~ ( member3697069702615009936_f_nat @ ( produc425345625175287255_f_nat @ ( produc5861087217798509725_f_nat @ A3 @ B3 ) @ ( produc5861087217798509725_f_nat @ Za @ Zb ) ) @ ( transi7343684842217456360_f_nat @ R ) ) ) ) ) ).
% converse_rtranclE2
thf(fact_88_converse__rtranclE2,axiom,
! [Xa: f,Xb: nat,Za: f,Zb: nat,R: set_Pr516226060960033221_f_nat] :
( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ ( product_Pair_f_nat @ Xa @ Xb ) @ ( product_Pair_f_nat @ Za @ Zb ) ) @ ( transi8903367562120364223_f_nat @ R ) )
=> ( ( ( product_Pair_f_nat @ Xa @ Xb )
!= ( product_Pair_f_nat @ Za @ Zb ) )
=> ~ ! [A3: f,B3: nat] :
( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ ( product_Pair_f_nat @ Xa @ Xb ) @ ( product_Pair_f_nat @ A3 @ B3 ) ) @ R )
=> ~ ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ ( product_Pair_f_nat @ A3 @ B3 ) @ ( product_Pair_f_nat @ Za @ Zb ) ) @ ( transi8903367562120364223_f_nat @ R ) ) ) ) ) ).
% converse_rtranclE2
thf(fact_89_converse__rtranclE2,axiom,
! [Xa: term_f_v,Xb: term_f_v,Za: term_f_v,Zb: term_f_v,R: set_Pr7897041799657265351rm_f_v] :
( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ ( produc2510790566102465375rm_f_v @ Xa @ Xb ) @ ( produc2510790566102465375rm_f_v @ Za @ Zb ) ) @ ( transi1062743709496199978rm_f_v @ R ) )
=> ( ( ( produc2510790566102465375rm_f_v @ Xa @ Xb )
!= ( produc2510790566102465375rm_f_v @ Za @ Zb ) )
=> ~ ! [A3: term_f_v,B3: term_f_v] :
( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ ( produc2510790566102465375rm_f_v @ Xa @ Xb ) @ ( produc2510790566102465375rm_f_v @ A3 @ B3 ) ) @ R )
=> ~ ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ ( produc2510790566102465375rm_f_v @ A3 @ B3 ) @ ( produc2510790566102465375rm_f_v @ Za @ Zb ) ) @ ( transi1062743709496199978rm_f_v @ R ) ) ) ) ) ).
% converse_rtranclE2
thf(fact_90_converse__rtrancl__induct2,axiom,
! [Ax: produc8027700073345061287rm_f_v,Ay: produc8027700073345061287rm_f_v,Bx: produc8027700073345061287rm_f_v,By: produc8027700073345061287rm_f_v,R: set_Pr4964659734097919175rm_f_v,P2: produc8027700073345061287rm_f_v > produc8027700073345061287rm_f_v > $o] :
( ( member1670422607875439376rm_f_v @ ( produc1558868356289833815rm_f_v @ ( produc3874942763110801239rm_f_v @ Ax @ Ay ) @ ( produc3874942763110801239rm_f_v @ Bx @ By ) ) @ ( transi4215105848801502698rm_f_v @ R ) )
=> ( ( P2 @ Bx @ By )
=> ( ! [A3: produc8027700073345061287rm_f_v,B3: produc8027700073345061287rm_f_v,Aa: produc8027700073345061287rm_f_v,Ba: produc8027700073345061287rm_f_v] :
( ( member1670422607875439376rm_f_v @ ( produc1558868356289833815rm_f_v @ ( produc3874942763110801239rm_f_v @ A3 @ B3 ) @ ( produc3874942763110801239rm_f_v @ Aa @ Ba ) ) @ R )
=> ( ( member1670422607875439376rm_f_v @ ( produc1558868356289833815rm_f_v @ ( produc3874942763110801239rm_f_v @ Aa @ Ba ) @ ( produc3874942763110801239rm_f_v @ Bx @ By ) ) @ ( transi4215105848801502698rm_f_v @ R ) )
=> ( ( P2 @ Aa @ Ba )
=> ( P2 @ A3 @ B3 ) ) ) )
=> ( P2 @ Ax @ Ay ) ) ) ) ).
% converse_rtrancl_induct2
thf(fact_91_converse__rtrancl__induct2,axiom,
! [Ax: product_prod_f_nat,Ay: product_prod_f_nat,Bx: product_prod_f_nat,By: product_prod_f_nat,R: set_Pr8434786782240690247_f_nat,P2: product_prod_f_nat > product_prod_f_nat > $o] :
( ( member3697069702615009936_f_nat @ ( produc425345625175287255_f_nat @ ( produc5861087217798509725_f_nat @ Ax @ Ay ) @ ( produc5861087217798509725_f_nat @ Bx @ By ) ) @ ( transi7343684842217456360_f_nat @ R ) )
=> ( ( P2 @ Bx @ By )
=> ( ! [A3: product_prod_f_nat,B3: product_prod_f_nat,Aa: product_prod_f_nat,Ba: product_prod_f_nat] :
( ( member3697069702615009936_f_nat @ ( produc425345625175287255_f_nat @ ( produc5861087217798509725_f_nat @ A3 @ B3 ) @ ( produc5861087217798509725_f_nat @ Aa @ Ba ) ) @ R )
=> ( ( member3697069702615009936_f_nat @ ( produc425345625175287255_f_nat @ ( produc5861087217798509725_f_nat @ Aa @ Ba ) @ ( produc5861087217798509725_f_nat @ Bx @ By ) ) @ ( transi7343684842217456360_f_nat @ R ) )
=> ( ( P2 @ Aa @ Ba )
=> ( P2 @ A3 @ B3 ) ) ) )
=> ( P2 @ Ax @ Ay ) ) ) ) ).
% converse_rtrancl_induct2
thf(fact_92_converse__rtrancl__induct2,axiom,
! [Ax: f,Ay: nat,Bx: f,By: nat,R: set_Pr516226060960033221_f_nat,P2: f > nat > $o] :
( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ ( product_Pair_f_nat @ Ax @ Ay ) @ ( product_Pair_f_nat @ Bx @ By ) ) @ ( transi8903367562120364223_f_nat @ R ) )
=> ( ( P2 @ Bx @ By )
=> ( ! [A3: f,B3: nat,Aa: f,Ba: nat] :
( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ ( product_Pair_f_nat @ A3 @ B3 ) @ ( product_Pair_f_nat @ Aa @ Ba ) ) @ R )
=> ( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ ( product_Pair_f_nat @ Aa @ Ba ) @ ( product_Pair_f_nat @ Bx @ By ) ) @ ( transi8903367562120364223_f_nat @ R ) )
=> ( ( P2 @ Aa @ Ba )
=> ( P2 @ A3 @ B3 ) ) ) )
=> ( P2 @ Ax @ Ay ) ) ) ) ).
% converse_rtrancl_induct2
thf(fact_93_converse__rtrancl__induct2,axiom,
! [Ax: term_f_v,Ay: term_f_v,Bx: term_f_v,By: term_f_v,R: set_Pr7897041799657265351rm_f_v,P2: term_f_v > term_f_v > $o] :
( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ ( produc2510790566102465375rm_f_v @ Ax @ Ay ) @ ( produc2510790566102465375rm_f_v @ Bx @ By ) ) @ ( transi1062743709496199978rm_f_v @ R ) )
=> ( ( P2 @ Bx @ By )
=> ( ! [A3: term_f_v,B3: term_f_v,Aa: term_f_v,Ba: term_f_v] :
( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ ( produc2510790566102465375rm_f_v @ A3 @ B3 ) @ ( produc2510790566102465375rm_f_v @ Aa @ Ba ) ) @ R )
=> ( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ ( produc2510790566102465375rm_f_v @ Aa @ Ba ) @ ( produc2510790566102465375rm_f_v @ Bx @ By ) ) @ ( transi1062743709496199978rm_f_v @ R ) )
=> ( ( P2 @ Aa @ Ba )
=> ( P2 @ A3 @ B3 ) ) ) )
=> ( P2 @ Ax @ Ay ) ) ) ) ).
% converse_rtrancl_induct2
thf(fact_94_rtrancl__subset__rtrancl,axiom,
! [R: set_Pr7897041799657265351rm_f_v,S: set_Pr7897041799657265351rm_f_v] :
( ( ord_le7518223742277120103rm_f_v @ R @ ( transi1062743709496199978rm_f_v @ S ) )
=> ( ord_le7518223742277120103rm_f_v @ ( transi1062743709496199978rm_f_v @ R ) @ ( transi1062743709496199978rm_f_v @ S ) ) ) ).
% rtrancl_subset_rtrancl
thf(fact_95_rtrancl__subset__rtrancl,axiom,
! [R: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ R @ ( transi2905341329935302413cl_nat @ S ) )
=> ( ord_le3146513528884898305at_nat @ ( transi2905341329935302413cl_nat @ R ) @ ( transi2905341329935302413cl_nat @ S ) ) ) ).
% rtrancl_subset_rtrancl
thf(fact_96_rtrancl__subset__rtrancl,axiom,
! [R: set_Product_prod_f_f,S: set_Product_prod_f_f] :
( ( ord_le7147000192471188839od_f_f @ R @ ( transitive_rtrancl_f @ S ) )
=> ( ord_le7147000192471188839od_f_f @ ( transitive_rtrancl_f @ R ) @ ( transitive_rtrancl_f @ S ) ) ) ).
% rtrancl_subset_rtrancl
thf(fact_97_rtrancl__subset__rtrancl,axiom,
! [R: set_Pr8827868859434726151rm_f_v,S: set_Pr8827868859434726151rm_f_v] :
( ( ord_le4559761987009501863rm_f_v @ R @ ( transi3028809021613410336rm_f_v @ S ) )
=> ( ord_le4559761987009501863rm_f_v @ ( transi3028809021613410336rm_f_v @ R ) @ ( transi3028809021613410336rm_f_v @ S ) ) ) ).
% rtrancl_subset_rtrancl
thf(fact_98_rtrancl__subset,axiom,
! [R3: set_Pr7897041799657265351rm_f_v,S2: set_Pr7897041799657265351rm_f_v] :
( ( ord_le7518223742277120103rm_f_v @ R3 @ S2 )
=> ( ( ord_le7518223742277120103rm_f_v @ S2 @ ( transi1062743709496199978rm_f_v @ R3 ) )
=> ( ( transi1062743709496199978rm_f_v @ S2 )
= ( transi1062743709496199978rm_f_v @ R3 ) ) ) ) ).
% rtrancl_subset
thf(fact_99_rtrancl__subset,axiom,
! [R3: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ R3 @ S2 )
=> ( ( ord_le3146513528884898305at_nat @ S2 @ ( transi2905341329935302413cl_nat @ R3 ) )
=> ( ( transi2905341329935302413cl_nat @ S2 )
= ( transi2905341329935302413cl_nat @ R3 ) ) ) ) ).
% rtrancl_subset
thf(fact_100_rtrancl__subset,axiom,
! [R3: set_Product_prod_f_f,S2: set_Product_prod_f_f] :
( ( ord_le7147000192471188839od_f_f @ R3 @ S2 )
=> ( ( ord_le7147000192471188839od_f_f @ S2 @ ( transitive_rtrancl_f @ R3 ) )
=> ( ( transitive_rtrancl_f @ S2 )
= ( transitive_rtrancl_f @ R3 ) ) ) ) ).
% rtrancl_subset
thf(fact_101_rtrancl__subset,axiom,
! [R3: set_Pr8827868859434726151rm_f_v,S2: set_Pr8827868859434726151rm_f_v] :
( ( ord_le4559761987009501863rm_f_v @ R3 @ S2 )
=> ( ( ord_le4559761987009501863rm_f_v @ S2 @ ( transi3028809021613410336rm_f_v @ R3 ) )
=> ( ( transi3028809021613410336rm_f_v @ S2 )
= ( transi3028809021613410336rm_f_v @ R3 ) ) ) ) ).
% rtrancl_subset
thf(fact_102_rtrancl__mono,axiom,
! [R: set_Pr7897041799657265351rm_f_v,S: set_Pr7897041799657265351rm_f_v] :
( ( ord_le7518223742277120103rm_f_v @ R @ S )
=> ( ord_le7518223742277120103rm_f_v @ ( transi1062743709496199978rm_f_v @ R ) @ ( transi1062743709496199978rm_f_v @ S ) ) ) ).
% rtrancl_mono
thf(fact_103_rtrancl__mono,axiom,
! [R: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ R @ S )
=> ( ord_le3146513528884898305at_nat @ ( transi2905341329935302413cl_nat @ R ) @ ( transi2905341329935302413cl_nat @ S ) ) ) ).
% rtrancl_mono
thf(fact_104_rtrancl__mono,axiom,
! [R: set_Product_prod_f_f,S: set_Product_prod_f_f] :
( ( ord_le7147000192471188839od_f_f @ R @ S )
=> ( ord_le7147000192471188839od_f_f @ ( transitive_rtrancl_f @ R ) @ ( transitive_rtrancl_f @ S ) ) ) ).
% rtrancl_mono
thf(fact_105_rtrancl__mono,axiom,
! [R: set_Pr8827868859434726151rm_f_v,S: set_Pr8827868859434726151rm_f_v] :
( ( ord_le4559761987009501863rm_f_v @ R @ S )
=> ( ord_le4559761987009501863rm_f_v @ ( transi3028809021613410336rm_f_v @ R ) @ ( transi3028809021613410336rm_f_v @ S ) ) ) ).
% rtrancl_mono
thf(fact_106_srstep__ctxt__closed,axiom,
! [C2: subterm_and_ctxt_f_v,F: set_Pr5245412377734449720_f_nat,S: term_f_v,T: term_f_v,R2: set_Pr8827868859434726151rm_f_v] :
( ( ord_le8976984241387448984_f_nat @ ( term_funas_ctxt_f_v @ C2 ) @ F )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ ( sig_step_f_v @ F @ ( rstep_f_v @ R2 ) ) )
=> ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ ( subter6886040689604212261rm_f_v @ C2 @ S ) @ ( subter6886040689604212261rm_f_v @ C2 @ T ) ) @ ( sig_step_f_v @ F @ ( rstep_f_v @ R2 ) ) ) ) ) ).
% srstep_ctxt_closed
thf(fact_107_srsteps__eq__ctxt__closed,axiom,
! [C2: subterm_and_ctxt_f_v,F: set_Pr5245412377734449720_f_nat,S: term_f_v,T: term_f_v,R2: set_Pr8827868859434726151rm_f_v] :
( ( ord_le8976984241387448984_f_nat @ ( term_funas_ctxt_f_v @ C2 ) @ F )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ ( transi3028809021613410336rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ R2 ) ) ) )
=> ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ ( subter6886040689604212261rm_f_v @ C2 @ S ) @ ( subter6886040689604212261rm_f_v @ C2 @ T ) ) @ ( transi3028809021613410336rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ R2 ) ) ) ) ) ) ).
% srsteps_eq_ctxt_closed
thf(fact_108_mem__Collect__eq,axiom,
! [A: produc3406425829894461415rm_f_v,P2: produc3406425829894461415rm_f_v > $o] :
( ( member7300054593328134928rm_f_v @ A @ ( collec227443146937027410rm_f_v @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_109_mem__Collect__eq,axiom,
! [A: term_f_v,P2: term_f_v > $o] :
( ( member_term_f_v @ A @ ( collect_term_f_v @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_110_mem__Collect__eq,axiom,
! [A: produc2676906022994773093_f_nat,P2: produc2676906022994773093_f_nat > $o] :
( ( member7030746907725424910_f_nat @ A @ ( collec8116569655616119888_f_nat @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_111_mem__Collect__eq,axiom,
! [A: product_prod_f_nat,P2: product_prod_f_nat > $o] :
( ( member2678038196313615385_f_nat @ A @ ( collec1417984143148964183_f_nat @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_112_mem__Collect__eq,axiom,
! [A: produc8027700073345061287rm_f_v,P2: produc8027700073345061287rm_f_v > $o] :
( ( member4316649861536941904rm_f_v @ A @ ( collec7769483783292507282rm_f_v @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_113_Collect__mem__eq,axiom,
! [A4: set_Pr7897041799657265351rm_f_v] :
( ( collec227443146937027410rm_f_v
@ ^ [X4: produc3406425829894461415rm_f_v] : ( member7300054593328134928rm_f_v @ X4 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_114_Collect__mem__eq,axiom,
! [A4: set_term_f_v] :
( ( collect_term_f_v
@ ^ [X4: term_f_v] : ( member_term_f_v @ X4 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_115_Collect__mem__eq,axiom,
! [A4: set_Pr516226060960033221_f_nat] :
( ( collec8116569655616119888_f_nat
@ ^ [X4: produc2676906022994773093_f_nat] : ( member7030746907725424910_f_nat @ X4 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_116_Collect__mem__eq,axiom,
! [A4: set_Pr5245412377734449720_f_nat] :
( ( collec1417984143148964183_f_nat
@ ^ [X4: product_prod_f_nat] : ( member2678038196313615385_f_nat @ X4 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_117_Collect__mem__eq,axiom,
! [A4: set_Pr8827868859434726151rm_f_v] :
( ( collec7769483783292507282rm_f_v
@ ^ [X4: produc8027700073345061287rm_f_v] : ( member4316649861536941904rm_f_v @ X4 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_118_Collect__cong,axiom,
! [P2: produc8027700073345061287rm_f_v > $o,Q: produc8027700073345061287rm_f_v > $o] :
( ! [X: produc8027700073345061287rm_f_v] :
( ( P2 @ X )
= ( Q @ X ) )
=> ( ( collec7769483783292507282rm_f_v @ P2 )
= ( collec7769483783292507282rm_f_v @ Q ) ) ) ).
% Collect_cong
thf(fact_119_Collect__cong,axiom,
! [P2: product_prod_f_nat > $o,Q: product_prod_f_nat > $o] :
( ! [X: product_prod_f_nat] :
( ( P2 @ X )
= ( Q @ X ) )
=> ( ( collec1417984143148964183_f_nat @ P2 )
= ( collec1417984143148964183_f_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_120_rtrancl_Ocases,axiom,
! [A1: product_prod_f_nat,A22: product_prod_f_nat,R: set_Pr516226060960033221_f_nat] :
( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ A1 @ A22 ) @ ( transi8903367562120364223_f_nat @ R ) )
=> ( ( A22 != A1 )
=> ~ ! [B3: product_prod_f_nat] :
( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ A1 @ B3 ) @ ( transi8903367562120364223_f_nat @ R ) )
=> ~ ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ B3 @ A22 ) @ R ) ) ) ) ).
% rtrancl.cases
thf(fact_121_rtrancl_Ocases,axiom,
! [A1: produc8027700073345061287rm_f_v,A22: produc8027700073345061287rm_f_v,R: set_Pr7897041799657265351rm_f_v] :
( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A1 @ A22 ) @ ( transi1062743709496199978rm_f_v @ R ) )
=> ( ( A22 != A1 )
=> ~ ! [B3: produc8027700073345061287rm_f_v] :
( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A1 @ B3 ) @ ( transi1062743709496199978rm_f_v @ R ) )
=> ~ ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ B3 @ A22 ) @ R ) ) ) ) ).
% rtrancl.cases
thf(fact_122_rtrancl_Ocases,axiom,
! [A1: f,A22: f,R: set_Product_prod_f_f] :
( ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ A1 @ A22 ) @ ( transitive_rtrancl_f @ R ) )
=> ( ( A22 != A1 )
=> ~ ! [B3: f] :
( ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ A1 @ B3 ) @ ( transitive_rtrancl_f @ R ) )
=> ~ ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ B3 @ A22 ) @ R ) ) ) ) ).
% rtrancl.cases
thf(fact_123_rtrancl_Ocases,axiom,
! [A1: nat,A22: nat,R: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A1 @ A22 ) @ ( transi2905341329935302413cl_nat @ R ) )
=> ( ( A22 != A1 )
=> ~ ! [B3: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A1 @ B3 ) @ ( transi2905341329935302413cl_nat @ R ) )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B3 @ A22 ) @ R ) ) ) ) ).
% rtrancl.cases
thf(fact_124_rtrancl_Ocases,axiom,
! [A1: term_f_v,A22: term_f_v,R: set_Pr8827868859434726151rm_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A1 @ A22 ) @ ( transi3028809021613410336rm_f_v @ R ) )
=> ( ( A22 != A1 )
=> ~ ! [B3: term_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A1 @ B3 ) @ ( transi3028809021613410336rm_f_v @ R ) )
=> ~ ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B3 @ A22 ) @ R ) ) ) ) ).
% rtrancl.cases
thf(fact_125_rtrancl_Osimps,axiom,
! [A1: product_prod_f_nat,A22: product_prod_f_nat,R: set_Pr516226060960033221_f_nat] :
( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ A1 @ A22 ) @ ( transi8903367562120364223_f_nat @ R ) )
= ( ? [A5: product_prod_f_nat] :
( ( A1 = A5 )
& ( A22 = A5 ) )
| ? [A5: product_prod_f_nat,B4: product_prod_f_nat,C4: product_prod_f_nat] :
( ( A1 = A5 )
& ( A22 = C4 )
& ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ A5 @ B4 ) @ ( transi8903367562120364223_f_nat @ R ) )
& ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ B4 @ C4 ) @ R ) ) ) ) ).
% rtrancl.simps
thf(fact_126_rtrancl_Osimps,axiom,
! [A1: produc8027700073345061287rm_f_v,A22: produc8027700073345061287rm_f_v,R: set_Pr7897041799657265351rm_f_v] :
( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A1 @ A22 ) @ ( transi1062743709496199978rm_f_v @ R ) )
= ( ? [A5: produc8027700073345061287rm_f_v] :
( ( A1 = A5 )
& ( A22 = A5 ) )
| ? [A5: produc8027700073345061287rm_f_v,B4: produc8027700073345061287rm_f_v,C4: produc8027700073345061287rm_f_v] :
( ( A1 = A5 )
& ( A22 = C4 )
& ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A5 @ B4 ) @ ( transi1062743709496199978rm_f_v @ R ) )
& ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ B4 @ C4 ) @ R ) ) ) ) ).
% rtrancl.simps
thf(fact_127_rtrancl_Osimps,axiom,
! [A1: f,A22: f,R: set_Product_prod_f_f] :
( ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ A1 @ A22 ) @ ( transitive_rtrancl_f @ R ) )
= ( ? [A5: f] :
( ( A1 = A5 )
& ( A22 = A5 ) )
| ? [A5: f,B4: f,C4: f] :
( ( A1 = A5 )
& ( A22 = C4 )
& ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ A5 @ B4 ) @ ( transitive_rtrancl_f @ R ) )
& ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ B4 @ C4 ) @ R ) ) ) ) ).
% rtrancl.simps
thf(fact_128_rtrancl_Osimps,axiom,
! [A1: nat,A22: nat,R: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A1 @ A22 ) @ ( transi2905341329935302413cl_nat @ R ) )
= ( ? [A5: nat] :
( ( A1 = A5 )
& ( A22 = A5 ) )
| ? [A5: nat,B4: nat,C4: nat] :
( ( A1 = A5 )
& ( A22 = C4 )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A5 @ B4 ) @ ( transi2905341329935302413cl_nat @ R ) )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B4 @ C4 ) @ R ) ) ) ) ).
% rtrancl.simps
thf(fact_129_rtrancl_Osimps,axiom,
! [A1: term_f_v,A22: term_f_v,R: set_Pr8827868859434726151rm_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A1 @ A22 ) @ ( transi3028809021613410336rm_f_v @ R ) )
= ( ? [A5: term_f_v] :
( ( A1 = A5 )
& ( A22 = A5 ) )
| ? [A5: term_f_v,B4: term_f_v,C4: term_f_v] :
( ( A1 = A5 )
& ( A22 = C4 )
& ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A5 @ B4 ) @ ( transi3028809021613410336rm_f_v @ R ) )
& ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B4 @ C4 ) @ R ) ) ) ) ).
% rtrancl.simps
thf(fact_130_rtrancl_Ortrancl__refl,axiom,
! [A: product_prod_f_nat,R: set_Pr516226060960033221_f_nat] : ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ A @ A ) @ ( transi8903367562120364223_f_nat @ R ) ) ).
% rtrancl.rtrancl_refl
thf(fact_131_rtrancl_Ortrancl__refl,axiom,
! [A: produc8027700073345061287rm_f_v,R: set_Pr7897041799657265351rm_f_v] : ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A @ A ) @ ( transi1062743709496199978rm_f_v @ R ) ) ).
% rtrancl.rtrancl_refl
thf(fact_132_rtrancl_Ortrancl__refl,axiom,
! [A: f,R: set_Product_prod_f_f] : ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ A @ A ) @ ( transitive_rtrancl_f @ R ) ) ).
% rtrancl.rtrancl_refl
thf(fact_133_rtrancl_Ortrancl__refl,axiom,
! [A: nat,R: set_Pr1261947904930325089at_nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ A ) @ ( transi2905341329935302413cl_nat @ R ) ) ).
% rtrancl.rtrancl_refl
thf(fact_134_rtrancl_Ortrancl__refl,axiom,
! [A: term_f_v,R: set_Pr8827868859434726151rm_f_v] : ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ A ) @ ( transi3028809021613410336rm_f_v @ R ) ) ).
% rtrancl.rtrancl_refl
thf(fact_135_subset__antisym,axiom,
! [A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ A4 @ B5 )
=> ( ( ord_le3146513528884898305at_nat @ B5 @ A4 )
=> ( A4 = B5 ) ) ) ).
% subset_antisym
thf(fact_136_subset__antisym,axiom,
! [A4: set_Pr4548512049067879902_nat_f,B5: set_Pr4548512049067879902_nat_f] :
( ( ord_le8280083912720879166_nat_f @ A4 @ B5 )
=> ( ( ord_le8280083912720879166_nat_f @ B5 @ A4 )
=> ( A4 = B5 ) ) ) ).
% subset_antisym
thf(fact_137_subset__antisym,axiom,
! [A4: set_Product_prod_f_f,B5: set_Product_prod_f_f] :
( ( ord_le7147000192471188839od_f_f @ A4 @ B5 )
=> ( ( ord_le7147000192471188839od_f_f @ B5 @ A4 )
=> ( A4 = B5 ) ) ) ).
% subset_antisym
thf(fact_138_subset__antisym,axiom,
! [A4: set_term_f_v,B5: set_term_f_v] :
( ( ord_le947722214152454265rm_f_v @ A4 @ B5 )
=> ( ( ord_le947722214152454265rm_f_v @ B5 @ A4 )
=> ( A4 = B5 ) ) ) ).
% subset_antisym
thf(fact_139_subset__antisym,axiom,
! [A4: set_Pr5245412377734449720_f_nat,B5: set_Pr5245412377734449720_f_nat] :
( ( ord_le8976984241387448984_f_nat @ A4 @ B5 )
=> ( ( ord_le8976984241387448984_f_nat @ B5 @ A4 )
=> ( A4 = B5 ) ) ) ).
% subset_antisym
thf(fact_140_subsetI,axiom,
! [A4: set_Pr7897041799657265351rm_f_v,B5: set_Pr7897041799657265351rm_f_v] :
( ! [X: produc3406425829894461415rm_f_v] :
( ( member7300054593328134928rm_f_v @ X @ A4 )
=> ( member7300054593328134928rm_f_v @ X @ B5 ) )
=> ( ord_le7518223742277120103rm_f_v @ A4 @ B5 ) ) ).
% subsetI
thf(fact_141_subsetI,axiom,
! [A4: set_Pr516226060960033221_f_nat,B5: set_Pr516226060960033221_f_nat] :
( ! [X: produc2676906022994773093_f_nat] :
( ( member7030746907725424910_f_nat @ X @ A4 )
=> ( member7030746907725424910_f_nat @ X @ B5 ) )
=> ( ord_le8946695678602162533_f_nat @ A4 @ B5 ) ) ).
% subsetI
thf(fact_142_subsetI,axiom,
! [A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
( ! [X: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X @ A4 )
=> ( member8440522571783428010at_nat @ X @ B5 ) )
=> ( ord_le3146513528884898305at_nat @ A4 @ B5 ) ) ).
% subsetI
thf(fact_143_subsetI,axiom,
! [A4: set_Pr4548512049067879902_nat_f,B5: set_Pr4548512049067879902_nat_f] :
( ! [X: product_prod_nat_f] :
( ( member8962352073626239679_nat_f @ X @ A4 )
=> ( member8962352073626239679_nat_f @ X @ B5 ) )
=> ( ord_le8280083912720879166_nat_f @ A4 @ B5 ) ) ).
% subsetI
thf(fact_144_subsetI,axiom,
! [A4: set_Product_prod_f_f,B5: set_Product_prod_f_f] :
( ! [X: product_prod_f_f] :
( ( member5935997662887750672od_f_f @ X @ A4 )
=> ( member5935997662887750672od_f_f @ X @ B5 ) )
=> ( ord_le7147000192471188839od_f_f @ A4 @ B5 ) ) ).
% subsetI
thf(fact_145_subsetI,axiom,
! [A4: set_term_f_v,B5: set_term_f_v] :
( ! [X: term_f_v] :
( ( member_term_f_v @ X @ A4 )
=> ( member_term_f_v @ X @ B5 ) )
=> ( ord_le947722214152454265rm_f_v @ A4 @ B5 ) ) ).
% subsetI
thf(fact_146_subsetI,axiom,
! [A4: set_Pr8827868859434726151rm_f_v,B5: set_Pr8827868859434726151rm_f_v] :
( ! [X: produc8027700073345061287rm_f_v] :
( ( member4316649861536941904rm_f_v @ X @ A4 )
=> ( member4316649861536941904rm_f_v @ X @ B5 ) )
=> ( ord_le4559761987009501863rm_f_v @ A4 @ B5 ) ) ).
% subsetI
thf(fact_147_subsetI,axiom,
! [A4: set_Pr5245412377734449720_f_nat,B5: set_Pr5245412377734449720_f_nat] :
( ! [X: product_prod_f_nat] :
( ( member2678038196313615385_f_nat @ X @ A4 )
=> ( member2678038196313615385_f_nat @ X @ B5 ) )
=> ( ord_le8976984241387448984_f_nat @ A4 @ B5 ) ) ).
% subsetI
thf(fact_148_dual__order_Orefl,axiom,
! [A: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ A @ A ) ).
% dual_order.refl
thf(fact_149_dual__order_Orefl,axiom,
! [A: set_Pr4548512049067879902_nat_f] : ( ord_le8280083912720879166_nat_f @ A @ A ) ).
% dual_order.refl
thf(fact_150_dual__order_Orefl,axiom,
! [A: set_Product_prod_f_f] : ( ord_le7147000192471188839od_f_f @ A @ A ) ).
% dual_order.refl
thf(fact_151_dual__order_Orefl,axiom,
! [A: set_term_f_v] : ( ord_le947722214152454265rm_f_v @ A @ A ) ).
% dual_order.refl
thf(fact_152_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_153_dual__order_Orefl,axiom,
! [A: set_Pr5245412377734449720_f_nat] : ( ord_le8976984241387448984_f_nat @ A @ A ) ).
% dual_order.refl
thf(fact_154_order__refl,axiom,
! [X3: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ X3 @ X3 ) ).
% order_refl
thf(fact_155_order__refl,axiom,
! [X3: set_Pr4548512049067879902_nat_f] : ( ord_le8280083912720879166_nat_f @ X3 @ X3 ) ).
% order_refl
thf(fact_156_order__refl,axiom,
! [X3: set_Product_prod_f_f] : ( ord_le7147000192471188839od_f_f @ X3 @ X3 ) ).
% order_refl
thf(fact_157_order__refl,axiom,
! [X3: set_term_f_v] : ( ord_le947722214152454265rm_f_v @ X3 @ X3 ) ).
% order_refl
thf(fact_158_order__refl,axiom,
! [X3: nat] : ( ord_less_eq_nat @ X3 @ X3 ) ).
% order_refl
thf(fact_159_order__refl,axiom,
! [X3: set_Pr5245412377734449720_f_nat] : ( ord_le8976984241387448984_f_nat @ X3 @ X3 ) ).
% order_refl
thf(fact_160_sig__steps__join__ctxt__closed,axiom,
! [C2: subterm_and_ctxt_f_v,F: set_Pr5245412377734449720_f_nat,S: term_f_v,T: term_f_v,R2: set_Pr8827868859434726151rm_f_v] :
( ( ord_le8976984241387448984_f_nat @ ( term_funas_ctxt_f_v @ C2 ) @ F )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ ( abstra8605546704478008873rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ R2 ) ) ) )
=> ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ ( subter6886040689604212261rm_f_v @ C2 @ S ) @ ( subter6886040689604212261rm_f_v @ C2 @ T ) ) @ ( abstra8605546704478008873rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ R2 ) ) ) ) ) ) ).
% sig_steps_join_ctxt_closed
thf(fact_161_subrelI,axiom,
! [R: set_Pr7897041799657265351rm_f_v,S: set_Pr7897041799657265351rm_f_v] :
( ! [X: produc8027700073345061287rm_f_v,Y3: produc8027700073345061287rm_f_v] :
( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ X @ Y3 ) @ R )
=> ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ X @ Y3 ) @ S ) )
=> ( ord_le7518223742277120103rm_f_v @ R @ S ) ) ).
% subrelI
thf(fact_162_subrelI,axiom,
! [R: set_Pr516226060960033221_f_nat,S: set_Pr516226060960033221_f_nat] :
( ! [X: product_prod_f_nat,Y3: product_prod_f_nat] :
( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ X @ Y3 ) @ R )
=> ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ X @ Y3 ) @ S ) )
=> ( ord_le8946695678602162533_f_nat @ R @ S ) ) ).
% subrelI
thf(fact_163_subrelI,axiom,
! [R: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
( ! [X: nat,Y3: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y3 ) @ R )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y3 ) @ S ) )
=> ( ord_le3146513528884898305at_nat @ R @ S ) ) ).
% subrelI
thf(fact_164_subrelI,axiom,
! [R: set_Pr4548512049067879902_nat_f,S: set_Pr4548512049067879902_nat_f] :
( ! [X: nat,Y3: f] :
( ( member8962352073626239679_nat_f @ ( product_Pair_nat_f @ X @ Y3 ) @ R )
=> ( member8962352073626239679_nat_f @ ( product_Pair_nat_f @ X @ Y3 ) @ S ) )
=> ( ord_le8280083912720879166_nat_f @ R @ S ) ) ).
% subrelI
thf(fact_165_subrelI,axiom,
! [R: set_Product_prod_f_f,S: set_Product_prod_f_f] :
( ! [X: f,Y3: f] :
( ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ X @ Y3 ) @ R )
=> ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ X @ Y3 ) @ S ) )
=> ( ord_le7147000192471188839od_f_f @ R @ S ) ) ).
% subrelI
thf(fact_166_subrelI,axiom,
! [R: set_Pr8827868859434726151rm_f_v,S: set_Pr8827868859434726151rm_f_v] :
( ! [X: term_f_v,Y3: term_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ Y3 ) @ R )
=> ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ Y3 ) @ S ) )
=> ( ord_le4559761987009501863rm_f_v @ R @ S ) ) ).
% subrelI
thf(fact_167_subrelI,axiom,
! [R: set_Pr5245412377734449720_f_nat,S: set_Pr5245412377734449720_f_nat] :
( ! [X: f,Y3: nat] :
( ( member2678038196313615385_f_nat @ ( product_Pair_f_nat @ X @ Y3 ) @ R )
=> ( member2678038196313615385_f_nat @ ( product_Pair_f_nat @ X @ Y3 ) @ S ) )
=> ( ord_le8976984241387448984_f_nat @ R @ S ) ) ).
% subrelI
thf(fact_168_srsteps__ctxt__closed,axiom,
! [C2: subterm_and_ctxt_f_v,F: set_Pr5245412377734449720_f_nat,S: term_f_v,T: term_f_v,R2: set_Pr8827868859434726151rm_f_v] :
( ( ord_le8976984241387448984_f_nat @ ( term_funas_ctxt_f_v @ C2 ) @ F )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ ( transi3208867851621559554rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ R2 ) ) ) )
=> ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ ( subter6886040689604212261rm_f_v @ C2 @ S ) @ ( subter6886040689604212261rm_f_v @ C2 @ T ) ) @ ( transi3208867851621559554rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ R2 ) ) ) ) ) ) ).
% srsteps_ctxt_closed
thf(fact_169_srsteps__eq__monp,axiom,
! [F: set_Pr5245412377734449720_f_nat,G: set_Pr5245412377734449720_f_nat,R2: set_Pr8827868859434726151rm_f_v] :
( ( ord_le8976984241387448984_f_nat @ F @ G )
=> ( ord_le4559761987009501863rm_f_v @ ( transi3028809021613410336rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ R2 ) ) ) @ ( transi3028809021613410336rm_f_v @ ( sig_step_f_v @ G @ ( rstep_f_v @ R2 ) ) ) ) ) ).
% srsteps_eq_monp
thf(fact_170_rstep_Ocases,axiom,
! [A1: term_f_v,A22: term_f_v,R2: set_Pr8827868859434726151rm_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A1 @ A22 ) @ ( rstep_f_v @ R2 ) )
=> ~ ! [S3: term_f_v,T2: term_f_v,C5: subterm_and_ctxt_f_v] :
( ( A1
= ( subter6886040689604212261rm_f_v @ C5 @ S3 ) )
=> ( ( A22
= ( subter6886040689604212261rm_f_v @ C5 @ T2 ) )
=> ~ ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S3 @ T2 ) @ ( rrstep_f_v @ R2 ) ) ) ) ) ).
% rstep.cases
thf(fact_171_rstep_Osimps,axiom,
! [A1: term_f_v,A22: term_f_v,R2: set_Pr8827868859434726151rm_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A1 @ A22 ) @ ( rstep_f_v @ R2 ) )
= ( ? [S4: term_f_v,T3: term_f_v,C6: subterm_and_ctxt_f_v] :
( ( A1
= ( subter6886040689604212261rm_f_v @ C6 @ S4 ) )
& ( A22
= ( subter6886040689604212261rm_f_v @ C6 @ T3 ) )
& ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S4 @ T3 ) @ ( rrstep_f_v @ R2 ) ) ) ) ) ).
% rstep.simps
thf(fact_172_rstep_Ointros,axiom,
! [S: term_f_v,T: term_f_v,R2: set_Pr8827868859434726151rm_f_v,C2: subterm_and_ctxt_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ ( rrstep_f_v @ R2 ) )
=> ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ ( subter6886040689604212261rm_f_v @ C2 @ S ) @ ( subter6886040689604212261rm_f_v @ C2 @ T ) ) @ ( rstep_f_v @ R2 ) ) ) ).
% rstep.intros
thf(fact_173_rtrancl__trancl__absorb,axiom,
! [R3: set_Pr7897041799657265351rm_f_v] :
( ( transi7658437189833989320rm_f_v @ ( transi1062743709496199978rm_f_v @ R3 ) )
= ( transi1062743709496199978rm_f_v @ R3 ) ) ).
% rtrancl_trancl_absorb
thf(fact_174_rtrancl__trancl__absorb,axiom,
! [R3: set_Product_prod_f_f] :
( ( transitive_trancl_f @ ( transitive_rtrancl_f @ R3 ) )
= ( transitive_rtrancl_f @ R3 ) ) ).
% rtrancl_trancl_absorb
thf(fact_175_rtrancl__trancl__absorb,axiom,
! [R3: set_Pr1261947904930325089at_nat] :
( ( transi6264000038957366511cl_nat @ ( transi2905341329935302413cl_nat @ R3 ) )
= ( transi2905341329935302413cl_nat @ R3 ) ) ).
% rtrancl_trancl_absorb
thf(fact_176_rtrancl__trancl__absorb,axiom,
! [R3: set_Pr8827868859434726151rm_f_v] :
( ( transi3208867851621559554rm_f_v @ ( transi3028809021613410336rm_f_v @ R3 ) )
= ( transi3028809021613410336rm_f_v @ R3 ) ) ).
% rtrancl_trancl_absorb
thf(fact_177_trancl__rtrancl__absorb,axiom,
! [R3: set_Pr7897041799657265351rm_f_v] :
( ( transi1062743709496199978rm_f_v @ ( transi7658437189833989320rm_f_v @ R3 ) )
= ( transi1062743709496199978rm_f_v @ R3 ) ) ).
% trancl_rtrancl_absorb
thf(fact_178_trancl__rtrancl__absorb,axiom,
! [R3: set_Product_prod_f_f] :
( ( transitive_rtrancl_f @ ( transitive_trancl_f @ R3 ) )
= ( transitive_rtrancl_f @ R3 ) ) ).
% trancl_rtrancl_absorb
thf(fact_179_trancl__rtrancl__absorb,axiom,
! [R3: set_Pr1261947904930325089at_nat] :
( ( transi2905341329935302413cl_nat @ ( transi6264000038957366511cl_nat @ R3 ) )
= ( transi2905341329935302413cl_nat @ R3 ) ) ).
% trancl_rtrancl_absorb
thf(fact_180_trancl__rtrancl__absorb,axiom,
! [R3: set_Pr8827868859434726151rm_f_v] :
( ( transi3028809021613410336rm_f_v @ ( transi3208867851621559554rm_f_v @ R3 ) )
= ( transi3028809021613410336rm_f_v @ R3 ) ) ).
% trancl_rtrancl_absorb
thf(fact_181_rrstep__basicI,axiom,
! [L: term_f_v,R: term_f_v,R2: set_Pr8827868859434726151rm_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ L @ R ) @ R2 )
=> ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ L @ R ) @ ( rrstep_f_v @ R2 ) ) ) ).
% rrstep_basicI
thf(fact_182_sig__stepI,axiom,
! [S: term_f_v,F: set_Pr5245412377734449720_f_nat,T: term_f_v,R2: set_Pr8827868859434726151rm_f_v] :
( ( ord_le8976984241387448984_f_nat @ ( term_funas_term_f_v @ S ) @ F )
=> ( ( ord_le8976984241387448984_f_nat @ ( term_funas_term_f_v @ T ) @ F )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ R2 )
=> ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ ( sig_step_f_v @ F @ R2 ) ) ) ) ) ).
% sig_stepI
thf(fact_183_r__into__trancl_H,axiom,
! [P: produc2676906022994773093_f_nat,R: set_Pr516226060960033221_f_nat] :
( ( member7030746907725424910_f_nat @ P @ R )
=> ( member7030746907725424910_f_nat @ P @ ( transi3759436115690491041_f_nat @ R ) ) ) ).
% r_into_trancl'
thf(fact_184_r__into__trancl_H,axiom,
! [P: produc3406425829894461415rm_f_v,R: set_Pr7897041799657265351rm_f_v] :
( ( member7300054593328134928rm_f_v @ P @ R )
=> ( member7300054593328134928rm_f_v @ P @ ( transi7658437189833989320rm_f_v @ R ) ) ) ).
% r_into_trancl'
thf(fact_185_r__into__trancl_H,axiom,
! [P: product_prod_f_f,R: set_Product_prod_f_f] :
( ( member5935997662887750672od_f_f @ P @ R )
=> ( member5935997662887750672od_f_f @ P @ ( transitive_trancl_f @ R ) ) ) ).
% r_into_trancl'
thf(fact_186_r__into__trancl_H,axiom,
! [P: product_prod_nat_nat,R: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ P @ R )
=> ( member8440522571783428010at_nat @ P @ ( transi6264000038957366511cl_nat @ R ) ) ) ).
% r_into_trancl'
thf(fact_187_r__into__trancl_H,axiom,
! [P: produc8027700073345061287rm_f_v,R: set_Pr8827868859434726151rm_f_v] :
( ( member4316649861536941904rm_f_v @ P @ R )
=> ( member4316649861536941904rm_f_v @ P @ ( transi3208867851621559554rm_f_v @ R ) ) ) ).
% r_into_trancl'
thf(fact_188_rrstep__rstep__mono,axiom,
! [R2: set_Pr8827868859434726151rm_f_v] : ( ord_le4559761987009501863rm_f_v @ ( rrstep_f_v @ R2 ) @ ( rstep_f_v @ R2 ) ) ).
% rrstep_rstep_mono
thf(fact_189_converse__trancl__induct,axiom,
! [A: product_prod_f_nat,B: product_prod_f_nat,R: set_Pr516226060960033221_f_nat,P2: product_prod_f_nat > $o] :
( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ A @ B ) @ ( transi3759436115690491041_f_nat @ R ) )
=> ( ! [Y3: product_prod_f_nat] :
( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ Y3 @ B ) @ R )
=> ( P2 @ Y3 ) )
=> ( ! [Y3: product_prod_f_nat,Z: product_prod_f_nat] :
( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ Y3 @ Z ) @ R )
=> ( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ Z @ B ) @ ( transi3759436115690491041_f_nat @ R ) )
=> ( ( P2 @ Z )
=> ( P2 @ Y3 ) ) ) )
=> ( P2 @ A ) ) ) ) ).
% converse_trancl_induct
thf(fact_190_converse__trancl__induct,axiom,
! [A: produc8027700073345061287rm_f_v,B: produc8027700073345061287rm_f_v,R: set_Pr7897041799657265351rm_f_v,P2: produc8027700073345061287rm_f_v > $o] :
( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A @ B ) @ ( transi7658437189833989320rm_f_v @ R ) )
=> ( ! [Y3: produc8027700073345061287rm_f_v] :
( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ Y3 @ B ) @ R )
=> ( P2 @ Y3 ) )
=> ( ! [Y3: produc8027700073345061287rm_f_v,Z: produc8027700073345061287rm_f_v] :
( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ Y3 @ Z ) @ R )
=> ( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ Z @ B ) @ ( transi7658437189833989320rm_f_v @ R ) )
=> ( ( P2 @ Z )
=> ( P2 @ Y3 ) ) ) )
=> ( P2 @ A ) ) ) ) ).
% converse_trancl_induct
thf(fact_191_converse__trancl__induct,axiom,
! [A: f,B: f,R: set_Product_prod_f_f,P2: f > $o] :
( ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ A @ B ) @ ( transitive_trancl_f @ R ) )
=> ( ! [Y3: f] :
( ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ Y3 @ B ) @ R )
=> ( P2 @ Y3 ) )
=> ( ! [Y3: f,Z: f] :
( ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ Y3 @ Z ) @ R )
=> ( ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ Z @ B ) @ ( transitive_trancl_f @ R ) )
=> ( ( P2 @ Z )
=> ( P2 @ Y3 ) ) ) )
=> ( P2 @ A ) ) ) ) ).
% converse_trancl_induct
thf(fact_192_converse__trancl__induct,axiom,
! [A: nat,B: nat,R: set_Pr1261947904930325089at_nat,P2: nat > $o] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( transi6264000038957366511cl_nat @ R ) )
=> ( ! [Y3: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ B ) @ R )
=> ( P2 @ Y3 ) )
=> ( ! [Y3: nat,Z: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ Z ) @ R )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Z @ B ) @ ( transi6264000038957366511cl_nat @ R ) )
=> ( ( P2 @ Z )
=> ( P2 @ Y3 ) ) ) )
=> ( P2 @ A ) ) ) ) ).
% converse_trancl_induct
thf(fact_193_converse__trancl__induct,axiom,
! [A: term_f_v,B: term_f_v,R: set_Pr8827868859434726151rm_f_v,P2: term_f_v > $o] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( transi3208867851621559554rm_f_v @ R ) )
=> ( ! [Y3: term_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y3 @ B ) @ R )
=> ( P2 @ Y3 ) )
=> ( ! [Y3: term_f_v,Z: term_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y3 @ Z ) @ R )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Z @ B ) @ ( transi3208867851621559554rm_f_v @ R ) )
=> ( ( P2 @ Z )
=> ( P2 @ Y3 ) ) ) )
=> ( P2 @ A ) ) ) ) ).
% converse_trancl_induct
thf(fact_194_trancl__trans__induct,axiom,
! [X3: product_prod_f_nat,Y: product_prod_f_nat,R: set_Pr516226060960033221_f_nat,P2: product_prod_f_nat > product_prod_f_nat > $o] :
( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ X3 @ Y ) @ ( transi3759436115690491041_f_nat @ R ) )
=> ( ! [X: product_prod_f_nat,Y3: product_prod_f_nat] :
( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ X @ Y3 ) @ R )
=> ( P2 @ X @ Y3 ) )
=> ( ! [X: product_prod_f_nat,Y3: product_prod_f_nat,Z: product_prod_f_nat] :
( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ X @ Y3 ) @ ( transi3759436115690491041_f_nat @ R ) )
=> ( ( P2 @ X @ Y3 )
=> ( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ Y3 @ Z ) @ ( transi3759436115690491041_f_nat @ R ) )
=> ( ( P2 @ Y3 @ Z )
=> ( P2 @ X @ Z ) ) ) ) )
=> ( P2 @ X3 @ Y ) ) ) ) ).
% trancl_trans_induct
thf(fact_195_trancl__trans__induct,axiom,
! [X3: produc8027700073345061287rm_f_v,Y: produc8027700073345061287rm_f_v,R: set_Pr7897041799657265351rm_f_v,P2: produc8027700073345061287rm_f_v > produc8027700073345061287rm_f_v > $o] :
( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ X3 @ Y ) @ ( transi7658437189833989320rm_f_v @ R ) )
=> ( ! [X: produc8027700073345061287rm_f_v,Y3: produc8027700073345061287rm_f_v] :
( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ X @ Y3 ) @ R )
=> ( P2 @ X @ Y3 ) )
=> ( ! [X: produc8027700073345061287rm_f_v,Y3: produc8027700073345061287rm_f_v,Z: produc8027700073345061287rm_f_v] :
( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ X @ Y3 ) @ ( transi7658437189833989320rm_f_v @ R ) )
=> ( ( P2 @ X @ Y3 )
=> ( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ Y3 @ Z ) @ ( transi7658437189833989320rm_f_v @ R ) )
=> ( ( P2 @ Y3 @ Z )
=> ( P2 @ X @ Z ) ) ) ) )
=> ( P2 @ X3 @ Y ) ) ) ) ).
% trancl_trans_induct
thf(fact_196_trancl__trans__induct,axiom,
! [X3: f,Y: f,R: set_Product_prod_f_f,P2: f > f > $o] :
( ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ X3 @ Y ) @ ( transitive_trancl_f @ R ) )
=> ( ! [X: f,Y3: f] :
( ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ X @ Y3 ) @ R )
=> ( P2 @ X @ Y3 ) )
=> ( ! [X: f,Y3: f,Z: f] :
( ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ X @ Y3 ) @ ( transitive_trancl_f @ R ) )
=> ( ( P2 @ X @ Y3 )
=> ( ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ Y3 @ Z ) @ ( transitive_trancl_f @ R ) )
=> ( ( P2 @ Y3 @ Z )
=> ( P2 @ X @ Z ) ) ) ) )
=> ( P2 @ X3 @ Y ) ) ) ) ).
% trancl_trans_induct
thf(fact_197_trancl__trans__induct,axiom,
! [X3: nat,Y: nat,R: set_Pr1261947904930325089at_nat,P2: nat > nat > $o] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y ) @ ( transi6264000038957366511cl_nat @ R ) )
=> ( ! [X: nat,Y3: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y3 ) @ R )
=> ( P2 @ X @ Y3 ) )
=> ( ! [X: nat,Y3: nat,Z: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y3 ) @ ( transi6264000038957366511cl_nat @ R ) )
=> ( ( P2 @ X @ Y3 )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ Z ) @ ( transi6264000038957366511cl_nat @ R ) )
=> ( ( P2 @ Y3 @ Z )
=> ( P2 @ X @ Z ) ) ) ) )
=> ( P2 @ X3 @ Y ) ) ) ) ).
% trancl_trans_induct
thf(fact_198_trancl__trans__induct,axiom,
! [X3: term_f_v,Y: term_f_v,R: set_Pr8827868859434726151rm_f_v,P2: term_f_v > term_f_v > $o] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Y ) @ ( transi3208867851621559554rm_f_v @ R ) )
=> ( ! [X: term_f_v,Y3: term_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ Y3 ) @ R )
=> ( P2 @ X @ Y3 ) )
=> ( ! [X: term_f_v,Y3: term_f_v,Z: term_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ Y3 ) @ ( transi3208867851621559554rm_f_v @ R ) )
=> ( ( P2 @ X @ Y3 )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y3 @ Z ) @ ( transi3208867851621559554rm_f_v @ R ) )
=> ( ( P2 @ Y3 @ Z )
=> ( P2 @ X @ Z ) ) ) ) )
=> ( P2 @ X3 @ Y ) ) ) ) ).
% trancl_trans_induct
thf(fact_199_trancl__into__trancl2,axiom,
! [A: product_prod_f_nat,B: product_prod_f_nat,R: set_Pr516226060960033221_f_nat,C3: product_prod_f_nat] :
( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ A @ B ) @ R )
=> ( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ B @ C3 ) @ ( transi3759436115690491041_f_nat @ R ) )
=> ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ A @ C3 ) @ ( transi3759436115690491041_f_nat @ R ) ) ) ) ).
% trancl_into_trancl2
thf(fact_200_trancl__into__trancl2,axiom,
! [A: produc8027700073345061287rm_f_v,B: produc8027700073345061287rm_f_v,R: set_Pr7897041799657265351rm_f_v,C3: produc8027700073345061287rm_f_v] :
( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A @ B ) @ R )
=> ( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ B @ C3 ) @ ( transi7658437189833989320rm_f_v @ R ) )
=> ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A @ C3 ) @ ( transi7658437189833989320rm_f_v @ R ) ) ) ) ).
% trancl_into_trancl2
thf(fact_201_trancl__into__trancl2,axiom,
! [A: f,B: f,R: set_Product_prod_f_f,C3: f] :
( ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ A @ B ) @ R )
=> ( ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ B @ C3 ) @ ( transitive_trancl_f @ R ) )
=> ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ A @ C3 ) @ ( transitive_trancl_f @ R ) ) ) ) ).
% trancl_into_trancl2
thf(fact_202_trancl__into__trancl2,axiom,
! [A: nat,B: nat,R: set_Pr1261947904930325089at_nat,C3: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ C3 ) @ ( transi6264000038957366511cl_nat @ R ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ C3 ) @ ( transi6264000038957366511cl_nat @ R ) ) ) ) ).
% trancl_into_trancl2
thf(fact_203_trancl__into__trancl2,axiom,
! [A: term_f_v,B: term_f_v,R: set_Pr8827868859434726151rm_f_v,C3: term_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ R )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B @ C3 ) @ ( transi3208867851621559554rm_f_v @ R ) )
=> ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ C3 ) @ ( transi3208867851621559554rm_f_v @ R ) ) ) ) ).
% trancl_into_trancl2
thf(fact_204_Transitive__Closure_Otrancl__into__trancl,axiom,
! [A: product_prod_f_nat,B: product_prod_f_nat,R: set_Pr516226060960033221_f_nat,C3: product_prod_f_nat] :
( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ A @ B ) @ ( transi3759436115690491041_f_nat @ R ) )
=> ( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ B @ C3 ) @ R )
=> ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ A @ C3 ) @ ( transi3759436115690491041_f_nat @ R ) ) ) ) ).
% Transitive_Closure.trancl_into_trancl
thf(fact_205_Transitive__Closure_Otrancl__into__trancl,axiom,
! [A: produc8027700073345061287rm_f_v,B: produc8027700073345061287rm_f_v,R: set_Pr7897041799657265351rm_f_v,C3: produc8027700073345061287rm_f_v] :
( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A @ B ) @ ( transi7658437189833989320rm_f_v @ R ) )
=> ( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ B @ C3 ) @ R )
=> ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A @ C3 ) @ ( transi7658437189833989320rm_f_v @ R ) ) ) ) ).
% Transitive_Closure.trancl_into_trancl
thf(fact_206_Transitive__Closure_Otrancl__into__trancl,axiom,
! [A: f,B: f,R: set_Product_prod_f_f,C3: f] :
( ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ A @ B ) @ ( transitive_trancl_f @ R ) )
=> ( ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ B @ C3 ) @ R )
=> ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ A @ C3 ) @ ( transitive_trancl_f @ R ) ) ) ) ).
% Transitive_Closure.trancl_into_trancl
thf(fact_207_Transitive__Closure_Otrancl__into__trancl,axiom,
! [A: nat,B: nat,R: set_Pr1261947904930325089at_nat,C3: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( transi6264000038957366511cl_nat @ R ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ C3 ) @ R )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ C3 ) @ ( transi6264000038957366511cl_nat @ R ) ) ) ) ).
% Transitive_Closure.trancl_into_trancl
thf(fact_208_Transitive__Closure_Otrancl__into__trancl,axiom,
! [A: term_f_v,B: term_f_v,R: set_Pr8827868859434726151rm_f_v,C3: term_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( transi3208867851621559554rm_f_v @ R ) )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B @ C3 ) @ R )
=> ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ C3 ) @ ( transi3208867851621559554rm_f_v @ R ) ) ) ) ).
% Transitive_Closure.trancl_into_trancl
thf(fact_209_irrefl__trancl__rD,axiom,
! [R: set_Pr516226060960033221_f_nat,X3: product_prod_f_nat,Y: product_prod_f_nat] :
( ! [X: product_prod_f_nat] :
~ ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ X @ X ) @ ( transi3759436115690491041_f_nat @ R ) )
=> ( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ X3 @ Y ) @ R )
=> ( X3 != Y ) ) ) ).
% irrefl_trancl_rD
thf(fact_210_irrefl__trancl__rD,axiom,
! [R: set_Pr7897041799657265351rm_f_v,X3: produc8027700073345061287rm_f_v,Y: produc8027700073345061287rm_f_v] :
( ! [X: produc8027700073345061287rm_f_v] :
~ ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ X @ X ) @ ( transi7658437189833989320rm_f_v @ R ) )
=> ( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ X3 @ Y ) @ R )
=> ( X3 != Y ) ) ) ).
% irrefl_trancl_rD
thf(fact_211_irrefl__trancl__rD,axiom,
! [R: set_Product_prod_f_f,X3: f,Y: f] :
( ! [X: f] :
~ ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ X @ X ) @ ( transitive_trancl_f @ R ) )
=> ( ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ X3 @ Y ) @ R )
=> ( X3 != Y ) ) ) ).
% irrefl_trancl_rD
thf(fact_212_irrefl__trancl__rD,axiom,
! [R: set_Pr1261947904930325089at_nat,X3: nat,Y: nat] :
( ! [X: nat] :
~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ X ) @ ( transi6264000038957366511cl_nat @ R ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y ) @ R )
=> ( X3 != Y ) ) ) ).
% irrefl_trancl_rD
thf(fact_213_irrefl__trancl__rD,axiom,
! [R: set_Pr8827868859434726151rm_f_v,X3: term_f_v,Y: term_f_v] :
( ! [X: term_f_v] :
~ ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ X ) @ ( transi3208867851621559554rm_f_v @ R ) )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Y ) @ R )
=> ( X3 != Y ) ) ) ).
% irrefl_trancl_rD
thf(fact_214_converse__tranclE,axiom,
! [X3: product_prod_f_nat,Z2: product_prod_f_nat,R: set_Pr516226060960033221_f_nat] :
( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ X3 @ Z2 ) @ ( transi3759436115690491041_f_nat @ R ) )
=> ( ~ ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ X3 @ Z2 ) @ R )
=> ~ ! [Y3: product_prod_f_nat] :
( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ X3 @ Y3 ) @ R )
=> ~ ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ Y3 @ Z2 ) @ ( transi3759436115690491041_f_nat @ R ) ) ) ) ) ).
% converse_tranclE
thf(fact_215_converse__tranclE,axiom,
! [X3: produc8027700073345061287rm_f_v,Z2: produc8027700073345061287rm_f_v,R: set_Pr7897041799657265351rm_f_v] :
( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ X3 @ Z2 ) @ ( transi7658437189833989320rm_f_v @ R ) )
=> ( ~ ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ X3 @ Z2 ) @ R )
=> ~ ! [Y3: produc8027700073345061287rm_f_v] :
( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ X3 @ Y3 ) @ R )
=> ~ ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ Y3 @ Z2 ) @ ( transi7658437189833989320rm_f_v @ R ) ) ) ) ) ).
% converse_tranclE
thf(fact_216_converse__tranclE,axiom,
! [X3: f,Z2: f,R: set_Product_prod_f_f] :
( ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ X3 @ Z2 ) @ ( transitive_trancl_f @ R ) )
=> ( ~ ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ X3 @ Z2 ) @ R )
=> ~ ! [Y3: f] :
( ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ X3 @ Y3 ) @ R )
=> ~ ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ Y3 @ Z2 ) @ ( transitive_trancl_f @ R ) ) ) ) ) ).
% converse_tranclE
thf(fact_217_converse__tranclE,axiom,
! [X3: nat,Z2: nat,R: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Z2 ) @ ( transi6264000038957366511cl_nat @ R ) )
=> ( ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Z2 ) @ R )
=> ~ ! [Y3: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ Z2 ) @ ( transi6264000038957366511cl_nat @ R ) ) ) ) ) ).
% converse_tranclE
thf(fact_218_converse__tranclE,axiom,
! [X3: term_f_v,Z2: term_f_v,R: set_Pr8827868859434726151rm_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Z2 ) @ ( transi3208867851621559554rm_f_v @ R ) )
=> ( ~ ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Z2 ) @ R )
=> ~ ! [Y3: term_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Y3 ) @ R )
=> ~ ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y3 @ Z2 ) @ ( transi3208867851621559554rm_f_v @ R ) ) ) ) ) ).
% converse_tranclE
thf(fact_219_r__r__into__trancl,axiom,
! [A: product_prod_f_nat,B: product_prod_f_nat,R3: set_Pr516226060960033221_f_nat,C3: product_prod_f_nat] :
( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ A @ B ) @ R3 )
=> ( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ B @ C3 ) @ R3 )
=> ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ A @ C3 ) @ ( transi3759436115690491041_f_nat @ R3 ) ) ) ) ).
% r_r_into_trancl
thf(fact_220_r__r__into__trancl,axiom,
! [A: produc8027700073345061287rm_f_v,B: produc8027700073345061287rm_f_v,R3: set_Pr7897041799657265351rm_f_v,C3: produc8027700073345061287rm_f_v] :
( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A @ B ) @ R3 )
=> ( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ B @ C3 ) @ R3 )
=> ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A @ C3 ) @ ( transi7658437189833989320rm_f_v @ R3 ) ) ) ) ).
% r_r_into_trancl
thf(fact_221_r__r__into__trancl,axiom,
! [A: f,B: f,R3: set_Product_prod_f_f,C3: f] :
( ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ A @ B ) @ R3 )
=> ( ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ B @ C3 ) @ R3 )
=> ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ A @ C3 ) @ ( transitive_trancl_f @ R3 ) ) ) ) ).
% r_r_into_trancl
thf(fact_222_r__r__into__trancl,axiom,
! [A: nat,B: nat,R3: set_Pr1261947904930325089at_nat,C3: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R3 )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ C3 ) @ R3 )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ C3 ) @ ( transi6264000038957366511cl_nat @ R3 ) ) ) ) ).
% r_r_into_trancl
thf(fact_223_r__r__into__trancl,axiom,
! [A: term_f_v,B: term_f_v,R3: set_Pr8827868859434726151rm_f_v,C3: term_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ R3 )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B @ C3 ) @ R3 )
=> ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ C3 ) @ ( transi3208867851621559554rm_f_v @ R3 ) ) ) ) ).
% r_r_into_trancl
thf(fact_224_trancl__induct,axiom,
! [A: product_prod_f_nat,B: product_prod_f_nat,R: set_Pr516226060960033221_f_nat,P2: product_prod_f_nat > $o] :
( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ A @ B ) @ ( transi3759436115690491041_f_nat @ R ) )
=> ( ! [Y3: product_prod_f_nat] :
( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ A @ Y3 ) @ R )
=> ( P2 @ Y3 ) )
=> ( ! [Y3: product_prod_f_nat,Z: product_prod_f_nat] :
( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ A @ Y3 ) @ ( transi3759436115690491041_f_nat @ R ) )
=> ( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ Y3 @ Z ) @ R )
=> ( ( P2 @ Y3 )
=> ( P2 @ Z ) ) ) )
=> ( P2 @ B ) ) ) ) ).
% trancl_induct
thf(fact_225_trancl__induct,axiom,
! [A: produc8027700073345061287rm_f_v,B: produc8027700073345061287rm_f_v,R: set_Pr7897041799657265351rm_f_v,P2: produc8027700073345061287rm_f_v > $o] :
( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A @ B ) @ ( transi7658437189833989320rm_f_v @ R ) )
=> ( ! [Y3: produc8027700073345061287rm_f_v] :
( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A @ Y3 ) @ R )
=> ( P2 @ Y3 ) )
=> ( ! [Y3: produc8027700073345061287rm_f_v,Z: produc8027700073345061287rm_f_v] :
( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A @ Y3 ) @ ( transi7658437189833989320rm_f_v @ R ) )
=> ( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ Y3 @ Z ) @ R )
=> ( ( P2 @ Y3 )
=> ( P2 @ Z ) ) ) )
=> ( P2 @ B ) ) ) ) ).
% trancl_induct
thf(fact_226_trancl__induct,axiom,
! [A: f,B: f,R: set_Product_prod_f_f,P2: f > $o] :
( ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ A @ B ) @ ( transitive_trancl_f @ R ) )
=> ( ! [Y3: f] :
( ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ A @ Y3 ) @ R )
=> ( P2 @ Y3 ) )
=> ( ! [Y3: f,Z: f] :
( ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ A @ Y3 ) @ ( transitive_trancl_f @ R ) )
=> ( ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ Y3 @ Z ) @ R )
=> ( ( P2 @ Y3 )
=> ( P2 @ Z ) ) ) )
=> ( P2 @ B ) ) ) ) ).
% trancl_induct
thf(fact_227_trancl__induct,axiom,
! [A: nat,B: nat,R: set_Pr1261947904930325089at_nat,P2: nat > $o] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( transi6264000038957366511cl_nat @ R ) )
=> ( ! [Y3: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ Y3 ) @ R )
=> ( P2 @ Y3 ) )
=> ( ! [Y3: nat,Z: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ Y3 ) @ ( transi6264000038957366511cl_nat @ R ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ Z ) @ R )
=> ( ( P2 @ Y3 )
=> ( P2 @ Z ) ) ) )
=> ( P2 @ B ) ) ) ) ).
% trancl_induct
thf(fact_228_trancl__induct,axiom,
! [A: term_f_v,B: term_f_v,R: set_Pr8827868859434726151rm_f_v,P2: term_f_v > $o] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( transi3208867851621559554rm_f_v @ R ) )
=> ( ! [Y3: term_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ Y3 ) @ R )
=> ( P2 @ Y3 ) )
=> ( ! [Y3: term_f_v,Z: term_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ Y3 ) @ ( transi3208867851621559554rm_f_v @ R ) )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y3 @ Z ) @ R )
=> ( ( P2 @ Y3 )
=> ( P2 @ Z ) ) ) )
=> ( P2 @ B ) ) ) ) ).
% trancl_induct
thf(fact_229_trancl__trans,axiom,
! [X3: product_prod_f_nat,Y: product_prod_f_nat,R: set_Pr516226060960033221_f_nat,Z2: product_prod_f_nat] :
( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ X3 @ Y ) @ ( transi3759436115690491041_f_nat @ R ) )
=> ( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ Y @ Z2 ) @ ( transi3759436115690491041_f_nat @ R ) )
=> ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ X3 @ Z2 ) @ ( transi3759436115690491041_f_nat @ R ) ) ) ) ).
% trancl_trans
thf(fact_230_trancl__trans,axiom,
! [X3: produc8027700073345061287rm_f_v,Y: produc8027700073345061287rm_f_v,R: set_Pr7897041799657265351rm_f_v,Z2: produc8027700073345061287rm_f_v] :
( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ X3 @ Y ) @ ( transi7658437189833989320rm_f_v @ R ) )
=> ( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ Y @ Z2 ) @ ( transi7658437189833989320rm_f_v @ R ) )
=> ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ X3 @ Z2 ) @ ( transi7658437189833989320rm_f_v @ R ) ) ) ) ).
% trancl_trans
thf(fact_231_trancl__trans,axiom,
! [X3: f,Y: f,R: set_Product_prod_f_f,Z2: f] :
( ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ X3 @ Y ) @ ( transitive_trancl_f @ R ) )
=> ( ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ Y @ Z2 ) @ ( transitive_trancl_f @ R ) )
=> ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ X3 @ Z2 ) @ ( transitive_trancl_f @ R ) ) ) ) ).
% trancl_trans
thf(fact_232_trancl__trans,axiom,
! [X3: nat,Y: nat,R: set_Pr1261947904930325089at_nat,Z2: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y ) @ ( transi6264000038957366511cl_nat @ R ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ Z2 ) @ ( transi6264000038957366511cl_nat @ R ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Z2 ) @ ( transi6264000038957366511cl_nat @ R ) ) ) ) ).
% trancl_trans
thf(fact_233_trancl__trans,axiom,
! [X3: term_f_v,Y: term_f_v,R: set_Pr8827868859434726151rm_f_v,Z2: term_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Y ) @ ( transi3208867851621559554rm_f_v @ R ) )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y @ Z2 ) @ ( transi3208867851621559554rm_f_v @ R ) )
=> ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Z2 ) @ ( transi3208867851621559554rm_f_v @ R ) ) ) ) ).
% trancl_trans
thf(fact_234_tranclE,axiom,
! [A: product_prod_f_nat,B: product_prod_f_nat,R: set_Pr516226060960033221_f_nat] :
( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ A @ B ) @ ( transi3759436115690491041_f_nat @ R ) )
=> ( ~ ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ A @ B ) @ R )
=> ~ ! [C: product_prod_f_nat] :
( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ A @ C ) @ ( transi3759436115690491041_f_nat @ R ) )
=> ~ ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ C @ B ) @ R ) ) ) ) ).
% tranclE
thf(fact_235_tranclE,axiom,
! [A: produc8027700073345061287rm_f_v,B: produc8027700073345061287rm_f_v,R: set_Pr7897041799657265351rm_f_v] :
( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A @ B ) @ ( transi7658437189833989320rm_f_v @ R ) )
=> ( ~ ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A @ B ) @ R )
=> ~ ! [C: produc8027700073345061287rm_f_v] :
( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A @ C ) @ ( transi7658437189833989320rm_f_v @ R ) )
=> ~ ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ C @ B ) @ R ) ) ) ) ).
% tranclE
thf(fact_236_tranclE,axiom,
! [A: f,B: f,R: set_Product_prod_f_f] :
( ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ A @ B ) @ ( transitive_trancl_f @ R ) )
=> ( ~ ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ A @ B ) @ R )
=> ~ ! [C: f] :
( ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ A @ C ) @ ( transitive_trancl_f @ R ) )
=> ~ ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ C @ B ) @ R ) ) ) ) ).
% tranclE
thf(fact_237_tranclE,axiom,
! [A: nat,B: nat,R: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( transi6264000038957366511cl_nat @ R ) )
=> ( ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R )
=> ~ ! [C: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ C ) @ ( transi6264000038957366511cl_nat @ R ) )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ C @ B ) @ R ) ) ) ) ).
% tranclE
thf(fact_238_tranclE,axiom,
! [A: term_f_v,B: term_f_v,R: set_Pr8827868859434726151rm_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( transi3208867851621559554rm_f_v @ R ) )
=> ( ~ ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ R )
=> ~ ! [C: term_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ C ) @ ( transi3208867851621559554rm_f_v @ R ) )
=> ~ ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ C @ B ) @ R ) ) ) ) ).
% tranclE
thf(fact_239_trancl_Or__into__trancl,axiom,
! [A: product_prod_f_nat,B: product_prod_f_nat,R: set_Pr516226060960033221_f_nat] :
( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ A @ B ) @ R )
=> ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ A @ B ) @ ( transi3759436115690491041_f_nat @ R ) ) ) ).
% trancl.r_into_trancl
thf(fact_240_trancl_Or__into__trancl,axiom,
! [A: produc8027700073345061287rm_f_v,B: produc8027700073345061287rm_f_v,R: set_Pr7897041799657265351rm_f_v] :
( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A @ B ) @ R )
=> ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A @ B ) @ ( transi7658437189833989320rm_f_v @ R ) ) ) ).
% trancl.r_into_trancl
thf(fact_241_trancl_Or__into__trancl,axiom,
! [A: f,B: f,R: set_Product_prod_f_f] :
( ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ A @ B ) @ R )
=> ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ A @ B ) @ ( transitive_trancl_f @ R ) ) ) ).
% trancl.r_into_trancl
thf(fact_242_trancl_Or__into__trancl,axiom,
! [A: nat,B: nat,R: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( transi6264000038957366511cl_nat @ R ) ) ) ).
