TPTP Problem File: SLH0170^1.p
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%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Rewrite_Properties_Reduction/0016_Rewriting/prob_00228_008029__13792544_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1539 ( 375 unt; 259 typ; 0 def)
% Number of atoms : 4132 ( 886 equ; 0 cnn)
% Maximal formula atoms : 12 ( 3 avg)
% Number of connectives : 14371 ( 454 ~; 43 |; 272 &;11513 @)
% ( 0 <=>;2089 =>; 0 <=; 0 <~>)
% Maximal formula depth : 22 ( 8 avg)
% Number of types : 25 ( 24 usr)
% Number of type conns : 637 ( 637 >; 0 *; 0 +; 0 <<)
% Number of symbols : 236 ( 235 usr; 15 con; 0-4 aty)
% Number of variables : 4152 ( 259 ^;3744 !; 149 ?;4152 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 14:18:24.499
%------------------------------------------------------------------------------
% Could-be-implicit typings (24)
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transi2615809358984392588rm_a_b: set_Pr2972776593051762503rm_a_b > set_Pr2972776593051762503rm_a_b ).
thf(sy_c_Transitive__Closure_Ortrancl_001t__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J,type,
transi2726145917338391738_a_nat: set_Pr1811044260758604347_a_nat > set_Pr1811044260758604347_a_nat ).
thf(sy_c_Transitive__Closure_Ortrancl_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
transi7742714808557438673rm_a_b: set_Pr4386577575007340137rm_a_b > set_Pr4386577575007340137rm_a_b ).
thf(sy_c_Transitive__Closure_Ortrancl_001tf__a,type,
transitive_rtrancl_a: set_Product_prod_a_a > set_Product_prod_a_a ).
thf(sy_c_Transitive__Closure_Otrancl_001t__Nat__Onat,type,
transi6264000038957366511cl_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).
thf(sy_c_Transitive__Closure_Otrancl_001t__Product____Type__Oprod_It__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__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J,type,
transi6805586507763294364_a_nat: set_Pr1811044260758604347_a_nat > set_Pr1811044260758604347_a_nat ).
thf(sy_c_Transitive__Closure_Otrancl_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
transi7922773638565587891rm_a_b: set_Pr4386577575007340137rm_a_b > set_Pr4386577575007340137rm_a_b ).
thf(sy_c_Transitive__Closure_Otrancl_001tf__a,type,
transitive_trancl_a: set_Product_prod_a_a > set_Product_prod_a_a ).
thf(sy_c_Wellfounded_Olex__prod_001t__Term__Oterm_Itf__a_Mtf__b_J_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
lex_pr6163557265797435481rm_a_b: set_Pr4386577575007340137rm_a_b > set_Pr4386577575007340137rm_a_b > set_Pr2972776593051762503rm_a_b ).
thf(sy_c_Wellfounded_Omax__ext_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
max_ext_term_a_b: set_Pr4386577575007340137rm_a_b > set_Pr7151917219565317845rm_a_b ).
thf(sy_c_Wellfounded_Omeasure_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
measure_term_a_b: ( term_a_b > nat ) > set_Pr4386577575007340137rm_a_b ).
thf(sy_c_Wellfounded_Omin__ext_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
min_ext_term_a_b: set_Pr4386577575007340137rm_a_b > set_Pr7151917219565317845rm_a_b ).
thf(sy_c_Wellfounded_Omlex__prod_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
mlex_prod_term_a_b: ( term_a_b > nat ) > set_Pr4386577575007340137rm_a_b > set_Pr4386577575007340137rm_a_b ).
thf(sy_c_Wellfounded_Owf_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
wf_Pro2335863617654816626rm_a_b: set_Pr2972776593051762503rm_a_b > $o ).
thf(sy_c_Wellfounded_Owf_001t__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J,type,
wf_Pro8109884346892000340_a_nat: set_Pr1811044260758604347_a_nat > $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_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
member8440522571783428010at_nat: product_prod_nat_nat > set_Pr1261947904930325089at_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mtf__a_J,type,
member8962352052110095674_nat_a: product_prod_nat_a > set_Pr4193341848836149977_nat_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_Mt__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
member8417600551952982416rm_a_b: produc4523971788519308903rm_a_b > set_Pr2972776593051762503rm_a_b > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_J,type,
member9062615507155100804_a_nat: produc4708774622424448987_a_nat > set_Pr1811044260758604347_a_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__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__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
member1426531477525435216od_a_a: product_prod_a_a > set_Product_prod_a_a > $o ).
thf(sy_c_member_001t__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 ).
% Relevant facts (1275)
thf(fact_0_rstep__ruleI,axiom,
! [L: term_a_b,R: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ L @ R ) @ R2 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ L @ R ) @ ( rstep_a_b @ R2 ) ) ) ).
% rstep_ruleI
thf(fact_1_sig__stepI,axiom,
! [S: term_a_b,F: set_Pr4934435412358123699_a_nat,T: term_a_b,R2: 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 ) @ R2 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( sig_step_a_b @ F @ R2 ) ) ) ) ) ).
% sig_stepI
thf(fact_2_srstepD,axiom,
! [S: term_a_b,T: term_a_b,F: set_Pr4934435412358123699_a_nat,R2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( sig_step_a_b @ F @ ( rstep_a_b @ R2 ) ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( rstep_a_b @ R2 ) )
& ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ S ) @ F )
& ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ T ) @ F ) ) ) ).
% srstepD
thf(fact_3_sig__stepE,axiom,
! [S: term_a_b,T: term_a_b,F: set_Pr4934435412358123699_a_nat,R2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( sig_step_a_b @ F @ R2 ) )
=> ~ ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ R2 )
=> ( ( 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_4_subsetI,axiom,
! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
( ! [X: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X @ A )
=> ( member8440522571783428010at_nat @ X @ B ) )
=> ( ord_le3146513528884898305at_nat @ A @ B ) ) ).
% subsetI
thf(fact_5_subsetI,axiom,
! [A: set_Pr4193341848836149977_nat_a,B: set_Pr4193341848836149977_nat_a] :
( ! [X: product_prod_nat_a] :
( ( member8962352052110095674_nat_a @ X @ A )
=> ( member8962352052110095674_nat_a @ X @ B ) )
=> ( ord_le7924913712489149241_nat_a @ A @ B ) ) ).
% subsetI
thf(fact_6_subsetI,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ! [X: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X @ A )
=> ( member1426531477525435216od_a_a @ X @ B ) )
=> ( ord_le746702958409616551od_a_a @ A @ B ) ) ).
% subsetI
thf(fact_7_subsetI,axiom,
! [A: set_term_a_b,B: set_term_a_b] :
( ! [X: term_a_b] :
( ( member_term_a_b @ X @ A )
=> ( member_term_a_b @ X @ B ) )
=> ( ord_le2705286416250468010rm_a_b @ A @ B ) ) ).
% subsetI
thf(fact_8_subsetI,axiom,
! [A: set_nat,B: set_nat] :
( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( member_nat @ X @ B ) )
=> ( ord_less_eq_set_nat @ A @ B ) ) ).
% subsetI
thf(fact_9_subsetI,axiom,
! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ! [X: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X @ A )
=> ( member5869715511025134514rm_a_b @ X @ B ) )
=> ( ord_le118470702582115849rm_a_b @ A @ B ) ) ).
% subsetI
thf(fact_10_subsetI,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ! [X: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ X @ A )
=> ( member5724188588386418708_a_nat @ X @ B ) )
=> ( ord_le8666007276011122963_a_nat @ A @ B ) ) ).
% subsetI
thf(fact_11_subset__antisym,axiom,
! [A: set_Pr4193341848836149977_nat_a,B: set_Pr4193341848836149977_nat_a] :
( ( ord_le7924913712489149241_nat_a @ A @ B )
=> ( ( ord_le7924913712489149241_nat_a @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_12_subset__antisym,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A @ B )
=> ( ( ord_le746702958409616551od_a_a @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_13_subset__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 ) ) ) ).
% subset_antisym
thf(fact_14_subset__antisym,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_15_subset__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 ) ) ) ).
% subset_antisym
thf(fact_16_subset__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 ) ) ) ).
% subset_antisym
thf(fact_17_prod_Oinject,axiom,
! [X1: set_term_a_b,X2: set_term_a_b,Y1: set_term_a_b,Y2: set_term_a_b] :
( ( ( produc8517751122659631533rm_a_b @ X1 @ X2 )
= ( produc8517751122659631533rm_a_b @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_18_prod_Oinject,axiom,
! [X1: nat,X2: nat,Y1: nat,Y2: nat] :
( ( ( product_Pair_nat_nat @ X1 @ X2 )
= ( product_Pair_nat_nat @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_19_prod_Oinject,axiom,
! [X1: nat,X2: a,Y1: nat,Y2: a] :
( ( ( product_Pair_nat_a @ X1 @ X2 )
= ( product_Pair_nat_a @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_20_prod_Oinject,axiom,
! [X1: a,X2: nat,Y1: a,Y2: nat] :
( ( ( product_Pair_a_nat @ X1 @ X2 )
= ( product_Pair_a_nat @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_21_prod_Oinject,axiom,
! [X1: a,X2: a,Y1: a,Y2: a] :
( ( ( product_Pair_a_a @ X1 @ X2 )
= ( product_Pair_a_a @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_22_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_23_old_Oprod_Oinject,axiom,
! [A2: set_term_a_b,B2: set_term_a_b,A3: set_term_a_b,B3: set_term_a_b] :
( ( ( produc8517751122659631533rm_a_b @ A2 @ B2 )
= ( produc8517751122659631533rm_a_b @ A3 @ B3 ) )
= ( ( A2 = A3 )
& ( B2 = B3 ) ) ) ).
% old.prod.inject
thf(fact_24_old_Oprod_Oinject,axiom,
! [A2: nat,B2: nat,A3: nat,B3: nat] :
( ( ( product_Pair_nat_nat @ A2 @ B2 )
= ( product_Pair_nat_nat @ A3 @ B3 ) )
= ( ( A2 = A3 )
& ( B2 = B3 ) ) ) ).
% old.prod.inject
thf(fact_25_old_Oprod_Oinject,axiom,
! [A2: nat,B2: a,A3: nat,B3: a] :
( ( ( product_Pair_nat_a @ A2 @ B2 )
= ( product_Pair_nat_a @ A3 @ B3 ) )
= ( ( A2 = A3 )
& ( B2 = B3 ) ) ) ).
% old.prod.inject
thf(fact_26_old_Oprod_Oinject,axiom,
! [A2: a,B2: nat,A3: a,B3: nat] :
( ( ( product_Pair_a_nat @ A2 @ B2 )
= ( product_Pair_a_nat @ A3 @ B3 ) )
= ( ( A2 = A3 )
& ( B2 = B3 ) ) ) ).
% old.prod.inject
thf(fact_27_old_Oprod_Oinject,axiom,
! [A2: a,B2: a,A3: a,B3: a] :
( ( ( product_Pair_a_a @ A2 @ B2 )
= ( product_Pair_a_a @ A3 @ B3 ) )
= ( ( A2 = A3 )
& ( B2 = B3 ) ) ) ).
% old.prod.inject
thf(fact_28_old_Oprod_Oinject,axiom,
! [A2: term_a_b,B2: term_a_b,A3: term_a_b,B3: term_a_b] :
( ( ( produc7020197800436672577rm_a_b @ A2 @ B2 )
= ( produc7020197800436672577rm_a_b @ A3 @ B3 ) )
= ( ( A2 = A3 )
& ( B2 = B3 ) ) ) ).
% old.prod.inject
thf(fact_29_order__refl,axiom,
! [X3: set_Pr4193341848836149977_nat_a] : ( ord_le7924913712489149241_nat_a @ X3 @ X3 ) ).
% order_refl
thf(fact_30_order__refl,axiom,
! [X3: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ X3 @ X3 ) ).
% order_refl
thf(fact_31_order__refl,axiom,
! [X3: set_term_a_b] : ( ord_le2705286416250468010rm_a_b @ X3 @ X3 ) ).
% order_refl
thf(fact_32_order__refl,axiom,
! [X3: set_nat] : ( ord_less_eq_set_nat @ X3 @ X3 ) ).
% order_refl
thf(fact_33_order__refl,axiom,
! [X3: nat] : ( ord_less_eq_nat @ X3 @ X3 ) ).
% order_refl
thf(fact_34_order__refl,axiom,
! [X3: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ X3 @ X3 ) ).
% order_refl
thf(fact_35_order__refl,axiom,
! [X3: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ X3 @ X3 ) ).
% order_refl
thf(fact_36_dual__order_Orefl,axiom,
! [A2: set_Pr4193341848836149977_nat_a] : ( ord_le7924913712489149241_nat_a @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_37_dual__order_Orefl,axiom,
! [A2: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_38_dual__order_Orefl,axiom,
! [A2: set_term_a_b] : ( ord_le2705286416250468010rm_a_b @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_39_dual__order_Orefl,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_40_dual__order_Orefl,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_41_dual__order_Orefl,axiom,
! [A2: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_42_dual__order_Orefl,axiom,
! [A2: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_43_rstep__term__to__sig__l,axiom,
! [S: term_a_b,T: term_a_b,R2: set_Pr4386577575007340137rm_a_b,F: set_Pr4934435412358123699_a_nat,V: b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( rstep_a_b @ R2 ) )
=> ( ( ord_le8666007276011122963_a_nat @ ( terms_7988297476397195622_a_b_b @ R2 ) @ 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 @ R2 ) ) ) ) ) ).
% rstep_term_to_sig_l
thf(fact_44_rstep__term__to__sig__r,axiom,
! [S: term_a_b,T: term_a_b,R2: set_Pr4386577575007340137rm_a_b,F: set_Pr4934435412358123699_a_nat,V: b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( rstep_a_b @ R2 ) )
=> ( ( ord_le8666007276011122963_a_nat @ ( terms_7988297476397195622_a_b_b @ R2 ) @ 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 @ R2 ) ) ) ) ) ).
% rstep_term_to_sig_r
thf(fact_45_subrelI,axiom,
! [R: set_Pr7151917219565317845rm_a_b,S: set_Pr7151917219565317845rm_a_b] :
( ! [X: set_term_a_b,Y: set_term_a_b] :
( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ X @ Y ) @ R )
=> ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ X @ Y ) @ S ) )
=> ( ord_le2663425029446080117rm_a_b @ R @ S ) ) ).
% subrelI
thf(fact_46_subrelI,axiom,
! [R: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
( ! [X: nat,Y: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ S ) )
=> ( ord_le3146513528884898305at_nat @ R @ S ) ) ).
% subrelI
thf(fact_47_subrelI,axiom,
! [R: set_Pr4193341848836149977_nat_a,S: set_Pr4193341848836149977_nat_a] :
( ! [X: nat,Y: a] :
( ( member8962352052110095674_nat_a @ ( product_Pair_nat_a @ X @ Y ) @ R )
=> ( member8962352052110095674_nat_a @ ( product_Pair_nat_a @ X @ Y ) @ S ) )
=> ( ord_le7924913712489149241_nat_a @ R @ S ) ) ).
% subrelI
thf(fact_48_subrelI,axiom,
! [R: set_Product_prod_a_a,S: set_Product_prod_a_a] :
( ! [X: a,Y: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ S ) )
=> ( ord_le746702958409616551od_a_a @ R @ S ) ) ).
% subrelI
thf(fact_49_subrelI,axiom,
! [R: set_Pr4934435412358123699_a_nat,S: set_Pr4934435412358123699_a_nat] :
( ! [X: a,Y: nat] :
( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ X @ Y ) @ R )
=> ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ X @ Y ) @ S ) )
=> ( ord_le8666007276011122963_a_nat @ R @ S ) ) ).
% subrelI
thf(fact_50_subrelI,axiom,
! [R: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
( ! [X: term_a_b,Y: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ R )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ S ) )
=> ( ord_le118470702582115849rm_a_b @ R @ S ) ) ).
% subrelI
thf(fact_51_srrstepD,axiom,
! [S: term_a_b,T: term_a_b,F: set_Pr4934435412358123699_a_nat,R2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( sig_step_a_b @ F @ ( rrstep_a_b @ R2 ) ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( rrstep_a_b @ R2 ) )
& ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ S ) @ F )
& ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ T ) @ F ) ) ) ).
% srrstepD
thf(fact_52_srstepsD,axiom,
! [S: term_a_b,T: term_a_b,F: set_Pr4934435412358123699_a_nat,R2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( transi7922773638565587891rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R2 ) ) ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( transi7922773638565587891rm_a_b @ ( rstep_a_b @ R2 ) ) )
& ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ S ) @ F )
& ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ T ) @ F ) ) ) ).
% srstepsD
thf(fact_53_relChain__def,axiom,
( bNF_Ca9222207453146494317_a_nat
= ( ^ [R3: 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 ) @ R3 )
=> ( ord_le8666007276011122963_a_nat @ ( As @ I ) @ ( As @ J ) ) ) ) ) ).
% relChain_def
thf(fact_54_relChain__def,axiom,
( bNF_Ca8501653506855686447rm_a_b
= ( ^ [R3: set_Pr4386577575007340137rm_a_b,As: term_a_b > set_Pr4386577575007340137rm_a_b] :
! [I: term_a_b,J: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ I @ J ) @ R3 )
=> ( ord_le118470702582115849rm_a_b @ ( As @ I ) @ ( As @ J ) ) ) ) ) ).
% relChain_def
thf(fact_55_relChain__def,axiom,
( bNF_Ca968750328013420230at_nat
= ( ^ [R3: set_Pr1261947904930325089at_nat,As: nat > nat] :
! [I: nat,J: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I @ J ) @ R3 )
=> ( ord_less_eq_nat @ ( As @ I ) @ ( As @ J ) ) ) ) ) ).
% relChain_def
thf(fact_56_relChain__def,axiom,
( bNF_Ca2675338678332661802_a_nat
= ( ^ [R3: set_Product_prod_a_a,As: a > nat] :
! [I: a,J: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ I @ J ) @ R3 )
=> ( ord_less_eq_nat @ ( As @ I ) @ ( As @ J ) ) ) ) ) ).
% relChain_def
thf(fact_57_relChain__def,axiom,
( bNF_Ca4389690986151823996et_nat
= ( ^ [R3: set_Pr1261947904930325089at_nat,As: nat > set_nat] :
! [I: nat,J: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I @ J ) @ R3 )
=> ( ord_less_eq_set_nat @ ( As @ I ) @ ( As @ J ) ) ) ) ) ).
% relChain_def
thf(fact_58_relChain__def,axiom,
( bNF_Ca5884907880858117088et_nat
= ( ^ [R3: set_Product_prod_a_a,As: a > set_nat] :
! [I: a,J: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ I @ J ) @ R3 )
=> ( ord_less_eq_set_nat @ ( As @ I ) @ ( As @ J ) ) ) ) ) ).
% relChain_def
thf(fact_59_relChain__def,axiom,
( bNF_Ca2413601583040309898_b_nat
= ( ^ [R3: 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 ) @ R3 )
=> ( ord_less_eq_nat @ ( As @ I ) @ ( As @ J ) ) ) ) ) ).
% relChain_def
thf(fact_60_relChain__def,axiom,
( bNF_Ca2301798522795029929_a_nat
= ( ^ [R3: set_Pr1261947904930325089at_nat,As: nat > set_Pr4934435412358123699_a_nat] :
! [I: nat,J: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I @ J ) @ R3 )
=> ( ord_le8666007276011122963_a_nat @ ( As @ I ) @ ( As @ J ) ) ) ) ) ).
% relChain_def
thf(fact_61_relChain__def,axiom,
( bNF_Ca6844405654680374029_a_nat
= ( ^ [R3: set_Product_prod_a_a,As: a > set_Pr4934435412358123699_a_nat] :
! [I: a,J: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ I @ J ) @ R3 )
=> ( ord_le8666007276011122963_a_nat @ ( As @ I ) @ ( As @ J ) ) ) ) ) ).
% relChain_def
thf(fact_62_relChain__def,axiom,
( bNF_Ca1560704959273056207_nat_a
= ( ^ [R3: set_Pr1261947904930325089at_nat,As: nat > set_Pr4193341848836149977_nat_a] :
! [I: nat,J: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I @ J ) @ R3 )
=> ( ord_le7924913712489149241_nat_a @ ( As @ I ) @ ( As @ J ) ) ) ) ) ).
% relChain_def
thf(fact_63_rrstep__basicI,axiom,
! [L: term_a_b,R: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ L @ R ) @ R2 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ L @ R ) @ ( rrstep_a_b @ R2 ) ) ) ).
% rrstep_basicI
thf(fact_64_rrstep__rstep__mono,axiom,
! [R2: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ ( rrstep_a_b @ R2 ) @ ( rstep_a_b @ R2 ) ) ).
% rrstep_rstep_mono
thf(fact_65_srrstep__monp,axiom,
! [F: set_Pr4934435412358123699_a_nat,G: set_Pr4934435412358123699_a_nat,R2: set_Pr4386577575007340137rm_a_b] :
( ( ord_le8666007276011122963_a_nat @ F @ G )
=> ( ord_le118470702582115849rm_a_b @ ( sig_step_a_b @ F @ ( rrstep_a_b @ R2 ) ) @ ( sig_step_a_b @ G @ ( rrstep_a_b @ R2 ) ) ) ) ).
% srrstep_monp
thf(fact_66_sig__step__mono2,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,L2: set_Pr4386577575007340137rm_a_b,F: set_Pr4934435412358123699_a_nat] :
( ( ord_le118470702582115849rm_a_b @ R2 @ L2 )
=> ( ord_le118470702582115849rm_a_b @ ( sig_step_a_b @ F @ R2 ) @ ( sig_step_a_b @ F @ L2 ) ) ) ).
% sig_step_mono2
thf(fact_67_srsteps__monp,axiom,
! [F: set_Pr4934435412358123699_a_nat,G: set_Pr4934435412358123699_a_nat,R2: 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 @ R2 ) ) ) @ ( transi7922773638565587891rm_a_b @ ( sig_step_a_b @ G @ ( rstep_a_b @ R2 ) ) ) ) ) ).
% srsteps_monp
thf(fact_68_sig__step__mono,axiom,
! [F: set_Pr4934435412358123699_a_nat,G: set_Pr4934435412358123699_a_nat,R2: set_Pr4386577575007340137rm_a_b] :
( ( ord_le8666007276011122963_a_nat @ F @ G )
=> ( ord_le118470702582115849rm_a_b @ ( sig_step_a_b @ F @ R2 ) @ ( sig_step_a_b @ G @ R2 ) ) ) ).
% sig_step_mono
thf(fact_69_order__antisym__conv,axiom,
! [Y3: set_Pr4193341848836149977_nat_a,X3: set_Pr4193341848836149977_nat_a] :
( ( ord_le7924913712489149241_nat_a @ Y3 @ X3 )
=> ( ( ord_le7924913712489149241_nat_a @ X3 @ Y3 )
= ( X3 = Y3 ) ) ) ).
% order_antisym_conv
thf(fact_70_order__antisym__conv,axiom,
! [Y3: set_Product_prod_a_a,X3: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ Y3 @ X3 )
=> ( ( ord_le746702958409616551od_a_a @ X3 @ Y3 )
= ( X3 = Y3 ) ) ) ).
% order_antisym_conv
thf(fact_71_order__antisym__conv,axiom,
! [Y3: set_term_a_b,X3: set_term_a_b] :
( ( ord_le2705286416250468010rm_a_b @ Y3 @ X3 )
=> ( ( ord_le2705286416250468010rm_a_b @ X3 @ Y3 )
= ( X3 = Y3 ) ) ) ).
% order_antisym_conv
thf(fact_72_order__antisym__conv,axiom,
! [Y3: set_nat,X3: set_nat] :
( ( ord_less_eq_set_nat @ Y3 @ X3 )
=> ( ( ord_less_eq_set_nat @ X3 @ Y3 )
= ( X3 = Y3 ) ) ) ).
% order_antisym_conv
thf(fact_73_order__antisym__conv,axiom,
! [Y3: nat,X3: nat] :
( ( ord_less_eq_nat @ Y3 @ X3 )
=> ( ( ord_less_eq_nat @ X3 @ Y3 )
= ( X3 = Y3 ) ) ) ).
% order_antisym_conv
thf(fact_74_order__antisym__conv,axiom,
! [Y3: set_Pr4934435412358123699_a_nat,X3: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ Y3 @ X3 )
=> ( ( ord_le8666007276011122963_a_nat @ X3 @ Y3 )
= ( X3 = Y3 ) ) ) ).
% order_antisym_conv
thf(fact_75_order__antisym__conv,axiom,
! [Y3: set_Pr4386577575007340137rm_a_b,X3: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ Y3 @ X3 )
=> ( ( ord_le118470702582115849rm_a_b @ X3 @ Y3 )
= ( X3 = Y3 ) ) ) ).
% order_antisym_conv
thf(fact_76_linorder__le__cases,axiom,
! [X3: nat,Y3: nat] :
( ~ ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ X3 ) ) ).
% linorder_le_cases
thf(fact_77_ord__le__eq__subst,axiom,
! [A2: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat,F2: set_Pr4934435412358123699_a_nat > set_Pr4934435412358123699_a_nat,C: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A2 @ B2 )
=> ( ( ( F2 @ B2 )
= C )
=> ( ! [X: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X @ Y )
=> ( ord_le8666007276011122963_a_nat @ ( F2 @ X ) @ ( F2 @ Y ) ) )
=> ( ord_le8666007276011122963_a_nat @ ( F2 @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_78_ord__le__eq__subst,axiom,
! [A2: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat,F2: set_Pr4934435412358123699_a_nat > set_Pr4386577575007340137rm_a_b,C: set_Pr4386577575007340137rm_a_b] :
( ( ord_le8666007276011122963_a_nat @ A2 @ B2 )
=> ( ( ( F2 @ B2 )
= C )
=> ( ! [X: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X @ Y )
=> ( ord_le118470702582115849rm_a_b @ ( F2 @ X ) @ ( F2 @ Y ) ) )
=> ( ord_le118470702582115849rm_a_b @ ( F2 @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_79_ord__le__eq__subst,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b,F2: set_Pr4386577575007340137rm_a_b > set_Pr4934435412358123699_a_nat,C: set_Pr4934435412358123699_a_nat] :
( ( ord_le118470702582115849rm_a_b @ A2 @ B2 )
=> ( ( ( F2 @ B2 )
= C )
=> ( ! [X: set_Pr4386577575007340137rm_a_b,Y: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ X @ Y )
=> ( ord_le8666007276011122963_a_nat @ ( F2 @ X ) @ ( F2 @ Y ) ) )
=> ( ord_le8666007276011122963_a_nat @ ( F2 @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_80_ord__le__eq__subst,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b,F2: set_Pr4386577575007340137rm_a_b > set_Pr4386577575007340137rm_a_b,C: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A2 @ B2 )
=> ( ( ( F2 @ B2 )
= C )
=> ( ! [X: set_Pr4386577575007340137rm_a_b,Y: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ X @ Y )
=> ( ord_le118470702582115849rm_a_b @ ( F2 @ X ) @ ( F2 @ Y ) ) )
=> ( ord_le118470702582115849rm_a_b @ ( F2 @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_81_ord__le__eq__subst,axiom,
! [A2: nat,B2: nat,F2: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ( F2 @ B2 )
= C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat @ ( F2 @ X ) @ ( F2 @ Y ) ) )
=> ( ord_less_eq_nat @ ( F2 @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_82_ord__le__eq__subst,axiom,
! [A2: set_nat,B2: set_nat,F2: set_nat > nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ( F2 @ B2 )
= C )
=> ( ! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ord_less_eq_nat @ ( F2 @ X ) @ ( F2 @ Y ) ) )
=> ( ord_less_eq_nat @ ( F2 @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_83_ord__le__eq__subst,axiom,
! [A2: nat,B2: nat,F2: nat > set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ( F2 @ B2 )
= C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_set_nat @ ( F2 @ X ) @ ( F2 @ Y ) ) )
=> ( ord_less_eq_set_nat @ ( F2 @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_84_ord__le__eq__subst,axiom,
! [A2: set_nat,B2: set_nat,F2: set_nat > set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ( F2 @ B2 )
= C )
=> ( ! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ord_less_eq_set_nat @ ( F2 @ X ) @ ( F2 @ Y ) ) )
=> ( ord_less_eq_set_nat @ ( F2 @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_85_ord__le__eq__subst,axiom,
! [A2: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat,F2: set_Pr4934435412358123699_a_nat > nat,C: nat] :
( ( ord_le8666007276011122963_a_nat @ A2 @ B2 )
=> ( ( ( F2 @ B2 )
= C )
=> ( ! [X: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X @ Y )
=> ( ord_less_eq_nat @ ( F2 @ X ) @ ( F2 @ Y ) ) )
=> ( ord_less_eq_nat @ ( F2 @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_86_ord__le__eq__subst,axiom,
! [A2: set_Pr4193341848836149977_nat_a,B2: set_Pr4193341848836149977_nat_a,F2: set_Pr4193341848836149977_nat_a > nat,C: nat] :
( ( ord_le7924913712489149241_nat_a @ A2 @ B2 )
=> ( ( ( F2 @ B2 )
= C )
=> ( ! [X: set_Pr4193341848836149977_nat_a,Y: set_Pr4193341848836149977_nat_a] :
( ( ord_le7924913712489149241_nat_a @ X @ Y )
=> ( ord_less_eq_nat @ ( F2 @ X ) @ ( F2 @ Y ) ) )
=> ( ord_less_eq_nat @ ( F2 @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_87_ord__eq__le__subst,axiom,
! [A2: set_Pr4934435412358123699_a_nat,F2: set_Pr4934435412358123699_a_nat > set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat,C: set_Pr4934435412358123699_a_nat] :
( ( A2
= ( F2 @ B2 ) )
=> ( ( ord_le8666007276011122963_a_nat @ B2 @ C )
=> ( ! [X: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X @ Y )
=> ( ord_le8666007276011122963_a_nat @ ( F2 @ X ) @ ( F2 @ Y ) ) )
=> ( ord_le8666007276011122963_a_nat @ A2 @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_88_ord__eq__le__subst,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,F2: set_Pr4934435412358123699_a_nat > set_Pr4386577575007340137rm_a_b,B2: set_Pr4934435412358123699_a_nat,C: set_Pr4934435412358123699_a_nat] :
( ( A2
= ( F2 @ B2 ) )
=> ( ( ord_le8666007276011122963_a_nat @ B2 @ C )
=> ( ! [X: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X @ Y )
=> ( ord_le118470702582115849rm_a_b @ ( F2 @ X ) @ ( F2 @ Y ) ) )
=> ( ord_le118470702582115849rm_a_b @ A2 @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_89_ord__eq__le__subst,axiom,
! [A2: set_Pr4934435412358123699_a_nat,F2: set_Pr4386577575007340137rm_a_b > set_Pr4934435412358123699_a_nat,B2: set_Pr4386577575007340137rm_a_b,C: set_Pr4386577575007340137rm_a_b] :
( ( A2
= ( F2 @ B2 ) )
=> ( ( ord_le118470702582115849rm_a_b @ B2 @ C )
=> ( ! [X: set_Pr4386577575007340137rm_a_b,Y: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ X @ Y )
=> ( ord_le8666007276011122963_a_nat @ ( F2 @ X ) @ ( F2 @ Y ) ) )
=> ( ord_le8666007276011122963_a_nat @ A2 @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_90_ord__eq__le__subst,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,F2: set_Pr4386577575007340137rm_a_b > set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b,C: set_Pr4386577575007340137rm_a_b] :
( ( A2
= ( F2 @ B2 ) )
=> ( ( ord_le118470702582115849rm_a_b @ B2 @ C )
=> ( ! [X: set_Pr4386577575007340137rm_a_b,Y: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ X @ Y )
=> ( ord_le118470702582115849rm_a_b @ ( F2 @ X ) @ ( F2 @ Y ) ) )
=> ( ord_le118470702582115849rm_a_b @ A2 @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_91_ord__eq__le__subst,axiom,
! [A2: nat,F2: nat > nat,B2: nat,C: nat] :
( ( A2
= ( F2 @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat @ ( F2 @ X ) @ ( F2 @ Y ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_92_ord__eq__le__subst,axiom,
! [A2: nat,F2: set_nat > nat,B2: set_nat,C: set_nat] :
( ( A2
= ( F2 @ B2 ) )
=> ( ( ord_less_eq_set_nat @ B2 @ C )
=> ( ! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ord_less_eq_nat @ ( F2 @ X ) @ ( F2 @ Y ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_93_ord__eq__le__subst,axiom,
! [A2: set_nat,F2: nat > set_nat,B2: nat,C: nat] :
( ( A2
= ( F2 @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_set_nat @ ( F2 @ X ) @ ( F2 @ Y ) ) )
=> ( ord_less_eq_set_nat @ A2 @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_94_ord__eq__le__subst,axiom,
! [A2: set_nat,F2: set_nat > set_nat,B2: set_nat,C: set_nat] :
( ( A2
= ( F2 @ B2 ) )
=> ( ( ord_less_eq_set_nat @ B2 @ C )
=> ( ! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ord_less_eq_set_nat @ ( F2 @ X ) @ ( F2 @ Y ) ) )
=> ( ord_less_eq_set_nat @ A2 @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_95_ord__eq__le__subst,axiom,
! [A2: nat,F2: set_Pr4934435412358123699_a_nat > nat,B2: set_Pr4934435412358123699_a_nat,C: set_Pr4934435412358123699_a_nat] :
( ( A2
= ( F2 @ B2 ) )
=> ( ( ord_le8666007276011122963_a_nat @ B2 @ C )
=> ( ! [X: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X @ Y )
=> ( ord_less_eq_nat @ ( F2 @ X ) @ ( F2 @ Y ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_96_ord__eq__le__subst,axiom,
! [A2: nat,F2: set_Pr4193341848836149977_nat_a > nat,B2: set_Pr4193341848836149977_nat_a,C: set_Pr4193341848836149977_nat_a] :
( ( A2
= ( F2 @ B2 ) )
=> ( ( ord_le7924913712489149241_nat_a @ B2 @ C )
=> ( ! [X: set_Pr4193341848836149977_nat_a,Y: set_Pr4193341848836149977_nat_a] :
( ( ord_le7924913712489149241_nat_a @ X @ Y )
=> ( ord_less_eq_nat @ ( F2 @ X ) @ ( F2 @ Y ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_97_linorder__linear,axiom,
! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
| ( ord_less_eq_nat @ Y3 @ X3 ) ) ).
% linorder_linear
thf(fact_98_order__eq__refl,axiom,
! [X3: set_Pr4193341848836149977_nat_a,Y3: set_Pr4193341848836149977_nat_a] :
( ( X3 = Y3 )
=> ( ord_le7924913712489149241_nat_a @ X3 @ Y3 ) ) ).
% order_eq_refl
thf(fact_99_order__eq__refl,axiom,
! [X3: set_Product_prod_a_a,Y3: set_Product_prod_a_a] :
( ( X3 = Y3 )
=> ( ord_le746702958409616551od_a_a @ X3 @ Y3 ) ) ).
% order_eq_refl
thf(fact_100_order__eq__refl,axiom,
! [X3: set_term_a_b,Y3: set_term_a_b] :
( ( X3 = Y3 )
=> ( ord_le2705286416250468010rm_a_b @ X3 @ Y3 ) ) ).
% order_eq_refl
thf(fact_101_order__eq__refl,axiom,
! [X3: set_nat,Y3: set_nat] :
( ( X3 = Y3 )
=> ( ord_less_eq_set_nat @ X3 @ Y3 ) ) ).
% order_eq_refl
thf(fact_102_order__eq__refl,axiom,
! [X3: nat,Y3: nat] :
( ( X3 = Y3 )
=> ( ord_less_eq_nat @ X3 @ Y3 ) ) ).
% order_eq_refl
thf(fact_103_order__eq__refl,axiom,
! [X3: set_Pr4934435412358123699_a_nat,Y3: set_Pr4934435412358123699_a_nat] :
( ( X3 = Y3 )
=> ( ord_le8666007276011122963_a_nat @ X3 @ Y3 ) ) ).
% order_eq_refl
thf(fact_104_order__eq__refl,axiom,
! [X3: set_Pr4386577575007340137rm_a_b,Y3: set_Pr4386577575007340137rm_a_b] :
( ( X3 = Y3 )
=> ( ord_le118470702582115849rm_a_b @ X3 @ Y3 ) ) ).
% order_eq_refl
thf(fact_105_order__subst2,axiom,
! [A2: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat,F2: set_Pr4934435412358123699_a_nat > set_Pr4934435412358123699_a_nat,C: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A2 @ B2 )
=> ( ( ord_le8666007276011122963_a_nat @ ( F2 @ B2 ) @ C )
=> ( ! [X: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X @ Y )
=> ( ord_le8666007276011122963_a_nat @ ( F2 @ X ) @ ( F2 @ Y ) ) )
=> ( ord_le8666007276011122963_a_nat @ ( F2 @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_106_order__subst2,axiom,
! [A2: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat,F2: set_Pr4934435412358123699_a_nat > set_Pr4386577575007340137rm_a_b,C: set_Pr4386577575007340137rm_a_b] :
( ( ord_le8666007276011122963_a_nat @ A2 @ B2 )
=> ( ( ord_le118470702582115849rm_a_b @ ( F2 @ B2 ) @ C )
=> ( ! [X: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X @ Y )
=> ( ord_le118470702582115849rm_a_b @ ( F2 @ X ) @ ( F2 @ Y ) ) )
=> ( ord_le118470702582115849rm_a_b @ ( F2 @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_107_order__subst2,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b,F2: set_Pr4386577575007340137rm_a_b > set_Pr4934435412358123699_a_nat,C: set_Pr4934435412358123699_a_nat] :
( ( ord_le118470702582115849rm_a_b @ A2 @ B2 )
=> ( ( ord_le8666007276011122963_a_nat @ ( F2 @ B2 ) @ C )
=> ( ! [X: set_Pr4386577575007340137rm_a_b,Y: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ X @ Y )
=> ( ord_le8666007276011122963_a_nat @ ( F2 @ X ) @ ( F2 @ Y ) ) )
=> ( ord_le8666007276011122963_a_nat @ ( F2 @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_108_order__subst2,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b,F2: set_Pr4386577575007340137rm_a_b > set_Pr4386577575007340137rm_a_b,C: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A2 @ B2 )
=> ( ( ord_le118470702582115849rm_a_b @ ( F2 @ B2 ) @ C )
=> ( ! [X: set_Pr4386577575007340137rm_a_b,Y: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ X @ Y )
=> ( ord_le118470702582115849rm_a_b @ ( F2 @ X ) @ ( F2 @ Y ) ) )
=> ( ord_le118470702582115849rm_a_b @ ( F2 @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_109_order__subst2,axiom,
! [A2: nat,B2: nat,F2: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F2 @ B2 ) @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat @ ( F2 @ X ) @ ( F2 @ Y ) ) )
=> ( ord_less_eq_nat @ ( F2 @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_110_order__subst2,axiom,
! [A2: set_nat,B2: set_nat,F2: set_nat > nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F2 @ B2 ) @ C )
=> ( ! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ord_less_eq_nat @ ( F2 @ X ) @ ( F2 @ Y ) ) )
=> ( ord_less_eq_nat @ ( F2 @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_111_order__subst2,axiom,
! [A2: nat,B2: nat,F2: nat > set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_nat @ ( F2 @ B2 ) @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_set_nat @ ( F2 @ X ) @ ( F2 @ Y ) ) )
=> ( ord_less_eq_set_nat @ ( F2 @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_112_order__subst2,axiom,
! [A2: set_nat,B2: set_nat,F2: set_nat > set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_nat @ ( F2 @ B2 ) @ C )
=> ( ! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ord_less_eq_set_nat @ ( F2 @ X ) @ ( F2 @ Y ) ) )
=> ( ord_less_eq_set_nat @ ( F2 @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_113_order__subst2,axiom,
! [A2: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat,F2: set_Pr4934435412358123699_a_nat > nat,C: nat] :
( ( ord_le8666007276011122963_a_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F2 @ B2 ) @ C )
=> ( ! [X: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X @ Y )
=> ( ord_less_eq_nat @ ( F2 @ X ) @ ( F2 @ Y ) ) )
=> ( ord_less_eq_nat @ ( F2 @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_114_order__subst2,axiom,
! [A2: set_Pr4193341848836149977_nat_a,B2: set_Pr4193341848836149977_nat_a,F2: set_Pr4193341848836149977_nat_a > nat,C: nat] :
( ( ord_le7924913712489149241_nat_a @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F2 @ B2 ) @ C )
=> ( ! [X: set_Pr4193341848836149977_nat_a,Y: set_Pr4193341848836149977_nat_a] :
( ( ord_le7924913712489149241_nat_a @ X @ Y )
=> ( ord_less_eq_nat @ ( F2 @ X ) @ ( F2 @ Y ) ) )
=> ( ord_less_eq_nat @ ( F2 @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_115_order__subst1,axiom,
! [A2: set_Pr4934435412358123699_a_nat,F2: set_Pr4934435412358123699_a_nat > set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat,C: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A2 @ ( F2 @ B2 ) )
=> ( ( ord_le8666007276011122963_a_nat @ B2 @ C )
=> ( ! [X: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X @ Y )
=> ( ord_le8666007276011122963_a_nat @ ( F2 @ X ) @ ( F2 @ Y ) ) )
=> ( ord_le8666007276011122963_a_nat @ A2 @ ( F2 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_116_order__subst1,axiom,
! [A2: set_Pr4934435412358123699_a_nat,F2: set_Pr4386577575007340137rm_a_b > set_Pr4934435412358123699_a_nat,B2: set_Pr4386577575007340137rm_a_b,C: set_Pr4386577575007340137rm_a_b] :
( ( ord_le8666007276011122963_a_nat @ A2 @ ( F2 @ B2 ) )
=> ( ( ord_le118470702582115849rm_a_b @ B2 @ C )
=> ( ! [X: set_Pr4386577575007340137rm_a_b,Y: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ X @ Y )
=> ( ord_le8666007276011122963_a_nat @ ( F2 @ X ) @ ( F2 @ Y ) ) )
=> ( ord_le8666007276011122963_a_nat @ A2 @ ( F2 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_117_order__subst1,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,F2: set_Pr4934435412358123699_a_nat > set_Pr4386577575007340137rm_a_b,B2: set_Pr4934435412358123699_a_nat,C: set_Pr4934435412358123699_a_nat] :
( ( ord_le118470702582115849rm_a_b @ A2 @ ( F2 @ B2 ) )
=> ( ( ord_le8666007276011122963_a_nat @ B2 @ C )
=> ( ! [X: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X @ Y )
=> ( ord_le118470702582115849rm_a_b @ ( F2 @ X ) @ ( F2 @ Y ) ) )
=> ( ord_le118470702582115849rm_a_b @ A2 @ ( F2 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_118_order__subst1,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,F2: set_Pr4386577575007340137rm_a_b > set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b,C: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A2 @ ( F2 @ B2 ) )
=> ( ( ord_le118470702582115849rm_a_b @ B2 @ C )
=> ( ! [X: set_Pr4386577575007340137rm_a_b,Y: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ X @ Y )
=> ( ord_le118470702582115849rm_a_b @ ( F2 @ X ) @ ( F2 @ Y ) ) )
=> ( ord_le118470702582115849rm_a_b @ A2 @ ( F2 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_119_order__subst1,axiom,
! [A2: nat,F2: nat > nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( F2 @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat @ ( F2 @ X ) @ ( F2 @ Y ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F2 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_120_order__subst1,axiom,
! [A2: set_nat,F2: nat > set_nat,B2: nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ ( F2 @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_set_nat @ ( F2 @ X ) @ ( F2 @ Y ) ) )
=> ( ord_less_eq_set_nat @ A2 @ ( F2 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_121_order__subst1,axiom,
! [A2: nat,F2: set_nat > nat,B2: set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A2 @ ( F2 @ B2 ) )
=> ( ( ord_less_eq_set_nat @ B2 @ C )
=> ( ! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ord_less_eq_nat @ ( F2 @ X ) @ ( F2 @ Y ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F2 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_122_order__subst1,axiom,
! [A2: set_nat,F2: set_nat > set_nat,B2: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ ( F2 @ B2 ) )
=> ( ( ord_less_eq_set_nat @ B2 @ C )
=> ( ! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ord_less_eq_set_nat @ ( F2 @ X ) @ ( F2 @ Y ) ) )
=> ( ord_less_eq_set_nat @ A2 @ ( F2 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_123_order__subst1,axiom,
! [A2: set_Pr4934435412358123699_a_nat,F2: nat > set_Pr4934435412358123699_a_nat,B2: nat,C: nat] :
( ( ord_le8666007276011122963_a_nat @ A2 @ ( F2 @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ord_le8666007276011122963_a_nat @ ( F2 @ X ) @ ( F2 @ Y ) ) )
=> ( ord_le8666007276011122963_a_nat @ A2 @ ( F2 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_124_order__subst1,axiom,
! [A2: set_Pr4193341848836149977_nat_a,F2: nat > set_Pr4193341848836149977_nat_a,B2: nat,C: nat] :
( ( ord_le7924913712489149241_nat_a @ A2 @ ( F2 @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ord_le7924913712489149241_nat_a @ ( F2 @ X ) @ ( F2 @ Y ) ) )
=> ( ord_le7924913712489149241_nat_a @ A2 @ ( F2 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_125_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_Pr4193341848836149977_nat_a,Z: set_Pr4193341848836149977_nat_a] : ( Y4 = Z ) )
= ( ^ [A4: set_Pr4193341848836149977_nat_a,B4: set_Pr4193341848836149977_nat_a] :
( ( ord_le7924913712489149241_nat_a @ A4 @ B4 )
& ( ord_le7924913712489149241_nat_a @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_126_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_Product_prod_a_a,Z: set_Product_prod_a_a] : ( Y4 = Z ) )
= ( ^ [A4: set_Product_prod_a_a,B4: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A4 @ B4 )
& ( ord_le746702958409616551od_a_a @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_127_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_term_a_b,Z: set_term_a_b] : ( Y4 = Z ) )
= ( ^ [A4: set_term_a_b,B4: set_term_a_b] :
( ( ord_le2705286416250468010rm_a_b @ A4 @ B4 )
& ( ord_le2705286416250468010rm_a_b @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_128_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_nat,Z: set_nat] : ( Y4 = Z ) )
= ( ^ [A4: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B4 )
& ( ord_less_eq_set_nat @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_129_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: nat,Z: nat] : ( Y4 = Z ) )
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ A4 @ B4 )
& ( ord_less_eq_nat @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_130_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_Pr4934435412358123699_a_nat,Z: set_Pr4934435412358123699_a_nat] : ( Y4 = Z ) )
= ( ^ [A4: set_Pr4934435412358123699_a_nat,B4: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A4 @ B4 )
& ( ord_le8666007276011122963_a_nat @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_131_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_Pr4386577575007340137rm_a_b,Z: set_Pr4386577575007340137rm_a_b] : ( Y4 = Z ) )
= ( ^ [A4: set_Pr4386577575007340137rm_a_b,B4: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A4 @ B4 )
& ( ord_le118470702582115849rm_a_b @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_132_antisym,axiom,
! [A2: set_Pr4193341848836149977_nat_a,B2: set_Pr4193341848836149977_nat_a] :
( ( ord_le7924913712489149241_nat_a @ A2 @ B2 )
=> ( ( ord_le7924913712489149241_nat_a @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_133_antisym,axiom,
! [A2: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A2 @ B2 )
=> ( ( ord_le746702958409616551od_a_a @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_134_antisym,axiom,
! [A2: set_term_a_b,B2: set_term_a_b] :
( ( ord_le2705286416250468010rm_a_b @ A2 @ B2 )
=> ( ( ord_le2705286416250468010rm_a_b @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_135_antisym,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_136_antisym,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_137_antisym,axiom,
! [A2: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A2 @ B2 )
=> ( ( ord_le8666007276011122963_a_nat @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_138_antisym,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A2 @ B2 )
=> ( ( ord_le118470702582115849rm_a_b @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_139_dual__order_Otrans,axiom,
! [B2: set_Pr4193341848836149977_nat_a,A2: set_Pr4193341848836149977_nat_a,C: set_Pr4193341848836149977_nat_a] :
( ( ord_le7924913712489149241_nat_a @ B2 @ A2 )
=> ( ( ord_le7924913712489149241_nat_a @ C @ B2 )
=> ( ord_le7924913712489149241_nat_a @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_140_dual__order_Otrans,axiom,
! [B2: set_Product_prod_a_a,A2: set_Product_prod_a_a,C: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ B2 @ A2 )
=> ( ( ord_le746702958409616551od_a_a @ C @ B2 )
=> ( ord_le746702958409616551od_a_a @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_141_dual__order_Otrans,axiom,
! [B2: set_term_a_b,A2: set_term_a_b,C: set_term_a_b] :
( ( ord_le2705286416250468010rm_a_b @ B2 @ A2 )
=> ( ( ord_le2705286416250468010rm_a_b @ C @ B2 )
=> ( ord_le2705286416250468010rm_a_b @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_142_dual__order_Otrans,axiom,
! [B2: set_nat,A2: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( ( ord_less_eq_set_nat @ C @ B2 )
=> ( ord_less_eq_set_nat @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_143_dual__order_Otrans,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( ord_less_eq_nat @ C @ B2 )
=> ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_144_dual__order_Otrans,axiom,
! [B2: set_Pr4934435412358123699_a_nat,A2: set_Pr4934435412358123699_a_nat,C: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ B2 @ A2 )
=> ( ( ord_le8666007276011122963_a_nat @ C @ B2 )
=> ( ord_le8666007276011122963_a_nat @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_145_dual__order_Otrans,axiom,
! [B2: set_Pr4386577575007340137rm_a_b,A2: set_Pr4386577575007340137rm_a_b,C: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ B2 @ A2 )
=> ( ( ord_le118470702582115849rm_a_b @ C @ B2 )
=> ( ord_le118470702582115849rm_a_b @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_146_dual__order_Oantisym,axiom,
! [B2: set_Pr4193341848836149977_nat_a,A2: set_Pr4193341848836149977_nat_a] :
( ( ord_le7924913712489149241_nat_a @ B2 @ A2 )
=> ( ( ord_le7924913712489149241_nat_a @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_147_dual__order_Oantisym,axiom,
! [B2: set_Product_prod_a_a,A2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ B2 @ A2 )
=> ( ( ord_le746702958409616551od_a_a @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_148_dual__order_Oantisym,axiom,
! [B2: set_term_a_b,A2: set_term_a_b] :
( ( ord_le2705286416250468010rm_a_b @ B2 @ A2 )
=> ( ( ord_le2705286416250468010rm_a_b @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_149_dual__order_Oantisym,axiom,
! [B2: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_150_dual__order_Oantisym,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( ord_less_eq_nat @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_151_dual__order_Oantisym,axiom,
! [B2: set_Pr4934435412358123699_a_nat,A2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ B2 @ A2 )
=> ( ( ord_le8666007276011122963_a_nat @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_152_dual__order_Oantisym,axiom,
! [B2: set_Pr4386577575007340137rm_a_b,A2: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ B2 @ A2 )
=> ( ( ord_le118470702582115849rm_a_b @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_153_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_Pr4193341848836149977_nat_a,Z: set_Pr4193341848836149977_nat_a] : ( Y4 = Z ) )
= ( ^ [A4: set_Pr4193341848836149977_nat_a,B4: set_Pr4193341848836149977_nat_a] :
( ( ord_le7924913712489149241_nat_a @ B4 @ A4 )
& ( ord_le7924913712489149241_nat_a @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_154_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_Product_prod_a_a,Z: set_Product_prod_a_a] : ( Y4 = Z ) )
= ( ^ [A4: set_Product_prod_a_a,B4: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ B4 @ A4 )
& ( ord_le746702958409616551od_a_a @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_155_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_term_a_b,Z: set_term_a_b] : ( Y4 = Z ) )
= ( ^ [A4: set_term_a_b,B4: set_term_a_b] :
( ( ord_le2705286416250468010rm_a_b @ B4 @ A4 )
& ( ord_le2705286416250468010rm_a_b @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_156_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_nat,Z: set_nat] : ( Y4 = Z ) )
= ( ^ [A4: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ A4 )
& ( ord_less_eq_set_nat @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_157_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: nat,Z: nat] : ( Y4 = Z ) )
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ B4 @ A4 )
& ( ord_less_eq_nat @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_158_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_Pr4934435412358123699_a_nat,Z: set_Pr4934435412358123699_a_nat] : ( Y4 = Z ) )
= ( ^ [A4: set_Pr4934435412358123699_a_nat,B4: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ B4 @ A4 )
& ( ord_le8666007276011122963_a_nat @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_159_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_Pr4386577575007340137rm_a_b,Z: set_Pr4386577575007340137rm_a_b] : ( Y4 = Z ) )
= ( ^ [A4: set_Pr4386577575007340137rm_a_b,B4: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ B4 @ A4 )
& ( ord_le118470702582115849rm_a_b @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_160_linorder__wlog,axiom,
! [P: nat > nat > $o,A2: nat,B2: nat] :
( ! [A5: nat,B5: nat] :
( ( ord_less_eq_nat @ A5 @ B5 )
=> ( P @ A5 @ B5 ) )
=> ( ! [A5: nat,B5: nat] :
( ( P @ B5 @ A5 )
=> ( P @ A5 @ B5 ) )
=> ( P @ A2 @ B2 ) ) ) ).
% linorder_wlog
thf(fact_161_order__trans,axiom,
! [X3: set_Pr4193341848836149977_nat_a,Y3: set_Pr4193341848836149977_nat_a,Z2: set_Pr4193341848836149977_nat_a] :
( ( ord_le7924913712489149241_nat_a @ X3 @ Y3 )
=> ( ( ord_le7924913712489149241_nat_a @ Y3 @ Z2 )
=> ( ord_le7924913712489149241_nat_a @ X3 @ Z2 ) ) ) ).
% order_trans
thf(fact_162_order__trans,axiom,
! [X3: set_Product_prod_a_a,Y3: set_Product_prod_a_a,Z2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ X3 @ Y3 )
=> ( ( ord_le746702958409616551od_a_a @ Y3 @ Z2 )
=> ( ord_le746702958409616551od_a_a @ X3 @ Z2 ) ) ) ).
% order_trans
thf(fact_163_order__trans,axiom,
! [X3: set_term_a_b,Y3: set_term_a_b,Z2: set_term_a_b] :
( ( ord_le2705286416250468010rm_a_b @ X3 @ Y3 )
=> ( ( ord_le2705286416250468010rm_a_b @ Y3 @ Z2 )
=> ( ord_le2705286416250468010rm_a_b @ X3 @ Z2 ) ) ) ).
% order_trans
thf(fact_164_order__trans,axiom,
! [X3: set_nat,Y3: set_nat,Z2: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y3 )
=> ( ( ord_less_eq_set_nat @ Y3 @ Z2 )
=> ( ord_less_eq_set_nat @ X3 @ Z2 ) ) ) ).
% order_trans
thf(fact_165_order__trans,axiom,
! [X3: nat,Y3: nat,Z2: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ( ord_less_eq_nat @ Y3 @ Z2 )
=> ( ord_less_eq_nat @ X3 @ Z2 ) ) ) ).
% order_trans
thf(fact_166_order__trans,axiom,
! [X3: set_Pr4934435412358123699_a_nat,Y3: set_Pr4934435412358123699_a_nat,Z2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X3 @ Y3 )
=> ( ( ord_le8666007276011122963_a_nat @ Y3 @ Z2 )
=> ( ord_le8666007276011122963_a_nat @ X3 @ Z2 ) ) ) ).
% order_trans
thf(fact_167_order__trans,axiom,
! [X3: set_Pr4386577575007340137rm_a_b,Y3: set_Pr4386577575007340137rm_a_b,Z2: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ X3 @ Y3 )
=> ( ( ord_le118470702582115849rm_a_b @ Y3 @ Z2 )
=> ( ord_le118470702582115849rm_a_b @ X3 @ Z2 ) ) ) ).
% order_trans
thf(fact_168_order_Otrans,axiom,
! [A2: set_Pr4193341848836149977_nat_a,B2: set_Pr4193341848836149977_nat_a,C: set_Pr4193341848836149977_nat_a] :
( ( ord_le7924913712489149241_nat_a @ A2 @ B2 )
=> ( ( ord_le7924913712489149241_nat_a @ B2 @ C )
=> ( ord_le7924913712489149241_nat_a @ A2 @ C ) ) ) ).
% order.trans
thf(fact_169_order_Otrans,axiom,
! [A2: set_Product_prod_a_a,B2: set_Product_prod_a_a,C: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A2 @ B2 )
=> ( ( ord_le746702958409616551od_a_a @ B2 @ C )
=> ( ord_le746702958409616551od_a_a @ A2 @ C ) ) ) ).
% order.trans
thf(fact_170_order_Otrans,axiom,
! [A2: set_term_a_b,B2: set_term_a_b,C: set_term_a_b] :
( ( ord_le2705286416250468010rm_a_b @ A2 @ B2 )
=> ( ( ord_le2705286416250468010rm_a_b @ B2 @ C )
=> ( ord_le2705286416250468010rm_a_b @ A2 @ C ) ) ) ).
% order.trans
thf(fact_171_order_Otrans,axiom,
! [A2: set_nat,B2: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ C )
=> ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).
% order.trans
thf(fact_172_order_Otrans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% order.trans
thf(fact_173_order_Otrans,axiom,
! [A2: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat,C: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A2 @ B2 )
=> ( ( ord_le8666007276011122963_a_nat @ B2 @ C )
=> ( ord_le8666007276011122963_a_nat @ A2 @ C ) ) ) ).
% order.trans
thf(fact_174_order_Otrans,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b,C: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A2 @ B2 )
=> ( ( ord_le118470702582115849rm_a_b @ B2 @ C )
=> ( ord_le118470702582115849rm_a_b @ A2 @ C ) ) ) ).
% order.trans
thf(fact_175_order__antisym,axiom,
! [X3: set_Pr4193341848836149977_nat_a,Y3: set_Pr4193341848836149977_nat_a] :
( ( ord_le7924913712489149241_nat_a @ X3 @ Y3 )
=> ( ( ord_le7924913712489149241_nat_a @ Y3 @ X3 )
=> ( X3 = Y3 ) ) ) ).
% order_antisym
thf(fact_176_order__antisym,axiom,
! [X3: set_Product_prod_a_a,Y3: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ X3 @ Y3 )
=> ( ( ord_le746702958409616551od_a_a @ Y3 @ X3 )
=> ( X3 = Y3 ) ) ) ).
% order_antisym
thf(fact_177_order__antisym,axiom,
! [X3: set_term_a_b,Y3: set_term_a_b] :
( ( ord_le2705286416250468010rm_a_b @ X3 @ Y3 )
=> ( ( ord_le2705286416250468010rm_a_b @ Y3 @ X3 )
=> ( X3 = Y3 ) ) ) ).
% order_antisym
thf(fact_178_order__antisym,axiom,
! [X3: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y3 )
=> ( ( ord_less_eq_set_nat @ Y3 @ X3 )
=> ( X3 = Y3 ) ) ) ).
% order_antisym
thf(fact_179_order__antisym,axiom,
! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ( ord_less_eq_nat @ Y3 @ X3 )
=> ( X3 = Y3 ) ) ) ).
% order_antisym
thf(fact_180_order__antisym,axiom,
! [X3: set_Pr4934435412358123699_a_nat,Y3: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X3 @ Y3 )
=> ( ( ord_le8666007276011122963_a_nat @ Y3 @ X3 )
=> ( X3 = Y3 ) ) ) ).
% order_antisym
thf(fact_181_order__antisym,axiom,
! [X3: set_Pr4386577575007340137rm_a_b,Y3: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ X3 @ Y3 )
=> ( ( ord_le118470702582115849rm_a_b @ Y3 @ X3 )
=> ( X3 = Y3 ) ) ) ).
% order_antisym
thf(fact_182_ord__le__eq__trans,axiom,
! [A2: set_Pr4193341848836149977_nat_a,B2: set_Pr4193341848836149977_nat_a,C: set_Pr4193341848836149977_nat_a] :
( ( ord_le7924913712489149241_nat_a @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_le7924913712489149241_nat_a @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_183_ord__le__eq__trans,axiom,
! [A2: set_Product_prod_a_a,B2: set_Product_prod_a_a,C: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_le746702958409616551od_a_a @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_184_ord__le__eq__trans,axiom,
! [A2: set_term_a_b,B2: set_term_a_b,C: set_term_a_b] :
( ( ord_le2705286416250468010rm_a_b @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_le2705286416250468010rm_a_b @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_185_ord__le__eq__trans,axiom,
! [A2: set_nat,B2: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_186_ord__le__eq__trans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_187_ord__le__eq__trans,axiom,
! [A2: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat,C: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_le8666007276011122963_a_nat @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_188_ord__le__eq__trans,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b,C: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_le118470702582115849rm_a_b @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_189_ord__eq__le__trans,axiom,
! [A2: set_Pr4193341848836149977_nat_a,B2: set_Pr4193341848836149977_nat_a,C: set_Pr4193341848836149977_nat_a] :
( ( A2 = B2 )
=> ( ( ord_le7924913712489149241_nat_a @ B2 @ C )
=> ( ord_le7924913712489149241_nat_a @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_190_ord__eq__le__trans,axiom,
! [A2: set_Product_prod_a_a,B2: set_Product_prod_a_a,C: set_Product_prod_a_a] :
( ( A2 = B2 )
=> ( ( ord_le746702958409616551od_a_a @ B2 @ C )
=> ( ord_le746702958409616551od_a_a @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_191_ord__eq__le__trans,axiom,
! [A2: set_term_a_b,B2: set_term_a_b,C: set_term_a_b] :
( ( A2 = B2 )
=> ( ( ord_le2705286416250468010rm_a_b @ B2 @ C )
=> ( ord_le2705286416250468010rm_a_b @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_192_ord__eq__le__trans,axiom,
! [A2: set_nat,B2: set_nat,C: set_nat] :
( ( A2 = B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ C )
=> ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_193_ord__eq__le__trans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( A2 = B2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_194_ord__eq__le__trans,axiom,
! [A2: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat,C: set_Pr4934435412358123699_a_nat] :
( ( A2 = B2 )
=> ( ( ord_le8666007276011122963_a_nat @ B2 @ C )
=> ( ord_le8666007276011122963_a_nat @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_195_ord__eq__le__trans,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b,C: set_Pr4386577575007340137rm_a_b] :
( ( A2 = B2 )
=> ( ( ord_le118470702582115849rm_a_b @ B2 @ C )
=> ( ord_le118470702582115849rm_a_b @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_196_mem__Collect__eq,axiom,
! [A2: product_prod_nat_nat,P: product_prod_nat_nat > $o] :
( ( member8440522571783428010at_nat @ A2 @ ( collec3392354462482085612at_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_197_mem__Collect__eq,axiom,
! [A2: product_prod_nat_a,P: product_prod_nat_a > $o] :
( ( member8962352052110095674_nat_a @ A2 @ ( collec7702297998945444472_nat_a @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_198_mem__Collect__eq,axiom,
! [A2: product_prod_a_a,P: product_prod_a_a > $o] :
( ( member1426531477525435216od_a_a @ A2 @ ( collec3336397797384452498od_a_a @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_199_mem__Collect__eq,axiom,
! [A2: term_a_b,P: term_a_b > $o] :
( ( member_term_a_b @ A2 @ ( collect_term_a_b @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_200_mem__Collect__eq,axiom,
! [A2: nat,P: nat > $o] :
( ( member_nat @ A2 @ ( collect_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_201_mem__Collect__eq,axiom,
! [A2: produc357393685978478089rm_a_b,P: produc357393685978478089rm_a_b > $o] :
( ( member5869715511025134514rm_a_b @ A2 @ ( collec99177395925924084rm_a_b @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_202_mem__Collect__eq,axiom,
! [A2: product_prod_a_nat,P: product_prod_a_nat > $o] :
( ( member5724188588386418708_a_nat @ A2 @ ( collec4464134535221767506_a_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_203_Collect__mem__eq,axiom,
! [A: set_Pr1261947904930325089at_nat] :
( ( collec3392354462482085612at_nat
@ ^ [X4: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X4 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_204_Collect__mem__eq,axiom,
! [A: set_Pr4193341848836149977_nat_a] :
( ( collec7702297998945444472_nat_a
@ ^ [X4: product_prod_nat_a] : ( member8962352052110095674_nat_a @ X4 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_205_Collect__mem__eq,axiom,
! [A: set_Product_prod_a_a] :
( ( collec3336397797384452498od_a_a
@ ^ [X4: product_prod_a_a] : ( member1426531477525435216od_a_a @ X4 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_206_Collect__mem__eq,axiom,
! [A: set_term_a_b] :
( ( collect_term_a_b
@ ^ [X4: term_a_b] : ( member_term_a_b @ X4 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_207_Collect__mem__eq,axiom,
! [A: set_nat] :
( ( collect_nat
@ ^ [X4: nat] : ( member_nat @ X4 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_208_Collect__mem__eq,axiom,
! [A: set_Pr4386577575007340137rm_a_b] :
( ( collec99177395925924084rm_a_b
@ ^ [X4: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ X4 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_209_Collect__mem__eq,axiom,
! [A: set_Pr4934435412358123699_a_nat] :
( ( collec4464134535221767506_a_nat
@ ^ [X4: product_prod_a_nat] : ( member5724188588386418708_a_nat @ X4 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_210_Collect__cong,axiom,
! [P: product_prod_a_nat > $o,Q: product_prod_a_nat > $o] :
( ! [X: product_prod_a_nat] :
( ( P @ X )
= ( Q @ X ) )
=> ( ( collec4464134535221767506_a_nat @ P )
= ( collec4464134535221767506_a_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_211_Collect__cong,axiom,
! [P: produc357393685978478089rm_a_b > $o,Q: produc357393685978478089rm_a_b > $o] :
( ! [X: produc357393685978478089rm_a_b] :
( ( P @ X )
= ( Q @ X ) )
=> ( ( collec99177395925924084rm_a_b @ P )
= ( collec99177395925924084rm_a_b @ Q ) ) ) ).
