TPTP Problem File: SLH0954^1.p
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%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Rewrite_Properties_Reduction/0016_Rewriting/prob_00080_002478__13755826_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1588 ( 395 unt; 310 typ; 0 def)
% Number of atoms : 3976 ( 853 equ; 0 cnn)
% Maximal formula atoms : 6 ( 3 avg)
% Number of connectives : 14533 ( 420 ~; 37 |; 244 &;11805 @)
% ( 0 <=>;2027 =>; 0 <=; 0 <~>)
% Maximal formula depth : 22 ( 8 avg)
% Number of types : 35 ( 34 usr)
% Number of type conns : 741 ( 741 >; 0 *; 0 +; 0 <<)
% Number of symbols : 279 ( 276 usr; 21 con; 0-3 aty)
% Number of variables : 4180 ( 166 ^;3873 !; 141 ?;4180 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 14:18:20.931
%------------------------------------------------------------------------------
% Could-be-implicit typings (34)
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thf(sy_v_t,type,
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% Relevant facts (1275)
thf(fact_0_subsetI,axiom,
! [A: set_Pr9093778441882193744at_nat,B: set_Pr9093778441882193744at_nat] :
( ! [X: produc8199716216217303280at_nat] :
( ( member7226740684066999833at_nat @ X @ A )
=> ( member7226740684066999833at_nat @ X @ B ) )
=> ( ord_le3678578370064672496at_nat @ A @ B ) ) ).
% subsetI
thf(fact_1_subsetI,axiom,
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( ! [X: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X @ A )
=> ( member5869715511025134514rm_a_b @ X @ B ) )
=> ( ord_le118470702582115849rm_a_b @ A @ B ) ) ).
% subsetI
thf(fact_2_subsetI,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ! [X: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ X @ A )
=> ( member5724188588386418708_a_nat @ X @ B ) )
=> ( ord_le8666007276011122963_a_nat @ A @ B ) ) ).
% subsetI
thf(fact_3_subset__antisym,axiom,
! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A @ B )
=> ( ( ord_le118470702582115849rm_a_b @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_4_subset__antisym,axiom,
! [A: set_Pr9093778441882193744at_nat,B: set_Pr9093778441882193744at_nat] :
( ( ord_le3678578370064672496at_nat @ A @ B )
=> ( ( ord_le3678578370064672496at_nat @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_5_subset__antisym,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A @ B )
=> ( ( ord_le8666007276011122963_a_nat @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_6_prod_Oinject,axiom,
! [X1: a,X2: nat,Y1: a,Y2: nat] :
( ( ( product_Pair_a_nat @ X1 @ X2 )
= ( product_Pair_a_nat @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_7_prod_Oinject,axiom,
! [X1: term_a_b,X2: term_a_b,Y1: term_a_b,Y2: term_a_b] :
( ( ( produc7020197800436672577rm_a_b @ X1 @ X2 )
= ( produc7020197800436672577rm_a_b @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_8_prod_Oinject,axiom,
! [X1: nat > nat,X2: nat,Y1: nat > nat,Y2: nat] :
( ( ( produc72220940542539688at_nat @ X1 @ X2 )
= ( produc72220940542539688at_nat @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_9_old_Oprod_Oinject,axiom,
! [A2: a,B2: nat,A3: a,B3: nat] :
( ( ( product_Pair_a_nat @ A2 @ B2 )
= ( product_Pair_a_nat @ A3 @ B3 ) )
= ( ( A2 = A3 )
& ( B2 = B3 ) ) ) ).
% old.prod.inject
thf(fact_10_old_Oprod_Oinject,axiom,
! [A2: term_a_b,B2: term_a_b,A3: term_a_b,B3: term_a_b] :
( ( ( produc7020197800436672577rm_a_b @ A2 @ B2 )
= ( produc7020197800436672577rm_a_b @ A3 @ B3 ) )
= ( ( A2 = A3 )
& ( B2 = B3 ) ) ) ).
% old.prod.inject
thf(fact_11_old_Oprod_Oinject,axiom,
! [A2: nat > nat,B2: nat,A3: nat > nat,B3: nat] :
( ( ( produc72220940542539688at_nat @ A2 @ B2 )
= ( produc72220940542539688at_nat @ A3 @ B3 ) )
= ( ( A2 = A3 )
& ( B2 = B3 ) ) ) ).
% old.prod.inject
thf(fact_12_order__refl,axiom,
! [X3: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ X3 @ X3 ) ).
% order_refl
thf(fact_13_order__refl,axiom,
! [X3: set_Pr9093778441882193744at_nat] : ( ord_le3678578370064672496at_nat @ X3 @ X3 ) ).
% order_refl
thf(fact_14_order__refl,axiom,
! [X3: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ X3 @ X3 ) ).
% order_refl
thf(fact_15_dual__order_Orefl,axiom,
! [A2: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_16_dual__order_Orefl,axiom,
! [A2: set_Pr9093778441882193744at_nat] : ( ord_le3678578370064672496at_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_17_dual__order_Orefl,axiom,
! [A2: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_18_subrelI,axiom,
! [R: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
( ! [X: term_a_b,Y: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ R )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ S ) )
=> ( ord_le118470702582115849rm_a_b @ R @ S ) ) ).
% subrelI
thf(fact_19_subrelI,axiom,
! [R: set_Pr9093778441882193744at_nat,S: set_Pr9093778441882193744at_nat] :
( ! [X: nat > nat,Y: nat] :
( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ X @ Y ) @ R )
=> ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ X @ Y ) @ S ) )
=> ( ord_le3678578370064672496at_nat @ R @ S ) ) ).
% subrelI
thf(fact_20_subrelI,axiom,
! [R: set_Pr4934435412358123699_a_nat,S: set_Pr4934435412358123699_a_nat] :
( ! [X: a,Y: nat] :
( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ X @ Y ) @ R )
=> ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ X @ Y ) @ S ) )
=> ( ord_le8666007276011122963_a_nat @ R @ S ) ) ).
% subrelI
thf(fact_21_relChain__def,axiom,
( bNF_Ca8501653506855686447rm_a_b
= ( ^ [R2: set_Pr4386577575007340137rm_a_b,As: term_a_b > set_Pr4386577575007340137rm_a_b] :
! [I: term_a_b,J: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ I @ J ) @ R2 )
=> ( ord_le118470702582115849rm_a_b @ ( As @ I ) @ ( As @ J ) ) ) ) ) ).
% relChain_def
thf(fact_22_relChain__def,axiom,
( bNF_Ca6076535606073093910at_nat
= ( ^ [R2: set_Pr4386577575007340137rm_a_b,As: term_a_b > set_Pr9093778441882193744at_nat] :
! [I: term_a_b,J: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ I @ J ) @ R2 )
=> ( ord_le3678578370064672496at_nat @ ( As @ I ) @ ( As @ J ) ) ) ) ) ).
% relChain_def
thf(fact_23_relChain__def,axiom,
( bNF_Ca9222207453146494317_a_nat
= ( ^ [R2: 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 ) @ R2 )
=> ( ord_le8666007276011122963_a_nat @ ( As @ I ) @ ( As @ J ) ) ) ) ) ).
% relChain_def
thf(fact_24_in__mono,axiom,
! [A: set_Pr9093778441882193744at_nat,B: set_Pr9093778441882193744at_nat,X3: produc8199716216217303280at_nat] :
( ( ord_le3678578370064672496at_nat @ A @ B )
=> ( ( member7226740684066999833at_nat @ X3 @ A )
=> ( member7226740684066999833at_nat @ X3 @ B ) ) ) ).
% in_mono
thf(fact_25_in__mono,axiom,
! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b,X3: produc357393685978478089rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A @ B )
=> ( ( member5869715511025134514rm_a_b @ X3 @ A )
=> ( member5869715511025134514rm_a_b @ X3 @ B ) ) ) ).
% in_mono
thf(fact_26_in__mono,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat,X3: product_prod_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A @ B )
=> ( ( member5724188588386418708_a_nat @ X3 @ A )
=> ( member5724188588386418708_a_nat @ X3 @ B ) ) ) ).
% in_mono
thf(fact_27_subsetD,axiom,
! [A: set_Pr9093778441882193744at_nat,B: set_Pr9093778441882193744at_nat,C: produc8199716216217303280at_nat] :
( ( ord_le3678578370064672496at_nat @ A @ B )
=> ( ( member7226740684066999833at_nat @ C @ A )
=> ( member7226740684066999833at_nat @ C @ B ) ) ) ).
% subsetD
thf(fact_28_subsetD,axiom,
! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b,C: produc357393685978478089rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A @ B )
=> ( ( member5869715511025134514rm_a_b @ C @ A )
=> ( member5869715511025134514rm_a_b @ C @ B ) ) ) ).
% subsetD
thf(fact_29_subsetD,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat,C: product_prod_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A @ B )
=> ( ( member5724188588386418708_a_nat @ C @ A )
=> ( member5724188588386418708_a_nat @ C @ B ) ) ) ).
% subsetD
thf(fact_30_equalityE,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( A = B )
=> ~ ( ( ord_le8666007276011122963_a_nat @ A @ B )
=> ~ ( ord_le8666007276011122963_a_nat @ B @ A ) ) ) ).
% equalityE
thf(fact_31_equalityE,axiom,
! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ( A = B )
=> ~ ( ( ord_le118470702582115849rm_a_b @ A @ B )
=> ~ ( ord_le118470702582115849rm_a_b @ B @ A ) ) ) ).
% equalityE
thf(fact_32_equalityE,axiom,
! [A: set_Pr9093778441882193744at_nat,B: set_Pr9093778441882193744at_nat] :
( ( A = B )
=> ~ ( ( ord_le3678578370064672496at_nat @ A @ B )
=> ~ ( ord_le3678578370064672496at_nat @ B @ A ) ) ) ).