% trancl.r_into_trancl
thf(fact_243_trancl_Or__into__trancl,axiom,
! [A: term_f_v,B: term_f_v,R: set_Pr8827868859434726151rm_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ R )
=> ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( transi3208867851621559554rm_f_v @ R ) ) ) ).
% trancl.r_into_trancl
thf(fact_244_trancl_Osimps,axiom,
! [A1: product_prod_f_nat,A22: product_prod_f_nat,R: set_Pr516226060960033221_f_nat] :
( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ A1 @ A22 ) @ ( transi3759436115690491041_f_nat @ R ) )
= ( ? [A5: product_prod_f_nat,B4: product_prod_f_nat] :
( ( A1 = A5 )
& ( A22 = B4 )
& ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ A5 @ B4 ) @ R ) )
| ? [A5: product_prod_f_nat,B4: product_prod_f_nat,C4: product_prod_f_nat] :
( ( A1 = A5 )
& ( A22 = C4 )
& ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ A5 @ B4 ) @ ( transi3759436115690491041_f_nat @ R ) )
& ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ B4 @ C4 ) @ R ) ) ) ) ).
% trancl.simps
thf(fact_245_trancl_Osimps,axiom,
! [A1: produc8027700073345061287rm_f_v,A22: produc8027700073345061287rm_f_v,R: set_Pr7897041799657265351rm_f_v] :
( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A1 @ A22 ) @ ( transi7658437189833989320rm_f_v @ R ) )
= ( ? [A5: produc8027700073345061287rm_f_v,B4: produc8027700073345061287rm_f_v] :
( ( A1 = A5 )
& ( A22 = B4 )
& ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A5 @ B4 ) @ R ) )
| ? [A5: produc8027700073345061287rm_f_v,B4: produc8027700073345061287rm_f_v,C4: produc8027700073345061287rm_f_v] :
( ( A1 = A5 )
& ( A22 = C4 )
& ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A5 @ B4 ) @ ( transi7658437189833989320rm_f_v @ R ) )
& ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ B4 @ C4 ) @ R ) ) ) ) ).
% trancl.simps
thf(fact_246_trancl_Osimps,axiom,
! [A1: f,A22: f,R: set_Product_prod_f_f] :
( ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ A1 @ A22 ) @ ( transitive_trancl_f @ R ) )
= ( ? [A5: f,B4: f] :
( ( A1 = A5 )
& ( A22 = B4 )
& ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ A5 @ B4 ) @ R ) )
| ? [A5: f,B4: f,C4: f] :
( ( A1 = A5 )
& ( A22 = C4 )
& ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ A5 @ B4 ) @ ( transitive_trancl_f @ R ) )
& ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ B4 @ C4 ) @ R ) ) ) ) ).
% trancl.simps
thf(fact_247_trancl_Osimps,axiom,
! [A1: nat,A22: nat,R: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A1 @ A22 ) @ ( transi6264000038957366511cl_nat @ R ) )
= ( ? [A5: nat,B4: nat] :
( ( A1 = A5 )
& ( A22 = B4 )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A5 @ B4 ) @ R ) )
| ? [A5: nat,B4: nat,C4: nat] :
( ( A1 = A5 )
& ( A22 = C4 )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A5 @ B4 ) @ ( transi6264000038957366511cl_nat @ R ) )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B4 @ C4 ) @ R ) ) ) ) ).
% trancl.simps
thf(fact_248_trancl_Osimps,axiom,
! [A1: term_f_v,A22: term_f_v,R: set_Pr8827868859434726151rm_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A1 @ A22 ) @ ( transi3208867851621559554rm_f_v @ R ) )
= ( ? [A5: term_f_v,B4: term_f_v] :
( ( A1 = A5 )
& ( A22 = B4 )
& ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A5 @ B4 ) @ R ) )
| ? [A5: term_f_v,B4: term_f_v,C4: term_f_v] :
( ( A1 = A5 )
& ( A22 = C4 )
& ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A5 @ B4 ) @ ( transi3208867851621559554rm_f_v @ R ) )
& ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B4 @ C4 ) @ R ) ) ) ) ).
% trancl.simps
thf(fact_249_trancl_Ocases,axiom,
! [A1: product_prod_f_nat,A22: product_prod_f_nat,R: set_Pr516226060960033221_f_nat] :
( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ A1 @ A22 ) @ ( transi3759436115690491041_f_nat @ R ) )
=> ( ~ ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ A1 @ A22 ) @ R )
=> ~ ! [B3: product_prod_f_nat] :
( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ A1 @ B3 ) @ ( transi3759436115690491041_f_nat @ R ) )
=> ~ ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ B3 @ A22 ) @ R ) ) ) ) ).
% trancl.cases
thf(fact_250_trancl_Ocases,axiom,
! [A1: produc8027700073345061287rm_f_v,A22: produc8027700073345061287rm_f_v,R: set_Pr7897041799657265351rm_f_v] :
( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A1 @ A22 ) @ ( transi7658437189833989320rm_f_v @ R ) )
=> ( ~ ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A1 @ A22 ) @ R )
=> ~ ! [B3: produc8027700073345061287rm_f_v] :
( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A1 @ B3 ) @ ( transi7658437189833989320rm_f_v @ R ) )
=> ~ ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ B3 @ A22 ) @ R ) ) ) ) ).
% trancl.cases
thf(fact_251_trancl_Ocases,axiom,
! [A1: f,A22: f,R: set_Product_prod_f_f] :
( ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ A1 @ A22 ) @ ( transitive_trancl_f @ R ) )
=> ( ~ ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ A1 @ A22 ) @ R )
=> ~ ! [B3: f] :
( ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ A1 @ B3 ) @ ( transitive_trancl_f @ R ) )
=> ~ ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ B3 @ A22 ) @ R ) ) ) ) ).
% trancl.cases
thf(fact_252_trancl_Ocases,axiom,
! [A1: nat,A22: nat,R: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A1 @ A22 ) @ ( transi6264000038957366511cl_nat @ R ) )
=> ( ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A1 @ A22 ) @ R )
=> ~ ! [B3: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A1 @ B3 ) @ ( transi6264000038957366511cl_nat @ R ) )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B3 @ A22 ) @ R ) ) ) ) ).
% trancl.cases
thf(fact_253_trancl_Ocases,axiom,
! [A1: term_f_v,A22: term_f_v,R: set_Pr8827868859434726151rm_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A1 @ A22 ) @ ( transi3208867851621559554rm_f_v @ R ) )
=> ( ~ ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A1 @ A22 ) @ R )
=> ~ ! [B3: term_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A1 @ B3 ) @ ( transi3208867851621559554rm_f_v @ R ) )
=> ~ ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B3 @ A22 ) @ R ) ) ) ) ).
% trancl.cases
thf(fact_254_trancl__mono,axiom,
! [P: produc2676906022994773093_f_nat,R: set_Pr516226060960033221_f_nat,S: set_Pr516226060960033221_f_nat] :
( ( member7030746907725424910_f_nat @ P @ ( transi3759436115690491041_f_nat @ R ) )
=> ( ( ord_le8946695678602162533_f_nat @ R @ S )
=> ( member7030746907725424910_f_nat @ P @ ( transi3759436115690491041_f_nat @ S ) ) ) ) ).
% trancl_mono
thf(fact_255_trancl__mono,axiom,
! [P: produc3406425829894461415rm_f_v,R: set_Pr7897041799657265351rm_f_v,S: set_Pr7897041799657265351rm_f_v] :
( ( member7300054593328134928rm_f_v @ P @ ( transi7658437189833989320rm_f_v @ R ) )
=> ( ( ord_le7518223742277120103rm_f_v @ R @ S )
=> ( member7300054593328134928rm_f_v @ P @ ( transi7658437189833989320rm_f_v @ S ) ) ) ) ).
% trancl_mono
thf(fact_256_trancl__mono,axiom,
! [P: product_prod_nat_nat,R: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ P @ ( transi6264000038957366511cl_nat @ R ) )
=> ( ( ord_le3146513528884898305at_nat @ R @ S )
=> ( member8440522571783428010at_nat @ P @ ( transi6264000038957366511cl_nat @ S ) ) ) ) ).
% trancl_mono
thf(fact_257_trancl__mono,axiom,
! [P: product_prod_f_f,R: set_Product_prod_f_f,S: set_Product_prod_f_f] :
( ( member5935997662887750672od_f_f @ P @ ( transitive_trancl_f @ R ) )
=> ( ( ord_le7147000192471188839od_f_f @ R @ S )
=> ( member5935997662887750672od_f_f @ P @ ( transitive_trancl_f @ S ) ) ) ) ).
% trancl_mono
thf(fact_258_trancl__mono,axiom,
! [P: produc8027700073345061287rm_f_v,R: set_Pr8827868859434726151rm_f_v,S: set_Pr8827868859434726151rm_f_v] :
( ( member4316649861536941904rm_f_v @ P @ ( transi3208867851621559554rm_f_v @ R ) )
=> ( ( ord_le4559761987009501863rm_f_v @ R @ S )
=> ( member4316649861536941904rm_f_v @ P @ ( transi3208867851621559554rm_f_v @ S ) ) ) ) ).
% trancl_mono
thf(fact_259_srrstep__monp,axiom,
! [F: set_Pr5245412377734449720_f_nat,G: set_Pr5245412377734449720_f_nat,R2: set_Pr8827868859434726151rm_f_v] :
( ( ord_le8976984241387448984_f_nat @ F @ G )
=> ( ord_le4559761987009501863rm_f_v @ ( sig_step_f_v @ F @ ( rrstep_f_v @ R2 ) ) @ ( sig_step_f_v @ G @ ( rrstep_f_v @ R2 ) ) ) ) ).
% srrstep_monp
thf(fact_260_sig__step__mono2,axiom,
! [R2: set_Pr8827868859434726151rm_f_v,L2: set_Pr8827868859434726151rm_f_v,F: set_Pr5245412377734449720_f_nat] :
( ( ord_le4559761987009501863rm_f_v @ R2 @ L2 )
=> ( ord_le4559761987009501863rm_f_v @ ( sig_step_f_v @ F @ R2 ) @ ( sig_step_f_v @ F @ L2 ) ) ) ).
% sig_step_mono2
thf(fact_261_srsteps__monp,axiom,
! [F: set_Pr5245412377734449720_f_nat,G: set_Pr5245412377734449720_f_nat,R2: set_Pr8827868859434726151rm_f_v] :
( ( ord_le8976984241387448984_f_nat @ F @ G )
=> ( ord_le4559761987009501863rm_f_v @ ( transi3208867851621559554rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ R2 ) ) ) @ ( transi3208867851621559554rm_f_v @ ( sig_step_f_v @ G @ ( rstep_f_v @ R2 ) ) ) ) ) ).
% srsteps_monp
thf(fact_262_srrstepD,axiom,
! [S: term_f_v,T: term_f_v,F: set_Pr5245412377734449720_f_nat,R2: set_Pr8827868859434726151rm_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ ( sig_step_f_v @ F @ ( rrstep_f_v @ R2 ) ) )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ ( rrstep_f_v @ R2 ) )
& ( ord_le8976984241387448984_f_nat @ ( term_funas_term_f_v @ S ) @ F )
& ( ord_le8976984241387448984_f_nat @ ( term_funas_term_f_v @ T ) @ F ) ) ) ).
% srrstepD
thf(fact_263_tranclD,axiom,
! [X3: product_prod_f_nat,Y: product_prod_f_nat,R3: set_Pr516226060960033221_f_nat] :
( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ X3 @ Y ) @ ( transi3759436115690491041_f_nat @ R3 ) )
=> ? [Z: product_prod_f_nat] :
( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ X3 @ Z ) @ R3 )
& ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ Z @ Y ) @ ( transi8903367562120364223_f_nat @ R3 ) ) ) ) ).
% tranclD
thf(fact_264_tranclD,axiom,
! [X3: produc8027700073345061287rm_f_v,Y: produc8027700073345061287rm_f_v,R3: set_Pr7897041799657265351rm_f_v] :
( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ X3 @ Y ) @ ( transi7658437189833989320rm_f_v @ R3 ) )
=> ? [Z: produc8027700073345061287rm_f_v] :
( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ X3 @ Z ) @ R3 )
& ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ Z @ Y ) @ ( transi1062743709496199978rm_f_v @ R3 ) ) ) ) ).
% tranclD
thf(fact_265_tranclD,axiom,
! [X3: f,Y: f,R3: set_Product_prod_f_f] :
( ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ X3 @ Y ) @ ( transitive_trancl_f @ R3 ) )
=> ? [Z: f] :
( ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ X3 @ Z ) @ R3 )
& ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ Z @ Y ) @ ( transitive_rtrancl_f @ R3 ) ) ) ) ).
% tranclD
thf(fact_266_tranclD,axiom,
! [X3: nat,Y: nat,R3: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y ) @ ( transi6264000038957366511cl_nat @ R3 ) )
=> ? [Z: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Z ) @ R3 )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Z @ Y ) @ ( transi2905341329935302413cl_nat @ R3 ) ) ) ) ).
% tranclD
thf(fact_267_tranclD,axiom,
! [X3: term_f_v,Y: term_f_v,R3: set_Pr8827868859434726151rm_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Y ) @ ( transi3208867851621559554rm_f_v @ R3 ) )
=> ? [Z: term_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Z ) @ R3 )
& ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Z @ Y ) @ ( transi3028809021613410336rm_f_v @ R3 ) ) ) ) ).
% tranclD
thf(fact_268_rtranclD,axiom,
! [A: product_prod_f_nat,B: product_prod_f_nat,R3: set_Pr516226060960033221_f_nat] :
( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ A @ B ) @ ( transi8903367562120364223_f_nat @ R3 ) )
=> ( ( A = B )
| ( ( A != B )
& ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ A @ B ) @ ( transi3759436115690491041_f_nat @ R3 ) ) ) ) ) ).
% rtranclD
thf(fact_269_rtranclD,axiom,
! [A: produc8027700073345061287rm_f_v,B: produc8027700073345061287rm_f_v,R3: set_Pr7897041799657265351rm_f_v] :
( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A @ B ) @ ( transi1062743709496199978rm_f_v @ R3 ) )
=> ( ( A = B )
| ( ( A != B )
& ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A @ B ) @ ( transi7658437189833989320rm_f_v @ R3 ) ) ) ) ) ).
% rtranclD
thf(fact_270_rtranclD,axiom,
! [A: f,B: f,R3: set_Product_prod_f_f] :
( ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ A @ B ) @ ( transitive_rtrancl_f @ R3 ) )
=> ( ( A = B )
| ( ( A != B )
& ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ A @ B ) @ ( transitive_trancl_f @ R3 ) ) ) ) ) ).
% rtranclD
thf(fact_271_rtranclD,axiom,
! [A: nat,B: nat,R3: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( transi2905341329935302413cl_nat @ R3 ) )
=> ( ( A = B )
| ( ( A != B )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( transi6264000038957366511cl_nat @ R3 ) ) ) ) ) ).
% rtranclD
thf(fact_272_rtranclD,axiom,
! [A: term_f_v,B: term_f_v,R3: set_Pr8827868859434726151rm_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( transi3028809021613410336rm_f_v @ R3 ) )
=> ( ( A = B )
| ( ( A != B )
& ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( transi3208867851621559554rm_f_v @ R3 ) ) ) ) ) ).
% rtranclD
thf(fact_273_tranclD2,axiom,
! [X3: product_prod_f_nat,Y: product_prod_f_nat,R3: set_Pr516226060960033221_f_nat] :
( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ X3 @ Y ) @ ( transi3759436115690491041_f_nat @ R3 ) )
=> ? [Z: product_prod_f_nat] :
( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ X3 @ Z ) @ ( transi8903367562120364223_f_nat @ R3 ) )
& ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ Z @ Y ) @ R3 ) ) ) ).
% tranclD2
thf(fact_274_tranclD2,axiom,
! [X3: produc8027700073345061287rm_f_v,Y: produc8027700073345061287rm_f_v,R3: set_Pr7897041799657265351rm_f_v] :
( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ X3 @ Y ) @ ( transi7658437189833989320rm_f_v @ R3 ) )
=> ? [Z: produc8027700073345061287rm_f_v] :
( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ X3 @ Z ) @ ( transi1062743709496199978rm_f_v @ R3 ) )
& ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ Z @ Y ) @ R3 ) ) ) ).
% tranclD2
thf(fact_275_tranclD2,axiom,
! [X3: f,Y: f,R3: set_Product_prod_f_f] :
( ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ X3 @ Y ) @ ( transitive_trancl_f @ R3 ) )
=> ? [Z: f] :
( ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ X3 @ Z ) @ ( transitive_rtrancl_f @ R3 ) )
& ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ Z @ Y ) @ R3 ) ) ) ).
% tranclD2
thf(fact_276_tranclD2,axiom,
! [X3: nat,Y: nat,R3: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y ) @ ( transi6264000038957366511cl_nat @ R3 ) )
=> ? [Z: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Z ) @ ( transi2905341329935302413cl_nat @ R3 ) )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Z @ Y ) @ R3 ) ) ) ).
% tranclD2
thf(fact_277_tranclD2,axiom,
! [X3: term_f_v,Y: term_f_v,R3: set_Pr8827868859434726151rm_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Y ) @ ( transi3208867851621559554rm_f_v @ R3 ) )
=> ? [Z: term_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Z ) @ ( transi3028809021613410336rm_f_v @ R3 ) )
& ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Z @ Y ) @ R3 ) ) ) ).
% tranclD2
thf(fact_278_trancl__into__rtrancl,axiom,
! [A: product_prod_f_nat,B: product_prod_f_nat,R: set_Pr516226060960033221_f_nat] :
( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ A @ B ) @ ( transi3759436115690491041_f_nat @ R ) )
=> ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ A @ B ) @ ( transi8903367562120364223_f_nat @ R ) ) ) ).
% trancl_into_rtrancl
thf(fact_279_trancl__into__rtrancl,axiom,
! [A: produc8027700073345061287rm_f_v,B: produc8027700073345061287rm_f_v,R: set_Pr7897041799657265351rm_f_v] :
( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A @ B ) @ ( transi7658437189833989320rm_f_v @ R ) )
=> ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A @ B ) @ ( transi1062743709496199978rm_f_v @ R ) ) ) ).
% trancl_into_rtrancl
thf(fact_280_trancl__into__rtrancl,axiom,
! [A: f,B: f,R: set_Product_prod_f_f] :
( ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ A @ B ) @ ( transitive_trancl_f @ R ) )
=> ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ A @ B ) @ ( transitive_rtrancl_f @ R ) ) ) ).
% trancl_into_rtrancl
thf(fact_281_trancl__into__rtrancl,axiom,
! [A: nat,B: nat,R: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( transi6264000038957366511cl_nat @ R ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( transi2905341329935302413cl_nat @ R ) ) ) ).
% trancl_into_rtrancl
thf(fact_282_trancl__into__rtrancl,axiom,
! [A: term_f_v,B: term_f_v,R: set_Pr8827868859434726151rm_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( transi3208867851621559554rm_f_v @ R ) )
=> ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( transi3028809021613410336rm_f_v @ R ) ) ) ).
% trancl_into_rtrancl
thf(fact_283_rtrancl__eq__or__trancl,axiom,
! [X3: product_prod_f_nat,Y: product_prod_f_nat,R3: set_Pr516226060960033221_f_nat] :
( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ X3 @ Y ) @ ( transi8903367562120364223_f_nat @ R3 ) )
= ( ( X3 = Y )
| ( ( X3 != Y )
& ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ X3 @ Y ) @ ( transi3759436115690491041_f_nat @ R3 ) ) ) ) ) ).
% rtrancl_eq_or_trancl
thf(fact_284_rtrancl__eq__or__trancl,axiom,
! [X3: produc8027700073345061287rm_f_v,Y: produc8027700073345061287rm_f_v,R3: set_Pr7897041799657265351rm_f_v] :
( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ X3 @ Y ) @ ( transi1062743709496199978rm_f_v @ R3 ) )
= ( ( X3 = Y )
| ( ( X3 != Y )
& ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ X3 @ Y ) @ ( transi7658437189833989320rm_f_v @ R3 ) ) ) ) ) ).
% rtrancl_eq_or_trancl
thf(fact_285_rtrancl__eq__or__trancl,axiom,
! [X3: f,Y: f,R3: set_Product_prod_f_f] :
( ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ X3 @ Y ) @ ( transitive_rtrancl_f @ R3 ) )
= ( ( X3 = Y )
| ( ( X3 != Y )
& ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ X3 @ Y ) @ ( transitive_trancl_f @ R3 ) ) ) ) ) ).
% rtrancl_eq_or_trancl
thf(fact_286_rtrancl__eq__or__trancl,axiom,
! [X3: nat,Y: nat,R3: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y ) @ ( transi2905341329935302413cl_nat @ R3 ) )
= ( ( X3 = Y )
| ( ( X3 != Y )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y ) @ ( transi6264000038957366511cl_nat @ R3 ) ) ) ) ) ).
% rtrancl_eq_or_trancl
thf(fact_287_rtrancl__eq__or__trancl,axiom,
! [X3: term_f_v,Y: term_f_v,R3: set_Pr8827868859434726151rm_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Y ) @ ( transi3028809021613410336rm_f_v @ R3 ) )
= ( ( X3 = Y )
| ( ( X3 != Y )
& ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Y ) @ ( transi3208867851621559554rm_f_v @ R3 ) ) ) ) ) ).
% rtrancl_eq_or_trancl
thf(fact_288_rtrancl__into__trancl1,axiom,
! [A: product_prod_f_nat,B: product_prod_f_nat,R: set_Pr516226060960033221_f_nat,C3: product_prod_f_nat] :
( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ A @ B ) @ ( transi8903367562120364223_f_nat @ R ) )
=> ( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ B @ C3 ) @ R )
=> ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ A @ C3 ) @ ( transi3759436115690491041_f_nat @ R ) ) ) ) ).
% rtrancl_into_trancl1
thf(fact_289_rtrancl__into__trancl1,axiom,
! [A: produc8027700073345061287rm_f_v,B: produc8027700073345061287rm_f_v,R: set_Pr7897041799657265351rm_f_v,C3: produc8027700073345061287rm_f_v] :
( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A @ B ) @ ( transi1062743709496199978rm_f_v @ R ) )
=> ( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ B @ C3 ) @ R )
=> ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A @ C3 ) @ ( transi7658437189833989320rm_f_v @ R ) ) ) ) ).
% rtrancl_into_trancl1
thf(fact_290_rtrancl__into__trancl1,axiom,
! [A: f,B: f,R: set_Product_prod_f_f,C3: f] :
( ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ A @ B ) @ ( transitive_rtrancl_f @ R ) )
=> ( ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ B @ C3 ) @ R )
=> ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ A @ C3 ) @ ( transitive_trancl_f @ R ) ) ) ) ).
% rtrancl_into_trancl1
thf(fact_291_rtrancl__into__trancl1,axiom,
! [A: nat,B: nat,R: set_Pr1261947904930325089at_nat,C3: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( transi2905341329935302413cl_nat @ R ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ C3 ) @ R )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ C3 ) @ ( transi6264000038957366511cl_nat @ R ) ) ) ) ).
% rtrancl_into_trancl1
thf(fact_292_rtrancl__into__trancl1,axiom,
! [A: term_f_v,B: term_f_v,R: set_Pr8827868859434726151rm_f_v,C3: term_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( transi3028809021613410336rm_f_v @ R ) )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B @ C3 ) @ R )
=> ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ C3 ) @ ( transi3208867851621559554rm_f_v @ R ) ) ) ) ).
% rtrancl_into_trancl1
thf(fact_293_rtrancl__into__trancl2,axiom,
! [A: product_prod_f_nat,B: product_prod_f_nat,R: set_Pr516226060960033221_f_nat,C3: product_prod_f_nat] :
( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ A @ B ) @ R )
=> ( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ B @ C3 ) @ ( transi8903367562120364223_f_nat @ R ) )
=> ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ A @ C3 ) @ ( transi3759436115690491041_f_nat @ R ) ) ) ) ).
% rtrancl_into_trancl2
thf(fact_294_rtrancl__into__trancl2,axiom,
! [A: produc8027700073345061287rm_f_v,B: produc8027700073345061287rm_f_v,R: set_Pr7897041799657265351rm_f_v,C3: produc8027700073345061287rm_f_v] :
( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A @ B ) @ R )
=> ( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ B @ C3 ) @ ( transi1062743709496199978rm_f_v @ R ) )
=> ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A @ C3 ) @ ( transi7658437189833989320rm_f_v @ R ) ) ) ) ).
% rtrancl_into_trancl2
thf(fact_295_rtrancl__into__trancl2,axiom,
! [A: f,B: f,R: set_Product_prod_f_f,C3: f] :
( ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ A @ B ) @ R )
=> ( ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ B @ C3 ) @ ( transitive_rtrancl_f @ R ) )
=> ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ A @ C3 ) @ ( transitive_trancl_f @ R ) ) ) ) ).
% rtrancl_into_trancl2
thf(fact_296_rtrancl__into__trancl2,axiom,
! [A: nat,B: nat,R: set_Pr1261947904930325089at_nat,C3: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ C3 ) @ ( transi2905341329935302413cl_nat @ R ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ C3 ) @ ( transi6264000038957366511cl_nat @ R ) ) ) ) ).
% rtrancl_into_trancl2
thf(fact_297_rtrancl__into__trancl2,axiom,
! [A: term_f_v,B: term_f_v,R: set_Pr8827868859434726151rm_f_v,C3: term_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ R )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B @ C3 ) @ ( transi3028809021613410336rm_f_v @ R ) )
=> ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ C3 ) @ ( transi3208867851621559554rm_f_v @ R ) ) ) ) ).
% rtrancl_into_trancl2
thf(fact_298_rtrancl__trancl__trancl,axiom,
! [X3: product_prod_f_nat,Y: product_prod_f_nat,R: set_Pr516226060960033221_f_nat,Z2: product_prod_f_nat] :
( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ X3 @ Y ) @ ( transi8903367562120364223_f_nat @ R ) )
=> ( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ Y @ Z2 ) @ ( transi3759436115690491041_f_nat @ R ) )
=> ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ X3 @ Z2 ) @ ( transi3759436115690491041_f_nat @ R ) ) ) ) ).
% rtrancl_trancl_trancl
thf(fact_299_rtrancl__trancl__trancl,axiom,
! [X3: produc8027700073345061287rm_f_v,Y: produc8027700073345061287rm_f_v,R: set_Pr7897041799657265351rm_f_v,Z2: produc8027700073345061287rm_f_v] :
( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ X3 @ Y ) @ ( transi1062743709496199978rm_f_v @ R ) )
=> ( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ Y @ Z2 ) @ ( transi7658437189833989320rm_f_v @ R ) )
=> ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ X3 @ Z2 ) @ ( transi7658437189833989320rm_f_v @ R ) ) ) ) ).
% rtrancl_trancl_trancl
thf(fact_300_rtrancl__trancl__trancl,axiom,
! [X3: f,Y: f,R: set_Product_prod_f_f,Z2: f] :
( ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ X3 @ Y ) @ ( transitive_rtrancl_f @ R ) )
=> ( ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ Y @ Z2 ) @ ( transitive_trancl_f @ R ) )
=> ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ X3 @ Z2 ) @ ( transitive_trancl_f @ R ) ) ) ) ).
% rtrancl_trancl_trancl
thf(fact_301_rtrancl__trancl__trancl,axiom,
! [X3: nat,Y: nat,R: set_Pr1261947904930325089at_nat,Z2: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y ) @ ( transi2905341329935302413cl_nat @ R ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ Z2 ) @ ( transi6264000038957366511cl_nat @ R ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Z2 ) @ ( transi6264000038957366511cl_nat @ R ) ) ) ) ).
% rtrancl_trancl_trancl
thf(fact_302_rtrancl__trancl__trancl,axiom,
! [X3: term_f_v,Y: term_f_v,R: set_Pr8827868859434726151rm_f_v,Z2: term_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Y ) @ ( transi3028809021613410336rm_f_v @ R ) )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y @ Z2 ) @ ( transi3208867851621559554rm_f_v @ R ) )
=> ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Z2 ) @ ( transi3208867851621559554rm_f_v @ R ) ) ) ) ).
% rtrancl_trancl_trancl
thf(fact_303_trancl__rtrancl__trancl,axiom,
! [A: product_prod_f_nat,B: product_prod_f_nat,R: set_Pr516226060960033221_f_nat,C3: product_prod_f_nat] :
( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ A @ B ) @ ( transi3759436115690491041_f_nat @ R ) )
=> ( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ B @ C3 ) @ ( transi8903367562120364223_f_nat @ R ) )
=> ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ A @ C3 ) @ ( transi3759436115690491041_f_nat @ R ) ) ) ) ).
% trancl_rtrancl_trancl
thf(fact_304_trancl__rtrancl__trancl,axiom,
! [A: produc8027700073345061287rm_f_v,B: produc8027700073345061287rm_f_v,R: set_Pr7897041799657265351rm_f_v,C3: produc8027700073345061287rm_f_v] :
( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A @ B ) @ ( transi7658437189833989320rm_f_v @ R ) )
=> ( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ B @ C3 ) @ ( transi1062743709496199978rm_f_v @ R ) )
=> ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A @ C3 ) @ ( transi7658437189833989320rm_f_v @ R ) ) ) ) ).
% trancl_rtrancl_trancl
thf(fact_305_trancl__rtrancl__trancl,axiom,
! [A: f,B: f,R: set_Product_prod_f_f,C3: f] :
( ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ A @ B ) @ ( transitive_trancl_f @ R ) )
=> ( ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ B @ C3 ) @ ( transitive_rtrancl_f @ R ) )
=> ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ A @ C3 ) @ ( transitive_trancl_f @ R ) ) ) ) ).
% trancl_rtrancl_trancl
thf(fact_306_trancl__rtrancl__trancl,axiom,
! [A: nat,B: nat,R: set_Pr1261947904930325089at_nat,C3: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( transi6264000038957366511cl_nat @ R ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ C3 ) @ ( transi2905341329935302413cl_nat @ R ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ C3 ) @ ( transi6264000038957366511cl_nat @ R ) ) ) ) ).
% trancl_rtrancl_trancl
thf(fact_307_trancl__rtrancl__trancl,axiom,
! [A: term_f_v,B: term_f_v,R: set_Pr8827868859434726151rm_f_v,C3: term_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( transi3208867851621559554rm_f_v @ R ) )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B @ C3 ) @ ( transi3028809021613410336rm_f_v @ R ) )
=> ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ C3 ) @ ( transi3208867851621559554rm_f_v @ R ) ) ) ) ).
% trancl_rtrancl_trancl
thf(fact_308_srstepsD,axiom,
! [S: term_f_v,T: term_f_v,F: set_Pr5245412377734449720_f_nat,R2: set_Pr8827868859434726151rm_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ ( transi3208867851621559554rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ R2 ) ) ) )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ ( transi3208867851621559554rm_f_v @ ( rstep_f_v @ R2 ) ) )
& ( ord_le8976984241387448984_f_nat @ ( term_funas_term_f_v @ S ) @ F )
& ( ord_le8976984241387448984_f_nat @ ( term_funas_term_f_v @ T ) @ F ) ) ) ).
% srstepsD
thf(fact_309_sig__step__mono,axiom,
! [F: set_Pr5245412377734449720_f_nat,G: set_Pr5245412377734449720_f_nat,R2: set_Pr8827868859434726151rm_f_v] :
( ( ord_le8976984241387448984_f_nat @ F @ G )
=> ( ord_le4559761987009501863rm_f_v @ ( sig_step_f_v @ F @ R2 ) @ ( sig_step_f_v @ G @ R2 ) ) ) ).
% sig_step_mono
thf(fact_310_nle__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_eq_nat @ A @ B ) )
= ( ( ord_less_eq_nat @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_311_le__cases3,axiom,
! [X3: nat,Y: nat,Z2: nat] :
( ( ( ord_less_eq_nat @ X3 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ Y @ X3 )
=> ~ ( ord_less_eq_nat @ X3 @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ X3 @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z2 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X3 ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ X3 ) )
=> ~ ( ( ord_less_eq_nat @ Z2 @ X3 )
=> ~ ( ord_less_eq_nat @ X3 @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_312_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_Pr1261947904930325089at_nat,Z3: set_Pr1261947904930325089at_nat] : ( Y4 = Z3 ) )
= ( ^ [X4: set_Pr1261947904930325089at_nat,Y5: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ X4 @ Y5 )
& ( ord_le3146513528884898305at_nat @ Y5 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_313_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_Pr4548512049067879902_nat_f,Z3: set_Pr4548512049067879902_nat_f] : ( Y4 = Z3 ) )
= ( ^ [X4: set_Pr4548512049067879902_nat_f,Y5: set_Pr4548512049067879902_nat_f] :
( ( ord_le8280083912720879166_nat_f @ X4 @ Y5 )
& ( ord_le8280083912720879166_nat_f @ Y5 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_314_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_Product_prod_f_f,Z3: set_Product_prod_f_f] : ( Y4 = Z3 ) )
= ( ^ [X4: set_Product_prod_f_f,Y5: set_Product_prod_f_f] :
( ( ord_le7147000192471188839od_f_f @ X4 @ Y5 )
& ( ord_le7147000192471188839od_f_f @ Y5 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_315_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_term_f_v,Z3: set_term_f_v] : ( Y4 = Z3 ) )
= ( ^ [X4: set_term_f_v,Y5: set_term_f_v] :
( ( ord_le947722214152454265rm_f_v @ X4 @ Y5 )
& ( ord_le947722214152454265rm_f_v @ Y5 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_316_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: nat,Z3: nat] : ( Y4 = Z3 ) )
= ( ^ [X4: nat,Y5: nat] :
( ( ord_less_eq_nat @ X4 @ Y5 )
& ( ord_less_eq_nat @ Y5 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_317_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_Pr5245412377734449720_f_nat,Z3: set_Pr5245412377734449720_f_nat] : ( Y4 = Z3 ) )
= ( ^ [X4: set_Pr5245412377734449720_f_nat,Y5: set_Pr5245412377734449720_f_nat] :
( ( ord_le8976984241387448984_f_nat @ X4 @ Y5 )
& ( ord_le8976984241387448984_f_nat @ Y5 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_318_ord__eq__le__trans,axiom,
! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,C3: set_Pr1261947904930325089at_nat] :
( ( A = B )
=> ( ( ord_le3146513528884898305at_nat @ B @ C3 )
=> ( ord_le3146513528884898305at_nat @ A @ C3 ) ) ) ).
% ord_eq_le_trans
thf(fact_319_ord__eq__le__trans,axiom,
! [A: set_Pr4548512049067879902_nat_f,B: set_Pr4548512049067879902_nat_f,C3: set_Pr4548512049067879902_nat_f] :
( ( A = B )
=> ( ( ord_le8280083912720879166_nat_f @ B @ C3 )
=> ( ord_le8280083912720879166_nat_f @ A @ C3 ) ) ) ).
% ord_eq_le_trans
thf(fact_320_ord__eq__le__trans,axiom,
! [A: set_Product_prod_f_f,B: set_Product_prod_f_f,C3: set_Product_prod_f_f] :
( ( A = B )
=> ( ( ord_le7147000192471188839od_f_f @ B @ C3 )
=> ( ord_le7147000192471188839od_f_f @ A @ C3 ) ) ) ).
% ord_eq_le_trans
thf(fact_321_ord__eq__le__trans,axiom,
! [A: set_term_f_v,B: set_term_f_v,C3: set_term_f_v] :
( ( A = B )
=> ( ( ord_le947722214152454265rm_f_v @ B @ C3 )
=> ( ord_le947722214152454265rm_f_v @ A @ C3 ) ) ) ).
% ord_eq_le_trans
thf(fact_322_ord__eq__le__trans,axiom,
! [A: nat,B: nat,C3: nat] :
( ( A = B )
=> ( ( ord_less_eq_nat @ B @ C3 )
=> ( ord_less_eq_nat @ A @ C3 ) ) ) ).
% ord_eq_le_trans
thf(fact_323_ord__eq__le__trans,axiom,
! [A: set_Pr5245412377734449720_f_nat,B: set_Pr5245412377734449720_f_nat,C3: set_Pr5245412377734449720_f_nat] :
( ( A = B )
=> ( ( ord_le8976984241387448984_f_nat @ B @ C3 )
=> ( ord_le8976984241387448984_f_nat @ A @ C3 ) ) ) ).
% ord_eq_le_trans
thf(fact_324_ord__le__eq__trans,axiom,
! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,C3: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ A @ B )
=> ( ( B = C3 )
=> ( ord_le3146513528884898305at_nat @ A @ C3 ) ) ) ).
% ord_le_eq_trans
thf(fact_325_ord__le__eq__trans,axiom,
! [A: set_Pr4548512049067879902_nat_f,B: set_Pr4548512049067879902_nat_f,C3: set_Pr4548512049067879902_nat_f] :
( ( ord_le8280083912720879166_nat_f @ A @ B )
=> ( ( B = C3 )
=> ( ord_le8280083912720879166_nat_f @ A @ C3 ) ) ) ).
% ord_le_eq_trans
thf(fact_326_ord__le__eq__trans,axiom,
! [A: set_Product_prod_f_f,B: set_Product_prod_f_f,C3: set_Product_prod_f_f] :
( ( ord_le7147000192471188839od_f_f @ A @ B )
=> ( ( B = C3 )
=> ( ord_le7147000192471188839od_f_f @ A @ C3 ) ) ) ).
% ord_le_eq_trans
thf(fact_327_ord__le__eq__trans,axiom,
! [A: set_term_f_v,B: set_term_f_v,C3: set_term_f_v] :
( ( ord_le947722214152454265rm_f_v @ A @ B )
=> ( ( B = C3 )
=> ( ord_le947722214152454265rm_f_v @ A @ C3 ) ) ) ).
% ord_le_eq_trans
thf(fact_328_ord__le__eq__trans,axiom,
! [A: nat,B: nat,C3: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( B = C3 )
=> ( ord_less_eq_nat @ A @ C3 ) ) ) ).
% ord_le_eq_trans
thf(fact_329_ord__le__eq__trans,axiom,
! [A: set_Pr5245412377734449720_f_nat,B: set_Pr5245412377734449720_f_nat,C3: set_Pr5245412377734449720_f_nat] :
( ( ord_le8976984241387448984_f_nat @ A @ B )
=> ( ( B = C3 )
=> ( ord_le8976984241387448984_f_nat @ A @ C3 ) ) ) ).
% ord_le_eq_trans
thf(fact_330_order__antisym,axiom,
! [X3: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ X3 @ Y )
=> ( ( ord_le3146513528884898305at_nat @ Y @ X3 )
=> ( X3 = Y ) ) ) ).
% order_antisym
thf(fact_331_order__antisym,axiom,
! [X3: set_Pr4548512049067879902_nat_f,Y: set_Pr4548512049067879902_nat_f] :
( ( ord_le8280083912720879166_nat_f @ X3 @ Y )
=> ( ( ord_le8280083912720879166_nat_f @ Y @ X3 )
=> ( X3 = Y ) ) ) ).
% order_antisym
thf(fact_332_order__antisym,axiom,
! [X3: set_Product_prod_f_f,Y: set_Product_prod_f_f] :
( ( ord_le7147000192471188839od_f_f @ X3 @ Y )
=> ( ( ord_le7147000192471188839od_f_f @ Y @ X3 )
=> ( X3 = Y ) ) ) ).
% order_antisym
thf(fact_333_order__antisym,axiom,
! [X3: set_term_f_v,Y: set_term_f_v] :
( ( ord_le947722214152454265rm_f_v @ X3 @ Y )
=> ( ( ord_le947722214152454265rm_f_v @ Y @ X3 )
=> ( X3 = Y ) ) ) ).
% order_antisym
thf(fact_334_order__antisym,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ( ord_less_eq_nat @ Y @ X3 )
=> ( X3 = Y ) ) ) ).
% order_antisym
thf(fact_335_order__antisym,axiom,
! [X3: set_Pr5245412377734449720_f_nat,Y: set_Pr5245412377734449720_f_nat] :
( ( ord_le8976984241387448984_f_nat @ X3 @ Y )
=> ( ( ord_le8976984241387448984_f_nat @ Y @ X3 )
=> ( X3 = Y ) ) ) ).
% order_antisym
thf(fact_336_order_Otrans,axiom,
! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,C3: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ A @ B )
=> ( ( ord_le3146513528884898305at_nat @ B @ C3 )
=> ( ord_le3146513528884898305at_nat @ A @ C3 ) ) ) ).
% order.trans
thf(fact_337_order_Otrans,axiom,
! [A: set_Pr4548512049067879902_nat_f,B: set_Pr4548512049067879902_nat_f,C3: set_Pr4548512049067879902_nat_f] :
( ( ord_le8280083912720879166_nat_f @ A @ B )
=> ( ( ord_le8280083912720879166_nat_f @ B @ C3 )
=> ( ord_le8280083912720879166_nat_f @ A @ C3 ) ) ) ).
% order.trans
thf(fact_338_order_Otrans,axiom,
! [A: set_Product_prod_f_f,B: set_Product_prod_f_f,C3: set_Product_prod_f_f] :
( ( ord_le7147000192471188839od_f_f @ A @ B )
=> ( ( ord_le7147000192471188839od_f_f @ B @ C3 )
=> ( ord_le7147000192471188839od_f_f @ A @ C3 ) ) ) ).
% order.trans
thf(fact_339_order_Otrans,axiom,
! [A: set_term_f_v,B: set_term_f_v,C3: set_term_f_v] :
( ( ord_le947722214152454265rm_f_v @ A @ B )
=> ( ( ord_le947722214152454265rm_f_v @ B @ C3 )
=> ( ord_le947722214152454265rm_f_v @ A @ C3 ) ) ) ).
% order.trans
thf(fact_340_order_Otrans,axiom,
! [A: nat,B: nat,C3: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C3 )
=> ( ord_less_eq_nat @ A @ C3 ) ) ) ).
% order.trans
thf(fact_341_order_Otrans,axiom,
! [A: set_Pr5245412377734449720_f_nat,B: set_Pr5245412377734449720_f_nat,C3: set_Pr5245412377734449720_f_nat] :
( ( ord_le8976984241387448984_f_nat @ A @ B )
=> ( ( ord_le8976984241387448984_f_nat @ B @ C3 )
=> ( ord_le8976984241387448984_f_nat @ A @ C3 ) ) ) ).
% order.trans
thf(fact_342_order__trans,axiom,
! [X3: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z2: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ X3 @ Y )
=> ( ( ord_le3146513528884898305at_nat @ Y @ Z2 )
=> ( ord_le3146513528884898305at_nat @ X3 @ Z2 ) ) ) ).
% order_trans
thf(fact_343_order__trans,axiom,
! [X3: set_Pr4548512049067879902_nat_f,Y: set_Pr4548512049067879902_nat_f,Z2: set_Pr4548512049067879902_nat_f] :
( ( ord_le8280083912720879166_nat_f @ X3 @ Y )
=> ( ( ord_le8280083912720879166_nat_f @ Y @ Z2 )
=> ( ord_le8280083912720879166_nat_f @ X3 @ Z2 ) ) ) ).
% order_trans
thf(fact_344_order__trans,axiom,
! [X3: set_Product_prod_f_f,Y: set_Product_prod_f_f,Z2: set_Product_prod_f_f] :
( ( ord_le7147000192471188839od_f_f @ X3 @ Y )
=> ( ( ord_le7147000192471188839od_f_f @ Y @ Z2 )
=> ( ord_le7147000192471188839od_f_f @ X3 @ Z2 ) ) ) ).
% order_trans
thf(fact_345_order__trans,axiom,
! [X3: set_term_f_v,Y: set_term_f_v,Z2: set_term_f_v] :
( ( ord_le947722214152454265rm_f_v @ X3 @ Y )
=> ( ( ord_le947722214152454265rm_f_v @ Y @ Z2 )
=> ( ord_le947722214152454265rm_f_v @ X3 @ Z2 ) ) ) ).
% order_trans
thf(fact_346_order__trans,axiom,
! [X3: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z2 )
=> ( ord_less_eq_nat @ X3 @ Z2 ) ) ) ).
% order_trans
thf(fact_347_order__trans,axiom,
! [X3: set_Pr5245412377734449720_f_nat,Y: set_Pr5245412377734449720_f_nat,Z2: set_Pr5245412377734449720_f_nat] :
( ( ord_le8976984241387448984_f_nat @ X3 @ Y )
=> ( ( ord_le8976984241387448984_f_nat @ Y @ Z2 )
=> ( ord_le8976984241387448984_f_nat @ X3 @ Z2 ) ) ) ).
% order_trans
thf(fact_348_linorder__wlog,axiom,
! [P2: nat > nat > $o,A: nat,B: nat] :
( ! [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( P2 @ A3 @ B3 ) )
=> ( ! [A3: nat,B3: nat] :
( ( P2 @ B3 @ A3 )
=> ( P2 @ A3 @ B3 ) )
=> ( P2 @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_349_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_Pr1261947904930325089at_nat,Z3: set_Pr1261947904930325089at_nat] : ( Y4 = Z3 ) )
= ( ^ [A5: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ B4 @ A5 )
& ( ord_le3146513528884898305at_nat @ A5 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_350_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_Pr4548512049067879902_nat_f,Z3: set_Pr4548512049067879902_nat_f] : ( Y4 = Z3 ) )
= ( ^ [A5: set_Pr4548512049067879902_nat_f,B4: set_Pr4548512049067879902_nat_f] :
( ( ord_le8280083912720879166_nat_f @ B4 @ A5 )
& ( ord_le8280083912720879166_nat_f @ A5 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_351_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_Product_prod_f_f,Z3: set_Product_prod_f_f] : ( Y4 = Z3 ) )
= ( ^ [A5: set_Product_prod_f_f,B4: set_Product_prod_f_f] :
( ( ord_le7147000192471188839od_f_f @ B4 @ A5 )
& ( ord_le7147000192471188839od_f_f @ A5 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_352_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_term_f_v,Z3: set_term_f_v] : ( Y4 = Z3 ) )
= ( ^ [A5: set_term_f_v,B4: set_term_f_v] :
( ( ord_le947722214152454265rm_f_v @ B4 @ A5 )
& ( ord_le947722214152454265rm_f_v @ A5 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_353_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: nat,Z3: nat] : ( Y4 = Z3 ) )
= ( ^ [A5: nat,B4: nat] :
( ( ord_less_eq_nat @ B4 @ A5 )
& ( ord_less_eq_nat @ A5 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_354_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_Pr5245412377734449720_f_nat,Z3: set_Pr5245412377734449720_f_nat] : ( Y4 = Z3 ) )
= ( ^ [A5: set_Pr5245412377734449720_f_nat,B4: set_Pr5245412377734449720_f_nat] :
( ( ord_le8976984241387448984_f_nat @ B4 @ A5 )
& ( ord_le8976984241387448984_f_nat @ A5 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_355_dual__order_Oantisym,axiom,
! [B: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ B @ A )
=> ( ( ord_le3146513528884898305at_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_356_dual__order_Oantisym,axiom,
! [B: set_Pr4548512049067879902_nat_f,A: set_Pr4548512049067879902_nat_f] :
( ( ord_le8280083912720879166_nat_f @ B @ A )
=> ( ( ord_le8280083912720879166_nat_f @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_357_dual__order_Oantisym,axiom,
! [B: set_Product_prod_f_f,A: set_Product_prod_f_f] :
( ( ord_le7147000192471188839od_f_f @ B @ A )
=> ( ( ord_le7147000192471188839od_f_f @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_358_dual__order_Oantisym,axiom,
! [B: set_term_f_v,A: set_term_f_v] :
( ( ord_le947722214152454265rm_f_v @ B @ A )
=> ( ( ord_le947722214152454265rm_f_v @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_359_dual__order_Oantisym,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_360_dual__order_Oantisym,axiom,
! [B: set_Pr5245412377734449720_f_nat,A: set_Pr5245412377734449720_f_nat] :
( ( ord_le8976984241387448984_f_nat @ B @ A )
=> ( ( ord_le8976984241387448984_f_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_361_dual__order_Otrans,axiom,
! [B: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat,C3: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ B @ A )
=> ( ( ord_le3146513528884898305at_nat @ C3 @ B )
=> ( ord_le3146513528884898305at_nat @ C3 @ A ) ) ) ).
% dual_order.trans
thf(fact_362_dual__order_Otrans,axiom,
! [B: set_Pr4548512049067879902_nat_f,A: set_Pr4548512049067879902_nat_f,C3: set_Pr4548512049067879902_nat_f] :
( ( ord_le8280083912720879166_nat_f @ B @ A )
=> ( ( ord_le8280083912720879166_nat_f @ C3 @ B )
=> ( ord_le8280083912720879166_nat_f @ C3 @ A ) ) ) ).
% dual_order.trans
thf(fact_363_dual__order_Otrans,axiom,
! [B: set_Product_prod_f_f,A: set_Product_prod_f_f,C3: set_Product_prod_f_f] :
( ( ord_le7147000192471188839od_f_f @ B @ A )
=> ( ( ord_le7147000192471188839od_f_f @ C3 @ B )
=> ( ord_le7147000192471188839od_f_f @ C3 @ A ) ) ) ).
% dual_order.trans
thf(fact_364_dual__order_Otrans,axiom,
! [B: set_term_f_v,A: set_term_f_v,C3: set_term_f_v] :
( ( ord_le947722214152454265rm_f_v @ B @ A )
=> ( ( ord_le947722214152454265rm_f_v @ C3 @ B )
=> ( ord_le947722214152454265rm_f_v @ C3 @ A ) ) ) ).
% dual_order.trans
thf(fact_365_dual__order_Otrans,axiom,
! [B: nat,A: nat,C3: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C3 @ B )
=> ( ord_less_eq_nat @ C3 @ A ) ) ) ).
% dual_order.trans
thf(fact_366_dual__order_Otrans,axiom,
! [B: set_Pr5245412377734449720_f_nat,A: set_Pr5245412377734449720_f_nat,C3: set_Pr5245412377734449720_f_nat] :
( ( ord_le8976984241387448984_f_nat @ B @ A )
=> ( ( ord_le8976984241387448984_f_nat @ C3 @ B )
=> ( ord_le8976984241387448984_f_nat @ C3 @ A ) ) ) ).
% dual_order.trans
thf(fact_367_antisym,axiom,
! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ A @ B )
=> ( ( ord_le3146513528884898305at_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_368_antisym,axiom,
! [A: set_Pr4548512049067879902_nat_f,B: set_Pr4548512049067879902_nat_f] :
( ( ord_le8280083912720879166_nat_f @ A @ B )
=> ( ( ord_le8280083912720879166_nat_f @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_369_antisym,axiom,
! [A: set_Product_prod_f_f,B: set_Product_prod_f_f] :
( ( ord_le7147000192471188839od_f_f @ A @ B )
=> ( ( ord_le7147000192471188839od_f_f @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_370_antisym,axiom,
! [A: set_term_f_v,B: set_term_f_v] :
( ( ord_le947722214152454265rm_f_v @ A @ B )
=> ( ( ord_le947722214152454265rm_f_v @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_371_antisym,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_372_antisym,axiom,
! [A: set_Pr5245412377734449720_f_nat,B: set_Pr5245412377734449720_f_nat] :
( ( ord_le8976984241387448984_f_nat @ A @ B )
=> ( ( ord_le8976984241387448984_f_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_373_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_Pr1261947904930325089at_nat,Z3: set_Pr1261947904930325089at_nat] : ( Y4 = Z3 ) )
= ( ^ [A5: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ A5 @ B4 )
& ( ord_le3146513528884898305at_nat @ B4 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_374_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_Pr4548512049067879902_nat_f,Z3: set_Pr4548512049067879902_nat_f] : ( Y4 = Z3 ) )
= ( ^ [A5: set_Pr4548512049067879902_nat_f,B4: set_Pr4548512049067879902_nat_f] :
( ( ord_le8280083912720879166_nat_f @ A5 @ B4 )
& ( ord_le8280083912720879166_nat_f @ B4 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_375_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_Product_prod_f_f,Z3: set_Product_prod_f_f] : ( Y4 = Z3 ) )
= ( ^ [A5: set_Product_prod_f_f,B4: set_Product_prod_f_f] :
( ( ord_le7147000192471188839od_f_f @ A5 @ B4 )
& ( ord_le7147000192471188839od_f_f @ B4 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_376_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_term_f_v,Z3: set_term_f_v] : ( Y4 = Z3 ) )
= ( ^ [A5: set_term_f_v,B4: set_term_f_v] :
( ( ord_le947722214152454265rm_f_v @ A5 @ B4 )
& ( ord_le947722214152454265rm_f_v @ B4 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_377_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: nat,Z3: nat] : ( Y4 = Z3 ) )
= ( ^ [A5: nat,B4: nat] :
( ( ord_less_eq_nat @ A5 @ B4 )
& ( ord_less_eq_nat @ B4 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_378_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_Pr5245412377734449720_f_nat,Z3: set_Pr5245412377734449720_f_nat] : ( Y4 = Z3 ) )
= ( ^ [A5: set_Pr5245412377734449720_f_nat,B4: set_Pr5245412377734449720_f_nat] :
( ( ord_le8976984241387448984_f_nat @ A5 @ B4 )
& ( ord_le8976984241387448984_f_nat @ B4 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_379_order__subst1,axiom,
! [A: set_Pr5245412377734449720_f_nat,F2: set_Pr5245412377734449720_f_nat > set_Pr5245412377734449720_f_nat,B: set_Pr5245412377734449720_f_nat,C3: set_Pr5245412377734449720_f_nat] :
( ( ord_le8976984241387448984_f_nat @ A @ ( F2 @ B ) )
=> ( ( ord_le8976984241387448984_f_nat @ B @ C3 )
=> ( ! [X: set_Pr5245412377734449720_f_nat,Y3: set_Pr5245412377734449720_f_nat] :
( ( ord_le8976984241387448984_f_nat @ X @ Y3 )
=> ( ord_le8976984241387448984_f_nat @ ( F2 @ X ) @ ( F2 @ Y3 ) ) )
=> ( ord_le8976984241387448984_f_nat @ A @ ( F2 @ C3 ) ) ) ) ) ).
% order_subst1
thf(fact_380_order__subst1,axiom,
! [A: nat,F2: nat > nat,B: nat,C3: nat] :
( ( ord_less_eq_nat @ A @ ( F2 @ B ) )
=> ( ( ord_less_eq_nat @ B @ C3 )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F2 @ C3 ) ) ) ) ) ).
% order_subst1
thf(fact_381_order__subst1,axiom,
! [A: set_Pr5245412377734449720_f_nat,F2: nat > set_Pr5245412377734449720_f_nat,B: nat,C3: nat] :
( ( ord_le8976984241387448984_f_nat @ A @ ( F2 @ B ) )
=> ( ( ord_less_eq_nat @ B @ C3 )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_le8976984241387448984_f_nat @ ( F2 @ X ) @ ( F2 @ Y3 ) ) )
=> ( ord_le8976984241387448984_f_nat @ A @ ( F2 @ C3 ) ) ) ) ) ).
% order_subst1
thf(fact_382_order__subst1,axiom,
! [A: set_Pr1261947904930325089at_nat,F2: nat > set_Pr1261947904930325089at_nat,B: nat,C3: nat] :
( ( ord_le3146513528884898305at_nat @ A @ ( F2 @ B ) )
=> ( ( ord_less_eq_nat @ B @ C3 )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_le3146513528884898305at_nat @ ( F2 @ X ) @ ( F2 @ Y3 ) ) )
=> ( ord_le3146513528884898305at_nat @ A @ ( F2 @ C3 ) ) ) ) ) ).
% order_subst1
thf(fact_383_order__subst1,axiom,
! [A: set_Pr4548512049067879902_nat_f,F2: nat > set_Pr4548512049067879902_nat_f,B: nat,C3: nat] :
( ( ord_le8280083912720879166_nat_f @ A @ ( F2 @ B ) )
=> ( ( ord_less_eq_nat @ B @ C3 )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_le8280083912720879166_nat_f @ ( F2 @ X ) @ ( F2 @ Y3 ) ) )
=> ( ord_le8280083912720879166_nat_f @ A @ ( F2 @ C3 ) ) ) ) ) ).
% order_subst1
thf(fact_384_order__subst1,axiom,
! [A: set_Product_prod_f_f,F2: nat > set_Product_prod_f_f,B: nat,C3: nat] :
( ( ord_le7147000192471188839od_f_f @ A @ ( F2 @ B ) )
=> ( ( ord_less_eq_nat @ B @ C3 )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_le7147000192471188839od_f_f @ ( F2 @ X ) @ ( F2 @ Y3 ) ) )
=> ( ord_le7147000192471188839od_f_f @ A @ ( F2 @ C3 ) ) ) ) ) ).
% order_subst1
thf(fact_385_order__subst1,axiom,
! [A: set_term_f_v,F2: nat > set_term_f_v,B: nat,C3: nat] :
( ( ord_le947722214152454265rm_f_v @ A @ ( F2 @ B ) )
=> ( ( ord_less_eq_nat @ B @ C3 )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_le947722214152454265rm_f_v @ ( F2 @ X ) @ ( F2 @ Y3 ) ) )
=> ( ord_le947722214152454265rm_f_v @ A @ ( F2 @ C3 ) ) ) ) ) ).
% order_subst1
thf(fact_386_order__subst1,axiom,
! [A: nat,F2: set_Pr5245412377734449720_f_nat > nat,B: set_Pr5245412377734449720_f_nat,C3: set_Pr5245412377734449720_f_nat] :
( ( ord_less_eq_nat @ A @ ( F2 @ B ) )
=> ( ( ord_le8976984241387448984_f_nat @ B @ C3 )
=> ( ! [X: set_Pr5245412377734449720_f_nat,Y3: set_Pr5245412377734449720_f_nat] :
( ( ord_le8976984241387448984_f_nat @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F2 @ C3 ) ) ) ) ) ).
% order_subst1
thf(fact_387_order__subst1,axiom,
! [A: nat,F2: set_Pr1261947904930325089at_nat > nat,B: set_Pr1261947904930325089at_nat,C3: set_Pr1261947904930325089at_nat] :
( ( ord_less_eq_nat @ A @ ( F2 @ B ) )
=> ( ( ord_le3146513528884898305at_nat @ B @ C3 )
=> ( ! [X: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F2 @ C3 ) ) ) ) ) ).
% order_subst1
thf(fact_388_order__subst1,axiom,
! [A: nat,F2: set_Pr4548512049067879902_nat_f > nat,B: set_Pr4548512049067879902_nat_f,C3: set_Pr4548512049067879902_nat_f] :
( ( ord_less_eq_nat @ A @ ( F2 @ B ) )
=> ( ( ord_le8280083912720879166_nat_f @ B @ C3 )
=> ( ! [X: set_Pr4548512049067879902_nat_f,Y3: set_Pr4548512049067879902_nat_f] :
( ( ord_le8280083912720879166_nat_f @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F2 @ C3 ) ) ) ) ) ).
% order_subst1
thf(fact_389_order__subst2,axiom,
! [A: set_Pr5245412377734449720_f_nat,B: set_Pr5245412377734449720_f_nat,F2: set_Pr5245412377734449720_f_nat > set_Pr5245412377734449720_f_nat,C3: set_Pr5245412377734449720_f_nat] :
( ( ord_le8976984241387448984_f_nat @ A @ B )
=> ( ( ord_le8976984241387448984_f_nat @ ( F2 @ B ) @ C3 )
=> ( ! [X: set_Pr5245412377734449720_f_nat,Y3: set_Pr5245412377734449720_f_nat] :
( ( ord_le8976984241387448984_f_nat @ X @ Y3 )
=> ( ord_le8976984241387448984_f_nat @ ( F2 @ X ) @ ( F2 @ Y3 ) ) )
=> ( ord_le8976984241387448984_f_nat @ ( F2 @ A ) @ C3 ) ) ) ) ).
% order_subst2
thf(fact_390_order__subst2,axiom,
! [A: nat,B: nat,F2: nat > nat,C3: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F2 @ B ) @ C3 )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F2 @ A ) @ C3 ) ) ) ) ).
% order_subst2
thf(fact_391_order__subst2,axiom,
! [A: set_Pr5245412377734449720_f_nat,B: set_Pr5245412377734449720_f_nat,F2: set_Pr5245412377734449720_f_nat > nat,C3: nat] :
( ( ord_le8976984241387448984_f_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F2 @ B ) @ C3 )
=> ( ! [X: set_Pr5245412377734449720_f_nat,Y3: set_Pr5245412377734449720_f_nat] :
( ( ord_le8976984241387448984_f_nat @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F2 @ A ) @ C3 ) ) ) ) ).
% order_subst2
thf(fact_392_order__subst2,axiom,
! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,F2: set_Pr1261947904930325089at_nat > nat,C3: nat] :
( ( ord_le3146513528884898305at_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F2 @ B ) @ C3 )
=> ( ! [X: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F2 @ A ) @ C3 ) ) ) ) ).
% order_subst2
thf(fact_393_order__subst2,axiom,
! [A: set_Pr4548512049067879902_nat_f,B: set_Pr4548512049067879902_nat_f,F2: set_Pr4548512049067879902_nat_f > nat,C3: nat] :
( ( ord_le8280083912720879166_nat_f @ A @ B )
=> ( ( ord_less_eq_nat @ ( F2 @ B ) @ C3 )
=> ( ! [X: set_Pr4548512049067879902_nat_f,Y3: set_Pr4548512049067879902_nat_f] :
( ( ord_le8280083912720879166_nat_f @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F2 @ A ) @ C3 ) ) ) ) ).
% order_subst2
thf(fact_394_order__subst2,axiom,
! [A: set_Product_prod_f_f,B: set_Product_prod_f_f,F2: set_Product_prod_f_f > nat,C3: nat] :
( ( ord_le7147000192471188839od_f_f @ A @ B )
=> ( ( ord_less_eq_nat @ ( F2 @ B ) @ C3 )
=> ( ! [X: set_Product_prod_f_f,Y3: set_Product_prod_f_f] :
( ( ord_le7147000192471188839od_f_f @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F2 @ A ) @ C3 ) ) ) ) ).
% order_subst2
thf(fact_395_order__subst2,axiom,
! [A: set_term_f_v,B: set_term_f_v,F2: set_term_f_v > nat,C3: nat] :
( ( ord_le947722214152454265rm_f_v @ A @ B )
=> ( ( ord_less_eq_nat @ ( F2 @ B ) @ C3 )
=> ( ! [X: set_term_f_v,Y3: set_term_f_v] :
( ( ord_le947722214152454265rm_f_v @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F2 @ A ) @ C3 ) ) ) ) ).
% order_subst2
thf(fact_396_order__subst2,axiom,
! [A: nat,B: nat,F2: nat > set_Pr5245412377734449720_f_nat,C3: set_Pr5245412377734449720_f_nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_le8976984241387448984_f_nat @ ( F2 @ B ) @ C3 )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_le8976984241387448984_f_nat @ ( F2 @ X ) @ ( F2 @ Y3 ) ) )
=> ( ord_le8976984241387448984_f_nat @ ( F2 @ A ) @ C3 ) ) ) ) ).
% order_subst2
thf(fact_397_order__subst2,axiom,
! [A: nat,B: nat,F2: nat > set_Pr1261947904930325089at_nat,C3: set_Pr1261947904930325089at_nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_le3146513528884898305at_nat @ ( F2 @ B ) @ C3 )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_le3146513528884898305at_nat @ ( F2 @ X ) @ ( F2 @ Y3 ) ) )
=> ( ord_le3146513528884898305at_nat @ ( F2 @ A ) @ C3 ) ) ) ) ).
% order_subst2
thf(fact_398_order__subst2,axiom,
! [A: nat,B: nat,F2: nat > set_Pr4548512049067879902_nat_f,C3: set_Pr4548512049067879902_nat_f] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_le8280083912720879166_nat_f @ ( F2 @ B ) @ C3 )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_le8280083912720879166_nat_f @ ( F2 @ X ) @ ( F2 @ Y3 ) ) )
=> ( ord_le8280083912720879166_nat_f @ ( F2 @ A ) @ C3 ) ) ) ) ).
% order_subst2
thf(fact_399_order__eq__refl,axiom,
! [X3: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
( ( X3 = Y )
=> ( ord_le3146513528884898305at_nat @ X3 @ Y ) ) ).
% order_eq_refl
thf(fact_400_order__eq__refl,axiom,
! [X3: set_Pr4548512049067879902_nat_f,Y: set_Pr4548512049067879902_nat_f] :
( ( X3 = Y )
=> ( ord_le8280083912720879166_nat_f @ X3 @ Y ) ) ).
% order_eq_refl
thf(fact_401_order__eq__refl,axiom,
! [X3: set_Product_prod_f_f,Y: set_Product_prod_f_f] :
( ( X3 = Y )
=> ( ord_le7147000192471188839od_f_f @ X3 @ Y ) ) ).
% order_eq_refl
thf(fact_402_order__eq__refl,axiom,
! [X3: set_term_f_v,Y: set_term_f_v] :
( ( X3 = Y )
=> ( ord_le947722214152454265rm_f_v @ X3 @ Y ) ) ).
% order_eq_refl
thf(fact_403_order__eq__refl,axiom,
! [X3: nat,Y: nat] :
( ( X3 = Y )
=> ( ord_less_eq_nat @ X3 @ Y ) ) ).
% order_eq_refl
thf(fact_404_order__eq__refl,axiom,
! [X3: set_Pr5245412377734449720_f_nat,Y: set_Pr5245412377734449720_f_nat] :
( ( X3 = Y )
=> ( ord_le8976984241387448984_f_nat @ X3 @ Y ) ) ).
% order_eq_refl
thf(fact_405_linorder__linear,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
| ( ord_less_eq_nat @ Y @ X3 ) ) ).
% linorder_linear
thf(fact_406_ord__eq__le__subst,axiom,
! [A: set_Pr5245412377734449720_f_nat,F2: set_Pr5245412377734449720_f_nat > set_Pr5245412377734449720_f_nat,B: set_Pr5245412377734449720_f_nat,C3: set_Pr5245412377734449720_f_nat] :
( ( A
= ( F2 @ B ) )
=> ( ( ord_le8976984241387448984_f_nat @ B @ C3 )
=> ( ! [X: set_Pr5245412377734449720_f_nat,Y3: set_Pr5245412377734449720_f_nat] :
( ( ord_le8976984241387448984_f_nat @ X @ Y3 )
=> ( ord_le8976984241387448984_f_nat @ ( F2 @ X ) @ ( F2 @ Y3 ) ) )
=> ( ord_le8976984241387448984_f_nat @ A @ ( F2 @ C3 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_407_ord__eq__le__subst,axiom,
! [A: nat,F2: nat > nat,B: nat,C3: nat] :
( ( A
= ( F2 @ B ) )
=> ( ( ord_less_eq_nat @ B @ C3 )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F2 @ C3 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_408_ord__eq__le__subst,axiom,
! [A: nat,F2: set_Pr5245412377734449720_f_nat > nat,B: set_Pr5245412377734449720_f_nat,C3: set_Pr5245412377734449720_f_nat] :
( ( A
= ( F2 @ B ) )
=> ( ( ord_le8976984241387448984_f_nat @ B @ C3 )
=> ( ! [X: set_Pr5245412377734449720_f_nat,Y3: set_Pr5245412377734449720_f_nat] :
( ( ord_le8976984241387448984_f_nat @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F2 @ C3 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_409_ord__eq__le__subst,axiom,
! [A: nat,F2: set_Pr1261947904930325089at_nat > nat,B: set_Pr1261947904930325089at_nat,C3: set_Pr1261947904930325089at_nat] :
( ( A
= ( F2 @ B ) )
=> ( ( ord_le3146513528884898305at_nat @ B @ C3 )
=> ( ! [X: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F2 @ C3 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_410_ord__eq__le__subst,axiom,
! [A: nat,F2: set_Pr4548512049067879902_nat_f > nat,B: set_Pr4548512049067879902_nat_f,C3: set_Pr4548512049067879902_nat_f] :
( ( A
= ( F2 @ B ) )
=> ( ( ord_le8280083912720879166_nat_f @ B @ C3 )
=> ( ! [X: set_Pr4548512049067879902_nat_f,Y3: set_Pr4548512049067879902_nat_f] :
( ( ord_le8280083912720879166_nat_f @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F2 @ C3 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_411_ord__eq__le__subst,axiom,
! [A: nat,F2: set_Product_prod_f_f > nat,B: set_Product_prod_f_f,C3: set_Product_prod_f_f] :
( ( A
= ( F2 @ B ) )
=> ( ( ord_le7147000192471188839od_f_f @ B @ C3 )
=> ( ! [X: set_Product_prod_f_f,Y3: set_Product_prod_f_f] :
( ( ord_le7147000192471188839od_f_f @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F2 @ C3 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_412_ord__eq__le__subst,axiom,
! [A: nat,F2: set_term_f_v > nat,B: set_term_f_v,C3: set_term_f_v] :
( ( A
= ( F2 @ B ) )
=> ( ( ord_le947722214152454265rm_f_v @ B @ C3 )
=> ( ! [X: set_term_f_v,Y3: set_term_f_v] :
( ( ord_le947722214152454265rm_f_v @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F2 @ C3 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_413_ord__eq__le__subst,axiom,
! [A: set_Pr5245412377734449720_f_nat,F2: nat > set_Pr5245412377734449720_f_nat,B: nat,C3: nat] :
( ( A
= ( F2 @ B ) )
=> ( ( ord_less_eq_nat @ B @ C3 )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_le8976984241387448984_f_nat @ ( F2 @ X ) @ ( F2 @ Y3 ) ) )
=> ( ord_le8976984241387448984_f_nat @ A @ ( F2 @ C3 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_414_ord__eq__le__subst,axiom,
! [A: set_Pr1261947904930325089at_nat,F2: nat > set_Pr1261947904930325089at_nat,B: nat,C3: nat] :
( ( A
= ( F2 @ B ) )
=> ( ( ord_less_eq_nat @ B @ C3 )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_le3146513528884898305at_nat @ ( F2 @ X ) @ ( F2 @ Y3 ) ) )
=> ( ord_le3146513528884898305at_nat @ A @ ( F2 @ C3 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_415_ord__eq__le__subst,axiom,
! [A: set_Pr4548512049067879902_nat_f,F2: nat > set_Pr4548512049067879902_nat_f,B: nat,C3: nat] :
( ( A
= ( F2 @ B ) )
=> ( ( ord_less_eq_nat @ B @ C3 )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_le8280083912720879166_nat_f @ ( F2 @ X ) @ ( F2 @ Y3 ) ) )
=> ( ord_le8280083912720879166_nat_f @ A @ ( F2 @ C3 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_416_ord__le__eq__subst,axiom,
! [A: set_Pr5245412377734449720_f_nat,B: set_Pr5245412377734449720_f_nat,F2: set_Pr5245412377734449720_f_nat > set_Pr5245412377734449720_f_nat,C3: set_Pr5245412377734449720_f_nat] :
( ( ord_le8976984241387448984_f_nat @ A @ B )
=> ( ( ( F2 @ B )
= C3 )
=> ( ! [X: set_Pr5245412377734449720_f_nat,Y3: set_Pr5245412377734449720_f_nat] :
( ( ord_le8976984241387448984_f_nat @ X @ Y3 )
=> ( ord_le8976984241387448984_f_nat @ ( F2 @ X ) @ ( F2 @ Y3 ) ) )
=> ( ord_le8976984241387448984_f_nat @ ( F2 @ A ) @ C3 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_417_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F2: nat > nat,C3: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F2 @ B )
= C3 )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F2 @ A ) @ C3 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_418_ord__le__eq__subst,axiom,
! [A: set_Pr5245412377734449720_f_nat,B: set_Pr5245412377734449720_f_nat,F2: set_Pr5245412377734449720_f_nat > nat,C3: nat] :
( ( ord_le8976984241387448984_f_nat @ A @ B )
=> ( ( ( F2 @ B )
= C3 )
=> ( ! [X: set_Pr5245412377734449720_f_nat,Y3: set_Pr5245412377734449720_f_nat] :
( ( ord_le8976984241387448984_f_nat @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F2 @ A ) @ C3 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_419_ord__le__eq__subst,axiom,
! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,F2: set_Pr1261947904930325089at_nat > nat,C3: nat] :
( ( ord_le3146513528884898305at_nat @ A @ B )
=> ( ( ( F2 @ B )
= C3 )
=> ( ! [X: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F2 @ A ) @ C3 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_420_ord__le__eq__subst,axiom,
! [A: set_Pr4548512049067879902_nat_f,B: set_Pr4548512049067879902_nat_f,F2: set_Pr4548512049067879902_nat_f > nat,C3: nat] :
( ( ord_le8280083912720879166_nat_f @ A @ B )
=> ( ( ( F2 @ B )
= C3 )
=> ( ! [X: set_Pr4548512049067879902_nat_f,Y3: set_Pr4548512049067879902_nat_f] :
( ( ord_le8280083912720879166_nat_f @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F2 @ A ) @ C3 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_421_ord__le__eq__subst,axiom,
! [A: set_Product_prod_f_f,B: set_Product_prod_f_f,F2: set_Product_prod_f_f > nat,C3: nat] :
( ( ord_le7147000192471188839od_f_f @ A @ B )
=> ( ( ( F2 @ B )
= C3 )
=> ( ! [X: set_Product_prod_f_f,Y3: set_Product_prod_f_f] :
( ( ord_le7147000192471188839od_f_f @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F2 @ A ) @ C3 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_422_ord__le__eq__subst,axiom,
! [A: set_term_f_v,B: set_term_f_v,F2: set_term_f_v > nat,C3: nat] :
( ( ord_le947722214152454265rm_f_v @ A @ B )
=> ( ( ( F2 @ B )
= C3 )
=> ( ! [X: set_term_f_v,Y3: set_term_f_v] :
( ( ord_le947722214152454265rm_f_v @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F2 @ A ) @ C3 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_423_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F2: nat > set_Pr5245412377734449720_f_nat,C3: set_Pr5245412377734449720_f_nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F2 @ B )
= C3 )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_le8976984241387448984_f_nat @ ( F2 @ X ) @ ( F2 @ Y3 ) ) )
=> ( ord_le8976984241387448984_f_nat @ ( F2 @ A ) @ C3 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_424_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F2: nat > set_Pr1261947904930325089at_nat,C3: set_Pr1261947904930325089at_nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F2 @ B )
= C3 )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_le3146513528884898305at_nat @ ( F2 @ X ) @ ( F2 @ Y3 ) ) )
=> ( ord_le3146513528884898305at_nat @ ( F2 @ A ) @ C3 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_425_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F2: nat > set_Pr4548512049067879902_nat_f,C3: set_Pr4548512049067879902_nat_f] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F2 @ B )
= C3 )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_le8280083912720879166_nat_f @ ( F2 @ X ) @ ( F2 @ Y3 ) ) )
=> ( ord_le8280083912720879166_nat_f @ ( F2 @ A ) @ C3 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_426_linorder__le__cases,axiom,
! [X3: nat,Y: nat] :
( ~ ( ord_less_eq_nat @ X3 @ Y )
=> ( ord_less_eq_nat @ Y @ X3 ) ) ).
