TPTP Problem File: SLH0422^1.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% 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_00661_032111__14227592_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1592 ( 368 unt; 315 typ; 0 def)
% Number of atoms : 4150 ( 877 equ; 0 cnn)
% Maximal formula atoms : 12 ( 3 avg)
% Number of connectives : 16365 ( 497 ~; 32 |; 234 &;13325 @)
% ( 0 <=>;2277 =>; 0 <=; 0 <~>)
% Maximal formula depth : 22 ( 9 avg)
% Number of types : 34 ( 33 usr)
% Number of type conns : 732 ( 732 >; 0 *; 0 +; 0 <<)
% Number of symbols : 283 ( 282 usr; 28 con; 0-5 aty)
% Number of variables : 4374 ( 233 ^;3999 !; 142 ?;4374 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 14:19:27.624
%------------------------------------------------------------------------------
% Could-be-implicit typings (33)
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_Mt__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_Mt__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J_J_J,type,
set_Pr6848273517546626247rm_a_b: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_Mt__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_Mt__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J_J,type,
produc2957090337049255143rm_a_b: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_Mt__List__Olist_It__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_Mt__List__Olist_It__Nat__Onat_J_J_J_J,type,
set_Pr2046385631809063665st_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J_J_J,type,
set_Pr2173775114972114941rm_a_b: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_Mt__List__Olist_It__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_Mt__List__Olist_It__Nat__Onat_J_J_J,type,
produc3734095815061405585st_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J_J,type,
produc8411532396478638749rm_a_b: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_Mt__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J_J,type,
set_Pr2972776593051762503rm_a_b: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J_J_J,type,
set_Pr915550674247830855st_nat: $tType ).
thf(ty_n_t__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,
produc4523971788519308903rm_a_b: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J_J,type,
produc1342688880896445543st_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_Mt__List__Olist_It__Nat__Onat_J_J_J,type,
set_Pr2659102613823815246st_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J_J,type,
set_Pr1829413238147913556rm_a_b: $tType ).
thf(ty_n_t__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__List__Olist_It__Nat__Onat_J_J,type,
produc2423145444804056984st_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
produc6373727562496775070rm_a_b: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Term__Oterm_Itf__a_Mtf__b_J_J_Mt__Set__Oset_It__Term__Oterm_Itf__a_Mtf__b_J_J_J_J,type,
set_Pr7151917219565317845rm_a_b: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Set__Oset_It__Term__Oterm_Itf__a_Mtf__b_J_J_Mt__Set__Oset_It__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
produc9001763001631046773rm_a_b: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
set_Pr4386577575007340137rm_a_b: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J_J,type,
set_Pr3451248702717554689st_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
produc357393685978478089rm_a_b: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
produc1828647624359046049st_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_J,type,
set_Pr4934435412358123699_a_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
set_set_term_a_b: $tType ).
thf(ty_n_t__Option__Ooption_It__Sum____Type__Osum_Itf__a_Mtf__b_J_J,type,
option_Sum_sum_a_b: $tType ).
thf(ty_n_t__Subterm____and____Context__Octxt_Itf__a_Mtf__b_J,type,
subterm_and_ctxt_a_b: $tType ).
thf(ty_n_t__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J,type,
product_prod_a_nat: $tType ).
thf(ty_n_t__List__Olist_It__Term__Oterm_Itf__a_Mtf__b_J_J,type,
list_term_a_b: $tType ).
thf(ty_n_t__Set__Oset_It__Term__Oterm_Itf__a_Mtf__b_J_J,type,
set_term_a_b: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
set_list_nat: $tType ).
thf(ty_n_t__Term__Oterm_Itf__a_Mtf__b_J,type,
term_a_b: $tType ).
thf(ty_n_t__List__Olist_It__Nat__Onat_J,type,
list_nat: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_tf__b,type,
b: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (282)
thf(sy_c_Abstract__Rewriting_OCR__on_001t__List__Olist_It__Nat__Onat_J,type,
abstra5991784817725020458st_nat: set_Pr3451248702717554689st_nat > set_list_nat > $o ).
thf(sy_c_Abstract__Rewriting_OCR__on_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
abstra7359694574113289279rm_a_b: set_Pr2972776593051762503rm_a_b > set_Pr4386577575007340137rm_a_b > $o ).
thf(sy_c_Abstract__Rewriting_OCR__on_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
abstra8448919418672941150rm_a_b: set_Pr4386577575007340137rm_a_b > set_term_a_b > $o ).
thf(sy_c_Abstract__Rewriting_ONF_001t__List__Olist_It__Nat__Onat_J,type,
abstract_NF_list_nat: set_Pr3451248702717554689st_nat > set_list_nat ).
thf(sy_c_Abstract__Rewriting_ONF_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
abstra1902783451776770197rm_a_b: set_Pr2972776593051762503rm_a_b > set_Pr4386577575007340137rm_a_b ).
thf(sy_c_Abstract__Rewriting_ONF_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
abstract_NF_term_a_b: set_Pr4386577575007340137rm_a_b > set_term_a_b ).
thf(sy_c_Abstract__Rewriting_OSN__on_001t__List__Olist_It__Nat__Onat_J,type,
abstra2262888740781824790st_nat: set_Pr3451248702717554689st_nat > set_list_nat > $o ).
thf(sy_c_Abstract__Rewriting_OSN__on_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
abstra2398554102055911763rm_a_b: set_Pr2972776593051762503rm_a_b > set_Pr4386577575007340137rm_a_b > $o ).
thf(sy_c_Abstract__Rewriting_OSN__on_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
abstra4720023341729745482rm_a_b: set_Pr4386577575007340137rm_a_b > set_term_a_b > $o ).
thf(sy_c_Abstract__Rewriting_OUNC_001t__List__Olist_It__Nat__Onat_J,type,
abstra2192636875637797680st_nat: set_Pr3451248702717554689st_nat > $o ).
thf(sy_c_Abstract__Rewriting_OUNC_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
abstra8480497335821202873rm_a_b: set_Pr2972776593051762503rm_a_b > $o ).
thf(sy_c_Abstract__Rewriting_OUNC_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
abstra4649771476585718372rm_a_b: set_Pr4386577575007340137rm_a_b > $o ).
thf(sy_c_Abstract__Rewriting_OUNF__on_001t__List__Olist_It__Nat__Onat_J,type,
abstra1162721134732043186st_nat: set_Pr3451248702717554689st_nat > set_list_nat > $o ).
thf(sy_c_Abstract__Rewriting_OUNF__on_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
abstra1303798319023342519rm_a_b: set_Pr2972776593051762503rm_a_b > set_Pr4386577575007340137rm_a_b > $o ).
thf(sy_c_Abstract__Rewriting_OUNF__on_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
abstra3619855735679963878rm_a_b: set_Pr4386577575007340137rm_a_b > set_term_a_b > $o ).
thf(sy_c_Abstract__Rewriting_OWCR__on_001t__List__Olist_It__Nat__Onat_J,type,
abstra2569714506102940299st_nat: set_Pr3451248702717554689st_nat > set_list_nat > $o ).
thf(sy_c_Abstract__Rewriting_OWCR__on_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
abstra2346411092049692598st_nat: set_Pr915550674247830855st_nat > set_Pr3451248702717554689st_nat > $o ).
thf(sy_c_Abstract__Rewriting_OWCR__on_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
abstra5817925157630498761rm_a_b: set_Pr2173775114972114941rm_a_b > set_Pr1829413238147913556rm_a_b > $o ).
thf(sy_c_Abstract__Rewriting_OWCR__on_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__List__Olist_It__Nat__Onat_J_J,type,
abstra1867343039937780675st_nat: set_Pr2046385631809063665st_nat > set_Pr2659102613823815246st_nat > $o ).
thf(sy_c_Abstract__Rewriting_OWCR__on_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
abstra875157153669124638rm_a_b: set_Pr2972776593051762503rm_a_b > set_Pr4386577575007340137rm_a_b > $o ).
thf(sy_c_Abstract__Rewriting_OWCR__on_001t__Set__Oset_It__Term__Oterm_Itf__a_Mtf__b_J_J,type,
abstra2457992723331821045rm_a_b: set_Pr7151917219565317845rm_a_b > set_set_term_a_b > $o ).
thf(sy_c_Abstract__Rewriting_OWCR__on_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
abstra5026849107050860991rm_a_b: set_Pr4386577575007340137rm_a_b > set_term_a_b > $o ).
thf(sy_c_Abstract__Rewriting_OWN__on_001t__List__Olist_It__Nat__Onat_J,type,
abstra2187600465986519570st_nat: set_Pr3451248702717554689st_nat > set_list_nat > $o ).
thf(sy_c_Abstract__Rewriting_OWN__on_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
abstra2721873542170697047rm_a_b: set_Pr2972776593051762503rm_a_b > set_Pr4386577575007340137rm_a_b > $o ).
thf(sy_c_Abstract__Rewriting_OWN__on_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
abstra4644735066934440262rm_a_b: set_Pr4386577575007340137rm_a_b > set_term_a_b > $o ).
thf(sy_c_Abstract__Rewriting_Ocommute_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
abstra455195320138238452rm_a_b: set_Pr4386577575007340137rm_a_b > set_Pr4386577575007340137rm_a_b > $o ).
thf(sy_c_Abstract__Rewriting_Oconversion_001t__List__Olist_It__Nat__Onat_J,type,
abstra7896806322651447026st_nat: set_Pr3451248702717554689st_nat > set_Pr3451248702717554689st_nat ).
thf(sy_c_Abstract__Rewriting_Oconversion_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
abstra6125656530262095991rm_a_b: set_Pr2972776593051762503rm_a_b > set_Pr2972776593051762503rm_a_b ).
thf(sy_c_Abstract__Rewriting_Oconversion_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
abstra1130568886744591910rm_a_b: set_Pr4386577575007340137rm_a_b > set_Pr4386577575007340137rm_a_b ).
thf(sy_c_Abstract__Rewriting_Odiamond_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
abstra8151087076625346926rm_a_b: set_Pr4386577575007340137rm_a_b > $o ).
thf(sy_c_Abstract__Rewriting_Ojoin_001t__List__Olist_It__Nat__Onat_J,type,
abstra1638945853619340710st_nat: set_Pr3451248702717554689st_nat > set_Pr3451248702717554689st_nat ).
thf(sy_c_Abstract__Rewriting_Ojoin_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
abstra6360936878533861467st_nat: set_Pr915550674247830855st_nat > set_Pr915550674247830855st_nat ).
thf(sy_c_Abstract__Rewriting_Ojoin_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
abstra4206929023845004132rm_a_b: set_Pr2173775114972114941rm_a_b > set_Pr2173775114972114941rm_a_b ).
thf(sy_c_Abstract__Rewriting_Ojoin_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__List__Olist_It__Nat__Onat_J_J,type,
abstra256346906152286046st_nat: set_Pr2046385631809063665st_nat > set_Pr2046385631809063665st_nat ).
thf(sy_c_Abstract__Rewriting_Ojoin_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
abstra4889682940153293507rm_a_b: set_Pr2972776593051762503rm_a_b > set_Pr2972776593051762503rm_a_b ).
thf(sy_c_Abstract__Rewriting_Ojoin_001t__Set__Oset_It__Term__Oterm_Itf__a_Mtf__b_J_J,type,
abstra2139004588645393680rm_a_b: set_Pr7151917219565317845rm_a_b > set_Pr7151917219565317845rm_a_b ).
thf(sy_c_Abstract__Rewriting_Ojoin_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
abstra4096080454567261402rm_a_b: set_Pr4386577575007340137rm_a_b > set_Pr4386577575007340137rm_a_b ).
thf(sy_c_Abstract__Rewriting_Omeet_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
abstra1093061187967292567rm_a_b: set_Pr4386577575007340137rm_a_b > set_Pr4386577575007340137rm_a_b ).
thf(sy_c_Abstract__Rewriting_Onormalizability_001t__List__Olist_It__Nat__Onat_J,type,
abstra515516630445031992st_nat: set_Pr3451248702717554689st_nat > set_Pr3451248702717554689st_nat ).
thf(sy_c_Abstract__Rewriting_Onormalizability_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
abstra1496011390211842225rm_a_b: set_Pr2972776593051762503rm_a_b > set_Pr2972776593051762503rm_a_b ).
thf(sy_c_Abstract__Rewriting_Onormalizability_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
abstra2972651231392952684rm_a_b: set_Pr4386577575007340137rm_a_b > set_Pr4386577575007340137rm_a_b ).
thf(sy_c_Abstract__Rewriting_Osemi__complete__on_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
abstra7162307272211697846rm_a_b: set_Pr4386577575007340137rm_a_b > set_term_a_b > $o ).
thf(sy_c_Abstract__Rewriting_Osome__NF_001t__List__Olist_It__Nat__Onat_J,type,
abstra132235892672370824st_nat: set_Pr3451248702717554689st_nat > list_nat > list_nat ).
thf(sy_c_Abstract__Rewriting_Osome__NF_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
abstra8203652949708803681rm_a_b: set_Pr2972776593051762503rm_a_b > produc357393685978478089rm_a_b > produc357393685978478089rm_a_b ).
thf(sy_c_Abstract__Rewriting_Osome__NF_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
abstra2589370493620291516rm_a_b: set_Pr4386577575007340137rm_a_b > term_a_b > term_a_b ).
thf(sy_c_Abstract__Rewriting_Ostrongly__confluent__on_001t__List__Olist_It__Nat__Onat_J,type,
abstra3206962040680252735st_nat: set_Pr3451248702717554689st_nat > set_list_nat > $o ).
thf(sy_c_Abstract__Rewriting_Ostrongly__confluent__on_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
abstra837640964651051114rm_a_b: set_Pr2972776593051762503rm_a_b > set_Pr4386577575007340137rm_a_b > $o ).
thf(sy_c_Abstract__Rewriting_Ostrongly__confluent__on_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
abstra5664096641628173427rm_a_b: set_Pr4386577575007340137rm_a_b > set_term_a_b > $o ).
thf(sy_c_Abstract__Rewriting_Othe__NF_001t__List__Olist_It__Nat__Onat_J,type,
abstra9009019697228978191st_nat: set_Pr3451248702717554689st_nat > list_nat > list_nat ).
thf(sy_c_Abstract__Rewriting_Othe__NF_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
abstra8972549633426078106rm_a_b: set_Pr2972776593051762503rm_a_b > produc357393685978478089rm_a_b > produc357393685978478089rm_a_b ).
thf(sy_c_Abstract__Rewriting_Othe__NF_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
abstra2242782261322123075rm_a_b: set_Pr4386577575007340137rm_a_b > term_a_b > term_a_b ).
thf(sy_c_BNF__Cardinal__Order__Relation_Ocofinal_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
bNF_Ca4789820777293777539rm_a_b: set_term_a_b > set_Pr4386577575007340137rm_a_b > $o ).
thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain_001t__List__Olist_It__Nat__Onat_J_001t__Nat__Onat,type,
bNF_Ca1233969902089702230at_nat: set_Pr3451248702717554689st_nat > ( list_nat > nat ) > $o ).
thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain_001t__List__Olist_It__Nat__Onat_J_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
bNF_Ca5626288062746064284st_nat: set_Pr3451248702717554689st_nat > ( list_nat > set_list_nat ) > $o ).
thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain_001t__List__Olist_It__Nat__Onat_J_001t__Set__Oset_It__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
bNF_Ca8662308032288398691rm_a_b: set_Pr3451248702717554689st_nat > ( list_nat > set_Pr4386577575007340137rm_a_b ) > $o ).
thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain_001t__List__Olist_It__Nat__Onat_J_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_J,type,
bNF_Ca3027480494327872057_a_nat: set_Pr3451248702717554689st_nat > ( list_nat > set_Pr4934435412358123699_a_nat ) > $o ).
thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain_001t__List__Olist_It__Nat__Onat_J_001t__Set__Oset_It__Term__Oterm_Itf__a_Mtf__b_J_J,type,
bNF_Ca2286008309882686160rm_a_b: set_Pr3451248702717554689st_nat > ( list_nat > set_term_a_b ) > $o ).
thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_001t__Nat__Onat,type,
bNF_Ca1184762318341890741_b_nat: set_Pr2972776593051762503rm_a_b > ( produc357393685978478089rm_a_b > nat ) > $o ).
thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain_001t__Term__Oterm_Itf__a_Mtf__b_J_001t__Nat__Onat,type,
bNF_Ca2413601583040309898_b_nat: set_Pr4386577575007340137rm_a_b > ( term_a_b > nat ) > $o ).
thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain_001t__Term__Oterm_Itf__a_Mtf__b_J_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
bNF_Ca7485710705408711376st_nat: set_Pr4386577575007340137rm_a_b > ( term_a_b > set_list_nat ) > $o ).
thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain_001t__Term__Oterm_Itf__a_Mtf__b_J_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_J,type,
bNF_Ca9222207453146494317_a_nat: set_Pr4386577575007340137rm_a_b > ( term_a_b > set_Pr4934435412358123699_a_nat ) > $o ).
thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain_001t__Term__Oterm_Itf__a_Mtf__b_J_001t__Set__Oset_It__Term__Oterm_Itf__a_Mtf__b_J_J,type,
bNF_Ca4145430952545333252rm_a_b: set_Pr4386577575007340137rm_a_b > ( term_a_b > set_term_a_b ) > $o ).
thf(sy_c_BNF__Wellorder__Relation_Owo__rel_001t__List__Olist_It__Nat__Onat_J,type,
bNF_We7018074364679882228st_nat: set_Pr3451248702717554689st_nat > $o ).
thf(sy_c_BNF__Wellorder__Relation_Owo__rel_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
bNF_We1688910813892497397rm_a_b: set_Pr2972776593051762503rm_a_b > $o ).
thf(sy_c_BNF__Wellorder__Relation_Owo__rel_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
bNF_We251836928773027112rm_a_b: set_Pr4386577575007340137rm_a_b > $o ).
thf(sy_c_BNF__Wellorder__Relation_Owo__rel_OisMinim_001t__List__Olist_It__Nat__Onat_J,type,
bNF_We7461629510496424106st_nat: set_Pr3451248702717554689st_nat > set_list_nat > list_nat > $o ).
thf(sy_c_BNF__Wellorder__Relation_Owo__rel_OisMinim_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
bNF_We8430659725849320639rm_a_b: set_Pr2972776593051762503rm_a_b > set_Pr4386577575007340137rm_a_b > produc357393685978478089rm_a_b > $o ).
thf(sy_c_BNF__Wellorder__Relation_Owo__rel_OisMinim_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
bNF_We695392074589568990rm_a_b: set_Pr4386577575007340137rm_a_b > set_term_a_b > term_a_b > $o ).
thf(sy_c_BNF__Wellorder__Relation_Owo__rel_Omax2_001t__List__Olist_It__Nat__Onat_J,type,
bNF_We2161624777392673042st_nat: set_Pr3451248702717554689st_nat > list_nat > list_nat > list_nat ).
thf(sy_c_BNF__Wellorder__Relation_Owo__rel_Omax2_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
bNF_We1369222903935127127rm_a_b: set_Pr2972776593051762503rm_a_b > produc357393685978478089rm_a_b > produc357393685978478089rm_a_b > produc357393685978478089rm_a_b ).
thf(sy_c_BNF__Wellorder__Relation_Owo__rel_Omax2_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
bNF_We4618759378340593734rm_a_b: set_Pr4386577575007340137rm_a_b > term_a_b > term_a_b > term_a_b ).
thf(sy_c_BNF__Wellorder__Relation_Owo__rel_Ominim_001t__List__Olist_It__Nat__Onat_J,type,
bNF_We3801768462575224852st_nat: set_Pr3451248702717554689st_nat > set_list_nat > list_nat ).
thf(sy_c_BNF__Wellorder__Relation_Owo__rel_Ominim_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
bNF_We1338740440619677141rm_a_b: set_Pr2972776593051762503rm_a_b > set_Pr4386577575007340137rm_a_b > produc357393685978478089rm_a_b ).
thf(sy_c_BNF__Wellorder__Relation_Owo__rel_Ominim_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
bNF_We6258903063523145544rm_a_b: set_Pr4386577575007340137rm_a_b > set_term_a_b > term_a_b ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
minus_7954133019191499631st_nat: set_list_nat > set_list_nat > set_list_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J_J,type,
minus_4256792079133151976st_nat: set_Pr3451248702717554689st_nat > set_Pr3451248702717554689st_nat > set_Pr3451248702717554689st_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_Mt__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J_J,type,
minus_45395518312058030rm_a_b: set_Pr2972776593051762503rm_a_b > set_Pr2972776593051762503rm_a_b > set_Pr2972776593051762503rm_a_b ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
minus_5192120951422937424rm_a_b: set_Pr4386577575007340137rm_a_b > set_Pr4386577575007340137rm_a_b > set_Pr4386577575007340137rm_a_b ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
inf_inf_set_list_nat: set_list_nat > set_list_nat > set_list_nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
inf_in1693821330460682071rm_a_b: set_Pr4386577575007340137rm_a_b > set_Pr4386577575007340137rm_a_b > set_Pr4386577575007340137rm_a_b ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
sup_sup_set_list_nat: set_list_nat > set_list_nat > set_list_nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J_J,type,
sup_su5841606568262889429st_nat: set_Pr3451248702717554689st_nat > set_Pr3451248702717554689st_nat > set_Pr3451248702717554689st_nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_Mt__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J_J,type,
sup_su1845815533690797339rm_a_b: set_Pr2972776593051762503rm_a_b > set_Pr2972776593051762503rm_a_b > set_Pr2972776593051762503rm_a_b ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
sup_su6776935440552674877rm_a_b: set_Pr4386577575007340137rm_a_b > set_Pr4386577575007340137rm_a_b > set_Pr4386577575007340137rm_a_b ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_J,type,
sup_su459911885395995103_a_nat: set_Pr4934435412358123699_a_nat > set_Pr4934435412358123699_a_nat > set_Pr4934435412358123699_a_nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Term__Oterm_Itf__a_Mtf__b_J_J,type,
sup_sup_set_term_a_b: set_term_a_b > set_term_a_b > set_term_a_b ).
thf(sy_c_List_Oappend_001t__Nat__Onat,type,
append_nat: list_nat > list_nat > list_nat ).
thf(sy_c_List_Olist_ONil_001t__Nat__Onat,type,
nil_nat: list_nat ).
thf(sy_c_List_Olist_ONil_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
nil_term_a_b: list_term_a_b ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Ocompow_001t__Set__Oset_It__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
compow4057154403645558940rm_a_b: nat > set_Pr4386577575007340137rm_a_b > set_Pr4386577575007340137rm_a_b ).
thf(sy_c_Order__Relation_Olinear__order__on_001t__List__Olist_It__Nat__Onat_J,type,
order_2931667645265552619st_nat: set_list_nat > set_Pr3451248702717554689st_nat > $o ).
thf(sy_c_Order__Relation_Olinear__order__on_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
order_681589137112254398rm_a_b: set_Pr4386577575007340137rm_a_b > set_Pr2972776593051762503rm_a_b > $o ).
thf(sy_c_Order__Relation_Olinear__order__on_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
order_5388802246213473311rm_a_b: set_term_a_b > set_Pr4386577575007340137rm_a_b > $o ).
thf(sy_c_Order__Relation_Opartial__order__on_001t__List__Olist_It__Nat__Onat_J,type,
order_1646981276062705451st_nat: set_list_nat > set_Pr3451248702717554689st_nat > $o ).
thf(sy_c_Order__Relation_Opartial__order__on_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
order_8762642138211079550rm_a_b: set_Pr4386577575007340137rm_a_b > set_Pr2972776593051762503rm_a_b > $o ).
thf(sy_c_Order__Relation_Opartial__order__on_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
order_4104115877010626143rm_a_b: set_term_a_b > set_Pr4386577575007340137rm_a_b > $o ).
thf(sy_c_Order__Relation_Opreorder__on_001t__List__Olist_It__Nat__Onat_J,type,
order_4596338698039041673st_nat: set_list_nat > set_Pr3451248702717554689st_nat > $o ).
thf(sy_c_Order__Relation_Opreorder__on_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
order_4024118566430455712rm_a_b: set_Pr4386577575007340137rm_a_b > set_Pr2972776593051762503rm_a_b > $o ).
thf(sy_c_Order__Relation_Opreorder__on_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
order_7053473298986962365rm_a_b: set_term_a_b > set_Pr4386577575007340137rm_a_b > $o ).
thf(sy_c_Order__Relation_OunderS_001t__List__Olist_It__Nat__Onat_J,type,
order_8386007542010951322st_nat: set_Pr3451248702717554689st_nat > list_nat > set_list_nat ).
thf(sy_c_Order__Relation_OunderS_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
order_8163520290499871439rm_a_b: set_Pr2972776593051762503rm_a_b > produc357393685978478089rm_a_b > set_Pr4386577575007340137rm_a_b ).
thf(sy_c_Order__Relation_OunderS_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
order_1619770106104096206rm_a_b: set_Pr4386577575007340137rm_a_b > term_a_b > set_term_a_b ).
thf(sy_c_Order__Relation_Ounder_001t__List__Olist_It__Nat__Onat_J,type,
order_under_list_nat: set_Pr3451248702717554689st_nat > list_nat > set_list_nat ).
thf(sy_c_Order__Relation_Ounder_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
order_346605463002971118rm_a_b: set_Pr2972776593051762503rm_a_b > produc357393685978478089rm_a_b > set_Pr4386577575007340137rm_a_b ).
thf(sy_c_Order__Relation_Ounder_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
order_under_term_a_b: set_Pr4386577575007340137rm_a_b > term_a_b > set_term_a_b ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__List__Olist_It__Nat__Onat_J_M_Eo_J,type,
bot_bot_list_nat_o: list_nat > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J_M_Eo_J,type,
bot_bo2074314105715143856_nat_o: produc1828647624359046049st_nat > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J_M_Eo_J,type,
bot_bo6896037246099296413_a_b_o: produc6373727562496775070rm_a_b > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_Mt__List__Olist_It__Nat__Onat_J_J_M_Eo_J,type,
bot_bo1794539809004266019_nat_o: produc2423145444804056984st_nat > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_M_Eo_J,type,
bot_bo23760184229683272_a_b_o: produc357393685978478089rm_a_b > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Set__Oset_It__Term__Oterm_Itf__a_Mtf__b_J_J_M_Eo_J,type,
bot_bo7197193537753692017_a_b_o: set_term_a_b > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Term__Oterm_Itf__a_Mtf__b_J_M_Eo_J,type,
bot_bot_term_a_b_o: term_a_b > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_Eo,type,
bot_bot_o: $o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
bot_bot_nat: nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
bot_bot_set_list_nat: set_list_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J_J,type,
bot_bo8485564385918328941st_nat: set_Pr3451248702717554689st_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J_J,type,
bot_bo6028291467071837672rm_a_b: set_Pr1829413238147913556rm_a_b ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_Mt__List__Olist_It__Nat__Onat_J_J_J,type,
bot_bo6857980842747739362st_nat: set_Pr2659102613823815246st_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_Mt__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J_J,type,
bot_bo5040873672931234227rm_a_b: set_Pr2972776593051762503rm_a_b ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
bot_bo197521221353338581rm_a_b: set_Pr4386577575007340137rm_a_b ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_J,type,
bot_bo9049108969261143879_a_nat: set_Pr4934435412358123699_a_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
bot_bo7738030261494536212rm_a_b: set_set_term_a_b ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Term__Oterm_Itf__a_Mtf__b_J_J,type,
bot_bot_set_term_a_b: set_term_a_b ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__List__Olist_It__Nat__Onat_J_M_Eo_J,type,
ord_le1520216061033275535_nat_o: ( list_nat > $o ) > ( list_nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_M_Eo_J,type,
ord_le6645574462093738516_a_b_o: ( produc357393685978478089rm_a_b > $o ) > ( produc357393685978478089rm_a_b > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
ord_le6045566169113846134st_nat: set_list_nat > set_list_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J_J,type,
ord_le8406513867147106209st_nat: set_Pr3451248702717554689st_nat > set_Pr3451248702717554689st_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J_J,type,
ord_le4361379431065024436rm_a_b: set_Pr1829413238147913556rm_a_b > set_Pr1829413238147913556rm_a_b > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_Mt__List__Olist_It__Nat__Onat_J_J_J,type,
ord_le5191068806740926126st_nat: set_Pr2659102613823815246st_nat > set_Pr2659102613823815246st_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_Mt__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J_J,type,
ord_le2593958535671617255rm_a_b: set_Pr2972776593051762503rm_a_b > set_Pr2972776593051762503rm_a_b > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
ord_le118470702582115849rm_a_b: set_Pr4386577575007340137rm_a_b > set_Pr4386577575007340137rm_a_b > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_J,type,
ord_le8666007276011122963_a_nat: set_Pr4934435412358123699_a_nat > set_Pr4934435412358123699_a_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
ord_le4920510218098530272rm_a_b: set_set_term_a_b > set_set_term_a_b > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Term__Oterm_Itf__a_Mtf__b_J_J,type,
ord_le2705286416250468010rm_a_b: set_term_a_b > set_term_a_b > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
top_top_set_list_nat: set_list_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
top_to1314267278846557113rm_a_b: set_Pr4386577575007340137rm_a_b ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Term__Oterm_Itf__a_Mtf__b_J_J,type,
top_top_set_term_a_b: set_term_a_b ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J,type,
produc2694037385005941721st_nat: list_nat > list_nat > produc1828647624359046049st_nat ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Nat__Onat_J_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
produc3015861401532503632rm_a_b: list_nat > produc357393685978478089rm_a_b > produc6373727562496775070rm_a_b ).
thf(sy_c_Product__Type_OPair_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
produc988773521843941591st_nat: produc1828647624359046049st_nat > produc1828647624359046049st_nat > produc1342688880896445543st_nat ).
thf(sy_c_Product__Type_OPair_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
produc8750439296677016789rm_a_b: produc6373727562496775070rm_a_b > produc6373727562496775070rm_a_b > produc8411532396478638749rm_a_b ).
thf(sy_c_Product__Type_OPair_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__List__Olist_It__Nat__Onat_J_J_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__List__Olist_It__Nat__Onat_J_J,type,
produc8516103258745939657st_nat: produc2423145444804056984st_nat > produc2423145444804056984st_nat > produc3734095815061405585st_nat ).
thf(sy_c_Product__Type_OPair_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_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,
produc2499998789532207959rm_a_b: produc4523971788519308903rm_a_b > produc4523971788519308903rm_a_b > produc2957090337049255143rm_a_b ).
thf(sy_c_Product__Type_OPair_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_001t__List__Olist_It__Nat__Onat_J,type,
produc5056965718246025362st_nat: produc357393685978478089rm_a_b > list_nat > produc2423145444804056984st_nat ).
thf(sy_c_Product__Type_OPair_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
produc1763473618796451543rm_a_b: produc357393685978478089rm_a_b > produc357393685978478089rm_a_b > produc4523971788519308903rm_a_b ).
thf(sy_c_Product__Type_OPair_001t__Set__Oset_It__Term__Oterm_Itf__a_Mtf__b_J_J_001t__Set__Oset_It__Term__Oterm_Itf__a_Mtf__b_J_J,type,
produc8517751122659631533rm_a_b: set_term_a_b > set_term_a_b > produc9001763001631046773rm_a_b ).
thf(sy_c_Product__Type_OPair_001t__Term__Oterm_Itf__a_Mtf__b_J_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
produc7020197800436672577rm_a_b: term_a_b > term_a_b > produc357393685978478089rm_a_b ).
thf(sy_c_Product__Type_OPair_001tf__a_001t__Nat__Onat,type,
product_Pair_a_nat: a > nat > product_prod_a_nat ).
thf(sy_c_Product__Type_Ocurry_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J_001_Eo,type,
produc6139401228252118475_nat_o: ( produc1828647624359046049st_nat > $o ) > list_nat > list_nat > $o ).
thf(sy_c_Product__Type_Ocurry_001t__List__Olist_It__Nat__Onat_J_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_001_Eo,type,
produc2875039139025540986_a_b_o: ( produc6373727562496775070rm_a_b > $o ) > list_nat > produc357393685978478089rm_a_b > $o ).
thf(sy_c_Product__Type_Ocurry_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_001t__List__Olist_It__Nat__Onat_J_001_Eo,type,
produc4172631894931527864_nat_o: ( produc2423145444804056984st_nat > $o ) > produc357393685978478089rm_a_b > list_nat > $o ).
thf(sy_c_Product__Type_Ocurry_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_001_Eo,type,
produc3713113261896612557_a_b_o: ( produc4523971788519308903rm_a_b > $o ) > produc357393685978478089rm_a_b > produc357393685978478089rm_a_b > $o ).
thf(sy_c_Product__Type_Ocurry_001t__Term__Oterm_Itf__a_Mtf__b_J_001t__Term__Oterm_Itf__a_Mtf__b_J_001_Eo,type,
produc6008550485034061667_a_b_o: ( produc357393685978478089rm_a_b > $o ) > term_a_b > term_a_b > $o ).
thf(sy_c_Relation_OField_001t__List__Olist_It__Nat__Onat_J,type,
field_list_nat: set_Pr3451248702717554689st_nat > set_list_nat ).
thf(sy_c_Relation_OField_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
field_6884932134483023318rm_a_b: set_Pr2972776593051762503rm_a_b > set_Pr4386577575007340137rm_a_b ).
thf(sy_c_Relation_OField_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
field_term_a_b: set_Pr4386577575007340137rm_a_b > set_term_a_b ).
thf(sy_c_Relation_OId_001t__List__Olist_It__Nat__Onat_J,type,
id_list_nat: set_Pr3451248702717554689st_nat ).
thf(sy_c_Relation_OId_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
id_Pro3488709451141449061rm_a_b: set_Pr2972776593051762503rm_a_b ).
thf(sy_c_Relation_OId_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
id_term_a_b: set_Pr4386577575007340137rm_a_b ).
thf(sy_c_Relation_OImage_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J,type,
image_2597627202720054805st_nat: set_Pr3451248702717554689st_nat > set_list_nat > set_list_nat ).
thf(sy_c_Relation_OImage_001t__List__Olist_It__Nat__Onat_J_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
image_6412611479114034964rm_a_b: set_Pr1829413238147913556rm_a_b > set_list_nat > set_Pr4386577575007340137rm_a_b ).
thf(sy_c_Relation_OImage_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_001t__List__Olist_It__Nat__Onat_J,type,
image_8453715795827556694st_nat: set_Pr2659102613823815246st_nat > set_Pr4386577575007340137rm_a_b > set_list_nat ).
thf(sy_c_Relation_OImage_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
image_871640889055226643rm_a_b: set_Pr2972776593051762503rm_a_b > set_Pr4386577575007340137rm_a_b > set_Pr4386577575007340137rm_a_b ).
thf(sy_c_Relation_OImage_001t__Term__Oterm_Itf__a_Mtf__b_J_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
image_6923787618150785661rm_a_b: set_Pr4386577575007340137rm_a_b > set_term_a_b > set_term_a_b ).
thf(sy_c_Relation_Oantisym__on_001t__List__Olist_It__Nat__Onat_J,type,
antisym_on_list_nat: set_list_nat > set_Pr3451248702717554689st_nat > $o ).
thf(sy_c_Relation_Oantisym__on_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
antisy5041284635856308844rm_a_b: set_Pr4386577575007340137rm_a_b > set_Pr2972776593051762503rm_a_b > $o ).
thf(sy_c_Relation_Oantisym__on_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
antisym_on_term_a_b: set_term_a_b > set_Pr4386577575007340137rm_a_b > $o ).
thf(sy_c_Relation_Oconverse_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J,type,
conver3282399561507114237st_nat: set_Pr3451248702717554689st_nat > set_Pr3451248702717554689st_nat ).
thf(sy_c_Relation_Oconverse_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
conver1575690562132235771rm_a_b: set_Pr2972776593051762503rm_a_b > set_Pr2972776593051762503rm_a_b ).
thf(sy_c_Relation_Oconverse_001t__Term__Oterm_Itf__a_Mtf__b_J_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
conver7608559976937845093rm_a_b: set_Pr4386577575007340137rm_a_b > set_Pr4386577575007340137rm_a_b ).
thf(sy_c_Relation_Orefl__on_001t__List__Olist_It__Nat__Onat_J,type,
refl_on_list_nat: set_list_nat > set_Pr3451248702717554689st_nat > $o ).
thf(sy_c_Relation_Orefl__on_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
refl_o5344404488616963464rm_a_b: set_Pr4386577575007340137rm_a_b > set_Pr2972776593051762503rm_a_b > $o ).
thf(sy_c_Relation_Orefl__on_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
refl_on_term_a_b: set_term_a_b > set_Pr4386577575007340137rm_a_b > $o ).
thf(sy_c_Relation_Orelcomp_001t__Term__Oterm_Itf__a_Mtf__b_J_001t__Term__Oterm_Itf__a_Mtf__b_J_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
relcom370159955682700863rm_a_b: set_Pr4386577575007340137rm_a_b > set_Pr4386577575007340137rm_a_b > set_Pr4386577575007340137rm_a_b ).
thf(sy_c_Relation_Ototal__on_001t__List__Olist_It__Nat__Onat_J,type,
total_on_list_nat: set_list_nat > set_Pr3451248702717554689st_nat > $o ).
thf(sy_c_Relation_Ototal__on_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
total_4010809703584253837rm_a_b: set_Pr4386577575007340137rm_a_b > set_Pr2972776593051762503rm_a_b > $o ).
thf(sy_c_Relation_Ototal__on_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
total_on_term_a_b: set_term_a_b > set_Pr4386577575007340137rm_a_b > $o ).
thf(sy_c_Relation_Otrans__on_001t__List__Olist_It__Nat__Onat_J,type,
trans_on_list_nat: set_list_nat > set_Pr3451248702717554689st_nat > $o ).
thf(sy_c_Relation_Otrans__on_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
trans_5404704112166290345rm_a_b: set_Pr4386577575007340137rm_a_b > set_Pr2972776593051762503rm_a_b > $o ).
thf(sy_c_Relation_Otrans__on_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
trans_on_term_a_b: set_term_a_b > set_Pr4386577575007340137rm_a_b > $o ).
thf(sy_c_Rewriting_Orrstep_001tf__a_001tf__b,type,
rrstep_a_b: set_Pr4386577575007340137rm_a_b > set_Pr4386577575007340137rm_a_b ).
thf(sy_c_Rewriting_Orstep_001tf__a_001tf__b,type,
rstep_a_b: set_Pr4386577575007340137rm_a_b > set_Pr4386577575007340137rm_a_b ).
thf(sy_c_Rewriting_Osig__step_001tf__a_001tf__b,type,
sig_step_a_b: set_Pr4934435412358123699_a_nat > set_Pr4386577575007340137rm_a_b > set_Pr4386577575007340137rm_a_b ).
thf(sy_c_Rewriting_Osrsteps__with__root__step_001tf__a_001tf__b,type,
srstep7844470518422762656ep_a_b: set_Pr4934435412358123699_a_nat > set_Pr4386577575007340137rm_a_b > set_Pr4386577575007340137rm_a_b ).
thf(sy_c_Rewriting__Properties_OCE__on_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
rewrit660295756789745446rm_a_b: set_Pr4386577575007340137rm_a_b > set_Pr4386577575007340137rm_a_b > set_term_a_b > $o ).
thf(sy_c_Rewriting__Properties_ONE__on_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
rewrit7370782028743737947rm_a_b: set_Pr4386577575007340137rm_a_b > set_Pr4386577575007340137rm_a_b > set_term_a_b > $o ).
thf(sy_c_Rewriting__Properties_ONE__redp_001tf__a_001tf__b,type,
rewrit3451484132325884455dp_a_b: set_Pr4934435412358123699_a_nat > set_Pr4386577575007340137rm_a_b > set_Pr4386577575007340137rm_a_b > term_a_b > term_a_b > $o ).
thf(sy_c_Rewriting__Properties_ONFP__on_001t__List__Olist_It__Nat__Onat_J,type,
rewrit3847575380561510056st_nat: set_Pr3451248702717554689st_nat > set_list_nat > $o ).
thf(sy_c_Rewriting__Properties_ONFP__on_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
rewrit2138790387785322561rm_a_b: set_Pr2972776593051762503rm_a_b > set_Pr4386577575007340137rm_a_b > $o ).
thf(sy_c_Rewriting__Properties_ONFP__on_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
rewrit6304709981509430748rm_a_b: set_Pr4386577575007340137rm_a_b > set_term_a_b > $o ).
thf(sy_c_Rewriting__Properties_ONFP__redp_001tf__a_001tf__b,type,
rewrit232733863534887208dp_a_b: set_Pr4934435412358123699_a_nat > set_Pr4386577575007340137rm_a_b > term_a_b > term_a_b > $o ).
thf(sy_c_Rewriting__Properties_OSCR__on_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
rewrit4966517460798004134rm_a_b: set_Pr4386577575007340137rm_a_b > set_term_a_b > $o ).
thf(sy_c_Rewriting__Properties_OUN__redp_001tf__a_001tf__b,type,
rewrit983855061078472681dp_a_b: set_Pr4934435412358123699_a_nat > set_Pr4386577575007340137rm_a_b > term_a_b > term_a_b > $o ).
thf(sy_c_Rewriting__Properties_Ocommute__redp_001tf__a_001tf__b,type,
rewrit8865997592796547114dp_a_b: set_Pr4934435412358123699_a_nat > set_Pr4386577575007340137rm_a_b > set_Pr4386577575007340137rm_a_b > term_a_b > term_a_b > $o ).
thf(sy_c_Set_OCollect_001t__List__Olist_It__Nat__Onat_J,type,
collect_list_nat: ( list_nat > $o ) > set_list_nat ).
thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
collec1570431334306492044st_nat: ( produc1828647624359046049st_nat > $o ) > set_Pr3451248702717554689st_nat ).
thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
collec2724650864453038195rm_a_b: ( produc6373727562496775070rm_a_b > $o ) > set_Pr1829413238147913556rm_a_b ).
thf(sy_c_Set_OCollect_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__List__Olist_It__Nat__Onat_J_J,type,
collec7997440783615095917st_nat: ( produc2423145444804056984st_nat > $o ) > set_Pr2659102613823815246st_nat ).
thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
collec99177395925924084rm_a_b: ( produc357393685978478089rm_a_b > $o ) > set_Pr4386577575007340137rm_a_b ).
thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J,type,
collec4464134535221767506_a_nat: ( product_prod_a_nat > $o ) > set_Pr4934435412358123699_a_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Term__Oterm_Itf__a_Mtf__b_J_J,type,
collect_set_term_a_b: ( set_term_a_b > $o ) > set_set_term_a_b ).
thf(sy_c_Set_OCollect_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
collect_term_a_b: ( term_a_b > $o ) > set_term_a_b ).
thf(sy_c_Set_Oinsert_001t__List__Olist_It__Nat__Onat_J,type,
insert_list_nat: list_nat > set_list_nat > set_list_nat ).
thf(sy_c_Set_Oinsert_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
insert7009541432154983385rm_a_b: produc357393685978478089rm_a_b > set_Pr4386577575007340137rm_a_b > set_Pr4386577575007340137rm_a_b ).
thf(sy_c_Set_Oinsert_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
insert_term_a_b: term_a_b > set_term_a_b > set_term_a_b ).
thf(sy_c_Set_Ois__empty_001t__List__Olist_It__Nat__Onat_J,type,
is_empty_list_nat: set_list_nat > $o ).
thf(sy_c_Set_Ois__empty_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
is_emp1328655501402583003rm_a_b: set_Pr4386577575007340137rm_a_b > $o ).
thf(sy_c_Set_Ois__empty_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
is_empty_term_a_b: set_term_a_b > $o ).
thf(sy_c_Set_Ois__singleton_001t__List__Olist_It__Nat__Onat_J,type,
is_sin2641923865335537900st_nat: set_list_nat > $o ).
thf(sy_c_Set_Ois__singleton_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
is_sin332545968047684349rm_a_b: set_Pr4386577575007340137rm_a_b > $o ).
thf(sy_c_Set_Oremove_001t__List__Olist_It__Nat__Onat_J,type,
remove_list_nat: list_nat > set_list_nat > set_list_nat ).
thf(sy_c_Set_Oremove_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
remove8523906646656265582rm_a_b: produc357393685978478089rm_a_b > set_Pr4386577575007340137rm_a_b > set_Pr4386577575007340137rm_a_b ).
thf(sy_c_Subterm__and__Context_Octxt__apply__term_001tf__a_001tf__b,type,
subter2376574525758040790rm_a_b: subterm_and_ctxt_a_b > term_a_b > term_a_b ).
thf(sy_c_Subterm__and__Context_Osubst__apply__ctxt_001tf__a_001tf__b_001tf__b,type,
subter4086027020339385547_a_b_b: subterm_and_ctxt_a_b > ( b > term_a_b ) > subterm_and_ctxt_a_b ).
thf(sy_c_Subterm__and__Context_Osupt_001tf__a_001tf__b,type,
subterm_and_supt_a_b: set_Pr4386577575007340137rm_a_b ).
thf(sy_c_Subterm__and__Context_Osupteq_001tf__a_001tf__b,type,
subter523971068842742411eq_a_b: set_Pr4386577575007340137rm_a_b ).
thf(sy_c_Term_Osubst_001tf__b_001tf__a,type,
subst_b_a: b > term_a_b > b > term_a_b ).
thf(sy_c_Term_Osubst__apply__term_001tf__a_001tf__b_001tf__b,type,
subst_7999470309526761004_a_b_b: term_a_b > ( b > term_a_b ) > term_a_b ).
thf(sy_c_Term_Oterm_OFun_001tf__a_001tf__b,type,
fun_a_b: a > list_term_a_b > term_a_b ).
thf(sy_c_Term__Context_Octxt__at__pos_001tf__a_001tf__b,type,
term_ctxt_at_pos_a_b: term_a_b > list_nat > subterm_and_ctxt_a_b ).
thf(sy_c_Term__Context_Ofun__at_001tf__a_001tf__b,type,
term_fun_at_a_b: term_a_b > list_nat > option_Sum_sum_a_b ).
thf(sy_c_Term__Context_Ofunas__ctxt_001tf__a_001tf__b,type,
term_funas_ctxt_a_b: subterm_and_ctxt_a_b > set_Pr4934435412358123699_a_nat ).
thf(sy_c_Term__Context_Ofunas__term_001tf__a_001tf__b,type,
term_funas_term_a_b: term_a_b > set_Pr4934435412358123699_a_nat ).
thf(sy_c_Term__Context_Ohole__pos_001tf__a_001tf__b,type,
term_hole_pos_a_b: subterm_and_ctxt_a_b > list_nat ).
thf(sy_c_Term__Context_Opos__diff_001t__Nat__Onat,type,
term_pos_diff_nat: list_nat > list_nat > list_nat ).
thf(sy_c_Term__Context_Oposition__less__eq_001t__Nat__Onat,type,
term_p3503116865373065078eq_nat: list_nat > list_nat > $o ).
thf(sy_c_Term__Context_Oposition__par_001t__Nat__Onat,type,
term_p5017330785391824242ar_nat: list_nat > list_nat > $o ).
thf(sy_c_Term__Context_Oposs_001tf__a_001tf__b,type,
term_poss_a_b: term_a_b > set_list_nat ).
thf(sy_c_Term__Context_Oreplace__term__at_001tf__a_001tf__b,type,
term_r6860082780075436317at_a_b: term_a_b > list_nat > term_a_b > term_a_b ).
thf(sy_c_Term__Context_Osubt__at_001tf__a_001tf__b,type,
term_subt_at_a_b: term_a_b > list_nat > term_a_b ).
thf(sy_c_Terms__Positions_Oall__ctxt__closed_001tf__a_001tf__b,type,
terms_5226143800768910156ed_a_b: set_Pr4934435412358123699_a_nat > set_Pr4386577575007340137rm_a_b > $o ).
thf(sy_c_Terms__Positions_Ofunas__rel_001tf__a_001tf__b_001tf__b,type,
terms_7988297476397195622_a_b_b: set_Pr4386577575007340137rm_a_b > set_Pr4934435412358123699_a_nat ).
thf(sy_c_Terms__Positions_Oinv__const__ctxt_001tf__a_001tf__b,type,
terms_3799363064701517935xt_a_b: set_Pr4934435412358123699_a_nat > b > subterm_and_ctxt_a_b > subterm_and_ctxt_a_b ).
thf(sy_c_Terms__Positions_Oposs__of__term_001tf__a_001tf__b,type,
terms_7168686267159881682rm_a_b: term_a_b > term_a_b > set_list_nat ).
thf(sy_c_Terms__Positions_Opossc_001tf__a_001tf__b,type,
terms_possc_a_b: subterm_and_ctxt_a_b > set_list_nat ).
thf(sy_c_Terms__Positions_Oreplace__term__context__at_001tf__a_001tf__b,type,
terms_4774307173741787698at_a_b: subterm_and_ctxt_a_b > list_nat > term_a_b > subterm_and_ctxt_a_b ).
thf(sy_c_Terms__Positions_Oterm__to__sig_001tf__a_001tf__b,type,
terms_8519481630511763164ig_a_b: set_Pr4934435412358123699_a_nat > b > term_a_b > term_a_b ).
thf(sy_c_Transitive__Closure_Oacyclic_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
transi5314701259734769157rm_a_b: set_Pr4386577575007340137rm_a_b > $o ).
thf(sy_c_Transitive__Closure_Ortrancl_001t__List__Olist_It__Nat__Onat_J,type,
transi5285580207609517981st_nat: set_Pr3451248702717554689st_nat > set_Pr3451248702717554689st_nat ).
thf(sy_c_Transitive__Closure_Ortrancl_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
transi4087063297364960548st_nat: set_Pr915550674247830855st_nat > set_Pr915550674247830855st_nat ).
thf(sy_c_Transitive__Closure_Ortrancl_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
transi2844406447684470747rm_a_b: set_Pr2173775114972114941rm_a_b > set_Pr2173775114972114941rm_a_b ).
thf(sy_c_Transitive__Closure_Ortrancl_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
transi8117196366846528469st_nat: set_Pr2046385631809063665st_nat > set_Pr2046385631809063665st_nat ).
thf(sy_c_Transitive__Closure_Ortrancl_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_Mt__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
transi5332651807426350186rm_a_b: set_Pr6848273517546626247rm_a_b > set_Pr6848273517546626247rm_a_b ).
thf(sy_c_Transitive__Closure_Ortrancl_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
transi2615809358984392588rm_a_b: set_Pr2972776593051762503rm_a_b > set_Pr2972776593051762503rm_a_b ).
thf(sy_c_Transitive__Closure_Ortrancl_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
transi7742714808557438673rm_a_b: set_Pr4386577575007340137rm_a_b > set_Pr4386577575007340137rm_a_b ).
thf(sy_c_Transitive__Closure_Otrancl_001t__List__Olist_It__Nat__Onat_J,type,
transi5465639037617667199st_nat: set_Pr3451248702717554689st_nat > set_Pr3451248702717554689st_nat ).
thf(sy_c_Transitive__Closure_Otrancl_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
transi1459384740847974082st_nat: set_Pr915550674247830855st_nat > set_Pr915550674247830855st_nat ).
thf(sy_c_Transitive__Closure_Otrancl_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
transi4457611186189819837rm_a_b: set_Pr2173775114972114941rm_a_b > set_Pr2173775114972114941rm_a_b ).
thf(sy_c_Transitive__Closure_Otrancl_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
transi507029068497101751st_nat: set_Pr2046385631809063665st_nat > set_Pr2046385631809063665st_nat ).
thf(sy_c_Transitive__Closure_Otrancl_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_Mt__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
transi4560671716973800200rm_a_b: set_Pr6848273517546626247rm_a_b > set_Pr6848273517546626247rm_a_b ).
thf(sy_c_Transitive__Closure_Otrancl_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
transi9211502839322181930rm_a_b: set_Pr2972776593051762503rm_a_b > set_Pr2972776593051762503rm_a_b ).
thf(sy_c_Transitive__Closure_Otrancl_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
transi7922773638565587891rm_a_b: set_Pr4386577575007340137rm_a_b > set_Pr4386577575007340137rm_a_b ).
thf(sy_c_Wellfounded_Olex__prod_001t__Term__Oterm_Itf__a_Mtf__b_J_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
lex_pr6163557265797435481rm_a_b: set_Pr4386577575007340137rm_a_b > set_Pr4386577575007340137rm_a_b > set_Pr2972776593051762503rm_a_b ).
thf(sy_c_Wellfounded_Owf_001t__List__Olist_It__Nat__Onat_J,type,
wf_list_nat: set_Pr3451248702717554689st_nat > $o ).
thf(sy_c_Wellfounded_Owf_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
wf_Pro2335863617654816626rm_a_b: set_Pr2972776593051762503rm_a_b > $o ).
thf(sy_c_Wellfounded_Owf_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
wf_term_a_b: set_Pr4386577575007340137rm_a_b > $o ).
thf(sy_c_Wfrec_Osame__fst_001t__Term__Oterm_Itf__a_Mtf__b_J_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
same_f843731131676418198rm_a_b: ( term_a_b > $o ) > ( term_a_b > set_Pr4386577575007340137rm_a_b ) > set_Pr2972776593051762503rm_a_b ).
thf(sy_c_Zorn_OChains_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
chains_term_a_b: set_Pr4386577575007340137rm_a_b > set_set_term_a_b ).
thf(sy_c_member_001t__List__Olist_It__Nat__Onat_J,type,
member_list_nat: list_nat > set_list_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
member7340969449405702474st_nat: produc1828647624359046049st_nat > set_Pr3451248702717554689st_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
member4140859439665155893rm_a_b: produc6373727562496775070rm_a_b > set_Pr1829413238147913556rm_a_b > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J_J,type,
member5236317644330119056st_nat: produc1342688880896445543st_nat > set_Pr915550674247830855st_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J_J,type,
member6865565617957190470rm_a_b: produc8411532396478638749rm_a_b > set_Pr2173775114972114941rm_a_b > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_Mt__List__Olist_It__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_Mt__List__Olist_It__Nat__Onat_J_J_J,type,
member2188129036539957306st_nat: produc3734095815061405585st_nat > set_Pr2046385631809063665st_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_Mt__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_Mt__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J_J,type,
member5793113912742238992rm_a_b: produc2957090337049255143rm_a_b > set_Pr6848273517546626247rm_a_b > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
member190277321972437807st_nat: produc2423145444804056984st_nat > set_Pr2659102613823815246st_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_Mt__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
member8417600551952982416rm_a_b: produc4523971788519308903rm_a_b > set_Pr2972776593051762503rm_a_b > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Term__Oterm_Itf__a_Mtf__b_J_J_Mt__Set__Oset_It__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
member8888773126115863070rm_a_b: produc9001763001631046773rm_a_b > set_Pr7151917219565317845rm_a_b > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
member5869715511025134514rm_a_b: produc357393685978478089rm_a_b > set_Pr4386577575007340137rm_a_b > $o ).
thf(sy_c_member_001t__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__Set__Oset_It__Term__Oterm_Itf__a_Mtf__b_J_J,type,
member_set_term_a_b: set_term_a_b > set_set_term_a_b > $o ).
thf(sy_c_member_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
member_term_a_b: term_a_b > set_term_a_b > $o ).
thf(sy_v__092_060F_062,type,
f: set_Pr4934435412358123699_a_nat ).
thf(sy_v__092_060R_062,type,
r: set_Pr4386577575007340137rm_a_b ).
thf(sy_v_s____,type,
s: term_a_b ).
thf(sy_v_t____,type,
t: term_a_b ).
thf(sy_v_thesis____,type,
thesis: $o ).
thf(sy_v_u____,type,
u: term_a_b ).
% Relevant facts (1275)
thf(fact_0__092_060open_062_Is_M_At_J_A_092_060in_062_Arstep_A_092_060R_062_A_092_060Longrightarrow_062_A_092_060exists_062p_Al_Ar_A_092_060sigma_062_O_Ap_A_092_060in_062_Aposs_As_A_092_060and_062_A_Il_M_Ar_J_A_092_060in_062_A_092_060R_062_A_092_060and_062_As_A_124___Ap_A_061_Al_A_092_060cdot_062_A_092_060sigma_062_A_092_060and_062_At_A_061_As_091p_A_092_060leftarrow_062_Ar_A_092_060cdot_062_A_092_060sigma_062_093_092_060close_062,axiom,
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ s @ t ) @ ( rstep_a_b @ r ) )
=> ? [P: list_nat,L: term_a_b,R: term_a_b,Sigma: b > term_a_b] :
( ( member_list_nat @ P @ ( term_poss_a_b @ s ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ L @ R ) @ r )
& ( ( term_subt_at_a_b @ s @ P )
= ( subst_7999470309526761004_a_b_b @ L @ Sigma ) )
& ( t
= ( term_r6860082780075436317at_a_b @ s @ P @ ( subst_7999470309526761004_a_b_b @ R @ Sigma ) ) ) ) ) ).
% \<open>(s, t) \<in> rstep \<R> \<Longrightarrow> \<exists>p l r \<sigma>. p \<in> poss s \<and> (l, r) \<in> \<R> \<and> s |_ p = l \<cdot> \<sigma> \<and> t = s[p \<leftarrow> r \<cdot> \<sigma>]\<close>
thf(fact_1_replace__term__at__subt__at__id,axiom,
! [S: term_a_b,P2: list_nat] :
( ( term_r6860082780075436317at_a_b @ S @ P2 @ ( term_subt_at_a_b @ S @ P2 ) )
= S ) ).
% replace_term_at_subt_at_id
thf(fact_2_replace__term__at__not__poss,axiom,
! [P2: list_nat,S: term_a_b,T: term_a_b] :
( ~ ( member_list_nat @ P2 @ ( term_poss_a_b @ S ) )
=> ( ( term_r6860082780075436317at_a_b @ S @ P2 @ T )
= S ) ) ).
% replace_term_at_not_poss
thf(fact_3_subst__subt__at__dist,axiom,
! [P2: list_nat,S: term_a_b,Sigma2: b > term_a_b] :
( ( member_list_nat @ P2 @ ( term_poss_a_b @ S ) )
=> ( ( term_subt_at_a_b @ ( subst_7999470309526761004_a_b_b @ S @ Sigma2 ) @ P2 )
= ( subst_7999470309526761004_a_b_b @ ( term_subt_at_a_b @ S @ P2 ) @ Sigma2 ) ) ) ).
% subst_subt_at_dist
thf(fact_4_local_Ostep_I1_J,axiom,
member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ s @ t ) @ ( sig_step_a_b @ f @ ( rstep_a_b @ r ) ) ).
% local.step(1)
thf(fact_5_replace__term__at__same__pos,axiom,
! [S: term_a_b,P2: list_nat,U: term_a_b,T: term_a_b] :
( ( term_r6860082780075436317at_a_b @ ( term_r6860082780075436317at_a_b @ S @ P2 @ U ) @ P2 @ T )
= ( term_r6860082780075436317at_a_b @ S @ P2 @ T ) ) ).
% replace_term_at_same_pos
thf(fact_6_pos__replace__to__rstep,axiom,
! [P2: list_nat,S: term_a_b,L2: term_a_b,R2: term_a_b,R3: set_Pr4386577575007340137rm_a_b,Sigma2: b > term_a_b,T: term_a_b] :
( ( member_list_nat @ P2 @ ( term_poss_a_b @ S ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ L2 @ R2 ) @ R3 )
=> ( ( ( term_subt_at_a_b @ S @ P2 )
= ( subst_7999470309526761004_a_b_b @ L2 @ Sigma2 ) )
=> ( ( T
= ( term_r6860082780075436317at_a_b @ S @ P2 @ ( subst_7999470309526761004_a_b_b @ R2 @ Sigma2 ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( rstep_a_b @ R3 ) ) ) ) ) ) ).
% pos_replace_to_rstep
thf(fact_7_rstep__to__pos__replace,axiom,
! [S: term_a_b,T: term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( rstep_a_b @ R3 ) )
=> ? [P: list_nat,L: term_a_b,R: term_a_b,Sigma: b > term_a_b] :
( ( member_list_nat @ P @ ( term_poss_a_b @ S ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ L @ R ) @ R3 )
& ( ( term_subt_at_a_b @ S @ P )
= ( subst_7999470309526761004_a_b_b @ L @ Sigma ) )
& ( T
= ( term_r6860082780075436317at_a_b @ S @ P @ ( subst_7999470309526761004_a_b_b @ R @ Sigma ) ) ) ) ) ).
% rstep_to_pos_replace
thf(fact_8_prod_Oinject,axiom,
! [X1: produc357393685978478089rm_a_b,X2: produc357393685978478089rm_a_b,Y1: produc357393685978478089rm_a_b,Y2: produc357393685978478089rm_a_b] :
( ( ( produc1763473618796451543rm_a_b @ X1 @ X2 )
= ( produc1763473618796451543rm_a_b @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_9_prod_Oinject,axiom,
! [X1: list_nat,X2: list_nat,Y1: list_nat,Y2: list_nat] :
( ( ( produc2694037385005941721st_nat @ X1 @ X2 )
= ( produc2694037385005941721st_nat @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_10_prod_Oinject,axiom,
! [X1: produc357393685978478089rm_a_b,X2: list_nat,Y1: produc357393685978478089rm_a_b,Y2: list_nat] :
( ( ( produc5056965718246025362st_nat @ X1 @ X2 )
= ( produc5056965718246025362st_nat @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_11_prod_Oinject,axiom,
! [X1: list_nat,X2: produc357393685978478089rm_a_b,Y1: list_nat,Y2: produc357393685978478089rm_a_b] :
( ( ( produc3015861401532503632rm_a_b @ X1 @ X2 )
= ( produc3015861401532503632rm_a_b @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_12_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_13_old_Oprod_Oinject,axiom,
! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,A2: produc357393685978478089rm_a_b,B2: produc357393685978478089rm_a_b] :
( ( ( produc1763473618796451543rm_a_b @ A @ B )
= ( produc1763473618796451543rm_a_b @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_14_old_Oprod_Oinject,axiom,
! [A: list_nat,B: list_nat,A2: list_nat,B2: list_nat] :
( ( ( produc2694037385005941721st_nat @ A @ B )
= ( produc2694037385005941721st_nat @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_15_old_Oprod_Oinject,axiom,
! [A: produc357393685978478089rm_a_b,B: list_nat,A2: produc357393685978478089rm_a_b,B2: list_nat] :
( ( ( produc5056965718246025362st_nat @ A @ B )
= ( produc5056965718246025362st_nat @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_16_old_Oprod_Oinject,axiom,
! [A: list_nat,B: produc357393685978478089rm_a_b,A2: list_nat,B2: produc357393685978478089rm_a_b] :
( ( ( produc3015861401532503632rm_a_b @ A @ B )
= ( produc3015861401532503632rm_a_b @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_17_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_18_pos__replace__at__pres,axiom,
! [P2: list_nat,S: term_a_b,T: term_a_b] :
( ( member_list_nat @ P2 @ ( term_poss_a_b @ S ) )
=> ( member_list_nat @ P2 @ ( term_poss_a_b @ ( term_r6860082780075436317at_a_b @ S @ P2 @ T ) ) ) ) ).
% pos_replace_at_pres
thf(fact_19_local_Ostep_I2_J,axiom,
member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ s @ u ) @ ( sig_step_a_b @ f @ ( rstep_a_b @ r ) ) ).
% local.step(2)
thf(fact_20_rstep__substI,axiom,
! [S: term_a_b,T: term_a_b,R3: set_Pr4386577575007340137rm_a_b,Sigma2: b > term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( rstep_a_b @ R3 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( subst_7999470309526761004_a_b_b @ S @ Sigma2 ) @ ( subst_7999470309526761004_a_b_b @ T @ Sigma2 ) ) @ ( rstep_a_b @ R3 ) ) ) ).
% rstep_substI
thf(fact_21_rstep__ruleI,axiom,
! [L2: term_a_b,R2: term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ L2 @ R2 ) @ R3 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ L2 @ R2 ) @ ( rstep_a_b @ R3 ) ) ) ).
% rstep_ruleI
thf(fact_22_prod__induct3,axiom,
! [P3: produc4523971788519308903rm_a_b > $o,X: produc4523971788519308903rm_a_b] :
( ! [A3: produc357393685978478089rm_a_b,B3: term_a_b,C: term_a_b] : ( P3 @ ( produc1763473618796451543rm_a_b @ A3 @ ( produc7020197800436672577rm_a_b @ B3 @ C ) ) )
=> ( P3 @ X ) ) ).
% prod_induct3
thf(fact_23_prod__induct3,axiom,
! [P3: produc6373727562496775070rm_a_b > $o,X: produc6373727562496775070rm_a_b] :
( ! [A3: list_nat,B3: term_a_b,C: term_a_b] : ( P3 @ ( produc3015861401532503632rm_a_b @ A3 @ ( produc7020197800436672577rm_a_b @ B3 @ C ) ) )
=> ( P3 @ X ) ) ).
% prod_induct3
thf(fact_24_prod__cases3,axiom,
! [Y: produc4523971788519308903rm_a_b] :
~ ! [A3: produc357393685978478089rm_a_b,B3: term_a_b,C: term_a_b] :
( Y
!= ( produc1763473618796451543rm_a_b @ A3 @ ( produc7020197800436672577rm_a_b @ B3 @ C ) ) ) ).
% prod_cases3
thf(fact_25_prod__cases3,axiom,
! [Y: produc6373727562496775070rm_a_b] :
~ ! [A3: list_nat,B3: term_a_b,C: term_a_b] :
( Y
!= ( produc3015861401532503632rm_a_b @ A3 @ ( produc7020197800436672577rm_a_b @ B3 @ C ) ) ) ).
% prod_cases3
thf(fact_26_Pair__inject,axiom,
! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,A2: produc357393685978478089rm_a_b,B2: produc357393685978478089rm_a_b] :
( ( ( produc1763473618796451543rm_a_b @ A @ B )
= ( produc1763473618796451543rm_a_b @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_27_Pair__inject,axiom,
! [A: list_nat,B: list_nat,A2: list_nat,B2: list_nat] :
( ( ( produc2694037385005941721st_nat @ A @ B )
= ( produc2694037385005941721st_nat @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_28_Pair__inject,axiom,
! [A: produc357393685978478089rm_a_b,B: list_nat,A2: produc357393685978478089rm_a_b,B2: list_nat] :
( ( ( produc5056965718246025362st_nat @ A @ B )
= ( produc5056965718246025362st_nat @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_29_Pair__inject,axiom,
! [A: list_nat,B: produc357393685978478089rm_a_b,A2: list_nat,B2: produc357393685978478089rm_a_b] :
( ( ( produc3015861401532503632rm_a_b @ A @ B )
= ( produc3015861401532503632rm_a_b @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_30_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_31_prod__cases,axiom,
! [P3: produc4523971788519308903rm_a_b > $o,P2: produc4523971788519308903rm_a_b] :
( ! [A3: produc357393685978478089rm_a_b,B3: produc357393685978478089rm_a_b] : ( P3 @ ( produc1763473618796451543rm_a_b @ A3 @ B3 ) )
=> ( P3 @ P2 ) ) ).
% prod_cases
thf(fact_32_prod__cases,axiom,
! [P3: produc1828647624359046049st_nat > $o,P2: produc1828647624359046049st_nat] :
( ! [A3: list_nat,B3: list_nat] : ( P3 @ ( produc2694037385005941721st_nat @ A3 @ B3 ) )
=> ( P3 @ P2 ) ) ).
% prod_cases
thf(fact_33_prod__cases,axiom,
! [P3: produc2423145444804056984st_nat > $o,P2: produc2423145444804056984st_nat] :
( ! [A3: produc357393685978478089rm_a_b,B3: list_nat] : ( P3 @ ( produc5056965718246025362st_nat @ A3 @ B3 ) )
=> ( P3 @ P2 ) ) ).
% prod_cases
thf(fact_34_prod__cases,axiom,
! [P3: produc6373727562496775070rm_a_b > $o,P2: produc6373727562496775070rm_a_b] :
( ! [A3: list_nat,B3: produc357393685978478089rm_a_b] : ( P3 @ ( produc3015861401532503632rm_a_b @ A3 @ B3 ) )
=> ( P3 @ P2 ) ) ).
% prod_cases
thf(fact_35_prod__cases,axiom,
! [P3: produc357393685978478089rm_a_b > $o,P2: produc357393685978478089rm_a_b] :
( ! [A3: term_a_b,B3: term_a_b] : ( P3 @ ( produc7020197800436672577rm_a_b @ A3 @ B3 ) )
=> ( P3 @ P2 ) ) ).
% prod_cases
thf(fact_36_surj__pair,axiom,
! [P2: produc4523971788519308903rm_a_b] :
? [X3: produc357393685978478089rm_a_b,Y3: produc357393685978478089rm_a_b] :
( P2
= ( produc1763473618796451543rm_a_b @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_37_surj__pair,axiom,
! [P2: produc1828647624359046049st_nat] :
? [X3: list_nat,Y3: list_nat] :
( P2
= ( produc2694037385005941721st_nat @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_38_surj__pair,axiom,
! [P2: produc2423145444804056984st_nat] :
? [X3: produc357393685978478089rm_a_b,Y3: list_nat] :
( P2
= ( produc5056965718246025362st_nat @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_39_surj__pair,axiom,
! [P2: produc6373727562496775070rm_a_b] :
? [X3: list_nat,Y3: produc357393685978478089rm_a_b] :
( P2
= ( produc3015861401532503632rm_a_b @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_40_surj__pair,axiom,
! [P2: produc357393685978478089rm_a_b] :
? [X3: term_a_b,Y3: term_a_b] :
( P2
= ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_41_old_Oprod_Oexhaust,axiom,
! [Y: produc4523971788519308903rm_a_b] :
~ ! [A3: produc357393685978478089rm_a_b,B3: produc357393685978478089rm_a_b] :
( Y
!= ( produc1763473618796451543rm_a_b @ A3 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_42_old_Oprod_Oexhaust,axiom,
! [Y: produc1828647624359046049st_nat] :
~ ! [A3: list_nat,B3: list_nat] :
( Y
!= ( produc2694037385005941721st_nat @ A3 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_43_old_Oprod_Oexhaust,axiom,
! [Y: produc2423145444804056984st_nat] :
~ ! [A3: produc357393685978478089rm_a_b,B3: list_nat] :
( Y
!= ( produc5056965718246025362st_nat @ A3 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_44_old_Oprod_Oexhaust,axiom,
! [Y: produc6373727562496775070rm_a_b] :
~ ! [A3: list_nat,B3: produc357393685978478089rm_a_b] :
( Y
!= ( produc3015861401532503632rm_a_b @ A3 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_45_old_Oprod_Oexhaust,axiom,
! [Y: produc357393685978478089rm_a_b] :
~ ! [A3: term_a_b,B3: term_a_b] :
( Y
!= ( produc7020197800436672577rm_a_b @ A3 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_46_assms,axiom,
! [S: term_a_b,T: term_a_b,U: term_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 @ U ) @ ( sig_step_a_b @ f @ ( rstep_a_b @ r ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T @ U ) @ ( abstra4096080454567261402rm_a_b @ ( sig_step_a_b @ f @ ( rstep_a_b @ r ) ) ) ) ) ) ).
% assms
thf(fact_47_poss__of__term__possI,axiom,
! [P2: list_nat,S: term_a_b,U: term_a_b] :
( ( member_list_nat @ P2 @ ( term_poss_a_b @ S ) )
=> ( ( ( term_subt_at_a_b @ S @ P2 )
= U )
=> ( member_list_nat @ P2 @ ( terms_7168686267159881682rm_a_b @ U @ S ) ) ) ) ).
