TPTP Problem File: SLH0961^1.p
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%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Rewrite_Properties_Reduction/0019_Rewriting_LLRG_LV_Mondaic/prob_00494_019189__14087268_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1615 ( 445 unt; 336 typ; 0 def)
% Number of atoms : 4146 ( 886 equ; 0 cnn)
% Maximal formula atoms : 8 ( 3 avg)
% Number of connectives : 14668 ( 368 ~; 40 |; 285 &;11997 @)
% ( 0 <=>;1978 =>; 0 <=; 0 <~>)
% Maximal formula depth : 22 ( 8 avg)
% Number of types : 34 ( 33 usr)
% Number of type conns : 1722 (1722 >; 0 *; 0 +; 0 <<)
% Number of symbols : 306 ( 303 usr; 22 con; 0-4 aty)
% Number of variables : 4611 ( 627 ^;3866 !; 118 ?;4611 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 14:19:08.271
%------------------------------------------------------------------------------
% Could-be-implicit typings (33)
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member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_Mt__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
member8417600551952982416rm_a_b: produc4523971788519308903rm_a_b > set_Pr2972776593051762503rm_a_b > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_Mtf__b_J,type,
member3184567406997941638_a_b_b: produc8054098559122065629_a_b_b > set_Pr584113377249817149_a_b_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__b_Mt__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
member8210134394708959770rm_a_b: produc3856293509978307953rm_a_b > set_Pr5162482618186357969rm_a_b > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_Itf__b_Mtf__b_J,type,
member7862447936710763792od_b_b: product_prod_b_b > set_Product_prod_b_b > $o ).
thf(sy_c_member_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
member_term_a_b: term_a_b > set_term_a_b > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_c_member_001tf__b,type,
member_b: b > set_b > $o ).
thf(sy_v__092_060F_062,type,
f: set_Pr4934435412358123699_a_nat ).
thf(sy_v__092_060R_062,type,
r: set_Pr4386577575007340137rm_a_b ).
thf(sy_v__092_060gamma_062____,type,
gamma: b > term_a_b ).
thf(sy_v__092_060sigma_062,type,
sigma: b > term_a_b ).
thf(sy_v__092_060tau_062,type,
tau: b > term_a_b ).
thf(sy_v_l____,type,
l: term_a_b ).
thf(sy_v_r____,type,
r2: term_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_assms_I1_J,axiom,
rewrit2869827417384121323_a_b_a @ r ).
% assms(1)
thf(fact_1_well_I2_J,axiom,
! [X: b] : ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ ( tau @ X ) ) @ f ) ).
% well(2)
thf(fact_2_well_I1_J,axiom,
! [X: b] : ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ ( sigma @ X ) ) @ f ) ).
% well(1)
thf(fact_3_mid_I3_J,axiom,
member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ l @ r2 ) @ r ).
% mid(3)
thf(fact_4_mid_I1_J,axiom,
( s
= ( subst_7999470309526761004_a_b_b @ l @ gamma ) ) ).
% mid(1)
thf(fact_5_mid_I2_J,axiom,
( t
= ( subst_7999470309526761004_a_b_b @ r2 @ gamma ) ) ).
% mid(2)
thf(fact_6_vs,axiom,
! [X: b] :
( ( member_b @ X @ ( vars_term_a_b @ l ) )
=> ~ ( member_b @ X @ ( vars_term_a_b @ r2 ) ) ) ).
% vs
thf(fact_7_assms_I2_J,axiom,
member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ s @ t ) @ ( sig_step_a_b @ f @ ( rrstep_a_b @ r ) ) ).
% assms(2)
thf(fact_8_local_Osubst_I1_J,axiom,
( ( subst_7999470309526761004_a_b_b @ s @ sigma )
= ( subst_7999470309526761004_a_b_b @ l
@ ^ [X2: b] : ( if_term_a_b @ ( member_b @ X2 @ ( vars_term_a_b @ l ) ) @ ( subst_7999470309526761004_a_b_b @ ( gamma @ X2 ) @ sigma ) @ ( subst_7999470309526761004_a_b_b @ ( gamma @ X2 ) @ tau ) ) ) ) ).
% local.subst(1)
thf(fact_9_local_Osubst_I2_J,axiom,
( ( subst_7999470309526761004_a_b_b @ t @ tau )
= ( subst_7999470309526761004_a_b_b @ r2
@ ^ [X2: b] : ( if_term_a_b @ ( member_b @ X2 @ ( vars_term_a_b @ l ) ) @ ( subst_7999470309526761004_a_b_b @ ( gamma @ X2 ) @ sigma ) @ ( subst_7999470309526761004_a_b_b @ ( gamma @ X2 ) @ tau ) ) ) ) ).
% local.subst(2)
thf(fact_10_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_11__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062l_Ar_A_092_060gamma_062_O_A_092_060lbrakk_062s_A_061_Al_A_092_060cdot_062_A_092_060gamma_062_059_At_A_061_Ar_A_092_060cdot_062_A_092_060gamma_062_059_A_Il_M_Ar_J_A_092_060in_062_A_092_060R_062_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [L2: term_a_b,R3: term_a_b,Gamma: b > term_a_b] :
( ( s
= ( subst_7999470309526761004_a_b_b @ L2 @ Gamma ) )
=> ( ( t
= ( subst_7999470309526761004_a_b_b @ R3 @ Gamma ) )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ L2 @ R3 ) @ r ) ) ) ).
% \<open>\<And>thesis. (\<And>l r \<gamma>. \<lbrakk>s = l \<cdot> \<gamma>; t = r \<cdot> \<gamma>; (l, r) \<in> \<R>\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_12_rrstep_Ocases,axiom,
! [A1: term_a_b,A2: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A2 ) @ ( rrstep_a_b @ R2 ) )
=> ~ ! [L2: term_a_b,R3: term_a_b,Sigma: b > term_a_b] :
( ( A1
= ( subst_7999470309526761004_a_b_b @ L2 @ Sigma ) )
=> ( ( A2
= ( subst_7999470309526761004_a_b_b @ R3 @ Sigma ) )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ L2 @ R3 ) @ R2 ) ) ) ) ).
% rrstep.cases
thf(fact_13_rrstep_Osimps,axiom,
! [A1: term_a_b,A2: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A2 ) @ ( rrstep_a_b @ R2 ) )
= ( ? [L3: term_a_b,R4: term_a_b,Sigma2: b > term_a_b] :
( ( A1
= ( subst_7999470309526761004_a_b_b @ L3 @ Sigma2 ) )
& ( A2
= ( subst_7999470309526761004_a_b_b @ R4 @ Sigma2 ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ L3 @ R4 ) @ R2 ) ) ) ) ).
% rrstep.simps
thf(fact_14_rrstep_Ointros,axiom,
! [L: term_a_b,R: term_a_b,R2: set_Pr4386577575007340137rm_a_b,Sigma3: 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 @ Sigma3 ) @ ( subst_7999470309526761004_a_b_b @ R @ Sigma3 ) ) @ ( rrstep_a_b @ R2 ) ) ) ).
% rrstep.intros
thf(fact_15_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 ) )
=> ~ ! [L2: term_a_b,R3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ L2 @ R3 ) @ R2 )
=> ! [Sigma: b > term_a_b] :
( ( S
= ( subst_7999470309526761004_a_b_b @ L2 @ Sigma ) )
=> ( T
!= ( subst_7999470309526761004_a_b_b @ R3 @ Sigma ) ) ) ) ) ).
% rrstep_subst
thf(fact_16_prod_Oinject,axiom,
! [X1: product_prod_a_nat,X22: product_prod_a_nat,Y1: product_prod_a_nat,Y2: product_prod_a_nat] :
( ( ( produc2026711137822539155_a_nat @ X1 @ X22 )
= ( produc2026711137822539155_a_nat @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X22 = Y2 ) ) ) ).
% prod.inject
thf(fact_17_prod_Oinject,axiom,
! [X1: set_term_a_b,X22: set_term_a_b,Y1: set_term_a_b,Y2: set_term_a_b] :
( ( ( produc8517751122659631533rm_a_b @ X1 @ X22 )
= ( produc8517751122659631533rm_a_b @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X22 = Y2 ) ) ) ).
% prod.inject
thf(fact_18_prod_Oinject,axiom,
! [X1: b,X22: produc357393685978478089rm_a_b,Y1: b,Y2: produc357393685978478089rm_a_b] :
( ( ( produc2673992483936284769rm_a_b @ X1 @ X22 )
= ( produc2673992483936284769rm_a_b @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X22 = Y2 ) ) ) ).
% prod.inject
thf(fact_19_prod_Oinject,axiom,
! [X1: b,X22: b,Y1: b,Y2: b] :
( ( ( product_Pair_b_b @ X1 @ X22 )
= ( product_Pair_b_b @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X22 = Y2 ) ) ) ).
% prod.inject
thf(fact_20_prod_Oinject,axiom,
! [X1: a,X22: nat,Y1: a,Y2: nat] :
( ( ( product_Pair_a_nat @ X1 @ X22 )
= ( product_Pair_a_nat @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X22 = Y2 ) ) ) ).
% prod.inject
thf(fact_21_prod_Oinject,axiom,
! [X1: term_a_b,X22: term_a_b,Y1: term_a_b,Y2: term_a_b] :
( ( ( produc7020197800436672577rm_a_b @ X1 @ X22 )
= ( produc7020197800436672577rm_a_b @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X22 = Y2 ) ) ) ).
% prod.inject
thf(fact_22_old_Oprod_Oinject,axiom,
! [A: product_prod_a_nat,B: product_prod_a_nat,A3: product_prod_a_nat,B2: product_prod_a_nat] :
( ( ( produc2026711137822539155_a_nat @ A @ B )
= ( produc2026711137822539155_a_nat @ A3 @ B2 ) )
= ( ( A = A3 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_23_old_Oprod_Oinject,axiom,
! [A: set_term_a_b,B: set_term_a_b,A3: set_term_a_b,B2: set_term_a_b] :
( ( ( produc8517751122659631533rm_a_b @ A @ B )
= ( produc8517751122659631533rm_a_b @ A3 @ B2 ) )
= ( ( A = A3 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_24_old_Oprod_Oinject,axiom,
! [A: b,B: produc357393685978478089rm_a_b,A3: b,B2: produc357393685978478089rm_a_b] :
( ( ( produc2673992483936284769rm_a_b @ A @ B )
= ( produc2673992483936284769rm_a_b @ A3 @ B2 ) )
= ( ( A = A3 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_25_old_Oprod_Oinject,axiom,
! [A: b,B: b,A3: b,B2: b] :
( ( ( product_Pair_b_b @ A @ B )
= ( product_Pair_b_b @ A3 @ B2 ) )
= ( ( A = A3 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_26_old_Oprod_Oinject,axiom,
! [A: a,B: nat,A3: a,B2: nat] :
( ( ( product_Pair_a_nat @ A @ B )
= ( product_Pair_a_nat @ A3 @ B2 ) )
= ( ( A = A3 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_27_old_Oprod_Oinject,axiom,
! [A: term_a_b,B: term_a_b,A3: term_a_b,B2: term_a_b] :
( ( ( produc7020197800436672577rm_a_b @ A @ B )
= ( produc7020197800436672577rm_a_b @ A3 @ B2 ) )
= ( ( A = A3 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_28_term__subst__eq,axiom,
! [T: term_a_b,Sigma3: b > term_a_b,Tau: b > term_a_b] :
( ! [X3: b] :
( ( member_b @ X3 @ ( vars_term_a_b @ T ) )
=> ( ( Sigma3 @ X3 )
= ( Tau @ X3 ) ) )
=> ( ( subst_7999470309526761004_a_b_b @ T @ Sigma3 )
= ( subst_7999470309526761004_a_b_b @ T @ Tau ) ) ) ).
% term_subst_eq
thf(fact_29_term__subst__eq__rev,axiom,
! [T: term_a_b,Sigma3: b > term_a_b,Tau: b > term_a_b] :
( ( ( subst_7999470309526761004_a_b_b @ T @ Sigma3 )
= ( subst_7999470309526761004_a_b_b @ T @ Tau ) )
=> ! [X4: b] :
( ( member_b @ X4 @ ( vars_term_a_b @ T ) )
=> ( ( Sigma3 @ X4 )
= ( Tau @ X4 ) ) ) ) ).
% term_subst_eq_rev
thf(fact_30_term__subst__eq__conv,axiom,
! [T: term_a_b,Sigma3: b > term_a_b,Tau: b > term_a_b] :
( ( ( subst_7999470309526761004_a_b_b @ T @ Sigma3 )
= ( subst_7999470309526761004_a_b_b @ T @ Tau ) )
= ( ! [X2: b] :
( ( member_b @ X2 @ ( vars_term_a_b @ T ) )
=> ( ( Sigma3 @ X2 )
= ( Tau @ X2 ) ) ) ) ) ).
% term_subst_eq_conv
thf(fact_31_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_32_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_33_prod__induct3,axiom,
! [P: produc4708774622424448987_a_nat > $o,X: produc4708774622424448987_a_nat] :
( ! [A4: product_prod_a_nat,B3: a,C: nat] : ( P @ ( produc2026711137822539155_a_nat @ A4 @ ( product_Pair_a_nat @ B3 @ C ) ) )
=> ( P @ X ) ) ).
% prod_induct3
thf(fact_34_prod__induct3,axiom,
! [P: produc3856293509978307953rm_a_b > $o,X: produc3856293509978307953rm_a_b] :
( ! [A4: b,B3: term_a_b,C: term_a_b] : ( P @ ( produc2673992483936284769rm_a_b @ A4 @ ( produc7020197800436672577rm_a_b @ B3 @ C ) ) )
=> ( P @ X ) ) ).
% prod_induct3
thf(fact_35_prod__cases3,axiom,
! [Y: produc4708774622424448987_a_nat] :
~ ! [A4: product_prod_a_nat,B3: a,C: nat] :
( Y
!= ( produc2026711137822539155_a_nat @ A4 @ ( product_Pair_a_nat @ B3 @ C ) ) ) ).
% prod_cases3
thf(fact_36_prod__cases3,axiom,
! [Y: produc3856293509978307953rm_a_b] :
~ ! [A4: b,B3: term_a_b,C: term_a_b] :
( Y
!= ( produc2673992483936284769rm_a_b @ A4 @ ( produc7020197800436672577rm_a_b @ B3 @ C ) ) ) ).
% prod_cases3
thf(fact_37_Pair__inject,axiom,
! [A: product_prod_a_nat,B: product_prod_a_nat,A3: product_prod_a_nat,B2: product_prod_a_nat] :
( ( ( produc2026711137822539155_a_nat @ A @ B )
= ( produc2026711137822539155_a_nat @ A3 @ B2 ) )
=> ~ ( ( A = A3 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_38_Pair__inject,axiom,
! [A: set_term_a_b,B: set_term_a_b,A3: set_term_a_b,B2: set_term_a_b] :
( ( ( produc8517751122659631533rm_a_b @ A @ B )
= ( produc8517751122659631533rm_a_b @ A3 @ B2 ) )
=> ~ ( ( A = A3 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_39_Pair__inject,axiom,
! [A: b,B: produc357393685978478089rm_a_b,A3: b,B2: produc357393685978478089rm_a_b] :
( ( ( produc2673992483936284769rm_a_b @ A @ B )
= ( produc2673992483936284769rm_a_b @ A3 @ B2 ) )
=> ~ ( ( A = A3 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_40_Pair__inject,axiom,
! [A: b,B: b,A3: b,B2: b] :
( ( ( product_Pair_b_b @ A @ B )
= ( product_Pair_b_b @ A3 @ B2 ) )
=> ~ ( ( A = A3 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_41_Pair__inject,axiom,
! [A: a,B: nat,A3: a,B2: nat] :
( ( ( product_Pair_a_nat @ A @ B )
= ( product_Pair_a_nat @ A3 @ B2 ) )
=> ~ ( ( A = A3 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_42_Pair__inject,axiom,
! [A: term_a_b,B: term_a_b,A3: term_a_b,B2: term_a_b] :
( ( ( produc7020197800436672577rm_a_b @ A @ B )
= ( produc7020197800436672577rm_a_b @ A3 @ B2 ) )
=> ~ ( ( A = A3 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_43_prod__cases,axiom,
! [P: produc4708774622424448987_a_nat > $o,P2: produc4708774622424448987_a_nat] :
( ! [A4: product_prod_a_nat,B3: product_prod_a_nat] : ( P @ ( produc2026711137822539155_a_nat @ A4 @ B3 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_44_prod__cases,axiom,
! [P: produc9001763001631046773rm_a_b > $o,P2: produc9001763001631046773rm_a_b] :
( ! [A4: set_term_a_b,B3: set_term_a_b] : ( P @ ( produc8517751122659631533rm_a_b @ A4 @ B3 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_45_prod__cases,axiom,
! [P: produc3856293509978307953rm_a_b > $o,P2: produc3856293509978307953rm_a_b] :
( ! [A4: b,B3: produc357393685978478089rm_a_b] : ( P @ ( produc2673992483936284769rm_a_b @ A4 @ B3 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_46_prod__cases,axiom,
! [P: product_prod_b_b > $o,P2: product_prod_b_b] :
( ! [A4: b,B3: b] : ( P @ ( product_Pair_b_b @ A4 @ B3 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_47_prod__cases,axiom,
! [P: product_prod_a_nat > $o,P2: product_prod_a_nat] :
( ! [A4: a,B3: nat] : ( P @ ( product_Pair_a_nat @ A4 @ B3 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_48_prod__cases,axiom,
! [P: produc357393685978478089rm_a_b > $o,P2: produc357393685978478089rm_a_b] :
( ! [A4: term_a_b,B3: term_a_b] : ( P @ ( produc7020197800436672577rm_a_b @ A4 @ B3 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_49_surj__pair,axiom,
! [P2: produc4708774622424448987_a_nat] :
? [X3: product_prod_a_nat,Y3: product_prod_a_nat] :
( P2
= ( produc2026711137822539155_a_nat @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_50_surj__pair,axiom,
! [P2: produc9001763001631046773rm_a_b] :
? [X3: set_term_a_b,Y3: set_term_a_b] :
( P2
= ( produc8517751122659631533rm_a_b @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_51_surj__pair,axiom,
! [P2: produc3856293509978307953rm_a_b] :
? [X3: b,Y3: produc357393685978478089rm_a_b] :
( P2
= ( produc2673992483936284769rm_a_b @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_52_surj__pair,axiom,
! [P2: product_prod_b_b] :
? [X3: b,Y3: b] :
( P2
= ( product_Pair_b_b @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_53_surj__pair,axiom,
! [P2: product_prod_a_nat] :
? [X3: a,Y3: nat] :
( P2
= ( product_Pair_a_nat @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_54_surj__pair,axiom,
! [P2: produc357393685978478089rm_a_b] :
? [X3: term_a_b,Y3: term_a_b] :
( P2
= ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_55_old_Oprod_Oexhaust,axiom,
! [Y: produc4708774622424448987_a_nat] :
~ ! [A4: product_prod_a_nat,B3: product_prod_a_nat] :
( Y
!= ( produc2026711137822539155_a_nat @ A4 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_56_old_Oprod_Oexhaust,axiom,
! [Y: produc9001763001631046773rm_a_b] :
~ ! [A4: set_term_a_b,B3: set_term_a_b] :
( Y
!= ( produc8517751122659631533rm_a_b @ A4 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_57_old_Oprod_Oexhaust,axiom,
! [Y: produc3856293509978307953rm_a_b] :
~ ! [A4: b,B3: produc357393685978478089rm_a_b] :
( Y
!= ( produc2673992483936284769rm_a_b @ A4 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_58_old_Oprod_Oexhaust,axiom,
! [Y: product_prod_b_b] :
~ ! [A4: b,B3: b] :
( Y
!= ( product_Pair_b_b @ A4 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_59_old_Oprod_Oexhaust,axiom,
! [Y: product_prod_a_nat] :
~ ! [A4: a,B3: nat] :
( Y
!= ( product_Pair_a_nat @ A4 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_60_old_Oprod_Oexhaust,axiom,
! [Y: produc357393685978478089rm_a_b] :
~ ! [A4: term_a_b,B3: term_a_b] :
( Y
!= ( produc7020197800436672577rm_a_b @ A4 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_61_subst__term__eqI,axiom,
! [Sigma3: b > term_a_b,Tau: b > term_a_b] :
( ! [T2: term_a_b] :
( ( subst_7999470309526761004_a_b_b @ T2 @ Sigma3 )
= ( subst_7999470309526761004_a_b_b @ T2 @ Tau ) )
=> ( Sigma3 = Tau ) ) ).
% subst_term_eqI
thf(fact_62_vars__term__subset__subst__eq,axiom,
! [T: term_a_b,S: term_a_b,Sigma3: b > term_a_b,Tau: b > term_a_b] :
( ( ord_less_eq_set_b @ ( vars_term_a_b @ T ) @ ( vars_term_a_b @ S ) )
=> ( ( ( subst_7999470309526761004_a_b_b @ S @ Sigma3 )
= ( subst_7999470309526761004_a_b_b @ S @ Tau ) )
=> ( ( subst_7999470309526761004_a_b_b @ T @ Sigma3 )
= ( subst_7999470309526761004_a_b_b @ T @ Tau ) ) ) ) ).
% vars_term_subset_subst_eq
thf(fact_63_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_64_subsetI,axiom,
! [A5: set_Product_prod_b_b,B4: set_Product_prod_b_b] :
( ! [X3: product_prod_b_b] :
( ( member7862447936710763792od_b_b @ X3 @ A5 )
=> ( member7862447936710763792od_b_b @ X3 @ B4 ) )
=> ( ord_le182087997850975847od_b_b @ A5 @ B4 ) ) ).
% subsetI
thf(fact_65_subsetI,axiom,
! [A5: set_term_a_b,B4: set_term_a_b] :
( ! [X3: term_a_b] :
( ( member_term_a_b @ X3 @ A5 )
=> ( member_term_a_b @ X3 @ B4 ) )
=> ( ord_le2705286416250468010rm_a_b @ A5 @ B4 ) ) ).
% subsetI
thf(fact_66_subsetI,axiom,
! [A5: set_nat,B4: set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A5 )
=> ( member_nat @ X3 @ B4 ) )
=> ( ord_less_eq_set_nat @ A5 @ B4 ) ) ).
% subsetI
thf(fact_67_subsetI,axiom,
! [A5: set_a,B4: set_a] :
( ! [X3: a] :
( ( member_a @ X3 @ A5 )
=> ( member_a @ X3 @ B4 ) )
=> ( ord_less_eq_set_a @ A5 @ B4 ) ) ).
% subsetI
thf(fact_68_subsetI,axiom,
! [A5: set_Pr4386577575007340137rm_a_b,B4: set_Pr4386577575007340137rm_a_b] :
( ! [X3: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X3 @ A5 )
=> ( member5869715511025134514rm_a_b @ X3 @ B4 ) )
=> ( ord_le118470702582115849rm_a_b @ A5 @ B4 ) ) ).
% subsetI
thf(fact_69_subsetI,axiom,
! [A5: set_b,B4: set_b] :
( ! [X3: b] :
( ( member_b @ X3 @ A5 )
=> ( member_b @ X3 @ B4 ) )
=> ( ord_less_eq_set_b @ A5 @ B4 ) ) ).
% subsetI
thf(fact_70_subsetI,axiom,
! [A5: set_Pr4934435412358123699_a_nat,B4: set_Pr4934435412358123699_a_nat] :
( ! [X3: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ X3 @ A5 )
=> ( member5724188588386418708_a_nat @ X3 @ B4 ) )
=> ( ord_le8666007276011122963_a_nat @ A5 @ B4 ) ) ).
% subsetI
thf(fact_71_subset__antisym,axiom,
! [A5: set_Pr4934435412358123699_a_nat,B4: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A5 @ B4 )
=> ( ( ord_le8666007276011122963_a_nat @ B4 @ A5 )
=> ( A5 = B4 ) ) ) ).
% subset_antisym
thf(fact_72_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_73_mem__Collect__eq,axiom,
! [A: a,P: a > $o] :
( ( member_a @ A @ ( collect_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_74_mem__Collect__eq,axiom,
! [A: produc9001763001631046773rm_a_b,P: produc9001763001631046773rm_a_b > $o] :
( ( member8888773126115863070rm_a_b @ A @ ( collec4780413679787665248rm_a_b @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_75_mem__Collect__eq,axiom,
! [A: product_prod_b_b,P: product_prod_b_b > $o] :
( ( member7862447936710763792od_b_b @ A @ ( collec548942219715005266od_b_b @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_76_mem__Collect__eq,axiom,
! [A: product_prod_a_nat,P: product_prod_a_nat > $o] :
( ( member5724188588386418708_a_nat @ A @ ( collec4464134535221767506_a_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_77_mem__Collect__eq,axiom,
! [A: term_a_b,P: term_a_b > $o] :
( ( member_term_a_b @ A @ ( collect_term_a_b @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_78_mem__Collect__eq,axiom,
! [A: produc357393685978478089rm_a_b,P: produc357393685978478089rm_a_b > $o] :
( ( member5869715511025134514rm_a_b @ A @ ( collec99177395925924084rm_a_b @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_79_mem__Collect__eq,axiom,
! [A: b,P: b > $o] :
( ( member_b @ A @ ( collect_b @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_80_Collect__mem__eq,axiom,
! [A5: set_nat] :
( ( collect_nat
@ ^ [X2: nat] : ( member_nat @ X2 @ A5 ) )
= A5 ) ).
% Collect_mem_eq
thf(fact_81_Collect__mem__eq,axiom,
! [A5: set_a] :
( ( collect_a
@ ^ [X2: a] : ( member_a @ X2 @ A5 ) )
= A5 ) ).
% Collect_mem_eq
thf(fact_82_Collect__mem__eq,axiom,
! [A5: set_Pr7151917219565317845rm_a_b] :
( ( collec4780413679787665248rm_a_b
@ ^ [X2: produc9001763001631046773rm_a_b] : ( member8888773126115863070rm_a_b @ X2 @ A5 ) )
= A5 ) ).
% Collect_mem_eq
thf(fact_83_Collect__mem__eq,axiom,
! [A5: set_Product_prod_b_b] :
( ( collec548942219715005266od_b_b
@ ^ [X2: product_prod_b_b] : ( member7862447936710763792od_b_b @ X2 @ A5 ) )
= A5 ) ).
% Collect_mem_eq
thf(fact_84_Collect__mem__eq,axiom,
! [A5: set_Pr4934435412358123699_a_nat] :
( ( collec4464134535221767506_a_nat
@ ^ [X2: product_prod_a_nat] : ( member5724188588386418708_a_nat @ X2 @ A5 ) )
= A5 ) ).
% Collect_mem_eq
thf(fact_85_Collect__mem__eq,axiom,
! [A5: set_term_a_b] :
( ( collect_term_a_b
@ ^ [X2: term_a_b] : ( member_term_a_b @ X2 @ A5 ) )
= A5 ) ).
% Collect_mem_eq
thf(fact_86_Collect__mem__eq,axiom,
! [A5: set_Pr4386577575007340137rm_a_b] :
( ( collec99177395925924084rm_a_b
@ ^ [X2: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ X2 @ A5 ) )
= A5 ) ).
% Collect_mem_eq
thf(fact_87_Collect__mem__eq,axiom,
! [A5: set_b] :
( ( collect_b
@ ^ [X2: b] : ( member_b @ X2 @ A5 ) )
= A5 ) ).
% Collect_mem_eq
thf(fact_88_Collect__cong,axiom,
! [P: produc9001763001631046773rm_a_b > $o,Q: produc9001763001631046773rm_a_b > $o] :
( ! [X3: produc9001763001631046773rm_a_b] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collec4780413679787665248rm_a_b @ P )
= ( collec4780413679787665248rm_a_b @ Q ) ) ) ).
% Collect_cong
thf(fact_89_Collect__cong,axiom,
! [P: product_prod_b_b > $o,Q: product_prod_b_b > $o] :
( ! [X3: product_prod_b_b] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collec548942219715005266od_b_b @ P )
= ( collec548942219715005266od_b_b @ Q ) ) ) ).
% Collect_cong
thf(fact_90_Collect__cong,axiom,
! [P: product_prod_a_nat > $o,Q: product_prod_a_nat > $o] :
( ! [X3: product_prod_a_nat] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collec4464134535221767506_a_nat @ P )
= ( collec4464134535221767506_a_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_91_Collect__cong,axiom,
! [P: term_a_b > $o,Q: term_a_b > $o] :
( ! [X3: term_a_b] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_term_a_b @ P )
= ( collect_term_a_b @ Q ) ) ) ).
% Collect_cong
thf(fact_92_Collect__cong,axiom,
! [P: b > $o,Q: b > $o] :
( ! [X3: b] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_b @ P )
= ( collect_b @ Q ) ) ) ).
% Collect_cong
thf(fact_93_order__refl,axiom,
! [X: produc357393685978478089rm_a_b > $o] : ( ord_le6645574462093738516_a_b_o @ X @ X ) ).
% order_refl
thf(fact_94_order__refl,axiom,
! [X: product_prod_a_nat > $o] : ( ord_le8950881081082477362_nat_o @ X @ X ) ).
% order_refl
thf(fact_95_order__refl,axiom,
! [X: term_a_b > term_a_b > $o] : ( ord_le8627286948385211390_a_b_o @ X @ X ) ).
% order_refl
thf(fact_96_order__refl,axiom,
! [X: b > $o] : ( ord_less_eq_b_o @ X @ X ) ).
% order_refl
thf(fact_97_order__refl,axiom,
! [X: a > nat > $o] : ( ord_less_eq_a_nat_o @ X @ X ) ).
% order_refl
thf(fact_98_order__refl,axiom,
! [X: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ X @ X ) ).
% order_refl
thf(fact_99_dual__order_Orefl,axiom,
! [A: produc357393685978478089rm_a_b > $o] : ( ord_le6645574462093738516_a_b_o @ A @ A ) ).
% dual_order.refl
thf(fact_100_dual__order_Orefl,axiom,
! [A: product_prod_a_nat > $o] : ( ord_le8950881081082477362_nat_o @ A @ A ) ).
% dual_order.refl
thf(fact_101_dual__order_Orefl,axiom,
! [A: term_a_b > term_a_b > $o] : ( ord_le8627286948385211390_a_b_o @ A @ A ) ).
% dual_order.refl
thf(fact_102_dual__order_Orefl,axiom,
! [A: b > $o] : ( ord_less_eq_b_o @ A @ A ) ).
% dual_order.refl
thf(fact_103_dual__order_Orefl,axiom,
! [A: a > nat > $o] : ( ord_less_eq_a_nat_o @ A @ A ) ).
% dual_order.refl
thf(fact_104_dual__order_Orefl,axiom,
! [A: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ A @ A ) ).
% dual_order.refl
thf(fact_105_subst__apply__term__restrict__subst__domain,axiom,
! [T: term_a_b,V: set_b,Sigma3: b > term_a_b] :
( ( ord_less_eq_set_b @ ( vars_term_a_b @ T ) @ V )
=> ( ( subst_7999470309526761004_a_b_b @ T @ ( restri22458263168500592in_b_a @ V @ Sigma3 ) )
= ( subst_7999470309526761004_a_b_b @ T @ Sigma3 ) ) ) ).
% subst_apply_term_restrict_subst_domain
thf(fact_106_subrelI,axiom,
! [R: set_Pr1811044260758604347_a_nat,S: set_Pr1811044260758604347_a_nat] :
( ! [X3: product_prod_a_nat,Y3: product_prod_a_nat] :
( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ X3 @ Y3 ) @ R )
=> ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ X3 @ Y3 ) @ S ) )
=> ( ord_le1018141841545957851_a_nat @ R @ S ) ) ).
% subrelI
thf(fact_107_subrelI,axiom,
! [R: set_Pr7151917219565317845rm_a_b,S: set_Pr7151917219565317845rm_a_b] :
( ! [X3: set_term_a_b,Y3: set_term_a_b] :
( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ X3 @ Y3 ) @ R )
=> ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ X3 @ Y3 ) @ S ) )
=> ( ord_le2663425029446080117rm_a_b @ R @ S ) ) ).
% subrelI
thf(fact_108_subrelI,axiom,
! [R: set_Pr5162482618186357969rm_a_b,S: set_Pr5162482618186357969rm_a_b] :
( ! [X3: b,Y3: produc357393685978478089rm_a_b] :
( ( member8210134394708959770rm_a_b @ ( produc2673992483936284769rm_a_b @ X3 @ Y3 ) @ R )
=> ( member8210134394708959770rm_a_b @ ( produc2673992483936284769rm_a_b @ X3 @ Y3 ) @ S ) )
=> ( ord_le4369580198973711473rm_a_b @ R @ S ) ) ).
% subrelI
thf(fact_109_subrelI,axiom,
! [R: set_Product_prod_b_b,S: set_Product_prod_b_b] :
( ! [X3: b,Y3: b] :
( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ X3 @ Y3 ) @ R )
=> ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ X3 @ Y3 ) @ S ) )
=> ( ord_le182087997850975847od_b_b @ R @ S ) ) ).
% subrelI
thf(fact_110_subrelI,axiom,
! [R: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
( ! [X3: term_a_b,Y3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ R )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ S ) )
=> ( ord_le118470702582115849rm_a_b @ R @ S ) ) ).
% subrelI
thf(fact_111_subrelI,axiom,
! [R: set_Pr4934435412358123699_a_nat,S: set_Pr4934435412358123699_a_nat] :
( ! [X3: a,Y3: nat] :
( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ X3 @ Y3 ) @ R )
=> ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ X3 @ Y3 ) @ S ) )
=> ( ord_le8666007276011122963_a_nat @ R @ S ) ) ).
% subrelI
thf(fact_112_subst__ident,axiom,
! [X: b,T: term_a_b,U: term_a_b] :
( ~ ( member_b @ X @ ( vars_term_a_b @ T ) )
=> ( ( subst_7999470309526761004_a_b_b @ T @ ( subst_b_a @ X @ U ) )
= T ) ) ).
% subst_ident
thf(fact_113_srstep__subst__closed,axiom,
! [S: term_a_b,T: term_a_b,F: set_Pr4934435412358123699_a_nat,R2: set_Pr4386577575007340137rm_a_b,Sigma3: b > term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( sig_step_a_b @ F @ ( rstep_a_b @ R2 ) ) )
=> ( ! [X3: b] : ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ ( Sigma3 @ X3 ) ) @ F )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( subst_7999470309526761004_a_b_b @ S @ Sigma3 ) @ ( subst_7999470309526761004_a_b_b @ T @ Sigma3 ) ) @ ( sig_step_a_b @ F @ ( rstep_a_b @ R2 ) ) ) ) ) ).
% srstep_subst_closed
thf(fact_114_rrstepp__rrstep__eq,axiom,
! [R2: set_Pr4386577575007340137rm_a_b] :
( ( rrstepp_a_b
@ ^ [X2: term_a_b,Y4: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y4 ) @ R2 ) )
= ( ^ [X2: term_a_b,Y4: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y4 ) @ ( rrstep_a_b @ R2 ) ) ) ) ).
% rrstepp_rrstep_eq
thf(fact_115_subset__Collect__iff,axiom,
! [B4: set_nat,A5: set_nat,P: nat > $o] :
( ( ord_less_eq_set_nat @ B4 @ A5 )
=> ( ( ord_less_eq_set_nat @ B4
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A5 )
& ( P @ X2 ) ) ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ B4 )
=> ( P @ X2 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_116_subset__Collect__iff,axiom,
! [B4: set_a,A5: set_a,P: a > $o] :
( ( ord_less_eq_set_a @ B4 @ A5 )
=> ( ( ord_less_eq_set_a @ B4
@ ( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ A5 )
& ( P @ X2 ) ) ) )
= ( ! [X2: a] :
( ( member_a @ X2 @ B4 )
=> ( P @ X2 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_117_subset__Collect__iff,axiom,
! [B4: set_Pr7151917219565317845rm_a_b,A5: set_Pr7151917219565317845rm_a_b,P: produc9001763001631046773rm_a_b > $o] :
( ( ord_le2663425029446080117rm_a_b @ B4 @ A5 )
=> ( ( ord_le2663425029446080117rm_a_b @ B4
@ ( collec4780413679787665248rm_a_b
@ ^ [X2: produc9001763001631046773rm_a_b] :
( ( member8888773126115863070rm_a_b @ X2 @ A5 )
& ( P @ X2 ) ) ) )
= ( ! [X2: produc9001763001631046773rm_a_b] :
( ( member8888773126115863070rm_a_b @ X2 @ B4 )
=> ( P @ X2 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_118_subset__Collect__iff,axiom,
! [B4: set_Product_prod_b_b,A5: set_Product_prod_b_b,P: product_prod_b_b > $o] :
( ( ord_le182087997850975847od_b_b @ B4 @ A5 )
=> ( ( ord_le182087997850975847od_b_b @ B4
@ ( collec548942219715005266od_b_b
@ ^ [X2: product_prod_b_b] :
( ( member7862447936710763792od_b_b @ X2 @ A5 )
& ( P @ X2 ) ) ) )
= ( ! [X2: product_prod_b_b] :
( ( member7862447936710763792od_b_b @ X2 @ B4 )
=> ( P @ X2 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_119_subset__Collect__iff,axiom,
! [B4: set_term_a_b,A5: set_term_a_b,P: term_a_b > $o] :
( ( ord_le2705286416250468010rm_a_b @ B4 @ A5 )
=> ( ( ord_le2705286416250468010rm_a_b @ B4
@ ( collect_term_a_b
@ ^ [X2: term_a_b] :
( ( member_term_a_b @ X2 @ A5 )
& ( P @ X2 ) ) ) )
= ( ! [X2: term_a_b] :
( ( member_term_a_b @ X2 @ B4 )
=> ( P @ X2 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_120_subset__Collect__iff,axiom,
! [B4: set_Pr4386577575007340137rm_a_b,A5: set_Pr4386577575007340137rm_a_b,P: produc357393685978478089rm_a_b > $o] :
( ( ord_le118470702582115849rm_a_b @ B4 @ A5 )
=> ( ( ord_le118470702582115849rm_a_b @ B4
@ ( collec99177395925924084rm_a_b
@ ^ [X2: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X2 @ A5 )
& ( P @ X2 ) ) ) )
= ( ! [X2: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X2 @ B4 )
=> ( P @ X2 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_121_subset__Collect__iff,axiom,
! [B4: set_b,A5: set_b,P: b > $o] :
( ( ord_less_eq_set_b @ B4 @ A5 )
=> ( ( ord_less_eq_set_b @ B4
@ ( collect_b
@ ^ [X2: b] :
( ( member_b @ X2 @ A5 )
& ( P @ X2 ) ) ) )
= ( ! [X2: b] :
( ( member_b @ X2 @ B4 )
=> ( P @ X2 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_122_subset__Collect__iff,axiom,
! [B4: set_Pr4934435412358123699_a_nat,A5: set_Pr4934435412358123699_a_nat,P: product_prod_a_nat > $o] :
( ( ord_le8666007276011122963_a_nat @ B4 @ A5 )
=> ( ( ord_le8666007276011122963_a_nat @ B4
@ ( collec4464134535221767506_a_nat
@ ^ [X2: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ X2 @ A5 )
& ( P @ X2 ) ) ) )
= ( ! [X2: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ X2 @ B4 )
=> ( P @ X2 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_123_subst__simps_I1_J,axiom,
! [X: b,T: term_a_b] :
( ( subst_b_a @ X @ T @ X )
= T ) ).
% subst_simps(1)
thf(fact_124_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_125_subst__apply__left__idemp,axiom,
! [Sigma3: b > term_a_b,X: b,T: term_a_b,S: term_a_b] :
( ( ( Sigma3 @ X )
= ( subst_7999470309526761004_a_b_b @ T @ Sigma3 ) )
=> ( ( subst_7999470309526761004_a_b_b @ ( subst_7999470309526761004_a_b_b @ S @ ( subst_b_a @ X @ T ) ) @ Sigma3 )
= ( subst_7999470309526761004_a_b_b @ S @ Sigma3 ) ) ) ).
% subst_apply_left_idemp
thf(fact_126_rstep__substI,axiom,
! [S: term_a_b,T: term_a_b,R2: set_Pr4386577575007340137rm_a_b,Sigma3: 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 @ Sigma3 ) @ ( subst_7999470309526761004_a_b_b @ T @ Sigma3 ) ) @ ( rstep_a_b @ R2 ) ) ) ).
% rstep_substI
thf(fact_127_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_128_pred__subset__eq,axiom,
! [R5: set_Product_prod_b_b,S2: set_Product_prod_b_b] :
( ( ord_le39139162152160566_b_b_o
@ ^ [X2: product_prod_b_b] : ( member7862447936710763792od_b_b @ X2 @ R5 )
@ ^ [X2: product_prod_b_b] : ( member7862447936710763792od_b_b @ X2 @ S2 ) )
= ( ord_le182087997850975847od_b_b @ R5 @ S2 ) ) ).
% pred_subset_eq
thf(fact_129_pred__subset__eq,axiom,
! [R5: set_term_a_b,S2: set_term_a_b] :
( ( ord_le3490609322949561435_a_b_o
@ ^ [X2: term_a_b] : ( member_term_a_b @ X2 @ R5 )
@ ^ [X2: term_a_b] : ( member_term_a_b @ X2 @ S2 ) )
= ( ord_le2705286416250468010rm_a_b @ R5 @ S2 ) ) ).
% pred_subset_eq
thf(fact_130_pred__subset__eq,axiom,
! [R5: set_nat,S2: set_nat] :
( ( ord_less_eq_nat_o
@ ^ [X2: nat] : ( member_nat @ X2 @ R5 )
@ ^ [X2: nat] : ( member_nat @ X2 @ S2 ) )
= ( ord_less_eq_set_nat @ R5 @ S2 ) ) ).
% pred_subset_eq
thf(fact_131_pred__subset__eq,axiom,
! [R5: set_a,S2: set_a] :
( ( ord_less_eq_a_o
@ ^ [X2: a] : ( member_a @ X2 @ R5 )
@ ^ [X2: a] : ( member_a @ X2 @ S2 ) )
= ( ord_less_eq_set_a @ R5 @ S2 ) ) ).
% pred_subset_eq
thf(fact_132_pred__subset__eq,axiom,
! [R5: set_Pr4386577575007340137rm_a_b,S2: set_Pr4386577575007340137rm_a_b] :
( ( ord_le6645574462093738516_a_b_o
@ ^ [X2: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ X2 @ R5 )
@ ^ [X2: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ X2 @ S2 ) )
= ( ord_le118470702582115849rm_a_b @ R5 @ S2 ) ) ).
% pred_subset_eq
thf(fact_133_pred__subset__eq,axiom,
! [R5: set_b,S2: set_b] :
( ( ord_less_eq_b_o
@ ^ [X2: b] : ( member_b @ X2 @ R5 )
@ ^ [X2: b] : ( member_b @ X2 @ S2 ) )
= ( ord_less_eq_set_b @ R5 @ S2 ) ) ).
% pred_subset_eq
thf(fact_134_pred__subset__eq,axiom,
! [R5: set_Pr4934435412358123699_a_nat,S2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8950881081082477362_nat_o
@ ^ [X2: product_prod_a_nat] : ( member5724188588386418708_a_nat @ X2 @ R5 )
@ ^ [X2: product_prod_a_nat] : ( member5724188588386418708_a_nat @ X2 @ S2 ) )
= ( ord_le8666007276011122963_a_nat @ R5 @ S2 ) ) ).
% pred_subset_eq
thf(fact_135_less__eq__set__def,axiom,
( ord_le182087997850975847od_b_b
= ( ^ [A6: set_Product_prod_b_b,B5: set_Product_prod_b_b] :
( ord_le39139162152160566_b_b_o
@ ^ [X2: product_prod_b_b] : ( member7862447936710763792od_b_b @ X2 @ A6 )
@ ^ [X2: product_prod_b_b] : ( member7862447936710763792od_b_b @ X2 @ B5 ) ) ) ) ).
% less_eq_set_def
thf(fact_136_less__eq__set__def,axiom,
( ord_le2705286416250468010rm_a_b
= ( ^ [A6: set_term_a_b,B5: set_term_a_b] :
( ord_le3490609322949561435_a_b_o
@ ^ [X2: term_a_b] : ( member_term_a_b @ X2 @ A6 )
@ ^ [X2: term_a_b] : ( member_term_a_b @ X2 @ B5 ) ) ) ) ).
% less_eq_set_def
thf(fact_137_less__eq__set__def,axiom,
( ord_less_eq_set_nat
= ( ^ [A6: set_nat,B5: set_nat] :
( ord_less_eq_nat_o
@ ^ [X2: nat] : ( member_nat @ X2 @ A6 )
@ ^ [X2: nat] : ( member_nat @ X2 @ B5 ) ) ) ) ).
% less_eq_set_def
thf(fact_138_less__eq__set__def,axiom,
( ord_less_eq_set_a
= ( ^ [A6: set_a,B5: set_a] :
( ord_less_eq_a_o
@ ^ [X2: a] : ( member_a @ X2 @ A6 )
@ ^ [X2: a] : ( member_a @ X2 @ B5 ) ) ) ) ).
% less_eq_set_def
thf(fact_139_less__eq__set__def,axiom,
( ord_le118470702582115849rm_a_b
= ( ^ [A6: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
( ord_le6645574462093738516_a_b_o
@ ^ [X2: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ X2 @ A6 )
@ ^ [X2: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ X2 @ B5 ) ) ) ) ).
% less_eq_set_def
thf(fact_140_less__eq__set__def,axiom,
( ord_less_eq_set_b
= ( ^ [A6: set_b,B5: set_b] :
( ord_less_eq_b_o
@ ^ [X2: b] : ( member_b @ X2 @ A6 )
@ ^ [X2: b] : ( member_b @ X2 @ B5 ) ) ) ) ).
% less_eq_set_def
thf(fact_141_less__eq__set__def,axiom,
( ord_le8666007276011122963_a_nat
= ( ^ [A6: set_Pr4934435412358123699_a_nat,B5: set_Pr4934435412358123699_a_nat] :
( ord_le8950881081082477362_nat_o
@ ^ [X2: product_prod_a_nat] : ( member5724188588386418708_a_nat @ X2 @ A6 )
@ ^ [X2: product_prod_a_nat] : ( member5724188588386418708_a_nat @ X2 @ B5 ) ) ) ) ).
% less_eq_set_def
thf(fact_142_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_143_sig__step__mono2,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,L4: set_Pr4386577575007340137rm_a_b,F: set_Pr4934435412358123699_a_nat] :
( ( ord_le118470702582115849rm_a_b @ R2 @ L4 )
=> ( ord_le118470702582115849rm_a_b @ ( sig_step_a_b @ F @ R2 ) @ ( sig_step_a_b @ F @ L4 ) ) ) ).
% sig_step_mono2
thf(fact_144_pred__subset__eq2,axiom,
! [R5: set_Pr1811044260758604347_a_nat,S2: set_Pr1811044260758604347_a_nat] :
( ( ord_le9075658533858151596_nat_o
@ ^ [X2: product_prod_a_nat,Y4: product_prod_a_nat] : ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ X2 @ Y4 ) @ R5 )
@ ^ [X2: product_prod_a_nat,Y4: product_prod_a_nat] : ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ X2 @ Y4 ) @ S2 ) )
= ( ord_le1018141841545957851_a_nat @ R5 @ S2 ) ) ).
% pred_subset_eq2
thf(fact_145_pred__subset__eq2,axiom,
! [R5: set_Pr7151917219565317845rm_a_b,S2: set_Pr7151917219565317845rm_a_b] :
( ( ord_le6917259649888057234_a_b_o
@ ^ [X2: set_term_a_b,Y4: set_term_a_b] : ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ X2 @ Y4 ) @ R5 )
@ ^ [X2: set_term_a_b,Y4: set_term_a_b] : ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ X2 @ Y4 ) @ S2 ) )
= ( ord_le2663425029446080117rm_a_b @ R5 @ S2 ) ) ).
% pred_subset_eq2
thf(fact_146_pred__subset__eq2,axiom,
! [R5: set_Pr5162482618186357969rm_a_b,S2: set_Pr5162482618186357969rm_a_b] :
( ( ord_le7684036864810614340_a_b_o
@ ^ [X2: b,Y4: produc357393685978478089rm_a_b] : ( member8210134394708959770rm_a_b @ ( produc2673992483936284769rm_a_b @ X2 @ Y4 ) @ R5 )
@ ^ [X2: b,Y4: produc357393685978478089rm_a_b] : ( member8210134394708959770rm_a_b @ ( produc2673992483936284769rm_a_b @ X2 @ Y4 ) @ S2 ) )
= ( ord_le4369580198973711473rm_a_b @ R5 @ S2 ) ) ).
% pred_subset_eq2
thf(fact_147_pred__subset__eq2,axiom,
! [R5: set_Product_prod_b_b,S2: set_Product_prod_b_b] :
( ( ord_less_eq_b_b_o
@ ^ [X2: b,Y4: b] : ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ X2 @ Y4 ) @ R5 )
@ ^ [X2: b,Y4: b] : ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ X2 @ Y4 ) @ S2 ) )
= ( ord_le182087997850975847od_b_b @ R5 @ S2 ) ) ).
% pred_subset_eq2
thf(fact_148_pred__subset__eq2,axiom,
! [R5: set_Pr4386577575007340137rm_a_b,S2: set_Pr4386577575007340137rm_a_b] :
( ( ord_le8627286948385211390_a_b_o
@ ^ [X2: term_a_b,Y4: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y4 ) @ R5 )
@ ^ [X2: term_a_b,Y4: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y4 ) @ S2 ) )
= ( ord_le118470702582115849rm_a_b @ R5 @ S2 ) ) ).
% pred_subset_eq2
thf(fact_149_pred__subset__eq2,axiom,
! [R5: set_Pr4934435412358123699_a_nat,S2: set_Pr4934435412358123699_a_nat] :
( ( ord_less_eq_a_nat_o
@ ^ [X2: a,Y4: nat] : ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ X2 @ Y4 ) @ R5 )
@ ^ [X2: a,Y4: nat] : ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ X2 @ Y4 ) @ S2 ) )
= ( ord_le8666007276011122963_a_nat @ R5 @ S2 ) ) ).
% pred_subset_eq2
thf(fact_150_rrstepp_Ocases,axiom,
! [R2: term_a_b > term_a_b > $o,A1: term_a_b,A2: term_a_b] :
( ( rrstepp_a_b @ R2 @ A1 @ A2 )
=> ~ ! [L2: term_a_b,R3: term_a_b,Sigma: b > term_a_b] :
( ( A1
= ( subst_7999470309526761004_a_b_b @ L2 @ Sigma ) )
=> ( ( A2
= ( subst_7999470309526761004_a_b_b @ R3 @ Sigma ) )
=> ~ ( R2 @ L2 @ R3 ) ) ) ) ).
% rrstepp.cases
thf(fact_151_rrstepp_Osimps,axiom,
( rrstepp_a_b
= ( ^ [R6: term_a_b > term_a_b > $o,A12: term_a_b,A22: term_a_b] :
? [L3: term_a_b,R4: term_a_b,Sigma2: b > term_a_b] :
( ( A12
= ( subst_7999470309526761004_a_b_b @ L3 @ Sigma2 ) )
& ( A22
= ( subst_7999470309526761004_a_b_b @ R4 @ Sigma2 ) )
& ( R6 @ L3 @ R4 ) ) ) ) ).
% rrstepp.simps
thf(fact_152_rrstepp_Ointros,axiom,
! [R2: term_a_b > term_a_b > $o,L: term_a_b,R: term_a_b,Sigma3: b > term_a_b] :
( ( R2 @ L @ R )
=> ( rrstepp_a_b @ R2 @ ( subst_7999470309526761004_a_b_b @ L @ Sigma3 ) @ ( subst_7999470309526761004_a_b_b @ R @ Sigma3 ) ) ) ).
% rrstepp.intros
thf(fact_153_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_154_order__antisym__conv,axiom,
! [Y: produc357393685978478089rm_a_b > $o,X: produc357393685978478089rm_a_b > $o] :
( ( ord_le6645574462093738516_a_b_o @ Y @ X )
=> ( ( ord_le6645574462093738516_a_b_o @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_155_order__antisym__conv,axiom,
! [Y: product_prod_a_nat > $o,X: product_prod_a_nat > $o] :
( ( ord_le8950881081082477362_nat_o @ Y @ X )
=> ( ( ord_le8950881081082477362_nat_o @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_156_order__antisym__conv,axiom,
! [Y: term_a_b > term_a_b > $o,X: term_a_b > term_a_b > $o] :
( ( ord_le8627286948385211390_a_b_o @ Y @ X )
=> ( ( ord_le8627286948385211390_a_b_o @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_157_order__antisym__conv,axiom,
! [Y: b > $o,X: b > $o] :
( ( ord_less_eq_b_o @ Y @ X )
=> ( ( ord_less_eq_b_o @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_158_order__antisym__conv,axiom,
! [Y: a > nat > $o,X: a > nat > $o] :
( ( ord_less_eq_a_nat_o @ Y @ X )
=> ( ( ord_less_eq_a_nat_o @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_159_order__antisym__conv,axiom,
! [Y: set_Pr4934435412358123699_a_nat,X: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ Y @ X )
=> ( ( ord_le8666007276011122963_a_nat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_160_ord__le__eq__subst,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat,F2: set_Pr4934435412358123699_a_nat > set_Pr4934435412358123699_a_nat,C2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A @ B )
=> ( ( ( F2 @ B )
= C2 )
=> ( ! [X3: set_Pr4934435412358123699_a_nat,Y3: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X3 @ Y3 )
=> ( ord_le8666007276011122963_a_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_le8666007276011122963_a_nat @ ( F2 @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_161_ord__le__eq__subst,axiom,
! [A: b > $o,B: b > $o,F2: ( b > $o ) > b > $o,C2: b > $o] :
( ( ord_less_eq_b_o @ A @ B )
=> ( ( ( F2 @ B )
= C2 )
=> ( ! [X3: b > $o,Y3: b > $o] :
( ( ord_less_eq_b_o @ X3 @ Y3 )
=> ( ord_less_eq_b_o @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_b_o @ ( F2 @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_162_ord__le__eq__subst,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat,F2: set_Pr4934435412358123699_a_nat > b > $o,C2: b > $o] :
( ( ord_le8666007276011122963_a_nat @ A @ B )
=> ( ( ( F2 @ B )
= C2 )
=> ( ! [X3: set_Pr4934435412358123699_a_nat,Y3: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X3 @ Y3 )
=> ( ord_less_eq_b_o @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_b_o @ ( F2 @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_163_ord__le__eq__subst,axiom,
! [A: b > $o,B: b > $o,F2: ( b > $o ) > set_Pr4934435412358123699_a_nat,C2: set_Pr4934435412358123699_a_nat] :
( ( ord_less_eq_b_o @ A @ B )
=> ( ( ( F2 @ B )
= C2 )
=> ( ! [X3: b > $o,Y3: b > $o] :
( ( ord_less_eq_b_o @ X3 @ Y3 )
=> ( ord_le8666007276011122963_a_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_le8666007276011122963_a_nat @ ( F2 @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_164_ord__le__eq__subst,axiom,
! [A: product_prod_a_nat > $o,B: product_prod_a_nat > $o,F2: ( product_prod_a_nat > $o ) > b > $o,C2: b > $o] :
( ( ord_le8950881081082477362_nat_o @ A @ B )
=> ( ( ( F2 @ B )
= C2 )
=> ( ! [X3: product_prod_a_nat > $o,Y3: product_prod_a_nat > $o] :
( ( ord_le8950881081082477362_nat_o @ X3 @ Y3 )
=> ( ord_less_eq_b_o @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_b_o @ ( F2 @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_165_ord__le__eq__subst,axiom,
! [A: b > $o,B: b > $o,F2: ( b > $o ) > product_prod_a_nat > $o,C2: product_prod_a_nat > $o] :
( ( ord_less_eq_b_o @ A @ B )
=> ( ( ( F2 @ B )
= C2 )
=> ( ! [X3: b > $o,Y3: b > $o] :
( ( ord_less_eq_b_o @ X3 @ Y3 )
=> ( ord_le8950881081082477362_nat_o @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_le8950881081082477362_nat_o @ ( F2 @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_166_ord__le__eq__subst,axiom,
! [A: b > $o,B: b > $o,F2: ( b > $o ) > a > nat > $o,C2: a > nat > $o] :
( ( ord_less_eq_b_o @ A @ B )
=> ( ( ( F2 @ B )
= C2 )
=> ( ! [X3: b > $o,Y3: b > $o] :
( ( ord_less_eq_b_o @ X3 @ Y3 )
=> ( ord_less_eq_a_nat_o @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_a_nat_o @ ( F2 @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_167_ord__le__eq__subst,axiom,
! [A: a > nat > $o,B: a > nat > $o,F2: ( a > nat > $o ) > b > $o,C2: b > $o] :
( ( ord_less_eq_a_nat_o @ A @ B )
=> ( ( ( F2 @ B )
= C2 )
=> ( ! [X3: a > nat > $o,Y3: a > nat > $o] :
( ( ord_less_eq_a_nat_o @ X3 @ Y3 )
=> ( ord_less_eq_b_o @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_b_o @ ( F2 @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_168_ord__le__eq__subst,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat,F2: set_Pr4934435412358123699_a_nat > product_prod_a_nat > $o,C2: product_prod_a_nat > $o] :
( ( ord_le8666007276011122963_a_nat @ A @ B )
=> ( ( ( F2 @ B )
= C2 )
=> ( ! [X3: set_Pr4934435412358123699_a_nat,Y3: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X3 @ Y3 )
=> ( ord_le8950881081082477362_nat_o @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_le8950881081082477362_nat_o @ ( F2 @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_169_ord__le__eq__subst,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat,F2: set_Pr4934435412358123699_a_nat > a > nat > $o,C2: a > nat > $o] :
( ( ord_le8666007276011122963_a_nat @ A @ B )
=> ( ( ( F2 @ B )
= C2 )
=> ( ! [X3: set_Pr4934435412358123699_a_nat,Y3: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X3 @ Y3 )
=> ( ord_less_eq_a_nat_o @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_a_nat_o @ ( F2 @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_170_ord__eq__le__subst,axiom,
! [A: set_Pr4934435412358123699_a_nat,F2: set_Pr4934435412358123699_a_nat > set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat,C2: set_Pr4934435412358123699_a_nat] :
( ( A
= ( F2 @ B ) )
=> ( ( ord_le8666007276011122963_a_nat @ B @ C2 )
=> ( ! [X3: set_Pr4934435412358123699_a_nat,Y3: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X3 @ Y3 )
=> ( ord_le8666007276011122963_a_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_le8666007276011122963_a_nat @ A @ ( F2 @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_171_ord__eq__le__subst,axiom,
! [A: b > $o,F2: ( b > $o ) > b > $o,B: b > $o,C2: b > $o] :
( ( A
= ( F2 @ B ) )
=> ( ( ord_less_eq_b_o @ B @ C2 )
=> ( ! [X3: b > $o,Y3: b > $o] :
( ( ord_less_eq_b_o @ X3 @ Y3 )
=> ( ord_less_eq_b_o @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_b_o @ A @ ( F2 @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_172_ord__eq__le__subst,axiom,
! [A: b > $o,F2: set_Pr4934435412358123699_a_nat > b > $o,B: set_Pr4934435412358123699_a_nat,C2: set_Pr4934435412358123699_a_nat] :
( ( A
= ( F2 @ B ) )
=> ( ( ord_le8666007276011122963_a_nat @ B @ C2 )
=> ( ! [X3: set_Pr4934435412358123699_a_nat,Y3: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X3 @ Y3 )
=> ( ord_less_eq_b_o @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_b_o @ A @ ( F2 @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_173_ord__eq__le__subst,axiom,
! [A: set_Pr4934435412358123699_a_nat,F2: ( b > $o ) > set_Pr4934435412358123699_a_nat,B: b > $o,C2: b > $o] :
( ( A
= ( F2 @ B ) )
=> ( ( ord_less_eq_b_o @ B @ C2 )
=> ( ! [X3: b > $o,Y3: b > $o] :
( ( ord_less_eq_b_o @ X3 @ Y3 )
=> ( ord_le8666007276011122963_a_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_le8666007276011122963_a_nat @ A @ ( F2 @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_174_ord__eq__le__subst,axiom,
! [A: b > $o,F2: ( product_prod_a_nat > $o ) > b > $o,B: product_prod_a_nat > $o,C2: product_prod_a_nat > $o] :
( ( A
= ( F2 @ B ) )
=> ( ( ord_le8950881081082477362_nat_o @ B @ C2 )
=> ( ! [X3: product_prod_a_nat > $o,Y3: product_prod_a_nat > $o] :
( ( ord_le8950881081082477362_nat_o @ X3 @ Y3 )
=> ( ord_less_eq_b_o @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_b_o @ A @ ( F2 @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_175_ord__eq__le__subst,axiom,
! [A: product_prod_a_nat > $o,F2: ( b > $o ) > product_prod_a_nat > $o,B: b > $o,C2: b > $o] :
( ( A
= ( F2 @ B ) )
=> ( ( ord_less_eq_b_o @ B @ C2 )
=> ( ! [X3: b > $o,Y3: b > $o] :
( ( ord_less_eq_b_o @ X3 @ Y3 )
=> ( ord_le8950881081082477362_nat_o @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_le8950881081082477362_nat_o @ A @ ( F2 @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_176_ord__eq__le__subst,axiom,
! [A: a > nat > $o,F2: ( b > $o ) > a > nat > $o,B: b > $o,C2: b > $o] :
( ( A
= ( F2 @ B ) )
=> ( ( ord_less_eq_b_o @ B @ C2 )
=> ( ! [X3: b > $o,Y3: b > $o] :
( ( ord_less_eq_b_o @ X3 @ Y3 )
=> ( ord_less_eq_a_nat_o @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_a_nat_o @ A @ ( F2 @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_177_ord__eq__le__subst,axiom,
! [A: b > $o,F2: ( a > nat > $o ) > b > $o,B: a > nat > $o,C2: a > nat > $o] :
( ( A
= ( F2 @ B ) )
=> ( ( ord_less_eq_a_nat_o @ B @ C2 )
=> ( ! [X3: a > nat > $o,Y3: a > nat > $o] :
( ( ord_less_eq_a_nat_o @ X3 @ Y3 )
=> ( ord_less_eq_b_o @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_b_o @ A @ ( F2 @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_178_ord__eq__le__subst,axiom,
! [A: product_prod_a_nat > $o,F2: set_Pr4934435412358123699_a_nat > product_prod_a_nat > $o,B: set_Pr4934435412358123699_a_nat,C2: set_Pr4934435412358123699_a_nat] :
( ( A
= ( F2 @ B ) )
=> ( ( ord_le8666007276011122963_a_nat @ B @ C2 )
=> ( ! [X3: set_Pr4934435412358123699_a_nat,Y3: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X3 @ Y3 )
=> ( ord_le8950881081082477362_nat_o @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_le8950881081082477362_nat_o @ A @ ( F2 @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_179_ord__eq__le__subst,axiom,
! [A: a > nat > $o,F2: set_Pr4934435412358123699_a_nat > a > nat > $o,B: set_Pr4934435412358123699_a_nat,C2: set_Pr4934435412358123699_a_nat] :
( ( A
= ( F2 @ B ) )
=> ( ( ord_le8666007276011122963_a_nat @ B @ C2 )
=> ( ! [X3: set_Pr4934435412358123699_a_nat,Y3: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X3 @ Y3 )
=> ( ord_less_eq_a_nat_o @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_a_nat_o @ A @ ( F2 @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_180_order__eq__refl,axiom,
! [X: produc357393685978478089rm_a_b > $o,Y: produc357393685978478089rm_a_b > $o] :
( ( X = Y )
=> ( ord_le6645574462093738516_a_b_o @ X @ Y ) ) ).
% order_eq_refl
thf(fact_181_order__eq__refl,axiom,
! [X: product_prod_a_nat > $o,Y: product_prod_a_nat > $o] :
( ( X = Y )
=> ( ord_le8950881081082477362_nat_o @ X @ Y ) ) ).
% order_eq_refl
thf(fact_182_order__eq__refl,axiom,
! [X: term_a_b > term_a_b > $o,Y: term_a_b > term_a_b > $o] :
( ( X = Y )
=> ( ord_le8627286948385211390_a_b_o @ X @ Y ) ) ).
% order_eq_refl
thf(fact_183_order__eq__refl,axiom,
! [X: b > $o,Y: b > $o] :
( ( X = Y )
=> ( ord_less_eq_b_o @ X @ Y ) ) ).
% order_eq_refl
thf(fact_184_order__eq__refl,axiom,
! [X: a > nat > $o,Y: a > nat > $o] :
( ( X = Y )
=> ( ord_less_eq_a_nat_o @ X @ Y ) ) ).