% linorder_le_cases
thf(fact_427_order__antisym__conv,axiom,
! [Y: set_Pr1261947904930325089at_nat,X3: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ Y @ X3 )
=> ( ( ord_le3146513528884898305at_nat @ X3 @ Y )
= ( X3 = Y ) ) ) ).
% order_antisym_conv
thf(fact_428_order__antisym__conv,axiom,
! [Y: set_Pr4548512049067879902_nat_f,X3: set_Pr4548512049067879902_nat_f] :
( ( ord_le8280083912720879166_nat_f @ Y @ X3 )
=> ( ( ord_le8280083912720879166_nat_f @ X3 @ Y )
= ( X3 = Y ) ) ) ).
% order_antisym_conv
thf(fact_429_order__antisym__conv,axiom,
! [Y: set_Product_prod_f_f,X3: set_Product_prod_f_f] :
( ( ord_le7147000192471188839od_f_f @ Y @ X3 )
=> ( ( ord_le7147000192471188839od_f_f @ X3 @ Y )
= ( X3 = Y ) ) ) ).
% order_antisym_conv
thf(fact_430_order__antisym__conv,axiom,
! [Y: set_term_f_v,X3: set_term_f_v] :
( ( ord_le947722214152454265rm_f_v @ Y @ X3 )
=> ( ( ord_le947722214152454265rm_f_v @ X3 @ Y )
= ( X3 = Y ) ) ) ).
% order_antisym_conv
thf(fact_431_order__antisym__conv,axiom,
! [Y: nat,X3: nat] :
( ( ord_less_eq_nat @ Y @ X3 )
=> ( ( ord_less_eq_nat @ X3 @ Y )
= ( X3 = Y ) ) ) ).
% order_antisym_conv
thf(fact_432_order__antisym__conv,axiom,
! [Y: set_Pr5245412377734449720_f_nat,X3: set_Pr5245412377734449720_f_nat] :
( ( ord_le8976984241387448984_f_nat @ Y @ X3 )
=> ( ( ord_le8976984241387448984_f_nat @ X3 @ Y )
= ( X3 = Y ) ) ) ).
% order_antisym_conv
thf(fact_433_sig__stepE,axiom,
! [S: term_f_v,T: term_f_v,F: set_Pr5245412377734449720_f_nat,R2: set_Pr8827868859434726151rm_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ ( sig_step_f_v @ F @ R2 ) )
=> ~ ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ R2 )
=> ( ( ord_le8976984241387448984_f_nat @ ( term_funas_term_f_v @ S ) @ F )
=> ~ ( ord_le8976984241387448984_f_nat @ ( term_funas_term_f_v @ T ) @ F ) ) ) ) ).
% sig_stepE
thf(fact_434_in__mono,axiom,
! [A4: set_Pr7897041799657265351rm_f_v,B5: set_Pr7897041799657265351rm_f_v,X3: produc3406425829894461415rm_f_v] :
( ( ord_le7518223742277120103rm_f_v @ A4 @ B5 )
=> ( ( member7300054593328134928rm_f_v @ X3 @ A4 )
=> ( member7300054593328134928rm_f_v @ X3 @ B5 ) ) ) ).
% in_mono
thf(fact_435_in__mono,axiom,
! [A4: set_Pr516226060960033221_f_nat,B5: set_Pr516226060960033221_f_nat,X3: produc2676906022994773093_f_nat] :
( ( ord_le8946695678602162533_f_nat @ A4 @ B5 )
=> ( ( member7030746907725424910_f_nat @ X3 @ A4 )
=> ( member7030746907725424910_f_nat @ X3 @ B5 ) ) ) ).
% in_mono
thf(fact_436_in__mono,axiom,
! [A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat,X3: product_prod_nat_nat] :
( ( ord_le3146513528884898305at_nat @ A4 @ B5 )
=> ( ( member8440522571783428010at_nat @ X3 @ A4 )
=> ( member8440522571783428010at_nat @ X3 @ B5 ) ) ) ).
% in_mono
thf(fact_437_in__mono,axiom,
! [A4: set_Pr4548512049067879902_nat_f,B5: set_Pr4548512049067879902_nat_f,X3: product_prod_nat_f] :
( ( ord_le8280083912720879166_nat_f @ A4 @ B5 )
=> ( ( member8962352073626239679_nat_f @ X3 @ A4 )
=> ( member8962352073626239679_nat_f @ X3 @ B5 ) ) ) ).
% in_mono
thf(fact_438_in__mono,axiom,
! [A4: set_Product_prod_f_f,B5: set_Product_prod_f_f,X3: product_prod_f_f] :
( ( ord_le7147000192471188839od_f_f @ A4 @ B5 )
=> ( ( member5935997662887750672od_f_f @ X3 @ A4 )
=> ( member5935997662887750672od_f_f @ X3 @ B5 ) ) ) ).
% in_mono
thf(fact_439_in__mono,axiom,
! [A4: set_term_f_v,B5: set_term_f_v,X3: term_f_v] :
( ( ord_le947722214152454265rm_f_v @ A4 @ B5 )
=> ( ( member_term_f_v @ X3 @ A4 )
=> ( member_term_f_v @ X3 @ B5 ) ) ) ).
% in_mono
thf(fact_440_in__mono,axiom,
! [A4: set_Pr8827868859434726151rm_f_v,B5: set_Pr8827868859434726151rm_f_v,X3: produc8027700073345061287rm_f_v] :
( ( ord_le4559761987009501863rm_f_v @ A4 @ B5 )
=> ( ( member4316649861536941904rm_f_v @ X3 @ A4 )
=> ( member4316649861536941904rm_f_v @ X3 @ B5 ) ) ) ).
% in_mono
thf(fact_441_in__mono,axiom,
! [A4: set_Pr5245412377734449720_f_nat,B5: set_Pr5245412377734449720_f_nat,X3: product_prod_f_nat] :
( ( ord_le8976984241387448984_f_nat @ A4 @ B5 )
=> ( ( member2678038196313615385_f_nat @ X3 @ A4 )
=> ( member2678038196313615385_f_nat @ X3 @ B5 ) ) ) ).
% in_mono
thf(fact_442_subsetD,axiom,
! [A4: set_Pr7897041799657265351rm_f_v,B5: set_Pr7897041799657265351rm_f_v,C3: produc3406425829894461415rm_f_v] :
( ( ord_le7518223742277120103rm_f_v @ A4 @ B5 )
=> ( ( member7300054593328134928rm_f_v @ C3 @ A4 )
=> ( member7300054593328134928rm_f_v @ C3 @ B5 ) ) ) ).
% subsetD
thf(fact_443_subsetD,axiom,
! [A4: set_Pr516226060960033221_f_nat,B5: set_Pr516226060960033221_f_nat,C3: produc2676906022994773093_f_nat] :
( ( ord_le8946695678602162533_f_nat @ A4 @ B5 )
=> ( ( member7030746907725424910_f_nat @ C3 @ A4 )
=> ( member7030746907725424910_f_nat @ C3 @ B5 ) ) ) ).
% subsetD
thf(fact_444_subsetD,axiom,
! [A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat,C3: product_prod_nat_nat] :
( ( ord_le3146513528884898305at_nat @ A4 @ B5 )
=> ( ( member8440522571783428010at_nat @ C3 @ A4 )
=> ( member8440522571783428010at_nat @ C3 @ B5 ) ) ) ).
% subsetD
thf(fact_445_subsetD,axiom,
! [A4: set_Pr4548512049067879902_nat_f,B5: set_Pr4548512049067879902_nat_f,C3: product_prod_nat_f] :
( ( ord_le8280083912720879166_nat_f @ A4 @ B5 )
=> ( ( member8962352073626239679_nat_f @ C3 @ A4 )
=> ( member8962352073626239679_nat_f @ C3 @ B5 ) ) ) ).
% subsetD
thf(fact_446_subsetD,axiom,
! [A4: set_Product_prod_f_f,B5: set_Product_prod_f_f,C3: product_prod_f_f] :
( ( ord_le7147000192471188839od_f_f @ A4 @ B5 )
=> ( ( member5935997662887750672od_f_f @ C3 @ A4 )
=> ( member5935997662887750672od_f_f @ C3 @ B5 ) ) ) ).
% subsetD
thf(fact_447_subsetD,axiom,
! [A4: set_term_f_v,B5: set_term_f_v,C3: term_f_v] :
( ( ord_le947722214152454265rm_f_v @ A4 @ B5 )
=> ( ( member_term_f_v @ C3 @ A4 )
=> ( member_term_f_v @ C3 @ B5 ) ) ) ).
% subsetD
thf(fact_448_subsetD,axiom,
! [A4: set_Pr8827868859434726151rm_f_v,B5: set_Pr8827868859434726151rm_f_v,C3: produc8027700073345061287rm_f_v] :
( ( ord_le4559761987009501863rm_f_v @ A4 @ B5 )
=> ( ( member4316649861536941904rm_f_v @ C3 @ A4 )
=> ( member4316649861536941904rm_f_v @ C3 @ B5 ) ) ) ).
% subsetD
thf(fact_449_subsetD,axiom,
! [A4: set_Pr5245412377734449720_f_nat,B5: set_Pr5245412377734449720_f_nat,C3: product_prod_f_nat] :
( ( ord_le8976984241387448984_f_nat @ A4 @ B5 )
=> ( ( member2678038196313615385_f_nat @ C3 @ A4 )
=> ( member2678038196313615385_f_nat @ C3 @ B5 ) ) ) ).
% subsetD
thf(fact_450_equalityE,axiom,
! [A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
( ( A4 = B5 )
=> ~ ( ( ord_le3146513528884898305at_nat @ A4 @ B5 )
=> ~ ( ord_le3146513528884898305at_nat @ B5 @ A4 ) ) ) ).
% equalityE
thf(fact_451_equalityE,axiom,
! [A4: set_Pr4548512049067879902_nat_f,B5: set_Pr4548512049067879902_nat_f] :
( ( A4 = B5 )
=> ~ ( ( ord_le8280083912720879166_nat_f @ A4 @ B5 )
=> ~ ( ord_le8280083912720879166_nat_f @ B5 @ A4 ) ) ) ).
% equalityE
thf(fact_452_equalityE,axiom,
! [A4: set_Product_prod_f_f,B5: set_Product_prod_f_f] :
( ( A4 = B5 )
=> ~ ( ( ord_le7147000192471188839od_f_f @ A4 @ B5 )
=> ~ ( ord_le7147000192471188839od_f_f @ B5 @ A4 ) ) ) ).
% equalityE
thf(fact_453_equalityE,axiom,
! [A4: set_term_f_v,B5: set_term_f_v] :
( ( A4 = B5 )
=> ~ ( ( ord_le947722214152454265rm_f_v @ A4 @ B5 )
=> ~ ( ord_le947722214152454265rm_f_v @ B5 @ A4 ) ) ) ).
% equalityE
thf(fact_454_equalityE,axiom,
! [A4: set_Pr5245412377734449720_f_nat,B5: set_Pr5245412377734449720_f_nat] :
( ( A4 = B5 )
=> ~ ( ( ord_le8976984241387448984_f_nat @ A4 @ B5 )
=> ~ ( ord_le8976984241387448984_f_nat @ B5 @ A4 ) ) ) ).
% equalityE
thf(fact_455_subset__eq,axiom,
( ord_le7518223742277120103rm_f_v
= ( ^ [A6: set_Pr7897041799657265351rm_f_v,B6: set_Pr7897041799657265351rm_f_v] :
! [X4: produc3406425829894461415rm_f_v] :
( ( member7300054593328134928rm_f_v @ X4 @ A6 )
=> ( member7300054593328134928rm_f_v @ X4 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_456_subset__eq,axiom,
( ord_le8946695678602162533_f_nat
= ( ^ [A6: set_Pr516226060960033221_f_nat,B6: set_Pr516226060960033221_f_nat] :
! [X4: produc2676906022994773093_f_nat] :
( ( member7030746907725424910_f_nat @ X4 @ A6 )
=> ( member7030746907725424910_f_nat @ X4 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_457_subset__eq,axiom,
( ord_le3146513528884898305at_nat
= ( ^ [A6: set_Pr1261947904930325089at_nat,B6: set_Pr1261947904930325089at_nat] :
! [X4: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X4 @ A6 )
=> ( member8440522571783428010at_nat @ X4 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_458_subset__eq,axiom,
( ord_le8280083912720879166_nat_f
= ( ^ [A6: set_Pr4548512049067879902_nat_f,B6: set_Pr4548512049067879902_nat_f] :
! [X4: product_prod_nat_f] :
( ( member8962352073626239679_nat_f @ X4 @ A6 )
=> ( member8962352073626239679_nat_f @ X4 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_459_subset__eq,axiom,
( ord_le7147000192471188839od_f_f
= ( ^ [A6: set_Product_prod_f_f,B6: set_Product_prod_f_f] :
! [X4: product_prod_f_f] :
( ( member5935997662887750672od_f_f @ X4 @ A6 )
=> ( member5935997662887750672od_f_f @ X4 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_460_subset__eq,axiom,
( ord_le947722214152454265rm_f_v
= ( ^ [A6: set_term_f_v,B6: set_term_f_v] :
! [X4: term_f_v] :
( ( member_term_f_v @ X4 @ A6 )
=> ( member_term_f_v @ X4 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_461_subset__eq,axiom,
( ord_le4559761987009501863rm_f_v
= ( ^ [A6: set_Pr8827868859434726151rm_f_v,B6: set_Pr8827868859434726151rm_f_v] :
! [X4: produc8027700073345061287rm_f_v] :
( ( member4316649861536941904rm_f_v @ X4 @ A6 )
=> ( member4316649861536941904rm_f_v @ X4 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_462_subset__eq,axiom,
( ord_le8976984241387448984_f_nat
= ( ^ [A6: set_Pr5245412377734449720_f_nat,B6: set_Pr5245412377734449720_f_nat] :
! [X4: product_prod_f_nat] :
( ( member2678038196313615385_f_nat @ X4 @ A6 )
=> ( member2678038196313615385_f_nat @ X4 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_463_equalityD1,axiom,
! [A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
( ( A4 = B5 )
=> ( ord_le3146513528884898305at_nat @ A4 @ B5 ) ) ).
% equalityD1
thf(fact_464_equalityD1,axiom,
! [A4: set_Pr4548512049067879902_nat_f,B5: set_Pr4548512049067879902_nat_f] :
( ( A4 = B5 )
=> ( ord_le8280083912720879166_nat_f @ A4 @ B5 ) ) ).
% equalityD1
thf(fact_465_equalityD1,axiom,
! [A4: set_Product_prod_f_f,B5: set_Product_prod_f_f] :
( ( A4 = B5 )
=> ( ord_le7147000192471188839od_f_f @ A4 @ B5 ) ) ).
% equalityD1
thf(fact_466_equalityD1,axiom,
! [A4: set_term_f_v,B5: set_term_f_v] :
( ( A4 = B5 )
=> ( ord_le947722214152454265rm_f_v @ A4 @ B5 ) ) ).
% equalityD1
thf(fact_467_equalityD1,axiom,
! [A4: set_Pr5245412377734449720_f_nat,B5: set_Pr5245412377734449720_f_nat] :
( ( A4 = B5 )
=> ( ord_le8976984241387448984_f_nat @ A4 @ B5 ) ) ).
% equalityD1
thf(fact_468_equalityD2,axiom,
! [A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
( ( A4 = B5 )
=> ( ord_le3146513528884898305at_nat @ B5 @ A4 ) ) ).
% equalityD2
thf(fact_469_equalityD2,axiom,
! [A4: set_Pr4548512049067879902_nat_f,B5: set_Pr4548512049067879902_nat_f] :
( ( A4 = B5 )
=> ( ord_le8280083912720879166_nat_f @ B5 @ A4 ) ) ).
% equalityD2
thf(fact_470_equalityD2,axiom,
! [A4: set_Product_prod_f_f,B5: set_Product_prod_f_f] :
( ( A4 = B5 )
=> ( ord_le7147000192471188839od_f_f @ B5 @ A4 ) ) ).
% equalityD2
thf(fact_471_equalityD2,axiom,
! [A4: set_term_f_v,B5: set_term_f_v] :
( ( A4 = B5 )
=> ( ord_le947722214152454265rm_f_v @ B5 @ A4 ) ) ).
% equalityD2
thf(fact_472_equalityD2,axiom,
! [A4: set_Pr5245412377734449720_f_nat,B5: set_Pr5245412377734449720_f_nat] :
( ( A4 = B5 )
=> ( ord_le8976984241387448984_f_nat @ B5 @ A4 ) ) ).
% equalityD2
thf(fact_473_subset__iff,axiom,
( ord_le7518223742277120103rm_f_v
= ( ^ [A6: set_Pr7897041799657265351rm_f_v,B6: set_Pr7897041799657265351rm_f_v] :
! [T3: produc3406425829894461415rm_f_v] :
( ( member7300054593328134928rm_f_v @ T3 @ A6 )
=> ( member7300054593328134928rm_f_v @ T3 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_474_subset__iff,axiom,
( ord_le8946695678602162533_f_nat
= ( ^ [A6: set_Pr516226060960033221_f_nat,B6: set_Pr516226060960033221_f_nat] :
! [T3: produc2676906022994773093_f_nat] :
( ( member7030746907725424910_f_nat @ T3 @ A6 )
=> ( member7030746907725424910_f_nat @ T3 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_475_subset__iff,axiom,
( ord_le3146513528884898305at_nat
= ( ^ [A6: set_Pr1261947904930325089at_nat,B6: set_Pr1261947904930325089at_nat] :
! [T3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ T3 @ A6 )
=> ( member8440522571783428010at_nat @ T3 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_476_subset__iff,axiom,
( ord_le8280083912720879166_nat_f
= ( ^ [A6: set_Pr4548512049067879902_nat_f,B6: set_Pr4548512049067879902_nat_f] :
! [T3: product_prod_nat_f] :
( ( member8962352073626239679_nat_f @ T3 @ A6 )
=> ( member8962352073626239679_nat_f @ T3 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_477_subset__iff,axiom,
( ord_le7147000192471188839od_f_f
= ( ^ [A6: set_Product_prod_f_f,B6: set_Product_prod_f_f] :
! [T3: product_prod_f_f] :
( ( member5935997662887750672od_f_f @ T3 @ A6 )
=> ( member5935997662887750672od_f_f @ T3 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_478_subset__iff,axiom,
( ord_le947722214152454265rm_f_v
= ( ^ [A6: set_term_f_v,B6: set_term_f_v] :
! [T3: term_f_v] :
( ( member_term_f_v @ T3 @ A6 )
=> ( member_term_f_v @ T3 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_479_subset__iff,axiom,
( ord_le4559761987009501863rm_f_v
= ( ^ [A6: set_Pr8827868859434726151rm_f_v,B6: set_Pr8827868859434726151rm_f_v] :
! [T3: produc8027700073345061287rm_f_v] :
( ( member4316649861536941904rm_f_v @ T3 @ A6 )
=> ( member4316649861536941904rm_f_v @ T3 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_480_subset__iff,axiom,
( ord_le8976984241387448984_f_nat
= ( ^ [A6: set_Pr5245412377734449720_f_nat,B6: set_Pr5245412377734449720_f_nat] :
! [T3: product_prod_f_nat] :
( ( member2678038196313615385_f_nat @ T3 @ A6 )
=> ( member2678038196313615385_f_nat @ T3 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_481_subset__refl,axiom,
! [A4: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ A4 @ A4 ) ).
% subset_refl
thf(fact_482_subset__refl,axiom,
! [A4: set_Pr4548512049067879902_nat_f] : ( ord_le8280083912720879166_nat_f @ A4 @ A4 ) ).
% subset_refl
thf(fact_483_subset__refl,axiom,
! [A4: set_Product_prod_f_f] : ( ord_le7147000192471188839od_f_f @ A4 @ A4 ) ).
% subset_refl
thf(fact_484_subset__refl,axiom,
! [A4: set_term_f_v] : ( ord_le947722214152454265rm_f_v @ A4 @ A4 ) ).
% subset_refl
thf(fact_485_subset__refl,axiom,
! [A4: set_Pr5245412377734449720_f_nat] : ( ord_le8976984241387448984_f_nat @ A4 @ A4 ) ).
% subset_refl
thf(fact_486_Collect__mono,axiom,
! [P2: produc8027700073345061287rm_f_v > $o,Q: produc8027700073345061287rm_f_v > $o] :
( ! [X: produc8027700073345061287rm_f_v] :
( ( P2 @ X )
=> ( Q @ X ) )
=> ( ord_le4559761987009501863rm_f_v @ ( collec7769483783292507282rm_f_v @ P2 ) @ ( collec7769483783292507282rm_f_v @ Q ) ) ) ).
% Collect_mono
thf(fact_487_Collect__mono,axiom,
! [P2: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
( ! [X: product_prod_nat_nat] :
( ( P2 @ X )
=> ( Q @ X ) )
=> ( ord_le3146513528884898305at_nat @ ( collec3392354462482085612at_nat @ P2 ) @ ( collec3392354462482085612at_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_488_Collect__mono,axiom,
! [P2: product_prod_nat_f > $o,Q: product_prod_nat_f > $o] :
( ! [X: product_prod_nat_f] :
( ( P2 @ X )
=> ( Q @ X ) )
=> ( ord_le8280083912720879166_nat_f @ ( collec7702298020461588477_nat_f @ P2 ) @ ( collec7702298020461588477_nat_f @ Q ) ) ) ).
% Collect_mono
thf(fact_489_Collect__mono,axiom,
! [P2: product_prod_f_f > $o,Q: product_prod_f_f > $o] :
( ! [X: product_prod_f_f] :
( ( P2 @ X )
=> ( Q @ X ) )
=> ( ord_le7147000192471188839od_f_f @ ( collec7845863982746767954od_f_f @ P2 ) @ ( collec7845863982746767954od_f_f @ Q ) ) ) ).
% Collect_mono
thf(fact_490_Collect__mono,axiom,
! [P2: term_f_v > $o,Q: term_f_v > $o] :
( ! [X: term_f_v] :
( ( P2 @ X )
=> ( Q @ X ) )
=> ( ord_le947722214152454265rm_f_v @ ( collect_term_f_v @ P2 ) @ ( collect_term_f_v @ Q ) ) ) ).
% Collect_mono
thf(fact_491_Collect__mono,axiom,
! [P2: product_prod_f_nat > $o,Q: product_prod_f_nat > $o] :
( ! [X: product_prod_f_nat] :
( ( P2 @ X )
=> ( Q @ X ) )
=> ( ord_le8976984241387448984_f_nat @ ( collec1417984143148964183_f_nat @ P2 ) @ ( collec1417984143148964183_f_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_492_subset__trans,axiom,
! [A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ A4 @ B5 )
=> ( ( ord_le3146513528884898305at_nat @ B5 @ C2 )
=> ( ord_le3146513528884898305at_nat @ A4 @ C2 ) ) ) ).
% subset_trans
thf(fact_493_subset__trans,axiom,
! [A4: set_Pr4548512049067879902_nat_f,B5: set_Pr4548512049067879902_nat_f,C2: set_Pr4548512049067879902_nat_f] :
( ( ord_le8280083912720879166_nat_f @ A4 @ B5 )
=> ( ( ord_le8280083912720879166_nat_f @ B5 @ C2 )
=> ( ord_le8280083912720879166_nat_f @ A4 @ C2 ) ) ) ).
% subset_trans
thf(fact_494_subset__trans,axiom,
! [A4: set_Product_prod_f_f,B5: set_Product_prod_f_f,C2: set_Product_prod_f_f] :
( ( ord_le7147000192471188839od_f_f @ A4 @ B5 )
=> ( ( ord_le7147000192471188839od_f_f @ B5 @ C2 )
=> ( ord_le7147000192471188839od_f_f @ A4 @ C2 ) ) ) ).
% subset_trans
thf(fact_495_subset__trans,axiom,
! [A4: set_term_f_v,B5: set_term_f_v,C2: set_term_f_v] :
( ( ord_le947722214152454265rm_f_v @ A4 @ B5 )
=> ( ( ord_le947722214152454265rm_f_v @ B5 @ C2 )
=> ( ord_le947722214152454265rm_f_v @ A4 @ C2 ) ) ) ).
% subset_trans
thf(fact_496_subset__trans,axiom,
! [A4: set_Pr5245412377734449720_f_nat,B5: set_Pr5245412377734449720_f_nat,C2: set_Pr5245412377734449720_f_nat] :
( ( ord_le8976984241387448984_f_nat @ A4 @ B5 )
=> ( ( ord_le8976984241387448984_f_nat @ B5 @ C2 )
=> ( ord_le8976984241387448984_f_nat @ A4 @ C2 ) ) ) ).
% subset_trans
thf(fact_497_set__eq__subset,axiom,
( ( ^ [Y4: set_Pr1261947904930325089at_nat,Z3: set_Pr1261947904930325089at_nat] : ( Y4 = Z3 ) )
= ( ^ [A6: set_Pr1261947904930325089at_nat,B6: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ A6 @ B6 )
& ( ord_le3146513528884898305at_nat @ B6 @ A6 ) ) ) ) ).
% set_eq_subset
thf(fact_498_set__eq__subset,axiom,
( ( ^ [Y4: set_Pr4548512049067879902_nat_f,Z3: set_Pr4548512049067879902_nat_f] : ( Y4 = Z3 ) )
= ( ^ [A6: set_Pr4548512049067879902_nat_f,B6: set_Pr4548512049067879902_nat_f] :
( ( ord_le8280083912720879166_nat_f @ A6 @ B6 )
& ( ord_le8280083912720879166_nat_f @ B6 @ A6 ) ) ) ) ).
% set_eq_subset
thf(fact_499_set__eq__subset,axiom,
( ( ^ [Y4: set_Product_prod_f_f,Z3: set_Product_prod_f_f] : ( Y4 = Z3 ) )
= ( ^ [A6: set_Product_prod_f_f,B6: set_Product_prod_f_f] :
( ( ord_le7147000192471188839od_f_f @ A6 @ B6 )
& ( ord_le7147000192471188839od_f_f @ B6 @ A6 ) ) ) ) ).
% set_eq_subset
thf(fact_500_set__eq__subset,axiom,
( ( ^ [Y4: set_term_f_v,Z3: set_term_f_v] : ( Y4 = Z3 ) )
= ( ^ [A6: set_term_f_v,B6: set_term_f_v] :
( ( ord_le947722214152454265rm_f_v @ A6 @ B6 )
& ( ord_le947722214152454265rm_f_v @ B6 @ A6 ) ) ) ) ).
% set_eq_subset
thf(fact_501_set__eq__subset,axiom,
( ( ^ [Y4: set_Pr5245412377734449720_f_nat,Z3: set_Pr5245412377734449720_f_nat] : ( Y4 = Z3 ) )
= ( ^ [A6: set_Pr5245412377734449720_f_nat,B6: set_Pr5245412377734449720_f_nat] :
( ( ord_le8976984241387448984_f_nat @ A6 @ B6 )
& ( ord_le8976984241387448984_f_nat @ B6 @ A6 ) ) ) ) ).
% set_eq_subset
thf(fact_502_Collect__mono__iff,axiom,
! [P2: produc8027700073345061287rm_f_v > $o,Q: produc8027700073345061287rm_f_v > $o] :
( ( ord_le4559761987009501863rm_f_v @ ( collec7769483783292507282rm_f_v @ P2 ) @ ( collec7769483783292507282rm_f_v @ Q ) )
= ( ! [X4: produc8027700073345061287rm_f_v] :
( ( P2 @ X4 )
=> ( Q @ X4 ) ) ) ) ).
% Collect_mono_iff
thf(fact_503_Collect__mono__iff,axiom,
! [P2: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
( ( ord_le3146513528884898305at_nat @ ( collec3392354462482085612at_nat @ P2 ) @ ( collec3392354462482085612at_nat @ Q ) )
= ( ! [X4: product_prod_nat_nat] :
( ( P2 @ X4 )
=> ( Q @ X4 ) ) ) ) ).
% Collect_mono_iff
thf(fact_504_Collect__mono__iff,axiom,
! [P2: product_prod_nat_f > $o,Q: product_prod_nat_f > $o] :
( ( ord_le8280083912720879166_nat_f @ ( collec7702298020461588477_nat_f @ P2 ) @ ( collec7702298020461588477_nat_f @ Q ) )
= ( ! [X4: product_prod_nat_f] :
( ( P2 @ X4 )
=> ( Q @ X4 ) ) ) ) ).
% Collect_mono_iff
thf(fact_505_Collect__mono__iff,axiom,
! [P2: product_prod_f_f > $o,Q: product_prod_f_f > $o] :
( ( ord_le7147000192471188839od_f_f @ ( collec7845863982746767954od_f_f @ P2 ) @ ( collec7845863982746767954od_f_f @ Q ) )
= ( ! [X4: product_prod_f_f] :
( ( P2 @ X4 )
=> ( Q @ X4 ) ) ) ) ).
% Collect_mono_iff
thf(fact_506_Collect__mono__iff,axiom,
! [P2: term_f_v > $o,Q: term_f_v > $o] :
( ( ord_le947722214152454265rm_f_v @ ( collect_term_f_v @ P2 ) @ ( collect_term_f_v @ Q ) )
= ( ! [X4: term_f_v] :
( ( P2 @ X4 )
=> ( Q @ X4 ) ) ) ) ).
% Collect_mono_iff
thf(fact_507_Collect__mono__iff,axiom,
! [P2: product_prod_f_nat > $o,Q: product_prod_f_nat > $o] :
( ( ord_le8976984241387448984_f_nat @ ( collec1417984143148964183_f_nat @ P2 ) @ ( collec1417984143148964183_f_nat @ Q ) )
= ( ! [X4: product_prod_f_nat] :
( ( P2 @ X4 )
=> ( Q @ X4 ) ) ) ) ).
% Collect_mono_iff
thf(fact_508_srstep__monp,axiom,
! [F: set_Pr5245412377734449720_f_nat,G: set_Pr5245412377734449720_f_nat,R2: set_Pr8827868859434726151rm_f_v] :
( ( ord_le8976984241387448984_f_nat @ F @ G )
=> ( ord_le4559761987009501863rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ R2 ) ) @ ( sig_step_f_v @ G @ ( rstep_f_v @ R2 ) ) ) ) ).
% srstep_monp
thf(fact_509_srstepD,axiom,
! [S: term_f_v,T: term_f_v,F: set_Pr5245412377734449720_f_nat,R2: set_Pr8827868859434726151rm_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ ( sig_step_f_v @ F @ ( rstep_f_v @ R2 ) ) )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ ( rstep_f_v @ R2 ) )
& ( ord_le8976984241387448984_f_nat @ ( term_funas_term_f_v @ S ) @ F )
& ( ord_le8976984241387448984_f_nat @ ( term_funas_term_f_v @ T ) @ F ) ) ) ).
% srstepD
thf(fact_510_joinI,axiom,
! [A: product_prod_f_nat,C3: product_prod_f_nat,A4: set_Pr516226060960033221_f_nat,B: product_prod_f_nat] :
( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ A @ C3 ) @ ( transi8903367562120364223_f_nat @ A4 ) )
=> ( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ B @ C3 ) @ ( transi8903367562120364223_f_nat @ A4 ) )
=> ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ A @ B ) @ ( abstra7058733697175803464_f_nat @ A4 ) ) ) ) ).
% joinI
thf(fact_511_joinI,axiom,
! [A: produc8027700073345061287rm_f_v,C3: produc8027700073345061287rm_f_v,A4: set_Pr7897041799657265351rm_f_v,B: produc8027700073345061287rm_f_v] :
( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A @ C3 ) @ ( transi1062743709496199978rm_f_v @ A4 ) )
=> ( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ B @ C3 ) @ ( transi1062743709496199978rm_f_v @ A4 ) )
=> ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A @ B ) @ ( abstra3336617290665100897rm_f_v @ A4 ) ) ) ) ).
% joinI
thf(fact_512_joinI,axiom,
! [A: f,C3: f,A4: set_Product_prod_f_f,B: f] :
( ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ A @ C3 ) @ ( transitive_rtrancl_f @ A4 ) )
=> ( ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ B @ C3 ) @ ( transitive_rtrancl_f @ A4 ) )
=> ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ A @ B ) @ ( abstract_join_f @ A4 ) ) ) ) ).
% joinI
thf(fact_513_joinI,axiom,
! [A: nat,C3: nat,A4: set_Pr1261947904930325089at_nat,B: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ C3 ) @ ( transi2905341329935302413cl_nat @ A4 ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ C3 ) @ ( transi2905341329935302413cl_nat @ A4 ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( abstract_join_nat @ A4 ) ) ) ) ).
% joinI
thf(fact_514_joinI,axiom,
! [A: term_f_v,C3: term_f_v,A4: set_Pr8827868859434726151rm_f_v,B: term_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ C3 ) @ ( transi3028809021613410336rm_f_v @ A4 ) )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B @ C3 ) @ ( transi3028809021613410336rm_f_v @ A4 ) )
=> ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( abstra8605546704478008873rm_f_v @ A4 ) ) ) ) ).
% joinI
thf(fact_515_joinI__left,axiom,
! [A: product_prod_f_nat,B: product_prod_f_nat,A4: set_Pr516226060960033221_f_nat] :
( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ A @ B ) @ ( transi8903367562120364223_f_nat @ A4 ) )
=> ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ A @ B ) @ ( abstra7058733697175803464_f_nat @ A4 ) ) ) ).
% joinI_left
thf(fact_516_joinI__left,axiom,
! [A: produc8027700073345061287rm_f_v,B: produc8027700073345061287rm_f_v,A4: set_Pr7897041799657265351rm_f_v] :
( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A @ B ) @ ( transi1062743709496199978rm_f_v @ A4 ) )
=> ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A @ B ) @ ( abstra3336617290665100897rm_f_v @ A4 ) ) ) ).
% joinI_left
thf(fact_517_joinI__left,axiom,
! [A: f,B: f,A4: set_Product_prod_f_f] :
( ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ A @ B ) @ ( transitive_rtrancl_f @ A4 ) )
=> ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ A @ B ) @ ( abstract_join_f @ A4 ) ) ) ).
% joinI_left
thf(fact_518_joinI__left,axiom,
! [A: nat,B: nat,A4: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( transi2905341329935302413cl_nat @ A4 ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( abstract_join_nat @ A4 ) ) ) ).
% joinI_left
thf(fact_519_joinI__left,axiom,
! [A: term_f_v,B: term_f_v,A4: set_Pr8827868859434726151rm_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( transi3028809021613410336rm_f_v @ A4 ) )
=> ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( abstra8605546704478008873rm_f_v @ A4 ) ) ) ).
% joinI_left
thf(fact_520_joinI__right,axiom,
! [B: product_prod_f_nat,A: product_prod_f_nat,A4: set_Pr516226060960033221_f_nat] :
( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ B @ A ) @ ( transi8903367562120364223_f_nat @ A4 ) )
=> ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ A @ B ) @ ( abstra7058733697175803464_f_nat @ A4 ) ) ) ).
% joinI_right
thf(fact_521_joinI__right,axiom,
! [B: produc8027700073345061287rm_f_v,A: produc8027700073345061287rm_f_v,A4: set_Pr7897041799657265351rm_f_v] :
( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ B @ A ) @ ( transi1062743709496199978rm_f_v @ A4 ) )
=> ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A @ B ) @ ( abstra3336617290665100897rm_f_v @ A4 ) ) ) ).
% joinI_right
thf(fact_522_joinI__right,axiom,
! [B: f,A: f,A4: set_Product_prod_f_f] :
( ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ B @ A ) @ ( transitive_rtrancl_f @ A4 ) )
=> ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ A @ B ) @ ( abstract_join_f @ A4 ) ) ) ).
% joinI_right
thf(fact_523_joinI__right,axiom,
! [B: nat,A: nat,A4: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ A ) @ ( transi2905341329935302413cl_nat @ A4 ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( abstract_join_nat @ A4 ) ) ) ).
% joinI_right
thf(fact_524_joinI__right,axiom,
! [B: term_f_v,A: term_f_v,A4: set_Pr8827868859434726151rm_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B @ A ) @ ( transi3028809021613410336rm_f_v @ A4 ) )
=> ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( abstra8605546704478008873rm_f_v @ A4 ) ) ) ).
% joinI_right
thf(fact_525_rsteps__srstepsI,axiom,
! [R2: set_Pr8827868859434726151rm_f_v,F: set_Pr5245412377734449720_f_nat,S: term_f_v,T: term_f_v] :
( ( ord_le8976984241387448984_f_nat @ ( terms_7890395023448655851_f_v_v @ R2 ) @ F )
=> ( ( ord_le8976984241387448984_f_nat @ ( term_funas_term_f_v @ S ) @ F )
=> ( ( ord_le8976984241387448984_f_nat @ ( term_funas_term_f_v @ T ) @ F )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ ( transi3208867851621559554rm_f_v @ ( rstep_f_v @ R2 ) ) )
=> ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ ( transi3208867851621559554rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ R2 ) ) ) ) ) ) ) ) ).
% rsteps_srstepsI
thf(fact_526_rsteps__eq__srsteps__eqI,axiom,
! [R2: set_Pr8827868859434726151rm_f_v,F: set_Pr5245412377734449720_f_nat,S: term_f_v,T: term_f_v] :
( ( ord_le8976984241387448984_f_nat @ ( terms_7890395023448655851_f_v_v @ R2 ) @ F )
=> ( ( ord_le8976984241387448984_f_nat @ ( term_funas_term_f_v @ S ) @ F )
=> ( ( ord_le8976984241387448984_f_nat @ ( term_funas_term_f_v @ T ) @ F )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ ( transi3028809021613410336rm_f_v @ ( rstep_f_v @ R2 ) ) )
=> ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ ( transi3028809021613410336rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ R2 ) ) ) ) ) ) ) ) ).
% rsteps_eq_srsteps_eqI
thf(fact_527_rstep__srstepI,axiom,
! [R2: set_Pr8827868859434726151rm_f_v,F: set_Pr5245412377734449720_f_nat,S: term_f_v,T: term_f_v] :
( ( ord_le8976984241387448984_f_nat @ ( terms_7890395023448655851_f_v_v @ R2 ) @ F )
=> ( ( ord_le8976984241387448984_f_nat @ ( term_funas_term_f_v @ S ) @ F )
=> ( ( ord_le8976984241387448984_f_nat @ ( term_funas_term_f_v @ T ) @ F )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ ( rstep_f_v @ R2 ) )
=> ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ ( sig_step_f_v @ F @ ( rstep_f_v @ R2 ) ) ) ) ) ) ) ).
% rstep_srstepI
thf(fact_528_joinD,axiom,
! [A: product_prod_f_nat,B: product_prod_f_nat,A4: set_Pr516226060960033221_f_nat] :
( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ A @ B ) @ ( abstra7058733697175803464_f_nat @ A4 ) )
=> ? [C: product_prod_f_nat] :
( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ A @ C ) @ ( transi8903367562120364223_f_nat @ A4 ) )
& ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ B @ C ) @ ( transi8903367562120364223_f_nat @ A4 ) ) ) ) ).
% joinD
thf(fact_529_joinD,axiom,
! [A: produc8027700073345061287rm_f_v,B: produc8027700073345061287rm_f_v,A4: set_Pr7897041799657265351rm_f_v] :
( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A @ B ) @ ( abstra3336617290665100897rm_f_v @ A4 ) )
=> ? [C: produc8027700073345061287rm_f_v] :
( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A @ C ) @ ( transi1062743709496199978rm_f_v @ A4 ) )
& ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ B @ C ) @ ( transi1062743709496199978rm_f_v @ A4 ) ) ) ) ).
% joinD
thf(fact_530_joinD,axiom,
! [A: f,B: f,A4: set_Product_prod_f_f] :
( ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ A @ B ) @ ( abstract_join_f @ A4 ) )
=> ? [C: f] :
( ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ A @ C ) @ ( transitive_rtrancl_f @ A4 ) )
& ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ B @ C ) @ ( transitive_rtrancl_f @ A4 ) ) ) ) ).
% joinD
thf(fact_531_joinD,axiom,
! [A: nat,B: nat,A4: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( abstract_join_nat @ A4 ) )
=> ? [C: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ C ) @ ( transi2905341329935302413cl_nat @ A4 ) )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ C ) @ ( transi2905341329935302413cl_nat @ A4 ) ) ) ) ).
% joinD
thf(fact_532_joinD,axiom,
! [A: term_f_v,B: term_f_v,A4: set_Pr8827868859434726151rm_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( abstra8605546704478008873rm_f_v @ A4 ) )
=> ? [C: term_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ C ) @ ( transi3028809021613410336rm_f_v @ A4 ) )
& ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B @ C ) @ ( transi3028809021613410336rm_f_v @ A4 ) ) ) ) ).
% joinD
thf(fact_533_joinE,axiom,
! [A: product_prod_f_nat,B: product_prod_f_nat,A4: set_Pr516226060960033221_f_nat] :
( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ A @ B ) @ ( abstra7058733697175803464_f_nat @ A4 ) )
=> ~ ! [C: product_prod_f_nat] :
( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ A @ C ) @ ( transi8903367562120364223_f_nat @ A4 ) )
=> ~ ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ B @ C ) @ ( transi8903367562120364223_f_nat @ A4 ) ) ) ) ).
% joinE
thf(fact_534_joinE,axiom,
! [A: produc8027700073345061287rm_f_v,B: produc8027700073345061287rm_f_v,A4: set_Pr7897041799657265351rm_f_v] :
( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A @ B ) @ ( abstra3336617290665100897rm_f_v @ A4 ) )
=> ~ ! [C: produc8027700073345061287rm_f_v] :
( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A @ C ) @ ( transi1062743709496199978rm_f_v @ A4 ) )
=> ~ ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ B @ C ) @ ( transi1062743709496199978rm_f_v @ A4 ) ) ) ) ).
% joinE
thf(fact_535_joinE,axiom,
! [A: f,B: f,A4: set_Product_prod_f_f] :
( ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ A @ B ) @ ( abstract_join_f @ A4 ) )
=> ~ ! [C: f] :
( ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ A @ C ) @ ( transitive_rtrancl_f @ A4 ) )
=> ~ ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ B @ C ) @ ( transitive_rtrancl_f @ A4 ) ) ) ) ).
% joinE
thf(fact_536_joinE,axiom,
! [A: nat,B: nat,A4: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( abstract_join_nat @ A4 ) )
=> ~ ! [C: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ C ) @ ( transi2905341329935302413cl_nat @ A4 ) )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ C ) @ ( transi2905341329935302413cl_nat @ A4 ) ) ) ) ).
% joinE
thf(fact_537_joinE,axiom,
! [A: term_f_v,B: term_f_v,A4: set_Pr8827868859434726151rm_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( abstra8605546704478008873rm_f_v @ A4 ) )
=> ~ ! [C: term_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ C ) @ ( transi3028809021613410336rm_f_v @ A4 ) )
=> ~ ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B @ C ) @ ( transi3028809021613410336rm_f_v @ A4 ) ) ) ) ).
% joinE
thf(fact_538_join__rtrancl__join,axiom,
! [A: product_prod_f_nat,B: product_prod_f_nat,A4: set_Pr516226060960033221_f_nat,C3: product_prod_f_nat] :
( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ A @ B ) @ ( abstra7058733697175803464_f_nat @ A4 ) )
=> ( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ C3 @ B ) @ ( transi8903367562120364223_f_nat @ A4 ) )
=> ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ A @ C3 ) @ ( abstra7058733697175803464_f_nat @ A4 ) ) ) ) ).
% join_rtrancl_join
thf(fact_539_join__rtrancl__join,axiom,
! [A: produc8027700073345061287rm_f_v,B: produc8027700073345061287rm_f_v,A4: set_Pr7897041799657265351rm_f_v,C3: produc8027700073345061287rm_f_v] :
( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A @ B ) @ ( abstra3336617290665100897rm_f_v @ A4 ) )
=> ( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ C3 @ B ) @ ( transi1062743709496199978rm_f_v @ A4 ) )
=> ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A @ C3 ) @ ( abstra3336617290665100897rm_f_v @ A4 ) ) ) ) ).
% join_rtrancl_join
thf(fact_540_join__rtrancl__join,axiom,
! [A: f,B: f,A4: set_Product_prod_f_f,C3: f] :
( ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ A @ B ) @ ( abstract_join_f @ A4 ) )
=> ( ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ C3 @ B ) @ ( transitive_rtrancl_f @ A4 ) )
=> ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ A @ C3 ) @ ( abstract_join_f @ A4 ) ) ) ) ).
% join_rtrancl_join
thf(fact_541_join__rtrancl__join,axiom,
! [A: nat,B: nat,A4: set_Pr1261947904930325089at_nat,C3: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( abstract_join_nat @ A4 ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ C3 @ B ) @ ( transi2905341329935302413cl_nat @ A4 ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ C3 ) @ ( abstract_join_nat @ A4 ) ) ) ) ).
% join_rtrancl_join
thf(fact_542_join__rtrancl__join,axiom,
! [A: term_f_v,B: term_f_v,A4: set_Pr8827868859434726151rm_f_v,C3: term_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( abstra8605546704478008873rm_f_v @ A4 ) )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ C3 @ B ) @ ( transi3028809021613410336rm_f_v @ A4 ) )
=> ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ C3 ) @ ( abstra8605546704478008873rm_f_v @ A4 ) ) ) ) ).
% join_rtrancl_join
thf(fact_543_rtrancl__join__join,axiom,
! [A: product_prod_f_nat,B: product_prod_f_nat,A4: set_Pr516226060960033221_f_nat,C3: product_prod_f_nat] :
( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ A @ B ) @ ( transi8903367562120364223_f_nat @ A4 ) )
=> ( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ B @ C3 ) @ ( abstra7058733697175803464_f_nat @ A4 ) )
=> ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ A @ C3 ) @ ( abstra7058733697175803464_f_nat @ A4 ) ) ) ) ).
% rtrancl_join_join
thf(fact_544_rtrancl__join__join,axiom,
! [A: produc8027700073345061287rm_f_v,B: produc8027700073345061287rm_f_v,A4: set_Pr7897041799657265351rm_f_v,C3: produc8027700073345061287rm_f_v] :
( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A @ B ) @ ( transi1062743709496199978rm_f_v @ A4 ) )
=> ( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ B @ C3 ) @ ( abstra3336617290665100897rm_f_v @ A4 ) )
=> ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A @ C3 ) @ ( abstra3336617290665100897rm_f_v @ A4 ) ) ) ) ).
% rtrancl_join_join
thf(fact_545_rtrancl__join__join,axiom,
! [A: f,B: f,A4: set_Product_prod_f_f,C3: f] :
( ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ A @ B ) @ ( transitive_rtrancl_f @ A4 ) )
=> ( ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ B @ C3 ) @ ( abstract_join_f @ A4 ) )
=> ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ A @ C3 ) @ ( abstract_join_f @ A4 ) ) ) ) ).
% rtrancl_join_join
thf(fact_546_rtrancl__join__join,axiom,
! [A: nat,B: nat,A4: set_Pr1261947904930325089at_nat,C3: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( transi2905341329935302413cl_nat @ A4 ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ C3 ) @ ( abstract_join_nat @ A4 ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ C3 ) @ ( abstract_join_nat @ A4 ) ) ) ) ).
% rtrancl_join_join
thf(fact_547_rtrancl__join__join,axiom,
! [A: term_f_v,B: term_f_v,A4: set_Pr8827868859434726151rm_f_v,C3: term_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( transi3028809021613410336rm_f_v @ A4 ) )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B @ C3 ) @ ( abstra8605546704478008873rm_f_v @ A4 ) )
=> ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ C3 ) @ ( abstra8605546704478008873rm_f_v @ A4 ) ) ) ) ).
% rtrancl_join_join
thf(fact_548_trancl__induct2,axiom,
! [Ax: produc8027700073345061287rm_f_v,Ay: produc8027700073345061287rm_f_v,Bx: produc8027700073345061287rm_f_v,By: produc8027700073345061287rm_f_v,R: set_Pr4964659734097919175rm_f_v,P2: produc8027700073345061287rm_f_v > produc8027700073345061287rm_f_v > $o] :
( ( member1670422607875439376rm_f_v @ ( produc1558868356289833815rm_f_v @ ( produc3874942763110801239rm_f_v @ Ax @ Ay ) @ ( produc3874942763110801239rm_f_v @ Bx @ By ) ) @ ( transi3443125758348952712rm_f_v @ R ) )
=> ( ! [A3: produc8027700073345061287rm_f_v,B3: produc8027700073345061287rm_f_v] :
( ( member1670422607875439376rm_f_v @ ( produc1558868356289833815rm_f_v @ ( produc3874942763110801239rm_f_v @ Ax @ Ay ) @ ( produc3874942763110801239rm_f_v @ A3 @ B3 ) ) @ R )
=> ( P2 @ A3 @ B3 ) )
=> ( ! [A3: produc8027700073345061287rm_f_v,B3: produc8027700073345061287rm_f_v,Aa: produc8027700073345061287rm_f_v,Ba: produc8027700073345061287rm_f_v] :
( ( member1670422607875439376rm_f_v @ ( produc1558868356289833815rm_f_v @ ( produc3874942763110801239rm_f_v @ Ax @ Ay ) @ ( produc3874942763110801239rm_f_v @ A3 @ B3 ) ) @ ( transi3443125758348952712rm_f_v @ R ) )
=> ( ( member1670422607875439376rm_f_v @ ( produc1558868356289833815rm_f_v @ ( produc3874942763110801239rm_f_v @ A3 @ B3 ) @ ( produc3874942763110801239rm_f_v @ Aa @ Ba ) ) @ R )
=> ( ( P2 @ A3 @ B3 )
=> ( P2 @ Aa @ Ba ) ) ) )
=> ( P2 @ Bx @ By ) ) ) ) ).
% trancl_induct2
thf(fact_549_trancl__induct2,axiom,
! [Ax: product_prod_f_nat,Ay: product_prod_f_nat,Bx: product_prod_f_nat,By: product_prod_f_nat,R: set_Pr8434786782240690247_f_nat,P2: product_prod_f_nat > product_prod_f_nat > $o] :
( ( member3697069702615009936_f_nat @ ( produc425345625175287255_f_nat @ ( produc5861087217798509725_f_nat @ Ax @ Ay ) @ ( produc5861087217798509725_f_nat @ Bx @ By ) ) @ ( transi650449548839634566_f_nat @ R ) )
=> ( ! [A3: product_prod_f_nat,B3: product_prod_f_nat] :
( ( member3697069702615009936_f_nat @ ( produc425345625175287255_f_nat @ ( produc5861087217798509725_f_nat @ Ax @ Ay ) @ ( produc5861087217798509725_f_nat @ A3 @ B3 ) ) @ R )
=> ( P2 @ A3 @ B3 ) )
=> ( ! [A3: product_prod_f_nat,B3: product_prod_f_nat,Aa: product_prod_f_nat,Ba: product_prod_f_nat] :
( ( member3697069702615009936_f_nat @ ( produc425345625175287255_f_nat @ ( produc5861087217798509725_f_nat @ Ax @ Ay ) @ ( produc5861087217798509725_f_nat @ A3 @ B3 ) ) @ ( transi650449548839634566_f_nat @ R ) )
=> ( ( member3697069702615009936_f_nat @ ( produc425345625175287255_f_nat @ ( produc5861087217798509725_f_nat @ A3 @ B3 ) @ ( produc5861087217798509725_f_nat @ Aa @ Ba ) ) @ R )
=> ( ( P2 @ A3 @ B3 )
=> ( P2 @ Aa @ Ba ) ) ) )
=> ( P2 @ Bx @ By ) ) ) ) ).
% trancl_induct2
thf(fact_550_trancl__induct2,axiom,
! [Ax: f,Ay: nat,Bx: f,By: nat,R: set_Pr516226060960033221_f_nat,P2: f > nat > $o] :
( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ ( product_Pair_f_nat @ Ax @ Ay ) @ ( product_Pair_f_nat @ Bx @ By ) ) @ ( transi3759436115690491041_f_nat @ R ) )
=> ( ! [A3: f,B3: nat] :
( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ ( product_Pair_f_nat @ Ax @ Ay ) @ ( product_Pair_f_nat @ A3 @ B3 ) ) @ R )
=> ( P2 @ A3 @ B3 ) )
=> ( ! [A3: f,B3: nat,Aa: f,Ba: nat] :
( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ ( product_Pair_f_nat @ Ax @ Ay ) @ ( product_Pair_f_nat @ A3 @ B3 ) ) @ ( transi3759436115690491041_f_nat @ R ) )
=> ( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ ( product_Pair_f_nat @ A3 @ B3 ) @ ( product_Pair_f_nat @ Aa @ Ba ) ) @ R )
=> ( ( P2 @ A3 @ B3 )
=> ( P2 @ Aa @ Ba ) ) ) )
=> ( P2 @ Bx @ By ) ) ) ) ).
% trancl_induct2
thf(fact_551_trancl__induct2,axiom,
! [Ax: term_f_v,Ay: term_f_v,Bx: term_f_v,By: term_f_v,R: set_Pr7897041799657265351rm_f_v,P2: term_f_v > term_f_v > $o] :
( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ ( produc2510790566102465375rm_f_v @ Ax @ Ay ) @ ( produc2510790566102465375rm_f_v @ Bx @ By ) ) @ ( transi7658437189833989320rm_f_v @ R ) )
=> ( ! [A3: term_f_v,B3: term_f_v] :
( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ ( produc2510790566102465375rm_f_v @ Ax @ Ay ) @ ( produc2510790566102465375rm_f_v @ A3 @ B3 ) ) @ R )
=> ( P2 @ A3 @ B3 ) )
=> ( ! [A3: term_f_v,B3: term_f_v,Aa: term_f_v,Ba: term_f_v] :
( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ ( produc2510790566102465375rm_f_v @ Ax @ Ay ) @ ( produc2510790566102465375rm_f_v @ A3 @ B3 ) ) @ ( transi7658437189833989320rm_f_v @ R ) )
=> ( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ ( produc2510790566102465375rm_f_v @ A3 @ B3 ) @ ( produc2510790566102465375rm_f_v @ Aa @ Ba ) ) @ R )
=> ( ( P2 @ A3 @ B3 )
=> ( P2 @ Aa @ Ba ) ) ) )
=> ( P2 @ Bx @ By ) ) ) ) ).
% trancl_induct2
thf(fact_552_trancl__mono__set,axiom,
! [R: set_Pr8827868859434726151rm_f_v,S: set_Pr8827868859434726151rm_f_v] :
( ( ord_le4559761987009501863rm_f_v @ R @ S )
=> ( ord_le4559761987009501863rm_f_v @ ( transi3208867851621559554rm_f_v @ R ) @ ( transi3208867851621559554rm_f_v @ S ) ) ) ).
% trancl_mono_set
thf(fact_553_trancl__mono__set,axiom,
! [R: set_Pr7897041799657265351rm_f_v,S: set_Pr7897041799657265351rm_f_v] :
( ( ord_le7518223742277120103rm_f_v @ R @ S )
=> ( ord_le7518223742277120103rm_f_v @ ( transi7658437189833989320rm_f_v @ R ) @ ( transi7658437189833989320rm_f_v @ S ) ) ) ).
% trancl_mono_set
thf(fact_554_trancl__mono__set,axiom,
! [R: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ R @ S )
=> ( ord_le3146513528884898305at_nat @ ( transi6264000038957366511cl_nat @ R ) @ ( transi6264000038957366511cl_nat @ S ) ) ) ).
% trancl_mono_set
thf(fact_555_trancl__mono__set,axiom,
! [R: set_Product_prod_f_f,S: set_Product_prod_f_f] :
( ( ord_le7147000192471188839od_f_f @ R @ S )
=> ( ord_le7147000192471188839od_f_f @ ( transitive_trancl_f @ R ) @ ( transitive_trancl_f @ S ) ) ) ).
% trancl_mono_set
thf(fact_556_sig__step__rsteps__dist,axiom,
! [R2: set_Pr8827868859434726151rm_f_v,F: set_Pr5245412377734449720_f_nat] :
( ( ord_le8976984241387448984_f_nat @ ( terms_7890395023448655851_f_v_v @ R2 ) @ F )
=> ( ( sig_step_f_v @ F @ ( transi3208867851621559554rm_f_v @ ( rstep_f_v @ R2 ) ) )
= ( transi3208867851621559554rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ R2 ) ) ) ) ) ).
% sig_step_rsteps_dist
thf(fact_557_join__sym,axiom,
! [S: product_prod_f_nat,T: product_prod_f_nat,A4: set_Pr516226060960033221_f_nat] :
( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ S @ T ) @ ( abstra7058733697175803464_f_nat @ A4 ) )
=> ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ T @ S ) @ ( abstra7058733697175803464_f_nat @ A4 ) ) ) ).
% join_sym
thf(fact_558_join__sym,axiom,
! [S: produc8027700073345061287rm_f_v,T: produc8027700073345061287rm_f_v,A4: set_Pr7897041799657265351rm_f_v] :
( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ S @ T ) @ ( abstra3336617290665100897rm_f_v @ A4 ) )
=> ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ T @ S ) @ ( abstra3336617290665100897rm_f_v @ A4 ) ) ) ).
% join_sym
thf(fact_559_join__sym,axiom,
! [S: term_f_v,T: term_f_v,A4: set_Pr8827868859434726151rm_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ ( abstra8605546704478008873rm_f_v @ A4 ) )
=> ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ T @ S ) @ ( abstra8605546704478008873rm_f_v @ A4 ) ) ) ).
% join_sym
thf(fact_560_join__mono,axiom,
! [R: set_Pr8827868859434726151rm_f_v,S: set_Pr8827868859434726151rm_f_v] :
( ( ord_le4559761987009501863rm_f_v @ R @ S )
=> ( ord_le4559761987009501863rm_f_v @ ( abstra8605546704478008873rm_f_v @ R ) @ ( abstra8605546704478008873rm_f_v @ S ) ) ) ).
% join_mono
thf(fact_561_join__mono,axiom,
! [R: set_Pr7897041799657265351rm_f_v,S: set_Pr7897041799657265351rm_f_v] :
( ( ord_le7518223742277120103rm_f_v @ R @ S )
=> ( ord_le7518223742277120103rm_f_v @ ( abstra3336617290665100897rm_f_v @ R ) @ ( abstra3336617290665100897rm_f_v @ S ) ) ) ).
% join_mono
thf(fact_562_join__mono,axiom,
! [R: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ R @ S )
=> ( ord_le3146513528884898305at_nat @ ( abstract_join_nat @ R ) @ ( abstract_join_nat @ S ) ) ) ).
% join_mono
thf(fact_563_join__mono,axiom,
! [R: set_Product_prod_f_f,S: set_Product_prod_f_f] :
( ( ord_le7147000192471188839od_f_f @ R @ S )
=> ( ord_le7147000192471188839od_f_f @ ( abstract_join_f @ R ) @ ( abstract_join_f @ S ) ) ) ).
% join_mono
thf(fact_564_rstep__trancl__sig__step__r,axiom,
! [S: term_f_v,T: term_f_v,R2: set_Pr8827868859434726151rm_f_v,F: set_Pr5245412377734449720_f_nat,V: v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ ( transi3208867851621559554rm_f_v @ ( rstep_f_v @ R2 ) ) )
=> ( ( ord_le8976984241387448984_f_nat @ ( terms_7890395023448655851_f_v_v @ R2 ) @ F )
=> ( ( ord_le8976984241387448984_f_nat @ ( term_funas_term_f_v @ S ) @ F )
=> ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ ( terms_3805575757503158827ig_f_v @ F @ V @ T ) ) @ ( transi3208867851621559554rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ R2 ) ) ) ) ) ) ) ).
% rstep_trancl_sig_step_r
thf(fact_565_rstep__trancl__sig__step__l,axiom,
! [S: term_f_v,T: term_f_v,R2: set_Pr8827868859434726151rm_f_v,F: set_Pr5245412377734449720_f_nat,V: v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ ( transi3208867851621559554rm_f_v @ ( rstep_f_v @ R2 ) ) )
=> ( ( ord_le8976984241387448984_f_nat @ ( terms_7890395023448655851_f_v_v @ R2 ) @ F )
=> ( ( ord_le8976984241387448984_f_nat @ ( term_funas_term_f_v @ T ) @ F )
=> ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ ( terms_3805575757503158827ig_f_v @ F @ V @ S ) @ T ) @ ( transi3208867851621559554rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ R2 ) ) ) ) ) ) ) ).
% rstep_trancl_sig_step_l
thf(fact_566_rsteps__eq__relcomp__srsteps__eq__relcompI,axiom,
! [R2: set_Pr8827868859434726151rm_f_v,F: set_Pr5245412377734449720_f_nat,S5: set_Pr8827868859434726151rm_f_v,S: term_f_v,T: term_f_v] :
( ( ord_le8976984241387448984_f_nat @ ( terms_7890395023448655851_f_v_v @ R2 ) @ F )
=> ( ( ord_le8976984241387448984_f_nat @ ( terms_7890395023448655851_f_v_v @ S5 ) @ F )
=> ( ( ord_le8976984241387448984_f_nat @ ( term_funas_term_f_v @ S ) @ F )
=> ( ( ord_le8976984241387448984_f_nat @ ( term_funas_term_f_v @ T ) @ F )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ ( relcom2934258169684288428rm_f_v @ ( transi3028809021613410336rm_f_v @ ( rstep_f_v @ R2 ) ) @ ( transi3028809021613410336rm_f_v @ ( rstep_f_v @ S5 ) ) ) )
=> ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ ( relcom2934258169684288428rm_f_v @ ( transi3028809021613410336rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ R2 ) ) ) @ ( transi3028809021613410336rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ S5 ) ) ) ) ) ) ) ) ) ) ).
% rsteps_eq_relcomp_srsteps_eq_relcompI
thf(fact_567_meetI,axiom,
! [A: product_prod_f_nat,B: product_prod_f_nat,A4: set_Pr516226060960033221_f_nat,C3: product_prod_f_nat] :
( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ A @ B ) @ ( transi8903367562120364223_f_nat @ A4 ) )
=> ( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ A @ C3 ) @ ( transi8903367562120364223_f_nat @ A4 ) )
=> ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ B @ C3 ) @ ( abstra2172256884198503301_f_nat @ A4 ) ) ) ) ).
% meetI
thf(fact_568_meetI,axiom,
! [A: produc8027700073345061287rm_f_v,B: produc8027700073345061287rm_f_v,A4: set_Pr7897041799657265351rm_f_v,C3: produc8027700073345061287rm_f_v] :
( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A @ B ) @ ( transi1062743709496199978rm_f_v @ A4 ) )
=> ( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A @ C3 ) @ ( transi1062743709496199978rm_f_v @ A4 ) )
=> ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ B @ C3 ) @ ( abstra6743905778397186276rm_f_v @ A4 ) ) ) ) ).
% meetI
thf(fact_569_meetI,axiom,
! [A: f,B: f,A4: set_Product_prod_f_f,C3: f] :
( ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ A @ B ) @ ( transitive_rtrancl_f @ A4 ) )
=> ( ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ A @ C3 ) @ ( transitive_rtrancl_f @ A4 ) )
=> ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ B @ C3 ) @ ( abstract_meet_f @ A4 ) ) ) ) ).
% meetI
thf(fact_570_meetI,axiom,
! [A: nat,B: nat,A4: set_Pr1261947904930325089at_nat,C3: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( transi2905341329935302413cl_nat @ A4 ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ C3 ) @ ( transi2905341329935302413cl_nat @ A4 ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ C3 ) @ ( abstract_meet_nat @ A4 ) ) ) ) ).
% meetI
thf(fact_571_meetI,axiom,
! [A: term_f_v,B: term_f_v,A4: set_Pr8827868859434726151rm_f_v,C3: term_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( transi3028809021613410336rm_f_v @ A4 ) )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ C3 ) @ ( transi3028809021613410336rm_f_v @ A4 ) )
=> ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B @ C3 ) @ ( abstra5602527437878040038rm_f_v @ A4 ) ) ) ) ).
% meetI
thf(fact_572_srsteps__with__root__step__sig__mono,axiom,
! [F: set_Pr5245412377734449720_f_nat,G: set_Pr5245412377734449720_f_nat,R2: set_Pr8827868859434726151rm_f_v] :
( ( ord_le8976984241387448984_f_nat @ F @ G )
=> ( ord_le4559761987009501863rm_f_v @ ( srstep3130564645414158319ep_f_v @ F @ R2 ) @ ( srstep3130564645414158319ep_f_v @ G @ R2 ) ) ) ).
% srsteps_with_root_step_sig_mono
thf(fact_573_restrict__rtrancl,axiom,
! [R: set_Pr7897041799657265351rm_f_v,S2: set_Pr8827868859434726151rm_f_v] : ( ord_le7518223742277120103rm_f_v @ ( transi1062743709496199978rm_f_v @ ( abstra801143999231238575rm_f_v @ R @ S2 ) ) @ ( transi1062743709496199978rm_f_v @ R ) ) ).
% restrict_rtrancl
thf(fact_574_restrict__rtrancl,axiom,
! [R: set_Pr1261947904930325089at_nat,S2: set_nat] : ( ord_le3146513528884898305at_nat @ ( transi2905341329935302413cl_nat @ ( abstra1219450721576609224ct_nat @ R @ S2 ) ) @ ( transi2905341329935302413cl_nat @ R ) ) ).
% restrict_rtrancl
thf(fact_575_restrict__rtrancl,axiom,
! [R: set_Product_prod_f_f,S2: set_f] : ( ord_le7147000192471188839od_f_f @ ( transitive_rtrancl_f @ ( abstract_restrict_f @ R @ S2 ) ) @ ( transitive_rtrancl_f @ R ) ) ).
% restrict_rtrancl
thf(fact_576_restrict__rtrancl,axiom,
! [R: set_Pr8827868859434726151rm_f_v,S2: set_term_f_v] : ( ord_le4559761987009501863rm_f_v @ ( transi3028809021613410336rm_f_v @ ( abstra1948659249064760667rm_f_v @ R @ S2 ) ) @ ( transi3028809021613410336rm_f_v @ R ) ) ).
% restrict_rtrancl
thf(fact_577_rstep__term__to__sig__r,axiom,
! [S: term_f_v,T: term_f_v,R2: set_Pr8827868859434726151rm_f_v,F: set_Pr5245412377734449720_f_nat,V: v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ ( rstep_f_v @ R2 ) )
=> ( ( ord_le8976984241387448984_f_nat @ ( terms_7890395023448655851_f_v_v @ R2 ) @ F )
=> ( ( ord_le8976984241387448984_f_nat @ ( term_funas_term_f_v @ S ) @ F )
=> ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ ( terms_3805575757503158827ig_f_v @ F @ V @ T ) ) @ ( rstep_f_v @ R2 ) ) ) ) ) ).
% rstep_term_to_sig_r
thf(fact_578_rstep__term__to__sig__l,axiom,
! [S: term_f_v,T: term_f_v,R2: set_Pr8827868859434726151rm_f_v,F: set_Pr5245412377734449720_f_nat,V: v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ ( rstep_f_v @ R2 ) )
=> ( ( ord_le8976984241387448984_f_nat @ ( terms_7890395023448655851_f_v_v @ R2 ) @ F )
=> ( ( ord_le8976984241387448984_f_nat @ ( term_funas_term_f_v @ T ) @ F )
=> ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ ( terms_3805575757503158827ig_f_v @ F @ V @ S ) @ T ) @ ( rstep_f_v @ R2 ) ) ) ) ) ).
% rstep_term_to_sig_l
thf(fact_579_trancl__map,axiom,
! [R: set_Pr8827868859434726151rm_f_v,F2: term_f_v > term_f_v,S: set_Pr8827868859434726151rm_f_v,X3: term_f_v,Y: term_f_v] :
( ! [X: term_f_v,Y3: term_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ Y3 ) @ R )
=> ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ ( F2 @ X ) @ ( F2 @ Y3 ) ) @ S ) )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Y ) @ ( transi3208867851621559554rm_f_v @ R ) )
=> ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ ( F2 @ X3 ) @ ( F2 @ Y ) ) @ ( transi3208867851621559554rm_f_v @ S ) ) ) ) ).
% trancl_map
thf(fact_580_trancl__map,axiom,
! [R: set_Product_prod_f_f,F2: f > f,S: set_Product_prod_f_f,X3: f,Y: f] :
( ! [X: f,Y3: f] :
( ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ X @ Y3 ) @ R )
=> ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ ( F2 @ X ) @ ( F2 @ Y3 ) ) @ S ) )
=> ( ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ X3 @ Y ) @ ( transitive_trancl_f @ R ) )
=> ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ ( F2 @ X3 ) @ ( F2 @ Y ) ) @ ( transitive_trancl_f @ S ) ) ) ) ).
% trancl_map
thf(fact_581_trancl__map,axiom,
! [R: set_Product_prod_f_f,F2: f > nat,S: set_Pr1261947904930325089at_nat,X3: f,Y: f] :
( ! [X: f,Y3: f] :
( ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ X @ Y3 ) @ R )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F2 @ X ) @ ( F2 @ Y3 ) ) @ S ) )
=> ( ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ X3 @ Y ) @ ( transitive_trancl_f @ R ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F2 @ X3 ) @ ( F2 @ Y ) ) @ ( transi6264000038957366511cl_nat @ S ) ) ) ) ).
% trancl_map
thf(fact_582_trancl__map,axiom,
! [R: set_Pr1261947904930325089at_nat,F2: nat > f,S: set_Product_prod_f_f,X3: nat,Y: nat] :
( ! [X: nat,Y3: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y3 ) @ R )
=> ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ ( F2 @ X ) @ ( F2 @ Y3 ) ) @ S ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y ) @ ( transi6264000038957366511cl_nat @ R ) )
=> ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ ( F2 @ X3 ) @ ( F2 @ Y ) ) @ ( transitive_trancl_f @ S ) ) ) ) ).
% trancl_map
thf(fact_583_trancl__map,axiom,
! [R: set_Pr1261947904930325089at_nat,F2: nat > nat,S: set_Pr1261947904930325089at_nat,X3: nat,Y: nat] :
( ! [X: nat,Y3: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y3 ) @ R )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F2 @ X ) @ ( F2 @ Y3 ) ) @ S ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y ) @ ( transi6264000038957366511cl_nat @ R ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F2 @ X3 ) @ ( F2 @ Y ) ) @ ( transi6264000038957366511cl_nat @ S ) ) ) ) ).