% Collect_cong
thf(fact_212_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_Pr4193341848836149977_nat_a,Z: set_Pr4193341848836149977_nat_a] : ( Y4 = Z ) )
= ( ^ [X4: set_Pr4193341848836149977_nat_a,Y5: set_Pr4193341848836149977_nat_a] :
( ( ord_le7924913712489149241_nat_a @ X4 @ Y5 )
& ( ord_le7924913712489149241_nat_a @ Y5 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_213_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_Product_prod_a_a,Z: set_Product_prod_a_a] : ( Y4 = Z ) )
= ( ^ [X4: set_Product_prod_a_a,Y5: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ X4 @ Y5 )
& ( ord_le746702958409616551od_a_a @ Y5 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_214_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_term_a_b,Z: set_term_a_b] : ( Y4 = Z ) )
= ( ^ [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_215_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_nat,Z: set_nat] : ( Y4 = Z ) )
= ( ^ [X4: set_nat,Y5: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y5 )
& ( ord_less_eq_set_nat @ Y5 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_216_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: nat,Z: nat] : ( Y4 = Z ) )
= ( ^ [X4: nat,Y5: nat] :
( ( ord_less_eq_nat @ X4 @ Y5 )
& ( ord_less_eq_nat @ Y5 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_217_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_Pr4934435412358123699_a_nat,Z: set_Pr4934435412358123699_a_nat] : ( Y4 = Z ) )
= ( ^ [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_218_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_Pr4386577575007340137rm_a_b,Z: set_Pr4386577575007340137rm_a_b] : ( Y4 = Z ) )
= ( ^ [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_219_le__cases3,axiom,
! [X3: nat,Y3: nat,Z2: nat] :
( ( ( ord_less_eq_nat @ X3 @ Y3 )
=> ~ ( ord_less_eq_nat @ Y3 @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ Y3 @ X3 )
=> ~ ( ord_less_eq_nat @ X3 @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ X3 @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ Y3 ) )
=> ( ( ( ord_less_eq_nat @ Z2 @ Y3 )
=> ~ ( ord_less_eq_nat @ Y3 @ X3 ) )
=> ( ( ( ord_less_eq_nat @ Y3 @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ X3 ) )
=> ~ ( ( ord_less_eq_nat @ Z2 @ X3 )
=> ~ ( ord_less_eq_nat @ X3 @ Y3 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_220_nle__le,axiom,
! [A2: nat,B2: nat] :
( ( ~ ( ord_less_eq_nat @ A2 @ B2 ) )
= ( ( ord_less_eq_nat @ B2 @ A2 )
& ( B2 != A2 ) ) ) ).
% nle_le
thf(fact_221_Pair__inject,axiom,
! [A2: set_term_a_b,B2: set_term_a_b,A3: set_term_a_b,B3: set_term_a_b] :
( ( ( produc8517751122659631533rm_a_b @ A2 @ B2 )
= ( produc8517751122659631533rm_a_b @ A3 @ B3 ) )
=> ~ ( ( A2 = A3 )
=> ( B2 != B3 ) ) ) ).
% Pair_inject
thf(fact_222_Pair__inject,axiom,
! [A2: nat,B2: nat,A3: nat,B3: nat] :
( ( ( product_Pair_nat_nat @ A2 @ B2 )
= ( product_Pair_nat_nat @ A3 @ B3 ) )
=> ~ ( ( A2 = A3 )
=> ( B2 != B3 ) ) ) ).
% Pair_inject
thf(fact_223_Pair__inject,axiom,
! [A2: nat,B2: a,A3: nat,B3: a] :
( ( ( product_Pair_nat_a @ A2 @ B2 )
= ( product_Pair_nat_a @ A3 @ B3 ) )
=> ~ ( ( A2 = A3 )
=> ( B2 != B3 ) ) ) ).
% Pair_inject
thf(fact_224_Pair__inject,axiom,
! [A2: a,B2: nat,A3: a,B3: nat] :
( ( ( product_Pair_a_nat @ A2 @ B2 )
= ( product_Pair_a_nat @ A3 @ B3 ) )
=> ~ ( ( A2 = A3 )
=> ( B2 != B3 ) ) ) ).
% Pair_inject
thf(fact_225_Pair__inject,axiom,
! [A2: a,B2: a,A3: a,B3: a] :
( ( ( product_Pair_a_a @ A2 @ B2 )
= ( product_Pair_a_a @ A3 @ B3 ) )
=> ~ ( ( A2 = A3 )
=> ( B2 != B3 ) ) ) ).
% Pair_inject
thf(fact_226_Pair__inject,axiom,
! [A2: term_a_b,B2: term_a_b,A3: term_a_b,B3: term_a_b] :
( ( ( produc7020197800436672577rm_a_b @ A2 @ B2 )
= ( produc7020197800436672577rm_a_b @ A3 @ B3 ) )
=> ~ ( ( A2 = A3 )
=> ( B2 != B3 ) ) ) ).
% Pair_inject
thf(fact_227_prod__cases,axiom,
! [P: produc9001763001631046773rm_a_b > $o,P2: produc9001763001631046773rm_a_b] :
( ! [A5: set_term_a_b,B5: set_term_a_b] : ( P @ ( produc8517751122659631533rm_a_b @ A5 @ B5 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_228_prod__cases,axiom,
! [P: product_prod_nat_nat > $o,P2: product_prod_nat_nat] :
( ! [A5: nat,B5: nat] : ( P @ ( product_Pair_nat_nat @ A5 @ B5 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_229_prod__cases,axiom,
! [P: product_prod_nat_a > $o,P2: product_prod_nat_a] :
( ! [A5: nat,B5: a] : ( P @ ( product_Pair_nat_a @ A5 @ B5 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_230_prod__cases,axiom,
! [P: product_prod_a_nat > $o,P2: product_prod_a_nat] :
( ! [A5: a,B5: nat] : ( P @ ( product_Pair_a_nat @ A5 @ B5 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_231_prod__cases,axiom,
! [P: product_prod_a_a > $o,P2: product_prod_a_a] :
( ! [A5: a,B5: a] : ( P @ ( product_Pair_a_a @ A5 @ B5 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_232_prod__cases,axiom,
! [P: produc357393685978478089rm_a_b > $o,P2: produc357393685978478089rm_a_b] :
( ! [A5: term_a_b,B5: term_a_b] : ( P @ ( produc7020197800436672577rm_a_b @ A5 @ B5 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_233_surj__pair,axiom,
! [P2: produc9001763001631046773rm_a_b] :
? [X: set_term_a_b,Y: set_term_a_b] :
( P2
= ( produc8517751122659631533rm_a_b @ X @ Y ) ) ).
% surj_pair
thf(fact_234_surj__pair,axiom,
! [P2: product_prod_nat_nat] :
? [X: nat,Y: nat] :
( P2
= ( product_Pair_nat_nat @ X @ Y ) ) ).
% surj_pair
thf(fact_235_surj__pair,axiom,
! [P2: product_prod_nat_a] :
? [X: nat,Y: a] :
( P2
= ( product_Pair_nat_a @ X @ Y ) ) ).
% surj_pair
thf(fact_236_surj__pair,axiom,
! [P2: product_prod_a_nat] :
? [X: a,Y: nat] :
( P2
= ( product_Pair_a_nat @ X @ Y ) ) ).
% surj_pair
thf(fact_237_surj__pair,axiom,
! [P2: product_prod_a_a] :
? [X: a,Y: a] :
( P2
= ( product_Pair_a_a @ X @ Y ) ) ).
% surj_pair
thf(fact_238_surj__pair,axiom,
! [P2: produc357393685978478089rm_a_b] :
? [X: term_a_b,Y: term_a_b] :
( P2
= ( produc7020197800436672577rm_a_b @ X @ Y ) ) ).
% surj_pair
thf(fact_239_old_Oprod_Oexhaust,axiom,
! [Y3: product_prod_a_nat] :
~ ! [A5: a,B5: nat] :
( Y3
!= ( product_Pair_a_nat @ A5 @ B5 ) ) ).
% old.prod.exhaust
thf(fact_240_old_Oprod_Oexhaust,axiom,
! [Y3: product_prod_a_a] :
~ ! [A5: a,B5: a] :
( Y3
!= ( product_Pair_a_a @ A5 @ B5 ) ) ).
% old.prod.exhaust
thf(fact_241_old_Oprod_Oexhaust,axiom,
! [Y3: produc357393685978478089rm_a_b] :
~ ! [A5: term_a_b,B5: term_a_b] :
( Y3
!= ( produc7020197800436672577rm_a_b @ A5 @ B5 ) ) ).
% old.prod.exhaust
thf(fact_242_Collect__mono__iff,axiom,
! [P: product_prod_a_nat > $o,Q: product_prod_a_nat > $o] :
( ( ord_le8666007276011122963_a_nat @ ( collec4464134535221767506_a_nat @ P ) @ ( collec4464134535221767506_a_nat @ Q ) )
= ( ! [X4: product_prod_a_nat] :
( ( P @ X4 )
=> ( Q @ X4 ) ) ) ) ).
% Collect_mono_iff
thf(fact_243_Collect__mono__iff,axiom,
! [P: produc357393685978478089rm_a_b > $o,Q: produc357393685978478089rm_a_b > $o] :
( ( ord_le118470702582115849rm_a_b @ ( collec99177395925924084rm_a_b @ P ) @ ( collec99177395925924084rm_a_b @ Q ) )
= ( ! [X4: produc357393685978478089rm_a_b] :
( ( P @ X4 )
=> ( Q @ X4 ) ) ) ) ).
% Collect_mono_iff
thf(fact_244_set__eq__subset,axiom,
( ( ^ [Y4: set_Pr4934435412358123699_a_nat,Z: set_Pr4934435412358123699_a_nat] : ( Y4 = Z ) )
= ( ^ [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_245_set__eq__subset,axiom,
( ( ^ [Y4: set_Pr4386577575007340137rm_a_b,Z: set_Pr4386577575007340137rm_a_b] : ( Y4 = Z ) )
= ( ^ [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_246_subset__trans,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 ) ) ) ).
% subset_trans
thf(fact_247_subset__trans,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 ) ) ) ).
% subset_trans
thf(fact_248_Collect__mono,axiom,
! [P: product_prod_a_nat > $o,Q: product_prod_a_nat > $o] :
( ! [X: product_prod_a_nat] :
( ( P @ X )
=> ( Q @ X ) )
=> ( ord_le8666007276011122963_a_nat @ ( collec4464134535221767506_a_nat @ P ) @ ( collec4464134535221767506_a_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_249_Collect__mono,axiom,
! [P: produc357393685978478089rm_a_b > $o,Q: produc357393685978478089rm_a_b > $o] :
( ! [X: produc357393685978478089rm_a_b] :
( ( P @ X )
=> ( Q @ X ) )
=> ( ord_le118470702582115849rm_a_b @ ( collec99177395925924084rm_a_b @ P ) @ ( collec99177395925924084rm_a_b @ Q ) ) ) ).
% Collect_mono
thf(fact_250_subset__refl,axiom,
! [A: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ A @ A ) ).
% subset_refl
thf(fact_251_subset__refl,axiom,
! [A: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ A @ A ) ).
% subset_refl
thf(fact_252_subset__iff,axiom,
( ord_le8666007276011122963_a_nat
= ( ^ [A6: set_Pr4934435412358123699_a_nat,B6: set_Pr4934435412358123699_a_nat] :
! [T2: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ T2 @ A6 )
=> ( member5724188588386418708_a_nat @ T2 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_253_subset__iff,axiom,
( ord_le118470702582115849rm_a_b
= ( ^ [A6: set_Pr4386577575007340137rm_a_b,B6: set_Pr4386577575007340137rm_a_b] :
! [T2: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ T2 @ A6 )
=> ( member5869715511025134514rm_a_b @ T2 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_254_equalityD2,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( A = B )
=> ( ord_le8666007276011122963_a_nat @ B @ A ) ) ).
% equalityD2
thf(fact_255_equalityD2,axiom,
! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ( A = B )
=> ( ord_le118470702582115849rm_a_b @ B @ A ) ) ).
% equalityD2
thf(fact_256_equalityD1,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( A = B )
=> ( ord_le8666007276011122963_a_nat @ A @ B ) ) ).
% equalityD1
thf(fact_257_equalityD1,axiom,
! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ( A = B )
=> ( ord_le118470702582115849rm_a_b @ A @ B ) ) ).
% equalityD1
thf(fact_258_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_259_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_260_equalityE,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( A = B )
=> ~ ( ( ord_le8666007276011122963_a_nat @ A @ B )
=> ~ ( ord_le8666007276011122963_a_nat @ B @ A ) ) ) ).
% equalityE
thf(fact_261_equalityE,axiom,
! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ( A = B )
=> ~ ( ( ord_le118470702582115849rm_a_b @ A @ B )
=> ~ ( ord_le118470702582115849rm_a_b @ B @ A ) ) ) ).
% equalityE
thf(fact_262_subsetD,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat,C: product_prod_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A @ B )
=> ( ( member5724188588386418708_a_nat @ C @ A )
=> ( member5724188588386418708_a_nat @ C @ B ) ) ) ).
% subsetD
thf(fact_263_subsetD,axiom,
! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b,C: produc357393685978478089rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A @ B )
=> ( ( member5869715511025134514rm_a_b @ C @ A )
=> ( member5869715511025134514rm_a_b @ C @ B ) ) ) ).
% subsetD
thf(fact_264_in__mono,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat,X3: product_prod_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A @ B )
=> ( ( member5724188588386418708_a_nat @ X3 @ A )
=> ( member5724188588386418708_a_nat @ X3 @ B ) ) ) ).
% in_mono
thf(fact_265_in__mono,axiom,
! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b,X3: produc357393685978478089rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A @ B )
=> ( ( member5869715511025134514rm_a_b @ X3 @ A )
=> ( member5869715511025134514rm_a_b @ X3 @ B ) ) ) ).
% in_mono
thf(fact_266_rstep__trancl__sig__step__r,axiom,
! [S: term_a_b,T: term_a_b,R2: 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 @ R2 ) ) )
=> ( ( ord_le8666007276011122963_a_nat @ ( terms_7988297476397195622_a_b_b @ R2 ) @ 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 @ R2 ) ) ) ) ) ) ) ).
% rstep_trancl_sig_step_r
thf(fact_267_rstep__trancl__sig__step__l,axiom,
! [S: term_a_b,T: term_a_b,R2: 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 @ R2 ) ) )
=> ( ( ord_le8666007276011122963_a_nat @ ( terms_7988297476397195622_a_b_b @ R2 ) @ 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 @ R2 ) ) ) ) ) ) ) ).
% rstep_trancl_sig_step_l
thf(fact_268_srstep__monp,axiom,
! [F: set_Pr4934435412358123699_a_nat,G: set_Pr4934435412358123699_a_nat,R2: set_Pr4386577575007340137rm_a_b] :
( ( ord_le8666007276011122963_a_nat @ F @ G )
=> ( ord_le118470702582115849rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R2 ) ) @ ( sig_step_a_b @ G @ ( rstep_a_b @ R2 ) ) ) ) ).
% srstep_monp
thf(fact_269_term__to__sig__id,axiom,
! [T: term_a_b,F: set_Pr4934435412358123699_a_nat,V: b] :
( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ T ) @ F )
=> ( ( terms_8519481630511763164ig_a_b @ F @ V @ T )
= T ) ) ).
% term_to_sig_id
thf(fact_270_fuans__term__term__to__sig,axiom,
! [F: set_Pr4934435412358123699_a_nat,V: b,T: term_a_b] : ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ ( terms_8519481630511763164ig_a_b @ F @ V @ T ) ) @ F ) ).
% fuans_term_term_to_sig
thf(fact_271_trancl__mono__set,axiom,
! [R: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ R @ S )
=> ( ord_le118470702582115849rm_a_b @ ( transi7922773638565587891rm_a_b @ R ) @ ( transi7922773638565587891rm_a_b @ S ) ) ) ).
% trancl_mono_set
thf(fact_272_trancl__mono,axiom,
! [P2: produc357393685978478089rm_a_b,R: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ P2 @ ( transi7922773638565587891rm_a_b @ R ) )
=> ( ( ord_le118470702582115849rm_a_b @ R @ S )
=> ( member5869715511025134514rm_a_b @ P2 @ ( transi7922773638565587891rm_a_b @ S ) ) ) ) ).
% trancl_mono
thf(fact_273_converse__trancl__induct,axiom,
! [A2: term_a_b,B2: term_a_b,R: set_Pr4386577575007340137rm_a_b,P: term_a_b > $o] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( transi7922773638565587891rm_a_b @ R ) )
=> ( ! [Y: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ B2 ) @ R )
=> ( P @ Y ) )
=> ( ! [Y: term_a_b,Z3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ Z3 ) @ R )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Z3 @ B2 ) @ ( transi7922773638565587891rm_a_b @ R ) )
=> ( ( P @ Z3 )
=> ( P @ Y ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% converse_trancl_induct
thf(fact_274_trancl__trans__induct,axiom,
! [X3: term_a_b,Y3: term_a_b,R: set_Pr4386577575007340137rm_a_b,P: term_a_b > term_a_b > $o] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ ( transi7922773638565587891rm_a_b @ R ) )
=> ( ! [X: term_a_b,Y: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ R )
=> ( P @ X @ Y ) )
=> ( ! [X: term_a_b,Y: term_a_b,Z3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7922773638565587891rm_a_b @ R ) )
=> ( ( P @ X @ Y )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ Z3 ) @ ( transi7922773638565587891rm_a_b @ R ) )
=> ( ( P @ Y @ Z3 )
=> ( P @ X @ Z3 ) ) ) ) )
=> ( P @ X3 @ Y3 ) ) ) ) ).
% trancl_trans_induct
thf(fact_275_trancl__into__trancl2,axiom,
! [A2: term_a_b,B2: term_a_b,R: set_Pr4386577575007340137rm_a_b,C: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ R )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ C ) @ ( transi7922773638565587891rm_a_b @ R ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ C ) @ ( transi7922773638565587891rm_a_b @ R ) ) ) ) ).
% trancl_into_trancl2
thf(fact_276_Transitive__Closure_Otrancl__into__trancl,axiom,
! [A2: term_a_b,B2: term_a_b,R: set_Pr4386577575007340137rm_a_b,C: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( transi7922773638565587891rm_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ C ) @ R )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ C ) @ ( transi7922773638565587891rm_a_b @ R ) ) ) ) ).
% Transitive_Closure.trancl_into_trancl
thf(fact_277_irrefl__trancl__rD,axiom,
! [R: set_Pr4386577575007340137rm_a_b,X3: term_a_b,Y3: term_a_b] :
( ! [X: term_a_b] :
~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ X ) @ ( transi7922773638565587891rm_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ R )
=> ( X3 != Y3 ) ) ) ).
% irrefl_trancl_rD
thf(fact_278_converse__tranclE,axiom,
! [X3: term_a_b,Z2: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Z2 ) @ ( transi7922773638565587891rm_a_b @ R ) )
=> ( ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Z2 ) @ R )
=> ~ ! [Y: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y ) @ R )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ Z2 ) @ ( transi7922773638565587891rm_a_b @ R ) ) ) ) ) ).
% converse_tranclE
thf(fact_279_r__r__into__trancl,axiom,
! [A2: term_a_b,B2: term_a_b,R4: set_Pr4386577575007340137rm_a_b,C: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ R4 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ C ) @ R4 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ C ) @ ( transi7922773638565587891rm_a_b @ R4 ) ) ) ) ).
% r_r_into_trancl
thf(fact_280_trancl__induct2,axiom,
! [Ax: term_a_b,Ay: term_a_b,Bx: term_a_b,By: term_a_b,R: set_Pr2972776593051762503rm_a_b,P: 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 @ R ) )
=> ( ! [A5: term_a_b,B5: term_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ Ax @ Ay ) @ ( produc7020197800436672577rm_a_b @ A5 @ B5 ) ) @ R )
=> ( P @ A5 @ B5 ) )
=> ( ! [A5: term_a_b,B5: 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 @ A5 @ B5 ) ) @ ( transi9211502839322181930rm_a_b @ R ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ A5 @ B5 ) @ ( produc7020197800436672577rm_a_b @ Aa @ Ba ) ) @ R )
=> ( ( P @ A5 @ B5 )
=> ( P @ Aa @ Ba ) ) ) )
=> ( P @ Bx @ By ) ) ) ) ).
% trancl_induct2
thf(fact_281_r__into__trancl_H,axiom,
! [P2: produc357393685978478089rm_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ P2 @ R )
=> ( member5869715511025134514rm_a_b @ P2 @ ( transi7922773638565587891rm_a_b @ R ) ) ) ).
% r_into_trancl'
thf(fact_282_trancl_Ocases,axiom,
! [A1: term_a_b,A22: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A22 ) @ ( transi7922773638565587891rm_a_b @ R ) )
=> ( ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A22 ) @ R )
=> ~ ! [B5: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ B5 ) @ ( transi7922773638565587891rm_a_b @ R ) )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B5 @ A22 ) @ R ) ) ) ) ).
% trancl.cases
thf(fact_283_trancl_Osimps,axiom,
! [A1: term_a_b,A22: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A22 ) @ ( transi7922773638565587891rm_a_b @ R ) )
= ( ? [A4: term_a_b,B4: term_a_b] :
( ( A1 = A4 )
& ( A22 = B4 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A4 @ B4 ) @ R ) )
| ? [A4: term_a_b,B4: term_a_b,C3: term_a_b] :
( ( A1 = A4 )
& ( A22 = C3 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A4 @ B4 ) @ ( transi7922773638565587891rm_a_b @ R ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B4 @ C3 ) @ R ) ) ) ) ).
% trancl.simps
thf(fact_284_trancl_Or__into__trancl,axiom,
! [A2: term_a_b,B2: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ R )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( transi7922773638565587891rm_a_b @ R ) ) ) ).
% trancl.r_into_trancl
thf(fact_285_tranclE,axiom,
! [A2: term_a_b,B2: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( transi7922773638565587891rm_a_b @ R ) )
=> ( ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ R )
=> ~ ! [C4: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ C4 ) @ ( transi7922773638565587891rm_a_b @ R ) )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ C4 @ B2 ) @ R ) ) ) ) ).
% tranclE
thf(fact_286_trancl__trans,axiom,
! [X3: term_a_b,Y3: term_a_b,R: set_Pr4386577575007340137rm_a_b,Z2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ ( transi7922773638565587891rm_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ Z2 ) @ ( transi7922773638565587891rm_a_b @ R ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Z2 ) @ ( transi7922773638565587891rm_a_b @ R ) ) ) ) ).
% trancl_trans
thf(fact_287_trancl__induct,axiom,
! [A2: term_a_b,B2: term_a_b,R: set_Pr4386577575007340137rm_a_b,P: term_a_b > $o] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( transi7922773638565587891rm_a_b @ R ) )
=> ( ! [Y: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ Y ) @ R )
=> ( P @ Y ) )
=> ( ! [Y: term_a_b,Z3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ Y ) @ ( transi7922773638565587891rm_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ Z3 ) @ R )
=> ( ( P @ Y )
=> ( P @ Z3 ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% trancl_induct
thf(fact_288_srsteps__with__root__step__sig__mono,axiom,
! [F: set_Pr4934435412358123699_a_nat,G: set_Pr4934435412358123699_a_nat,R2: set_Pr4386577575007340137rm_a_b] :
( ( ord_le8666007276011122963_a_nat @ F @ G )
=> ( ord_le118470702582115849rm_a_b @ ( srstep7844470518422762656ep_a_b @ F @ R2 ) @ ( srstep7844470518422762656ep_a_b @ G @ R2 ) ) ) ).
% srsteps_with_root_step_sig_mono
thf(fact_289_trancl__map,axiom,
! [R: set_Pr4386577575007340137rm_a_b,F2: term_a_b > term_a_b,S: set_Pr4386577575007340137rm_a_b,X3: term_a_b,Y3: term_a_b] :
( ! [X: term_a_b,Y: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ R )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F2 @ X ) @ ( F2 @ Y ) ) @ S ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ ( transi7922773638565587891rm_a_b @ R ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) @ ( transi7922773638565587891rm_a_b @ S ) ) ) ) ).
% trancl_map
thf(fact_290_srsteps__eq__monp,axiom,
! [F: set_Pr4934435412358123699_a_nat,G: set_Pr4934435412358123699_a_nat,R2: 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 @ R2 ) ) ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ G @ ( rstep_a_b @ R2 ) ) ) ) ) ).
% srsteps_eq_monp
thf(fact_291_GreatestI2__order,axiom,
! [P: set_Pr4934435412358123699_a_nat > $o,X3: set_Pr4934435412358123699_a_nat,Q: set_Pr4934435412358123699_a_nat > $o] :
( ( P @ X3 )
=> ( ! [Y: set_Pr4934435412358123699_a_nat] :
( ( P @ Y )
=> ( ord_le8666007276011122963_a_nat @ Y @ X3 ) )
=> ( ! [X: set_Pr4934435412358123699_a_nat] :
( ( P @ X )
=> ( ! [Y6: set_Pr4934435412358123699_a_nat] :
( ( P @ Y6 )
=> ( ord_le8666007276011122963_a_nat @ Y6 @ X ) )
=> ( Q @ X ) ) )
=> ( Q @ ( order_1399416596184996762_a_nat @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_292_GreatestI2__order,axiom,
! [P: set_Pr4386577575007340137rm_a_b > $o,X3: set_Pr4386577575007340137rm_a_b,Q: set_Pr4386577575007340137rm_a_b > $o] :
( ( P @ X3 )
=> ( ! [Y: set_Pr4386577575007340137rm_a_b] :
( ( P @ Y )
=> ( ord_le118470702582115849rm_a_b @ Y @ X3 ) )
=> ( ! [X: set_Pr4386577575007340137rm_a_b] :
( ( P @ X )
=> ( ! [Y6: set_Pr4386577575007340137rm_a_b] :
( ( P @ Y6 )
=> ( ord_le118470702582115849rm_a_b @ Y6 @ X ) )
=> ( Q @ X ) ) )
=> ( Q @ ( order_4999756673592925890rm_a_b @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_293_Greatest__equality,axiom,
! [P: set_Pr4934435412358123699_a_nat > $o,X3: set_Pr4934435412358123699_a_nat] :
( ( P @ X3 )
=> ( ! [Y: set_Pr4934435412358123699_a_nat] :
( ( P @ Y )
=> ( ord_le8666007276011122963_a_nat @ Y @ X3 ) )
=> ( ( order_1399416596184996762_a_nat @ P )
= X3 ) ) ) ).
% Greatest_equality
thf(fact_294_Greatest__equality,axiom,
! [P: set_Pr4386577575007340137rm_a_b > $o,X3: set_Pr4386577575007340137rm_a_b] :
( ( P @ X3 )
=> ( ! [Y: set_Pr4386577575007340137rm_a_b] :
( ( P @ Y )
=> ( ord_le118470702582115849rm_a_b @ Y @ X3 ) )
=> ( ( order_4999756673592925890rm_a_b @ P )
= X3 ) ) ) ).
% Greatest_equality
thf(fact_295_rstep_Ocases,axiom,
! [A1: term_a_b,A22: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A22 ) @ ( rstep_a_b @ R2 ) )
=> ~ ! [S2: term_a_b,T3: term_a_b,C5: subterm_and_ctxt_a_b] :
( ( A1
= ( subter2376574525758040790rm_a_b @ C5 @ S2 ) )
=> ( ( A22
= ( subter2376574525758040790rm_a_b @ C5 @ T3 ) )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S2 @ T3 ) @ ( rrstep_a_b @ R2 ) ) ) ) ) ).
% rstep.cases
thf(fact_296_rstep_Osimps,axiom,
! [A1: term_a_b,A22: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A22 ) @ ( rstep_a_b @ R2 ) )
= ( ? [S3: term_a_b,T2: term_a_b,C6: subterm_and_ctxt_a_b] :
( ( A1
= ( subter2376574525758040790rm_a_b @ C6 @ S3 ) )
& ( A22
= ( subter2376574525758040790rm_a_b @ C6 @ T2 ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S3 @ T2 ) @ ( rrstep_a_b @ R2 ) ) ) ) ) ).
% rstep.simps
thf(fact_297_r__into__rtrancl,axiom,
! [P2: produc357393685978478089rm_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ P2 @ R )
=> ( member5869715511025134514rm_a_b @ P2 @ ( transi7742714808557438673rm_a_b @ R ) ) ) ).
% r_into_rtrancl
thf(fact_298_rstep__ctxtI,axiom,
! [S: term_a_b,T: term_a_b,R2: set_Pr4386577575007340137rm_a_b,C2: subterm_and_ctxt_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( rstep_a_b @ R2 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( subter2376574525758040790rm_a_b @ C2 @ S ) @ ( subter2376574525758040790rm_a_b @ C2 @ T ) ) @ ( rstep_a_b @ R2 ) ) ) ).
% rstep_ctxtI
thf(fact_299_converse__rtrancl__into__rtrancl,axiom,
! [A2: term_a_b,B2: term_a_b,R: set_Pr4386577575007340137rm_a_b,C: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ R )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ C ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ C ) @ ( transi7742714808557438673rm_a_b @ R ) ) ) ) ).
% converse_rtrancl_into_rtrancl
thf(fact_300_converse__rtrancl__induct,axiom,
! [A2: term_a_b,B2: term_a_b,R: set_Pr4386577575007340137rm_a_b,P: term_a_b > $o] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ( ( P @ B2 )
=> ( ! [Y: term_a_b,Z3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ Z3 ) @ R )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Z3 @ B2 ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ( ( P @ Z3 )
=> ( P @ Y ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% converse_rtrancl_induct
thf(fact_301_converse__rtranclE,axiom,
! [X3: term_a_b,Z2: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Z2 ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ( ( X3 != Z2 )
=> ~ ! [Y: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y ) @ R )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ Z2 ) @ ( transi7742714808557438673rm_a_b @ R ) ) ) ) ) ).
% converse_rtranclE
thf(fact_302_rtrancl__induct,axiom,
! [A2: term_a_b,B2: term_a_b,R: set_Pr4386577575007340137rm_a_b,P: term_a_b > $o] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ( ( P @ A2 )
=> ( ! [Y: term_a_b,Z3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ Y ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ Z3 ) @ R )
=> ( ( P @ Y )
=> ( P @ Z3 ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% rtrancl_induct
thf(fact_303_rtrancl__trans,axiom,
! [X3: term_a_b,Y3: term_a_b,R: set_Pr4386577575007340137rm_a_b,Z2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ Z2 ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Z2 ) @ ( transi7742714808557438673rm_a_b @ R ) ) ) ) ).
% rtrancl_trans
thf(fact_304_rtranclE,axiom,
! [A2: term_a_b,B2: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ( ( A2 != B2 )
=> ~ ! [Y: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ Y ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ B2 ) @ R ) ) ) ) ).
% rtranclE
thf(fact_305_rtrancl_Ortrancl__into__rtrancl,axiom,
! [A2: term_a_b,B2: term_a_b,R: set_Pr4386577575007340137rm_a_b,C: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ C ) @ R )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ C ) @ ( transi7742714808557438673rm_a_b @ R ) ) ) ) ).
% rtrancl.rtrancl_into_rtrancl
thf(fact_306_rtrancl_Ortrancl__refl,axiom,
! [A2: term_a_b,R: set_Pr4386577575007340137rm_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ A2 ) @ ( transi7742714808557438673rm_a_b @ R ) ) ).
% rtrancl.rtrancl_refl
thf(fact_307_rtrancl_Osimps,axiom,
! [A1: term_a_b,A22: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A22 ) @ ( transi7742714808557438673rm_a_b @ R ) )
= ( ? [A4: term_a_b] :
( ( A1 = A4 )
& ( A22 = A4 ) )
| ? [A4: term_a_b,B4: term_a_b,C3: term_a_b] :
( ( A1 = A4 )
& ( A22 = C3 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A4 @ B4 ) @ ( transi7742714808557438673rm_a_b @ R ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B4 @ C3 ) @ R ) ) ) ) ).
% rtrancl.simps
thf(fact_308_rtrancl_Ocases,axiom,
! [A1: term_a_b,A22: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A22 ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ( ( A22 != A1 )
=> ~ ! [B5: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ B5 ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B5 @ A22 ) @ R ) ) ) ) ).
% rtrancl.cases
thf(fact_309_rtrancl__subset__rtrancl,axiom,
! [R: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ R @ ( transi7742714808557438673rm_a_b @ S ) )
=> ( ord_le118470702582115849rm_a_b @ ( transi7742714808557438673rm_a_b @ R ) @ ( transi7742714808557438673rm_a_b @ S ) ) ) ).
% rtrancl_subset_rtrancl
thf(fact_310_rtrancl__subset,axiom,
! [R4: set_Pr4386577575007340137rm_a_b,S4: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ R4 @ S4 )
=> ( ( ord_le118470702582115849rm_a_b @ S4 @ ( transi7742714808557438673rm_a_b @ R4 ) )
=> ( ( transi7742714808557438673rm_a_b @ S4 )
= ( transi7742714808557438673rm_a_b @ R4 ) ) ) ) ).
% rtrancl_subset
thf(fact_311_rtrancl__mono,axiom,
! [R: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ R @ S )
=> ( ord_le118470702582115849rm_a_b @ ( transi7742714808557438673rm_a_b @ R ) @ ( transi7742714808557438673rm_a_b @ S ) ) ) ).
% rtrancl_mono
thf(fact_312_tranclD,axiom,
! [X3: term_a_b,Y3: term_a_b,R4: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ ( transi7922773638565587891rm_a_b @ R4 ) )
=> ? [Z3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Z3 ) @ R4 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Z3 @ Y3 ) @ ( transi7742714808557438673rm_a_b @ R4 ) ) ) ) ).
% tranclD
thf(fact_313_rtranclD,axiom,
! [A2: term_a_b,B2: term_a_b,R4: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( transi7742714808557438673rm_a_b @ R4 ) )
=> ( ( A2 = B2 )
| ( ( A2 != B2 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( transi7922773638565587891rm_a_b @ R4 ) ) ) ) ) ).
% rtranclD
thf(fact_314_tranclD2,axiom,
! [X3: term_a_b,Y3: term_a_b,R4: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ ( transi7922773638565587891rm_a_b @ R4 ) )
=> ? [Z3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Z3 ) @ ( transi7742714808557438673rm_a_b @ R4 ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Z3 @ Y3 ) @ R4 ) ) ) ).
% tranclD2
thf(fact_315_trancl__into__rtrancl,axiom,
! [A2: term_a_b,B2: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( transi7922773638565587891rm_a_b @ R ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( transi7742714808557438673rm_a_b @ R ) ) ) ).
% trancl_into_rtrancl
thf(fact_316_rtrancl__eq__or__trancl,axiom,
! [X3: term_a_b,Y3: term_a_b,R4: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ ( transi7742714808557438673rm_a_b @ R4 ) )
= ( ( X3 = Y3 )
| ( ( X3 != Y3 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ ( transi7922773638565587891rm_a_b @ R4 ) ) ) ) ) ).
% rtrancl_eq_or_trancl
thf(fact_317_rtrancl__into__trancl1,axiom,
! [A2: term_a_b,B2: term_a_b,R: set_Pr4386577575007340137rm_a_b,C: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ C ) @ R )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ C ) @ ( transi7922773638565587891rm_a_b @ R ) ) ) ) ).
% rtrancl_into_trancl1
thf(fact_318_rtrancl__into__trancl2,axiom,
! [A2: term_a_b,B2: term_a_b,R: set_Pr4386577575007340137rm_a_b,C: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ R )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ C ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ C ) @ ( transi7922773638565587891rm_a_b @ R ) ) ) ) ).
% rtrancl_into_trancl2
thf(fact_319_rtrancl__trancl__trancl,axiom,
! [X3: term_a_b,Y3: term_a_b,R: set_Pr4386577575007340137rm_a_b,Z2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ Z2 ) @ ( transi7922773638565587891rm_a_b @ R ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Z2 ) @ ( transi7922773638565587891rm_a_b @ R ) ) ) ) ).
% rtrancl_trancl_trancl
thf(fact_320_trancl__rtrancl__trancl,axiom,
! [A2: term_a_b,B2: term_a_b,R: set_Pr4386577575007340137rm_a_b,C: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( transi7922773638565587891rm_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ C ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ C ) @ ( transi7922773638565587891rm_a_b @ R ) ) ) ) ).
% trancl_rtrancl_trancl
thf(fact_321_restrict__SN__subset,axiom,
! [R4: set_Pr4386577575007340137rm_a_b,S4: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ ( abstra4933154510623179783rm_a_b @ R4 @ S4 ) @ R4 ) ).
% restrict_SN_subset
thf(fact_322_rstep_Ointros,axiom,
! [S: term_a_b,T: term_a_b,R2: set_Pr4386577575007340137rm_a_b,C2: subterm_and_ctxt_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( rrstep_a_b @ R2 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( subter2376574525758040790rm_a_b @ C2 @ S ) @ ( subter2376574525758040790rm_a_b @ C2 @ T ) ) @ ( rstep_a_b @ R2 ) ) ) ).
% rstep.intros
thf(fact_323_meetI,axiom,
! [A2: term_a_b,B2: term_a_b,A: set_Pr4386577575007340137rm_a_b,C: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( transi7742714808557438673rm_a_b @ A ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ C ) @ ( transi7742714808557438673rm_a_b @ A ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ C ) @ ( abstra1093061187967292567rm_a_b @ A ) ) ) ) ).
% meetI
thf(fact_324_restrict__rtrancl,axiom,
! [R: set_Pr4386577575007340137rm_a_b,S4: set_term_a_b] : ( ord_le118470702582115849rm_a_b @ ( transi7742714808557438673rm_a_b @ ( abstra6662565036008789004rm_a_b @ R @ S4 ) ) @ ( transi7742714808557438673rm_a_b @ R ) ) ).
% restrict_rtrancl
thf(fact_325_srsteps__with__root__step__def,axiom,
( srstep7844470518422762656ep_a_b
= ( ^ [F3: set_Pr4934435412358123699_a_nat,R5: set_Pr4386577575007340137rm_a_b] : ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F3 @ ( rstep_a_b @ R5 ) ) ) @ ( relcom370159955682700863rm_a_b @ ( sig_step_a_b @ F3 @ ( rrstep_a_b @ R5 ) ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F3 @ ( rstep_a_b @ R5 ) ) ) ) ) ) ) ).
% srsteps_with_root_step_def
thf(fact_326_meetE,axiom,
! [B2: term_a_b,C: term_a_b,A: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ C ) @ ( abstra1093061187967292567rm_a_b @ A ) )
=> ~ ! [A5: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A5 @ B2 ) @ ( transi7742714808557438673rm_a_b @ A ) )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A5 @ C ) @ ( transi7742714808557438673rm_a_b @ A ) ) ) ) ).
% meetE
thf(fact_327_meetD,axiom,
! [B2: term_a_b,C: term_a_b,A: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ C ) @ ( abstra1093061187967292567rm_a_b @ A ) )
=> ? [A5: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A5 @ B2 ) @ ( transi7742714808557438673rm_a_b @ A ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A5 @ C ) @ ( transi7742714808557438673rm_a_b @ A ) ) ) ) ).
% meetD
thf(fact_328_same__fstI,axiom,
! [P: term_a_b > $o,X3: term_a_b,Y7: term_a_b,Y3: term_a_b,R4: term_a_b > set_Pr4386577575007340137rm_a_b] :
( ( P @ X3 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y7 @ Y3 ) @ ( R4 @ X3 ) )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y7 ) @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) ) @ ( same_f843731131676418198rm_a_b @ P @ R4 ) ) ) ) ).
% same_fstI
thf(fact_329_normalizability__I_H,axiom,
! [A2: term_a_b,B2: term_a_b,A: set_Pr4386577575007340137rm_a_b,C: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( transi7742714808557438673rm_a_b @ A ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ C ) @ ( abstra2972651231392952684rm_a_b @ A ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ C ) @ ( abstra2972651231392952684rm_a_b @ A ) ) ) ) ).
% normalizability_I'
thf(fact_330_rstepI,axiom,
! [L: term_a_b,R: term_a_b,R2: set_Pr4386577575007340137rm_a_b,S: term_a_b,C2: subterm_and_ctxt_a_b,Sigma: b > term_a_b,T: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ L @ R ) @ R2 )
=> ( ( S
= ( subter2376574525758040790rm_a_b @ C2 @ ( subst_7999470309526761004_a_b_b @ L @ Sigma ) ) )
=> ( ( T
= ( subter2376574525758040790rm_a_b @ C2 @ ( subst_7999470309526761004_a_b_b @ R @ Sigma ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( rstep_a_b @ R2 ) ) ) ) ) ).
% rstepI
thf(fact_331_rstep__substI,axiom,
! [S: term_a_b,T: term_a_b,R2: set_Pr4386577575007340137rm_a_b,Sigma: b > term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( rstep_a_b @ R2 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( subst_7999470309526761004_a_b_b @ S @ Sigma ) @ ( subst_7999470309526761004_a_b_b @ T @ Sigma ) ) @ ( rstep_a_b @ R2 ) ) ) ).
% rstep_substI
thf(fact_332_relcomp3__I,axiom,
! [T: nat,U: a,A: set_Pr4193341848836149977_nat_a,S: a,B: set_Pr4934435412358123699_a_nat,V: nat] :
( ( member8962352052110095674_nat_a @ ( product_Pair_nat_a @ T @ U ) @ A )
=> ( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ S @ T ) @ B )
=> ( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ U @ V ) @ B )
=> ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ S @ V ) @ ( relcomp_a_nat_nat @ B @ ( relcomp_nat_a_nat @ A @ B ) ) ) ) ) ) ).
% relcomp3_I
thf(fact_333_relcomp3__I,axiom,
! [T: a,U: nat,A: set_Pr4934435412358123699_a_nat,S: nat,B: set_Pr4193341848836149977_nat_a,V: a] :
( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ T @ U ) @ A )
=> ( ( member8962352052110095674_nat_a @ ( product_Pair_nat_a @ S @ T ) @ B )
=> ( ( member8962352052110095674_nat_a @ ( product_Pair_nat_a @ U @ V ) @ B )
=> ( member8962352052110095674_nat_a @ ( product_Pair_nat_a @ S @ V ) @ ( relcomp_nat_a_a @ B @ ( relcomp_a_nat_a @ A @ B ) ) ) ) ) ) ).
% relcomp3_I
thf(fact_334_relcomp3__I,axiom,
! [T: term_a_b,U: term_a_b,A: set_Pr4386577575007340137rm_a_b,S: term_a_b,B: set_Pr4386577575007340137rm_a_b,V: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T @ U ) @ A )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ B )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ U @ V ) @ B )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ V ) @ ( relcom370159955682700863rm_a_b @ B @ ( relcom370159955682700863rm_a_b @ A @ B ) ) ) ) ) ) ).
% relcomp3_I
thf(fact_335_relcomp_Ocases,axiom,
! [A1: a,A22: nat,R: set_Product_prod_a_a,S: set_Pr4934435412358123699_a_nat] :
( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ A1 @ A22 ) @ ( relcomp_a_a_nat @ R @ S ) )
=> ~ ! [B5: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A1 @ B5 ) @ R )
=> ~ ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ B5 @ A22 ) @ S ) ) ) ).
% relcomp.cases
thf(fact_336_relcomp_Ocases,axiom,
! [A1: a,A22: nat,R: set_Pr4934435412358123699_a_nat,S: set_Pr1261947904930325089at_nat] :
( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ A1 @ A22 ) @ ( relcomp_a_nat_nat @ R @ S ) )
=> ~ ! [B5: nat] :
( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ A1 @ B5 ) @ R )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B5 @ A22 ) @ S ) ) ) ).
% relcomp.cases
thf(fact_337_relcomp_Ocases,axiom,
! [A1: term_a_b,A22: term_a_b,R: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A22 ) @ ( relcom370159955682700863rm_a_b @ R @ S ) )
=> ~ ! [B5: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ B5 ) @ R )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B5 @ A22 ) @ S ) ) ) ).
% relcomp.cases
thf(fact_338_relcomp_Osimps,axiom,
! [A1: a,A22: nat,R: set_Product_prod_a_a,S: set_Pr4934435412358123699_a_nat] :
( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ A1 @ A22 ) @ ( relcomp_a_a_nat @ R @ S ) )
= ( ? [A4: a,B4: a,C3: nat] :
( ( A1 = A4 )
& ( A22 = C3 )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A4 @ B4 ) @ R )
& ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ B4 @ C3 ) @ S ) ) ) ) ).
% relcomp.simps
thf(fact_339_relcomp_Osimps,axiom,
! [A1: a,A22: nat,R: set_Pr4934435412358123699_a_nat,S: set_Pr1261947904930325089at_nat] :
( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ A1 @ A22 ) @ ( relcomp_a_nat_nat @ R @ S ) )
= ( ? [A4: a,B4: nat,C3: nat] :
( ( A1 = A4 )
& ( A22 = C3 )
& ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ A4 @ B4 ) @ R )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B4 @ C3 ) @ S ) ) ) ) ).
% relcomp.simps
thf(fact_340_relcomp_Osimps,axiom,
! [A1: term_a_b,A22: term_a_b,R: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A22 ) @ ( relcom370159955682700863rm_a_b @ R @ S ) )
= ( ? [A4: term_a_b,B4: term_a_b,C3: term_a_b] :
( ( A1 = A4 )
& ( A22 = C3 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A4 @ B4 ) @ R )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B4 @ C3 ) @ S ) ) ) ) ).
% relcomp.simps
thf(fact_341_relcomp_OrelcompI,axiom,
! [A2: a,B2: a,R: set_Product_prod_a_a,C: nat,S: set_Pr4934435412358123699_a_nat] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ B2 ) @ R )
=> ( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ B2 @ C ) @ S )
=> ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ A2 @ C ) @ ( relcomp_a_a_nat @ R @ S ) ) ) ) ).
% relcomp.relcompI
thf(fact_342_relcomp_OrelcompI,axiom,
! [A2: a,B2: nat,R: set_Pr4934435412358123699_a_nat,C: nat,S: set_Pr1261947904930325089at_nat] :
( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ A2 @ B2 ) @ R )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B2 @ C ) @ S )
=> ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ A2 @ C ) @ ( relcomp_a_nat_nat @ R @ S ) ) ) ) ).
% relcomp.relcompI
thf(fact_343_relcomp_OrelcompI,axiom,
! [A2: term_a_b,B2: term_a_b,R: set_Pr4386577575007340137rm_a_b,C: term_a_b,S: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ R )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ C ) @ S )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ C ) @ ( relcom370159955682700863rm_a_b @ R @ S ) ) ) ) ).
% relcomp.relcompI
thf(fact_344_relcompE,axiom,
! [Xz: product_prod_a_nat,R: set_Product_prod_a_a,S: set_Pr4934435412358123699_a_nat] :
( ( member5724188588386418708_a_nat @ Xz @ ( relcomp_a_a_nat @ R @ S ) )
=> ~ ! [X: a,Y: a,Z3: nat] :
( ( Xz
= ( product_Pair_a_nat @ X @ Z3 ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R )
=> ~ ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ Y @ Z3 ) @ S ) ) ) ) ).
% relcompE
thf(fact_345_relcompE,axiom,
! [Xz: product_prod_a_nat,R: set_Pr4934435412358123699_a_nat,S: set_Pr1261947904930325089at_nat] :
( ( member5724188588386418708_a_nat @ Xz @ ( relcomp_a_nat_nat @ R @ S ) )
=> ~ ! [X: a,Y: nat,Z3: nat] :
( ( Xz
= ( product_Pair_a_nat @ X @ Z3 ) )
=> ( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ X @ Y ) @ R )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ Z3 ) @ S ) ) ) ) ).
% relcompE
thf(fact_346_relcompE,axiom,
! [Xz: produc357393685978478089rm_a_b,R: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ Xz @ ( relcom370159955682700863rm_a_b @ R @ S ) )
=> ~ ! [X: term_a_b,Y: term_a_b,Z3: term_a_b] :
( ( Xz
= ( produc7020197800436672577rm_a_b @ X @ Z3 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ R )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ Z3 ) @ S ) ) ) ) ).
% relcompE
thf(fact_347_relcompEpair,axiom,
! [A2: a,C: nat,R: set_Product_prod_a_a,S: set_Pr4934435412358123699_a_nat] :
( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ A2 @ C ) @ ( relcomp_a_a_nat @ R @ S ) )
=> ~ ! [B5: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ B5 ) @ R )
=> ~ ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ B5 @ C ) @ S ) ) ) ).
% relcompEpair
thf(fact_348_relcompEpair,axiom,
! [A2: a,C: nat,R: set_Pr4934435412358123699_a_nat,S: set_Pr1261947904930325089at_nat] :
( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ A2 @ C ) @ ( relcomp_a_nat_nat @ R @ S ) )
=> ~ ! [B5: nat] :
( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ A2 @ B5 ) @ R )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B5 @ C ) @ S ) ) ) ).
% relcompEpair
thf(fact_349_relcompEpair,axiom,
! [A2: term_a_b,C: term_a_b,R: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ C ) @ ( relcom370159955682700863rm_a_b @ R @ S ) )
=> ~ ! [B5: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B5 ) @ R )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B5 @ C ) @ S ) ) ) ).
% relcompEpair
thf(fact_350_O__mono1,axiom,
! [R4: set_Pr4934435412358123699_a_nat,R6: set_Pr4934435412358123699_a_nat,S4: set_Product_prod_a_a] :
( ( ord_le8666007276011122963_a_nat @ R4 @ R6 )
=> ( ord_le8666007276011122963_a_nat @ ( relcomp_a_a_nat @ S4 @ R4 ) @ ( relcomp_a_a_nat @ S4 @ R6 ) ) ) ).
% O_mono1
thf(fact_351_O__mono1,axiom,
! [R4: set_Pr4386577575007340137rm_a_b,R6: set_Pr4386577575007340137rm_a_b,S4: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ R4 @ R6 )
=> ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ S4 @ R4 ) @ ( relcom370159955682700863rm_a_b @ S4 @ R6 ) ) ) ).
% O_mono1
thf(fact_352_O__mono2,axiom,
! [R4: set_Pr4934435412358123699_a_nat,R6: set_Pr4934435412358123699_a_nat,T4: set_Pr1261947904930325089at_nat] :
( ( ord_le8666007276011122963_a_nat @ R4 @ R6 )
=> ( ord_le8666007276011122963_a_nat @ ( relcomp_a_nat_nat @ R4 @ T4 ) @ ( relcomp_a_nat_nat @ R6 @ T4 ) ) ) ).
% O_mono2
thf(fact_353_O__mono2,axiom,
! [R4: set_Pr4386577575007340137rm_a_b,R6: set_Pr4386577575007340137rm_a_b,T4: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ R4 @ R6 )
=> ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ R4 @ T4 ) @ ( relcom370159955682700863rm_a_b @ R6 @ T4 ) ) ) ).
% O_mono2
thf(fact_354_relcomp__mono,axiom,
! [R7: set_Product_prod_a_a,R: set_Product_prod_a_a,S5: set_Pr4934435412358123699_a_nat,S: set_Pr4934435412358123699_a_nat] :
( ( ord_le746702958409616551od_a_a @ R7 @ R )
=> ( ( ord_le8666007276011122963_a_nat @ S5 @ S )
=> ( ord_le8666007276011122963_a_nat @ ( relcomp_a_a_nat @ R7 @ S5 ) @ ( relcomp_a_a_nat @ R @ S ) ) ) ) ).
% relcomp_mono
thf(fact_355_relcomp__mono,axiom,
! [R7: set_Pr4934435412358123699_a_nat,R: set_Pr4934435412358123699_a_nat,S5: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
( ( ord_le8666007276011122963_a_nat @ R7 @ R )
=> ( ( ord_le3146513528884898305at_nat @ S5 @ S )
=> ( ord_le8666007276011122963_a_nat @ ( relcomp_a_nat_nat @ R7 @ S5 ) @ ( relcomp_a_nat_nat @ R @ S ) ) ) ) ).
% relcomp_mono
thf(fact_356_relcomp__mono,axiom,
! [R7: set_Pr4386577575007340137rm_a_b,R: set_Pr4386577575007340137rm_a_b,S5: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ R7 @ R )
=> ( ( ord_le118470702582115849rm_a_b @ S5 @ S )
=> ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ R7 @ S5 ) @ ( relcom370159955682700863rm_a_b @ R @ S ) ) ) ) ).
% relcomp_mono
thf(fact_357_steps__map,axiom,
! [P: term_a_b > $o,Q: set_Pr4386577575007340137rm_a_b > $o,F2: term_a_b > term_a_b,G2: set_Pr4386577575007340137rm_a_b > set_Pr4386577575007340137rm_a_b,T: term_a_b,R4: set_Pr4386577575007340137rm_a_b,S4: set_Pr4386577575007340137rm_a_b,U: term_a_b] :
( ! [T3: term_a_b,U2: term_a_b,R8: set_Pr4386577575007340137rm_a_b] :
( ( P @ T3 )
=> ( ( Q @ R8 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T3 @ U2 ) @ R8 )
=> ( ( P @ U2 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F2 @ T3 ) @ ( F2 @ U2 ) ) @ ( G2 @ R8 ) ) ) ) ) )
=> ( ( P @ T )
=> ( ( Q @ R4 )
=> ( ( Q @ S4 )
=> ( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T @ U ) @ ( transi7742714808557438673rm_a_b @ R4 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F2 @ T ) @ ( F2 @ U ) ) @ ( transi7742714808557438673rm_a_b @ ( G2 @ R4 ) ) ) )
& ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T @ U ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ R4 ) @ ( relcom370159955682700863rm_a_b @ S4 @ ( transi7742714808557438673rm_a_b @ R4 ) ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F2 @ T ) @ ( F2 @ U ) ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( G2 @ R4 ) ) @ ( relcom370159955682700863rm_a_b @ ( G2 @ S4 ) @ ( transi7742714808557438673rm_a_b @ ( G2 @ R4 ) ) ) ) ) ) ) ) ) ) ) ).
% steps_map
thf(fact_358_compat__tr__compat,axiom,
! [NS: set_Product_prod_a_a,S4: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ ( relcomp_a_a_nat @ NS @ S4 ) @ S4 )
=> ( ord_le8666007276011122963_a_nat @ ( relcomp_a_a_nat @ ( transitive_rtrancl_a @ NS ) @ S4 ) @ S4 ) ) ).
% compat_tr_compat
thf(fact_359_compat__tr__compat,axiom,
! [NS: set_Pr4386577575007340137rm_a_b,S4: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ NS @ S4 ) @ S4 )
=> ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ NS ) @ S4 ) @ S4 ) ) ).
% compat_tr_compat
thf(fact_360_rtrancl__O__push,axiom,
! [S4: set_Pr4386577575007340137rm_a_b,R4: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ S4 @ R4 ) @ ( relcom370159955682700863rm_a_b @ R4 @ ( transi7742714808557438673rm_a_b @ S4 ) ) )
=> ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ S4 ) @ R4 ) @ ( relcom370159955682700863rm_a_b @ R4 @ ( transi7742714808557438673rm_a_b @ S4 ) ) ) ) ).
% rtrancl_O_push
thf(fact_361_kleene__induct,axiom,
! [A: set_Pr4934435412358123699_a_nat,X5: set_Pr4934435412358123699_a_nat,B: set_Product_prod_a_a,C2: set_Pr1261947904930325089at_nat] :
( ( ord_le8666007276011122963_a_nat @ A @ X5 )
=> ( ( ord_le8666007276011122963_a_nat @ ( relcomp_a_a_nat @ B @ X5 ) @ X5 )
=> ( ( ord_le8666007276011122963_a_nat @ ( relcomp_a_nat_nat @ X5 @ C2 ) @ X5 )
=> ( ord_le8666007276011122963_a_nat @ ( relcomp_a_a_nat @ ( transitive_rtrancl_a @ B ) @ ( relcomp_a_nat_nat @ A @ ( transi2905341329935302413cl_nat @ C2 ) ) ) @ X5 ) ) ) ) ).
% kleene_induct
thf(fact_362_kleene__induct,axiom,
! [A: set_Pr4386577575007340137rm_a_b,X5: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b,C2: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A @ X5 )
=> ( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ B @ X5 ) @ X5 )
=> ( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ X5 @ C2 ) @ X5 )
=> ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ B ) @ ( relcom370159955682700863rm_a_b @ A @ ( transi7742714808557438673rm_a_b @ C2 ) ) ) @ X5 ) ) ) ) ).
% kleene_induct
thf(fact_363_rrstep_Ocases,axiom,
! [A1: term_a_b,A22: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A22 ) @ ( rrstep_a_b @ R2 ) )
=> ~ ! [L3: term_a_b,R9: term_a_b,Sigma2: b > term_a_b] :
( ( A1
= ( subst_7999470309526761004_a_b_b @ L3 @ Sigma2 ) )
=> ( ( A22
= ( subst_7999470309526761004_a_b_b @ R9 @ Sigma2 ) )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ L3 @ R9 ) @ R2 ) ) ) ) ).
% rrstep.cases
thf(fact_364_rrstep_Osimps,axiom,
! [A1: term_a_b,A22: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A22 ) @ ( rrstep_a_b @ R2 ) )
= ( ? [L4: term_a_b,R3: term_a_b,Sigma3: b > term_a_b] :
( ( A1
= ( subst_7999470309526761004_a_b_b @ L4 @ Sigma3 ) )
& ( A22
= ( subst_7999470309526761004_a_b_b @ R3 @ Sigma3 ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ L4 @ R3 ) @ R2 ) ) ) ) ).
% rrstep.simps
thf(fact_365_rrstep_Ointros,axiom,
! [L: term_a_b,R: term_a_b,R2: set_Pr4386577575007340137rm_a_b,Sigma: b > term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ L @ R ) @ R2 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( subst_7999470309526761004_a_b_b @ L @ Sigma ) @ ( subst_7999470309526761004_a_b_b @ R @ Sigma ) ) @ ( rrstep_a_b @ R2 ) ) ) ).
% rrstep.intros
thf(fact_366_rrstep__subst,axiom,
! [S: term_a_b,T: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( rrstep_a_b @ R2 ) )
=> ~ ! [L3: term_a_b,R9: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ L3 @ R9 ) @ R2 )
=> ! [Sigma2: b > term_a_b] :
( ( S
= ( subst_7999470309526761004_a_b_b @ L3 @ Sigma2 ) )
=> ( T
!= ( subst_7999470309526761004_a_b_b @ R9 @ Sigma2 ) ) ) ) ) ).
% rrstep_subst
thf(fact_367_comp__trancl,axiom,
! [R4: set_Pr4386577575007340137rm_a_b,S4: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ R4 @ S4 ) @ S4 )
=> ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ R4 @ ( transi7922773638565587891rm_a_b @ S4 ) ) @ ( transi7922773638565587891rm_a_b @ S4 ) ) ) ).
% comp_trancl
thf(fact_368_kleene__trancl__induct,axiom,
! [A: set_Pr4934435412358123699_a_nat,X5: set_Pr4934435412358123699_a_nat,B: set_Product_prod_a_a,C2: set_Pr1261947904930325089at_nat] :
( ( ord_le8666007276011122963_a_nat @ A @ X5 )
=> ( ( ord_le8666007276011122963_a_nat @ ( relcomp_a_a_nat @ B @ X5 ) @ X5 )
=> ( ( ord_le8666007276011122963_a_nat @ ( relcomp_a_nat_nat @ X5 @ C2 ) @ X5 )
=> ( ord_le8666007276011122963_a_nat @ ( relcomp_a_a_nat @ ( transitive_trancl_a @ B ) @ ( relcomp_a_nat_nat @ A @ ( transi6264000038957366511cl_nat @ C2 ) ) ) @ X5 ) ) ) ) ).
% kleene_trancl_induct
thf(fact_369_kleene__trancl__induct,axiom,
! [A: set_Pr4386577575007340137rm_a_b,X5: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b,C2: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A @ X5 )
=> ( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ B @ X5 ) @ X5 )
=> ( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ X5 @ C2 ) @ X5 )
=> ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ ( transi7922773638565587891rm_a_b @ B ) @ ( relcom370159955682700863rm_a_b @ A @ ( transi7922773638565587891rm_a_b @ C2 ) ) ) @ X5 ) ) ) ) ).
% kleene_trancl_induct
thf(fact_370_rtrancl__induct2,axiom,
! [Ax: term_a_b,Ay: term_a_b,Bx: term_a_b,By: term_a_b,R: set_Pr2972776593051762503rm_a_b,P: 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 @ R ) )
=> ( ( P @ Ax @ Ay )
=> ( ! [A5: term_a_b,B5: 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 @ A5 @ B5 ) ) @ ( transi2615809358984392588rm_a_b @ R ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ A5 @ B5 ) @ ( produc7020197800436672577rm_a_b @ Aa @ Ba ) ) @ R )
=> ( ( P @ A5 @ B5 )
=> ( P @ Aa @ Ba ) ) ) )
=> ( P @ Bx @ By ) ) ) ) ).
% rtrancl_induct2
thf(fact_371_converse__rtranclE2,axiom,
! [Xa: term_a_b,Xb: term_a_b,Za: term_a_b,Zb: term_a_b,R: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ Xa @ Xb ) @ ( produc7020197800436672577rm_a_b @ Za @ Zb ) ) @ ( transi2615809358984392588rm_a_b @ R ) )
=> ( ( ( produc7020197800436672577rm_a_b @ Xa @ Xb )
!= ( produc7020197800436672577rm_a_b @ Za @ Zb ) )
=> ~ ! [A5: term_a_b,B5: term_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ Xa @ Xb ) @ ( produc7020197800436672577rm_a_b @ A5 @ B5 ) ) @ R )
=> ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ A5 @ B5 ) @ ( produc7020197800436672577rm_a_b @ Za @ Zb ) ) @ ( transi2615809358984392588rm_a_b @ R ) ) ) ) ) ).