% equalityE
thf(fact_33_subset__eq,axiom,
( ord_le8666007276011122963_a_nat
= ( ^ [A4: set_Pr4934435412358123699_a_nat,B4: set_Pr4934435412358123699_a_nat] :
! [X4: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ X4 @ A4 )
=> ( member5724188588386418708_a_nat @ X4 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_34_subset__eq,axiom,
( ord_le118470702582115849rm_a_b
= ( ^ [A4: set_Pr4386577575007340137rm_a_b,B4: set_Pr4386577575007340137rm_a_b] :
! [X4: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X4 @ A4 )
=> ( member5869715511025134514rm_a_b @ X4 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_35_subset__eq,axiom,
( ord_le3678578370064672496at_nat
= ( ^ [A4: set_Pr9093778441882193744at_nat,B4: set_Pr9093778441882193744at_nat] :
! [X4: produc8199716216217303280at_nat] :
( ( member7226740684066999833at_nat @ X4 @ A4 )
=> ( member7226740684066999833at_nat @ X4 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_36_sig__step__mono2,axiom,
! [R3: set_Pr4386577575007340137rm_a_b,L: set_Pr4386577575007340137rm_a_b,F: set_Pr4934435412358123699_a_nat] :
( ( ord_le118470702582115849rm_a_b @ R3 @ L )
=> ( ord_le118470702582115849rm_a_b @ ( sig_step_a_b @ F @ R3 ) @ ( sig_step_a_b @ F @ L ) ) ) ).
% sig_step_mono2
thf(fact_37_sig__step__mono,axiom,
! [F: set_Pr4934435412358123699_a_nat,G: set_Pr4934435412358123699_a_nat,R3: set_Pr4386577575007340137rm_a_b] :
( ( ord_le8666007276011122963_a_nat @ F @ G )
=> ( ord_le118470702582115849rm_a_b @ ( sig_step_a_b @ F @ R3 ) @ ( sig_step_a_b @ G @ R3 ) ) ) ).
% sig_step_mono
thf(fact_38_order__antisym__conv,axiom,
! [Y3: set_Pr4934435412358123699_a_nat,X3: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ Y3 @ X3 )
=> ( ( ord_le8666007276011122963_a_nat @ X3 @ Y3 )
= ( X3 = Y3 ) ) ) ).
% order_antisym_conv
thf(fact_39_order__antisym__conv,axiom,
! [Y3: set_Pr4386577575007340137rm_a_b,X3: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ Y3 @ X3 )
=> ( ( ord_le118470702582115849rm_a_b @ X3 @ Y3 )
= ( X3 = Y3 ) ) ) ).
% order_antisym_conv
thf(fact_40_order__antisym__conv,axiom,
! [Y3: set_Pr9093778441882193744at_nat,X3: set_Pr9093778441882193744at_nat] :
( ( ord_le3678578370064672496at_nat @ Y3 @ X3 )
=> ( ( ord_le3678578370064672496at_nat @ X3 @ Y3 )
= ( X3 = Y3 ) ) ) ).
% order_antisym_conv
thf(fact_41_ord__le__eq__subst,axiom,
! [A2: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat,F2: set_Pr4934435412358123699_a_nat > set_Pr4934435412358123699_a_nat,C: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A2 @ B2 )
=> ( ( ( F2 @ B2 )
= C )
=> ( ! [X: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X @ Y )
=> ( ord_le8666007276011122963_a_nat @ ( F2 @ X ) @ ( F2 @ Y ) ) )
=> ( ord_le8666007276011122963_a_nat @ ( F2 @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_42_ord__le__eq__subst,axiom,
! [A2: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat,F2: set_Pr4934435412358123699_a_nat > set_Pr4386577575007340137rm_a_b,C: set_Pr4386577575007340137rm_a_b] :
( ( ord_le8666007276011122963_a_nat @ A2 @ B2 )
=> ( ( ( F2 @ B2 )
= C )
=> ( ! [X: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X @ Y )
=> ( ord_le118470702582115849rm_a_b @ ( F2 @ X ) @ ( F2 @ Y ) ) )
=> ( ord_le118470702582115849rm_a_b @ ( F2 @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_43_ord__le__eq__subst,axiom,
! [A2: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat,F2: set_Pr4934435412358123699_a_nat > set_Pr9093778441882193744at_nat,C: set_Pr9093778441882193744at_nat] :
( ( ord_le8666007276011122963_a_nat @ A2 @ B2 )
=> ( ( ( F2 @ B2 )
= C )
=> ( ! [X: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X @ Y )
=> ( ord_le3678578370064672496at_nat @ ( F2 @ X ) @ ( F2 @ Y ) ) )
=> ( ord_le3678578370064672496at_nat @ ( F2 @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_44_ord__le__eq__subst,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b,F2: set_Pr4386577575007340137rm_a_b > set_Pr4934435412358123699_a_nat,C: set_Pr4934435412358123699_a_nat] :
( ( ord_le118470702582115849rm_a_b @ A2 @ B2 )
=> ( ( ( F2 @ B2 )
= C )
=> ( ! [X: set_Pr4386577575007340137rm_a_b,Y: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ X @ Y )
=> ( ord_le8666007276011122963_a_nat @ ( F2 @ X ) @ ( F2 @ Y ) ) )
=> ( ord_le8666007276011122963_a_nat @ ( F2 @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_45_ord__le__eq__subst,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b,F2: set_Pr4386577575007340137rm_a_b > set_Pr4386577575007340137rm_a_b,C: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A2 @ B2 )
=> ( ( ( F2 @ B2 )
= C )
=> ( ! [X: set_Pr4386577575007340137rm_a_b,Y: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ X @ Y )
=> ( ord_le118470702582115849rm_a_b @ ( F2 @ X ) @ ( F2 @ Y ) ) )
=> ( ord_le118470702582115849rm_a_b @ ( F2 @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_46_ord__le__eq__subst,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b,F2: set_Pr4386577575007340137rm_a_b > set_Pr9093778441882193744at_nat,C: set_Pr9093778441882193744at_nat] :
( ( ord_le118470702582115849rm_a_b @ A2 @ B2 )
=> ( ( ( F2 @ B2 )
= C )
=> ( ! [X: set_Pr4386577575007340137rm_a_b,Y: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ X @ Y )
=> ( ord_le3678578370064672496at_nat @ ( F2 @ X ) @ ( F2 @ Y ) ) )
=> ( ord_le3678578370064672496at_nat @ ( F2 @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_47_ord__le__eq__subst,axiom,
! [A2: set_Pr9093778441882193744at_nat,B2: set_Pr9093778441882193744at_nat,F2: set_Pr9093778441882193744at_nat > set_Pr4934435412358123699_a_nat,C: set_Pr4934435412358123699_a_nat] :
( ( ord_le3678578370064672496at_nat @ A2 @ B2 )
=> ( ( ( F2 @ B2 )
= C )
=> ( ! [X: set_Pr9093778441882193744at_nat,Y: set_Pr9093778441882193744at_nat] :
( ( ord_le3678578370064672496at_nat @ X @ Y )
=> ( ord_le8666007276011122963_a_nat @ ( F2 @ X ) @ ( F2 @ Y ) ) )
=> ( ord_le8666007276011122963_a_nat @ ( F2 @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_48_ord__le__eq__subst,axiom,
! [A2: set_Pr9093778441882193744at_nat,B2: set_Pr9093778441882193744at_nat,F2: set_Pr9093778441882193744at_nat > set_Pr4386577575007340137rm_a_b,C: set_Pr4386577575007340137rm_a_b] :
( ( ord_le3678578370064672496at_nat @ A2 @ B2 )
=> ( ( ( F2 @ B2 )
= C )
=> ( ! [X: set_Pr9093778441882193744at_nat,Y: set_Pr9093778441882193744at_nat] :
( ( ord_le3678578370064672496at_nat @ X @ Y )
=> ( ord_le118470702582115849rm_a_b @ ( F2 @ X ) @ ( F2 @ Y ) ) )
=> ( ord_le118470702582115849rm_a_b @ ( F2 @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_49_ord__le__eq__subst,axiom,
! [A2: set_Pr9093778441882193744at_nat,B2: set_Pr9093778441882193744at_nat,F2: set_Pr9093778441882193744at_nat > set_Pr9093778441882193744at_nat,C: set_Pr9093778441882193744at_nat] :
( ( ord_le3678578370064672496at_nat @ A2 @ B2 )
=> ( ( ( F2 @ B2 )
= C )
=> ( ! [X: set_Pr9093778441882193744at_nat,Y: set_Pr9093778441882193744at_nat] :
( ( ord_le3678578370064672496at_nat @ X @ Y )
=> ( ord_le3678578370064672496at_nat @ ( F2 @ X ) @ ( F2 @ Y ) ) )
=> ( ord_le3678578370064672496at_nat @ ( F2 @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_50_ord__eq__le__subst,axiom,
! [A2: set_Pr4934435412358123699_a_nat,F2: set_Pr4934435412358123699_a_nat > set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat,C: set_Pr4934435412358123699_a_nat] :
( ( A2
= ( F2 @ B2 ) )
=> ( ( ord_le8666007276011122963_a_nat @ B2 @ C )
=> ( ! [X: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X @ Y )
=> ( ord_le8666007276011122963_a_nat @ ( F2 @ X ) @ ( F2 @ Y ) ) )
=> ( ord_le8666007276011122963_a_nat @ A2 @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_51_ord__eq__le__subst,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,F2: set_Pr4934435412358123699_a_nat > set_Pr4386577575007340137rm_a_b,B2: set_Pr4934435412358123699_a_nat,C: set_Pr4934435412358123699_a_nat] :
( ( A2
= ( F2 @ B2 ) )
=> ( ( ord_le8666007276011122963_a_nat @ B2 @ C )
=> ( ! [X: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X @ Y )
=> ( ord_le118470702582115849rm_a_b @ ( F2 @ X ) @ ( F2 @ Y ) ) )
=> ( ord_le118470702582115849rm_a_b @ A2 @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_52_ord__eq__le__subst,axiom,
! [A2: set_Pr9093778441882193744at_nat,F2: set_Pr4934435412358123699_a_nat > set_Pr9093778441882193744at_nat,B2: set_Pr4934435412358123699_a_nat,C: set_Pr4934435412358123699_a_nat] :
( ( A2
= ( F2 @ B2 ) )
=> ( ( ord_le8666007276011122963_a_nat @ B2 @ C )
=> ( ! [X: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X @ Y )