% order_eq_refl
thf(fact_185_order__eq__refl,axiom,
! [X: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat] :
( ( X = Y )
=> ( ord_le8666007276011122963_a_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_186_order__subst2,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat,F2: set_Pr4934435412358123699_a_nat > set_Pr4934435412358123699_a_nat,C2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A @ B )
=> ( ( ord_le8666007276011122963_a_nat @ ( F2 @ B ) @ C2 )
=> ( ! [X3: set_Pr4934435412358123699_a_nat,Y3: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X3 @ Y3 )
=> ( ord_le8666007276011122963_a_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_le8666007276011122963_a_nat @ ( F2 @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_187_order__subst2,axiom,
! [A: b > $o,B: b > $o,F2: ( b > $o ) > b > $o,C2: b > $o] :
( ( ord_less_eq_b_o @ A @ B )
=> ( ( ord_less_eq_b_o @ ( F2 @ B ) @ C2 )
=> ( ! [X3: b > $o,Y3: b > $o] :
( ( ord_less_eq_b_o @ X3 @ Y3 )
=> ( ord_less_eq_b_o @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_b_o @ ( F2 @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_188_order__subst2,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat,F2: set_Pr4934435412358123699_a_nat > b > $o,C2: b > $o] :
( ( ord_le8666007276011122963_a_nat @ A @ B )
=> ( ( ord_less_eq_b_o @ ( F2 @ B ) @ C2 )
=> ( ! [X3: set_Pr4934435412358123699_a_nat,Y3: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X3 @ Y3 )
=> ( ord_less_eq_b_o @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_b_o @ ( F2 @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_189_order__subst2,axiom,
! [A: b > $o,B: b > $o,F2: ( b > $o ) > set_Pr4934435412358123699_a_nat,C2: set_Pr4934435412358123699_a_nat] :
( ( ord_less_eq_b_o @ A @ B )
=> ( ( ord_le8666007276011122963_a_nat @ ( F2 @ B ) @ C2 )
=> ( ! [X3: b > $o,Y3: b > $o] :
( ( ord_less_eq_b_o @ X3 @ Y3 )
=> ( ord_le8666007276011122963_a_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_le8666007276011122963_a_nat @ ( F2 @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_190_order__subst2,axiom,
! [A: product_prod_a_nat > $o,B: product_prod_a_nat > $o,F2: ( product_prod_a_nat > $o ) > b > $o,C2: b > $o] :
( ( ord_le8950881081082477362_nat_o @ A @ B )
=> ( ( ord_less_eq_b_o @ ( F2 @ B ) @ C2 )
=> ( ! [X3: product_prod_a_nat > $o,Y3: product_prod_a_nat > $o] :
( ( ord_le8950881081082477362_nat_o @ X3 @ Y3 )
=> ( ord_less_eq_b_o @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_b_o @ ( F2 @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_191_order__subst2,axiom,
! [A: b > $o,B: b > $o,F2: ( b > $o ) > product_prod_a_nat > $o,C2: product_prod_a_nat > $o] :
( ( ord_less_eq_b_o @ A @ B )
=> ( ( ord_le8950881081082477362_nat_o @ ( F2 @ B ) @ C2 )
=> ( ! [X3: b > $o,Y3: b > $o] :
( ( ord_less_eq_b_o @ X3 @ Y3 )
=> ( ord_le8950881081082477362_nat_o @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_le8950881081082477362_nat_o @ ( F2 @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_192_order__subst2,axiom,
! [A: b > $o,B: b > $o,F2: ( b > $o ) > a > nat > $o,C2: a > nat > $o] :
( ( ord_less_eq_b_o @ A @ B )
=> ( ( ord_less_eq_a_nat_o @ ( F2 @ B ) @ C2 )
=> ( ! [X3: b > $o,Y3: b > $o] :
( ( ord_less_eq_b_o @ X3 @ Y3 )
=> ( ord_less_eq_a_nat_o @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_a_nat_o @ ( F2 @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_193_order__subst2,axiom,
! [A: a > nat > $o,B: a > nat > $o,F2: ( a > nat > $o ) > b > $o,C2: b > $o] :
( ( ord_less_eq_a_nat_o @ A @ B )
=> ( ( ord_less_eq_b_o @ ( F2 @ B ) @ C2 )
=> ( ! [X3: a > nat > $o,Y3: a > nat > $o] :
( ( ord_less_eq_a_nat_o @ X3 @ Y3 )
=> ( ord_less_eq_b_o @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_b_o @ ( F2 @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_194_order__subst2,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat,F2: set_Pr4934435412358123699_a_nat > product_prod_a_nat > $o,C2: product_prod_a_nat > $o] :
( ( ord_le8666007276011122963_a_nat @ A @ B )
=> ( ( ord_le8950881081082477362_nat_o @ ( F2 @ B ) @ C2 )
=> ( ! [X3: set_Pr4934435412358123699_a_nat,Y3: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X3 @ Y3 )
=> ( ord_le8950881081082477362_nat_o @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_le8950881081082477362_nat_o @ ( F2 @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_195_order__subst2,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat,F2: set_Pr4934435412358123699_a_nat > a > nat > $o,C2: a > nat > $o] :
( ( ord_le8666007276011122963_a_nat @ A @ B )
=> ( ( ord_less_eq_a_nat_o @ ( F2 @ B ) @ C2 )
=> ( ! [X3: set_Pr4934435412358123699_a_nat,Y3: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X3 @ Y3 )
=> ( ord_less_eq_a_nat_o @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_a_nat_o @ ( F2 @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_196_order__subst1,axiom,
! [A: set_Pr4934435412358123699_a_nat,F2: set_Pr4934435412358123699_a_nat > set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat,C2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A @ ( F2 @ B ) )
=> ( ( ord_le8666007276011122963_a_nat @ B @ C2 )
=> ( ! [X3: set_Pr4934435412358123699_a_nat,Y3: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X3 @ Y3 )
=> ( ord_le8666007276011122963_a_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_le8666007276011122963_a_nat @ A @ ( F2 @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_197_order__subst1,axiom,
! [A: b > $o,F2: ( b > $o ) > b > $o,B: b > $o,C2: b > $o] :
( ( ord_less_eq_b_o @ A @ ( F2 @ B ) )
=> ( ( ord_less_eq_b_o @ B @ C2 )
=> ( ! [X3: b > $o,Y3: b > $o] :
( ( ord_less_eq_b_o @ X3 @ Y3 )
=> ( ord_less_eq_b_o @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_b_o @ A @ ( F2 @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_198_order__subst1,axiom,
! [A: set_Pr4934435412358123699_a_nat,F2: ( b > $o ) > set_Pr4934435412358123699_a_nat,B: b > $o,C2: b > $o] :
( ( ord_le8666007276011122963_a_nat @ A @ ( F2 @ B ) )
=> ( ( ord_less_eq_b_o @ B @ C2 )
=> ( ! [X3: b > $o,Y3: b > $o] :
( ( ord_less_eq_b_o @ X3 @ Y3 )
=> ( ord_le8666007276011122963_a_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_le8666007276011122963_a_nat @ A @ ( F2 @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_199_order__subst1,axiom,
! [A: b > $o,F2: set_Pr4934435412358123699_a_nat > b > $o,B: set_Pr4934435412358123699_a_nat,C2: set_Pr4934435412358123699_a_nat] :
( ( ord_less_eq_b_o @ A @ ( F2 @ B ) )
=> ( ( ord_le8666007276011122963_a_nat @ B @ C2 )
=> ( ! [X3: set_Pr4934435412358123699_a_nat,Y3: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X3 @ Y3 )
=> ( ord_less_eq_b_o @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_b_o @ A @ ( F2 @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_200_order__subst1,axiom,
! [A: product_prod_a_nat > $o,F2: ( b > $o ) > product_prod_a_nat > $o,B: b > $o,C2: b > $o] :
( ( ord_le8950881081082477362_nat_o @ A @ ( F2 @ B ) )
=> ( ( ord_less_eq_b_o @ B @ C2 )
=> ( ! [X3: b > $o,Y3: b > $o] :
( ( ord_less_eq_b_o @ X3 @ Y3 )
=> ( ord_le8950881081082477362_nat_o @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_le8950881081082477362_nat_o @ A @ ( F2 @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_201_order__subst1,axiom,
! [A: b > $o,F2: ( product_prod_a_nat > $o ) > b > $o,B: product_prod_a_nat > $o,C2: product_prod_a_nat > $o] :
( ( ord_less_eq_b_o @ A @ ( F2 @ B ) )
=> ( ( ord_le8950881081082477362_nat_o @ B @ C2 )
=> ( ! [X3: product_prod_a_nat > $o,Y3: product_prod_a_nat > $o] :
( ( ord_le8950881081082477362_nat_o @ X3 @ Y3 )
=> ( ord_less_eq_b_o @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_b_o @ A @ ( F2 @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_202_order__subst1,axiom,
! [A: b > $o,F2: ( a > nat > $o ) > b > $o,B: a > nat > $o,C2: a > nat > $o] :
( ( ord_less_eq_b_o @ A @ ( F2 @ B ) )
=> ( ( ord_less_eq_a_nat_o @ B @ C2 )
=> ( ! [X3: a > nat > $o,Y3: a > nat > $o] :
( ( ord_less_eq_a_nat_o @ X3 @ Y3 )
=> ( ord_less_eq_b_o @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_b_o @ A @ ( F2 @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_203_order__subst1,axiom,
! [A: a > nat > $o,F2: ( b > $o ) > a > nat > $o,B: b > $o,C2: b > $o] :
( ( ord_less_eq_a_nat_o @ A @ ( F2 @ B ) )
=> ( ( ord_less_eq_b_o @ B @ C2 )
=> ( ! [X3: b > $o,Y3: b > $o] :
( ( ord_less_eq_b_o @ X3 @ Y3 )
=> ( ord_less_eq_a_nat_o @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_a_nat_o @ A @ ( F2 @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_204_order__subst1,axiom,
! [A: set_Pr4934435412358123699_a_nat,F2: ( product_prod_a_nat > $o ) > set_Pr4934435412358123699_a_nat,B: product_prod_a_nat > $o,C2: product_prod_a_nat > $o] :
( ( ord_le8666007276011122963_a_nat @ A @ ( F2 @ B ) )
=> ( ( ord_le8950881081082477362_nat_o @ B @ C2 )
=> ( ! [X3: product_prod_a_nat > $o,Y3: product_prod_a_nat > $o] :
( ( ord_le8950881081082477362_nat_o @ X3 @ Y3 )
=> ( ord_le8666007276011122963_a_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_le8666007276011122963_a_nat @ A @ ( F2 @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_205_order__subst1,axiom,
! [A: set_Pr4934435412358123699_a_nat,F2: ( a > nat > $o ) > set_Pr4934435412358123699_a_nat,B: a > nat > $o,C2: a > nat > $o] :
( ( ord_le8666007276011122963_a_nat @ A @ ( F2 @ B ) )
=> ( ( ord_less_eq_a_nat_o @ B @ C2 )
=> ( ! [X3: a > nat > $o,Y3: a > nat > $o] :
( ( ord_less_eq_a_nat_o @ X3 @ Y3 )
=> ( ord_le8666007276011122963_a_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_le8666007276011122963_a_nat @ A @ ( F2 @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_206_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: produc357393685978478089rm_a_b > $o,Z: produc357393685978478089rm_a_b > $o] : ( Y5 = Z ) )
= ( ^ [A7: produc357393685978478089rm_a_b > $o,B6: produc357393685978478089rm_a_b > $o] :
( ( ord_le6645574462093738516_a_b_o @ A7 @ B6 )
& ( ord_le6645574462093738516_a_b_o @ B6 @ A7 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_207_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: product_prod_a_nat > $o,Z: product_prod_a_nat > $o] : ( Y5 = Z ) )
= ( ^ [A7: product_prod_a_nat > $o,B6: product_prod_a_nat > $o] :
( ( ord_le8950881081082477362_nat_o @ A7 @ B6 )
& ( ord_le8950881081082477362_nat_o @ B6 @ A7 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_208_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: term_a_b > term_a_b > $o,Z: term_a_b > term_a_b > $o] : ( Y5 = Z ) )
= ( ^ [A7: term_a_b > term_a_b > $o,B6: term_a_b > term_a_b > $o] :
( ( ord_le8627286948385211390_a_b_o @ A7 @ B6 )
& ( ord_le8627286948385211390_a_b_o @ B6 @ A7 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_209_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: b > $o,Z: b > $o] : ( Y5 = Z ) )
= ( ^ [A7: b > $o,B6: b > $o] :
( ( ord_less_eq_b_o @ A7 @ B6 )
& ( ord_less_eq_b_o @ B6 @ A7 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_210_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: a > nat > $o,Z: a > nat > $o] : ( Y5 = Z ) )
= ( ^ [A7: a > nat > $o,B6: a > nat > $o] :
( ( ord_less_eq_a_nat_o @ A7 @ B6 )
& ( ord_less_eq_a_nat_o @ B6 @ A7 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_211_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_Pr4934435412358123699_a_nat,Z: set_Pr4934435412358123699_a_nat] : ( Y5 = Z ) )
= ( ^ [A7: set_Pr4934435412358123699_a_nat,B6: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A7 @ B6 )
& ( ord_le8666007276011122963_a_nat @ B6 @ A7 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_212_le__fun__def,axiom,
( ord_le6645574462093738516_a_b_o
= ( ^ [F3: produc357393685978478089rm_a_b > $o,G2: produc357393685978478089rm_a_b > $o] :
! [X2: produc357393685978478089rm_a_b] : ( ord_less_eq_o @ ( F3 @ X2 ) @ ( G2 @ X2 ) ) ) ) ).
% le_fun_def
thf(fact_213_le__fun__def,axiom,
( ord_le8950881081082477362_nat_o
= ( ^ [F3: product_prod_a_nat > $o,G2: product_prod_a_nat > $o] :
! [X2: product_prod_a_nat] : ( ord_less_eq_o @ ( F3 @ X2 ) @ ( G2 @ X2 ) ) ) ) ).
% le_fun_def
thf(fact_214_le__fun__def,axiom,
( ord_le8627286948385211390_a_b_o
= ( ^ [F3: term_a_b > term_a_b > $o,G2: term_a_b > term_a_b > $o] :
! [X2: term_a_b] : ( ord_le3490609322949561435_a_b_o @ ( F3 @ X2 ) @ ( G2 @ X2 ) ) ) ) ).
% le_fun_def
thf(fact_215_le__fun__def,axiom,
( ord_less_eq_b_o
= ( ^ [F3: b > $o,G2: b > $o] :
! [X2: b] : ( ord_less_eq_o @ ( F3 @ X2 ) @ ( G2 @ X2 ) ) ) ) ).
% le_fun_def
thf(fact_216_le__fun__def,axiom,
( ord_less_eq_a_nat_o
= ( ^ [F3: a > nat > $o,G2: a > nat > $o] :
! [X2: a] : ( ord_less_eq_nat_o @ ( F3 @ X2 ) @ ( G2 @ X2 ) ) ) ) ).
% le_fun_def
thf(fact_217_le__funI,axiom,
! [F2: produc357393685978478089rm_a_b > $o,G3: produc357393685978478089rm_a_b > $o] :
( ! [X3: produc357393685978478089rm_a_b] : ( ord_less_eq_o @ ( F2 @ X3 ) @ ( G3 @ X3 ) )
=> ( ord_le6645574462093738516_a_b_o @ F2 @ G3 ) ) ).
% le_funI
thf(fact_218_le__funI,axiom,
! [F2: product_prod_a_nat > $o,G3: product_prod_a_nat > $o] :
( ! [X3: product_prod_a_nat] : ( ord_less_eq_o @ ( F2 @ X3 ) @ ( G3 @ X3 ) )
=> ( ord_le8950881081082477362_nat_o @ F2 @ G3 ) ) ).
% le_funI
thf(fact_219_le__funI,axiom,
! [F2: term_a_b > term_a_b > $o,G3: term_a_b > term_a_b > $o] :
( ! [X3: term_a_b] : ( ord_le3490609322949561435_a_b_o @ ( F2 @ X3 ) @ ( G3 @ X3 ) )
=> ( ord_le8627286948385211390_a_b_o @ F2 @ G3 ) ) ).
% le_funI
thf(fact_220_le__funI,axiom,
! [F2: b > $o,G3: b > $o] :
( ! [X3: b] : ( ord_less_eq_o @ ( F2 @ X3 ) @ ( G3 @ X3 ) )
=> ( ord_less_eq_b_o @ F2 @ G3 ) ) ).
% le_funI
thf(fact_221_le__funI,axiom,
! [F2: a > nat > $o,G3: a > nat > $o] :
( ! [X3: a] : ( ord_less_eq_nat_o @ ( F2 @ X3 ) @ ( G3 @ X3 ) )
=> ( ord_less_eq_a_nat_o @ F2 @ G3 ) ) ).
% le_funI
thf(fact_222_le__funE,axiom,
! [F2: produc357393685978478089rm_a_b > $o,G3: produc357393685978478089rm_a_b > $o,X: produc357393685978478089rm_a_b] :
( ( ord_le6645574462093738516_a_b_o @ F2 @ G3 )
=> ( ord_less_eq_o @ ( F2 @ X ) @ ( G3 @ X ) ) ) ).
% le_funE
thf(fact_223_le__funE,axiom,
! [F2: product_prod_a_nat > $o,G3: product_prod_a_nat > $o,X: product_prod_a_nat] :
( ( ord_le8950881081082477362_nat_o @ F2 @ G3 )
=> ( ord_less_eq_o @ ( F2 @ X ) @ ( G3 @ X ) ) ) ).
% le_funE
thf(fact_224_le__funE,axiom,
! [F2: term_a_b > term_a_b > $o,G3: term_a_b > term_a_b > $o,X: term_a_b] :
( ( ord_le8627286948385211390_a_b_o @ F2 @ G3 )
=> ( ord_le3490609322949561435_a_b_o @ ( F2 @ X ) @ ( G3 @ X ) ) ) ).
% le_funE
thf(fact_225_le__funE,axiom,
! [F2: b > $o,G3: b > $o,X: b] :
( ( ord_less_eq_b_o @ F2 @ G3 )
=> ( ord_less_eq_o @ ( F2 @ X ) @ ( G3 @ X ) ) ) ).
% le_funE
thf(fact_226_le__funE,axiom,
! [F2: a > nat > $o,G3: a > nat > $o,X: a] :
( ( ord_less_eq_a_nat_o @ F2 @ G3 )
=> ( ord_less_eq_nat_o @ ( F2 @ X ) @ ( G3 @ X ) ) ) ).
% le_funE
thf(fact_227_le__funD,axiom,
! [F2: produc357393685978478089rm_a_b > $o,G3: produc357393685978478089rm_a_b > $o,X: produc357393685978478089rm_a_b] :
( ( ord_le6645574462093738516_a_b_o @ F2 @ G3 )
=> ( ord_less_eq_o @ ( F2 @ X ) @ ( G3 @ X ) ) ) ).
% le_funD
thf(fact_228_le__funD,axiom,
! [F2: product_prod_a_nat > $o,G3: product_prod_a_nat > $o,X: product_prod_a_nat] :
( ( ord_le8950881081082477362_nat_o @ F2 @ G3 )
=> ( ord_less_eq_o @ ( F2 @ X ) @ ( G3 @ X ) ) ) ).
% le_funD
thf(fact_229_le__funD,axiom,
! [F2: term_a_b > term_a_b > $o,G3: term_a_b > term_a_b > $o,X: term_a_b] :
( ( ord_le8627286948385211390_a_b_o @ F2 @ G3 )
=> ( ord_le3490609322949561435_a_b_o @ ( F2 @ X ) @ ( G3 @ X ) ) ) ).
% le_funD
thf(fact_230_le__funD,axiom,
! [F2: b > $o,G3: b > $o,X: b] :
( ( ord_less_eq_b_o @ F2 @ G3 )
=> ( ord_less_eq_o @ ( F2 @ X ) @ ( G3 @ X ) ) ) ).
% le_funD
thf(fact_231_le__funD,axiom,
! [F2: a > nat > $o,G3: a > nat > $o,X: a] :
( ( ord_less_eq_a_nat_o @ F2 @ G3 )
=> ( ord_less_eq_nat_o @ ( F2 @ X ) @ ( G3 @ X ) ) ) ).
% le_funD
thf(fact_232_antisym,axiom,
! [A: produc357393685978478089rm_a_b > $o,B: produc357393685978478089rm_a_b > $o] :
( ( ord_le6645574462093738516_a_b_o @ A @ B )
=> ( ( ord_le6645574462093738516_a_b_o @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_233_antisym,axiom,
! [A: product_prod_a_nat > $o,B: product_prod_a_nat > $o] :
( ( ord_le8950881081082477362_nat_o @ A @ B )
=> ( ( ord_le8950881081082477362_nat_o @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_234_antisym,axiom,
! [A: term_a_b > term_a_b > $o,B: term_a_b > term_a_b > $o] :
( ( ord_le8627286948385211390_a_b_o @ A @ B )
=> ( ( ord_le8627286948385211390_a_b_o @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_235_antisym,axiom,
! [A: b > $o,B: b > $o] :
( ( ord_less_eq_b_o @ A @ B )
=> ( ( ord_less_eq_b_o @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_236_antisym,axiom,
! [A: a > nat > $o,B: a > nat > $o] :
( ( ord_less_eq_a_nat_o @ A @ B )
=> ( ( ord_less_eq_a_nat_o @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_237_antisym,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A @ B )
=> ( ( ord_le8666007276011122963_a_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_238_dual__order_Otrans,axiom,
! [B: produc357393685978478089rm_a_b > $o,A: produc357393685978478089rm_a_b > $o,C2: produc357393685978478089rm_a_b > $o] :
( ( ord_le6645574462093738516_a_b_o @ B @ A )
=> ( ( ord_le6645574462093738516_a_b_o @ C2 @ B )
=> ( ord_le6645574462093738516_a_b_o @ C2 @ A ) ) ) ).
% dual_order.trans
thf(fact_239_dual__order_Otrans,axiom,
! [B: product_prod_a_nat > $o,A: product_prod_a_nat > $o,C2: product_prod_a_nat > $o] :
( ( ord_le8950881081082477362_nat_o @ B @ A )
=> ( ( ord_le8950881081082477362_nat_o @ C2 @ B )
=> ( ord_le8950881081082477362_nat_o @ C2 @ A ) ) ) ).
% dual_order.trans
thf(fact_240_dual__order_Otrans,axiom,
! [B: term_a_b > term_a_b > $o,A: term_a_b > term_a_b > $o,C2: term_a_b > term_a_b > $o] :
( ( ord_le8627286948385211390_a_b_o @ B @ A )
=> ( ( ord_le8627286948385211390_a_b_o @ C2 @ B )
=> ( ord_le8627286948385211390_a_b_o @ C2 @ A ) ) ) ).
% dual_order.trans
thf(fact_241_dual__order_Otrans,axiom,
! [B: b > $o,A: b > $o,C2: b > $o] :
( ( ord_less_eq_b_o @ B @ A )
=> ( ( ord_less_eq_b_o @ C2 @ B )
=> ( ord_less_eq_b_o @ C2 @ A ) ) ) ).
% dual_order.trans
thf(fact_242_dual__order_Otrans,axiom,
! [B: a > nat > $o,A: a > nat > $o,C2: a > nat > $o] :
( ( ord_less_eq_a_nat_o @ B @ A )
=> ( ( ord_less_eq_a_nat_o @ C2 @ B )
=> ( ord_less_eq_a_nat_o @ C2 @ A ) ) ) ).
% dual_order.trans
thf(fact_243_dual__order_Otrans,axiom,
! [B: set_Pr4934435412358123699_a_nat,A: set_Pr4934435412358123699_a_nat,C2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ B @ A )
=> ( ( ord_le8666007276011122963_a_nat @ C2 @ B )
=> ( ord_le8666007276011122963_a_nat @ C2 @ A ) ) ) ).
% dual_order.trans
thf(fact_244_dual__order_Oantisym,axiom,
! [B: produc357393685978478089rm_a_b > $o,A: produc357393685978478089rm_a_b > $o] :
( ( ord_le6645574462093738516_a_b_o @ B @ A )
=> ( ( ord_le6645574462093738516_a_b_o @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_245_dual__order_Oantisym,axiom,
! [B: product_prod_a_nat > $o,A: product_prod_a_nat > $o] :
( ( ord_le8950881081082477362_nat_o @ B @ A )
=> ( ( ord_le8950881081082477362_nat_o @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_246_dual__order_Oantisym,axiom,
! [B: term_a_b > term_a_b > $o,A: term_a_b > term_a_b > $o] :
( ( ord_le8627286948385211390_a_b_o @ B @ A )
=> ( ( ord_le8627286948385211390_a_b_o @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_247_dual__order_Oantisym,axiom,
! [B: b > $o,A: b > $o] :
( ( ord_less_eq_b_o @ B @ A )
=> ( ( ord_less_eq_b_o @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_248_dual__order_Oantisym,axiom,
! [B: a > nat > $o,A: a > nat > $o] :
( ( ord_less_eq_a_nat_o @ B @ A )
=> ( ( ord_less_eq_a_nat_o @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_249_dual__order_Oantisym,axiom,
! [B: set_Pr4934435412358123699_a_nat,A: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ B @ A )
=> ( ( ord_le8666007276011122963_a_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_250_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: produc357393685978478089rm_a_b > $o,Z: produc357393685978478089rm_a_b > $o] : ( Y5 = Z ) )
= ( ^ [A7: produc357393685978478089rm_a_b > $o,B6: produc357393685978478089rm_a_b > $o] :
( ( ord_le6645574462093738516_a_b_o @ B6 @ A7 )
& ( ord_le6645574462093738516_a_b_o @ A7 @ B6 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_251_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: product_prod_a_nat > $o,Z: product_prod_a_nat > $o] : ( Y5 = Z ) )
= ( ^ [A7: product_prod_a_nat > $o,B6: product_prod_a_nat > $o] :
( ( ord_le8950881081082477362_nat_o @ B6 @ A7 )
& ( ord_le8950881081082477362_nat_o @ A7 @ B6 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_252_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: term_a_b > term_a_b > $o,Z: term_a_b > term_a_b > $o] : ( Y5 = Z ) )
= ( ^ [A7: term_a_b > term_a_b > $o,B6: term_a_b > term_a_b > $o] :
( ( ord_le8627286948385211390_a_b_o @ B6 @ A7 )
& ( ord_le8627286948385211390_a_b_o @ A7 @ B6 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_253_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: b > $o,Z: b > $o] : ( Y5 = Z ) )
= ( ^ [A7: b > $o,B6: b > $o] :
( ( ord_less_eq_b_o @ B6 @ A7 )
& ( ord_less_eq_b_o @ A7 @ B6 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_254_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: a > nat > $o,Z: a > nat > $o] : ( Y5 = Z ) )
= ( ^ [A7: a > nat > $o,B6: a > nat > $o] :
( ( ord_less_eq_a_nat_o @ B6 @ A7 )
& ( ord_less_eq_a_nat_o @ A7 @ B6 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_255_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: set_Pr4934435412358123699_a_nat,Z: set_Pr4934435412358123699_a_nat] : ( Y5 = Z ) )
= ( ^ [A7: set_Pr4934435412358123699_a_nat,B6: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ B6 @ A7 )
& ( ord_le8666007276011122963_a_nat @ A7 @ B6 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_256_order__trans,axiom,
! [X: produc357393685978478089rm_a_b > $o,Y: produc357393685978478089rm_a_b > $o,Z2: produc357393685978478089rm_a_b > $o] :
( ( ord_le6645574462093738516_a_b_o @ X @ Y )
=> ( ( ord_le6645574462093738516_a_b_o @ Y @ Z2 )
=> ( ord_le6645574462093738516_a_b_o @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_257_order__trans,axiom,
! [X: product_prod_a_nat > $o,Y: product_prod_a_nat > $o,Z2: product_prod_a_nat > $o] :
( ( ord_le8950881081082477362_nat_o @ X @ Y )
=> ( ( ord_le8950881081082477362_nat_o @ Y @ Z2 )
=> ( ord_le8950881081082477362_nat_o @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_258_order__trans,axiom,
! [X: term_a_b > term_a_b > $o,Y: term_a_b > term_a_b > $o,Z2: term_a_b > term_a_b > $o] :
( ( ord_le8627286948385211390_a_b_o @ X @ Y )
=> ( ( ord_le8627286948385211390_a_b_o @ Y @ Z2 )
=> ( ord_le8627286948385211390_a_b_o @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_259_order__trans,axiom,
! [X: b > $o,Y: b > $o,Z2: b > $o] :
( ( ord_less_eq_b_o @ X @ Y )
=> ( ( ord_less_eq_b_o @ Y @ Z2 )
=> ( ord_less_eq_b_o @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_260_order__trans,axiom,
! [X: a > nat > $o,Y: a > nat > $o,Z2: a > nat > $o] :
( ( ord_less_eq_a_nat_o @ X @ Y )
=> ( ( ord_less_eq_a_nat_o @ Y @ Z2 )
=> ( ord_less_eq_a_nat_o @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_261_order__trans,axiom,
! [X: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat,Z2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X @ Y )
=> ( ( ord_le8666007276011122963_a_nat @ Y @ Z2 )
=> ( ord_le8666007276011122963_a_nat @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_262_order_Otrans,axiom,
! [A: produc357393685978478089rm_a_b > $o,B: produc357393685978478089rm_a_b > $o,C2: produc357393685978478089rm_a_b > $o] :
( ( ord_le6645574462093738516_a_b_o @ A @ B )
=> ( ( ord_le6645574462093738516_a_b_o @ B @ C2 )
=> ( ord_le6645574462093738516_a_b_o @ A @ C2 ) ) ) ).
% order.trans
thf(fact_263_order_Otrans,axiom,
! [A: product_prod_a_nat > $o,B: product_prod_a_nat > $o,C2: product_prod_a_nat > $o] :
( ( ord_le8950881081082477362_nat_o @ A @ B )
=> ( ( ord_le8950881081082477362_nat_o @ B @ C2 )
=> ( ord_le8950881081082477362_nat_o @ A @ C2 ) ) ) ).
% order.trans
thf(fact_264_order_Otrans,axiom,
! [A: term_a_b > term_a_b > $o,B: term_a_b > term_a_b > $o,C2: term_a_b > term_a_b > $o] :
( ( ord_le8627286948385211390_a_b_o @ A @ B )
=> ( ( ord_le8627286948385211390_a_b_o @ B @ C2 )
=> ( ord_le8627286948385211390_a_b_o @ A @ C2 ) ) ) ).
% order.trans
thf(fact_265_order_Otrans,axiom,
! [A: b > $o,B: b > $o,C2: b > $o] :
( ( ord_less_eq_b_o @ A @ B )
=> ( ( ord_less_eq_b_o @ B @ C2 )
=> ( ord_less_eq_b_o @ A @ C2 ) ) ) ).
% order.trans
thf(fact_266_order_Otrans,axiom,
! [A: a > nat > $o,B: a > nat > $o,C2: a > nat > $o] :
( ( ord_less_eq_a_nat_o @ A @ B )
=> ( ( ord_less_eq_a_nat_o @ B @ C2 )
=> ( ord_less_eq_a_nat_o @ A @ C2 ) ) ) ).
% order.trans
thf(fact_267_order_Otrans,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat,C2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A @ B )
=> ( ( ord_le8666007276011122963_a_nat @ B @ C2 )
=> ( ord_le8666007276011122963_a_nat @ A @ C2 ) ) ) ).
% order.trans
thf(fact_268_order__antisym,axiom,
! [X: produc357393685978478089rm_a_b > $o,Y: produc357393685978478089rm_a_b > $o] :
( ( ord_le6645574462093738516_a_b_o @ X @ Y )
=> ( ( ord_le6645574462093738516_a_b_o @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_269_order__antisym,axiom,
! [X: product_prod_a_nat > $o,Y: product_prod_a_nat > $o] :
( ( ord_le8950881081082477362_nat_o @ X @ Y )
=> ( ( ord_le8950881081082477362_nat_o @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_270_order__antisym,axiom,
! [X: term_a_b > term_a_b > $o,Y: term_a_b > term_a_b > $o] :
( ( ord_le8627286948385211390_a_b_o @ X @ Y )
=> ( ( ord_le8627286948385211390_a_b_o @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_271_order__antisym,axiom,
! [X: b > $o,Y: b > $o] :
( ( ord_less_eq_b_o @ X @ Y )
=> ( ( ord_less_eq_b_o @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_272_order__antisym,axiom,
! [X: a > nat > $o,Y: a > nat > $o] :
( ( ord_less_eq_a_nat_o @ X @ Y )
=> ( ( ord_less_eq_a_nat_o @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_273_order__antisym,axiom,
! [X: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X @ Y )
=> ( ( ord_le8666007276011122963_a_nat @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_274_ord__le__eq__trans,axiom,
! [A: produc357393685978478089rm_a_b > $o,B: produc357393685978478089rm_a_b > $o,C2: produc357393685978478089rm_a_b > $o] :
( ( ord_le6645574462093738516_a_b_o @ A @ B )
=> ( ( B = C2 )
=> ( ord_le6645574462093738516_a_b_o @ A @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_275_ord__le__eq__trans,axiom,
! [A: product_prod_a_nat > $o,B: product_prod_a_nat > $o,C2: product_prod_a_nat > $o] :
( ( ord_le8950881081082477362_nat_o @ A @ B )
=> ( ( B = C2 )
=> ( ord_le8950881081082477362_nat_o @ A @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_276_ord__le__eq__trans,axiom,
! [A: term_a_b > term_a_b > $o,B: term_a_b > term_a_b > $o,C2: term_a_b > term_a_b > $o] :
( ( ord_le8627286948385211390_a_b_o @ A @ B )
=> ( ( B = C2 )
=> ( ord_le8627286948385211390_a_b_o @ A @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_277_ord__le__eq__trans,axiom,
! [A: b > $o,B: b > $o,C2: b > $o] :
( ( ord_less_eq_b_o @ A @ B )
=> ( ( B = C2 )
=> ( ord_less_eq_b_o @ A @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_278_ord__le__eq__trans,axiom,
! [A: a > nat > $o,B: a > nat > $o,C2: a > nat > $o] :
( ( ord_less_eq_a_nat_o @ A @ B )
=> ( ( B = C2 )
=> ( ord_less_eq_a_nat_o @ A @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_279_ord__le__eq__trans,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat,C2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A @ B )
=> ( ( B = C2 )
=> ( ord_le8666007276011122963_a_nat @ A @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_280_ord__eq__le__trans,axiom,
! [A: produc357393685978478089rm_a_b > $o,B: produc357393685978478089rm_a_b > $o,C2: produc357393685978478089rm_a_b > $o] :
( ( A = B )
=> ( ( ord_le6645574462093738516_a_b_o @ B @ C2 )
=> ( ord_le6645574462093738516_a_b_o @ A @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_281_ord__eq__le__trans,axiom,
! [A: product_prod_a_nat > $o,B: product_prod_a_nat > $o,C2: product_prod_a_nat > $o] :
( ( A = B )
=> ( ( ord_le8950881081082477362_nat_o @ B @ C2 )
=> ( ord_le8950881081082477362_nat_o @ A @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_282_ord__eq__le__trans,axiom,
! [A: term_a_b > term_a_b > $o,B: term_a_b > term_a_b > $o,C2: term_a_b > term_a_b > $o] :
( ( A = B )
=> ( ( ord_le8627286948385211390_a_b_o @ B @ C2 )
=> ( ord_le8627286948385211390_a_b_o @ A @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_283_ord__eq__le__trans,axiom,
! [A: b > $o,B: b > $o,C2: b > $o] :
( ( A = B )
=> ( ( ord_less_eq_b_o @ B @ C2 )
=> ( ord_less_eq_b_o @ A @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_284_ord__eq__le__trans,axiom,
! [A: a > nat > $o,B: a > nat > $o,C2: a > nat > $o] :
( ( A = B )
=> ( ( ord_less_eq_a_nat_o @ B @ C2 )
=> ( ord_less_eq_a_nat_o @ A @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_285_ord__eq__le__trans,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat,C2: set_Pr4934435412358123699_a_nat] :
( ( A = B )
=> ( ( ord_le8666007276011122963_a_nat @ B @ C2 )
=> ( ord_le8666007276011122963_a_nat @ A @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_286_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: produc357393685978478089rm_a_b > $o,Z: produc357393685978478089rm_a_b > $o] : ( Y5 = Z ) )
= ( ^ [X2: produc357393685978478089rm_a_b > $o,Y4: produc357393685978478089rm_a_b > $o] :
( ( ord_le6645574462093738516_a_b_o @ X2 @ Y4 )
& ( ord_le6645574462093738516_a_b_o @ Y4 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_287_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: product_prod_a_nat > $o,Z: product_prod_a_nat > $o] : ( Y5 = Z ) )
= ( ^ [X2: product_prod_a_nat > $o,Y4: product_prod_a_nat > $o] :
( ( ord_le8950881081082477362_nat_o @ X2 @ Y4 )
& ( ord_le8950881081082477362_nat_o @ Y4 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_288_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: term_a_b > term_a_b > $o,Z: term_a_b > term_a_b > $o] : ( Y5 = Z ) )
= ( ^ [X2: term_a_b > term_a_b > $o,Y4: term_a_b > term_a_b > $o] :
( ( ord_le8627286948385211390_a_b_o @ X2 @ Y4 )
& ( ord_le8627286948385211390_a_b_o @ Y4 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_289_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: b > $o,Z: b > $o] : ( Y5 = Z ) )
= ( ^ [X2: b > $o,Y4: b > $o] :
( ( ord_less_eq_b_o @ X2 @ Y4 )
& ( ord_less_eq_b_o @ Y4 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_290_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: a > nat > $o,Z: a > nat > $o] : ( Y5 = Z ) )
= ( ^ [X2: a > nat > $o,Y4: a > nat > $o] :
( ( ord_less_eq_a_nat_o @ X2 @ Y4 )
& ( ord_less_eq_a_nat_o @ Y4 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_291_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_Pr4934435412358123699_a_nat,Z: set_Pr4934435412358123699_a_nat] : ( Y5 = Z ) )
= ( ^ [X2: set_Pr4934435412358123699_a_nat,Y4: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X2 @ Y4 )
& ( ord_le8666007276011122963_a_nat @ Y4 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_292_Collect__mono__iff,axiom,
! [P: produc9001763001631046773rm_a_b > $o,Q: produc9001763001631046773rm_a_b > $o] :
( ( ord_le2663425029446080117rm_a_b @ ( collec4780413679787665248rm_a_b @ P ) @ ( collec4780413679787665248rm_a_b @ Q ) )
= ( ! [X2: produc9001763001631046773rm_a_b] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_293_Collect__mono__iff,axiom,
! [P: product_prod_b_b > $o,Q: product_prod_b_b > $o] :
( ( ord_le182087997850975847od_b_b @ ( collec548942219715005266od_b_b @ P ) @ ( collec548942219715005266od_b_b @ Q ) )
= ( ! [X2: product_prod_b_b] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_294_Collect__mono__iff,axiom,
! [P: term_a_b > $o,Q: term_a_b > $o] :
( ( ord_le2705286416250468010rm_a_b @ ( collect_term_a_b @ P ) @ ( collect_term_a_b @ Q ) )
= ( ! [X2: term_a_b] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_295_Collect__mono__iff,axiom,
! [P: b > $o,Q: b > $o] :
( ( ord_less_eq_set_b @ ( collect_b @ P ) @ ( collect_b @ Q ) )
= ( ! [X2: b] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_296_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 ) )
= ( ! [X2: product_prod_a_nat] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_297_set__eq__subset,axiom,
( ( ^ [Y5: set_Pr4934435412358123699_a_nat,Z: set_Pr4934435412358123699_a_nat] : ( Y5 = Z ) )
= ( ^ [A6: set_Pr4934435412358123699_a_nat,B5: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A6 @ B5 )
& ( ord_le8666007276011122963_a_nat @ B5 @ A6 ) ) ) ) ).
% set_eq_subset
thf(fact_298_subset__trans,axiom,
! [A5: set_Pr4934435412358123699_a_nat,B4: set_Pr4934435412358123699_a_nat,C3: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A5 @ B4 )
=> ( ( ord_le8666007276011122963_a_nat @ B4 @ C3 )
=> ( ord_le8666007276011122963_a_nat @ A5 @ C3 ) ) ) ).
% subset_trans
thf(fact_299_Collect__mono,axiom,
! [P: produc9001763001631046773rm_a_b > $o,Q: produc9001763001631046773rm_a_b > $o] :
( ! [X3: produc9001763001631046773rm_a_b] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_le2663425029446080117rm_a_b @ ( collec4780413679787665248rm_a_b @ P ) @ ( collec4780413679787665248rm_a_b @ Q ) ) ) ).
% Collect_mono
thf(fact_300_Collect__mono,axiom,
! [P: product_prod_b_b > $o,Q: product_prod_b_b > $o] :
( ! [X3: product_prod_b_b] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_le182087997850975847od_b_b @ ( collec548942219715005266od_b_b @ P ) @ ( collec548942219715005266od_b_b @ Q ) ) ) ).
% Collect_mono
thf(fact_301_Collect__mono,axiom,
! [P: term_a_b > $o,Q: term_a_b > $o] :
( ! [X3: term_a_b] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_le2705286416250468010rm_a_b @ ( collect_term_a_b @ P ) @ ( collect_term_a_b @ Q ) ) ) ).
% Collect_mono
thf(fact_302_Collect__mono,axiom,
! [P: b > $o,Q: b > $o] :
( ! [X3: b] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq_set_b @ ( collect_b @ P ) @ ( collect_b @ Q ) ) ) ).
% Collect_mono
thf(fact_303_Collect__mono,axiom,
! [P: product_prod_a_nat > $o,Q: product_prod_a_nat > $o] :
( ! [X3: product_prod_a_nat] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_le8666007276011122963_a_nat @ ( collec4464134535221767506_a_nat @ P ) @ ( collec4464134535221767506_a_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_304_subset__refl,axiom,
! [A5: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ A5 @ A5 ) ).
% subset_refl
thf(fact_305_subset__iff,axiom,
( ord_le182087997850975847od_b_b
= ( ^ [A6: set_Product_prod_b_b,B5: set_Product_prod_b_b] :
! [T3: product_prod_b_b] :
( ( member7862447936710763792od_b_b @ T3 @ A6 )
=> ( member7862447936710763792od_b_b @ T3 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_306_subset__iff,axiom,
( ord_le2705286416250468010rm_a_b
= ( ^ [A6: set_term_a_b,B5: set_term_a_b] :
! [T3: term_a_b] :
( ( member_term_a_b @ T3 @ A6 )
=> ( member_term_a_b @ T3 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_307_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A6: set_nat,B5: set_nat] :
! [T3: nat] :
( ( member_nat @ T3 @ A6 )
=> ( member_nat @ T3 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_308_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A6: set_a,B5: set_a] :
! [T3: a] :
( ( member_a @ T3 @ A6 )
=> ( member_a @ T3 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_309_subset__iff,axiom,
( ord_le118470702582115849rm_a_b
= ( ^ [A6: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
! [T3: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ T3 @ A6 )
=> ( member5869715511025134514rm_a_b @ T3 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_310_subset__iff,axiom,
( ord_less_eq_set_b
= ( ^ [A6: set_b,B5: set_b] :
! [T3: b] :
( ( member_b @ T3 @ A6 )
=> ( member_b @ T3 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_311_subset__iff,axiom,
( ord_le8666007276011122963_a_nat
= ( ^ [A6: set_Pr4934435412358123699_a_nat,B5: set_Pr4934435412358123699_a_nat] :
! [T3: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ T3 @ A6 )
=> ( member5724188588386418708_a_nat @ T3 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_312_equalityD2,axiom,
! [A5: set_Pr4934435412358123699_a_nat,B4: set_Pr4934435412358123699_a_nat] :
( ( A5 = B4 )
=> ( ord_le8666007276011122963_a_nat @ B4 @ A5 ) ) ).
% equalityD2
thf(fact_313_equalityD1,axiom,
! [A5: set_Pr4934435412358123699_a_nat,B4: set_Pr4934435412358123699_a_nat] :
( ( A5 = B4 )
=> ( ord_le8666007276011122963_a_nat @ A5 @ B4 ) ) ).
% equalityD1
thf(fact_314_subset__eq,axiom,
( ord_le182087997850975847od_b_b
= ( ^ [A6: set_Product_prod_b_b,B5: set_Product_prod_b_b] :
! [X2: product_prod_b_b] :
( ( member7862447936710763792od_b_b @ X2 @ A6 )
=> ( member7862447936710763792od_b_b @ X2 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_315_subset__eq,axiom,
( ord_le2705286416250468010rm_a_b
= ( ^ [A6: set_term_a_b,B5: set_term_a_b] :
! [X2: term_a_b] :
( ( member_term_a_b @ X2 @ A6 )
=> ( member_term_a_b @ X2 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_316_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A6: set_nat,B5: set_nat] :
! [X2: nat] :
( ( member_nat @ X2 @ A6 )
=> ( member_nat @ X2 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_317_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A6: set_a,B5: set_a] :
! [X2: a] :
( ( member_a @ X2 @ A6 )
=> ( member_a @ X2 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_318_subset__eq,axiom,
( ord_le118470702582115849rm_a_b
= ( ^ [A6: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
! [X2: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X2 @ A6 )
=> ( member5869715511025134514rm_a_b @ X2 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_319_subset__eq,axiom,
( ord_less_eq_set_b
= ( ^ [A6: set_b,B5: set_b] :
! [X2: b] :
( ( member_b @ X2 @ A6 )
=> ( member_b @ X2 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_320_subset__eq,axiom,
( ord_le8666007276011122963_a_nat
= ( ^ [A6: set_Pr4934435412358123699_a_nat,B5: set_Pr4934435412358123699_a_nat] :
! [X2: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ X2 @ A6 )
=> ( member5724188588386418708_a_nat @ X2 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_321_equalityE,axiom,
! [A5: set_Pr4934435412358123699_a_nat,B4: set_Pr4934435412358123699_a_nat] :
( ( A5 = B4 )
=> ~ ( ( ord_le8666007276011122963_a_nat @ A5 @ B4 )
=> ~ ( ord_le8666007276011122963_a_nat @ B4 @ A5 ) ) ) ).
% equalityE
thf(fact_322_subsetD,axiom,
! [A5: set_Product_prod_b_b,B4: set_Product_prod_b_b,C2: product_prod_b_b] :
( ( ord_le182087997850975847od_b_b @ A5 @ B4 )
=> ( ( member7862447936710763792od_b_b @ C2 @ A5 )
=> ( member7862447936710763792od_b_b @ C2 @ B4 ) ) ) ).
% subsetD
thf(fact_323_subsetD,axiom,
! [A5: set_term_a_b,B4: set_term_a_b,C2: term_a_b] :
( ( ord_le2705286416250468010rm_a_b @ A5 @ B4 )
=> ( ( member_term_a_b @ C2 @ A5 )
=> ( member_term_a_b @ C2 @ B4 ) ) ) ).
% subsetD
thf(fact_324_subsetD,axiom,
! [A5: set_nat,B4: set_nat,C2: nat] :
( ( ord_less_eq_set_nat @ A5 @ B4 )
=> ( ( member_nat @ C2 @ A5 )
=> ( member_nat @ C2 @ B4 ) ) ) ).
% subsetD
thf(fact_325_subsetD,axiom,
! [A5: set_a,B4: set_a,C2: a] :
( ( ord_less_eq_set_a @ A5 @ B4 )
=> ( ( member_a @ C2 @ A5 )
=> ( member_a @ C2 @ B4 ) ) ) ).
% subsetD
thf(fact_326_subsetD,axiom,
! [A5: set_Pr4386577575007340137rm_a_b,B4: set_Pr4386577575007340137rm_a_b,C2: produc357393685978478089rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A5 @ B4 )
=> ( ( member5869715511025134514rm_a_b @ C2 @ A5 )
=> ( member5869715511025134514rm_a_b @ C2 @ B4 ) ) ) ).
% subsetD
thf(fact_327_subsetD,axiom,
! [A5: set_b,B4: set_b,C2: b] :
( ( ord_less_eq_set_b @ A5 @ B4 )
=> ( ( member_b @ C2 @ A5 )
=> ( member_b @ C2 @ B4 ) ) ) ).
% subsetD
thf(fact_328_subsetD,axiom,
! [A5: set_Pr4934435412358123699_a_nat,B4: set_Pr4934435412358123699_a_nat,C2: product_prod_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A5 @ B4 )
=> ( ( member5724188588386418708_a_nat @ C2 @ A5 )
=> ( member5724188588386418708_a_nat @ C2 @ B4 ) ) ) ).
% subsetD
thf(fact_329_in__mono,axiom,
! [A5: set_Product_prod_b_b,B4: set_Product_prod_b_b,X: product_prod_b_b] :
( ( ord_le182087997850975847od_b_b @ A5 @ B4 )
=> ( ( member7862447936710763792od_b_b @ X @ A5 )
=> ( member7862447936710763792od_b_b @ X @ B4 ) ) ) ).
% in_mono
thf(fact_330_in__mono,axiom,
! [A5: set_term_a_b,B4: set_term_a_b,X: term_a_b] :
( ( ord_le2705286416250468010rm_a_b @ A5 @ B4 )
=> ( ( member_term_a_b @ X @ A5 )
=> ( member_term_a_b @ X @ B4 ) ) ) ).
% in_mono
thf(fact_331_in__mono,axiom,
! [A5: set_nat,B4: set_nat,X: nat] :
( ( ord_less_eq_set_nat @ A5 @ B4 )
=> ( ( member_nat @ X @ A5 )
=> ( member_nat @ X @ B4 ) ) ) ).
% in_mono
thf(fact_332_in__mono,axiom,
! [A5: set_a,B4: set_a,X: a] :
( ( ord_less_eq_set_a @ A5 @ B4 )
=> ( ( member_a @ X @ A5 )
=> ( member_a @ X @ B4 ) ) ) ).
% in_mono
thf(fact_333_in__mono,axiom,
! [A5: set_Pr4386577575007340137rm_a_b,B4: set_Pr4386577575007340137rm_a_b,X: produc357393685978478089rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A5 @ B4 )
=> ( ( member5869715511025134514rm_a_b @ X @ A5 )
=> ( member5869715511025134514rm_a_b @ X @ B4 ) ) ) ).
% in_mono
thf(fact_334_in__mono,axiom,
! [A5: set_b,B4: set_b,X: b] :
( ( ord_less_eq_set_b @ A5 @ B4 )
=> ( ( member_b @ X @ A5 )
=> ( member_b @ X @ B4 ) ) ) ).
% in_mono
thf(fact_335_in__mono,axiom,
! [A5: set_Pr4934435412358123699_a_nat,B4: set_Pr4934435412358123699_a_nat,X: product_prod_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A5 @ B4 )
=> ( ( member5724188588386418708_a_nat @ X @ A5 )
=> ( member5724188588386418708_a_nat @ X @ B4 ) ) ) ).
% in_mono
thf(fact_336_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_337_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_338_pred__equals__eq2,axiom,
! [R5: set_Pr1811044260758604347_a_nat,S2: set_Pr1811044260758604347_a_nat] :
( ( ( ^ [X2: product_prod_a_nat,Y4: product_prod_a_nat] : ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ X2 @ Y4 ) @ R5 ) )
= ( ^ [X2: product_prod_a_nat,Y4: product_prod_a_nat] : ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ X2 @ Y4 ) @ S2 ) ) )
= ( R5 = S2 ) ) ).
% pred_equals_eq2
thf(fact_339_pred__equals__eq2,axiom,
! [R5: set_Pr7151917219565317845rm_a_b,S2: set_Pr7151917219565317845rm_a_b] :
( ( ( ^ [X2: set_term_a_b,Y4: set_term_a_b] : ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ X2 @ Y4 ) @ R5 ) )
= ( ^ [X2: set_term_a_b,Y4: set_term_a_b] : ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ X2 @ Y4 ) @ S2 ) ) )
= ( R5 = S2 ) ) ).
% pred_equals_eq2
thf(fact_340_pred__equals__eq2,axiom,
! [R5: set_Pr5162482618186357969rm_a_b,S2: set_Pr5162482618186357969rm_a_b] :
( ( ( ^ [X2: b,Y4: produc357393685978478089rm_a_b] : ( member8210134394708959770rm_a_b @ ( produc2673992483936284769rm_a_b @ X2 @ Y4 ) @ R5 ) )
= ( ^ [X2: b,Y4: produc357393685978478089rm_a_b] : ( member8210134394708959770rm_a_b @ ( produc2673992483936284769rm_a_b @ X2 @ Y4 ) @ S2 ) ) )
= ( R5 = S2 ) ) ).
% pred_equals_eq2
thf(fact_341_pred__equals__eq2,axiom,
! [R5: set_Product_prod_b_b,S2: set_Product_prod_b_b] :
( ( ( ^ [X2: b,Y4: b] : ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ X2 @ Y4 ) @ R5 ) )
= ( ^ [X2: b,Y4: b] : ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ X2 @ Y4 ) @ S2 ) ) )
= ( R5 = S2 ) ) ).
% pred_equals_eq2
thf(fact_342_pred__equals__eq2,axiom,
! [R5: set_Pr4934435412358123699_a_nat,S2: set_Pr4934435412358123699_a_nat] :
( ( ( ^ [X2: a,Y4: nat] : ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ X2 @ Y4 ) @ R5 ) )
= ( ^ [X2: a,Y4: nat] : ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ X2 @ Y4 ) @ S2 ) ) )
= ( R5 = S2 ) ) ).
% pred_equals_eq2
thf(fact_343_pred__equals__eq2,axiom,
! [R5: set_Pr4386577575007340137rm_a_b,S2: set_Pr4386577575007340137rm_a_b] :
( ( ( ^ [X2: term_a_b,Y4: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y4 ) @ R5 ) )
= ( ^ [X2: term_a_b,Y4: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y4 ) @ S2 ) ) )
= ( R5 = S2 ) ) ).
% pred_equals_eq2
thf(fact_344_Collect__subset,axiom,
! [A5: set_nat,P: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A5 )
& ( P @ X2 ) ) )
@ A5 ) ).
% Collect_subset
thf(fact_345_Collect__subset,axiom,
! [A5: set_a,P: a > $o] :
( ord_less_eq_set_a
@ ( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ A5 )
& ( P @ X2 ) ) )
@ A5 ) ).
% Collect_subset
thf(fact_346_Collect__subset,axiom,
! [A5: set_Pr7151917219565317845rm_a_b,P: produc9001763001631046773rm_a_b > $o] :
( ord_le2663425029446080117rm_a_b
@ ( collec4780413679787665248rm_a_b
@ ^ [X2: produc9001763001631046773rm_a_b] :
( ( member8888773126115863070rm_a_b @ X2 @ A5 )
& ( P @ X2 ) ) )
@ A5 ) ).
% Collect_subset
thf(fact_347_Collect__subset,axiom,
! [A5: set_Product_prod_b_b,P: product_prod_b_b > $o] :
( ord_le182087997850975847od_b_b
@ ( collec548942219715005266od_b_b
@ ^ [X2: product_prod_b_b] :
( ( member7862447936710763792od_b_b @ X2 @ A5 )
& ( P @ X2 ) ) )
@ A5 ) ).
% Collect_subset
thf(fact_348_Collect__subset,axiom,
! [A5: set_term_a_b,P: term_a_b > $o] :
( ord_le2705286416250468010rm_a_b
@ ( collect_term_a_b
@ ^ [X2: term_a_b] :
( ( member_term_a_b @ X2 @ A5 )
& ( P @ X2 ) ) )
@ A5 ) ).
% Collect_subset
thf(fact_349_Collect__subset,axiom,
! [A5: set_Pr4386577575007340137rm_a_b,P: produc357393685978478089rm_a_b > $o] :
( ord_le118470702582115849rm_a_b
@ ( collec99177395925924084rm_a_b
@ ^ [X2: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X2 @ A5 )
& ( P @ X2 ) ) )
@ A5 ) ).
% Collect_subset
thf(fact_350_Collect__subset,axiom,
! [A5: set_b,P: b > $o] :
( ord_less_eq_set_b
@ ( collect_b
@ ^ [X2: b] :
( ( member_b @ X2 @ A5 )
& ( P @ X2 ) ) )
@ A5 ) ).
% Collect_subset
thf(fact_351_Collect__subset,axiom,
! [A5: set_Pr4934435412358123699_a_nat,P: product_prod_a_nat > $o] :
( ord_le8666007276011122963_a_nat
@ ( collec4464134535221767506_a_nat
@ ^ [X2: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ X2 @ A5 )
& ( P @ X2 ) ) )
@ A5 ) ).
% Collect_subset
thf(fact_352_subset__CollectI,axiom,
! [B4: set_nat,A5: set_nat,Q: nat > $o,P: nat > $o] :
( ( ord_less_eq_set_nat @ B4 @ A5 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B4 )
=> ( ( Q @ X3 )
=> ( P @ X3 ) ) )
=> ( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ B4 )
& ( Q @ X2 ) ) )
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A5 )
& ( P @ X2 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_353_subset__CollectI,axiom,
! [B4: set_a,A5: set_a,Q: a > $o,P: a > $o] :
( ( ord_less_eq_set_a @ B4 @ A5 )
=> ( ! [X3: a] :
( ( member_a @ X3 @ B4 )
=> ( ( Q @ X3 )
=> ( P @ X3 ) ) )
=> ( ord_less_eq_set_a
@ ( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ B4 )
& ( Q @ X2 ) ) )
@ ( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ A5 )
& ( P @ X2 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_354_subset__CollectI,axiom,
! [B4: set_Pr7151917219565317845rm_a_b,A5: set_Pr7151917219565317845rm_a_b,Q: produc9001763001631046773rm_a_b > $o,P: produc9001763001631046773rm_a_b > $o] :
( ( ord_le2663425029446080117rm_a_b @ B4 @ A5 )
=> ( ! [X3: produc9001763001631046773rm_a_b] :
( ( member8888773126115863070rm_a_b @ X3 @ B4 )
=> ( ( Q @ X3 )
=> ( P @ X3 ) ) )
=> ( ord_le2663425029446080117rm_a_b
@ ( collec4780413679787665248rm_a_b
@ ^ [X2: produc9001763001631046773rm_a_b] :
( ( member8888773126115863070rm_a_b @ X2 @ B4 )
& ( Q @ X2 ) ) )
@ ( collec4780413679787665248rm_a_b
@ ^ [X2: produc9001763001631046773rm_a_b] :
( ( member8888773126115863070rm_a_b @ X2 @ A5 )
& ( P @ X2 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_355_subset__CollectI,axiom,
! [B4: set_Product_prod_b_b,A5: set_Product_prod_b_b,Q: product_prod_b_b > $o,P: product_prod_b_b > $o] :
( ( ord_le182087997850975847od_b_b @ B4 @ A5 )
=> ( ! [X3: product_prod_b_b] :
( ( member7862447936710763792od_b_b @ X3 @ B4 )
=> ( ( Q @ X3 )
=> ( P @ X3 ) ) )
=> ( ord_le182087997850975847od_b_b
@ ( collec548942219715005266od_b_b
@ ^ [X2: product_prod_b_b] :
( ( member7862447936710763792od_b_b @ X2 @ B4 )
& ( Q @ X2 ) ) )
@ ( collec548942219715005266od_b_b
@ ^ [X2: product_prod_b_b] :
( ( member7862447936710763792od_b_b @ X2 @ A5 )
& ( P @ X2 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_356_subset__CollectI,axiom,
! [B4: set_term_a_b,A5: set_term_a_b,Q: term_a_b > $o,P: term_a_b > $o] :
( ( ord_le2705286416250468010rm_a_b @ B4 @ A5 )
=> ( ! [X3: term_a_b] :
( ( member_term_a_b @ X3 @ B4 )
=> ( ( Q @ X3 )
=> ( P @ X3 ) ) )
=> ( ord_le2705286416250468010rm_a_b
@ ( collect_term_a_b
@ ^ [X2: term_a_b] :
( ( member_term_a_b @ X2 @ B4 )
& ( Q @ X2 ) ) )
@ ( collect_term_a_b
@ ^ [X2: term_a_b] :
( ( member_term_a_b @ X2 @ A5 )
& ( P @ X2 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_357_subset__CollectI,axiom,
! [B4: set_Pr4386577575007340137rm_a_b,A5: set_Pr4386577575007340137rm_a_b,Q: produc357393685978478089rm_a_b > $o,P: produc357393685978478089rm_a_b > $o] :
( ( ord_le118470702582115849rm_a_b @ B4 @ A5 )
=> ( ! [X3: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X3 @ B4 )
=> ( ( Q @ X3 )
=> ( P @ X3 ) ) )
=> ( ord_le118470702582115849rm_a_b
@ ( collec99177395925924084rm_a_b
@ ^ [X2: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X2 @ B4 )
& ( Q @ X2 ) ) )
@ ( collec99177395925924084rm_a_b
@ ^ [X2: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X2 @ A5 )
& ( P @ X2 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_358_subset__CollectI,axiom,
! [B4: set_b,A5: set_b,Q: b > $o,P: b > $o] :
( ( ord_less_eq_set_b @ B4 @ A5 )
=> ( ! [X3: b] :
( ( member_b @ X3 @ B4 )
=> ( ( Q @ X3 )
=> ( P @ X3 ) ) )
=> ( ord_less_eq_set_b
@ ( collect_b
@ ^ [X2: b] :
( ( member_b @ X2 @ B4 )
& ( Q @ X2 ) ) )
@ ( collect_b
@ ^ [X2: b] :
( ( member_b @ X2 @ A5 )
& ( P @ X2 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_359_subset__CollectI,axiom,
! [B4: set_Pr4934435412358123699_a_nat,A5: set_Pr4934435412358123699_a_nat,Q: product_prod_a_nat > $o,P: product_prod_a_nat > $o] :
( ( ord_le8666007276011122963_a_nat @ B4 @ A5 )
=> ( ! [X3: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ X3 @ B4 )
=> ( ( Q @ X3 )
=> ( P @ X3 ) ) )
=> ( ord_le8666007276011122963_a_nat
@ ( collec4464134535221767506_a_nat
@ ^ [X2: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ X2 @ B4 )
& ( Q @ X2 ) ) )
@ ( collec4464134535221767506_a_nat
@ ^ [X2: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ X2 @ A5 )
& ( P @ X2 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_360_rstep__srstepI,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,F: set_Pr4934435412358123699_a_nat,S: term_a_b,T: term_a_b] :
( ( ord_le8666007276011122963_a_nat @ ( terms_7988297476397195622_a_b_b @ R2 ) @ F )
=> ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ S ) @ F )
=> ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ T ) @ F )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( rstep_a_b @ R2 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( sig_step_a_b @ F @ ( rstep_a_b @ R2 ) ) ) ) ) ) ) ).
% rstep_srstepI
thf(fact_361_srsteps__eq__subst__closed,axiom,
! [S: term_a_b,T: term_a_b,F: set_Pr4934435412358123699_a_nat,R2: set_Pr4386577575007340137rm_a_b,Sigma3: b > term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R2 ) ) ) )
=> ( ! [X3: b] : ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ ( Sigma3 @ X3 ) ) @ F )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( subst_7999470309526761004_a_b_b @ S @ Sigma3 ) @ ( subst_7999470309526761004_a_b_b @ T @ Sigma3 ) ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R2 ) ) ) ) ) ) ).
% srsteps_eq_subst_closed
thf(fact_362_srsteps__subst__closed,axiom,
! [S: term_a_b,T: term_a_b,F: set_Pr4934435412358123699_a_nat,R2: set_Pr4386577575007340137rm_a_b,Sigma3: b > term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( transi7922773638565587891rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R2 ) ) ) )
=> ( ! [X3: b] : ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ ( Sigma3 @ X3 ) ) @ F )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( subst_7999470309526761004_a_b_b @ S @ Sigma3 ) @ ( subst_7999470309526761004_a_b_b @ T @ Sigma3 ) ) @ ( transi7922773638565587891rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R2 ) ) ) ) ) ) ).
% srsteps_subst_closed
thf(fact_363_rstepp__rstep__eq,axiom,
! [R2: set_Pr4386577575007340137rm_a_b] :
( ( rstepp_a_b
@ ^ [X2: term_a_b,Y4: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y4 ) @ R2 ) )
= ( ^ [X2: term_a_b,Y4: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y4 ) @ ( rstep_a_b @ R2 ) ) ) ) ).
% rstepp_rstep_eq
thf(fact_364_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_365_conj__subset__def,axiom,
! [A5: set_Pr7151917219565317845rm_a_b,P: produc9001763001631046773rm_a_b > $o,Q: produc9001763001631046773rm_a_b > $o] :
( ( ord_le2663425029446080117rm_a_b @ A5
@ ( collec4780413679787665248rm_a_b
@ ^ [X2: produc9001763001631046773rm_a_b] :
( ( P @ X2 )
& ( Q @ X2 ) ) ) )
= ( ( ord_le2663425029446080117rm_a_b @ A5 @ ( collec4780413679787665248rm_a_b @ P ) )
& ( ord_le2663425029446080117rm_a_b @ A5 @ ( collec4780413679787665248rm_a_b @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_366_conj__subset__def,axiom,
! [A5: set_Product_prod_b_b,P: product_prod_b_b > $o,Q: product_prod_b_b > $o] :
( ( ord_le182087997850975847od_b_b @ A5
@ ( collec548942219715005266od_b_b
@ ^ [X2: product_prod_b_b] :
( ( P @ X2 )
& ( Q @ X2 ) ) ) )
= ( ( ord_le182087997850975847od_b_b @ A5 @ ( collec548942219715005266od_b_b @ P ) )
& ( ord_le182087997850975847od_b_b @ A5 @ ( collec548942219715005266od_b_b @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_367_conj__subset__def,axiom,
! [A5: set_term_a_b,P: term_a_b > $o,Q: term_a_b > $o] :
( ( ord_le2705286416250468010rm_a_b @ A5
@ ( collect_term_a_b
@ ^ [X2: term_a_b] :
( ( P @ X2 )
& ( Q @ X2 ) ) ) )
= ( ( ord_le2705286416250468010rm_a_b @ A5 @ ( collect_term_a_b @ P ) )
& ( ord_le2705286416250468010rm_a_b @ A5 @ ( collect_term_a_b @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_368_conj__subset__def,axiom,
! [A5: set_b,P: b > $o,Q: b > $o] :
( ( ord_less_eq_set_b @ A5
@ ( collect_b
@ ^ [X2: b] :
( ( P @ X2 )
& ( Q @ X2 ) ) ) )
= ( ( ord_less_eq_set_b @ A5 @ ( collect_b @ P ) )
& ( ord_less_eq_set_b @ A5 @ ( collect_b @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_369_conj__subset__def,axiom,
! [A5: set_Pr4934435412358123699_a_nat,P: product_prod_a_nat > $o,Q: product_prod_a_nat > $o] :
( ( ord_le8666007276011122963_a_nat @ A5
@ ( collec4464134535221767506_a_nat
@ ^ [X2: product_prod_a_nat] :
( ( P @ X2 )
& ( Q @ X2 ) ) ) )
= ( ( ord_le8666007276011122963_a_nat @ A5 @ ( collec4464134535221767506_a_nat @ P ) )
& ( ord_le8666007276011122963_a_nat @ A5 @ ( collec4464134535221767506_a_nat @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_370_prop__restrict,axiom,
! [X: nat,Z3: set_nat,X5: set_nat,P: nat > $o] :
( ( member_nat @ X @ Z3 )
=> ( ( ord_less_eq_set_nat @ Z3
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ X5 )
& ( P @ X2 ) ) ) )
=> ( P @ X ) ) ) ).