% converse_rtranclE2
thf(fact_372_converse__rtrancl__induct2,axiom,
! [Ax: term_a_b,Ay: term_a_b,Bx: term_a_b,By: term_a_b,R: set_Pr2972776593051762503rm_a_b,P: 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 @ R ) )
=> ( ( P @ Bx @ By )
=> ( ! [A5: term_a_b,B5: term_a_b,Aa: term_a_b,Ba: term_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ A5 @ B5 ) @ ( produc7020197800436672577rm_a_b @ Aa @ Ba ) ) @ R )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ Aa @ Ba ) @ ( produc7020197800436672577rm_a_b @ Bx @ By ) ) @ ( transi2615809358984392588rm_a_b @ R ) )
=> ( ( P @ Aa @ Ba )
=> ( P @ A5 @ B5 ) ) ) )
=> ( P @ Ax @ Ay ) ) ) ) ).
% converse_rtrancl_induct2
thf(fact_373_rstep__imp__C__s__r,axiom,
! [S: term_a_b,T: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( rstep_a_b @ R2 ) )
=> ? [C5: subterm_and_ctxt_a_b,Sigma2: b > term_a_b,L3: term_a_b,R9: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ L3 @ R9 ) @ R2 )
& ( S
= ( subter2376574525758040790rm_a_b @ C5 @ ( subst_7999470309526761004_a_b_b @ L3 @ Sigma2 ) ) )
& ( T
= ( subter2376574525758040790rm_a_b @ C5 @ ( subst_7999470309526761004_a_b_b @ R9 @ Sigma2 ) ) ) ) ) ).
% rstep_imp_C_s_r
thf(fact_374_rstep__imp__C__s__r_H,axiom,
! [S: term_a_b,T: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( rstep_a_b @ R2 ) )
=> ~ ! [C5: subterm_and_ctxt_a_b,L3: term_a_b,R9: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ L3 @ R9 ) @ R2 )
=> ! [Sigma2: b > term_a_b] :
( ( S
= ( subter2376574525758040790rm_a_b @ C5 @ ( subst_7999470309526761004_a_b_b @ L3 @ Sigma2 ) ) )
=> ( T
!= ( subter2376574525758040790rm_a_b @ C5 @ ( subst_7999470309526761004_a_b_b @ R9 @ Sigma2 ) ) ) ) ) ) ).
% rstep_imp_C_s_r'
thf(fact_375_non__strict__into__strict,axiom,
! [NS: set_Product_prod_a_a,S4: set_Pr4934435412358123699_a_nat,S: a,T: nat] :
( ( ord_le8666007276011122963_a_nat @ ( relcomp_a_a_nat @ NS @ S4 ) @ S4 )
=> ( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ S @ T ) @ ( relcomp_a_a_nat @ ( transitive_rtrancl_a @ NS ) @ S4 ) )
=> ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ S @ T ) @ S4 ) ) ) ).
% non_strict_into_strict
thf(fact_376_non__strict__into__strict,axiom,
! [NS: set_Pr4386577575007340137rm_a_b,S4: set_Pr4386577575007340137rm_a_b,S: term_a_b,T: term_a_b] :
( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ NS @ S4 ) @ S4 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ NS ) @ S4 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ S4 ) ) ) ).
% non_strict_into_strict
thf(fact_377_UNF__onI,axiom,
! [A: set_Pr4386577575007340137rm_a_b,R: set_Pr2972776593051762503rm_a_b] :
( ! [A5: produc357393685978478089rm_a_b,B5: produc357393685978478089rm_a_b,C4: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ A5 @ A )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A5 @ B5 ) @ ( abstra1496011390211842225rm_a_b @ R ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A5 @ C4 ) @ ( abstra1496011390211842225rm_a_b @ R ) )
=> ( B5 = C4 ) ) ) )
=> ( abstra1303798319023342519rm_a_b @ R @ A ) ) ).
% UNF_onI
thf(fact_378_UNF__onI,axiom,
! [A: set_Pr4934435412358123699_a_nat,R: set_Pr1811044260758604347_a_nat] :
( ! [A5: product_prod_a_nat,B5: product_prod_a_nat,C4: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ A5 @ A )
=> ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A5 @ B5 ) @ ( abstra3832961281062565333_a_nat @ R ) )
=> ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A5 @ C4 ) @ ( abstra3832961281062565333_a_nat @ R ) )
=> ( B5 = C4 ) ) ) )
=> ( abstra7833083522531381839_a_nat @ R @ A ) ) ).
% UNF_onI
thf(fact_379_UNF__onI,axiom,
! [A: set_term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ! [A5: term_a_b,B5: term_a_b,C4: term_a_b] :
( ( member_term_a_b @ A5 @ A )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A5 @ B5 ) @ ( abstra2972651231392952684rm_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A5 @ C4 ) @ ( abstra2972651231392952684rm_a_b @ R ) )
=> ( B5 = C4 ) ) ) )
=> ( abstra3619855735679963878rm_a_b @ R @ A ) ) ).
% UNF_onI
thf(fact_380_WN__onI,axiom,
! [A: set_Pr4386577575007340137rm_a_b,R: set_Pr2972776593051762503rm_a_b] :
( ! [A5: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ A5 @ A )
=> ? [B7: produc357393685978478089rm_a_b] : ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A5 @ B7 ) @ ( abstra1496011390211842225rm_a_b @ R ) ) )
=> ( abstra2721873542170697047rm_a_b @ R @ A ) ) ).
% WN_onI
thf(fact_381_WN__onI,axiom,
! [A: set_Pr4934435412358123699_a_nat,R: set_Pr1811044260758604347_a_nat] :
( ! [A5: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ A5 @ A )
=> ? [B7: product_prod_a_nat] : ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A5 @ B7 ) @ ( abstra3832961281062565333_a_nat @ R ) ) )
=> ( abstra4582696096554303663_a_nat @ R @ A ) ) ).
% WN_onI
thf(fact_382_WN__onI,axiom,
! [A: set_term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ! [A5: term_a_b] :
( ( member_term_a_b @ A5 @ A )
=> ? [B7: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A5 @ B7 ) @ ( abstra2972651231392952684rm_a_b @ R ) ) )
=> ( abstra4644735066934440262rm_a_b @ R @ A ) ) ).
% WN_onI
thf(fact_383_comp__rtrancl__trancl,axiom,
! [R4: set_Pr4386577575007340137rm_a_b,S4: set_Pr4386577575007340137rm_a_b,S: term_a_b,T: term_a_b] :
( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ R4 @ S4 ) @ S4 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ R4 @ S4 ) ) @ S4 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( transi7922773638565587891rm_a_b @ S4 ) ) ) ) ).
% comp_rtrancl_trancl
thf(fact_384_quasi__commute__def,axiom,
( abstra4317384144805385751rm_a_b
= ( ^ [R3: set_Pr4386577575007340137rm_a_b,S3: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ S3 @ R3 ) @ ( relcom370159955682700863rm_a_b @ R3 @ ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ R3 @ S3 ) ) ) ) ) ) ).
% quasi_commute_def
thf(fact_385_normalizability__I,axiom,
! [A2: produc357393685978478089rm_a_b,B2: produc357393685978478089rm_a_b,A: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A2 @ B2 ) @ ( transi2615809358984392588rm_a_b @ A ) )
=> ( ( member5869715511025134514rm_a_b @ B2 @ ( abstra1902783451776770197rm_a_b @ A ) )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A2 @ B2 ) @ ( abstra1496011390211842225rm_a_b @ A ) ) ) ) ).
% normalizability_I
thf(fact_386_normalizability__I,axiom,
! [A2: product_prod_a_nat,B2: product_prod_a_nat,A: set_Pr1811044260758604347_a_nat] :
( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A2 @ B2 ) @ ( transi2726145917338391738_a_nat @ A ) )
=> ( ( member5724188588386418708_a_nat @ B2 @ ( abstra2685462378305987313_a_nat @ A ) )
=> ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A2 @ B2 ) @ ( abstra3832961281062565333_a_nat @ A ) ) ) ) ).
% normalizability_I
thf(fact_387_normalizability__I,axiom,
! [A2: term_a_b,B2: term_a_b,A: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( transi7742714808557438673rm_a_b @ A ) )
=> ( ( member_term_a_b @ B2 @ ( abstract_NF_term_a_b @ A ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( abstra2972651231392952684rm_a_b @ A ) ) ) ) ).
% normalizability_I
thf(fact_388_UNF__onD,axiom,
! [R: set_Pr2972776593051762503rm_a_b,A: set_Pr4386577575007340137rm_a_b,A2: produc357393685978478089rm_a_b,B2: produc357393685978478089rm_a_b,C: produc357393685978478089rm_a_b] :
( ( abstra1303798319023342519rm_a_b @ R @ A )
=> ( ( member5869715511025134514rm_a_b @ A2 @ A )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A2 @ B2 ) @ ( abstra1496011390211842225rm_a_b @ R ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A2 @ C ) @ ( abstra1496011390211842225rm_a_b @ R ) )
=> ( B2 = C ) ) ) ) ) ).
% UNF_onD
thf(fact_389_UNF__onD,axiom,
! [R: set_Pr1811044260758604347_a_nat,A: set_Pr4934435412358123699_a_nat,A2: product_prod_a_nat,B2: product_prod_a_nat,C: product_prod_a_nat] :
( ( abstra7833083522531381839_a_nat @ R @ A )
=> ( ( member5724188588386418708_a_nat @ A2 @ A )
=> ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A2 @ B2 ) @ ( abstra3832961281062565333_a_nat @ R ) )
=> ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A2 @ C ) @ ( abstra3832961281062565333_a_nat @ R ) )
=> ( B2 = C ) ) ) ) ) ).
% UNF_onD
thf(fact_390_UNF__onD,axiom,
! [R: set_Pr4386577575007340137rm_a_b,A: set_term_a_b,A2: term_a_b,B2: term_a_b,C: term_a_b] :
( ( abstra3619855735679963878rm_a_b @ R @ A )
=> ( ( member_term_a_b @ A2 @ A )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( abstra2972651231392952684rm_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ C ) @ ( abstra2972651231392952684rm_a_b @ R ) )
=> ( B2 = C ) ) ) ) ) ).
% UNF_onD
thf(fact_391_UNF__onE,axiom,
! [R: set_Pr2972776593051762503rm_a_b,A: set_Pr4386577575007340137rm_a_b,A2: produc357393685978478089rm_a_b,B2: produc357393685978478089rm_a_b,C: produc357393685978478089rm_a_b] :
( ( abstra1303798319023342519rm_a_b @ R @ A )
=> ( ( member5869715511025134514rm_a_b @ A2 @ A )
=> ( ( B2 != C )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A2 @ B2 ) @ ( abstra1496011390211842225rm_a_b @ R ) )
=> ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A2 @ C ) @ ( abstra1496011390211842225rm_a_b @ R ) ) ) ) ) ) ).
% UNF_onE
thf(fact_392_UNF__onE,axiom,
! [R: set_Pr1811044260758604347_a_nat,A: set_Pr4934435412358123699_a_nat,A2: product_prod_a_nat,B2: product_prod_a_nat,C: product_prod_a_nat] :
( ( abstra7833083522531381839_a_nat @ R @ A )
=> ( ( member5724188588386418708_a_nat @ A2 @ A )
=> ( ( B2 != C )
=> ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A2 @ B2 ) @ ( abstra3832961281062565333_a_nat @ R ) )
=> ~ ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A2 @ C ) @ ( abstra3832961281062565333_a_nat @ R ) ) ) ) ) ) ).
% UNF_onE
thf(fact_393_UNF__onE,axiom,
! [R: set_Pr4386577575007340137rm_a_b,A: set_term_a_b,A2: term_a_b,B2: term_a_b,C: term_a_b] :
( ( abstra3619855735679963878rm_a_b @ R @ A )
=> ( ( member_term_a_b @ A2 @ A )
=> ( ( B2 != C )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( abstra2972651231392952684rm_a_b @ R ) )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ C ) @ ( abstra2972651231392952684rm_a_b @ R ) ) ) ) ) ) ).
% UNF_onE
thf(fact_394_UNF__on__def,axiom,
( abstra3619855735679963878rm_a_b
= ( ^ [R3: 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 @ R3 ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X4 @ C3 ) @ ( abstra2972651231392952684rm_a_b @ R3 ) ) )
=> ( B4 = C3 ) ) ) ) ) ).
% UNF_on_def
thf(fact_395_WN__onD,axiom,
! [R: set_Pr2972776593051762503rm_a_b,A: set_Pr4386577575007340137rm_a_b,A2: produc357393685978478089rm_a_b] :
( ( abstra2721873542170697047rm_a_b @ R @ A )
=> ( ( member5869715511025134514rm_a_b @ A2 @ A )
=> ? [B5: produc357393685978478089rm_a_b] : ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A2 @ B5 ) @ ( abstra1496011390211842225rm_a_b @ R ) ) ) ) ).
% WN_onD
thf(fact_396_WN__onD,axiom,
! [R: set_Pr1811044260758604347_a_nat,A: set_Pr4934435412358123699_a_nat,A2: product_prod_a_nat] :
( ( abstra4582696096554303663_a_nat @ R @ A )
=> ( ( member5724188588386418708_a_nat @ A2 @ A )
=> ? [B5: product_prod_a_nat] : ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A2 @ B5 ) @ ( abstra3832961281062565333_a_nat @ R ) ) ) ) ).
% WN_onD
thf(fact_397_WN__onD,axiom,
! [R: set_Pr4386577575007340137rm_a_b,A: set_term_a_b,A2: term_a_b] :
( ( abstra4644735066934440262rm_a_b @ R @ A )
=> ( ( member_term_a_b @ A2 @ A )
=> ? [B5: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B5 ) @ ( abstra2972651231392952684rm_a_b @ R ) ) ) ) ).
% WN_onD
thf(fact_398_Un__iff,axiom,
! [C: produc357393685978478089rm_a_b,A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ C @ ( sup_su6776935440552674877rm_a_b @ A @ B ) )
= ( ( member5869715511025134514rm_a_b @ C @ A )
| ( member5869715511025134514rm_a_b @ C @ B ) ) ) ).
% Un_iff
thf(fact_399_Un__iff,axiom,
! [C: product_prod_a_nat,A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( member5724188588386418708_a_nat @ C @ ( sup_su459911885395995103_a_nat @ A @ B ) )
= ( ( member5724188588386418708_a_nat @ C @ A )
| ( member5724188588386418708_a_nat @ C @ B ) ) ) ).
% Un_iff
thf(fact_400_UnCI,axiom,
! [C: produc357393685978478089rm_a_b,B: set_Pr4386577575007340137rm_a_b,A: set_Pr4386577575007340137rm_a_b] :
( ( ~ ( member5869715511025134514rm_a_b @ C @ B )
=> ( member5869715511025134514rm_a_b @ C @ A ) )
=> ( member5869715511025134514rm_a_b @ C @ ( sup_su6776935440552674877rm_a_b @ A @ B ) ) ) ).
% UnCI
thf(fact_401_UnCI,axiom,
! [C: product_prod_a_nat,B: set_Pr4934435412358123699_a_nat,A: set_Pr4934435412358123699_a_nat] :
( ( ~ ( member5724188588386418708_a_nat @ C @ B )
=> ( member5724188588386418708_a_nat @ C @ A ) )
=> ( member5724188588386418708_a_nat @ C @ ( sup_su459911885395995103_a_nat @ A @ B ) ) ) ).
% UnCI
thf(fact_402_Un__subset__iff,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat,C2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ ( sup_su459911885395995103_a_nat @ A @ B ) @ C2 )
= ( ( ord_le8666007276011122963_a_nat @ A @ C2 )
& ( ord_le8666007276011122963_a_nat @ B @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_403_Un__subset__iff,axiom,
! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b,C2: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ ( sup_su6776935440552674877rm_a_b @ A @ B ) @ C2 )
= ( ( ord_le118470702582115849rm_a_b @ A @ C2 )
& ( ord_le118470702582115849rm_a_b @ B @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_404_NF__I,axiom,
! [A2: produc357393685978478089rm_a_b,A: set_Pr2972776593051762503rm_a_b] :
( ! [B5: produc357393685978478089rm_a_b] :
~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A2 @ B5 ) @ A )
=> ( member5869715511025134514rm_a_b @ A2 @ ( abstra1902783451776770197rm_a_b @ A ) ) ) ).
% NF_I
thf(fact_405_NF__I,axiom,
! [A2: product_prod_a_nat,A: set_Pr1811044260758604347_a_nat] :
( ! [B5: product_prod_a_nat] :
~ ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A2 @ B5 ) @ A )
=> ( member5724188588386418708_a_nat @ A2 @ ( abstra2685462378305987313_a_nat @ A ) ) ) ).
% NF_I
thf(fact_406_NF__I,axiom,
! [A2: term_a_b,A: set_Pr4386577575007340137rm_a_b] :
( ! [B5: term_a_b] :
~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B5 ) @ A )
=> ( member_term_a_b @ A2 @ ( abstract_NF_term_a_b @ A ) ) ) ).
% NF_I
thf(fact_407_UnI2,axiom,
! [C: produc357393685978478089rm_a_b,B: set_Pr4386577575007340137rm_a_b,A: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ C @ B )
=> ( member5869715511025134514rm_a_b @ C @ ( sup_su6776935440552674877rm_a_b @ A @ B ) ) ) ).
% UnI2
thf(fact_408_UnI2,axiom,
! [C: product_prod_a_nat,B: set_Pr4934435412358123699_a_nat,A: set_Pr4934435412358123699_a_nat] :
( ( member5724188588386418708_a_nat @ C @ B )
=> ( member5724188588386418708_a_nat @ C @ ( sup_su459911885395995103_a_nat @ A @ B ) ) ) ).
% UnI2
thf(fact_409_UnI1,axiom,
! [C: produc357393685978478089rm_a_b,A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ C @ A )
=> ( member5869715511025134514rm_a_b @ C @ ( sup_su6776935440552674877rm_a_b @ A @ B ) ) ) ).
% UnI1
thf(fact_410_UnI1,axiom,
! [C: product_prod_a_nat,A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( member5724188588386418708_a_nat @ C @ A )
=> ( member5724188588386418708_a_nat @ C @ ( sup_su459911885395995103_a_nat @ A @ B ) ) ) ).
% UnI1
thf(fact_411_UnE,axiom,
! [C: produc357393685978478089rm_a_b,A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ C @ ( sup_su6776935440552674877rm_a_b @ A @ B ) )
=> ( ~ ( member5869715511025134514rm_a_b @ C @ A )
=> ( member5869715511025134514rm_a_b @ C @ B ) ) ) ).
% UnE
thf(fact_412_UnE,axiom,
! [C: product_prod_a_nat,A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( member5724188588386418708_a_nat @ C @ ( sup_su459911885395995103_a_nat @ A @ B ) )
=> ( ~ ( member5724188588386418708_a_nat @ C @ A )
=> ( member5724188588386418708_a_nat @ C @ B ) ) ) ).
% UnE
thf(fact_413_subset__Un__eq,axiom,
( ord_le8666007276011122963_a_nat
= ( ^ [A6: set_Pr4934435412358123699_a_nat,B6: set_Pr4934435412358123699_a_nat] :
( ( sup_su459911885395995103_a_nat @ A6 @ B6 )
= B6 ) ) ) ).
% subset_Un_eq
thf(fact_414_subset__Un__eq,axiom,
( ord_le118470702582115849rm_a_b
= ( ^ [A6: set_Pr4386577575007340137rm_a_b,B6: set_Pr4386577575007340137rm_a_b] :
( ( sup_su6776935440552674877rm_a_b @ A6 @ B6 )
= B6 ) ) ) ).
% subset_Un_eq
thf(fact_415_subset__UnE,axiom,
! [C2: set_Pr4934435412358123699_a_nat,A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ C2 @ ( sup_su459911885395995103_a_nat @ A @ B ) )
=> ~ ! [A7: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A7 @ A )
=> ! [B8: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ B8 @ B )
=> ( C2
!= ( sup_su459911885395995103_a_nat @ A7 @ B8 ) ) ) ) ) ).
% subset_UnE
thf(fact_416_subset__UnE,axiom,
! [C2: set_Pr4386577575007340137rm_a_b,A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ C2 @ ( sup_su6776935440552674877rm_a_b @ A @ B ) )
=> ~ ! [A7: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A7 @ A )
=> ! [B8: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ B8 @ B )
=> ( C2
!= ( sup_su6776935440552674877rm_a_b @ A7 @ B8 ) ) ) ) ) ).
% subset_UnE
thf(fact_417_Un__absorb2,axiom,
! [B: set_Pr4934435412358123699_a_nat,A: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ B @ A )
=> ( ( sup_su459911885395995103_a_nat @ A @ B )
= A ) ) ).
% Un_absorb2
thf(fact_418_Un__absorb2,axiom,
! [B: set_Pr4386577575007340137rm_a_b,A: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ B @ A )
=> ( ( sup_su6776935440552674877rm_a_b @ A @ B )
= A ) ) ).
% Un_absorb2
thf(fact_419_Un__absorb1,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A @ B )
=> ( ( sup_su459911885395995103_a_nat @ A @ B )
= B ) ) ).
% Un_absorb1
thf(fact_420_Un__absorb1,axiom,
! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A @ B )
=> ( ( sup_su6776935440552674877rm_a_b @ A @ B )
= B ) ) ).
% Un_absorb1
thf(fact_421_Un__upper2,axiom,
! [B: set_Pr4934435412358123699_a_nat,A: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ B @ ( sup_su459911885395995103_a_nat @ A @ B ) ) ).
% Un_upper2
thf(fact_422_Un__upper2,axiom,
! [B: set_Pr4386577575007340137rm_a_b,A: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ B @ ( sup_su6776935440552674877rm_a_b @ A @ B ) ) ).
% Un_upper2
thf(fact_423_Un__upper1,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ A @ ( sup_su459911885395995103_a_nat @ A @ B ) ) ).
% Un_upper1
thf(fact_424_Un__upper1,axiom,
! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ A @ ( sup_su6776935440552674877rm_a_b @ A @ B ) ) ).
% Un_upper1
thf(fact_425_Un__least,axiom,
! [A: set_Pr4934435412358123699_a_nat,C2: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A @ C2 )
=> ( ( ord_le8666007276011122963_a_nat @ B @ C2 )
=> ( ord_le8666007276011122963_a_nat @ ( sup_su459911885395995103_a_nat @ A @ B ) @ C2 ) ) ) ).
% Un_least
thf(fact_426_Un__least,axiom,
! [A: set_Pr4386577575007340137rm_a_b,C2: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A @ C2 )
=> ( ( ord_le118470702582115849rm_a_b @ B @ C2 )
=> ( ord_le118470702582115849rm_a_b @ ( sup_su6776935440552674877rm_a_b @ A @ B ) @ C2 ) ) ) ).
% Un_least
thf(fact_427_Un__mono,axiom,
! [A: set_Pr4934435412358123699_a_nat,C2: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat,D: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A @ C2 )
=> ( ( ord_le8666007276011122963_a_nat @ B @ D )
=> ( ord_le8666007276011122963_a_nat @ ( sup_su459911885395995103_a_nat @ A @ B ) @ ( sup_su459911885395995103_a_nat @ C2 @ D ) ) ) ) ).
% Un_mono
thf(fact_428_Un__mono,axiom,
! [A: set_Pr4386577575007340137rm_a_b,C2: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b,D: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A @ C2 )
=> ( ( ord_le118470702582115849rm_a_b @ B @ D )
=> ( ord_le118470702582115849rm_a_b @ ( sup_su6776935440552674877rm_a_b @ A @ B ) @ ( sup_su6776935440552674877rm_a_b @ C2 @ D ) ) ) ) ).
% Un_mono
thf(fact_429_in__rtrancl__UnI,axiom,
! [X3: produc357393685978478089rm_a_b,R4: set_Pr4386577575007340137rm_a_b,S4: set_Pr4386577575007340137rm_a_b] :
( ( ( member5869715511025134514rm_a_b @ X3 @ ( transi7742714808557438673rm_a_b @ R4 ) )
| ( member5869715511025134514rm_a_b @ X3 @ ( transi7742714808557438673rm_a_b @ S4 ) ) )
=> ( member5869715511025134514rm_a_b @ X3 @ ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ R4 @ S4 ) ) ) ) ).
% in_rtrancl_UnI
thf(fact_430_NF__iff__no__step,axiom,
! [A2: produc357393685978478089rm_a_b,A: set_Pr2972776593051762503rm_a_b] :
( ( member5869715511025134514rm_a_b @ A2 @ ( abstra1902783451776770197rm_a_b @ A ) )
= ( ! [B4: produc357393685978478089rm_a_b] :
~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A2 @ B4 ) @ A ) ) ) ).
% NF_iff_no_step
thf(fact_431_NF__iff__no__step,axiom,
! [A2: product_prod_a_nat,A: set_Pr1811044260758604347_a_nat] :
( ( member5724188588386418708_a_nat @ A2 @ ( abstra2685462378305987313_a_nat @ A ) )
= ( ! [B4: product_prod_a_nat] :
~ ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A2 @ B4 ) @ A ) ) ) ).
% NF_iff_no_step
thf(fact_432_NF__iff__no__step,axiom,
! [A2: term_a_b,A: set_Pr4386577575007340137rm_a_b] :
( ( member_term_a_b @ A2 @ ( abstract_NF_term_a_b @ A ) )
= ( ! [B4: term_a_b] :
~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B4 ) @ A ) ) ) ).
% NF_iff_no_step
thf(fact_433_NF__no__step,axiom,
! [A2: produc357393685978478089rm_a_b,A: set_Pr2972776593051762503rm_a_b] :
( ( member5869715511025134514rm_a_b @ A2 @ ( abstra1902783451776770197rm_a_b @ A ) )
=> ! [B7: produc357393685978478089rm_a_b] :
~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A2 @ B7 ) @ A ) ) ).
% NF_no_step
thf(fact_434_NF__no__step,axiom,
! [A2: product_prod_a_nat,A: set_Pr1811044260758604347_a_nat] :
( ( member5724188588386418708_a_nat @ A2 @ ( abstra2685462378305987313_a_nat @ A ) )
=> ! [B7: product_prod_a_nat] :
~ ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A2 @ B7 ) @ A ) ) ).
% NF_no_step
thf(fact_435_NF__no__step,axiom,
! [A2: term_a_b,A: set_Pr4386577575007340137rm_a_b] :
( ( member_term_a_b @ A2 @ ( abstract_NF_term_a_b @ A ) )
=> ! [B7: term_a_b] :
~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B7 ) @ A ) ) ).
% NF_no_step
thf(fact_436_NF__E,axiom,
! [A2: produc357393685978478089rm_a_b,A: set_Pr2972776593051762503rm_a_b,B2: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ A2 @ ( abstra1902783451776770197rm_a_b @ A ) )
=> ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A2 @ B2 ) @ A ) ) ).
% NF_E
thf(fact_437_NF__E,axiom,
! [A2: product_prod_a_nat,A: set_Pr1811044260758604347_a_nat,B2: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ A2 @ ( abstra2685462378305987313_a_nat @ A ) )
=> ~ ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A2 @ B2 ) @ A ) ) ).
% NF_E
thf(fact_438_NF__E,axiom,
! [A2: term_a_b,A: set_Pr4386577575007340137rm_a_b,B2: term_a_b] :
( ( member_term_a_b @ A2 @ ( abstract_NF_term_a_b @ A ) )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ A ) ) ).
% NF_E
thf(fact_439_rtrancl__Un__separatorE,axiom,
! [A2: term_a_b,B2: term_a_b,P: set_Pr4386577575007340137rm_a_b,Q: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ P @ Q ) ) )
=> ( ! [X: term_a_b,Y: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ X ) @ ( transi7742714808557438673rm_a_b @ P ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ Q )
=> ( X = Y ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( transi7742714808557438673rm_a_b @ P ) ) ) ) ).
% rtrancl_Un_separatorE
thf(fact_440_rtrancl__Un__separator__converseE,axiom,
! [A2: term_a_b,B2: term_a_b,P: set_Pr4386577575007340137rm_a_b,Q: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ P @ Q ) ) )
=> ( ! [X: term_a_b,Y: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ B2 ) @ ( transi7742714808557438673rm_a_b @ P ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X ) @ Q )
=> ( Y = X ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( transi7742714808557438673rm_a_b @ P ) ) ) ) ).
% rtrancl_Un_separator_converseE
thf(fact_441_rtrancl__Un__subset,axiom,
! [R4: set_Pr4386577575007340137rm_a_b,S4: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ ( sup_su6776935440552674877rm_a_b @ ( transi7742714808557438673rm_a_b @ R4 ) @ ( transi7742714808557438673rm_a_b @ S4 ) ) @ ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ R4 @ S4 ) ) ) ).
% rtrancl_Un_subset
thf(fact_442_trancl__union__right,axiom,
! [R: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ ( transi7922773638565587891rm_a_b @ R ) @ ( transi7922773638565587891rm_a_b @ ( sup_su6776935440552674877rm_a_b @ S @ R ) ) ) ).
% trancl_union_right
thf(fact_443_NF__not__suc,axiom,
! [X3: produc357393685978478089rm_a_b,Y3: produc357393685978478089rm_a_b,R: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X3 @ Y3 ) @ ( transi2615809358984392588rm_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ X3 @ ( abstra1902783451776770197rm_a_b @ R ) )
=> ( X3 = Y3 ) ) ) ).
% NF_not_suc
thf(fact_444_NF__not__suc,axiom,
! [X3: product_prod_a_nat,Y3: product_prod_a_nat,R: set_Pr1811044260758604347_a_nat] :
( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ X3 @ Y3 ) @ ( transi2726145917338391738_a_nat @ R ) )
=> ( ( member5724188588386418708_a_nat @ X3 @ ( abstra2685462378305987313_a_nat @ R ) )
=> ( X3 = Y3 ) ) ) ).
% NF_not_suc
thf(fact_445_NF__not__suc,axiom,
! [X3: term_a_b,Y3: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ( ( member_term_a_b @ X3 @ ( abstract_NF_term_a_b @ R ) )
=> ( X3 = Y3 ) ) ) ).
% NF_not_suc
thf(fact_446_NF__anti__mono,axiom,
! [A: set_Pr1811044260758604347_a_nat,B: set_Pr1811044260758604347_a_nat] :
( ( ord_le1018141841545957851_a_nat @ A @ B )
=> ( ord_le8666007276011122963_a_nat @ ( abstra2685462378305987313_a_nat @ B ) @ ( abstra2685462378305987313_a_nat @ A ) ) ) ).
% NF_anti_mono
thf(fact_447_NF__anti__mono,axiom,
! [A: set_Pr2972776593051762503rm_a_b,B: set_Pr2972776593051762503rm_a_b] :
( ( ord_le2593958535671617255rm_a_b @ A @ B )
=> ( ord_le118470702582115849rm_a_b @ ( abstra1902783451776770197rm_a_b @ B ) @ ( abstra1902783451776770197rm_a_b @ A ) ) ) ).
% NF_anti_mono
thf(fact_448_NF__anti__mono,axiom,
! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A @ B )
=> ( ord_le2705286416250468010rm_a_b @ ( abstract_NF_term_a_b @ B ) @ ( abstract_NF_term_a_b @ A ) ) ) ).
% NF_anti_mono
thf(fact_449_NF__no__trancl__step,axiom,
! [A2: produc357393685978478089rm_a_b,A: set_Pr2972776593051762503rm_a_b] :
( ( member5869715511025134514rm_a_b @ A2 @ ( abstra1902783451776770197rm_a_b @ A ) )
=> ! [B7: produc357393685978478089rm_a_b] :
~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A2 @ B7 ) @ ( transi9211502839322181930rm_a_b @ A ) ) ) ).
% NF_no_trancl_step
thf(fact_450_NF__no__trancl__step,axiom,
! [A2: product_prod_a_nat,A: set_Pr1811044260758604347_a_nat] :
( ( member5724188588386418708_a_nat @ A2 @ ( abstra2685462378305987313_a_nat @ A ) )
=> ! [B7: product_prod_a_nat] :
~ ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A2 @ B7 ) @ ( transi6805586507763294364_a_nat @ A ) ) ) ).
% NF_no_trancl_step
thf(fact_451_NF__no__trancl__step,axiom,
! [A2: term_a_b,A: set_Pr4386577575007340137rm_a_b] :
( ( member_term_a_b @ A2 @ ( abstract_NF_term_a_b @ A ) )
=> ! [B7: term_a_b] :
~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B7 ) @ ( transi7922773638565587891rm_a_b @ A ) ) ) ).
% NF_no_trancl_step
thf(fact_452_firstStep,axiom,
! [L5: set_Pr4386577575007340137rm_a_b,S4: set_Pr4386577575007340137rm_a_b,R4: set_Pr4386577575007340137rm_a_b,X3: term_a_b,Y3: term_a_b] :
( ( L5
= ( sup_su6776935440552674877rm_a_b @ S4 @ R4 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ ( transi7742714808557438673rm_a_b @ L5 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ ( transi7742714808557438673rm_a_b @ R4 ) )
| ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ R4 ) @ ( relcom370159955682700863rm_a_b @ S4 @ ( transi7742714808557438673rm_a_b @ L5 ) ) ) ) ) ) ) ).
% firstStep
thf(fact_453_first__step,axiom,
! [C2: set_Pr4386577575007340137rm_a_b,A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b,X3: term_a_b,Y3: term_a_b,Z2: term_a_b] :
( ( C2
= ( sup_su6776935440552674877rm_a_b @ A @ B ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ ( transi7742714808557438673rm_a_b @ C2 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ Z2 ) @ B )
=> ? [Y: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ A ) @ B ) ) ) ) ) ).
% first_step
thf(fact_454_first__step__O,axiom,
! [C2: set_Pr4386577575007340137rm_a_b,A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b,X3: term_a_b,Y3: term_a_b] :
( ( C2
= ( sup_su6776935440552674877rm_a_b @ A @ B ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ C2 ) @ B ) )
=> ? [Y: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ A ) @ B ) ) ) ) ).
% first_step_O
thf(fact_455_right__comp__S,axiom,
! [X3: term_a_b,Y3: term_a_b,S4: set_Pr4386577575007340137rm_a_b,NS: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ ( relcom370159955682700863rm_a_b @ S4 @ ( sup_su6776935440552674877rm_a_b @ ( relcom370159955682700863rm_a_b @ S4 @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ S4 ) @ ( transi7742714808557438673rm_a_b @ NS ) ) ) @ ( transi7742714808557438673rm_a_b @ NS ) ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ ( relcom370159955682700863rm_a_b @ S4 @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ S4 ) @ ( transi7742714808557438673rm_a_b @ NS ) ) ) ) ) ).
% right_comp_S
thf(fact_456_rtrancl__U__push,axiom,
! [S4: set_Pr4386577575007340137rm_a_b,R4: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ S4 @ R4 ) @ ( relcom370159955682700863rm_a_b @ R4 @ ( transi7742714808557438673rm_a_b @ S4 ) ) )
=> ( ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ S4 @ R4 ) )
= ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ R4 ) @ ( transi7742714808557438673rm_a_b @ S4 ) ) ) ) ).
% rtrancl_U_push
thf(fact_457_compatible__conv,axiom,
! [NS: set_Pr4386577575007340137rm_a_b,S4: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ NS @ S4 ) @ S4 )
=> ( ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ NS @ S4 ) ) @ ( relcom370159955682700863rm_a_b @ S4 @ ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ NS @ S4 ) ) ) )
= ( relcom370159955682700863rm_a_b @ S4 @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ S4 ) @ ( transi7742714808557438673rm_a_b @ NS ) ) ) ) ) ).
% compatible_conv
thf(fact_458_normalizability__E,axiom,
! [A2: produc357393685978478089rm_a_b,B2: produc357393685978478089rm_a_b,A: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A2 @ B2 ) @ ( abstra1496011390211842225rm_a_b @ A ) )
=> ~ ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A2 @ B2 ) @ ( transi2615809358984392588rm_a_b @ A ) )
=> ~ ( member5869715511025134514rm_a_b @ B2 @ ( abstra1902783451776770197rm_a_b @ A ) ) ) ) ).
% normalizability_E
thf(fact_459_normalizability__E,axiom,
! [A2: product_prod_a_nat,B2: product_prod_a_nat,A: set_Pr1811044260758604347_a_nat] :
( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A2 @ B2 ) @ ( abstra3832961281062565333_a_nat @ A ) )
=> ~ ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A2 @ B2 ) @ ( transi2726145917338391738_a_nat @ A ) )
=> ~ ( member5724188588386418708_a_nat @ B2 @ ( abstra2685462378305987313_a_nat @ A ) ) ) ) ).
% normalizability_E
thf(fact_460_normalizability__E,axiom,
! [A2: term_a_b,B2: term_a_b,A: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( abstra2972651231392952684rm_a_b @ A ) )
=> ~ ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( transi7742714808557438673rm_a_b @ A ) )
=> ~ ( member_term_a_b @ B2 @ ( abstract_NF_term_a_b @ A ) ) ) ) ).
% normalizability_E
thf(fact_461_compatible__rtrancl__split,axiom,
! [NS: set_Pr4386577575007340137rm_a_b,S4: set_Pr4386577575007340137rm_a_b,X3: term_a_b,Y3: term_a_b] :
( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ NS @ S4 ) @ S4 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ NS @ S4 ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ ( sup_su6776935440552674877rm_a_b @ ( relcom370159955682700863rm_a_b @ S4 @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ S4 ) @ ( transi7742714808557438673rm_a_b @ NS ) ) ) @ ( transi7742714808557438673rm_a_b @ NS ) ) ) ) ) ).
% compatible_rtrancl_split
thf(fact_462_WN__on__def,axiom,
( abstra4644735066934440262rm_a_b
= ( ^ [R3: 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 @ R3 ) ) ) ) ) ).
% WN_on_def
thf(fact_463_WN__onE,axiom,
! [R: set_Pr2972776593051762503rm_a_b,A: set_Pr4386577575007340137rm_a_b,A2: produc357393685978478089rm_a_b] :
( ( abstra2721873542170697047rm_a_b @ R @ A )
=> ( ( member5869715511025134514rm_a_b @ A2 @ A )
=> ~ ! [B5: produc357393685978478089rm_a_b] :
~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A2 @ B5 ) @ ( abstra1496011390211842225rm_a_b @ R ) ) ) ) ).
% WN_onE
thf(fact_464_WN__onE,axiom,
! [R: set_Pr1811044260758604347_a_nat,A: set_Pr4934435412358123699_a_nat,A2: product_prod_a_nat] :
( ( abstra4582696096554303663_a_nat @ R @ A )
=> ( ( member5724188588386418708_a_nat @ A2 @ A )
=> ~ ! [B5: product_prod_a_nat] :
~ ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A2 @ B5 ) @ ( abstra3832961281062565333_a_nat @ R ) ) ) ) ).
% WN_onE
thf(fact_465_WN__onE,axiom,
! [R: set_Pr4386577575007340137rm_a_b,A: set_term_a_b,A2: term_a_b] :
( ( abstra4644735066934440262rm_a_b @ R @ A )
=> ( ( member_term_a_b @ A2 @ A )
=> ~ ! [B5: term_a_b] :
~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B5 ) @ ( abstra2972651231392952684rm_a_b @ R ) ) ) ) ).
% WN_onE
thf(fact_466_le__sup__iff,axiom,
! [X3: set_Pr4934435412358123699_a_nat,Y3: set_Pr4934435412358123699_a_nat,Z2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ ( sup_su459911885395995103_a_nat @ X3 @ Y3 ) @ Z2 )
= ( ( ord_le8666007276011122963_a_nat @ X3 @ Z2 )
& ( ord_le8666007276011122963_a_nat @ Y3 @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_467_le__sup__iff,axiom,
! [X3: set_Pr4386577575007340137rm_a_b,Y3: set_Pr4386577575007340137rm_a_b,Z2: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ ( sup_su6776935440552674877rm_a_b @ X3 @ Y3 ) @ Z2 )
= ( ( ord_le118470702582115849rm_a_b @ X3 @ Z2 )
& ( ord_le118470702582115849rm_a_b @ Y3 @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_468_sup_Obounded__iff,axiom,
! [B2: set_Pr4934435412358123699_a_nat,C: set_Pr4934435412358123699_a_nat,A2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ ( sup_su459911885395995103_a_nat @ B2 @ C ) @ A2 )
= ( ( ord_le8666007276011122963_a_nat @ B2 @ A2 )
& ( ord_le8666007276011122963_a_nat @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_469_sup_Obounded__iff,axiom,
! [B2: set_Pr4386577575007340137rm_a_b,C: set_Pr4386577575007340137rm_a_b,A2: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ ( sup_su6776935440552674877rm_a_b @ B2 @ C ) @ A2 )
= ( ( ord_le118470702582115849rm_a_b @ B2 @ A2 )
& ( ord_le118470702582115849rm_a_b @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_470_sup_OcoboundedI2,axiom,
! [C: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat,A2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ C @ B2 )
=> ( ord_le8666007276011122963_a_nat @ C @ ( sup_su459911885395995103_a_nat @ A2 @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_471_sup_OcoboundedI2,axiom,
! [C: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b,A2: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ C @ B2 )
=> ( ord_le118470702582115849rm_a_b @ C @ ( sup_su6776935440552674877rm_a_b @ A2 @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_472_sup_OcoboundedI1,axiom,
! [C: set_Pr4934435412358123699_a_nat,A2: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ C @ A2 )
=> ( ord_le8666007276011122963_a_nat @ C @ ( sup_su459911885395995103_a_nat @ A2 @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_473_sup_OcoboundedI1,axiom,
! [C: set_Pr4386577575007340137rm_a_b,A2: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ C @ A2 )
=> ( ord_le118470702582115849rm_a_b @ C @ ( sup_su6776935440552674877rm_a_b @ A2 @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_474_sup_Oabsorb__iff2,axiom,
( ord_le8666007276011122963_a_nat
= ( ^ [A4: set_Pr4934435412358123699_a_nat,B4: set_Pr4934435412358123699_a_nat] :
( ( sup_su459911885395995103_a_nat @ A4 @ B4 )
= B4 ) ) ) ).
% sup.absorb_iff2
thf(fact_475_sup_Oabsorb__iff2,axiom,
( ord_le118470702582115849rm_a_b
= ( ^ [A4: set_Pr4386577575007340137rm_a_b,B4: set_Pr4386577575007340137rm_a_b] :
( ( sup_su6776935440552674877rm_a_b @ A4 @ B4 )
= B4 ) ) ) ).
% sup.absorb_iff2
thf(fact_476_sup_Oabsorb__iff1,axiom,
( ord_le8666007276011122963_a_nat
= ( ^ [B4: set_Pr4934435412358123699_a_nat,A4: set_Pr4934435412358123699_a_nat] :
( ( sup_su459911885395995103_a_nat @ A4 @ B4 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_477_sup_Oabsorb__iff1,axiom,
( ord_le118470702582115849rm_a_b
= ( ^ [B4: set_Pr4386577575007340137rm_a_b,A4: set_Pr4386577575007340137rm_a_b] :
( ( sup_su6776935440552674877rm_a_b @ A4 @ B4 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_478_sup_Ocobounded2,axiom,
! [B2: set_Pr4934435412358123699_a_nat,A2: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ B2 @ ( sup_su459911885395995103_a_nat @ A2 @ B2 ) ) ).
% sup.cobounded2
thf(fact_479_sup_Ocobounded2,axiom,
! [B2: set_Pr4386577575007340137rm_a_b,A2: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ B2 @ ( sup_su6776935440552674877rm_a_b @ A2 @ B2 ) ) ).
% sup.cobounded2
thf(fact_480_sup_Ocobounded1,axiom,
! [A2: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ A2 @ ( sup_su459911885395995103_a_nat @ A2 @ B2 ) ) ).
% sup.cobounded1
thf(fact_481_sup_Ocobounded1,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ A2 @ ( sup_su6776935440552674877rm_a_b @ A2 @ B2 ) ) ).
% sup.cobounded1
thf(fact_482_sup_Oorder__iff,axiom,
( ord_le8666007276011122963_a_nat
= ( ^ [B4: set_Pr4934435412358123699_a_nat,A4: set_Pr4934435412358123699_a_nat] :
( A4
= ( sup_su459911885395995103_a_nat @ A4 @ B4 ) ) ) ) ).
% sup.order_iff
thf(fact_483_sup_Oorder__iff,axiom,
( ord_le118470702582115849rm_a_b
= ( ^ [B4: set_Pr4386577575007340137rm_a_b,A4: set_Pr4386577575007340137rm_a_b] :
( A4
= ( sup_su6776935440552674877rm_a_b @ A4 @ B4 ) ) ) ) ).
% sup.order_iff
thf(fact_484_sup_OboundedI,axiom,
! [B2: set_Pr4934435412358123699_a_nat,A2: set_Pr4934435412358123699_a_nat,C: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ B2 @ A2 )
=> ( ( ord_le8666007276011122963_a_nat @ C @ A2 )
=> ( ord_le8666007276011122963_a_nat @ ( sup_su459911885395995103_a_nat @ B2 @ C ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_485_sup_OboundedI,axiom,
! [B2: set_Pr4386577575007340137rm_a_b,A2: set_Pr4386577575007340137rm_a_b,C: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ B2 @ A2 )
=> ( ( ord_le118470702582115849rm_a_b @ C @ A2 )
=> ( ord_le118470702582115849rm_a_b @ ( sup_su6776935440552674877rm_a_b @ B2 @ C ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_486_sup_OboundedE,axiom,
! [B2: set_Pr4934435412358123699_a_nat,C: set_Pr4934435412358123699_a_nat,A2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ ( sup_su459911885395995103_a_nat @ B2 @ C ) @ A2 )
=> ~ ( ( ord_le8666007276011122963_a_nat @ B2 @ A2 )
=> ~ ( ord_le8666007276011122963_a_nat @ C @ A2 ) ) ) ).
% sup.boundedE
thf(fact_487_sup_OboundedE,axiom,
! [B2: set_Pr4386577575007340137rm_a_b,C: set_Pr4386577575007340137rm_a_b,A2: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ ( sup_su6776935440552674877rm_a_b @ B2 @ C ) @ A2 )
=> ~ ( ( ord_le118470702582115849rm_a_b @ B2 @ A2 )
=> ~ ( ord_le118470702582115849rm_a_b @ C @ A2 ) ) ) ).
% sup.boundedE
thf(fact_488_sup__absorb2,axiom,
! [X3: set_Pr4934435412358123699_a_nat,Y3: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X3 @ Y3 )
=> ( ( sup_su459911885395995103_a_nat @ X3 @ Y3 )
= Y3 ) ) ).
% sup_absorb2
thf(fact_489_sup__absorb2,axiom,
! [X3: set_Pr4386577575007340137rm_a_b,Y3: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ X3 @ Y3 )
=> ( ( sup_su6776935440552674877rm_a_b @ X3 @ Y3 )
= Y3 ) ) ).
% sup_absorb2
thf(fact_490_sup__absorb1,axiom,
! [Y3: set_Pr4934435412358123699_a_nat,X3: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ Y3 @ X3 )
=> ( ( sup_su459911885395995103_a_nat @ X3 @ Y3 )
= X3 ) ) ).
% sup_absorb1
thf(fact_491_sup__absorb1,axiom,
! [Y3: set_Pr4386577575007340137rm_a_b,X3: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ Y3 @ X3 )
=> ( ( sup_su6776935440552674877rm_a_b @ X3 @ Y3 )
= X3 ) ) ).
% sup_absorb1
thf(fact_492_sup_Oabsorb2,axiom,
! [A2: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A2 @ B2 )
=> ( ( sup_su459911885395995103_a_nat @ A2 @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_493_sup_Oabsorb2,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A2 @ B2 )
=> ( ( sup_su6776935440552674877rm_a_b @ A2 @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_494_sup_Oabsorb1,axiom,
! [B2: set_Pr4934435412358123699_a_nat,A2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ B2 @ A2 )
=> ( ( sup_su459911885395995103_a_nat @ A2 @ B2 )
= A2 ) ) ).
% sup.absorb1
thf(fact_495_sup_Oabsorb1,axiom,
! [B2: set_Pr4386577575007340137rm_a_b,A2: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ B2 @ A2 )
=> ( ( sup_su6776935440552674877rm_a_b @ A2 @ B2 )
= A2 ) ) ).
% sup.absorb1
thf(fact_496_sup__unique,axiom,
! [F2: set_Pr4934435412358123699_a_nat > set_Pr4934435412358123699_a_nat > set_Pr4934435412358123699_a_nat,X3: set_Pr4934435412358123699_a_nat,Y3: set_Pr4934435412358123699_a_nat] :
( ! [X: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ X @ ( F2 @ X @ Y ) )
=> ( ! [X: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ Y @ ( F2 @ X @ Y ) )
=> ( ! [X: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat,Z3: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ Y @ X )
=> ( ( ord_le8666007276011122963_a_nat @ Z3 @ X )
=> ( ord_le8666007276011122963_a_nat @ ( F2 @ Y @ Z3 ) @ X ) ) )
=> ( ( sup_su459911885395995103_a_nat @ X3 @ Y3 )
= ( F2 @ X3 @ Y3 ) ) ) ) ) ).
% sup_unique
thf(fact_497_sup__unique,axiom,
! [F2: set_Pr4386577575007340137rm_a_b > set_Pr4386577575007340137rm_a_b > set_Pr4386577575007340137rm_a_b,X3: set_Pr4386577575007340137rm_a_b,Y3: set_Pr4386577575007340137rm_a_b] :
( ! [X: set_Pr4386577575007340137rm_a_b,Y: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ X @ ( F2 @ X @ Y ) )
=> ( ! [X: set_Pr4386577575007340137rm_a_b,Y: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ Y @ ( F2 @ X @ Y ) )
=> ( ! [X: set_Pr4386577575007340137rm_a_b,Y: set_Pr4386577575007340137rm_a_b,Z3: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ Y @ X )
=> ( ( ord_le118470702582115849rm_a_b @ Z3 @ X )
=> ( ord_le118470702582115849rm_a_b @ ( F2 @ Y @ Z3 ) @ X ) ) )
=> ( ( sup_su6776935440552674877rm_a_b @ X3 @ Y3 )
= ( F2 @ X3 @ Y3 ) ) ) ) ) ).
% sup_unique
thf(fact_498_sup_OorderI,axiom,
! [A2: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat] :
( ( A2
= ( sup_su459911885395995103_a_nat @ A2 @ B2 ) )
=> ( ord_le8666007276011122963_a_nat @ B2 @ A2 ) ) ).
% sup.orderI
thf(fact_499_sup_OorderI,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b] :
( ( A2
= ( sup_su6776935440552674877rm_a_b @ A2 @ B2 ) )
=> ( ord_le118470702582115849rm_a_b @ B2 @ A2 ) ) ).
% sup.orderI
thf(fact_500_sup_OorderE,axiom,
! [B2: set_Pr4934435412358123699_a_nat,A2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ B2 @ A2 )
=> ( A2
= ( sup_su459911885395995103_a_nat @ A2 @ B2 ) ) ) ).
% sup.orderE
thf(fact_501_sup_OorderE,axiom,
! [B2: set_Pr4386577575007340137rm_a_b,A2: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ B2 @ A2 )
=> ( A2
= ( sup_su6776935440552674877rm_a_b @ A2 @ B2 ) ) ) ).
% sup.orderE
thf(fact_502_le__iff__sup,axiom,
( ord_le8666007276011122963_a_nat
= ( ^ [X4: set_Pr4934435412358123699_a_nat,Y5: set_Pr4934435412358123699_a_nat] :
( ( sup_su459911885395995103_a_nat @ X4 @ Y5 )
= Y5 ) ) ) ).
% le_iff_sup
thf(fact_503_le__iff__sup,axiom,
( ord_le118470702582115849rm_a_b
= ( ^ [X4: set_Pr4386577575007340137rm_a_b,Y5: set_Pr4386577575007340137rm_a_b] :
( ( sup_su6776935440552674877rm_a_b @ X4 @ Y5 )
= Y5 ) ) ) ).
% le_iff_sup
thf(fact_504_sup__least,axiom,
! [Y3: set_Pr4934435412358123699_a_nat,X3: set_Pr4934435412358123699_a_nat,Z2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ Y3 @ X3 )
=> ( ( ord_le8666007276011122963_a_nat @ Z2 @ X3 )
=> ( ord_le8666007276011122963_a_nat @ ( sup_su459911885395995103_a_nat @ Y3 @ Z2 ) @ X3 ) ) ) ).
% sup_least
thf(fact_505_sup__least,axiom,
! [Y3: set_Pr4386577575007340137rm_a_b,X3: set_Pr4386577575007340137rm_a_b,Z2: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ Y3 @ X3 )
=> ( ( ord_le118470702582115849rm_a_b @ Z2 @ X3 )
=> ( ord_le118470702582115849rm_a_b @ ( sup_su6776935440552674877rm_a_b @ Y3 @ Z2 ) @ X3 ) ) ) ).
% sup_least
thf(fact_506_sup__mono,axiom,
! [A2: set_Pr4934435412358123699_a_nat,C: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat,D2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A2 @ C )
=> ( ( ord_le8666007276011122963_a_nat @ B2 @ D2 )
=> ( ord_le8666007276011122963_a_nat @ ( sup_su459911885395995103_a_nat @ A2 @ B2 ) @ ( sup_su459911885395995103_a_nat @ C @ D2 ) ) ) ) ).
% sup_mono
thf(fact_507_sup__mono,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,C: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b,D2: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A2 @ C )
=> ( ( ord_le118470702582115849rm_a_b @ B2 @ D2 )
=> ( ord_le118470702582115849rm_a_b @ ( sup_su6776935440552674877rm_a_b @ A2 @ B2 ) @ ( sup_su6776935440552674877rm_a_b @ C @ D2 ) ) ) ) ).
% sup_mono
thf(fact_508_sup_Omono,axiom,
! [C: set_Pr4934435412358123699_a_nat,A2: set_Pr4934435412358123699_a_nat,D2: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ C @ A2 )
=> ( ( ord_le8666007276011122963_a_nat @ D2 @ B2 )
=> ( ord_le8666007276011122963_a_nat @ ( sup_su459911885395995103_a_nat @ C @ D2 ) @ ( sup_su459911885395995103_a_nat @ A2 @ B2 ) ) ) ) ).
% sup.mono
thf(fact_509_sup_Omono,axiom,
! [C: set_Pr4386577575007340137rm_a_b,A2: set_Pr4386577575007340137rm_a_b,D2: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ C @ A2 )
=> ( ( ord_le118470702582115849rm_a_b @ D2 @ B2 )
=> ( ord_le118470702582115849rm_a_b @ ( sup_su6776935440552674877rm_a_b @ C @ D2 ) @ ( sup_su6776935440552674877rm_a_b @ A2 @ B2 ) ) ) ) ).
% sup.mono
thf(fact_510_le__supI2,axiom,
! [X3: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat,A2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X3 @ B2 )
=> ( ord_le8666007276011122963_a_nat @ X3 @ ( sup_su459911885395995103_a_nat @ A2 @ B2 ) ) ) ).
% le_supI2
thf(fact_511_le__supI2,axiom,
! [X3: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b,A2: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ X3 @ B2 )
=> ( ord_le118470702582115849rm_a_b @ X3 @ ( sup_su6776935440552674877rm_a_b @ A2 @ B2 ) ) ) ).
% le_supI2
thf(fact_512_le__supI1,axiom,
! [X3: set_Pr4934435412358123699_a_nat,A2: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X3 @ A2 )
=> ( ord_le8666007276011122963_a_nat @ X3 @ ( sup_su459911885395995103_a_nat @ A2 @ B2 ) ) ) ).
% le_supI1
thf(fact_513_le__supI1,axiom,
! [X3: set_Pr4386577575007340137rm_a_b,A2: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ X3 @ A2 )
=> ( ord_le118470702582115849rm_a_b @ X3 @ ( sup_su6776935440552674877rm_a_b @ A2 @ B2 ) ) ) ).
% le_supI1
thf(fact_514_sup__ge2,axiom,
! [Y3: set_Pr4934435412358123699_a_nat,X3: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ Y3 @ ( sup_su459911885395995103_a_nat @ X3 @ Y3 ) ) ).
% sup_ge2
thf(fact_515_sup__ge2,axiom,
! [Y3: set_Pr4386577575007340137rm_a_b,X3: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ Y3 @ ( sup_su6776935440552674877rm_a_b @ X3 @ Y3 ) ) ).
% sup_ge2
thf(fact_516_sup__ge1,axiom,
! [X3: set_Pr4934435412358123699_a_nat,Y3: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ X3 @ ( sup_su459911885395995103_a_nat @ X3 @ Y3 ) ) ).
% sup_ge1
thf(fact_517_sup__ge1,axiom,
! [X3: set_Pr4386577575007340137rm_a_b,Y3: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ X3 @ ( sup_su6776935440552674877rm_a_b @ X3 @ Y3 ) ) ).
% sup_ge1
thf(fact_518_le__supI,axiom,
! [A2: set_Pr4934435412358123699_a_nat,X3: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A2 @ X3 )
=> ( ( ord_le8666007276011122963_a_nat @ B2 @ X3 )
=> ( ord_le8666007276011122963_a_nat @ ( sup_su459911885395995103_a_nat @ A2 @ B2 ) @ X3 ) ) ) ).
% le_supI
thf(fact_519_le__supI,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,X3: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A2 @ X3 )
=> ( ( ord_le118470702582115849rm_a_b @ B2 @ X3 )
=> ( ord_le118470702582115849rm_a_b @ ( sup_su6776935440552674877rm_a_b @ A2 @ B2 ) @ X3 ) ) ) ).
% le_supI
thf(fact_520_le__supE,axiom,
! [A2: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat,X3: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ ( sup_su459911885395995103_a_nat @ A2 @ B2 ) @ X3 )
=> ~ ( ( ord_le8666007276011122963_a_nat @ A2 @ X3 )
=> ~ ( ord_le8666007276011122963_a_nat @ B2 @ X3 ) ) ) ).
% le_supE
thf(fact_521_le__supE,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b,X3: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ ( sup_su6776935440552674877rm_a_b @ A2 @ B2 ) @ X3 )
=> ~ ( ( ord_le118470702582115849rm_a_b @ A2 @ X3 )
=> ~ ( ord_le118470702582115849rm_a_b @ B2 @ X3 ) ) ) ).
% le_supE
thf(fact_522_inf__sup__ord_I3_J,axiom,
! [X3: set_Pr4934435412358123699_a_nat,Y3: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ X3 @ ( sup_su459911885395995103_a_nat @ X3 @ Y3 ) ) ).
% inf_sup_ord(3)
thf(fact_523_inf__sup__ord_I3_J,axiom,
! [X3: set_Pr4386577575007340137rm_a_b,Y3: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ X3 @ ( sup_su6776935440552674877rm_a_b @ X3 @ Y3 ) ) ).
% inf_sup_ord(3)
thf(fact_524_inf__sup__ord_I4_J,axiom,
! [Y3: set_Pr4934435412358123699_a_nat,X3: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ Y3 @ ( sup_su459911885395995103_a_nat @ X3 @ Y3 ) ) ).
% inf_sup_ord(4)
thf(fact_525_inf__sup__ord_I4_J,axiom,
! [Y3: set_Pr4386577575007340137rm_a_b,X3: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ Y3 @ ( sup_su6776935440552674877rm_a_b @ X3 @ Y3 ) ) ).
% inf_sup_ord(4)
thf(fact_526_NF__join__imp__reach,axiom,
! [Y3: produc357393685978478089rm_a_b,A: set_Pr2972776593051762503rm_a_b,X3: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ Y3 @ ( abstra1902783451776770197rm_a_b @ A ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X3 @ Y3 ) @ ( abstra4889682940153293507rm_a_b @ A ) )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X3 @ Y3 ) @ ( transi2615809358984392588rm_a_b @ A ) ) ) ) ).
% NF_join_imp_reach
thf(fact_527_NF__join__imp__reach,axiom,
! [Y3: product_prod_a_nat,A: set_Pr1811044260758604347_a_nat,X3: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ Y3 @ ( abstra2685462378305987313_a_nat @ A ) )
=> ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ X3 @ Y3 ) @ ( abstra881512052393830979_a_nat @ A ) )
=> ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ X3 @ Y3 ) @ ( transi2726145917338391738_a_nat @ A ) ) ) ) ).
% NF_join_imp_reach
thf(fact_528_NF__join__imp__reach,axiom,
! [Y3: term_a_b,A: set_Pr4386577575007340137rm_a_b,X3: term_a_b] :
( ( member_term_a_b @ Y3 @ ( abstract_NF_term_a_b @ A ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ ( abstra4096080454567261402rm_a_b @ A ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ ( transi7742714808557438673rm_a_b @ A ) ) ) ) ).
% NF_join_imp_reach
thf(fact_529_all__not__in__conv,axiom,
! [A: set_Pr4386577575007340137rm_a_b] :
( ( ! [X4: produc357393685978478089rm_a_b] :
~ ( member5869715511025134514rm_a_b @ X4 @ A ) )
= ( A = bot_bo197521221353338581rm_a_b ) ) ).
% all_not_in_conv
thf(fact_530_all__not__in__conv,axiom,
! [A: set_Pr4934435412358123699_a_nat] :
( ( ! [X4: product_prod_a_nat] :
~ ( member5724188588386418708_a_nat @ X4 @ A ) )
= ( A = bot_bo9049108969261143879_a_nat ) ) ).
% all_not_in_conv
thf(fact_531_empty__iff,axiom,
! [C: produc357393685978478089rm_a_b] :
~ ( member5869715511025134514rm_a_b @ C @ bot_bo197521221353338581rm_a_b ) ).
% empty_iff
thf(fact_532_empty__iff,axiom,
! [C: product_prod_a_nat] :
~ ( member5724188588386418708_a_nat @ C @ bot_bo9049108969261143879_a_nat ) ).
% empty_iff
thf(fact_533_empty__subsetI,axiom,
! [A: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ bot_bo9049108969261143879_a_nat @ A ) ).
% empty_subsetI
thf(fact_534_empty__subsetI,axiom,
! [A: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ bot_bo197521221353338581rm_a_b @ A ) ).
% empty_subsetI
thf(fact_535_subset__empty,axiom,
! [A: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A @ bot_bo9049108969261143879_a_nat )
= ( A = bot_bo9049108969261143879_a_nat ) ) ).
% subset_empty
thf(fact_536_subset__empty,axiom,
! [A: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A @ bot_bo197521221353338581rm_a_b )
= ( A = bot_bo197521221353338581rm_a_b ) ) ).
% subset_empty
thf(fact_537_pair__in__Id__conv,axiom,
! [A2: term_a_b,B2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ id_term_a_b )
= ( A2 = B2 ) ) ).
% pair_in_Id_conv
thf(fact_538_IdI,axiom,
! [A2: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ A2 ) @ id_term_a_b ) ).
% IdI
thf(fact_539_joinI__right,axiom,
! [B2: term_a_b,A2: term_a_b,A: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ A2 ) @ ( transi7742714808557438673rm_a_b @ A ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( abstra4096080454567261402rm_a_b @ A ) ) ) ).
% joinI_right
thf(fact_540_joinI__left,axiom,
! [A2: term_a_b,B2: term_a_b,A: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( transi7742714808557438673rm_a_b @ A ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( abstra4096080454567261402rm_a_b @ A ) ) ) ).
% joinI_left
thf(fact_541_joinI,axiom,
! [A2: term_a_b,C: term_a_b,A: set_Pr4386577575007340137rm_a_b,B2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ C ) @ ( transi7742714808557438673rm_a_b @ A ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ C ) @ ( transi7742714808557438673rm_a_b @ A ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( abstra4096080454567261402rm_a_b @ A ) ) ) ) ).
% joinI
thf(fact_542_ex__in__conv,axiom,
! [A: set_Pr4386577575007340137rm_a_b] :
( ( ? [X4: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ X4 @ A ) )
= ( A != bot_bo197521221353338581rm_a_b ) ) ).
