TPTP Problem File: SLH0356^1.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_00179_007111__14100094_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1598 ( 402 unt; 344 typ; 0 def)
% Number of atoms : 3704 (1045 equ; 0 cnn)
% Maximal formula atoms : 20 ( 2 avg)
% Number of connectives : 15001 ( 490 ~; 62 |; 254 &;12160 @)
% ( 0 <=>;2035 =>; 0 <=; 0 <~>)
% Maximal formula depth : 29 ( 8 avg)
% Number of types : 69 ( 68 usr)
% Number of type conns : 755 ( 755 >; 0 *; 0 +; 0 <<)
% Number of symbols : 279 ( 276 usr; 21 con; 0-3 aty)
% Number of variables : 4391 ( 211 ^;3964 !; 216 ?;4391 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 14:19:10.248
%------------------------------------------------------------------------------
% Could-be-implicit typings (68)
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thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mtf__a_J,type,
member8962352052110095674_nat_a: product_prod_nat_a > set_Pr4193341848836149977_nat_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
member2837775823723636880rm_a_b: produc8167519097144739175rm_a_b > set_Pr3371896974081115719rm_a_b > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
member8206827879206165904at_nat: produc859450856879609959at_nat > set_Pr8693737435421807431at_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_Itf__b_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J_Mt__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_Itf__b_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J_J,type,
member4955916718216803920rm_a_b: produc8033117147056819751rm_a_b > set_Pr3349378317018664967rm_a_b > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_Mt__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
member8417600551952982416rm_a_b: produc4523971788519308903rm_a_b > set_Pr2972776593051762503rm_a_b > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_J,type,
member9062615507155100804_a_nat: produc4708774622424448987_a_nat > set_Pr1811044260758604347_a_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__b_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_Mt__Product____Type__Oprod_Itf__b_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
member7542685802389602574rm_a_b: produc4700046383489396069rm_a_b > set_Pr5233514982410851781rm_a_b > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__b_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_Mt__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_J_J,type,
member4521035315342701695_a_nat: produc5464860344867494998_a_nat > set_Pr6288720886220400310_a_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_Itf__b_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
member4335681162695099391rm_a_b: produc5279506192219892694rm_a_b > set_Pr107789759150765622rm_a_b > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
member9209995263888512638st_nat: produc3697673438841856213st_nat > set_Pr534774883469167413st_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
member5869715511025134514rm_a_b: produc357393685978478089rm_a_b > set_Pr4386577575007340137rm_a_b > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mtf__b_J,type,
member6959280461167840375_a_b_b: produc6510227487322075744_a_b_b > set_Pr3369833000147915158_a_b_b > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J,type,
member5724188588386418708_a_nat: product_prod_a_nat > set_Pr4934435412358123699_a_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
member1426531477525435216od_a_a: product_prod_a_a > set_Product_prod_a_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_Itf__b_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
member7000133282412810073rm_a_b: produc6551080308567045442rm_a_b > set_Pr1357592988741419896rm_a_b > $o ).
thf(sy_c_member_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
member_term_a_b: term_a_b > set_term_a_b > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_P,type,
p: term_a_b > term_a_b > $o ).
thf(sy_v__092_060F_062,type,
f: set_Pr4934435412358123699_a_nat ).
thf(sy_v__092_060R_062,type,
r: set_Pr4386577575007340137rm_a_b ).
thf(sy_v_s,type,
s: term_a_b ).
thf(sy_v_t,type,
t: term_a_b ).
thf(sy_v_ta____,type,
ta: term_a_b ).
thf(sy_v_x____,type,
x: b ).
% Relevant facts (1249)
thf(fact_0_assms_I1_J,axiom,
! [S: term_a_b,T: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( srstep7844470518422762656ep_a_b @ f @ r ) )
=> ( p @ S @ T ) ) ).
% assms(1)
thf(fact_1_nsrsteps__with__root__step__step__on__args,axiom,
! [S: term_a_b,T: term_a_b,F: set_Pr4934435412358123699_a_nat,R: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( transi7922773638565587891rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R ) ) ) )
=> ( ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( srstep7844470518422762656ep_a_b @ F @ R ) )
=> ? [F2: a,Ss: list_term_a_b,Ts: list_term_a_b] :
( ( S
= ( fun_a_b @ F2 @ Ss ) )
& ( T
= ( fun_a_b @ F2 @ Ts ) )
& ( ( size_s8906293707977694520rm_a_b @ Ss )
= ( size_s8906293707977694520rm_a_b @ Ts ) )
& ! [I: nat] :
( ( ord_less_nat @ I @ ( size_s8906293707977694520rm_a_b @ Ts ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( nth_term_a_b @ Ss @ I ) @ ( nth_term_a_b @ Ts @ I ) ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R ) ) ) ) ) ) ) ) ).
% nsrsteps_with_root_step_step_on_args
thf(fact_2_Var_Oprems_I1_J,axiom,
member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( var_b_a @ x ) @ ta ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ f @ ( rstep_a_b @ r ) ) ) ).
% Var.prems(1)
thf(fact_3_srsteps__with__root__step__srstepsD,axiom,
! [S: term_a_b,T: term_a_b,F: set_Pr4934435412358123699_a_nat,R: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( srstep7844470518422762656ep_a_b @ F @ R ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( transi7922773638565587891rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R ) ) ) ) ) ).
% srsteps_with_root_step_srstepsD
thf(fact_4_srsteps__with__root__step__sresteps__eqD,axiom,
! [S: term_a_b,T: term_a_b,F: set_Pr4934435412358123699_a_nat,R: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( srstep7844470518422762656ep_a_b @ F @ R ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R ) ) ) ) ) ).
% srsteps_with_root_step_sresteps_eqD
thf(fact_5_rstep__ruleI,axiom,
! [L: term_a_b,R2: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ L @ R2 ) @ R )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ L @ R2 ) @ ( rstep_a_b @ R ) ) ) ).
% rstep_ruleI
thf(fact_6_steps,axiom,
member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ s @ t ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ f @ ( rstep_a_b @ r ) ) ) ).
% steps
thf(fact_7_term_Oinject_I1_J,axiom,
! [X1: b,Y1: b] :
( ( ( var_b_a @ X1 )
= ( var_b_a @ Y1 ) )
= ( X1 = Y1 ) ) ).
% term.inject(1)
thf(fact_8_Var_Oprems_I2_J,axiom,
ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ ( var_b_a @ x ) ) @ f ).
% Var.prems(2)
thf(fact_9_prod_Oinject,axiom,
! [X1: list_nat,X2: term_a_b,Y1: list_nat,Y2: term_a_b] :
( ( ( produc5151171985953862413rm_a_b @ X1 @ X2 )
= ( produc5151171985953862413rm_a_b @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_10_prod_Oinject,axiom,
! [X1: set_Pr4934435412358123699_a_nat,X2: produc6551080308567045442rm_a_b,Y1: set_Pr4934435412358123699_a_nat,Y2: produc6551080308567045442rm_a_b] :
( ( ( produc8030969961714872974rm_a_b @ X1 @ X2 )
= ( produc8030969961714872974rm_a_b @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_11_prod_Oinject,axiom,
! [X1: nat,X2: nat,Y1: nat,Y2: nat] :
( ( ( product_Pair_nat_nat @ X1 @ X2 )
= ( product_Pair_nat_nat @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_12_prod_Oinject,axiom,
! [X1: b,X2: term_a_b,Y1: b,Y2: term_a_b] :
( ( ( produc1437816968797971900rm_a_b @ X1 @ X2 )
= ( produc1437816968797971900rm_a_b @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_13_prod_Oinject,axiom,
! [X1: a,X2: nat,Y1: a,Y2: nat] :
( ( ( product_Pair_a_nat @ X1 @ X2 )
= ( product_Pair_a_nat @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_14_prod_Oinject,axiom,
! [X1: term_a_b,X2: term_a_b,Y1: term_a_b,Y2: term_a_b] :
( ( ( produc7020197800436672577rm_a_b @ X1 @ X2 )
= ( produc7020197800436672577rm_a_b @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_15_old_Oprod_Oinject,axiom,
! [A: list_nat,B: term_a_b,A2: list_nat,B2: term_a_b] :
( ( ( produc5151171985953862413rm_a_b @ A @ B )
= ( produc5151171985953862413rm_a_b @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_16_old_Oprod_Oinject,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: produc6551080308567045442rm_a_b,A2: set_Pr4934435412358123699_a_nat,B2: produc6551080308567045442rm_a_b] :
( ( ( produc8030969961714872974rm_a_b @ A @ B )
= ( produc8030969961714872974rm_a_b @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_17_old_Oprod_Oinject,axiom,
! [A: nat,B: nat,A2: nat,B2: nat] :
( ( ( product_Pair_nat_nat @ A @ B )
= ( product_Pair_nat_nat @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_18_old_Oprod_Oinject,axiom,
! [A: b,B: term_a_b,A2: b,B2: term_a_b] :
( ( ( produc1437816968797971900rm_a_b @ A @ B )
= ( produc1437816968797971900rm_a_b @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_19_old_Oprod_Oinject,axiom,
! [A: a,B: nat,A2: a,B2: nat] :
( ( ( product_Pair_a_nat @ A @ B )
= ( product_Pair_a_nat @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_20_old_Oprod_Oinject,axiom,
! [A: term_a_b,B: term_a_b,A2: term_a_b,B2: term_a_b] :
( ( ( produc7020197800436672577rm_a_b @ A @ B )
= ( produc7020197800436672577rm_a_b @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_21_trancl_Ocases,axiom,
! [A1: produc357393685978478089rm_a_b,A22: produc357393685978478089rm_a_b,R2: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A1 @ A22 ) @ ( transi9211502839322181930rm_a_b @ R2 ) )
=> ( ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A1 @ A22 ) @ R2 )
=> ~ ! [B3: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A1 @ B3 ) @ ( transi9211502839322181930rm_a_b @ R2 ) )
=> ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B3 @ A22 ) @ R2 ) ) ) ) ).
% trancl.cases
thf(fact_22_trancl_Ocases,axiom,
! [A1: a,A22: a,R2: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A1 @ A22 ) @ ( transitive_trancl_a @ R2 ) )
=> ( ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A1 @ A22 ) @ R2 )
=> ~ ! [B3: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A1 @ B3 ) @ ( transitive_trancl_a @ R2 ) )
=> ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B3 @ A22 ) @ R2 ) ) ) ) ).
% trancl.cases
thf(fact_23_trancl_Ocases,axiom,
! [A1: nat,A22: nat,R2: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A1 @ A22 ) @ ( transi6264000038957366511cl_nat @ R2 ) )
=> ( ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A1 @ A22 ) @ R2 )
=> ~ ! [B3: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A1 @ B3 ) @ ( transi6264000038957366511cl_nat @ R2 ) )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B3 @ A22 ) @ R2 ) ) ) ) ).
% trancl.cases
thf(fact_24_trancl_Ocases,axiom,
! [A1: term_a_b,A22: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A22 ) @ ( transi7922773638565587891rm_a_b @ R2 ) )
=> ( ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A22 ) @ R2 )
=> ~ ! [B3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ B3 ) @ ( transi7922773638565587891rm_a_b @ R2 ) )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B3 @ A22 ) @ R2 ) ) ) ) ).
% trancl.cases
thf(fact_25_trancl_Osimps,axiom,
! [A1: produc357393685978478089rm_a_b,A22: produc357393685978478089rm_a_b,R2: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A1 @ A22 ) @ ( transi9211502839322181930rm_a_b @ R2 ) )
= ( ? [A3: produc357393685978478089rm_a_b,B4: produc357393685978478089rm_a_b] :
( ( A1 = A3 )
& ( A22 = B4 )
& ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A3 @ B4 ) @ R2 ) )
| ? [A3: produc357393685978478089rm_a_b,B4: produc357393685978478089rm_a_b,C: produc357393685978478089rm_a_b] :
( ( A1 = A3 )
& ( A22 = C )
& ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A3 @ B4 ) @ ( transi9211502839322181930rm_a_b @ R2 ) )
& ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B4 @ C ) @ R2 ) ) ) ) ).
% trancl.simps
thf(fact_26_trancl_Osimps,axiom,
! [A1: a,A22: a,R2: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A1 @ A22 ) @ ( transitive_trancl_a @ R2 ) )
= ( ? [A3: a,B4: a] :
( ( A1 = A3 )
& ( A22 = B4 )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A3 @ B4 ) @ R2 ) )
| ? [A3: a,B4: a,C: a] :
( ( A1 = A3 )
& ( A22 = C )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A3 @ B4 ) @ ( transitive_trancl_a @ R2 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B4 @ C ) @ R2 ) ) ) ) ).
% trancl.simps
thf(fact_27_trancl_Osimps,axiom,
! [A1: nat,A22: nat,R2: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A1 @ A22 ) @ ( transi6264000038957366511cl_nat @ R2 ) )
= ( ? [A3: nat,B4: nat] :
( ( A1 = A3 )
& ( A22 = B4 )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A3 @ B4 ) @ R2 ) )
| ? [A3: nat,B4: nat,C: nat] :
( ( A1 = A3 )
& ( A22 = C )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A3 @ B4 ) @ ( transi6264000038957366511cl_nat @ R2 ) )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B4 @ C ) @ R2 ) ) ) ) ).
% trancl.simps
thf(fact_28_trancl_Osimps,axiom,
! [A1: term_a_b,A22: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A22 ) @ ( transi7922773638565587891rm_a_b @ R2 ) )
= ( ? [A3: term_a_b,B4: term_a_b] :
( ( A1 = A3 )
& ( A22 = B4 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B4 ) @ R2 ) )
| ? [A3: term_a_b,B4: term_a_b,C: term_a_b] :
( ( A1 = A3 )
& ( A22 = C )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B4 ) @ ( transi7922773638565587891rm_a_b @ R2 ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B4 @ C ) @ R2 ) ) ) ) ).
% trancl.simps
thf(fact_29_trancl_Or__into__trancl,axiom,
! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,R2: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ R2 )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( transi9211502839322181930rm_a_b @ R2 ) ) ) ).
% trancl.r_into_trancl
thf(fact_30_trancl_Or__into__trancl,axiom,
! [A: a,B: a,R2: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ R2 )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ ( transitive_trancl_a @ R2 ) ) ) ).
% trancl.r_into_trancl
thf(fact_31_trancl_Or__into__trancl,axiom,
! [A: nat,B: nat,R2: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R2 )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( transi6264000038957366511cl_nat @ R2 ) ) ) ).
% trancl.r_into_trancl
thf(fact_32_trancl_Or__into__trancl,axiom,
! [A: term_a_b,B: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R2 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7922773638565587891rm_a_b @ R2 ) ) ) ).
% trancl.r_into_trancl
thf(fact_33_trancl__map,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,F3: term_a_b > term_a_b,S: set_Pr4386577575007340137rm_a_b,X: term_a_b,Y: term_a_b] :
( ! [X3: term_a_b,Y3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ R2 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) @ S ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7922773638565587891rm_a_b @ R2 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F3 @ X ) @ ( F3 @ Y ) ) @ ( transi7922773638565587891rm_a_b @ S ) ) ) ) ).
% trancl_map
thf(fact_34_trancl__map,axiom,
! [R2: set_Product_prod_a_a,F3: a > a,S: set_Product_prod_a_a,X: a,Y: a] :
( ! [X3: a,Y3: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y3 ) @ R2 )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) @ S ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ ( transitive_trancl_a @ R2 ) )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( F3 @ X ) @ ( F3 @ Y ) ) @ ( transitive_trancl_a @ S ) ) ) ) ).
% trancl_map
thf(fact_35_trancl__map,axiom,
! [R2: set_Product_prod_a_a,F3: a > nat,S: set_Pr1261947904930325089at_nat,X: a,Y: a] :
( ! [X3: a,Y3: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y3 ) @ R2 )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) @ S ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ ( transitive_trancl_a @ R2 ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F3 @ X ) @ ( F3 @ Y ) ) @ ( transi6264000038957366511cl_nat @ S ) ) ) ) ).
% trancl_map
thf(fact_36_trancl__map,axiom,
! [R2: set_Pr1261947904930325089at_nat,F3: nat > a,S: set_Product_prod_a_a,X: nat,Y: nat] :
( ! [X3: nat,Y3: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R2 )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) @ S ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( transi6264000038957366511cl_nat @ R2 ) )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( F3 @ X ) @ ( F3 @ Y ) ) @ ( transitive_trancl_a @ S ) ) ) ) ).
% trancl_map
thf(fact_37_trancl__map,axiom,
! [R2: set_Pr1261947904930325089at_nat,F3: nat > nat,S: set_Pr1261947904930325089at_nat,X: nat,Y: nat] :
( ! [X3: nat,Y3: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R2 )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) @ S ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( transi6264000038957366511cl_nat @ R2 ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F3 @ X ) @ ( F3 @ Y ) ) @ ( transi6264000038957366511cl_nat @ S ) ) ) ) ).
% trancl_map
thf(fact_38_trancl__map,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,F3: term_a_b > a,S: set_Product_prod_a_a,X: term_a_b,Y: term_a_b] :
( ! [X3: term_a_b,Y3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ R2 )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) @ S ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7922773638565587891rm_a_b @ R2 ) )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( F3 @ X ) @ ( F3 @ Y ) ) @ ( transitive_trancl_a @ S ) ) ) ) ).
% trancl_map
thf(fact_39_trancl__map,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,F3: term_a_b > nat,S: set_Pr1261947904930325089at_nat,X: term_a_b,Y: term_a_b] :
( ! [X3: term_a_b,Y3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ R2 )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) @ S ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7922773638565587891rm_a_b @ R2 ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F3 @ X ) @ ( F3 @ Y ) ) @ ( transi6264000038957366511cl_nat @ S ) ) ) ) ).
% trancl_map
thf(fact_40_trancl__map,axiom,
! [R2: set_Product_prod_a_a,F3: a > term_a_b,S: set_Pr4386577575007340137rm_a_b,X: a,Y: a] :
( ! [X3: a,Y3: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y3 ) @ R2 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) @ S ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ ( transitive_trancl_a @ R2 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F3 @ X ) @ ( F3 @ Y ) ) @ ( transi7922773638565587891rm_a_b @ S ) ) ) ) ).
% trancl_map
thf(fact_41_trancl__map,axiom,
! [R2: set_Pr1261947904930325089at_nat,F3: nat > term_a_b,S: set_Pr4386577575007340137rm_a_b,X: nat,Y: nat] :
( ! [X3: nat,Y3: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R2 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) @ S ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( transi6264000038957366511cl_nat @ R2 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F3 @ X ) @ ( F3 @ Y ) ) @ ( transi7922773638565587891rm_a_b @ S ) ) ) ) ).
% trancl_map
thf(fact_42_trancl__map,axiom,
! [R2: set_Pr2972776593051762503rm_a_b,F3: produc357393685978478089rm_a_b > a,S: set_Product_prod_a_a,X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b] :
( ! [X3: produc357393685978478089rm_a_b,Y3: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X3 @ Y3 ) @ R2 )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) @ S ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ ( transi9211502839322181930rm_a_b @ R2 ) )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( F3 @ X ) @ ( F3 @ Y ) ) @ ( transitive_trancl_a @ S ) ) ) ) ).
% trancl_map
thf(fact_43_well_I1_J,axiom,
ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ s ) @ f ).
% well(1)
thf(fact_44_well_I2_J,axiom,
ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ t ) @ f ).
% well(2)
thf(fact_45_Var_Oprems_I3_J,axiom,
ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ ta ) @ f ).
% Var.prems(3)
thf(fact_46_rtrancl__idemp,axiom,
! [R2: set_Pr2972776593051762503rm_a_b] :
( ( transi2615809358984392588rm_a_b @ ( transi2615809358984392588rm_a_b @ R2 ) )
= ( transi2615809358984392588rm_a_b @ R2 ) ) ).
% rtrancl_idemp
thf(fact_47_rtrancl__idemp,axiom,
! [R2: set_Product_prod_a_a] :
( ( transitive_rtrancl_a @ ( transitive_rtrancl_a @ R2 ) )
= ( transitive_rtrancl_a @ R2 ) ) ).
% rtrancl_idemp
thf(fact_48_rtrancl__idemp,axiom,
! [R2: set_Pr1261947904930325089at_nat] :
( ( transi2905341329935302413cl_nat @ ( transi2905341329935302413cl_nat @ R2 ) )
= ( transi2905341329935302413cl_nat @ R2 ) ) ).
% rtrancl_idemp
thf(fact_49_rtrancl__idemp,axiom,
! [R2: set_Pr8564414093027780873rm_a_b] :
( ( transi911495901722478305rm_a_b @ ( transi911495901722478305rm_a_b @ R2 ) )
= ( transi911495901722478305rm_a_b @ R2 ) ) ).
% rtrancl_idemp
thf(fact_50_rtrancl__idemp,axiom,
! [R2: set_Pr4386577575007340137rm_a_b] :
( ( transi7742714808557438673rm_a_b @ ( transi7742714808557438673rm_a_b @ R2 ) )
= ( transi7742714808557438673rm_a_b @ R2 ) ) ).
% rtrancl_idemp
thf(fact_51_r__into__rtrancl,axiom,
! [P: produc4523971788519308903rm_a_b,R2: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ P @ R2 )
=> ( member8417600551952982416rm_a_b @ P @ ( transi2615809358984392588rm_a_b @ R2 ) ) ) ).
% r_into_rtrancl
thf(fact_52_r__into__rtrancl,axiom,
! [P: product_prod_a_a,R2: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ P @ R2 )
=> ( member1426531477525435216od_a_a @ P @ ( transitive_rtrancl_a @ R2 ) ) ) ).
% r_into_rtrancl
thf(fact_53_r__into__rtrancl,axiom,
! [P: product_prod_nat_nat,R2: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ P @ R2 )
=> ( member8440522571783428010at_nat @ P @ ( transi2905341329935302413cl_nat @ R2 ) ) ) ).
% r_into_rtrancl
thf(fact_54_r__into__rtrancl,axiom,
! [P: produc51424535725745577rm_a_b,R2: set_Pr8564414093027780873rm_a_b] :
( ( member4405265420456397394rm_a_b @ P @ R2 )
=> ( member4405265420456397394rm_a_b @ P @ ( transi911495901722478305rm_a_b @ R2 ) ) ) ).
% r_into_rtrancl
thf(fact_55_r__into__rtrancl,axiom,
! [P: produc357393685978478089rm_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ P @ R2 )
=> ( member5869715511025134514rm_a_b @ P @ ( transi7742714808557438673rm_a_b @ R2 ) ) ) ).
% r_into_rtrancl
thf(fact_56_term_Oinject_I2_J,axiom,
! [X21: a,X22: list_term_a_b,Y21: a,Y22: list_term_a_b] :
( ( ( fun_a_b @ X21 @ X22 )
= ( fun_a_b @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% term.inject(2)
thf(fact_57_trancl__rtrancl__absorb,axiom,
! [R3: set_Pr2972776593051762503rm_a_b] :
( ( transi2615809358984392588rm_a_b @ ( transi9211502839322181930rm_a_b @ R3 ) )
= ( transi2615809358984392588rm_a_b @ R3 ) ) ).
% trancl_rtrancl_absorb
thf(fact_58_trancl__rtrancl__absorb,axiom,
! [R3: set_Product_prod_a_a] :
( ( transitive_rtrancl_a @ ( transitive_trancl_a @ R3 ) )
= ( transitive_rtrancl_a @ R3 ) ) ).
% trancl_rtrancl_absorb
thf(fact_59_trancl__rtrancl__absorb,axiom,
! [R3: set_Pr1261947904930325089at_nat] :
( ( transi2905341329935302413cl_nat @ ( transi6264000038957366511cl_nat @ R3 ) )
= ( transi2905341329935302413cl_nat @ R3 ) ) ).
% trancl_rtrancl_absorb
thf(fact_60_trancl__rtrancl__absorb,axiom,
! [R3: set_Pr8564414093027780873rm_a_b] :
( ( transi911495901722478305rm_a_b @ ( transi4990936492147380931rm_a_b @ R3 ) )
= ( transi911495901722478305rm_a_b @ R3 ) ) ).
% trancl_rtrancl_absorb
thf(fact_61_trancl__rtrancl__absorb,axiom,
! [R3: set_Pr4386577575007340137rm_a_b] :
( ( transi7742714808557438673rm_a_b @ ( transi7922773638565587891rm_a_b @ R3 ) )
= ( transi7742714808557438673rm_a_b @ R3 ) ) ).
% trancl_rtrancl_absorb
thf(fact_62_rtrancl__trancl__absorb,axiom,
! [R3: set_Pr2972776593051762503rm_a_b] :
( ( transi9211502839322181930rm_a_b @ ( transi2615809358984392588rm_a_b @ R3 ) )
= ( transi2615809358984392588rm_a_b @ R3 ) ) ).
% rtrancl_trancl_absorb
thf(fact_63_rtrancl__trancl__absorb,axiom,
! [R3: set_Product_prod_a_a] :
( ( transitive_trancl_a @ ( transitive_rtrancl_a @ R3 ) )
= ( transitive_rtrancl_a @ R3 ) ) ).
% rtrancl_trancl_absorb
thf(fact_64_rtrancl__trancl__absorb,axiom,
! [R3: set_Pr1261947904930325089at_nat] :
( ( transi6264000038957366511cl_nat @ ( transi2905341329935302413cl_nat @ R3 ) )
= ( transi2905341329935302413cl_nat @ R3 ) ) ).
% rtrancl_trancl_absorb
thf(fact_65_rtrancl__trancl__absorb,axiom,
! [R3: set_Pr8564414093027780873rm_a_b] :
( ( transi4990936492147380931rm_a_b @ ( transi911495901722478305rm_a_b @ R3 ) )
= ( transi911495901722478305rm_a_b @ R3 ) ) ).
% rtrancl_trancl_absorb
thf(fact_66_rtrancl__trancl__absorb,axiom,
! [R3: set_Pr4386577575007340137rm_a_b] :
( ( transi7922773638565587891rm_a_b @ ( transi7742714808557438673rm_a_b @ R3 ) )
= ( transi7742714808557438673rm_a_b @ R3 ) ) ).
% rtrancl_trancl_absorb
thf(fact_67_sig__stepI,axiom,
! [S: term_a_b,F: set_Pr4934435412358123699_a_nat,T: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ S ) @ F )
=> ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ T ) @ F )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ R )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( sig_step_a_b @ F @ R ) ) ) ) ) ).
% sig_stepI
thf(fact_68_rtrancl__mono,axiom,
! [R2: set_Pr2972776593051762503rm_a_b,S: set_Pr2972776593051762503rm_a_b] :
( ( ord_le2593958535671617255rm_a_b @ R2 @ S )
=> ( ord_le2593958535671617255rm_a_b @ ( transi2615809358984392588rm_a_b @ R2 ) @ ( transi2615809358984392588rm_a_b @ S ) ) ) ).
% rtrancl_mono
thf(fact_69_rtrancl__mono,axiom,
! [R2: set_Pr8564414093027780873rm_a_b,S: set_Pr8564414093027780873rm_a_b] :
( ( ord_le7771511673815134377rm_a_b @ R2 @ S )
=> ( ord_le7771511673815134377rm_a_b @ ( transi911495901722478305rm_a_b @ R2 ) @ ( transi911495901722478305rm_a_b @ S ) ) ) ).
% rtrancl_mono
thf(fact_70_rtrancl__mono,axiom,
! [R2: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ R2 @ S )
=> ( ord_le3146513528884898305at_nat @ ( transi2905341329935302413cl_nat @ R2 ) @ ( transi2905341329935302413cl_nat @ S ) ) ) ).
% rtrancl_mono
thf(fact_71_rtrancl__mono,axiom,
! [R2: set_Product_prod_a_a,S: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ R2 @ S )
=> ( ord_le746702958409616551od_a_a @ ( transitive_rtrancl_a @ R2 ) @ ( transitive_rtrancl_a @ S ) ) ) ).
% rtrancl_mono
thf(fact_72_rtrancl__mono,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ R2 @ S )
=> ( ord_le118470702582115849rm_a_b @ ( transi7742714808557438673rm_a_b @ R2 ) @ ( transi7742714808557438673rm_a_b @ S ) ) ) ).
% rtrancl_mono
thf(fact_73_rtrancl__subset,axiom,
! [R3: set_Pr2972776593051762503rm_a_b,S2: set_Pr2972776593051762503rm_a_b] :
( ( ord_le2593958535671617255rm_a_b @ R3 @ S2 )
=> ( ( ord_le2593958535671617255rm_a_b @ S2 @ ( transi2615809358984392588rm_a_b @ R3 ) )
=> ( ( transi2615809358984392588rm_a_b @ S2 )
= ( transi2615809358984392588rm_a_b @ R3 ) ) ) ) ).
% rtrancl_subset
thf(fact_74_rtrancl__subset,axiom,
! [R3: set_Pr8564414093027780873rm_a_b,S2: set_Pr8564414093027780873rm_a_b] :
( ( ord_le7771511673815134377rm_a_b @ R3 @ S2 )
=> ( ( ord_le7771511673815134377rm_a_b @ S2 @ ( transi911495901722478305rm_a_b @ R3 ) )
=> ( ( transi911495901722478305rm_a_b @ S2 )
= ( transi911495901722478305rm_a_b @ R3 ) ) ) ) ).
% rtrancl_subset
thf(fact_75_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_76_rtrancl__subset,axiom,
! [R3: set_Product_prod_a_a,S2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ R3 @ S2 )
=> ( ( ord_le746702958409616551od_a_a @ S2 @ ( transitive_rtrancl_a @ R3 ) )
=> ( ( transitive_rtrancl_a @ S2 )
= ( transitive_rtrancl_a @ R3 ) ) ) ) ).
% rtrancl_subset
thf(fact_77_rtrancl__subset,axiom,
! [R3: set_Pr4386577575007340137rm_a_b,S2: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ R3 @ S2 )
=> ( ( ord_le118470702582115849rm_a_b @ S2 @ ( transi7742714808557438673rm_a_b @ R3 ) )
=> ( ( transi7742714808557438673rm_a_b @ S2 )
= ( transi7742714808557438673rm_a_b @ R3 ) ) ) ) ).
% rtrancl_subset
thf(fact_78_rtrancl__subset__rtrancl,axiom,
! [R2: set_Pr2972776593051762503rm_a_b,S: set_Pr2972776593051762503rm_a_b] :
( ( ord_le2593958535671617255rm_a_b @ R2 @ ( transi2615809358984392588rm_a_b @ S ) )
=> ( ord_le2593958535671617255rm_a_b @ ( transi2615809358984392588rm_a_b @ R2 ) @ ( transi2615809358984392588rm_a_b @ S ) ) ) ).
% rtrancl_subset_rtrancl
thf(fact_79_rtrancl__subset__rtrancl,axiom,
! [R2: set_Pr8564414093027780873rm_a_b,S: set_Pr8564414093027780873rm_a_b] :
( ( ord_le7771511673815134377rm_a_b @ R2 @ ( transi911495901722478305rm_a_b @ S ) )
=> ( ord_le7771511673815134377rm_a_b @ ( transi911495901722478305rm_a_b @ R2 ) @ ( transi911495901722478305rm_a_b @ S ) ) ) ).
% rtrancl_subset_rtrancl
thf(fact_80_rtrancl__subset__rtrancl,axiom,
! [R2: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ R2 @ ( transi2905341329935302413cl_nat @ S ) )
=> ( ord_le3146513528884898305at_nat @ ( transi2905341329935302413cl_nat @ R2 ) @ ( transi2905341329935302413cl_nat @ S ) ) ) ).
% rtrancl_subset_rtrancl
thf(fact_81_rtrancl__subset__rtrancl,axiom,
! [R2: set_Product_prod_a_a,S: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ R2 @ ( transitive_rtrancl_a @ S ) )
=> ( ord_le746702958409616551od_a_a @ ( transitive_rtrancl_a @ R2 ) @ ( transitive_rtrancl_a @ S ) ) ) ).
% rtrancl_subset_rtrancl
thf(fact_82_rtrancl__subset__rtrancl,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ R2 @ ( transi7742714808557438673rm_a_b @ S ) )
=> ( ord_le118470702582115849rm_a_b @ ( transi7742714808557438673rm_a_b @ R2 ) @ ( transi7742714808557438673rm_a_b @ S ) ) ) ).
% rtrancl_subset_rtrancl
thf(fact_83_trancl__induct2,axiom,
! [Ax: list_nat,Ay: term_a_b,Bx: list_nat,By: term_a_b,R2: set_Pr3371896974081115719rm_a_b,P2: list_nat > term_a_b > $o] :
( ( member2837775823723636880rm_a_b @ ( produc621521293690684375rm_a_b @ ( produc5151171985953862413rm_a_b @ Ax @ Ay ) @ ( produc5151171985953862413rm_a_b @ Bx @ By ) ) @ ( transi7342477024839371766rm_a_b @ R2 ) )
=> ( ! [A4: list_nat,B3: term_a_b] :
( ( member2837775823723636880rm_a_b @ ( produc621521293690684375rm_a_b @ ( produc5151171985953862413rm_a_b @ Ax @ Ay ) @ ( produc5151171985953862413rm_a_b @ A4 @ B3 ) ) @ R2 )
=> ( P2 @ A4 @ B3 ) )
=> ( ! [A4: list_nat,B3: term_a_b,Aa: list_nat,Ba: term_a_b] :
( ( member2837775823723636880rm_a_b @ ( produc621521293690684375rm_a_b @ ( produc5151171985953862413rm_a_b @ Ax @ Ay ) @ ( produc5151171985953862413rm_a_b @ A4 @ B3 ) ) @ ( transi7342477024839371766rm_a_b @ R2 ) )
=> ( ( member2837775823723636880rm_a_b @ ( produc621521293690684375rm_a_b @ ( produc5151171985953862413rm_a_b @ A4 @ B3 ) @ ( produc5151171985953862413rm_a_b @ Aa @ Ba ) ) @ R2 )
=> ( ( P2 @ A4 @ B3 )
=> ( P2 @ Aa @ Ba ) ) ) )
=> ( P2 @ Bx @ By ) ) ) ) ).
% trancl_induct2
thf(fact_84_trancl__induct2,axiom,
! [Ax: set_Pr4934435412358123699_a_nat,Ay: produc6551080308567045442rm_a_b,Bx: set_Pr4934435412358123699_a_nat,By: produc6551080308567045442rm_a_b,R2: set_Pr3349378317018664967rm_a_b,P2: set_Pr4934435412358123699_a_nat > produc6551080308567045442rm_a_b > $o] :
( ( member4955916718216803920rm_a_b @ ( produc3065199688681838103rm_a_b @ ( produc8030969961714872974rm_a_b @ Ax @ Ay ) @ ( produc8030969961714872974rm_a_b @ Bx @ By ) ) @ ( transi2902453399402325367rm_a_b @ R2 ) )
=> ( ! [A4: set_Pr4934435412358123699_a_nat,B3: produc6551080308567045442rm_a_b] :
( ( member4955916718216803920rm_a_b @ ( produc3065199688681838103rm_a_b @ ( produc8030969961714872974rm_a_b @ Ax @ Ay ) @ ( produc8030969961714872974rm_a_b @ A4 @ B3 ) ) @ R2 )
=> ( P2 @ A4 @ B3 ) )
=> ( ! [A4: set_Pr4934435412358123699_a_nat,B3: produc6551080308567045442rm_a_b,Aa: set_Pr4934435412358123699_a_nat,Ba: produc6551080308567045442rm_a_b] :
( ( member4955916718216803920rm_a_b @ ( produc3065199688681838103rm_a_b @ ( produc8030969961714872974rm_a_b @ Ax @ Ay ) @ ( produc8030969961714872974rm_a_b @ A4 @ B3 ) ) @ ( transi2902453399402325367rm_a_b @ R2 ) )
=> ( ( member4955916718216803920rm_a_b @ ( produc3065199688681838103rm_a_b @ ( produc8030969961714872974rm_a_b @ A4 @ B3 ) @ ( produc8030969961714872974rm_a_b @ Aa @ Ba ) ) @ R2 )
=> ( ( P2 @ A4 @ B3 )
=> ( P2 @ Aa @ Ba ) ) ) )
=> ( P2 @ Bx @ By ) ) ) ) ).
% trancl_induct2
thf(fact_85_trancl__induct2,axiom,
! [Ax: nat,Ay: nat,Bx: nat,By: nat,R2: set_Pr8693737435421807431at_nat,P2: nat > nat > $o] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ Ax @ Ay ) @ ( product_Pair_nat_nat @ Bx @ By ) ) @ ( transi243908449541399842at_nat @ R2 ) )
=> ( ! [A4: nat,B3: nat] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ Ax @ Ay ) @ ( product_Pair_nat_nat @ A4 @ B3 ) ) @ R2 )
=> ( P2 @ A4 @ B3 ) )
=> ( ! [A4: nat,B3: nat,Aa: nat,Ba: nat] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ Ax @ Ay ) @ ( product_Pair_nat_nat @ A4 @ B3 ) ) @ ( transi243908449541399842at_nat @ R2 ) )
=> ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A4 @ B3 ) @ ( product_Pair_nat_nat @ Aa @ Ba ) ) @ R2 )
=> ( ( P2 @ A4 @ B3 )
=> ( P2 @ Aa @ Ba ) ) ) )
=> ( P2 @ Bx @ By ) ) ) ) ).
% trancl_induct2
thf(fact_86_trancl__induct2,axiom,
! [Ax: b,Ay: term_a_b,Bx: b,By: term_a_b,R2: set_Pr5233514982410851781rm_a_b,P2: b > term_a_b > $o] :
( ( member7542685802389602574rm_a_b @ ( produc411707225774285981rm_a_b @ ( produc1437816968797971900rm_a_b @ Ax @ Ay ) @ ( produc1437816968797971900rm_a_b @ Bx @ By ) ) @ ( transi7411496497035545569rm_a_b @ R2 ) )
=> ( ! [A4: b,B3: term_a_b] :
( ( member7542685802389602574rm_a_b @ ( produc411707225774285981rm_a_b @ ( produc1437816968797971900rm_a_b @ Ax @ Ay ) @ ( produc1437816968797971900rm_a_b @ A4 @ B3 ) ) @ R2 )
=> ( P2 @ A4 @ B3 ) )
=> ( ! [A4: b,B3: term_a_b,Aa: b,Ba: term_a_b] :
( ( member7542685802389602574rm_a_b @ ( produc411707225774285981rm_a_b @ ( produc1437816968797971900rm_a_b @ Ax @ Ay ) @ ( produc1437816968797971900rm_a_b @ A4 @ B3 ) ) @ ( transi7411496497035545569rm_a_b @ R2 ) )
=> ( ( member7542685802389602574rm_a_b @ ( produc411707225774285981rm_a_b @ ( produc1437816968797971900rm_a_b @ A4 @ B3 ) @ ( produc1437816968797971900rm_a_b @ Aa @ Ba ) ) @ R2 )
=> ( ( P2 @ A4 @ B3 )
=> ( P2 @ Aa @ Ba ) ) ) )
=> ( P2 @ Bx @ By ) ) ) ) ).
% trancl_induct2
thf(fact_87_trancl__induct2,axiom,
! [Ax: a,Ay: nat,Bx: a,By: nat,R2: set_Pr1811044260758604347_a_nat,P2: a > nat > $o] :
( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ ( product_Pair_a_nat @ Ax @ Ay ) @ ( product_Pair_a_nat @ Bx @ By ) ) @ ( transi6805586507763294364_a_nat @ R2 ) )
=> ( ! [A4: a,B3: nat] :
( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ ( product_Pair_a_nat @ Ax @ Ay ) @ ( product_Pair_a_nat @ A4 @ B3 ) ) @ R2 )
=> ( P2 @ A4 @ B3 ) )
=> ( ! [A4: a,B3: nat,Aa: a,Ba: nat] :
( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ ( product_Pair_a_nat @ Ax @ Ay ) @ ( product_Pair_a_nat @ A4 @ B3 ) ) @ ( transi6805586507763294364_a_nat @ R2 ) )
=> ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ ( product_Pair_a_nat @ A4 @ B3 ) @ ( product_Pair_a_nat @ Aa @ Ba ) ) @ R2 )
=> ( ( P2 @ A4 @ B3 )
=> ( P2 @ Aa @ Ba ) ) ) )
=> ( P2 @ Bx @ By ) ) ) ) ).
% trancl_induct2
thf(fact_88_trancl__induct2,axiom,
! [Ax: term_a_b,Ay: term_a_b,Bx: term_a_b,By: term_a_b,R2: set_Pr2972776593051762503rm_a_b,P2: term_a_b > term_a_b > $o] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ Ax @ Ay ) @ ( produc7020197800436672577rm_a_b @ Bx @ By ) ) @ ( transi9211502839322181930rm_a_b @ R2 ) )
=> ( ! [A4: term_a_b,B3: term_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ Ax @ Ay ) @ ( produc7020197800436672577rm_a_b @ A4 @ B3 ) ) @ R2 )
=> ( P2 @ A4 @ B3 ) )
=> ( ! [A4: term_a_b,B3: term_a_b,Aa: term_a_b,Ba: term_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ Ax @ Ay ) @ ( produc7020197800436672577rm_a_b @ A4 @ B3 ) ) @ ( transi9211502839322181930rm_a_b @ R2 ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ A4 @ B3 ) @ ( produc7020197800436672577rm_a_b @ Aa @ Ba ) ) @ R2 )
=> ( ( P2 @ A4 @ B3 )
=> ( P2 @ Aa @ Ba ) ) ) )
=> ( P2 @ Bx @ By ) ) ) ) ).
% trancl_induct2
thf(fact_89_trancl__mono,axiom,
! [P: produc4523971788519308903rm_a_b,R2: set_Pr2972776593051762503rm_a_b,S: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ P @ ( transi9211502839322181930rm_a_b @ R2 ) )
=> ( ( ord_le2593958535671617255rm_a_b @ R2 @ S )
=> ( member8417600551952982416rm_a_b @ P @ ( transi9211502839322181930rm_a_b @ S ) ) ) ) ).
% trancl_mono
thf(fact_90_trancl__mono,axiom,
! [P: produc51424535725745577rm_a_b,R2: set_Pr8564414093027780873rm_a_b,S: set_Pr8564414093027780873rm_a_b] :
( ( member4405265420456397394rm_a_b @ P @ ( transi4990936492147380931rm_a_b @ R2 ) )
=> ( ( ord_le7771511673815134377rm_a_b @ R2 @ S )
=> ( member4405265420456397394rm_a_b @ P @ ( transi4990936492147380931rm_a_b @ S ) ) ) ) ).
% trancl_mono
thf(fact_91_trancl__mono,axiom,
! [P: product_prod_nat_nat,R2: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ P @ ( transi6264000038957366511cl_nat @ R2 ) )
=> ( ( ord_le3146513528884898305at_nat @ R2 @ S )
=> ( member8440522571783428010at_nat @ P @ ( transi6264000038957366511cl_nat @ S ) ) ) ) ).
% trancl_mono
thf(fact_92_trancl__mono,axiom,
! [P: product_prod_a_a,R2: set_Product_prod_a_a,S: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ P @ ( transitive_trancl_a @ R2 ) )
=> ( ( ord_le746702958409616551od_a_a @ R2 @ S )
=> ( member1426531477525435216od_a_a @ P @ ( transitive_trancl_a @ S ) ) ) ) ).
% trancl_mono
thf(fact_93_trancl__mono,axiom,
! [P: produc357393685978478089rm_a_b,R2: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ P @ ( transi7922773638565587891rm_a_b @ R2 ) )
=> ( ( ord_le118470702582115849rm_a_b @ R2 @ S )
=> ( member5869715511025134514rm_a_b @ P @ ( transi7922773638565587891rm_a_b @ S ) ) ) ) ).
% trancl_mono
thf(fact_94_sig__stepE,axiom,
! [S: term_a_b,T: term_a_b,F: set_Pr4934435412358123699_a_nat,R: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( sig_step_a_b @ F @ R ) )
=> ~ ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ R )
=> ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ S ) @ F )
=> ~ ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ T ) @ F ) ) ) ) ).
% sig_stepE
thf(fact_95_srstepD,axiom,
! [S: term_a_b,T: term_a_b,F: set_Pr4934435412358123699_a_nat,R: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( sig_step_a_b @ F @ ( rstep_a_b @ R ) ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( rstep_a_b @ R ) )
& ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ S ) @ F )
& ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ T ) @ F ) ) ) ).
% srstepD
thf(fact_96_sigstep__trancl__funas_I1_J,axiom,
! [S: term_a_b,T: term_a_b,F: set_Pr4934435412358123699_a_nat,S3: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ S3 ) ) ) )
=> ( ( S != T )
=> ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ S ) @ F ) ) ) ).
% sigstep_trancl_funas(1)
thf(fact_97_sigstep__trancl__funas_I2_J,axiom,
! [S: term_a_b,T: term_a_b,F: set_Pr4934435412358123699_a_nat,S3: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ S3 ) ) ) )
=> ( ( S != T )
=> ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ T ) @ F ) ) ) ).
% sigstep_trancl_funas(2)
thf(fact_98_converse__rtrancl__into__rtrancl,axiom,
! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,R2: set_Pr2972776593051762503rm_a_b,C2: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ R2 )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B @ C2 ) @ ( transi2615809358984392588rm_a_b @ R2 ) )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ C2 ) @ ( transi2615809358984392588rm_a_b @ R2 ) ) ) ) ).
% converse_rtrancl_into_rtrancl
thf(fact_99_converse__rtrancl__into__rtrancl,axiom,
! [A: a,B: a,R2: set_Product_prod_a_a,C2: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ R2 )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B @ C2 ) @ ( transitive_rtrancl_a @ R2 ) )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ C2 ) @ ( transitive_rtrancl_a @ R2 ) ) ) ) ).
% converse_rtrancl_into_rtrancl
thf(fact_100_converse__rtrancl__into__rtrancl,axiom,
! [A: nat,B: nat,R2: set_Pr1261947904930325089at_nat,C2: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R2 )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ C2 ) @ ( transi2905341329935302413cl_nat @ R2 ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ C2 ) @ ( transi2905341329935302413cl_nat @ R2 ) ) ) ) ).
% converse_rtrancl_into_rtrancl
thf(fact_101_converse__rtrancl__into__rtrancl,axiom,
! [A: list_term_a_b,B: list_term_a_b,R2: set_Pr8564414093027780873rm_a_b,C2: list_term_a_b] :
( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ A @ B ) @ R2 )
=> ( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ B @ C2 ) @ ( transi911495901722478305rm_a_b @ R2 ) )
=> ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ A @ C2 ) @ ( transi911495901722478305rm_a_b @ R2 ) ) ) ) ).
% converse_rtrancl_into_rtrancl
thf(fact_102_converse__rtrancl__into__rtrancl,axiom,
! [A: term_a_b,B: term_a_b,R2: set_Pr4386577575007340137rm_a_b,C2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R2 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C2 ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C2 ) @ ( transi7742714808557438673rm_a_b @ R2 ) ) ) ) ).
% converse_rtrancl_into_rtrancl
thf(fact_103_converse__rtrancl__induct,axiom,
! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,R2: set_Pr2972776593051762503rm_a_b,P2: produc357393685978478089rm_a_b > $o] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( transi2615809358984392588rm_a_b @ R2 ) )
=> ( ( P2 @ B )
=> ( ! [Y3: produc357393685978478089rm_a_b,Z: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y3 @ Z ) @ R2 )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Z @ B ) @ ( transi2615809358984392588rm_a_b @ R2 ) )
=> ( ( P2 @ Z )
=> ( P2 @ Y3 ) ) ) )
=> ( P2 @ A ) ) ) ) ).
% converse_rtrancl_induct
thf(fact_104_converse__rtrancl__induct,axiom,
! [A: a,B: a,R2: set_Product_prod_a_a,P2: a > $o] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ ( transitive_rtrancl_a @ R2 ) )
=> ( ( P2 @ B )
=> ( ! [Y3: a,Z: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y3 @ Z ) @ R2 )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Z @ B ) @ ( transitive_rtrancl_a @ R2 ) )
=> ( ( P2 @ Z )
=> ( P2 @ Y3 ) ) ) )
=> ( P2 @ A ) ) ) ) ).
% converse_rtrancl_induct
thf(fact_105_converse__rtrancl__induct,axiom,
! [A: nat,B: nat,R2: set_Pr1261947904930325089at_nat,P2: nat > $o] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( transi2905341329935302413cl_nat @ R2 ) )
=> ( ( P2 @ B )
=> ( ! [Y3: nat,Z: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ Z ) @ R2 )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Z @ B ) @ ( transi2905341329935302413cl_nat @ R2 ) )
=> ( ( P2 @ Z )
=> ( P2 @ Y3 ) ) ) )
=> ( P2 @ A ) ) ) ) ).
% converse_rtrancl_induct
thf(fact_106_converse__rtrancl__induct,axiom,
! [A: list_term_a_b,B: list_term_a_b,R2: set_Pr8564414093027780873rm_a_b,P2: list_term_a_b > $o] :
( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ A @ B ) @ ( transi911495901722478305rm_a_b @ R2 ) )
=> ( ( P2 @ B )
=> ( ! [Y3: list_term_a_b,Z: list_term_a_b] :
( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Y3 @ Z ) @ R2 )
=> ( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Z @ B ) @ ( transi911495901722478305rm_a_b @ R2 ) )
=> ( ( P2 @ Z )
=> ( P2 @ Y3 ) ) ) )
=> ( P2 @ A ) ) ) ) ).
% converse_rtrancl_induct
thf(fact_107_converse__rtrancl__induct,axiom,
! [A: term_a_b,B: term_a_b,R2: set_Pr4386577575007340137rm_a_b,P2: term_a_b > $o] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
=> ( ( P2 @ B )
=> ( ! [Y3: term_a_b,Z: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ Z ) @ R2 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Z @ B ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
=> ( ( P2 @ Z )
=> ( P2 @ Y3 ) ) ) )
=> ( P2 @ A ) ) ) ) ).
% converse_rtrancl_induct
thf(fact_108_converse__rtranclE,axiom,
! [X: produc357393685978478089rm_a_b,Z2: produc357393685978478089rm_a_b,R2: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Z2 ) @ ( transi2615809358984392588rm_a_b @ R2 ) )
=> ( ( X != Z2 )
=> ~ ! [Y3: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y3 ) @ R2 )
=> ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y3 @ Z2 ) @ ( transi2615809358984392588rm_a_b @ R2 ) ) ) ) ) ).
% converse_rtranclE
thf(fact_109_converse__rtranclE,axiom,
! [X: a,Z2: a,R2: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Z2 ) @ ( transitive_rtrancl_a @ R2 ) )
=> ( ( X != Z2 )
=> ~ ! [Y3: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y3 ) @ R2 )
=> ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y3 @ Z2 ) @ ( transitive_rtrancl_a @ R2 ) ) ) ) ) ).
% converse_rtranclE
thf(fact_110_converse__rtranclE,axiom,
! [X: nat,Z2: nat,R2: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Z2 ) @ ( transi2905341329935302413cl_nat @ R2 ) )
=> ( ( X != Z2 )
=> ~ ! [Y3: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y3 ) @ R2 )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ Z2 ) @ ( transi2905341329935302413cl_nat @ R2 ) ) ) ) ) ).
% converse_rtranclE
thf(fact_111_converse__rtranclE,axiom,
! [X: list_term_a_b,Z2: list_term_a_b,R2: set_Pr8564414093027780873rm_a_b] :
( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ X @ Z2 ) @ ( transi911495901722478305rm_a_b @ R2 ) )
=> ( ( X != Z2 )
=> ~ ! [Y3: list_term_a_b] :
( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ X @ Y3 ) @ R2 )
=> ~ ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Y3 @ Z2 ) @ ( transi911495901722478305rm_a_b @ R2 ) ) ) ) ) ).
% converse_rtranclE
thf(fact_112_converse__rtranclE,axiom,
! [X: term_a_b,Z2: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z2 ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
=> ( ( X != Z2 )
=> ~ ! [Y3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y3 ) @ R2 )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ Z2 ) @ ( transi7742714808557438673rm_a_b @ R2 ) ) ) ) ) ).
% converse_rtranclE
thf(fact_113_rtrancl__induct,axiom,
! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,R2: set_Pr2972776593051762503rm_a_b,P2: produc357393685978478089rm_a_b > $o] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( transi2615809358984392588rm_a_b @ R2 ) )
=> ( ( P2 @ A )
=> ( ! [Y3: produc357393685978478089rm_a_b,Z: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ Y3 ) @ ( transi2615809358984392588rm_a_b @ R2 ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y3 @ Z ) @ R2 )
=> ( ( P2 @ Y3 )
=> ( P2 @ Z ) ) ) )
=> ( P2 @ B ) ) ) ) ).
% rtrancl_induct
thf(fact_114_rtrancl__induct,axiom,
! [A: a,B: a,R2: set_Product_prod_a_a,P2: a > $o] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ ( transitive_rtrancl_a @ R2 ) )
=> ( ( P2 @ A )
=> ( ! [Y3: a,Z: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ Y3 ) @ ( transitive_rtrancl_a @ R2 ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y3 @ Z ) @ R2 )
=> ( ( P2 @ Y3 )
=> ( P2 @ Z ) ) ) )
=> ( P2 @ B ) ) ) ) ).
% rtrancl_induct
thf(fact_115_rtrancl__induct,axiom,
! [A: nat,B: nat,R2: set_Pr1261947904930325089at_nat,P2: nat > $o] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( transi2905341329935302413cl_nat @ R2 ) )
=> ( ( P2 @ A )
=> ( ! [Y3: nat,Z: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ Y3 ) @ ( transi2905341329935302413cl_nat @ R2 ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ Z ) @ R2 )
=> ( ( P2 @ Y3 )
=> ( P2 @ Z ) ) ) )
=> ( P2 @ B ) ) ) ) ).
% rtrancl_induct
thf(fact_116_rtrancl__induct,axiom,
! [A: list_term_a_b,B: list_term_a_b,R2: set_Pr8564414093027780873rm_a_b,P2: list_term_a_b > $o] :
( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ A @ B ) @ ( transi911495901722478305rm_a_b @ R2 ) )
=> ( ( P2 @ A )
=> ( ! [Y3: list_term_a_b,Z: list_term_a_b] :
( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ A @ Y3 ) @ ( transi911495901722478305rm_a_b @ R2 ) )
=> ( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Y3 @ Z ) @ R2 )
=> ( ( P2 @ Y3 )
=> ( P2 @ Z ) ) ) )
=> ( P2 @ B ) ) ) ) ).
% rtrancl_induct
thf(fact_117_rtrancl__induct,axiom,
! [A: term_a_b,B: term_a_b,R2: set_Pr4386577575007340137rm_a_b,P2: term_a_b > $o] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
=> ( ( P2 @ A )
=> ( ! [Y3: term_a_b,Z: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ Y3 ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ Z ) @ R2 )
=> ( ( P2 @ Y3 )
=> ( P2 @ Z ) ) ) )
=> ( P2 @ B ) ) ) ) ).
% rtrancl_induct
thf(fact_118_rtrancl__trans,axiom,
! [X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b,R2: set_Pr2972776593051762503rm_a_b,Z2: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ ( transi2615809358984392588rm_a_b @ R2 ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y @ Z2 ) @ ( transi2615809358984392588rm_a_b @ R2 ) )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Z2 ) @ ( transi2615809358984392588rm_a_b @ R2 ) ) ) ) ).
% rtrancl_trans
thf(fact_119_rtrancl__trans,axiom,
! [X: a,Y: a,R2: set_Product_prod_a_a,Z2: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ ( transitive_rtrancl_a @ R2 ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y @ Z2 ) @ ( transitive_rtrancl_a @ R2 ) )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Z2 ) @ ( transitive_rtrancl_a @ R2 ) ) ) ) ).
% rtrancl_trans
thf(fact_120_rtrancl__trans,axiom,
! [X: nat,Y: nat,R2: set_Pr1261947904930325089at_nat,Z2: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( transi2905341329935302413cl_nat @ R2 ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ Z2 ) @ ( transi2905341329935302413cl_nat @ R2 ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Z2 ) @ ( transi2905341329935302413cl_nat @ R2 ) ) ) ) ).
% rtrancl_trans
thf(fact_121_rtrancl__trans,axiom,
! [X: list_term_a_b,Y: list_term_a_b,R2: set_Pr8564414093027780873rm_a_b,Z2: list_term_a_b] :
( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ X @ Y ) @ ( transi911495901722478305rm_a_b @ R2 ) )
=> ( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Y @ Z2 ) @ ( transi911495901722478305rm_a_b @ R2 ) )
=> ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ X @ Z2 ) @ ( transi911495901722478305rm_a_b @ R2 ) ) ) ) ).
% rtrancl_trans
thf(fact_122_rtrancl__trans,axiom,
! [X: term_a_b,Y: term_a_b,R2: set_Pr4386577575007340137rm_a_b,Z2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ Z2 ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z2 ) @ ( transi7742714808557438673rm_a_b @ R2 ) ) ) ) ).
% rtrancl_trans
thf(fact_123_rtranclE,axiom,
! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,R2: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( transi2615809358984392588rm_a_b @ R2 ) )
=> ( ( A != B )
=> ~ ! [Y3: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ Y3 ) @ ( transi2615809358984392588rm_a_b @ R2 ) )
=> ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y3 @ B ) @ R2 ) ) ) ) ).
% rtranclE
thf(fact_124_rtranclE,axiom,
! [A: a,B: a,R2: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ ( transitive_rtrancl_a @ R2 ) )
=> ( ( A != B )
=> ~ ! [Y3: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ Y3 ) @ ( transitive_rtrancl_a @ R2 ) )
=> ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y3 @ B ) @ R2 ) ) ) ) ).
% rtranclE
thf(fact_125_rtranclE,axiom,
! [A: nat,B: nat,R2: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( transi2905341329935302413cl_nat @ R2 ) )
=> ( ( A != B )
=> ~ ! [Y3: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ Y3 ) @ ( transi2905341329935302413cl_nat @ R2 ) )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ B ) @ R2 ) ) ) ) ).
% rtranclE
thf(fact_126_rtranclE,axiom,
! [A: list_term_a_b,B: list_term_a_b,R2: set_Pr8564414093027780873rm_a_b] :
( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ A @ B ) @ ( transi911495901722478305rm_a_b @ R2 ) )
=> ( ( A != B )
=> ~ ! [Y3: list_term_a_b] :
( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ A @ Y3 ) @ ( transi911495901722478305rm_a_b @ R2 ) )
=> ~ ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Y3 @ B ) @ R2 ) ) ) ) ).
% rtranclE
thf(fact_127_rtranclE,axiom,
! [A: term_a_b,B: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
=> ( ( A != B )
=> ~ ! [Y3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ Y3 ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ B ) @ R2 ) ) ) ) ).
% rtranclE
thf(fact_128_rtrancl_Ortrancl__into__rtrancl,axiom,
! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,R2: set_Pr2972776593051762503rm_a_b,C2: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( transi2615809358984392588rm_a_b @ R2 ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B @ C2 ) @ R2 )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ C2 ) @ ( transi2615809358984392588rm_a_b @ R2 ) ) ) ) ).
% rtrancl.rtrancl_into_rtrancl
thf(fact_129_rtrancl_Ortrancl__into__rtrancl,axiom,
! [A: a,B: a,R2: set_Product_prod_a_a,C2: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ ( transitive_rtrancl_a @ R2 ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B @ C2 ) @ R2 )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ C2 ) @ ( transitive_rtrancl_a @ R2 ) ) ) ) ).
% rtrancl.rtrancl_into_rtrancl
thf(fact_130_rtrancl_Ortrancl__into__rtrancl,axiom,
! [A: nat,B: nat,R2: set_Pr1261947904930325089at_nat,C2: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( transi2905341329935302413cl_nat @ R2 ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ C2 ) @ R2 )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ C2 ) @ ( transi2905341329935302413cl_nat @ R2 ) ) ) ) ).
% rtrancl.rtrancl_into_rtrancl
thf(fact_131_rtrancl_Ortrancl__into__rtrancl,axiom,
! [A: list_term_a_b,B: list_term_a_b,R2: set_Pr8564414093027780873rm_a_b,C2: list_term_a_b] :
( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ A @ B ) @ ( transi911495901722478305rm_a_b @ R2 ) )
=> ( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ B @ C2 ) @ R2 )
=> ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ A @ C2 ) @ ( transi911495901722478305rm_a_b @ R2 ) ) ) ) ).
% rtrancl.rtrancl_into_rtrancl
thf(fact_132_rtrancl_Ortrancl__into__rtrancl,axiom,
! [A: term_a_b,B: term_a_b,R2: set_Pr4386577575007340137rm_a_b,C2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C2 ) @ R2 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C2 ) @ ( transi7742714808557438673rm_a_b @ R2 ) ) ) ) ).
% rtrancl.rtrancl_into_rtrancl
thf(fact_133_rtrancl_Ortrancl__refl,axiom,
! [A: produc357393685978478089rm_a_b,R2: set_Pr2972776593051762503rm_a_b] : ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ A ) @ ( transi2615809358984392588rm_a_b @ R2 ) ) ).
% rtrancl.rtrancl_refl
thf(fact_134_rtrancl_Ortrancl__refl,axiom,
! [A: a,R2: set_Product_prod_a_a] : ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ A ) @ ( transitive_rtrancl_a @ R2 ) ) ).
% rtrancl.rtrancl_refl
thf(fact_135_rtrancl_Ortrancl__refl,axiom,
! [A: nat,R2: set_Pr1261947904930325089at_nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ A ) @ ( transi2905341329935302413cl_nat @ R2 ) ) ).