% poss_of_term_possI
thf(fact_48_par__pos__replace__term__at,axiom,
! [P2: list_nat,S: term_a_b,Q: list_nat,T: term_a_b] :
( ( member_list_nat @ P2 @ ( term_poss_a_b @ S ) )
=> ( ( term_p5017330785391824242ar_nat @ P2 @ Q )
=> ( ( term_subt_at_a_b @ ( term_r6860082780075436317at_a_b @ S @ Q @ T ) @ P2 )
= ( term_subt_at_a_b @ S @ P2 ) ) ) ) ).
% par_pos_replace_term_at
thf(fact_49_poss__of__term__replace__term__at,axiom,
! [P2: list_nat,S: term_a_b,U: term_a_b] :
( ( member_list_nat @ P2 @ ( term_poss_a_b @ S ) )
=> ( member_list_nat @ P2 @ ( terms_7168686267159881682rm_a_b @ U @ ( term_r6860082780075436317at_a_b @ S @ P2 @ U ) ) ) ) ).
% poss_of_term_replace_term_at
thf(fact_50_poss__of__termE,axiom,
! [P2: list_nat,U: term_a_b,S: term_a_b] :
( ( member_list_nat @ P2 @ ( terms_7168686267159881682rm_a_b @ U @ S ) )
=> ~ ( ( member_list_nat @ P2 @ ( term_poss_a_b @ S ) )
=> ( ( term_subt_at_a_b @ S @ P2 )
!= U ) ) ) ).
% poss_of_termE
thf(fact_51_srrstep__to__srestep,axiom,
! [S: term_a_b,T: term_a_b,F: set_Pr4934435412358123699_a_nat,R3: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( sig_step_a_b @ F @ ( rrstep_a_b @ R3 ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( sig_step_a_b @ F @ ( rstep_a_b @ R3 ) ) ) ) ).
% srrstep_to_srestep
thf(fact_52_subt__at__imp__supteq,axiom,
! [P2: list_nat,S: term_a_b] :
( ( member_list_nat @ P2 @ ( term_poss_a_b @ S ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ ( term_subt_at_a_b @ S @ P2 ) ) @ subter523971068842742411eq_a_b ) ) ).
% subt_at_imp_supteq
thf(fact_53_subt__at__subterm__eq,axiom,
! [P2: list_nat,T: term_a_b] :
( ( member_list_nat @ P2 @ ( term_poss_a_b @ T ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T @ ( term_subt_at_a_b @ T @ P2 ) ) @ subter523971068842742411eq_a_b ) ) ).
% subt_at_subterm_eq
thf(fact_54_par__pos__replace__pres,axiom,
! [P2: list_nat,S: term_a_b,Q: list_nat,T: term_a_b] :
( ( member_list_nat @ P2 @ ( term_poss_a_b @ S ) )
=> ( ( term_p5017330785391824242ar_nat @ P2 @ Q )
=> ( member_list_nat @ P2 @ ( term_poss_a_b @ ( term_r6860082780075436317at_a_b @ S @ Q @ T ) ) ) ) ) ).
% par_pos_replace_pres
thf(fact_55_subst__term__eqI,axiom,
! [Sigma2: b > term_a_b,Tau: b > term_a_b] :
( ! [T2: term_a_b] :
( ( subst_7999470309526761004_a_b_b @ T2 @ Sigma2 )
= ( subst_7999470309526761004_a_b_b @ T2 @ Tau ) )
=> ( Sigma2 = Tau ) ) ).
% subst_term_eqI
thf(fact_56_par__not__refl,axiom,
! [P2: list_nat] :
~ ( term_p5017330785391824242ar_nat @ P2 @ P2 ) ).
% par_not_refl
thf(fact_57_rrstep__basicI,axiom,
! [L2: term_a_b,R2: term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ L2 @ R2 ) @ R3 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ L2 @ R2 ) @ ( rrstep_a_b @ R3 ) ) ) ).
% rrstep_basicI
thf(fact_58_parallel__replace__term__commute,axiom,
! [P2: list_nat,Q: list_nat,S: term_a_b,T: term_a_b,U: term_a_b] :
( ( term_p5017330785391824242ar_nat @ P2 @ Q )
=> ( ( term_r6860082780075436317at_a_b @ ( term_r6860082780075436317at_a_b @ S @ P2 @ T ) @ Q @ U )
= ( term_r6860082780075436317at_a_b @ ( term_r6860082780075436317at_a_b @ S @ Q @ U ) @ P2 @ T ) ) ) ).
% parallel_replace_term_commute
thf(fact_59_rrstep_Ocases,axiom,
! [A1: term_a_b,A22: term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A22 ) @ ( rrstep_a_b @ R3 ) )
=> ~ ! [L: term_a_b,R: term_a_b,Sigma: b > term_a_b] :
( ( A1
= ( subst_7999470309526761004_a_b_b @ L @ Sigma ) )
=> ( ( A22
= ( subst_7999470309526761004_a_b_b @ R @ Sigma ) )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ L @ R ) @ R3 ) ) ) ) ).
% rrstep.cases
thf(fact_60_mem__Collect__eq,axiom,
! [A: produc2423145444804056984st_nat,P3: produc2423145444804056984st_nat > $o] :
( ( member190277321972437807st_nat @ A @ ( collec7997440783615095917st_nat @ P3 ) )
= ( P3 @ A ) ) ).
% mem_Collect_eq
thf(fact_61_mem__Collect__eq,axiom,
! [A: produc6373727562496775070rm_a_b,P3: produc6373727562496775070rm_a_b > $o] :
( ( member4140859439665155893rm_a_b @ A @ ( collec2724650864453038195rm_a_b @ P3 ) )
= ( P3 @ A ) ) ).
% mem_Collect_eq
thf(fact_62_mem__Collect__eq,axiom,
! [A: produc1828647624359046049st_nat,P3: produc1828647624359046049st_nat > $o] :
( ( member7340969449405702474st_nat @ A @ ( collec1570431334306492044st_nat @ P3 ) )
= ( P3 @ A ) ) ).
% mem_Collect_eq
thf(fact_63_mem__Collect__eq,axiom,
! [A: term_a_b,P3: term_a_b > $o] :
( ( member_term_a_b @ A @ ( collect_term_a_b @ P3 ) )
= ( P3 @ A ) ) ).
% mem_Collect_eq
thf(fact_64_mem__Collect__eq,axiom,
! [A: set_term_a_b,P3: set_term_a_b > $o] :
( ( member_set_term_a_b @ A @ ( collect_set_term_a_b @ P3 ) )
= ( P3 @ A ) ) ).
% mem_Collect_eq
thf(fact_65_mem__Collect__eq,axiom,
! [A: list_nat,P3: list_nat > $o] :
( ( member_list_nat @ A @ ( collect_list_nat @ P3 ) )
= ( P3 @ A ) ) ).
% mem_Collect_eq
thf(fact_66_mem__Collect__eq,axiom,
! [A: produc357393685978478089rm_a_b,P3: produc357393685978478089rm_a_b > $o] :
( ( member5869715511025134514rm_a_b @ A @ ( collec99177395925924084rm_a_b @ P3 ) )
= ( P3 @ A ) ) ).
% mem_Collect_eq
thf(fact_67_Collect__mem__eq,axiom,
! [A4: set_Pr2659102613823815246st_nat] :
( ( collec7997440783615095917st_nat
@ ^ [X4: produc2423145444804056984st_nat] : ( member190277321972437807st_nat @ X4 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_68_Collect__mem__eq,axiom,
! [A4: set_Pr1829413238147913556rm_a_b] :
( ( collec2724650864453038195rm_a_b
@ ^ [X4: produc6373727562496775070rm_a_b] : ( member4140859439665155893rm_a_b @ X4 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_69_Collect__mem__eq,axiom,
! [A4: set_Pr3451248702717554689st_nat] :
( ( collec1570431334306492044st_nat
@ ^ [X4: produc1828647624359046049st_nat] : ( member7340969449405702474st_nat @ X4 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_70_Collect__mem__eq,axiom,
! [A4: set_term_a_b] :
( ( collect_term_a_b
@ ^ [X4: term_a_b] : ( member_term_a_b @ X4 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_71_Collect__mem__eq,axiom,
! [A4: set_set_term_a_b] :
( ( collect_set_term_a_b
@ ^ [X4: set_term_a_b] : ( member_set_term_a_b @ X4 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_72_Collect__mem__eq,axiom,
! [A4: set_list_nat] :
( ( collect_list_nat
@ ^ [X4: list_nat] : ( member_list_nat @ X4 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_73_Collect__mem__eq,axiom,
! [A4: set_Pr4386577575007340137rm_a_b] :
( ( collec99177395925924084rm_a_b
@ ^ [X4: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ X4 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_74_Collect__cong,axiom,
! [P3: produc357393685978478089rm_a_b > $o,Q2: produc357393685978478089rm_a_b > $o] :
( ! [X3: produc357393685978478089rm_a_b] :
( ( P3 @ X3 )
= ( Q2 @ X3 ) )
=> ( ( collec99177395925924084rm_a_b @ P3 )
= ( collec99177395925924084rm_a_b @ Q2 ) ) ) ).
% Collect_cong
thf(fact_75_Collect__cong,axiom,
! [P3: list_nat > $o,Q2: list_nat > $o] :
( ! [X3: list_nat] :
( ( P3 @ X3 )
= ( Q2 @ X3 ) )
=> ( ( collect_list_nat @ P3 )
= ( collect_list_nat @ Q2 ) ) ) ).
% Collect_cong
thf(fact_76_rrstep_Osimps,axiom,
! [A1: term_a_b,A22: term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A22 ) @ ( rrstep_a_b @ R3 ) )
= ( ? [L3: term_a_b,R4: term_a_b,Sigma3: b > term_a_b] :
( ( A1
= ( subst_7999470309526761004_a_b_b @ L3 @ Sigma3 ) )
& ( A22
= ( subst_7999470309526761004_a_b_b @ R4 @ Sigma3 ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ L3 @ R4 ) @ R3 ) ) ) ) ).
% rrstep.simps
thf(fact_77_rrstep_Ointros,axiom,
! [L2: term_a_b,R2: term_a_b,R3: set_Pr4386577575007340137rm_a_b,Sigma2: b > term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ L2 @ R2 ) @ R3 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( subst_7999470309526761004_a_b_b @ L2 @ Sigma2 ) @ ( subst_7999470309526761004_a_b_b @ R2 @ Sigma2 ) ) @ ( rrstep_a_b @ R3 ) ) ) ).
% rrstep.intros
thf(fact_78_rrstep__subst,axiom,
! [S: term_a_b,T: term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( rrstep_a_b @ R3 ) )
=> ~ ! [L: term_a_b,R: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ L @ R ) @ R3 )
=> ! [Sigma: b > term_a_b] :
( ( S
= ( subst_7999470309526761004_a_b_b @ L @ Sigma ) )
=> ( T
!= ( subst_7999470309526761004_a_b_b @ R @ Sigma ) ) ) ) ) ).
% rrstep_subst
thf(fact_79_subt__at__imp__supteq_H,axiom,
! [P2: list_nat,S: term_a_b,T: term_a_b] :
( ( member_list_nat @ P2 @ ( term_poss_a_b @ S ) )
=> ( ( ( term_subt_at_a_b @ S @ P2 )
= T )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ subter523971068842742411eq_a_b ) ) ) ).
% subt_at_imp_supteq'
thf(fact_80_supteq__subst,axiom,
! [S: term_a_b,T: term_a_b,Sigma2: b > term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ subter523971068842742411eq_a_b )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( subst_7999470309526761004_a_b_b @ S @ Sigma2 ) @ ( subst_7999470309526761004_a_b_b @ T @ Sigma2 ) ) @ subter523971068842742411eq_a_b ) ) ).
% supteq_subst
thf(fact_81_join__sym,axiom,
! [S: produc357393685978478089rm_a_b,T: produc357393685978478089rm_a_b,A4: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ S @ T ) @ ( abstra4889682940153293507rm_a_b @ A4 ) )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ T @ S ) @ ( abstra4889682940153293507rm_a_b @ A4 ) ) ) ).
% join_sym
thf(fact_82_join__sym,axiom,
! [S: list_nat,T: list_nat,A4: set_Pr3451248702717554689st_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ S @ T ) @ ( abstra1638945853619340710st_nat @ A4 ) )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ T @ S ) @ ( abstra1638945853619340710st_nat @ A4 ) ) ) ).
% join_sym
thf(fact_83_join__sym,axiom,
! [S: term_a_b,T: term_a_b,A4: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( abstra4096080454567261402rm_a_b @ A4 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T @ S ) @ ( abstra4096080454567261402rm_a_b @ A4 ) ) ) ).
% join_sym
thf(fact_84_supteq_Orefl,axiom,
! [T: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T @ T ) @ subter523971068842742411eq_a_b ) ).
% supteq.refl
thf(fact_85_subterm_Oeq__refl,axiom,
! [X: term_a_b,Y: term_a_b] :
( ( X = Y )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X ) @ subter523971068842742411eq_a_b ) ) ).
% subterm.eq_refl
thf(fact_86_subterm_Oorder__refl,axiom,
! [X: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ X ) @ subter523971068842742411eq_a_b ) ).
% subterm.order_refl
thf(fact_87_subterm_Oorder_Otrans,axiom,
! [B: term_a_b,A: term_a_b,C2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ subter523971068842742411eq_a_b )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ C2 @ B ) @ subter523971068842742411eq_a_b )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ C2 @ A ) @ subter523971068842742411eq_a_b ) ) ) ).
% subterm.order.trans
thf(fact_88_subterm_Oorder__trans,axiom,
! [Y: term_a_b,X: term_a_b,Z: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X ) @ subter523971068842742411eq_a_b )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Z @ Y ) @ subter523971068842742411eq_a_b )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Z @ X ) @ subter523971068842742411eq_a_b ) ) ) ).
% subterm.order_trans
thf(fact_89_subterm_Oantisym__conv,axiom,
! [X: term_a_b,Y: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ subter523971068842742411eq_a_b )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X ) @ subter523971068842742411eq_a_b )
= ( X = Y ) ) ) ).
% subterm.antisym_conv
thf(fact_90_subterm_Oorder_Oeq__iff,axiom,
( ( ^ [Y4: term_a_b,Z2: term_a_b] : ( Y4 = Z2 ) )
= ( ^ [A5: term_a_b,B4: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B4 @ A5 ) @ subter523971068842742411eq_a_b )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A5 @ B4 ) @ subter523971068842742411eq_a_b ) ) ) ) ).
% subterm.order.eq_iff
thf(fact_91_subterm_Oorder__eq__iff,axiom,
( ( ^ [Y4: term_a_b,Z2: term_a_b] : ( Y4 = Z2 ) )
= ( ^ [X4: term_a_b,Y5: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y5 @ X4 ) @ subter523971068842742411eq_a_b )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X4 @ Y5 ) @ subter523971068842742411eq_a_b ) ) ) ) ).
% subterm.order_eq_iff
thf(fact_92_subterm_Oorder_Oantisym,axiom,
! [B: term_a_b,A: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ subter523971068842742411eq_a_b )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ subter523971068842742411eq_a_b )
=> ( A = B ) ) ) ).
% subterm.order.antisym
thf(fact_93_supteq__antisym,axiom,
! [S: term_a_b,T: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ subter523971068842742411eq_a_b )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T @ S ) @ subter523971068842742411eq_a_b )
=> ( S = T ) ) ) ).
% supteq_antisym
thf(fact_94_supteq__trans,axiom,
! [S: term_a_b,T: term_a_b,U: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ subter523971068842742411eq_a_b )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T @ U ) @ subter523971068842742411eq_a_b )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ U ) @ subter523971068842742411eq_a_b ) ) ) ).
% supteq_trans
thf(fact_95_eq__supteq,axiom,
! [S: term_a_b,T: term_a_b] :
( ( S = T )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ subter523971068842742411eq_a_b ) ) ).
% eq_supteq
thf(fact_96_subterm_Odual__order_Oantisym,axiom,
! [A: term_a_b,B: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ subter523971068842742411eq_a_b )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ subter523971068842742411eq_a_b )
=> ( A = B ) ) ) ).
% subterm.dual_order.antisym
thf(fact_97_subterm_Odual__order_Oeq__iff,axiom,
( ( ^ [Y4: term_a_b,Z2: term_a_b] : ( Y4 = Z2 ) )
= ( ^ [A5: term_a_b,B4: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A5 @ B4 ) @ subter523971068842742411eq_a_b )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B4 @ A5 ) @ subter523971068842742411eq_a_b ) ) ) ) ).
% subterm.dual_order.eq_iff
thf(fact_98_subterm_Odual__order_Otrans,axiom,
! [A: term_a_b,B: term_a_b,C2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ subter523971068842742411eq_a_b )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C2 ) @ subter523971068842742411eq_a_b )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C2 ) @ subter523971068842742411eq_a_b ) ) ) ).
% subterm.dual_order.trans
thf(fact_99_subterm_Oord__le__eq__trans,axiom,
! [B: term_a_b,A: term_a_b,C2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ subter523971068842742411eq_a_b )
=> ( ( B = C2 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ C2 @ A ) @ subter523971068842742411eq_a_b ) ) ) ).
% subterm.ord_le_eq_trans
thf(fact_100_subterm_Oord__eq__le__trans,axiom,
! [A: term_a_b,B: term_a_b,C2: term_a_b] :
( ( A = B )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ C2 @ B ) @ subter523971068842742411eq_a_b )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ C2 @ A ) @ subter523971068842742411eq_a_b ) ) ) ).
% subterm.ord_eq_le_trans
thf(fact_101_subterm_Odual__order_Orefl,axiom,
! [A: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ A ) @ subter523971068842742411eq_a_b ) ).
% subterm.dual_order.refl
thf(fact_102_subterm_Oorder__antisym,axiom,
! [Y: term_a_b,X: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X ) @ subter523971068842742411eq_a_b )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ subter523971068842742411eq_a_b )
=> ( X = Y ) ) ) ).
% subterm.order_antisym
thf(fact_103_WCR__onI,axiom,
! [A4: set_Pr2659102613823815246st_nat,R2: set_Pr2046385631809063665st_nat] :
( ! [A3: produc2423145444804056984st_nat,B3: produc2423145444804056984st_nat,C: produc2423145444804056984st_nat] :
( ( member190277321972437807st_nat @ A3 @ A4 )
=> ( ( member2188129036539957306st_nat @ ( produc8516103258745939657st_nat @ A3 @ B3 ) @ R2 )
=> ( ( member2188129036539957306st_nat @ ( produc8516103258745939657st_nat @ A3 @ C ) @ R2 )
=> ( member2188129036539957306st_nat @ ( produc8516103258745939657st_nat @ B3 @ C ) @ ( abstra256346906152286046st_nat @ R2 ) ) ) ) )
=> ( abstra1867343039937780675st_nat @ R2 @ A4 ) ) ).
% WCR_onI
thf(fact_104_WCR__onI,axiom,
! [A4: set_Pr1829413238147913556rm_a_b,R2: set_Pr2173775114972114941rm_a_b] :
( ! [A3: produc6373727562496775070rm_a_b,B3: produc6373727562496775070rm_a_b,C: produc6373727562496775070rm_a_b] :
( ( member4140859439665155893rm_a_b @ A3 @ A4 )
=> ( ( member6865565617957190470rm_a_b @ ( produc8750439296677016789rm_a_b @ A3 @ B3 ) @ R2 )
=> ( ( member6865565617957190470rm_a_b @ ( produc8750439296677016789rm_a_b @ A3 @ C ) @ R2 )
=> ( member6865565617957190470rm_a_b @ ( produc8750439296677016789rm_a_b @ B3 @ C ) @ ( abstra4206929023845004132rm_a_b @ R2 ) ) ) ) )
=> ( abstra5817925157630498761rm_a_b @ R2 @ A4 ) ) ).
% WCR_onI
thf(fact_105_WCR__onI,axiom,
! [A4: set_Pr3451248702717554689st_nat,R2: set_Pr915550674247830855st_nat] :
( ! [A3: produc1828647624359046049st_nat,B3: produc1828647624359046049st_nat,C: produc1828647624359046049st_nat] :
( ( member7340969449405702474st_nat @ A3 @ A4 )
=> ( ( member5236317644330119056st_nat @ ( produc988773521843941591st_nat @ A3 @ B3 ) @ R2 )
=> ( ( member5236317644330119056st_nat @ ( produc988773521843941591st_nat @ A3 @ C ) @ R2 )
=> ( member5236317644330119056st_nat @ ( produc988773521843941591st_nat @ B3 @ C ) @ ( abstra6360936878533861467st_nat @ R2 ) ) ) ) )
=> ( abstra2346411092049692598st_nat @ R2 @ A4 ) ) ).
% WCR_onI
thf(fact_106_WCR__onI,axiom,
! [A4: set_set_term_a_b,R2: set_Pr7151917219565317845rm_a_b] :
( ! [A3: set_term_a_b,B3: set_term_a_b,C: set_term_a_b] :
( ( member_set_term_a_b @ A3 @ A4 )
=> ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A3 @ B3 ) @ R2 )
=> ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A3 @ C ) @ R2 )
=> ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ B3 @ C ) @ ( abstra2139004588645393680rm_a_b @ R2 ) ) ) ) )
=> ( abstra2457992723331821045rm_a_b @ R2 @ A4 ) ) ).
% WCR_onI
thf(fact_107_WCR__onI,axiom,
! [A4: set_list_nat,R2: set_Pr3451248702717554689st_nat] :
( ! [A3: list_nat,B3: list_nat,C: list_nat] :
( ( member_list_nat @ A3 @ A4 )
=> ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A3 @ B3 ) @ R2 )
=> ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A3 @ C ) @ R2 )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ B3 @ C ) @ ( abstra1638945853619340710st_nat @ R2 ) ) ) ) )
=> ( abstra2569714506102940299st_nat @ R2 @ A4 ) ) ).
% WCR_onI
thf(fact_108_WCR__onI,axiom,
! [A4: set_Pr4386577575007340137rm_a_b,R2: set_Pr2972776593051762503rm_a_b] :
( ! [A3: produc357393685978478089rm_a_b,B3: produc357393685978478089rm_a_b,C: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ A3 @ A4 )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A3 @ B3 ) @ R2 )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A3 @ C ) @ R2 )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B3 @ C ) @ ( abstra4889682940153293507rm_a_b @ R2 ) ) ) ) )
=> ( abstra875157153669124638rm_a_b @ R2 @ A4 ) ) ).
% WCR_onI
thf(fact_109_WCR__onI,axiom,
! [A4: set_term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ! [A3: term_a_b,B3: term_a_b,C: term_a_b] :
( ( member_term_a_b @ A3 @ A4 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B3 ) @ R2 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ C ) @ R2 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B3 @ C ) @ ( abstra4096080454567261402rm_a_b @ R2 ) ) ) ) )
=> ( abstra5026849107050860991rm_a_b @ R2 @ A4 ) ) ).
% WCR_onI
thf(fact_110_WCR__on__def,axiom,
( abstra875157153669124638rm_a_b
= ( ^ [R4: set_Pr2972776593051762503rm_a_b,A6: set_Pr4386577575007340137rm_a_b] :
! [X4: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X4 @ A6 )
=> ! [B4: produc357393685978478089rm_a_b,C3: produc357393685978478089rm_a_b] :
( ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X4 @ B4 ) @ R4 )
& ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X4 @ C3 ) @ R4 ) )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B4 @ C3 ) @ ( abstra4889682940153293507rm_a_b @ R4 ) ) ) ) ) ) ).
% WCR_on_def
thf(fact_111_WCR__on__def,axiom,
( abstra2569714506102940299st_nat
= ( ^ [R4: set_Pr3451248702717554689st_nat,A6: set_list_nat] :
! [X4: list_nat] :
( ( member_list_nat @ X4 @ A6 )
=> ! [B4: list_nat,C3: list_nat] :
( ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X4 @ B4 ) @ R4 )
& ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X4 @ C3 ) @ R4 ) )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ B4 @ C3 ) @ ( abstra1638945853619340710st_nat @ R4 ) ) ) ) ) ) ).
% WCR_on_def
thf(fact_112_WCR__on__def,axiom,
( abstra5026849107050860991rm_a_b
= ( ^ [R4: set_Pr4386577575007340137rm_a_b,A6: set_term_a_b] :
! [X4: term_a_b] :
( ( member_term_a_b @ X4 @ A6 )
=> ! [B4: term_a_b,C3: term_a_b] :
( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X4 @ B4 ) @ R4 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X4 @ C3 ) @ R4 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B4 @ C3 ) @ ( abstra4096080454567261402rm_a_b @ R4 ) ) ) ) ) ) ).
% WCR_on_def
thf(fact_113_WCR__onE,axiom,
! [R2: set_Pr2046385631809063665st_nat,A4: set_Pr2659102613823815246st_nat,A: produc2423145444804056984st_nat,B: produc2423145444804056984st_nat,C2: produc2423145444804056984st_nat] :
( ( abstra1867343039937780675st_nat @ R2 @ A4 )
=> ( ( member190277321972437807st_nat @ A @ A4 )
=> ( ~ ( member2188129036539957306st_nat @ ( produc8516103258745939657st_nat @ B @ C2 ) @ ( abstra256346906152286046st_nat @ R2 ) )
=> ( ( member2188129036539957306st_nat @ ( produc8516103258745939657st_nat @ A @ B ) @ R2 )
=> ~ ( member2188129036539957306st_nat @ ( produc8516103258745939657st_nat @ A @ C2 ) @ R2 ) ) ) ) ) ).
% WCR_onE
thf(fact_114_WCR__onE,axiom,
! [R2: set_Pr2173775114972114941rm_a_b,A4: set_Pr1829413238147913556rm_a_b,A: produc6373727562496775070rm_a_b,B: produc6373727562496775070rm_a_b,C2: produc6373727562496775070rm_a_b] :
( ( abstra5817925157630498761rm_a_b @ R2 @ A4 )
=> ( ( member4140859439665155893rm_a_b @ A @ A4 )
=> ( ~ ( member6865565617957190470rm_a_b @ ( produc8750439296677016789rm_a_b @ B @ C2 ) @ ( abstra4206929023845004132rm_a_b @ R2 ) )
=> ( ( member6865565617957190470rm_a_b @ ( produc8750439296677016789rm_a_b @ A @ B ) @ R2 )
=> ~ ( member6865565617957190470rm_a_b @ ( produc8750439296677016789rm_a_b @ A @ C2 ) @ R2 ) ) ) ) ) ).
% WCR_onE
thf(fact_115_WCR__onE,axiom,
! [R2: set_Pr915550674247830855st_nat,A4: set_Pr3451248702717554689st_nat,A: produc1828647624359046049st_nat,B: produc1828647624359046049st_nat,C2: produc1828647624359046049st_nat] :
( ( abstra2346411092049692598st_nat @ R2 @ A4 )
=> ( ( member7340969449405702474st_nat @ A @ A4 )
=> ( ~ ( member5236317644330119056st_nat @ ( produc988773521843941591st_nat @ B @ C2 ) @ ( abstra6360936878533861467st_nat @ R2 ) )
=> ( ( member5236317644330119056st_nat @ ( produc988773521843941591st_nat @ A @ B ) @ R2 )
=> ~ ( member5236317644330119056st_nat @ ( produc988773521843941591st_nat @ A @ C2 ) @ R2 ) ) ) ) ) ).
% WCR_onE
thf(fact_116_WCR__onE,axiom,
! [R2: set_Pr7151917219565317845rm_a_b,A4: set_set_term_a_b,A: set_term_a_b,B: set_term_a_b,C2: set_term_a_b] :
( ( abstra2457992723331821045rm_a_b @ R2 @ A4 )
=> ( ( member_set_term_a_b @ A @ A4 )
=> ( ~ ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ B @ C2 ) @ ( abstra2139004588645393680rm_a_b @ R2 ) )
=> ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A @ B ) @ R2 )
=> ~ ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A @ C2 ) @ R2 ) ) ) ) ) ).
% WCR_onE
thf(fact_117_WCR__onE,axiom,
! [R2: set_Pr3451248702717554689st_nat,A4: set_list_nat,A: list_nat,B: list_nat,C2: list_nat] :
( ( abstra2569714506102940299st_nat @ R2 @ A4 )
=> ( ( member_list_nat @ A @ A4 )
=> ( ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ B @ C2 ) @ ( abstra1638945853619340710st_nat @ R2 ) )
=> ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A @ B ) @ R2 )
=> ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A @ C2 ) @ R2 ) ) ) ) ) ).
% WCR_onE
thf(fact_118_WCR__onE,axiom,
! [R2: set_Pr2972776593051762503rm_a_b,A4: set_Pr4386577575007340137rm_a_b,A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,C2: produc357393685978478089rm_a_b] :
( ( abstra875157153669124638rm_a_b @ R2 @ A4 )
=> ( ( member5869715511025134514rm_a_b @ A @ A4 )
=> ( ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B @ C2 ) @ ( abstra4889682940153293507rm_a_b @ R2 ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ R2 )
=> ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ C2 ) @ R2 ) ) ) ) ) ).
% WCR_onE
thf(fact_119_WCR__onE,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,A4: set_term_a_b,A: term_a_b,B: term_a_b,C2: term_a_b] :
( ( abstra5026849107050860991rm_a_b @ R2 @ A4 )
=> ( ( member_term_a_b @ A @ A4 )
=> ( ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C2 ) @ ( abstra4096080454567261402rm_a_b @ R2 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R2 )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C2 ) @ R2 ) ) ) ) ) ).
% WCR_onE
thf(fact_120_WCR__onD,axiom,
! [R2: set_Pr2046385631809063665st_nat,A4: set_Pr2659102613823815246st_nat,A: produc2423145444804056984st_nat,B: produc2423145444804056984st_nat,C2: produc2423145444804056984st_nat] :
( ( abstra1867343039937780675st_nat @ R2 @ A4 )
=> ( ( member190277321972437807st_nat @ A @ A4 )
=> ( ( member2188129036539957306st_nat @ ( produc8516103258745939657st_nat @ A @ B ) @ R2 )
=> ( ( member2188129036539957306st_nat @ ( produc8516103258745939657st_nat @ A @ C2 ) @ R2 )
=> ( member2188129036539957306st_nat @ ( produc8516103258745939657st_nat @ B @ C2 ) @ ( abstra256346906152286046st_nat @ R2 ) ) ) ) ) ) ).
% WCR_onD
thf(fact_121_WCR__onD,axiom,
! [R2: set_Pr2173775114972114941rm_a_b,A4: set_Pr1829413238147913556rm_a_b,A: produc6373727562496775070rm_a_b,B: produc6373727562496775070rm_a_b,C2: produc6373727562496775070rm_a_b] :
( ( abstra5817925157630498761rm_a_b @ R2 @ A4 )
=> ( ( member4140859439665155893rm_a_b @ A @ A4 )
=> ( ( member6865565617957190470rm_a_b @ ( produc8750439296677016789rm_a_b @ A @ B ) @ R2 )
=> ( ( member6865565617957190470rm_a_b @ ( produc8750439296677016789rm_a_b @ A @ C2 ) @ R2 )
=> ( member6865565617957190470rm_a_b @ ( produc8750439296677016789rm_a_b @ B @ C2 ) @ ( abstra4206929023845004132rm_a_b @ R2 ) ) ) ) ) ) ).
% WCR_onD
thf(fact_122_WCR__onD,axiom,
! [R2: set_Pr915550674247830855st_nat,A4: set_Pr3451248702717554689st_nat,A: produc1828647624359046049st_nat,B: produc1828647624359046049st_nat,C2: produc1828647624359046049st_nat] :
( ( abstra2346411092049692598st_nat @ R2 @ A4 )
=> ( ( member7340969449405702474st_nat @ A @ A4 )
=> ( ( member5236317644330119056st_nat @ ( produc988773521843941591st_nat @ A @ B ) @ R2 )
=> ( ( member5236317644330119056st_nat @ ( produc988773521843941591st_nat @ A @ C2 ) @ R2 )
=> ( member5236317644330119056st_nat @ ( produc988773521843941591st_nat @ B @ C2 ) @ ( abstra6360936878533861467st_nat @ R2 ) ) ) ) ) ) ).
% WCR_onD
thf(fact_123_WCR__onD,axiom,
! [R2: set_Pr7151917219565317845rm_a_b,A4: set_set_term_a_b,A: set_term_a_b,B: set_term_a_b,C2: set_term_a_b] :
( ( abstra2457992723331821045rm_a_b @ R2 @ A4 )
=> ( ( member_set_term_a_b @ A @ A4 )
=> ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A @ B ) @ R2 )
=> ( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ A @ C2 ) @ R2 )
=> ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ B @ C2 ) @ ( abstra2139004588645393680rm_a_b @ R2 ) ) ) ) ) ) ).
% WCR_onD
thf(fact_124_WCR__onD,axiom,
! [R2: set_Pr3451248702717554689st_nat,A4: set_list_nat,A: list_nat,B: list_nat,C2: list_nat] :
( ( abstra2569714506102940299st_nat @ R2 @ A4 )
=> ( ( member_list_nat @ A @ A4 )
=> ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A @ B ) @ R2 )
=> ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A @ C2 ) @ R2 )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ B @ C2 ) @ ( abstra1638945853619340710st_nat @ R2 ) ) ) ) ) ) ).
% WCR_onD
thf(fact_125_WCR__onD,axiom,
! [R2: set_Pr2972776593051762503rm_a_b,A4: set_Pr4386577575007340137rm_a_b,A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,C2: produc357393685978478089rm_a_b] :
( ( abstra875157153669124638rm_a_b @ R2 @ A4 )
=> ( ( member5869715511025134514rm_a_b @ A @ A4 )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ R2 )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ C2 ) @ R2 )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B @ C2 ) @ ( abstra4889682940153293507rm_a_b @ R2 ) ) ) ) ) ) ).
% WCR_onD
thf(fact_126_WCR__onD,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,A4: set_term_a_b,A: term_a_b,B: term_a_b,C2: term_a_b] :
( ( abstra5026849107050860991rm_a_b @ R2 @ A4 )
=> ( ( member_term_a_b @ A @ A4 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R2 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C2 ) @ R2 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C2 ) @ ( abstra4096080454567261402rm_a_b @ R2 ) ) ) ) ) ) ).
% WCR_onD
thf(fact_127_poss__of__term__poss__emptyD,axiom,
! [U: term_a_b,S: term_a_b,P2: list_nat] :
( ( ( terms_7168686267159881682rm_a_b @ U @ S )
= bot_bot_set_list_nat )
=> ( ( member_list_nat @ P2 @ ( term_poss_a_b @ S ) )
=> ( ( term_subt_at_a_b @ S @ P2 )
!= U ) ) ) ).
% poss_of_term_poss_emptyD
thf(fact_128_rstepI,axiom,
! [L2: term_a_b,R2: term_a_b,R3: set_Pr4386577575007340137rm_a_b,S: term_a_b,C4: subterm_and_ctxt_a_b,Sigma2: b > term_a_b,T: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ L2 @ R2 ) @ R3 )
=> ( ( S
= ( subter2376574525758040790rm_a_b @ C4 @ ( subst_7999470309526761004_a_b_b @ L2 @ Sigma2 ) ) )
=> ( ( T
= ( subter2376574525758040790rm_a_b @ C4 @ ( subst_7999470309526761004_a_b_b @ R2 @ Sigma2 ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( rstep_a_b @ R3 ) ) ) ) ) ).
% rstepI
thf(fact_129_subst__at__ctxt__at__eq__termI,axiom,
! [P2: list_nat,S: term_a_b,T: term_a_b] :
( ( member_list_nat @ P2 @ ( term_poss_a_b @ S ) )
=> ( ( member_list_nat @ P2 @ ( term_poss_a_b @ T ) )
=> ( ( ( term_subt_at_a_b @ S @ P2 )
= ( term_subt_at_a_b @ T @ P2 ) )
=> ( ( ( term_ctxt_at_pos_a_b @ S @ P2 )
= ( term_ctxt_at_pos_a_b @ T @ P2 ) )
=> ( S = T ) ) ) ) ) ).
% subst_at_ctxt_at_eq_termI
thf(fact_130_subst__at__ctxt__at__eq__termD,axiom,
! [S: term_a_b,T: term_a_b,P2: list_nat] :
( ( S = T )
=> ( ( member_list_nat @ P2 @ ( term_poss_a_b @ T ) )
=> ( ( ( term_subt_at_a_b @ S @ P2 )
= ( term_subt_at_a_b @ T @ P2 ) )
& ( ( term_ctxt_at_pos_a_b @ S @ P2 )
= ( term_ctxt_at_pos_a_b @ T @ P2 ) ) ) ) ) ).
% subst_at_ctxt_at_eq_termD
thf(fact_131_ssubst__Pair__rhs,axiom,
! [R2: produc357393685978478089rm_a_b,S: produc357393685978478089rm_a_b,R5: set_Pr2972776593051762503rm_a_b,S2: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ R2 @ S ) @ R5 )
=> ( ( S2 = S )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ R2 @ S2 ) @ R5 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_132_ssubst__Pair__rhs,axiom,
! [R2: list_nat,S: list_nat,R5: set_Pr3451248702717554689st_nat,S2: list_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ R2 @ S ) @ R5 )
=> ( ( S2 = S )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ R2 @ S2 ) @ R5 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_133_ssubst__Pair__rhs,axiom,
! [R2: produc357393685978478089rm_a_b,S: list_nat,R5: set_Pr2659102613823815246st_nat,S2: list_nat] :
( ( member190277321972437807st_nat @ ( produc5056965718246025362st_nat @ R2 @ S ) @ R5 )
=> ( ( S2 = S )
=> ( member190277321972437807st_nat @ ( produc5056965718246025362st_nat @ R2 @ S2 ) @ R5 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_134_ssubst__Pair__rhs,axiom,
! [R2: list_nat,S: produc357393685978478089rm_a_b,R5: set_Pr1829413238147913556rm_a_b,S2: produc357393685978478089rm_a_b] :
( ( member4140859439665155893rm_a_b @ ( produc3015861401532503632rm_a_b @ R2 @ S ) @ R5 )
=> ( ( S2 = S )
=> ( member4140859439665155893rm_a_b @ ( produc3015861401532503632rm_a_b @ R2 @ S2 ) @ R5 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_135_ssubst__Pair__rhs,axiom,
! [R2: term_a_b,S: term_a_b,R5: set_Pr4386577575007340137rm_a_b,S2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ R2 @ S ) @ R5 )
=> ( ( S2 = S )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ R2 @ S2 ) @ R5 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_136_curry__conv,axiom,
( produc6008550485034061667_a_b_o
= ( ^ [F2: produc357393685978478089rm_a_b > $o,A5: term_a_b,B4: term_a_b] : ( F2 @ ( produc7020197800436672577rm_a_b @ A5 @ B4 ) ) ) ) ).
% curry_conv
thf(fact_137_curryI,axiom,
! [F3: produc4523971788519308903rm_a_b > $o,A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b] :
( ( F3 @ ( produc1763473618796451543rm_a_b @ A @ B ) )
=> ( produc3713113261896612557_a_b_o @ F3 @ A @ B ) ) ).
% curryI
thf(fact_138_curryI,axiom,
! [F3: produc1828647624359046049st_nat > $o,A: list_nat,B: list_nat] :
( ( F3 @ ( produc2694037385005941721st_nat @ A @ B ) )
=> ( produc6139401228252118475_nat_o @ F3 @ A @ B ) ) ).
% curryI
thf(fact_139_curryI,axiom,
! [F3: produc2423145444804056984st_nat > $o,A: produc357393685978478089rm_a_b,B: list_nat] :
( ( F3 @ ( produc5056965718246025362st_nat @ A @ B ) )
=> ( produc4172631894931527864_nat_o @ F3 @ A @ B ) ) ).
% curryI
thf(fact_140_curryI,axiom,
! [F3: produc6373727562496775070rm_a_b > $o,A: list_nat,B: produc357393685978478089rm_a_b] :
( ( F3 @ ( produc3015861401532503632rm_a_b @ A @ B ) )
=> ( produc2875039139025540986_a_b_o @ F3 @ A @ B ) ) ).
% curryI
thf(fact_141_curryI,axiom,
! [F3: produc357393685978478089rm_a_b > $o,A: term_a_b,B: term_a_b] :
( ( F3 @ ( produc7020197800436672577rm_a_b @ A @ B ) )
=> ( produc6008550485034061667_a_b_o @ F3 @ A @ B ) ) ).
% curryI
thf(fact_142_ctxt__eq,axiom,
! [C4: subterm_and_ctxt_a_b,S: term_a_b,T: term_a_b] :
( ( ( subter2376574525758040790rm_a_b @ C4 @ S )
= ( subter2376574525758040790rm_a_b @ C4 @ T ) )
= ( S = T ) ) ).
% ctxt_eq
thf(fact_143_ctxt__supteq,axiom,
! [S: term_a_b,C4: subterm_and_ctxt_a_b,T: term_a_b] :
( ( S
= ( subter2376574525758040790rm_a_b @ C4 @ T ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ subter523971068842742411eq_a_b ) ) ).
% ctxt_supteq
thf(fact_144_rstep__ctxtI,axiom,
! [S: term_a_b,T: term_a_b,R3: set_Pr4386577575007340137rm_a_b,C4: subterm_and_ctxt_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( rstep_a_b @ R3 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( subter2376574525758040790rm_a_b @ C4 @ S ) @ ( subter2376574525758040790rm_a_b @ C4 @ T ) ) @ ( rstep_a_b @ R3 ) ) ) ).
% rstep_ctxtI
thf(fact_145_ctxt__at__pos__subt__at__id,axiom,
! [P2: list_nat,T: term_a_b] :
( ( member_list_nat @ P2 @ ( term_poss_a_b @ T ) )
=> ( ( subter2376574525758040790rm_a_b @ ( term_ctxt_at_pos_a_b @ T @ P2 ) @ ( term_subt_at_a_b @ T @ P2 ) )
= T ) ) ).
% ctxt_at_pos_subt_at_id
thf(fact_146_ctxt__at__pos__subt__at__pos,axiom,
! [P2: list_nat,T: term_a_b,U: term_a_b] :
( ( member_list_nat @ P2 @ ( term_poss_a_b @ T ) )
=> ( ( term_subt_at_a_b @ ( subter2376574525758040790rm_a_b @ ( term_ctxt_at_pos_a_b @ T @ P2 ) @ U ) @ P2 )
= U ) ) ).
% ctxt_at_pos_subt_at_pos
thf(fact_147_ctxt__of__pos__term__apply__replace__at__ident,axiom,
! [P2: list_nat,S: term_a_b,T: term_a_b] :
( ( member_list_nat @ P2 @ ( term_poss_a_b @ S ) )
=> ( ( subter2376574525758040790rm_a_b @ ( term_ctxt_at_pos_a_b @ S @ P2 ) @ T )
= ( term_r6860082780075436317at_a_b @ S @ P2 @ T ) ) ) ).
% ctxt_of_pos_term_apply_replace_at_ident
thf(fact_148_curryD,axiom,
! [F3: produc4523971788519308903rm_a_b > $o,A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b] :
( ( produc3713113261896612557_a_b_o @ F3 @ A @ B )
=> ( F3 @ ( produc1763473618796451543rm_a_b @ A @ B ) ) ) ).
% curryD
thf(fact_149_curryD,axiom,
! [F3: produc1828647624359046049st_nat > $o,A: list_nat,B: list_nat] :
( ( produc6139401228252118475_nat_o @ F3 @ A @ B )
=> ( F3 @ ( produc2694037385005941721st_nat @ A @ B ) ) ) ).
% curryD
thf(fact_150_curryD,axiom,
! [F3: produc2423145444804056984st_nat > $o,A: produc357393685978478089rm_a_b,B: list_nat] :
( ( produc4172631894931527864_nat_o @ F3 @ A @ B )
=> ( F3 @ ( produc5056965718246025362st_nat @ A @ B ) ) ) ).
% curryD
thf(fact_151_curryD,axiom,
! [F3: produc6373727562496775070rm_a_b > $o,A: list_nat,B: produc357393685978478089rm_a_b] :
( ( produc2875039139025540986_a_b_o @ F3 @ A @ B )
=> ( F3 @ ( produc3015861401532503632rm_a_b @ A @ B ) ) ) ).
% curryD
thf(fact_152_curryD,axiom,
! [F3: produc357393685978478089rm_a_b > $o,A: term_a_b,B: term_a_b] :
( ( produc6008550485034061667_a_b_o @ F3 @ A @ B )
=> ( F3 @ ( produc7020197800436672577rm_a_b @ A @ B ) ) ) ).
% curryD
thf(fact_153_curryE,axiom,
! [F3: produc4523971788519308903rm_a_b > $o,A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b] :
( ( produc3713113261896612557_a_b_o @ F3 @ A @ B )
=> ( F3 @ ( produc1763473618796451543rm_a_b @ A @ B ) ) ) ).
% curryE
thf(fact_154_curryE,axiom,
! [F3: produc1828647624359046049st_nat > $o,A: list_nat,B: list_nat] :
( ( produc6139401228252118475_nat_o @ F3 @ A @ B )
=> ( F3 @ ( produc2694037385005941721st_nat @ A @ B ) ) ) ).
% curryE
thf(fact_155_curryE,axiom,
! [F3: produc2423145444804056984st_nat > $o,A: produc357393685978478089rm_a_b,B: list_nat] :
( ( produc4172631894931527864_nat_o @ F3 @ A @ B )
=> ( F3 @ ( produc5056965718246025362st_nat @ A @ B ) ) ) ).
% curryE
thf(fact_156_curryE,axiom,
! [F3: produc6373727562496775070rm_a_b > $o,A: list_nat,B: produc357393685978478089rm_a_b] :
( ( produc2875039139025540986_a_b_o @ F3 @ A @ B )
=> ( F3 @ ( produc3015861401532503632rm_a_b @ A @ B ) ) ) ).
% curryE
thf(fact_157_curryE,axiom,
! [F3: produc357393685978478089rm_a_b > $o,A: term_a_b,B: term_a_b] :
( ( produc6008550485034061667_a_b_o @ F3 @ A @ B )
=> ( F3 @ ( produc7020197800436672577rm_a_b @ A @ B ) ) ) ).
% curryE
thf(fact_158_supteq__ctxtE,axiom,
! [S: term_a_b,T: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ subter523971068842742411eq_a_b )
=> ~ ! [C5: subterm_and_ctxt_a_b] :
( S
!= ( subter2376574525758040790rm_a_b @ C5 @ T ) ) ) ).
% supteq_ctxtE
thf(fact_159_ctxt__imp__supteq,axiom,
! [C4: subterm_and_ctxt_a_b,T: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( subter2376574525758040790rm_a_b @ C4 @ T ) @ T ) @ subter523971068842742411eq_a_b ) ).
% ctxt_imp_supteq
thf(fact_160_supteq__ctxt__conv,axiom,
! [S: term_a_b,T: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ subter523971068842742411eq_a_b )
= ( ? [C6: subterm_and_ctxt_a_b] :
( S
= ( subter2376574525758040790rm_a_b @ C6 @ T ) ) ) ) ).
% supteq_ctxt_conv
thf(fact_161_rstep__imp__C__s__r,axiom,
! [S: term_a_b,T: term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( rstep_a_b @ R3 ) )
=> ? [C5: subterm_and_ctxt_a_b,Sigma: b > term_a_b,L: term_a_b,R: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ L @ R ) @ R3 )
& ( S
= ( subter2376574525758040790rm_a_b @ C5 @ ( subst_7999470309526761004_a_b_b @ L @ Sigma ) ) )
& ( T
= ( subter2376574525758040790rm_a_b @ C5 @ ( subst_7999470309526761004_a_b_b @ R @ Sigma ) ) ) ) ) ).
% rstep_imp_C_s_r
thf(fact_162_rstep__imp__C__s__r_H,axiom,
! [S: term_a_b,T: term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( rstep_a_b @ R3 ) )
=> ~ ! [C5: subterm_and_ctxt_a_b,L: term_a_b,R: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ L @ R ) @ R3 )
=> ! [Sigma: b > term_a_b] :
( ( S
= ( subter2376574525758040790rm_a_b @ C5 @ ( subst_7999470309526761004_a_b_b @ L @ Sigma ) ) )
=> ( T
!= ( subter2376574525758040790rm_a_b @ C5 @ ( subst_7999470309526761004_a_b_b @ R @ Sigma ) ) ) ) ) ) ).
% rstep_imp_C_s_r'
thf(fact_163_rstep_Ointros,axiom,
! [S: term_a_b,T: term_a_b,R3: set_Pr4386577575007340137rm_a_b,C4: subterm_and_ctxt_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( rrstep_a_b @ R3 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( subter2376574525758040790rm_a_b @ C4 @ S ) @ ( subter2376574525758040790rm_a_b @ C4 @ T ) ) @ ( rstep_a_b @ R3 ) ) ) ).
% rstep.intros
thf(fact_164_rstep_Osimps,axiom,
! [A1: term_a_b,A22: term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A22 ) @ ( rstep_a_b @ R3 ) )
= ( ? [S3: term_a_b,T3: term_a_b,C6: subterm_and_ctxt_a_b] :
( ( A1
= ( subter2376574525758040790rm_a_b @ C6 @ S3 ) )
& ( A22
= ( subter2376574525758040790rm_a_b @ C6 @ T3 ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S3 @ T3 ) @ ( rrstep_a_b @ R3 ) ) ) ) ) ).
% rstep.simps
thf(fact_165_rstep_Ocases,axiom,
! [A1: term_a_b,A22: term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A22 ) @ ( rstep_a_b @ R3 ) )
=> ~ ! [S4: term_a_b,T2: term_a_b,C5: subterm_and_ctxt_a_b] :
( ( A1
= ( subter2376574525758040790rm_a_b @ C5 @ S4 ) )
=> ( ( A22
= ( subter2376574525758040790rm_a_b @ C5 @ T2 ) )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S4 @ T2 ) @ ( rrstep_a_b @ R3 ) ) ) ) ) ).
% rstep.cases
thf(fact_166_empty__Collect__eq,axiom,
! [P3: list_nat > $o] :
( ( bot_bot_set_list_nat
= ( collect_list_nat @ P3 ) )
= ( ! [X4: list_nat] :
~ ( P3 @ X4 ) ) ) ).
% empty_Collect_eq
thf(fact_167_empty__Collect__eq,axiom,
! [P3: produc357393685978478089rm_a_b > $o] :
( ( bot_bo197521221353338581rm_a_b
= ( collec99177395925924084rm_a_b @ P3 ) )
= ( ! [X4: produc357393685978478089rm_a_b] :
~ ( P3 @ X4 ) ) ) ).
% empty_Collect_eq
thf(fact_168_empty__Collect__eq,axiom,
! [P3: term_a_b > $o] :
( ( bot_bot_set_term_a_b
= ( collect_term_a_b @ P3 ) )
= ( ! [X4: term_a_b] :
~ ( P3 @ X4 ) ) ) ).
% empty_Collect_eq
thf(fact_169_Collect__empty__eq,axiom,
! [P3: list_nat > $o] :
( ( ( collect_list_nat @ P3 )
= bot_bot_set_list_nat )
= ( ! [X4: list_nat] :
~ ( P3 @ X4 ) ) ) ).
% Collect_empty_eq
thf(fact_170_Collect__empty__eq,axiom,
! [P3: produc357393685978478089rm_a_b > $o] :
( ( ( collec99177395925924084rm_a_b @ P3 )
= bot_bo197521221353338581rm_a_b )
= ( ! [X4: produc357393685978478089rm_a_b] :
~ ( P3 @ X4 ) ) ) ).
% Collect_empty_eq
thf(fact_171_Collect__empty__eq,axiom,
! [P3: term_a_b > $o] :
( ( ( collect_term_a_b @ P3 )
= bot_bot_set_term_a_b )
= ( ! [X4: term_a_b] :
~ ( P3 @ X4 ) ) ) ).
% Collect_empty_eq
thf(fact_172_all__not__in__conv,axiom,
! [A4: set_Pr2659102613823815246st_nat] :
( ( ! [X4: produc2423145444804056984st_nat] :
~ ( member190277321972437807st_nat @ X4 @ A4 ) )
= ( A4 = bot_bo6857980842747739362st_nat ) ) ).
% all_not_in_conv
thf(fact_173_all__not__in__conv,axiom,
! [A4: set_Pr1829413238147913556rm_a_b] :
( ( ! [X4: produc6373727562496775070rm_a_b] :
~ ( member4140859439665155893rm_a_b @ X4 @ A4 ) )
= ( A4 = bot_bo6028291467071837672rm_a_b ) ) ).
% all_not_in_conv
thf(fact_174_all__not__in__conv,axiom,
! [A4: set_Pr3451248702717554689st_nat] :
( ( ! [X4: produc1828647624359046049st_nat] :
~ ( member7340969449405702474st_nat @ X4 @ A4 ) )
= ( A4 = bot_bo8485564385918328941st_nat ) ) ).
% all_not_in_conv
thf(fact_175_all__not__in__conv,axiom,
! [A4: set_set_term_a_b] :
( ( ! [X4: set_term_a_b] :
~ ( member_set_term_a_b @ X4 @ A4 ) )
= ( A4 = bot_bo7738030261494536212rm_a_b ) ) ).
% all_not_in_conv
thf(fact_176_all__not__in__conv,axiom,
! [A4: set_term_a_b] :
( ( ! [X4: term_a_b] :
~ ( member_term_a_b @ X4 @ A4 ) )
= ( A4 = bot_bot_set_term_a_b ) ) ).
% all_not_in_conv
thf(fact_177_all__not__in__conv,axiom,
! [A4: set_list_nat] :
( ( ! [X4: list_nat] :
~ ( member_list_nat @ X4 @ A4 ) )
= ( A4 = bot_bot_set_list_nat ) ) ).
% all_not_in_conv
thf(fact_178_all__not__in__conv,axiom,
! [A4: set_Pr4386577575007340137rm_a_b] :
( ( ! [X4: produc357393685978478089rm_a_b] :
~ ( member5869715511025134514rm_a_b @ X4 @ A4 ) )
= ( A4 = bot_bo197521221353338581rm_a_b ) ) ).
% all_not_in_conv
thf(fact_179_empty__iff,axiom,
! [C2: produc2423145444804056984st_nat] :
~ ( member190277321972437807st_nat @ C2 @ bot_bo6857980842747739362st_nat ) ).
% empty_iff
thf(fact_180_empty__iff,axiom,
! [C2: produc6373727562496775070rm_a_b] :
~ ( member4140859439665155893rm_a_b @ C2 @ bot_bo6028291467071837672rm_a_b ) ).
% empty_iff
thf(fact_181_empty__iff,axiom,
! [C2: produc1828647624359046049st_nat] :
~ ( member7340969449405702474st_nat @ C2 @ bot_bo8485564385918328941st_nat ) ).
% empty_iff
thf(fact_182_empty__iff,axiom,
! [C2: set_term_a_b] :
~ ( member_set_term_a_b @ C2 @ bot_bo7738030261494536212rm_a_b ) ).
% empty_iff
thf(fact_183_empty__iff,axiom,
! [C2: term_a_b] :
~ ( member_term_a_b @ C2 @ bot_bot_set_term_a_b ) ).
% empty_iff
thf(fact_184_empty__iff,axiom,
! [C2: list_nat] :
~ ( member_list_nat @ C2 @ bot_bot_set_list_nat ) ).
% empty_iff
thf(fact_185_empty__iff,axiom,
! [C2: produc357393685978478089rm_a_b] :
~ ( member5869715511025134514rm_a_b @ C2 @ bot_bo197521221353338581rm_a_b ) ).
% empty_iff
thf(fact_186_bot__apply,axiom,
( bot_bo23760184229683272_a_b_o
= ( ^ [X4: produc357393685978478089rm_a_b] : bot_bot_o ) ) ).
% bot_apply
thf(fact_187_bot__apply,axiom,
( bot_bot_list_nat_o
= ( ^ [X4: list_nat] : bot_bot_o ) ) ).
% bot_apply
thf(fact_188_ctxt__at__pos__hole__pos,axiom,
! [C4: subterm_and_ctxt_a_b,S: term_a_b] :
( ( term_ctxt_at_pos_a_b @ ( subter2376574525758040790rm_a_b @ C4 @ S ) @ ( term_hole_pos_a_b @ C4 ) )
= C4 ) ).
% ctxt_at_pos_hole_pos
thf(fact_189_subst__apply__term__ctxt__apply__distrib,axiom,
! [C4: subterm_and_ctxt_a_b,T: term_a_b,Mu: b > term_a_b] :
( ( subst_7999470309526761004_a_b_b @ ( subter2376574525758040790rm_a_b @ C4 @ T ) @ Mu )
= ( subter2376574525758040790rm_a_b @ ( subter4086027020339385547_a_b_b @ C4 @ Mu ) @ ( subst_7999470309526761004_a_b_b @ T @ Mu ) ) ) ).
% subst_apply_term_ctxt_apply_distrib
thf(fact_190_replace__at__hole__pos,axiom,
! [C4: subterm_and_ctxt_a_b,S: term_a_b,T: term_a_b] :
( ( term_r6860082780075436317at_a_b @ ( subter2376574525758040790rm_a_b @ C4 @ S ) @ ( term_hole_pos_a_b @ C4 ) @ T )
= ( subter2376574525758040790rm_a_b @ C4 @ T ) ) ).
% replace_at_hole_pos
thf(fact_191_ctxt__apply__subt__at__hole__pos,axiom,
! [C4: subterm_and_ctxt_a_b,S: term_a_b] :
( ( term_subt_at_a_b @ ( subter2376574525758040790rm_a_b @ C4 @ S ) @ ( term_hole_pos_a_b @ C4 ) )
= S ) ).
% ctxt_apply_subt_at_hole_pos
thf(fact_192_possc__subt__at__ctxt__apply,axiom,
! [P2: list_nat,C4: subterm_and_ctxt_a_b,S: term_a_b,T: term_a_b] :
( ( member_list_nat @ P2 @ ( terms_possc_a_b @ C4 ) )
=> ( ( term_p5017330785391824242ar_nat @ P2 @ ( term_hole_pos_a_b @ C4 ) )
=> ( ( term_subt_at_a_b @ ( subter2376574525758040790rm_a_b @ C4 @ S ) @ P2 )
= ( term_subt_at_a_b @ ( subter2376574525758040790rm_a_b @ C4 @ T ) @ P2 ) ) ) ) ).
% possc_subt_at_ctxt_apply
thf(fact_193_bot__set__def,axiom,
( bot_bot_set_list_nat
= ( collect_list_nat @ bot_bot_list_nat_o ) ) ).
% bot_set_def
thf(fact_194_bot__set__def,axiom,
( bot_bo197521221353338581rm_a_b
= ( collec99177395925924084rm_a_b @ bot_bo23760184229683272_a_b_o ) ) ).
% bot_set_def
thf(fact_195_bot__set__def,axiom,
( bot_bot_set_term_a_b
= ( collect_term_a_b @ bot_bot_term_a_b_o ) ) ).
% bot_set_def
thf(fact_196_hole__pos__in__ctxt__apply,axiom,
! [C4: subterm_and_ctxt_a_b,U: term_a_b] : ( member_list_nat @ ( term_hole_pos_a_b @ C4 ) @ ( term_poss_a_b @ ( subter2376574525758040790rm_a_b @ C4 @ U ) ) ) ).
% hole_pos_in_ctxt_apply
thf(fact_197_bot__fun__def,axiom,
( bot_bo23760184229683272_a_b_o
= ( ^ [X4: produc357393685978478089rm_a_b] : bot_bot_o ) ) ).
% bot_fun_def
thf(fact_198_bot__fun__def,axiom,
( bot_bot_list_nat_o
= ( ^ [X4: list_nat] : bot_bot_o ) ) ).
% bot_fun_def
thf(fact_199_emptyE,axiom,
! [A: produc2423145444804056984st_nat] :
~ ( member190277321972437807st_nat @ A @ bot_bo6857980842747739362st_nat ) ).
% emptyE
thf(fact_200_emptyE,axiom,
! [A: produc6373727562496775070rm_a_b] :
~ ( member4140859439665155893rm_a_b @ A @ bot_bo6028291467071837672rm_a_b ) ).
% emptyE
thf(fact_201_emptyE,axiom,
! [A: produc1828647624359046049st_nat] :
~ ( member7340969449405702474st_nat @ A @ bot_bo8485564385918328941st_nat ) ).
% emptyE
thf(fact_202_emptyE,axiom,
! [A: set_term_a_b] :
~ ( member_set_term_a_b @ A @ bot_bo7738030261494536212rm_a_b ) ).
% emptyE
thf(fact_203_emptyE,axiom,
! [A: term_a_b] :
~ ( member_term_a_b @ A @ bot_bot_set_term_a_b ) ).
% emptyE
thf(fact_204_emptyE,axiom,
! [A: list_nat] :
~ ( member_list_nat @ A @ bot_bot_set_list_nat ) ).
% emptyE
thf(fact_205_emptyE,axiom,
! [A: produc357393685978478089rm_a_b] :
~ ( member5869715511025134514rm_a_b @ A @ bot_bo197521221353338581rm_a_b ) ).
% emptyE
thf(fact_206_equals0D,axiom,
! [A4: set_Pr2659102613823815246st_nat,A: produc2423145444804056984st_nat] :
( ( A4 = bot_bo6857980842747739362st_nat )
=> ~ ( member190277321972437807st_nat @ A @ A4 ) ) ).
% equals0D
thf(fact_207_equals0D,axiom,
! [A4: set_Pr1829413238147913556rm_a_b,A: produc6373727562496775070rm_a_b] :
( ( A4 = bot_bo6028291467071837672rm_a_b )
=> ~ ( member4140859439665155893rm_a_b @ A @ A4 ) ) ).
% equals0D
thf(fact_208_equals0D,axiom,
! [A4: set_Pr3451248702717554689st_nat,A: produc1828647624359046049st_nat] :
( ( A4 = bot_bo8485564385918328941st_nat )
=> ~ ( member7340969449405702474st_nat @ A @ A4 ) ) ).
% equals0D
thf(fact_209_equals0D,axiom,
! [A4: set_set_term_a_b,A: set_term_a_b] :
( ( A4 = bot_bo7738030261494536212rm_a_b )
=> ~ ( member_set_term_a_b @ A @ A4 ) ) ).
% equals0D
thf(fact_210_equals0D,axiom,
! [A4: set_term_a_b,A: term_a_b] :
( ( A4 = bot_bot_set_term_a_b )
=> ~ ( member_term_a_b @ A @ A4 ) ) ).
% equals0D
thf(fact_211_equals0D,axiom,
! [A4: set_list_nat,A: list_nat] :
( ( A4 = bot_bot_set_list_nat )
=> ~ ( member_list_nat @ A @ A4 ) ) ).
% equals0D
thf(fact_212_equals0D,axiom,
! [A4: set_Pr4386577575007340137rm_a_b,A: produc357393685978478089rm_a_b] :
( ( A4 = bot_bo197521221353338581rm_a_b )
=> ~ ( member5869715511025134514rm_a_b @ A @ A4 ) ) ).
% equals0D
thf(fact_213_equals0I,axiom,
! [A4: set_Pr2659102613823815246st_nat] :
( ! [Y3: produc2423145444804056984st_nat] :
~ ( member190277321972437807st_nat @ Y3 @ A4 )
=> ( A4 = bot_bo6857980842747739362st_nat ) ) ).
% equals0I
thf(fact_214_equals0I,axiom,
! [A4: set_Pr1829413238147913556rm_a_b] :
( ! [Y3: produc6373727562496775070rm_a_b] :
~ ( member4140859439665155893rm_a_b @ Y3 @ A4 )
=> ( A4 = bot_bo6028291467071837672rm_a_b ) ) ).
% equals0I
thf(fact_215_equals0I,axiom,
! [A4: set_Pr3451248702717554689st_nat] :
( ! [Y3: produc1828647624359046049st_nat] :
~ ( member7340969449405702474st_nat @ Y3 @ A4 )
=> ( A4 = bot_bo8485564385918328941st_nat ) ) ).
% equals0I
thf(fact_216_equals0I,axiom,
! [A4: set_set_term_a_b] :
( ! [Y3: set_term_a_b] :
~ ( member_set_term_a_b @ Y3 @ A4 )
=> ( A4 = bot_bo7738030261494536212rm_a_b ) ) ).
% equals0I
thf(fact_217_equals0I,axiom,
! [A4: set_term_a_b] :
( ! [Y3: term_a_b] :
~ ( member_term_a_b @ Y3 @ A4 )
=> ( A4 = bot_bot_set_term_a_b ) ) ).
% equals0I
thf(fact_218_equals0I,axiom,
! [A4: set_list_nat] :
( ! [Y3: list_nat] :
~ ( member_list_nat @ Y3 @ A4 )
=> ( A4 = bot_bot_set_list_nat ) ) ).
% equals0I
thf(fact_219_equals0I,axiom,
! [A4: set_Pr4386577575007340137rm_a_b] :
( ! [Y3: produc357393685978478089rm_a_b] :
~ ( member5869715511025134514rm_a_b @ Y3 @ A4 )
=> ( A4 = bot_bo197521221353338581rm_a_b ) ) ).
% equals0I
thf(fact_220_ex__in__conv,axiom,
! [A4: set_Pr2659102613823815246st_nat] :
( ( ? [X4: produc2423145444804056984st_nat] : ( member190277321972437807st_nat @ X4 @ A4 ) )
= ( A4 != bot_bo6857980842747739362st_nat ) ) ).
% ex_in_conv
thf(fact_221_ex__in__conv,axiom,
! [A4: set_Pr1829413238147913556rm_a_b] :
( ( ? [X4: produc6373727562496775070rm_a_b] : ( member4140859439665155893rm_a_b @ X4 @ A4 ) )
= ( A4 != bot_bo6028291467071837672rm_a_b ) ) ).
% ex_in_conv
thf(fact_222_ex__in__conv,axiom,
! [A4: set_Pr3451248702717554689st_nat] :
( ( ? [X4: produc1828647624359046049st_nat] : ( member7340969449405702474st_nat @ X4 @ A4 ) )
= ( A4 != bot_bo8485564385918328941st_nat ) ) ).
% ex_in_conv
thf(fact_223_ex__in__conv,axiom,
! [A4: set_set_term_a_b] :
( ( ? [X4: set_term_a_b] : ( member_set_term_a_b @ X4 @ A4 ) )
= ( A4 != bot_bo7738030261494536212rm_a_b ) ) ).
% ex_in_conv
thf(fact_224_ex__in__conv,axiom,
! [A4: set_term_a_b] :
( ( ? [X4: term_a_b] : ( member_term_a_b @ X4 @ A4 ) )
= ( A4 != bot_bot_set_term_a_b ) ) ).
% ex_in_conv
thf(fact_225_ex__in__conv,axiom,
! [A4: set_list_nat] :
( ( ? [X4: list_nat] : ( member_list_nat @ X4 @ A4 ) )
= ( A4 != bot_bot_set_list_nat ) ) ).
% ex_in_conv
thf(fact_226_ex__in__conv,axiom,
! [A4: set_Pr4386577575007340137rm_a_b] :
( ( ? [X4: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ X4 @ A4 ) )
= ( A4 != bot_bo197521221353338581rm_a_b ) ) ).
% ex_in_conv
thf(fact_227_par__hole__pos__replace__term__context__at,axiom,
! [P2: list_nat,C4: subterm_and_ctxt_a_b,S: term_a_b,U: term_a_b] :
( ( term_p5017330785391824242ar_nat @ P2 @ ( term_hole_pos_a_b @ C4 ) )
=> ( ( term_r6860082780075436317at_a_b @ ( subter2376574525758040790rm_a_b @ C4 @ S ) @ P2 @ U )
= ( subter2376574525758040790rm_a_b @ ( terms_4774307173741787698at_a_b @ C4 @ P2 @ U ) @ S ) ) ) ).
% par_hole_pos_replace_term_context_at
thf(fact_228_Set_Ois__empty__def,axiom,
( is_empty_list_nat
= ( ^ [A6: set_list_nat] : ( A6 = bot_bot_set_list_nat ) ) ) ).
% Set.is_empty_def
thf(fact_229_Set_Ois__empty__def,axiom,
( is_emp1328655501402583003rm_a_b
= ( ^ [A6: set_Pr4386577575007340137rm_a_b] : ( A6 = bot_bo197521221353338581rm_a_b ) ) ) ).
% Set.is_empty_def
thf(fact_230_Set_Ois__empty__def,axiom,
( is_empty_term_a_b
= ( ^ [A6: set_term_a_b] : ( A6 = bot_bot_set_term_a_b ) ) ) ).
% Set.is_empty_def
thf(fact_231_ctxt__apply__term__replace__term__hole__pos,axiom,
! [C4: subterm_and_ctxt_a_b,S: term_a_b,Q: list_nat,U: term_a_b] :
( ( term_r6860082780075436317at_a_b @ ( subter2376574525758040790rm_a_b @ C4 @ S ) @ ( append_nat @ ( term_hole_pos_a_b @ C4 ) @ Q ) @ U )
= ( subter2376574525758040790rm_a_b @ C4 @ ( term_r6860082780075436317at_a_b @ S @ Q @ U ) ) ) ).
% ctxt_apply_term_replace_term_hole_pos
thf(fact_232_ctxt__apply__term__subt__at__hole__pos,axiom,
! [C4: subterm_and_ctxt_a_b,S: term_a_b,Q: list_nat] :
( ( term_subt_at_a_b @ ( subter2376574525758040790rm_a_b @ C4 @ S ) @ ( append_nat @ ( term_hole_pos_a_b @ C4 ) @ Q ) )
= ( term_subt_at_a_b @ S @ Q ) ) ).
% ctxt_apply_term_subt_at_hole_pos
thf(fact_233_poss__ctxt__apply,axiom,
! [C4: subterm_and_ctxt_a_b,P2: list_nat,S: term_a_b] :
( ( member_list_nat @ ( append_nat @ ( term_hole_pos_a_b @ C4 ) @ P2 ) @ ( term_poss_a_b @ ( subter2376574525758040790rm_a_b @ C4 @ S ) ) )
= ( member_list_nat @ P2 @ ( term_poss_a_b @ S ) ) ) ).
% poss_ctxt_apply
thf(fact_234_sig__steps__join__ctxt__closed,axiom,
! [C4: subterm_and_ctxt_a_b,F: set_Pr4934435412358123699_a_nat,S: term_a_b,T: term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
( ( ord_le8666007276011122963_a_nat @ ( term_funas_ctxt_a_b @ C4 ) @ F )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( abstra4096080454567261402rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R3 ) ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( subter2376574525758040790rm_a_b @ C4 @ S ) @ ( subter2376574525758040790rm_a_b @ C4 @ T ) ) @ ( abstra4096080454567261402rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R3 ) ) ) ) ) ) ).
% sig_steps_join_ctxt_closed
thf(fact_235_bot__empty__eq,axiom,
( bot_bo1794539809004266019_nat_o
= ( ^ [X4: produc2423145444804056984st_nat] : ( member190277321972437807st_nat @ X4 @ bot_bo6857980842747739362st_nat ) ) ) ).
% bot_empty_eq
thf(fact_236_bot__empty__eq,axiom,
( bot_bo6896037246099296413_a_b_o
= ( ^ [X4: produc6373727562496775070rm_a_b] : ( member4140859439665155893rm_a_b @ X4 @ bot_bo6028291467071837672rm_a_b ) ) ) ).
% bot_empty_eq
thf(fact_237_bot__empty__eq,axiom,
( bot_bo2074314105715143856_nat_o
= ( ^ [X4: produc1828647624359046049st_nat] : ( member7340969449405702474st_nat @ X4 @ bot_bo8485564385918328941st_nat ) ) ) ).