% ex_in_conv
thf(fact_543_ex__in__conv,axiom,
! [A: set_Pr4934435412358123699_a_nat] :
( ( ? [X4: product_prod_a_nat] : ( member5724188588386418708_a_nat @ X4 @ A ) )
= ( A != bot_bo9049108969261143879_a_nat ) ) ).
% ex_in_conv
thf(fact_544_equals0I,axiom,
! [A: set_Pr4386577575007340137rm_a_b] :
( ! [Y: produc357393685978478089rm_a_b] :
~ ( member5869715511025134514rm_a_b @ Y @ A )
=> ( A = bot_bo197521221353338581rm_a_b ) ) ).
% equals0I
thf(fact_545_equals0I,axiom,
! [A: set_Pr4934435412358123699_a_nat] :
( ! [Y: product_prod_a_nat] :
~ ( member5724188588386418708_a_nat @ Y @ A )
=> ( A = bot_bo9049108969261143879_a_nat ) ) ).
% equals0I
thf(fact_546_equals0D,axiom,
! [A: set_Pr4386577575007340137rm_a_b,A2: produc357393685978478089rm_a_b] :
( ( A = bot_bo197521221353338581rm_a_b )
=> ~ ( member5869715511025134514rm_a_b @ A2 @ A ) ) ).
% equals0D
thf(fact_547_equals0D,axiom,
! [A: set_Pr4934435412358123699_a_nat,A2: product_prod_a_nat] :
( ( A = bot_bo9049108969261143879_a_nat )
=> ~ ( member5724188588386418708_a_nat @ A2 @ A ) ) ).
% equals0D
thf(fact_548_emptyE,axiom,
! [A2: produc357393685978478089rm_a_b] :
~ ( member5869715511025134514rm_a_b @ A2 @ bot_bo197521221353338581rm_a_b ) ).
% emptyE
thf(fact_549_emptyE,axiom,
! [A2: product_prod_a_nat] :
~ ( member5724188588386418708_a_nat @ A2 @ bot_bo9049108969261143879_a_nat ) ).
% emptyE
thf(fact_550_bot_Oextremum__uniqueI,axiom,
! [A2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A2 @ bot_bo9049108969261143879_a_nat )
=> ( A2 = bot_bo9049108969261143879_a_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_551_bot_Oextremum__uniqueI,axiom,
! [A2: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A2 @ bot_bo197521221353338581rm_a_b )
=> ( A2 = bot_bo197521221353338581rm_a_b ) ) ).
% bot.extremum_uniqueI
thf(fact_552_bot_Oextremum__unique,axiom,
! [A2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A2 @ bot_bo9049108969261143879_a_nat )
= ( A2 = bot_bo9049108969261143879_a_nat ) ) ).
% bot.extremum_unique
thf(fact_553_bot_Oextremum__unique,axiom,
! [A2: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A2 @ bot_bo197521221353338581rm_a_b )
= ( A2 = bot_bo197521221353338581rm_a_b ) ) ).
% bot.extremum_unique
thf(fact_554_bot_Oextremum,axiom,
! [A2: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ bot_bo9049108969261143879_a_nat @ A2 ) ).
% bot.extremum
thf(fact_555_bot_Oextremum,axiom,
! [A2: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ bot_bo197521221353338581rm_a_b @ A2 ) ).
% bot.extremum
thf(fact_556_IdE,axiom,
! [P2: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ P2 @ id_term_a_b )
=> ~ ! [X: term_a_b] :
( P2
!= ( produc7020197800436672577rm_a_b @ X @ X ) ) ) ).
% IdE
thf(fact_557_join__sym,axiom,
! [S: term_a_b,T: term_a_b,A: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( abstra4096080454567261402rm_a_b @ A ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T @ S ) @ ( abstra4096080454567261402rm_a_b @ A ) ) ) ).
% join_sym
thf(fact_558_join__mono,axiom,
! [R: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ R @ S )
=> ( ord_le118470702582115849rm_a_b @ ( abstra4096080454567261402rm_a_b @ R ) @ ( abstra4096080454567261402rm_a_b @ S ) ) ) ).
% join_mono
thf(fact_559_joinD,axiom,
! [A2: term_a_b,B2: term_a_b,A: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( abstra4096080454567261402rm_a_b @ A ) )
=> ? [C4: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ C4 ) @ ( transi7742714808557438673rm_a_b @ A ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ C4 ) @ ( transi7742714808557438673rm_a_b @ A ) ) ) ) ).
% joinD
thf(fact_560_joinE,axiom,
! [A2: term_a_b,B2: term_a_b,A: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( abstra4096080454567261402rm_a_b @ A ) )
=> ~ ! [C4: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ C4 ) @ ( transi7742714808557438673rm_a_b @ A ) )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ C4 ) @ ( transi7742714808557438673rm_a_b @ A ) ) ) ) ).
% joinE
thf(fact_561_join__rtrancl__join,axiom,
! [A2: term_a_b,B2: term_a_b,A: set_Pr4386577575007340137rm_a_b,C: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( abstra4096080454567261402rm_a_b @ A ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ C @ B2 ) @ ( transi7742714808557438673rm_a_b @ A ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ C ) @ ( abstra4096080454567261402rm_a_b @ A ) ) ) ) ).
% join_rtrancl_join
thf(fact_562_rtrancl__join__join,axiom,
! [A2: term_a_b,B2: term_a_b,A: set_Pr4386577575007340137rm_a_b,C: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( transi7742714808557438673rm_a_b @ A ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ C ) @ ( abstra4096080454567261402rm_a_b @ A ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ C ) @ ( abstra4096080454567261402rm_a_b @ A ) ) ) ) ).
% rtrancl_join_join
thf(fact_563_join__NF__imp__eq,axiom,
! [X3: produc357393685978478089rm_a_b,Y3: produc357393685978478089rm_a_b,R: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X3 @ Y3 ) @ ( abstra4889682940153293507rm_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ X3 @ ( abstra1902783451776770197rm_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ Y3 @ ( abstra1902783451776770197rm_a_b @ R ) )
=> ( X3 = Y3 ) ) ) ) ).
% join_NF_imp_eq
thf(fact_564_join__NF__imp__eq,axiom,
! [X3: product_prod_a_nat,Y3: product_prod_a_nat,R: set_Pr1811044260758604347_a_nat] :
( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ X3 @ Y3 ) @ ( abstra881512052393830979_a_nat @ R ) )
=> ( ( member5724188588386418708_a_nat @ X3 @ ( abstra2685462378305987313_a_nat @ R ) )
=> ( ( member5724188588386418708_a_nat @ Y3 @ ( abstra2685462378305987313_a_nat @ R ) )
=> ( X3 = Y3 ) ) ) ) ).
% join_NF_imp_eq
thf(fact_565_join__NF__imp__eq,axiom,
! [X3: term_a_b,Y3: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ ( abstra4096080454567261402rm_a_b @ R ) )
=> ( ( member_term_a_b @ X3 @ ( abstract_NF_term_a_b @ R ) )
=> ( ( member_term_a_b @ Y3 @ ( abstract_NF_term_a_b @ R ) )
=> ( X3 = Y3 ) ) ) ) ).
% join_NF_imp_eq
thf(fact_566_strongly__confluent__on__E11,axiom,
! [R: set_Pr2972776593051762503rm_a_b,A: set_Pr4386577575007340137rm_a_b,X3: produc357393685978478089rm_a_b,Y3: produc357393685978478089rm_a_b,Z2: produc357393685978478089rm_a_b] :
( ( abstra837640964651051114rm_a_b @ R @ A )
=> ( ( member5869715511025134514rm_a_b @ X3 @ A )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X3 @ Y3 ) @ R )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X3 @ Z2 ) @ R )
=> ? [U2: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y3 @ U2 ) @ ( transi2615809358984392588rm_a_b @ R ) )
& ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Z2 @ U2 ) @ ( sup_su1845815533690797339rm_a_b @ R @ id_Pro3488709451141449061rm_a_b ) ) ) ) ) ) ) ).
% strongly_confluent_on_E11
thf(fact_567_strongly__confluent__on__E11,axiom,
! [R: set_Pr1811044260758604347_a_nat,A: set_Pr4934435412358123699_a_nat,X3: product_prod_a_nat,Y3: product_prod_a_nat,Z2: product_prod_a_nat] :
( ( abstra5315942894939654492_a_nat @ R @ A )
=> ( ( member5724188588386418708_a_nat @ X3 @ A )
=> ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ X3 @ Y3 ) @ R )
=> ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ X3 @ Z2 ) @ R )
=> ? [U2: product_prod_a_nat] :
( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ Y3 @ U2 ) @ ( transi2726145917338391738_a_nat @ R ) )
& ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ Z2 @ U2 ) @ ( sup_su958340139570419215_a_nat @ R @ id_Pro5207055338379199009_a_nat ) ) ) ) ) ) ) ).
% strongly_confluent_on_E11
thf(fact_568_strongly__confluent__on__E11,axiom,
! [R: set_Pr4386577575007340137rm_a_b,A: set_term_a_b,X3: term_a_b,Y3: term_a_b,Z2: term_a_b] :
( ( abstra5664096641628173427rm_a_b @ R @ A )
=> ( ( member_term_a_b @ X3 @ A )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ R )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Z2 ) @ R )
=> ? [U2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ U2 ) @ ( transi7742714808557438673rm_a_b @ R ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Z2 @ U2 ) @ ( sup_su6776935440552674877rm_a_b @ R @ id_term_a_b ) ) ) ) ) ) ) ).
% strongly_confluent_on_E11
thf(fact_569_strongly__confluent__on__def,axiom,
( abstra5664096641628173427rm_a_b
= ( ^ [R3: 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 ) @ R3 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X4 @ Z4 ) @ R3 ) )
=> ? [U3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y5 @ U3 ) @ ( transi7742714808557438673rm_a_b @ R3 ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Z4 @ U3 ) @ ( sup_su6776935440552674877rm_a_b @ R3 @ id_term_a_b ) ) ) ) ) ) ) ).
% strongly_confluent_on_def
thf(fact_570_less__by__empty,axiom,
! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ( A = bot_bo197521221353338581rm_a_b )
=> ( ord_le118470702582115849rm_a_b @ A @ B ) ) ).
% less_by_empty
thf(fact_571_IdD,axiom,
! [A2: term_a_b,B2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ id_term_a_b )
=> ( A2 = B2 ) ) ).
% IdD
thf(fact_572_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_573_bot__empty__eq,axiom,
( bot_bo1388487281569869310_nat_o
= ( ^ [X4: product_prod_a_nat] : ( member5724188588386418708_a_nat @ X4 @ bot_bo9049108969261143879_a_nat ) ) ) ).
% bot_empty_eq
thf(fact_574_strongly__confluentI,axiom,
! [R: set_Pr4386577575007340137rm_a_b] :
( ! [X: term_a_b,Y: term_a_b,Z3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ R )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z3 ) @ R )
=> ? [U4: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ U4 ) @ ( transi7742714808557438673rm_a_b @ R ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Z3 @ U4 ) @ ( sup_su6776935440552674877rm_a_b @ R @ id_term_a_b ) ) ) ) )
=> ( abstra5664096641628173427rm_a_b @ R @ top_top_set_term_a_b ) ) ).
% strongly_confluentI
thf(fact_575_subset__emptyI,axiom,
! [A: set_Pr4934435412358123699_a_nat] :
( ! [X: product_prod_a_nat] :
~ ( member5724188588386418708_a_nat @ X @ A )
=> ( ord_le8666007276011122963_a_nat @ A @ bot_bo9049108969261143879_a_nat ) ) ).
% subset_emptyI
thf(fact_576_subset__emptyI,axiom,
! [A: set_Pr4386577575007340137rm_a_b] :
( ! [X: produc357393685978478089rm_a_b] :
~ ( member5869715511025134514rm_a_b @ X @ A )
=> ( ord_le118470702582115849rm_a_b @ A @ bot_bo197521221353338581rm_a_b ) ) ).
% subset_emptyI
thf(fact_577_WCR__onI,axiom,
! [A: set_Pr4386577575007340137rm_a_b,R: set_Pr2972776593051762503rm_a_b] :
( ! [A5: produc357393685978478089rm_a_b,B5: produc357393685978478089rm_a_b,C4: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ A5 @ A )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A5 @ B5 ) @ R )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A5 @ C4 ) @ R )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B5 @ C4 ) @ ( abstra4889682940153293507rm_a_b @ R ) ) ) ) )
=> ( abstra875157153669124638rm_a_b @ R @ A ) ) ).
% WCR_onI
thf(fact_578_WCR__onI,axiom,
! [A: set_Pr4934435412358123699_a_nat,R: set_Pr1811044260758604347_a_nat] :
( ! [A5: product_prod_a_nat,B5: product_prod_a_nat,C4: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ A5 @ A )
=> ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A5 @ B5 ) @ R )
=> ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A5 @ C4 ) @ R )
=> ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ B5 @ C4 ) @ ( abstra881512052393830979_a_nat @ R ) ) ) ) )
=> ( abstra6648775111202918568_a_nat @ R @ A ) ) ).
% WCR_onI
thf(fact_579_WCR__onI,axiom,
! [A: set_term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ! [A5: term_a_b,B5: term_a_b,C4: term_a_b] :
( ( member_term_a_b @ A5 @ A )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A5 @ B5 ) @ R )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A5 @ C4 ) @ R )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B5 @ C4 ) @ ( abstra4096080454567261402rm_a_b @ R ) ) ) ) )
=> ( abstra5026849107050860991rm_a_b @ R @ A ) ) ).
% WCR_onI
thf(fact_580_CR__onI,axiom,
! [A: set_Pr4386577575007340137rm_a_b,R: set_Pr2972776593051762503rm_a_b] :
( ! [A5: produc357393685978478089rm_a_b,B5: produc357393685978478089rm_a_b,C4: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ A5 @ A )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A5 @ B5 ) @ ( transi2615809358984392588rm_a_b @ R ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A5 @ C4 ) @ ( transi2615809358984392588rm_a_b @ R ) )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B5 @ C4 ) @ ( abstra4889682940153293507rm_a_b @ R ) ) ) ) )
=> ( abstra7359694574113289279rm_a_b @ R @ A ) ) ).
% CR_onI
thf(fact_581_CR__onI,axiom,
! [A: set_Pr4934435412358123699_a_nat,R: set_Pr1811044260758604347_a_nat] :
( ! [A5: product_prod_a_nat,B5: product_prod_a_nat,C4: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ A5 @ A )
=> ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A5 @ B5 ) @ ( transi2726145917338391738_a_nat @ R ) )
=> ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A5 @ C4 ) @ ( transi2726145917338391738_a_nat @ R ) )
=> ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ B5 @ C4 ) @ ( abstra881512052393830979_a_nat @ R ) ) ) ) )
=> ( abstra7964079670522473415_a_nat @ R @ A ) ) ).
% CR_onI
thf(fact_582_CR__onI,axiom,
! [A: set_term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ! [A5: term_a_b,B5: term_a_b,C4: term_a_b] :
( ( member_term_a_b @ A5 @ A )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A5 @ B5 ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A5 @ C4 ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B5 @ C4 ) @ ( abstra4096080454567261402rm_a_b @ R ) ) ) ) )
=> ( abstra8448919418672941150rm_a_b @ R @ A ) ) ).
% CR_onI
thf(fact_583_UNIV__I,axiom,
! [X3: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ X3 @ top_to1314267278846557113rm_a_b ) ).
% UNIV_I
thf(fact_584_UNIV__I,axiom,
! [X3: product_prod_a_nat] : ( member5724188588386418708_a_nat @ X3 @ top_to3353692345378799459_a_nat ) ).
% UNIV_I
thf(fact_585_UNIV__witness,axiom,
? [X: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ X @ top_to1314267278846557113rm_a_b ) ).
% UNIV_witness
thf(fact_586_UNIV__witness,axiom,
? [X: product_prod_a_nat] : ( member5724188588386418708_a_nat @ X @ top_to3353692345378799459_a_nat ) ).
% UNIV_witness
thf(fact_587_UNIV__eq__I,axiom,
! [A: set_Pr4386577575007340137rm_a_b] :
( ! [X: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ X @ A )
=> ( top_to1314267278846557113rm_a_b = A ) ) ).
% UNIV_eq_I
thf(fact_588_UNIV__eq__I,axiom,
! [A: set_Pr4934435412358123699_a_nat] :
( ! [X: product_prod_a_nat] : ( member5724188588386418708_a_nat @ X @ A )
=> ( top_to3353692345378799459_a_nat = A ) ) ).
% UNIV_eq_I
thf(fact_589_top_Oextremum__uniqueI,axiom,
! [A2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ top_to3353692345378799459_a_nat @ A2 )
=> ( A2 = top_to3353692345378799459_a_nat ) ) ).
% top.extremum_uniqueI
thf(fact_590_top_Oextremum__uniqueI,axiom,
! [A2: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ top_to1314267278846557113rm_a_b @ A2 )
=> ( A2 = top_to1314267278846557113rm_a_b ) ) ).
% top.extremum_uniqueI
thf(fact_591_top_Oextremum__unique,axiom,
! [A2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ top_to3353692345378799459_a_nat @ A2 )
= ( A2 = top_to3353692345378799459_a_nat ) ) ).
% top.extremum_unique
thf(fact_592_top_Oextremum__unique,axiom,
! [A2: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ top_to1314267278846557113rm_a_b @ A2 )
= ( A2 = top_to1314267278846557113rm_a_b ) ) ).
% top.extremum_unique
thf(fact_593_top__greatest,axiom,
! [A2: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ A2 @ top_to3353692345378799459_a_nat ) ).
% top_greatest
thf(fact_594_top__greatest,axiom,
! [A2: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ A2 @ top_to1314267278846557113rm_a_b ) ).
% top_greatest
thf(fact_595_subset__UNIV,axiom,
! [A: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ A @ top_to3353692345378799459_a_nat ) ).
% subset_UNIV
thf(fact_596_subset__UNIV,axiom,
! [A: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ A @ top_to1314267278846557113rm_a_b ) ).
% subset_UNIV
thf(fact_597_CR__divergence__imp__join,axiom,
! [R: set_Pr4386577575007340137rm_a_b,X3: term_a_b,Y3: term_a_b,Z2: term_a_b] :
( ( abstra8448919418672941150rm_a_b @ R @ top_top_set_term_a_b )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Z2 ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ Z2 ) @ ( abstra4096080454567261402rm_a_b @ R ) ) ) ) ) ).
% CR_divergence_imp_join
thf(fact_598_partially__localize__CR,axiom,
! [R: set_Pr4386577575007340137rm_a_b] :
( ( abstra8448919418672941150rm_a_b @ R @ 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 ) @ R )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X4 @ Z4 ) @ ( transi7742714808557438673rm_a_b @ R ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y5 @ Z4 ) @ ( abstra4096080454567261402rm_a_b @ R ) ) ) ) ) ).
% partially_localize_CR
thf(fact_599_CR__join__right__I,axiom,
! [R: set_Pr4386577575007340137rm_a_b,X3: term_a_b,Y3: term_a_b,Z2: term_a_b] :
( ( abstra8448919418672941150rm_a_b @ R @ top_top_set_term_a_b )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ ( abstra4096080454567261402rm_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ Z2 ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Z2 ) @ ( abstra4096080454567261402rm_a_b @ R ) ) ) ) ) ).
% CR_join_right_I
thf(fact_600_CR__join__left__I,axiom,
! [R: set_Pr4386577575007340137rm_a_b,X3: term_a_b,Y3: term_a_b,Z2: term_a_b] :
( ( abstra8448919418672941150rm_a_b @ R @ top_top_set_term_a_b )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Z2 ) @ ( abstra4096080454567261402rm_a_b @ R ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ Z2 ) @ ( abstra4096080454567261402rm_a_b @ R ) ) ) ) ) ).
% CR_join_left_I
thf(fact_601_CR__iff__meet__subset__join,axiom,
! [R: set_Pr4386577575007340137rm_a_b] :
( ( abstra8448919418672941150rm_a_b @ R @ top_top_set_term_a_b )
= ( ord_le118470702582115849rm_a_b @ ( abstra1093061187967292567rm_a_b @ R ) @ ( abstra4096080454567261402rm_a_b @ R ) ) ) ).
% CR_iff_meet_subset_join
thf(fact_602_ssubst__Pair__rhs,axiom,
! [R: a,S: nat,R4: set_Pr4934435412358123699_a_nat,S5: nat] :
( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ R @ S ) @ R4 )
=> ( ( S5 = S )
=> ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ R @ S5 ) @ R4 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_603_ssubst__Pair__rhs,axiom,
! [R: term_a_b,S: term_a_b,R4: set_Pr4386577575007340137rm_a_b,S5: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ R @ S ) @ R4 )
=> ( ( S5 = S )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ R @ S5 ) @ R4 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_604_CR__onD,axiom,
! [R: set_Pr2972776593051762503rm_a_b,A: set_Pr4386577575007340137rm_a_b,A2: produc357393685978478089rm_a_b,B2: produc357393685978478089rm_a_b,C: produc357393685978478089rm_a_b] :
( ( abstra7359694574113289279rm_a_b @ R @ A )
=> ( ( member5869715511025134514rm_a_b @ A2 @ A )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A2 @ B2 ) @ ( transi2615809358984392588rm_a_b @ R ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A2 @ C ) @ ( transi2615809358984392588rm_a_b @ R ) )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B2 @ C ) @ ( abstra4889682940153293507rm_a_b @ R ) ) ) ) ) ) ).
% CR_onD
thf(fact_605_CR__onD,axiom,
! [R: set_Pr1811044260758604347_a_nat,A: set_Pr4934435412358123699_a_nat,A2: product_prod_a_nat,B2: product_prod_a_nat,C: product_prod_a_nat] :
( ( abstra7964079670522473415_a_nat @ R @ A )
=> ( ( member5724188588386418708_a_nat @ A2 @ A )
=> ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A2 @ B2 ) @ ( transi2726145917338391738_a_nat @ R ) )
=> ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A2 @ C ) @ ( transi2726145917338391738_a_nat @ R ) )
=> ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ B2 @ C ) @ ( abstra881512052393830979_a_nat @ R ) ) ) ) ) ) ).
% CR_onD
thf(fact_606_CR__onD,axiom,
! [R: set_Pr4386577575007340137rm_a_b,A: set_term_a_b,A2: term_a_b,B2: term_a_b,C: term_a_b] :
( ( abstra8448919418672941150rm_a_b @ R @ A )
=> ( ( member_term_a_b @ A2 @ A )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ C ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ C ) @ ( abstra4096080454567261402rm_a_b @ R ) ) ) ) ) ) ).
% CR_onD
thf(fact_607_CR__onE,axiom,
! [R: set_Pr2972776593051762503rm_a_b,A: set_Pr4386577575007340137rm_a_b,A2: produc357393685978478089rm_a_b,B2: produc357393685978478089rm_a_b,C: produc357393685978478089rm_a_b] :
( ( abstra7359694574113289279rm_a_b @ R @ A )
=> ( ( member5869715511025134514rm_a_b @ A2 @ A )
=> ( ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B2 @ C ) @ ( abstra4889682940153293507rm_a_b @ R ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A2 @ B2 ) @ ( transi2615809358984392588rm_a_b @ R ) )
=> ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A2 @ C ) @ ( transi2615809358984392588rm_a_b @ R ) ) ) ) ) ) ).
% CR_onE
thf(fact_608_CR__onE,axiom,
! [R: set_Pr1811044260758604347_a_nat,A: set_Pr4934435412358123699_a_nat,A2: product_prod_a_nat,B2: product_prod_a_nat,C: product_prod_a_nat] :
( ( abstra7964079670522473415_a_nat @ R @ A )
=> ( ( member5724188588386418708_a_nat @ A2 @ A )
=> ( ~ ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ B2 @ C ) @ ( abstra881512052393830979_a_nat @ R ) )
=> ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A2 @ B2 ) @ ( transi2726145917338391738_a_nat @ R ) )
=> ~ ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A2 @ C ) @ ( transi2726145917338391738_a_nat @ R ) ) ) ) ) ) ).
% CR_onE
thf(fact_609_CR__onE,axiom,
! [R: set_Pr4386577575007340137rm_a_b,A: set_term_a_b,A2: term_a_b,B2: term_a_b,C: term_a_b] :
( ( abstra8448919418672941150rm_a_b @ R @ A )
=> ( ( member_term_a_b @ A2 @ A )
=> ( ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ C ) @ ( abstra4096080454567261402rm_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ C ) @ ( transi7742714808557438673rm_a_b @ R ) ) ) ) ) ) ).
% CR_onE
thf(fact_610_CR__on__def,axiom,
( abstra8448919418672941150rm_a_b
= ( ^ [R3: 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 @ R3 ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X4 @ C3 ) @ ( transi7742714808557438673rm_a_b @ R3 ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B4 @ C3 ) @ ( abstra4096080454567261402rm_a_b @ R3 ) ) ) ) ) ) ).
% CR_on_def
thf(fact_611_WCR__onD,axiom,
! [R: set_Pr2972776593051762503rm_a_b,A: set_Pr4386577575007340137rm_a_b,A2: produc357393685978478089rm_a_b,B2: produc357393685978478089rm_a_b,C: produc357393685978478089rm_a_b] :
( ( abstra875157153669124638rm_a_b @ R @ A )
=> ( ( member5869715511025134514rm_a_b @ A2 @ A )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A2 @ B2 ) @ R )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A2 @ C ) @ R )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B2 @ C ) @ ( abstra4889682940153293507rm_a_b @ R ) ) ) ) ) ) ).
% WCR_onD
thf(fact_612_WCR__onD,axiom,
! [R: set_Pr1811044260758604347_a_nat,A: set_Pr4934435412358123699_a_nat,A2: product_prod_a_nat,B2: product_prod_a_nat,C: product_prod_a_nat] :
( ( abstra6648775111202918568_a_nat @ R @ A )
=> ( ( member5724188588386418708_a_nat @ A2 @ A )
=> ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A2 @ B2 ) @ R )
=> ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A2 @ C ) @ R )
=> ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ B2 @ C ) @ ( abstra881512052393830979_a_nat @ R ) ) ) ) ) ) ).
% WCR_onD
thf(fact_613_WCR__onD,axiom,
! [R: set_Pr4386577575007340137rm_a_b,A: set_term_a_b,A2: term_a_b,B2: term_a_b,C: term_a_b] :
( ( abstra5026849107050860991rm_a_b @ R @ A )
=> ( ( member_term_a_b @ A2 @ A )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ R )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ C ) @ R )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ C ) @ ( abstra4096080454567261402rm_a_b @ R ) ) ) ) ) ) ).
% WCR_onD
thf(fact_614_WCR__onE,axiom,
! [R: set_Pr2972776593051762503rm_a_b,A: set_Pr4386577575007340137rm_a_b,A2: produc357393685978478089rm_a_b,B2: produc357393685978478089rm_a_b,C: produc357393685978478089rm_a_b] :
( ( abstra875157153669124638rm_a_b @ R @ A )
=> ( ( member5869715511025134514rm_a_b @ A2 @ A )
=> ( ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B2 @ C ) @ ( abstra4889682940153293507rm_a_b @ R ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A2 @ B2 ) @ R )
=> ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A2 @ C ) @ R ) ) ) ) ) ).
% WCR_onE
thf(fact_615_WCR__onE,axiom,
! [R: set_Pr1811044260758604347_a_nat,A: set_Pr4934435412358123699_a_nat,A2: product_prod_a_nat,B2: product_prod_a_nat,C: product_prod_a_nat] :
( ( abstra6648775111202918568_a_nat @ R @ A )
=> ( ( member5724188588386418708_a_nat @ A2 @ A )
=> ( ~ ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ B2 @ C ) @ ( abstra881512052393830979_a_nat @ R ) )
=> ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A2 @ B2 ) @ R )
=> ~ ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A2 @ C ) @ R ) ) ) ) ) ).
% WCR_onE
thf(fact_616_WCR__onE,axiom,
! [R: set_Pr4386577575007340137rm_a_b,A: set_term_a_b,A2: term_a_b,B2: term_a_b,C: term_a_b] :
( ( abstra5026849107050860991rm_a_b @ R @ A )
=> ( ( member_term_a_b @ A2 @ A )
=> ( ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ C ) @ ( abstra4096080454567261402rm_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ R )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ C ) @ R ) ) ) ) ) ).
% WCR_onE
thf(fact_617_WCR__on__def,axiom,
( abstra5026849107050860991rm_a_b
= ( ^ [R3: 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 ) @ R3 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X4 @ C3 ) @ R3 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B4 @ C3 ) @ ( abstra4096080454567261402rm_a_b @ R3 ) ) ) ) ) ) ).
% WCR_on_def
thf(fact_618_some__NF__UNF,axiom,
! [R: set_Pr2972776593051762503rm_a_b,X3: produc357393685978478089rm_a_b,Y3: produc357393685978478089rm_a_b] :
( ( abstra1303798319023342519rm_a_b @ R @ top_to1314267278846557113rm_a_b )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X3 @ Y3 ) @ ( transi2615809358984392588rm_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ Y3 @ ( abstra1902783451776770197rm_a_b @ R ) )
=> ( Y3
= ( abstra8203652949708803681rm_a_b @ R @ X3 ) ) ) ) ) ).
% some_NF_UNF
thf(fact_619_some__NF__UNF,axiom,
! [R: set_Pr1811044260758604347_a_nat,X3: product_prod_a_nat,Y3: product_prod_a_nat] :
( ( abstra7833083522531381839_a_nat @ R @ top_to3353692345378799459_a_nat )
=> ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ X3 @ Y3 ) @ ( transi2726145917338391738_a_nat @ R ) )
=> ( ( member5724188588386418708_a_nat @ Y3 @ ( abstra2685462378305987313_a_nat @ R ) )
=> ( Y3
= ( abstra5913684503434863653_a_nat @ R @ X3 ) ) ) ) ) ).
% some_NF_UNF
thf(fact_620_some__NF__UNF,axiom,
! [R: set_Pr4386577575007340137rm_a_b,X3: term_a_b,Y3: term_a_b] :
( ( abstra3619855735679963878rm_a_b @ R @ top_top_set_term_a_b )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ( ( member_term_a_b @ Y3 @ ( abstract_NF_term_a_b @ R ) )
=> ( Y3
= ( abstra2589370493620291516rm_a_b @ R @ X3 ) ) ) ) ) ).
% some_NF_UNF
thf(fact_621_diamond__imp__CR_H,axiom,
! [S: set_Pr4386577575007340137rm_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( abstra8151087076625346926rm_a_b @ S )
=> ( ( ord_le118470702582115849rm_a_b @ R @ S )
=> ( ( ord_le118470702582115849rm_a_b @ S @ ( transi7742714808557438673rm_a_b @ R ) )
=> ( abstra8448919418672941150rm_a_b @ R @ top_top_set_term_a_b ) ) ) ) ).
% diamond_imp_CR'
thf(fact_622_top__empty__eq,axiom,
( top_to1262317115727556708_a_b_o
= ( ^ [X4: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ X4 @ top_to1314267278846557113rm_a_b ) ) ) ).
% top_empty_eq
thf(fact_623_top__empty__eq,axiom,
( top_to1985590801843794146_nat_o
= ( ^ [X4: product_prod_a_nat] : ( member5724188588386418708_a_nat @ X4 @ top_to3353692345378799459_a_nat ) ) ) ).
% top_empty_eq
thf(fact_624_CR__NF__conv,axiom,
! [R: set_Pr2972776593051762503rm_a_b,T: produc357393685978478089rm_a_b,U: produc357393685978478089rm_a_b] :
( ( abstra7359694574113289279rm_a_b @ R @ top_to1314267278846557113rm_a_b )
=> ( ( member5869715511025134514rm_a_b @ T @ ( abstra1902783451776770197rm_a_b @ R ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ U @ T ) @ ( abstra6125656530262095991rm_a_b @ R ) )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ U @ T ) @ ( abstra1496011390211842225rm_a_b @ R ) ) ) ) ) ).
% CR_NF_conv
thf(fact_625_CR__NF__conv,axiom,
! [R: set_Pr1811044260758604347_a_nat,T: product_prod_a_nat,U: product_prod_a_nat] :
( ( abstra7964079670522473415_a_nat @ R @ top_to3353692345378799459_a_nat )
=> ( ( member5724188588386418708_a_nat @ T @ ( abstra2685462378305987313_a_nat @ R ) )
=> ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ U @ T ) @ ( abstra3054561309482028431_a_nat @ R ) )
=> ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ U @ T ) @ ( abstra3832961281062565333_a_nat @ R ) ) ) ) ) ).
% CR_NF_conv
thf(fact_626_CR__NF__conv,axiom,
! [R: set_Pr4386577575007340137rm_a_b,T: term_a_b,U: term_a_b] :
( ( abstra8448919418672941150rm_a_b @ R @ top_top_set_term_a_b )
=> ( ( member_term_a_b @ T @ ( abstract_NF_term_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ U @ T ) @ ( abstra1130568886744591910rm_a_b @ R ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ U @ T ) @ ( abstra2972651231392952684rm_a_b @ R ) ) ) ) ) ).
% CR_NF_conv
thf(fact_627_strongly__confluent__E1n,axiom,
! [R: set_Pr4386577575007340137rm_a_b,X3: term_a_b,Y3: term_a_b,Z2: term_a_b,N: nat] :
( ( abstra5664096641628173427rm_a_b @ R @ top_top_set_term_a_b )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ ( sup_su6776935440552674877rm_a_b @ R @ id_term_a_b ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Z2 ) @ ( compow4057154403645558940rm_a_b @ N @ R ) )
=> ? [U2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ U2 ) @ ( transi7742714808557438673rm_a_b @ R ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Z2 @ U2 ) @ ( sup_su6776935440552674877rm_a_b @ R @ id_term_a_b ) ) ) ) ) ) ).
% strongly_confluent_E1n
thf(fact_628_CR__on__singletonI,axiom,
! [A2: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ! [B5: term_a_b,C4: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B5 ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ C4 ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B5 @ C4 ) @ ( abstra4096080454567261402rm_a_b @ R ) ) ) )
=> ( abstra8448919418672941150rm_a_b @ R @ ( insert_term_a_b @ A2 @ bot_bot_set_term_a_b ) ) ) ).
% CR_on_singletonI
thf(fact_629_semi__confluence__imp__CR,axiom,
! [R: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ ( conver7608559976937845093rm_a_b @ R ) @ ( transi7742714808557438673rm_a_b @ R ) ) @ ( abstra4096080454567261402rm_a_b @ R ) )
=> ( abstra8448919418672941150rm_a_b @ R @ top_top_set_term_a_b ) ) ).
% semi_confluence_imp_CR
thf(fact_630_insert__iff,axiom,
! [A2: produc357393685978478089rm_a_b,B2: produc357393685978478089rm_a_b,A: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ A2 @ ( insert7009541432154983385rm_a_b @ B2 @ A ) )
= ( ( A2 = B2 )
| ( member5869715511025134514rm_a_b @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_631_insert__iff,axiom,
! [A2: product_prod_a_nat,B2: product_prod_a_nat,A: set_Pr4934435412358123699_a_nat] :
( ( member5724188588386418708_a_nat @ A2 @ ( insert8054603423593749677_a_nat @ B2 @ A ) )
= ( ( A2 = B2 )
| ( member5724188588386418708_a_nat @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_632_insertCI,axiom,
! [A2: produc357393685978478089rm_a_b,B: set_Pr4386577575007340137rm_a_b,B2: produc357393685978478089rm_a_b] :
( ( ~ ( member5869715511025134514rm_a_b @ A2 @ B )
=> ( A2 = B2 ) )
=> ( member5869715511025134514rm_a_b @ A2 @ ( insert7009541432154983385rm_a_b @ B2 @ B ) ) ) ).
% insertCI
thf(fact_633_insertCI,axiom,
! [A2: product_prod_a_nat,B: set_Pr4934435412358123699_a_nat,B2: product_prod_a_nat] :
( ( ~ ( member5724188588386418708_a_nat @ A2 @ B )
=> ( A2 = B2 ) )
=> ( member5724188588386418708_a_nat @ A2 @ ( insert8054603423593749677_a_nat @ B2 @ B ) ) ) ).
% insertCI
thf(fact_634_singletonI,axiom,
! [A2: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ A2 @ ( insert7009541432154983385rm_a_b @ A2 @ bot_bo197521221353338581rm_a_b ) ) ).
% singletonI
thf(fact_635_singletonI,axiom,
! [A2: product_prod_a_nat] : ( member5724188588386418708_a_nat @ A2 @ ( insert8054603423593749677_a_nat @ A2 @ bot_bo9049108969261143879_a_nat ) ) ).
% singletonI
thf(fact_636_insert__subset,axiom,
! [X3: product_prod_a_nat,A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ ( insert8054603423593749677_a_nat @ X3 @ A ) @ B )
= ( ( member5724188588386418708_a_nat @ X3 @ B )
& ( ord_le8666007276011122963_a_nat @ A @ B ) ) ) ).
% insert_subset
thf(fact_637_insert__subset,axiom,
! [X3: produc357393685978478089rm_a_b,A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ ( insert7009541432154983385rm_a_b @ X3 @ A ) @ B )
= ( ( member5869715511025134514rm_a_b @ X3 @ B )
& ( ord_le118470702582115849rm_a_b @ A @ B ) ) ) ).
% insert_subset
thf(fact_638_converse__iff,axiom,
! [A2: nat,B2: a,R: set_Pr4934435412358123699_a_nat] :
( ( member8962352052110095674_nat_a @ ( product_Pair_nat_a @ A2 @ B2 ) @ ( converse_a_nat @ R ) )
= ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ B2 @ A2 ) @ R ) ) ).
% converse_iff
thf(fact_639_converse__iff,axiom,
! [A2: a,B2: nat,R: set_Pr4193341848836149977_nat_a] :
( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ A2 @ B2 ) @ ( converse_nat_a @ R ) )
= ( member8962352052110095674_nat_a @ ( product_Pair_nat_a @ B2 @ A2 ) @ R ) ) ).
% converse_iff
thf(fact_640_converse__iff,axiom,
! [A2: term_a_b,B2: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( conver7608559976937845093rm_a_b @ R ) )
= ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ A2 ) @ R ) ) ).
% converse_iff
thf(fact_641_converse__mono,axiom,
! [R: set_Pr4934435412358123699_a_nat,S: set_Pr4934435412358123699_a_nat] :
( ( ord_le7924913712489149241_nat_a @ ( converse_a_nat @ R ) @ ( converse_a_nat @ S ) )
= ( ord_le8666007276011122963_a_nat @ R @ S ) ) ).
% converse_mono
thf(fact_642_converse__mono,axiom,
! [R: set_Pr4193341848836149977_nat_a,S: set_Pr4193341848836149977_nat_a] :
( ( ord_le8666007276011122963_a_nat @ ( converse_nat_a @ R ) @ ( converse_nat_a @ S ) )
= ( ord_le7924913712489149241_nat_a @ R @ S ) ) ).
% converse_mono
thf(fact_643_converse__mono,axiom,
! [R: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ ( conver7608559976937845093rm_a_b @ R ) @ ( conver7608559976937845093rm_a_b @ S ) )
= ( ord_le118470702582115849rm_a_b @ R @ S ) ) ).
% converse_mono
thf(fact_644_singleton__insert__inj__eq_H,axiom,
! [A2: product_prod_a_nat,A: set_Pr4934435412358123699_a_nat,B2: product_prod_a_nat] :
( ( ( insert8054603423593749677_a_nat @ A2 @ A )
= ( insert8054603423593749677_a_nat @ B2 @ bot_bo9049108969261143879_a_nat ) )
= ( ( A2 = B2 )
& ( ord_le8666007276011122963_a_nat @ A @ ( insert8054603423593749677_a_nat @ B2 @ bot_bo9049108969261143879_a_nat ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_645_singleton__insert__inj__eq_H,axiom,
! [A2: produc357393685978478089rm_a_b,A: set_Pr4386577575007340137rm_a_b,B2: produc357393685978478089rm_a_b] :
( ( ( insert7009541432154983385rm_a_b @ A2 @ A )
= ( insert7009541432154983385rm_a_b @ B2 @ bot_bo197521221353338581rm_a_b ) )
= ( ( A2 = B2 )
& ( ord_le118470702582115849rm_a_b @ A @ ( insert7009541432154983385rm_a_b @ B2 @ bot_bo197521221353338581rm_a_b ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_646_singleton__insert__inj__eq,axiom,
! [B2: product_prod_a_nat,A2: product_prod_a_nat,A: set_Pr4934435412358123699_a_nat] :
( ( ( insert8054603423593749677_a_nat @ B2 @ bot_bo9049108969261143879_a_nat )
= ( insert8054603423593749677_a_nat @ A2 @ A ) )
= ( ( A2 = B2 )
& ( ord_le8666007276011122963_a_nat @ A @ ( insert8054603423593749677_a_nat @ B2 @ bot_bo9049108969261143879_a_nat ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_647_singleton__insert__inj__eq,axiom,
! [B2: produc357393685978478089rm_a_b,A2: produc357393685978478089rm_a_b,A: set_Pr4386577575007340137rm_a_b] :
( ( ( insert7009541432154983385rm_a_b @ B2 @ bot_bo197521221353338581rm_a_b )
= ( insert7009541432154983385rm_a_b @ A2 @ A ) )
= ( ( A2 = B2 )
& ( ord_le118470702582115849rm_a_b @ A @ ( insert7009541432154983385rm_a_b @ B2 @ bot_bo197521221353338581rm_a_b ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_648_conversionI_H,axiom,
! [A2: term_a_b,B2: term_a_b,A: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( transi7742714808557438673rm_a_b @ A ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( abstra1130568886744591910rm_a_b @ A ) ) ) ).
% conversionI'
thf(fact_649_diamond__I,axiom,
! [R: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ ( conver7608559976937845093rm_a_b @ R ) @ R ) @ ( relcom370159955682700863rm_a_b @ R @ ( conver7608559976937845093rm_a_b @ R ) ) )
=> ( abstra8151087076625346926rm_a_b @ R ) ) ).
% diamond_I
thf(fact_650_conversionI,axiom,
! [A2: term_a_b,B2: term_a_b,A: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ A @ ( conver7608559976937845093rm_a_b @ A ) ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( abstra1130568886744591910rm_a_b @ A ) ) ) ).
% conversionI
thf(fact_651_relpow__image,axiom,
! [R: set_Pr4386577575007340137rm_a_b,F2: term_a_b > term_a_b,R7: set_Pr4386577575007340137rm_a_b,S: term_a_b,T: term_a_b,N: nat] :
( ! [S2: term_a_b,T3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S2 @ T3 ) @ R )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F2 @ S2 ) @ ( F2 @ T3 ) ) @ R7 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( compow4057154403645558940rm_a_b @ N @ R ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F2 @ S ) @ ( F2 @ T ) ) @ ( compow4057154403645558940rm_a_b @ N @ R7 ) ) ) ) ).
% relpow_image
thf(fact_652_relpow__Suc__D2_H,axiom,
! [N: nat,R4: set_Pr4386577575007340137rm_a_b,X6: term_a_b,Y6: term_a_b,Z5: term_a_b] :
( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X6 @ Y6 ) @ ( compow4057154403645558940rm_a_b @ N @ R4 ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y6 @ Z5 ) @ R4 ) )
=> ? [W: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X6 @ W ) @ R4 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ W @ Z5 ) @ ( compow4057154403645558940rm_a_b @ N @ R4 ) ) ) ) ).
% relpow_Suc_D2'
thf(fact_653_relpow__refl__mono,axiom,
! [Rel: set_Pr4386577575007340137rm_a_b,M: nat,N: nat,A2: term_a_b,B2: term_a_b] :
( ! [X: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ X ) @ Rel )
=> ( ( ord_less_eq_nat @ M @ N )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( compow4057154403645558940rm_a_b @ M @ Rel ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( compow4057154403645558940rm_a_b @ N @ Rel ) ) ) ) ) ).
% relpow_refl_mono
thf(fact_654_converse__subset__swap,axiom,
! [R: set_Pr4193341848836149977_nat_a,S: set_Pr4934435412358123699_a_nat] :
( ( ord_le7924913712489149241_nat_a @ R @ ( converse_a_nat @ S ) )
= ( ord_le8666007276011122963_a_nat @ ( converse_nat_a @ R ) @ S ) ) ).
% converse_subset_swap
thf(fact_655_converse__subset__swap,axiom,
! [R: set_Pr4934435412358123699_a_nat,S: set_Pr4193341848836149977_nat_a] :
( ( ord_le8666007276011122963_a_nat @ R @ ( converse_nat_a @ S ) )
= ( ord_le7924913712489149241_nat_a @ ( converse_a_nat @ R ) @ S ) ) ).
% converse_subset_swap
thf(fact_656_converse__subset__swap,axiom,
! [R: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ R @ ( conver7608559976937845093rm_a_b @ S ) )
= ( ord_le118470702582115849rm_a_b @ ( conver7608559976937845093rm_a_b @ R ) @ S ) ) ).
% converse_subset_swap
thf(fact_657_singletonD,axiom,
! [B2: produc357393685978478089rm_a_b,A2: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ B2 @ ( insert7009541432154983385rm_a_b @ A2 @ bot_bo197521221353338581rm_a_b ) )
=> ( B2 = A2 ) ) ).
% singletonD
thf(fact_658_singletonD,axiom,
! [B2: product_prod_a_nat,A2: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ B2 @ ( insert8054603423593749677_a_nat @ A2 @ bot_bo9049108969261143879_a_nat ) )
=> ( B2 = A2 ) ) ).
% singletonD
thf(fact_659_singleton__iff,axiom,
! [B2: produc357393685978478089rm_a_b,A2: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ B2 @ ( insert7009541432154983385rm_a_b @ A2 @ bot_bo197521221353338581rm_a_b ) )
= ( B2 = A2 ) ) ).
% singleton_iff
thf(fact_660_singleton__iff,axiom,
! [B2: product_prod_a_nat,A2: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ B2 @ ( insert8054603423593749677_a_nat @ A2 @ bot_bo9049108969261143879_a_nat ) )
= ( B2 = A2 ) ) ).
% singleton_iff
thf(fact_661_insert__mono,axiom,
! [C2: set_Pr4934435412358123699_a_nat,D: set_Pr4934435412358123699_a_nat,A2: product_prod_a_nat] :
( ( ord_le8666007276011122963_a_nat @ C2 @ D )
=> ( ord_le8666007276011122963_a_nat @ ( insert8054603423593749677_a_nat @ A2 @ C2 ) @ ( insert8054603423593749677_a_nat @ A2 @ D ) ) ) ).
% insert_mono
thf(fact_662_insert__mono,axiom,
! [C2: set_Pr4386577575007340137rm_a_b,D: set_Pr4386577575007340137rm_a_b,A2: produc357393685978478089rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ C2 @ D )
=> ( ord_le118470702582115849rm_a_b @ ( insert7009541432154983385rm_a_b @ A2 @ C2 ) @ ( insert7009541432154983385rm_a_b @ A2 @ D ) ) ) ).
% insert_mono
thf(fact_663_subset__insert,axiom,
! [X3: product_prod_a_nat,A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ~ ( member5724188588386418708_a_nat @ X3 @ A )
=> ( ( ord_le8666007276011122963_a_nat @ A @ ( insert8054603423593749677_a_nat @ X3 @ B ) )
= ( ord_le8666007276011122963_a_nat @ A @ B ) ) ) ).
% subset_insert
thf(fact_664_subset__insert,axiom,
! [X3: produc357393685978478089rm_a_b,A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ~ ( member5869715511025134514rm_a_b @ X3 @ A )
=> ( ( ord_le118470702582115849rm_a_b @ A @ ( insert7009541432154983385rm_a_b @ X3 @ B ) )
= ( ord_le118470702582115849rm_a_b @ A @ B ) ) ) ).
% subset_insert
thf(fact_665_subset__insertI,axiom,
! [B: set_Pr4934435412358123699_a_nat,A2: product_prod_a_nat] : ( ord_le8666007276011122963_a_nat @ B @ ( insert8054603423593749677_a_nat @ A2 @ B ) ) ).
% subset_insertI
thf(fact_666_subset__insertI,axiom,
! [B: set_Pr4386577575007340137rm_a_b,A2: produc357393685978478089rm_a_b] : ( ord_le118470702582115849rm_a_b @ B @ ( insert7009541432154983385rm_a_b @ A2 @ B ) ) ).
% subset_insertI
thf(fact_667_subset__insertI2,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat,B2: product_prod_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A @ B )
=> ( ord_le8666007276011122963_a_nat @ A @ ( insert8054603423593749677_a_nat @ B2 @ B ) ) ) ).
% subset_insertI2
thf(fact_668_subset__insertI2,axiom,
! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b,B2: produc357393685978478089rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A @ B )
=> ( ord_le118470702582115849rm_a_b @ A @ ( insert7009541432154983385rm_a_b @ B2 @ B ) ) ) ).
% subset_insertI2
thf(fact_669_insert__subsetI,axiom,
! [X3: product_prod_a_nat,A: set_Pr4934435412358123699_a_nat,X5: set_Pr4934435412358123699_a_nat] :
( ( member5724188588386418708_a_nat @ X3 @ A )
=> ( ( ord_le8666007276011122963_a_nat @ X5 @ A )
=> ( ord_le8666007276011122963_a_nat @ ( insert8054603423593749677_a_nat @ X3 @ X5 ) @ A ) ) ) ).
% insert_subsetI
thf(fact_670_insert__subsetI,axiom,
! [X3: produc357393685978478089rm_a_b,A: set_Pr4386577575007340137rm_a_b,X5: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ X3 @ A )
=> ( ( ord_le118470702582115849rm_a_b @ X5 @ A )
=> ( ord_le118470702582115849rm_a_b @ ( insert7009541432154983385rm_a_b @ X3 @ X5 ) @ A ) ) ) ).
% insert_subsetI
thf(fact_671_converse_Ocases,axiom,
! [A1: nat,A22: a,R: set_Pr4934435412358123699_a_nat] :
( ( member8962352052110095674_nat_a @ ( product_Pair_nat_a @ A1 @ A22 ) @ ( converse_a_nat @ R ) )
=> ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ A22 @ A1 ) @ R ) ) ).
% converse.cases
thf(fact_672_converse_Ocases,axiom,
! [A1: a,A22: nat,R: set_Pr4193341848836149977_nat_a] :
( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ A1 @ A22 ) @ ( converse_nat_a @ R ) )
=> ( member8962352052110095674_nat_a @ ( product_Pair_nat_a @ A22 @ A1 ) @ R ) ) ).
% converse.cases
thf(fact_673_converse_Ocases,axiom,
! [A1: term_a_b,A22: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A22 ) @ ( conver7608559976937845093rm_a_b @ R ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A22 @ A1 ) @ R ) ) ).
% converse.cases
thf(fact_674_converse_Osimps,axiom,
! [A1: nat,A22: a,R: set_Pr4934435412358123699_a_nat] :
( ( member8962352052110095674_nat_a @ ( product_Pair_nat_a @ A1 @ A22 ) @ ( converse_a_nat @ R ) )
= ( ? [A4: a,B4: nat] :
( ( A1 = B4 )
& ( A22 = A4 )
& ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ A4 @ B4 ) @ R ) ) ) ) ).
% converse.simps
thf(fact_675_converse_Osimps,axiom,
! [A1: a,A22: nat,R: set_Pr4193341848836149977_nat_a] :
( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ A1 @ A22 ) @ ( converse_nat_a @ R ) )
= ( ? [A4: nat,B4: a] :
( ( A1 = B4 )
& ( A22 = A4 )
& ( member8962352052110095674_nat_a @ ( product_Pair_nat_a @ A4 @ B4 ) @ R ) ) ) ) ).
% converse.simps
thf(fact_676_converse_Osimps,axiom,
! [A1: term_a_b,A22: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A22 ) @ ( conver7608559976937845093rm_a_b @ R ) )
= ( ? [A4: term_a_b,B4: term_a_b] :
( ( A1 = B4 )
& ( A22 = A4 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A4 @ B4 ) @ R ) ) ) ) ).
% converse.simps
thf(fact_677_converseD,axiom,
! [A2: nat,B2: a,R: set_Pr4934435412358123699_a_nat] :
( ( member8962352052110095674_nat_a @ ( product_Pair_nat_a @ A2 @ B2 ) @ ( converse_a_nat @ R ) )
=> ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ B2 @ A2 ) @ R ) ) ).
% converseD
thf(fact_678_converseD,axiom,
! [A2: a,B2: nat,R: set_Pr4193341848836149977_nat_a] :
( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ A2 @ B2 ) @ ( converse_nat_a @ R ) )
=> ( member8962352052110095674_nat_a @ ( product_Pair_nat_a @ B2 @ A2 ) @ R ) ) ).
% converseD
thf(fact_679_converseD,axiom,
! [A2: term_a_b,B2: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( conver7608559976937845093rm_a_b @ R ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ A2 ) @ R ) ) ).
% converseD
thf(fact_680_converseE,axiom,
! [Yx: product_prod_nat_a,R: set_Pr4934435412358123699_a_nat] :
( ( member8962352052110095674_nat_a @ Yx @ ( converse_a_nat @ R ) )
=> ~ ! [X: a,Y: nat] :
( ( Yx
= ( product_Pair_nat_a @ Y @ X ) )
=> ~ ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ X @ Y ) @ R ) ) ) ).
% converseE
thf(fact_681_converseE,axiom,
! [Yx: product_prod_a_nat,R: set_Pr4193341848836149977_nat_a] :
( ( member5724188588386418708_a_nat @ Yx @ ( converse_nat_a @ R ) )
=> ~ ! [X: nat,Y: a] :
( ( Yx
= ( product_Pair_a_nat @ Y @ X ) )
=> ~ ( member8962352052110095674_nat_a @ ( product_Pair_nat_a @ X @ Y ) @ R ) ) ) ).
% converseE
thf(fact_682_converseE,axiom,
! [Yx: produc357393685978478089rm_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ Yx @ ( conver7608559976937845093rm_a_b @ R ) )
=> ~ ! [X: term_a_b,Y: term_a_b] :
( ( Yx
= ( produc7020197800436672577rm_a_b @ Y @ X ) )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ R ) ) ) ).
% converseE
thf(fact_683_converseI,axiom,
! [A2: nat,B2: a,R: set_Pr4193341848836149977_nat_a] :
( ( member8962352052110095674_nat_a @ ( product_Pair_nat_a @ A2 @ B2 ) @ R )
=> ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ B2 @ A2 ) @ ( converse_nat_a @ R ) ) ) ).
% converseI
thf(fact_684_converseI,axiom,
! [A2: a,B2: nat,R: set_Pr4934435412358123699_a_nat] :
( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ A2 @ B2 ) @ R )
=> ( member8962352052110095674_nat_a @ ( product_Pair_nat_a @ B2 @ A2 ) @ ( converse_a_nat @ R ) ) ) ).
% converseI
thf(fact_685_converseI,axiom,
! [A2: term_a_b,B2: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ R )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ A2 ) @ ( conver7608559976937845093rm_a_b @ R ) ) ) ).
% converseI
thf(fact_686_mk__disjoint__insert,axiom,
! [A2: produc357393685978478089rm_a_b,A: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ A2 @ A )
=> ? [B9: set_Pr4386577575007340137rm_a_b] :
( ( A
= ( insert7009541432154983385rm_a_b @ A2 @ B9 ) )
& ~ ( member5869715511025134514rm_a_b @ A2 @ B9 ) ) ) ).
% mk_disjoint_insert
thf(fact_687_mk__disjoint__insert,axiom,
! [A2: product_prod_a_nat,A: set_Pr4934435412358123699_a_nat] :
( ( member5724188588386418708_a_nat @ A2 @ A )
=> ? [B9: set_Pr4934435412358123699_a_nat] :
( ( A
= ( insert8054603423593749677_a_nat @ A2 @ B9 ) )
& ~ ( member5724188588386418708_a_nat @ A2 @ B9 ) ) ) ).
% mk_disjoint_insert
thf(fact_688_insert__eq__iff,axiom,
! [A2: produc357393685978478089rm_a_b,A: set_Pr4386577575007340137rm_a_b,B2: produc357393685978478089rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ~ ( member5869715511025134514rm_a_b @ A2 @ A )
=> ( ~ ( member5869715511025134514rm_a_b @ B2 @ B )
=> ( ( ( insert7009541432154983385rm_a_b @ A2 @ A )
= ( insert7009541432154983385rm_a_b @ B2 @ B ) )
= ( ( ( A2 = B2 )
=> ( A = B ) )
& ( ( A2 != B2 )
=> ? [C6: set_Pr4386577575007340137rm_a_b] :
( ( A
= ( insert7009541432154983385rm_a_b @ B2 @ C6 ) )
& ~ ( member5869715511025134514rm_a_b @ B2 @ C6 )
& ( B
= ( insert7009541432154983385rm_a_b @ A2 @ C6 ) )
& ~ ( member5869715511025134514rm_a_b @ A2 @ C6 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_689_insert__eq__iff,axiom,
! [A2: product_prod_a_nat,A: set_Pr4934435412358123699_a_nat,B2: product_prod_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ~ ( member5724188588386418708_a_nat @ A2 @ A )
=> ( ~ ( member5724188588386418708_a_nat @ B2 @ B )
=> ( ( ( insert8054603423593749677_a_nat @ A2 @ A )
= ( insert8054603423593749677_a_nat @ B2 @ B ) )
= ( ( ( A2 = B2 )
=> ( A = B ) )
& ( ( A2 != B2 )
=> ? [C6: set_Pr4934435412358123699_a_nat] :
( ( A
= ( insert8054603423593749677_a_nat @ B2 @ C6 ) )
& ~ ( member5724188588386418708_a_nat @ B2 @ C6 )
& ( B
= ( insert8054603423593749677_a_nat @ A2 @ C6 ) )
& ~ ( member5724188588386418708_a_nat @ A2 @ C6 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_690_insert__absorb,axiom,
! [A2: produc357393685978478089rm_a_b,A: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ A2 @ A )
=> ( ( insert7009541432154983385rm_a_b @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_691_insert__absorb,axiom,
! [A2: product_prod_a_nat,A: set_Pr4934435412358123699_a_nat] :
( ( member5724188588386418708_a_nat @ A2 @ A )
=> ( ( insert8054603423593749677_a_nat @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_692_insert__ident,axiom,
! [X3: produc357393685978478089rm_a_b,A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ~ ( member5869715511025134514rm_a_b @ X3 @ A )
=> ( ~ ( member5869715511025134514rm_a_b @ X3 @ B )
=> ( ( ( insert7009541432154983385rm_a_b @ X3 @ A )
= ( insert7009541432154983385rm_a_b @ X3 @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_693_insert__ident,axiom,
! [X3: product_prod_a_nat,A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ~ ( member5724188588386418708_a_nat @ X3 @ A )
=> ( ~ ( member5724188588386418708_a_nat @ X3 @ B )
=> ( ( ( insert8054603423593749677_a_nat @ X3 @ A )
= ( insert8054603423593749677_a_nat @ X3 @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_694_Set_Oset__insert,axiom,
! [X3: produc357393685978478089rm_a_b,A: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ X3 @ A )
=> ~ ! [B9: set_Pr4386577575007340137rm_a_b] :
( ( A
= ( insert7009541432154983385rm_a_b @ X3 @ B9 ) )
=> ( member5869715511025134514rm_a_b @ X3 @ B9 ) ) ) ).
% Set.set_insert
thf(fact_695_Set_Oset__insert,axiom,
! [X3: product_prod_a_nat,A: set_Pr4934435412358123699_a_nat] :
( ( member5724188588386418708_a_nat @ X3 @ A )
=> ~ ! [B9: set_Pr4934435412358123699_a_nat] :
( ( A
= ( insert8054603423593749677_a_nat @ X3 @ B9 ) )
=> ( member5724188588386418708_a_nat @ X3 @ B9 ) ) ) ).
% Set.set_insert
thf(fact_696_insertI2,axiom,
! [A2: produc357393685978478089rm_a_b,B: set_Pr4386577575007340137rm_a_b,B2: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ A2 @ B )
=> ( member5869715511025134514rm_a_b @ A2 @ ( insert7009541432154983385rm_a_b @ B2 @ B ) ) ) ).
% insertI2
thf(fact_697_insertI2,axiom,
! [A2: product_prod_a_nat,B: set_Pr4934435412358123699_a_nat,B2: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ A2 @ B )
=> ( member5724188588386418708_a_nat @ A2 @ ( insert8054603423593749677_a_nat @ B2 @ B ) ) ) ).
% insertI2
thf(fact_698_insertI1,axiom,
! [A2: produc357393685978478089rm_a_b,B: set_Pr4386577575007340137rm_a_b] : ( member5869715511025134514rm_a_b @ A2 @ ( insert7009541432154983385rm_a_b @ A2 @ B ) ) ).
% insertI1
thf(fact_699_insertI1,axiom,
! [A2: product_prod_a_nat,B: set_Pr4934435412358123699_a_nat] : ( member5724188588386418708_a_nat @ A2 @ ( insert8054603423593749677_a_nat @ A2 @ B ) ) ).
% insertI1
thf(fact_700_insertE,axiom,
! [A2: produc357393685978478089rm_a_b,B2: produc357393685978478089rm_a_b,A: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ A2 @ ( insert7009541432154983385rm_a_b @ B2 @ A ) )
=> ( ( A2 != B2 )
=> ( member5869715511025134514rm_a_b @ A2 @ A ) ) ) ).
% insertE
thf(fact_701_insertE,axiom,
! [A2: product_prod_a_nat,B2: product_prod_a_nat,A: set_Pr4934435412358123699_a_nat] :
( ( member5724188588386418708_a_nat @ A2 @ ( insert8054603423593749677_a_nat @ B2 @ A ) )
=> ( ( A2 != B2 )
=> ( member5724188588386418708_a_nat @ A2 @ A ) ) ) ).
% insertE
thf(fact_702_rtrancl__imp__relpow,axiom,
! [P2: produc357393685978478089rm_a_b,R4: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ P2 @ ( transi7742714808557438673rm_a_b @ R4 ) )
=> ? [N2: nat] : ( member5869715511025134514rm_a_b @ P2 @ ( compow4057154403645558940rm_a_b @ N2 @ R4 ) ) ) ).
% rtrancl_imp_relpow
thf(fact_703_relpow__imp__rtrancl,axiom,
! [P2: produc357393685978478089rm_a_b,N: nat,R4: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ P2 @ ( compow4057154403645558940rm_a_b @ N @ R4 ) )
=> ( member5869715511025134514rm_a_b @ P2 @ ( transi7742714808557438673rm_a_b @ R4 ) ) ) ).
% relpow_imp_rtrancl
thf(fact_704_rtrancl__power,axiom,
! [P2: produc357393685978478089rm_a_b,R4: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ P2 @ ( transi7742714808557438673rm_a_b @ R4 ) )
= ( ? [N3: nat] : ( member5869715511025134514rm_a_b @ P2 @ ( compow4057154403645558940rm_a_b @ N3 @ R4 ) ) ) ) ).
% rtrancl_power
thf(fact_705_relpow__mono,axiom,
! [R: set_Pr4386577575007340137rm_a_b,R7: set_Pr4386577575007340137rm_a_b,N: nat] :
( ( ord_le118470702582115849rm_a_b @ R @ R7 )
=> ( ord_le118470702582115849rm_a_b @ ( compow4057154403645558940rm_a_b @ N @ R ) @ ( compow4057154403645558940rm_a_b @ N @ R7 ) ) ) ).
% relpow_mono
thf(fact_706_conversion__refl,axiom,
! [A2: term_a_b,A: set_Pr4386577575007340137rm_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ A2 ) @ ( abstra1130568886744591910rm_a_b @ A ) ) ).
% conversion_refl
thf(fact_707_conversion__inv,axiom,
! [X3: term_a_b,Y3: term_a_b,R4: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ ( abstra1130568886744591910rm_a_b @ R4 ) )
= ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ X3 ) @ ( abstra1130568886744591910rm_a_b @ R4 ) ) ) ).