% trancl_map
thf(fact_584_trancl__map,axiom,
! [R: set_Pr516226060960033221_f_nat,F2: product_prod_f_nat > f,S: set_Product_prod_f_f,X3: product_prod_f_nat,Y: product_prod_f_nat] :
( ! [X: product_prod_f_nat,Y3: product_prod_f_nat] :
( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ X @ Y3 ) @ R )
=> ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ ( F2 @ X ) @ ( F2 @ Y3 ) ) @ S ) )
=> ( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ X3 @ Y ) @ ( transi3759436115690491041_f_nat @ R ) )
=> ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ ( F2 @ X3 ) @ ( F2 @ Y ) ) @ ( transitive_trancl_f @ S ) ) ) ) ).
% trancl_map
thf(fact_585_trancl__map,axiom,
! [R: set_Pr516226060960033221_f_nat,F2: product_prod_f_nat > nat,S: set_Pr1261947904930325089at_nat,X3: product_prod_f_nat,Y: product_prod_f_nat] :
( ! [X: product_prod_f_nat,Y3: product_prod_f_nat] :
( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ X @ Y3 ) @ R )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F2 @ X ) @ ( F2 @ Y3 ) ) @ S ) )
=> ( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ X3 @ Y ) @ ( transi3759436115690491041_f_nat @ R ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F2 @ X3 ) @ ( F2 @ Y ) ) @ ( transi6264000038957366511cl_nat @ S ) ) ) ) ).
% trancl_map
thf(fact_586_trancl__map,axiom,
! [R: set_Pr8827868859434726151rm_f_v,F2: term_f_v > f,S: set_Product_prod_f_f,X3: term_f_v,Y: term_f_v] :
( ! [X: term_f_v,Y3: term_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ Y3 ) @ R )
=> ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ ( F2 @ X ) @ ( F2 @ Y3 ) ) @ S ) )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Y ) @ ( transi3208867851621559554rm_f_v @ R ) )
=> ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ ( F2 @ X3 ) @ ( F2 @ Y ) ) @ ( transitive_trancl_f @ S ) ) ) ) ).
% trancl_map
thf(fact_587_trancl__map,axiom,
! [R: set_Pr8827868859434726151rm_f_v,F2: term_f_v > nat,S: set_Pr1261947904930325089at_nat,X3: term_f_v,Y: term_f_v] :
( ! [X: term_f_v,Y3: term_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ Y3 ) @ R )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F2 @ X ) @ ( F2 @ Y3 ) ) @ S ) )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Y ) @ ( transi3208867851621559554rm_f_v @ R ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F2 @ X3 ) @ ( F2 @ Y ) ) @ ( transi6264000038957366511cl_nat @ S ) ) ) ) ).
% trancl_map
thf(fact_588_trancl__map,axiom,
! [R: set_Product_prod_f_f,F2: f > product_prod_f_nat,S: set_Pr516226060960033221_f_nat,X3: f,Y: f] :
( ! [X: f,Y3: f] :
( ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ X @ Y3 ) @ R )
=> ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ ( F2 @ X ) @ ( F2 @ Y3 ) ) @ S ) )
=> ( ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ X3 @ Y ) @ ( transitive_trancl_f @ R ) )
=> ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ ( F2 @ X3 ) @ ( F2 @ Y ) ) @ ( transi3759436115690491041_f_nat @ S ) ) ) ) ).
% trancl_map
thf(fact_589_rtrancl__idemp__self__comp,axiom,
! [R3: set_Pr7897041799657265351rm_f_v] :
( ( relcom7124267630914856742rm_f_v @ ( transi1062743709496199978rm_f_v @ R3 ) @ ( transi1062743709496199978rm_f_v @ R3 ) )
= ( transi1062743709496199978rm_f_v @ R3 ) ) ).
% rtrancl_idemp_self_comp
thf(fact_590_rtrancl__idemp__self__comp,axiom,
! [R3: set_Product_prod_f_f] :
( ( relcomp_f_f_f @ ( transitive_rtrancl_f @ R3 ) @ ( transitive_rtrancl_f @ R3 ) )
= ( transitive_rtrancl_f @ R3 ) ) ).
% rtrancl_idemp_self_comp
thf(fact_591_rtrancl__idemp__self__comp,axiom,
! [R3: set_Pr1261947904930325089at_nat] :
( ( relcomp_nat_nat_nat @ ( transi2905341329935302413cl_nat @ R3 ) @ ( transi2905341329935302413cl_nat @ R3 ) )
= ( transi2905341329935302413cl_nat @ R3 ) ) ).
% rtrancl_idemp_self_comp
thf(fact_592_rtrancl__idemp__self__comp,axiom,
! [R3: set_Pr8827868859434726151rm_f_v] :
( ( relcom2934258169684288428rm_f_v @ ( transi3028809021613410336rm_f_v @ R3 ) @ ( transi3028809021613410336rm_f_v @ R3 ) )
= ( transi3028809021613410336rm_f_v @ R3 ) ) ).
% rtrancl_idemp_self_comp
thf(fact_593_O__assoc,axiom,
! [R3: set_Pr8827868859434726151rm_f_v,S2: set_Pr8827868859434726151rm_f_v,T4: set_Pr8827868859434726151rm_f_v] :
( ( relcom2934258169684288428rm_f_v @ ( relcom2934258169684288428rm_f_v @ R3 @ S2 ) @ T4 )
= ( relcom2934258169684288428rm_f_v @ R3 @ ( relcom2934258169684288428rm_f_v @ S2 @ T4 ) ) ) ).
% O_assoc
thf(fact_594_O__assoc,axiom,
! [R3: set_Pr5245412377734449720_f_nat,S2: set_Pr1261947904930325089at_nat,T4: set_Pr1261947904930325089at_nat] :
( ( relcomp_f_nat_nat @ ( relcomp_f_nat_nat @ R3 @ S2 ) @ T4 )
= ( relcomp_f_nat_nat @ R3 @ ( relcomp_nat_nat_nat @ S2 @ T4 ) ) ) ).
% O_assoc
thf(fact_595_O__assoc,axiom,
! [R3: set_Product_prod_f_f,S2: set_Pr5245412377734449720_f_nat,T4: set_Pr1261947904930325089at_nat] :
( ( relcomp_f_nat_nat @ ( relcomp_f_f_nat @ R3 @ S2 ) @ T4 )
= ( relcomp_f_f_nat @ R3 @ ( relcomp_f_nat_nat @ S2 @ T4 ) ) ) ).
% O_assoc
thf(fact_596_O__assoc,axiom,
! [R3: set_Pr5245412377734449720_f_nat,S2: set_Pr4548512049067879902_nat_f,T4: set_Pr5245412377734449720_f_nat] :
( ( relcomp_f_f_nat @ ( relcomp_f_nat_f @ R3 @ S2 ) @ T4 )
= ( relcomp_f_nat_nat @ R3 @ ( relcomp_nat_f_nat @ S2 @ T4 ) ) ) ).
% O_assoc
thf(fact_597_O__assoc,axiom,
! [R3: set_Product_prod_f_f,S2: set_Product_prod_f_f,T4: set_Pr5245412377734449720_f_nat] :
( ( relcomp_f_f_nat @ ( relcomp_f_f_f @ R3 @ S2 ) @ T4 )
= ( relcomp_f_f_nat @ R3 @ ( relcomp_f_f_nat @ S2 @ T4 ) ) ) ).
% O_assoc
thf(fact_598_relcompEpair,axiom,
! [A: produc8027700073345061287rm_f_v,C3: produc8027700073345061287rm_f_v,R: set_Pr7897041799657265351rm_f_v,S: set_Pr7897041799657265351rm_f_v] :
( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A @ C3 ) @ ( relcom7124267630914856742rm_f_v @ R @ S ) )
=> ~ ! [B3: produc8027700073345061287rm_f_v] :
( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A @ B3 ) @ R )
=> ~ ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ B3 @ C3 ) @ S ) ) ) ).
% relcompEpair
thf(fact_599_relcompEpair,axiom,
! [A: f,C3: nat,R: set_Product_prod_f_f,S: set_Pr5245412377734449720_f_nat] :
( ( member2678038196313615385_f_nat @ ( product_Pair_f_nat @ A @ C3 ) @ ( relcomp_f_f_nat @ R @ S ) )
=> ~ ! [B3: f] :
( ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ A @ B3 ) @ R )
=> ~ ( member2678038196313615385_f_nat @ ( product_Pair_f_nat @ B3 @ C3 ) @ S ) ) ) ).
% relcompEpair
thf(fact_600_relcompEpair,axiom,
! [A: f,C3: nat,R: set_Pr5245412377734449720_f_nat,S: set_Pr1261947904930325089at_nat] :
( ( member2678038196313615385_f_nat @ ( product_Pair_f_nat @ A @ C3 ) @ ( relcomp_f_nat_nat @ R @ S ) )
=> ~ ! [B3: nat] :
( ( member2678038196313615385_f_nat @ ( product_Pair_f_nat @ A @ B3 ) @ R )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B3 @ C3 ) @ S ) ) ) ).
% relcompEpair
thf(fact_601_relcompEpair,axiom,
! [A: product_prod_f_nat,C3: product_prod_f_nat,R: set_Pr516226060960033221_f_nat,S: set_Pr516226060960033221_f_nat] :
( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ A @ C3 ) @ ( relcom3086153608568277129_f_nat @ R @ S ) )
=> ~ ! [B3: product_prod_f_nat] :
( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ A @ B3 ) @ R )
=> ~ ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ B3 @ C3 ) @ S ) ) ) ).
% relcompEpair
thf(fact_602_relcompEpair,axiom,
! [A: term_f_v,C3: term_f_v,R: set_Pr8827868859434726151rm_f_v,S: set_Pr8827868859434726151rm_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ C3 ) @ ( relcom2934258169684288428rm_f_v @ R @ S ) )
=> ~ ! [B3: term_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B3 ) @ R )
=> ~ ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B3 @ C3 ) @ S ) ) ) ).
% relcompEpair
thf(fact_603_relcompE,axiom,
! [Xz: produc3406425829894461415rm_f_v,R: set_Pr7897041799657265351rm_f_v,S: set_Pr7897041799657265351rm_f_v] :
( ( member7300054593328134928rm_f_v @ Xz @ ( relcom7124267630914856742rm_f_v @ R @ S ) )
=> ~ ! [X: produc8027700073345061287rm_f_v,Y3: produc8027700073345061287rm_f_v,Z: produc8027700073345061287rm_f_v] :
( ( Xz
= ( produc3874942763110801239rm_f_v @ X @ Z ) )
=> ( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ X @ Y3 ) @ R )
=> ~ ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ Y3 @ Z ) @ S ) ) ) ) ).
% relcompE
thf(fact_604_relcompE,axiom,
! [Xz: product_prod_f_nat,R: set_Product_prod_f_f,S: set_Pr5245412377734449720_f_nat] :
( ( member2678038196313615385_f_nat @ Xz @ ( relcomp_f_f_nat @ R @ S ) )
=> ~ ! [X: f,Y3: f,Z: nat] :
( ( Xz
= ( product_Pair_f_nat @ X @ Z ) )
=> ( ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ X @ Y3 ) @ R )
=> ~ ( member2678038196313615385_f_nat @ ( product_Pair_f_nat @ Y3 @ Z ) @ S ) ) ) ) ).
% relcompE
thf(fact_605_relcompE,axiom,
! [Xz: product_prod_f_nat,R: set_Pr5245412377734449720_f_nat,S: set_Pr1261947904930325089at_nat] :
( ( member2678038196313615385_f_nat @ Xz @ ( relcomp_f_nat_nat @ R @ S ) )
=> ~ ! [X: f,Y3: nat,Z: nat] :
( ( Xz
= ( product_Pair_f_nat @ X @ Z ) )
=> ( ( member2678038196313615385_f_nat @ ( product_Pair_f_nat @ X @ Y3 ) @ R )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ Z ) @ S ) ) ) ) ).
% relcompE
thf(fact_606_relcompE,axiom,
! [Xz: produc2676906022994773093_f_nat,R: set_Pr516226060960033221_f_nat,S: set_Pr516226060960033221_f_nat] :
( ( member7030746907725424910_f_nat @ Xz @ ( relcom3086153608568277129_f_nat @ R @ S ) )
=> ~ ! [X: product_prod_f_nat,Y3: product_prod_f_nat,Z: product_prod_f_nat] :
( ( Xz
= ( produc5861087217798509725_f_nat @ X @ Z ) )
=> ( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ X @ Y3 ) @ R )
=> ~ ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ Y3 @ Z ) @ S ) ) ) ) ).
% relcompE
thf(fact_607_relcompE,axiom,
! [Xz: produc8027700073345061287rm_f_v,R: set_Pr8827868859434726151rm_f_v,S: set_Pr8827868859434726151rm_f_v] :
( ( member4316649861536941904rm_f_v @ Xz @ ( relcom2934258169684288428rm_f_v @ R @ S ) )
=> ~ ! [X: term_f_v,Y3: term_f_v,Z: term_f_v] :
( ( Xz
= ( produc2510790566102465375rm_f_v @ X @ Z ) )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ Y3 ) @ R )
=> ~ ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y3 @ Z ) @ S ) ) ) ) ).
% relcompE
thf(fact_608_relcomp_OrelcompI,axiom,
! [A: f,B: f,R: set_Product_prod_f_f,C3: nat,S: set_Pr5245412377734449720_f_nat] :
( ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ A @ B ) @ R )
=> ( ( member2678038196313615385_f_nat @ ( product_Pair_f_nat @ B @ C3 ) @ S )
=> ( member2678038196313615385_f_nat @ ( product_Pair_f_nat @ A @ C3 ) @ ( relcomp_f_f_nat @ R @ S ) ) ) ) ).
% relcomp.relcompI
thf(fact_609_relcomp_OrelcompI,axiom,
! [A: produc8027700073345061287rm_f_v,B: produc8027700073345061287rm_f_v,R: set_Pr7897041799657265351rm_f_v,C3: produc8027700073345061287rm_f_v,S: set_Pr7897041799657265351rm_f_v] :
( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A @ B ) @ R )
=> ( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ B @ C3 ) @ S )
=> ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A @ C3 ) @ ( relcom7124267630914856742rm_f_v @ R @ S ) ) ) ) ).
% relcomp.relcompI
thf(fact_610_relcomp_OrelcompI,axiom,
! [A: f,B: nat,R: set_Pr5245412377734449720_f_nat,C3: nat,S: set_Pr1261947904930325089at_nat] :
( ( member2678038196313615385_f_nat @ ( product_Pair_f_nat @ A @ B ) @ R )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ C3 ) @ S )
=> ( member2678038196313615385_f_nat @ ( product_Pair_f_nat @ A @ C3 ) @ ( relcomp_f_nat_nat @ R @ S ) ) ) ) ).
% relcomp.relcompI
thf(fact_611_relcomp_OrelcompI,axiom,
! [A: product_prod_f_nat,B: product_prod_f_nat,R: set_Pr516226060960033221_f_nat,C3: product_prod_f_nat,S: set_Pr516226060960033221_f_nat] :
( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ A @ B ) @ R )
=> ( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ B @ C3 ) @ S )
=> ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ A @ C3 ) @ ( relcom3086153608568277129_f_nat @ R @ S ) ) ) ) ).
% relcomp.relcompI
thf(fact_612_relcomp_OrelcompI,axiom,
! [A: term_f_v,B: term_f_v,R: set_Pr8827868859434726151rm_f_v,C3: term_f_v,S: set_Pr8827868859434726151rm_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ R )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B @ C3 ) @ S )
=> ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ C3 ) @ ( relcom2934258169684288428rm_f_v @ R @ S ) ) ) ) ).
% relcomp.relcompI
thf(fact_613_relcomp_Osimps,axiom,
! [A1: produc8027700073345061287rm_f_v,A22: produc8027700073345061287rm_f_v,R: set_Pr7897041799657265351rm_f_v,S: set_Pr7897041799657265351rm_f_v] :
( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A1 @ A22 ) @ ( relcom7124267630914856742rm_f_v @ R @ S ) )
= ( ? [A5: produc8027700073345061287rm_f_v,B4: produc8027700073345061287rm_f_v,C4: produc8027700073345061287rm_f_v] :
( ( A1 = A5 )
& ( A22 = C4 )
& ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A5 @ B4 ) @ R )
& ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ B4 @ C4 ) @ S ) ) ) ) ).
% relcomp.simps
thf(fact_614_relcomp_Osimps,axiom,
! [A1: f,A22: nat,R: set_Product_prod_f_f,S: set_Pr5245412377734449720_f_nat] :
( ( member2678038196313615385_f_nat @ ( product_Pair_f_nat @ A1 @ A22 ) @ ( relcomp_f_f_nat @ R @ S ) )
= ( ? [A5: f,B4: f,C4: nat] :
( ( A1 = A5 )
& ( A22 = C4 )
& ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ A5 @ B4 ) @ R )
& ( member2678038196313615385_f_nat @ ( product_Pair_f_nat @ B4 @ C4 ) @ S ) ) ) ) ).
% relcomp.simps
thf(fact_615_relcomp_Osimps,axiom,
! [A1: f,A22: nat,R: set_Pr5245412377734449720_f_nat,S: set_Pr1261947904930325089at_nat] :
( ( member2678038196313615385_f_nat @ ( product_Pair_f_nat @ A1 @ A22 ) @ ( relcomp_f_nat_nat @ R @ S ) )
= ( ? [A5: f,B4: nat,C4: nat] :
( ( A1 = A5 )
& ( A22 = C4 )
& ( member2678038196313615385_f_nat @ ( product_Pair_f_nat @ A5 @ B4 ) @ R )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B4 @ C4 ) @ S ) ) ) ) ).
% relcomp.simps
thf(fact_616_relcomp_Osimps,axiom,
! [A1: product_prod_f_nat,A22: product_prod_f_nat,R: set_Pr516226060960033221_f_nat,S: set_Pr516226060960033221_f_nat] :
( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ A1 @ A22 ) @ ( relcom3086153608568277129_f_nat @ R @ S ) )
= ( ? [A5: product_prod_f_nat,B4: product_prod_f_nat,C4: product_prod_f_nat] :
( ( A1 = A5 )
& ( A22 = C4 )
& ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ A5 @ B4 ) @ R )
& ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ B4 @ C4 ) @ S ) ) ) ) ).
% relcomp.simps
thf(fact_617_relcomp_Osimps,axiom,
! [A1: term_f_v,A22: term_f_v,R: set_Pr8827868859434726151rm_f_v,S: set_Pr8827868859434726151rm_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A1 @ A22 ) @ ( relcom2934258169684288428rm_f_v @ R @ S ) )
= ( ? [A5: term_f_v,B4: term_f_v,C4: term_f_v] :
( ( A1 = A5 )
& ( A22 = C4 )
& ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A5 @ B4 ) @ R )
& ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B4 @ C4 ) @ S ) ) ) ) ).
% relcomp.simps
thf(fact_618_relcomp_Ocases,axiom,
! [A1: produc8027700073345061287rm_f_v,A22: produc8027700073345061287rm_f_v,R: set_Pr7897041799657265351rm_f_v,S: set_Pr7897041799657265351rm_f_v] :
( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A1 @ A22 ) @ ( relcom7124267630914856742rm_f_v @ R @ S ) )
=> ~ ! [B3: produc8027700073345061287rm_f_v] :
( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A1 @ B3 ) @ R )
=> ~ ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ B3 @ A22 ) @ S ) ) ) ).
% relcomp.cases
thf(fact_619_relcomp_Ocases,axiom,
! [A1: f,A22: nat,R: set_Product_prod_f_f,S: set_Pr5245412377734449720_f_nat] :
( ( member2678038196313615385_f_nat @ ( product_Pair_f_nat @ A1 @ A22 ) @ ( relcomp_f_f_nat @ R @ S ) )
=> ~ ! [B3: f] :
( ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ A1 @ B3 ) @ R )
=> ~ ( member2678038196313615385_f_nat @ ( product_Pair_f_nat @ B3 @ A22 ) @ S ) ) ) ).
% relcomp.cases
thf(fact_620_relcomp_Ocases,axiom,
! [A1: f,A22: nat,R: set_Pr5245412377734449720_f_nat,S: set_Pr1261947904930325089at_nat] :
( ( member2678038196313615385_f_nat @ ( product_Pair_f_nat @ A1 @ A22 ) @ ( relcomp_f_nat_nat @ R @ S ) )
=> ~ ! [B3: nat] :
( ( member2678038196313615385_f_nat @ ( product_Pair_f_nat @ A1 @ B3 ) @ R )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B3 @ A22 ) @ S ) ) ) ).
% relcomp.cases
thf(fact_621_relcomp_Ocases,axiom,
! [A1: product_prod_f_nat,A22: product_prod_f_nat,R: set_Pr516226060960033221_f_nat,S: set_Pr516226060960033221_f_nat] :
( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ A1 @ A22 ) @ ( relcom3086153608568277129_f_nat @ R @ S ) )
=> ~ ! [B3: product_prod_f_nat] :
( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ A1 @ B3 ) @ R )
=> ~ ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ B3 @ A22 ) @ S ) ) ) ).
% relcomp.cases
thf(fact_622_relcomp_Ocases,axiom,
! [A1: term_f_v,A22: term_f_v,R: set_Pr8827868859434726151rm_f_v,S: set_Pr8827868859434726151rm_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A1 @ A22 ) @ ( relcom2934258169684288428rm_f_v @ R @ S ) )
=> ~ ! [B3: term_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A1 @ B3 ) @ R )
=> ~ ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B3 @ A22 ) @ S ) ) ) ).
% relcomp.cases
thf(fact_623_relcomp3__I,axiom,
! [T: nat,U: f,A4: set_Pr4548512049067879902_nat_f,S: f,B5: set_Pr5245412377734449720_f_nat,V: nat] :
( ( member8962352073626239679_nat_f @ ( product_Pair_nat_f @ T @ U ) @ A4 )
=> ( ( member2678038196313615385_f_nat @ ( product_Pair_f_nat @ S @ T ) @ B5 )
=> ( ( member2678038196313615385_f_nat @ ( product_Pair_f_nat @ U @ V ) @ B5 )
=> ( member2678038196313615385_f_nat @ ( product_Pair_f_nat @ S @ V ) @ ( relcomp_f_nat_nat @ B5 @ ( relcomp_nat_f_nat @ A4 @ B5 ) ) ) ) ) ) ).
% relcomp3_I
thf(fact_624_relcomp3__I,axiom,
! [T: produc8027700073345061287rm_f_v,U: produc8027700073345061287rm_f_v,A4: set_Pr7897041799657265351rm_f_v,S: produc8027700073345061287rm_f_v,B5: set_Pr7897041799657265351rm_f_v,V: produc8027700073345061287rm_f_v] :
( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ T @ U ) @ A4 )
=> ( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ S @ T ) @ B5 )
=> ( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ U @ V ) @ B5 )
=> ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ S @ V ) @ ( relcom7124267630914856742rm_f_v @ B5 @ ( relcom7124267630914856742rm_f_v @ A4 @ B5 ) ) ) ) ) ) ).
% relcomp3_I
thf(fact_625_relcomp3__I,axiom,
! [T: f,U: nat,A4: set_Pr5245412377734449720_f_nat,S: nat,B5: set_Pr4548512049067879902_nat_f,V: f] :
( ( member2678038196313615385_f_nat @ ( product_Pair_f_nat @ T @ U ) @ A4 )
=> ( ( member8962352073626239679_nat_f @ ( product_Pair_nat_f @ S @ T ) @ B5 )
=> ( ( member8962352073626239679_nat_f @ ( product_Pair_nat_f @ U @ V ) @ B5 )
=> ( member8962352073626239679_nat_f @ ( product_Pair_nat_f @ S @ V ) @ ( relcomp_nat_f_f @ B5 @ ( relcomp_f_nat_f @ A4 @ B5 ) ) ) ) ) ) ).
% relcomp3_I
thf(fact_626_relcomp3__I,axiom,
! [T: product_prod_f_nat,U: product_prod_f_nat,A4: set_Pr516226060960033221_f_nat,S: product_prod_f_nat,B5: set_Pr516226060960033221_f_nat,V: product_prod_f_nat] :
( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ T @ U ) @ A4 )
=> ( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ S @ T ) @ B5 )
=> ( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ U @ V ) @ B5 )
=> ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ S @ V ) @ ( relcom3086153608568277129_f_nat @ B5 @ ( relcom3086153608568277129_f_nat @ A4 @ B5 ) ) ) ) ) ) ).
% relcomp3_I
thf(fact_627_relcomp3__I,axiom,
! [T: term_f_v,U: term_f_v,A4: set_Pr8827868859434726151rm_f_v,S: term_f_v,B5: set_Pr8827868859434726151rm_f_v,V: term_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ T @ U ) @ A4 )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ B5 )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ U @ V ) @ B5 )
=> ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ V ) @ ( relcom2934258169684288428rm_f_v @ B5 @ ( relcom2934258169684288428rm_f_v @ A4 @ B5 ) ) ) ) ) ) ).
% relcomp3_I
thf(fact_628_r__comp__rtrancl__eq,axiom,
! [R: set_Pr7897041799657265351rm_f_v] :
( ( relcom7124267630914856742rm_f_v @ R @ ( transi1062743709496199978rm_f_v @ R ) )
= ( relcom7124267630914856742rm_f_v @ ( transi1062743709496199978rm_f_v @ R ) @ R ) ) ).
% r_comp_rtrancl_eq
thf(fact_629_r__comp__rtrancl__eq,axiom,
! [R: set_Product_prod_f_f] :
( ( relcomp_f_f_f @ R @ ( transitive_rtrancl_f @ R ) )
= ( relcomp_f_f_f @ ( transitive_rtrancl_f @ R ) @ R ) ) ).
% r_comp_rtrancl_eq
thf(fact_630_r__comp__rtrancl__eq,axiom,
! [R: set_Pr1261947904930325089at_nat] :
( ( relcomp_nat_nat_nat @ R @ ( transi2905341329935302413cl_nat @ R ) )
= ( relcomp_nat_nat_nat @ ( transi2905341329935302413cl_nat @ R ) @ R ) ) ).
% r_comp_rtrancl_eq
thf(fact_631_r__comp__rtrancl__eq,axiom,
! [R: set_Pr8827868859434726151rm_f_v] :
( ( relcom2934258169684288428rm_f_v @ R @ ( transi3028809021613410336rm_f_v @ R ) )
= ( relcom2934258169684288428rm_f_v @ ( transi3028809021613410336rm_f_v @ R ) @ R ) ) ).
% r_comp_rtrancl_eq
thf(fact_632_rtrancl__O__shift,axiom,
! [S2: set_Pr8827868859434726151rm_f_v,R3: set_Pr8827868859434726151rm_f_v] :
( ( relcom2934258169684288428rm_f_v @ ( transi3028809021613410336rm_f_v @ ( relcom2934258169684288428rm_f_v @ S2 @ R3 ) ) @ S2 )
= ( relcom2934258169684288428rm_f_v @ S2 @ ( transi3028809021613410336rm_f_v @ ( relcom2934258169684288428rm_f_v @ R3 @ S2 ) ) ) ) ).
% rtrancl_O_shift
thf(fact_633_rtrancl__O__shift,axiom,
! [S2: set_Product_prod_f_f,R3: set_Product_prod_f_f] :
( ( relcomp_f_f_f @ ( transitive_rtrancl_f @ ( relcomp_f_f_f @ S2 @ R3 ) ) @ S2 )
= ( relcomp_f_f_f @ S2 @ ( transitive_rtrancl_f @ ( relcomp_f_f_f @ R3 @ S2 ) ) ) ) ).
% rtrancl_O_shift
thf(fact_634_rtrancl__O__shift,axiom,
! [S2: set_Pr5245412377734449720_f_nat,R3: set_Pr4548512049067879902_nat_f] :
( ( relcomp_f_f_nat @ ( transitive_rtrancl_f @ ( relcomp_f_nat_f @ S2 @ R3 ) ) @ S2 )
= ( relcomp_f_nat_nat @ S2 @ ( transi2905341329935302413cl_nat @ ( relcomp_nat_f_nat @ R3 @ S2 ) ) ) ) ).
% rtrancl_O_shift
thf(fact_635_rtrancl__O__shift,axiom,
! [S2: set_Pr4548512049067879902_nat_f,R3: set_Pr5245412377734449720_f_nat] :
( ( relcomp_nat_nat_f @ ( transi2905341329935302413cl_nat @ ( relcomp_nat_f_nat @ S2 @ R3 ) ) @ S2 )
= ( relcomp_nat_f_f @ S2 @ ( transitive_rtrancl_f @ ( relcomp_f_nat_f @ R3 @ S2 ) ) ) ) ).
% rtrancl_O_shift
thf(fact_636_rtrancl__O__shift,axiom,
! [S2: set_Pr1261947904930325089at_nat,R3: set_Pr1261947904930325089at_nat] :
( ( relcomp_nat_nat_nat @ ( transi2905341329935302413cl_nat @ ( relcomp_nat_nat_nat @ S2 @ R3 ) ) @ S2 )
= ( relcomp_nat_nat_nat @ S2 @ ( transi2905341329935302413cl_nat @ ( relcomp_nat_nat_nat @ R3 @ S2 ) ) ) ) ).
% rtrancl_O_shift
thf(fact_637_rtrancl__O__shift,axiom,
! [S2: set_Pr5880368083585996171_f_v_f,R3: set_Pr3748340825032421835rm_f_v] :
( ( relcom6773260102509987450_f_v_f @ ( transi3028809021613410336rm_f_v @ ( relcom889343255288228738rm_f_v @ S2 @ R3 ) ) @ S2 )
= ( relcomp_term_f_v_f_f @ S2 @ ( transitive_rtrancl_f @ ( relcomp_f_term_f_v_f @ R3 @ S2 ) ) ) ) ).
% rtrancl_O_shift
thf(fact_638_rtrancl__O__shift,axiom,
! [S2: set_Pr4690941997361092980_v_nat,R3: set_Pr4272985574496756084rm_f_v] :
( ( relcom6549496499929616537_v_nat @ ( transi3028809021613410336rm_f_v @ ( relcom2131275577122648089rm_f_v @ S2 @ R3 ) ) @ S2 )
= ( relcom5993431013869582214at_nat @ S2 @ ( transi2905341329935302413cl_nat @ ( relcom8119003054590121734_v_nat @ R3 @ S2 ) ) ) ) ).
% rtrancl_O_shift
thf(fact_639_rtrancl__O__shift,axiom,
! [S2: set_Pr3748340825032421835rm_f_v,R3: set_Pr5880368083585996171_f_v_f] :
( ( relcomp_f_f_term_f_v @ ( transitive_rtrancl_f @ ( relcomp_f_term_f_v_f @ S2 @ R3 ) ) @ S2 )
= ( relcom487563801099439242rm_f_v @ S2 @ ( transi3028809021613410336rm_f_v @ ( relcom889343255288228738rm_f_v @ R3 @ S2 ) ) ) ) ).
% rtrancl_O_shift
thf(fact_640_rtrancl__O__shift,axiom,
! [S2: set_Pr4272985574496756084rm_f_v,R3: set_Pr4690941997361092980_v_nat] :
( ( relcom3700782131783153286rm_f_v @ ( transi2905341329935302413cl_nat @ ( relcom8119003054590121734_v_nat @ S2 @ R3 ) ) @ S2 )
= ( relcom2056057532694990745rm_f_v @ S2 @ ( transi3028809021613410336rm_f_v @ ( relcom2131275577122648089rm_f_v @ R3 @ S2 ) ) ) ) ).
% rtrancl_O_shift
thf(fact_641_rtrancl__O__shift,axiom,
! [S2: set_Pr479911000340079907_f_v_f,R3: set_Pr1266492636989324139rm_f_v] :
( ( relcom7746504459861906178_f_v_f @ ( transi1062743709496199978rm_f_v @ ( relcom4668453842596808010rm_f_v @ S2 @ R3 ) ) @ S2 )
= ( relcom3151732783321806630_v_f_f @ S2 @ ( transitive_rtrancl_f @ ( relcom9088990414662444766_f_v_f @ R3 @ S2 ) ) ) ) ).
% rtrancl_O_shift
thf(fact_642_relcomp__mono,axiom,
! [R4: set_Pr8827868859434726151rm_f_v,R: set_Pr8827868859434726151rm_f_v,S6: set_Pr8827868859434726151rm_f_v,S: set_Pr8827868859434726151rm_f_v] :
( ( ord_le4559761987009501863rm_f_v @ R4 @ R )
=> ( ( ord_le4559761987009501863rm_f_v @ S6 @ S )
=> ( ord_le4559761987009501863rm_f_v @ ( relcom2934258169684288428rm_f_v @ R4 @ S6 ) @ ( relcom2934258169684288428rm_f_v @ R @ S ) ) ) ) ).
% relcomp_mono
thf(fact_643_relcomp__mono,axiom,
! [R4: set_Pr5245412377734449720_f_nat,R: set_Pr5245412377734449720_f_nat,S6: set_Pr4548512049067879902_nat_f,S: set_Pr4548512049067879902_nat_f] :
( ( ord_le8976984241387448984_f_nat @ R4 @ R )
=> ( ( ord_le8280083912720879166_nat_f @ S6 @ S )
=> ( ord_le7147000192471188839od_f_f @ ( relcomp_f_nat_f @ R4 @ S6 ) @ ( relcomp_f_nat_f @ R @ S ) ) ) ) ).
% relcomp_mono
thf(fact_644_relcomp__mono,axiom,
! [R4: set_Pr1261947904930325089at_nat,R: set_Pr1261947904930325089at_nat,S6: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ R4 @ R )
=> ( ( ord_le3146513528884898305at_nat @ S6 @ S )
=> ( ord_le3146513528884898305at_nat @ ( relcomp_nat_nat_nat @ R4 @ S6 ) @ ( relcomp_nat_nat_nat @ R @ S ) ) ) ) ).
% relcomp_mono
thf(fact_645_relcomp__mono,axiom,
! [R4: set_Pr1261947904930325089at_nat,R: set_Pr1261947904930325089at_nat,S6: set_Pr4548512049067879902_nat_f,S: set_Pr4548512049067879902_nat_f] :
( ( ord_le3146513528884898305at_nat @ R4 @ R )
=> ( ( ord_le8280083912720879166_nat_f @ S6 @ S )
=> ( ord_le8280083912720879166_nat_f @ ( relcomp_nat_nat_f @ R4 @ S6 ) @ ( relcomp_nat_nat_f @ R @ S ) ) ) ) ).
% relcomp_mono
thf(fact_646_relcomp__mono,axiom,
! [R4: set_Pr4548512049067879902_nat_f,R: set_Pr4548512049067879902_nat_f,S6: set_Pr5245412377734449720_f_nat,S: set_Pr5245412377734449720_f_nat] :
( ( ord_le8280083912720879166_nat_f @ R4 @ R )
=> ( ( ord_le8976984241387448984_f_nat @ S6 @ S )
=> ( ord_le3146513528884898305at_nat @ ( relcomp_nat_f_nat @ R4 @ S6 ) @ ( relcomp_nat_f_nat @ R @ S ) ) ) ) ).
% relcomp_mono
thf(fact_647_relcomp__mono,axiom,
! [R4: set_Pr4548512049067879902_nat_f,R: set_Pr4548512049067879902_nat_f,S6: set_Product_prod_f_f,S: set_Product_prod_f_f] :
( ( ord_le8280083912720879166_nat_f @ R4 @ R )
=> ( ( ord_le7147000192471188839od_f_f @ S6 @ S )
=> ( ord_le8280083912720879166_nat_f @ ( relcomp_nat_f_f @ R4 @ S6 ) @ ( relcomp_nat_f_f @ R @ S ) ) ) ) ).
% relcomp_mono
thf(fact_648_relcomp__mono,axiom,
! [R4: set_Product_prod_f_f,R: set_Product_prod_f_f,S6: set_Product_prod_f_f,S: set_Product_prod_f_f] :
( ( ord_le7147000192471188839od_f_f @ R4 @ R )
=> ( ( ord_le7147000192471188839od_f_f @ S6 @ S )
=> ( ord_le7147000192471188839od_f_f @ ( relcomp_f_f_f @ R4 @ S6 ) @ ( relcomp_f_f_f @ R @ S ) ) ) ) ).
% relcomp_mono
thf(fact_649_relcomp__mono,axiom,
! [R4: set_Product_prod_f_f,R: set_Product_prod_f_f,S6: set_Pr5245412377734449720_f_nat,S: set_Pr5245412377734449720_f_nat] :
( ( ord_le7147000192471188839od_f_f @ R4 @ R )
=> ( ( ord_le8976984241387448984_f_nat @ S6 @ S )
=> ( ord_le8976984241387448984_f_nat @ ( relcomp_f_f_nat @ R4 @ S6 ) @ ( relcomp_f_f_nat @ R @ S ) ) ) ) ).
% relcomp_mono
thf(fact_650_relcomp__mono,axiom,
! [R4: set_Pr5245412377734449720_f_nat,R: set_Pr5245412377734449720_f_nat,S6: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
( ( ord_le8976984241387448984_f_nat @ R4 @ R )
=> ( ( ord_le3146513528884898305at_nat @ S6 @ S )
=> ( ord_le8976984241387448984_f_nat @ ( relcomp_f_nat_nat @ R4 @ S6 ) @ ( relcomp_f_nat_nat @ R @ S ) ) ) ) ).
% relcomp_mono
thf(fact_651_O__mono1,axiom,
! [R3: set_Product_prod_f_f,R5: set_Product_prod_f_f,S2: set_Product_prod_f_f] :
( ( ord_le7147000192471188839od_f_f @ R3 @ R5 )
=> ( ord_le7147000192471188839od_f_f @ ( relcomp_f_f_f @ S2 @ R3 ) @ ( relcomp_f_f_f @ S2 @ R5 ) ) ) ).
% O_mono1
thf(fact_652_O__mono1,axiom,
! [R3: set_Pr5245412377734449720_f_nat,R5: set_Pr5245412377734449720_f_nat,S2: set_Product_prod_f_f] :
( ( ord_le8976984241387448984_f_nat @ R3 @ R5 )
=> ( ord_le8976984241387448984_f_nat @ ( relcomp_f_f_nat @ S2 @ R3 ) @ ( relcomp_f_f_nat @ S2 @ R5 ) ) ) ).
% O_mono1
thf(fact_653_O__mono2,axiom,
! [R3: set_Pr5245412377734449720_f_nat,R5: set_Pr5245412377734449720_f_nat,T4: set_Pr1261947904930325089at_nat] :
( ( ord_le8976984241387448984_f_nat @ R3 @ R5 )
=> ( ord_le8976984241387448984_f_nat @ ( relcomp_f_nat_nat @ R3 @ T4 ) @ ( relcomp_f_nat_nat @ R5 @ T4 ) ) ) ).
% O_mono2
thf(fact_654_kleene__induct,axiom,
! [A4: set_Pr8827868859434726151rm_f_v,X5: set_Pr8827868859434726151rm_f_v,B5: set_Pr8827868859434726151rm_f_v,C2: set_Pr8827868859434726151rm_f_v] :
( ( ord_le4559761987009501863rm_f_v @ A4 @ X5 )
=> ( ( ord_le4559761987009501863rm_f_v @ ( relcom2934258169684288428rm_f_v @ B5 @ X5 ) @ X5 )
=> ( ( ord_le4559761987009501863rm_f_v @ ( relcom2934258169684288428rm_f_v @ X5 @ C2 ) @ X5 )
=> ( ord_le4559761987009501863rm_f_v @ ( relcom2934258169684288428rm_f_v @ ( transi3028809021613410336rm_f_v @ B5 ) @ ( relcom2934258169684288428rm_f_v @ A4 @ ( transi3028809021613410336rm_f_v @ C2 ) ) ) @ X5 ) ) ) ) ).
% kleene_induct
thf(fact_655_kleene__induct,axiom,
! [A4: set_Pr5245412377734449720_f_nat,X5: set_Pr5245412377734449720_f_nat,B5: set_Product_prod_f_f,C2: set_Pr1261947904930325089at_nat] :
( ( ord_le8976984241387448984_f_nat @ A4 @ X5 )
=> ( ( ord_le8976984241387448984_f_nat @ ( relcomp_f_f_nat @ B5 @ X5 ) @ X5 )
=> ( ( ord_le8976984241387448984_f_nat @ ( relcomp_f_nat_nat @ X5 @ C2 ) @ X5 )
=> ( ord_le8976984241387448984_f_nat @ ( relcomp_f_f_nat @ ( transitive_rtrancl_f @ B5 ) @ ( relcomp_f_nat_nat @ A4 @ ( transi2905341329935302413cl_nat @ C2 ) ) ) @ X5 ) ) ) ) ).
% kleene_induct
thf(fact_656_kleene__trancl__induct,axiom,
! [A4: set_Pr5245412377734449720_f_nat,X5: set_Pr5245412377734449720_f_nat,B5: set_Product_prod_f_f,C2: set_Pr1261947904930325089at_nat] :
( ( ord_le8976984241387448984_f_nat @ A4 @ X5 )
=> ( ( ord_le8976984241387448984_f_nat @ ( relcomp_f_f_nat @ B5 @ X5 ) @ X5 )
=> ( ( ord_le8976984241387448984_f_nat @ ( relcomp_f_nat_nat @ X5 @ C2 ) @ X5 )
=> ( ord_le8976984241387448984_f_nat @ ( relcomp_f_f_nat @ ( transitive_trancl_f @ B5 ) @ ( relcomp_f_nat_nat @ A4 @ ( transi6264000038957366511cl_nat @ C2 ) ) ) @ X5 ) ) ) ) ).
% kleene_trancl_induct
thf(fact_657_steps__map,axiom,
! [P2: term_f_v > $o,Q: set_Pr8827868859434726151rm_f_v > $o,F2: term_f_v > term_f_v,G2: set_Pr8827868859434726151rm_f_v > set_Pr8827868859434726151rm_f_v,T: term_f_v,R3: set_Pr8827868859434726151rm_f_v,S2: set_Pr8827868859434726151rm_f_v,U: term_f_v] :
( ! [T2: term_f_v,U2: term_f_v,R6: set_Pr8827868859434726151rm_f_v] :
( ( P2 @ T2 )
=> ( ( Q @ R6 )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ T2 @ U2 ) @ R6 )
=> ( ( P2 @ U2 )
& ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ ( F2 @ T2 ) @ ( F2 @ U2 ) ) @ ( G2 @ R6 ) ) ) ) ) )
=> ( ( P2 @ T )
=> ( ( Q @ R3 )
=> ( ( Q @ S2 )
=> ( ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ T @ U ) @ ( transi3028809021613410336rm_f_v @ R3 ) )
=> ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ ( F2 @ T ) @ ( F2 @ U ) ) @ ( transi3028809021613410336rm_f_v @ ( G2 @ R3 ) ) ) )
& ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ T @ U ) @ ( relcom2934258169684288428rm_f_v @ ( transi3028809021613410336rm_f_v @ R3 ) @ ( relcom2934258169684288428rm_f_v @ S2 @ ( transi3028809021613410336rm_f_v @ R3 ) ) ) )
=> ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ ( F2 @ T ) @ ( F2 @ U ) ) @ ( relcom2934258169684288428rm_f_v @ ( transi3028809021613410336rm_f_v @ ( G2 @ R3 ) ) @ ( relcom2934258169684288428rm_f_v @ ( G2 @ S2 ) @ ( transi3028809021613410336rm_f_v @ ( G2 @ R3 ) ) ) ) ) ) ) ) ) ) ) ).
% steps_map
thf(fact_658_compat__tr__compat,axiom,
! [NS: set_Product_prod_f_f,S2: set_Pr5245412377734449720_f_nat] :
( ( ord_le8976984241387448984_f_nat @ ( relcomp_f_f_nat @ NS @ S2 ) @ S2 )
=> ( ord_le8976984241387448984_f_nat @ ( relcomp_f_f_nat @ ( transitive_rtrancl_f @ NS ) @ S2 ) @ S2 ) ) ).
% compat_tr_compat
thf(fact_659_rtrancl__O__push,axiom,
! [S2: set_Pr8827868859434726151rm_f_v,R3: set_Pr8827868859434726151rm_f_v] :
( ( ord_le4559761987009501863rm_f_v @ ( relcom2934258169684288428rm_f_v @ S2 @ R3 ) @ ( relcom2934258169684288428rm_f_v @ R3 @ ( transi3028809021613410336rm_f_v @ S2 ) ) )
=> ( ord_le4559761987009501863rm_f_v @ ( relcom2934258169684288428rm_f_v @ ( transi3028809021613410336rm_f_v @ S2 ) @ R3 ) @ ( relcom2934258169684288428rm_f_v @ R3 @ ( transi3028809021613410336rm_f_v @ S2 ) ) ) ) ).
% rtrancl_O_push
thf(fact_660_trancl__unfold__left,axiom,
( transi3208867851621559554rm_f_v
= ( ^ [R7: set_Pr8827868859434726151rm_f_v] : ( relcom2934258169684288428rm_f_v @ R7 @ ( transi3028809021613410336rm_f_v @ R7 ) ) ) ) ).
% trancl_unfold_left
thf(fact_661_rtrancl__comp__trancl__conv,axiom,
! [R: set_Pr8827868859434726151rm_f_v] :
( ( relcom2934258169684288428rm_f_v @ ( transi3028809021613410336rm_f_v @ R ) @ R )
= ( transi3208867851621559554rm_f_v @ R ) ) ).
% rtrancl_comp_trancl_conv
thf(fact_662_non__strict__into__strict,axiom,
! [NS: set_Pr8827868859434726151rm_f_v,S2: set_Pr8827868859434726151rm_f_v,S: term_f_v,T: term_f_v] :
( ( ord_le4559761987009501863rm_f_v @ ( relcom2934258169684288428rm_f_v @ NS @ S2 ) @ S2 )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ ( relcom2934258169684288428rm_f_v @ ( transi3028809021613410336rm_f_v @ NS ) @ S2 ) )
=> ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ S2 ) ) ) ).
% non_strict_into_strict
thf(fact_663_non__strict__into__strict,axiom,
! [NS: set_Product_prod_f_f,S2: set_Pr5245412377734449720_f_nat,S: f,T: nat] :
( ( ord_le8976984241387448984_f_nat @ ( relcomp_f_f_nat @ NS @ S2 ) @ S2 )
=> ( ( member2678038196313615385_f_nat @ ( product_Pair_f_nat @ S @ T ) @ ( relcomp_f_f_nat @ ( transitive_rtrancl_f @ NS ) @ S2 ) )
=> ( member2678038196313615385_f_nat @ ( product_Pair_f_nat @ S @ T ) @ S2 ) ) ) ).
% non_strict_into_strict
thf(fact_664_srsteps__with__root__step__def,axiom,
( srstep3130564645414158319ep_f_v
= ( ^ [F3: set_Pr5245412377734449720_f_nat,R8: set_Pr8827868859434726151rm_f_v] : ( relcom2934258169684288428rm_f_v @ ( transi3028809021613410336rm_f_v @ ( sig_step_f_v @ F3 @ ( rstep_f_v @ R8 ) ) ) @ ( relcom2934258169684288428rm_f_v @ ( sig_step_f_v @ F3 @ ( rrstep_f_v @ R8 ) ) @ ( transi3028809021613410336rm_f_v @ ( sig_step_f_v @ F3 @ ( rstep_f_v @ R8 ) ) ) ) ) ) ) ).
% srsteps_with_root_step_def
thf(fact_665_meetD,axiom,
! [B: term_f_v,C3: term_f_v,A4: set_Pr8827868859434726151rm_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B @ C3 ) @ ( abstra5602527437878040038rm_f_v @ A4 ) )
=> ? [A3: term_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A3 @ B ) @ ( transi3028809021613410336rm_f_v @ A4 ) )
& ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A3 @ C3 ) @ ( transi3028809021613410336rm_f_v @ A4 ) ) ) ) ).
% meetD
thf(fact_666_meetE,axiom,
! [B: term_f_v,C3: term_f_v,A4: set_Pr8827868859434726151rm_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B @ C3 ) @ ( abstra5602527437878040038rm_f_v @ A4 ) )
=> ~ ! [A3: term_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A3 @ B ) @ ( transi3028809021613410336rm_f_v @ A4 ) )
=> ~ ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A3 @ C3 ) @ ( transi3028809021613410336rm_f_v @ A4 ) ) ) ) ).
% meetE
thf(fact_667_term__to__sig__id,axiom,
! [T: term_f_v,F: set_Pr5245412377734449720_f_nat,V: v] :
( ( ord_le8976984241387448984_f_nat @ ( term_funas_term_f_v @ T ) @ F )
=> ( ( terms_3805575757503158827ig_f_v @ F @ V @ T )
= T ) ) ).
% term_to_sig_id
thf(fact_668_fuans__term__term__to__sig,axiom,
! [F: set_Pr5245412377734449720_f_nat,V: v,T: term_f_v] : ( ord_le8976984241387448984_f_nat @ ( term_funas_term_f_v @ ( terms_3805575757503158827ig_f_v @ F @ V @ T ) ) @ F ) ).
% fuans_term_term_to_sig
thf(fact_669_srsteps__eq__subst__relcomp__closed,axiom,
! [S: term_f_v,T: term_f_v,F: set_Pr5245412377734449720_f_nat,R2: set_Pr8827868859434726151rm_f_v,S5: set_Pr8827868859434726151rm_f_v,Sigma: v > term_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ ( relcom2934258169684288428rm_f_v @ ( transi3028809021613410336rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ R2 ) ) ) @ ( transi3028809021613410336rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ S5 ) ) ) ) )
=> ( ! [X: v] : ( ord_le8976984241387448984_f_nat @ ( term_funas_term_f_v @ ( Sigma @ X ) ) @ F )
=> ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ ( subst_7901567856578221233_f_v_v @ S @ Sigma ) @ ( subst_7901567856578221233_f_v_v @ T @ Sigma ) ) @ ( relcom2934258169684288428rm_f_v @ ( transi3028809021613410336rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ R2 ) ) ) @ ( transi3028809021613410336rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ S5 ) ) ) ) ) ) ) ).
% srsteps_eq_subst_relcomp_closed
thf(fact_670_srsteps__subst__closed,axiom,
! [S: term_f_v,T: term_f_v,F: set_Pr5245412377734449720_f_nat,R2: set_Pr8827868859434726151rm_f_v,Sigma: v > term_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ ( transi3208867851621559554rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ R2 ) ) ) )
=> ( ! [X: v] : ( ord_le8976984241387448984_f_nat @ ( term_funas_term_f_v @ ( Sigma @ X ) ) @ F )
=> ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ ( subst_7901567856578221233_f_v_v @ S @ Sigma ) @ ( subst_7901567856578221233_f_v_v @ T @ Sigma ) ) @ ( transi3208867851621559554rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ R2 ) ) ) ) ) ) ).
% srsteps_subst_closed
thf(fact_671_srsteps__eq__subst__closed,axiom,
! [S: term_f_v,T: term_f_v,F: set_Pr5245412377734449720_f_nat,R2: set_Pr8827868859434726151rm_f_v,Sigma: v > term_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ ( transi3028809021613410336rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ R2 ) ) ) )
=> ( ! [X: v] : ( ord_le8976984241387448984_f_nat @ ( term_funas_term_f_v @ ( Sigma @ X ) ) @ F )
=> ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ ( subst_7901567856578221233_f_v_v @ S @ Sigma ) @ ( subst_7901567856578221233_f_v_v @ T @ Sigma ) ) @ ( transi3028809021613410336rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ R2 ) ) ) ) ) ) ).
% srsteps_eq_subst_closed
thf(fact_672_WCR__onI,axiom,
! [A4: set_Pr8827868859434726151rm_f_v,R: set_Pr7897041799657265351rm_f_v] :
( ! [A3: produc8027700073345061287rm_f_v,B3: produc8027700073345061287rm_f_v,C: produc8027700073345061287rm_f_v] :
( ( member4316649861536941904rm_f_v @ A3 @ A4 )
=> ( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A3 @ B3 ) @ R )
=> ( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A3 @ C ) @ R )
=> ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ B3 @ C ) @ ( abstra3336617290665100897rm_f_v @ R ) ) ) ) )
=> ( abstra8545463541035707836rm_f_v @ R @ A4 ) ) ).
% WCR_onI
thf(fact_673_WCR__onI,axiom,
! [A4: set_term_f_v,R: set_Pr8827868859434726151rm_f_v] :
( ! [A3: term_f_v,B3: term_f_v,C: term_f_v] :
( ( member_term_f_v @ A3 @ A4 )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A3 @ B3 ) @ R )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A3 @ C ) @ R )
=> ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B3 @ C ) @ ( abstra8605546704478008873rm_f_v @ R ) ) ) ) )
=> ( abstra312943320106832654rm_f_v @ R @ A4 ) ) ).
% WCR_onI
thf(fact_674_comp__rtrancl__trancl,axiom,
! [R3: set_Pr8827868859434726151rm_f_v,S2: set_Pr8827868859434726151rm_f_v,S: term_f_v,T: term_f_v] :
( ( ord_le4559761987009501863rm_f_v @ ( relcom2934258169684288428rm_f_v @ R3 @ S2 ) @ S2 )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ ( relcom2934258169684288428rm_f_v @ ( transi3028809021613410336rm_f_v @ ( sup_su1994854688125285083rm_f_v @ R3 @ S2 ) ) @ S2 ) )
=> ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ ( transi3208867851621559554rm_f_v @ S2 ) ) ) ) ).
% comp_rtrancl_trancl
thf(fact_675_UnCI,axiom,
! [C3: produc8027700073345061287rm_f_v,B5: set_Pr8827868859434726151rm_f_v,A4: set_Pr8827868859434726151rm_f_v] :
( ( ~ ( member4316649861536941904rm_f_v @ C3 @ B5 )
=> ( member4316649861536941904rm_f_v @ C3 @ A4 ) )
=> ( member4316649861536941904rm_f_v @ C3 @ ( sup_su1994854688125285083rm_f_v @ A4 @ B5 ) ) ) ).
% UnCI
thf(fact_676_Un__iff,axiom,
! [C3: produc8027700073345061287rm_f_v,A4: set_Pr8827868859434726151rm_f_v,B5: set_Pr8827868859434726151rm_f_v] :
( ( member4316649861536941904rm_f_v @ C3 @ ( sup_su1994854688125285083rm_f_v @ A4 @ B5 ) )
= ( ( member4316649861536941904rm_f_v @ C3 @ A4 )
| ( member4316649861536941904rm_f_v @ C3 @ B5 ) ) ) ).
% Un_iff
thf(fact_677_Un__subset__iff,axiom,
! [A4: set_Pr5245412377734449720_f_nat,B5: set_Pr5245412377734449720_f_nat,C2: set_Pr5245412377734449720_f_nat] :
( ( ord_le8976984241387448984_f_nat @ ( sup_su770888850772321124_f_nat @ A4 @ B5 ) @ C2 )
= ( ( ord_le8976984241387448984_f_nat @ A4 @ C2 )
& ( ord_le8976984241387448984_f_nat @ B5 @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_678_rstep__substI,axiom,
! [S: term_f_v,T: term_f_v,R2: set_Pr8827868859434726151rm_f_v,Sigma: v > term_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ ( rstep_f_v @ R2 ) )
=> ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ ( subst_7901567856578221233_f_v_v @ S @ Sigma ) @ ( subst_7901567856578221233_f_v_v @ T @ Sigma ) ) @ ( rstep_f_v @ R2 ) ) ) ).
% rstep_substI
thf(fact_679_rstepI,axiom,
! [L: term_f_v,R: term_f_v,R2: set_Pr8827868859434726151rm_f_v,S: term_f_v,C2: subterm_and_ctxt_f_v,Sigma: v > term_f_v,T: term_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ L @ R ) @ R2 )
=> ( ( S
= ( subter6886040689604212261rm_f_v @ C2 @ ( subst_7901567856578221233_f_v_v @ L @ Sigma ) ) )
=> ( ( T
= ( subter6886040689604212261rm_f_v @ C2 @ ( subst_7901567856578221233_f_v_v @ R @ Sigma ) ) )
=> ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ ( rstep_f_v @ R2 ) ) ) ) ) ).
% rstepI
thf(fact_680_UnE,axiom,
! [C3: produc8027700073345061287rm_f_v,A4: set_Pr8827868859434726151rm_f_v,B5: set_Pr8827868859434726151rm_f_v] :
( ( member4316649861536941904rm_f_v @ C3 @ ( sup_su1994854688125285083rm_f_v @ A4 @ B5 ) )
=> ( ~ ( member4316649861536941904rm_f_v @ C3 @ A4 )
=> ( member4316649861536941904rm_f_v @ C3 @ B5 ) ) ) ).
% UnE
thf(fact_681_UnI1,axiom,
! [C3: produc8027700073345061287rm_f_v,A4: set_Pr8827868859434726151rm_f_v,B5: set_Pr8827868859434726151rm_f_v] :
( ( member4316649861536941904rm_f_v @ C3 @ A4 )
=> ( member4316649861536941904rm_f_v @ C3 @ ( sup_su1994854688125285083rm_f_v @ A4 @ B5 ) ) ) ).
% UnI1
thf(fact_682_UnI2,axiom,
! [C3: produc8027700073345061287rm_f_v,B5: set_Pr8827868859434726151rm_f_v,A4: set_Pr8827868859434726151rm_f_v] :
( ( member4316649861536941904rm_f_v @ C3 @ B5 )
=> ( member4316649861536941904rm_f_v @ C3 @ ( sup_su1994854688125285083rm_f_v @ A4 @ B5 ) ) ) ).
% UnI2
thf(fact_683_subset__Un__eq,axiom,
( ord_le8976984241387448984_f_nat
= ( ^ [A6: set_Pr5245412377734449720_f_nat,B6: set_Pr5245412377734449720_f_nat] :
( ( sup_su770888850772321124_f_nat @ A6 @ B6 )
= B6 ) ) ) ).
% subset_Un_eq
thf(fact_684_subset__UnE,axiom,
! [C2: set_Pr5245412377734449720_f_nat,A4: set_Pr5245412377734449720_f_nat,B5: set_Pr5245412377734449720_f_nat] :
( ( ord_le8976984241387448984_f_nat @ C2 @ ( sup_su770888850772321124_f_nat @ A4 @ B5 ) )
=> ~ ! [A7: set_Pr5245412377734449720_f_nat] :
( ( ord_le8976984241387448984_f_nat @ A7 @ A4 )
=> ! [B7: set_Pr5245412377734449720_f_nat] :
( ( ord_le8976984241387448984_f_nat @ B7 @ B5 )
=> ( C2
!= ( sup_su770888850772321124_f_nat @ A7 @ B7 ) ) ) ) ) ).
% subset_UnE
thf(fact_685_Un__absorb2,axiom,
! [B5: set_Pr5245412377734449720_f_nat,A4: set_Pr5245412377734449720_f_nat] :
( ( ord_le8976984241387448984_f_nat @ B5 @ A4 )
=> ( ( sup_su770888850772321124_f_nat @ A4 @ B5 )
= A4 ) ) ).
% Un_absorb2
thf(fact_686_Un__absorb1,axiom,
! [A4: set_Pr5245412377734449720_f_nat,B5: set_Pr5245412377734449720_f_nat] :
( ( ord_le8976984241387448984_f_nat @ A4 @ B5 )
=> ( ( sup_su770888850772321124_f_nat @ A4 @ B5 )
= B5 ) ) ).
% Un_absorb1
thf(fact_687_Un__upper2,axiom,
! [B5: set_Pr5245412377734449720_f_nat,A4: set_Pr5245412377734449720_f_nat] : ( ord_le8976984241387448984_f_nat @ B5 @ ( sup_su770888850772321124_f_nat @ A4 @ B5 ) ) ).
% Un_upper2
thf(fact_688_Un__upper1,axiom,
! [A4: set_Pr5245412377734449720_f_nat,B5: set_Pr5245412377734449720_f_nat] : ( ord_le8976984241387448984_f_nat @ A4 @ ( sup_su770888850772321124_f_nat @ A4 @ B5 ) ) ).
% Un_upper1
thf(fact_689_Un__least,axiom,
! [A4: set_Pr5245412377734449720_f_nat,C2: set_Pr5245412377734449720_f_nat,B5: set_Pr5245412377734449720_f_nat] :
( ( ord_le8976984241387448984_f_nat @ A4 @ C2 )
=> ( ( ord_le8976984241387448984_f_nat @ B5 @ C2 )
=> ( ord_le8976984241387448984_f_nat @ ( sup_su770888850772321124_f_nat @ A4 @ B5 ) @ C2 ) ) ) ).
% Un_least
thf(fact_690_Un__mono,axiom,
! [A4: set_Pr5245412377734449720_f_nat,C2: set_Pr5245412377734449720_f_nat,B5: set_Pr5245412377734449720_f_nat,D: set_Pr5245412377734449720_f_nat] :
( ( ord_le8976984241387448984_f_nat @ A4 @ C2 )
=> ( ( ord_le8976984241387448984_f_nat @ B5 @ D )
=> ( ord_le8976984241387448984_f_nat @ ( sup_su770888850772321124_f_nat @ A4 @ B5 ) @ ( sup_su770888850772321124_f_nat @ C2 @ D ) ) ) ) ).
% Un_mono
thf(fact_691_rtrancl__Un__rtrancl,axiom,
! [R3: set_Pr8827868859434726151rm_f_v,S2: set_Pr8827868859434726151rm_f_v] :
( ( transi3028809021613410336rm_f_v @ ( sup_su1994854688125285083rm_f_v @ ( transi3028809021613410336rm_f_v @ R3 ) @ ( transi3028809021613410336rm_f_v @ S2 ) ) )
= ( transi3028809021613410336rm_f_v @ ( sup_su1994854688125285083rm_f_v @ R3 @ S2 ) ) ) ).
% rtrancl_Un_rtrancl
thf(fact_692_in__rtrancl__UnI,axiom,
! [X3: produc8027700073345061287rm_f_v,R3: set_Pr8827868859434726151rm_f_v,S2: set_Pr8827868859434726151rm_f_v] :
( ( ( member4316649861536941904rm_f_v @ X3 @ ( transi3028809021613410336rm_f_v @ R3 ) )
| ( member4316649861536941904rm_f_v @ X3 @ ( transi3028809021613410336rm_f_v @ S2 ) ) )
=> ( member4316649861536941904rm_f_v @ X3 @ ( transi3028809021613410336rm_f_v @ ( sup_su1994854688125285083rm_f_v @ R3 @ S2 ) ) ) ) ).
% in_rtrancl_UnI
thf(fact_693_rrstep__subst,axiom,
! [S: term_f_v,T: term_f_v,R2: set_Pr8827868859434726151rm_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ ( rrstep_f_v @ R2 ) )
=> ~ ! [L3: term_f_v,R9: term_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ L3 @ R9 ) @ R2 )
=> ! [Sigma2: v > term_f_v] :
( ( S
= ( subst_7901567856578221233_f_v_v @ L3 @ Sigma2 ) )
=> ( T
!= ( subst_7901567856578221233_f_v_v @ R9 @ Sigma2 ) ) ) ) ) ).
% rrstep_subst
thf(fact_694_rrstep_Ointros,axiom,
! [L: term_f_v,R: term_f_v,R2: set_Pr8827868859434726151rm_f_v,Sigma: v > term_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ L @ R ) @ R2 )
=> ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ ( subst_7901567856578221233_f_v_v @ L @ Sigma ) @ ( subst_7901567856578221233_f_v_v @ R @ Sigma ) ) @ ( rrstep_f_v @ R2 ) ) ) ).
% rrstep.intros
thf(fact_695_rrstep_Osimps,axiom,
! [A1: term_f_v,A22: term_f_v,R2: set_Pr8827868859434726151rm_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A1 @ A22 ) @ ( rrstep_f_v @ R2 ) )
= ( ? [L4: term_f_v,R7: term_f_v,Sigma3: v > term_f_v] :
( ( A1
= ( subst_7901567856578221233_f_v_v @ L4 @ Sigma3 ) )
& ( A22
= ( subst_7901567856578221233_f_v_v @ R7 @ Sigma3 ) )
& ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ L4 @ R7 ) @ R2 ) ) ) ) ).
% rrstep.simps
thf(fact_696_rrstep_Ocases,axiom,
! [A1: term_f_v,A22: term_f_v,R2: set_Pr8827868859434726151rm_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A1 @ A22 ) @ ( rrstep_f_v @ R2 ) )
=> ~ ! [L3: term_f_v,R9: term_f_v,Sigma2: v > term_f_v] :
( ( A1
= ( subst_7901567856578221233_f_v_v @ L3 @ Sigma2 ) )
=> ( ( A22
= ( subst_7901567856578221233_f_v_v @ R9 @ Sigma2 ) )
=> ~ ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ L3 @ R9 ) @ R2 ) ) ) ) ).
% rrstep.cases
thf(fact_697_rtrancl__Un__separator__converseE,axiom,
! [A: term_f_v,B: term_f_v,P2: set_Pr8827868859434726151rm_f_v,Q: set_Pr8827868859434726151rm_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( transi3028809021613410336rm_f_v @ ( sup_su1994854688125285083rm_f_v @ P2 @ Q ) ) )
=> ( ! [X: term_f_v,Y3: term_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ B ) @ ( transi3028809021613410336rm_f_v @ P2 ) )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y3 @ X ) @ Q )
=> ( Y3 = X ) ) )
=> ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( transi3028809021613410336rm_f_v @ P2 ) ) ) ) ).
% rtrancl_Un_separator_converseE
thf(fact_698_rtrancl__Un__separatorE,axiom,
! [A: term_f_v,B: term_f_v,P2: set_Pr8827868859434726151rm_f_v,Q: set_Pr8827868859434726151rm_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( transi3028809021613410336rm_f_v @ ( sup_su1994854688125285083rm_f_v @ P2 @ Q ) ) )
=> ( ! [X: term_f_v,Y3: term_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ X ) @ ( transi3028809021613410336rm_f_v @ P2 ) )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ Y3 ) @ Q )
=> ( X = Y3 ) ) )
=> ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( transi3028809021613410336rm_f_v @ P2 ) ) ) ) ).
% rtrancl_Un_separatorE
thf(fact_699_rtrancl__Un__subset,axiom,
! [R3: set_Pr8827868859434726151rm_f_v,S2: set_Pr8827868859434726151rm_f_v] : ( ord_le4559761987009501863rm_f_v @ ( sup_su1994854688125285083rm_f_v @ ( transi3028809021613410336rm_f_v @ R3 ) @ ( transi3028809021613410336rm_f_v @ S2 ) ) @ ( transi3028809021613410336rm_f_v @ ( sup_su1994854688125285083rm_f_v @ R3 @ S2 ) ) ) ).
% rtrancl_Un_subset
thf(fact_700_rtrancl__union__subset__rtrancl__union__trancl,axiom,
! [R: set_Pr8827868859434726151rm_f_v,S: set_Pr8827868859434726151rm_f_v] :
( ( transi3028809021613410336rm_f_v @ ( sup_su1994854688125285083rm_f_v @ R @ ( transi3208867851621559554rm_f_v @ S ) ) )
= ( transi3028809021613410336rm_f_v @ ( sup_su1994854688125285083rm_f_v @ R @ S ) ) ) ).
% rtrancl_union_subset_rtrancl_union_trancl
thf(fact_701_rstep__imp__C__s__r,axiom,
! [S: term_f_v,T: term_f_v,R2: set_Pr8827868859434726151rm_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ ( rstep_f_v @ R2 ) )
=> ? [C5: subterm_and_ctxt_f_v,Sigma2: v > term_f_v,L3: term_f_v,R9: term_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ L3 @ R9 ) @ R2 )
& ( S
= ( subter6886040689604212261rm_f_v @ C5 @ ( subst_7901567856578221233_f_v_v @ L3 @ Sigma2 ) ) )
& ( T
= ( subter6886040689604212261rm_f_v @ C5 @ ( subst_7901567856578221233_f_v_v @ R9 @ Sigma2 ) ) ) ) ) ).
% rstep_imp_C_s_r
thf(fact_702_rstep__imp__C__s__r_H,axiom,
! [S: term_f_v,T: term_f_v,R2: set_Pr8827868859434726151rm_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ ( rstep_f_v @ R2 ) )
=> ~ ! [C5: subterm_and_ctxt_f_v,L3: term_f_v,R9: term_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ L3 @ R9 ) @ R2 )
=> ! [Sigma2: v > term_f_v] :
( ( S
= ( subter6886040689604212261rm_f_v @ C5 @ ( subst_7901567856578221233_f_v_v @ L3 @ Sigma2 ) ) )
=> ( T
!= ( subter6886040689604212261rm_f_v @ C5 @ ( subst_7901567856578221233_f_v_v @ R9 @ Sigma2 ) ) ) ) ) ) ).
% rstep_imp_C_s_r'
thf(fact_703_right__comp__S,axiom,
! [X3: term_f_v,Y: term_f_v,S2: set_Pr8827868859434726151rm_f_v,NS: set_Pr8827868859434726151rm_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Y ) @ ( relcom2934258169684288428rm_f_v @ S2 @ ( sup_su1994854688125285083rm_f_v @ ( relcom2934258169684288428rm_f_v @ S2 @ ( relcom2934258169684288428rm_f_v @ ( transi3028809021613410336rm_f_v @ S2 ) @ ( transi3028809021613410336rm_f_v @ NS ) ) ) @ ( transi3028809021613410336rm_f_v @ NS ) ) ) )
=> ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Y ) @ ( relcom2934258169684288428rm_f_v @ S2 @ ( relcom2934258169684288428rm_f_v @ ( transi3028809021613410336rm_f_v @ S2 ) @ ( transi3028809021613410336rm_f_v @ NS ) ) ) ) ) ).
% right_comp_S
thf(fact_704_first__step__O,axiom,
! [C2: set_Pr8827868859434726151rm_f_v,A4: set_Pr8827868859434726151rm_f_v,B5: set_Pr8827868859434726151rm_f_v,X3: term_f_v,Y: term_f_v] :
( ( C2
= ( sup_su1994854688125285083rm_f_v @ A4 @ B5 ) )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Y ) @ ( relcom2934258169684288428rm_f_v @ ( transi3028809021613410336rm_f_v @ C2 ) @ B5 ) )
=> ? [Y3: term_f_v] : ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Y3 ) @ ( relcom2934258169684288428rm_f_v @ ( transi3028809021613410336rm_f_v @ A4 ) @ B5 ) ) ) ) ).
% first_step_O
thf(fact_705_first__step,axiom,
! [C2: set_Pr8827868859434726151rm_f_v,A4: set_Pr8827868859434726151rm_f_v,B5: set_Pr8827868859434726151rm_f_v,X3: term_f_v,Y: term_f_v,Z2: term_f_v] :
( ( C2
= ( sup_su1994854688125285083rm_f_v @ A4 @ B5 ) )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Y ) @ ( transi3028809021613410336rm_f_v @ C2 ) )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y @ Z2 ) @ B5 )
=> ? [Y3: term_f_v] : ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Y3 ) @ ( relcom2934258169684288428rm_f_v @ ( transi3028809021613410336rm_f_v @ A4 ) @ B5 ) ) ) ) ) ).
% first_step
thf(fact_706_firstStep,axiom,
! [L5: set_Pr8827868859434726151rm_f_v,S2: set_Pr8827868859434726151rm_f_v,R3: set_Pr8827868859434726151rm_f_v,X3: term_f_v,Y: term_f_v] :
( ( L5
= ( sup_su1994854688125285083rm_f_v @ S2 @ R3 ) )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Y ) @ ( transi3028809021613410336rm_f_v @ L5 ) )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Y ) @ ( transi3028809021613410336rm_f_v @ R3 ) )
| ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Y ) @ ( relcom2934258169684288428rm_f_v @ ( transi3028809021613410336rm_f_v @ R3 ) @ ( relcom2934258169684288428rm_f_v @ S2 @ ( transi3028809021613410336rm_f_v @ L5 ) ) ) ) ) ) ) ).
% firstStep
thf(fact_707_rtrancl__U__push,axiom,
! [S2: set_Pr8827868859434726151rm_f_v,R3: set_Pr8827868859434726151rm_f_v] :
( ( ord_le4559761987009501863rm_f_v @ ( relcom2934258169684288428rm_f_v @ S2 @ R3 ) @ ( relcom2934258169684288428rm_f_v @ R3 @ ( transi3028809021613410336rm_f_v @ S2 ) ) )
=> ( ( transi3028809021613410336rm_f_v @ ( sup_su1994854688125285083rm_f_v @ S2 @ R3 ) )
= ( relcom2934258169684288428rm_f_v @ ( transi3028809021613410336rm_f_v @ R3 ) @ ( transi3028809021613410336rm_f_v @ S2 ) ) ) ) ).