% rtrancl.rtrancl_refl
thf(fact_136_rtrancl_Ortrancl__refl,axiom,
! [A: list_term_a_b,R2: set_Pr8564414093027780873rm_a_b] : ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ A @ A ) @ ( transi911495901722478305rm_a_b @ R2 ) ) ).
% rtrancl.rtrancl_refl
thf(fact_137_rtrancl_Ortrancl__refl,axiom,
! [A: term_a_b,R2: set_Pr4386577575007340137rm_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ A ) @ ( transi7742714808557438673rm_a_b @ R2 ) ) ).
% rtrancl.rtrancl_refl
thf(fact_138_rtrancl_Osimps,axiom,
! [A1: produc357393685978478089rm_a_b,A22: produc357393685978478089rm_a_b,R2: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A1 @ A22 ) @ ( transi2615809358984392588rm_a_b @ R2 ) )
= ( ? [A3: produc357393685978478089rm_a_b] :
( ( A1 = A3 )
& ( A22 = A3 ) )
| ? [A3: produc357393685978478089rm_a_b,B4: produc357393685978478089rm_a_b,C: produc357393685978478089rm_a_b] :
( ( A1 = A3 )
& ( A22 = C )
& ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A3 @ B4 ) @ ( transi2615809358984392588rm_a_b @ R2 ) )
& ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B4 @ C ) @ R2 ) ) ) ) ).
% rtrancl.simps
thf(fact_139_rtrancl_Osimps,axiom,
! [A1: a,A22: a,R2: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A1 @ A22 ) @ ( transitive_rtrancl_a @ R2 ) )
= ( ? [A3: a] :
( ( A1 = A3 )
& ( A22 = A3 ) )
| ? [A3: a,B4: a,C: a] :
( ( A1 = A3 )
& ( A22 = C )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A3 @ B4 ) @ ( transitive_rtrancl_a @ R2 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B4 @ C ) @ R2 ) ) ) ) ).
% rtrancl.simps
thf(fact_140_rtrancl_Osimps,axiom,
! [A1: nat,A22: nat,R2: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A1 @ A22 ) @ ( transi2905341329935302413cl_nat @ R2 ) )
= ( ? [A3: nat] :
( ( A1 = A3 )
& ( A22 = A3 ) )
| ? [A3: nat,B4: nat,C: nat] :
( ( A1 = A3 )
& ( A22 = C )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A3 @ B4 ) @ ( transi2905341329935302413cl_nat @ R2 ) )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B4 @ C ) @ R2 ) ) ) ) ).
% rtrancl.simps
thf(fact_141_rtrancl_Osimps,axiom,
! [A1: list_term_a_b,A22: list_term_a_b,R2: set_Pr8564414093027780873rm_a_b] :
( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ A1 @ A22 ) @ ( transi911495901722478305rm_a_b @ R2 ) )
= ( ? [A3: list_term_a_b] :
( ( A1 = A3 )
& ( A22 = A3 ) )
| ? [A3: list_term_a_b,B4: list_term_a_b,C: list_term_a_b] :
( ( A1 = A3 )
& ( A22 = C )
& ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ A3 @ B4 ) @ ( transi911495901722478305rm_a_b @ R2 ) )
& ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ B4 @ C ) @ R2 ) ) ) ) ).
% rtrancl.simps
thf(fact_142_rtrancl_Osimps,axiom,
! [A1: term_a_b,A22: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A22 ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
= ( ? [A3: term_a_b] :
( ( A1 = A3 )
& ( A22 = A3 ) )
| ? [A3: term_a_b,B4: term_a_b,C: term_a_b] :
( ( A1 = A3 )
& ( A22 = C )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B4 ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B4 @ C ) @ R2 ) ) ) ) ).
% rtrancl.simps
thf(fact_143_mem__Collect__eq,axiom,
! [A: term_a_b,P2: term_a_b > $o] :
( ( member_term_a_b @ A @ ( collect_term_a_b @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_144_mem__Collect__eq,axiom,
! [A: list_term_a_b,P2: list_term_a_b > $o] :
( ( member_list_term_a_b @ A @ ( collec2649484519605854073rm_a_b @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_145_mem__Collect__eq,axiom,
! [A: list_nat,P2: list_nat > $o] :
( ( member_list_nat @ A @ ( collect_list_nat @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_146_mem__Collect__eq,axiom,
! [A: nat,P2: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_147_mem__Collect__eq,axiom,
! [A: product_prod_a_nat,P2: product_prod_a_nat > $o] :
( ( member5724188588386418708_a_nat @ A @ ( collec4464134535221767506_a_nat @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_148_mem__Collect__eq,axiom,
! [A: produc357393685978478089rm_a_b,P2: produc357393685978478089rm_a_b > $o] :
( ( member5869715511025134514rm_a_b @ A @ ( collec99177395925924084rm_a_b @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_149_Collect__mem__eq,axiom,
! [A5: set_term_a_b] :
( ( collect_term_a_b
@ ^ [X4: term_a_b] : ( member_term_a_b @ X4 @ A5 ) )
= A5 ) ).
% Collect_mem_eq
thf(fact_150_Collect__mem__eq,axiom,
! [A5: set_list_term_a_b] :
( ( collec2649484519605854073rm_a_b
@ ^ [X4: list_term_a_b] : ( member_list_term_a_b @ X4 @ A5 ) )
= A5 ) ).
% Collect_mem_eq
thf(fact_151_Collect__mem__eq,axiom,
! [A5: set_list_nat] :
( ( collect_list_nat
@ ^ [X4: list_nat] : ( member_list_nat @ X4 @ A5 ) )
= A5 ) ).
% Collect_mem_eq
thf(fact_152_Collect__mem__eq,axiom,
! [A5: set_nat] :
( ( collect_nat
@ ^ [X4: nat] : ( member_nat @ X4 @ A5 ) )
= A5 ) ).
% Collect_mem_eq
thf(fact_153_Collect__mem__eq,axiom,
! [A5: set_Pr4934435412358123699_a_nat] :
( ( collec4464134535221767506_a_nat
@ ^ [X4: product_prod_a_nat] : ( member5724188588386418708_a_nat @ X4 @ A5 ) )
= A5 ) ).
% Collect_mem_eq
thf(fact_154_Collect__mem__eq,axiom,
! [A5: set_Pr4386577575007340137rm_a_b] :
( ( collec99177395925924084rm_a_b
@ ^ [X4: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ X4 @ A5 ) )
= A5 ) ).
% Collect_mem_eq
thf(fact_155_Collect__cong,axiom,
! [P2: produc357393685978478089rm_a_b > $o,Q: produc357393685978478089rm_a_b > $o] :
( ! [X3: produc357393685978478089rm_a_b] :
( ( P2 @ X3 )
= ( Q @ X3 ) )
=> ( ( collec99177395925924084rm_a_b @ P2 )
= ( collec99177395925924084rm_a_b @ Q ) ) ) ).
% Collect_cong
thf(fact_156_Collect__cong,axiom,
! [P2: product_prod_a_nat > $o,Q: product_prod_a_nat > $o] :
( ! [X3: product_prod_a_nat] :
( ( P2 @ X3 )
= ( Q @ X3 ) )
=> ( ( collec4464134535221767506_a_nat @ P2 )
= ( collec4464134535221767506_a_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_157_rtrancl_Ocases,axiom,
! [A1: produc357393685978478089rm_a_b,A22: produc357393685978478089rm_a_b,R2: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A1 @ A22 ) @ ( transi2615809358984392588rm_a_b @ R2 ) )
=> ( ( A22 != A1 )
=> ~ ! [B3: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A1 @ B3 ) @ ( transi2615809358984392588rm_a_b @ R2 ) )
=> ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B3 @ A22 ) @ R2 ) ) ) ) ).
% rtrancl.cases
thf(fact_158_rtrancl_Ocases,axiom,
! [A1: a,A22: a,R2: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A1 @ A22 ) @ ( transitive_rtrancl_a @ R2 ) )
=> ( ( A22 != A1 )
=> ~ ! [B3: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A1 @ B3 ) @ ( transitive_rtrancl_a @ R2 ) )
=> ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B3 @ A22 ) @ R2 ) ) ) ) ).
% rtrancl.cases
thf(fact_159_rtrancl_Ocases,axiom,
! [A1: nat,A22: nat,R2: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A1 @ A22 ) @ ( transi2905341329935302413cl_nat @ R2 ) )
=> ( ( A22 != A1 )
=> ~ ! [B3: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A1 @ B3 ) @ ( transi2905341329935302413cl_nat @ R2 ) )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B3 @ A22 ) @ R2 ) ) ) ) ).
% rtrancl.cases
thf(fact_160_rtrancl_Ocases,axiom,
! [A1: list_term_a_b,A22: list_term_a_b,R2: set_Pr8564414093027780873rm_a_b] :
( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ A1 @ A22 ) @ ( transi911495901722478305rm_a_b @ R2 ) )
=> ( ( A22 != A1 )
=> ~ ! [B3: list_term_a_b] :
( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ A1 @ B3 ) @ ( transi911495901722478305rm_a_b @ R2 ) )
=> ~ ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ B3 @ A22 ) @ R2 ) ) ) ) ).
% rtrancl.cases
thf(fact_161_rtrancl_Ocases,axiom,
! [A1: term_a_b,A22: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A22 ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
=> ( ( A22 != A1 )
=> ~ ! [B3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ B3 ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B3 @ A22 ) @ R2 ) ) ) ) ).
% rtrancl.cases
thf(fact_162_Term_Oterm_Oexhaust,axiom,
! [Y: term_a_b] :
( ! [X12: b] :
( Y
!= ( var_b_a @ X12 ) )
=> ~ ! [X212: a,X222: list_term_a_b] :
( Y
!= ( fun_a_b @ X212 @ X222 ) ) ) ).
% Term.term.exhaust
thf(fact_163_term_Odistinct_I1_J,axiom,
! [X1: b,X21: a,X22: list_term_a_b] :
( ( var_b_a @ X1 )
!= ( fun_a_b @ X21 @ X22 ) ) ).
% term.distinct(1)
thf(fact_164_srstepsD,axiom,
! [S: term_a_b,T: term_a_b,F: set_Pr4934435412358123699_a_nat,R: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( transi7922773638565587891rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R ) ) ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( transi7922773638565587891rm_a_b @ ( rstep_a_b @ R ) ) )
& ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ S ) @ F )
& ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ T ) @ F ) ) ) ).
% srstepsD
thf(fact_165_prod__induct3,axiom,
! [P2: produc5279506192219892694rm_a_b > $o,X: produc5279506192219892694rm_a_b] :
( ! [A4: set_Pr4934435412358123699_a_nat,B3: b,C3: term_a_b] : ( P2 @ ( produc8030969961714872974rm_a_b @ A4 @ ( produc1437816968797971900rm_a_b @ B3 @ C3 ) ) )
=> ( P2 @ X ) ) ).
% prod_induct3
thf(fact_166_prod__cases3,axiom,
! [Y: produc5279506192219892694rm_a_b] :
~ ! [A4: set_Pr4934435412358123699_a_nat,B3: b,C3: term_a_b] :
( Y
!= ( produc8030969961714872974rm_a_b @ A4 @ ( produc1437816968797971900rm_a_b @ B3 @ C3 ) ) ) ).
% prod_cases3
thf(fact_167_Pair__inject,axiom,
! [A: list_nat,B: term_a_b,A2: list_nat,B2: term_a_b] :
( ( ( produc5151171985953862413rm_a_b @ A @ B )
= ( produc5151171985953862413rm_a_b @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_168_Pair__inject,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: produc6551080308567045442rm_a_b,A2: set_Pr4934435412358123699_a_nat,B2: produc6551080308567045442rm_a_b] :
( ( ( produc8030969961714872974rm_a_b @ A @ B )
= ( produc8030969961714872974rm_a_b @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_169_Pair__inject,axiom,
! [A: nat,B: nat,A2: nat,B2: nat] :
( ( ( product_Pair_nat_nat @ A @ B )
= ( product_Pair_nat_nat @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_170_Pair__inject,axiom,
! [A: b,B: term_a_b,A2: b,B2: term_a_b] :
( ( ( produc1437816968797971900rm_a_b @ A @ B )
= ( produc1437816968797971900rm_a_b @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_171_Pair__inject,axiom,
! [A: a,B: nat,A2: a,B2: nat] :
( ( ( product_Pair_a_nat @ A @ B )
= ( product_Pair_a_nat @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_172_Pair__inject,axiom,
! [A: term_a_b,B: term_a_b,A2: term_a_b,B2: term_a_b] :
( ( ( produc7020197800436672577rm_a_b @ A @ B )
= ( produc7020197800436672577rm_a_b @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_173_prod__cases,axiom,
! [P2: produc7711739908350443733rm_a_b > $o,P: produc7711739908350443733rm_a_b] :
( ! [A4: list_nat,B3: term_a_b] : ( P2 @ ( produc5151171985953862413rm_a_b @ A4 @ B3 ) )
=> ( P2 @ P ) ) ).
% prod_cases
thf(fact_174_prod__cases,axiom,
! [P2: produc5279506192219892694rm_a_b > $o,P: produc5279506192219892694rm_a_b] :
( ! [A4: set_Pr4934435412358123699_a_nat,B3: produc6551080308567045442rm_a_b] : ( P2 @ ( produc8030969961714872974rm_a_b @ A4 @ B3 ) )
=> ( P2 @ P ) ) ).
% prod_cases
thf(fact_175_prod__cases,axiom,
! [P2: product_prod_nat_nat > $o,P: product_prod_nat_nat] :
( ! [A4: nat,B3: nat] : ( P2 @ ( product_Pair_nat_nat @ A4 @ B3 ) )
=> ( P2 @ P ) ) ).
% prod_cases
thf(fact_176_prod__cases,axiom,
! [P2: produc6551080308567045442rm_a_b > $o,P: produc6551080308567045442rm_a_b] :
( ! [A4: b,B3: term_a_b] : ( P2 @ ( produc1437816968797971900rm_a_b @ A4 @ B3 ) )
=> ( P2 @ P ) ) ).
% prod_cases
thf(fact_177_prod__cases,axiom,
! [P2: product_prod_a_nat > $o,P: product_prod_a_nat] :
( ! [A4: a,B3: nat] : ( P2 @ ( product_Pair_a_nat @ A4 @ B3 ) )
=> ( P2 @ P ) ) ).
% prod_cases
thf(fact_178_prod__cases,axiom,
! [P2: produc357393685978478089rm_a_b > $o,P: produc357393685978478089rm_a_b] :
( ! [A4: term_a_b,B3: term_a_b] : ( P2 @ ( produc7020197800436672577rm_a_b @ A4 @ B3 ) )
=> ( P2 @ P ) ) ).
% prod_cases
thf(fact_179_surj__pair,axiom,
! [P: produc7711739908350443733rm_a_b] :
? [X3: list_nat,Y3: term_a_b] :
( P
= ( produc5151171985953862413rm_a_b @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_180_surj__pair,axiom,
! [P: produc5279506192219892694rm_a_b] :
? [X3: set_Pr4934435412358123699_a_nat,Y3: produc6551080308567045442rm_a_b] :
( P
= ( produc8030969961714872974rm_a_b @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_181_surj__pair,axiom,
! [P: product_prod_nat_nat] :
? [X3: nat,Y3: nat] :
( P
= ( product_Pair_nat_nat @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_182_surj__pair,axiom,
! [P: produc6551080308567045442rm_a_b] :
? [X3: b,Y3: term_a_b] :
( P
= ( produc1437816968797971900rm_a_b @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_183_surj__pair,axiom,
! [P: product_prod_a_nat] :
? [X3: a,Y3: nat] :
( P
= ( product_Pair_a_nat @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_184_surj__pair,axiom,
! [P: produc357393685978478089rm_a_b] :
? [X3: term_a_b,Y3: term_a_b] :
( P
= ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_185_old_Oprod_Oexhaust,axiom,
! [Y: produc7711739908350443733rm_a_b] :
~ ! [A4: list_nat,B3: term_a_b] :
( Y
!= ( produc5151171985953862413rm_a_b @ A4 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_186_old_Oprod_Oexhaust,axiom,
! [Y: produc5279506192219892694rm_a_b] :
~ ! [A4: set_Pr4934435412358123699_a_nat,B3: produc6551080308567045442rm_a_b] :
( Y
!= ( produc8030969961714872974rm_a_b @ A4 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_187_old_Oprod_Oexhaust,axiom,
! [Y: product_prod_nat_nat] :
~ ! [A4: nat,B3: nat] :
( Y
!= ( product_Pair_nat_nat @ A4 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_188_old_Oprod_Oexhaust,axiom,
! [Y: produc6551080308567045442rm_a_b] :
~ ! [A4: b,B3: term_a_b] :
( Y
!= ( produc1437816968797971900rm_a_b @ A4 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_189_old_Oprod_Oexhaust,axiom,
! [Y: product_prod_a_nat] :
~ ! [A4: a,B3: nat] :
( Y
!= ( product_Pair_a_nat @ A4 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_190_old_Oprod_Oexhaust,axiom,
! [Y: produc357393685978478089rm_a_b] :
~ ! [A4: term_a_b,B3: term_a_b] :
( Y
!= ( produc7020197800436672577rm_a_b @ A4 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_191_trancl__rtrancl__trancl,axiom,
! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,R2: set_Pr2972776593051762503rm_a_b,C2: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( transi9211502839322181930rm_a_b @ R2 ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B @ C2 ) @ ( transi2615809358984392588rm_a_b @ R2 ) )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ C2 ) @ ( transi9211502839322181930rm_a_b @ R2 ) ) ) ) ).
% trancl_rtrancl_trancl
thf(fact_192_trancl__rtrancl__trancl,axiom,
! [A: a,B: a,R2: set_Product_prod_a_a,C2: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ ( transitive_trancl_a @ R2 ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B @ C2 ) @ ( transitive_rtrancl_a @ R2 ) )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ C2 ) @ ( transitive_trancl_a @ R2 ) ) ) ) ).
% trancl_rtrancl_trancl
thf(fact_193_trancl__rtrancl__trancl,axiom,
! [A: nat,B: nat,R2: set_Pr1261947904930325089at_nat,C2: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( transi6264000038957366511cl_nat @ R2 ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ C2 ) @ ( transi2905341329935302413cl_nat @ R2 ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ C2 ) @ ( transi6264000038957366511cl_nat @ R2 ) ) ) ) ).
% trancl_rtrancl_trancl
thf(fact_194_trancl__rtrancl__trancl,axiom,
! [A: list_term_a_b,B: list_term_a_b,R2: set_Pr8564414093027780873rm_a_b,C2: list_term_a_b] :
( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ A @ B ) @ ( transi4990936492147380931rm_a_b @ R2 ) )
=> ( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ B @ C2 ) @ ( transi911495901722478305rm_a_b @ R2 ) )
=> ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ A @ C2 ) @ ( transi4990936492147380931rm_a_b @ R2 ) ) ) ) ).
% trancl_rtrancl_trancl
thf(fact_195_trancl__rtrancl__trancl,axiom,
! [A: term_a_b,B: term_a_b,R2: set_Pr4386577575007340137rm_a_b,C2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7922773638565587891rm_a_b @ R2 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C2 ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C2 ) @ ( transi7922773638565587891rm_a_b @ R2 ) ) ) ) ).
% trancl_rtrancl_trancl
thf(fact_196_rtrancl__trancl__trancl,axiom,
! [X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b,R2: set_Pr2972776593051762503rm_a_b,Z2: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ ( transi2615809358984392588rm_a_b @ R2 ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y @ Z2 ) @ ( transi9211502839322181930rm_a_b @ R2 ) )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Z2 ) @ ( transi9211502839322181930rm_a_b @ R2 ) ) ) ) ).
% rtrancl_trancl_trancl
thf(fact_197_rtrancl__trancl__trancl,axiom,
! [X: a,Y: a,R2: set_Product_prod_a_a,Z2: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ ( transitive_rtrancl_a @ R2 ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y @ Z2 ) @ ( transitive_trancl_a @ R2 ) )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Z2 ) @ ( transitive_trancl_a @ R2 ) ) ) ) ).
% rtrancl_trancl_trancl
thf(fact_198_rtrancl__trancl__trancl,axiom,
! [X: nat,Y: nat,R2: set_Pr1261947904930325089at_nat,Z2: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( transi2905341329935302413cl_nat @ R2 ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ Z2 ) @ ( transi6264000038957366511cl_nat @ R2 ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Z2 ) @ ( transi6264000038957366511cl_nat @ R2 ) ) ) ) ).
% rtrancl_trancl_trancl
thf(fact_199_rtrancl__trancl__trancl,axiom,
! [X: list_term_a_b,Y: list_term_a_b,R2: set_Pr8564414093027780873rm_a_b,Z2: list_term_a_b] :
( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ X @ Y ) @ ( transi911495901722478305rm_a_b @ R2 ) )
=> ( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Y @ Z2 ) @ ( transi4990936492147380931rm_a_b @ R2 ) )
=> ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ X @ Z2 ) @ ( transi4990936492147380931rm_a_b @ R2 ) ) ) ) ).
% rtrancl_trancl_trancl
thf(fact_200_rtrancl__trancl__trancl,axiom,
! [X: term_a_b,Y: term_a_b,R2: set_Pr4386577575007340137rm_a_b,Z2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ Z2 ) @ ( transi7922773638565587891rm_a_b @ R2 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z2 ) @ ( transi7922773638565587891rm_a_b @ R2 ) ) ) ) ).
% rtrancl_trancl_trancl
thf(fact_201_rtrancl__into__trancl2,axiom,
! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,R2: set_Pr2972776593051762503rm_a_b,C2: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ R2 )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B @ C2 ) @ ( transi2615809358984392588rm_a_b @ R2 ) )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ C2 ) @ ( transi9211502839322181930rm_a_b @ R2 ) ) ) ) ).
% rtrancl_into_trancl2
thf(fact_202_rtrancl__into__trancl2,axiom,
! [A: a,B: a,R2: set_Product_prod_a_a,C2: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ R2 )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B @ C2 ) @ ( transitive_rtrancl_a @ R2 ) )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ C2 ) @ ( transitive_trancl_a @ R2 ) ) ) ) ).
% rtrancl_into_trancl2
thf(fact_203_rtrancl__into__trancl2,axiom,
! [A: nat,B: nat,R2: set_Pr1261947904930325089at_nat,C2: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R2 )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ C2 ) @ ( transi2905341329935302413cl_nat @ R2 ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ C2 ) @ ( transi6264000038957366511cl_nat @ R2 ) ) ) ) ).
% rtrancl_into_trancl2
thf(fact_204_rtrancl__into__trancl2,axiom,
! [A: list_term_a_b,B: list_term_a_b,R2: set_Pr8564414093027780873rm_a_b,C2: list_term_a_b] :
( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ A @ B ) @ R2 )
=> ( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ B @ C2 ) @ ( transi911495901722478305rm_a_b @ R2 ) )
=> ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ A @ C2 ) @ ( transi4990936492147380931rm_a_b @ R2 ) ) ) ) ).
% rtrancl_into_trancl2
thf(fact_205_rtrancl__into__trancl2,axiom,
! [A: term_a_b,B: term_a_b,R2: set_Pr4386577575007340137rm_a_b,C2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R2 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C2 ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C2 ) @ ( transi7922773638565587891rm_a_b @ R2 ) ) ) ) ).
% rtrancl_into_trancl2
thf(fact_206_rtrancl__into__trancl1,axiom,
! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,R2: set_Pr2972776593051762503rm_a_b,C2: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( transi2615809358984392588rm_a_b @ R2 ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B @ C2 ) @ R2 )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ C2 ) @ ( transi9211502839322181930rm_a_b @ R2 ) ) ) ) ).
% rtrancl_into_trancl1
thf(fact_207_rtrancl__into__trancl1,axiom,
! [A: a,B: a,R2: set_Product_prod_a_a,C2: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ ( transitive_rtrancl_a @ R2 ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B @ C2 ) @ R2 )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ C2 ) @ ( transitive_trancl_a @ R2 ) ) ) ) ).
% rtrancl_into_trancl1
thf(fact_208_rtrancl__into__trancl1,axiom,
! [A: nat,B: nat,R2: set_Pr1261947904930325089at_nat,C2: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( transi2905341329935302413cl_nat @ R2 ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ C2 ) @ R2 )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ C2 ) @ ( transi6264000038957366511cl_nat @ R2 ) ) ) ) ).
% rtrancl_into_trancl1
thf(fact_209_rtrancl__into__trancl1,axiom,
! [A: list_term_a_b,B: list_term_a_b,R2: set_Pr8564414093027780873rm_a_b,C2: list_term_a_b] :
( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ A @ B ) @ ( transi911495901722478305rm_a_b @ R2 ) )
=> ( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ B @ C2 ) @ R2 )
=> ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ A @ C2 ) @ ( transi4990936492147380931rm_a_b @ R2 ) ) ) ) ).
% rtrancl_into_trancl1
thf(fact_210_rtrancl__into__trancl1,axiom,
! [A: term_a_b,B: term_a_b,R2: set_Pr4386577575007340137rm_a_b,C2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C2 ) @ R2 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C2 ) @ ( transi7922773638565587891rm_a_b @ R2 ) ) ) ) ).
% rtrancl_into_trancl1
thf(fact_211_rtrancl__eq__or__trancl,axiom,
! [X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b,R3: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ ( transi2615809358984392588rm_a_b @ R3 ) )
= ( ( X = Y )
| ( ( X != Y )
& ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ ( transi9211502839322181930rm_a_b @ R3 ) ) ) ) ) ).
% rtrancl_eq_or_trancl
thf(fact_212_rtrancl__eq__or__trancl,axiom,
! [X: a,Y: a,R3: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ ( transitive_rtrancl_a @ R3 ) )
= ( ( X = Y )
| ( ( X != Y )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ ( transitive_trancl_a @ R3 ) ) ) ) ) ).
% rtrancl_eq_or_trancl
thf(fact_213_rtrancl__eq__or__trancl,axiom,
! [X: nat,Y: nat,R3: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( transi2905341329935302413cl_nat @ R3 ) )
= ( ( X = Y )
| ( ( X != Y )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( transi6264000038957366511cl_nat @ R3 ) ) ) ) ) ).
% rtrancl_eq_or_trancl
thf(fact_214_rtrancl__eq__or__trancl,axiom,
! [X: list_term_a_b,Y: list_term_a_b,R3: set_Pr8564414093027780873rm_a_b] :
( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ X @ Y ) @ ( transi911495901722478305rm_a_b @ R3 ) )
= ( ( X = Y )
| ( ( X != Y )
& ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ X @ Y ) @ ( transi4990936492147380931rm_a_b @ R3 ) ) ) ) ) ).
% rtrancl_eq_or_trancl
thf(fact_215_rtrancl__eq__or__trancl,axiom,
! [X: term_a_b,Y: term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ R3 ) )
= ( ( X = Y )
| ( ( X != Y )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7922773638565587891rm_a_b @ R3 ) ) ) ) ) ).
% rtrancl_eq_or_trancl
thf(fact_216_trancl__into__rtrancl,axiom,
! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,R2: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( transi9211502839322181930rm_a_b @ R2 ) )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( transi2615809358984392588rm_a_b @ R2 ) ) ) ).
% trancl_into_rtrancl
thf(fact_217_trancl__into__rtrancl,axiom,
! [A: a,B: a,R2: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ ( transitive_trancl_a @ R2 ) )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ ( transitive_rtrancl_a @ R2 ) ) ) ).
% trancl_into_rtrancl
thf(fact_218_trancl__into__rtrancl,axiom,
! [A: nat,B: nat,R2: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( transi6264000038957366511cl_nat @ R2 ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( transi2905341329935302413cl_nat @ R2 ) ) ) ).
% trancl_into_rtrancl
thf(fact_219_trancl__into__rtrancl,axiom,
! [A: list_term_a_b,B: list_term_a_b,R2: set_Pr8564414093027780873rm_a_b] :
( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ A @ B ) @ ( transi4990936492147380931rm_a_b @ R2 ) )
=> ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ A @ B ) @ ( transi911495901722478305rm_a_b @ R2 ) ) ) ).
% trancl_into_rtrancl
thf(fact_220_trancl__into__rtrancl,axiom,
! [A: term_a_b,B: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7922773638565587891rm_a_b @ R2 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ R2 ) ) ) ).
% trancl_into_rtrancl
thf(fact_221_tranclD2,axiom,
! [X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b,R3: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ ( transi9211502839322181930rm_a_b @ R3 ) )
=> ? [Z: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Z ) @ ( transi2615809358984392588rm_a_b @ R3 ) )
& ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Z @ Y ) @ R3 ) ) ) ).
% tranclD2
thf(fact_222_tranclD2,axiom,
! [X: a,Y: a,R3: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ ( transitive_trancl_a @ R3 ) )
=> ? [Z: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Z ) @ ( transitive_rtrancl_a @ R3 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Z @ Y ) @ R3 ) ) ) ).
% tranclD2
thf(fact_223_tranclD2,axiom,
! [X: nat,Y: nat,R3: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( transi6264000038957366511cl_nat @ R3 ) )
=> ? [Z: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Z ) @ ( transi2905341329935302413cl_nat @ R3 ) )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Z @ Y ) @ R3 ) ) ) ).
% tranclD2
thf(fact_224_tranclD2,axiom,
! [X: list_term_a_b,Y: list_term_a_b,R3: set_Pr8564414093027780873rm_a_b] :
( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ X @ Y ) @ ( transi4990936492147380931rm_a_b @ R3 ) )
=> ? [Z: list_term_a_b] :
( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ X @ Z ) @ ( transi911495901722478305rm_a_b @ R3 ) )
& ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Z @ Y ) @ R3 ) ) ) ).
% tranclD2
thf(fact_225_tranclD2,axiom,
! [X: term_a_b,Y: term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7922773638565587891rm_a_b @ R3 ) )
=> ? [Z: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z ) @ ( transi7742714808557438673rm_a_b @ R3 ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Z @ Y ) @ R3 ) ) ) ).
% tranclD2
thf(fact_226_rtranclD,axiom,
! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,R3: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( transi2615809358984392588rm_a_b @ R3 ) )
=> ( ( A = B )
| ( ( A != B )
& ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( transi9211502839322181930rm_a_b @ R3 ) ) ) ) ) ).
% rtranclD
thf(fact_227_rtranclD,axiom,
! [A: a,B: a,R3: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ ( transitive_rtrancl_a @ R3 ) )
=> ( ( A = B )
| ( ( A != B )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ ( transitive_trancl_a @ R3 ) ) ) ) ) ).
% rtranclD
thf(fact_228_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_229_rtranclD,axiom,
! [A: list_term_a_b,B: list_term_a_b,R3: set_Pr8564414093027780873rm_a_b] :
( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ A @ B ) @ ( transi911495901722478305rm_a_b @ R3 ) )
=> ( ( A = B )
| ( ( A != B )
& ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ A @ B ) @ ( transi4990936492147380931rm_a_b @ R3 ) ) ) ) ) ).
% rtranclD
thf(fact_230_rtranclD,axiom,
! [A: term_a_b,B: term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ R3 ) )
=> ( ( A = B )
| ( ( A != B )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7922773638565587891rm_a_b @ R3 ) ) ) ) ) ).
% rtranclD
thf(fact_231_tranclD,axiom,
! [X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b,R3: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ ( transi9211502839322181930rm_a_b @ R3 ) )
=> ? [Z: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Z ) @ R3 )
& ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Z @ Y ) @ ( transi2615809358984392588rm_a_b @ R3 ) ) ) ) ).
% tranclD
thf(fact_232_tranclD,axiom,
! [X: a,Y: a,R3: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ ( transitive_trancl_a @ R3 ) )
=> ? [Z: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Z ) @ R3 )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Z @ Y ) @ ( transitive_rtrancl_a @ R3 ) ) ) ) ).
% tranclD
thf(fact_233_tranclD,axiom,
! [X: nat,Y: nat,R3: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( transi6264000038957366511cl_nat @ R3 ) )
=> ? [Z: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Z ) @ R3 )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Z @ Y ) @ ( transi2905341329935302413cl_nat @ R3 ) ) ) ) ).
% tranclD
thf(fact_234_tranclD,axiom,
! [X: list_term_a_b,Y: list_term_a_b,R3: set_Pr8564414093027780873rm_a_b] :
( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ X @ Y ) @ ( transi4990936492147380931rm_a_b @ R3 ) )
=> ? [Z: list_term_a_b] :
( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ X @ Z ) @ R3 )
& ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Z @ Y ) @ ( transi911495901722478305rm_a_b @ R3 ) ) ) ) ).
% tranclD
thf(fact_235_tranclD,axiom,
! [X: term_a_b,Y: term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7922773638565587891rm_a_b @ R3 ) )
=> ? [Z: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z ) @ R3 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Z @ Y ) @ ( transi7742714808557438673rm_a_b @ R3 ) ) ) ) ).
% tranclD
thf(fact_236_r__into__trancl_H,axiom,
! [P: produc4523971788519308903rm_a_b,R2: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ P @ R2 )
=> ( member8417600551952982416rm_a_b @ P @ ( transi9211502839322181930rm_a_b @ R2 ) ) ) ).
% r_into_trancl'
thf(fact_237_r__into__trancl_H,axiom,
! [P: product_prod_a_a,R2: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ P @ R2 )
=> ( member1426531477525435216od_a_a @ P @ ( transitive_trancl_a @ R2 ) ) ) ).
% r_into_trancl'
thf(fact_238_r__into__trancl_H,axiom,
! [P: product_prod_nat_nat,R2: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ P @ R2 )
=> ( member8440522571783428010at_nat @ P @ ( transi6264000038957366511cl_nat @ R2 ) ) ) ).
% r_into_trancl'
thf(fact_239_r__into__trancl_H,axiom,
! [P: produc357393685978478089rm_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ P @ R2 )
=> ( member5869715511025134514rm_a_b @ P @ ( transi7922773638565587891rm_a_b @ R2 ) ) ) ).
% r_into_trancl'
thf(fact_240_converse__trancl__induct,axiom,
! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,R2: set_Pr2972776593051762503rm_a_b,P2: produc357393685978478089rm_a_b > $o] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( transi9211502839322181930rm_a_b @ R2 ) )
=> ( ! [Y3: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y3 @ B ) @ R2 )
=> ( P2 @ Y3 ) )
=> ( ! [Y3: produc357393685978478089rm_a_b,Z: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y3 @ Z ) @ R2 )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Z @ B ) @ ( transi9211502839322181930rm_a_b @ R2 ) )
=> ( ( P2 @ Z )
=> ( P2 @ Y3 ) ) ) )
=> ( P2 @ A ) ) ) ) ).
% converse_trancl_induct
thf(fact_241_converse__trancl__induct,axiom,
! [A: a,B: a,R2: set_Product_prod_a_a,P2: a > $o] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ ( transitive_trancl_a @ R2 ) )
=> ( ! [Y3: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y3 @ B ) @ R2 )
=> ( P2 @ Y3 ) )
=> ( ! [Y3: a,Z: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y3 @ Z ) @ R2 )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Z @ B ) @ ( transitive_trancl_a @ R2 ) )
=> ( ( P2 @ Z )
=> ( P2 @ Y3 ) ) ) )
=> ( P2 @ A ) ) ) ) ).
% converse_trancl_induct
thf(fact_242_converse__trancl__induct,axiom,
! [A: nat,B: nat,R2: set_Pr1261947904930325089at_nat,P2: nat > $o] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( transi6264000038957366511cl_nat @ R2 ) )
=> ( ! [Y3: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ B ) @ R2 )
=> ( P2 @ Y3 ) )
=> ( ! [Y3: nat,Z: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ Z ) @ R2 )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Z @ B ) @ ( transi6264000038957366511cl_nat @ R2 ) )
=> ( ( P2 @ Z )
=> ( P2 @ Y3 ) ) ) )
=> ( P2 @ A ) ) ) ) ).
% converse_trancl_induct
thf(fact_243_converse__trancl__induct,axiom,
! [A: term_a_b,B: term_a_b,R2: set_Pr4386577575007340137rm_a_b,P2: term_a_b > $o] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7922773638565587891rm_a_b @ R2 ) )
=> ( ! [Y3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ B ) @ R2 )
=> ( P2 @ Y3 ) )
=> ( ! [Y3: term_a_b,Z: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ Z ) @ R2 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Z @ B ) @ ( transi7922773638565587891rm_a_b @ R2 ) )
=> ( ( P2 @ Z )
=> ( P2 @ Y3 ) ) ) )
=> ( P2 @ A ) ) ) ) ).
% converse_trancl_induct
thf(fact_244_trancl__trans__induct,axiom,
! [X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b,R2: set_Pr2972776593051762503rm_a_b,P2: produc357393685978478089rm_a_b > produc357393685978478089rm_a_b > $o] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ ( transi9211502839322181930rm_a_b @ R2 ) )
=> ( ! [X3: produc357393685978478089rm_a_b,Y3: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X3 @ Y3 ) @ R2 )
=> ( P2 @ X3 @ Y3 ) )
=> ( ! [X3: produc357393685978478089rm_a_b,Y3: produc357393685978478089rm_a_b,Z: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X3 @ Y3 ) @ ( transi9211502839322181930rm_a_b @ R2 ) )
=> ( ( P2 @ X3 @ Y3 )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y3 @ Z ) @ ( transi9211502839322181930rm_a_b @ R2 ) )
=> ( ( P2 @ Y3 @ Z )
=> ( P2 @ X3 @ Z ) ) ) ) )
=> ( P2 @ X @ Y ) ) ) ) ).
% trancl_trans_induct
thf(fact_245_trancl__trans__induct,axiom,
! [X: a,Y: a,R2: set_Product_prod_a_a,P2: a > a > $o] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ ( transitive_trancl_a @ R2 ) )
=> ( ! [X3: a,Y3: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y3 ) @ R2 )
=> ( P2 @ X3 @ Y3 ) )
=> ( ! [X3: a,Y3: a,Z: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y3 ) @ ( transitive_trancl_a @ R2 ) )
=> ( ( P2 @ X3 @ Y3 )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y3 @ Z ) @ ( transitive_trancl_a @ R2 ) )
=> ( ( P2 @ Y3 @ Z )
=> ( P2 @ X3 @ Z ) ) ) ) )
=> ( P2 @ X @ Y ) ) ) ) ).
% trancl_trans_induct
thf(fact_246_trancl__trans__induct,axiom,
! [X: nat,Y: nat,R2: set_Pr1261947904930325089at_nat,P2: nat > nat > $o] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( transi6264000038957366511cl_nat @ R2 ) )
=> ( ! [X3: nat,Y3: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R2 )
=> ( P2 @ X3 @ Y3 ) )
=> ( ! [X3: nat,Y3: nat,Z: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ ( transi6264000038957366511cl_nat @ R2 ) )
=> ( ( P2 @ X3 @ Y3 )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ Z ) @ ( transi6264000038957366511cl_nat @ R2 ) )
=> ( ( P2 @ Y3 @ Z )
=> ( P2 @ X3 @ Z ) ) ) ) )
=> ( P2 @ X @ Y ) ) ) ) ).
% trancl_trans_induct
thf(fact_247_trancl__trans__induct,axiom,
! [X: term_a_b,Y: term_a_b,R2: set_Pr4386577575007340137rm_a_b,P2: term_a_b > term_a_b > $o] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7922773638565587891rm_a_b @ R2 ) )
=> ( ! [X3: term_a_b,Y3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ R2 )
=> ( P2 @ X3 @ Y3 ) )
=> ( ! [X3: term_a_b,Y3: term_a_b,Z: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ ( transi7922773638565587891rm_a_b @ R2 ) )
=> ( ( P2 @ X3 @ Y3 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ Z ) @ ( transi7922773638565587891rm_a_b @ R2 ) )
=> ( ( P2 @ Y3 @ Z )
=> ( P2 @ X3 @ Z ) ) ) ) )
=> ( P2 @ X @ Y ) ) ) ) ).
% trancl_trans_induct
thf(fact_248_trancl__into__trancl2,axiom,
! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,R2: set_Pr2972776593051762503rm_a_b,C2: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ R2 )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B @ C2 ) @ ( transi9211502839322181930rm_a_b @ R2 ) )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ C2 ) @ ( transi9211502839322181930rm_a_b @ R2 ) ) ) ) ).
% trancl_into_trancl2
thf(fact_249_trancl__into__trancl2,axiom,
! [A: a,B: a,R2: set_Product_prod_a_a,C2: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ R2 )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B @ C2 ) @ ( transitive_trancl_a @ R2 ) )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ C2 ) @ ( transitive_trancl_a @ R2 ) ) ) ) ).
% trancl_into_trancl2
thf(fact_250_trancl__into__trancl2,axiom,
! [A: nat,B: nat,R2: set_Pr1261947904930325089at_nat,C2: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R2 )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ C2 ) @ ( transi6264000038957366511cl_nat @ R2 ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ C2 ) @ ( transi6264000038957366511cl_nat @ R2 ) ) ) ) ).
% trancl_into_trancl2
thf(fact_251_trancl__into__trancl2,axiom,
! [A: term_a_b,B: term_a_b,R2: set_Pr4386577575007340137rm_a_b,C2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R2 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C2 ) @ ( transi7922773638565587891rm_a_b @ R2 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C2 ) @ ( transi7922773638565587891rm_a_b @ R2 ) ) ) ) ).
% trancl_into_trancl2
thf(fact_252_Transitive__Closure_Otrancl__into__trancl,axiom,
! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,R2: set_Pr2972776593051762503rm_a_b,C2: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( transi9211502839322181930rm_a_b @ R2 ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B @ C2 ) @ R2 )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ C2 ) @ ( transi9211502839322181930rm_a_b @ R2 ) ) ) ) ).
% Transitive_Closure.trancl_into_trancl
thf(fact_253_Transitive__Closure_Otrancl__into__trancl,axiom,
! [A: a,B: a,R2: set_Product_prod_a_a,C2: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ ( transitive_trancl_a @ R2 ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B @ C2 ) @ R2 )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ C2 ) @ ( transitive_trancl_a @ R2 ) ) ) ) ).
% Transitive_Closure.trancl_into_trancl
thf(fact_254_Transitive__Closure_Otrancl__into__trancl,axiom,
! [A: nat,B: nat,R2: set_Pr1261947904930325089at_nat,C2: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( transi6264000038957366511cl_nat @ R2 ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ C2 ) @ R2 )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ C2 ) @ ( transi6264000038957366511cl_nat @ R2 ) ) ) ) ).
% Transitive_Closure.trancl_into_trancl
thf(fact_255_Transitive__Closure_Otrancl__into__trancl,axiom,
! [A: term_a_b,B: term_a_b,R2: set_Pr4386577575007340137rm_a_b,C2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7922773638565587891rm_a_b @ R2 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C2 ) @ R2 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C2 ) @ ( transi7922773638565587891rm_a_b @ R2 ) ) ) ) ).
% Transitive_Closure.trancl_into_trancl
thf(fact_256_irrefl__trancl__rD,axiom,
! [R2: set_Pr2972776593051762503rm_a_b,X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b] :
( ! [X3: produc357393685978478089rm_a_b] :
~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X3 @ X3 ) @ ( transi9211502839322181930rm_a_b @ R2 ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ R2 )
=> ( X != Y ) ) ) ).
% irrefl_trancl_rD
thf(fact_257_irrefl__trancl__rD,axiom,
! [R2: set_Product_prod_a_a,X: a,Y: a] :
( ! [X3: a] :
~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ X3 ) @ ( transitive_trancl_a @ R2 ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R2 )
=> ( X != Y ) ) ) ).
% irrefl_trancl_rD
thf(fact_258_irrefl__trancl__rD,axiom,
! [R2: set_Pr1261947904930325089at_nat,X: nat,Y: nat] :
( ! [X3: nat] :
~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ X3 ) @ ( transi6264000038957366511cl_nat @ R2 ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R2 )
=> ( X != Y ) ) ) ).
% irrefl_trancl_rD
thf(fact_259_irrefl__trancl__rD,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,X: term_a_b,Y: term_a_b] :
( ! [X3: term_a_b] :
~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ X3 ) @ ( transi7922773638565587891rm_a_b @ R2 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ R2 )
=> ( X != Y ) ) ) ).
% irrefl_trancl_rD
thf(fact_260_converse__tranclE,axiom,
! [X: produc357393685978478089rm_a_b,Z2: produc357393685978478089rm_a_b,R2: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Z2 ) @ ( transi9211502839322181930rm_a_b @ R2 ) )
=> ( ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Z2 ) @ R2 )
=> ~ ! [Y3: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y3 ) @ R2 )
=> ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y3 @ Z2 ) @ ( transi9211502839322181930rm_a_b @ R2 ) ) ) ) ) ).
% converse_tranclE
thf(fact_261_converse__tranclE,axiom,
! [X: a,Z2: a,R2: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Z2 ) @ ( transitive_trancl_a @ R2 ) )
=> ( ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Z2 ) @ R2 )
=> ~ ! [Y3: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y3 ) @ R2 )
=> ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y3 @ Z2 ) @ ( transitive_trancl_a @ R2 ) ) ) ) ) ).
% converse_tranclE
thf(fact_262_converse__tranclE,axiom,
! [X: nat,Z2: nat,R2: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Z2 ) @ ( transi6264000038957366511cl_nat @ R2 ) )
=> ( ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Z2 ) @ R2 )
=> ~ ! [Y3: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y3 ) @ R2 )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ Z2 ) @ ( transi6264000038957366511cl_nat @ R2 ) ) ) ) ) ).
% converse_tranclE
thf(fact_263_converse__tranclE,axiom,
! [X: term_a_b,Z2: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z2 ) @ ( transi7922773638565587891rm_a_b @ R2 ) )
=> ( ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z2 ) @ R2 )
=> ~ ! [Y3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y3 ) @ R2 )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ Z2 ) @ ( transi7922773638565587891rm_a_b @ R2 ) ) ) ) ) ).
% converse_tranclE
thf(fact_264_r__r__into__trancl,axiom,
! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,R3: set_Pr2972776593051762503rm_a_b,C2: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ R3 )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B @ C2 ) @ R3 )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ C2 ) @ ( transi9211502839322181930rm_a_b @ R3 ) ) ) ) ).
% r_r_into_trancl
thf(fact_265_r__r__into__trancl,axiom,
! [A: a,B: a,R3: set_Product_prod_a_a,C2: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ R3 )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B @ C2 ) @ R3 )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ C2 ) @ ( transitive_trancl_a @ R3 ) ) ) ) ).
% r_r_into_trancl
thf(fact_266_r__r__into__trancl,axiom,
! [A: nat,B: nat,R3: set_Pr1261947904930325089at_nat,C2: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R3 )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ C2 ) @ R3 )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ C2 ) @ ( transi6264000038957366511cl_nat @ R3 ) ) ) ) ).
% r_r_into_trancl
thf(fact_267_r__r__into__trancl,axiom,
! [A: term_a_b,B: term_a_b,R3: set_Pr4386577575007340137rm_a_b,C2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R3 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C2 ) @ R3 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C2 ) @ ( transi7922773638565587891rm_a_b @ R3 ) ) ) ) ).
% r_r_into_trancl
thf(fact_268_trancl__induct,axiom,
! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,R2: set_Pr2972776593051762503rm_a_b,P2: produc357393685978478089rm_a_b > $o] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( transi9211502839322181930rm_a_b @ R2 ) )
=> ( ! [Y3: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ Y3 ) @ R2 )
=> ( P2 @ Y3 ) )
=> ( ! [Y3: produc357393685978478089rm_a_b,Z: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ Y3 ) @ ( transi9211502839322181930rm_a_b @ R2 ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y3 @ Z ) @ R2 )
=> ( ( P2 @ Y3 )
=> ( P2 @ Z ) ) ) )
=> ( P2 @ B ) ) ) ) ).
% trancl_induct
thf(fact_269_trancl__induct,axiom,
! [A: a,B: a,R2: set_Product_prod_a_a,P2: a > $o] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ ( transitive_trancl_a @ R2 ) )
=> ( ! [Y3: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ Y3 ) @ R2 )
=> ( P2 @ Y3 ) )
=> ( ! [Y3: a,Z: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ Y3 ) @ ( transitive_trancl_a @ R2 ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y3 @ Z ) @ R2 )
=> ( ( P2 @ Y3 )
=> ( P2 @ Z ) ) ) )
=> ( P2 @ B ) ) ) ) ).
% trancl_induct
thf(fact_270_trancl__induct,axiom,
! [A: nat,B: nat,R2: set_Pr1261947904930325089at_nat,P2: nat > $o] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( transi6264000038957366511cl_nat @ R2 ) )
=> ( ! [Y3: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ Y3 ) @ R2 )
=> ( P2 @ Y3 ) )
=> ( ! [Y3: nat,Z: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ Y3 ) @ ( transi6264000038957366511cl_nat @ R2 ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ Z ) @ R2 )
=> ( ( P2 @ Y3 )
=> ( P2 @ Z ) ) ) )
=> ( P2 @ B ) ) ) ) ).
% trancl_induct
thf(fact_271_trancl__induct,axiom,
! [A: term_a_b,B: term_a_b,R2: set_Pr4386577575007340137rm_a_b,P2: term_a_b > $o] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7922773638565587891rm_a_b @ R2 ) )
=> ( ! [Y3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ Y3 ) @ R2 )
=> ( P2 @ Y3 ) )
=> ( ! [Y3: term_a_b,Z: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ Y3 ) @ ( transi7922773638565587891rm_a_b @ R2 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ Z ) @ R2 )
=> ( ( P2 @ Y3 )
=> ( P2 @ Z ) ) ) )
=> ( P2 @ B ) ) ) ) ).
% trancl_induct
thf(fact_272_trancl__trans,axiom,
! [X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b,R2: set_Pr2972776593051762503rm_a_b,Z2: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ ( transi9211502839322181930rm_a_b @ R2 ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y @ Z2 ) @ ( transi9211502839322181930rm_a_b @ R2 ) )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Z2 ) @ ( transi9211502839322181930rm_a_b @ R2 ) ) ) ) ).
% trancl_trans
thf(fact_273_trancl__trans,axiom,
! [X: a,Y: a,R2: set_Product_prod_a_a,Z2: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ ( transitive_trancl_a @ R2 ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y @ Z2 ) @ ( transitive_trancl_a @ R2 ) )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Z2 ) @ ( transitive_trancl_a @ R2 ) ) ) ) ).
% trancl_trans
thf(fact_274_trancl__trans,axiom,
! [X: nat,Y: nat,R2: set_Pr1261947904930325089at_nat,Z2: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( transi6264000038957366511cl_nat @ R2 ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ Z2 ) @ ( transi6264000038957366511cl_nat @ R2 ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Z2 ) @ ( transi6264000038957366511cl_nat @ R2 ) ) ) ) ).
% trancl_trans
thf(fact_275_trancl__trans,axiom,
! [X: term_a_b,Y: term_a_b,R2: set_Pr4386577575007340137rm_a_b,Z2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7922773638565587891rm_a_b @ R2 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ Z2 ) @ ( transi7922773638565587891rm_a_b @ R2 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z2 ) @ ( transi7922773638565587891rm_a_b @ R2 ) ) ) ) ).
% trancl_trans
thf(fact_276_tranclE,axiom,
! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,R2: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( transi9211502839322181930rm_a_b @ R2 ) )
=> ( ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ R2 )
=> ~ ! [C3: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ C3 ) @ ( transi9211502839322181930rm_a_b @ R2 ) )
=> ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ C3 @ B ) @ R2 ) ) ) ) ).
% tranclE
thf(fact_277_tranclE,axiom,
! [A: a,B: a,R2: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ ( transitive_trancl_a @ R2 ) )
=> ( ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ R2 )
=> ~ ! [C3: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ C3 ) @ ( transitive_trancl_a @ R2 ) )
=> ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ C3 @ B ) @ R2 ) ) ) ) ).
% tranclE
thf(fact_278_tranclE,axiom,
! [A: nat,B: nat,R2: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( transi6264000038957366511cl_nat @ R2 ) )
=> ( ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R2 )
=> ~ ! [C3: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ C3 ) @ ( transi6264000038957366511cl_nat @ R2 ) )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ C3 @ B ) @ R2 ) ) ) ) ).
% tranclE
thf(fact_279_tranclE,axiom,
! [A: term_a_b,B: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7922773638565587891rm_a_b @ R2 ) )
=> ( ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R2 )
=> ~ ! [C3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C3 ) @ ( transi7922773638565587891rm_a_b @ R2 ) )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ C3 @ B ) @ R2 ) ) ) ) ).
% tranclE
thf(fact_280_rsteps__srstepsI,axiom,
! [R: set_Pr4386577575007340137rm_a_b,F: set_Pr4934435412358123699_a_nat,S: term_a_b,T: term_a_b] :
( ( ord_le8666007276011122963_a_nat @ ( terms_7988297476397195622_a_b_b @ R ) @ F )
=> ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ S ) @ F )
=> ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ T ) @ F )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( transi7922773638565587891rm_a_b @ ( rstep_a_b @ R ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( transi7922773638565587891rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R ) ) ) ) ) ) ) ) ).
% rsteps_srstepsI
thf(fact_281_rsteps__eq__srsteps__eqI,axiom,
! [R: set_Pr4386577575007340137rm_a_b,F: set_Pr4934435412358123699_a_nat,S: term_a_b,T: term_a_b] :
( ( ord_le8666007276011122963_a_nat @ ( terms_7988297476397195622_a_b_b @ R ) @ F )
=> ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ S ) @ F )
=> ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ T ) @ F )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( transi7742714808557438673rm_a_b @ ( rstep_a_b @ R ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R ) ) ) ) ) ) ) ) ).
% rsteps_eq_srsteps_eqI
thf(fact_282_rstep__srstepI,axiom,
! [R: set_Pr4386577575007340137rm_a_b,F: set_Pr4934435412358123699_a_nat,S: term_a_b,T: term_a_b] :
( ( ord_le8666007276011122963_a_nat @ ( terms_7988297476397195622_a_b_b @ R ) @ F )
=> ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ S ) @ F )
=> ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ T ) @ F )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( rstep_a_b @ R ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( sig_step_a_b @ F @ ( rstep_a_b @ R ) ) ) ) ) ) ) ).
% rstep_srstepI
thf(fact_283_term__to__sig_Ocases,axiom,
! [X: produc5279506192219892694rm_a_b] :
( ! [F4: set_Pr4934435412358123699_a_nat,V: b,X3: b] :
( X
!= ( produc8030969961714872974rm_a_b @ F4 @ ( produc1437816968797971900rm_a_b @ V @ ( var_b_a @ X3 ) ) ) )
=> ~ ! [F4: set_Pr4934435412358123699_a_nat,V: b,F2: a,Ts: list_term_a_b] :
( X
!= ( produc8030969961714872974rm_a_b @ F4 @ ( produc1437816968797971900rm_a_b @ V @ ( fun_a_b @ F2 @ Ts ) ) ) ) ) ).
% term_to_sig.cases
thf(fact_284_nth__equalityI,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ Xs @ I2 )
= ( nth_nat @ Ys @ I2 ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_285_nth__equalityI,axiom,
! [Xs: list_list_term_a_b,Ys: list_list_term_a_b] :
( ( ( size_s877380706853472072rm_a_b @ Xs )
= ( size_s877380706853472072rm_a_b @ Ys ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s877380706853472072rm_a_b @ Xs ) )
=> ( ( nth_list_term_a_b @ Xs @ I2 )
= ( nth_list_term_a_b @ Ys @ I2 ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_286_nth__equalityI,axiom,
! [Xs: list_P8875379029341186191rm_a_b,Ys: list_P8875379029341186191rm_a_b] :
( ( ( size_s8600120110137556219rm_a_b @ Xs )
= ( size_s8600120110137556219rm_a_b @ Ys ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s8600120110137556219rm_a_b @ Xs ) )
=> ( ( nth_Pr1098598711982373392rm_a_b @ Xs @ I2 )
= ( nth_Pr1098598711982373392rm_a_b @ Ys @ I2 ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_287_nth__equalityI,axiom,
! [Xs: list_P3592885314253461005_a_nat,Ys: list_P3592885314253461005_a_nat] :
( ( ( size_s984997627204368545_a_nat @ Xs )
= ( size_s984997627204368545_a_nat @ Ys ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s984997627204368545_a_nat @ Xs ) )
=> ( ( nth_Pr8461465654520414006_a_nat @ Xs @ I2 )
= ( nth_Pr8461465654520414006_a_nat @ Ys @ I2 ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_288_nth__equalityI,axiom,
! [Xs: list_term_a_b,Ys: list_term_a_b] :
( ( ( size_s8906293707977694520rm_a_b @ Xs )
= ( size_s8906293707977694520rm_a_b @ Ys ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s8906293707977694520rm_a_b @ Xs ) )
=> ( ( nth_term_a_b @ Xs @ I2 )
= ( nth_term_a_b @ Ys @ I2 ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_289_Skolem__list__nth,axiom,
! [K: nat,P2: nat > nat > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ? [X5: nat] : ( P2 @ I3 @ X5 ) ) )
= ( ? [Xs2: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= K )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ( P2 @ I3 @ ( nth_nat @ Xs2 @ I3 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_290_Skolem__list__nth,axiom,
! [K: nat,P2: nat > list_term_a_b > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ? [X5: list_term_a_b] : ( P2 @ I3 @ X5 ) ) )
= ( ? [Xs2: list_list_term_a_b] :
( ( ( size_s877380706853472072rm_a_b @ Xs2 )
= K )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ( P2 @ I3 @ ( nth_list_term_a_b @ Xs2 @ I3 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_291_Skolem__list__nth,axiom,
! [K: nat,P2: nat > produc357393685978478089rm_a_b > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ? [X5: produc357393685978478089rm_a_b] : ( P2 @ I3 @ X5 ) ) )
= ( ? [Xs2: list_P8875379029341186191rm_a_b] :
( ( ( size_s8600120110137556219rm_a_b @ Xs2 )
= K )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ( P2 @ I3 @ ( nth_Pr1098598711982373392rm_a_b @ Xs2 @ I3 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_292_Skolem__list__nth,axiom,
! [K: nat,P2: nat > product_prod_a_nat > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ? [X5: product_prod_a_nat] : ( P2 @ I3 @ X5 ) ) )
= ( ? [Xs2: list_P3592885314253461005_a_nat] :
( ( ( size_s984997627204368545_a_nat @ Xs2 )
= K )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ( P2 @ I3 @ ( nth_Pr8461465654520414006_a_nat @ Xs2 @ I3 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_293_Skolem__list__nth,axiom,
! [K: nat,P2: nat > term_a_b > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ? [X5: term_a_b] : ( P2 @ I3 @ X5 ) ) )
= ( ? [Xs2: list_term_a_b] :
( ( ( size_s8906293707977694520rm_a_b @ Xs2 )
= K )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ( P2 @ I3 @ ( nth_term_a_b @ Xs2 @ I3 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_294_list__eq__iff__nth__eq,axiom,
( ( ^ [Y4: list_nat,Z3: list_nat] : ( Y4 = Z3 ) )
= ( ^ [Xs2: list_nat,Ys2: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys2 ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs2 ) )
=> ( ( nth_nat @ Xs2 @ I3 )
= ( nth_nat @ Ys2 @ I3 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_295_list__eq__iff__nth__eq,axiom,
( ( ^ [Y4: list_list_term_a_b,Z3: list_list_term_a_b] : ( Y4 = Z3 ) )
= ( ^ [Xs2: list_list_term_a_b,Ys2: list_list_term_a_b] :
( ( ( size_s877380706853472072rm_a_b @ Xs2 )
= ( size_s877380706853472072rm_a_b @ Ys2 ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s877380706853472072rm_a_b @ Xs2 ) )
=> ( ( nth_list_term_a_b @ Xs2 @ I3 )
= ( nth_list_term_a_b @ Ys2 @ I3 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_296_list__eq__iff__nth__eq,axiom,
( ( ^ [Y4: list_P8875379029341186191rm_a_b,Z3: list_P8875379029341186191rm_a_b] : ( Y4 = Z3 ) )
= ( ^ [Xs2: list_P8875379029341186191rm_a_b,Ys2: list_P8875379029341186191rm_a_b] :
( ( ( size_s8600120110137556219rm_a_b @ Xs2 )
= ( size_s8600120110137556219rm_a_b @ Ys2 ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s8600120110137556219rm_a_b @ Xs2 ) )
=> ( ( nth_Pr1098598711982373392rm_a_b @ Xs2 @ I3 )
= ( nth_Pr1098598711982373392rm_a_b @ Ys2 @ I3 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_297_list__eq__iff__nth__eq,axiom,
( ( ^ [Y4: list_P3592885314253461005_a_nat,Z3: list_P3592885314253461005_a_nat] : ( Y4 = Z3 ) )
= ( ^ [Xs2: list_P3592885314253461005_a_nat,Ys2: list_P3592885314253461005_a_nat] :
( ( ( size_s984997627204368545_a_nat @ Xs2 )
= ( size_s984997627204368545_a_nat @ Ys2 ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s984997627204368545_a_nat @ Xs2 ) )
=> ( ( nth_Pr8461465654520414006_a_nat @ Xs2 @ I3 )
= ( nth_Pr8461465654520414006_a_nat @ Ys2 @ I3 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_298_list__eq__iff__nth__eq,axiom,
( ( ^ [Y4: list_term_a_b,Z3: list_term_a_b] : ( Y4 = Z3 ) )
= ( ^ [Xs2: list_term_a_b,Ys2: list_term_a_b] :
( ( ( size_s8906293707977694520rm_a_b @ Xs2 )
= ( size_s8906293707977694520rm_a_b @ Ys2 ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s8906293707977694520rm_a_b @ Xs2 ) )
=> ( ( nth_term_a_b @ Xs2 @ I3 )
= ( nth_term_a_b @ Ys2 @ I3 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_299_Ex__list__of__length__P,axiom,
! [N: nat,P2: nat > nat > $o] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ? [X6: nat] : ( P2 @ X6 @ I2 ) )
=> ? [Xs3: list_nat] :
( ( ( size_size_list_nat @ Xs3 )
= N )
& ! [I: nat] :
( ( ord_less_nat @ I @ N )
=> ( P2 @ ( nth_nat @ Xs3 @ I ) @ I ) ) ) ) ).