% conversion_inv
thf(fact_708_conversion__mono,axiom,
! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A @ B )
=> ( ord_le118470702582115849rm_a_b @ ( abstra1130568886744591910rm_a_b @ A ) @ ( abstra1130568886744591910rm_a_b @ B ) ) ) ).
% conversion_mono
thf(fact_709_conversionE,axiom,
! [A2: term_a_b,B2: term_a_b,A: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( abstra1130568886744591910rm_a_b @ A ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ A @ ( conver7608559976937845093rm_a_b @ A ) ) ) ) ) ).
% conversionE
thf(fact_710_UNC__def,axiom,
( abstra8480497335821202873rm_a_b
= ( ^ [A6: set_Pr2972776593051762503rm_a_b] :
! [A4: produc357393685978478089rm_a_b,B4: produc357393685978478089rm_a_b] :
( ( ( member5869715511025134514rm_a_b @ A4 @ ( abstra1902783451776770197rm_a_b @ A6 ) )
& ( member5869715511025134514rm_a_b @ B4 @ ( abstra1902783451776770197rm_a_b @ A6 ) )
& ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A4 @ B4 ) @ ( abstra6125656530262095991rm_a_b @ A6 ) ) )
=> ( A4 = B4 ) ) ) ) ).
% UNC_def
thf(fact_711_UNC__def,axiom,
( abstra7179253994116303565_a_nat
= ( ^ [A6: set_Pr1811044260758604347_a_nat] :
! [A4: product_prod_a_nat,B4: product_prod_a_nat] :
( ( ( member5724188588386418708_a_nat @ A4 @ ( abstra2685462378305987313_a_nat @ A6 ) )
& ( member5724188588386418708_a_nat @ B4 @ ( abstra2685462378305987313_a_nat @ A6 ) )
& ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A4 @ B4 ) @ ( abstra3054561309482028431_a_nat @ A6 ) ) )
=> ( A4 = B4 ) ) ) ) ).
% UNC_def
thf(fact_712_UNC__def,axiom,
( abstra4649771476585718372rm_a_b
= ( ^ [A6: set_Pr4386577575007340137rm_a_b] :
! [A4: term_a_b,B4: term_a_b] :
( ( ( member_term_a_b @ A4 @ ( abstract_NF_term_a_b @ A6 ) )
& ( member_term_a_b @ B4 @ ( abstract_NF_term_a_b @ A6 ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A4 @ B4 ) @ ( abstra1130568886744591910rm_a_b @ A6 ) ) )
=> ( A4 = B4 ) ) ) ) ).
% UNC_def
thf(fact_713_UNC__I,axiom,
! [A: set_Pr2972776593051762503rm_a_b] :
( ! [A5: produc357393685978478089rm_a_b,B5: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ A5 @ ( abstra1902783451776770197rm_a_b @ A ) )
=> ( ( member5869715511025134514rm_a_b @ B5 @ ( abstra1902783451776770197rm_a_b @ A ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A5 @ B5 ) @ ( abstra6125656530262095991rm_a_b @ A ) )
=> ( A5 = B5 ) ) ) )
=> ( abstra8480497335821202873rm_a_b @ A ) ) ).
% UNC_I
thf(fact_714_UNC__I,axiom,
! [A: set_Pr1811044260758604347_a_nat] :
( ! [A5: product_prod_a_nat,B5: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ A5 @ ( abstra2685462378305987313_a_nat @ A ) )
=> ( ( member5724188588386418708_a_nat @ B5 @ ( abstra2685462378305987313_a_nat @ A ) )
=> ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A5 @ B5 ) @ ( abstra3054561309482028431_a_nat @ A ) )
=> ( A5 = B5 ) ) ) )
=> ( abstra7179253994116303565_a_nat @ A ) ) ).
% UNC_I
thf(fact_715_UNC__I,axiom,
! [A: set_Pr4386577575007340137rm_a_b] :
( ! [A5: term_a_b,B5: term_a_b] :
( ( member_term_a_b @ A5 @ ( abstract_NF_term_a_b @ A ) )
=> ( ( member_term_a_b @ B5 @ ( abstract_NF_term_a_b @ A ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A5 @ B5 ) @ ( abstra1130568886744591910rm_a_b @ A ) )
=> ( A5 = B5 ) ) ) )
=> ( abstra4649771476585718372rm_a_b @ A ) ) ).
% UNC_I
thf(fact_716_UNC__E,axiom,
! [A: set_Pr2972776593051762503rm_a_b,A2: produc357393685978478089rm_a_b,B2: produc357393685978478089rm_a_b] :
( ( abstra8480497335821202873rm_a_b @ A )
=> ( ( A2 != B2 )
=> ( ( member5869715511025134514rm_a_b @ A2 @ ( abstra1902783451776770197rm_a_b @ A ) )
=> ( ( member5869715511025134514rm_a_b @ B2 @ ( abstra1902783451776770197rm_a_b @ A ) )
=> ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A2 @ B2 ) @ ( abstra6125656530262095991rm_a_b @ A ) ) ) ) ) ) ).
% UNC_E
thf(fact_717_UNC__E,axiom,
! [A: set_Pr1811044260758604347_a_nat,A2: product_prod_a_nat,B2: product_prod_a_nat] :
( ( abstra7179253994116303565_a_nat @ A )
=> ( ( A2 != B2 )
=> ( ( member5724188588386418708_a_nat @ A2 @ ( abstra2685462378305987313_a_nat @ A ) )
=> ( ( member5724188588386418708_a_nat @ B2 @ ( abstra2685462378305987313_a_nat @ A ) )
=> ~ ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A2 @ B2 ) @ ( abstra3054561309482028431_a_nat @ A ) ) ) ) ) ) ).
% UNC_E
thf(fact_718_UNC__E,axiom,
! [A: set_Pr4386577575007340137rm_a_b,A2: term_a_b,B2: term_a_b] :
( ( abstra4649771476585718372rm_a_b @ A )
=> ( ( A2 != B2 )
=> ( ( member_term_a_b @ A2 @ ( abstract_NF_term_a_b @ A ) )
=> ( ( member_term_a_b @ B2 @ ( abstract_NF_term_a_b @ A ) )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( abstra1130568886744591910rm_a_b @ A ) ) ) ) ) ) ).
% UNC_E
thf(fact_719_rtrancl__converseD,axiom,
! [X3: term_a_b,Y3: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ ( transi7742714808557438673rm_a_b @ ( conver7608559976937845093rm_a_b @ R ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ X3 ) @ ( transi7742714808557438673rm_a_b @ R ) ) ) ).
% rtrancl_converseD
thf(fact_720_rtrancl__converseI,axiom,
! [Y3: term_a_b,X3: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ X3 ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ ( transi7742714808557438673rm_a_b @ ( conver7608559976937845093rm_a_b @ R ) ) ) ) ).
% rtrancl_converseI
thf(fact_721_peak__iff,axiom,
! [X3: nat,Y3: nat,A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ ( relcomp_nat_a_nat @ ( converse_a_nat @ A ) @ B ) )
= ( ? [U3: a] :
( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ U3 @ X3 ) @ A )
& ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ U3 @ Y3 ) @ B ) ) ) ) ).
% peak_iff
thf(fact_722_peak__iff,axiom,
! [X3: a,Y3: nat,A: set_Product_prod_a_a,B: set_Pr4934435412358123699_a_nat] :
( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ X3 @ Y3 ) @ ( relcomp_a_a_nat @ ( converse_a_a @ A ) @ B ) )
= ( ? [U3: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ U3 @ X3 ) @ A )
& ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ U3 @ Y3 ) @ B ) ) ) ) ).
% peak_iff
thf(fact_723_peak__iff,axiom,
! [X3: term_a_b,Y3: term_a_b,A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ ( relcom370159955682700863rm_a_b @ ( conver7608559976937845093rm_a_b @ A ) @ B ) )
= ( ? [U3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ U3 @ X3 ) @ A )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ U3 @ Y3 ) @ B ) ) ) ) ).
% peak_iff
thf(fact_724_trancl__converseD,axiom,
! [X3: term_a_b,Y3: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ ( transi7922773638565587891rm_a_b @ ( conver7608559976937845093rm_a_b @ R ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ ( conver7608559976937845093rm_a_b @ ( transi7922773638565587891rm_a_b @ R ) ) ) ) ).
% trancl_converseD
thf(fact_725_trancl__converseI,axiom,
! [X3: term_a_b,Y3: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ ( conver7608559976937845093rm_a_b @ ( transi7922773638565587891rm_a_b @ R ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ ( transi7922773638565587891rm_a_b @ ( conver7608559976937845093rm_a_b @ R ) ) ) ) ).
% trancl_converseI
thf(fact_726_rtrancl__len__E,axiom,
! [X3: term_a_b,Y3: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ~ ! [N2: nat] :
~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ ( compow4057154403645558940rm_a_b @ N2 @ R ) ) ) ).
% rtrancl_len_E
thf(fact_727_subset__singleton__iff,axiom,
! [X5: set_Pr4934435412358123699_a_nat,A2: product_prod_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X5 @ ( insert8054603423593749677_a_nat @ A2 @ bot_bo9049108969261143879_a_nat ) )
= ( ( X5 = bot_bo9049108969261143879_a_nat )
| ( X5
= ( insert8054603423593749677_a_nat @ A2 @ bot_bo9049108969261143879_a_nat ) ) ) ) ).
% subset_singleton_iff
thf(fact_728_subset__singleton__iff,axiom,
! [X5: set_Pr4386577575007340137rm_a_b,A2: produc357393685978478089rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ X5 @ ( insert7009541432154983385rm_a_b @ A2 @ bot_bo197521221353338581rm_a_b ) )
= ( ( X5 = bot_bo197521221353338581rm_a_b )
| ( X5
= ( insert7009541432154983385rm_a_b @ A2 @ bot_bo197521221353338581rm_a_b ) ) ) ) ).
% subset_singleton_iff
thf(fact_729_subset__singletonD,axiom,
! [A: set_Pr4934435412358123699_a_nat,X3: product_prod_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A @ ( insert8054603423593749677_a_nat @ X3 @ bot_bo9049108969261143879_a_nat ) )
=> ( ( A = bot_bo9049108969261143879_a_nat )
| ( A
= ( insert8054603423593749677_a_nat @ X3 @ bot_bo9049108969261143879_a_nat ) ) ) ) ).
% subset_singletonD
thf(fact_730_subset__singletonD,axiom,
! [A: set_Pr4386577575007340137rm_a_b,X3: produc357393685978478089rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A @ ( insert7009541432154983385rm_a_b @ X3 @ bot_bo197521221353338581rm_a_b ) )
=> ( ( A = bot_bo197521221353338581rm_a_b )
| ( A
= ( insert7009541432154983385rm_a_b @ X3 @ bot_bo197521221353338581rm_a_b ) ) ) ) ).
% subset_singletonD
thf(fact_731_funas__rel__converse,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,F: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ ( terms_7988297476397195622_a_b_b @ R2 ) @ F )
=> ( ord_le8666007276011122963_a_nat @ ( terms_7988297476397195622_a_b_b @ ( conver7608559976937845093rm_a_b @ R2 ) ) @ F ) ) ).
% funas_rel_converse
thf(fact_732_join__imp__conversion,axiom,
! [R: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ ( abstra4096080454567261402rm_a_b @ R ) @ ( abstra1130568886744591910rm_a_b @ R ) ) ).
% join_imp_conversion
thf(fact_733_trancl__steps__relpow,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b,X3: term_a_b,Y3: term_a_b,N: nat] :
( ( ord_le118470702582115849rm_a_b @ A2 @ ( transi7922773638565587891rm_a_b @ B2 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ ( compow4057154403645558940rm_a_b @ N @ A2 ) )
=> ? [M2: nat] :
( ( ord_less_eq_nat @ N @ M2 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ ( compow4057154403645558940rm_a_b @ M2 @ B2 ) ) ) ) ) ).
% trancl_steps_relpow
thf(fact_734_meet__imp__conversion,axiom,
! [R: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ ( abstra1093061187967292567rm_a_b @ R ) @ ( abstra1130568886744591910rm_a_b @ R ) ) ).
% meet_imp_conversion
thf(fact_735_diamond__E,axiom,
! [R: set_Pr4386577575007340137rm_a_b] :
( ( abstra8151087076625346926rm_a_b @ R )
=> ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ ( conver7608559976937845093rm_a_b @ R ) @ R ) @ ( relcom370159955682700863rm_a_b @ R @ ( conver7608559976937845093rm_a_b @ R ) ) ) ) ).
% diamond_E
thf(fact_736_diamond__def,axiom,
( abstra8151087076625346926rm_a_b
= ( ^ [R3: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ ( conver7608559976937845093rm_a_b @ R3 ) @ R3 ) @ ( relcom370159955682700863rm_a_b @ R3 @ ( conver7608559976937845093rm_a_b @ R3 ) ) ) ) ) ).
% diamond_def
thf(fact_737_commute__def,axiom,
( abstra455195320138238452rm_a_b
= ( ^ [R3: set_Pr4386577575007340137rm_a_b,S3: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( conver7608559976937845093rm_a_b @ R3 ) ) @ ( transi7742714808557438673rm_a_b @ S3 ) ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ S3 ) @ ( transi7742714808557438673rm_a_b @ ( conver7608559976937845093rm_a_b @ R3 ) ) ) ) ) ) ).
% commute_def
thf(fact_738_WCR__alt__def,axiom,
! [A: set_Pr4386577575007340137rm_a_b] :
( ( abstra5026849107050860991rm_a_b @ A @ top_top_set_term_a_b )
= ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ ( conver7608559976937845093rm_a_b @ A ) @ A ) @ ( abstra4096080454567261402rm_a_b @ A ) ) ) ).
% WCR_alt_def
thf(fact_739_CR__iff__conversion__imp__join,axiom,
! [R: set_Pr4386577575007340137rm_a_b] :
( ( abstra8448919418672941150rm_a_b @ R @ top_top_set_term_a_b )
= ( ord_le118470702582115849rm_a_b @ ( abstra1130568886744591910rm_a_b @ R ) @ ( abstra4096080454567261402rm_a_b @ R ) ) ) ).
% CR_iff_conversion_imp_join
thf(fact_740_diamond__imp__semi__confluence,axiom,
! [R: set_Pr4386577575007340137rm_a_b] :
( ( abstra8151087076625346926rm_a_b @ R )
=> ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ ( conver7608559976937845093rm_a_b @ R ) @ ( transi7742714808557438673rm_a_b @ R ) ) @ ( abstra4096080454567261402rm_a_b @ R ) ) ) ).
% diamond_imp_semi_confluence
thf(fact_741_semi__complete__imp__conversionIff__same__NF,axiom,
! [R: set_Pr4386577575007340137rm_a_b,X3: term_a_b,Y3: term_a_b] :
( ( abstra7162307272211697846rm_a_b @ R @ top_top_set_term_a_b )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ ( abstra1130568886744591910rm_a_b @ R ) )
= ( ! [U3: term_a_b,V2: term_a_b] :
( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ U3 ) @ ( abstra2972651231392952684rm_a_b @ R ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ V2 ) @ ( abstra2972651231392952684rm_a_b @ R ) ) )
=> ( U3 = V2 ) ) ) ) ) ).
% semi_complete_imp_conversionIff_same_NF
thf(fact_742_min__ext__compat,axiom,
! [R4: set_Pr4386577575007340137rm_a_b,S4: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ R4 @ S4 ) @ R4 )
=> ( ord_le2663425029446080117rm_a_b @ ( relcom3320975603336820193rm_a_b @ ( min_ext_term_a_b @ R4 ) @ ( sup_su1246761269818691241rm_a_b @ ( min_ext_term_a_b @ S4 ) @ ( insert807748097412076101rm_a_b @ ( produc8517751122659631533rm_a_b @ bot_bot_set_term_a_b @ bot_bot_set_term_a_b ) @ bot_bo6544111711586975553rm_a_b ) ) ) @ ( min_ext_term_a_b @ R4 ) ) ) ).
% min_ext_compat
thf(fact_743_weak__diamond__def,axiom,
( abstra5190899124789103926rm_a_b
= ( ^ [R3: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ ( minus_5192120951422937424rm_a_b @ ( relcom370159955682700863rm_a_b @ ( conver7608559976937845093rm_a_b @ R3 ) @ R3 ) @ id_term_a_b ) @ ( relcom370159955682700863rm_a_b @ R3 @ ( conver7608559976937845093rm_a_b @ R3 ) ) ) ) ) ).
% weak_diamond_def
thf(fact_744_DiffI,axiom,
! [C: produc357393685978478089rm_a_b,A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ C @ A )
=> ( ~ ( member5869715511025134514rm_a_b @ C @ B )
=> ( member5869715511025134514rm_a_b @ C @ ( minus_5192120951422937424rm_a_b @ A @ B ) ) ) ) ).
% DiffI
thf(fact_745_DiffI,axiom,
! [C: product_prod_a_nat,A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( member5724188588386418708_a_nat @ C @ A )
=> ( ~ ( member5724188588386418708_a_nat @ C @ B )
=> ( member5724188588386418708_a_nat @ C @ ( minus_8630901905721465484_a_nat @ A @ B ) ) ) ) ).
% DiffI
thf(fact_746_Diff__iff,axiom,
! [C: produc357393685978478089rm_a_b,A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ C @ ( minus_5192120951422937424rm_a_b @ A @ B ) )
= ( ( member5869715511025134514rm_a_b @ C @ A )
& ~ ( member5869715511025134514rm_a_b @ C @ B ) ) ) ).
% Diff_iff
thf(fact_747_Diff__iff,axiom,
! [C: product_prod_a_nat,A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( member5724188588386418708_a_nat @ C @ ( minus_8630901905721465484_a_nat @ A @ B ) )
= ( ( member5724188588386418708_a_nat @ C @ A )
& ~ ( member5724188588386418708_a_nat @ C @ B ) ) ) ).
% Diff_iff
thf(fact_748_Diff__insert0,axiom,
! [X3: produc357393685978478089rm_a_b,A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ~ ( member5869715511025134514rm_a_b @ X3 @ A )
=> ( ( minus_5192120951422937424rm_a_b @ A @ ( insert7009541432154983385rm_a_b @ X3 @ B ) )
= ( minus_5192120951422937424rm_a_b @ A @ B ) ) ) ).
% Diff_insert0
thf(fact_749_Diff__insert0,axiom,
! [X3: product_prod_a_nat,A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ~ ( member5724188588386418708_a_nat @ X3 @ A )
=> ( ( minus_8630901905721465484_a_nat @ A @ ( insert8054603423593749677_a_nat @ X3 @ B ) )
= ( minus_8630901905721465484_a_nat @ A @ B ) ) ) ).
% Diff_insert0
thf(fact_750_insert__Diff1,axiom,
! [X3: produc357393685978478089rm_a_b,B: set_Pr4386577575007340137rm_a_b,A: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ X3 @ B )
=> ( ( minus_5192120951422937424rm_a_b @ ( insert7009541432154983385rm_a_b @ X3 @ A ) @ B )
= ( minus_5192120951422937424rm_a_b @ A @ B ) ) ) ).
% insert_Diff1
thf(fact_751_insert__Diff1,axiom,
! [X3: product_prod_a_nat,B: set_Pr4934435412358123699_a_nat,A: set_Pr4934435412358123699_a_nat] :
( ( member5724188588386418708_a_nat @ X3 @ B )
=> ( ( minus_8630901905721465484_a_nat @ ( insert8054603423593749677_a_nat @ X3 @ A ) @ B )
= ( minus_8630901905721465484_a_nat @ A @ B ) ) ) ).
% insert_Diff1
thf(fact_752_Diff__eq__empty__iff,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( ( minus_8630901905721465484_a_nat @ A @ B )
= bot_bo9049108969261143879_a_nat )
= ( ord_le8666007276011122963_a_nat @ A @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_753_Diff__eq__empty__iff,axiom,
! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ( ( minus_5192120951422937424rm_a_b @ A @ B )
= bot_bo197521221353338581rm_a_b )
= ( ord_le118470702582115849rm_a_b @ A @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_754_insert__Diff__if,axiom,
! [X3: produc357393685978478089rm_a_b,B: set_Pr4386577575007340137rm_a_b,A: set_Pr4386577575007340137rm_a_b] :
( ( ( member5869715511025134514rm_a_b @ X3 @ B )
=> ( ( minus_5192120951422937424rm_a_b @ ( insert7009541432154983385rm_a_b @ X3 @ A ) @ B )
= ( minus_5192120951422937424rm_a_b @ A @ B ) ) )
& ( ~ ( member5869715511025134514rm_a_b @ X3 @ B )
=> ( ( minus_5192120951422937424rm_a_b @ ( insert7009541432154983385rm_a_b @ X3 @ A ) @ B )
= ( insert7009541432154983385rm_a_b @ X3 @ ( minus_5192120951422937424rm_a_b @ A @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_755_insert__Diff__if,axiom,
! [X3: product_prod_a_nat,B: set_Pr4934435412358123699_a_nat,A: set_Pr4934435412358123699_a_nat] :
( ( ( member5724188588386418708_a_nat @ X3 @ B )
=> ( ( minus_8630901905721465484_a_nat @ ( insert8054603423593749677_a_nat @ X3 @ A ) @ B )
= ( minus_8630901905721465484_a_nat @ A @ B ) ) )
& ( ~ ( member5724188588386418708_a_nat @ X3 @ B )
=> ( ( minus_8630901905721465484_a_nat @ ( insert8054603423593749677_a_nat @ X3 @ A ) @ B )
= ( insert8054603423593749677_a_nat @ X3 @ ( minus_8630901905721465484_a_nat @ A @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_756_DiffE,axiom,
! [C: produc357393685978478089rm_a_b,A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ C @ ( minus_5192120951422937424rm_a_b @ A @ B ) )
=> ~ ( ( member5869715511025134514rm_a_b @ C @ A )
=> ( member5869715511025134514rm_a_b @ C @ B ) ) ) ).
% DiffE
thf(fact_757_DiffE,axiom,
! [C: product_prod_a_nat,A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( member5724188588386418708_a_nat @ C @ ( minus_8630901905721465484_a_nat @ A @ B ) )
=> ~ ( ( member5724188588386418708_a_nat @ C @ A )
=> ( member5724188588386418708_a_nat @ C @ B ) ) ) ).
% DiffE
thf(fact_758_DiffD1,axiom,
! [C: produc357393685978478089rm_a_b,A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ C @ ( minus_5192120951422937424rm_a_b @ A @ B ) )
=> ( member5869715511025134514rm_a_b @ C @ A ) ) ).
% DiffD1
thf(fact_759_DiffD1,axiom,
! [C: product_prod_a_nat,A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( member5724188588386418708_a_nat @ C @ ( minus_8630901905721465484_a_nat @ A @ B ) )
=> ( member5724188588386418708_a_nat @ C @ A ) ) ).
% DiffD1
thf(fact_760_DiffD2,axiom,
! [C: produc357393685978478089rm_a_b,A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ C @ ( minus_5192120951422937424rm_a_b @ A @ B ) )
=> ~ ( member5869715511025134514rm_a_b @ C @ B ) ) ).
% DiffD2
thf(fact_761_DiffD2,axiom,
! [C: product_prod_a_nat,A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( member5724188588386418708_a_nat @ C @ ( minus_8630901905721465484_a_nat @ A @ B ) )
=> ~ ( member5724188588386418708_a_nat @ C @ B ) ) ).
% DiffD2
thf(fact_762_double__diff,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 )
=> ( ( minus_8630901905721465484_a_nat @ B @ ( minus_8630901905721465484_a_nat @ C2 @ A ) )
= A ) ) ) ).
% double_diff
thf(fact_763_double__diff,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 )
=> ( ( minus_5192120951422937424rm_a_b @ B @ ( minus_5192120951422937424rm_a_b @ C2 @ A ) )
= A ) ) ) ).
% double_diff
thf(fact_764_Diff__subset,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ ( minus_8630901905721465484_a_nat @ A @ B ) @ A ) ).
% Diff_subset
thf(fact_765_Diff__subset,axiom,
! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ ( minus_5192120951422937424rm_a_b @ A @ B ) @ A ) ).
% Diff_subset
thf(fact_766_Diff__mono,axiom,
! [A: set_Pr4934435412358123699_a_nat,C2: set_Pr4934435412358123699_a_nat,D: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A @ C2 )
=> ( ( ord_le8666007276011122963_a_nat @ D @ B )
=> ( ord_le8666007276011122963_a_nat @ ( minus_8630901905721465484_a_nat @ A @ B ) @ ( minus_8630901905721465484_a_nat @ C2 @ D ) ) ) ) ).
% Diff_mono
thf(fact_767_Diff__mono,axiom,
! [A: set_Pr4386577575007340137rm_a_b,C2: set_Pr4386577575007340137rm_a_b,D: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A @ C2 )
=> ( ( ord_le118470702582115849rm_a_b @ D @ B )
=> ( ord_le118470702582115849rm_a_b @ ( minus_5192120951422937424rm_a_b @ A @ B ) @ ( minus_5192120951422937424rm_a_b @ C2 @ D ) ) ) ) ).
% Diff_mono
thf(fact_768_diff__shunt__var,axiom,
! [X3: set_Pr4934435412358123699_a_nat,Y3: set_Pr4934435412358123699_a_nat] :
( ( ( minus_8630901905721465484_a_nat @ X3 @ Y3 )
= bot_bo9049108969261143879_a_nat )
= ( ord_le8666007276011122963_a_nat @ X3 @ Y3 ) ) ).
% diff_shunt_var
thf(fact_769_diff__shunt__var,axiom,
! [X3: set_Pr4386577575007340137rm_a_b,Y3: set_Pr4386577575007340137rm_a_b] :
( ( ( minus_5192120951422937424rm_a_b @ X3 @ Y3 )
= bot_bo197521221353338581rm_a_b )
= ( ord_le118470702582115849rm_a_b @ X3 @ Y3 ) ) ).
% diff_shunt_var
thf(fact_770_Diff__insert__absorb,axiom,
! [X3: produc357393685978478089rm_a_b,A: set_Pr4386577575007340137rm_a_b] :
( ~ ( member5869715511025134514rm_a_b @ X3 @ A )
=> ( ( minus_5192120951422937424rm_a_b @ ( insert7009541432154983385rm_a_b @ X3 @ A ) @ ( insert7009541432154983385rm_a_b @ X3 @ bot_bo197521221353338581rm_a_b ) )
= A ) ) ).
% Diff_insert_absorb
thf(fact_771_Diff__insert__absorb,axiom,
! [X3: product_prod_a_nat,A: set_Pr4934435412358123699_a_nat] :
( ~ ( member5724188588386418708_a_nat @ X3 @ A )
=> ( ( minus_8630901905721465484_a_nat @ ( insert8054603423593749677_a_nat @ X3 @ A ) @ ( insert8054603423593749677_a_nat @ X3 @ bot_bo9049108969261143879_a_nat ) )
= A ) ) ).
% Diff_insert_absorb
thf(fact_772_insert__Diff,axiom,
! [A2: produc357393685978478089rm_a_b,A: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ A2 @ A )
=> ( ( insert7009541432154983385rm_a_b @ A2 @ ( minus_5192120951422937424rm_a_b @ A @ ( insert7009541432154983385rm_a_b @ A2 @ bot_bo197521221353338581rm_a_b ) ) )
= A ) ) ).
% insert_Diff
thf(fact_773_insert__Diff,axiom,
! [A2: product_prod_a_nat,A: set_Pr4934435412358123699_a_nat] :
( ( member5724188588386418708_a_nat @ A2 @ A )
=> ( ( insert8054603423593749677_a_nat @ A2 @ ( minus_8630901905721465484_a_nat @ A @ ( insert8054603423593749677_a_nat @ A2 @ bot_bo9049108969261143879_a_nat ) ) )
= A ) ) ).
% insert_Diff
thf(fact_774_subset__Diff__insert,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat,X3: product_prod_a_nat,C2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A @ ( minus_8630901905721465484_a_nat @ B @ ( insert8054603423593749677_a_nat @ X3 @ C2 ) ) )
= ( ( ord_le8666007276011122963_a_nat @ A @ ( minus_8630901905721465484_a_nat @ B @ C2 ) )
& ~ ( member5724188588386418708_a_nat @ X3 @ A ) ) ) ).
% subset_Diff_insert
thf(fact_775_subset__Diff__insert,axiom,
! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b,X3: produc357393685978478089rm_a_b,C2: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A @ ( minus_5192120951422937424rm_a_b @ B @ ( insert7009541432154983385rm_a_b @ X3 @ C2 ) ) )
= ( ( ord_le118470702582115849rm_a_b @ A @ ( minus_5192120951422937424rm_a_b @ B @ C2 ) )
& ~ ( member5869715511025134514rm_a_b @ X3 @ A ) ) ) ).
% subset_Diff_insert
thf(fact_776_Diff__partition,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A @ B )
=> ( ( sup_su459911885395995103_a_nat @ A @ ( minus_8630901905721465484_a_nat @ B @ A ) )
= B ) ) ).
% Diff_partition
thf(fact_777_Diff__partition,axiom,
! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A @ B )
=> ( ( sup_su6776935440552674877rm_a_b @ A @ ( minus_5192120951422937424rm_a_b @ B @ A ) )
= B ) ) ).
% Diff_partition
thf(fact_778_Diff__subset__conv,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat,C2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ ( minus_8630901905721465484_a_nat @ A @ B ) @ C2 )
= ( ord_le8666007276011122963_a_nat @ A @ ( sup_su459911885395995103_a_nat @ B @ C2 ) ) ) ).
% Diff_subset_conv
thf(fact_779_Diff__subset__conv,axiom,
! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b,C2: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ ( minus_5192120951422937424rm_a_b @ A @ B ) @ C2 )
= ( ord_le118470702582115849rm_a_b @ A @ ( sup_su6776935440552674877rm_a_b @ B @ C2 ) ) ) ).
% Diff_subset_conv
thf(fact_780_is__singletonI_H,axiom,
! [A: set_Pr4386577575007340137rm_a_b] :
( ( A != bot_bo197521221353338581rm_a_b )
=> ( ! [X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X @ A )
=> ( ( member5869715511025134514rm_a_b @ Y @ A )
=> ( X = Y ) ) )
=> ( is_sin332545968047684349rm_a_b @ A ) ) ) ).
% is_singletonI'
thf(fact_781_is__singletonI_H,axiom,
! [A: set_Pr4934435412358123699_a_nat] :
( ( A != bot_bo9049108969261143879_a_nat )
=> ( ! [X: product_prod_a_nat,Y: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ X @ A )
=> ( ( member5724188588386418708_a_nat @ Y @ A )
=> ( X = Y ) ) )
=> ( is_sin7638514652509285001_a_nat @ A ) ) ) ).
% is_singletonI'
thf(fact_782_Diff__single__insert,axiom,
! [A: set_Pr4934435412358123699_a_nat,X3: product_prod_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ ( minus_8630901905721465484_a_nat @ A @ ( insert8054603423593749677_a_nat @ X3 @ bot_bo9049108969261143879_a_nat ) ) @ B )
=> ( ord_le8666007276011122963_a_nat @ A @ ( insert8054603423593749677_a_nat @ X3 @ B ) ) ) ).
% Diff_single_insert
thf(fact_783_Diff__single__insert,axiom,
! [A: set_Pr4386577575007340137rm_a_b,X3: produc357393685978478089rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ ( minus_5192120951422937424rm_a_b @ A @ ( insert7009541432154983385rm_a_b @ X3 @ bot_bo197521221353338581rm_a_b ) ) @ B )
=> ( ord_le118470702582115849rm_a_b @ A @ ( insert7009541432154983385rm_a_b @ X3 @ B ) ) ) ).
% Diff_single_insert
thf(fact_784_subset__insert__iff,axiom,
! [A: set_Pr4934435412358123699_a_nat,X3: product_prod_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A @ ( insert8054603423593749677_a_nat @ X3 @ B ) )
= ( ( ( member5724188588386418708_a_nat @ X3 @ A )
=> ( ord_le8666007276011122963_a_nat @ ( minus_8630901905721465484_a_nat @ A @ ( insert8054603423593749677_a_nat @ X3 @ bot_bo9049108969261143879_a_nat ) ) @ B ) )
& ( ~ ( member5724188588386418708_a_nat @ X3 @ A )
=> ( ord_le8666007276011122963_a_nat @ A @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_785_subset__insert__iff,axiom,
! [A: set_Pr4386577575007340137rm_a_b,X3: produc357393685978478089rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A @ ( insert7009541432154983385rm_a_b @ X3 @ B ) )
= ( ( ( member5869715511025134514rm_a_b @ X3 @ A )
=> ( ord_le118470702582115849rm_a_b @ ( minus_5192120951422937424rm_a_b @ A @ ( insert7009541432154983385rm_a_b @ X3 @ bot_bo197521221353338581rm_a_b ) ) @ B ) )
& ( ~ ( member5869715511025134514rm_a_b @ X3 @ A )
=> ( ord_le118470702582115849rm_a_b @ A @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_786_rtrancl__diff__decomp,axiom,
! [X3: term_a_b,Y3: term_a_b,A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ ( minus_5192120951422937424rm_a_b @ ( transi7742714808557438673rm_a_b @ A ) @ ( transi7742714808557438673rm_a_b @ B ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ A ) @ ( relcom370159955682700863rm_a_b @ ( minus_5192120951422937424rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ A ) ) ) ) ) ).
% rtrancl_diff_decomp
thf(fact_787_max__ext__compat,axiom,
! [R4: set_Pr4386577575007340137rm_a_b,S4: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ R4 @ S4 ) @ R4 )
=> ( ord_le2663425029446080117rm_a_b @ ( relcom3320975603336820193rm_a_b @ ( max_ext_term_a_b @ R4 ) @ ( sup_su1246761269818691241rm_a_b @ ( max_ext_term_a_b @ S4 ) @ ( insert807748097412076101rm_a_b @ ( produc8517751122659631533rm_a_b @ bot_bot_set_term_a_b @ bot_bot_set_term_a_b ) @ bot_bo6544111711586975553rm_a_b ) ) ) @ ( max_ext_term_a_b @ R4 ) ) ) ).
% max_ext_compat
thf(fact_788_some__NF,axiom,
! [R: set_Pr2972776593051762503rm_a_b,X3: produc357393685978478089rm_a_b] :
( ( abstra2398554102055911763rm_a_b @ R @ ( insert7009541432154983385rm_a_b @ X3 @ bot_bo197521221353338581rm_a_b ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X3 @ ( abstra8203652949708803681rm_a_b @ R @ X3 ) ) @ ( transi2615809358984392588rm_a_b @ R ) )
& ( member5869715511025134514rm_a_b @ ( abstra8203652949708803681rm_a_b @ R @ X3 ) @ ( abstra1902783451776770197rm_a_b @ R ) ) ) ) ).
% some_NF
thf(fact_789_some__NF,axiom,
! [R: set_Pr1811044260758604347_a_nat,X3: product_prod_a_nat] :
( ( abstra8653715922312955827_a_nat @ R @ ( insert8054603423593749677_a_nat @ X3 @ bot_bo9049108969261143879_a_nat ) )
=> ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ X3 @ ( abstra5913684503434863653_a_nat @ R @ X3 ) ) @ ( transi2726145917338391738_a_nat @ R ) )
& ( member5724188588386418708_a_nat @ ( abstra5913684503434863653_a_nat @ R @ X3 ) @ ( abstra2685462378305987313_a_nat @ R ) ) ) ) ).
% some_NF
thf(fact_790_some__NF,axiom,
! [R: set_Pr4386577575007340137rm_a_b,X3: term_a_b] :
( ( abstra4720023341729745482rm_a_b @ R @ ( insert_term_a_b @ X3 @ bot_bot_set_term_a_b ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ ( abstra2589370493620291516rm_a_b @ R @ X3 ) ) @ ( transi7742714808557438673rm_a_b @ R ) )
& ( member_term_a_b @ ( abstra2589370493620291516rm_a_b @ R @ X3 ) @ ( abstract_NF_term_a_b @ R ) ) ) ) ).
% some_NF
thf(fact_791_Field__insert,axiom,
! [A2: term_a_b,B2: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( field_term_a_b @ ( insert7009541432154983385rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ R ) )
= ( sup_sup_set_term_a_b @ ( insert_term_a_b @ A2 @ ( insert_term_a_b @ B2 @ bot_bot_set_term_a_b ) ) @ ( field_term_a_b @ R ) ) ) ).
% Field_insert
thf(fact_792_member__remove,axiom,
! [X3: produc357393685978478089rm_a_b,Y3: produc357393685978478089rm_a_b,A: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ X3 @ ( remove8523906646656265582rm_a_b @ Y3 @ A ) )
= ( ( member5869715511025134514rm_a_b @ X3 @ A )
& ( X3 != Y3 ) ) ) ).
% member_remove
thf(fact_793_member__remove,axiom,
! [X3: product_prod_a_nat,Y3: product_prod_a_nat,A: set_Pr4934435412358123699_a_nat] :
( ( member5724188588386418708_a_nat @ X3 @ ( remove5305571144230967640_a_nat @ Y3 @ A ) )
= ( ( member5724188588386418708_a_nat @ X3 @ A )
& ( X3 != Y3 ) ) ) ).
% member_remove
thf(fact_794_SN__on__subset1,axiom,
! [R: set_Pr4386577575007340137rm_a_b,A: set_term_a_b,S: set_Pr4386577575007340137rm_a_b] :
( ( abstra4720023341729745482rm_a_b @ R @ A )
=> ( ( ord_le118470702582115849rm_a_b @ S @ R )
=> ( abstra4720023341729745482rm_a_b @ S @ A ) ) ) ).
% SN_on_subset1
thf(fact_795_FieldI1,axiom,
! [I2: produc357393685978478089rm_a_b,J2: produc357393685978478089rm_a_b,R4: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ I2 @ J2 ) @ R4 )
=> ( member5869715511025134514rm_a_b @ I2 @ ( field_6884932134483023318rm_a_b @ R4 ) ) ) ).
% FieldI1
thf(fact_796_FieldI1,axiom,
! [I2: product_prod_a_nat,J2: product_prod_a_nat,R4: set_Pr1811044260758604347_a_nat] :
( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ I2 @ J2 ) @ R4 )
=> ( member5724188588386418708_a_nat @ I2 @ ( field_8954927560578634480_a_nat @ R4 ) ) ) ).
% FieldI1
thf(fact_797_FieldI1,axiom,
! [I2: term_a_b,J2: term_a_b,R4: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ I2 @ J2 ) @ R4 )
=> ( member_term_a_b @ I2 @ ( field_term_a_b @ R4 ) ) ) ).
% FieldI1
thf(fact_798_FieldI2,axiom,
! [I2: produc357393685978478089rm_a_b,J2: produc357393685978478089rm_a_b,R4: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ I2 @ J2 ) @ R4 )
=> ( member5869715511025134514rm_a_b @ J2 @ ( field_6884932134483023318rm_a_b @ R4 ) ) ) ).
% FieldI2
thf(fact_799_FieldI2,axiom,
! [I2: product_prod_a_nat,J2: product_prod_a_nat,R4: set_Pr1811044260758604347_a_nat] :
( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ I2 @ J2 ) @ R4 )
=> ( member5724188588386418708_a_nat @ J2 @ ( field_8954927560578634480_a_nat @ R4 ) ) ) ).
% FieldI2
thf(fact_800_FieldI2,axiom,
! [I2: term_a_b,J2: term_a_b,R4: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ I2 @ J2 ) @ R4 )
=> ( member_term_a_b @ J2 @ ( field_term_a_b @ R4 ) ) ) ).
% FieldI2
thf(fact_801_SN__on__induct,axiom,
! [R4: set_Pr2972776593051762503rm_a_b,A: set_Pr4386577575007340137rm_a_b,S: produc357393685978478089rm_a_b,P: produc357393685978478089rm_a_b > $o] :
( ( abstra2398554102055911763rm_a_b @ R4 @ A )
=> ( ( member5869715511025134514rm_a_b @ S @ A )
=> ( ! [T3: produc357393685978478089rm_a_b] :
( ! [U4: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ T3 @ U4 ) @ R4 )
=> ( P @ U4 ) )
=> ( P @ T3 ) )
=> ( P @ S ) ) ) ) ).
% SN_on_induct
thf(fact_802_SN__on__induct,axiom,
! [R4: set_Pr1811044260758604347_a_nat,A: set_Pr4934435412358123699_a_nat,S: product_prod_a_nat,P: product_prod_a_nat > $o] :
( ( abstra8653715922312955827_a_nat @ R4 @ A )
=> ( ( member5724188588386418708_a_nat @ S @ A )
=> ( ! [T3: product_prod_a_nat] :
( ! [U4: product_prod_a_nat] :
( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ T3 @ U4 ) @ R4 )
=> ( P @ U4 ) )
=> ( P @ T3 ) )
=> ( P @ S ) ) ) ) ).
% SN_on_induct
thf(fact_803_SN__on__induct,axiom,
! [R4: set_Pr4386577575007340137rm_a_b,A: set_term_a_b,S: term_a_b,P: term_a_b > $o] :
( ( abstra4720023341729745482rm_a_b @ R4 @ A )
=> ( ( member_term_a_b @ S @ A )
=> ( ! [T3: term_a_b] :
( ! [U4: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T3 @ U4 ) @ R4 )
=> ( P @ U4 ) )
=> ( P @ T3 ) )
=> ( P @ S ) ) ) ) ).
% SN_on_induct
thf(fact_804_SN__on__irrefl,axiom,
! [R: set_Pr4386577575007340137rm_a_b,A: set_term_a_b] :
( ( abstra4720023341729745482rm_a_b @ R @ A )
=> ! [X6: term_a_b] :
( ( member_term_a_b @ X6 @ A )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X6 @ X6 ) @ R ) ) ) ).
% SN_on_irrefl
thf(fact_805_SN__on__subset2,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat,R: set_Pr1811044260758604347_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A @ B )
=> ( ( abstra8653715922312955827_a_nat @ R @ B )
=> ( abstra8653715922312955827_a_nat @ R @ A ) ) ) ).
% SN_on_subset2
thf(fact_806_SN__on__subset2,axiom,
! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b,R: set_Pr2972776593051762503rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A @ B )
=> ( ( abstra2398554102055911763rm_a_b @ R @ B )
=> ( abstra2398554102055911763rm_a_b @ R @ A ) ) ) ).
% SN_on_subset2
thf(fact_807_refl__not__SN,axiom,
! [X3: term_a_b,R4: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ X3 ) @ R4 )
=> ~ ( abstra4720023341729745482rm_a_b @ R4 @ top_top_set_term_a_b ) ) ).
% refl_not_SN
thf(fact_808_SN__imp__minimal,axiom,
! [A: set_Pr2972776593051762503rm_a_b] :
( ( abstra2398554102055911763rm_a_b @ A @ top_to1314267278846557113rm_a_b )
=> ! [Q2: set_Pr4386577575007340137rm_a_b] :
( ? [X6: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ X6 @ Q2 )
=> ? [X: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X @ Q2 )
& ! [Y6: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y6 ) @ A )
=> ~ ( member5869715511025134514rm_a_b @ Y6 @ Q2 ) ) ) ) ) ).
% SN_imp_minimal
thf(fact_809_SN__imp__minimal,axiom,
! [A: set_Pr1811044260758604347_a_nat] :
( ( abstra8653715922312955827_a_nat @ A @ top_to3353692345378799459_a_nat )
=> ! [Q2: set_Pr4934435412358123699_a_nat] :
( ? [X6: product_prod_a_nat] : ( member5724188588386418708_a_nat @ X6 @ Q2 )
=> ? [X: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ X @ Q2 )
& ! [Y6: product_prod_a_nat] :
( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ X @ Y6 ) @ A )
=> ~ ( member5724188588386418708_a_nat @ Y6 @ Q2 ) ) ) ) ) ).
% SN_imp_minimal
thf(fact_810_SN__imp__minimal,axiom,
! [A: set_Pr4386577575007340137rm_a_b] :
( ( abstra4720023341729745482rm_a_b @ A @ top_top_set_term_a_b )
=> ! [Q2: set_term_a_b] :
( ? [X6: term_a_b] : ( member_term_a_b @ X6 @ Q2 )
=> ? [X: term_a_b] :
( ( member_term_a_b @ X @ Q2 )
& ! [Y6: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y6 ) @ A )
=> ~ ( member_term_a_b @ Y6 @ Q2 ) ) ) ) ) ).
% SN_imp_minimal
thf(fact_811_SN__induct__rule,axiom,
! [R: set_Pr4386577575007340137rm_a_b,P: term_a_b > $o,A2: term_a_b] :
( ( abstra4720023341729745482rm_a_b @ R @ top_top_set_term_a_b )
=> ( ! [A5: term_a_b] :
( ! [B7: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A5 @ B7 ) @ R )
=> ( P @ B7 ) )
=> ( P @ A5 ) )
=> ( P @ A2 ) ) ) ).
% SN_induct_rule
thf(fact_812_SN__subset,axiom,
! [R4: set_Pr4386577575007340137rm_a_b,R6: set_Pr4386577575007340137rm_a_b] :
( ( abstra4720023341729745482rm_a_b @ R4 @ top_top_set_term_a_b )
=> ( ( ord_le118470702582115849rm_a_b @ R6 @ R4 )
=> ( abstra4720023341729745482rm_a_b @ R6 @ top_top_set_term_a_b ) ) ) ).
% SN_subset
thf(fact_813_mono__Field,axiom,
! [R: set_Pr1811044260758604347_a_nat,S: set_Pr1811044260758604347_a_nat] :
( ( ord_le1018141841545957851_a_nat @ R @ S )
=> ( ord_le8666007276011122963_a_nat @ ( field_8954927560578634480_a_nat @ R ) @ ( field_8954927560578634480_a_nat @ S ) ) ) ).
% mono_Field
thf(fact_814_mono__Field,axiom,
! [R: set_Pr2972776593051762503rm_a_b,S: set_Pr2972776593051762503rm_a_b] :
( ( ord_le2593958535671617255rm_a_b @ R @ S )
=> ( ord_le118470702582115849rm_a_b @ ( field_6884932134483023318rm_a_b @ R ) @ ( field_6884932134483023318rm_a_b @ S ) ) ) ).
% mono_Field
thf(fact_815_mono__Field,axiom,
! [R: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ R @ S )
=> ( ord_le2705286416250468010rm_a_b @ ( field_term_a_b @ R ) @ ( field_term_a_b @ S ) ) ) ).
% mono_Field
thf(fact_816_SN__on__all__reducts__SN__on__conv,axiom,
! [R: set_Pr4386577575007340137rm_a_b,A2: term_a_b] :
( ( abstra4720023341729745482rm_a_b @ R @ ( insert_term_a_b @ A2 @ bot_bot_set_term_a_b ) )
= ( ! [B4: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B4 ) @ R )
=> ( abstra4720023341729745482rm_a_b @ R @ ( insert_term_a_b @ B4 @ bot_bot_set_term_a_b ) ) ) ) ) ).
% SN_on_all_reducts_SN_on_conv
thf(fact_817_lower__set__imp__not__SN__on,axiom,
! [S: produc357393685978478089rm_a_b,X5: set_Pr4386577575007340137rm_a_b,R4: set_Pr2972776593051762503rm_a_b] :
( ( member5869715511025134514rm_a_b @ S @ X5 )
=> ( ! [X: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X @ X5 )
=> ? [Xa2: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ Xa2 @ X5 )
& ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Xa2 ) @ R4 ) ) )
=> ~ ( abstra2398554102055911763rm_a_b @ R4 @ ( insert7009541432154983385rm_a_b @ S @ bot_bo197521221353338581rm_a_b ) ) ) ) ).
% lower_set_imp_not_SN_on
thf(fact_818_lower__set__imp__not__SN__on,axiom,
! [S: product_prod_a_nat,X5: set_Pr4934435412358123699_a_nat,R4: set_Pr1811044260758604347_a_nat] :
( ( member5724188588386418708_a_nat @ S @ X5 )
=> ( ! [X: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ X @ X5 )
=> ? [Xa2: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ Xa2 @ X5 )
& ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ X @ Xa2 ) @ R4 ) ) )
=> ~ ( abstra8653715922312955827_a_nat @ R4 @ ( insert8054603423593749677_a_nat @ S @ bot_bo9049108969261143879_a_nat ) ) ) ) ).
% lower_set_imp_not_SN_on
thf(fact_819_lower__set__imp__not__SN__on,axiom,
! [S: term_a_b,X5: set_term_a_b,R4: set_Pr4386577575007340137rm_a_b] :
( ( member_term_a_b @ S @ X5 )
=> ( ! [X: term_a_b] :
( ( member_term_a_b @ X @ X5 )
=> ? [Xa2: term_a_b] :
( ( member_term_a_b @ Xa2 @ X5 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Xa2 ) @ R4 ) ) )
=> ~ ( abstra4720023341729745482rm_a_b @ R4 @ ( insert_term_a_b @ S @ bot_bot_set_term_a_b ) ) ) ) ).
% lower_set_imp_not_SN_on
thf(fact_820_SN__on__induct__acc__style,axiom,
! [R4: set_Pr4386577575007340137rm_a_b,A2: term_a_b,P: term_a_b > $o] :
( ( abstra4720023341729745482rm_a_b @ R4 @ ( insert_term_a_b @ A2 @ bot_bot_set_term_a_b ) )
=> ( ! [X: term_a_b] :
( ( abstra4720023341729745482rm_a_b @ R4 @ ( insert_term_a_b @ X @ bot_bot_set_term_a_b ) )
=> ( ! [Y6: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y6 ) @ R4 )
=> ( P @ Y6 ) )
=> ( P @ X ) ) )
=> ( P @ A2 ) ) ) ).
% SN_on_induct_acc_style
thf(fact_821_step__preserves__SN__on,axiom,
! [A2: term_a_b,B2: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ R )
=> ( ( abstra4720023341729745482rm_a_b @ R @ ( insert_term_a_b @ A2 @ bot_bot_set_term_a_b ) )
=> ( abstra4720023341729745482rm_a_b @ R @ ( insert_term_a_b @ B2 @ bot_bot_set_term_a_b ) ) ) ) ).
% step_preserves_SN_on
thf(fact_822_SN__on__imp__on__minimal,axiom,
! [R: set_Pr2972776593051762503rm_a_b,X3: produc357393685978478089rm_a_b] :
( ( abstra2398554102055911763rm_a_b @ R @ ( insert7009541432154983385rm_a_b @ X3 @ bot_bo197521221353338581rm_a_b ) )
=> ! [Q2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ X3 @ Q2 )
=> ? [X: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X @ Q2 )
& ! [Y6: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y6 ) @ R )
=> ~ ( member5869715511025134514rm_a_b @ Y6 @ Q2 ) ) ) ) ) ).
% SN_on_imp_on_minimal
thf(fact_823_SN__on__imp__on__minimal,axiom,
! [R: set_Pr1811044260758604347_a_nat,X3: product_prod_a_nat] :
( ( abstra8653715922312955827_a_nat @ R @ ( insert8054603423593749677_a_nat @ X3 @ bot_bo9049108969261143879_a_nat ) )
=> ! [Q2: set_Pr4934435412358123699_a_nat] :
( ( member5724188588386418708_a_nat @ X3 @ Q2 )
=> ? [X: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ X @ Q2 )
& ! [Y6: product_prod_a_nat] :
( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ X @ Y6 ) @ R )
=> ~ ( member5724188588386418708_a_nat @ Y6 @ Q2 ) ) ) ) ) ).
% SN_on_imp_on_minimal
thf(fact_824_SN__on__imp__on__minimal,axiom,
! [R: set_Pr4386577575007340137rm_a_b,X3: term_a_b] :
( ( abstra4720023341729745482rm_a_b @ R @ ( insert_term_a_b @ X3 @ bot_bot_set_term_a_b ) )
=> ! [Q2: set_term_a_b] :
( ( member_term_a_b @ X3 @ Q2 )
=> ? [X: term_a_b] :
( ( member_term_a_b @ X @ Q2 )
& ! [Y6: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y6 ) @ R )
=> ~ ( member_term_a_b @ Y6 @ Q2 ) ) ) ) ) ).
% SN_on_imp_on_minimal
thf(fact_825_step__reflects__SN__on,axiom,
! [A2: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ! [B5: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B5 ) @ R )
=> ( abstra4720023341729745482rm_a_b @ R @ ( insert_term_a_b @ B5 @ bot_bot_set_term_a_b ) ) )
=> ( abstra4720023341729745482rm_a_b @ R @ ( insert_term_a_b @ A2 @ bot_bot_set_term_a_b ) ) ) ).
% step_reflects_SN_on
thf(fact_826_SN__on__induct_H,axiom,
! [R4: set_Pr4386577575007340137rm_a_b,S: term_a_b,P: term_a_b > $o] :
( ( abstra4720023341729745482rm_a_b @ R4 @ ( insert_term_a_b @ S @ bot_bot_set_term_a_b ) )
=> ( ! [T3: term_a_b] :
( ! [U4: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T3 @ U4 ) @ R4 )
=> ( P @ U4 ) )
=> ( P @ T3 ) )
=> ( P @ S ) ) ) ).
% SN_on_induct'
thf(fact_827_SN__on__O__push,axiom,
! [S4: set_Pr4386577575007340137rm_a_b,R4: set_Pr4386577575007340137rm_a_b,X5: set_term_a_b] :
( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ S4 @ R4 ) @ ( relcom370159955682700863rm_a_b @ R4 @ ( transi7742714808557438673rm_a_b @ S4 ) ) )
=> ( ( abstra4720023341729745482rm_a_b @ R4 @ X5 )
=> ( abstra4720023341729745482rm_a_b @ ( relcom370159955682700863rm_a_b @ R4 @ ( transi7742714808557438673rm_a_b @ S4 ) ) @ X5 ) ) ) ).
% SN_on_O_push
thf(fact_828_NF__imp__SN__on,axiom,
! [A2: produc357393685978478089rm_a_b,R4: set_Pr2972776593051762503rm_a_b] :
( ( member5869715511025134514rm_a_b @ A2 @ ( abstra1902783451776770197rm_a_b @ R4 ) )
=> ( abstra2398554102055911763rm_a_b @ R4 @ ( insert7009541432154983385rm_a_b @ A2 @ bot_bo197521221353338581rm_a_b ) ) ) ).
% NF_imp_SN_on
thf(fact_829_NF__imp__SN__on,axiom,
! [A2: product_prod_a_nat,R4: set_Pr1811044260758604347_a_nat] :
( ( member5724188588386418708_a_nat @ A2 @ ( abstra2685462378305987313_a_nat @ R4 ) )
=> ( abstra8653715922312955827_a_nat @ R4 @ ( insert8054603423593749677_a_nat @ A2 @ bot_bo9049108969261143879_a_nat ) ) ) ).
% NF_imp_SN_on
thf(fact_830_steps__preserve__SN__on,axiom,
! [A2: term_a_b,B2: term_a_b,A: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( transi7742714808557438673rm_a_b @ A ) )
=> ( ( abstra4720023341729745482rm_a_b @ A @ ( insert_term_a_b @ A2 @ bot_bot_set_term_a_b ) )
=> ( abstra4720023341729745482rm_a_b @ A @ ( insert_term_a_b @ B2 @ bot_bot_set_term_a_b ) ) ) ) ).
% steps_preserve_SN_on
thf(fact_831_steps__reflect__SN__on,axiom,
! [R: set_Pr4386577575007340137rm_a_b,B2: term_a_b,A2: term_a_b] :
( ~ ( abstra4720023341729745482rm_a_b @ R @ ( insert_term_a_b @ B2 @ bot_bot_set_term_a_b ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ~ ( abstra4720023341729745482rm_a_b @ R @ ( insert_term_a_b @ A2 @ bot_bot_set_term_a_b ) ) ) ) ).
% steps_reflect_SN_on
thf(fact_832_compatible__SN,axiom,
! [S4: set_Pr4386577575007340137rm_a_b,NS: set_Pr4386577575007340137rm_a_b] :
( ( abstra4720023341729745482rm_a_b @ S4 @ top_top_set_term_a_b )
=> ( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ NS @ S4 ) @ S4 )
=> ( abstra4720023341729745482rm_a_b @ ( relcom370159955682700863rm_a_b @ S4 @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ S4 ) @ ( transi7742714808557438673rm_a_b @ NS ) ) ) @ top_top_set_term_a_b ) ) ) ).
% compatible_SN
thf(fact_833_SN__on__imp__normalizability,axiom,
! [R: set_Pr4386577575007340137rm_a_b,A2: term_a_b] :
( ( abstra4720023341729745482rm_a_b @ R @ ( insert_term_a_b @ A2 @ bot_bot_set_term_a_b ) )
=> ? [B5: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B5 ) @ ( abstra2972651231392952684rm_a_b @ R ) ) ) ).
% SN_on_imp_normalizability
thf(fact_834_SN__reaches__NF,axiom,
! [R: set_Pr2972776593051762503rm_a_b,X3: produc357393685978478089rm_a_b] :
( ( abstra2398554102055911763rm_a_b @ R @ ( insert7009541432154983385rm_a_b @ X3 @ bot_bo197521221353338581rm_a_b ) )
=> ? [Y: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X3 @ Y ) @ ( transi2615809358984392588rm_a_b @ R ) )
& ( member5869715511025134514rm_a_b @ Y @ ( abstra1902783451776770197rm_a_b @ R ) ) ) ) ).
% SN_reaches_NF
thf(fact_835_SN__reaches__NF,axiom,
! [R: set_Pr1811044260758604347_a_nat,X3: product_prod_a_nat] :
( ( abstra8653715922312955827_a_nat @ R @ ( insert8054603423593749677_a_nat @ X3 @ bot_bo9049108969261143879_a_nat ) )
=> ? [Y: product_prod_a_nat] :
( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ X3 @ Y ) @ ( transi2726145917338391738_a_nat @ R ) )
& ( member5724188588386418708_a_nat @ Y @ ( abstra2685462378305987313_a_nat @ R ) ) ) ) ).
% SN_reaches_NF
thf(fact_836_SN__reaches__NF,axiom,
! [R: set_Pr4386577575007340137rm_a_b,X3: term_a_b] :
( ( abstra4720023341729745482rm_a_b @ R @ ( insert_term_a_b @ X3 @ bot_bot_set_term_a_b ) )
=> ? [Y: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y ) @ ( transi7742714808557438673rm_a_b @ R ) )
& ( member_term_a_b @ Y @ ( abstract_NF_term_a_b @ R ) ) ) ) ).
% SN_reaches_NF
thf(fact_837_compatible__SN_H,axiom,
! [NS: set_Pr4386577575007340137rm_a_b,S4: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ NS @ S4 ) @ S4 )
=> ( ( abstra4720023341729745482rm_a_b @ S4 @ top_top_set_term_a_b )
=> ( abstra4720023341729745482rm_a_b @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ NS @ S4 ) ) @ ( relcom370159955682700863rm_a_b @ S4 @ ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ NS @ S4 ) ) ) ) @ top_top_set_term_a_b ) ) ) ).
% compatible_SN'
thf(fact_838_qc__SN__relto__iff,axiom,
! [R: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ R @ S ) @ ( relcom370159955682700863rm_a_b @ S @ ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ S @ R ) ) ) )
=> ( ( abstra4720023341729745482rm_a_b @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ R ) @ ( relcom370159955682700863rm_a_b @ S @ ( transi7742714808557438673rm_a_b @ R ) ) ) @ top_top_set_term_a_b )
= ( abstra4720023341729745482rm_a_b @ S @ top_top_set_term_a_b ) ) ) ).
% qc_SN_relto_iff
thf(fact_839_the__NF,axiom,
! [A: set_Pr4386577575007340137rm_a_b,A2: term_a_b] :
( ( abstra4720023341729745482rm_a_b @ A @ top_top_set_term_a_b )
=> ( ( abstra8448919418672941150rm_a_b @ A @ top_top_set_term_a_b )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ ( abstra2242782261322123075rm_a_b @ A @ A2 ) ) @ ( abstra2972651231392952684rm_a_b @ A ) ) ) ) ).
% the_NF
thf(fact_840_the__NF__conv,axiom,
! [A: set_Pr4386577575007340137rm_a_b,A2: term_a_b,B2: term_a_b] :
( ( abstra4720023341729745482rm_a_b @ A @ top_top_set_term_a_b )
=> ( ( abstra8448919418672941150rm_a_b @ A @ top_top_set_term_a_b )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( abstra1130568886744591910rm_a_b @ A ) )
=> ( ( abstra2242782261322123075rm_a_b @ A @ A2 )
= ( abstra2242782261322123075rm_a_b @ A @ B2 ) ) ) ) ) ).
% the_NF_conv
thf(fact_841_the__NF__steps,axiom,
! [A: set_Pr4386577575007340137rm_a_b,A2: term_a_b,B2: term_a_b] :
( ( abstra4720023341729745482rm_a_b @ A @ top_top_set_term_a_b )
=> ( ( abstra8448919418672941150rm_a_b @ A @ top_top_set_term_a_b )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( transi7742714808557438673rm_a_b @ A ) )
=> ( ( abstra2242782261322123075rm_a_b @ A @ A2 )
= ( abstra2242782261322123075rm_a_b @ A @ B2 ) ) ) ) ) ).
% the_NF_steps
thf(fact_842_the__NF__step,axiom,
! [A: set_Pr4386577575007340137rm_a_b,A2: term_a_b,B2: term_a_b] :
( ( abstra4720023341729745482rm_a_b @ A @ top_top_set_term_a_b )
=> ( ( abstra8448919418672941150rm_a_b @ A @ top_top_set_term_a_b )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ A )
=> ( ( abstra2242782261322123075rm_a_b @ A @ A2 )
= ( abstra2242782261322123075rm_a_b @ A @ B2 ) ) ) ) ) ).
% the_NF_step
thf(fact_843_the__NF__NF,axiom,
! [A: set_Pr2972776593051762503rm_a_b,A2: produc357393685978478089rm_a_b] :
( ( abstra2398554102055911763rm_a_b @ A @ top_to1314267278846557113rm_a_b )
=> ( ( abstra7359694574113289279rm_a_b @ A @ top_to1314267278846557113rm_a_b )
=> ( member5869715511025134514rm_a_b @ ( abstra8972549633426078106rm_a_b @ A @ A2 ) @ ( abstra1902783451776770197rm_a_b @ A ) ) ) ) ).
% the_NF_NF
thf(fact_844_the__NF__NF,axiom,
! [A: set_Pr1811044260758604347_a_nat,A2: product_prod_a_nat] :
( ( abstra8653715922312955827_a_nat @ A @ top_to3353692345378799459_a_nat )
=> ( ( abstra7964079670522473415_a_nat @ A @ top_to3353692345378799459_a_nat )
=> ( member5724188588386418708_a_nat @ ( abstra6900011479857116204_a_nat @ A @ A2 ) @ ( abstra2685462378305987313_a_nat @ A ) ) ) ) ).
% the_NF_NF
thf(fact_845_cofinal__def,axiom,
( bNF_Ca4789820777293777539rm_a_b
= ( ^ [A6: set_term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
! [X4: term_a_b] :
( ( member_term_a_b @ X4 @ ( field_term_a_b @ R3 ) )
=> ? [Y5: term_a_b] :
( ( member_term_a_b @ Y5 @ A6 )
& ( X4 != Y5 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X4 @ Y5 ) @ R3 ) ) ) ) ) ).
% cofinal_def
thf(fact_846_acyclic__insert,axiom,
! [Y3: term_a_b,X3: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( transi5314701259734769157rm_a_b @ ( insert7009541432154983385rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ X3 ) @ R ) )
= ( ( transi5314701259734769157rm_a_b @ R )
& ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ ( transi7742714808557438673rm_a_b @ R ) ) ) ) ).