% rtrancl_U_push
thf(fact_708_compatible__conv,axiom,
! [NS: set_Pr8827868859434726151rm_f_v,S2: set_Pr8827868859434726151rm_f_v] :
( ( ord_le4559761987009501863rm_f_v @ ( relcom2934258169684288428rm_f_v @ NS @ S2 ) @ S2 )
=> ( ( relcom2934258169684288428rm_f_v @ ( transi3028809021613410336rm_f_v @ ( sup_su1994854688125285083rm_f_v @ NS @ S2 ) ) @ ( relcom2934258169684288428rm_f_v @ S2 @ ( transi3028809021613410336rm_f_v @ ( sup_su1994854688125285083rm_f_v @ NS @ S2 ) ) ) )
= ( relcom2934258169684288428rm_f_v @ S2 @ ( relcom2934258169684288428rm_f_v @ ( transi3028809021613410336rm_f_v @ S2 ) @ ( transi3028809021613410336rm_f_v @ NS ) ) ) ) ) ).
% compatible_conv
thf(fact_709_WCR__on__def,axiom,
( abstra312943320106832654rm_f_v
= ( ^ [R7: set_Pr8827868859434726151rm_f_v,A6: set_term_f_v] :
! [X4: term_f_v] :
( ( member_term_f_v @ X4 @ A6 )
=> ! [B4: term_f_v,C4: term_f_v] :
( ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X4 @ B4 ) @ R7 )
& ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X4 @ C4 ) @ R7 ) )
=> ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B4 @ C4 ) @ ( abstra8605546704478008873rm_f_v @ R7 ) ) ) ) ) ) ).
% WCR_on_def
thf(fact_710_WCR__onE,axiom,
! [R: set_Pr7897041799657265351rm_f_v,A4: set_Pr8827868859434726151rm_f_v,A: produc8027700073345061287rm_f_v,B: produc8027700073345061287rm_f_v,C3: produc8027700073345061287rm_f_v] :
( ( abstra8545463541035707836rm_f_v @ R @ A4 )
=> ( ( member4316649861536941904rm_f_v @ A @ A4 )
=> ( ~ ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ B @ C3 ) @ ( abstra3336617290665100897rm_f_v @ R ) )
=> ( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A @ B ) @ R )
=> ~ ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A @ C3 ) @ R ) ) ) ) ) ).
% WCR_onE
thf(fact_711_WCR__onE,axiom,
! [R: set_Pr8827868859434726151rm_f_v,A4: set_term_f_v,A: term_f_v,B: term_f_v,C3: term_f_v] :
( ( abstra312943320106832654rm_f_v @ R @ A4 )
=> ( ( member_term_f_v @ A @ A4 )
=> ( ~ ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B @ C3 ) @ ( abstra8605546704478008873rm_f_v @ R ) )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ R )
=> ~ ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ C3 ) @ R ) ) ) ) ) ).
% WCR_onE
thf(fact_712_WCR__onD,axiom,
! [R: set_Pr7897041799657265351rm_f_v,A4: set_Pr8827868859434726151rm_f_v,A: produc8027700073345061287rm_f_v,B: produc8027700073345061287rm_f_v,C3: produc8027700073345061287rm_f_v] :
( ( abstra8545463541035707836rm_f_v @ R @ A4 )
=> ( ( member4316649861536941904rm_f_v @ A @ A4 )
=> ( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A @ B ) @ R )
=> ( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A @ C3 ) @ R )
=> ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ B @ C3 ) @ ( abstra3336617290665100897rm_f_v @ R ) ) ) ) ) ) ).
% WCR_onD
thf(fact_713_WCR__onD,axiom,
! [R: set_Pr8827868859434726151rm_f_v,A4: set_term_f_v,A: term_f_v,B: term_f_v,C3: term_f_v] :
( ( abstra312943320106832654rm_f_v @ R @ A4 )
=> ( ( member_term_f_v @ A @ A4 )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ R )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ C3 ) @ R )
=> ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B @ C3 ) @ ( abstra8605546704478008873rm_f_v @ R ) ) ) ) ) ) ).
% WCR_onD
thf(fact_714_compatible__rtrancl__split,axiom,
! [NS: set_Pr8827868859434726151rm_f_v,S2: set_Pr8827868859434726151rm_f_v,X3: term_f_v,Y: term_f_v] :
( ( ord_le4559761987009501863rm_f_v @ ( relcom2934258169684288428rm_f_v @ NS @ S2 ) @ S2 )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Y ) @ ( transi3028809021613410336rm_f_v @ ( sup_su1994854688125285083rm_f_v @ NS @ S2 ) ) )
=> ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Y ) @ ( sup_su1994854688125285083rm_f_v @ ( relcom2934258169684288428rm_f_v @ S2 @ ( relcom2934258169684288428rm_f_v @ ( transi3028809021613410336rm_f_v @ S2 ) @ ( transi3028809021613410336rm_f_v @ NS ) ) ) @ ( transi3028809021613410336rm_f_v @ NS ) ) ) ) ) ).
% compatible_rtrancl_split
thf(fact_715_srstep__subst__closed,axiom,
! [S: term_f_v,T: term_f_v,F: set_Pr5245412377734449720_f_nat,R2: set_Pr8827868859434726151rm_f_v,Sigma: v > term_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ ( sig_step_f_v @ F @ ( rstep_f_v @ R2 ) ) )
=> ( ! [X: v] : ( ord_le8976984241387448984_f_nat @ ( term_funas_term_f_v @ ( Sigma @ X ) ) @ F )
=> ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ ( subst_7901567856578221233_f_v_v @ S @ Sigma ) @ ( subst_7901567856578221233_f_v_v @ T @ Sigma ) ) @ ( sig_step_f_v @ F @ ( rstep_f_v @ R2 ) ) ) ) ) ).
% srstep_subst_closed
thf(fact_716_funas__ctxt__apply,axiom,
! [C2: subterm_and_ctxt_f_v,T: term_f_v] :
( ( term_funas_term_f_v @ ( subter6886040689604212261rm_f_v @ C2 @ T ) )
= ( sup_su770888850772321124_f_nat @ ( term_funas_ctxt_f_v @ C2 ) @ ( term_funas_term_f_v @ T ) ) ) ).
% funas_ctxt_apply
thf(fact_717_le__sup__iff,axiom,
! [X3: set_Pr5245412377734449720_f_nat,Y: set_Pr5245412377734449720_f_nat,Z2: set_Pr5245412377734449720_f_nat] :
( ( ord_le8976984241387448984_f_nat @ ( sup_su770888850772321124_f_nat @ X3 @ Y ) @ Z2 )
= ( ( ord_le8976984241387448984_f_nat @ X3 @ Z2 )
& ( ord_le8976984241387448984_f_nat @ Y @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_718_sup_Obounded__iff,axiom,
! [B: set_Pr5245412377734449720_f_nat,C3: set_Pr5245412377734449720_f_nat,A: set_Pr5245412377734449720_f_nat] :
( ( ord_le8976984241387448984_f_nat @ ( sup_su770888850772321124_f_nat @ B @ C3 ) @ A )
= ( ( ord_le8976984241387448984_f_nat @ B @ A )
& ( ord_le8976984241387448984_f_nat @ C3 @ A ) ) ) ).
% sup.bounded_iff
thf(fact_719_quasi__commute__def,axiom,
( abstra8826850394716133222rm_f_v
= ( ^ [R7: set_Pr8827868859434726151rm_f_v,S4: set_Pr8827868859434726151rm_f_v] : ( ord_le4559761987009501863rm_f_v @ ( relcom2934258169684288428rm_f_v @ S4 @ R7 ) @ ( relcom2934258169684288428rm_f_v @ R7 @ ( transi3028809021613410336rm_f_v @ ( sup_su1994854688125285083rm_f_v @ R7 @ S4 ) ) ) ) ) ) ).
% quasi_commute_def
thf(fact_720_sig__step__rsteps__eq__dist,axiom,
! [R2: set_Pr8827868859434726151rm_f_v,F: set_Pr5245412377734449720_f_nat] :
( ( ord_le8976984241387448984_f_nat @ ( terms_7890395023448655851_f_v_v @ R2 ) @ F )
=> ( ( sup_su1994854688125285083rm_f_v @ ( sig_step_f_v @ F @ ( transi3208867851621559554rm_f_v @ ( rstep_f_v @ R2 ) ) ) @ id_term_f_v )
= ( transi3028809021613410336rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ R2 ) ) ) ) ) ).
% sig_step_rsteps_eq_dist
thf(fact_721_funas__ctxt__ctxt__inv__const__ctxt__ind,axiom,
! [C2: subterm_and_ctxt_f_v,F: set_Pr5245412377734449720_f_nat,V: v] :
( ( ord_le8976984241387448984_f_nat @ ( term_funas_ctxt_f_v @ C2 ) @ F )
=> ( ( terms_8308829228547689406xt_f_v @ F @ V @ C2 )
= C2 ) ) ).
% funas_ctxt_ctxt_inv_const_ctxt_ind
thf(fact_722_inf__sup__ord_I4_J,axiom,
! [Y: set_Pr5245412377734449720_f_nat,X3: set_Pr5245412377734449720_f_nat] : ( ord_le8976984241387448984_f_nat @ Y @ ( sup_su770888850772321124_f_nat @ X3 @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_723_inf__sup__ord_I3_J,axiom,
! [X3: set_Pr5245412377734449720_f_nat,Y: set_Pr5245412377734449720_f_nat] : ( ord_le8976984241387448984_f_nat @ X3 @ ( sup_su770888850772321124_f_nat @ X3 @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_724_pair__in__Id__conv,axiom,
! [A: term_f_v,B: term_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ id_term_f_v )
= ( A = B ) ) ).
% pair_in_Id_conv
thf(fact_725_IdI,axiom,
! [A: term_f_v] : ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ A ) @ id_term_f_v ) ).
% IdI
thf(fact_726_rtrancl__reflcl,axiom,
! [R3: set_Pr8827868859434726151rm_f_v] :
( ( transi3028809021613410336rm_f_v @ ( sup_su1994854688125285083rm_f_v @ R3 @ id_term_f_v ) )
= ( transi3028809021613410336rm_f_v @ R3 ) ) ).
% rtrancl_reflcl
thf(fact_727_rtrancl__reflcl__absorb,axiom,
! [R3: set_Pr8827868859434726151rm_f_v] :
( ( sup_su1994854688125285083rm_f_v @ ( transi3028809021613410336rm_f_v @ R3 ) @ id_term_f_v )
= ( transi3028809021613410336rm_f_v @ R3 ) ) ).
% rtrancl_reflcl_absorb
thf(fact_728_trancl__reflcl,axiom,
! [R: set_Pr8827868859434726151rm_f_v] :
( ( transi3208867851621559554rm_f_v @ ( sup_su1994854688125285083rm_f_v @ R @ id_term_f_v ) )
= ( transi3028809021613410336rm_f_v @ R ) ) ).
% trancl_reflcl
thf(fact_729_IdE,axiom,
! [P: produc8027700073345061287rm_f_v] :
( ( member4316649861536941904rm_f_v @ P @ id_term_f_v )
=> ~ ! [X: term_f_v] :
( P
!= ( produc2510790566102465375rm_f_v @ X @ X ) ) ) ).
% IdE
thf(fact_730_rtrancl__unfold,axiom,
( transi3028809021613410336rm_f_v
= ( ^ [R7: set_Pr8827868859434726151rm_f_v] : ( sup_su1994854688125285083rm_f_v @ id_term_f_v @ ( relcom2934258169684288428rm_f_v @ ( transi3028809021613410336rm_f_v @ R7 ) @ R7 ) ) ) ) ).
% rtrancl_unfold
thf(fact_731_rtrancl__trancl__reflcl,axiom,
( transi3028809021613410336rm_f_v
= ( ^ [R7: set_Pr8827868859434726151rm_f_v] : ( sup_su1994854688125285083rm_f_v @ ( transi3208867851621559554rm_f_v @ R7 ) @ id_term_f_v ) ) ) ).
% rtrancl_trancl_reflcl
thf(fact_732_sup_OcoboundedI2,axiom,
! [C3: set_Pr5245412377734449720_f_nat,B: set_Pr5245412377734449720_f_nat,A: set_Pr5245412377734449720_f_nat] :
( ( ord_le8976984241387448984_f_nat @ C3 @ B )
=> ( ord_le8976984241387448984_f_nat @ C3 @ ( sup_su770888850772321124_f_nat @ A @ B ) ) ) ).
% sup.coboundedI2
thf(fact_733_sup_OcoboundedI1,axiom,
! [C3: set_Pr5245412377734449720_f_nat,A: set_Pr5245412377734449720_f_nat,B: set_Pr5245412377734449720_f_nat] :
( ( ord_le8976984241387448984_f_nat @ C3 @ A )
=> ( ord_le8976984241387448984_f_nat @ C3 @ ( sup_su770888850772321124_f_nat @ A @ B ) ) ) ).
% sup.coboundedI1
thf(fact_734_sup_Oabsorb__iff2,axiom,
( ord_le8976984241387448984_f_nat
= ( ^ [A5: set_Pr5245412377734449720_f_nat,B4: set_Pr5245412377734449720_f_nat] :
( ( sup_su770888850772321124_f_nat @ A5 @ B4 )
= B4 ) ) ) ).
% sup.absorb_iff2
thf(fact_735_sup_Oabsorb__iff1,axiom,
( ord_le8976984241387448984_f_nat
= ( ^ [B4: set_Pr5245412377734449720_f_nat,A5: set_Pr5245412377734449720_f_nat] :
( ( sup_su770888850772321124_f_nat @ A5 @ B4 )
= A5 ) ) ) ).
% sup.absorb_iff1
thf(fact_736_sup_Ocobounded2,axiom,
! [B: set_Pr5245412377734449720_f_nat,A: set_Pr5245412377734449720_f_nat] : ( ord_le8976984241387448984_f_nat @ B @ ( sup_su770888850772321124_f_nat @ A @ B ) ) ).
% sup.cobounded2
thf(fact_737_sup_Ocobounded1,axiom,
! [A: set_Pr5245412377734449720_f_nat,B: set_Pr5245412377734449720_f_nat] : ( ord_le8976984241387448984_f_nat @ A @ ( sup_su770888850772321124_f_nat @ A @ B ) ) ).
% sup.cobounded1
thf(fact_738_sup_Oorder__iff,axiom,
( ord_le8976984241387448984_f_nat
= ( ^ [B4: set_Pr5245412377734449720_f_nat,A5: set_Pr5245412377734449720_f_nat] :
( A5
= ( sup_su770888850772321124_f_nat @ A5 @ B4 ) ) ) ) ).
% sup.order_iff
thf(fact_739_sup_OboundedI,axiom,
! [B: set_Pr5245412377734449720_f_nat,A: set_Pr5245412377734449720_f_nat,C3: set_Pr5245412377734449720_f_nat] :
( ( ord_le8976984241387448984_f_nat @ B @ A )
=> ( ( ord_le8976984241387448984_f_nat @ C3 @ A )
=> ( ord_le8976984241387448984_f_nat @ ( sup_su770888850772321124_f_nat @ B @ C3 ) @ A ) ) ) ).
% sup.boundedI
thf(fact_740_sup_OboundedE,axiom,
! [B: set_Pr5245412377734449720_f_nat,C3: set_Pr5245412377734449720_f_nat,A: set_Pr5245412377734449720_f_nat] :
( ( ord_le8976984241387448984_f_nat @ ( sup_su770888850772321124_f_nat @ B @ C3 ) @ A )
=> ~ ( ( ord_le8976984241387448984_f_nat @ B @ A )
=> ~ ( ord_le8976984241387448984_f_nat @ C3 @ A ) ) ) ).
% sup.boundedE
thf(fact_741_sup__absorb2,axiom,
! [X3: set_Pr5245412377734449720_f_nat,Y: set_Pr5245412377734449720_f_nat] :
( ( ord_le8976984241387448984_f_nat @ X3 @ Y )
=> ( ( sup_su770888850772321124_f_nat @ X3 @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_742_sup__absorb1,axiom,
! [Y: set_Pr5245412377734449720_f_nat,X3: set_Pr5245412377734449720_f_nat] :
( ( ord_le8976984241387448984_f_nat @ Y @ X3 )
=> ( ( sup_su770888850772321124_f_nat @ X3 @ Y )
= X3 ) ) ).
% sup_absorb1
thf(fact_743_sup_Oabsorb2,axiom,
! [A: set_Pr5245412377734449720_f_nat,B: set_Pr5245412377734449720_f_nat] :
( ( ord_le8976984241387448984_f_nat @ A @ B )
=> ( ( sup_su770888850772321124_f_nat @ A @ B )
= B ) ) ).
% sup.absorb2
thf(fact_744_sup_Oabsorb1,axiom,
! [B: set_Pr5245412377734449720_f_nat,A: set_Pr5245412377734449720_f_nat] :
( ( ord_le8976984241387448984_f_nat @ B @ A )
=> ( ( sup_su770888850772321124_f_nat @ A @ B )
= A ) ) ).
% sup.absorb1
thf(fact_745_sup__unique,axiom,
! [F2: set_Pr5245412377734449720_f_nat > set_Pr5245412377734449720_f_nat > set_Pr5245412377734449720_f_nat,X3: set_Pr5245412377734449720_f_nat,Y: set_Pr5245412377734449720_f_nat] :
( ! [X: set_Pr5245412377734449720_f_nat,Y3: set_Pr5245412377734449720_f_nat] : ( ord_le8976984241387448984_f_nat @ X @ ( F2 @ X @ Y3 ) )
=> ( ! [X: set_Pr5245412377734449720_f_nat,Y3: set_Pr5245412377734449720_f_nat] : ( ord_le8976984241387448984_f_nat @ Y3 @ ( F2 @ X @ Y3 ) )
=> ( ! [X: set_Pr5245412377734449720_f_nat,Y3: set_Pr5245412377734449720_f_nat,Z: set_Pr5245412377734449720_f_nat] :
( ( ord_le8976984241387448984_f_nat @ Y3 @ X )
=> ( ( ord_le8976984241387448984_f_nat @ Z @ X )
=> ( ord_le8976984241387448984_f_nat @ ( F2 @ Y3 @ Z ) @ X ) ) )
=> ( ( sup_su770888850772321124_f_nat @ X3 @ Y )
= ( F2 @ X3 @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_746_sup_OorderI,axiom,
! [A: set_Pr5245412377734449720_f_nat,B: set_Pr5245412377734449720_f_nat] :
( ( A
= ( sup_su770888850772321124_f_nat @ A @ B ) )
=> ( ord_le8976984241387448984_f_nat @ B @ A ) ) ).
% sup.orderI
thf(fact_747_sup_OorderE,axiom,
! [B: set_Pr5245412377734449720_f_nat,A: set_Pr5245412377734449720_f_nat] :
( ( ord_le8976984241387448984_f_nat @ B @ A )
=> ( A
= ( sup_su770888850772321124_f_nat @ A @ B ) ) ) ).
% sup.orderE
thf(fact_748_le__iff__sup,axiom,
( ord_le8976984241387448984_f_nat
= ( ^ [X4: set_Pr5245412377734449720_f_nat,Y5: set_Pr5245412377734449720_f_nat] :
( ( sup_su770888850772321124_f_nat @ X4 @ Y5 )
= Y5 ) ) ) ).
% le_iff_sup
thf(fact_749_sup__least,axiom,
! [Y: set_Pr5245412377734449720_f_nat,X3: set_Pr5245412377734449720_f_nat,Z2: set_Pr5245412377734449720_f_nat] :
( ( ord_le8976984241387448984_f_nat @ Y @ X3 )
=> ( ( ord_le8976984241387448984_f_nat @ Z2 @ X3 )
=> ( ord_le8976984241387448984_f_nat @ ( sup_su770888850772321124_f_nat @ Y @ Z2 ) @ X3 ) ) ) ).
% sup_least
thf(fact_750_sup__mono,axiom,
! [A: set_Pr5245412377734449720_f_nat,C3: set_Pr5245412377734449720_f_nat,B: set_Pr5245412377734449720_f_nat,D2: set_Pr5245412377734449720_f_nat] :
( ( ord_le8976984241387448984_f_nat @ A @ C3 )
=> ( ( ord_le8976984241387448984_f_nat @ B @ D2 )
=> ( ord_le8976984241387448984_f_nat @ ( sup_su770888850772321124_f_nat @ A @ B ) @ ( sup_su770888850772321124_f_nat @ C3 @ D2 ) ) ) ) ).
% sup_mono
thf(fact_751_sup_Omono,axiom,
! [C3: set_Pr5245412377734449720_f_nat,A: set_Pr5245412377734449720_f_nat,D2: set_Pr5245412377734449720_f_nat,B: set_Pr5245412377734449720_f_nat] :
( ( ord_le8976984241387448984_f_nat @ C3 @ A )
=> ( ( ord_le8976984241387448984_f_nat @ D2 @ B )
=> ( ord_le8976984241387448984_f_nat @ ( sup_su770888850772321124_f_nat @ C3 @ D2 ) @ ( sup_su770888850772321124_f_nat @ A @ B ) ) ) ) ).
% sup.mono
thf(fact_752_le__supI2,axiom,
! [X3: set_Pr5245412377734449720_f_nat,B: set_Pr5245412377734449720_f_nat,A: set_Pr5245412377734449720_f_nat] :
( ( ord_le8976984241387448984_f_nat @ X3 @ B )
=> ( ord_le8976984241387448984_f_nat @ X3 @ ( sup_su770888850772321124_f_nat @ A @ B ) ) ) ).
% le_supI2
thf(fact_753_le__supI1,axiom,
! [X3: set_Pr5245412377734449720_f_nat,A: set_Pr5245412377734449720_f_nat,B: set_Pr5245412377734449720_f_nat] :
( ( ord_le8976984241387448984_f_nat @ X3 @ A )
=> ( ord_le8976984241387448984_f_nat @ X3 @ ( sup_su770888850772321124_f_nat @ A @ B ) ) ) ).
% le_supI1
thf(fact_754_sup__ge2,axiom,
! [Y: set_Pr5245412377734449720_f_nat,X3: set_Pr5245412377734449720_f_nat] : ( ord_le8976984241387448984_f_nat @ Y @ ( sup_su770888850772321124_f_nat @ X3 @ Y ) ) ).
% sup_ge2
thf(fact_755_sup__ge1,axiom,
! [X3: set_Pr5245412377734449720_f_nat,Y: set_Pr5245412377734449720_f_nat] : ( ord_le8976984241387448984_f_nat @ X3 @ ( sup_su770888850772321124_f_nat @ X3 @ Y ) ) ).
% sup_ge1
thf(fact_756_le__supI,axiom,
! [A: set_Pr5245412377734449720_f_nat,X3: set_Pr5245412377734449720_f_nat,B: set_Pr5245412377734449720_f_nat] :
( ( ord_le8976984241387448984_f_nat @ A @ X3 )
=> ( ( ord_le8976984241387448984_f_nat @ B @ X3 )
=> ( ord_le8976984241387448984_f_nat @ ( sup_su770888850772321124_f_nat @ A @ B ) @ X3 ) ) ) ).
% le_supI
thf(fact_757_le__supE,axiom,
! [A: set_Pr5245412377734449720_f_nat,B: set_Pr5245412377734449720_f_nat,X3: set_Pr5245412377734449720_f_nat] :
( ( ord_le8976984241387448984_f_nat @ ( sup_su770888850772321124_f_nat @ A @ B ) @ X3 )
=> ~ ( ( ord_le8976984241387448984_f_nat @ A @ X3 )
=> ~ ( ord_le8976984241387448984_f_nat @ B @ X3 ) ) ) ).
% le_supE
thf(fact_758_SCR__on__def,axiom,
( rewrit252611673853975797rm_f_v
= ( ^ [R7: set_Pr8827868859434726151rm_f_v,A6: set_term_f_v] :
! [X4: term_f_v] :
( ( member_term_f_v @ X4 @ A6 )
=> ! [B4: term_f_v,C4: term_f_v] :
( ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X4 @ B4 ) @ R7 )
& ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X4 @ C4 ) @ R7 ) )
=> ? [D3: term_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B4 @ D3 ) @ ( sup_su1994854688125285083rm_f_v @ R7 @ id_term_f_v ) )
& ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ C4 @ D3 ) @ ( transi3028809021613410336rm_f_v @ R7 ) ) ) ) ) ) ) ).
% SCR_on_def
thf(fact_759_strongly__confluent__on__E11,axiom,
! [R: set_Pr7897041799657265351rm_f_v,A4: set_Pr8827868859434726151rm_f_v,X3: produc8027700073345061287rm_f_v,Y: produc8027700073345061287rm_f_v,Z2: produc8027700073345061287rm_f_v] :
( ( abstra8507947352017634312rm_f_v @ R @ A4 )
=> ( ( member4316649861536941904rm_f_v @ X3 @ A4 )
=> ( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ X3 @ Y ) @ R )
=> ( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ X3 @ Z2 ) @ R )
=> ? [U2: produc8027700073345061287rm_f_v] :
( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ Y @ U2 ) @ ( transi1062743709496199978rm_f_v @ R ) )
& ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ Z2 @ U2 ) @ ( sup_su6770080740296300187rm_f_v @ R @ id_Pro1935643801653256451rm_f_v ) ) ) ) ) ) ) ).
% strongly_confluent_on_E11
thf(fact_760_strongly__confluent__on__E11,axiom,
! [R: set_Pr8827868859434726151rm_f_v,A4: set_term_f_v,X3: term_f_v,Y: term_f_v,Z2: term_f_v] :
( ( abstra950190854684145090rm_f_v @ R @ A4 )
=> ( ( member_term_f_v @ X3 @ A4 )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Y ) @ R )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Z2 ) @ R )
=> ? [U2: term_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y @ U2 ) @ ( transi3028809021613410336rm_f_v @ R ) )
& ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Z2 @ U2 ) @ ( sup_su1994854688125285083rm_f_v @ R @ id_term_f_v ) ) ) ) ) ) ) ).
% strongly_confluent_on_E11
thf(fact_761_strongly__confluent__on__def,axiom,
( abstra950190854684145090rm_f_v
= ( ^ [R7: set_Pr8827868859434726151rm_f_v,A6: set_term_f_v] :
! [X4: term_f_v] :
( ( member_term_f_v @ X4 @ A6 )
=> ! [Y5: term_f_v,Z4: term_f_v] :
( ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X4 @ Y5 ) @ R7 )
& ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X4 @ Z4 ) @ R7 ) )
=> ? [U3: term_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y5 @ U3 ) @ ( transi3028809021613410336rm_f_v @ R7 ) )
& ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Z4 @ U3 ) @ ( sup_su1994854688125285083rm_f_v @ R7 @ id_term_f_v ) ) ) ) ) ) ) ).
% strongly_confluent_on_def
thf(fact_762_term__to__sig__ctxt__apply,axiom,
! [F: set_Pr5245412377734449720_f_nat,C2: subterm_and_ctxt_f_v,V: v,S: term_f_v] :
( ( terms_3660607981918322800th_f_v @ F @ C2 )
=> ( ( terms_3805575757503158827ig_f_v @ F @ V @ ( subter6886040689604212261rm_f_v @ C2 @ S ) )
= ( subter6886040689604212261rm_f_v @ ( terms_8308829228547689406xt_f_v @ F @ V @ C2 ) @ ( terms_3805575757503158827ig_f_v @ F @ V @ S ) ) ) ) ).
% term_to_sig_ctxt_apply
thf(fact_763_IdD,axiom,
! [A: term_f_v,B: term_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ id_term_f_v )
=> ( A = B ) ) ).
% IdD
thf(fact_764_CR__onI,axiom,
! [A4: set_Pr8827868859434726151rm_f_v,R: set_Pr7897041799657265351rm_f_v] :
( ! [A3: produc8027700073345061287rm_f_v,B3: produc8027700073345061287rm_f_v,C: produc8027700073345061287rm_f_v] :
( ( member4316649861536941904rm_f_v @ A3 @ A4 )
=> ( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A3 @ B3 ) @ ( transi1062743709496199978rm_f_v @ R ) )
=> ( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A3 @ C ) @ ( transi1062743709496199978rm_f_v @ R ) )
=> ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ B3 @ C ) @ ( abstra3336617290665100897rm_f_v @ R ) ) ) ) )
=> ( abstra5806628924625096669rm_f_v @ R @ A4 ) ) ).
% CR_onI
thf(fact_765_CR__onI,axiom,
! [A4: set_term_f_v,R: set_Pr8827868859434726151rm_f_v] :
( ! [A3: term_f_v,B3: term_f_v,C: term_f_v] :
( ( member_term_f_v @ A3 @ A4 )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A3 @ B3 ) @ ( transi3028809021613410336rm_f_v @ R ) )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A3 @ C ) @ ( transi3028809021613410336rm_f_v @ R ) )
=> ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B3 @ C ) @ ( abstra8605546704478008873rm_f_v @ R ) ) ) ) )
=> ( abstra3735013631728912813rm_f_v @ R @ A4 ) ) ).
% CR_onI
thf(fact_766_funas__ctxt__ctxt__well__def__hole__path,axiom,
! [C2: subterm_and_ctxt_f_v,F: set_Pr5245412377734449720_f_nat] :
( ( ord_le8976984241387448984_f_nat @ ( term_funas_ctxt_f_v @ C2 ) @ F )
=> ( terms_3660607981918322800th_f_v @ F @ C2 ) ) ).
% funas_ctxt_ctxt_well_def_hole_path
thf(fact_767_CR__on__def,axiom,
( abstra3735013631728912813rm_f_v
= ( ^ [R7: set_Pr8827868859434726151rm_f_v,A6: set_term_f_v] :
! [X4: term_f_v] :
( ( member_term_f_v @ X4 @ A6 )
=> ! [B4: term_f_v,C4: term_f_v] :
( ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X4 @ B4 ) @ ( transi3028809021613410336rm_f_v @ R7 ) )
& ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X4 @ C4 ) @ ( transi3028809021613410336rm_f_v @ R7 ) ) )
=> ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B4 @ C4 ) @ ( abstra8605546704478008873rm_f_v @ R7 ) ) ) ) ) ) ).
% CR_on_def
thf(fact_768_CR__onE,axiom,
! [R: set_Pr7897041799657265351rm_f_v,A4: set_Pr8827868859434726151rm_f_v,A: produc8027700073345061287rm_f_v,B: produc8027700073345061287rm_f_v,C3: produc8027700073345061287rm_f_v] :
( ( abstra5806628924625096669rm_f_v @ R @ A4 )
=> ( ( member4316649861536941904rm_f_v @ A @ A4 )
=> ( ~ ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ B @ C3 ) @ ( abstra3336617290665100897rm_f_v @ R ) )
=> ( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A @ B ) @ ( transi1062743709496199978rm_f_v @ R ) )
=> ~ ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A @ C3 ) @ ( transi1062743709496199978rm_f_v @ R ) ) ) ) ) ) ).
% CR_onE
thf(fact_769_CR__onE,axiom,
! [R: set_Pr8827868859434726151rm_f_v,A4: set_term_f_v,A: term_f_v,B: term_f_v,C3: term_f_v] :
( ( abstra3735013631728912813rm_f_v @ R @ A4 )
=> ( ( member_term_f_v @ A @ A4 )
=> ( ~ ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B @ C3 ) @ ( abstra8605546704478008873rm_f_v @ R ) )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( transi3028809021613410336rm_f_v @ R ) )
=> ~ ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ C3 ) @ ( transi3028809021613410336rm_f_v @ R ) ) ) ) ) ) ).
% CR_onE
thf(fact_770_CR__onD,axiom,
! [R: set_Pr7897041799657265351rm_f_v,A4: set_Pr8827868859434726151rm_f_v,A: produc8027700073345061287rm_f_v,B: produc8027700073345061287rm_f_v,C3: produc8027700073345061287rm_f_v] :
( ( abstra5806628924625096669rm_f_v @ R @ A4 )
=> ( ( member4316649861536941904rm_f_v @ A @ A4 )
=> ( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A @ B ) @ ( transi1062743709496199978rm_f_v @ R ) )
=> ( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A @ C3 ) @ ( transi1062743709496199978rm_f_v @ R ) )
=> ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ B @ C3 ) @ ( abstra3336617290665100897rm_f_v @ R ) ) ) ) ) ) ).
% CR_onD
thf(fact_771_CR__onD,axiom,
! [R: set_Pr8827868859434726151rm_f_v,A4: set_term_f_v,A: term_f_v,B: term_f_v,C3: term_f_v] :
( ( abstra3735013631728912813rm_f_v @ R @ A4 )
=> ( ( member_term_f_v @ A @ A4 )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( transi3028809021613410336rm_f_v @ R ) )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ C3 ) @ ( transi3028809021613410336rm_f_v @ R ) )
=> ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B @ C3 ) @ ( abstra8605546704478008873rm_f_v @ R ) ) ) ) ) ) ).
% CR_onD
thf(fact_772_term__to__sig__ctxt__apply_H,axiom,
! [F: set_Pr5245412377734449720_f_nat,C2: subterm_and_ctxt_f_v,V: v,S: term_f_v] :
( ~ ( terms_3660607981918322800th_f_v @ F @ C2 )
=> ( ( terms_3805575757503158827ig_f_v @ F @ V @ ( subter6886040689604212261rm_f_v @ C2 @ S ) )
= ( terms_4639549856110724071xt_f_v @ F @ V @ C2 ) ) ) ).
% term_to_sig_ctxt_apply'
thf(fact_773_strongly__confluentI,axiom,
! [R: set_Pr8827868859434726151rm_f_v] :
( ! [X: term_f_v,Y3: term_f_v,Z: term_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ Y3 ) @ R )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ Z ) @ R )
=> ? [U4: term_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y3 @ U4 ) @ ( transi3028809021613410336rm_f_v @ R ) )
& ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Z @ U4 ) @ ( sup_su1994854688125285083rm_f_v @ R @ id_term_f_v ) ) ) ) )
=> ( abstra950190854684145090rm_f_v @ R @ top_top_set_term_f_v ) ) ).
% strongly_confluentI
thf(fact_774_CR__divergence__imp__join,axiom,
! [R: set_Pr8827868859434726151rm_f_v,X3: term_f_v,Y: term_f_v,Z2: term_f_v] :
( ( abstra3735013631728912813rm_f_v @ R @ top_top_set_term_f_v )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Y ) @ ( transi3028809021613410336rm_f_v @ R ) )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Z2 ) @ ( transi3028809021613410336rm_f_v @ R ) )
=> ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y @ Z2 ) @ ( abstra8605546704478008873rm_f_v @ R ) ) ) ) ) ).
% CR_divergence_imp_join
thf(fact_775_partially__localize__CR,axiom,
! [R: set_Pr8827868859434726151rm_f_v] :
( ( abstra3735013631728912813rm_f_v @ R @ top_top_set_term_f_v )
= ( ! [X4: term_f_v,Y5: term_f_v,Z4: term_f_v] :
( ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X4 @ Y5 ) @ R )
& ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X4 @ Z4 ) @ ( transi3028809021613410336rm_f_v @ R ) ) )
=> ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y5 @ Z4 ) @ ( abstra8605546704478008873rm_f_v @ R ) ) ) ) ) ).
% partially_localize_CR
thf(fact_776_CR__join__right__I,axiom,
! [R: set_Pr8827868859434726151rm_f_v,X3: term_f_v,Y: term_f_v,Z2: term_f_v] :
( ( abstra3735013631728912813rm_f_v @ R @ top_top_set_term_f_v )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Y ) @ ( abstra8605546704478008873rm_f_v @ R ) )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y @ Z2 ) @ ( transi3028809021613410336rm_f_v @ R ) )
=> ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Z2 ) @ ( abstra8605546704478008873rm_f_v @ R ) ) ) ) ) ).
% CR_join_right_I
thf(fact_777_UNIV__I,axiom,
! [X3: produc8027700073345061287rm_f_v] : ( member4316649861536941904rm_f_v @ X3 @ top_to5755558563273943127rm_f_v ) ).
% UNIV_I
thf(fact_778_UNIV__witness,axiom,
? [X: produc8027700073345061287rm_f_v] : ( member4316649861536941904rm_f_v @ X @ top_to5755558563273943127rm_f_v ) ).
% UNIV_witness
thf(fact_779_UNIV__eq__I,axiom,
! [A4: set_Pr8827868859434726151rm_f_v] :
( ! [X: produc8027700073345061287rm_f_v] : ( member4316649861536941904rm_f_v @ X @ A4 )
=> ( top_to5755558563273943127rm_f_v = A4 ) ) ).
% UNIV_eq_I
thf(fact_780_top__greatest,axiom,
! [A: set_Pr5245412377734449720_f_nat] : ( ord_le8976984241387448984_f_nat @ A @ top_to3664669310755125480_f_nat ) ).
% top_greatest
thf(fact_781_top_Oextremum__unique,axiom,
! [A: set_Pr5245412377734449720_f_nat] :
( ( ord_le8976984241387448984_f_nat @ top_to3664669310755125480_f_nat @ A )
= ( A = top_to3664669310755125480_f_nat ) ) ).
% top.extremum_unique
thf(fact_782_top_Oextremum__uniqueI,axiom,
! [A: set_Pr5245412377734449720_f_nat] :
( ( ord_le8976984241387448984_f_nat @ top_to3664669310755125480_f_nat @ A )
=> ( A = top_to3664669310755125480_f_nat ) ) ).
% top.extremum_uniqueI
thf(fact_783_subset__UNIV,axiom,
! [A4: set_Pr5245412377734449720_f_nat] : ( ord_le8976984241387448984_f_nat @ A4 @ top_to3664669310755125480_f_nat ) ).
% subset_UNIV
thf(fact_784_CR__join__left__I,axiom,
! [R: set_Pr8827868859434726151rm_f_v,X3: term_f_v,Y: term_f_v,Z2: term_f_v] :
( ( abstra3735013631728912813rm_f_v @ R @ top_top_set_term_f_v )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Y ) @ ( transi3028809021613410336rm_f_v @ R ) )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Z2 ) @ ( abstra8605546704478008873rm_f_v @ R ) )
=> ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y @ Z2 ) @ ( abstra8605546704478008873rm_f_v @ R ) ) ) ) ) ).
% CR_join_left_I
thf(fact_785_diamond__imp__CR_H,axiom,
! [S: set_Pr8827868859434726151rm_f_v,R: set_Pr8827868859434726151rm_f_v] :
( ( abstra3437181289681318589rm_f_v @ S )
=> ( ( ord_le4559761987009501863rm_f_v @ R @ S )
=> ( ( ord_le4559761987009501863rm_f_v @ S @ ( transi3028809021613410336rm_f_v @ R ) )
=> ( abstra3735013631728912813rm_f_v @ R @ top_top_set_term_f_v ) ) ) ) ).
% diamond_imp_CR'
thf(fact_786_strongly__confluent__E1n,axiom,
! [R: set_Pr8827868859434726151rm_f_v,X3: term_f_v,Y: term_f_v,Z2: term_f_v,N: nat] :
( ( abstra950190854684145090rm_f_v @ R @ top_top_set_term_f_v )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Y ) @ ( sup_su1994854688125285083rm_f_v @ R @ id_term_f_v ) )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Z2 ) @ ( compow8498445688072944954rm_f_v @ N @ R ) )
=> ? [U2: term_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y @ U2 ) @ ( transi3028809021613410336rm_f_v @ R ) )
& ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Z2 @ U2 ) @ ( sup_su1994854688125285083rm_f_v @ R @ id_term_f_v ) ) ) ) ) ) ).
% strongly_confluent_E1n
thf(fact_787_semi__confluence__imp__CR,axiom,
! [R: set_Pr8827868859434726151rm_f_v] :
( ( ord_le4559761987009501863rm_f_v @ ( relcom2934258169684288428rm_f_v @ ( conver3099152742603637891rm_f_v @ R ) @ ( transi3028809021613410336rm_f_v @ R ) ) @ ( abstra8605546704478008873rm_f_v @ R ) )
=> ( abstra3735013631728912813rm_f_v @ R @ top_top_set_term_f_v ) ) ).
% semi_confluence_imp_CR
thf(fact_788_converse__iff,axiom,
! [A: term_f_v,B: term_f_v,R: set_Pr8827868859434726151rm_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( conver3099152742603637891rm_f_v @ R ) )
= ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B @ A ) @ R ) ) ).
% converse_iff
thf(fact_789_converse__mono,axiom,
! [R: set_Pr5245412377734449720_f_nat,S: set_Pr5245412377734449720_f_nat] :
( ( ord_le8280083912720879166_nat_f @ ( converse_f_nat @ R ) @ ( converse_f_nat @ S ) )
= ( ord_le8976984241387448984_f_nat @ R @ S ) ) ).
% converse_mono
thf(fact_790_converse__mono,axiom,
! [R: set_Pr4548512049067879902_nat_f,S: set_Pr4548512049067879902_nat_f] :
( ( ord_le8976984241387448984_f_nat @ ( converse_nat_f @ R ) @ ( converse_nat_f @ S ) )
= ( ord_le8280083912720879166_nat_f @ R @ S ) ) ).
% converse_mono
thf(fact_791_converse_Ocases,axiom,
! [A1: term_f_v,A22: term_f_v,R: set_Pr8827868859434726151rm_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A1 @ A22 ) @ ( conver3099152742603637891rm_f_v @ R ) )
=> ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A22 @ A1 ) @ R ) ) ).
% converse.cases
thf(fact_792_converse_Osimps,axiom,
! [A1: term_f_v,A22: term_f_v,R: set_Pr8827868859434726151rm_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A1 @ A22 ) @ ( conver3099152742603637891rm_f_v @ R ) )
= ( ? [A5: term_f_v,B4: term_f_v] :
( ( A1 = B4 )
& ( A22 = A5 )
& ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A5 @ B4 ) @ R ) ) ) ) ).
% converse.simps
thf(fact_793_converseD,axiom,
! [A: term_f_v,B: term_f_v,R: set_Pr8827868859434726151rm_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( conver3099152742603637891rm_f_v @ R ) )
=> ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B @ A ) @ R ) ) ).
% converseD
thf(fact_794_converseE,axiom,
! [Yx: produc8027700073345061287rm_f_v,R: set_Pr8827868859434726151rm_f_v] :
( ( member4316649861536941904rm_f_v @ Yx @ ( conver3099152742603637891rm_f_v @ R ) )
=> ~ ! [X: term_f_v,Y3: term_f_v] :
( ( Yx
= ( produc2510790566102465375rm_f_v @ Y3 @ X ) )
=> ~ ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ Y3 ) @ R ) ) ) ).
% converseE
thf(fact_795_converseI,axiom,
! [A: term_f_v,B: term_f_v,R: set_Pr8827868859434726151rm_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ R )
=> ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B @ A ) @ ( conver3099152742603637891rm_f_v @ R ) ) ) ).
% converseI
thf(fact_796_top__empty__eq,axiom,
( top_to8437117464082123078_f_v_o
= ( ^ [X4: produc8027700073345061287rm_f_v] : ( member4316649861536941904rm_f_v @ X4 @ top_to5755558563273943127rm_f_v ) ) ) ).
% top_empty_eq
thf(fact_797_rtrancl__converse,axiom,
! [R: set_Pr8827868859434726151rm_f_v] :
( ( transi3028809021613410336rm_f_v @ ( conver3099152742603637891rm_f_v @ R ) )
= ( conver3099152742603637891rm_f_v @ ( transi3028809021613410336rm_f_v @ R ) ) ) ).
% rtrancl_converse
thf(fact_798_relpow__image,axiom,
! [R: set_Pr8827868859434726151rm_f_v,F2: term_f_v > term_f_v,R4: set_Pr8827868859434726151rm_f_v,S: term_f_v,T: term_f_v,N: nat] :
( ! [S3: term_f_v,T2: term_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S3 @ T2 ) @ R )
=> ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ ( F2 @ S3 ) @ ( F2 @ T2 ) ) @ R4 ) )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ ( compow8498445688072944954rm_f_v @ N @ R ) )
=> ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ ( F2 @ S ) @ ( F2 @ T ) ) @ ( compow8498445688072944954rm_f_v @ N @ R4 ) ) ) ) ).
% relpow_image
thf(fact_799_relpow__refl__mono,axiom,
! [Rel: set_Pr8827868859434726151rm_f_v,M: nat,N: nat,A: term_f_v,B: term_f_v] :
( ! [X: term_f_v] : ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ X ) @ Rel )
=> ( ( ord_less_eq_nat @ M @ N )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( compow8498445688072944954rm_f_v @ M @ Rel ) )
=> ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( compow8498445688072944954rm_f_v @ N @ Rel ) ) ) ) ) ).
% relpow_refl_mono
thf(fact_800_relpow__Suc__D2_H,axiom,
! [N: nat,R3: set_Pr8827868859434726151rm_f_v,X6: term_f_v,Y6: term_f_v,Z5: term_f_v] :
( ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X6 @ Y6 ) @ ( compow8498445688072944954rm_f_v @ N @ R3 ) )
& ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y6 @ Z5 ) @ R3 ) )
=> ? [W: term_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X6 @ W ) @ R3 )
& ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ W @ Z5 ) @ ( compow8498445688072944954rm_f_v @ N @ R3 ) ) ) ) ).
% relpow_Suc_D2'
thf(fact_801_converse__subset__swap,axiom,
! [R: set_Pr4548512049067879902_nat_f,S: set_Pr5245412377734449720_f_nat] :
( ( ord_le8280083912720879166_nat_f @ R @ ( converse_f_nat @ S ) )
= ( ord_le8976984241387448984_f_nat @ ( converse_nat_f @ R ) @ S ) ) ).
% converse_subset_swap
thf(fact_802_converse__subset__swap,axiom,
! [R: set_Pr5245412377734449720_f_nat,S: set_Pr4548512049067879902_nat_f] :
( ( ord_le8976984241387448984_f_nat @ R @ ( converse_nat_f @ S ) )
= ( ord_le8280083912720879166_nat_f @ ( converse_f_nat @ R ) @ S ) ) ).
% converse_subset_swap
thf(fact_803_rtrancl__imp__relpow,axiom,
! [P: produc8027700073345061287rm_f_v,R3: set_Pr8827868859434726151rm_f_v] :
( ( member4316649861536941904rm_f_v @ P @ ( transi3028809021613410336rm_f_v @ R3 ) )
=> ? [N2: nat] : ( member4316649861536941904rm_f_v @ P @ ( compow8498445688072944954rm_f_v @ N2 @ R3 ) ) ) ).
% rtrancl_imp_relpow
thf(fact_804_relpow__imp__rtrancl,axiom,
! [P: produc8027700073345061287rm_f_v,N: nat,R3: set_Pr8827868859434726151rm_f_v] :
( ( member4316649861536941904rm_f_v @ P @ ( compow8498445688072944954rm_f_v @ N @ R3 ) )
=> ( member4316649861536941904rm_f_v @ P @ ( transi3028809021613410336rm_f_v @ R3 ) ) ) ).
% relpow_imp_rtrancl
thf(fact_805_rtrancl__power,axiom,
! [P: produc8027700073345061287rm_f_v,R3: set_Pr8827868859434726151rm_f_v] :
( ( member4316649861536941904rm_f_v @ P @ ( transi3028809021613410336rm_f_v @ R3 ) )
= ( ? [N3: nat] : ( member4316649861536941904rm_f_v @ P @ ( compow8498445688072944954rm_f_v @ N3 @ R3 ) ) ) ) ).
% rtrancl_power
thf(fact_806_rstep__converse__dist,axiom,
! [R2: set_Pr8827868859434726151rm_f_v] :
( ( conver3099152742603637891rm_f_v @ ( rstep_f_v @ R2 ) )
= ( rstep_f_v @ ( conver3099152742603637891rm_f_v @ R2 ) ) ) ).
% rstep_converse_dist
thf(fact_807_rtrancl__converseI,axiom,
! [Y: term_f_v,X3: term_f_v,R: set_Pr8827868859434726151rm_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y @ X3 ) @ ( transi3028809021613410336rm_f_v @ R ) )
=> ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Y ) @ ( transi3028809021613410336rm_f_v @ ( conver3099152742603637891rm_f_v @ R ) ) ) ) ).
% rtrancl_converseI
thf(fact_808_rtrancl__converseD,axiom,
! [X3: term_f_v,Y: term_f_v,R: set_Pr8827868859434726151rm_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Y ) @ ( transi3028809021613410336rm_f_v @ ( conver3099152742603637891rm_f_v @ R ) ) )
=> ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y @ X3 ) @ ( transi3028809021613410336rm_f_v @ R ) ) ) ).
% rtrancl_converseD
thf(fact_809_peak__iff,axiom,
! [X3: term_f_v,Y: term_f_v,A4: set_Pr8827868859434726151rm_f_v,B5: set_Pr8827868859434726151rm_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Y ) @ ( relcom2934258169684288428rm_f_v @ ( conver3099152742603637891rm_f_v @ A4 ) @ B5 ) )
= ( ? [U3: term_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ U3 @ X3 ) @ A4 )
& ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ U3 @ Y ) @ B5 ) ) ) ) ).
% peak_iff
thf(fact_810_trancl__converseI,axiom,
! [X3: term_f_v,Y: term_f_v,R: set_Pr8827868859434726151rm_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Y ) @ ( conver3099152742603637891rm_f_v @ ( transi3208867851621559554rm_f_v @ R ) ) )
=> ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Y ) @ ( transi3208867851621559554rm_f_v @ ( conver3099152742603637891rm_f_v @ R ) ) ) ) ).
% trancl_converseI
thf(fact_811_trancl__converseD,axiom,
! [X3: term_f_v,Y: term_f_v,R: set_Pr8827868859434726151rm_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Y ) @ ( transi3208867851621559554rm_f_v @ ( conver3099152742603637891rm_f_v @ R ) ) )
=> ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Y ) @ ( conver3099152742603637891rm_f_v @ ( transi3208867851621559554rm_f_v @ R ) ) ) ) ).
% trancl_converseD
thf(fact_812_rtrancl__len__E,axiom,
! [X3: term_f_v,Y: term_f_v,R: set_Pr8827868859434726151rm_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Y ) @ ( transi3028809021613410336rm_f_v @ R ) )
=> ~ ! [N2: nat] :
~ ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Y ) @ ( compow8498445688072944954rm_f_v @ N2 @ R ) ) ) ).
% rtrancl_len_E
thf(fact_813_srstep__converse__dist,axiom,
! [F: set_Pr5245412377734449720_f_nat,R2: set_Pr8827868859434726151rm_f_v] :
( ( conver3099152742603637891rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ R2 ) ) )
= ( sig_step_f_v @ F @ ( rstep_f_v @ ( conver3099152742603637891rm_f_v @ R2 ) ) ) ) ).
% srstep_converse_dist
thf(fact_814_sig__step__converse__rstep,axiom,
! [F: set_Pr5245412377734449720_f_nat,R2: set_Pr8827868859434726151rm_f_v] :
( ( conver3099152742603637891rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ R2 ) ) )
= ( sig_step_f_v @ F @ ( conver3099152742603637891rm_f_v @ ( rstep_f_v @ R2 ) ) ) ) ).
% sig_step_converse_rstep
thf(fact_815_srrstep__converse__dist,axiom,
! [F: set_Pr5245412377734449720_f_nat,R2: set_Pr8827868859434726151rm_f_v] :
( ( conver3099152742603637891rm_f_v @ ( sig_step_f_v @ F @ ( rrstep_f_v @ R2 ) ) )
= ( sig_step_f_v @ F @ ( rrstep_f_v @ ( conver3099152742603637891rm_f_v @ R2 ) ) ) ) ).
% srrstep_converse_dist
thf(fact_816_commute__def,axiom,
( abstra4964661570048985923rm_f_v
= ( ^ [R7: set_Pr8827868859434726151rm_f_v,S4: set_Pr8827868859434726151rm_f_v] : ( ord_le4559761987009501863rm_f_v @ ( relcom2934258169684288428rm_f_v @ ( transi3028809021613410336rm_f_v @ ( conver3099152742603637891rm_f_v @ R7 ) ) @ ( transi3028809021613410336rm_f_v @ S4 ) ) @ ( relcom2934258169684288428rm_f_v @ ( transi3028809021613410336rm_f_v @ S4 ) @ ( transi3028809021613410336rm_f_v @ ( conver3099152742603637891rm_f_v @ R7 ) ) ) ) ) ) ).
% commute_def
thf(fact_817_rstep__smycl__dist,axiom,
! [R2: set_Pr8827868859434726151rm_f_v] :
( ( sup_su1994854688125285083rm_f_v @ ( rstep_f_v @ R2 ) @ ( conver3099152742603637891rm_f_v @ ( rstep_f_v @ R2 ) ) )
= ( rstep_f_v @ ( sup_su1994854688125285083rm_f_v @ R2 @ ( conver3099152742603637891rm_f_v @ R2 ) ) ) ) ).
% rstep_smycl_dist
thf(fact_818_sig__step__symcl__dist,axiom,
! [F: set_Pr5245412377734449720_f_nat,R2: set_Pr8827868859434726151rm_f_v] :
( ( sup_su1994854688125285083rm_f_v @ ( sig_step_f_v @ F @ R2 ) @ ( conver3099152742603637891rm_f_v @ ( sig_step_f_v @ F @ R2 ) ) )
= ( sig_step_f_v @ F @ ( sup_su1994854688125285083rm_f_v @ R2 @ ( conver3099152742603637891rm_f_v @ R2 ) ) ) ) ).
% sig_step_symcl_dist
thf(fact_819_diamond__imp__semi__confluence,axiom,
! [R: set_Pr8827868859434726151rm_f_v] :
( ( abstra3437181289681318589rm_f_v @ R )
=> ( ord_le4559761987009501863rm_f_v @ ( relcom2934258169684288428rm_f_v @ ( conver3099152742603637891rm_f_v @ R ) @ ( transi3028809021613410336rm_f_v @ R ) ) @ ( abstra8605546704478008873rm_f_v @ R ) ) ) ).
% diamond_imp_semi_confluence
thf(fact_820_trancl__steps__relpow,axiom,
! [A: set_Pr8827868859434726151rm_f_v,B: set_Pr8827868859434726151rm_f_v,X3: term_f_v,Y: term_f_v,N: nat] :
( ( ord_le4559761987009501863rm_f_v @ A @ ( transi3208867851621559554rm_f_v @ B ) )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Y ) @ ( compow8498445688072944954rm_f_v @ N @ A ) )
=> ? [M2: nat] :
( ( ord_less_eq_nat @ N @ M2 )
& ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Y ) @ ( compow8498445688072944954rm_f_v @ M2 @ B ) ) ) ) ) ).
% trancl_steps_relpow
thf(fact_821_join__def,axiom,
( abstra8605546704478008873rm_f_v
= ( ^ [A6: set_Pr8827868859434726151rm_f_v] : ( relcom2934258169684288428rm_f_v @ ( transi3028809021613410336rm_f_v @ A6 ) @ ( transi3028809021613410336rm_f_v @ ( conver3099152742603637891rm_f_v @ A6 ) ) ) ) ) ).
% join_def
thf(fact_822_meet__def,axiom,
( abstra5602527437878040038rm_f_v
= ( ^ [A6: set_Pr8827868859434726151rm_f_v] : ( relcom2934258169684288428rm_f_v @ ( transi3028809021613410336rm_f_v @ ( conver3099152742603637891rm_f_v @ A6 ) ) @ ( transi3028809021613410336rm_f_v @ A6 ) ) ) ) ).
% meet_def
thf(fact_823_srstep__symcl__dist,axiom,
! [F: set_Pr5245412377734449720_f_nat,R2: set_Pr8827868859434726151rm_f_v] :
( ( sup_su1994854688125285083rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ R2 ) ) @ ( conver3099152742603637891rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ R2 ) ) ) )
= ( sig_step_f_v @ F @ ( rstep_f_v @ ( sup_su1994854688125285083rm_f_v @ R2 @ ( conver3099152742603637891rm_f_v @ R2 ) ) ) ) ) ).
% srstep_symcl_dist
thf(fact_824_conversionI,axiom,
! [A: term_f_v,B: term_f_v,A4: set_Pr8827868859434726151rm_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( transi3028809021613410336rm_f_v @ ( sup_su1994854688125285083rm_f_v @ A4 @ ( conver3099152742603637891rm_f_v @ A4 ) ) ) )
=> ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( abstra5640035136655339381rm_f_v @ A4 ) ) ) ).
% conversionI
thf(fact_825_sig__step__conversion__dist,axiom,
! [F: set_Pr5245412377734449720_f_nat,R2: set_Pr8827868859434726151rm_f_v] :
( ( abstra5640035136655339381rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ R2 ) ) )
= ( transi3028809021613410336rm_f_v @ ( sig_step_f_v @ F @ ( rstep_f_v @ ( sup_su1994854688125285083rm_f_v @ R2 @ ( conver3099152742603637891rm_f_v @ R2 ) ) ) ) ) ) ).
% sig_step_conversion_dist
thf(fact_826_Diff__iff,axiom,
! [C3: produc8027700073345061287rm_f_v,A4: set_Pr8827868859434726151rm_f_v,B5: set_Pr8827868859434726151rm_f_v] :
( ( member4316649861536941904rm_f_v @ C3 @ ( minus_410040198995547630rm_f_v @ A4 @ B5 ) )
= ( ( member4316649861536941904rm_f_v @ C3 @ A4 )
& ~ ( member4316649861536941904rm_f_v @ C3 @ B5 ) ) ) ).
% Diff_iff
thf(fact_827_DiffI,axiom,
! [C3: produc8027700073345061287rm_f_v,A4: set_Pr8827868859434726151rm_f_v,B5: set_Pr8827868859434726151rm_f_v] :
( ( member4316649861536941904rm_f_v @ C3 @ A4 )
=> ( ~ ( member4316649861536941904rm_f_v @ C3 @ B5 )
=> ( member4316649861536941904rm_f_v @ C3 @ ( minus_410040198995547630rm_f_v @ A4 @ B5 ) ) ) ) ).
% DiffI
thf(fact_828_conversion__rtrancl,axiom,
! [A4: set_Pr8827868859434726151rm_f_v] :
( ( transi3028809021613410336rm_f_v @ ( abstra5640035136655339381rm_f_v @ A4 ) )
= ( abstra5640035136655339381rm_f_v @ A4 ) ) ).
% conversion_rtrancl
thf(fact_829_conversionI_H,axiom,
! [A: term_f_v,B: term_f_v,A4: set_Pr8827868859434726151rm_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( transi3028809021613410336rm_f_v @ A4 ) )
=> ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( abstra5640035136655339381rm_f_v @ A4 ) ) ) ).
% conversionI'
thf(fact_830_DiffD2,axiom,
! [C3: produc8027700073345061287rm_f_v,A4: set_Pr8827868859434726151rm_f_v,B5: set_Pr8827868859434726151rm_f_v] :
( ( member4316649861536941904rm_f_v @ C3 @ ( minus_410040198995547630rm_f_v @ A4 @ B5 ) )
=> ~ ( member4316649861536941904rm_f_v @ C3 @ B5 ) ) ).
% DiffD2
thf(fact_831_DiffD1,axiom,
! [C3: produc8027700073345061287rm_f_v,A4: set_Pr8827868859434726151rm_f_v,B5: set_Pr8827868859434726151rm_f_v] :
( ( member4316649861536941904rm_f_v @ C3 @ ( minus_410040198995547630rm_f_v @ A4 @ B5 ) )
=> ( member4316649861536941904rm_f_v @ C3 @ A4 ) ) ).
% DiffD1
thf(fact_832_DiffE,axiom,
! [C3: produc8027700073345061287rm_f_v,A4: set_Pr8827868859434726151rm_f_v,B5: set_Pr8827868859434726151rm_f_v] :
( ( member4316649861536941904rm_f_v @ C3 @ ( minus_410040198995547630rm_f_v @ A4 @ B5 ) )
=> ~ ( ( member4316649861536941904rm_f_v @ C3 @ A4 )
=> ( member4316649861536941904rm_f_v @ C3 @ B5 ) ) ) ).
% DiffE
thf(fact_833_Diff__mono,axiom,
! [A4: set_Pr5245412377734449720_f_nat,C2: set_Pr5245412377734449720_f_nat,D: set_Pr5245412377734449720_f_nat,B5: set_Pr5245412377734449720_f_nat] :
( ( ord_le8976984241387448984_f_nat @ A4 @ C2 )
=> ( ( ord_le8976984241387448984_f_nat @ D @ B5 )
=> ( ord_le8976984241387448984_f_nat @ ( minus_8941878871097791505_f_nat @ A4 @ B5 ) @ ( minus_8941878871097791505_f_nat @ C2 @ D ) ) ) ) ).
% Diff_mono
thf(fact_834_Diff__subset,axiom,
! [A4: set_Pr5245412377734449720_f_nat,B5: set_Pr5245412377734449720_f_nat] : ( ord_le8976984241387448984_f_nat @ ( minus_8941878871097791505_f_nat @ A4 @ B5 ) @ A4 ) ).
% Diff_subset
thf(fact_835_double__diff,axiom,
! [A4: set_Pr5245412377734449720_f_nat,B5: set_Pr5245412377734449720_f_nat,C2: set_Pr5245412377734449720_f_nat] :
( ( ord_le8976984241387448984_f_nat @ A4 @ B5 )
=> ( ( ord_le8976984241387448984_f_nat @ B5 @ C2 )
=> ( ( minus_8941878871097791505_f_nat @ B5 @ ( minus_8941878871097791505_f_nat @ C2 @ A4 ) )
= A4 ) ) ) ).
% double_diff
thf(fact_836_conversion__refl,axiom,
! [A: term_f_v,A4: set_Pr8827868859434726151rm_f_v] : ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ A ) @ ( abstra5640035136655339381rm_f_v @ A4 ) ) ).
% conversion_refl
thf(fact_837_conversion__inv,axiom,
! [X3: term_f_v,Y: term_f_v,R3: set_Pr8827868859434726151rm_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Y ) @ ( abstra5640035136655339381rm_f_v @ R3 ) )
= ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y @ X3 ) @ ( abstra5640035136655339381rm_f_v @ R3 ) ) ) ).
% conversion_inv
thf(fact_838_Diff__partition,axiom,
! [A4: set_Pr5245412377734449720_f_nat,B5: set_Pr5245412377734449720_f_nat] :
( ( ord_le8976984241387448984_f_nat @ A4 @ B5 )
=> ( ( sup_su770888850772321124_f_nat @ A4 @ ( minus_8941878871097791505_f_nat @ B5 @ A4 ) )
= B5 ) ) ).
% Diff_partition
thf(fact_839_Diff__subset__conv,axiom,
! [A4: set_Pr5245412377734449720_f_nat,B5: set_Pr5245412377734449720_f_nat,C2: set_Pr5245412377734449720_f_nat] :
( ( ord_le8976984241387448984_f_nat @ ( minus_8941878871097791505_f_nat @ A4 @ B5 ) @ C2 )
= ( ord_le8976984241387448984_f_nat @ A4 @ ( sup_su770888850772321124_f_nat @ B5 @ C2 ) ) ) ).
% Diff_subset_conv
thf(fact_840_rtrancl__r__diff__Id,axiom,
! [R: set_Pr8827868859434726151rm_f_v] :
( ( transi3028809021613410336rm_f_v @ ( minus_410040198995547630rm_f_v @ R @ id_term_f_v ) )
= ( transi3028809021613410336rm_f_v @ R ) ) ).
% rtrancl_r_diff_Id
thf(fact_841_rtrancl__diff__decomp,axiom,
! [X3: term_f_v,Y: term_f_v,A4: set_Pr8827868859434726151rm_f_v,B5: set_Pr8827868859434726151rm_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Y ) @ ( minus_410040198995547630rm_f_v @ ( transi3028809021613410336rm_f_v @ A4 ) @ ( transi3028809021613410336rm_f_v @ B5 ) ) )
=> ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Y ) @ ( relcom2934258169684288428rm_f_v @ ( transi3028809021613410336rm_f_v @ A4 ) @ ( relcom2934258169684288428rm_f_v @ ( minus_410040198995547630rm_f_v @ A4 @ B5 ) @ ( transi3028809021613410336rm_f_v @ A4 ) ) ) ) ) ).
% rtrancl_diff_decomp
thf(fact_842_conversion__def,axiom,
( abstra5640035136655339381rm_f_v
= ( ^ [A6: set_Pr8827868859434726151rm_f_v] : ( transi3028809021613410336rm_f_v @ ( sup_su1994854688125285083rm_f_v @ A6 @ ( conver3099152742603637891rm_f_v @ A6 ) ) ) ) ) ).
% conversion_def
thf(fact_843_conversionE,axiom,
! [A: term_f_v,B: term_f_v,A4: set_Pr8827868859434726151rm_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( abstra5640035136655339381rm_f_v @ A4 ) )
=> ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( transi3028809021613410336rm_f_v @ ( sup_su1994854688125285083rm_f_v @ A4 @ ( conver3099152742603637891rm_f_v @ A4 ) ) ) ) ) ).
% conversionE
thf(fact_844_CE__on__def,axiom,
( rewrit5169762006700492917rm_f_v
= ( ^ [R7: set_Pr8827868859434726151rm_f_v,S4: set_Pr8827868859434726151rm_f_v,A6: set_term_f_v] :
! [X4: term_f_v] :
( ( member_term_f_v @ X4 @ A6 )
=> ! [B4: term_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X4 @ B4 ) @ ( abstra5640035136655339381rm_f_v @ R7 ) )
= ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X4 @ B4 ) @ ( abstra5640035136655339381rm_f_v @ S4 ) ) ) ) ) ) ).
% CE_on_def
thf(fact_845_qc__SN__relto__iff,axiom,
! [R: set_Pr8827868859434726151rm_f_v,S: set_Pr8827868859434726151rm_f_v] :
( ( ord_le4559761987009501863rm_f_v @ ( relcom2934258169684288428rm_f_v @ R @ S ) @ ( relcom2934258169684288428rm_f_v @ S @ ( transi3028809021613410336rm_f_v @ ( sup_su1994854688125285083rm_f_v @ S @ R ) ) ) )
=> ( ( abstra6117554785717145rm_f_v @ ( relcom2934258169684288428rm_f_v @ ( transi3028809021613410336rm_f_v @ R ) @ ( relcom2934258169684288428rm_f_v @ S @ ( transi3028809021613410336rm_f_v @ R ) ) ) @ top_top_set_term_f_v )
= ( abstra6117554785717145rm_f_v @ S @ top_top_set_term_f_v ) ) ) ).
% qc_SN_relto_iff
thf(fact_846_compatible__SN_H,axiom,
! [NS: set_Pr8827868859434726151rm_f_v,S2: set_Pr8827868859434726151rm_f_v] :
( ( ord_le4559761987009501863rm_f_v @ ( relcom2934258169684288428rm_f_v @ NS @ S2 ) @ S2 )
=> ( ( abstra6117554785717145rm_f_v @ S2 @ top_top_set_term_f_v )
=> ( abstra6117554785717145rm_f_v @ ( relcom2934258169684288428rm_f_v @ ( transi3028809021613410336rm_f_v @ ( sup_su1994854688125285083rm_f_v @ NS @ S2 ) ) @ ( relcom2934258169684288428rm_f_v @ S2 @ ( transi3028809021613410336rm_f_v @ ( sup_su1994854688125285083rm_f_v @ NS @ S2 ) ) ) ) @ top_top_set_term_f_v ) ) ) ).
% compatible_SN'
thf(fact_847_SN__on__subset2,axiom,
! [A4: set_Pr5245412377734449720_f_nat,B5: set_Pr5245412377734449720_f_nat,R: set_Pr516226060960033221_f_nat] :
( ( ord_le8976984241387448984_f_nat @ A4 @ B5 )
=> ( ( abstra5607565530240152504_f_nat @ R @ B5 )
=> ( abstra5607565530240152504_f_nat @ R @ A4 ) ) ) ).
% SN_on_subset2
thf(fact_848_SN__on__induct,axiom,
! [R3: set_Pr7897041799657265351rm_f_v,A4: set_Pr8827868859434726151rm_f_v,S: produc8027700073345061287rm_f_v,P2: produc8027700073345061287rm_f_v > $o] :
( ( abstra845488452567719153rm_f_v @ R3 @ A4 )
=> ( ( member4316649861536941904rm_f_v @ S @ A4 )
=> ( ! [T2: produc8027700073345061287rm_f_v] :
( ! [U4: produc8027700073345061287rm_f_v] :
( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ T2 @ U4 ) @ R3 )
=> ( P2 @ U4 ) )
=> ( P2 @ T2 ) )
=> ( P2 @ S ) ) ) ) ).
% SN_on_induct
thf(fact_849_SN__on__induct,axiom,
! [R3: set_Pr8827868859434726151rm_f_v,A4: set_term_f_v,S: term_f_v,P2: term_f_v > $o] :
( ( abstra6117554785717145rm_f_v @ R3 @ A4 )
=> ( ( member_term_f_v @ S @ A4 )
=> ( ! [T2: term_f_v] :
( ! [U4: term_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ T2 @ U4 ) @ R3 )
=> ( P2 @ U4 ) )
=> ( P2 @ T2 ) )
=> ( P2 @ S ) ) ) ) ).
% SN_on_induct
thf(fact_850_SN__on__irrefl,axiom,
! [R: set_Pr8827868859434726151rm_f_v,A4: set_term_f_v] :
( ( abstra6117554785717145rm_f_v @ R @ A4 )
=> ! [X6: term_f_v] :
( ( member_term_f_v @ X6 @ A4 )
=> ~ ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X6 @ X6 ) @ R ) ) ) ).
% SN_on_irrefl
thf(fact_851_refl__not__SN,axiom,
! [X3: term_f_v,R3: set_Pr8827868859434726151rm_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ X3 ) @ R3 )
=> ~ ( abstra6117554785717145rm_f_v @ R3 @ top_top_set_term_f_v ) ) ).
% refl_not_SN
thf(fact_852_SN__imp__minimal,axiom,
! [A4: set_Pr7897041799657265351rm_f_v] :
( ( abstra845488452567719153rm_f_v @ A4 @ top_to5755558563273943127rm_f_v )
=> ! [Q2: set_Pr8827868859434726151rm_f_v] :
( ? [X6: produc8027700073345061287rm_f_v] : ( member4316649861536941904rm_f_v @ X6 @ Q2 )
=> ? [X: produc8027700073345061287rm_f_v] :
( ( member4316649861536941904rm_f_v @ X @ Q2 )
& ! [Y6: produc8027700073345061287rm_f_v] :
( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ X @ Y6 ) @ A4 )
=> ~ ( member4316649861536941904rm_f_v @ Y6 @ Q2 ) ) ) ) ) ).
% SN_imp_minimal
thf(fact_853_SN__imp__minimal,axiom,
! [A4: set_Pr8827868859434726151rm_f_v] :
( ( abstra6117554785717145rm_f_v @ A4 @ top_top_set_term_f_v )
=> ! [Q2: set_term_f_v] :
( ? [X6: term_f_v] : ( member_term_f_v @ X6 @ Q2 )
=> ? [X: term_f_v] :
( ( member_term_f_v @ X @ Q2 )
& ! [Y6: term_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ Y6 ) @ A4 )
=> ~ ( member_term_f_v @ Y6 @ Q2 ) ) ) ) ) ).
% SN_imp_minimal
thf(fact_854_SN__induct__rule,axiom,
! [R: set_Pr8827868859434726151rm_f_v,P2: term_f_v > $o,A: term_f_v] :
( ( abstra6117554785717145rm_f_v @ R @ top_top_set_term_f_v )
=> ( ! [A3: term_f_v] :
( ! [B8: term_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A3 @ B8 ) @ R )
=> ( P2 @ B8 ) )
=> ( P2 @ A3 ) )
=> ( P2 @ A ) ) ) ).
% SN_induct_rule
thf(fact_855_SN__on__O__push,axiom,
! [S2: set_Pr8827868859434726151rm_f_v,R3: set_Pr8827868859434726151rm_f_v,X5: set_term_f_v] :
( ( ord_le4559761987009501863rm_f_v @ ( relcom2934258169684288428rm_f_v @ S2 @ R3 ) @ ( relcom2934258169684288428rm_f_v @ R3 @ ( transi3028809021613410336rm_f_v @ S2 ) ) )
=> ( ( abstra6117554785717145rm_f_v @ R3 @ X5 )
=> ( abstra6117554785717145rm_f_v @ ( relcom2934258169684288428rm_f_v @ R3 @ ( transi3028809021613410336rm_f_v @ S2 ) ) @ X5 ) ) ) ).
% SN_on_O_push
thf(fact_856_compatible__SN,axiom,
! [S2: set_Pr8827868859434726151rm_f_v,NS: set_Pr8827868859434726151rm_f_v] :
( ( abstra6117554785717145rm_f_v @ S2 @ top_top_set_term_f_v )
=> ( ( ord_le4559761987009501863rm_f_v @ ( relcom2934258169684288428rm_f_v @ NS @ S2 ) @ S2 )
=> ( abstra6117554785717145rm_f_v @ ( relcom2934258169684288428rm_f_v @ S2 @ ( relcom2934258169684288428rm_f_v @ ( transi3028809021613410336rm_f_v @ S2 ) @ ( transi3028809021613410336rm_f_v @ NS ) ) ) @ top_top_set_term_f_v ) ) ) ).