% prop_restrict
thf(fact_371_prop__restrict,axiom,
! [X: a,Z3: set_a,X5: set_a,P: a > $o] :
( ( member_a @ X @ Z3 )
=> ( ( ord_less_eq_set_a @ Z3
@ ( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ X5 )
& ( P @ X2 ) ) ) )
=> ( P @ X ) ) ) ).
% prop_restrict
thf(fact_372_prop__restrict,axiom,
! [X: produc9001763001631046773rm_a_b,Z3: set_Pr7151917219565317845rm_a_b,X5: set_Pr7151917219565317845rm_a_b,P: produc9001763001631046773rm_a_b > $o] :
( ( member8888773126115863070rm_a_b @ X @ Z3 )
=> ( ( ord_le2663425029446080117rm_a_b @ Z3
@ ( collec4780413679787665248rm_a_b
@ ^ [X2: produc9001763001631046773rm_a_b] :
( ( member8888773126115863070rm_a_b @ X2 @ X5 )
& ( P @ X2 ) ) ) )
=> ( P @ X ) ) ) ).
% prop_restrict
thf(fact_373_prop__restrict,axiom,
! [X: product_prod_b_b,Z3: set_Product_prod_b_b,X5: set_Product_prod_b_b,P: product_prod_b_b > $o] :
( ( member7862447936710763792od_b_b @ X @ Z3 )
=> ( ( ord_le182087997850975847od_b_b @ Z3
@ ( collec548942219715005266od_b_b
@ ^ [X2: product_prod_b_b] :
( ( member7862447936710763792od_b_b @ X2 @ X5 )
& ( P @ X2 ) ) ) )
=> ( P @ X ) ) ) ).
% prop_restrict
thf(fact_374_prop__restrict,axiom,
! [X: term_a_b,Z3: set_term_a_b,X5: set_term_a_b,P: term_a_b > $o] :
( ( member_term_a_b @ X @ Z3 )
=> ( ( ord_le2705286416250468010rm_a_b @ Z3
@ ( collect_term_a_b
@ ^ [X2: term_a_b] :
( ( member_term_a_b @ X2 @ X5 )
& ( P @ X2 ) ) ) )
=> ( P @ X ) ) ) ).
% prop_restrict
thf(fact_375_prop__restrict,axiom,
! [X: produc357393685978478089rm_a_b,Z3: set_Pr4386577575007340137rm_a_b,X5: set_Pr4386577575007340137rm_a_b,P: produc357393685978478089rm_a_b > $o] :
( ( member5869715511025134514rm_a_b @ X @ Z3 )
=> ( ( ord_le118470702582115849rm_a_b @ Z3
@ ( collec99177395925924084rm_a_b
@ ^ [X2: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X2 @ X5 )
& ( P @ X2 ) ) ) )
=> ( P @ X ) ) ) ).
% prop_restrict
thf(fact_376_prop__restrict,axiom,
! [X: b,Z3: set_b,X5: set_b,P: b > $o] :
( ( member_b @ X @ Z3 )
=> ( ( ord_less_eq_set_b @ Z3
@ ( collect_b
@ ^ [X2: b] :
( ( member_b @ X2 @ X5 )
& ( P @ X2 ) ) ) )
=> ( P @ X ) ) ) ).
% prop_restrict
thf(fact_377_prop__restrict,axiom,
! [X: product_prod_a_nat,Z3: set_Pr4934435412358123699_a_nat,X5: set_Pr4934435412358123699_a_nat,P: product_prod_a_nat > $o] :
( ( member5724188588386418708_a_nat @ X @ Z3 )
=> ( ( ord_le8666007276011122963_a_nat @ Z3
@ ( collec4464134535221767506_a_nat
@ ^ [X2: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ X2 @ X5 )
& ( P @ X2 ) ) ) )
=> ( P @ X ) ) ) ).
% prop_restrict
thf(fact_378_Collect__restrict,axiom,
! [X5: set_nat,P: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ X5 )
& ( P @ X2 ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_379_Collect__restrict,axiom,
! [X5: set_a,P: a > $o] :
( ord_less_eq_set_a
@ ( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ X5 )
& ( P @ X2 ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_380_Collect__restrict,axiom,
! [X5: set_Pr7151917219565317845rm_a_b,P: produc9001763001631046773rm_a_b > $o] :
( ord_le2663425029446080117rm_a_b
@ ( collec4780413679787665248rm_a_b
@ ^ [X2: produc9001763001631046773rm_a_b] :
( ( member8888773126115863070rm_a_b @ X2 @ X5 )
& ( P @ X2 ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_381_Collect__restrict,axiom,
! [X5: set_Product_prod_b_b,P: product_prod_b_b > $o] :
( ord_le182087997850975847od_b_b
@ ( collec548942219715005266od_b_b
@ ^ [X2: product_prod_b_b] :
( ( member7862447936710763792od_b_b @ X2 @ X5 )
& ( P @ X2 ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_382_Collect__restrict,axiom,
! [X5: set_term_a_b,P: term_a_b > $o] :
( ord_le2705286416250468010rm_a_b
@ ( collect_term_a_b
@ ^ [X2: term_a_b] :
( ( member_term_a_b @ X2 @ X5 )
& ( P @ X2 ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_383_Collect__restrict,axiom,
! [X5: set_Pr4386577575007340137rm_a_b,P: produc357393685978478089rm_a_b > $o] :
( ord_le118470702582115849rm_a_b
@ ( collec99177395925924084rm_a_b
@ ^ [X2: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X2 @ X5 )
& ( P @ X2 ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_384_Collect__restrict,axiom,
! [X5: set_b,P: b > $o] :
( ord_less_eq_set_b
@ ( collect_b
@ ^ [X2: b] :
( ( member_b @ X2 @ X5 )
& ( P @ X2 ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_385_Collect__restrict,axiom,
! [X5: set_Pr4934435412358123699_a_nat,P: product_prod_a_nat > $o] :
( ord_le8666007276011122963_a_nat
@ ( collec4464134535221767506_a_nat
@ ^ [X2: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ X2 @ X5 )
& ( P @ X2 ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_386_relChain__def,axiom,
( bNF_Ca9222207453146494317_a_nat
= ( ^ [R4: set_Pr4386577575007340137rm_a_b,As: term_a_b > set_Pr4934435412358123699_a_nat] :
! [I: term_a_b,J: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ I @ J ) @ R4 )
=> ( ord_le8666007276011122963_a_nat @ ( As @ I ) @ ( As @ J ) ) ) ) ) ).
% relChain_def
thf(fact_387_relChain__def,axiom,
( bNF_Ca7350012948471701913_b_b_o
= ( ^ [R4: set_Product_prod_b_b,As: b > b > $o] :
! [I: b,J: b] :
( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ I @ J ) @ R4 )
=> ( ord_less_eq_b_o @ ( As @ I ) @ ( As @ J ) ) ) ) ) ).
% relChain_def
thf(fact_388_relChain__def,axiom,
( bNF_Ca6375836541719173646_a_nat
= ( ^ [R4: set_Product_prod_b_b,As: b > set_Pr4934435412358123699_a_nat] :
! [I: b,J: b] :
( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ I @ J ) @ R4 )
=> ( ord_le8666007276011122963_a_nat @ ( As @ I ) @ ( As @ J ) ) ) ) ) ).
% relChain_def
thf(fact_389_relChain__def,axiom,
( bNF_Ca7216400615022343671_nat_o
= ( ^ [R4: set_Product_prod_b_b,As: b > product_prod_a_nat > $o] :
! [I: b,J: b] :
( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ I @ J ) @ R4 )
=> ( ord_le8950881081082477362_nat_o @ ( As @ I ) @ ( As @ J ) ) ) ) ) ).
% relChain_def
thf(fact_390_relChain__def,axiom,
( bNF_Ca7295911705182073208_b_b_o
= ( ^ [R4: set_Pr4386577575007340137rm_a_b,As: term_a_b > b > $o] :
! [I: term_a_b,J: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ I @ J ) @ R4 )
=> ( ord_less_eq_b_o @ ( As @ I ) @ ( As @ J ) ) ) ) ) ).
% relChain_def
thf(fact_391_relChain__def,axiom,
( bNF_Ca2796605332057631201at_b_o
= ( ^ [R4: set_Pr1811044260758604347_a_nat,As: product_prod_a_nat > b > $o] :
! [I: product_prod_a_nat,J: product_prod_a_nat] :
( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ I @ J ) @ R4 )
=> ( ord_less_eq_b_o @ ( As @ I ) @ ( As @ J ) ) ) ) ) ).
% relChain_def
thf(fact_392_relChain__def,axiom,
( bNF_Ca911492713003348673_nat_o
= ( ^ [R4: set_Product_prod_b_b,As: b > a > nat > $o] :
! [I: b,J: b] :
( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ I @ J ) @ R4 )
=> ( ord_less_eq_a_nat_o @ ( As @ I ) @ ( As @ J ) ) ) ) ) ).
% relChain_def
thf(fact_393_relChain__def,axiom,
( bNF_Ca8408884726352799318_a_nat
= ( ^ [R4: set_Pr1811044260758604347_a_nat,As: product_prod_a_nat > set_Pr4934435412358123699_a_nat] :
! [I: product_prod_a_nat,J: product_prod_a_nat] :
( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ I @ J ) @ R4 )
=> ( ord_le8666007276011122963_a_nat @ ( As @ I ) @ ( As @ J ) ) ) ) ) ).
% relChain_def
thf(fact_394_relChain__def,axiom,
( bNF_Ca2324036474760502446_b_b_o
= ( ^ [R4: set_Pr7151917219565317845rm_a_b,As: set_term_a_b > b > $o] :
! [I: set_term_a_b,J: set_term_a_b] :
( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ I @ J ) @ R4 )
=> ( ord_less_eq_b_o @ ( As @ I ) @ ( As @ J ) ) ) ) ) ).
% relChain_def
thf(fact_395_relChain__def,axiom,
( bNF_Ca6092810557366899619_a_nat
= ( ^ [R4: set_Pr7151917219565317845rm_a_b,As: set_term_a_b > set_Pr4934435412358123699_a_nat] :
! [I: set_term_a_b,J: set_term_a_b] :
( ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ I @ J ) @ R4 )
=> ( ord_le8666007276011122963_a_nat @ ( As @ I ) @ ( As @ J ) ) ) ) ) ).
% relChain_def
thf(fact_396_rrstep__def,axiom,
( rrstep_a_b
= ( ^ [R6: set_Pr4386577575007340137rm_a_b] :
( collec99177395925924084rm_a_b
@ ( produc6251846659952913346_a_b_o
@ ( rrstepp_a_b
@ ^ [X2: term_a_b,Y4: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y4 ) @ R6 ) ) ) ) ) ) ).
% rrstep_def
thf(fact_397_predicate2I,axiom,
! [P: term_a_b > term_a_b > $o,Q: term_a_b > term_a_b > $o] :
( ! [X3: term_a_b,Y3: term_a_b] :
( ( P @ X3 @ Y3 )
=> ( Q @ X3 @ Y3 ) )
=> ( ord_le8627286948385211390_a_b_o @ P @ Q ) ) ).
% predicate2I
thf(fact_398_predicate2I,axiom,
! [P: a > nat > $o,Q: a > nat > $o] :
( ! [X3: a,Y3: nat] :
( ( P @ X3 @ Y3 )
=> ( Q @ X3 @ Y3 ) )
=> ( ord_less_eq_a_nat_o @ P @ Q ) ) ).
% predicate2I
thf(fact_399_predicate1I,axiom,
! [P: produc357393685978478089rm_a_b > $o,Q: produc357393685978478089rm_a_b > $o] :
( ! [X3: produc357393685978478089rm_a_b] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_le6645574462093738516_a_b_o @ P @ Q ) ) ).
% predicate1I
thf(fact_400_predicate1I,axiom,
! [P: product_prod_a_nat > $o,Q: product_prod_a_nat > $o] :
( ! [X3: product_prod_a_nat] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_le8950881081082477362_nat_o @ P @ Q ) ) ).
% predicate1I
thf(fact_401_predicate1I,axiom,
! [P: b > $o,Q: b > $o] :
( ! [X3: b] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq_b_o @ P @ Q ) ) ).
% predicate1I
thf(fact_402_split__part,axiom,
! [P: $o,Q: set_term_a_b > set_term_a_b > $o] :
( ( produc4628319930418777942_a_b_o
@ ^ [A7: set_term_a_b,B6: set_term_a_b] :
( P
& ( Q @ A7 @ B6 ) ) )
= ( ^ [Ab: produc9001763001631046773rm_a_b] :
( P
& ( produc4628319930418777942_a_b_o @ Q @ Ab ) ) ) ) ).
% split_part
thf(fact_403_split__part,axiom,
! [P: $o,Q: b > b > $o] :
( ( produc4772849988445941548_b_b_o
@ ^ [A7: b,B6: b] :
( P
& ( Q @ A7 @ B6 ) ) )
= ( ^ [Ab: product_prod_b_b] :
( P
& ( produc4772849988445941548_b_b_o @ Q @ Ab ) ) ) ) ).
% split_part
thf(fact_404_split__part,axiom,
! [P: $o,Q: a > nat > $o] :
( ( produc3680711911437148916_nat_o
@ ^ [A7: a,B6: nat] :
( P
& ( Q @ A7 @ B6 ) ) )
= ( ^ [Ab: product_prod_a_nat] :
( P
& ( produc3680711911437148916_nat_o @ Q @ Ab ) ) ) ) ).
% split_part
thf(fact_405_case__prod__conv,axiom,
! [F2: term_a_b > term_a_b > set_b,A: term_a_b,B: term_a_b] :
( ( produc5762022541116952329_set_b @ F2 @ ( produc7020197800436672577rm_a_b @ A @ B ) )
= ( F2 @ A @ B ) ) ).
% case_prod_conv
thf(fact_406_case__prod__conv,axiom,
! [F2: term_a_b > term_a_b > produc357393685978478089rm_a_b > $o,A: term_a_b,B: term_a_b] :
( ( produc4352215818379867658_a_b_o @ F2 @ ( produc7020197800436672577rm_a_b @ A @ B ) )
= ( F2 @ A @ B ) ) ).
% case_prod_conv
thf(fact_407_case__prod__conv,axiom,
! [F2: set_term_a_b > set_term_a_b > $o,A: set_term_a_b,B: set_term_a_b] :
( ( produc4628319930418777942_a_b_o @ F2 @ ( produc8517751122659631533rm_a_b @ A @ B ) )
= ( F2 @ A @ B ) ) ).
% case_prod_conv
thf(fact_408_case__prod__conv,axiom,
! [F2: b > b > $o,A: b,B: b] :
( ( produc4772849988445941548_b_b_o @ F2 @ ( product_Pair_b_b @ A @ B ) )
= ( F2 @ A @ B ) ) ).
% case_prod_conv
thf(fact_409_case__prod__conv,axiom,
! [F2: a > nat > $o,A: a,B: nat] :
( ( produc3680711911437148916_nat_o @ F2 @ ( product_Pair_a_nat @ A @ B ) )
= ( F2 @ A @ B ) ) ).
% case_prod_conv
thf(fact_410_case__prodI2,axiom,
! [P2: produc4708774622424448987_a_nat,C2: product_prod_a_nat > product_prod_a_nat > $o] :
( ! [A4: product_prod_a_nat,B3: product_prod_a_nat] :
( ( P2
= ( produc2026711137822539155_a_nat @ A4 @ B3 ) )
=> ( C2 @ A4 @ B3 ) )
=> ( produc292522255390808560_nat_o @ C2 @ P2 ) ) ).
% case_prodI2
thf(fact_411_case__prodI2,axiom,
! [P2: produc3856293509978307953rm_a_b,C2: b > produc357393685978478089rm_a_b > $o] :
( ! [A4: b,B3: produc357393685978478089rm_a_b] :
( ( P2
= ( produc2673992483936284769rm_a_b @ A4 @ B3 ) )
=> ( C2 @ A4 @ B3 ) )
=> ( produc8008124475385245602_a_b_o @ C2 @ P2 ) ) ).
% case_prodI2
thf(fact_412_case__prodI2,axiom,
! [P2: produc9001763001631046773rm_a_b,C2: set_term_a_b > set_term_a_b > $o] :
( ! [A4: set_term_a_b,B3: set_term_a_b] :
( ( P2
= ( produc8517751122659631533rm_a_b @ A4 @ B3 ) )
=> ( C2 @ A4 @ B3 ) )
=> ( produc4628319930418777942_a_b_o @ C2 @ P2 ) ) ).
% case_prodI2
thf(fact_413_case__prodI2,axiom,
! [P2: product_prod_b_b,C2: b > b > $o] :
( ! [A4: b,B3: b] :
( ( P2
= ( product_Pair_b_b @ A4 @ B3 ) )
=> ( C2 @ A4 @ B3 ) )
=> ( produc4772849988445941548_b_b_o @ C2 @ P2 ) ) ).
% case_prodI2
thf(fact_414_case__prodI2,axiom,
! [P2: product_prod_a_nat,C2: a > nat > $o] :
( ! [A4: a,B3: nat] :
( ( P2
= ( product_Pair_a_nat @ A4 @ B3 ) )
=> ( C2 @ A4 @ B3 ) )
=> ( produc3680711911437148916_nat_o @ C2 @ P2 ) ) ).
% case_prodI2
thf(fact_415_case__prodI2,axiom,
! [P2: produc357393685978478089rm_a_b,C2: term_a_b > term_a_b > $o] :
( ! [A4: term_a_b,B3: term_a_b] :
( ( P2
= ( produc7020197800436672577rm_a_b @ A4 @ B3 ) )
=> ( C2 @ A4 @ B3 ) )
=> ( produc6251846659952913346_a_b_o @ C2 @ P2 ) ) ).
% case_prodI2
thf(fact_416_case__prodI,axiom,
! [F2: product_prod_a_nat > product_prod_a_nat > $o,A: product_prod_a_nat,B: product_prod_a_nat] :
( ( F2 @ A @ B )
=> ( produc292522255390808560_nat_o @ F2 @ ( produc2026711137822539155_a_nat @ A @ B ) ) ) ).
% case_prodI
thf(fact_417_case__prodI,axiom,
! [F2: b > produc357393685978478089rm_a_b > $o,A: b,B: produc357393685978478089rm_a_b] :
( ( F2 @ A @ B )
=> ( produc8008124475385245602_a_b_o @ F2 @ ( produc2673992483936284769rm_a_b @ A @ B ) ) ) ).
% case_prodI
thf(fact_418_case__prodI,axiom,
! [F2: set_term_a_b > set_term_a_b > $o,A: set_term_a_b,B: set_term_a_b] :
( ( F2 @ A @ B )
=> ( produc4628319930418777942_a_b_o @ F2 @ ( produc8517751122659631533rm_a_b @ A @ B ) ) ) ).
% case_prodI
thf(fact_419_case__prodI,axiom,
! [F2: b > b > $o,A: b,B: b] :
( ( F2 @ A @ B )
=> ( produc4772849988445941548_b_b_o @ F2 @ ( product_Pair_b_b @ A @ B ) ) ) ).
% case_prodI
thf(fact_420_case__prodI,axiom,
! [F2: a > nat > $o,A: a,B: nat] :
( ( F2 @ A @ B )
=> ( produc3680711911437148916_nat_o @ F2 @ ( product_Pair_a_nat @ A @ B ) ) ) ).
% case_prodI
thf(fact_421_case__prodI,axiom,
! [F2: term_a_b > term_a_b > $o,A: term_a_b,B: term_a_b] :
( ( F2 @ A @ B )
=> ( produc6251846659952913346_a_b_o @ F2 @ ( produc7020197800436672577rm_a_b @ A @ B ) ) ) ).
% case_prodI
thf(fact_422_rsteps__eq__srsteps__eqI,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,F: set_Pr4934435412358123699_a_nat,S: term_a_b,T: term_a_b] :
( ( ord_le8666007276011122963_a_nat @ ( terms_7988297476397195622_a_b_b @ R2 ) @ F )
=> ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ S ) @ F )
=> ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ T ) @ F )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( transi7742714808557438673rm_a_b @ ( rstep_a_b @ R2 ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R2 ) ) ) ) ) ) ) ) ).
% rsteps_eq_srsteps_eqI
thf(fact_423_rsteps__srstepsI,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,F: set_Pr4934435412358123699_a_nat,S: term_a_b,T: term_a_b] :
( ( ord_le8666007276011122963_a_nat @ ( terms_7988297476397195622_a_b_b @ R2 ) @ F )
=> ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ S ) @ F )
=> ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ T ) @ F )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( transi7922773638565587891rm_a_b @ ( rstep_a_b @ R2 ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( transi7922773638565587891rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R2 ) ) ) ) ) ) ) ) ).
% rsteps_srstepsI
thf(fact_424_predicate1D,axiom,
! [P: produc357393685978478089rm_a_b > $o,Q: produc357393685978478089rm_a_b > $o,X: produc357393685978478089rm_a_b] :
( ( ord_le6645574462093738516_a_b_o @ P @ Q )
=> ( ( P @ X )
=> ( Q @ X ) ) ) ).
% predicate1D
thf(fact_425_predicate1D,axiom,
! [P: product_prod_a_nat > $o,Q: product_prod_a_nat > $o,X: product_prod_a_nat] :
( ( ord_le8950881081082477362_nat_o @ P @ Q )
=> ( ( P @ X )
=> ( Q @ X ) ) ) ).
% predicate1D
thf(fact_426_predicate1D,axiom,
! [P: b > $o,Q: b > $o,X: b] :
( ( ord_less_eq_b_o @ P @ Q )
=> ( ( P @ X )
=> ( Q @ X ) ) ) ).
% predicate1D
thf(fact_427_predicate2D,axiom,
! [P: term_a_b > term_a_b > $o,Q: term_a_b > term_a_b > $o,X: term_a_b,Y: term_a_b] :
( ( ord_le8627286948385211390_a_b_o @ P @ Q )
=> ( ( P @ X @ Y )
=> ( Q @ X @ Y ) ) ) ).
% predicate2D
thf(fact_428_predicate2D,axiom,
! [P: a > nat > $o,Q: a > nat > $o,X: a,Y: nat] :
( ( ord_less_eq_a_nat_o @ P @ Q )
=> ( ( P @ X @ Y )
=> ( Q @ X @ Y ) ) ) ).
% predicate2D
thf(fact_429_rev__predicate1D,axiom,
! [P: produc357393685978478089rm_a_b > $o,X: produc357393685978478089rm_a_b,Q: produc357393685978478089rm_a_b > $o] :
( ( P @ X )
=> ( ( ord_le6645574462093738516_a_b_o @ P @ Q )
=> ( Q @ X ) ) ) ).
% rev_predicate1D
thf(fact_430_rev__predicate1D,axiom,
! [P: product_prod_a_nat > $o,X: product_prod_a_nat,Q: product_prod_a_nat > $o] :
( ( P @ X )
=> ( ( ord_le8950881081082477362_nat_o @ P @ Q )
=> ( Q @ X ) ) ) ).
% rev_predicate1D
thf(fact_431_rev__predicate1D,axiom,
! [P: b > $o,X: b,Q: b > $o] :
( ( P @ X )
=> ( ( ord_less_eq_b_o @ P @ Q )
=> ( Q @ X ) ) ) ).
% rev_predicate1D
thf(fact_432_rev__predicate2D,axiom,
! [P: term_a_b > term_a_b > $o,X: term_a_b,Y: term_a_b,Q: term_a_b > term_a_b > $o] :
( ( P @ X @ Y )
=> ( ( ord_le8627286948385211390_a_b_o @ P @ Q )
=> ( Q @ X @ Y ) ) ) ).
% rev_predicate2D
thf(fact_433_rev__predicate2D,axiom,
! [P: a > nat > $o,X: a,Y: nat,Q: a > nat > $o] :
( ( P @ X @ Y )
=> ( ( ord_less_eq_a_nat_o @ P @ Q )
=> ( Q @ X @ Y ) ) ) ).
% rev_predicate2D
thf(fact_434_internal__case__prod__def,axiom,
produc2340987516086830771_set_b = produc5762022541116952329_set_b ).
% internal_case_prod_def
thf(fact_435_internal__case__prod__def,axiom,
produc6297341409162782176_a_b_o = produc4352215818379867658_a_b_o ).
% internal_case_prod_def
thf(fact_436_internal__case__prod__def,axiom,
produc2281327766331315200_a_b_o = produc4628319930418777942_a_b_o ).
% internal_case_prod_def
thf(fact_437_internal__case__prod__def,axiom,
produc5852870737868468438_b_b_o = produc4772849988445941548_b_b_o ).
% internal_case_prod_def
thf(fact_438_internal__case__prod__def,axiom,
produc4802101623976553162_nat_o = produc3680711911437148916_nat_o ).
% internal_case_prod_def
thf(fact_439_prod_Odisc__eq__case,axiom,
! [Prod: produc9001763001631046773rm_a_b] :
( produc4628319930418777942_a_b_o
@ ^ [Uu: set_term_a_b,Uv: set_term_a_b] : $true
@ Prod ) ).
% prod.disc_eq_case
thf(fact_440_prod_Odisc__eq__case,axiom,
! [Prod: product_prod_b_b] :
( produc4772849988445941548_b_b_o
@ ^ [Uu: b,Uv: b] : $true
@ Prod ) ).
% prod.disc_eq_case
thf(fact_441_prod_Odisc__eq__case,axiom,
! [Prod: product_prod_a_nat] :
( produc3680711911437148916_nat_o
@ ^ [Uu: a,Uv: nat] : $true
@ Prod ) ).
% prod.disc_eq_case
thf(fact_442_prod_Ocase__distrib,axiom,
! [H: set_b > set_b,F2: term_a_b > term_a_b > set_b,Prod: produc357393685978478089rm_a_b] :
( ( H @ ( produc5762022541116952329_set_b @ F2 @ Prod ) )
= ( produc5762022541116952329_set_b
@ ^ [X12: term_a_b,X23: term_a_b] : ( H @ ( F2 @ X12 @ X23 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_443_prod_Ocase__distrib,axiom,
! [H: set_b > produc357393685978478089rm_a_b > $o,F2: term_a_b > term_a_b > set_b,Prod: produc357393685978478089rm_a_b] :
( ( H @ ( produc5762022541116952329_set_b @ F2 @ Prod ) )
= ( produc4352215818379867658_a_b_o
@ ^ [X12: term_a_b,X23: term_a_b] : ( H @ ( F2 @ X12 @ X23 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_444_prod_Ocase__distrib,axiom,
! [H: ( produc357393685978478089rm_a_b > $o ) > set_b,F2: term_a_b > term_a_b > produc357393685978478089rm_a_b > $o,Prod: produc357393685978478089rm_a_b] :
( ( H @ ( produc4352215818379867658_a_b_o @ F2 @ Prod ) )
= ( produc5762022541116952329_set_b
@ ^ [X12: term_a_b,X23: term_a_b] : ( H @ ( F2 @ X12 @ X23 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_445_prod_Ocase__distrib,axiom,
! [H: ( produc357393685978478089rm_a_b > $o ) > produc357393685978478089rm_a_b > $o,F2: term_a_b > term_a_b > produc357393685978478089rm_a_b > $o,Prod: produc357393685978478089rm_a_b] :
( ( H @ ( produc4352215818379867658_a_b_o @ F2 @ Prod ) )
= ( produc4352215818379867658_a_b_o
@ ^ [X12: term_a_b,X23: term_a_b] : ( H @ ( F2 @ X12 @ X23 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_446_prod_Ocase__distrib,axiom,
! [H: $o > $o,F2: set_term_a_b > set_term_a_b > $o,Prod: produc9001763001631046773rm_a_b] :
( ( H @ ( produc4628319930418777942_a_b_o @ F2 @ Prod ) )
= ( produc4628319930418777942_a_b_o
@ ^ [X12: set_term_a_b,X23: set_term_a_b] : ( H @ ( F2 @ X12 @ X23 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_447_prod_Ocase__distrib,axiom,
! [H: $o > $o,F2: b > b > $o,Prod: product_prod_b_b] :
( ( H @ ( produc4772849988445941548_b_b_o @ F2 @ Prod ) )
= ( produc4772849988445941548_b_b_o
@ ^ [X12: b,X23: b] : ( H @ ( F2 @ X12 @ X23 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_448_prod_Ocase__distrib,axiom,
! [H: $o > $o,F2: a > nat > $o,Prod: product_prod_a_nat] :
( ( H @ ( produc3680711911437148916_nat_o @ F2 @ Prod ) )
= ( produc3680711911437148916_nat_o
@ ^ [X12: a,X23: nat] : ( H @ ( F2 @ X12 @ X23 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_449_Collect__case__prod__mono,axiom,
! [A5: term_a_b > term_a_b > $o,B4: term_a_b > term_a_b > $o] :
( ( ord_le8627286948385211390_a_b_o @ A5 @ B4 )
=> ( ord_le118470702582115849rm_a_b @ ( collec99177395925924084rm_a_b @ ( produc6251846659952913346_a_b_o @ A5 ) ) @ ( collec99177395925924084rm_a_b @ ( produc6251846659952913346_a_b_o @ B4 ) ) ) ) ).
% Collect_case_prod_mono
thf(fact_450_Collect__case__prod__mono,axiom,
! [A5: set_term_a_b > set_term_a_b > $o,B4: set_term_a_b > set_term_a_b > $o] :
( ( ord_le6917259649888057234_a_b_o @ A5 @ B4 )
=> ( ord_le2663425029446080117rm_a_b @ ( collec4780413679787665248rm_a_b @ ( produc4628319930418777942_a_b_o @ A5 ) ) @ ( collec4780413679787665248rm_a_b @ ( produc4628319930418777942_a_b_o @ B4 ) ) ) ) ).
% Collect_case_prod_mono
thf(fact_451_Collect__case__prod__mono,axiom,
! [A5: b > b > $o,B4: b > b > $o] :
( ( ord_less_eq_b_b_o @ A5 @ B4 )
=> ( ord_le182087997850975847od_b_b @ ( collec548942219715005266od_b_b @ ( produc4772849988445941548_b_b_o @ A5 ) ) @ ( collec548942219715005266od_b_b @ ( produc4772849988445941548_b_b_o @ B4 ) ) ) ) ).
% Collect_case_prod_mono
thf(fact_452_Collect__case__prod__mono,axiom,
! [A5: a > nat > $o,B4: a > nat > $o] :
( ( ord_less_eq_a_nat_o @ A5 @ B4 )
=> ( ord_le8666007276011122963_a_nat @ ( collec4464134535221767506_a_nat @ ( produc3680711911437148916_nat_o @ A5 ) ) @ ( collec4464134535221767506_a_nat @ ( produc3680711911437148916_nat_o @ B4 ) ) ) ) ).
% Collect_case_prod_mono
thf(fact_453_old_Oprod_Ocase,axiom,
! [F2: term_a_b > term_a_b > set_b,X1: term_a_b,X22: term_a_b] :
( ( produc5762022541116952329_set_b @ F2 @ ( produc7020197800436672577rm_a_b @ X1 @ X22 ) )
= ( F2 @ X1 @ X22 ) ) ).
% old.prod.case
thf(fact_454_old_Oprod_Ocase,axiom,
! [F2: term_a_b > term_a_b > produc357393685978478089rm_a_b > $o,X1: term_a_b,X22: term_a_b] :
( ( produc4352215818379867658_a_b_o @ F2 @ ( produc7020197800436672577rm_a_b @ X1 @ X22 ) )
= ( F2 @ X1 @ X22 ) ) ).
% old.prod.case
thf(fact_455_old_Oprod_Ocase,axiom,
! [F2: set_term_a_b > set_term_a_b > $o,X1: set_term_a_b,X22: set_term_a_b] :
( ( produc4628319930418777942_a_b_o @ F2 @ ( produc8517751122659631533rm_a_b @ X1 @ X22 ) )
= ( F2 @ X1 @ X22 ) ) ).
% old.prod.case
thf(fact_456_old_Oprod_Ocase,axiom,
! [F2: b > b > $o,X1: b,X22: b] :
( ( produc4772849988445941548_b_b_o @ F2 @ ( product_Pair_b_b @ X1 @ X22 ) )
= ( F2 @ X1 @ X22 ) ) ).
% old.prod.case
thf(fact_457_old_Oprod_Ocase,axiom,
! [F2: a > nat > $o,X1: a,X22: nat] :
( ( produc3680711911437148916_nat_o @ F2 @ ( product_Pair_a_nat @ X1 @ X22 ) )
= ( F2 @ X1 @ X22 ) ) ).
% old.prod.case
thf(fact_458_case__prodE,axiom,
! [C2: product_prod_a_nat > product_prod_a_nat > $o,P2: produc4708774622424448987_a_nat] :
( ( produc292522255390808560_nat_o @ C2 @ P2 )
=> ~ ! [X3: product_prod_a_nat,Y3: product_prod_a_nat] :
( ( P2
= ( produc2026711137822539155_a_nat @ X3 @ Y3 ) )
=> ~ ( C2 @ X3 @ Y3 ) ) ) ).
% case_prodE
thf(fact_459_case__prodE,axiom,
! [C2: b > produc357393685978478089rm_a_b > $o,P2: produc3856293509978307953rm_a_b] :
( ( produc8008124475385245602_a_b_o @ C2 @ P2 )
=> ~ ! [X3: b,Y3: produc357393685978478089rm_a_b] :
( ( P2
= ( produc2673992483936284769rm_a_b @ X3 @ Y3 ) )
=> ~ ( C2 @ X3 @ Y3 ) ) ) ).
% case_prodE
thf(fact_460_case__prodE,axiom,
! [C2: set_term_a_b > set_term_a_b > $o,P2: produc9001763001631046773rm_a_b] :
( ( produc4628319930418777942_a_b_o @ C2 @ P2 )
=> ~ ! [X3: set_term_a_b,Y3: set_term_a_b] :
( ( P2
= ( produc8517751122659631533rm_a_b @ X3 @ Y3 ) )
=> ~ ( C2 @ X3 @ Y3 ) ) ) ).
% case_prodE
thf(fact_461_case__prodE,axiom,
! [C2: b > b > $o,P2: product_prod_b_b] :
( ( produc4772849988445941548_b_b_o @ C2 @ P2 )
=> ~ ! [X3: b,Y3: b] :
( ( P2
= ( product_Pair_b_b @ X3 @ Y3 ) )
=> ~ ( C2 @ X3 @ Y3 ) ) ) ).
% case_prodE
thf(fact_462_case__prodE,axiom,
! [C2: a > nat > $o,P2: product_prod_a_nat] :
( ( produc3680711911437148916_nat_o @ C2 @ P2 )
=> ~ ! [X3: a,Y3: nat] :
( ( P2
= ( product_Pair_a_nat @ X3 @ Y3 ) )
=> ~ ( C2 @ X3 @ Y3 ) ) ) ).
% case_prodE
thf(fact_463_case__prodE,axiom,
! [C2: term_a_b > term_a_b > $o,P2: produc357393685978478089rm_a_b] :
( ( produc6251846659952913346_a_b_o @ C2 @ P2 )
=> ~ ! [X3: term_a_b,Y3: term_a_b] :
( ( P2
= ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) )
=> ~ ( C2 @ X3 @ Y3 ) ) ) ).
% case_prodE
thf(fact_464_case__prodD,axiom,
! [F2: product_prod_a_nat > product_prod_a_nat > $o,A: product_prod_a_nat,B: product_prod_a_nat] :
( ( produc292522255390808560_nat_o @ F2 @ ( produc2026711137822539155_a_nat @ A @ B ) )
=> ( F2 @ A @ B ) ) ).
% case_prodD
thf(fact_465_case__prodD,axiom,
! [F2: b > produc357393685978478089rm_a_b > $o,A: b,B: produc357393685978478089rm_a_b] :
( ( produc8008124475385245602_a_b_o @ F2 @ ( produc2673992483936284769rm_a_b @ A @ B ) )
=> ( F2 @ A @ B ) ) ).
% case_prodD
thf(fact_466_case__prodD,axiom,
! [F2: set_term_a_b > set_term_a_b > $o,A: set_term_a_b,B: set_term_a_b] :
( ( produc4628319930418777942_a_b_o @ F2 @ ( produc8517751122659631533rm_a_b @ A @ B ) )
=> ( F2 @ A @ B ) ) ).
% case_prodD
thf(fact_467_case__prodD,axiom,
! [F2: b > b > $o,A: b,B: b] :
( ( produc4772849988445941548_b_b_o @ F2 @ ( product_Pair_b_b @ A @ B ) )
=> ( F2 @ A @ B ) ) ).
% case_prodD
thf(fact_468_case__prodD,axiom,
! [F2: a > nat > $o,A: a,B: nat] :
( ( produc3680711911437148916_nat_o @ F2 @ ( product_Pair_a_nat @ A @ B ) )
=> ( F2 @ A @ B ) ) ).
% case_prodD
thf(fact_469_case__prodD,axiom,
! [F2: term_a_b > term_a_b > $o,A: term_a_b,B: term_a_b] :
( ( produc6251846659952913346_a_b_o @ F2 @ ( produc7020197800436672577rm_a_b @ A @ B ) )
=> ( F2 @ A @ B ) ) ).
% case_prodD
thf(fact_470_cond__case__prod__eta,axiom,
! [F2: term_a_b > term_a_b > set_b,G3: produc357393685978478089rm_a_b > set_b] :
( ! [X3: term_a_b,Y3: term_a_b] :
( ( F2 @ X3 @ Y3 )
= ( G3 @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) ) )
=> ( ( produc5762022541116952329_set_b @ F2 )
= G3 ) ) ).
% cond_case_prod_eta
thf(fact_471_cond__case__prod__eta,axiom,
! [F2: term_a_b > term_a_b > produc357393685978478089rm_a_b > $o,G3: produc357393685978478089rm_a_b > produc357393685978478089rm_a_b > $o] :
( ! [X3: term_a_b,Y3: term_a_b] :
( ( F2 @ X3 @ Y3 )
= ( G3 @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) ) )
=> ( ( produc4352215818379867658_a_b_o @ F2 )
= G3 ) ) ).
% cond_case_prod_eta
thf(fact_472_cond__case__prod__eta,axiom,
! [F2: set_term_a_b > set_term_a_b > $o,G3: produc9001763001631046773rm_a_b > $o] :
( ! [X3: set_term_a_b,Y3: set_term_a_b] :
( ( F2 @ X3 @ Y3 )
= ( G3 @ ( produc8517751122659631533rm_a_b @ X3 @ Y3 ) ) )
=> ( ( produc4628319930418777942_a_b_o @ F2 )
= G3 ) ) ).
% cond_case_prod_eta
thf(fact_473_cond__case__prod__eta,axiom,
! [F2: b > b > $o,G3: product_prod_b_b > $o] :
( ! [X3: b,Y3: b] :
( ( F2 @ X3 @ Y3 )
= ( G3 @ ( product_Pair_b_b @ X3 @ Y3 ) ) )
=> ( ( produc4772849988445941548_b_b_o @ F2 )
= G3 ) ) ).
% cond_case_prod_eta
thf(fact_474_cond__case__prod__eta,axiom,
! [F2: a > nat > $o,G3: product_prod_a_nat > $o] :
( ! [X3: a,Y3: nat] :
( ( F2 @ X3 @ Y3 )
= ( G3 @ ( product_Pair_a_nat @ X3 @ Y3 ) ) )
=> ( ( produc3680711911437148916_nat_o @ F2 )
= G3 ) ) ).
% cond_case_prod_eta
thf(fact_475_case__prod__eta,axiom,
! [F2: produc357393685978478089rm_a_b > set_b] :
( ( produc5762022541116952329_set_b
@ ^ [X2: term_a_b,Y4: term_a_b] : ( F2 @ ( produc7020197800436672577rm_a_b @ X2 @ Y4 ) ) )
= F2 ) ).
% case_prod_eta
thf(fact_476_case__prod__eta,axiom,
! [F2: produc357393685978478089rm_a_b > produc357393685978478089rm_a_b > $o] :
( ( produc4352215818379867658_a_b_o
@ ^ [X2: term_a_b,Y4: term_a_b] : ( F2 @ ( produc7020197800436672577rm_a_b @ X2 @ Y4 ) ) )
= F2 ) ).
% case_prod_eta
thf(fact_477_case__prod__eta,axiom,
! [F2: produc9001763001631046773rm_a_b > $o] :
( ( produc4628319930418777942_a_b_o
@ ^ [X2: set_term_a_b,Y4: set_term_a_b] : ( F2 @ ( produc8517751122659631533rm_a_b @ X2 @ Y4 ) ) )
= F2 ) ).
% case_prod_eta
thf(fact_478_case__prod__eta,axiom,
! [F2: product_prod_b_b > $o] :
( ( produc4772849988445941548_b_b_o
@ ^ [X2: b,Y4: b] : ( F2 @ ( product_Pair_b_b @ X2 @ Y4 ) ) )
= F2 ) ).
% case_prod_eta
thf(fact_479_case__prod__eta,axiom,
! [F2: product_prod_a_nat > $o] :
( ( produc3680711911437148916_nat_o
@ ^ [X2: a,Y4: nat] : ( F2 @ ( product_Pair_a_nat @ X2 @ Y4 ) ) )
= F2 ) ).
% case_prod_eta
thf(fact_480_case__prodE2,axiom,
! [Q: set_b > $o,P: term_a_b > term_a_b > set_b,Z2: produc357393685978478089rm_a_b] :
( ( Q @ ( produc5762022541116952329_set_b @ P @ Z2 ) )
=> ~ ! [X3: term_a_b,Y3: term_a_b] :
( ( Z2
= ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) )
=> ~ ( Q @ ( P @ X3 @ Y3 ) ) ) ) ).
% case_prodE2
thf(fact_481_case__prodE2,axiom,
! [Q: ( produc357393685978478089rm_a_b > $o ) > $o,P: term_a_b > term_a_b > produc357393685978478089rm_a_b > $o,Z2: produc357393685978478089rm_a_b] :
( ( Q @ ( produc4352215818379867658_a_b_o @ P @ Z2 ) )
=> ~ ! [X3: term_a_b,Y3: term_a_b] :
( ( Z2
= ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) )
=> ~ ( Q @ ( P @ X3 @ Y3 ) ) ) ) ).
% case_prodE2
thf(fact_482_case__prodE2,axiom,
! [Q: $o > $o,P: set_term_a_b > set_term_a_b > $o,Z2: produc9001763001631046773rm_a_b] :
( ( Q @ ( produc4628319930418777942_a_b_o @ P @ Z2 ) )
=> ~ ! [X3: set_term_a_b,Y3: set_term_a_b] :
( ( Z2
= ( produc8517751122659631533rm_a_b @ X3 @ Y3 ) )
=> ~ ( Q @ ( P @ X3 @ Y3 ) ) ) ) ).
% case_prodE2
thf(fact_483_case__prodE2,axiom,
! [Q: $o > $o,P: b > b > $o,Z2: product_prod_b_b] :
( ( Q @ ( produc4772849988445941548_b_b_o @ P @ Z2 ) )
=> ~ ! [X3: b,Y3: b] :
( ( Z2
= ( product_Pair_b_b @ X3 @ Y3 ) )
=> ~ ( Q @ ( P @ X3 @ Y3 ) ) ) ) ).
% case_prodE2
thf(fact_484_case__prodE2,axiom,
! [Q: $o > $o,P: a > nat > $o,Z2: product_prod_a_nat] :
( ( Q @ ( produc3680711911437148916_nat_o @ P @ Z2 ) )
=> ~ ! [X3: a,Y3: nat] :
( ( Z2
= ( product_Pair_a_nat @ X3 @ Y3 ) )
=> ~ ( Q @ ( P @ X3 @ Y3 ) ) ) ) ).
% case_prodE2
thf(fact_485_rstep__def,axiom,
( rstep_a_b
= ( ^ [R6: set_Pr4386577575007340137rm_a_b] :
( collec99177395925924084rm_a_b
@ ( produc6251846659952913346_a_b_o
@ ( rstepp_a_b
@ ^ [X2: term_a_b,Y4: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y4 ) @ R6 ) ) ) ) ) ) ).
% rstep_def
thf(fact_486_sig__step__rsteps__dist,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,F: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ ( terms_7988297476397195622_a_b_b @ R2 ) @ F )
=> ( ( sig_step_a_b @ F @ ( transi7922773638565587891rm_a_b @ ( rstep_a_b @ R2 ) ) )
= ( transi7922773638565587891rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R2 ) ) ) ) ) ).
% sig_step_rsteps_dist
thf(fact_487_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_488_ssubst__Pair__rhs,axiom,
! [R: a,S: nat,R5: set_Pr4934435412358123699_a_nat,S3: nat] :
( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ R @ S ) @ R5 )
=> ( ( S3 = S )
=> ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ R @ S3 ) @ R5 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_489_ssubst__Pair__rhs,axiom,
! [R: term_a_b,S: term_a_b,R5: set_Pr4386577575007340137rm_a_b,S3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ R @ S ) @ R5 )
=> ( ( S3 = S )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ R @ S3 ) @ R5 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_490_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_491_tranclD,axiom,
! [X: term_a_b,Y: term_a_b,R5: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7922773638565587891rm_a_b @ R5 ) )
=> ? [Z4: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z4 ) @ R5 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Z4 @ Y ) @ ( transi7742714808557438673rm_a_b @ R5 ) ) ) ) ).
% tranclD
thf(fact_492_rtranclD,axiom,
! [A: term_a_b,B: term_a_b,R5: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ R5 ) )
=> ( ( A = B )
| ( ( A != B )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7922773638565587891rm_a_b @ R5 ) ) ) ) ) ).
% rtranclD
thf(fact_493_tranclD2,axiom,
! [X: term_a_b,Y: term_a_b,R5: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7922773638565587891rm_a_b @ R5 ) )
=> ? [Z4: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z4 ) @ ( transi7742714808557438673rm_a_b @ R5 ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Z4 @ Y ) @ R5 ) ) ) ).
% tranclD2
thf(fact_494_trancl__into__rtrancl,axiom,
! [A: term_a_b,B: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7922773638565587891rm_a_b @ R ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ R ) ) ) ).
% trancl_into_rtrancl
thf(fact_495_rtrancl__eq__or__trancl,axiom,
! [X: term_a_b,Y: term_a_b,R5: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ R5 ) )
= ( ( X = Y )
| ( ( X != Y )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7922773638565587891rm_a_b @ R5 ) ) ) ) ) ).
% rtrancl_eq_or_trancl
thf(fact_496_rtrancl__into__trancl1,axiom,
! [A: term_a_b,B: term_a_b,R: set_Pr4386577575007340137rm_a_b,C2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C2 ) @ R )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C2 ) @ ( transi7922773638565587891rm_a_b @ R ) ) ) ) ).
% rtrancl_into_trancl1
thf(fact_497_rtrancl__into__trancl2,axiom,
! [A: term_a_b,B: term_a_b,R: set_Pr4386577575007340137rm_a_b,C2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C2 ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C2 ) @ ( transi7922773638565587891rm_a_b @ R ) ) ) ) ).
% rtrancl_into_trancl2
thf(fact_498_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_499_mem__case__prodI,axiom,
! [Z2: produc357393685978478089rm_a_b,C2: term_a_b > term_a_b > set_Pr4386577575007340137rm_a_b,A: term_a_b,B: term_a_b] :
( ( member5869715511025134514rm_a_b @ Z2 @ ( C2 @ A @ B ) )
=> ( member5869715511025134514rm_a_b @ Z2 @ ( produc782875466020721299rm_a_b @ C2 @ ( produc7020197800436672577rm_a_b @ A @ B ) ) ) ) ).
% mem_case_prodI
thf(fact_500_mem__case__prodI,axiom,
! [Z2: b,C2: term_a_b > term_a_b > set_b,A: term_a_b,B: term_a_b] :
( ( member_b @ Z2 @ ( C2 @ A @ B ) )
=> ( member_b @ Z2 @ ( produc5762022541116952329_set_b @ C2 @ ( produc7020197800436672577rm_a_b @ A @ B ) ) ) ) ).
% mem_case_prodI
thf(fact_501_mem__case__prodI2,axiom,
! [P2: produc357393685978478089rm_a_b,Z2: produc357393685978478089rm_a_b,C2: term_a_b > term_a_b > set_Pr4386577575007340137rm_a_b] :
( ! [A4: term_a_b,B3: term_a_b] :
( ( P2
= ( produc7020197800436672577rm_a_b @ A4 @ B3 ) )
=> ( member5869715511025134514rm_a_b @ Z2 @ ( C2 @ A4 @ B3 ) ) )
=> ( member5869715511025134514rm_a_b @ Z2 @ ( produc782875466020721299rm_a_b @ C2 @ P2 ) ) ) ).
% mem_case_prodI2
thf(fact_502_mem__case__prodI2,axiom,
! [P2: produc357393685978478089rm_a_b,Z2: b,C2: term_a_b > term_a_b > set_b] :
( ! [A4: term_a_b,B3: term_a_b] :
( ( P2
= ( produc7020197800436672577rm_a_b @ A4 @ B3 ) )
=> ( member_b @ Z2 @ ( C2 @ A4 @ B3 ) ) )
=> ( member_b @ Z2 @ ( produc5762022541116952329_set_b @ C2 @ P2 ) ) ) ).
% mem_case_prodI2
thf(fact_503_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 ) )
=> ( ! [A4: term_a_b,B3: term_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ Ax @ Ay ) @ ( produc7020197800436672577rm_a_b @ A4 @ B3 ) ) @ R )
=> ( P @ A4 @ B3 ) )
=> ( ! [A4: term_a_b,B3: term_a_b,Aa: term_a_b,Ba: term_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ Ax @ Ay ) @ ( produc7020197800436672577rm_a_b @ A4 @ B3 ) ) @ ( transi9211502839322181930rm_a_b @ R ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ A4 @ B3 ) @ ( produc7020197800436672577rm_a_b @ Aa @ Ba ) ) @ R )
=> ( ( P @ A4 @ B3 )
=> ( P @ Aa @ Ba ) ) ) )
=> ( P @ Bx @ By ) ) ) ) ).
% trancl_induct2
thf(fact_504_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 )
=> ( ! [A4: term_a_b,B3: term_a_b,Aa: term_a_b,Ba: term_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ A4 @ B3 ) @ ( produc7020197800436672577rm_a_b @ Aa @ Ba ) ) @ 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 @ A4 @ B3 ) ) ) )
=> ( P @ Ax @ Ay ) ) ) ) ).
% converse_rtrancl_induct2
thf(fact_505_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 ) )
=> ~ ! [A4: term_a_b,B3: term_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ Xa @ Xb ) @ ( produc7020197800436672577rm_a_b @ A4 @ B3 ) ) @ R )
=> ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ A4 @ B3 ) @ ( produc7020197800436672577rm_a_b @ Za @ Zb ) ) @ ( transi2615809358984392588rm_a_b @ R ) ) ) ) ) ).
% converse_rtranclE2
thf(fact_506_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 )
=> ( ! [A4: term_a_b,B3: term_a_b,Aa: term_a_b,Ba: term_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ Ax @ Ay ) @ ( produc7020197800436672577rm_a_b @ A4 @ B3 ) ) @ ( transi2615809358984392588rm_a_b @ R ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ A4 @ B3 ) @ ( produc7020197800436672577rm_a_b @ Aa @ Ba ) ) @ R )
=> ( ( P @ A4 @ B3 )
=> ( P @ Aa @ Ba ) ) ) )
=> ( P @ Bx @ By ) ) ) ) ).
% rtrancl_induct2
thf(fact_507_mem__case__prodE,axiom,
! [Z2: produc357393685978478089rm_a_b,C2: term_a_b > term_a_b > set_Pr4386577575007340137rm_a_b,P2: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ Z2 @ ( produc782875466020721299rm_a_b @ C2 @ P2 ) )
=> ~ ! [X3: term_a_b,Y3: term_a_b] :
( ( P2
= ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) )
=> ~ ( member5869715511025134514rm_a_b @ Z2 @ ( C2 @ X3 @ Y3 ) ) ) ) ).
% mem_case_prodE
thf(fact_508_mem__case__prodE,axiom,
! [Z2: b,C2: term_a_b > term_a_b > set_b,P2: produc357393685978478089rm_a_b] :
( ( member_b @ Z2 @ ( produc5762022541116952329_set_b @ C2 @ P2 ) )
=> ~ ! [X3: term_a_b,Y3: term_a_b] :
( ( P2
= ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) )
=> ~ ( member_b @ Z2 @ ( C2 @ X3 @ Y3 ) ) ) ) ).
% mem_case_prodE
thf(fact_509_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_510_rtrancl_Ocases,axiom,
! [A1: term_a_b,A2: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A2 ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ( ( A2 != A1 )
=> ~ ! [B3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ B3 ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B3 @ A2 ) @ R ) ) ) ) ).
% rtrancl.cases
thf(fact_511_rtrancl_Osimps,axiom,
! [A1: term_a_b,A2: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A2 ) @ ( transi7742714808557438673rm_a_b @ R ) )
= ( ? [A7: term_a_b] :
( ( A1 = A7 )
& ( A2 = A7 ) )
| ? [A7: term_a_b,B6: term_a_b,C4: term_a_b] :
( ( A1 = A7 )
& ( A2 = C4 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A7 @ B6 ) @ ( transi7742714808557438673rm_a_b @ R ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B6 @ C4 ) @ R ) ) ) ) ).
% rtrancl.simps
thf(fact_512_rtrancl_Ortrancl__refl,axiom,
! [A: term_a_b,R: set_Pr4386577575007340137rm_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ A ) @ ( transi7742714808557438673rm_a_b @ R ) ) ).
% rtrancl.rtrancl_refl
thf(fact_513_rtrancl_Ortrancl__into__rtrancl,axiom,
! [A: term_a_b,B: term_a_b,R: set_Pr4386577575007340137rm_a_b,C2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C2 ) @ R )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C2 ) @ ( transi7742714808557438673rm_a_b @ R ) ) ) ) ).
% rtrancl.rtrancl_into_rtrancl
thf(fact_514_rtranclE,axiom,
! [A: term_a_b,B: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ( ( A != B )
=> ~ ! [Y3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ Y3 ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ B ) @ R ) ) ) ) ).
% rtranclE
thf(fact_515_rtrancl__trans,axiom,
! [X: term_a_b,Y: term_a_b,R: set_Pr4386577575007340137rm_a_b,Z2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ Z2 ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z2 ) @ ( transi7742714808557438673rm_a_b @ R ) ) ) ) ).
% rtrancl_trans
thf(fact_516_rtrancl__induct,axiom,
! [A: term_a_b,B: term_a_b,R: set_Pr4386577575007340137rm_a_b,P: term_a_b > $o] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ( ( P @ A )
=> ( ! [Y3: term_a_b,Z4: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ Y3 ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ Z4 ) @ R )
=> ( ( P @ Y3 )
=> ( P @ Z4 ) ) ) )
=> ( P @ B ) ) ) ) ).
% rtrancl_induct
thf(fact_517_converse__rtranclE,axiom,
! [X: term_a_b,Z2: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z2 ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ( ( X != Z2 )
=> ~ ! [Y3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y3 ) @ R )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ Z2 ) @ ( transi7742714808557438673rm_a_b @ R ) ) ) ) ) ).
% converse_rtranclE
thf(fact_518_converse__rtrancl__induct,axiom,
! [A: term_a_b,B: term_a_b,R: set_Pr4386577575007340137rm_a_b,P: term_a_b > $o] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ( ( P @ B )
=> ( ! [Y3: term_a_b,Z4: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ Z4 ) @ R )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Z4 @ B ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ( ( P @ Z4 )
=> ( P @ Y3 ) ) ) )
=> ( P @ A ) ) ) ) ).
% converse_rtrancl_induct
thf(fact_519_converse__rtrancl__into__rtrancl,axiom,
! [A: term_a_b,B: term_a_b,R: set_Pr4386577575007340137rm_a_b,C2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C2 ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C2 ) @ ( transi7742714808557438673rm_a_b @ R ) ) ) ) ).
% converse_rtrancl_into_rtrancl
thf(fact_520_trancl_Ocases,axiom,
! [A1: term_a_b,A2: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A2 ) @ ( transi7922773638565587891rm_a_b @ R ) )
=> ( ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A2 ) @ R )
=> ~ ! [B3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ B3 ) @ ( transi7922773638565587891rm_a_b @ R ) )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B3 @ A2 ) @ R ) ) ) ) ).
% trancl.cases
thf(fact_521_trancl_Osimps,axiom,
! [A1: term_a_b,A2: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A2 ) @ ( transi7922773638565587891rm_a_b @ R ) )
= ( ? [A7: term_a_b,B6: term_a_b] :
( ( A1 = A7 )
& ( A2 = B6 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A7 @ B6 ) @ R ) )
| ? [A7: term_a_b,B6: term_a_b,C4: term_a_b] :
( ( A1 = A7 )
& ( A2 = C4 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A7 @ B6 ) @ ( transi7922773638565587891rm_a_b @ R ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B6 @ C4 ) @ R ) ) ) ) ).
% trancl.simps
thf(fact_522_trancl_Or__into__trancl,axiom,
! [A: term_a_b,B: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7922773638565587891rm_a_b @ R ) ) ) ).
% trancl.r_into_trancl
thf(fact_523_tranclE,axiom,
! [A: term_a_b,B: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7922773638565587891rm_a_b @ R ) )
=> ( ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R )
=> ~ ! [C: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C ) @ ( transi7922773638565587891rm_a_b @ R ) )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ C @ B ) @ R ) ) ) ) ).
% tranclE
thf(fact_524_trancl__trans,axiom,
! [X: term_a_b,Y: term_a_b,R: set_Pr4386577575007340137rm_a_b,Z2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7922773638565587891rm_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ Z2 ) @ ( transi7922773638565587891rm_a_b @ R ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z2 ) @ ( transi7922773638565587891rm_a_b @ R ) ) ) ) ).
% trancl_trans
thf(fact_525_trancl__induct,axiom,
! [A: term_a_b,B: term_a_b,R: set_Pr4386577575007340137rm_a_b,P: term_a_b > $o] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7922773638565587891rm_a_b @ R ) )
=> ( ! [Y3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ Y3 ) @ R )
=> ( P @ Y3 ) )
=> ( ! [Y3: term_a_b,Z4: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ Y3 ) @ ( transi7922773638565587891rm_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ Z4 ) @ R )
=> ( ( P @ Y3 )
=> ( P @ Z4 ) ) ) )
=> ( P @ B ) ) ) ) ).
% trancl_induct
thf(fact_526_r__r__into__trancl,axiom,
! [A: term_a_b,B: term_a_b,R5: set_Pr4386577575007340137rm_a_b,C2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R5 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C2 ) @ R5 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C2 ) @ ( transi7922773638565587891rm_a_b @ R5 ) ) ) ) ).
% r_r_into_trancl
thf(fact_527_converse__tranclE,axiom,
! [X: term_a_b,Z2: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z2 ) @ ( transi7922773638565587891rm_a_b @ R ) )
=> ( ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z2 ) @ R )
=> ~ ! [Y3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y3 ) @ R )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ Z2 ) @ ( transi7922773638565587891rm_a_b @ R ) ) ) ) ) ).
% converse_tranclE
thf(fact_528_irrefl__trancl__rD,axiom,
! [R: set_Pr4386577575007340137rm_a_b,X: term_a_b,Y: term_a_b] :
( ! [X3: term_a_b] :
~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ X3 ) @ ( transi7922773638565587891rm_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ R )
=> ( X != Y ) ) ) ).
% irrefl_trancl_rD
thf(fact_529_Transitive__Closure_Otrancl__into__trancl,axiom,
! [A: term_a_b,B: term_a_b,R: set_Pr4386577575007340137rm_a_b,C2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7922773638565587891rm_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C2 ) @ R )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C2 ) @ ( transi7922773638565587891rm_a_b @ R ) ) ) ) ).
% Transitive_Closure.trancl_into_trancl
thf(fact_530_trancl__into__trancl2,axiom,
! [A: term_a_b,B: term_a_b,R: set_Pr4386577575007340137rm_a_b,C2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C2 ) @ ( transi7922773638565587891rm_a_b @ R ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C2 ) @ ( transi7922773638565587891rm_a_b @ R ) ) ) ) ).
% trancl_into_trancl2
thf(fact_531_trancl__trans__induct,axiom,
! [X: term_a_b,Y: term_a_b,R: set_Pr4386577575007340137rm_a_b,P: term_a_b > term_a_b > $o] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7922773638565587891rm_a_b @ R ) )
=> ( ! [X3: term_a_b,Y3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ R )
=> ( P @ X3 @ Y3 ) )
=> ( ! [X3: term_a_b,Y3: term_a_b,Z4: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ ( transi7922773638565587891rm_a_b @ R ) )
=> ( ( P @ X3 @ Y3 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ Z4 ) @ ( transi7922773638565587891rm_a_b @ R ) )
=> ( ( P @ Y3 @ Z4 )
=> ( P @ X3 @ Z4 ) ) ) ) )
=> ( P @ X @ Y ) ) ) ) ).
% trancl_trans_induct
thf(fact_532_converse__trancl__induct,axiom,
! [A: term_a_b,B: term_a_b,R: set_Pr4386577575007340137rm_a_b,P: term_a_b > $o] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7922773638565587891rm_a_b @ R ) )
=> ( ! [Y3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ B ) @ R )
=> ( P @ Y3 ) )
=> ( ! [Y3: term_a_b,Z4: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ Z4 ) @ R )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Z4 @ B ) @ ( transi7922773638565587891rm_a_b @ R ) )
=> ( ( P @ Z4 )
=> ( P @ Y3 ) ) ) )
=> ( P @ A ) ) ) ) ).
% converse_trancl_induct
thf(fact_533_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_534_trancl__rtrancl__trancl,axiom,
! [A: term_a_b,B: term_a_b,R: set_Pr4386577575007340137rm_a_b,C2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7922773638565587891rm_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C2 ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C2 ) @ ( transi7922773638565587891rm_a_b @ R ) ) ) ) ).
% trancl_rtrancl_trancl
thf(fact_535_rtrancl__trancl__trancl,axiom,
! [X: term_a_b,Y: term_a_b,R: set_Pr4386577575007340137rm_a_b,Z2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ Z2 ) @ ( transi7922773638565587891rm_a_b @ R ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z2 ) @ ( transi7922773638565587891rm_a_b @ R ) ) ) ) ).
% rtrancl_trancl_trancl
thf(fact_536_rstep__trancl__sig__step__r,axiom,
! [S: term_a_b,T: term_a_b,R2: set_Pr4386577575007340137rm_a_b,F: set_Pr4934435412358123699_a_nat,V2: b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( transi7922773638565587891rm_a_b @ ( rstep_a_b @ 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 @ V2 @ T ) ) @ ( transi7922773638565587891rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R2 ) ) ) ) ) ) ) ).
% rstep_trancl_sig_step_r
thf(fact_537_rstep__trancl__sig__step__l,axiom,
! [S: term_a_b,T: term_a_b,R2: set_Pr4386577575007340137rm_a_b,F: set_Pr4934435412358123699_a_nat,V2: b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( transi7922773638565587891rm_a_b @ ( rstep_a_b @ 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 @ V2 @ S ) @ T ) @ ( transi7922773638565587891rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R2 ) ) ) ) ) ) ) ).
% rstep_trancl_sig_step_l
thf(fact_538_rsteps__eq__relcomp__srsteps__eq__relcompI,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,F: set_Pr4934435412358123699_a_nat,S4: set_Pr4386577575007340137rm_a_b,S: term_a_b,T: term_a_b] :
( ( ord_le8666007276011122963_a_nat @ ( terms_7988297476397195622_a_b_b @ R2 ) @ F )
=> ( ( ord_le8666007276011122963_a_nat @ ( terms_7988297476397195622_a_b_b @ S4 ) @ F )
=> ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ S ) @ F )
=> ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ T ) @ F )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( rstep_a_b @ R2 ) ) @ ( transi7742714808557438673rm_a_b @ ( rstep_a_b @ S4 ) ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R2 ) ) ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ S4 ) ) ) ) ) ) ) ) ) ) ).