% acyclic_insert
thf(fact_847_wo__rel_Ocases__Total3,axiom,
! [R: set_Pr4386577575007340137rm_a_b,A2: term_a_b,B2: term_a_b,Phi: term_a_b > term_a_b > $o] :
( ( bNF_We251836928773027112rm_a_b @ R )
=> ( ( ord_le2705286416250468010rm_a_b @ ( insert_term_a_b @ A2 @ ( insert_term_a_b @ B2 @ bot_bot_set_term_a_b ) ) @ ( field_term_a_b @ R ) )
=> ( ( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( minus_5192120951422937424rm_a_b @ R @ id_term_a_b ) )
| ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ A2 ) @ ( minus_5192120951422937424rm_a_b @ R @ id_term_a_b ) ) )
=> ( Phi @ A2 @ B2 ) )
=> ( ( ( A2 = B2 )
=> ( Phi @ A2 @ B2 ) )
=> ( Phi @ A2 @ B2 ) ) ) ) ) ).
% wo_rel.cases_Total3
thf(fact_848_wo__rel_Ocases__Total3,axiom,
! [R: set_Pr1811044260758604347_a_nat,A2: product_prod_a_nat,B2: product_prod_a_nat,Phi: product_prod_a_nat > product_prod_a_nat > $o] :
( ( bNF_We4298404315170990993_a_nat @ R )
=> ( ( ord_le8666007276011122963_a_nat @ ( insert8054603423593749677_a_nat @ A2 @ ( insert8054603423593749677_a_nat @ B2 @ bot_bo9049108969261143879_a_nat ) ) @ ( field_8954927560578634480_a_nat @ R ) )
=> ( ( ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A2 @ B2 ) @ ( minus_1341117088723627554_a_nat @ R @ id_Pro5207055338379199009_a_nat ) )
| ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ B2 @ A2 ) @ ( minus_1341117088723627554_a_nat @ R @ id_Pro5207055338379199009_a_nat ) ) )
=> ( Phi @ A2 @ B2 ) )
=> ( ( ( A2 = B2 )
=> ( Phi @ A2 @ B2 ) )
=> ( Phi @ A2 @ B2 ) ) ) ) ) ).
% wo_rel.cases_Total3
thf(fact_849_wo__rel_Ocases__Total3,axiom,
! [R: set_Pr2972776593051762503rm_a_b,A2: produc357393685978478089rm_a_b,B2: produc357393685978478089rm_a_b,Phi: produc357393685978478089rm_a_b > produc357393685978478089rm_a_b > $o] :
( ( bNF_We1688910813892497397rm_a_b @ R )
=> ( ( ord_le118470702582115849rm_a_b @ ( insert7009541432154983385rm_a_b @ A2 @ ( insert7009541432154983385rm_a_b @ B2 @ bot_bo197521221353338581rm_a_b ) ) @ ( field_6884932134483023318rm_a_b @ R ) )
=> ( ( ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A2 @ B2 ) @ ( minus_45395518312058030rm_a_b @ R @ id_Pro3488709451141449061rm_a_b ) )
| ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B2 @ A2 ) @ ( minus_45395518312058030rm_a_b @ R @ id_Pro3488709451141449061rm_a_b ) ) )
=> ( Phi @ A2 @ B2 ) )
=> ( ( ( A2 = B2 )
=> ( Phi @ A2 @ B2 ) )
=> ( Phi @ A2 @ B2 ) ) ) ) ) ).
% wo_rel.cases_Total3
thf(fact_850_acyclic__subset,axiom,
! [S: set_Pr4386577575007340137rm_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( transi5314701259734769157rm_a_b @ S )
=> ( ( ord_le118470702582115849rm_a_b @ R @ S )
=> ( transi5314701259734769157rm_a_b @ R ) ) ) ).
% acyclic_subset
thf(fact_851_wo__rel_Owell__order__induct,axiom,
! [R: set_Pr4386577575007340137rm_a_b,P: term_a_b > $o,A2: term_a_b] :
( ( bNF_We251836928773027112rm_a_b @ R )
=> ( ! [X: term_a_b] :
( ! [Y6: term_a_b] :
( ( ( Y6 != X )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y6 @ X ) @ R ) )
=> ( P @ Y6 ) )
=> ( P @ X ) )
=> ( P @ A2 ) ) ) ).
% wo_rel.well_order_induct
thf(fact_852_wo__rel_OTOTALS,axiom,
! [R: set_Pr4386577575007340137rm_a_b] :
( ( bNF_We251836928773027112rm_a_b @ R )
=> ! [X6: term_a_b] :
( ( member_term_a_b @ X6 @ ( field_term_a_b @ R ) )
=> ! [Xa2: term_a_b] :
( ( member_term_a_b @ Xa2 @ ( field_term_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X6 @ Xa2 ) @ R )
| ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Xa2 @ X6 ) @ R ) ) ) ) ) ).
% wo_rel.TOTALS
thf(fact_853_wo__rel_Oin__notinI,axiom,
! [R: set_Pr2972776593051762503rm_a_b,J2: produc357393685978478089rm_a_b,I2: produc357393685978478089rm_a_b] :
( ( bNF_We1688910813892497397rm_a_b @ R )
=> ( ( ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ J2 @ I2 ) @ R )
| ( J2 = I2 ) )
=> ( ( member5869715511025134514rm_a_b @ I2 @ ( field_6884932134483023318rm_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ J2 @ ( field_6884932134483023318rm_a_b @ R ) )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ I2 @ J2 ) @ R ) ) ) ) ) ).
% wo_rel.in_notinI
thf(fact_854_wo__rel_Oin__notinI,axiom,
! [R: set_Pr1811044260758604347_a_nat,J2: product_prod_a_nat,I2: product_prod_a_nat] :
( ( bNF_We4298404315170990993_a_nat @ R )
=> ( ( ~ ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ J2 @ I2 ) @ R )
| ( J2 = I2 ) )
=> ( ( member5724188588386418708_a_nat @ I2 @ ( field_8954927560578634480_a_nat @ R ) )
=> ( ( member5724188588386418708_a_nat @ J2 @ ( field_8954927560578634480_a_nat @ R ) )
=> ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ I2 @ J2 ) @ R ) ) ) ) ) ).
% wo_rel.in_notinI
thf(fact_855_wo__rel_Oin__notinI,axiom,
! [R: set_Pr4386577575007340137rm_a_b,J2: term_a_b,I2: term_a_b] :
( ( bNF_We251836928773027112rm_a_b @ R )
=> ( ( ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ J2 @ I2 ) @ R )
| ( J2 = I2 ) )
=> ( ( member_term_a_b @ I2 @ ( field_term_a_b @ R ) )
=> ( ( member_term_a_b @ J2 @ ( field_term_a_b @ R ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ I2 @ J2 ) @ R ) ) ) ) ) ).
% wo_rel.in_notinI
thf(fact_856_well__order__induct__imp,axiom,
! [R: set_Pr2972776593051762503rm_a_b,P: produc357393685978478089rm_a_b > $o,A2: produc357393685978478089rm_a_b] :
( ( bNF_We1688910813892497397rm_a_b @ R )
=> ( ! [X: produc357393685978478089rm_a_b] :
( ! [Y6: produc357393685978478089rm_a_b] :
( ( ( Y6 != X )
& ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y6 @ X ) @ R ) )
=> ( ( member5869715511025134514rm_a_b @ Y6 @ ( field_6884932134483023318rm_a_b @ R ) )
=> ( P @ Y6 ) ) )
=> ( ( member5869715511025134514rm_a_b @ X @ ( field_6884932134483023318rm_a_b @ R ) )
=> ( P @ X ) ) )
=> ( ( member5869715511025134514rm_a_b @ A2 @ ( field_6884932134483023318rm_a_b @ R ) )
=> ( P @ A2 ) ) ) ) ).
% well_order_induct_imp
thf(fact_857_well__order__induct__imp,axiom,
! [R: set_Pr1811044260758604347_a_nat,P: product_prod_a_nat > $o,A2: product_prod_a_nat] :
( ( bNF_We4298404315170990993_a_nat @ R )
=> ( ! [X: product_prod_a_nat] :
( ! [Y6: product_prod_a_nat] :
( ( ( Y6 != X )
& ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ Y6 @ X ) @ R ) )
=> ( ( member5724188588386418708_a_nat @ Y6 @ ( field_8954927560578634480_a_nat @ R ) )
=> ( P @ Y6 ) ) )
=> ( ( member5724188588386418708_a_nat @ X @ ( field_8954927560578634480_a_nat @ R ) )
=> ( P @ X ) ) )
=> ( ( member5724188588386418708_a_nat @ A2 @ ( field_8954927560578634480_a_nat @ R ) )
=> ( P @ A2 ) ) ) ) ).
% well_order_induct_imp
thf(fact_858_well__order__induct__imp,axiom,
! [R: set_Pr4386577575007340137rm_a_b,P: term_a_b > $o,A2: term_a_b] :
( ( bNF_We251836928773027112rm_a_b @ R )
=> ( ! [X: term_a_b] :
( ! [Y6: term_a_b] :
( ( ( Y6 != X )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y6 @ X ) @ R ) )
=> ( ( member_term_a_b @ Y6 @ ( field_term_a_b @ R ) )
=> ( P @ Y6 ) ) )
=> ( ( member_term_a_b @ X @ ( field_term_a_b @ R ) )
=> ( P @ X ) ) )
=> ( ( member_term_a_b @ A2 @ ( field_term_a_b @ R ) )
=> ( P @ A2 ) ) ) ) ).
% well_order_induct_imp
thf(fact_859_acyclic__def,axiom,
( transi5314701259734769157rm_a_b
= ( ^ [R3: set_Pr4386577575007340137rm_a_b] :
! [X4: term_a_b] :
~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X4 @ X4 ) @ ( transi7922773638565587891rm_a_b @ R3 ) ) ) ) ).
% acyclic_def
thf(fact_860_acyclicI,axiom,
! [R: set_Pr4386577575007340137rm_a_b] :
( ! [X: term_a_b] :
~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ X ) @ ( transi7922773638565587891rm_a_b @ R ) )
=> ( transi5314701259734769157rm_a_b @ R ) ) ).
% acyclicI
thf(fact_861_wo__rel_Ocases__Total,axiom,
! [R: set_Pr4386577575007340137rm_a_b,A2: term_a_b,B2: term_a_b,Phi: term_a_b > term_a_b > $o] :
( ( bNF_We251836928773027112rm_a_b @ R )
=> ( ( ord_le2705286416250468010rm_a_b @ ( insert_term_a_b @ A2 @ ( insert_term_a_b @ B2 @ bot_bot_set_term_a_b ) ) @ ( field_term_a_b @ R ) )
=> ( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ R )
=> ( Phi @ A2 @ B2 ) )
=> ( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ A2 ) @ R )
=> ( Phi @ A2 @ B2 ) )
=> ( Phi @ A2 @ B2 ) ) ) ) ) ).
% wo_rel.cases_Total
thf(fact_862_wo__rel_Ocases__Total,axiom,
! [R: set_Pr1811044260758604347_a_nat,A2: product_prod_a_nat,B2: product_prod_a_nat,Phi: product_prod_a_nat > product_prod_a_nat > $o] :
( ( bNF_We4298404315170990993_a_nat @ R )
=> ( ( ord_le8666007276011122963_a_nat @ ( insert8054603423593749677_a_nat @ A2 @ ( insert8054603423593749677_a_nat @ B2 @ bot_bo9049108969261143879_a_nat ) ) @ ( field_8954927560578634480_a_nat @ R ) )
=> ( ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A2 @ B2 ) @ R )
=> ( Phi @ A2 @ B2 ) )
=> ( ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ B2 @ A2 ) @ R )
=> ( Phi @ A2 @ B2 ) )
=> ( Phi @ A2 @ B2 ) ) ) ) ) ).
% wo_rel.cases_Total
thf(fact_863_wo__rel_Ocases__Total,axiom,
! [R: set_Pr2972776593051762503rm_a_b,A2: produc357393685978478089rm_a_b,B2: produc357393685978478089rm_a_b,Phi: produc357393685978478089rm_a_b > produc357393685978478089rm_a_b > $o] :
( ( bNF_We1688910813892497397rm_a_b @ R )
=> ( ( ord_le118470702582115849rm_a_b @ ( insert7009541432154983385rm_a_b @ A2 @ ( insert7009541432154983385rm_a_b @ B2 @ bot_bo197521221353338581rm_a_b ) ) @ ( field_6884932134483023318rm_a_b @ R ) )
=> ( ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A2 @ B2 ) @ R )
=> ( Phi @ A2 @ B2 ) )
=> ( ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B2 @ A2 ) @ R )
=> ( Phi @ A2 @ B2 ) )
=> ( Phi @ A2 @ B2 ) ) ) ) ) ).
% wo_rel.cases_Total
thf(fact_864_wo__rel_OWell__order__isMinim__exists,axiom,
! [R: set_Pr1811044260758604347_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( bNF_We4298404315170990993_a_nat @ R )
=> ( ( ord_le8666007276011122963_a_nat @ B @ ( field_8954927560578634480_a_nat @ R ) )
=> ( ( B != bot_bo9049108969261143879_a_nat )
=> ? [X_1: product_prod_a_nat] : ( bNF_We4498669257762480967_a_nat @ R @ B @ X_1 ) ) ) ) ).
% wo_rel.Well_order_isMinim_exists
thf(fact_865_wo__rel_OWell__order__isMinim__exists,axiom,
! [R: set_Pr2972776593051762503rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ( bNF_We1688910813892497397rm_a_b @ R )
=> ( ( ord_le118470702582115849rm_a_b @ B @ ( field_6884932134483023318rm_a_b @ R ) )
=> ( ( B != bot_bo197521221353338581rm_a_b )
=> ? [X_1: produc357393685978478089rm_a_b] : ( bNF_We8430659725849320639rm_a_b @ R @ B @ X_1 ) ) ) ) ).
% wo_rel.Well_order_isMinim_exists
thf(fact_866_wo__rel_Ominim__in,axiom,
! [R: set_Pr1811044260758604347_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( bNF_We4298404315170990993_a_nat @ R )
=> ( ( ord_le8666007276011122963_a_nat @ B @ ( field_8954927560578634480_a_nat @ R ) )
=> ( ( B != bot_bo9049108969261143879_a_nat )
=> ( member5724188588386418708_a_nat @ ( bNF_We3895015349946260913_a_nat @ R @ B ) @ B ) ) ) ) ).
% wo_rel.minim_in
thf(fact_867_wo__rel_Ominim__in,axiom,
! [R: set_Pr2972776593051762503rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ( bNF_We1688910813892497397rm_a_b @ R )
=> ( ( ord_le118470702582115849rm_a_b @ B @ ( field_6884932134483023318rm_a_b @ R ) )
=> ( ( B != bot_bo197521221353338581rm_a_b )
=> ( member5869715511025134514rm_a_b @ ( bNF_We1338740440619677141rm_a_b @ R @ B ) @ B ) ) ) ) ).
% wo_rel.minim_in
thf(fact_868_wo__rel_Ominim__inField,axiom,
! [R: set_Pr1811044260758604347_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( bNF_We4298404315170990993_a_nat @ R )
=> ( ( ord_le8666007276011122963_a_nat @ B @ ( field_8954927560578634480_a_nat @ R ) )
=> ( ( B != bot_bo9049108969261143879_a_nat )
=> ( member5724188588386418708_a_nat @ ( bNF_We3895015349946260913_a_nat @ R @ B ) @ ( field_8954927560578634480_a_nat @ R ) ) ) ) ) ).
% wo_rel.minim_inField
thf(fact_869_wo__rel_Ominim__inField,axiom,
! [R: set_Pr2972776593051762503rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ( bNF_We1688910813892497397rm_a_b @ R )
=> ( ( ord_le118470702582115849rm_a_b @ B @ ( field_6884932134483023318rm_a_b @ R ) )
=> ( ( B != bot_bo197521221353338581rm_a_b )
=> ( member5869715511025134514rm_a_b @ ( bNF_We1338740440619677141rm_a_b @ R @ B ) @ ( field_6884932134483023318rm_a_b @ R ) ) ) ) ) ).
% wo_rel.minim_inField
thf(fact_870_wo__rel_Ominim__isMinim,axiom,
! [R: set_Pr1811044260758604347_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( bNF_We4298404315170990993_a_nat @ R )
=> ( ( ord_le8666007276011122963_a_nat @ B @ ( field_8954927560578634480_a_nat @ R ) )
=> ( ( B != bot_bo9049108969261143879_a_nat )
=> ( bNF_We4498669257762480967_a_nat @ R @ B @ ( bNF_We3895015349946260913_a_nat @ R @ B ) ) ) ) ) ).
% wo_rel.minim_isMinim
thf(fact_871_wo__rel_Ominim__isMinim,axiom,
! [R: set_Pr2972776593051762503rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ( bNF_We1688910813892497397rm_a_b @ R )
=> ( ( ord_le118470702582115849rm_a_b @ B @ ( field_6884932134483023318rm_a_b @ R ) )
=> ( ( B != bot_bo197521221353338581rm_a_b )
=> ( bNF_We8430659725849320639rm_a_b @ R @ B @ ( bNF_We1338740440619677141rm_a_b @ R @ B ) ) ) ) ) ).
% wo_rel.minim_isMinim
thf(fact_872_wo__rel_OisMinim__def,axiom,
! [R: set_Pr2972776593051762503rm_a_b,A: set_Pr4386577575007340137rm_a_b,B2: produc357393685978478089rm_a_b] :
( ( bNF_We1688910813892497397rm_a_b @ R )
=> ( ( bNF_We8430659725849320639rm_a_b @ R @ A @ B2 )
= ( ( member5869715511025134514rm_a_b @ B2 @ A )
& ! [X4: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X4 @ A )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B2 @ X4 ) @ R ) ) ) ) ) ).
% wo_rel.isMinim_def
thf(fact_873_wo__rel_OisMinim__def,axiom,
! [R: set_Pr1811044260758604347_a_nat,A: set_Pr4934435412358123699_a_nat,B2: product_prod_a_nat] :
( ( bNF_We4298404315170990993_a_nat @ R )
=> ( ( bNF_We4498669257762480967_a_nat @ R @ A @ B2 )
= ( ( member5724188588386418708_a_nat @ B2 @ A )
& ! [X4: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ X4 @ A )
=> ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ B2 @ X4 ) @ R ) ) ) ) ) ).
% wo_rel.isMinim_def
thf(fact_874_wo__rel_OisMinim__def,axiom,
! [R: set_Pr4386577575007340137rm_a_b,A: set_term_a_b,B2: term_a_b] :
( ( bNF_We251836928773027112rm_a_b @ R )
=> ( ( bNF_We695392074589568990rm_a_b @ R @ A @ B2 )
= ( ( member_term_a_b @ B2 @ A )
& ! [X4: term_a_b] :
( ( member_term_a_b @ X4 @ A )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ X4 ) @ R ) ) ) ) ) ).
% wo_rel.isMinim_def
thf(fact_875_wo__rel_Oequals__minim,axiom,
! [R: set_Pr4386577575007340137rm_a_b,B: set_term_a_b,A2: term_a_b] :
( ( bNF_We251836928773027112rm_a_b @ R )
=> ( ( ord_le2705286416250468010rm_a_b @ B @ ( field_term_a_b @ R ) )
=> ( ( member_term_a_b @ A2 @ B )
=> ( ! [B5: term_a_b] :
( ( member_term_a_b @ B5 @ B )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B5 ) @ R ) )
=> ( A2
= ( bNF_We6258903063523145544rm_a_b @ R @ B ) ) ) ) ) ) ).
% wo_rel.equals_minim
thf(fact_876_wo__rel_Oequals__minim,axiom,
! [R: set_Pr1811044260758604347_a_nat,B: set_Pr4934435412358123699_a_nat,A2: product_prod_a_nat] :
( ( bNF_We4298404315170990993_a_nat @ R )
=> ( ( ord_le8666007276011122963_a_nat @ B @ ( field_8954927560578634480_a_nat @ R ) )
=> ( ( member5724188588386418708_a_nat @ A2 @ B )
=> ( ! [B5: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ B5 @ B )
=> ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A2 @ B5 ) @ R ) )
=> ( A2
= ( bNF_We3895015349946260913_a_nat @ R @ B ) ) ) ) ) ) ).
% wo_rel.equals_minim
thf(fact_877_wo__rel_Oequals__minim,axiom,
! [R: set_Pr2972776593051762503rm_a_b,B: set_Pr4386577575007340137rm_a_b,A2: produc357393685978478089rm_a_b] :
( ( bNF_We1688910813892497397rm_a_b @ R )
=> ( ( ord_le118470702582115849rm_a_b @ B @ ( field_6884932134483023318rm_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ A2 @ B )
=> ( ! [B5: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ B5 @ B )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A2 @ B5 ) @ R ) )
=> ( A2
= ( bNF_We1338740440619677141rm_a_b @ R @ B ) ) ) ) ) ) ).
% wo_rel.equals_minim
thf(fact_878_wo__rel_Ominim__least,axiom,
! [R: set_Pr4386577575007340137rm_a_b,B: set_term_a_b,B2: term_a_b] :
( ( bNF_We251836928773027112rm_a_b @ R )
=> ( ( ord_le2705286416250468010rm_a_b @ B @ ( field_term_a_b @ R ) )
=> ( ( member_term_a_b @ B2 @ B )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( bNF_We6258903063523145544rm_a_b @ R @ B ) @ B2 ) @ R ) ) ) ) ).
% wo_rel.minim_least
thf(fact_879_wo__rel_Ominim__least,axiom,
! [R: set_Pr1811044260758604347_a_nat,B: set_Pr4934435412358123699_a_nat,B2: product_prod_a_nat] :
( ( bNF_We4298404315170990993_a_nat @ R )
=> ( ( ord_le8666007276011122963_a_nat @ B @ ( field_8954927560578634480_a_nat @ R ) )
=> ( ( member5724188588386418708_a_nat @ B2 @ B )
=> ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ ( bNF_We3895015349946260913_a_nat @ R @ B ) @ B2 ) @ R ) ) ) ) ).
% wo_rel.minim_least
thf(fact_880_wo__rel_Ominim__least,axiom,
! [R: set_Pr2972776593051762503rm_a_b,B: set_Pr4386577575007340137rm_a_b,B2: produc357393685978478089rm_a_b] :
( ( bNF_We1688910813892497397rm_a_b @ R )
=> ( ( ord_le118470702582115849rm_a_b @ B @ ( field_6884932134483023318rm_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ B2 @ B )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( bNF_We1338740440619677141rm_a_b @ R @ B ) @ B2 ) @ R ) ) ) ) ).
% wo_rel.minim_least
thf(fact_881_wo__rel_Omax2__greater__among,axiom,
! [R: set_Pr2972776593051762503rm_a_b,A2: produc357393685978478089rm_a_b,B2: produc357393685978478089rm_a_b] :
( ( bNF_We1688910813892497397rm_a_b @ R )
=> ( ( member5869715511025134514rm_a_b @ A2 @ ( field_6884932134483023318rm_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ B2 @ ( field_6884932134483023318rm_a_b @ R ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A2 @ ( bNF_We1369222903935127127rm_a_b @ R @ A2 @ B2 ) ) @ R )
& ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B2 @ ( bNF_We1369222903935127127rm_a_b @ R @ A2 @ B2 ) ) @ R )
& ( member5869715511025134514rm_a_b @ ( bNF_We1369222903935127127rm_a_b @ R @ A2 @ B2 ) @ ( insert7009541432154983385rm_a_b @ A2 @ ( insert7009541432154983385rm_a_b @ B2 @ bot_bo197521221353338581rm_a_b ) ) ) ) ) ) ) ).
% wo_rel.max2_greater_among
thf(fact_882_wo__rel_Omax2__greater__among,axiom,
! [R: set_Pr1811044260758604347_a_nat,A2: product_prod_a_nat,B2: product_prod_a_nat] :
( ( bNF_We4298404315170990993_a_nat @ R )
=> ( ( member5724188588386418708_a_nat @ A2 @ ( field_8954927560578634480_a_nat @ R ) )
=> ( ( member5724188588386418708_a_nat @ B2 @ ( field_8954927560578634480_a_nat @ R ) )
=> ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A2 @ ( bNF_We7882469860449324975_a_nat @ R @ A2 @ B2 ) ) @ R )
& ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ B2 @ ( bNF_We7882469860449324975_a_nat @ R @ A2 @ B2 ) ) @ R )
& ( member5724188588386418708_a_nat @ ( bNF_We7882469860449324975_a_nat @ R @ A2 @ B2 ) @ ( insert8054603423593749677_a_nat @ A2 @ ( insert8054603423593749677_a_nat @ B2 @ bot_bo9049108969261143879_a_nat ) ) ) ) ) ) ) ).
% wo_rel.max2_greater_among
thf(fact_883_wo__rel_Omax2__greater__among,axiom,
! [R: set_Pr4386577575007340137rm_a_b,A2: term_a_b,B2: term_a_b] :
( ( bNF_We251836928773027112rm_a_b @ R )
=> ( ( member_term_a_b @ A2 @ ( field_term_a_b @ R ) )
=> ( ( member_term_a_b @ B2 @ ( field_term_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ ( bNF_We4618759378340593734rm_a_b @ R @ A2 @ B2 ) ) @ R )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ ( bNF_We4618759378340593734rm_a_b @ R @ A2 @ B2 ) ) @ R )
& ( member_term_a_b @ ( bNF_We4618759378340593734rm_a_b @ R @ A2 @ B2 ) @ ( insert_term_a_b @ A2 @ ( insert_term_a_b @ B2 @ bot_bot_set_term_a_b ) ) ) ) ) ) ) ).
% wo_rel.max2_greater_among
thf(fact_884_refl__on__singleton,axiom,
! [X3: term_a_b] : ( refl_on_term_a_b @ ( insert_term_a_b @ X3 @ bot_bot_set_term_a_b ) @ ( insert7009541432154983385rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ X3 ) @ bot_bo197521221353338581rm_a_b ) ) ).
% refl_on_singleton
thf(fact_885_irrefl__tranclI,axiom,
! [R: set_Pr4386577575007340137rm_a_b,X3: term_a_b] :
( ( ( inf_in1693821330460682071rm_a_b @ ( conver7608559976937845093rm_a_b @ R ) @ ( transi7742714808557438673rm_a_b @ R ) )
= bot_bo197521221353338581rm_a_b )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ X3 ) @ ( transi7922773638565587891rm_a_b @ R ) ) ) ).
% irrefl_tranclI
thf(fact_886_IntI,axiom,
! [C: produc357393685978478089rm_a_b,A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ C @ A )
=> ( ( member5869715511025134514rm_a_b @ C @ B )
=> ( member5869715511025134514rm_a_b @ C @ ( inf_in1693821330460682071rm_a_b @ A @ B ) ) ) ) ).
% IntI
thf(fact_887_IntI,axiom,
! [C: product_prod_a_nat,A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( member5724188588386418708_a_nat @ C @ A )
=> ( ( member5724188588386418708_a_nat @ C @ B )
=> ( member5724188588386418708_a_nat @ C @ ( inf_in3813033988066259269_a_nat @ A @ B ) ) ) ) ).
% IntI
thf(fact_888_Int__iff,axiom,
! [C: produc357393685978478089rm_a_b,A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ C @ ( inf_in1693821330460682071rm_a_b @ A @ B ) )
= ( ( member5869715511025134514rm_a_b @ C @ A )
& ( member5869715511025134514rm_a_b @ C @ B ) ) ) ).
% Int_iff
thf(fact_889_Int__iff,axiom,
! [C: product_prod_a_nat,A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( member5724188588386418708_a_nat @ C @ ( inf_in3813033988066259269_a_nat @ A @ B ) )
= ( ( member5724188588386418708_a_nat @ C @ A )
& ( member5724188588386418708_a_nat @ C @ B ) ) ) ).
% Int_iff
thf(fact_890_le__inf__iff,axiom,
! [X3: set_Pr4934435412358123699_a_nat,Y3: set_Pr4934435412358123699_a_nat,Z2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X3 @ ( inf_in3813033988066259269_a_nat @ Y3 @ Z2 ) )
= ( ( ord_le8666007276011122963_a_nat @ X3 @ Y3 )
& ( ord_le8666007276011122963_a_nat @ X3 @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_891_le__inf__iff,axiom,
! [X3: set_Pr4386577575007340137rm_a_b,Y3: set_Pr4386577575007340137rm_a_b,Z2: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ X3 @ ( inf_in1693821330460682071rm_a_b @ Y3 @ Z2 ) )
= ( ( ord_le118470702582115849rm_a_b @ X3 @ Y3 )
& ( ord_le118470702582115849rm_a_b @ X3 @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_892_inf_Obounded__iff,axiom,
! [A2: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat,C: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A2 @ ( inf_in3813033988066259269_a_nat @ B2 @ C ) )
= ( ( ord_le8666007276011122963_a_nat @ A2 @ B2 )
& ( ord_le8666007276011122963_a_nat @ A2 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_893_inf_Obounded__iff,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b,C: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A2 @ ( inf_in1693821330460682071rm_a_b @ B2 @ C ) )
= ( ( ord_le118470702582115849rm_a_b @ A2 @ B2 )
& ( ord_le118470702582115849rm_a_b @ A2 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_894_Int__subset__iff,axiom,
! [C2: set_Pr4934435412358123699_a_nat,A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ C2 @ ( inf_in3813033988066259269_a_nat @ A @ B ) )
= ( ( ord_le8666007276011122963_a_nat @ C2 @ A )
& ( ord_le8666007276011122963_a_nat @ C2 @ B ) ) ) ).
% Int_subset_iff
thf(fact_895_Int__subset__iff,axiom,
! [C2: set_Pr4386577575007340137rm_a_b,A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ C2 @ ( inf_in1693821330460682071rm_a_b @ A @ B ) )
= ( ( ord_le118470702582115849rm_a_b @ C2 @ A )
& ( ord_le118470702582115849rm_a_b @ C2 @ B ) ) ) ).
% Int_subset_iff
thf(fact_896_Int__insert__left__if0,axiom,
! [A2: produc357393685978478089rm_a_b,C2: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ~ ( member5869715511025134514rm_a_b @ A2 @ C2 )
=> ( ( inf_in1693821330460682071rm_a_b @ ( insert7009541432154983385rm_a_b @ A2 @ B ) @ C2 )
= ( inf_in1693821330460682071rm_a_b @ B @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_897_Int__insert__left__if0,axiom,
! [A2: product_prod_a_nat,C2: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ~ ( member5724188588386418708_a_nat @ A2 @ C2 )
=> ( ( inf_in3813033988066259269_a_nat @ ( insert8054603423593749677_a_nat @ A2 @ B ) @ C2 )
= ( inf_in3813033988066259269_a_nat @ B @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_898_Int__insert__left__if1,axiom,
! [A2: produc357393685978478089rm_a_b,C2: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ A2 @ C2 )
=> ( ( inf_in1693821330460682071rm_a_b @ ( insert7009541432154983385rm_a_b @ A2 @ B ) @ C2 )
= ( insert7009541432154983385rm_a_b @ A2 @ ( inf_in1693821330460682071rm_a_b @ B @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_899_Int__insert__left__if1,axiom,
! [A2: product_prod_a_nat,C2: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( member5724188588386418708_a_nat @ A2 @ C2 )
=> ( ( inf_in3813033988066259269_a_nat @ ( insert8054603423593749677_a_nat @ A2 @ B ) @ C2 )
= ( insert8054603423593749677_a_nat @ A2 @ ( inf_in3813033988066259269_a_nat @ B @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_900_Int__insert__right__if0,axiom,
! [A2: produc357393685978478089rm_a_b,A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ~ ( member5869715511025134514rm_a_b @ A2 @ A )
=> ( ( inf_in1693821330460682071rm_a_b @ A @ ( insert7009541432154983385rm_a_b @ A2 @ B ) )
= ( inf_in1693821330460682071rm_a_b @ A @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_901_Int__insert__right__if0,axiom,
! [A2: product_prod_a_nat,A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ~ ( member5724188588386418708_a_nat @ A2 @ A )
=> ( ( inf_in3813033988066259269_a_nat @ A @ ( insert8054603423593749677_a_nat @ A2 @ B ) )
= ( inf_in3813033988066259269_a_nat @ A @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_902_Int__insert__right__if1,axiom,
! [A2: produc357393685978478089rm_a_b,A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ A2 @ A )
=> ( ( inf_in1693821330460682071rm_a_b @ A @ ( insert7009541432154983385rm_a_b @ A2 @ B ) )
= ( insert7009541432154983385rm_a_b @ A2 @ ( inf_in1693821330460682071rm_a_b @ A @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_903_Int__insert__right__if1,axiom,
! [A2: product_prod_a_nat,A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( member5724188588386418708_a_nat @ A2 @ A )
=> ( ( inf_in3813033988066259269_a_nat @ A @ ( insert8054603423593749677_a_nat @ A2 @ B ) )
= ( insert8054603423593749677_a_nat @ A2 @ ( inf_in3813033988066259269_a_nat @ A @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_904_insert__disjoint_I1_J,axiom,
! [A2: produc357393685978478089rm_a_b,A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ( ( inf_in1693821330460682071rm_a_b @ ( insert7009541432154983385rm_a_b @ A2 @ A ) @ B )
= bot_bo197521221353338581rm_a_b )
= ( ~ ( member5869715511025134514rm_a_b @ A2 @ B )
& ( ( inf_in1693821330460682071rm_a_b @ A @ B )
= bot_bo197521221353338581rm_a_b ) ) ) ).
% insert_disjoint(1)
thf(fact_905_insert__disjoint_I1_J,axiom,
! [A2: product_prod_a_nat,A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( ( inf_in3813033988066259269_a_nat @ ( insert8054603423593749677_a_nat @ A2 @ A ) @ B )
= bot_bo9049108969261143879_a_nat )
= ( ~ ( member5724188588386418708_a_nat @ A2 @ B )
& ( ( inf_in3813033988066259269_a_nat @ A @ B )
= bot_bo9049108969261143879_a_nat ) ) ) ).
% insert_disjoint(1)
thf(fact_906_insert__disjoint_I2_J,axiom,
! [A2: produc357393685978478089rm_a_b,A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ( bot_bo197521221353338581rm_a_b
= ( inf_in1693821330460682071rm_a_b @ ( insert7009541432154983385rm_a_b @ A2 @ A ) @ B ) )
= ( ~ ( member5869715511025134514rm_a_b @ A2 @ B )
& ( bot_bo197521221353338581rm_a_b
= ( inf_in1693821330460682071rm_a_b @ A @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_907_insert__disjoint_I2_J,axiom,
! [A2: product_prod_a_nat,A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( bot_bo9049108969261143879_a_nat
= ( inf_in3813033988066259269_a_nat @ ( insert8054603423593749677_a_nat @ A2 @ A ) @ B ) )
= ( ~ ( member5724188588386418708_a_nat @ A2 @ B )
& ( bot_bo9049108969261143879_a_nat
= ( inf_in3813033988066259269_a_nat @ A @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_908_disjoint__insert_I1_J,axiom,
! [B: set_Pr4386577575007340137rm_a_b,A2: produc357393685978478089rm_a_b,A: set_Pr4386577575007340137rm_a_b] :
( ( ( inf_in1693821330460682071rm_a_b @ B @ ( insert7009541432154983385rm_a_b @ A2 @ A ) )
= bot_bo197521221353338581rm_a_b )
= ( ~ ( member5869715511025134514rm_a_b @ A2 @ B )
& ( ( inf_in1693821330460682071rm_a_b @ B @ A )
= bot_bo197521221353338581rm_a_b ) ) ) ).
% disjoint_insert(1)
thf(fact_909_disjoint__insert_I1_J,axiom,
! [B: set_Pr4934435412358123699_a_nat,A2: product_prod_a_nat,A: set_Pr4934435412358123699_a_nat] :
( ( ( inf_in3813033988066259269_a_nat @ B @ ( insert8054603423593749677_a_nat @ A2 @ A ) )
= bot_bo9049108969261143879_a_nat )
= ( ~ ( member5724188588386418708_a_nat @ A2 @ B )
& ( ( inf_in3813033988066259269_a_nat @ B @ A )
= bot_bo9049108969261143879_a_nat ) ) ) ).
% disjoint_insert(1)
thf(fact_910_disjoint__insert_I2_J,axiom,
! [A: set_Pr4386577575007340137rm_a_b,B2: produc357393685978478089rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ( bot_bo197521221353338581rm_a_b
= ( inf_in1693821330460682071rm_a_b @ A @ ( insert7009541432154983385rm_a_b @ B2 @ B ) ) )
= ( ~ ( member5869715511025134514rm_a_b @ B2 @ A )
& ( bot_bo197521221353338581rm_a_b
= ( inf_in1693821330460682071rm_a_b @ A @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_911_disjoint__insert_I2_J,axiom,
! [A: set_Pr4934435412358123699_a_nat,B2: product_prod_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( bot_bo9049108969261143879_a_nat
= ( inf_in3813033988066259269_a_nat @ A @ ( insert8054603423593749677_a_nat @ B2 @ B ) ) )
= ( ~ ( member5724188588386418708_a_nat @ B2 @ A )
& ( bot_bo9049108969261143879_a_nat
= ( inf_in3813033988066259269_a_nat @ A @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_912_IntE,axiom,
! [C: produc357393685978478089rm_a_b,A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ C @ ( inf_in1693821330460682071rm_a_b @ A @ B ) )
=> ~ ( ( member5869715511025134514rm_a_b @ C @ A )
=> ~ ( member5869715511025134514rm_a_b @ C @ B ) ) ) ).
% IntE
thf(fact_913_IntE,axiom,
! [C: product_prod_a_nat,A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( member5724188588386418708_a_nat @ C @ ( inf_in3813033988066259269_a_nat @ A @ B ) )
=> ~ ( ( member5724188588386418708_a_nat @ C @ A )
=> ~ ( member5724188588386418708_a_nat @ C @ B ) ) ) ).
% IntE
thf(fact_914_IntD1,axiom,
! [C: produc357393685978478089rm_a_b,A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ C @ ( inf_in1693821330460682071rm_a_b @ A @ B ) )
=> ( member5869715511025134514rm_a_b @ C @ A ) ) ).
% IntD1
thf(fact_915_IntD1,axiom,
! [C: product_prod_a_nat,A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( member5724188588386418708_a_nat @ C @ ( inf_in3813033988066259269_a_nat @ A @ B ) )
=> ( member5724188588386418708_a_nat @ C @ A ) ) ).
% IntD1
thf(fact_916_IntD2,axiom,
! [C: produc357393685978478089rm_a_b,A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ C @ ( inf_in1693821330460682071rm_a_b @ A @ B ) )
=> ( member5869715511025134514rm_a_b @ C @ B ) ) ).
% IntD2
thf(fact_917_IntD2,axiom,
! [C: product_prod_a_nat,A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( member5724188588386418708_a_nat @ C @ ( inf_in3813033988066259269_a_nat @ A @ B ) )
=> ( member5724188588386418708_a_nat @ C @ B ) ) ).
% IntD2
thf(fact_918_Int__emptyI,axiom,
! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ! [X: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X @ A )
=> ~ ( member5869715511025134514rm_a_b @ X @ B ) )
=> ( ( inf_in1693821330460682071rm_a_b @ A @ B )
= bot_bo197521221353338581rm_a_b ) ) ).
% Int_emptyI
thf(fact_919_Int__emptyI,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ! [X: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ X @ A )
=> ~ ( member5724188588386418708_a_nat @ X @ B ) )
=> ( ( inf_in3813033988066259269_a_nat @ A @ B )
= bot_bo9049108969261143879_a_nat ) ) ).
% Int_emptyI
thf(fact_920_disjoint__iff,axiom,
! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ( ( inf_in1693821330460682071rm_a_b @ A @ B )
= bot_bo197521221353338581rm_a_b )
= ( ! [X4: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X4 @ A )
=> ~ ( member5869715511025134514rm_a_b @ X4 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_921_disjoint__iff,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( ( inf_in3813033988066259269_a_nat @ A @ B )
= bot_bo9049108969261143879_a_nat )
= ( ! [X4: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ X4 @ A )
=> ~ ( member5724188588386418708_a_nat @ X4 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_922_Int__mono,axiom,
! [A: set_Pr4934435412358123699_a_nat,C2: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat,D: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A @ C2 )
=> ( ( ord_le8666007276011122963_a_nat @ B @ D )
=> ( ord_le8666007276011122963_a_nat @ ( inf_in3813033988066259269_a_nat @ A @ B ) @ ( inf_in3813033988066259269_a_nat @ C2 @ D ) ) ) ) ).
% Int_mono
thf(fact_923_Int__mono,axiom,
! [A: set_Pr4386577575007340137rm_a_b,C2: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b,D: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A @ C2 )
=> ( ( ord_le118470702582115849rm_a_b @ B @ D )
=> ( ord_le118470702582115849rm_a_b @ ( inf_in1693821330460682071rm_a_b @ A @ B ) @ ( inf_in1693821330460682071rm_a_b @ C2 @ D ) ) ) ) ).
% Int_mono
thf(fact_924_Int__lower1,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ ( inf_in3813033988066259269_a_nat @ A @ B ) @ A ) ).
% Int_lower1
thf(fact_925_Int__lower1,axiom,
! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ ( inf_in1693821330460682071rm_a_b @ A @ B ) @ A ) ).
% Int_lower1
thf(fact_926_Int__lower2,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ ( inf_in3813033988066259269_a_nat @ A @ B ) @ B ) ).
% Int_lower2
thf(fact_927_Int__lower2,axiom,
! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ ( inf_in1693821330460682071rm_a_b @ A @ B ) @ B ) ).
% Int_lower2
thf(fact_928_Int__absorb1,axiom,
! [B: set_Pr4934435412358123699_a_nat,A: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ B @ A )
=> ( ( inf_in3813033988066259269_a_nat @ A @ B )
= B ) ) ).
% Int_absorb1
thf(fact_929_Int__absorb1,axiom,
! [B: set_Pr4386577575007340137rm_a_b,A: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ B @ A )
=> ( ( inf_in1693821330460682071rm_a_b @ A @ B )
= B ) ) ).
% Int_absorb1
thf(fact_930_Int__absorb2,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A @ B )
=> ( ( inf_in3813033988066259269_a_nat @ A @ B )
= A ) ) ).
% Int_absorb2
thf(fact_931_Int__absorb2,axiom,
! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A @ B )
=> ( ( inf_in1693821330460682071rm_a_b @ A @ B )
= A ) ) ).
% Int_absorb2
thf(fact_932_Int__greatest,axiom,
! [C2: set_Pr4934435412358123699_a_nat,A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ C2 @ A )
=> ( ( ord_le8666007276011122963_a_nat @ C2 @ B )
=> ( ord_le8666007276011122963_a_nat @ C2 @ ( inf_in3813033988066259269_a_nat @ A @ B ) ) ) ) ).
% Int_greatest
thf(fact_933_Int__greatest,axiom,
! [C2: set_Pr4386577575007340137rm_a_b,A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ C2 @ A )
=> ( ( ord_le118470702582115849rm_a_b @ C2 @ B )
=> ( ord_le118470702582115849rm_a_b @ C2 @ ( inf_in1693821330460682071rm_a_b @ A @ B ) ) ) ) ).
% Int_greatest
thf(fact_934_Int__Collect__mono,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat,P: product_prod_a_nat > $o,Q: product_prod_a_nat > $o] :
( ( ord_le8666007276011122963_a_nat @ A @ B )
=> ( ! [X: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ X @ A )
=> ( ( P @ X )
=> ( Q @ X ) ) )
=> ( ord_le8666007276011122963_a_nat @ ( inf_in3813033988066259269_a_nat @ A @ ( collec4464134535221767506_a_nat @ P ) ) @ ( inf_in3813033988066259269_a_nat @ B @ ( collec4464134535221767506_a_nat @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_935_Int__Collect__mono,axiom,
! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b,P: produc357393685978478089rm_a_b > $o,Q: produc357393685978478089rm_a_b > $o] :
( ( ord_le118470702582115849rm_a_b @ A @ B )
=> ( ! [X: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X @ A )
=> ( ( P @ X )
=> ( Q @ X ) ) )
=> ( ord_le118470702582115849rm_a_b @ ( inf_in1693821330460682071rm_a_b @ A @ ( collec99177395925924084rm_a_b @ P ) ) @ ( inf_in1693821330460682071rm_a_b @ B @ ( collec99177395925924084rm_a_b @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_936_inf_OcoboundedI2,axiom,
! [B2: set_Pr4934435412358123699_a_nat,C: set_Pr4934435412358123699_a_nat,A2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ B2 @ C )
=> ( ord_le8666007276011122963_a_nat @ ( inf_in3813033988066259269_a_nat @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI2
thf(fact_937_inf_OcoboundedI2,axiom,
! [B2: set_Pr4386577575007340137rm_a_b,C: set_Pr4386577575007340137rm_a_b,A2: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ B2 @ C )
=> ( ord_le118470702582115849rm_a_b @ ( inf_in1693821330460682071rm_a_b @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI2
thf(fact_938_inf_OcoboundedI1,axiom,
! [A2: set_Pr4934435412358123699_a_nat,C: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A2 @ C )
=> ( ord_le8666007276011122963_a_nat @ ( inf_in3813033988066259269_a_nat @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI1
thf(fact_939_inf_OcoboundedI1,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,C: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A2 @ C )
=> ( ord_le118470702582115849rm_a_b @ ( inf_in1693821330460682071rm_a_b @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI1
thf(fact_940_inf_Oabsorb__iff2,axiom,
( ord_le8666007276011122963_a_nat
= ( ^ [B4: set_Pr4934435412358123699_a_nat,A4: set_Pr4934435412358123699_a_nat] :
( ( inf_in3813033988066259269_a_nat @ A4 @ B4 )
= B4 ) ) ) ).
% inf.absorb_iff2
thf(fact_941_inf_Oabsorb__iff2,axiom,
( ord_le118470702582115849rm_a_b
= ( ^ [B4: set_Pr4386577575007340137rm_a_b,A4: set_Pr4386577575007340137rm_a_b] :
( ( inf_in1693821330460682071rm_a_b @ A4 @ B4 )
= B4 ) ) ) ).
% inf.absorb_iff2
thf(fact_942_inf_Oabsorb__iff1,axiom,
( ord_le8666007276011122963_a_nat
= ( ^ [A4: set_Pr4934435412358123699_a_nat,B4: set_Pr4934435412358123699_a_nat] :
( ( inf_in3813033988066259269_a_nat @ A4 @ B4 )
= A4 ) ) ) ).
% inf.absorb_iff1
thf(fact_943_inf_Oabsorb__iff1,axiom,
( ord_le118470702582115849rm_a_b
= ( ^ [A4: set_Pr4386577575007340137rm_a_b,B4: set_Pr4386577575007340137rm_a_b] :
( ( inf_in1693821330460682071rm_a_b @ A4 @ B4 )
= A4 ) ) ) ).
% inf.absorb_iff1
thf(fact_944_inf_Ocobounded2,axiom,
! [A2: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ ( inf_in3813033988066259269_a_nat @ A2 @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_945_inf_Ocobounded2,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ ( inf_in1693821330460682071rm_a_b @ A2 @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_946_inf_Ocobounded1,axiom,
! [A2: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ ( inf_in3813033988066259269_a_nat @ A2 @ B2 ) @ A2 ) ).
% inf.cobounded1
thf(fact_947_inf_Ocobounded1,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ ( inf_in1693821330460682071rm_a_b @ A2 @ B2 ) @ A2 ) ).
% inf.cobounded1
thf(fact_948_inf_Oorder__iff,axiom,
( ord_le8666007276011122963_a_nat
= ( ^ [A4: set_Pr4934435412358123699_a_nat,B4: set_Pr4934435412358123699_a_nat] :
( A4
= ( inf_in3813033988066259269_a_nat @ A4 @ B4 ) ) ) ) ).
% inf.order_iff
thf(fact_949_inf_Oorder__iff,axiom,
( ord_le118470702582115849rm_a_b
= ( ^ [A4: set_Pr4386577575007340137rm_a_b,B4: set_Pr4386577575007340137rm_a_b] :
( A4
= ( inf_in1693821330460682071rm_a_b @ A4 @ B4 ) ) ) ) ).
% inf.order_iff
thf(fact_950_inf__greatest,axiom,
! [X3: set_Pr4934435412358123699_a_nat,Y3: set_Pr4934435412358123699_a_nat,Z2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X3 @ Y3 )
=> ( ( ord_le8666007276011122963_a_nat @ X3 @ Z2 )
=> ( ord_le8666007276011122963_a_nat @ X3 @ ( inf_in3813033988066259269_a_nat @ Y3 @ Z2 ) ) ) ) ).
% inf_greatest
thf(fact_951_inf__greatest,axiom,
! [X3: set_Pr4386577575007340137rm_a_b,Y3: set_Pr4386577575007340137rm_a_b,Z2: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ X3 @ Y3 )
=> ( ( ord_le118470702582115849rm_a_b @ X3 @ Z2 )
=> ( ord_le118470702582115849rm_a_b @ X3 @ ( inf_in1693821330460682071rm_a_b @ Y3 @ Z2 ) ) ) ) ).
% inf_greatest
thf(fact_952_inf_OboundedI,axiom,
! [A2: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat,C: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A2 @ B2 )
=> ( ( ord_le8666007276011122963_a_nat @ A2 @ C )
=> ( ord_le8666007276011122963_a_nat @ A2 @ ( inf_in3813033988066259269_a_nat @ B2 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_953_inf_OboundedI,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b,C: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A2 @ B2 )
=> ( ( ord_le118470702582115849rm_a_b @ A2 @ C )
=> ( ord_le118470702582115849rm_a_b @ A2 @ ( inf_in1693821330460682071rm_a_b @ B2 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_954_inf_OboundedE,axiom,
! [A2: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat,C: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A2 @ ( inf_in3813033988066259269_a_nat @ B2 @ C ) )
=> ~ ( ( ord_le8666007276011122963_a_nat @ A2 @ B2 )
=> ~ ( ord_le8666007276011122963_a_nat @ A2 @ C ) ) ) ).
% inf.boundedE
thf(fact_955_inf_OboundedE,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b,C: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A2 @ ( inf_in1693821330460682071rm_a_b @ B2 @ C ) )
=> ~ ( ( ord_le118470702582115849rm_a_b @ A2 @ B2 )
=> ~ ( ord_le118470702582115849rm_a_b @ A2 @ C ) ) ) ).
% inf.boundedE
thf(fact_956_inf__absorb2,axiom,
! [Y3: set_Pr4934435412358123699_a_nat,X3: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ Y3 @ X3 )
=> ( ( inf_in3813033988066259269_a_nat @ X3 @ Y3 )
= Y3 ) ) ).
% inf_absorb2
thf(fact_957_inf__absorb2,axiom,
! [Y3: set_Pr4386577575007340137rm_a_b,X3: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ Y3 @ X3 )
=> ( ( inf_in1693821330460682071rm_a_b @ X3 @ Y3 )
= Y3 ) ) ).
% inf_absorb2
thf(fact_958_inf__absorb1,axiom,
! [X3: set_Pr4934435412358123699_a_nat,Y3: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X3 @ Y3 )
=> ( ( inf_in3813033988066259269_a_nat @ X3 @ Y3 )
= X3 ) ) ).
% inf_absorb1
thf(fact_959_inf__absorb1,axiom,
! [X3: set_Pr4386577575007340137rm_a_b,Y3: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ X3 @ Y3 )
=> ( ( inf_in1693821330460682071rm_a_b @ X3 @ Y3 )
= X3 ) ) ).
% inf_absorb1
thf(fact_960_inf_Oabsorb2,axiom,
! [B2: set_Pr4934435412358123699_a_nat,A2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ B2 @ A2 )
=> ( ( inf_in3813033988066259269_a_nat @ A2 @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_961_inf_Oabsorb2,axiom,
! [B2: set_Pr4386577575007340137rm_a_b,A2: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ B2 @ A2 )
=> ( ( inf_in1693821330460682071rm_a_b @ A2 @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_962_inf_Oabsorb1,axiom,
! [A2: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A2 @ B2 )
=> ( ( inf_in3813033988066259269_a_nat @ A2 @ B2 )
= A2 ) ) ).
% inf.absorb1
thf(fact_963_inf_Oabsorb1,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A2 @ B2 )
=> ( ( inf_in1693821330460682071rm_a_b @ A2 @ B2 )
= A2 ) ) ).
% inf.absorb1
thf(fact_964_le__iff__inf,axiom,
( ord_le8666007276011122963_a_nat
= ( ^ [X4: set_Pr4934435412358123699_a_nat,Y5: set_Pr4934435412358123699_a_nat] :
( ( inf_in3813033988066259269_a_nat @ X4 @ Y5 )
= X4 ) ) ) ).
% le_iff_inf
thf(fact_965_le__iff__inf,axiom,
( ord_le118470702582115849rm_a_b
= ( ^ [X4: set_Pr4386577575007340137rm_a_b,Y5: set_Pr4386577575007340137rm_a_b] :
( ( inf_in1693821330460682071rm_a_b @ X4 @ Y5 )
= X4 ) ) ) ).
% le_iff_inf
thf(fact_966_inf__unique,axiom,
! [F2: set_Pr4934435412358123699_a_nat > set_Pr4934435412358123699_a_nat > set_Pr4934435412358123699_a_nat,X3: set_Pr4934435412358123699_a_nat,Y3: set_Pr4934435412358123699_a_nat] :
( ! [X: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ ( F2 @ X @ Y ) @ X )
=> ( ! [X: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ ( F2 @ X @ Y ) @ Y )
=> ( ! [X: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat,Z3: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X @ Y )
=> ( ( ord_le8666007276011122963_a_nat @ X @ Z3 )
=> ( ord_le8666007276011122963_a_nat @ X @ ( F2 @ Y @ Z3 ) ) ) )
=> ( ( inf_in3813033988066259269_a_nat @ X3 @ Y3 )
= ( F2 @ X3 @ Y3 ) ) ) ) ) ).
% inf_unique
thf(fact_967_inf__unique,axiom,
! [F2: set_Pr4386577575007340137rm_a_b > set_Pr4386577575007340137rm_a_b > set_Pr4386577575007340137rm_a_b,X3: set_Pr4386577575007340137rm_a_b,Y3: set_Pr4386577575007340137rm_a_b] :
( ! [X: set_Pr4386577575007340137rm_a_b,Y: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ ( F2 @ X @ Y ) @ X )
=> ( ! [X: set_Pr4386577575007340137rm_a_b,Y: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ ( F2 @ X @ Y ) @ Y )
=> ( ! [X: set_Pr4386577575007340137rm_a_b,Y: set_Pr4386577575007340137rm_a_b,Z3: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ X @ Y )
=> ( ( ord_le118470702582115849rm_a_b @ X @ Z3 )
=> ( ord_le118470702582115849rm_a_b @ X @ ( F2 @ Y @ Z3 ) ) ) )
=> ( ( inf_in1693821330460682071rm_a_b @ X3 @ Y3 )
= ( F2 @ X3 @ Y3 ) ) ) ) ) ).
% inf_unique
thf(fact_968_inf_OorderI,axiom,
! [A2: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat] :
( ( A2
= ( inf_in3813033988066259269_a_nat @ A2 @ B2 ) )
=> ( ord_le8666007276011122963_a_nat @ A2 @ B2 ) ) ).
% inf.orderI
thf(fact_969_inf_OorderI,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b] :
( ( A2
= ( inf_in1693821330460682071rm_a_b @ A2 @ B2 ) )
=> ( ord_le118470702582115849rm_a_b @ A2 @ B2 ) ) ).
% inf.orderI
thf(fact_970_inf_OorderE,axiom,
! [A2: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A2 @ B2 )
=> ( A2
= ( inf_in3813033988066259269_a_nat @ A2 @ B2 ) ) ) ).
% inf.orderE
thf(fact_971_inf_OorderE,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A2 @ B2 )
=> ( A2
= ( inf_in1693821330460682071rm_a_b @ A2 @ B2 ) ) ) ).
% inf.orderE
thf(fact_972_le__infI2,axiom,
! [B2: set_Pr4934435412358123699_a_nat,X3: set_Pr4934435412358123699_a_nat,A2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ B2 @ X3 )
=> ( ord_le8666007276011122963_a_nat @ ( inf_in3813033988066259269_a_nat @ A2 @ B2 ) @ X3 ) ) ).
% le_infI2
thf(fact_973_le__infI2,axiom,
! [B2: set_Pr4386577575007340137rm_a_b,X3: set_Pr4386577575007340137rm_a_b,A2: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ B2 @ X3 )
=> ( ord_le118470702582115849rm_a_b @ ( inf_in1693821330460682071rm_a_b @ A2 @ B2 ) @ X3 ) ) ).
% le_infI2
thf(fact_974_le__infI1,axiom,
! [A2: set_Pr4934435412358123699_a_nat,X3: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A2 @ X3 )
=> ( ord_le8666007276011122963_a_nat @ ( inf_in3813033988066259269_a_nat @ A2 @ B2 ) @ X3 ) ) ).
% le_infI1
thf(fact_975_le__infI1,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,X3: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A2 @ X3 )
=> ( ord_le118470702582115849rm_a_b @ ( inf_in1693821330460682071rm_a_b @ A2 @ B2 ) @ X3 ) ) ).
% le_infI1
thf(fact_976_inf__mono,axiom,
! [A2: set_Pr4934435412358123699_a_nat,C: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat,D2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A2 @ C )
=> ( ( ord_le8666007276011122963_a_nat @ B2 @ D2 )
=> ( ord_le8666007276011122963_a_nat @ ( inf_in3813033988066259269_a_nat @ A2 @ B2 ) @ ( inf_in3813033988066259269_a_nat @ C @ D2 ) ) ) ) ).
% inf_mono
thf(fact_977_inf__mono,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,C: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b,D2: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A2 @ C )
=> ( ( ord_le118470702582115849rm_a_b @ B2 @ D2 )
=> ( ord_le118470702582115849rm_a_b @ ( inf_in1693821330460682071rm_a_b @ A2 @ B2 ) @ ( inf_in1693821330460682071rm_a_b @ C @ D2 ) ) ) ) ).
% inf_mono
thf(fact_978_le__infI,axiom,
! [X3: set_Pr4934435412358123699_a_nat,A2: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X3 @ A2 )
=> ( ( ord_le8666007276011122963_a_nat @ X3 @ B2 )
=> ( ord_le8666007276011122963_a_nat @ X3 @ ( inf_in3813033988066259269_a_nat @ A2 @ B2 ) ) ) ) ).
% le_infI
thf(fact_979_le__infI,axiom,
! [X3: set_Pr4386577575007340137rm_a_b,A2: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ X3 @ A2 )
=> ( ( ord_le118470702582115849rm_a_b @ X3 @ B2 )
=> ( ord_le118470702582115849rm_a_b @ X3 @ ( inf_in1693821330460682071rm_a_b @ A2 @ B2 ) ) ) ) ).
% le_infI
thf(fact_980_le__infE,axiom,
! [X3: set_Pr4934435412358123699_a_nat,A2: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X3 @ ( inf_in3813033988066259269_a_nat @ A2 @ B2 ) )
=> ~ ( ( ord_le8666007276011122963_a_nat @ X3 @ A2 )
=> ~ ( ord_le8666007276011122963_a_nat @ X3 @ B2 ) ) ) ).
% le_infE
thf(fact_981_le__infE,axiom,
! [X3: set_Pr4386577575007340137rm_a_b,A2: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ X3 @ ( inf_in1693821330460682071rm_a_b @ A2 @ B2 ) )
=> ~ ( ( ord_le118470702582115849rm_a_b @ X3 @ A2 )
=> ~ ( ord_le118470702582115849rm_a_b @ X3 @ B2 ) ) ) ).
% le_infE
thf(fact_982_inf__le2,axiom,
! [X3: set_Pr4934435412358123699_a_nat,Y3: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ ( inf_in3813033988066259269_a_nat @ X3 @ Y3 ) @ Y3 ) ).
% inf_le2
thf(fact_983_inf__le2,axiom,
! [X3: set_Pr4386577575007340137rm_a_b,Y3: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ ( inf_in1693821330460682071rm_a_b @ X3 @ Y3 ) @ Y3 ) ).
% inf_le2
thf(fact_984_inf__le1,axiom,
! [X3: set_Pr4934435412358123699_a_nat,Y3: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ ( inf_in3813033988066259269_a_nat @ X3 @ Y3 ) @ X3 ) ).
% inf_le1
thf(fact_985_inf__le1,axiom,
! [X3: set_Pr4386577575007340137rm_a_b,Y3: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ ( inf_in1693821330460682071rm_a_b @ X3 @ Y3 ) @ X3 ) ).
% inf_le1
thf(fact_986_inf__sup__ord_I1_J,axiom,
! [X3: set_Pr4934435412358123699_a_nat,Y3: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ ( inf_in3813033988066259269_a_nat @ X3 @ Y3 ) @ X3 ) ).
% inf_sup_ord(1)
thf(fact_987_inf__sup__ord_I1_J,axiom,
! [X3: set_Pr4386577575007340137rm_a_b,Y3: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ ( inf_in1693821330460682071rm_a_b @ X3 @ Y3 ) @ X3 ) ).
% inf_sup_ord(1)
thf(fact_988_inf__sup__ord_I2_J,axiom,
! [X3: set_Pr4934435412358123699_a_nat,Y3: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ ( inf_in3813033988066259269_a_nat @ X3 @ Y3 ) @ Y3 ) ).
% inf_sup_ord(2)
thf(fact_989_inf__sup__ord_I2_J,axiom,
! [X3: set_Pr4386577575007340137rm_a_b,Y3: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ ( inf_in1693821330460682071rm_a_b @ X3 @ Y3 ) @ Y3 ) ).
% inf_sup_ord(2)
thf(fact_990_refl__onD,axiom,
! [A: set_Pr4386577575007340137rm_a_b,R: set_Pr2972776593051762503rm_a_b,A2: produc357393685978478089rm_a_b] :
( ( refl_o5344404488616963464rm_a_b @ A @ R )
=> ( ( member5869715511025134514rm_a_b @ A2 @ A )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A2 @ A2 ) @ R ) ) ) ).
% refl_onD
thf(fact_991_refl__onD,axiom,
! [A: set_Pr4934435412358123699_a_nat,R: set_Pr1811044260758604347_a_nat,A2: product_prod_a_nat] :
( ( refl_o8237544196271505854_a_nat @ A @ R )
=> ( ( member5724188588386418708_a_nat @ A2 @ A )
=> ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A2 @ A2 ) @ R ) ) ) ).
% refl_onD
thf(fact_992_refl__onD,axiom,
! [A: set_term_a_b,R: set_Pr4386577575007340137rm_a_b,A2: term_a_b] :
( ( refl_on_term_a_b @ A @ R )
=> ( ( member_term_a_b @ A2 @ A )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ A2 ) @ R ) ) ) ).
% refl_onD
thf(fact_993_refl__onD1,axiom,
! [A: set_Pr4386577575007340137rm_a_b,R: set_Pr2972776593051762503rm_a_b,X3: produc357393685978478089rm_a_b,Y3: produc357393685978478089rm_a_b] :
( ( refl_o5344404488616963464rm_a_b @ A @ R )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X3 @ Y3 ) @ R )
=> ( member5869715511025134514rm_a_b @ X3 @ A ) ) ) ).
% refl_onD1
thf(fact_994_refl__onD1,axiom,
! [A: set_Pr4934435412358123699_a_nat,R: set_Pr1811044260758604347_a_nat,X3: product_prod_a_nat,Y3: product_prod_a_nat] :
( ( refl_o8237544196271505854_a_nat @ A @ R )
=> ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ X3 @ Y3 ) @ R )
=> ( member5724188588386418708_a_nat @ X3 @ A ) ) ) ).
% refl_onD1
thf(fact_995_refl__onD1,axiom,
! [A: set_term_a_b,R: set_Pr4386577575007340137rm_a_b,X3: term_a_b,Y3: term_a_b] :
( ( refl_on_term_a_b @ A @ R )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ R )
=> ( member_term_a_b @ X3 @ A ) ) ) ).