% Ex_list_of_length_P
thf(fact_300_Ex__list__of__length__P,axiom,
! [N: nat,P2: list_term_a_b > nat > $o] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ? [X6: list_term_a_b] : ( P2 @ X6 @ I2 ) )
=> ? [Xs3: list_list_term_a_b] :
( ( ( size_s877380706853472072rm_a_b @ Xs3 )
= N )
& ! [I: nat] :
( ( ord_less_nat @ I @ N )
=> ( P2 @ ( nth_list_term_a_b @ Xs3 @ I ) @ I ) ) ) ) ).
% Ex_list_of_length_P
thf(fact_301_Ex__list__of__length__P,axiom,
! [N: nat,P2: produc357393685978478089rm_a_b > nat > $o] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ? [X6: produc357393685978478089rm_a_b] : ( P2 @ X6 @ I2 ) )
=> ? [Xs3: list_P8875379029341186191rm_a_b] :
( ( ( size_s8600120110137556219rm_a_b @ Xs3 )
= N )
& ! [I: nat] :
( ( ord_less_nat @ I @ N )
=> ( P2 @ ( nth_Pr1098598711982373392rm_a_b @ Xs3 @ I ) @ I ) ) ) ) ).
% Ex_list_of_length_P
thf(fact_302_Ex__list__of__length__P,axiom,
! [N: nat,P2: product_prod_a_nat > nat > $o] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ? [X6: product_prod_a_nat] : ( P2 @ X6 @ I2 ) )
=> ? [Xs3: list_P3592885314253461005_a_nat] :
( ( ( size_s984997627204368545_a_nat @ Xs3 )
= N )
& ! [I: nat] :
( ( ord_less_nat @ I @ N )
=> ( P2 @ ( nth_Pr8461465654520414006_a_nat @ Xs3 @ I ) @ I ) ) ) ) ).
% Ex_list_of_length_P
thf(fact_303_Ex__list__of__length__P,axiom,
! [N: nat,P2: term_a_b > nat > $o] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ? [X6: term_a_b] : ( P2 @ X6 @ I2 ) )
=> ? [Xs3: list_term_a_b] :
( ( ( size_s8906293707977694520rm_a_b @ Xs3 )
= N )
& ! [I: nat] :
( ( ord_less_nat @ I @ N )
=> ( P2 @ ( nth_term_a_b @ Xs3 @ I ) @ I ) ) ) ) ).
% Ex_list_of_length_P
thf(fact_304_NF__srstep__args,axiom,
! [F3: a,Ss2: list_term_a_b,F: set_Pr4934435412358123699_a_nat,R: set_Pr4386577575007340137rm_a_b,I4: nat] :
( ( member_term_a_b @ ( fun_a_b @ F3 @ Ss2 ) @ ( abstract_NF_term_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R ) ) ) )
=> ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ ( fun_a_b @ F3 @ Ss2 ) ) @ F )
=> ( ( ord_less_nat @ I4 @ ( size_s8906293707977694520rm_a_b @ Ss2 ) )
=> ( member_term_a_b @ ( nth_term_a_b @ Ss2 @ I4 ) @ ( abstract_NF_term_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R ) ) ) ) ) ) ) ).
% NF_srstep_args
thf(fact_305_subsetI,axiom,
! [A5: set_list_term_a_b,B5: set_list_term_a_b] :
( ! [X3: list_term_a_b] :
( ( member_list_term_a_b @ X3 @ A5 )
=> ( member_list_term_a_b @ X3 @ B5 ) )
=> ( ord_le8558390355660226490rm_a_b @ A5 @ B5 ) ) ).
% subsetI
thf(fact_306_subsetI,axiom,
! [A5: set_list_nat,B5: set_list_nat] :
( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ A5 )
=> ( member_list_nat @ X3 @ B5 ) )
=> ( ord_le6045566169113846134st_nat @ A5 @ B5 ) ) ).
% subsetI
thf(fact_307_subsetI,axiom,
! [A5: set_nat,B5: set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A5 )
=> ( member_nat @ X3 @ B5 ) )
=> ( ord_less_eq_set_nat @ A5 @ B5 ) ) ).
% subsetI
thf(fact_308_subsetI,axiom,
! [A5: set_Pr8564414093027780873rm_a_b,B5: set_Pr8564414093027780873rm_a_b] :
( ! [X3: produc51424535725745577rm_a_b] :
( ( member4405265420456397394rm_a_b @ X3 @ A5 )
=> ( member4405265420456397394rm_a_b @ X3 @ B5 ) )
=> ( ord_le7771511673815134377rm_a_b @ A5 @ B5 ) ) ).
% subsetI
thf(fact_309_subsetI,axiom,
! [A5: set_Pr5046312416420021961st_nat,B5: set_Pr5046312416420021961st_nat] :
( ! [X3: produc432399132543013523st_nat] :
( ( member4851138774834033962st_nat @ X3 @ A5 )
=> ( member4851138774834033962st_nat @ X3 @ B5 ) )
=> ( ord_le1912520282604406569st_nat @ A5 @ B5 ) ) ).
% subsetI
thf(fact_310_subsetI,axiom,
! [A5: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
( ! [X3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X3 @ A5 )
=> ( member8440522571783428010at_nat @ X3 @ B5 ) )
=> ( ord_le3146513528884898305at_nat @ A5 @ B5 ) ) ).
% subsetI
thf(fact_311_subsetI,axiom,
! [A5: set_Product_prod_a_a,B5: set_Product_prod_a_a] :
( ! [X3: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X3 @ A5 )
=> ( member1426531477525435216od_a_a @ X3 @ B5 ) )
=> ( ord_le746702958409616551od_a_a @ A5 @ B5 ) ) ).
% subsetI
thf(fact_312_subsetI,axiom,
! [A5: set_term_a_b,B5: set_term_a_b] :
( ! [X3: term_a_b] :
( ( member_term_a_b @ X3 @ A5 )
=> ( member_term_a_b @ X3 @ B5 ) )
=> ( ord_le2705286416250468010rm_a_b @ A5 @ B5 ) ) ).
% subsetI
thf(fact_313_subsetI,axiom,
! [A5: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
( ! [X3: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X3 @ A5 )
=> ( member5869715511025134514rm_a_b @ X3 @ B5 ) )
=> ( ord_le118470702582115849rm_a_b @ A5 @ B5 ) ) ).
% subsetI
thf(fact_314_subsetI,axiom,
! [A5: set_Pr4934435412358123699_a_nat,B5: set_Pr4934435412358123699_a_nat] :
( ! [X3: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ X3 @ A5 )
=> ( member5724188588386418708_a_nat @ X3 @ B5 ) )
=> ( ord_le8666007276011122963_a_nat @ A5 @ B5 ) ) ).
% subsetI
thf(fact_315_subset__antisym,axiom,
! [A5: set_Pr8564414093027780873rm_a_b,B5: set_Pr8564414093027780873rm_a_b] :
( ( ord_le7771511673815134377rm_a_b @ A5 @ B5 )
=> ( ( ord_le7771511673815134377rm_a_b @ B5 @ A5 )
=> ( A5 = B5 ) ) ) ).
% subset_antisym
thf(fact_316_subset__antisym,axiom,
! [A5: set_Pr5046312416420021961st_nat,B5: set_Pr5046312416420021961st_nat] :
( ( ord_le1912520282604406569st_nat @ A5 @ B5 )
=> ( ( ord_le1912520282604406569st_nat @ B5 @ A5 )
=> ( A5 = B5 ) ) ) ).
% subset_antisym
thf(fact_317_subset__antisym,axiom,
! [A5: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ A5 @ B5 )
=> ( ( ord_le3146513528884898305at_nat @ B5 @ A5 )
=> ( A5 = B5 ) ) ) ).
% subset_antisym
thf(fact_318_subset__antisym,axiom,
! [A5: set_Product_prod_a_a,B5: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A5 @ B5 )
=> ( ( ord_le746702958409616551od_a_a @ B5 @ A5 )
=> ( A5 = B5 ) ) ) ).
% subset_antisym
thf(fact_319_subset__antisym,axiom,
! [A5: set_term_a_b,B5: set_term_a_b] :
( ( ord_le2705286416250468010rm_a_b @ A5 @ B5 )
=> ( ( ord_le2705286416250468010rm_a_b @ B5 @ A5 )
=> ( A5 = B5 ) ) ) ).
% subset_antisym
thf(fact_320_subset__antisym,axiom,
! [A5: set_Pr4934435412358123699_a_nat,B5: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A5 @ B5 )
=> ( ( ord_le8666007276011122963_a_nat @ B5 @ A5 )
=> ( A5 = B5 ) ) ) ).
% subset_antisym
thf(fact_321_rtrancl__induct2,axiom,
! [Ax: list_nat,Ay: term_a_b,Bx: list_nat,By: term_a_b,R2: set_Pr3371896974081115719rm_a_b,P2: list_nat > term_a_b > $o] :
( ( member2837775823723636880rm_a_b @ ( produc621521293690684375rm_a_b @ ( produc5151171985953862413rm_a_b @ Ax @ Ay ) @ ( produc5151171985953862413rm_a_b @ Bx @ By ) ) @ ( transi746783544501582424rm_a_b @ R2 ) )
=> ( ( P2 @ Ax @ Ay )
=> ( ! [A4: list_nat,B3: term_a_b,Aa: list_nat,Ba: term_a_b] :
( ( member2837775823723636880rm_a_b @ ( produc621521293690684375rm_a_b @ ( produc5151171985953862413rm_a_b @ Ax @ Ay ) @ ( produc5151171985953862413rm_a_b @ A4 @ B3 ) ) @ ( transi746783544501582424rm_a_b @ R2 ) )
=> ( ( member2837775823723636880rm_a_b @ ( produc621521293690684375rm_a_b @ ( produc5151171985953862413rm_a_b @ A4 @ B3 ) @ ( produc5151171985953862413rm_a_b @ Aa @ Ba ) ) @ R2 )
=> ( ( P2 @ A4 @ B3 )
=> ( P2 @ Aa @ Ba ) ) ) )
=> ( P2 @ Bx @ By ) ) ) ) ).
% rtrancl_induct2
thf(fact_322_rtrancl__induct2,axiom,
! [Ax: set_Pr4934435412358123699_a_nat,Ay: produc6551080308567045442rm_a_b,Bx: set_Pr4934435412358123699_a_nat,By: produc6551080308567045442rm_a_b,R2: set_Pr3349378317018664967rm_a_b,P2: set_Pr4934435412358123699_a_nat > produc6551080308567045442rm_a_b > $o] :
( ( member4955916718216803920rm_a_b @ ( produc3065199688681838103rm_a_b @ ( produc8030969961714872974rm_a_b @ Ax @ Ay ) @ ( produc8030969961714872974rm_a_b @ Bx @ By ) ) @ ( transi5649879975317962969rm_a_b @ R2 ) )
=> ( ( P2 @ Ax @ Ay )
=> ( ! [A4: set_Pr4934435412358123699_a_nat,B3: produc6551080308567045442rm_a_b,Aa: set_Pr4934435412358123699_a_nat,Ba: produc6551080308567045442rm_a_b] :
( ( member4955916718216803920rm_a_b @ ( produc3065199688681838103rm_a_b @ ( produc8030969961714872974rm_a_b @ Ax @ Ay ) @ ( produc8030969961714872974rm_a_b @ A4 @ B3 ) ) @ ( transi5649879975317962969rm_a_b @ R2 ) )
=> ( ( member4955916718216803920rm_a_b @ ( produc3065199688681838103rm_a_b @ ( produc8030969961714872974rm_a_b @ A4 @ B3 ) @ ( produc8030969961714872974rm_a_b @ Aa @ Ba ) ) @ R2 )
=> ( ( P2 @ A4 @ B3 )
=> ( P2 @ Aa @ Ba ) ) ) )
=> ( P2 @ Bx @ By ) ) ) ) ).
% rtrancl_induct2
thf(fact_323_rtrancl__induct2,axiom,
! [Ax: nat,Ay: nat,Bx: nat,By: nat,R2: set_Pr8693737435421807431at_nat,P2: nat > nat > $o] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ Ax @ Ay ) @ ( product_Pair_nat_nat @ Bx @ By ) ) @ ( transi8609417484261754244at_nat @ R2 ) )
=> ( ( P2 @ Ax @ Ay )
=> ( ! [A4: nat,B3: nat,Aa: nat,Ba: nat] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ Ax @ Ay ) @ ( product_Pair_nat_nat @ A4 @ B3 ) ) @ ( transi8609417484261754244at_nat @ R2 ) )
=> ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A4 @ B3 ) @ ( product_Pair_nat_nat @ Aa @ Ba ) ) @ R2 )
=> ( ( P2 @ A4 @ B3 )
=> ( P2 @ Aa @ Ba ) ) ) )
=> ( P2 @ Bx @ By ) ) ) ) ).
% rtrancl_induct2
thf(fact_324_rtrancl__induct2,axiom,
! [Ax: b,Ay: term_a_b,Bx: b,By: term_a_b,R2: set_Pr5233514982410851781rm_a_b,P2: b > term_a_b > $o] :
( ( member7542685802389602574rm_a_b @ ( produc411707225774285981rm_a_b @ ( produc1437816968797971900rm_a_b @ Ax @ Ay ) @ ( produc1437816968797971900rm_a_b @ Bx @ By ) ) @ ( transi2027978691874478335rm_a_b @ R2 ) )
=> ( ( P2 @ Ax @ Ay )
=> ( ! [A4: b,B3: term_a_b,Aa: b,Ba: term_a_b] :
( ( member7542685802389602574rm_a_b @ ( produc411707225774285981rm_a_b @ ( produc1437816968797971900rm_a_b @ Ax @ Ay ) @ ( produc1437816968797971900rm_a_b @ A4 @ B3 ) ) @ ( transi2027978691874478335rm_a_b @ R2 ) )
=> ( ( member7542685802389602574rm_a_b @ ( produc411707225774285981rm_a_b @ ( produc1437816968797971900rm_a_b @ A4 @ B3 ) @ ( produc1437816968797971900rm_a_b @ Aa @ Ba ) ) @ R2 )
=> ( ( P2 @ A4 @ B3 )
=> ( P2 @ Aa @ Ba ) ) ) )
=> ( P2 @ Bx @ By ) ) ) ) ).
% rtrancl_induct2
thf(fact_325_rtrancl__induct2,axiom,
! [Ax: a,Ay: nat,Bx: a,By: nat,R2: set_Pr1811044260758604347_a_nat,P2: a > nat > $o] :
( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ ( product_Pair_a_nat @ Ax @ Ay ) @ ( product_Pair_a_nat @ Bx @ By ) ) @ ( transi2726145917338391738_a_nat @ R2 ) )
=> ( ( P2 @ Ax @ Ay )
=> ( ! [A4: a,B3: nat,Aa: a,Ba: nat] :
( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ ( product_Pair_a_nat @ Ax @ Ay ) @ ( product_Pair_a_nat @ A4 @ B3 ) ) @ ( transi2726145917338391738_a_nat @ R2 ) )
=> ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ ( product_Pair_a_nat @ A4 @ B3 ) @ ( product_Pair_a_nat @ Aa @ Ba ) ) @ R2 )
=> ( ( P2 @ A4 @ B3 )
=> ( P2 @ Aa @ Ba ) ) ) )
=> ( P2 @ Bx @ By ) ) ) ) ).
% rtrancl_induct2
thf(fact_326_rtrancl__induct2,axiom,
! [Ax: term_a_b,Ay: term_a_b,Bx: term_a_b,By: term_a_b,R2: set_Pr2972776593051762503rm_a_b,P2: term_a_b > term_a_b > $o] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ Ax @ Ay ) @ ( produc7020197800436672577rm_a_b @ Bx @ By ) ) @ ( transi2615809358984392588rm_a_b @ R2 ) )
=> ( ( P2 @ Ax @ Ay )
=> ( ! [A4: term_a_b,B3: term_a_b,Aa: term_a_b,Ba: term_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ Ax @ Ay ) @ ( produc7020197800436672577rm_a_b @ A4 @ B3 ) ) @ ( transi2615809358984392588rm_a_b @ R2 ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ A4 @ B3 ) @ ( produc7020197800436672577rm_a_b @ Aa @ Ba ) ) @ R2 )
=> ( ( P2 @ A4 @ B3 )
=> ( P2 @ Aa @ Ba ) ) ) )
=> ( P2 @ Bx @ By ) ) ) ) ).
% rtrancl_induct2
thf(fact_327_converse__rtranclE2,axiom,
! [Xa: list_nat,Xb: term_a_b,Za: list_nat,Zb: term_a_b,R2: set_Pr3371896974081115719rm_a_b] :
( ( member2837775823723636880rm_a_b @ ( produc621521293690684375rm_a_b @ ( produc5151171985953862413rm_a_b @ Xa @ Xb ) @ ( produc5151171985953862413rm_a_b @ Za @ Zb ) ) @ ( transi746783544501582424rm_a_b @ R2 ) )
=> ( ( ( produc5151171985953862413rm_a_b @ Xa @ Xb )
!= ( produc5151171985953862413rm_a_b @ Za @ Zb ) )
=> ~ ! [A4: list_nat,B3: term_a_b] :
( ( member2837775823723636880rm_a_b @ ( produc621521293690684375rm_a_b @ ( produc5151171985953862413rm_a_b @ Xa @ Xb ) @ ( produc5151171985953862413rm_a_b @ A4 @ B3 ) ) @ R2 )
=> ~ ( member2837775823723636880rm_a_b @ ( produc621521293690684375rm_a_b @ ( produc5151171985953862413rm_a_b @ A4 @ B3 ) @ ( produc5151171985953862413rm_a_b @ Za @ Zb ) ) @ ( transi746783544501582424rm_a_b @ R2 ) ) ) ) ) ).
% converse_rtranclE2
thf(fact_328_converse__rtranclE2,axiom,
! [Xa: set_Pr4934435412358123699_a_nat,Xb: produc6551080308567045442rm_a_b,Za: set_Pr4934435412358123699_a_nat,Zb: produc6551080308567045442rm_a_b,R2: set_Pr3349378317018664967rm_a_b] :
( ( member4955916718216803920rm_a_b @ ( produc3065199688681838103rm_a_b @ ( produc8030969961714872974rm_a_b @ Xa @ Xb ) @ ( produc8030969961714872974rm_a_b @ Za @ Zb ) ) @ ( transi5649879975317962969rm_a_b @ R2 ) )
=> ( ( ( produc8030969961714872974rm_a_b @ Xa @ Xb )
!= ( produc8030969961714872974rm_a_b @ Za @ Zb ) )
=> ~ ! [A4: set_Pr4934435412358123699_a_nat,B3: produc6551080308567045442rm_a_b] :
( ( member4955916718216803920rm_a_b @ ( produc3065199688681838103rm_a_b @ ( produc8030969961714872974rm_a_b @ Xa @ Xb ) @ ( produc8030969961714872974rm_a_b @ A4 @ B3 ) ) @ R2 )
=> ~ ( member4955916718216803920rm_a_b @ ( produc3065199688681838103rm_a_b @ ( produc8030969961714872974rm_a_b @ A4 @ B3 ) @ ( produc8030969961714872974rm_a_b @ Za @ Zb ) ) @ ( transi5649879975317962969rm_a_b @ R2 ) ) ) ) ) ).
% converse_rtranclE2
thf(fact_329_converse__rtranclE2,axiom,
! [Xa: nat,Xb: nat,Za: nat,Zb: nat,R2: set_Pr8693737435421807431at_nat] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ Xa @ Xb ) @ ( product_Pair_nat_nat @ Za @ Zb ) ) @ ( transi8609417484261754244at_nat @ R2 ) )
=> ( ( ( product_Pair_nat_nat @ Xa @ Xb )
!= ( product_Pair_nat_nat @ Za @ Zb ) )
=> ~ ! [A4: nat,B3: nat] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ Xa @ Xb ) @ ( product_Pair_nat_nat @ A4 @ B3 ) ) @ R2 )
=> ~ ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A4 @ B3 ) @ ( product_Pair_nat_nat @ Za @ Zb ) ) @ ( transi8609417484261754244at_nat @ R2 ) ) ) ) ) ).
% converse_rtranclE2
thf(fact_330_converse__rtranclE2,axiom,
! [Xa: b,Xb: term_a_b,Za: b,Zb: term_a_b,R2: set_Pr5233514982410851781rm_a_b] :
( ( member7542685802389602574rm_a_b @ ( produc411707225774285981rm_a_b @ ( produc1437816968797971900rm_a_b @ Xa @ Xb ) @ ( produc1437816968797971900rm_a_b @ Za @ Zb ) ) @ ( transi2027978691874478335rm_a_b @ R2 ) )
=> ( ( ( produc1437816968797971900rm_a_b @ Xa @ Xb )
!= ( produc1437816968797971900rm_a_b @ Za @ Zb ) )
=> ~ ! [A4: b,B3: term_a_b] :
( ( member7542685802389602574rm_a_b @ ( produc411707225774285981rm_a_b @ ( produc1437816968797971900rm_a_b @ Xa @ Xb ) @ ( produc1437816968797971900rm_a_b @ A4 @ B3 ) ) @ R2 )
=> ~ ( member7542685802389602574rm_a_b @ ( produc411707225774285981rm_a_b @ ( produc1437816968797971900rm_a_b @ A4 @ B3 ) @ ( produc1437816968797971900rm_a_b @ Za @ Zb ) ) @ ( transi2027978691874478335rm_a_b @ R2 ) ) ) ) ) ).
% converse_rtranclE2
thf(fact_331_converse__rtranclE2,axiom,
! [Xa: a,Xb: nat,Za: a,Zb: nat,R2: set_Pr1811044260758604347_a_nat] :
( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ ( product_Pair_a_nat @ Xa @ Xb ) @ ( product_Pair_a_nat @ Za @ Zb ) ) @ ( transi2726145917338391738_a_nat @ R2 ) )
=> ( ( ( product_Pair_a_nat @ Xa @ Xb )
!= ( product_Pair_a_nat @ Za @ Zb ) )
=> ~ ! [A4: a,B3: nat] :
( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ ( product_Pair_a_nat @ Xa @ Xb ) @ ( product_Pair_a_nat @ A4 @ B3 ) ) @ R2 )
=> ~ ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ ( product_Pair_a_nat @ A4 @ B3 ) @ ( product_Pair_a_nat @ Za @ Zb ) ) @ ( transi2726145917338391738_a_nat @ R2 ) ) ) ) ) ).
% converse_rtranclE2
thf(fact_332_converse__rtranclE2,axiom,
! [Xa: term_a_b,Xb: term_a_b,Za: term_a_b,Zb: term_a_b,R2: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ Xa @ Xb ) @ ( produc7020197800436672577rm_a_b @ Za @ Zb ) ) @ ( transi2615809358984392588rm_a_b @ R2 ) )
=> ( ( ( produc7020197800436672577rm_a_b @ Xa @ Xb )
!= ( produc7020197800436672577rm_a_b @ Za @ Zb ) )
=> ~ ! [A4: term_a_b,B3: term_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ Xa @ Xb ) @ ( produc7020197800436672577rm_a_b @ A4 @ B3 ) ) @ R2 )
=> ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ A4 @ B3 ) @ ( produc7020197800436672577rm_a_b @ Za @ Zb ) ) @ ( transi2615809358984392588rm_a_b @ R2 ) ) ) ) ) ).
% converse_rtranclE2
thf(fact_333_converse__rtrancl__induct2,axiom,
! [Ax: list_nat,Ay: term_a_b,Bx: list_nat,By: term_a_b,R2: set_Pr3371896974081115719rm_a_b,P2: list_nat > term_a_b > $o] :
( ( member2837775823723636880rm_a_b @ ( produc621521293690684375rm_a_b @ ( produc5151171985953862413rm_a_b @ Ax @ Ay ) @ ( produc5151171985953862413rm_a_b @ Bx @ By ) ) @ ( transi746783544501582424rm_a_b @ R2 ) )
=> ( ( P2 @ Bx @ By )
=> ( ! [A4: list_nat,B3: term_a_b,Aa: list_nat,Ba: term_a_b] :
( ( member2837775823723636880rm_a_b @ ( produc621521293690684375rm_a_b @ ( produc5151171985953862413rm_a_b @ A4 @ B3 ) @ ( produc5151171985953862413rm_a_b @ Aa @ Ba ) ) @ R2 )
=> ( ( member2837775823723636880rm_a_b @ ( produc621521293690684375rm_a_b @ ( produc5151171985953862413rm_a_b @ Aa @ Ba ) @ ( produc5151171985953862413rm_a_b @ Bx @ By ) ) @ ( transi746783544501582424rm_a_b @ R2 ) )
=> ( ( P2 @ Aa @ Ba )
=> ( P2 @ A4 @ B3 ) ) ) )
=> ( P2 @ Ax @ Ay ) ) ) ) ).
% converse_rtrancl_induct2
thf(fact_334_converse__rtrancl__induct2,axiom,
! [Ax: set_Pr4934435412358123699_a_nat,Ay: produc6551080308567045442rm_a_b,Bx: set_Pr4934435412358123699_a_nat,By: produc6551080308567045442rm_a_b,R2: set_Pr3349378317018664967rm_a_b,P2: set_Pr4934435412358123699_a_nat > produc6551080308567045442rm_a_b > $o] :
( ( member4955916718216803920rm_a_b @ ( produc3065199688681838103rm_a_b @ ( produc8030969961714872974rm_a_b @ Ax @ Ay ) @ ( produc8030969961714872974rm_a_b @ Bx @ By ) ) @ ( transi5649879975317962969rm_a_b @ R2 ) )
=> ( ( P2 @ Bx @ By )
=> ( ! [A4: set_Pr4934435412358123699_a_nat,B3: produc6551080308567045442rm_a_b,Aa: set_Pr4934435412358123699_a_nat,Ba: produc6551080308567045442rm_a_b] :
( ( member4955916718216803920rm_a_b @ ( produc3065199688681838103rm_a_b @ ( produc8030969961714872974rm_a_b @ A4 @ B3 ) @ ( produc8030969961714872974rm_a_b @ Aa @ Ba ) ) @ R2 )
=> ( ( member4955916718216803920rm_a_b @ ( produc3065199688681838103rm_a_b @ ( produc8030969961714872974rm_a_b @ Aa @ Ba ) @ ( produc8030969961714872974rm_a_b @ Bx @ By ) ) @ ( transi5649879975317962969rm_a_b @ R2 ) )
=> ( ( P2 @ Aa @ Ba )
=> ( P2 @ A4 @ B3 ) ) ) )
=> ( P2 @ Ax @ Ay ) ) ) ) ).
% converse_rtrancl_induct2
thf(fact_335_converse__rtrancl__induct2,axiom,
! [Ax: nat,Ay: nat,Bx: nat,By: nat,R2: set_Pr8693737435421807431at_nat,P2: nat > nat > $o] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ Ax @ Ay ) @ ( product_Pair_nat_nat @ Bx @ By ) ) @ ( transi8609417484261754244at_nat @ R2 ) )
=> ( ( P2 @ Bx @ By )
=> ( ! [A4: nat,B3: nat,Aa: nat,Ba: nat] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A4 @ B3 ) @ ( product_Pair_nat_nat @ Aa @ Ba ) ) @ R2 )
=> ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ Aa @ Ba ) @ ( product_Pair_nat_nat @ Bx @ By ) ) @ ( transi8609417484261754244at_nat @ R2 ) )
=> ( ( P2 @ Aa @ Ba )
=> ( P2 @ A4 @ B3 ) ) ) )
=> ( P2 @ Ax @ Ay ) ) ) ) ).
% converse_rtrancl_induct2
thf(fact_336_converse__rtrancl__induct2,axiom,
! [Ax: b,Ay: term_a_b,Bx: b,By: term_a_b,R2: set_Pr5233514982410851781rm_a_b,P2: b > term_a_b > $o] :
( ( member7542685802389602574rm_a_b @ ( produc411707225774285981rm_a_b @ ( produc1437816968797971900rm_a_b @ Ax @ Ay ) @ ( produc1437816968797971900rm_a_b @ Bx @ By ) ) @ ( transi2027978691874478335rm_a_b @ R2 ) )
=> ( ( P2 @ Bx @ By )
=> ( ! [A4: b,B3: term_a_b,Aa: b,Ba: term_a_b] :
( ( member7542685802389602574rm_a_b @ ( produc411707225774285981rm_a_b @ ( produc1437816968797971900rm_a_b @ A4 @ B3 ) @ ( produc1437816968797971900rm_a_b @ Aa @ Ba ) ) @ R2 )
=> ( ( member7542685802389602574rm_a_b @ ( produc411707225774285981rm_a_b @ ( produc1437816968797971900rm_a_b @ Aa @ Ba ) @ ( produc1437816968797971900rm_a_b @ Bx @ By ) ) @ ( transi2027978691874478335rm_a_b @ R2 ) )
=> ( ( P2 @ Aa @ Ba )
=> ( P2 @ A4 @ B3 ) ) ) )
=> ( P2 @ Ax @ Ay ) ) ) ) ).
% converse_rtrancl_induct2
thf(fact_337_converse__rtrancl__induct2,axiom,
! [Ax: a,Ay: nat,Bx: a,By: nat,R2: set_Pr1811044260758604347_a_nat,P2: a > nat > $o] :
( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ ( product_Pair_a_nat @ Ax @ Ay ) @ ( product_Pair_a_nat @ Bx @ By ) ) @ ( transi2726145917338391738_a_nat @ R2 ) )
=> ( ( P2 @ Bx @ By )
=> ( ! [A4: a,B3: nat,Aa: a,Ba: nat] :
( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ ( product_Pair_a_nat @ A4 @ B3 ) @ ( product_Pair_a_nat @ Aa @ Ba ) ) @ R2 )
=> ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ ( product_Pair_a_nat @ Aa @ Ba ) @ ( product_Pair_a_nat @ Bx @ By ) ) @ ( transi2726145917338391738_a_nat @ R2 ) )
=> ( ( P2 @ Aa @ Ba )
=> ( P2 @ A4 @ B3 ) ) ) )
=> ( P2 @ Ax @ Ay ) ) ) ) ).
% converse_rtrancl_induct2
thf(fact_338_converse__rtrancl__induct2,axiom,
! [Ax: term_a_b,Ay: term_a_b,Bx: term_a_b,By: term_a_b,R2: set_Pr2972776593051762503rm_a_b,P2: term_a_b > term_a_b > $o] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ Ax @ Ay ) @ ( produc7020197800436672577rm_a_b @ Bx @ By ) ) @ ( transi2615809358984392588rm_a_b @ R2 ) )
=> ( ( P2 @ Bx @ By )
=> ( ! [A4: term_a_b,B3: term_a_b,Aa: term_a_b,Ba: term_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ A4 @ B3 ) @ ( produc7020197800436672577rm_a_b @ Aa @ Ba ) ) @ R2 )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ Aa @ Ba ) @ ( produc7020197800436672577rm_a_b @ Bx @ By ) ) @ ( transi2615809358984392588rm_a_b @ R2 ) )
=> ( ( P2 @ Aa @ Ba )
=> ( P2 @ A4 @ B3 ) ) ) )
=> ( P2 @ Ax @ Ay ) ) ) ) ).
% converse_rtrancl_induct2
thf(fact_339_sig__step__mono2,axiom,
! [R: set_Pr4386577575007340137rm_a_b,L2: set_Pr4386577575007340137rm_a_b,F: set_Pr4934435412358123699_a_nat] :
( ( ord_le118470702582115849rm_a_b @ R @ L2 )
=> ( ord_le118470702582115849rm_a_b @ ( sig_step_a_b @ F @ R ) @ ( sig_step_a_b @ F @ L2 ) ) ) ).
% sig_step_mono2
thf(fact_340_sig__step__mono,axiom,
! [F: set_Pr4934435412358123699_a_nat,G: set_Pr4934435412358123699_a_nat,R: set_Pr4386577575007340137rm_a_b] :
( ( ord_le8666007276011122963_a_nat @ F @ G )
=> ( ord_le118470702582115849rm_a_b @ ( sig_step_a_b @ F @ R ) @ ( sig_step_a_b @ G @ R ) ) ) ).
% sig_step_mono
thf(fact_341_srsteps__with__root__step__sig__mono,axiom,
! [F: set_Pr4934435412358123699_a_nat,G: set_Pr4934435412358123699_a_nat,R: set_Pr4386577575007340137rm_a_b] :
( ( ord_le8666007276011122963_a_nat @ F @ G )
=> ( ord_le118470702582115849rm_a_b @ ( srstep7844470518422762656ep_a_b @ F @ R ) @ ( srstep7844470518422762656ep_a_b @ G @ R ) ) ) ).
% srsteps_with_root_step_sig_mono
thf(fact_342_Collect__mono__iff,axiom,
! [P2: produc357393685978478089rm_a_b > $o,Q: produc357393685978478089rm_a_b > $o] :
( ( ord_le118470702582115849rm_a_b @ ( collec99177395925924084rm_a_b @ P2 ) @ ( collec99177395925924084rm_a_b @ Q ) )
= ( ! [X4: produc357393685978478089rm_a_b] :
( ( P2 @ X4 )
=> ( Q @ X4 ) ) ) ) ).
% Collect_mono_iff
thf(fact_343_Collect__mono__iff,axiom,
! [P2: produc51424535725745577rm_a_b > $o,Q: produc51424535725745577rm_a_b > $o] :
( ( ord_le7771511673815134377rm_a_b @ ( collec5491088168347092372rm_a_b @ P2 ) @ ( collec5491088168347092372rm_a_b @ Q ) )
= ( ! [X4: produc51424535725745577rm_a_b] :
( ( P2 @ X4 )
=> ( Q @ X4 ) ) ) ) ).
% Collect_mono_iff
thf(fact_344_Collect__mono__iff,axiom,
! [P2: produc432399132543013523st_nat > $o,Q: produc432399132543013523st_nat > $o] :
( ( ord_le1912520282604406569st_nat @ ( collec3417396070205469800st_nat @ P2 ) @ ( collec3417396070205469800st_nat @ Q ) )
= ( ! [X4: produc432399132543013523st_nat] :
( ( P2 @ X4 )
=> ( Q @ X4 ) ) ) ) ).
% Collect_mono_iff
thf(fact_345_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_346_Collect__mono__iff,axiom,
! [P2: product_prod_a_a > $o,Q: product_prod_a_a > $o] :
( ( ord_le746702958409616551od_a_a @ ( collec3336397797384452498od_a_a @ P2 ) @ ( collec3336397797384452498od_a_a @ Q ) )
= ( ! [X4: product_prod_a_a] :
( ( P2 @ X4 )
=> ( Q @ X4 ) ) ) ) ).
% Collect_mono_iff
thf(fact_347_Collect__mono__iff,axiom,
! [P2: term_a_b > $o,Q: term_a_b > $o] :
( ( ord_le2705286416250468010rm_a_b @ ( collect_term_a_b @ P2 ) @ ( collect_term_a_b @ Q ) )
= ( ! [X4: term_a_b] :
( ( P2 @ X4 )
=> ( Q @ X4 ) ) ) ) ).
% Collect_mono_iff
thf(fact_348_Collect__mono__iff,axiom,
! [P2: product_prod_a_nat > $o,Q: product_prod_a_nat > $o] :
( ( ord_le8666007276011122963_a_nat @ ( collec4464134535221767506_a_nat @ P2 ) @ ( collec4464134535221767506_a_nat @ Q ) )
= ( ! [X4: product_prod_a_nat] :
( ( P2 @ X4 )
=> ( Q @ X4 ) ) ) ) ).
% Collect_mono_iff
thf(fact_349_set__eq__subset,axiom,
( ( ^ [Y4: set_Pr8564414093027780873rm_a_b,Z3: set_Pr8564414093027780873rm_a_b] : ( Y4 = Z3 ) )
= ( ^ [A6: set_Pr8564414093027780873rm_a_b,B6: set_Pr8564414093027780873rm_a_b] :
( ( ord_le7771511673815134377rm_a_b @ A6 @ B6 )
& ( ord_le7771511673815134377rm_a_b @ B6 @ A6 ) ) ) ) ).
% set_eq_subset
thf(fact_350_set__eq__subset,axiom,
( ( ^ [Y4: set_Pr5046312416420021961st_nat,Z3: set_Pr5046312416420021961st_nat] : ( Y4 = Z3 ) )
= ( ^ [A6: set_Pr5046312416420021961st_nat,B6: set_Pr5046312416420021961st_nat] :
( ( ord_le1912520282604406569st_nat @ A6 @ B6 )
& ( ord_le1912520282604406569st_nat @ B6 @ A6 ) ) ) ) ).
% set_eq_subset
thf(fact_351_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_352_set__eq__subset,axiom,
( ( ^ [Y4: set_Product_prod_a_a,Z3: set_Product_prod_a_a] : ( Y4 = Z3 ) )
= ( ^ [A6: set_Product_prod_a_a,B6: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A6 @ B6 )
& ( ord_le746702958409616551od_a_a @ B6 @ A6 ) ) ) ) ).
% set_eq_subset
thf(fact_353_set__eq__subset,axiom,
( ( ^ [Y4: set_term_a_b,Z3: set_term_a_b] : ( Y4 = Z3 ) )
= ( ^ [A6: set_term_a_b,B6: set_term_a_b] :
( ( ord_le2705286416250468010rm_a_b @ A6 @ B6 )
& ( ord_le2705286416250468010rm_a_b @ B6 @ A6 ) ) ) ) ).
% set_eq_subset
thf(fact_354_set__eq__subset,axiom,
( ( ^ [Y4: set_Pr4934435412358123699_a_nat,Z3: set_Pr4934435412358123699_a_nat] : ( Y4 = Z3 ) )
= ( ^ [A6: set_Pr4934435412358123699_a_nat,B6: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A6 @ B6 )
& ( ord_le8666007276011122963_a_nat @ B6 @ A6 ) ) ) ) ).
% set_eq_subset
thf(fact_355_subset__trans,axiom,
! [A5: set_Pr8564414093027780873rm_a_b,B5: set_Pr8564414093027780873rm_a_b,C4: set_Pr8564414093027780873rm_a_b] :
( ( ord_le7771511673815134377rm_a_b @ A5 @ B5 )
=> ( ( ord_le7771511673815134377rm_a_b @ B5 @ C4 )
=> ( ord_le7771511673815134377rm_a_b @ A5 @ C4 ) ) ) ).
% subset_trans
thf(fact_356_subset__trans,axiom,
! [A5: set_Pr5046312416420021961st_nat,B5: set_Pr5046312416420021961st_nat,C4: set_Pr5046312416420021961st_nat] :
( ( ord_le1912520282604406569st_nat @ A5 @ B5 )
=> ( ( ord_le1912520282604406569st_nat @ B5 @ C4 )
=> ( ord_le1912520282604406569st_nat @ A5 @ C4 ) ) ) ).
% subset_trans
thf(fact_357_subset__trans,axiom,
! [A5: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat,C4: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ A5 @ B5 )
=> ( ( ord_le3146513528884898305at_nat @ B5 @ C4 )
=> ( ord_le3146513528884898305at_nat @ A5 @ C4 ) ) ) ).
% subset_trans
thf(fact_358_subset__trans,axiom,
! [A5: set_Product_prod_a_a,B5: set_Product_prod_a_a,C4: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A5 @ B5 )
=> ( ( ord_le746702958409616551od_a_a @ B5 @ C4 )
=> ( ord_le746702958409616551od_a_a @ A5 @ C4 ) ) ) ).
% subset_trans
thf(fact_359_subset__trans,axiom,
! [A5: set_term_a_b,B5: set_term_a_b,C4: set_term_a_b] :
( ( ord_le2705286416250468010rm_a_b @ A5 @ B5 )
=> ( ( ord_le2705286416250468010rm_a_b @ B5 @ C4 )
=> ( ord_le2705286416250468010rm_a_b @ A5 @ C4 ) ) ) ).
% subset_trans
thf(fact_360_subset__trans,axiom,
! [A5: set_Pr4934435412358123699_a_nat,B5: set_Pr4934435412358123699_a_nat,C4: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A5 @ B5 )
=> ( ( ord_le8666007276011122963_a_nat @ B5 @ C4 )
=> ( ord_le8666007276011122963_a_nat @ A5 @ C4 ) ) ) ).
% subset_trans
thf(fact_361_Collect__mono,axiom,
! [P2: produc357393685978478089rm_a_b > $o,Q: produc357393685978478089rm_a_b > $o] :
( ! [X3: produc357393685978478089rm_a_b] :
( ( P2 @ X3 )
=> ( Q @ X3 ) )
=> ( ord_le118470702582115849rm_a_b @ ( collec99177395925924084rm_a_b @ P2 ) @ ( collec99177395925924084rm_a_b @ Q ) ) ) ).
% Collect_mono
thf(fact_362_Collect__mono,axiom,
! [P2: produc51424535725745577rm_a_b > $o,Q: produc51424535725745577rm_a_b > $o] :
( ! [X3: produc51424535725745577rm_a_b] :
( ( P2 @ X3 )
=> ( Q @ X3 ) )
=> ( ord_le7771511673815134377rm_a_b @ ( collec5491088168347092372rm_a_b @ P2 ) @ ( collec5491088168347092372rm_a_b @ Q ) ) ) ).
% Collect_mono
thf(fact_363_Collect__mono,axiom,
! [P2: produc432399132543013523st_nat > $o,Q: produc432399132543013523st_nat > $o] :
( ! [X3: produc432399132543013523st_nat] :
( ( P2 @ X3 )
=> ( Q @ X3 ) )
=> ( ord_le1912520282604406569st_nat @ ( collec3417396070205469800st_nat @ P2 ) @ ( collec3417396070205469800st_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_364_Collect__mono,axiom,
! [P2: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
( ! [X3: product_prod_nat_nat] :
( ( P2 @ X3 )
=> ( Q @ X3 ) )
=> ( ord_le3146513528884898305at_nat @ ( collec3392354462482085612at_nat @ P2 ) @ ( collec3392354462482085612at_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_365_Collect__mono,axiom,
! [P2: product_prod_a_a > $o,Q: product_prod_a_a > $o] :
( ! [X3: product_prod_a_a] :
( ( P2 @ X3 )
=> ( Q @ X3 ) )
=> ( ord_le746702958409616551od_a_a @ ( collec3336397797384452498od_a_a @ P2 ) @ ( collec3336397797384452498od_a_a @ Q ) ) ) ).
% Collect_mono
thf(fact_366_Collect__mono,axiom,
! [P2: term_a_b > $o,Q: term_a_b > $o] :
( ! [X3: term_a_b] :
( ( P2 @ X3 )
=> ( Q @ X3 ) )
=> ( ord_le2705286416250468010rm_a_b @ ( collect_term_a_b @ P2 ) @ ( collect_term_a_b @ Q ) ) ) ).
% Collect_mono
thf(fact_367_Collect__mono,axiom,
! [P2: product_prod_a_nat > $o,Q: product_prod_a_nat > $o] :
( ! [X3: product_prod_a_nat] :
( ( P2 @ X3 )
=> ( Q @ X3 ) )
=> ( ord_le8666007276011122963_a_nat @ ( collec4464134535221767506_a_nat @ P2 ) @ ( collec4464134535221767506_a_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_368_subset__refl,axiom,
! [A5: set_Pr8564414093027780873rm_a_b] : ( ord_le7771511673815134377rm_a_b @ A5 @ A5 ) ).
% subset_refl
thf(fact_369_subset__refl,axiom,
! [A5: set_Pr5046312416420021961st_nat] : ( ord_le1912520282604406569st_nat @ A5 @ A5 ) ).
% subset_refl
thf(fact_370_subset__refl,axiom,
! [A5: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ A5 @ A5 ) ).
% subset_refl
thf(fact_371_subset__refl,axiom,
! [A5: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ A5 @ A5 ) ).
% subset_refl
thf(fact_372_subset__refl,axiom,
! [A5: set_term_a_b] : ( ord_le2705286416250468010rm_a_b @ A5 @ A5 ) ).
% subset_refl
thf(fact_373_subset__refl,axiom,
! [A5: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ A5 @ A5 ) ).
% subset_refl
thf(fact_374_subset__iff,axiom,
( ord_le8558390355660226490rm_a_b
= ( ^ [A6: set_list_term_a_b,B6: set_list_term_a_b] :
! [T2: list_term_a_b] :
( ( member_list_term_a_b @ T2 @ A6 )
=> ( member_list_term_a_b @ T2 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_375_subset__iff,axiom,
( ord_le6045566169113846134st_nat
= ( ^ [A6: set_list_nat,B6: set_list_nat] :
! [T2: list_nat] :
( ( member_list_nat @ T2 @ A6 )
=> ( member_list_nat @ T2 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_376_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A6: set_nat,B6: set_nat] :
! [T2: nat] :
( ( member_nat @ T2 @ A6 )
=> ( member_nat @ T2 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_377_subset__iff,axiom,
( ord_le7771511673815134377rm_a_b
= ( ^ [A6: set_Pr8564414093027780873rm_a_b,B6: set_Pr8564414093027780873rm_a_b] :
! [T2: produc51424535725745577rm_a_b] :
( ( member4405265420456397394rm_a_b @ T2 @ A6 )
=> ( member4405265420456397394rm_a_b @ T2 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_378_subset__iff,axiom,
( ord_le1912520282604406569st_nat
= ( ^ [A6: set_Pr5046312416420021961st_nat,B6: set_Pr5046312416420021961st_nat] :
! [T2: produc432399132543013523st_nat] :
( ( member4851138774834033962st_nat @ T2 @ A6 )
=> ( member4851138774834033962st_nat @ T2 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_379_subset__iff,axiom,
( ord_le3146513528884898305at_nat
= ( ^ [A6: set_Pr1261947904930325089at_nat,B6: set_Pr1261947904930325089at_nat] :
! [T2: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ T2 @ A6 )
=> ( member8440522571783428010at_nat @ T2 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_380_subset__iff,axiom,
( ord_le746702958409616551od_a_a
= ( ^ [A6: set_Product_prod_a_a,B6: set_Product_prod_a_a] :
! [T2: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ T2 @ A6 )
=> ( member1426531477525435216od_a_a @ T2 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_381_subset__iff,axiom,
( ord_le2705286416250468010rm_a_b
= ( ^ [A6: set_term_a_b,B6: set_term_a_b] :
! [T2: term_a_b] :
( ( member_term_a_b @ T2 @ A6 )
=> ( member_term_a_b @ T2 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_382_subset__iff,axiom,
( ord_le118470702582115849rm_a_b
= ( ^ [A6: set_Pr4386577575007340137rm_a_b,B6: set_Pr4386577575007340137rm_a_b] :
! [T2: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ T2 @ A6 )
=> ( member5869715511025134514rm_a_b @ T2 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_383_subset__iff,axiom,
( ord_le8666007276011122963_a_nat
= ( ^ [A6: set_Pr4934435412358123699_a_nat,B6: set_Pr4934435412358123699_a_nat] :
! [T2: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ T2 @ A6 )
=> ( member5724188588386418708_a_nat @ T2 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_384_equalityD2,axiom,
! [A5: set_Pr8564414093027780873rm_a_b,B5: set_Pr8564414093027780873rm_a_b] :
( ( A5 = B5 )
=> ( ord_le7771511673815134377rm_a_b @ B5 @ A5 ) ) ).
% equalityD2
thf(fact_385_equalityD2,axiom,
! [A5: set_Pr5046312416420021961st_nat,B5: set_Pr5046312416420021961st_nat] :
( ( A5 = B5 )
=> ( ord_le1912520282604406569st_nat @ B5 @ A5 ) ) ).
% equalityD2
thf(fact_386_equalityD2,axiom,
! [A5: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
( ( A5 = B5 )
=> ( ord_le3146513528884898305at_nat @ B5 @ A5 ) ) ).
% equalityD2
thf(fact_387_equalityD2,axiom,
! [A5: set_Product_prod_a_a,B5: set_Product_prod_a_a] :
( ( A5 = B5 )
=> ( ord_le746702958409616551od_a_a @ B5 @ A5 ) ) ).
% equalityD2
thf(fact_388_equalityD2,axiom,
! [A5: set_term_a_b,B5: set_term_a_b] :
( ( A5 = B5 )
=> ( ord_le2705286416250468010rm_a_b @ B5 @ A5 ) ) ).
% equalityD2
thf(fact_389_equalityD2,axiom,
! [A5: set_Pr4934435412358123699_a_nat,B5: set_Pr4934435412358123699_a_nat] :
( ( A5 = B5 )
=> ( ord_le8666007276011122963_a_nat @ B5 @ A5 ) ) ).
% equalityD2
thf(fact_390_equalityD1,axiom,
! [A5: set_Pr8564414093027780873rm_a_b,B5: set_Pr8564414093027780873rm_a_b] :
( ( A5 = B5 )
=> ( ord_le7771511673815134377rm_a_b @ A5 @ B5 ) ) ).
% equalityD1
thf(fact_391_equalityD1,axiom,
! [A5: set_Pr5046312416420021961st_nat,B5: set_Pr5046312416420021961st_nat] :
( ( A5 = B5 )
=> ( ord_le1912520282604406569st_nat @ A5 @ B5 ) ) ).
% equalityD1
thf(fact_392_equalityD1,axiom,
! [A5: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
( ( A5 = B5 )
=> ( ord_le3146513528884898305at_nat @ A5 @ B5 ) ) ).
% equalityD1
thf(fact_393_equalityD1,axiom,
! [A5: set_Product_prod_a_a,B5: set_Product_prod_a_a] :
( ( A5 = B5 )
=> ( ord_le746702958409616551od_a_a @ A5 @ B5 ) ) ).
% equalityD1
thf(fact_394_equalityD1,axiom,
! [A5: set_term_a_b,B5: set_term_a_b] :
( ( A5 = B5 )
=> ( ord_le2705286416250468010rm_a_b @ A5 @ B5 ) ) ).
% equalityD1
thf(fact_395_equalityD1,axiom,
! [A5: set_Pr4934435412358123699_a_nat,B5: set_Pr4934435412358123699_a_nat] :
( ( A5 = B5 )
=> ( ord_le8666007276011122963_a_nat @ A5 @ B5 ) ) ).
% equalityD1
thf(fact_396_subset__eq,axiom,
( ord_le8558390355660226490rm_a_b
= ( ^ [A6: set_list_term_a_b,B6: set_list_term_a_b] :
! [X4: list_term_a_b] :
( ( member_list_term_a_b @ X4 @ A6 )
=> ( member_list_term_a_b @ X4 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_397_subset__eq,axiom,
( ord_le6045566169113846134st_nat
= ( ^ [A6: set_list_nat,B6: set_list_nat] :
! [X4: list_nat] :
( ( member_list_nat @ X4 @ A6 )
=> ( member_list_nat @ X4 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_398_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A6: set_nat,B6: set_nat] :
! [X4: nat] :
( ( member_nat @ X4 @ A6 )
=> ( member_nat @ X4 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_399_subset__eq,axiom,
( ord_le7771511673815134377rm_a_b
= ( ^ [A6: set_Pr8564414093027780873rm_a_b,B6: set_Pr8564414093027780873rm_a_b] :
! [X4: produc51424535725745577rm_a_b] :
( ( member4405265420456397394rm_a_b @ X4 @ A6 )
=> ( member4405265420456397394rm_a_b @ X4 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_400_subset__eq,axiom,
( ord_le1912520282604406569st_nat
= ( ^ [A6: set_Pr5046312416420021961st_nat,B6: set_Pr5046312416420021961st_nat] :
! [X4: produc432399132543013523st_nat] :
( ( member4851138774834033962st_nat @ X4 @ A6 )
=> ( member4851138774834033962st_nat @ X4 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_401_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_402_subset__eq,axiom,
( ord_le746702958409616551od_a_a
= ( ^ [A6: set_Product_prod_a_a,B6: set_Product_prod_a_a] :
! [X4: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X4 @ A6 )
=> ( member1426531477525435216od_a_a @ X4 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_403_subset__eq,axiom,
( ord_le2705286416250468010rm_a_b
= ( ^ [A6: set_term_a_b,B6: set_term_a_b] :
! [X4: term_a_b] :
( ( member_term_a_b @ X4 @ A6 )
=> ( member_term_a_b @ X4 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_404_subset__eq,axiom,
( ord_le118470702582115849rm_a_b
= ( ^ [A6: set_Pr4386577575007340137rm_a_b,B6: set_Pr4386577575007340137rm_a_b] :
! [X4: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X4 @ A6 )
=> ( member5869715511025134514rm_a_b @ X4 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_405_subset__eq,axiom,
( ord_le8666007276011122963_a_nat
= ( ^ [A6: set_Pr4934435412358123699_a_nat,B6: set_Pr4934435412358123699_a_nat] :
! [X4: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ X4 @ A6 )
=> ( member5724188588386418708_a_nat @ X4 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_406_equalityE,axiom,
! [A5: set_Pr8564414093027780873rm_a_b,B5: set_Pr8564414093027780873rm_a_b] :
( ( A5 = B5 )
=> ~ ( ( ord_le7771511673815134377rm_a_b @ A5 @ B5 )
=> ~ ( ord_le7771511673815134377rm_a_b @ B5 @ A5 ) ) ) ).
% equalityE
thf(fact_407_equalityE,axiom,
! [A5: set_Pr5046312416420021961st_nat,B5: set_Pr5046312416420021961st_nat] :
( ( A5 = B5 )
=> ~ ( ( ord_le1912520282604406569st_nat @ A5 @ B5 )
=> ~ ( ord_le1912520282604406569st_nat @ B5 @ A5 ) ) ) ).
% equalityE
thf(fact_408_equalityE,axiom,
! [A5: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
( ( A5 = B5 )
=> ~ ( ( ord_le3146513528884898305at_nat @ A5 @ B5 )
=> ~ ( ord_le3146513528884898305at_nat @ B5 @ A5 ) ) ) ).
% equalityE
thf(fact_409_equalityE,axiom,
! [A5: set_Product_prod_a_a,B5: set_Product_prod_a_a] :
( ( A5 = B5 )
=> ~ ( ( ord_le746702958409616551od_a_a @ A5 @ B5 )
=> ~ ( ord_le746702958409616551od_a_a @ B5 @ A5 ) ) ) ).
% equalityE
thf(fact_410_equalityE,axiom,
! [A5: set_term_a_b,B5: set_term_a_b] :
( ( A5 = B5 )
=> ~ ( ( ord_le2705286416250468010rm_a_b @ A5 @ B5 )
=> ~ ( ord_le2705286416250468010rm_a_b @ B5 @ A5 ) ) ) ).
% equalityE
thf(fact_411_equalityE,axiom,
! [A5: set_Pr4934435412358123699_a_nat,B5: set_Pr4934435412358123699_a_nat] :
( ( A5 = B5 )
=> ~ ( ( ord_le8666007276011122963_a_nat @ A5 @ B5 )
=> ~ ( ord_le8666007276011122963_a_nat @ B5 @ A5 ) ) ) ).
% equalityE
thf(fact_412_subsetD,axiom,
! [A5: set_list_term_a_b,B5: set_list_term_a_b,C2: list_term_a_b] :
( ( ord_le8558390355660226490rm_a_b @ A5 @ B5 )
=> ( ( member_list_term_a_b @ C2 @ A5 )
=> ( member_list_term_a_b @ C2 @ B5 ) ) ) ).
% subsetD
thf(fact_413_subsetD,axiom,
! [A5: set_list_nat,B5: set_list_nat,C2: list_nat] :
( ( ord_le6045566169113846134st_nat @ A5 @ B5 )
=> ( ( member_list_nat @ C2 @ A5 )
=> ( member_list_nat @ C2 @ B5 ) ) ) ).
% subsetD
thf(fact_414_subsetD,axiom,
! [A5: set_nat,B5: set_nat,C2: nat] :
( ( ord_less_eq_set_nat @ A5 @ B5 )
=> ( ( member_nat @ C2 @ A5 )
=> ( member_nat @ C2 @ B5 ) ) ) ).
% subsetD
thf(fact_415_subsetD,axiom,
! [A5: set_Pr8564414093027780873rm_a_b,B5: set_Pr8564414093027780873rm_a_b,C2: produc51424535725745577rm_a_b] :
( ( ord_le7771511673815134377rm_a_b @ A5 @ B5 )
=> ( ( member4405265420456397394rm_a_b @ C2 @ A5 )
=> ( member4405265420456397394rm_a_b @ C2 @ B5 ) ) ) ).
% subsetD
thf(fact_416_subsetD,axiom,
! [A5: set_Pr5046312416420021961st_nat,B5: set_Pr5046312416420021961st_nat,C2: produc432399132543013523st_nat] :
( ( ord_le1912520282604406569st_nat @ A5 @ B5 )
=> ( ( member4851138774834033962st_nat @ C2 @ A5 )
=> ( member4851138774834033962st_nat @ C2 @ B5 ) ) ) ).
% subsetD
thf(fact_417_subsetD,axiom,
! [A5: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat,C2: product_prod_nat_nat] :
( ( ord_le3146513528884898305at_nat @ A5 @ B5 )
=> ( ( member8440522571783428010at_nat @ C2 @ A5 )
=> ( member8440522571783428010at_nat @ C2 @ B5 ) ) ) ).
% subsetD
thf(fact_418_subsetD,axiom,
! [A5: set_Product_prod_a_a,B5: set_Product_prod_a_a,C2: product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A5 @ B5 )
=> ( ( member1426531477525435216od_a_a @ C2 @ A5 )
=> ( member1426531477525435216od_a_a @ C2 @ B5 ) ) ) ).
% subsetD
thf(fact_419_subsetD,axiom,
! [A5: set_term_a_b,B5: set_term_a_b,C2: term_a_b] :
( ( ord_le2705286416250468010rm_a_b @ A5 @ B5 )
=> ( ( member_term_a_b @ C2 @ A5 )
=> ( member_term_a_b @ C2 @ B5 ) ) ) ).
% subsetD
thf(fact_420_subsetD,axiom,
! [A5: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b,C2: produc357393685978478089rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A5 @ B5 )
=> ( ( member5869715511025134514rm_a_b @ C2 @ A5 )
=> ( member5869715511025134514rm_a_b @ C2 @ B5 ) ) ) ).
% subsetD
thf(fact_421_subsetD,axiom,
! [A5: set_Pr4934435412358123699_a_nat,B5: set_Pr4934435412358123699_a_nat,C2: product_prod_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A5 @ B5 )
=> ( ( member5724188588386418708_a_nat @ C2 @ A5 )
=> ( member5724188588386418708_a_nat @ C2 @ B5 ) ) ) ).
% subsetD
thf(fact_422_in__mono,axiom,
! [A5: set_list_term_a_b,B5: set_list_term_a_b,X: list_term_a_b] :
( ( ord_le8558390355660226490rm_a_b @ A5 @ B5 )
=> ( ( member_list_term_a_b @ X @ A5 )
=> ( member_list_term_a_b @ X @ B5 ) ) ) ).
% in_mono
thf(fact_423_in__mono,axiom,
! [A5: set_list_nat,B5: set_list_nat,X: list_nat] :
( ( ord_le6045566169113846134st_nat @ A5 @ B5 )
=> ( ( member_list_nat @ X @ A5 )
=> ( member_list_nat @ X @ B5 ) ) ) ).
% in_mono
thf(fact_424_in__mono,axiom,
! [A5: set_nat,B5: set_nat,X: nat] :
( ( ord_less_eq_set_nat @ A5 @ B5 )
=> ( ( member_nat @ X @ A5 )
=> ( member_nat @ X @ B5 ) ) ) ).
% in_mono
thf(fact_425_in__mono,axiom,
! [A5: set_Pr8564414093027780873rm_a_b,B5: set_Pr8564414093027780873rm_a_b,X: produc51424535725745577rm_a_b] :
( ( ord_le7771511673815134377rm_a_b @ A5 @ B5 )
=> ( ( member4405265420456397394rm_a_b @ X @ A5 )
=> ( member4405265420456397394rm_a_b @ X @ B5 ) ) ) ).
% in_mono
thf(fact_426_in__mono,axiom,
! [A5: set_Pr5046312416420021961st_nat,B5: set_Pr5046312416420021961st_nat,X: produc432399132543013523st_nat] :
( ( ord_le1912520282604406569st_nat @ A5 @ B5 )
=> ( ( member4851138774834033962st_nat @ X @ A5 )
=> ( member4851138774834033962st_nat @ X @ B5 ) ) ) ).
% in_mono
thf(fact_427_in__mono,axiom,
! [A5: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat] :
( ( ord_le3146513528884898305at_nat @ A5 @ B5 )
=> ( ( member8440522571783428010at_nat @ X @ A5 )
=> ( member8440522571783428010at_nat @ X @ B5 ) ) ) ).
% in_mono
thf(fact_428_in__mono,axiom,
! [A5: set_Product_prod_a_a,B5: set_Product_prod_a_a,X: product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A5 @ B5 )
=> ( ( member1426531477525435216od_a_a @ X @ A5 )
=> ( member1426531477525435216od_a_a @ X @ B5 ) ) ) ).