% bot_empty_eq
thf(fact_238_bot__empty__eq,axiom,
( bot_bo7197193537753692017_a_b_o
= ( ^ [X4: set_term_a_b] : ( member_set_term_a_b @ X4 @ bot_bo7738030261494536212rm_a_b ) ) ) ).
% bot_empty_eq
thf(fact_239_bot__empty__eq,axiom,
( bot_bot_term_a_b_o
= ( ^ [X4: term_a_b] : ( member_term_a_b @ X4 @ bot_bot_set_term_a_b ) ) ) ).
% bot_empty_eq
thf(fact_240_bot__empty__eq,axiom,
( bot_bot_list_nat_o
= ( ^ [X4: list_nat] : ( member_list_nat @ X4 @ bot_bot_set_list_nat ) ) ) ).
% bot_empty_eq
thf(fact_241_bot__empty__eq,axiom,
( bot_bo23760184229683272_a_b_o
= ( ^ [X4: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ X4 @ bot_bo197521221353338581rm_a_b ) ) ) ).
% bot_empty_eq
thf(fact_242_Collect__empty__eq__bot,axiom,
! [P3: list_nat > $o] :
( ( ( collect_list_nat @ P3 )
= bot_bot_set_list_nat )
= ( P3 = bot_bot_list_nat_o ) ) ).
% Collect_empty_eq_bot
thf(fact_243_Collect__empty__eq__bot,axiom,
! [P3: produc357393685978478089rm_a_b > $o] :
( ( ( collec99177395925924084rm_a_b @ P3 )
= bot_bo197521221353338581rm_a_b )
= ( P3 = bot_bo23760184229683272_a_b_o ) ) ).
% Collect_empty_eq_bot
thf(fact_244_Collect__empty__eq__bot,axiom,
! [P3: term_a_b > $o] :
( ( ( collect_term_a_b @ P3 )
= bot_bot_set_term_a_b )
= ( P3 = bot_bot_term_a_b_o ) ) ).
% Collect_empty_eq_bot
thf(fact_245_subst__apply__left__idemp,axiom,
! [Sigma2: b > term_a_b,X: b,T: term_a_b,S: term_a_b] :
( ( ( Sigma2 @ X )
= ( subst_7999470309526761004_a_b_b @ T @ Sigma2 ) )
=> ( ( subst_7999470309526761004_a_b_b @ ( subst_7999470309526761004_a_b_b @ S @ ( subst_b_a @ X @ T ) ) @ Sigma2 )
= ( subst_7999470309526761004_a_b_b @ S @ Sigma2 ) ) ) ).
% subst_apply_left_idemp
thf(fact_246_order__refl,axiom,
! [X: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ X @ X ) ).
% order_refl
thf(fact_247_order__refl,axiom,
! [X: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ X @ X ) ).
% order_refl
thf(fact_248_order__refl,axiom,
! [X: set_list_nat] : ( ord_le6045566169113846134st_nat @ X @ X ) ).
% order_refl
thf(fact_249_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_250_order__refl,axiom,
! [X: set_term_a_b] : ( ord_le2705286416250468010rm_a_b @ X @ X ) ).
% order_refl
thf(fact_251_dual__order_Orefl,axiom,
! [A: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ A @ A ) ).
% dual_order.refl
thf(fact_252_dual__order_Orefl,axiom,
! [A: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ A @ A ) ).
% dual_order.refl
thf(fact_253_dual__order_Orefl,axiom,
! [A: set_list_nat] : ( ord_le6045566169113846134st_nat @ A @ A ) ).
% dual_order.refl
thf(fact_254_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_255_dual__order_Orefl,axiom,
! [A: set_term_a_b] : ( ord_le2705286416250468010rm_a_b @ A @ A ) ).
% dual_order.refl
thf(fact_256_subsetI,axiom,
! [A4: set_Pr2659102613823815246st_nat,B5: set_Pr2659102613823815246st_nat] :
( ! [X3: produc2423145444804056984st_nat] :
( ( member190277321972437807st_nat @ X3 @ A4 )
=> ( member190277321972437807st_nat @ X3 @ B5 ) )
=> ( ord_le5191068806740926126st_nat @ A4 @ B5 ) ) ).
% subsetI
thf(fact_257_subsetI,axiom,
! [A4: set_Pr1829413238147913556rm_a_b,B5: set_Pr1829413238147913556rm_a_b] :
( ! [X3: produc6373727562496775070rm_a_b] :
( ( member4140859439665155893rm_a_b @ X3 @ A4 )
=> ( member4140859439665155893rm_a_b @ X3 @ B5 ) )
=> ( ord_le4361379431065024436rm_a_b @ A4 @ B5 ) ) ).
% subsetI
thf(fact_258_subsetI,axiom,
! [A4: set_Pr3451248702717554689st_nat,B5: set_Pr3451248702717554689st_nat] :
( ! [X3: produc1828647624359046049st_nat] :
( ( member7340969449405702474st_nat @ X3 @ A4 )
=> ( member7340969449405702474st_nat @ X3 @ B5 ) )
=> ( ord_le8406513867147106209st_nat @ A4 @ B5 ) ) ).
% subsetI
thf(fact_259_subsetI,axiom,
! [A4: set_set_term_a_b,B5: set_set_term_a_b] :
( ! [X3: set_term_a_b] :
( ( member_set_term_a_b @ X3 @ A4 )
=> ( member_set_term_a_b @ X3 @ B5 ) )
=> ( ord_le4920510218098530272rm_a_b @ A4 @ B5 ) ) ).
% subsetI
thf(fact_260_subsetI,axiom,
! [A4: set_Pr4934435412358123699_a_nat,B5: set_Pr4934435412358123699_a_nat] :
( ! [X3: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ X3 @ A4 )
=> ( member5724188588386418708_a_nat @ X3 @ B5 ) )
=> ( ord_le8666007276011122963_a_nat @ A4 @ B5 ) ) ).
% subsetI
thf(fact_261_subsetI,axiom,
! [A4: set_term_a_b,B5: set_term_a_b] :
( ! [X3: term_a_b] :
( ( member_term_a_b @ X3 @ A4 )
=> ( member_term_a_b @ X3 @ B5 ) )
=> ( ord_le2705286416250468010rm_a_b @ A4 @ B5 ) ) ).
% subsetI
thf(fact_262_subsetI,axiom,
! [A4: set_list_nat,B5: set_list_nat] :
( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ A4 )
=> ( member_list_nat @ X3 @ B5 ) )
=> ( ord_le6045566169113846134st_nat @ A4 @ B5 ) ) ).
% subsetI
thf(fact_263_subsetI,axiom,
! [A4: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
( ! [X3: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X3 @ A4 )
=> ( member5869715511025134514rm_a_b @ X3 @ B5 ) )
=> ( ord_le118470702582115849rm_a_b @ A4 @ B5 ) ) ).
% subsetI
thf(fact_264_subset__antisym,axiom,
! [A4: set_Pr4934435412358123699_a_nat,B5: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A4 @ B5 )
=> ( ( ord_le8666007276011122963_a_nat @ B5 @ A4 )
=> ( A4 = B5 ) ) ) ).
% subset_antisym
thf(fact_265_subset__antisym,axiom,
! [A4: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A4 @ B5 )
=> ( ( ord_le118470702582115849rm_a_b @ B5 @ A4 )
=> ( A4 = B5 ) ) ) ).
% subset_antisym
thf(fact_266_subset__antisym,axiom,
! [A4: set_list_nat,B5: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A4 @ B5 )
=> ( ( ord_le6045566169113846134st_nat @ B5 @ A4 )
=> ( A4 = B5 ) ) ) ).
% subset_antisym
thf(fact_267_subset__antisym,axiom,
! [A4: set_term_a_b,B5: set_term_a_b] :
( ( ord_le2705286416250468010rm_a_b @ A4 @ B5 )
=> ( ( ord_le2705286416250468010rm_a_b @ B5 @ A4 )
=> ( A4 = B5 ) ) ) ).
% subset_antisym
thf(fact_268_subst__simps_I1_J,axiom,
! [X: b,T: term_a_b] :
( ( subst_b_a @ X @ T @ X )
= T ) ).
% subst_simps(1)
thf(fact_269_empty__subsetI,axiom,
! [A4: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ bot_bo9049108969261143879_a_nat @ A4 ) ).
% empty_subsetI
thf(fact_270_empty__subsetI,axiom,
! [A4: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ bot_bo197521221353338581rm_a_b @ A4 ) ).
% empty_subsetI
thf(fact_271_empty__subsetI,axiom,
! [A4: set_list_nat] : ( ord_le6045566169113846134st_nat @ bot_bot_set_list_nat @ A4 ) ).
% empty_subsetI
thf(fact_272_empty__subsetI,axiom,
! [A4: set_term_a_b] : ( ord_le2705286416250468010rm_a_b @ bot_bot_set_term_a_b @ A4 ) ).
% empty_subsetI
thf(fact_273_subset__empty,axiom,
! [A4: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A4 @ bot_bo9049108969261143879_a_nat )
= ( A4 = bot_bo9049108969261143879_a_nat ) ) ).
% subset_empty
thf(fact_274_subset__empty,axiom,
! [A4: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A4 @ bot_bo197521221353338581rm_a_b )
= ( A4 = bot_bo197521221353338581rm_a_b ) ) ).
% subset_empty
thf(fact_275_subset__empty,axiom,
! [A4: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A4 @ bot_bot_set_list_nat )
= ( A4 = bot_bot_set_list_nat ) ) ).
% subset_empty
thf(fact_276_subset__empty,axiom,
! [A4: set_term_a_b] :
( ( ord_le2705286416250468010rm_a_b @ A4 @ bot_bot_set_term_a_b )
= ( A4 = bot_bot_set_term_a_b ) ) ).
% subset_empty
thf(fact_277_replace__subterm__at__itself,axiom,
! [S: term_a_b,P2: list_nat,Q: list_nat,T: term_a_b] :
( ( term_r6860082780075436317at_a_b @ S @ P2 @ ( term_r6860082780075436317at_a_b @ ( term_subt_at_a_b @ S @ P2 ) @ Q @ T ) )
= ( term_r6860082780075436317at_a_b @ S @ ( append_nat @ P2 @ Q ) @ T ) ) ).
% replace_subterm_at_itself
thf(fact_278_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_279_le__cases3,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ( ord_less_eq_nat @ X @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z ) )
=> ( ( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_eq_nat @ X @ Z ) )
=> ( ( ( ord_less_eq_nat @ X @ Z )
=> ~ ( ord_less_eq_nat @ Z @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z )
=> ~ ( ord_less_eq_nat @ Z @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z @ X )
=> ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_280_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_Pr4934435412358123699_a_nat,Z2: set_Pr4934435412358123699_a_nat] : ( Y4 = Z2 ) )
= ( ^ [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_281_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_Pr4386577575007340137rm_a_b,Z2: set_Pr4386577575007340137rm_a_b] : ( Y4 = Z2 ) )
= ( ^ [X4: set_Pr4386577575007340137rm_a_b,Y5: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ X4 @ Y5 )
& ( ord_le118470702582115849rm_a_b @ Y5 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_282_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_list_nat,Z2: set_list_nat] : ( Y4 = Z2 ) )
= ( ^ [X4: set_list_nat,Y5: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ X4 @ Y5 )
& ( ord_le6045566169113846134st_nat @ Y5 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_283_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 ) )
= ( ^ [X4: nat,Y5: nat] :
( ( ord_less_eq_nat @ X4 @ Y5 )
& ( ord_less_eq_nat @ Y5 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_284_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_term_a_b,Z2: set_term_a_b] : ( Y4 = Z2 ) )
= ( ^ [X4: set_term_a_b,Y5: set_term_a_b] :
( ( ord_le2705286416250468010rm_a_b @ X4 @ Y5 )
& ( ord_le2705286416250468010rm_a_b @ Y5 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_285_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_286_ord__eq__le__trans,axiom,
! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b,C2: set_Pr4386577575007340137rm_a_b] :
( ( A = B )
=> ( ( ord_le118470702582115849rm_a_b @ B @ C2 )
=> ( ord_le118470702582115849rm_a_b @ A @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_287_ord__eq__le__trans,axiom,
! [A: set_list_nat,B: set_list_nat,C2: set_list_nat] :
( ( A = B )
=> ( ( ord_le6045566169113846134st_nat @ B @ C2 )
=> ( ord_le6045566169113846134st_nat @ A @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_288_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_289_ord__eq__le__trans,axiom,
! [A: set_term_a_b,B: set_term_a_b,C2: set_term_a_b] :
( ( A = B )
=> ( ( ord_le2705286416250468010rm_a_b @ B @ C2 )
=> ( ord_le2705286416250468010rm_a_b @ A @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_290_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_291_ord__le__eq__trans,axiom,
! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b,C2: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A @ B )
=> ( ( B = C2 )
=> ( ord_le118470702582115849rm_a_b @ A @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_292_ord__le__eq__trans,axiom,
! [A: set_list_nat,B: set_list_nat,C2: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A @ B )
=> ( ( B = C2 )
=> ( ord_le6045566169113846134st_nat @ A @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_293_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_294_ord__le__eq__trans,axiom,
! [A: set_term_a_b,B: set_term_a_b,C2: set_term_a_b] :
( ( ord_le2705286416250468010rm_a_b @ A @ B )
=> ( ( B = C2 )
=> ( ord_le2705286416250468010rm_a_b @ A @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_295_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_296_order__antisym,axiom,
! [X: set_Pr4386577575007340137rm_a_b,Y: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ X @ Y )
=> ( ( ord_le118470702582115849rm_a_b @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_297_order__antisym,axiom,
! [X: set_list_nat,Y: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ X @ Y )
=> ( ( ord_le6045566169113846134st_nat @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_298_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_299_order__antisym,axiom,
! [X: set_term_a_b,Y: set_term_a_b] :
( ( ord_le2705286416250468010rm_a_b @ X @ Y )
=> ( ( ord_le2705286416250468010rm_a_b @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_300_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_301_order_Otrans,axiom,
! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b,C2: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A @ B )
=> ( ( ord_le118470702582115849rm_a_b @ B @ C2 )
=> ( ord_le118470702582115849rm_a_b @ A @ C2 ) ) ) ).
% order.trans
thf(fact_302_order_Otrans,axiom,
! [A: set_list_nat,B: set_list_nat,C2: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A @ B )
=> ( ( ord_le6045566169113846134st_nat @ B @ C2 )
=> ( ord_le6045566169113846134st_nat @ A @ C2 ) ) ) ).
% order.trans
thf(fact_303_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_304_order_Otrans,axiom,
! [A: set_term_a_b,B: set_term_a_b,C2: set_term_a_b] :
( ( ord_le2705286416250468010rm_a_b @ A @ B )
=> ( ( ord_le2705286416250468010rm_a_b @ B @ C2 )
=> ( ord_le2705286416250468010rm_a_b @ A @ C2 ) ) ) ).
% order.trans
thf(fact_305_order__trans,axiom,
! [X: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat,Z: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X @ Y )
=> ( ( ord_le8666007276011122963_a_nat @ Y @ Z )
=> ( ord_le8666007276011122963_a_nat @ X @ Z ) ) ) ).
% order_trans
thf(fact_306_order__trans,axiom,
! [X: set_Pr4386577575007340137rm_a_b,Y: set_Pr4386577575007340137rm_a_b,Z: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ X @ Y )
=> ( ( ord_le118470702582115849rm_a_b @ Y @ Z )
=> ( ord_le118470702582115849rm_a_b @ X @ Z ) ) ) ).
% order_trans
thf(fact_307_order__trans,axiom,
! [X: set_list_nat,Y: set_list_nat,Z: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ X @ Y )
=> ( ( ord_le6045566169113846134st_nat @ Y @ Z )
=> ( ord_le6045566169113846134st_nat @ X @ Z ) ) ) ).
% order_trans
thf(fact_308_order__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z )
=> ( ord_less_eq_nat @ X @ Z ) ) ) ).
% order_trans
thf(fact_309_order__trans,axiom,
! [X: set_term_a_b,Y: set_term_a_b,Z: set_term_a_b] :
( ( ord_le2705286416250468010rm_a_b @ X @ Y )
=> ( ( ord_le2705286416250468010rm_a_b @ Y @ Z )
=> ( ord_le2705286416250468010rm_a_b @ X @ Z ) ) ) ).
% order_trans
thf(fact_310_linorder__wlog,axiom,
! [P3: nat > nat > $o,A: nat,B: nat] :
( ! [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( P3 @ A3 @ B3 ) )
=> ( ! [A3: nat,B3: nat] :
( ( P3 @ B3 @ A3 )
=> ( P3 @ A3 @ B3 ) )
=> ( P3 @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_311_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_Pr4934435412358123699_a_nat,Z2: set_Pr4934435412358123699_a_nat] : ( Y4 = Z2 ) )
= ( ^ [A5: set_Pr4934435412358123699_a_nat,B4: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ B4 @ A5 )
& ( ord_le8666007276011122963_a_nat @ A5 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_312_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_Pr4386577575007340137rm_a_b,Z2: set_Pr4386577575007340137rm_a_b] : ( Y4 = Z2 ) )
= ( ^ [A5: set_Pr4386577575007340137rm_a_b,B4: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ B4 @ A5 )
& ( ord_le118470702582115849rm_a_b @ A5 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_313_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_list_nat,Z2: set_list_nat] : ( Y4 = Z2 ) )
= ( ^ [A5: set_list_nat,B4: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ B4 @ A5 )
& ( ord_le6045566169113846134st_nat @ A5 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_314_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 ) )
= ( ^ [A5: nat,B4: nat] :
( ( ord_less_eq_nat @ B4 @ A5 )
& ( ord_less_eq_nat @ A5 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_315_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_term_a_b,Z2: set_term_a_b] : ( Y4 = Z2 ) )
= ( ^ [A5: set_term_a_b,B4: set_term_a_b] :
( ( ord_le2705286416250468010rm_a_b @ B4 @ A5 )
& ( ord_le2705286416250468010rm_a_b @ A5 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_316_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_317_dual__order_Oantisym,axiom,
! [B: set_Pr4386577575007340137rm_a_b,A: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ B @ A )
=> ( ( ord_le118470702582115849rm_a_b @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_318_dual__order_Oantisym,axiom,
! [B: set_list_nat,A: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ B @ A )
=> ( ( ord_le6045566169113846134st_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_319_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_320_dual__order_Oantisym,axiom,
! [B: set_term_a_b,A: set_term_a_b] :
( ( ord_le2705286416250468010rm_a_b @ B @ A )
=> ( ( ord_le2705286416250468010rm_a_b @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_321_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_322_dual__order_Otrans,axiom,
! [B: set_Pr4386577575007340137rm_a_b,A: set_Pr4386577575007340137rm_a_b,C2: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ B @ A )
=> ( ( ord_le118470702582115849rm_a_b @ C2 @ B )
=> ( ord_le118470702582115849rm_a_b @ C2 @ A ) ) ) ).
% dual_order.trans
thf(fact_323_dual__order_Otrans,axiom,
! [B: set_list_nat,A: set_list_nat,C2: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ B @ A )
=> ( ( ord_le6045566169113846134st_nat @ C2 @ B )
=> ( ord_le6045566169113846134st_nat @ C2 @ A ) ) ) ).
% dual_order.trans
thf(fact_324_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_325_dual__order_Otrans,axiom,
! [B: set_term_a_b,A: set_term_a_b,C2: set_term_a_b] :
( ( ord_le2705286416250468010rm_a_b @ B @ A )
=> ( ( ord_le2705286416250468010rm_a_b @ C2 @ B )
=> ( ord_le2705286416250468010rm_a_b @ C2 @ A ) ) ) ).
% dual_order.trans
thf(fact_326_in__mono,axiom,
! [A4: set_Pr2659102613823815246st_nat,B5: set_Pr2659102613823815246st_nat,X: produc2423145444804056984st_nat] :
( ( ord_le5191068806740926126st_nat @ A4 @ B5 )
=> ( ( member190277321972437807st_nat @ X @ A4 )
=> ( member190277321972437807st_nat @ X @ B5 ) ) ) ).
% in_mono
thf(fact_327_in__mono,axiom,
! [A4: set_Pr1829413238147913556rm_a_b,B5: set_Pr1829413238147913556rm_a_b,X: produc6373727562496775070rm_a_b] :
( ( ord_le4361379431065024436rm_a_b @ A4 @ B5 )
=> ( ( member4140859439665155893rm_a_b @ X @ A4 )
=> ( member4140859439665155893rm_a_b @ X @ B5 ) ) ) ).
% in_mono
thf(fact_328_in__mono,axiom,
! [A4: set_Pr3451248702717554689st_nat,B5: set_Pr3451248702717554689st_nat,X: produc1828647624359046049st_nat] :
( ( ord_le8406513867147106209st_nat @ A4 @ B5 )
=> ( ( member7340969449405702474st_nat @ X @ A4 )
=> ( member7340969449405702474st_nat @ X @ B5 ) ) ) ).
% in_mono
thf(fact_329_in__mono,axiom,
! [A4: set_set_term_a_b,B5: set_set_term_a_b,X: set_term_a_b] :
( ( ord_le4920510218098530272rm_a_b @ A4 @ B5 )
=> ( ( member_set_term_a_b @ X @ A4 )
=> ( member_set_term_a_b @ X @ B5 ) ) ) ).
% in_mono
thf(fact_330_in__mono,axiom,
! [A4: set_Pr4934435412358123699_a_nat,B5: set_Pr4934435412358123699_a_nat,X: product_prod_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A4 @ B5 )
=> ( ( member5724188588386418708_a_nat @ X @ A4 )
=> ( member5724188588386418708_a_nat @ X @ B5 ) ) ) ).
% in_mono
thf(fact_331_in__mono,axiom,
! [A4: set_term_a_b,B5: set_term_a_b,X: term_a_b] :
( ( ord_le2705286416250468010rm_a_b @ A4 @ B5 )
=> ( ( member_term_a_b @ X @ A4 )
=> ( member_term_a_b @ X @ B5 ) ) ) ).
% in_mono
thf(fact_332_in__mono,axiom,
! [A4: set_list_nat,B5: set_list_nat,X: list_nat] :
( ( ord_le6045566169113846134st_nat @ A4 @ B5 )
=> ( ( member_list_nat @ X @ A4 )
=> ( member_list_nat @ X @ B5 ) ) ) ).
% in_mono
thf(fact_333_in__mono,axiom,
! [A4: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b,X: produc357393685978478089rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A4 @ B5 )
=> ( ( member5869715511025134514rm_a_b @ X @ A4 )
=> ( member5869715511025134514rm_a_b @ X @ B5 ) ) ) ).
% in_mono
thf(fact_334_subsetD,axiom,
! [A4: set_Pr2659102613823815246st_nat,B5: set_Pr2659102613823815246st_nat,C2: produc2423145444804056984st_nat] :
( ( ord_le5191068806740926126st_nat @ A4 @ B5 )
=> ( ( member190277321972437807st_nat @ C2 @ A4 )
=> ( member190277321972437807st_nat @ C2 @ B5 ) ) ) ).
% subsetD
thf(fact_335_subsetD,axiom,
! [A4: set_Pr1829413238147913556rm_a_b,B5: set_Pr1829413238147913556rm_a_b,C2: produc6373727562496775070rm_a_b] :
( ( ord_le4361379431065024436rm_a_b @ A4 @ B5 )
=> ( ( member4140859439665155893rm_a_b @ C2 @ A4 )
=> ( member4140859439665155893rm_a_b @ C2 @ B5 ) ) ) ).
% subsetD
thf(fact_336_subsetD,axiom,
! [A4: set_Pr3451248702717554689st_nat,B5: set_Pr3451248702717554689st_nat,C2: produc1828647624359046049st_nat] :
( ( ord_le8406513867147106209st_nat @ A4 @ B5 )
=> ( ( member7340969449405702474st_nat @ C2 @ A4 )
=> ( member7340969449405702474st_nat @ C2 @ B5 ) ) ) ).
% subsetD
thf(fact_337_subsetD,axiom,
! [A4: set_set_term_a_b,B5: set_set_term_a_b,C2: set_term_a_b] :
( ( ord_le4920510218098530272rm_a_b @ A4 @ B5 )
=> ( ( member_set_term_a_b @ C2 @ A4 )
=> ( member_set_term_a_b @ C2 @ B5 ) ) ) ).
% subsetD
thf(fact_338_subsetD,axiom,
! [A4: set_Pr4934435412358123699_a_nat,B5: set_Pr4934435412358123699_a_nat,C2: product_prod_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A4 @ B5 )
=> ( ( member5724188588386418708_a_nat @ C2 @ A4 )
=> ( member5724188588386418708_a_nat @ C2 @ B5 ) ) ) ).
% subsetD
thf(fact_339_subsetD,axiom,
! [A4: set_term_a_b,B5: set_term_a_b,C2: term_a_b] :
( ( ord_le2705286416250468010rm_a_b @ A4 @ B5 )
=> ( ( member_term_a_b @ C2 @ A4 )
=> ( member_term_a_b @ C2 @ B5 ) ) ) ).
% subsetD
thf(fact_340_subsetD,axiom,
! [A4: set_list_nat,B5: set_list_nat,C2: list_nat] :
( ( ord_le6045566169113846134st_nat @ A4 @ B5 )
=> ( ( member_list_nat @ C2 @ A4 )
=> ( member_list_nat @ C2 @ B5 ) ) ) ).
% subsetD
thf(fact_341_subsetD,axiom,
! [A4: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b,C2: produc357393685978478089rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A4 @ B5 )
=> ( ( member5869715511025134514rm_a_b @ C2 @ A4 )
=> ( member5869715511025134514rm_a_b @ C2 @ B5 ) ) ) ).
% subsetD
thf(fact_342_equalityE,axiom,
! [A4: set_Pr4934435412358123699_a_nat,B5: set_Pr4934435412358123699_a_nat] :
( ( A4 = B5 )
=> ~ ( ( ord_le8666007276011122963_a_nat @ A4 @ B5 )
=> ~ ( ord_le8666007276011122963_a_nat @ B5 @ A4 ) ) ) ).
% equalityE
thf(fact_343_equalityE,axiom,
! [A4: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
( ( A4 = B5 )
=> ~ ( ( ord_le118470702582115849rm_a_b @ A4 @ B5 )
=> ~ ( ord_le118470702582115849rm_a_b @ B5 @ A4 ) ) ) ).
% equalityE
thf(fact_344_equalityE,axiom,
! [A4: set_list_nat,B5: set_list_nat] :
( ( A4 = B5 )
=> ~ ( ( ord_le6045566169113846134st_nat @ A4 @ B5 )
=> ~ ( ord_le6045566169113846134st_nat @ B5 @ A4 ) ) ) ).
% equalityE
thf(fact_345_equalityE,axiom,
! [A4: set_term_a_b,B5: set_term_a_b] :
( ( A4 = B5 )
=> ~ ( ( ord_le2705286416250468010rm_a_b @ A4 @ B5 )
=> ~ ( ord_le2705286416250468010rm_a_b @ B5 @ A4 ) ) ) ).
% equalityE
thf(fact_346_subset__eq,axiom,
( ord_le5191068806740926126st_nat
= ( ^ [A6: set_Pr2659102613823815246st_nat,B6: set_Pr2659102613823815246st_nat] :
! [X4: produc2423145444804056984st_nat] :
( ( member190277321972437807st_nat @ X4 @ A6 )
=> ( member190277321972437807st_nat @ X4 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_347_subset__eq,axiom,
( ord_le4361379431065024436rm_a_b
= ( ^ [A6: set_Pr1829413238147913556rm_a_b,B6: set_Pr1829413238147913556rm_a_b] :
! [X4: produc6373727562496775070rm_a_b] :
( ( member4140859439665155893rm_a_b @ X4 @ A6 )
=> ( member4140859439665155893rm_a_b @ X4 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_348_subset__eq,axiom,
( ord_le8406513867147106209st_nat
= ( ^ [A6: set_Pr3451248702717554689st_nat,B6: set_Pr3451248702717554689st_nat] :
! [X4: produc1828647624359046049st_nat] :
( ( member7340969449405702474st_nat @ X4 @ A6 )
=> ( member7340969449405702474st_nat @ X4 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_349_subset__eq,axiom,
( ord_le4920510218098530272rm_a_b
= ( ^ [A6: set_set_term_a_b,B6: set_set_term_a_b] :
! [X4: set_term_a_b] :
( ( member_set_term_a_b @ X4 @ A6 )
=> ( member_set_term_a_b @ X4 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_350_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_351_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_352_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_353_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_354_equalityD1,axiom,
! [A4: set_Pr4934435412358123699_a_nat,B5: set_Pr4934435412358123699_a_nat] :
( ( A4 = B5 )
=> ( ord_le8666007276011122963_a_nat @ A4 @ B5 ) ) ).
% equalityD1
thf(fact_355_equalityD1,axiom,
! [A4: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
( ( A4 = B5 )
=> ( ord_le118470702582115849rm_a_b @ A4 @ B5 ) ) ).
% equalityD1
thf(fact_356_equalityD1,axiom,
! [A4: set_list_nat,B5: set_list_nat] :
( ( A4 = B5 )
=> ( ord_le6045566169113846134st_nat @ A4 @ B5 ) ) ).
% equalityD1
thf(fact_357_equalityD1,axiom,
! [A4: set_term_a_b,B5: set_term_a_b] :
( ( A4 = B5 )
=> ( ord_le2705286416250468010rm_a_b @ A4 @ B5 ) ) ).
% equalityD1
thf(fact_358_equalityD2,axiom,
! [A4: set_Pr4934435412358123699_a_nat,B5: set_Pr4934435412358123699_a_nat] :
( ( A4 = B5 )
=> ( ord_le8666007276011122963_a_nat @ B5 @ A4 ) ) ).
% equalityD2
thf(fact_359_equalityD2,axiom,
! [A4: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
( ( A4 = B5 )
=> ( ord_le118470702582115849rm_a_b @ B5 @ A4 ) ) ).
% equalityD2
thf(fact_360_equalityD2,axiom,
! [A4: set_list_nat,B5: set_list_nat] :
( ( A4 = B5 )
=> ( ord_le6045566169113846134st_nat @ B5 @ A4 ) ) ).
% equalityD2
thf(fact_361_equalityD2,axiom,
! [A4: set_term_a_b,B5: set_term_a_b] :
( ( A4 = B5 )
=> ( ord_le2705286416250468010rm_a_b @ B5 @ A4 ) ) ).
% equalityD2
thf(fact_362_subset__iff,axiom,
( ord_le5191068806740926126st_nat
= ( ^ [A6: set_Pr2659102613823815246st_nat,B6: set_Pr2659102613823815246st_nat] :
! [T3: produc2423145444804056984st_nat] :
( ( member190277321972437807st_nat @ T3 @ A6 )
=> ( member190277321972437807st_nat @ T3 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_363_subset__iff,axiom,
( ord_le4361379431065024436rm_a_b
= ( ^ [A6: set_Pr1829413238147913556rm_a_b,B6: set_Pr1829413238147913556rm_a_b] :
! [T3: produc6373727562496775070rm_a_b] :
( ( member4140859439665155893rm_a_b @ T3 @ A6 )
=> ( member4140859439665155893rm_a_b @ T3 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_364_subset__iff,axiom,
( ord_le8406513867147106209st_nat
= ( ^ [A6: set_Pr3451248702717554689st_nat,B6: set_Pr3451248702717554689st_nat] :
! [T3: produc1828647624359046049st_nat] :
( ( member7340969449405702474st_nat @ T3 @ A6 )
=> ( member7340969449405702474st_nat @ T3 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_365_subset__iff,axiom,
( ord_le4920510218098530272rm_a_b
= ( ^ [A6: set_set_term_a_b,B6: set_set_term_a_b] :
! [T3: set_term_a_b] :
( ( member_set_term_a_b @ T3 @ A6 )
=> ( member_set_term_a_b @ T3 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_366_subset__iff,axiom,
( ord_le8666007276011122963_a_nat
= ( ^ [A6: set_Pr4934435412358123699_a_nat,B6: set_Pr4934435412358123699_a_nat] :
! [T3: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ T3 @ A6 )
=> ( member5724188588386418708_a_nat @ T3 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_367_subset__iff,axiom,
( ord_le2705286416250468010rm_a_b
= ( ^ [A6: set_term_a_b,B6: set_term_a_b] :
! [T3: term_a_b] :
( ( member_term_a_b @ T3 @ A6 )
=> ( member_term_a_b @ T3 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_368_subset__iff,axiom,
( ord_le6045566169113846134st_nat
= ( ^ [A6: set_list_nat,B6: set_list_nat] :
! [T3: list_nat] :
( ( member_list_nat @ T3 @ A6 )
=> ( member_list_nat @ T3 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_369_subset__iff,axiom,
( ord_le118470702582115849rm_a_b
= ( ^ [A6: set_Pr4386577575007340137rm_a_b,B6: set_Pr4386577575007340137rm_a_b] :
! [T3: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ T3 @ A6 )
=> ( member5869715511025134514rm_a_b @ T3 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_370_subset__refl,axiom,
! [A4: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ A4 @ A4 ) ).
% subset_refl
thf(fact_371_subset__refl,axiom,
! [A4: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ A4 @ A4 ) ).
% subset_refl
thf(fact_372_subset__refl,axiom,
! [A4: set_list_nat] : ( ord_le6045566169113846134st_nat @ A4 @ A4 ) ).
% subset_refl
thf(fact_373_subset__refl,axiom,
! [A4: set_term_a_b] : ( ord_le2705286416250468010rm_a_b @ A4 @ A4 ) ).
% subset_refl
thf(fact_374_Collect__mono,axiom,
! [P3: product_prod_a_nat > $o,Q2: product_prod_a_nat > $o] :
( ! [X3: product_prod_a_nat] :
( ( P3 @ X3 )
=> ( Q2 @ X3 ) )
=> ( ord_le8666007276011122963_a_nat @ ( collec4464134535221767506_a_nat @ P3 ) @ ( collec4464134535221767506_a_nat @ Q2 ) ) ) ).
% Collect_mono
thf(fact_375_Collect__mono,axiom,
! [P3: produc357393685978478089rm_a_b > $o,Q2: produc357393685978478089rm_a_b > $o] :
( ! [X3: produc357393685978478089rm_a_b] :
( ( P3 @ X3 )
=> ( Q2 @ X3 ) )
=> ( ord_le118470702582115849rm_a_b @ ( collec99177395925924084rm_a_b @ P3 ) @ ( collec99177395925924084rm_a_b @ Q2 ) ) ) ).
% Collect_mono
thf(fact_376_Collect__mono,axiom,
! [P3: list_nat > $o,Q2: list_nat > $o] :
( ! [X3: list_nat] :
( ( P3 @ X3 )
=> ( Q2 @ X3 ) )
=> ( ord_le6045566169113846134st_nat @ ( collect_list_nat @ P3 ) @ ( collect_list_nat @ Q2 ) ) ) ).
% Collect_mono
thf(fact_377_Collect__mono,axiom,
! [P3: term_a_b > $o,Q2: term_a_b > $o] :
( ! [X3: term_a_b] :
( ( P3 @ X3 )
=> ( Q2 @ X3 ) )
=> ( ord_le2705286416250468010rm_a_b @ ( collect_term_a_b @ P3 ) @ ( collect_term_a_b @ Q2 ) ) ) ).
% Collect_mono
thf(fact_378_subset__trans,axiom,
! [A4: set_Pr4934435412358123699_a_nat,B5: set_Pr4934435412358123699_a_nat,C4: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A4 @ B5 )
=> ( ( ord_le8666007276011122963_a_nat @ B5 @ C4 )
=> ( ord_le8666007276011122963_a_nat @ A4 @ C4 ) ) ) ).
% subset_trans
thf(fact_379_subset__trans,axiom,
! [A4: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b,C4: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A4 @ B5 )
=> ( ( ord_le118470702582115849rm_a_b @ B5 @ C4 )
=> ( ord_le118470702582115849rm_a_b @ A4 @ C4 ) ) ) ).
% subset_trans
thf(fact_380_subset__trans,axiom,
! [A4: set_list_nat,B5: set_list_nat,C4: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A4 @ B5 )
=> ( ( ord_le6045566169113846134st_nat @ B5 @ C4 )
=> ( ord_le6045566169113846134st_nat @ A4 @ C4 ) ) ) ).
% subset_trans
thf(fact_381_subset__trans,axiom,
! [A4: set_term_a_b,B5: set_term_a_b,C4: set_term_a_b] :
( ( ord_le2705286416250468010rm_a_b @ A4 @ B5 )
=> ( ( ord_le2705286416250468010rm_a_b @ B5 @ C4 )
=> ( ord_le2705286416250468010rm_a_b @ A4 @ C4 ) ) ) ).
% subset_trans
thf(fact_382_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_383_antisym,axiom,
! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A @ B )
=> ( ( ord_le118470702582115849rm_a_b @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_384_antisym,axiom,
! [A: set_list_nat,B: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A @ B )
=> ( ( ord_le6045566169113846134st_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_385_antisym,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_386_antisym,axiom,
! [A: set_term_a_b,B: set_term_a_b] :
( ( ord_le2705286416250468010rm_a_b @ A @ B )
=> ( ( ord_le2705286416250468010rm_a_b @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_387_set__eq__subset,axiom,
( ( ^ [Y4: set_Pr4934435412358123699_a_nat,Z2: set_Pr4934435412358123699_a_nat] : ( Y4 = Z2 ) )
= ( ^ [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_388_set__eq__subset,axiom,
( ( ^ [Y4: set_Pr4386577575007340137rm_a_b,Z2: set_Pr4386577575007340137rm_a_b] : ( Y4 = Z2 ) )
= ( ^ [A6: set_Pr4386577575007340137rm_a_b,B6: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A6 @ B6 )
& ( ord_le118470702582115849rm_a_b @ B6 @ A6 ) ) ) ) ).
% set_eq_subset
thf(fact_389_set__eq__subset,axiom,
( ( ^ [Y4: set_list_nat,Z2: set_list_nat] : ( Y4 = Z2 ) )
= ( ^ [A6: set_list_nat,B6: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A6 @ B6 )
& ( ord_le6045566169113846134st_nat @ B6 @ A6 ) ) ) ) ).
% set_eq_subset
thf(fact_390_set__eq__subset,axiom,
( ( ^ [Y4: set_term_a_b,Z2: set_term_a_b] : ( Y4 = Z2 ) )
= ( ^ [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_391_Collect__mono__iff,axiom,
! [P3: product_prod_a_nat > $o,Q2: product_prod_a_nat > $o] :
( ( ord_le8666007276011122963_a_nat @ ( collec4464134535221767506_a_nat @ P3 ) @ ( collec4464134535221767506_a_nat @ Q2 ) )
= ( ! [X4: product_prod_a_nat] :
( ( P3 @ X4 )
=> ( Q2 @ X4 ) ) ) ) ).
% Collect_mono_iff
thf(fact_392_Collect__mono__iff,axiom,
! [P3: produc357393685978478089rm_a_b > $o,Q2: produc357393685978478089rm_a_b > $o] :
( ( ord_le118470702582115849rm_a_b @ ( collec99177395925924084rm_a_b @ P3 ) @ ( collec99177395925924084rm_a_b @ Q2 ) )
= ( ! [X4: produc357393685978478089rm_a_b] :
( ( P3 @ X4 )
=> ( Q2 @ X4 ) ) ) ) ).
% Collect_mono_iff
thf(fact_393_Collect__mono__iff,axiom,
! [P3: list_nat > $o,Q2: list_nat > $o] :
( ( ord_le6045566169113846134st_nat @ ( collect_list_nat @ P3 ) @ ( collect_list_nat @ Q2 ) )
= ( ! [X4: list_nat] :
( ( P3 @ X4 )
=> ( Q2 @ X4 ) ) ) ) ).
% Collect_mono_iff
thf(fact_394_Collect__mono__iff,axiom,
! [P3: term_a_b > $o,Q2: term_a_b > $o] :
( ( ord_le2705286416250468010rm_a_b @ ( collect_term_a_b @ P3 ) @ ( collect_term_a_b @ Q2 ) )
= ( ! [X4: term_a_b] :
( ( P3 @ X4 )
=> ( Q2 @ X4 ) ) ) ) ).
% Collect_mono_iff
thf(fact_395_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_Pr4934435412358123699_a_nat,Z2: set_Pr4934435412358123699_a_nat] : ( Y4 = Z2 ) )
= ( ^ [A5: set_Pr4934435412358123699_a_nat,B4: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A5 @ B4 )
& ( ord_le8666007276011122963_a_nat @ B4 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_396_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_Pr4386577575007340137rm_a_b,Z2: set_Pr4386577575007340137rm_a_b] : ( Y4 = Z2 ) )
= ( ^ [A5: set_Pr4386577575007340137rm_a_b,B4: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A5 @ B4 )
& ( ord_le118470702582115849rm_a_b @ B4 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_397_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_list_nat,Z2: set_list_nat] : ( Y4 = Z2 ) )
= ( ^ [A5: set_list_nat,B4: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A5 @ B4 )
& ( ord_le6045566169113846134st_nat @ B4 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_398_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 ) )
= ( ^ [A5: nat,B4: nat] :
( ( ord_less_eq_nat @ A5 @ B4 )
& ( ord_less_eq_nat @ B4 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_399_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_term_a_b,Z2: set_term_a_b] : ( Y4 = Z2 ) )
= ( ^ [A5: set_term_a_b,B4: set_term_a_b] :
( ( ord_le2705286416250468010rm_a_b @ A5 @ B4 )
& ( ord_le2705286416250468010rm_a_b @ B4 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_400_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_401_order__subst1,axiom,
! [A: set_list_nat,F3: nat > set_list_nat,B: nat,C2: nat] :
( ( ord_le6045566169113846134st_nat @ A @ ( F3 @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_le6045566169113846134st_nat @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ord_le6045566169113846134st_nat @ A @ ( F3 @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_402_order__subst1,axiom,
! [A: nat,F3: set_list_nat > nat,B: set_list_nat,C2: set_list_nat] :
( ( ord_less_eq_nat @ A @ ( F3 @ B ) )
=> ( ( ord_le6045566169113846134st_nat @ B @ C2 )
=> ( ! [X3: set_list_nat,Y3: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F3 @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_403_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_404_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_405_order__subst1,axiom,
! [A: nat,F3: set_term_a_b > nat,B: set_term_a_b,C2: set_term_a_b] :
( ( ord_less_eq_nat @ A @ ( F3 @ B ) )
=> ( ( ord_le2705286416250468010rm_a_b @ B @ C2 )
=> ( ! [X3: set_term_a_b,Y3: set_term_a_b] :
( ( ord_le2705286416250468010rm_a_b @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F3 @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_406_order__subst1,axiom,
! [A: set_term_a_b,F3: nat > set_term_a_b,B: nat,C2: nat] :
( ( ord_le2705286416250468010rm_a_b @ A @ ( F3 @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_le2705286416250468010rm_a_b @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ord_le2705286416250468010rm_a_b @ A @ ( F3 @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_407_order__subst1,axiom,
! [A: set_list_nat,F3: set_list_nat > set_list_nat,B: set_list_nat,C2: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A @ ( F3 @ B ) )
=> ( ( ord_le6045566169113846134st_nat @ B @ C2 )
=> ( ! [X3: set_list_nat,Y3: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ X3 @ Y3 )
=> ( ord_le6045566169113846134st_nat @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ord_le6045566169113846134st_nat @ A @ ( F3 @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_408_order__subst1,axiom,
! [A: set_Pr4934435412358123699_a_nat,F3: set_list_nat > set_Pr4934435412358123699_a_nat,B: set_list_nat,C2: set_list_nat] :
( ( ord_le8666007276011122963_a_nat @ A @ ( F3 @ B ) )
=> ( ( ord_le6045566169113846134st_nat @ B @ C2 )
=> ( ! [X3: set_list_nat,Y3: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ X3 @ Y3 )
=> ( ord_le8666007276011122963_a_nat @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ord_le8666007276011122963_a_nat @ A @ ( F3 @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_409_order__subst1,axiom,
! [A: set_list_nat,F3: set_Pr4934435412358123699_a_nat > set_list_nat,B: set_Pr4934435412358123699_a_nat,C2: set_Pr4934435412358123699_a_nat] :
( ( ord_le6045566169113846134st_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_le6045566169113846134st_nat @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ord_le6045566169113846134st_nat @ A @ ( F3 @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_410_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_411_order__subst2,axiom,
! [A: set_list_nat,B: set_list_nat,F3: set_list_nat > nat,C2: nat] :
( ( ord_le6045566169113846134st_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F3 @ B ) @ C2 )
=> ( ! [X3: set_list_nat,Y3: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F3 @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_412_order__subst2,axiom,
! [A: nat,B: nat,F3: nat > set_list_nat,C2: set_list_nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_le6045566169113846134st_nat @ ( F3 @ B ) @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_le6045566169113846134st_nat @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ord_le6045566169113846134st_nat @ ( F3 @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_413_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_414_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_415_order__subst2,axiom,
! [A: nat,B: nat,F3: nat > set_term_a_b,C2: set_term_a_b] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_le2705286416250468010rm_a_b @ ( F3 @ B ) @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_le2705286416250468010rm_a_b @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ord_le2705286416250468010rm_a_b @ ( F3 @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_416_order__subst2,axiom,
! [A: set_term_a_b,B: set_term_a_b,F3: set_term_a_b > nat,C2: nat] :
( ( ord_le2705286416250468010rm_a_b @ A @ B )
=> ( ( ord_less_eq_nat @ ( F3 @ B ) @ C2 )
=> ( ! [X3: set_term_a_b,Y3: set_term_a_b] :
( ( ord_le2705286416250468010rm_a_b @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F3 @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_417_order__subst2,axiom,
! [A: set_list_nat,B: set_list_nat,F3: set_list_nat > set_list_nat,C2: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A @ B )
=> ( ( ord_le6045566169113846134st_nat @ ( F3 @ B ) @ C2 )
=> ( ! [X3: set_list_nat,Y3: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ X3 @ Y3 )
=> ( ord_le6045566169113846134st_nat @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ord_le6045566169113846134st_nat @ ( F3 @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_418_order__subst2,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat,F3: set_Pr4934435412358123699_a_nat > set_list_nat,C2: set_list_nat] :
( ( ord_le8666007276011122963_a_nat @ A @ B )
=> ( ( ord_le6045566169113846134st_nat @ ( F3 @ B ) @ C2 )
=> ( ! [X3: set_Pr4934435412358123699_a_nat,Y3: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X3 @ Y3 )
=> ( ord_le6045566169113846134st_nat @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ord_le6045566169113846134st_nat @ ( F3 @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_419_order__subst2,axiom,
! [A: set_list_nat,B: set_list_nat,F3: set_list_nat > set_Pr4934435412358123699_a_nat,C2: set_Pr4934435412358123699_a_nat] :
( ( ord_le6045566169113846134st_nat @ A @ B )
=> ( ( ord_le8666007276011122963_a_nat @ ( F3 @ B ) @ C2 )
=> ( ! [X3: set_list_nat,Y3: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ X3 @ Y3 )
=> ( ord_le8666007276011122963_a_nat @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ord_le8666007276011122963_a_nat @ ( F3 @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_420_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_421_order__eq__refl,axiom,
! [X: set_Pr4386577575007340137rm_a_b,Y: set_Pr4386577575007340137rm_a_b] :
( ( X = Y )
=> ( ord_le118470702582115849rm_a_b @ X @ Y ) ) ).
% order_eq_refl
thf(fact_422_order__eq__refl,axiom,
! [X: set_list_nat,Y: set_list_nat] :
( ( X = Y )
=> ( ord_le6045566169113846134st_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_423_order__eq__refl,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_424_order__eq__refl,axiom,
! [X: set_term_a_b,Y: set_term_a_b] :
( ( X = Y )
=> ( ord_le2705286416250468010rm_a_b @ X @ Y ) ) ).
% order_eq_refl
thf(fact_425_linorder__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_linear
thf(fact_426_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_427_ord__eq__le__subst,axiom,
! [A: nat,F3: set_list_nat > nat,B: set_list_nat,C2: set_list_nat] :
( ( A
= ( F3 @ B ) )
=> ( ( ord_le6045566169113846134st_nat @ B @ C2 )
=> ( ! [X3: set_list_nat,Y3: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F3 @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_428_ord__eq__le__subst,axiom,
! [A: set_list_nat,F3: nat > set_list_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_le6045566169113846134st_nat @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ord_le6045566169113846134st_nat @ A @ ( F3 @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_429_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_430_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_431_ord__eq__le__subst,axiom,
! [A: set_term_a_b,F3: nat > set_term_a_b,B: nat,C2: nat] :
( ( A
= ( F3 @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_le2705286416250468010rm_a_b @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ord_le2705286416250468010rm_a_b @ A @ ( F3 @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_432_ord__eq__le__subst,axiom,
! [A: nat,F3: set_term_a_b > nat,B: set_term_a_b,C2: set_term_a_b] :
( ( A
= ( F3 @ B ) )
=> ( ( ord_le2705286416250468010rm_a_b @ B @ C2 )
=> ( ! [X3: set_term_a_b,Y3: set_term_a_b] :
( ( ord_le2705286416250468010rm_a_b @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F3 @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_433_ord__eq__le__subst,axiom,
! [A: set_list_nat,F3: set_list_nat > set_list_nat,B: set_list_nat,C2: set_list_nat] :
( ( A
= ( F3 @ B ) )
=> ( ( ord_le6045566169113846134st_nat @ B @ C2 )
=> ( ! [X3: set_list_nat,Y3: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ X3 @ Y3 )
=> ( ord_le6045566169113846134st_nat @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ord_le6045566169113846134st_nat @ A @ ( F3 @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_434_ord__eq__le__subst,axiom,
! [A: set_list_nat,F3: set_Pr4934435412358123699_a_nat > set_list_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_le6045566169113846134st_nat @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ord_le6045566169113846134st_nat @ A @ ( F3 @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_435_ord__eq__le__subst,axiom,
! [A: set_Pr4934435412358123699_a_nat,F3: set_list_nat > set_Pr4934435412358123699_a_nat,B: set_list_nat,C2: set_list_nat] :
( ( A
= ( F3 @ B ) )
=> ( ( ord_le6045566169113846134st_nat @ B @ C2 )
=> ( ! [X3: set_list_nat,Y3: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ X3 @ Y3 )
=> ( ord_le8666007276011122963_a_nat @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ord_le8666007276011122963_a_nat @ A @ ( F3 @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_436_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_437_ord__le__eq__subst,axiom,
! [A: set_list_nat,B: set_list_nat,F3: set_list_nat > nat,C2: nat] :
( ( ord_le6045566169113846134st_nat @ A @ B )
=> ( ( ( F3 @ B )
= C2 )
=> ( ! [X3: set_list_nat,Y3: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F3 @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_438_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F3: nat > set_list_nat,C2: set_list_nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F3 @ B )
= C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_le6045566169113846134st_nat @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ord_le6045566169113846134st_nat @ ( F3 @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_439_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_440_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_441_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F3: nat > set_term_a_b,C2: set_term_a_b] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F3 @ B )
= C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_le2705286416250468010rm_a_b @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ord_le2705286416250468010rm_a_b @ ( F3 @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_442_ord__le__eq__subst,axiom,
! [A: set_term_a_b,B: set_term_a_b,F3: set_term_a_b > nat,C2: nat] :
( ( ord_le2705286416250468010rm_a_b @ A @ B )
=> ( ( ( F3 @ B )
= C2 )
=> ( ! [X3: set_term_a_b,Y3: set_term_a_b] :
( ( ord_le2705286416250468010rm_a_b @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F3 @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_443_ord__le__eq__subst,axiom,
! [A: set_list_nat,B: set_list_nat,F3: set_list_nat > set_list_nat,C2: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A @ B )
=> ( ( ( F3 @ B )
= C2 )
=> ( ! [X3: set_list_nat,Y3: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ X3 @ Y3 )
=> ( ord_le6045566169113846134st_nat @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ord_le6045566169113846134st_nat @ ( F3 @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_444_ord__le__eq__subst,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat,F3: set_Pr4934435412358123699_a_nat > set_list_nat,C2: set_list_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_le6045566169113846134st_nat @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ord_le6045566169113846134st_nat @ ( F3 @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_445_ord__le__eq__subst,axiom,
! [A: set_list_nat,B: set_list_nat,F3: set_list_nat > set_Pr4934435412358123699_a_nat,C2: set_Pr4934435412358123699_a_nat] :
( ( ord_le6045566169113846134st_nat @ A @ B )
=> ( ( ( F3 @ B )
= C2 )
=> ( ! [X3: set_list_nat,Y3: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ X3 @ Y3 )
=> ( ord_le8666007276011122963_a_nat @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ord_le8666007276011122963_a_nat @ ( F3 @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_446_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_447_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_448_order__antisym__conv,axiom,
! [Y: set_Pr4386577575007340137rm_a_b,X: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ Y @ X )
=> ( ( ord_le118470702582115849rm_a_b @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_449_order__antisym__conv,axiom,
! [Y: set_list_nat,X: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ Y @ X )
=> ( ( ord_le6045566169113846134st_nat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_450_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_451_order__antisym__conv,axiom,
! [Y: set_term_a_b,X: set_term_a_b] :
( ( ord_le2705286416250468010rm_a_b @ Y @ X )
=> ( ( ord_le2705286416250468010rm_a_b @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_452_subrelI,axiom,
! [R2: set_Pr2972776593051762503rm_a_b,S: set_Pr2972776593051762503rm_a_b] :
( ! [X3: produc357393685978478089rm_a_b,Y3: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X3 @ Y3 ) @ R2 )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X3 @ Y3 ) @ S ) )
=> ( ord_le2593958535671617255rm_a_b @ R2 @ S ) ) ).
% subrelI
thf(fact_453_subrelI,axiom,
! [R2: set_Pr3451248702717554689st_nat,S: set_Pr3451248702717554689st_nat] :
( ! [X3: list_nat,Y3: list_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X3 @ Y3 ) @ R2 )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X3 @ Y3 ) @ S ) )
=> ( ord_le8406513867147106209st_nat @ R2 @ S ) ) ).
% subrelI
thf(fact_454_subrelI,axiom,
! [R2: set_Pr2659102613823815246st_nat,S: set_Pr2659102613823815246st_nat] :
( ! [X3: produc357393685978478089rm_a_b,Y3: list_nat] :
( ( member190277321972437807st_nat @ ( produc5056965718246025362st_nat @ X3 @ Y3 ) @ R2 )
=> ( member190277321972437807st_nat @ ( produc5056965718246025362st_nat @ X3 @ Y3 ) @ S ) )
=> ( ord_le5191068806740926126st_nat @ R2 @ S ) ) ).
% subrelI
thf(fact_455_subrelI,axiom,
! [R2: set_Pr1829413238147913556rm_a_b,S: set_Pr1829413238147913556rm_a_b] :
( ! [X3: list_nat,Y3: produc357393685978478089rm_a_b] :
( ( member4140859439665155893rm_a_b @ ( produc3015861401532503632rm_a_b @ X3 @ Y3 ) @ R2 )
=> ( member4140859439665155893rm_a_b @ ( produc3015861401532503632rm_a_b @ X3 @ Y3 ) @ S ) )
=> ( ord_le4361379431065024436rm_a_b @ R2 @ S ) ) ).
% subrelI
thf(fact_456_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_457_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_458_bot_Oextremum__uniqueI,axiom,
! [A: produc357393685978478089rm_a_b > $o] :
( ( ord_le6645574462093738516_a_b_o @ A @ bot_bo23760184229683272_a_b_o )
=> ( A = bot_bo23760184229683272_a_b_o ) ) ).
% bot.extremum_uniqueI
thf(fact_459_bot_Oextremum__uniqueI,axiom,
! [A: list_nat > $o] :
( ( ord_le1520216061033275535_nat_o @ A @ bot_bot_list_nat_o )
=> ( A = bot_bot_list_nat_o ) ) ).
% bot.extremum_uniqueI
thf(fact_460_bot_Oextremum__uniqueI,axiom,
! [A: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A @ bot_bo9049108969261143879_a_nat )
=> ( A = bot_bo9049108969261143879_a_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_461_bot_Oextremum__uniqueI,axiom,
! [A: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A @ bot_bo197521221353338581rm_a_b )
=> ( A = bot_bo197521221353338581rm_a_b ) ) ).
% bot.extremum_uniqueI
thf(fact_462_bot_Oextremum__uniqueI,axiom,
! [A: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A @ bot_bot_set_list_nat )
=> ( A = bot_bot_set_list_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_463_bot_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
=> ( A = bot_bot_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_464_bot_Oextremum__uniqueI,axiom,
! [A: set_term_a_b] :
( ( ord_le2705286416250468010rm_a_b @ A @ bot_bot_set_term_a_b )
=> ( A = bot_bot_set_term_a_b ) ) ).
% bot.extremum_uniqueI
thf(fact_465_bot_Oextremum__unique,axiom,
! [A: produc357393685978478089rm_a_b > $o] :
( ( ord_le6645574462093738516_a_b_o @ A @ bot_bo23760184229683272_a_b_o )
= ( A = bot_bo23760184229683272_a_b_o ) ) ).
% bot.extremum_unique
thf(fact_466_bot_Oextremum__unique,axiom,
! [A: list_nat > $o] :
( ( ord_le1520216061033275535_nat_o @ A @ bot_bot_list_nat_o )
= ( A = bot_bot_list_nat_o ) ) ).
% bot.extremum_unique
thf(fact_467_bot_Oextremum__unique,axiom,
! [A: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A @ bot_bo9049108969261143879_a_nat )
= ( A = bot_bo9049108969261143879_a_nat ) ) ).
% bot.extremum_unique
thf(fact_468_bot_Oextremum__unique,axiom,
! [A: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A @ bot_bo197521221353338581rm_a_b )
= ( A = bot_bo197521221353338581rm_a_b ) ) ).
% bot.extremum_unique
thf(fact_469_bot_Oextremum__unique,axiom,
! [A: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A @ bot_bot_set_list_nat )
= ( A = bot_bot_set_list_nat ) ) ).
% bot.extremum_unique
thf(fact_470_bot_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
= ( A = bot_bot_nat ) ) ).
% bot.extremum_unique
thf(fact_471_bot_Oextremum__unique,axiom,
! [A: set_term_a_b] :
( ( ord_le2705286416250468010rm_a_b @ A @ bot_bot_set_term_a_b )
= ( A = bot_bot_set_term_a_b ) ) ).
% bot.extremum_unique
thf(fact_472_bot_Oextremum,axiom,
! [A: produc357393685978478089rm_a_b > $o] : ( ord_le6645574462093738516_a_b_o @ bot_bo23760184229683272_a_b_o @ A ) ).
% bot.extremum
thf(fact_473_bot_Oextremum,axiom,
! [A: list_nat > $o] : ( ord_le1520216061033275535_nat_o @ bot_bot_list_nat_o @ A ) ).
% bot.extremum
thf(fact_474_bot_Oextremum,axiom,
! [A: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ bot_bo9049108969261143879_a_nat @ A ) ).
% bot.extremum
thf(fact_475_bot_Oextremum,axiom,
! [A: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ bot_bo197521221353338581rm_a_b @ A ) ).
% bot.extremum
thf(fact_476_bot_Oextremum,axiom,
! [A: set_list_nat] : ( ord_le6045566169113846134st_nat @ bot_bot_set_list_nat @ A ) ).
% bot.extremum
thf(fact_477_bot_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A ) ).
% bot.extremum
thf(fact_478_bot_Oextremum,axiom,
! [A: set_term_a_b] : ( ord_le2705286416250468010rm_a_b @ bot_bot_set_term_a_b @ A ) ).
% bot.extremum
thf(fact_479_subset__emptyI,axiom,
! [A4: set_Pr2659102613823815246st_nat] :
( ! [X3: produc2423145444804056984st_nat] :
~ ( member190277321972437807st_nat @ X3 @ A4 )
=> ( ord_le5191068806740926126st_nat @ A4 @ bot_bo6857980842747739362st_nat ) ) ).
% subset_emptyI
thf(fact_480_subset__emptyI,axiom,
! [A4: set_Pr1829413238147913556rm_a_b] :
( ! [X3: produc6373727562496775070rm_a_b] :
~ ( member4140859439665155893rm_a_b @ X3 @ A4 )
=> ( ord_le4361379431065024436rm_a_b @ A4 @ bot_bo6028291467071837672rm_a_b ) ) ).
% subset_emptyI
thf(fact_481_subset__emptyI,axiom,
! [A4: set_Pr3451248702717554689st_nat] :
( ! [X3: produc1828647624359046049st_nat] :
~ ( member7340969449405702474st_nat @ X3 @ A4 )
=> ( ord_le8406513867147106209st_nat @ A4 @ bot_bo8485564385918328941st_nat ) ) ).
% subset_emptyI
thf(fact_482_subset__emptyI,axiom,
! [A4: set_set_term_a_b] :
( ! [X3: set_term_a_b] :
~ ( member_set_term_a_b @ X3 @ A4 )
=> ( ord_le4920510218098530272rm_a_b @ A4 @ bot_bo7738030261494536212rm_a_b ) ) ).
% subset_emptyI
thf(fact_483_subset__emptyI,axiom,
! [A4: set_Pr4934435412358123699_a_nat] :
( ! [X3: product_prod_a_nat] :
~ ( member5724188588386418708_a_nat @ X3 @ A4 )
=> ( ord_le8666007276011122963_a_nat @ A4 @ bot_bo9049108969261143879_a_nat ) ) ).
% subset_emptyI
thf(fact_484_subset__emptyI,axiom,
! [A4: set_term_a_b] :
( ! [X3: term_a_b] :
~ ( member_term_a_b @ X3 @ A4 )
=> ( ord_le2705286416250468010rm_a_b @ A4 @ bot_bot_set_term_a_b ) ) ).
% subset_emptyI
thf(fact_485_subset__emptyI,axiom,
! [A4: set_list_nat] :
( ! [X3: list_nat] :
~ ( member_list_nat @ X3 @ A4 )
=> ( ord_le6045566169113846134st_nat @ A4 @ bot_bot_set_list_nat ) ) ).
% subset_emptyI
thf(fact_486_subset__emptyI,axiom,
! [A4: set_Pr4386577575007340137rm_a_b] :
( ! [X3: produc357393685978478089rm_a_b] :
~ ( member5869715511025134514rm_a_b @ X3 @ A4 )
=> ( ord_le118470702582115849rm_a_b @ A4 @ bot_bo197521221353338581rm_a_b ) ) ).
% subset_emptyI
thf(fact_487_join__mono,axiom,
! [R2: set_Pr2972776593051762503rm_a_b,S: set_Pr2972776593051762503rm_a_b] :
( ( ord_le2593958535671617255rm_a_b @ R2 @ S )
=> ( ord_le2593958535671617255rm_a_b @ ( abstra4889682940153293507rm_a_b @ R2 ) @ ( abstra4889682940153293507rm_a_b @ S ) ) ) ).
% join_mono
thf(fact_488_join__mono,axiom,
! [R2: set_Pr3451248702717554689st_nat,S: set_Pr3451248702717554689st_nat] :
( ( ord_le8406513867147106209st_nat @ R2 @ S )
=> ( ord_le8406513867147106209st_nat @ ( abstra1638945853619340710st_nat @ R2 ) @ ( abstra1638945853619340710st_nat @ S ) ) ) ).
% join_mono
thf(fact_489_join__mono,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ R2 @ S )
=> ( ord_le118470702582115849rm_a_b @ ( abstra4096080454567261402rm_a_b @ R2 ) @ ( abstra4096080454567261402rm_a_b @ S ) ) ) ).
% join_mono
thf(fact_490_poss__append__poss,axiom,
! [P2: list_nat,Q: list_nat,T: term_a_b] :
( ( member_list_nat @ ( append_nat @ P2 @ Q ) @ ( term_poss_a_b @ T ) )
= ( ( member_list_nat @ P2 @ ( term_poss_a_b @ T ) )
& ( member_list_nat @ Q @ ( term_poss_a_b @ ( term_subt_at_a_b @ T @ P2 ) ) ) ) ) ).
% poss_append_poss
thf(fact_491_subt__at__append__dist,axiom,
! [P2: list_nat,Q: list_nat,S: term_a_b] :
( ( member_list_nat @ ( append_nat @ P2 @ Q ) @ ( term_poss_a_b @ S ) )
=> ( ( term_subt_at_a_b @ S @ ( append_nat @ P2 @ Q ) )
= ( term_subt_at_a_b @ ( term_subt_at_a_b @ S @ P2 ) @ Q ) ) ) ).
% subt_at_append_dist
thf(fact_492_srstep__ctxt__closed,axiom,
! [C4: subterm_and_ctxt_a_b,F: set_Pr4934435412358123699_a_nat,S: term_a_b,T: term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
( ( ord_le8666007276011122963_a_nat @ ( term_funas_ctxt_a_b @ C4 ) @ F )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( sig_step_a_b @ F @ ( rstep_a_b @ R3 ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( subter2376574525758040790rm_a_b @ C4 @ S ) @ ( subter2376574525758040790rm_a_b @ C4 @ T ) ) @ ( sig_step_a_b @ F @ ( rstep_a_b @ R3 ) ) ) ) ) ).
% srstep_ctxt_closed
thf(fact_493_funas__ctxt__ctxt__inv__const__ctxt__ind,axiom,
! [C4: subterm_and_ctxt_a_b,F: set_Pr4934435412358123699_a_nat,V: b] :
( ( ord_le8666007276011122963_a_nat @ ( term_funas_ctxt_a_b @ C4 ) @ F )
=> ( ( terms_3799363064701517935xt_a_b @ F @ V @ C4 )
= C4 ) ) ).
% funas_ctxt_ctxt_inv_const_ctxt_ind
thf(fact_494_srsteps__eq__ctxt__closed,axiom,
! [C4: subterm_and_ctxt_a_b,F: set_Pr4934435412358123699_a_nat,S: term_a_b,T: term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
( ( ord_le8666007276011122963_a_nat @ ( term_funas_ctxt_a_b @ C4 ) @ F )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R3 ) ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( subter2376574525758040790rm_a_b @ C4 @ S ) @ ( subter2376574525758040790rm_a_b @ C4 @ T ) ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R3 ) ) ) ) ) ) ).
% srsteps_eq_ctxt_closed
thf(fact_495_srsteps__ctxt__closed,axiom,
! [C4: subterm_and_ctxt_a_b,F: set_Pr4934435412358123699_a_nat,S: term_a_b,T: term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
( ( ord_le8666007276011122963_a_nat @ ( term_funas_ctxt_a_b @ C4 ) @ F )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( transi7922773638565587891rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R3 ) ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( subter2376574525758040790rm_a_b @ C4 @ S ) @ ( subter2376574525758040790rm_a_b @ C4 @ T ) ) @ ( transi7922773638565587891rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R3 ) ) ) ) ) ) ).
% srsteps_ctxt_closed
thf(fact_496_srstep__subst__closed,axiom,
! [S: term_a_b,T: term_a_b,F: set_Pr4934435412358123699_a_nat,R3: set_Pr4386577575007340137rm_a_b,Sigma2: b > term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( sig_step_a_b @ F @ ( rstep_a_b @ R3 ) ) )
=> ( ! [X3: b] : ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ ( Sigma2 @ X3 ) ) @ F )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( subst_7999470309526761004_a_b_b @ S @ Sigma2 ) @ ( subst_7999470309526761004_a_b_b @ T @ Sigma2 ) ) @ ( sig_step_a_b @ F @ ( rstep_a_b @ R3 ) ) ) ) ) ).
% srstep_subst_closed
thf(fact_497_srrstepD,axiom,
! [S: term_a_b,T: term_a_b,F: set_Pr4934435412358123699_a_nat,R3: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( sig_step_a_b @ F @ ( rrstep_a_b @ R3 ) ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( rrstep_a_b @ R3 ) )
& ( 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_498_srstepD,axiom,
! [S: term_a_b,T: term_a_b,F: set_Pr4934435412358123699_a_nat,R3: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( sig_step_a_b @ F @ ( rstep_a_b @ R3 ) ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( rstep_a_b @ R3 ) )
& ( 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_499_relChain__def,axiom,
( bNF_Ca1233969902089702230at_nat
= ( ^ [R4: set_Pr3451248702717554689st_nat,As: list_nat > nat] :
! [I: list_nat,J: list_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ I @ J ) @ R4 )
=> ( ord_less_eq_nat @ ( As @ I ) @ ( As @ J ) ) ) ) ) ).
% relChain_def
thf(fact_500_relChain__def,axiom,
( bNF_Ca2413601583040309898_b_nat
= ( ^ [R4: set_Pr4386577575007340137rm_a_b,As: term_a_b > nat] :
! [I: term_a_b,J: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ I @ J ) @ R4 )
=> ( ord_less_eq_nat @ ( As @ I ) @ ( As @ J ) ) ) ) ) ).
% relChain_def
thf(fact_501_relChain__def,axiom,
( bNF_Ca5626288062746064284st_nat
= ( ^ [R4: set_Pr3451248702717554689st_nat,As: list_nat > set_list_nat] :
! [I: list_nat,J: list_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ I @ J ) @ R4 )
=> ( ord_le6045566169113846134st_nat @ ( As @ I ) @ ( As @ J ) ) ) ) ) ).
% relChain_def
thf(fact_502_relChain__def,axiom,
( bNF_Ca3027480494327872057_a_nat
= ( ^ [R4: set_Pr3451248702717554689st_nat,As: list_nat > set_Pr4934435412358123699_a_nat] :
! [I: list_nat,J: list_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ I @ J ) @ R4 )
=> ( ord_le8666007276011122963_a_nat @ ( As @ I ) @ ( As @ J ) ) ) ) ) ).
% relChain_def
thf(fact_503_relChain__def,axiom,
( bNF_Ca7485710705408711376st_nat
= ( ^ [R4: set_Pr4386577575007340137rm_a_b,As: term_a_b > set_list_nat] :
! [I: term_a_b,J: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ I @ J ) @ R4 )
=> ( ord_le6045566169113846134st_nat @ ( As @ I ) @ ( As @ J ) ) ) ) ) ).
% relChain_def
thf(fact_504_relChain__def,axiom,
( bNF_Ca2286008309882686160rm_a_b
= ( ^ [R4: set_Pr3451248702717554689st_nat,As: list_nat > set_term_a_b] :
! [I: list_nat,J: list_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ I @ J ) @ R4 )
=> ( ord_le2705286416250468010rm_a_b @ ( As @ I ) @ ( As @ J ) ) ) ) ) ).
% relChain_def
thf(fact_505_relChain__def,axiom,
( bNF_Ca9222207453146494317_a_nat
= ( ^ [R4: set_Pr4386577575007340137rm_a_b,As: term_a_b > set_Pr4934435412358123699_a_nat] :
! [I: term_a_b,J: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ I @ J ) @ R4 )
=> ( ord_le8666007276011122963_a_nat @ ( As @ I ) @ ( As @ J ) ) ) ) ) ).
% relChain_def
thf(fact_506_relChain__def,axiom,
( bNF_Ca4145430952545333252rm_a_b
= ( ^ [R4: set_Pr4386577575007340137rm_a_b,As: term_a_b > set_term_a_b] :
! [I: term_a_b,J: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ I @ J ) @ R4 )
=> ( ord_le2705286416250468010rm_a_b @ ( As @ I ) @ ( As @ J ) ) ) ) ) ).
% relChain_def
thf(fact_507_relChain__def,axiom,
( bNF_Ca1184762318341890741_b_nat
= ( ^ [R4: set_Pr2972776593051762503rm_a_b,As: produc357393685978478089rm_a_b > nat] :
! [I: produc357393685978478089rm_a_b,J: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ I @ J ) @ R4 )
=> ( ord_less_eq_nat @ ( As @ I ) @ ( As @ J ) ) ) ) ) ).