% refl_onD1
thf(fact_996_refl__onD2,axiom,
! [A: set_Pr4386577575007340137rm_a_b,R: set_Pr2972776593051762503rm_a_b,X3: produc357393685978478089rm_a_b,Y3: produc357393685978478089rm_a_b] :
( ( refl_o5344404488616963464rm_a_b @ A @ R )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X3 @ Y3 ) @ R )
=> ( member5869715511025134514rm_a_b @ Y3 @ A ) ) ) ).
% refl_onD2
thf(fact_997_refl__onD2,axiom,
! [A: set_Pr4934435412358123699_a_nat,R: set_Pr1811044260758604347_a_nat,X3: product_prod_a_nat,Y3: product_prod_a_nat] :
( ( refl_o8237544196271505854_a_nat @ A @ R )
=> ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ X3 @ Y3 ) @ R )
=> ( member5724188588386418708_a_nat @ Y3 @ A ) ) ) ).
% refl_onD2
thf(fact_998_refl__onD2,axiom,
! [A: set_term_a_b,R: set_Pr4386577575007340137rm_a_b,X3: term_a_b,Y3: term_a_b] :
( ( refl_on_term_a_b @ A @ R )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ R )
=> ( member_term_a_b @ Y3 @ A ) ) ) ).
% refl_onD2
thf(fact_999_Int__insert__right,axiom,
! [A2: produc357393685978478089rm_a_b,A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ( ( member5869715511025134514rm_a_b @ A2 @ A )
=> ( ( inf_in1693821330460682071rm_a_b @ A @ ( insert7009541432154983385rm_a_b @ A2 @ B ) )
= ( insert7009541432154983385rm_a_b @ A2 @ ( inf_in1693821330460682071rm_a_b @ A @ B ) ) ) )
& ( ~ ( member5869715511025134514rm_a_b @ A2 @ A )
=> ( ( inf_in1693821330460682071rm_a_b @ A @ ( insert7009541432154983385rm_a_b @ A2 @ B ) )
= ( inf_in1693821330460682071rm_a_b @ A @ B ) ) ) ) ).
% Int_insert_right
thf(fact_1000_Int__insert__right,axiom,
! [A2: product_prod_a_nat,A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( ( member5724188588386418708_a_nat @ A2 @ A )
=> ( ( inf_in3813033988066259269_a_nat @ A @ ( insert8054603423593749677_a_nat @ A2 @ B ) )
= ( insert8054603423593749677_a_nat @ A2 @ ( inf_in3813033988066259269_a_nat @ A @ B ) ) ) )
& ( ~ ( member5724188588386418708_a_nat @ A2 @ A )
=> ( ( inf_in3813033988066259269_a_nat @ A @ ( insert8054603423593749677_a_nat @ A2 @ B ) )
= ( inf_in3813033988066259269_a_nat @ A @ B ) ) ) ) ).
% Int_insert_right
thf(fact_1001_Int__insert__left,axiom,
! [A2: produc357393685978478089rm_a_b,C2: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ( ( member5869715511025134514rm_a_b @ A2 @ C2 )
=> ( ( inf_in1693821330460682071rm_a_b @ ( insert7009541432154983385rm_a_b @ A2 @ B ) @ C2 )
= ( insert7009541432154983385rm_a_b @ A2 @ ( inf_in1693821330460682071rm_a_b @ B @ C2 ) ) ) )
& ( ~ ( member5869715511025134514rm_a_b @ A2 @ C2 )
=> ( ( inf_in1693821330460682071rm_a_b @ ( insert7009541432154983385rm_a_b @ A2 @ B ) @ C2 )
= ( inf_in1693821330460682071rm_a_b @ B @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_1002_Int__insert__left,axiom,
! [A2: product_prod_a_nat,C2: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( ( member5724188588386418708_a_nat @ A2 @ C2 )
=> ( ( inf_in3813033988066259269_a_nat @ ( insert8054603423593749677_a_nat @ A2 @ B ) @ C2 )
= ( insert8054603423593749677_a_nat @ A2 @ ( inf_in3813033988066259269_a_nat @ B @ C2 ) ) ) )
& ( ~ ( member5724188588386418708_a_nat @ A2 @ C2 )
=> ( ( inf_in3813033988066259269_a_nat @ ( insert8054603423593749677_a_nat @ A2 @ B ) @ C2 )
= ( inf_in3813033988066259269_a_nat @ B @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_1003_distrib__sup__le,axiom,
! [X3: set_Pr4934435412358123699_a_nat,Y3: set_Pr4934435412358123699_a_nat,Z2: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ ( sup_su459911885395995103_a_nat @ X3 @ ( inf_in3813033988066259269_a_nat @ Y3 @ Z2 ) ) @ ( inf_in3813033988066259269_a_nat @ ( sup_su459911885395995103_a_nat @ X3 @ Y3 ) @ ( sup_su459911885395995103_a_nat @ X3 @ Z2 ) ) ) ).
% distrib_sup_le
thf(fact_1004_distrib__sup__le,axiom,
! [X3: set_Pr4386577575007340137rm_a_b,Y3: set_Pr4386577575007340137rm_a_b,Z2: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ ( sup_su6776935440552674877rm_a_b @ X3 @ ( inf_in1693821330460682071rm_a_b @ Y3 @ Z2 ) ) @ ( inf_in1693821330460682071rm_a_b @ ( sup_su6776935440552674877rm_a_b @ X3 @ Y3 ) @ ( sup_su6776935440552674877rm_a_b @ X3 @ Z2 ) ) ) ).
% distrib_sup_le
thf(fact_1005_distrib__inf__le,axiom,
! [X3: set_Pr4934435412358123699_a_nat,Y3: set_Pr4934435412358123699_a_nat,Z2: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ ( sup_su459911885395995103_a_nat @ ( inf_in3813033988066259269_a_nat @ X3 @ Y3 ) @ ( inf_in3813033988066259269_a_nat @ X3 @ Z2 ) ) @ ( inf_in3813033988066259269_a_nat @ X3 @ ( sup_su459911885395995103_a_nat @ Y3 @ Z2 ) ) ) ).
% distrib_inf_le
thf(fact_1006_distrib__inf__le,axiom,
! [X3: set_Pr4386577575007340137rm_a_b,Y3: set_Pr4386577575007340137rm_a_b,Z2: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ ( sup_su6776935440552674877rm_a_b @ ( inf_in1693821330460682071rm_a_b @ X3 @ Y3 ) @ ( inf_in1693821330460682071rm_a_b @ X3 @ Z2 ) ) @ ( inf_in1693821330460682071rm_a_b @ X3 @ ( sup_su6776935440552674877rm_a_b @ Y3 @ Z2 ) ) ) ).
% distrib_inf_le
thf(fact_1007_Un__Int__assoc__eq,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat,C2: set_Pr4934435412358123699_a_nat] :
( ( ( sup_su459911885395995103_a_nat @ ( inf_in3813033988066259269_a_nat @ A @ B ) @ C2 )
= ( inf_in3813033988066259269_a_nat @ A @ ( sup_su459911885395995103_a_nat @ B @ C2 ) ) )
= ( ord_le8666007276011122963_a_nat @ C2 @ A ) ) ).
% Un_Int_assoc_eq
thf(fact_1008_Un__Int__assoc__eq,axiom,
! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b,C2: set_Pr4386577575007340137rm_a_b] :
( ( ( sup_su6776935440552674877rm_a_b @ ( inf_in1693821330460682071rm_a_b @ A @ B ) @ C2 )
= ( inf_in1693821330460682071rm_a_b @ A @ ( sup_su6776935440552674877rm_a_b @ B @ C2 ) ) )
= ( ord_le118470702582115849rm_a_b @ C2 @ A ) ) ).
% Un_Int_assoc_eq
thf(fact_1009_reflI,axiom,
! [R: set_Pr4386577575007340137rm_a_b] :
( ! [X: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ X ) @ R )
=> ( refl_on_term_a_b @ top_top_set_term_a_b @ R ) ) ).
% reflI
thf(fact_1010_reflD,axiom,
! [R: set_Pr4386577575007340137rm_a_b,A2: term_a_b] :
( ( refl_on_term_a_b @ top_top_set_term_a_b @ R )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ A2 ) @ R ) ) ).
% reflD
thf(fact_1011_wo__rel_Omax2__def,axiom,
! [R: set_Pr4386577575007340137rm_a_b,A2: term_a_b,B2: term_a_b] :
( ( bNF_We251836928773027112rm_a_b @ R )
=> ( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ R )
=> ( ( bNF_We4618759378340593734rm_a_b @ R @ A2 @ B2 )
= B2 ) )
& ( ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ R )
=> ( ( bNF_We4618759378340593734rm_a_b @ R @ A2 @ B2 )
= A2 ) ) ) ) ).
% wo_rel.max2_def
thf(fact_1012_trancl__Int__subset,axiom,
! [R: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ R @ S )
=> ( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ ( inf_in1693821330460682071rm_a_b @ ( transi7922773638565587891rm_a_b @ R ) @ S ) @ R ) @ S )
=> ( ord_le118470702582115849rm_a_b @ ( transi7922773638565587891rm_a_b @ R ) @ S ) ) ) ).
% trancl_Int_subset
thf(fact_1013_refl__O__iff,axiom,
! [A: set_Pr4386577575007340137rm_a_b] :
( ( refl_on_term_a_b @ top_top_set_term_a_b @ A )
= ( ord_le118470702582115849rm_a_b @ id_term_a_b @ A ) ) ).
% refl_O_iff
thf(fact_1014_rtrancl__Int__subset,axiom,
! [S: set_Pr4386577575007340137rm_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ id_term_a_b @ S )
=> ( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ ( inf_in1693821330460682071rm_a_b @ ( transi7742714808557438673rm_a_b @ R ) @ S ) @ R ) @ S )
=> ( ord_le118470702582115849rm_a_b @ ( transi7742714808557438673rm_a_b @ R ) @ S ) ) ) ).
% rtrancl_Int_subset
thf(fact_1015_wo__rel_Omax2__equals1,axiom,
! [R: set_Pr2972776593051762503rm_a_b,A2: produc357393685978478089rm_a_b,B2: produc357393685978478089rm_a_b] :
( ( bNF_We1688910813892497397rm_a_b @ R )
=> ( ( member5869715511025134514rm_a_b @ A2 @ ( field_6884932134483023318rm_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ B2 @ ( field_6884932134483023318rm_a_b @ R ) )
=> ( ( ( bNF_We1369222903935127127rm_a_b @ R @ A2 @ B2 )
= A2 )
= ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B2 @ A2 ) @ R ) ) ) ) ) ).
% wo_rel.max2_equals1
thf(fact_1016_wo__rel_Omax2__equals1,axiom,
! [R: set_Pr1811044260758604347_a_nat,A2: product_prod_a_nat,B2: product_prod_a_nat] :
( ( bNF_We4298404315170990993_a_nat @ R )
=> ( ( member5724188588386418708_a_nat @ A2 @ ( field_8954927560578634480_a_nat @ R ) )
=> ( ( member5724188588386418708_a_nat @ B2 @ ( field_8954927560578634480_a_nat @ R ) )
=> ( ( ( bNF_We7882469860449324975_a_nat @ R @ A2 @ B2 )
= A2 )
= ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ B2 @ A2 ) @ R ) ) ) ) ) ).
% wo_rel.max2_equals1
thf(fact_1017_wo__rel_Omax2__equals1,axiom,
! [R: set_Pr4386577575007340137rm_a_b,A2: term_a_b,B2: term_a_b] :
( ( bNF_We251836928773027112rm_a_b @ R )
=> ( ( member_term_a_b @ A2 @ ( field_term_a_b @ R ) )
=> ( ( member_term_a_b @ B2 @ ( field_term_a_b @ R ) )
=> ( ( ( bNF_We4618759378340593734rm_a_b @ R @ A2 @ B2 )
= A2 )
= ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ A2 ) @ R ) ) ) ) ) ).
% wo_rel.max2_equals1
thf(fact_1018_wo__rel_Omax2__equals2,axiom,
! [R: set_Pr2972776593051762503rm_a_b,A2: produc357393685978478089rm_a_b,B2: produc357393685978478089rm_a_b] :
( ( bNF_We1688910813892497397rm_a_b @ R )
=> ( ( member5869715511025134514rm_a_b @ A2 @ ( field_6884932134483023318rm_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ B2 @ ( field_6884932134483023318rm_a_b @ R ) )
=> ( ( ( bNF_We1369222903935127127rm_a_b @ R @ A2 @ B2 )
= B2 )
= ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A2 @ B2 ) @ R ) ) ) ) ) ).
% wo_rel.max2_equals2
thf(fact_1019_wo__rel_Omax2__equals2,axiom,
! [R: set_Pr1811044260758604347_a_nat,A2: product_prod_a_nat,B2: product_prod_a_nat] :
( ( bNF_We4298404315170990993_a_nat @ R )
=> ( ( member5724188588386418708_a_nat @ A2 @ ( field_8954927560578634480_a_nat @ R ) )
=> ( ( member5724188588386418708_a_nat @ B2 @ ( field_8954927560578634480_a_nat @ R ) )
=> ( ( ( bNF_We7882469860449324975_a_nat @ R @ A2 @ B2 )
= B2 )
= ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A2 @ B2 ) @ R ) ) ) ) ) ).
% wo_rel.max2_equals2
thf(fact_1020_wo__rel_Omax2__equals2,axiom,
! [R: set_Pr4386577575007340137rm_a_b,A2: term_a_b,B2: term_a_b] :
( ( bNF_We251836928773027112rm_a_b @ R )
=> ( ( member_term_a_b @ A2 @ ( field_term_a_b @ R ) )
=> ( ( member_term_a_b @ B2 @ ( field_term_a_b @ R ) )
=> ( ( ( bNF_We4618759378340593734rm_a_b @ R @ A2 @ B2 )
= B2 )
= ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ R ) ) ) ) ) ).
% wo_rel.max2_equals2
thf(fact_1021_wo__rel_Omax2__greater,axiom,
! [R: set_Pr2972776593051762503rm_a_b,A2: produc357393685978478089rm_a_b,B2: produc357393685978478089rm_a_b] :
( ( bNF_We1688910813892497397rm_a_b @ R )
=> ( ( member5869715511025134514rm_a_b @ A2 @ ( field_6884932134483023318rm_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ B2 @ ( field_6884932134483023318rm_a_b @ R ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A2 @ ( bNF_We1369222903935127127rm_a_b @ R @ A2 @ B2 ) ) @ R )
& ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B2 @ ( bNF_We1369222903935127127rm_a_b @ R @ A2 @ B2 ) ) @ R ) ) ) ) ) ).
% wo_rel.max2_greater
thf(fact_1022_wo__rel_Omax2__greater,axiom,
! [R: set_Pr1811044260758604347_a_nat,A2: product_prod_a_nat,B2: product_prod_a_nat] :
( ( bNF_We4298404315170990993_a_nat @ R )
=> ( ( member5724188588386418708_a_nat @ A2 @ ( field_8954927560578634480_a_nat @ R ) )
=> ( ( member5724188588386418708_a_nat @ B2 @ ( field_8954927560578634480_a_nat @ R ) )
=> ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A2 @ ( bNF_We7882469860449324975_a_nat @ R @ A2 @ B2 ) ) @ R )
& ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ B2 @ ( bNF_We7882469860449324975_a_nat @ R @ A2 @ B2 ) ) @ R ) ) ) ) ) ).
% wo_rel.max2_greater
thf(fact_1023_wo__rel_Omax2__greater,axiom,
! [R: set_Pr4386577575007340137rm_a_b,A2: term_a_b,B2: term_a_b] :
( ( bNF_We251836928773027112rm_a_b @ R )
=> ( ( member_term_a_b @ A2 @ ( field_term_a_b @ R ) )
=> ( ( member_term_a_b @ B2 @ ( field_term_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ ( bNF_We4618759378340593734rm_a_b @ R @ A2 @ B2 ) ) @ R )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ ( bNF_We4618759378340593734rm_a_b @ R @ A2 @ B2 ) ) @ R ) ) ) ) ) ).
% wo_rel.max2_greater
thf(fact_1024_refl__on__comp__subset,axiom,
! [A: set_term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( refl_on_term_a_b @ A @ R )
=> ( ord_le118470702582115849rm_a_b @ R @ ( relcom370159955682700863rm_a_b @ ( conver7608559976937845093rm_a_b @ R ) @ R ) ) ) ).
% refl_on_comp_subset
thf(fact_1025_ideriv__SN,axiom,
! [S4: set_Pr4386577575007340137rm_a_b,NS: set_Pr4386577575007340137rm_a_b,R4: set_Pr4386577575007340137rm_a_b,As2: nat > term_a_b] :
( ( abstra4720023341729745482rm_a_b @ S4 @ top_top_set_term_a_b )
=> ( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ NS @ S4 ) @ S4 )
=> ( ( ord_le118470702582115849rm_a_b @ R4 @ ( sup_su6776935440552674877rm_a_b @ NS @ S4 ) )
=> ~ ( abstra5386884192082944997rm_a_b @ ( inf_in1693821330460682071rm_a_b @ S4 @ R4 ) @ ( minus_5192120951422937424rm_a_b @ R4 @ S4 ) @ As2 ) ) ) ) ).
% ideriv_SN
thf(fact_1026_refl__on__domain,axiom,
! [A: set_Pr4386577575007340137rm_a_b,R: set_Pr2972776593051762503rm_a_b,A2: produc357393685978478089rm_a_b,B2: produc357393685978478089rm_a_b] :
( ( refl_o5344404488616963464rm_a_b @ A @ R )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A2 @ B2 ) @ R )
=> ( ( member5869715511025134514rm_a_b @ A2 @ A )
& ( member5869715511025134514rm_a_b @ B2 @ A ) ) ) ) ).
% refl_on_domain
thf(fact_1027_refl__on__domain,axiom,
! [A: set_Pr4934435412358123699_a_nat,R: set_Pr1811044260758604347_a_nat,A2: product_prod_a_nat,B2: product_prod_a_nat] :
( ( refl_o8237544196271505854_a_nat @ A @ R )
=> ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A2 @ B2 ) @ R )
=> ( ( member5724188588386418708_a_nat @ A2 @ A )
& ( member5724188588386418708_a_nat @ B2 @ A ) ) ) ) ).
% refl_on_domain
thf(fact_1028_refl__on__domain,axiom,
! [A: set_term_a_b,R: set_Pr4386577575007340137rm_a_b,A2: term_a_b,B2: term_a_b] :
( ( refl_on_term_a_b @ A @ R )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ R )
=> ( ( member_term_a_b @ A2 @ A )
& ( member_term_a_b @ B2 @ A ) ) ) ) ).
% refl_on_domain
thf(fact_1029_ideriv__mono,axiom,
! [R4: set_Pr4386577575007340137rm_a_b,R6: set_Pr4386577575007340137rm_a_b,S4: set_Pr4386577575007340137rm_a_b,S6: set_Pr4386577575007340137rm_a_b,As2: nat > term_a_b] :
( ( ord_le118470702582115849rm_a_b @ R4 @ R6 )
=> ( ( ord_le118470702582115849rm_a_b @ S4 @ S6 )
=> ( ( abstra5386884192082944997rm_a_b @ R4 @ S4 @ As2 )
=> ( abstra5386884192082944997rm_a_b @ R6 @ S6 @ As2 ) ) ) ) ).
% ideriv_mono
thf(fact_1030_Total__subset__Id,axiom,
! [R: set_Pr4386577575007340137rm_a_b] :
( ( total_on_term_a_b @ ( field_term_a_b @ R ) @ R )
=> ( ( ord_le118470702582115849rm_a_b @ R @ id_term_a_b )
=> ( ( R = bot_bo197521221353338581rm_a_b )
| ? [A5: term_a_b] :
( R
= ( insert7009541432154983385rm_a_b @ ( produc7020197800436672577rm_a_b @ A5 @ A5 ) @ bot_bo197521221353338581rm_a_b ) ) ) ) ) ).
% Total_subset_Id
thf(fact_1031_total__on__subset,axiom,
! [A: set_Pr4934435412358123699_a_nat,R: set_Pr1811044260758604347_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( total_9110697612923228665_a_nat @ A @ R )
=> ( ( ord_le8666007276011122963_a_nat @ B @ A )
=> ( total_9110697612923228665_a_nat @ B @ R ) ) ) ).
% total_on_subset
thf(fact_1032_total__on__subset,axiom,
! [A: set_Pr4386577575007340137rm_a_b,R: set_Pr2972776593051762503rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ( total_4010809703584253837rm_a_b @ A @ R )
=> ( ( ord_le118470702582115849rm_a_b @ B @ A )
=> ( total_4010809703584253837rm_a_b @ B @ R ) ) ) ).
% total_on_subset
thf(fact_1033_total__on__def,axiom,
( total_on_term_a_b
= ( ^ [A6: set_term_a_b,R3: 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 ) @ R3 )
| ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y5 @ X4 ) @ R3 ) ) ) ) ) ) ) ).
% total_on_def
thf(fact_1034_total__onI,axiom,
! [A: set_Pr4386577575007340137rm_a_b,R: set_Pr2972776593051762503rm_a_b] :
( ! [X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X @ A )
=> ( ( member5869715511025134514rm_a_b @ Y @ A )
=> ( ( X != Y )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ R )
| ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y @ X ) @ R ) ) ) ) )
=> ( total_4010809703584253837rm_a_b @ A @ R ) ) ).
% total_onI
thf(fact_1035_total__onI,axiom,
! [A: set_Pr4934435412358123699_a_nat,R: set_Pr1811044260758604347_a_nat] :
( ! [X: product_prod_a_nat,Y: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ X @ A )
=> ( ( member5724188588386418708_a_nat @ Y @ A )
=> ( ( X != Y )
=> ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ X @ Y ) @ R )
| ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ Y @ X ) @ R ) ) ) ) )
=> ( total_9110697612923228665_a_nat @ A @ R ) ) ).
% total_onI
thf(fact_1036_total__onI,axiom,
! [A: set_term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ! [X: term_a_b,Y: term_a_b] :
( ( member_term_a_b @ X @ A )
=> ( ( member_term_a_b @ Y @ A )
=> ( ( X != Y )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ R )
| ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X ) @ R ) ) ) ) )
=> ( total_on_term_a_b @ A @ R ) ) ).
% total_onI
thf(fact_1037_totalI,axiom,
! [R: set_Pr4386577575007340137rm_a_b] :
( ! [X: term_a_b,Y: term_a_b] :
( ( X != Y )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ R )
| ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X ) @ R ) ) )
=> ( total_on_term_a_b @ top_top_set_term_a_b @ R ) ) ).
% totalI
thf(fact_1038_Total__Id__Field,axiom,
! [R: set_Pr4386577575007340137rm_a_b] :
( ( total_on_term_a_b @ ( field_term_a_b @ R ) @ R )
=> ( ~ ( ord_le118470702582115849rm_a_b @ R @ id_term_a_b )
=> ( ( field_term_a_b @ R )
= ( field_term_a_b @ ( minus_5192120951422937424rm_a_b @ R @ id_term_a_b ) ) ) ) ) ).
% Total_Id_Field
thf(fact_1039_linear__order__on__singleton,axiom,
! [X3: term_a_b] : ( order_5388802246213473311rm_a_b @ ( insert_term_a_b @ X3 @ bot_bot_set_term_a_b ) @ ( insert7009541432154983385rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ X3 ) @ bot_bo197521221353338581rm_a_b ) ) ).
% linear_order_on_singleton
thf(fact_1040_Linear__order__in__diff__Id,axiom,
! [R: set_Pr2972776593051762503rm_a_b,A2: produc357393685978478089rm_a_b,B2: produc357393685978478089rm_a_b] :
( ( order_681589137112254398rm_a_b @ ( field_6884932134483023318rm_a_b @ R ) @ R )
=> ( ( member5869715511025134514rm_a_b @ A2 @ ( field_6884932134483023318rm_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ B2 @ ( field_6884932134483023318rm_a_b @ R ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A2 @ B2 ) @ R )
= ( ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B2 @ A2 ) @ ( minus_45395518312058030rm_a_b @ R @ id_Pro3488709451141449061rm_a_b ) ) ) ) ) ) ) ).
% Linear_order_in_diff_Id
thf(fact_1041_Linear__order__in__diff__Id,axiom,
! [R: set_Pr1811044260758604347_a_nat,A2: product_prod_a_nat,B2: product_prod_a_nat] :
( ( order_6388386362885710600_a_nat @ ( field_8954927560578634480_a_nat @ R ) @ R )
=> ( ( member5724188588386418708_a_nat @ A2 @ ( field_8954927560578634480_a_nat @ R ) )
=> ( ( member5724188588386418708_a_nat @ B2 @ ( field_8954927560578634480_a_nat @ R ) )
=> ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A2 @ B2 ) @ R )
= ( ~ ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ B2 @ A2 ) @ ( minus_1341117088723627554_a_nat @ R @ id_Pro5207055338379199009_a_nat ) ) ) ) ) ) ) ).
% Linear_order_in_diff_Id
thf(fact_1042_Linear__order__in__diff__Id,axiom,
! [R: set_Pr4386577575007340137rm_a_b,A2: term_a_b,B2: term_a_b] :
( ( order_5388802246213473311rm_a_b @ ( field_term_a_b @ R ) @ R )
=> ( ( member_term_a_b @ A2 @ ( field_term_a_b @ R ) )
=> ( ( member_term_a_b @ B2 @ ( field_term_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ R )
= ( ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ A2 ) @ ( minus_5192120951422937424rm_a_b @ R @ id_term_a_b ) ) ) ) ) ) ) ).
% Linear_order_in_diff_Id
thf(fact_1043_Linear__order__wf__diff__Id,axiom,
! [R: set_Pr4386577575007340137rm_a_b] :
( ( order_5388802246213473311rm_a_b @ ( field_term_a_b @ R ) @ R )
=> ( ( wf_term_a_b @ ( minus_5192120951422937424rm_a_b @ R @ id_term_a_b ) )
= ( ! [A6: set_term_a_b] :
( ( ord_le2705286416250468010rm_a_b @ A6 @ ( field_term_a_b @ R ) )
=> ( ( 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 ) @ R ) ) ) ) ) ) ) ) ).
% Linear_order_wf_diff_Id
thf(fact_1044_Linear__order__wf__diff__Id,axiom,
! [R: set_Pr1811044260758604347_a_nat] :
( ( order_6388386362885710600_a_nat @ ( field_8954927560578634480_a_nat @ R ) @ R )
=> ( ( wf_Pro8109884346892000340_a_nat @ ( minus_1341117088723627554_a_nat @ R @ id_Pro5207055338379199009_a_nat ) )
= ( ! [A6: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A6 @ ( field_8954927560578634480_a_nat @ R ) )
=> ( ( A6 != bot_bo9049108969261143879_a_nat )
=> ? [X4: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ X4 @ A6 )
& ! [Y5: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ Y5 @ A6 )
=> ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ X4 @ Y5 ) @ R ) ) ) ) ) ) ) ) ).
% Linear_order_wf_diff_Id
thf(fact_1045_Linear__order__wf__diff__Id,axiom,
! [R: set_Pr2972776593051762503rm_a_b] :
( ( order_681589137112254398rm_a_b @ ( field_6884932134483023318rm_a_b @ R ) @ R )
=> ( ( wf_Pro2335863617654816626rm_a_b @ ( minus_45395518312058030rm_a_b @ R @ id_Pro3488709451141449061rm_a_b ) )
= ( ! [A6: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A6 @ ( field_6884932134483023318rm_a_b @ R ) )
=> ( ( A6 != bot_bo197521221353338581rm_a_b )
=> ? [X4: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X4 @ A6 )
& ! [Y5: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ Y5 @ A6 )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X4 @ Y5 ) @ R ) ) ) ) ) ) ) ) ).
% Linear_order_wf_diff_Id
thf(fact_1046_in__lex__prod,axiom,
! [A2: term_a_b,B2: term_a_b,A3: term_a_b,B3: term_a_b,R: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( produc7020197800436672577rm_a_b @ A3 @ B3 ) ) @ ( lex_pr6163557265797435481rm_a_b @ R @ S ) )
= ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ A3 ) @ R )
| ( ( A2 = A3 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ B3 ) @ S ) ) ) ) ).
% in_lex_prod
thf(fact_1047_wf__insert,axiom,
! [Y3: term_a_b,X3: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( wf_term_a_b @ ( insert7009541432154983385rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ X3 ) @ R ) )
= ( ( wf_term_a_b @ R )
& ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ ( transi7742714808557438673rm_a_b @ R ) ) ) ) ).
% wf_insert
thf(fact_1048_wfE__min_H,axiom,
! [R4: set_Pr2972776593051762503rm_a_b,Q: set_Pr4386577575007340137rm_a_b] :
( ( wf_Pro2335863617654816626rm_a_b @ R4 )
=> ( ( Q != bot_bo197521221353338581rm_a_b )
=> ~ ! [Z3: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ Z3 @ Q )
=> ~ ! [Y6: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y6 @ Z3 ) @ R4 )
=> ~ ( member5869715511025134514rm_a_b @ Y6 @ Q ) ) ) ) ) ).
% wfE_min'
thf(fact_1049_wfE__min_H,axiom,
! [R4: set_Pr1811044260758604347_a_nat,Q: set_Pr4934435412358123699_a_nat] :
( ( wf_Pro8109884346892000340_a_nat @ R4 )
=> ( ( Q != bot_bo9049108969261143879_a_nat )
=> ~ ! [Z3: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ Z3 @ Q )
=> ~ ! [Y6: product_prod_a_nat] :
( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ Y6 @ Z3 ) @ R4 )
=> ~ ( member5724188588386418708_a_nat @ Y6 @ Q ) ) ) ) ) ).
% wfE_min'
thf(fact_1050_wfE__min_H,axiom,
! [R4: set_Pr4386577575007340137rm_a_b,Q: set_term_a_b] :
( ( wf_term_a_b @ R4 )
=> ( ( Q != bot_bot_set_term_a_b )
=> ~ ! [Z3: term_a_b] :
( ( member_term_a_b @ Z3 @ Q )
=> ~ ! [Y6: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y6 @ Z3 ) @ R4 )
=> ~ ( member_term_a_b @ Y6 @ Q ) ) ) ) ) ).
% wfE_min'
thf(fact_1051_minimal__imp__wf,axiom,
! [R: set_Pr2972776593051762503rm_a_b] :
( ! [Q3: set_Pr4386577575007340137rm_a_b] :
( ? [X: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ X @ Q3 )
=> ? [X6: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X6 @ Q3 )
& ! [Y: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X6 @ Y ) @ R )
=> ~ ( member5869715511025134514rm_a_b @ Y @ Q3 ) ) ) )
=> ( wf_Pro2335863617654816626rm_a_b @ ( conver1575690562132235771rm_a_b @ R ) ) ) ).
% minimal_imp_wf
thf(fact_1052_minimal__imp__wf,axiom,
! [R: set_Pr1811044260758604347_a_nat] :
( ! [Q3: set_Pr4934435412358123699_a_nat] :
( ? [X: product_prod_a_nat] : ( member5724188588386418708_a_nat @ X @ Q3 )
=> ? [X6: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ X6 @ Q3 )
& ! [Y: product_prod_a_nat] :
( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ X6 @ Y ) @ R )
=> ~ ( member5724188588386418708_a_nat @ Y @ Q3 ) ) ) )
=> ( wf_Pro8109884346892000340_a_nat @ ( conver4024954229950350007_a_nat @ R ) ) ) ).
% minimal_imp_wf
thf(fact_1053_minimal__imp__wf,axiom,
! [R: set_Pr4386577575007340137rm_a_b] :
( ! [Q3: set_term_a_b] :
( ? [X: term_a_b] : ( member_term_a_b @ X @ Q3 )
=> ? [X6: term_a_b] :
( ( member_term_a_b @ X6 @ Q3 )
& ! [Y: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X6 @ Y ) @ R )
=> ~ ( member_term_a_b @ Y @ Q3 ) ) ) )
=> ( wf_term_a_b @ ( conver7608559976937845093rm_a_b @ R ) ) ) ).
% minimal_imp_wf
thf(fact_1054_wf__relcomp__compatible,axiom,
! [R4: set_Pr4386577575007340137rm_a_b,S4: set_Pr4386577575007340137rm_a_b] :
( ( wf_term_a_b @ R4 )
=> ( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ R4 @ S4 ) @ ( relcom370159955682700863rm_a_b @ S4 @ R4 ) )
=> ( wf_term_a_b @ ( relcom370159955682700863rm_a_b @ S4 @ R4 ) ) ) ) ).
% wf_relcomp_compatible
thf(fact_1055_wf__def,axiom,
( wf_term_a_b
= ( ^ [R3: set_Pr4386577575007340137rm_a_b] :
! [P3: term_a_b > $o] :
( ! [X4: term_a_b] :
( ! [Y5: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y5 @ X4 ) @ R3 )
=> ( P3 @ Y5 ) )
=> ( P3 @ X4 ) )
=> ! [X7: term_a_b] : ( P3 @ X7 ) ) ) ) ).
% wf_def
thf(fact_1056_wfE__min,axiom,
! [R4: set_Pr2972776593051762503rm_a_b,X3: produc357393685978478089rm_a_b,Q: set_Pr4386577575007340137rm_a_b] :
( ( wf_Pro2335863617654816626rm_a_b @ R4 )
=> ( ( member5869715511025134514rm_a_b @ X3 @ Q )
=> ~ ! [Z3: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ Z3 @ Q )
=> ~ ! [Y6: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y6 @ Z3 ) @ R4 )
=> ~ ( member5869715511025134514rm_a_b @ Y6 @ Q ) ) ) ) ) ).
% wfE_min
thf(fact_1057_wfE__min,axiom,
! [R4: set_Pr1811044260758604347_a_nat,X3: product_prod_a_nat,Q: set_Pr4934435412358123699_a_nat] :
( ( wf_Pro8109884346892000340_a_nat @ R4 )
=> ( ( member5724188588386418708_a_nat @ X3 @ Q )
=> ~ ! [Z3: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ Z3 @ Q )
=> ~ ! [Y6: product_prod_a_nat] :
( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ Y6 @ Z3 ) @ R4 )
=> ~ ( member5724188588386418708_a_nat @ Y6 @ Q ) ) ) ) ) ).
% wfE_min
thf(fact_1058_wfE__min,axiom,
! [R4: set_Pr4386577575007340137rm_a_b,X3: term_a_b,Q: set_term_a_b] :
( ( wf_term_a_b @ R4 )
=> ( ( member_term_a_b @ X3 @ Q )
=> ~ ! [Z3: term_a_b] :
( ( member_term_a_b @ Z3 @ Q )
=> ~ ! [Y6: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y6 @ Z3 ) @ R4 )
=> ~ ( member_term_a_b @ Y6 @ Q ) ) ) ) ) ).
% wfE_min
thf(fact_1059_wfI__min,axiom,
! [R4: set_Pr2972776593051762503rm_a_b] :
( ! [X: produc357393685978478089rm_a_b,Q3: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ X @ Q3 )
=> ? [Xa2: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ Xa2 @ Q3 )
& ! [Y: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y @ Xa2 ) @ R4 )
=> ~ ( member5869715511025134514rm_a_b @ Y @ Q3 ) ) ) )
=> ( wf_Pro2335863617654816626rm_a_b @ R4 ) ) ).
% wfI_min
thf(fact_1060_wfI__min,axiom,
! [R4: set_Pr1811044260758604347_a_nat] :
( ! [X: product_prod_a_nat,Q3: set_Pr4934435412358123699_a_nat] :
( ( member5724188588386418708_a_nat @ X @ Q3 )
=> ? [Xa2: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ Xa2 @ Q3 )
& ! [Y: product_prod_a_nat] :
( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ Y @ Xa2 ) @ R4 )
=> ~ ( member5724188588386418708_a_nat @ Y @ Q3 ) ) ) )
=> ( wf_Pro8109884346892000340_a_nat @ R4 ) ) ).
% wfI_min
thf(fact_1061_wfI__min,axiom,
! [R4: set_Pr4386577575007340137rm_a_b] :
( ! [X: term_a_b,Q3: set_term_a_b] :
( ( member_term_a_b @ X @ Q3 )
=> ? [Xa2: term_a_b] :
( ( member_term_a_b @ Xa2 @ Q3 )
& ! [Y: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ Xa2 ) @ R4 )
=> ~ ( member_term_a_b @ Y @ Q3 ) ) ) )
=> ( wf_term_a_b @ R4 ) ) ).
% wfI_min
thf(fact_1062_wfUNIVI,axiom,
! [R: set_Pr4386577575007340137rm_a_b] :
( ! [P4: term_a_b > $o,X: term_a_b] :
( ! [Xa2: term_a_b] :
( ! [Y: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ Xa2 ) @ R )
=> ( P4 @ Y ) )
=> ( P4 @ Xa2 ) )
=> ( P4 @ X ) )
=> ( wf_term_a_b @ R ) ) ).
% wfUNIVI
thf(fact_1063_wf__asym,axiom,
! [R: set_Pr4386577575007340137rm_a_b,A2: term_a_b,X3: term_a_b] :
( ( wf_term_a_b @ R )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ X3 ) @ R )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ A2 ) @ R ) ) ) ).
% wf_asym
thf(fact_1064_wf__induct,axiom,
! [R: set_Pr4386577575007340137rm_a_b,P: term_a_b > $o,A2: term_a_b] :
( ( wf_term_a_b @ R )
=> ( ! [X: term_a_b] :
( ! [Y6: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y6 @ X ) @ R )
=> ( P @ Y6 ) )
=> ( P @ X ) )
=> ( P @ A2 ) ) ) ).
% wf_induct
thf(fact_1065_wf__irrefl,axiom,
! [R: set_Pr4386577575007340137rm_a_b,A2: term_a_b] :
( ( wf_term_a_b @ R )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ A2 ) @ R ) ) ).
% wf_irrefl
thf(fact_1066_wf__not__sym,axiom,
! [R: set_Pr4386577575007340137rm_a_b,A2: term_a_b,X3: term_a_b] :
( ( wf_term_a_b @ R )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ X3 ) @ R )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ A2 ) @ R ) ) ) ).
% wf_not_sym
thf(fact_1067_wf__not__refl,axiom,
! [R: set_Pr4386577575007340137rm_a_b,A2: term_a_b] :
( ( wf_term_a_b @ R )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ A2 ) @ R ) ) ).
% wf_not_refl
thf(fact_1068_wf__eq__minimal,axiom,
( wf_Pro2335863617654816626rm_a_b
= ( ^ [R3: set_Pr2972776593051762503rm_a_b] :
! [Q4: set_Pr4386577575007340137rm_a_b] :
( ? [X4: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ X4 @ Q4 )
=> ? [X4: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X4 @ Q4 )
& ! [Y5: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y5 @ X4 ) @ R3 )
=> ~ ( member5869715511025134514rm_a_b @ Y5 @ Q4 ) ) ) ) ) ) ).
% wf_eq_minimal
thf(fact_1069_wf__eq__minimal,axiom,
( wf_Pro8109884346892000340_a_nat
= ( ^ [R3: set_Pr1811044260758604347_a_nat] :
! [Q4: set_Pr4934435412358123699_a_nat] :
( ? [X4: product_prod_a_nat] : ( member5724188588386418708_a_nat @ X4 @ Q4 )
=> ? [X4: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ X4 @ Q4 )
& ! [Y5: product_prod_a_nat] :
( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ Y5 @ X4 ) @ R3 )
=> ~ ( member5724188588386418708_a_nat @ Y5 @ Q4 ) ) ) ) ) ) ).
% wf_eq_minimal
thf(fact_1070_wf__eq__minimal,axiom,
( wf_term_a_b
= ( ^ [R3: set_Pr4386577575007340137rm_a_b] :
! [Q4: set_term_a_b] :
( ? [X4: term_a_b] : ( member_term_a_b @ X4 @ Q4 )
=> ? [X4: term_a_b] :
( ( member_term_a_b @ X4 @ Q4 )
& ! [Y5: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y5 @ X4 ) @ R3 )
=> ~ ( member_term_a_b @ Y5 @ Q4 ) ) ) ) ) ) ).
% wf_eq_minimal
thf(fact_1071_wf__induct__rule,axiom,
! [R: set_Pr4386577575007340137rm_a_b,P: term_a_b > $o,A2: term_a_b] :
( ( wf_term_a_b @ R )
=> ( ! [X: term_a_b] :
( ! [Y6: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y6 @ X ) @ R )
=> ( P @ Y6 ) )
=> ( P @ X ) )
=> ( P @ A2 ) ) ) ).
% wf_induct_rule
thf(fact_1072_wf__if__convertible__to__wf,axiom,
! [S: set_Pr4386577575007340137rm_a_b,R: set_Pr4386577575007340137rm_a_b,F2: term_a_b > term_a_b] :
( ( wf_term_a_b @ S )
=> ( ! [X: term_a_b,Y: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ R )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F2 @ X ) @ ( F2 @ Y ) ) @ S ) )
=> ( wf_term_a_b @ R ) ) ) ).
% wf_if_convertible_to_wf
thf(fact_1073_wf__subset,axiom,
! [R: set_Pr4386577575007340137rm_a_b,P2: set_Pr4386577575007340137rm_a_b] :
( ( wf_term_a_b @ R )
=> ( ( ord_le118470702582115849rm_a_b @ P2 @ R )
=> ( wf_term_a_b @ P2 ) ) ) ).
% wf_subset
thf(fact_1074_wf__union__compatible,axiom,
! [R4: set_Pr4386577575007340137rm_a_b,S4: set_Pr4386577575007340137rm_a_b] :
( ( wf_term_a_b @ R4 )
=> ( ( wf_term_a_b @ S4 )
=> ( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ R4 @ S4 ) @ R4 )
=> ( wf_term_a_b @ ( sup_su6776935440552674877rm_a_b @ R4 @ S4 ) ) ) ) ) ).
% wf_union_compatible
thf(fact_1075_wf__eq__minimal2,axiom,
( wf_term_a_b
= ( ^ [R3: set_Pr4386577575007340137rm_a_b] :
! [A6: set_term_a_b] :
( ( ( ord_le2705286416250468010rm_a_b @ A6 @ ( field_term_a_b @ R3 ) )
& ( 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 ) @ R3 ) ) ) ) ) ) ).
% wf_eq_minimal2
thf(fact_1076_wf__eq__minimal2,axiom,
( wf_Pro8109884346892000340_a_nat
= ( ^ [R3: set_Pr1811044260758604347_a_nat] :
! [A6: set_Pr4934435412358123699_a_nat] :
( ( ( ord_le8666007276011122963_a_nat @ A6 @ ( field_8954927560578634480_a_nat @ R3 ) )
& ( A6 != bot_bo9049108969261143879_a_nat ) )
=> ? [X4: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ X4 @ A6 )
& ! [Y5: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ Y5 @ A6 )
=> ~ ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ Y5 @ X4 ) @ R3 ) ) ) ) ) ) ).
% wf_eq_minimal2
thf(fact_1077_wf__eq__minimal2,axiom,
( wf_Pro2335863617654816626rm_a_b
= ( ^ [R3: set_Pr2972776593051762503rm_a_b] :
! [A6: set_Pr4386577575007340137rm_a_b] :
( ( ( ord_le118470702582115849rm_a_b @ A6 @ ( field_6884932134483023318rm_a_b @ R3 ) )
& ( A6 != bot_bo197521221353338581rm_a_b ) )
=> ? [X4: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X4 @ A6 )
& ! [Y5: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ Y5 @ A6 )
=> ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y5 @ X4 ) @ R3 ) ) ) ) ) ) ).
% wf_eq_minimal2
thf(fact_1078_qc__wf__relto__iff,axiom,
! [R4: set_Pr4386577575007340137rm_a_b,S4: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ R4 @ S4 ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ R4 @ S4 ) ) @ R4 ) )
=> ( ( wf_term_a_b @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ S4 ) @ ( relcom370159955682700863rm_a_b @ R4 @ ( transi7742714808557438673rm_a_b @ S4 ) ) ) )
= ( wf_term_a_b @ R4 ) ) ) ).
% qc_wf_relto_iff
thf(fact_1079_reduction__pairI,axiom,
! [R4: set_Pr4386577575007340137rm_a_b,S4: set_Pr4386577575007340137rm_a_b] :
( ( wf_term_a_b @ R4 )
=> ( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ R4 @ S4 ) @ R4 )
=> ( fun_re2623890006011866704rm_a_b @ ( produc3977138704197412567rm_a_b @ R4 @ S4 ) ) ) ) ).
% reduction_pairI
thf(fact_1080_non__strict__ending,axiom,
! [T: nat > term_a_b,R4: set_Pr4386577575007340137rm_a_b,S4: set_Pr4386577575007340137rm_a_b] :
( ! [I3: nat] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( T @ I3 ) @ ( T @ ( suc @ I3 ) ) ) @ ( sup_su6776935440552674877rm_a_b @ R4 @ S4 ) )
=> ( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ R4 @ S4 ) @ S4 )
=> ( ( abstra4720023341729745482rm_a_b @ S4 @ ( insert_term_a_b @ ( T @ zero_zero_nat ) @ bot_bot_set_term_a_b ) )
=> ? [J3: nat] :
! [I4: nat] :
( ( ord_less_eq_nat @ J3 @ I4 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( T @ I4 ) @ ( T @ ( suc @ I4 ) ) ) @ ( minus_5192120951422937424rm_a_b @ R4 @ S4 ) ) ) ) ) ) ).
% non_strict_ending
thf(fact_1081_not__SN__onI,axiom,
! [F2: nat > produc357393685978478089rm_a_b,X5: set_Pr4386577575007340137rm_a_b,R4: set_Pr2972776593051762503rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( F2 @ zero_zero_nat ) @ X5 )
=> ( ! [I3: nat] : ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( F2 @ I3 ) @ ( F2 @ ( suc @ I3 ) ) ) @ R4 )
=> ~ ( abstra2398554102055911763rm_a_b @ R4 @ X5 ) ) ) ).
% not_SN_onI
thf(fact_1082_not__SN__onI,axiom,
! [F2: nat > product_prod_a_nat,X5: set_Pr4934435412358123699_a_nat,R4: set_Pr1811044260758604347_a_nat] :
( ( member5724188588386418708_a_nat @ ( F2 @ zero_zero_nat ) @ X5 )
=> ( ! [I3: nat] : ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ ( F2 @ I3 ) @ ( F2 @ ( suc @ I3 ) ) ) @ R4 )
=> ~ ( abstra8653715922312955827_a_nat @ R4 @ X5 ) ) ) ).
% not_SN_onI
thf(fact_1083_not__SN__onI,axiom,
! [F2: nat > term_a_b,X5: set_term_a_b,R4: set_Pr4386577575007340137rm_a_b] :
( ( member_term_a_b @ ( F2 @ zero_zero_nat ) @ X5 )
=> ( ! [I3: nat] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F2 @ I3 ) @ ( F2 @ ( suc @ I3 ) ) ) @ R4 )
=> ~ ( abstra4720023341729745482rm_a_b @ R4 @ X5 ) ) ) ).
% not_SN_onI
thf(fact_1084_SN__onI,axiom,
! [A: set_Pr4386577575007340137rm_a_b,R: set_Pr2972776593051762503rm_a_b] :
( ! [F4: nat > produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( F4 @ zero_zero_nat ) @ A )
=> ~ ! [I4: nat] : ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( F4 @ I4 ) @ ( F4 @ ( suc @ I4 ) ) ) @ R ) )
=> ( abstra2398554102055911763rm_a_b @ R @ A ) ) ).
% SN_onI
thf(fact_1085_SN__onI,axiom,
! [A: set_Pr4934435412358123699_a_nat,R: set_Pr1811044260758604347_a_nat] :
( ! [F4: nat > product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ ( F4 @ zero_zero_nat ) @ A )
=> ~ ! [I4: nat] : ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ ( F4 @ I4 ) @ ( F4 @ ( suc @ I4 ) ) ) @ R ) )
=> ( abstra8653715922312955827_a_nat @ R @ A ) ) ).
% SN_onI
thf(fact_1086_SN__onI,axiom,
! [A: set_term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ! [F4: nat > term_a_b] :
( ( member_term_a_b @ ( F4 @ zero_zero_nat ) @ A )
=> ~ ! [I4: nat] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F4 @ I4 ) @ ( F4 @ ( suc @ I4 ) ) ) @ R ) )
=> ( abstra4720023341729745482rm_a_b @ R @ A ) ) ).
% SN_onI
thf(fact_1087_not__SN__onE,axiom,
! [R: set_Pr2972776593051762503rm_a_b,A: set_Pr4386577575007340137rm_a_b] :
( ~ ( abstra2398554102055911763rm_a_b @ R @ A )
=> ~ ! [F4: nat > produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( F4 @ zero_zero_nat ) @ A )
=> ~ ! [I4: nat] : ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( F4 @ I4 ) @ ( F4 @ ( suc @ I4 ) ) ) @ R ) ) ) ).
% not_SN_onE
thf(fact_1088_not__SN__onE,axiom,
! [R: set_Pr1811044260758604347_a_nat,A: set_Pr4934435412358123699_a_nat] :
( ~ ( abstra8653715922312955827_a_nat @ R @ A )
=> ~ ! [F4: nat > product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ ( F4 @ zero_zero_nat ) @ A )
=> ~ ! [I4: nat] : ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ ( F4 @ I4 ) @ ( F4 @ ( suc @ I4 ) ) ) @ R ) ) ) ).
% not_SN_onE
thf(fact_1089_not__SN__onE,axiom,
! [R: set_Pr4386577575007340137rm_a_b,A: set_term_a_b] :
( ~ ( abstra4720023341729745482rm_a_b @ R @ A )
=> ~ ! [F4: nat > term_a_b] :
( ( member_term_a_b @ ( F4 @ zero_zero_nat ) @ A )
=> ~ ! [I4: nat] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F4 @ I4 ) @ ( F4 @ ( suc @ I4 ) ) ) @ R ) ) ) ).
% not_SN_onE
thf(fact_1090_SN__on__def,axiom,
( abstra2398554102055911763rm_a_b
= ( ^ [R3: set_Pr2972776593051762503rm_a_b,A6: set_Pr4386577575007340137rm_a_b] :
~ ? [F5: nat > produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( F5 @ zero_zero_nat ) @ A6 )
& ! [I: nat] : ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( F5 @ I ) @ ( F5 @ ( suc @ I ) ) ) @ R3 ) ) ) ) ).
% SN_on_def
thf(fact_1091_SN__on__def,axiom,
( abstra8653715922312955827_a_nat
= ( ^ [R3: set_Pr1811044260758604347_a_nat,A6: set_Pr4934435412358123699_a_nat] :
~ ? [F5: nat > product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ ( F5 @ zero_zero_nat ) @ A6 )
& ! [I: nat] : ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ ( F5 @ I ) @ ( F5 @ ( suc @ I ) ) ) @ R3 ) ) ) ) ).
% SN_on_def
thf(fact_1092_SN__on__def,axiom,
( abstra4720023341729745482rm_a_b
= ( ^ [R3: set_Pr4386577575007340137rm_a_b,A6: set_term_a_b] :
~ ? [F5: nat > term_a_b] :
( ( member_term_a_b @ ( F5 @ zero_zero_nat ) @ A6 )
& ! [I: nat] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F5 @ I ) @ ( F5 @ ( suc @ I ) ) ) @ R3 ) ) ) ) ).
% SN_on_def
thf(fact_1093_SN__onE,axiom,
! [R: set_Pr2972776593051762503rm_a_b,A: set_Pr4386577575007340137rm_a_b] :
( ( abstra2398554102055911763rm_a_b @ R @ A )
=> ~ ? [F6: nat > produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( F6 @ zero_zero_nat ) @ A )
& ! [I3: nat] : ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( F6 @ I3 ) @ ( F6 @ ( suc @ I3 ) ) ) @ R ) ) ) ).
% SN_onE
thf(fact_1094_SN__onE,axiom,
! [R: set_Pr1811044260758604347_a_nat,A: set_Pr4934435412358123699_a_nat] :
( ( abstra8653715922312955827_a_nat @ R @ A )
=> ~ ? [F6: nat > product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ ( F6 @ zero_zero_nat ) @ A )
& ! [I3: nat] : ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ ( F6 @ I3 ) @ ( F6 @ ( suc @ I3 ) ) ) @ R ) ) ) ).
% SN_onE
thf(fact_1095_SN__onE,axiom,
! [R: set_Pr4386577575007340137rm_a_b,A: set_term_a_b] :
( ( abstra4720023341729745482rm_a_b @ R @ A )
=> ~ ? [F6: nat > term_a_b] :
( ( member_term_a_b @ ( F6 @ zero_zero_nat ) @ A )
& ! [I3: nat] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F6 @ I3 ) @ ( F6 @ ( suc @ I3 ) ) ) @ R ) ) ) ).
% SN_onE
thf(fact_1096_relpow__E2,axiom,
! [X3: term_a_b,Z2: term_a_b,N: nat,R4: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Z2 ) @ ( compow4057154403645558940rm_a_b @ N @ R4 ) )
=> ( ( ( N = zero_zero_nat )
=> ( X3 != Z2 ) )
=> ~ ! [Y: term_a_b,M2: nat] :
( ( N
= ( suc @ M2 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y ) @ R4 )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ Z2 ) @ ( compow4057154403645558940rm_a_b @ M2 @ R4 ) ) ) ) ) ) ).
% relpow_E2
thf(fact_1097_relpow__E,axiom,
! [X3: term_a_b,Z2: term_a_b,N: nat,R4: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Z2 ) @ ( compow4057154403645558940rm_a_b @ N @ R4 ) )
=> ( ( ( N = zero_zero_nat )
=> ( X3 != Z2 ) )
=> ~ ! [Y: term_a_b,M2: nat] :
( ( N
= ( suc @ M2 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y ) @ ( compow4057154403645558940rm_a_b @ M2 @ R4 ) )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ Z2 ) @ R4 ) ) ) ) ) ).
% relpow_E
thf(fact_1098_relpow__Suc__E,axiom,
! [X3: term_a_b,Z2: term_a_b,N: nat,R4: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Z2 ) @ ( compow4057154403645558940rm_a_b @ ( suc @ N ) @ R4 ) )
=> ~ ! [Y: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y ) @ ( compow4057154403645558940rm_a_b @ N @ R4 ) )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ Z2 ) @ R4 ) ) ) ).
% relpow_Suc_E
thf(fact_1099_relpow__Suc__I,axiom,
! [X3: term_a_b,Y3: term_a_b,N: nat,R4: set_Pr4386577575007340137rm_a_b,Z2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ ( compow4057154403645558940rm_a_b @ N @ R4 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ Z2 ) @ R4 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Z2 ) @ ( compow4057154403645558940rm_a_b @ ( suc @ N ) @ R4 ) ) ) ) ).
% relpow_Suc_I
thf(fact_1100_relpow__Suc__D2,axiom,
! [X3: term_a_b,Z2: term_a_b,N: nat,R4: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Z2 ) @ ( compow4057154403645558940rm_a_b @ ( suc @ N ) @ R4 ) )
=> ? [Y: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y ) @ R4 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ Z2 ) @ ( compow4057154403645558940rm_a_b @ N @ R4 ) ) ) ) ).
% relpow_Suc_D2
thf(fact_1101_relpow__Suc__E2,axiom,
! [X3: term_a_b,Z2: term_a_b,N: nat,R4: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Z2 ) @ ( compow4057154403645558940rm_a_b @ ( suc @ N ) @ R4 ) )
=> ~ ! [Y: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y ) @ R4 )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ Z2 ) @ ( compow4057154403645558940rm_a_b @ N @ R4 ) ) ) ) ).
% relpow_Suc_E2
thf(fact_1102_relpow__Suc__I2,axiom,
! [X3: term_a_b,Y3: term_a_b,R4: set_Pr4386577575007340137rm_a_b,Z2: term_a_b,N: nat] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ R4 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ Z2 ) @ ( compow4057154403645558940rm_a_b @ N @ R4 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Z2 ) @ ( compow4057154403645558940rm_a_b @ ( suc @ N ) @ R4 ) ) ) ) ).
% relpow_Suc_I2
thf(fact_1103_Abstract__Rewriting_Ochain__mono,axiom,
! [R6: set_Pr4386577575007340137rm_a_b,R4: set_Pr4386577575007340137rm_a_b,Seq: nat > term_a_b] :
( ( ord_le118470702582115849rm_a_b @ R6 @ R4 )
=> ( ! [I3: nat] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( Seq @ I3 ) @ ( Seq @ ( suc @ I3 ) ) ) @ R6 )
=> ! [I4: nat] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( Seq @ I4 ) @ ( Seq @ ( suc @ I4 ) ) ) @ R4 ) ) ) ).
% Abstract_Rewriting.chain_mono
thf(fact_1104_relpow__0__I,axiom,
! [X3: term_a_b,R4: set_Pr4386577575007340137rm_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ X3 ) @ ( compow4057154403645558940rm_a_b @ zero_zero_nat @ R4 ) ) ).
% relpow_0_I
thf(fact_1105_relpow__0__E,axiom,
! [X3: term_a_b,Y3: term_a_b,R4: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ ( compow4057154403645558940rm_a_b @ zero_zero_nat @ R4 ) )
=> ( X3 = Y3 ) ) ).
% relpow_0_E
thf(fact_1106_relpow__Suc__E2_H,axiom,
! [X3: term_a_b,Z2: term_a_b,N: nat,A: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Z2 ) @ ( compow4057154403645558940rm_a_b @ ( suc @ N ) @ A ) )
=> ~ ! [Y: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y ) @ A )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ Z2 ) @ ( transi7742714808557438673rm_a_b @ A ) ) ) ) ).
% relpow_Suc_E2'
thf(fact_1107_pow__Suc__subset__trancl,axiom,
! [N: nat,R4: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ ( compow4057154403645558940rm_a_b @ ( suc @ N ) @ R4 ) @ ( transi7922773638565587891rm_a_b @ R4 ) ) ).
% pow_Suc_subset_trancl
thf(fact_1108_chain__imp__not__SN__on,axiom,
! [F2: nat > term_a_b,R: set_Pr4386577575007340137rm_a_b,I2: nat] :
( ! [I3: nat] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F2 @ I3 ) @ ( F2 @ ( suc @ I3 ) ) ) @ R )
=> ~ ( abstra4720023341729745482rm_a_b @ R @ ( insert_term_a_b @ ( F2 @ I2 ) @ bot_bot_set_term_a_b ) ) ) ).
% chain_imp_not_SN_on
thf(fact_1109_chain__imp__rtrancl,axiom,
! [S4: nat > term_a_b,R: set_Pr4386577575007340137rm_a_b,I2: nat,J2: nat] :
( ! [I3: nat] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( S4 @ I3 ) @ ( S4 @ ( suc @ I3 ) ) ) @ R )
=> ( ( ord_less_eq_nat @ I2 @ J2 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( S4 @ I2 ) @ ( S4 @ J2 ) ) @ ( transi7742714808557438673rm_a_b @ R ) ) ) ) ).
% chain_imp_rtrancl
thf(fact_1110_wf__no__infinite__down__chainE,axiom,
! [R: set_Pr4386577575007340137rm_a_b,F2: nat > term_a_b] :
( ( wf_term_a_b @ R )
=> ~ ! [K: nat] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F2 @ ( suc @ K ) ) @ ( F2 @ K ) ) @ R ) ) ).
% wf_no_infinite_down_chainE
thf(fact_1111_wf__iff__no__infinite__down__chain,axiom,
( wf_term_a_b
= ( ^ [R3: set_Pr4386577575007340137rm_a_b] :
~ ? [F5: nat > term_a_b] :
! [I: nat] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F5 @ ( suc @ I ) ) @ ( F5 @ I ) ) @ R3 ) ) ) ).
% wf_iff_no_infinite_down_chain
thf(fact_1112_lift__Suc__mono__le,axiom,
! [F2: nat > set_Pr4934435412358123699_a_nat,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_le8666007276011122963_a_nat @ ( F2 @ N2 ) @ ( F2 @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_le8666007276011122963_a_nat @ ( F2 @ N ) @ ( F2 @ N4 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_1113_lift__Suc__mono__le,axiom,
! [F2: nat > set_Pr4386577575007340137rm_a_b,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_le118470702582115849rm_a_b @ ( F2 @ N2 ) @ ( F2 @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_le118470702582115849rm_a_b @ ( F2 @ N ) @ ( F2 @ N4 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_1114_lift__Suc__antimono__le,axiom,
! [F2: nat > set_Pr4934435412358123699_a_nat,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_le8666007276011122963_a_nat @ ( F2 @ ( suc @ N2 ) ) @ ( F2 @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_le8666007276011122963_a_nat @ ( F2 @ N4 ) @ ( F2 @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_1115_lift__Suc__antimono__le,axiom,
! [F2: nat > set_Pr4386577575007340137rm_a_b,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_le118470702582115849rm_a_b @ ( F2 @ ( suc @ N2 ) ) @ ( F2 @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_le118470702582115849rm_a_b @ ( F2 @ N4 ) @ ( F2 @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_1116_chain__Un__SN__on__imp__first__step,axiom,
! [T: nat > term_a_b,R4: set_Pr4386577575007340137rm_a_b,S4: set_Pr4386577575007340137rm_a_b] :
( ! [I3: nat] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( T @ I3 ) @ ( T @ ( suc @ I3 ) ) ) @ ( sup_su6776935440552674877rm_a_b @ R4 @ S4 ) )
=> ( ( abstra4720023341729745482rm_a_b @ S4 @ ( 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 ) ) ) @ R4 )
& ! [J4: nat] :
( ( ord_less_nat @ J4 @ I3 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( T @ J4 ) @ ( T @ ( suc @ J4 ) ) ) @ S4 )
& ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( T @ J4 ) @ ( T @ ( suc @ J4 ) ) ) @ R4 ) ) ) ) ) ) ).
% chain_Un_SN_on_imp_first_step
thf(fact_1117_leD,axiom,
! [Y3: set_Pr4934435412358123699_a_nat,X3: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ Y3 @ X3 )
=> ~ ( ord_le2482536469199093511_a_nat @ X3 @ Y3 ) ) ).
% leD
thf(fact_1118_leD,axiom,
! [Y3: set_Pr4386577575007340137rm_a_b,X3: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ Y3 @ X3 )
=> ~ ( ord_le4883060154188259093rm_a_b @ X3 @ Y3 ) ) ).
% leD
thf(fact_1119_nless__le,axiom,
! [A2: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat] :
( ( ~ ( ord_le2482536469199093511_a_nat @ A2 @ B2 ) )
= ( ~ ( ord_le8666007276011122963_a_nat @ A2 @ B2 )
| ( A2 = B2 ) ) ) ).
% nless_le
thf(fact_1120_nless__le,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b] :
( ( ~ ( ord_le4883060154188259093rm_a_b @ A2 @ B2 ) )
= ( ~ ( ord_le118470702582115849rm_a_b @ A2 @ B2 )
| ( A2 = B2 ) ) ) ).
% nless_le
thf(fact_1121_antisym__conv1,axiom,
! [X3: set_Pr4934435412358123699_a_nat,Y3: set_Pr4934435412358123699_a_nat] :
( ~ ( ord_le2482536469199093511_a_nat @ X3 @ Y3 )
=> ( ( ord_le8666007276011122963_a_nat @ X3 @ Y3 )
= ( X3 = Y3 ) ) ) ).
% antisym_conv1
thf(fact_1122_antisym__conv1,axiom,
! [X3: set_Pr4386577575007340137rm_a_b,Y3: set_Pr4386577575007340137rm_a_b] :
( ~ ( ord_le4883060154188259093rm_a_b @ X3 @ Y3 )
=> ( ( ord_le118470702582115849rm_a_b @ X3 @ Y3 )
= ( X3 = Y3 ) ) ) ).
% antisym_conv1
thf(fact_1123_antisym__conv2,axiom,
! [X3: set_Pr4934435412358123699_a_nat,Y3: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X3 @ Y3 )
=> ( ( ~ ( ord_le2482536469199093511_a_nat @ X3 @ Y3 ) )
= ( X3 = Y3 ) ) ) ).