% in_mono
thf(fact_429_in__mono,axiom,
! [A5: set_term_a_b,B5: set_term_a_b,X: term_a_b] :
( ( ord_le2705286416250468010rm_a_b @ A5 @ B5 )
=> ( ( member_term_a_b @ X @ A5 )
=> ( member_term_a_b @ X @ B5 ) ) ) ).
% in_mono
thf(fact_430_in__mono,axiom,
! [A5: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b,X: produc357393685978478089rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A5 @ B5 )
=> ( ( member5869715511025134514rm_a_b @ X @ A5 )
=> ( member5869715511025134514rm_a_b @ X @ B5 ) ) ) ).
% in_mono
thf(fact_431_in__mono,axiom,
! [A5: set_Pr4934435412358123699_a_nat,B5: set_Pr4934435412358123699_a_nat,X: product_prod_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A5 @ B5 )
=> ( ( member5724188588386418708_a_nat @ X @ A5 )
=> ( member5724188588386418708_a_nat @ X @ B5 ) ) ) ).
% in_mono
thf(fact_432_neq__if__length__neq,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( size_size_list_nat @ Xs )
!= ( size_size_list_nat @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_433_neq__if__length__neq,axiom,
! [Xs: list_list_term_a_b,Ys: list_list_term_a_b] :
( ( ( size_s877380706853472072rm_a_b @ Xs )
!= ( size_s877380706853472072rm_a_b @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_434_neq__if__length__neq,axiom,
! [Xs: list_P8875379029341186191rm_a_b,Ys: list_P8875379029341186191rm_a_b] :
( ( ( size_s8600120110137556219rm_a_b @ Xs )
!= ( size_s8600120110137556219rm_a_b @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_435_neq__if__length__neq,axiom,
! [Xs: list_P3592885314253461005_a_nat,Ys: list_P3592885314253461005_a_nat] :
( ( ( size_s984997627204368545_a_nat @ Xs )
!= ( size_s984997627204368545_a_nat @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_436_neq__if__length__neq,axiom,
! [Xs: list_term_a_b,Ys: list_term_a_b] :
( ( ( size_s8906293707977694520rm_a_b @ Xs )
!= ( size_s8906293707977694520rm_a_b @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_437_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs3: list_nat] :
( ( size_size_list_nat @ Xs3 )
= N ) ).
% Ex_list_of_length
thf(fact_438_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs3: list_list_term_a_b] :
( ( size_s877380706853472072rm_a_b @ Xs3 )
= N ) ).
% Ex_list_of_length
thf(fact_439_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs3: list_P8875379029341186191rm_a_b] :
( ( size_s8600120110137556219rm_a_b @ Xs3 )
= N ) ).
% Ex_list_of_length
thf(fact_440_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs3: list_P3592885314253461005_a_nat] :
( ( size_s984997627204368545_a_nat @ Xs3 )
= N ) ).
% Ex_list_of_length
thf(fact_441_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs3: list_term_a_b] :
( ( size_s8906293707977694520rm_a_b @ Xs3 )
= N ) ).
% Ex_list_of_length
thf(fact_442_srstep__monp,axiom,
! [F: set_Pr4934435412358123699_a_nat,G: set_Pr4934435412358123699_a_nat,R: set_Pr4386577575007340137rm_a_b] :
( ( ord_le8666007276011122963_a_nat @ F @ G )
=> ( ord_le118470702582115849rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R ) ) @ ( sig_step_a_b @ G @ ( rstep_a_b @ R ) ) ) ) ).
% srstep_monp
thf(fact_443_srsteps__eq__monp,axiom,
! [F: set_Pr4934435412358123699_a_nat,G: set_Pr4934435412358123699_a_nat,R: set_Pr4386577575007340137rm_a_b] :
( ( ord_le8666007276011122963_a_nat @ F @ G )
=> ( ord_le118470702582115849rm_a_b @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R ) ) ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ G @ ( rstep_a_b @ R ) ) ) ) ) ).
% srsteps_eq_monp
thf(fact_444_srsteps__monp,axiom,
! [F: set_Pr4934435412358123699_a_nat,G: set_Pr4934435412358123699_a_nat,R: set_Pr4386577575007340137rm_a_b] :
( ( ord_le8666007276011122963_a_nat @ F @ G )
=> ( ord_le118470702582115849rm_a_b @ ( transi7922773638565587891rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R ) ) ) @ ( transi7922773638565587891rm_a_b @ ( sig_step_a_b @ G @ ( rstep_a_b @ R ) ) ) ) ) ).
% srsteps_monp
thf(fact_445_length__induct,axiom,
! [P2: list_nat > $o,Xs: list_nat] :
( ! [Xs3: list_nat] :
( ! [Ys3: list_nat] :
( ( ord_less_nat @ ( size_size_list_nat @ Ys3 ) @ ( size_size_list_nat @ Xs3 ) )
=> ( P2 @ Ys3 ) )
=> ( P2 @ Xs3 ) )
=> ( P2 @ Xs ) ) ).
% length_induct
thf(fact_446_length__induct,axiom,
! [P2: list_list_term_a_b > $o,Xs: list_list_term_a_b] :
( ! [Xs3: list_list_term_a_b] :
( ! [Ys3: list_list_term_a_b] :
( ( ord_less_nat @ ( size_s877380706853472072rm_a_b @ Ys3 ) @ ( size_s877380706853472072rm_a_b @ Xs3 ) )
=> ( P2 @ Ys3 ) )
=> ( P2 @ Xs3 ) )
=> ( P2 @ Xs ) ) ).
% length_induct
thf(fact_447_length__induct,axiom,
! [P2: list_P8875379029341186191rm_a_b > $o,Xs: list_P8875379029341186191rm_a_b] :
( ! [Xs3: list_P8875379029341186191rm_a_b] :
( ! [Ys3: list_P8875379029341186191rm_a_b] :
( ( ord_less_nat @ ( size_s8600120110137556219rm_a_b @ Ys3 ) @ ( size_s8600120110137556219rm_a_b @ Xs3 ) )
=> ( P2 @ Ys3 ) )
=> ( P2 @ Xs3 ) )
=> ( P2 @ Xs ) ) ).
% length_induct
thf(fact_448_length__induct,axiom,
! [P2: list_P3592885314253461005_a_nat > $o,Xs: list_P3592885314253461005_a_nat] :
( ! [Xs3: list_P3592885314253461005_a_nat] :
( ! [Ys3: list_P3592885314253461005_a_nat] :
( ( ord_less_nat @ ( size_s984997627204368545_a_nat @ Ys3 ) @ ( size_s984997627204368545_a_nat @ Xs3 ) )
=> ( P2 @ Ys3 ) )
=> ( P2 @ Xs3 ) )
=> ( P2 @ Xs ) ) ).
% length_induct
thf(fact_449_length__induct,axiom,
! [P2: list_term_a_b > $o,Xs: list_term_a_b] :
( ! [Xs3: list_term_a_b] :
( ! [Ys3: list_term_a_b] :
( ( ord_less_nat @ ( size_s8906293707977694520rm_a_b @ Ys3 ) @ ( size_s8906293707977694520rm_a_b @ Xs3 ) )
=> ( P2 @ Ys3 ) )
=> ( P2 @ Xs3 ) )
=> ( P2 @ Xs ) ) ).
% length_induct
thf(fact_450_linear__term_Ocases,axiom,
! [X: term_a_b] :
( ! [Uu: b] :
( X
!= ( var_b_a @ Uu ) )
=> ~ ! [Uv: a,Ts: list_term_a_b] :
( X
!= ( fun_a_b @ Uv @ Ts ) ) ) ).
% linear_term.cases
thf(fact_451_sig__step__rsteps__dist,axiom,
! [R: set_Pr4386577575007340137rm_a_b,F: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ ( terms_7988297476397195622_a_b_b @ R ) @ F )
=> ( ( sig_step_a_b @ F @ ( transi7922773638565587891rm_a_b @ ( rstep_a_b @ R ) ) )
= ( transi7922773638565587891rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R ) ) ) ) ) ).
% sig_step_rsteps_dist
thf(fact_452_NF__I,axiom,
! [A: list_term_a_b,A5: set_Pr8564414093027780873rm_a_b] :
( ! [B3: list_term_a_b] :
~ ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ A @ B3 ) @ A5 )
=> ( member_list_term_a_b @ A @ ( abstra870812362690073880rm_a_b @ A5 ) ) ) ).
% NF_I
thf(fact_453_NF__I,axiom,
! [A: list_nat,A5: set_Pr3451248702717554689st_nat] :
( ! [B3: list_nat] :
~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A @ B3 ) @ A5 )
=> ( member_list_nat @ A @ ( abstract_NF_list_nat @ A5 ) ) ) ).
% NF_I
thf(fact_454_NF__I,axiom,
! [A: nat,A5: set_Pr1261947904930325089at_nat] :
( ! [B3: nat] :
~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B3 ) @ A5 )
=> ( member_nat @ A @ ( abstract_NF_nat @ A5 ) ) ) ).
% NF_I
thf(fact_455_NF__I,axiom,
! [A: product_prod_a_nat,A5: set_Pr1811044260758604347_a_nat] :
( ! [B3: product_prod_a_nat] :
~ ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A @ B3 ) @ A5 )
=> ( member5724188588386418708_a_nat @ A @ ( abstra2685462378305987313_a_nat @ A5 ) ) ) ).
% NF_I
thf(fact_456_NF__I,axiom,
! [A: produc357393685978478089rm_a_b,A5: set_Pr2972776593051762503rm_a_b] :
( ! [B3: produc357393685978478089rm_a_b] :
~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B3 ) @ A5 )
=> ( member5869715511025134514rm_a_b @ A @ ( abstra1902783451776770197rm_a_b @ A5 ) ) ) ).
% NF_I
thf(fact_457_NF__I,axiom,
! [A: term_a_b,A5: set_Pr4386577575007340137rm_a_b] :
( ! [B3: term_a_b] :
~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B3 ) @ A5 )
=> ( member_term_a_b @ A @ ( abstract_NF_term_a_b @ A5 ) ) ) ).
% NF_I
thf(fact_458_all__ctxt__closedD,axiom,
! [F5: set_Pr4934435412358123699_a_nat,R2: set_Pr4386577575007340137rm_a_b,F3: a,Ss2: list_term_a_b,Ts2: list_term_a_b] :
( ( terms_5226143800768910156ed_a_b @ F5 @ R2 )
=> ( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ F3 @ ( size_s8906293707977694520rm_a_b @ Ss2 ) ) @ F5 )
=> ( ( ( size_s8906293707977694520rm_a_b @ Ts2 )
= ( size_s8906293707977694520rm_a_b @ Ss2 ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s8906293707977694520rm_a_b @ Ts2 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( nth_term_a_b @ Ts2 @ I2 ) @ ( nth_term_a_b @ Ss2 @ I2 ) ) @ R2 ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s8906293707977694520rm_a_b @ Ts2 ) )
=> ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ ( nth_term_a_b @ Ts2 @ I2 ) ) @ F5 ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s8906293707977694520rm_a_b @ Ts2 ) )
=> ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ ( nth_term_a_b @ Ss2 @ I2 ) ) @ F5 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( fun_a_b @ F3 @ Ts2 ) @ ( fun_a_b @ F3 @ Ss2 ) ) @ R2 ) ) ) ) ) ) ) ).
% all_ctxt_closedD
thf(fact_459_NF__no__trancl__step,axiom,
! [A: list_term_a_b,A5: set_Pr8564414093027780873rm_a_b] :
( ( member_list_term_a_b @ A @ ( abstra870812362690073880rm_a_b @ A5 ) )
=> ! [B7: list_term_a_b] :
~ ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ A @ B7 ) @ ( transi4990936492147380931rm_a_b @ A5 ) ) ) ).
% NF_no_trancl_step
thf(fact_460_NF__no__trancl__step,axiom,
! [A: list_nat,A5: set_Pr3451248702717554689st_nat] :
( ( member_list_nat @ A @ ( abstract_NF_list_nat @ A5 ) )
=> ! [B7: list_nat] :
~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A @ B7 ) @ ( transi5465639037617667199st_nat @ A5 ) ) ) ).
% NF_no_trancl_step
thf(fact_461_NF__no__trancl__step,axiom,
! [A: product_prod_a_nat,A5: set_Pr1811044260758604347_a_nat] :
( ( member5724188588386418708_a_nat @ A @ ( abstra2685462378305987313_a_nat @ A5 ) )
=> ! [B7: product_prod_a_nat] :
~ ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A @ B7 ) @ ( transi6805586507763294364_a_nat @ A5 ) ) ) ).
% NF_no_trancl_step
thf(fact_462_NF__no__trancl__step,axiom,
! [A: a,A5: set_Product_prod_a_a] :
( ( member_a @ A @ ( abstract_NF_a @ A5 ) )
=> ! [B7: a] :
~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B7 ) @ ( transitive_trancl_a @ A5 ) ) ) ).
% NF_no_trancl_step
thf(fact_463_NF__no__trancl__step,axiom,
! [A: nat,A5: set_Pr1261947904930325089at_nat] :
( ( member_nat @ A @ ( abstract_NF_nat @ A5 ) )
=> ! [B7: nat] :
~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B7 ) @ ( transi6264000038957366511cl_nat @ A5 ) ) ) ).
% NF_no_trancl_step
thf(fact_464_NF__no__trancl__step,axiom,
! [A: produc357393685978478089rm_a_b,A5: set_Pr2972776593051762503rm_a_b] :
( ( member5869715511025134514rm_a_b @ A @ ( abstra1902783451776770197rm_a_b @ A5 ) )
=> ! [B7: produc357393685978478089rm_a_b] :
~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B7 ) @ ( transi9211502839322181930rm_a_b @ A5 ) ) ) ).
% NF_no_trancl_step
thf(fact_465_NF__no__trancl__step,axiom,
! [A: term_a_b,A5: set_Pr4386577575007340137rm_a_b] :
( ( member_term_a_b @ A @ ( abstract_NF_term_a_b @ A5 ) )
=> ! [B7: term_a_b] :
~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B7 ) @ ( transi7922773638565587891rm_a_b @ A5 ) ) ) ).
% NF_no_trancl_step
thf(fact_466_NF__anti__mono,axiom,
! [A5: set_Pr2972776593051762503rm_a_b,B5: set_Pr2972776593051762503rm_a_b] :
( ( ord_le2593958535671617255rm_a_b @ A5 @ B5 )
=> ( ord_le118470702582115849rm_a_b @ ( abstra1902783451776770197rm_a_b @ B5 ) @ ( abstra1902783451776770197rm_a_b @ A5 ) ) ) ).
% NF_anti_mono
thf(fact_467_NF__anti__mono,axiom,
! [A5: set_Pr2032521619243639111rm_a_b,B5: set_Pr2032521619243639111rm_a_b] :
( ( ord_le1207157771189532903rm_a_b @ A5 @ B5 )
=> ( ord_le7771511673815134377rm_a_b @ ( abstra6589260069541782581rm_a_b @ B5 ) @ ( abstra6589260069541782581rm_a_b @ A5 ) ) ) ).
% NF_anti_mono
thf(fact_468_NF__anti__mono,axiom,
! [A5: set_Pr2100845345442473831st_nat,B5: set_Pr2100845345442473831st_nat] :
( ( ord_le4267885275771350791st_nat @ A5 @ B5 )
=> ( ord_le1912520282604406569st_nat @ ( abstra510088194364535559st_nat @ B5 ) @ ( abstra510088194364535559st_nat @ A5 ) ) ) ).
% NF_anti_mono
thf(fact_469_NF__anti__mono,axiom,
! [A5: set_Pr8693737435421807431at_nat,B5: set_Pr8693737435421807431at_nat] :
( ( ord_le3000389064537975527at_nat @ A5 @ B5 )
=> ( ord_le3146513528884898305at_nat @ ( abstra3751743084180187021at_nat @ B5 ) @ ( abstra3751743084180187021at_nat @ A5 ) ) ) ).
% NF_anti_mono
thf(fact_470_NF__anti__mono,axiom,
! [A5: set_Pr8600417178894128327od_a_a,B5: set_Pr8600417178894128327od_a_a] :
( ( ord_le3469131294019144807od_a_a @ A5 @ B5 )
=> ( ord_le746702958409616551od_a_a @ ( abstra3815842924459568563od_a_a @ B5 ) @ ( abstra3815842924459568563od_a_a @ A5 ) ) ) ).
% NF_anti_mono
thf(fact_471_NF__anti__mono,axiom,
! [A5: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A5 @ B5 )
=> ( ord_le2705286416250468010rm_a_b @ ( abstract_NF_term_a_b @ B5 ) @ ( abstract_NF_term_a_b @ A5 ) ) ) ).
% NF_anti_mono
thf(fact_472_NF__anti__mono,axiom,
! [A5: set_Pr8564414093027780873rm_a_b,B5: set_Pr8564414093027780873rm_a_b] :
( ( ord_le7771511673815134377rm_a_b @ A5 @ B5 )
=> ( ord_le8558390355660226490rm_a_b @ ( abstra870812362690073880rm_a_b @ B5 ) @ ( abstra870812362690073880rm_a_b @ A5 ) ) ) ).
% NF_anti_mono
thf(fact_473_NF__anti__mono,axiom,
! [A5: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ A5 @ B5 )
=> ( ord_less_eq_set_nat @ ( abstract_NF_nat @ B5 ) @ ( abstract_NF_nat @ A5 ) ) ) ).
% NF_anti_mono
thf(fact_474_NF__anti__mono,axiom,
! [A5: set_Product_prod_a_a,B5: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A5 @ B5 )
=> ( ord_less_eq_set_a @ ( abstract_NF_a @ B5 ) @ ( abstract_NF_a @ A5 ) ) ) ).
% NF_anti_mono
thf(fact_475_NF__anti__mono,axiom,
! [A5: set_Pr1811044260758604347_a_nat,B5: set_Pr1811044260758604347_a_nat] :
( ( ord_le1018141841545957851_a_nat @ A5 @ B5 )
=> ( ord_le8666007276011122963_a_nat @ ( abstra2685462378305987313_a_nat @ B5 ) @ ( abstra2685462378305987313_a_nat @ A5 ) ) ) ).
% NF_anti_mono
thf(fact_476_NF__not__suc,axiom,
! [X: list_nat,Y: list_nat,R2: set_Pr3451248702717554689st_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X @ Y ) @ ( transi5285580207609517981st_nat @ R2 ) )
=> ( ( member_list_nat @ X @ ( abstract_NF_list_nat @ R2 ) )
=> ( X = Y ) ) ) ).
% NF_not_suc
thf(fact_477_NF__not__suc,axiom,
! [X: product_prod_a_nat,Y: product_prod_a_nat,R2: set_Pr1811044260758604347_a_nat] :
( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ X @ Y ) @ ( transi2726145917338391738_a_nat @ R2 ) )
=> ( ( member5724188588386418708_a_nat @ X @ ( abstra2685462378305987313_a_nat @ R2 ) )
=> ( X = Y ) ) ) ).
% NF_not_suc
thf(fact_478_NF__not__suc,axiom,
! [X: a,Y: a,R2: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ ( transitive_rtrancl_a @ R2 ) )
=> ( ( member_a @ X @ ( abstract_NF_a @ R2 ) )
=> ( X = Y ) ) ) ).
% NF_not_suc
thf(fact_479_NF__not__suc,axiom,
! [X: nat,Y: nat,R2: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( transi2905341329935302413cl_nat @ R2 ) )
=> ( ( member_nat @ X @ ( abstract_NF_nat @ R2 ) )
=> ( X = Y ) ) ) ).
% NF_not_suc
thf(fact_480_NF__not__suc,axiom,
! [X: list_term_a_b,Y: list_term_a_b,R2: set_Pr8564414093027780873rm_a_b] :
( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ X @ Y ) @ ( transi911495901722478305rm_a_b @ R2 ) )
=> ( ( member_list_term_a_b @ X @ ( abstra870812362690073880rm_a_b @ R2 ) )
=> ( X = Y ) ) ) ).
% NF_not_suc
thf(fact_481_NF__not__suc,axiom,
! [X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b,R2: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ ( transi2615809358984392588rm_a_b @ R2 ) )
=> ( ( member5869715511025134514rm_a_b @ X @ ( abstra1902783451776770197rm_a_b @ R2 ) )
=> ( X = Y ) ) ) ).
% NF_not_suc
thf(fact_482_NF__not__suc,axiom,
! [X: term_a_b,Y: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
=> ( ( member_term_a_b @ X @ ( abstract_NF_term_a_b @ R2 ) )
=> ( X = Y ) ) ) ).
% NF_not_suc
thf(fact_483_rstep__trancl__sig__step__l,axiom,
! [S: term_a_b,T: term_a_b,R: set_Pr4386577575007340137rm_a_b,F: set_Pr4934435412358123699_a_nat,V2: b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( transi7922773638565587891rm_a_b @ ( rstep_a_b @ R ) ) )
=> ( ( ord_le8666007276011122963_a_nat @ ( terms_7988297476397195622_a_b_b @ R ) @ F )
=> ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ T ) @ F )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( terms_8519481630511763164ig_a_b @ F @ V2 @ S ) @ T ) @ ( transi7922773638565587891rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R ) ) ) ) ) ) ) ).
% rstep_trancl_sig_step_l
thf(fact_484_rstep__trancl__sig__step__r,axiom,
! [S: term_a_b,T: term_a_b,R: set_Pr4386577575007340137rm_a_b,F: set_Pr4934435412358123699_a_nat,V2: b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( transi7922773638565587891rm_a_b @ ( rstep_a_b @ R ) ) )
=> ( ( ord_le8666007276011122963_a_nat @ ( terms_7988297476397195622_a_b_b @ R ) @ F )
=> ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ S ) @ F )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ ( terms_8519481630511763164ig_a_b @ F @ V2 @ T ) ) @ ( transi7922773638565587891rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R ) ) ) ) ) ) ) ).
% rstep_trancl_sig_step_r
thf(fact_485_permut__sound,axiom,
! [I4: nat,As: list_nat,F3: nat > nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ As ) )
=> ( ( nth_nat @ ( missing_permut_nat @ As @ F3 ) @ I4 )
= ( nth_nat @ As @ ( F3 @ I4 ) ) ) ) ).
% permut_sound
thf(fact_486_permut__sound,axiom,
! [I4: nat,As: list_list_term_a_b,F3: nat > nat] :
( ( ord_less_nat @ I4 @ ( size_s877380706853472072rm_a_b @ As ) )
=> ( ( nth_list_term_a_b @ ( missin6533443037858246860rm_a_b @ As @ F3 ) @ I4 )
= ( nth_list_term_a_b @ As @ ( F3 @ I4 ) ) ) ) ).
% permut_sound
thf(fact_487_permut__sound,axiom,
! [I4: nat,As: list_P8875379029341186191rm_a_b,F3: nat > nat] :
( ( ord_less_nat @ I4 @ ( size_s8600120110137556219rm_a_b @ As ) )
=> ( ( nth_Pr1098598711982373392rm_a_b @ ( missin2997301442730742753rm_a_b @ As @ F3 ) @ I4 )
= ( nth_Pr1098598711982373392rm_a_b @ As @ ( F3 @ I4 ) ) ) ) ).
% permut_sound
thf(fact_488_permut__sound,axiom,
! [I4: nat,As: list_P3592885314253461005_a_nat,F3: nat > nat] :
( ( ord_less_nat @ I4 @ ( size_s984997627204368545_a_nat @ As ) )
=> ( ( nth_Pr8461465654520414006_a_nat @ ( missin8348093053474160293_a_nat @ As @ F3 ) @ I4 )
= ( nth_Pr8461465654520414006_a_nat @ As @ ( F3 @ I4 ) ) ) ) ).
% permut_sound
thf(fact_489_permut__sound,axiom,
! [I4: nat,As: list_term_a_b,F3: nat > nat] :
( ( ord_less_nat @ I4 @ ( size_s8906293707977694520rm_a_b @ As ) )
=> ( ( nth_term_a_b @ ( missin5417461770552926780rm_a_b @ As @ F3 ) @ I4 )
= ( nth_term_a_b @ As @ ( F3 @ I4 ) ) ) ) ).
% permut_sound
thf(fact_490_rsteps__eq__relcomp__srsteps__eq__relcompI,axiom,
! [R: set_Pr4386577575007340137rm_a_b,F: set_Pr4934435412358123699_a_nat,S3: set_Pr4386577575007340137rm_a_b,S: term_a_b,T: term_a_b] :
( ( ord_le8666007276011122963_a_nat @ ( terms_7988297476397195622_a_b_b @ R ) @ F )
=> ( ( ord_le8666007276011122963_a_nat @ ( terms_7988297476397195622_a_b_b @ S3 ) @ F )
=> ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ S ) @ F )
=> ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ T ) @ F )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( rstep_a_b @ R ) ) @ ( transi7742714808557438673rm_a_b @ ( rstep_a_b @ S3 ) ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R ) ) ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ S3 ) ) ) ) ) ) ) ) ) ) ).
% rsteps_eq_relcomp_srsteps_eq_relcompI
thf(fact_491_dual__order_Orefl,axiom,
! [A: set_Pr8564414093027780873rm_a_b] : ( ord_le7771511673815134377rm_a_b @ A @ A ) ).
% dual_order.refl
thf(fact_492_dual__order_Orefl,axiom,
! [A: set_Pr5046312416420021961st_nat] : ( ord_le1912520282604406569st_nat @ A @ A ) ).
% dual_order.refl
thf(fact_493_dual__order_Orefl,axiom,
! [A: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ A @ A ) ).
% dual_order.refl
thf(fact_494_dual__order_Orefl,axiom,
! [A: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ A @ A ) ).
% dual_order.refl
thf(fact_495_dual__order_Orefl,axiom,
! [A: set_term_a_b] : ( ord_le2705286416250468010rm_a_b @ A @ A ) ).
% dual_order.refl
thf(fact_496_dual__order_Orefl,axiom,
! [A: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ A @ A ) ).
% dual_order.refl
thf(fact_497_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_498_order__refl,axiom,
! [X: set_Pr8564414093027780873rm_a_b] : ( ord_le7771511673815134377rm_a_b @ X @ X ) ).
% order_refl
thf(fact_499_order__refl,axiom,
! [X: set_Pr5046312416420021961st_nat] : ( ord_le1912520282604406569st_nat @ X @ X ) ).
% order_refl
thf(fact_500_order__refl,axiom,
! [X: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ X @ X ) ).
% order_refl
thf(fact_501_order__refl,axiom,
! [X: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ X @ X ) ).
% order_refl
thf(fact_502_order__refl,axiom,
! [X: set_term_a_b] : ( ord_le2705286416250468010rm_a_b @ X @ X ) ).
% order_refl
thf(fact_503_order__refl,axiom,
! [X: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ X @ X ) ).
% order_refl
thf(fact_504_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_505_rtrancl__idemp__self__comp,axiom,
! [R3: set_Pr2972776593051762503rm_a_b] :
( ( relcom6159724393391289352rm_a_b @ ( transi2615809358984392588rm_a_b @ R3 ) @ ( transi2615809358984392588rm_a_b @ R3 ) )
= ( transi2615809358984392588rm_a_b @ R3 ) ) ).
% rtrancl_idemp_self_comp
thf(fact_506_rtrancl__idemp__self__comp,axiom,
! [R3: set_Product_prod_a_a] :
( ( relcomp_a_a_a @ ( transitive_rtrancl_a @ R3 ) @ ( transitive_rtrancl_a @ R3 ) )
= ( transitive_rtrancl_a @ R3 ) ) ).
% rtrancl_idemp_self_comp
thf(fact_507_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_508_rtrancl__idemp__self__comp,axiom,
! [R3: set_Pr8564414093027780873rm_a_b] :
( ( relcom7449828557252701423rm_a_b @ ( transi911495901722478305rm_a_b @ R3 ) @ ( transi911495901722478305rm_a_b @ R3 ) )
= ( transi911495901722478305rm_a_b @ R3 ) ) ).
% rtrancl_idemp_self_comp
thf(fact_509_rtrancl__idemp__self__comp,axiom,
! [R3: set_Pr4386577575007340137rm_a_b] :
( ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ R3 ) @ ( transi7742714808557438673rm_a_b @ R3 ) )
= ( transi7742714808557438673rm_a_b @ R3 ) ) ).
% rtrancl_idemp_self_comp
thf(fact_510_term__to__sig__id,axiom,
! [T: term_a_b,F: set_Pr4934435412358123699_a_nat,V2: b] :
( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ T ) @ F )
=> ( ( terms_8519481630511763164ig_a_b @ F @ V2 @ T )
= T ) ) ).
% term_to_sig_id
thf(fact_511_all__ctxt__closed__sig__rsteps,axiom,
! [F: set_Pr4934435412358123699_a_nat,R: set_Pr4386577575007340137rm_a_b] : ( terms_5226143800768910156ed_a_b @ F @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R ) ) ) ) ).
% all_ctxt_closed_sig_rsteps
thf(fact_512_all__ctxt__closed__relcomp,axiom,
! [R: set_Pr4386577575007340137rm_a_b,F: set_Pr4934435412358123699_a_nat,S3: set_Pr4386577575007340137rm_a_b] :
( ! [S4: term_a_b,T3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S4 @ T3 ) @ R )
=> ( ( S4 != T3 )
=> ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ S4 ) @ F )
& ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ T3 ) @ F ) ) ) )
=> ( ! [S4: term_a_b,T3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S4 @ T3 ) @ S3 )
=> ( ( S4 != T3 )
=> ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ S4 ) @ F )
& ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ T3 ) @ F ) ) ) )
=> ( ( terms_5226143800768910156ed_a_b @ F @ R )
=> ( ( terms_5226143800768910156ed_a_b @ F @ S3 )
=> ( terms_5226143800768910156ed_a_b @ F @ ( relcom370159955682700863rm_a_b @ R @ S3 ) ) ) ) ) ) ).
% all_ctxt_closed_relcomp
thf(fact_513_relcomp3__I,axiom,
! [T: term_a_b,U: list_nat,A5: set_Pr534774883469167413st_nat,S: list_nat,B5: set_Pr7303051394255727413rm_a_b,V2: term_a_b] :
( ( member9209995263888512638st_nat @ ( produc4563063199488751885st_nat @ T @ U ) @ A5 )
=> ( ( member4000689696542324350rm_a_b @ ( produc5151171985953862413rm_a_b @ S @ T ) @ B5 )
=> ( ( member4000689696542324350rm_a_b @ ( produc5151171985953862413rm_a_b @ U @ V2 ) @ B5 )
=> ( member4000689696542324350rm_a_b @ ( produc5151171985953862413rm_a_b @ S @ V2 ) @ ( relcom3398805033718854411rm_a_b @ B5 @ ( relcom7724506178054666507rm_a_b @ A5 @ B5 ) ) ) ) ) ) ).
% relcomp3_I
thf(fact_514_relcomp3__I,axiom,
! [T: produc6551080308567045442rm_a_b,U: set_Pr4934435412358123699_a_nat,A5: set_Pr6288720886220400310_a_nat,S: set_Pr4934435412358123699_a_nat,B5: set_Pr107789759150765622rm_a_b,V2: produc6551080308567045442rm_a_b] :
( ( member4521035315342701695_a_nat @ ( produc6197040100472290702_a_nat @ T @ U ) @ A5 )
=> ( ( member4335681162695099391rm_a_b @ ( produc8030969961714872974rm_a_b @ S @ T ) @ B5 )
=> ( ( member4335681162695099391rm_a_b @ ( produc8030969961714872974rm_a_b @ U @ V2 ) @ B5 )
=> ( member4335681162695099391rm_a_b @ ( produc8030969961714872974rm_a_b @ S @ V2 ) @ ( relcom7287775472681716666rm_a_b @ B5 @ ( relcom4247060615391516346rm_a_b @ A5 @ B5 ) ) ) ) ) ) ).
% relcomp3_I
thf(fact_515_relcomp3__I,axiom,
! [T: term_a_b,U: b,A5: set_Pr3369833000147915158_a_b_b,S: b,B5: set_Pr1357592988741419896rm_a_b,V2: term_a_b] :
( ( member6959280461167840375_a_b_b @ ( produc1666658391573147538_a_b_b @ T @ U ) @ A5 )
=> ( ( member7000133282412810073rm_a_b @ ( produc1437816968797971900rm_a_b @ S @ T ) @ B5 )
=> ( ( member7000133282412810073rm_a_b @ ( produc1437816968797971900rm_a_b @ U @ V2 ) @ B5 )
=> ( member7000133282412810073rm_a_b @ ( produc1437816968797971900rm_a_b @ S @ V2 ) @ ( relcom6871247487278447976rm_a_b @ B5 @ ( relcom5423879875627813950rm_a_b @ A5 @ B5 ) ) ) ) ) ) ).
% relcomp3_I
thf(fact_516_relcomp3__I,axiom,
! [T: nat,U: a,A5: set_Pr4193341848836149977_nat_a,S: a,B5: set_Pr4934435412358123699_a_nat,V2: nat] :
( ( member8962352052110095674_nat_a @ ( product_Pair_nat_a @ T @ U ) @ A5 )
=> ( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ S @ T ) @ B5 )
=> ( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ U @ V2 ) @ B5 )
=> ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ S @ V2 ) @ ( relcomp_a_nat_nat @ B5 @ ( relcomp_nat_a_nat @ A5 @ B5 ) ) ) ) ) ) ).
% relcomp3_I
thf(fact_517_relcomp3__I,axiom,
! [T: list_nat,U: term_a_b,A5: set_Pr7303051394255727413rm_a_b,S: term_a_b,B5: set_Pr534774883469167413st_nat,V2: list_nat] :
( ( member4000689696542324350rm_a_b @ ( produc5151171985953862413rm_a_b @ T @ U ) @ A5 )
=> ( ( member9209995263888512638st_nat @ ( produc4563063199488751885st_nat @ S @ T ) @ B5 )
=> ( ( member9209995263888512638st_nat @ ( produc4563063199488751885st_nat @ U @ V2 ) @ B5 )
=> ( member9209995263888512638st_nat @ ( produc4563063199488751885st_nat @ S @ V2 ) @ ( relcom5267371577106745815st_nat @ B5 @ ( relcom941670432770933719st_nat @ A5 @ B5 ) ) ) ) ) ) ).
% relcomp3_I
thf(fact_518_relcomp3__I,axiom,
! [T: set_Pr4934435412358123699_a_nat,U: produc6551080308567045442rm_a_b,A5: set_Pr107789759150765622rm_a_b,S: produc6551080308567045442rm_a_b,B5: set_Pr6288720886220400310_a_nat,V2: set_Pr4934435412358123699_a_nat] :
( ( member4335681162695099391rm_a_b @ ( produc8030969961714872974rm_a_b @ T @ U ) @ A5 )
=> ( ( member4521035315342701695_a_nat @ ( produc6197040100472290702_a_nat @ S @ T ) @ B5 )
=> ( ( member4521035315342701695_a_nat @ ( produc6197040100472290702_a_nat @ U @ V2 ) @ B5 )
=> ( member4521035315342701695_a_nat @ ( produc6197040100472290702_a_nat @ S @ V2 ) @ ( relcom6433560692955196971_a_nat @ B5 @ ( relcom4736086928405177131_a_nat @ A5 @ B5 ) ) ) ) ) ) ).
% relcomp3_I
thf(fact_519_relcomp3__I,axiom,
! [T: nat,U: nat,A5: set_Pr1261947904930325089at_nat,S: nat,B5: set_Pr1261947904930325089at_nat,V2: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ T @ U ) @ A5 )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ S @ T ) @ B5 )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ U @ V2 ) @ B5 )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ S @ V2 ) @ ( relcomp_nat_nat_nat @ B5 @ ( relcomp_nat_nat_nat @ A5 @ B5 ) ) ) ) ) ) ).
% relcomp3_I
thf(fact_520_relcomp3__I,axiom,
! [T: b,U: term_a_b,A5: set_Pr1357592988741419896rm_a_b,S: term_a_b,B5: set_Pr3369833000147915158_a_b_b,V2: b] :
( ( member7000133282412810073rm_a_b @ ( produc1437816968797971900rm_a_b @ T @ U ) @ A5 )
=> ( ( member6959280461167840375_a_b_b @ ( produc1666658391573147538_a_b_b @ S @ T ) @ B5 )
=> ( ( member6959280461167840375_a_b_b @ ( produc1666658391573147538_a_b_b @ U @ V2 ) @ B5 )
=> ( member6959280461167840375_a_b_b @ ( produc1666658391573147538_a_b_b @ S @ V2 ) @ ( relcomp_term_a_b_b_b @ B5 @ ( relcomp_b_term_a_b_b @ A5 @ B5 ) ) ) ) ) ) ).
% relcomp3_I
thf(fact_521_relcomp3__I,axiom,
! [T: a,U: nat,A5: set_Pr4934435412358123699_a_nat,S: nat,B5: set_Pr4193341848836149977_nat_a,V2: a] :
( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ T @ U ) @ A5 )
=> ( ( member8962352052110095674_nat_a @ ( product_Pair_nat_a @ S @ T ) @ B5 )
=> ( ( member8962352052110095674_nat_a @ ( product_Pair_nat_a @ U @ V2 ) @ B5 )
=> ( member8962352052110095674_nat_a @ ( product_Pair_nat_a @ S @ V2 ) @ ( relcomp_nat_a_a @ B5 @ ( relcomp_a_nat_a @ A5 @ B5 ) ) ) ) ) ) ).
% relcomp3_I
thf(fact_522_relcomp3__I,axiom,
! [T: term_a_b,U: term_a_b,A5: set_Pr4386577575007340137rm_a_b,S: term_a_b,B5: set_Pr4386577575007340137rm_a_b,V2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T @ U ) @ A5 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ B5 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ U @ V2 ) @ B5 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ V2 ) @ ( relcom370159955682700863rm_a_b @ B5 @ ( relcom370159955682700863rm_a_b @ A5 @ B5 ) ) ) ) ) ) ).
% relcomp3_I
thf(fact_523_rtrancl__O__shift,axiom,
! [S2: set_Pr4386577575007340137rm_a_b,R3: set_Pr4386577575007340137rm_a_b] :
( ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( relcom370159955682700863rm_a_b @ S2 @ R3 ) ) @ S2 )
= ( relcom370159955682700863rm_a_b @ S2 @ ( transi7742714808557438673rm_a_b @ ( relcom370159955682700863rm_a_b @ R3 @ S2 ) ) ) ) ).
% rtrancl_O_shift
thf(fact_524_rtrancl__O__shift,axiom,
! [S2: set_Product_prod_a_a,R3: set_Product_prod_a_a] :
( ( relcomp_a_a_a @ ( transitive_rtrancl_a @ ( relcomp_a_a_a @ S2 @ R3 ) ) @ S2 )
= ( relcomp_a_a_a @ S2 @ ( transitive_rtrancl_a @ ( relcomp_a_a_a @ R3 @ S2 ) ) ) ) ).
% rtrancl_O_shift
thf(fact_525_rtrancl__O__shift,axiom,
! [S2: set_Pr4934435412358123699_a_nat,R3: set_Pr4193341848836149977_nat_a] :
( ( relcomp_a_a_nat @ ( transitive_rtrancl_a @ ( relcomp_a_nat_a @ S2 @ R3 ) ) @ S2 )
= ( relcomp_a_nat_nat @ S2 @ ( transi2905341329935302413cl_nat @ ( relcomp_nat_a_nat @ R3 @ S2 ) ) ) ) ).
% rtrancl_O_shift
thf(fact_526_rtrancl__O__shift,axiom,
! [S2: set_Pr4193341848836149977_nat_a,R3: set_Pr4934435412358123699_a_nat] :
( ( relcomp_nat_nat_a @ ( transi2905341329935302413cl_nat @ ( relcomp_nat_a_nat @ S2 @ R3 ) ) @ S2 )
= ( relcomp_nat_a_a @ S2 @ ( transitive_rtrancl_a @ ( relcomp_a_nat_a @ R3 @ S2 ) ) ) ) ).
% rtrancl_O_shift
thf(fact_527_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_528_rtrancl__O__shift,axiom,
! [S2: set_Pr3298798960101569173_a_b_a,R3: set_Pr690693513201511543rm_a_b] :
( ( relcom5652721298402989587_a_b_a @ ( transi7742714808557438673rm_a_b @ ( relcom4147987028256916669rm_a_b @ S2 @ R3 ) ) @ S2 )
= ( relcomp_term_a_b_a_a @ S2 @ ( transitive_rtrancl_a @ ( relcomp_a_term_a_b_a @ R3 @ S2 ) ) ) ) ).
% rtrancl_O_shift
thf(fact_529_rtrancl__O__shift,axiom,
! [S2: set_Pr6708357783813671845_b_nat,R3: set_Pr4549835640365387557rm_a_b] :
( ( relcom2286245697325151611_b_nat @ ( transi7742714808557438673rm_a_b @ ( relcom1324791570160179451rm_a_b @ S2 @ R3 ) ) @ S2 )
= ( relcom2781103321250786871at_nat @ S2 @ ( transi2905341329935302413cl_nat @ ( relcom152770009037377975_b_nat @ R3 @ S2 ) ) ) ) ).
% rtrancl_O_shift
thf(fact_530_rtrancl__O__shift,axiom,
! [S2: set_Pr690693513201511543rm_a_b,R3: set_Pr3298798960101569173_a_b_a] :
( ( relcomp_a_a_term_a_b @ ( transitive_rtrancl_a @ ( relcomp_a_term_a_b_a @ S2 @ R3 ) ) @ S2 )
= ( relcom7339816600239648359rm_a_b @ S2 @ ( transi7742714808557438673rm_a_b @ ( relcom4147987028256916669rm_a_b @ R3 @ S2 ) ) ) ) ).
% rtrancl_O_shift
thf(fact_531_rtrancl__O__shift,axiom,
! [S2: set_Pr4549835640365387557rm_a_b,R3: set_Pr6708357783813671845_b_nat] :
( ( relcom8414687918727181623rm_a_b @ ( transi2905341329935302413cl_nat @ ( relcom152770009037377975_b_nat @ S2 @ R3 ) ) @ S2 )
= ( relcom6565464767029197947rm_a_b @ S2 @ ( transi7742714808557438673rm_a_b @ ( relcom1324791570160179451rm_a_b @ R3 @ S2 ) ) ) ) ).
% rtrancl_O_shift
thf(fact_532_rtrancl__O__shift,axiom,
! [S2: set_Pr1610664696791676039rm_a_b,R3: set_Pr718184761707768581_a_b_a] :
( ( relcom6356982103834089637rm_a_b @ ( transitive_rtrancl_a @ ( relcom4491530472489256027_a_b_a @ S2 @ R3 ) ) @ S2 )
= ( relcom4772511747376838407rm_a_b @ S2 @ ( transi911495901722478305rm_a_b @ ( relcom1481946070270747837rm_a_b @ R3 @ S2 ) ) ) ) ).
% rtrancl_O_shift
thf(fact_533_O__mono1,axiom,
! [R3: set_Pr4386577575007340137rm_a_b,R4: set_Pr4386577575007340137rm_a_b,S2: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ R3 @ R4 )
=> ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ S2 @ R3 ) @ ( relcom370159955682700863rm_a_b @ S2 @ R4 ) ) ) ).
% O_mono1
thf(fact_534_O__mono1,axiom,
! [R3: set_Pr4934435412358123699_a_nat,R4: set_Pr4934435412358123699_a_nat,S2: set_Pr4193341848836149977_nat_a] :
( ( ord_le8666007276011122963_a_nat @ R3 @ R4 )
=> ( ord_le3146513528884898305at_nat @ ( relcomp_nat_a_nat @ S2 @ R3 ) @ ( relcomp_nat_a_nat @ S2 @ R4 ) ) ) ).
% O_mono1
thf(fact_535_O__mono1,axiom,
! [R3: set_Pr8564414093027780873rm_a_b,R4: set_Pr8564414093027780873rm_a_b,S2: set_Pr8564414093027780873rm_a_b] :
( ( ord_le7771511673815134377rm_a_b @ R3 @ R4 )
=> ( ord_le7771511673815134377rm_a_b @ ( relcom7449828557252701423rm_a_b @ S2 @ R3 ) @ ( relcom7449828557252701423rm_a_b @ S2 @ R4 ) ) ) ).
% O_mono1
thf(fact_536_O__mono1,axiom,
! [R3: set_Pr5046312416420021961st_nat,R4: set_Pr5046312416420021961st_nat,S2: set_Pr4048851178543822343list_a] :
( ( ord_le1912520282604406569st_nat @ R3 @ R4 )
=> ( ord_le1912520282604406569st_nat @ ( relcom5586209314952451681st_nat @ S2 @ R3 ) @ ( relcom5586209314952451681st_nat @ S2 @ R4 ) ) ) ).
% O_mono1
thf(fact_537_O__mono1,axiom,
! [R3: set_Pr1261947904930325089at_nat,R4: set_Pr1261947904930325089at_nat,S2: set_Pr4934435412358123699_a_nat] :
( ( ord_le3146513528884898305at_nat @ R3 @ R4 )
=> ( ord_le8666007276011122963_a_nat @ ( relcomp_a_nat_nat @ S2 @ R3 ) @ ( relcomp_a_nat_nat @ S2 @ R4 ) ) ) ).
% O_mono1
thf(fact_538_O__mono1,axiom,
! [R3: set_Pr1261947904930325089at_nat,R4: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ R3 @ R4 )
=> ( ord_le3146513528884898305at_nat @ ( relcomp_nat_nat_nat @ S2 @ R3 ) @ ( relcomp_nat_nat_nat @ S2 @ R4 ) ) ) ).
% O_mono1
thf(fact_539_O__mono1,axiom,
! [R3: set_Product_prod_a_a,R4: set_Product_prod_a_a,S2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ R3 @ R4 )
=> ( ord_le746702958409616551od_a_a @ ( relcomp_a_a_a @ S2 @ R3 ) @ ( relcomp_a_a_a @ S2 @ R4 ) ) ) ).
% O_mono1
thf(fact_540_O__mono1,axiom,
! [R3: set_Pr4934435412358123699_a_nat,R4: set_Pr4934435412358123699_a_nat,S2: set_Product_prod_a_a] :
( ( ord_le8666007276011122963_a_nat @ R3 @ R4 )
=> ( ord_le8666007276011122963_a_nat @ ( relcomp_a_a_nat @ S2 @ R3 ) @ ( relcomp_a_a_nat @ S2 @ R4 ) ) ) ).
% O_mono1
thf(fact_541_O__mono2,axiom,
! [R3: set_Pr4934435412358123699_a_nat,R4: set_Pr4934435412358123699_a_nat,T4: set_Pr4193341848836149977_nat_a] :
( ( ord_le8666007276011122963_a_nat @ R3 @ R4 )
=> ( ord_le746702958409616551od_a_a @ ( relcomp_a_nat_a @ R3 @ T4 ) @ ( relcomp_a_nat_a @ R4 @ T4 ) ) ) ).
% O_mono2
thf(fact_542_O__mono2,axiom,
! [R3: set_Pr8564414093027780873rm_a_b,R4: set_Pr8564414093027780873rm_a_b,T4: set_Pr8564414093027780873rm_a_b] :
( ( ord_le7771511673815134377rm_a_b @ R3 @ R4 )
=> ( ord_le7771511673815134377rm_a_b @ ( relcom7449828557252701423rm_a_b @ R3 @ T4 ) @ ( relcom7449828557252701423rm_a_b @ R4 @ T4 ) ) ) ).
% O_mono2
thf(fact_543_O__mono2,axiom,
! [R3: set_Pr5046312416420021961st_nat,R4: set_Pr5046312416420021961st_nat,T4: set_Pr3451248702717554689st_nat] :
( ( ord_le1912520282604406569st_nat @ R3 @ R4 )
=> ( ord_le1912520282604406569st_nat @ ( relcom2496265949195007237st_nat @ R3 @ T4 ) @ ( relcom2496265949195007237st_nat @ R4 @ T4 ) ) ) ).
% O_mono2
thf(fact_544_O__mono2,axiom,
! [R3: set_Pr1261947904930325089at_nat,R4: set_Pr1261947904930325089at_nat,T4: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ R3 @ R4 )
=> ( ord_le3146513528884898305at_nat @ ( relcomp_nat_nat_nat @ R3 @ T4 ) @ ( relcomp_nat_nat_nat @ R4 @ T4 ) ) ) ).
% O_mono2
thf(fact_545_O__mono2,axiom,
! [R3: set_Product_prod_a_a,R4: set_Product_prod_a_a,T4: set_Pr4934435412358123699_a_nat] :
( ( ord_le746702958409616551od_a_a @ R3 @ R4 )
=> ( ord_le8666007276011122963_a_nat @ ( relcomp_a_a_nat @ R3 @ T4 ) @ ( relcomp_a_a_nat @ R4 @ T4 ) ) ) ).
% O_mono2
thf(fact_546_O__mono2,axiom,
! [R3: set_Product_prod_a_a,R4: set_Product_prod_a_a,T4: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ R3 @ R4 )
=> ( ord_le746702958409616551od_a_a @ ( relcomp_a_a_a @ R3 @ T4 ) @ ( relcomp_a_a_a @ R4 @ T4 ) ) ) ).
% O_mono2
thf(fact_547_O__mono2,axiom,
! [R3: set_Pr4934435412358123699_a_nat,R4: set_Pr4934435412358123699_a_nat,T4: set_Pr1261947904930325089at_nat] :
( ( ord_le8666007276011122963_a_nat @ R3 @ R4 )
=> ( ord_le8666007276011122963_a_nat @ ( relcomp_a_nat_nat @ R3 @ T4 ) @ ( relcomp_a_nat_nat @ R4 @ T4 ) ) ) ).
% O_mono2
thf(fact_548_r__comp__rtrancl__eq,axiom,
! [R2: set_Pr4386577575007340137rm_a_b] :
( ( relcom370159955682700863rm_a_b @ R2 @ ( transi7742714808557438673rm_a_b @ R2 ) )
= ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ R2 ) @ R2 ) ) ).
% r_comp_rtrancl_eq
thf(fact_549_steps__map,axiom,
! [P2: term_a_b > $o,Q: set_Pr4386577575007340137rm_a_b > $o,F3: term_a_b > term_a_b,G2: set_Pr4386577575007340137rm_a_b > set_Pr4386577575007340137rm_a_b,T: term_a_b,R3: set_Pr4386577575007340137rm_a_b,S2: set_Pr4386577575007340137rm_a_b,U: term_a_b] :
( ! [T3: term_a_b,U2: term_a_b,R5: set_Pr4386577575007340137rm_a_b] :
( ( P2 @ T3 )
=> ( ( Q @ R5 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T3 @ U2 ) @ R5 )
=> ( ( P2 @ U2 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F3 @ T3 ) @ ( F3 @ U2 ) ) @ ( G2 @ R5 ) ) ) ) ) )
=> ( ( P2 @ T )
=> ( ( Q @ R3 )
=> ( ( Q @ S2 )
=> ( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T @ U ) @ ( transi7742714808557438673rm_a_b @ R3 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F3 @ T ) @ ( F3 @ U ) ) @ ( transi7742714808557438673rm_a_b @ ( G2 @ R3 ) ) ) )
& ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T @ U ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ R3 ) @ ( relcom370159955682700863rm_a_b @ S2 @ ( transi7742714808557438673rm_a_b @ R3 ) ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F3 @ T ) @ ( F3 @ U ) ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( G2 @ R3 ) ) @ ( relcom370159955682700863rm_a_b @ ( G2 @ S2 ) @ ( transi7742714808557438673rm_a_b @ ( G2 @ R3 ) ) ) ) ) ) ) ) ) ) ) ).
% steps_map
thf(fact_550_rtrancl__O__push,axiom,
! [S2: set_Pr4386577575007340137rm_a_b,R3: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ S2 @ R3 ) @ ( relcom370159955682700863rm_a_b @ R3 @ ( transi7742714808557438673rm_a_b @ S2 ) ) )
=> ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ S2 ) @ R3 ) @ ( relcom370159955682700863rm_a_b @ R3 @ ( transi7742714808557438673rm_a_b @ S2 ) ) ) ) ).
% rtrancl_O_push
thf(fact_551_compat__tr__compat,axiom,
! [NS: set_Product_prod_a_a,S2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ ( relcomp_a_a_nat @ NS @ S2 ) @ S2 )
=> ( ord_le8666007276011122963_a_nat @ ( relcomp_a_a_nat @ ( transitive_rtrancl_a @ NS ) @ S2 ) @ S2 ) ) ).
% compat_tr_compat
thf(fact_552_rtrancl__comp__trancl__conv,axiom,
! [R2: set_Pr4386577575007340137rm_a_b] :
( ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ R2 ) @ R2 )
= ( transi7922773638565587891rm_a_b @ R2 ) ) ).
% rtrancl_comp_trancl_conv
thf(fact_553_O__rtrancl__O__O,axiom,
! [R3: set_Pr4386577575007340137rm_a_b,S2: set_Pr4386577575007340137rm_a_b] :
( ( relcom370159955682700863rm_a_b @ R3 @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( relcom370159955682700863rm_a_b @ S2 @ R3 ) ) @ S2 ) )
= ( transi7922773638565587891rm_a_b @ ( relcom370159955682700863rm_a_b @ R3 @ S2 ) ) ) ).
% O_rtrancl_O_O
thf(fact_554_comp__trancl,axiom,
! [R3: set_Pr4386577575007340137rm_a_b,S2: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ R3 @ S2 ) @ S2 )
=> ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ R3 @ ( transi7922773638565587891rm_a_b @ S2 ) ) @ ( transi7922773638565587891rm_a_b @ S2 ) ) ) ).
% comp_trancl
thf(fact_555_term__to__sig_Osimps_I1_J,axiom,
! [F: set_Pr4934435412358123699_a_nat,V2: b,X: b] :
( ( terms_8519481630511763164ig_a_b @ F @ V2 @ ( var_b_a @ X ) )
= ( var_b_a @ X ) ) ).
% term_to_sig.simps(1)
thf(fact_556_non__strict__into__strict,axiom,
! [NS: set_Pr4386577575007340137rm_a_b,S2: set_Pr4386577575007340137rm_a_b,S: term_a_b,T: term_a_b] :
( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ NS @ S2 ) @ S2 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ NS ) @ S2 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ S2 ) ) ) ).
% non_strict_into_strict
thf(fact_557_non__strict__into__strict,axiom,
! [NS: set_Product_prod_a_a,S2: set_Pr4934435412358123699_a_nat,S: a,T: nat] :
( ( ord_le8666007276011122963_a_nat @ ( relcomp_a_a_nat @ NS @ S2 ) @ S2 )
=> ( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ S @ T ) @ ( relcomp_a_a_nat @ ( transitive_rtrancl_a @ NS ) @ S2 ) )
=> ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ S @ T ) @ S2 ) ) ) ).
% non_strict_into_strict
thf(fact_558_kleene__induct,axiom,
! [A5: set_Pr4386577575007340137rm_a_b,X7: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b,C4: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A5 @ X7 )
=> ( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ B5 @ X7 ) @ X7 )
=> ( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ X7 @ C4 ) @ X7 )
=> ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ B5 ) @ ( relcom370159955682700863rm_a_b @ A5 @ ( transi7742714808557438673rm_a_b @ C4 ) ) ) @ X7 ) ) ) ) ).
% kleene_induct
thf(fact_559_kleene__induct,axiom,
! [A5: set_Pr4934435412358123699_a_nat,X7: set_Pr4934435412358123699_a_nat,B5: set_Product_prod_a_a,C4: set_Pr1261947904930325089at_nat] :
( ( ord_le8666007276011122963_a_nat @ A5 @ X7 )
=> ( ( ord_le8666007276011122963_a_nat @ ( relcomp_a_a_nat @ B5 @ X7 ) @ X7 )
=> ( ( ord_le8666007276011122963_a_nat @ ( relcomp_a_nat_nat @ X7 @ C4 ) @ X7 )
=> ( ord_le8666007276011122963_a_nat @ ( relcomp_a_a_nat @ ( transitive_rtrancl_a @ B5 ) @ ( relcomp_a_nat_nat @ A5 @ ( transi2905341329935302413cl_nat @ C4 ) ) ) @ X7 ) ) ) ) ).
% kleene_induct
thf(fact_560_trancl__unfold__left,axiom,
( transi7922773638565587891rm_a_b
= ( ^ [R6: set_Pr4386577575007340137rm_a_b] : ( relcom370159955682700863rm_a_b @ R6 @ ( transi7742714808557438673rm_a_b @ R6 ) ) ) ) ).
% trancl_unfold_left
thf(fact_561_kleene__trancl__induct,axiom,
! [A5: set_Pr4386577575007340137rm_a_b,X7: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b,C4: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A5 @ X7 )
=> ( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ B5 @ X7 ) @ X7 )
=> ( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ X7 @ C4 ) @ X7 )
=> ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ ( transi7922773638565587891rm_a_b @ B5 ) @ ( relcom370159955682700863rm_a_b @ A5 @ ( transi7922773638565587891rm_a_b @ C4 ) ) ) @ X7 ) ) ) ) ).
% kleene_trancl_induct
thf(fact_562_kleene__trancl__induct,axiom,
! [A5: set_Pr4934435412358123699_a_nat,X7: set_Pr4934435412358123699_a_nat,B5: set_Product_prod_a_a,C4: set_Pr1261947904930325089at_nat] :
( ( ord_le8666007276011122963_a_nat @ A5 @ X7 )
=> ( ( ord_le8666007276011122963_a_nat @ ( relcomp_a_a_nat @ B5 @ X7 ) @ X7 )
=> ( ( ord_le8666007276011122963_a_nat @ ( relcomp_a_nat_nat @ X7 @ C4 ) @ X7 )
=> ( ord_le8666007276011122963_a_nat @ ( relcomp_a_a_nat @ ( transitive_trancl_a @ B5 ) @ ( relcomp_a_nat_nat @ A5 @ ( transi6264000038957366511cl_nat @ C4 ) ) ) @ X7 ) ) ) ) ).
% kleene_trancl_induct
thf(fact_563_fuans__term__term__to__sig,axiom,
! [F: set_Pr4934435412358123699_a_nat,V2: b,T: term_a_b] : ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ ( terms_8519481630511763164ig_a_b @ F @ V2 @ T ) ) @ F ) ).
% fuans_term_term_to_sig
thf(fact_564_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_565_le__cases3,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ( ord_less_eq_nat @ X @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_eq_nat @ X @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ X @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z2 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z2 @ X )
=> ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_566_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_Pr4934435412358123699_a_nat,Z3: set_Pr4934435412358123699_a_nat] : ( Y4 = Z3 ) )
= ( ^ [X4: set_Pr4934435412358123699_a_nat,Y5: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X4 @ Y5 )
& ( ord_le8666007276011122963_a_nat @ Y5 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_567_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_568_ord__eq__le__trans,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat,C2: set_Pr4934435412358123699_a_nat] :
( ( A = B )
=> ( ( ord_le8666007276011122963_a_nat @ B @ C2 )
=> ( ord_le8666007276011122963_a_nat @ A @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_569_ord__eq__le__trans,axiom,
! [A: nat,B: nat,C2: nat] :
( ( A = B )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ord_less_eq_nat @ A @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_570_ord__le__eq__trans,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat,C2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A @ B )
=> ( ( B = C2 )
=> ( ord_le8666007276011122963_a_nat @ A @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_571_ord__le__eq__trans,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( B = C2 )
=> ( ord_less_eq_nat @ A @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_572_order__antisym,axiom,
! [X: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X @ Y )
=> ( ( ord_le8666007276011122963_a_nat @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_573_order__antisym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_574_order_Otrans,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat,C2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A @ B )
=> ( ( ord_le8666007276011122963_a_nat @ B @ C2 )
=> ( ord_le8666007276011122963_a_nat @ A @ C2 ) ) ) ).
% order.trans
thf(fact_575_order_Otrans,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ord_less_eq_nat @ A @ C2 ) ) ) ).
% order.trans
thf(fact_576_order__trans,axiom,
! [X: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat,Z2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X @ Y )
=> ( ( ord_le8666007276011122963_a_nat @ Y @ Z2 )
=> ( ord_le8666007276011122963_a_nat @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_577_order__trans,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z2 )
=> ( ord_less_eq_nat @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_578_linorder__wlog,axiom,
! [P2: nat > nat > $o,A: nat,B: nat] :
( ! [A4: nat,B3: nat] :
( ( ord_less_eq_nat @ A4 @ B3 )
=> ( P2 @ A4 @ B3 ) )
=> ( ! [A4: nat,B3: nat] :
( ( P2 @ B3 @ A4 )
=> ( P2 @ A4 @ B3 ) )
=> ( P2 @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_579_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_Pr4934435412358123699_a_nat,Z3: set_Pr4934435412358123699_a_nat] : ( Y4 = Z3 ) )
= ( ^ [A3: set_Pr4934435412358123699_a_nat,B4: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ B4 @ A3 )
& ( ord_le8666007276011122963_a_nat @ A3 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_580_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: nat,Z3: nat] : ( Y4 = Z3 ) )
= ( ^ [A3: nat,B4: nat] :
( ( ord_less_eq_nat @ B4 @ A3 )
& ( ord_less_eq_nat @ A3 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_581_dual__order_Oantisym,axiom,
! [B: set_Pr4934435412358123699_a_nat,A: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ B @ A )
=> ( ( ord_le8666007276011122963_a_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_582_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_583_dual__order_Otrans,axiom,
! [B: set_Pr4934435412358123699_a_nat,A: set_Pr4934435412358123699_a_nat,C2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ B @ A )
=> ( ( ord_le8666007276011122963_a_nat @ C2 @ B )
=> ( ord_le8666007276011122963_a_nat @ C2 @ A ) ) ) ).