=> ( ord_le3678578370064672496at_nat @ ( F2 @ X ) @ ( F2 @ Y ) ) )
=> ( ord_le3678578370064672496at_nat @ A2 @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_53_ord__eq__le__subst,axiom,
! [A2: set_Pr4934435412358123699_a_nat,F2: set_Pr4386577575007340137rm_a_b > set_Pr4934435412358123699_a_nat,B2: set_Pr4386577575007340137rm_a_b,C: set_Pr4386577575007340137rm_a_b] :
( ( A2
= ( F2 @ B2 ) )
=> ( ( ord_le118470702582115849rm_a_b @ B2 @ C )
=> ( ! [X: set_Pr4386577575007340137rm_a_b,Y: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ X @ Y )
=> ( ord_le8666007276011122963_a_nat @ ( F2 @ X ) @ ( F2 @ Y ) ) )
=> ( ord_le8666007276011122963_a_nat @ A2 @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_54_ord__eq__le__subst,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,F2: set_Pr4386577575007340137rm_a_b > set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b,C: set_Pr4386577575007340137rm_a_b] :
( ( A2
= ( F2 @ B2 ) )
=> ( ( ord_le118470702582115849rm_a_b @ B2 @ C )
=> ( ! [X: set_Pr4386577575007340137rm_a_b,Y: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ X @ Y )
=> ( ord_le118470702582115849rm_a_b @ ( F2 @ X ) @ ( F2 @ Y ) ) )
=> ( ord_le118470702582115849rm_a_b @ A2 @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_55_ord__eq__le__subst,axiom,
! [A2: set_Pr9093778441882193744at_nat,F2: set_Pr4386577575007340137rm_a_b > set_Pr9093778441882193744at_nat,B2: set_Pr4386577575007340137rm_a_b,C: set_Pr4386577575007340137rm_a_b] :
( ( A2
= ( F2 @ B2 ) )
=> ( ( ord_le118470702582115849rm_a_b @ B2 @ C )
=> ( ! [X: set_Pr4386577575007340137rm_a_b,Y: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ X @ Y )
=> ( ord_le3678578370064672496at_nat @ ( F2 @ X ) @ ( F2 @ Y ) ) )
=> ( ord_le3678578370064672496at_nat @ A2 @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_56_ord__eq__le__subst,axiom,
! [A2: set_Pr4934435412358123699_a_nat,F2: set_Pr9093778441882193744at_nat > set_Pr4934435412358123699_a_nat,B2: set_Pr9093778441882193744at_nat,C: set_Pr9093778441882193744at_nat] :
( ( A2
= ( F2 @ B2 ) )
=> ( ( ord_le3678578370064672496at_nat @ B2 @ C )
=> ( ! [X: set_Pr9093778441882193744at_nat,Y: set_Pr9093778441882193744at_nat] :
( ( ord_le3678578370064672496at_nat @ X @ Y )
=> ( ord_le8666007276011122963_a_nat @ ( F2 @ X ) @ ( F2 @ Y ) ) )
=> ( ord_le8666007276011122963_a_nat @ A2 @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_57_ord__eq__le__subst,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,F2: set_Pr9093778441882193744at_nat > set_Pr4386577575007340137rm_a_b,B2: set_Pr9093778441882193744at_nat,C: set_Pr9093778441882193744at_nat] :
( ( A2
= ( F2 @ B2 ) )
=> ( ( ord_le3678578370064672496at_nat @ B2 @ C )
=> ( ! [X: set_Pr9093778441882193744at_nat,Y: set_Pr9093778441882193744at_nat] :
( ( ord_le3678578370064672496at_nat @ X @ Y )
=> ( ord_le118470702582115849rm_a_b @ ( F2 @ X ) @ ( F2 @ Y ) ) )
=> ( ord_le118470702582115849rm_a_b @ A2 @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_58_ord__eq__le__subst,axiom,
! [A2: set_Pr9093778441882193744at_nat,F2: set_Pr9093778441882193744at_nat > set_Pr9093778441882193744at_nat,B2: set_Pr9093778441882193744at_nat,C: set_Pr9093778441882193744at_nat] :
( ( A2
= ( F2 @ B2 ) )
=> ( ( ord_le3678578370064672496at_nat @ B2 @ C )
=> ( ! [X: set_Pr9093778441882193744at_nat,Y: set_Pr9093778441882193744at_nat] :
( ( ord_le3678578370064672496at_nat @ X @ Y )
=> ( ord_le3678578370064672496at_nat @ ( F2 @ X ) @ ( F2 @ Y ) ) )
=> ( ord_le3678578370064672496at_nat @ A2 @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_59_order__eq__refl,axiom,
! [X3: set_Pr4934435412358123699_a_nat,Y3: set_Pr4934435412358123699_a_nat] :
( ( X3 = Y3 )
=> ( ord_le8666007276011122963_a_nat @ X3 @ Y3 ) ) ).
% order_eq_refl
thf(fact_60_order__eq__refl,axiom,
! [X3: set_Pr4386577575007340137rm_a_b,Y3: set_Pr4386577575007340137rm_a_b] :
( ( X3 = Y3 )
=> ( ord_le118470702582115849rm_a_b @ X3 @ Y3 ) ) ).
% order_eq_refl
thf(fact_61_order__eq__refl,axiom,
! [X3: set_Pr9093778441882193744at_nat,Y3: set_Pr9093778441882193744at_nat] :
( ( X3 = Y3 )
=> ( ord_le3678578370064672496at_nat @ X3 @ Y3 ) ) ).
% order_eq_refl
thf(fact_62_order__subst2,axiom,
! [A2: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat,F2: set_Pr4934435412358123699_a_nat > set_Pr4934435412358123699_a_nat,C: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A2 @ B2 )
=> ( ( ord_le8666007276011122963_a_nat @ ( F2 @ B2 ) @ C )
=> ( ! [X: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X @ Y )
=> ( ord_le8666007276011122963_a_nat @ ( F2 @ X ) @ ( F2 @ Y ) ) )
=> ( ord_le8666007276011122963_a_nat @ ( F2 @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_63_order__subst2,axiom,
! [A2: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat,F2: set_Pr4934435412358123699_a_nat > set_Pr4386577575007340137rm_a_b,C: set_Pr4386577575007340137rm_a_b] :
( ( ord_le8666007276011122963_a_nat @ A2 @ B2 )
=> ( ( ord_le118470702582115849rm_a_b @ ( F2 @ B2 ) @ C )
=> ( ! [X: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X @ Y )
=> ( ord_le118470702582115849rm_a_b @ ( F2 @ X ) @ ( F2 @ Y ) ) )
=> ( ord_le118470702582115849rm_a_b @ ( F2 @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_64_order__subst2,axiom,
! [A2: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat,F2: set_Pr4934435412358123699_a_nat > set_Pr9093778441882193744at_nat,C: set_Pr9093778441882193744at_nat] :
( ( ord_le8666007276011122963_a_nat @ A2 @ B2 )
=> ( ( ord_le3678578370064672496at_nat @ ( F2 @ B2 ) @ C )
=> ( ! [X: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X @ Y )
=> ( ord_le3678578370064672496at_nat @ ( F2 @ X ) @ ( F2 @ Y ) ) )
=> ( ord_le3678578370064672496at_nat @ ( F2 @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_65_order__subst2,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b,F2: set_Pr4386577575007340137rm_a_b > set_Pr4934435412358123699_a_nat,C: set_Pr4934435412358123699_a_nat] :
( ( ord_le118470702582115849rm_a_b @ A2 @ B2 )
=> ( ( ord_le8666007276011122963_a_nat @ ( F2 @ B2 ) @ C )
=> ( ! [X: set_Pr4386577575007340137rm_a_b,Y: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ X @ Y )
=> ( ord_le8666007276011122963_a_nat @ ( F2 @ X ) @ ( F2 @ Y ) ) )
=> ( ord_le8666007276011122963_a_nat @ ( F2 @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_66_order__subst2,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b,F2: set_Pr4386577575007340137rm_a_b > set_Pr4386577575007340137rm_a_b,C: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A2 @ B2 )
=> ( ( ord_le118470702582115849rm_a_b @ ( F2 @ B2 ) @ C )
=> ( ! [X: set_Pr4386577575007340137rm_a_b,Y: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ X @ Y )
=> ( ord_le118470702582115849rm_a_b @ ( F2 @ X ) @ ( F2 @ Y ) ) )
=> ( ord_le118470702582115849rm_a_b @ ( F2 @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_67_order__subst2,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b,F2: set_Pr4386577575007340137rm_a_b > set_Pr9093778441882193744at_nat,C: set_Pr9093778441882193744at_nat] :
( ( ord_le118470702582115849rm_a_b @ A2 @ B2 )
=> ( ( ord_le3678578370064672496at_nat @ ( F2 @ B2 ) @ C )
=> ( ! [X: set_Pr4386577575007340137rm_a_b,Y: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ X @ Y )
=> ( ord_le3678578370064672496at_nat @ ( F2 @ X ) @ ( F2 @ Y ) ) )
=> ( ord_le3678578370064672496at_nat @ ( F2 @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_68_order__subst2,axiom,
! [A2: set_Pr9093778441882193744at_nat,B2: set_Pr9093778441882193744at_nat,F2: set_Pr9093778441882193744at_nat > set_Pr4934435412358123699_a_nat,C: set_Pr4934435412358123699_a_nat] :
( ( ord_le3678578370064672496at_nat @ A2 @ B2 )
=> ( ( ord_le8666007276011122963_a_nat @ ( F2 @ B2 ) @ C )
=> ( ! [X: set_Pr9093778441882193744at_nat,Y: set_Pr9093778441882193744at_nat] :
( ( ord_le3678578370064672496at_nat @ X @ Y )
=> ( ord_le8666007276011122963_a_nat @ ( F2 @ X ) @ ( F2 @ Y ) ) )
=> ( ord_le8666007276011122963_a_nat @ ( F2 @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_69_order__subst2,axiom,
! [A2: set_Pr9093778441882193744at_nat,B2: set_Pr9093778441882193744at_nat,F2: set_Pr9093778441882193744at_nat > set_Pr4386577575007340137rm_a_b,C: set_Pr4386577575007340137rm_a_b] :
( ( ord_le3678578370064672496at_nat @ A2 @ B2 )
=> ( ( ord_le118470702582115849rm_a_b @ ( F2 @ B2 ) @ C )
=> ( ! [X: set_Pr9093778441882193744at_nat,Y: set_Pr9093778441882193744at_nat] :
( ( ord_le3678578370064672496at_nat @ X @ Y )