% rsteps_eq_relcomp_srsteps_eq_relcompI
thf(fact_539_srsteps__eq__subst__relcomp__closed,axiom,
! [S: term_a_b,T: term_a_b,F: set_Pr4934435412358123699_a_nat,R2: set_Pr4386577575007340137rm_a_b,S4: set_Pr4386577575007340137rm_a_b,Sigma3: b > term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R2 ) ) ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ S4 ) ) ) ) )
=> ( ! [X3: b] : ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ ( Sigma3 @ X3 ) ) @ F )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( subst_7999470309526761004_a_b_b @ S @ Sigma3 ) @ ( subst_7999470309526761004_a_b_b @ T @ Sigma3 ) ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R2 ) ) ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ S4 ) ) ) ) ) ) ) ).
% srsteps_eq_subst_relcomp_closed
thf(fact_540_trancl__map,axiom,
! [R: set_Pr4386577575007340137rm_a_b,F2: term_a_b > term_a_b,S: set_Pr4386577575007340137rm_a_b,X: term_a_b,Y: term_a_b] :
( ! [X3: term_a_b,Y3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ R )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) @ S ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7922773638565587891rm_a_b @ R ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F2 @ X ) @ ( F2 @ Y ) ) @ ( transi7922773638565587891rm_a_b @ S ) ) ) ) ).
% trancl_map
thf(fact_541_rstep__term__to__sig__r,axiom,
! [S: term_a_b,T: term_a_b,R2: set_Pr4386577575007340137rm_a_b,F: set_Pr4934435412358123699_a_nat,V2: 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 @ V2 @ T ) ) @ ( rstep_a_b @ R2 ) ) ) ) ) ).
% rstep_term_to_sig_r
thf(fact_542_O__mono2,axiom,
! [R5: set_Pr4934435412358123699_a_nat,R7: set_Pr4934435412358123699_a_nat,T4: set_Pr1261947904930325089at_nat] :
( ( ord_le8666007276011122963_a_nat @ R5 @ R7 )
=> ( ord_le8666007276011122963_a_nat @ ( relcomp_a_nat_nat @ R5 @ T4 ) @ ( relcomp_a_nat_nat @ R7 @ T4 ) ) ) ).
% O_mono2
thf(fact_543_O__mono1,axiom,
! [R5: set_Pr4934435412358123699_a_nat,R7: set_Pr4934435412358123699_a_nat,S2: set_Product_prod_a_a] :
( ( ord_le8666007276011122963_a_nat @ R5 @ R7 )
=> ( ord_le8666007276011122963_a_nat @ ( relcomp_a_a_nat @ S2 @ R5 ) @ ( relcomp_a_a_nat @ S2 @ R7 ) ) ) ).
% O_mono1
thf(fact_544_relcomp3__I,axiom,
! [T: term_a_b,U: term_a_b,A5: set_Pr4386577575007340137rm_a_b,S: term_a_b,B4: set_Pr4386577575007340137rm_a_b,V2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T @ U ) @ A5 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ B4 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ U @ V2 ) @ B4 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ V2 ) @ ( relcom370159955682700863rm_a_b @ B4 @ ( relcom370159955682700863rm_a_b @ A5 @ B4 ) ) ) ) ) ) ).
% relcomp3_I
thf(fact_545_relcompEpair,axiom,
! [A: term_a_b,C2: term_a_b,R: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C2 ) @ ( relcom370159955682700863rm_a_b @ R @ S ) )
=> ~ ! [B3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B3 ) @ R )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B3 @ C2 ) @ S ) ) ) ).
% relcompEpair
thf(fact_546_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 ) )
=> ~ ! [X3: term_a_b,Y3: term_a_b,Z4: term_a_b] :
( ( Xz
= ( produc7020197800436672577rm_a_b @ X3 @ Z4 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ R )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ Z4 ) @ S ) ) ) ) ).
% relcompE
thf(fact_547_relcomp_OrelcompI,axiom,
! [A: term_a_b,B: term_a_b,R: set_Pr4386577575007340137rm_a_b,C2: term_a_b,S: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C2 ) @ S )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C2 ) @ ( relcom370159955682700863rm_a_b @ R @ S ) ) ) ) ).
% relcomp.relcompI
thf(fact_548_relcomp_Osimps,axiom,
! [A1: term_a_b,A2: term_a_b,R: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A2 ) @ ( relcom370159955682700863rm_a_b @ R @ S ) )
= ( ? [A7: term_a_b,B6: term_a_b,C4: term_a_b] :
( ( A1 = A7 )
& ( A2 = C4 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A7 @ B6 ) @ R )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B6 @ C4 ) @ S ) ) ) ) ).
% relcomp.simps
thf(fact_549_relcomp_Ocases,axiom,
! [A1: term_a_b,A2: term_a_b,R: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A2 ) @ ( relcom370159955682700863rm_a_b @ R @ S ) )
=> ~ ! [B3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ B3 ) @ R )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B3 @ A2 ) @ S ) ) ) ).
% relcomp.cases
thf(fact_550_relcomp__mono,axiom,
! [R8: set_Product_prod_a_a,R: set_Product_prod_a_a,S3: set_Pr4934435412358123699_a_nat,S: set_Pr4934435412358123699_a_nat] :
( ( ord_le746702958409616551od_a_a @ R8 @ R )
=> ( ( ord_le8666007276011122963_a_nat @ S3 @ S )
=> ( ord_le8666007276011122963_a_nat @ ( relcomp_a_a_nat @ R8 @ S3 ) @ ( relcomp_a_a_nat @ R @ S ) ) ) ) ).
% relcomp_mono
thf(fact_551_relcomp__mono,axiom,
! [R8: set_Pr4934435412358123699_a_nat,R: set_Pr4934435412358123699_a_nat,S3: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
( ( ord_le8666007276011122963_a_nat @ R8 @ R )
=> ( ( ord_le3146513528884898305at_nat @ S3 @ S )
=> ( ord_le8666007276011122963_a_nat @ ( relcomp_a_nat_nat @ R8 @ S3 ) @ ( relcomp_a_nat_nat @ R @ S ) ) ) ) ).
% relcomp_mono
thf(fact_552_steps__map,axiom,
! [P: term_a_b > $o,Q: set_Pr4386577575007340137rm_a_b > $o,F2: term_a_b > term_a_b,G3: set_Pr4386577575007340137rm_a_b > set_Pr4386577575007340137rm_a_b,T: term_a_b,R5: set_Pr4386577575007340137rm_a_b,S2: set_Pr4386577575007340137rm_a_b,U: term_a_b] :
( ! [T2: term_a_b,U2: term_a_b,R9: set_Pr4386577575007340137rm_a_b] :
( ( P @ T2 )
=> ( ( Q @ R9 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T2 @ U2 ) @ R9 )
=> ( ( P @ U2 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F2 @ T2 ) @ ( F2 @ U2 ) ) @ ( G3 @ R9 ) ) ) ) ) )
=> ( ( P @ T )
=> ( ( Q @ R5 )
=> ( ( Q @ S2 )
=> ( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T @ U ) @ ( transi7742714808557438673rm_a_b @ R5 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F2 @ T ) @ ( F2 @ U ) ) @ ( transi7742714808557438673rm_a_b @ ( G3 @ R5 ) ) ) )
& ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T @ U ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ R5 ) @ ( relcom370159955682700863rm_a_b @ S2 @ ( transi7742714808557438673rm_a_b @ R5 ) ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F2 @ T ) @ ( F2 @ U ) ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( G3 @ R5 ) ) @ ( relcom370159955682700863rm_a_b @ ( G3 @ S2 ) @ ( transi7742714808557438673rm_a_b @ ( G3 @ R5 ) ) ) ) ) ) ) ) ) ) ) ).
% steps_map
thf(fact_553_kleene__induct,axiom,
! [A5: set_Pr4934435412358123699_a_nat,X5: set_Pr4934435412358123699_a_nat,B4: set_Product_prod_a_a,C3: set_Pr1261947904930325089at_nat] :
( ( ord_le8666007276011122963_a_nat @ A5 @ X5 )
=> ( ( ord_le8666007276011122963_a_nat @ ( relcomp_a_a_nat @ B4 @ X5 ) @ X5 )
=> ( ( ord_le8666007276011122963_a_nat @ ( relcomp_a_nat_nat @ X5 @ C3 ) @ X5 )
=> ( ord_le8666007276011122963_a_nat @ ( relcomp_a_a_nat @ ( transitive_rtrancl_a @ B4 ) @ ( relcomp_a_nat_nat @ A5 @ ( transi2905341329935302413cl_nat @ C3 ) ) ) @ X5 ) ) ) ) ).
% kleene_induct
thf(fact_554_compat__tr__compat,axiom,
! [NS: set_Product_prod_a_a,S2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ ( relcomp_a_a_nat @ NS @ S2 ) @ S2 )
=> ( ord_le8666007276011122963_a_nat @ ( relcomp_a_a_nat @ ( transitive_rtrancl_a @ NS ) @ S2 ) @ S2 ) ) ).
% compat_tr_compat
thf(fact_555_kleene__trancl__induct,axiom,
! [A5: set_Pr4934435412358123699_a_nat,X5: set_Pr4934435412358123699_a_nat,B4: set_Product_prod_a_a,C3: set_Pr1261947904930325089at_nat] :
( ( ord_le8666007276011122963_a_nat @ A5 @ X5 )
=> ( ( ord_le8666007276011122963_a_nat @ ( relcomp_a_a_nat @ B4 @ X5 ) @ X5 )
=> ( ( ord_le8666007276011122963_a_nat @ ( relcomp_a_nat_nat @ X5 @ C3 ) @ X5 )
=> ( ord_le8666007276011122963_a_nat @ ( relcomp_a_a_nat @ ( transitive_trancl_a @ B4 ) @ ( relcomp_a_nat_nat @ A5 @ ( transi6264000038957366511cl_nat @ C3 ) ) ) @ X5 ) ) ) ) ).
% kleene_trancl_induct
thf(fact_556_non__strict__into__strict,axiom,
! [NS: set_Pr4386577575007340137rm_a_b,S2: set_Pr4386577575007340137rm_a_b,S: term_a_b,T: term_a_b] :
( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ NS @ S2 ) @ S2 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ NS ) @ S2 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ S2 ) ) ) ).
% non_strict_into_strict
thf(fact_557_non__strict__into__strict,axiom,
! [NS: set_Product_prod_a_a,S2: set_Pr4934435412358123699_a_nat,S: a,T: nat] :
( ( ord_le8666007276011122963_a_nat @ ( relcomp_a_a_nat @ NS @ S2 ) @ S2 )
=> ( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ S @ T ) @ ( relcomp_a_a_nat @ ( transitive_rtrancl_a @ NS ) @ S2 ) )
=> ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ S @ T ) @ S2 ) ) ) ).
% non_strict_into_strict
thf(fact_558_rstep__term__to__sig__l,axiom,
! [S: term_a_b,T: term_a_b,R2: set_Pr4386577575007340137rm_a_b,F: set_Pr4934435412358123699_a_nat,V2: 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 @ V2 @ S ) @ T ) @ ( rstep_a_b @ R2 ) ) ) ) ) ).
% rstep_term_to_sig_l
thf(fact_559_term__to__sig__subst__sig,axiom,
! [T: term_a_b,F: set_Pr4934435412358123699_a_nat,V2: b,Sigma3: b > term_a_b] :
( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ T ) @ F )
=> ( ( terms_8519481630511763164ig_a_b @ F @ V2 @ ( subst_7999470309526761004_a_b_b @ T @ Sigma3 ) )
= ( subst_7999470309526761004_a_b_b @ T
@ ^ [X2: b] : ( terms_8519481630511763164ig_a_b @ F @ V2 @ ( Sigma3 @ X2 ) ) ) ) ) ).
% term_to_sig_subst_sig
thf(fact_560_term__to__sig__id,axiom,
! [T: term_a_b,F: set_Pr4934435412358123699_a_nat,V2: b] :
( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ T ) @ F )
=> ( ( terms_8519481630511763164ig_a_b @ F @ V2 @ T )
= T ) ) ).
% term_to_sig_id
thf(fact_561_srsteps__with__root__step__def,axiom,
( srstep7844470518422762656ep_a_b
= ( ^ [F4: set_Pr4934435412358123699_a_nat,R6: set_Pr4386577575007340137rm_a_b] : ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F4 @ ( rstep_a_b @ R6 ) ) ) @ ( relcom370159955682700863rm_a_b @ ( sig_step_a_b @ F4 @ ( rrstep_a_b @ R6 ) ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F4 @ ( rstep_a_b @ R6 ) ) ) ) ) ) ) ).
% srsteps_with_root_step_def
thf(fact_562_fuans__term__term__to__sig,axiom,
! [F: set_Pr4934435412358123699_a_nat,V2: b,T: term_a_b] : ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ ( terms_8519481630511763164ig_a_b @ F @ V2 @ T ) ) @ F ) ).
% fuans_term_term_to_sig
thf(fact_563_restrict__def,axiom,
( abstra2354209648719431185rm_a_b
= ( ^ [R4: set_Pr2972776593051762503rm_a_b,S5: set_Pr4386577575007340137rm_a_b] :
( collec1344989105561874898rm_a_b
@ ( produc161078424057476524_a_b_o
@ ^ [X2: produc357393685978478089rm_a_b,Y4: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X2 @ S5 )
& ( member5869715511025134514rm_a_b @ Y4 @ S5 )
& ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X2 @ Y4 ) @ R4 ) ) ) ) ) ) ).
% restrict_def
thf(fact_564_restrict__def,axiom,
( abstract_restrict_b
= ( ^ [R4: set_Product_prod_b_b,S5: set_b] :
( collec548942219715005266od_b_b
@ ( produc4772849988445941548_b_b_o
@ ^ [X2: b,Y4: b] :
( ( member_b @ X2 @ S5 )
& ( member_b @ Y4 @ S5 )
& ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ X2 @ Y4 ) @ R4 ) ) ) ) ) ) ).
% restrict_def
thf(fact_565_restrict__def,axiom,
( abstra6662565036008789004rm_a_b
= ( ^ [R4: set_Pr4386577575007340137rm_a_b,S5: set_term_a_b] :
( collec99177395925924084rm_a_b
@ ( produc6251846659952913346_a_b_o
@ ^ [X2: term_a_b,Y4: term_a_b] :
( ( member_term_a_b @ X2 @ S5 )
& ( member_term_a_b @ Y4 @ S5 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y4 ) @ R4 ) ) ) ) ) ) ).
% restrict_def
thf(fact_566_meetI,axiom,
! [A: term_a_b,B: term_a_b,A5: set_Pr4386577575007340137rm_a_b,C2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ A5 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C2 ) @ ( transi7742714808557438673rm_a_b @ A5 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C2 ) @ ( abstra1093061187967292567rm_a_b @ A5 ) ) ) ) ).
% meetI
thf(fact_567_case__prod__Pair__iden,axiom,
! [P2: produc357393685978478089rm_a_b] :
( ( produc2089504961790700083rm_a_b @ produc7020197800436672577rm_a_b @ P2 )
= P2 ) ).
% case_prod_Pair_iden
thf(fact_568_meetE,axiom,
! [B: term_a_b,C2: term_a_b,A5: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C2 ) @ ( abstra1093061187967292567rm_a_b @ A5 ) )
=> ~ ! [A4: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A4 @ B ) @ ( transi7742714808557438673rm_a_b @ A5 ) )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A4 @ C2 ) @ ( transi7742714808557438673rm_a_b @ A5 ) ) ) ) ).
% meetE
thf(fact_569_meetD,axiom,
! [B: term_a_b,C2: term_a_b,A5: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C2 ) @ ( abstra1093061187967292567rm_a_b @ A5 ) )
=> ? [A4: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A4 @ B ) @ ( transi7742714808557438673rm_a_b @ A5 ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A4 @ C2 ) @ ( transi7742714808557438673rm_a_b @ A5 ) ) ) ) ).
% meetD
thf(fact_570_comp__rtrancl__trancl,axiom,
! [R5: set_Pr4386577575007340137rm_a_b,S2: set_Pr4386577575007340137rm_a_b,S: term_a_b,T: term_a_b] :
( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ R5 @ S2 ) @ S2 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ R5 @ S2 ) ) @ S2 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( transi7922773638565587891rm_a_b @ S2 ) ) ) ) ).
% comp_rtrancl_trancl
thf(fact_571_same__fstI,axiom,
! [P: term_a_b > $o,X: term_a_b,Y6: term_a_b,Y: term_a_b,R5: term_a_b > set_Pr4386577575007340137rm_a_b] :
( ( P @ X )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y6 @ Y ) @ ( R5 @ X ) )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y6 ) @ ( produc7020197800436672577rm_a_b @ X @ Y ) ) @ ( same_f843731131676418198rm_a_b @ P @ R5 ) ) ) ) ).
% same_fstI
thf(fact_572_compatible__rtrancl__split,axiom,
! [NS: set_Pr4386577575007340137rm_a_b,S2: set_Pr4386577575007340137rm_a_b,X: term_a_b,Y: term_a_b] :
( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ NS @ S2 ) @ S2 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ NS @ S2 ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( sup_su6776935440552674877rm_a_b @ ( relcom370159955682700863rm_a_b @ S2 @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ S2 ) @ ( transi7742714808557438673rm_a_b @ NS ) ) ) @ ( transi7742714808557438673rm_a_b @ NS ) ) ) ) ) ).
% compatible_rtrancl_split
thf(fact_573_llrg__funas__term__steps__pres,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,S: term_a_b,T: term_a_b] :
( ( rewrit5000278123254648670_b_a_b @ R2 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( transi7742714808557438673rm_a_b @ ( rstep_a_b @ R2 ) ) )
=> ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ T ) @ ( sup_su459911885395995103_a_nat @ ( terms_7988297476397195622_a_b_b @ R2 ) @ ( term_funas_term_a_b @ S ) ) ) ) ) ).
% llrg_funas_term_steps_pres
thf(fact_574_Collect__case__prod__in__rel__leE,axiom,
! [X5: set_Pr4934435412358123699_a_nat,Y7: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X5 @ ( collec4464134535221767506_a_nat @ ( produc3680711911437148916_nat_o @ ( fun_in_rel_a_nat @ Y7 ) ) ) )
=> ( ord_le8666007276011122963_a_nat @ X5 @ Y7 ) ) ).
% Collect_case_prod_in_rel_leE
thf(fact_575_Un__iff,axiom,
! [C2: produc357393685978478089rm_a_b,A5: set_Pr4386577575007340137rm_a_b,B4: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ C2 @ ( sup_su6776935440552674877rm_a_b @ A5 @ B4 ) )
= ( ( member5869715511025134514rm_a_b @ C2 @ A5 )
| ( member5869715511025134514rm_a_b @ C2 @ B4 ) ) ) ).
% Un_iff
thf(fact_576_Un__iff,axiom,
! [C2: b,A5: set_b,B4: set_b] :
( ( member_b @ C2 @ ( sup_sup_set_b @ A5 @ B4 ) )
= ( ( member_b @ C2 @ A5 )
| ( member_b @ C2 @ B4 ) ) ) ).
% Un_iff
thf(fact_577_UnCI,axiom,
! [C2: produc357393685978478089rm_a_b,B4: set_Pr4386577575007340137rm_a_b,A5: set_Pr4386577575007340137rm_a_b] :
( ( ~ ( member5869715511025134514rm_a_b @ C2 @ B4 )
=> ( member5869715511025134514rm_a_b @ C2 @ A5 ) )
=> ( member5869715511025134514rm_a_b @ C2 @ ( sup_su6776935440552674877rm_a_b @ A5 @ B4 ) ) ) ).
% UnCI
thf(fact_578_UnCI,axiom,
! [C2: b,B4: set_b,A5: set_b] :
( ( ~ ( member_b @ C2 @ B4 )
=> ( member_b @ C2 @ A5 ) )
=> ( member_b @ C2 @ ( sup_sup_set_b @ A5 @ B4 ) ) ) ).
% UnCI
thf(fact_579_Un__subset__iff,axiom,
! [A5: set_Pr4934435412358123699_a_nat,B4: set_Pr4934435412358123699_a_nat,C3: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ ( sup_su459911885395995103_a_nat @ A5 @ B4 ) @ C3 )
= ( ( ord_le8666007276011122963_a_nat @ A5 @ C3 )
& ( ord_le8666007276011122963_a_nat @ B4 @ C3 ) ) ) ).
% Un_subset_iff
thf(fact_580_in__inv__image,axiom,
! [X: term_a_b,Y: term_a_b,R: set_Pr4386577575007340137rm_a_b,F2: term_a_b > term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( inv_im3004224898869642620rm_a_b @ R @ F2 ) )
= ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F2 @ X ) @ ( F2 @ Y ) ) @ R ) ) ).
% in_inv_image
thf(fact_581_UnI2,axiom,
! [C2: produc357393685978478089rm_a_b,B4: set_Pr4386577575007340137rm_a_b,A5: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ C2 @ B4 )
=> ( member5869715511025134514rm_a_b @ C2 @ ( sup_su6776935440552674877rm_a_b @ A5 @ B4 ) ) ) ).
% UnI2
thf(fact_582_UnI2,axiom,
! [C2: b,B4: set_b,A5: set_b] :
( ( member_b @ C2 @ B4 )
=> ( member_b @ C2 @ ( sup_sup_set_b @ A5 @ B4 ) ) ) ).
% UnI2
thf(fact_583_UnI1,axiom,
! [C2: produc357393685978478089rm_a_b,A5: set_Pr4386577575007340137rm_a_b,B4: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ C2 @ A5 )
=> ( member5869715511025134514rm_a_b @ C2 @ ( sup_su6776935440552674877rm_a_b @ A5 @ B4 ) ) ) ).
% UnI1
thf(fact_584_UnI1,axiom,
! [C2: b,A5: set_b,B4: set_b] :
( ( member_b @ C2 @ A5 )
=> ( member_b @ C2 @ ( sup_sup_set_b @ A5 @ B4 ) ) ) ).
% UnI1
thf(fact_585_UnE,axiom,
! [C2: produc357393685978478089rm_a_b,A5: set_Pr4386577575007340137rm_a_b,B4: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ C2 @ ( sup_su6776935440552674877rm_a_b @ A5 @ B4 ) )
=> ( ~ ( member5869715511025134514rm_a_b @ C2 @ A5 )
=> ( member5869715511025134514rm_a_b @ C2 @ B4 ) ) ) ).
% UnE
thf(fact_586_UnE,axiom,
! [C2: b,A5: set_b,B4: set_b] :
( ( member_b @ C2 @ ( sup_sup_set_b @ A5 @ B4 ) )
=> ( ~ ( member_b @ C2 @ A5 )
=> ( member_b @ C2 @ B4 ) ) ) ).
% UnE
thf(fact_587_Un__def,axiom,
( sup_su6776935440552674877rm_a_b
= ( ^ [A6: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
( collec99177395925924084rm_a_b
@ ^ [X2: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X2 @ A6 )
| ( member5869715511025134514rm_a_b @ X2 @ B5 ) ) ) ) ) ).
% Un_def
thf(fact_588_Un__def,axiom,
( sup_sup_set_b
= ( ^ [A6: set_b,B5: set_b] :
( collect_b
@ ^ [X2: b] :
( ( member_b @ X2 @ A6 )
| ( member_b @ X2 @ B5 ) ) ) ) ) ).
% Un_def
thf(fact_589_Un__mono,axiom,
! [A5: set_Pr4934435412358123699_a_nat,C3: set_Pr4934435412358123699_a_nat,B4: set_Pr4934435412358123699_a_nat,D: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A5 @ C3 )
=> ( ( ord_le8666007276011122963_a_nat @ B4 @ D )
=> ( ord_le8666007276011122963_a_nat @ ( sup_su459911885395995103_a_nat @ A5 @ B4 ) @ ( sup_su459911885395995103_a_nat @ C3 @ D ) ) ) ) ).
% Un_mono
thf(fact_590_Un__least,axiom,
! [A5: set_Pr4934435412358123699_a_nat,C3: set_Pr4934435412358123699_a_nat,B4: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A5 @ C3 )
=> ( ( ord_le8666007276011122963_a_nat @ B4 @ C3 )
=> ( ord_le8666007276011122963_a_nat @ ( sup_su459911885395995103_a_nat @ A5 @ B4 ) @ C3 ) ) ) ).
% Un_least
thf(fact_591_Un__upper1,axiom,
! [A5: set_Pr4934435412358123699_a_nat,B4: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ A5 @ ( sup_su459911885395995103_a_nat @ A5 @ B4 ) ) ).
% Un_upper1
thf(fact_592_Un__upper2,axiom,
! [B4: set_Pr4934435412358123699_a_nat,A5: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ B4 @ ( sup_su459911885395995103_a_nat @ A5 @ B4 ) ) ).
% Un_upper2
thf(fact_593_Un__absorb1,axiom,
! [A5: set_Pr4934435412358123699_a_nat,B4: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A5 @ B4 )
=> ( ( sup_su459911885395995103_a_nat @ A5 @ B4 )
= B4 ) ) ).
% Un_absorb1
thf(fact_594_Un__absorb2,axiom,
! [B4: set_Pr4934435412358123699_a_nat,A5: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ B4 @ A5 )
=> ( ( sup_su459911885395995103_a_nat @ A5 @ B4 )
= A5 ) ) ).
% Un_absorb2
thf(fact_595_subset__UnE,axiom,
! [C3: set_Pr4934435412358123699_a_nat,A5: set_Pr4934435412358123699_a_nat,B4: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ C3 @ ( sup_su459911885395995103_a_nat @ A5 @ B4 ) )
=> ~ ! [A8: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A8 @ A5 )
=> ! [B7: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ B7 @ B4 )
=> ( C3
!= ( sup_su459911885395995103_a_nat @ A8 @ B7 ) ) ) ) ) ).
% subset_UnE
thf(fact_596_subset__Un__eq,axiom,
( ord_le8666007276011122963_a_nat
= ( ^ [A6: set_Pr4934435412358123699_a_nat,B5: set_Pr4934435412358123699_a_nat] :
( ( sup_su459911885395995103_a_nat @ A6 @ B5 )
= B5 ) ) ) ).
% subset_Un_eq
thf(fact_597_in__rtrancl__UnI,axiom,
! [X: produc357393685978478089rm_a_b,R5: set_Pr4386577575007340137rm_a_b,S2: set_Pr4386577575007340137rm_a_b] :
( ( ( member5869715511025134514rm_a_b @ X @ ( transi7742714808557438673rm_a_b @ R5 ) )
| ( member5869715511025134514rm_a_b @ X @ ( transi7742714808557438673rm_a_b @ S2 ) ) )
=> ( member5869715511025134514rm_a_b @ X @ ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ R5 @ S2 ) ) ) ) ).
% in_rtrancl_UnI
thf(fact_598_in__rel__def,axiom,
( fun_in4743132711034513644rm_a_b
= ( ^ [R10: set_Pr4386577575007340137rm_a_b,X2: term_a_b,Y4: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y4 ) @ R10 ) ) ) ).
% in_rel_def
thf(fact_599_rtrancl__Un__separatorE,axiom,
! [A: term_a_b,B: term_a_b,P: set_Pr4386577575007340137rm_a_b,Q: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ P @ Q ) ) )
=> ( ! [X3: term_a_b,Y3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ X3 ) @ ( transi7742714808557438673rm_a_b @ P ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ Q )
=> ( X3 = Y3 ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ P ) ) ) ) ).
% rtrancl_Un_separatorE
thf(fact_600_rtrancl__Un__separator__converseE,axiom,
! [A: term_a_b,B: term_a_b,P: set_Pr4386577575007340137rm_a_b,Q: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ P @ Q ) ) )
=> ( ! [X3: term_a_b,Y3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ B ) @ ( transi7742714808557438673rm_a_b @ P ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ X3 ) @ Q )
=> ( Y3 = X3 ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ P ) ) ) ) ).
% rtrancl_Un_separator_converseE
thf(fact_601_firstStep,axiom,
! [L5: set_Pr4386577575007340137rm_a_b,S2: set_Pr4386577575007340137rm_a_b,R5: set_Pr4386577575007340137rm_a_b,X: term_a_b,Y: term_a_b] :
( ( L5
= ( sup_su6776935440552674877rm_a_b @ S2 @ R5 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ L5 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ R5 ) )
| ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ R5 ) @ ( relcom370159955682700863rm_a_b @ S2 @ ( transi7742714808557438673rm_a_b @ L5 ) ) ) ) ) ) ) ).
% firstStep
thf(fact_602_first__step,axiom,
! [C3: set_Pr4386577575007340137rm_a_b,A5: set_Pr4386577575007340137rm_a_b,B4: set_Pr4386577575007340137rm_a_b,X: term_a_b,Y: term_a_b,Z2: term_a_b] :
( ( C3
= ( sup_su6776935440552674877rm_a_b @ A5 @ B4 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ C3 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ Z2 ) @ B4 )
=> ? [Y3: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y3 ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ A5 ) @ B4 ) ) ) ) ) ).
% first_step
thf(fact_603_first__step__O,axiom,
! [C3: set_Pr4386577575007340137rm_a_b,A5: set_Pr4386577575007340137rm_a_b,B4: set_Pr4386577575007340137rm_a_b,X: term_a_b,Y: term_a_b] :
( ( C3
= ( sup_su6776935440552674877rm_a_b @ A5 @ B4 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ C3 ) @ B4 ) )
=> ? [Y3: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y3 ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ A5 ) @ B4 ) ) ) ) ).
% first_step_O
thf(fact_604_right__comp__S,axiom,
! [X: term_a_b,Y: term_a_b,S2: set_Pr4386577575007340137rm_a_b,NS: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( relcom370159955682700863rm_a_b @ S2 @ ( sup_su6776935440552674877rm_a_b @ ( relcom370159955682700863rm_a_b @ S2 @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ S2 ) @ ( transi7742714808557438673rm_a_b @ NS ) ) ) @ ( transi7742714808557438673rm_a_b @ NS ) ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( relcom370159955682700863rm_a_b @ S2 @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ S2 ) @ ( transi7742714808557438673rm_a_b @ NS ) ) ) ) ) ).
% right_comp_S
thf(fact_605_llrg__funas__term__step__pres,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,S: term_a_b,T: term_a_b] :
( ( rewrit5000278123254648670_b_a_b @ R2 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( rstep_a_b @ R2 ) )
=> ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ T ) @ ( sup_su459911885395995103_a_nat @ ( terms_7988297476397195622_a_b_b @ R2 ) @ ( term_funas_term_a_b @ S ) ) ) ) ) ).
% llrg_funas_term_step_pres
thf(fact_606_Collect__case__prod__in__rel__leI,axiom,
! [X5: set_Pr4934435412358123699_a_nat,Y7: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X5 @ Y7 )
=> ( ord_le8666007276011122963_a_nat @ X5 @ ( collec4464134535221767506_a_nat @ ( produc3680711911437148916_nat_o @ ( fun_in_rel_a_nat @ Y7 ) ) ) ) ) ).
% Collect_case_prod_in_rel_leI
thf(fact_607_sup_Obounded__iff,axiom,
! [B: set_Pr4934435412358123699_a_nat,C2: set_Pr4934435412358123699_a_nat,A: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ ( sup_su459911885395995103_a_nat @ B @ C2 ) @ A )
= ( ( ord_le8666007276011122963_a_nat @ B @ A )
& ( ord_le8666007276011122963_a_nat @ C2 @ A ) ) ) ).
% sup.bounded_iff
thf(fact_608_le__sup__iff,axiom,
! [X: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat,Z2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ ( sup_su459911885395995103_a_nat @ X @ Y ) @ Z2 )
= ( ( ord_le8666007276011122963_a_nat @ X @ Z2 )
& ( ord_le8666007276011122963_a_nat @ Y @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_609_sig__step__rsteps__eq__dist,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,F: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ ( terms_7988297476397195622_a_b_b @ R2 ) @ F )
=> ( ( sup_su6776935440552674877rm_a_b @ ( sig_step_a_b @ F @ ( transi7922773638565587891rm_a_b @ ( rstep_a_b @ R2 ) ) ) @ id_term_a_b )
= ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R2 ) ) ) ) ) ).
% sig_step_rsteps_eq_dist
thf(fact_610_sup_OcoboundedI2,axiom,
! [C2: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat,A: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ C2 @ B )
=> ( ord_le8666007276011122963_a_nat @ C2 @ ( sup_su459911885395995103_a_nat @ A @ B ) ) ) ).
% sup.coboundedI2
thf(fact_611_sup_OcoboundedI1,axiom,
! [C2: set_Pr4934435412358123699_a_nat,A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ C2 @ A )
=> ( ord_le8666007276011122963_a_nat @ C2 @ ( sup_su459911885395995103_a_nat @ A @ B ) ) ) ).
% sup.coboundedI1
thf(fact_612_IdI,axiom,
! [A: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ A ) @ id_term_a_b ) ).
% IdI
thf(fact_613_pair__in__Id__conv,axiom,
! [A: term_a_b,B: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ id_term_a_b )
= ( A = B ) ) ).
% pair_in_Id_conv
thf(fact_614_reflcl__set__eq,axiom,
! [R: set_Pr4386577575007340137rm_a_b] :
( ( sup_su8172761420313695282_a_b_o
@ ^ [X2: term_a_b,Y4: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y4 ) @ R )
@ ^ [Y5: term_a_b,Z: term_a_b] : ( Y5 = Z ) )
= ( ^ [X2: term_a_b,Y4: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y4 ) @ ( sup_su6776935440552674877rm_a_b @ R @ id_term_a_b ) ) ) ) ).
% reflcl_set_eq
thf(fact_615_IdE,axiom,
! [P2: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ P2 @ id_term_a_b )
=> ~ ! [X3: term_a_b] :
( P2
!= ( produc7020197800436672577rm_a_b @ X3 @ X3 ) ) ) ).
% IdE
thf(fact_616_IdD,axiom,
! [A: term_a_b,B: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ id_term_a_b )
=> ( A = B ) ) ).
% IdD
thf(fact_617_sup__set__def,axiom,
( sup_su6776935440552674877rm_a_b
= ( ^ [A6: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
( collec99177395925924084rm_a_b
@ ( sup_su2803897115920335584_a_b_o
@ ^ [X2: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ X2 @ A6 )
@ ^ [X2: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ X2 @ B5 ) ) ) ) ) ).
% sup_set_def
thf(fact_618_sup__set__def,axiom,
( sup_sup_set_b
= ( ^ [A6: set_b,B5: set_b] :
( collect_b
@ ( sup_sup_b_o
@ ^ [X2: b] : ( member_b @ X2 @ A6 )
@ ^ [X2: b] : ( member_b @ X2 @ B5 ) ) ) ) ) ).
% sup_set_def
thf(fact_619_sup__Un__eq,axiom,
! [R5: set_Pr4386577575007340137rm_a_b,S2: set_Pr4386577575007340137rm_a_b] :
( ( sup_su2803897115920335584_a_b_o
@ ^ [X2: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ X2 @ R5 )
@ ^ [X2: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ X2 @ S2 ) )
= ( ^ [X2: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ X2 @ ( sup_su6776935440552674877rm_a_b @ R5 @ S2 ) ) ) ) ).
% sup_Un_eq
thf(fact_620_sup__Un__eq,axiom,
! [R5: set_b,S2: set_b] :
( ( sup_sup_b_o
@ ^ [X2: b] : ( member_b @ X2 @ R5 )
@ ^ [X2: b] : ( member_b @ X2 @ S2 ) )
= ( ^ [X2: b] : ( member_b @ X2 @ ( sup_sup_set_b @ R5 @ S2 ) ) ) ) ).
% sup_Un_eq
thf(fact_621_sup__Un__eq2,axiom,
! [R5: set_Pr4386577575007340137rm_a_b,S2: set_Pr4386577575007340137rm_a_b] :
( ( sup_su8172761420313695282_a_b_o
@ ^ [X2: term_a_b,Y4: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y4 ) @ R5 )
@ ^ [X2: term_a_b,Y4: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y4 ) @ S2 ) )
= ( ^ [X2: term_a_b,Y4: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y4 ) @ ( sup_su6776935440552674877rm_a_b @ R5 @ S2 ) ) ) ) ).
% sup_Un_eq2
thf(fact_622_inf__sup__ord_I4_J,axiom,
! [Y: set_Pr4934435412358123699_a_nat,X: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ Y @ ( sup_su459911885395995103_a_nat @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_623_inf__sup__ord_I3_J,axiom,
! [X: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ X @ ( sup_su459911885395995103_a_nat @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_624_le__supE,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat,X: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ ( sup_su459911885395995103_a_nat @ A @ B ) @ X )
=> ~ ( ( ord_le8666007276011122963_a_nat @ A @ X )
=> ~ ( ord_le8666007276011122963_a_nat @ B @ X ) ) ) ).
% le_supE
thf(fact_625_le__supI,axiom,
! [A: set_Pr4934435412358123699_a_nat,X: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A @ X )
=> ( ( ord_le8666007276011122963_a_nat @ B @ X )
=> ( ord_le8666007276011122963_a_nat @ ( sup_su459911885395995103_a_nat @ A @ B ) @ X ) ) ) ).
% le_supI
thf(fact_626_sup__ge1,axiom,
! [X: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ X @ ( sup_su459911885395995103_a_nat @ X @ Y ) ) ).
% sup_ge1
thf(fact_627_sup__ge2,axiom,
! [Y: set_Pr4934435412358123699_a_nat,X: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ Y @ ( sup_su459911885395995103_a_nat @ X @ Y ) ) ).
% sup_ge2
thf(fact_628_le__supI1,axiom,
! [X: set_Pr4934435412358123699_a_nat,A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X @ A )
=> ( ord_le8666007276011122963_a_nat @ X @ ( sup_su459911885395995103_a_nat @ A @ B ) ) ) ).
% le_supI1
thf(fact_629_le__supI2,axiom,
! [X: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat,A: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X @ B )
=> ( ord_le8666007276011122963_a_nat @ X @ ( sup_su459911885395995103_a_nat @ A @ B ) ) ) ).
% le_supI2
thf(fact_630_sup_Omono,axiom,
! [C2: set_Pr4934435412358123699_a_nat,A: set_Pr4934435412358123699_a_nat,D2: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ C2 @ A )
=> ( ( ord_le8666007276011122963_a_nat @ D2 @ B )
=> ( ord_le8666007276011122963_a_nat @ ( sup_su459911885395995103_a_nat @ C2 @ D2 ) @ ( sup_su459911885395995103_a_nat @ A @ B ) ) ) ) ).
% sup.mono
thf(fact_631_sup__mono,axiom,
! [A: set_Pr4934435412358123699_a_nat,C2: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat,D2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A @ C2 )
=> ( ( ord_le8666007276011122963_a_nat @ B @ D2 )
=> ( ord_le8666007276011122963_a_nat @ ( sup_su459911885395995103_a_nat @ A @ B ) @ ( sup_su459911885395995103_a_nat @ C2 @ D2 ) ) ) ) ).
% sup_mono
thf(fact_632_sup__least,axiom,
! [Y: set_Pr4934435412358123699_a_nat,X: set_Pr4934435412358123699_a_nat,Z2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ Y @ X )
=> ( ( ord_le8666007276011122963_a_nat @ Z2 @ X )
=> ( ord_le8666007276011122963_a_nat @ ( sup_su459911885395995103_a_nat @ Y @ Z2 ) @ X ) ) ) ).
% sup_least
thf(fact_633_le__iff__sup,axiom,
( ord_le8666007276011122963_a_nat
= ( ^ [X2: set_Pr4934435412358123699_a_nat,Y4: set_Pr4934435412358123699_a_nat] :
( ( sup_su459911885395995103_a_nat @ X2 @ Y4 )
= Y4 ) ) ) ).
% le_iff_sup
thf(fact_634_sup_OorderE,axiom,
! [B: set_Pr4934435412358123699_a_nat,A: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ B @ A )
=> ( A
= ( sup_su459911885395995103_a_nat @ A @ B ) ) ) ).
% sup.orderE
thf(fact_635_sup_OorderI,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( A
= ( sup_su459911885395995103_a_nat @ A @ B ) )
=> ( ord_le8666007276011122963_a_nat @ B @ A ) ) ).
% sup.orderI
thf(fact_636_sup__unique,axiom,
! [F2: set_Pr4934435412358123699_a_nat > set_Pr4934435412358123699_a_nat > set_Pr4934435412358123699_a_nat,X: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat] :
( ! [X3: set_Pr4934435412358123699_a_nat,Y3: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ X3 @ ( F2 @ X3 @ Y3 ) )
=> ( ! [X3: set_Pr4934435412358123699_a_nat,Y3: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ Y3 @ ( F2 @ X3 @ Y3 ) )
=> ( ! [X3: set_Pr4934435412358123699_a_nat,Y3: set_Pr4934435412358123699_a_nat,Z4: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ Y3 @ X3 )
=> ( ( ord_le8666007276011122963_a_nat @ Z4 @ X3 )
=> ( ord_le8666007276011122963_a_nat @ ( F2 @ Y3 @ Z4 ) @ X3 ) ) )
=> ( ( sup_su459911885395995103_a_nat @ X @ Y )
= ( F2 @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_637_sup_Oabsorb1,axiom,
! [B: set_Pr4934435412358123699_a_nat,A: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ B @ A )
=> ( ( sup_su459911885395995103_a_nat @ A @ B )
= A ) ) ).
% sup.absorb1
thf(fact_638_sup_Oabsorb2,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A @ B )
=> ( ( sup_su459911885395995103_a_nat @ A @ B )
= B ) ) ).
% sup.absorb2
thf(fact_639_sup__absorb1,axiom,
! [Y: set_Pr4934435412358123699_a_nat,X: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ Y @ X )
=> ( ( sup_su459911885395995103_a_nat @ X @ Y )
= X ) ) ).
% sup_absorb1
thf(fact_640_sup__absorb2,axiom,
! [X: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X @ Y )
=> ( ( sup_su459911885395995103_a_nat @ X @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_641_sup_OboundedE,axiom,
! [B: set_Pr4934435412358123699_a_nat,C2: set_Pr4934435412358123699_a_nat,A: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ ( sup_su459911885395995103_a_nat @ B @ C2 ) @ A )
=> ~ ( ( ord_le8666007276011122963_a_nat @ B @ A )
=> ~ ( ord_le8666007276011122963_a_nat @ C2 @ A ) ) ) ).
% sup.boundedE
thf(fact_642_sup_OboundedI,axiom,
! [B: set_Pr4934435412358123699_a_nat,A: set_Pr4934435412358123699_a_nat,C2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ B @ A )
=> ( ( ord_le8666007276011122963_a_nat @ C2 @ A )
=> ( ord_le8666007276011122963_a_nat @ ( sup_su459911885395995103_a_nat @ B @ C2 ) @ A ) ) ) ).
% sup.boundedI
thf(fact_643_sup_Oorder__iff,axiom,
( ord_le8666007276011122963_a_nat
= ( ^ [B6: set_Pr4934435412358123699_a_nat,A7: set_Pr4934435412358123699_a_nat] :
( A7
= ( sup_su459911885395995103_a_nat @ A7 @ B6 ) ) ) ) ).
% sup.order_iff
thf(fact_644_sup_Ocobounded1,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ A @ ( sup_su459911885395995103_a_nat @ A @ B ) ) ).
% sup.cobounded1
thf(fact_645_sup_Ocobounded2,axiom,
! [B: set_Pr4934435412358123699_a_nat,A: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ B @ ( sup_su459911885395995103_a_nat @ A @ B ) ) ).
% sup.cobounded2
thf(fact_646_sup_Oabsorb__iff1,axiom,
( ord_le8666007276011122963_a_nat
= ( ^ [B6: set_Pr4934435412358123699_a_nat,A7: set_Pr4934435412358123699_a_nat] :
( ( sup_su459911885395995103_a_nat @ A7 @ B6 )
= A7 ) ) ) ).
% sup.absorb_iff1
thf(fact_647_sup_Oabsorb__iff2,axiom,
( ord_le8666007276011122963_a_nat
= ( ^ [A7: set_Pr4934435412358123699_a_nat,B6: set_Pr4934435412358123699_a_nat] :
( ( sup_su459911885395995103_a_nat @ A7 @ B6 )
= B6 ) ) ) ).
% sup.absorb_iff2
thf(fact_648_strongly__confluent__on__E11,axiom,
! [R: set_Pr2972776593051762503rm_a_b,A5: set_Pr4386577575007340137rm_a_b,X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b,Z2: produc357393685978478089rm_a_b] :
( ( abstra837640964651051114rm_a_b @ R @ A5 )
=> ( ( member5869715511025134514rm_a_b @ X @ A5 )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ R )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Z2 ) @ R )
=> ? [U2: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y @ 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_649_strongly__confluent__on__E11,axiom,
! [R: set_Product_prod_b_b,A5: set_b,X: b,Y: b,Z2: b] :
( ( abstra4021214168631544544t_on_b @ R @ A5 )
=> ( ( member_b @ X @ A5 )
=> ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ X @ Y ) @ R )
=> ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ X @ Z2 ) @ R )
=> ? [U2: b] :
( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ Y @ U2 ) @ ( transitive_rtrancl_b @ R ) )
& ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ Z2 @ U2 ) @ ( sup_su2483643821041016987od_b_b @ R @ id_b ) ) ) ) ) ) ) ).
% strongly_confluent_on_E11
thf(fact_650_strongly__confluent__on__E11,axiom,
! [R: set_Pr4386577575007340137rm_a_b,A5: set_term_a_b,X: term_a_b,Y: term_a_b,Z2: term_a_b] :
( ( abstra5664096641628173427rm_a_b @ R @ A5 )
=> ( ( member_term_a_b @ X @ A5 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ R )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z2 ) @ R )
=> ? [U2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ 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_651_strongly__confluent__on__def,axiom,
( abstra5664096641628173427rm_a_b
= ( ^ [R4: set_Pr4386577575007340137rm_a_b,A6: set_term_a_b] :
! [X2: term_a_b] :
( ( member_term_a_b @ X2 @ A6 )
=> ! [Y4: term_a_b,Z5: term_a_b] :
( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y4 ) @ R4 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Z5 ) @ R4 ) )
=> ? [U3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y4 @ U3 ) @ ( transi7742714808557438673rm_a_b @ R4 ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Z5 @ U3 ) @ ( sup_su6776935440552674877rm_a_b @ R4 @ id_term_a_b ) ) ) ) ) ) ) ).
% strongly_confluent_on_def
thf(fact_652_lex__prod__def,axiom,
( lex_pr6163557265797435481rm_a_b
= ( ^ [Ra: set_Pr4386577575007340137rm_a_b,Rb: set_Pr4386577575007340137rm_a_b] :
( collec1344989105561874898rm_a_b
@ ( produc161078424057476524_a_b_o
@ ( produc4352215818379867658_a_b_o
@ ^ [A7: term_a_b,B6: term_a_b] :
( produc6251846659952913346_a_b_o
@ ^ [A9: term_a_b,B8: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A7 @ A9 ) @ Ra )
| ( ( A7 = A9 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B6 @ B8 ) @ Rb ) ) ) ) ) ) ) ) ) ).
% lex_prod_def
thf(fact_653_trancl__insert,axiom,
! [Y: term_a_b,X: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( transi7922773638565587891rm_a_b @ ( insert7009541432154983385rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X ) @ R ) )
= ( sup_su6776935440552674877rm_a_b @ ( transi7922773638565587891rm_a_b @ R )
@ ( collec99177395925924084rm_a_b
@ ( produc6251846659952913346_a_b_o
@ ^ [A7: term_a_b,B6: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A7 @ Y ) @ ( transi7742714808557438673rm_a_b @ R ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ B6 ) @ ( transi7742714808557438673rm_a_b @ R ) ) ) ) ) ) ) ).
% trancl_insert
thf(fact_654_llrg__rsteps__pres__groundness,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,S: term_a_b,T: term_a_b,F: set_Pr4934435412358123699_a_nat] :
( ( rewrit5000278123254648670_b_a_b @ R2 )
=> ( ( term_ground_a_b @ S )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R2 ) ) ) )
=> ( term_ground_a_b @ T ) ) ) ) ).
% llrg_rsteps_pres_groundness
thf(fact_655_insert__iff,axiom,
! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,A5: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ A @ ( insert7009541432154983385rm_a_b @ B @ A5 ) )
= ( ( A = B )
| ( member5869715511025134514rm_a_b @ A @ A5 ) ) ) ).
% insert_iff
thf(fact_656_insert__iff,axiom,
! [A: b,B: b,A5: set_b] :
( ( member_b @ A @ ( insert_b @ B @ A5 ) )
= ( ( A = B )
| ( member_b @ A @ A5 ) ) ) ).
% insert_iff
thf(fact_657_insertCI,axiom,
! [A: produc357393685978478089rm_a_b,B4: set_Pr4386577575007340137rm_a_b,B: produc357393685978478089rm_a_b] :
( ( ~ ( member5869715511025134514rm_a_b @ A @ B4 )
=> ( A = B ) )
=> ( member5869715511025134514rm_a_b @ A @ ( insert7009541432154983385rm_a_b @ B @ B4 ) ) ) ).
% insertCI
thf(fact_658_insertCI,axiom,
! [A: b,B4: set_b,B: b] :
( ( ~ ( member_b @ A @ B4 )
=> ( A = B ) )
=> ( member_b @ A @ ( insert_b @ B @ B4 ) ) ) ).
% insertCI
thf(fact_659_insert__subset,axiom,
! [X: produc357393685978478089rm_a_b,A5: set_Pr4386577575007340137rm_a_b,B4: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ ( insert7009541432154983385rm_a_b @ X @ A5 ) @ B4 )
= ( ( member5869715511025134514rm_a_b @ X @ B4 )
& ( ord_le118470702582115849rm_a_b @ A5 @ B4 ) ) ) ).
% insert_subset
thf(fact_660_insert__subset,axiom,
! [X: b,A5: set_b,B4: set_b] :
( ( ord_less_eq_set_b @ ( insert_b @ X @ A5 ) @ B4 )
= ( ( member_b @ X @ B4 )
& ( ord_less_eq_set_b @ A5 @ B4 ) ) ) ).
% insert_subset
thf(fact_661_insert__subset,axiom,
! [X: product_prod_a_nat,A5: set_Pr4934435412358123699_a_nat,B4: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ ( insert8054603423593749677_a_nat @ X @ A5 ) @ B4 )
= ( ( member5724188588386418708_a_nat @ X @ B4 )
& ( ord_le8666007276011122963_a_nat @ A5 @ B4 ) ) ) ).
% insert_subset
thf(fact_662_in__lex__prod,axiom,
! [A: term_a_b,B: term_a_b,A3: term_a_b,B2: term_a_b,R: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( produc7020197800436672577rm_a_b @ A3 @ B2 ) ) @ ( lex_pr6163557265797435481rm_a_b @ R @ S ) )
= ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ A3 ) @ R )
| ( ( A = A3 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ B2 ) @ S ) ) ) ) ).
% in_lex_prod
thf(fact_663_insert__compr,axiom,
( insert7009541432154983385rm_a_b
= ( ^ [A7: produc357393685978478089rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
( collec99177395925924084rm_a_b
@ ^ [X2: produc357393685978478089rm_a_b] :
( ( X2 = A7 )
| ( member5869715511025134514rm_a_b @ X2 @ B5 ) ) ) ) ) ).
% insert_compr
thf(fact_664_insert__compr,axiom,
( insert_b
= ( ^ [A7: b,B5: set_b] :
( collect_b
@ ^ [X2: b] :
( ( X2 = A7 )
| ( member_b @ X2 @ B5 ) ) ) ) ) ).
% insert_compr
thf(fact_665_mk__disjoint__insert,axiom,
! [A: produc357393685978478089rm_a_b,A5: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ A @ A5 )
=> ? [B9: set_Pr4386577575007340137rm_a_b] :
( ( A5
= ( insert7009541432154983385rm_a_b @ A @ B9 ) )
& ~ ( member5869715511025134514rm_a_b @ A @ B9 ) ) ) ).
% mk_disjoint_insert
thf(fact_666_mk__disjoint__insert,axiom,
! [A: b,A5: set_b] :
( ( member_b @ A @ A5 )
=> ? [B9: set_b] :
( ( A5
= ( insert_b @ A @ B9 ) )
& ~ ( member_b @ A @ B9 ) ) ) ).
% mk_disjoint_insert
thf(fact_667_insert__eq__iff,axiom,
! [A: produc357393685978478089rm_a_b,A5: set_Pr4386577575007340137rm_a_b,B: produc357393685978478089rm_a_b,B4: set_Pr4386577575007340137rm_a_b] :
( ~ ( member5869715511025134514rm_a_b @ A @ A5 )
=> ( ~ ( member5869715511025134514rm_a_b @ B @ B4 )
=> ( ( ( insert7009541432154983385rm_a_b @ A @ A5 )
= ( insert7009541432154983385rm_a_b @ B @ B4 ) )
= ( ( ( A = B )
=> ( A5 = B4 ) )
& ( ( A != B )
=> ? [C5: set_Pr4386577575007340137rm_a_b] :
( ( A5
= ( insert7009541432154983385rm_a_b @ B @ C5 ) )
& ~ ( member5869715511025134514rm_a_b @ B @ C5 )
& ( B4
= ( insert7009541432154983385rm_a_b @ A @ C5 ) )
& ~ ( member5869715511025134514rm_a_b @ A @ C5 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_668_insert__eq__iff,axiom,
! [A: b,A5: set_b,B: b,B4: set_b] :
( ~ ( member_b @ A @ A5 )
=> ( ~ ( member_b @ B @ B4 )
=> ( ( ( insert_b @ A @ A5 )
= ( insert_b @ B @ B4 ) )
= ( ( ( A = B )
=> ( A5 = B4 ) )
& ( ( A != B )
=> ? [C5: set_b] :
( ( A5
= ( insert_b @ B @ C5 ) )
& ~ ( member_b @ B @ C5 )
& ( B4
= ( insert_b @ A @ C5 ) )
& ~ ( member_b @ A @ C5 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_669_insert__absorb,axiom,
! [A: produc357393685978478089rm_a_b,A5: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ A @ A5 )
=> ( ( insert7009541432154983385rm_a_b @ A @ A5 )
= A5 ) ) ).
% insert_absorb
thf(fact_670_insert__absorb,axiom,
! [A: b,A5: set_b] :
( ( member_b @ A @ A5 )
=> ( ( insert_b @ A @ A5 )
= A5 ) ) ).
% insert_absorb
thf(fact_671_insert__ident,axiom,
! [X: produc357393685978478089rm_a_b,A5: set_Pr4386577575007340137rm_a_b,B4: set_Pr4386577575007340137rm_a_b] :
( ~ ( member5869715511025134514rm_a_b @ X @ A5 )
=> ( ~ ( member5869715511025134514rm_a_b @ X @ B4 )
=> ( ( ( insert7009541432154983385rm_a_b @ X @ A5 )
= ( insert7009541432154983385rm_a_b @ X @ B4 ) )
= ( A5 = B4 ) ) ) ) ).
% insert_ident
thf(fact_672_insert__ident,axiom,
! [X: b,A5: set_b,B4: set_b] :
( ~ ( member_b @ X @ A5 )
=> ( ~ ( member_b @ X @ B4 )
=> ( ( ( insert_b @ X @ A5 )
= ( insert_b @ X @ B4 ) )
= ( A5 = B4 ) ) ) ) ).
% insert_ident
thf(fact_673_Set_Oset__insert,axiom,
! [X: produc357393685978478089rm_a_b,A5: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ X @ A5 )
=> ~ ! [B9: set_Pr4386577575007340137rm_a_b] :
( ( A5
= ( insert7009541432154983385rm_a_b @ X @ B9 ) )
=> ( member5869715511025134514rm_a_b @ X @ B9 ) ) ) ).
% Set.set_insert
thf(fact_674_Set_Oset__insert,axiom,
! [X: b,A5: set_b] :
( ( member_b @ X @ A5 )
=> ~ ! [B9: set_b] :
( ( A5
= ( insert_b @ X @ B9 ) )
=> ( member_b @ X @ B9 ) ) ) ).
% Set.set_insert
thf(fact_675_insertI2,axiom,
! [A: produc357393685978478089rm_a_b,B4: set_Pr4386577575007340137rm_a_b,B: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ A @ B4 )
=> ( member5869715511025134514rm_a_b @ A @ ( insert7009541432154983385rm_a_b @ B @ B4 ) ) ) ).
% insertI2
thf(fact_676_insertI2,axiom,
! [A: b,B4: set_b,B: b] :
( ( member_b @ A @ B4 )
=> ( member_b @ A @ ( insert_b @ B @ B4 ) ) ) ).
% insertI2
thf(fact_677_insertI1,axiom,
! [A: produc357393685978478089rm_a_b,B4: set_Pr4386577575007340137rm_a_b] : ( member5869715511025134514rm_a_b @ A @ ( insert7009541432154983385rm_a_b @ A @ B4 ) ) ).
% insertI1
thf(fact_678_insertI1,axiom,
! [A: b,B4: set_b] : ( member_b @ A @ ( insert_b @ A @ B4 ) ) ).
% insertI1
thf(fact_679_insertE,axiom,
! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,A5: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ A @ ( insert7009541432154983385rm_a_b @ B @ A5 ) )
=> ( ( A != B )
=> ( member5869715511025134514rm_a_b @ A @ A5 ) ) ) ).
% insertE
thf(fact_680_insertE,axiom,
! [A: b,B: b,A5: set_b] :
( ( member_b @ A @ ( insert_b @ B @ A5 ) )
=> ( ( A != B )
=> ( member_b @ A @ A5 ) ) ) ).
% insertE
thf(fact_681_ground__subst__apply,axiom,
! [T: term_a_b,Sigma3: b > term_a_b] :
( ( term_ground_a_b @ T )
=> ( ( subst_7999470309526761004_a_b_b @ T @ Sigma3 )
= T ) ) ).
% ground_subst_apply
thf(fact_682_insert__subsetI,axiom,
! [X: produc357393685978478089rm_a_b,A5: set_Pr4386577575007340137rm_a_b,X5: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ X @ A5 )
=> ( ( ord_le118470702582115849rm_a_b @ X5 @ A5 )
=> ( ord_le118470702582115849rm_a_b @ ( insert7009541432154983385rm_a_b @ X @ X5 ) @ A5 ) ) ) ).
% insert_subsetI
thf(fact_683_insert__subsetI,axiom,
! [X: b,A5: set_b,X5: set_b] :
( ( member_b @ X @ A5 )
=> ( ( ord_less_eq_set_b @ X5 @ A5 )
=> ( ord_less_eq_set_b @ ( insert_b @ X @ X5 ) @ A5 ) ) ) ).
% insert_subsetI
thf(fact_684_insert__subsetI,axiom,
! [X: product_prod_a_nat,A5: set_Pr4934435412358123699_a_nat,X5: set_Pr4934435412358123699_a_nat] :
( ( member5724188588386418708_a_nat @ X @ A5 )
=> ( ( ord_le8666007276011122963_a_nat @ X5 @ A5 )
=> ( ord_le8666007276011122963_a_nat @ ( insert8054603423593749677_a_nat @ X @ X5 ) @ A5 ) ) ) ).
% insert_subsetI
thf(fact_685_insert__mono,axiom,
! [C3: set_Pr4934435412358123699_a_nat,D: set_Pr4934435412358123699_a_nat,A: product_prod_a_nat] :
( ( ord_le8666007276011122963_a_nat @ C3 @ D )
=> ( ord_le8666007276011122963_a_nat @ ( insert8054603423593749677_a_nat @ A @ C3 ) @ ( insert8054603423593749677_a_nat @ A @ D ) ) ) ).
% insert_mono
thf(fact_686_subset__insert,axiom,
! [X: produc357393685978478089rm_a_b,A5: set_Pr4386577575007340137rm_a_b,B4: set_Pr4386577575007340137rm_a_b] :
( ~ ( member5869715511025134514rm_a_b @ X @ A5 )
=> ( ( ord_le118470702582115849rm_a_b @ A5 @ ( insert7009541432154983385rm_a_b @ X @ B4 ) )
= ( ord_le118470702582115849rm_a_b @ A5 @ B4 ) ) ) ).
% subset_insert
thf(fact_687_subset__insert,axiom,
! [X: b,A5: set_b,B4: set_b] :
( ~ ( member_b @ X @ A5 )
=> ( ( ord_less_eq_set_b @ A5 @ ( insert_b @ X @ B4 ) )
= ( ord_less_eq_set_b @ A5 @ B4 ) ) ) ).
% subset_insert
thf(fact_688_subset__insert,axiom,
! [X: product_prod_a_nat,A5: set_Pr4934435412358123699_a_nat,B4: set_Pr4934435412358123699_a_nat] :
( ~ ( member5724188588386418708_a_nat @ X @ A5 )
=> ( ( ord_le8666007276011122963_a_nat @ A5 @ ( insert8054603423593749677_a_nat @ X @ B4 ) )
= ( ord_le8666007276011122963_a_nat @ A5 @ B4 ) ) ) ).
% subset_insert
thf(fact_689_subset__insertI,axiom,
! [B4: set_Pr4934435412358123699_a_nat,A: product_prod_a_nat] : ( ord_le8666007276011122963_a_nat @ B4 @ ( insert8054603423593749677_a_nat @ A @ B4 ) ) ).
% subset_insertI
thf(fact_690_subset__insertI2,axiom,
! [A5: set_Pr4934435412358123699_a_nat,B4: set_Pr4934435412358123699_a_nat,B: product_prod_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A5 @ B4 )
=> ( ord_le8666007276011122963_a_nat @ A5 @ ( insert8054603423593749677_a_nat @ B @ B4 ) ) ) ).
% subset_insertI2
thf(fact_691_llrg__ground__rhs,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,L: term_a_b,R: term_a_b] :
( ( rewrit5000278123254648670_b_a_b @ R2 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ L @ R ) @ R2 )
=> ( term_ground_a_b @ R ) ) ) ).
% llrg_ground_rhs
thf(fact_692_llrg__srsteps__with__root__step__ground,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,S: term_a_b,T: term_a_b,F: set_Pr4934435412358123699_a_nat] :
( ( rewrit5000278123254648670_b_a_b @ R2 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( srstep7844470518422762656ep_a_b @ F @ R2 ) )
=> ( term_ground_a_b @ T ) ) ) ).
% llrg_srsteps_with_root_step_ground
thf(fact_693_llrg__rrsteps__groundness,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,S: term_a_b,T: term_a_b,F: set_Pr4934435412358123699_a_nat] :
( ( rewrit5000278123254648670_b_a_b @ R2 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( sig_step_a_b @ F @ ( rrstep_a_b @ R2 ) ) )
=> ( term_ground_a_b @ T ) ) ) ).
% llrg_rrsteps_groundness
thf(fact_694_rtrancl__insert,axiom,
! [A: term_a_b,B: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( transi7742714808557438673rm_a_b @ ( insert7009541432154983385rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R ) )
= ( sup_su6776935440552674877rm_a_b @ ( transi7742714808557438673rm_a_b @ R )
@ ( collec99177395925924084rm_a_b
@ ( produc6251846659952913346_a_b_o
@ ^ [X2: term_a_b,Y4: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ A ) @ ( transi7742714808557438673rm_a_b @ R ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ Y4 ) @ ( transi7742714808557438673rm_a_b @ R ) ) ) ) ) ) ) ).
% rtrancl_insert
thf(fact_695_trancl__insert2,axiom,
! [A: term_a_b,B: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( transi7922773638565587891rm_a_b @ ( insert7009541432154983385rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R ) )
= ( sup_su6776935440552674877rm_a_b @ ( transi7922773638565587891rm_a_b @ R )
@ ( collec99177395925924084rm_a_b
@ ( produc6251846659952913346_a_b_o
@ ^ [X2: term_a_b,Y4: term_a_b] :
( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ A ) @ ( transi7922773638565587891rm_a_b @ R ) )
| ( X2 = A ) )
& ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ Y4 ) @ ( transi7922773638565587891rm_a_b @ R ) )
| ( Y4 = B ) ) ) ) ) ) ) ).
% trancl_insert2
thf(fact_696_ground__substI,axiom,
! [S: term_a_b,Sigma3: b > term_a_b] :
( ! [X3: b] :
( ( member_b @ X3 @ ( vars_term_a_b @ S ) )
=> ( term_ground_a_b @ ( Sigma3 @ X3 ) ) )
=> ( term_ground_a_b @ ( subst_7999470309526761004_a_b_b @ S @ Sigma3 ) ) ) ).
% ground_substI
thf(fact_697_llrg__monadic__rsteps__groundness,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,F: set_Pr4934435412358123699_a_nat,S: term_a_b,T: term_a_b] :
( ( rewrit5000278123254648670_b_a_b @ R2 )
=> ( ( rewrit3916899077429870363adic_a @ F )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( transi7922773638565587891rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R2 ) ) ) )
=> ( term_ground_a_b @ T ) ) ) ) ).
% llrg_monadic_rsteps_groundness
thf(fact_698_ground__substD,axiom,
! [L: term_a_b,Sigma3: b > term_a_b,X: b] :
( ( term_ground_a_b @ ( subst_7999470309526761004_a_b_b @ L @ Sigma3 ) )
=> ( ( member_b @ X @ ( vars_term_a_b @ L ) )
=> ( term_ground_a_b @ ( Sigma3 @ X ) ) ) ) ).
% ground_substD
thf(fact_699_llrg__monadic__rstep__pres__groundness,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,F: set_Pr4934435412358123699_a_nat,S: term_a_b,T: term_a_b] :
( ( rewrit5000278123254648670_b_a_b @ R2 )
=> ( ( rewrit3916899077429870363adic_a @ F )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( sig_step_a_b @ F @ ( rstep_a_b @ R2 ) ) )
=> ( term_ground_a_b @ T ) ) ) ) ).