% dual_order.trans
thf(fact_584_dual__order_Otrans,axiom,
! [B: nat,A: nat,C2: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C2 @ B )
=> ( ord_less_eq_nat @ C2 @ A ) ) ) ).
% dual_order.trans
thf(fact_585_antisym,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A @ B )
=> ( ( ord_le8666007276011122963_a_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_586_antisym,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_587_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_Pr4934435412358123699_a_nat,Z3: set_Pr4934435412358123699_a_nat] : ( Y4 = Z3 ) )
= ( ^ [A3: set_Pr4934435412358123699_a_nat,B4: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A3 @ B4 )
& ( ord_le8666007276011122963_a_nat @ B4 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_588_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: nat,Z3: nat] : ( Y4 = Z3 ) )
= ( ^ [A3: nat,B4: nat] :
( ( ord_less_eq_nat @ A3 @ B4 )
& ( ord_less_eq_nat @ B4 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_589_order__subst1,axiom,
! [A: set_Pr4934435412358123699_a_nat,F3: set_Pr4934435412358123699_a_nat > set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat,C2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A @ ( F3 @ B ) )
=> ( ( ord_le8666007276011122963_a_nat @ B @ C2 )
=> ( ! [X3: set_Pr4934435412358123699_a_nat,Y3: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X3 @ Y3 )
=> ( ord_le8666007276011122963_a_nat @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ord_le8666007276011122963_a_nat @ A @ ( F3 @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_590_order__subst1,axiom,
! [A: set_Pr4934435412358123699_a_nat,F3: nat > set_Pr4934435412358123699_a_nat,B: nat,C2: nat] :
( ( ord_le8666007276011122963_a_nat @ A @ ( F3 @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_le8666007276011122963_a_nat @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ord_le8666007276011122963_a_nat @ A @ ( F3 @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_591_order__subst1,axiom,
! [A: nat,F3: set_Pr4934435412358123699_a_nat > nat,B: set_Pr4934435412358123699_a_nat,C2: set_Pr4934435412358123699_a_nat] :
( ( ord_less_eq_nat @ A @ ( F3 @ B ) )
=> ( ( ord_le8666007276011122963_a_nat @ B @ C2 )
=> ( ! [X3: set_Pr4934435412358123699_a_nat,Y3: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F3 @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_592_order__subst1,axiom,
! [A: nat,F3: nat > nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ ( F3 @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F3 @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_593_order__subst2,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat,F3: set_Pr4934435412358123699_a_nat > set_Pr4934435412358123699_a_nat,C2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A @ B )
=> ( ( ord_le8666007276011122963_a_nat @ ( F3 @ B ) @ C2 )
=> ( ! [X3: set_Pr4934435412358123699_a_nat,Y3: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X3 @ Y3 )
=> ( ord_le8666007276011122963_a_nat @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ord_le8666007276011122963_a_nat @ ( F3 @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_594_order__subst2,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat,F3: set_Pr4934435412358123699_a_nat > nat,C2: nat] :
( ( ord_le8666007276011122963_a_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F3 @ B ) @ C2 )
=> ( ! [X3: set_Pr4934435412358123699_a_nat,Y3: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F3 @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_595_order__subst2,axiom,
! [A: nat,B: nat,F3: nat > set_Pr4934435412358123699_a_nat,C2: set_Pr4934435412358123699_a_nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_le8666007276011122963_a_nat @ ( F3 @ B ) @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_le8666007276011122963_a_nat @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ord_le8666007276011122963_a_nat @ ( F3 @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_596_order__subst2,axiom,
! [A: nat,B: nat,F3: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F3 @ B ) @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F3 @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_597_order__eq__refl,axiom,
! [X: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat] :
( ( X = Y )
=> ( ord_le8666007276011122963_a_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_598_order__eq__refl,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_599_linorder__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_linear
thf(fact_600_ord__eq__le__subst,axiom,
! [A: set_Pr4934435412358123699_a_nat,F3: set_Pr4934435412358123699_a_nat > set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat,C2: set_Pr4934435412358123699_a_nat] :
( ( A
= ( F3 @ B ) )
=> ( ( ord_le8666007276011122963_a_nat @ B @ C2 )
=> ( ! [X3: set_Pr4934435412358123699_a_nat,Y3: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X3 @ Y3 )
=> ( ord_le8666007276011122963_a_nat @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ord_le8666007276011122963_a_nat @ A @ ( F3 @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_601_ord__eq__le__subst,axiom,
! [A: nat,F3: set_Pr4934435412358123699_a_nat > nat,B: set_Pr4934435412358123699_a_nat,C2: set_Pr4934435412358123699_a_nat] :
( ( A
= ( F3 @ B ) )
=> ( ( ord_le8666007276011122963_a_nat @ B @ C2 )
=> ( ! [X3: set_Pr4934435412358123699_a_nat,Y3: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F3 @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_602_ord__eq__le__subst,axiom,
! [A: set_Pr4934435412358123699_a_nat,F3: nat > set_Pr4934435412358123699_a_nat,B: nat,C2: nat] :
( ( A
= ( F3 @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_le8666007276011122963_a_nat @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ord_le8666007276011122963_a_nat @ A @ ( F3 @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_603_ord__eq__le__subst,axiom,
! [A: nat,F3: nat > nat,B: nat,C2: nat] :
( ( A
= ( F3 @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F3 @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_604_ord__le__eq__subst,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat,F3: set_Pr4934435412358123699_a_nat > set_Pr4934435412358123699_a_nat,C2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A @ B )
=> ( ( ( F3 @ B )
= C2 )
=> ( ! [X3: set_Pr4934435412358123699_a_nat,Y3: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X3 @ Y3 )
=> ( ord_le8666007276011122963_a_nat @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ord_le8666007276011122963_a_nat @ ( F3 @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_605_ord__le__eq__subst,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat,F3: set_Pr4934435412358123699_a_nat > nat,C2: nat] :
( ( ord_le8666007276011122963_a_nat @ A @ B )
=> ( ( ( F3 @ B )
= C2 )
=> ( ! [X3: set_Pr4934435412358123699_a_nat,Y3: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F3 @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_606_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F3: nat > set_Pr4934435412358123699_a_nat,C2: set_Pr4934435412358123699_a_nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F3 @ B )
= C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_le8666007276011122963_a_nat @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ord_le8666007276011122963_a_nat @ ( F3 @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_607_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F3: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F3 @ B )
= C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F3 @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_608_linorder__le__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_609_order__antisym__conv,axiom,
! [Y: set_Pr4934435412358123699_a_nat,X: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ Y @ X )
=> ( ( ord_le8666007276011122963_a_nat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_610_order__antisym__conv,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_611_gt__ex,axiom,
! [X: nat] :
? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% gt_ex
thf(fact_612_less__imp__neq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_613_order_Oasym,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order.asym
thf(fact_614_ord__eq__less__trans,axiom,
! [A: nat,B: nat,C2: nat] :
( ( A = B )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ord_less_nat @ A @ C2 ) ) ) ).
% ord_eq_less_trans
thf(fact_615_ord__less__eq__trans,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( B = C2 )
=> ( ord_less_nat @ A @ C2 ) ) ) ).
% ord_less_eq_trans
thf(fact_616_less__induct,axiom,
! [P2: nat > $o,A: nat] :
( ! [X3: nat] :
( ! [Y6: nat] :
( ( ord_less_nat @ Y6 @ X3 )
=> ( P2 @ Y6 ) )
=> ( P2 @ X3 ) )
=> ( P2 @ A ) ) ).
% less_induct
thf(fact_617_antisym__conv3,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_nat @ Y @ X )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_618_linorder__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_619_dual__order_Oasym,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ A @ B ) ) ).
% dual_order.asym
thf(fact_620_dual__order_Oirrefl,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_621_exists__least__iff,axiom,
( ( ^ [P3: nat > $o] :
? [X8: nat] : ( P3 @ X8 ) )
= ( ^ [P4: nat > $o] :
? [N2: nat] :
( ( P4 @ N2 )
& ! [M: nat] :
( ( ord_less_nat @ M @ N2 )
=> ~ ( P4 @ M ) ) ) ) ) ).
% exists_least_iff
thf(fact_622_linorder__less__wlog,axiom,
! [P2: nat > nat > $o,A: nat,B: nat] :
( ! [A4: nat,B3: nat] :
( ( ord_less_nat @ A4 @ B3 )
=> ( P2 @ A4 @ B3 ) )
=> ( ! [A4: nat] : ( P2 @ A4 @ A4 )
=> ( ! [A4: nat,B3: nat] :
( ( P2 @ B3 @ A4 )
=> ( P2 @ A4 @ B3 ) )
=> ( P2 @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_623_order_Ostrict__trans,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ord_less_nat @ A @ C2 ) ) ) ).
% order.strict_trans
thf(fact_624_not__less__iff__gr__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ( ord_less_nat @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_625_dual__order_Ostrict__trans,axiom,
! [B: nat,A: nat,C2: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_nat @ C2 @ B )
=> ( ord_less_nat @ C2 @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_626_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_627_dual__order_Ostrict__implies__not__eq,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_628_linorder__neqE,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_629_order__less__asym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_asym
thf(fact_630_linorder__neq__iff,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
= ( ( ord_less_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_631_order__less__asym_H,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order_less_asym'
thf(fact_632_order__less__trans,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z2 )
=> ( ord_less_nat @ X @ Z2 ) ) ) ).
% order_less_trans
thf(fact_633_ord__eq__less__subst,axiom,
! [A: nat,F3: nat > nat,B: nat,C2: nat] :
( ( A
= ( F3 @ B ) )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F3 @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_634_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F3: nat > nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F3 @ B )
= C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_nat @ ( F3 @ A ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_635_order__less__irrefl,axiom,
! [X: nat] :
~ ( ord_less_nat @ X @ X ) ).
% order_less_irrefl
thf(fact_636_order__less__subst1,axiom,
! [A: nat,F3: nat > nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ ( F3 @ B ) )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F3 @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_637_order__less__subst2,axiom,
! [A: nat,B: nat,F3: nat > nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ ( F3 @ B ) @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_nat @ ( F3 @ A ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_638_order__less__not__sym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_639_order__less__imp__triv,axiom,
! [X: nat,Y: nat,P2: $o] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ X )
=> P2 ) ) ).
% order_less_imp_triv
thf(fact_640_linorder__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
| ( X = Y )
| ( ord_less_nat @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_641_order__less__imp__not__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_642_order__less__imp__not__eq2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_643_order__less__imp__not__less,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_644_all__ctxt__closed__sig__reflE,axiom,
! [F: set_Pr4934435412358123699_a_nat,R: set_Pr4386577575007340137rm_a_b,T: term_a_b] :
( ( terms_5226143800768910156ed_a_b @ F @ R )
=> ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ T ) @ F )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T @ T ) @ R ) ) ) ).
% all_ctxt_closed_sig_reflE
thf(fact_645_leD,axiom,
! [Y: set_Pr4934435412358123699_a_nat,X: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ Y @ X )
=> ~ ( ord_le2482536469199093511_a_nat @ X @ Y ) ) ).
% leD
thf(fact_646_leD,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_nat @ X @ Y ) ) ).
% leD
thf(fact_647_leI,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% leI
thf(fact_648_nless__le,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( ~ ( ord_le2482536469199093511_a_nat @ A @ B ) )
= ( ~ ( ord_le8666007276011122963_a_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_649_nless__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_nat @ A @ B ) )
= ( ~ ( ord_less_eq_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_650_antisym__conv1,axiom,
! [X: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat] :
( ~ ( ord_le2482536469199093511_a_nat @ X @ Y )
=> ( ( ord_le8666007276011122963_a_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_651_antisym__conv1,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_652_antisym__conv2,axiom,
! [X: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X @ Y )
=> ( ( ~ ( ord_le2482536469199093511_a_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_653_antisym__conv2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_654_less__le__not__le,axiom,
( ord_le2482536469199093511_a_nat
= ( ^ [X4: set_Pr4934435412358123699_a_nat,Y5: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X4 @ Y5 )
& ~ ( ord_le8666007276011122963_a_nat @ Y5 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_655_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X4: nat,Y5: nat] :
( ( ord_less_eq_nat @ X4 @ Y5 )
& ~ ( ord_less_eq_nat @ Y5 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_656_not__le__imp__less,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_eq_nat @ Y @ X )
=> ( ord_less_nat @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_657_order_Oorder__iff__strict,axiom,
( ord_le8666007276011122963_a_nat
= ( ^ [A3: set_Pr4934435412358123699_a_nat,B4: set_Pr4934435412358123699_a_nat] :
( ( ord_le2482536469199093511_a_nat @ A3 @ B4 )
| ( A3 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_658_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B4: nat] :
( ( ord_less_nat @ A3 @ B4 )
| ( A3 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_659_order_Ostrict__iff__order,axiom,
( ord_le2482536469199093511_a_nat
= ( ^ [A3: set_Pr4934435412358123699_a_nat,B4: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A3 @ B4 )
& ( A3 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_660_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A3: nat,B4: nat] :
( ( ord_less_eq_nat @ A3 @ B4 )
& ( A3 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_661_order_Ostrict__trans1,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat,C2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A @ B )
=> ( ( ord_le2482536469199093511_a_nat @ B @ C2 )
=> ( ord_le2482536469199093511_a_nat @ A @ C2 ) ) ) ).
% order.strict_trans1
thf(fact_662_order_Ostrict__trans1,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ord_less_nat @ A @ C2 ) ) ) ).
% order.strict_trans1
thf(fact_663_order_Ostrict__trans2,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat,C2: set_Pr4934435412358123699_a_nat] :
( ( ord_le2482536469199093511_a_nat @ A @ B )
=> ( ( ord_le8666007276011122963_a_nat @ B @ C2 )
=> ( ord_le2482536469199093511_a_nat @ A @ C2 ) ) ) ).
% order.strict_trans2
thf(fact_664_order_Ostrict__trans2,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ord_less_nat @ A @ C2 ) ) ) ).
% order.strict_trans2
thf(fact_665_order_Ostrict__iff__not,axiom,
( ord_le2482536469199093511_a_nat
= ( ^ [A3: set_Pr4934435412358123699_a_nat,B4: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A3 @ B4 )
& ~ ( ord_le8666007276011122963_a_nat @ B4 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_666_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A3: nat,B4: nat] :
( ( ord_less_eq_nat @ A3 @ B4 )
& ~ ( ord_less_eq_nat @ B4 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_667_dual__order_Oorder__iff__strict,axiom,
( ord_le8666007276011122963_a_nat
= ( ^ [B4: set_Pr4934435412358123699_a_nat,A3: set_Pr4934435412358123699_a_nat] :
( ( ord_le2482536469199093511_a_nat @ B4 @ A3 )
| ( A3 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_668_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B4: nat,A3: nat] :
( ( ord_less_nat @ B4 @ A3 )
| ( A3 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_669_dual__order_Ostrict__iff__order,axiom,
( ord_le2482536469199093511_a_nat
= ( ^ [B4: set_Pr4934435412358123699_a_nat,A3: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ B4 @ A3 )
& ( A3 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_670_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B4: nat,A3: nat] :
( ( ord_less_eq_nat @ B4 @ A3 )
& ( A3 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_671_dual__order_Ostrict__trans1,axiom,
! [B: set_Pr4934435412358123699_a_nat,A: set_Pr4934435412358123699_a_nat,C2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ B @ A )
=> ( ( ord_le2482536469199093511_a_nat @ C2 @ B )
=> ( ord_le2482536469199093511_a_nat @ C2 @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_672_dual__order_Ostrict__trans1,axiom,
! [B: nat,A: nat,C2: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_nat @ C2 @ B )
=> ( ord_less_nat @ C2 @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_673_dual__order_Ostrict__trans2,axiom,
! [B: set_Pr4934435412358123699_a_nat,A: set_Pr4934435412358123699_a_nat,C2: set_Pr4934435412358123699_a_nat] :
( ( ord_le2482536469199093511_a_nat @ B @ A )
=> ( ( ord_le8666007276011122963_a_nat @ C2 @ B )
=> ( ord_le2482536469199093511_a_nat @ C2 @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_674_dual__order_Ostrict__trans2,axiom,
! [B: nat,A: nat,C2: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C2 @ B )
=> ( ord_less_nat @ C2 @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_675_dual__order_Ostrict__iff__not,axiom,
( ord_le2482536469199093511_a_nat
= ( ^ [B4: set_Pr4934435412358123699_a_nat,A3: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ B4 @ A3 )
& ~ ( ord_le8666007276011122963_a_nat @ A3 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_676_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B4: nat,A3: nat] :
( ( ord_less_eq_nat @ B4 @ A3 )
& ~ ( ord_less_eq_nat @ A3 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_677_order_Ostrict__implies__order,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( ord_le2482536469199093511_a_nat @ A @ B )
=> ( ord_le8666007276011122963_a_nat @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_678_order_Ostrict__implies__order,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_679_dual__order_Ostrict__implies__order,axiom,
! [B: set_Pr4934435412358123699_a_nat,A: set_Pr4934435412358123699_a_nat] :
( ( ord_le2482536469199093511_a_nat @ B @ A )
=> ( ord_le8666007276011122963_a_nat @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_680_dual__order_Ostrict__implies__order,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_681_order__le__less,axiom,
( ord_le8666007276011122963_a_nat
= ( ^ [X4: set_Pr4934435412358123699_a_nat,Y5: set_Pr4934435412358123699_a_nat] :
( ( ord_le2482536469199093511_a_nat @ X4 @ Y5 )
| ( X4 = Y5 ) ) ) ) ).
% order_le_less
thf(fact_682_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X4: nat,Y5: nat] :
( ( ord_less_nat @ X4 @ Y5 )
| ( X4 = Y5 ) ) ) ) ).
% order_le_less
thf(fact_683_order__less__le,axiom,
( ord_le2482536469199093511_a_nat
= ( ^ [X4: set_Pr4934435412358123699_a_nat,Y5: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X4 @ Y5 )
& ( X4 != Y5 ) ) ) ) ).
% order_less_le
thf(fact_684_order__less__le,axiom,
( ord_less_nat
= ( ^ [X4: nat,Y5: nat] :
( ( ord_less_eq_nat @ X4 @ Y5 )
& ( X4 != Y5 ) ) ) ) ).
% order_less_le
thf(fact_685_linorder__not__le,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X @ Y ) )
= ( ord_less_nat @ Y @ X ) ) ).
% linorder_not_le
thf(fact_686_linorder__not__less,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_not_less
thf(fact_687_order__less__imp__le,axiom,
! [X: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat] :
( ( ord_le2482536469199093511_a_nat @ X @ Y )
=> ( ord_le8666007276011122963_a_nat @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_688_order__less__imp__le,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_689_order__le__neq__trans,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A @ B )
=> ( ( A != B )
=> ( ord_le2482536469199093511_a_nat @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_690_order__le__neq__trans,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_691_order__neq__le__trans,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( A != B )
=> ( ( ord_le8666007276011122963_a_nat @ A @ B )
=> ( ord_le2482536469199093511_a_nat @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_692_order__neq__le__trans,axiom,
! [A: nat,B: nat] :
( ( A != B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_693_order__le__less__trans,axiom,
! [X: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat,Z2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X @ Y )
=> ( ( ord_le2482536469199093511_a_nat @ Y @ Z2 )
=> ( ord_le2482536469199093511_a_nat @ X @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_694_order__le__less__trans,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z2 )
=> ( ord_less_nat @ X @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_695_order__less__le__trans,axiom,
! [X: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat,Z2: set_Pr4934435412358123699_a_nat] :
( ( ord_le2482536469199093511_a_nat @ X @ Y )
=> ( ( ord_le8666007276011122963_a_nat @ Y @ Z2 )
=> ( ord_le2482536469199093511_a_nat @ X @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_696_order__less__le__trans,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z2 )
=> ( ord_less_nat @ X @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_697_order__le__less__subst1,axiom,
! [A: set_Pr4934435412358123699_a_nat,F3: nat > set_Pr4934435412358123699_a_nat,B: nat,C2: nat] :
( ( ord_le8666007276011122963_a_nat @ A @ ( F3 @ B ) )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_le2482536469199093511_a_nat @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ord_le2482536469199093511_a_nat @ A @ ( F3 @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_698_order__le__less__subst1,axiom,
! [A: nat,F3: nat > nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ ( F3 @ B ) )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F3 @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_699_order__le__less__subst2,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat,F3: set_Pr4934435412358123699_a_nat > set_Pr4934435412358123699_a_nat,C2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A @ B )
=> ( ( ord_le2482536469199093511_a_nat @ ( F3 @ B ) @ C2 )
=> ( ! [X3: set_Pr4934435412358123699_a_nat,Y3: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X3 @ Y3 )
=> ( ord_le8666007276011122963_a_nat @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ord_le2482536469199093511_a_nat @ ( F3 @ A ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_700_order__le__less__subst2,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat,F3: set_Pr4934435412358123699_a_nat > nat,C2: nat] :
( ( ord_le8666007276011122963_a_nat @ A @ B )
=> ( ( ord_less_nat @ ( F3 @ B ) @ C2 )
=> ( ! [X3: set_Pr4934435412358123699_a_nat,Y3: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_nat @ ( F3 @ A ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_701_order__le__less__subst2,axiom,
! [A: nat,B: nat,F3: nat > set_Pr4934435412358123699_a_nat,C2: set_Pr4934435412358123699_a_nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_le2482536469199093511_a_nat @ ( F3 @ B ) @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_le8666007276011122963_a_nat @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ord_le2482536469199093511_a_nat @ ( F3 @ A ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_702_order__le__less__subst2,axiom,
! [A: nat,B: nat,F3: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ ( F3 @ B ) @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_nat @ ( F3 @ A ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_703_order__less__le__subst1,axiom,
! [A: set_Pr4934435412358123699_a_nat,F3: set_Pr4934435412358123699_a_nat > set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat,C2: set_Pr4934435412358123699_a_nat] :
( ( ord_le2482536469199093511_a_nat @ A @ ( F3 @ B ) )
=> ( ( ord_le8666007276011122963_a_nat @ B @ C2 )
=> ( ! [X3: set_Pr4934435412358123699_a_nat,Y3: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X3 @ Y3 )
=> ( ord_le8666007276011122963_a_nat @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ord_le2482536469199093511_a_nat @ A @ ( F3 @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_704_order__less__le__subst1,axiom,
! [A: nat,F3: set_Pr4934435412358123699_a_nat > nat,B: set_Pr4934435412358123699_a_nat,C2: set_Pr4934435412358123699_a_nat] :
( ( ord_less_nat @ A @ ( F3 @ B ) )
=> ( ( ord_le8666007276011122963_a_nat @ B @ C2 )
=> ( ! [X3: set_Pr4934435412358123699_a_nat,Y3: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F3 @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_705_order__less__le__subst1,axiom,
! [A: set_Pr4934435412358123699_a_nat,F3: nat > set_Pr4934435412358123699_a_nat,B: nat,C2: nat] :
( ( ord_le2482536469199093511_a_nat @ A @ ( F3 @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_le8666007276011122963_a_nat @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ord_le2482536469199093511_a_nat @ A @ ( F3 @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_706_order__less__le__subst1,axiom,
! [A: nat,F3: nat > nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ ( F3 @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F3 @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_707_order__less__le__subst2,axiom,
! [A: nat,B: nat,F3: nat > set_Pr4934435412358123699_a_nat,C2: set_Pr4934435412358123699_a_nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_le8666007276011122963_a_nat @ ( F3 @ B ) @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_le2482536469199093511_a_nat @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ord_le2482536469199093511_a_nat @ ( F3 @ A ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_708_order__less__le__subst2,axiom,
! [A: nat,B: nat,F3: nat > nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F3 @ B ) @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_nat @ ( F3 @ A ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_709_linorder__le__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_710_order__le__imp__less__or__eq,axiom,
! [X: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X @ Y )
=> ( ( ord_le2482536469199093511_a_nat @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_711_order__le__imp__less__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_712_rstep__term__to__sig__l,axiom,
! [S: term_a_b,T: term_a_b,R: set_Pr4386577575007340137rm_a_b,F: set_Pr4934435412358123699_a_nat,V2: b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( rstep_a_b @ R ) )
=> ( ( ord_le8666007276011122963_a_nat @ ( terms_7988297476397195622_a_b_b @ R ) @ F )
=> ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ T ) @ F )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( terms_8519481630511763164ig_a_b @ F @ V2 @ S ) @ T ) @ ( rstep_a_b @ R ) ) ) ) ) ).
% rstep_term_to_sig_l
thf(fact_713_rstep__term__to__sig__r,axiom,
! [S: term_a_b,T: term_a_b,R: set_Pr4386577575007340137rm_a_b,F: set_Pr4934435412358123699_a_nat,V2: b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( rstep_a_b @ R ) )
=> ( ( ord_le8666007276011122963_a_nat @ ( terms_7988297476397195622_a_b_b @ R ) @ F )
=> ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ S ) @ F )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ ( terms_8519481630511763164ig_a_b @ F @ V2 @ T ) ) @ ( rstep_a_b @ R ) ) ) ) ) ).
% rstep_term_to_sig_r
thf(fact_714_trancl__mono__set,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ R2 @ S )
=> ( ord_le118470702582115849rm_a_b @ ( transi7922773638565587891rm_a_b @ R2 ) @ ( transi7922773638565587891rm_a_b @ S ) ) ) ).
% trancl_mono_set
thf(fact_715_NF__iff__no__step,axiom,
! [A: produc357393685978478089rm_a_b,A5: set_Pr2972776593051762503rm_a_b] :
( ( member5869715511025134514rm_a_b @ A @ ( abstra1902783451776770197rm_a_b @ A5 ) )
= ( ! [B4: produc357393685978478089rm_a_b] :
~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B4 ) @ A5 ) ) ) ).
% NF_iff_no_step
thf(fact_716_NF__iff__no__step,axiom,
! [A: term_a_b,A5: set_Pr4386577575007340137rm_a_b] :
( ( member_term_a_b @ A @ ( abstract_NF_term_a_b @ A5 ) )
= ( ! [B4: term_a_b] :
~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B4 ) @ A5 ) ) ) ).
% NF_iff_no_step
thf(fact_717_NF__no__step,axiom,
! [A: produc357393685978478089rm_a_b,A5: set_Pr2972776593051762503rm_a_b] :
( ( member5869715511025134514rm_a_b @ A @ ( abstra1902783451776770197rm_a_b @ A5 ) )
=> ! [B7: produc357393685978478089rm_a_b] :
~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B7 ) @ A5 ) ) ).
% NF_no_step
thf(fact_718_NF__no__step,axiom,
! [A: term_a_b,A5: set_Pr4386577575007340137rm_a_b] :
( ( member_term_a_b @ A @ ( abstract_NF_term_a_b @ A5 ) )
=> ! [B7: term_a_b] :
~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B7 ) @ A5 ) ) ).
% NF_no_step
thf(fact_719_NF__E,axiom,
! [A: produc357393685978478089rm_a_b,A5: set_Pr2972776593051762503rm_a_b,B: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ A @ ( abstra1902783451776770197rm_a_b @ A5 ) )
=> ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ A5 ) ) ).
% NF_E
thf(fact_720_NF__E,axiom,
! [A: term_a_b,A5: set_Pr4386577575007340137rm_a_b,B: term_a_b] :
( ( member_term_a_b @ A @ ( abstract_NF_term_a_b @ A5 ) )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ A5 ) ) ).
% NF_E
thf(fact_721_all__ctxt__closed__def,axiom,
( terms_5226143800768910156ed_a_b
= ( ^ [F6: set_Pr4934435412358123699_a_nat,R6: set_Pr4386577575007340137rm_a_b] :
( ! [F7: a,Ts3: list_term_a_b,Ss3: list_term_a_b] :
( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ F7 @ ( size_s8906293707977694520rm_a_b @ Ss3 ) ) @ F6 )
=> ( ( ( size_s8906293707977694520rm_a_b @ Ts3 )
= ( size_s8906293707977694520rm_a_b @ Ss3 ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s8906293707977694520rm_a_b @ Ts3 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( nth_term_a_b @ Ts3 @ I3 ) @ ( nth_term_a_b @ Ss3 @ I3 ) ) @ R6 ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s8906293707977694520rm_a_b @ Ts3 ) )
=> ( ord_le8666007276011122963_a_nat @ ( sup_su459911885395995103_a_nat @ ( term_funas_term_a_b @ ( nth_term_a_b @ Ts3 @ I3 ) ) @ ( term_funas_term_a_b @ ( nth_term_a_b @ Ss3 @ I3 ) ) ) @ F6 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( fun_a_b @ F7 @ Ts3 ) @ ( fun_a_b @ F7 @ Ss3 ) ) @ R6 ) ) ) ) )
& ! [X4: b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( var_b_a @ X4 ) @ ( var_b_a @ X4 ) ) @ R6 ) ) ) ) ).
% all_ctxt_closed_def
thf(fact_722_srsteps__with__root__step__def,axiom,
( srstep7844470518422762656ep_a_b
= ( ^ [F8: set_Pr4934435412358123699_a_nat,R7: set_Pr4386577575007340137rm_a_b] : ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F8 @ ( rstep_a_b @ R7 ) ) ) @ ( relcom370159955682700863rm_a_b @ ( sig_step_a_b @ F8 @ ( rrstep_a_b @ R7 ) ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F8 @ ( rstep_a_b @ R7 ) ) ) ) ) ) ) ).
% srsteps_with_root_step_def
thf(fact_723_meetI,axiom,
! [A: term_a_b,B: term_a_b,A5: set_Pr4386577575007340137rm_a_b,C2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ A5 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C2 ) @ ( transi7742714808557438673rm_a_b @ A5 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C2 ) @ ( abstra1093061187967292567rm_a_b @ A5 ) ) ) ) ).
% meetI
thf(fact_724_gterm__of__term_Ocases,axiom,
! [X: term_a_b] :
( ! [F2: a,Ts: list_term_a_b] :
( X
!= ( fun_a_b @ F2 @ Ts ) )
=> ~ ! [V: b] :
( X
!= ( var_b_a @ V ) ) ) ).
% gterm_of_term.cases
thf(fact_725_funas__term_Ocases,axiom,
! [X: term_a_b] :
( ! [X3: b] :
( X
!= ( var_b_a @ X3 ) )
=> ~ ! [F2: a,Ts: list_term_a_b] :
( X
!= ( fun_a_b @ F2 @ Ts ) ) ) ).
% funas_term.cases
thf(fact_726_relcomp__mono,axiom,
! [R8: set_Product_prod_a_a,R2: set_Product_prod_a_a,S5: set_Pr4934435412358123699_a_nat,S: set_Pr4934435412358123699_a_nat] :
( ( ord_le746702958409616551od_a_a @ R8 @ R2 )
=> ( ( ord_le8666007276011122963_a_nat @ S5 @ S )
=> ( ord_le8666007276011122963_a_nat @ ( relcomp_a_a_nat @ R8 @ S5 ) @ ( relcomp_a_a_nat @ R2 @ S ) ) ) ) ).
% relcomp_mono
thf(fact_727_relcomp__mono,axiom,
! [R8: set_Pr4934435412358123699_a_nat,R2: set_Pr4934435412358123699_a_nat,S5: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
( ( ord_le8666007276011122963_a_nat @ R8 @ R2 )
=> ( ( ord_le3146513528884898305at_nat @ S5 @ S )
=> ( ord_le8666007276011122963_a_nat @ ( relcomp_a_nat_nat @ R8 @ S5 ) @ ( relcomp_a_nat_nat @ R2 @ S ) ) ) ) ).
% relcomp_mono
thf(fact_728_srsteps__eq__subst__relcomp__closed,axiom,
! [S: term_a_b,T: term_a_b,F: set_Pr4934435412358123699_a_nat,R: set_Pr4386577575007340137rm_a_b,S3: set_Pr4386577575007340137rm_a_b,Sigma: b > term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R ) ) ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ S3 ) ) ) ) )
=> ( ! [X3: b] : ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ ( Sigma @ X3 ) ) @ F )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( subst_7999470309526761004_a_b_b @ S @ Sigma ) @ ( subst_7999470309526761004_a_b_b @ T @ Sigma ) ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R ) ) ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ S3 ) ) ) ) ) ) ) ).
% srsteps_eq_subst_relcomp_closed
thf(fact_729_restrict__rtrancl,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,S2: set_term_a_b] : ( ord_le118470702582115849rm_a_b @ ( transi7742714808557438673rm_a_b @ ( abstra6662565036008789004rm_a_b @ R2 @ S2 ) ) @ ( transi7742714808557438673rm_a_b @ R2 ) ) ).
% restrict_rtrancl
thf(fact_730_relcomp_Ocases,axiom,
! [A1: term_a_b,A22: term_a_b,R2: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A22 ) @ ( relcom370159955682700863rm_a_b @ R2 @ S ) )
=> ~ ! [B3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ B3 ) @ R2 )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B3 @ A22 ) @ S ) ) ) ).
% relcomp.cases
thf(fact_731_psubsetI,axiom,
! [A5: set_Pr4934435412358123699_a_nat,B5: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A5 @ B5 )
=> ( ( A5 != B5 )
=> ( ord_le2482536469199093511_a_nat @ A5 @ B5 ) ) ) ).
% psubsetI
thf(fact_732_UnCI,axiom,
! [C2: produc357393685978478089rm_a_b,B5: set_Pr4386577575007340137rm_a_b,A5: set_Pr4386577575007340137rm_a_b] :
( ( ~ ( member5869715511025134514rm_a_b @ C2 @ B5 )
=> ( member5869715511025134514rm_a_b @ C2 @ A5 ) )
=> ( member5869715511025134514rm_a_b @ C2 @ ( sup_su6776935440552674877rm_a_b @ A5 @ B5 ) ) ) ).
% UnCI
thf(fact_733_Un__iff,axiom,
! [C2: produc357393685978478089rm_a_b,A5: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ C2 @ ( sup_su6776935440552674877rm_a_b @ A5 @ B5 ) )
= ( ( member5869715511025134514rm_a_b @ C2 @ A5 )
| ( member5869715511025134514rm_a_b @ C2 @ B5 ) ) ) ).
% Un_iff
thf(fact_734_Un__subset__iff,axiom,
! [A5: set_Pr4934435412358123699_a_nat,B5: set_Pr4934435412358123699_a_nat,C4: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ ( sup_su459911885395995103_a_nat @ A5 @ B5 ) @ C4 )
= ( ( ord_le8666007276011122963_a_nat @ A5 @ C4 )
& ( ord_le8666007276011122963_a_nat @ B5 @ C4 ) ) ) ).
% Un_subset_iff
thf(fact_735_subst__apply__term__empty,axiom,
! [T: term_a_b] :
( ( subst_7999470309526761004_a_b_b @ T @ var_b_a )
= T ) ).
% subst_apply_term_empty
thf(fact_736_rrstep__basicI,axiom,
! [L: term_a_b,R2: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ L @ R2 ) @ R )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ L @ R2 ) @ ( rrstep_a_b @ R ) ) ) ).
% rrstep_basicI
thf(fact_737_rstep__substI,axiom,
! [S: term_a_b,T: term_a_b,R: set_Pr4386577575007340137rm_a_b,Sigma: b > term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( rstep_a_b @ R ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( subst_7999470309526761004_a_b_b @ S @ Sigma ) @ ( subst_7999470309526761004_a_b_b @ T @ Sigma ) ) @ ( rstep_a_b @ R ) ) ) ).
% rstep_substI
thf(fact_738_rrstep_Ocases,axiom,
! [A1: term_a_b,A22: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A22 ) @ ( rrstep_a_b @ R ) )
=> ~ ! [L3: term_a_b,R9: term_a_b,Sigma2: b > term_a_b] :
( ( A1
= ( subst_7999470309526761004_a_b_b @ L3 @ Sigma2 ) )
=> ( ( A22
= ( subst_7999470309526761004_a_b_b @ R9 @ Sigma2 ) )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ L3 @ R9 ) @ R ) ) ) ) ).
% rrstep.cases
thf(fact_739_rrstep_Osimps,axiom,
! [A1: term_a_b,A22: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A22 ) @ ( rrstep_a_b @ R ) )
= ( ? [L4: term_a_b,R6: term_a_b,Sigma3: b > term_a_b] :
( ( A1
= ( subst_7999470309526761004_a_b_b @ L4 @ Sigma3 ) )
& ( A22
= ( subst_7999470309526761004_a_b_b @ R6 @ Sigma3 ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ L4 @ R6 ) @ R ) ) ) ) ).
% rrstep.simps
thf(fact_740_rrstep_Ointros,axiom,
! [L: term_a_b,R2: term_a_b,R: set_Pr4386577575007340137rm_a_b,Sigma: b > term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ L @ R2 ) @ R )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( subst_7999470309526761004_a_b_b @ L @ Sigma ) @ ( subst_7999470309526761004_a_b_b @ R2 @ Sigma ) ) @ ( rrstep_a_b @ R ) ) ) ).
% rrstep.intros
thf(fact_741_rrstep__subst,axiom,
! [S: term_a_b,T: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( rrstep_a_b @ R ) )
=> ~ ! [L3: term_a_b,R9: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ L3 @ R9 ) @ R )
=> ! [Sigma2: b > term_a_b] :
( ( S
= ( subst_7999470309526761004_a_b_b @ L3 @ Sigma2 ) )
=> ( T
!= ( subst_7999470309526761004_a_b_b @ R9 @ Sigma2 ) ) ) ) ) ).
% rrstep_subst
thf(fact_742_UnE,axiom,
! [C2: produc357393685978478089rm_a_b,A5: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ C2 @ ( sup_su6776935440552674877rm_a_b @ A5 @ B5 ) )
=> ( ~ ( member5869715511025134514rm_a_b @ C2 @ A5 )
=> ( member5869715511025134514rm_a_b @ C2 @ B5 ) ) ) ).
% UnE
thf(fact_743_UnI1,axiom,
! [C2: produc357393685978478089rm_a_b,A5: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ C2 @ A5 )
=> ( member5869715511025134514rm_a_b @ C2 @ ( sup_su6776935440552674877rm_a_b @ A5 @ B5 ) ) ) ).
% UnI1
thf(fact_744_UnI2,axiom,
! [C2: produc357393685978478089rm_a_b,B5: set_Pr4386577575007340137rm_a_b,A5: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ C2 @ B5 )
=> ( member5869715511025134514rm_a_b @ C2 @ ( sup_su6776935440552674877rm_a_b @ A5 @ B5 ) ) ) ).
% UnI2
thf(fact_745_subst__apply__eq__Var,axiom,
! [S: term_a_b,Sigma: b > term_a_b,X: b] :
( ( ( subst_7999470309526761004_a_b_b @ S @ Sigma )
= ( var_b_a @ X ) )
=> ~ ! [Y3: b] :
( ( S
= ( var_b_a @ Y3 ) )
=> ( ( Sigma @ Y3 )
!= ( var_b_a @ X ) ) ) ) ).
% subst_apply_eq_Var
thf(fact_746_subset__Un__eq,axiom,
( ord_le8666007276011122963_a_nat
= ( ^ [A6: set_Pr4934435412358123699_a_nat,B6: set_Pr4934435412358123699_a_nat] :
( ( sup_su459911885395995103_a_nat @ A6 @ B6 )
= B6 ) ) ) ).
% subset_Un_eq
thf(fact_747_subset__UnE,axiom,
! [C4: set_Pr4934435412358123699_a_nat,A5: set_Pr4934435412358123699_a_nat,B5: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ C4 @ ( sup_su459911885395995103_a_nat @ A5 @ B5 ) )
=> ~ ! [A7: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A7 @ A5 )
=> ! [B8: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ B8 @ B5 )
=> ( C4
!= ( sup_su459911885395995103_a_nat @ A7 @ B8 ) ) ) ) ) ).
% subset_UnE
thf(fact_748_Un__absorb2,axiom,
! [B5: set_Pr4934435412358123699_a_nat,A5: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ B5 @ A5 )
=> ( ( sup_su459911885395995103_a_nat @ A5 @ B5 )
= A5 ) ) ).
% Un_absorb2
thf(fact_749_Un__absorb1,axiom,
! [A5: set_Pr4934435412358123699_a_nat,B5: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A5 @ B5 )
=> ( ( sup_su459911885395995103_a_nat @ A5 @ B5 )
= B5 ) ) ).
% Un_absorb1
thf(fact_750_Un__upper2,axiom,
! [B5: set_Pr4934435412358123699_a_nat,A5: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ B5 @ ( sup_su459911885395995103_a_nat @ A5 @ B5 ) ) ).
% Un_upper2
thf(fact_751_Un__upper1,axiom,
! [A5: set_Pr4934435412358123699_a_nat,B5: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ A5 @ ( sup_su459911885395995103_a_nat @ A5 @ B5 ) ) ).
% Un_upper1
thf(fact_752_Un__least,axiom,
! [A5: set_Pr4934435412358123699_a_nat,C4: set_Pr4934435412358123699_a_nat,B5: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A5 @ C4 )
=> ( ( ord_le8666007276011122963_a_nat @ B5 @ C4 )
=> ( ord_le8666007276011122963_a_nat @ ( sup_su459911885395995103_a_nat @ A5 @ B5 ) @ C4 ) ) ) ).
% Un_least
thf(fact_753_Un__mono,axiom,
! [A5: set_Pr4934435412358123699_a_nat,C4: set_Pr4934435412358123699_a_nat,B5: set_Pr4934435412358123699_a_nat,D: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A5 @ C4 )
=> ( ( ord_le8666007276011122963_a_nat @ B5 @ D )
=> ( ord_le8666007276011122963_a_nat @ ( sup_su459911885395995103_a_nat @ A5 @ B5 ) @ ( sup_su459911885395995103_a_nat @ C4 @ D ) ) ) ) ).
% Un_mono
thf(fact_754_subset__iff__psubset__eq,axiom,
( ord_le8666007276011122963_a_nat
= ( ^ [A6: set_Pr4934435412358123699_a_nat,B6: set_Pr4934435412358123699_a_nat] :
( ( ord_le2482536469199093511_a_nat @ A6 @ B6 )
| ( A6 = B6 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_755_subset__psubset__trans,axiom,
! [A5: set_Pr4934435412358123699_a_nat,B5: set_Pr4934435412358123699_a_nat,C4: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A5 @ B5 )
=> ( ( ord_le2482536469199093511_a_nat @ B5 @ C4 )
=> ( ord_le2482536469199093511_a_nat @ A5 @ C4 ) ) ) ).
% subset_psubset_trans
thf(fact_756_subset__not__subset__eq,axiom,
( ord_le2482536469199093511_a_nat
= ( ^ [A6: set_Pr4934435412358123699_a_nat,B6: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A6 @ B6 )
& ~ ( ord_le8666007276011122963_a_nat @ B6 @ A6 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_757_psubset__subset__trans,axiom,
! [A5: set_Pr4934435412358123699_a_nat,B5: set_Pr4934435412358123699_a_nat,C4: set_Pr4934435412358123699_a_nat] :
( ( ord_le2482536469199093511_a_nat @ A5 @ B5 )
=> ( ( ord_le8666007276011122963_a_nat @ B5 @ C4 )
=> ( ord_le2482536469199093511_a_nat @ A5 @ C4 ) ) ) ).
% psubset_subset_trans
thf(fact_758_psubset__imp__subset,axiom,
! [A5: set_Pr4934435412358123699_a_nat,B5: set_Pr4934435412358123699_a_nat] :
( ( ord_le2482536469199093511_a_nat @ A5 @ B5 )
=> ( ord_le8666007276011122963_a_nat @ A5 @ B5 ) ) ).
% psubset_imp_subset
thf(fact_759_psubset__eq,axiom,
( ord_le2482536469199093511_a_nat
= ( ^ [A6: set_Pr4934435412358123699_a_nat,B6: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A6 @ B6 )
& ( A6 != B6 ) ) ) ) ).
% psubset_eq
thf(fact_760_psubsetE,axiom,
! [A5: set_Pr4934435412358123699_a_nat,B5: set_Pr4934435412358123699_a_nat] :
( ( ord_le2482536469199093511_a_nat @ A5 @ B5 )
=> ~ ( ( ord_le8666007276011122963_a_nat @ A5 @ B5 )
=> ( ord_le8666007276011122963_a_nat @ B5 @ A5 ) ) ) ).
% psubsetE
thf(fact_761_rtrancl__Un__rtrancl,axiom,
! [R3: set_Pr4386577575007340137rm_a_b,S2: set_Pr4386577575007340137rm_a_b] :
( ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ ( transi7742714808557438673rm_a_b @ R3 ) @ ( transi7742714808557438673rm_a_b @ S2 ) ) )
= ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ R3 @ S2 ) ) ) ).
% rtrancl_Un_rtrancl
thf(fact_762_in__rtrancl__UnI,axiom,
! [X: produc357393685978478089rm_a_b,R3: set_Pr4386577575007340137rm_a_b,S2: set_Pr4386577575007340137rm_a_b] :
( ( ( member5869715511025134514rm_a_b @ X @ ( transi7742714808557438673rm_a_b @ R3 ) )
| ( member5869715511025134514rm_a_b @ X @ ( transi7742714808557438673rm_a_b @ S2 ) ) )
=> ( member5869715511025134514rm_a_b @ X @ ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ R3 @ S2 ) ) ) ) ).
% in_rtrancl_UnI
thf(fact_763_rtrancl__Un__separatorE,axiom,
! [A: term_a_b,B: term_a_b,P2: set_Pr4386577575007340137rm_a_b,Q: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ P2 @ Q ) ) )
=> ( ! [X3: term_a_b,Y3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ X3 ) @ ( transi7742714808557438673rm_a_b @ P2 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ Q )
=> ( X3 = Y3 ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ P2 ) ) ) ) ).
% rtrancl_Un_separatorE
thf(fact_764_rtrancl__Un__separator__converseE,axiom,
! [A: term_a_b,B: term_a_b,P2: set_Pr4386577575007340137rm_a_b,Q: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ P2 @ Q ) ) )
=> ( ! [X3: term_a_b,Y3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ B ) @ ( transi7742714808557438673rm_a_b @ P2 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ X3 ) @ Q )
=> ( Y3 = X3 ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ P2 ) ) ) ) ).
% rtrancl_Un_separator_converseE
thf(fact_765_rtrancl__Un__subset,axiom,
! [R3: set_Pr4386577575007340137rm_a_b,S2: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ ( sup_su6776935440552674877rm_a_b @ ( transi7742714808557438673rm_a_b @ R3 ) @ ( transi7742714808557438673rm_a_b @ S2 ) ) @ ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ R3 @ S2 ) ) ) ).
% rtrancl_Un_subset
thf(fact_766_rtrancl__union__subset__rtrancl__union__trancl,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
( ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ R2 @ ( transi7922773638565587891rm_a_b @ S ) ) )
= ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ R2 @ S ) ) ) ).
% rtrancl_union_subset_rtrancl_union_trancl
thf(fact_767_trancl__union__right,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ ( transi7922773638565587891rm_a_b @ R2 ) @ ( transi7922773638565587891rm_a_b @ ( sup_su6776935440552674877rm_a_b @ S @ R2 ) ) ) ).
% trancl_union_right
thf(fact_768_trancl__unfold,axiom,
( transi7922773638565587891rm_a_b
= ( ^ [R6: set_Pr4386577575007340137rm_a_b] : ( sup_su6776935440552674877rm_a_b @ R6 @ ( relcom370159955682700863rm_a_b @ ( transi7922773638565587891rm_a_b @ R6 ) @ R6 ) ) ) ) ).
% trancl_unfold
thf(fact_769_right__comp__S,axiom,
! [X: term_a_b,Y: term_a_b,S2: set_Pr4386577575007340137rm_a_b,NS: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( relcom370159955682700863rm_a_b @ S2 @ ( sup_su6776935440552674877rm_a_b @ ( relcom370159955682700863rm_a_b @ S2 @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ S2 ) @ ( transi7742714808557438673rm_a_b @ NS ) ) ) @ ( transi7742714808557438673rm_a_b @ NS ) ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( relcom370159955682700863rm_a_b @ S2 @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ S2 ) @ ( transi7742714808557438673rm_a_b @ NS ) ) ) ) ) ).
% right_comp_S
thf(fact_770_first__step__O,axiom,
! [C4: set_Pr4386577575007340137rm_a_b,A5: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b,X: term_a_b,Y: term_a_b] :
( ( C4
= ( sup_su6776935440552674877rm_a_b @ A5 @ B5 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ C4 ) @ B5 ) )
=> ? [Y3: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y3 ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ A5 ) @ B5 ) ) ) ) ).
% first_step_O
thf(fact_771_first__step,axiom,
! [C4: set_Pr4386577575007340137rm_a_b,A5: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b,X: term_a_b,Y: term_a_b,Z2: term_a_b] :
( ( C4
= ( sup_su6776935440552674877rm_a_b @ A5 @ B5 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ C4 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ Z2 ) @ B5 )
=> ? [Y3: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y3 ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ A5 ) @ B5 ) ) ) ) ) ).
% first_step
thf(fact_772_firstStep,axiom,
! [L5: set_Pr4386577575007340137rm_a_b,S2: set_Pr4386577575007340137rm_a_b,R3: set_Pr4386577575007340137rm_a_b,X: term_a_b,Y: term_a_b] :
( ( L5
= ( sup_su6776935440552674877rm_a_b @ S2 @ R3 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ L5 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ R3 ) )
| ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ R3 ) @ ( relcom370159955682700863rm_a_b @ S2 @ ( transi7742714808557438673rm_a_b @ L5 ) ) ) ) ) ) ) ).
% firstStep
thf(fact_773_compatible__conv,axiom,
! [NS: set_Pr4386577575007340137rm_a_b,S2: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ NS @ S2 ) @ S2 )
=> ( ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ NS @ S2 ) ) @ ( relcom370159955682700863rm_a_b @ S2 @ ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ NS @ S2 ) ) ) )
= ( relcom370159955682700863rm_a_b @ S2 @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ S2 ) @ ( transi7742714808557438673rm_a_b @ NS ) ) ) ) ) ).
% compatible_conv
thf(fact_774_rtrancl__U__push,axiom,
! [S2: set_Pr4386577575007340137rm_a_b,R3: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ S2 @ R3 ) @ ( relcom370159955682700863rm_a_b @ R3 @ ( transi7742714808557438673rm_a_b @ S2 ) ) )
=> ( ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ S2 @ R3 ) )
= ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ R3 ) @ ( transi7742714808557438673rm_a_b @ S2 ) ) ) ) ).
% rtrancl_U_push
thf(fact_775_rrstep__rstep__mono,axiom,
! [R: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ ( rrstep_a_b @ R ) @ ( rstep_a_b @ R ) ) ).
% rrstep_rstep_mono
thf(fact_776_compatible__rtrancl__split,axiom,
! [NS: set_Pr4386577575007340137rm_a_b,S2: set_Pr4386577575007340137rm_a_b,X: term_a_b,Y: term_a_b] :
( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ NS @ S2 ) @ S2 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ NS @ S2 ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( sup_su6776935440552674877rm_a_b @ ( relcom370159955682700863rm_a_b @ S2 @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ S2 ) @ ( transi7742714808557438673rm_a_b @ NS ) ) ) @ ( transi7742714808557438673rm_a_b @ NS ) ) ) ) ) ).
% compatible_rtrancl_split
thf(fact_777_srrstep__to__srestep,axiom,
! [S: term_a_b,T: term_a_b,F: set_Pr4934435412358123699_a_nat,R: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( sig_step_a_b @ F @ ( rrstep_a_b @ R ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( sig_step_a_b @ F @ ( rstep_a_b @ R ) ) ) ) ).
% srrstep_to_srestep
thf(fact_778_srstep__subst__closed,axiom,
! [S: term_a_b,T: term_a_b,F: set_Pr4934435412358123699_a_nat,R: set_Pr4386577575007340137rm_a_b,Sigma: b > term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( sig_step_a_b @ F @ ( rstep_a_b @ R ) ) )
=> ( ! [X3: b] : ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ ( Sigma @ X3 ) ) @ F )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( subst_7999470309526761004_a_b_b @ S @ Sigma ) @ ( subst_7999470309526761004_a_b_b @ T @ Sigma ) ) @ ( sig_step_a_b @ F @ ( rstep_a_b @ R ) ) ) ) ) ).
% srstep_subst_closed
thf(fact_779_meetD,axiom,
! [B: term_a_b,C2: term_a_b,A5: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C2 ) @ ( abstra1093061187967292567rm_a_b @ A5 ) )
=> ? [A4: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A4 @ B ) @ ( transi7742714808557438673rm_a_b @ A5 ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A4 @ C2 ) @ ( transi7742714808557438673rm_a_b @ A5 ) ) ) ) ).
% meetD
thf(fact_780_meetE,axiom,
! [B: term_a_b,C2: term_a_b,A5: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C2 ) @ ( abstra1093061187967292567rm_a_b @ A5 ) )
=> ~ ! [A4: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A4 @ B ) @ ( transi7742714808557438673rm_a_b @ A5 ) )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A4 @ C2 ) @ ( transi7742714808557438673rm_a_b @ A5 ) ) ) ) ).
% meetE
thf(fact_781_srrstep__monp,axiom,
! [F: set_Pr4934435412358123699_a_nat,G: set_Pr4934435412358123699_a_nat,R: set_Pr4386577575007340137rm_a_b] :
( ( ord_le8666007276011122963_a_nat @ F @ G )
=> ( ord_le118470702582115849rm_a_b @ ( sig_step_a_b @ F @ ( rrstep_a_b @ R ) ) @ ( sig_step_a_b @ G @ ( rrstep_a_b @ R ) ) ) ) ).
% srrstep_monp
thf(fact_782_comp__rtrancl__trancl,axiom,
! [R3: set_Pr4386577575007340137rm_a_b,S2: set_Pr4386577575007340137rm_a_b,S: term_a_b,T: term_a_b] :
( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ R3 @ S2 ) @ S2 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ R3 @ S2 ) ) @ S2 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( transi7922773638565587891rm_a_b @ S2 ) ) ) ) ).
% comp_rtrancl_trancl
thf(fact_783_srsteps__eq__subst__closed,axiom,
! [S: term_a_b,T: term_a_b,F: set_Pr4934435412358123699_a_nat,R: set_Pr4386577575007340137rm_a_b,Sigma: b > term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R ) ) ) )
=> ( ! [X3: b] : ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ ( Sigma @ X3 ) ) @ F )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( subst_7999470309526761004_a_b_b @ S @ Sigma ) @ ( subst_7999470309526761004_a_b_b @ T @ Sigma ) ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R ) ) ) ) ) ) ).
% srsteps_eq_subst_closed
thf(fact_784_srsteps__subst__closed,axiom,
! [S: term_a_b,T: term_a_b,F: set_Pr4934435412358123699_a_nat,R: set_Pr4386577575007340137rm_a_b,Sigma: b > term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( transi7922773638565587891rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R ) ) ) )
=> ( ! [X3: b] : ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ ( Sigma @ X3 ) ) @ F )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( subst_7999470309526761004_a_b_b @ S @ Sigma ) @ ( subst_7999470309526761004_a_b_b @ T @ Sigma ) ) @ ( transi7922773638565587891rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R ) ) ) ) ) ) ).
% srsteps_subst_closed
thf(fact_785_srrstepD,axiom,
! [S: term_a_b,T: term_a_b,F: set_Pr4934435412358123699_a_nat,R: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( sig_step_a_b @ F @ ( rrstep_a_b @ R ) ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( rrstep_a_b @ R ) )
& ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ S ) @ F )
& ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ T ) @ F ) ) ) ).
% srrstepD
thf(fact_786_subrelI,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
( ! [X3: term_a_b,Y3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ R2 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ S ) )
=> ( ord_le118470702582115849rm_a_b @ R2 @ S ) ) ).
% subrelI
thf(fact_787_subrelI,axiom,
! [R2: set_Pr4934435412358123699_a_nat,S: set_Pr4934435412358123699_a_nat] :
( ! [X3: a,Y3: nat] :
( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ X3 @ Y3 ) @ R2 )
=> ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ X3 @ Y3 ) @ S ) )
=> ( ord_le8666007276011122963_a_nat @ R2 @ S ) ) ).
% subrelI
thf(fact_788_relcompEpair,axiom,
! [A: term_a_b,C2: term_a_b,R2: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C2 ) @ ( relcom370159955682700863rm_a_b @ R2 @ S ) )
=> ~ ! [B3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B3 ) @ R2 )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B3 @ C2 ) @ S ) ) ) ).
% relcompEpair
thf(fact_789_relcompE,axiom,
! [Xz: produc357393685978478089rm_a_b,R2: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ Xz @ ( relcom370159955682700863rm_a_b @ R2 @ S ) )
=> ~ ! [X3: term_a_b,Y3: term_a_b,Z: term_a_b] :
( ( Xz
= ( produc7020197800436672577rm_a_b @ X3 @ Z ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ R2 )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ Z ) @ S ) ) ) ) ).
% relcompE
thf(fact_790_relcomp_OrelcompI,axiom,
! [A: term_a_b,B: term_a_b,R2: set_Pr4386577575007340137rm_a_b,C2: term_a_b,S: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R2 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C2 ) @ S )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C2 ) @ ( relcom370159955682700863rm_a_b @ R2 @ S ) ) ) ) ).
% relcomp.relcompI
thf(fact_791_relcomp_Osimps,axiom,
! [A1: term_a_b,A22: term_a_b,R2: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A22 ) @ ( relcom370159955682700863rm_a_b @ R2 @ S ) )
= ( ? [A3: term_a_b,B4: term_a_b,C: term_a_b] :
( ( A1 = A3 )
& ( A22 = C )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B4 ) @ R2 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B4 @ C ) @ S ) ) ) ) ).
% relcomp.simps
thf(fact_792_sup_Obounded__iff,axiom,
! [B: set_Pr4934435412358123699_a_nat,C2: set_Pr4934435412358123699_a_nat,A: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ ( sup_su459911885395995103_a_nat @ B @ C2 ) @ A )
= ( ( ord_le8666007276011122963_a_nat @ B @ A )
& ( ord_le8666007276011122963_a_nat @ C2 @ A ) ) ) ).
% sup.bounded_iff
thf(fact_793_sup_Obounded__iff,axiom,
! [B: nat,C2: nat,A: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C2 ) @ A )
= ( ( ord_less_eq_nat @ B @ A )
& ( ord_less_eq_nat @ C2 @ A ) ) ) ).
% sup.bounded_iff
thf(fact_794_le__sup__iff,axiom,
! [X: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat,Z2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ ( sup_su459911885395995103_a_nat @ X @ Y ) @ Z2 )
= ( ( ord_le8666007276011122963_a_nat @ X @ Z2 )
& ( ord_le8666007276011122963_a_nat @ Y @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_795_le__sup__iff,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ X @ Y ) @ Z2 )
= ( ( ord_less_eq_nat @ X @ Z2 )
& ( ord_less_eq_nat @ Y @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_796_quasi__commute__def,axiom,
( abstra4317384144805385751rm_a_b
= ( ^ [R6: set_Pr4386577575007340137rm_a_b,S6: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ S6 @ R6 ) @ ( relcom370159955682700863rm_a_b @ R6 @ ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ R6 @ S6 ) ) ) ) ) ) ).
% quasi_commute_def
thf(fact_797_psubsetD,axiom,
! [A5: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b,C2: produc357393685978478089rm_a_b] :
( ( ord_le4883060154188259093rm_a_b @ A5 @ B5 )
=> ( ( member5869715511025134514rm_a_b @ C2 @ A5 )
=> ( member5869715511025134514rm_a_b @ C2 @ B5 ) ) ) ).
% psubsetD
thf(fact_798_qc__imp__qc__trancl,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
( ( abstra4317384144805385751rm_a_b @ R2 @ S )
=> ( abstra4317384144805385751rm_a_b @ R2 @ ( transi7922773638565587891rm_a_b @ S ) ) ) ).
% qc_imp_qc_trancl
thf(fact_799_inf__sup__ord_I4_J,axiom,
! [Y: set_Pr4934435412358123699_a_nat,X: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ Y @ ( sup_su459911885395995103_a_nat @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_800_inf__sup__ord_I4_J,axiom,
! [Y: nat,X: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_801_inf__sup__ord_I3_J,axiom,
! [X: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ X @ ( sup_su459911885395995103_a_nat @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_802_inf__sup__ord_I3_J,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ X @ ( sup_sup_nat @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_803_le__supE,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat,X: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ ( sup_su459911885395995103_a_nat @ A @ B ) @ X )
=> ~ ( ( ord_le8666007276011122963_a_nat @ A @ X )
=> ~ ( ord_le8666007276011122963_a_nat @ B @ X ) ) ) ).
% le_supE
thf(fact_804_le__supE,axiom,
! [A: nat,B: nat,X: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B ) @ X )
=> ~ ( ( ord_less_eq_nat @ A @ X )
=> ~ ( ord_less_eq_nat @ B @ X ) ) ) ).
% le_supE
thf(fact_805_le__supI,axiom,
! [A: set_Pr4934435412358123699_a_nat,X: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A @ X )
=> ( ( ord_le8666007276011122963_a_nat @ B @ X )
=> ( ord_le8666007276011122963_a_nat @ ( sup_su459911885395995103_a_nat @ A @ B ) @ X ) ) ) ).
% le_supI
thf(fact_806_le__supI,axiom,
! [A: nat,X: nat,B: nat] :
( ( ord_less_eq_nat @ A @ X )
=> ( ( ord_less_eq_nat @ B @ X )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B ) @ X ) ) ) ).