=> ( ord_le118470702582115849rm_a_b @ ( F2 @ X ) @ ( F2 @ Y ) ) )
=> ( ord_le118470702582115849rm_a_b @ ( F2 @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_70_order__subst2,axiom,
! [A2: set_Pr9093778441882193744at_nat,B2: set_Pr9093778441882193744at_nat,F2: set_Pr9093778441882193744at_nat > set_Pr9093778441882193744at_nat,C: set_Pr9093778441882193744at_nat] :
( ( ord_le3678578370064672496at_nat @ A2 @ B2 )
=> ( ( ord_le3678578370064672496at_nat @ ( F2 @ B2 ) @ C )
=> ( ! [X: set_Pr9093778441882193744at_nat,Y: set_Pr9093778441882193744at_nat] :
( ( ord_le3678578370064672496at_nat @ X @ Y )
=> ( ord_le3678578370064672496at_nat @ ( F2 @ X ) @ ( F2 @ Y ) ) )
=> ( ord_le3678578370064672496at_nat @ ( F2 @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_71_order__subst1,axiom,
! [A2: set_Pr4934435412358123699_a_nat,F2: set_Pr4934435412358123699_a_nat > set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat,C: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A2 @ ( F2 @ B2 ) )
=> ( ( ord_le8666007276011122963_a_nat @ B2 @ C )
=> ( ! [X: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X @ Y )
=> ( ord_le8666007276011122963_a_nat @ ( F2 @ X ) @ ( F2 @ Y ) ) )
=> ( ord_le8666007276011122963_a_nat @ A2 @ ( F2 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_72_order__subst1,axiom,
! [A2: set_Pr4934435412358123699_a_nat,F2: set_Pr4386577575007340137rm_a_b > set_Pr4934435412358123699_a_nat,B2: set_Pr4386577575007340137rm_a_b,C: set_Pr4386577575007340137rm_a_b] :
( ( ord_le8666007276011122963_a_nat @ A2 @ ( F2 @ B2 ) )
=> ( ( ord_le118470702582115849rm_a_b @ B2 @ C )
=> ( ! [X: set_Pr4386577575007340137rm_a_b,Y: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ X @ Y )
=> ( ord_le8666007276011122963_a_nat @ ( F2 @ X ) @ ( F2 @ Y ) ) )
=> ( ord_le8666007276011122963_a_nat @ A2 @ ( F2 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_73_order__subst1,axiom,
! [A2: set_Pr4934435412358123699_a_nat,F2: set_Pr9093778441882193744at_nat > set_Pr4934435412358123699_a_nat,B2: set_Pr9093778441882193744at_nat,C: set_Pr9093778441882193744at_nat] :
( ( ord_le8666007276011122963_a_nat @ A2 @ ( F2 @ B2 ) )
=> ( ( ord_le3678578370064672496at_nat @ B2 @ C )
=> ( ! [X: set_Pr9093778441882193744at_nat,Y: set_Pr9093778441882193744at_nat] :
( ( ord_le3678578370064672496at_nat @ X @ Y )
=> ( ord_le8666007276011122963_a_nat @ ( F2 @ X ) @ ( F2 @ Y ) ) )
=> ( ord_le8666007276011122963_a_nat @ A2 @ ( F2 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_74_order__subst1,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,F2: set_Pr4934435412358123699_a_nat > set_Pr4386577575007340137rm_a_b,B2: set_Pr4934435412358123699_a_nat,C: set_Pr4934435412358123699_a_nat] :
( ( ord_le118470702582115849rm_a_b @ A2 @ ( F2 @ B2 ) )
=> ( ( ord_le8666007276011122963_a_nat @ B2 @ C )
=> ( ! [X: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X @ Y )
=> ( ord_le118470702582115849rm_a_b @ ( F2 @ X ) @ ( F2 @ Y ) ) )
=> ( ord_le118470702582115849rm_a_b @ A2 @ ( F2 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_75_order__subst1,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,F2: set_Pr4386577575007340137rm_a_b > set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b,C: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A2 @ ( F2 @ B2 ) )
=> ( ( ord_le118470702582115849rm_a_b @ B2 @ C )
=> ( ! [X: set_Pr4386577575007340137rm_a_b,Y: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ X @ Y )
=> ( ord_le118470702582115849rm_a_b @ ( F2 @ X ) @ ( F2 @ Y ) ) )
=> ( ord_le118470702582115849rm_a_b @ A2 @ ( F2 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_76_order__subst1,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,F2: set_Pr9093778441882193744at_nat > set_Pr4386577575007340137rm_a_b,B2: set_Pr9093778441882193744at_nat,C: set_Pr9093778441882193744at_nat] :
( ( ord_le118470702582115849rm_a_b @ A2 @ ( F2 @ B2 ) )
=> ( ( ord_le3678578370064672496at_nat @ B2 @ C )
=> ( ! [X: set_Pr9093778441882193744at_nat,Y: set_Pr9093778441882193744at_nat] :
( ( ord_le3678578370064672496at_nat @ X @ Y )
=> ( ord_le118470702582115849rm_a_b @ ( F2 @ X ) @ ( F2 @ Y ) ) )
=> ( ord_le118470702582115849rm_a_b @ A2 @ ( F2 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_77_order__subst1,axiom,
! [A2: set_Pr9093778441882193744at_nat,F2: set_Pr4934435412358123699_a_nat > set_Pr9093778441882193744at_nat,B2: set_Pr4934435412358123699_a_nat,C: set_Pr4934435412358123699_a_nat] :
( ( ord_le3678578370064672496at_nat @ A2 @ ( F2 @ B2 ) )
=> ( ( ord_le8666007276011122963_a_nat @ B2 @ C )
=> ( ! [X: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X @ Y )
=> ( ord_le3678578370064672496at_nat @ ( F2 @ X ) @ ( F2 @ Y ) ) )
=> ( ord_le3678578370064672496at_nat @ A2 @ ( F2 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_78_order__subst1,axiom,
! [A2: set_Pr9093778441882193744at_nat,F2: set_Pr4386577575007340137rm_a_b > set_Pr9093778441882193744at_nat,B2: set_Pr4386577575007340137rm_a_b,C: set_Pr4386577575007340137rm_a_b] :
( ( ord_le3678578370064672496at_nat @ A2 @ ( F2 @ B2 ) )
=> ( ( ord_le118470702582115849rm_a_b @ B2 @ C )
=> ( ! [X: set_Pr4386577575007340137rm_a_b,Y: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ X @ Y )
=> ( ord_le3678578370064672496at_nat @ ( F2 @ X ) @ ( F2 @ Y ) ) )
=> ( ord_le3678578370064672496at_nat @ A2 @ ( F2 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_79_order__subst1,axiom,
! [A2: set_Pr9093778441882193744at_nat,F2: set_Pr9093778441882193744at_nat > set_Pr9093778441882193744at_nat,B2: set_Pr9093778441882193744at_nat,C: set_Pr9093778441882193744at_nat] :
( ( ord_le3678578370064672496at_nat @ A2 @ ( F2 @ B2 ) )
=> ( ( ord_le3678578370064672496at_nat @ B2 @ C )
=> ( ! [X: set_Pr9093778441882193744at_nat,Y: set_Pr9093778441882193744at_nat] :
( ( ord_le3678578370064672496at_nat @ X @ Y )
=> ( ord_le3678578370064672496at_nat @ ( F2 @ X ) @ ( F2 @ Y ) ) )
=> ( ord_le3678578370064672496at_nat @ A2 @ ( F2 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_80_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_Pr4934435412358123699_a_nat,Z: set_Pr4934435412358123699_a_nat] : ( Y4 = Z ) )
= ( ^ [A5: set_Pr4934435412358123699_a_nat,B5: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A5 @ B5 )
& ( ord_le8666007276011122963_a_nat @ B5 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_81_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_Pr4386577575007340137rm_a_b,Z: set_Pr4386577575007340137rm_a_b] : ( Y4 = Z ) )
= ( ^ [A5: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A5 @ B5 )
& ( ord_le118470702582115849rm_a_b @ B5 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_82_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_Pr9093778441882193744at_nat,Z: set_Pr9093778441882193744at_nat] : ( Y4 = Z ) )
= ( ^ [A5: set_Pr9093778441882193744at_nat,B5: set_Pr9093778441882193744at_nat] :
( ( ord_le3678578370064672496at_nat @ A5 @ B5 )
& ( ord_le3678578370064672496at_nat @ B5 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_83_antisym,axiom,
! [A2: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A2 @ B2 )
=> ( ( ord_le8666007276011122963_a_nat @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_84_antisym,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A2 @ B2 )
=> ( ( ord_le118470702582115849rm_a_b @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_85_antisym,axiom,
! [A2: set_Pr9093778441882193744at_nat,B2: set_Pr9093778441882193744at_nat] :
( ( ord_le3678578370064672496at_nat @ A2 @ B2 )
=> ( ( ord_le3678578370064672496at_nat @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_86_dual__order_Otrans,axiom,
! [B2: set_Pr4934435412358123699_a_nat,A2: set_Pr4934435412358123699_a_nat,C: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ B2 @ A2 )
=> ( ( ord_le8666007276011122963_a_nat @ C @ B2 )
=> ( ord_le8666007276011122963_a_nat @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_87_dual__order_Otrans,axiom,
! [B2: set_Pr4386577575007340137rm_a_b,A2: set_Pr4386577575007340137rm_a_b,C: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ B2 @ A2 )
=> ( ( ord_le118470702582115849rm_a_b @ C @ B2 )
=> ( ord_le118470702582115849rm_a_b @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_88_dual__order_Otrans,axiom,
! [B2: set_Pr9093778441882193744at_nat,A2: set_Pr9093778441882193744at_nat,C: set_Pr9093778441882193744at_nat] :
( ( ord_le3678578370064672496at_nat @ B2 @ A2 )
=> ( ( ord_le3678578370064672496at_nat @ C @ B2 )