% llrg_monadic_rstep_pres_groundness
thf(fact_700_strongly__confluentI,axiom,
! [R: set_Pr4386577575007340137rm_a_b] :
( ! [X3: term_a_b,Y3: term_a_b,Z4: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ R )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Z4 ) @ R )
=> ? [U4: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ U4 ) @ ( transi7742714808557438673rm_a_b @ R ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Z4 @ U4 ) @ ( sup_su6776935440552674877rm_a_b @ R @ id_term_a_b ) ) ) ) )
=> ( abstra5664096641628173427rm_a_b @ R @ top_top_set_term_a_b ) ) ).
% strongly_confluentI
thf(fact_701_UNIV__I,axiom,
! [X: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ X @ top_to1314267278846557113rm_a_b ) ).
% UNIV_I
thf(fact_702_UNIV__I,axiom,
! [X: b] : ( member_b @ X @ top_top_set_b ) ).
% UNIV_I
thf(fact_703_top__greatest,axiom,
! [A: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ A @ top_to3353692345378799459_a_nat ) ).
% top_greatest
thf(fact_704_top_Oextremum__unique,axiom,
! [A: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ top_to3353692345378799459_a_nat @ A )
= ( A = top_to3353692345378799459_a_nat ) ) ).
% top.extremum_unique
thf(fact_705_top_Oextremum__uniqueI,axiom,
! [A: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ top_to3353692345378799459_a_nat @ A )
=> ( A = top_to3353692345378799459_a_nat ) ) ).
% top.extremum_uniqueI
thf(fact_706_UNIV__eq__I,axiom,
! [A5: set_Pr4386577575007340137rm_a_b] :
( ! [X3: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ X3 @ A5 )
=> ( top_to1314267278846557113rm_a_b = A5 ) ) ).
% UNIV_eq_I
thf(fact_707_UNIV__eq__I,axiom,
! [A5: set_b] :
( ! [X3: b] : ( member_b @ X3 @ A5 )
=> ( top_top_set_b = A5 ) ) ).
% UNIV_eq_I
thf(fact_708_UNIV__witness,axiom,
? [X3: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ X3 @ top_to1314267278846557113rm_a_b ) ).
% UNIV_witness
thf(fact_709_UNIV__witness,axiom,
? [X3: b] : ( member_b @ X3 @ top_top_set_b ) ).
% UNIV_witness
thf(fact_710_subset__UNIV,axiom,
! [A5: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ A5 @ top_to3353692345378799459_a_nat ) ).
% subset_UNIV
thf(fact_711_strongly__confluent__E1n,axiom,
! [R: set_Pr4386577575007340137rm_a_b,X: term_a_b,Y: 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 @ X @ Y ) @ ( sup_su6776935440552674877rm_a_b @ R @ id_term_a_b ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z2 ) @ ( compow4057154403645558940rm_a_b @ N @ R ) )
=> ? [U2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ 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_712_acyclic__insert,axiom,
! [Y: term_a_b,X: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( transi5314701259734769157rm_a_b @ ( insert7009541432154983385rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X ) @ R ) )
= ( ( transi5314701259734769157rm_a_b @ R )
& ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ R ) ) ) ) ).
% acyclic_insert
thf(fact_713_insert__Times__insert,axiom,
! [A: term_a_b,A5: set_term_a_b,B: term_a_b,B4: set_term_a_b] :
( ( produc1953217797448512882rm_a_b @ ( insert_term_a_b @ A @ A5 )
@ ^ [Uu: term_a_b] : ( insert_term_a_b @ B @ B4 ) )
= ( insert7009541432154983385rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B )
@ ( sup_su6776935440552674877rm_a_b
@ ( produc1953217797448512882rm_a_b @ A5
@ ^ [Uu: term_a_b] : ( insert_term_a_b @ B @ B4 ) )
@ ( produc1953217797448512882rm_a_b @ ( insert_term_a_b @ A @ A5 )
@ ^ [Uu: term_a_b] : B4 ) ) ) ) ).
% insert_Times_insert
thf(fact_714_monadic__sig__funas__term__mt,axiom,
! [F: set_Pr4934435412358123699_a_nat,S: term_a_b] :
( ( rewrit3916899077429870363adic_a @ F )
=> ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ S ) @ F )
=> ( rewrit3151024789171632623rm_a_b @ S ) ) ) ).
% monadic_sig_funas_term_mt
thf(fact_715_mem__Sigma__iff,axiom,
! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,A5: set_Pr4386577575007340137rm_a_b,B4: produc357393685978478089rm_a_b > set_Pr4386577575007340137rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( produc5745020764710804872rm_a_b @ A5 @ B4 ) )
= ( ( member5869715511025134514rm_a_b @ A @ A5 )
& ( member5869715511025134514rm_a_b @ B @ ( B4 @ A ) ) ) ) ).
% mem_Sigma_iff
thf(fact_716_mem__Sigma__iff,axiom,
! [A: produc357393685978478089rm_a_b,B: b,A5: set_Pr4386577575007340137rm_a_b,B4: produc357393685978478089rm_a_b > set_b] :
( ( member3184567406997941638_a_b_b @ ( produc483848491608174797_a_b_b @ A @ B ) @ ( produc2289974294304433406_a_b_b @ A5 @ B4 ) )
= ( ( member5869715511025134514rm_a_b @ A @ A5 )
& ( member_b @ B @ ( B4 @ A ) ) ) ) ).
% mem_Sigma_iff
thf(fact_717_mem__Sigma__iff,axiom,
! [A: b,B: produc357393685978478089rm_a_b,A5: set_b,B4: b > set_Pr4386577575007340137rm_a_b] :
( ( member8210134394708959770rm_a_b @ ( produc2673992483936284769rm_a_b @ A @ B ) @ ( produc4480118286632543378rm_a_b @ A5 @ B4 ) )
= ( ( member_b @ A @ A5 )
& ( member5869715511025134514rm_a_b @ B @ ( B4 @ A ) ) ) ) ).
% mem_Sigma_iff
thf(fact_718_mem__Sigma__iff,axiom,
! [A: b,B: b,A5: set_b,B4: b > set_b] :
( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ A @ B ) @ ( product_Sigma_b_b @ A5 @ B4 ) )
= ( ( member_b @ A @ A5 )
& ( member_b @ B @ ( B4 @ A ) ) ) ) ).
% mem_Sigma_iff
thf(fact_719_mem__Sigma__iff,axiom,
! [A: term_a_b,B: term_a_b,A5: set_term_a_b,B4: term_a_b > set_term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( produc1953217797448512882rm_a_b @ A5 @ B4 ) )
= ( ( member_term_a_b @ A @ A5 )
& ( member_term_a_b @ B @ ( B4 @ A ) ) ) ) ).
% mem_Sigma_iff
thf(fact_720_SigmaI,axiom,
! [A: produc357393685978478089rm_a_b,A5: set_Pr4386577575007340137rm_a_b,B: produc357393685978478089rm_a_b,B4: produc357393685978478089rm_a_b > set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ A @ A5 )
=> ( ( member5869715511025134514rm_a_b @ B @ ( B4 @ A ) )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( produc5745020764710804872rm_a_b @ A5 @ B4 ) ) ) ) ).
% SigmaI
thf(fact_721_SigmaI,axiom,
! [A: produc357393685978478089rm_a_b,A5: set_Pr4386577575007340137rm_a_b,B: b,B4: produc357393685978478089rm_a_b > set_b] :
( ( member5869715511025134514rm_a_b @ A @ A5 )
=> ( ( member_b @ B @ ( B4 @ A ) )
=> ( member3184567406997941638_a_b_b @ ( produc483848491608174797_a_b_b @ A @ B ) @ ( produc2289974294304433406_a_b_b @ A5 @ B4 ) ) ) ) ).
% SigmaI
thf(fact_722_SigmaI,axiom,
! [A: b,A5: set_b,B: produc357393685978478089rm_a_b,B4: b > set_Pr4386577575007340137rm_a_b] :
( ( member_b @ A @ A5 )
=> ( ( member5869715511025134514rm_a_b @ B @ ( B4 @ A ) )
=> ( member8210134394708959770rm_a_b @ ( produc2673992483936284769rm_a_b @ A @ B ) @ ( produc4480118286632543378rm_a_b @ A5 @ B4 ) ) ) ) ).
% SigmaI
thf(fact_723_SigmaI,axiom,
! [A: b,A5: set_b,B: b,B4: b > set_b] :
( ( member_b @ A @ A5 )
=> ( ( member_b @ B @ ( B4 @ A ) )
=> ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ A @ B ) @ ( product_Sigma_b_b @ A5 @ B4 ) ) ) ) ).
% SigmaI
thf(fact_724_SigmaI,axiom,
! [A: term_a_b,A5: set_term_a_b,B: term_a_b,B4: term_a_b > set_term_a_b] :
( ( member_term_a_b @ A @ A5 )
=> ( ( member_term_a_b @ B @ ( B4 @ A ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( produc1953217797448512882rm_a_b @ A5 @ B4 ) ) ) ) ).
% SigmaI
thf(fact_725_wf__insert,axiom,
! [Y: term_a_b,X: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( wf_term_a_b @ ( insert7009541432154983385rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X ) @ R ) )
= ( ( wf_term_a_b @ R )
& ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ R ) ) ) ) ).
% wf_insert
thf(fact_726_top__empty__eq,axiom,
( top_to1262317115727556708_a_b_o
= ( ^ [X2: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ X2 @ top_to1314267278846557113rm_a_b ) ) ) ).
% top_empty_eq
thf(fact_727_top__empty__eq,axiom,
( top_top_b_o
= ( ^ [X2: b] : ( member_b @ X2 @ top_top_set_b ) ) ) ).
% top_empty_eq
thf(fact_728_relpow__Suc__D2_H,axiom,
! [N: nat,R5: set_Pr4386577575007340137rm_a_b,X4: term_a_b,Y8: term_a_b,Z6: term_a_b] :
( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X4 @ Y8 ) @ ( compow4057154403645558940rm_a_b @ N @ R5 ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y8 @ Z6 ) @ R5 ) )
=> ? [W: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X4 @ W ) @ R5 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ W @ Z6 ) @ ( compow4057154403645558940rm_a_b @ N @ R5 ) ) ) ) ).
% relpow_Suc_D2'
thf(fact_729_relpow__image,axiom,
! [R: set_Pr4386577575007340137rm_a_b,F2: term_a_b > term_a_b,R8: set_Pr4386577575007340137rm_a_b,S: term_a_b,T: term_a_b,N: nat] :
( ! [S6: term_a_b,T2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S6 @ T2 ) @ R )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F2 @ S6 ) @ ( F2 @ T2 ) ) @ R8 ) )
=> ( ( 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 @ R8 ) ) ) ) ).
% relpow_image
thf(fact_730_relpow__refl__mono,axiom,
! [Rel: set_Pr4386577575007340137rm_a_b,M: nat,N: nat,A: term_a_b,B: term_a_b] :
( ! [X3: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ X3 ) @ Rel )
=> ( ( ord_less_eq_nat @ M @ N )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( compow4057154403645558940rm_a_b @ M @ Rel ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( compow4057154403645558940rm_a_b @ N @ Rel ) ) ) ) ) ).
% relpow_refl_mono
thf(fact_731_SigmaE,axiom,
! [C2: produc4523971788519308903rm_a_b,A5: set_Pr4386577575007340137rm_a_b,B4: produc357393685978478089rm_a_b > set_Pr4386577575007340137rm_a_b] :
( ( member8417600551952982416rm_a_b @ C2 @ ( produc5745020764710804872rm_a_b @ A5 @ B4 ) )
=> ~ ! [X3: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X3 @ A5 )
=> ! [Y3: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ Y3 @ ( B4 @ X3 ) )
=> ( C2
!= ( produc1763473618796451543rm_a_b @ X3 @ Y3 ) ) ) ) ) ).
% SigmaE
thf(fact_732_SigmaE,axiom,
! [C2: produc8054098559122065629_a_b_b,A5: set_Pr4386577575007340137rm_a_b,B4: produc357393685978478089rm_a_b > set_b] :
( ( member3184567406997941638_a_b_b @ C2 @ ( produc2289974294304433406_a_b_b @ A5 @ B4 ) )
=> ~ ! [X3: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X3 @ A5 )
=> ! [Y3: b] :
( ( member_b @ Y3 @ ( B4 @ X3 ) )
=> ( C2
!= ( produc483848491608174797_a_b_b @ X3 @ Y3 ) ) ) ) ) ).
% SigmaE
thf(fact_733_SigmaE,axiom,
! [C2: produc3856293509978307953rm_a_b,A5: set_b,B4: b > set_Pr4386577575007340137rm_a_b] :
( ( member8210134394708959770rm_a_b @ C2 @ ( produc4480118286632543378rm_a_b @ A5 @ B4 ) )
=> ~ ! [X3: b] :
( ( member_b @ X3 @ A5 )
=> ! [Y3: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ Y3 @ ( B4 @ X3 ) )
=> ( C2
!= ( produc2673992483936284769rm_a_b @ X3 @ Y3 ) ) ) ) ) ).
% SigmaE
thf(fact_734_SigmaE,axiom,
! [C2: product_prod_b_b,A5: set_b,B4: b > set_b] :
( ( member7862447936710763792od_b_b @ C2 @ ( product_Sigma_b_b @ A5 @ B4 ) )
=> ~ ! [X3: b] :
( ( member_b @ X3 @ A5 )
=> ! [Y3: b] :
( ( member_b @ Y3 @ ( B4 @ X3 ) )
=> ( C2
!= ( product_Pair_b_b @ X3 @ Y3 ) ) ) ) ) ).
% SigmaE
thf(fact_735_SigmaE,axiom,
! [C2: produc357393685978478089rm_a_b,A5: set_term_a_b,B4: term_a_b > set_term_a_b] :
( ( member5869715511025134514rm_a_b @ C2 @ ( produc1953217797448512882rm_a_b @ A5 @ B4 ) )
=> ~ ! [X3: term_a_b] :
( ( member_term_a_b @ X3 @ A5 )
=> ! [Y3: term_a_b] :
( ( member_term_a_b @ Y3 @ ( B4 @ X3 ) )
=> ( C2
!= ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) ) ) ) ) ).
% SigmaE
thf(fact_736_SigmaD1,axiom,
! [A: term_a_b,B: term_a_b,A5: set_term_a_b,B4: term_a_b > set_term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( produc1953217797448512882rm_a_b @ A5 @ B4 ) )
=> ( member_term_a_b @ A @ A5 ) ) ).
% SigmaD1
thf(fact_737_SigmaD2,axiom,
! [A: term_a_b,B: term_a_b,A5: set_term_a_b,B4: term_a_b > set_term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( produc1953217797448512882rm_a_b @ A5 @ B4 ) )
=> ( member_term_a_b @ B @ ( B4 @ A ) ) ) ).
% SigmaD2
thf(fact_738_SigmaE2,axiom,
! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,A5: set_Pr4386577575007340137rm_a_b,B4: produc357393685978478089rm_a_b > set_Pr4386577575007340137rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( produc5745020764710804872rm_a_b @ A5 @ B4 ) )
=> ~ ( ( member5869715511025134514rm_a_b @ A @ A5 )
=> ~ ( member5869715511025134514rm_a_b @ B @ ( B4 @ A ) ) ) ) ).
% SigmaE2
thf(fact_739_SigmaE2,axiom,
! [A: produc357393685978478089rm_a_b,B: b,A5: set_Pr4386577575007340137rm_a_b,B4: produc357393685978478089rm_a_b > set_b] :
( ( member3184567406997941638_a_b_b @ ( produc483848491608174797_a_b_b @ A @ B ) @ ( produc2289974294304433406_a_b_b @ A5 @ B4 ) )
=> ~ ( ( member5869715511025134514rm_a_b @ A @ A5 )
=> ~ ( member_b @ B @ ( B4 @ A ) ) ) ) ).
% SigmaE2
thf(fact_740_SigmaE2,axiom,
! [A: b,B: produc357393685978478089rm_a_b,A5: set_b,B4: b > set_Pr4386577575007340137rm_a_b] :
( ( member8210134394708959770rm_a_b @ ( produc2673992483936284769rm_a_b @ A @ B ) @ ( produc4480118286632543378rm_a_b @ A5 @ B4 ) )
=> ~ ( ( member_b @ A @ A5 )
=> ~ ( member5869715511025134514rm_a_b @ B @ ( B4 @ A ) ) ) ) ).
% SigmaE2
thf(fact_741_SigmaE2,axiom,
! [A: b,B: b,A5: set_b,B4: b > set_b] :
( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ A @ B ) @ ( product_Sigma_b_b @ A5 @ B4 ) )
=> ~ ( ( member_b @ A @ A5 )
=> ~ ( member_b @ B @ ( B4 @ A ) ) ) ) ).
% SigmaE2
thf(fact_742_SigmaE2,axiom,
! [A: term_a_b,B: term_a_b,A5: set_term_a_b,B4: term_a_b > set_term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( produc1953217797448512882rm_a_b @ A5 @ B4 ) )
=> ~ ( ( member_term_a_b @ A @ A5 )
=> ~ ( member_term_a_b @ B @ ( B4 @ A ) ) ) ) ).
% SigmaE2
thf(fact_743_wfI,axiom,
! [R: set_Pr2972776593051762503rm_a_b,A5: set_Pr4386577575007340137rm_a_b,B4: set_Pr4386577575007340137rm_a_b] :
( ( ord_le2593958535671617255rm_a_b @ R
@ ( produc5745020764710804872rm_a_b @ A5
@ ^ [Uu: produc357393685978478089rm_a_b] : B4 ) )
=> ( ! [X3: produc357393685978478089rm_a_b,P3: produc357393685978478089rm_a_b > $o] :
( ! [Xa2: produc357393685978478089rm_a_b] :
( ! [Y3: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y3 @ Xa2 ) @ R )
=> ( P3 @ Y3 ) )
=> ( P3 @ Xa2 ) )
=> ( ( member5869715511025134514rm_a_b @ X3 @ A5 )
=> ( ( member5869715511025134514rm_a_b @ X3 @ B4 )
=> ( P3 @ X3 ) ) ) )
=> ( wf_Pro2335863617654816626rm_a_b @ R ) ) ) ).
% wfI
thf(fact_744_wfI,axiom,
! [R: set_Product_prod_b_b,A5: set_b,B4: set_b] :
( ( ord_le182087997850975847od_b_b @ R
@ ( product_Sigma_b_b @ A5
@ ^ [Uu: b] : B4 ) )
=> ( ! [X3: b,P3: b > $o] :
( ! [Xa2: b] :
( ! [Y3: b] :
( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ Y3 @ Xa2 ) @ R )
=> ( P3 @ Y3 ) )
=> ( P3 @ Xa2 ) )
=> ( ( member_b @ X3 @ A5 )
=> ( ( member_b @ X3 @ B4 )
=> ( P3 @ X3 ) ) ) )
=> ( wf_b @ R ) ) ) ).
% wfI
thf(fact_745_wfI,axiom,
! [R: set_Pr4386577575007340137rm_a_b,A5: set_term_a_b,B4: set_term_a_b] :
( ( ord_le118470702582115849rm_a_b @ R
@ ( produc1953217797448512882rm_a_b @ A5
@ ^ [Uu: term_a_b] : B4 ) )
=> ( ! [X3: term_a_b,P3: term_a_b > $o] :
( ! [Xa2: term_a_b] :
( ! [Y3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ Xa2 ) @ R )
=> ( P3 @ Y3 ) )
=> ( P3 @ Xa2 ) )
=> ( ( member_term_a_b @ X3 @ A5 )
=> ( ( member_term_a_b @ X3 @ B4 )
=> ( P3 @ X3 ) ) ) )
=> ( wf_term_a_b @ R ) ) ) ).
% wfI
thf(fact_746_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 )
=> ( ! [X3: term_a_b,Y3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ R )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) @ S ) )
=> ( wf_term_a_b @ R ) ) ) ).
% wf_if_convertible_to_wf
thf(fact_747_wf__induct__rule,axiom,
! [R: set_Pr4386577575007340137rm_a_b,P: term_a_b > $o,A: term_a_b] :
( ( wf_term_a_b @ R )
=> ( ! [X3: term_a_b] :
( ! [Y8: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y8 @ X3 ) @ R )
=> ( P @ Y8 ) )
=> ( P @ X3 ) )
=> ( P @ A ) ) ) ).
% wf_induct_rule
thf(fact_748_wf__eq__minimal,axiom,
( wf_Pro2335863617654816626rm_a_b
= ( ^ [R4: set_Pr2972776593051762503rm_a_b] :
! [Q2: set_Pr4386577575007340137rm_a_b] :
( ? [X2: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ X2 @ Q2 )
=> ? [X2: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X2 @ Q2 )
& ! [Y4: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y4 @ X2 ) @ R4 )
=> ~ ( member5869715511025134514rm_a_b @ Y4 @ Q2 ) ) ) ) ) ) ).
% wf_eq_minimal
thf(fact_749_wf__eq__minimal,axiom,
( wf_b
= ( ^ [R4: set_Product_prod_b_b] :
! [Q2: set_b] :
( ? [X2: b] : ( member_b @ X2 @ Q2 )
=> ? [X2: b] :
( ( member_b @ X2 @ Q2 )
& ! [Y4: b] :
( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ Y4 @ X2 ) @ R4 )
=> ~ ( member_b @ Y4 @ Q2 ) ) ) ) ) ) ).
% wf_eq_minimal
thf(fact_750_wf__eq__minimal,axiom,
( wf_term_a_b
= ( ^ [R4: set_Pr4386577575007340137rm_a_b] :
! [Q2: set_term_a_b] :
( ? [X2: term_a_b] : ( member_term_a_b @ X2 @ Q2 )
=> ? [X2: term_a_b] :
( ( member_term_a_b @ X2 @ Q2 )
& ! [Y4: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y4 @ X2 ) @ R4 )
=> ~ ( member_term_a_b @ Y4 @ Q2 ) ) ) ) ) ) ).
% wf_eq_minimal
thf(fact_751_wf__not__refl,axiom,
! [R: set_Pr4386577575007340137rm_a_b,A: term_a_b] :
( ( wf_term_a_b @ R )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ A ) @ R ) ) ).
% wf_not_refl
thf(fact_752_wf__not__sym,axiom,
! [R: set_Pr4386577575007340137rm_a_b,A: term_a_b,X: term_a_b] :
( ( wf_term_a_b @ R )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ X ) @ R )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ A ) @ R ) ) ) ).
% wf_not_sym
thf(fact_753_wf__irrefl,axiom,
! [R: set_Pr4386577575007340137rm_a_b,A: term_a_b] :
( ( wf_term_a_b @ R )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ A ) @ R ) ) ).
% wf_irrefl
thf(fact_754_wf__induct,axiom,
! [R: set_Pr4386577575007340137rm_a_b,P: term_a_b > $o,A: term_a_b] :
( ( wf_term_a_b @ R )
=> ( ! [X3: term_a_b] :
( ! [Y8: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y8 @ X3 ) @ R )
=> ( P @ Y8 ) )
=> ( P @ X3 ) )
=> ( P @ A ) ) ) ).
% wf_induct
thf(fact_755_wf__asym,axiom,
! [R: set_Pr4386577575007340137rm_a_b,A: term_a_b,X: term_a_b] :
( ( wf_term_a_b @ R )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ X ) @ R )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ A ) @ R ) ) ) ).
% wf_asym
thf(fact_756_wfUNIVI,axiom,
! [R: set_Pr4386577575007340137rm_a_b] :
( ! [P3: term_a_b > $o,X3: term_a_b] :
( ! [Xa2: term_a_b] :
( ! [Y3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ Xa2 ) @ R )
=> ( P3 @ Y3 ) )
=> ( P3 @ Xa2 ) )
=> ( P3 @ X3 ) )
=> ( wf_term_a_b @ R ) ) ).
% wfUNIVI
thf(fact_757_wfI__min,axiom,
! [R5: set_Pr2972776593051762503rm_a_b] :
( ! [X3: produc357393685978478089rm_a_b,Q3: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ X3 @ Q3 )
=> ? [Xa2: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ Xa2 @ Q3 )
& ! [Y3: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y3 @ Xa2 ) @ R5 )
=> ~ ( member5869715511025134514rm_a_b @ Y3 @ Q3 ) ) ) )
=> ( wf_Pro2335863617654816626rm_a_b @ R5 ) ) ).
% wfI_min
thf(fact_758_wfI__min,axiom,
! [R5: set_Product_prod_b_b] :
( ! [X3: b,Q3: set_b] :
( ( member_b @ X3 @ Q3 )
=> ? [Xa2: b] :
( ( member_b @ Xa2 @ Q3 )
& ! [Y3: b] :
( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ Y3 @ Xa2 ) @ R5 )
=> ~ ( member_b @ Y3 @ Q3 ) ) ) )
=> ( wf_b @ R5 ) ) ).
% wfI_min
thf(fact_759_wfI__min,axiom,
! [R5: set_Pr4386577575007340137rm_a_b] :
( ! [X3: term_a_b,Q3: set_term_a_b] :
( ( member_term_a_b @ X3 @ Q3 )
=> ? [Xa2: term_a_b] :
( ( member_term_a_b @ Xa2 @ Q3 )
& ! [Y3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ Xa2 ) @ R5 )
=> ~ ( member_term_a_b @ Y3 @ Q3 ) ) ) )
=> ( wf_term_a_b @ R5 ) ) ).
% wfI_min
thf(fact_760_wfE__min,axiom,
! [R5: set_Pr2972776593051762503rm_a_b,X: produc357393685978478089rm_a_b,Q: set_Pr4386577575007340137rm_a_b] :
( ( wf_Pro2335863617654816626rm_a_b @ R5 )
=> ( ( member5869715511025134514rm_a_b @ X @ Q )
=> ~ ! [Z4: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ Z4 @ Q )
=> ~ ! [Y8: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y8 @ Z4 ) @ R5 )
=> ~ ( member5869715511025134514rm_a_b @ Y8 @ Q ) ) ) ) ) ).
% wfE_min
thf(fact_761_wfE__min,axiom,
! [R5: set_Product_prod_b_b,X: b,Q: set_b] :
( ( wf_b @ R5 )
=> ( ( member_b @ X @ Q )
=> ~ ! [Z4: b] :
( ( member_b @ Z4 @ Q )
=> ~ ! [Y8: b] :
( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ Y8 @ Z4 ) @ R5 )
=> ~ ( member_b @ Y8 @ Q ) ) ) ) ) ).
% wfE_min
thf(fact_762_wfE__min,axiom,
! [R5: set_Pr4386577575007340137rm_a_b,X: term_a_b,Q: set_term_a_b] :
( ( wf_term_a_b @ R5 )
=> ( ( member_term_a_b @ X @ Q )
=> ~ ! [Z4: term_a_b] :
( ( member_term_a_b @ Z4 @ Q )
=> ~ ! [Y8: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y8 @ Z4 ) @ R5 )
=> ~ ( member_term_a_b @ Y8 @ Q ) ) ) ) ) ).
% wfE_min
thf(fact_763_wf__def,axiom,
( wf_term_a_b
= ( ^ [R4: set_Pr4386577575007340137rm_a_b] :
! [P4: term_a_b > $o] :
( ! [X2: term_a_b] :
( ! [Y4: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y4 @ X2 ) @ R4 )
=> ( P4 @ Y4 ) )
=> ( P4 @ X2 ) )
=> ! [X6: term_a_b] : ( P4 @ X6 ) ) ) ) ).
% wf_def
thf(fact_764_rtrancl__power,axiom,
! [P2: produc357393685978478089rm_a_b,R5: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ P2 @ ( transi7742714808557438673rm_a_b @ R5 ) )
= ( ? [N2: nat] : ( member5869715511025134514rm_a_b @ P2 @ ( compow4057154403645558940rm_a_b @ N2 @ R5 ) ) ) ) ).
% rtrancl_power
thf(fact_765_relpow__imp__rtrancl,axiom,
! [P2: produc357393685978478089rm_a_b,N: nat,R5: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ P2 @ ( compow4057154403645558940rm_a_b @ N @ R5 ) )
=> ( member5869715511025134514rm_a_b @ P2 @ ( transi7742714808557438673rm_a_b @ R5 ) ) ) ).
% relpow_imp_rtrancl
thf(fact_766_rtrancl__imp__relpow,axiom,
! [P2: produc357393685978478089rm_a_b,R5: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ P2 @ ( transi7742714808557438673rm_a_b @ R5 ) )
=> ? [N3: nat] : ( member5869715511025134514rm_a_b @ P2 @ ( compow4057154403645558940rm_a_b @ N3 @ R5 ) ) ) ).
% rtrancl_imp_relpow
thf(fact_767_top__empty__eq2,axiom,
( top_to7072192486108705198_a_b_o
= ( ^ [X2: term_a_b,Y4: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y4 ) @ top_to1314267278846557113rm_a_b ) ) ) ).
% top_empty_eq2
thf(fact_768_Sigma__mono,axiom,
! [A5: set_a,C3: set_a,B4: a > set_nat,D: a > set_nat] :
( ( ord_less_eq_set_a @ A5 @ C3 )
=> ( ! [X3: a] :
( ( member_a @ X3 @ A5 )
=> ( ord_less_eq_set_nat @ ( B4 @ X3 ) @ ( D @ X3 ) ) )
=> ( ord_le8666007276011122963_a_nat @ ( product_Sigma_a_nat @ A5 @ B4 ) @ ( product_Sigma_a_nat @ C3 @ D ) ) ) ) ).
% Sigma_mono
thf(fact_769_Sigma__mono,axiom,
! [A5: set_Pr4386577575007340137rm_a_b,C3: set_Pr4386577575007340137rm_a_b,B4: produc357393685978478089rm_a_b > set_Pr4934435412358123699_a_nat,D: produc357393685978478089rm_a_b > set_Pr4934435412358123699_a_nat] :
( ( ord_le118470702582115849rm_a_b @ A5 @ C3 )
=> ( ! [X3: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X3 @ A5 )
=> ( ord_le8666007276011122963_a_nat @ ( B4 @ X3 ) @ ( D @ X3 ) ) )
=> ( ord_le3843850037797373259_a_nat @ ( produc3981687593757823422_a_nat @ A5 @ B4 ) @ ( produc3981687593757823422_a_nat @ C3 @ D ) ) ) ) ).
% Sigma_mono
thf(fact_770_Sigma__mono,axiom,
! [A5: set_b,C3: set_b,B4: b > set_Pr4934435412358123699_a_nat,D: b > set_Pr4934435412358123699_a_nat] :
( ( ord_less_eq_set_b @ A5 @ C3 )
=> ( ! [X3: b] :
( ( member_b @ X3 @ A5 )
=> ( ord_le8666007276011122963_a_nat @ ( B4 @ X3 ) @ ( D @ X3 ) ) )
=> ( ord_le8567420702850673729_a_nat @ ( produc4149643547837505332_a_nat @ A5 @ B4 ) @ ( produc4149643547837505332_a_nat @ C3 @ D ) ) ) ) ).
% Sigma_mono
thf(fact_771_Sigma__mono,axiom,
! [A5: set_Pr4934435412358123699_a_nat,C3: set_Pr4934435412358123699_a_nat,B4: product_prod_a_nat > set_Pr4934435412358123699_a_nat,D: product_prod_a_nat > set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A5 @ C3 )
=> ( ! [X3: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ X3 @ A5 )
=> ( ord_le8666007276011122963_a_nat @ ( B4 @ X3 ) @ ( D @ X3 ) ) )
=> ( ord_le1018141841545957851_a_nat @ ( produc3832836940518797764_a_nat @ A5 @ B4 ) @ ( produc3832836940518797764_a_nat @ C3 @ D ) ) ) ) ).
% Sigma_mono
thf(fact_772_rtrancl__len__E,axiom,
! [X: term_a_b,Y: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ~ ! [N3: nat] :
~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( compow4057154403645558940rm_a_b @ N3 @ R ) ) ) ).
% rtrancl_len_E
thf(fact_773_Times__subset__cancel2,axiom,
! [X: produc357393685978478089rm_a_b,C3: set_Pr4386577575007340137rm_a_b,A5: set_Pr4934435412358123699_a_nat,B4: set_Pr4934435412358123699_a_nat] :
( ( member5869715511025134514rm_a_b @ X @ C3 )
=> ( ( ord_le3089357705885668951rm_a_b
@ ( produc4180933298386230786rm_a_b @ A5
@ ^ [Uu: product_prod_a_nat] : C3 )
@ ( produc4180933298386230786rm_a_b @ B4
@ ^ [Uu: product_prod_a_nat] : C3 ) )
= ( ord_le8666007276011122963_a_nat @ A5 @ B4 ) ) ) ).
% Times_subset_cancel2
thf(fact_774_Times__subset__cancel2,axiom,
! [X: b,C3: set_b,A5: set_Pr4934435412358123699_a_nat,B4: set_Pr4934435412358123699_a_nat] :
( ( member_b @ X @ C3 )
=> ( ( ord_le7516853312212197709_nat_b
@ ( produc7877981437133276280_nat_b @ A5
@ ^ [Uu: product_prod_a_nat] : C3 )
@ ( produc7877981437133276280_nat_b @ B4
@ ^ [Uu: product_prod_a_nat] : C3 ) )
= ( ord_le8666007276011122963_a_nat @ A5 @ B4 ) ) ) ).
% Times_subset_cancel2
thf(fact_775_Times__subset__cancel2,axiom,
! [X: nat,C3: set_nat,A5: set_a,B4: set_a] :
( ( member_nat @ X @ C3 )
=> ( ( ord_le8666007276011122963_a_nat
@ ( product_Sigma_a_nat @ A5
@ ^ [Uu: a] : C3 )
@ ( product_Sigma_a_nat @ B4
@ ^ [Uu: a] : C3 ) )
= ( ord_less_eq_set_a @ A5 @ B4 ) ) ) ).
% Times_subset_cancel2
thf(fact_776_relcomp__subset__Sigma,axiom,
! [R: set_Product_prod_a_a,A5: set_a,B4: set_a,S: set_Pr4934435412358123699_a_nat,C3: set_nat] :
( ( ord_le746702958409616551od_a_a @ R
@ ( product_Sigma_a_a @ A5
@ ^ [Uu: a] : B4 ) )
=> ( ( ord_le8666007276011122963_a_nat @ S
@ ( product_Sigma_a_nat @ B4
@ ^ [Uu: a] : C3 ) )
=> ( ord_le8666007276011122963_a_nat @ ( relcomp_a_a_nat @ R @ S )
@ ( product_Sigma_a_nat @ A5
@ ^ [Uu: a] : C3 ) ) ) ) ).
% relcomp_subset_Sigma
thf(fact_777_relcomp__subset__Sigma,axiom,
! [R: set_Pr4934435412358123699_a_nat,A5: set_a,B4: set_nat,S: set_Pr1261947904930325089at_nat,C3: set_nat] :
( ( ord_le8666007276011122963_a_nat @ R
@ ( product_Sigma_a_nat @ A5
@ ^ [Uu: a] : B4 ) )
=> ( ( ord_le3146513528884898305at_nat @ S
@ ( produc457027306803732586at_nat @ B4
@ ^ [Uu: nat] : C3 ) )
=> ( ord_le8666007276011122963_a_nat @ ( relcomp_a_nat_nat @ R @ S )
@ ( product_Sigma_a_nat @ A5
@ ^ [Uu: a] : C3 ) ) ) ) ).
% relcomp_subset_Sigma
thf(fact_778_acyclicI,axiom,
! [R: set_Pr4386577575007340137rm_a_b] :
( ! [X3: term_a_b] :
~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ X3 ) @ ( transi7922773638565587891rm_a_b @ R ) )
=> ( transi5314701259734769157rm_a_b @ R ) ) ).
% acyclicI
thf(fact_779_acyclic__def,axiom,
( transi5314701259734769157rm_a_b
= ( ^ [R4: set_Pr4386577575007340137rm_a_b] :
! [X2: term_a_b] :
~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ X2 ) @ ( transi7922773638565587891rm_a_b @ R4 ) ) ) ) ).
% acyclic_def
thf(fact_780_trancl__steps__relpow,axiom,
! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b,X: term_a_b,Y: term_a_b,N: nat] :
( ( ord_le118470702582115849rm_a_b @ A @ ( transi7922773638565587891rm_a_b @ B ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( compow4057154403645558940rm_a_b @ N @ A ) )
=> ? [M2: nat] :
( ( ord_less_eq_nat @ N @ M2 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( compow4057154403645558940rm_a_b @ M2 @ B ) ) ) ) ) ).
% trancl_steps_relpow
thf(fact_781_trancl__subset__Sigma__aux,axiom,
! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,R: set_Pr2972776593051762503rm_a_b,A5: set_Pr4386577575007340137rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( transi2615809358984392588rm_a_b @ R ) )
=> ( ( ord_le2593958535671617255rm_a_b @ R
@ ( produc5745020764710804872rm_a_b @ A5
@ ^ [Uu: produc357393685978478089rm_a_b] : A5 ) )
=> ( ( A = B )
| ( member5869715511025134514rm_a_b @ A @ A5 ) ) ) ) ).
% trancl_subset_Sigma_aux
thf(fact_782_trancl__subset__Sigma__aux,axiom,
! [A: b,B: b,R: set_Product_prod_b_b,A5: set_b] :
( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ A @ B ) @ ( transitive_rtrancl_b @ R ) )
=> ( ( ord_le182087997850975847od_b_b @ R
@ ( product_Sigma_b_b @ A5
@ ^ [Uu: b] : A5 ) )
=> ( ( A = B )
| ( member_b @ A @ A5 ) ) ) ) ).
% trancl_subset_Sigma_aux
thf(fact_783_trancl__subset__Sigma__aux,axiom,
! [A: term_a_b,B: term_a_b,R: set_Pr4386577575007340137rm_a_b,A5: set_term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ( ( ord_le118470702582115849rm_a_b @ R
@ ( produc1953217797448512882rm_a_b @ A5
@ ^ [Uu: term_a_b] : A5 ) )
=> ( ( A = B )
| ( member_term_a_b @ A @ A5 ) ) ) ) ).
% trancl_subset_Sigma_aux
thf(fact_784_gsrsteps__eq__relcomp__to__rsteps__relcomp,axiom,
! [S: term_a_b,T: term_a_b,F: set_Pr4934435412358123699_a_nat,R2: set_Pr4386577575007340137rm_a_b,S4: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T )
@ ( relcom370159955682700863rm_a_b
@ ( transi7742714808557438673rm_a_b
@ ( inf_in1693821330460682071rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R2 ) )
@ ( produc1953217797448512882rm_a_b @ ( collect_term_a_b @ term_ground_a_b )
@ ^ [Uu: term_a_b] : ( collect_term_a_b @ term_ground_a_b ) ) ) )
@ ( transi7742714808557438673rm_a_b
@ ( inf_in1693821330460682071rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ S4 ) )
@ ( produc1953217797448512882rm_a_b @ ( collect_term_a_b @ term_ground_a_b )
@ ^ [Uu: term_a_b] : ( collect_term_a_b @ term_ground_a_b ) ) ) ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( rstep_a_b @ R2 ) ) @ ( transi7742714808557438673rm_a_b @ ( rstep_a_b @ S4 ) ) ) ) ) ).
% gsrsteps_eq_relcomp_to_rsteps_relcomp
thf(fact_785_gsrsteps__eq__relcomp__srsteps__relcompD,axiom,
! [S: term_a_b,T: term_a_b,F: set_Pr4934435412358123699_a_nat,R2: set_Pr4386577575007340137rm_a_b,S4: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T )
@ ( relcom370159955682700863rm_a_b
@ ( transi7742714808557438673rm_a_b
@ ( inf_in1693821330460682071rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R2 ) )
@ ( produc1953217797448512882rm_a_b @ ( collect_term_a_b @ term_ground_a_b )
@ ^ [Uu: term_a_b] : ( collect_term_a_b @ term_ground_a_b ) ) ) )
@ ( transi7742714808557438673rm_a_b
@ ( inf_in1693821330460682071rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ S4 ) )
@ ( produc1953217797448512882rm_a_b @ ( collect_term_a_b @ term_ground_a_b )
@ ^ [Uu: term_a_b] : ( collect_term_a_b @ term_ground_a_b ) ) ) ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R2 ) ) ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ S4 ) ) ) ) ) ) ).
% gsrsteps_eq_relcomp_srsteps_relcompD
thf(fact_786_srsteps__eq__relcomp__gsrsteps__relcomp,axiom,
! [S: term_a_b,T: term_a_b,F: set_Pr4934435412358123699_a_nat,R2: set_Pr4386577575007340137rm_a_b,S4: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R2 ) ) ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ S4 ) ) ) ) )
=> ( ( term_ground_a_b @ S )
=> ( ( term_ground_a_b @ T )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T )
@ ( relcom370159955682700863rm_a_b
@ ( transi7742714808557438673rm_a_b
@ ( inf_in1693821330460682071rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R2 ) )
@ ( produc1953217797448512882rm_a_b @ ( collect_term_a_b @ term_ground_a_b )
@ ^ [Uu: term_a_b] : ( collect_term_a_b @ term_ground_a_b ) ) ) )
@ ( transi7742714808557438673rm_a_b
@ ( inf_in1693821330460682071rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ S4 ) )
@ ( produc1953217797448512882rm_a_b @ ( collect_term_a_b @ term_ground_a_b )
@ ^ [Uu: term_a_b] : ( collect_term_a_b @ term_ground_a_b ) ) ) ) ) ) ) ) ) ).
% srsteps_eq_relcomp_gsrsteps_relcomp
thf(fact_787_IntI,axiom,
! [C2: produc357393685978478089rm_a_b,A5: set_Pr4386577575007340137rm_a_b,B4: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ C2 @ A5 )
=> ( ( member5869715511025134514rm_a_b @ C2 @ B4 )
=> ( member5869715511025134514rm_a_b @ C2 @ ( inf_in1693821330460682071rm_a_b @ A5 @ B4 ) ) ) ) ).
% IntI
thf(fact_788_IntI,axiom,
! [C2: b,A5: set_b,B4: set_b] :
( ( member_b @ C2 @ A5 )
=> ( ( member_b @ C2 @ B4 )
=> ( member_b @ C2 @ ( inf_inf_set_b @ A5 @ B4 ) ) ) ) ).
% IntI
thf(fact_789_Int__iff,axiom,
! [C2: produc357393685978478089rm_a_b,A5: set_Pr4386577575007340137rm_a_b,B4: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ C2 @ ( inf_in1693821330460682071rm_a_b @ A5 @ B4 ) )
= ( ( member5869715511025134514rm_a_b @ C2 @ A5 )
& ( member5869715511025134514rm_a_b @ C2 @ B4 ) ) ) ).
% Int_iff
thf(fact_790_Int__iff,axiom,
! [C2: b,A5: set_b,B4: set_b] :
( ( member_b @ C2 @ ( inf_inf_set_b @ A5 @ B4 ) )
= ( ( member_b @ C2 @ A5 )
& ( member_b @ C2 @ B4 ) ) ) ).
% Int_iff
thf(fact_791_le__inf__iff,axiom,
! [X: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat,Z2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X @ ( inf_in3813033988066259269_a_nat @ Y @ Z2 ) )
= ( ( ord_le8666007276011122963_a_nat @ X @ Y )
& ( ord_le8666007276011122963_a_nat @ X @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_792_inf_Obounded__iff,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat,C2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A @ ( inf_in3813033988066259269_a_nat @ B @ C2 ) )
= ( ( ord_le8666007276011122963_a_nat @ A @ B )
& ( ord_le8666007276011122963_a_nat @ A @ C2 ) ) ) ).
% inf.bounded_iff
thf(fact_793_Int__subset__iff,axiom,
! [C3: set_Pr4934435412358123699_a_nat,A5: set_Pr4934435412358123699_a_nat,B4: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ C3 @ ( inf_in3813033988066259269_a_nat @ A5 @ B4 ) )
= ( ( ord_le8666007276011122963_a_nat @ C3 @ A5 )
& ( ord_le8666007276011122963_a_nat @ C3 @ B4 ) ) ) ).
% Int_subset_iff
thf(fact_794_Int__insert__left__if0,axiom,
! [A: produc357393685978478089rm_a_b,C3: set_Pr4386577575007340137rm_a_b,B4: set_Pr4386577575007340137rm_a_b] :
( ~ ( member5869715511025134514rm_a_b @ A @ C3 )
=> ( ( inf_in1693821330460682071rm_a_b @ ( insert7009541432154983385rm_a_b @ A @ B4 ) @ C3 )
= ( inf_in1693821330460682071rm_a_b @ B4 @ C3 ) ) ) ).
% Int_insert_left_if0
thf(fact_795_Int__insert__left__if0,axiom,
! [A: b,C3: set_b,B4: set_b] :
( ~ ( member_b @ A @ C3 )
=> ( ( inf_inf_set_b @ ( insert_b @ A @ B4 ) @ C3 )
= ( inf_inf_set_b @ B4 @ C3 ) ) ) ).
% Int_insert_left_if0
thf(fact_796_Int__insert__left__if1,axiom,
! [A: produc357393685978478089rm_a_b,C3: set_Pr4386577575007340137rm_a_b,B4: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ A @ C3 )
=> ( ( inf_in1693821330460682071rm_a_b @ ( insert7009541432154983385rm_a_b @ A @ B4 ) @ C3 )
= ( insert7009541432154983385rm_a_b @ A @ ( inf_in1693821330460682071rm_a_b @ B4 @ C3 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_797_Int__insert__left__if1,axiom,
! [A: b,C3: set_b,B4: set_b] :
( ( member_b @ A @ C3 )
=> ( ( inf_inf_set_b @ ( insert_b @ A @ B4 ) @ C3 )
= ( insert_b @ A @ ( inf_inf_set_b @ B4 @ C3 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_798_Int__insert__right__if0,axiom,
! [A: produc357393685978478089rm_a_b,A5: set_Pr4386577575007340137rm_a_b,B4: set_Pr4386577575007340137rm_a_b] :
( ~ ( member5869715511025134514rm_a_b @ A @ A5 )
=> ( ( inf_in1693821330460682071rm_a_b @ A5 @ ( insert7009541432154983385rm_a_b @ A @ B4 ) )
= ( inf_in1693821330460682071rm_a_b @ A5 @ B4 ) ) ) ).
% Int_insert_right_if0
thf(fact_799_Int__insert__right__if0,axiom,
! [A: b,A5: set_b,B4: set_b] :
( ~ ( member_b @ A @ A5 )
=> ( ( inf_inf_set_b @ A5 @ ( insert_b @ A @ B4 ) )
= ( inf_inf_set_b @ A5 @ B4 ) ) ) ).
% Int_insert_right_if0
thf(fact_800_Int__insert__right__if1,axiom,
! [A: produc357393685978478089rm_a_b,A5: set_Pr4386577575007340137rm_a_b,B4: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ A @ A5 )
=> ( ( inf_in1693821330460682071rm_a_b @ A5 @ ( insert7009541432154983385rm_a_b @ A @ B4 ) )
= ( insert7009541432154983385rm_a_b @ A @ ( inf_in1693821330460682071rm_a_b @ A5 @ B4 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_801_Int__insert__right__if1,axiom,
! [A: b,A5: set_b,B4: set_b] :
( ( member_b @ A @ A5 )
=> ( ( inf_inf_set_b @ A5 @ ( insert_b @ A @ B4 ) )
= ( insert_b @ A @ ( inf_inf_set_b @ A5 @ B4 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_802_inf_OcoboundedI2,axiom,
! [B: set_Pr4934435412358123699_a_nat,C2: set_Pr4934435412358123699_a_nat,A: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ B @ C2 )
=> ( ord_le8666007276011122963_a_nat @ ( inf_in3813033988066259269_a_nat @ A @ B ) @ C2 ) ) ).
% inf.coboundedI2
thf(fact_803_inf_OcoboundedI1,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 @ ( inf_in3813033988066259269_a_nat @ A @ B ) @ C2 ) ) ).
% inf.coboundedI1
thf(fact_804_inf_Oabsorb__iff2,axiom,
( ord_le8666007276011122963_a_nat
= ( ^ [B6: set_Pr4934435412358123699_a_nat,A7: set_Pr4934435412358123699_a_nat] :
( ( inf_in3813033988066259269_a_nat @ A7 @ B6 )
= B6 ) ) ) ).
% inf.absorb_iff2
thf(fact_805_inf_Oabsorb__iff1,axiom,
( ord_le8666007276011122963_a_nat
= ( ^ [A7: set_Pr4934435412358123699_a_nat,B6: set_Pr4934435412358123699_a_nat] :
( ( inf_in3813033988066259269_a_nat @ A7 @ B6 )
= A7 ) ) ) ).
% inf.absorb_iff1
thf(fact_806_inf_Ocobounded2,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ ( inf_in3813033988066259269_a_nat @ A @ B ) @ B ) ).
% inf.cobounded2
thf(fact_807_inf_Ocobounded1,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ ( inf_in3813033988066259269_a_nat @ A @ B ) @ A ) ).
% inf.cobounded1
thf(fact_808_inf_Oorder__iff,axiom,
( ord_le8666007276011122963_a_nat
= ( ^ [A7: set_Pr4934435412358123699_a_nat,B6: set_Pr4934435412358123699_a_nat] :
( A7
= ( inf_in3813033988066259269_a_nat @ A7 @ B6 ) ) ) ) ).
% inf.order_iff
thf(fact_809_inf__greatest,axiom,
! [X: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat,Z2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X @ Y )
=> ( ( ord_le8666007276011122963_a_nat @ X @ Z2 )
=> ( ord_le8666007276011122963_a_nat @ X @ ( inf_in3813033988066259269_a_nat @ Y @ Z2 ) ) ) ) ).
% inf_greatest
thf(fact_810_inf_OboundedI,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 @ A @ C2 )
=> ( ord_le8666007276011122963_a_nat @ A @ ( inf_in3813033988066259269_a_nat @ B @ C2 ) ) ) ) ).
% inf.boundedI
thf(fact_811_inf_OboundedE,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat,C2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A @ ( inf_in3813033988066259269_a_nat @ B @ C2 ) )
=> ~ ( ( ord_le8666007276011122963_a_nat @ A @ B )
=> ~ ( ord_le8666007276011122963_a_nat @ A @ C2 ) ) ) ).
% inf.boundedE
thf(fact_812_inf__absorb2,axiom,
! [Y: set_Pr4934435412358123699_a_nat,X: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ Y @ X )
=> ( ( inf_in3813033988066259269_a_nat @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_813_inf__absorb1,axiom,
! [X: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X @ Y )
=> ( ( inf_in3813033988066259269_a_nat @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_814_inf_Oabsorb2,axiom,
! [B: set_Pr4934435412358123699_a_nat,A: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ B @ A )
=> ( ( inf_in3813033988066259269_a_nat @ A @ B )
= B ) ) ).
% inf.absorb2
thf(fact_815_inf_Oabsorb1,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A @ B )
=> ( ( inf_in3813033988066259269_a_nat @ A @ B )
= A ) ) ).
% inf.absorb1
thf(fact_816_le__iff__inf,axiom,
( ord_le8666007276011122963_a_nat
= ( ^ [X2: set_Pr4934435412358123699_a_nat,Y4: set_Pr4934435412358123699_a_nat] :
( ( inf_in3813033988066259269_a_nat @ X2 @ Y4 )
= X2 ) ) ) ).
% le_iff_inf
thf(fact_817_inf__unique,axiom,
! [F2: set_Pr4934435412358123699_a_nat > set_Pr4934435412358123699_a_nat > set_Pr4934435412358123699_a_nat,X: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat] :
( ! [X3: set_Pr4934435412358123699_a_nat,Y3: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ ( F2 @ X3 @ Y3 ) @ X3 )
=> ( ! [X3: set_Pr4934435412358123699_a_nat,Y3: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ ( F2 @ X3 @ Y3 ) @ Y3 )
=> ( ! [X3: set_Pr4934435412358123699_a_nat,Y3: set_Pr4934435412358123699_a_nat,Z4: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X3 @ Y3 )
=> ( ( ord_le8666007276011122963_a_nat @ X3 @ Z4 )
=> ( ord_le8666007276011122963_a_nat @ X3 @ ( F2 @ Y3 @ Z4 ) ) ) )
=> ( ( inf_in3813033988066259269_a_nat @ X @ Y )
= ( F2 @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_818_inf_OorderI,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( A
= ( inf_in3813033988066259269_a_nat @ A @ B ) )
=> ( ord_le8666007276011122963_a_nat @ A @ B ) ) ).
% inf.orderI
thf(fact_819_inf_OorderE,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A @ B )
=> ( A
= ( inf_in3813033988066259269_a_nat @ A @ B ) ) ) ).
% inf.orderE
thf(fact_820_le__infI2,axiom,
! [B: set_Pr4934435412358123699_a_nat,X: set_Pr4934435412358123699_a_nat,A: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ B @ X )
=> ( ord_le8666007276011122963_a_nat @ ( inf_in3813033988066259269_a_nat @ A @ B ) @ X ) ) ).
% le_infI2
thf(fact_821_le__infI1,axiom,
! [A: set_Pr4934435412358123699_a_nat,X: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A @ X )
=> ( ord_le8666007276011122963_a_nat @ ( inf_in3813033988066259269_a_nat @ A @ B ) @ X ) ) ).
% le_infI1
thf(fact_822_inf__mono,axiom,
! [A: set_Pr4934435412358123699_a_nat,C2: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat,D2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A @ C2 )
=> ( ( ord_le8666007276011122963_a_nat @ B @ D2 )
=> ( ord_le8666007276011122963_a_nat @ ( inf_in3813033988066259269_a_nat @ A @ B ) @ ( inf_in3813033988066259269_a_nat @ C2 @ D2 ) ) ) ) ).
% inf_mono
thf(fact_823_le__infI,axiom,
! [X: set_Pr4934435412358123699_a_nat,A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X @ A )
=> ( ( ord_le8666007276011122963_a_nat @ X @ B )
=> ( ord_le8666007276011122963_a_nat @ X @ ( inf_in3813033988066259269_a_nat @ A @ B ) ) ) ) ).
% le_infI
thf(fact_824_le__infE,axiom,
! [X: set_Pr4934435412358123699_a_nat,A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X @ ( inf_in3813033988066259269_a_nat @ A @ B ) )
=> ~ ( ( ord_le8666007276011122963_a_nat @ X @ A )
=> ~ ( ord_le8666007276011122963_a_nat @ X @ B ) ) ) ).
% le_infE
thf(fact_825_inf__le2,axiom,
! [X: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ ( inf_in3813033988066259269_a_nat @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_826_inf__le1,axiom,
! [X: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ ( inf_in3813033988066259269_a_nat @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_827_inf__sup__ord_I1_J,axiom,
! [X: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ ( inf_in3813033988066259269_a_nat @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_828_inf__sup__ord_I2_J,axiom,
! [X: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ ( inf_in3813033988066259269_a_nat @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_829_IntE,axiom,
! [C2: produc357393685978478089rm_a_b,A5: set_Pr4386577575007340137rm_a_b,B4: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ C2 @ ( inf_in1693821330460682071rm_a_b @ A5 @ B4 ) )
=> ~ ( ( member5869715511025134514rm_a_b @ C2 @ A5 )
=> ~ ( member5869715511025134514rm_a_b @ C2 @ B4 ) ) ) ).
% IntE
thf(fact_830_IntE,axiom,
! [C2: b,A5: set_b,B4: set_b] :
( ( member_b @ C2 @ ( inf_inf_set_b @ A5 @ B4 ) )
=> ~ ( ( member_b @ C2 @ A5 )
=> ~ ( member_b @ C2 @ B4 ) ) ) ).
% IntE
thf(fact_831_IntD1,axiom,
! [C2: produc357393685978478089rm_a_b,A5: set_Pr4386577575007340137rm_a_b,B4: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ C2 @ ( inf_in1693821330460682071rm_a_b @ A5 @ B4 ) )
=> ( member5869715511025134514rm_a_b @ C2 @ A5 ) ) ).
% IntD1
thf(fact_832_IntD1,axiom,
! [C2: b,A5: set_b,B4: set_b] :
( ( member_b @ C2 @ ( inf_inf_set_b @ A5 @ B4 ) )
=> ( member_b @ C2 @ A5 ) ) ).
% IntD1
thf(fact_833_IntD2,axiom,
! [C2: produc357393685978478089rm_a_b,A5: set_Pr4386577575007340137rm_a_b,B4: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ C2 @ ( inf_in1693821330460682071rm_a_b @ A5 @ B4 ) )
=> ( member5869715511025134514rm_a_b @ C2 @ B4 ) ) ).
% IntD2
thf(fact_834_IntD2,axiom,
! [C2: b,A5: set_b,B4: set_b] :
( ( member_b @ C2 @ ( inf_inf_set_b @ A5 @ B4 ) )
=> ( member_b @ C2 @ B4 ) ) ).
% IntD2
thf(fact_835_Int__Collect,axiom,
! [X: produc357393685978478089rm_a_b,A5: set_Pr4386577575007340137rm_a_b,P: produc357393685978478089rm_a_b > $o] :
( ( member5869715511025134514rm_a_b @ X @ ( inf_in1693821330460682071rm_a_b @ A5 @ ( collec99177395925924084rm_a_b @ P ) ) )
= ( ( member5869715511025134514rm_a_b @ X @ A5 )
& ( P @ X ) ) ) ).
% Int_Collect
thf(fact_836_Int__Collect,axiom,
! [X: b,A5: set_b,P: b > $o] :
( ( member_b @ X @ ( inf_inf_set_b @ A5 @ ( collect_b @ P ) ) )
= ( ( member_b @ X @ A5 )
& ( P @ X ) ) ) ).
% Int_Collect
thf(fact_837_Int__def,axiom,
( inf_in1693821330460682071rm_a_b
= ( ^ [A6: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
( collec99177395925924084rm_a_b
@ ^ [X2: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X2 @ A6 )
& ( member5869715511025134514rm_a_b @ X2 @ B5 ) ) ) ) ) ).
% Int_def
thf(fact_838_Int__def,axiom,
( inf_inf_set_b
= ( ^ [A6: set_b,B5: set_b] :
( collect_b
@ ^ [X2: b] :
( ( member_b @ X2 @ A6 )
& ( member_b @ X2 @ B5 ) ) ) ) ) ).
% Int_def
thf(fact_839_Int__insert__right,axiom,
! [A: produc357393685978478089rm_a_b,A5: set_Pr4386577575007340137rm_a_b,B4: set_Pr4386577575007340137rm_a_b] :
( ( ( member5869715511025134514rm_a_b @ A @ A5 )
=> ( ( inf_in1693821330460682071rm_a_b @ A5 @ ( insert7009541432154983385rm_a_b @ A @ B4 ) )
= ( insert7009541432154983385rm_a_b @ A @ ( inf_in1693821330460682071rm_a_b @ A5 @ B4 ) ) ) )
& ( ~ ( member5869715511025134514rm_a_b @ A @ A5 )
=> ( ( inf_in1693821330460682071rm_a_b @ A5 @ ( insert7009541432154983385rm_a_b @ A @ B4 ) )
= ( inf_in1693821330460682071rm_a_b @ A5 @ B4 ) ) ) ) ).
% Int_insert_right
thf(fact_840_Int__insert__right,axiom,
! [A: b,A5: set_b,B4: set_b] :
( ( ( member_b @ A @ A5 )
=> ( ( inf_inf_set_b @ A5 @ ( insert_b @ A @ B4 ) )
= ( insert_b @ A @ ( inf_inf_set_b @ A5 @ B4 ) ) ) )
& ( ~ ( member_b @ A @ A5 )
=> ( ( inf_inf_set_b @ A5 @ ( insert_b @ A @ B4 ) )
= ( inf_inf_set_b @ A5 @ B4 ) ) ) ) ).
% Int_insert_right
thf(fact_841_Int__insert__left,axiom,
! [A: produc357393685978478089rm_a_b,C3: set_Pr4386577575007340137rm_a_b,B4: set_Pr4386577575007340137rm_a_b] :
( ( ( member5869715511025134514rm_a_b @ A @ C3 )
=> ( ( inf_in1693821330460682071rm_a_b @ ( insert7009541432154983385rm_a_b @ A @ B4 ) @ C3 )
= ( insert7009541432154983385rm_a_b @ A @ ( inf_in1693821330460682071rm_a_b @ B4 @ C3 ) ) ) )
& ( ~ ( member5869715511025134514rm_a_b @ A @ C3 )
=> ( ( inf_in1693821330460682071rm_a_b @ ( insert7009541432154983385rm_a_b @ A @ B4 ) @ C3 )
= ( inf_in1693821330460682071rm_a_b @ B4 @ C3 ) ) ) ) ).
% Int_insert_left
thf(fact_842_Int__insert__left,axiom,
! [A: b,C3: set_b,B4: set_b] :
( ( ( member_b @ A @ C3 )
=> ( ( inf_inf_set_b @ ( insert_b @ A @ B4 ) @ C3 )
= ( insert_b @ A @ ( inf_inf_set_b @ B4 @ C3 ) ) ) )
& ( ~ ( member_b @ A @ C3 )
=> ( ( inf_inf_set_b @ ( insert_b @ A @ B4 ) @ C3 )
= ( inf_inf_set_b @ B4 @ C3 ) ) ) ) ).
% Int_insert_left
thf(fact_843_Int__mono,axiom,
! [A5: set_Pr4934435412358123699_a_nat,C3: set_Pr4934435412358123699_a_nat,B4: set_Pr4934435412358123699_a_nat,D: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A5 @ C3 )
=> ( ( ord_le8666007276011122963_a_nat @ B4 @ D )
=> ( ord_le8666007276011122963_a_nat @ ( inf_in3813033988066259269_a_nat @ A5 @ B4 ) @ ( inf_in3813033988066259269_a_nat @ C3 @ D ) ) ) ) ).
% Int_mono
thf(fact_844_Int__lower1,axiom,
! [A5: set_Pr4934435412358123699_a_nat,B4: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ ( inf_in3813033988066259269_a_nat @ A5 @ B4 ) @ A5 ) ).
% Int_lower1
thf(fact_845_Int__lower2,axiom,
! [A5: set_Pr4934435412358123699_a_nat,B4: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ ( inf_in3813033988066259269_a_nat @ A5 @ B4 ) @ B4 ) ).
% Int_lower2
thf(fact_846_Int__absorb1,axiom,
! [B4: set_Pr4934435412358123699_a_nat,A5: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ B4 @ A5 )
=> ( ( inf_in3813033988066259269_a_nat @ A5 @ B4 )
= B4 ) ) ).
% Int_absorb1
thf(fact_847_Int__absorb2,axiom,
! [A5: set_Pr4934435412358123699_a_nat,B4: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A5 @ B4 )
=> ( ( inf_in3813033988066259269_a_nat @ A5 @ B4 )
= A5 ) ) ).
% Int_absorb2
thf(fact_848_Int__greatest,axiom,
! [C3: set_Pr4934435412358123699_a_nat,A5: set_Pr4934435412358123699_a_nat,B4: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ C3 @ A5 )
=> ( ( ord_le8666007276011122963_a_nat @ C3 @ B4 )
=> ( ord_le8666007276011122963_a_nat @ C3 @ ( inf_in3813033988066259269_a_nat @ A5 @ B4 ) ) ) ) ).
% Int_greatest
thf(fact_849_Int__Collect__mono,axiom,
! [A5: set_Pr4386577575007340137rm_a_b,B4: set_Pr4386577575007340137rm_a_b,P: produc357393685978478089rm_a_b > $o,Q: produc357393685978478089rm_a_b > $o] :
( ( ord_le118470702582115849rm_a_b @ A5 @ B4 )
=> ( ! [X3: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X3 @ A5 )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_le118470702582115849rm_a_b @ ( inf_in1693821330460682071rm_a_b @ A5 @ ( collec99177395925924084rm_a_b @ P ) ) @ ( inf_in1693821330460682071rm_a_b @ B4 @ ( collec99177395925924084rm_a_b @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_850_Int__Collect__mono,axiom,
! [A5: set_b,B4: set_b,P: b > $o,Q: b > $o] :
( ( ord_less_eq_set_b @ A5 @ B4 )
=> ( ! [X3: b] :
( ( member_b @ X3 @ A5 )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_less_eq_set_b @ ( inf_inf_set_b @ A5 @ ( collect_b @ P ) ) @ ( inf_inf_set_b @ B4 @ ( collect_b @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_851_Int__Collect__mono,axiom,
! [A5: set_Pr4934435412358123699_a_nat,B4: set_Pr4934435412358123699_a_nat,P: product_prod_a_nat > $o,Q: product_prod_a_nat > $o] :
( ( ord_le8666007276011122963_a_nat @ A5 @ B4 )
=> ( ! [X3: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ X3 @ A5 )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_le8666007276011122963_a_nat @ ( inf_in3813033988066259269_a_nat @ A5 @ ( collec4464134535221767506_a_nat @ P ) ) @ ( inf_in3813033988066259269_a_nat @ B4 @ ( collec4464134535221767506_a_nat @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_852_distrib__inf__le,axiom,
! [X: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat,Z2: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ ( sup_su459911885395995103_a_nat @ ( inf_in3813033988066259269_a_nat @ X @ Y ) @ ( inf_in3813033988066259269_a_nat @ X @ Z2 ) ) @ ( inf_in3813033988066259269_a_nat @ X @ ( sup_su459911885395995103_a_nat @ Y @ Z2 ) ) ) ).
% distrib_inf_le
thf(fact_853_distrib__sup__le,axiom,
! [X: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat,Z2: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ ( sup_su459911885395995103_a_nat @ X @ ( inf_in3813033988066259269_a_nat @ Y @ Z2 ) ) @ ( inf_in3813033988066259269_a_nat @ ( sup_su459911885395995103_a_nat @ X @ Y ) @ ( sup_su459911885395995103_a_nat @ X @ Z2 ) ) ) ).