% le_supI
thf(fact_807_sup__ge1,axiom,
! [X: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ X @ ( sup_su459911885395995103_a_nat @ X @ Y ) ) ).
% sup_ge1
thf(fact_808_sup__ge1,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ X @ ( sup_sup_nat @ X @ Y ) ) ).
% sup_ge1
thf(fact_809_sup__ge2,axiom,
! [Y: set_Pr4934435412358123699_a_nat,X: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ Y @ ( sup_su459911885395995103_a_nat @ X @ Y ) ) ).
% sup_ge2
thf(fact_810_sup__ge2,axiom,
! [Y: nat,X: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X @ Y ) ) ).
% sup_ge2
thf(fact_811_le__supI1,axiom,
! [X: set_Pr4934435412358123699_a_nat,A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X @ A )
=> ( ord_le8666007276011122963_a_nat @ X @ ( sup_su459911885395995103_a_nat @ A @ B ) ) ) ).
% le_supI1
thf(fact_812_le__supI1,axiom,
! [X: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ X @ A )
=> ( ord_less_eq_nat @ X @ ( sup_sup_nat @ A @ B ) ) ) ).
% le_supI1
thf(fact_813_le__supI2,axiom,
! [X: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat,A: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X @ B )
=> ( ord_le8666007276011122963_a_nat @ X @ ( sup_su459911885395995103_a_nat @ A @ B ) ) ) ).
% le_supI2
thf(fact_814_le__supI2,axiom,
! [X: nat,B: nat,A: nat] :
( ( ord_less_eq_nat @ X @ B )
=> ( ord_less_eq_nat @ X @ ( sup_sup_nat @ A @ B ) ) ) ).
% le_supI2
thf(fact_815_sup_Omono,axiom,
! [C2: set_Pr4934435412358123699_a_nat,A: set_Pr4934435412358123699_a_nat,D2: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ C2 @ A )
=> ( ( ord_le8666007276011122963_a_nat @ D2 @ B )
=> ( ord_le8666007276011122963_a_nat @ ( sup_su459911885395995103_a_nat @ C2 @ D2 ) @ ( sup_su459911885395995103_a_nat @ A @ B ) ) ) ) ).
% sup.mono
thf(fact_816_sup_Omono,axiom,
! [C2: nat,A: nat,D2: nat,B: nat] :
( ( ord_less_eq_nat @ C2 @ A )
=> ( ( ord_less_eq_nat @ D2 @ B )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ C2 @ D2 ) @ ( sup_sup_nat @ A @ B ) ) ) ) ).
% sup.mono
thf(fact_817_sup__mono,axiom,
! [A: set_Pr4934435412358123699_a_nat,C2: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat,D2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A @ C2 )
=> ( ( ord_le8666007276011122963_a_nat @ B @ D2 )
=> ( ord_le8666007276011122963_a_nat @ ( sup_su459911885395995103_a_nat @ A @ B ) @ ( sup_su459911885395995103_a_nat @ C2 @ D2 ) ) ) ) ).
% sup_mono
thf(fact_818_sup__mono,axiom,
! [A: nat,C2: nat,B: nat,D2: nat] :
( ( ord_less_eq_nat @ A @ C2 )
=> ( ( ord_less_eq_nat @ B @ D2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B ) @ ( sup_sup_nat @ C2 @ D2 ) ) ) ) ).
% sup_mono
thf(fact_819_sup__least,axiom,
! [Y: set_Pr4934435412358123699_a_nat,X: set_Pr4934435412358123699_a_nat,Z2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ Y @ X )
=> ( ( ord_le8666007276011122963_a_nat @ Z2 @ X )
=> ( ord_le8666007276011122963_a_nat @ ( sup_su459911885395995103_a_nat @ Y @ Z2 ) @ X ) ) ) ).
% sup_least
thf(fact_820_sup__least,axiom,
! [Y: nat,X: nat,Z2: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_less_eq_nat @ Z2 @ X )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ Y @ Z2 ) @ X ) ) ) ).
% sup_least
thf(fact_821_le__iff__sup,axiom,
( ord_le8666007276011122963_a_nat
= ( ^ [X4: set_Pr4934435412358123699_a_nat,Y5: set_Pr4934435412358123699_a_nat] :
( ( sup_su459911885395995103_a_nat @ X4 @ Y5 )
= Y5 ) ) ) ).
% le_iff_sup
thf(fact_822_le__iff__sup,axiom,
( ord_less_eq_nat
= ( ^ [X4: nat,Y5: nat] :
( ( sup_sup_nat @ X4 @ Y5 )
= Y5 ) ) ) ).
% le_iff_sup
thf(fact_823_sup_OorderE,axiom,
! [B: set_Pr4934435412358123699_a_nat,A: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ B @ A )
=> ( A
= ( sup_su459911885395995103_a_nat @ A @ B ) ) ) ).
% sup.orderE
thf(fact_824_sup_OorderE,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( A
= ( sup_sup_nat @ A @ B ) ) ) ).
% sup.orderE
thf(fact_825_sup_OorderI,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( A
= ( sup_su459911885395995103_a_nat @ A @ B ) )
=> ( ord_le8666007276011122963_a_nat @ B @ A ) ) ).
% sup.orderI
thf(fact_826_sup_OorderI,axiom,
! [A: nat,B: nat] :
( ( A
= ( sup_sup_nat @ A @ B ) )
=> ( ord_less_eq_nat @ B @ A ) ) ).
% sup.orderI
thf(fact_827_sup__unique,axiom,
! [F3: set_Pr4934435412358123699_a_nat > set_Pr4934435412358123699_a_nat > set_Pr4934435412358123699_a_nat,X: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat] :
( ! [X3: set_Pr4934435412358123699_a_nat,Y3: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ X3 @ ( F3 @ X3 @ Y3 ) )
=> ( ! [X3: set_Pr4934435412358123699_a_nat,Y3: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ Y3 @ ( F3 @ X3 @ Y3 ) )
=> ( ! [X3: set_Pr4934435412358123699_a_nat,Y3: set_Pr4934435412358123699_a_nat,Z: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ Y3 @ X3 )
=> ( ( ord_le8666007276011122963_a_nat @ Z @ X3 )
=> ( ord_le8666007276011122963_a_nat @ ( F3 @ Y3 @ Z ) @ X3 ) ) )
=> ( ( sup_su459911885395995103_a_nat @ X @ Y )
= ( F3 @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_828_sup__unique,axiom,
! [F3: nat > nat > nat,X: nat,Y: nat] :
( ! [X3: nat,Y3: nat] : ( ord_less_eq_nat @ X3 @ ( F3 @ X3 @ Y3 ) )
=> ( ! [X3: nat,Y3: nat] : ( ord_less_eq_nat @ Y3 @ ( F3 @ X3 @ Y3 ) )
=> ( ! [X3: nat,Y3: nat,Z: nat] :
( ( ord_less_eq_nat @ Y3 @ X3 )
=> ( ( ord_less_eq_nat @ Z @ X3 )
=> ( ord_less_eq_nat @ ( F3 @ Y3 @ Z ) @ X3 ) ) )
=> ( ( sup_sup_nat @ X @ Y )
= ( F3 @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_829_sup_Oabsorb1,axiom,
! [B: set_Pr4934435412358123699_a_nat,A: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ B @ A )
=> ( ( sup_su459911885395995103_a_nat @ A @ B )
= A ) ) ).
% sup.absorb1
thf(fact_830_sup_Oabsorb1,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( sup_sup_nat @ A @ B )
= A ) ) ).
% sup.absorb1
thf(fact_831_sup_Oabsorb2,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A @ B )
=> ( ( sup_su459911885395995103_a_nat @ A @ B )
= B ) ) ).
% sup.absorb2
thf(fact_832_sup_Oabsorb2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( sup_sup_nat @ A @ B )
= B ) ) ).
% sup.absorb2
thf(fact_833_sup__absorb1,axiom,
! [Y: set_Pr4934435412358123699_a_nat,X: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ Y @ X )
=> ( ( sup_su459911885395995103_a_nat @ X @ Y )
= X ) ) ).
% sup_absorb1
thf(fact_834_sup__absorb1,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( sup_sup_nat @ X @ Y )
= X ) ) ).
% sup_absorb1
thf(fact_835_sup__absorb2,axiom,
! [X: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X @ Y )
=> ( ( sup_su459911885395995103_a_nat @ X @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_836_sup__absorb2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( sup_sup_nat @ X @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_837_sup_OboundedE,axiom,
! [B: set_Pr4934435412358123699_a_nat,C2: set_Pr4934435412358123699_a_nat,A: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ ( sup_su459911885395995103_a_nat @ B @ C2 ) @ A )
=> ~ ( ( ord_le8666007276011122963_a_nat @ B @ A )
=> ~ ( ord_le8666007276011122963_a_nat @ C2 @ A ) ) ) ).
% sup.boundedE
thf(fact_838_sup_OboundedE,axiom,
! [B: nat,C2: nat,A: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C2 ) @ A )
=> ~ ( ( ord_less_eq_nat @ B @ A )
=> ~ ( ord_less_eq_nat @ C2 @ A ) ) ) ).
% sup.boundedE
thf(fact_839_sup_OboundedI,axiom,
! [B: set_Pr4934435412358123699_a_nat,A: set_Pr4934435412358123699_a_nat,C2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ B @ A )
=> ( ( ord_le8666007276011122963_a_nat @ C2 @ A )
=> ( ord_le8666007276011122963_a_nat @ ( sup_su459911885395995103_a_nat @ B @ C2 ) @ A ) ) ) ).
% sup.boundedI
thf(fact_840_sup_OboundedI,axiom,
! [B: nat,A: nat,C2: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C2 @ A )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C2 ) @ A ) ) ) ).
% sup.boundedI
thf(fact_841_sup_Oorder__iff,axiom,
( ord_le8666007276011122963_a_nat
= ( ^ [B4: set_Pr4934435412358123699_a_nat,A3: set_Pr4934435412358123699_a_nat] :
( A3
= ( sup_su459911885395995103_a_nat @ A3 @ B4 ) ) ) ) ).
% sup.order_iff
thf(fact_842_sup_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [B4: nat,A3: nat] :
( A3
= ( sup_sup_nat @ A3 @ B4 ) ) ) ) ).
% sup.order_iff
thf(fact_843_sup_Ocobounded1,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ A @ ( sup_su459911885395995103_a_nat @ A @ B ) ) ).
% sup.cobounded1
thf(fact_844_sup_Ocobounded1,axiom,
! [A: nat,B: nat] : ( ord_less_eq_nat @ A @ ( sup_sup_nat @ A @ B ) ) ).
% sup.cobounded1
thf(fact_845_sup_Ocobounded2,axiom,
! [B: set_Pr4934435412358123699_a_nat,A: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ B @ ( sup_su459911885395995103_a_nat @ A @ B ) ) ).
% sup.cobounded2
thf(fact_846_sup_Ocobounded2,axiom,
! [B: nat,A: nat] : ( ord_less_eq_nat @ B @ ( sup_sup_nat @ A @ B ) ) ).
% sup.cobounded2
thf(fact_847_sup_Oabsorb__iff1,axiom,
( ord_le8666007276011122963_a_nat
= ( ^ [B4: set_Pr4934435412358123699_a_nat,A3: set_Pr4934435412358123699_a_nat] :
( ( sup_su459911885395995103_a_nat @ A3 @ B4 )
= A3 ) ) ) ).
% sup.absorb_iff1
thf(fact_848_sup_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [B4: nat,A3: nat] :
( ( sup_sup_nat @ A3 @ B4 )
= A3 ) ) ) ).
% sup.absorb_iff1
thf(fact_849_sup_Oabsorb__iff2,axiom,
( ord_le8666007276011122963_a_nat
= ( ^ [A3: set_Pr4934435412358123699_a_nat,B4: set_Pr4934435412358123699_a_nat] :
( ( sup_su459911885395995103_a_nat @ A3 @ B4 )
= B4 ) ) ) ).
% sup.absorb_iff2
thf(fact_850_sup_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B4: nat] :
( ( sup_sup_nat @ A3 @ B4 )
= B4 ) ) ) ).
% sup.absorb_iff2
thf(fact_851_sup_OcoboundedI1,axiom,
! [C2: set_Pr4934435412358123699_a_nat,A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ C2 @ A )
=> ( ord_le8666007276011122963_a_nat @ C2 @ ( sup_su459911885395995103_a_nat @ A @ B ) ) ) ).
% sup.coboundedI1
thf(fact_852_sup_OcoboundedI1,axiom,
! [C2: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ C2 @ A )
=> ( ord_less_eq_nat @ C2 @ ( sup_sup_nat @ A @ B ) ) ) ).
% sup.coboundedI1
thf(fact_853_sup_OcoboundedI2,axiom,
! [C2: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat,A: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ C2 @ B )
=> ( ord_le8666007276011122963_a_nat @ C2 @ ( sup_su459911885395995103_a_nat @ A @ B ) ) ) ).
% sup.coboundedI2
thf(fact_854_sup_OcoboundedI2,axiom,
! [C2: nat,B: nat,A: nat] :
( ( ord_less_eq_nat @ C2 @ B )
=> ( ord_less_eq_nat @ C2 @ ( sup_sup_nat @ A @ B ) ) ) ).
% sup.coboundedI2
thf(fact_855_sup_Ostrict__coboundedI2,axiom,
! [C2: nat,B: nat,A: nat] :
( ( ord_less_nat @ C2 @ B )
=> ( ord_less_nat @ C2 @ ( sup_sup_nat @ A @ B ) ) ) ).
% sup.strict_coboundedI2
thf(fact_856_sup_Ostrict__coboundedI1,axiom,
! [C2: nat,A: nat,B: nat] :
( ( ord_less_nat @ C2 @ A )
=> ( ord_less_nat @ C2 @ ( sup_sup_nat @ A @ B ) ) ) ).
% sup.strict_coboundedI1
thf(fact_857_sup_Ostrict__order__iff,axiom,
( ord_less_nat
= ( ^ [B4: nat,A3: nat] :
( ( A3
= ( sup_sup_nat @ A3 @ B4 ) )
& ( A3 != B4 ) ) ) ) ).
% sup.strict_order_iff
thf(fact_858_sup_Ostrict__boundedE,axiom,
! [B: nat,C2: nat,A: nat] :
( ( ord_less_nat @ ( sup_sup_nat @ B @ C2 ) @ A )
=> ~ ( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ C2 @ A ) ) ) ).
% sup.strict_boundedE
thf(fact_859_sup_Oabsorb4,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( sup_sup_nat @ A @ B )
= B ) ) ).
% sup.absorb4
thf(fact_860_sup_Oabsorb3,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( sup_sup_nat @ A @ B )
= A ) ) ).
% sup.absorb3
thf(fact_861_less__supI2,axiom,
! [X: nat,B: nat,A: nat] :
( ( ord_less_nat @ X @ B )
=> ( ord_less_nat @ X @ ( sup_sup_nat @ A @ B ) ) ) ).
% less_supI2
thf(fact_862_less__supI1,axiom,
! [X: nat,A: nat,B: nat] :
( ( ord_less_nat @ X @ A )
=> ( ord_less_nat @ X @ ( sup_sup_nat @ A @ B ) ) ) ).
% less_supI1
thf(fact_863_sig__step__rsteps__eq__dist,axiom,
! [R: set_Pr4386577575007340137rm_a_b,F: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ ( terms_7988297476397195622_a_b_b @ R ) @ F )
=> ( ( sup_su6776935440552674877rm_a_b @ ( sig_step_a_b @ F @ ( transi7922773638565587891rm_a_b @ ( rstep_a_b @ R ) ) ) @ id_term_a_b )
= ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R ) ) ) ) ) ).
% sig_step_rsteps_eq_dist
thf(fact_864_minf_I8_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z )
=> ~ ( ord_less_eq_nat @ T @ X6 ) ) ).
% minf(8)
thf(fact_865_minf_I6_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z )
=> ( ord_less_eq_nat @ X6 @ T ) ) ).
% minf(6)
thf(fact_866_pinf_I8_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z @ X6 )
=> ( ord_less_eq_nat @ T @ X6 ) ) ).
% pinf(8)
thf(fact_867_pinf_I6_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z @ X6 )
=> ~ ( ord_less_eq_nat @ X6 @ T ) ) ).
% pinf(6)
thf(fact_868_verit__comp__simplify1_I3_J,axiom,
! [B2: nat,A2: nat] :
( ( ~ ( ord_less_eq_nat @ B2 @ A2 ) )
= ( ord_less_nat @ A2 @ B2 ) ) ).
% verit_comp_simplify1(3)
thf(fact_869_IdI,axiom,
! [A: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ A ) @ id_term_a_b ) ).
% IdI
thf(fact_870_pair__in__Id__conv,axiom,
! [A: term_a_b,B: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ id_term_a_b )
= ( A = B ) ) ).
% pair_in_Id_conv
thf(fact_871_rtrancl__reflcl,axiom,
! [R3: set_Pr4386577575007340137rm_a_b] :
( ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ R3 @ id_term_a_b ) )
= ( transi7742714808557438673rm_a_b @ R3 ) ) ).
% rtrancl_reflcl
thf(fact_872_rtrancl__reflcl__absorb,axiom,
! [R3: set_Pr4386577575007340137rm_a_b] :
( ( sup_su6776935440552674877rm_a_b @ ( transi7742714808557438673rm_a_b @ R3 ) @ id_term_a_b )
= ( transi7742714808557438673rm_a_b @ R3 ) ) ).
% rtrancl_reflcl_absorb
thf(fact_873_trancl__reflcl,axiom,
! [R2: set_Pr4386577575007340137rm_a_b] :
( ( transi7922773638565587891rm_a_b @ ( sup_su6776935440552674877rm_a_b @ R2 @ id_term_a_b ) )
= ( transi7742714808557438673rm_a_b @ R2 ) ) ).
% trancl_reflcl
thf(fact_874_IdE,axiom,
! [P: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ P @ id_term_a_b )
=> ~ ! [X3: term_a_b] :
( P
!= ( produc7020197800436672577rm_a_b @ X3 @ X3 ) ) ) ).
% IdE
thf(fact_875_verit__la__disequality,axiom,
! [A: nat,B: nat] :
( ( A = B )
| ~ ( ord_less_eq_nat @ A @ B )
| ~ ( ord_less_eq_nat @ B @ A ) ) ).
% verit_la_disequality
thf(fact_876_verit__comp__simplify1_I2_J,axiom,
! [A: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_877_verit__comp__simplify1_I2_J,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_878_verit__comp__simplify1_I1_J,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_879_pinf_I1_J,axiom,
! [P2: nat > $o,P5: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z4 @ X3 )
=> ( ( P2 @ X3 )
= ( P5 @ X3 ) ) )
=> ( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z4 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z @ X6 )
=> ( ( ( P2 @ X6 )
& ( Q @ X6 ) )
= ( ( P5 @ X6 )
& ( Q2 @ X6 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_880_pinf_I2_J,axiom,
! [P2: nat > $o,P5: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z4 @ X3 )
=> ( ( P2 @ X3 )
= ( P5 @ X3 ) ) )
=> ( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z4 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z @ X6 )
=> ( ( ( P2 @ X6 )
| ( Q @ X6 ) )
= ( ( P5 @ X6 )
| ( Q2 @ X6 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_881_pinf_I3_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z @ X6 )
=> ( X6 != T ) ) ).
% pinf(3)
thf(fact_882_pinf_I4_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z @ X6 )
=> ( X6 != T ) ) ).
% pinf(4)
thf(fact_883_pinf_I5_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z @ X6 )
=> ~ ( ord_less_nat @ X6 @ T ) ) ).
% pinf(5)
thf(fact_884_pinf_I7_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z @ X6 )
=> ( ord_less_nat @ T @ X6 ) ) ).
% pinf(7)
thf(fact_885_minf_I1_J,axiom,
! [P2: nat > $o,P5: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z4 )
=> ( ( P2 @ X3 )
= ( P5 @ X3 ) ) )
=> ( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z4 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z )
=> ( ( ( P2 @ X6 )
& ( Q @ X6 ) )
= ( ( P5 @ X6 )
& ( Q2 @ X6 ) ) ) ) ) ) ).
% minf(1)
thf(fact_886_minf_I2_J,axiom,
! [P2: nat > $o,P5: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z4 )
=> ( ( P2 @ X3 )
= ( P5 @ X3 ) ) )
=> ( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z4 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z )
=> ( ( ( P2 @ X6 )
| ( Q @ X6 ) )
= ( ( P5 @ X6 )
| ( Q2 @ X6 ) ) ) ) ) ) ).
% minf(2)
thf(fact_887_minf_I3_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z )
=> ( X6 != T ) ) ).
% minf(3)
thf(fact_888_minf_I4_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z )
=> ( X6 != T ) ) ).
% minf(4)
thf(fact_889_minf_I5_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z )
=> ( ord_less_nat @ X6 @ T ) ) ).
% minf(5)
thf(fact_890_minf_I7_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z )
=> ~ ( ord_less_nat @ T @ X6 ) ) ).
% minf(7)
thf(fact_891_rtrancl__unfold,axiom,
( transi7742714808557438673rm_a_b
= ( ^ [R6: set_Pr4386577575007340137rm_a_b] : ( sup_su6776935440552674877rm_a_b @ id_term_a_b @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ R6 ) @ R6 ) ) ) ) ).
% rtrancl_unfold
thf(fact_892_rtrancl__trancl__reflcl,axiom,
( transi7742714808557438673rm_a_b
= ( ^ [R6: set_Pr4386577575007340137rm_a_b] : ( sup_su6776935440552674877rm_a_b @ ( transi7922773638565587891rm_a_b @ R6 ) @ id_term_a_b ) ) ) ).
% rtrancl_trancl_reflcl
thf(fact_893_trancl__o__refl__is__trancl,axiom,
! [R2: set_Pr4386577575007340137rm_a_b] :
( ( relcom370159955682700863rm_a_b @ ( transi7922773638565587891rm_a_b @ R2 ) @ ( sup_su6776935440552674877rm_a_b @ R2 @ id_term_a_b ) )
= ( transi7922773638565587891rm_a_b @ R2 ) ) ).
% trancl_o_refl_is_trancl
thf(fact_894_SCR__on__def,axiom,
( rewrit4966517460798004134rm_a_b
= ( ^ [R6: set_Pr4386577575007340137rm_a_b,A6: set_term_a_b] :
! [X4: term_a_b] :
( ( member_term_a_b @ X4 @ A6 )
=> ! [B4: term_a_b,C: term_a_b] :
( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X4 @ B4 ) @ R6 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X4 @ C ) @ R6 ) )
=> ? [D3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B4 @ D3 ) @ ( sup_su6776935440552674877rm_a_b @ R6 @ id_term_a_b ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ C @ D3 ) @ ( transi7742714808557438673rm_a_b @ R6 ) ) ) ) ) ) ) ).
% SCR_on_def
thf(fact_895_strongly__confluent__on__def,axiom,
( abstra5664096641628173427rm_a_b
= ( ^ [R6: set_Pr4386577575007340137rm_a_b,A6: set_term_a_b] :
! [X4: term_a_b] :
( ( member_term_a_b @ X4 @ A6 )
=> ! [Y5: term_a_b,Z5: term_a_b] :
( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X4 @ Y5 ) @ R6 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X4 @ Z5 ) @ R6 ) )
=> ? [U3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y5 @ U3 ) @ ( transi7742714808557438673rm_a_b @ R6 ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Z5 @ U3 ) @ ( sup_su6776935440552674877rm_a_b @ R6 @ id_term_a_b ) ) ) ) ) ) ) ).
% strongly_confluent_on_def
thf(fact_896_strongly__confluent__on__E11,axiom,
! [R2: set_Pr2972776593051762503rm_a_b,A5: set_Pr4386577575007340137rm_a_b,X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b,Z2: produc357393685978478089rm_a_b] :
( ( abstra837640964651051114rm_a_b @ R2 @ A5 )
=> ( ( member5869715511025134514rm_a_b @ X @ A5 )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ R2 )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Z2 ) @ R2 )
=> ? [U2: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y @ U2 ) @ ( transi2615809358984392588rm_a_b @ R2 ) )
& ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Z2 @ U2 ) @ ( sup_su1845815533690797339rm_a_b @ R2 @ id_Pro3488709451141449061rm_a_b ) ) ) ) ) ) ) ).
% strongly_confluent_on_E11
thf(fact_897_strongly__confluent__on__E11,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,A5: set_term_a_b,X: term_a_b,Y: term_a_b,Z2: term_a_b] :
( ( abstra5664096641628173427rm_a_b @ R2 @ A5 )
=> ( ( member_term_a_b @ X @ A5 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ R2 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z2 ) @ R2 )
=> ? [U2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ U2 ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Z2 @ U2 ) @ ( sup_su6776935440552674877rm_a_b @ R2 @ id_term_a_b ) ) ) ) ) ) ) ).
% strongly_confluent_on_E11
thf(fact_898_IdD,axiom,
! [A: term_a_b,B: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ id_term_a_b )
=> ( A = B ) ) ).
% IdD
thf(fact_899_complete__interval,axiom,
! [A: nat,B: nat,P2: nat > $o] :
( ( ord_less_nat @ A @ B )
=> ( ( P2 @ A )
=> ( ~ ( P2 @ B )
=> ? [C3: nat] :
( ( ord_less_eq_nat @ A @ C3 )
& ( ord_less_eq_nat @ C3 @ B )
& ! [X6: nat] :
( ( ( ord_less_eq_nat @ A @ X6 )
& ( ord_less_nat @ X6 @ C3 ) )
=> ( P2 @ X6 ) )
& ! [D4: nat] :
( ! [X3: nat] :
( ( ( ord_less_eq_nat @ A @ X3 )
& ( ord_less_nat @ X3 @ D4 ) )
=> ( P2 @ X3 ) )
=> ( ord_less_eq_nat @ D4 @ C3 ) ) ) ) ) ) ).
% complete_interval
thf(fact_900_strongly__confluentI,axiom,
! [R2: set_Pr4386577575007340137rm_a_b] :
( ! [X3: term_a_b,Y3: term_a_b,Z: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ R2 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Z ) @ R2 )
=> ? [U4: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ U4 ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Z @ U4 ) @ ( sup_su6776935440552674877rm_a_b @ R2 @ id_term_a_b ) ) ) ) )
=> ( abstra5664096641628173427rm_a_b @ R2 @ top_top_set_term_a_b ) ) ).
% strongly_confluentI
thf(fact_901_permut__aux__sound,axiom,
! [I4: nat,As: list_term_a_b,F3: nat > nat,Bs: list_term_a_b] :
( ( ord_less_nat @ I4 @ ( size_s8906293707977694520rm_a_b @ As ) )
=> ( ( nth_term_a_b @ ( missin5661735679256376914rm_a_b @ As @ F3 @ Bs ) @ I4 )
= ( nth_term_a_b @ Bs @ ( F3 @ I4 ) ) ) ) ).
% permut_aux_sound
thf(fact_902_min__list__nth,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Ys ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs @ I2 ) @ ( nth_nat @ Ys @ I2 ) ) )
=> ( ord_less_eq_nat @ ( missing_min_list_nat @ Xs ) @ ( missing_min_list_nat @ Ys ) ) ) ) ).
% min_list_nth
thf(fact_903_normalizability__I,axiom,
! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,A5: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( transi2615809358984392588rm_a_b @ A5 ) )
=> ( ( member5869715511025134514rm_a_b @ B @ ( abstra1902783451776770197rm_a_b @ A5 ) )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( abstra1496011390211842225rm_a_b @ A5 ) ) ) ) ).
% normalizability_I
thf(fact_904_normalizability__I,axiom,
! [A: term_a_b,B: term_a_b,A5: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ A5 ) )
=> ( ( member_term_a_b @ B @ ( abstract_NF_term_a_b @ A5 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( abstra2972651231392952684rm_a_b @ A5 ) ) ) ) ).
% normalizability_I
thf(fact_905_UNIV__I,axiom,
! [X: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ X @ top_to1314267278846557113rm_a_b ) ).
% UNIV_I
thf(fact_906_normalizability__I_H,axiom,
! [A: term_a_b,B: term_a_b,A5: set_Pr4386577575007340137rm_a_b,C2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ A5 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C2 ) @ ( abstra2972651231392952684rm_a_b @ A5 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C2 ) @ ( abstra2972651231392952684rm_a_b @ A5 ) ) ) ) ).
% normalizability_I'
thf(fact_907_subset__UNIV,axiom,
! [A5: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ A5 @ top_to3353692345378799459_a_nat ) ).
% subset_UNIV
thf(fact_908_UNIV__eq__I,axiom,
! [A5: set_Pr4386577575007340137rm_a_b] :
( ! [X3: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ X3 @ A5 )
=> ( top_to1314267278846557113rm_a_b = A5 ) ) ).
% UNIV_eq_I
thf(fact_909_UNIV__witness,axiom,
? [X3: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ X3 @ top_to1314267278846557113rm_a_b ) ).
% UNIV_witness
thf(fact_910_impossible__Cons,axiom,
! [Xs: list_term_a_b,Ys: list_term_a_b,X: term_a_b] :
( ( ord_less_eq_nat @ ( size_s8906293707977694520rm_a_b @ Xs ) @ ( size_s8906293707977694520rm_a_b @ Ys ) )
=> ( Xs
!= ( cons_term_a_b @ X @ Ys ) ) ) ).
% impossible_Cons
thf(fact_911_min__list__Cons,axiom,
! [X: nat,Y: nat,Xs: list_nat,Ys: list_nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ord_less_eq_nat @ ( missing_min_list_nat @ Xs ) @ ( missing_min_list_nat @ Ys ) )
=> ( ord_less_eq_nat @ ( missing_min_list_nat @ ( cons_nat @ X @ Xs ) ) @ ( missing_min_list_nat @ ( cons_nat @ Y @ Ys ) ) ) ) ) ) ).
% min_list_Cons
thf(fact_912_top_Oextremum__uniqueI,axiom,
! [A: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ top_to3353692345378799459_a_nat @ A )
=> ( A = top_to3353692345378799459_a_nat ) ) ).
% top.extremum_uniqueI
thf(fact_913_top_Oextremum__unique,axiom,
! [A: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ top_to3353692345378799459_a_nat @ A )
= ( A = top_to3353692345378799459_a_nat ) ) ).
% top.extremum_unique
thf(fact_914_top__greatest,axiom,
! [A: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ A @ top_to3353692345378799459_a_nat ) ).
% top_greatest
thf(fact_915_normalizability__E,axiom,
! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,A5: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( abstra1496011390211842225rm_a_b @ A5 ) )
=> ~ ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( transi2615809358984392588rm_a_b @ A5 ) )
=> ~ ( member5869715511025134514rm_a_b @ B @ ( abstra1902783451776770197rm_a_b @ A5 ) ) ) ) ).
% normalizability_E
thf(fact_916_normalizability__E,axiom,
! [A: term_a_b,B: term_a_b,A5: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( abstra2972651231392952684rm_a_b @ A5 ) )
=> ~ ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ A5 ) )
=> ~ ( member_term_a_b @ B @ ( abstract_NF_term_a_b @ A5 ) ) ) ) ).
% normalizability_E
thf(fact_917_NFP__stepD,axiom,
! [R2: set_Pr2972776593051762503rm_a_b,A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,C2: produc357393685978478089rm_a_b] :
( ( rewrit2138790387785322561rm_a_b @ R2 @ top_to1314267278846557113rm_a_b )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( transi2615809358984392588rm_a_b @ R2 ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ C2 ) @ ( transi2615809358984392588rm_a_b @ R2 ) )
=> ( ( member5869715511025134514rm_a_b @ C2 @ ( abstra1902783451776770197rm_a_b @ R2 ) )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B @ C2 ) @ ( transi2615809358984392588rm_a_b @ R2 ) ) ) ) ) ) ).
% NFP_stepD
thf(fact_918_NFP__stepD,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,A: term_a_b,B: term_a_b,C2: term_a_b] :
( ( rewrit6304709981509430748rm_a_b @ R2 @ top_top_set_term_a_b )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C2 ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
=> ( ( member_term_a_b @ C2 @ ( abstract_NF_term_a_b @ R2 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C2 ) @ ( transi7742714808557438673rm_a_b @ R2 ) ) ) ) ) ) ).
% NFP_stepD
thf(fact_919_NFP__on__def,axiom,
( rewrit6304709981509430748rm_a_b
= ( ^ [R6: set_Pr4386577575007340137rm_a_b,A6: set_term_a_b] :
! [X4: term_a_b] :
( ( member_term_a_b @ X4 @ A6 )
=> ! [B4: term_a_b,C: term_a_b] :
( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X4 @ B4 ) @ ( transi7742714808557438673rm_a_b @ R6 ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X4 @ C ) @ ( abstra2972651231392952684rm_a_b @ R6 ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B4 @ C ) @ ( transi7742714808557438673rm_a_b @ R6 ) ) ) ) ) ) ).
% NFP_on_def
thf(fact_920_NE__on__def,axiom,
( rewrit7370782028743737947rm_a_b
= ( ^ [R6: set_Pr4386577575007340137rm_a_b,S6: set_Pr4386577575007340137rm_a_b,A6: set_term_a_b] :
! [X4: term_a_b] :
( ( member_term_a_b @ X4 @ A6 )
=> ! [B4: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X4 @ B4 ) @ ( abstra2972651231392952684rm_a_b @ R6 ) )
= ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X4 @ B4 ) @ ( abstra2972651231392952684rm_a_b @ S6 ) ) ) ) ) ) ).
% NE_on_def
thf(fact_921_top__empty__eq,axiom,
( top_to1262317115727556708_a_b_o
= ( ^ [X4: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ X4 @ top_to1314267278846557113rm_a_b ) ) ) ).
% top_empty_eq
thf(fact_922_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M: nat,N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
& ( M != N2 ) ) ) ) ).
% nat_less_le
thf(fact_923_less__imp__le__nat,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% less_imp_le_nat
thf(fact_924_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
| ( M = N2 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_925_less__or__eq__imp__le,axiom,
! [M2: nat,N: nat] :
( ( ( ord_less_nat @ M2 @ N )
| ( M2 = N ) )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% less_or_eq_imp_le
thf(fact_926_linorder__neqE__nat,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_927_infinite__descent,axiom,
! [P2: nat > $o,N: nat] :
( ! [N3: nat] :
( ~ ( P2 @ N3 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
& ~ ( P2 @ M3 ) ) )
=> ( P2 @ N ) ) ).
% infinite_descent
thf(fact_928_nat__less__induct,axiom,
! [P2: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
=> ( P2 @ M3 ) )
=> ( P2 @ N3 ) )
=> ( P2 @ N ) ) ).
% nat_less_induct
thf(fact_929_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_930_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_931_less__not__refl2,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ N @ M2 )
=> ( M2 != N ) ) ).
% less_not_refl2
thf(fact_932_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_933_nat__neq__iff,axiom,
! [M2: nat,N: nat] :
( ( M2 != N )
= ( ( ord_less_nat @ M2 @ N )
| ( ord_less_nat @ N @ M2 ) ) ) ).
% nat_neq_iff
thf(fact_934_size__neq__size__imp__neq,axiom,
! [X: list_term_a_b,Y: list_term_a_b] :
( ( ( size_s8906293707977694520rm_a_b @ X )
!= ( size_s8906293707977694520rm_a_b @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_935_less__mono__imp__le__mono,axiom,
! [F3: nat > nat,I4: nat,J: nat] :
( ! [I2: nat,J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ord_less_nat @ ( F3 @ I2 ) @ ( F3 @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I4 @ J )
=> ( ord_less_eq_nat @ ( F3 @ I4 ) @ ( F3 @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_936_le__neq__implies__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( M2 != N )
=> ( ord_less_nat @ M2 @ N ) ) ) ).
% le_neq_implies_less
thf(fact_937_inf__pigeonhole__principle,axiom,
! [N: nat,F3: nat > nat > $o] :
( ! [K2: nat] :
? [I: nat] :
( ( ord_less_nat @ I @ N )
& ( F3 @ K2 @ I ) )
=> ? [I2: nat] :
( ( ord_less_nat @ I2 @ N )
& ! [K3: nat] :
? [K4: nat] :
( ( ord_less_eq_nat @ K3 @ K4 )
& ( F3 @ K4 @ I2 ) ) ) ) ).
% inf_pigeonhole_principle
thf(fact_938_nat__descend__induct,axiom,
! [N: nat,P2: nat > $o,M2: nat] :
( ! [K2: nat] :
( ( ord_less_nat @ N @ K2 )
=> ( P2 @ K2 ) )
=> ( ! [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
=> ( ! [I: nat] :
( ( ord_less_nat @ K2 @ I )
=> ( P2 @ I ) )
=> ( P2 @ K2 ) ) )
=> ( P2 @ M2 ) ) ) ).
% nat_descend_induct
thf(fact_939_list__induct2,axiom,
! [Xs: list_term_a_b,Ys: list_term_a_b,P2: list_term_a_b > list_term_a_b > $o] :
( ( ( size_s8906293707977694520rm_a_b @ Xs )
= ( size_s8906293707977694520rm_a_b @ Ys ) )
=> ( ( P2 @ nil_term_a_b @ nil_term_a_b )
=> ( ! [X3: term_a_b,Xs3: list_term_a_b,Y3: term_a_b,Ys4: list_term_a_b] :
( ( ( size_s8906293707977694520rm_a_b @ Xs3 )
= ( size_s8906293707977694520rm_a_b @ Ys4 ) )
=> ( ( P2 @ Xs3 @ Ys4 )
=> ( P2 @ ( cons_term_a_b @ X3 @ Xs3 ) @ ( cons_term_a_b @ Y3 @ Ys4 ) ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_940_list__induct3,axiom,
! [Xs: list_term_a_b,Ys: list_term_a_b,Zs: list_term_a_b,P2: list_term_a_b > list_term_a_b > list_term_a_b > $o] :
( ( ( size_s8906293707977694520rm_a_b @ Xs )
= ( size_s8906293707977694520rm_a_b @ Ys ) )
=> ( ( ( size_s8906293707977694520rm_a_b @ Ys )
= ( size_s8906293707977694520rm_a_b @ Zs ) )
=> ( ( P2 @ nil_term_a_b @ nil_term_a_b @ nil_term_a_b )
=> ( ! [X3: term_a_b,Xs3: list_term_a_b,Y3: term_a_b,Ys4: list_term_a_b,Z: term_a_b,Zs2: list_term_a_b] :
( ( ( size_s8906293707977694520rm_a_b @ Xs3 )
= ( size_s8906293707977694520rm_a_b @ Ys4 ) )
=> ( ( ( size_s8906293707977694520rm_a_b @ Ys4 )
= ( size_s8906293707977694520rm_a_b @ Zs2 ) )
=> ( ( P2 @ Xs3 @ Ys4 @ Zs2 )
=> ( P2 @ ( cons_term_a_b @ X3 @ Xs3 ) @ ( cons_term_a_b @ Y3 @ Ys4 ) @ ( cons_term_a_b @ Z @ Zs2 ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_941_list__induct4,axiom,
! [Xs: list_term_a_b,Ys: list_term_a_b,Zs: list_term_a_b,Ws: list_term_a_b,P2: list_term_a_b > list_term_a_b > list_term_a_b > list_term_a_b > $o] :
( ( ( size_s8906293707977694520rm_a_b @ Xs )
= ( size_s8906293707977694520rm_a_b @ Ys ) )
=> ( ( ( size_s8906293707977694520rm_a_b @ Ys )
= ( size_s8906293707977694520rm_a_b @ Zs ) )
=> ( ( ( size_s8906293707977694520rm_a_b @ Zs )
= ( size_s8906293707977694520rm_a_b @ Ws ) )
=> ( ( P2 @ nil_term_a_b @ nil_term_a_b @ nil_term_a_b @ nil_term_a_b )
=> ( ! [X3: term_a_b,Xs3: list_term_a_b,Y3: term_a_b,Ys4: list_term_a_b,Z: term_a_b,Zs2: list_term_a_b,W: term_a_b,Ws2: list_term_a_b] :
( ( ( size_s8906293707977694520rm_a_b @ Xs3 )
= ( size_s8906293707977694520rm_a_b @ Ys4 ) )
=> ( ( ( size_s8906293707977694520rm_a_b @ Ys4 )
= ( size_s8906293707977694520rm_a_b @ Zs2 ) )
=> ( ( ( size_s8906293707977694520rm_a_b @ Zs2 )
= ( size_s8906293707977694520rm_a_b @ Ws2 ) )
=> ( ( P2 @ Xs3 @ Ys4 @ Zs2 @ Ws2 )
=> ( P2 @ ( cons_term_a_b @ X3 @ Xs3 ) @ ( cons_term_a_b @ Y3 @ Ys4 ) @ ( cons_term_a_b @ Z @ Zs2 ) @ ( cons_term_a_b @ W @ Ws2 ) ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_942_replace__term__at_Ocases,axiom,
! [X: produc2732850333517536310rm_a_b] :
( ! [S4: term_a_b,T3: term_a_b] :
( X
!= ( produc3812856575676843240rm_a_b @ S4 @ ( produc5151171985953862413rm_a_b @ nil_nat @ T3 ) ) )
=> ( ! [X3: b,V: nat,Va: list_nat,T3: term_a_b] :
( X
!= ( produc3812856575676843240rm_a_b @ ( var_b_a @ X3 ) @ ( produc5151171985953862413rm_a_b @ ( cons_nat @ V @ Va ) @ T3 ) ) )
=> ~ ! [F2: a,Ts: list_term_a_b,I2: nat,Ps: list_nat,T3: term_a_b] :
( X
!= ( produc3812856575676843240rm_a_b @ ( fun_a_b @ F2 @ Ts ) @ ( produc5151171985953862413rm_a_b @ ( cons_nat @ I2 @ Ps ) @ T3 ) ) ) ) ) ).
% replace_term_at.cases
thf(fact_943_ctxt__at__pos_Ocases,axiom,
! [X: produc3697673438841856213st_nat] :
( ! [S4: term_a_b] :
( X
!= ( produc4563063199488751885st_nat @ S4 @ nil_nat ) )
=> ( ! [F2: a,Ss: list_term_a_b,I2: nat,P6: list_nat] :
( X
!= ( produc4563063199488751885st_nat @ ( fun_a_b @ F2 @ Ss ) @ ( cons_nat @ I2 @ P6 ) ) )
=> ~ ! [X3: b,V: nat,Va: list_nat] :
( X
!= ( produc4563063199488751885st_nat @ ( var_b_a @ X3 ) @ ( cons_nat @ V @ Va ) ) ) ) ) ).
% ctxt_at_pos.cases
thf(fact_944_subt__at_Ocases,axiom,
! [X: produc3697673438841856213st_nat] :
( ! [S4: term_a_b] :
( X
!= ( produc4563063199488751885st_nat @ S4 @ nil_nat ) )
=> ( ! [F2: a,Ss: list_term_a_b,I2: nat,P6: list_nat] :
( X
!= ( produc4563063199488751885st_nat @ ( fun_a_b @ F2 @ Ss ) @ ( cons_nat @ I2 @ P6 ) ) )
=> ~ ! [X3: b,V: nat,Va: list_nat] :
( X
!= ( produc4563063199488751885st_nat @ ( var_b_a @ X3 ) @ ( cons_nat @ V @ Va ) ) ) ) ) ).
% subt_at.cases
thf(fact_945_fun__at_Ocases,axiom,
! [X: produc3697673438841856213st_nat] :
( ! [X3: b] :
( X
!= ( produc4563063199488751885st_nat @ ( var_b_a @ X3 ) @ nil_nat ) )
=> ( ! [F2: a,Ts: list_term_a_b] :
( X
!= ( produc4563063199488751885st_nat @ ( fun_a_b @ F2 @ Ts ) @ nil_nat ) )
=> ( ! [F2: a,Ts: list_term_a_b,I2: nat,P6: list_nat] :
( X
!= ( produc4563063199488751885st_nat @ ( fun_a_b @ F2 @ Ts ) @ ( cons_nat @ I2 @ P6 ) ) )
=> ~ ! [Vb: b,V: nat,Va: list_nat] :
( X
!= ( produc4563063199488751885st_nat @ ( var_b_a @ Vb ) @ ( cons_nat @ V @ Va ) ) ) ) ) ) ).
% fun_at.cases
thf(fact_946_UNF__onI,axiom,
! [A5: set_Pr4386577575007340137rm_a_b,R2: set_Pr2972776593051762503rm_a_b] :
( ! [A4: produc357393685978478089rm_a_b,B3: produc357393685978478089rm_a_b,C3: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ A4 @ A5 )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A4 @ B3 ) @ ( abstra1496011390211842225rm_a_b @ R2 ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A4 @ C3 ) @ ( abstra1496011390211842225rm_a_b @ R2 ) )
=> ( B3 = C3 ) ) ) )
=> ( abstra1303798319023342519rm_a_b @ R2 @ A5 ) ) ).
% UNF_onI
thf(fact_947_UNF__onI,axiom,
! [A5: set_term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ! [A4: term_a_b,B3: term_a_b,C3: term_a_b] :
( ( member_term_a_b @ A4 @ A5 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A4 @ B3 ) @ ( abstra2972651231392952684rm_a_b @ R2 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A4 @ C3 ) @ ( abstra2972651231392952684rm_a_b @ R2 ) )
=> ( B3 = C3 ) ) ) )
=> ( abstra3619855735679963878rm_a_b @ R2 @ A5 ) ) ).
% UNF_onI
thf(fact_948_strongly__confluent__E1n,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,X: term_a_b,Y: term_a_b,Z2: term_a_b,N: nat] :
( ( abstra5664096641628173427rm_a_b @ R2 @ top_top_set_term_a_b )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( sup_su6776935440552674877rm_a_b @ R2 @ id_term_a_b ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z2 ) @ ( compow4057154403645558940rm_a_b @ N @ R2 ) )
=> ? [U2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ U2 ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Z2 @ U2 ) @ ( sup_su6776935440552674877rm_a_b @ R2 @ id_term_a_b ) ) ) ) ) ) ).
% strongly_confluent_E1n
thf(fact_949_poss__of__term__Cons,axiom,
! [I4: nat,P: list_nat,U: term_a_b,F3: a,Ts2: list_term_a_b] :
( ( member_list_nat @ ( cons_nat @ I4 @ P ) @ ( terms_7168686267159881682rm_a_b @ U @ ( fun_a_b @ F3 @ Ts2 ) ) )
=> ( member_list_nat @ P @ ( terms_7168686267159881682rm_a_b @ U @ ( nth_term_a_b @ Ts2 @ I4 ) ) ) ) ).
% poss_of_term_Cons
thf(fact_950_WN__onI,axiom,
! [A5: set_Pr4386577575007340137rm_a_b,R2: set_Pr2972776593051762503rm_a_b] :
( ! [A4: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ A4 @ A5 )
=> ? [B7: produc357393685978478089rm_a_b] : ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A4 @ B7 ) @ ( abstra1496011390211842225rm_a_b @ R2 ) ) )
=> ( abstra2721873542170697047rm_a_b @ R2 @ A5 ) ) ).
% WN_onI
thf(fact_951_WN__onI,axiom,
! [A5: set_term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ! [A4: term_a_b] :
( ( member_term_a_b @ A4 @ A5 )
=> ? [B7: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A4 @ B7 ) @ ( abstra2972651231392952684rm_a_b @ R2 ) ) )
=> ( abstra4644735066934440262rm_a_b @ R2 @ A5 ) ) ).
% WN_onI
thf(fact_952_rtrancl__power,axiom,
! [P: produc357393685978478089rm_a_b,R3: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ P @ ( transi7742714808557438673rm_a_b @ R3 ) )
= ( ? [N2: nat] : ( member5869715511025134514rm_a_b @ P @ ( compow4057154403645558940rm_a_b @ N2 @ R3 ) ) ) ) ).
% rtrancl_power
thf(fact_953_relpow__imp__rtrancl,axiom,
! [P: produc357393685978478089rm_a_b,N: nat,R3: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ P @ ( compow4057154403645558940rm_a_b @ N @ R3 ) )
=> ( member5869715511025134514rm_a_b @ P @ ( transi7742714808557438673rm_a_b @ R3 ) ) ) ).
% relpow_imp_rtrancl
thf(fact_954_rtrancl__imp__relpow,axiom,
! [P: produc357393685978478089rm_a_b,R3: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ P @ ( transi7742714808557438673rm_a_b @ R3 ) )
=> ? [N3: nat] : ( member5869715511025134514rm_a_b @ P @ ( compow4057154403645558940rm_a_b @ N3 @ R3 ) ) ) ).
% rtrancl_imp_relpow
thf(fact_955_relpow__image,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,F3: term_a_b > term_a_b,R8: set_Pr4386577575007340137rm_a_b,S: term_a_b,T: term_a_b,N: nat] :
( ! [S4: term_a_b,T3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S4 @ T3 ) @ R2 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F3 @ S4 ) @ ( F3 @ T3 ) ) @ R8 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( compow4057154403645558940rm_a_b @ N @ R2 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F3 @ S ) @ ( F3 @ T ) ) @ ( compow4057154403645558940rm_a_b @ N @ R8 ) ) ) ) ).
% relpow_image
thf(fact_956_relpow__Suc__D2_H,axiom,
! [N: nat,R3: set_Pr4386577575007340137rm_a_b,X6: term_a_b,Y6: term_a_b,Z4: term_a_b] :
( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X6 @ Y6 ) @ ( compow4057154403645558940rm_a_b @ N @ R3 ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y6 @ Z4 ) @ R3 ) )
=> ? [W: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X6 @ W ) @ R3 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ W @ Z4 ) @ ( compow4057154403645558940rm_a_b @ N @ R3 ) ) ) ) ).
% relpow_Suc_D2'
thf(fact_957_relpow__refl__mono,axiom,
! [Rel: set_Pr4386577575007340137rm_a_b,M2: nat,N: nat,A: term_a_b,B: term_a_b] :
( ! [X3: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ X3 ) @ Rel )
=> ( ( ord_less_eq_nat @ M2 @ N )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( compow4057154403645558940rm_a_b @ M2 @ Rel ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( compow4057154403645558940rm_a_b @ N @ Rel ) ) ) ) ) ).
% relpow_refl_mono
thf(fact_958_rtrancl__len__E,axiom,
! [X: term_a_b,Y: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
=> ~ ! [N3: nat] :
~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( compow4057154403645558940rm_a_b @ N3 @ R2 ) ) ) ).
% rtrancl_len_E
thf(fact_959_depth_Ocases,axiom,
! [X: term_a_b] :
( ! [X3: b] :
( X
!= ( var_b_a @ X3 ) )
=> ( ! [F2: a] :
( X
!= ( fun_a_b @ F2 @ nil_term_a_b ) )
=> ~ ! [F2: a,V: term_a_b,Va: list_term_a_b] :
( X
!= ( fun_a_b @ F2 @ ( cons_term_a_b @ V @ Va ) ) ) ) ) ).
% depth.cases
thf(fact_960_trancl__steps__relpow,axiom,
! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b,X: term_a_b,Y: term_a_b,N: nat] :
( ( ord_le118470702582115849rm_a_b @ A @ ( transi7922773638565587891rm_a_b @ B ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( compow4057154403645558940rm_a_b @ N @ A ) )
=> ? [M4: nat] :
( ( ord_less_eq_nat @ N @ M4 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( compow4057154403645558940rm_a_b @ M4 @ B ) ) ) ) ) ).
% trancl_steps_relpow
thf(fact_961_WN__on__def,axiom,
( abstra4644735066934440262rm_a_b
= ( ^ [R6: set_Pr4386577575007340137rm_a_b,A6: set_term_a_b] :
! [X4: term_a_b] :
( ( member_term_a_b @ X4 @ A6 )
=> ? [B4: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X4 @ B4 ) @ ( abstra2972651231392952684rm_a_b @ R6 ) ) ) ) ) ).
% WN_on_def
thf(fact_962_WN__onE,axiom,
! [R2: set_Pr2972776593051762503rm_a_b,A5: set_Pr4386577575007340137rm_a_b,A: produc357393685978478089rm_a_b] :
( ( abstra2721873542170697047rm_a_b @ R2 @ A5 )
=> ( ( member5869715511025134514rm_a_b @ A @ A5 )
=> ~ ! [B3: produc357393685978478089rm_a_b] :
~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B3 ) @ ( abstra1496011390211842225rm_a_b @ R2 ) ) ) ) ).
% WN_onE
thf(fact_963_WN__onE,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,A5: set_term_a_b,A: term_a_b] :
( ( abstra4644735066934440262rm_a_b @ R2 @ A5 )
=> ( ( member_term_a_b @ A @ A5 )
=> ~ ! [B3: term_a_b] :
~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B3 ) @ ( abstra2972651231392952684rm_a_b @ R2 ) ) ) ) ).
% WN_onE
thf(fact_964_WN__onD,axiom,
! [R2: set_Pr2972776593051762503rm_a_b,A5: set_Pr4386577575007340137rm_a_b,A: produc357393685978478089rm_a_b] :
( ( abstra2721873542170697047rm_a_b @ R2 @ A5 )
=> ( ( member5869715511025134514rm_a_b @ A @ A5 )
=> ? [B3: produc357393685978478089rm_a_b] : ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B3 ) @ ( abstra1496011390211842225rm_a_b @ R2 ) ) ) ) ).
% WN_onD
thf(fact_965_WN__onD,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,A5: set_term_a_b,A: term_a_b] :
( ( abstra4644735066934440262rm_a_b @ R2 @ A5 )
=> ( ( member_term_a_b @ A @ A5 )
=> ? [B3: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B3 ) @ ( abstra2972651231392952684rm_a_b @ R2 ) ) ) ) ).
% WN_onD
thf(fact_966_UNF__onD,axiom,
! [R2: set_Pr2972776593051762503rm_a_b,A5: set_Pr4386577575007340137rm_a_b,A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,C2: produc357393685978478089rm_a_b] :
( ( abstra1303798319023342519rm_a_b @ R2 @ A5 )
=> ( ( member5869715511025134514rm_a_b @ A @ A5 )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( abstra1496011390211842225rm_a_b @ R2 ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ C2 ) @ ( abstra1496011390211842225rm_a_b @ R2 ) )
=> ( B = C2 ) ) ) ) ) ).
% UNF_onD
thf(fact_967_UNF__onD,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,A5: set_term_a_b,A: term_a_b,B: term_a_b,C2: term_a_b] :
( ( abstra3619855735679963878rm_a_b @ R2 @ A5 )
=> ( ( member_term_a_b @ A @ A5 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( abstra2972651231392952684rm_a_b @ R2 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C2 ) @ ( abstra2972651231392952684rm_a_b @ R2 ) )
=> ( B = C2 ) ) ) ) ) ).
% UNF_onD
thf(fact_968_UNF__onE,axiom,
! [R2: set_Pr2972776593051762503rm_a_b,A5: set_Pr4386577575007340137rm_a_b,A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,C2: produc357393685978478089rm_a_b] :
( ( abstra1303798319023342519rm_a_b @ R2 @ A5 )
=> ( ( member5869715511025134514rm_a_b @ A @ A5 )
=> ( ( B != C2 )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( abstra1496011390211842225rm_a_b @ R2 ) )
=> ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ C2 ) @ ( abstra1496011390211842225rm_a_b @ R2 ) ) ) ) ) ) ).
% UNF_onE
thf(fact_969_UNF__onE,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,A5: set_term_a_b,A: term_a_b,B: term_a_b,C2: term_a_b] :
( ( abstra3619855735679963878rm_a_b @ R2 @ A5 )
=> ( ( member_term_a_b @ A @ A5 )
=> ( ( B != C2 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( abstra2972651231392952684rm_a_b @ R2 ) )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C2 ) @ ( abstra2972651231392952684rm_a_b @ R2 ) ) ) ) ) ) ).
% UNF_onE
thf(fact_970_UNF__on__def,axiom,
( abstra3619855735679963878rm_a_b
= ( ^ [R6: set_Pr4386577575007340137rm_a_b,A6: set_term_a_b] :
! [X4: term_a_b] :
( ( member_term_a_b @ X4 @ A6 )
=> ! [B4: term_a_b,C: term_a_b] :
( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X4 @ B4 ) @ ( abstra2972651231392952684rm_a_b @ R6 ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X4 @ C ) @ ( abstra2972651231392952684rm_a_b @ R6 ) ) )
=> ( B4 = C ) ) ) ) ) ).
% UNF_on_def
thf(fact_971_some__NF__UNF,axiom,
! [R2: set_Pr2972776593051762503rm_a_b,X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b] :
( ( abstra1303798319023342519rm_a_b @ R2 @ top_to1314267278846557113rm_a_b )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ ( transi2615809358984392588rm_a_b @ R2 ) )
=> ( ( member5869715511025134514rm_a_b @ Y @ ( abstra1902783451776770197rm_a_b @ R2 ) )
=> ( Y
= ( abstra8203652949708803681rm_a_b @ R2 @ X ) ) ) ) ) ).
% some_NF_UNF
thf(fact_972_some__NF__UNF,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,X: term_a_b,Y: term_a_b] :
( ( abstra3619855735679963878rm_a_b @ R2 @ top_top_set_term_a_b )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
=> ( ( member_term_a_b @ Y @ ( abstract_NF_term_a_b @ R2 ) )
=> ( Y
= ( abstra2589370493620291516rm_a_b @ R2 @ X ) ) ) ) ) ).
% some_NF_UNF
thf(fact_973_Cons__lenlex__iff,axiom,
! [M2: term_a_b,Ms: list_term_a_b,N: term_a_b,Ns: list_term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ ( cons_term_a_b @ M2 @ Ms ) @ ( cons_term_a_b @ N @ Ns ) ) @ ( lenlex_term_a_b @ R2 ) )
= ( ( ord_less_nat @ ( size_s8906293707977694520rm_a_b @ Ms ) @ ( size_s8906293707977694520rm_a_b @ Ns ) )
| ( ( ( size_s8906293707977694520rm_a_b @ Ms )
= ( size_s8906293707977694520rm_a_b @ Ns ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ M2 @ N ) @ R2 ) )
| ( ( M2 = N )
& ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Ms @ Ns ) @ ( lenlex_term_a_b @ R2 ) ) ) ) ) ).
% Cons_lenlex_iff
thf(fact_974_lenlex__irreflexive,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,Xs: list_term_a_b] :
( ! [X3: term_a_b] :
~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ X3 ) @ R2 )
=> ~ ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Xs @ Xs ) @ ( lenlex_term_a_b @ R2 ) ) ) ).
% lenlex_irreflexive
thf(fact_975_lenlex__length,axiom,
! [Ms: list_term_a_b,Ns: list_term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Ms @ Ns ) @ ( lenlex_term_a_b @ R2 ) )
=> ( ord_less_eq_nat @ ( size_s8906293707977694520rm_a_b @ Ms ) @ ( size_s8906293707977694520rm_a_b @ Ns ) ) ) ).
% lenlex_length
thf(fact_976_Cons__in__lex,axiom,
! [X: term_a_b,Xs: list_term_a_b,Y: term_a_b,Ys: list_term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ ( cons_term_a_b @ X @ Xs ) @ ( cons_term_a_b @ Y @ Ys ) ) @ ( lex_term_a_b @ R2 ) )
= ( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ R2 )
& ( ( size_s8906293707977694520rm_a_b @ Xs )
= ( size_s8906293707977694520rm_a_b @ Ys ) ) )
| ( ( X = Y )
& ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Xs @ Ys ) @ ( lex_term_a_b @ R2 ) ) ) ) ) ).
% Cons_in_lex
thf(fact_977_listrel__iff__nth,axiom,
! [Xs: list_term_a_b,Ys: list_term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Xs @ Ys ) @ ( listre1194016999521814427rm_a_b @ R2 ) )
= ( ( ( size_s8906293707977694520rm_a_b @ Xs )
= ( size_s8906293707977694520rm_a_b @ Ys ) )
& ! [N2: nat] :
( ( ord_less_nat @ N2 @ ( size_s8906293707977694520rm_a_b @ Xs ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( nth_term_a_b @ Xs @ N2 ) @ ( nth_term_a_b @ Ys @ N2 ) ) @ R2 ) ) ) ) ).
% listrel_iff_nth
thf(fact_978_lex__take__index,axiom,
! [Xs: list_term_a_b,Ys: list_term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Xs @ Ys ) @ ( lex_term_a_b @ R2 ) )
=> ~ ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s8906293707977694520rm_a_b @ Xs ) )
=> ( ( ord_less_nat @ I2 @ ( size_s8906293707977694520rm_a_b @ Ys ) )
=> ( ( ( take_term_a_b @ I2 @ Xs )
= ( take_term_a_b @ I2 @ Ys ) )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( nth_term_a_b @ Xs @ I2 ) @ ( nth_term_a_b @ Ys @ I2 ) ) @ R2 ) ) ) ) ) ).
% lex_take_index
thf(fact_979_take__all__iff,axiom,
! [N: nat,Xs: list_term_a_b] :
( ( ( take_term_a_b @ N @ Xs )
= Xs )
= ( ord_less_eq_nat @ ( size_s8906293707977694520rm_a_b @ Xs ) @ N ) ) ).
% take_all_iff
thf(fact_980_take__all,axiom,
! [Xs: list_term_a_b,N: nat] :
( ( ord_less_eq_nat @ ( size_s8906293707977694520rm_a_b @ Xs ) @ N )
=> ( ( take_term_a_b @ N @ Xs )
= Xs ) ) ).
% take_all
thf(fact_981_nth__take,axiom,
! [I4: nat,N: nat,Xs: list_term_a_b] :
( ( ord_less_nat @ I4 @ N )
=> ( ( nth_term_a_b @ ( take_term_a_b @ N @ Xs ) @ I4 )
= ( nth_term_a_b @ Xs @ I4 ) ) ) ).