% relChain_def
thf(fact_508_relChain__def,axiom,
( bNF_Ca8662308032288398691rm_a_b
= ( ^ [R4: set_Pr3451248702717554689st_nat,As: list_nat > set_Pr4386577575007340137rm_a_b] :
! [I: list_nat,J: list_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ I @ J ) @ R4 )
=> ( ord_le118470702582115849rm_a_b @ ( As @ I ) @ ( As @ J ) ) ) ) ) ).
% relChain_def
thf(fact_509_pos__diff__append__itself,axiom,
! [P2: list_nat,Q: list_nat] :
( ( term_pos_diff_nat @ ( append_nat @ P2 @ Q ) @ P2 )
= Q ) ).
% pos_diff_append_itself
thf(fact_510_funas__term__subterm__atI,axiom,
! [P2: list_nat,S: term_a_b,F: set_Pr4934435412358123699_a_nat] :
( ( member_list_nat @ P2 @ ( term_poss_a_b @ S ) )
=> ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ S ) @ F )
=> ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ ( term_subt_at_a_b @ S @ P2 ) ) @ F ) ) ) ).
% funas_term_subterm_atI
thf(fact_511_rtrancl__supteq,axiom,
( ( transi7742714808557438673rm_a_b @ subter523971068842742411eq_a_b )
= subter523971068842742411eq_a_b ) ).
% rtrancl_supteq
thf(fact_512_trancl__supteq,axiom,
( ( transi7922773638565587891rm_a_b @ subter523971068842742411eq_a_b )
= subter523971068842742411eq_a_b ) ).
% trancl_supteq
thf(fact_513_joinI__right,axiom,
! [B: produc357393685978478089rm_a_b,A: produc357393685978478089rm_a_b,A4: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B @ A ) @ ( transi2615809358984392588rm_a_b @ A4 ) )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( abstra4889682940153293507rm_a_b @ A4 ) ) ) ).
% joinI_right
thf(fact_514_joinI__right,axiom,
! [B: list_nat,A: list_nat,A4: set_Pr3451248702717554689st_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ B @ A ) @ ( transi5285580207609517981st_nat @ A4 ) )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A @ B ) @ ( abstra1638945853619340710st_nat @ A4 ) ) ) ).
% joinI_right
thf(fact_515_joinI__right,axiom,
! [B: term_a_b,A: term_a_b,A4: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ ( transi7742714808557438673rm_a_b @ A4 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( abstra4096080454567261402rm_a_b @ A4 ) ) ) ).
% joinI_right
thf(fact_516_joinI__left,axiom,
! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,A4: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( transi2615809358984392588rm_a_b @ A4 ) )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( abstra4889682940153293507rm_a_b @ A4 ) ) ) ).
% joinI_left
thf(fact_517_joinI__left,axiom,
! [A: list_nat,B: list_nat,A4: set_Pr3451248702717554689st_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A @ B ) @ ( transi5285580207609517981st_nat @ A4 ) )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A @ B ) @ ( abstra1638945853619340710st_nat @ A4 ) ) ) ).
% joinI_left
thf(fact_518_joinI__left,axiom,
! [A: term_a_b,B: term_a_b,A4: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ A4 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( abstra4096080454567261402rm_a_b @ A4 ) ) ) ).
% joinI_left
thf(fact_519_joinI,axiom,
! [A: produc357393685978478089rm_a_b,C2: produc357393685978478089rm_a_b,A4: set_Pr2972776593051762503rm_a_b,B: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ C2 ) @ ( transi2615809358984392588rm_a_b @ A4 ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B @ C2 ) @ ( transi2615809358984392588rm_a_b @ A4 ) )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( abstra4889682940153293507rm_a_b @ A4 ) ) ) ) ).
% joinI
thf(fact_520_joinI,axiom,
! [A: list_nat,C2: list_nat,A4: set_Pr3451248702717554689st_nat,B: list_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A @ C2 ) @ ( transi5285580207609517981st_nat @ A4 ) )
=> ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ B @ C2 ) @ ( transi5285580207609517981st_nat @ A4 ) )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A @ B ) @ ( abstra1638945853619340710st_nat @ A4 ) ) ) ) ).
% joinI
thf(fact_521_joinI,axiom,
! [A: term_a_b,C2: term_a_b,A4: set_Pr4386577575007340137rm_a_b,B: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C2 ) @ ( transi7742714808557438673rm_a_b @ A4 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C2 ) @ ( transi7742714808557438673rm_a_b @ A4 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( abstra4096080454567261402rm_a_b @ A4 ) ) ) ) ).
% joinI
thf(fact_522_sig__stepI,axiom,
! [S: term_a_b,F: set_Pr4934435412358123699_a_nat,T: term_a_b,R3: 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 ) @ R3 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( sig_step_a_b @ F @ R3 ) ) ) ) ) ).
% sig_stepI
thf(fact_523_trancl__mono__set,axiom,
! [R2: set_Pr2972776593051762503rm_a_b,S: set_Pr2972776593051762503rm_a_b] :
( ( ord_le2593958535671617255rm_a_b @ R2 @ S )
=> ( ord_le2593958535671617255rm_a_b @ ( transi9211502839322181930rm_a_b @ R2 ) @ ( transi9211502839322181930rm_a_b @ S ) ) ) ).
% trancl_mono_set
thf(fact_524_trancl__mono__set,axiom,
! [R2: set_Pr3451248702717554689st_nat,S: set_Pr3451248702717554689st_nat] :
( ( ord_le8406513867147106209st_nat @ R2 @ S )
=> ( ord_le8406513867147106209st_nat @ ( transi5465639037617667199st_nat @ R2 ) @ ( transi5465639037617667199st_nat @ S ) ) ) ).
% trancl_mono_set
thf(fact_525_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_526_sig__step__mono2,axiom,
! [R3: set_Pr4386577575007340137rm_a_b,L4: set_Pr4386577575007340137rm_a_b,F: set_Pr4934435412358123699_a_nat] :
( ( ord_le118470702582115849rm_a_b @ R3 @ L4 )
=> ( ord_le118470702582115849rm_a_b @ ( sig_step_a_b @ F @ R3 ) @ ( sig_step_a_b @ F @ L4 ) ) ) ).
% sig_step_mono2
thf(fact_527_srsteps__monp,axiom,
! [F: set_Pr4934435412358123699_a_nat,G: set_Pr4934435412358123699_a_nat,R3: 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 @ R3 ) ) ) @ ( transi7922773638565587891rm_a_b @ ( sig_step_a_b @ G @ ( rstep_a_b @ R3 ) ) ) ) ) ).
% srsteps_monp
thf(fact_528_srsteps__eq__monp,axiom,
! [F: set_Pr4934435412358123699_a_nat,G: set_Pr4934435412358123699_a_nat,R3: 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 @ R3 ) ) ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ G @ ( rstep_a_b @ R3 ) ) ) ) ) ).
% srsteps_eq_monp
thf(fact_529_srstepsD,axiom,
! [S: term_a_b,T: term_a_b,F: set_Pr4934435412358123699_a_nat,R3: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( transi7922773638565587891rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R3 ) ) ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( transi7922773638565587891rm_a_b @ ( rstep_a_b @ R3 ) ) )
& ( 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_530_sigstep__trancl__funas_I2_J,axiom,
! [S: term_a_b,T: term_a_b,F: set_Pr4934435412358123699_a_nat,S5: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ S5 ) ) ) )
=> ( ( S != T )
=> ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ T ) @ F ) ) ) ).
% sigstep_trancl_funas(2)
thf(fact_531_sigstep__trancl__funas_I1_J,axiom,
! [S: term_a_b,T: term_a_b,F: set_Pr4934435412358123699_a_nat,S5: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ S5 ) ) ) )
=> ( ( S != T )
=> ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ S ) @ F ) ) ) ).
% sigstep_trancl_funas(1)
thf(fact_532_rtrancl__join__join,axiom,
! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,A4: set_Pr2972776593051762503rm_a_b,C2: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( transi2615809358984392588rm_a_b @ A4 ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B @ C2 ) @ ( abstra4889682940153293507rm_a_b @ A4 ) )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ C2 ) @ ( abstra4889682940153293507rm_a_b @ A4 ) ) ) ) ).
% rtrancl_join_join
thf(fact_533_rtrancl__join__join,axiom,
! [A: list_nat,B: list_nat,A4: set_Pr3451248702717554689st_nat,C2: list_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A @ B ) @ ( transi5285580207609517981st_nat @ A4 ) )
=> ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ B @ C2 ) @ ( abstra1638945853619340710st_nat @ A4 ) )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A @ C2 ) @ ( abstra1638945853619340710st_nat @ A4 ) ) ) ) ).
% rtrancl_join_join
thf(fact_534_rtrancl__join__join,axiom,
! [A: term_a_b,B: term_a_b,A4: set_Pr4386577575007340137rm_a_b,C2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ A4 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C2 ) @ ( abstra4096080454567261402rm_a_b @ A4 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C2 ) @ ( abstra4096080454567261402rm_a_b @ A4 ) ) ) ) ).
% rtrancl_join_join
thf(fact_535_join__rtrancl__join,axiom,
! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,A4: set_Pr2972776593051762503rm_a_b,C2: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( abstra4889682940153293507rm_a_b @ A4 ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ C2 @ B ) @ ( transi2615809358984392588rm_a_b @ A4 ) )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ C2 ) @ ( abstra4889682940153293507rm_a_b @ A4 ) ) ) ) ).
% join_rtrancl_join
thf(fact_536_join__rtrancl__join,axiom,
! [A: list_nat,B: list_nat,A4: set_Pr3451248702717554689st_nat,C2: list_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A @ B ) @ ( abstra1638945853619340710st_nat @ A4 ) )
=> ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ C2 @ B ) @ ( transi5285580207609517981st_nat @ A4 ) )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A @ C2 ) @ ( abstra1638945853619340710st_nat @ A4 ) ) ) ) ).
% join_rtrancl_join
thf(fact_537_join__rtrancl__join,axiom,
! [A: term_a_b,B: term_a_b,A4: set_Pr4386577575007340137rm_a_b,C2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( abstra4096080454567261402rm_a_b @ A4 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ C2 @ B ) @ ( transi7742714808557438673rm_a_b @ A4 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C2 ) @ ( abstra4096080454567261402rm_a_b @ A4 ) ) ) ) ).
% join_rtrancl_join
thf(fact_538_joinE,axiom,
! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,A4: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( abstra4889682940153293507rm_a_b @ A4 ) )
=> ~ ! [C: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ C ) @ ( transi2615809358984392588rm_a_b @ A4 ) )
=> ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B @ C ) @ ( transi2615809358984392588rm_a_b @ A4 ) ) ) ) ).
% joinE
thf(fact_539_joinE,axiom,
! [A: list_nat,B: list_nat,A4: set_Pr3451248702717554689st_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A @ B ) @ ( abstra1638945853619340710st_nat @ A4 ) )
=> ~ ! [C: list_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A @ C ) @ ( transi5285580207609517981st_nat @ A4 ) )
=> ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ B @ C ) @ ( transi5285580207609517981st_nat @ A4 ) ) ) ) ).
% joinE
thf(fact_540_joinE,axiom,
! [A: term_a_b,B: term_a_b,A4: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( abstra4096080454567261402rm_a_b @ A4 ) )
=> ~ ! [C: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C ) @ ( transi7742714808557438673rm_a_b @ A4 ) )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C ) @ ( transi7742714808557438673rm_a_b @ A4 ) ) ) ) ).
% joinE
thf(fact_541_joinD,axiom,
! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,A4: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( abstra4889682940153293507rm_a_b @ A4 ) )
=> ? [C: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ C ) @ ( transi2615809358984392588rm_a_b @ A4 ) )
& ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B @ C ) @ ( transi2615809358984392588rm_a_b @ A4 ) ) ) ) ).
% joinD
thf(fact_542_joinD,axiom,
! [A: list_nat,B: list_nat,A4: set_Pr3451248702717554689st_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A @ B ) @ ( abstra1638945853619340710st_nat @ A4 ) )
=> ? [C: list_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A @ C ) @ ( transi5285580207609517981st_nat @ A4 ) )
& ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ B @ C ) @ ( transi5285580207609517981st_nat @ A4 ) ) ) ) ).
% joinD
thf(fact_543_joinD,axiom,
! [A: term_a_b,B: term_a_b,A4: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( abstra4096080454567261402rm_a_b @ A4 ) )
=> ? [C: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C ) @ ( transi7742714808557438673rm_a_b @ A4 ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C ) @ ( transi7742714808557438673rm_a_b @ A4 ) ) ) ) ).
% joinD
thf(fact_544_srsteps__subst__closed,axiom,
! [S: term_a_b,T: term_a_b,F: set_Pr4934435412358123699_a_nat,R3: set_Pr4386577575007340137rm_a_b,Sigma2: b > term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( transi7922773638565587891rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R3 ) ) ) )
=> ( ! [X3: b] : ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ ( Sigma2 @ X3 ) ) @ F )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( subst_7999470309526761004_a_b_b @ S @ Sigma2 ) @ ( subst_7999470309526761004_a_b_b @ T @ Sigma2 ) ) @ ( transi7922773638565587891rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R3 ) ) ) ) ) ) ).
% srsteps_subst_closed
thf(fact_545_sig__step__mono,axiom,
! [F: set_Pr4934435412358123699_a_nat,G: set_Pr4934435412358123699_a_nat,R3: set_Pr4386577575007340137rm_a_b] :
( ( ord_le8666007276011122963_a_nat @ F @ G )
=> ( ord_le118470702582115849rm_a_b @ ( sig_step_a_b @ F @ R3 ) @ ( sig_step_a_b @ G @ R3 ) ) ) ).
% sig_step_mono
thf(fact_546_subst__poss__mono,axiom,
! [S: term_a_b,Sigma2: b > term_a_b] : ( ord_le6045566169113846134st_nat @ ( term_poss_a_b @ S ) @ ( term_poss_a_b @ ( subst_7999470309526761004_a_b_b @ S @ Sigma2 ) ) ) ).
% subst_poss_mono
thf(fact_547_srsteps__eq__subst__closed,axiom,
! [S: term_a_b,T: term_a_b,F: set_Pr4934435412358123699_a_nat,R3: set_Pr4386577575007340137rm_a_b,Sigma2: b > term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R3 ) ) ) )
=> ( ! [X3: b] : ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ ( Sigma2 @ X3 ) ) @ F )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( subst_7999470309526761004_a_b_b @ S @ Sigma2 ) @ ( subst_7999470309526761004_a_b_b @ T @ Sigma2 ) ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R3 ) ) ) ) ) ) ).
% srsteps_eq_subst_closed
thf(fact_548_rrstep__rstep__mono,axiom,
! [R3: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ ( rrstep_a_b @ R3 ) @ ( rstep_a_b @ R3 ) ) ).
% rrstep_rstep_mono
thf(fact_549_sig__stepE,axiom,
! [S: term_a_b,T: term_a_b,F: set_Pr4934435412358123699_a_nat,R3: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( sig_step_a_b @ F @ R3 ) )
=> ~ ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ R3 )
=> ( ( 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_550_funas__term__replace__at__lower,axiom,
! [P2: list_nat,S: term_a_b,T: term_a_b] :
( ( member_list_nat @ P2 @ ( term_poss_a_b @ S ) )
=> ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ T ) @ ( term_funas_term_a_b @ ( term_r6860082780075436317at_a_b @ S @ P2 @ T ) ) ) ) ).
% funas_term_replace_at_lower
thf(fact_551_srstep__monp,axiom,
! [F: set_Pr4934435412358123699_a_nat,G: set_Pr4934435412358123699_a_nat,R3: set_Pr4386577575007340137rm_a_b] :
( ( ord_le8666007276011122963_a_nat @ F @ G )
=> ( ord_le118470702582115849rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R3 ) ) @ ( sig_step_a_b @ G @ ( rstep_a_b @ R3 ) ) ) ) ).
% srstep_monp
thf(fact_552_srrstep__monp,axiom,
! [F: set_Pr4934435412358123699_a_nat,G: set_Pr4934435412358123699_a_nat,R3: set_Pr4386577575007340137rm_a_b] :
( ( ord_le8666007276011122963_a_nat @ F @ G )
=> ( ord_le118470702582115849rm_a_b @ ( sig_step_a_b @ F @ ( rrstep_a_b @ R3 ) ) @ ( sig_step_a_b @ G @ ( rrstep_a_b @ R3 ) ) ) ) ).
% srrstep_monp
thf(fact_553_rsteps__srstepsI,axiom,
! [R3: 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 @ R3 ) @ 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 @ R3 ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( transi7922773638565587891rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R3 ) ) ) ) ) ) ) ) ).
% rsteps_srstepsI
thf(fact_554_rsteps__eq__srsteps__eqI,axiom,
! [R3: 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 @ R3 ) @ 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 @ R3 ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R3 ) ) ) ) ) ) ) ) ).
% rsteps_eq_srsteps_eqI
thf(fact_555_rstep__srstepI,axiom,
! [R3: 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 @ R3 ) @ 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 @ R3 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( sig_step_a_b @ F @ ( rstep_a_b @ R3 ) ) ) ) ) ) ) ).
% rstep_srstepI
thf(fact_556_tranclD,axiom,
! [X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b,R5: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ ( transi9211502839322181930rm_a_b @ R5 ) )
=> ? [Z3: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Z3 ) @ R5 )
& ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Z3 @ Y ) @ ( transi2615809358984392588rm_a_b @ R5 ) ) ) ) ).
% tranclD
thf(fact_557_tranclD,axiom,
! [X: list_nat,Y: list_nat,R5: set_Pr3451248702717554689st_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X @ Y ) @ ( transi5465639037617667199st_nat @ R5 ) )
=> ? [Z3: list_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X @ Z3 ) @ R5 )
& ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Z3 @ Y ) @ ( transi5285580207609517981st_nat @ R5 ) ) ) ) ).
% tranclD
thf(fact_558_tranclD,axiom,
! [X: term_a_b,Y: term_a_b,R5: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7922773638565587891rm_a_b @ R5 ) )
=> ? [Z3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z3 ) @ R5 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Z3 @ Y ) @ ( transi7742714808557438673rm_a_b @ R5 ) ) ) ) ).
% tranclD
thf(fact_559_rtranclD,axiom,
! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,R5: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( transi2615809358984392588rm_a_b @ R5 ) )
=> ( ( A = B )
| ( ( A != B )
& ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( transi9211502839322181930rm_a_b @ R5 ) ) ) ) ) ).
% rtranclD
thf(fact_560_rtranclD,axiom,
! [A: list_nat,B: list_nat,R5: set_Pr3451248702717554689st_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A @ B ) @ ( transi5285580207609517981st_nat @ R5 ) )
=> ( ( A = B )
| ( ( A != B )
& ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A @ B ) @ ( transi5465639037617667199st_nat @ R5 ) ) ) ) ) ).
% rtranclD
thf(fact_561_rtranclD,axiom,
! [A: term_a_b,B: term_a_b,R5: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ R5 ) )
=> ( ( A = B )
| ( ( A != B )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7922773638565587891rm_a_b @ R5 ) ) ) ) ) ).
% rtranclD
thf(fact_562_tranclD2,axiom,
! [X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b,R5: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ ( transi9211502839322181930rm_a_b @ R5 ) )
=> ? [Z3: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Z3 ) @ ( transi2615809358984392588rm_a_b @ R5 ) )
& ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Z3 @ Y ) @ R5 ) ) ) ).
% tranclD2
thf(fact_563_tranclD2,axiom,
! [X: list_nat,Y: list_nat,R5: set_Pr3451248702717554689st_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X @ Y ) @ ( transi5465639037617667199st_nat @ R5 ) )
=> ? [Z3: list_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X @ Z3 ) @ ( transi5285580207609517981st_nat @ R5 ) )
& ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Z3 @ Y ) @ R5 ) ) ) ).
% tranclD2
thf(fact_564_tranclD2,axiom,
! [X: term_a_b,Y: term_a_b,R5: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7922773638565587891rm_a_b @ R5 ) )
=> ? [Z3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z3 ) @ ( transi7742714808557438673rm_a_b @ R5 ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Z3 @ Y ) @ R5 ) ) ) ).
% tranclD2
thf(fact_565_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_566_trancl__into__rtrancl,axiom,
! [A: list_nat,B: list_nat,R2: set_Pr3451248702717554689st_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A @ B ) @ ( transi5465639037617667199st_nat @ R2 ) )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A @ B ) @ ( transi5285580207609517981st_nat @ R2 ) ) ) ).
% trancl_into_rtrancl
thf(fact_567_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_568_rtrancl__eq__or__trancl,axiom,
! [X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b,R5: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ ( transi2615809358984392588rm_a_b @ R5 ) )
= ( ( X = Y )
| ( ( X != Y )
& ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ ( transi9211502839322181930rm_a_b @ R5 ) ) ) ) ) ).
% rtrancl_eq_or_trancl
thf(fact_569_rtrancl__eq__or__trancl,axiom,
! [X: list_nat,Y: list_nat,R5: set_Pr3451248702717554689st_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X @ Y ) @ ( transi5285580207609517981st_nat @ R5 ) )
= ( ( X = Y )
| ( ( X != Y )
& ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X @ Y ) @ ( transi5465639037617667199st_nat @ R5 ) ) ) ) ) ).
% rtrancl_eq_or_trancl
thf(fact_570_rtrancl__eq__or__trancl,axiom,
! [X: term_a_b,Y: term_a_b,R5: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ R5 ) )
= ( ( X = Y )
| ( ( X != Y )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7922773638565587891rm_a_b @ R5 ) ) ) ) ) ).
% rtrancl_eq_or_trancl
thf(fact_571_trancl__empty,axiom,
( ( transi7922773638565587891rm_a_b @ bot_bo197521221353338581rm_a_b )
= bot_bo197521221353338581rm_a_b ) ).
% trancl_empty
thf(fact_572_trancl__empty,axiom,
( ( transi9211502839322181930rm_a_b @ bot_bo5040873672931234227rm_a_b )
= bot_bo5040873672931234227rm_a_b ) ).
% trancl_empty
thf(fact_573_trancl__empty,axiom,
( ( transi5465639037617667199st_nat @ bot_bo8485564385918328941st_nat )
= bot_bo8485564385918328941st_nat ) ).
% trancl_empty
thf(fact_574_trancl__induct2,axiom,
! [Ax: produc357393685978478089rm_a_b,Ay: produc357393685978478089rm_a_b,Bx: produc357393685978478089rm_a_b,By: produc357393685978478089rm_a_b,R2: set_Pr6848273517546626247rm_a_b,P3: produc357393685978478089rm_a_b > produc357393685978478089rm_a_b > $o] :
( ( member5793113912742238992rm_a_b @ ( produc2499998789532207959rm_a_b @ ( produc1763473618796451543rm_a_b @ Ax @ Ay ) @ ( produc1763473618796451543rm_a_b @ Bx @ By ) ) @ ( transi4560671716973800200rm_a_b @ R2 ) )
=> ( ! [A3: produc357393685978478089rm_a_b,B3: produc357393685978478089rm_a_b] :
( ( member5793113912742238992rm_a_b @ ( produc2499998789532207959rm_a_b @ ( produc1763473618796451543rm_a_b @ Ax @ Ay ) @ ( produc1763473618796451543rm_a_b @ A3 @ B3 ) ) @ R2 )
=> ( P3 @ A3 @ B3 ) )
=> ( ! [A3: produc357393685978478089rm_a_b,B3: produc357393685978478089rm_a_b,Aa: produc357393685978478089rm_a_b,Ba: produc357393685978478089rm_a_b] :
( ( member5793113912742238992rm_a_b @ ( produc2499998789532207959rm_a_b @ ( produc1763473618796451543rm_a_b @ Ax @ Ay ) @ ( produc1763473618796451543rm_a_b @ A3 @ B3 ) ) @ ( transi4560671716973800200rm_a_b @ R2 ) )
=> ( ( member5793113912742238992rm_a_b @ ( produc2499998789532207959rm_a_b @ ( produc1763473618796451543rm_a_b @ A3 @ B3 ) @ ( produc1763473618796451543rm_a_b @ Aa @ Ba ) ) @ R2 )
=> ( ( P3 @ A3 @ B3 )
=> ( P3 @ Aa @ Ba ) ) ) )
=> ( P3 @ Bx @ By ) ) ) ) ).
% trancl_induct2
thf(fact_575_trancl__induct2,axiom,
! [Ax: list_nat,Ay: list_nat,Bx: list_nat,By: list_nat,R2: set_Pr915550674247830855st_nat,P3: list_nat > list_nat > $o] :
( ( member5236317644330119056st_nat @ ( produc988773521843941591st_nat @ ( produc2694037385005941721st_nat @ Ax @ Ay ) @ ( produc2694037385005941721st_nat @ Bx @ By ) ) @ ( transi1459384740847974082st_nat @ R2 ) )
=> ( ! [A3: list_nat,B3: list_nat] :
( ( member5236317644330119056st_nat @ ( produc988773521843941591st_nat @ ( produc2694037385005941721st_nat @ Ax @ Ay ) @ ( produc2694037385005941721st_nat @ A3 @ B3 ) ) @ R2 )
=> ( P3 @ A3 @ B3 ) )
=> ( ! [A3: list_nat,B3: list_nat,Aa: list_nat,Ba: list_nat] :
( ( member5236317644330119056st_nat @ ( produc988773521843941591st_nat @ ( produc2694037385005941721st_nat @ Ax @ Ay ) @ ( produc2694037385005941721st_nat @ A3 @ B3 ) ) @ ( transi1459384740847974082st_nat @ R2 ) )
=> ( ( member5236317644330119056st_nat @ ( produc988773521843941591st_nat @ ( produc2694037385005941721st_nat @ A3 @ B3 ) @ ( produc2694037385005941721st_nat @ Aa @ Ba ) ) @ R2 )
=> ( ( P3 @ A3 @ B3 )
=> ( P3 @ Aa @ Ba ) ) ) )
=> ( P3 @ Bx @ By ) ) ) ) ).
% trancl_induct2
thf(fact_576_trancl__induct2,axiom,
! [Ax: produc357393685978478089rm_a_b,Ay: list_nat,Bx: produc357393685978478089rm_a_b,By: list_nat,R2: set_Pr2046385631809063665st_nat,P3: produc357393685978478089rm_a_b > list_nat > $o] :
( ( member2188129036539957306st_nat @ ( produc8516103258745939657st_nat @ ( produc5056965718246025362st_nat @ Ax @ Ay ) @ ( produc5056965718246025362st_nat @ Bx @ By ) ) @ ( transi507029068497101751st_nat @ R2 ) )
=> ( ! [A3: produc357393685978478089rm_a_b,B3: list_nat] :
( ( member2188129036539957306st_nat @ ( produc8516103258745939657st_nat @ ( produc5056965718246025362st_nat @ Ax @ Ay ) @ ( produc5056965718246025362st_nat @ A3 @ B3 ) ) @ R2 )
=> ( P3 @ A3 @ B3 ) )
=> ( ! [A3: produc357393685978478089rm_a_b,B3: list_nat,Aa: produc357393685978478089rm_a_b,Ba: list_nat] :
( ( member2188129036539957306st_nat @ ( produc8516103258745939657st_nat @ ( produc5056965718246025362st_nat @ Ax @ Ay ) @ ( produc5056965718246025362st_nat @ A3 @ B3 ) ) @ ( transi507029068497101751st_nat @ R2 ) )
=> ( ( member2188129036539957306st_nat @ ( produc8516103258745939657st_nat @ ( produc5056965718246025362st_nat @ A3 @ B3 ) @ ( produc5056965718246025362st_nat @ Aa @ Ba ) ) @ R2 )
=> ( ( P3 @ A3 @ B3 )
=> ( P3 @ Aa @ Ba ) ) ) )
=> ( P3 @ Bx @ By ) ) ) ) ).
% trancl_induct2
thf(fact_577_trancl__induct2,axiom,
! [Ax: list_nat,Ay: produc357393685978478089rm_a_b,Bx: list_nat,By: produc357393685978478089rm_a_b,R2: set_Pr2173775114972114941rm_a_b,P3: list_nat > produc357393685978478089rm_a_b > $o] :
( ( member6865565617957190470rm_a_b @ ( produc8750439296677016789rm_a_b @ ( produc3015861401532503632rm_a_b @ Ax @ Ay ) @ ( produc3015861401532503632rm_a_b @ Bx @ By ) ) @ ( transi4457611186189819837rm_a_b @ R2 ) )
=> ( ! [A3: list_nat,B3: produc357393685978478089rm_a_b] :
( ( member6865565617957190470rm_a_b @ ( produc8750439296677016789rm_a_b @ ( produc3015861401532503632rm_a_b @ Ax @ Ay ) @ ( produc3015861401532503632rm_a_b @ A3 @ B3 ) ) @ R2 )
=> ( P3 @ A3 @ B3 ) )
=> ( ! [A3: list_nat,B3: produc357393685978478089rm_a_b,Aa: list_nat,Ba: produc357393685978478089rm_a_b] :
( ( member6865565617957190470rm_a_b @ ( produc8750439296677016789rm_a_b @ ( produc3015861401532503632rm_a_b @ Ax @ Ay ) @ ( produc3015861401532503632rm_a_b @ A3 @ B3 ) ) @ ( transi4457611186189819837rm_a_b @ R2 ) )
=> ( ( member6865565617957190470rm_a_b @ ( produc8750439296677016789rm_a_b @ ( produc3015861401532503632rm_a_b @ A3 @ B3 ) @ ( produc3015861401532503632rm_a_b @ Aa @ Ba ) ) @ R2 )
=> ( ( P3 @ A3 @ B3 )
=> ( P3 @ Aa @ Ba ) ) ) )
=> ( P3 @ Bx @ By ) ) ) ) ).
% trancl_induct2
thf(fact_578_trancl__induct2,axiom,
! [Ax: term_a_b,Ay: term_a_b,Bx: term_a_b,By: term_a_b,R2: set_Pr2972776593051762503rm_a_b,P3: 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 ) )
=> ( ! [A3: term_a_b,B3: term_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ Ax @ Ay ) @ ( produc7020197800436672577rm_a_b @ A3 @ B3 ) ) @ R2 )
=> ( P3 @ A3 @ B3 ) )
=> ( ! [A3: term_a_b,B3: term_a_b,Aa: term_a_b,Ba: term_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ Ax @ Ay ) @ ( produc7020197800436672577rm_a_b @ A3 @ B3 ) ) @ ( transi9211502839322181930rm_a_b @ R2 ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B3 ) @ ( produc7020197800436672577rm_a_b @ Aa @ Ba ) ) @ R2 )
=> ( ( P3 @ A3 @ B3 )
=> ( P3 @ Aa @ Ba ) ) ) )
=> ( P3 @ Bx @ By ) ) ) ) ).
% trancl_induct2
thf(fact_579_rtrancl__induct2,axiom,
! [Ax: produc357393685978478089rm_a_b,Ay: produc357393685978478089rm_a_b,Bx: produc357393685978478089rm_a_b,By: produc357393685978478089rm_a_b,R2: set_Pr6848273517546626247rm_a_b,P3: produc357393685978478089rm_a_b > produc357393685978478089rm_a_b > $o] :
( ( member5793113912742238992rm_a_b @ ( produc2499998789532207959rm_a_b @ ( produc1763473618796451543rm_a_b @ Ax @ Ay ) @ ( produc1763473618796451543rm_a_b @ Bx @ By ) ) @ ( transi5332651807426350186rm_a_b @ R2 ) )
=> ( ( P3 @ Ax @ Ay )
=> ( ! [A3: produc357393685978478089rm_a_b,B3: produc357393685978478089rm_a_b,Aa: produc357393685978478089rm_a_b,Ba: produc357393685978478089rm_a_b] :
( ( member5793113912742238992rm_a_b @ ( produc2499998789532207959rm_a_b @ ( produc1763473618796451543rm_a_b @ Ax @ Ay ) @ ( produc1763473618796451543rm_a_b @ A3 @ B3 ) ) @ ( transi5332651807426350186rm_a_b @ R2 ) )
=> ( ( member5793113912742238992rm_a_b @ ( produc2499998789532207959rm_a_b @ ( produc1763473618796451543rm_a_b @ A3 @ B3 ) @ ( produc1763473618796451543rm_a_b @ Aa @ Ba ) ) @ R2 )
=> ( ( P3 @ A3 @ B3 )
=> ( P3 @ Aa @ Ba ) ) ) )
=> ( P3 @ Bx @ By ) ) ) ) ).
% rtrancl_induct2
thf(fact_580_rtrancl__induct2,axiom,
! [Ax: list_nat,Ay: list_nat,Bx: list_nat,By: list_nat,R2: set_Pr915550674247830855st_nat,P3: list_nat > list_nat > $o] :
( ( member5236317644330119056st_nat @ ( produc988773521843941591st_nat @ ( produc2694037385005941721st_nat @ Ax @ Ay ) @ ( produc2694037385005941721st_nat @ Bx @ By ) ) @ ( transi4087063297364960548st_nat @ R2 ) )
=> ( ( P3 @ Ax @ Ay )
=> ( ! [A3: list_nat,B3: list_nat,Aa: list_nat,Ba: list_nat] :
( ( member5236317644330119056st_nat @ ( produc988773521843941591st_nat @ ( produc2694037385005941721st_nat @ Ax @ Ay ) @ ( produc2694037385005941721st_nat @ A3 @ B3 ) ) @ ( transi4087063297364960548st_nat @ R2 ) )
=> ( ( member5236317644330119056st_nat @ ( produc988773521843941591st_nat @ ( produc2694037385005941721st_nat @ A3 @ B3 ) @ ( produc2694037385005941721st_nat @ Aa @ Ba ) ) @ R2 )
=> ( ( P3 @ A3 @ B3 )
=> ( P3 @ Aa @ Ba ) ) ) )
=> ( P3 @ Bx @ By ) ) ) ) ).
% rtrancl_induct2
thf(fact_581_rtrancl__induct2,axiom,
! [Ax: produc357393685978478089rm_a_b,Ay: list_nat,Bx: produc357393685978478089rm_a_b,By: list_nat,R2: set_Pr2046385631809063665st_nat,P3: produc357393685978478089rm_a_b > list_nat > $o] :
( ( member2188129036539957306st_nat @ ( produc8516103258745939657st_nat @ ( produc5056965718246025362st_nat @ Ax @ Ay ) @ ( produc5056965718246025362st_nat @ Bx @ By ) ) @ ( transi8117196366846528469st_nat @ R2 ) )
=> ( ( P3 @ Ax @ Ay )
=> ( ! [A3: produc357393685978478089rm_a_b,B3: list_nat,Aa: produc357393685978478089rm_a_b,Ba: list_nat] :
( ( member2188129036539957306st_nat @ ( produc8516103258745939657st_nat @ ( produc5056965718246025362st_nat @ Ax @ Ay ) @ ( produc5056965718246025362st_nat @ A3 @ B3 ) ) @ ( transi8117196366846528469st_nat @ R2 ) )
=> ( ( member2188129036539957306st_nat @ ( produc8516103258745939657st_nat @ ( produc5056965718246025362st_nat @ A3 @ B3 ) @ ( produc5056965718246025362st_nat @ Aa @ Ba ) ) @ R2 )
=> ( ( P3 @ A3 @ B3 )
=> ( P3 @ Aa @ Ba ) ) ) )
=> ( P3 @ Bx @ By ) ) ) ) ).
% rtrancl_induct2
thf(fact_582_rtrancl__induct2,axiom,
! [Ax: list_nat,Ay: produc357393685978478089rm_a_b,Bx: list_nat,By: produc357393685978478089rm_a_b,R2: set_Pr2173775114972114941rm_a_b,P3: list_nat > produc357393685978478089rm_a_b > $o] :
( ( member6865565617957190470rm_a_b @ ( produc8750439296677016789rm_a_b @ ( produc3015861401532503632rm_a_b @ Ax @ Ay ) @ ( produc3015861401532503632rm_a_b @ Bx @ By ) ) @ ( transi2844406447684470747rm_a_b @ R2 ) )
=> ( ( P3 @ Ax @ Ay )
=> ( ! [A3: list_nat,B3: produc357393685978478089rm_a_b,Aa: list_nat,Ba: produc357393685978478089rm_a_b] :
( ( member6865565617957190470rm_a_b @ ( produc8750439296677016789rm_a_b @ ( produc3015861401532503632rm_a_b @ Ax @ Ay ) @ ( produc3015861401532503632rm_a_b @ A3 @ B3 ) ) @ ( transi2844406447684470747rm_a_b @ R2 ) )
=> ( ( member6865565617957190470rm_a_b @ ( produc8750439296677016789rm_a_b @ ( produc3015861401532503632rm_a_b @ A3 @ B3 ) @ ( produc3015861401532503632rm_a_b @ Aa @ Ba ) ) @ R2 )
=> ( ( P3 @ A3 @ B3 )
=> ( P3 @ Aa @ Ba ) ) ) )
=> ( P3 @ Bx @ By ) ) ) ) ).
% rtrancl_induct2
thf(fact_583_rtrancl__induct2,axiom,
! [Ax: term_a_b,Ay: term_a_b,Bx: term_a_b,By: term_a_b,R2: set_Pr2972776593051762503rm_a_b,P3: 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 ) )
=> ( ( P3 @ Ax @ Ay )
=> ( ! [A3: term_a_b,B3: term_a_b,Aa: term_a_b,Ba: term_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ Ax @ Ay ) @ ( produc7020197800436672577rm_a_b @ A3 @ B3 ) ) @ ( transi2615809358984392588rm_a_b @ R2 ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B3 ) @ ( produc7020197800436672577rm_a_b @ Aa @ Ba ) ) @ R2 )
=> ( ( P3 @ A3 @ B3 )
=> ( P3 @ Aa @ Ba ) ) ) )
=> ( P3 @ Bx @ By ) ) ) ) ).
% rtrancl_induct2
thf(fact_584_converse__rtranclE2,axiom,
! [Xa: produc357393685978478089rm_a_b,Xb: produc357393685978478089rm_a_b,Za: produc357393685978478089rm_a_b,Zb: produc357393685978478089rm_a_b,R2: set_Pr6848273517546626247rm_a_b] :
( ( member5793113912742238992rm_a_b @ ( produc2499998789532207959rm_a_b @ ( produc1763473618796451543rm_a_b @ Xa @ Xb ) @ ( produc1763473618796451543rm_a_b @ Za @ Zb ) ) @ ( transi5332651807426350186rm_a_b @ R2 ) )
=> ( ( ( produc1763473618796451543rm_a_b @ Xa @ Xb )
!= ( produc1763473618796451543rm_a_b @ Za @ Zb ) )
=> ~ ! [A3: produc357393685978478089rm_a_b,B3: produc357393685978478089rm_a_b] :
( ( member5793113912742238992rm_a_b @ ( produc2499998789532207959rm_a_b @ ( produc1763473618796451543rm_a_b @ Xa @ Xb ) @ ( produc1763473618796451543rm_a_b @ A3 @ B3 ) ) @ R2 )
=> ~ ( member5793113912742238992rm_a_b @ ( produc2499998789532207959rm_a_b @ ( produc1763473618796451543rm_a_b @ A3 @ B3 ) @ ( produc1763473618796451543rm_a_b @ Za @ Zb ) ) @ ( transi5332651807426350186rm_a_b @ R2 ) ) ) ) ) ).
% converse_rtranclE2
thf(fact_585_converse__rtranclE2,axiom,
! [Xa: list_nat,Xb: list_nat,Za: list_nat,Zb: list_nat,R2: set_Pr915550674247830855st_nat] :
( ( member5236317644330119056st_nat @ ( produc988773521843941591st_nat @ ( produc2694037385005941721st_nat @ Xa @ Xb ) @ ( produc2694037385005941721st_nat @ Za @ Zb ) ) @ ( transi4087063297364960548st_nat @ R2 ) )
=> ( ( ( produc2694037385005941721st_nat @ Xa @ Xb )
!= ( produc2694037385005941721st_nat @ Za @ Zb ) )
=> ~ ! [A3: list_nat,B3: list_nat] :
( ( member5236317644330119056st_nat @ ( produc988773521843941591st_nat @ ( produc2694037385005941721st_nat @ Xa @ Xb ) @ ( produc2694037385005941721st_nat @ A3 @ B3 ) ) @ R2 )
=> ~ ( member5236317644330119056st_nat @ ( produc988773521843941591st_nat @ ( produc2694037385005941721st_nat @ A3 @ B3 ) @ ( produc2694037385005941721st_nat @ Za @ Zb ) ) @ ( transi4087063297364960548st_nat @ R2 ) ) ) ) ) ).
% converse_rtranclE2
thf(fact_586_converse__rtranclE2,axiom,
! [Xa: produc357393685978478089rm_a_b,Xb: list_nat,Za: produc357393685978478089rm_a_b,Zb: list_nat,R2: set_Pr2046385631809063665st_nat] :
( ( member2188129036539957306st_nat @ ( produc8516103258745939657st_nat @ ( produc5056965718246025362st_nat @ Xa @ Xb ) @ ( produc5056965718246025362st_nat @ Za @ Zb ) ) @ ( transi8117196366846528469st_nat @ R2 ) )
=> ( ( ( produc5056965718246025362st_nat @ Xa @ Xb )
!= ( produc5056965718246025362st_nat @ Za @ Zb ) )
=> ~ ! [A3: produc357393685978478089rm_a_b,B3: list_nat] :
( ( member2188129036539957306st_nat @ ( produc8516103258745939657st_nat @ ( produc5056965718246025362st_nat @ Xa @ Xb ) @ ( produc5056965718246025362st_nat @ A3 @ B3 ) ) @ R2 )
=> ~ ( member2188129036539957306st_nat @ ( produc8516103258745939657st_nat @ ( produc5056965718246025362st_nat @ A3 @ B3 ) @ ( produc5056965718246025362st_nat @ Za @ Zb ) ) @ ( transi8117196366846528469st_nat @ R2 ) ) ) ) ) ).
% converse_rtranclE2
thf(fact_587_converse__rtranclE2,axiom,
! [Xa: list_nat,Xb: produc357393685978478089rm_a_b,Za: list_nat,Zb: produc357393685978478089rm_a_b,R2: set_Pr2173775114972114941rm_a_b] :
( ( member6865565617957190470rm_a_b @ ( produc8750439296677016789rm_a_b @ ( produc3015861401532503632rm_a_b @ Xa @ Xb ) @ ( produc3015861401532503632rm_a_b @ Za @ Zb ) ) @ ( transi2844406447684470747rm_a_b @ R2 ) )
=> ( ( ( produc3015861401532503632rm_a_b @ Xa @ Xb )
!= ( produc3015861401532503632rm_a_b @ Za @ Zb ) )
=> ~ ! [A3: list_nat,B3: produc357393685978478089rm_a_b] :
( ( member6865565617957190470rm_a_b @ ( produc8750439296677016789rm_a_b @ ( produc3015861401532503632rm_a_b @ Xa @ Xb ) @ ( produc3015861401532503632rm_a_b @ A3 @ B3 ) ) @ R2 )
=> ~ ( member6865565617957190470rm_a_b @ ( produc8750439296677016789rm_a_b @ ( produc3015861401532503632rm_a_b @ A3 @ B3 ) @ ( produc3015861401532503632rm_a_b @ Za @ Zb ) ) @ ( transi2844406447684470747rm_a_b @ R2 ) ) ) ) ) ).
% converse_rtranclE2
thf(fact_588_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 ) )
=> ~ ! [A3: term_a_b,B3: term_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ Xa @ Xb ) @ ( produc7020197800436672577rm_a_b @ A3 @ B3 ) ) @ R2 )
=> ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B3 ) @ ( produc7020197800436672577rm_a_b @ Za @ Zb ) ) @ ( transi2615809358984392588rm_a_b @ R2 ) ) ) ) ) ).
% converse_rtranclE2
thf(fact_589_converse__rtrancl__induct2,axiom,
! [Ax: produc357393685978478089rm_a_b,Ay: produc357393685978478089rm_a_b,Bx: produc357393685978478089rm_a_b,By: produc357393685978478089rm_a_b,R2: set_Pr6848273517546626247rm_a_b,P3: produc357393685978478089rm_a_b > produc357393685978478089rm_a_b > $o] :
( ( member5793113912742238992rm_a_b @ ( produc2499998789532207959rm_a_b @ ( produc1763473618796451543rm_a_b @ Ax @ Ay ) @ ( produc1763473618796451543rm_a_b @ Bx @ By ) ) @ ( transi5332651807426350186rm_a_b @ R2 ) )
=> ( ( P3 @ Bx @ By )
=> ( ! [A3: produc357393685978478089rm_a_b,B3: produc357393685978478089rm_a_b,Aa: produc357393685978478089rm_a_b,Ba: produc357393685978478089rm_a_b] :
( ( member5793113912742238992rm_a_b @ ( produc2499998789532207959rm_a_b @ ( produc1763473618796451543rm_a_b @ A3 @ B3 ) @ ( produc1763473618796451543rm_a_b @ Aa @ Ba ) ) @ R2 )
=> ( ( member5793113912742238992rm_a_b @ ( produc2499998789532207959rm_a_b @ ( produc1763473618796451543rm_a_b @ Aa @ Ba ) @ ( produc1763473618796451543rm_a_b @ Bx @ By ) ) @ ( transi5332651807426350186rm_a_b @ R2 ) )
=> ( ( P3 @ Aa @ Ba )
=> ( P3 @ A3 @ B3 ) ) ) )
=> ( P3 @ Ax @ Ay ) ) ) ) ).
% converse_rtrancl_induct2
thf(fact_590_converse__rtrancl__induct2,axiom,
! [Ax: list_nat,Ay: list_nat,Bx: list_nat,By: list_nat,R2: set_Pr915550674247830855st_nat,P3: list_nat > list_nat > $o] :
( ( member5236317644330119056st_nat @ ( produc988773521843941591st_nat @ ( produc2694037385005941721st_nat @ Ax @ Ay ) @ ( produc2694037385005941721st_nat @ Bx @ By ) ) @ ( transi4087063297364960548st_nat @ R2 ) )
=> ( ( P3 @ Bx @ By )
=> ( ! [A3: list_nat,B3: list_nat,Aa: list_nat,Ba: list_nat] :
( ( member5236317644330119056st_nat @ ( produc988773521843941591st_nat @ ( produc2694037385005941721st_nat @ A3 @ B3 ) @ ( produc2694037385005941721st_nat @ Aa @ Ba ) ) @ R2 )
=> ( ( member5236317644330119056st_nat @ ( produc988773521843941591st_nat @ ( produc2694037385005941721st_nat @ Aa @ Ba ) @ ( produc2694037385005941721st_nat @ Bx @ By ) ) @ ( transi4087063297364960548st_nat @ R2 ) )
=> ( ( P3 @ Aa @ Ba )
=> ( P3 @ A3 @ B3 ) ) ) )
=> ( P3 @ Ax @ Ay ) ) ) ) ).
% converse_rtrancl_induct2
thf(fact_591_converse__rtrancl__induct2,axiom,
! [Ax: produc357393685978478089rm_a_b,Ay: list_nat,Bx: produc357393685978478089rm_a_b,By: list_nat,R2: set_Pr2046385631809063665st_nat,P3: produc357393685978478089rm_a_b > list_nat > $o] :
( ( member2188129036539957306st_nat @ ( produc8516103258745939657st_nat @ ( produc5056965718246025362st_nat @ Ax @ Ay ) @ ( produc5056965718246025362st_nat @ Bx @ By ) ) @ ( transi8117196366846528469st_nat @ R2 ) )
=> ( ( P3 @ Bx @ By )
=> ( ! [A3: produc357393685978478089rm_a_b,B3: list_nat,Aa: produc357393685978478089rm_a_b,Ba: list_nat] :
( ( member2188129036539957306st_nat @ ( produc8516103258745939657st_nat @ ( produc5056965718246025362st_nat @ A3 @ B3 ) @ ( produc5056965718246025362st_nat @ Aa @ Ba ) ) @ R2 )
=> ( ( member2188129036539957306st_nat @ ( produc8516103258745939657st_nat @ ( produc5056965718246025362st_nat @ Aa @ Ba ) @ ( produc5056965718246025362st_nat @ Bx @ By ) ) @ ( transi8117196366846528469st_nat @ R2 ) )
=> ( ( P3 @ Aa @ Ba )
=> ( P3 @ A3 @ B3 ) ) ) )
=> ( P3 @ Ax @ Ay ) ) ) ) ).
% converse_rtrancl_induct2
thf(fact_592_converse__rtrancl__induct2,axiom,
! [Ax: list_nat,Ay: produc357393685978478089rm_a_b,Bx: list_nat,By: produc357393685978478089rm_a_b,R2: set_Pr2173775114972114941rm_a_b,P3: list_nat > produc357393685978478089rm_a_b > $o] :
( ( member6865565617957190470rm_a_b @ ( produc8750439296677016789rm_a_b @ ( produc3015861401532503632rm_a_b @ Ax @ Ay ) @ ( produc3015861401532503632rm_a_b @ Bx @ By ) ) @ ( transi2844406447684470747rm_a_b @ R2 ) )
=> ( ( P3 @ Bx @ By )
=> ( ! [A3: list_nat,B3: produc357393685978478089rm_a_b,Aa: list_nat,Ba: produc357393685978478089rm_a_b] :
( ( member6865565617957190470rm_a_b @ ( produc8750439296677016789rm_a_b @ ( produc3015861401532503632rm_a_b @ A3 @ B3 ) @ ( produc3015861401532503632rm_a_b @ Aa @ Ba ) ) @ R2 )
=> ( ( member6865565617957190470rm_a_b @ ( produc8750439296677016789rm_a_b @ ( produc3015861401532503632rm_a_b @ Aa @ Ba ) @ ( produc3015861401532503632rm_a_b @ Bx @ By ) ) @ ( transi2844406447684470747rm_a_b @ R2 ) )
=> ( ( P3 @ Aa @ Ba )
=> ( P3 @ A3 @ B3 ) ) ) )
=> ( P3 @ Ax @ Ay ) ) ) ) ).
% converse_rtrancl_induct2
thf(fact_593_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,P3: 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 ) )
=> ( ( P3 @ Bx @ By )
=> ( ! [A3: term_a_b,B3: term_a_b,Aa: term_a_b,Ba: term_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B3 ) @ ( produc7020197800436672577rm_a_b @ Aa @ Ba ) ) @ R2 )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ Aa @ Ba ) @ ( produc7020197800436672577rm_a_b @ Bx @ By ) ) @ ( transi2615809358984392588rm_a_b @ R2 ) )
=> ( ( P3 @ Aa @ Ba )
=> ( P3 @ A3 @ B3 ) ) ) )
=> ( P3 @ Ax @ Ay ) ) ) ) ).
% converse_rtrancl_induct2
thf(fact_594_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_595_converse__rtrancl__into__rtrancl,axiom,
! [A: list_nat,B: list_nat,R2: set_Pr3451248702717554689st_nat,C2: list_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A @ B ) @ R2 )
=> ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ B @ C2 ) @ ( transi5285580207609517981st_nat @ R2 ) )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A @ C2 ) @ ( transi5285580207609517981st_nat @ R2 ) ) ) ) ).
% converse_rtrancl_into_rtrancl
thf(fact_596_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_597_converse__rtrancl__induct,axiom,
! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,R2: set_Pr2972776593051762503rm_a_b,P3: produc357393685978478089rm_a_b > $o] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( transi2615809358984392588rm_a_b @ R2 ) )
=> ( ( P3 @ B )
=> ( ! [Y3: produc357393685978478089rm_a_b,Z3: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y3 @ Z3 ) @ R2 )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Z3 @ B ) @ ( transi2615809358984392588rm_a_b @ R2 ) )
=> ( ( P3 @ Z3 )
=> ( P3 @ Y3 ) ) ) )
=> ( P3 @ A ) ) ) ) ).
% converse_rtrancl_induct
thf(fact_598_converse__rtrancl__induct,axiom,
! [A: list_nat,B: list_nat,R2: set_Pr3451248702717554689st_nat,P3: list_nat > $o] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A @ B ) @ ( transi5285580207609517981st_nat @ R2 ) )
=> ( ( P3 @ B )
=> ( ! [Y3: list_nat,Z3: list_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Y3 @ Z3 ) @ R2 )
=> ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Z3 @ B ) @ ( transi5285580207609517981st_nat @ R2 ) )
=> ( ( P3 @ Z3 )
=> ( P3 @ Y3 ) ) ) )
=> ( P3 @ A ) ) ) ) ).
% converse_rtrancl_induct
thf(fact_599_converse__rtrancl__induct,axiom,
! [A: term_a_b,B: term_a_b,R2: set_Pr4386577575007340137rm_a_b,P3: term_a_b > $o] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
=> ( ( P3 @ B )
=> ( ! [Y3: term_a_b,Z3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ Z3 ) @ R2 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Z3 @ B ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
=> ( ( P3 @ Z3 )
=> ( P3 @ Y3 ) ) ) )
=> ( P3 @ A ) ) ) ) ).
% converse_rtrancl_induct
thf(fact_600_converse__rtranclE,axiom,
! [X: produc357393685978478089rm_a_b,Z: produc357393685978478089rm_a_b,R2: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Z ) @ ( transi2615809358984392588rm_a_b @ R2 ) )
=> ( ( X != Z )
=> ~ ! [Y3: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y3 ) @ R2 )
=> ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y3 @ Z ) @ ( transi2615809358984392588rm_a_b @ R2 ) ) ) ) ) ).
% converse_rtranclE
thf(fact_601_converse__rtranclE,axiom,
! [X: list_nat,Z: list_nat,R2: set_Pr3451248702717554689st_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X @ Z ) @ ( transi5285580207609517981st_nat @ R2 ) )
=> ( ( X != Z )
=> ~ ! [Y3: list_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X @ Y3 ) @ R2 )
=> ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Y3 @ Z ) @ ( transi5285580207609517981st_nat @ R2 ) ) ) ) ) ).
% converse_rtranclE
thf(fact_602_converse__rtranclE,axiom,
! [X: term_a_b,Z: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
=> ( ( X != Z )
=> ~ ! [Y3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y3 ) @ R2 )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ Z ) @ ( transi7742714808557438673rm_a_b @ R2 ) ) ) ) ) ).
% converse_rtranclE
thf(fact_603_rtrancl__induct,axiom,
! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,R2: set_Pr2972776593051762503rm_a_b,P3: produc357393685978478089rm_a_b > $o] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( transi2615809358984392588rm_a_b @ R2 ) )
=> ( ( P3 @ A )
=> ( ! [Y3: produc357393685978478089rm_a_b,Z3: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ Y3 ) @ ( transi2615809358984392588rm_a_b @ R2 ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y3 @ Z3 ) @ R2 )
=> ( ( P3 @ Y3 )
=> ( P3 @ Z3 ) ) ) )
=> ( P3 @ B ) ) ) ) ).
% rtrancl_induct
thf(fact_604_rtrancl__induct,axiom,
! [A: list_nat,B: list_nat,R2: set_Pr3451248702717554689st_nat,P3: list_nat > $o] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A @ B ) @ ( transi5285580207609517981st_nat @ R2 ) )
=> ( ( P3 @ A )
=> ( ! [Y3: list_nat,Z3: list_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A @ Y3 ) @ ( transi5285580207609517981st_nat @ R2 ) )
=> ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Y3 @ Z3 ) @ R2 )
=> ( ( P3 @ Y3 )
=> ( P3 @ Z3 ) ) ) )
=> ( P3 @ B ) ) ) ) ).
% rtrancl_induct
thf(fact_605_rtrancl__induct,axiom,
! [A: term_a_b,B: term_a_b,R2: set_Pr4386577575007340137rm_a_b,P3: term_a_b > $o] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
=> ( ( P3 @ A )
=> ( ! [Y3: term_a_b,Z3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ Y3 ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ Z3 ) @ R2 )
=> ( ( P3 @ Y3 )
=> ( P3 @ Z3 ) ) ) )
=> ( P3 @ B ) ) ) ) ).
% rtrancl_induct
thf(fact_606_rtrancl__trans,axiom,
! [X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b,R2: set_Pr2972776593051762503rm_a_b,Z: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ ( transi2615809358984392588rm_a_b @ R2 ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y @ Z ) @ ( transi2615809358984392588rm_a_b @ R2 ) )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Z ) @ ( transi2615809358984392588rm_a_b @ R2 ) ) ) ) ).
% rtrancl_trans
thf(fact_607_rtrancl__trans,axiom,
! [X: list_nat,Y: list_nat,R2: set_Pr3451248702717554689st_nat,Z: list_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X @ Y ) @ ( transi5285580207609517981st_nat @ R2 ) )
=> ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Y @ Z ) @ ( transi5285580207609517981st_nat @ R2 ) )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X @ Z ) @ ( transi5285580207609517981st_nat @ R2 ) ) ) ) ).
% rtrancl_trans
thf(fact_608_rtrancl__trans,axiom,
! [X: term_a_b,Y: term_a_b,R2: set_Pr4386577575007340137rm_a_b,Z: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ Z ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z ) @ ( transi7742714808557438673rm_a_b @ R2 ) ) ) ) ).
% rtrancl_trans
thf(fact_609_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_610_rtranclE,axiom,
! [A: list_nat,B: list_nat,R2: set_Pr3451248702717554689st_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A @ B ) @ ( transi5285580207609517981st_nat @ R2 ) )
=> ( ( A != B )
=> ~ ! [Y3: list_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A @ Y3 ) @ ( transi5285580207609517981st_nat @ R2 ) )
=> ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Y3 @ B ) @ R2 ) ) ) ) ).
% rtranclE
thf(fact_611_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_612_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_613_rtrancl_Ortrancl__into__rtrancl,axiom,
! [A: list_nat,B: list_nat,R2: set_Pr3451248702717554689st_nat,C2: list_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A @ B ) @ ( transi5285580207609517981st_nat @ R2 ) )
=> ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ B @ C2 ) @ R2 )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A @ C2 ) @ ( transi5285580207609517981st_nat @ R2 ) ) ) ) ).
% rtrancl.rtrancl_into_rtrancl
thf(fact_614_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_615_rtrancl_Ortrancl__refl,axiom,
! [A: list_nat,R2: set_Pr3451248702717554689st_nat] : ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A @ A ) @ ( transi5285580207609517981st_nat @ R2 ) ) ).
% rtrancl.rtrancl_refl
thf(fact_616_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_617_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 ) )
= ( ? [A5: term_a_b] :
( ( A1 = A5 )
& ( A22 = A5 ) )
| ? [A5: term_a_b,B4: term_a_b,C3: term_a_b] :
( ( A1 = A5 )
& ( A22 = C3 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A5 @ B4 ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B4 @ C3 ) @ R2 ) ) ) ) ).
% rtrancl.simps
thf(fact_618_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_619_converse__trancl__induct,axiom,
! [A: term_a_b,B: term_a_b,R2: set_Pr4386577575007340137rm_a_b,P3: 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 )
=> ( P3 @ Y3 ) )
=> ( ! [Y3: term_a_b,Z3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ Z3 ) @ R2 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Z3 @ B ) @ ( transi7922773638565587891rm_a_b @ R2 ) )
=> ( ( P3 @ Z3 )
=> ( P3 @ Y3 ) ) ) )
=> ( P3 @ A ) ) ) ) ).
% converse_trancl_induct
thf(fact_620_trancl__trans__induct,axiom,
! [X: term_a_b,Y: term_a_b,R2: set_Pr4386577575007340137rm_a_b,P3: 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 )
=> ( P3 @ X3 @ Y3 ) )
=> ( ! [X3: term_a_b,Y3: term_a_b,Z3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ ( transi7922773638565587891rm_a_b @ R2 ) )
=> ( ( P3 @ X3 @ Y3 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ Z3 ) @ ( transi7922773638565587891rm_a_b @ R2 ) )
=> ( ( P3 @ Y3 @ Z3 )
=> ( P3 @ X3 @ Z3 ) ) ) ) )
=> ( P3 @ X @ Y ) ) ) ) ).
% trancl_trans_induct
thf(fact_621_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_622_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_623_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_624_converse__tranclE,axiom,
! [X: term_a_b,Z: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z ) @ ( transi7922773638565587891rm_a_b @ R2 ) )
=> ( ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z ) @ R2 )
=> ~ ! [Y3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y3 ) @ R2 )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ Z ) @ ( transi7922773638565587891rm_a_b @ R2 ) ) ) ) ) ).
% converse_tranclE
thf(fact_625_r__r__into__trancl,axiom,
! [A: term_a_b,B: term_a_b,R5: set_Pr4386577575007340137rm_a_b,C2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R5 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C2 ) @ R5 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C2 ) @ ( transi7922773638565587891rm_a_b @ R5 ) ) ) ) ).
% r_r_into_trancl
thf(fact_626_trancl__induct,axiom,
! [A: term_a_b,B: term_a_b,R2: set_Pr4386577575007340137rm_a_b,P3: 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 )
=> ( P3 @ Y3 ) )
=> ( ! [Y3: term_a_b,Z3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ Y3 ) @ ( transi7922773638565587891rm_a_b @ R2 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ Z3 ) @ R2 )
=> ( ( P3 @ Y3 )
=> ( P3 @ Z3 ) ) ) )
=> ( P3 @ B ) ) ) ) ).
% trancl_induct
thf(fact_627_trancl__trans,axiom,
! [X: term_a_b,Y: term_a_b,R2: set_Pr4386577575007340137rm_a_b,Z: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7922773638565587891rm_a_b @ R2 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ Z ) @ ( transi7922773638565587891rm_a_b @ R2 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z ) @ ( transi7922773638565587891rm_a_b @ R2 ) ) ) ) ).
% trancl_trans
thf(fact_628_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 )
=> ~ ! [C: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C ) @ ( transi7922773638565587891rm_a_b @ R2 ) )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ C @ B ) @ R2 ) ) ) ) ).
% tranclE
thf(fact_629_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_630_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 ) )
= ( ? [A5: term_a_b,B4: term_a_b] :
( ( A1 = A5 )
& ( A22 = B4 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A5 @ B4 ) @ R2 ) )
| ? [A5: term_a_b,B4: term_a_b,C3: term_a_b] :
( ( A1 = A5 )
& ( A22 = C3 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A5 @ B4 ) @ ( transi7922773638565587891rm_a_b @ R2 ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B4 @ C3 ) @ R2 ) ) ) ) ).
% trancl.simps
thf(fact_631_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_632_sig__step__rsteps__dist,axiom,
! [R3: set_Pr4386577575007340137rm_a_b,F: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ ( terms_7988297476397195622_a_b_b @ R3 ) @ F )
=> ( ( sig_step_a_b @ F @ ( transi7922773638565587891rm_a_b @ ( rstep_a_b @ R3 ) ) )
= ( transi7922773638565587891rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R3 ) ) ) ) ) ).
% sig_step_rsteps_dist
thf(fact_633_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_634_rtrancl__trancl__trancl,axiom,
! [X: term_a_b,Y: term_a_b,R2: set_Pr4386577575007340137rm_a_b,Z: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ Z ) @ ( transi7922773638565587891rm_a_b @ R2 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z ) @ ( transi7922773638565587891rm_a_b @ R2 ) ) ) ) ).
% rtrancl_trancl_trancl
thf(fact_635_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_636_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_637_rstep__trancl__sig__step__r,axiom,
! [S: term_a_b,T: term_a_b,R3: set_Pr4386577575007340137rm_a_b,F: set_Pr4934435412358123699_a_nat,V: b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( transi7922773638565587891rm_a_b @ ( rstep_a_b @ R3 ) ) )
=> ( ( ord_le8666007276011122963_a_nat @ ( terms_7988297476397195622_a_b_b @ R3 ) @ F )
=> ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ S ) @ F )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ ( terms_8519481630511763164ig_a_b @ F @ V @ T ) ) @ ( transi7922773638565587891rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R3 ) ) ) ) ) ) ) ).
% rstep_trancl_sig_step_r
thf(fact_638_rstep__trancl__sig__step__l,axiom,
! [S: term_a_b,T: term_a_b,R3: set_Pr4386577575007340137rm_a_b,F: set_Pr4934435412358123699_a_nat,V: b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( transi7922773638565587891rm_a_b @ ( rstep_a_b @ R3 ) ) )
=> ( ( ord_le8666007276011122963_a_nat @ ( terms_7988297476397195622_a_b_b @ R3 ) @ F )
=> ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ T ) @ F )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( terms_8519481630511763164ig_a_b @ F @ V @ S ) @ T ) @ ( transi7922773638565587891rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R3 ) ) ) ) ) ) ) ).
% rstep_trancl_sig_step_l
thf(fact_639_rsteps__eq__relcomp__srsteps__eq__relcompI,axiom,
! [R3: set_Pr4386577575007340137rm_a_b,F: set_Pr4934435412358123699_a_nat,S5: set_Pr4386577575007340137rm_a_b,S: term_a_b,T: term_a_b] :
( ( ord_le8666007276011122963_a_nat @ ( terms_7988297476397195622_a_b_b @ R3 ) @ F )
=> ( ( ord_le8666007276011122963_a_nat @ ( terms_7988297476397195622_a_b_b @ S5 ) @ 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 @ R3 ) ) @ ( transi7742714808557438673rm_a_b @ ( rstep_a_b @ S5 ) ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R3 ) ) ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ S5 ) ) ) ) ) ) ) ) ) ) ).
% rsteps_eq_relcomp_srsteps_eq_relcompI
thf(fact_640_meetI,axiom,
! [A: term_a_b,B: term_a_b,A4: set_Pr4386577575007340137rm_a_b,C2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ A4 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C2 ) @ ( transi7742714808557438673rm_a_b @ A4 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C2 ) @ ( abstra1093061187967292567rm_a_b @ A4 ) ) ) ) ).
% meetI
thf(fact_641_srsteps__eq__subst__relcomp__closed,axiom,
! [S: term_a_b,T: term_a_b,F: set_Pr4934435412358123699_a_nat,R3: set_Pr4386577575007340137rm_a_b,S5: set_Pr4386577575007340137rm_a_b,Sigma2: 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 @ R3 ) ) ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ S5 ) ) ) ) )
=> ( ! [X3: b] : ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ ( Sigma2 @ X3 ) ) @ F )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( subst_7999470309526761004_a_b_b @ S @ Sigma2 ) @ ( subst_7999470309526761004_a_b_b @ T @ Sigma2 ) ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R3 ) ) ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ S5 ) ) ) ) ) ) ) ).
% srsteps_eq_subst_relcomp_closed
thf(fact_642_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_643_relcomp3__I,axiom,
! [T: term_a_b,U: term_a_b,A4: set_Pr4386577575007340137rm_a_b,S: term_a_b,B5: set_Pr4386577575007340137rm_a_b,V: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T @ U ) @ A4 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ B5 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ U @ V ) @ B5 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ V ) @ ( relcom370159955682700863rm_a_b @ B5 @ ( relcom370159955682700863rm_a_b @ A4 @ B5 ) ) ) ) ) ) ).
% relcomp3_I
thf(fact_644_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_645_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 ) )
= ( ? [A5: term_a_b,B4: term_a_b,C3: term_a_b] :
( ( A1 = A5 )
& ( A22 = C3 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A5 @ B4 ) @ R2 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B4 @ C3 ) @ S ) ) ) ) ).
% relcomp.simps
thf(fact_646_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_647_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,Z3: term_a_b] :
( ( Xz
= ( produc7020197800436672577rm_a_b @ X3 @ Z3 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ R2 )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ Z3 ) @ S ) ) ) ) ).
% relcompE
thf(fact_648_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_649_steps__map,axiom,
! [P3: term_a_b > $o,Q2: 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,R5: set_Pr4386577575007340137rm_a_b,S6: set_Pr4386577575007340137rm_a_b,U: term_a_b] :
( ! [T2: term_a_b,U2: term_a_b,R6: set_Pr4386577575007340137rm_a_b] :
( ( P3 @ T2 )
=> ( ( Q2 @ R6 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T2 @ U2 ) @ R6 )
=> ( ( P3 @ U2 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F3 @ T2 ) @ ( F3 @ U2 ) ) @ ( G2 @ R6 ) ) ) ) ) )
=> ( ( P3 @ T )
=> ( ( Q2 @ R5 )
=> ( ( Q2 @ S6 )
=> ( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T @ U ) @ ( transi7742714808557438673rm_a_b @ R5 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F3 @ T ) @ ( F3 @ U ) ) @ ( transi7742714808557438673rm_a_b @ ( G2 @ R5 ) ) ) )
& ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T @ U ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ R5 ) @ ( relcom370159955682700863rm_a_b @ S6 @ ( transi7742714808557438673rm_a_b @ R5 ) ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F3 @ T ) @ ( F3 @ U ) ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( G2 @ R5 ) ) @ ( relcom370159955682700863rm_a_b @ ( G2 @ S6 ) @ ( transi7742714808557438673rm_a_b @ ( G2 @ R5 ) ) ) ) ) ) ) ) ) ) ) ).
% steps_map
thf(fact_650_non__strict__into__strict,axiom,
! [NS: set_Pr4386577575007340137rm_a_b,S6: set_Pr4386577575007340137rm_a_b,S: term_a_b,T: term_a_b] :
( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ NS @ S6 ) @ S6 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ NS ) @ S6 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ S6 ) ) ) ).
% non_strict_into_strict
thf(fact_651_meetD,axiom,
! [B: term_a_b,C2: term_a_b,A4: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C2 ) @ ( abstra1093061187967292567rm_a_b @ A4 ) )
=> ? [A3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B ) @ ( transi7742714808557438673rm_a_b @ A4 ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ C2 ) @ ( transi7742714808557438673rm_a_b @ A4 ) ) ) ) ).
% meetD
thf(fact_652_meetE,axiom,
! [B: term_a_b,C2: term_a_b,A4: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C2 ) @ ( abstra1093061187967292567rm_a_b @ A4 ) )
=> ~ ! [A3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B ) @ ( transi7742714808557438673rm_a_b @ A4 ) )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ C2 ) @ ( transi7742714808557438673rm_a_b @ A4 ) ) ) ) ).
% meetE
thf(fact_653_rstep__term__to__sig__r,axiom,
! [S: term_a_b,T: term_a_b,R3: set_Pr4386577575007340137rm_a_b,F: set_Pr4934435412358123699_a_nat,V: b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( rstep_a_b @ R3 ) )
=> ( ( ord_le8666007276011122963_a_nat @ ( terms_7988297476397195622_a_b_b @ R3 ) @ F )
=> ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ S ) @ F )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ ( terms_8519481630511763164ig_a_b @ F @ V @ T ) ) @ ( rstep_a_b @ R3 ) ) ) ) ) ).
% rstep_term_to_sig_r
thf(fact_654_rstep__term__to__sig__l,axiom,
! [S: term_a_b,T: term_a_b,R3: set_Pr4386577575007340137rm_a_b,F: set_Pr4934435412358123699_a_nat,V: b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( rstep_a_b @ R3 ) )
=> ( ( ord_le8666007276011122963_a_nat @ ( terms_7988297476397195622_a_b_b @ R3 ) @ F )
=> ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ T ) @ F )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( terms_8519481630511763164ig_a_b @ F @ V @ S ) @ T ) @ ( rstep_a_b @ R3 ) ) ) ) ) ).