% distrib_sup_le
thf(fact_854_Un__Int__assoc__eq,axiom,
! [A5: set_Pr4934435412358123699_a_nat,B4: set_Pr4934435412358123699_a_nat,C3: set_Pr4934435412358123699_a_nat] :
( ( ( sup_su459911885395995103_a_nat @ ( inf_in3813033988066259269_a_nat @ A5 @ B4 ) @ C3 )
= ( inf_in3813033988066259269_a_nat @ A5 @ ( sup_su459911885395995103_a_nat @ B4 @ C3 ) ) )
= ( ord_le8666007276011122963_a_nat @ C3 @ A5 ) ) ).
% Un_Int_assoc_eq
thf(fact_855_rtrancl__Restr,axiom,
! [X: term_a_b,Y: term_a_b,R: set_Pr4386577575007340137rm_a_b,A5: set_term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y )
@ ( transi7742714808557438673rm_a_b
@ ( inf_in1693821330460682071rm_a_b @ R
@ ( produc1953217797448512882rm_a_b @ A5
@ ^ [Uu: term_a_b] : A5 ) ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ R ) ) ) ).
% rtrancl_Restr
thf(fact_856_sig__step__grstep__dist,axiom,
! [F: set_Pr4934435412358123699_a_nat,R2: set_Pr4386577575007340137rm_a_b] :
( ( inf_in1693821330460682071rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R2 ) )
@ ( produc1953217797448512882rm_a_b @ ( collect_term_a_b @ term_ground_a_b )
@ ^ [Uu: term_a_b] : ( collect_term_a_b @ term_ground_a_b ) ) )
= ( sig_step_a_b @ F
@ ( inf_in1693821330460682071rm_a_b @ ( rstep_a_b @ R2 )
@ ( produc1953217797448512882rm_a_b @ ( collect_term_a_b @ term_ground_a_b )
@ ^ [Uu: term_a_b] : ( collect_term_a_b @ term_ground_a_b ) ) ) ) ) ).
% sig_step_grstep_dist
thf(fact_857_sig__step__def,axiom,
( sig_step_a_b
= ( ^ [F4: set_Pr4934435412358123699_a_nat,R6: set_Pr4386577575007340137rm_a_b] :
( inf_in1693821330460682071rm_a_b @ R6
@ ( produc1953217797448512882rm_a_b
@ ( collect_term_a_b
@ ^ [S7: term_a_b] : ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ S7 ) @ F4 ) )
@ ^ [Uu: term_a_b] :
( collect_term_a_b
@ ^ [S7: term_a_b] : ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ S7 ) @ F4 ) ) ) ) ) ) ).
% sig_step_def
thf(fact_858_gsrsteps__eq__to__rsteps__eq,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 )
@ ( transi7742714808557438673rm_a_b
@ ( inf_in1693821330460682071rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R2 ) )
@ ( produc1953217797448512882rm_a_b @ ( collect_term_a_b @ term_ground_a_b )
@ ^ [Uu: term_a_b] : ( collect_term_a_b @ term_ground_a_b ) ) ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( transi7742714808557438673rm_a_b @ ( rstep_a_b @ R2 ) ) ) ) ).
% gsrsteps_eq_to_rsteps_eq
thf(fact_859_gsrsteps__eq__to__srsteps__eq,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 )
@ ( transi7742714808557438673rm_a_b
@ ( inf_in1693821330460682071rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R2 ) )
@ ( produc1953217797448512882rm_a_b @ ( collect_term_a_b @ term_ground_a_b )
@ ^ [Uu: term_a_b] : ( collect_term_a_b @ term_ground_a_b ) ) ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R2 ) ) ) ) ) ).
% gsrsteps_eq_to_srsteps_eq
thf(fact_860_ground__srsteps__eq__gsrsteps__eq,axiom,
! [S: term_a_b,T: term_a_b,F: set_Pr4934435412358123699_a_nat,R2: set_Pr4386577575007340137rm_a_b] :
( ( term_ground_a_b @ S )
=> ( ( term_ground_a_b @ T )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R2 ) ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T )
@ ( transi7742714808557438673rm_a_b
@ ( inf_in1693821330460682071rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R2 ) )
@ ( produc1953217797448512882rm_a_b @ ( collect_term_a_b @ term_ground_a_b )
@ ^ [Uu: term_a_b] : ( collect_term_a_b @ term_ground_a_b ) ) ) ) ) ) ) ) ).
% ground_srsteps_eq_gsrsteps_eq
thf(fact_861_gsrsteps__to__rsteps,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
@ ( inf_in1693821330460682071rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R2 ) )
@ ( produc1953217797448512882rm_a_b @ ( collect_term_a_b @ term_ground_a_b )
@ ^ [Uu: term_a_b] : ( collect_term_a_b @ term_ground_a_b ) ) ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( transi7922773638565587891rm_a_b @ ( rstep_a_b @ R2 ) ) ) ) ).
% gsrsteps_to_rsteps
thf(fact_862_gsrsteps__to__srsteps,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
@ ( inf_in1693821330460682071rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R2 ) )
@ ( produc1953217797448512882rm_a_b @ ( collect_term_a_b @ term_ground_a_b )
@ ^ [Uu: term_a_b] : ( collect_term_a_b @ term_ground_a_b ) ) ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( transi7922773638565587891rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R2 ) ) ) ) ) ).
% gsrsteps_to_srsteps
thf(fact_863_ground__srsteps__gsrsteps,axiom,
! [S: term_a_b,T: term_a_b,F: set_Pr4934435412358123699_a_nat,R2: set_Pr4386577575007340137rm_a_b] :
( ( term_ground_a_b @ S )
=> ( ( term_ground_a_b @ T )
=> ( ( 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
@ ( inf_in1693821330460682071rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R2 ) )
@ ( produc1953217797448512882rm_a_b @ ( collect_term_a_b @ term_ground_a_b )
@ ^ [Uu: term_a_b] : ( collect_term_a_b @ term_ground_a_b ) ) ) ) ) ) ) ) ).
% ground_srsteps_gsrsteps
thf(fact_864_Restr__subset,axiom,
! [A5: set_Pr4934435412358123699_a_nat,B4: set_Pr4934435412358123699_a_nat,R: set_Pr1811044260758604347_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A5 @ B4 )
=> ( ( inf_in8678195602585455913_a_nat
@ ( inf_in8678195602585455913_a_nat @ R
@ ( produc3832836940518797764_a_nat @ B4
@ ^ [Uu: product_prod_a_nat] : B4 ) )
@ ( produc3832836940518797764_a_nat @ A5
@ ^ [Uu: product_prod_a_nat] : A5 ) )
= ( inf_in8678195602585455913_a_nat @ R
@ ( produc3832836940518797764_a_nat @ A5
@ ^ [Uu: product_prod_a_nat] : A5 ) ) ) ) ).
% Restr_subset
thf(fact_865_ground__NF__srstep__gsrstep,axiom,
! [S: term_a_b,F: set_Pr4934435412358123699_a_nat,R2: set_Pr4386577575007340137rm_a_b] :
( ( term_ground_a_b @ S )
=> ( ( member_term_a_b @ S @ ( abstract_NF_term_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R2 ) ) ) )
=> ( member_term_a_b @ S
@ ( abstract_NF_term_a_b
@ ( inf_in1693821330460682071rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R2 ) )
@ ( produc1953217797448512882rm_a_b @ ( collect_term_a_b @ term_ground_a_b )
@ ^ [Uu: term_a_b] : ( collect_term_a_b @ term_ground_a_b ) ) ) ) ) ) ) ).
% ground_NF_srstep_gsrstep
thf(fact_866_monadic__replace__leave__subst,axiom,
! [S: term_a_b,Sigma3: b > term_a_b,T: term_a_b] :
( ( rewrit3151024789171632623rm_a_b @ ( subst_7999470309526761004_a_b_b @ S @ Sigma3 ) )
=> ( ~ ( term_ground_a_b @ S )
=> ( ( rewrit2102430416898327096_a_b_b @ T @ ( subst_7999470309526761004_a_b_b @ S @ Sigma3 ) )
= ( subst_7999470309526761004_a_b_b @ S
@ ^ [X2: b] : ( rewrit2102430416898327096_a_b_b @ T @ ( Sigma3 @ X2 ) ) ) ) ) ) ).
% monadic_replace_leave_subst
thf(fact_867_NF__I,axiom,
! [A: produc357393685978478089rm_a_b,A5: set_Pr2972776593051762503rm_a_b] :
( ! [B3: produc357393685978478089rm_a_b] :
~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B3 ) @ A5 )
=> ( member5869715511025134514rm_a_b @ A @ ( abstra1902783451776770197rm_a_b @ A5 ) ) ) ).
% NF_I
thf(fact_868_NF__I,axiom,
! [A: b,A5: set_Product_prod_b_b] :
( ! [B3: b] :
~ ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ A @ B3 ) @ A5 )
=> ( member_b @ A @ ( abstract_NF_b @ A5 ) ) ) ).
% NF_I
thf(fact_869_NF__I,axiom,
! [A: term_a_b,A5: set_Pr4386577575007340137rm_a_b] :
( ! [B3: term_a_b] :
~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B3 ) @ A5 )
=> ( member_term_a_b @ A @ ( abstract_NF_term_a_b @ A5 ) ) ) ).
% NF_I
thf(fact_870_NF__iff__no__step,axiom,
! [A: produc357393685978478089rm_a_b,A5: set_Pr2972776593051762503rm_a_b] :
( ( member5869715511025134514rm_a_b @ A @ ( abstra1902783451776770197rm_a_b @ A5 ) )
= ( ! [B6: produc357393685978478089rm_a_b] :
~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B6 ) @ A5 ) ) ) ).
% NF_iff_no_step
thf(fact_871_NF__iff__no__step,axiom,
! [A: b,A5: set_Product_prod_b_b] :
( ( member_b @ A @ ( abstract_NF_b @ A5 ) )
= ( ! [B6: b] :
~ ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ A @ B6 ) @ A5 ) ) ) ).
% NF_iff_no_step
thf(fact_872_NF__iff__no__step,axiom,
! [A: term_a_b,A5: set_Pr4386577575007340137rm_a_b] :
( ( member_term_a_b @ A @ ( abstract_NF_term_a_b @ A5 ) )
= ( ! [B6: term_a_b] :
~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B6 ) @ A5 ) ) ) ).
% NF_iff_no_step
thf(fact_873_NF__no__step,axiom,
! [A: produc357393685978478089rm_a_b,A5: set_Pr2972776593051762503rm_a_b] :
( ( member5869715511025134514rm_a_b @ A @ ( abstra1902783451776770197rm_a_b @ A5 ) )
=> ! [B10: produc357393685978478089rm_a_b] :
~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B10 ) @ A5 ) ) ).
% NF_no_step
thf(fact_874_NF__no__step,axiom,
! [A: b,A5: set_Product_prod_b_b] :
( ( member_b @ A @ ( abstract_NF_b @ A5 ) )
=> ! [B10: b] :
~ ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ A @ B10 ) @ A5 ) ) ).
% NF_no_step
thf(fact_875_NF__no__step,axiom,
! [A: term_a_b,A5: set_Pr4386577575007340137rm_a_b] :
( ( member_term_a_b @ A @ ( abstract_NF_term_a_b @ A5 ) )
=> ! [B10: term_a_b] :
~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B10 ) @ A5 ) ) ).
% NF_no_step
thf(fact_876_NF__E,axiom,
! [A: produc357393685978478089rm_a_b,A5: set_Pr2972776593051762503rm_a_b,B: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ A @ ( abstra1902783451776770197rm_a_b @ A5 ) )
=> ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ A5 ) ) ).
% NF_E
thf(fact_877_NF__E,axiom,
! [A: b,A5: set_Product_prod_b_b,B: b] :
( ( member_b @ A @ ( abstract_NF_b @ A5 ) )
=> ~ ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ A @ B ) @ A5 ) ) ).
% NF_E
thf(fact_878_NF__E,axiom,
! [A: term_a_b,A5: set_Pr4386577575007340137rm_a_b,B: term_a_b] :
( ( member_term_a_b @ A @ ( abstract_NF_term_a_b @ A5 ) )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ A5 ) ) ).
% NF_E
thf(fact_879_inf__set__def,axiom,
( inf_in1693821330460682071rm_a_b
= ( ^ [A6: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
( collec99177395925924084rm_a_b
@ ( inf_in7523581892642934342_a_b_o
@ ^ [X2: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ X2 @ A6 )
@ ^ [X2: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ X2 @ B5 ) ) ) ) ) ).
% inf_set_def
thf(fact_880_inf__set__def,axiom,
( inf_inf_set_b
= ( ^ [A6: set_b,B5: set_b] :
( collect_b
@ ( inf_inf_b_o
@ ^ [X2: b] : ( member_b @ X2 @ A6 )
@ ^ [X2: b] : ( member_b @ X2 @ B5 ) ) ) ) ) ).
% inf_set_def
thf(fact_881_inf__Int__eq,axiom,
! [R5: set_Pr4386577575007340137rm_a_b,S2: set_Pr4386577575007340137rm_a_b] :
( ( inf_in7523581892642934342_a_b_o
@ ^ [X2: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ X2 @ R5 )
@ ^ [X2: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ X2 @ S2 ) )
= ( ^ [X2: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ X2 @ ( inf_in1693821330460682071rm_a_b @ R5 @ S2 ) ) ) ) ).
% inf_Int_eq
thf(fact_882_inf__Int__eq,axiom,
! [R5: set_b,S2: set_b] :
( ( inf_inf_b_o
@ ^ [X2: b] : ( member_b @ X2 @ R5 )
@ ^ [X2: b] : ( member_b @ X2 @ S2 ) )
= ( ^ [X2: b] : ( member_b @ X2 @ ( inf_inf_set_b @ R5 @ S2 ) ) ) ) ).
% inf_Int_eq
thf(fact_883_inf__Int__eq2,axiom,
! [R5: set_Pr4386577575007340137rm_a_b,S2: set_Pr4386577575007340137rm_a_b] :
( ( inf_in2005149299588960588_a_b_o
@ ^ [X2: term_a_b,Y4: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y4 ) @ R5 )
@ ^ [X2: term_a_b,Y4: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y4 ) @ S2 ) )
= ( ^ [X2: term_a_b,Y4: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y4 ) @ ( inf_in1693821330460682071rm_a_b @ R5 @ S2 ) ) ) ) ).
% inf_Int_eq2
thf(fact_884_NF__not__suc,axiom,
! [X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b,R: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ ( transi2615809358984392588rm_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ X @ ( abstra1902783451776770197rm_a_b @ R ) )
=> ( X = Y ) ) ) ).
% NF_not_suc
thf(fact_885_NF__not__suc,axiom,
! [X: b,Y: b,R: set_Product_prod_b_b] :
( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ X @ Y ) @ ( transitive_rtrancl_b @ R ) )
=> ( ( member_b @ X @ ( abstract_NF_b @ R ) )
=> ( X = Y ) ) ) ).
% NF_not_suc
thf(fact_886_NF__not__suc,axiom,
! [X: term_a_b,Y: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ( ( member_term_a_b @ X @ ( abstract_NF_term_a_b @ R ) )
=> ( X = Y ) ) ) ).
% NF_not_suc
thf(fact_887_NF__anti__mono,axiom,
! [A5: set_Pr1811044260758604347_a_nat,B4: set_Pr1811044260758604347_a_nat] :
( ( ord_le1018141841545957851_a_nat @ A5 @ B4 )
=> ( ord_le8666007276011122963_a_nat @ ( abstra2685462378305987313_a_nat @ B4 ) @ ( abstra2685462378305987313_a_nat @ A5 ) ) ) ).
% NF_anti_mono
thf(fact_888_NF__no__trancl__step,axiom,
! [A: produc357393685978478089rm_a_b,A5: set_Pr2972776593051762503rm_a_b] :
( ( member5869715511025134514rm_a_b @ A @ ( abstra1902783451776770197rm_a_b @ A5 ) )
=> ! [B10: produc357393685978478089rm_a_b] :
~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B10 ) @ ( transi9211502839322181930rm_a_b @ A5 ) ) ) ).
% NF_no_trancl_step
thf(fact_889_NF__no__trancl__step,axiom,
! [A: b,A5: set_Product_prod_b_b] :
( ( member_b @ A @ ( abstract_NF_b @ A5 ) )
=> ! [B10: b] :
~ ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ A @ B10 ) @ ( transitive_trancl_b @ A5 ) ) ) ).
% NF_no_trancl_step
thf(fact_890_NF__no__trancl__step,axiom,
! [A: term_a_b,A5: set_Pr4386577575007340137rm_a_b] :
( ( member_term_a_b @ A @ ( abstract_NF_term_a_b @ A5 ) )
=> ! [B10: term_a_b] :
~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B10 ) @ ( transi7922773638565587891rm_a_b @ A5 ) ) ) ).
% NF_no_trancl_step
thf(fact_891_normalizability__def,axiom,
( abstra1496011390211842225rm_a_b
= ( ^ [A6: set_Pr2972776593051762503rm_a_b] :
( collec1344989105561874898rm_a_b
@ ( produc161078424057476524_a_b_o
@ ^ [A7: produc357393685978478089rm_a_b,B6: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A7 @ B6 ) @ ( transi2615809358984392588rm_a_b @ A6 ) )
& ( member5869715511025134514rm_a_b @ B6 @ ( abstra1902783451776770197rm_a_b @ A6 ) ) ) ) ) ) ) ).
% normalizability_def
thf(fact_892_normalizability__def,axiom,
( abstra380364156431640999lity_b
= ( ^ [A6: set_Product_prod_b_b] :
( collec548942219715005266od_b_b
@ ( produc4772849988445941548_b_b_o
@ ^ [A7: b,B6: b] :
( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ A7 @ B6 ) @ ( transitive_rtrancl_b @ A6 ) )
& ( member_b @ B6 @ ( abstract_NF_b @ A6 ) ) ) ) ) ) ) ).
% normalizability_def
thf(fact_893_normalizability__def,axiom,
( abstra2972651231392952684rm_a_b
= ( ^ [A6: set_Pr4386577575007340137rm_a_b] :
( collec99177395925924084rm_a_b
@ ( produc6251846659952913346_a_b_o
@ ^ [A7: term_a_b,B6: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A7 @ B6 ) @ ( transi7742714808557438673rm_a_b @ A6 ) )
& ( member_term_a_b @ B6 @ ( abstract_NF_term_a_b @ A6 ) ) ) ) ) ) ) ).
% normalizability_def
thf(fact_894_normalizability__I,axiom,
! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,A5: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( transi2615809358984392588rm_a_b @ A5 ) )
=> ( ( member5869715511025134514rm_a_b @ B @ ( abstra1902783451776770197rm_a_b @ A5 ) )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( abstra1496011390211842225rm_a_b @ A5 ) ) ) ) ).
% normalizability_I
thf(fact_895_normalizability__I,axiom,
! [A: b,B: b,A5: set_Product_prod_b_b] :
( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ A @ B ) @ ( transitive_rtrancl_b @ A5 ) )
=> ( ( member_b @ B @ ( abstract_NF_b @ A5 ) )
=> ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ A @ B ) @ ( abstra380364156431640999lity_b @ A5 ) ) ) ) ).
% normalizability_I
thf(fact_896_normalizability__I,axiom,
! [A: term_a_b,B: term_a_b,A5: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ A5 ) )
=> ( ( member_term_a_b @ B @ ( abstract_NF_term_a_b @ A5 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( abstra2972651231392952684rm_a_b @ A5 ) ) ) ) ).
% normalizability_I
thf(fact_897_member__product,axiom,
! [X: produc357393685978478089rm_a_b,A5: set_term_a_b,B4: set_term_a_b] :
( ( member5869715511025134514rm_a_b @ X @ ( produc3503722073536325500rm_a_b @ A5 @ B4 ) )
= ( member5869715511025134514rm_a_b @ X
@ ( produc1953217797448512882rm_a_b @ A5
@ ^ [Uu: term_a_b] : B4 ) ) ) ).
% member_product
thf(fact_898_normalizability__I_H,axiom,
! [A: term_a_b,B: term_a_b,A5: set_Pr4386577575007340137rm_a_b,C2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ A5 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C2 ) @ ( abstra2972651231392952684rm_a_b @ A5 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C2 ) @ ( abstra2972651231392952684rm_a_b @ A5 ) ) ) ) ).
% normalizability_I'
thf(fact_899_normalizability__E,axiom,
! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,A5: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( abstra1496011390211842225rm_a_b @ A5 ) )
=> ~ ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( transi2615809358984392588rm_a_b @ A5 ) )
=> ~ ( member5869715511025134514rm_a_b @ B @ ( abstra1902783451776770197rm_a_b @ A5 ) ) ) ) ).
% normalizability_E
thf(fact_900_normalizability__E,axiom,
! [A: b,B: b,A5: set_Product_prod_b_b] :
( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ A @ B ) @ ( abstra380364156431640999lity_b @ A5 ) )
=> ~ ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ A @ B ) @ ( transitive_rtrancl_b @ A5 ) )
=> ~ ( member_b @ B @ ( abstract_NF_b @ A5 ) ) ) ) ).
% normalizability_E
thf(fact_901_normalizability__E,axiom,
! [A: term_a_b,B: term_a_b,A5: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( abstra2972651231392952684rm_a_b @ A5 ) )
=> ~ ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ A5 ) )
=> ~ ( member_term_a_b @ B @ ( abstract_NF_term_a_b @ A5 ) ) ) ) ).
% normalizability_E
thf(fact_902_UNF__onI,axiom,
! [A5: set_Pr4386577575007340137rm_a_b,R: set_Pr2972776593051762503rm_a_b] :
( ! [A4: produc357393685978478089rm_a_b,B3: produc357393685978478089rm_a_b,C: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ A4 @ A5 )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A4 @ B3 ) @ ( abstra1496011390211842225rm_a_b @ R ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A4 @ C ) @ ( abstra1496011390211842225rm_a_b @ R ) )
=> ( B3 = C ) ) ) )
=> ( abstra1303798319023342519rm_a_b @ R @ A5 ) ) ).
% UNF_onI
thf(fact_903_UNF__onI,axiom,
! [A5: set_b,R: set_Product_prod_b_b] :
( ! [A4: b,B3: b,C: b] :
( ( member_b @ A4 @ A5 )
=> ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ A4 @ B3 ) @ ( abstra380364156431640999lity_b @ R ) )
=> ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ A4 @ C ) @ ( abstra380364156431640999lity_b @ R ) )
=> ( B3 = C ) ) ) )
=> ( abstract_UNF_on_b @ R @ A5 ) ) ).
% UNF_onI
thf(fact_904_UNF__onI,axiom,
! [A5: set_term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ! [A4: term_a_b,B3: term_a_b,C: term_a_b] :
( ( member_term_a_b @ A4 @ A5 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A4 @ B3 ) @ ( abstra2972651231392952684rm_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A4 @ C ) @ ( abstra2972651231392952684rm_a_b @ R ) )
=> ( B3 = C ) ) ) )
=> ( abstra3619855735679963878rm_a_b @ R @ A5 ) ) ).
% UNF_onI
thf(fact_905_WN__onI,axiom,
! [A5: set_Pr4386577575007340137rm_a_b,R: set_Pr2972776593051762503rm_a_b] :
( ! [A4: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ A4 @ A5 )
=> ? [B10: produc357393685978478089rm_a_b] : ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A4 @ B10 ) @ ( abstra1496011390211842225rm_a_b @ R ) ) )
=> ( abstra2721873542170697047rm_a_b @ R @ A5 ) ) ).
% WN_onI
thf(fact_906_WN__onI,axiom,
! [A5: set_b,R: set_Product_prod_b_b] :
( ! [A4: b] :
( ( member_b @ A4 @ A5 )
=> ? [B10: b] : ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ A4 @ B10 ) @ ( abstra380364156431640999lity_b @ R ) ) )
=> ( abstract_WN_on_b @ R @ A5 ) ) ).
% WN_onI
thf(fact_907_WN__onI,axiom,
! [A5: set_term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ! [A4: term_a_b] :
( ( member_term_a_b @ A4 @ A5 )
=> ? [B10: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A4 @ B10 ) @ ( abstra2972651231392952684rm_a_b @ R ) ) )
=> ( abstra4644735066934440262rm_a_b @ R @ A5 ) ) ).
% WN_onI
thf(fact_908_SN__on__subset2,axiom,
! [A5: set_Pr4934435412358123699_a_nat,B4: set_Pr4934435412358123699_a_nat,R: set_Pr1811044260758604347_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A5 @ B4 )
=> ( ( abstra8653715922312955827_a_nat @ R @ B4 )
=> ( abstra8653715922312955827_a_nat @ R @ A5 ) ) ) ).
% SN_on_subset2
thf(fact_909_SN__on__irrefl,axiom,
! [R: set_Pr4386577575007340137rm_a_b,A5: set_term_a_b] :
( ( abstra4720023341729745482rm_a_b @ R @ A5 )
=> ! [X4: term_a_b] :
( ( member_term_a_b @ X4 @ A5 )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X4 @ X4 ) @ R ) ) ) ).
% SN_on_irrefl
thf(fact_910_SN__on__induct,axiom,
! [R5: set_Pr2972776593051762503rm_a_b,A5: set_Pr4386577575007340137rm_a_b,S: produc357393685978478089rm_a_b,P: produc357393685978478089rm_a_b > $o] :
( ( abstra2398554102055911763rm_a_b @ R5 @ A5 )
=> ( ( member5869715511025134514rm_a_b @ S @ A5 )
=> ( ! [T2: produc357393685978478089rm_a_b] :
( ! [U4: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ T2 @ U4 ) @ R5 )
=> ( P @ U4 ) )
=> ( P @ T2 ) )
=> ( P @ S ) ) ) ) ).
% SN_on_induct
thf(fact_911_SN__on__induct,axiom,
! [R5: set_Product_prod_b_b,A5: set_b,S: b,P: b > $o] :
( ( abstract_SN_on_b @ R5 @ A5 )
=> ( ( member_b @ S @ A5 )
=> ( ! [T2: b] :
( ! [U4: b] :
( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ T2 @ U4 ) @ R5 )
=> ( P @ U4 ) )
=> ( P @ T2 ) )
=> ( P @ S ) ) ) ) ).
% SN_on_induct
thf(fact_912_SN__on__induct,axiom,
! [R5: set_Pr4386577575007340137rm_a_b,A5: set_term_a_b,S: term_a_b,P: term_a_b > $o] :
( ( abstra4720023341729745482rm_a_b @ R5 @ A5 )
=> ( ( member_term_a_b @ S @ A5 )
=> ( ! [T2: term_a_b] :
( ! [U4: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T2 @ U4 ) @ R5 )
=> ( P @ U4 ) )
=> ( P @ T2 ) )
=> ( P @ S ) ) ) ) ).
% SN_on_induct
thf(fact_913_refl__not__SN,axiom,
! [X: term_a_b,R5: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ X ) @ R5 )
=> ~ ( abstra4720023341729745482rm_a_b @ R5 @ top_top_set_term_a_b ) ) ).
% refl_not_SN
thf(fact_914_SN__imp__minimal,axiom,
! [A5: set_Pr2972776593051762503rm_a_b] :
( ( abstra2398554102055911763rm_a_b @ A5 @ top_to1314267278846557113rm_a_b )
=> ! [Q4: set_Pr4386577575007340137rm_a_b] :
( ? [X4: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ X4 @ Q4 )
=> ? [X3: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X3 @ Q4 )
& ! [Y8: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X3 @ Y8 ) @ A5 )
=> ~ ( member5869715511025134514rm_a_b @ Y8 @ Q4 ) ) ) ) ) ).
% SN_imp_minimal
thf(fact_915_SN__imp__minimal,axiom,
! [A5: set_Product_prod_b_b] :
( ( abstract_SN_on_b @ A5 @ top_top_set_b )
=> ! [Q4: set_b] :
( ? [X4: b] : ( member_b @ X4 @ Q4 )
=> ? [X3: b] :
( ( member_b @ X3 @ Q4 )
& ! [Y8: b] :
( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ X3 @ Y8 ) @ A5 )
=> ~ ( member_b @ Y8 @ Q4 ) ) ) ) ) ).
% SN_imp_minimal
thf(fact_916_SN__imp__minimal,axiom,
! [A5: set_Pr4386577575007340137rm_a_b] :
( ( abstra4720023341729745482rm_a_b @ A5 @ top_top_set_term_a_b )
=> ! [Q4: set_term_a_b] :
( ? [X4: term_a_b] : ( member_term_a_b @ X4 @ Q4 )
=> ? [X3: term_a_b] :
( ( member_term_a_b @ X3 @ Q4 )
& ! [Y8: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y8 ) @ A5 )
=> ~ ( member_term_a_b @ Y8 @ Q4 ) ) ) ) ) ).
% SN_imp_minimal
thf(fact_917_SN__induct__rule,axiom,
! [R: set_Pr4386577575007340137rm_a_b,P: term_a_b > $o,A: term_a_b] :
( ( abstra4720023341729745482rm_a_b @ R @ top_top_set_term_a_b )
=> ( ! [A4: term_a_b] :
( ! [B10: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A4 @ B10 ) @ R )
=> ( P @ B10 ) )
=> ( P @ A4 ) )
=> ( P @ A ) ) ) ).
% SN_induct_rule
thf(fact_918_WN__on__def,axiom,
( abstra4644735066934440262rm_a_b
= ( ^ [R4: set_Pr4386577575007340137rm_a_b,A6: set_term_a_b] :
! [X2: term_a_b] :
( ( member_term_a_b @ X2 @ A6 )
=> ? [B6: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ B6 ) @ ( abstra2972651231392952684rm_a_b @ R4 ) ) ) ) ) ).
% WN_on_def
thf(fact_919_WN__onE,axiom,
! [R: set_Pr2972776593051762503rm_a_b,A5: set_Pr4386577575007340137rm_a_b,A: produc357393685978478089rm_a_b] :
( ( abstra2721873542170697047rm_a_b @ R @ A5 )
=> ( ( member5869715511025134514rm_a_b @ A @ A5 )
=> ~ ! [B3: produc357393685978478089rm_a_b] :
~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B3 ) @ ( abstra1496011390211842225rm_a_b @ R ) ) ) ) ).
% WN_onE
thf(fact_920_WN__onE,axiom,
! [R: set_Product_prod_b_b,A5: set_b,A: b] :
( ( abstract_WN_on_b @ R @ A5 )
=> ( ( member_b @ A @ A5 )
=> ~ ! [B3: b] :
~ ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ A @ B3 ) @ ( abstra380364156431640999lity_b @ R ) ) ) ) ).
% WN_onE
thf(fact_921_WN__onE,axiom,
! [R: set_Pr4386577575007340137rm_a_b,A5: set_term_a_b,A: term_a_b] :
( ( abstra4644735066934440262rm_a_b @ R @ A5 )
=> ( ( member_term_a_b @ A @ A5 )
=> ~ ! [B3: term_a_b] :
~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B3 ) @ ( abstra2972651231392952684rm_a_b @ R ) ) ) ) ).
% WN_onE
thf(fact_922_WN__onD,axiom,
! [R: set_Pr2972776593051762503rm_a_b,A5: set_Pr4386577575007340137rm_a_b,A: produc357393685978478089rm_a_b] :
( ( abstra2721873542170697047rm_a_b @ R @ A5 )
=> ( ( member5869715511025134514rm_a_b @ A @ A5 )
=> ? [B3: produc357393685978478089rm_a_b] : ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B3 ) @ ( abstra1496011390211842225rm_a_b @ R ) ) ) ) ).
% WN_onD
thf(fact_923_WN__onD,axiom,
! [R: set_Product_prod_b_b,A5: set_b,A: b] :
( ( abstract_WN_on_b @ R @ A5 )
=> ( ( member_b @ A @ A5 )
=> ? [B3: b] : ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ A @ B3 ) @ ( abstra380364156431640999lity_b @ R ) ) ) ) ).
% WN_onD
thf(fact_924_WN__onD,axiom,
! [R: set_Pr4386577575007340137rm_a_b,A5: set_term_a_b,A: term_a_b] :
( ( abstra4644735066934440262rm_a_b @ R @ A5 )
=> ( ( member_term_a_b @ A @ A5 )
=> ? [B3: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B3 ) @ ( abstra2972651231392952684rm_a_b @ R ) ) ) ) ).
% WN_onD
thf(fact_925_UNF__onD,axiom,
! [R: set_Pr2972776593051762503rm_a_b,A5: set_Pr4386577575007340137rm_a_b,A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,C2: produc357393685978478089rm_a_b] :
( ( abstra1303798319023342519rm_a_b @ R @ A5 )
=> ( ( member5869715511025134514rm_a_b @ A @ A5 )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( abstra1496011390211842225rm_a_b @ R ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ C2 ) @ ( abstra1496011390211842225rm_a_b @ R ) )
=> ( B = C2 ) ) ) ) ) ).
% UNF_onD
thf(fact_926_UNF__onD,axiom,
! [R: set_Product_prod_b_b,A5: set_b,A: b,B: b,C2: b] :
( ( abstract_UNF_on_b @ R @ A5 )
=> ( ( member_b @ A @ A5 )
=> ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ A @ B ) @ ( abstra380364156431640999lity_b @ R ) )
=> ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ A @ C2 ) @ ( abstra380364156431640999lity_b @ R ) )
=> ( B = C2 ) ) ) ) ) ).
% UNF_onD
thf(fact_927_UNF__onD,axiom,
! [R: set_Pr4386577575007340137rm_a_b,A5: set_term_a_b,A: term_a_b,B: term_a_b,C2: term_a_b] :
( ( abstra3619855735679963878rm_a_b @ R @ A5 )
=> ( ( member_term_a_b @ A @ A5 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( abstra2972651231392952684rm_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C2 ) @ ( abstra2972651231392952684rm_a_b @ R ) )
=> ( B = C2 ) ) ) ) ) ).
% UNF_onD
thf(fact_928_UNF__onE,axiom,
! [R: set_Pr2972776593051762503rm_a_b,A5: set_Pr4386577575007340137rm_a_b,A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,C2: produc357393685978478089rm_a_b] :
( ( abstra1303798319023342519rm_a_b @ R @ A5 )
=> ( ( member5869715511025134514rm_a_b @ A @ A5 )
=> ( ( B != C2 )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( abstra1496011390211842225rm_a_b @ R ) )
=> ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ C2 ) @ ( abstra1496011390211842225rm_a_b @ R ) ) ) ) ) ) ).
% UNF_onE
thf(fact_929_UNF__onE,axiom,
! [R: set_Product_prod_b_b,A5: set_b,A: b,B: b,C2: b] :
( ( abstract_UNF_on_b @ R @ A5 )
=> ( ( member_b @ A @ A5 )
=> ( ( B != C2 )
=> ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ A @ B ) @ ( abstra380364156431640999lity_b @ R ) )
=> ~ ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ A @ C2 ) @ ( abstra380364156431640999lity_b @ R ) ) ) ) ) ) ).
% UNF_onE
thf(fact_930_UNF__onE,axiom,
! [R: set_Pr4386577575007340137rm_a_b,A5: set_term_a_b,A: term_a_b,B: term_a_b,C2: term_a_b] :
( ( abstra3619855735679963878rm_a_b @ R @ A5 )
=> ( ( member_term_a_b @ A @ A5 )
=> ( ( B != C2 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( abstra2972651231392952684rm_a_b @ R ) )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C2 ) @ ( abstra2972651231392952684rm_a_b @ R ) ) ) ) ) ) ).
% UNF_onE
thf(fact_931_UNF__on__def,axiom,
( abstra3619855735679963878rm_a_b
= ( ^ [R4: set_Pr4386577575007340137rm_a_b,A6: set_term_a_b] :
! [X2: term_a_b] :
( ( member_term_a_b @ X2 @ A6 )
=> ! [B6: term_a_b,C4: term_a_b] :
( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ B6 ) @ ( abstra2972651231392952684rm_a_b @ R4 ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ C4 ) @ ( abstra2972651231392952684rm_a_b @ R4 ) ) )
=> ( B6 = C4 ) ) ) ) ) ).
% UNF_on_def
thf(fact_932_some__NF__UNF,axiom,
! [R: set_Pr2972776593051762503rm_a_b,X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b] :
( ( abstra1303798319023342519rm_a_b @ R @ top_to1314267278846557113rm_a_b )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ ( transi2615809358984392588rm_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ Y @ ( abstra1902783451776770197rm_a_b @ R ) )
=> ( Y
= ( abstra8203652949708803681rm_a_b @ R @ X ) ) ) ) ) ).
% some_NF_UNF
thf(fact_933_some__NF__UNF,axiom,
! [R: set_Product_prod_b_b,X: b,Y: b] :
( ( abstract_UNF_on_b @ R @ top_top_set_b )
=> ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ X @ Y ) @ ( transitive_rtrancl_b @ R ) )
=> ( ( member_b @ Y @ ( abstract_NF_b @ R ) )
=> ( Y
= ( abstract_some_NF_b @ R @ X ) ) ) ) ) ).
% some_NF_UNF
thf(fact_934_some__NF__UNF,axiom,
! [R: set_Pr4386577575007340137rm_a_b,X: term_a_b,Y: term_a_b] :
( ( abstra3619855735679963878rm_a_b @ R @ top_top_set_term_a_b )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ( ( member_term_a_b @ Y @ ( abstract_NF_term_a_b @ R ) )
=> ( Y
= ( abstra2589370493620291516rm_a_b @ R @ X ) ) ) ) ) ).
% some_NF_UNF
thf(fact_935_SN__reaches__NF,axiom,
! [R: set_Pr2972776593051762503rm_a_b,X: produc357393685978478089rm_a_b] :
( ( abstra2398554102055911763rm_a_b @ R @ ( insert7009541432154983385rm_a_b @ X @ bot_bo197521221353338581rm_a_b ) )
=> ? [Y3: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y3 ) @ ( transi2615809358984392588rm_a_b @ R ) )
& ( member5869715511025134514rm_a_b @ Y3 @ ( abstra1902783451776770197rm_a_b @ R ) ) ) ) ).
% SN_reaches_NF
thf(fact_936_SN__reaches__NF,axiom,
! [R: set_Product_prod_b_b,X: b] :
( ( abstract_SN_on_b @ R @ ( insert_b @ X @ bot_bot_set_b ) )
=> ? [Y3: b] :
( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ X @ Y3 ) @ ( transitive_rtrancl_b @ R ) )
& ( member_b @ Y3 @ ( abstract_NF_b @ R ) ) ) ) ).
% SN_reaches_NF
thf(fact_937_SN__reaches__NF,axiom,
! [R: set_Pr4386577575007340137rm_a_b,X: term_a_b] :
( ( abstra4720023341729745482rm_a_b @ R @ ( insert_term_a_b @ X @ bot_bot_set_term_a_b ) )
=> ? [Y3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y3 ) @ ( transi7742714808557438673rm_a_b @ R ) )
& ( member_term_a_b @ Y3 @ ( abstract_NF_term_a_b @ R ) ) ) ) ).
% SN_reaches_NF
thf(fact_938_empty__iff,axiom,
! [C2: produc357393685978478089rm_a_b] :
~ ( member5869715511025134514rm_a_b @ C2 @ bot_bo197521221353338581rm_a_b ) ).
% empty_iff
thf(fact_939_empty__iff,axiom,
! [C2: b] :
~ ( member_b @ C2 @ bot_bot_set_b ) ).
% empty_iff
thf(fact_940_all__not__in__conv,axiom,
! [A5: set_Pr4386577575007340137rm_a_b] :
( ( ! [X2: produc357393685978478089rm_a_b] :
~ ( member5869715511025134514rm_a_b @ X2 @ A5 ) )
= ( A5 = bot_bo197521221353338581rm_a_b ) ) ).
% all_not_in_conv
thf(fact_941_all__not__in__conv,axiom,
! [A5: set_b] :
( ( ! [X2: b] :
~ ( member_b @ X2 @ A5 ) )
= ( A5 = bot_bot_set_b ) ) ).
% all_not_in_conv
thf(fact_942_subset__empty,axiom,
! [A5: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A5 @ bot_bo9049108969261143879_a_nat )
= ( A5 = bot_bo9049108969261143879_a_nat ) ) ).
% subset_empty
thf(fact_943_empty__subsetI,axiom,
! [A5: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ bot_bo9049108969261143879_a_nat @ A5 ) ).
% empty_subsetI
thf(fact_944_singletonI,axiom,
! [A: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ A @ ( insert7009541432154983385rm_a_b @ A @ bot_bo197521221353338581rm_a_b ) ) ).
% singletonI
thf(fact_945_singletonI,axiom,
! [A: b] : ( member_b @ A @ ( insert_b @ A @ bot_bot_set_b ) ) ).
% singletonI
thf(fact_946_singleton__insert__inj__eq_H,axiom,
! [A: product_prod_a_nat,A5: set_Pr4934435412358123699_a_nat,B: product_prod_a_nat] :
( ( ( insert8054603423593749677_a_nat @ A @ A5 )
= ( insert8054603423593749677_a_nat @ B @ bot_bo9049108969261143879_a_nat ) )
= ( ( A = B )
& ( ord_le8666007276011122963_a_nat @ A5 @ ( insert8054603423593749677_a_nat @ B @ bot_bo9049108969261143879_a_nat ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_947_singleton__insert__inj__eq,axiom,
! [B: product_prod_a_nat,A: product_prod_a_nat,A5: set_Pr4934435412358123699_a_nat] :
( ( ( insert8054603423593749677_a_nat @ B @ bot_bo9049108969261143879_a_nat )
= ( insert8054603423593749677_a_nat @ A @ A5 ) )
= ( ( A = B )
& ( ord_le8666007276011122963_a_nat @ A5 @ ( insert8054603423593749677_a_nat @ B @ bot_bo9049108969261143879_a_nat ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_948_disjoint__insert_I2_J,axiom,
! [A5: set_Pr4386577575007340137rm_a_b,B: produc357393685978478089rm_a_b,B4: set_Pr4386577575007340137rm_a_b] :
( ( bot_bo197521221353338581rm_a_b
= ( inf_in1693821330460682071rm_a_b @ A5 @ ( insert7009541432154983385rm_a_b @ B @ B4 ) ) )
= ( ~ ( member5869715511025134514rm_a_b @ B @ A5 )
& ( bot_bo197521221353338581rm_a_b
= ( inf_in1693821330460682071rm_a_b @ A5 @ B4 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_949_disjoint__insert_I2_J,axiom,
! [A5: set_b,B: b,B4: set_b] :
( ( bot_bot_set_b
= ( inf_inf_set_b @ A5 @ ( insert_b @ B @ B4 ) ) )
= ( ~ ( member_b @ B @ A5 )
& ( bot_bot_set_b
= ( inf_inf_set_b @ A5 @ B4 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_950_disjoint__insert_I1_J,axiom,
! [B4: set_Pr4386577575007340137rm_a_b,A: produc357393685978478089rm_a_b,A5: set_Pr4386577575007340137rm_a_b] :
( ( ( inf_in1693821330460682071rm_a_b @ B4 @ ( insert7009541432154983385rm_a_b @ A @ A5 ) )
= bot_bo197521221353338581rm_a_b )
= ( ~ ( member5869715511025134514rm_a_b @ A @ B4 )
& ( ( inf_in1693821330460682071rm_a_b @ B4 @ A5 )
= bot_bo197521221353338581rm_a_b ) ) ) ).
% disjoint_insert(1)
thf(fact_951_disjoint__insert_I1_J,axiom,
! [B4: set_b,A: b,A5: set_b] :
( ( ( inf_inf_set_b @ B4 @ ( insert_b @ A @ A5 ) )
= bot_bot_set_b )
= ( ~ ( member_b @ A @ B4 )
& ( ( inf_inf_set_b @ B4 @ A5 )
= bot_bot_set_b ) ) ) ).
% disjoint_insert(1)
thf(fact_952_insert__disjoint_I2_J,axiom,
! [A: produc357393685978478089rm_a_b,A5: set_Pr4386577575007340137rm_a_b,B4: set_Pr4386577575007340137rm_a_b] :
( ( bot_bo197521221353338581rm_a_b
= ( inf_in1693821330460682071rm_a_b @ ( insert7009541432154983385rm_a_b @ A @ A5 ) @ B4 ) )
= ( ~ ( member5869715511025134514rm_a_b @ A @ B4 )
& ( bot_bo197521221353338581rm_a_b
= ( inf_in1693821330460682071rm_a_b @ A5 @ B4 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_953_insert__disjoint_I2_J,axiom,
! [A: b,A5: set_b,B4: set_b] :
( ( bot_bot_set_b
= ( inf_inf_set_b @ ( insert_b @ A @ A5 ) @ B4 ) )
= ( ~ ( member_b @ A @ B4 )
& ( bot_bot_set_b
= ( inf_inf_set_b @ A5 @ B4 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_954_insert__disjoint_I1_J,axiom,
! [A: produc357393685978478089rm_a_b,A5: set_Pr4386577575007340137rm_a_b,B4: set_Pr4386577575007340137rm_a_b] :
( ( ( inf_in1693821330460682071rm_a_b @ ( insert7009541432154983385rm_a_b @ A @ A5 ) @ B4 )
= bot_bo197521221353338581rm_a_b )
= ( ~ ( member5869715511025134514rm_a_b @ A @ B4 )
& ( ( inf_in1693821330460682071rm_a_b @ A5 @ B4 )
= bot_bo197521221353338581rm_a_b ) ) ) ).
% insert_disjoint(1)
thf(fact_955_insert__disjoint_I1_J,axiom,
! [A: b,A5: set_b,B4: set_b] :
( ( ( inf_inf_set_b @ ( insert_b @ A @ A5 ) @ B4 )
= bot_bot_set_b )
= ( ~ ( member_b @ A @ B4 )
& ( ( inf_inf_set_b @ A5 @ B4 )
= bot_bot_set_b ) ) ) ).
% insert_disjoint(1)
thf(fact_956_ground__vars__term__empty,axiom,
! [T: term_a_b] :
( ( term_ground_a_b @ T )
=> ( ( vars_term_a_b @ T )
= bot_bot_set_b ) ) ).
% ground_vars_term_empty
thf(fact_957_emptyE,axiom,
! [A: produc357393685978478089rm_a_b] :
~ ( member5869715511025134514rm_a_b @ A @ bot_bo197521221353338581rm_a_b ) ).
% emptyE
thf(fact_958_emptyE,axiom,
! [A: b] :
~ ( member_b @ A @ bot_bot_set_b ) ).
% emptyE
thf(fact_959_equals0D,axiom,
! [A5: set_Pr4386577575007340137rm_a_b,A: produc357393685978478089rm_a_b] :
( ( A5 = bot_bo197521221353338581rm_a_b )
=> ~ ( member5869715511025134514rm_a_b @ A @ A5 ) ) ).
% equals0D
thf(fact_960_equals0D,axiom,
! [A5: set_b,A: b] :
( ( A5 = bot_bot_set_b )
=> ~ ( member_b @ A @ A5 ) ) ).
% equals0D
thf(fact_961_equals0I,axiom,
! [A5: set_Pr4386577575007340137rm_a_b] :
( ! [Y3: produc357393685978478089rm_a_b] :
~ ( member5869715511025134514rm_a_b @ Y3 @ A5 )
=> ( A5 = bot_bo197521221353338581rm_a_b ) ) ).
% equals0I
thf(fact_962_equals0I,axiom,
! [A5: set_b] :
( ! [Y3: b] :
~ ( member_b @ Y3 @ A5 )
=> ( A5 = bot_bot_set_b ) ) ).
% equals0I
thf(fact_963_ex__in__conv,axiom,
! [A5: set_Pr4386577575007340137rm_a_b] :
( ( ? [X2: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ X2 @ A5 ) )
= ( A5 != bot_bo197521221353338581rm_a_b ) ) ).
% ex_in_conv
thf(fact_964_ex__in__conv,axiom,
! [A5: set_b] :
( ( ? [X2: b] : ( member_b @ X2 @ A5 ) )
= ( A5 != bot_bot_set_b ) ) ).
% ex_in_conv
thf(fact_965_singleton__iff,axiom,
! [B: produc357393685978478089rm_a_b,A: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ B @ ( insert7009541432154983385rm_a_b @ A @ bot_bo197521221353338581rm_a_b ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_966_singleton__iff,axiom,
! [B: b,A: b] :
( ( member_b @ B @ ( insert_b @ A @ bot_bot_set_b ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_967_singletonD,axiom,
! [B: produc357393685978478089rm_a_b,A: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ B @ ( insert7009541432154983385rm_a_b @ A @ bot_bo197521221353338581rm_a_b ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_968_singletonD,axiom,
! [B: b,A: b] :
( ( member_b @ B @ ( insert_b @ A @ bot_bot_set_b ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_969_subset__emptyI,axiom,
! [A5: set_Pr4386577575007340137rm_a_b] :
( ! [X3: produc357393685978478089rm_a_b] :
~ ( member5869715511025134514rm_a_b @ X3 @ A5 )
=> ( ord_le118470702582115849rm_a_b @ A5 @ bot_bo197521221353338581rm_a_b ) ) ).
% subset_emptyI
thf(fact_970_subset__emptyI,axiom,
! [A5: set_b] :
( ! [X3: b] :
~ ( member_b @ X3 @ A5 )
=> ( ord_less_eq_set_b @ A5 @ bot_bot_set_b ) ) ).
% subset_emptyI
thf(fact_971_subset__emptyI,axiom,
! [A5: set_Pr4934435412358123699_a_nat] :
( ! [X3: product_prod_a_nat] :
~ ( member5724188588386418708_a_nat @ X3 @ A5 )
=> ( ord_le8666007276011122963_a_nat @ A5 @ bot_bo9049108969261143879_a_nat ) ) ).
% subset_emptyI
thf(fact_972_bot_Oextremum__uniqueI,axiom,
! [A: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A @ bot_bo9049108969261143879_a_nat )
=> ( A = bot_bo9049108969261143879_a_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_973_bot_Oextremum__unique,axiom,
! [A: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A @ bot_bo9049108969261143879_a_nat )
= ( A = bot_bo9049108969261143879_a_nat ) ) ).
% bot.extremum_unique
thf(fact_974_bot_Oextremum,axiom,
! [A: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ bot_bo9049108969261143879_a_nat @ A ) ).
% bot.extremum
thf(fact_975_disjoint__iff,axiom,
! [A5: set_Pr4386577575007340137rm_a_b,B4: set_Pr4386577575007340137rm_a_b] :
( ( ( inf_in1693821330460682071rm_a_b @ A5 @ B4 )
= bot_bo197521221353338581rm_a_b )
= ( ! [X2: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X2 @ A5 )
=> ~ ( member5869715511025134514rm_a_b @ X2 @ B4 ) ) ) ) ).
% disjoint_iff
thf(fact_976_disjoint__iff,axiom,
! [A5: set_b,B4: set_b] :
( ( ( inf_inf_set_b @ A5 @ B4 )
= bot_bot_set_b )
= ( ! [X2: b] :
( ( member_b @ X2 @ A5 )
=> ~ ( member_b @ X2 @ B4 ) ) ) ) ).
% disjoint_iff
thf(fact_977_Int__emptyI,axiom,
! [A5: set_Pr4386577575007340137rm_a_b,B4: set_Pr4386577575007340137rm_a_b] :
( ! [X3: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X3 @ A5 )
=> ~ ( member5869715511025134514rm_a_b @ X3 @ B4 ) )
=> ( ( inf_in1693821330460682071rm_a_b @ A5 @ B4 )
= bot_bo197521221353338581rm_a_b ) ) ).
% Int_emptyI
thf(fact_978_Int__emptyI,axiom,
! [A5: set_b,B4: set_b] :
( ! [X3: b] :
( ( member_b @ X3 @ A5 )
=> ~ ( member_b @ X3 @ B4 ) )
=> ( ( inf_inf_set_b @ A5 @ B4 )
= bot_bot_set_b ) ) ).
% Int_emptyI
thf(fact_979_subset__singleton__iff,axiom,
! [X5: set_Pr4934435412358123699_a_nat,A: product_prod_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X5 @ ( insert8054603423593749677_a_nat @ A @ bot_bo9049108969261143879_a_nat ) )
= ( ( X5 = bot_bo9049108969261143879_a_nat )
| ( X5
= ( insert8054603423593749677_a_nat @ A @ bot_bo9049108969261143879_a_nat ) ) ) ) ).
% subset_singleton_iff
thf(fact_980_subset__singletonD,axiom,
! [A5: set_Pr4934435412358123699_a_nat,X: product_prod_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A5 @ ( insert8054603423593749677_a_nat @ X @ bot_bo9049108969261143879_a_nat ) )
=> ( ( A5 = bot_bo9049108969261143879_a_nat )
| ( A5
= ( insert8054603423593749677_a_nat @ X @ bot_bo9049108969261143879_a_nat ) ) ) ) ).
% subset_singletonD
thf(fact_981_wfE__min_H,axiom,
! [R5: set_Pr2972776593051762503rm_a_b,Q: set_Pr4386577575007340137rm_a_b] :
( ( wf_Pro2335863617654816626rm_a_b @ R5 )
=> ( ( Q != bot_bo197521221353338581rm_a_b )
=> ~ ! [Z4: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ Z4 @ Q )
=> ~ ! [Y8: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y8 @ Z4 ) @ R5 )
=> ~ ( member5869715511025134514rm_a_b @ Y8 @ Q ) ) ) ) ) ).
% wfE_min'
thf(fact_982_wfE__min_H,axiom,
! [R5: set_Product_prod_b_b,Q: set_b] :
( ( wf_b @ R5 )
=> ( ( Q != bot_bot_set_b )
=> ~ ! [Z4: b] :
( ( member_b @ Z4 @ Q )
=> ~ ! [Y8: b] :
( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ Y8 @ Z4 ) @ R5 )
=> ~ ( member_b @ Y8 @ Q ) ) ) ) ) ).
% wfE_min'
thf(fact_983_wfE__min_H,axiom,
! [R5: set_Pr4386577575007340137rm_a_b,Q: set_term_a_b] :
( ( wf_term_a_b @ R5 )
=> ( ( Q != bot_bot_set_term_a_b )
=> ~ ! [Z4: term_a_b] :
( ( member_term_a_b @ Z4 @ Q )
=> ~ ! [Y8: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y8 @ Z4 ) @ R5 )
=> ~ ( member_term_a_b @ Y8 @ Q ) ) ) ) ) ).
% wfE_min'
thf(fact_984_vars__term__empty__ground,axiom,
! [S: term_a_b] :
( ( ( vars_term_a_b @ S )
= bot_bot_set_b )
=> ( term_ground_a_b @ S ) ) ).
% vars_term_empty_ground
thf(fact_985_SN__on__all__reducts__SN__on__conv,axiom,
! [R: set_Pr4386577575007340137rm_a_b,A: term_a_b] :
( ( abstra4720023341729745482rm_a_b @ R @ ( insert_term_a_b @ A @ bot_bot_set_term_a_b ) )
= ( ! [B6: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B6 ) @ R )
=> ( abstra4720023341729745482rm_a_b @ R @ ( insert_term_a_b @ B6 @ bot_bot_set_term_a_b ) ) ) ) ) ).
% SN_on_all_reducts_SN_on_conv
thf(fact_986_lower__set__imp__not__SN__on,axiom,
! [S: produc357393685978478089rm_a_b,X5: set_Pr4386577575007340137rm_a_b,R5: set_Pr2972776593051762503rm_a_b] :
( ( member5869715511025134514rm_a_b @ S @ X5 )
=> ( ! [X3: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X3 @ X5 )
=> ? [Xa2: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ Xa2 @ X5 )
& ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X3 @ Xa2 ) @ R5 ) ) )
=> ~ ( abstra2398554102055911763rm_a_b @ R5 @ ( insert7009541432154983385rm_a_b @ S @ bot_bo197521221353338581rm_a_b ) ) ) ) ).
% lower_set_imp_not_SN_on
thf(fact_987_lower__set__imp__not__SN__on,axiom,
! [S: b,X5: set_b,R5: set_Product_prod_b_b] :
( ( member_b @ S @ X5 )
=> ( ! [X3: b] :
( ( member_b @ X3 @ X5 )
=> ? [Xa2: b] :
( ( member_b @ Xa2 @ X5 )
& ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ X3 @ Xa2 ) @ R5 ) ) )
=> ~ ( abstract_SN_on_b @ R5 @ ( insert_b @ S @ bot_bot_set_b ) ) ) ) ).
% lower_set_imp_not_SN_on
thf(fact_988_lower__set__imp__not__SN__on,axiom,
! [S: term_a_b,X5: set_term_a_b,R5: set_Pr4386577575007340137rm_a_b] :
( ( member_term_a_b @ S @ X5 )
=> ( ! [X3: term_a_b] :
( ( member_term_a_b @ X3 @ X5 )
=> ? [Xa2: term_a_b] :
( ( member_term_a_b @ Xa2 @ X5 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Xa2 ) @ R5 ) ) )
=> ~ ( abstra4720023341729745482rm_a_b @ R5 @ ( insert_term_a_b @ S @ bot_bot_set_term_a_b ) ) ) ) ).
% lower_set_imp_not_SN_on
thf(fact_989_SN__on__induct__acc__style,axiom,
! [R5: set_Pr4386577575007340137rm_a_b,A: term_a_b,P: term_a_b > $o] :
( ( abstra4720023341729745482rm_a_b @ R5 @ ( insert_term_a_b @ A @ bot_bot_set_term_a_b ) )
=> ( ! [X3: term_a_b] :
( ( abstra4720023341729745482rm_a_b @ R5 @ ( insert_term_a_b @ X3 @ bot_bot_set_term_a_b ) )
=> ( ! [Y8: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y8 ) @ R5 )
=> ( P @ Y8 ) )
=> ( P @ X3 ) ) )
=> ( P @ A ) ) ) ).
% SN_on_induct_acc_style
thf(fact_990_step__preserves__SN__on,axiom,
! [A: term_a_b,B: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R )
=> ( ( abstra4720023341729745482rm_a_b @ R @ ( insert_term_a_b @ A @ bot_bot_set_term_a_b ) )
=> ( abstra4720023341729745482rm_a_b @ R @ ( insert_term_a_b @ B @ bot_bot_set_term_a_b ) ) ) ) ).
% step_preserves_SN_on
thf(fact_991_SN__on__imp__on__minimal,axiom,
! [R: set_Pr2972776593051762503rm_a_b,X: produc357393685978478089rm_a_b] :
( ( abstra2398554102055911763rm_a_b @ R @ ( insert7009541432154983385rm_a_b @ X @ bot_bo197521221353338581rm_a_b ) )
=> ! [Q4: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ X @ Q4 )
=> ? [X3: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X3 @ Q4 )
& ! [Y8: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X3 @ Y8 ) @ R )
=> ~ ( member5869715511025134514rm_a_b @ Y8 @ Q4 ) ) ) ) ) ).
% SN_on_imp_on_minimal
thf(fact_992_SN__on__imp__on__minimal,axiom,
! [R: set_Product_prod_b_b,X: b] :
( ( abstract_SN_on_b @ R @ ( insert_b @ X @ bot_bot_set_b ) )
=> ! [Q4: set_b] :
( ( member_b @ X @ Q4 )
=> ? [X3: b] :
( ( member_b @ X3 @ Q4 )
& ! [Y8: b] :
( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ X3 @ Y8 ) @ R )
=> ~ ( member_b @ Y8 @ Q4 ) ) ) ) ) ).
% SN_on_imp_on_minimal
thf(fact_993_SN__on__imp__on__minimal,axiom,
! [R: set_Pr4386577575007340137rm_a_b,X: term_a_b] :
( ( abstra4720023341729745482rm_a_b @ R @ ( insert_term_a_b @ X @ bot_bot_set_term_a_b ) )
=> ! [Q4: set_term_a_b] :
( ( member_term_a_b @ X @ Q4 )
=> ? [X3: term_a_b] :
( ( member_term_a_b @ X3 @ Q4 )
& ! [Y8: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y8 ) @ R )
=> ~ ( member_term_a_b @ Y8 @ Q4 ) ) ) ) ) ).
% SN_on_imp_on_minimal
thf(fact_994_step__reflects__SN__on,axiom,
! [A: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ! [B3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B3 ) @ R )
=> ( abstra4720023341729745482rm_a_b @ R @ ( insert_term_a_b @ B3 @ bot_bot_set_term_a_b ) ) )
=> ( abstra4720023341729745482rm_a_b @ R @ ( insert_term_a_b @ A @ bot_bot_set_term_a_b ) ) ) ).
% step_reflects_SN_on
thf(fact_995_SN__on__induct_H,axiom,
! [R5: set_Pr4386577575007340137rm_a_b,S: term_a_b,P: term_a_b > $o] :
( ( abstra4720023341729745482rm_a_b @ R5 @ ( insert_term_a_b @ S @ bot_bot_set_term_a_b ) )
=> ( ! [T2: term_a_b] :
( ! [U4: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T2 @ U4 ) @ R5 )
=> ( P @ U4 ) )
=> ( P @ T2 ) )
=> ( P @ S ) ) ) ).
% SN_on_induct'
thf(fact_996_vars__term__disjoint__imp__unifier,axiom,
! [S: term_a_b,T: term_a_b,Sigma3: b > term_a_b,Tau: b > term_a_b] :
( ( ( inf_inf_set_b @ ( vars_term_a_b @ S ) @ ( vars_term_a_b @ T ) )
= bot_bot_set_b )
=> ( ( ( subst_7999470309526761004_a_b_b @ S @ Sigma3 )
= ( subst_7999470309526761004_a_b_b @ T @ Tau ) )
=> ? [Mu: b > term_a_b] :
( ( subst_7999470309526761004_a_b_b @ S @ Mu )
= ( subst_7999470309526761004_a_b_b @ T @ Mu ) ) ) ) ).
% vars_term_disjoint_imp_unifier
thf(fact_997_NF__imp__SN__on,axiom,
! [A: produc357393685978478089rm_a_b,R5: set_Pr2972776593051762503rm_a_b] :
( ( member5869715511025134514rm_a_b @ A @ ( abstra1902783451776770197rm_a_b @ R5 ) )
=> ( abstra2398554102055911763rm_a_b @ R5 @ ( insert7009541432154983385rm_a_b @ A @ bot_bo197521221353338581rm_a_b ) ) ) ).
% NF_imp_SN_on
thf(fact_998_NF__imp__SN__on,axiom,
! [A: b,R5: set_Product_prod_b_b] :
( ( member_b @ A @ ( abstract_NF_b @ R5 ) )
=> ( abstract_SN_on_b @ R5 @ ( insert_b @ A @ bot_bot_set_b ) ) ) ).
% NF_imp_SN_on
thf(fact_999_times__subset__iff,axiom,
! [A5: set_Pr4934435412358123699_a_nat,C3: set_Pr4934435412358123699_a_nat,B4: set_Pr4934435412358123699_a_nat,D: set_Pr4934435412358123699_a_nat] :
( ( ord_le1018141841545957851_a_nat
@ ( produc3832836940518797764_a_nat @ A5
@ ^ [Uu: product_prod_a_nat] : C3 )
@ ( produc3832836940518797764_a_nat @ B4
@ ^ [Uu: product_prod_a_nat] : D ) )
= ( ( A5 = bot_bo9049108969261143879_a_nat )
| ( C3 = bot_bo9049108969261143879_a_nat )
| ( ( ord_le8666007276011122963_a_nat @ A5 @ B4 )
& ( ord_le8666007276011122963_a_nat @ C3 @ D ) ) ) ) ).
% times_subset_iff
thf(fact_1000_times__subset__iff,axiom,
! [A5: set_a,C3: set_nat,B4: set_a,D: set_nat] :
( ( ord_le8666007276011122963_a_nat
@ ( product_Sigma_a_nat @ A5
@ ^ [Uu: a] : C3 )
@ ( product_Sigma_a_nat @ B4
@ ^ [Uu: a] : D ) )
= ( ( A5 = bot_bot_set_a )
| ( C3 = bot_bot_set_nat )
| ( ( ord_less_eq_set_a @ A5 @ B4 )
& ( ord_less_eq_set_nat @ C3 @ D ) ) ) ) ).
% times_subset_iff
thf(fact_1001_SN__on__subset__SN__terms,axiom,
! [R5: set_Pr1811044260758604347_a_nat,X5: set_Pr4934435412358123699_a_nat] :
( ( abstra8653715922312955827_a_nat @ R5 @ X5 )
=> ( ord_le8666007276011122963_a_nat @ X5
@ ( collec4464134535221767506_a_nat
@ ^ [X2: product_prod_a_nat] : ( abstra8653715922312955827_a_nat @ R5 @ ( insert8054603423593749677_a_nat @ X2 @ bot_bo9049108969261143879_a_nat ) ) ) ) ) ).
% SN_on_subset_SN_terms
thf(fact_1002_some__NF,axiom,
! [R: set_Pr2972776593051762503rm_a_b,X: produc357393685978478089rm_a_b] :
( ( abstra2398554102055911763rm_a_b @ R @ ( insert7009541432154983385rm_a_b @ X @ bot_bo197521221353338581rm_a_b ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ ( abstra8203652949708803681rm_a_b @ R @ X ) ) @ ( transi2615809358984392588rm_a_b @ R ) )
& ( member5869715511025134514rm_a_b @ ( abstra8203652949708803681rm_a_b @ R @ X ) @ ( abstra1902783451776770197rm_a_b @ R ) ) ) ) ).