% nth_take
thf(fact_982_listrel__rtrancl__refl,axiom,
! [Xs: list_term_a_b,R2: set_Pr4386577575007340137rm_a_b] : ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Xs @ Xs ) @ ( listre1194016999521814427rm_a_b @ ( transi7742714808557438673rm_a_b @ R2 ) ) ) ).
% listrel_rtrancl_refl
thf(fact_983_listrel__mono,axiom,
! [R2: set_Pr4934435412358123699_a_nat,S: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ R2 @ S )
=> ( ord_le1912520282604406569st_nat @ ( listrel_a_nat @ R2 ) @ ( listrel_a_nat @ S ) ) ) ).
% listrel_mono
thf(fact_984_listrel__eq__len,axiom,
! [Xs: list_term_a_b,Ys: list_term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Xs @ Ys ) @ ( listre1194016999521814427rm_a_b @ R2 ) )
=> ( ( size_s8906293707977694520rm_a_b @ Xs )
= ( size_s8906293707977694520rm_a_b @ Ys ) ) ) ).
% listrel_eq_len
thf(fact_985_listrel__rtrancl__trans,axiom,
! [Xs: list_term_a_b,Ys: list_term_a_b,R2: set_Pr4386577575007340137rm_a_b,Zs: list_term_a_b] :
( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Xs @ Ys ) @ ( listre1194016999521814427rm_a_b @ ( transi7742714808557438673rm_a_b @ R2 ) ) )
=> ( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Ys @ Zs ) @ ( listre1194016999521814427rm_a_b @ ( transi7742714808557438673rm_a_b @ R2 ) ) )
=> ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Xs @ Zs ) @ ( listre1194016999521814427rm_a_b @ ( transi7742714808557438673rm_a_b @ R2 ) ) ) ) ) ).
% listrel_rtrancl_trans
thf(fact_986_listrel__Cons2,axiom,
! [Xs: list_term_a_b,Y: term_a_b,Ys: list_term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Xs @ ( cons_term_a_b @ Y @ Ys ) ) @ ( listre1194016999521814427rm_a_b @ R2 ) )
=> ~ ! [X3: term_a_b,Xs3: list_term_a_b] :
( ( Xs
= ( cons_term_a_b @ X3 @ Xs3 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y ) @ R2 )
=> ~ ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Xs3 @ Ys ) @ ( listre1194016999521814427rm_a_b @ R2 ) ) ) ) ) ).
% listrel_Cons2
thf(fact_987_listrel__Cons1,axiom,
! [Y: term_a_b,Ys: list_term_a_b,Xs: list_term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ ( cons_term_a_b @ Y @ Ys ) @ Xs ) @ ( listre1194016999521814427rm_a_b @ R2 ) )
=> ~ ! [Y3: term_a_b,Ys4: list_term_a_b] :
( ( Xs
= ( cons_term_a_b @ Y3 @ Ys4 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ Y3 ) @ R2 )
=> ~ ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Ys @ Ys4 ) @ ( listre1194016999521814427rm_a_b @ R2 ) ) ) ) ) ).
% listrel_Cons1
thf(fact_988_listrel_OCons,axiom,
! [X: term_a_b,Y: term_a_b,R2: set_Pr4386577575007340137rm_a_b,Xs: list_term_a_b,Ys: list_term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ R2 )
=> ( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Xs @ Ys ) @ ( listre1194016999521814427rm_a_b @ R2 ) )
=> ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ ( cons_term_a_b @ X @ Xs ) @ ( cons_term_a_b @ Y @ Ys ) ) @ ( listre1194016999521814427rm_a_b @ R2 ) ) ) ) ).
% listrel.Cons
thf(fact_989_nth__take__lemma,axiom,
! [K: nat,Xs: list_term_a_b,Ys: list_term_a_b] :
( ( ord_less_eq_nat @ K @ ( size_s8906293707977694520rm_a_b @ Xs ) )
=> ( ( ord_less_eq_nat @ K @ ( size_s8906293707977694520rm_a_b @ Ys ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ K )
=> ( ( nth_term_a_b @ Xs @ I2 )
= ( nth_term_a_b @ Ys @ I2 ) ) )
=> ( ( take_term_a_b @ K @ Xs )
= ( take_term_a_b @ K @ Ys ) ) ) ) ) ).
% nth_take_lemma
thf(fact_990_nth__take__prefix,axiom,
! [Ys: list_term_a_b,Xs: list_term_a_b] :
( ( ord_less_eq_nat @ ( size_s8906293707977694520rm_a_b @ Ys ) @ ( size_s8906293707977694520rm_a_b @ Xs ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s8906293707977694520rm_a_b @ Ys ) )
=> ( ( nth_term_a_b @ Xs @ I2 )
= ( nth_term_a_b @ Ys @ I2 ) ) )
=> ( ( take_term_a_b @ ( size_s8906293707977694520rm_a_b @ Ys ) @ Xs )
= Ys ) ) ) ).
% nth_take_prefix
thf(fact_991_listrel_Osimps,axiom,
! [A1: list_term_a_b,A22: list_term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ A1 @ A22 ) @ ( listre1194016999521814427rm_a_b @ R2 ) )
= ( ( ( A1 = nil_term_a_b )
& ( A22 = nil_term_a_b ) )
| ? [X4: term_a_b,Y5: term_a_b,Xs2: list_term_a_b,Ys2: list_term_a_b] :
( ( A1
= ( cons_term_a_b @ X4 @ Xs2 ) )
& ( A22
= ( cons_term_a_b @ Y5 @ Ys2 ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X4 @ Y5 ) @ R2 )
& ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Xs2 @ Ys2 ) @ ( listre1194016999521814427rm_a_b @ R2 ) ) ) ) ) ).
% listrel.simps
thf(fact_992_listrel_Ocases,axiom,
! [A1: list_term_a_b,A22: list_term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ A1 @ A22 ) @ ( listre1194016999521814427rm_a_b @ R2 ) )
=> ( ( ( A1 = nil_term_a_b )
=> ( A22 != nil_term_a_b ) )
=> ~ ! [X3: term_a_b,Y3: term_a_b,Xs3: list_term_a_b] :
( ( A1
= ( cons_term_a_b @ X3 @ Xs3 ) )
=> ! [Ys4: list_term_a_b] :
( ( A22
= ( cons_term_a_b @ Y3 @ Ys4 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ R2 )
=> ~ ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Xs3 @ Ys4 ) @ ( listre1194016999521814427rm_a_b @ R2 ) ) ) ) ) ) ) ).
% listrel.cases
thf(fact_993_in__measures_I2_J,axiom,
! [X: term_a_b,Y: term_a_b,F3: term_a_b > nat,Fs: list_term_a_b_nat] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( measures_term_a_b @ ( cons_term_a_b_nat @ F3 @ Fs ) ) )
= ( ( ord_less_nat @ ( F3 @ X ) @ ( F3 @ Y ) )
| ( ( ( F3 @ X )
= ( F3 @ Y ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( measures_term_a_b @ Fs ) ) ) ) ) ).
% in_measures(2)
thf(fact_994_restrict__SN__trancl__simp,axiom,
! [A5: set_Pr4386577575007340137rm_a_b] :
( ( transi7922773638565587891rm_a_b @ ( abstra4933154510623179783rm_a_b @ A5 @ A5 ) )
= ( abstra4933154510623179783rm_a_b @ ( transi7922773638565587891rm_a_b @ A5 ) @ A5 ) ) ).
% restrict_SN_trancl_simp
thf(fact_995_term__to__sig_Osimps_I2_J,axiom,
! [F3: a,Ts2: list_term_a_b,F: set_Pr4934435412358123699_a_nat,V2: b] :
( ( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ F3 @ ( size_s8906293707977694520rm_a_b @ Ts2 ) ) @ F )
=> ( ( terms_8519481630511763164ig_a_b @ F @ V2 @ ( fun_a_b @ F3 @ Ts2 ) )
= ( fun_a_b @ F3 @ ( map_te2328734355998148206rm_a_b @ ( terms_8519481630511763164ig_a_b @ F @ V2 ) @ Ts2 ) ) ) )
& ( ~ ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ F3 @ ( size_s8906293707977694520rm_a_b @ Ts2 ) ) @ F )
=> ( ( terms_8519481630511763164ig_a_b @ F @ V2 @ ( fun_a_b @ F3 @ Ts2 ) )
= ( var_b_a @ V2 ) ) ) ) ).
% term_to_sig.simps(2)
thf(fact_996_length__map,axiom,
! [F3: term_a_b > term_a_b,Xs: list_term_a_b] :
( ( size_s8906293707977694520rm_a_b @ ( map_te2328734355998148206rm_a_b @ F3 @ Xs ) )
= ( size_s8906293707977694520rm_a_b @ Xs ) ) ).
% length_map
thf(fact_997_in__measures_I1_J,axiom,
! [X: term_a_b,Y: term_a_b] :
~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( measures_term_a_b @ nil_term_a_b_nat ) ) ).
% in_measures(1)
thf(fact_998_nth__map,axiom,
! [N: nat,Xs: list_term_a_b,F3: term_a_b > term_a_b] :
( ( ord_less_nat @ N @ ( size_s8906293707977694520rm_a_b @ Xs ) )
=> ( ( nth_term_a_b @ ( map_te2328734355998148206rm_a_b @ F3 @ Xs ) @ N )
= ( F3 @ ( nth_term_a_b @ Xs @ N ) ) ) ) ).
% nth_map
thf(fact_999_map__nth__eq__conv,axiom,
! [Xs: list_term_a_b,Ys: list_term_a_b,F3: term_a_b > term_a_b] :
( ( ( size_s8906293707977694520rm_a_b @ Xs )
= ( size_s8906293707977694520rm_a_b @ Ys ) )
=> ( ( ( map_te2328734355998148206rm_a_b @ F3 @ Xs )
= Ys )
= ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s8906293707977694520rm_a_b @ Ys ) )
=> ( ( F3 @ ( nth_term_a_b @ Xs @ I3 ) )
= ( nth_term_a_b @ Ys @ I3 ) ) ) ) ) ) ).
% map_nth_eq_conv
thf(fact_1000_measures__less,axiom,
! [F3: term_a_b > nat,X: term_a_b,Y: term_a_b,Fs: list_term_a_b_nat] :
( ( ord_less_nat @ ( F3 @ X ) @ ( F3 @ Y ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( measures_term_a_b @ ( cons_term_a_b_nat @ F3 @ Fs ) ) ) ) ).
% measures_less
thf(fact_1001_measures__lesseq,axiom,
! [F3: term_a_b > nat,X: term_a_b,Y: term_a_b,Fs: list_term_a_b_nat] :
( ( ord_less_eq_nat @ ( F3 @ X ) @ ( F3 @ Y ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( measures_term_a_b @ Fs ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( measures_term_a_b @ ( cons_term_a_b_nat @ F3 @ Fs ) ) ) ) ) ).
% measures_lesseq
thf(fact_1002_term__to__sig_Oelims,axiom,
! [X: set_Pr4934435412358123699_a_nat,Xa: b,Xb: term_a_b,Y: term_a_b] :
( ( ( terms_8519481630511763164ig_a_b @ X @ Xa @ Xb )
= Y )
=> ( ! [X3: b] :
( ( Xb
= ( var_b_a @ X3 ) )
=> ( Y
!= ( var_b_a @ X3 ) ) )
=> ~ ! [F2: a,Ts: list_term_a_b] :
( ( Xb
= ( fun_a_b @ F2 @ Ts ) )
=> ~ ( ( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ F2 @ ( size_s8906293707977694520rm_a_b @ Ts ) ) @ X )
=> ( Y
= ( fun_a_b @ F2 @ ( map_te2328734355998148206rm_a_b @ ( terms_8519481630511763164ig_a_b @ X @ Xa ) @ Ts ) ) ) )
& ( ~ ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ F2 @ ( size_s8906293707977694520rm_a_b @ Ts ) ) @ X )
=> ( Y
= ( var_b_a @ Xa ) ) ) ) ) ) ) ).
% term_to_sig.elims
thf(fact_1003_term__to__sig_Opelims,axiom,
! [X: set_Pr4934435412358123699_a_nat,Xa: b,Xb: term_a_b,Y: term_a_b] :
( ( ( terms_8519481630511763164ig_a_b @ X @ Xa @ Xb )
= Y )
=> ( ( accp_P3366084933459579103rm_a_b @ terms_8023737178250165727el_a_b @ ( produc8030969961714872974rm_a_b @ X @ ( produc1437816968797971900rm_a_b @ Xa @ Xb ) ) )
=> ( ! [X3: b] :
( ( Xb
= ( var_b_a @ X3 ) )
=> ( ( Y
= ( var_b_a @ X3 ) )
=> ~ ( accp_P3366084933459579103rm_a_b @ terms_8023737178250165727el_a_b @ ( produc8030969961714872974rm_a_b @ X @ ( produc1437816968797971900rm_a_b @ Xa @ ( var_b_a @ X3 ) ) ) ) ) )
=> ~ ! [F2: a,Ts: list_term_a_b] :
( ( Xb
= ( fun_a_b @ F2 @ Ts ) )
=> ( ( ( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ F2 @ ( size_s8906293707977694520rm_a_b @ Ts ) ) @ X )
=> ( Y
= ( fun_a_b @ F2 @ ( map_te2328734355998148206rm_a_b @ ( terms_8519481630511763164ig_a_b @ X @ Xa ) @ Ts ) ) ) )
& ( ~ ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ F2 @ ( size_s8906293707977694520rm_a_b @ Ts ) ) @ X )
=> ( Y
= ( var_b_a @ Xa ) ) ) )
=> ~ ( accp_P3366084933459579103rm_a_b @ terms_8023737178250165727el_a_b @ ( produc8030969961714872974rm_a_b @ X @ ( produc1437816968797971900rm_a_b @ Xa @ ( fun_a_b @ F2 @ Ts ) ) ) ) ) ) ) ) ) ).
% term_to_sig.pelims
thf(fact_1004_nth__append__take__is__nth__conv,axiom,
! [I4: nat,J: nat,Xs: list_term_a_b,Ys: list_term_a_b] :
( ( ord_less_nat @ I4 @ J )
=> ( ( ord_less_eq_nat @ J @ ( size_s8906293707977694520rm_a_b @ Xs ) )
=> ( ( nth_term_a_b @ ( append_term_a_b @ ( take_term_a_b @ J @ Xs ) @ Ys ) @ I4 )
= ( nth_term_a_b @ Xs @ I4 ) ) ) ) ).
% nth_append_take_is_nth_conv
thf(fact_1005_nth__append__take,axiom,
! [I4: nat,Xs: list_term_a_b,Y: term_a_b,Ys: list_term_a_b] :
( ( ord_less_eq_nat @ I4 @ ( size_s8906293707977694520rm_a_b @ Xs ) )
=> ( ( nth_term_a_b @ ( append_term_a_b @ ( take_term_a_b @ I4 @ Xs ) @ ( cons_term_a_b @ Y @ Ys ) ) @ I4 )
= Y ) ) ).
% nth_append_take
thf(fact_1006_append__eq__append__conv,axiom,
! [Xs: list_term_a_b,Ys: list_term_a_b,Us: list_term_a_b,Vs: list_term_a_b] :
( ( ( ( size_s8906293707977694520rm_a_b @ Xs )
= ( size_s8906293707977694520rm_a_b @ Ys ) )
| ( ( size_s8906293707977694520rm_a_b @ Us )
= ( size_s8906293707977694520rm_a_b @ Vs ) ) )
=> ( ( ( append_term_a_b @ Xs @ Us )
= ( append_term_a_b @ Ys @ Vs ) )
= ( ( Xs = Ys )
& ( Us = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_1007_nth__append__length,axiom,
! [Xs: list_term_a_b,X: term_a_b,Ys: list_term_a_b] :
( ( nth_term_a_b @ ( append_term_a_b @ Xs @ ( cons_term_a_b @ X @ Ys ) ) @ ( size_s8906293707977694520rm_a_b @ Xs ) )
= X ) ).
% nth_append_length
thf(fact_1008_same__length__different,axiom,
! [Xs: list_term_a_b,Ys: list_term_a_b] :
( ( Xs != Ys )
=> ( ( ( size_s8906293707977694520rm_a_b @ Xs )
= ( size_s8906293707977694520rm_a_b @ Ys ) )
=> ? [Pre: list_term_a_b,X3: term_a_b,Xs4: list_term_a_b,Y3: term_a_b,Ys5: list_term_a_b] :
( ( X3 != Y3 )
& ( Xs
= ( append_term_a_b @ Pre @ ( append_term_a_b @ ( cons_term_a_b @ X3 @ nil_term_a_b ) @ Xs4 ) ) )
& ( Ys
= ( append_term_a_b @ Pre @ ( append_term_a_b @ ( cons_term_a_b @ Y3 @ nil_term_a_b ) @ Ys5 ) ) ) ) ) ) ).
% same_length_different
thf(fact_1009_append__Cons__nth__not__middle,axiom,
! [I4: nat,Xs: list_term_a_b,U: term_a_b,Ys: list_term_a_b,Z2: term_a_b] :
( ( I4
!= ( size_s8906293707977694520rm_a_b @ Xs ) )
=> ( ( nth_term_a_b @ ( append_term_a_b @ Xs @ ( cons_term_a_b @ U @ Ys ) ) @ I4 )
= ( nth_term_a_b @ ( append_term_a_b @ Xs @ ( cons_term_a_b @ Z2 @ Ys ) ) @ I4 ) ) ) ).
% append_Cons_nth_not_middle
thf(fact_1010_append__Cons__nth__middle,axiom,
! [I4: nat,Xs: list_term_a_b,Y: term_a_b,Zs: list_term_a_b] :
( ( I4
= ( size_s8906293707977694520rm_a_b @ Xs ) )
=> ( ( nth_term_a_b @ ( append_term_a_b @ Xs @ ( cons_term_a_b @ Y @ Zs ) ) @ I4 )
= Y ) ) ).
% append_Cons_nth_middle
thf(fact_1011_P__as__bs__extend,axiom,
! [As: list_term_a_b,Bs: list_term_a_b,Cs: list_term_a_b,Ds: list_term_a_b,P2: term_a_b > term_a_b > $o] :
( ( ( size_s8906293707977694520rm_a_b @ As )
= ( size_s8906293707977694520rm_a_b @ Bs ) )
=> ( ( ( size_s8906293707977694520rm_a_b @ Cs )
= ( size_s8906293707977694520rm_a_b @ Ds ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s8906293707977694520rm_a_b @ Bs ) )
=> ( P2 @ ( nth_term_a_b @ As @ I2 ) @ ( nth_term_a_b @ Bs @ I2 ) ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s8906293707977694520rm_a_b @ Ds ) )
=> ( P2 @ ( nth_term_a_b @ Cs @ I2 ) @ ( nth_term_a_b @ Ds @ I2 ) ) )
=> ! [I: nat] :
( ( ord_less_nat @ I @ ( size_s8906293707977694520rm_a_b @ ( append_term_a_b @ Bs @ Ds ) ) )
=> ( P2 @ ( nth_term_a_b @ ( append_term_a_b @ As @ Cs ) @ I ) @ ( nth_term_a_b @ ( append_term_a_b @ Bs @ Ds ) @ I ) ) ) ) ) ) ) ).
% P_as_bs_extend
thf(fact_1012_lex__append__left__iff,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,Xs: list_term_a_b,Ys: list_term_a_b,Zs: list_term_a_b] :
( ! [X3: term_a_b] :
~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ X3 ) @ R2 )
=> ( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ ( append_term_a_b @ Xs @ Ys ) @ ( append_term_a_b @ Xs @ Zs ) ) @ ( lex_term_a_b @ R2 ) )
= ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Ys @ Zs ) @ ( lex_term_a_b @ R2 ) ) ) ) ).
% lex_append_left_iff
thf(fact_1013_lex__append__leftD,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,Xs: list_term_a_b,Ys: list_term_a_b,Zs: list_term_a_b] :
( ! [X3: term_a_b] :
~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ X3 ) @ R2 )
=> ( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ ( append_term_a_b @ Xs @ Ys ) @ ( append_term_a_b @ Xs @ Zs ) ) @ ( lex_term_a_b @ R2 ) )
=> ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Ys @ Zs ) @ ( lex_term_a_b @ R2 ) ) ) ) ).
% lex_append_leftD
thf(fact_1014_lex__append__rightI,axiom,
! [Xs: list_term_a_b,Ys: list_term_a_b,R2: set_Pr4386577575007340137rm_a_b,Vs: list_term_a_b,Us: list_term_a_b] :
( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Xs @ Ys ) @ ( lex_term_a_b @ R2 ) )
=> ( ( ( size_s8906293707977694520rm_a_b @ Vs )
= ( size_s8906293707977694520rm_a_b @ Us ) )
=> ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ ( append_term_a_b @ Xs @ Us ) @ ( append_term_a_b @ Ys @ Vs ) ) @ ( lex_term_a_b @ R2 ) ) ) ) ).
% lex_append_rightI
thf(fact_1015_lenlex__append1,axiom,
! [Us: list_term_a_b,Xs: list_term_a_b,R3: set_Pr4386577575007340137rm_a_b,Vs: list_term_a_b,Ys: list_term_a_b] :
( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Us @ Xs ) @ ( lenlex_term_a_b @ R3 ) )
=> ( ( ( size_s8906293707977694520rm_a_b @ Vs )
= ( size_s8906293707977694520rm_a_b @ Ys ) )
=> ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ ( append_term_a_b @ Us @ Vs ) @ ( append_term_a_b @ Xs @ Ys ) ) @ ( lenlex_term_a_b @ R3 ) ) ) ) ).
% lenlex_append1
thf(fact_1016_append__Cons__nth__left,axiom,
! [I4: nat,Xs: list_term_a_b,U: term_a_b,Ys: list_term_a_b] :
( ( ord_less_nat @ I4 @ ( size_s8906293707977694520rm_a_b @ Xs ) )
=> ( ( nth_term_a_b @ ( append_term_a_b @ Xs @ ( cons_term_a_b @ U @ Ys ) ) @ I4 )
= ( nth_term_a_b @ Xs @ I4 ) ) ) ).
% append_Cons_nth_left
thf(fact_1017_append__Cons__nth__right,axiom,
! [Xs: list_term_a_b,I4: nat,U: term_a_b,Ys: list_term_a_b,Z2: term_a_b] :
( ( ord_less_nat @ ( size_s8906293707977694520rm_a_b @ Xs ) @ I4 )
=> ( ( nth_term_a_b @ ( append_term_a_b @ Xs @ ( cons_term_a_b @ U @ Ys ) ) @ I4 )
= ( nth_term_a_b @ ( append_term_a_b @ Xs @ ( cons_term_a_b @ Z2 @ Ys ) ) @ I4 ) ) ) ).
% append_Cons_nth_right
thf(fact_1018_take__Suc__conv__app__nth,axiom,
! [I4: nat,Xs: list_term_a_b] :
( ( ord_less_nat @ I4 @ ( size_s8906293707977694520rm_a_b @ Xs ) )
=> ( ( take_term_a_b @ ( suc @ I4 ) @ Xs )
= ( append_term_a_b @ ( take_term_a_b @ I4 @ Xs ) @ ( cons_term_a_b @ ( nth_term_a_b @ Xs @ I4 ) @ nil_term_a_b ) ) ) ) ).
% take_Suc_conv_app_nth
thf(fact_1019_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_1020_Suc__mono,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_1021_Suc__less__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% Suc_less_eq
thf(fact_1022_nth__Cons__Suc,axiom,
! [X: term_a_b,Xs: list_term_a_b,N: nat] :
( ( nth_term_a_b @ ( cons_term_a_b @ X @ Xs ) @ ( suc @ N ) )
= ( nth_term_a_b @ Xs @ N ) ) ).
% nth_Cons_Suc
thf(fact_1023_not__less__less__Suc__eq,axiom,
! [N: nat,M2: nat] :
( ~ ( ord_less_nat @ N @ M2 )
=> ( ( ord_less_nat @ N @ ( suc @ M2 ) )
= ( N = M2 ) ) ) ).
% not_less_less_Suc_eq
thf(fact_1024_strict__inc__induct,axiom,
! [I4: nat,J: nat,P2: nat > $o] :
( ( ord_less_nat @ I4 @ J )
=> ( ! [I2: nat] :
( ( J
= ( suc @ I2 ) )
=> ( P2 @ I2 ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( P2 @ ( suc @ I2 ) )
=> ( P2 @ I2 ) ) )
=> ( P2 @ I4 ) ) ) ) ).
% strict_inc_induct
thf(fact_1025_less__Suc__induct,axiom,
! [I4: nat,J: nat,P2: nat > nat > $o] :
( ( ord_less_nat @ I4 @ J )
=> ( ! [I2: nat] : ( P2 @ I2 @ ( suc @ I2 ) )
=> ( ! [I2: nat,J2: nat,K2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ( ord_less_nat @ J2 @ K2 )
=> ( ( P2 @ I2 @ J2 )
=> ( ( P2 @ J2 @ K2 )
=> ( P2 @ I2 @ K2 ) ) ) ) )
=> ( P2 @ I4 @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_1026_less__trans__Suc,axiom,
! [I4: nat,J: nat,K: nat] :
( ( ord_less_nat @ I4 @ J )
=> ( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( suc @ I4 ) @ K ) ) ) ).
% less_trans_Suc
thf(fact_1027_Suc__less__SucD,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ).
% Suc_less_SucD
thf(fact_1028_less__antisym,axiom,
! [N: nat,M2: nat] :
( ~ ( ord_less_nat @ N @ M2 )
=> ( ( ord_less_nat @ N @ ( suc @ M2 ) )
=> ( M2 = N ) ) ) ).
% less_antisym
thf(fact_1029_Suc__less__eq2,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ ( suc @ N ) @ M2 )
= ( ? [M5: nat] :
( ( M2
= ( suc @ M5 ) )
& ( ord_less_nat @ N @ M5 ) ) ) ) ).
% Suc_less_eq2
thf(fact_1030_All__less__Suc,axiom,
! [N: nat,P2: nat > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
=> ( P2 @ I3 ) ) )
= ( ( P2 @ N )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ( P2 @ I3 ) ) ) ) ).
% All_less_Suc
thf(fact_1031_not__less__eq,axiom,
! [M2: nat,N: nat] :
( ( ~ ( ord_less_nat @ M2 @ N ) )
= ( ord_less_nat @ N @ ( suc @ M2 ) ) ) ).
% not_less_eq
thf(fact_1032_less__Suc__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N ) )
= ( ( ord_less_nat @ M2 @ N )
| ( M2 = N ) ) ) ).
% less_Suc_eq
thf(fact_1033_Ex__less__Suc,axiom,
! [N: nat,P2: nat > $o] :
( ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
& ( P2 @ I3 ) ) )
= ( ( P2 @ N )
| ? [I3: nat] :
( ( ord_less_nat @ I3 @ N )
& ( P2 @ I3 ) ) ) ) ).
% Ex_less_Suc
thf(fact_1034_less__SucI,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_nat @ M2 @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_1035_less__SucE,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M2 @ N )
=> ( M2 = N ) ) ) ).
% less_SucE
thf(fact_1036_Suc__lessI,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ( ( suc @ M2 )
!= N )
=> ( ord_less_nat @ ( suc @ M2 ) @ N ) ) ) ).
% Suc_lessI
thf(fact_1037_Suc__lessE,axiom,
! [I4: nat,K: nat] :
( ( ord_less_nat @ ( suc @ I4 ) @ K )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I4 @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ).
% Suc_lessE
thf(fact_1038_Suc__lessD,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ N )
=> ( ord_less_nat @ M2 @ N ) ) ).
% Suc_lessD
thf(fact_1039_Nat_OlessE,axiom,
! [I4: nat,K: nat] :
( ( ord_less_nat @ I4 @ K )
=> ( ( K
!= ( suc @ I4 ) )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I4 @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ) ).
% Nat.lessE
thf(fact_1040_irrefl__onD,axiom,
! [A5: set_Pr4386577575007340137rm_a_b,R2: set_Pr2972776593051762503rm_a_b,A: produc357393685978478089rm_a_b] :
( ( irrefl6620156600162922239rm_a_b @ A5 @ R2 )
=> ( ( member5869715511025134514rm_a_b @ A @ A5 )
=> ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ A ) @ R2 ) ) ) ).
% irrefl_onD
thf(fact_1041_irrefl__onD,axiom,
! [A5: set_term_a_b,R2: set_Pr4386577575007340137rm_a_b,A: term_a_b] :
( ( irrefl_on_term_a_b @ A5 @ R2 )
=> ( ( member_term_a_b @ A @ A5 )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ A ) @ R2 ) ) ) ).
% irrefl_onD
thf(fact_1042_irrefl__onI,axiom,
! [A5: set_Pr4386577575007340137rm_a_b,R2: set_Pr2972776593051762503rm_a_b] :
( ! [A4: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ A4 @ A5 )
=> ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A4 @ A4 ) @ R2 ) )
=> ( irrefl6620156600162922239rm_a_b @ A5 @ R2 ) ) ).
% irrefl_onI
thf(fact_1043_irrefl__onI,axiom,
! [A5: set_term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ! [A4: term_a_b] :
( ( member_term_a_b @ A4 @ A5 )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A4 @ A4 ) @ R2 ) )
=> ( irrefl_on_term_a_b @ A5 @ R2 ) ) ).
% irrefl_onI
thf(fact_1044_irrefl__on__def,axiom,
( irrefl_on_term_a_b
= ( ^ [A6: set_term_a_b,R6: set_Pr4386577575007340137rm_a_b] :
! [X4: term_a_b] :
( ( member_term_a_b @ X4 @ A6 )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X4 @ X4 ) @ R6 ) ) ) ) ).
% irrefl_on_def
thf(fact_1045_irrefl__on__subset,axiom,
! [A5: set_Pr4934435412358123699_a_nat,R2: set_Pr1811044260758604347_a_nat,B5: set_Pr4934435412358123699_a_nat] :
( ( irrefl177138428007194887_a_nat @ A5 @ R2 )
=> ( ( ord_le8666007276011122963_a_nat @ B5 @ A5 )
=> ( irrefl177138428007194887_a_nat @ B5 @ R2 ) ) ) ).
% irrefl_on_subset
thf(fact_1046_lift__Suc__mono__less,axiom,
! [F3: nat > nat,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_nat @ ( F3 @ N3 ) @ ( F3 @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ N @ N4 )
=> ( ord_less_nat @ ( F3 @ N ) @ ( F3 @ N4 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_1047_lift__Suc__mono__less__iff,axiom,
! [F3: nat > nat,N: nat,M2: nat] :
( ! [N3: nat] : ( ord_less_nat @ ( F3 @ N3 ) @ ( F3 @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ ( F3 @ N ) @ ( F3 @ M2 ) )
= ( ord_less_nat @ N @ M2 ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_1048_lift__Suc__mono__le,axiom,
! [F3: nat > set_Pr4934435412358123699_a_nat,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_le8666007276011122963_a_nat @ ( F3 @ N3 ) @ ( F3 @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_le8666007276011122963_a_nat @ ( F3 @ N ) @ ( F3 @ N4 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_1049_lift__Suc__mono__le,axiom,
! [F3: nat > nat,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ ( F3 @ N3 ) @ ( F3 @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_nat @ ( F3 @ N ) @ ( F3 @ N4 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_1050_lift__Suc__antimono__le,axiom,
! [F3: nat > set_Pr4934435412358123699_a_nat,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_le8666007276011122963_a_nat @ ( F3 @ ( suc @ N3 ) ) @ ( F3 @ N3 ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_le8666007276011122963_a_nat @ ( F3 @ N4 ) @ ( F3 @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_1051_lift__Suc__antimono__le,axiom,
! [F3: nat > nat,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ ( F3 @ ( suc @ N3 ) ) @ ( F3 @ N3 ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_nat @ ( F3 @ N4 ) @ ( F3 @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_1052_length__Cons,axiom,
! [X: term_a_b,Xs: list_term_a_b] :
( ( size_s8906293707977694520rm_a_b @ ( cons_term_a_b @ X @ Xs ) )
= ( suc @ ( size_s8906293707977694520rm_a_b @ Xs ) ) ) ).
% length_Cons
thf(fact_1053_length__Suc__conv,axiom,
! [Xs: list_term_a_b,N: nat] :
( ( ( size_s8906293707977694520rm_a_b @ Xs )
= ( suc @ N ) )
= ( ? [Y5: term_a_b,Ys2: list_term_a_b] :
( ( Xs
= ( cons_term_a_b @ Y5 @ Ys2 ) )
& ( ( size_s8906293707977694520rm_a_b @ Ys2 )
= N ) ) ) ) ).
% length_Suc_conv
thf(fact_1054_Suc__length__conv,axiom,
! [N: nat,Xs: list_term_a_b] :
( ( ( suc @ N )
= ( size_s8906293707977694520rm_a_b @ Xs ) )
= ( ? [Y5: term_a_b,Ys2: list_term_a_b] :
( ( Xs
= ( cons_term_a_b @ Y5 @ Ys2 ) )
& ( ( size_s8906293707977694520rm_a_b @ Ys2 )
= N ) ) ) ) ).
% Suc_length_conv
thf(fact_1055_Abstract__Rewriting_Ochain__mono,axiom,
! [R4: set_Pr4386577575007340137rm_a_b,R3: set_Pr4386577575007340137rm_a_b,Seq: nat > term_a_b] :
( ( ord_le118470702582115849rm_a_b @ R4 @ R3 )
=> ( ! [I2: nat] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( Seq @ I2 ) @ ( Seq @ ( suc @ I2 ) ) ) @ R4 )
=> ! [I: nat] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( Seq @ I ) @ ( Seq @ ( suc @ I ) ) ) @ R3 ) ) ) ).
% Abstract_Rewriting.chain_mono
thf(fact_1056_le__imp__less__Suc,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_nat @ M2 @ ( suc @ N ) ) ) ).
% le_imp_less_Suc
thf(fact_1057_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N2: nat] : ( ord_less_eq_nat @ ( suc @ N2 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_1058_less__Suc__eq__le,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% less_Suc_eq_le
thf(fact_1059_le__less__Suc__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( ord_less_nat @ N @ ( suc @ M2 ) )
= ( N = M2 ) ) ) ).
% le_less_Suc_eq
thf(fact_1060_Suc__le__lessD,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N )
=> ( ord_less_nat @ M2 @ N ) ) ).
% Suc_le_lessD
thf(fact_1061_inc__induct,axiom,
! [I4: nat,J: nat,P2: nat > $o] :
( ( ord_less_eq_nat @ I4 @ J )
=> ( ( P2 @ J )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I4 @ N3 )
=> ( ( ord_less_nat @ N3 @ J )
=> ( ( P2 @ ( suc @ N3 ) )
=> ( P2 @ N3 ) ) ) )
=> ( P2 @ I4 ) ) ) ) ).
% inc_induct
thf(fact_1062_dec__induct,axiom,
! [I4: nat,J: nat,P2: nat > $o] :
( ( ord_less_eq_nat @ I4 @ J )
=> ( ( P2 @ I4 )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I4 @ N3 )
=> ( ( ord_less_nat @ N3 @ J )
=> ( ( P2 @ N3 )
=> ( P2 @ ( suc @ N3 ) ) ) ) )
=> ( P2 @ J ) ) ) ) ).
% dec_induct
thf(fact_1063_Suc__le__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N )
= ( ord_less_nat @ M2 @ N ) ) ).
% Suc_le_eq
thf(fact_1064_Suc__leI,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_eq_nat @ ( suc @ M2 ) @ N ) ) ).
% Suc_leI
thf(fact_1065_relpow__Suc__I2,axiom,
! [X: term_a_b,Y: term_a_b,R3: set_Pr4386577575007340137rm_a_b,Z2: term_a_b,N: nat] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ R3 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ Z2 ) @ ( compow4057154403645558940rm_a_b @ N @ R3 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z2 ) @ ( compow4057154403645558940rm_a_b @ ( suc @ N ) @ R3 ) ) ) ) ).
% relpow_Suc_I2
thf(fact_1066_relpow__Suc__E2,axiom,
! [X: term_a_b,Z2: term_a_b,N: nat,R3: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z2 ) @ ( compow4057154403645558940rm_a_b @ ( suc @ N ) @ R3 ) )
=> ~ ! [Y3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y3 ) @ R3 )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ Z2 ) @ ( compow4057154403645558940rm_a_b @ N @ R3 ) ) ) ) ).
% relpow_Suc_E2
thf(fact_1067_relpow__Suc__D2,axiom,
! [X: term_a_b,Z2: term_a_b,N: nat,R3: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z2 ) @ ( compow4057154403645558940rm_a_b @ ( suc @ N ) @ R3 ) )
=> ? [Y3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y3 ) @ R3 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ Z2 ) @ ( compow4057154403645558940rm_a_b @ N @ R3 ) ) ) ) ).
% relpow_Suc_D2
thf(fact_1068_relpow__Suc__I,axiom,
! [X: term_a_b,Y: term_a_b,N: nat,R3: set_Pr4386577575007340137rm_a_b,Z2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( compow4057154403645558940rm_a_b @ N @ R3 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ Z2 ) @ R3 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z2 ) @ ( compow4057154403645558940rm_a_b @ ( suc @ N ) @ R3 ) ) ) ) ).
% relpow_Suc_I
thf(fact_1069_relpow__Suc__E,axiom,
! [X: term_a_b,Z2: term_a_b,N: nat,R3: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z2 ) @ ( compow4057154403645558940rm_a_b @ ( suc @ N ) @ R3 ) )
=> ~ ! [Y3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y3 ) @ ( compow4057154403645558940rm_a_b @ N @ R3 ) )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ Z2 ) @ R3 ) ) ) ).
% relpow_Suc_E
thf(fact_1070_irreflI,axiom,
! [R2: set_Pr4386577575007340137rm_a_b] :
( ! [A4: term_a_b] :
~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A4 @ A4 ) @ R2 )
=> ( irrefl_on_term_a_b @ top_top_set_term_a_b @ R2 ) ) ).
% irreflI
thf(fact_1071_irreflD,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,X: term_a_b] :
( ( irrefl_on_term_a_b @ top_top_set_term_a_b @ R2 )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ X ) @ R2 ) ) ).
% irreflD
thf(fact_1072_Suc__le__length__iff,axiom,
! [N: nat,Xs: list_term_a_b] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_s8906293707977694520rm_a_b @ Xs ) )
= ( ? [X4: term_a_b,Ys2: list_term_a_b] :
( ( Xs
= ( cons_term_a_b @ X4 @ Ys2 ) )
& ( ord_less_eq_nat @ N @ ( size_s8906293707977694520rm_a_b @ Ys2 ) ) ) ) ) ).
% Suc_le_length_iff
thf(fact_1073_relpow__Suc__E2_H,axiom,
! [X: term_a_b,Z2: term_a_b,N: nat,A5: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z2 ) @ ( compow4057154403645558940rm_a_b @ ( suc @ N ) @ A5 ) )
=> ~ ! [Y3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y3 ) @ A5 )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ Z2 ) @ ( transi7742714808557438673rm_a_b @ A5 ) ) ) ) ).
% relpow_Suc_E2'
thf(fact_1074_pow__Suc__subset__trancl,axiom,
! [N: nat,R3: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ ( compow4057154403645558940rm_a_b @ ( suc @ N ) @ R3 ) @ ( transi7922773638565587891rm_a_b @ R3 ) ) ).
% pow_Suc_subset_trancl
thf(fact_1075_length__append__singleton,axiom,
! [Xs: list_term_a_b,X: term_a_b] :
( ( size_s8906293707977694520rm_a_b @ ( append_term_a_b @ Xs @ ( cons_term_a_b @ X @ nil_term_a_b ) ) )
= ( suc @ ( size_s8906293707977694520rm_a_b @ Xs ) ) ) ).
% length_append_singleton
thf(fact_1076_length__Suc__conv__rev,axiom,
! [Xs: list_term_a_b,N: nat] :
( ( ( size_s8906293707977694520rm_a_b @ Xs )
= ( suc @ N ) )
= ( ? [Y5: term_a_b,Ys2: list_term_a_b] :
( ( Xs
= ( append_term_a_b @ Ys2 @ ( cons_term_a_b @ Y5 @ nil_term_a_b ) ) )
& ( ( size_s8906293707977694520rm_a_b @ Ys2 )
= N ) ) ) ) ).
% length_Suc_conv_rev
thf(fact_1077_chain__imp__trancl,axiom,
! [S2: nat > term_a_b,R2: set_Pr4386577575007340137rm_a_b,I4: nat,J: nat] :
( ! [I2: nat] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( S2 @ I2 ) @ ( S2 @ ( suc @ I2 ) ) ) @ R2 )
=> ( ( ord_less_nat @ I4 @ J )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( S2 @ I4 ) @ ( S2 @ J ) ) @ ( transi7922773638565587891rm_a_b @ R2 ) ) ) ) ).
% chain_imp_trancl
thf(fact_1078_chain__imp__rtrancl,axiom,
! [S2: nat > term_a_b,R2: set_Pr4386577575007340137rm_a_b,I4: nat,J: nat] :
( ! [I2: nat] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( S2 @ I2 ) @ ( S2 @ ( suc @ I2 ) ) ) @ R2 )
=> ( ( ord_less_eq_nat @ I4 @ J )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( S2 @ I4 ) @ ( S2 @ J ) ) @ ( transi7742714808557438673rm_a_b @ R2 ) ) ) ) ).
% chain_imp_rtrancl
thf(fact_1079_remove__nth__sound__r,axiom,
! [N: nat,P: nat,Xs: list_term_a_b] :
( ( ord_less_eq_nat @ N @ P )
=> ( ( ord_less_nat @ P @ ( size_s8906293707977694520rm_a_b @ Xs ) )
=> ( ( nth_term_a_b @ ( missin7590947408886171483rm_a_b @ N @ Xs ) @ P )
= ( nth_term_a_b @ Xs @ ( suc @ P ) ) ) ) ) ).
% remove_nth_sound_r
thf(fact_1080_rtrancl__list__conv,axiom,
! [S: term_a_b,T: term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( transi7742714808557438673rm_a_b @ R3 ) )
= ( ? [Ts3: list_term_a_b] :
( ( ( last_term_a_b @ ( cons_term_a_b @ S @ Ts3 ) )
= T )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s8906293707977694520rm_a_b @ Ts3 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( nth_term_a_b @ ( cons_term_a_b @ S @ Ts3 ) @ I3 ) @ ( nth_term_a_b @ ( cons_term_a_b @ S @ Ts3 ) @ ( suc @ I3 ) ) ) @ R3 ) ) ) ) ) ).
% rtrancl_list_conv
thf(fact_1081_remove__nth__id,axiom,
! [Xs: list_term_a_b,N: nat] :
( ( ord_less_eq_nat @ ( size_s8906293707977694520rm_a_b @ Xs ) @ N )
=> ( ( missin7590947408886171483rm_a_b @ N @ Xs )
= Xs ) ) ).
% remove_nth_id
thf(fact_1082_remove__nth__sound__l,axiom,
! [P: nat,N: nat,Xs: list_term_a_b] :
( ( ord_less_nat @ P @ N )
=> ( ( nth_term_a_b @ ( missin7590947408886171483rm_a_b @ N @ Xs ) @ P )
= ( nth_term_a_b @ Xs @ P ) ) ) ).
% remove_nth_sound_l
thf(fact_1083_remove__nth__len,axiom,
! [I4: nat,Xs: list_term_a_b] :
( ( ord_less_nat @ I4 @ ( size_s8906293707977694520rm_a_b @ Xs ) )
=> ( ( size_s8906293707977694520rm_a_b @ Xs )
= ( suc @ ( size_s8906293707977694520rm_a_b @ ( missin7590947408886171483rm_a_b @ I4 @ Xs ) ) ) ) ) ).
% remove_nth_len
thf(fact_1084_remove__nth__P__compat,axiom,
! [As: list_term_a_b,Bs: list_term_a_b,P2: term_a_b > term_a_b > $o,P: nat] :
( ( ( size_s8906293707977694520rm_a_b @ As )
= ( size_s8906293707977694520rm_a_b @ Bs ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s8906293707977694520rm_a_b @ As ) )
=> ( P2 @ ( nth_term_a_b @ As @ I2 ) @ ( nth_term_a_b @ Bs @ I2 ) ) )
=> ! [I: nat] :
( ( ord_less_nat @ I @ ( size_s8906293707977694520rm_a_b @ ( missin7590947408886171483rm_a_b @ P @ As ) ) )
=> ( P2 @ ( nth_term_a_b @ ( missin7590947408886171483rm_a_b @ P @ As ) @ I ) @ ( nth_term_a_b @ ( missin7590947408886171483rm_a_b @ P @ Bs ) @ I ) ) ) ) ) ).
% remove_nth_P_compat
thf(fact_1085_nth__remove__nth__conv,axiom,
! [I4: nat,N: nat,Xs: list_term_a_b] :
( ( ord_less_nat @ I4 @ ( size_s8906293707977694520rm_a_b @ ( missin7590947408886171483rm_a_b @ N @ Xs ) ) )
=> ( ( nth_term_a_b @ ( missin7590947408886171483rm_a_b @ N @ Xs ) @ I4 )
= ( nth_term_a_b @ Xs @ ( if_nat @ ( ord_less_nat @ I4 @ N ) @ I4 @ ( suc @ I4 ) ) ) ) ) ).
% nth_remove_nth_conv
thf(fact_1086_adjust__idx__rev__nth,axiom,
! [I4: nat,Xs: list_term_a_b,J: nat] :
( ( ord_less_nat @ I4 @ ( size_s8906293707977694520rm_a_b @ Xs ) )
=> ( ( J != I4 )
=> ( ( nth_term_a_b @ ( missin7590947408886171483rm_a_b @ I4 @ Xs ) @ ( missin3815256168798769645dx_rev @ I4 @ J ) )
= ( nth_term_a_b @ Xs @ J ) ) ) ) ).
% adjust_idx_rev_nth
thf(fact_1087_adjust__idx__nth,axiom,
! [I4: nat,Xs: list_term_a_b,J: nat] :
( ( ord_less_nat @ I4 @ ( size_s8906293707977694520rm_a_b @ Xs ) )
=> ( ( nth_term_a_b @ ( missin7590947408886171483rm_a_b @ I4 @ Xs ) @ J )
= ( nth_term_a_b @ Xs @ ( missing_adjust_idx @ I4 @ J ) ) ) ) ).
% adjust_idx_nth
thf(fact_1088_adjust__idx__i,axiom,
! [I4: nat,J: nat] :
( ( missing_adjust_idx @ I4 @ J )
!= I4 ) ).
% adjust_idx_i
thf(fact_1089_adjust__idx__rev2,axiom,
! [J: nat,I4: nat] :
( ( J != I4 )
=> ( ( missing_adjust_idx @ I4 @ ( missin3815256168798769645dx_rev @ I4 @ J ) )
= J ) ) ).
% adjust_idx_rev2
thf(fact_1090_adjust__idx__rev1,axiom,
! [I4: nat,J: nat] :
( ( missin3815256168798769645dx_rev @ I4 @ ( missing_adjust_idx @ I4 @ J ) )
= J ) ).
% adjust_idx_rev1
thf(fact_1091_adjust__idx__def,axiom,
( missing_adjust_idx
= ( ^ [I3: nat,J3: nat] : ( if_nat @ ( ord_less_nat @ J3 @ I3 ) @ J3 @ ( suc @ J3 ) ) ) ) ).
% adjust_idx_def
thf(fact_1092_adjust__idx__length,axiom,
! [I4: nat,Xs: list_term_a_b,J: nat] :
( ( ord_less_nat @ I4 @ ( size_s8906293707977694520rm_a_b @ Xs ) )
=> ( ( ord_less_nat @ J @ ( size_s8906293707977694520rm_a_b @ ( missin7590947408886171483rm_a_b @ I4 @ Xs ) ) )
=> ( ord_less_nat @ ( missing_adjust_idx @ I4 @ J ) @ ( size_s8906293707977694520rm_a_b @ Xs ) ) ) ) ).
% adjust_idx_length
thf(fact_1093_adjust__idx__rev__length,axiom,
! [I4: nat,Xs: list_term_a_b,J: nat] :
( ( ord_less_nat @ I4 @ ( size_s8906293707977694520rm_a_b @ Xs ) )
=> ( ( ord_less_nat @ J @ ( size_s8906293707977694520rm_a_b @ Xs ) )
=> ( ( J != I4 )
=> ( ord_less_nat @ ( missin3815256168798769645dx_rev @ I4 @ J ) @ ( size_s8906293707977694520rm_a_b @ ( missin7590947408886171483rm_a_b @ I4 @ Xs ) ) ) ) ) ) ).
% adjust_idx_rev_length
thf(fact_1094_nth__append__drop__is__nth__conv,axiom,
! [J: nat,I4: nat,Xs: list_term_a_b,Y: term_a_b] :
( ( ord_less_nat @ J @ I4 )
=> ( ( ord_less_eq_nat @ J @ ( size_s8906293707977694520rm_a_b @ Xs ) )
=> ( ( ord_less_eq_nat @ I4 @ ( size_s8906293707977694520rm_a_b @ Xs ) )
=> ( ( nth_term_a_b @ ( append_term_a_b @ ( take_term_a_b @ J @ Xs ) @ ( cons_term_a_b @ Y @ ( drop_term_a_b @ ( suc @ J ) @ Xs ) ) ) @ I4 )
= ( nth_term_a_b @ Xs @ I4 ) ) ) ) ) ).
% nth_append_drop_is_nth_conv
thf(fact_1095_nth__append__take__drop__is__nth__conv,axiom,
! [I4: nat,Xs: list_term_a_b,J: nat,Y: term_a_b] :
( ( ord_less_eq_nat @ I4 @ ( size_s8906293707977694520rm_a_b @ Xs ) )
=> ( ( ord_less_eq_nat @ J @ ( size_s8906293707977694520rm_a_b @ Xs ) )
=> ( ( I4 != J )
=> ( ( nth_term_a_b @ ( append_term_a_b @ ( take_term_a_b @ J @ Xs ) @ ( cons_term_a_b @ Y @ ( drop_term_a_b @ ( suc @ J ) @ Xs ) ) ) @ I4 )
= ( nth_term_a_b @ Xs @ I4 ) ) ) ) ) ).
% nth_append_take_drop_is_nth_conv
thf(fact_1096_drop__all,axiom,
! [Xs: list_term_a_b,N: nat] :
( ( ord_less_eq_nat @ ( size_s8906293707977694520rm_a_b @ Xs ) @ N )
=> ( ( drop_term_a_b @ N @ Xs )
= nil_term_a_b ) ) ).
% drop_all
thf(fact_1097_drop__eq__Nil,axiom,
! [N: nat,Xs: list_term_a_b] :
( ( ( drop_term_a_b @ N @ Xs )
= nil_term_a_b )
= ( ord_less_eq_nat @ ( size_s8906293707977694520rm_a_b @ Xs ) @ N ) ) ).
% drop_eq_Nil
thf(fact_1098_drop__eq__Nil2,axiom,
! [N: nat,Xs: list_term_a_b] :
( ( nil_term_a_b
= ( drop_term_a_b @ N @ Xs ) )
= ( ord_less_eq_nat @ ( size_s8906293707977694520rm_a_b @ Xs ) @ N ) ) ).
% drop_eq_Nil2
thf(fact_1099_last__drop,axiom,
! [N: nat,Xs: list_term_a_b] :
( ( ord_less_nat @ N @ ( size_s8906293707977694520rm_a_b @ Xs ) )
=> ( ( last_term_a_b @ ( drop_term_a_b @ N @ Xs ) )
= ( last_term_a_b @ Xs ) ) ) ).
% last_drop
thf(fact_1100_nth__via__drop,axiom,
! [N: nat,Xs: list_term_a_b,Y: term_a_b,Ys: list_term_a_b] :
( ( ( drop_term_a_b @ N @ Xs )
= ( cons_term_a_b @ Y @ Ys ) )
=> ( ( nth_term_a_b @ Xs @ N )
= Y ) ) ).
% nth_via_drop
thf(fact_1101_append__eq__conv__conj,axiom,
! [Xs: list_term_a_b,Ys: list_term_a_b,Zs: list_term_a_b] :
( ( ( append_term_a_b @ Xs @ Ys )
= Zs )
= ( ( Xs
= ( take_term_a_b @ ( size_s8906293707977694520rm_a_b @ Xs ) @ Zs ) )
& ( Ys
= ( drop_term_a_b @ ( size_s8906293707977694520rm_a_b @ Xs ) @ Zs ) ) ) ) ).
% append_eq_conv_conj
thf(fact_1102_append__eq__append__conv__if,axiom,
! [Xs_1: list_term_a_b,Xs_2: list_term_a_b,Ys_1: list_term_a_b,Ys_2: list_term_a_b] :
( ( ( append_term_a_b @ Xs_1 @ Xs_2 )
= ( append_term_a_b @ Ys_1 @ Ys_2 ) )
= ( ( ( ord_less_eq_nat @ ( size_s8906293707977694520rm_a_b @ Xs_1 ) @ ( size_s8906293707977694520rm_a_b @ Ys_1 ) )
=> ( ( Xs_1
= ( take_term_a_b @ ( size_s8906293707977694520rm_a_b @ Xs_1 ) @ Ys_1 ) )
& ( Xs_2
= ( append_term_a_b @ ( drop_term_a_b @ ( size_s8906293707977694520rm_a_b @ Xs_1 ) @ Ys_1 ) @ Ys_2 ) ) ) )
& ( ~ ( ord_less_eq_nat @ ( size_s8906293707977694520rm_a_b @ Xs_1 ) @ ( size_s8906293707977694520rm_a_b @ Ys_1 ) )
=> ( ( ( take_term_a_b @ ( size_s8906293707977694520rm_a_b @ Ys_1 ) @ Xs_1 )
= Ys_1 )
& ( ( append_term_a_b @ ( drop_term_a_b @ ( size_s8906293707977694520rm_a_b @ Ys_1 ) @ Xs_1 ) @ Xs_2 )
= Ys_2 ) ) ) ) ) ).
% append_eq_append_conv_if
thf(fact_1103_Cons__nth__drop__Suc,axiom,
! [I4: nat,Xs: list_term_a_b] :
( ( ord_less_nat @ I4 @ ( size_s8906293707977694520rm_a_b @ Xs ) )
=> ( ( cons_term_a_b @ ( nth_term_a_b @ Xs @ I4 ) @ ( drop_term_a_b @ ( suc @ I4 ) @ Xs ) )
= ( drop_term_a_b @ I4 @ Xs ) ) ) ).
% Cons_nth_drop_Suc
thf(fact_1104_take__drop__imp__nth,axiom,
! [I4: nat,Ss2: list_term_a_b,X: term_a_b] :
( ( ( append_term_a_b @ ( take_term_a_b @ I4 @ Ss2 ) @ ( cons_term_a_b @ X @ ( drop_term_a_b @ ( suc @ I4 ) @ Ss2 ) ) )
= Ss2 )
=> ( X
= ( nth_term_a_b @ Ss2 @ I4 ) ) ) ).
% take_drop_imp_nth
thf(fact_1105_id__take__nth__drop,axiom,
! [I4: nat,Xs: list_term_a_b] :
( ( ord_less_nat @ I4 @ ( size_s8906293707977694520rm_a_b @ Xs ) )
=> ( Xs
= ( append_term_a_b @ ( take_term_a_b @ I4 @ Xs ) @ ( cons_term_a_b @ ( nth_term_a_b @ Xs @ I4 ) @ ( drop_term_a_b @ ( suc @ I4 ) @ Xs ) ) ) ) ) ).
% id_take_nth_drop
thf(fact_1106_take__hd__drop,axiom,
! [N: nat,Xs: list_term_a_b] :
( ( ord_less_nat @ N @ ( size_s8906293707977694520rm_a_b @ Xs ) )
=> ( ( append_term_a_b @ ( take_term_a_b @ N @ Xs ) @ ( cons_term_a_b @ ( hd_term_a_b @ ( drop_term_a_b @ N @ Xs ) ) @ nil_term_a_b ) )
= ( take_term_a_b @ ( suc @ N ) @ Xs ) ) ) ).
% take_hd_drop
thf(fact_1107_add__elem__list__listsE,axiom,
! [Ys: list_term_a_b,X: term_a_b,Xs: list_term_a_b] :
( ( member_list_term_a_b @ Ys @ ( set_list_term_a_b2 @ ( basic_1593220722155286443rm_a_b @ X @ Xs ) ) )
=> ? [N3: nat] :
( ( ord_less_eq_nat @ N3 @ ( size_s8906293707977694520rm_a_b @ Xs ) )
& ( Ys
= ( append_term_a_b @ ( take_term_a_b @ N3 @ Xs ) @ ( cons_term_a_b @ X @ ( drop_term_a_b @ N3 @ Xs ) ) ) ) ) ) ).
% add_elem_list_listsE
thf(fact_1108_in__set__dropD,axiom,
! [X: produc357393685978478089rm_a_b,N: nat,Xs: list_P8875379029341186191rm_a_b] :
( ( member5869715511025134514rm_a_b @ X @ ( set_Pr2370880052973118494rm_a_b @ ( drop_P2146130716461001295rm_a_b @ N @ Xs ) ) )
=> ( member5869715511025134514rm_a_b @ X @ ( set_Pr2370880052973118494rm_a_b @ Xs ) ) ) ).
% in_set_dropD
thf(fact_1109_set__drop__subset,axiom,
! [N: nat,Xs: list_P3592885314253461005_a_nat] : ( ord_le8666007276011122963_a_nat @ ( set_Pr924983374503034536_a_nat @ ( drop_P2883665741211355575_a_nat @ N @ Xs ) ) @ ( set_Pr924983374503034536_a_nat @ Xs ) ) ).
% set_drop_subset
thf(fact_1110_last__in__set,axiom,
! [As: list_P8875379029341186191rm_a_b] :
( ( As != nil_Pr6942174756412032271rm_a_b )
=> ( member5869715511025134514rm_a_b @ ( last_P6746431593072707880rm_a_b @ As ) @ ( set_Pr2370880052973118494rm_a_b @ As ) ) ) ).
% last_in_set
thf(fact_1111_list_Omap__ident__strong,axiom,
! [T: list_P8875379029341186191rm_a_b,F3: produc357393685978478089rm_a_b > produc357393685978478089rm_a_b] :
( ! [Z: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ Z @ ( set_Pr2370880052973118494rm_a_b @ T ) )
=> ( ( F3 @ Z )
= Z ) )
=> ( ( map_Pr3281474655358167172rm_a_b @ F3 @ T )
= T ) ) ).
% list.map_ident_strong
thf(fact_1112_map__idI,axiom,
! [Xs: list_P8875379029341186191rm_a_b,F3: produc357393685978478089rm_a_b > produc357393685978478089rm_a_b] :
( ! [X3: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X3 @ ( set_Pr2370880052973118494rm_a_b @ Xs ) )
=> ( ( F3 @ X3 )
= X3 ) )
=> ( ( map_Pr3281474655358167172rm_a_b @ F3 @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_1113_list_Oset__intros_I2_J,axiom,
! [Y: produc357393685978478089rm_a_b,X22: list_P8875379029341186191rm_a_b,X21: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ Y @ ( set_Pr2370880052973118494rm_a_b @ X22 ) )
=> ( member5869715511025134514rm_a_b @ Y @ ( set_Pr2370880052973118494rm_a_b @ ( cons_P3536305108106557631rm_a_b @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_1114_list_Oset__intros_I1_J,axiom,
! [X21: produc357393685978478089rm_a_b,X22: list_P8875379029341186191rm_a_b] : ( member5869715511025134514rm_a_b @ X21 @ ( set_Pr2370880052973118494rm_a_b @ ( cons_P3536305108106557631rm_a_b @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_1115_list_Oset__cases,axiom,
! [E: produc357393685978478089rm_a_b,A: list_P8875379029341186191rm_a_b] :
( ( member5869715511025134514rm_a_b @ E @ ( set_Pr2370880052973118494rm_a_b @ A ) )
=> ( ! [Z22: list_P8875379029341186191rm_a_b] :
( A
!= ( cons_P3536305108106557631rm_a_b @ E @ Z22 ) )
=> ~ ! [Z1: produc357393685978478089rm_a_b,Z22: list_P8875379029341186191rm_a_b] :
( ( A
= ( cons_P3536305108106557631rm_a_b @ Z1 @ Z22 ) )
=> ~ ( member5869715511025134514rm_a_b @ E @ ( set_Pr2370880052973118494rm_a_b @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_1116_set__ConsD,axiom,
! [Y: produc357393685978478089rm_a_b,X: produc357393685978478089rm_a_b,Xs: list_P8875379029341186191rm_a_b] :
( ( member5869715511025134514rm_a_b @ Y @ ( set_Pr2370880052973118494rm_a_b @ ( cons_P3536305108106557631rm_a_b @ X @ Xs ) ) )
=> ( ( Y = X )
| ( member5869715511025134514rm_a_b @ Y @ ( set_Pr2370880052973118494rm_a_b @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_1117_subset__code_I1_J,axiom,
! [Xs: list_P8875379029341186191rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ ( set_Pr2370880052973118494rm_a_b @ Xs ) @ B5 )
= ( ! [X4: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X4 @ ( set_Pr2370880052973118494rm_a_b @ Xs ) )
=> ( member5869715511025134514rm_a_b @ X4 @ B5 ) ) ) ) ).