% rstep_term_to_sig_l
thf(fact_655_all__ctxt__closed__relcomp,axiom,
! [R3: set_Pr4386577575007340137rm_a_b,F: set_Pr4934435412358123699_a_nat,S5: set_Pr4386577575007340137rm_a_b] :
( ! [S4: term_a_b,T2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S4 @ T2 ) @ R3 )
=> ( ( S4 != T2 )
=> ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ S4 ) @ F )
& ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ T2 ) @ F ) ) ) )
=> ( ! [S4: term_a_b,T2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S4 @ T2 ) @ S5 )
=> ( ( S4 != T2 )
=> ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ S4 ) @ F )
& ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ T2 ) @ F ) ) ) )
=> ( ( terms_5226143800768910156ed_a_b @ F @ R3 )
=> ( ( terms_5226143800768910156ed_a_b @ F @ S5 )
=> ( terms_5226143800768910156ed_a_b @ F @ ( relcom370159955682700863rm_a_b @ R3 @ S5 ) ) ) ) ) ) ).
% all_ctxt_closed_relcomp
thf(fact_656_srsteps__with__root__step__def,axiom,
( srstep7844470518422762656ep_a_b
= ( ^ [F4: set_Pr4934435412358123699_a_nat,R7: set_Pr4386577575007340137rm_a_b] : ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F4 @ ( rstep_a_b @ R7 ) ) ) @ ( relcom370159955682700863rm_a_b @ ( sig_step_a_b @ F4 @ ( rrstep_a_b @ R7 ) ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F4 @ ( rstep_a_b @ R7 ) ) ) ) ) ) ) ).
% srsteps_with_root_step_def
thf(fact_657_all__ctxt__closed__sig__rsteps,axiom,
! [F: set_Pr4934435412358123699_a_nat,R3: set_Pr4386577575007340137rm_a_b] : ( terms_5226143800768910156ed_a_b @ F @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R3 ) ) ) ) ).
% all_ctxt_closed_sig_rsteps
thf(fact_658_hole__pos__replace__term__at,axiom,
! [C4: subterm_and_ctxt_a_b,P2: list_nat,S: term_a_b,U: term_a_b] :
( ( term_p3503116865373065078eq_nat @ ( term_hole_pos_a_b @ C4 ) @ P2 )
=> ( ( term_r6860082780075436317at_a_b @ ( subter2376574525758040790rm_a_b @ C4 @ S ) @ P2 @ U )
= ( subter2376574525758040790rm_a_b @ C4 @ ( term_r6860082780075436317at_a_b @ S @ ( term_pos_diff_nat @ P2 @ ( term_hole_pos_a_b @ C4 ) ) @ U ) ) ) ) ).
% hole_pos_replace_term_at
thf(fact_659_replace__term__at__above,axiom,
! [P2: list_nat,Q: list_nat,S: term_a_b,T: term_a_b,U: term_a_b] :
( ( term_p3503116865373065078eq_nat @ P2 @ Q )
=> ( ( term_r6860082780075436317at_a_b @ ( term_r6860082780075436317at_a_b @ S @ Q @ T ) @ P2 @ U )
= ( term_r6860082780075436317at_a_b @ S @ P2 @ U ) ) ) ).
% replace_term_at_above
thf(fact_660_replace__term__at__below,axiom,
! [P2: list_nat,Q: list_nat,S: term_a_b,T: term_a_b,U: term_a_b] :
( ( ( P2 != Q )
& ( term_p3503116865373065078eq_nat @ P2 @ Q ) )
=> ( ( term_r6860082780075436317at_a_b @ ( term_r6860082780075436317at_a_b @ S @ P2 @ T ) @ Q @ U )
= ( term_r6860082780075436317at_a_b @ S @ P2 @ ( term_r6860082780075436317at_a_b @ T @ ( term_pos_diff_nat @ Q @ P2 ) @ U ) ) ) ) ).
% replace_term_at_below
thf(fact_661_poss__pos__diffI,axiom,
! [P2: list_nat,Q: list_nat,S: term_a_b] :
( ( term_p3503116865373065078eq_nat @ P2 @ Q )
=> ( ( member_list_nat @ Q @ ( term_poss_a_b @ S ) )
=> ( member_list_nat @ ( term_pos_diff_nat @ Q @ P2 ) @ ( term_poss_a_b @ ( term_subt_at_a_b @ S @ P2 ) ) ) ) ) ).
% poss_pos_diffI
thf(fact_662_less__eq__subt__at__replace,axiom,
! [P2: list_nat,S: term_a_b,Q: list_nat,T: term_a_b] :
( ( member_list_nat @ P2 @ ( term_poss_a_b @ S ) )
=> ( ( term_p3503116865373065078eq_nat @ P2 @ Q )
=> ( ( term_subt_at_a_b @ ( term_r6860082780075436317at_a_b @ S @ Q @ T ) @ P2 )
= ( term_r6860082780075436317at_a_b @ ( term_subt_at_a_b @ S @ P2 ) @ ( term_pos_diff_nat @ Q @ P2 ) @ T ) ) ) ) ).
% less_eq_subt_at_replace
thf(fact_663_greater__eq__subt__at__replace,axiom,
! [P2: list_nat,S: term_a_b,Q: list_nat,T: term_a_b] :
( ( member_list_nat @ P2 @ ( term_poss_a_b @ S ) )
=> ( ( term_p3503116865373065078eq_nat @ Q @ P2 )
=> ( ( term_subt_at_a_b @ ( term_r6860082780075436317at_a_b @ S @ Q @ T ) @ P2 )
= ( term_subt_at_a_b @ T @ ( term_pos_diff_nat @ P2 @ Q ) ) ) ) ) ).
% greater_eq_subt_at_replace
thf(fact_664_all__ctxt__closed__sig__reflE,axiom,
! [F: set_Pr4934435412358123699_a_nat,R3: set_Pr4386577575007340137rm_a_b,T: term_a_b] :
( ( terms_5226143800768910156ed_a_b @ F @ R3 )
=> ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ T ) @ F )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T @ T ) @ R3 ) ) ) ).
% all_ctxt_closed_sig_reflE
thf(fact_665_srsteps__with__root__step__sresteps__eqD,axiom,
! [S: term_a_b,T: term_a_b,F: set_Pr4934435412358123699_a_nat,R3: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( srstep7844470518422762656ep_a_b @ F @ R3 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R3 ) ) ) ) ) ).
% srsteps_with_root_step_sresteps_eqD
thf(fact_666_srsteps__with__root__step__srstepsD,axiom,
! [S: term_a_b,T: term_a_b,F: set_Pr4934435412358123699_a_nat,R3: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( srstep7844470518422762656ep_a_b @ F @ R3 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( transi7922773638565587891rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R3 ) ) ) ) ) ).
% srsteps_with_root_step_srstepsD
thf(fact_667_eq__ctxt__at__pos__by__poss,axiom,
! [P2: list_nat,S: term_a_b,T: term_a_b] :
( ( member_list_nat @ P2 @ ( term_poss_a_b @ S ) )
=> ( ( member_list_nat @ P2 @ ( term_poss_a_b @ T ) )
=> ( ! [Q3: list_nat] :
( ~ ( term_p3503116865373065078eq_nat @ P2 @ Q3 )
=> ( ( member_list_nat @ Q3 @ ( term_poss_a_b @ S ) )
= ( member_list_nat @ Q3 @ ( term_poss_a_b @ T ) ) ) )
=> ( ! [Q3: list_nat] :
( ( member_list_nat @ Q3 @ ( term_poss_a_b @ S ) )
=> ( ~ ( term_p3503116865373065078eq_nat @ P2 @ Q3 )
=> ( ( term_fun_at_a_b @ S @ Q3 )
= ( term_fun_at_a_b @ T @ Q3 ) ) ) )
=> ( ( term_ctxt_at_pos_a_b @ S @ P2 )
= ( term_ctxt_at_pos_a_b @ T @ P2 ) ) ) ) ) ) ).
% eq_ctxt_at_pos_by_poss
thf(fact_668_comp__rtrancl__trancl,axiom,
! [R5: set_Pr4386577575007340137rm_a_b,S6: set_Pr4386577575007340137rm_a_b,S: term_a_b,T: term_a_b] :
( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ R5 @ S6 ) @ S6 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ R5 @ S6 ) ) @ S6 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( transi7922773638565587891rm_a_b @ S6 ) ) ) ) ).
% comp_rtrancl_trancl
thf(fact_669_CR__onI,axiom,
! [A4: set_list_nat,R2: set_Pr3451248702717554689st_nat] :
( ! [A3: list_nat,B3: list_nat,C: list_nat] :
( ( member_list_nat @ A3 @ A4 )
=> ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A3 @ B3 ) @ ( transi5285580207609517981st_nat @ R2 ) )
=> ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A3 @ C ) @ ( transi5285580207609517981st_nat @ R2 ) )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ B3 @ C ) @ ( abstra1638945853619340710st_nat @ R2 ) ) ) ) )
=> ( abstra5991784817725020458st_nat @ R2 @ A4 ) ) ).
% CR_onI
thf(fact_670_CR__onI,axiom,
! [A4: set_Pr4386577575007340137rm_a_b,R2: set_Pr2972776593051762503rm_a_b] :
( ! [A3: produc357393685978478089rm_a_b,B3: produc357393685978478089rm_a_b,C: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ A3 @ A4 )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A3 @ B3 ) @ ( transi2615809358984392588rm_a_b @ R2 ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A3 @ C ) @ ( transi2615809358984392588rm_a_b @ R2 ) )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B3 @ C ) @ ( abstra4889682940153293507rm_a_b @ R2 ) ) ) ) )
=> ( abstra7359694574113289279rm_a_b @ R2 @ A4 ) ) ).
% CR_onI
thf(fact_671_CR__onI,axiom,
! [A4: set_term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ! [A3: term_a_b,B3: term_a_b,C: term_a_b] :
( ( member_term_a_b @ A3 @ A4 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B3 ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ C ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B3 @ C ) @ ( abstra4096080454567261402rm_a_b @ R2 ) ) ) ) )
=> ( abstra8448919418672941150rm_a_b @ R2 @ A4 ) ) ).
% CR_onI
thf(fact_672_same__fstI,axiom,
! [P3: term_a_b > $o,X: term_a_b,Y6: term_a_b,Y: term_a_b,R5: term_a_b > set_Pr4386577575007340137rm_a_b] :
( ( P3 @ X )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y6 @ Y ) @ ( R5 @ X ) )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y6 ) @ ( produc7020197800436672577rm_a_b @ X @ Y ) ) @ ( same_f843731131676418198rm_a_b @ P3 @ R5 ) ) ) ) ).
% same_fstI
thf(fact_673_Un__iff,axiom,
! [C2: list_nat,A4: set_list_nat,B5: set_list_nat] :
( ( member_list_nat @ C2 @ ( sup_sup_set_list_nat @ A4 @ B5 ) )
= ( ( member_list_nat @ C2 @ A4 )
| ( member_list_nat @ C2 @ B5 ) ) ) ).
% Un_iff
thf(fact_674_Un__iff,axiom,
! [C2: produc357393685978478089rm_a_b,A4: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ C2 @ ( sup_su6776935440552674877rm_a_b @ A4 @ B5 ) )
= ( ( member5869715511025134514rm_a_b @ C2 @ A4 )
| ( member5869715511025134514rm_a_b @ C2 @ B5 ) ) ) ).
% Un_iff
thf(fact_675_UnCI,axiom,
! [C2: list_nat,B5: set_list_nat,A4: set_list_nat] :
( ( ~ ( member_list_nat @ C2 @ B5 )
=> ( member_list_nat @ C2 @ A4 ) )
=> ( member_list_nat @ C2 @ ( sup_sup_set_list_nat @ A4 @ B5 ) ) ) ).
% UnCI
thf(fact_676_UnCI,axiom,
! [C2: produc357393685978478089rm_a_b,B5: set_Pr4386577575007340137rm_a_b,A4: set_Pr4386577575007340137rm_a_b] :
( ( ~ ( member5869715511025134514rm_a_b @ C2 @ B5 )
=> ( member5869715511025134514rm_a_b @ C2 @ A4 ) )
=> ( member5869715511025134514rm_a_b @ C2 @ ( sup_su6776935440552674877rm_a_b @ A4 @ B5 ) ) ) ).
% UnCI
thf(fact_677_UnI2,axiom,
! [C2: list_nat,B5: set_list_nat,A4: set_list_nat] :
( ( member_list_nat @ C2 @ B5 )
=> ( member_list_nat @ C2 @ ( sup_sup_set_list_nat @ A4 @ B5 ) ) ) ).
% UnI2
thf(fact_678_UnI2,axiom,
! [C2: produc357393685978478089rm_a_b,B5: set_Pr4386577575007340137rm_a_b,A4: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ C2 @ B5 )
=> ( member5869715511025134514rm_a_b @ C2 @ ( sup_su6776935440552674877rm_a_b @ A4 @ B5 ) ) ) ).
% UnI2
thf(fact_679_UnI1,axiom,
! [C2: list_nat,A4: set_list_nat,B5: set_list_nat] :
( ( member_list_nat @ C2 @ A4 )
=> ( member_list_nat @ C2 @ ( sup_sup_set_list_nat @ A4 @ B5 ) ) ) ).
% UnI1
thf(fact_680_UnI1,axiom,
! [C2: produc357393685978478089rm_a_b,A4: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ C2 @ A4 )
=> ( member5869715511025134514rm_a_b @ C2 @ ( sup_su6776935440552674877rm_a_b @ A4 @ B5 ) ) ) ).
% UnI1
thf(fact_681_UnE,axiom,
! [C2: list_nat,A4: set_list_nat,B5: set_list_nat] :
( ( member_list_nat @ C2 @ ( sup_sup_set_list_nat @ A4 @ B5 ) )
=> ( ~ ( member_list_nat @ C2 @ A4 )
=> ( member_list_nat @ C2 @ B5 ) ) ) ).
% UnE
thf(fact_682_UnE,axiom,
! [C2: produc357393685978478089rm_a_b,A4: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ C2 @ ( sup_su6776935440552674877rm_a_b @ A4 @ B5 ) )
=> ( ~ ( member5869715511025134514rm_a_b @ C2 @ A4 )
=> ( member5869715511025134514rm_a_b @ C2 @ B5 ) ) ) ).
% UnE
thf(fact_683_eq__term__by__poss__fun__at,axiom,
! [S: term_a_b,T: term_a_b] :
( ( ( term_poss_a_b @ S )
= ( term_poss_a_b @ T ) )
=> ( ! [P: list_nat] :
( ( member_list_nat @ P @ ( term_poss_a_b @ S ) )
=> ( ( term_fun_at_a_b @ S @ P )
= ( term_fun_at_a_b @ T @ P ) ) )
=> ( S = T ) ) ) ).
% eq_term_by_poss_fun_at
thf(fact_684_rtrancl__Un__separatorE,axiom,
! [A: term_a_b,B: term_a_b,P3: set_Pr4386577575007340137rm_a_b,Q2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ P3 @ Q2 ) ) )
=> ( ! [X3: term_a_b,Y3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ X3 ) @ ( transi7742714808557438673rm_a_b @ P3 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ Q2 )
=> ( X3 = Y3 ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ P3 ) ) ) ) ).
% rtrancl_Un_separatorE
thf(fact_685_rtrancl__Un__separator__converseE,axiom,
! [A: term_a_b,B: term_a_b,P3: set_Pr4386577575007340137rm_a_b,Q2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ P3 @ Q2 ) ) )
=> ( ! [X3: term_a_b,Y3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ B ) @ ( transi7742714808557438673rm_a_b @ P3 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ X3 ) @ Q2 )
=> ( Y3 = X3 ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ P3 ) ) ) ) ).
% rtrancl_Un_separator_converseE
thf(fact_686_right__comp__S,axiom,
! [X: term_a_b,Y: term_a_b,S6: set_Pr4386577575007340137rm_a_b,NS: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( relcom370159955682700863rm_a_b @ S6 @ ( sup_su6776935440552674877rm_a_b @ ( relcom370159955682700863rm_a_b @ S6 @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ S6 ) @ ( transi7742714808557438673rm_a_b @ NS ) ) ) @ ( transi7742714808557438673rm_a_b @ NS ) ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( relcom370159955682700863rm_a_b @ S6 @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ S6 ) @ ( transi7742714808557438673rm_a_b @ NS ) ) ) ) ) ).
% right_comp_S
thf(fact_687_first__step__O,axiom,
! [C4: set_Pr4386577575007340137rm_a_b,A4: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b,X: term_a_b,Y: term_a_b] :
( ( C4
= ( sup_su6776935440552674877rm_a_b @ A4 @ 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 @ A4 ) @ B5 ) ) ) ) ).
% first_step_O
thf(fact_688_first__step,axiom,
! [C4: set_Pr4386577575007340137rm_a_b,A4: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b,X: term_a_b,Y: term_a_b,Z: term_a_b] :
( ( C4
= ( sup_su6776935440552674877rm_a_b @ A4 @ B5 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ C4 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ Z ) @ B5 )
=> ? [Y3: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y3 ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ A4 ) @ B5 ) ) ) ) ) ).
% first_step
thf(fact_689_firstStep,axiom,
! [L5: set_Pr4386577575007340137rm_a_b,S6: set_Pr4386577575007340137rm_a_b,R5: set_Pr4386577575007340137rm_a_b,X: term_a_b,Y: term_a_b] :
( ( L5
= ( sup_su6776935440552674877rm_a_b @ S6 @ R5 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ L5 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ R5 ) )
| ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ R5 ) @ ( relcom370159955682700863rm_a_b @ S6 @ ( transi7742714808557438673rm_a_b @ L5 ) ) ) ) ) ) ) ).
% firstStep
thf(fact_690_funas__term__replace__at__upper,axiom,
! [S: term_a_b,P2: list_nat,T: term_a_b] : ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ ( term_r6860082780075436317at_a_b @ S @ P2 @ T ) ) @ ( sup_su459911885395995103_a_nat @ ( term_funas_term_a_b @ S ) @ ( term_funas_term_a_b @ T ) ) ) ).
% funas_term_replace_at_upper
thf(fact_691_CR__on__def,axiom,
( abstra8448919418672941150rm_a_b
= ( ^ [R4: set_Pr4386577575007340137rm_a_b,A6: set_term_a_b] :
! [X4: term_a_b] :
( ( member_term_a_b @ X4 @ A6 )
=> ! [B4: term_a_b,C3: term_a_b] :
( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X4 @ B4 ) @ ( transi7742714808557438673rm_a_b @ R4 ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X4 @ C3 ) @ ( transi7742714808557438673rm_a_b @ R4 ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B4 @ C3 ) @ ( abstra4096080454567261402rm_a_b @ R4 ) ) ) ) ) ) ).
% CR_on_def
thf(fact_692_CR__onE,axiom,
! [R2: set_Pr3451248702717554689st_nat,A4: set_list_nat,A: list_nat,B: list_nat,C2: list_nat] :
( ( abstra5991784817725020458st_nat @ R2 @ A4 )
=> ( ( member_list_nat @ A @ A4 )
=> ( ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ B @ C2 ) @ ( abstra1638945853619340710st_nat @ R2 ) )
=> ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A @ B ) @ ( transi5285580207609517981st_nat @ R2 ) )
=> ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A @ C2 ) @ ( transi5285580207609517981st_nat @ R2 ) ) ) ) ) ) ).
% CR_onE
thf(fact_693_CR__onE,axiom,
! [R2: set_Pr2972776593051762503rm_a_b,A4: set_Pr4386577575007340137rm_a_b,A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,C2: produc357393685978478089rm_a_b] :
( ( abstra7359694574113289279rm_a_b @ R2 @ A4 )
=> ( ( member5869715511025134514rm_a_b @ A @ A4 )
=> ( ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B @ C2 ) @ ( abstra4889682940153293507rm_a_b @ R2 ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( transi2615809358984392588rm_a_b @ R2 ) )
=> ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ C2 ) @ ( transi2615809358984392588rm_a_b @ R2 ) ) ) ) ) ) ).
% CR_onE
thf(fact_694_CR__onE,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,A4: set_term_a_b,A: term_a_b,B: term_a_b,C2: term_a_b] :
( ( abstra8448919418672941150rm_a_b @ R2 @ A4 )
=> ( ( member_term_a_b @ A @ A4 )
=> ( ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C2 ) @ ( abstra4096080454567261402rm_a_b @ R2 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C2 ) @ ( transi7742714808557438673rm_a_b @ R2 ) ) ) ) ) ) ).
% CR_onE
thf(fact_695_CR__onD,axiom,
! [R2: set_Pr3451248702717554689st_nat,A4: set_list_nat,A: list_nat,B: list_nat,C2: list_nat] :
( ( abstra5991784817725020458st_nat @ R2 @ A4 )
=> ( ( member_list_nat @ A @ A4 )
=> ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A @ B ) @ ( transi5285580207609517981st_nat @ R2 ) )
=> ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A @ C2 ) @ ( transi5285580207609517981st_nat @ R2 ) )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ B @ C2 ) @ ( abstra1638945853619340710st_nat @ R2 ) ) ) ) ) ) ).
% CR_onD
thf(fact_696_CR__onD,axiom,
! [R2: set_Pr2972776593051762503rm_a_b,A4: set_Pr4386577575007340137rm_a_b,A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,C2: produc357393685978478089rm_a_b] :
( ( abstra7359694574113289279rm_a_b @ R2 @ A4 )
=> ( ( member5869715511025134514rm_a_b @ A @ A4 )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( transi2615809358984392588rm_a_b @ R2 ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ C2 ) @ ( transi2615809358984392588rm_a_b @ R2 ) )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B @ C2 ) @ ( abstra4889682940153293507rm_a_b @ R2 ) ) ) ) ) ) ).
% CR_onD
thf(fact_697_CR__onD,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,A4: set_term_a_b,A: term_a_b,B: term_a_b,C2: term_a_b] :
( ( abstra8448919418672941150rm_a_b @ R2 @ A4 )
=> ( ( member_term_a_b @ A @ A4 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C2 ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C2 ) @ ( abstra4096080454567261402rm_a_b @ R2 ) ) ) ) ) ) ).
% CR_onD
thf(fact_698_compatible__rtrancl__split,axiom,
! [NS: set_Pr4386577575007340137rm_a_b,S6: set_Pr4386577575007340137rm_a_b,X: term_a_b,Y: term_a_b] :
( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ NS @ S6 ) @ S6 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ NS @ S6 ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( sup_su6776935440552674877rm_a_b @ ( relcom370159955682700863rm_a_b @ S6 @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ S6 ) @ ( transi7742714808557438673rm_a_b @ NS ) ) ) @ ( transi7742714808557438673rm_a_b @ NS ) ) ) ) ) ).
% compatible_rtrancl_split
thf(fact_699_sig__step__rsteps__eq__dist,axiom,
! [R3: set_Pr4386577575007340137rm_a_b,F: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ ( terms_7988297476397195622_a_b_b @ R3 ) @ F )
=> ( ( sup_su6776935440552674877rm_a_b @ ( sig_step_a_b @ F @ ( transi7922773638565587891rm_a_b @ ( rstep_a_b @ R3 ) ) ) @ id_term_a_b )
= ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R3 ) ) ) ) ) ).
% sig_step_rsteps_eq_dist
thf(fact_700_CR__on__singletonI,axiom,
! [A: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ! [B3: term_a_b,C: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B3 ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B3 @ C ) @ ( abstra4096080454567261402rm_a_b @ R2 ) ) ) )
=> ( abstra8448919418672941150rm_a_b @ R2 @ ( insert_term_a_b @ A @ bot_bot_set_term_a_b ) ) ) ).
% CR_on_singletonI
thf(fact_701_CR__iff__meet__subset__join,axiom,
! [R2: set_Pr4386577575007340137rm_a_b] :
( ( abstra8448919418672941150rm_a_b @ R2 @ top_top_set_term_a_b )
= ( ord_le118470702582115849rm_a_b @ ( abstra1093061187967292567rm_a_b @ R2 ) @ ( abstra4096080454567261402rm_a_b @ R2 ) ) ) ).
% CR_iff_meet_subset_join
thf(fact_702_UNIV__I,axiom,
! [X: list_nat] : ( member_list_nat @ X @ top_top_set_list_nat ) ).
% UNIV_I
thf(fact_703_UNIV__I,axiom,
! [X: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ X @ top_to1314267278846557113rm_a_b ) ).
% UNIV_I
thf(fact_704_insert__iff,axiom,
! [A: list_nat,B: list_nat,A4: set_list_nat] :
( ( member_list_nat @ A @ ( insert_list_nat @ B @ A4 ) )
= ( ( A = B )
| ( member_list_nat @ A @ A4 ) ) ) ).
% insert_iff
thf(fact_705_insert__iff,axiom,
! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,A4: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ A @ ( insert7009541432154983385rm_a_b @ B @ A4 ) )
= ( ( A = B )
| ( member5869715511025134514rm_a_b @ A @ A4 ) ) ) ).
% insert_iff
thf(fact_706_insertCI,axiom,
! [A: list_nat,B5: set_list_nat,B: list_nat] :
( ( ~ ( member_list_nat @ A @ B5 )
=> ( A = B ) )
=> ( member_list_nat @ A @ ( insert_list_nat @ B @ B5 ) ) ) ).
% insertCI
thf(fact_707_insertCI,axiom,
! [A: produc357393685978478089rm_a_b,B5: set_Pr4386577575007340137rm_a_b,B: produc357393685978478089rm_a_b] :
( ( ~ ( member5869715511025134514rm_a_b @ A @ B5 )
=> ( A = B ) )
=> ( member5869715511025134514rm_a_b @ A @ ( insert7009541432154983385rm_a_b @ B @ B5 ) ) ) ).
% insertCI
thf(fact_708_singletonI,axiom,
! [A: list_nat] : ( member_list_nat @ A @ ( insert_list_nat @ A @ bot_bot_set_list_nat ) ) ).
% singletonI
thf(fact_709_singletonI,axiom,
! [A: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ A @ ( insert7009541432154983385rm_a_b @ A @ bot_bo197521221353338581rm_a_b ) ) ).
% singletonI
thf(fact_710_insert__subset,axiom,
! [X: list_nat,A4: set_list_nat,B5: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ ( insert_list_nat @ X @ A4 ) @ B5 )
= ( ( member_list_nat @ X @ B5 )
& ( ord_le6045566169113846134st_nat @ A4 @ B5 ) ) ) ).
% insert_subset
thf(fact_711_insert__subset,axiom,
! [X: produc357393685978478089rm_a_b,A4: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ ( insert7009541432154983385rm_a_b @ X @ A4 ) @ B5 )
= ( ( member5869715511025134514rm_a_b @ X @ B5 )
& ( ord_le118470702582115849rm_a_b @ A4 @ B5 ) ) ) ).
% insert_subset
thf(fact_712_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_713_IdI,axiom,
! [A: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ A ) @ id_term_a_b ) ).
% IdI
thf(fact_714_mk__disjoint__insert,axiom,
! [A: list_nat,A4: set_list_nat] :
( ( member_list_nat @ A @ A4 )
=> ? [B7: set_list_nat] :
( ( A4
= ( insert_list_nat @ A @ B7 ) )
& ~ ( member_list_nat @ A @ B7 ) ) ) ).
% mk_disjoint_insert
thf(fact_715_mk__disjoint__insert,axiom,
! [A: produc357393685978478089rm_a_b,A4: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ A @ A4 )
=> ? [B7: set_Pr4386577575007340137rm_a_b] :
( ( A4
= ( insert7009541432154983385rm_a_b @ A @ B7 ) )
& ~ ( member5869715511025134514rm_a_b @ A @ B7 ) ) ) ).
% mk_disjoint_insert
thf(fact_716_insert__eq__iff,axiom,
! [A: list_nat,A4: set_list_nat,B: list_nat,B5: set_list_nat] :
( ~ ( member_list_nat @ A @ A4 )
=> ( ~ ( member_list_nat @ B @ B5 )
=> ( ( ( insert_list_nat @ A @ A4 )
= ( insert_list_nat @ B @ B5 ) )
= ( ( ( A = B )
=> ( A4 = B5 ) )
& ( ( A != B )
=> ? [C6: set_list_nat] :
( ( A4
= ( insert_list_nat @ B @ C6 ) )
& ~ ( member_list_nat @ B @ C6 )
& ( B5
= ( insert_list_nat @ A @ C6 ) )
& ~ ( member_list_nat @ A @ C6 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_717_insert__eq__iff,axiom,
! [A: produc357393685978478089rm_a_b,A4: set_Pr4386577575007340137rm_a_b,B: produc357393685978478089rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
( ~ ( member5869715511025134514rm_a_b @ A @ A4 )
=> ( ~ ( member5869715511025134514rm_a_b @ B @ B5 )
=> ( ( ( insert7009541432154983385rm_a_b @ A @ A4 )
= ( insert7009541432154983385rm_a_b @ B @ B5 ) )
= ( ( ( A = B )
=> ( A4 = B5 ) )
& ( ( A != B )
=> ? [C6: set_Pr4386577575007340137rm_a_b] :
( ( A4
= ( insert7009541432154983385rm_a_b @ B @ C6 ) )
& ~ ( member5869715511025134514rm_a_b @ B @ C6 )
& ( B5
= ( insert7009541432154983385rm_a_b @ A @ C6 ) )
& ~ ( member5869715511025134514rm_a_b @ A @ C6 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_718_insert__absorb,axiom,
! [A: list_nat,A4: set_list_nat] :
( ( member_list_nat @ A @ A4 )
=> ( ( insert_list_nat @ A @ A4 )
= A4 ) ) ).
% insert_absorb
thf(fact_719_insert__absorb,axiom,
! [A: produc357393685978478089rm_a_b,A4: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ A @ A4 )
=> ( ( insert7009541432154983385rm_a_b @ A @ A4 )
= A4 ) ) ).
% insert_absorb
thf(fact_720_insert__ident,axiom,
! [X: list_nat,A4: set_list_nat,B5: set_list_nat] :
( ~ ( member_list_nat @ X @ A4 )
=> ( ~ ( member_list_nat @ X @ B5 )
=> ( ( ( insert_list_nat @ X @ A4 )
= ( insert_list_nat @ X @ B5 ) )
= ( A4 = B5 ) ) ) ) ).
% insert_ident
thf(fact_721_insert__ident,axiom,
! [X: produc357393685978478089rm_a_b,A4: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
( ~ ( member5869715511025134514rm_a_b @ X @ A4 )
=> ( ~ ( member5869715511025134514rm_a_b @ X @ B5 )
=> ( ( ( insert7009541432154983385rm_a_b @ X @ A4 )
= ( insert7009541432154983385rm_a_b @ X @ B5 ) )
= ( A4 = B5 ) ) ) ) ).
% insert_ident
thf(fact_722_UNIV__witness,axiom,
? [X3: list_nat] : ( member_list_nat @ X3 @ top_top_set_list_nat ) ).
% UNIV_witness
thf(fact_723_UNIV__witness,axiom,
? [X3: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ X3 @ top_to1314267278846557113rm_a_b ) ).
% UNIV_witness
thf(fact_724_Set_Oset__insert,axiom,
! [X: list_nat,A4: set_list_nat] :
( ( member_list_nat @ X @ A4 )
=> ~ ! [B7: set_list_nat] :
( ( A4
= ( insert_list_nat @ X @ B7 ) )
=> ( member_list_nat @ X @ B7 ) ) ) ).
% Set.set_insert
thf(fact_725_Set_Oset__insert,axiom,
! [X: produc357393685978478089rm_a_b,A4: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ X @ A4 )
=> ~ ! [B7: set_Pr4386577575007340137rm_a_b] :
( ( A4
= ( insert7009541432154983385rm_a_b @ X @ B7 ) )
=> ( member5869715511025134514rm_a_b @ X @ B7 ) ) ) ).
% Set.set_insert
thf(fact_726_UNIV__eq__I,axiom,
! [A4: set_list_nat] :
( ! [X3: list_nat] : ( member_list_nat @ X3 @ A4 )
=> ( top_top_set_list_nat = A4 ) ) ).
% UNIV_eq_I
thf(fact_727_UNIV__eq__I,axiom,
! [A4: set_Pr4386577575007340137rm_a_b] :
( ! [X3: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ X3 @ A4 )
=> ( top_to1314267278846557113rm_a_b = A4 ) ) ).
% UNIV_eq_I
thf(fact_728_insertI2,axiom,
! [A: list_nat,B5: set_list_nat,B: list_nat] :
( ( member_list_nat @ A @ B5 )
=> ( member_list_nat @ A @ ( insert_list_nat @ B @ B5 ) ) ) ).
% insertI2
thf(fact_729_insertI2,axiom,
! [A: produc357393685978478089rm_a_b,B5: set_Pr4386577575007340137rm_a_b,B: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ A @ B5 )
=> ( member5869715511025134514rm_a_b @ A @ ( insert7009541432154983385rm_a_b @ B @ B5 ) ) ) ).
% insertI2
thf(fact_730_insertI1,axiom,
! [A: list_nat,B5: set_list_nat] : ( member_list_nat @ A @ ( insert_list_nat @ A @ B5 ) ) ).
% insertI1
thf(fact_731_insertI1,axiom,
! [A: produc357393685978478089rm_a_b,B5: set_Pr4386577575007340137rm_a_b] : ( member5869715511025134514rm_a_b @ A @ ( insert7009541432154983385rm_a_b @ A @ B5 ) ) ).
% insertI1
thf(fact_732_insertE,axiom,
! [A: list_nat,B: list_nat,A4: set_list_nat] :
( ( member_list_nat @ A @ ( insert_list_nat @ B @ A4 ) )
=> ( ( A != B )
=> ( member_list_nat @ A @ A4 ) ) ) ).
% insertE
thf(fact_733_insertE,axiom,
! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,A4: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ A @ ( insert7009541432154983385rm_a_b @ B @ A4 ) )
=> ( ( A != B )
=> ( member5869715511025134514rm_a_b @ A @ A4 ) ) ) ).
% insertE
thf(fact_734_singletonD,axiom,
! [B: list_nat,A: list_nat] :
( ( member_list_nat @ B @ ( insert_list_nat @ A @ bot_bot_set_list_nat ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_735_singletonD,axiom,
! [B: produc357393685978478089rm_a_b,A: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ B @ ( insert7009541432154983385rm_a_b @ A @ bot_bo197521221353338581rm_a_b ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_736_singleton__iff,axiom,
! [B: list_nat,A: list_nat] :
( ( member_list_nat @ B @ ( insert_list_nat @ A @ bot_bot_set_list_nat ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_737_singleton__iff,axiom,
! [B: produc357393685978478089rm_a_b,A: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ B @ ( insert7009541432154983385rm_a_b @ A @ bot_bo197521221353338581rm_a_b ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_738_insert__subsetI,axiom,
! [X: list_nat,A4: set_list_nat,X5: set_list_nat] :
( ( member_list_nat @ X @ A4 )
=> ( ( ord_le6045566169113846134st_nat @ X5 @ A4 )
=> ( ord_le6045566169113846134st_nat @ ( insert_list_nat @ X @ X5 ) @ A4 ) ) ) ).
% insert_subsetI
thf(fact_739_insert__subsetI,axiom,
! [X: produc357393685978478089rm_a_b,A4: set_Pr4386577575007340137rm_a_b,X5: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ X @ A4 )
=> ( ( ord_le118470702582115849rm_a_b @ X5 @ A4 )
=> ( ord_le118470702582115849rm_a_b @ ( insert7009541432154983385rm_a_b @ X @ X5 ) @ A4 ) ) ) ).
% insert_subsetI
thf(fact_740_subset__insert,axiom,
! [X: list_nat,A4: set_list_nat,B5: set_list_nat] :
( ~ ( member_list_nat @ X @ A4 )
=> ( ( ord_le6045566169113846134st_nat @ A4 @ ( insert_list_nat @ X @ B5 ) )
= ( ord_le6045566169113846134st_nat @ A4 @ B5 ) ) ) ).
% subset_insert
thf(fact_741_subset__insert,axiom,
! [X: produc357393685978478089rm_a_b,A4: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
( ~ ( member5869715511025134514rm_a_b @ X @ A4 )
=> ( ( ord_le118470702582115849rm_a_b @ A4 @ ( insert7009541432154983385rm_a_b @ X @ B5 ) )
= ( ord_le118470702582115849rm_a_b @ A4 @ B5 ) ) ) ).
% subset_insert
thf(fact_742_IdE,axiom,
! [P2: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ P2 @ id_term_a_b )
=> ~ ! [X3: term_a_b] :
( P2
!= ( produc7020197800436672577rm_a_b @ X3 @ X3 ) ) ) ).
% IdE
thf(fact_743_CR__divergence__imp__join,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,X: term_a_b,Y: term_a_b,Z: term_a_b] :
( ( abstra8448919418672941150rm_a_b @ R2 @ top_top_set_term_a_b )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ Z ) @ ( abstra4096080454567261402rm_a_b @ R2 ) ) ) ) ) ).
% CR_divergence_imp_join
thf(fact_744_partially__localize__CR,axiom,
! [R2: set_Pr4386577575007340137rm_a_b] :
( ( abstra8448919418672941150rm_a_b @ R2 @ top_top_set_term_a_b )
= ( ! [X4: term_a_b,Y5: term_a_b,Z4: term_a_b] :
( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X4 @ Y5 ) @ R2 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X4 @ Z4 ) @ ( transi7742714808557438673rm_a_b @ R2 ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y5 @ Z4 ) @ ( abstra4096080454567261402rm_a_b @ R2 ) ) ) ) ) ).
% partially_localize_CR
thf(fact_745_CR__join__right__I,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,X: term_a_b,Y: term_a_b,Z: term_a_b] :
( ( abstra8448919418672941150rm_a_b @ R2 @ top_top_set_term_a_b )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( abstra4096080454567261402rm_a_b @ R2 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ Z ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z ) @ ( abstra4096080454567261402rm_a_b @ R2 ) ) ) ) ) ).
% CR_join_right_I
thf(fact_746_CR__join__left__I,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,X: term_a_b,Y: term_a_b,Z: term_a_b] :
( ( abstra8448919418672941150rm_a_b @ R2 @ top_top_set_term_a_b )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z ) @ ( abstra4096080454567261402rm_a_b @ R2 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ Z ) @ ( abstra4096080454567261402rm_a_b @ R2 ) ) ) ) ) ).
% CR_join_left_I
thf(fact_747_SCR__on__def,axiom,
( rewrit4966517460798004134rm_a_b
= ( ^ [R4: set_Pr4386577575007340137rm_a_b,A6: set_term_a_b] :
! [X4: term_a_b] :
( ( member_term_a_b @ X4 @ A6 )
=> ! [B4: term_a_b,C3: term_a_b] :
( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X4 @ B4 ) @ R4 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X4 @ C3 ) @ R4 ) )
=> ? [D: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B4 @ D ) @ ( sup_su6776935440552674877rm_a_b @ R4 @ id_term_a_b ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ C3 @ D ) @ ( transi7742714808557438673rm_a_b @ R4 ) ) ) ) ) ) ) ).
% SCR_on_def
thf(fact_748_SCE__to__rrstep_I2_J,axiom,
! [F: set_Pr4934435412358123699_a_nat,R3: set_Pr4386577575007340137rm_a_b,S: term_a_b,T: term_a_b,U: term_a_b] :
( ( rewrit4966517460798004134rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R3 ) ) @ top_top_set_term_a_b )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( sig_step_a_b @ F @ ( rstep_a_b @ R3 ) ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ U ) @ ( sig_step_a_b @ F @ ( rrstep_a_b @ R3 ) ) )
=> ? [V2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T @ V2 ) @ ( sup_su6776935440552674877rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R3 ) ) @ id_term_a_b ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ U @ V2 ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R3 ) ) ) ) ) ) ) ) ).
% SCE_to_rrstep(2)
thf(fact_749_SCE__to__rrstep_I1_J,axiom,
! [F: set_Pr4934435412358123699_a_nat,R3: set_Pr4386577575007340137rm_a_b,S: term_a_b,T: term_a_b,U: term_a_b] :
( ( rewrit4966517460798004134rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R3 ) ) @ top_top_set_term_a_b )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( sig_step_a_b @ F @ ( rrstep_a_b @ R3 ) ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ U ) @ ( sig_step_a_b @ F @ ( rstep_a_b @ R3 ) ) )
=> ? [V2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T @ V2 ) @ ( sup_su6776935440552674877rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R3 ) ) @ id_term_a_b ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ U @ V2 ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R3 ) ) ) ) ) ) ) ) ).
% SCE_to_rrstep(1)
thf(fact_750_SCR__rrstep__intro,axiom,
! [F: set_Pr4934435412358123699_a_nat,R3: set_Pr4386577575007340137rm_a_b] :
( ! [S4: term_a_b,T2: term_a_b,U2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S4 @ T2 ) @ ( sig_step_a_b @ F @ ( rrstep_a_b @ R3 ) ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S4 @ U2 ) @ ( sig_step_a_b @ F @ ( rstep_a_b @ R3 ) ) )
=> ? [V3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T2 @ V3 ) @ ( sup_su6776935440552674877rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R3 ) ) @ id_term_a_b ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ U2 @ V3 ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R3 ) ) ) ) ) ) )
=> ( ! [S4: term_a_b,T2: term_a_b,U2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S4 @ T2 ) @ ( sig_step_a_b @ F @ ( rstep_a_b @ R3 ) ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S4 @ U2 ) @ ( sig_step_a_b @ F @ ( rrstep_a_b @ R3 ) ) )
=> ? [V3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T2 @ V3 ) @ ( sup_su6776935440552674877rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R3 ) ) @ id_term_a_b ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ U2 @ V3 ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R3 ) ) ) ) ) ) )
=> ( rewrit4966517460798004134rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R3 ) ) @ top_top_set_term_a_b ) ) ) ).
% SCR_rrstep_intro
thf(fact_751_strongly__confluentI,axiom,
! [R2: set_Pr4386577575007340137rm_a_b] :
( ! [X3: term_a_b,Y3: term_a_b,Z3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ R2 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Z3 ) @ R2 )
=> ? [U3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ U3 ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Z3 @ U3 ) @ ( sup_su6776935440552674877rm_a_b @ R2 @ id_term_a_b ) ) ) ) )
=> ( abstra5664096641628173427rm_a_b @ R2 @ top_top_set_term_a_b ) ) ).
% strongly_confluentI
thf(fact_752_strongly__confluent__on__def,axiom,
( abstra5664096641628173427rm_a_b
= ( ^ [R4: set_Pr4386577575007340137rm_a_b,A6: set_term_a_b] :
! [X4: term_a_b] :
( ( member_term_a_b @ X4 @ A6 )
=> ! [Y5: term_a_b,Z4: term_a_b] :
( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X4 @ Y5 ) @ R4 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X4 @ Z4 ) @ R4 ) )
=> ? [U4: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y5 @ U4 ) @ ( transi7742714808557438673rm_a_b @ R4 ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Z4 @ U4 ) @ ( sup_su6776935440552674877rm_a_b @ R4 @ id_term_a_b ) ) ) ) ) ) ) ).
% strongly_confluent_on_def
thf(fact_753_strongly__confluent__on__E11,axiom,
! [R2: set_Pr3451248702717554689st_nat,A4: set_list_nat,X: list_nat,Y: list_nat,Z: list_nat] :
( ( abstra3206962040680252735st_nat @ R2 @ A4 )
=> ( ( member_list_nat @ X @ A4 )
=> ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X @ Y ) @ R2 )
=> ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X @ Z ) @ R2 )
=> ? [U2: list_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Y @ U2 ) @ ( transi5285580207609517981st_nat @ R2 ) )
& ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Z @ U2 ) @ ( sup_su5841606568262889429st_nat @ R2 @ id_list_nat ) ) ) ) ) ) ) ).
% strongly_confluent_on_E11
thf(fact_754_strongly__confluent__on__E11,axiom,
! [R2: set_Pr2972776593051762503rm_a_b,A4: set_Pr4386577575007340137rm_a_b,X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b,Z: produc357393685978478089rm_a_b] :
( ( abstra837640964651051114rm_a_b @ R2 @ A4 )
=> ( ( member5869715511025134514rm_a_b @ X @ A4 )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ R2 )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Z ) @ R2 )
=> ? [U2: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y @ U2 ) @ ( transi2615809358984392588rm_a_b @ R2 ) )
& ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Z @ U2 ) @ ( sup_su1845815533690797339rm_a_b @ R2 @ id_Pro3488709451141449061rm_a_b ) ) ) ) ) ) ) ).
% strongly_confluent_on_E11
thf(fact_755_strongly__confluent__on__E11,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,A4: set_term_a_b,X: term_a_b,Y: term_a_b,Z: term_a_b] :
( ( abstra5664096641628173427rm_a_b @ R2 @ A4 )
=> ( ( member_term_a_b @ X @ A4 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ R2 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z ) @ R2 )
=> ? [U2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ U2 ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Z @ U2 ) @ ( sup_su6776935440552674877rm_a_b @ R2 @ id_term_a_b ) ) ) ) ) ) ) ).
% strongly_confluent_on_E11
thf(fact_756_trans__ctxt__sig__imp__all__ctxt__closed,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,F5: set_Pr4934435412358123699_a_nat] :
( ( trans_on_term_a_b @ top_top_set_term_a_b @ R2 )
=> ( ! [T2: term_a_b] :
( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ T2 ) @ F5 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T2 @ T2 ) @ R2 ) )
=> ( ! [C5: subterm_and_ctxt_a_b,S4: term_a_b,T2: term_a_b] :
( ( ord_le8666007276011122963_a_nat @ ( term_funas_ctxt_a_b @ C5 ) @ F5 )
=> ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ S4 ) @ F5 )
=> ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ T2 ) @ F5 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S4 @ T2 ) @ R2 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( subter2376574525758040790rm_a_b @ C5 @ S4 ) @ ( subter2376574525758040790rm_a_b @ C5 @ T2 ) ) @ R2 ) ) ) ) )
=> ( terms_5226143800768910156ed_a_b @ F5 @ R2 ) ) ) ) ).
% trans_ctxt_sig_imp_all_ctxt_closed
thf(fact_757_trans__on__def,axiom,
( trans_on_term_a_b
= ( ^ [A6: set_term_a_b,R4: set_Pr4386577575007340137rm_a_b] :
! [X4: term_a_b] :
( ( member_term_a_b @ X4 @ A6 )
=> ! [Y5: term_a_b] :
( ( member_term_a_b @ Y5 @ A6 )
=> ! [Z4: term_a_b] :
( ( member_term_a_b @ Z4 @ A6 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X4 @ Y5 ) @ R4 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y5 @ Z4 ) @ R4 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X4 @ Z4 ) @ R4 ) ) ) ) ) ) ) ) ).
% trans_on_def
thf(fact_758_trans__onI,axiom,
! [A4: set_list_nat,R2: set_Pr3451248702717554689st_nat] :
( ! [X3: list_nat,Y3: list_nat,Z3: list_nat] :
( ( member_list_nat @ X3 @ A4 )
=> ( ( member_list_nat @ Y3 @ A4 )
=> ( ( member_list_nat @ Z3 @ A4 )
=> ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X3 @ Y3 ) @ R2 )
=> ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Y3 @ Z3 ) @ R2 )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X3 @ Z3 ) @ R2 ) ) ) ) ) )
=> ( trans_on_list_nat @ A4 @ R2 ) ) ).
% trans_onI
thf(fact_759_trans__onI,axiom,
! [A4: set_Pr4386577575007340137rm_a_b,R2: set_Pr2972776593051762503rm_a_b] :
( ! [X3: produc357393685978478089rm_a_b,Y3: produc357393685978478089rm_a_b,Z3: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X3 @ A4 )
=> ( ( member5869715511025134514rm_a_b @ Y3 @ A4 )
=> ( ( member5869715511025134514rm_a_b @ Z3 @ A4 )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X3 @ Y3 ) @ R2 )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y3 @ Z3 ) @ R2 )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X3 @ Z3 ) @ R2 ) ) ) ) ) )
=> ( trans_5404704112166290345rm_a_b @ A4 @ R2 ) ) ).
% trans_onI
thf(fact_760_trans__onI,axiom,
! [A4: set_term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ! [X3: term_a_b,Y3: term_a_b,Z3: term_a_b] :
( ( member_term_a_b @ X3 @ A4 )
=> ( ( member_term_a_b @ Y3 @ A4 )
=> ( ( member_term_a_b @ Z3 @ A4 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ R2 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ Z3 ) @ R2 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Z3 ) @ R2 ) ) ) ) ) )
=> ( trans_on_term_a_b @ A4 @ R2 ) ) ).
% trans_onI
thf(fact_761_trans__onD,axiom,
! [A4: set_list_nat,R2: set_Pr3451248702717554689st_nat,X: list_nat,Y: list_nat,Z: list_nat] :
( ( trans_on_list_nat @ A4 @ R2 )
=> ( ( member_list_nat @ X @ A4 )
=> ( ( member_list_nat @ Y @ A4 )
=> ( ( member_list_nat @ Z @ A4 )
=> ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X @ Y ) @ R2 )
=> ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Y @ Z ) @ R2 )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X @ Z ) @ R2 ) ) ) ) ) ) ) ).
% trans_onD
thf(fact_762_trans__onD,axiom,
! [A4: set_Pr4386577575007340137rm_a_b,R2: set_Pr2972776593051762503rm_a_b,X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b,Z: produc357393685978478089rm_a_b] :
( ( trans_5404704112166290345rm_a_b @ A4 @ R2 )
=> ( ( member5869715511025134514rm_a_b @ X @ A4 )
=> ( ( member5869715511025134514rm_a_b @ Y @ A4 )
=> ( ( member5869715511025134514rm_a_b @ Z @ A4 )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ R2 )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y @ Z ) @ R2 )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Z ) @ R2 ) ) ) ) ) ) ) ).
% trans_onD
thf(fact_763_trans__onD,axiom,
! [A4: set_term_a_b,R2: set_Pr4386577575007340137rm_a_b,X: term_a_b,Y: term_a_b,Z: term_a_b] :
( ( trans_on_term_a_b @ A4 @ R2 )
=> ( ( member_term_a_b @ X @ A4 )
=> ( ( member_term_a_b @ Y @ A4 )
=> ( ( member_term_a_b @ Z @ A4 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ R2 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ Z ) @ R2 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z ) @ R2 ) ) ) ) ) ) ) ).
% trans_onD
thf(fact_764_transD,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,X: term_a_b,Y: term_a_b,Z: term_a_b] :
( ( trans_on_term_a_b @ top_top_set_term_a_b @ R2 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ R2 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ Z ) @ R2 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z ) @ R2 ) ) ) ) ).
% transD
thf(fact_765_transE,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,X: term_a_b,Y: term_a_b,Z: term_a_b] :
( ( trans_on_term_a_b @ top_top_set_term_a_b @ R2 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ R2 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ Z ) @ R2 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z ) @ R2 ) ) ) ) ).
% transE
thf(fact_766_transI,axiom,
! [R2: set_Pr4386577575007340137rm_a_b] :
( ! [X3: term_a_b,Y3: term_a_b,Z3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ R2 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ Z3 ) @ R2 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Z3 ) @ R2 ) ) )
=> ( trans_on_term_a_b @ top_top_set_term_a_b @ R2 ) ) ).
% transI
thf(fact_767_relcomp3__transI,axiom,
! [B5: set_Pr4386577575007340137rm_a_b,T: term_a_b,U: term_a_b,A4: set_Pr4386577575007340137rm_a_b,S: term_a_b,V: term_a_b] :
( ( trans_on_term_a_b @ top_top_set_term_a_b @ B5 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T @ U ) @ ( relcom370159955682700863rm_a_b @ B5 @ ( relcom370159955682700863rm_a_b @ A4 @ B5 ) ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ B5 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ U @ V ) @ B5 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ V ) @ ( relcom370159955682700863rm_a_b @ B5 @ ( relcom370159955682700863rm_a_b @ A4 @ B5 ) ) ) ) ) ) ) ).
% relcomp3_transI
thf(fact_768_trans__singleton,axiom,
! [A: term_a_b] : ( trans_on_term_a_b @ top_top_set_term_a_b @ ( insert7009541432154983385rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ A ) @ bot_bo197521221353338581rm_a_b ) ) ).
% trans_singleton
thf(fact_769_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_770_strongly__confluent__E1n,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,X: term_a_b,Y: term_a_b,Z: 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 @ Z ) @ ( 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 @ Z @ U2 ) @ ( sup_su6776935440552674877rm_a_b @ R2 @ id_term_a_b ) ) ) ) ) ) ).
% strongly_confluent_E1n
thf(fact_771_relpow__refl__mono,axiom,
! [Rel: set_Pr4386577575007340137rm_a_b,M: nat,N: nat,A: term_a_b,B: term_a_b] :
( ! [X3: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ X3 ) @ Rel )
=> ( ( ord_less_eq_nat @ M @ N )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( compow4057154403645558940rm_a_b @ M @ Rel ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( compow4057154403645558940rm_a_b @ N @ Rel ) ) ) ) ) ).
% relpow_refl_mono
thf(fact_772_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,T2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S4 @ T2 ) @ R2 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F3 @ S4 ) @ ( F3 @ T2 ) ) @ 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_773_relpow__Suc__D2_H,axiom,
! [N: nat,R5: set_Pr4386577575007340137rm_a_b,X6: term_a_b,Y7: term_a_b,Z5: term_a_b] :
( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X6 @ Y7 ) @ ( compow4057154403645558940rm_a_b @ N @ R5 ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y7 @ Z5 ) @ R5 ) )
=> ? [W: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X6 @ W ) @ R5 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ W @ Z5 ) @ ( compow4057154403645558940rm_a_b @ N @ R5 ) ) ) ) ).
% relpow_Suc_D2'
thf(fact_774_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 ) )
=> ~ ! [N2: nat] :
~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( compow4057154403645558940rm_a_b @ N2 @ R2 ) ) ) ).
% rtrancl_len_E
thf(fact_775_is__singletonI_H,axiom,
! [A4: set_list_nat] :
( ( A4 != bot_bot_set_list_nat )
=> ( ! [X3: list_nat,Y3: list_nat] :
( ( member_list_nat @ X3 @ A4 )
=> ( ( member_list_nat @ Y3 @ A4 )
=> ( X3 = Y3 ) ) )
=> ( is_sin2641923865335537900st_nat @ A4 ) ) ) ).
% is_singletonI'
thf(fact_776_is__singletonI_H,axiom,
! [A4: set_Pr4386577575007340137rm_a_b] :
( ( A4 != bot_bo197521221353338581rm_a_b )
=> ( ! [X3: produc357393685978478089rm_a_b,Y3: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X3 @ A4 )
=> ( ( member5869715511025134514rm_a_b @ Y3 @ A4 )
=> ( X3 = Y3 ) ) )
=> ( is_sin332545968047684349rm_a_b @ A4 ) ) ) ).
% is_singletonI'
thf(fact_777_trancl__steps__relpow,axiom,
! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b,X: term_a_b,Y: term_a_b,N: nat] :
( ( ord_le118470702582115849rm_a_b @ A @ ( transi7922773638565587891rm_a_b @ B ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( compow4057154403645558940rm_a_b @ N @ A ) )
=> ? [M2: nat] :
( ( ord_less_eq_nat @ N @ M2 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( compow4057154403645558940rm_a_b @ M2 @ B ) ) ) ) ) ).
% trancl_steps_relpow
thf(fact_778_Field__insert,axiom,
! [A: term_a_b,B: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ( field_term_a_b @ ( insert7009541432154983385rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R2 ) )
= ( sup_sup_set_term_a_b @ ( insert_term_a_b @ A @ ( insert_term_a_b @ B @ bot_bot_set_term_a_b ) ) @ ( field_term_a_b @ R2 ) ) ) ).
% Field_insert
thf(fact_779_semi__confluence__imp__CR,axiom,
! [R2: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ ( conver7608559976937845093rm_a_b @ R2 ) @ ( transi7742714808557438673rm_a_b @ R2 ) ) @ ( abstra4096080454567261402rm_a_b @ R2 ) )
=> ( abstra8448919418672941150rm_a_b @ R2 @ top_top_set_term_a_b ) ) ).
% semi_confluence_imp_CR
thf(fact_780_NE__to__rrstep_I1_J,axiom,
! [F: set_Pr4934435412358123699_a_nat,R3: set_Pr4386577575007340137rm_a_b,S5: set_Pr4386577575007340137rm_a_b,S: term_a_b,T: term_a_b] :
( ( rewrit7370782028743737947rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R3 ) ) @ ( sig_step_a_b @ F @ ( rstep_a_b @ S5 ) ) @ top_top_set_term_a_b )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( srstep7844470518422762656ep_a_b @ F @ R3 ) )
=> ( rewrit3451484132325884455dp_a_b @ F @ R3 @ S5 @ S @ T ) ) ) ).
% NE_to_rrstep(1)
thf(fact_781_converse__iff,axiom,
! [A: term_a_b,B: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( conver7608559976937845093rm_a_b @ R2 ) )
= ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ R2 ) ) ).
% converse_iff
thf(fact_782_converse_Ocases,axiom,
! [A1: term_a_b,A22: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A22 ) @ ( conver7608559976937845093rm_a_b @ R2 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A22 @ A1 ) @ R2 ) ) ).
% converse.cases
thf(fact_783_converse_Osimps,axiom,
! [A1: term_a_b,A22: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A22 ) @ ( conver7608559976937845093rm_a_b @ R2 ) )
= ( ? [A5: term_a_b,B4: term_a_b] :
( ( A1 = B4 )
& ( A22 = A5 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A5 @ B4 ) @ R2 ) ) ) ) ).
% converse.simps
thf(fact_784_converseD,axiom,
! [A: term_a_b,B: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( conver7608559976937845093rm_a_b @ R2 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ R2 ) ) ).
% converseD
thf(fact_785_converseE,axiom,
! [Yx: produc357393685978478089rm_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ Yx @ ( conver7608559976937845093rm_a_b @ R2 ) )
=> ~ ! [X3: term_a_b,Y3: term_a_b] :
( ( Yx
= ( produc7020197800436672577rm_a_b @ Y3 @ X3 ) )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ R2 ) ) ) ).
% converseE
thf(fact_786_converseI,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 @ B @ A ) @ ( conver7608559976937845093rm_a_b @ R2 ) ) ) ).
% converseI
thf(fact_787_FieldI1,axiom,
! [I2: list_nat,J2: list_nat,R5: set_Pr3451248702717554689st_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ I2 @ J2 ) @ R5 )
=> ( member_list_nat @ I2 @ ( field_list_nat @ R5 ) ) ) ).
% FieldI1
thf(fact_788_FieldI1,axiom,
! [I2: produc357393685978478089rm_a_b,J2: produc357393685978478089rm_a_b,R5: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ I2 @ J2 ) @ R5 )
=> ( member5869715511025134514rm_a_b @ I2 @ ( field_6884932134483023318rm_a_b @ R5 ) ) ) ).
% FieldI1
thf(fact_789_FieldI1,axiom,
! [I2: term_a_b,J2: term_a_b,R5: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ I2 @ J2 ) @ R5 )
=> ( member_term_a_b @ I2 @ ( field_term_a_b @ R5 ) ) ) ).
% FieldI1
thf(fact_790_FieldI2,axiom,
! [I2: list_nat,J2: list_nat,R5: set_Pr3451248702717554689st_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ I2 @ J2 ) @ R5 )
=> ( member_list_nat @ J2 @ ( field_list_nat @ R5 ) ) ) ).
% FieldI2
thf(fact_791_FieldI2,axiom,
! [I2: produc357393685978478089rm_a_b,J2: produc357393685978478089rm_a_b,R5: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ I2 @ J2 ) @ R5 )
=> ( member5869715511025134514rm_a_b @ J2 @ ( field_6884932134483023318rm_a_b @ R5 ) ) ) ).
% FieldI2
thf(fact_792_FieldI2,axiom,
! [I2: term_a_b,J2: term_a_b,R5: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ I2 @ J2 ) @ R5 )
=> ( member_term_a_b @ J2 @ ( field_term_a_b @ R5 ) ) ) ).
% FieldI2
thf(fact_793_rstep__converse__dist,axiom,
! [R3: set_Pr4386577575007340137rm_a_b] :
( ( conver7608559976937845093rm_a_b @ ( rstep_a_b @ R3 ) )
= ( rstep_a_b @ ( conver7608559976937845093rm_a_b @ R3 ) ) ) ).
% rstep_converse_dist
thf(fact_794_rtrancl__converseD,axiom,
! [X: term_a_b,Y: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ ( conver7608559976937845093rm_a_b @ R2 ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X ) @ ( transi7742714808557438673rm_a_b @ R2 ) ) ) ).
% rtrancl_converseD
thf(fact_795_rtrancl__converseI,axiom,
! [Y: term_a_b,X: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ ( conver7608559976937845093rm_a_b @ R2 ) ) ) ) ).
% rtrancl_converseI
thf(fact_796_peak__iff,axiom,
! [X: term_a_b,Y: term_a_b,A4: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( relcom370159955682700863rm_a_b @ ( conver7608559976937845093rm_a_b @ A4 ) @ B5 ) )
= ( ? [U4: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ U4 @ X ) @ A4 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ U4 @ Y ) @ B5 ) ) ) ) ).
% peak_iff
thf(fact_797_trancl__converseD,axiom,
! [X: term_a_b,Y: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7922773638565587891rm_a_b @ ( conver7608559976937845093rm_a_b @ R2 ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( conver7608559976937845093rm_a_b @ ( transi7922773638565587891rm_a_b @ R2 ) ) ) ) ).
% trancl_converseD
thf(fact_798_trancl__converseI,axiom,
! [X: term_a_b,Y: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( conver7608559976937845093rm_a_b @ ( transi7922773638565587891rm_a_b @ R2 ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7922773638565587891rm_a_b @ ( conver7608559976937845093rm_a_b @ R2 ) ) ) ) ).
% trancl_converseI
thf(fact_799_srstep__converse__dist,axiom,
! [F: set_Pr4934435412358123699_a_nat,R3: set_Pr4386577575007340137rm_a_b] :
( ( conver7608559976937845093rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R3 ) ) )
= ( sig_step_a_b @ F @ ( rstep_a_b @ ( conver7608559976937845093rm_a_b @ R3 ) ) ) ) ).
% srstep_converse_dist
thf(fact_800_sig__step__converse__rstep,axiom,
! [F: set_Pr4934435412358123699_a_nat,R3: set_Pr4386577575007340137rm_a_b] :
( ( conver7608559976937845093rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R3 ) ) )
= ( sig_step_a_b @ F @ ( conver7608559976937845093rm_a_b @ ( rstep_a_b @ R3 ) ) ) ) ).
% sig_step_converse_rstep
thf(fact_801_rstep__smycl__dist,axiom,
! [R3: set_Pr4386577575007340137rm_a_b] :
( ( sup_su6776935440552674877rm_a_b @ ( rstep_a_b @ R3 ) @ ( conver7608559976937845093rm_a_b @ ( rstep_a_b @ R3 ) ) )
= ( rstep_a_b @ ( sup_su6776935440552674877rm_a_b @ R3 @ ( conver7608559976937845093rm_a_b @ R3 ) ) ) ) ).
% rstep_smycl_dist
thf(fact_802_sig__step__symcl__dist,axiom,
! [F: set_Pr4934435412358123699_a_nat,R3: set_Pr4386577575007340137rm_a_b] :
( ( sup_su6776935440552674877rm_a_b @ ( sig_step_a_b @ F @ R3 ) @ ( conver7608559976937845093rm_a_b @ ( sig_step_a_b @ F @ R3 ) ) )
= ( sig_step_a_b @ F @ ( sup_su6776935440552674877rm_a_b @ R3 @ ( conver7608559976937845093rm_a_b @ R3 ) ) ) ) ).
% sig_step_symcl_dist
thf(fact_803_srrstep__converse__dist,axiom,
! [F: set_Pr4934435412358123699_a_nat,R3: set_Pr4386577575007340137rm_a_b] :
( ( conver7608559976937845093rm_a_b @ ( sig_step_a_b @ F @ ( rrstep_a_b @ R3 ) ) )
= ( sig_step_a_b @ F @ ( rrstep_a_b @ ( conver7608559976937845093rm_a_b @ R3 ) ) ) ) ).
% srrstep_converse_dist
thf(fact_804_join__def,axiom,
( abstra4096080454567261402rm_a_b
= ( ^ [A6: set_Pr4386577575007340137rm_a_b] : ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ A6 ) @ ( transi7742714808557438673rm_a_b @ ( conver7608559976937845093rm_a_b @ A6 ) ) ) ) ) ).
% join_def
thf(fact_805_srstep__symcl__dist,axiom,
! [F: set_Pr4934435412358123699_a_nat,R3: set_Pr4386577575007340137rm_a_b] :
( ( sup_su6776935440552674877rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R3 ) ) @ ( conver7608559976937845093rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R3 ) ) ) )
= ( sig_step_a_b @ F @ ( rstep_a_b @ ( sup_su6776935440552674877rm_a_b @ R3 @ ( conver7608559976937845093rm_a_b @ R3 ) ) ) ) ) ).
% srstep_symcl_dist
thf(fact_806_WCR__alt__def,axiom,
! [A4: set_Pr4386577575007340137rm_a_b] :
( ( abstra5026849107050860991rm_a_b @ A4 @ top_top_set_term_a_b )
= ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ ( conver7608559976937845093rm_a_b @ A4 ) @ A4 ) @ ( abstra4096080454567261402rm_a_b @ A4 ) ) ) ).
% WCR_alt_def
thf(fact_807_diamond__imp__semi__confluence,axiom,
! [R2: set_Pr4386577575007340137rm_a_b] :
( ( abstra8151087076625346926rm_a_b @ R2 )
=> ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ ( conver7608559976937845093rm_a_b @ R2 ) @ ( transi7742714808557438673rm_a_b @ R2 ) ) @ ( abstra4096080454567261402rm_a_b @ R2 ) ) ) ).
% diamond_imp_semi_confluence
thf(fact_808_NE__to__rrstep_I2_J,axiom,
! [F: set_Pr4934435412358123699_a_nat,R3: set_Pr4386577575007340137rm_a_b,S5: set_Pr4386577575007340137rm_a_b,S: term_a_b,T: term_a_b] :
( ( rewrit7370782028743737947rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R3 ) ) @ ( sig_step_a_b @ F @ ( rstep_a_b @ S5 ) ) @ top_top_set_term_a_b )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( srstep7844470518422762656ep_a_b @ F @ S5 ) )
=> ( rewrit3451484132325884455dp_a_b @ F @ S5 @ R3 @ S @ T ) ) ) ).
% NE_to_rrstep(2)
thf(fact_809_CR__to__rrstep,axiom,
! [F: set_Pr4934435412358123699_a_nat,R3: set_Pr4386577575007340137rm_a_b,S: term_a_b,T: term_a_b] :
( ( abstra8448919418672941150rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R3 ) ) @ top_top_set_term_a_b )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( sup_su6776935440552674877rm_a_b @ ( relcom370159955682700863rm_a_b @ ( srstep7844470518422762656ep_a_b @ F @ ( conver7608559976937845093rm_a_b @ R3 ) ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R3 ) ) ) ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ ( conver7608559976937845093rm_a_b @ R3 ) ) ) ) @ ( srstep7844470518422762656ep_a_b @ F @ R3 ) ) ) )
=> ( rewrit8865997592796547114dp_a_b @ F @ R3 @ R3 @ S @ T ) ) ) ).
% CR_to_rrstep
thf(fact_810_commute__rrstep__intro,axiom,
! [F: set_Pr4934435412358123699_a_nat,R3: set_Pr4386577575007340137rm_a_b,S5: set_Pr4386577575007340137rm_a_b] :
( ! [S4: term_a_b,T2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S4 @ T2 ) @ ( sup_su6776935440552674877rm_a_b @ ( relcom370159955682700863rm_a_b @ ( srstep7844470518422762656ep_a_b @ F @ ( conver7608559976937845093rm_a_b @ R3 ) ) @ ( transi7922773638565587891rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ S5 ) ) ) ) @ ( relcom370159955682700863rm_a_b @ ( transi7922773638565587891rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ ( conver7608559976937845093rm_a_b @ R3 ) ) ) ) @ ( srstep7844470518422762656ep_a_b @ F @ S5 ) ) ) )
=> ( rewrit8865997592796547114dp_a_b @ F @ R3 @ S5 @ S4 @ T2 ) )
=> ( abstra455195320138238452rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R3 ) ) @ ( sig_step_a_b @ F @ ( rstep_a_b @ S5 ) ) ) ) ).
% commute_rrstep_intro
thf(fact_811_CR__rrstep__intro,axiom,
! [F: set_Pr4934435412358123699_a_nat,R3: set_Pr4386577575007340137rm_a_b] :
( ! [S4: term_a_b,T2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S4 @ T2 ) @ ( relcom370159955682700863rm_a_b @ ( transi7922773638565587891rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ ( conver7608559976937845093rm_a_b @ R3 ) ) ) ) @ ( srstep7844470518422762656ep_a_b @ F @ R3 ) ) )
=> ( rewrit8865997592796547114dp_a_b @ F @ R3 @ R3 @ S4 @ T2 ) )
=> ( abstra8448919418672941150rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R3 ) ) @ top_top_set_term_a_b ) ) ).
% CR_rrstep_intro
thf(fact_812_commute__to__rrstep,axiom,
! [F: set_Pr4934435412358123699_a_nat,R3: set_Pr4386577575007340137rm_a_b,S5: set_Pr4386577575007340137rm_a_b,S: term_a_b,T: term_a_b] :
( ( abstra455195320138238452rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R3 ) ) @ ( sig_step_a_b @ F @ ( rstep_a_b @ S5 ) ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( sup_su6776935440552674877rm_a_b @ ( relcom370159955682700863rm_a_b @ ( srstep7844470518422762656ep_a_b @ F @ ( conver7608559976937845093rm_a_b @ R3 ) ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ S5 ) ) ) ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ ( conver7608559976937845093rm_a_b @ R3 ) ) ) ) @ ( srstep7844470518422762656ep_a_b @ F @ S5 ) ) ) )
=> ( rewrit8865997592796547114dp_a_b @ F @ R3 @ S5 @ S @ T ) ) ) ).
% commute_to_rrstep
thf(fact_813_commute__redp__def,axiom,
( rewrit8865997592796547114dp_a_b
= ( ^ [F4: set_Pr4934435412358123699_a_nat,R7: set_Pr4386577575007340137rm_a_b,S7: set_Pr4386577575007340137rm_a_b,S3: term_a_b,T3: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S3 @ T3 ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F4 @ ( rstep_a_b @ S7 ) ) ) @ ( transi7742714808557438673rm_a_b @ ( conver7608559976937845093rm_a_b @ ( sig_step_a_b @ F4 @ ( rstep_a_b @ R7 ) ) ) ) ) ) ) ) ).
% commute_redp_def
thf(fact_814_CR__Aux,axiom,
! [F: set_Pr4934435412358123699_a_nat,R3: set_Pr4386577575007340137rm_a_b,S: term_a_b,T: term_a_b] :
( ! [S4: term_a_b,T2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S4 @ T2 ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ ( conver7608559976937845093rm_a_b @ R3 ) ) ) ) @ ( srstep7844470518422762656ep_a_b @ F @ R3 ) ) )
=> ( rewrit8865997592796547114dp_a_b @ F @ R3 @ R3 @ S4 @ T2 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( sup_su6776935440552674877rm_a_b @ ( relcom370159955682700863rm_a_b @ ( srstep7844470518422762656ep_a_b @ F @ ( conver7608559976937845093rm_a_b @ R3 ) ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R3 ) ) ) ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ ( conver7608559976937845093rm_a_b @ R3 ) ) ) ) @ ( srstep7844470518422762656ep_a_b @ F @ R3 ) ) ) )
=> ( rewrit8865997592796547114dp_a_b @ F @ R3 @ R3 @ S @ T ) ) ) ).