% compatible_SN
thf(fact_857_the__NF__conv,axiom,
! [A4: set_Pr8827868859434726151rm_f_v,A: term_f_v,B: term_f_v] :
( ( abstra6117554785717145rm_f_v @ A4 @ top_top_set_term_f_v )
=> ( ( abstra3735013631728912813rm_f_v @ A4 @ top_top_set_term_f_v )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( abstra5640035136655339381rm_f_v @ A4 ) )
=> ( ( abstra6752248511232870546rm_f_v @ A4 @ A )
= ( abstra6752248511232870546rm_f_v @ A4 @ B ) ) ) ) ) ).
% the_NF_conv
thf(fact_858_the__NF__steps,axiom,
! [A4: set_Pr8827868859434726151rm_f_v,A: term_f_v,B: term_f_v] :
( ( abstra6117554785717145rm_f_v @ A4 @ top_top_set_term_f_v )
=> ( ( abstra3735013631728912813rm_f_v @ A4 @ top_top_set_term_f_v )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( transi3028809021613410336rm_f_v @ A4 ) )
=> ( ( abstra6752248511232870546rm_f_v @ A4 @ A )
= ( abstra6752248511232870546rm_f_v @ A4 @ B ) ) ) ) ) ).
% the_NF_steps
thf(fact_859_the__NF__step,axiom,
! [A4: set_Pr8827868859434726151rm_f_v,A: term_f_v,B: term_f_v] :
( ( abstra6117554785717145rm_f_v @ A4 @ top_top_set_term_f_v )
=> ( ( abstra3735013631728912813rm_f_v @ A4 @ top_top_set_term_f_v )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ A4 )
=> ( ( abstra6752248511232870546rm_f_v @ A4 @ A )
= ( abstra6752248511232870546rm_f_v @ A4 @ B ) ) ) ) ) ).
% the_NF_step
thf(fact_860_the__NF,axiom,
! [A4: set_Pr8827868859434726151rm_f_v,A: term_f_v] :
( ( abstra6117554785717145rm_f_v @ A4 @ top_top_set_term_f_v )
=> ( ( abstra3735013631728912813rm_f_v @ A4 @ top_top_set_term_f_v )
=> ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ ( abstra6752248511232870546rm_f_v @ A4 @ A ) ) @ ( abstra7482117481303700155rm_f_v @ A4 ) ) ) ) ).
% the_NF
thf(fact_861_semi__complete__imp__conversionIff__same__NF,axiom,
! [R: set_Pr8827868859434726151rm_f_v,X3: term_f_v,Y: term_f_v] :
( ( abstra2448401485267669509rm_f_v @ R @ top_top_set_term_f_v )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Y ) @ ( abstra5640035136655339381rm_f_v @ R ) )
= ( ! [U3: term_f_v,V2: term_f_v] :
( ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ U3 ) @ ( abstra7482117481303700155rm_f_v @ R ) )
& ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y @ V2 ) @ ( abstra7482117481303700155rm_f_v @ R ) ) )
=> ( U3 = V2 ) ) ) ) ) ).
% semi_complete_imp_conversionIff_same_NF
thf(fact_862_the__NF__NF,axiom,
! [A4: set_Pr7897041799657265351rm_f_v,A: produc8027700073345061287rm_f_v] :
( ( abstra845488452567719153rm_f_v @ A4 @ top_to5755558563273943127rm_f_v )
=> ( ( abstra5806628924625096669rm_f_v @ A4 @ top_to5755558563273943127rm_f_v )
=> ( member4316649861536941904rm_f_v @ ( abstra7419483983937885496rm_f_v @ A4 @ A ) @ ( abstra349717802288577587rm_f_v @ A4 ) ) ) ) ).
% the_NF_NF
thf(fact_863_UNC__def,axiom,
( abstra6927431686333010263rm_f_v
= ( ^ [A6: set_Pr7897041799657265351rm_f_v] :
! [A5: produc8027700073345061287rm_f_v,B4: produc8027700073345061287rm_f_v] :
( ( ( member4316649861536941904rm_f_v @ A5 @ ( abstra349717802288577587rm_f_v @ A6 ) )
& ( member4316649861536941904rm_f_v @ B4 @ ( abstra349717802288577587rm_f_v @ A6 ) )
& ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A5 @ B4 ) @ ( abstra4572590880773903381rm_f_v @ A6 ) ) )
=> ( A5 = B4 ) ) ) ) ).
% UNC_def
thf(fact_864_UNC__def,axiom,
( abstra9159237726496465843rm_f_v
= ( ^ [A6: set_Pr8827868859434726151rm_f_v] :
! [A5: term_f_v,B4: term_f_v] :
( ( ( member_term_f_v @ A5 @ ( abstract_NF_term_f_v @ A6 ) )
& ( member_term_f_v @ B4 @ ( abstract_NF_term_f_v @ A6 ) )
& ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A5 @ B4 ) @ ( abstra5640035136655339381rm_f_v @ A6 ) ) )
=> ( A5 = B4 ) ) ) ) ).
% UNC_def
thf(fact_865_NF__I,axiom,
! [A: produc8027700073345061287rm_f_v,A4: set_Pr7897041799657265351rm_f_v] :
( ! [B3: produc8027700073345061287rm_f_v] :
~ ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A @ B3 ) @ A4 )
=> ( member4316649861536941904rm_f_v @ A @ ( abstra349717802288577587rm_f_v @ A4 ) ) ) ).
% NF_I
thf(fact_866_NF__I,axiom,
! [A: term_f_v,A4: set_Pr8827868859434726151rm_f_v] :
( ! [B3: term_f_v] :
~ ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B3 ) @ A4 )
=> ( member_term_f_v @ A @ ( abstract_NF_term_f_v @ A4 ) ) ) ).
% NF_I
thf(fact_867_normalizability__I_H,axiom,
! [A: term_f_v,B: term_f_v,A4: set_Pr8827868859434726151rm_f_v,C3: term_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( transi3028809021613410336rm_f_v @ A4 ) )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B @ C3 ) @ ( abstra7482117481303700155rm_f_v @ A4 ) )
=> ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ C3 ) @ ( abstra7482117481303700155rm_f_v @ A4 ) ) ) ) ).
% normalizability_I'
thf(fact_868_WN__onI,axiom,
! [A4: set_Pr8827868859434726151rm_f_v,R: set_Pr7897041799657265351rm_f_v] :
( ! [A3: produc8027700073345061287rm_f_v] :
( ( member4316649861536941904rm_f_v @ A3 @ A4 )
=> ? [B8: produc8027700073345061287rm_f_v] : ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A3 @ B8 ) @ ( abstra9166317777578425423rm_f_v @ R ) ) )
=> ( abstra1168807892682504437rm_f_v @ R @ A4 ) ) ).
% WN_onI
thf(fact_869_WN__onI,axiom,
! [A4: set_term_f_v,R: set_Pr8827868859434726151rm_f_v] :
( ! [A3: term_f_v] :
( ( member_term_f_v @ A3 @ A4 )
=> ? [B8: term_f_v] : ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A3 @ B8 ) @ ( abstra7482117481303700155rm_f_v @ R ) ) )
=> ( abstra9154201316845187733rm_f_v @ R @ A4 ) ) ).
% WN_onI
thf(fact_870_UNF__onI,axiom,
! [A4: set_Pr8827868859434726151rm_f_v,R: set_Pr7897041799657265351rm_f_v] :
( ! [A3: produc8027700073345061287rm_f_v,B3: produc8027700073345061287rm_f_v,C: produc8027700073345061287rm_f_v] :
( ( member4316649861536941904rm_f_v @ A3 @ A4 )
=> ( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A3 @ B3 ) @ ( abstra9166317777578425423rm_f_v @ R ) )
=> ( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A3 @ C ) @ ( abstra9166317777578425423rm_f_v @ R ) )
=> ( B3 = C ) ) ) )
=> ( abstra8974104706389925717rm_f_v @ R @ A4 ) ) ).
% UNF_onI
thf(fact_871_UNF__onI,axiom,
! [A4: set_term_f_v,R: set_Pr8827868859434726151rm_f_v] :
( ! [A3: term_f_v,B3: term_f_v,C: term_f_v] :
( ( member_term_f_v @ A3 @ A4 )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A3 @ B3 ) @ ( abstra7482117481303700155rm_f_v @ R ) )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A3 @ C ) @ ( abstra7482117481303700155rm_f_v @ R ) )
=> ( B3 = C ) ) ) )
=> ( abstra8129321985590711349rm_f_v @ R @ A4 ) ) ).
% UNF_onI
thf(fact_872_normalizability__I,axiom,
! [A: produc8027700073345061287rm_f_v,B: produc8027700073345061287rm_f_v,A4: set_Pr7897041799657265351rm_f_v] :
( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A @ B ) @ ( transi1062743709496199978rm_f_v @ A4 ) )
=> ( ( member4316649861536941904rm_f_v @ B @ ( abstra349717802288577587rm_f_v @ A4 ) )
=> ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A @ B ) @ ( abstra9166317777578425423rm_f_v @ A4 ) ) ) ) ).
% normalizability_I
thf(fact_873_normalizability__I,axiom,
! [A: term_f_v,B: term_f_v,A4: set_Pr8827868859434726151rm_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( transi3028809021613410336rm_f_v @ A4 ) )
=> ( ( member_term_f_v @ B @ ( abstract_NF_term_f_v @ A4 ) )
=> ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( abstra7482117481303700155rm_f_v @ A4 ) ) ) ) ).
% normalizability_I
thf(fact_874_NF__E,axiom,
! [A: produc8027700073345061287rm_f_v,A4: set_Pr7897041799657265351rm_f_v,B: produc8027700073345061287rm_f_v] :
( ( member4316649861536941904rm_f_v @ A @ ( abstra349717802288577587rm_f_v @ A4 ) )
=> ~ ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A @ B ) @ A4 ) ) ).
% NF_E
thf(fact_875_NF__E,axiom,
! [A: term_f_v,A4: set_Pr8827868859434726151rm_f_v,B: term_f_v] :
( ( member_term_f_v @ A @ ( abstract_NF_term_f_v @ A4 ) )
=> ~ ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ A4 ) ) ).
% NF_E
thf(fact_876_NF__no__step,axiom,
! [A: produc8027700073345061287rm_f_v,A4: set_Pr7897041799657265351rm_f_v] :
( ( member4316649861536941904rm_f_v @ A @ ( abstra349717802288577587rm_f_v @ A4 ) )
=> ! [B8: produc8027700073345061287rm_f_v] :
~ ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A @ B8 ) @ A4 ) ) ).
% NF_no_step
thf(fact_877_NF__no__step,axiom,
! [A: term_f_v,A4: set_Pr8827868859434726151rm_f_v] :
( ( member_term_f_v @ A @ ( abstract_NF_term_f_v @ A4 ) )
=> ! [B8: term_f_v] :
~ ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B8 ) @ A4 ) ) ).
% NF_no_step
thf(fact_878_NF__iff__no__step,axiom,
! [A: produc8027700073345061287rm_f_v,A4: set_Pr7897041799657265351rm_f_v] :
( ( member4316649861536941904rm_f_v @ A @ ( abstra349717802288577587rm_f_v @ A4 ) )
= ( ! [B4: produc8027700073345061287rm_f_v] :
~ ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A @ B4 ) @ A4 ) ) ) ).
% NF_iff_no_step
thf(fact_879_NF__iff__no__step,axiom,
! [A: term_f_v,A4: set_Pr8827868859434726151rm_f_v] :
( ( member_term_f_v @ A @ ( abstract_NF_term_f_v @ A4 ) )
= ( ! [B4: term_f_v] :
~ ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B4 ) @ A4 ) ) ) ).
% NF_iff_no_step
thf(fact_880_normalizability__E,axiom,
! [A: produc8027700073345061287rm_f_v,B: produc8027700073345061287rm_f_v,A4: set_Pr7897041799657265351rm_f_v] :
( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A @ B ) @ ( abstra9166317777578425423rm_f_v @ A4 ) )
=> ~ ( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A @ B ) @ ( transi1062743709496199978rm_f_v @ A4 ) )
=> ~ ( member4316649861536941904rm_f_v @ B @ ( abstra349717802288577587rm_f_v @ A4 ) ) ) ) ).
% normalizability_E
thf(fact_881_normalizability__E,axiom,
! [A: term_f_v,B: term_f_v,A4: set_Pr8827868859434726151rm_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( abstra7482117481303700155rm_f_v @ A4 ) )
=> ~ ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( transi3028809021613410336rm_f_v @ A4 ) )
=> ~ ( member_term_f_v @ B @ ( abstract_NF_term_f_v @ A4 ) ) ) ) ).
% normalizability_E
thf(fact_882_NF__not__suc,axiom,
! [X3: produc8027700073345061287rm_f_v,Y: produc8027700073345061287rm_f_v,R: set_Pr7897041799657265351rm_f_v] :
( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ X3 @ Y ) @ ( transi1062743709496199978rm_f_v @ R ) )
=> ( ( member4316649861536941904rm_f_v @ X3 @ ( abstra349717802288577587rm_f_v @ R ) )
=> ( X3 = Y ) ) ) ).
% NF_not_suc
thf(fact_883_NF__not__suc,axiom,
! [X3: term_f_v,Y: term_f_v,R: set_Pr8827868859434726151rm_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Y ) @ ( transi3028809021613410336rm_f_v @ R ) )
=> ( ( member_term_f_v @ X3 @ ( abstract_NF_term_f_v @ R ) )
=> ( X3 = Y ) ) ) ).
% NF_not_suc
thf(fact_884_NF__anti__mono,axiom,
! [A4: set_Pr516226060960033221_f_nat,B5: set_Pr516226060960033221_f_nat] :
( ( ord_le8946695678602162533_f_nat @ A4 @ B5 )
=> ( ord_le8976984241387448984_f_nat @ ( abstra8862684023087959798_f_nat @ B5 ) @ ( abstra8862684023087959798_f_nat @ A4 ) ) ) ).
% NF_anti_mono
thf(fact_885_NF__no__trancl__step,axiom,
! [A: produc8027700073345061287rm_f_v,A4: set_Pr7897041799657265351rm_f_v] :
( ( member4316649861536941904rm_f_v @ A @ ( abstra349717802288577587rm_f_v @ A4 ) )
=> ! [B8: produc8027700073345061287rm_f_v] :
~ ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A @ B8 ) @ ( transi7658437189833989320rm_f_v @ A4 ) ) ) ).
% NF_no_trancl_step
thf(fact_886_NF__no__trancl__step,axiom,
! [A: term_f_v,A4: set_Pr8827868859434726151rm_f_v] :
( ( member_term_f_v @ A @ ( abstract_NF_term_f_v @ A4 ) )
=> ! [B8: term_f_v] :
~ ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B8 ) @ ( transi3208867851621559554rm_f_v @ A4 ) ) ) ).
% NF_no_trancl_step
thf(fact_887_CR__NF__conv,axiom,
! [R: set_Pr7897041799657265351rm_f_v,T: produc8027700073345061287rm_f_v,U: produc8027700073345061287rm_f_v] :
( ( abstra5806628924625096669rm_f_v @ R @ top_to5755558563273943127rm_f_v )
=> ( ( member4316649861536941904rm_f_v @ T @ ( abstra349717802288577587rm_f_v @ R ) )
=> ( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ U @ T ) @ ( abstra4572590880773903381rm_f_v @ R ) )
=> ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ U @ T ) @ ( abstra9166317777578425423rm_f_v @ R ) ) ) ) ) ).
% CR_NF_conv
thf(fact_888_CR__NF__conv,axiom,
! [R: set_Pr8827868859434726151rm_f_v,T: term_f_v,U: term_f_v] :
( ( abstra3735013631728912813rm_f_v @ R @ top_top_set_term_f_v )
=> ( ( member_term_f_v @ T @ ( abstract_NF_term_f_v @ R ) )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ U @ T ) @ ( abstra5640035136655339381rm_f_v @ R ) )
=> ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ U @ T ) @ ( abstra7482117481303700155rm_f_v @ R ) ) ) ) ) ).
% CR_NF_conv
thf(fact_889_join__NF__imp__eq,axiom,
! [X3: produc8027700073345061287rm_f_v,Y: produc8027700073345061287rm_f_v,R: set_Pr7897041799657265351rm_f_v] :
( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ X3 @ Y ) @ ( abstra3336617290665100897rm_f_v @ R ) )
=> ( ( member4316649861536941904rm_f_v @ X3 @ ( abstra349717802288577587rm_f_v @ R ) )
=> ( ( member4316649861536941904rm_f_v @ Y @ ( abstra349717802288577587rm_f_v @ R ) )
=> ( X3 = Y ) ) ) ) ).
% join_NF_imp_eq
thf(fact_890_join__NF__imp__eq,axiom,
! [X3: term_f_v,Y: term_f_v,R: set_Pr8827868859434726151rm_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Y ) @ ( abstra8605546704478008873rm_f_v @ R ) )
=> ( ( member_term_f_v @ X3 @ ( abstract_NF_term_f_v @ R ) )
=> ( ( member_term_f_v @ Y @ ( abstract_NF_term_f_v @ R ) )
=> ( X3 = Y ) ) ) ) ).
% join_NF_imp_eq
thf(fact_891_WN__on__def,axiom,
( abstra9154201316845187733rm_f_v
= ( ^ [R7: set_Pr8827868859434726151rm_f_v,A6: set_term_f_v] :
! [X4: term_f_v] :
( ( member_term_f_v @ X4 @ A6 )
=> ? [B4: term_f_v] : ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X4 @ B4 ) @ ( abstra7482117481303700155rm_f_v @ R7 ) ) ) ) ) ).
% WN_on_def
thf(fact_892_WN__onE,axiom,
! [R: set_Pr7897041799657265351rm_f_v,A4: set_Pr8827868859434726151rm_f_v,A: produc8027700073345061287rm_f_v] :
( ( abstra1168807892682504437rm_f_v @ R @ A4 )
=> ( ( member4316649861536941904rm_f_v @ A @ A4 )
=> ~ ! [B3: produc8027700073345061287rm_f_v] :
~ ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A @ B3 ) @ ( abstra9166317777578425423rm_f_v @ R ) ) ) ) ).
% WN_onE
thf(fact_893_WN__onE,axiom,
! [R: set_Pr8827868859434726151rm_f_v,A4: set_term_f_v,A: term_f_v] :
( ( abstra9154201316845187733rm_f_v @ R @ A4 )
=> ( ( member_term_f_v @ A @ A4 )
=> ~ ! [B3: term_f_v] :
~ ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B3 ) @ ( abstra7482117481303700155rm_f_v @ R ) ) ) ) ).
% WN_onE
thf(fact_894_WN__onD,axiom,
! [R: set_Pr7897041799657265351rm_f_v,A4: set_Pr8827868859434726151rm_f_v,A: produc8027700073345061287rm_f_v] :
( ( abstra1168807892682504437rm_f_v @ R @ A4 )
=> ( ( member4316649861536941904rm_f_v @ A @ A4 )
=> ? [B3: produc8027700073345061287rm_f_v] : ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A @ B3 ) @ ( abstra9166317777578425423rm_f_v @ R ) ) ) ) ).
% WN_onD
thf(fact_895_WN__onD,axiom,
! [R: set_Pr8827868859434726151rm_f_v,A4: set_term_f_v,A: term_f_v] :
( ( abstra9154201316845187733rm_f_v @ R @ A4 )
=> ( ( member_term_f_v @ A @ A4 )
=> ? [B3: term_f_v] : ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B3 ) @ ( abstra7482117481303700155rm_f_v @ R ) ) ) ) ).
% WN_onD
thf(fact_896_UNF__onD,axiom,
! [R: set_Pr7897041799657265351rm_f_v,A4: set_Pr8827868859434726151rm_f_v,A: produc8027700073345061287rm_f_v,B: produc8027700073345061287rm_f_v,C3: produc8027700073345061287rm_f_v] :
( ( abstra8974104706389925717rm_f_v @ R @ A4 )
=> ( ( member4316649861536941904rm_f_v @ A @ A4 )
=> ( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A @ B ) @ ( abstra9166317777578425423rm_f_v @ R ) )
=> ( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A @ C3 ) @ ( abstra9166317777578425423rm_f_v @ R ) )
=> ( B = C3 ) ) ) ) ) ).
% UNF_onD
thf(fact_897_UNF__onD,axiom,
! [R: set_Pr8827868859434726151rm_f_v,A4: set_term_f_v,A: term_f_v,B: term_f_v,C3: term_f_v] :
( ( abstra8129321985590711349rm_f_v @ R @ A4 )
=> ( ( member_term_f_v @ A @ A4 )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( abstra7482117481303700155rm_f_v @ R ) )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ C3 ) @ ( abstra7482117481303700155rm_f_v @ R ) )
=> ( B = C3 ) ) ) ) ) ).
% UNF_onD
thf(fact_898_UNF__onE,axiom,
! [R: set_Pr7897041799657265351rm_f_v,A4: set_Pr8827868859434726151rm_f_v,A: produc8027700073345061287rm_f_v,B: produc8027700073345061287rm_f_v,C3: produc8027700073345061287rm_f_v] :
( ( abstra8974104706389925717rm_f_v @ R @ A4 )
=> ( ( member4316649861536941904rm_f_v @ A @ A4 )
=> ( ( B != C3 )
=> ( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A @ B ) @ ( abstra9166317777578425423rm_f_v @ R ) )
=> ~ ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A @ C3 ) @ ( abstra9166317777578425423rm_f_v @ R ) ) ) ) ) ) ).
% UNF_onE
thf(fact_899_UNF__onE,axiom,
! [R: set_Pr8827868859434726151rm_f_v,A4: set_term_f_v,A: term_f_v,B: term_f_v,C3: term_f_v] :
( ( abstra8129321985590711349rm_f_v @ R @ A4 )
=> ( ( member_term_f_v @ A @ A4 )
=> ( ( B != C3 )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( abstra7482117481303700155rm_f_v @ R ) )
=> ~ ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ C3 ) @ ( abstra7482117481303700155rm_f_v @ R ) ) ) ) ) ) ).
% UNF_onE
thf(fact_900_UNF__on__def,axiom,
( abstra8129321985590711349rm_f_v
= ( ^ [R7: set_Pr8827868859434726151rm_f_v,A6: set_term_f_v] :
! [X4: term_f_v] :
( ( member_term_f_v @ X4 @ A6 )
=> ! [B4: term_f_v,C4: term_f_v] :
( ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X4 @ B4 ) @ ( abstra7482117481303700155rm_f_v @ R7 ) )
& ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X4 @ C4 ) @ ( abstra7482117481303700155rm_f_v @ R7 ) ) )
=> ( B4 = C4 ) ) ) ) ) ).
% UNF_on_def
thf(fact_901_NF__join__imp__reach,axiom,
! [Y: produc8027700073345061287rm_f_v,A4: set_Pr7897041799657265351rm_f_v,X3: produc8027700073345061287rm_f_v] :
( ( member4316649861536941904rm_f_v @ Y @ ( abstra349717802288577587rm_f_v @ A4 ) )
=> ( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ X3 @ Y ) @ ( abstra3336617290665100897rm_f_v @ A4 ) )
=> ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ X3 @ Y ) @ ( transi1062743709496199978rm_f_v @ A4 ) ) ) ) ).
% NF_join_imp_reach
thf(fact_902_NF__join__imp__reach,axiom,
! [Y: term_f_v,A4: set_Pr8827868859434726151rm_f_v,X3: term_f_v] :
( ( member_term_f_v @ Y @ ( abstract_NF_term_f_v @ A4 ) )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Y ) @ ( abstra8605546704478008873rm_f_v @ A4 ) )
=> ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Y ) @ ( transi3028809021613410336rm_f_v @ A4 ) ) ) ) ).
% NF_join_imp_reach
thf(fact_903_UNC__E,axiom,
! [A4: set_Pr7897041799657265351rm_f_v,A: produc8027700073345061287rm_f_v,B: produc8027700073345061287rm_f_v] :
( ( abstra6927431686333010263rm_f_v @ A4 )
=> ( ( A != B )
=> ( ( member4316649861536941904rm_f_v @ A @ ( abstra349717802288577587rm_f_v @ A4 ) )
=> ( ( member4316649861536941904rm_f_v @ B @ ( abstra349717802288577587rm_f_v @ A4 ) )
=> ~ ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A @ B ) @ ( abstra4572590880773903381rm_f_v @ A4 ) ) ) ) ) ) ).
% UNC_E
thf(fact_904_UNC__E,axiom,
! [A4: set_Pr8827868859434726151rm_f_v,A: term_f_v,B: term_f_v] :
( ( abstra9159237726496465843rm_f_v @ A4 )
=> ( ( A != B )
=> ( ( member_term_f_v @ A @ ( abstract_NF_term_f_v @ A4 ) )
=> ( ( member_term_f_v @ B @ ( abstract_NF_term_f_v @ A4 ) )
=> ~ ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( abstra5640035136655339381rm_f_v @ A4 ) ) ) ) ) ) ).
% UNC_E
thf(fact_905_UNC__I,axiom,
! [A4: set_Pr7897041799657265351rm_f_v] :
( ! [A3: produc8027700073345061287rm_f_v,B3: produc8027700073345061287rm_f_v] :
( ( member4316649861536941904rm_f_v @ A3 @ ( abstra349717802288577587rm_f_v @ A4 ) )
=> ( ( member4316649861536941904rm_f_v @ B3 @ ( abstra349717802288577587rm_f_v @ A4 ) )
=> ( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A3 @ B3 ) @ ( abstra4572590880773903381rm_f_v @ A4 ) )
=> ( A3 = B3 ) ) ) )
=> ( abstra6927431686333010263rm_f_v @ A4 ) ) ).
% UNC_I
thf(fact_906_UNC__I,axiom,
! [A4: set_Pr8827868859434726151rm_f_v] :
( ! [A3: term_f_v,B3: term_f_v] :
( ( member_term_f_v @ A3 @ ( abstract_NF_term_f_v @ A4 ) )
=> ( ( member_term_f_v @ B3 @ ( abstract_NF_term_f_v @ A4 ) )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A3 @ B3 ) @ ( abstra5640035136655339381rm_f_v @ A4 ) )
=> ( A3 = B3 ) ) ) )
=> ( abstra9159237726496465843rm_f_v @ A4 ) ) ).
% UNC_I
thf(fact_907_NFP__on__def,axiom,
( rewrit1590804194565402411rm_f_v
= ( ^ [R7: set_Pr8827868859434726151rm_f_v,A6: set_term_f_v] :
! [X4: term_f_v] :
( ( member_term_f_v @ X4 @ A6 )
=> ! [B4: term_f_v,C4: term_f_v] :
( ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X4 @ B4 ) @ ( transi3028809021613410336rm_f_v @ R7 ) )
& ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X4 @ C4 ) @ ( abstra7482117481303700155rm_f_v @ R7 ) ) )
=> ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B4 @ C4 ) @ ( transi3028809021613410336rm_f_v @ R7 ) ) ) ) ) ) ).
% NFP_on_def
thf(fact_908_NE__on__def,axiom,
( rewrit2656876241799709610rm_f_v
= ( ^ [R7: set_Pr8827868859434726151rm_f_v,S4: set_Pr8827868859434726151rm_f_v,A6: set_term_f_v] :
! [X4: term_f_v] :
( ( member_term_f_v @ X4 @ A6 )
=> ! [B4: term_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X4 @ B4 ) @ ( abstra7482117481303700155rm_f_v @ R7 ) )
= ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X4 @ B4 ) @ ( abstra7482117481303700155rm_f_v @ S4 ) ) ) ) ) ) ).
% NE_on_def
thf(fact_909_some__NF__UNF,axiom,
! [R: set_Pr7897041799657265351rm_f_v,X3: produc8027700073345061287rm_f_v,Y: produc8027700073345061287rm_f_v] :
( ( abstra8974104706389925717rm_f_v @ R @ top_to5755558563273943127rm_f_v )
=> ( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ X3 @ Y ) @ ( transi1062743709496199978rm_f_v @ R ) )
=> ( ( member4316649861536941904rm_f_v @ Y @ ( abstra349717802288577587rm_f_v @ R ) )
=> ( Y
= ( abstra6650587300220611071rm_f_v @ R @ X3 ) ) ) ) ) ).
% some_NF_UNF
thf(fact_910_some__NF__UNF,axiom,
! [R: set_Pr8827868859434726151rm_f_v,X3: term_f_v,Y: term_f_v] :
( ( abstra8129321985590711349rm_f_v @ R @ top_top_set_term_f_v )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Y ) @ ( transi3028809021613410336rm_f_v @ R ) )
=> ( ( member_term_f_v @ Y @ ( abstract_NF_term_f_v @ R ) )
=> ( Y
= ( abstra7098836743531038987rm_f_v @ R @ X3 ) ) ) ) ) ).
% some_NF_UNF
thf(fact_911_same__fstI,axiom,
! [P2: term_f_v > $o,X3: term_f_v,Y7: term_f_v,Y: term_f_v,R3: term_f_v > set_Pr8827868859434726151rm_f_v] :
( ( P2 @ X3 )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y7 @ Y ) @ ( R3 @ X3 ) )
=> ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Y7 ) @ ( produc2510790566102465375rm_f_v @ X3 @ Y ) ) @ ( same_f5557695934196986804rm_f_v @ P2 @ R3 ) ) ) ) ).
% same_fstI
thf(fact_912_rtrancl__Int__subset,axiom,
! [S: set_Pr8827868859434726151rm_f_v,R: set_Pr8827868859434726151rm_f_v] :
( ( ord_le4559761987009501863rm_f_v @ id_term_f_v @ S )
=> ( ( ord_le4559761987009501863rm_f_v @ ( relcom2934258169684288428rm_f_v @ ( inf_in6135112614888068085rm_f_v @ ( transi3028809021613410336rm_f_v @ R ) @ S ) @ R ) @ S )
=> ( ord_le4559761987009501863rm_f_v @ ( transi3028809021613410336rm_f_v @ R ) @ S ) ) ) ).
% rtrancl_Int_subset
thf(fact_913_subst__apply__term__ctxt__apply__distrib,axiom,
! [C2: subterm_and_ctxt_f_v,T: term_f_v,Mu: v > term_f_v] :
( ( subst_7901567856578221233_f_v_v @ ( subter6886040689604212261rm_f_v @ C2 @ T ) @ Mu )
= ( subter6886040689604212261rm_f_v @ ( subter3988124567390845776_f_v_v @ C2 @ Mu ) @ ( subst_7901567856578221233_f_v_v @ T @ Mu ) ) ) ).
% subst_apply_term_ctxt_apply_distrib
thf(fact_914_Int__iff,axiom,
! [C3: produc8027700073345061287rm_f_v,A4: set_Pr8827868859434726151rm_f_v,B5: set_Pr8827868859434726151rm_f_v] :
( ( member4316649861536941904rm_f_v @ C3 @ ( inf_in6135112614888068085rm_f_v @ A4 @ B5 ) )
= ( ( member4316649861536941904rm_f_v @ C3 @ A4 )
& ( member4316649861536941904rm_f_v @ C3 @ B5 ) ) ) ).
% Int_iff
thf(fact_915_IntI,axiom,
! [C3: produc8027700073345061287rm_f_v,A4: set_Pr8827868859434726151rm_f_v,B5: set_Pr8827868859434726151rm_f_v] :
( ( member4316649861536941904rm_f_v @ C3 @ A4 )
=> ( ( member4316649861536941904rm_f_v @ C3 @ B5 )
=> ( member4316649861536941904rm_f_v @ C3 @ ( inf_in6135112614888068085rm_f_v @ A4 @ B5 ) ) ) ) ).
% IntI
thf(fact_916_le__inf__iff,axiom,
! [X3: set_Pr5245412377734449720_f_nat,Y: set_Pr5245412377734449720_f_nat,Z2: set_Pr5245412377734449720_f_nat] :
( ( ord_le8976984241387448984_f_nat @ X3 @ ( inf_in4124010953442585290_f_nat @ Y @ Z2 ) )
= ( ( ord_le8976984241387448984_f_nat @ X3 @ Y )
& ( ord_le8976984241387448984_f_nat @ X3 @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_917_inf_Obounded__iff,axiom,
! [A: set_Pr5245412377734449720_f_nat,B: set_Pr5245412377734449720_f_nat,C3: set_Pr5245412377734449720_f_nat] :
( ( ord_le8976984241387448984_f_nat @ A @ ( inf_in4124010953442585290_f_nat @ B @ C3 ) )
= ( ( ord_le8976984241387448984_f_nat @ A @ B )
& ( ord_le8976984241387448984_f_nat @ A @ C3 ) ) ) ).
% inf.bounded_iff
thf(fact_918_Int__subset__iff,axiom,
! [C2: set_Pr5245412377734449720_f_nat,A4: set_Pr5245412377734449720_f_nat,B5: set_Pr5245412377734449720_f_nat] :
( ( ord_le8976984241387448984_f_nat @ C2 @ ( inf_in4124010953442585290_f_nat @ A4 @ B5 ) )
= ( ( ord_le8976984241387448984_f_nat @ C2 @ A4 )
& ( ord_le8976984241387448984_f_nat @ C2 @ B5 ) ) ) ).
% Int_subset_iff
thf(fact_919_irrefl__onD,axiom,
! [A4: set_Pr8827868859434726151rm_f_v,R: set_Pr7897041799657265351rm_f_v,A: produc8027700073345061287rm_f_v] :
( ( irrefl5067090950674729629rm_f_v @ A4 @ R )
=> ( ( member4316649861536941904rm_f_v @ A @ A4 )
=> ~ ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A @ A ) @ R ) ) ) ).
% irrefl_onD
thf(fact_920_irrefl__onD,axiom,
! [A4: set_term_f_v,R: set_Pr8827868859434726151rm_f_v,A: term_f_v] :
( ( irrefl_on_term_f_v @ A4 @ R )
=> ( ( member_term_f_v @ A @ A4 )
=> ~ ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ A ) @ R ) ) ) ).
% irrefl_onD
thf(fact_921_irrefl__onI,axiom,
! [A4: set_Pr8827868859434726151rm_f_v,R: set_Pr7897041799657265351rm_f_v] :
( ! [A3: produc8027700073345061287rm_f_v] :
( ( member4316649861536941904rm_f_v @ A3 @ A4 )
=> ~ ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A3 @ A3 ) @ R ) )
=> ( irrefl5067090950674729629rm_f_v @ A4 @ R ) ) ).
% irrefl_onI
thf(fact_922_irrefl__onI,axiom,
! [A4: set_term_f_v,R: set_Pr8827868859434726151rm_f_v] :
( ! [A3: term_f_v] :
( ( member_term_f_v @ A3 @ A4 )
=> ~ ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A3 @ A3 ) @ R ) )
=> ( irrefl_on_term_f_v @ A4 @ R ) ) ).
% irrefl_onI
thf(fact_923_irrefl__on__def,axiom,
( irrefl_on_term_f_v
= ( ^ [A6: set_term_f_v,R7: set_Pr8827868859434726151rm_f_v] :
! [X4: term_f_v] :
( ( member_term_f_v @ X4 @ A6 )
=> ~ ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X4 @ X4 ) @ R7 ) ) ) ) ).
% irrefl_on_def
thf(fact_924_IntD2,axiom,
! [C3: produc8027700073345061287rm_f_v,A4: set_Pr8827868859434726151rm_f_v,B5: set_Pr8827868859434726151rm_f_v] :
( ( member4316649861536941904rm_f_v @ C3 @ ( inf_in6135112614888068085rm_f_v @ A4 @ B5 ) )
=> ( member4316649861536941904rm_f_v @ C3 @ B5 ) ) ).
% IntD2
thf(fact_925_IntD1,axiom,
! [C3: produc8027700073345061287rm_f_v,A4: set_Pr8827868859434726151rm_f_v,B5: set_Pr8827868859434726151rm_f_v] :
( ( member4316649861536941904rm_f_v @ C3 @ ( inf_in6135112614888068085rm_f_v @ A4 @ B5 ) )
=> ( member4316649861536941904rm_f_v @ C3 @ A4 ) ) ).
% IntD1
thf(fact_926_IntE,axiom,
! [C3: produc8027700073345061287rm_f_v,A4: set_Pr8827868859434726151rm_f_v,B5: set_Pr8827868859434726151rm_f_v] :
( ( member4316649861536941904rm_f_v @ C3 @ ( inf_in6135112614888068085rm_f_v @ A4 @ B5 ) )
=> ~ ( ( member4316649861536941904rm_f_v @ C3 @ A4 )
=> ~ ( member4316649861536941904rm_f_v @ C3 @ B5 ) ) ) ).
% IntE
thf(fact_927_Int__Collect__mono,axiom,
! [A4: set_Pr8827868859434726151rm_f_v,B5: set_Pr8827868859434726151rm_f_v,P2: produc8027700073345061287rm_f_v > $o,Q: produc8027700073345061287rm_f_v > $o] :
( ( ord_le4559761987009501863rm_f_v @ A4 @ B5 )
=> ( ! [X: produc8027700073345061287rm_f_v] :
( ( member4316649861536941904rm_f_v @ X @ A4 )
=> ( ( P2 @ X )
=> ( Q @ X ) ) )
=> ( ord_le4559761987009501863rm_f_v @ ( inf_in6135112614888068085rm_f_v @ A4 @ ( collec7769483783292507282rm_f_v @ P2 ) ) @ ( inf_in6135112614888068085rm_f_v @ B5 @ ( collec7769483783292507282rm_f_v @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_928_Int__Collect__mono,axiom,
! [A4: set_Pr5245412377734449720_f_nat,B5: set_Pr5245412377734449720_f_nat,P2: product_prod_f_nat > $o,Q: product_prod_f_nat > $o] :
( ( ord_le8976984241387448984_f_nat @ A4 @ B5 )
=> ( ! [X: product_prod_f_nat] :
( ( member2678038196313615385_f_nat @ X @ A4 )
=> ( ( P2 @ X )
=> ( Q @ X ) ) )
=> ( ord_le8976984241387448984_f_nat @ ( inf_in4124010953442585290_f_nat @ A4 @ ( collec1417984143148964183_f_nat @ P2 ) ) @ ( inf_in4124010953442585290_f_nat @ B5 @ ( collec1417984143148964183_f_nat @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_929_Int__greatest,axiom,
! [C2: set_Pr5245412377734449720_f_nat,A4: set_Pr5245412377734449720_f_nat,B5: set_Pr5245412377734449720_f_nat] :
( ( ord_le8976984241387448984_f_nat @ C2 @ A4 )
=> ( ( ord_le8976984241387448984_f_nat @ C2 @ B5 )
=> ( ord_le8976984241387448984_f_nat @ C2 @ ( inf_in4124010953442585290_f_nat @ A4 @ B5 ) ) ) ) ).
% Int_greatest
thf(fact_930_Int__absorb2,axiom,
! [A4: set_Pr5245412377734449720_f_nat,B5: set_Pr5245412377734449720_f_nat] :
( ( ord_le8976984241387448984_f_nat @ A4 @ B5 )
=> ( ( inf_in4124010953442585290_f_nat @ A4 @ B5 )
= A4 ) ) ).
% Int_absorb2
thf(fact_931_Int__absorb1,axiom,
! [B5: set_Pr5245412377734449720_f_nat,A4: set_Pr5245412377734449720_f_nat] :
( ( ord_le8976984241387448984_f_nat @ B5 @ A4 )
=> ( ( inf_in4124010953442585290_f_nat @ A4 @ B5 )
= B5 ) ) ).
% Int_absorb1
thf(fact_932_Int__lower2,axiom,
! [A4: set_Pr5245412377734449720_f_nat,B5: set_Pr5245412377734449720_f_nat] : ( ord_le8976984241387448984_f_nat @ ( inf_in4124010953442585290_f_nat @ A4 @ B5 ) @ B5 ) ).
% Int_lower2
thf(fact_933_Int__lower1,axiom,
! [A4: set_Pr5245412377734449720_f_nat,B5: set_Pr5245412377734449720_f_nat] : ( ord_le8976984241387448984_f_nat @ ( inf_in4124010953442585290_f_nat @ A4 @ B5 ) @ A4 ) ).
% Int_lower1
thf(fact_934_Int__mono,axiom,
! [A4: set_Pr5245412377734449720_f_nat,C2: set_Pr5245412377734449720_f_nat,B5: set_Pr5245412377734449720_f_nat,D: set_Pr5245412377734449720_f_nat] :
( ( ord_le8976984241387448984_f_nat @ A4 @ C2 )
=> ( ( ord_le8976984241387448984_f_nat @ B5 @ D )
=> ( ord_le8976984241387448984_f_nat @ ( inf_in4124010953442585290_f_nat @ A4 @ B5 ) @ ( inf_in4124010953442585290_f_nat @ C2 @ D ) ) ) ) ).
% Int_mono
thf(fact_935_inf__sup__ord_I2_J,axiom,
! [X3: set_Pr5245412377734449720_f_nat,Y: set_Pr5245412377734449720_f_nat] : ( ord_le8976984241387448984_f_nat @ ( inf_in4124010953442585290_f_nat @ X3 @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_936_inf__sup__ord_I1_J,axiom,
! [X3: set_Pr5245412377734449720_f_nat,Y: set_Pr5245412377734449720_f_nat] : ( ord_le8976984241387448984_f_nat @ ( inf_in4124010953442585290_f_nat @ X3 @ Y ) @ X3 ) ).
% inf_sup_ord(1)
thf(fact_937_inf__le1,axiom,
! [X3: set_Pr5245412377734449720_f_nat,Y: set_Pr5245412377734449720_f_nat] : ( ord_le8976984241387448984_f_nat @ ( inf_in4124010953442585290_f_nat @ X3 @ Y ) @ X3 ) ).
% inf_le1
thf(fact_938_inf__le2,axiom,
! [X3: set_Pr5245412377734449720_f_nat,Y: set_Pr5245412377734449720_f_nat] : ( ord_le8976984241387448984_f_nat @ ( inf_in4124010953442585290_f_nat @ X3 @ Y ) @ Y ) ).
% inf_le2
thf(fact_939_le__infE,axiom,
! [X3: set_Pr5245412377734449720_f_nat,A: set_Pr5245412377734449720_f_nat,B: set_Pr5245412377734449720_f_nat] :
( ( ord_le8976984241387448984_f_nat @ X3 @ ( inf_in4124010953442585290_f_nat @ A @ B ) )
=> ~ ( ( ord_le8976984241387448984_f_nat @ X3 @ A )
=> ~ ( ord_le8976984241387448984_f_nat @ X3 @ B ) ) ) ).
% le_infE
thf(fact_940_le__infI,axiom,
! [X3: set_Pr5245412377734449720_f_nat,A: set_Pr5245412377734449720_f_nat,B: set_Pr5245412377734449720_f_nat] :
( ( ord_le8976984241387448984_f_nat @ X3 @ A )
=> ( ( ord_le8976984241387448984_f_nat @ X3 @ B )
=> ( ord_le8976984241387448984_f_nat @ X3 @ ( inf_in4124010953442585290_f_nat @ A @ B ) ) ) ) ).
% le_infI
thf(fact_941_inf__mono,axiom,
! [A: set_Pr5245412377734449720_f_nat,C3: set_Pr5245412377734449720_f_nat,B: set_Pr5245412377734449720_f_nat,D2: set_Pr5245412377734449720_f_nat] :
( ( ord_le8976984241387448984_f_nat @ A @ C3 )
=> ( ( ord_le8976984241387448984_f_nat @ B @ D2 )
=> ( ord_le8976984241387448984_f_nat @ ( inf_in4124010953442585290_f_nat @ A @ B ) @ ( inf_in4124010953442585290_f_nat @ C3 @ D2 ) ) ) ) ).
% inf_mono
thf(fact_942_le__infI1,axiom,
! [A: set_Pr5245412377734449720_f_nat,X3: set_Pr5245412377734449720_f_nat,B: set_Pr5245412377734449720_f_nat] :
( ( ord_le8976984241387448984_f_nat @ A @ X3 )
=> ( ord_le8976984241387448984_f_nat @ ( inf_in4124010953442585290_f_nat @ A @ B ) @ X3 ) ) ).
% le_infI1
thf(fact_943_le__infI2,axiom,
! [B: set_Pr5245412377734449720_f_nat,X3: set_Pr5245412377734449720_f_nat,A: set_Pr5245412377734449720_f_nat] :
( ( ord_le8976984241387448984_f_nat @ B @ X3 )
=> ( ord_le8976984241387448984_f_nat @ ( inf_in4124010953442585290_f_nat @ A @ B ) @ X3 ) ) ).
% le_infI2
thf(fact_944_inf_OorderE,axiom,
! [A: set_Pr5245412377734449720_f_nat,B: set_Pr5245412377734449720_f_nat] :
( ( ord_le8976984241387448984_f_nat @ A @ B )
=> ( A
= ( inf_in4124010953442585290_f_nat @ A @ B ) ) ) ).
% inf.orderE
thf(fact_945_inf_OorderI,axiom,
! [A: set_Pr5245412377734449720_f_nat,B: set_Pr5245412377734449720_f_nat] :
( ( A
= ( inf_in4124010953442585290_f_nat @ A @ B ) )
=> ( ord_le8976984241387448984_f_nat @ A @ B ) ) ).
% inf.orderI
thf(fact_946_inf__unique,axiom,
! [F2: set_Pr5245412377734449720_f_nat > set_Pr5245412377734449720_f_nat > set_Pr5245412377734449720_f_nat,X3: set_Pr5245412377734449720_f_nat,Y: set_Pr5245412377734449720_f_nat] :
( ! [X: set_Pr5245412377734449720_f_nat,Y3: set_Pr5245412377734449720_f_nat] : ( ord_le8976984241387448984_f_nat @ ( F2 @ X @ Y3 ) @ X )
=> ( ! [X: set_Pr5245412377734449720_f_nat,Y3: set_Pr5245412377734449720_f_nat] : ( ord_le8976984241387448984_f_nat @ ( F2 @ X @ Y3 ) @ Y3 )
=> ( ! [X: set_Pr5245412377734449720_f_nat,Y3: set_Pr5245412377734449720_f_nat,Z: set_Pr5245412377734449720_f_nat] :
( ( ord_le8976984241387448984_f_nat @ X @ Y3 )
=> ( ( ord_le8976984241387448984_f_nat @ X @ Z )
=> ( ord_le8976984241387448984_f_nat @ X @ ( F2 @ Y3 @ Z ) ) ) )
=> ( ( inf_in4124010953442585290_f_nat @ X3 @ Y )
= ( F2 @ X3 @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_947_le__iff__inf,axiom,
( ord_le8976984241387448984_f_nat
= ( ^ [X4: set_Pr5245412377734449720_f_nat,Y5: set_Pr5245412377734449720_f_nat] :
( ( inf_in4124010953442585290_f_nat @ X4 @ Y5 )
= X4 ) ) ) ).
% le_iff_inf
thf(fact_948_inf_Oabsorb1,axiom,
! [A: set_Pr5245412377734449720_f_nat,B: set_Pr5245412377734449720_f_nat] :
( ( ord_le8976984241387448984_f_nat @ A @ B )
=> ( ( inf_in4124010953442585290_f_nat @ A @ B )
= A ) ) ).
% inf.absorb1
thf(fact_949_inf_Oabsorb2,axiom,
! [B: set_Pr5245412377734449720_f_nat,A: set_Pr5245412377734449720_f_nat] :
( ( ord_le8976984241387448984_f_nat @ B @ A )
=> ( ( inf_in4124010953442585290_f_nat @ A @ B )
= B ) ) ).
% inf.absorb2
thf(fact_950_inf__absorb1,axiom,
! [X3: set_Pr5245412377734449720_f_nat,Y: set_Pr5245412377734449720_f_nat] :
( ( ord_le8976984241387448984_f_nat @ X3 @ Y )
=> ( ( inf_in4124010953442585290_f_nat @ X3 @ Y )
= X3 ) ) ).
% inf_absorb1
thf(fact_951_inf__absorb2,axiom,
! [Y: set_Pr5245412377734449720_f_nat,X3: set_Pr5245412377734449720_f_nat] :
( ( ord_le8976984241387448984_f_nat @ Y @ X3 )
=> ( ( inf_in4124010953442585290_f_nat @ X3 @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_952_inf_OboundedE,axiom,
! [A: set_Pr5245412377734449720_f_nat,B: set_Pr5245412377734449720_f_nat,C3: set_Pr5245412377734449720_f_nat] :
( ( ord_le8976984241387448984_f_nat @ A @ ( inf_in4124010953442585290_f_nat @ B @ C3 ) )
=> ~ ( ( ord_le8976984241387448984_f_nat @ A @ B )
=> ~ ( ord_le8976984241387448984_f_nat @ A @ C3 ) ) ) ).
% inf.boundedE
thf(fact_953_inf_OboundedI,axiom,
! [A: set_Pr5245412377734449720_f_nat,B: set_Pr5245412377734449720_f_nat,C3: set_Pr5245412377734449720_f_nat] :
( ( ord_le8976984241387448984_f_nat @ A @ B )
=> ( ( ord_le8976984241387448984_f_nat @ A @ C3 )
=> ( ord_le8976984241387448984_f_nat @ A @ ( inf_in4124010953442585290_f_nat @ B @ C3 ) ) ) ) ).
% inf.boundedI
thf(fact_954_inf__greatest,axiom,
! [X3: set_Pr5245412377734449720_f_nat,Y: set_Pr5245412377734449720_f_nat,Z2: set_Pr5245412377734449720_f_nat] :
( ( ord_le8976984241387448984_f_nat @ X3 @ Y )
=> ( ( ord_le8976984241387448984_f_nat @ X3 @ Z2 )
=> ( ord_le8976984241387448984_f_nat @ X3 @ ( inf_in4124010953442585290_f_nat @ Y @ Z2 ) ) ) ) ).
% inf_greatest
thf(fact_955_inf_Oorder__iff,axiom,
( ord_le8976984241387448984_f_nat
= ( ^ [A5: set_Pr5245412377734449720_f_nat,B4: set_Pr5245412377734449720_f_nat] :
( A5
= ( inf_in4124010953442585290_f_nat @ A5 @ B4 ) ) ) ) ).
% inf.order_iff
thf(fact_956_inf_Ocobounded1,axiom,
! [A: set_Pr5245412377734449720_f_nat,B: set_Pr5245412377734449720_f_nat] : ( ord_le8976984241387448984_f_nat @ ( inf_in4124010953442585290_f_nat @ A @ B ) @ A ) ).
% inf.cobounded1
thf(fact_957_inf_Ocobounded2,axiom,
! [A: set_Pr5245412377734449720_f_nat,B: set_Pr5245412377734449720_f_nat] : ( ord_le8976984241387448984_f_nat @ ( inf_in4124010953442585290_f_nat @ A @ B ) @ B ) ).
% inf.cobounded2
thf(fact_958_inf_Oabsorb__iff1,axiom,
( ord_le8976984241387448984_f_nat
= ( ^ [A5: set_Pr5245412377734449720_f_nat,B4: set_Pr5245412377734449720_f_nat] :
( ( inf_in4124010953442585290_f_nat @ A5 @ B4 )
= A5 ) ) ) ).
% inf.absorb_iff1
thf(fact_959_inf_Oabsorb__iff2,axiom,
( ord_le8976984241387448984_f_nat
= ( ^ [B4: set_Pr5245412377734449720_f_nat,A5: set_Pr5245412377734449720_f_nat] :
( ( inf_in4124010953442585290_f_nat @ A5 @ B4 )
= B4 ) ) ) ).
% inf.absorb_iff2
thf(fact_960_inf_OcoboundedI1,axiom,
! [A: set_Pr5245412377734449720_f_nat,C3: set_Pr5245412377734449720_f_nat,B: set_Pr5245412377734449720_f_nat] :
( ( ord_le8976984241387448984_f_nat @ A @ C3 )
=> ( ord_le8976984241387448984_f_nat @ ( inf_in4124010953442585290_f_nat @ A @ B ) @ C3 ) ) ).
% inf.coboundedI1
thf(fact_961_inf_OcoboundedI2,axiom,
! [B: set_Pr5245412377734449720_f_nat,C3: set_Pr5245412377734449720_f_nat,A: set_Pr5245412377734449720_f_nat] :
( ( ord_le8976984241387448984_f_nat @ B @ C3 )
=> ( ord_le8976984241387448984_f_nat @ ( inf_in4124010953442585290_f_nat @ A @ B ) @ C3 ) ) ).
% inf.coboundedI2
thf(fact_962_irrefl__on__subset,axiom,
! [A4: set_Pr5245412377734449720_f_nat,R: set_Pr516226060960033221_f_nat,B5: set_Pr5245412377734449720_f_nat] :
( ( irrefl6354360072789167372_f_nat @ A4 @ R )
=> ( ( ord_le8976984241387448984_f_nat @ B5 @ A4 )
=> ( irrefl6354360072789167372_f_nat @ B5 @ R ) ) ) ).
% irrefl_on_subset
thf(fact_963_distrib__sup__le,axiom,
! [X3: set_Pr5245412377734449720_f_nat,Y: set_Pr5245412377734449720_f_nat,Z2: set_Pr5245412377734449720_f_nat] : ( ord_le8976984241387448984_f_nat @ ( sup_su770888850772321124_f_nat @ X3 @ ( inf_in4124010953442585290_f_nat @ Y @ Z2 ) ) @ ( inf_in4124010953442585290_f_nat @ ( sup_su770888850772321124_f_nat @ X3 @ Y ) @ ( sup_su770888850772321124_f_nat @ X3 @ Z2 ) ) ) ).
% distrib_sup_le
thf(fact_964_distrib__inf__le,axiom,
! [X3: set_Pr5245412377734449720_f_nat,Y: set_Pr5245412377734449720_f_nat,Z2: set_Pr5245412377734449720_f_nat] : ( ord_le8976984241387448984_f_nat @ ( sup_su770888850772321124_f_nat @ ( inf_in4124010953442585290_f_nat @ X3 @ Y ) @ ( inf_in4124010953442585290_f_nat @ X3 @ Z2 ) ) @ ( inf_in4124010953442585290_f_nat @ X3 @ ( sup_su770888850772321124_f_nat @ Y @ Z2 ) ) ) ).
% distrib_inf_le
thf(fact_965_Un__Int__assoc__eq,axiom,
! [A4: set_Pr5245412377734449720_f_nat,B5: set_Pr5245412377734449720_f_nat,C2: set_Pr5245412377734449720_f_nat] :
( ( ( sup_su770888850772321124_f_nat @ ( inf_in4124010953442585290_f_nat @ A4 @ B5 ) @ C2 )
= ( inf_in4124010953442585290_f_nat @ A4 @ ( sup_su770888850772321124_f_nat @ B5 @ C2 ) ) )
= ( ord_le8976984241387448984_f_nat @ C2 @ A4 ) ) ).
% Un_Int_assoc_eq
thf(fact_966_irreflI,axiom,
! [R: set_Pr8827868859434726151rm_f_v] :
( ! [A3: term_f_v] :
~ ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A3 @ A3 ) @ R )
=> ( irrefl_on_term_f_v @ top_top_set_term_f_v @ R ) ) ).
% irreflI
thf(fact_967_irreflD,axiom,
! [R: set_Pr8827868859434726151rm_f_v,X3: term_f_v] :
( ( irrefl_on_term_f_v @ top_top_set_term_f_v @ R )
=> ~ ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ X3 ) @ R ) ) ).
% irreflD
thf(fact_968_irrefl__tranclI,axiom,
! [R: set_Pr8827868859434726151rm_f_v,X3: term_f_v] :
( ( ( inf_in6135112614888068085rm_f_v @ ( conver3099152742603637891rm_f_v @ R ) @ ( transi3028809021613410336rm_f_v @ R ) )
= bot_bo4638812505780724595rm_f_v )
=> ~ ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ X3 ) @ ( transi3208867851621559554rm_f_v @ R ) ) ) ).
% irrefl_tranclI
thf(fact_969_trans__ctxt__sig__imp__all__ctxt__closed,axiom,
! [R: set_Pr8827868859434726151rm_f_v,F4: set_Pr5245412377734449720_f_nat] :
( ( trans_on_term_f_v @ top_top_set_term_f_v @ R )
=> ( ! [T2: term_f_v] :
( ( ord_le8976984241387448984_f_nat @ ( term_funas_term_f_v @ T2 ) @ F4 )
=> ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ T2 @ T2 ) @ R ) )
=> ( ! [C5: subterm_and_ctxt_f_v,S3: term_f_v,T2: term_f_v] :
( ( ord_le8976984241387448984_f_nat @ ( term_funas_ctxt_f_v @ C5 ) @ F4 )
=> ( ( ord_le8976984241387448984_f_nat @ ( term_funas_term_f_v @ S3 ) @ F4 )
=> ( ( ord_le8976984241387448984_f_nat @ ( term_funas_term_f_v @ T2 ) @ F4 )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S3 @ T2 ) @ R )
=> ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ ( subter6886040689604212261rm_f_v @ C5 @ S3 ) @ ( subter6886040689604212261rm_f_v @ C5 @ T2 ) ) @ R ) ) ) ) )
=> ( terms_512237927760305819ed_f_v @ F4 @ R ) ) ) ) ).
% trans_ctxt_sig_imp_all_ctxt_closed
thf(fact_970_empty__iff,axiom,
! [C3: produc8027700073345061287rm_f_v] :
~ ( member4316649861536941904rm_f_v @ C3 @ bot_bo4638812505780724595rm_f_v ) ).
% empty_iff
thf(fact_971_all__not__in__conv,axiom,
! [A4: set_Pr8827868859434726151rm_f_v] :
( ( ! [X4: produc8027700073345061287rm_f_v] :
~ ( member4316649861536941904rm_f_v @ X4 @ A4 ) )
= ( A4 = bot_bo4638812505780724595rm_f_v ) ) ).
% all_not_in_conv
thf(fact_972_empty__subsetI,axiom,
! [A4: set_Pr5245412377734449720_f_nat] : ( ord_le8976984241387448984_f_nat @ bot_bo136713897782694092_f_nat @ A4 ) ).
% empty_subsetI
thf(fact_973_subset__empty,axiom,
! [A4: set_Pr5245412377734449720_f_nat] :
( ( ord_le8976984241387448984_f_nat @ A4 @ bot_bo136713897782694092_f_nat )
= ( A4 = bot_bo136713897782694092_f_nat ) ) ).
% subset_empty
thf(fact_974_Diff__eq__empty__iff,axiom,
! [A4: set_Pr5245412377734449720_f_nat,B5: set_Pr5245412377734449720_f_nat] :
( ( ( minus_8941878871097791505_f_nat @ A4 @ B5 )
= bot_bo136713897782694092_f_nat )
= ( ord_le8976984241387448984_f_nat @ A4 @ B5 ) ) ).
% Diff_eq_empty_iff
thf(fact_975_rtrancl__empty,axiom,
( ( transi3028809021613410336rm_f_v @ bot_bo4638812505780724595rm_f_v )
= id_term_f_v ) ).
% rtrancl_empty
thf(fact_976_disjoint__iff,axiom,
! [A4: set_Pr8827868859434726151rm_f_v,B5: set_Pr8827868859434726151rm_f_v] :
( ( ( inf_in6135112614888068085rm_f_v @ A4 @ B5 )
= bot_bo4638812505780724595rm_f_v )
= ( ! [X4: produc8027700073345061287rm_f_v] :
( ( member4316649861536941904rm_f_v @ X4 @ A4 )
=> ~ ( member4316649861536941904rm_f_v @ X4 @ B5 ) ) ) ) ).
% disjoint_iff
thf(fact_977_Int__emptyI,axiom,
! [A4: set_Pr8827868859434726151rm_f_v,B5: set_Pr8827868859434726151rm_f_v] :
( ! [X: produc8027700073345061287rm_f_v] :
( ( member4316649861536941904rm_f_v @ X @ A4 )
=> ~ ( member4316649861536941904rm_f_v @ X @ B5 ) )
=> ( ( inf_in6135112614888068085rm_f_v @ A4 @ B5 )
= bot_bo4638812505780724595rm_f_v ) ) ).
% Int_emptyI
thf(fact_978_bot_Oextremum,axiom,
! [A: set_Pr5245412377734449720_f_nat] : ( ord_le8976984241387448984_f_nat @ bot_bo136713897782694092_f_nat @ A ) ).
% bot.extremum
thf(fact_979_bot_Oextremum__unique,axiom,
! [A: set_Pr5245412377734449720_f_nat] :
( ( ord_le8976984241387448984_f_nat @ A @ bot_bo136713897782694092_f_nat )
= ( A = bot_bo136713897782694092_f_nat ) ) ).
% bot.extremum_unique
thf(fact_980_bot_Oextremum__uniqueI,axiom,
! [A: set_Pr5245412377734449720_f_nat] :
( ( ord_le8976984241387448984_f_nat @ A @ bot_bo136713897782694092_f_nat )
=> ( A = bot_bo136713897782694092_f_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_981_trans__on__def,axiom,
( trans_on_term_f_v
= ( ^ [A6: set_term_f_v,R7: set_Pr8827868859434726151rm_f_v] :
! [X4: term_f_v] :
( ( member_term_f_v @ X4 @ A6 )
=> ! [Y5: term_f_v] :
( ( member_term_f_v @ Y5 @ A6 )
=> ! [Z4: term_f_v] :
( ( member_term_f_v @ Z4 @ A6 )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X4 @ Y5 ) @ R7 )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y5 @ Z4 ) @ R7 )
=> ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X4 @ Z4 ) @ R7 ) ) ) ) ) ) ) ) ).
% trans_on_def
thf(fact_982_trans__onI,axiom,
! [A4: set_Pr8827868859434726151rm_f_v,R: set_Pr7897041799657265351rm_f_v] :
( ! [X: produc8027700073345061287rm_f_v,Y3: produc8027700073345061287rm_f_v,Z: produc8027700073345061287rm_f_v] :
( ( member4316649861536941904rm_f_v @ X @ A4 )
=> ( ( member4316649861536941904rm_f_v @ Y3 @ A4 )
=> ( ( member4316649861536941904rm_f_v @ Z @ A4 )
=> ( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ X @ Y3 ) @ R )
=> ( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ Y3 @ Z ) @ R )
=> ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ X @ Z ) @ R ) ) ) ) ) )
=> ( trans_3851638462678097735rm_f_v @ A4 @ R ) ) ).
% trans_onI
thf(fact_983_trans__onI,axiom,
! [A4: set_term_f_v,R: set_Pr8827868859434726151rm_f_v] :
( ! [X: term_f_v,Y3: term_f_v,Z: term_f_v] :
( ( member_term_f_v @ X @ A4 )
=> ( ( member_term_f_v @ Y3 @ A4 )
=> ( ( member_term_f_v @ Z @ A4 )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ Y3 ) @ R )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y3 @ Z ) @ R )
=> ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ Z ) @ R ) ) ) ) ) )
=> ( trans_on_term_f_v @ A4 @ R ) ) ).
% trans_onI
thf(fact_984_trans__onD,axiom,
! [A4: set_Pr8827868859434726151rm_f_v,R: set_Pr7897041799657265351rm_f_v,X3: produc8027700073345061287rm_f_v,Y: produc8027700073345061287rm_f_v,Z2: produc8027700073345061287rm_f_v] :
( ( trans_3851638462678097735rm_f_v @ A4 @ R )
=> ( ( member4316649861536941904rm_f_v @ X3 @ A4 )
=> ( ( member4316649861536941904rm_f_v @ Y @ A4 )
=> ( ( member4316649861536941904rm_f_v @ Z2 @ A4 )
=> ( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ X3 @ Y ) @ R )
=> ( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ Y @ Z2 ) @ R )
=> ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ X3 @ Z2 ) @ R ) ) ) ) ) ) ) ).
% trans_onD
thf(fact_985_trans__onD,axiom,
! [A4: set_term_f_v,R: set_Pr8827868859434726151rm_f_v,X3: term_f_v,Y: term_f_v,Z2: term_f_v] :
( ( trans_on_term_f_v @ A4 @ R )
=> ( ( member_term_f_v @ X3 @ A4 )
=> ( ( member_term_f_v @ Y @ A4 )
=> ( ( member_term_f_v @ Z2 @ A4 )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Y ) @ R )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y @ Z2 ) @ R )
=> ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Z2 ) @ R ) ) ) ) ) ) ) ).
% trans_onD
thf(fact_986_trans__on__subset,axiom,
! [A4: set_Pr5245412377734449720_f_nat,R: set_Pr516226060960033221_f_nat,B5: set_Pr5245412377734449720_f_nat] :
( ( trans_2699862687909549282_f_nat @ A4 @ R )
=> ( ( ord_le8976984241387448984_f_nat @ B5 @ A4 )
=> ( trans_2699862687909549282_f_nat @ B5 @ R ) ) ) ).
% trans_on_subset
thf(fact_987_emptyE,axiom,
! [A: produc8027700073345061287rm_f_v] :
~ ( member4316649861536941904rm_f_v @ A @ bot_bo4638812505780724595rm_f_v ) ).
% emptyE
thf(fact_988_equals0D,axiom,
! [A4: set_Pr8827868859434726151rm_f_v,A: produc8027700073345061287rm_f_v] :
( ( A4 = bot_bo4638812505780724595rm_f_v )
=> ~ ( member4316649861536941904rm_f_v @ A @ A4 ) ) ).
% equals0D
thf(fact_989_equals0I,axiom,
! [A4: set_Pr8827868859434726151rm_f_v] :
( ! [Y3: produc8027700073345061287rm_f_v] :
~ ( member4316649861536941904rm_f_v @ Y3 @ A4 )
=> ( A4 = bot_bo4638812505780724595rm_f_v ) ) ).
% equals0I
thf(fact_990_ex__in__conv,axiom,
! [A4: set_Pr8827868859434726151rm_f_v] :
( ( ? [X4: produc8027700073345061287rm_f_v] : ( member4316649861536941904rm_f_v @ X4 @ A4 ) )
= ( A4 != bot_bo4638812505780724595rm_f_v ) ) ).
% ex_in_conv
thf(fact_991_transD,axiom,
! [R: set_Pr8827868859434726151rm_f_v,X3: term_f_v,Y: term_f_v,Z2: term_f_v] :
( ( trans_on_term_f_v @ top_top_set_term_f_v @ R )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Y ) @ R )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y @ Z2 ) @ R )
=> ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Z2 ) @ R ) ) ) ) ).
% transD
thf(fact_992_transE,axiom,
! [R: set_Pr8827868859434726151rm_f_v,X3: term_f_v,Y: term_f_v,Z2: term_f_v] :
( ( trans_on_term_f_v @ top_top_set_term_f_v @ R )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Y ) @ R )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y @ Z2 ) @ R )
=> ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Z2 ) @ R ) ) ) ) ).
% transE
thf(fact_993_transI,axiom,
! [R: set_Pr8827868859434726151rm_f_v] :
( ! [X: term_f_v,Y3: term_f_v,Z: term_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ Y3 ) @ R )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y3 @ Z ) @ R )
=> ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ Z ) @ R ) ) )
=> ( trans_on_term_f_v @ top_top_set_term_f_v @ R ) ) ).
% transI
thf(fact_994_trans__rtrancl,axiom,
! [R: set_Pr8827868859434726151rm_f_v] : ( trans_on_term_f_v @ top_top_set_term_f_v @ ( transi3028809021613410336rm_f_v @ R ) ) ).
% trans_rtrancl
thf(fact_995_acyclicI,axiom,
! [R: set_Pr8827868859434726151rm_f_v] :
( ! [X: term_f_v] :
~ ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ X ) @ ( transi3208867851621559554rm_f_v @ R ) )
=> ( transi600795472790740820rm_f_v @ R ) ) ).
% acyclicI
thf(fact_996_acyclic__def,axiom,
( transi600795472790740820rm_f_v
= ( ^ [R7: set_Pr8827868859434726151rm_f_v] :
! [X4: term_f_v] :
~ ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X4 @ X4 ) @ ( transi3208867851621559554rm_f_v @ R7 ) ) ) ) ).
% acyclic_def
thf(fact_997_relcomp3__transI,axiom,
! [B5: set_Pr8827868859434726151rm_f_v,T: term_f_v,U: term_f_v,A4: set_Pr8827868859434726151rm_f_v,S: term_f_v,V: term_f_v] :
( ( trans_on_term_f_v @ top_top_set_term_f_v @ B5 )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ T @ U ) @ ( relcom2934258169684288428rm_f_v @ B5 @ ( relcom2934258169684288428rm_f_v @ A4 @ B5 ) ) )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ B5 )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ U @ V ) @ B5 )
=> ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ V ) @ ( relcom2934258169684288428rm_f_v @ B5 @ ( relcom2934258169684288428rm_f_v @ A4 @ B5 ) ) ) ) ) ) ) ).
% relcomp3_transI
thf(fact_998_rtrancl__Un2__separatorE,axiom,
! [B5: set_Pr8827868859434726151rm_f_v,A4: set_Pr8827868859434726151rm_f_v] :
( ( ( relcom2934258169684288428rm_f_v @ B5 @ A4 )
= bot_bo4638812505780724595rm_f_v )
=> ( ( transi3028809021613410336rm_f_v @ ( sup_su1994854688125285083rm_f_v @ A4 @ B5 ) )
= ( sup_su1994854688125285083rm_f_v @ ( transi3028809021613410336rm_f_v @ A4 ) @ ( relcom2934258169684288428rm_f_v @ ( transi3028809021613410336rm_f_v @ A4 ) @ ( transi3028809021613410336rm_f_v @ B5 ) ) ) ) ) ).
% rtrancl_Un2_separatorE
thf(fact_999_diff__shunt__var,axiom,
! [X3: set_Pr5245412377734449720_f_nat,Y: set_Pr5245412377734449720_f_nat] :
( ( ( minus_8941878871097791505_f_nat @ X3 @ Y )
= bot_bo136713897782694092_f_nat )
= ( ord_le8976984241387448984_f_nat @ X3 @ Y ) ) ).
% diff_shunt_var
thf(fact_1000_bot__empty__eq,axiom,
( bot_bo7198560532584249642_f_v_o
= ( ^ [X4: produc8027700073345061287rm_f_v] : ( member4316649861536941904rm_f_v @ X4 @ bot_bo4638812505780724595rm_f_v ) ) ) ).
% bot_empty_eq
thf(fact_1001_some__NF,axiom,
! [R: set_Pr7897041799657265351rm_f_v,X3: produc8027700073345061287rm_f_v] :
( ( abstra845488452567719153rm_f_v @ R @ ( insert5456475782666790775rm_f_v @ X3 @ bot_bo4638812505780724595rm_f_v ) )
=> ( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ X3 @ ( abstra6650587300220611071rm_f_v @ R @ X3 ) ) @ ( transi1062743709496199978rm_f_v @ R ) )
& ( member4316649861536941904rm_f_v @ ( abstra6650587300220611071rm_f_v @ R @ X3 ) @ ( abstra349717802288577587rm_f_v @ R ) ) ) ) ).
% some_NF
thf(fact_1002_some__NF,axiom,
! [R: set_Pr8827868859434726151rm_f_v,X3: term_f_v] :
( ( abstra6117554785717145rm_f_v @ R @ ( insert_term_f_v @ X3 @ bot_bot_set_term_f_v ) )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ ( abstra7098836743531038987rm_f_v @ R @ X3 ) ) @ ( transi3028809021613410336rm_f_v @ R ) )
& ( member_term_f_v @ ( abstra7098836743531038987rm_f_v @ R @ X3 ) @ ( abstract_NF_term_f_v @ R ) ) ) ) ).
% some_NF
thf(fact_1003_subset__emptyI,axiom,
! [A4: set_Pr8827868859434726151rm_f_v] :
( ! [X: produc8027700073345061287rm_f_v] :
~ ( member4316649861536941904rm_f_v @ X @ A4 )
=> ( ord_le4559761987009501863rm_f_v @ A4 @ bot_bo4638812505780724595rm_f_v ) ) ).
% subset_emptyI
thf(fact_1004_subset__emptyI,axiom,
! [A4: set_Pr5245412377734449720_f_nat] :
( ! [X: product_prod_f_nat] :
~ ( member2678038196313615385_f_nat @ X @ A4 )
=> ( ord_le8976984241387448984_f_nat @ A4 @ bot_bo136713897782694092_f_nat ) ) ).
% subset_emptyI
thf(fact_1005_CR__on__singletonI,axiom,
! [A: term_f_v,R: set_Pr8827868859434726151rm_f_v] :
( ! [B3: term_f_v,C: term_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B3 ) @ ( transi3028809021613410336rm_f_v @ R ) )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ C ) @ ( transi3028809021613410336rm_f_v @ R ) )
=> ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B3 @ C ) @ ( abstra8605546704478008873rm_f_v @ R ) ) ) )
=> ( abstra3735013631728912813rm_f_v @ R @ ( insert_term_f_v @ A @ bot_bot_set_term_f_v ) ) ) ).