=> ( ord_le3678578370064672496at_nat @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_89_dual__order_Oantisym,axiom,
! [B2: set_Pr4934435412358123699_a_nat,A2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ B2 @ A2 )
=> ( ( ord_le8666007276011122963_a_nat @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_90_dual__order_Oantisym,axiom,
! [B2: set_Pr4386577575007340137rm_a_b,A2: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ B2 @ A2 )
=> ( ( ord_le118470702582115849rm_a_b @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_91_dual__order_Oantisym,axiom,
! [B2: set_Pr9093778441882193744at_nat,A2: set_Pr9093778441882193744at_nat] :
( ( ord_le3678578370064672496at_nat @ B2 @ A2 )
=> ( ( ord_le3678578370064672496at_nat @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_92_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_Pr4934435412358123699_a_nat,Z: set_Pr4934435412358123699_a_nat] : ( Y4 = Z ) )
= ( ^ [A5: set_Pr4934435412358123699_a_nat,B5: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ B5 @ A5 )
& ( ord_le8666007276011122963_a_nat @ A5 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_93_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_Pr4386577575007340137rm_a_b,Z: set_Pr4386577575007340137rm_a_b] : ( Y4 = Z ) )
= ( ^ [A5: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ B5 @ A5 )
& ( ord_le118470702582115849rm_a_b @ A5 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_94_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_Pr9093778441882193744at_nat,Z: set_Pr9093778441882193744at_nat] : ( Y4 = Z ) )
= ( ^ [A5: set_Pr9093778441882193744at_nat,B5: set_Pr9093778441882193744at_nat] :
( ( ord_le3678578370064672496at_nat @ B5 @ A5 )
& ( ord_le3678578370064672496at_nat @ A5 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_95_order__trans,axiom,
! [X3: set_Pr4934435412358123699_a_nat,Y3: set_Pr4934435412358123699_a_nat,Z2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X3 @ Y3 )
=> ( ( ord_le8666007276011122963_a_nat @ Y3 @ Z2 )
=> ( ord_le8666007276011122963_a_nat @ X3 @ Z2 ) ) ) ).
% order_trans
thf(fact_96_order__trans,axiom,
! [X3: set_Pr4386577575007340137rm_a_b,Y3: set_Pr4386577575007340137rm_a_b,Z2: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ X3 @ Y3 )
=> ( ( ord_le118470702582115849rm_a_b @ Y3 @ Z2 )
=> ( ord_le118470702582115849rm_a_b @ X3 @ Z2 ) ) ) ).
% order_trans
thf(fact_97_order__trans,axiom,
! [X3: set_Pr9093778441882193744at_nat,Y3: set_Pr9093778441882193744at_nat,Z2: set_Pr9093778441882193744at_nat] :
( ( ord_le3678578370064672496at_nat @ X3 @ Y3 )
=> ( ( ord_le3678578370064672496at_nat @ Y3 @ Z2 )
=> ( ord_le3678578370064672496at_nat @ X3 @ Z2 ) ) ) ).
% order_trans
thf(fact_98_order_Otrans,axiom,
! [A2: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat,C: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A2 @ B2 )
=> ( ( ord_le8666007276011122963_a_nat @ B2 @ C )
=> ( ord_le8666007276011122963_a_nat @ A2 @ C ) ) ) ).
% order.trans
thf(fact_99_order_Otrans,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b,C: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A2 @ B2 )
=> ( ( ord_le118470702582115849rm_a_b @ B2 @ C )
=> ( ord_le118470702582115849rm_a_b @ A2 @ C ) ) ) ).
% order.trans
thf(fact_100_order_Otrans,axiom,
! [A2: set_Pr9093778441882193744at_nat,B2: set_Pr9093778441882193744at_nat,C: set_Pr9093778441882193744at_nat] :
( ( ord_le3678578370064672496at_nat @ A2 @ B2 )
=> ( ( ord_le3678578370064672496at_nat @ B2 @ C )
=> ( ord_le3678578370064672496at_nat @ A2 @ C ) ) ) ).
% order.trans
thf(fact_101_order__antisym,axiom,
! [X3: set_Pr4934435412358123699_a_nat,Y3: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X3 @ Y3 )
=> ( ( ord_le8666007276011122963_a_nat @ Y3 @ X3 )
=> ( X3 = Y3 ) ) ) ).
% order_antisym
thf(fact_102_order__antisym,axiom,
! [X3: set_Pr4386577575007340137rm_a_b,Y3: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ X3 @ Y3 )
=> ( ( ord_le118470702582115849rm_a_b @ Y3 @ X3 )
=> ( X3 = Y3 ) ) ) ).
% order_antisym
thf(fact_103_order__antisym,axiom,
! [X3: set_Pr9093778441882193744at_nat,Y3: set_Pr9093778441882193744at_nat] :
( ( ord_le3678578370064672496at_nat @ X3 @ Y3 )
=> ( ( ord_le3678578370064672496at_nat @ Y3 @ X3 )
=> ( X3 = Y3 ) ) ) ).
% order_antisym
thf(fact_104_ord__le__eq__trans,axiom,
! [A2: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat,C: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_le8666007276011122963_a_nat @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_105_ord__le__eq__trans,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b,C: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_le118470702582115849rm_a_b @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_106_ord__le__eq__trans,axiom,
! [A2: set_Pr9093778441882193744at_nat,B2: set_Pr9093778441882193744at_nat,C: set_Pr9093778441882193744at_nat] :
( ( ord_le3678578370064672496at_nat @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_le3678578370064672496at_nat @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_107_ord__eq__le__trans,axiom,
! [A2: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat,C: set_Pr4934435412358123699_a_nat] :
( ( A2 = B2 )
=> ( ( ord_le8666007276011122963_a_nat @ B2 @ C )
=> ( ord_le8666007276011122963_a_nat @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_108_ord__eq__le__trans,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b,C: set_Pr4386577575007340137rm_a_b] :
( ( A2 = B2 )
=> ( ( ord_le118470702582115849rm_a_b @ B2 @ C )
=> ( ord_le118470702582115849rm_a_b @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_109_ord__eq__le__trans,axiom,
! [A2: set_Pr9093778441882193744at_nat,B2: set_Pr9093778441882193744at_nat,C: set_Pr9093778441882193744at_nat] :
( ( A2 = B2 )
=> ( ( ord_le3678578370064672496at_nat @ B2 @ C )
=> ( ord_le3678578370064672496at_nat @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_110_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_Pr4934435412358123699_a_nat,Z: set_Pr4934435412358123699_a_nat] : ( Y4 = Z ) )
= ( ^ [X4: set_Pr4934435412358123699_a_nat,Y5: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X4 @ Y5 )
& ( ord_le8666007276011122963_a_nat @ Y5 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_111_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_Pr4386577575007340137rm_a_b,Z: set_Pr4386577575007340137rm_a_b] : ( Y4 = Z ) )
= ( ^ [X4: set_Pr4386577575007340137rm_a_b,Y5: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ X4 @ Y5 )
& ( ord_le118470702582115849rm_a_b @ Y5 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_112_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_Pr9093778441882193744at_nat,Z: set_Pr9093778441882193744at_nat] : ( Y4 = Z ) )
= ( ^ [X4: set_Pr9093778441882193744at_nat,Y5: set_Pr9093778441882193744at_nat] :
( ( ord_le3678578370064672496at_nat @ X4 @ Y5 )
& ( ord_le3678578370064672496at_nat @ Y5 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_113_mem__Collect__eq,axiom,
! [A2: produc357393685978478089rm_a_b,P: produc357393685978478089rm_a_b > $o] :
( ( member5869715511025134514rm_a_b @ A2 @ ( collec99177395925924084rm_a_b @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_114_mem__Collect__eq,axiom,
! [A2: product_prod_a_nat,P: product_prod_a_nat > $o] :
( ( member5724188588386418708_a_nat @ A2 @ ( collec4464134535221767506_a_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_115_mem__Collect__eq,axiom,
! [A2: produc8199716216217303280at_nat,P: produc8199716216217303280at_nat > $o] :
( ( member7226740684066999833at_nat @ A2 @ ( collec9201399625632817755at_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_116_Collect__mem__eq,axiom,
! [A: set_Pr4386577575007340137rm_a_b] :
( ( collec99177395925924084rm_a_b
@ ^ [X4: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ X4 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_117_Collect__mem__eq,axiom,
! [A: set_Pr4934435412358123699_a_nat] :
( ( collec4464134535221767506_a_nat
@ ^ [X4: product_prod_a_nat] : ( member5724188588386418708_a_nat @ X4 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_118_Collect__mem__eq,axiom,
! [A: set_Pr9093778441882193744at_nat] :
( ( collec9201399625632817755at_nat
@ ^ [X4: produc8199716216217303280at_nat] : ( member7226740684066999833at_nat @ X4 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_119_Pair__inject,axiom,
! [A2: term_a_b,B2: term_a_b,A3: term_a_b,B3: term_a_b] :
( ( ( produc7020197800436672577rm_a_b @ A2 @ B2 )
= ( produc7020197800436672577rm_a_b @ A3 @ B3 ) )
=> ~ ( ( A2 = A3 )
=> ( B2 != B3 ) ) ) ).