% CR_Aux
thf(fact_815_NFP__rrstep__intro,axiom,
! [F: set_Pr4934435412358123699_a_nat,R3: set_Pr4386577575007340137rm_a_b] :
( ! [S4: term_a_b,T2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S4 @ T2 ) @ ( sup_su6776935440552674877rm_a_b @ ( relcom370159955682700863rm_a_b @ ( srstep7844470518422762656ep_a_b @ F @ ( conver7608559976937845093rm_a_b @ R3 ) ) @ ( transi7922773638565587891rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R3 ) ) ) ) @ ( relcom370159955682700863rm_a_b @ ( transi7922773638565587891rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ ( conver7608559976937845093rm_a_b @ R3 ) ) ) ) @ ( srstep7844470518422762656ep_a_b @ F @ R3 ) ) ) )
=> ( rewrit232733863534887208dp_a_b @ F @ R3 @ S4 @ T2 ) )
=> ( rewrit6304709981509430748rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R3 ) ) @ top_top_set_term_a_b ) ) ).
% NFP_rrstep_intro
thf(fact_816_UNF__rrstep__intro,axiom,
! [F: set_Pr4934435412358123699_a_nat,R3: set_Pr4386577575007340137rm_a_b] :
( ! [T2: term_a_b,U2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T2 @ U2 ) @ ( sup_su6776935440552674877rm_a_b @ ( relcom370159955682700863rm_a_b @ ( srstep7844470518422762656ep_a_b @ F @ ( conver7608559976937845093rm_a_b @ R3 ) ) @ ( transi7922773638565587891rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R3 ) ) ) ) @ ( relcom370159955682700863rm_a_b @ ( transi7922773638565587891rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ ( conver7608559976937845093rm_a_b @ R3 ) ) ) ) @ ( srstep7844470518422762656ep_a_b @ F @ R3 ) ) ) )
=> ( rewrit983855061078472681dp_a_b @ F @ R3 @ T2 @ U2 ) )
=> ( abstra3619855735679963878rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R3 ) ) @ top_top_set_term_a_b ) ) ).
% UNF_rrstep_intro
thf(fact_817_UNF__to__rrstep,axiom,
! [F: set_Pr4934435412358123699_a_nat,R3: set_Pr4386577575007340137rm_a_b,S: term_a_b,T: term_a_b] :
( ( abstra3619855735679963878rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R3 ) ) @ top_top_set_term_a_b )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( sup_su6776935440552674877rm_a_b @ ( relcom370159955682700863rm_a_b @ ( srstep7844470518422762656ep_a_b @ F @ ( conver7608559976937845093rm_a_b @ R3 ) ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R3 ) ) ) ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ ( conver7608559976937845093rm_a_b @ R3 ) ) ) ) @ ( srstep7844470518422762656ep_a_b @ F @ R3 ) ) ) )
=> ( rewrit983855061078472681dp_a_b @ F @ R3 @ S @ T ) ) ) ).
% UNF_to_rrstep
thf(fact_818_NFP__to__rrstep,axiom,
! [F: set_Pr4934435412358123699_a_nat,R3: set_Pr4386577575007340137rm_a_b,S: term_a_b,T: term_a_b] :
( ( rewrit6304709981509430748rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R3 ) ) @ top_top_set_term_a_b )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( srstep7844470518422762656ep_a_b @ F @ ( sup_su6776935440552674877rm_a_b @ R3 @ ( conver7608559976937845093rm_a_b @ R3 ) ) ) )
=> ( rewrit232733863534887208dp_a_b @ F @ R3 @ S @ T ) ) ) ).
% NFP_to_rrstep
thf(fact_819_UNC__to__rrstep,axiom,
! [F: set_Pr4934435412358123699_a_nat,R3: set_Pr4386577575007340137rm_a_b,S: term_a_b,T: term_a_b] :
( ( abstra4649771476585718372rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R3 ) ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( srstep7844470518422762656ep_a_b @ F @ ( sup_su6776935440552674877rm_a_b @ R3 @ ( conver7608559976937845093rm_a_b @ R3 ) ) ) )
=> ( rewrit983855061078472681dp_a_b @ F @ R3 @ S @ T ) ) ) ).
% UNC_to_rrstep
thf(fact_820_UNC__rrstep__intro,axiom,
! [F: set_Pr4934435412358123699_a_nat,R3: set_Pr4386577575007340137rm_a_b] :
( ! [S4: term_a_b,T2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S4 @ T2 ) @ ( srstep7844470518422762656ep_a_b @ F @ ( sup_su6776935440552674877rm_a_b @ R3 @ ( conver7608559976937845093rm_a_b @ R3 ) ) ) )
=> ( rewrit983855061078472681dp_a_b @ F @ R3 @ S4 @ T2 ) )
=> ( abstra4649771476585718372rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R3 ) ) ) ) ).
% UNC_rrstep_intro
thf(fact_821_NE__rrstep__intro,axiom,
! [F: set_Pr4934435412358123699_a_nat,R3: set_Pr4386577575007340137rm_a_b,S5: set_Pr4386577575007340137rm_a_b] :
( ! [S4: term_a_b,T2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S4 @ T2 ) @ ( srstep7844470518422762656ep_a_b @ F @ R3 ) )
=> ( rewrit3451484132325884455dp_a_b @ F @ R3 @ S5 @ S4 @ T2 ) )
=> ( ! [S4: term_a_b,T2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S4 @ T2 ) @ ( srstep7844470518422762656ep_a_b @ F @ S5 ) )
=> ( rewrit3451484132325884455dp_a_b @ F @ S5 @ R3 @ S4 @ T2 ) )
=> ( ( ( abstract_NF_term_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R3 ) ) )
= ( abstract_NF_term_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ S5 ) ) ) )
=> ( rewrit7370782028743737947rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R3 ) ) @ ( sig_step_a_b @ F @ ( rstep_a_b @ S5 ) ) @ top_top_set_term_a_b ) ) ) ) ).
% NE_rrstep_intro
thf(fact_822_cofinal__def,axiom,
( bNF_Ca4789820777293777539rm_a_b
= ( ^ [A6: set_term_a_b,R4: set_Pr4386577575007340137rm_a_b] :
! [X4: term_a_b] :
( ( member_term_a_b @ X4 @ ( field_term_a_b @ R4 ) )
=> ? [Y5: term_a_b] :
( ( member_term_a_b @ Y5 @ A6 )
& ( X4 != Y5 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X4 @ Y5 ) @ R4 ) ) ) ) ) ).
% cofinal_def
thf(fact_823_NF__I,axiom,
! [A: list_nat,A4: set_Pr3451248702717554689st_nat] :
( ! [B3: list_nat] :
~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A @ B3 ) @ A4 )
=> ( member_list_nat @ A @ ( abstract_NF_list_nat @ A4 ) ) ) ).
% NF_I
thf(fact_824_NF__I,axiom,
! [A: produc357393685978478089rm_a_b,A4: set_Pr2972776593051762503rm_a_b] :
( ! [B3: produc357393685978478089rm_a_b] :
~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B3 ) @ A4 )
=> ( member5869715511025134514rm_a_b @ A @ ( abstra1902783451776770197rm_a_b @ A4 ) ) ) ).
% NF_I
thf(fact_825_NF__I,axiom,
! [A: term_a_b,A4: set_Pr4386577575007340137rm_a_b] :
( ! [B3: term_a_b] :
~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B3 ) @ A4 )
=> ( member_term_a_b @ A @ ( abstract_NF_term_a_b @ A4 ) ) ) ).
% NF_I
thf(fact_826_NF__E,axiom,
! [A: list_nat,A4: set_Pr3451248702717554689st_nat,B: list_nat] :
( ( member_list_nat @ A @ ( abstract_NF_list_nat @ A4 ) )
=> ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A @ B ) @ A4 ) ) ).
% NF_E
thf(fact_827_NF__E,axiom,
! [A: produc357393685978478089rm_a_b,A4: set_Pr2972776593051762503rm_a_b,B: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ A @ ( abstra1902783451776770197rm_a_b @ A4 ) )
=> ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ A4 ) ) ).
% NF_E
thf(fact_828_NF__E,axiom,
! [A: term_a_b,A4: set_Pr4386577575007340137rm_a_b,B: term_a_b] :
( ( member_term_a_b @ A @ ( abstract_NF_term_a_b @ A4 ) )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ A4 ) ) ).
% NF_E
thf(fact_829_NF__no__step,axiom,
! [A: list_nat,A4: set_Pr3451248702717554689st_nat] :
( ( member_list_nat @ A @ ( abstract_NF_list_nat @ A4 ) )
=> ! [B8: list_nat] :
~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A @ B8 ) @ A4 ) ) ).
% NF_no_step
thf(fact_830_NF__no__step,axiom,
! [A: produc357393685978478089rm_a_b,A4: set_Pr2972776593051762503rm_a_b] :
( ( member5869715511025134514rm_a_b @ A @ ( abstra1902783451776770197rm_a_b @ A4 ) )
=> ! [B8: produc357393685978478089rm_a_b] :
~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B8 ) @ A4 ) ) ).
% NF_no_step
thf(fact_831_NF__no__step,axiom,
! [A: term_a_b,A4: set_Pr4386577575007340137rm_a_b] :
( ( member_term_a_b @ A @ ( abstract_NF_term_a_b @ A4 ) )
=> ! [B8: term_a_b] :
~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B8 ) @ A4 ) ) ).
% NF_no_step
thf(fact_832_NF__iff__no__step,axiom,
! [A: list_nat,A4: set_Pr3451248702717554689st_nat] :
( ( member_list_nat @ A @ ( abstract_NF_list_nat @ A4 ) )
= ( ! [B4: list_nat] :
~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A @ B4 ) @ A4 ) ) ) ).
% NF_iff_no_step
thf(fact_833_NF__iff__no__step,axiom,
! [A: produc357393685978478089rm_a_b,A4: set_Pr2972776593051762503rm_a_b] :
( ( member5869715511025134514rm_a_b @ A @ ( abstra1902783451776770197rm_a_b @ A4 ) )
= ( ! [B4: produc357393685978478089rm_a_b] :
~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B4 ) @ A4 ) ) ) ).
% NF_iff_no_step
thf(fact_834_NF__iff__no__step,axiom,
! [A: term_a_b,A4: set_Pr4386577575007340137rm_a_b] :
( ( member_term_a_b @ A @ ( abstract_NF_term_a_b @ A4 ) )
= ( ! [B4: term_a_b] :
~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B4 ) @ A4 ) ) ) ).
% NF_iff_no_step
thf(fact_835_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_836_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_837_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_838_NF__no__trancl__step,axiom,
! [A: list_nat,A4: set_Pr3451248702717554689st_nat] :
( ( member_list_nat @ A @ ( abstract_NF_list_nat @ A4 ) )
=> ! [B8: list_nat] :
~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A @ B8 ) @ ( transi5465639037617667199st_nat @ A4 ) ) ) ).
% NF_no_trancl_step
thf(fact_839_NF__no__trancl__step,axiom,
! [A: produc357393685978478089rm_a_b,A4: set_Pr2972776593051762503rm_a_b] :
( ( member5869715511025134514rm_a_b @ A @ ( abstra1902783451776770197rm_a_b @ A4 ) )
=> ! [B8: produc357393685978478089rm_a_b] :
~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B8 ) @ ( transi9211502839322181930rm_a_b @ A4 ) ) ) ).
% NF_no_trancl_step
thf(fact_840_NF__no__trancl__step,axiom,
! [A: term_a_b,A4: set_Pr4386577575007340137rm_a_b] :
( ( member_term_a_b @ A @ ( abstract_NF_term_a_b @ A4 ) )
=> ! [B8: term_a_b] :
~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B8 ) @ ( transi7922773638565587891rm_a_b @ A4 ) ) ) ).
% NF_no_trancl_step
thf(fact_841_join__NF__imp__eq,axiom,
! [X: list_nat,Y: list_nat,R2: set_Pr3451248702717554689st_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X @ Y ) @ ( abstra1638945853619340710st_nat @ R2 ) )
=> ( ( member_list_nat @ X @ ( abstract_NF_list_nat @ R2 ) )
=> ( ( member_list_nat @ Y @ ( abstract_NF_list_nat @ R2 ) )
=> ( X = Y ) ) ) ) ).
% join_NF_imp_eq
thf(fact_842_join__NF__imp__eq,axiom,
! [X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b,R2: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ ( abstra4889682940153293507rm_a_b @ R2 ) )
=> ( ( member5869715511025134514rm_a_b @ X @ ( abstra1902783451776770197rm_a_b @ R2 ) )
=> ( ( member5869715511025134514rm_a_b @ Y @ ( abstra1902783451776770197rm_a_b @ R2 ) )
=> ( X = Y ) ) ) ) ).
% join_NF_imp_eq
thf(fact_843_join__NF__imp__eq,axiom,
! [X: term_a_b,Y: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( abstra4096080454567261402rm_a_b @ R2 ) )
=> ( ( member_term_a_b @ X @ ( abstract_NF_term_a_b @ R2 ) )
=> ( ( member_term_a_b @ Y @ ( abstract_NF_term_a_b @ R2 ) )
=> ( X = Y ) ) ) ) ).
% join_NF_imp_eq
thf(fact_844_NF__join__imp__reach,axiom,
! [Y: list_nat,A4: set_Pr3451248702717554689st_nat,X: list_nat] :
( ( member_list_nat @ Y @ ( abstract_NF_list_nat @ A4 ) )
=> ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X @ Y ) @ ( abstra1638945853619340710st_nat @ A4 ) )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X @ Y ) @ ( transi5285580207609517981st_nat @ A4 ) ) ) ) ).
% NF_join_imp_reach
thf(fact_845_NF__join__imp__reach,axiom,
! [Y: produc357393685978478089rm_a_b,A4: set_Pr2972776593051762503rm_a_b,X: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ Y @ ( abstra1902783451776770197rm_a_b @ A4 ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ ( abstra4889682940153293507rm_a_b @ A4 ) )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ ( transi2615809358984392588rm_a_b @ A4 ) ) ) ) ).
% NF_join_imp_reach
thf(fact_846_NF__join__imp__reach,axiom,
! [Y: term_a_b,A4: set_Pr4386577575007340137rm_a_b,X: term_a_b] :
( ( member_term_a_b @ Y @ ( abstract_NF_term_a_b @ A4 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( abstra4096080454567261402rm_a_b @ A4 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ A4 ) ) ) ) ).
% NF_join_imp_reach
thf(fact_847_UN__redp__def,axiom,
( rewrit983855061078472681dp_a_b
= ( ^ [F4: set_Pr4934435412358123699_a_nat,R7: set_Pr4386577575007340137rm_a_b,S3: term_a_b,T3: term_a_b] :
( ( ( member_term_a_b @ S3 @ ( abstract_NF_term_a_b @ ( sig_step_a_b @ F4 @ ( rstep_a_b @ R7 ) ) ) )
& ( member_term_a_b @ T3 @ ( abstract_NF_term_a_b @ ( sig_step_a_b @ F4 @ ( rstep_a_b @ R7 ) ) ) ) )
=> ( S3 = T3 ) ) ) ) ).
% UN_redp_def
thf(fact_848_NFP__stepD,axiom,
! [R2: set_Pr3451248702717554689st_nat,A: list_nat,B: list_nat,C2: list_nat] :
( ( rewrit3847575380561510056st_nat @ R2 @ top_top_set_list_nat )
=> ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A @ B ) @ ( transi5285580207609517981st_nat @ R2 ) )
=> ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A @ C2 ) @ ( transi5285580207609517981st_nat @ R2 ) )
=> ( ( member_list_nat @ C2 @ ( abstract_NF_list_nat @ R2 ) )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ B @ C2 ) @ ( transi5285580207609517981st_nat @ R2 ) ) ) ) ) ) ).
% NFP_stepD
thf(fact_849_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_850_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_851_NE__redp__def,axiom,
( rewrit3451484132325884455dp_a_b
= ( ^ [F4: set_Pr4934435412358123699_a_nat,R7: set_Pr4386577575007340137rm_a_b,S7: set_Pr4386577575007340137rm_a_b,S3: term_a_b,T3: term_a_b] :
( ( member_term_a_b @ T3 @ ( abstract_NF_term_a_b @ ( sig_step_a_b @ F4 @ ( rstep_a_b @ R7 ) ) ) )
=> ( ( member_term_a_b @ T3 @ ( abstract_NF_term_a_b @ ( sig_step_a_b @ F4 @ ( rstep_a_b @ R7 ) ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S3 @ T3 ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F4 @ ( rstep_a_b @ S7 ) ) ) ) ) ) ) ) ).
% NE_redp_def
thf(fact_852_NFP__redp__def,axiom,
( rewrit232733863534887208dp_a_b
= ( ^ [F4: set_Pr4934435412358123699_a_nat,R7: set_Pr4386577575007340137rm_a_b,S3: term_a_b,T3: term_a_b] :
( ( member_term_a_b @ T3 @ ( abstract_NF_term_a_b @ ( sig_step_a_b @ F4 @ ( rstep_a_b @ R7 ) ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S3 @ T3 ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F4 @ ( rstep_a_b @ R7 ) ) ) ) ) ) ) ).
% NFP_redp_def
thf(fact_853_NFP__lift__to__conversion,axiom,
! [R2: set_Pr3451248702717554689st_nat,S: list_nat,T: list_nat] :
( ( rewrit3847575380561510056st_nat @ R2 @ top_top_set_list_nat )
=> ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ S @ T ) @ ( transi5285580207609517981st_nat @ ( sup_su5841606568262889429st_nat @ R2 @ ( conver3282399561507114237st_nat @ R2 ) ) ) )
=> ( ( member_list_nat @ T @ ( abstract_NF_list_nat @ R2 ) )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ S @ T ) @ ( transi5285580207609517981st_nat @ R2 ) ) ) ) ) ).
% NFP_lift_to_conversion
thf(fact_854_NFP__lift__to__conversion,axiom,
! [R2: set_Pr2972776593051762503rm_a_b,S: produc357393685978478089rm_a_b,T: produc357393685978478089rm_a_b] :
( ( rewrit2138790387785322561rm_a_b @ R2 @ top_to1314267278846557113rm_a_b )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ S @ T ) @ ( transi2615809358984392588rm_a_b @ ( sup_su1845815533690797339rm_a_b @ R2 @ ( conver1575690562132235771rm_a_b @ R2 ) ) ) )
=> ( ( member5869715511025134514rm_a_b @ T @ ( abstra1902783451776770197rm_a_b @ R2 ) )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ S @ T ) @ ( transi2615809358984392588rm_a_b @ R2 ) ) ) ) ) ).
% NFP_lift_to_conversion
thf(fact_855_NFP__lift__to__conversion,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,S: term_a_b,T: term_a_b] :
( ( rewrit6304709981509430748rm_a_b @ R2 @ top_top_set_term_a_b )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ R2 @ ( conver7608559976937845093rm_a_b @ R2 ) ) ) )
=> ( ( member_term_a_b @ T @ ( abstract_NF_term_a_b @ R2 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( transi7742714808557438673rm_a_b @ R2 ) ) ) ) ) ).
% NFP_lift_to_conversion
thf(fact_856_some__NF__UNF,axiom,
! [R2: set_Pr3451248702717554689st_nat,X: list_nat,Y: list_nat] :
( ( abstra1162721134732043186st_nat @ R2 @ top_top_set_list_nat )
=> ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X @ Y ) @ ( transi5285580207609517981st_nat @ R2 ) )
=> ( ( member_list_nat @ Y @ ( abstract_NF_list_nat @ R2 ) )
=> ( Y
= ( abstra132235892672370824st_nat @ R2 @ X ) ) ) ) ) ).
% some_NF_UNF
thf(fact_857_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_858_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_859_symcl__srsteps__conversion,axiom,
! [S: term_a_b,T: term_a_b,F: set_Pr4934435412358123699_a_nat,R3: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ ( sup_su6776935440552674877rm_a_b @ R3 @ ( conver7608559976937845093rm_a_b @ R3 ) ) ) ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( abstra1130568886744591910rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R3 ) ) ) ) ) ).
% symcl_srsteps_conversion
thf(fact_860_Diff__iff,axiom,
! [C2: list_nat,A4: set_list_nat,B5: set_list_nat] :
( ( member_list_nat @ C2 @ ( minus_7954133019191499631st_nat @ A4 @ B5 ) )
= ( ( member_list_nat @ C2 @ A4 )
& ~ ( member_list_nat @ C2 @ B5 ) ) ) ).
% Diff_iff
thf(fact_861_Diff__iff,axiom,
! [C2: produc357393685978478089rm_a_b,A4: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ C2 @ ( minus_5192120951422937424rm_a_b @ A4 @ B5 ) )
= ( ( member5869715511025134514rm_a_b @ C2 @ A4 )
& ~ ( member5869715511025134514rm_a_b @ C2 @ B5 ) ) ) ).
% Diff_iff
thf(fact_862_DiffI,axiom,
! [C2: list_nat,A4: set_list_nat,B5: set_list_nat] :
( ( member_list_nat @ C2 @ A4 )
=> ( ~ ( member_list_nat @ C2 @ B5 )
=> ( member_list_nat @ C2 @ ( minus_7954133019191499631st_nat @ A4 @ B5 ) ) ) ) ).
% DiffI
thf(fact_863_DiffI,axiom,
! [C2: produc357393685978478089rm_a_b,A4: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ C2 @ A4 )
=> ( ~ ( member5869715511025134514rm_a_b @ C2 @ B5 )
=> ( member5869715511025134514rm_a_b @ C2 @ ( minus_5192120951422937424rm_a_b @ A4 @ B5 ) ) ) ) ).
% DiffI
thf(fact_864_Diff__insert0,axiom,
! [X: list_nat,A4: set_list_nat,B5: set_list_nat] :
( ~ ( member_list_nat @ X @ A4 )
=> ( ( minus_7954133019191499631st_nat @ A4 @ ( insert_list_nat @ X @ B5 ) )
= ( minus_7954133019191499631st_nat @ A4 @ B5 ) ) ) ).
% Diff_insert0
thf(fact_865_Diff__insert0,axiom,
! [X: produc357393685978478089rm_a_b,A4: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
( ~ ( member5869715511025134514rm_a_b @ X @ A4 )
=> ( ( minus_5192120951422937424rm_a_b @ A4 @ ( insert7009541432154983385rm_a_b @ X @ B5 ) )
= ( minus_5192120951422937424rm_a_b @ A4 @ B5 ) ) ) ).
% Diff_insert0
thf(fact_866_insert__Diff1,axiom,
! [X: list_nat,B5: set_list_nat,A4: set_list_nat] :
( ( member_list_nat @ X @ B5 )
=> ( ( minus_7954133019191499631st_nat @ ( insert_list_nat @ X @ A4 ) @ B5 )
= ( minus_7954133019191499631st_nat @ A4 @ B5 ) ) ) ).
% insert_Diff1
thf(fact_867_insert__Diff1,axiom,
! [X: produc357393685978478089rm_a_b,B5: set_Pr4386577575007340137rm_a_b,A4: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ X @ B5 )
=> ( ( minus_5192120951422937424rm_a_b @ ( insert7009541432154983385rm_a_b @ X @ A4 ) @ B5 )
= ( minus_5192120951422937424rm_a_b @ A4 @ B5 ) ) ) ).
% insert_Diff1
thf(fact_868_conversionI_H,axiom,
! [A: term_a_b,B: term_a_b,A4: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ A4 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( abstra1130568886744591910rm_a_b @ A4 ) ) ) ).
% conversionI'
thf(fact_869_conversionI,axiom,
! [A: term_a_b,B: term_a_b,A4: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ A4 @ ( conver7608559976937845093rm_a_b @ A4 ) ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( abstra1130568886744591910rm_a_b @ A4 ) ) ) ).
% conversionI
thf(fact_870_insert__Diff__if,axiom,
! [X: list_nat,B5: set_list_nat,A4: set_list_nat] :
( ( ( member_list_nat @ X @ B5 )
=> ( ( minus_7954133019191499631st_nat @ ( insert_list_nat @ X @ A4 ) @ B5 )
= ( minus_7954133019191499631st_nat @ A4 @ B5 ) ) )
& ( ~ ( member_list_nat @ X @ B5 )
=> ( ( minus_7954133019191499631st_nat @ ( insert_list_nat @ X @ A4 ) @ B5 )
= ( insert_list_nat @ X @ ( minus_7954133019191499631st_nat @ A4 @ B5 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_871_insert__Diff__if,axiom,
! [X: produc357393685978478089rm_a_b,B5: set_Pr4386577575007340137rm_a_b,A4: set_Pr4386577575007340137rm_a_b] :
( ( ( member5869715511025134514rm_a_b @ X @ B5 )
=> ( ( minus_5192120951422937424rm_a_b @ ( insert7009541432154983385rm_a_b @ X @ A4 ) @ B5 )
= ( minus_5192120951422937424rm_a_b @ A4 @ B5 ) ) )
& ( ~ ( member5869715511025134514rm_a_b @ X @ B5 )
=> ( ( minus_5192120951422937424rm_a_b @ ( insert7009541432154983385rm_a_b @ X @ A4 ) @ B5 )
= ( insert7009541432154983385rm_a_b @ X @ ( minus_5192120951422937424rm_a_b @ A4 @ B5 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_872_DiffD2,axiom,
! [C2: list_nat,A4: set_list_nat,B5: set_list_nat] :
( ( member_list_nat @ C2 @ ( minus_7954133019191499631st_nat @ A4 @ B5 ) )
=> ~ ( member_list_nat @ C2 @ B5 ) ) ).
% DiffD2
thf(fact_873_DiffD2,axiom,
! [C2: produc357393685978478089rm_a_b,A4: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ C2 @ ( minus_5192120951422937424rm_a_b @ A4 @ B5 ) )
=> ~ ( member5869715511025134514rm_a_b @ C2 @ B5 ) ) ).
% DiffD2
thf(fact_874_DiffD1,axiom,
! [C2: list_nat,A4: set_list_nat,B5: set_list_nat] :
( ( member_list_nat @ C2 @ ( minus_7954133019191499631st_nat @ A4 @ B5 ) )
=> ( member_list_nat @ C2 @ A4 ) ) ).
% DiffD1
thf(fact_875_DiffD1,axiom,
! [C2: produc357393685978478089rm_a_b,A4: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ C2 @ ( minus_5192120951422937424rm_a_b @ A4 @ B5 ) )
=> ( member5869715511025134514rm_a_b @ C2 @ A4 ) ) ).
% DiffD1
thf(fact_876_DiffE,axiom,
! [C2: list_nat,A4: set_list_nat,B5: set_list_nat] :
( ( member_list_nat @ C2 @ ( minus_7954133019191499631st_nat @ A4 @ B5 ) )
=> ~ ( ( member_list_nat @ C2 @ A4 )
=> ( member_list_nat @ C2 @ B5 ) ) ) ).
% DiffE
thf(fact_877_DiffE,axiom,
! [C2: produc357393685978478089rm_a_b,A4: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ C2 @ ( minus_5192120951422937424rm_a_b @ A4 @ B5 ) )
=> ~ ( ( member5869715511025134514rm_a_b @ C2 @ A4 )
=> ( member5869715511025134514rm_a_b @ C2 @ B5 ) ) ) ).
% DiffE
thf(fact_878_conversion__inv,axiom,
! [X: term_a_b,Y: term_a_b,R5: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( abstra1130568886744591910rm_a_b @ R5 ) )
= ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X ) @ ( abstra1130568886744591910rm_a_b @ R5 ) ) ) ).
% conversion_inv
thf(fact_879_conversion__refl,axiom,
! [A: term_a_b,A4: set_Pr4386577575007340137rm_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ A ) @ ( abstra1130568886744591910rm_a_b @ A4 ) ) ).
% conversion_refl
thf(fact_880_insert__Diff,axiom,
! [A: list_nat,A4: set_list_nat] :
( ( member_list_nat @ A @ A4 )
=> ( ( insert_list_nat @ A @ ( minus_7954133019191499631st_nat @ A4 @ ( insert_list_nat @ A @ bot_bot_set_list_nat ) ) )
= A4 ) ) ).
% insert_Diff
thf(fact_881_insert__Diff,axiom,
! [A: produc357393685978478089rm_a_b,A4: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ A @ A4 )
=> ( ( insert7009541432154983385rm_a_b @ A @ ( minus_5192120951422937424rm_a_b @ A4 @ ( insert7009541432154983385rm_a_b @ A @ bot_bo197521221353338581rm_a_b ) ) )
= A4 ) ) ).
% insert_Diff
thf(fact_882_Diff__insert__absorb,axiom,
! [X: list_nat,A4: set_list_nat] :
( ~ ( member_list_nat @ X @ A4 )
=> ( ( minus_7954133019191499631st_nat @ ( insert_list_nat @ X @ A4 ) @ ( insert_list_nat @ X @ bot_bot_set_list_nat ) )
= A4 ) ) ).
% Diff_insert_absorb
thf(fact_883_Diff__insert__absorb,axiom,
! [X: produc357393685978478089rm_a_b,A4: set_Pr4386577575007340137rm_a_b] :
( ~ ( member5869715511025134514rm_a_b @ X @ A4 )
=> ( ( minus_5192120951422937424rm_a_b @ ( insert7009541432154983385rm_a_b @ X @ A4 ) @ ( insert7009541432154983385rm_a_b @ X @ bot_bo197521221353338581rm_a_b ) )
= A4 ) ) ).
% Diff_insert_absorb
thf(fact_884_subset__Diff__insert,axiom,
! [A4: set_list_nat,B5: set_list_nat,X: list_nat,C4: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A4 @ ( minus_7954133019191499631st_nat @ B5 @ ( insert_list_nat @ X @ C4 ) ) )
= ( ( ord_le6045566169113846134st_nat @ A4 @ ( minus_7954133019191499631st_nat @ B5 @ C4 ) )
& ~ ( member_list_nat @ X @ A4 ) ) ) ).
% subset_Diff_insert
thf(fact_885_subset__Diff__insert,axiom,
! [A4: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b,X: produc357393685978478089rm_a_b,C4: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A4 @ ( minus_5192120951422937424rm_a_b @ B5 @ ( insert7009541432154983385rm_a_b @ X @ C4 ) ) )
= ( ( ord_le118470702582115849rm_a_b @ A4 @ ( minus_5192120951422937424rm_a_b @ B5 @ C4 ) )
& ~ ( member5869715511025134514rm_a_b @ X @ A4 ) ) ) ).
% subset_Diff_insert
thf(fact_886_join__imp__conversion,axiom,
! [R2: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ ( abstra4096080454567261402rm_a_b @ R2 ) @ ( abstra1130568886744591910rm_a_b @ R2 ) ) ).
% join_imp_conversion
thf(fact_887_subset__insert__iff,axiom,
! [A4: set_list_nat,X: list_nat,B5: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A4 @ ( insert_list_nat @ X @ B5 ) )
= ( ( ( member_list_nat @ X @ A4 )
=> ( ord_le6045566169113846134st_nat @ ( minus_7954133019191499631st_nat @ A4 @ ( insert_list_nat @ X @ bot_bot_set_list_nat ) ) @ B5 ) )
& ( ~ ( member_list_nat @ X @ A4 )
=> ( ord_le6045566169113846134st_nat @ A4 @ B5 ) ) ) ) ).
% subset_insert_iff
thf(fact_888_subset__insert__iff,axiom,
! [A4: set_Pr4386577575007340137rm_a_b,X: produc357393685978478089rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A4 @ ( insert7009541432154983385rm_a_b @ X @ B5 ) )
= ( ( ( member5869715511025134514rm_a_b @ X @ A4 )
=> ( ord_le118470702582115849rm_a_b @ ( minus_5192120951422937424rm_a_b @ A4 @ ( insert7009541432154983385rm_a_b @ X @ bot_bo197521221353338581rm_a_b ) ) @ B5 ) )
& ( ~ ( member5869715511025134514rm_a_b @ X @ A4 )
=> ( ord_le118470702582115849rm_a_b @ A4 @ B5 ) ) ) ) ).
% subset_insert_iff
thf(fact_889_rtrancl__diff__decomp,axiom,
! [X: term_a_b,Y: term_a_b,A4: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( minus_5192120951422937424rm_a_b @ ( transi7742714808557438673rm_a_b @ A4 ) @ ( transi7742714808557438673rm_a_b @ B5 ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ A4 ) @ ( relcom370159955682700863rm_a_b @ ( minus_5192120951422937424rm_a_b @ A4 @ B5 ) @ ( transi7742714808557438673rm_a_b @ A4 ) ) ) ) ) ).
% rtrancl_diff_decomp
thf(fact_890_all__ctxt__closed__srstep__conversions,axiom,
! [F: set_Pr4934435412358123699_a_nat,R3: set_Pr4386577575007340137rm_a_b] : ( terms_5226143800768910156ed_a_b @ F @ ( abstra1130568886744591910rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R3 ) ) ) ) ).
% all_ctxt_closed_srstep_conversions
thf(fact_891_CR__imp__conversionIff__join,axiom,
! [R2: set_Pr4386577575007340137rm_a_b] :
( ( abstra8448919418672941150rm_a_b @ R2 @ top_top_set_term_a_b )
=> ( ( abstra1130568886744591910rm_a_b @ R2 )
= ( abstra4096080454567261402rm_a_b @ R2 ) ) ) ).
% CR_imp_conversionIff_join
thf(fact_892_UNC__def,axiom,
( abstra2192636875637797680st_nat
= ( ^ [A6: set_Pr3451248702717554689st_nat] :
! [A5: list_nat,B4: list_nat] :
( ( ( member_list_nat @ A5 @ ( abstract_NF_list_nat @ A6 ) )
& ( member_list_nat @ B4 @ ( abstract_NF_list_nat @ A6 ) )
& ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A5 @ B4 ) @ ( abstra7896806322651447026st_nat @ A6 ) ) )
=> ( A5 = B4 ) ) ) ) ).
% UNC_def
thf(fact_893_UNC__def,axiom,
( abstra8480497335821202873rm_a_b
= ( ^ [A6: set_Pr2972776593051762503rm_a_b] :
! [A5: produc357393685978478089rm_a_b,B4: produc357393685978478089rm_a_b] :
( ( ( member5869715511025134514rm_a_b @ A5 @ ( abstra1902783451776770197rm_a_b @ A6 ) )
& ( member5869715511025134514rm_a_b @ B4 @ ( abstra1902783451776770197rm_a_b @ A6 ) )
& ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A5 @ B4 ) @ ( abstra6125656530262095991rm_a_b @ A6 ) ) )
=> ( A5 = B4 ) ) ) ) ).
% UNC_def
thf(fact_894_UNC__def,axiom,
( abstra4649771476585718372rm_a_b
= ( ^ [A6: set_Pr4386577575007340137rm_a_b] :
! [A5: term_a_b,B4: term_a_b] :
( ( ( member_term_a_b @ A5 @ ( abstract_NF_term_a_b @ A6 ) )
& ( member_term_a_b @ B4 @ ( abstract_NF_term_a_b @ A6 ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A5 @ B4 ) @ ( abstra1130568886744591910rm_a_b @ A6 ) ) )
=> ( A5 = B4 ) ) ) ) ).
% UNC_def
thf(fact_895_UNC__I,axiom,
! [A4: set_Pr3451248702717554689st_nat] :
( ! [A3: list_nat,B3: list_nat] :
( ( member_list_nat @ A3 @ ( abstract_NF_list_nat @ A4 ) )
=> ( ( member_list_nat @ B3 @ ( abstract_NF_list_nat @ A4 ) )
=> ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A3 @ B3 ) @ ( abstra7896806322651447026st_nat @ A4 ) )
=> ( A3 = B3 ) ) ) )
=> ( abstra2192636875637797680st_nat @ A4 ) ) ).
% UNC_I
thf(fact_896_UNC__I,axiom,
! [A4: set_Pr2972776593051762503rm_a_b] :
( ! [A3: produc357393685978478089rm_a_b,B3: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ A3 @ ( abstra1902783451776770197rm_a_b @ A4 ) )
=> ( ( member5869715511025134514rm_a_b @ B3 @ ( abstra1902783451776770197rm_a_b @ A4 ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A3 @ B3 ) @ ( abstra6125656530262095991rm_a_b @ A4 ) )
=> ( A3 = B3 ) ) ) )
=> ( abstra8480497335821202873rm_a_b @ A4 ) ) ).
% UNC_I
thf(fact_897_UNC__I,axiom,
! [A4: set_Pr4386577575007340137rm_a_b] :
( ! [A3: term_a_b,B3: term_a_b] :
( ( member_term_a_b @ A3 @ ( abstract_NF_term_a_b @ A4 ) )
=> ( ( member_term_a_b @ B3 @ ( abstract_NF_term_a_b @ A4 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B3 ) @ ( abstra1130568886744591910rm_a_b @ A4 ) )
=> ( A3 = B3 ) ) ) )
=> ( abstra4649771476585718372rm_a_b @ A4 ) ) ).
% UNC_I
thf(fact_898_UNC__E,axiom,
! [A4: set_Pr3451248702717554689st_nat,A: list_nat,B: list_nat] :
( ( abstra2192636875637797680st_nat @ A4 )
=> ( ( A != B )
=> ( ( member_list_nat @ A @ ( abstract_NF_list_nat @ A4 ) )
=> ( ( member_list_nat @ B @ ( abstract_NF_list_nat @ A4 ) )
=> ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A @ B ) @ ( abstra7896806322651447026st_nat @ A4 ) ) ) ) ) ) ).
% UNC_E
thf(fact_899_UNC__E,axiom,
! [A4: set_Pr2972776593051762503rm_a_b,A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b] :
( ( abstra8480497335821202873rm_a_b @ A4 )
=> ( ( A != B )
=> ( ( member5869715511025134514rm_a_b @ A @ ( abstra1902783451776770197rm_a_b @ A4 ) )
=> ( ( member5869715511025134514rm_a_b @ B @ ( abstra1902783451776770197rm_a_b @ A4 ) )
=> ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( abstra6125656530262095991rm_a_b @ A4 ) ) ) ) ) ) ).
% UNC_E
thf(fact_900_UNC__E,axiom,
! [A4: set_Pr4386577575007340137rm_a_b,A: term_a_b,B: term_a_b] :
( ( abstra4649771476585718372rm_a_b @ A4 )
=> ( ( A != B )
=> ( ( member_term_a_b @ A @ ( abstract_NF_term_a_b @ A4 ) )
=> ( ( member_term_a_b @ B @ ( abstract_NF_term_a_b @ A4 ) )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( abstra1130568886744591910rm_a_b @ A4 ) ) ) ) ) ) ).
% UNC_E
thf(fact_901_conversionE,axiom,
! [A: term_a_b,B: term_a_b,A4: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( abstra1130568886744591910rm_a_b @ A4 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ A4 @ ( conver7608559976937845093rm_a_b @ A4 ) ) ) ) ) ).
% conversionE
thf(fact_902_CR__iff__conversion__imp__join,axiom,
! [R2: set_Pr4386577575007340137rm_a_b] :
( ( abstra8448919418672941150rm_a_b @ R2 @ top_top_set_term_a_b )
= ( ord_le118470702582115849rm_a_b @ ( abstra1130568886744591910rm_a_b @ R2 ) @ ( abstra4096080454567261402rm_a_b @ R2 ) ) ) ).
% CR_iff_conversion_imp_join
thf(fact_903_symcl__srstep__conversion,axiom,
! [S: term_a_b,T: term_a_b,F: set_Pr4934435412358123699_a_nat,R3: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( sig_step_a_b @ F @ ( rstep_a_b @ ( sup_su6776935440552674877rm_a_b @ R3 @ ( conver7608559976937845093rm_a_b @ R3 ) ) ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( abstra1130568886744591910rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R3 ) ) ) ) ) ).
% symcl_srstep_conversion
thf(fact_904_sig__step__conversion__dist,axiom,
! [F: set_Pr4934435412358123699_a_nat,R3: set_Pr4386577575007340137rm_a_b] :
( ( abstra1130568886744591910rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R3 ) ) )
= ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ ( sup_su6776935440552674877rm_a_b @ R3 @ ( conver7608559976937845093rm_a_b @ R3 ) ) ) ) ) ) ).
% sig_step_conversion_dist
thf(fact_905_CE__rrstep__intro,axiom,
! [F: set_Pr4934435412358123699_a_nat,R3: set_Pr4386577575007340137rm_a_b,S5: set_Pr4386577575007340137rm_a_b] :
( ! [S4: term_a_b,T2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S4 @ T2 ) @ ( srstep7844470518422762656ep_a_b @ F @ ( sup_su6776935440552674877rm_a_b @ R3 @ ( conver7608559976937845093rm_a_b @ R3 ) ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S4 @ T2 ) @ ( abstra1130568886744591910rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ S5 ) ) ) ) )
=> ( ! [S4: term_a_b,T2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S4 @ T2 ) @ ( srstep7844470518422762656ep_a_b @ F @ ( sup_su6776935440552674877rm_a_b @ S5 @ ( conver7608559976937845093rm_a_b @ S5 ) ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S4 @ T2 ) @ ( abstra1130568886744591910rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R3 ) ) ) ) )
=> ( rewrit660295756789745446rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R3 ) ) @ ( sig_step_a_b @ F @ ( rstep_a_b @ S5 ) ) @ top_top_set_term_a_b ) ) ) ).
% CE_rrstep_intro
thf(fact_906_CE__to__rrstep_I1_J,axiom,
! [F: set_Pr4934435412358123699_a_nat,R3: set_Pr4386577575007340137rm_a_b,S5: set_Pr4386577575007340137rm_a_b,S: term_a_b,T: term_a_b] :
( ( rewrit660295756789745446rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R3 ) ) @ ( sig_step_a_b @ F @ ( rstep_a_b @ S5 ) ) @ top_top_set_term_a_b )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( srstep7844470518422762656ep_a_b @ F @ ( sup_su6776935440552674877rm_a_b @ R3 @ ( conver7608559976937845093rm_a_b @ R3 ) ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( abstra1130568886744591910rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ S5 ) ) ) ) ) ) ).
% CE_to_rrstep(1)
thf(fact_907_CE__to__rrstep_I2_J,axiom,
! [F: set_Pr4934435412358123699_a_nat,R3: set_Pr4386577575007340137rm_a_b,S5: set_Pr4386577575007340137rm_a_b,S: term_a_b,T: term_a_b] :
( ( rewrit660295756789745446rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R3 ) ) @ ( sig_step_a_b @ F @ ( rstep_a_b @ S5 ) ) @ top_top_set_term_a_b )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( srstep7844470518422762656ep_a_b @ F @ ( sup_su6776935440552674877rm_a_b @ S5 @ ( conver7608559976937845093rm_a_b @ S5 ) ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( abstra1130568886744591910rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R3 ) ) ) ) ) ) ).
% CE_to_rrstep(2)
thf(fact_908_CE__on__def,axiom,
( rewrit660295756789745446rm_a_b
= ( ^ [R4: set_Pr4386577575007340137rm_a_b,S3: 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 ) @ ( abstra1130568886744591910rm_a_b @ R4 ) )
= ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X4 @ B4 ) @ ( abstra1130568886744591910rm_a_b @ S3 ) ) ) ) ) ) ).
% CE_on_def
thf(fact_909_CR__NF__conv,axiom,
! [R2: set_Pr3451248702717554689st_nat,T: list_nat,U: list_nat] :
( ( abstra5991784817725020458st_nat @ R2 @ top_top_set_list_nat )
=> ( ( member_list_nat @ T @ ( abstract_NF_list_nat @ R2 ) )
=> ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ U @ T ) @ ( abstra7896806322651447026st_nat @ R2 ) )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ U @ T ) @ ( abstra515516630445031992st_nat @ R2 ) ) ) ) ) ).
% CR_NF_conv
thf(fact_910_CR__NF__conv,axiom,
! [R2: set_Pr2972776593051762503rm_a_b,T: produc357393685978478089rm_a_b,U: produc357393685978478089rm_a_b] :
( ( abstra7359694574113289279rm_a_b @ R2 @ top_to1314267278846557113rm_a_b )
=> ( ( member5869715511025134514rm_a_b @ T @ ( abstra1902783451776770197rm_a_b @ R2 ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ U @ T ) @ ( abstra6125656530262095991rm_a_b @ R2 ) )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ U @ T ) @ ( abstra1496011390211842225rm_a_b @ R2 ) ) ) ) ) ).
% CR_NF_conv
thf(fact_911_CR__NF__conv,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,T: term_a_b,U: term_a_b] :
( ( abstra8448919418672941150rm_a_b @ R2 @ top_top_set_term_a_b )
=> ( ( member_term_a_b @ T @ ( abstract_NF_term_a_b @ R2 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ U @ T ) @ ( abstra1130568886744591910rm_a_b @ R2 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ U @ T ) @ ( abstra2972651231392952684rm_a_b @ R2 ) ) ) ) ) ).
% CR_NF_conv
thf(fact_912_some__NF,axiom,
! [R2: set_Pr3451248702717554689st_nat,X: list_nat] :
( ( abstra2262888740781824790st_nat @ R2 @ ( insert_list_nat @ X @ bot_bot_set_list_nat ) )
=> ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X @ ( abstra132235892672370824st_nat @ R2 @ X ) ) @ ( transi5285580207609517981st_nat @ R2 ) )
& ( member_list_nat @ ( abstra132235892672370824st_nat @ R2 @ X ) @ ( abstract_NF_list_nat @ R2 ) ) ) ) ).
% some_NF
thf(fact_913_some__NF,axiom,
! [R2: set_Pr2972776593051762503rm_a_b,X: produc357393685978478089rm_a_b] :
( ( abstra2398554102055911763rm_a_b @ R2 @ ( insert7009541432154983385rm_a_b @ X @ bot_bo197521221353338581rm_a_b ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ ( abstra8203652949708803681rm_a_b @ R2 @ X ) ) @ ( transi2615809358984392588rm_a_b @ R2 ) )
& ( member5869715511025134514rm_a_b @ ( abstra8203652949708803681rm_a_b @ R2 @ X ) @ ( abstra1902783451776770197rm_a_b @ R2 ) ) ) ) ).
% some_NF
thf(fact_914_some__NF,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,X: term_a_b] :
( ( abstra4720023341729745482rm_a_b @ R2 @ ( insert_term_a_b @ X @ bot_bot_set_term_a_b ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ ( abstra2589370493620291516rm_a_b @ R2 @ X ) ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
& ( member_term_a_b @ ( abstra2589370493620291516rm_a_b @ R2 @ X ) @ ( abstract_NF_term_a_b @ R2 ) ) ) ) ).
% some_NF
thf(fact_915_semi__complete__imp__conversionIff__same__NF,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,X: term_a_b,Y: term_a_b] :
( ( abstra7162307272211697846rm_a_b @ R2 @ top_top_set_term_a_b )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( abstra1130568886744591910rm_a_b @ R2 ) )
= ( ! [U4: term_a_b,V4: term_a_b] :
( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ U4 ) @ ( abstra2972651231392952684rm_a_b @ R2 ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ V4 ) @ ( abstra2972651231392952684rm_a_b @ R2 ) ) )
=> ( U4 = V4 ) ) ) ) ) ).
% semi_complete_imp_conversionIff_same_NF
thf(fact_916_normalizability__I_H,axiom,
! [A: term_a_b,B: term_a_b,A4: set_Pr4386577575007340137rm_a_b,C2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ A4 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C2 ) @ ( abstra2972651231392952684rm_a_b @ A4 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C2 ) @ ( abstra2972651231392952684rm_a_b @ A4 ) ) ) ) ).
% normalizability_I'
thf(fact_917_UNF__onI,axiom,
! [A4: set_list_nat,R2: set_Pr3451248702717554689st_nat] :
( ! [A3: list_nat,B3: list_nat,C: list_nat] :
( ( member_list_nat @ A3 @ A4 )
=> ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A3 @ B3 ) @ ( abstra515516630445031992st_nat @ R2 ) )
=> ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A3 @ C ) @ ( abstra515516630445031992st_nat @ R2 ) )
=> ( B3 = C ) ) ) )
=> ( abstra1162721134732043186st_nat @ R2 @ A4 ) ) ).
% UNF_onI
thf(fact_918_UNF__onI,axiom,
! [A4: set_Pr4386577575007340137rm_a_b,R2: set_Pr2972776593051762503rm_a_b] :
( ! [A3: produc357393685978478089rm_a_b,B3: produc357393685978478089rm_a_b,C: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ A3 @ A4 )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A3 @ B3 ) @ ( abstra1496011390211842225rm_a_b @ R2 ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A3 @ C ) @ ( abstra1496011390211842225rm_a_b @ R2 ) )
=> ( B3 = C ) ) ) )
=> ( abstra1303798319023342519rm_a_b @ R2 @ A4 ) ) ).
% UNF_onI
thf(fact_919_UNF__onI,axiom,
! [A4: set_term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ! [A3: term_a_b,B3: term_a_b,C: term_a_b] :
( ( member_term_a_b @ A3 @ A4 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B3 ) @ ( abstra2972651231392952684rm_a_b @ R2 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ C ) @ ( abstra2972651231392952684rm_a_b @ R2 ) )
=> ( B3 = C ) ) ) )
=> ( abstra3619855735679963878rm_a_b @ R2 @ A4 ) ) ).
% UNF_onI
thf(fact_920_WN__onI,axiom,
! [A4: set_list_nat,R2: set_Pr3451248702717554689st_nat] :
( ! [A3: list_nat] :
( ( member_list_nat @ A3 @ A4 )
=> ? [B8: list_nat] : ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A3 @ B8 ) @ ( abstra515516630445031992st_nat @ R2 ) ) )
=> ( abstra2187600465986519570st_nat @ R2 @ A4 ) ) ).
% WN_onI
thf(fact_921_WN__onI,axiom,
! [A4: set_Pr4386577575007340137rm_a_b,R2: set_Pr2972776593051762503rm_a_b] :
( ! [A3: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ A3 @ A4 )
=> ? [B8: produc357393685978478089rm_a_b] : ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A3 @ B8 ) @ ( abstra1496011390211842225rm_a_b @ R2 ) ) )
=> ( abstra2721873542170697047rm_a_b @ R2 @ A4 ) ) ).
% WN_onI
thf(fact_922_WN__onI,axiom,
! [A4: set_term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ! [A3: term_a_b] :
( ( member_term_a_b @ A3 @ A4 )
=> ? [B8: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B8 ) @ ( abstra2972651231392952684rm_a_b @ R2 ) ) )
=> ( abstra4644735066934440262rm_a_b @ R2 @ A4 ) ) ).
% WN_onI
thf(fact_923_normalizability__I,axiom,
! [A: list_nat,B: list_nat,A4: set_Pr3451248702717554689st_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A @ B ) @ ( transi5285580207609517981st_nat @ A4 ) )
=> ( ( member_list_nat @ B @ ( abstract_NF_list_nat @ A4 ) )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A @ B ) @ ( abstra515516630445031992st_nat @ A4 ) ) ) ) ).
% normalizability_I
thf(fact_924_normalizability__I,axiom,
! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,A4: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( transi2615809358984392588rm_a_b @ A4 ) )
=> ( ( member5869715511025134514rm_a_b @ B @ ( abstra1902783451776770197rm_a_b @ A4 ) )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( abstra1496011390211842225rm_a_b @ A4 ) ) ) ) ).
% normalizability_I
thf(fact_925_normalizability__I,axiom,
! [A: term_a_b,B: term_a_b,A4: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ A4 ) )
=> ( ( member_term_a_b @ B @ ( abstract_NF_term_a_b @ A4 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( abstra2972651231392952684rm_a_b @ A4 ) ) ) ) ).
% normalizability_I
thf(fact_926_SN__on__irrefl,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,A4: set_term_a_b] :
( ( abstra4720023341729745482rm_a_b @ R2 @ A4 )
=> ! [X6: term_a_b] :
( ( member_term_a_b @ X6 @ A4 )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X6 @ X6 ) @ R2 ) ) ) ).
% SN_on_irrefl
thf(fact_927_SN__on__induct,axiom,
! [R5: set_Pr3451248702717554689st_nat,A4: set_list_nat,S: list_nat,P3: list_nat > $o] :
( ( abstra2262888740781824790st_nat @ R5 @ A4 )
=> ( ( member_list_nat @ S @ A4 )
=> ( ! [T2: list_nat] :
( ! [U3: list_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ T2 @ U3 ) @ R5 )
=> ( P3 @ U3 ) )
=> ( P3 @ T2 ) )
=> ( P3 @ S ) ) ) ) ).
% SN_on_induct
thf(fact_928_SN__on__induct,axiom,
! [R5: set_Pr2972776593051762503rm_a_b,A4: set_Pr4386577575007340137rm_a_b,S: produc357393685978478089rm_a_b,P3: produc357393685978478089rm_a_b > $o] :
( ( abstra2398554102055911763rm_a_b @ R5 @ A4 )
=> ( ( member5869715511025134514rm_a_b @ S @ A4 )
=> ( ! [T2: produc357393685978478089rm_a_b] :
( ! [U3: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ T2 @ U3 ) @ R5 )
=> ( P3 @ U3 ) )
=> ( P3 @ T2 ) )
=> ( P3 @ S ) ) ) ) ).
% SN_on_induct
thf(fact_929_SN__on__induct,axiom,
! [R5: set_Pr4386577575007340137rm_a_b,A4: set_term_a_b,S: term_a_b,P3: term_a_b > $o] :
( ( abstra4720023341729745482rm_a_b @ R5 @ A4 )
=> ( ( member_term_a_b @ S @ A4 )
=> ( ! [T2: term_a_b] :
( ! [U3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T2 @ U3 ) @ R5 )
=> ( P3 @ U3 ) )
=> ( P3 @ T2 ) )
=> ( P3 @ S ) ) ) ) ).
% SN_on_induct
thf(fact_930_SN__on__imp__normalizability,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,A: term_a_b] :
( ( abstra4720023341729745482rm_a_b @ R2 @ ( insert_term_a_b @ A @ bot_bot_set_term_a_b ) )
=> ? [B3: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B3 ) @ ( abstra2972651231392952684rm_a_b @ R2 ) ) ) ).
% SN_on_imp_normalizability
thf(fact_931_refl__not__SN,axiom,
! [X: term_a_b,R5: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ X ) @ R5 )
=> ~ ( abstra4720023341729745482rm_a_b @ R5 @ top_top_set_term_a_b ) ) ).
% refl_not_SN
thf(fact_932_SN__imp__minimal,axiom,
! [A4: set_Pr3451248702717554689st_nat] :
( ( abstra2262888740781824790st_nat @ A4 @ top_top_set_list_nat )
=> ! [Q4: set_list_nat] :
( ? [X6: list_nat] : ( member_list_nat @ X6 @ Q4 )
=> ? [X3: list_nat] :
( ( member_list_nat @ X3 @ Q4 )
& ! [Y7: list_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X3 @ Y7 ) @ A4 )
=> ~ ( member_list_nat @ Y7 @ Q4 ) ) ) ) ) ).
% SN_imp_minimal
thf(fact_933_SN__imp__minimal,axiom,
! [A4: set_Pr2972776593051762503rm_a_b] :
( ( abstra2398554102055911763rm_a_b @ A4 @ top_to1314267278846557113rm_a_b )
=> ! [Q4: set_Pr4386577575007340137rm_a_b] :
( ? [X6: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ X6 @ Q4 )
=> ? [X3: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X3 @ Q4 )
& ! [Y7: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X3 @ Y7 ) @ A4 )
=> ~ ( member5869715511025134514rm_a_b @ Y7 @ Q4 ) ) ) ) ) ).
% SN_imp_minimal
thf(fact_934_SN__imp__minimal,axiom,
! [A4: set_Pr4386577575007340137rm_a_b] :
( ( abstra4720023341729745482rm_a_b @ A4 @ top_top_set_term_a_b )
=> ! [Q4: set_term_a_b] :
( ? [X6: term_a_b] : ( member_term_a_b @ X6 @ Q4 )
=> ? [X3: term_a_b] :
( ( member_term_a_b @ X3 @ Q4 )
& ! [Y7: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y7 ) @ A4 )
=> ~ ( member_term_a_b @ Y7 @ Q4 ) ) ) ) ) ).
% SN_imp_minimal
thf(fact_935_SN__induct__rule,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,P3: term_a_b > $o,A: term_a_b] :
( ( abstra4720023341729745482rm_a_b @ R2 @ top_top_set_term_a_b )
=> ( ! [A3: term_a_b] :
( ! [B8: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B8 ) @ R2 )
=> ( P3 @ B8 ) )
=> ( P3 @ A3 ) )
=> ( P3 @ A ) ) ) ).
% SN_induct_rule
thf(fact_936_SN__on__induct_H,axiom,
! [R5: set_Pr4386577575007340137rm_a_b,S: term_a_b,P3: term_a_b > $o] :
( ( abstra4720023341729745482rm_a_b @ R5 @ ( insert_term_a_b @ S @ bot_bot_set_term_a_b ) )
=> ( ! [T2: term_a_b] :
( ! [U3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T2 @ U3 ) @ R5 )
=> ( P3 @ U3 ) )
=> ( P3 @ T2 ) )
=> ( P3 @ S ) ) ) ).
% SN_on_induct'
thf(fact_937_step__reflects__SN__on,axiom,
! [A: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ! [B3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B3 ) @ R2 )
=> ( abstra4720023341729745482rm_a_b @ R2 @ ( insert_term_a_b @ B3 @ bot_bot_set_term_a_b ) ) )
=> ( abstra4720023341729745482rm_a_b @ R2 @ ( insert_term_a_b @ A @ bot_bot_set_term_a_b ) ) ) ).
% step_reflects_SN_on
thf(fact_938_SN__on__imp__on__minimal,axiom,
! [R2: set_Pr3451248702717554689st_nat,X: list_nat] :
( ( abstra2262888740781824790st_nat @ R2 @ ( insert_list_nat @ X @ bot_bot_set_list_nat ) )
=> ! [Q4: set_list_nat] :
( ( member_list_nat @ X @ Q4 )
=> ? [X3: list_nat] :
( ( member_list_nat @ X3 @ Q4 )
& ! [Y7: list_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X3 @ Y7 ) @ R2 )
=> ~ ( member_list_nat @ Y7 @ Q4 ) ) ) ) ) ).
% SN_on_imp_on_minimal
thf(fact_939_SN__on__imp__on__minimal,axiom,
! [R2: set_Pr2972776593051762503rm_a_b,X: produc357393685978478089rm_a_b] :
( ( abstra2398554102055911763rm_a_b @ R2 @ ( insert7009541432154983385rm_a_b @ X @ bot_bo197521221353338581rm_a_b ) )
=> ! [Q4: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ X @ Q4 )
=> ? [X3: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X3 @ Q4 )
& ! [Y7: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X3 @ Y7 ) @ R2 )
=> ~ ( member5869715511025134514rm_a_b @ Y7 @ Q4 ) ) ) ) ) ).
% SN_on_imp_on_minimal
thf(fact_940_SN__on__imp__on__minimal,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,X: term_a_b] :
( ( abstra4720023341729745482rm_a_b @ R2 @ ( insert_term_a_b @ X @ bot_bot_set_term_a_b ) )
=> ! [Q4: set_term_a_b] :
( ( member_term_a_b @ X @ Q4 )
=> ? [X3: term_a_b] :
( ( member_term_a_b @ X3 @ Q4 )
& ! [Y7: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y7 ) @ R2 )
=> ~ ( member_term_a_b @ Y7 @ Q4 ) ) ) ) ) ).
% SN_on_imp_on_minimal
thf(fact_941_step__preserves__SN__on,axiom,
! [A: term_a_b,B: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R2 )
=> ( ( abstra4720023341729745482rm_a_b @ R2 @ ( insert_term_a_b @ A @ bot_bot_set_term_a_b ) )
=> ( abstra4720023341729745482rm_a_b @ R2 @ ( insert_term_a_b @ B @ bot_bot_set_term_a_b ) ) ) ) ).
% step_preserves_SN_on
thf(fact_942_SN__on__induct__acc__style,axiom,
! [R5: set_Pr4386577575007340137rm_a_b,A: term_a_b,P3: term_a_b > $o] :
( ( abstra4720023341729745482rm_a_b @ R5 @ ( insert_term_a_b @ A @ bot_bot_set_term_a_b ) )
=> ( ! [X3: term_a_b] :
( ( abstra4720023341729745482rm_a_b @ R5 @ ( insert_term_a_b @ X3 @ bot_bot_set_term_a_b ) )
=> ( ! [Y7: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y7 ) @ R5 )
=> ( P3 @ Y7 ) )
=> ( P3 @ X3 ) ) )
=> ( P3 @ A ) ) ) ).
% SN_on_induct_acc_style
thf(fact_943_lower__set__imp__not__SN__on,axiom,
! [S: list_nat,X5: set_list_nat,R5: set_Pr3451248702717554689st_nat] :
( ( member_list_nat @ S @ X5 )
=> ( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ X5 )
=> ? [Xa2: list_nat] :
( ( member_list_nat @ Xa2 @ X5 )
& ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X3 @ Xa2 ) @ R5 ) ) )
=> ~ ( abstra2262888740781824790st_nat @ R5 @ ( insert_list_nat @ S @ bot_bot_set_list_nat ) ) ) ) ).
% lower_set_imp_not_SN_on
thf(fact_944_lower__set__imp__not__SN__on,axiom,
! [S: produc357393685978478089rm_a_b,X5: set_Pr4386577575007340137rm_a_b,R5: set_Pr2972776593051762503rm_a_b] :
( ( member5869715511025134514rm_a_b @ S @ X5 )
=> ( ! [X3: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X3 @ X5 )
=> ? [Xa2: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ Xa2 @ X5 )
& ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X3 @ Xa2 ) @ R5 ) ) )
=> ~ ( abstra2398554102055911763rm_a_b @ R5 @ ( insert7009541432154983385rm_a_b @ S @ bot_bo197521221353338581rm_a_b ) ) ) ) ).
% lower_set_imp_not_SN_on
thf(fact_945_lower__set__imp__not__SN__on,axiom,
! [S: term_a_b,X5: set_term_a_b,R5: set_Pr4386577575007340137rm_a_b] :
( ( member_term_a_b @ S @ X5 )
=> ( ! [X3: term_a_b] :
( ( member_term_a_b @ X3 @ X5 )
=> ? [Xa2: term_a_b] :
( ( member_term_a_b @ Xa2 @ X5 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Xa2 ) @ R5 ) ) )
=> ~ ( abstra4720023341729745482rm_a_b @ R5 @ ( insert_term_a_b @ S @ bot_bot_set_term_a_b ) ) ) ) ).
% lower_set_imp_not_SN_on
thf(fact_946_SN__on__all__reducts__SN__on__conv,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,A: term_a_b] :
( ( abstra4720023341729745482rm_a_b @ R2 @ ( insert_term_a_b @ A @ bot_bot_set_term_a_b ) )
= ( ! [B4: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B4 ) @ R2 )
=> ( abstra4720023341729745482rm_a_b @ R2 @ ( insert_term_a_b @ B4 @ bot_bot_set_term_a_b ) ) ) ) ) ).
% SN_on_all_reducts_SN_on_conv
thf(fact_947_NF__imp__SN__on,axiom,
! [A: list_nat,R5: set_Pr3451248702717554689st_nat] :
( ( member_list_nat @ A @ ( abstract_NF_list_nat @ R5 ) )
=> ( abstra2262888740781824790st_nat @ R5 @ ( insert_list_nat @ A @ bot_bot_set_list_nat ) ) ) ).
% NF_imp_SN_on
thf(fact_948_NF__imp__SN__on,axiom,
! [A: produc357393685978478089rm_a_b,R5: set_Pr2972776593051762503rm_a_b] :
( ( member5869715511025134514rm_a_b @ A @ ( abstra1902783451776770197rm_a_b @ R5 ) )
=> ( abstra2398554102055911763rm_a_b @ R5 @ ( insert7009541432154983385rm_a_b @ A @ bot_bo197521221353338581rm_a_b ) ) ) ).
% NF_imp_SN_on
thf(fact_949_UNF__onD,axiom,
! [R2: set_Pr3451248702717554689st_nat,A4: set_list_nat,A: list_nat,B: list_nat,C2: list_nat] :
( ( abstra1162721134732043186st_nat @ R2 @ A4 )
=> ( ( member_list_nat @ A @ A4 )
=> ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A @ B ) @ ( abstra515516630445031992st_nat @ R2 ) )
=> ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A @ C2 ) @ ( abstra515516630445031992st_nat @ R2 ) )
=> ( B = C2 ) ) ) ) ) ).
% UNF_onD
thf(fact_950_UNF__onD,axiom,
! [R2: set_Pr2972776593051762503rm_a_b,A4: set_Pr4386577575007340137rm_a_b,A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,C2: produc357393685978478089rm_a_b] :
( ( abstra1303798319023342519rm_a_b @ R2 @ A4 )
=> ( ( member5869715511025134514rm_a_b @ A @ A4 )
=> ( ( 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_951_UNF__onD,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,A4: set_term_a_b,A: term_a_b,B: term_a_b,C2: term_a_b] :
( ( abstra3619855735679963878rm_a_b @ R2 @ A4 )
=> ( ( member_term_a_b @ A @ A4 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( abstra2972651231392952684rm_a_b @ R2 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C2 ) @ ( abstra2972651231392952684rm_a_b @ R2 ) )
=> ( B = C2 ) ) ) ) ) ).
% UNF_onD
thf(fact_952_UNF__onE,axiom,
! [R2: set_Pr3451248702717554689st_nat,A4: set_list_nat,A: list_nat,B: list_nat,C2: list_nat] :
( ( abstra1162721134732043186st_nat @ R2 @ A4 )
=> ( ( member_list_nat @ A @ A4 )
=> ( ( B != C2 )
=> ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A @ B ) @ ( abstra515516630445031992st_nat @ R2 ) )
=> ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A @ C2 ) @ ( abstra515516630445031992st_nat @ R2 ) ) ) ) ) ) ).
% UNF_onE
thf(fact_953_UNF__onE,axiom,
! [R2: set_Pr2972776593051762503rm_a_b,A4: set_Pr4386577575007340137rm_a_b,A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,C2: produc357393685978478089rm_a_b] :
( ( abstra1303798319023342519rm_a_b @ R2 @ A4 )
=> ( ( member5869715511025134514rm_a_b @ A @ A4 )
=> ( ( 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_954_UNF__onE,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,A4: set_term_a_b,A: term_a_b,B: term_a_b,C2: term_a_b] :
( ( abstra3619855735679963878rm_a_b @ R2 @ A4 )
=> ( ( member_term_a_b @ A @ A4 )
=> ( ( 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_955_UNF__on__def,axiom,
( abstra3619855735679963878rm_a_b
= ( ^ [R4: set_Pr4386577575007340137rm_a_b,A6: set_term_a_b] :
! [X4: term_a_b] :
( ( member_term_a_b @ X4 @ A6 )
=> ! [B4: term_a_b,C3: term_a_b] :
( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X4 @ B4 ) @ ( abstra2972651231392952684rm_a_b @ R4 ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X4 @ C3 ) @ ( abstra2972651231392952684rm_a_b @ R4 ) ) )
=> ( B4 = C3 ) ) ) ) ) ).
% UNF_on_def
thf(fact_956_WN__on__def,axiom,
( abstra4644735066934440262rm_a_b
= ( ^ [R4: 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 @ R4 ) ) ) ) ) ).
% WN_on_def
thf(fact_957_WN__onE,axiom,
! [R2: set_Pr3451248702717554689st_nat,A4: set_list_nat,A: list_nat] :
( ( abstra2187600465986519570st_nat @ R2 @ A4 )
=> ( ( member_list_nat @ A @ A4 )
=> ~ ! [B3: list_nat] :
~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A @ B3 ) @ ( abstra515516630445031992st_nat @ R2 ) ) ) ) ).
% WN_onE
thf(fact_958_WN__onE,axiom,
! [R2: set_Pr2972776593051762503rm_a_b,A4: set_Pr4386577575007340137rm_a_b,A: produc357393685978478089rm_a_b] :
( ( abstra2721873542170697047rm_a_b @ R2 @ A4 )
=> ( ( member5869715511025134514rm_a_b @ A @ A4 )
=> ~ ! [B3: produc357393685978478089rm_a_b] :
~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B3 ) @ ( abstra1496011390211842225rm_a_b @ R2 ) ) ) ) ).
% WN_onE
thf(fact_959_WN__onE,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,A4: set_term_a_b,A: term_a_b] :
( ( abstra4644735066934440262rm_a_b @ R2 @ A4 )
=> ( ( member_term_a_b @ A @ A4 )
=> ~ ! [B3: term_a_b] :
~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B3 ) @ ( abstra2972651231392952684rm_a_b @ R2 ) ) ) ) ).
% WN_onE
thf(fact_960_WN__onD,axiom,
! [R2: set_Pr3451248702717554689st_nat,A4: set_list_nat,A: list_nat] :
( ( abstra2187600465986519570st_nat @ R2 @ A4 )
=> ( ( member_list_nat @ A @ A4 )
=> ? [B3: list_nat] : ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A @ B3 ) @ ( abstra515516630445031992st_nat @ R2 ) ) ) ) ).
% WN_onD
thf(fact_961_WN__onD,axiom,
! [R2: set_Pr2972776593051762503rm_a_b,A4: set_Pr4386577575007340137rm_a_b,A: produc357393685978478089rm_a_b] :
( ( abstra2721873542170697047rm_a_b @ R2 @ A4 )
=> ( ( member5869715511025134514rm_a_b @ A @ A4 )
=> ? [B3: produc357393685978478089rm_a_b] : ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B3 ) @ ( abstra1496011390211842225rm_a_b @ R2 ) ) ) ) ).
% WN_onD
thf(fact_962_WN__onD,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,A4: set_term_a_b,A: term_a_b] :
( ( abstra4644735066934440262rm_a_b @ R2 @ A4 )
=> ( ( member_term_a_b @ A @ A4 )
=> ? [B3: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B3 ) @ ( abstra2972651231392952684rm_a_b @ R2 ) ) ) ) ).
% WN_onD
thf(fact_963_NE__on__def,axiom,
( rewrit7370782028743737947rm_a_b
= ( ^ [R4: set_Pr4386577575007340137rm_a_b,S3: 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 @ R4 ) )
= ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X4 @ B4 ) @ ( abstra2972651231392952684rm_a_b @ S3 ) ) ) ) ) ) ).
% NE_on_def
thf(fact_964_steps__preserve__SN__on,axiom,
! [A: term_a_b,B: term_a_b,A4: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ A4 ) )
=> ( ( abstra4720023341729745482rm_a_b @ A4 @ ( insert_term_a_b @ A @ bot_bot_set_term_a_b ) )
=> ( abstra4720023341729745482rm_a_b @ A4 @ ( insert_term_a_b @ B @ bot_bot_set_term_a_b ) ) ) ) ).
% steps_preserve_SN_on
thf(fact_965_steps__reflect__SN__on,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,B: term_a_b,A: term_a_b] :
( ~ ( abstra4720023341729745482rm_a_b @ R2 @ ( insert_term_a_b @ B @ bot_bot_set_term_a_b ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
=> ~ ( abstra4720023341729745482rm_a_b @ R2 @ ( insert_term_a_b @ A @ bot_bot_set_term_a_b ) ) ) ) ).
% steps_reflect_SN_on
thf(fact_966_normalizability__E,axiom,
! [A: list_nat,B: list_nat,A4: set_Pr3451248702717554689st_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A @ B ) @ ( abstra515516630445031992st_nat @ A4 ) )
=> ~ ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A @ B ) @ ( transi5285580207609517981st_nat @ A4 ) )
=> ~ ( member_list_nat @ B @ ( abstract_NF_list_nat @ A4 ) ) ) ) ).