% subset_code(1)
thf(fact_1118_subset__code_I1_J,axiom,
! [Xs: list_P3592885314253461005_a_nat,B5: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ ( set_Pr924983374503034536_a_nat @ Xs ) @ B5 )
= ( ! [X4: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ X4 @ ( set_Pr924983374503034536_a_nat @ Xs ) )
=> ( member5724188588386418708_a_nat @ X4 @ B5 ) ) ) ) ).
% subset_code(1)
thf(fact_1119_in__set__takeD,axiom,
! [X: produc357393685978478089rm_a_b,N: nat,Xs: list_P8875379029341186191rm_a_b] :
( ( member5869715511025134514rm_a_b @ X @ ( set_Pr2370880052973118494rm_a_b @ ( take_P7698341771522494711rm_a_b @ N @ Xs ) ) )
=> ( member5869715511025134514rm_a_b @ X @ ( set_Pr2370880052973118494rm_a_b @ Xs ) ) ) ).
% in_set_takeD
thf(fact_1120_hd__in__set,axiom,
! [Xs: list_P8875379029341186191rm_a_b] :
( ( Xs != nil_Pr6942174756412032271rm_a_b )
=> ( member5869715511025134514rm_a_b @ ( hd_Pro6001814652670619252rm_a_b @ Xs ) @ ( set_Pr2370880052973118494rm_a_b @ Xs ) ) ) ).
% hd_in_set
thf(fact_1121_list_Oset__sel_I1_J,axiom,
! [A: list_P8875379029341186191rm_a_b] :
( ( A != nil_Pr6942174756412032271rm_a_b )
=> ( member5869715511025134514rm_a_b @ ( hd_Pro6001814652670619252rm_a_b @ A ) @ ( set_Pr2370880052973118494rm_a_b @ A ) ) ) ).
% list.set_sel(1)
thf(fact_1122_set__take__subset,axiom,
! [N: nat,Xs: list_P3592885314253461005_a_nat] : ( ord_le8666007276011122963_a_nat @ ( set_Pr924983374503034536_a_nat @ ( take_P7249722199902868751_a_nat @ N @ Xs ) ) @ ( set_Pr924983374503034536_a_nat @ Xs ) ) ).
% set_take_subset
thf(fact_1123_in__set__conv__decomp__first,axiom,
! [X: produc357393685978478089rm_a_b,Xs: list_P8875379029341186191rm_a_b] :
( ( member5869715511025134514rm_a_b @ X @ ( set_Pr2370880052973118494rm_a_b @ Xs ) )
= ( ? [Ys2: list_P8875379029341186191rm_a_b,Zs3: list_P8875379029341186191rm_a_b] :
( ( Xs
= ( append1152621114427393060rm_a_b @ Ys2 @ ( cons_P3536305108106557631rm_a_b @ X @ Zs3 ) ) )
& ~ ( member5869715511025134514rm_a_b @ X @ ( set_Pr2370880052973118494rm_a_b @ Ys2 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_1124_in__set__conv__decomp__last,axiom,
! [X: produc357393685978478089rm_a_b,Xs: list_P8875379029341186191rm_a_b] :
( ( member5869715511025134514rm_a_b @ X @ ( set_Pr2370880052973118494rm_a_b @ Xs ) )
= ( ? [Ys2: list_P8875379029341186191rm_a_b,Zs3: list_P8875379029341186191rm_a_b] :
( ( Xs
= ( append1152621114427393060rm_a_b @ Ys2 @ ( cons_P3536305108106557631rm_a_b @ X @ Zs3 ) ) )
& ~ ( member5869715511025134514rm_a_b @ X @ ( set_Pr2370880052973118494rm_a_b @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_1125_in__set__conv__decomp,axiom,
! [X: produc357393685978478089rm_a_b,Xs: list_P8875379029341186191rm_a_b] :
( ( member5869715511025134514rm_a_b @ X @ ( set_Pr2370880052973118494rm_a_b @ Xs ) )
= ( ? [Ys2: list_P8875379029341186191rm_a_b,Zs3: list_P8875379029341186191rm_a_b] :
( Xs
= ( append1152621114427393060rm_a_b @ Ys2 @ ( cons_P3536305108106557631rm_a_b @ X @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_1126_append__Cons__eq__iff,axiom,
! [X: produc357393685978478089rm_a_b,Xs: list_P8875379029341186191rm_a_b,Ys: list_P8875379029341186191rm_a_b,Xs5: list_P8875379029341186191rm_a_b,Ys6: list_P8875379029341186191rm_a_b] :
( ~ ( member5869715511025134514rm_a_b @ X @ ( set_Pr2370880052973118494rm_a_b @ Xs ) )
=> ( ~ ( member5869715511025134514rm_a_b @ X @ ( set_Pr2370880052973118494rm_a_b @ Ys ) )
=> ( ( ( append1152621114427393060rm_a_b @ Xs @ ( cons_P3536305108106557631rm_a_b @ X @ Ys ) )
= ( append1152621114427393060rm_a_b @ Xs5 @ ( cons_P3536305108106557631rm_a_b @ X @ Ys6 ) ) )
= ( ( Xs = Xs5 )
& ( Ys = Ys6 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_1127_split__list__first,axiom,
! [X: produc357393685978478089rm_a_b,Xs: list_P8875379029341186191rm_a_b] :
( ( member5869715511025134514rm_a_b @ X @ ( set_Pr2370880052973118494rm_a_b @ Xs ) )
=> ? [Ys4: list_P8875379029341186191rm_a_b,Zs2: list_P8875379029341186191rm_a_b] :
( ( Xs
= ( append1152621114427393060rm_a_b @ Ys4 @ ( cons_P3536305108106557631rm_a_b @ X @ Zs2 ) ) )
& ~ ( member5869715511025134514rm_a_b @ X @ ( set_Pr2370880052973118494rm_a_b @ Ys4 ) ) ) ) ).
% split_list_first
thf(fact_1128_split__list__last,axiom,
! [X: produc357393685978478089rm_a_b,Xs: list_P8875379029341186191rm_a_b] :
( ( member5869715511025134514rm_a_b @ X @ ( set_Pr2370880052973118494rm_a_b @ Xs ) )
=> ? [Ys4: list_P8875379029341186191rm_a_b,Zs2: list_P8875379029341186191rm_a_b] :
( ( Xs
= ( append1152621114427393060rm_a_b @ Ys4 @ ( cons_P3536305108106557631rm_a_b @ X @ Zs2 ) ) )
& ~ ( member5869715511025134514rm_a_b @ X @ ( set_Pr2370880052973118494rm_a_b @ Zs2 ) ) ) ) ).
% split_list_last
thf(fact_1129_split__list,axiom,
! [X: produc357393685978478089rm_a_b,Xs: list_P8875379029341186191rm_a_b] :
( ( member5869715511025134514rm_a_b @ X @ ( set_Pr2370880052973118494rm_a_b @ Xs ) )
=> ? [Ys4: list_P8875379029341186191rm_a_b,Zs2: list_P8875379029341186191rm_a_b] :
( Xs
= ( append1152621114427393060rm_a_b @ Ys4 @ ( cons_P3536305108106557631rm_a_b @ X @ Zs2 ) ) ) ) ).
% split_list
thf(fact_1130_set__subset__Cons,axiom,
! [Xs: list_P3592885314253461005_a_nat,X: product_prod_a_nat] : ( ord_le8666007276011122963_a_nat @ ( set_Pr924983374503034536_a_nat @ Xs ) @ ( set_Pr924983374503034536_a_nat @ ( cons_P5205166803686508359_a_nat @ X @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_1131_min__list,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ( ord_less_eq_nat @ ( missing_min_list_nat @ Xs ) @ X ) ) ).
% min_list
thf(fact_1132_in__set__product__lists__length,axiom,
! [Xs: list_term_a_b,Xss: list_list_term_a_b] :
( ( member_list_term_a_b @ Xs @ ( set_list_term_a_b2 @ ( produc17669015410068453rm_a_b @ Xss ) ) )
=> ( ( size_s8906293707977694520rm_a_b @ Xs )
= ( size_s877380706853472072rm_a_b @ Xss ) ) ) ).
% in_set_product_lists_length
thf(fact_1133_in__set__idx,axiom,
! [X: produc357393685978478089rm_a_b,Xs: list_P8875379029341186191rm_a_b] :
( ( member5869715511025134514rm_a_b @ X @ ( set_Pr2370880052973118494rm_a_b @ Xs ) )
=> ? [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s8600120110137556219rm_a_b @ Xs ) )
& ( ( nth_Pr1098598711982373392rm_a_b @ Xs @ I2 )
= X ) ) ) ).
% in_set_idx
thf(fact_1134_in__set__idx,axiom,
! [X: term_a_b,Xs: list_term_a_b] :
( ( member_term_a_b @ X @ ( set_term_a_b2 @ Xs ) )
=> ? [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s8906293707977694520rm_a_b @ Xs ) )
& ( ( nth_term_a_b @ Xs @ I2 )
= X ) ) ) ).
% in_set_idx
thf(fact_1135_nth__mem,axiom,
! [N: nat,Xs: list_P8875379029341186191rm_a_b] :
( ( ord_less_nat @ N @ ( size_s8600120110137556219rm_a_b @ Xs ) )
=> ( member5869715511025134514rm_a_b @ ( nth_Pr1098598711982373392rm_a_b @ Xs @ N ) @ ( set_Pr2370880052973118494rm_a_b @ Xs ) ) ) ).
% nth_mem
thf(fact_1136_nth__mem,axiom,
! [N: nat,Xs: list_term_a_b] :
( ( ord_less_nat @ N @ ( size_s8906293707977694520rm_a_b @ Xs ) )
=> ( member_term_a_b @ ( nth_term_a_b @ Xs @ N ) @ ( set_term_a_b2 @ Xs ) ) ) ).
% nth_mem
thf(fact_1137_list__ball__nth,axiom,
! [N: nat,Xs: list_term_a_b,P2: term_a_b > $o] :
( ( ord_less_nat @ N @ ( size_s8906293707977694520rm_a_b @ Xs ) )
=> ( ! [X3: term_a_b] :
( ( member_term_a_b @ X3 @ ( set_term_a_b2 @ Xs ) )
=> ( P2 @ X3 ) )
=> ( P2 @ ( nth_term_a_b @ Xs @ N ) ) ) ) ).
% list_ball_nth
thf(fact_1138_in__set__conv__nth,axiom,
! [X: produc357393685978478089rm_a_b,Xs: list_P8875379029341186191rm_a_b] :
( ( member5869715511025134514rm_a_b @ X @ ( set_Pr2370880052973118494rm_a_b @ Xs ) )
= ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s8600120110137556219rm_a_b @ Xs ) )
& ( ( nth_Pr1098598711982373392rm_a_b @ Xs @ I3 )
= X ) ) ) ) ).
% in_set_conv_nth
thf(fact_1139_in__set__conv__nth,axiom,
! [X: term_a_b,Xs: list_term_a_b] :
( ( member_term_a_b @ X @ ( set_term_a_b2 @ Xs ) )
= ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s8906293707977694520rm_a_b @ Xs ) )
& ( ( nth_term_a_b @ Xs @ I3 )
= X ) ) ) ) ).
% in_set_conv_nth
thf(fact_1140_all__nth__imp__all__set,axiom,
! [Xs: list_P8875379029341186191rm_a_b,P2: produc357393685978478089rm_a_b > $o,X: produc357393685978478089rm_a_b] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s8600120110137556219rm_a_b @ Xs ) )
=> ( P2 @ ( nth_Pr1098598711982373392rm_a_b @ Xs @ I2 ) ) )
=> ( ( member5869715511025134514rm_a_b @ X @ ( set_Pr2370880052973118494rm_a_b @ Xs ) )
=> ( P2 @ X ) ) ) ).
% all_nth_imp_all_set
thf(fact_1141_all__nth__imp__all__set,axiom,
! [Xs: list_term_a_b,P2: term_a_b > $o,X: term_a_b] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s8906293707977694520rm_a_b @ Xs ) )
=> ( P2 @ ( nth_term_a_b @ Xs @ I2 ) ) )
=> ( ( member_term_a_b @ X @ ( set_term_a_b2 @ Xs ) )
=> ( P2 @ X ) ) ) ).
% all_nth_imp_all_set
thf(fact_1142_all__set__conv__all__nth,axiom,
! [Xs: list_term_a_b,P2: term_a_b > $o] :
( ( ! [X4: term_a_b] :
( ( member_term_a_b @ X4 @ ( set_term_a_b2 @ Xs ) )
=> ( P2 @ X4 ) ) )
= ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s8906293707977694520rm_a_b @ Xs ) )
=> ( P2 @ ( nth_term_a_b @ Xs @ I3 ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_1143_ex__set__conv__ex__nth,axiom,
! [Xs: list_term_a_b,P2: term_a_b > $o] :
( ( ? [X4: term_a_b] :
( ( member_term_a_b @ X4 @ ( set_term_a_b2 @ Xs ) )
& ( P2 @ X4 ) ) )
= ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s8906293707977694520rm_a_b @ Xs ) )
& ( P2 @ ( nth_term_a_b @ Xs @ I3 ) ) ) ) ) ).
% ex_set_conv_ex_nth
thf(fact_1144_set__take__subset__set__take,axiom,
! [M2: nat,N: nat,Xs: list_P3592885314253461005_a_nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_le8666007276011122963_a_nat @ ( set_Pr924983374503034536_a_nat @ ( take_P7249722199902868751_a_nat @ M2 @ Xs ) ) @ ( set_Pr924983374503034536_a_nat @ ( take_P7249722199902868751_a_nat @ N @ Xs ) ) ) ) ).
% set_take_subset_set_take
thf(fact_1145_set__drop__subset__set__drop,axiom,
! [N: nat,M2: nat,Xs: list_P3592885314253461005_a_nat] :
( ( ord_less_eq_nat @ N @ M2 )
=> ( ord_le8666007276011122963_a_nat @ ( set_Pr924983374503034536_a_nat @ ( drop_P2883665741211355575_a_nat @ M2 @ Xs ) ) @ ( set_Pr924983374503034536_a_nat @ ( drop_P2883665741211355575_a_nat @ N @ Xs ) ) ) ) ).
% set_drop_subset_set_drop
thf(fact_1146_length__add__elem__list__lists,axiom,
! [Ys: list_term_a_b,X: term_a_b,Xs: list_term_a_b] :
( ( member_list_term_a_b @ Ys @ ( set_list_term_a_b2 @ ( basic_1593220722155286443rm_a_b @ X @ Xs ) ) )
=> ( ( size_s8906293707977694520rm_a_b @ Ys )
= ( suc @ ( size_s8906293707977694520rm_a_b @ Xs ) ) ) ) ).
% length_add_elem_list_lists
thf(fact_1147_set__list__subset__eq__nth__conv,axiom,
! [Xs: list_P8875379029341186191rm_a_b,A5: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ ( set_Pr2370880052973118494rm_a_b @ Xs ) @ A5 )
= ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s8600120110137556219rm_a_b @ Xs ) )
=> ( member5869715511025134514rm_a_b @ ( nth_Pr1098598711982373392rm_a_b @ Xs @ I3 ) @ A5 ) ) ) ) ).
% set_list_subset_eq_nth_conv
thf(fact_1148_set__list__subset__eq__nth__conv,axiom,
! [Xs: list_term_a_b,A5: set_term_a_b] :
( ( ord_le2705286416250468010rm_a_b @ ( set_term_a_b2 @ Xs ) @ A5 )
= ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s8906293707977694520rm_a_b @ Xs ) )
=> ( member_term_a_b @ ( nth_term_a_b @ Xs @ I3 ) @ A5 ) ) ) ) ).
% set_list_subset_eq_nth_conv
thf(fact_1149_set__list__subset__eq__nth__conv,axiom,
! [Xs: list_P3592885314253461005_a_nat,A5: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ ( set_Pr924983374503034536_a_nat @ Xs ) @ A5 )
= ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s984997627204368545_a_nat @ Xs ) )
=> ( member5724188588386418708_a_nat @ ( nth_Pr8461465654520414006_a_nat @ Xs @ I3 ) @ A5 ) ) ) ) ).
% set_list_subset_eq_nth_conv
thf(fact_1150_hd__drop__conv__nth,axiom,
! [N: nat,Xs: list_term_a_b] :
( ( ord_less_nat @ N @ ( size_s8906293707977694520rm_a_b @ Xs ) )
=> ( ( hd_term_a_b @ ( drop_term_a_b @ N @ Xs ) )
= ( nth_term_a_b @ Xs @ N ) ) ) ).
% hd_drop_conv_nth
thf(fact_1151_add__elem__list__listsI,axiom,
! [N: nat,Xs: list_term_a_b,Ys: list_term_a_b,X: term_a_b] :
( ( ord_less_eq_nat @ N @ ( size_s8906293707977694520rm_a_b @ Xs ) )
=> ( ( Ys
= ( append_term_a_b @ ( take_term_a_b @ N @ Xs ) @ ( cons_term_a_b @ X @ ( drop_term_a_b @ N @ Xs ) ) ) )
=> ( member_list_term_a_b @ Ys @ ( set_list_term_a_b2 @ ( basic_1593220722155286443rm_a_b @ X @ Xs ) ) ) ) ) ).
% add_elem_list_listsI
thf(fact_1152_take__nth__drop__concat,axiom,
! [I4: nat,Xss: list_list_term_a_b,Ys: list_term_a_b,J: nat,Z2: term_a_b] :
( ( ord_less_nat @ I4 @ ( size_s877380706853472072rm_a_b @ Xss ) )
=> ( ( ( nth_list_term_a_b @ Xss @ I4 )
= Ys )
=> ( ( ord_less_nat @ J @ ( size_s8906293707977694520rm_a_b @ Ys ) )
=> ( ( ( nth_term_a_b @ Ys @ J )
= Z2 )
=> ? [K2: nat] :
( ( ord_less_nat @ K2 @ ( size_s8906293707977694520rm_a_b @ ( concat_term_a_b @ Xss ) ) )
& ( ( take_term_a_b @ K2 @ ( concat_term_a_b @ Xss ) )
= ( append_term_a_b @ ( concat_term_a_b @ ( take_list_term_a_b @ I4 @ Xss ) ) @ ( take_term_a_b @ J @ Ys ) ) )
& ( ( nth_term_a_b @ ( concat_term_a_b @ Xss ) @ K2 )
= ( nth_term_a_b @ ( nth_list_term_a_b @ Xss @ I4 ) @ J ) )
& ( ( drop_term_a_b @ ( suc @ K2 ) @ ( concat_term_a_b @ Xss ) )
= ( append_term_a_b @ ( drop_term_a_b @ ( suc @ J ) @ Ys ) @ ( concat_term_a_b @ ( drop_list_term_a_b @ ( suc @ I4 ) @ Xss ) ) ) ) ) ) ) ) ) ).
% take_nth_drop_concat
thf(fact_1153_eq__length__concat__nth,axiom,
! [Xs: list_list_term_a_b,Ys: list_list_term_a_b] :
( ( ( size_s877380706853472072rm_a_b @ Xs )
= ( size_s877380706853472072rm_a_b @ Ys ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s877380706853472072rm_a_b @ Xs ) )
=> ( ( size_s8906293707977694520rm_a_b @ ( nth_list_term_a_b @ Xs @ I2 ) )
= ( size_s8906293707977694520rm_a_b @ ( nth_list_term_a_b @ Ys @ I2 ) ) ) )
=> ( ( size_s8906293707977694520rm_a_b @ ( concat_term_a_b @ Xs ) )
= ( size_s8906293707977694520rm_a_b @ ( concat_term_a_b @ Ys ) ) ) ) ) ).
% eq_length_concat_nth
thf(fact_1154_concat__all__nth,axiom,
! [Xs: list_list_term_a_b,Ys: list_list_term_a_b,P2: term_a_b > term_a_b > $o] :
( ( ( size_s877380706853472072rm_a_b @ Xs )
= ( size_s877380706853472072rm_a_b @ Ys ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s877380706853472072rm_a_b @ Xs ) )
=> ( ( size_s8906293707977694520rm_a_b @ ( nth_list_term_a_b @ Xs @ I2 ) )
= ( size_s8906293707977694520rm_a_b @ ( nth_list_term_a_b @ Ys @ I2 ) ) ) )
=> ( ! [I2: nat,J2: nat] :
( ( ord_less_nat @ I2 @ ( size_s877380706853472072rm_a_b @ Xs ) )
=> ( ( ord_less_nat @ J2 @ ( size_s8906293707977694520rm_a_b @ ( nth_list_term_a_b @ Xs @ I2 ) ) )
=> ( P2 @ ( nth_term_a_b @ ( nth_list_term_a_b @ Xs @ I2 ) @ J2 ) @ ( nth_term_a_b @ ( nth_list_term_a_b @ Ys @ I2 ) @ J2 ) ) ) )
=> ! [K3: nat] :
( ( ord_less_nat @ K3 @ ( size_s8906293707977694520rm_a_b @ ( concat_term_a_b @ Xs ) ) )
=> ( P2 @ ( nth_term_a_b @ ( concat_term_a_b @ Xs ) @ K3 ) @ ( nth_term_a_b @ ( concat_term_a_b @ Ys ) @ K3 ) ) ) ) ) ) ).
% concat_all_nth
thf(fact_1155_nth__concat__diff,axiom,
! [I1: nat,Xs: list_list_term_a_b,I22: nat] :
( ( ord_less_nat @ I1 @ ( size_s8906293707977694520rm_a_b @ ( concat_term_a_b @ Xs ) ) )
=> ( ( ord_less_nat @ I22 @ ( size_s8906293707977694520rm_a_b @ ( concat_term_a_b @ Xs ) ) )
=> ( ( I1 != I22 )
=> ? [J1: nat,K1: nat,J22: nat,K22: nat] :
( ( ( product_Pair_nat_nat @ J1 @ K1 )
!= ( product_Pair_nat_nat @ J22 @ K22 ) )
& ( ord_less_nat @ J1 @ ( size_s877380706853472072rm_a_b @ Xs ) )
& ( ord_less_nat @ J22 @ ( size_s877380706853472072rm_a_b @ Xs ) )
& ( ord_less_nat @ K1 @ ( size_s8906293707977694520rm_a_b @ ( nth_list_term_a_b @ Xs @ J1 ) ) )
& ( ord_less_nat @ K22 @ ( size_s8906293707977694520rm_a_b @ ( nth_list_term_a_b @ Xs @ J22 ) ) )
& ( ( nth_term_a_b @ ( concat_term_a_b @ Xs ) @ I1 )
= ( nth_term_a_b @ ( nth_list_term_a_b @ Xs @ J1 ) @ K1 ) )
& ( ( nth_term_a_b @ ( concat_term_a_b @ Xs ) @ I22 )
= ( nth_term_a_b @ ( nth_list_term_a_b @ Xs @ J22 ) @ K22 ) ) ) ) ) ) ).
% nth_concat_diff
thf(fact_1156_nth__concat__split,axiom,
! [I4: nat,Xs: list_list_term_a_b] :
( ( ord_less_nat @ I4 @ ( size_s8906293707977694520rm_a_b @ ( concat_term_a_b @ Xs ) ) )
=> ? [J2: nat,K2: nat] :
( ( ord_less_nat @ J2 @ ( size_s877380706853472072rm_a_b @ Xs ) )
& ( ord_less_nat @ K2 @ ( size_s8906293707977694520rm_a_b @ ( nth_list_term_a_b @ Xs @ J2 ) ) )
& ( ( nth_term_a_b @ ( concat_term_a_b @ Xs ) @ I4 )
= ( nth_term_a_b @ ( nth_list_term_a_b @ Xs @ J2 ) @ K2 ) ) ) ) ).
% nth_concat_split
thf(fact_1157_nth__concat__two__lists,axiom,
! [I4: nat,Xs: list_list_term_a_b,Ys: list_list_term_a_b] :
( ( ord_less_nat @ I4 @ ( size_s8906293707977694520rm_a_b @ ( concat_term_a_b @ Xs ) ) )
=> ( ( ( size_s877380706853472072rm_a_b @ Ys )
= ( size_s877380706853472072rm_a_b @ Xs ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s877380706853472072rm_a_b @ Xs ) )
=> ( ( size_s8906293707977694520rm_a_b @ ( nth_list_term_a_b @ Ys @ I2 ) )
= ( size_s8906293707977694520rm_a_b @ ( nth_list_term_a_b @ Xs @ I2 ) ) ) )
=> ? [J2: nat,K2: nat] :
( ( ord_less_nat @ J2 @ ( size_s877380706853472072rm_a_b @ Xs ) )
& ( ord_less_nat @ K2 @ ( size_s8906293707977694520rm_a_b @ ( nth_list_term_a_b @ Xs @ J2 ) ) )
& ( ( nth_term_a_b @ ( concat_term_a_b @ Xs ) @ I4 )
= ( nth_term_a_b @ ( nth_list_term_a_b @ Xs @ J2 ) @ K2 ) )
& ( ( nth_term_a_b @ ( concat_term_a_b @ Ys ) @ I4 )
= ( nth_term_a_b @ ( nth_list_term_a_b @ Ys @ J2 ) @ K2 ) ) ) ) ) ) ).
% nth_concat_two_lists
thf(fact_1158_term__not__arg,axiom,
! [F3: a,Ss2: list_term_a_b] :
~ ( member_term_a_b @ ( fun_a_b @ F3 @ Ss2 ) @ ( set_term_a_b2 @ Ss2 ) ) ).
% term_not_arg
thf(fact_1159_take__drop__update__first,axiom,
! [J: nat,Ds: list_term_a_b,Cs: list_term_a_b] :
( ( ord_less_nat @ J @ ( size_s8906293707977694520rm_a_b @ Ds ) )
=> ( ( ( size_s8906293707977694520rm_a_b @ Cs )
= ( size_s8906293707977694520rm_a_b @ Ds ) )
=> ( ( list_update_term_a_b @ ( append_term_a_b @ ( take_term_a_b @ J @ Ds ) @ ( drop_term_a_b @ J @ Cs ) ) @ J @ ( nth_term_a_b @ Ds @ J ) )
= ( append_term_a_b @ ( take_term_a_b @ ( suc @ J ) @ Ds ) @ ( drop_term_a_b @ ( suc @ J ) @ Cs ) ) ) ) ) ).
% take_drop_update_first
thf(fact_1160_upd__conv__take__nth__drop,axiom,
! [I4: nat,Xs: list_term_a_b,A: term_a_b] :
( ( ord_less_nat @ I4 @ ( size_s8906293707977694520rm_a_b @ Xs ) )
=> ( ( list_update_term_a_b @ Xs @ I4 @ A )
= ( append_term_a_b @ ( take_term_a_b @ I4 @ Xs ) @ ( cons_term_a_b @ A @ ( drop_term_a_b @ ( suc @ I4 ) @ Xs ) ) ) ) ) ).
% upd_conv_take_nth_drop
thf(fact_1161_length__list__update,axiom,
! [Xs: list_term_a_b,I4: nat,X: term_a_b] :
( ( size_s8906293707977694520rm_a_b @ ( list_update_term_a_b @ Xs @ I4 @ X ) )
= ( size_s8906293707977694520rm_a_b @ Xs ) ) ).
% length_list_update
thf(fact_1162_list__update__id,axiom,
! [Xs: list_term_a_b,I4: nat] :
( ( list_update_term_a_b @ Xs @ I4 @ ( nth_term_a_b @ Xs @ I4 ) )
= Xs ) ).
% list_update_id
thf(fact_1163_nth__list__update__neq,axiom,
! [I4: nat,J: nat,Xs: list_term_a_b,X: term_a_b] :
( ( I4 != J )
=> ( ( nth_term_a_b @ ( list_update_term_a_b @ Xs @ I4 @ X ) @ J )
= ( nth_term_a_b @ Xs @ J ) ) ) ).
% nth_list_update_neq
thf(fact_1164_list__update__beyond,axiom,
! [Xs: list_term_a_b,I4: nat,X: term_a_b] :
( ( ord_less_eq_nat @ ( size_s8906293707977694520rm_a_b @ Xs ) @ I4 )
=> ( ( list_update_term_a_b @ Xs @ I4 @ X )
= Xs ) ) ).
% list_update_beyond
thf(fact_1165_list__update__length,axiom,
! [Xs: list_term_a_b,X: term_a_b,Ys: list_term_a_b,Y: term_a_b] :
( ( list_update_term_a_b @ ( append_term_a_b @ Xs @ ( cons_term_a_b @ X @ Ys ) ) @ ( size_s8906293707977694520rm_a_b @ Xs ) @ Y )
= ( append_term_a_b @ Xs @ ( cons_term_a_b @ Y @ Ys ) ) ) ).
% list_update_length
thf(fact_1166_nth__list__update__eq,axiom,
! [I4: nat,Xs: list_term_a_b,X: term_a_b] :
( ( ord_less_nat @ I4 @ ( size_s8906293707977694520rm_a_b @ Xs ) )
=> ( ( nth_term_a_b @ ( list_update_term_a_b @ Xs @ I4 @ X ) @ I4 )
= X ) ) ).
% nth_list_update_eq
thf(fact_1167_set__swap,axiom,
! [I4: nat,Xs: list_term_a_b,J: nat] :
( ( ord_less_nat @ I4 @ ( size_s8906293707977694520rm_a_b @ Xs ) )
=> ( ( ord_less_nat @ J @ ( size_s8906293707977694520rm_a_b @ Xs ) )
=> ( ( set_term_a_b2 @ ( list_update_term_a_b @ ( list_update_term_a_b @ Xs @ I4 @ ( nth_term_a_b @ Xs @ J ) ) @ J @ ( nth_term_a_b @ Xs @ I4 ) ) )
= ( set_term_a_b2 @ Xs ) ) ) ) ).
% set_swap
thf(fact_1168_set__update__subsetI,axiom,
! [Xs: list_P8875379029341186191rm_a_b,A5: set_Pr4386577575007340137rm_a_b,X: produc357393685978478089rm_a_b,I4: nat] :
( ( ord_le118470702582115849rm_a_b @ ( set_Pr2370880052973118494rm_a_b @ Xs ) @ A5 )
=> ( ( member5869715511025134514rm_a_b @ X @ A5 )
=> ( ord_le118470702582115849rm_a_b @ ( set_Pr2370880052973118494rm_a_b @ ( list_u4211539095652206455rm_a_b @ Xs @ I4 @ X ) ) @ A5 ) ) ) ).
% set_update_subsetI
thf(fact_1169_set__update__subsetI,axiom,
! [Xs: list_P3592885314253461005_a_nat,A5: set_Pr4934435412358123699_a_nat,X: product_prod_a_nat,I4: nat] :
( ( ord_le8666007276011122963_a_nat @ ( set_Pr924983374503034536_a_nat @ Xs ) @ A5 )
=> ( ( member5724188588386418708_a_nat @ X @ A5 )
=> ( ord_le8666007276011122963_a_nat @ ( set_Pr924983374503034536_a_nat @ ( list_u4318310736832020111_a_nat @ Xs @ I4 @ X ) ) @ A5 ) ) ) ).
% set_update_subsetI
thf(fact_1170_set__update__memI,axiom,
! [N: nat,Xs: list_P8875379029341186191rm_a_b,X: produc357393685978478089rm_a_b] :
( ( ord_less_nat @ N @ ( size_s8600120110137556219rm_a_b @ Xs ) )
=> ( member5869715511025134514rm_a_b @ X @ ( set_Pr2370880052973118494rm_a_b @ ( list_u4211539095652206455rm_a_b @ Xs @ N @ X ) ) ) ) ).
% set_update_memI
thf(fact_1171_set__update__memI,axiom,
! [N: nat,Xs: list_term_a_b,X: term_a_b] :
( ( ord_less_nat @ N @ ( size_s8906293707977694520rm_a_b @ Xs ) )
=> ( member_term_a_b @ X @ ( set_term_a_b2 @ ( list_update_term_a_b @ Xs @ N @ X ) ) ) ) ).
% set_update_memI
thf(fact_1172_list__update__append1,axiom,
! [I4: nat,Xs: list_term_a_b,Ys: list_term_a_b,X: term_a_b] :
( ( ord_less_nat @ I4 @ ( size_s8906293707977694520rm_a_b @ Xs ) )
=> ( ( list_update_term_a_b @ ( append_term_a_b @ Xs @ Ys ) @ I4 @ X )
= ( append_term_a_b @ ( list_update_term_a_b @ Xs @ I4 @ X ) @ Ys ) ) ) ).
% list_update_append1
thf(fact_1173_nth__list__update,axiom,
! [I4: nat,Xs: list_term_a_b,J: nat,X: term_a_b] :
( ( ord_less_nat @ I4 @ ( size_s8906293707977694520rm_a_b @ Xs ) )
=> ( ( ( I4 = J )
=> ( ( nth_term_a_b @ ( list_update_term_a_b @ Xs @ I4 @ X ) @ J )
= X ) )
& ( ( I4 != J )
=> ( ( nth_term_a_b @ ( list_update_term_a_b @ Xs @ I4 @ X ) @ J )
= ( nth_term_a_b @ Xs @ J ) ) ) ) ) ).
% nth_list_update
thf(fact_1174_list__update__same__conv,axiom,
! [I4: nat,Xs: list_term_a_b,X: term_a_b] :
( ( ord_less_nat @ I4 @ ( size_s8906293707977694520rm_a_b @ Xs ) )
=> ( ( ( list_update_term_a_b @ Xs @ I4 @ X )
= Xs )
= ( ( nth_term_a_b @ Xs @ I4 )
= X ) ) ) ).
% list_update_same_conv
thf(fact_1175_parallel__list__update,axiom,
! [N: nat,R2: term_a_b > term_a_b > $o,P: list_term_a_b > $o,Xs: list_term_a_b,Ys: list_term_a_b] :
( ! [Xs3: list_term_a_b,I2: nat,Y3: term_a_b] :
( ( ( size_s8906293707977694520rm_a_b @ Xs3 )
= N )
=> ( ( ord_less_nat @ I2 @ N )
=> ( ( R2 @ ( nth_term_a_b @ Xs3 @ I2 ) @ Y3 )
=> ( ( P @ Xs3 )
=> ( P @ ( list_update_term_a_b @ Xs3 @ I2 @ Y3 ) ) ) ) ) )
=> ( ( ( size_s8906293707977694520rm_a_b @ Xs )
= N )
=> ( ( P @ Xs )
=> ( ( ( size_s8906293707977694520rm_a_b @ Ys )
= N )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( R2 @ ( nth_term_a_b @ Xs @ I2 ) @ ( nth_term_a_b @ Ys @ I2 ) ) )
=> ( P @ Ys ) ) ) ) ) ) ).
% parallel_list_update
thf(fact_1176_take__drop__update__second,axiom,
! [J: nat,Ds: list_term_a_b,Cs: list_term_a_b] :
( ( ord_less_nat @ J @ ( size_s8906293707977694520rm_a_b @ Ds ) )
=> ( ( ( size_s8906293707977694520rm_a_b @ Cs )
= ( size_s8906293707977694520rm_a_b @ Ds ) )
=> ( ( list_update_term_a_b @ ( append_term_a_b @ ( take_term_a_b @ J @ Ds ) @ ( drop_term_a_b @ J @ Cs ) ) @ J @ ( nth_term_a_b @ Cs @ J ) )
= ( append_term_a_b @ ( take_term_a_b @ J @ Ds ) @ ( drop_term_a_b @ J @ Cs ) ) ) ) ) ).
% take_drop_update_second
thf(fact_1177_replace__term__at_Oelims,axiom,
! [X: term_a_b,Xa: list_nat,Xb: term_a_b,Y: term_a_b] :
( ( ( term_r6860082780075436317at_a_b @ X @ Xa @ Xb )
= Y )
=> ( ( ( Xa = nil_nat )
=> ( Y != Xb ) )
=> ( ! [X3: b] :
( ( X
= ( var_b_a @ X3 ) )
=> ( ? [V: nat,Va: list_nat] :
( Xa
= ( cons_nat @ V @ Va ) )
=> ( Y
!= ( var_b_a @ X3 ) ) ) )
=> ~ ! [F2: a,Ts: list_term_a_b] :
( ( X
= ( fun_a_b @ F2 @ Ts ) )
=> ! [I2: nat,Ps: list_nat] :
( ( Xa
= ( cons_nat @ I2 @ Ps ) )
=> ~ ( ( ( ord_less_nat @ I2 @ ( size_s8906293707977694520rm_a_b @ Ts ) )
=> ( Y
= ( fun_a_b @ F2 @ ( list_update_term_a_b @ Ts @ I2 @ ( term_r6860082780075436317at_a_b @ ( nth_term_a_b @ Ts @ I2 ) @ Ps @ Xb ) ) ) ) )
& ( ~ ( ord_less_nat @ I2 @ ( size_s8906293707977694520rm_a_b @ Ts ) )
=> ( Y
= ( fun_a_b @ F2 @ Ts ) ) ) ) ) ) ) ) ) ).
% replace_term_at.elims
thf(fact_1178_listrel1__iff__update,axiom,
! [Xs: list_term_a_b,Ys: list_term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Xs @ Ys ) @ ( listrel1_term_a_b @ R2 ) )
= ( ? [Y5: term_a_b,N2: nat] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( nth_term_a_b @ Xs @ N2 ) @ Y5 ) @ R2 )
& ( ord_less_nat @ N2 @ ( size_s8906293707977694520rm_a_b @ Xs ) )
& ( Ys
= ( list_update_term_a_b @ Xs @ N2 @ Y5 ) ) ) ) ) ).
% listrel1_iff_update
thf(fact_1179_Cons__listrel1__Cons,axiom,
! [X: term_a_b,Xs: list_term_a_b,Y: term_a_b,Ys: list_term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ ( cons_term_a_b @ X @ Xs ) @ ( cons_term_a_b @ Y @ Ys ) ) @ ( listrel1_term_a_b @ R2 ) )
= ( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ R2 )
& ( Xs = Ys ) )
| ( ( X = Y )
& ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Xs @ Ys ) @ ( listrel1_term_a_b @ R2 ) ) ) ) ) ).
% Cons_listrel1_Cons
thf(fact_1180_replace__term__at_Osimps_I2_J,axiom,
! [X: b,V2: nat,Va2: list_nat,T: term_a_b] :
( ( term_r6860082780075436317at_a_b @ ( var_b_a @ X ) @ ( cons_nat @ V2 @ Va2 ) @ T )
= ( var_b_a @ X ) ) ).
% replace_term_at.simps(2)
thf(fact_1181_listrel__rtrancl__eq__rtrancl__listrel1,axiom,
! [R2: set_Pr4386577575007340137rm_a_b] :
( ( listre1194016999521814427rm_a_b @ ( transi7742714808557438673rm_a_b @ R2 ) )
= ( transi911495901722478305rm_a_b @ ( listrel1_term_a_b @ R2 ) ) ) ).
% listrel_rtrancl_eq_rtrancl_listrel1
thf(fact_1182_listrel1__eq__len,axiom,
! [Xs: list_term_a_b,Ys: list_term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Xs @ Ys ) @ ( listrel1_term_a_b @ R2 ) )
=> ( ( size_s8906293707977694520rm_a_b @ Xs )
= ( size_s8906293707977694520rm_a_b @ Ys ) ) ) ).
% listrel1_eq_len
thf(fact_1183_rtrancl__listrel1__eq__len,axiom,
! [X: list_term_a_b,Y: list_term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ X @ Y ) @ ( transi911495901722478305rm_a_b @ ( listrel1_term_a_b @ R2 ) ) )
=> ( ( size_s8906293707977694520rm_a_b @ X )
= ( size_s8906293707977694520rm_a_b @ Y ) ) ) ).
% rtrancl_listrel1_eq_len
thf(fact_1184_listrel1__rtrancl__subset__rtrancl__listrel1,axiom,
! [R2: set_Pr4386577575007340137rm_a_b] : ( ord_le7771511673815134377rm_a_b @ ( listrel1_term_a_b @ ( transi7742714808557438673rm_a_b @ R2 ) ) @ ( transi911495901722478305rm_a_b @ ( listrel1_term_a_b @ R2 ) ) ) ).
% listrel1_rtrancl_subset_rtrancl_listrel1
thf(fact_1185_listrel1I1,axiom,
! [X: term_a_b,Y: term_a_b,R2: set_Pr4386577575007340137rm_a_b,Xs: list_term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ R2 )
=> ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ ( cons_term_a_b @ X @ Xs ) @ ( cons_term_a_b @ Y @ Xs ) ) @ ( listrel1_term_a_b @ R2 ) ) ) ).
% listrel1I1
thf(fact_1186_Cons__listrel1E1,axiom,
! [X: term_a_b,Xs: list_term_a_b,Ys: list_term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ ( cons_term_a_b @ X @ Xs ) @ Ys ) @ ( listrel1_term_a_b @ R2 ) )
=> ( ! [Y3: term_a_b] :
( ( Ys
= ( cons_term_a_b @ Y3 @ Xs ) )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y3 ) @ R2 ) )
=> ~ ! [Zs2: list_term_a_b] :
( ( Ys
= ( cons_term_a_b @ X @ Zs2 ) )
=> ~ ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Xs @ Zs2 ) @ ( listrel1_term_a_b @ R2 ) ) ) ) ) ).
% Cons_listrel1E1
thf(fact_1187_Cons__listrel1E2,axiom,
! [Xs: list_term_a_b,Y: term_a_b,Ys: list_term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Xs @ ( cons_term_a_b @ Y @ Ys ) ) @ ( listrel1_term_a_b @ R2 ) )
=> ( ! [X3: term_a_b] :
( ( Xs
= ( cons_term_a_b @ X3 @ Ys ) )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y ) @ R2 ) )
=> ~ ! [Zs2: list_term_a_b] :
( ( Xs
= ( cons_term_a_b @ Y @ Zs2 ) )
=> ~ ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Zs2 @ Ys ) @ ( listrel1_term_a_b @ R2 ) ) ) ) ) ).
% Cons_listrel1E2
thf(fact_1188_listrel__reflcl__if__listrel1,axiom,
! [Xs: list_term_a_b,Ys: list_term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Xs @ Ys ) @ ( listrel1_term_a_b @ R2 ) )
=> ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Xs @ Ys ) @ ( listre1194016999521814427rm_a_b @ ( transi7742714808557438673rm_a_b @ R2 ) ) ) ) ).
% listrel_reflcl_if_listrel1
thf(fact_1189_funas__term__replace__at__upper,axiom,
! [S: term_a_b,P: list_nat,T: term_a_b] : ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ ( term_r6860082780075436317at_a_b @ S @ P @ T ) ) @ ( sup_su459911885395995103_a_nat @ ( term_funas_term_a_b @ S ) @ ( term_funas_term_a_b @ T ) ) ) ).
% funas_term_replace_at_upper
thf(fact_1190_listrel1I,axiom,
! [X: term_a_b,Y: term_a_b,R2: set_Pr4386577575007340137rm_a_b,Xs: list_term_a_b,Us: list_term_a_b,Vs: list_term_a_b,Ys: list_term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ R2 )
=> ( ( Xs
= ( append_term_a_b @ Us @ ( cons_term_a_b @ X @ Vs ) ) )
=> ( ( Ys
= ( append_term_a_b @ Us @ ( cons_term_a_b @ Y @ Vs ) ) )
=> ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Xs @ Ys ) @ ( listrel1_term_a_b @ R2 ) ) ) ) ) ).
% listrel1I
thf(fact_1191_listrel1E,axiom,
! [Xs: list_term_a_b,Ys: list_term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Xs @ Ys ) @ ( listrel1_term_a_b @ R2 ) )
=> ~ ! [X3: term_a_b,Y3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ R2 )
=> ! [Us2: list_term_a_b,Vs2: list_term_a_b] :
( ( Xs
= ( append_term_a_b @ Us2 @ ( cons_term_a_b @ X3 @ Vs2 ) ) )
=> ( Ys
!= ( append_term_a_b @ Us2 @ ( cons_term_a_b @ Y3 @ Vs2 ) ) ) ) ) ) ).
% listrel1E
thf(fact_1192_rtrancl__listrel1__ConsI2,axiom,
! [X: term_a_b,Y: term_a_b,R2: set_Pr4386577575007340137rm_a_b,Xs: list_term_a_b,Ys: list_term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
=> ( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Xs @ Ys ) @ ( transi911495901722478305rm_a_b @ ( listrel1_term_a_b @ R2 ) ) )
=> ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ ( cons_term_a_b @ X @ Xs ) @ ( cons_term_a_b @ Y @ Ys ) ) @ ( transi911495901722478305rm_a_b @ ( listrel1_term_a_b @ R2 ) ) ) ) ) ).
% rtrancl_listrel1_ConsI2
thf(fact_1193_snoc__listrel1__snoc__iff,axiom,
! [Xs: list_term_a_b,X: term_a_b,Ys: list_term_a_b,Y: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ ( append_term_a_b @ Xs @ ( cons_term_a_b @ X @ nil_term_a_b ) ) @ ( append_term_a_b @ Ys @ ( cons_term_a_b @ Y @ nil_term_a_b ) ) ) @ ( listrel1_term_a_b @ R2 ) )
= ( ( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Xs @ Ys ) @ ( listrel1_term_a_b @ R2 ) )
& ( X = Y ) )
| ( ( Xs = Ys )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ R2 ) ) ) ) ).
% snoc_listrel1_snoc_iff
thf(fact_1194_replace__term__at_Osimps_I3_J,axiom,
! [I4: nat,Ts2: list_term_a_b,F3: a,Ps2: list_nat,T: term_a_b] :
( ( ( ord_less_nat @ I4 @ ( size_s8906293707977694520rm_a_b @ Ts2 ) )
=> ( ( term_r6860082780075436317at_a_b @ ( fun_a_b @ F3 @ Ts2 ) @ ( cons_nat @ I4 @ Ps2 ) @ T )
= ( fun_a_b @ F3 @ ( list_update_term_a_b @ Ts2 @ I4 @ ( term_r6860082780075436317at_a_b @ ( nth_term_a_b @ Ts2 @ I4 ) @ Ps2 @ T ) ) ) ) )
& ( ~ ( ord_less_nat @ I4 @ ( size_s8906293707977694520rm_a_b @ Ts2 ) )
=> ( ( term_r6860082780075436317at_a_b @ ( fun_a_b @ F3 @ Ts2 ) @ ( cons_nat @ I4 @ Ps2 ) @ T )
= ( fun_a_b @ F3 @ Ts2 ) ) ) ) ).
% replace_term_at.simps(3)
thf(fact_1195_replace__term__at_Opelims,axiom,
! [X: term_a_b,Xa: list_nat,Xb: term_a_b,Y: term_a_b] :
( ( ( term_r6860082780075436317at_a_b @ X @ Xa @ Xb )
= Y )
=> ( ( accp_P2729577386226225901rm_a_b @ term_r1280879029893354718el_a_b @ ( produc3812856575676843240rm_a_b @ X @ ( produc5151171985953862413rm_a_b @ Xa @ Xb ) ) )
=> ( ( ( Xa = nil_nat )
=> ( ( Y = Xb )
=> ~ ( accp_P2729577386226225901rm_a_b @ term_r1280879029893354718el_a_b @ ( produc3812856575676843240rm_a_b @ X @ ( produc5151171985953862413rm_a_b @ nil_nat @ Xb ) ) ) ) )
=> ( ! [X3: b] :
( ( X
= ( var_b_a @ X3 ) )
=> ! [V: nat,Va: list_nat] :
( ( Xa
= ( cons_nat @ V @ Va ) )
=> ( ( Y
= ( var_b_a @ X3 ) )
=> ~ ( accp_P2729577386226225901rm_a_b @ term_r1280879029893354718el_a_b @ ( produc3812856575676843240rm_a_b @ ( var_b_a @ X3 ) @ ( produc5151171985953862413rm_a_b @ ( cons_nat @ V @ Va ) @ Xb ) ) ) ) ) )
=> ~ ! [F2: a,Ts: list_term_a_b] :
( ( X
= ( fun_a_b @ F2 @ Ts ) )
=> ! [I2: nat,Ps: list_nat] :
( ( Xa
= ( cons_nat @ I2 @ Ps ) )
=> ( ( ( ( ord_less_nat @ I2 @ ( size_s8906293707977694520rm_a_b @ Ts ) )
=> ( Y
= ( fun_a_b @ F2 @ ( list_update_term_a_b @ Ts @ I2 @ ( term_r6860082780075436317at_a_b @ ( nth_term_a_b @ Ts @ I2 ) @ Ps @ Xb ) ) ) ) )
& ( ~ ( ord_less_nat @ I2 @ ( size_s8906293707977694520rm_a_b @ Ts ) )
=> ( Y
= ( fun_a_b @ F2 @ Ts ) ) ) )
=> ~ ( accp_P2729577386226225901rm_a_b @ term_r1280879029893354718el_a_b @ ( produc3812856575676843240rm_a_b @ ( fun_a_b @ F2 @ Ts ) @ ( produc5151171985953862413rm_a_b @ ( cons_nat @ I2 @ Ps ) @ Xb ) ) ) ) ) ) ) ) ) ) ).
% replace_term_at.pelims
thf(fact_1196_nth__drop__0,axiom,
! [Ss2: list_term_a_b] :
( ( ord_less_nat @ zero_zero_nat @ ( size_s8906293707977694520rm_a_b @ Ss2 ) )
=> ( ( cons_term_a_b @ ( nth_term_a_b @ Ss2 @ zero_zero_nat ) @ ( drop_term_a_b @ ( suc @ zero_zero_nat ) @ Ss2 ) )
= Ss2 ) ) ).
% nth_drop_0
thf(fact_1197_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_1198_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_1199_bot__nat__0_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_1200_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_1201_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_1202_length__0__conv,axiom,
! [Xs: list_term_a_b] :
( ( ( size_s8906293707977694520rm_a_b @ Xs )
= zero_zero_nat )
= ( Xs = nil_term_a_b ) ) ).
% length_0_conv
thf(fact_1203_nth__Cons__0,axiom,
! [X: term_a_b,Xs: list_term_a_b] :
( ( nth_term_a_b @ ( cons_term_a_b @ X @ Xs ) @ zero_zero_nat )
= X ) ).
% nth_Cons_0
thf(fact_1204_length__greater__0__conv,axiom,
! [Xs: list_term_a_b] :
( ( ord_less_nat @ zero_zero_nat @ ( size_s8906293707977694520rm_a_b @ Xs ) )
= ( Xs != nil_term_a_b ) ) ).
% length_greater_0_conv
thf(fact_1205_relpow__0__E,axiom,
! [X: term_a_b,Y: term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( compow4057154403645558940rm_a_b @ zero_zero_nat @ R3 ) )
=> ( X = Y ) ) ).
% relpow_0_E
thf(fact_1206_relpow__0__I,axiom,
! [X: term_a_b,R3: set_Pr4386577575007340137rm_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ X ) @ ( compow4057154403645558940rm_a_b @ zero_zero_nat @ R3 ) ) ).
% relpow_0_I
thf(fact_1207_list_Osize_I3_J,axiom,
( ( size_s8906293707977694520rm_a_b @ nil_term_a_b )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_1208_term_Osize_I3_J,axiom,
! [X1: b] :
( ( size_size_term_a_b @ ( var_b_a @ X1 ) )
= ( suc @ zero_zero_nat ) ) ).
% term.size(3)
thf(fact_1209_infinite__descent0,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ~ ( P2 @ N3 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
& ~ ( P2 @ M3 ) ) ) )
=> ( P2 @ N ) ) ) ).
% infinite_descent0
thf(fact_1210_gr__implies__not0,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_1211_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_1212_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_1213_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_1214_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_1215_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_1216_ex__least__nat__le,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ N )
=> ( ~ ( P2 @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
& ! [I: nat] :
( ( ord_less_nat @ I @ K2 )
=> ~ ( P2 @ I ) )
& ( P2 @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_1217_all__less__two,axiom,
! [P2: nat > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ ( suc @ zero_zero_nat ) ) )
=> ( P2 @ I3 ) ) )
= ( ( P2 @ zero_zero_nat )
& ( P2 @ ( suc @ zero_zero_nat ) ) ) ) ).
% all_less_two
thf(fact_1218_all__Suc__conv,axiom,
! [N: nat,P2: nat > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
=> ( P2 @ I3 ) ) )
= ( ( P2 @ zero_zero_nat )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ( P2 @ ( suc @ I3 ) ) ) ) ) ).
% all_Suc_conv
thf(fact_1219_ex__Suc__conv,axiom,
! [N: nat,P2: nat > $o] :
( ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
& ( P2 @ I3 ) ) )
= ( ( P2 @ zero_zero_nat )
| ? [I3: nat] :
( ( ord_less_nat @ I3 @ N )
& ( P2 @ ( suc @ I3 ) ) ) ) ) ).
% ex_Suc_conv
thf(fact_1220_less__Suc__eq__0__disj,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N ) )
= ( ( M2 = zero_zero_nat )
| ? [J3: nat] :
( ( M2
= ( suc @ J3 ) )
& ( ord_less_nat @ J3 @ N ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_1221_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M4: nat] :
( N
= ( suc @ M4 ) ) ) ).
% gr0_implies_Suc
thf(fact_1222_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( ? [M: nat] :
( N
= ( suc @ M ) ) ) ) ).
% gr0_conv_Suc
thf(fact_1223_length__pos__if__in__set,axiom,
! [X: produc357393685978478089rm_a_b,Xs: list_P8875379029341186191rm_a_b] :
( ( member5869715511025134514rm_a_b @ X @ ( set_Pr2370880052973118494rm_a_b @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_s8600120110137556219rm_a_b @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_1224_length__pos__if__in__set,axiom,
! [X: term_a_b,Xs: list_term_a_b] :
( ( member_term_a_b @ X @ ( set_term_a_b2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_s8906293707977694520rm_a_b @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_1225_ex__least__nat__less,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ N )
=> ( ~ ( P2 @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_nat @ K2 @ N )
& ! [I: nat] :
( ( ord_less_eq_nat @ I @ K2 )
=> ~ ( P2 @ I ) )
& ( P2 @ ( suc @ K2 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_1226_relpow__E,axiom,
! [X: term_a_b,Z2: term_a_b,N: nat,R3: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z2 ) @ ( compow4057154403645558940rm_a_b @ N @ R3 ) )
=> ( ( ( N = zero_zero_nat )
=> ( X != Z2 ) )
=> ~ ! [Y3: term_a_b,M4: nat] :
( ( N
= ( suc @ M4 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y3 ) @ ( compow4057154403645558940rm_a_b @ M4 @ R3 ) )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ Z2 ) @ R3 ) ) ) ) ) ).
% relpow_E
thf(fact_1227_relpow__E2,axiom,
! [X: term_a_b,Z2: term_a_b,N: nat,R3: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z2 ) @ ( compow4057154403645558940rm_a_b @ N @ R3 ) )
=> ( ( ( N = zero_zero_nat )
=> ( X != Z2 ) )
=> ~ ! [Y3: term_a_b,M4: nat] :
( ( N
= ( suc @ M4 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y3 ) @ R3 )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ Z2 ) @ ( compow4057154403645558940rm_a_b @ M4 @ R3 ) ) ) ) ) ) ).
% relpow_E2
thf(fact_1228_hd__conv__nth,axiom,
! [Xs: list_term_a_b] :
( ( Xs != nil_term_a_b )
=> ( ( hd_term_a_b @ Xs )
= ( nth_term_a_b @ Xs @ zero_zero_nat ) ) ) ).
% hd_conv_nth
thf(fact_1229_trancl__power,axiom,
! [P: produc357393685978478089rm_a_b,R3: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ P @ ( transi7922773638565587891rm_a_b @ R3 ) )
= ( ? [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
& ( member5869715511025134514rm_a_b @ P @ ( compow4057154403645558940rm_a_b @ N2 @ R3 ) ) ) ) ) ).
% trancl_power
thf(fact_1230_rtrancl__fun__conv,axiom,
! [S: term_a_b,T: term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( transi7742714808557438673rm_a_b @ R3 ) )
= ( ? [F7: nat > term_a_b,N2: nat] :
( ( ( F7 @ zero_zero_nat )
= S )
& ( ( F7 @ N2 )
= T )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ N2 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F7 @ I3 ) @ ( F7 @ ( suc @ I3 ) ) ) @ R3 ) ) ) ) ) ).
% rtrancl_fun_conv
thf(fact_1231_rtrancl__imp__seq,axiom,
! [X: term_a_b,Y: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
=> ? [F2: nat > term_a_b,N3: nat] :
( ( ( F2 @ zero_zero_nat )
= X )
& ( ( F2 @ N3 )
= Y )
& ! [I: nat] :
( ( ord_less_nat @ I @ N3 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F2 @ I ) @ ( F2 @ ( suc @ I ) ) ) @ R2 ) ) ) ) ).
% rtrancl_imp_seq
thf(fact_1232_relpow__seq,axiom,
! [X: term_a_b,Y: term_a_b,N: nat,R2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( compow4057154403645558940rm_a_b @ N @ R2 ) )
=> ? [F2: nat > term_a_b] :
( ( ( F2 @ zero_zero_nat )
= X )
& ( ( F2 @ N )
= Y )
& ! [I: nat] :
( ( ord_less_nat @ I @ N )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F2 @ I ) @ ( F2 @ ( suc @ I ) ) ) @ R2 ) ) ) ) ).
% relpow_seq
thf(fact_1233_relpow__fun__conv,axiom,
! [A: term_a_b,B: term_a_b,N: nat,R3: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( compow4057154403645558940rm_a_b @ N @ R3 ) )
= ( ? [F7: nat > term_a_b] :
( ( ( F7 @ zero_zero_nat )
= A )
& ( ( F7 @ N )
= B )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F7 @ I3 ) @ ( F7 @ ( suc @ I3 ) ) ) @ R3 ) ) ) ) ) ).
% relpow_fun_conv
thf(fact_1234_nth__equal__first__eq,axiom,
! [X: produc357393685978478089rm_a_b,Xs: list_P8875379029341186191rm_a_b,N: nat] :
( ~ ( member5869715511025134514rm_a_b @ X @ ( set_Pr2370880052973118494rm_a_b @ Xs ) )
=> ( ( ord_less_eq_nat @ N @ ( size_s8600120110137556219rm_a_b @ Xs ) )
=> ( ( ( nth_Pr1098598711982373392rm_a_b @ ( cons_P3536305108106557631rm_a_b @ X @ Xs ) @ N )
= X )
= ( N = zero_zero_nat ) ) ) ) ).
% nth_equal_first_eq
thf(fact_1235_nth__equal__first__eq,axiom,
! [X: term_a_b,Xs: list_term_a_b,N: nat] :
( ~ ( member_term_a_b @ X @ ( set_term_a_b2 @ Xs ) )
=> ( ( ord_less_eq_nat @ N @ ( size_s8906293707977694520rm_a_b @ Xs ) )
=> ( ( ( nth_term_a_b @ ( cons_term_a_b @ X @ Xs ) @ N )
= X )
= ( N = zero_zero_nat ) ) ) ) ).
% nth_equal_first_eq
thf(fact_1236_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_1237_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_1238_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_1239_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_1240_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_1241_gr__implies__not__zero,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_1242_zero__less__iff__neq__zero,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( N != zero_zero_nat ) ) ).
% zero_less_iff_neq_zero
thf(fact_1243_length__n__lists__elem,axiom,
! [Ys: list_term_a_b,N: nat,Xs: list_term_a_b] :
( ( member_list_term_a_b @ Ys @ ( set_list_term_a_b2 @ ( n_lists_term_a_b @ N @ Xs ) ) )
=> ( ( size_s8906293707977694520rm_a_b @ Ys )
= N ) ) ).
% length_n_lists_elem
thf(fact_1244_constT__nfunas__term__poss__of__term__empty,axiom,
! [C2: a,T: term_a_b] :
( ( ~ ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ C2 @ zero_zero_nat ) @ ( term_funas_term_a_b @ T ) ) )
= ( ( terms_7168686267159881682rm_a_b @ ( fun_a_b @ C2 @ nil_term_a_b ) @ T )
= bot_bot_set_list_nat ) ) ).
% constT_nfunas_term_poss_of_term_empty
thf(fact_1245_empty__iff,axiom,
! [C2: produc357393685978478089rm_a_b] :
~ ( member5869715511025134514rm_a_b @ C2 @ bot_bo197521221353338581rm_a_b ) ).
% empty_iff
thf(fact_1246_all__not__in__conv,axiom,
! [A5: set_Pr4386577575007340137rm_a_b] :
( ( ! [X4: produc357393685978478089rm_a_b] :
~ ( member5869715511025134514rm_a_b @ X4 @ A5 ) )
= ( A5 = bot_bo197521221353338581rm_a_b ) ) ).
% all_not_in_conv
thf(fact_1247_empty__subsetI,axiom,
! [A5: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ bot_bo9049108969261143879_a_nat @ A5 ) ).
% empty_subsetI
thf(fact_1248_subset__empty,axiom,
! [A5: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A5 @ bot_bo9049108969261143879_a_nat )
= ( A5 = bot_bo9049108969261143879_a_nat ) ) ).
% subset_empty
% Helper facts (3)
thf(help_If_3_1_If_001t__Nat__Onat_T,axiom,
! [P2: $o] :
( ( P2 = $true )
| ( P2 = $false ) ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $true @ X @ Y )
= X ) ).
% Conjectures (2)
thf(conj_0,hypothesis,
member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( var_b_a @ x ) @ ta ) @ ( transi7922773638565587891rm_a_b @ ( sig_step_a_b @ f @ ( rstep_a_b @ r ) ) ) ).
thf(conj_1,conjecture,
member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( var_b_a @ x ) @ ta ) @ ( srstep7844470518422762656ep_a_b @ f @ r ) ).
%------------------------------------------------------------------------------