% Pair_inject
thf(fact_120_Pair__inject,axiom,
! [A2: nat > nat,B2: nat,A3: nat > nat,B3: nat] :
( ( ( produc72220940542539688at_nat @ A2 @ B2 )
= ( produc72220940542539688at_nat @ A3 @ B3 ) )
=> ~ ( ( A2 = A3 )
=> ( B2 != B3 ) ) ) ).
% Pair_inject
thf(fact_121_Pair__inject,axiom,
! [A2: a,B2: nat,A3: a,B3: nat] :
( ( ( product_Pair_a_nat @ A2 @ B2 )
= ( product_Pair_a_nat @ A3 @ B3 ) )
=> ~ ( ( A2 = A3 )
=> ( B2 != B3 ) ) ) ).
% Pair_inject
thf(fact_122_prod__cases,axiom,
! [P: produc357393685978478089rm_a_b > $o,P2: produc357393685978478089rm_a_b] :
( ! [A6: term_a_b,B6: term_a_b] : ( P @ ( produc7020197800436672577rm_a_b @ A6 @ B6 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_123_prod__cases,axiom,
! [P: produc8199716216217303280at_nat > $o,P2: produc8199716216217303280at_nat] :
( ! [A6: nat > nat,B6: nat] : ( P @ ( produc72220940542539688at_nat @ A6 @ B6 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_124_prod__cases,axiom,
! [P: product_prod_a_nat > $o,P2: product_prod_a_nat] :
( ! [A6: a,B6: nat] : ( P @ ( product_Pair_a_nat @ A6 @ B6 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_125_surj__pair,axiom,
! [P2: produc357393685978478089rm_a_b] :
? [X: term_a_b,Y: term_a_b] :
( P2
= ( produc7020197800436672577rm_a_b @ X @ Y ) ) ).
% surj_pair
thf(fact_126_surj__pair,axiom,
! [P2: produc8199716216217303280at_nat] :
? [X: nat > nat,Y: nat] :
( P2
= ( produc72220940542539688at_nat @ X @ Y ) ) ).
% surj_pair
thf(fact_127_surj__pair,axiom,
! [P2: product_prod_a_nat] :
? [X: a,Y: nat] :
( P2
= ( product_Pair_a_nat @ X @ Y ) ) ).
% surj_pair
thf(fact_128_old_Oprod_Oexhaust,axiom,
! [Y3: produc357393685978478089rm_a_b] :
~ ! [A6: term_a_b,B6: term_a_b] :
( Y3
!= ( produc7020197800436672577rm_a_b @ A6 @ B6 ) ) ).
% old.prod.exhaust
thf(fact_129_old_Oprod_Oexhaust,axiom,
! [Y3: produc8199716216217303280at_nat] :
~ ! [A6: nat > nat,B6: nat] :
( Y3
!= ( produc72220940542539688at_nat @ A6 @ B6 ) ) ).
% old.prod.exhaust
thf(fact_130_old_Oprod_Oexhaust,axiom,
! [Y3: product_prod_a_nat] :
~ ! [A6: a,B6: nat] :
( Y3
!= ( product_Pair_a_nat @ A6 @ B6 ) ) ).
% old.prod.exhaust
thf(fact_131_Collect__mono__iff,axiom,
! [P: product_prod_a_nat > $o,Q: product_prod_a_nat > $o] :
( ( ord_le8666007276011122963_a_nat @ ( collec4464134535221767506_a_nat @ P ) @ ( collec4464134535221767506_a_nat @ Q ) )
= ( ! [X4: product_prod_a_nat] :
( ( P @ X4 )
=> ( Q @ X4 ) ) ) ) ).
% Collect_mono_iff
thf(fact_132_Collect__mono__iff,axiom,
! [P: produc357393685978478089rm_a_b > $o,Q: produc357393685978478089rm_a_b > $o] :
( ( ord_le118470702582115849rm_a_b @ ( collec99177395925924084rm_a_b @ P ) @ ( collec99177395925924084rm_a_b @ Q ) )
= ( ! [X4: produc357393685978478089rm_a_b] :
( ( P @ X4 )
=> ( Q @ X4 ) ) ) ) ).
% Collect_mono_iff
thf(fact_133_Collect__mono__iff,axiom,
! [P: produc8199716216217303280at_nat > $o,Q: produc8199716216217303280at_nat > $o] :
( ( ord_le3678578370064672496at_nat @ ( collec9201399625632817755at_nat @ P ) @ ( collec9201399625632817755at_nat @ Q ) )
= ( ! [X4: produc8199716216217303280at_nat] :
( ( P @ X4 )
=> ( Q @ X4 ) ) ) ) ).
% Collect_mono_iff
thf(fact_134_set__eq__subset,axiom,
( ( ^ [Y4: set_Pr4934435412358123699_a_nat,Z: set_Pr4934435412358123699_a_nat] : ( Y4 = Z ) )
= ( ^ [A4: set_Pr4934435412358123699_a_nat,B4: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A4 @ B4 )
& ( ord_le8666007276011122963_a_nat @ B4 @ A4 ) ) ) ) ).
% set_eq_subset
thf(fact_135_set__eq__subset,axiom,
( ( ^ [Y4: set_Pr4386577575007340137rm_a_b,Z: set_Pr4386577575007340137rm_a_b] : ( Y4 = Z ) )
= ( ^ [A4: set_Pr4386577575007340137rm_a_b,B4: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A4 @ B4 )
& ( ord_le118470702582115849rm_a_b @ B4 @ A4 ) ) ) ) ).
% set_eq_subset
thf(fact_136_set__eq__subset,axiom,
( ( ^ [Y4: set_Pr9093778441882193744at_nat,Z: set_Pr9093778441882193744at_nat] : ( Y4 = Z ) )
= ( ^ [A4: set_Pr9093778441882193744at_nat,B4: set_Pr9093778441882193744at_nat] :
( ( ord_le3678578370064672496at_nat @ A4 @ B4 )
& ( ord_le3678578370064672496at_nat @ B4 @ A4 ) ) ) ) ).
% set_eq_subset
thf(fact_137_subset__trans,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat,C2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A @ B )
=> ( ( ord_le8666007276011122963_a_nat @ B @ C2 )
=> ( ord_le8666007276011122963_a_nat @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_138_subset__trans,axiom,
! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b,C2: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A @ B )
=> ( ( ord_le118470702582115849rm_a_b @ B @ C2 )
=> ( ord_le118470702582115849rm_a_b @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_139_subset__trans,axiom,
! [A: set_Pr9093778441882193744at_nat,B: set_Pr9093778441882193744at_nat,C2: set_Pr9093778441882193744at_nat] :
( ( ord_le3678578370064672496at_nat @ A @ B )
=> ( ( ord_le3678578370064672496at_nat @ B @ C2 )
=> ( ord_le3678578370064672496at_nat @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_140_Collect__mono,axiom,
! [P: product_prod_a_nat > $o,Q: product_prod_a_nat > $o] :
( ! [X: product_prod_a_nat] :
( ( P @ X )
=> ( Q @ X ) )
=> ( ord_le8666007276011122963_a_nat @ ( collec4464134535221767506_a_nat @ P ) @ ( collec4464134535221767506_a_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_141_Collect__mono,axiom,
! [P: produc357393685978478089rm_a_b > $o,Q: produc357393685978478089rm_a_b > $o] :
( ! [X: produc357393685978478089rm_a_b] :
( ( P @ X )
=> ( Q @ X ) )
=> ( ord_le118470702582115849rm_a_b @ ( collec99177395925924084rm_a_b @ P ) @ ( collec99177395925924084rm_a_b @ Q ) ) ) ).
% Collect_mono
thf(fact_142_Collect__mono,axiom,
! [P: produc8199716216217303280at_nat > $o,Q: produc8199716216217303280at_nat > $o] :
( ! [X: produc8199716216217303280at_nat] :
( ( P @ X )
=> ( Q @ X ) )
=> ( ord_le3678578370064672496at_nat @ ( collec9201399625632817755at_nat @ P ) @ ( collec9201399625632817755at_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_143_subset__refl,axiom,
! [A: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ A @ A ) ).
% subset_refl
thf(fact_144_subset__refl,axiom,
! [A: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ A @ A ) ).
% subset_refl
thf(fact_145_subset__refl,axiom,
! [A: set_Pr9093778441882193744at_nat] : ( ord_le3678578370064672496at_nat @ A @ A ) ).
% subset_refl
thf(fact_146_subset__iff,axiom,
( ord_le8666007276011122963_a_nat
= ( ^ [A4: set_Pr4934435412358123699_a_nat,B4: set_Pr4934435412358123699_a_nat] :
! [T: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ T @ A4 )
=> ( member5724188588386418708_a_nat @ T @ B4 ) ) ) ) ).
% subset_iff
thf(fact_147_subset__iff,axiom,
( ord_le118470702582115849rm_a_b
= ( ^ [A4: set_Pr4386577575007340137rm_a_b,B4: set_Pr4386577575007340137rm_a_b] :
! [T: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ T @ A4 )
=> ( member5869715511025134514rm_a_b @ T @ B4 ) ) ) ) ).