% normalizability_E
thf(fact_967_normalizability__E,axiom,
! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,A4: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( abstra1496011390211842225rm_a_b @ A4 ) )
=> ~ ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( transi2615809358984392588rm_a_b @ A4 ) )
=> ~ ( member5869715511025134514rm_a_b @ B @ ( abstra1902783451776770197rm_a_b @ A4 ) ) ) ) ).
% normalizability_E
thf(fact_968_normalizability__E,axiom,
! [A: term_a_b,B: term_a_b,A4: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( abstra2972651231392952684rm_a_b @ A4 ) )
=> ~ ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ A4 ) )
=> ~ ( member_term_a_b @ B @ ( abstract_NF_term_a_b @ A4 ) ) ) ) ).
% normalizability_E
thf(fact_969_NFP__on__def,axiom,
( rewrit6304709981509430748rm_a_b
= ( ^ [R4: set_Pr4386577575007340137rm_a_b,A6: set_term_a_b] :
! [X4: term_a_b] :
( ( member_term_a_b @ X4 @ A6 )
=> ! [B4: term_a_b,C3: term_a_b] :
( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X4 @ B4 ) @ ( transi7742714808557438673rm_a_b @ R4 ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X4 @ C3 ) @ ( abstra2972651231392952684rm_a_b @ R4 ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B4 @ C3 ) @ ( transi7742714808557438673rm_a_b @ R4 ) ) ) ) ) ) ).
% NFP_on_def
thf(fact_970_SN__reaches__NF,axiom,
! [R2: set_Pr3451248702717554689st_nat,X: list_nat] :
( ( abstra2262888740781824790st_nat @ R2 @ ( insert_list_nat @ X @ bot_bot_set_list_nat ) )
=> ? [Y3: list_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X @ Y3 ) @ ( transi5285580207609517981st_nat @ R2 ) )
& ( member_list_nat @ Y3 @ ( abstract_NF_list_nat @ R2 ) ) ) ) ).
% SN_reaches_NF
thf(fact_971_SN__reaches__NF,axiom,
! [R2: set_Pr2972776593051762503rm_a_b,X: produc357393685978478089rm_a_b] :
( ( abstra2398554102055911763rm_a_b @ R2 @ ( insert7009541432154983385rm_a_b @ X @ bot_bo197521221353338581rm_a_b ) )
=> ? [Y3: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y3 ) @ ( transi2615809358984392588rm_a_b @ R2 ) )
& ( member5869715511025134514rm_a_b @ Y3 @ ( abstra1902783451776770197rm_a_b @ R2 ) ) ) ) ).
% SN_reaches_NF
thf(fact_972_SN__reaches__NF,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,X: term_a_b] :
( ( abstra4720023341729745482rm_a_b @ R2 @ ( insert_term_a_b @ X @ bot_bot_set_term_a_b ) )
=> ? [Y3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y3 ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
& ( member_term_a_b @ Y3 @ ( abstract_NF_term_a_b @ R2 ) ) ) ) ).
% SN_reaches_NF
thf(fact_973_the__NF,axiom,
! [A4: set_Pr4386577575007340137rm_a_b,A: term_a_b] :
( ( abstra4720023341729745482rm_a_b @ A4 @ top_top_set_term_a_b )
=> ( ( abstra8448919418672941150rm_a_b @ A4 @ top_top_set_term_a_b )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ ( abstra2242782261322123075rm_a_b @ A4 @ A ) ) @ ( abstra2972651231392952684rm_a_b @ A4 ) ) ) ) ).
% the_NF
thf(fact_974_the__NF__conv,axiom,
! [A4: set_Pr4386577575007340137rm_a_b,A: term_a_b,B: term_a_b] :
( ( abstra4720023341729745482rm_a_b @ A4 @ top_top_set_term_a_b )
=> ( ( abstra8448919418672941150rm_a_b @ A4 @ top_top_set_term_a_b )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( abstra1130568886744591910rm_a_b @ A4 ) )
=> ( ( abstra2242782261322123075rm_a_b @ A4 @ A )
= ( abstra2242782261322123075rm_a_b @ A4 @ B ) ) ) ) ) ).
% the_NF_conv
thf(fact_975_the__NF__step,axiom,
! [A4: set_Pr4386577575007340137rm_a_b,A: term_a_b,B: term_a_b] :
( ( abstra4720023341729745482rm_a_b @ A4 @ top_top_set_term_a_b )
=> ( ( abstra8448919418672941150rm_a_b @ A4 @ top_top_set_term_a_b )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ A4 )
=> ( ( abstra2242782261322123075rm_a_b @ A4 @ A )
= ( abstra2242782261322123075rm_a_b @ A4 @ B ) ) ) ) ) ).
% the_NF_step
thf(fact_976_the__NF__NF,axiom,
! [A4: set_Pr3451248702717554689st_nat,A: list_nat] :
( ( abstra2262888740781824790st_nat @ A4 @ top_top_set_list_nat )
=> ( ( abstra5991784817725020458st_nat @ A4 @ top_top_set_list_nat )
=> ( member_list_nat @ ( abstra9009019697228978191st_nat @ A4 @ A ) @ ( abstract_NF_list_nat @ A4 ) ) ) ) ).
% the_NF_NF
thf(fact_977_the__NF__NF,axiom,
! [A4: set_Pr2972776593051762503rm_a_b,A: produc357393685978478089rm_a_b] :
( ( abstra2398554102055911763rm_a_b @ A4 @ top_to1314267278846557113rm_a_b )
=> ( ( abstra7359694574113289279rm_a_b @ A4 @ top_to1314267278846557113rm_a_b )
=> ( member5869715511025134514rm_a_b @ ( abstra8972549633426078106rm_a_b @ A4 @ A ) @ ( abstra1902783451776770197rm_a_b @ A4 ) ) ) ) ).
% the_NF_NF
thf(fact_978_the__NF__steps,axiom,
! [A4: set_Pr4386577575007340137rm_a_b,A: term_a_b,B: term_a_b] :
( ( abstra4720023341729745482rm_a_b @ A4 @ top_top_set_term_a_b )
=> ( ( abstra8448919418672941150rm_a_b @ A4 @ top_top_set_term_a_b )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ A4 ) )
=> ( ( abstra2242782261322123075rm_a_b @ A4 @ A )
= ( abstra2242782261322123075rm_a_b @ A4 @ B ) ) ) ) ) ).
% the_NF_steps
thf(fact_979_acyclic__insert,axiom,
! [Y: term_a_b,X: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ( transi5314701259734769157rm_a_b @ ( insert7009541432154983385rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X ) @ R2 ) )
= ( ( transi5314701259734769157rm_a_b @ R2 )
& ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ R2 ) ) ) ) ).
% acyclic_insert
thf(fact_980_wo__rel_Ocases__Total3,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,A: term_a_b,B: term_a_b,Phi: term_a_b > term_a_b > $o] :
( ( bNF_We251836928773027112rm_a_b @ R2 )
=> ( ( ord_le2705286416250468010rm_a_b @ ( insert_term_a_b @ A @ ( insert_term_a_b @ B @ bot_bot_set_term_a_b ) ) @ ( field_term_a_b @ R2 ) )
=> ( ( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( minus_5192120951422937424rm_a_b @ R2 @ id_term_a_b ) )
| ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ ( minus_5192120951422937424rm_a_b @ R2 @ id_term_a_b ) ) )
=> ( Phi @ A @ B ) )
=> ( ( ( A = B )
=> ( Phi @ A @ B ) )
=> ( Phi @ A @ B ) ) ) ) ) ).
% wo_rel.cases_Total3
thf(fact_981_member__remove,axiom,
! [X: list_nat,Y: list_nat,A4: set_list_nat] :
( ( member_list_nat @ X @ ( remove_list_nat @ Y @ A4 ) )
= ( ( member_list_nat @ X @ A4 )
& ( X != Y ) ) ) ).
% member_remove
thf(fact_982_member__remove,axiom,
! [X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b,A4: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ X @ ( remove8523906646656265582rm_a_b @ Y @ A4 ) )
= ( ( member5869715511025134514rm_a_b @ X @ A4 )
& ( X != Y ) ) ) ).
% member_remove
thf(fact_983_wo__rel_Oin__notinI,axiom,
! [R2: set_Pr3451248702717554689st_nat,J2: list_nat,I2: list_nat] :
( ( bNF_We7018074364679882228st_nat @ R2 )
=> ( ( ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ J2 @ I2 ) @ R2 )
| ( J2 = I2 ) )
=> ( ( member_list_nat @ I2 @ ( field_list_nat @ R2 ) )
=> ( ( member_list_nat @ J2 @ ( field_list_nat @ R2 ) )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ I2 @ J2 ) @ R2 ) ) ) ) ) ).
% wo_rel.in_notinI
thf(fact_984_wo__rel_Oin__notinI,axiom,
! [R2: set_Pr2972776593051762503rm_a_b,J2: produc357393685978478089rm_a_b,I2: produc357393685978478089rm_a_b] :
( ( bNF_We1688910813892497397rm_a_b @ R2 )
=> ( ( ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ J2 @ I2 ) @ R2 )
| ( J2 = I2 ) )
=> ( ( member5869715511025134514rm_a_b @ I2 @ ( field_6884932134483023318rm_a_b @ R2 ) )
=> ( ( member5869715511025134514rm_a_b @ J2 @ ( field_6884932134483023318rm_a_b @ R2 ) )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ I2 @ J2 ) @ R2 ) ) ) ) ) ).
% wo_rel.in_notinI
thf(fact_985_wo__rel_Oin__notinI,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,J2: term_a_b,I2: term_a_b] :
( ( bNF_We251836928773027112rm_a_b @ R2 )
=> ( ( ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ J2 @ I2 ) @ R2 )
| ( J2 = I2 ) )
=> ( ( member_term_a_b @ I2 @ ( field_term_a_b @ R2 ) )
=> ( ( member_term_a_b @ J2 @ ( field_term_a_b @ R2 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ I2 @ J2 ) @ R2 ) ) ) ) ) ).
% wo_rel.in_notinI
thf(fact_986_wo__rel_OTOTALS,axiom,
! [R2: set_Pr4386577575007340137rm_a_b] :
( ( bNF_We251836928773027112rm_a_b @ R2 )
=> ! [X6: term_a_b] :
( ( member_term_a_b @ X6 @ ( field_term_a_b @ R2 ) )
=> ! [Xa2: term_a_b] :
( ( member_term_a_b @ Xa2 @ ( field_term_a_b @ R2 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X6 @ Xa2 ) @ R2 )
| ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Xa2 @ X6 ) @ R2 ) ) ) ) ) ).
% wo_rel.TOTALS
thf(fact_987_wo__rel_Owell__order__induct,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,P3: term_a_b > $o,A: term_a_b] :
( ( bNF_We251836928773027112rm_a_b @ R2 )
=> ( ! [X3: term_a_b] :
( ! [Y7: term_a_b] :
( ( ( Y7 != X3 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y7 @ X3 ) @ R2 ) )
=> ( P3 @ Y7 ) )
=> ( P3 @ X3 ) )
=> ( P3 @ A ) ) ) ).
% wo_rel.well_order_induct
thf(fact_988_well__order__induct__imp,axiom,
! [R2: set_Pr3451248702717554689st_nat,P3: list_nat > $o,A: list_nat] :
( ( bNF_We7018074364679882228st_nat @ R2 )
=> ( ! [X3: list_nat] :
( ! [Y7: list_nat] :
( ( ( Y7 != X3 )
& ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Y7 @ X3 ) @ R2 ) )
=> ( ( member_list_nat @ Y7 @ ( field_list_nat @ R2 ) )
=> ( P3 @ Y7 ) ) )
=> ( ( member_list_nat @ X3 @ ( field_list_nat @ R2 ) )
=> ( P3 @ X3 ) ) )
=> ( ( member_list_nat @ A @ ( field_list_nat @ R2 ) )
=> ( P3 @ A ) ) ) ) ).
% well_order_induct_imp
thf(fact_989_well__order__induct__imp,axiom,
! [R2: set_Pr2972776593051762503rm_a_b,P3: produc357393685978478089rm_a_b > $o,A: produc357393685978478089rm_a_b] :
( ( bNF_We1688910813892497397rm_a_b @ R2 )
=> ( ! [X3: produc357393685978478089rm_a_b] :
( ! [Y7: produc357393685978478089rm_a_b] :
( ( ( Y7 != X3 )
& ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y7 @ X3 ) @ R2 ) )
=> ( ( member5869715511025134514rm_a_b @ Y7 @ ( field_6884932134483023318rm_a_b @ R2 ) )
=> ( P3 @ Y7 ) ) )
=> ( ( member5869715511025134514rm_a_b @ X3 @ ( field_6884932134483023318rm_a_b @ R2 ) )
=> ( P3 @ X3 ) ) )
=> ( ( member5869715511025134514rm_a_b @ A @ ( field_6884932134483023318rm_a_b @ R2 ) )
=> ( P3 @ A ) ) ) ) ).
% well_order_induct_imp
thf(fact_990_well__order__induct__imp,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,P3: term_a_b > $o,A: term_a_b] :
( ( bNF_We251836928773027112rm_a_b @ R2 )
=> ( ! [X3: term_a_b] :
( ! [Y7: term_a_b] :
( ( ( Y7 != X3 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y7 @ X3 ) @ R2 ) )
=> ( ( member_term_a_b @ Y7 @ ( field_term_a_b @ R2 ) )
=> ( P3 @ Y7 ) ) )
=> ( ( member_term_a_b @ X3 @ ( field_term_a_b @ R2 ) )
=> ( P3 @ X3 ) ) )
=> ( ( member_term_a_b @ A @ ( field_term_a_b @ R2 ) )
=> ( P3 @ A ) ) ) ) ).
% well_order_induct_imp
thf(fact_991_acyclicI,axiom,
! [R2: set_Pr4386577575007340137rm_a_b] :
( ! [X3: term_a_b] :
~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ X3 ) @ ( transi7922773638565587891rm_a_b @ R2 ) )
=> ( transi5314701259734769157rm_a_b @ R2 ) ) ).
% acyclicI
thf(fact_992_acyclic__def,axiom,
( transi5314701259734769157rm_a_b
= ( ^ [R4: set_Pr4386577575007340137rm_a_b] :
! [X4: term_a_b] :
~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X4 @ X4 ) @ ( transi7922773638565587891rm_a_b @ R4 ) ) ) ) ).
% acyclic_def
thf(fact_993_wo__rel_Ocases__Total,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,A: term_a_b,B: term_a_b,Phi: term_a_b > term_a_b > $o] :
( ( bNF_We251836928773027112rm_a_b @ R2 )
=> ( ( ord_le2705286416250468010rm_a_b @ ( insert_term_a_b @ A @ ( insert_term_a_b @ B @ bot_bot_set_term_a_b ) ) @ ( field_term_a_b @ R2 ) )
=> ( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R2 )
=> ( Phi @ A @ B ) )
=> ( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ R2 )
=> ( Phi @ A @ B ) )
=> ( Phi @ A @ B ) ) ) ) ) ).
% wo_rel.cases_Total
thf(fact_994_wo__rel_Ominim__in,axiom,
! [R2: set_Pr3451248702717554689st_nat,B5: set_list_nat] :
( ( bNF_We7018074364679882228st_nat @ R2 )
=> ( ( ord_le6045566169113846134st_nat @ B5 @ ( field_list_nat @ R2 ) )
=> ( ( B5 != bot_bot_set_list_nat )
=> ( member_list_nat @ ( bNF_We3801768462575224852st_nat @ R2 @ B5 ) @ B5 ) ) ) ) ).
% wo_rel.minim_in
thf(fact_995_wo__rel_Ominim__in,axiom,
! [R2: set_Pr2972776593051762503rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
( ( bNF_We1688910813892497397rm_a_b @ R2 )
=> ( ( ord_le118470702582115849rm_a_b @ B5 @ ( field_6884932134483023318rm_a_b @ R2 ) )
=> ( ( B5 != bot_bo197521221353338581rm_a_b )
=> ( member5869715511025134514rm_a_b @ ( bNF_We1338740440619677141rm_a_b @ R2 @ B5 ) @ B5 ) ) ) ) ).
% wo_rel.minim_in
thf(fact_996_wo__rel_Ominim__inField,axiom,
! [R2: set_Pr3451248702717554689st_nat,B5: set_list_nat] :
( ( bNF_We7018074364679882228st_nat @ R2 )
=> ( ( ord_le6045566169113846134st_nat @ B5 @ ( field_list_nat @ R2 ) )
=> ( ( B5 != bot_bot_set_list_nat )
=> ( member_list_nat @ ( bNF_We3801768462575224852st_nat @ R2 @ B5 ) @ ( field_list_nat @ R2 ) ) ) ) ) ).
% wo_rel.minim_inField
thf(fact_997_wo__rel_Ominim__inField,axiom,
! [R2: set_Pr2972776593051762503rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
( ( bNF_We1688910813892497397rm_a_b @ R2 )
=> ( ( ord_le118470702582115849rm_a_b @ B5 @ ( field_6884932134483023318rm_a_b @ R2 ) )
=> ( ( B5 != bot_bo197521221353338581rm_a_b )
=> ( member5869715511025134514rm_a_b @ ( bNF_We1338740440619677141rm_a_b @ R2 @ B5 ) @ ( field_6884932134483023318rm_a_b @ R2 ) ) ) ) ) ).
% wo_rel.minim_inField
thf(fact_998_wo__rel_OisMinim__def,axiom,
! [R2: set_Pr3451248702717554689st_nat,A4: set_list_nat,B: list_nat] :
( ( bNF_We7018074364679882228st_nat @ R2 )
=> ( ( bNF_We7461629510496424106st_nat @ R2 @ A4 @ B )
= ( ( member_list_nat @ B @ A4 )
& ! [X4: list_nat] :
( ( member_list_nat @ X4 @ A4 )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ B @ X4 ) @ R2 ) ) ) ) ) ).
% wo_rel.isMinim_def
thf(fact_999_wo__rel_OisMinim__def,axiom,
! [R2: set_Pr2972776593051762503rm_a_b,A4: set_Pr4386577575007340137rm_a_b,B: produc357393685978478089rm_a_b] :
( ( bNF_We1688910813892497397rm_a_b @ R2 )
=> ( ( bNF_We8430659725849320639rm_a_b @ R2 @ A4 @ B )
= ( ( member5869715511025134514rm_a_b @ B @ A4 )
& ! [X4: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X4 @ A4 )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B @ X4 ) @ R2 ) ) ) ) ) ).
% wo_rel.isMinim_def
thf(fact_1000_wo__rel_OisMinim__def,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,A4: set_term_a_b,B: term_a_b] :
( ( bNF_We251836928773027112rm_a_b @ R2 )
=> ( ( bNF_We695392074589568990rm_a_b @ R2 @ A4 @ B )
= ( ( member_term_a_b @ B @ A4 )
& ! [X4: term_a_b] :
( ( member_term_a_b @ X4 @ A4 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ X4 ) @ R2 ) ) ) ) ) ).
% wo_rel.isMinim_def
thf(fact_1001_wo__rel_Oequals__minim,axiom,
! [R2: set_Pr3451248702717554689st_nat,B5: set_list_nat,A: list_nat] :
( ( bNF_We7018074364679882228st_nat @ R2 )
=> ( ( ord_le6045566169113846134st_nat @ B5 @ ( field_list_nat @ R2 ) )
=> ( ( member_list_nat @ A @ B5 )
=> ( ! [B3: list_nat] :
( ( member_list_nat @ B3 @ B5 )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A @ B3 ) @ R2 ) )
=> ( A
= ( bNF_We3801768462575224852st_nat @ R2 @ B5 ) ) ) ) ) ) ).
% wo_rel.equals_minim
thf(fact_1002_wo__rel_Oequals__minim,axiom,
! [R2: set_Pr2972776593051762503rm_a_b,B5: set_Pr4386577575007340137rm_a_b,A: produc357393685978478089rm_a_b] :
( ( bNF_We1688910813892497397rm_a_b @ R2 )
=> ( ( ord_le118470702582115849rm_a_b @ B5 @ ( field_6884932134483023318rm_a_b @ R2 ) )
=> ( ( member5869715511025134514rm_a_b @ A @ B5 )
=> ( ! [B3: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ B3 @ B5 )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B3 ) @ R2 ) )
=> ( A
= ( bNF_We1338740440619677141rm_a_b @ R2 @ B5 ) ) ) ) ) ) ).
% wo_rel.equals_minim
thf(fact_1003_wo__rel_Oequals__minim,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,B5: set_term_a_b,A: term_a_b] :
( ( bNF_We251836928773027112rm_a_b @ R2 )
=> ( ( ord_le2705286416250468010rm_a_b @ B5 @ ( field_term_a_b @ R2 ) )
=> ( ( member_term_a_b @ A @ B5 )
=> ( ! [B3: term_a_b] :
( ( member_term_a_b @ B3 @ B5 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B3 ) @ R2 ) )
=> ( A
= ( bNF_We6258903063523145544rm_a_b @ R2 @ B5 ) ) ) ) ) ) ).
% wo_rel.equals_minim
thf(fact_1004_wo__rel_Ominim__least,axiom,
! [R2: set_Pr3451248702717554689st_nat,B5: set_list_nat,B: list_nat] :
( ( bNF_We7018074364679882228st_nat @ R2 )
=> ( ( ord_le6045566169113846134st_nat @ B5 @ ( field_list_nat @ R2 ) )
=> ( ( member_list_nat @ B @ B5 )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( bNF_We3801768462575224852st_nat @ R2 @ B5 ) @ B ) @ R2 ) ) ) ) ).
% wo_rel.minim_least
thf(fact_1005_wo__rel_Ominim__least,axiom,
! [R2: set_Pr2972776593051762503rm_a_b,B5: set_Pr4386577575007340137rm_a_b,B: produc357393685978478089rm_a_b] :
( ( bNF_We1688910813892497397rm_a_b @ R2 )
=> ( ( ord_le118470702582115849rm_a_b @ B5 @ ( field_6884932134483023318rm_a_b @ R2 ) )
=> ( ( member5869715511025134514rm_a_b @ B @ B5 )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( bNF_We1338740440619677141rm_a_b @ R2 @ B5 ) @ B ) @ R2 ) ) ) ) ).
% wo_rel.minim_least
thf(fact_1006_wo__rel_Ominim__least,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,B5: set_term_a_b,B: term_a_b] :
( ( bNF_We251836928773027112rm_a_b @ R2 )
=> ( ( ord_le2705286416250468010rm_a_b @ B5 @ ( field_term_a_b @ R2 ) )
=> ( ( member_term_a_b @ B @ B5 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( bNF_We6258903063523145544rm_a_b @ R2 @ B5 ) @ B ) @ R2 ) ) ) ) ).
% wo_rel.minim_least
thf(fact_1007_wo__rel_Omax2__greater__among,axiom,
! [R2: set_Pr3451248702717554689st_nat,A: list_nat,B: list_nat] :
( ( bNF_We7018074364679882228st_nat @ R2 )
=> ( ( member_list_nat @ A @ ( field_list_nat @ R2 ) )
=> ( ( member_list_nat @ B @ ( field_list_nat @ R2 ) )
=> ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A @ ( bNF_We2161624777392673042st_nat @ R2 @ A @ B ) ) @ R2 )
& ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ B @ ( bNF_We2161624777392673042st_nat @ R2 @ A @ B ) ) @ R2 )
& ( member_list_nat @ ( bNF_We2161624777392673042st_nat @ R2 @ A @ B ) @ ( insert_list_nat @ A @ ( insert_list_nat @ B @ bot_bot_set_list_nat ) ) ) ) ) ) ) ).
% wo_rel.max2_greater_among
thf(fact_1008_wo__rel_Omax2__greater__among,axiom,
! [R2: set_Pr2972776593051762503rm_a_b,A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b] :
( ( bNF_We1688910813892497397rm_a_b @ R2 )
=> ( ( member5869715511025134514rm_a_b @ A @ ( field_6884932134483023318rm_a_b @ R2 ) )
=> ( ( member5869715511025134514rm_a_b @ B @ ( field_6884932134483023318rm_a_b @ R2 ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ ( bNF_We1369222903935127127rm_a_b @ R2 @ A @ B ) ) @ R2 )
& ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B @ ( bNF_We1369222903935127127rm_a_b @ R2 @ A @ B ) ) @ R2 )
& ( member5869715511025134514rm_a_b @ ( bNF_We1369222903935127127rm_a_b @ R2 @ A @ B ) @ ( insert7009541432154983385rm_a_b @ A @ ( insert7009541432154983385rm_a_b @ B @ bot_bo197521221353338581rm_a_b ) ) ) ) ) ) ) ).
% wo_rel.max2_greater_among
thf(fact_1009_wo__rel_Omax2__greater__among,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,A: term_a_b,B: term_a_b] :
( ( bNF_We251836928773027112rm_a_b @ R2 )
=> ( ( member_term_a_b @ A @ ( field_term_a_b @ R2 ) )
=> ( ( member_term_a_b @ B @ ( field_term_a_b @ R2 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ ( bNF_We4618759378340593734rm_a_b @ R2 @ A @ B ) ) @ R2 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ ( bNF_We4618759378340593734rm_a_b @ R2 @ A @ B ) ) @ R2 )
& ( member_term_a_b @ ( bNF_We4618759378340593734rm_a_b @ R2 @ A @ B ) @ ( insert_term_a_b @ A @ ( insert_term_a_b @ B @ bot_bot_set_term_a_b ) ) ) ) ) ) ) ).
% wo_rel.max2_greater_among
thf(fact_1010_wo__rel_Omax2__among,axiom,
! [R2: set_Pr3451248702717554689st_nat,A: list_nat,B: list_nat] :
( ( bNF_We7018074364679882228st_nat @ R2 )
=> ( ( member_list_nat @ A @ ( field_list_nat @ R2 ) )
=> ( ( member_list_nat @ B @ ( field_list_nat @ R2 ) )
=> ( member_list_nat @ ( bNF_We2161624777392673042st_nat @ R2 @ A @ B ) @ ( insert_list_nat @ A @ ( insert_list_nat @ B @ bot_bot_set_list_nat ) ) ) ) ) ) ).
% wo_rel.max2_among
thf(fact_1011_wo__rel_Omax2__among,axiom,
! [R2: set_Pr2972776593051762503rm_a_b,A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b] :
( ( bNF_We1688910813892497397rm_a_b @ R2 )
=> ( ( member5869715511025134514rm_a_b @ A @ ( field_6884932134483023318rm_a_b @ R2 ) )
=> ( ( member5869715511025134514rm_a_b @ B @ ( field_6884932134483023318rm_a_b @ R2 ) )
=> ( member5869715511025134514rm_a_b @ ( bNF_We1369222903935127127rm_a_b @ R2 @ A @ B ) @ ( insert7009541432154983385rm_a_b @ A @ ( insert7009541432154983385rm_a_b @ B @ bot_bo197521221353338581rm_a_b ) ) ) ) ) ) ).
% wo_rel.max2_among
thf(fact_1012_refl__on__singleton,axiom,
! [X: term_a_b] : ( refl_on_term_a_b @ ( insert_term_a_b @ X @ bot_bot_set_term_a_b ) @ ( insert7009541432154983385rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ X ) @ bot_bo197521221353338581rm_a_b ) ) ).
% refl_on_singleton
thf(fact_1013_refl__onD2,axiom,
! [A4: set_list_nat,R2: set_Pr3451248702717554689st_nat,X: list_nat,Y: list_nat] :
( ( refl_on_list_nat @ A4 @ R2 )
=> ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X @ Y ) @ R2 )
=> ( member_list_nat @ Y @ A4 ) ) ) ).
% refl_onD2
thf(fact_1014_refl__onD2,axiom,
! [A4: set_Pr4386577575007340137rm_a_b,R2: set_Pr2972776593051762503rm_a_b,X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b] :
( ( refl_o5344404488616963464rm_a_b @ A4 @ R2 )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ R2 )
=> ( member5869715511025134514rm_a_b @ Y @ A4 ) ) ) ).
% refl_onD2
thf(fact_1015_refl__onD2,axiom,
! [A4: set_term_a_b,R2: set_Pr4386577575007340137rm_a_b,X: term_a_b,Y: term_a_b] :
( ( refl_on_term_a_b @ A4 @ R2 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ R2 )
=> ( member_term_a_b @ Y @ A4 ) ) ) ).
% refl_onD2
thf(fact_1016_refl__onD1,axiom,
! [A4: set_list_nat,R2: set_Pr3451248702717554689st_nat,X: list_nat,Y: list_nat] :
( ( refl_on_list_nat @ A4 @ R2 )
=> ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X @ Y ) @ R2 )
=> ( member_list_nat @ X @ A4 ) ) ) ).
% refl_onD1
thf(fact_1017_refl__onD1,axiom,
! [A4: set_Pr4386577575007340137rm_a_b,R2: set_Pr2972776593051762503rm_a_b,X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b] :
( ( refl_o5344404488616963464rm_a_b @ A4 @ R2 )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ R2 )
=> ( member5869715511025134514rm_a_b @ X @ A4 ) ) ) ).
% refl_onD1
thf(fact_1018_refl__onD1,axiom,
! [A4: set_term_a_b,R2: set_Pr4386577575007340137rm_a_b,X: term_a_b,Y: term_a_b] :
( ( refl_on_term_a_b @ A4 @ R2 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ R2 )
=> ( member_term_a_b @ X @ A4 ) ) ) ).
% refl_onD1
thf(fact_1019_refl__onD,axiom,
! [A4: set_list_nat,R2: set_Pr3451248702717554689st_nat,A: list_nat] :
( ( refl_on_list_nat @ A4 @ R2 )
=> ( ( member_list_nat @ A @ A4 )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A @ A ) @ R2 ) ) ) ).
% refl_onD
thf(fact_1020_refl__onD,axiom,
! [A4: set_Pr4386577575007340137rm_a_b,R2: set_Pr2972776593051762503rm_a_b,A: produc357393685978478089rm_a_b] :
( ( refl_o5344404488616963464rm_a_b @ A4 @ R2 )
=> ( ( member5869715511025134514rm_a_b @ A @ A4 )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ A ) @ R2 ) ) ) ).
% refl_onD
thf(fact_1021_refl__onD,axiom,
! [A4: set_term_a_b,R2: set_Pr4386577575007340137rm_a_b,A: term_a_b] :
( ( refl_on_term_a_b @ A4 @ R2 )
=> ( ( member_term_a_b @ A @ A4 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ A ) @ R2 ) ) ) ).
% refl_onD
thf(fact_1022_reflD,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,A: term_a_b] :
( ( refl_on_term_a_b @ top_top_set_term_a_b @ R2 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ A ) @ R2 ) ) ).
% reflD
thf(fact_1023_reflI,axiom,
! [R2: set_Pr4386577575007340137rm_a_b] :
( ! [X3: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ X3 ) @ R2 )
=> ( refl_on_term_a_b @ top_top_set_term_a_b @ R2 ) ) ).
% reflI
thf(fact_1024_wo__rel_Omax2__def,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,A: term_a_b,B: term_a_b] :
( ( bNF_We251836928773027112rm_a_b @ R2 )
=> ( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R2 )
=> ( ( bNF_We4618759378340593734rm_a_b @ R2 @ A @ B )
= B ) )
& ( ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R2 )
=> ( ( bNF_We4618759378340593734rm_a_b @ R2 @ A @ B )
= A ) ) ) ) ).
% wo_rel.max2_def
thf(fact_1025_wo__rel_Omax2__greater,axiom,
! [R2: set_Pr3451248702717554689st_nat,A: list_nat,B: list_nat] :
( ( bNF_We7018074364679882228st_nat @ R2 )
=> ( ( member_list_nat @ A @ ( field_list_nat @ R2 ) )
=> ( ( member_list_nat @ B @ ( field_list_nat @ R2 ) )
=> ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A @ ( bNF_We2161624777392673042st_nat @ R2 @ A @ B ) ) @ R2 )
& ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ B @ ( bNF_We2161624777392673042st_nat @ R2 @ A @ B ) ) @ R2 ) ) ) ) ) ).
% wo_rel.max2_greater
thf(fact_1026_wo__rel_Omax2__greater,axiom,
! [R2: set_Pr2972776593051762503rm_a_b,A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b] :
( ( bNF_We1688910813892497397rm_a_b @ R2 )
=> ( ( member5869715511025134514rm_a_b @ A @ ( field_6884932134483023318rm_a_b @ R2 ) )
=> ( ( member5869715511025134514rm_a_b @ B @ ( field_6884932134483023318rm_a_b @ R2 ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ ( bNF_We1369222903935127127rm_a_b @ R2 @ A @ B ) ) @ R2 )
& ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B @ ( bNF_We1369222903935127127rm_a_b @ R2 @ A @ B ) ) @ R2 ) ) ) ) ) ).
% wo_rel.max2_greater
thf(fact_1027_wo__rel_Omax2__greater,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,A: term_a_b,B: term_a_b] :
( ( bNF_We251836928773027112rm_a_b @ R2 )
=> ( ( member_term_a_b @ A @ ( field_term_a_b @ R2 ) )
=> ( ( member_term_a_b @ B @ ( field_term_a_b @ R2 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ ( bNF_We4618759378340593734rm_a_b @ R2 @ A @ B ) ) @ R2 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ ( bNF_We4618759378340593734rm_a_b @ R2 @ A @ B ) ) @ R2 ) ) ) ) ) ).
% wo_rel.max2_greater
thf(fact_1028_wo__rel_Omax2__equals2,axiom,
! [R2: set_Pr3451248702717554689st_nat,A: list_nat,B: list_nat] :
( ( bNF_We7018074364679882228st_nat @ R2 )
=> ( ( member_list_nat @ A @ ( field_list_nat @ R2 ) )
=> ( ( member_list_nat @ B @ ( field_list_nat @ R2 ) )
=> ( ( ( bNF_We2161624777392673042st_nat @ R2 @ A @ B )
= B )
= ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A @ B ) @ R2 ) ) ) ) ) ).
% wo_rel.max2_equals2
thf(fact_1029_wo__rel_Omax2__equals2,axiom,
! [R2: set_Pr2972776593051762503rm_a_b,A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b] :
( ( bNF_We1688910813892497397rm_a_b @ R2 )
=> ( ( member5869715511025134514rm_a_b @ A @ ( field_6884932134483023318rm_a_b @ R2 ) )
=> ( ( member5869715511025134514rm_a_b @ B @ ( field_6884932134483023318rm_a_b @ R2 ) )
=> ( ( ( bNF_We1369222903935127127rm_a_b @ R2 @ A @ B )
= B )
= ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ R2 ) ) ) ) ) ).
% wo_rel.max2_equals2
thf(fact_1030_wo__rel_Omax2__equals2,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,A: term_a_b,B: term_a_b] :
( ( bNF_We251836928773027112rm_a_b @ R2 )
=> ( ( member_term_a_b @ A @ ( field_term_a_b @ R2 ) )
=> ( ( member_term_a_b @ B @ ( field_term_a_b @ R2 ) )
=> ( ( ( bNF_We4618759378340593734rm_a_b @ R2 @ A @ B )
= B )
= ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R2 ) ) ) ) ) ).
% wo_rel.max2_equals2
thf(fact_1031_wo__rel_Omax2__equals1,axiom,
! [R2: set_Pr3451248702717554689st_nat,A: list_nat,B: list_nat] :
( ( bNF_We7018074364679882228st_nat @ R2 )
=> ( ( member_list_nat @ A @ ( field_list_nat @ R2 ) )
=> ( ( member_list_nat @ B @ ( field_list_nat @ R2 ) )
=> ( ( ( bNF_We2161624777392673042st_nat @ R2 @ A @ B )
= A )
= ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ B @ A ) @ R2 ) ) ) ) ) ).
% wo_rel.max2_equals1
thf(fact_1032_wo__rel_Omax2__equals1,axiom,
! [R2: set_Pr2972776593051762503rm_a_b,A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b] :
( ( bNF_We1688910813892497397rm_a_b @ R2 )
=> ( ( member5869715511025134514rm_a_b @ A @ ( field_6884932134483023318rm_a_b @ R2 ) )
=> ( ( member5869715511025134514rm_a_b @ B @ ( field_6884932134483023318rm_a_b @ R2 ) )
=> ( ( ( bNF_We1369222903935127127rm_a_b @ R2 @ A @ B )
= A )
= ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B @ A ) @ R2 ) ) ) ) ) ).
% wo_rel.max2_equals1
thf(fact_1033_wo__rel_Omax2__equals1,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,A: term_a_b,B: term_a_b] :
( ( bNF_We251836928773027112rm_a_b @ R2 )
=> ( ( member_term_a_b @ A @ ( field_term_a_b @ R2 ) )
=> ( ( member_term_a_b @ B @ ( field_term_a_b @ R2 ) )
=> ( ( ( bNF_We4618759378340593734rm_a_b @ R2 @ A @ B )
= A )
= ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ R2 ) ) ) ) ) ).
% wo_rel.max2_equals1
thf(fact_1034_refl__on__domain,axiom,
! [A4: set_list_nat,R2: set_Pr3451248702717554689st_nat,A: list_nat,B: list_nat] :
( ( refl_on_list_nat @ A4 @ R2 )
=> ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A @ B ) @ R2 )
=> ( ( member_list_nat @ A @ A4 )
& ( member_list_nat @ B @ A4 ) ) ) ) ).
% refl_on_domain
thf(fact_1035_refl__on__domain,axiom,
! [A4: set_Pr4386577575007340137rm_a_b,R2: set_Pr2972776593051762503rm_a_b,A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b] :
( ( refl_o5344404488616963464rm_a_b @ A4 @ R2 )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ R2 )
=> ( ( member5869715511025134514rm_a_b @ A @ A4 )
& ( member5869715511025134514rm_a_b @ B @ A4 ) ) ) ) ).
% refl_on_domain
thf(fact_1036_refl__on__domain,axiom,
! [A4: set_term_a_b,R2: set_Pr4386577575007340137rm_a_b,A: term_a_b,B: term_a_b] :
( ( refl_on_term_a_b @ A4 @ R2 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R2 )
=> ( ( member_term_a_b @ A @ A4 )
& ( member_term_a_b @ B @ A4 ) ) ) ) ).
% refl_on_domain
thf(fact_1037_irrefl__tranclI,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,X: term_a_b] :
( ( ( inf_in1693821330460682071rm_a_b @ ( conver7608559976937845093rm_a_b @ R2 ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
= bot_bo197521221353338581rm_a_b )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ X ) @ ( transi7922773638565587891rm_a_b @ R2 ) ) ) ).
% irrefl_tranclI
thf(fact_1038_Total__subset__Id,axiom,
! [R2: set_Pr4386577575007340137rm_a_b] :
( ( total_on_term_a_b @ ( field_term_a_b @ R2 ) @ R2 )
=> ( ( ord_le118470702582115849rm_a_b @ R2 @ id_term_a_b )
=> ( ( R2 = bot_bo197521221353338581rm_a_b )
| ? [A3: term_a_b] :
( R2
= ( insert7009541432154983385rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ A3 ) @ bot_bo197521221353338581rm_a_b ) ) ) ) ) ).
% Total_subset_Id
thf(fact_1039_IntI,axiom,
! [C2: list_nat,A4: set_list_nat,B5: set_list_nat] :
( ( member_list_nat @ C2 @ A4 )
=> ( ( member_list_nat @ C2 @ B5 )
=> ( member_list_nat @ C2 @ ( inf_inf_set_list_nat @ A4 @ B5 ) ) ) ) ).
% IntI
thf(fact_1040_IntI,axiom,
! [C2: produc357393685978478089rm_a_b,A4: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ C2 @ A4 )
=> ( ( member5869715511025134514rm_a_b @ C2 @ B5 )
=> ( member5869715511025134514rm_a_b @ C2 @ ( inf_in1693821330460682071rm_a_b @ A4 @ B5 ) ) ) ) ).
% IntI
thf(fact_1041_Int__iff,axiom,
! [C2: list_nat,A4: set_list_nat,B5: set_list_nat] :
( ( member_list_nat @ C2 @ ( inf_inf_set_list_nat @ A4 @ B5 ) )
= ( ( member_list_nat @ C2 @ A4 )
& ( member_list_nat @ C2 @ B5 ) ) ) ).
% Int_iff
thf(fact_1042_Int__iff,axiom,
! [C2: produc357393685978478089rm_a_b,A4: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ C2 @ ( inf_in1693821330460682071rm_a_b @ A4 @ B5 ) )
= ( ( member5869715511025134514rm_a_b @ C2 @ A4 )
& ( member5869715511025134514rm_a_b @ C2 @ B5 ) ) ) ).
% Int_iff
thf(fact_1043_Int__insert__right__if1,axiom,
! [A: list_nat,A4: set_list_nat,B5: set_list_nat] :
( ( member_list_nat @ A @ A4 )
=> ( ( inf_inf_set_list_nat @ A4 @ ( insert_list_nat @ A @ B5 ) )
= ( insert_list_nat @ A @ ( inf_inf_set_list_nat @ A4 @ B5 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_1044_Int__insert__right__if1,axiom,
! [A: produc357393685978478089rm_a_b,A4: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ A @ A4 )
=> ( ( inf_in1693821330460682071rm_a_b @ A4 @ ( insert7009541432154983385rm_a_b @ A @ B5 ) )
= ( insert7009541432154983385rm_a_b @ A @ ( inf_in1693821330460682071rm_a_b @ A4 @ B5 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_1045_Int__insert__right__if0,axiom,
! [A: list_nat,A4: set_list_nat,B5: set_list_nat] :
( ~ ( member_list_nat @ A @ A4 )
=> ( ( inf_inf_set_list_nat @ A4 @ ( insert_list_nat @ A @ B5 ) )
= ( inf_inf_set_list_nat @ A4 @ B5 ) ) ) ).
% Int_insert_right_if0
thf(fact_1046_Int__insert__right__if0,axiom,
! [A: produc357393685978478089rm_a_b,A4: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
( ~ ( member5869715511025134514rm_a_b @ A @ A4 )
=> ( ( inf_in1693821330460682071rm_a_b @ A4 @ ( insert7009541432154983385rm_a_b @ A @ B5 ) )
= ( inf_in1693821330460682071rm_a_b @ A4 @ B5 ) ) ) ).
% Int_insert_right_if0
thf(fact_1047_Int__insert__left__if1,axiom,
! [A: list_nat,C4: set_list_nat,B5: set_list_nat] :
( ( member_list_nat @ A @ C4 )
=> ( ( inf_inf_set_list_nat @ ( insert_list_nat @ A @ B5 ) @ C4 )
= ( insert_list_nat @ A @ ( inf_inf_set_list_nat @ B5 @ C4 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_1048_Int__insert__left__if1,axiom,
! [A: produc357393685978478089rm_a_b,C4: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ A @ C4 )
=> ( ( inf_in1693821330460682071rm_a_b @ ( insert7009541432154983385rm_a_b @ A @ B5 ) @ C4 )
= ( insert7009541432154983385rm_a_b @ A @ ( inf_in1693821330460682071rm_a_b @ B5 @ C4 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_1049_Int__insert__left__if0,axiom,
! [A: list_nat,C4: set_list_nat,B5: set_list_nat] :
( ~ ( member_list_nat @ A @ C4 )
=> ( ( inf_inf_set_list_nat @ ( insert_list_nat @ A @ B5 ) @ C4 )
= ( inf_inf_set_list_nat @ B5 @ C4 ) ) ) ).
% Int_insert_left_if0
thf(fact_1050_Int__insert__left__if0,axiom,
! [A: produc357393685978478089rm_a_b,C4: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
( ~ ( member5869715511025134514rm_a_b @ A @ C4 )
=> ( ( inf_in1693821330460682071rm_a_b @ ( insert7009541432154983385rm_a_b @ A @ B5 ) @ C4 )
= ( inf_in1693821330460682071rm_a_b @ B5 @ C4 ) ) ) ).
% Int_insert_left_if0
thf(fact_1051_insert__disjoint_I1_J,axiom,
! [A: list_nat,A4: set_list_nat,B5: set_list_nat] :
( ( ( inf_inf_set_list_nat @ ( insert_list_nat @ A @ A4 ) @ B5 )
= bot_bot_set_list_nat )
= ( ~ ( member_list_nat @ A @ B5 )
& ( ( inf_inf_set_list_nat @ A4 @ B5 )
= bot_bot_set_list_nat ) ) ) ).
% insert_disjoint(1)
thf(fact_1052_insert__disjoint_I1_J,axiom,
! [A: produc357393685978478089rm_a_b,A4: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
( ( ( inf_in1693821330460682071rm_a_b @ ( insert7009541432154983385rm_a_b @ A @ A4 ) @ B5 )
= bot_bo197521221353338581rm_a_b )
= ( ~ ( member5869715511025134514rm_a_b @ A @ B5 )
& ( ( inf_in1693821330460682071rm_a_b @ A4 @ B5 )
= bot_bo197521221353338581rm_a_b ) ) ) ).
% insert_disjoint(1)
thf(fact_1053_insert__disjoint_I2_J,axiom,
! [A: list_nat,A4: set_list_nat,B5: set_list_nat] :
( ( bot_bot_set_list_nat
= ( inf_inf_set_list_nat @ ( insert_list_nat @ A @ A4 ) @ B5 ) )
= ( ~ ( member_list_nat @ A @ B5 )
& ( bot_bot_set_list_nat
= ( inf_inf_set_list_nat @ A4 @ B5 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_1054_insert__disjoint_I2_J,axiom,
! [A: produc357393685978478089rm_a_b,A4: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
( ( bot_bo197521221353338581rm_a_b
= ( inf_in1693821330460682071rm_a_b @ ( insert7009541432154983385rm_a_b @ A @ A4 ) @ B5 ) )
= ( ~ ( member5869715511025134514rm_a_b @ A @ B5 )
& ( bot_bo197521221353338581rm_a_b
= ( inf_in1693821330460682071rm_a_b @ A4 @ B5 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_1055_disjoint__insert_I1_J,axiom,
! [B5: set_list_nat,A: list_nat,A4: set_list_nat] :
( ( ( inf_inf_set_list_nat @ B5 @ ( insert_list_nat @ A @ A4 ) )
= bot_bot_set_list_nat )
= ( ~ ( member_list_nat @ A @ B5 )
& ( ( inf_inf_set_list_nat @ B5 @ A4 )
= bot_bot_set_list_nat ) ) ) ).
% disjoint_insert(1)
thf(fact_1056_disjoint__insert_I1_J,axiom,
! [B5: set_Pr4386577575007340137rm_a_b,A: produc357393685978478089rm_a_b,A4: set_Pr4386577575007340137rm_a_b] :
( ( ( inf_in1693821330460682071rm_a_b @ B5 @ ( insert7009541432154983385rm_a_b @ A @ A4 ) )
= bot_bo197521221353338581rm_a_b )
= ( ~ ( member5869715511025134514rm_a_b @ A @ B5 )
& ( ( inf_in1693821330460682071rm_a_b @ B5 @ A4 )
= bot_bo197521221353338581rm_a_b ) ) ) ).
% disjoint_insert(1)
thf(fact_1057_disjoint__insert_I2_J,axiom,
! [A4: set_list_nat,B: list_nat,B5: set_list_nat] :
( ( bot_bot_set_list_nat
= ( inf_inf_set_list_nat @ A4 @ ( insert_list_nat @ B @ B5 ) ) )
= ( ~ ( member_list_nat @ B @ A4 )
& ( bot_bot_set_list_nat
= ( inf_inf_set_list_nat @ A4 @ B5 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_1058_disjoint__insert_I2_J,axiom,
! [A4: set_Pr4386577575007340137rm_a_b,B: produc357393685978478089rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
( ( bot_bo197521221353338581rm_a_b
= ( inf_in1693821330460682071rm_a_b @ A4 @ ( insert7009541432154983385rm_a_b @ B @ B5 ) ) )
= ( ~ ( member5869715511025134514rm_a_b @ B @ A4 )
& ( bot_bo197521221353338581rm_a_b
= ( inf_in1693821330460682071rm_a_b @ A4 @ B5 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_1059_Int__Collect__mono,axiom,
! [A4: set_list_nat,B5: set_list_nat,P3: list_nat > $o,Q2: list_nat > $o] :
( ( ord_le6045566169113846134st_nat @ A4 @ B5 )
=> ( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ A4 )
=> ( ( P3 @ X3 )
=> ( Q2 @ X3 ) ) )
=> ( ord_le6045566169113846134st_nat @ ( inf_inf_set_list_nat @ A4 @ ( collect_list_nat @ P3 ) ) @ ( inf_inf_set_list_nat @ B5 @ ( collect_list_nat @ Q2 ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_1060_Int__Collect__mono,axiom,
! [A4: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b,P3: produc357393685978478089rm_a_b > $o,Q2: produc357393685978478089rm_a_b > $o] :
( ( ord_le118470702582115849rm_a_b @ A4 @ B5 )
=> ( ! [X3: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X3 @ A4 )
=> ( ( P3 @ X3 )
=> ( Q2 @ X3 ) ) )
=> ( ord_le118470702582115849rm_a_b @ ( inf_in1693821330460682071rm_a_b @ A4 @ ( collec99177395925924084rm_a_b @ P3 ) ) @ ( inf_in1693821330460682071rm_a_b @ B5 @ ( collec99177395925924084rm_a_b @ Q2 ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_1061_IntE,axiom,
! [C2: list_nat,A4: set_list_nat,B5: set_list_nat] :
( ( member_list_nat @ C2 @ ( inf_inf_set_list_nat @ A4 @ B5 ) )
=> ~ ( ( member_list_nat @ C2 @ A4 )
=> ~ ( member_list_nat @ C2 @ B5 ) ) ) ).
% IntE
thf(fact_1062_IntE,axiom,
! [C2: produc357393685978478089rm_a_b,A4: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ C2 @ ( inf_in1693821330460682071rm_a_b @ A4 @ B5 ) )
=> ~ ( ( member5869715511025134514rm_a_b @ C2 @ A4 )
=> ~ ( member5869715511025134514rm_a_b @ C2 @ B5 ) ) ) ).
% IntE
thf(fact_1063_IntD1,axiom,
! [C2: list_nat,A4: set_list_nat,B5: set_list_nat] :
( ( member_list_nat @ C2 @ ( inf_inf_set_list_nat @ A4 @ B5 ) )
=> ( member_list_nat @ C2 @ A4 ) ) ).
% IntD1
thf(fact_1064_IntD1,axiom,
! [C2: produc357393685978478089rm_a_b,A4: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ C2 @ ( inf_in1693821330460682071rm_a_b @ A4 @ B5 ) )
=> ( member5869715511025134514rm_a_b @ C2 @ A4 ) ) ).
% IntD1
thf(fact_1065_IntD2,axiom,
! [C2: list_nat,A4: set_list_nat,B5: set_list_nat] :
( ( member_list_nat @ C2 @ ( inf_inf_set_list_nat @ A4 @ B5 ) )
=> ( member_list_nat @ C2 @ B5 ) ) ).
% IntD2
thf(fact_1066_IntD2,axiom,
! [C2: produc357393685978478089rm_a_b,A4: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ C2 @ ( inf_in1693821330460682071rm_a_b @ A4 @ B5 ) )
=> ( member5869715511025134514rm_a_b @ C2 @ B5 ) ) ).
% IntD2
thf(fact_1067_total__on__def,axiom,
( total_on_term_a_b
= ( ^ [A6: set_term_a_b,R4: set_Pr4386577575007340137rm_a_b] :
! [X4: term_a_b] :
( ( member_term_a_b @ X4 @ A6 )
=> ! [Y5: term_a_b] :
( ( member_term_a_b @ Y5 @ A6 )
=> ( ( X4 != Y5 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X4 @ Y5 ) @ R4 )
| ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y5 @ X4 ) @ R4 ) ) ) ) ) ) ) ).
% total_on_def
thf(fact_1068_total__onI,axiom,
! [A4: set_list_nat,R2: set_Pr3451248702717554689st_nat] :
( ! [X3: list_nat,Y3: list_nat] :
( ( member_list_nat @ X3 @ A4 )
=> ( ( member_list_nat @ Y3 @ A4 )
=> ( ( X3 != Y3 )
=> ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X3 @ Y3 ) @ R2 )
| ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Y3 @ X3 ) @ R2 ) ) ) ) )
=> ( total_on_list_nat @ A4 @ R2 ) ) ).
% total_onI
thf(fact_1069_total__onI,axiom,
! [A4: set_Pr4386577575007340137rm_a_b,R2: set_Pr2972776593051762503rm_a_b] :
( ! [X3: produc357393685978478089rm_a_b,Y3: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X3 @ A4 )
=> ( ( member5869715511025134514rm_a_b @ Y3 @ A4 )
=> ( ( X3 != Y3 )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X3 @ Y3 ) @ R2 )
| ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y3 @ X3 ) @ R2 ) ) ) ) )
=> ( total_4010809703584253837rm_a_b @ A4 @ R2 ) ) ).
% total_onI
thf(fact_1070_total__onI,axiom,
! [A4: set_term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ! [X3: term_a_b,Y3: term_a_b] :
( ( member_term_a_b @ X3 @ A4 )
=> ( ( member_term_a_b @ Y3 @ A4 )
=> ( ( X3 != Y3 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ R2 )
| ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ X3 ) @ R2 ) ) ) ) )
=> ( total_on_term_a_b @ A4 @ R2 ) ) ).
% total_onI
thf(fact_1071_disjoint__iff,axiom,
! [A4: set_list_nat,B5: set_list_nat] :
( ( ( inf_inf_set_list_nat @ A4 @ B5 )
= bot_bot_set_list_nat )
= ( ! [X4: list_nat] :
( ( member_list_nat @ X4 @ A4 )
=> ~ ( member_list_nat @ X4 @ B5 ) ) ) ) ).
% disjoint_iff
thf(fact_1072_disjoint__iff,axiom,
! [A4: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
( ( ( inf_in1693821330460682071rm_a_b @ A4 @ B5 )
= bot_bo197521221353338581rm_a_b )
= ( ! [X4: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X4 @ A4 )
=> ~ ( member5869715511025134514rm_a_b @ X4 @ B5 ) ) ) ) ).
% disjoint_iff
thf(fact_1073_Int__emptyI,axiom,
! [A4: set_list_nat,B5: set_list_nat] :
( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ A4 )
=> ~ ( member_list_nat @ X3 @ B5 ) )
=> ( ( inf_inf_set_list_nat @ A4 @ B5 )
= bot_bot_set_list_nat ) ) ).
% Int_emptyI
thf(fact_1074_Int__emptyI,axiom,
! [A4: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
( ! [X3: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X3 @ A4 )
=> ~ ( member5869715511025134514rm_a_b @ X3 @ B5 ) )
=> ( ( inf_in1693821330460682071rm_a_b @ A4 @ B5 )
= bot_bo197521221353338581rm_a_b ) ) ).
% Int_emptyI
thf(fact_1075_Int__insert__left,axiom,
! [A: list_nat,C4: set_list_nat,B5: set_list_nat] :
( ( ( member_list_nat @ A @ C4 )
=> ( ( inf_inf_set_list_nat @ ( insert_list_nat @ A @ B5 ) @ C4 )
= ( insert_list_nat @ A @ ( inf_inf_set_list_nat @ B5 @ C4 ) ) ) )
& ( ~ ( member_list_nat @ A @ C4 )
=> ( ( inf_inf_set_list_nat @ ( insert_list_nat @ A @ B5 ) @ C4 )
= ( inf_inf_set_list_nat @ B5 @ C4 ) ) ) ) ).
% Int_insert_left
thf(fact_1076_Int__insert__left,axiom,
! [A: produc357393685978478089rm_a_b,C4: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
( ( ( member5869715511025134514rm_a_b @ A @ C4 )
=> ( ( inf_in1693821330460682071rm_a_b @ ( insert7009541432154983385rm_a_b @ A @ B5 ) @ C4 )
= ( insert7009541432154983385rm_a_b @ A @ ( inf_in1693821330460682071rm_a_b @ B5 @ C4 ) ) ) )
& ( ~ ( member5869715511025134514rm_a_b @ A @ C4 )
=> ( ( inf_in1693821330460682071rm_a_b @ ( insert7009541432154983385rm_a_b @ A @ B5 ) @ C4 )
= ( inf_in1693821330460682071rm_a_b @ B5 @ C4 ) ) ) ) ).
% Int_insert_left
thf(fact_1077_Int__insert__right,axiom,
! [A: list_nat,A4: set_list_nat,B5: set_list_nat] :
( ( ( member_list_nat @ A @ A4 )
=> ( ( inf_inf_set_list_nat @ A4 @ ( insert_list_nat @ A @ B5 ) )
= ( insert_list_nat @ A @ ( inf_inf_set_list_nat @ A4 @ B5 ) ) ) )
& ( ~ ( member_list_nat @ A @ A4 )
=> ( ( inf_inf_set_list_nat @ A4 @ ( insert_list_nat @ A @ B5 ) )
= ( inf_inf_set_list_nat @ A4 @ B5 ) ) ) ) ).
% Int_insert_right
thf(fact_1078_Int__insert__right,axiom,
! [A: produc357393685978478089rm_a_b,A4: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
( ( ( member5869715511025134514rm_a_b @ A @ A4 )
=> ( ( inf_in1693821330460682071rm_a_b @ A4 @ ( insert7009541432154983385rm_a_b @ A @ B5 ) )
= ( insert7009541432154983385rm_a_b @ A @ ( inf_in1693821330460682071rm_a_b @ A4 @ B5 ) ) ) )
& ( ~ ( member5869715511025134514rm_a_b @ A @ A4 )
=> ( ( inf_in1693821330460682071rm_a_b @ A4 @ ( insert7009541432154983385rm_a_b @ A @ B5 ) )
= ( inf_in1693821330460682071rm_a_b @ A4 @ B5 ) ) ) ) ).
% Int_insert_right
thf(fact_1079_totalI,axiom,
! [R2: set_Pr4386577575007340137rm_a_b] :
( ! [X3: term_a_b,Y3: term_a_b] :
( ( X3 != Y3 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ R2 )
| ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ X3 ) @ R2 ) ) )
=> ( total_on_term_a_b @ top_top_set_term_a_b @ R2 ) ) ).
% totalI
thf(fact_1080_linear__order__on__singleton,axiom,
! [X: term_a_b] : ( order_5388802246213473311rm_a_b @ ( insert_term_a_b @ X @ bot_bot_set_term_a_b ) @ ( insert7009541432154983385rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ X ) @ bot_bo197521221353338581rm_a_b ) ) ).
% linear_order_on_singleton
thf(fact_1081_Linear__order__in__diff__Id,axiom,
! [R2: set_Pr3451248702717554689st_nat,A: list_nat,B: list_nat] :
( ( order_2931667645265552619st_nat @ ( field_list_nat @ R2 ) @ R2 )
=> ( ( member_list_nat @ A @ ( field_list_nat @ R2 ) )
=> ( ( member_list_nat @ B @ ( field_list_nat @ R2 ) )
=> ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A @ B ) @ R2 )
= ( ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ B @ A ) @ ( minus_4256792079133151976st_nat @ R2 @ id_list_nat ) ) ) ) ) ) ) ).
% Linear_order_in_diff_Id
thf(fact_1082_Linear__order__in__diff__Id,axiom,
! [R2: set_Pr2972776593051762503rm_a_b,A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b] :
( ( order_681589137112254398rm_a_b @ ( field_6884932134483023318rm_a_b @ R2 ) @ R2 )
=> ( ( member5869715511025134514rm_a_b @ A @ ( field_6884932134483023318rm_a_b @ R2 ) )
=> ( ( member5869715511025134514rm_a_b @ B @ ( field_6884932134483023318rm_a_b @ R2 ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ R2 )
= ( ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B @ A ) @ ( minus_45395518312058030rm_a_b @ R2 @ id_Pro3488709451141449061rm_a_b ) ) ) ) ) ) ) ).
% Linear_order_in_diff_Id
thf(fact_1083_Linear__order__in__diff__Id,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,A: term_a_b,B: term_a_b] :
( ( order_5388802246213473311rm_a_b @ ( field_term_a_b @ R2 ) @ R2 )
=> ( ( member_term_a_b @ A @ ( field_term_a_b @ R2 ) )
=> ( ( member_term_a_b @ B @ ( field_term_a_b @ R2 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R2 )
= ( ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ ( minus_5192120951422937424rm_a_b @ R2 @ id_term_a_b ) ) ) ) ) ) ) ).
% Linear_order_in_diff_Id
thf(fact_1084_Linear__order__wf__diff__Id,axiom,
! [R2: set_Pr4386577575007340137rm_a_b] :
( ( order_5388802246213473311rm_a_b @ ( field_term_a_b @ R2 ) @ R2 )
=> ( ( wf_term_a_b @ ( minus_5192120951422937424rm_a_b @ R2 @ id_term_a_b ) )
= ( ! [A6: set_term_a_b] :
( ( ord_le2705286416250468010rm_a_b @ A6 @ ( field_term_a_b @ R2 ) )
=> ( ( A6 != bot_bot_set_term_a_b )
=> ? [X4: term_a_b] :
( ( member_term_a_b @ X4 @ A6 )
& ! [Y5: term_a_b] :
( ( member_term_a_b @ Y5 @ A6 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X4 @ Y5 ) @ R2 ) ) ) ) ) ) ) ) ).
% Linear_order_wf_diff_Id
thf(fact_1085_minimal__imp__wf,axiom,
! [R2: set_Pr3451248702717554689st_nat] :
( ! [Q5: set_list_nat] :
( ? [X3: list_nat] : ( member_list_nat @ X3 @ Q5 )
=> ? [X6: list_nat] :
( ( member_list_nat @ X6 @ Q5 )
& ! [Y3: list_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X6 @ Y3 ) @ R2 )
=> ~ ( member_list_nat @ Y3 @ Q5 ) ) ) )
=> ( wf_list_nat @ ( conver3282399561507114237st_nat @ R2 ) ) ) ).
% minimal_imp_wf
thf(fact_1086_minimal__imp__wf,axiom,
! [R2: set_Pr2972776593051762503rm_a_b] :
( ! [Q5: set_Pr4386577575007340137rm_a_b] :
( ? [X3: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ X3 @ Q5 )
=> ? [X6: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X6 @ Q5 )
& ! [Y3: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X6 @ Y3 ) @ R2 )
=> ~ ( member5869715511025134514rm_a_b @ Y3 @ Q5 ) ) ) )
=> ( wf_Pro2335863617654816626rm_a_b @ ( conver1575690562132235771rm_a_b @ R2 ) ) ) ).
% minimal_imp_wf
thf(fact_1087_minimal__imp__wf,axiom,
! [R2: set_Pr4386577575007340137rm_a_b] :
( ! [Q5: set_term_a_b] :
( ? [X3: term_a_b] : ( member_term_a_b @ X3 @ Q5 )
=> ? [X6: term_a_b] :
( ( member_term_a_b @ X6 @ Q5 )
& ! [Y3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X6 @ Y3 ) @ R2 )
=> ~ ( member_term_a_b @ Y3 @ Q5 ) ) ) )
=> ( wf_term_a_b @ ( conver7608559976937845093rm_a_b @ R2 ) ) ) ).
% minimal_imp_wf
thf(fact_1088_wf__eq__minimal2,axiom,
( wf_term_a_b
= ( ^ [R4: set_Pr4386577575007340137rm_a_b] :
! [A6: set_term_a_b] :
( ( ( ord_le2705286416250468010rm_a_b @ A6 @ ( field_term_a_b @ R4 ) )
& ( A6 != bot_bot_set_term_a_b ) )
=> ? [X4: term_a_b] :
( ( member_term_a_b @ X4 @ A6 )
& ! [Y5: term_a_b] :
( ( member_term_a_b @ Y5 @ A6 )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y5 @ X4 ) @ R4 ) ) ) ) ) ) ).
% wf_eq_minimal2
thf(fact_1089_wf__insert,axiom,
! [Y: term_a_b,X: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ( wf_term_a_b @ ( insert7009541432154983385rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X ) @ R2 ) )
= ( ( wf_term_a_b @ R2 )
& ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ R2 ) ) ) ) ).
% wf_insert
thf(fact_1090_wf__def,axiom,
( wf_term_a_b
= ( ^ [R4: set_Pr4386577575007340137rm_a_b] :
! [P4: term_a_b > $o] :
( ! [X4: term_a_b] :
( ! [Y5: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y5 @ X4 ) @ R4 )
=> ( P4 @ Y5 ) )
=> ( P4 @ X4 ) )
=> ! [X7: term_a_b] : ( P4 @ X7 ) ) ) ) ).
% wf_def
thf(fact_1091_wfE__min,axiom,
! [R5: set_Pr3451248702717554689st_nat,X: list_nat,Q2: set_list_nat] :
( ( wf_list_nat @ R5 )
=> ( ( member_list_nat @ X @ Q2 )
=> ~ ! [Z3: list_nat] :
( ( member_list_nat @ Z3 @ Q2 )
=> ~ ! [Y7: list_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Y7 @ Z3 ) @ R5 )
=> ~ ( member_list_nat @ Y7 @ Q2 ) ) ) ) ) ).
% wfE_min
thf(fact_1092_wfE__min,axiom,
! [R5: set_Pr2972776593051762503rm_a_b,X: produc357393685978478089rm_a_b,Q2: set_Pr4386577575007340137rm_a_b] :
( ( wf_Pro2335863617654816626rm_a_b @ R5 )
=> ( ( member5869715511025134514rm_a_b @ X @ Q2 )
=> ~ ! [Z3: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ Z3 @ Q2 )
=> ~ ! [Y7: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y7 @ Z3 ) @ R5 )
=> ~ ( member5869715511025134514rm_a_b @ Y7 @ Q2 ) ) ) ) ) ).
% wfE_min
thf(fact_1093_wfE__min,axiom,
! [R5: set_Pr4386577575007340137rm_a_b,X: term_a_b,Q2: set_term_a_b] :
( ( wf_term_a_b @ R5 )
=> ( ( member_term_a_b @ X @ Q2 )
=> ~ ! [Z3: term_a_b] :
( ( member_term_a_b @ Z3 @ Q2 )
=> ~ ! [Y7: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y7 @ Z3 ) @ R5 )
=> ~ ( member_term_a_b @ Y7 @ Q2 ) ) ) ) ) ).
% wfE_min
thf(fact_1094_wfI__min,axiom,
! [R5: set_Pr3451248702717554689st_nat] :
( ! [X3: list_nat,Q5: set_list_nat] :
( ( member_list_nat @ X3 @ Q5 )
=> ? [Xa2: list_nat] :
( ( member_list_nat @ Xa2 @ Q5 )
& ! [Y3: list_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Y3 @ Xa2 ) @ R5 )
=> ~ ( member_list_nat @ Y3 @ Q5 ) ) ) )
=> ( wf_list_nat @ R5 ) ) ).
% wfI_min
thf(fact_1095_wfI__min,axiom,
! [R5: set_Pr2972776593051762503rm_a_b] :
( ! [X3: produc357393685978478089rm_a_b,Q5: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ X3 @ Q5 )
=> ? [Xa2: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ Xa2 @ Q5 )
& ! [Y3: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y3 @ Xa2 ) @ R5 )
=> ~ ( member5869715511025134514rm_a_b @ Y3 @ Q5 ) ) ) )
=> ( wf_Pro2335863617654816626rm_a_b @ R5 ) ) ).
% wfI_min
thf(fact_1096_wfI__min,axiom,
! [R5: set_Pr4386577575007340137rm_a_b] :
( ! [X3: term_a_b,Q5: set_term_a_b] :
( ( member_term_a_b @ X3 @ Q5 )
=> ? [Xa2: term_a_b] :
( ( member_term_a_b @ Xa2 @ Q5 )
& ! [Y3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ Xa2 ) @ R5 )
=> ~ ( member_term_a_b @ Y3 @ Q5 ) ) ) )
=> ( wf_term_a_b @ R5 ) ) ).
% wfI_min
thf(fact_1097_wfUNIVI,axiom,
! [R2: set_Pr4386577575007340137rm_a_b] :
( ! [P5: term_a_b > $o,X3: term_a_b] :
( ! [Xa2: term_a_b] :
( ! [Y3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ Xa2 ) @ R2 )
=> ( P5 @ Y3 ) )
=> ( P5 @ Xa2 ) )
=> ( P5 @ X3 ) )
=> ( wf_term_a_b @ R2 ) ) ).
% wfUNIVI
thf(fact_1098_wf__asym,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,A: term_a_b,X: term_a_b] :
( ( wf_term_a_b @ R2 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ X ) @ R2 )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ A ) @ R2 ) ) ) ).
% wf_asym
thf(fact_1099_wf__induct,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,P3: term_a_b > $o,A: term_a_b] :
( ( wf_term_a_b @ R2 )
=> ( ! [X3: term_a_b] :
( ! [Y7: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y7 @ X3 ) @ R2 )
=> ( P3 @ Y7 ) )
=> ( P3 @ X3 ) )
=> ( P3 @ A ) ) ) ).
% wf_induct
thf(fact_1100_wf__irrefl,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,A: term_a_b] :
( ( wf_term_a_b @ R2 )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ A ) @ R2 ) ) ).
% wf_irrefl
thf(fact_1101_wf__not__sym,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,A: term_a_b,X: term_a_b] :
( ( wf_term_a_b @ R2 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ X ) @ R2 )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ A ) @ R2 ) ) ) ).
% wf_not_sym
thf(fact_1102_wf__not__refl,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,A: term_a_b] :
( ( wf_term_a_b @ R2 )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ A ) @ R2 ) ) ).
% wf_not_refl
thf(fact_1103_wf__eq__minimal,axiom,
( wf_list_nat
= ( ^ [R4: set_Pr3451248702717554689st_nat] :
! [Q6: set_list_nat] :
( ? [X4: list_nat] : ( member_list_nat @ X4 @ Q6 )
=> ? [X4: list_nat] :
( ( member_list_nat @ X4 @ Q6 )
& ! [Y5: list_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Y5 @ X4 ) @ R4 )
=> ~ ( member_list_nat @ Y5 @ Q6 ) ) ) ) ) ) ).
% wf_eq_minimal
thf(fact_1104_wf__eq__minimal,axiom,
( wf_Pro2335863617654816626rm_a_b
= ( ^ [R4: set_Pr2972776593051762503rm_a_b] :
! [Q6: set_Pr4386577575007340137rm_a_b] :
( ? [X4: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ X4 @ Q6 )
=> ? [X4: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X4 @ Q6 )
& ! [Y5: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y5 @ X4 ) @ R4 )
=> ~ ( member5869715511025134514rm_a_b @ Y5 @ Q6 ) ) ) ) ) ) ).
% wf_eq_minimal
thf(fact_1105_wf__eq__minimal,axiom,
( wf_term_a_b
= ( ^ [R4: set_Pr4386577575007340137rm_a_b] :
! [Q6: set_term_a_b] :
( ? [X4: term_a_b] : ( member_term_a_b @ X4 @ Q6 )
=> ? [X4: term_a_b] :
( ( member_term_a_b @ X4 @ Q6 )
& ! [Y5: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y5 @ X4 ) @ R4 )
=> ~ ( member_term_a_b @ Y5 @ Q6 ) ) ) ) ) ) ).
% wf_eq_minimal
thf(fact_1106_wf__induct__rule,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,P3: term_a_b > $o,A: term_a_b] :
( ( wf_term_a_b @ R2 )
=> ( ! [X3: term_a_b] :
( ! [Y7: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y7 @ X3 ) @ R2 )
=> ( P3 @ Y7 ) )
=> ( P3 @ X3 ) )
=> ( P3 @ A ) ) ) ).
% wf_induct_rule
thf(fact_1107_wf__if__convertible__to__wf,axiom,
! [S: set_Pr4386577575007340137rm_a_b,R2: set_Pr4386577575007340137rm_a_b,F3: term_a_b > term_a_b] :
( ( wf_term_a_b @ S )
=> ( ! [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 ) )
=> ( wf_term_a_b @ R2 ) ) ) ).