% CR_on_singletonI
thf(fact_1006_insertCI,axiom,
! [A: produc8027700073345061287rm_f_v,B5: set_Pr8827868859434726151rm_f_v,B: produc8027700073345061287rm_f_v] :
( ( ~ ( member4316649861536941904rm_f_v @ A @ B5 )
=> ( A = B ) )
=> ( member4316649861536941904rm_f_v @ A @ ( insert5456475782666790775rm_f_v @ B @ B5 ) ) ) ).
% insertCI
thf(fact_1007_insert__iff,axiom,
! [A: produc8027700073345061287rm_f_v,B: produc8027700073345061287rm_f_v,A4: set_Pr8827868859434726151rm_f_v] :
( ( member4316649861536941904rm_f_v @ A @ ( insert5456475782666790775rm_f_v @ B @ A4 ) )
= ( ( A = B )
| ( member4316649861536941904rm_f_v @ A @ A4 ) ) ) ).
% insert_iff
thf(fact_1008_singletonI,axiom,
! [A: produc8027700073345061287rm_f_v] : ( member4316649861536941904rm_f_v @ A @ ( insert5456475782666790775rm_f_v @ A @ bot_bo4638812505780724595rm_f_v ) ) ).
% singletonI
thf(fact_1009_insert__subset,axiom,
! [X3: produc8027700073345061287rm_f_v,A4: set_Pr8827868859434726151rm_f_v,B5: set_Pr8827868859434726151rm_f_v] :
( ( ord_le4559761987009501863rm_f_v @ ( insert5456475782666790775rm_f_v @ X3 @ A4 ) @ B5 )
= ( ( member4316649861536941904rm_f_v @ X3 @ B5 )
& ( ord_le4559761987009501863rm_f_v @ A4 @ B5 ) ) ) ).
% insert_subset
thf(fact_1010_insert__subset,axiom,
! [X3: product_prod_f_nat,A4: set_Pr5245412377734449720_f_nat,B5: set_Pr5245412377734449720_f_nat] :
( ( ord_le8976984241387448984_f_nat @ ( insert5008453031520946354_f_nat @ X3 @ A4 ) @ B5 )
= ( ( member2678038196313615385_f_nat @ X3 @ B5 )
& ( ord_le8976984241387448984_f_nat @ A4 @ B5 ) ) ) ).
% insert_subset
thf(fact_1011_Int__insert__right__if1,axiom,
! [A: produc8027700073345061287rm_f_v,A4: set_Pr8827868859434726151rm_f_v,B5: set_Pr8827868859434726151rm_f_v] :
( ( member4316649861536941904rm_f_v @ A @ A4 )
=> ( ( inf_in6135112614888068085rm_f_v @ A4 @ ( insert5456475782666790775rm_f_v @ A @ B5 ) )
= ( insert5456475782666790775rm_f_v @ A @ ( inf_in6135112614888068085rm_f_v @ A4 @ B5 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_1012_Int__insert__right__if0,axiom,
! [A: produc8027700073345061287rm_f_v,A4: set_Pr8827868859434726151rm_f_v,B5: set_Pr8827868859434726151rm_f_v] :
( ~ ( member4316649861536941904rm_f_v @ A @ A4 )
=> ( ( inf_in6135112614888068085rm_f_v @ A4 @ ( insert5456475782666790775rm_f_v @ A @ B5 ) )
= ( inf_in6135112614888068085rm_f_v @ A4 @ B5 ) ) ) ).
% Int_insert_right_if0
thf(fact_1013_Int__insert__left__if1,axiom,
! [A: produc8027700073345061287rm_f_v,C2: set_Pr8827868859434726151rm_f_v,B5: set_Pr8827868859434726151rm_f_v] :
( ( member4316649861536941904rm_f_v @ A @ C2 )
=> ( ( inf_in6135112614888068085rm_f_v @ ( insert5456475782666790775rm_f_v @ A @ B5 ) @ C2 )
= ( insert5456475782666790775rm_f_v @ A @ ( inf_in6135112614888068085rm_f_v @ B5 @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_1014_Int__insert__left__if0,axiom,
! [A: produc8027700073345061287rm_f_v,C2: set_Pr8827868859434726151rm_f_v,B5: set_Pr8827868859434726151rm_f_v] :
( ~ ( member4316649861536941904rm_f_v @ A @ C2 )
=> ( ( inf_in6135112614888068085rm_f_v @ ( insert5456475782666790775rm_f_v @ A @ B5 ) @ C2 )
= ( inf_in6135112614888068085rm_f_v @ B5 @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_1015_insert__Diff1,axiom,
! [X3: produc8027700073345061287rm_f_v,B5: set_Pr8827868859434726151rm_f_v,A4: set_Pr8827868859434726151rm_f_v] :
( ( member4316649861536941904rm_f_v @ X3 @ B5 )
=> ( ( minus_410040198995547630rm_f_v @ ( insert5456475782666790775rm_f_v @ X3 @ A4 ) @ B5 )
= ( minus_410040198995547630rm_f_v @ A4 @ B5 ) ) ) ).
% insert_Diff1
thf(fact_1016_Diff__insert0,axiom,
! [X3: produc8027700073345061287rm_f_v,A4: set_Pr8827868859434726151rm_f_v,B5: set_Pr8827868859434726151rm_f_v] :
( ~ ( member4316649861536941904rm_f_v @ X3 @ A4 )
=> ( ( minus_410040198995547630rm_f_v @ A4 @ ( insert5456475782666790775rm_f_v @ X3 @ B5 ) )
= ( minus_410040198995547630rm_f_v @ A4 @ B5 ) ) ) ).
% Diff_insert0
thf(fact_1017_singleton__insert__inj__eq,axiom,
! [B: product_prod_f_nat,A: product_prod_f_nat,A4: set_Pr5245412377734449720_f_nat] :
( ( ( insert5008453031520946354_f_nat @ B @ bot_bo136713897782694092_f_nat )
= ( insert5008453031520946354_f_nat @ A @ A4 ) )
= ( ( A = B )
& ( ord_le8976984241387448984_f_nat @ A4 @ ( insert5008453031520946354_f_nat @ B @ bot_bo136713897782694092_f_nat ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_1018_singleton__insert__inj__eq_H,axiom,
! [A: product_prod_f_nat,A4: set_Pr5245412377734449720_f_nat,B: product_prod_f_nat] :
( ( ( insert5008453031520946354_f_nat @ A @ A4 )
= ( insert5008453031520946354_f_nat @ B @ bot_bo136713897782694092_f_nat ) )
= ( ( A = B )
& ( ord_le8976984241387448984_f_nat @ A4 @ ( insert5008453031520946354_f_nat @ B @ bot_bo136713897782694092_f_nat ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_1019_insert__disjoint_I1_J,axiom,
! [A: produc8027700073345061287rm_f_v,A4: set_Pr8827868859434726151rm_f_v,B5: set_Pr8827868859434726151rm_f_v] :
( ( ( inf_in6135112614888068085rm_f_v @ ( insert5456475782666790775rm_f_v @ A @ A4 ) @ B5 )
= bot_bo4638812505780724595rm_f_v )
= ( ~ ( member4316649861536941904rm_f_v @ A @ B5 )
& ( ( inf_in6135112614888068085rm_f_v @ A4 @ B5 )
= bot_bo4638812505780724595rm_f_v ) ) ) ).
% insert_disjoint(1)
thf(fact_1020_insert__disjoint_I2_J,axiom,
! [A: produc8027700073345061287rm_f_v,A4: set_Pr8827868859434726151rm_f_v,B5: set_Pr8827868859434726151rm_f_v] :
( ( bot_bo4638812505780724595rm_f_v
= ( inf_in6135112614888068085rm_f_v @ ( insert5456475782666790775rm_f_v @ A @ A4 ) @ B5 ) )
= ( ~ ( member4316649861536941904rm_f_v @ A @ B5 )
& ( bot_bo4638812505780724595rm_f_v
= ( inf_in6135112614888068085rm_f_v @ A4 @ B5 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_1021_disjoint__insert_I1_J,axiom,
! [B5: set_Pr8827868859434726151rm_f_v,A: produc8027700073345061287rm_f_v,A4: set_Pr8827868859434726151rm_f_v] :
( ( ( inf_in6135112614888068085rm_f_v @ B5 @ ( insert5456475782666790775rm_f_v @ A @ A4 ) )
= bot_bo4638812505780724595rm_f_v )
= ( ~ ( member4316649861536941904rm_f_v @ A @ B5 )
& ( ( inf_in6135112614888068085rm_f_v @ B5 @ A4 )
= bot_bo4638812505780724595rm_f_v ) ) ) ).
% disjoint_insert(1)
thf(fact_1022_disjoint__insert_I2_J,axiom,
! [A4: set_Pr8827868859434726151rm_f_v,B: produc8027700073345061287rm_f_v,B5: set_Pr8827868859434726151rm_f_v] :
( ( bot_bo4638812505780724595rm_f_v
= ( inf_in6135112614888068085rm_f_v @ A4 @ ( insert5456475782666790775rm_f_v @ B @ B5 ) ) )
= ( ~ ( member4316649861536941904rm_f_v @ B @ A4 )
& ( bot_bo4638812505780724595rm_f_v
= ( inf_in6135112614888068085rm_f_v @ A4 @ B5 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_1023_acyclic__insert,axiom,
! [Y: term_f_v,X3: term_f_v,R: set_Pr8827868859434726151rm_f_v] :
( ( transi600795472790740820rm_f_v @ ( insert5456475782666790775rm_f_v @ ( produc2510790566102465375rm_f_v @ Y @ X3 ) @ R ) )
= ( ( transi600795472790740820rm_f_v @ R )
& ~ ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Y ) @ ( transi3028809021613410336rm_f_v @ R ) ) ) ) ).
% acyclic_insert
thf(fact_1024_singletonD,axiom,
! [B: produc8027700073345061287rm_f_v,A: produc8027700073345061287rm_f_v] :
( ( member4316649861536941904rm_f_v @ B @ ( insert5456475782666790775rm_f_v @ A @ bot_bo4638812505780724595rm_f_v ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_1025_singleton__iff,axiom,
! [B: produc8027700073345061287rm_f_v,A: produc8027700073345061287rm_f_v] :
( ( member4316649861536941904rm_f_v @ B @ ( insert5456475782666790775rm_f_v @ A @ bot_bo4638812505780724595rm_f_v ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_1026_insert__subsetI,axiom,
! [X3: produc8027700073345061287rm_f_v,A4: set_Pr8827868859434726151rm_f_v,X5: set_Pr8827868859434726151rm_f_v] :
( ( member4316649861536941904rm_f_v @ X3 @ A4 )
=> ( ( ord_le4559761987009501863rm_f_v @ X5 @ A4 )
=> ( ord_le4559761987009501863rm_f_v @ ( insert5456475782666790775rm_f_v @ X3 @ X5 ) @ A4 ) ) ) ).
% insert_subsetI
thf(fact_1027_insert__subsetI,axiom,
! [X3: product_prod_f_nat,A4: set_Pr5245412377734449720_f_nat,X5: set_Pr5245412377734449720_f_nat] :
( ( member2678038196313615385_f_nat @ X3 @ A4 )
=> ( ( ord_le8976984241387448984_f_nat @ X5 @ A4 )
=> ( ord_le8976984241387448984_f_nat @ ( insert5008453031520946354_f_nat @ X3 @ X5 ) @ A4 ) ) ) ).
% insert_subsetI
thf(fact_1028_insert__mono,axiom,
! [C2: set_Pr5245412377734449720_f_nat,D: set_Pr5245412377734449720_f_nat,A: product_prod_f_nat] :
( ( ord_le8976984241387448984_f_nat @ C2 @ D )
=> ( ord_le8976984241387448984_f_nat @ ( insert5008453031520946354_f_nat @ A @ C2 ) @ ( insert5008453031520946354_f_nat @ A @ D ) ) ) ).
% insert_mono
thf(fact_1029_subset__insert,axiom,
! [X3: produc8027700073345061287rm_f_v,A4: set_Pr8827868859434726151rm_f_v,B5: set_Pr8827868859434726151rm_f_v] :
( ~ ( member4316649861536941904rm_f_v @ X3 @ A4 )
=> ( ( ord_le4559761987009501863rm_f_v @ A4 @ ( insert5456475782666790775rm_f_v @ X3 @ B5 ) )
= ( ord_le4559761987009501863rm_f_v @ A4 @ B5 ) ) ) ).
% subset_insert
thf(fact_1030_subset__insert,axiom,
! [X3: product_prod_f_nat,A4: set_Pr5245412377734449720_f_nat,B5: set_Pr5245412377734449720_f_nat] :
( ~ ( member2678038196313615385_f_nat @ X3 @ A4 )
=> ( ( ord_le8976984241387448984_f_nat @ A4 @ ( insert5008453031520946354_f_nat @ X3 @ B5 ) )
= ( ord_le8976984241387448984_f_nat @ A4 @ B5 ) ) ) ).
% subset_insert
thf(fact_1031_subset__insertI,axiom,
! [B5: set_Pr5245412377734449720_f_nat,A: product_prod_f_nat] : ( ord_le8976984241387448984_f_nat @ B5 @ ( insert5008453031520946354_f_nat @ A @ B5 ) ) ).
% subset_insertI
thf(fact_1032_subset__insertI2,axiom,
! [A4: set_Pr5245412377734449720_f_nat,B5: set_Pr5245412377734449720_f_nat,B: product_prod_f_nat] :
( ( ord_le8976984241387448984_f_nat @ A4 @ B5 )
=> ( ord_le8976984241387448984_f_nat @ A4 @ ( insert5008453031520946354_f_nat @ B @ B5 ) ) ) ).
% subset_insertI2
thf(fact_1033_insertE,axiom,
! [A: produc8027700073345061287rm_f_v,B: produc8027700073345061287rm_f_v,A4: set_Pr8827868859434726151rm_f_v] :
( ( member4316649861536941904rm_f_v @ A @ ( insert5456475782666790775rm_f_v @ B @ A4 ) )
=> ( ( A != B )
=> ( member4316649861536941904rm_f_v @ A @ A4 ) ) ) ).
% insertE
thf(fact_1034_insertI1,axiom,
! [A: produc8027700073345061287rm_f_v,B5: set_Pr8827868859434726151rm_f_v] : ( member4316649861536941904rm_f_v @ A @ ( insert5456475782666790775rm_f_v @ A @ B5 ) ) ).
% insertI1
thf(fact_1035_insertI2,axiom,
! [A: produc8027700073345061287rm_f_v,B5: set_Pr8827868859434726151rm_f_v,B: produc8027700073345061287rm_f_v] :
( ( member4316649861536941904rm_f_v @ A @ B5 )
=> ( member4316649861536941904rm_f_v @ A @ ( insert5456475782666790775rm_f_v @ B @ B5 ) ) ) ).
% insertI2
thf(fact_1036_Set_Oset__insert,axiom,
! [X3: produc8027700073345061287rm_f_v,A4: set_Pr8827868859434726151rm_f_v] :
( ( member4316649861536941904rm_f_v @ X3 @ A4 )
=> ~ ! [B9: set_Pr8827868859434726151rm_f_v] :
( ( A4
= ( insert5456475782666790775rm_f_v @ X3 @ B9 ) )
=> ( member4316649861536941904rm_f_v @ X3 @ B9 ) ) ) ).
% Set.set_insert
thf(fact_1037_insert__ident,axiom,
! [X3: produc8027700073345061287rm_f_v,A4: set_Pr8827868859434726151rm_f_v,B5: set_Pr8827868859434726151rm_f_v] :
( ~ ( member4316649861536941904rm_f_v @ X3 @ A4 )
=> ( ~ ( member4316649861536941904rm_f_v @ X3 @ B5 )
=> ( ( ( insert5456475782666790775rm_f_v @ X3 @ A4 )
= ( insert5456475782666790775rm_f_v @ X3 @ B5 ) )
= ( A4 = B5 ) ) ) ) ).
% insert_ident
thf(fact_1038_insert__absorb,axiom,
! [A: produc8027700073345061287rm_f_v,A4: set_Pr8827868859434726151rm_f_v] :
( ( member4316649861536941904rm_f_v @ A @ A4 )
=> ( ( insert5456475782666790775rm_f_v @ A @ A4 )
= A4 ) ) ).
% insert_absorb
thf(fact_1039_insert__eq__iff,axiom,
! [A: produc8027700073345061287rm_f_v,A4: set_Pr8827868859434726151rm_f_v,B: produc8027700073345061287rm_f_v,B5: set_Pr8827868859434726151rm_f_v] :
( ~ ( member4316649861536941904rm_f_v @ A @ A4 )
=> ( ~ ( member4316649861536941904rm_f_v @ B @ B5 )
=> ( ( ( insert5456475782666790775rm_f_v @ A @ A4 )
= ( insert5456475782666790775rm_f_v @ B @ B5 ) )
= ( ( ( A = B )
=> ( A4 = B5 ) )
& ( ( A != B )
=> ? [C6: set_Pr8827868859434726151rm_f_v] :
( ( A4
= ( insert5456475782666790775rm_f_v @ B @ C6 ) )
& ~ ( member4316649861536941904rm_f_v @ B @ C6 )
& ( B5
= ( insert5456475782666790775rm_f_v @ A @ C6 ) )
& ~ ( member4316649861536941904rm_f_v @ A @ C6 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_1040_mk__disjoint__insert,axiom,
! [A: produc8027700073345061287rm_f_v,A4: set_Pr8827868859434726151rm_f_v] :
( ( member4316649861536941904rm_f_v @ A @ A4 )
=> ? [B9: set_Pr8827868859434726151rm_f_v] :
( ( A4
= ( insert5456475782666790775rm_f_v @ A @ B9 ) )
& ~ ( member4316649861536941904rm_f_v @ A @ B9 ) ) ) ).
% mk_disjoint_insert
thf(fact_1041_insert__Diff__if,axiom,
! [X3: produc8027700073345061287rm_f_v,B5: set_Pr8827868859434726151rm_f_v,A4: set_Pr8827868859434726151rm_f_v] :
( ( ( member4316649861536941904rm_f_v @ X3 @ B5 )
=> ( ( minus_410040198995547630rm_f_v @ ( insert5456475782666790775rm_f_v @ X3 @ A4 ) @ B5 )
= ( minus_410040198995547630rm_f_v @ A4 @ B5 ) ) )
& ( ~ ( member4316649861536941904rm_f_v @ X3 @ B5 )
=> ( ( minus_410040198995547630rm_f_v @ ( insert5456475782666790775rm_f_v @ X3 @ A4 ) @ B5 )
= ( insert5456475782666790775rm_f_v @ X3 @ ( minus_410040198995547630rm_f_v @ A4 @ B5 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_1042_Int__insert__left,axiom,
! [A: produc8027700073345061287rm_f_v,C2: set_Pr8827868859434726151rm_f_v,B5: set_Pr8827868859434726151rm_f_v] :
( ( ( member4316649861536941904rm_f_v @ A @ C2 )
=> ( ( inf_in6135112614888068085rm_f_v @ ( insert5456475782666790775rm_f_v @ A @ B5 ) @ C2 )
= ( insert5456475782666790775rm_f_v @ A @ ( inf_in6135112614888068085rm_f_v @ B5 @ C2 ) ) ) )
& ( ~ ( member4316649861536941904rm_f_v @ A @ C2 )
=> ( ( inf_in6135112614888068085rm_f_v @ ( insert5456475782666790775rm_f_v @ A @ B5 ) @ C2 )
= ( inf_in6135112614888068085rm_f_v @ B5 @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_1043_Int__insert__right,axiom,
! [A: produc8027700073345061287rm_f_v,A4: set_Pr8827868859434726151rm_f_v,B5: set_Pr8827868859434726151rm_f_v] :
( ( ( member4316649861536941904rm_f_v @ A @ A4 )
=> ( ( inf_in6135112614888068085rm_f_v @ A4 @ ( insert5456475782666790775rm_f_v @ A @ B5 ) )
= ( insert5456475782666790775rm_f_v @ A @ ( inf_in6135112614888068085rm_f_v @ A4 @ B5 ) ) ) )
& ( ~ ( member4316649861536941904rm_f_v @ A @ A4 )
=> ( ( inf_in6135112614888068085rm_f_v @ A4 @ ( insert5456475782666790775rm_f_v @ A @ B5 ) )
= ( inf_in6135112614888068085rm_f_v @ A4 @ B5 ) ) ) ) ).
% Int_insert_right
thf(fact_1044_subset__singletonD,axiom,
! [A4: set_Pr5245412377734449720_f_nat,X3: product_prod_f_nat] :
( ( ord_le8976984241387448984_f_nat @ A4 @ ( insert5008453031520946354_f_nat @ X3 @ bot_bo136713897782694092_f_nat ) )
=> ( ( A4 = bot_bo136713897782694092_f_nat )
| ( A4
= ( insert5008453031520946354_f_nat @ X3 @ bot_bo136713897782694092_f_nat ) ) ) ) ).
% subset_singletonD
thf(fact_1045_subset__singleton__iff,axiom,
! [X5: set_Pr5245412377734449720_f_nat,A: product_prod_f_nat] :
( ( ord_le8976984241387448984_f_nat @ X5 @ ( insert5008453031520946354_f_nat @ A @ bot_bo136713897782694092_f_nat ) )
= ( ( X5 = bot_bo136713897782694092_f_nat )
| ( X5
= ( insert5008453031520946354_f_nat @ A @ bot_bo136713897782694092_f_nat ) ) ) ) ).
% subset_singleton_iff
thf(fact_1046_insert__Diff,axiom,
! [A: produc8027700073345061287rm_f_v,A4: set_Pr8827868859434726151rm_f_v] :
( ( member4316649861536941904rm_f_v @ A @ A4 )
=> ( ( insert5456475782666790775rm_f_v @ A @ ( minus_410040198995547630rm_f_v @ A4 @ ( insert5456475782666790775rm_f_v @ A @ bot_bo4638812505780724595rm_f_v ) ) )
= A4 ) ) ).
% insert_Diff
thf(fact_1047_Diff__insert__absorb,axiom,
! [X3: produc8027700073345061287rm_f_v,A4: set_Pr8827868859434726151rm_f_v] :
( ~ ( member4316649861536941904rm_f_v @ X3 @ A4 )
=> ( ( minus_410040198995547630rm_f_v @ ( insert5456475782666790775rm_f_v @ X3 @ A4 ) @ ( insert5456475782666790775rm_f_v @ X3 @ bot_bo4638812505780724595rm_f_v ) )
= A4 ) ) ).
% Diff_insert_absorb
thf(fact_1048_subset__Diff__insert,axiom,
! [A4: set_Pr8827868859434726151rm_f_v,B5: set_Pr8827868859434726151rm_f_v,X3: produc8027700073345061287rm_f_v,C2: set_Pr8827868859434726151rm_f_v] :
( ( ord_le4559761987009501863rm_f_v @ A4 @ ( minus_410040198995547630rm_f_v @ B5 @ ( insert5456475782666790775rm_f_v @ X3 @ C2 ) ) )
= ( ( ord_le4559761987009501863rm_f_v @ A4 @ ( minus_410040198995547630rm_f_v @ B5 @ C2 ) )
& ~ ( member4316649861536941904rm_f_v @ X3 @ A4 ) ) ) ).
% subset_Diff_insert
thf(fact_1049_subset__Diff__insert,axiom,
! [A4: set_Pr5245412377734449720_f_nat,B5: set_Pr5245412377734449720_f_nat,X3: product_prod_f_nat,C2: set_Pr5245412377734449720_f_nat] :
( ( ord_le8976984241387448984_f_nat @ A4 @ ( minus_8941878871097791505_f_nat @ B5 @ ( insert5008453031520946354_f_nat @ X3 @ C2 ) ) )
= ( ( ord_le8976984241387448984_f_nat @ A4 @ ( minus_8941878871097791505_f_nat @ B5 @ C2 ) )
& ~ ( member2678038196313615385_f_nat @ X3 @ A4 ) ) ) ).
% subset_Diff_insert
thf(fact_1050_subset__insert__iff,axiom,
! [A4: set_Pr8827868859434726151rm_f_v,X3: produc8027700073345061287rm_f_v,B5: set_Pr8827868859434726151rm_f_v] :
( ( ord_le4559761987009501863rm_f_v @ A4 @ ( insert5456475782666790775rm_f_v @ X3 @ B5 ) )
= ( ( ( member4316649861536941904rm_f_v @ X3 @ A4 )
=> ( ord_le4559761987009501863rm_f_v @ ( minus_410040198995547630rm_f_v @ A4 @ ( insert5456475782666790775rm_f_v @ X3 @ bot_bo4638812505780724595rm_f_v ) ) @ B5 ) )
& ( ~ ( member4316649861536941904rm_f_v @ X3 @ A4 )
=> ( ord_le4559761987009501863rm_f_v @ A4 @ B5 ) ) ) ) ).
% subset_insert_iff
thf(fact_1051_subset__insert__iff,axiom,
! [A4: set_Pr5245412377734449720_f_nat,X3: product_prod_f_nat,B5: set_Pr5245412377734449720_f_nat] :
( ( ord_le8976984241387448984_f_nat @ A4 @ ( insert5008453031520946354_f_nat @ X3 @ B5 ) )
= ( ( ( member2678038196313615385_f_nat @ X3 @ A4 )
=> ( ord_le8976984241387448984_f_nat @ ( minus_8941878871097791505_f_nat @ A4 @ ( insert5008453031520946354_f_nat @ X3 @ bot_bo136713897782694092_f_nat ) ) @ B5 ) )
& ( ~ ( member2678038196313615385_f_nat @ X3 @ A4 )
=> ( ord_le8976984241387448984_f_nat @ A4 @ B5 ) ) ) ) ).
% subset_insert_iff
thf(fact_1052_Diff__single__insert,axiom,
! [A4: set_Pr5245412377734449720_f_nat,X3: product_prod_f_nat,B5: set_Pr5245412377734449720_f_nat] :
( ( ord_le8976984241387448984_f_nat @ ( minus_8941878871097791505_f_nat @ A4 @ ( insert5008453031520946354_f_nat @ X3 @ bot_bo136713897782694092_f_nat ) ) @ B5 )
=> ( ord_le8976984241387448984_f_nat @ A4 @ ( insert5008453031520946354_f_nat @ X3 @ B5 ) ) ) ).
% Diff_single_insert
thf(fact_1053_SN__on__induct_H,axiom,
! [R3: set_Pr8827868859434726151rm_f_v,S: term_f_v,P2: term_f_v > $o] :
( ( abstra6117554785717145rm_f_v @ R3 @ ( insert_term_f_v @ S @ bot_bot_set_term_f_v ) )
=> ( ! [T2: term_f_v] :
( ! [U4: term_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ T2 @ U4 ) @ R3 )
=> ( P2 @ U4 ) )
=> ( P2 @ T2 ) )
=> ( P2 @ S ) ) ) ).
% SN_on_induct'
thf(fact_1054_step__reflects__SN__on,axiom,
! [A: term_f_v,R: set_Pr8827868859434726151rm_f_v] :
( ! [B3: term_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B3 ) @ R )
=> ( abstra6117554785717145rm_f_v @ R @ ( insert_term_f_v @ B3 @ bot_bot_set_term_f_v ) ) )
=> ( abstra6117554785717145rm_f_v @ R @ ( insert_term_f_v @ A @ bot_bot_set_term_f_v ) ) ) ).
% step_reflects_SN_on
thf(fact_1055_SN__on__imp__on__minimal,axiom,
! [R: set_Pr7897041799657265351rm_f_v,X3: produc8027700073345061287rm_f_v] :
( ( abstra845488452567719153rm_f_v @ R @ ( insert5456475782666790775rm_f_v @ X3 @ bot_bo4638812505780724595rm_f_v ) )
=> ! [Q2: set_Pr8827868859434726151rm_f_v] :
( ( member4316649861536941904rm_f_v @ X3 @ Q2 )
=> ? [X: produc8027700073345061287rm_f_v] :
( ( member4316649861536941904rm_f_v @ X @ Q2 )
& ! [Y6: produc8027700073345061287rm_f_v] :
( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ X @ Y6 ) @ R )
=> ~ ( member4316649861536941904rm_f_v @ Y6 @ Q2 ) ) ) ) ) ).
% SN_on_imp_on_minimal
thf(fact_1056_SN__on__imp__on__minimal,axiom,
! [R: set_Pr8827868859434726151rm_f_v,X3: term_f_v] :
( ( abstra6117554785717145rm_f_v @ R @ ( insert_term_f_v @ X3 @ bot_bot_set_term_f_v ) )
=> ! [Q2: set_term_f_v] :
( ( member_term_f_v @ X3 @ Q2 )
=> ? [X: term_f_v] :
( ( member_term_f_v @ X @ Q2 )
& ! [Y6: term_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ Y6 ) @ R )
=> ~ ( member_term_f_v @ Y6 @ Q2 ) ) ) ) ) ).
% SN_on_imp_on_minimal
thf(fact_1057_step__preserves__SN__on,axiom,
! [A: term_f_v,B: term_f_v,R: set_Pr8827868859434726151rm_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ R )
=> ( ( abstra6117554785717145rm_f_v @ R @ ( insert_term_f_v @ A @ bot_bot_set_term_f_v ) )
=> ( abstra6117554785717145rm_f_v @ R @ ( insert_term_f_v @ B @ bot_bot_set_term_f_v ) ) ) ) ).
% step_preserves_SN_on
thf(fact_1058_SN__on__induct__acc__style,axiom,
! [R3: set_Pr8827868859434726151rm_f_v,A: term_f_v,P2: term_f_v > $o] :
( ( abstra6117554785717145rm_f_v @ R3 @ ( insert_term_f_v @ A @ bot_bot_set_term_f_v ) )
=> ( ! [X: term_f_v] :
( ( abstra6117554785717145rm_f_v @ R3 @ ( insert_term_f_v @ X @ bot_bot_set_term_f_v ) )
=> ( ! [Y6: term_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ Y6 ) @ R3 )
=> ( P2 @ Y6 ) )
=> ( P2 @ X ) ) )
=> ( P2 @ A ) ) ) ).
% SN_on_induct_acc_style
thf(fact_1059_lower__set__imp__not__SN__on,axiom,
! [S: produc8027700073345061287rm_f_v,X5: set_Pr8827868859434726151rm_f_v,R3: set_Pr7897041799657265351rm_f_v] :
( ( member4316649861536941904rm_f_v @ S @ X5 )
=> ( ! [X: produc8027700073345061287rm_f_v] :
( ( member4316649861536941904rm_f_v @ X @ X5 )
=> ? [Xa2: produc8027700073345061287rm_f_v] :
( ( member4316649861536941904rm_f_v @ Xa2 @ X5 )
& ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ X @ Xa2 ) @ R3 ) ) )
=> ~ ( abstra845488452567719153rm_f_v @ R3 @ ( insert5456475782666790775rm_f_v @ S @ bot_bo4638812505780724595rm_f_v ) ) ) ) ).
% lower_set_imp_not_SN_on
thf(fact_1060_lower__set__imp__not__SN__on,axiom,
! [S: term_f_v,X5: set_term_f_v,R3: set_Pr8827868859434726151rm_f_v] :
( ( member_term_f_v @ S @ X5 )
=> ( ! [X: term_f_v] :
( ( member_term_f_v @ X @ X5 )
=> ? [Xa2: term_f_v] :
( ( member_term_f_v @ Xa2 @ X5 )
& ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ Xa2 ) @ R3 ) ) )
=> ~ ( abstra6117554785717145rm_f_v @ R3 @ ( insert_term_f_v @ S @ bot_bot_set_term_f_v ) ) ) ) ).
% lower_set_imp_not_SN_on
thf(fact_1061_SN__on__all__reducts__SN__on__conv,axiom,
! [R: set_Pr8827868859434726151rm_f_v,A: term_f_v] :
( ( abstra6117554785717145rm_f_v @ R @ ( insert_term_f_v @ A @ bot_bot_set_term_f_v ) )
= ( ! [B4: term_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B4 ) @ R )
=> ( abstra6117554785717145rm_f_v @ R @ ( insert_term_f_v @ B4 @ bot_bot_set_term_f_v ) ) ) ) ) ).
% SN_on_all_reducts_SN_on_conv
thf(fact_1062_ssubst__Pair__rhs,axiom,
! [R: term_f_v,S: term_f_v,R3: set_Pr8827868859434726151rm_f_v,S6: term_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ R @ S ) @ R3 )
=> ( ( S6 = S )
=> ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ R @ S6 ) @ R3 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_1063_NF__imp__SN__on,axiom,
! [A: produc8027700073345061287rm_f_v,R3: set_Pr7897041799657265351rm_f_v] :
( ( member4316649861536941904rm_f_v @ A @ ( abstra349717802288577587rm_f_v @ R3 ) )
=> ( abstra845488452567719153rm_f_v @ R3 @ ( insert5456475782666790775rm_f_v @ A @ bot_bo4638812505780724595rm_f_v ) ) ) ).
% NF_imp_SN_on
thf(fact_1064_steps__reflect__SN__on,axiom,
! [R: set_Pr8827868859434726151rm_f_v,B: term_f_v,A: term_f_v] :
( ~ ( abstra6117554785717145rm_f_v @ R @ ( insert_term_f_v @ B @ bot_bot_set_term_f_v ) )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( transi3028809021613410336rm_f_v @ R ) )
=> ~ ( abstra6117554785717145rm_f_v @ R @ ( insert_term_f_v @ A @ bot_bot_set_term_f_v ) ) ) ) ).
% steps_reflect_SN_on
thf(fact_1065_steps__preserve__SN__on,axiom,
! [A: term_f_v,B: term_f_v,A4: set_Pr8827868859434726151rm_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( transi3028809021613410336rm_f_v @ A4 ) )
=> ( ( abstra6117554785717145rm_f_v @ A4 @ ( insert_term_f_v @ A @ bot_bot_set_term_f_v ) )
=> ( abstra6117554785717145rm_f_v @ A4 @ ( insert_term_f_v @ B @ bot_bot_set_term_f_v ) ) ) ) ).
% steps_preserve_SN_on
thf(fact_1066_trans__singleton,axiom,
! [A: term_f_v] : ( trans_on_term_f_v @ top_top_set_term_f_v @ ( insert5456475782666790775rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ A ) @ bot_bo4638812505780724595rm_f_v ) ) ).
% trans_singleton
thf(fact_1067_SN__on__imp__normalizability,axiom,
! [R: set_Pr8827868859434726151rm_f_v,A: term_f_v] :
( ( abstra6117554785717145rm_f_v @ R @ ( insert_term_f_v @ A @ bot_bot_set_term_f_v ) )
=> ? [B3: term_f_v] : ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B3 ) @ ( abstra7482117481303700155rm_f_v @ R ) ) ) ).
% SN_on_imp_normalizability
thf(fact_1068_SN__reaches__NF,axiom,
! [R: set_Pr7897041799657265351rm_f_v,X3: produc8027700073345061287rm_f_v] :
( ( abstra845488452567719153rm_f_v @ R @ ( insert5456475782666790775rm_f_v @ X3 @ bot_bo4638812505780724595rm_f_v ) )
=> ? [Y3: produc8027700073345061287rm_f_v] :
( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ X3 @ Y3 ) @ ( transi1062743709496199978rm_f_v @ R ) )
& ( member4316649861536941904rm_f_v @ Y3 @ ( abstra349717802288577587rm_f_v @ R ) ) ) ) ).
% SN_reaches_NF
thf(fact_1069_SN__reaches__NF,axiom,
! [R: set_Pr8827868859434726151rm_f_v,X3: term_f_v] :
( ( abstra6117554785717145rm_f_v @ R @ ( insert_term_f_v @ X3 @ bot_bot_set_term_f_v ) )
=> ? [Y3: term_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Y3 ) @ ( transi3028809021613410336rm_f_v @ R ) )
& ( member_term_f_v @ Y3 @ ( abstract_NF_term_f_v @ R ) ) ) ) ).
% SN_reaches_NF
thf(fact_1070_is__singletonI_H,axiom,
! [A4: set_Pr8827868859434726151rm_f_v] :
( ( A4 != bot_bo4638812505780724595rm_f_v )
=> ( ! [X: produc8027700073345061287rm_f_v,Y3: produc8027700073345061287rm_f_v] :
( ( member4316649861536941904rm_f_v @ X @ A4 )
=> ( ( member4316649861536941904rm_f_v @ Y3 @ A4 )
=> ( X = Y3 ) ) )
=> ( is_sin8002852355414267547rm_f_v @ A4 ) ) ) ).
% is_singletonI'
thf(fact_1071_Field__insert,axiom,
! [A: term_f_v,B: term_f_v,R: set_Pr8827868859434726151rm_f_v] :
( ( field_term_f_v @ ( insert5456475782666790775rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ R ) )
= ( sup_sup_set_term_f_v @ ( insert_term_f_v @ A @ ( insert_term_f_v @ B @ bot_bot_set_term_f_v ) ) @ ( field_term_f_v @ R ) ) ) ).
% Field_insert
thf(fact_1072_member__remove,axiom,
! [X3: produc8027700073345061287rm_f_v,Y: produc8027700073345061287rm_f_v,A4: set_Pr8827868859434726151rm_f_v] :
( ( member4316649861536941904rm_f_v @ X3 @ ( remove6970840997168072972rm_f_v @ Y @ A4 ) )
= ( ( member4316649861536941904rm_f_v @ X3 @ A4 )
& ( X3 != Y ) ) ) ).
% member_remove
thf(fact_1073_FieldI1,axiom,
! [I: produc8027700073345061287rm_f_v,J: produc8027700073345061287rm_f_v,R3: set_Pr7897041799657265351rm_f_v] :
( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ I @ J ) @ R3 )
=> ( member4316649861536941904rm_f_v @ I @ ( field_5331866484994830708rm_f_v @ R3 ) ) ) ).
% FieldI1
thf(fact_1074_FieldI1,axiom,
! [I: term_f_v,J: term_f_v,R3: set_Pr8827868859434726151rm_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ I @ J ) @ R3 )
=> ( member_term_f_v @ I @ ( field_term_f_v @ R3 ) ) ) ).
% FieldI1
thf(fact_1075_FieldI2,axiom,
! [I: produc8027700073345061287rm_f_v,J: produc8027700073345061287rm_f_v,R3: set_Pr7897041799657265351rm_f_v] :
( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ I @ J ) @ R3 )
=> ( member4316649861536941904rm_f_v @ J @ ( field_5331866484994830708rm_f_v @ R3 ) ) ) ).
% FieldI2
thf(fact_1076_FieldI2,axiom,
! [I: term_f_v,J: term_f_v,R3: set_Pr8827868859434726151rm_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ I @ J ) @ R3 )
=> ( member_term_f_v @ J @ ( field_term_f_v @ R3 ) ) ) ).
% FieldI2
thf(fact_1077_mono__Field,axiom,
! [R: set_Pr516226060960033221_f_nat,S: set_Pr516226060960033221_f_nat] :
( ( ord_le8946695678602162533_f_nat @ R @ S )
=> ( ord_le8976984241387448984_f_nat @ ( field_5908777168505831157_f_nat @ R ) @ ( field_5908777168505831157_f_nat @ S ) ) ) ).
% mono_Field
thf(fact_1078_wo__rel_Ocases__Total3,axiom,
! [R: set_Pr8827868859434726151rm_f_v,A: term_f_v,B: term_f_v,Phi: term_f_v > term_f_v > $o] :
( ( bNF_We4761303178683774583rm_f_v @ R )
=> ( ( ord_le947722214152454265rm_f_v @ ( insert_term_f_v @ A @ ( insert_term_f_v @ B @ bot_bot_set_term_f_v ) ) @ ( field_term_f_v @ R ) )
=> ( ( ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( minus_410040198995547630rm_f_v @ R @ id_term_f_v ) )
| ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B @ A ) @ ( minus_410040198995547630rm_f_v @ R @ id_term_f_v ) ) )
=> ( Phi @ A @ B ) )
=> ( ( ( A = B )
=> ( Phi @ A @ B ) )
=> ( Phi @ A @ B ) ) ) ) ) ).
% wo_rel.cases_Total3
thf(fact_1079_wo__rel_Ocases__Total3,axiom,
! [R: set_Pr516226060960033221_f_nat,A: product_prod_f_nat,B: product_prod_f_nat,Phi: product_prod_f_nat > product_prod_f_nat > $o] :
( ( bNF_We1252253923098187670_f_nat @ R )
=> ( ( ord_le8976984241387448984_f_nat @ ( insert5008453031520946354_f_nat @ A @ ( insert5008453031520946354_f_nat @ B @ bot_bo136713897782694092_f_nat ) ) @ ( field_5908777168505831157_f_nat @ R ) )
=> ( ( ( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ A @ B ) @ ( minus_46298888925056428_f_nat @ R @ id_Pro2160904946306395686_f_nat ) )
| ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ B @ A ) @ ( minus_46298888925056428_f_nat @ R @ id_Pro2160904946306395686_f_nat ) ) )
=> ( Phi @ A @ B ) )
=> ( ( ( A = B )
=> ( Phi @ A @ B ) )
=> ( Phi @ A @ B ) ) ) ) ) ).
% wo_rel.cases_Total3
thf(fact_1080_refl__on__singleton,axiom,
! [X3: term_f_v] : ( refl_on_term_f_v @ ( insert_term_f_v @ X3 @ bot_bot_set_term_f_v ) @ ( insert5456475782666790775rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ X3 ) @ bot_bo4638812505780724595rm_f_v ) ) ).
% refl_on_singleton
thf(fact_1081_well__order__induct__imp,axiom,
! [R: set_Pr7897041799657265351rm_f_v,P2: produc8027700073345061287rm_f_v > $o,A: produc8027700073345061287rm_f_v] :
( ( bNF_We135845164404304787rm_f_v @ R )
=> ( ! [X: produc8027700073345061287rm_f_v] :
( ! [Y6: produc8027700073345061287rm_f_v] :
( ( ( Y6 != X )
& ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ Y6 @ X ) @ R ) )
=> ( ( member4316649861536941904rm_f_v @ Y6 @ ( field_5331866484994830708rm_f_v @ R ) )
=> ( P2 @ Y6 ) ) )
=> ( ( member4316649861536941904rm_f_v @ X @ ( field_5331866484994830708rm_f_v @ R ) )
=> ( P2 @ X ) ) )
=> ( ( member4316649861536941904rm_f_v @ A @ ( field_5331866484994830708rm_f_v @ R ) )
=> ( P2 @ A ) ) ) ) ).
% well_order_induct_imp
thf(fact_1082_well__order__induct__imp,axiom,
! [R: set_Pr8827868859434726151rm_f_v,P2: term_f_v > $o,A: term_f_v] :
( ( bNF_We4761303178683774583rm_f_v @ R )
=> ( ! [X: term_f_v] :
( ! [Y6: term_f_v] :
( ( ( Y6 != X )
& ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y6 @ X ) @ R ) )
=> ( ( member_term_f_v @ Y6 @ ( field_term_f_v @ R ) )
=> ( P2 @ Y6 ) ) )
=> ( ( member_term_f_v @ X @ ( field_term_f_v @ R ) )
=> ( P2 @ X ) ) )
=> ( ( member_term_f_v @ A @ ( field_term_f_v @ R ) )
=> ( P2 @ A ) ) ) ) ).
% well_order_induct_imp
thf(fact_1083_wo__rel_Oin__notinI,axiom,
! [R: set_Pr7897041799657265351rm_f_v,J: produc8027700073345061287rm_f_v,I: produc8027700073345061287rm_f_v] :
( ( bNF_We135845164404304787rm_f_v @ R )
=> ( ( ~ ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ J @ I ) @ R )
| ( J = I ) )
=> ( ( member4316649861536941904rm_f_v @ I @ ( field_5331866484994830708rm_f_v @ R ) )
=> ( ( member4316649861536941904rm_f_v @ J @ ( field_5331866484994830708rm_f_v @ R ) )
=> ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ I @ J ) @ R ) ) ) ) ) ).
% wo_rel.in_notinI
thf(fact_1084_wo__rel_Oin__notinI,axiom,
! [R: set_Pr8827868859434726151rm_f_v,J: term_f_v,I: term_f_v] :
( ( bNF_We4761303178683774583rm_f_v @ R )
=> ( ( ~ ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ J @ I ) @ R )
| ( J = I ) )
=> ( ( member_term_f_v @ I @ ( field_term_f_v @ R ) )
=> ( ( member_term_f_v @ J @ ( field_term_f_v @ R ) )
=> ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ I @ J ) @ R ) ) ) ) ) ).
% wo_rel.in_notinI
thf(fact_1085_wo__rel_OTOTALS,axiom,
! [R: set_Pr8827868859434726151rm_f_v] :
( ( bNF_We4761303178683774583rm_f_v @ R )
=> ! [X6: term_f_v] :
( ( member_term_f_v @ X6 @ ( field_term_f_v @ R ) )
=> ! [Xa2: term_f_v] :
( ( member_term_f_v @ Xa2 @ ( field_term_f_v @ R ) )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X6 @ Xa2 ) @ R )
| ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Xa2 @ X6 ) @ R ) ) ) ) ) ).
% wo_rel.TOTALS
thf(fact_1086_refl__rtrancl,axiom,
! [R: set_Pr8827868859434726151rm_f_v] : ( refl_on_term_f_v @ top_top_set_term_f_v @ ( transi3028809021613410336rm_f_v @ R ) ) ).
% refl_rtrancl
thf(fact_1087_reflD,axiom,
! [R: set_Pr8827868859434726151rm_f_v,A: term_f_v] :
( ( refl_on_term_f_v @ top_top_set_term_f_v @ R )
=> ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ A ) @ R ) ) ).
% reflD
thf(fact_1088_reflI,axiom,
! [R: set_Pr8827868859434726151rm_f_v] :
( ! [X: term_f_v] : ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ X ) @ R )
=> ( refl_on_term_f_v @ top_top_set_term_f_v @ R ) ) ).
% reflI
thf(fact_1089_wo__rel_Owell__order__induct,axiom,
! [R: set_Pr8827868859434726151rm_f_v,P2: term_f_v > $o,A: term_f_v] :
( ( bNF_We4761303178683774583rm_f_v @ R )
=> ( ! [X: term_f_v] :
( ! [Y6: term_f_v] :
( ( ( Y6 != X )
& ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y6 @ X ) @ R ) )
=> ( P2 @ Y6 ) )
=> ( P2 @ X ) )
=> ( P2 @ A ) ) ) ).
% wo_rel.well_order_induct
thf(fact_1090_refl__onD2,axiom,
! [A4: set_Pr8827868859434726151rm_f_v,R: set_Pr7897041799657265351rm_f_v,X3: produc8027700073345061287rm_f_v,Y: produc8027700073345061287rm_f_v] :
( ( refl_o3791338839128770854rm_f_v @ A4 @ R )
=> ( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ X3 @ Y ) @ R )
=> ( member4316649861536941904rm_f_v @ Y @ A4 ) ) ) ).
% refl_onD2
thf(fact_1091_refl__onD2,axiom,
! [A4: set_term_f_v,R: set_Pr8827868859434726151rm_f_v,X3: term_f_v,Y: term_f_v] :
( ( refl_on_term_f_v @ A4 @ R )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Y ) @ R )
=> ( member_term_f_v @ Y @ A4 ) ) ) ).
% refl_onD2
thf(fact_1092_refl__onD1,axiom,
! [A4: set_Pr8827868859434726151rm_f_v,R: set_Pr7897041799657265351rm_f_v,X3: produc8027700073345061287rm_f_v,Y: produc8027700073345061287rm_f_v] :
( ( refl_o3791338839128770854rm_f_v @ A4 @ R )
=> ( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ X3 @ Y ) @ R )
=> ( member4316649861536941904rm_f_v @ X3 @ A4 ) ) ) ).
% refl_onD1
thf(fact_1093_refl__onD1,axiom,
! [A4: set_term_f_v,R: set_Pr8827868859434726151rm_f_v,X3: term_f_v,Y: term_f_v] :
( ( refl_on_term_f_v @ A4 @ R )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Y ) @ R )
=> ( member_term_f_v @ X3 @ A4 ) ) ) ).
% refl_onD1
thf(fact_1094_refl__onD,axiom,
! [A4: set_Pr8827868859434726151rm_f_v,R: set_Pr7897041799657265351rm_f_v,A: produc8027700073345061287rm_f_v] :
( ( refl_o3791338839128770854rm_f_v @ A4 @ R )
=> ( ( member4316649861536941904rm_f_v @ A @ A4 )
=> ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A @ A ) @ R ) ) ) ).
% refl_onD
thf(fact_1095_refl__onD,axiom,
! [A4: set_term_f_v,R: set_Pr8827868859434726151rm_f_v,A: term_f_v] :
( ( refl_on_term_f_v @ A4 @ R )
=> ( ( member_term_f_v @ A @ A4 )
=> ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ A ) @ R ) ) ) ).
% refl_onD
thf(fact_1096_trans__refl__imp__rtrancl__id,axiom,
! [R: set_Pr8827868859434726151rm_f_v] :
( ( trans_on_term_f_v @ top_top_set_term_f_v @ R )
=> ( ( refl_on_term_f_v @ top_top_set_term_f_v @ R )
=> ( ( transi3028809021613410336rm_f_v @ R )
= R ) ) ) ).
% trans_refl_imp_rtrancl_id
thf(fact_1097_wo__rel_Ocases__Total,axiom,
! [R: set_Pr8827868859434726151rm_f_v,A: term_f_v,B: term_f_v,Phi: term_f_v > term_f_v > $o] :
( ( bNF_We4761303178683774583rm_f_v @ R )
=> ( ( ord_le947722214152454265rm_f_v @ ( insert_term_f_v @ A @ ( insert_term_f_v @ B @ bot_bot_set_term_f_v ) ) @ ( field_term_f_v @ R ) )
=> ( ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ R )
=> ( Phi @ A @ B ) )
=> ( ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B @ A ) @ R )
=> ( Phi @ A @ B ) )
=> ( Phi @ A @ B ) ) ) ) ) ).
% wo_rel.cases_Total
thf(fact_1098_wo__rel_Ocases__Total,axiom,
! [R: set_Pr516226060960033221_f_nat,A: product_prod_f_nat,B: product_prod_f_nat,Phi: product_prod_f_nat > product_prod_f_nat > $o] :
( ( bNF_We1252253923098187670_f_nat @ R )
=> ( ( ord_le8976984241387448984_f_nat @ ( insert5008453031520946354_f_nat @ A @ ( insert5008453031520946354_f_nat @ B @ bot_bo136713897782694092_f_nat ) ) @ ( field_5908777168505831157_f_nat @ R ) )
=> ( ( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ A @ B ) @ R )
=> ( Phi @ A @ B ) )
=> ( ( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ B @ A ) @ R )
=> ( Phi @ A @ B ) )
=> ( Phi @ A @ B ) ) ) ) ) ).
% wo_rel.cases_Total
thf(fact_1099_wo__rel_OWell__order__isMinim__exists,axiom,
! [R: set_Pr516226060960033221_f_nat,B5: set_Pr5245412377734449720_f_nat] :
( ( bNF_We1252253923098187670_f_nat @ R )
=> ( ( ord_le8976984241387448984_f_nat @ B5 @ ( field_5908777168505831157_f_nat @ R ) )
=> ( ( B5 != bot_bo136713897782694092_f_nat )
=> ? [X_1: product_prod_f_nat] : ( bNF_We1452518865689677644_f_nat @ R @ B5 @ X_1 ) ) ) ) ).
% wo_rel.Well_order_isMinim_exists
thf(fact_1100_wo__rel_OisMinim__def,axiom,
! [R: set_Pr7897041799657265351rm_f_v,A4: set_Pr8827868859434726151rm_f_v,B: produc8027700073345061287rm_f_v] :
( ( bNF_We135845164404304787rm_f_v @ R )
=> ( ( bNF_We6877594076361128029rm_f_v @ R @ A4 @ B )
= ( ( member4316649861536941904rm_f_v @ B @ A4 )
& ! [X4: produc8027700073345061287rm_f_v] :
( ( member4316649861536941904rm_f_v @ X4 @ A4 )
=> ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ B @ X4 ) @ R ) ) ) ) ) ).
% wo_rel.isMinim_def
thf(fact_1101_wo__rel_OisMinim__def,axiom,
! [R: set_Pr8827868859434726151rm_f_v,A4: set_term_f_v,B: term_f_v] :
( ( bNF_We4761303178683774583rm_f_v @ R )
=> ( ( bNF_We5204858324500316461rm_f_v @ R @ A4 @ B )
= ( ( member_term_f_v @ B @ A4 )
& ! [X4: term_f_v] :
( ( member_term_f_v @ X4 @ A4 )
=> ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B @ X4 ) @ R ) ) ) ) ) ).
% wo_rel.isMinim_def
thf(fact_1102_wo__rel_Ominim__isMinim,axiom,
! [R: set_Pr516226060960033221_f_nat,B5: set_Pr5245412377734449720_f_nat] :
( ( bNF_We1252253923098187670_f_nat @ R )
=> ( ( ord_le8976984241387448984_f_nat @ B5 @ ( field_5908777168505831157_f_nat @ R ) )
=> ( ( B5 != bot_bo136713897782694092_f_nat )
=> ( bNF_We1452518865689677644_f_nat @ R @ B5 @ ( bNF_We848864957873457590_f_nat @ R @ B5 ) ) ) ) ) ).
% wo_rel.minim_isMinim
thf(fact_1103_wo__rel_Ominim__in,axiom,
! [R: set_Pr7897041799657265351rm_f_v,B5: set_Pr8827868859434726151rm_f_v] :
( ( bNF_We135845164404304787rm_f_v @ R )
=> ( ( ord_le4559761987009501863rm_f_v @ B5 @ ( field_5331866484994830708rm_f_v @ R ) )
=> ( ( B5 != bot_bo4638812505780724595rm_f_v )
=> ( member4316649861536941904rm_f_v @ ( bNF_We9009046827986260339rm_f_v @ R @ B5 ) @ B5 ) ) ) ) ).
% wo_rel.minim_in
thf(fact_1104_wo__rel_Ominim__in,axiom,
! [R: set_Pr516226060960033221_f_nat,B5: set_Pr5245412377734449720_f_nat] :
( ( bNF_We1252253923098187670_f_nat @ R )
=> ( ( ord_le8976984241387448984_f_nat @ B5 @ ( field_5908777168505831157_f_nat @ R ) )
=> ( ( B5 != bot_bo136713897782694092_f_nat )
=> ( member2678038196313615385_f_nat @ ( bNF_We848864957873457590_f_nat @ R @ B5 ) @ B5 ) ) ) ) ).
% wo_rel.minim_in
thf(fact_1105_wo__rel_Oequals__minim,axiom,
! [R: set_Pr7897041799657265351rm_f_v,B5: set_Pr8827868859434726151rm_f_v,A: produc8027700073345061287rm_f_v] :
( ( bNF_We135845164404304787rm_f_v @ R )
=> ( ( ord_le4559761987009501863rm_f_v @ B5 @ ( field_5331866484994830708rm_f_v @ R ) )
=> ( ( member4316649861536941904rm_f_v @ A @ B5 )
=> ( ! [B3: produc8027700073345061287rm_f_v] :
( ( member4316649861536941904rm_f_v @ B3 @ B5 )
=> ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A @ B3 ) @ R ) )
=> ( A
= ( bNF_We9009046827986260339rm_f_v @ R @ B5 ) ) ) ) ) ) ).
% wo_rel.equals_minim
thf(fact_1106_wo__rel_Oequals__minim,axiom,
! [R: set_Pr8827868859434726151rm_f_v,B5: set_term_f_v,A: term_f_v] :
( ( bNF_We4761303178683774583rm_f_v @ R )
=> ( ( ord_le947722214152454265rm_f_v @ B5 @ ( field_term_f_v @ R ) )
=> ( ( member_term_f_v @ A @ B5 )
=> ( ! [B3: term_f_v] :
( ( member_term_f_v @ B3 @ B5 )
=> ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B3 ) @ R ) )
=> ( A
= ( bNF_We1544997276579117207rm_f_v @ R @ B5 ) ) ) ) ) ) ).
% wo_rel.equals_minim
thf(fact_1107_wo__rel_Oequals__minim,axiom,
! [R: set_Pr516226060960033221_f_nat,B5: set_Pr5245412377734449720_f_nat,A: product_prod_f_nat] :
( ( bNF_We1252253923098187670_f_nat @ R )
=> ( ( ord_le8976984241387448984_f_nat @ B5 @ ( field_5908777168505831157_f_nat @ R ) )
=> ( ( member2678038196313615385_f_nat @ A @ B5 )
=> ( ! [B3: product_prod_f_nat] :
( ( member2678038196313615385_f_nat @ B3 @ B5 )
=> ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ A @ B3 ) @ R ) )
=> ( A
= ( bNF_We848864957873457590_f_nat @ R @ B5 ) ) ) ) ) ) ).
% wo_rel.equals_minim
thf(fact_1108_wo__rel_Ominim__least,axiom,
! [R: set_Pr7897041799657265351rm_f_v,B5: set_Pr8827868859434726151rm_f_v,B: produc8027700073345061287rm_f_v] :
( ( bNF_We135845164404304787rm_f_v @ R )
=> ( ( ord_le4559761987009501863rm_f_v @ B5 @ ( field_5331866484994830708rm_f_v @ R ) )
=> ( ( member4316649861536941904rm_f_v @ B @ B5 )
=> ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ ( bNF_We9009046827986260339rm_f_v @ R @ B5 ) @ B ) @ R ) ) ) ) ).
% wo_rel.minim_least
thf(fact_1109_wo__rel_Ominim__least,axiom,
! [R: set_Pr8827868859434726151rm_f_v,B5: set_term_f_v,B: term_f_v] :
( ( bNF_We4761303178683774583rm_f_v @ R )
=> ( ( ord_le947722214152454265rm_f_v @ B5 @ ( field_term_f_v @ R ) )
=> ( ( member_term_f_v @ B @ B5 )
=> ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ ( bNF_We1544997276579117207rm_f_v @ R @ B5 ) @ B ) @ R ) ) ) ) ).
% wo_rel.minim_least
thf(fact_1110_wo__rel_Ominim__least,axiom,
! [R: set_Pr516226060960033221_f_nat,B5: set_Pr5245412377734449720_f_nat,B: product_prod_f_nat] :
( ( bNF_We1252253923098187670_f_nat @ R )
=> ( ( ord_le8976984241387448984_f_nat @ B5 @ ( field_5908777168505831157_f_nat @ R ) )
=> ( ( member2678038196313615385_f_nat @ B @ B5 )
=> ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ ( bNF_We848864957873457590_f_nat @ R @ B5 ) @ B ) @ R ) ) ) ) ).
% wo_rel.minim_least
thf(fact_1111_wo__rel_Ominim__inField,axiom,
! [R: set_Pr7897041799657265351rm_f_v,B5: set_Pr8827868859434726151rm_f_v] :
( ( bNF_We135845164404304787rm_f_v @ R )
=> ( ( ord_le4559761987009501863rm_f_v @ B5 @ ( field_5331866484994830708rm_f_v @ R ) )
=> ( ( B5 != bot_bo4638812505780724595rm_f_v )
=> ( member4316649861536941904rm_f_v @ ( bNF_We9009046827986260339rm_f_v @ R @ B5 ) @ ( field_5331866484994830708rm_f_v @ R ) ) ) ) ) ).
% wo_rel.minim_inField
thf(fact_1112_wo__rel_Ominim__inField,axiom,
! [R: set_Pr516226060960033221_f_nat,B5: set_Pr5245412377734449720_f_nat] :
( ( bNF_We1252253923098187670_f_nat @ R )
=> ( ( ord_le8976984241387448984_f_nat @ B5 @ ( field_5908777168505831157_f_nat @ R ) )
=> ( ( B5 != bot_bo136713897782694092_f_nat )
=> ( member2678038196313615385_f_nat @ ( bNF_We848864957873457590_f_nat @ R @ B5 ) @ ( field_5908777168505831157_f_nat @ R ) ) ) ) ) ).
% wo_rel.minim_inField
thf(fact_1113_wo__rel_Omax2__greater__among,axiom,
! [R: set_Pr7897041799657265351rm_f_v,A: produc8027700073345061287rm_f_v,B: produc8027700073345061287rm_f_v] :
( ( bNF_We135845164404304787rm_f_v @ R )
=> ( ( member4316649861536941904rm_f_v @ A @ ( field_5331866484994830708rm_f_v @ R ) )
=> ( ( member4316649861536941904rm_f_v @ B @ ( field_5331866484994830708rm_f_v @ R ) )
=> ( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A @ ( bNF_We9039529291301710325rm_f_v @ R @ A @ B ) ) @ R )
& ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ B @ ( bNF_We9039529291301710325rm_f_v @ R @ A @ B ) ) @ R )
& ( member4316649861536941904rm_f_v @ ( bNF_We9039529291301710325rm_f_v @ R @ A @ B ) @ ( insert5456475782666790775rm_f_v @ A @ ( insert5456475782666790775rm_f_v @ B @ bot_bo4638812505780724595rm_f_v ) ) ) ) ) ) ) ).
% wo_rel.max2_greater_among
thf(fact_1114_wo__rel_Omax2__greater__among,axiom,
! [R: set_Pr8827868859434726151rm_f_v,A: term_f_v,B: term_f_v] :
( ( bNF_We4761303178683774583rm_f_v @ R )
=> ( ( member_term_f_v @ A @ ( field_term_f_v @ R ) )
=> ( ( member_term_f_v @ B @ ( field_term_f_v @ R ) )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ ( bNF_We9128225628251341205rm_f_v @ R @ A @ B ) ) @ R )
& ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B @ ( bNF_We9128225628251341205rm_f_v @ R @ A @ B ) ) @ R )
& ( member_term_f_v @ ( bNF_We9128225628251341205rm_f_v @ R @ A @ B ) @ ( insert_term_f_v @ A @ ( insert_term_f_v @ B @ bot_bot_set_term_f_v ) ) ) ) ) ) ) ).
% wo_rel.max2_greater_among
thf(fact_1115_refl__on__domain,axiom,
! [A4: set_Pr8827868859434726151rm_f_v,R: set_Pr7897041799657265351rm_f_v,A: produc8027700073345061287rm_f_v,B: produc8027700073345061287rm_f_v] :
( ( refl_o3791338839128770854rm_f_v @ A4 @ R )
=> ( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A @ B ) @ R )
=> ( ( member4316649861536941904rm_f_v @ A @ A4 )
& ( member4316649861536941904rm_f_v @ B @ A4 ) ) ) ) ).
% refl_on_domain
thf(fact_1116_refl__on__domain,axiom,
! [A4: set_term_f_v,R: set_Pr8827868859434726151rm_f_v,A: term_f_v,B: term_f_v] :
( ( refl_on_term_f_v @ A4 @ R )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ R )
=> ( ( member_term_f_v @ A @ A4 )
& ( member_term_f_v @ B @ A4 ) ) ) ) ).
% refl_on_domain
thf(fact_1117_wo__rel_Omax2__def,axiom,
! [R: set_Pr8827868859434726151rm_f_v,A: term_f_v,B: term_f_v] :
( ( bNF_We4761303178683774583rm_f_v @ R )
=> ( ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ R )
=> ( ( bNF_We9128225628251341205rm_f_v @ R @ A @ B )
= B ) )
& ( ~ ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ R )
=> ( ( bNF_We9128225628251341205rm_f_v @ R @ A @ B )
= A ) ) ) ) ).
% wo_rel.max2_def
thf(fact_1118_wo__rel_Omax2__equals1,axiom,
! [R: set_Pr7897041799657265351rm_f_v,A: produc8027700073345061287rm_f_v,B: produc8027700073345061287rm_f_v] :
( ( bNF_We135845164404304787rm_f_v @ R )
=> ( ( member4316649861536941904rm_f_v @ A @ ( field_5331866484994830708rm_f_v @ R ) )
=> ( ( member4316649861536941904rm_f_v @ B @ ( field_5331866484994830708rm_f_v @ R ) )
=> ( ( ( bNF_We9039529291301710325rm_f_v @ R @ A @ B )
= A )
= ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ B @ A ) @ R ) ) ) ) ) ).
% wo_rel.max2_equals1
thf(fact_1119_wo__rel_Omax2__equals1,axiom,
! [R: set_Pr8827868859434726151rm_f_v,A: term_f_v,B: term_f_v] :
( ( bNF_We4761303178683774583rm_f_v @ R )
=> ( ( member_term_f_v @ A @ ( field_term_f_v @ R ) )
=> ( ( member_term_f_v @ B @ ( field_term_f_v @ R ) )
=> ( ( ( bNF_We9128225628251341205rm_f_v @ R @ A @ B )
= A )
= ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B @ A ) @ R ) ) ) ) ) ).
% wo_rel.max2_equals1
thf(fact_1120_wo__rel_Omax2__equals2,axiom,
! [R: set_Pr7897041799657265351rm_f_v,A: produc8027700073345061287rm_f_v,B: produc8027700073345061287rm_f_v] :
( ( bNF_We135845164404304787rm_f_v @ R )
=> ( ( member4316649861536941904rm_f_v @ A @ ( field_5331866484994830708rm_f_v @ R ) )
=> ( ( member4316649861536941904rm_f_v @ B @ ( field_5331866484994830708rm_f_v @ R ) )
=> ( ( ( bNF_We9039529291301710325rm_f_v @ R @ A @ B )
= B )
= ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A @ B ) @ R ) ) ) ) ) ).
% wo_rel.max2_equals2
thf(fact_1121_wo__rel_Omax2__equals2,axiom,
! [R: set_Pr8827868859434726151rm_f_v,A: term_f_v,B: term_f_v] :
( ( bNF_We4761303178683774583rm_f_v @ R )
=> ( ( member_term_f_v @ A @ ( field_term_f_v @ R ) )
=> ( ( member_term_f_v @ B @ ( field_term_f_v @ R ) )
=> ( ( ( bNF_We9128225628251341205rm_f_v @ R @ A @ B )
= B )
= ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ R ) ) ) ) ) ).
% wo_rel.max2_equals2
thf(fact_1122_wo__rel_Omax2__greater,axiom,
! [R: set_Pr7897041799657265351rm_f_v,A: produc8027700073345061287rm_f_v,B: produc8027700073345061287rm_f_v] :
( ( bNF_We135845164404304787rm_f_v @ R )
=> ( ( member4316649861536941904rm_f_v @ A @ ( field_5331866484994830708rm_f_v @ R ) )
=> ( ( member4316649861536941904rm_f_v @ B @ ( field_5331866484994830708rm_f_v @ R ) )
=> ( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A @ ( bNF_We9039529291301710325rm_f_v @ R @ A @ B ) ) @ R )
& ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ B @ ( bNF_We9039529291301710325rm_f_v @ R @ A @ B ) ) @ R ) ) ) ) ) ).
% wo_rel.max2_greater
thf(fact_1123_wo__rel_Omax2__greater,axiom,
! [R: set_Pr8827868859434726151rm_f_v,A: term_f_v,B: term_f_v] :
( ( bNF_We4761303178683774583rm_f_v @ R )
=> ( ( member_term_f_v @ A @ ( field_term_f_v @ R ) )
=> ( ( member_term_f_v @ B @ ( field_term_f_v @ R ) )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ ( bNF_We9128225628251341205rm_f_v @ R @ A @ B ) ) @ R )
& ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B @ ( bNF_We9128225628251341205rm_f_v @ R @ A @ B ) ) @ R ) ) ) ) ) ).
% wo_rel.max2_greater
thf(fact_1124_Total__subset__Id,axiom,
! [R: set_Pr8827868859434726151rm_f_v] :
( ( total_on_term_f_v @ ( field_term_f_v @ R ) @ R )
=> ( ( ord_le4559761987009501863rm_f_v @ R @ id_term_f_v )
=> ( ( R = bot_bo4638812505780724595rm_f_v )
| ? [A3: term_f_v] :
( R
= ( insert5456475782666790775rm_f_v @ ( produc2510790566102465375rm_f_v @ A3 @ A3 ) @ bot_bo4638812505780724595rm_f_v ) ) ) ) ) ).
% Total_subset_Id
thf(fact_1125_linear__order__on__singleton,axiom,
! [X3: term_f_v] : ( order_674896459269444974rm_f_v @ ( insert_term_f_v @ X3 @ bot_bot_set_term_f_v ) @ ( insert5456475782666790775rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ X3 ) @ bot_bo4638812505780724595rm_f_v ) ) ).
% linear_order_on_singleton
thf(fact_1126_totalI,axiom,
! [R: set_Pr8827868859434726151rm_f_v] :
( ! [X: term_f_v,Y3: term_f_v] :
( ( X != Y3 )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ Y3 ) @ R )
| ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y3 @ X ) @ R ) ) )
=> ( total_on_term_f_v @ top_top_set_term_f_v @ R ) ) ).
% totalI
thf(fact_1127_total__on__def,axiom,
( total_on_term_f_v
= ( ^ [A6: set_term_f_v,R7: set_Pr8827868859434726151rm_f_v] :
! [X4: term_f_v] :
( ( member_term_f_v @ X4 @ A6 )
=> ! [Y5: term_f_v] :
( ( member_term_f_v @ Y5 @ A6 )
=> ( ( X4 != Y5 )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X4 @ Y5 ) @ R7 )
| ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y5 @ X4 ) @ R7 ) ) ) ) ) ) ) ).
% total_on_def
thf(fact_1128_total__onI,axiom,
! [A4: set_Pr8827868859434726151rm_f_v,R: set_Pr7897041799657265351rm_f_v] :
( ! [X: produc8027700073345061287rm_f_v,Y3: produc8027700073345061287rm_f_v] :
( ( member4316649861536941904rm_f_v @ X @ A4 )
=> ( ( member4316649861536941904rm_f_v @ Y3 @ A4 )
=> ( ( X != Y3 )
=> ( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ X @ Y3 ) @ R )
| ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ Y3 @ X ) @ R ) ) ) ) )
=> ( total_2457744054096061227rm_f_v @ A4 @ R ) ) ).
% total_onI
thf(fact_1129_total__onI,axiom,
! [A4: set_term_f_v,R: set_Pr8827868859434726151rm_f_v] :
( ! [X: term_f_v,Y3: term_f_v] :
( ( member_term_f_v @ X @ A4 )
=> ( ( member_term_f_v @ Y3 @ A4 )
=> ( ( X != Y3 )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ Y3 ) @ R )
| ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y3 @ X ) @ R ) ) ) ) )
=> ( total_on_term_f_v @ A4 @ R ) ) ).
% total_onI
thf(fact_1130_total__on__subset,axiom,
! [A4: set_Pr5245412377734449720_f_nat,R: set_Pr516226060960033221_f_nat,B5: set_Pr5245412377734449720_f_nat] :
( ( total_6064547220850425342_f_nat @ A4 @ R )
=> ( ( ord_le8976984241387448984_f_nat @ B5 @ A4 )
=> ( total_6064547220850425342_f_nat @ B5 @ R ) ) ) ).
% total_on_subset
thf(fact_1131_Linear__order__in__diff__Id,axiom,
! [R: set_Pr7897041799657265351rm_f_v,A: produc8027700073345061287rm_f_v,B: produc8027700073345061287rm_f_v] :
( ( order_8351895524478837596rm_f_v @ ( field_5331866484994830708rm_f_v @ R ) @ R )
=> ( ( member4316649861536941904rm_f_v @ A @ ( field_5331866484994830708rm_f_v @ R ) )
=> ( ( member4316649861536941904rm_f_v @ B @ ( field_5331866484994830708rm_f_v @ R ) )
=> ( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A @ B ) @ R )
= ( ~ ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ B @ A ) @ ( minus_4969660724917560878rm_f_v @ R @ id_Pro1935643801653256451rm_f_v ) ) ) ) ) ) ) ).
% Linear_order_in_diff_Id
thf(fact_1132_Linear__order__in__diff__Id,axiom,
! [R: set_Pr8827868859434726151rm_f_v,A: term_f_v,B: term_f_v] :
( ( order_674896459269444974rm_f_v @ ( field_term_f_v @ R ) @ R )
=> ( ( member_term_f_v @ A @ ( field_term_f_v @ R ) )
=> ( ( member_term_f_v @ B @ ( field_term_f_v @ R ) )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ R )
= ( ~ ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B @ A ) @ ( minus_410040198995547630rm_f_v @ R @ id_term_f_v ) ) ) ) ) ) ) ).
% Linear_order_in_diff_Id
thf(fact_1133_Linear__order__wf__diff__Id,axiom,
! [R: set_Pr8827868859434726151rm_f_v] :
( ( order_674896459269444974rm_f_v @ ( field_term_f_v @ R ) @ R )
=> ( ( wf_term_f_v @ ( minus_410040198995547630rm_f_v @ R @ id_term_f_v ) )
= ( ! [A6: set_term_f_v] :
( ( ord_le947722214152454265rm_f_v @ A6 @ ( field_term_f_v @ R ) )
=> ( ( A6 != bot_bot_set_term_f_v )
=> ? [X4: term_f_v] :
( ( member_term_f_v @ X4 @ A6 )
& ! [Y5: term_f_v] :
( ( member_term_f_v @ Y5 @ A6 )
=> ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X4 @ Y5 ) @ R ) ) ) ) ) ) ) ) ).