% subset_iff
thf(fact_148_subset__iff,axiom,
( ord_le3678578370064672496at_nat
= ( ^ [A4: set_Pr9093778441882193744at_nat,B4: set_Pr9093778441882193744at_nat] :
! [T: produc8199716216217303280at_nat] :
( ( member7226740684066999833at_nat @ T @ A4 )
=> ( member7226740684066999833at_nat @ T @ B4 ) ) ) ) ).
% subset_iff
thf(fact_149_equalityD2,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( A = B )
=> ( ord_le8666007276011122963_a_nat @ B @ A ) ) ).
% equalityD2
thf(fact_150_equalityD2,axiom,
! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ( A = B )
=> ( ord_le118470702582115849rm_a_b @ B @ A ) ) ).
% equalityD2
thf(fact_151_equalityD2,axiom,
! [A: set_Pr9093778441882193744at_nat,B: set_Pr9093778441882193744at_nat] :
( ( A = B )
=> ( ord_le3678578370064672496at_nat @ B @ A ) ) ).
% equalityD2
thf(fact_152_equalityD1,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( A = B )
=> ( ord_le8666007276011122963_a_nat @ A @ B ) ) ).
% equalityD1
thf(fact_153_equalityD1,axiom,
! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ( A = B )
=> ( ord_le118470702582115849rm_a_b @ A @ B ) ) ).
% equalityD1
thf(fact_154_equalityD1,axiom,
! [A: set_Pr9093778441882193744at_nat,B: set_Pr9093778441882193744at_nat] :
( ( A = B )
=> ( ord_le3678578370064672496at_nat @ A @ B ) ) ).
% equalityD1
thf(fact_155_srsteps__with__root__step__sig__mono,axiom,
! [F: set_Pr4934435412358123699_a_nat,G: set_Pr4934435412358123699_a_nat,R3: set_Pr4386577575007340137rm_a_b] :
( ( ord_le8666007276011122963_a_nat @ F @ G )
=> ( ord_le118470702582115849rm_a_b @ ( srstep7844470518422762656ep_a_b @ F @ R3 ) @ ( srstep7844470518422762656ep_a_b @ G @ R3 ) ) ) ).
% srsteps_with_root_step_sig_mono
thf(fact_156_srrstep__monp,axiom,
! [F: set_Pr4934435412358123699_a_nat,G: set_Pr4934435412358123699_a_nat,R3: set_Pr4386577575007340137rm_a_b] :
( ( ord_le8666007276011122963_a_nat @ F @ G )
=> ( ord_le118470702582115849rm_a_b @ ( sig_step_a_b @ F @ ( rrstep_a_b @ R3 ) ) @ ( sig_step_a_b @ G @ ( rrstep_a_b @ R3 ) ) ) ) ).
% srrstep_monp
thf(fact_157_srstep__monp,axiom,
! [F: set_Pr4934435412358123699_a_nat,G: set_Pr4934435412358123699_a_nat,R3: set_Pr4386577575007340137rm_a_b] :
( ( ord_le8666007276011122963_a_nat @ F @ G )
=> ( ord_le118470702582115849rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R3 ) ) @ ( sig_step_a_b @ G @ ( rstep_a_b @ R3 ) ) ) ) ).
% srstep_monp
thf(fact_158_Greatest__equality,axiom,
! [P: set_Pr4934435412358123699_a_nat > $o,X3: set_Pr4934435412358123699_a_nat] :
( ( P @ X3 )
=> ( ! [Y: set_Pr4934435412358123699_a_nat] :
( ( P @ Y )
=> ( ord_le8666007276011122963_a_nat @ Y @ X3 ) )
=> ( ( order_1399416596184996762_a_nat @ P )
= X3 ) ) ) ).
% Greatest_equality
thf(fact_159_Greatest__equality,axiom,
! [P: set_Pr4386577575007340137rm_a_b > $o,X3: set_Pr4386577575007340137rm_a_b] :
( ( P @ X3 )
=> ( ! [Y: set_Pr4386577575007340137rm_a_b] :
( ( P @ Y )
=> ( ord_le118470702582115849rm_a_b @ Y @ X3 ) )
=> ( ( order_4999756673592925890rm_a_b @ P )
= X3 ) ) ) ).
% Greatest_equality
thf(fact_160_Greatest__equality,axiom,
! [P: set_Pr9093778441882193744at_nat > $o,X3: set_Pr9093778441882193744at_nat] :
( ( P @ X3 )
=> ( ! [Y: set_Pr9093778441882193744at_nat] :
( ( P @ Y )
=> ( ord_le3678578370064672496at_nat @ Y @ X3 ) )
=> ( ( order_2998882744634502953at_nat @ P )
= X3 ) ) ) ).
% Greatest_equality
thf(fact_161_GreatestI2__order,axiom,
! [P: set_Pr4934435412358123699_a_nat > $o,X3: set_Pr4934435412358123699_a_nat,Q: set_Pr4934435412358123699_a_nat > $o] :
( ( P @ X3 )
=> ( ! [Y: set_Pr4934435412358123699_a_nat] :
( ( P @ Y )
=> ( ord_le8666007276011122963_a_nat @ Y @ X3 ) )
=> ( ! [X: set_Pr4934435412358123699_a_nat] :
( ( P @ X )
=> ( ! [Y6: set_Pr4934435412358123699_a_nat] :
( ( P @ Y6 )
=> ( ord_le8666007276011122963_a_nat @ Y6 @ X ) )
=> ( Q @ X ) ) )
=> ( Q @ ( order_1399416596184996762_a_nat @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_162_GreatestI2__order,axiom,
! [P: set_Pr4386577575007340137rm_a_b > $o,X3: set_Pr4386577575007340137rm_a_b,Q: set_Pr4386577575007340137rm_a_b > $o] :
( ( P @ X3 )
=> ( ! [Y: set_Pr4386577575007340137rm_a_b] :
( ( P @ Y )
=> ( ord_le118470702582115849rm_a_b @ Y @ X3 ) )
=> ( ! [X: set_Pr4386577575007340137rm_a_b] :
( ( P @ X )
=> ( ! [Y6: set_Pr4386577575007340137rm_a_b] :
( ( P @ Y6 )
=> ( ord_le118470702582115849rm_a_b @ Y6 @ X ) )
=> ( Q @ X ) ) )
=> ( Q @ ( order_4999756673592925890rm_a_b @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_163_GreatestI2__order,axiom,
! [P: set_Pr9093778441882193744at_nat > $o,X3: set_Pr9093778441882193744at_nat,Q: set_Pr9093778441882193744at_nat > $o] :
( ( P @ X3 )
=> ( ! [Y: set_Pr9093778441882193744at_nat] :
( ( P @ Y )
=> ( ord_le3678578370064672496at_nat @ Y @ X3 ) )
=> ( ! [X: set_Pr9093778441882193744at_nat] :
( ( P @ X )
=> ( ! [Y6: set_Pr9093778441882193744at_nat] :
( ( P @ Y6 )
=> ( ord_le3678578370064672496at_nat @ Y6 @ X ) )
=> ( Q @ X ) ) )
=> ( Q @ ( order_2998882744634502953at_nat @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_164_ssubst__Pair__rhs,axiom,
! [R: term_a_b,S: term_a_b,R4: set_Pr4386577575007340137rm_a_b,S2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ R @ S ) @ R4 )
=> ( ( S2 = S )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ R @ S2 ) @ R4 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_165_ssubst__Pair__rhs,axiom,
! [R: nat > nat,S: nat,R4: set_Pr9093778441882193744at_nat,S2: nat] :
( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ R @ S ) @ R4 )
=> ( ( S2 = S )
=> ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ R @ S2 ) @ R4 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_166_ssubst__Pair__rhs,axiom,
! [R: a,S: nat,R4: set_Pr4934435412358123699_a_nat,S2: nat] :
( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ R @ S ) @ R4 )
=> ( ( S2 = S )
=> ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ R @ S2 ) @ R4 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_167_le__rel__bool__arg__iff,axiom,
( ord_le7949481618631132124_a_nat
= ( ^ [X5: $o > set_Pr4934435412358123699_a_nat,Y7: $o > set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ ( X5 @ $false ) @ ( Y7 @ $false ) )
& ( ord_le8666007276011122963_a_nat @ ( X5 @ $true ) @ ( Y7 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_168_le__rel__bool__arg__iff,axiom,
( ord_le7287365424611327766rm_a_b
= ( ^ [X5: $o > set_Pr4386577575007340137rm_a_b,Y7: $o > set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ ( X5 @ $false ) @ ( Y7 @ $false ) )
& ( ord_le118470702582115849rm_a_b @ ( X5 @ $true ) @ ( Y7 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_169_le__rel__bool__arg__iff,axiom,
( ord_le6507832023268780925at_nat
= ( ^ [X5: $o > set_Pr9093778441882193744at_nat,Y7: $o > set_Pr9093778441882193744at_nat] :
( ( ord_le3678578370064672496at_nat @ ( X5 @ $false ) @ ( Y7 @ $false ) )
& ( ord_le3678578370064672496at_nat @ ( X5 @ $true ) @ ( Y7 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_170_curryI,axiom,
! [F2: produc357393685978478089rm_a_b > $o,A2: term_a_b,B2: term_a_b] :
( ( F2 @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) )
=> ( produc6008550485034061667_a_b_o @ F2 @ A2 @ B2 ) ) ).
% curryI
thf(fact_171_curryI,axiom,
! [F2: produc8199716216217303280at_nat > $o,A2: nat > nat,B2: nat] :
( ( F2 @ ( produc72220940542539688at_nat @ A2 @ B2 ) )
=> ( produc1969786197626499516_nat_o @ F2 @ A2 @ B2 ) ) ).