% wf_if_convertible_to_wf
thf(fact_1108_wfE__min_H,axiom,
! [R5: set_Pr3451248702717554689st_nat,Q2: set_list_nat] :
( ( wf_list_nat @ R5 )
=> ( ( Q2 != bot_bot_set_list_nat )
=> ~ ! [Z3: list_nat] :
( ( member_list_nat @ Z3 @ Q2 )
=> ~ ! [Y7: list_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Y7 @ Z3 ) @ R5 )
=> ~ ( member_list_nat @ Y7 @ Q2 ) ) ) ) ) ).
% wfE_min'
thf(fact_1109_wfE__min_H,axiom,
! [R5: set_Pr2972776593051762503rm_a_b,Q2: set_Pr4386577575007340137rm_a_b] :
( ( wf_Pro2335863617654816626rm_a_b @ R5 )
=> ( ( Q2 != bot_bo197521221353338581rm_a_b )
=> ~ ! [Z3: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ Z3 @ Q2 )
=> ~ ! [Y7: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y7 @ Z3 ) @ R5 )
=> ~ ( member5869715511025134514rm_a_b @ Y7 @ Q2 ) ) ) ) ) ).
% wfE_min'
thf(fact_1110_wfE__min_H,axiom,
! [R5: set_Pr4386577575007340137rm_a_b,Q2: set_term_a_b] :
( ( wf_term_a_b @ R5 )
=> ( ( Q2 != bot_bot_set_term_a_b )
=> ~ ! [Z3: term_a_b] :
( ( member_term_a_b @ Z3 @ Q2 )
=> ~ ! [Y7: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y7 @ Z3 ) @ R5 )
=> ~ ( member_term_a_b @ Y7 @ Q2 ) ) ) ) ) ).
% wfE_min'
thf(fact_1111_in__lex__prod,axiom,
! [A: term_a_b,B: term_a_b,A2: term_a_b,B2: term_a_b,R2: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) ) @ ( lex_pr6163557265797435481rm_a_b @ R2 @ S ) )
= ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ A2 ) @ R2 )
| ( ( A = A2 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ B2 ) @ S ) ) ) ) ).
% in_lex_prod
thf(fact_1112_under__incr,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,A: term_a_b,B: term_a_b] :
( ( trans_on_term_a_b @ top_top_set_term_a_b @ R2 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R2 )
=> ( ord_le2705286416250468010rm_a_b @ ( order_under_term_a_b @ R2 @ A ) @ ( order_under_term_a_b @ R2 @ B ) ) ) ) ).
% under_incr
thf(fact_1113_Refl__under__underS,axiom,
! [R2: set_Pr3451248702717554689st_nat,A: list_nat] :
( ( refl_on_list_nat @ ( field_list_nat @ R2 ) @ R2 )
=> ( ( member_list_nat @ A @ ( field_list_nat @ R2 ) )
=> ( ( order_under_list_nat @ R2 @ A )
= ( sup_sup_set_list_nat @ ( order_8386007542010951322st_nat @ R2 @ A ) @ ( insert_list_nat @ A @ bot_bot_set_list_nat ) ) ) ) ) ).
% Refl_under_underS
thf(fact_1114_Refl__under__underS,axiom,
! [R2: set_Pr2972776593051762503rm_a_b,A: produc357393685978478089rm_a_b] :
( ( refl_o5344404488616963464rm_a_b @ ( field_6884932134483023318rm_a_b @ R2 ) @ R2 )
=> ( ( member5869715511025134514rm_a_b @ A @ ( field_6884932134483023318rm_a_b @ R2 ) )
=> ( ( order_346605463002971118rm_a_b @ R2 @ A )
= ( sup_su6776935440552674877rm_a_b @ ( order_8163520290499871439rm_a_b @ R2 @ A ) @ ( insert7009541432154983385rm_a_b @ A @ bot_bo197521221353338581rm_a_b ) ) ) ) ) ).
% Refl_under_underS
thf(fact_1115_underS__empty,axiom,
! [A: list_nat,R2: set_Pr3451248702717554689st_nat] :
( ~ ( member_list_nat @ A @ ( field_list_nat @ R2 ) )
=> ( ( order_8386007542010951322st_nat @ R2 @ A )
= bot_bot_set_list_nat ) ) ).
% underS_empty
thf(fact_1116_underS__empty,axiom,
! [A: produc357393685978478089rm_a_b,R2: set_Pr2972776593051762503rm_a_b] :
( ~ ( member5869715511025134514rm_a_b @ A @ ( field_6884932134483023318rm_a_b @ R2 ) )
=> ( ( order_8163520290499871439rm_a_b @ R2 @ A )
= bot_bo197521221353338581rm_a_b ) ) ).
% underS_empty
thf(fact_1117_BNF__Least__Fixpoint_OunderS__Field,axiom,
! [I2: list_nat,R5: set_Pr3451248702717554689st_nat,J2: list_nat] :
( ( member_list_nat @ I2 @ ( order_8386007542010951322st_nat @ R5 @ J2 ) )
=> ( member_list_nat @ I2 @ ( field_list_nat @ R5 ) ) ) ).
% BNF_Least_Fixpoint.underS_Field
thf(fact_1118_BNF__Least__Fixpoint_OunderS__Field,axiom,
! [I2: produc357393685978478089rm_a_b,R5: set_Pr2972776593051762503rm_a_b,J2: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ I2 @ ( order_8163520290499871439rm_a_b @ R5 @ J2 ) )
=> ( member5869715511025134514rm_a_b @ I2 @ ( field_6884932134483023318rm_a_b @ R5 ) ) ) ).
% BNF_Least_Fixpoint.underS_Field
thf(fact_1119_Nil__in__poss,axiom,
! [T: term_a_b] : ( member_list_nat @ nil_nat @ ( term_poss_a_b @ T ) ) ).
% Nil_in_poss
thf(fact_1120_underS__E,axiom,
! [I2: list_nat,R5: set_Pr3451248702717554689st_nat,J2: list_nat] :
( ( member_list_nat @ I2 @ ( order_8386007542010951322st_nat @ R5 @ J2 ) )
=> ( ( I2 != J2 )
& ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ I2 @ J2 ) @ R5 ) ) ) ).
% underS_E
thf(fact_1121_underS__E,axiom,
! [I2: produc357393685978478089rm_a_b,R5: set_Pr2972776593051762503rm_a_b,J2: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ I2 @ ( order_8163520290499871439rm_a_b @ R5 @ J2 ) )
=> ( ( I2 != J2 )
& ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ I2 @ J2 ) @ R5 ) ) ) ).
% underS_E
thf(fact_1122_underS__E,axiom,
! [I2: term_a_b,R5: set_Pr4386577575007340137rm_a_b,J2: term_a_b] :
( ( member_term_a_b @ I2 @ ( order_1619770106104096206rm_a_b @ R5 @ J2 ) )
=> ( ( I2 != J2 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ I2 @ J2 ) @ R5 ) ) ) ).
% underS_E
thf(fact_1123_underS__I,axiom,
! [I2: list_nat,J2: list_nat,R5: set_Pr3451248702717554689st_nat] :
( ( I2 != J2 )
=> ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ I2 @ J2 ) @ R5 )
=> ( member_list_nat @ I2 @ ( order_8386007542010951322st_nat @ R5 @ J2 ) ) ) ) ).
% underS_I
thf(fact_1124_underS__I,axiom,
! [I2: produc357393685978478089rm_a_b,J2: produc357393685978478089rm_a_b,R5: set_Pr2972776593051762503rm_a_b] :
( ( I2 != J2 )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ I2 @ J2 ) @ R5 )
=> ( member5869715511025134514rm_a_b @ I2 @ ( order_8163520290499871439rm_a_b @ R5 @ J2 ) ) ) ) ).
% underS_I
thf(fact_1125_underS__I,axiom,
! [I2: term_a_b,J2: term_a_b,R5: set_Pr4386577575007340137rm_a_b] :
( ( I2 != J2 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ I2 @ J2 ) @ R5 )
=> ( member_term_a_b @ I2 @ ( order_1619770106104096206rm_a_b @ R5 @ J2 ) ) ) ) ).
% underS_I
thf(fact_1126_replace__term__at_Osimps_I1_J,axiom,
! [S: term_a_b,T: term_a_b] :
( ( term_r6860082780075436317at_a_b @ S @ nil_nat @ T )
= T ) ).
% replace_term_at.simps(1)
thf(fact_1127_subt__at_Osimps_I1_J,axiom,
! [S: term_a_b] :
( ( term_subt_at_a_b @ S @ nil_nat )
= S ) ).
% subt_at.simps(1)
thf(fact_1128_underS__incl__iff,axiom,
! [R2: set_Pr3451248702717554689st_nat,A: list_nat,B: list_nat] :
( ( order_2931667645265552619st_nat @ ( field_list_nat @ R2 ) @ R2 )
=> ( ( member_list_nat @ A @ ( field_list_nat @ R2 ) )
=> ( ( member_list_nat @ B @ ( field_list_nat @ R2 ) )
=> ( ( ord_le6045566169113846134st_nat @ ( order_8386007542010951322st_nat @ R2 @ A ) @ ( order_8386007542010951322st_nat @ R2 @ B ) )
= ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A @ B ) @ R2 ) ) ) ) ) ).
% underS_incl_iff
thf(fact_1129_underS__incl__iff,axiom,
! [R2: set_Pr2972776593051762503rm_a_b,A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b] :
( ( order_681589137112254398rm_a_b @ ( field_6884932134483023318rm_a_b @ R2 ) @ R2 )
=> ( ( member5869715511025134514rm_a_b @ A @ ( field_6884932134483023318rm_a_b @ R2 ) )
=> ( ( member5869715511025134514rm_a_b @ B @ ( field_6884932134483023318rm_a_b @ R2 ) )
=> ( ( ord_le118470702582115849rm_a_b @ ( order_8163520290499871439rm_a_b @ R2 @ A ) @ ( order_8163520290499871439rm_a_b @ R2 @ B ) )
= ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ R2 ) ) ) ) ) ).
% underS_incl_iff
thf(fact_1130_underS__incl__iff,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,A: term_a_b,B: term_a_b] :
( ( order_5388802246213473311rm_a_b @ ( field_term_a_b @ R2 ) @ R2 )
=> ( ( member_term_a_b @ A @ ( field_term_a_b @ R2 ) )
=> ( ( member_term_a_b @ B @ ( field_term_a_b @ R2 ) )
=> ( ( ord_le2705286416250468010rm_a_b @ ( order_1619770106104096206rm_a_b @ R2 @ A ) @ ( order_1619770106104096206rm_a_b @ R2 @ B ) )
= ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R2 ) ) ) ) ) ).
% underS_incl_iff
thf(fact_1131_underS__incr,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,A: term_a_b,B: term_a_b] :
( ( trans_on_term_a_b @ top_top_set_term_a_b @ R2 )
=> ( ( antisym_on_term_a_b @ top_top_set_term_a_b @ R2 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R2 )
=> ( ord_le2705286416250468010rm_a_b @ ( order_1619770106104096206rm_a_b @ R2 @ A ) @ ( order_1619770106104096206rm_a_b @ R2 @ B ) ) ) ) ) ).
% underS_incr
thf(fact_1132_subt__at__subterm,axiom,
! [P2: list_nat,T: term_a_b] :
( ( member_list_nat @ P2 @ ( term_poss_a_b @ T ) )
=> ( ( P2 != nil_nat )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T @ ( term_subt_at_a_b @ T @ P2 ) ) @ subterm_and_supt_a_b ) ) ) ).
% subt_at_subterm
thf(fact_1133_suptI,axiom,
! [A: term_a_b,B: term_a_b] :
( ( A != B )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ subter523971068842742411eq_a_b )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ subterm_and_supt_a_b ) ) ) ).
% suptI
thf(fact_1134_supt__subst,axiom,
! [S: term_a_b,T: term_a_b,Sigma2: b > term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ subterm_and_supt_a_b )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( subst_7999470309526761004_a_b_b @ S @ Sigma2 ) @ ( subst_7999470309526761004_a_b_b @ T @ Sigma2 ) ) @ subterm_and_supt_a_b ) ) ).
% supt_subst
thf(fact_1135_subterm__induct,axiom,
! [P3: term_a_b > $o,T: term_a_b] :
( ! [T2: term_a_b] :
( ! [S8: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T2 @ S8 ) @ subterm_and_supt_a_b )
=> ( P3 @ S8 ) )
=> ( P3 @ T2 ) )
=> ( P3 @ T ) ) ).
% subterm_induct
thf(fact_1136_supt__not__refl,axiom,
! [T: term_a_b] :
~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T @ T ) @ subterm_and_supt_a_b ) ).
% supt_not_refl
thf(fact_1137_supt__not__sym,axiom,
! [S: term_a_b,T: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ subterm_and_supt_a_b )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T @ S ) @ subterm_and_supt_a_b ) ) ).
% supt_not_sym
thf(fact_1138_supt__irrefl,axiom,
! [T: term_a_b] :
~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T @ T ) @ subterm_and_supt_a_b ) ).
% supt_irrefl
thf(fact_1139_supt__trans,axiom,
! [S: term_a_b,T: term_a_b,U: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ subterm_and_supt_a_b )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T @ U ) @ subterm_and_supt_a_b )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ U ) @ subterm_and_supt_a_b ) ) ) ).
% supt_trans
thf(fact_1140_supt__neqD,axiom,
! [S: term_a_b,T: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ subterm_and_supt_a_b )
=> ( S != T ) ) ).
% supt_neqD
thf(fact_1141_subterm_Odual__order_Ostrict__implies__not__eq,axiom,
! [A: term_a_b,B: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ subterm_and_supt_a_b )
=> ( A != B ) ) ).
% subterm.dual_order.strict_implies_not_eq
thf(fact_1142_subterm_Oorder_Ostrict__implies__not__eq,axiom,
! [B: term_a_b,A: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ subterm_and_supt_a_b )
=> ( A != B ) ) ).
% subterm.order.strict_implies_not_eq
thf(fact_1143_subterm_Odual__order_Ostrict__trans,axiom,
! [A: term_a_b,B: term_a_b,C2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ subterm_and_supt_a_b )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C2 ) @ subterm_and_supt_a_b )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C2 ) @ subterm_and_supt_a_b ) ) ) ).
% subterm.dual_order.strict_trans
thf(fact_1144_subterm_Oorder_Ostrict__trans,axiom,
! [B: term_a_b,A: term_a_b,C2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ subterm_and_supt_a_b )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ C2 @ B ) @ subterm_and_supt_a_b )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ C2 @ A ) @ subterm_and_supt_a_b ) ) ) ).
% subterm.order.strict_trans
thf(fact_1145_subterm_Oord__less__eq__trans,axiom,
! [B: term_a_b,A: term_a_b,C2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ subterm_and_supt_a_b )
=> ( ( B = C2 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ C2 @ A ) @ subterm_and_supt_a_b ) ) ) ).
% subterm.ord_less_eq_trans
thf(fact_1146_subterm_Oord__eq__less__trans,axiom,
! [A: term_a_b,B: term_a_b,C2: term_a_b] :
( ( A = B )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ C2 @ B ) @ subterm_and_supt_a_b )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ C2 @ A ) @ subterm_and_supt_a_b ) ) ) ).
% subterm.ord_eq_less_trans
thf(fact_1147_subterm_Oless__imp__not__less,axiom,
! [Y: term_a_b,X: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X ) @ subterm_and_supt_a_b )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ subterm_and_supt_a_b ) ) ).
% subterm.less_imp_not_less
thf(fact_1148_subterm_Odual__order_Oirrefl,axiom,
! [A: term_a_b] :
~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ A ) @ subterm_and_supt_a_b ) ).
% subterm.dual_order.irrefl
thf(fact_1149_subterm_Oless__imp__not__eq2,axiom,
! [Y: term_a_b,X: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X ) @ subterm_and_supt_a_b )
=> ( Y != X ) ) ).
% subterm.less_imp_not_eq2
thf(fact_1150_subterm_Oless__imp__not__eq,axiom,
! [Y: term_a_b,X: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X ) @ subterm_and_supt_a_b )
=> ( X != Y ) ) ).
% subterm.less_imp_not_eq
thf(fact_1151_subterm_Odual__order_Oasym,axiom,
! [A: term_a_b,B: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ subterm_and_supt_a_b )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ subterm_and_supt_a_b ) ) ).
% subterm.dual_order.asym
thf(fact_1152_subterm_Oless__imp__triv,axiom,
! [Y: term_a_b,X: term_a_b,P3: $o] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X ) @ subterm_and_supt_a_b )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ subterm_and_supt_a_b )
=> P3 ) ) ).
% subterm.less_imp_triv
thf(fact_1153_subterm_Oless__not__sym,axiom,
! [Y: term_a_b,X: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X ) @ subterm_and_supt_a_b )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ subterm_and_supt_a_b ) ) ).
% subterm.less_not_sym
thf(fact_1154_subterm_Oless__imp__neq,axiom,
! [Y: term_a_b,X: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X ) @ subterm_and_supt_a_b )
=> ( X != Y ) ) ).
% subterm.less_imp_neq
thf(fact_1155_subterm_Oless__irrefl,axiom,
! [X: term_a_b] :
~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ X ) @ subterm_and_supt_a_b ) ).
% subterm.less_irrefl
thf(fact_1156_subterm_Oorder_Oasym,axiom,
! [B: term_a_b,A: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ subterm_and_supt_a_b )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ subterm_and_supt_a_b ) ) ).
% subterm.order.asym
thf(fact_1157_subterm_Oless__trans,axiom,
! [Y: term_a_b,X: term_a_b,Z: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X ) @ subterm_and_supt_a_b )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Z @ Y ) @ subterm_and_supt_a_b )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Z @ X ) @ subterm_and_supt_a_b ) ) ) ).
% subterm.less_trans
thf(fact_1158_subterm_Oless__asym_H,axiom,
! [B: term_a_b,A: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ subterm_and_supt_a_b )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ subterm_and_supt_a_b ) ) ).
% subterm.less_asym'
thf(fact_1159_subterm_Oless__asym,axiom,
! [Y: term_a_b,X: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X ) @ subterm_and_supt_a_b )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ subterm_and_supt_a_b ) ) ).
% subterm.less_asym
thf(fact_1160_antisym__onD,axiom,
! [A4: set_list_nat,R2: set_Pr3451248702717554689st_nat,X: list_nat,Y: list_nat] :
( ( antisym_on_list_nat @ A4 @ R2 )
=> ( ( member_list_nat @ X @ A4 )
=> ( ( member_list_nat @ Y @ A4 )
=> ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X @ Y ) @ R2 )
=> ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Y @ X ) @ R2 )
=> ( X = Y ) ) ) ) ) ) ).
% antisym_onD
thf(fact_1161_antisym__onD,axiom,
! [A4: set_Pr4386577575007340137rm_a_b,R2: set_Pr2972776593051762503rm_a_b,X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b] :
( ( antisy5041284635856308844rm_a_b @ A4 @ R2 )
=> ( ( member5869715511025134514rm_a_b @ X @ A4 )
=> ( ( member5869715511025134514rm_a_b @ Y @ A4 )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ R2 )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y @ X ) @ R2 )
=> ( X = Y ) ) ) ) ) ) ).
% antisym_onD
thf(fact_1162_antisym__onD,axiom,
! [A4: set_term_a_b,R2: set_Pr4386577575007340137rm_a_b,X: term_a_b,Y: term_a_b] :
( ( antisym_on_term_a_b @ A4 @ R2 )
=> ( ( member_term_a_b @ X @ A4 )
=> ( ( member_term_a_b @ Y @ A4 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ R2 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X ) @ R2 )
=> ( X = Y ) ) ) ) ) ) ).
% antisym_onD
thf(fact_1163_antisym__onI,axiom,
! [A4: set_list_nat,R2: set_Pr3451248702717554689st_nat] :
( ! [X3: list_nat,Y3: list_nat] :
( ( member_list_nat @ X3 @ A4 )
=> ( ( member_list_nat @ Y3 @ A4 )
=> ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X3 @ Y3 ) @ R2 )
=> ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Y3 @ X3 ) @ R2 )
=> ( X3 = Y3 ) ) ) ) )
=> ( antisym_on_list_nat @ A4 @ R2 ) ) ).
% antisym_onI
thf(fact_1164_antisym__onI,axiom,
! [A4: set_Pr4386577575007340137rm_a_b,R2: set_Pr2972776593051762503rm_a_b] :
( ! [X3: produc357393685978478089rm_a_b,Y3: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X3 @ A4 )
=> ( ( member5869715511025134514rm_a_b @ Y3 @ A4 )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X3 @ Y3 ) @ R2 )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y3 @ X3 ) @ R2 )
=> ( X3 = Y3 ) ) ) ) )
=> ( antisy5041284635856308844rm_a_b @ A4 @ R2 ) ) ).
% antisym_onI
thf(fact_1165_antisym__onI,axiom,
! [A4: set_term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ! [X3: term_a_b,Y3: term_a_b] :
( ( member_term_a_b @ X3 @ A4 )
=> ( ( member_term_a_b @ Y3 @ A4 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ R2 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ X3 ) @ R2 )
=> ( X3 = Y3 ) ) ) ) )
=> ( antisym_on_term_a_b @ A4 @ R2 ) ) ).
% antisym_onI
thf(fact_1166_antisym__on__def,axiom,
( antisym_on_term_a_b
= ( ^ [A6: set_term_a_b,R4: set_Pr4386577575007340137rm_a_b] :
! [X4: term_a_b] :
( ( member_term_a_b @ X4 @ A6 )
=> ! [Y5: term_a_b] :
( ( member_term_a_b @ Y5 @ A6 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X4 @ Y5 ) @ R4 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y5 @ X4 ) @ R4 )
=> ( X4 = Y5 ) ) ) ) ) ) ) ).
% antisym_on_def
thf(fact_1167_subterm_OleD,axiom,
! [X: term_a_b,Y: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ subter523971068842742411eq_a_b )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X ) @ subterm_and_supt_a_b ) ) ).
% subterm.leD
thf(fact_1168_subterm_Ole__less,axiom,
! [Y: term_a_b,X: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X ) @ subter523971068842742411eq_a_b )
= ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X ) @ subterm_and_supt_a_b )
| ( X = Y ) ) ) ).
% subterm.le_less
thf(fact_1169_subterm_Oless__le,axiom,
! [Y: term_a_b,X: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X ) @ subterm_and_supt_a_b )
= ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X ) @ subter523971068842742411eq_a_b )
& ( X != Y ) ) ) ).
% subterm.less_le
thf(fact_1170_subterm_Onless__le,axiom,
! [B: term_a_b,A: term_a_b] :
( ( ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ subterm_and_supt_a_b ) )
= ( ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ subter523971068842742411eq_a_b )
| ( A = B ) ) ) ).
% subterm.nless_le
thf(fact_1171_subterm_Oless__imp__le,axiom,
! [Y: term_a_b,X: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X ) @ subterm_and_supt_a_b )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X ) @ subter523971068842742411eq_a_b ) ) ).
% subterm.less_imp_le
thf(fact_1172_subterm_Ole__neq__trans,axiom,
! [B: term_a_b,A: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ subter523971068842742411eq_a_b )
=> ( ( A != B )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ subterm_and_supt_a_b ) ) ) ).
% subterm.le_neq_trans
thf(fact_1173_subterm_Oantisym__conv1,axiom,
! [Y: term_a_b,X: term_a_b] :
( ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X ) @ subterm_and_supt_a_b )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X ) @ subter523971068842742411eq_a_b )
= ( X = Y ) ) ) ).
% subterm.antisym_conv1
thf(fact_1174_subterm_Oantisym__conv2,axiom,
! [Y: term_a_b,X: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X ) @ subter523971068842742411eq_a_b )
=> ( ( ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X ) @ subterm_and_supt_a_b ) )
= ( X = Y ) ) ) ).
% subterm.antisym_conv2
thf(fact_1175_subterm_Ole__less__trans,axiom,
! [Y: term_a_b,X: term_a_b,Z: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X ) @ subter523971068842742411eq_a_b )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Z @ Y ) @ subterm_and_supt_a_b )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Z @ X ) @ subterm_and_supt_a_b ) ) ) ).
% subterm.le_less_trans
thf(fact_1176_subterm_Oless__le__trans,axiom,
! [Y: term_a_b,X: term_a_b,Z: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X ) @ subterm_and_supt_a_b )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Z @ Y ) @ subter523971068842742411eq_a_b )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Z @ X ) @ subterm_and_supt_a_b ) ) ) ).
% subterm.less_le_trans
thf(fact_1177_subterm_Oless__le__not__le,axiom,
! [Y: term_a_b,X: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X ) @ subterm_and_supt_a_b )
= ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X ) @ subter523971068842742411eq_a_b )
& ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ subter523971068842742411eq_a_b ) ) ) ).
% subterm.less_le_not_le
thf(fact_1178_subterm_Ole__imp__less__or__eq,axiom,
! [Y: term_a_b,X: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X ) @ subter523971068842742411eq_a_b )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X ) @ subterm_and_supt_a_b )
| ( X = Y ) ) ) ).
% subterm.le_imp_less_or_eq
thf(fact_1179_subterm_Oorder_Ostrict__trans1,axiom,
! [B: term_a_b,A: term_a_b,C2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ subter523971068842742411eq_a_b )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ C2 @ B ) @ subterm_and_supt_a_b )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ C2 @ A ) @ subterm_and_supt_a_b ) ) ) ).
% subterm.order.strict_trans1
thf(fact_1180_subterm_Oorder_Ostrict__trans2,axiom,
! [B: term_a_b,A: term_a_b,C2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ subterm_and_supt_a_b )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ C2 @ B ) @ subter523971068842742411eq_a_b )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ C2 @ A ) @ subterm_and_supt_a_b ) ) ) ).
% subterm.order.strict_trans2
thf(fact_1181_subterm_Oorder_Ostrict__iff__not,axiom,
! [B: term_a_b,A: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ subterm_and_supt_a_b )
= ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ subter523971068842742411eq_a_b )
& ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ subter523971068842742411eq_a_b ) ) ) ).
% subterm.order.strict_iff_not
thf(fact_1182_subterm_Oorder_Oorder__iff__strict,axiom,
! [B: term_a_b,A: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ subter523971068842742411eq_a_b )
= ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ subterm_and_supt_a_b )
| ( A = B ) ) ) ).
% subterm.order.order_iff_strict
thf(fact_1183_subterm_Oorder_Ostrict__iff__order,axiom,
! [B: term_a_b,A: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ subterm_and_supt_a_b )
= ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ subter523971068842742411eq_a_b )
& ( A != B ) ) ) ).
% subterm.order.strict_iff_order
thf(fact_1184_subterm_Odual__order_Ostrict__trans1,axiom,
! [A: term_a_b,B: term_a_b,C2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ subter523971068842742411eq_a_b )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C2 ) @ subterm_and_supt_a_b )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C2 ) @ subterm_and_supt_a_b ) ) ) ).
% subterm.dual_order.strict_trans1
thf(fact_1185_subterm_Odual__order_Ostrict__trans2,axiom,
! [A: term_a_b,B: term_a_b,C2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ subterm_and_supt_a_b )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C2 ) @ subter523971068842742411eq_a_b )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C2 ) @ subterm_and_supt_a_b ) ) ) ).
% subterm.dual_order.strict_trans2
thf(fact_1186_subterm_Odual__order_Ostrict__iff__not,axiom,
! [A: term_a_b,B: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ subterm_and_supt_a_b )
= ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ subter523971068842742411eq_a_b )
& ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ subter523971068842742411eq_a_b ) ) ) ).
% subterm.dual_order.strict_iff_not
thf(fact_1187_subterm_Oorder_Ostrict__implies__order,axiom,
! [B: term_a_b,A: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ subterm_and_supt_a_b )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ subter523971068842742411eq_a_b ) ) ).
% subterm.order.strict_implies_order
thf(fact_1188_subterm_Odual__order_Oorder__iff__strict,axiom,
! [A: term_a_b,B: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ subter523971068842742411eq_a_b )
= ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ subterm_and_supt_a_b )
| ( A = B ) ) ) ).
% subterm.dual_order.order_iff_strict
thf(fact_1189_subterm_Odual__order_Ostrict__iff__order,axiom,
! [A: term_a_b,B: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ subterm_and_supt_a_b )
= ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ subter523971068842742411eq_a_b )
& ( A != B ) ) ) ).
% subterm.dual_order.strict_iff_order
thf(fact_1190_subterm_Odual__order_Ostrict__implies__order,axiom,
! [A: term_a_b,B: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ subterm_and_supt_a_b )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ subter523971068842742411eq_a_b ) ) ).
% subterm.dual_order.strict_implies_order
thf(fact_1191_subterm_Oorder_Onot__eq__order__implies__strict,axiom,
! [A: term_a_b,B: term_a_b] :
( ( A != B )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ subter523971068842742411eq_a_b )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ subterm_and_supt_a_b ) ) ) ).
% subterm.order.not_eq_order_implies_strict
thf(fact_1192_suptE,axiom,
! [S: term_a_b,T: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ subterm_and_supt_a_b )
=> ( ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ subter523971068842742411eq_a_b )
=> ( S != T ) ) ) ).
% suptE
thf(fact_1193_supt__imp__supteq,axiom,
! [S: term_a_b,T: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ subterm_and_supt_a_b )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ subter523971068842742411eq_a_b ) ) ).
% supt_imp_supteq
thf(fact_1194_supteq__not__supt,axiom,
! [S: term_a_b,T: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ subter523971068842742411eq_a_b )
=> ( ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ subterm_and_supt_a_b )
=> ( S = T ) ) ) ).
% supteq_not_supt
thf(fact_1195_supt__supteq__conv,axiom,
! [S: term_a_b,T: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ subterm_and_supt_a_b )
= ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ subter523971068842742411eq_a_b )
& ( S != T ) ) ) ).
% supt_supteq_conv
thf(fact_1196_supteq__supt__conv,axiom,
! [S: term_a_b,T: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ subter523971068842742411eq_a_b )
= ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ subterm_and_supt_a_b )
| ( S = T ) ) ) ).
% supteq_supt_conv
thf(fact_1197_supt__supteq__trans,axiom,
! [S: term_a_b,T: term_a_b,U: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ subterm_and_supt_a_b )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T @ U ) @ subter523971068842742411eq_a_b )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ U ) @ subterm_and_supt_a_b ) ) ) ).
% supt_supteq_trans
thf(fact_1198_supteq__supt__trans,axiom,
! [S: term_a_b,T: term_a_b,U: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ subter523971068842742411eq_a_b )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T @ U ) @ subterm_and_supt_a_b )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ U ) @ subterm_and_supt_a_b ) ) ) ).
% supteq_supt_trans
thf(fact_1199_supt__supteq__not__supteq,axiom,
! [S: term_a_b,T: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ subterm_and_supt_a_b )
= ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ subter523971068842742411eq_a_b )
& ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T @ S ) @ subter523971068842742411eq_a_b ) ) ) ).
% supt_supteq_not_supteq
thf(fact_1200_antisymI,axiom,
! [R2: 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 @ Y3 @ X3 ) @ R2 )
=> ( X3 = Y3 ) ) )
=> ( antisym_on_term_a_b @ top_top_set_term_a_b @ R2 ) ) ).
% antisymI
thf(fact_1201_antisymD,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,X: term_a_b,Y: term_a_b] :
( ( antisym_on_term_a_b @ top_top_set_term_a_b @ R2 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ R2 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X ) @ R2 )
=> ( X = Y ) ) ) ) ).
% antisymD
thf(fact_1202_subterm_OacyclicI__order,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,F3: term_a_b > term_a_b] :
( ! [A3: term_a_b,B3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B3 ) @ R2 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F3 @ A3 ) @ ( F3 @ B3 ) ) @ subterm_and_supt_a_b ) )
=> ( transi5314701259734769157rm_a_b @ R2 ) ) ).
% subterm.acyclicI_order
thf(fact_1203_Refl__antisym__eq__Image1__Image1__iff,axiom,
! [R2: set_Pr3451248702717554689st_nat,A: list_nat,B: list_nat] :
( ( refl_on_list_nat @ ( field_list_nat @ R2 ) @ R2 )
=> ( ( antisym_on_list_nat @ top_top_set_list_nat @ R2 )
=> ( ( member_list_nat @ A @ ( field_list_nat @ R2 ) )
=> ( ( member_list_nat @ B @ ( field_list_nat @ R2 ) )
=> ( ( ( image_2597627202720054805st_nat @ R2 @ ( insert_list_nat @ A @ bot_bot_set_list_nat ) )
= ( image_2597627202720054805st_nat @ R2 @ ( insert_list_nat @ B @ bot_bot_set_list_nat ) ) )
= ( A = B ) ) ) ) ) ) ).
% Refl_antisym_eq_Image1_Image1_iff
thf(fact_1204_Refl__antisym__eq__Image1__Image1__iff,axiom,
! [R2: set_Pr2972776593051762503rm_a_b,A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b] :
( ( refl_o5344404488616963464rm_a_b @ ( field_6884932134483023318rm_a_b @ R2 ) @ R2 )
=> ( ( antisy5041284635856308844rm_a_b @ top_to1314267278846557113rm_a_b @ R2 )
=> ( ( member5869715511025134514rm_a_b @ A @ ( field_6884932134483023318rm_a_b @ R2 ) )
=> ( ( member5869715511025134514rm_a_b @ B @ ( field_6884932134483023318rm_a_b @ R2 ) )
=> ( ( ( image_871640889055226643rm_a_b @ R2 @ ( insert7009541432154983385rm_a_b @ A @ bot_bo197521221353338581rm_a_b ) )
= ( image_871640889055226643rm_a_b @ R2 @ ( insert7009541432154983385rm_a_b @ B @ bot_bo197521221353338581rm_a_b ) ) )
= ( A = B ) ) ) ) ) ) ).
% Refl_antisym_eq_Image1_Image1_iff
thf(fact_1205_ImageI,axiom,
! [A: list_nat,B: list_nat,R2: set_Pr3451248702717554689st_nat,A4: set_list_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A @ B ) @ R2 )
=> ( ( member_list_nat @ A @ A4 )
=> ( member_list_nat @ B @ ( image_2597627202720054805st_nat @ R2 @ A4 ) ) ) ) ).
% ImageI
thf(fact_1206_ImageI,axiom,
! [A: list_nat,B: produc357393685978478089rm_a_b,R2: set_Pr1829413238147913556rm_a_b,A4: set_list_nat] :
( ( member4140859439665155893rm_a_b @ ( produc3015861401532503632rm_a_b @ A @ B ) @ R2 )
=> ( ( member_list_nat @ A @ A4 )
=> ( member5869715511025134514rm_a_b @ B @ ( image_6412611479114034964rm_a_b @ R2 @ A4 ) ) ) ) ).
% ImageI
thf(fact_1207_ImageI,axiom,
! [A: produc357393685978478089rm_a_b,B: list_nat,R2: set_Pr2659102613823815246st_nat,A4: set_Pr4386577575007340137rm_a_b] :
( ( member190277321972437807st_nat @ ( produc5056965718246025362st_nat @ A @ B ) @ R2 )
=> ( ( member5869715511025134514rm_a_b @ A @ A4 )
=> ( member_list_nat @ B @ ( image_8453715795827556694st_nat @ R2 @ A4 ) ) ) ) ).
% ImageI
thf(fact_1208_ImageI,axiom,
! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,R2: set_Pr2972776593051762503rm_a_b,A4: set_Pr4386577575007340137rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ R2 )
=> ( ( member5869715511025134514rm_a_b @ A @ A4 )
=> ( member5869715511025134514rm_a_b @ B @ ( image_871640889055226643rm_a_b @ R2 @ A4 ) ) ) ) ).
% ImageI
thf(fact_1209_ImageI,axiom,
! [A: term_a_b,B: term_a_b,R2: set_Pr4386577575007340137rm_a_b,A4: set_term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R2 )
=> ( ( member_term_a_b @ A @ A4 )
=> ( member_term_a_b @ B @ ( image_6923787618150785661rm_a_b @ R2 @ A4 ) ) ) ) ).
% ImageI
thf(fact_1210_Image__singleton__iff,axiom,
! [B: term_a_b,R2: set_Pr4386577575007340137rm_a_b,A: term_a_b] :
( ( member_term_a_b @ B @ ( image_6923787618150785661rm_a_b @ R2 @ ( insert_term_a_b @ A @ bot_bot_set_term_a_b ) ) )
= ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R2 ) ) ).
% Image_singleton_iff
thf(fact_1211_supt__const,axiom,
! [F3: a,U: term_a_b] :
~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( fun_a_b @ F3 @ nil_term_a_b ) @ U ) @ subterm_and_supt_a_b ) ).
% supt_const
thf(fact_1212_SN__on__Image__normalizable,axiom,
! [R2: set_Pr3451248702717554689st_nat,A4: set_list_nat] :
( ( abstra2262888740781824790st_nat @ R2 @ A4 )
=> ( ? [X6: list_nat] : ( member_list_nat @ X6 @ A4 )
=> ? [B3: list_nat] : ( member_list_nat @ B3 @ ( image_2597627202720054805st_nat @ ( abstra515516630445031992st_nat @ R2 ) @ A4 ) ) ) ) ).
% SN_on_Image_normalizable
thf(fact_1213_SN__on__Image__normalizable,axiom,
! [R2: set_Pr2972776593051762503rm_a_b,A4: set_Pr4386577575007340137rm_a_b] :
( ( abstra2398554102055911763rm_a_b @ R2 @ A4 )
=> ( ? [X6: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ X6 @ A4 )
=> ? [B3: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ B3 @ ( image_871640889055226643rm_a_b @ ( abstra1496011390211842225rm_a_b @ R2 ) @ A4 ) ) ) ) ).
% SN_on_Image_normalizable
thf(fact_1214_rev__ImageI,axiom,
! [A: list_nat,A4: set_list_nat,B: list_nat,R2: set_Pr3451248702717554689st_nat] :
( ( member_list_nat @ A @ A4 )
=> ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A @ B ) @ R2 )
=> ( member_list_nat @ B @ ( image_2597627202720054805st_nat @ R2 @ A4 ) ) ) ) ).
% rev_ImageI
thf(fact_1215_rev__ImageI,axiom,
! [A: list_nat,A4: set_list_nat,B: produc357393685978478089rm_a_b,R2: set_Pr1829413238147913556rm_a_b] :
( ( member_list_nat @ A @ A4 )
=> ( ( member4140859439665155893rm_a_b @ ( produc3015861401532503632rm_a_b @ A @ B ) @ R2 )
=> ( member5869715511025134514rm_a_b @ B @ ( image_6412611479114034964rm_a_b @ R2 @ A4 ) ) ) ) ).
% rev_ImageI
thf(fact_1216_rev__ImageI,axiom,
! [A: produc357393685978478089rm_a_b,A4: set_Pr4386577575007340137rm_a_b,B: list_nat,R2: set_Pr2659102613823815246st_nat] :
( ( member5869715511025134514rm_a_b @ A @ A4 )
=> ( ( member190277321972437807st_nat @ ( produc5056965718246025362st_nat @ A @ B ) @ R2 )
=> ( member_list_nat @ B @ ( image_8453715795827556694st_nat @ R2 @ A4 ) ) ) ) ).
% rev_ImageI
thf(fact_1217_rev__ImageI,axiom,
! [A: produc357393685978478089rm_a_b,A4: set_Pr4386577575007340137rm_a_b,B: produc357393685978478089rm_a_b,R2: set_Pr2972776593051762503rm_a_b] :
( ( member5869715511025134514rm_a_b @ A @ A4 )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ R2 )
=> ( member5869715511025134514rm_a_b @ B @ ( image_871640889055226643rm_a_b @ R2 @ A4 ) ) ) ) ).
% rev_ImageI
thf(fact_1218_rev__ImageI,axiom,
! [A: term_a_b,A4: set_term_a_b,B: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ( member_term_a_b @ A @ A4 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R2 )
=> ( member_term_a_b @ B @ ( image_6923787618150785661rm_a_b @ R2 @ A4 ) ) ) ) ).
% rev_ImageI
thf(fact_1219_Image__iff,axiom,
! [B: term_a_b,R2: set_Pr4386577575007340137rm_a_b,A4: set_term_a_b] :
( ( member_term_a_b @ B @ ( image_6923787618150785661rm_a_b @ R2 @ A4 ) )
= ( ? [X4: term_a_b] :
( ( member_term_a_b @ X4 @ A4 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X4 @ B ) @ R2 ) ) ) ) ).
% Image_iff
thf(fact_1220_ImageE,axiom,
! [B: list_nat,R2: set_Pr3451248702717554689st_nat,A4: set_list_nat] :
( ( member_list_nat @ B @ ( image_2597627202720054805st_nat @ R2 @ A4 ) )
=> ~ ! [X3: list_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X3 @ B ) @ R2 )
=> ~ ( member_list_nat @ X3 @ A4 ) ) ) ).
% ImageE
thf(fact_1221_ImageE,axiom,
! [B: list_nat,R2: set_Pr2659102613823815246st_nat,A4: set_Pr4386577575007340137rm_a_b] :
( ( member_list_nat @ B @ ( image_8453715795827556694st_nat @ R2 @ A4 ) )
=> ~ ! [X3: produc357393685978478089rm_a_b] :
( ( member190277321972437807st_nat @ ( produc5056965718246025362st_nat @ X3 @ B ) @ R2 )
=> ~ ( member5869715511025134514rm_a_b @ X3 @ A4 ) ) ) ).
% ImageE
thf(fact_1222_ImageE,axiom,
! [B: produc357393685978478089rm_a_b,R2: set_Pr1829413238147913556rm_a_b,A4: set_list_nat] :
( ( member5869715511025134514rm_a_b @ B @ ( image_6412611479114034964rm_a_b @ R2 @ A4 ) )
=> ~ ! [X3: list_nat] :
( ( member4140859439665155893rm_a_b @ ( produc3015861401532503632rm_a_b @ X3 @ B ) @ R2 )
=> ~ ( member_list_nat @ X3 @ A4 ) ) ) ).
% ImageE
thf(fact_1223_ImageE,axiom,
! [B: produc357393685978478089rm_a_b,R2: set_Pr2972776593051762503rm_a_b,A4: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ B @ ( image_871640889055226643rm_a_b @ R2 @ A4 ) )
=> ~ ! [X3: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X3 @ B ) @ R2 )
=> ~ ( member5869715511025134514rm_a_b @ X3 @ A4 ) ) ) ).
% ImageE
thf(fact_1224_ImageE,axiom,
! [B: term_a_b,R2: set_Pr4386577575007340137rm_a_b,A4: set_term_a_b] :
( ( member_term_a_b @ B @ ( image_6923787618150785661rm_a_b @ R2 @ A4 ) )
=> ~ ! [X3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ B ) @ R2 )
=> ~ ( member_term_a_b @ X3 @ A4 ) ) ) ).
% ImageE
thf(fact_1225_rtrancl__Image__step,axiom,
! [A: list_nat,R2: set_Pr3451248702717554689st_nat,A4: set_list_nat,B: list_nat] :
( ( member_list_nat @ A @ ( image_2597627202720054805st_nat @ ( transi5285580207609517981st_nat @ R2 ) @ A4 ) )
=> ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A @ B ) @ ( transi5285580207609517981st_nat @ R2 ) )
=> ( member_list_nat @ B @ ( image_2597627202720054805st_nat @ ( transi5285580207609517981st_nat @ R2 ) @ A4 ) ) ) ) ).
% rtrancl_Image_step
thf(fact_1226_rtrancl__Image__step,axiom,
! [A: produc357393685978478089rm_a_b,R2: set_Pr2972776593051762503rm_a_b,A4: set_Pr4386577575007340137rm_a_b,B: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ A @ ( image_871640889055226643rm_a_b @ ( transi2615809358984392588rm_a_b @ R2 ) @ A4 ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( transi2615809358984392588rm_a_b @ R2 ) )
=> ( member5869715511025134514rm_a_b @ B @ ( image_871640889055226643rm_a_b @ ( transi2615809358984392588rm_a_b @ R2 ) @ A4 ) ) ) ) ).
% rtrancl_Image_step
thf(fact_1227_rtrancl__Image__step,axiom,
! [A: term_a_b,R2: set_Pr4386577575007340137rm_a_b,A4: set_term_a_b,B: term_a_b] :
( ( member_term_a_b @ A @ ( image_6923787618150785661rm_a_b @ ( transi7742714808557438673rm_a_b @ R2 ) @ A4 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
=> ( member_term_a_b @ B @ ( image_6923787618150785661rm_a_b @ ( transi7742714808557438673rm_a_b @ R2 ) @ A4 ) ) ) ) ).
% rtrancl_Image_step
thf(fact_1228_no__step,axiom,
! [A4: set_Pr3451248702717554689st_nat,A: list_nat] :
( ( ( image_2597627202720054805st_nat @ A4 @ ( insert_list_nat @ A @ bot_bot_set_list_nat ) )
= bot_bot_set_list_nat )
=> ( member_list_nat @ A @ ( abstract_NF_list_nat @ A4 ) ) ) ).
% no_step
thf(fact_1229_no__step,axiom,
! [A4: set_Pr2972776593051762503rm_a_b,A: produc357393685978478089rm_a_b] :
( ( ( image_871640889055226643rm_a_b @ A4 @ ( insert7009541432154983385rm_a_b @ A @ bot_bo197521221353338581rm_a_b ) )
= bot_bo197521221353338581rm_a_b )
=> ( member5869715511025134514rm_a_b @ A @ ( abstra1902783451776770197rm_a_b @ A4 ) ) ) ).
% no_step
thf(fact_1230_subset__Image1__Image1__iff,axiom,
! [R2: set_Pr3451248702717554689st_nat,A: list_nat,B: list_nat] :
( ( order_4596338698039041673st_nat @ ( field_list_nat @ R2 ) @ R2 )
=> ( ( member_list_nat @ A @ ( field_list_nat @ R2 ) )
=> ( ( member_list_nat @ B @ ( field_list_nat @ R2 ) )
=> ( ( ord_le6045566169113846134st_nat @ ( image_2597627202720054805st_nat @ R2 @ ( insert_list_nat @ A @ bot_bot_set_list_nat ) ) @ ( image_2597627202720054805st_nat @ R2 @ ( insert_list_nat @ B @ bot_bot_set_list_nat ) ) )
= ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ B @ A ) @ R2 ) ) ) ) ) ).
% subset_Image1_Image1_iff
thf(fact_1231_subset__Image1__Image1__iff,axiom,
! [R2: set_Pr2972776593051762503rm_a_b,A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b] :
( ( order_4024118566430455712rm_a_b @ ( field_6884932134483023318rm_a_b @ R2 ) @ R2 )
=> ( ( member5869715511025134514rm_a_b @ A @ ( field_6884932134483023318rm_a_b @ R2 ) )
=> ( ( member5869715511025134514rm_a_b @ B @ ( field_6884932134483023318rm_a_b @ R2 ) )
=> ( ( ord_le118470702582115849rm_a_b @ ( image_871640889055226643rm_a_b @ R2 @ ( insert7009541432154983385rm_a_b @ A @ bot_bo197521221353338581rm_a_b ) ) @ ( image_871640889055226643rm_a_b @ R2 @ ( insert7009541432154983385rm_a_b @ B @ bot_bo197521221353338581rm_a_b ) ) )
= ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B @ A ) @ R2 ) ) ) ) ) ).
% subset_Image1_Image1_iff
thf(fact_1232_subset__Image1__Image1__iff,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,A: term_a_b,B: term_a_b] :
( ( order_7053473298986962365rm_a_b @ ( field_term_a_b @ R2 ) @ R2 )
=> ( ( member_term_a_b @ A @ ( field_term_a_b @ R2 ) )
=> ( ( member_term_a_b @ B @ ( field_term_a_b @ R2 ) )
=> ( ( ord_le2705286416250468010rm_a_b @ ( image_6923787618150785661rm_a_b @ R2 @ ( insert_term_a_b @ A @ bot_bot_set_term_a_b ) ) @ ( image_6923787618150785661rm_a_b @ R2 @ ( insert_term_a_b @ B @ bot_bot_set_term_a_b ) ) )
= ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ R2 ) ) ) ) ) ).
% subset_Image1_Image1_iff
thf(fact_1233_subset__Image__Image__iff,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,A4: set_term_a_b,B5: set_term_a_b] :
( ( order_7053473298986962365rm_a_b @ ( field_term_a_b @ R2 ) @ R2 )
=> ( ( ord_le2705286416250468010rm_a_b @ A4 @ ( field_term_a_b @ R2 ) )
=> ( ( ord_le2705286416250468010rm_a_b @ B5 @ ( field_term_a_b @ R2 ) )
=> ( ( ord_le2705286416250468010rm_a_b @ ( image_6923787618150785661rm_a_b @ R2 @ A4 ) @ ( image_6923787618150785661rm_a_b @ R2 @ B5 ) )
= ( ! [X4: term_a_b] :
( ( member_term_a_b @ X4 @ A4 )
=> ? [Y5: term_a_b] :
( ( member_term_a_b @ Y5 @ B5 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y5 @ X4 ) @ R2 ) ) ) ) ) ) ) ) ).
% subset_Image_Image_iff
thf(fact_1234_Partial__order__eq__Image1__Image1__iff,axiom,
! [R2: set_Pr3451248702717554689st_nat,A: list_nat,B: list_nat] :
( ( order_1646981276062705451st_nat @ ( field_list_nat @ R2 ) @ R2 )
=> ( ( member_list_nat @ A @ ( field_list_nat @ R2 ) )
=> ( ( member_list_nat @ B @ ( field_list_nat @ R2 ) )
=> ( ( ( image_2597627202720054805st_nat @ R2 @ ( insert_list_nat @ A @ bot_bot_set_list_nat ) )
= ( image_2597627202720054805st_nat @ R2 @ ( insert_list_nat @ B @ bot_bot_set_list_nat ) ) )
= ( A = B ) ) ) ) ) ).
% Partial_order_eq_Image1_Image1_iff
thf(fact_1235_Partial__order__eq__Image1__Image1__iff,axiom,
! [R2: set_Pr2972776593051762503rm_a_b,A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b] :
( ( order_8762642138211079550rm_a_b @ ( field_6884932134483023318rm_a_b @ R2 ) @ R2 )
=> ( ( member5869715511025134514rm_a_b @ A @ ( field_6884932134483023318rm_a_b @ R2 ) )
=> ( ( member5869715511025134514rm_a_b @ B @ ( field_6884932134483023318rm_a_b @ R2 ) )
=> ( ( ( image_871640889055226643rm_a_b @ R2 @ ( insert7009541432154983385rm_a_b @ A @ bot_bo197521221353338581rm_a_b ) )
= ( image_871640889055226643rm_a_b @ R2 @ ( insert7009541432154983385rm_a_b @ B @ bot_bo197521221353338581rm_a_b ) ) )
= ( A = B ) ) ) ) ) ).
% Partial_order_eq_Image1_Image1_iff
thf(fact_1236_relpow__0__E,axiom,
! [X: term_a_b,Y: term_a_b,R5: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( compow4057154403645558940rm_a_b @ zero_zero_nat @ R5 ) )
=> ( X = Y ) ) ).
% relpow_0_E
thf(fact_1237_relpow__0__I,axiom,
! [X: term_a_b,R5: set_Pr4386577575007340137rm_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ X ) @ ( compow4057154403645558940rm_a_b @ zero_zero_nat @ R5 ) ) ).
% relpow_0_I
thf(fact_1238_non__strict__ending,axiom,
! [T: nat > term_a_b,R5: set_Pr4386577575007340137rm_a_b,S6: set_Pr4386577575007340137rm_a_b] :
( ! [I3: nat] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( T @ I3 ) @ ( T @ ( suc @ I3 ) ) ) @ ( sup_su6776935440552674877rm_a_b @ R5 @ S6 ) )
=> ( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ R5 @ S6 ) @ S6 )
=> ( ( abstra4720023341729745482rm_a_b @ S6 @ ( insert_term_a_b @ ( T @ zero_zero_nat ) @ bot_bot_set_term_a_b ) )
=> ? [J3: nat] :
! [I4: nat] :
( ( ord_less_eq_nat @ J3 @ I4 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( T @ I4 ) @ ( T @ ( suc @ I4 ) ) ) @ ( minus_5192120951422937424rm_a_b @ R5 @ S6 ) ) ) ) ) ) ).
% non_strict_ending
thf(fact_1239_not__SN__onI,axiom,
! [F3: nat > list_nat,X5: set_list_nat,R5: set_Pr3451248702717554689st_nat] :
( ( member_list_nat @ ( F3 @ zero_zero_nat ) @ X5 )
=> ( ! [I3: nat] : ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( F3 @ I3 ) @ ( F3 @ ( suc @ I3 ) ) ) @ R5 )
=> ~ ( abstra2262888740781824790st_nat @ R5 @ X5 ) ) ) ).
% not_SN_onI
thf(fact_1240_not__SN__onI,axiom,
! [F3: nat > produc357393685978478089rm_a_b,X5: set_Pr4386577575007340137rm_a_b,R5: set_Pr2972776593051762503rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( F3 @ zero_zero_nat ) @ X5 )
=> ( ! [I3: nat] : ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( F3 @ I3 ) @ ( F3 @ ( suc @ I3 ) ) ) @ R5 )
=> ~ ( abstra2398554102055911763rm_a_b @ R5 @ X5 ) ) ) ).
% not_SN_onI
thf(fact_1241_not__SN__onI,axiom,
! [F3: nat > term_a_b,X5: set_term_a_b,R5: set_Pr4386577575007340137rm_a_b] :
( ( member_term_a_b @ ( F3 @ zero_zero_nat ) @ X5 )
=> ( ! [I3: nat] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F3 @ I3 ) @ ( F3 @ ( suc @ I3 ) ) ) @ R5 )
=> ~ ( abstra4720023341729745482rm_a_b @ R5 @ X5 ) ) ) ).
% not_SN_onI
thf(fact_1242_SN__onI,axiom,
! [A4: set_list_nat,R2: set_Pr3451248702717554689st_nat] :
( ! [F6: nat > list_nat] :
( ( member_list_nat @ ( F6 @ zero_zero_nat ) @ A4 )
=> ~ ! [I4: nat] : ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( F6 @ I4 ) @ ( F6 @ ( suc @ I4 ) ) ) @ R2 ) )
=> ( abstra2262888740781824790st_nat @ R2 @ A4 ) ) ).
% SN_onI
thf(fact_1243_SN__onI,axiom,
! [A4: set_Pr4386577575007340137rm_a_b,R2: set_Pr2972776593051762503rm_a_b] :
( ! [F6: nat > produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( F6 @ zero_zero_nat ) @ A4 )
=> ~ ! [I4: nat] : ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( F6 @ I4 ) @ ( F6 @ ( suc @ I4 ) ) ) @ R2 ) )
=> ( abstra2398554102055911763rm_a_b @ R2 @ A4 ) ) ).
% SN_onI
thf(fact_1244_SN__onI,axiom,
! [A4: set_term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ! [F6: nat > term_a_b] :
( ( member_term_a_b @ ( F6 @ zero_zero_nat ) @ A4 )
=> ~ ! [I4: nat] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F6 @ I4 ) @ ( F6 @ ( suc @ I4 ) ) ) @ R2 ) )
=> ( abstra4720023341729745482rm_a_b @ R2 @ A4 ) ) ).
% SN_onI
thf(fact_1245_relpow__Suc__E,axiom,
! [X: term_a_b,Z: term_a_b,N: nat,R5: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z ) @ ( compow4057154403645558940rm_a_b @ ( suc @ N ) @ R5 ) )
=> ~ ! [Y3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y3 ) @ ( compow4057154403645558940rm_a_b @ N @ R5 ) )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ Z ) @ R5 ) ) ) ).
% relpow_Suc_E
thf(fact_1246_relpow__Suc__I,axiom,
! [X: term_a_b,Y: term_a_b,N: nat,R5: set_Pr4386577575007340137rm_a_b,Z: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( compow4057154403645558940rm_a_b @ N @ R5 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ Z ) @ R5 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z ) @ ( compow4057154403645558940rm_a_b @ ( suc @ N ) @ R5 ) ) ) ) ).
% relpow_Suc_I
thf(fact_1247_relpow__Suc__D2,axiom,
! [X: term_a_b,Z: term_a_b,N: nat,R5: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z ) @ ( compow4057154403645558940rm_a_b @ ( suc @ N ) @ R5 ) )
=> ? [Y3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y3 ) @ R5 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ Z ) @ ( compow4057154403645558940rm_a_b @ N @ R5 ) ) ) ) ).
% relpow_Suc_D2
thf(fact_1248_relpow__Suc__E2,axiom,
! [X: term_a_b,Z: term_a_b,N: nat,R5: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z ) @ ( compow4057154403645558940rm_a_b @ ( suc @ N ) @ R5 ) )
=> ~ ! [Y3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y3 ) @ R5 )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ Z ) @ ( compow4057154403645558940rm_a_b @ N @ R5 ) ) ) ) ).
% relpow_Suc_E2
thf(fact_1249_relpow__Suc__I2,axiom,
! [X: term_a_b,Y: term_a_b,R5: set_Pr4386577575007340137rm_a_b,Z: term_a_b,N: nat] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ R5 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ Z ) @ ( compow4057154403645558940rm_a_b @ N @ R5 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z ) @ ( compow4057154403645558940rm_a_b @ ( suc @ N ) @ R5 ) ) ) ) ).
% relpow_Suc_I2
thf(fact_1250_Abstract__Rewriting_Ochain__mono,axiom,
! [R9: set_Pr4386577575007340137rm_a_b,R5: set_Pr4386577575007340137rm_a_b,Seq: nat > term_a_b] :
( ( ord_le118470702582115849rm_a_b @ R9 @ R5 )
=> ( ! [I3: nat] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( Seq @ I3 ) @ ( Seq @ ( suc @ I3 ) ) ) @ R9 )
=> ! [I4: nat] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( Seq @ I4 ) @ ( Seq @ ( suc @ I4 ) ) ) @ R5 ) ) ) ).
% Abstract_Rewriting.chain_mono
thf(fact_1251_not__SN__onE,axiom,
! [R2: set_Pr3451248702717554689st_nat,A4: set_list_nat] :
( ~ ( abstra2262888740781824790st_nat @ R2 @ A4 )
=> ~ ! [F6: nat > list_nat] :
( ( member_list_nat @ ( F6 @ zero_zero_nat ) @ A4 )
=> ~ ! [I4: nat] : ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( F6 @ I4 ) @ ( F6 @ ( suc @ I4 ) ) ) @ R2 ) ) ) ).
% not_SN_onE
thf(fact_1252_not__SN__onE,axiom,
! [R2: set_Pr2972776593051762503rm_a_b,A4: set_Pr4386577575007340137rm_a_b] :
( ~ ( abstra2398554102055911763rm_a_b @ R2 @ A4 )
=> ~ ! [F6: nat > produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( F6 @ zero_zero_nat ) @ A4 )
=> ~ ! [I4: nat] : ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( F6 @ I4 ) @ ( F6 @ ( suc @ I4 ) ) ) @ R2 ) ) ) ).
% not_SN_onE
thf(fact_1253_not__SN__onE,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,A4: set_term_a_b] :
( ~ ( abstra4720023341729745482rm_a_b @ R2 @ A4 )
=> ~ ! [F6: nat > term_a_b] :
( ( member_term_a_b @ ( F6 @ zero_zero_nat ) @ A4 )
=> ~ ! [I4: nat] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F6 @ I4 ) @ ( F6 @ ( suc @ I4 ) ) ) @ R2 ) ) ) ).
% not_SN_onE
thf(fact_1254_SN__on__def,axiom,
( abstra2262888740781824790st_nat
= ( ^ [R4: set_Pr3451248702717554689st_nat,A6: set_list_nat] :
~ ? [F2: nat > list_nat] :
( ( member_list_nat @ ( F2 @ zero_zero_nat ) @ A6 )
& ! [I: nat] : ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( F2 @ I ) @ ( F2 @ ( suc @ I ) ) ) @ R4 ) ) ) ) ).
% SN_on_def
thf(fact_1255_SN__on__def,axiom,
( abstra2398554102055911763rm_a_b
= ( ^ [R4: set_Pr2972776593051762503rm_a_b,A6: set_Pr4386577575007340137rm_a_b] :
~ ? [F2: nat > produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( F2 @ zero_zero_nat ) @ A6 )
& ! [I: nat] : ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( F2 @ I ) @ ( F2 @ ( suc @ I ) ) ) @ R4 ) ) ) ) ).
% SN_on_def
thf(fact_1256_SN__on__def,axiom,
( abstra4720023341729745482rm_a_b
= ( ^ [R4: set_Pr4386577575007340137rm_a_b,A6: set_term_a_b] :
~ ? [F2: nat > term_a_b] :
( ( member_term_a_b @ ( F2 @ zero_zero_nat ) @ A6 )
& ! [I: nat] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F2 @ I ) @ ( F2 @ ( suc @ I ) ) ) @ R4 ) ) ) ) ).
% SN_on_def
thf(fact_1257_SN__onE,axiom,
! [R2: set_Pr3451248702717554689st_nat,A4: set_list_nat] :
( ( abstra2262888740781824790st_nat @ R2 @ A4 )
=> ~ ? [F7: nat > list_nat] :
( ( member_list_nat @ ( F7 @ zero_zero_nat ) @ A4 )
& ! [I3: nat] : ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( F7 @ I3 ) @ ( F7 @ ( suc @ I3 ) ) ) @ R2 ) ) ) ).
% SN_onE
thf(fact_1258_SN__onE,axiom,
! [R2: set_Pr2972776593051762503rm_a_b,A4: set_Pr4386577575007340137rm_a_b] :
( ( abstra2398554102055911763rm_a_b @ R2 @ A4 )
=> ~ ? [F7: nat > produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( F7 @ zero_zero_nat ) @ A4 )
& ! [I3: nat] : ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( F7 @ I3 ) @ ( F7 @ ( suc @ I3 ) ) ) @ R2 ) ) ) ).
% SN_onE
thf(fact_1259_SN__onE,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,A4: set_term_a_b] :
( ( abstra4720023341729745482rm_a_b @ R2 @ A4 )
=> ~ ? [F7: nat > term_a_b] :
( ( member_term_a_b @ ( F7 @ zero_zero_nat ) @ A4 )
& ! [I3: nat] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F7 @ I3 ) @ ( F7 @ ( suc @ I3 ) ) ) @ R2 ) ) ) ).
% SN_onE
thf(fact_1260_subterm_Olift__Suc__antimono__le,axiom,
! [F3: nat > term_a_b,N: nat,N3: nat] :
( ! [N2: nat] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F3 @ N2 ) @ ( F3 @ ( suc @ N2 ) ) ) @ subter523971068842742411eq_a_b )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F3 @ N ) @ ( F3 @ N3 ) ) @ subter523971068842742411eq_a_b ) ) ) ).
% subterm.lift_Suc_antimono_le
thf(fact_1261_subterm_Olift__Suc__mono__le,axiom,
! [F3: nat > term_a_b,N: nat,N3: nat] :
( ! [N2: nat] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F3 @ ( suc @ N2 ) ) @ ( F3 @ N2 ) ) @ subter523971068842742411eq_a_b )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F3 @ N3 ) @ ( F3 @ N ) ) @ subter523971068842742411eq_a_b ) ) ) ).
% subterm.lift_Suc_mono_le
thf(fact_1262_relpow__E2,axiom,
! [X: term_a_b,Z: term_a_b,N: nat,R5: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z ) @ ( compow4057154403645558940rm_a_b @ N @ R5 ) )
=> ( ( ( N = zero_zero_nat )
=> ( X != Z ) )
=> ~ ! [Y3: term_a_b,M2: nat] :
( ( N
= ( suc @ M2 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y3 ) @ R5 )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ Z ) @ ( compow4057154403645558940rm_a_b @ M2 @ R5 ) ) ) ) ) ) ).
% relpow_E2
thf(fact_1263_relpow__E,axiom,
! [X: term_a_b,Z: term_a_b,N: nat,R5: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z ) @ ( compow4057154403645558940rm_a_b @ N @ R5 ) )
=> ( ( ( N = zero_zero_nat )
=> ( X != Z ) )
=> ~ ! [Y3: term_a_b,M2: nat] :
( ( N
= ( suc @ M2 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y3 ) @ ( compow4057154403645558940rm_a_b @ M2 @ R5 ) )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ Z ) @ R5 ) ) ) ) ) ).
% relpow_E
thf(fact_1264_relpow__Suc__E2_H,axiom,
! [X: term_a_b,Z: term_a_b,N: nat,A4: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z ) @ ( compow4057154403645558940rm_a_b @ ( suc @ N ) @ A4 ) )
=> ~ ! [Y3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y3 ) @ A4 )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ Z ) @ ( transi7742714808557438673rm_a_b @ A4 ) ) ) ) ).
% relpow_Suc_E2'
thf(fact_1265_chain__imp__not__SN__on,axiom,
! [F3: nat > term_a_b,R2: set_Pr4386577575007340137rm_a_b,I2: nat] :
( ! [I3: nat] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F3 @ I3 ) @ ( F3 @ ( suc @ I3 ) ) ) @ R2 )
=> ~ ( abstra4720023341729745482rm_a_b @ R2 @ ( insert_term_a_b @ ( F3 @ I2 ) @ bot_bot_set_term_a_b ) ) ) ).
% chain_imp_not_SN_on
thf(fact_1266_chain__imp__rtrancl,axiom,
! [S6: nat > term_a_b,R2: set_Pr4386577575007340137rm_a_b,I2: nat,J2: nat] :
( ! [I3: nat] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( S6 @ I3 ) @ ( S6 @ ( suc @ I3 ) ) ) @ R2 )
=> ( ( ord_less_eq_nat @ I2 @ J2 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( S6 @ I2 ) @ ( S6 @ J2 ) ) @ ( transi7742714808557438673rm_a_b @ R2 ) ) ) ) ).
% chain_imp_rtrancl
thf(fact_1267_wf__no__infinite__down__chainE,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,F3: nat > term_a_b] :
( ( wf_term_a_b @ R2 )
=> ~ ! [K: nat] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F3 @ ( suc @ K ) ) @ ( F3 @ K ) ) @ R2 ) ) ).
% wf_no_infinite_down_chainE
thf(fact_1268_wf__iff__no__infinite__down__chain,axiom,
( wf_term_a_b
= ( ^ [R4: set_Pr4386577575007340137rm_a_b] :
~ ? [F2: nat > term_a_b] :
! [I: nat] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F2 @ ( suc @ I ) ) @ ( F2 @ I ) ) @ R4 ) ) ) ).
% wf_iff_no_infinite_down_chain
thf(fact_1269_chain__Un__SN__on__imp__first__step,axiom,
! [T: nat > term_a_b,R5: set_Pr4386577575007340137rm_a_b,S6: set_Pr4386577575007340137rm_a_b] :
( ! [I3: nat] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( T @ I3 ) @ ( T @ ( suc @ I3 ) ) ) @ ( sup_su6776935440552674877rm_a_b @ R5 @ S6 ) )
=> ( ( abstra4720023341729745482rm_a_b @ S6 @ ( insert_term_a_b @ ( T @ zero_zero_nat ) @ bot_bot_set_term_a_b ) )
=> ? [I3: nat] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( T @ I3 ) @ ( T @ ( suc @ I3 ) ) ) @ R5 )
& ! [J4: nat] :
( ( ord_less_nat @ J4 @ I3 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( T @ J4 ) @ ( T @ ( suc @ J4 ) ) ) @ S6 )
& ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( T @ J4 ) @ ( T @ ( suc @ J4 ) ) ) @ R5 ) ) ) ) ) ) ).
% chain_Un_SN_on_imp_first_step
thf(fact_1270_Zorns__po__lemma,axiom,
! [R2: set_Pr4386577575007340137rm_a_b] :
( ( order_4104115877010626143rm_a_b @ ( field_term_a_b @ R2 ) @ R2 )
=> ( ! [C5: set_term_a_b] :
( ( member_set_term_a_b @ C5 @ ( chains_term_a_b @ R2 ) )
=> ? [X6: term_a_b] :
( ( member_term_a_b @ X6 @ ( field_term_a_b @ R2 ) )
& ! [Xa3: term_a_b] :
( ( member_term_a_b @ Xa3 @ C5 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Xa3 @ X6 ) @ R2 ) ) ) )
=> ? [X3: term_a_b] :
( ( member_term_a_b @ X3 @ ( field_term_a_b @ R2 ) )
& ! [Xa2: term_a_b] :
( ( member_term_a_b @ Xa2 @ ( field_term_a_b @ R2 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Xa2 ) @ R2 )
=> ( Xa2 = X3 ) ) ) ) ) ) ).
% Zorns_po_lemma
thf(fact_1271_chain__imp__trancl,axiom,
! [S6: nat > term_a_b,R2: set_Pr4386577575007340137rm_a_b,I2: nat,J2: nat] :
( ! [I3: nat] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( S6 @ I3 ) @ ( S6 @ ( suc @ I3 ) ) ) @ R2 )
=> ( ( ord_less_nat @ I2 @ J2 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( S6 @ I2 ) @ ( S6 @ J2 ) ) @ ( transi7922773638565587891rm_a_b @ R2 ) ) ) ) ).
% chain_imp_trancl
thf(fact_1272_subterm_Olift__Suc__mono__less,axiom,
! [F3: nat > term_a_b,N: nat,N3: nat] :
( ! [N2: nat] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F3 @ ( suc @ N2 ) ) @ ( F3 @ N2 ) ) @ subterm_and_supt_a_b )
=> ( ( ord_less_nat @ N @ N3 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F3 @ N3 ) @ ( F3 @ N ) ) @ subterm_and_supt_a_b ) ) ) ).
% subterm.lift_Suc_mono_less
thf(fact_1273_subterm_Olift__Suc__mono__less__iff,axiom,
! [F3: nat > term_a_b,M: nat,N: nat] :
( ! [N2: nat] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F3 @ ( suc @ N2 ) ) @ ( F3 @ N2 ) ) @ subterm_and_supt_a_b )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F3 @ M ) @ ( F3 @ N ) ) @ subterm_and_supt_a_b )
= ( ord_less_nat @ N @ M ) ) ) ).
% subterm.lift_Suc_mono_less_iff
thf(fact_1274_wf__bounded__measure,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,Ub: term_a_b > nat,F3: term_a_b > nat] :
( ! [A3: term_a_b,B3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B3 @ A3 ) @ R2 )
=> ( ( ord_less_eq_nat @ ( Ub @ B3 ) @ ( Ub @ A3 ) )
& ( ord_less_eq_nat @ ( F3 @ B3 ) @ ( Ub @ A3 ) )
& ( ord_less_nat @ ( F3 @ A3 ) @ ( F3 @ B3 ) ) ) )
=> ( wf_term_a_b @ R2 ) ) ).
% wf_bounded_measure
% Conjectures (2)
thf(conj_0,hypothesis,
! [P6: list_nat,L6: term_a_b,R10: term_a_b,Sigma4: b > term_a_b] :
( ( member_list_nat @ P6 @ ( term_poss_a_b @ s ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ L6 @ R10 ) @ r )
=> ( ( ( term_subt_at_a_b @ s @ P6 )
= ( subst_7999470309526761004_a_b_b @ L6 @ Sigma4 ) )
=> ( ( t
= ( term_r6860082780075436317at_a_b @ s @ P6 @ ( subst_7999470309526761004_a_b_b @ R10 @ Sigma4 ) ) )
=> thesis ) ) ) ) ).
thf(conj_1,conjecture,
thesis ).
%------------------------------------------------------------------------------