% Linear_order_wf_diff_Id
thf(fact_1134_Linear__order__wf__diff__Id,axiom,
! [R: set_Pr516226060960033221_f_nat] :
( ( order_3342235970812907277_f_nat @ ( field_5908777168505831157_f_nat @ R ) @ R )
=> ( ( wf_Pro5063733954819197017_f_nat @ ( minus_46298888925056428_f_nat @ R @ id_Pro2160904946306395686_f_nat ) )
= ( ! [A6: set_Pr5245412377734449720_f_nat] :
( ( ord_le8976984241387448984_f_nat @ A6 @ ( field_5908777168505831157_f_nat @ R ) )
=> ( ( A6 != bot_bo136713897782694092_f_nat )
=> ? [X4: product_prod_f_nat] :
( ( member2678038196313615385_f_nat @ X4 @ A6 )
& ! [Y5: product_prod_f_nat] :
( ( member2678038196313615385_f_nat @ Y5 @ A6 )
=> ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ X4 @ Y5 ) @ R ) ) ) ) ) ) ) ) ).
% Linear_order_wf_diff_Id
thf(fact_1135_symI,axiom,
! [R: set_Pr8827868859434726151rm_f_v] :
( ! [X: term_f_v,Y3: term_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ Y3 ) @ R )
=> ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y3 @ X ) @ R ) )
=> ( sym_on_term_f_v @ top_top_set_term_f_v @ R ) ) ).
% symI
thf(fact_1136_symE,axiom,
! [R: set_Pr8827868859434726151rm_f_v,B: term_f_v,A: term_f_v] :
( ( sym_on_term_f_v @ top_top_set_term_f_v @ R )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B @ A ) @ R )
=> ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ R ) ) ) ).
% symE
thf(fact_1137_symD,axiom,
! [R: set_Pr8827868859434726151rm_f_v,X3: term_f_v,Y: term_f_v] :
( ( sym_on_term_f_v @ top_top_set_term_f_v @ R )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Y ) @ R )
=> ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y @ X3 ) @ R ) ) ) ).
% symD
thf(fact_1138_sym__on__subset,axiom,
! [A4: set_Pr5245412377734449720_f_nat,R: set_Pr516226060960033221_f_nat,B5: set_Pr5245412377734449720_f_nat] :
( ( sym_on4385837785859991329_f_nat @ A4 @ R )
=> ( ( ord_le8976984241387448984_f_nat @ B5 @ A4 )
=> ( sym_on4385837785859991329_f_nat @ B5 @ R ) ) ) ).
% sym_on_subset
thf(fact_1139_sym__onD,axiom,
! [A4: set_Pr8827868859434726151rm_f_v,R: set_Pr7897041799657265351rm_f_v,X3: produc8027700073345061287rm_f_v,Y: produc8027700073345061287rm_f_v] :
( ( sym_on3936082784489890376rm_f_v @ A4 @ R )
=> ( ( member4316649861536941904rm_f_v @ X3 @ A4 )
=> ( ( member4316649861536941904rm_f_v @ Y @ A4 )
=> ( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ X3 @ Y ) @ R )
=> ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ Y @ X3 ) @ R ) ) ) ) ) ).
% sym_onD
thf(fact_1140_sym__onD,axiom,
! [A4: set_term_f_v,R: set_Pr8827868859434726151rm_f_v,X3: term_f_v,Y: term_f_v] :
( ( sym_on_term_f_v @ A4 @ R )
=> ( ( member_term_f_v @ X3 @ A4 )
=> ( ( member_term_f_v @ Y @ A4 )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Y ) @ R )
=> ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y @ X3 ) @ R ) ) ) ) ) ).
% sym_onD
thf(fact_1141_sym__onI,axiom,
! [A4: set_Pr8827868859434726151rm_f_v,R: set_Pr7897041799657265351rm_f_v] :
( ! [X: produc8027700073345061287rm_f_v,Y3: produc8027700073345061287rm_f_v] :
( ( member4316649861536941904rm_f_v @ X @ A4 )
=> ( ( member4316649861536941904rm_f_v @ Y3 @ A4 )
=> ( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ X @ Y3 ) @ R )
=> ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ Y3 @ X ) @ R ) ) ) )
=> ( sym_on3936082784489890376rm_f_v @ A4 @ R ) ) ).
% sym_onI
thf(fact_1142_sym__onI,axiom,
! [A4: set_term_f_v,R: set_Pr8827868859434726151rm_f_v] :
( ! [X: term_f_v,Y3: term_f_v] :
( ( member_term_f_v @ X @ A4 )
=> ( ( member_term_f_v @ Y3 @ A4 )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ Y3 ) @ R )
=> ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y3 @ X ) @ R ) ) ) )
=> ( sym_on_term_f_v @ A4 @ R ) ) ).
% sym_onI
thf(fact_1143_sym__on__def,axiom,
( sym_on_term_f_v
= ( ^ [A6: set_term_f_v,R7: set_Pr8827868859434726151rm_f_v] :
! [X4: term_f_v] :
( ( member_term_f_v @ X4 @ A6 )
=> ! [Y5: term_f_v] :
( ( member_term_f_v @ Y5 @ A6 )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X4 @ Y5 ) @ R7 )
=> ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y5 @ X4 ) @ R7 ) ) ) ) ) ) ).
% sym_on_def
thf(fact_1144_minimal__imp__wf,axiom,
! [R: set_Pr7897041799657265351rm_f_v] :
( ! [Q3: set_Pr8827868859434726151rm_f_v] :
( ? [X: produc8027700073345061287rm_f_v] : ( member4316649861536941904rm_f_v @ X @ Q3 )
=> ? [X6: produc8027700073345061287rm_f_v] :
( ( member4316649861536941904rm_f_v @ X6 @ Q3 )
& ! [Y3: produc8027700073345061287rm_f_v] :
( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ X6 @ Y3 ) @ R )
=> ~ ( member4316649861536941904rm_f_v @ Y3 @ Q3 ) ) ) )
=> ( wf_Pro782797968166624016rm_f_v @ ( conver3687159706446585467rm_f_v @ R ) ) ) ).
% minimal_imp_wf
thf(fact_1145_minimal__imp__wf,axiom,
! [R: set_Pr8827868859434726151rm_f_v] :
( ! [Q3: set_term_f_v] :
( ? [X: term_f_v] : ( member_term_f_v @ X @ Q3 )
=> ? [X6: term_f_v] :
( ( member_term_f_v @ X6 @ Q3 )
& ! [Y3: term_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X6 @ Y3 ) @ R )
=> ~ ( member_term_f_v @ Y3 @ Q3 ) ) ) )
=> ( wf_term_f_v @ ( conver3099152742603637891rm_f_v @ R ) ) ) ).
% minimal_imp_wf
thf(fact_1146_sym__rtrancl,axiom,
! [R: set_Pr8827868859434726151rm_f_v] :
( ( sym_on_term_f_v @ top_top_set_term_f_v @ R )
=> ( sym_on_term_f_v @ top_top_set_term_f_v @ ( transi3028809021613410336rm_f_v @ R ) ) ) ).
% sym_rtrancl
thf(fact_1147_wf__eq__minimal2,axiom,
( wf_term_f_v
= ( ^ [R7: set_Pr8827868859434726151rm_f_v] :
! [A6: set_term_f_v] :
( ( ( ord_le947722214152454265rm_f_v @ A6 @ ( field_term_f_v @ R7 ) )
& ( A6 != bot_bot_set_term_f_v ) )
=> ? [X4: term_f_v] :
( ( member_term_f_v @ X4 @ A6 )
& ! [Y5: term_f_v] :
( ( member_term_f_v @ Y5 @ A6 )
=> ~ ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y5 @ X4 ) @ R7 ) ) ) ) ) ) ).
% wf_eq_minimal2
thf(fact_1148_wf__eq__minimal2,axiom,
( wf_Pro5063733954819197017_f_nat
= ( ^ [R7: set_Pr516226060960033221_f_nat] :
! [A6: set_Pr5245412377734449720_f_nat] :
( ( ( ord_le8976984241387448984_f_nat @ A6 @ ( field_5908777168505831157_f_nat @ R7 ) )
& ( A6 != bot_bo136713897782694092_f_nat ) )
=> ? [X4: product_prod_f_nat] :
( ( member2678038196313615385_f_nat @ X4 @ A6 )
& ! [Y5: product_prod_f_nat] :
( ( member2678038196313615385_f_nat @ Y5 @ A6 )
=> ~ ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ Y5 @ X4 ) @ R7 ) ) ) ) ) ) ).
% wf_eq_minimal2
thf(fact_1149_wf__insert,axiom,
! [Y: term_f_v,X3: term_f_v,R: set_Pr8827868859434726151rm_f_v] :
( ( wf_term_f_v @ ( insert5456475782666790775rm_f_v @ ( produc2510790566102465375rm_f_v @ Y @ X3 ) @ R ) )
= ( ( wf_term_f_v @ R )
& ~ ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Y ) @ ( transi3028809021613410336rm_f_v @ R ) ) ) ) ).
% wf_insert
thf(fact_1150_qc__wf__relto__iff,axiom,
! [R3: set_Pr8827868859434726151rm_f_v,S2: set_Pr8827868859434726151rm_f_v] :
( ( ord_le4559761987009501863rm_f_v @ ( relcom2934258169684288428rm_f_v @ R3 @ S2 ) @ ( relcom2934258169684288428rm_f_v @ ( transi3028809021613410336rm_f_v @ ( sup_su1994854688125285083rm_f_v @ R3 @ S2 ) ) @ R3 ) )
=> ( ( wf_term_f_v @ ( relcom2934258169684288428rm_f_v @ ( transi3028809021613410336rm_f_v @ S2 ) @ ( relcom2934258169684288428rm_f_v @ R3 @ ( transi3028809021613410336rm_f_v @ S2 ) ) ) )
= ( wf_term_f_v @ R3 ) ) ) ).
% qc_wf_relto_iff
thf(fact_1151_wf__if__convertible__to__wf,axiom,
! [S: set_Pr8827868859434726151rm_f_v,R: set_Pr8827868859434726151rm_f_v,F2: term_f_v > term_f_v] :
( ( wf_term_f_v @ S )
=> ( ! [X: term_f_v,Y3: term_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ Y3 ) @ R )
=> ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ ( F2 @ X ) @ ( F2 @ Y3 ) ) @ S ) )
=> ( wf_term_f_v @ R ) ) ) ).
% wf_if_convertible_to_wf
thf(fact_1152_wf__induct__rule,axiom,
! [R: set_Pr8827868859434726151rm_f_v,P2: term_f_v > $o,A: term_f_v] :
( ( wf_term_f_v @ R )
=> ( ! [X: term_f_v] :
( ! [Y6: term_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y6 @ X ) @ R )
=> ( P2 @ Y6 ) )
=> ( P2 @ X ) )
=> ( P2 @ A ) ) ) ).
% wf_induct_rule
thf(fact_1153_wf__eq__minimal,axiom,
( wf_Pro782797968166624016rm_f_v
= ( ^ [R7: set_Pr7897041799657265351rm_f_v] :
! [Q4: set_Pr8827868859434726151rm_f_v] :
( ? [X4: produc8027700073345061287rm_f_v] : ( member4316649861536941904rm_f_v @ X4 @ Q4 )
=> ? [X4: produc8027700073345061287rm_f_v] :
( ( member4316649861536941904rm_f_v @ X4 @ Q4 )
& ! [Y5: produc8027700073345061287rm_f_v] :
( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ Y5 @ X4 ) @ R7 )
=> ~ ( member4316649861536941904rm_f_v @ Y5 @ Q4 ) ) ) ) ) ) ).
% wf_eq_minimal
thf(fact_1154_wf__eq__minimal,axiom,
( wf_term_f_v
= ( ^ [R7: set_Pr8827868859434726151rm_f_v] :
! [Q4: set_term_f_v] :
( ? [X4: term_f_v] : ( member_term_f_v @ X4 @ Q4 )
=> ? [X4: term_f_v] :
( ( member_term_f_v @ X4 @ Q4 )
& ! [Y5: term_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y5 @ X4 ) @ R7 )
=> ~ ( member_term_f_v @ Y5 @ Q4 ) ) ) ) ) ) ).
% wf_eq_minimal
thf(fact_1155_wf__not__refl,axiom,
! [R: set_Pr8827868859434726151rm_f_v,A: term_f_v] :
( ( wf_term_f_v @ R )
=> ~ ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ A ) @ R ) ) ).
% wf_not_refl
thf(fact_1156_wf__not__sym,axiom,
! [R: set_Pr8827868859434726151rm_f_v,A: term_f_v,X3: term_f_v] :
( ( wf_term_f_v @ R )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ X3 ) @ R )
=> ~ ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ A ) @ R ) ) ) ).
% wf_not_sym
thf(fact_1157_wf__irrefl,axiom,
! [R: set_Pr8827868859434726151rm_f_v,A: term_f_v] :
( ( wf_term_f_v @ R )
=> ~ ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ A ) @ R ) ) ).
% wf_irrefl
thf(fact_1158_wf__induct,axiom,
! [R: set_Pr8827868859434726151rm_f_v,P2: term_f_v > $o,A: term_f_v] :
( ( wf_term_f_v @ R )
=> ( ! [X: term_f_v] :
( ! [Y6: term_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y6 @ X ) @ R )
=> ( P2 @ Y6 ) )
=> ( P2 @ X ) )
=> ( P2 @ A ) ) ) ).
% wf_induct
thf(fact_1159_wf__asym,axiom,
! [R: set_Pr8827868859434726151rm_f_v,A: term_f_v,X3: term_f_v] :
( ( wf_term_f_v @ R )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ X3 ) @ R )
=> ~ ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ A ) @ R ) ) ) ).
% wf_asym
thf(fact_1160_wfUNIVI,axiom,
! [R: set_Pr8827868859434726151rm_f_v] :
( ! [P3: term_f_v > $o,X: term_f_v] :
( ! [Xa2: term_f_v] :
( ! [Y3: term_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y3 @ Xa2 ) @ R )
=> ( P3 @ Y3 ) )
=> ( P3 @ Xa2 ) )
=> ( P3 @ X ) )
=> ( wf_term_f_v @ R ) ) ).
% wfUNIVI
thf(fact_1161_wfI__min,axiom,
! [R3: set_Pr7897041799657265351rm_f_v] :
( ! [X: produc8027700073345061287rm_f_v,Q3: set_Pr8827868859434726151rm_f_v] :
( ( member4316649861536941904rm_f_v @ X @ Q3 )
=> ? [Xa2: produc8027700073345061287rm_f_v] :
( ( member4316649861536941904rm_f_v @ Xa2 @ Q3 )
& ! [Y3: produc8027700073345061287rm_f_v] :
( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ Y3 @ Xa2 ) @ R3 )
=> ~ ( member4316649861536941904rm_f_v @ Y3 @ Q3 ) ) ) )
=> ( wf_Pro782797968166624016rm_f_v @ R3 ) ) ).
% wfI_min
thf(fact_1162_wfI__min,axiom,
! [R3: set_Pr8827868859434726151rm_f_v] :
( ! [X: term_f_v,Q3: set_term_f_v] :
( ( member_term_f_v @ X @ Q3 )
=> ? [Xa2: term_f_v] :
( ( member_term_f_v @ Xa2 @ Q3 )
& ! [Y3: term_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y3 @ Xa2 ) @ R3 )
=> ~ ( member_term_f_v @ Y3 @ Q3 ) ) ) )
=> ( wf_term_f_v @ R3 ) ) ).
% wfI_min
thf(fact_1163_wfE__min,axiom,
! [R3: set_Pr7897041799657265351rm_f_v,X3: produc8027700073345061287rm_f_v,Q: set_Pr8827868859434726151rm_f_v] :
( ( wf_Pro782797968166624016rm_f_v @ R3 )
=> ( ( member4316649861536941904rm_f_v @ X3 @ Q )
=> ~ ! [Z: produc8027700073345061287rm_f_v] :
( ( member4316649861536941904rm_f_v @ Z @ Q )
=> ~ ! [Y6: produc8027700073345061287rm_f_v] :
( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ Y6 @ Z ) @ R3 )
=> ~ ( member4316649861536941904rm_f_v @ Y6 @ Q ) ) ) ) ) ).
% wfE_min
thf(fact_1164_wfE__min,axiom,
! [R3: set_Pr8827868859434726151rm_f_v,X3: term_f_v,Q: set_term_f_v] :
( ( wf_term_f_v @ R3 )
=> ( ( member_term_f_v @ X3 @ Q )
=> ~ ! [Z: term_f_v] :
( ( member_term_f_v @ Z @ Q )
=> ~ ! [Y6: term_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y6 @ Z ) @ R3 )
=> ~ ( member_term_f_v @ Y6 @ Q ) ) ) ) ) ).
% wfE_min
thf(fact_1165_wf__def,axiom,
( wf_term_f_v
= ( ^ [R7: set_Pr8827868859434726151rm_f_v] :
! [P4: term_f_v > $o] :
( ! [X4: term_f_v] :
( ! [Y5: term_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y5 @ X4 ) @ R7 )
=> ( P4 @ Y5 ) )
=> ( P4 @ X4 ) )
=> ! [X7: term_f_v] : ( P4 @ X7 ) ) ) ) ).
% wf_def
thf(fact_1166_wfE__min_H,axiom,
! [R3: set_Pr7897041799657265351rm_f_v,Q: set_Pr8827868859434726151rm_f_v] :
( ( wf_Pro782797968166624016rm_f_v @ R3 )
=> ( ( Q != bot_bo4638812505780724595rm_f_v )
=> ~ ! [Z: produc8027700073345061287rm_f_v] :
( ( member4316649861536941904rm_f_v @ Z @ Q )
=> ~ ! [Y6: produc8027700073345061287rm_f_v] :
( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ Y6 @ Z ) @ R3 )
=> ~ ( member4316649861536941904rm_f_v @ Y6 @ Q ) ) ) ) ) ).
% wfE_min'
thf(fact_1167_wfE__min_H,axiom,
! [R3: set_Pr8827868859434726151rm_f_v,Q: set_term_f_v] :
( ( wf_term_f_v @ R3 )
=> ( ( Q != bot_bot_set_term_f_v )
=> ~ ! [Z: term_f_v] :
( ( member_term_f_v @ Z @ Q )
=> ~ ! [Y6: term_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y6 @ Z ) @ R3 )
=> ~ ( member_term_f_v @ Y6 @ Q ) ) ) ) ) ).
% wfE_min'
thf(fact_1168_antisym__onD,axiom,
! [A4: set_Pr8827868859434726151rm_f_v,R: set_Pr7897041799657265351rm_f_v,X3: produc8027700073345061287rm_f_v,Y: produc8027700073345061287rm_f_v] :
( ( antisy3488218986368116234rm_f_v @ A4 @ R )
=> ( ( member4316649861536941904rm_f_v @ X3 @ A4 )
=> ( ( member4316649861536941904rm_f_v @ Y @ A4 )
=> ( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ X3 @ Y ) @ R )
=> ( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ Y @ X3 ) @ R )
=> ( X3 = Y ) ) ) ) ) ) ).
% antisym_onD
thf(fact_1169_antisym__onD,axiom,
! [A4: set_term_f_v,R: set_Pr8827868859434726151rm_f_v,X3: term_f_v,Y: term_f_v] :
( ( antisym_on_term_f_v @ A4 @ R )
=> ( ( member_term_f_v @ X3 @ A4 )
=> ( ( member_term_f_v @ Y @ A4 )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Y ) @ R )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y @ X3 ) @ R )
=> ( X3 = Y ) ) ) ) ) ) ).
% antisym_onD
thf(fact_1170_antisym__onI,axiom,
! [A4: set_Pr8827868859434726151rm_f_v,R: set_Pr7897041799657265351rm_f_v] :
( ! [X: produc8027700073345061287rm_f_v,Y3: produc8027700073345061287rm_f_v] :
( ( member4316649861536941904rm_f_v @ X @ A4 )
=> ( ( member4316649861536941904rm_f_v @ Y3 @ A4 )
=> ( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ X @ Y3 ) @ R )
=> ( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ Y3 @ X ) @ R )
=> ( X = Y3 ) ) ) ) )
=> ( antisy3488218986368116234rm_f_v @ A4 @ R ) ) ).
% antisym_onI
thf(fact_1171_antisym__onI,axiom,
! [A4: set_term_f_v,R: set_Pr8827868859434726151rm_f_v] :
( ! [X: term_f_v,Y3: term_f_v] :
( ( member_term_f_v @ X @ A4 )
=> ( ( member_term_f_v @ Y3 @ A4 )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ Y3 ) @ R )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y3 @ X ) @ R )
=> ( X = Y3 ) ) ) ) )
=> ( antisym_on_term_f_v @ A4 @ R ) ) ).
% antisym_onI
thf(fact_1172_antisym__on__def,axiom,
( antisym_on_term_f_v
= ( ^ [A6: set_term_f_v,R7: set_Pr8827868859434726151rm_f_v] :
! [X4: term_f_v] :
( ( member_term_f_v @ X4 @ A6 )
=> ! [Y5: term_f_v] :
( ( member_term_f_v @ Y5 @ A6 )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X4 @ Y5 ) @ R7 )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y5 @ X4 ) @ R7 )
=> ( X4 = Y5 ) ) ) ) ) ) ) ).
% antisym_on_def
thf(fact_1173_antisymI,axiom,
! [R: set_Pr8827868859434726151rm_f_v] :
( ! [X: term_f_v,Y3: term_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X @ Y3 ) @ R )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y3 @ X ) @ R )
=> ( X = Y3 ) ) )
=> ( antisym_on_term_f_v @ top_top_set_term_f_v @ R ) ) ).
% antisymI
thf(fact_1174_antisymD,axiom,
! [R: set_Pr8827868859434726151rm_f_v,X3: term_f_v,Y: term_f_v] :
( ( antisym_on_term_f_v @ top_top_set_term_f_v @ R )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Y ) @ R )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y @ X3 ) @ R )
=> ( X3 = Y ) ) ) ) ).
% antisymD
thf(fact_1175_antisym__on__subset,axiom,
! [A4: set_Pr5245412377734449720_f_nat,R: set_Pr516226060960033221_f_nat,B5: set_Pr5245412377734449720_f_nat] :
( ( antisy7095432349626482143_f_nat @ A4 @ R )
=> ( ( ord_le8976984241387448984_f_nat @ B5 @ A4 )
=> ( antisy7095432349626482143_f_nat @ B5 @ R ) ) ) ).
% antisym_on_subset
thf(fact_1176_acyclic__impl__antisym__rtrancl,axiom,
! [R: set_Pr8827868859434726151rm_f_v] :
( ( transi600795472790740820rm_f_v @ R )
=> ( antisym_on_term_f_v @ top_top_set_term_f_v @ ( transi3028809021613410336rm_f_v @ R ) ) ) ).
% acyclic_impl_antisym_rtrancl
thf(fact_1177_underS__incr,axiom,
! [R: set_Pr8827868859434726151rm_f_v,A: term_f_v,B: term_f_v] :
( ( trans_on_term_f_v @ top_top_set_term_f_v @ R )
=> ( ( antisym_on_term_f_v @ top_top_set_term_f_v @ R )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ R )
=> ( ord_le947722214152454265rm_f_v @ ( order_6129236356014843677rm_f_v @ R @ A ) @ ( order_6129236356014843677rm_f_v @ R @ B ) ) ) ) ) ).
% underS_incr
thf(fact_1178_underS__incr,axiom,
! [R: set_Pr516226060960033221_f_nat,A: product_prod_f_nat,B: product_prod_f_nat] :
( ( trans_2699862687909549282_f_nat @ top_to3664669310755125480_f_nat @ R )
=> ( ( antisy7095432349626482143_f_nat @ top_to3664669310755125480_f_nat @ R )
=> ( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ A @ B ) @ R )
=> ( ord_le8976984241387448984_f_nat @ ( order_8397142099418545468_f_nat @ R @ A ) @ ( order_8397142099418545468_f_nat @ R @ B ) ) ) ) ) ).
% underS_incr
thf(fact_1179_in__lex__prod,axiom,
! [A: term_f_v,B: term_f_v,A2: term_f_v,B2: term_f_v,R: set_Pr8827868859434726151rm_f_v,S: set_Pr8827868859434726151rm_f_v] :
( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( produc2510790566102465375rm_f_v @ A2 @ B2 ) ) @ ( lex_pr1654150031463228279rm_f_v @ R @ S ) )
= ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ A2 ) @ R )
| ( ( A = A2 )
& ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B @ B2 ) @ S ) ) ) ) ).
% in_lex_prod
thf(fact_1180_Order__Relation_OunderS__Field,axiom,
! [R: set_Pr516226060960033221_f_nat,A: product_prod_f_nat] : ( ord_le8976984241387448984_f_nat @ ( order_8397142099418545468_f_nat @ R @ A ) @ ( field_5908777168505831157_f_nat @ R ) ) ).
% Order_Relation.underS_Field
thf(fact_1181_underS__E,axiom,
! [I: produc8027700073345061287rm_f_v,R3: set_Pr7897041799657265351rm_f_v,J: produc8027700073345061287rm_f_v] :
( ( member4316649861536941904rm_f_v @ I @ ( order_6610454641011678829rm_f_v @ R3 @ J ) )
=> ( ( I != J )
& ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ I @ J ) @ R3 ) ) ) ).
% underS_E
thf(fact_1182_underS__E,axiom,
! [I: term_f_v,R3: set_Pr8827868859434726151rm_f_v,J: term_f_v] :
( ( member_term_f_v @ I @ ( order_6129236356014843677rm_f_v @ R3 @ J ) )
=> ( ( I != J )
& ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ I @ J ) @ R3 ) ) ) ).
% underS_E
thf(fact_1183_underS__I,axiom,
! [I: produc8027700073345061287rm_f_v,J: produc8027700073345061287rm_f_v,R3: set_Pr7897041799657265351rm_f_v] :
( ( I != J )
=> ( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ I @ J ) @ R3 )
=> ( member4316649861536941904rm_f_v @ I @ ( order_6610454641011678829rm_f_v @ R3 @ J ) ) ) ) ).
% underS_I
thf(fact_1184_underS__I,axiom,
! [I: term_f_v,J: term_f_v,R3: set_Pr8827868859434726151rm_f_v] :
( ( I != J )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ I @ J ) @ R3 )
=> ( member_term_f_v @ I @ ( order_6129236356014843677rm_f_v @ R3 @ J ) ) ) ) ).
% underS_I
thf(fact_1185_underS__incl__iff,axiom,
! [R: set_Pr7897041799657265351rm_f_v,A: produc8027700073345061287rm_f_v,B: produc8027700073345061287rm_f_v] :
( ( order_8351895524478837596rm_f_v @ ( field_5331866484994830708rm_f_v @ R ) @ R )
=> ( ( member4316649861536941904rm_f_v @ A @ ( field_5331866484994830708rm_f_v @ R ) )
=> ( ( member4316649861536941904rm_f_v @ B @ ( field_5331866484994830708rm_f_v @ R ) )
=> ( ( ord_le4559761987009501863rm_f_v @ ( order_6610454641011678829rm_f_v @ R @ A ) @ ( order_6610454641011678829rm_f_v @ R @ B ) )
= ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A @ B ) @ R ) ) ) ) ) ).
% underS_incl_iff
thf(fact_1186_underS__incl__iff,axiom,
! [R: set_Pr8827868859434726151rm_f_v,A: term_f_v,B: term_f_v] :
( ( order_674896459269444974rm_f_v @ ( field_term_f_v @ R ) @ R )
=> ( ( member_term_f_v @ A @ ( field_term_f_v @ R ) )
=> ( ( member_term_f_v @ B @ ( field_term_f_v @ R ) )
=> ( ( ord_le947722214152454265rm_f_v @ ( order_6129236356014843677rm_f_v @ R @ A ) @ ( order_6129236356014843677rm_f_v @ R @ B ) )
= ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ R ) ) ) ) ) ).
% underS_incl_iff
thf(fact_1187_underS__incl__iff,axiom,
! [R: set_Pr516226060960033221_f_nat,A: product_prod_f_nat,B: product_prod_f_nat] :
( ( order_3342235970812907277_f_nat @ ( field_5908777168505831157_f_nat @ R ) @ R )
=> ( ( member2678038196313615385_f_nat @ A @ ( field_5908777168505831157_f_nat @ R ) )
=> ( ( member2678038196313615385_f_nat @ B @ ( field_5908777168505831157_f_nat @ R ) )
=> ( ( ord_le8976984241387448984_f_nat @ ( order_8397142099418545468_f_nat @ R @ A ) @ ( order_8397142099418545468_f_nat @ R @ B ) )
= ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ A @ B ) @ R ) ) ) ) ) ).
% underS_incl_iff
thf(fact_1188_under__incr,axiom,
! [R: set_Pr8827868859434726151rm_f_v,A: term_f_v,B: term_f_v] :
( ( trans_on_term_f_v @ top_top_set_term_f_v @ R )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ R )
=> ( ord_le947722214152454265rm_f_v @ ( order_under_term_f_v @ R @ A ) @ ( order_under_term_f_v @ R @ B ) ) ) ) ).
% under_incr
thf(fact_1189_under__incr,axiom,
! [R: set_Pr516226060960033221_f_nat,A: product_prod_f_nat,B: product_prod_f_nat] :
( ( trans_2699862687909549282_f_nat @ top_to3664669310755125480_f_nat @ R )
=> ( ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ A @ B ) @ R )
=> ( ord_le8976984241387448984_f_nat @ ( order_3489069451034613469_f_nat @ R @ A ) @ ( order_3489069451034613469_f_nat @ R @ B ) ) ) ) ).
% under_incr
thf(fact_1190_underS__subset__under,axiom,
! [R: set_Pr516226060960033221_f_nat,A: product_prod_f_nat] : ( ord_le8976984241387448984_f_nat @ ( order_8397142099418545468_f_nat @ R @ A ) @ ( order_3489069451034613469_f_nat @ R @ A ) ) ).
% underS_subset_under
thf(fact_1191_under__Field,axiom,
! [R: set_Pr516226060960033221_f_nat,A: product_prod_f_nat] : ( ord_le8976984241387448984_f_nat @ ( order_3489069451034613469_f_nat @ R @ A ) @ ( field_5908777168505831157_f_nat @ R ) ) ).
% under_Field
thf(fact_1192_wo__rel_Oofilter__def,axiom,
! [R: set_Pr516226060960033221_f_nat,A4: set_Pr5245412377734449720_f_nat] :
( ( bNF_We1252253923098187670_f_nat @ R )
=> ( ( order_4248866431976349292_f_nat @ R @ A4 )
= ( ( ord_le8976984241387448984_f_nat @ A4 @ ( field_5908777168505831157_f_nat @ R ) )
& ! [X4: product_prod_f_nat] :
( ( member2678038196313615385_f_nat @ X4 @ A4 )
=> ( ord_le8976984241387448984_f_nat @ ( order_3489069451034613469_f_nat @ R @ X4 ) @ A4 ) ) ) ) ) ).
% wo_rel.ofilter_def
thf(fact_1193_Linear__order__Well__order__iff,axiom,
! [R: set_Pr8827868859434726151rm_f_v] :
( ( order_674896459269444974rm_f_v @ ( field_term_f_v @ R ) @ R )
=> ( ( order_1005990429947353157rm_f_v @ ( field_term_f_v @ R ) @ R )
= ( ! [A6: set_term_f_v] :
( ( ord_le947722214152454265rm_f_v @ A6 @ ( field_term_f_v @ R ) )
=> ( ( A6 != bot_bot_set_term_f_v )
=> ? [X4: term_f_v] :
( ( member_term_f_v @ X4 @ A6 )
& ! [Y5: term_f_v] :
( ( member_term_f_v @ Y5 @ A6 )
=> ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X4 @ Y5 ) @ R ) ) ) ) ) ) ) ) ).
% Linear_order_Well_order_iff
thf(fact_1194_Linear__order__Well__order__iff,axiom,
! [R: set_Pr516226060960033221_f_nat] :
( ( order_3342235970812907277_f_nat @ ( field_5908777168505831157_f_nat @ R ) @ R )
=> ( ( order_4707705119997360740_f_nat @ ( field_5908777168505831157_f_nat @ R ) @ R )
= ( ! [A6: set_Pr5245412377734449720_f_nat] :
( ( ord_le8976984241387448984_f_nat @ A6 @ ( field_5908777168505831157_f_nat @ R ) )
=> ( ( A6 != bot_bo136713897782694092_f_nat )
=> ? [X4: product_prod_f_nat] :
( ( member2678038196313615385_f_nat @ X4 @ A6 )
& ! [Y5: product_prod_f_nat] :
( ( member2678038196313615385_f_nat @ Y5 @ A6 )
=> ( member7030746907725424910_f_nat @ ( produc5861087217798509725_f_nat @ X4 @ Y5 ) @ R ) ) ) ) ) ) ) ) ).
% Linear_order_Well_order_iff
thf(fact_1195_wo__rel_Oofilter__linord,axiom,
! [R: set_Pr516226060960033221_f_nat,A4: set_Pr5245412377734449720_f_nat,B5: set_Pr5245412377734449720_f_nat] :
( ( bNF_We1252253923098187670_f_nat @ R )
=> ( ( order_4248866431976349292_f_nat @ R @ A4 )
=> ( ( order_4248866431976349292_f_nat @ R @ B5 )
=> ( ( ord_le8976984241387448984_f_nat @ A4 @ B5 )
| ( ord_le8976984241387448984_f_nat @ B5 @ A4 ) ) ) ) ) ).
% wo_rel.ofilter_linord
thf(fact_1196_well__order__on__domain,axiom,
! [A4: set_Pr8827868859434726151rm_f_v,R: set_Pr7897041799657265351rm_f_v,A: produc8027700073345061287rm_f_v,B: produc8027700073345061287rm_f_v] :
( ( order_7084804957670756165rm_f_v @ A4 @ R )
=> ( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ A @ B ) @ R )
=> ( ( member4316649861536941904rm_f_v @ A @ A4 )
& ( member4316649861536941904rm_f_v @ B @ A4 ) ) ) ) ).
% well_order_on_domain
thf(fact_1197_well__order__on__domain,axiom,
! [A4: set_term_f_v,R: set_Pr8827868859434726151rm_f_v,A: term_f_v,B: term_f_v] :
( ( order_1005990429947353157rm_f_v @ A4 @ R )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ R )
=> ( ( member_term_f_v @ A @ A4 )
& ( member_term_f_v @ B @ A4 ) ) ) ) ).
% well_order_on_domain
thf(fact_1198_ofilter__def,axiom,
( order_4248866431976349292_f_nat
= ( ^ [R7: set_Pr516226060960033221_f_nat,A6: set_Pr5245412377734449720_f_nat] :
( ( ord_le8976984241387448984_f_nat @ A6 @ ( field_5908777168505831157_f_nat @ R7 ) )
& ! [X4: product_prod_f_nat] :
( ( member2678038196313615385_f_nat @ X4 @ A6 )
=> ( ord_le8976984241387448984_f_nat @ ( order_3489069451034613469_f_nat @ R7 @ X4 ) @ A6 ) ) ) ) ) ).
% ofilter_def
thf(fact_1199_bsqr__max2,axiom,
! [R: set_Pr8827868859434726151rm_f_v,A1: term_f_v,A22: term_f_v,B1: term_f_v,B22: term_f_v] :
( ( order_1005990429947353157rm_f_v @ ( field_term_f_v @ R ) @ R )
=> ( ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ ( produc2510790566102465375rm_f_v @ A1 @ A22 ) @ ( produc2510790566102465375rm_f_v @ B1 @ B22 ) ) @ ( bNF_We3808192248168317830rm_f_v @ R ) )
=> ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ ( bNF_We9128225628251341205rm_f_v @ R @ A1 @ A22 ) @ ( bNF_We9128225628251341205rm_f_v @ R @ B1 @ B22 ) ) @ R ) ) ) ).
% bsqr_max2
thf(fact_1200_non__strict__ending,axiom,
! [T: nat > term_f_v,R3: set_Pr8827868859434726151rm_f_v,S2: set_Pr8827868859434726151rm_f_v] :
( ! [I2: nat] : ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ ( T @ I2 ) @ ( T @ ( suc @ I2 ) ) ) @ ( sup_su1994854688125285083rm_f_v @ R3 @ S2 ) )
=> ( ( ord_le4559761987009501863rm_f_v @ ( relcom2934258169684288428rm_f_v @ R3 @ S2 ) @ S2 )
=> ( ( abstra6117554785717145rm_f_v @ S2 @ ( insert_term_f_v @ ( T @ zero_zero_nat ) @ bot_bot_set_term_f_v ) )
=> ? [J2: nat] :
! [I3: nat] :
( ( ord_less_eq_nat @ J2 @ I3 )
=> ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ ( T @ I3 ) @ ( T @ ( suc @ I3 ) ) ) @ ( minus_410040198995547630rm_f_v @ R3 @ S2 ) ) ) ) ) ) ).
% non_strict_ending
thf(fact_1201_not__SN__onI,axiom,
! [F2: nat > produc8027700073345061287rm_f_v,X5: set_Pr8827868859434726151rm_f_v,R3: set_Pr7897041799657265351rm_f_v] :
( ( member4316649861536941904rm_f_v @ ( F2 @ zero_zero_nat ) @ X5 )
=> ( ! [I2: nat] : ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ ( F2 @ I2 ) @ ( F2 @ ( suc @ I2 ) ) ) @ R3 )
=> ~ ( abstra845488452567719153rm_f_v @ R3 @ X5 ) ) ) ).
% not_SN_onI
thf(fact_1202_not__SN__onI,axiom,
! [F2: nat > term_f_v,X5: set_term_f_v,R3: set_Pr8827868859434726151rm_f_v] :
( ( member_term_f_v @ ( F2 @ zero_zero_nat ) @ X5 )
=> ( ! [I2: nat] : ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ ( F2 @ I2 ) @ ( F2 @ ( suc @ I2 ) ) ) @ R3 )
=> ~ ( abstra6117554785717145rm_f_v @ R3 @ X5 ) ) ) ).
% not_SN_onI
thf(fact_1203_SN__onI,axiom,
! [A4: set_Pr8827868859434726151rm_f_v,R: set_Pr7897041799657265351rm_f_v] :
( ! [F5: nat > produc8027700073345061287rm_f_v] :
( ( member4316649861536941904rm_f_v @ ( F5 @ zero_zero_nat ) @ A4 )
=> ~ ! [I3: nat] : ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ ( F5 @ I3 ) @ ( F5 @ ( suc @ I3 ) ) ) @ R ) )
=> ( abstra845488452567719153rm_f_v @ R @ A4 ) ) ).
% SN_onI
thf(fact_1204_SN__onI,axiom,
! [A4: set_term_f_v,R: set_Pr8827868859434726151rm_f_v] :
( ! [F5: nat > term_f_v] :
( ( member_term_f_v @ ( F5 @ zero_zero_nat ) @ A4 )
=> ~ ! [I3: nat] : ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ ( F5 @ I3 ) @ ( F5 @ ( suc @ I3 ) ) ) @ R ) )
=> ( abstra6117554785717145rm_f_v @ R @ A4 ) ) ).
% SN_onI
thf(fact_1205_relpow__0__E,axiom,
! [X3: term_f_v,Y: term_f_v,R3: set_Pr8827868859434726151rm_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Y ) @ ( compow8498445688072944954rm_f_v @ zero_zero_nat @ R3 ) )
=> ( X3 = Y ) ) ).
% relpow_0_E
thf(fact_1206_relpow__0__I,axiom,
! [X3: term_f_v,R3: set_Pr8827868859434726151rm_f_v] : ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ X3 ) @ ( compow8498445688072944954rm_f_v @ zero_zero_nat @ R3 ) ) ).
% relpow_0_I
thf(fact_1207_relpow__E,axiom,
! [X3: term_f_v,Z2: term_f_v,N: nat,R3: set_Pr8827868859434726151rm_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Z2 ) @ ( compow8498445688072944954rm_f_v @ N @ R3 ) )
=> ( ( ( N = zero_zero_nat )
=> ( X3 != Z2 ) )
=> ~ ! [Y3: term_f_v,M2: nat] :
( ( N
= ( suc @ M2 ) )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Y3 ) @ ( compow8498445688072944954rm_f_v @ M2 @ R3 ) )
=> ~ ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y3 @ Z2 ) @ R3 ) ) ) ) ) ).
% relpow_E
thf(fact_1208_relpow__E2,axiom,
! [X3: term_f_v,Z2: term_f_v,N: nat,R3: set_Pr8827868859434726151rm_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Z2 ) @ ( compow8498445688072944954rm_f_v @ N @ R3 ) )
=> ( ( ( N = zero_zero_nat )
=> ( X3 != Z2 ) )
=> ~ ! [Y3: term_f_v,M2: nat] :
( ( N
= ( suc @ M2 ) )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Y3 ) @ R3 )
=> ~ ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y3 @ Z2 ) @ ( compow8498445688072944954rm_f_v @ M2 @ R3 ) ) ) ) ) ) ).
% relpow_E2
thf(fact_1209_not__SN__onE,axiom,
! [R: set_Pr7897041799657265351rm_f_v,A4: set_Pr8827868859434726151rm_f_v] :
( ~ ( abstra845488452567719153rm_f_v @ R @ A4 )
=> ~ ! [F5: nat > produc8027700073345061287rm_f_v] :
( ( member4316649861536941904rm_f_v @ ( F5 @ zero_zero_nat ) @ A4 )
=> ~ ! [I3: nat] : ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ ( F5 @ I3 ) @ ( F5 @ ( suc @ I3 ) ) ) @ R ) ) ) ).
% not_SN_onE
thf(fact_1210_not__SN__onE,axiom,
! [R: set_Pr8827868859434726151rm_f_v,A4: set_term_f_v] :
( ~ ( abstra6117554785717145rm_f_v @ R @ A4 )
=> ~ ! [F5: nat > term_f_v] :
( ( member_term_f_v @ ( F5 @ zero_zero_nat ) @ A4 )
=> ~ ! [I3: nat] : ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ ( F5 @ I3 ) @ ( F5 @ ( suc @ I3 ) ) ) @ R ) ) ) ).
% not_SN_onE
thf(fact_1211_SN__on__def,axiom,
( abstra845488452567719153rm_f_v
= ( ^ [R7: set_Pr7897041799657265351rm_f_v,A6: set_Pr8827868859434726151rm_f_v] :
~ ? [F6: nat > produc8027700073345061287rm_f_v] :
( ( member4316649861536941904rm_f_v @ ( F6 @ zero_zero_nat ) @ A6 )
& ! [I4: nat] : ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ ( F6 @ I4 ) @ ( F6 @ ( suc @ I4 ) ) ) @ R7 ) ) ) ) ).
% SN_on_def
thf(fact_1212_SN__on__def,axiom,
( abstra6117554785717145rm_f_v
= ( ^ [R7: set_Pr8827868859434726151rm_f_v,A6: set_term_f_v] :
~ ? [F6: nat > term_f_v] :
( ( member_term_f_v @ ( F6 @ zero_zero_nat ) @ A6 )
& ! [I4: nat] : ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ ( F6 @ I4 ) @ ( F6 @ ( suc @ I4 ) ) ) @ R7 ) ) ) ) ).
% SN_on_def
thf(fact_1213_SN__onE,axiom,
! [R: set_Pr7897041799657265351rm_f_v,A4: set_Pr8827868859434726151rm_f_v] :
( ( abstra845488452567719153rm_f_v @ R @ A4 )
=> ~ ? [F7: nat > produc8027700073345061287rm_f_v] :
( ( member4316649861536941904rm_f_v @ ( F7 @ zero_zero_nat ) @ A4 )
& ! [I2: nat] : ( member7300054593328134928rm_f_v @ ( produc3874942763110801239rm_f_v @ ( F7 @ I2 ) @ ( F7 @ ( suc @ I2 ) ) ) @ R ) ) ) ).
% SN_onE
thf(fact_1214_SN__onE,axiom,
! [R: set_Pr8827868859434726151rm_f_v,A4: set_term_f_v] :
( ( abstra6117554785717145rm_f_v @ R @ A4 )
=> ~ ? [F7: nat > term_f_v] :
( ( member_term_f_v @ ( F7 @ zero_zero_nat ) @ A4 )
& ! [I2: nat] : ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ ( F7 @ I2 ) @ ( F7 @ ( suc @ I2 ) ) ) @ R ) ) ) ).
% SN_onE
thf(fact_1215_Abstract__Rewriting_Ochain__mono,axiom,
! [R5: set_Pr8827868859434726151rm_f_v,R3: set_Pr8827868859434726151rm_f_v,Seq: nat > term_f_v] :
( ( ord_le4559761987009501863rm_f_v @ R5 @ R3 )
=> ( ! [I2: nat] : ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ ( Seq @ I2 ) @ ( Seq @ ( suc @ I2 ) ) ) @ R5 )
=> ! [I3: nat] : ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ ( Seq @ I3 ) @ ( Seq @ ( suc @ I3 ) ) ) @ R3 ) ) ) ).
% Abstract_Rewriting.chain_mono
thf(fact_1216_relpow__Suc__E,axiom,
! [X3: term_f_v,Z2: term_f_v,N: nat,R3: set_Pr8827868859434726151rm_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Z2 ) @ ( compow8498445688072944954rm_f_v @ ( suc @ N ) @ R3 ) )
=> ~ ! [Y3: term_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Y3 ) @ ( compow8498445688072944954rm_f_v @ N @ R3 ) )
=> ~ ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y3 @ Z2 ) @ R3 ) ) ) ).
% relpow_Suc_E
thf(fact_1217_relpow__Suc__I,axiom,
! [X3: term_f_v,Y: term_f_v,N: nat,R3: set_Pr8827868859434726151rm_f_v,Z2: term_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Y ) @ ( compow8498445688072944954rm_f_v @ N @ R3 ) )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y @ Z2 ) @ R3 )
=> ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Z2 ) @ ( compow8498445688072944954rm_f_v @ ( suc @ N ) @ R3 ) ) ) ) ).
% relpow_Suc_I
thf(fact_1218_relpow__Suc__D2,axiom,
! [X3: term_f_v,Z2: term_f_v,N: nat,R3: set_Pr8827868859434726151rm_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Z2 ) @ ( compow8498445688072944954rm_f_v @ ( suc @ N ) @ R3 ) )
=> ? [Y3: term_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Y3 ) @ R3 )
& ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y3 @ Z2 ) @ ( compow8498445688072944954rm_f_v @ N @ R3 ) ) ) ) ).
% relpow_Suc_D2
thf(fact_1219_relpow__Suc__E2,axiom,
! [X3: term_f_v,Z2: term_f_v,N: nat,R3: set_Pr8827868859434726151rm_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Z2 ) @ ( compow8498445688072944954rm_f_v @ ( suc @ N ) @ R3 ) )
=> ~ ! [Y3: term_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Y3 ) @ R3 )
=> ~ ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y3 @ Z2 ) @ ( compow8498445688072944954rm_f_v @ N @ R3 ) ) ) ) ).
% relpow_Suc_E2
thf(fact_1220_relpow__Suc__I2,axiom,
! [X3: term_f_v,Y: term_f_v,R3: set_Pr8827868859434726151rm_f_v,Z2: term_f_v,N: nat] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Y ) @ R3 )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y @ Z2 ) @ ( compow8498445688072944954rm_f_v @ N @ R3 ) )
=> ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Z2 ) @ ( compow8498445688072944954rm_f_v @ ( suc @ N ) @ R3 ) ) ) ) ).
% relpow_Suc_I2
thf(fact_1221_relpow__Suc__E2_H,axiom,
! [X3: term_f_v,Z2: term_f_v,N: nat,A4: set_Pr8827868859434726151rm_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Z2 ) @ ( compow8498445688072944954rm_f_v @ ( suc @ N ) @ A4 ) )
=> ~ ! [Y3: term_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Y3 ) @ A4 )
=> ~ ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ Y3 @ Z2 ) @ ( transi3028809021613410336rm_f_v @ A4 ) ) ) ) ).
% relpow_Suc_E2'
thf(fact_1222_chain__imp__not__SN__on,axiom,
! [F2: nat > term_f_v,R: set_Pr8827868859434726151rm_f_v,I: nat] :
( ! [I2: nat] : ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ ( F2 @ I2 ) @ ( F2 @ ( suc @ I2 ) ) ) @ R )
=> ~ ( abstra6117554785717145rm_f_v @ R @ ( insert_term_f_v @ ( F2 @ I ) @ bot_bot_set_term_f_v ) ) ) ).
% chain_imp_not_SN_on
thf(fact_1223_chain__imp__rtrancl,axiom,
! [S2: nat > term_f_v,R: set_Pr8827868859434726151rm_f_v,I: nat,J: nat] :
( ! [I2: nat] : ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ ( S2 @ I2 ) @ ( S2 @ ( suc @ I2 ) ) ) @ R )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ ( S2 @ I ) @ ( S2 @ J ) ) @ ( transi3028809021613410336rm_f_v @ R ) ) ) ) ).
% chain_imp_rtrancl
thf(fact_1224_wf__iff__no__infinite__down__chain,axiom,
( wf_term_f_v
= ( ^ [R7: set_Pr8827868859434726151rm_f_v] :
~ ? [F6: nat > term_f_v] :
! [I4: nat] : ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ ( F6 @ ( suc @ I4 ) ) @ ( F6 @ I4 ) ) @ R7 ) ) ) ).
% wf_iff_no_infinite_down_chain
thf(fact_1225_wf__no__infinite__down__chainE,axiom,
! [R: set_Pr8827868859434726151rm_f_v,F2: nat > term_f_v] :
( ( wf_term_f_v @ R )
=> ~ ! [K: nat] : ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ ( F2 @ ( suc @ K ) ) @ ( F2 @ K ) ) @ R ) ) ).
% wf_no_infinite_down_chainE
thf(fact_1226_lift__Suc__antimono__le,axiom,
! [F2: nat > set_Pr5245412377734449720_f_nat,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_le8976984241387448984_f_nat @ ( F2 @ ( suc @ N2 ) ) @ ( F2 @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_le8976984241387448984_f_nat @ ( F2 @ N4 ) @ ( F2 @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_1227_lift__Suc__mono__le,axiom,
! [F2: nat > set_Pr5245412377734449720_f_nat,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_le8976984241387448984_f_nat @ ( F2 @ N2 ) @ ( F2 @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_le8976984241387448984_f_nat @ ( F2 @ N ) @ ( F2 @ N4 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_1228_chain__Un__SN__on__imp__first__step,axiom,
! [T: nat > term_f_v,R3: set_Pr8827868859434726151rm_f_v,S2: set_Pr8827868859434726151rm_f_v] :
( ! [I2: nat] : ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ ( T @ I2 ) @ ( T @ ( suc @ I2 ) ) ) @ ( sup_su1994854688125285083rm_f_v @ R3 @ S2 ) )
=> ( ( abstra6117554785717145rm_f_v @ S2 @ ( insert_term_f_v @ ( T @ zero_zero_nat ) @ bot_bot_set_term_f_v ) )
=> ? [I2: nat] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ ( T @ I2 ) @ ( T @ ( suc @ I2 ) ) ) @ R3 )
& ! [J3: nat] :
( ( ord_less_nat @ J3 @ I2 )
=> ( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ ( T @ J3 ) @ ( T @ ( suc @ J3 ) ) ) @ S2 )
& ~ ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ ( T @ J3 ) @ ( T @ ( suc @ J3 ) ) ) @ R3 ) ) ) ) ) ) ).
% chain_Un_SN_on_imp_first_step
thf(fact_1229_leD,axiom,
! [Y: set_Pr5245412377734449720_f_nat,X3: set_Pr5245412377734449720_f_nat] :
( ( ord_le8976984241387448984_f_nat @ Y @ X3 )
=> ~ ( ord_le2793513434575419532_f_nat @ X3 @ Y ) ) ).
% leD
thf(fact_1230_nless__le,axiom,
! [A: set_Pr5245412377734449720_f_nat,B: set_Pr5245412377734449720_f_nat] :
( ( ~ ( ord_le2793513434575419532_f_nat @ A @ B ) )
= ( ~ ( ord_le8976984241387448984_f_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_1231_antisym__conv1,axiom,
! [X3: set_Pr5245412377734449720_f_nat,Y: set_Pr5245412377734449720_f_nat] :
( ~ ( ord_le2793513434575419532_f_nat @ X3 @ Y )
=> ( ( ord_le8976984241387448984_f_nat @ X3 @ Y )
= ( X3 = Y ) ) ) ).
% antisym_conv1
thf(fact_1232_antisym__conv2,axiom,
! [X3: set_Pr5245412377734449720_f_nat,Y: set_Pr5245412377734449720_f_nat] :
( ( ord_le8976984241387448984_f_nat @ X3 @ Y )
=> ( ( ~ ( ord_le2793513434575419532_f_nat @ X3 @ Y ) )
= ( X3 = Y ) ) ) ).
% antisym_conv2
thf(fact_1233_less__le__not__le,axiom,
( ord_le2793513434575419532_f_nat
= ( ^ [X4: set_Pr5245412377734449720_f_nat,Y5: set_Pr5245412377734449720_f_nat] :
( ( ord_le8976984241387448984_f_nat @ X4 @ Y5 )
& ~ ( ord_le8976984241387448984_f_nat @ Y5 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_1234_order_Oorder__iff__strict,axiom,
( ord_le8976984241387448984_f_nat
= ( ^ [A5: set_Pr5245412377734449720_f_nat,B4: set_Pr5245412377734449720_f_nat] :
( ( ord_le2793513434575419532_f_nat @ A5 @ B4 )
| ( A5 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_1235_order_Ostrict__iff__order,axiom,
( ord_le2793513434575419532_f_nat
= ( ^ [A5: set_Pr5245412377734449720_f_nat,B4: set_Pr5245412377734449720_f_nat] :
( ( ord_le8976984241387448984_f_nat @ A5 @ B4 )
& ( A5 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_1236_order_Ostrict__trans1,axiom,
! [A: set_Pr5245412377734449720_f_nat,B: set_Pr5245412377734449720_f_nat,C3: set_Pr5245412377734449720_f_nat] :
( ( ord_le8976984241387448984_f_nat @ A @ B )
=> ( ( ord_le2793513434575419532_f_nat @ B @ C3 )
=> ( ord_le2793513434575419532_f_nat @ A @ C3 ) ) ) ).
% order.strict_trans1
thf(fact_1237_order_Ostrict__trans2,axiom,
! [A: set_Pr5245412377734449720_f_nat,B: set_Pr5245412377734449720_f_nat,C3: set_Pr5245412377734449720_f_nat] :
( ( ord_le2793513434575419532_f_nat @ A @ B )
=> ( ( ord_le8976984241387448984_f_nat @ B @ C3 )
=> ( ord_le2793513434575419532_f_nat @ A @ C3 ) ) ) ).
% order.strict_trans2
thf(fact_1238_order_Ostrict__iff__not,axiom,
( ord_le2793513434575419532_f_nat
= ( ^ [A5: set_Pr5245412377734449720_f_nat,B4: set_Pr5245412377734449720_f_nat] :
( ( ord_le8976984241387448984_f_nat @ A5 @ B4 )
& ~ ( ord_le8976984241387448984_f_nat @ B4 @ A5 ) ) ) ) ).
% order.strict_iff_not
thf(fact_1239_dual__order_Oorder__iff__strict,axiom,
( ord_le8976984241387448984_f_nat
= ( ^ [B4: set_Pr5245412377734449720_f_nat,A5: set_Pr5245412377734449720_f_nat] :
( ( ord_le2793513434575419532_f_nat @ B4 @ A5 )
| ( A5 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_1240_dual__order_Ostrict__iff__order,axiom,
( ord_le2793513434575419532_f_nat
= ( ^ [B4: set_Pr5245412377734449720_f_nat,A5: set_Pr5245412377734449720_f_nat] :
( ( ord_le8976984241387448984_f_nat @ B4 @ A5 )
& ( A5 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_1241_dual__order_Ostrict__trans1,axiom,
! [B: set_Pr5245412377734449720_f_nat,A: set_Pr5245412377734449720_f_nat,C3: set_Pr5245412377734449720_f_nat] :
( ( ord_le8976984241387448984_f_nat @ B @ A )
=> ( ( ord_le2793513434575419532_f_nat @ C3 @ B )
=> ( ord_le2793513434575419532_f_nat @ C3 @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_1242_dual__order_Ostrict__trans2,axiom,
! [B: set_Pr5245412377734449720_f_nat,A: set_Pr5245412377734449720_f_nat,C3: set_Pr5245412377734449720_f_nat] :
( ( ord_le2793513434575419532_f_nat @ B @ A )
=> ( ( ord_le8976984241387448984_f_nat @ C3 @ B )
=> ( ord_le2793513434575419532_f_nat @ C3 @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_1243_dual__order_Ostrict__iff__not,axiom,
( ord_le2793513434575419532_f_nat
= ( ^ [B4: set_Pr5245412377734449720_f_nat,A5: set_Pr5245412377734449720_f_nat] :
( ( ord_le8976984241387448984_f_nat @ B4 @ A5 )
& ~ ( ord_le8976984241387448984_f_nat @ A5 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_1244_order_Ostrict__implies__order,axiom,
! [A: set_Pr5245412377734449720_f_nat,B: set_Pr5245412377734449720_f_nat] :
( ( ord_le2793513434575419532_f_nat @ A @ B )
=> ( ord_le8976984241387448984_f_nat @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_1245_dual__order_Ostrict__implies__order,axiom,
! [B: set_Pr5245412377734449720_f_nat,A: set_Pr5245412377734449720_f_nat] :
( ( ord_le2793513434575419532_f_nat @ B @ A )
=> ( ord_le8976984241387448984_f_nat @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_1246_order__le__less,axiom,
( ord_le8976984241387448984_f_nat
= ( ^ [X4: set_Pr5245412377734449720_f_nat,Y5: set_Pr5245412377734449720_f_nat] :
( ( ord_le2793513434575419532_f_nat @ X4 @ Y5 )
| ( X4 = Y5 ) ) ) ) ).
% order_le_less
thf(fact_1247_order__less__le,axiom,
( ord_le2793513434575419532_f_nat
= ( ^ [X4: set_Pr5245412377734449720_f_nat,Y5: set_Pr5245412377734449720_f_nat] :
( ( ord_le8976984241387448984_f_nat @ X4 @ Y5 )
& ( X4 != Y5 ) ) ) ) ).
% order_less_le
thf(fact_1248_order__less__imp__le,axiom,
! [X3: set_Pr5245412377734449720_f_nat,Y: set_Pr5245412377734449720_f_nat] :
( ( ord_le2793513434575419532_f_nat @ X3 @ Y )
=> ( ord_le8976984241387448984_f_nat @ X3 @ Y ) ) ).
% order_less_imp_le
thf(fact_1249_order__le__neq__trans,axiom,
! [A: set_Pr5245412377734449720_f_nat,B: set_Pr5245412377734449720_f_nat] :
( ( ord_le8976984241387448984_f_nat @ A @ B )
=> ( ( A != B )
=> ( ord_le2793513434575419532_f_nat @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_1250_order__neq__le__trans,axiom,
! [A: set_Pr5245412377734449720_f_nat,B: set_Pr5245412377734449720_f_nat] :
( ( A != B )
=> ( ( ord_le8976984241387448984_f_nat @ A @ B )
=> ( ord_le2793513434575419532_f_nat @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_1251_order__le__less__trans,axiom,
! [X3: set_Pr5245412377734449720_f_nat,Y: set_Pr5245412377734449720_f_nat,Z2: set_Pr5245412377734449720_f_nat] :
( ( ord_le8976984241387448984_f_nat @ X3 @ Y )
=> ( ( ord_le2793513434575419532_f_nat @ Y @ Z2 )
=> ( ord_le2793513434575419532_f_nat @ X3 @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_1252_order__less__le__trans,axiom,
! [X3: set_Pr5245412377734449720_f_nat,Y: set_Pr5245412377734449720_f_nat,Z2: set_Pr5245412377734449720_f_nat] :
( ( ord_le2793513434575419532_f_nat @ X3 @ Y )
=> ( ( ord_le8976984241387448984_f_nat @ Y @ Z2 )
=> ( ord_le2793513434575419532_f_nat @ X3 @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_1253_order__le__less__subst2,axiom,
! [A: set_Pr5245412377734449720_f_nat,B: set_Pr5245412377734449720_f_nat,F2: set_Pr5245412377734449720_f_nat > set_Pr5245412377734449720_f_nat,C3: set_Pr5245412377734449720_f_nat] :
( ( ord_le8976984241387448984_f_nat @ A @ B )
=> ( ( ord_le2793513434575419532_f_nat @ ( F2 @ B ) @ C3 )
=> ( ! [X: set_Pr5245412377734449720_f_nat,Y3: set_Pr5245412377734449720_f_nat] :
( ( ord_le8976984241387448984_f_nat @ X @ Y3 )
=> ( ord_le8976984241387448984_f_nat @ ( F2 @ X ) @ ( F2 @ Y3 ) ) )
=> ( ord_le2793513434575419532_f_nat @ ( F2 @ A ) @ C3 ) ) ) ) ).
% order_le_less_subst2
thf(fact_1254_order__less__le__subst1,axiom,
! [A: set_Pr5245412377734449720_f_nat,F2: set_Pr5245412377734449720_f_nat > set_Pr5245412377734449720_f_nat,B: set_Pr5245412377734449720_f_nat,C3: set_Pr5245412377734449720_f_nat] :
( ( ord_le2793513434575419532_f_nat @ A @ ( F2 @ B ) )
=> ( ( ord_le8976984241387448984_f_nat @ B @ C3 )
=> ( ! [X: set_Pr5245412377734449720_f_nat,Y3: set_Pr5245412377734449720_f_nat] :
( ( ord_le8976984241387448984_f_nat @ X @ Y3 )
=> ( ord_le8976984241387448984_f_nat @ ( F2 @ X ) @ ( F2 @ Y3 ) ) )
=> ( ord_le2793513434575419532_f_nat @ A @ ( F2 @ C3 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1255_order__le__imp__less__or__eq,axiom,
! [X3: set_Pr5245412377734449720_f_nat,Y: set_Pr5245412377734449720_f_nat] :
( ( ord_le8976984241387448984_f_nat @ X3 @ Y )
=> ( ( ord_le2793513434575419532_f_nat @ X3 @ Y )
| ( X3 = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_1256_chain__imp__trancl,axiom,
! [S2: nat > term_f_v,R: set_Pr8827868859434726151rm_f_v,I: nat,J: nat] :
( ! [I2: nat] : ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ ( S2 @ I2 ) @ ( S2 @ ( suc @ I2 ) ) ) @ R )
=> ( ( ord_less_nat @ I @ J )
=> ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ ( S2 @ I ) @ ( S2 @ J ) ) @ ( transi3208867851621559554rm_f_v @ R ) ) ) ) ).
% chain_imp_trancl
thf(fact_1257_wf__bounded__measure,axiom,
! [R: set_Pr8827868859434726151rm_f_v,Ub: term_f_v > nat,F2: term_f_v > nat] :
( ! [A3: term_f_v,B3: term_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ B3 @ A3 ) @ R )
=> ( ( ord_less_eq_nat @ ( Ub @ B3 ) @ ( Ub @ A3 ) )
& ( ord_less_eq_nat @ ( F2 @ B3 ) @ ( Ub @ A3 ) )
& ( ord_less_nat @ ( F2 @ A3 ) @ ( F2 @ B3 ) ) ) )
=> ( wf_term_f_v @ R ) ) ).
% wf_bounded_measure
thf(fact_1258_trancl__power,axiom,
! [P: produc8027700073345061287rm_f_v,R3: set_Pr8827868859434726151rm_f_v] :
( ( member4316649861536941904rm_f_v @ P @ ( transi3208867851621559554rm_f_v @ R3 ) )
= ( ? [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
& ( member4316649861536941904rm_f_v @ P @ ( compow8498445688072944954rm_f_v @ N3 @ R3 ) ) ) ) ) ).
% trancl_power
thf(fact_1259_rtrancl__imp__seq,axiom,
! [X3: term_f_v,Y: term_f_v,R: set_Pr8827868859434726151rm_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Y ) @ ( transi3028809021613410336rm_f_v @ R ) )
=> ? [F5: nat > term_f_v,N2: nat] :
( ( ( F5 @ zero_zero_nat )
= X3 )
& ( ( F5 @ N2 )
= Y )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ N2 )
=> ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ ( F5 @ I3 ) @ ( F5 @ ( suc @ I3 ) ) ) @ R ) ) ) ) ).
% rtrancl_imp_seq
thf(fact_1260_rtrancl__fun__conv,axiom,
! [S: term_f_v,T: term_f_v,R3: set_Pr8827868859434726151rm_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ S @ T ) @ ( transi3028809021613410336rm_f_v @ R3 ) )
= ( ? [F6: nat > term_f_v,N3: nat] :
( ( ( F6 @ zero_zero_nat )
= S )
& ( ( F6 @ N3 )
= T )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ N3 )
=> ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ ( F6 @ I4 ) @ ( F6 @ ( suc @ I4 ) ) ) @ R3 ) ) ) ) ) ).
% rtrancl_fun_conv
thf(fact_1261_relpow__fun__conv,axiom,
! [A: term_f_v,B: term_f_v,N: nat,R3: set_Pr8827868859434726151rm_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ A @ B ) @ ( compow8498445688072944954rm_f_v @ N @ R3 ) )
= ( ? [F6: nat > term_f_v] :
( ( ( F6 @ zero_zero_nat )
= A )
& ( ( F6 @ N )
= B )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ N )
=> ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ ( F6 @ I4 ) @ ( F6 @ ( suc @ I4 ) ) ) @ R3 ) ) ) ) ) ).
% relpow_fun_conv
thf(fact_1262_relpow__seq,axiom,
! [X3: term_f_v,Y: term_f_v,N: nat,R: set_Pr8827868859434726151rm_f_v] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Y ) @ ( compow8498445688072944954rm_f_v @ N @ R ) )
=> ? [F5: nat > term_f_v] :
( ( ( F5 @ zero_zero_nat )
= X3 )
& ( ( F5 @ N )
= Y )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ ( F5 @ I3 ) @ ( F5 @ ( suc @ I3 ) ) ) @ R ) ) ) ) ).
% relpow_seq
thf(fact_1263_in__measure,axiom,
! [X3: term_f_v,Y: term_f_v,F2: term_f_v > nat] :
( ( member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ X3 @ Y ) @ ( measure_term_f_v @ F2 ) )
= ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y ) ) ) ).
% in_measure
thf(fact_1264_psubsetI,axiom,
! [A4: set_Pr5245412377734449720_f_nat,B5: set_Pr5245412377734449720_f_nat] :
( ( ord_le8976984241387448984_f_nat @ A4 @ B5 )
=> ( ( A4 != B5 )
=> ( ord_le2793513434575419532_f_nat @ A4 @ B5 ) ) ) ).
% psubsetI
thf(fact_1265_psubset__imp__ex__mem,axiom,
! [A4: set_Pr8827868859434726151rm_f_v,B5: set_Pr8827868859434726151rm_f_v] :
( ( ord_le100979401760869299rm_f_v @ A4 @ B5 )
=> ? [B3: produc8027700073345061287rm_f_v] : ( member4316649861536941904rm_f_v @ B3 @ ( minus_410040198995547630rm_f_v @ B5 @ A4 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_1266_psubsetE,axiom,
! [A4: set_Pr5245412377734449720_f_nat,B5: set_Pr5245412377734449720_f_nat] :
( ( ord_le2793513434575419532_f_nat @ A4 @ B5 )
=> ~ ( ( ord_le8976984241387448984_f_nat @ A4 @ B5 )
=> ( ord_le8976984241387448984_f_nat @ B5 @ A4 ) ) ) ).
% psubsetE
thf(fact_1267_psubset__eq,axiom,
( ord_le2793513434575419532_f_nat
= ( ^ [A6: set_Pr5245412377734449720_f_nat,B6: set_Pr5245412377734449720_f_nat] :
( ( ord_le8976984241387448984_f_nat @ A6 @ B6 )
& ( A6 != B6 ) ) ) ) ).
% psubset_eq
thf(fact_1268_psubset__imp__subset,axiom,
! [A4: set_Pr5245412377734449720_f_nat,B5: set_Pr5245412377734449720_f_nat] :
( ( ord_le2793513434575419532_f_nat @ A4 @ B5 )
=> ( ord_le8976984241387448984_f_nat @ A4 @ B5 ) ) ).
% psubset_imp_subset
thf(fact_1269_psubset__subset__trans,axiom,
! [A4: set_Pr5245412377734449720_f_nat,B5: set_Pr5245412377734449720_f_nat,C2: set_Pr5245412377734449720_f_nat] :
( ( ord_le2793513434575419532_f_nat @ A4 @ B5 )
=> ( ( ord_le8976984241387448984_f_nat @ B5 @ C2 )
=> ( ord_le2793513434575419532_f_nat @ A4 @ C2 ) ) ) ).
% psubset_subset_trans
thf(fact_1270_subset__not__subset__eq,axiom,
( ord_le2793513434575419532_f_nat
= ( ^ [A6: set_Pr5245412377734449720_f_nat,B6: set_Pr5245412377734449720_f_nat] :
( ( ord_le8976984241387448984_f_nat @ A6 @ B6 )
& ~ ( ord_le8976984241387448984_f_nat @ B6 @ A6 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_1271_subset__psubset__trans,axiom,
! [A4: set_Pr5245412377734449720_f_nat,B5: set_Pr5245412377734449720_f_nat,C2: set_Pr5245412377734449720_f_nat] :
( ( ord_le8976984241387448984_f_nat @ A4 @ B5 )
=> ( ( ord_le2793513434575419532_f_nat @ B5 @ C2 )
=> ( ord_le2793513434575419532_f_nat @ A4 @ C2 ) ) ) ).
% subset_psubset_trans
thf(fact_1272_subset__iff__psubset__eq,axiom,
( ord_le8976984241387448984_f_nat
= ( ^ [A6: set_Pr5245412377734449720_f_nat,B6: set_Pr5245412377734449720_f_nat] :
( ( ord_le2793513434575419532_f_nat @ A6 @ B6 )
| ( A6 = B6 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_1273_psubset__insert__iff,axiom,
! [A4: set_Pr8827868859434726151rm_f_v,X3: produc8027700073345061287rm_f_v,B5: set_Pr8827868859434726151rm_f_v] :
( ( ord_le100979401760869299rm_f_v @ A4 @ ( insert5456475782666790775rm_f_v @ X3 @ B5 ) )
= ( ( ( member4316649861536941904rm_f_v @ X3 @ B5 )
=> ( ord_le100979401760869299rm_f_v @ A4 @ B5 ) )
& ( ~ ( member4316649861536941904rm_f_v @ X3 @ B5 )
=> ( ( ( member4316649861536941904rm_f_v @ X3 @ A4 )
=> ( ord_le100979401760869299rm_f_v @ ( minus_410040198995547630rm_f_v @ A4 @ ( insert5456475782666790775rm_f_v @ X3 @ bot_bo4638812505780724595rm_f_v ) ) @ B5 ) )
& ( ~ ( member4316649861536941904rm_f_v @ X3 @ A4 )
=> ( ord_le4559761987009501863rm_f_v @ A4 @ B5 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_1274_psubset__insert__iff,axiom,
! [A4: set_Pr5245412377734449720_f_nat,X3: product_prod_f_nat,B5: set_Pr5245412377734449720_f_nat] :
( ( ord_le2793513434575419532_f_nat @ A4 @ ( insert5008453031520946354_f_nat @ X3 @ B5 ) )
= ( ( ( member2678038196313615385_f_nat @ X3 @ B5 )
=> ( ord_le2793513434575419532_f_nat @ A4 @ B5 ) )
& ( ~ ( member2678038196313615385_f_nat @ X3 @ B5 )
=> ( ( ( member2678038196313615385_f_nat @ X3 @ A4 )
=> ( ord_le2793513434575419532_f_nat @ ( minus_8941878871097791505_f_nat @ A4 @ ( insert5008453031520946354_f_nat @ X3 @ bot_bo136713897782694092_f_nat ) ) @ B5 ) )
& ( ~ ( member2678038196313615385_f_nat @ X3 @ A4 )
=> ( ord_le8976984241387448984_f_nat @ A4 @ B5 ) ) ) ) ) ) ).
% psubset_insert_iff
% Conjectures (1)
thf(conj_0,conjecture,
member4316649861536941904rm_f_v @ ( produc2510790566102465375rm_f_v @ ta @ u ) @ ( rstep_f_v @ r ) ).
%------------------------------------------------------------------------------