% antisym_conv2
thf(fact_1124_antisym__conv2,axiom,
! [X3: set_Pr4386577575007340137rm_a_b,Y3: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ X3 @ Y3 )
=> ( ( ~ ( ord_le4883060154188259093rm_a_b @ X3 @ Y3 ) )
= ( X3 = Y3 ) ) ) ).
% antisym_conv2
thf(fact_1125_less__le__not__le,axiom,
( ord_le2482536469199093511_a_nat
= ( ^ [X4: set_Pr4934435412358123699_a_nat,Y5: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X4 @ Y5 )
& ~ ( ord_le8666007276011122963_a_nat @ Y5 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_1126_less__le__not__le,axiom,
( ord_le4883060154188259093rm_a_b
= ( ^ [X4: set_Pr4386577575007340137rm_a_b,Y5: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ X4 @ Y5 )
& ~ ( ord_le118470702582115849rm_a_b @ Y5 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_1127_order_Oorder__iff__strict,axiom,
( ord_le8666007276011122963_a_nat
= ( ^ [A4: set_Pr4934435412358123699_a_nat,B4: set_Pr4934435412358123699_a_nat] :
( ( ord_le2482536469199093511_a_nat @ A4 @ B4 )
| ( A4 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_1128_order_Oorder__iff__strict,axiom,
( ord_le118470702582115849rm_a_b
= ( ^ [A4: set_Pr4386577575007340137rm_a_b,B4: set_Pr4386577575007340137rm_a_b] :
( ( ord_le4883060154188259093rm_a_b @ A4 @ B4 )
| ( A4 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_1129_order_Ostrict__iff__order,axiom,
( ord_le2482536469199093511_a_nat
= ( ^ [A4: set_Pr4934435412358123699_a_nat,B4: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A4 @ B4 )
& ( A4 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_1130_order_Ostrict__iff__order,axiom,
( ord_le4883060154188259093rm_a_b
= ( ^ [A4: set_Pr4386577575007340137rm_a_b,B4: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A4 @ B4 )
& ( A4 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_1131_order_Ostrict__trans1,axiom,
! [A2: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat,C: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A2 @ B2 )
=> ( ( ord_le2482536469199093511_a_nat @ B2 @ C )
=> ( ord_le2482536469199093511_a_nat @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_1132_order_Ostrict__trans1,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b,C: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A2 @ B2 )
=> ( ( ord_le4883060154188259093rm_a_b @ B2 @ C )
=> ( ord_le4883060154188259093rm_a_b @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_1133_order_Ostrict__trans2,axiom,
! [A2: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat,C: set_Pr4934435412358123699_a_nat] :
( ( ord_le2482536469199093511_a_nat @ A2 @ B2 )
=> ( ( ord_le8666007276011122963_a_nat @ B2 @ C )
=> ( ord_le2482536469199093511_a_nat @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_1134_order_Ostrict__trans2,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b,C: set_Pr4386577575007340137rm_a_b] :
( ( ord_le4883060154188259093rm_a_b @ A2 @ B2 )
=> ( ( ord_le118470702582115849rm_a_b @ B2 @ C )
=> ( ord_le4883060154188259093rm_a_b @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_1135_order_Ostrict__iff__not,axiom,
( ord_le2482536469199093511_a_nat
= ( ^ [A4: set_Pr4934435412358123699_a_nat,B4: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A4 @ B4 )
& ~ ( ord_le8666007276011122963_a_nat @ B4 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_1136_order_Ostrict__iff__not,axiom,
( ord_le4883060154188259093rm_a_b
= ( ^ [A4: set_Pr4386577575007340137rm_a_b,B4: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A4 @ B4 )
& ~ ( ord_le118470702582115849rm_a_b @ B4 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_1137_dual__order_Oorder__iff__strict,axiom,
( ord_le8666007276011122963_a_nat
= ( ^ [B4: set_Pr4934435412358123699_a_nat,A4: set_Pr4934435412358123699_a_nat] :
( ( ord_le2482536469199093511_a_nat @ B4 @ A4 )
| ( A4 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_1138_dual__order_Oorder__iff__strict,axiom,
( ord_le118470702582115849rm_a_b
= ( ^ [B4: set_Pr4386577575007340137rm_a_b,A4: set_Pr4386577575007340137rm_a_b] :
( ( ord_le4883060154188259093rm_a_b @ B4 @ A4 )
| ( A4 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_1139_dual__order_Ostrict__iff__order,axiom,
( ord_le2482536469199093511_a_nat
= ( ^ [B4: set_Pr4934435412358123699_a_nat,A4: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ B4 @ A4 )
& ( A4 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_1140_dual__order_Ostrict__iff__order,axiom,
( ord_le4883060154188259093rm_a_b
= ( ^ [B4: set_Pr4386577575007340137rm_a_b,A4: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ B4 @ A4 )
& ( A4 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_1141_dual__order_Ostrict__trans1,axiom,
! [B2: set_Pr4934435412358123699_a_nat,A2: set_Pr4934435412358123699_a_nat,C: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ B2 @ A2 )
=> ( ( ord_le2482536469199093511_a_nat @ C @ B2 )
=> ( ord_le2482536469199093511_a_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_1142_dual__order_Ostrict__trans1,axiom,
! [B2: set_Pr4386577575007340137rm_a_b,A2: set_Pr4386577575007340137rm_a_b,C: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ B2 @ A2 )
=> ( ( ord_le4883060154188259093rm_a_b @ C @ B2 )
=> ( ord_le4883060154188259093rm_a_b @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_1143_dual__order_Ostrict__trans2,axiom,
! [B2: set_Pr4934435412358123699_a_nat,A2: set_Pr4934435412358123699_a_nat,C: set_Pr4934435412358123699_a_nat] :
( ( ord_le2482536469199093511_a_nat @ B2 @ A2 )
=> ( ( ord_le8666007276011122963_a_nat @ C @ B2 )
=> ( ord_le2482536469199093511_a_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_1144_dual__order_Ostrict__trans2,axiom,
! [B2: set_Pr4386577575007340137rm_a_b,A2: set_Pr4386577575007340137rm_a_b,C: set_Pr4386577575007340137rm_a_b] :
( ( ord_le4883060154188259093rm_a_b @ B2 @ A2 )
=> ( ( ord_le118470702582115849rm_a_b @ C @ B2 )
=> ( ord_le4883060154188259093rm_a_b @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_1145_dual__order_Ostrict__iff__not,axiom,
( ord_le2482536469199093511_a_nat
= ( ^ [B4: set_Pr4934435412358123699_a_nat,A4: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ B4 @ A4 )
& ~ ( ord_le8666007276011122963_a_nat @ A4 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_1146_dual__order_Ostrict__iff__not,axiom,
( ord_le4883060154188259093rm_a_b
= ( ^ [B4: set_Pr4386577575007340137rm_a_b,A4: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ B4 @ A4 )
& ~ ( ord_le118470702582115849rm_a_b @ A4 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_1147_order_Ostrict__implies__order,axiom,
! [A2: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat] :
( ( ord_le2482536469199093511_a_nat @ A2 @ B2 )
=> ( ord_le8666007276011122963_a_nat @ A2 @ B2 ) ) ).
% order.strict_implies_order
thf(fact_1148_order_Ostrict__implies__order,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b] :
( ( ord_le4883060154188259093rm_a_b @ A2 @ B2 )
=> ( ord_le118470702582115849rm_a_b @ A2 @ B2 ) ) ).
% order.strict_implies_order
thf(fact_1149_dual__order_Ostrict__implies__order,axiom,
! [B2: set_Pr4934435412358123699_a_nat,A2: set_Pr4934435412358123699_a_nat] :
( ( ord_le2482536469199093511_a_nat @ B2 @ A2 )
=> ( ord_le8666007276011122963_a_nat @ B2 @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_1150_dual__order_Ostrict__implies__order,axiom,
! [B2: set_Pr4386577575007340137rm_a_b,A2: set_Pr4386577575007340137rm_a_b] :
( ( ord_le4883060154188259093rm_a_b @ B2 @ A2 )
=> ( ord_le118470702582115849rm_a_b @ B2 @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_1151_order__le__less,axiom,
( ord_le8666007276011122963_a_nat
= ( ^ [X4: set_Pr4934435412358123699_a_nat,Y5: set_Pr4934435412358123699_a_nat] :
( ( ord_le2482536469199093511_a_nat @ X4 @ Y5 )
| ( X4 = Y5 ) ) ) ) ).
% order_le_less
thf(fact_1152_order__le__less,axiom,
( ord_le118470702582115849rm_a_b
= ( ^ [X4: set_Pr4386577575007340137rm_a_b,Y5: set_Pr4386577575007340137rm_a_b] :
( ( ord_le4883060154188259093rm_a_b @ X4 @ Y5 )
| ( X4 = Y5 ) ) ) ) ).
% order_le_less
thf(fact_1153_order__less__le,axiom,
( ord_le2482536469199093511_a_nat
= ( ^ [X4: set_Pr4934435412358123699_a_nat,Y5: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X4 @ Y5 )
& ( X4 != Y5 ) ) ) ) ).
% order_less_le
thf(fact_1154_order__less__le,axiom,
( ord_le4883060154188259093rm_a_b
= ( ^ [X4: set_Pr4386577575007340137rm_a_b,Y5: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ X4 @ Y5 )
& ( X4 != Y5 ) ) ) ) ).
% order_less_le
thf(fact_1155_order__less__imp__le,axiom,
! [X3: set_Pr4934435412358123699_a_nat,Y3: set_Pr4934435412358123699_a_nat] :
( ( ord_le2482536469199093511_a_nat @ X3 @ Y3 )
=> ( ord_le8666007276011122963_a_nat @ X3 @ Y3 ) ) ).
% order_less_imp_le
thf(fact_1156_order__less__imp__le,axiom,
! [X3: set_Pr4386577575007340137rm_a_b,Y3: set_Pr4386577575007340137rm_a_b] :
( ( ord_le4883060154188259093rm_a_b @ X3 @ Y3 )
=> ( ord_le118470702582115849rm_a_b @ X3 @ Y3 ) ) ).
% order_less_imp_le
thf(fact_1157_order__le__neq__trans,axiom,
! [A2: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_le2482536469199093511_a_nat @ A2 @ B2 ) ) ) ).
% order_le_neq_trans
thf(fact_1158_order__le__neq__trans,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_le4883060154188259093rm_a_b @ A2 @ B2 ) ) ) ).
% order_le_neq_trans
thf(fact_1159_order__neq__le__trans,axiom,
! [A2: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat] :
( ( A2 != B2 )
=> ( ( ord_le8666007276011122963_a_nat @ A2 @ B2 )
=> ( ord_le2482536469199093511_a_nat @ A2 @ B2 ) ) ) ).
% order_neq_le_trans
thf(fact_1160_order__neq__le__trans,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b] :
( ( A2 != B2 )
=> ( ( ord_le118470702582115849rm_a_b @ A2 @ B2 )
=> ( ord_le4883060154188259093rm_a_b @ A2 @ B2 ) ) ) ).
% order_neq_le_trans
thf(fact_1161_order__le__less__trans,axiom,
! [X3: set_Pr4934435412358123699_a_nat,Y3: set_Pr4934435412358123699_a_nat,Z2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X3 @ Y3 )
=> ( ( ord_le2482536469199093511_a_nat @ Y3 @ Z2 )
=> ( ord_le2482536469199093511_a_nat @ X3 @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_1162_order__le__less__trans,axiom,
! [X3: set_Pr4386577575007340137rm_a_b,Y3: set_Pr4386577575007340137rm_a_b,Z2: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ X3 @ Y3 )
=> ( ( ord_le4883060154188259093rm_a_b @ Y3 @ Z2 )
=> ( ord_le4883060154188259093rm_a_b @ X3 @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_1163_order__less__le__trans,axiom,
! [X3: set_Pr4934435412358123699_a_nat,Y3: set_Pr4934435412358123699_a_nat,Z2: set_Pr4934435412358123699_a_nat] :
( ( ord_le2482536469199093511_a_nat @ X3 @ Y3 )
=> ( ( ord_le8666007276011122963_a_nat @ Y3 @ Z2 )
=> ( ord_le2482536469199093511_a_nat @ X3 @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_1164_order__less__le__trans,axiom,
! [X3: set_Pr4386577575007340137rm_a_b,Y3: set_Pr4386577575007340137rm_a_b,Z2: set_Pr4386577575007340137rm_a_b] :
( ( ord_le4883060154188259093rm_a_b @ X3 @ Y3 )
=> ( ( ord_le118470702582115849rm_a_b @ Y3 @ Z2 )
=> ( ord_le4883060154188259093rm_a_b @ X3 @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_1165_order__le__less__subst2,axiom,
! [A2: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat,F2: set_Pr4934435412358123699_a_nat > set_Pr4934435412358123699_a_nat,C: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A2 @ B2 )
=> ( ( ord_le2482536469199093511_a_nat @ ( F2 @ B2 ) @ C )
=> ( ! [X: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X @ Y )
=> ( ord_le8666007276011122963_a_nat @ ( F2 @ X ) @ ( F2 @ Y ) ) )
=> ( ord_le2482536469199093511_a_nat @ ( F2 @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_1166_order__le__less__subst2,axiom,
! [A2: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat,F2: set_Pr4934435412358123699_a_nat > set_Pr4386577575007340137rm_a_b,C: set_Pr4386577575007340137rm_a_b] :
( ( ord_le8666007276011122963_a_nat @ A2 @ B2 )
=> ( ( ord_le4883060154188259093rm_a_b @ ( F2 @ B2 ) @ C )
=> ( ! [X: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X @ Y )
=> ( ord_le118470702582115849rm_a_b @ ( F2 @ X ) @ ( F2 @ Y ) ) )
=> ( ord_le4883060154188259093rm_a_b @ ( F2 @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_1167_order__le__less__subst2,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b,F2: set_Pr4386577575007340137rm_a_b > set_Pr4934435412358123699_a_nat,C: set_Pr4934435412358123699_a_nat] :
( ( ord_le118470702582115849rm_a_b @ A2 @ B2 )
=> ( ( ord_le2482536469199093511_a_nat @ ( F2 @ B2 ) @ C )
=> ( ! [X: set_Pr4386577575007340137rm_a_b,Y: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ X @ Y )
=> ( ord_le8666007276011122963_a_nat @ ( F2 @ X ) @ ( F2 @ Y ) ) )
=> ( ord_le2482536469199093511_a_nat @ ( F2 @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_1168_order__le__less__subst2,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b,F2: set_Pr4386577575007340137rm_a_b > set_Pr4386577575007340137rm_a_b,C: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A2 @ B2 )
=> ( ( ord_le4883060154188259093rm_a_b @ ( F2 @ B2 ) @ C )
=> ( ! [X: set_Pr4386577575007340137rm_a_b,Y: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ X @ Y )
=> ( ord_le118470702582115849rm_a_b @ ( F2 @ X ) @ ( F2 @ Y ) ) )
=> ( ord_le4883060154188259093rm_a_b @ ( F2 @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_1169_order__less__le__subst1,axiom,
! [A2: set_Pr4934435412358123699_a_nat,F2: set_Pr4934435412358123699_a_nat > set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat,C: set_Pr4934435412358123699_a_nat] :
( ( ord_le2482536469199093511_a_nat @ A2 @ ( F2 @ B2 ) )
=> ( ( ord_le8666007276011122963_a_nat @ B2 @ C )
=> ( ! [X: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X @ Y )
=> ( ord_le8666007276011122963_a_nat @ ( F2 @ X ) @ ( F2 @ Y ) ) )
=> ( ord_le2482536469199093511_a_nat @ A2 @ ( F2 @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1170_order__less__le__subst1,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,F2: set_Pr4934435412358123699_a_nat > set_Pr4386577575007340137rm_a_b,B2: set_Pr4934435412358123699_a_nat,C: set_Pr4934435412358123699_a_nat] :
( ( ord_le4883060154188259093rm_a_b @ A2 @ ( F2 @ B2 ) )
=> ( ( ord_le8666007276011122963_a_nat @ B2 @ C )
=> ( ! [X: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X @ Y )
=> ( ord_le118470702582115849rm_a_b @ ( F2 @ X ) @ ( F2 @ Y ) ) )
=> ( ord_le4883060154188259093rm_a_b @ A2 @ ( F2 @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1171_order__less__le__subst1,axiom,
! [A2: set_Pr4934435412358123699_a_nat,F2: set_Pr4386577575007340137rm_a_b > set_Pr4934435412358123699_a_nat,B2: set_Pr4386577575007340137rm_a_b,C: set_Pr4386577575007340137rm_a_b] :
( ( ord_le2482536469199093511_a_nat @ A2 @ ( F2 @ B2 ) )
=> ( ( ord_le118470702582115849rm_a_b @ B2 @ C )
=> ( ! [X: set_Pr4386577575007340137rm_a_b,Y: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ X @ Y )
=> ( ord_le8666007276011122963_a_nat @ ( F2 @ X ) @ ( F2 @ Y ) ) )
=> ( ord_le2482536469199093511_a_nat @ A2 @ ( F2 @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1172_order__less__le__subst1,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,F2: set_Pr4386577575007340137rm_a_b > set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b,C: set_Pr4386577575007340137rm_a_b] :
( ( ord_le4883060154188259093rm_a_b @ A2 @ ( F2 @ B2 ) )
=> ( ( ord_le118470702582115849rm_a_b @ B2 @ C )
=> ( ! [X: set_Pr4386577575007340137rm_a_b,Y: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ X @ Y )
=> ( ord_le118470702582115849rm_a_b @ ( F2 @ X ) @ ( F2 @ Y ) ) )
=> ( ord_le4883060154188259093rm_a_b @ A2 @ ( F2 @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1173_order__le__imp__less__or__eq,axiom,
! [X3: set_Pr4934435412358123699_a_nat,Y3: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X3 @ Y3 )
=> ( ( ord_le2482536469199093511_a_nat @ X3 @ Y3 )
| ( X3 = Y3 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_1174_order__le__imp__less__or__eq,axiom,
! [X3: set_Pr4386577575007340137rm_a_b,Y3: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ X3 @ Y3 )
=> ( ( ord_le4883060154188259093rm_a_b @ X3 @ Y3 )
| ( X3 = Y3 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_1175_chain__imp__trancl,axiom,
! [S4: nat > term_a_b,R: set_Pr4386577575007340137rm_a_b,I2: nat,J2: nat] :
( ! [I3: nat] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( S4 @ I3 ) @ ( S4 @ ( suc @ I3 ) ) ) @ R )
=> ( ( ord_less_nat @ I2 @ J2 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( S4 @ I2 ) @ ( S4 @ J2 ) ) @ ( transi7922773638565587891rm_a_b @ R ) ) ) ) ).
% chain_imp_trancl
thf(fact_1176_wf__bounded__measure,axiom,
! [R: set_Pr4386577575007340137rm_a_b,Ub: term_a_b > nat,F2: term_a_b > nat] :
( ! [A5: term_a_b,B5: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B5 @ A5 ) @ R )
=> ( ( ord_less_eq_nat @ ( Ub @ B5 ) @ ( Ub @ A5 ) )
& ( ord_less_eq_nat @ ( F2 @ B5 ) @ ( Ub @ A5 ) )
& ( ord_less_nat @ ( F2 @ A5 ) @ ( F2 @ B5 ) ) ) )
=> ( wf_term_a_b @ R ) ) ).
% wf_bounded_measure
thf(fact_1177_trancl__power,axiom,
! [P2: produc357393685978478089rm_a_b,R4: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ P2 @ ( transi7922773638565587891rm_a_b @ R4 ) )
= ( ? [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
& ( member5869715511025134514rm_a_b @ P2 @ ( compow4057154403645558940rm_a_b @ N3 @ R4 ) ) ) ) ) ).
% trancl_power
thf(fact_1178_rtrancl__fun__conv,axiom,
! [S: term_a_b,T: term_a_b,R4: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( transi7742714808557438673rm_a_b @ R4 ) )
= ( ? [F5: nat > term_a_b,N3: nat] :
( ( ( F5 @ zero_zero_nat )
= S )
& ( ( F5 @ N3 )
= T )
& ! [I: nat] :
( ( ord_less_nat @ I @ N3 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F5 @ I ) @ ( F5 @ ( suc @ I ) ) ) @ R4 ) ) ) ) ) ).
% rtrancl_fun_conv
thf(fact_1179_rtrancl__imp__seq,axiom,
! [X3: term_a_b,Y3: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ? [F4: nat > term_a_b,N2: nat] :
( ( ( F4 @ zero_zero_nat )
= X3 )
& ( ( F4 @ N2 )
= Y3 )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ N2 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F4 @ I4 ) @ ( F4 @ ( suc @ I4 ) ) ) @ R ) ) ) ) ).
% rtrancl_imp_seq
thf(fact_1180_relpow__fun__conv,axiom,
! [A2: term_a_b,B2: term_a_b,N: nat,R4: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( compow4057154403645558940rm_a_b @ N @ R4 ) )
= ( ? [F5: nat > term_a_b] :
( ( ( F5 @ zero_zero_nat )
= A2 )
& ( ( F5 @ N )
= B2 )
& ! [I: nat] :
( ( ord_less_nat @ I @ N )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F5 @ I ) @ ( F5 @ ( suc @ I ) ) ) @ R4 ) ) ) ) ) ).
% relpow_fun_conv
thf(fact_1181_relpow__seq,axiom,
! [X3: term_a_b,Y3: term_a_b,N: nat,R: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ ( compow4057154403645558940rm_a_b @ N @ R ) )
=> ? [F4: nat > term_a_b] :
( ( ( F4 @ zero_zero_nat )
= X3 )
& ( ( F4 @ N )
= Y3 )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ N )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F4 @ I4 ) @ ( F4 @ ( suc @ I4 ) ) ) @ R ) ) ) ) ).
% relpow_seq
thf(fact_1182_in__measure,axiom,
! [X3: term_a_b,Y3: term_a_b,F2: term_a_b > nat] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ ( measure_term_a_b @ F2 ) )
= ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) ) ).
% in_measure
thf(fact_1183_psubsetI,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A @ B )
=> ( ( A != B )
=> ( ord_le2482536469199093511_a_nat @ A @ B ) ) ) ).
% psubsetI
thf(fact_1184_psubsetI,axiom,
! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A @ B )
=> ( ( A != B )
=> ( ord_le4883060154188259093rm_a_b @ A @ B ) ) ) ).
% psubsetI
thf(fact_1185_psubset__imp__ex__mem,axiom,
! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ( ord_le4883060154188259093rm_a_b @ A @ B )
=> ? [B5: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ B5 @ ( minus_5192120951422937424rm_a_b @ B @ A ) ) ) ).
% psubset_imp_ex_mem
thf(fact_1186_psubset__imp__ex__mem,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( ord_le2482536469199093511_a_nat @ A @ B )
=> ? [B5: product_prod_a_nat] : ( member5724188588386418708_a_nat @ B5 @ ( minus_8630901905721465484_a_nat @ B @ A ) ) ) ).
% psubset_imp_ex_mem
thf(fact_1187_psubsetE,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( ord_le2482536469199093511_a_nat @ A @ B )
=> ~ ( ( ord_le8666007276011122963_a_nat @ A @ B )
=> ( ord_le8666007276011122963_a_nat @ B @ A ) ) ) ).
% psubsetE
thf(fact_1188_psubsetE,axiom,
! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ( ord_le4883060154188259093rm_a_b @ A @ B )
=> ~ ( ( ord_le118470702582115849rm_a_b @ A @ B )
=> ( ord_le118470702582115849rm_a_b @ B @ A ) ) ) ).
% psubsetE
thf(fact_1189_psubset__eq,axiom,
( ord_le2482536469199093511_a_nat
= ( ^ [A6: set_Pr4934435412358123699_a_nat,B6: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A6 @ B6 )
& ( A6 != B6 ) ) ) ) ).
% psubset_eq
thf(fact_1190_psubset__eq,axiom,
( ord_le4883060154188259093rm_a_b
= ( ^ [A6: set_Pr4386577575007340137rm_a_b,B6: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A6 @ B6 )
& ( A6 != B6 ) ) ) ) ).
% psubset_eq
thf(fact_1191_psubset__imp__subset,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( ord_le2482536469199093511_a_nat @ A @ B )
=> ( ord_le8666007276011122963_a_nat @ A @ B ) ) ).
% psubset_imp_subset
thf(fact_1192_psubset__imp__subset,axiom,
! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ( ord_le4883060154188259093rm_a_b @ A @ B )
=> ( ord_le118470702582115849rm_a_b @ A @ B ) ) ).
% psubset_imp_subset
thf(fact_1193_psubset__subset__trans,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat,C2: set_Pr4934435412358123699_a_nat] :
( ( ord_le2482536469199093511_a_nat @ A @ B )
=> ( ( ord_le8666007276011122963_a_nat @ B @ C2 )
=> ( ord_le2482536469199093511_a_nat @ A @ C2 ) ) ) ).
% psubset_subset_trans
thf(fact_1194_psubset__subset__trans,axiom,
! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b,C2: set_Pr4386577575007340137rm_a_b] :
( ( ord_le4883060154188259093rm_a_b @ A @ B )
=> ( ( ord_le118470702582115849rm_a_b @ B @ C2 )
=> ( ord_le4883060154188259093rm_a_b @ A @ C2 ) ) ) ).
% psubset_subset_trans
thf(fact_1195_subset__not__subset__eq,axiom,
( ord_le2482536469199093511_a_nat
= ( ^ [A6: set_Pr4934435412358123699_a_nat,B6: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A6 @ B6 )
& ~ ( ord_le8666007276011122963_a_nat @ B6 @ A6 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_1196_subset__not__subset__eq,axiom,
( ord_le4883060154188259093rm_a_b
= ( ^ [A6: set_Pr4386577575007340137rm_a_b,B6: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A6 @ B6 )
& ~ ( ord_le118470702582115849rm_a_b @ B6 @ A6 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_1197_subset__psubset__trans,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat,C2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A @ B )
=> ( ( ord_le2482536469199093511_a_nat @ B @ C2 )
=> ( ord_le2482536469199093511_a_nat @ A @ C2 ) ) ) ).
% subset_psubset_trans
thf(fact_1198_subset__psubset__trans,axiom,
! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b,C2: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A @ B )
=> ( ( ord_le4883060154188259093rm_a_b @ B @ C2 )
=> ( ord_le4883060154188259093rm_a_b @ A @ C2 ) ) ) ).
% subset_psubset_trans
thf(fact_1199_subset__iff__psubset__eq,axiom,
( ord_le8666007276011122963_a_nat
= ( ^ [A6: set_Pr4934435412358123699_a_nat,B6: set_Pr4934435412358123699_a_nat] :
( ( ord_le2482536469199093511_a_nat @ A6 @ B6 )
| ( A6 = B6 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_1200_subset__iff__psubset__eq,axiom,
( ord_le118470702582115849rm_a_b
= ( ^ [A6: set_Pr4386577575007340137rm_a_b,B6: set_Pr4386577575007340137rm_a_b] :
( ( ord_le4883060154188259093rm_a_b @ A6 @ B6 )
| ( A6 = B6 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_1201_psubset__insert__iff,axiom,
! [A: set_Pr4934435412358123699_a_nat,X3: product_prod_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( ord_le2482536469199093511_a_nat @ A @ ( insert8054603423593749677_a_nat @ X3 @ B ) )
= ( ( ( member5724188588386418708_a_nat @ X3 @ B )
=> ( ord_le2482536469199093511_a_nat @ A @ B ) )
& ( ~ ( member5724188588386418708_a_nat @ X3 @ B )
=> ( ( ( member5724188588386418708_a_nat @ X3 @ A )
=> ( ord_le2482536469199093511_a_nat @ ( minus_8630901905721465484_a_nat @ A @ ( insert8054603423593749677_a_nat @ X3 @ bot_bo9049108969261143879_a_nat ) ) @ B ) )
& ( ~ ( member5724188588386418708_a_nat @ X3 @ A )
=> ( ord_le8666007276011122963_a_nat @ A @ B ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_1202_psubset__insert__iff,axiom,
! [A: set_Pr4386577575007340137rm_a_b,X3: produc357393685978478089rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ( ord_le4883060154188259093rm_a_b @ A @ ( insert7009541432154983385rm_a_b @ X3 @ B ) )
= ( ( ( member5869715511025134514rm_a_b @ X3 @ B )
=> ( ord_le4883060154188259093rm_a_b @ A @ B ) )
& ( ~ ( member5869715511025134514rm_a_b @ X3 @ B )
=> ( ( ( member5869715511025134514rm_a_b @ X3 @ A )
=> ( ord_le4883060154188259093rm_a_b @ ( minus_5192120951422937424rm_a_b @ A @ ( insert7009541432154983385rm_a_b @ X3 @ bot_bo197521221353338581rm_a_b ) ) @ B ) )
& ( ~ ( member5869715511025134514rm_a_b @ X3 @ A )
=> ( ord_le118470702582115849rm_a_b @ A @ B ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_1203_psubsetD,axiom,
! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b,C: produc357393685978478089rm_a_b] :
( ( ord_le4883060154188259093rm_a_b @ A @ B )
=> ( ( member5869715511025134514rm_a_b @ C @ A )
=> ( member5869715511025134514rm_a_b @ C @ B ) ) ) ).
% psubsetD
thf(fact_1204_psubsetD,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat,C: product_prod_a_nat] :
( ( ord_le2482536469199093511_a_nat @ A @ B )
=> ( ( member5724188588386418708_a_nat @ C @ A )
=> ( member5724188588386418708_a_nat @ C @ B ) ) ) ).
% psubsetD
thf(fact_1205_verit__comp__simplify1_I2_J,axiom,
! [A2: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_1206_verit__comp__simplify1_I2_J,axiom,
! [A2: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_1207_inf__top_Osemilattice__neutr__order__axioms,axiom,
semila3720225504097595269_a_nat @ inf_in3813033988066259269_a_nat @ top_to3353692345378799459_a_nat @ ord_le8666007276011122963_a_nat @ ord_le2482536469199093511_a_nat ).
% inf_top.semilattice_neutr_order_axioms
thf(fact_1208_inf__top_Osemilattice__neutr__order__axioms,axiom,
semila1747947753510541591rm_a_b @ inf_in1693821330460682071rm_a_b @ top_to1314267278846557113rm_a_b @ ord_le118470702582115849rm_a_b @ ord_le4883060154188259093rm_a_b ).
% inf_top.semilattice_neutr_order_axioms
thf(fact_1209_top_Oordering__top__axioms,axiom,
orderi2411975122467621314_a_nat @ ord_le8666007276011122963_a_nat @ ord_le2482536469199093511_a_nat @ top_to3353692345378799459_a_nat ).
% top.ordering_top_axioms
thf(fact_1210_top_Oordering__top__axioms,axiom,
orderi3923411991008206490rm_a_b @ ord_le118470702582115849rm_a_b @ ord_le4883060154188259093rm_a_b @ top_to1314267278846557113rm_a_b ).
% top.ordering_top_axioms
thf(fact_1211_Linear__order__Well__order__iff,axiom,
! [R: set_Pr4386577575007340137rm_a_b] :
( ( order_5388802246213473311rm_a_b @ ( field_term_a_b @ R ) @ R )
=> ( ( order_5719896216891381494rm_a_b @ ( field_term_a_b @ R ) @ R )
= ( ! [A6: set_term_a_b] :
( ( ord_le2705286416250468010rm_a_b @ A6 @ ( field_term_a_b @ R ) )
=> ( ( 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 ) @ R ) ) ) ) ) ) ) ) ).
% Linear_order_Well_order_iff
thf(fact_1212_Linear__order__Well__order__iff,axiom,
! [R: set_Pr1811044260758604347_a_nat] :
( ( order_6388386362885710600_a_nat @ ( field_8954927560578634480_a_nat @ R ) @ R )
=> ( ( order_7753855512070164063_a_nat @ ( field_8954927560578634480_a_nat @ R ) @ R )
= ( ! [A6: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A6 @ ( field_8954927560578634480_a_nat @ R ) )
=> ( ( A6 != bot_bo9049108969261143879_a_nat )
=> ? [X4: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ X4 @ A6 )
& ! [Y5: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ Y5 @ A6 )
=> ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ X4 @ Y5 ) @ R ) ) ) ) ) ) ) ) ).
% Linear_order_Well_order_iff
thf(fact_1213_Linear__order__Well__order__iff,axiom,
! [R: set_Pr2972776593051762503rm_a_b] :
( ( order_681589137112254398rm_a_b @ ( field_6884932134483023318rm_a_b @ R ) @ R )
=> ( ( order_8637870607158948775rm_a_b @ ( field_6884932134483023318rm_a_b @ R ) @ R )
= ( ! [A6: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A6 @ ( field_6884932134483023318rm_a_b @ R ) )
=> ( ( A6 != bot_bo197521221353338581rm_a_b )
=> ? [X4: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X4 @ A6 )
& ! [Y5: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ Y5 @ A6 )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X4 @ Y5 ) @ R ) ) ) ) ) ) ) ) ).
% Linear_order_Well_order_iff
thf(fact_1214_underS__incl__iff,axiom,
! [R: set_Pr4386577575007340137rm_a_b,A2: term_a_b,B2: term_a_b] :
( ( order_5388802246213473311rm_a_b @ ( field_term_a_b @ R ) @ R )
=> ( ( member_term_a_b @ A2 @ ( field_term_a_b @ R ) )
=> ( ( member_term_a_b @ B2 @ ( field_term_a_b @ R ) )
=> ( ( ord_le2705286416250468010rm_a_b @ ( order_1619770106104096206rm_a_b @ R @ A2 ) @ ( order_1619770106104096206rm_a_b @ R @ B2 ) )
= ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ R ) ) ) ) ) ).
% underS_incl_iff
thf(fact_1215_underS__incl__iff,axiom,
! [R: set_Pr1811044260758604347_a_nat,A2: product_prod_a_nat,B2: product_prod_a_nat] :
( ( order_6388386362885710600_a_nat @ ( field_8954927560578634480_a_nat @ R ) @ R )
=> ( ( member5724188588386418708_a_nat @ A2 @ ( field_8954927560578634480_a_nat @ R ) )
=> ( ( member5724188588386418708_a_nat @ B2 @ ( field_8954927560578634480_a_nat @ R ) )
=> ( ( ord_le8666007276011122963_a_nat @ ( order_2219920454636572983_a_nat @ R @ A2 ) @ ( order_2219920454636572983_a_nat @ R @ B2 ) )
= ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A2 @ B2 ) @ R ) ) ) ) ) ).
% underS_incl_iff
thf(fact_1216_underS__incl__iff,axiom,
! [R: set_Pr2972776593051762503rm_a_b,A2: produc357393685978478089rm_a_b,B2: produc357393685978478089rm_a_b] :
( ( order_681589137112254398rm_a_b @ ( field_6884932134483023318rm_a_b @ R ) @ R )
=> ( ( member5869715511025134514rm_a_b @ A2 @ ( field_6884932134483023318rm_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ B2 @ ( field_6884932134483023318rm_a_b @ R ) )
=> ( ( ord_le118470702582115849rm_a_b @ ( order_8163520290499871439rm_a_b @ R @ A2 ) @ ( order_8163520290499871439rm_a_b @ R @ B2 ) )
= ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A2 @ B2 ) @ R ) ) ) ) ) ).
% underS_incl_iff
thf(fact_1217_well__order__on__domain,axiom,
! [A: set_Pr4386577575007340137rm_a_b,R: set_Pr2972776593051762503rm_a_b,A2: produc357393685978478089rm_a_b,B2: produc357393685978478089rm_a_b] :
( ( order_8637870607158948775rm_a_b @ A @ R )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A2 @ B2 ) @ R )
=> ( ( member5869715511025134514rm_a_b @ A2 @ A )
& ( member5869715511025134514rm_a_b @ B2 @ A ) ) ) ) ).
% well_order_on_domain
thf(fact_1218_well__order__on__domain,axiom,
! [A: set_Pr4934435412358123699_a_nat,R: set_Pr1811044260758604347_a_nat,A2: product_prod_a_nat,B2: product_prod_a_nat] :
( ( order_7753855512070164063_a_nat @ A @ R )
=> ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A2 @ B2 ) @ R )
=> ( ( member5724188588386418708_a_nat @ A2 @ A )
& ( member5724188588386418708_a_nat @ B2 @ A ) ) ) ) ).
% well_order_on_domain
thf(fact_1219_well__order__on__domain,axiom,
! [A: set_term_a_b,R: set_Pr4386577575007340137rm_a_b,A2: term_a_b,B2: term_a_b] :
( ( order_5719896216891381494rm_a_b @ A @ R )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ R )
=> ( ( member_term_a_b @ A2 @ A )
& ( member_term_a_b @ B2 @ A ) ) ) ) ).
% well_order_on_domain
thf(fact_1220_underS__E,axiom,
! [I2: produc357393685978478089rm_a_b,R4: set_Pr2972776593051762503rm_a_b,J2: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ I2 @ ( order_8163520290499871439rm_a_b @ R4 @ J2 ) )
=> ( ( I2 != J2 )
& ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ I2 @ J2 ) @ R4 ) ) ) ).
% underS_E
thf(fact_1221_underS__E,axiom,
! [I2: product_prod_a_nat,R4: set_Pr1811044260758604347_a_nat,J2: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ I2 @ ( order_2219920454636572983_a_nat @ R4 @ J2 ) )
=> ( ( I2 != J2 )
& ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ I2 @ J2 ) @ R4 ) ) ) ).
% underS_E
thf(fact_1222_underS__E,axiom,
! [I2: term_a_b,R4: set_Pr4386577575007340137rm_a_b,J2: term_a_b] :
( ( member_term_a_b @ I2 @ ( order_1619770106104096206rm_a_b @ R4 @ J2 ) )
=> ( ( I2 != J2 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ I2 @ J2 ) @ R4 ) ) ) ).
% underS_E
thf(fact_1223_underS__I,axiom,
! [I2: produc357393685978478089rm_a_b,J2: produc357393685978478089rm_a_b,R4: set_Pr2972776593051762503rm_a_b] :
( ( I2 != J2 )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ I2 @ J2 ) @ R4 )
=> ( member5869715511025134514rm_a_b @ I2 @ ( order_8163520290499871439rm_a_b @ R4 @ J2 ) ) ) ) ).
% underS_I
thf(fact_1224_underS__I,axiom,
! [I2: product_prod_a_nat,J2: product_prod_a_nat,R4: set_Pr1811044260758604347_a_nat] :
( ( I2 != J2 )
=> ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ I2 @ J2 ) @ R4 )
=> ( member5724188588386418708_a_nat @ I2 @ ( order_2219920454636572983_a_nat @ R4 @ J2 ) ) ) ) ).
% underS_I
thf(fact_1225_underS__I,axiom,
! [I2: term_a_b,J2: term_a_b,R4: set_Pr4386577575007340137rm_a_b] :
( ( I2 != J2 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ I2 @ J2 ) @ R4 )
=> ( member_term_a_b @ I2 @ ( order_1619770106104096206rm_a_b @ R4 @ J2 ) ) ) ) ).
% underS_I
thf(fact_1226_Order__Relation_OunderS__Field,axiom,
! [R: set_Pr1811044260758604347_a_nat,A2: product_prod_a_nat] : ( ord_le8666007276011122963_a_nat @ ( order_2219920454636572983_a_nat @ R @ A2 ) @ ( field_8954927560578634480_a_nat @ R ) ) ).
% Order_Relation.underS_Field
thf(fact_1227_Order__Relation_OunderS__Field,axiom,
! [R: set_Pr2972776593051762503rm_a_b,A2: produc357393685978478089rm_a_b] : ( ord_le118470702582115849rm_a_b @ ( order_8163520290499871439rm_a_b @ R @ A2 ) @ ( field_6884932134483023318rm_a_b @ R ) ) ).
% Order_Relation.underS_Field
thf(fact_1228_Refl__under__underS,axiom,
! [R: set_Pr2972776593051762503rm_a_b,A2: produc357393685978478089rm_a_b] :
( ( refl_o5344404488616963464rm_a_b @ ( field_6884932134483023318rm_a_b @ R ) @ R )
=> ( ( member5869715511025134514rm_a_b @ A2 @ ( field_6884932134483023318rm_a_b @ R ) )
=> ( ( order_346605463002971118rm_a_b @ R @ A2 )
= ( sup_su6776935440552674877rm_a_b @ ( order_8163520290499871439rm_a_b @ R @ A2 ) @ ( insert7009541432154983385rm_a_b @ A2 @ bot_bo197521221353338581rm_a_b ) ) ) ) ) ).
% Refl_under_underS
thf(fact_1229_Refl__under__underS,axiom,
! [R: set_Pr1811044260758604347_a_nat,A2: product_prod_a_nat] :
( ( refl_o8237544196271505854_a_nat @ ( field_8954927560578634480_a_nat @ R ) @ R )
=> ( ( member5724188588386418708_a_nat @ A2 @ ( field_8954927560578634480_a_nat @ R ) )
=> ( ( order_6535219843107416792_a_nat @ R @ A2 )
= ( sup_su459911885395995103_a_nat @ ( order_2219920454636572983_a_nat @ R @ A2 ) @ ( insert8054603423593749677_a_nat @ A2 @ bot_bo9049108969261143879_a_nat ) ) ) ) ) ).
% Refl_under_underS
thf(fact_1230_bsqr__max2,axiom,
! [R: set_Pr4386577575007340137rm_a_b,A1: term_a_b,A22: term_a_b,B1: term_a_b,B22: term_a_b] :
( ( order_5719896216891381494rm_a_b @ ( field_term_a_b @ R ) @ R )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A22 ) @ ( produc7020197800436672577rm_a_b @ B1 @ B22 ) ) @ ( bNF_We8522098035112346167rm_a_b @ R ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( bNF_We4618759378340593734rm_a_b @ R @ A1 @ A22 ) @ ( bNF_We4618759378340593734rm_a_b @ R @ B1 @ B22 ) ) @ R ) ) ) ).
% bsqr_max2
thf(fact_1231_underS__subset__under,axiom,
! [R: set_Pr1811044260758604347_a_nat,A2: product_prod_a_nat] : ( ord_le8666007276011122963_a_nat @ ( order_2219920454636572983_a_nat @ R @ A2 ) @ ( order_6535219843107416792_a_nat @ R @ A2 ) ) ).
% underS_subset_under
thf(fact_1232_underS__subset__under,axiom,
! [R: set_Pr2972776593051762503rm_a_b,A2: produc357393685978478089rm_a_b] : ( ord_le118470702582115849rm_a_b @ ( order_8163520290499871439rm_a_b @ R @ A2 ) @ ( order_346605463002971118rm_a_b @ R @ A2 ) ) ).
% underS_subset_under
thf(fact_1233_under__Field,axiom,
! [R: set_Pr1811044260758604347_a_nat,A2: product_prod_a_nat] : ( ord_le8666007276011122963_a_nat @ ( order_6535219843107416792_a_nat @ R @ A2 ) @ ( field_8954927560578634480_a_nat @ R ) ) ).
% under_Field
thf(fact_1234_under__Field,axiom,
! [R: set_Pr2972776593051762503rm_a_b,A2: produc357393685978478089rm_a_b] : ( ord_le118470702582115849rm_a_b @ ( order_346605463002971118rm_a_b @ R @ A2 ) @ ( field_6884932134483023318rm_a_b @ R ) ) ).
% under_Field
thf(fact_1235_wo__rel_Oofilter__def,axiom,
! [R: set_Pr1811044260758604347_a_nat,A: set_Pr4934435412358123699_a_nat] :
( ( bNF_We4298404315170990993_a_nat @ R )
=> ( ( order_7295016824049152615_a_nat @ R @ A )
= ( ( ord_le8666007276011122963_a_nat @ A @ ( field_8954927560578634480_a_nat @ R ) )
& ! [X4: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ X4 @ A )
=> ( ord_le8666007276011122963_a_nat @ ( order_6535219843107416792_a_nat @ R @ X4 ) @ A ) ) ) ) ) ).
% wo_rel.ofilter_def
thf(fact_1236_wo__rel_Oofilter__def,axiom,
! [R: set_Pr2972776593051762503rm_a_b,A: set_Pr4386577575007340137rm_a_b] :
( ( bNF_We1688910813892497397rm_a_b @ R )
=> ( ( order_7982088000704734623rm_a_b @ R @ A )
= ( ( ord_le118470702582115849rm_a_b @ A @ ( field_6884932134483023318rm_a_b @ R ) )
& ! [X4: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X4 @ A )
=> ( ord_le118470702582115849rm_a_b @ ( order_346605463002971118rm_a_b @ R @ X4 ) @ A ) ) ) ) ) ).
% wo_rel.ofilter_def
thf(fact_1237_Inf__fin_Osemilattice__order__set__axioms,axiom,
lattic4070998268164056665_a_nat @ inf_in3813033988066259269_a_nat @ ord_le8666007276011122963_a_nat @ ord_le2482536469199093511_a_nat ).
% Inf_fin.semilattice_order_set_axioms
thf(fact_1238_Inf__fin_Osemilattice__order__set__axioms,axiom,
lattic4620910111349311811rm_a_b @ inf_in1693821330460682071rm_a_b @ ord_le118470702582115849rm_a_b @ ord_le4883060154188259093rm_a_b ).
% Inf_fin.semilattice_order_set_axioms
thf(fact_1239_wo__rel_Oofilter__linord,axiom,
! [R: set_Pr1811044260758604347_a_nat,A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( bNF_We4298404315170990993_a_nat @ R )
=> ( ( order_7295016824049152615_a_nat @ R @ A )
=> ( ( order_7295016824049152615_a_nat @ R @ B )
=> ( ( ord_le8666007276011122963_a_nat @ A @ B )
| ( ord_le8666007276011122963_a_nat @ B @ A ) ) ) ) ) ).
% wo_rel.ofilter_linord
thf(fact_1240_wo__rel_Oofilter__linord,axiom,
! [R: set_Pr2972776593051762503rm_a_b,A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ( bNF_We1688910813892497397rm_a_b @ R )
=> ( ( order_7982088000704734623rm_a_b @ R @ A )
=> ( ( order_7982088000704734623rm_a_b @ R @ B )
=> ( ( ord_le118470702582115849rm_a_b @ A @ B )
| ( ord_le118470702582115849rm_a_b @ B @ A ) ) ) ) ) ).
% wo_rel.ofilter_linord
thf(fact_1241_ofilter__def,axiom,
( order_7295016824049152615_a_nat
= ( ^ [R3: set_Pr1811044260758604347_a_nat,A6: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A6 @ ( field_8954927560578634480_a_nat @ R3 ) )
& ! [X4: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ X4 @ A6 )
=> ( ord_le8666007276011122963_a_nat @ ( order_6535219843107416792_a_nat @ R3 @ X4 ) @ A6 ) ) ) ) ) ).
% ofilter_def
thf(fact_1242_ofilter__def,axiom,
( order_7982088000704734623rm_a_b
= ( ^ [R3: set_Pr2972776593051762503rm_a_b,A6: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A6 @ ( field_6884932134483023318rm_a_b @ R3 ) )
& ! [X4: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X4 @ A6 )
=> ( ord_le118470702582115849rm_a_b @ ( order_346605463002971118rm_a_b @ R3 @ X4 ) @ A6 ) ) ) ) ) ).
% ofilter_def
thf(fact_1243_Range__insert,axiom,
! [A2: term_a_b,B2: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( range_8688746797885257759rm_a_b @ ( insert7009541432154983385rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ R ) )
= ( insert_term_a_b @ B2 @ ( range_8688746797885257759rm_a_b @ R ) ) ) ).
% Range_insert
thf(fact_1244_AboveS__Field,axiom,
! [R: set_Pr1811044260758604347_a_nat,A: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ ( order_5164045672468085234_a_nat @ R @ A ) @ ( field_8954927560578634480_a_nat @ R ) ) ).
% AboveS_Field
thf(fact_1245_AboveS__Field,axiom,
! [R: set_Pr2972776593051762503rm_a_b,A: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ ( order_7177823041100145748rm_a_b @ R @ A ) @ ( field_6884932134483023318rm_a_b @ R ) ) ).
% AboveS_Field
thf(fact_1246_Range__mono,axiom,
! [R: set_Pr4934435412358123699_a_nat,S: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ R @ S )
=> ( ord_less_eq_set_nat @ ( range_a_nat @ R ) @ ( range_a_nat @ S ) ) ) ).
% Range_mono
thf(fact_1247_Range__mono,axiom,
! [R: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ R @ S )
=> ( ord_le2705286416250468010rm_a_b @ ( range_8688746797885257759rm_a_b @ R ) @ ( range_8688746797885257759rm_a_b @ S ) ) ) ).
% Range_mono
thf(fact_1248_Range_Ocases,axiom,
! [A2: nat,R: set_Pr4934435412358123699_a_nat] :
( ( member_nat @ A2 @ ( range_a_nat @ R ) )
=> ~ ! [A5: a] :
~ ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ A5 @ A2 ) @ R ) ) ).
% Range.cases
thf(fact_1249_Range_Ocases,axiom,
! [A2: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( member_term_a_b @ A2 @ ( range_8688746797885257759rm_a_b @ R ) )
=> ~ ! [A5: term_a_b] :
~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A5 @ A2 ) @ R ) ) ).
% Range.cases
thf(fact_1250_Range_Osimps,axiom,
! [A2: nat,R: set_Pr4934435412358123699_a_nat] :
( ( member_nat @ A2 @ ( range_a_nat @ R ) )
= ( ? [A4: a,B4: nat] :
( ( A2 = B4 )
& ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ A4 @ B4 ) @ R ) ) ) ) ).
% Range.simps
thf(fact_1251_Range_Osimps,axiom,
! [A2: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( member_term_a_b @ A2 @ ( range_8688746797885257759rm_a_b @ R ) )
= ( ? [A4: term_a_b,B4: term_a_b] :
( ( A2 = B4 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A4 @ B4 ) @ R ) ) ) ) ).
% Range.simps
thf(fact_1252_Range_Ointros,axiom,
! [A2: a,B2: nat,R: set_Pr4934435412358123699_a_nat] :
( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ A2 @ B2 ) @ R )
=> ( member_nat @ B2 @ ( range_a_nat @ R ) ) ) ).
% Range.intros
thf(fact_1253_Range_Ointros,axiom,
! [A2: term_a_b,B2: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ R )
=> ( member_term_a_b @ B2 @ ( range_8688746797885257759rm_a_b @ R ) ) ) ).
% Range.intros
thf(fact_1254_RangeE,axiom,
! [B2: nat,R: set_Pr4934435412358123699_a_nat] :
( ( member_nat @ B2 @ ( range_a_nat @ R ) )
=> ~ ! [A5: a] :
~ ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ A5 @ B2 ) @ R ) ) ).
% RangeE
thf(fact_1255_RangeE,axiom,
! [B2: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( member_term_a_b @ B2 @ ( range_8688746797885257759rm_a_b @ R ) )
=> ~ ! [A5: term_a_b] :
~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A5 @ B2 ) @ R ) ) ).
% RangeE
thf(fact_1256_Range__iff,axiom,
! [A2: nat,R: set_Pr4934435412358123699_a_nat] :
( ( member_nat @ A2 @ ( range_a_nat @ R ) )
= ( ? [Y5: a] : ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ Y5 @ A2 ) @ R ) ) ) ).
% Range_iff
thf(fact_1257_Range__iff,axiom,
! [A2: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( member_term_a_b @ A2 @ ( range_8688746797885257759rm_a_b @ R ) )
= ( ? [Y5: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y5 @ A2 ) @ R ) ) ) ).
% Range_iff
thf(fact_1258_wo__rel_Osuc__greater,axiom,
! [R: set_Pr4386577575007340137rm_a_b,B: set_term_a_b,B2: term_a_b] :
( ( bNF_We251836928773027112rm_a_b @ R )
=> ( ( ord_le2705286416250468010rm_a_b @ B @ ( field_term_a_b @ R ) )
=> ( ( ( order_8684397134004054025rm_a_b @ R @ B )
!= bot_bot_set_term_a_b )
=> ( ( member_term_a_b @ B2 @ B )
=> ( ( ( bNF_We5270766611192919795rm_a_b @ R @ B )
!= B2 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ ( bNF_We5270766611192919795rm_a_b @ R @ B ) ) @ R ) ) ) ) ) ) ).
% wo_rel.suc_greater
thf(fact_1259_wo__rel_Osuc__greater,axiom,
! [R: set_Pr1811044260758604347_a_nat,B: set_Pr4934435412358123699_a_nat,B2: product_prod_a_nat] :
( ( bNF_We4298404315170990993_a_nat @ R )
=> ( ( ord_le8666007276011122963_a_nat @ B @ ( field_8954927560578634480_a_nat @ R ) )
=> ( ( ( order_5164045672468085234_a_nat @ R @ B )
!= bot_bo9049108969261143879_a_nat )
=> ( ( member5724188588386418708_a_nat @ B2 @ B )
=> ( ( ( bNF_We8244576997016751068_a_nat @ R @ B )
!= B2 )
& ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ B2 @ ( bNF_We8244576997016751068_a_nat @ R @ B ) ) @ R ) ) ) ) ) ) ).
% wo_rel.suc_greater
thf(fact_1260_wo__rel_Osuc__greater,axiom,
! [R: set_Pr2972776593051762503rm_a_b,B: set_Pr4386577575007340137rm_a_b,B2: produc357393685978478089rm_a_b] :
( ( bNF_We1688910813892497397rm_a_b @ R )
=> ( ( ord_le118470702582115849rm_a_b @ B @ ( field_6884932134483023318rm_a_b @ R ) )
=> ( ( ( order_7177823041100145748rm_a_b @ R @ B )
!= bot_bo197521221353338581rm_a_b )
=> ( ( member5869715511025134514rm_a_b @ B2 @ B )
=> ( ( ( bNF_We1496752033444906474rm_a_b @ R @ B )
!= B2 )
& ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B2 @ ( bNF_We1496752033444906474rm_a_b @ R @ B ) ) @ R ) ) ) ) ) ) ).
% wo_rel.suc_greater
thf(fact_1261_wo__rel_Osuc__ofilter__in,axiom,
! [R: set_Pr2972776593051762503rm_a_b,A: set_Pr4386577575007340137rm_a_b,B2: produc357393685978478089rm_a_b] :
( ( bNF_We1688910813892497397rm_a_b @ R )
=> ( ( order_7982088000704734623rm_a_b @ R @ A )
=> ( ( ( order_7177823041100145748rm_a_b @ R @ A )
!= bot_bo197521221353338581rm_a_b )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B2 @ ( bNF_We1496752033444906474rm_a_b @ R @ A ) ) @ R )
=> ( ( B2
!= ( bNF_We1496752033444906474rm_a_b @ R @ A ) )
=> ( member5869715511025134514rm_a_b @ B2 @ A ) ) ) ) ) ) ).
% wo_rel.suc_ofilter_in
thf(fact_1262_wo__rel_Osuc__ofilter__in,axiom,
! [R: set_Pr1811044260758604347_a_nat,A: set_Pr4934435412358123699_a_nat,B2: product_prod_a_nat] :
( ( bNF_We4298404315170990993_a_nat @ R )
=> ( ( order_7295016824049152615_a_nat @ R @ A )
=> ( ( ( order_5164045672468085234_a_nat @ R @ A )
!= bot_bo9049108969261143879_a_nat )
=> ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ B2 @ ( bNF_We8244576997016751068_a_nat @ R @ A ) ) @ R )
=> ( ( B2
!= ( bNF_We8244576997016751068_a_nat @ R @ A ) )
=> ( member5724188588386418708_a_nat @ B2 @ A ) ) ) ) ) ) ).
% wo_rel.suc_ofilter_in
thf(fact_1263_wo__rel_Osuc__ofilter__in,axiom,
! [R: set_Pr4386577575007340137rm_a_b,A: set_term_a_b,B2: term_a_b] :
( ( bNF_We251836928773027112rm_a_b @ R )
=> ( ( order_4225516472610201342rm_a_b @ R @ A )
=> ( ( ( order_8684397134004054025rm_a_b @ R @ A )
!= bot_bot_set_term_a_b )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ ( bNF_We5270766611192919795rm_a_b @ R @ A ) ) @ R )
=> ( ( B2
!= ( bNF_We5270766611192919795rm_a_b @ R @ A ) )
=> ( member_term_a_b @ B2 @ A ) ) ) ) ) ) ).
% wo_rel.suc_ofilter_in
thf(fact_1264_wo__rel_Osuc__least__AboveS,axiom,
! [R: set_Pr2972776593051762503rm_a_b,A2: produc357393685978478089rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ( bNF_We1688910813892497397rm_a_b @ R )
=> ( ( member5869715511025134514rm_a_b @ A2 @ ( order_7177823041100145748rm_a_b @ R @ B ) )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( bNF_We1496752033444906474rm_a_b @ R @ B ) @ A2 ) @ R ) ) ) ).
% wo_rel.suc_least_AboveS
thf(fact_1265_wo__rel_Osuc__least__AboveS,axiom,
! [R: set_Pr1811044260758604347_a_nat,A2: product_prod_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( bNF_We4298404315170990993_a_nat @ R )
=> ( ( member5724188588386418708_a_nat @ A2 @ ( order_5164045672468085234_a_nat @ R @ B ) )
=> ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ ( bNF_We8244576997016751068_a_nat @ R @ B ) @ A2 ) @ R ) ) ) ).
% wo_rel.suc_least_AboveS
thf(fact_1266_wo__rel_Osuc__least__AboveS,axiom,
! [R: set_Pr4386577575007340137rm_a_b,A2: term_a_b,B: set_term_a_b] :
( ( bNF_We251836928773027112rm_a_b @ R )
=> ( ( member_term_a_b @ A2 @ ( order_8684397134004054025rm_a_b @ R @ B ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( bNF_We5270766611192919795rm_a_b @ R @ B ) @ A2 ) @ R ) ) ) ).
% wo_rel.suc_least_AboveS
thf(fact_1267_wo__rel_Oequals__suc__AboveS,axiom,
! [R: set_Pr4386577575007340137rm_a_b,B: set_term_a_b,A2: term_a_b] :
( ( bNF_We251836928773027112rm_a_b @ R )
=> ( ( ord_le2705286416250468010rm_a_b @ B @ ( field_term_a_b @ R ) )
=> ( ( member_term_a_b @ A2 @ ( order_8684397134004054025rm_a_b @ R @ B ) )
=> ( ! [A8: term_a_b] :
( ( member_term_a_b @ A8 @ ( order_8684397134004054025rm_a_b @ R @ B ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ A8 ) @ R ) )
=> ( A2
= ( bNF_We5270766611192919795rm_a_b @ R @ B ) ) ) ) ) ) ).
% wo_rel.equals_suc_AboveS
thf(fact_1268_wo__rel_Oequals__suc__AboveS,axiom,
! [R: set_Pr1811044260758604347_a_nat,B: set_Pr4934435412358123699_a_nat,A2: product_prod_a_nat] :
( ( bNF_We4298404315170990993_a_nat @ R )
=> ( ( ord_le8666007276011122963_a_nat @ B @ ( field_8954927560578634480_a_nat @ R ) )
=> ( ( member5724188588386418708_a_nat @ A2 @ ( order_5164045672468085234_a_nat @ R @ B ) )
=> ( ! [A8: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ A8 @ ( order_5164045672468085234_a_nat @ R @ B ) )
=> ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A2 @ A8 ) @ R ) )
=> ( A2
= ( bNF_We8244576997016751068_a_nat @ R @ B ) ) ) ) ) ) ).
% wo_rel.equals_suc_AboveS
thf(fact_1269_wo__rel_Oequals__suc__AboveS,axiom,
! [R: set_Pr2972776593051762503rm_a_b,B: set_Pr4386577575007340137rm_a_b,A2: produc357393685978478089rm_a_b] :
( ( bNF_We1688910813892497397rm_a_b @ R )
=> ( ( ord_le118470702582115849rm_a_b @ B @ ( field_6884932134483023318rm_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ A2 @ ( order_7177823041100145748rm_a_b @ R @ B ) )
=> ( ! [A8: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ A8 @ ( order_7177823041100145748rm_a_b @ R @ B ) )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A2 @ A8 ) @ R ) )
=> ( A2
= ( bNF_We1496752033444906474rm_a_b @ R @ B ) ) ) ) ) ) ).
% wo_rel.equals_suc_AboveS
thf(fact_1270_wo__rel_Osuc__inField,axiom,
! [R: set_Pr1811044260758604347_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( bNF_We4298404315170990993_a_nat @ R )
=> ( ( ord_le8666007276011122963_a_nat @ B @ ( field_8954927560578634480_a_nat @ R ) )
=> ( ( ( order_5164045672468085234_a_nat @ R @ B )
!= bot_bo9049108969261143879_a_nat )
=> ( member5724188588386418708_a_nat @ ( bNF_We8244576997016751068_a_nat @ R @ B ) @ ( field_8954927560578634480_a_nat @ R ) ) ) ) ) ).
% wo_rel.suc_inField
thf(fact_1271_wo__rel_Osuc__inField,axiom,
! [R: set_Pr2972776593051762503rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ( bNF_We1688910813892497397rm_a_b @ R )
=> ( ( ord_le118470702582115849rm_a_b @ B @ ( field_6884932134483023318rm_a_b @ R ) )
=> ( ( ( order_7177823041100145748rm_a_b @ R @ B )
!= bot_bo197521221353338581rm_a_b )
=> ( member5869715511025134514rm_a_b @ ( bNF_We1496752033444906474rm_a_b @ R @ B ) @ ( field_6884932134483023318rm_a_b @ R ) ) ) ) ) ).
% wo_rel.suc_inField
thf(fact_1272_wo__rel_Osuc__AboveS,axiom,
! [R: set_Pr1811044260758604347_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( bNF_We4298404315170990993_a_nat @ R )
=> ( ( ord_le8666007276011122963_a_nat @ B @ ( field_8954927560578634480_a_nat @ R ) )
=> ( ( ( order_5164045672468085234_a_nat @ R @ B )
!= bot_bo9049108969261143879_a_nat )
=> ( member5724188588386418708_a_nat @ ( bNF_We8244576997016751068_a_nat @ R @ B ) @ ( order_5164045672468085234_a_nat @ R @ B ) ) ) ) ) ).
% wo_rel.suc_AboveS
thf(fact_1273_wo__rel_Osuc__AboveS,axiom,
! [R: set_Pr2972776593051762503rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ( bNF_We1688910813892497397rm_a_b @ R )
=> ( ( ord_le118470702582115849rm_a_b @ B @ ( field_6884932134483023318rm_a_b @ R ) )
=> ( ( ( order_7177823041100145748rm_a_b @ R @ B )
!= bot_bo197521221353338581rm_a_b )
=> ( member5869715511025134514rm_a_b @ ( bNF_We1496752033444906474rm_a_b @ R @ B ) @ ( order_7177823041100145748rm_a_b @ R @ B ) ) ) ) ) ).
% wo_rel.suc_AboveS
thf(fact_1274_mlex__leq,axiom,
! [F2: term_a_b > nat,X3: term_a_b,Y3: term_a_b,R4: set_Pr4386577575007340137rm_a_b] :
( ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ R4 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ ( mlex_prod_term_a_b @ F2 @ R4 ) ) ) ) ).
% mlex_leq
% Conjectures (5)
thf(conj_0,hypothesis,
ord_le8666007276011122963_a_nat @ ( terms_7988297476397195622_a_b_b @ r ) @ f ).
thf(conj_1,hypothesis,
ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ s ) @ f ).
thf(conj_2,hypothesis,
ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ t ) @ f ).
thf(conj_3,hypothesis,
member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ s @ t ) @ ( rstep_a_b @ r ) ).
thf(conj_4,conjecture,
member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ s @ t ) @ ( sig_step_a_b @ f @ ( rstep_a_b @ r ) ) ).
%------------------------------------------------------------------------------