% some_NF
thf(fact_1003_some__NF,axiom,
! [R: set_Product_prod_b_b,X: b] :
( ( abstract_SN_on_b @ R @ ( insert_b @ X @ bot_bot_set_b ) )
=> ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ X @ ( abstract_some_NF_b @ R @ X ) ) @ ( transitive_rtrancl_b @ R ) )
& ( member_b @ ( abstract_some_NF_b @ R @ X ) @ ( abstract_NF_b @ R ) ) ) ) ).
% some_NF
thf(fact_1004_some__NF,axiom,
! [R: set_Pr4386577575007340137rm_a_b,X: term_a_b] :
( ( abstra4720023341729745482rm_a_b @ R @ ( insert_term_a_b @ X @ bot_bot_set_term_a_b ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ ( abstra2589370493620291516rm_a_b @ R @ X ) ) @ ( transi7742714808557438673rm_a_b @ R ) )
& ( member_term_a_b @ ( abstra2589370493620291516rm_a_b @ R @ X ) @ ( abstract_NF_term_a_b @ R ) ) ) ) ).
% some_NF
thf(fact_1005_steps__reflect__SN__on,axiom,
! [R: set_Pr4386577575007340137rm_a_b,B: term_a_b,A: term_a_b] :
( ~ ( abstra4720023341729745482rm_a_b @ R @ ( insert_term_a_b @ B @ bot_bot_set_term_a_b ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ~ ( abstra4720023341729745482rm_a_b @ R @ ( insert_term_a_b @ A @ bot_bot_set_term_a_b ) ) ) ) ).
% steps_reflect_SN_on
thf(fact_1006_steps__preserve__SN__on,axiom,
! [A: term_a_b,B: term_a_b,A5: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ A5 ) )
=> ( ( abstra4720023341729745482rm_a_b @ A5 @ ( insert_term_a_b @ A @ bot_bot_set_term_a_b ) )
=> ( abstra4720023341729745482rm_a_b @ A5 @ ( insert_term_a_b @ B @ bot_bot_set_term_a_b ) ) ) ) ).
% steps_preserve_SN_on
thf(fact_1007_SN__on__imp__normalizability,axiom,
! [R: set_Pr4386577575007340137rm_a_b,A: term_a_b] :
( ( abstra4720023341729745482rm_a_b @ R @ ( insert_term_a_b @ A @ bot_bot_set_term_a_b ) )
=> ? [B3: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B3 ) @ ( abstra2972651231392952684rm_a_b @ R ) ) ) ).
% SN_on_imp_normalizability
thf(fact_1008_some__NF__WCR,axiom,
! [R: set_Pr2972776593051762503rm_a_b,X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b] :
( ( abstra2398554102055911763rm_a_b @ R @ ( insert7009541432154983385rm_a_b @ X @ bot_bo197521221353338581rm_a_b ) )
=> ( ( abstra875157153669124638rm_a_b @ R
@ ( collec99177395925924084rm_a_b
@ ^ [X2: produc357393685978478089rm_a_b] : ( abstra2398554102055911763rm_a_b @ R @ ( insert7009541432154983385rm_a_b @ X2 @ bot_bo197521221353338581rm_a_b ) ) ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ ( transi2615809358984392588rm_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ Y @ ( abstra1902783451776770197rm_a_b @ R ) )
=> ( Y
= ( abstra8203652949708803681rm_a_b @ R @ X ) ) ) ) ) ) ).
% some_NF_WCR
thf(fact_1009_some__NF__WCR,axiom,
! [R: set_Product_prod_b_b,X: b,Y: b] :
( ( abstract_SN_on_b @ R @ ( insert_b @ X @ bot_bot_set_b ) )
=> ( ( abstract_WCR_on_b @ R
@ ( collect_b
@ ^ [X2: b] : ( abstract_SN_on_b @ R @ ( insert_b @ X2 @ bot_bot_set_b ) ) ) )
=> ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ X @ Y ) @ ( transitive_rtrancl_b @ R ) )
=> ( ( member_b @ Y @ ( abstract_NF_b @ R ) )
=> ( Y
= ( abstract_some_NF_b @ R @ X ) ) ) ) ) ) ).
% some_NF_WCR
thf(fact_1010_some__NF__WCR,axiom,
! [R: set_Pr4386577575007340137rm_a_b,X: term_a_b,Y: term_a_b] :
( ( abstra4720023341729745482rm_a_b @ R @ ( insert_term_a_b @ X @ bot_bot_set_term_a_b ) )
=> ( ( abstra5026849107050860991rm_a_b @ R
@ ( collect_term_a_b
@ ^ [X2: term_a_b] : ( abstra4720023341729745482rm_a_b @ R @ ( insert_term_a_b @ X2 @ bot_bot_set_term_a_b ) ) ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ( ( member_term_a_b @ Y @ ( abstract_NF_term_a_b @ R ) )
=> ( Y
= ( abstra2589370493620291516rm_a_b @ R @ X ) ) ) ) ) ) ).
% some_NF_WCR
thf(fact_1011_SN__WCR__reaches__NF,axiom,
! [R: set_Pr2972776593051762503rm_a_b,X: produc357393685978478089rm_a_b] :
( ( abstra2398554102055911763rm_a_b @ R @ ( insert7009541432154983385rm_a_b @ X @ bot_bo197521221353338581rm_a_b ) )
=> ( ( abstra875157153669124638rm_a_b @ R
@ ( collec99177395925924084rm_a_b
@ ^ [X2: produc357393685978478089rm_a_b] : ( abstra2398554102055911763rm_a_b @ R @ ( insert7009541432154983385rm_a_b @ X2 @ bot_bo197521221353338581rm_a_b ) ) ) )
=> ? [X3: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ X3 ) @ ( transi2615809358984392588rm_a_b @ R ) )
& ( member5869715511025134514rm_a_b @ X3 @ ( abstra1902783451776770197rm_a_b @ R ) )
& ! [Y8: produc357393685978478089rm_a_b] :
( ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y8 ) @ ( transi2615809358984392588rm_a_b @ R ) )
& ( member5869715511025134514rm_a_b @ Y8 @ ( abstra1902783451776770197rm_a_b @ R ) ) )
=> ( Y8 = X3 ) ) ) ) ) ).
% SN_WCR_reaches_NF
thf(fact_1012_SN__WCR__reaches__NF,axiom,
! [R: set_Product_prod_b_b,X: b] :
( ( abstract_SN_on_b @ R @ ( insert_b @ X @ bot_bot_set_b ) )
=> ( ( abstract_WCR_on_b @ R
@ ( collect_b
@ ^ [X2: b] : ( abstract_SN_on_b @ R @ ( insert_b @ X2 @ bot_bot_set_b ) ) ) )
=> ? [X3: b] :
( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ X @ X3 ) @ ( transitive_rtrancl_b @ R ) )
& ( member_b @ X3 @ ( abstract_NF_b @ R ) )
& ! [Y8: b] :
( ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ X @ Y8 ) @ ( transitive_rtrancl_b @ R ) )
& ( member_b @ Y8 @ ( abstract_NF_b @ R ) ) )
=> ( Y8 = X3 ) ) ) ) ) ).
% SN_WCR_reaches_NF
thf(fact_1013_SN__WCR__reaches__NF,axiom,
! [R: set_Pr4386577575007340137rm_a_b,X: term_a_b] :
( ( abstra4720023341729745482rm_a_b @ R @ ( insert_term_a_b @ X @ bot_bot_set_term_a_b ) )
=> ( ( abstra5026849107050860991rm_a_b @ R
@ ( collect_term_a_b
@ ^ [X2: term_a_b] : ( abstra4720023341729745482rm_a_b @ R @ ( insert_term_a_b @ X2 @ bot_bot_set_term_a_b ) ) ) )
=> ? [X3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ X3 ) @ ( transi7742714808557438673rm_a_b @ R ) )
& ( member_term_a_b @ X3 @ ( abstract_NF_term_a_b @ R ) )
& ! [Y8: term_a_b] :
( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y8 ) @ ( transi7742714808557438673rm_a_b @ R ) )
& ( member_term_a_b @ Y8 @ ( abstract_NF_term_a_b @ R ) ) )
=> ( Y8 = X3 ) ) ) ) ) ).
% SN_WCR_reaches_NF
thf(fact_1014_bot__empty__eq,axiom,
( bot_bo23760184229683272_a_b_o
= ( ^ [X2: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ X2 @ bot_bo197521221353338581rm_a_b ) ) ) ).
% bot_empty_eq
thf(fact_1015_bot__empty__eq,axiom,
( bot_bot_b_o
= ( ^ [X2: b] : ( member_b @ X2 @ bot_bot_set_b ) ) ) ).
% bot_empty_eq
thf(fact_1016_bot__empty__eq2,axiom,
( bot_bo1413036057729848522_a_b_o
= ( ^ [X2: term_a_b,Y4: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y4 ) @ bot_bo197521221353338581rm_a_b ) ) ) ).
% bot_empty_eq2
thf(fact_1017_is__singletonI_H,axiom,
! [A5: set_Pr4386577575007340137rm_a_b] :
( ( A5 != bot_bo197521221353338581rm_a_b )
=> ( ! [X3: produc357393685978478089rm_a_b,Y3: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X3 @ A5 )
=> ( ( member5869715511025134514rm_a_b @ Y3 @ A5 )
=> ( X3 = Y3 ) ) )
=> ( is_sin332545968047684349rm_a_b @ A5 ) ) ) ).
% is_singletonI'
thf(fact_1018_is__singletonI_H,axiom,
! [A5: set_b] :
( ( A5 != bot_bot_set_b )
=> ( ! [X3: b,Y3: b] :
( ( member_b @ X3 @ A5 )
=> ( ( member_b @ Y3 @ A5 )
=> ( X3 = Y3 ) ) )
=> ( is_singleton_b @ A5 ) ) ) ).
% is_singletonI'
thf(fact_1019_max__extp__max__ext__eq,axiom,
! [R5: set_Pr4386577575007340137rm_a_b] :
( ( max_extp_term_a_b
@ ^ [X2: term_a_b,Y4: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y4 ) @ R5 ) )
= ( ^ [X2: set_term_a_b,Y4: set_term_a_b] : ( member8888773126115863070rm_a_b @ ( produc8517751122659631533rm_a_b @ X2 @ Y4 ) @ ( max_ext_term_a_b @ R5 ) ) ) ) ).
% max_extp_max_ext_eq
thf(fact_1020_SN__on__imp__acc,axiom,
! [R: set_Pr2972776593051762503rm_a_b,X: produc357393685978478089rm_a_b] :
( ( abstra2398554102055911763rm_a_b
@ ( collec1344989105561874898rm_a_b
@ ( produc161078424057476524_a_b_o
@ ^ [Y4: produc357393685978478089rm_a_b,Z5: produc357393685978478089rm_a_b] : ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Z5 @ Y4 ) @ R ) ) )
@ ( insert7009541432154983385rm_a_b @ X @ bot_bo197521221353338581rm_a_b ) )
=> ( member5869715511025134514rm_a_b @ X @ ( acc_Pr4364474097874652878rm_a_b @ R ) ) ) ).
% SN_on_imp_acc
thf(fact_1021_SN__on__imp__acc,axiom,
! [R: set_Product_prod_b_b,X: b] :
( ( abstract_SN_on_b
@ ( collec548942219715005266od_b_b
@ ( produc4772849988445941548_b_b_o
@ ^ [Y4: b,Z5: b] : ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ Z5 @ Y4 ) @ R ) ) )
@ ( insert_b @ X @ bot_bot_set_b ) )
=> ( member_b @ X @ ( acc_b @ R ) ) ) ).
% SN_on_imp_acc
thf(fact_1022_SN__on__imp__acc,axiom,
! [R: set_Pr4386577575007340137rm_a_b,X: term_a_b] :
( ( abstra4720023341729745482rm_a_b
@ ( collec99177395925924084rm_a_b
@ ( produc6251846659952913346_a_b_o
@ ^ [Y4: term_a_b,Z5: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Z5 @ Y4 ) @ R ) ) )
@ ( insert_term_a_b @ X @ bot_bot_set_term_a_b ) )
=> ( member_term_a_b @ X @ ( acc_term_a_b @ R ) ) ) ).
% SN_on_imp_acc
thf(fact_1023_acc__imp__SN__on,axiom,
! [X: produc357393685978478089rm_a_b,R: set_Pr2972776593051762503rm_a_b] :
( ( member5869715511025134514rm_a_b @ X @ ( acc_Pr4364474097874652878rm_a_b @ R ) )
=> ( abstra2398554102055911763rm_a_b
@ ( collec1344989105561874898rm_a_b
@ ( produc161078424057476524_a_b_o
@ ^ [Y4: produc357393685978478089rm_a_b,Z5: produc357393685978478089rm_a_b] : ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Z5 @ Y4 ) @ R ) ) )
@ ( insert7009541432154983385rm_a_b @ X @ bot_bo197521221353338581rm_a_b ) ) ) ).
% acc_imp_SN_on
thf(fact_1024_acc__imp__SN__on,axiom,
! [X: b,R: set_Product_prod_b_b] :
( ( member_b @ X @ ( acc_b @ R ) )
=> ( abstract_SN_on_b
@ ( collec548942219715005266od_b_b
@ ( produc4772849988445941548_b_b_o
@ ^ [Y4: b,Z5: b] : ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ Z5 @ Y4 ) @ R ) ) )
@ ( insert_b @ X @ bot_bot_set_b ) ) ) ).
% acc_imp_SN_on
thf(fact_1025_acc__imp__SN__on,axiom,
! [X: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( member_term_a_b @ X @ ( acc_term_a_b @ R ) )
=> ( abstra4720023341729745482rm_a_b
@ ( collec99177395925924084rm_a_b
@ ( produc6251846659952913346_a_b_o
@ ^ [Y4: term_a_b,Z5: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Z5 @ Y4 ) @ R ) ) )
@ ( insert_term_a_b @ X @ bot_bot_set_term_a_b ) ) ) ).
% acc_imp_SN_on
thf(fact_1026_acc_Ocases,axiom,
! [A: produc357393685978478089rm_a_b,R: set_Pr2972776593051762503rm_a_b] :
( ( member5869715511025134514rm_a_b @ A @ ( acc_Pr4364474097874652878rm_a_b @ R ) )
=> ! [Y8: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y8 @ A ) @ R )
=> ( member5869715511025134514rm_a_b @ Y8 @ ( acc_Pr4364474097874652878rm_a_b @ R ) ) ) ) ).
% acc.cases
thf(fact_1027_acc_Ocases,axiom,
! [A: b,R: set_Product_prod_b_b] :
( ( member_b @ A @ ( acc_b @ R ) )
=> ! [Y8: b] :
( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ Y8 @ A ) @ R )
=> ( member_b @ Y8 @ ( acc_b @ R ) ) ) ) ).
% acc.cases
thf(fact_1028_acc_Ocases,axiom,
! [A: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( member_term_a_b @ A @ ( acc_term_a_b @ R ) )
=> ! [Y8: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y8 @ A ) @ R )
=> ( member_term_a_b @ Y8 @ ( acc_term_a_b @ R ) ) ) ) ).
% acc.cases
thf(fact_1029_acc_Osimps,axiom,
! [A: produc357393685978478089rm_a_b,R: set_Pr2972776593051762503rm_a_b] :
( ( member5869715511025134514rm_a_b @ A @ ( acc_Pr4364474097874652878rm_a_b @ R ) )
= ( ? [X2: produc357393685978478089rm_a_b] :
( ( A = X2 )
& ! [Y4: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y4 @ X2 ) @ R )
=> ( member5869715511025134514rm_a_b @ Y4 @ ( acc_Pr4364474097874652878rm_a_b @ R ) ) ) ) ) ) ).
% acc.simps
thf(fact_1030_acc_Osimps,axiom,
! [A: b,R: set_Product_prod_b_b] :
( ( member_b @ A @ ( acc_b @ R ) )
= ( ? [X2: b] :
( ( A = X2 )
& ! [Y4: b] :
( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ Y4 @ X2 ) @ R )
=> ( member_b @ Y4 @ ( acc_b @ R ) ) ) ) ) ) ).
% acc.simps
thf(fact_1031_acc_Osimps,axiom,
! [A: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( member_term_a_b @ A @ ( acc_term_a_b @ R ) )
= ( ? [X2: term_a_b] :
( ( A = X2 )
& ! [Y4: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y4 @ X2 ) @ R )
=> ( member_term_a_b @ Y4 @ ( acc_term_a_b @ R ) ) ) ) ) ) ).
% acc.simps
thf(fact_1032_acc_Ointros,axiom,
! [X: produc357393685978478089rm_a_b,R: set_Pr2972776593051762503rm_a_b] :
( ! [Y3: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y3 @ X ) @ R )
=> ( member5869715511025134514rm_a_b @ Y3 @ ( acc_Pr4364474097874652878rm_a_b @ R ) ) )
=> ( member5869715511025134514rm_a_b @ X @ ( acc_Pr4364474097874652878rm_a_b @ R ) ) ) ).
% acc.intros
thf(fact_1033_acc_Ointros,axiom,
! [X: b,R: set_Product_prod_b_b] :
( ! [Y3: b] :
( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ Y3 @ X ) @ R )
=> ( member_b @ Y3 @ ( acc_b @ R ) ) )
=> ( member_b @ X @ ( acc_b @ R ) ) ) ).
% acc.intros
thf(fact_1034_acc_Ointros,axiom,
! [X: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ! [Y3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ X ) @ R )
=> ( member_term_a_b @ Y3 @ ( acc_term_a_b @ R ) ) )
=> ( member_term_a_b @ X @ ( acc_term_a_b @ R ) ) ) ).
% acc.intros
thf(fact_1035_acc__induct,axiom,
! [A: produc357393685978478089rm_a_b,R: set_Pr2972776593051762503rm_a_b,P: produc357393685978478089rm_a_b > $o] :
( ( member5869715511025134514rm_a_b @ A @ ( acc_Pr4364474097874652878rm_a_b @ R ) )
=> ( ! [X3: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X3 @ ( acc_Pr4364474097874652878rm_a_b @ R ) )
=> ( ! [Y8: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y8 @ X3 ) @ R )
=> ( P @ Y8 ) )
=> ( P @ X3 ) ) )
=> ( P @ A ) ) ) ).
% acc_induct
thf(fact_1036_acc__induct,axiom,
! [A: b,R: set_Product_prod_b_b,P: b > $o] :
( ( member_b @ A @ ( acc_b @ R ) )
=> ( ! [X3: b] :
( ( member_b @ X3 @ ( acc_b @ R ) )
=> ( ! [Y8: b] :
( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ Y8 @ X3 ) @ R )
=> ( P @ Y8 ) )
=> ( P @ X3 ) ) )
=> ( P @ A ) ) ) ).
% acc_induct
thf(fact_1037_acc__induct,axiom,
! [A: term_a_b,R: set_Pr4386577575007340137rm_a_b,P: term_a_b > $o] :
( ( member_term_a_b @ A @ ( acc_term_a_b @ R ) )
=> ( ! [X3: term_a_b] :
( ( member_term_a_b @ X3 @ ( acc_term_a_b @ R ) )
=> ( ! [Y8: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y8 @ X3 ) @ R )
=> ( P @ Y8 ) )
=> ( P @ X3 ) ) )
=> ( P @ A ) ) ) ).
% acc_induct
thf(fact_1038_acc__downward,axiom,
! [B: produc357393685978478089rm_a_b,R: set_Pr2972776593051762503rm_a_b,A: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ B @ ( acc_Pr4364474097874652878rm_a_b @ R ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ R )
=> ( member5869715511025134514rm_a_b @ A @ ( acc_Pr4364474097874652878rm_a_b @ R ) ) ) ) ).
% acc_downward
thf(fact_1039_acc__downward,axiom,
! [B: b,R: set_Product_prod_b_b,A: b] :
( ( member_b @ B @ ( acc_b @ R ) )
=> ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ A @ B ) @ R )
=> ( member_b @ A @ ( acc_b @ R ) ) ) ) ).
% acc_downward
thf(fact_1040_acc__downward,axiom,
! [B: term_a_b,R: set_Pr4386577575007340137rm_a_b,A: term_a_b] :
( ( member_term_a_b @ B @ ( acc_term_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R )
=> ( member_term_a_b @ A @ ( acc_term_a_b @ R ) ) ) ) ).
% acc_downward
thf(fact_1041_not__acc__down,axiom,
! [X: produc357393685978478089rm_a_b,R5: set_Pr2972776593051762503rm_a_b] :
( ~ ( member5869715511025134514rm_a_b @ X @ ( acc_Pr4364474097874652878rm_a_b @ R5 ) )
=> ~ ! [Z4: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Z4 @ X ) @ R5 )
=> ( member5869715511025134514rm_a_b @ Z4 @ ( acc_Pr4364474097874652878rm_a_b @ R5 ) ) ) ) ).
% not_acc_down
thf(fact_1042_not__acc__down,axiom,
! [X: b,R5: set_Product_prod_b_b] :
( ~ ( member_b @ X @ ( acc_b @ R5 ) )
=> ~ ! [Z4: b] :
( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ Z4 @ X ) @ R5 )
=> ( member_b @ Z4 @ ( acc_b @ R5 ) ) ) ) ).
% not_acc_down
thf(fact_1043_not__acc__down,axiom,
! [X: term_a_b,R5: set_Pr4386577575007340137rm_a_b] :
( ~ ( member_term_a_b @ X @ ( acc_term_a_b @ R5 ) )
=> ~ ! [Z4: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Z4 @ X ) @ R5 )
=> ( member_term_a_b @ Z4 @ ( acc_term_a_b @ R5 ) ) ) ) ).
% not_acc_down
thf(fact_1044_acc__induct__rule,axiom,
! [A: produc357393685978478089rm_a_b,R: set_Pr2972776593051762503rm_a_b,P: produc357393685978478089rm_a_b > $o] :
( ( member5869715511025134514rm_a_b @ A @ ( acc_Pr4364474097874652878rm_a_b @ R ) )
=> ( ! [X3: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X3 @ ( acc_Pr4364474097874652878rm_a_b @ R ) )
=> ( ! [Y8: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y8 @ X3 ) @ R )
=> ( P @ Y8 ) )
=> ( P @ X3 ) ) )
=> ( P @ A ) ) ) ).
% acc_induct_rule
thf(fact_1045_acc__induct__rule,axiom,
! [A: b,R: set_Product_prod_b_b,P: b > $o] :
( ( member_b @ A @ ( acc_b @ R ) )
=> ( ! [X3: b] :
( ( member_b @ X3 @ ( acc_b @ R ) )
=> ( ! [Y8: b] :
( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ Y8 @ X3 ) @ R )
=> ( P @ Y8 ) )
=> ( P @ X3 ) ) )
=> ( P @ A ) ) ) ).
% acc_induct_rule
thf(fact_1046_acc__induct__rule,axiom,
! [A: term_a_b,R: set_Pr4386577575007340137rm_a_b,P: term_a_b > $o] :
( ( member_term_a_b @ A @ ( acc_term_a_b @ R ) )
=> ( ! [X3: term_a_b] :
( ( member_term_a_b @ X3 @ ( acc_term_a_b @ R ) )
=> ( ! [Y8: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y8 @ X3 ) @ R )
=> ( P @ Y8 ) )
=> ( P @ X3 ) ) )
=> ( P @ A ) ) ) ).
% acc_induct_rule
thf(fact_1047_acc__subset__induct,axiom,
! [D: set_Pr4386577575007340137rm_a_b,R5: set_Pr2972776593051762503rm_a_b,X: produc357393685978478089rm_a_b,P: produc357393685978478089rm_a_b > $o] :
( ( ord_le118470702582115849rm_a_b @ D @ ( acc_Pr4364474097874652878rm_a_b @ R5 ) )
=> ( ! [X3: produc357393685978478089rm_a_b,Z4: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X3 @ D )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Z4 @ X3 ) @ R5 )
=> ( member5869715511025134514rm_a_b @ Z4 @ D ) ) )
=> ( ( member5869715511025134514rm_a_b @ X @ D )
=> ( ! [X3: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X3 @ D )
=> ( ! [Z6: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Z6 @ X3 ) @ R5 )
=> ( P @ Z6 ) )
=> ( P @ X3 ) ) )
=> ( P @ X ) ) ) ) ) ).
% acc_subset_induct
thf(fact_1048_acc__subset__induct,axiom,
! [D: set_b,R5: set_Product_prod_b_b,X: b,P: b > $o] :
( ( ord_less_eq_set_b @ D @ ( acc_b @ R5 ) )
=> ( ! [X3: b,Z4: b] :
( ( member_b @ X3 @ D )
=> ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ Z4 @ X3 ) @ R5 )
=> ( member_b @ Z4 @ D ) ) )
=> ( ( member_b @ X @ D )
=> ( ! [X3: b] :
( ( member_b @ X3 @ D )
=> ( ! [Z6: b] :
( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ Z6 @ X3 ) @ R5 )
=> ( P @ Z6 ) )
=> ( P @ X3 ) ) )
=> ( P @ X ) ) ) ) ) ).
% acc_subset_induct
thf(fact_1049_acc__subset__induct,axiom,
! [D: set_term_a_b,R5: set_Pr4386577575007340137rm_a_b,X: term_a_b,P: term_a_b > $o] :
( ( ord_le2705286416250468010rm_a_b @ D @ ( acc_term_a_b @ R5 ) )
=> ( ! [X3: term_a_b,Z4: term_a_b] :
( ( member_term_a_b @ X3 @ D )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Z4 @ X3 ) @ R5 )
=> ( member_term_a_b @ Z4 @ D ) ) )
=> ( ( member_term_a_b @ X @ D )
=> ( ! [X3: term_a_b] :
( ( member_term_a_b @ X3 @ D )
=> ( ! [Z6: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Z6 @ X3 ) @ R5 )
=> ( P @ Z6 ) )
=> ( P @ X3 ) ) )
=> ( P @ X ) ) ) ) ) ).
% acc_subset_induct
thf(fact_1050_acc__subset__induct,axiom,
! [D: set_Pr4934435412358123699_a_nat,R5: set_Pr1811044260758604347_a_nat,X: product_prod_a_nat,P: product_prod_a_nat > $o] :
( ( ord_le8666007276011122963_a_nat @ D @ ( acc_Pr3113470191960567800_a_nat @ R5 ) )
=> ( ! [X3: product_prod_a_nat,Z4: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ X3 @ D )
=> ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ Z4 @ X3 ) @ R5 )
=> ( member5724188588386418708_a_nat @ Z4 @ D ) ) )
=> ( ( member5724188588386418708_a_nat @ X @ D )
=> ( ! [X3: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ X3 @ D )
=> ( ! [Z6: product_prod_a_nat] :
( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ Z6 @ X3 ) @ R5 )
=> ( P @ Z6 ) )
=> ( P @ X3 ) ) )
=> ( P @ X ) ) ) ) ) ).
% acc_subset_induct
thf(fact_1051_acc__downwards,axiom,
! [A: produc357393685978478089rm_a_b,R: set_Pr2972776593051762503rm_a_b,B: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ A @ ( acc_Pr4364474097874652878rm_a_b @ R ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B @ A ) @ ( transi2615809358984392588rm_a_b @ R ) )
=> ( member5869715511025134514rm_a_b @ B @ ( acc_Pr4364474097874652878rm_a_b @ R ) ) ) ) ).
% acc_downwards
thf(fact_1052_acc__downwards,axiom,
! [A: b,R: set_Product_prod_b_b,B: b] :
( ( member_b @ A @ ( acc_b @ R ) )
=> ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ B @ A ) @ ( transitive_rtrancl_b @ R ) )
=> ( member_b @ B @ ( acc_b @ R ) ) ) ) ).
% acc_downwards
thf(fact_1053_acc__downwards,axiom,
! [A: term_a_b,R: set_Pr4386577575007340137rm_a_b,B: term_a_b] :
( ( member_term_a_b @ A @ ( acc_term_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ( member_term_a_b @ B @ ( acc_term_a_b @ R ) ) ) ) ).
% acc_downwards
thf(fact_1054_acc__downwards__aux,axiom,
! [B: produc357393685978478089rm_a_b,A: produc357393685978478089rm_a_b,R: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B @ A ) @ ( transi2615809358984392588rm_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ A @ ( acc_Pr4364474097874652878rm_a_b @ R ) )
=> ( member5869715511025134514rm_a_b @ B @ ( acc_Pr4364474097874652878rm_a_b @ R ) ) ) ) ).
% acc_downwards_aux
thf(fact_1055_acc__downwards__aux,axiom,
! [B: b,A: b,R: set_Product_prod_b_b] :
( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ B @ A ) @ ( transitive_rtrancl_b @ R ) )
=> ( ( member_b @ A @ ( acc_b @ R ) )
=> ( member_b @ B @ ( acc_b @ R ) ) ) ) ).
% acc_downwards_aux
thf(fact_1056_acc__downwards__aux,axiom,
! [B: term_a_b,A: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ( ( member_term_a_b @ A @ ( acc_term_a_b @ R ) )
=> ( member_term_a_b @ B @ ( acc_term_a_b @ R ) ) ) ) ).
% acc_downwards_aux
thf(fact_1057_acc__subset,axiom,
! [R1: set_Pr1811044260758604347_a_nat,R22: set_Pr1811044260758604347_a_nat] :
( ( ord_le1018141841545957851_a_nat @ R1 @ R22 )
=> ( ord_le8666007276011122963_a_nat @ ( acc_Pr3113470191960567800_a_nat @ R22 ) @ ( acc_Pr3113470191960567800_a_nat @ R1 ) ) ) ).
% acc_subset
thf(fact_1058_max__ext__def,axiom,
( max_ext_term_a_b
= ( ^ [R10: set_Pr4386577575007340137rm_a_b] :
( collec4780413679787665248rm_a_b
@ ( produc4628319930418777942_a_b_o
@ ( max_extp_term_a_b
@ ^ [X2: term_a_b,Y4: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y4 ) @ R10 ) ) ) ) ) ) ).
% max_ext_def
thf(fact_1059_SN__on__conv__acc,axiom,
! [R: set_Pr2972776593051762503rm_a_b,X: produc357393685978478089rm_a_b] :
( ( abstra2398554102055911763rm_a_b
@ ( collec1344989105561874898rm_a_b
@ ( produc161078424057476524_a_b_o
@ ^ [Y4: produc357393685978478089rm_a_b,Z5: produc357393685978478089rm_a_b] : ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Z5 @ Y4 ) @ R ) ) )
@ ( insert7009541432154983385rm_a_b @ X @ bot_bo197521221353338581rm_a_b ) )
= ( member5869715511025134514rm_a_b @ X @ ( acc_Pr4364474097874652878rm_a_b @ R ) ) ) ).
% SN_on_conv_acc
thf(fact_1060_SN__on__conv__acc,axiom,
! [R: set_Product_prod_b_b,X: b] :
( ( abstract_SN_on_b
@ ( collec548942219715005266od_b_b
@ ( produc4772849988445941548_b_b_o
@ ^ [Y4: b,Z5: b] : ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ Z5 @ Y4 ) @ R ) ) )
@ ( insert_b @ X @ bot_bot_set_b ) )
= ( member_b @ X @ ( acc_b @ R ) ) ) ).
% SN_on_conv_acc
thf(fact_1061_SN__on__conv__acc,axiom,
! [R: set_Pr4386577575007340137rm_a_b,X: term_a_b] :
( ( abstra4720023341729745482rm_a_b
@ ( collec99177395925924084rm_a_b
@ ( produc6251846659952913346_a_b_o
@ ^ [Y4: term_a_b,Z5: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Z5 @ Y4 ) @ R ) ) )
@ ( insert_term_a_b @ X @ bot_bot_set_term_a_b ) )
= ( member_term_a_b @ X @ ( acc_term_a_b @ R ) ) ) ).
% SN_on_conv_acc
thf(fact_1062_Field__insert,axiom,
! [A: term_a_b,B: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( field_term_a_b @ ( insert7009541432154983385rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R ) )
= ( sup_sup_set_term_a_b @ ( insert_term_a_b @ A @ ( insert_term_a_b @ B @ bot_bot_set_term_a_b ) ) @ ( field_term_a_b @ R ) ) ) ).
% Field_insert
thf(fact_1063_FieldI1,axiom,
! [I2: produc357393685978478089rm_a_b,J2: produc357393685978478089rm_a_b,R5: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ I2 @ J2 ) @ R5 )
=> ( member5869715511025134514rm_a_b @ I2 @ ( field_6884932134483023318rm_a_b @ R5 ) ) ) ).
% FieldI1
thf(fact_1064_FieldI1,axiom,
! [I2: b,J2: b,R5: set_Product_prod_b_b] :
( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ I2 @ J2 ) @ R5 )
=> ( member_b @ I2 @ ( field_b @ R5 ) ) ) ).
% FieldI1
thf(fact_1065_FieldI1,axiom,
! [I2: term_a_b,J2: term_a_b,R5: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ I2 @ J2 ) @ R5 )
=> ( member_term_a_b @ I2 @ ( field_term_a_b @ R5 ) ) ) ).
% FieldI1
thf(fact_1066_FieldI2,axiom,
! [I2: produc357393685978478089rm_a_b,J2: produc357393685978478089rm_a_b,R5: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ I2 @ J2 ) @ R5 )
=> ( member5869715511025134514rm_a_b @ J2 @ ( field_6884932134483023318rm_a_b @ R5 ) ) ) ).
% FieldI2
thf(fact_1067_FieldI2,axiom,
! [I2: b,J2: b,R5: set_Product_prod_b_b] :
( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ I2 @ J2 ) @ R5 )
=> ( member_b @ J2 @ ( field_b @ R5 ) ) ) ).
% FieldI2
thf(fact_1068_FieldI2,axiom,
! [I2: term_a_b,J2: term_a_b,R5: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ I2 @ J2 ) @ R5 )
=> ( member_term_a_b @ J2 @ ( field_term_a_b @ R5 ) ) ) ).
% FieldI2
thf(fact_1069_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_1070_Field__Restr__subset,axiom,
! [R: set_Pr1811044260758604347_a_nat,A5: set_Pr4934435412358123699_a_nat] :
( ord_le8666007276011122963_a_nat
@ ( field_8954927560578634480_a_nat
@ ( inf_in8678195602585455913_a_nat @ R
@ ( produc3832836940518797764_a_nat @ A5
@ ^ [Uu: product_prod_a_nat] : A5 ) ) )
@ A5 ) ).
% Field_Restr_subset
thf(fact_1071_wf__eq__minimal2,axiom,
( wf_term_a_b
= ( ^ [R4: set_Pr4386577575007340137rm_a_b] :
! [A6: set_term_a_b] :
( ( ( ord_le2705286416250468010rm_a_b @ A6 @ ( field_term_a_b @ R4 ) )
& ( A6 != bot_bot_set_term_a_b ) )
=> ? [X2: term_a_b] :
( ( member_term_a_b @ X2 @ A6 )
& ! [Y4: term_a_b] :
( ( member_term_a_b @ Y4 @ A6 )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y4 @ X2 ) @ R4 ) ) ) ) ) ) ).
% wf_eq_minimal2
thf(fact_1072_wf__eq__minimal2,axiom,
( wf_Pro8109884346892000340_a_nat
= ( ^ [R4: set_Pr1811044260758604347_a_nat] :
! [A6: set_Pr4934435412358123699_a_nat] :
( ( ( ord_le8666007276011122963_a_nat @ A6 @ ( field_8954927560578634480_a_nat @ R4 ) )
& ( A6 != bot_bo9049108969261143879_a_nat ) )
=> ? [X2: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ X2 @ A6 )
& ! [Y4: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ Y4 @ A6 )
=> ~ ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ Y4 @ X2 ) @ R4 ) ) ) ) ) ) ).
% wf_eq_minimal2
thf(fact_1073_the__NF,axiom,
! [A5: set_Pr4386577575007340137rm_a_b,A: term_a_b] :
( ( abstra4720023341729745482rm_a_b @ A5 @ top_top_set_term_a_b )
=> ( ( abstra8448919418672941150rm_a_b @ A5 @ top_top_set_term_a_b )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ ( abstra2242782261322123075rm_a_b @ A5 @ A ) ) @ ( abstra2972651231392952684rm_a_b @ A5 ) ) ) ) ).
% the_NF
thf(fact_1074_the__NF__step,axiom,
! [A5: set_Pr4386577575007340137rm_a_b,A: term_a_b,B: term_a_b] :
( ( abstra4720023341729745482rm_a_b @ A5 @ top_top_set_term_a_b )
=> ( ( abstra8448919418672941150rm_a_b @ A5 @ top_top_set_term_a_b )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ A5 )
=> ( ( abstra2242782261322123075rm_a_b @ A5 @ A )
= ( abstra2242782261322123075rm_a_b @ A5 @ B ) ) ) ) ) ).
% the_NF_step
thf(fact_1075_the__NF__NF,axiom,
! [A5: set_Pr2972776593051762503rm_a_b,A: produc357393685978478089rm_a_b] :
( ( abstra2398554102055911763rm_a_b @ A5 @ top_to1314267278846557113rm_a_b )
=> ( ( abstra7359694574113289279rm_a_b @ A5 @ top_to1314267278846557113rm_a_b )
=> ( member5869715511025134514rm_a_b @ ( abstra8972549633426078106rm_a_b @ A5 @ A ) @ ( abstra1902783451776770197rm_a_b @ A5 ) ) ) ) ).
% the_NF_NF
thf(fact_1076_the__NF__NF,axiom,
! [A5: set_Product_prod_b_b,A: b] :
( ( abstract_SN_on_b @ A5 @ top_top_set_b )
=> ( ( abstract_CR_on_b @ A5 @ top_top_set_b )
=> ( member_b @ ( abstract_the_NF_b @ A5 @ A ) @ ( abstract_NF_b @ A5 ) ) ) ) ).
% the_NF_NF
thf(fact_1077_the__NF__steps,axiom,
! [A5: set_Pr4386577575007340137rm_a_b,A: term_a_b,B: term_a_b] :
( ( abstra4720023341729745482rm_a_b @ A5 @ top_top_set_term_a_b )
=> ( ( abstra8448919418672941150rm_a_b @ A5 @ top_top_set_term_a_b )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ A5 ) )
=> ( ( abstra2242782261322123075rm_a_b @ A5 @ A )
= ( abstra2242782261322123075rm_a_b @ A5 @ B ) ) ) ) ) ).
% the_NF_steps
thf(fact_1078_cofinal__def,axiom,
( bNF_Ca4789820777293777539rm_a_b
= ( ^ [A6: set_term_a_b,R4: set_Pr4386577575007340137rm_a_b] :
! [X2: term_a_b] :
( ( member_term_a_b @ X2 @ ( field_term_a_b @ R4 ) )
=> ? [Y4: term_a_b] :
( ( member_term_a_b @ Y4 @ A6 )
& ( X2 != Y4 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y4 ) @ R4 ) ) ) ) ) ).
% cofinal_def
thf(fact_1079_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 )
| ? [A4: term_a_b] :
( R
= ( insert7009541432154983385rm_a_b @ ( produc7020197800436672577rm_a_b @ A4 @ A4 ) @ bot_bo197521221353338581rm_a_b ) ) ) ) ) ).
% Total_subset_Id
thf(fact_1080_CR__on__singletonI,axiom,
! [A: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ! [B3: term_a_b,C: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B3 ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B3 @ C ) @ ( abstra4096080454567261402rm_a_b @ R ) ) ) )
=> ( abstra8448919418672941150rm_a_b @ R @ ( insert_term_a_b @ A @ bot_bot_set_term_a_b ) ) ) ).
% CR_on_singletonI
thf(fact_1081_joinI,axiom,
! [A: term_a_b,C2: term_a_b,A5: set_Pr4386577575007340137rm_a_b,B: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C2 ) @ ( transi7742714808557438673rm_a_b @ A5 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C2 ) @ ( transi7742714808557438673rm_a_b @ A5 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( abstra4096080454567261402rm_a_b @ A5 ) ) ) ) ).
% joinI
thf(fact_1082_joinI__left,axiom,
! [A: term_a_b,B: term_a_b,A5: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ A5 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( abstra4096080454567261402rm_a_b @ A5 ) ) ) ).
% joinI_left
thf(fact_1083_joinI__right,axiom,
! [B: term_a_b,A: term_a_b,A5: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ ( transi7742714808557438673rm_a_b @ A5 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( abstra4096080454567261402rm_a_b @ A5 ) ) ) ).
% joinI_right
thf(fact_1084_WCR__onI,axiom,
! [A5: set_Pr4386577575007340137rm_a_b,R: set_Pr2972776593051762503rm_a_b] :
( ! [A4: produc357393685978478089rm_a_b,B3: produc357393685978478089rm_a_b,C: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ A4 @ A5 )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A4 @ B3 ) @ R )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A4 @ C ) @ R )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B3 @ C ) @ ( abstra4889682940153293507rm_a_b @ R ) ) ) ) )
=> ( abstra875157153669124638rm_a_b @ R @ A5 ) ) ).
% WCR_onI
thf(fact_1085_WCR__onI,axiom,
! [A5: set_b,R: set_Product_prod_b_b] :
( ! [A4: b,B3: b,C: b] :
( ( member_b @ A4 @ A5 )
=> ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ A4 @ B3 ) @ R )
=> ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ A4 @ C ) @ R )
=> ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ B3 @ C ) @ ( abstract_join_b @ R ) ) ) ) )
=> ( abstract_WCR_on_b @ R @ A5 ) ) ).
% WCR_onI
thf(fact_1086_WCR__onI,axiom,
! [A5: set_term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ! [A4: term_a_b,B3: term_a_b,C: term_a_b] :
( ( member_term_a_b @ A4 @ A5 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A4 @ B3 ) @ R )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A4 @ C ) @ R )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B3 @ C ) @ ( abstra4096080454567261402rm_a_b @ R ) ) ) ) )
=> ( abstra5026849107050860991rm_a_b @ R @ A5 ) ) ).
% WCR_onI
thf(fact_1087_CR__onI,axiom,
! [A5: set_Pr4386577575007340137rm_a_b,R: set_Pr2972776593051762503rm_a_b] :
( ! [A4: produc357393685978478089rm_a_b,B3: produc357393685978478089rm_a_b,C: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ A4 @ A5 )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A4 @ B3 ) @ ( transi2615809358984392588rm_a_b @ R ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A4 @ C ) @ ( transi2615809358984392588rm_a_b @ R ) )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B3 @ C ) @ ( abstra4889682940153293507rm_a_b @ R ) ) ) ) )
=> ( abstra7359694574113289279rm_a_b @ R @ A5 ) ) ).
% CR_onI
thf(fact_1088_CR__onI,axiom,
! [A5: set_b,R: set_Product_prod_b_b] :
( ! [A4: b,B3: b,C: b] :
( ( member_b @ A4 @ A5 )
=> ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ A4 @ B3 ) @ ( transitive_rtrancl_b @ R ) )
=> ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ A4 @ C ) @ ( transitive_rtrancl_b @ R ) )
=> ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ B3 @ C ) @ ( abstract_join_b @ R ) ) ) ) )
=> ( abstract_CR_on_b @ R @ A5 ) ) ).
% CR_onI
thf(fact_1089_CR__onI,axiom,
! [A5: set_term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ! [A4: term_a_b,B3: term_a_b,C: term_a_b] :
( ( member_term_a_b @ A4 @ A5 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A4 @ B3 ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A4 @ C ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B3 @ C ) @ ( abstra4096080454567261402rm_a_b @ R ) ) ) ) )
=> ( abstra8448919418672941150rm_a_b @ R @ A5 ) ) ).
% CR_onI
thf(fact_1090_total__on__subset,axiom,
! [A5: set_Pr4934435412358123699_a_nat,R: set_Pr1811044260758604347_a_nat,B4: set_Pr4934435412358123699_a_nat] :
( ( total_9110697612923228665_a_nat @ A5 @ R )
=> ( ( ord_le8666007276011122963_a_nat @ B4 @ A5 )
=> ( total_9110697612923228665_a_nat @ B4 @ R ) ) ) ).
% total_on_subset
thf(fact_1091_total__on__def,axiom,
( total_on_term_a_b
= ( ^ [A6: set_term_a_b,R4: set_Pr4386577575007340137rm_a_b] :
! [X2: term_a_b] :
( ( member_term_a_b @ X2 @ A6 )
=> ! [Y4: term_a_b] :
( ( member_term_a_b @ Y4 @ A6 )
=> ( ( X2 != Y4 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y4 ) @ R4 )
| ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y4 @ X2 ) @ R4 ) ) ) ) ) ) ) ).
% total_on_def
thf(fact_1092_total__onI,axiom,
! [A5: set_Pr4386577575007340137rm_a_b,R: set_Pr2972776593051762503rm_a_b] :
( ! [X3: produc357393685978478089rm_a_b,Y3: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X3 @ A5 )
=> ( ( member5869715511025134514rm_a_b @ Y3 @ A5 )
=> ( ( X3 != Y3 )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X3 @ Y3 ) @ R )
| ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y3 @ X3 ) @ R ) ) ) ) )
=> ( total_4010809703584253837rm_a_b @ A5 @ R ) ) ).
% total_onI
thf(fact_1093_total__onI,axiom,
! [A5: set_b,R: set_Product_prod_b_b] :
( ! [X3: b,Y3: b] :
( ( member_b @ X3 @ A5 )
=> ( ( member_b @ Y3 @ A5 )
=> ( ( X3 != Y3 )
=> ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ X3 @ Y3 ) @ R )
| ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ Y3 @ X3 ) @ R ) ) ) ) )
=> ( total_on_b @ A5 @ R ) ) ).
% total_onI
thf(fact_1094_total__onI,axiom,
! [A5: set_term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ! [X3: term_a_b,Y3: term_a_b] :
( ( member_term_a_b @ X3 @ A5 )
=> ( ( member_term_a_b @ Y3 @ A5 )
=> ( ( X3 != Y3 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ R )
| ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ X3 ) @ R ) ) ) ) )
=> ( total_on_term_a_b @ A5 @ R ) ) ).
% total_onI
thf(fact_1095_join__sym,axiom,
! [S: term_a_b,T: term_a_b,A5: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( abstra4096080454567261402rm_a_b @ A5 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T @ S ) @ ( abstra4096080454567261402rm_a_b @ A5 ) ) ) ).
% join_sym
thf(fact_1096_totalI,axiom,
! [R: set_Pr4386577575007340137rm_a_b] :
( ! [X3: term_a_b,Y3: term_a_b] :
( ( X3 != Y3 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ R )
| ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ X3 ) @ R ) ) )
=> ( total_on_term_a_b @ top_top_set_term_a_b @ R ) ) ).
% totalI
thf(fact_1097_joinD,axiom,
! [A: term_a_b,B: term_a_b,A5: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( abstra4096080454567261402rm_a_b @ A5 ) )
=> ? [C: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C ) @ ( transi7742714808557438673rm_a_b @ A5 ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C ) @ ( transi7742714808557438673rm_a_b @ A5 ) ) ) ) ).
% joinD
thf(fact_1098_joinE,axiom,
! [A: term_a_b,B: term_a_b,A5: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( abstra4096080454567261402rm_a_b @ A5 ) )
=> ~ ! [C: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C ) @ ( transi7742714808557438673rm_a_b @ A5 ) )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C ) @ ( transi7742714808557438673rm_a_b @ A5 ) ) ) ) ).
% joinE
thf(fact_1099_join__rtrancl__join,axiom,
! [A: term_a_b,B: term_a_b,A5: set_Pr4386577575007340137rm_a_b,C2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( abstra4096080454567261402rm_a_b @ A5 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ C2 @ B ) @ ( transi7742714808557438673rm_a_b @ A5 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C2 ) @ ( abstra4096080454567261402rm_a_b @ A5 ) ) ) ) ).
% join_rtrancl_join
thf(fact_1100_rtrancl__join__join,axiom,
! [A: term_a_b,B: term_a_b,A5: set_Pr4386577575007340137rm_a_b,C2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ A5 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C2 ) @ ( abstra4096080454567261402rm_a_b @ A5 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C2 ) @ ( abstra4096080454567261402rm_a_b @ A5 ) ) ) ) ).
% rtrancl_join_join
thf(fact_1101_join__NF__imp__eq,axiom,
! [X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b,R: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ ( abstra4889682940153293507rm_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ X @ ( abstra1902783451776770197rm_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ Y @ ( abstra1902783451776770197rm_a_b @ R ) )
=> ( X = Y ) ) ) ) ).
% join_NF_imp_eq
thf(fact_1102_join__NF__imp__eq,axiom,
! [X: b,Y: b,R: set_Product_prod_b_b] :
( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ X @ Y ) @ ( abstract_join_b @ R ) )
=> ( ( member_b @ X @ ( abstract_NF_b @ R ) )
=> ( ( member_b @ Y @ ( abstract_NF_b @ R ) )
=> ( X = Y ) ) ) ) ).
% join_NF_imp_eq
thf(fact_1103_join__NF__imp__eq,axiom,
! [X: term_a_b,Y: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( abstra4096080454567261402rm_a_b @ R ) )
=> ( ( member_term_a_b @ X @ ( abstract_NF_term_a_b @ R ) )
=> ( ( member_term_a_b @ Y @ ( abstract_NF_term_a_b @ R ) )
=> ( X = Y ) ) ) ) ).
% join_NF_imp_eq
thf(fact_1104_WCR__onD,axiom,
! [R: set_Pr2972776593051762503rm_a_b,A5: set_Pr4386577575007340137rm_a_b,A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,C2: produc357393685978478089rm_a_b] :
( ( abstra875157153669124638rm_a_b @ R @ A5 )
=> ( ( member5869715511025134514rm_a_b @ A @ A5 )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ R )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ C2 ) @ R )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B @ C2 ) @ ( abstra4889682940153293507rm_a_b @ R ) ) ) ) ) ) ).
% WCR_onD
thf(fact_1105_WCR__onD,axiom,
! [R: set_Product_prod_b_b,A5: set_b,A: b,B: b,C2: b] :
( ( abstract_WCR_on_b @ R @ A5 )
=> ( ( member_b @ A @ A5 )
=> ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ A @ B ) @ R )
=> ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ A @ C2 ) @ R )
=> ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ B @ C2 ) @ ( abstract_join_b @ R ) ) ) ) ) ) ).
% WCR_onD
thf(fact_1106_WCR__onD,axiom,
! [R: set_Pr4386577575007340137rm_a_b,A5: set_term_a_b,A: term_a_b,B: term_a_b,C2: term_a_b] :
( ( abstra5026849107050860991rm_a_b @ R @ A5 )
=> ( ( member_term_a_b @ A @ A5 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C2 ) @ R )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C2 ) @ ( abstra4096080454567261402rm_a_b @ R ) ) ) ) ) ) ).
% WCR_onD
thf(fact_1107_WCR__onE,axiom,
! [R: set_Pr2972776593051762503rm_a_b,A5: set_Pr4386577575007340137rm_a_b,A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,C2: produc357393685978478089rm_a_b] :
( ( abstra875157153669124638rm_a_b @ R @ A5 )
=> ( ( member5869715511025134514rm_a_b @ A @ A5 )
=> ( ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B @ C2 ) @ ( abstra4889682940153293507rm_a_b @ R ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ R )
=> ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ C2 ) @ R ) ) ) ) ) ).
% WCR_onE
thf(fact_1108_WCR__onE,axiom,
! [R: set_Product_prod_b_b,A5: set_b,A: b,B: b,C2: b] :
( ( abstract_WCR_on_b @ R @ A5 )
=> ( ( member_b @ A @ A5 )
=> ( ~ ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ B @ C2 ) @ ( abstract_join_b @ R ) )
=> ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ A @ B ) @ R )
=> ~ ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ A @ C2 ) @ R ) ) ) ) ) ).
% WCR_onE
thf(fact_1109_WCR__onE,axiom,
! [R: set_Pr4386577575007340137rm_a_b,A5: set_term_a_b,A: term_a_b,B: term_a_b,C2: term_a_b] :
( ( abstra5026849107050860991rm_a_b @ R @ A5 )
=> ( ( member_term_a_b @ A @ A5 )
=> ( ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C2 ) @ ( abstra4096080454567261402rm_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C2 ) @ R ) ) ) ) ) ).
% WCR_onE
thf(fact_1110_WCR__on__def,axiom,
( abstra5026849107050860991rm_a_b
= ( ^ [R4: set_Pr4386577575007340137rm_a_b,A6: set_term_a_b] :
! [X2: term_a_b] :
( ( member_term_a_b @ X2 @ A6 )
=> ! [B6: term_a_b,C4: term_a_b] :
( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ B6 ) @ R4 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ C4 ) @ R4 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B6 @ C4 ) @ ( abstra4096080454567261402rm_a_b @ R4 ) ) ) ) ) ) ).
% WCR_on_def
thf(fact_1111_NF__join__imp__reach,axiom,
! [Y: produc357393685978478089rm_a_b,A5: set_Pr2972776593051762503rm_a_b,X: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ Y @ ( abstra1902783451776770197rm_a_b @ A5 ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ ( abstra4889682940153293507rm_a_b @ A5 ) )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ ( transi2615809358984392588rm_a_b @ A5 ) ) ) ) ).
% NF_join_imp_reach
thf(fact_1112_NF__join__imp__reach,axiom,
! [Y: b,A5: set_Product_prod_b_b,X: b] :
( ( member_b @ Y @ ( abstract_NF_b @ A5 ) )
=> ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ X @ Y ) @ ( abstract_join_b @ A5 ) )
=> ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ X @ Y ) @ ( transitive_rtrancl_b @ A5 ) ) ) ) ).
% NF_join_imp_reach
thf(fact_1113_NF__join__imp__reach,axiom,
! [Y: term_a_b,A5: set_Pr4386577575007340137rm_a_b,X: term_a_b] :
( ( member_term_a_b @ Y @ ( abstract_NF_term_a_b @ A5 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( abstra4096080454567261402rm_a_b @ A5 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ A5 ) ) ) ) ).
% NF_join_imp_reach
thf(fact_1114_CR__onD,axiom,
! [R: set_Pr2972776593051762503rm_a_b,A5: set_Pr4386577575007340137rm_a_b,A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,C2: produc357393685978478089rm_a_b] :
( ( abstra7359694574113289279rm_a_b @ R @ A5 )
=> ( ( member5869715511025134514rm_a_b @ A @ A5 )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( transi2615809358984392588rm_a_b @ R ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ C2 ) @ ( transi2615809358984392588rm_a_b @ R ) )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B @ C2 ) @ ( abstra4889682940153293507rm_a_b @ R ) ) ) ) ) ) ).
% CR_onD
thf(fact_1115_CR__onD,axiom,
! [R: set_Product_prod_b_b,A5: set_b,A: b,B: b,C2: b] :
( ( abstract_CR_on_b @ R @ A5 )
=> ( ( member_b @ A @ A5 )
=> ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ A @ B ) @ ( transitive_rtrancl_b @ R ) )
=> ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ A @ C2 ) @ ( transitive_rtrancl_b @ R ) )
=> ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ B @ C2 ) @ ( abstract_join_b @ R ) ) ) ) ) ) ).
% CR_onD
thf(fact_1116_CR__onD,axiom,
! [R: set_Pr4386577575007340137rm_a_b,A5: set_term_a_b,A: term_a_b,B: term_a_b,C2: term_a_b] :
( ( abstra8448919418672941150rm_a_b @ R @ A5 )
=> ( ( member_term_a_b @ A @ A5 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C2 ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C2 ) @ ( abstra4096080454567261402rm_a_b @ R ) ) ) ) ) ) ).
% CR_onD
thf(fact_1117_CR__onE,axiom,
! [R: set_Pr2972776593051762503rm_a_b,A5: set_Pr4386577575007340137rm_a_b,A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,C2: produc357393685978478089rm_a_b] :
( ( abstra7359694574113289279rm_a_b @ R @ A5 )
=> ( ( member5869715511025134514rm_a_b @ A @ A5 )
=> ( ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B @ C2 ) @ ( abstra4889682940153293507rm_a_b @ R ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( transi2615809358984392588rm_a_b @ R ) )
=> ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ C2 ) @ ( transi2615809358984392588rm_a_b @ R ) ) ) ) ) ) ).
% CR_onE
thf(fact_1118_CR__onE,axiom,
! [R: set_Product_prod_b_b,A5: set_b,A: b,B: b,C2: b] :
( ( abstract_CR_on_b @ R @ A5 )
=> ( ( member_b @ A @ A5 )
=> ( ~ ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ B @ C2 ) @ ( abstract_join_b @ R ) )
=> ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ A @ B ) @ ( transitive_rtrancl_b @ R ) )
=> ~ ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ A @ C2 ) @ ( transitive_rtrancl_b @ R ) ) ) ) ) ) ).
% CR_onE
thf(fact_1119_CR__onE,axiom,
! [R: set_Pr4386577575007340137rm_a_b,A5: set_term_a_b,A: term_a_b,B: term_a_b,C2: term_a_b] :
( ( abstra8448919418672941150rm_a_b @ R @ A5 )
=> ( ( member_term_a_b @ A @ A5 )
=> ( ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C2 ) @ ( abstra4096080454567261402rm_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C2 ) @ ( transi7742714808557438673rm_a_b @ R ) ) ) ) ) ) ).
% CR_onE
thf(fact_1120_CR__on__def,axiom,
( abstra8448919418672941150rm_a_b
= ( ^ [R4: set_Pr4386577575007340137rm_a_b,A6: set_term_a_b] :
! [X2: term_a_b] :
( ( member_term_a_b @ X2 @ A6 )
=> ! [B6: term_a_b,C4: term_a_b] :
( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ B6 ) @ ( transi7742714808557438673rm_a_b @ R4 ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ C4 ) @ ( transi7742714808557438673rm_a_b @ R4 ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B6 @ C4 ) @ ( abstra4096080454567261402rm_a_b @ R4 ) ) ) ) ) ) ).
% CR_on_def
thf(fact_1121_CR__join__left__I,axiom,
! [R: set_Pr4386577575007340137rm_a_b,X: term_a_b,Y: term_a_b,Z2: term_a_b] :
( ( abstra8448919418672941150rm_a_b @ R @ top_top_set_term_a_b )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z2 ) @ ( abstra4096080454567261402rm_a_b @ R ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ Z2 ) @ ( abstra4096080454567261402rm_a_b @ R ) ) ) ) ) ).
% CR_join_left_I
thf(fact_1122_CR__join__right__I,axiom,
! [R: set_Pr4386577575007340137rm_a_b,X: term_a_b,Y: term_a_b,Z2: term_a_b] :
( ( abstra8448919418672941150rm_a_b @ R @ top_top_set_term_a_b )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( abstra4096080454567261402rm_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ Z2 ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z2 ) @ ( abstra4096080454567261402rm_a_b @ R ) ) ) ) ) ).
% CR_join_right_I
thf(fact_1123_partially__localize__CR,axiom,
! [R: set_Pr4386577575007340137rm_a_b] :
( ( abstra8448919418672941150rm_a_b @ R @ top_top_set_term_a_b )
= ( ! [X2: term_a_b,Y4: term_a_b,Z5: term_a_b] :
( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y4 ) @ R )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Z5 ) @ ( transi7742714808557438673rm_a_b @ R ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y4 @ Z5 ) @ ( abstra4096080454567261402rm_a_b @ R ) ) ) ) ) ).
% partially_localize_CR
thf(fact_1124_CR__divergence__imp__join,axiom,
! [R: set_Pr4386577575007340137rm_a_b,X: term_a_b,Y: term_a_b,Z2: term_a_b] :
( ( abstra8448919418672941150rm_a_b @ R @ top_top_set_term_a_b )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z2 ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ Z2 ) @ ( abstra4096080454567261402rm_a_b @ R ) ) ) ) ) ).
% CR_divergence_imp_join
thf(fact_1125_linear__order__on__singleton,axiom,
! [X: term_a_b] : ( order_5388802246213473311rm_a_b @ ( insert_term_a_b @ X @ bot_bot_set_term_a_b ) @ ( insert7009541432154983385rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ X ) @ bot_bo197521221353338581rm_a_b ) ) ).
% linear_order_on_singleton
thf(fact_1126_the__NF__conv,axiom,
! [A5: set_Pr4386577575007340137rm_a_b,A: term_a_b,B: term_a_b] :
( ( abstra4720023341729745482rm_a_b @ A5 @ top_top_set_term_a_b )
=> ( ( abstra8448919418672941150rm_a_b @ A5 @ top_top_set_term_a_b )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( abstra1130568886744591910rm_a_b @ A5 ) )
=> ( ( abstra2242782261322123075rm_a_b @ A5 @ A )
= ( abstra2242782261322123075rm_a_b @ A5 @ B ) ) ) ) ) ).
% the_NF_conv
thf(fact_1127_conversionI_H,axiom,
! [A: term_a_b,B: term_a_b,A5: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ A5 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( abstra1130568886744591910rm_a_b @ A5 ) ) ) ).
% conversionI'
thf(fact_1128_conversion__inv,axiom,
! [X: term_a_b,Y: term_a_b,R5: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( abstra1130568886744591910rm_a_b @ R5 ) )
= ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X ) @ ( abstra1130568886744591910rm_a_b @ R5 ) ) ) ).
% conversion_inv
thf(fact_1129_conversion__refl,axiom,
! [A: term_a_b,A5: set_Pr4386577575007340137rm_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ A ) @ ( abstra1130568886744591910rm_a_b @ A5 ) ) ).
% conversion_refl
thf(fact_1130_UNC__def,axiom,
( abstra8480497335821202873rm_a_b
= ( ^ [A6: set_Pr2972776593051762503rm_a_b] :
! [A7: produc357393685978478089rm_a_b,B6: produc357393685978478089rm_a_b] :
( ( ( member5869715511025134514rm_a_b @ A7 @ ( abstra1902783451776770197rm_a_b @ A6 ) )
& ( member5869715511025134514rm_a_b @ B6 @ ( abstra1902783451776770197rm_a_b @ A6 ) )
& ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A7 @ B6 ) @ ( abstra6125656530262095991rm_a_b @ A6 ) ) )
=> ( A7 = B6 ) ) ) ) ).
% UNC_def
thf(fact_1131_UNC__def,axiom,
( abstract_UNC_b
= ( ^ [A6: set_Product_prod_b_b] :
! [A7: b,B6: b] :
( ( ( member_b @ A7 @ ( abstract_NF_b @ A6 ) )
& ( member_b @ B6 @ ( abstract_NF_b @ A6 ) )
& ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ A7 @ B6 ) @ ( abstra3918519255145857645sion_b @ A6 ) ) )
=> ( A7 = B6 ) ) ) ) ).
% UNC_def
thf(fact_1132_UNC__def,axiom,
( abstra4649771476585718372rm_a_b
= ( ^ [A6: set_Pr4386577575007340137rm_a_b] :
! [A7: term_a_b,B6: term_a_b] :
( ( ( member_term_a_b @ A7 @ ( abstract_NF_term_a_b @ A6 ) )
& ( member_term_a_b @ B6 @ ( abstract_NF_term_a_b @ A6 ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A7 @ B6 ) @ ( abstra1130568886744591910rm_a_b @ A6 ) ) )
=> ( A7 = B6 ) ) ) ) ).
% UNC_def
thf(fact_1133_UNC__I,axiom,
! [A5: set_Pr2972776593051762503rm_a_b] :
( ! [A4: produc357393685978478089rm_a_b,B3: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ A4 @ ( abstra1902783451776770197rm_a_b @ A5 ) )
=> ( ( member5869715511025134514rm_a_b @ B3 @ ( abstra1902783451776770197rm_a_b @ A5 ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A4 @ B3 ) @ ( abstra6125656530262095991rm_a_b @ A5 ) )
=> ( A4 = B3 ) ) ) )
=> ( abstra8480497335821202873rm_a_b @ A5 ) ) ).
% UNC_I
thf(fact_1134_UNC__I,axiom,
! [A5: set_Product_prod_b_b] :
( ! [A4: b,B3: b] :
( ( member_b @ A4 @ ( abstract_NF_b @ A5 ) )
=> ( ( member_b @ B3 @ ( abstract_NF_b @ A5 ) )
=> ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ A4 @ B3 ) @ ( abstra3918519255145857645sion_b @ A5 ) )
=> ( A4 = B3 ) ) ) )
=> ( abstract_UNC_b @ A5 ) ) ).
% UNC_I
thf(fact_1135_UNC__I,axiom,
! [A5: set_Pr4386577575007340137rm_a_b] :
( ! [A4: term_a_b,B3: term_a_b] :
( ( member_term_a_b @ A4 @ ( abstract_NF_term_a_b @ A5 ) )
=> ( ( member_term_a_b @ B3 @ ( abstract_NF_term_a_b @ A5 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A4 @ B3 ) @ ( abstra1130568886744591910rm_a_b @ A5 ) )
=> ( A4 = B3 ) ) ) )
=> ( abstra4649771476585718372rm_a_b @ A5 ) ) ).
% UNC_I
thf(fact_1136_UNC__E,axiom,
! [A5: set_Pr2972776593051762503rm_a_b,A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b] :
( ( abstra8480497335821202873rm_a_b @ A5 )
=> ( ( A != B )
=> ( ( member5869715511025134514rm_a_b @ A @ ( abstra1902783451776770197rm_a_b @ A5 ) )
=> ( ( member5869715511025134514rm_a_b @ B @ ( abstra1902783451776770197rm_a_b @ A5 ) )
=> ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( abstra6125656530262095991rm_a_b @ A5 ) ) ) ) ) ) ).
% UNC_E
thf(fact_1137_UNC__E,axiom,
! [A5: set_Product_prod_b_b,A: b,B: b] :
( ( abstract_UNC_b @ A5 )
=> ( ( A != B )
=> ( ( member_b @ A @ ( abstract_NF_b @ A5 ) )
=> ( ( member_b @ B @ ( abstract_NF_b @ A5 ) )
=> ~ ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ A @ B ) @ ( abstra3918519255145857645sion_b @ A5 ) ) ) ) ) ) ).
% UNC_E
thf(fact_1138_UNC__E,axiom,
! [A5: set_Pr4386577575007340137rm_a_b,A: term_a_b,B: term_a_b] :
( ( abstra4649771476585718372rm_a_b @ A5 )
=> ( ( A != B )
=> ( ( member_term_a_b @ A @ ( abstract_NF_term_a_b @ A5 ) )
=> ( ( member_term_a_b @ B @ ( abstract_NF_term_a_b @ A5 ) )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( abstra1130568886744591910rm_a_b @ A5 ) ) ) ) ) ) ).
% UNC_E
thf(fact_1139_semi__complete__imp__conversionIff__same__NF,axiom,
! [R: set_Pr4386577575007340137rm_a_b,X: term_a_b,Y: term_a_b] :
( ( abstra7162307272211697846rm_a_b @ R @ top_top_set_term_a_b )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( abstra1130568886744591910rm_a_b @ R ) )
= ( ! [U3: term_a_b,V3: term_a_b] :
( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ U3 ) @ ( abstra2972651231392952684rm_a_b @ R ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ V3 ) @ ( abstra2972651231392952684rm_a_b @ R ) ) )
=> ( U3 = V3 ) ) ) ) ) ).
% semi_complete_imp_conversionIff_same_NF
thf(fact_1140_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_1141_CR__NF__conv,axiom,
! [R: set_Product_prod_b_b,T: b,U: b] :
( ( abstract_CR_on_b @ R @ top_top_set_b )
=> ( ( member_b @ T @ ( abstract_NF_b @ R ) )
=> ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ U @ T ) @ ( abstra3918519255145857645sion_b @ R ) )
=> ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ U @ T ) @ ( abstra380364156431640999lity_b @ R ) ) ) ) ) ).
% CR_NF_conv
thf(fact_1142_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_1143_gsrstep__conversion__dist,axiom,
! [F: set_Pr4934435412358123699_a_nat,R2: set_Pr4386577575007340137rm_a_b] :
( ( abstra1130568886744591910rm_a_b
@ ( inf_in1693821330460682071rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R2 ) )
@ ( produc1953217797448512882rm_a_b @ ( collect_term_a_b @ term_ground_a_b )
@ ^ [Uu: term_a_b] : ( collect_term_a_b @ term_ground_a_b ) ) ) )
= ( transi7742714808557438673rm_a_b
@ ( inf_in1693821330460682071rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ ( sup_su6776935440552674877rm_a_b @ R2 @ ( conver7608559976937845093rm_a_b @ R2 ) ) ) )
@ ( produc1953217797448512882rm_a_b @ ( collect_term_a_b @ term_ground_a_b )
@ ^ [Uu: term_a_b] : ( collect_term_a_b @ term_ground_a_b ) ) ) ) ) ).
% gsrstep_conversion_dist
thf(fact_1144_converse__iff,axiom,
! [A: term_a_b,B: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( conver7608559976937845093rm_a_b @ R ) )
= ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ R ) ) ).
% converse_iff
thf(fact_1145_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_1146_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_1147_conversionI,axiom,
! [A: term_a_b,B: term_a_b,A5: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ A5 @ ( conver7608559976937845093rm_a_b @ A5 ) ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( abstra1130568886744591910rm_a_b @ A5 ) ) ) ).
% conversionI
thf(fact_1148_sig__step__symcl__dist,axiom,
! [F: set_Pr4934435412358123699_a_nat,R2: set_Pr4386577575007340137rm_a_b] :
( ( sup_su6776935440552674877rm_a_b @ ( sig_step_a_b @ F @ R2 ) @ ( conver7608559976937845093rm_a_b @ ( sig_step_a_b @ F @ R2 ) ) )
= ( sig_step_a_b @ F @ ( sup_su6776935440552674877rm_a_b @ R2 @ ( conver7608559976937845093rm_a_b @ R2 ) ) ) ) ).
% sig_step_symcl_dist
thf(fact_1149_sig__step__converse__rstep,axiom,
! [F: set_Pr4934435412358123699_a_nat,R2: set_Pr4386577575007340137rm_a_b] :
( ( conver7608559976937845093rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R2 ) ) )
= ( sig_step_a_b @ F @ ( conver7608559976937845093rm_a_b @ ( rstep_a_b @ R2 ) ) ) ) ).
% sig_step_converse_rstep
thf(fact_1150_srstep__converse__dist,axiom,
! [F: set_Pr4934435412358123699_a_nat,R2: set_Pr4386577575007340137rm_a_b] :
( ( conver7608559976937845093rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R2 ) ) )
= ( sig_step_a_b @ F @ ( rstep_a_b @ ( conver7608559976937845093rm_a_b @ R2 ) ) ) ) ).
% srstep_converse_dist
thf(fact_1151_minimal__imp__wf,axiom,
! [R: set_Pr2972776593051762503rm_a_b] :
( ! [Q3: set_Pr4386577575007340137rm_a_b] :
( ? [X3: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ X3 @ Q3 )
=> ? [X4: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X4 @ Q3 )
& ! [Y3: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X4 @ Y3 ) @ R )
=> ~ ( member5869715511025134514rm_a_b @ Y3 @ Q3 ) ) ) )
=> ( wf_Pro2335863617654816626rm_a_b @ ( conver1575690562132235771rm_a_b @ R ) ) ) ).
% minimal_imp_wf
thf(fact_1152_minimal__imp__wf,axiom,
! [R: set_Product_prod_b_b] :
( ! [Q3: set_b] :
( ? [X3: b] : ( member_b @ X3 @ Q3 )
=> ? [X4: b] :
( ( member_b @ X4 @ Q3 )
& ! [Y3: b] :
( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ X4 @ Y3 ) @ R )
=> ~ ( member_b @ Y3 @ Q3 ) ) ) )
=> ( wf_b @ ( converse_b_b @ R ) ) ) ).
% minimal_imp_wf
thf(fact_1153_minimal__imp__wf,axiom,
! [R: set_Pr4386577575007340137rm_a_b] :
( ! [Q3: set_term_a_b] :
( ? [X3: term_a_b] : ( member_term_a_b @ X3 @ Q3 )
=> ? [X4: term_a_b] :
( ( member_term_a_b @ X4 @ Q3 )
& ! [Y3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X4 @ Y3 ) @ R )
=> ~ ( member_term_a_b @ Y3 @ Q3 ) ) ) )
=> ( wf_term_a_b @ ( conver7608559976937845093rm_a_b @ R ) ) ) ).
% minimal_imp_wf
thf(fact_1154_trancl__converseI,axiom,
! [X: term_a_b,Y: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( conver7608559976937845093rm_a_b @ ( transi7922773638565587891rm_a_b @ R ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7922773638565587891rm_a_b @ ( conver7608559976937845093rm_a_b @ R ) ) ) ) ).
% trancl_converseI
thf(fact_1155_trancl__converseD,axiom,
! [X: term_a_b,Y: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7922773638565587891rm_a_b @ ( conver7608559976937845093rm_a_b @ R ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( conver7608559976937845093rm_a_b @ ( transi7922773638565587891rm_a_b @ R ) ) ) ) ).
% trancl_converseD
thf(fact_1156_peak__iff,axiom,
! [X: term_a_b,Y: term_a_b,A5: set_Pr4386577575007340137rm_a_b,B4: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( relcom370159955682700863rm_a_b @ ( conver7608559976937845093rm_a_b @ A5 ) @ B4 ) )
= ( ? [U3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ U3 @ X ) @ A5 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ U3 @ Y ) @ B4 ) ) ) ) ).
% peak_iff
thf(fact_1157_rtrancl__converseI,axiom,
! [Y: term_a_b,X: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ ( conver7608559976937845093rm_a_b @ R ) ) ) ) ).
% rtrancl_converseI
thf(fact_1158_rtrancl__converseD,axiom,
! [X: term_a_b,Y: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ ( conver7608559976937845093rm_a_b @ R ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X ) @ ( transi7742714808557438673rm_a_b @ R ) ) ) ).
% rtrancl_converseD
thf(fact_1159_converseI,axiom,
! [A: term_a_b,B: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ ( conver7608559976937845093rm_a_b @ R ) ) ) ).
% converseI
thf(fact_1160_converseE,axiom,
! [Yx: produc357393685978478089rm_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ Yx @ ( conver7608559976937845093rm_a_b @ R ) )
=> ~ ! [X3: term_a_b,Y3: term_a_b] :
( ( Yx
= ( produc7020197800436672577rm_a_b @ Y3 @ X3 ) )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ R ) ) ) ).
% converseE
thf(fact_1161_converseD,axiom,
! [A: term_a_b,B: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( conver7608559976937845093rm_a_b @ R ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ R ) ) ).
% converseD
thf(fact_1162_converse_Osimps,axiom,
! [A1: term_a_b,A2: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A2 ) @ ( conver7608559976937845093rm_a_b @ R ) )
= ( ? [A7: term_a_b,B6: term_a_b] :
( ( A1 = B6 )
& ( A2 = A7 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A7 @ B6 ) @ R ) ) ) ) ).
% converse.simps
thf(fact_1163_converse_Ocases,axiom,
! [A1: term_a_b,A2: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A2 ) @ ( conver7608559976937845093rm_a_b @ R ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ A1 ) @ R ) ) ).
% converse.cases
thf(fact_1164_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_1165_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_1166_converse__unfold,axiom,
( conver7608559976937845093rm_a_b
= ( ^ [R4: set_Pr4386577575007340137rm_a_b] :
( collec99177395925924084rm_a_b
@ ( produc6251846659952913346_a_b_o
@ ^ [Y4: term_a_b,X2: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y4 ) @ R4 ) ) ) ) ) ).
% converse_unfold
thf(fact_1167_srrstep__converse__dist,axiom,
! [F: set_Pr4934435412358123699_a_nat,R2: set_Pr4386577575007340137rm_a_b] :
( ( conver7608559976937845093rm_a_b @ ( sig_step_a_b @ F @ ( rrstep_a_b @ R2 ) ) )
= ( sig_step_a_b @ F @ ( rrstep_a_b @ ( conver7608559976937845093rm_a_b @ R2 ) ) ) ) ).
% srrstep_converse_dist
thf(fact_1168_srstep__symcl__dist,axiom,
! [F: set_Pr4934435412358123699_a_nat,R2: set_Pr4386577575007340137rm_a_b] :
( ( sup_su6776935440552674877rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R2 ) ) @ ( conver7608559976937845093rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R2 ) ) ) )
= ( sig_step_a_b @ F @ ( rstep_a_b @ ( sup_su6776935440552674877rm_a_b @ R2 @ ( conver7608559976937845093rm_a_b @ R2 ) ) ) ) ) ).
% srstep_symcl_dist
thf(fact_1169_conversionE,axiom,
! [A: term_a_b,B: term_a_b,A5: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( abstra1130568886744591910rm_a_b @ A5 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ A5 @ ( conver7608559976937845093rm_a_b @ A5 ) ) ) ) ) ).
% conversionE
thf(fact_1170_above__def,axiom,
( order_above_term_a_b
= ( ^ [R4: set_Pr4386577575007340137rm_a_b,A7: term_a_b] :
( collect_term_a_b
@ ^ [B6: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A7 @ B6 ) @ R4 ) ) ) ) ).
% above_def
thf(fact_1171_sig__step__conversion__dist,axiom,
! [F: set_Pr4934435412358123699_a_nat,R2: set_Pr4386577575007340137rm_a_b] :
( ( abstra1130568886744591910rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R2 ) ) )
= ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ ( sup_su6776935440552674877rm_a_b @ R2 @ ( conver7608559976937845093rm_a_b @ R2 ) ) ) ) ) ) ).
% sig_step_conversion_dist
thf(fact_1172_llrg__srsteps__with__root__step__inv__ground,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,S: term_a_b,T: term_a_b,F: set_Pr4934435412358123699_a_nat] :
( ( rewrit5000278123254648670_b_a_b @ R2 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( srstep7844470518422762656ep_a_b @ F @ ( conver7608559976937845093rm_a_b @ R2 ) ) )
=> ( term_ground_a_b @ S ) ) ) ).
% llrg_srsteps_with_root_step_inv_ground
thf(fact_1173_irrefl__tranclI,axiom,
! [R: set_Pr4386577575007340137rm_a_b,X: 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 @ X @ X ) @ ( transi7922773638565587891rm_a_b @ R ) ) ) ).
% irrefl_tranclI
thf(fact_1174_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 )
=> ? [X2: term_a_b] :
( ( member_term_a_b @ X2 @ A6 )
& ! [Y4: term_a_b] :
( ( member_term_a_b @ Y4 @ A6 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y4 ) @ R ) ) ) ) ) ) ) ) ).
% Linear_order_wf_diff_Id
thf(fact_1175_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 )
=> ? [X2: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ X2 @ A6 )
& ! [Y4: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ Y4 @ A6 )
=> ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ X2 @ Y4 ) @ R ) ) ) ) ) ) ) ) ).
% Linear_order_wf_diff_Id
thf(fact_1176_DiffI,axiom,
! [C2: produc357393685978478089rm_a_b,A5: set_Pr4386577575007340137rm_a_b,B4: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ C2 @ A5 )
=> ( ~ ( member5869715511025134514rm_a_b @ C2 @ B4 )
=> ( member5869715511025134514rm_a_b @ C2 @ ( minus_5192120951422937424rm_a_b @ A5 @ B4 ) ) ) ) ).
% DiffI
thf(fact_1177_DiffI,axiom,
! [C2: b,A5: set_b,B4: set_b] :
( ( member_b @ C2 @ A5 )
=> ( ~ ( member_b @ C2 @ B4 )
=> ( member_b @ C2 @ ( minus_minus_set_b @ A5 @ B4 ) ) ) ) ).
% DiffI
thf(fact_1178_Diff__iff,axiom,
! [C2: produc357393685978478089rm_a_b,A5: set_Pr4386577575007340137rm_a_b,B4: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ C2 @ ( minus_5192120951422937424rm_a_b @ A5 @ B4 ) )
= ( ( member5869715511025134514rm_a_b @ C2 @ A5 )
& ~ ( member5869715511025134514rm_a_b @ C2 @ B4 ) ) ) ).
% Diff_iff
thf(fact_1179_Diff__iff,axiom,
! [C2: b,A5: set_b,B4: set_b] :
( ( member_b @ C2 @ ( minus_minus_set_b @ A5 @ B4 ) )
= ( ( member_b @ C2 @ A5 )
& ~ ( member_b @ C2 @ B4 ) ) ) ).
% Diff_iff
thf(fact_1180_Diff__insert0,axiom,
! [X: produc357393685978478089rm_a_b,A5: set_Pr4386577575007340137rm_a_b,B4: set_Pr4386577575007340137rm_a_b] :
( ~ ( member5869715511025134514rm_a_b @ X @ A5 )
=> ( ( minus_5192120951422937424rm_a_b @ A5 @ ( insert7009541432154983385rm_a_b @ X @ B4 ) )
= ( minus_5192120951422937424rm_a_b @ A5 @ B4 ) ) ) ).
% Diff_insert0
thf(fact_1181_Diff__insert0,axiom,
! [X: b,A5: set_b,B4: set_b] :
( ~ ( member_b @ X @ A5 )
=> ( ( minus_minus_set_b @ A5 @ ( insert_b @ X @ B4 ) )
= ( minus_minus_set_b @ A5 @ B4 ) ) ) ).
% Diff_insert0
thf(fact_1182_insert__Diff1,axiom,
! [X: produc357393685978478089rm_a_b,B4: set_Pr4386577575007340137rm_a_b,A5: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ X @ B4 )
=> ( ( minus_5192120951422937424rm_a_b @ ( insert7009541432154983385rm_a_b @ X @ A5 ) @ B4 )
= ( minus_5192120951422937424rm_a_b @ A5 @ B4 ) ) ) ).
% insert_Diff1
thf(fact_1183_insert__Diff1,axiom,
! [X: b,B4: set_b,A5: set_b] :
( ( member_b @ X @ B4 )
=> ( ( minus_minus_set_b @ ( insert_b @ X @ A5 ) @ B4 )
= ( minus_minus_set_b @ A5 @ B4 ) ) ) ).
% insert_Diff1
thf(fact_1184_Diff__eq__empty__iff,axiom,
! [A5: set_Pr4934435412358123699_a_nat,B4: set_Pr4934435412358123699_a_nat] :
( ( ( minus_8630901905721465484_a_nat @ A5 @ B4 )
= bot_bo9049108969261143879_a_nat )
= ( ord_le8666007276011122963_a_nat @ A5 @ B4 ) ) ).
% Diff_eq_empty_iff
thf(fact_1185_Diff__mono,axiom,
! [A5: set_Pr4934435412358123699_a_nat,C3: set_Pr4934435412358123699_a_nat,D: set_Pr4934435412358123699_a_nat,B4: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A5 @ C3 )
=> ( ( ord_le8666007276011122963_a_nat @ D @ B4 )
=> ( ord_le8666007276011122963_a_nat @ ( minus_8630901905721465484_a_nat @ A5 @ B4 ) @ ( minus_8630901905721465484_a_nat @ C3 @ D ) ) ) ) ).
% Diff_mono
thf(fact_1186_Diff__subset,axiom,
! [A5: set_Pr4934435412358123699_a_nat,B4: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ ( minus_8630901905721465484_a_nat @ A5 @ B4 ) @ A5 ) ).
% Diff_subset
thf(fact_1187_double__diff,axiom,
! [A5: set_Pr4934435412358123699_a_nat,B4: set_Pr4934435412358123699_a_nat,C3: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A5 @ B4 )
=> ( ( ord_le8666007276011122963_a_nat @ B4 @ C3 )
=> ( ( minus_8630901905721465484_a_nat @ B4 @ ( minus_8630901905721465484_a_nat @ C3 @ A5 ) )
= A5 ) ) ) ).
% double_diff
thf(fact_1188_insert__Diff__if,axiom,
! [X: produc357393685978478089rm_a_b,B4: set_Pr4386577575007340137rm_a_b,A5: set_Pr4386577575007340137rm_a_b] :
( ( ( member5869715511025134514rm_a_b @ X @ B4 )
=> ( ( minus_5192120951422937424rm_a_b @ ( insert7009541432154983385rm_a_b @ X @ A5 ) @ B4 )
= ( minus_5192120951422937424rm_a_b @ A5 @ B4 ) ) )
& ( ~ ( member5869715511025134514rm_a_b @ X @ B4 )
=> ( ( minus_5192120951422937424rm_a_b @ ( insert7009541432154983385rm_a_b @ X @ A5 ) @ B4 )
= ( insert7009541432154983385rm_a_b @ X @ ( minus_5192120951422937424rm_a_b @ A5 @ B4 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_1189_insert__Diff__if,axiom,
! [X: b,B4: set_b,A5: set_b] :
( ( ( member_b @ X @ B4 )
=> ( ( minus_minus_set_b @ ( insert_b @ X @ A5 ) @ B4 )
= ( minus_minus_set_b @ A5 @ B4 ) ) )
& ( ~ ( member_b @ X @ B4 )
=> ( ( minus_minus_set_b @ ( insert_b @ X @ A5 ) @ B4 )
= ( insert_b @ X @ ( minus_minus_set_b @ A5 @ B4 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_1190_subset__Diff__insert,axiom,
! [A5: set_Pr4386577575007340137rm_a_b,B4: set_Pr4386577575007340137rm_a_b,X: produc357393685978478089rm_a_b,C3: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A5 @ ( minus_5192120951422937424rm_a_b @ B4 @ ( insert7009541432154983385rm_a_b @ X @ C3 ) ) )
= ( ( ord_le118470702582115849rm_a_b @ A5 @ ( minus_5192120951422937424rm_a_b @ B4 @ C3 ) )
& ~ ( member5869715511025134514rm_a_b @ X @ A5 ) ) ) ).
% subset_Diff_insert
thf(fact_1191_subset__Diff__insert,axiom,
! [A5: set_b,B4: set_b,X: b,C3: set_b] :
( ( ord_less_eq_set_b @ A5 @ ( minus_minus_set_b @ B4 @ ( insert_b @ X @ C3 ) ) )
= ( ( ord_less_eq_set_b @ A5 @ ( minus_minus_set_b @ B4 @ C3 ) )
& ~ ( member_b @ X @ A5 ) ) ) ).
% subset_Diff_insert
thf(fact_1192_subset__Diff__insert,axiom,
! [A5: set_Pr4934435412358123699_a_nat,B4: set_Pr4934435412358123699_a_nat,X: product_prod_a_nat,C3: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A5 @ ( minus_8630901905721465484_a_nat @ B4 @ ( insert8054603423593749677_a_nat @ X @ C3 ) ) )
= ( ( ord_le8666007276011122963_a_nat @ A5 @ ( minus_8630901905721465484_a_nat @ B4 @ C3 ) )
& ~ ( member5724188588386418708_a_nat @ X @ A5 ) ) ) ).
% subset_Diff_insert
thf(fact_1193_Diff__partition,axiom,
! [A5: set_Pr4934435412358123699_a_nat,B4: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A5 @ B4 )
=> ( ( sup_su459911885395995103_a_nat @ A5 @ ( minus_8630901905721465484_a_nat @ B4 @ A5 ) )
= B4 ) ) ).
% Diff_partition
thf(fact_1194_Diff__subset__conv,axiom,
! [A5: set_Pr4934435412358123699_a_nat,B4: set_Pr4934435412358123699_a_nat,C3: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ ( minus_8630901905721465484_a_nat @ A5 @ B4 ) @ C3 )
= ( ord_le8666007276011122963_a_nat @ A5 @ ( sup_su459911885395995103_a_nat @ B4 @ C3 ) ) ) ).
% Diff_subset_conv
thf(fact_1195_DiffE,axiom,
! [C2: produc357393685978478089rm_a_b,A5: set_Pr4386577575007340137rm_a_b,B4: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ C2 @ ( minus_5192120951422937424rm_a_b @ A5 @ B4 ) )
=> ~ ( ( member5869715511025134514rm_a_b @ C2 @ A5 )
=> ( member5869715511025134514rm_a_b @ C2 @ B4 ) ) ) ).
% DiffE
thf(fact_1196_DiffE,axiom,
! [C2: b,A5: set_b,B4: set_b] :
( ( member_b @ C2 @ ( minus_minus_set_b @ A5 @ B4 ) )
=> ~ ( ( member_b @ C2 @ A5 )
=> ( member_b @ C2 @ B4 ) ) ) ).
% DiffE
thf(fact_1197_DiffD1,axiom,
! [C2: produc357393685978478089rm_a_b,A5: set_Pr4386577575007340137rm_a_b,B4: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ C2 @ ( minus_5192120951422937424rm_a_b @ A5 @ B4 ) )
=> ( member5869715511025134514rm_a_b @ C2 @ A5 ) ) ).
% DiffD1
thf(fact_1198_DiffD1,axiom,
! [C2: b,A5: set_b,B4: set_b] :
( ( member_b @ C2 @ ( minus_minus_set_b @ A5 @ B4 ) )
=> ( member_b @ C2 @ A5 ) ) ).
% DiffD1
thf(fact_1199_DiffD2,axiom,
! [C2: produc357393685978478089rm_a_b,A5: set_Pr4386577575007340137rm_a_b,B4: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ C2 @ ( minus_5192120951422937424rm_a_b @ A5 @ B4 ) )
=> ~ ( member5869715511025134514rm_a_b @ C2 @ B4 ) ) ).
% DiffD2
thf(fact_1200_DiffD2,axiom,
! [C2: b,A5: set_b,B4: set_b] :
( ( member_b @ C2 @ ( minus_minus_set_b @ A5 @ B4 ) )
=> ~ ( member_b @ C2 @ B4 ) ) ).
% DiffD2
thf(fact_1201_set__diff__eq,axiom,
( minus_5192120951422937424rm_a_b
= ( ^ [A6: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
( collec99177395925924084rm_a_b
@ ^ [X2: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X2 @ A6 )
& ~ ( member5869715511025134514rm_a_b @ X2 @ B5 ) ) ) ) ) ).
% set_diff_eq
thf(fact_1202_set__diff__eq,axiom,
( minus_minus_set_b
= ( ^ [A6: set_b,B5: set_b] :
( collect_b
@ ^ [X2: b] :
( ( member_b @ X2 @ A6 )
& ~ ( member_b @ X2 @ B5 ) ) ) ) ) ).
% set_diff_eq
thf(fact_1203_insert__Diff,axiom,
! [A: produc357393685978478089rm_a_b,A5: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ A @ A5 )
=> ( ( insert7009541432154983385rm_a_b @ A @ ( minus_5192120951422937424rm_a_b @ A5 @ ( insert7009541432154983385rm_a_b @ A @ bot_bo197521221353338581rm_a_b ) ) )
= A5 ) ) ).
% insert_Diff
thf(fact_1204_insert__Diff,axiom,
! [A: b,A5: set_b] :
( ( member_b @ A @ A5 )
=> ( ( insert_b @ A @ ( minus_minus_set_b @ A5 @ ( insert_b @ A @ bot_bot_set_b ) ) )
= A5 ) ) ).
% insert_Diff
thf(fact_1205_Diff__insert__absorb,axiom,
! [X: produc357393685978478089rm_a_b,A5: set_Pr4386577575007340137rm_a_b] :
( ~ ( member5869715511025134514rm_a_b @ X @ A5 )
=> ( ( minus_5192120951422937424rm_a_b @ ( insert7009541432154983385rm_a_b @ X @ A5 ) @ ( insert7009541432154983385rm_a_b @ X @ bot_bo197521221353338581rm_a_b ) )
= A5 ) ) ).
% Diff_insert_absorb
thf(fact_1206_Diff__insert__absorb,axiom,
! [X: b,A5: set_b] :
( ~ ( member_b @ X @ A5 )
=> ( ( minus_minus_set_b @ ( insert_b @ X @ A5 ) @ ( insert_b @ X @ bot_bot_set_b ) )
= A5 ) ) ).
% Diff_insert_absorb
thf(fact_1207_Diff__single__insert,axiom,
! [A5: set_Pr4934435412358123699_a_nat,X: product_prod_a_nat,B4: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ ( minus_8630901905721465484_a_nat @ A5 @ ( insert8054603423593749677_a_nat @ X @ bot_bo9049108969261143879_a_nat ) ) @ B4 )
=> ( ord_le8666007276011122963_a_nat @ A5 @ ( insert8054603423593749677_a_nat @ X @ B4 ) ) ) ).
% Diff_single_insert
thf(fact_1208_subset__insert__iff,axiom,
! [A5: set_Pr4386577575007340137rm_a_b,X: produc357393685978478089rm_a_b,B4: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A5 @ ( insert7009541432154983385rm_a_b @ X @ B4 ) )
= ( ( ( member5869715511025134514rm_a_b @ X @ A5 )
=> ( ord_le118470702582115849rm_a_b @ ( minus_5192120951422937424rm_a_b @ A5 @ ( insert7009541432154983385rm_a_b @ X @ bot_bo197521221353338581rm_a_b ) ) @ B4 ) )
& ( ~ ( member5869715511025134514rm_a_b @ X @ A5 )
=> ( ord_le118470702582115849rm_a_b @ A5 @ B4 ) ) ) ) ).
% subset_insert_iff
thf(fact_1209_subset__insert__iff,axiom,
! [A5: set_b,X: b,B4: set_b] :
( ( ord_less_eq_set_b @ A5 @ ( insert_b @ X @ B4 ) )
= ( ( ( member_b @ X @ A5 )
=> ( ord_less_eq_set_b @ ( minus_minus_set_b @ A5 @ ( insert_b @ X @ bot_bot_set_b ) ) @ B4 ) )
& ( ~ ( member_b @ X @ A5 )
=> ( ord_less_eq_set_b @ A5 @ B4 ) ) ) ) ).
% subset_insert_iff
thf(fact_1210_subset__insert__iff,axiom,
! [A5: set_Pr4934435412358123699_a_nat,X: product_prod_a_nat,B4: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A5 @ ( insert8054603423593749677_a_nat @ X @ B4 ) )
= ( ( ( member5724188588386418708_a_nat @ X @ A5 )
=> ( ord_le8666007276011122963_a_nat @ ( minus_8630901905721465484_a_nat @ A5 @ ( insert8054603423593749677_a_nat @ X @ bot_bo9049108969261143879_a_nat ) ) @ B4 ) )
& ( ~ ( member5724188588386418708_a_nat @ X @ A5 )
=> ( ord_le8666007276011122963_a_nat @ A5 @ B4 ) ) ) ) ).
% subset_insert_iff
thf(fact_1211_rtrancl__diff__decomp,axiom,
! [X: term_a_b,Y: term_a_b,A5: set_Pr4386577575007340137rm_a_b,B4: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( minus_5192120951422937424rm_a_b @ ( transi7742714808557438673rm_a_b @ A5 ) @ ( transi7742714808557438673rm_a_b @ B4 ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ A5 ) @ ( relcom370159955682700863rm_a_b @ ( minus_5192120951422937424rm_a_b @ A5 @ B4 ) @ ( transi7742714808557438673rm_a_b @ A5 ) ) ) ) ) ).
% rtrancl_diff_decomp
thf(fact_1212_Linear__order__in__diff__Id,axiom,
! [R: set_Pr2972776593051762503rm_a_b,A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b] :
( ( order_681589137112254398rm_a_b @ ( field_6884932134483023318rm_a_b @ R ) @ R )
=> ( ( member5869715511025134514rm_a_b @ A @ ( field_6884932134483023318rm_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ B @ ( field_6884932134483023318rm_a_b @ R ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ R )
= ( ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B @ A ) @ ( minus_45395518312058030rm_a_b @ R @ id_Pro3488709451141449061rm_a_b ) ) ) ) ) ) ) ).
% Linear_order_in_diff_Id
thf(fact_1213_Linear__order__in__diff__Id,axiom,
! [R: set_Product_prod_b_b,A: b,B: b] :
( ( order_8768733634509060148r_on_b @ ( field_b @ R ) @ R )
=> ( ( member_b @ A @ ( field_b @ R ) )
=> ( ( member_b @ B @ ( field_b @ R ) )
=> ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ A @ B ) @ R )
= ( ~ ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ B @ A ) @ ( minus_6252421959248544046od_b_b @ R @ id_b ) ) ) ) ) ) ) ).
% Linear_order_in_diff_Id
thf(fact_1214_Linear__order__in__diff__Id,axiom,
! [R: set_Pr4386577575007340137rm_a_b,A: term_a_b,B: term_a_b] :
( ( order_5388802246213473311rm_a_b @ ( field_term_a_b @ R ) @ R )
=> ( ( member_term_a_b @ A @ ( field_term_a_b @ R ) )
=> ( ( member_term_a_b @ B @ ( field_term_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R )
= ( ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ ( minus_5192120951422937424rm_a_b @ R @ id_term_a_b ) ) ) ) ) ) ) ).
% Linear_order_in_diff_Id
thf(fact_1215_diff__shunt__var,axiom,
! [X: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat] :
( ( ( minus_8630901905721465484_a_nat @ X @ Y )
= bot_bo9049108969261143879_a_nat )
= ( ord_le8666007276011122963_a_nat @ X @ Y ) ) ).
% diff_shunt_var
thf(fact_1216_minus__set__def,axiom,
( minus_5192120951422937424rm_a_b
= ( ^ [A6: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
( collec99177395925924084rm_a_b
@ ( minus_496813532085645709_a_b_o
@ ^ [X2: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ X2 @ A6 )
@ ^ [X2: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ X2 @ B5 ) ) ) ) ) ).
% minus_set_def
thf(fact_1217_minus__set__def,axiom,
( minus_minus_set_b
= ( ^ [A6: set_b,B5: set_b] :
( collect_b
@ ( minus_minus_b_o
@ ^ [X2: b] : ( member_b @ X2 @ A6 )
@ ^ [X2: b] : ( member_b @ X2 @ B5 ) ) ) ) ) ).
% minus_set_def
thf(fact_1218_bsqr__def,axiom,
( bNF_We8522098035112346167rm_a_b
= ( ^ [R4: set_Pr4386577575007340137rm_a_b] :
( collec1344989105561874898rm_a_b
@ ( produc161078424057476524_a_b_o
@ ( produc4352215818379867658_a_b_o
@ ^ [A12: term_a_b,A22: term_a_b] :
( produc6251846659952913346_a_b_o
@ ^ [B1: term_a_b,B22: term_a_b] :
( ( ord_le2705286416250468010rm_a_b @ ( insert_term_a_b @ A12 @ ( insert_term_a_b @ A22 @ ( insert_term_a_b @ B1 @ ( insert_term_a_b @ B22 @ bot_bot_set_term_a_b ) ) ) ) @ ( field_term_a_b @ R4 ) )
& ( ( ( A12 = B1 )
& ( A22 = B22 ) )
| ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( bNF_We4618759378340593734rm_a_b @ R4 @ A12 @ A22 ) @ ( bNF_We4618759378340593734rm_a_b @ R4 @ B1 @ B22 ) ) @ ( minus_5192120951422937424rm_a_b @ R4 @ id_term_a_b ) )
| ( ( ( bNF_We4618759378340593734rm_a_b @ R4 @ A12 @ A22 )
= ( bNF_We4618759378340593734rm_a_b @ R4 @ B1 @ B22 ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A12 @ B1 ) @ ( minus_5192120951422937424rm_a_b @ R4 @ id_term_a_b ) ) )
| ( ( ( bNF_We4618759378340593734rm_a_b @ R4 @ A12 @ A22 )
= ( bNF_We4618759378340593734rm_a_b @ R4 @ B1 @ B22 ) )
& ( A12 = B1 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A22 @ B22 ) @ ( minus_5192120951422937424rm_a_b @ R4 @ id_term_a_b ) ) ) ) ) ) ) ) ) ) ) ).
% bsqr_def
thf(fact_1219_bsqr__def,axiom,
( bNF_We4271928086518441248_a_nat
= ( ^ [R4: set_Pr1811044260758604347_a_nat] :
( collec909746108454458450_a_nat
@ ( produc4182344056952305196_nat_o
@ ( produc5851009877341970442_nat_o
@ ^ [A12: product_prod_a_nat,A22: product_prod_a_nat] :
( produc292522255390808560_nat_o
@ ^ [B1: product_prod_a_nat,B22: product_prod_a_nat] :
( ( ord_le8666007276011122963_a_nat @ ( insert8054603423593749677_a_nat @ A12 @ ( insert8054603423593749677_a_nat @ A22 @ ( insert8054603423593749677_a_nat @ B1 @ ( insert8054603423593749677_a_nat @ B22 @ bot_bo9049108969261143879_a_nat ) ) ) ) @ ( field_8954927560578634480_a_nat @ R4 ) )
& ( ( ( A12 = B1 )
& ( A22 = B22 ) )
| ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ ( bNF_We7882469860449324975_a_nat @ R4 @ A12 @ A22 ) @ ( bNF_We7882469860449324975_a_nat @ R4 @ B1 @ B22 ) ) @ ( minus_1341117088723627554_a_nat @ R4 @ id_Pro5207055338379199009_a_nat ) )
| ( ( ( bNF_We7882469860449324975_a_nat @ R4 @ A12 @ A22 )
= ( bNF_We7882469860449324975_a_nat @ R4 @ B1 @ B22 ) )
& ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A12 @ B1 ) @ ( minus_1341117088723627554_a_nat @ R4 @ id_Pro5207055338379199009_a_nat ) ) )
| ( ( ( bNF_We7882469860449324975_a_nat @ R4 @ A12 @ A22 )
= ( bNF_We7882469860449324975_a_nat @ R4 @ B1 @ B22 ) )
& ( A12 = B1 )
& ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A22 @ B22 ) @ ( minus_1341117088723627554_a_nat @ R4 @ id_Pro5207055338379199009_a_nat ) ) ) ) ) ) ) ) ) ) ) ).
% bsqr_def
thf(fact_1220_member__remove,axiom,
! [X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b,A5: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ X @ ( remove8523906646656265582rm_a_b @ Y @ A5 ) )
= ( ( member5869715511025134514rm_a_b @ X @ A5 )
& ( X != Y ) ) ) ).
% member_remove
thf(fact_1221_member__remove,axiom,
! [X: b,Y: b,A5: set_b] :
( ( member_b @ X @ ( remove_b @ Y @ A5 ) )
= ( ( member_b @ X @ A5 )
& ( X != Y ) ) ) ).
% member_remove
thf(fact_1222_relation__of__def,axiom,
( order_1678919936295849798rm_a_b
= ( ^ [P4: term_a_b > term_a_b > $o,A6: set_term_a_b] :
( collec99177395925924084rm_a_b
@ ( produc6251846659952913346_a_b_o
@ ^ [A7: term_a_b,B6: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A7 @ B6 )
@ ( produc1953217797448512882rm_a_b @ A6
@ ^ [Uu: term_a_b] : A6 ) )
& ( P4 @ A7 @ B6 ) ) ) ) ) ) ).
% relation_of_def
thf(fact_1223_subset__Image1__Image1__iff,axiom,
! [R: set_Pr2972776593051762503rm_a_b,A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b] :
( ( order_4024118566430455712rm_a_b @ ( field_6884932134483023318rm_a_b @ R ) @ R )
=> ( ( member5869715511025134514rm_a_b @ A @ ( field_6884932134483023318rm_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ B @ ( field_6884932134483023318rm_a_b @ R ) )
=> ( ( ord_le118470702582115849rm_a_b @ ( image_871640889055226643rm_a_b @ R @ ( insert7009541432154983385rm_a_b @ A @ bot_bo197521221353338581rm_a_b ) ) @ ( image_871640889055226643rm_a_b @ R @ ( insert7009541432154983385rm_a_b @ B @ bot_bo197521221353338581rm_a_b ) ) )
= ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B @ A ) @ R ) ) ) ) ) ).
% subset_Image1_Image1_iff
thf(fact_1224_subset__Image1__Image1__iff,axiom,
! [R: set_Product_prod_b_b,A: b,B: b] :
( ( order_preorder_on_b @ ( field_b @ R ) @ R )
=> ( ( member_b @ A @ ( field_b @ R ) )
=> ( ( member_b @ B @ ( field_b @ R ) )
=> ( ( ord_less_eq_set_b @ ( image_b_b @ R @ ( insert_b @ A @ bot_bot_set_b ) ) @ ( image_b_b @ R @ ( insert_b @ B @ bot_bot_set_b ) ) )
= ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ B @ A ) @ R ) ) ) ) ) ).
% subset_Image1_Image1_iff
thf(fact_1225_subset__Image1__Image1__iff,axiom,
! [R: set_Pr4386577575007340137rm_a_b,A: term_a_b,B: term_a_b] :
( ( order_7053473298986962365rm_a_b @ ( field_term_a_b @ R ) @ R )
=> ( ( member_term_a_b @ A @ ( field_term_a_b @ R ) )
=> ( ( member_term_a_b @ B @ ( field_term_a_b @ R ) )
=> ( ( ord_le2705286416250468010rm_a_b @ ( image_6923787618150785661rm_a_b @ R @ ( insert_term_a_b @ A @ bot_bot_set_term_a_b ) ) @ ( image_6923787618150785661rm_a_b @ R @ ( insert_term_a_b @ B @ bot_bot_set_term_a_b ) ) )
= ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A ) @ R ) ) ) ) ) ).
% subset_Image1_Image1_iff
thf(fact_1226_subset__Image1__Image1__iff,axiom,
! [R: set_Pr1811044260758604347_a_nat,A: product_prod_a_nat,B: product_prod_a_nat] :
( ( order_3959669267349681574_a_nat @ ( field_8954927560578634480_a_nat @ R ) @ R )
=> ( ( member5724188588386418708_a_nat @ A @ ( field_8954927560578634480_a_nat @ R ) )
=> ( ( member5724188588386418708_a_nat @ B @ ( field_8954927560578634480_a_nat @ R ) )
=> ( ( ord_le8666007276011122963_a_nat @ ( image_4918098389706103759_a_nat @ R @ ( insert8054603423593749677_a_nat @ A @ bot_bo9049108969261143879_a_nat ) ) @ ( image_4918098389706103759_a_nat @ R @ ( insert8054603423593749677_a_nat @ B @ bot_bo9049108969261143879_a_nat ) ) )
= ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ B @ A ) @ R ) ) ) ) ) ).
% subset_Image1_Image1_iff
thf(fact_1227_ImageI,axiom,
! [A: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,R: set_Pr2972776593051762503rm_a_b,A5: set_Pr4386577575007340137rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ R )
=> ( ( member5869715511025134514rm_a_b @ A @ A5 )
=> ( member5869715511025134514rm_a_b @ B @ ( image_871640889055226643rm_a_b @ R @ A5 ) ) ) ) ).
% ImageI
thf(fact_1228_ImageI,axiom,
! [A: produc357393685978478089rm_a_b,B: b,R: set_Pr584113377249817149_a_b_b,A5: set_Pr4386577575007340137rm_a_b] :
( ( member3184567406997941638_a_b_b @ ( produc483848491608174797_a_b_b @ A @ B ) @ R )
=> ( ( member5869715511025134514rm_a_b @ A @ A5 )
=> ( member_b @ B @ ( image_3375235743491739401_a_b_b @ R @ A5 ) ) ) ) ).
% ImageI
thf(fact_1229_ImageI,axiom,
! [A: b,B: produc357393685978478089rm_a_b,R: set_Pr5162482618186357969rm_a_b,A5: set_b] :
( ( member8210134394708959770rm_a_b @ ( produc2673992483936284769rm_a_b @ A @ B ) @ R )
=> ( ( member_b @ A @ A5 )
=> ( member5869715511025134514rm_a_b @ B @ ( image_5565379735819849373rm_a_b @ R @ A5 ) ) ) ) ).
% ImageI
thf(fact_1230_ImageI,axiom,
! [A: b,B: b,R: set_Product_prod_b_b,A5: set_b] :
( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ A @ B ) @ R )
=> ( ( member_b @ A @ A5 )
=> ( member_b @ B @ ( image_b_b @ R @ A5 ) ) ) ) ).
% ImageI
thf(fact_1231_ImageI,axiom,
! [A: term_a_b,B: term_a_b,R: set_Pr4386577575007340137rm_a_b,A5: set_term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R )
=> ( ( member_term_a_b @ A @ A5 )
=> ( member_term_a_b @ B @ ( image_6923787618150785661rm_a_b @ R @ A5 ) ) ) ) ).
% ImageI
thf(fact_1232_Image__singleton__iff,axiom,
! [B: term_a_b,R: set_Pr4386577575007340137rm_a_b,A: term_a_b] :
( ( member_term_a_b @ B @ ( image_6923787618150785661rm_a_b @ R @ ( insert_term_a_b @ A @ bot_bot_set_term_a_b ) ) )
= ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R ) ) ).
% Image_singleton_iff
thf(fact_1233_Image__subset,axiom,
! [R: set_Pr4934435412358123699_a_nat,A5: set_a,B4: set_nat,C3: set_a] :
( ( ord_le8666007276011122963_a_nat @ R
@ ( product_Sigma_a_nat @ A5
@ ^ [Uu: a] : B4 ) )
=> ( ord_less_eq_set_nat @ ( image_a_nat @ R @ C3 ) @ B4 ) ) ).
% Image_subset
thf(fact_1234_wfI__pf,axiom,
! [R5: set_Pr1811044260758604347_a_nat] :
( ! [A10: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A10 @ ( image_4918098389706103759_a_nat @ R5 @ A10 ) )
=> ( A10 = bot_bo9049108969261143879_a_nat ) )
=> ( wf_Pro8109884346892000340_a_nat @ R5 ) ) ).
% wfI_pf
thf(fact_1235_wfE__pf,axiom,
! [R5: set_Pr1811044260758604347_a_nat,A5: set_Pr4934435412358123699_a_nat] :
( ( wf_Pro8109884346892000340_a_nat @ R5 )
=> ( ( ord_le8666007276011122963_a_nat @ A5 @ ( image_4918098389706103759_a_nat @ R5 @ A5 ) )
=> ( A5 = bot_bo9049108969261143879_a_nat ) ) ) ).
% wfE_pf
thf(fact_1236_SN__on__Image__normalizable,axiom,
! [R: set_Pr2972776593051762503rm_a_b,A5: set_Pr4386577575007340137rm_a_b] :
( ( abstra2398554102055911763rm_a_b @ R @ A5 )
=> ( ? [X4: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ X4 @ A5 )
=> ? [B3: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ B3 @ ( image_871640889055226643rm_a_b @ ( abstra1496011390211842225rm_a_b @ R ) @ A5 ) ) ) ) ).
% SN_on_Image_normalizable
thf(fact_1237_SN__on__Image__normalizable,axiom,
! [R: set_Product_prod_b_b,A5: set_b] :
( ( abstract_SN_on_b @ R @ A5 )
=> ( ? [X4: b] : ( member_b @ X4 @ A5 )
=> ? [B3: b] : ( member_b @ B3 @ ( image_b_b @ ( abstra380364156431640999lity_b @ R ) @ A5 ) ) ) ) ).
% SN_on_Image_normalizable
thf(fact_1238_Image__mono,axiom,
! [R8: set_Pr1811044260758604347_a_nat,R: set_Pr1811044260758604347_a_nat,A11: set_Pr4934435412358123699_a_nat,A5: set_Pr4934435412358123699_a_nat] :
( ( ord_le1018141841545957851_a_nat @ R8 @ R )
=> ( ( ord_le8666007276011122963_a_nat @ A11 @ A5 )
=> ( ord_le8666007276011122963_a_nat @ ( image_4918098389706103759_a_nat @ R8 @ A11 ) @ ( image_4918098389706103759_a_nat @ R @ A5 ) ) ) ) ).
% Image_mono
thf(fact_1239_Image__mono,axiom,
! [R8: set_Pr4934435412358123699_a_nat,R: set_Pr4934435412358123699_a_nat,A11: set_a,A5: set_a] :
( ( ord_le8666007276011122963_a_nat @ R8 @ R )
=> ( ( ord_less_eq_set_a @ A11 @ A5 )
=> ( ord_less_eq_set_nat @ ( image_a_nat @ R8 @ A11 ) @ ( image_a_nat @ R @ A5 ) ) ) ) ).
% Image_mono
thf(fact_1240_Image__subsetI,axiom,
! [R5: set_Pr4934435412358123699_a_nat,R7: set_Pr4934435412358123699_a_nat,X5: set_a] :
( ( ord_le8666007276011122963_a_nat @ R5 @ R7 )
=> ( ord_less_eq_set_nat @ ( image_a_nat @ R5 @ X5 ) @ ( image_a_nat @ R7 @ X5 ) ) ) ).
% Image_subsetI
thf(fact_1241_Image__closed__trancl,axiom,
! [R: set_Pr1811044260758604347_a_nat,X5: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ ( image_4918098389706103759_a_nat @ R @ X5 ) @ X5 )
=> ( ( image_4918098389706103759_a_nat @ ( transi2726145917338391738_a_nat @ R ) @ X5 )
= X5 ) ) ).
% Image_closed_trancl
thf(fact_1242_rtrancl__Image__step,axiom,
! [A: produc357393685978478089rm_a_b,R: set_Pr2972776593051762503rm_a_b,A5: set_Pr4386577575007340137rm_a_b,B: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ A @ ( image_871640889055226643rm_a_b @ ( transi2615809358984392588rm_a_b @ R ) @ A5 ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ ( transi2615809358984392588rm_a_b @ R ) )
=> ( member5869715511025134514rm_a_b @ B @ ( image_871640889055226643rm_a_b @ ( transi2615809358984392588rm_a_b @ R ) @ A5 ) ) ) ) ).
% rtrancl_Image_step
thf(fact_1243_rtrancl__Image__step,axiom,
! [A: b,R: set_Product_prod_b_b,A5: set_b,B: b] :
( ( member_b @ A @ ( image_b_b @ ( transitive_rtrancl_b @ R ) @ A5 ) )
=> ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ A @ B ) @ ( transitive_rtrancl_b @ R ) )
=> ( member_b @ B @ ( image_b_b @ ( transitive_rtrancl_b @ R ) @ A5 ) ) ) ) ).
% rtrancl_Image_step
thf(fact_1244_rtrancl__Image__step,axiom,
! [A: term_a_b,R: set_Pr4386577575007340137rm_a_b,A5: set_term_a_b,B: term_a_b] :
( ( member_term_a_b @ A @ ( image_6923787618150785661rm_a_b @ ( transi7742714808557438673rm_a_b @ R ) @ A5 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ( member_term_a_b @ B @ ( image_6923787618150785661rm_a_b @ ( transi7742714808557438673rm_a_b @ R ) @ A5 ) ) ) ) ).
% rtrancl_Image_step
thf(fact_1245_rev__ImageI,axiom,
! [A: produc357393685978478089rm_a_b,A5: set_Pr4386577575007340137rm_a_b,B: produc357393685978478089rm_a_b,R: set_Pr2972776593051762503rm_a_b] :
( ( member5869715511025134514rm_a_b @ A @ A5 )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A @ B ) @ R )
=> ( member5869715511025134514rm_a_b @ B @ ( image_871640889055226643rm_a_b @ R @ A5 ) ) ) ) ).
% rev_ImageI
thf(fact_1246_rev__ImageI,axiom,
! [A: produc357393685978478089rm_a_b,A5: set_Pr4386577575007340137rm_a_b,B: b,R: set_Pr584113377249817149_a_b_b] :
( ( member5869715511025134514rm_a_b @ A @ A5 )
=> ( ( member3184567406997941638_a_b_b @ ( produc483848491608174797_a_b_b @ A @ B ) @ R )
=> ( member_b @ B @ ( image_3375235743491739401_a_b_b @ R @ A5 ) ) ) ) ).
% rev_ImageI
thf(fact_1247_rev__ImageI,axiom,
! [A: b,A5: set_b,B: produc357393685978478089rm_a_b,R: set_Pr5162482618186357969rm_a_b] :
( ( member_b @ A @ A5 )
=> ( ( member8210134394708959770rm_a_b @ ( produc2673992483936284769rm_a_b @ A @ B ) @ R )
=> ( member5869715511025134514rm_a_b @ B @ ( image_5565379735819849373rm_a_b @ R @ A5 ) ) ) ) ).
% rev_ImageI
thf(fact_1248_rev__ImageI,axiom,
! [A: b,A5: set_b,B: b,R: set_Product_prod_b_b] :
( ( member_b @ A @ A5 )
=> ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ A @ B ) @ R )
=> ( member_b @ B @ ( image_b_b @ R @ A5 ) ) ) ) ).
% rev_ImageI
thf(fact_1249_rev__ImageI,axiom,
! [A: term_a_b,A5: set_term_a_b,B: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( member_term_a_b @ A @ A5 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R )
=> ( member_term_a_b @ B @ ( image_6923787618150785661rm_a_b @ R @ A5 ) ) ) ) ).
% rev_ImageI
thf(fact_1250_Image__iff,axiom,
! [B: term_a_b,R: set_Pr4386577575007340137rm_a_b,A5: set_term_a_b] :
( ( member_term_a_b @ B @ ( image_6923787618150785661rm_a_b @ R @ A5 ) )
= ( ? [X2: term_a_b] :
( ( member_term_a_b @ X2 @ A5 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ B ) @ R ) ) ) ) ).
% Image_iff
thf(fact_1251_ImageE,axiom,
! [B: produc357393685978478089rm_a_b,R: set_Pr2972776593051762503rm_a_b,A5: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ B @ ( image_871640889055226643rm_a_b @ R @ A5 ) )
=> ~ ! [X3: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X3 @ B ) @ R )
=> ~ ( member5869715511025134514rm_a_b @ X3 @ A5 ) ) ) ).
% ImageE
thf(fact_1252_ImageE,axiom,
! [B: produc357393685978478089rm_a_b,R: set_Pr5162482618186357969rm_a_b,A5: set_b] :
( ( member5869715511025134514rm_a_b @ B @ ( image_5565379735819849373rm_a_b @ R @ A5 ) )
=> ~ ! [X3: b] :
( ( member8210134394708959770rm_a_b @ ( produc2673992483936284769rm_a_b @ X3 @ B ) @ R )
=> ~ ( member_b @ X3 @ A5 ) ) ) ).
% ImageE
thf(fact_1253_ImageE,axiom,
! [B: b,R: set_Pr584113377249817149_a_b_b,A5: set_Pr4386577575007340137rm_a_b] :
( ( member_b @ B @ ( image_3375235743491739401_a_b_b @ R @ A5 ) )
=> ~ ! [X3: produc357393685978478089rm_a_b] :
( ( member3184567406997941638_a_b_b @ ( produc483848491608174797_a_b_b @ X3 @ B ) @ R )
=> ~ ( member5869715511025134514rm_a_b @ X3 @ A5 ) ) ) ).
% ImageE
thf(fact_1254_ImageE,axiom,
! [B: b,R: set_Product_prod_b_b,A5: set_b] :
( ( member_b @ B @ ( image_b_b @ R @ A5 ) )
=> ~ ! [X3: b] :
( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ X3 @ B ) @ R )
=> ~ ( member_b @ X3 @ A5 ) ) ) ).
% ImageE
thf(fact_1255_ImageE,axiom,
! [B: term_a_b,R: set_Pr4386577575007340137rm_a_b,A5: set_term_a_b] :
( ( member_term_a_b @ B @ ( image_6923787618150785661rm_a_b @ R @ A5 ) )
=> ~ ! [X3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ B ) @ R )
=> ~ ( member_term_a_b @ X3 @ A5 ) ) ) ).
% ImageE
thf(fact_1256_Image__singleton,axiom,
! [R: set_Pr4386577575007340137rm_a_b,A: term_a_b] :
( ( image_6923787618150785661rm_a_b @ R @ ( insert_term_a_b @ A @ bot_bot_set_term_a_b ) )
= ( collect_term_a_b
@ ^ [B6: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B6 ) @ R ) ) ) ).
% Image_singleton
thf(fact_1257_no__step,axiom,
! [A5: set_Pr2972776593051762503rm_a_b,A: produc357393685978478089rm_a_b] :
( ( ( image_871640889055226643rm_a_b @ A5 @ ( insert7009541432154983385rm_a_b @ A @ bot_bo197521221353338581rm_a_b ) )
= bot_bo197521221353338581rm_a_b )
=> ( member5869715511025134514rm_a_b @ A @ ( abstra1902783451776770197rm_a_b @ A5 ) ) ) ).
% no_step
thf(fact_1258_no__step,axiom,
! [A5: set_Product_prod_b_b,A: b] :
( ( ( image_b_b @ A5 @ ( insert_b @ A @ bot_bot_set_b ) )
= bot_bot_set_b )
=> ( member_b @ A @ ( abstract_NF_b @ A5 ) ) ) ).
% no_step
thf(fact_1259_subset__Image__Image__iff,axiom,
! [R: set_Pr4386577575007340137rm_a_b,A5: set_term_a_b,B4: set_term_a_b] :
( ( order_7053473298986962365rm_a_b @ ( field_term_a_b @ R ) @ R )
=> ( ( ord_le2705286416250468010rm_a_b @ A5 @ ( field_term_a_b @ R ) )
=> ( ( ord_le2705286416250468010rm_a_b @ B4 @ ( field_term_a_b @ R ) )
=> ( ( ord_le2705286416250468010rm_a_b @ ( image_6923787618150785661rm_a_b @ R @ A5 ) @ ( image_6923787618150785661rm_a_b @ R @ B4 ) )
= ( ! [X2: term_a_b] :
( ( member_term_a_b @ X2 @ A5 )
=> ? [Y4: term_a_b] :
( ( member_term_a_b @ Y4 @ B4 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y4 @ X2 ) @ R ) ) ) ) ) ) ) ) ).
% subset_Image_Image_iff
thf(fact_1260_subset__Image__Image__iff,axiom,
! [R: set_Pr1811044260758604347_a_nat,A5: set_Pr4934435412358123699_a_nat,B4: set_Pr4934435412358123699_a_nat] :
( ( order_3959669267349681574_a_nat @ ( field_8954927560578634480_a_nat @ R ) @ R )
=> ( ( ord_le8666007276011122963_a_nat @ A5 @ ( field_8954927560578634480_a_nat @ R ) )
=> ( ( ord_le8666007276011122963_a_nat @ B4 @ ( field_8954927560578634480_a_nat @ R ) )
=> ( ( ord_le8666007276011122963_a_nat @ ( image_4918098389706103759_a_nat @ R @ A5 ) @ ( image_4918098389706103759_a_nat @ R @ B4 ) )
= ( ! [X2: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ X2 @ A5 )
=> ? [Y4: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ Y4 @ B4 )
& ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ Y4 @ X2 ) @ R ) ) ) ) ) ) ) ) ).
% subset_Image_Image_iff
thf(fact_1261_restrict__SN__def,axiom,
( abstra4933154510623179783rm_a_b
= ( ^ [R4: set_Pr4386577575007340137rm_a_b,S7: set_Pr4386577575007340137rm_a_b] :
( collec99177395925924084rm_a_b
@ ^ [Uu: produc357393685978478089rm_a_b] :
? [A7: term_a_b,B6: term_a_b] :
( ( Uu
= ( produc7020197800436672577rm_a_b @ A7 @ B6 ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A7 @ B6 ) @ R4 )
& ( abstra4720023341729745482rm_a_b @ S7 @ ( insert_term_a_b @ A7 @ bot_bot_set_term_a_b ) ) ) ) ) ) ).
% restrict_SN_def
thf(fact_1262_relcomp3__transI,axiom,
! [B4: set_Pr4386577575007340137rm_a_b,T: term_a_b,U: term_a_b,A5: set_Pr4386577575007340137rm_a_b,S: term_a_b,V2: term_a_b] :
( ( trans_on_term_a_b @ top_top_set_term_a_b @ B4 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T @ U ) @ ( relcom370159955682700863rm_a_b @ B4 @ ( relcom370159955682700863rm_a_b @ A5 @ B4 ) ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ B4 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ U @ V2 ) @ B4 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ V2 ) @ ( relcom370159955682700863rm_a_b @ B4 @ ( relcom370159955682700863rm_a_b @ A5 @ B4 ) ) ) ) ) ) ) ).
% relcomp3_transI
thf(fact_1263_trans__on__subset,axiom,
! [A5: set_Pr4934435412358123699_a_nat,R: set_Pr1811044260758604347_a_nat,B4: set_Pr4934435412358123699_a_nat] :
( ( trans_5746013079982352605_a_nat @ A5 @ R )
=> ( ( ord_le8666007276011122963_a_nat @ B4 @ A5 )
=> ( trans_5746013079982352605_a_nat @ B4 @ R ) ) ) ).
% trans_on_subset
thf(fact_1264_trans__onD,axiom,
! [A5: set_Pr4386577575007340137rm_a_b,R: set_Pr2972776593051762503rm_a_b,X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b,Z2: produc357393685978478089rm_a_b] :
( ( trans_5404704112166290345rm_a_b @ A5 @ R )
=> ( ( member5869715511025134514rm_a_b @ X @ A5 )
=> ( ( member5869715511025134514rm_a_b @ Y @ A5 )
=> ( ( member5869715511025134514rm_a_b @ Z2 @ A5 )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ R )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y @ Z2 ) @ R )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Z2 ) @ R ) ) ) ) ) ) ) ).
% trans_onD
thf(fact_1265_trans__onD,axiom,
! [A5: set_b,R: set_Product_prod_b_b,X: b,Y: b,Z2: b] :
( ( trans_on_b @ A5 @ R )
=> ( ( member_b @ X @ A5 )
=> ( ( member_b @ Y @ A5 )
=> ( ( member_b @ Z2 @ A5 )
=> ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ X @ Y ) @ R )
=> ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ Y @ Z2 ) @ R )
=> ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ X @ Z2 ) @ R ) ) ) ) ) ) ) ).
% trans_onD
thf(fact_1266_trans__onD,axiom,
! [A5: set_term_a_b,R: set_Pr4386577575007340137rm_a_b,X: term_a_b,Y: term_a_b,Z2: term_a_b] :
( ( trans_on_term_a_b @ A5 @ R )
=> ( ( member_term_a_b @ X @ A5 )
=> ( ( member_term_a_b @ Y @ A5 )
=> ( ( member_term_a_b @ Z2 @ A5 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ R )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ Z2 ) @ R )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z2 ) @ R ) ) ) ) ) ) ) ).
% trans_onD
thf(fact_1267_trans__onI,axiom,
! [A5: set_Pr4386577575007340137rm_a_b,R: set_Pr2972776593051762503rm_a_b] :
( ! [X3: produc357393685978478089rm_a_b,Y3: produc357393685978478089rm_a_b,Z4: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X3 @ A5 )
=> ( ( member5869715511025134514rm_a_b @ Y3 @ A5 )
=> ( ( member5869715511025134514rm_a_b @ Z4 @ A5 )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X3 @ Y3 ) @ R )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y3 @ Z4 ) @ R )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X3 @ Z4 ) @ R ) ) ) ) ) )
=> ( trans_5404704112166290345rm_a_b @ A5 @ R ) ) ).
% trans_onI
thf(fact_1268_trans__onI,axiom,
! [A5: set_b,R: set_Product_prod_b_b] :
( ! [X3: b,Y3: b,Z4: b] :
( ( member_b @ X3 @ A5 )
=> ( ( member_b @ Y3 @ A5 )
=> ( ( member_b @ Z4 @ A5 )
=> ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ X3 @ Y3 ) @ R )
=> ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ Y3 @ Z4 ) @ R )
=> ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ X3 @ Z4 ) @ R ) ) ) ) ) )
=> ( trans_on_b @ A5 @ R ) ) ).
% trans_onI
thf(fact_1269_trans__onI,axiom,
! [A5: set_term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ! [X3: term_a_b,Y3: term_a_b,Z4: term_a_b] :
( ( member_term_a_b @ X3 @ A5 )
=> ( ( member_term_a_b @ Y3 @ A5 )
=> ( ( member_term_a_b @ Z4 @ A5 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ R )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ Z4 ) @ R )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Z4 ) @ R ) ) ) ) ) )
=> ( trans_on_term_a_b @ A5 @ R ) ) ).
% trans_onI
thf(fact_1270_trans__on__def,axiom,
( trans_on_term_a_b
= ( ^ [A6: set_term_a_b,R4: set_Pr4386577575007340137rm_a_b] :
! [X2: term_a_b] :
( ( member_term_a_b @ X2 @ A6 )
=> ! [Y4: term_a_b] :
( ( member_term_a_b @ Y4 @ A6 )
=> ! [Z5: term_a_b] :
( ( member_term_a_b @ Z5 @ A6 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y4 ) @ R4 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y4 @ Z5 ) @ R4 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Z5 ) @ R4 ) ) ) ) ) ) ) ) ).
% trans_on_def
thf(fact_1271_transD,axiom,
! [R: set_Pr4386577575007340137rm_a_b,X: term_a_b,Y: term_a_b,Z2: term_a_b] :
( ( trans_on_term_a_b @ top_top_set_term_a_b @ R )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ R )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ Z2 ) @ R )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z2 ) @ R ) ) ) ) ).
% transD
thf(fact_1272_transE,axiom,
! [R: set_Pr4386577575007340137rm_a_b,X: term_a_b,Y: term_a_b,Z2: term_a_b] :
( ( trans_on_term_a_b @ top_top_set_term_a_b @ R )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ R )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ Z2 ) @ R )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z2 ) @ R ) ) ) ) ).
% transE
thf(fact_1273_transI,axiom,
! [R: set_Pr4386577575007340137rm_a_b] :
( ! [X3: term_a_b,Y3: term_a_b,Z4: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ R )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ Z4 ) @ R )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Z4 ) @ R ) ) )
=> ( trans_on_term_a_b @ top_top_set_term_a_b @ R ) ) ).
% transI
thf(fact_1274_Id__def,axiom,
( id_term_a_b
= ( collec99177395925924084rm_a_b
@ ^ [P5: produc357393685978478089rm_a_b] :
? [X2: term_a_b] :
( P5
= ( produc7020197800436672577rm_a_b @ X2 @ X2 ) ) ) ) ).
% Id_def
% Helper facts (3)
thf(help_If_3_1_If_001t__Term__Oterm_Itf__a_Mtf__b_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Term__Oterm_Itf__a_Mtf__b_J_T,axiom,
! [X: term_a_b,Y: term_a_b] :
( ( if_term_a_b @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Term__Oterm_Itf__a_Mtf__b_J_T,axiom,
! [X: term_a_b,Y: term_a_b] :
( ( if_term_a_b @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( member5869715511025134514rm_a_b
@ ( produc7020197800436672577rm_a_b
@ ( subst_7999470309526761004_a_b_b @ l
@ ^ [X2: b] : ( if_term_a_b @ ( member_b @ X2 @ ( vars_term_a_b @ l ) ) @ ( subst_7999470309526761004_a_b_b @ ( gamma @ X2 ) @ sigma ) @ ( subst_7999470309526761004_a_b_b @ ( gamma @ X2 ) @ tau ) ) )
@ ( subst_7999470309526761004_a_b_b @ r2
@ ^ [X2: b] : ( if_term_a_b @ ( member_b @ X2 @ ( vars_term_a_b @ l ) ) @ ( subst_7999470309526761004_a_b_b @ ( gamma @ X2 ) @ sigma ) @ ( subst_7999470309526761004_a_b_b @ ( gamma @ X2 ) @ tau ) ) ) )
@ ( sig_step_a_b @ f @ ( rrstep_a_b @ r ) ) ) ).
%------------------------------------------------------------------------------