% curryI
thf(fact_172_curryI,axiom,
! [F2: product_prod_a_nat > $o,A2: a,B2: nat] :
( ( F2 @ ( product_Pair_a_nat @ A2 @ B2 ) )
=> ( produc5346504556365831315_nat_o @ F2 @ A2 @ B2 ) ) ).
% curryI
thf(fact_173_rrstep__rstep__mono,axiom,
! [R3: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ ( rrstep_a_b @ R3 ) @ ( rstep_a_b @ R3 ) ) ).
% rrstep_rstep_mono
thf(fact_174_curryD,axiom,
! [F2: produc357393685978478089rm_a_b > $o,A2: term_a_b,B2: term_a_b] :
( ( produc6008550485034061667_a_b_o @ F2 @ A2 @ B2 )
=> ( F2 @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) ) ) ).
% curryD
thf(fact_175_curryD,axiom,
! [F2: produc8199716216217303280at_nat > $o,A2: nat > nat,B2: nat] :
( ( produc1969786197626499516_nat_o @ F2 @ A2 @ B2 )
=> ( F2 @ ( produc72220940542539688at_nat @ A2 @ B2 ) ) ) ).
% curryD
thf(fact_176_curryD,axiom,
! [F2: product_prod_a_nat > $o,A2: a,B2: nat] :
( ( produc5346504556365831315_nat_o @ F2 @ A2 @ B2 )
=> ( F2 @ ( product_Pair_a_nat @ A2 @ B2 ) ) ) ).
% curryD
thf(fact_177_curryE,axiom,
! [F2: produc357393685978478089rm_a_b > $o,A2: term_a_b,B2: term_a_b] :
( ( produc6008550485034061667_a_b_o @ F2 @ A2 @ B2 )
=> ( F2 @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) ) ) ).
% curryE
thf(fact_178_curryE,axiom,
! [F2: produc8199716216217303280at_nat > $o,A2: nat > nat,B2: nat] :
( ( produc1969786197626499516_nat_o @ F2 @ A2 @ B2 )
=> ( F2 @ ( produc72220940542539688at_nat @ A2 @ B2 ) ) ) ).
% curryE
thf(fact_179_curryE,axiom,
! [F2: product_prod_a_nat > $o,A2: a,B2: nat] :
( ( produc5346504556365831315_nat_o @ F2 @ A2 @ B2 )
=> ( F2 @ ( product_Pair_a_nat @ A2 @ B2 ) ) ) ).
% curryE
thf(fact_180_verit__comp__simplify1_I2_J,axiom,
! [A2: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_181_verit__comp__simplify1_I2_J,axiom,
! [A2: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_182_verit__comp__simplify1_I2_J,axiom,
! [A2: set_Pr9093778441882193744at_nat] : ( ord_le3678578370064672496at_nat @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_183_srsteps__eq__monp,axiom,
! [F: set_Pr4934435412358123699_a_nat,G: set_Pr4934435412358123699_a_nat,R3: set_Pr4386577575007340137rm_a_b] :
( ( ord_le8666007276011122963_a_nat @ F @ G )
=> ( ord_le118470702582115849rm_a_b @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R3 ) ) ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ G @ ( rstep_a_b @ R3 ) ) ) ) ) ).
% srsteps_eq_monp
thf(fact_184_srsteps__monp,axiom,
! [F: set_Pr4934435412358123699_a_nat,G: set_Pr4934435412358123699_a_nat,R3: set_Pr4386577575007340137rm_a_b] :
( ( ord_le8666007276011122963_a_nat @ F @ G )
=> ( ord_le118470702582115849rm_a_b @ ( transi7922773638565587891rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R3 ) ) ) @ ( transi7922773638565587891rm_a_b @ ( sig_step_a_b @ G @ ( rstep_a_b @ R3 ) ) ) ) ) ).
% srsteps_monp
thf(fact_185_rstep_Ocases,axiom,
! [A1: term_a_b,A22: term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A22 ) @ ( rstep_a_b @ R3 ) )
=> ~ ! [S3: term_a_b,T2: term_a_b,C3: subterm_and_ctxt_a_b] :
( ( A1
= ( subter2376574525758040790rm_a_b @ C3 @ S3 ) )
=> ( ( A22
= ( subter2376574525758040790rm_a_b @ C3 @ T2 ) )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S3 @ T2 ) @ ( rrstep_a_b @ R3 ) ) ) ) ) ).
% rstep.cases
thf(fact_186_rstep_Osimps,axiom,
! [A1: term_a_b,A22: term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A22 ) @ ( rstep_a_b @ R3 ) )
= ( ? [S4: term_a_b,T: term_a_b,C4: subterm_and_ctxt_a_b] :
( ( A1
= ( subter2376574525758040790rm_a_b @ C4 @ S4 ) )
& ( A22
= ( subter2376574525758040790rm_a_b @ C4 @ T ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S4 @ T ) @ ( rrstep_a_b @ R3 ) ) ) ) ) ).
% rstep.simps
thf(fact_187_rstep_Ointros,axiom,
! [S: term_a_b,T3: term_a_b,R3: set_Pr4386577575007340137rm_a_b,C2: subterm_and_ctxt_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T3 ) @ ( rrstep_a_b @ R3 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( subter2376574525758040790rm_a_b @ C2 @ S ) @ ( subter2376574525758040790rm_a_b @ C2 @ T3 ) ) @ ( rstep_a_b @ R3 ) ) ) ).
% rstep.intros
thf(fact_188_srsteps__with__root__step__def,axiom,
( srstep7844470518422762656ep_a_b
= ( ^ [F3: set_Pr4934435412358123699_a_nat,R5: set_Pr4386577575007340137rm_a_b] : ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F3 @ ( rstep_a_b @ R5 ) ) ) @ ( relcom370159955682700863rm_a_b @ ( sig_step_a_b @ F3 @ ( rrstep_a_b @ R5 ) ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F3 @ ( rstep_a_b @ R5 ) ) ) ) ) ) ) ).
% srsteps_with_root_step_def
thf(fact_189_Range__mono,axiom,
! [R: set_Pr4934435412358123699_a_nat,S: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ R @ S )
=> ( ord_less_eq_set_nat @ ( range_a_nat @ R ) @ ( range_a_nat @ S ) ) ) ).
% Range_mono
thf(fact_190_Range__mono,axiom,
! [R: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ R @ S )
=> ( ord_le2705286416250468010rm_a_b @ ( range_8688746797885257759rm_a_b @ R ) @ ( range_8688746797885257759rm_a_b @ S ) ) ) ).
% Range_mono
thf(fact_191_Range__mono,axiom,
! [R: set_Pr9093778441882193744at_nat,S: set_Pr9093778441882193744at_nat] :
( ( ord_le3678578370064672496at_nat @ R @ S )
=> ( ord_less_eq_set_nat @ ( range_nat_nat_nat @ R ) @ ( range_nat_nat_nat @ S ) ) ) ).
% Range_mono
thf(fact_192_in__inv__image,axiom,
! [X3: term_a_b,Y3: term_a_b,R: set_Pr4386577575007340137rm_a_b,F2: term_a_b > term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ ( inv_im3004224898869642620rm_a_b @ R @ F2 ) )
= ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) @ R ) ) ).
% in_inv_image
thf(fact_193_swap__simp,axiom,
! [X3: nat,Y3: nat > nat] :
( ( produc3547366087443502024at_nat @ ( produc7839516862119294504at_nat @ X3 @ Y3 ) )
= ( produc72220940542539688at_nat @ Y3 @ X3 ) ) ).
% swap_simp
thf(fact_194_swap__simp,axiom,
! [X3: nat,Y3: a] :
( ( product_swap_nat_a @ ( product_Pair_nat_a @ X3 @ Y3 ) )
= ( product_Pair_a_nat @ Y3 @ X3 ) ) ).
% swap_simp
thf(fact_195_swap__simp,axiom,
! [X3: term_a_b,Y3: term_a_b] :
( ( produc2541846788875228641rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) )
= ( produc7020197800436672577rm_a_b @ Y3 @ X3 ) ) ).
% swap_simp
thf(fact_196_swap__simp,axiom,
! [X3: nat > nat,Y3: nat] :
( ( produc5003442202721523016at_nat @ ( produc72220940542539688at_nat @ X3 @ Y3 ) )
= ( produc7839516862119294504at_nat @ Y3 @ X3 ) ) ).
% swap_simp
thf(fact_197_swap__simp,axiom,
! [X3: a,Y3: nat] :
( ( product_swap_a_nat @ ( product_Pair_a_nat @ X3 @ Y3 ) )
= ( product_Pair_nat_a @ Y3 @ X3 ) ) ).
% swap_simp
thf(fact_198_relcompEpair,axiom,
! [A2: term_a_b,C: term_a_b,R: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ C ) @ ( relcom370159955682700863rm_a_b @ R @ S ) )
=> ~ ! [B6: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B6 ) @ R )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B6 @ C ) @ S ) ) ) ).
% relcompEpair
thf(fact_199_relcompEpair,axiom,
! [A2: nat > nat,C: nat,R: set_Pr7682762132356531903at_nat,S: set_Pr9093778441882193744at_nat] :
( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ A2 @ C ) @ ( relcom9054258243621454929at_nat @ R @ S ) )
=> ~ ! [B6: nat > nat] :
( ( member1096249278104964872at_nat @ ( produc5770335208449155351at_nat @ A2 @ B6 ) @ R )
=> ~ ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ B6 @ C ) @ S ) ) ) ).
% relcompEpair
thf(fact_200_relcompEpair,axiom,
! [A2: nat > nat,C: nat,R: set_Pr3254233319718257578_nat_a,S: set_Pr4934435412358123699_a_nat] :
( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ A2 @ C ) @ ( relcom1163327185954966926_a_nat @ R @ S ) )
=> ~ ! [B6: a] :
( ( member7754445317155421323_nat_a @ ( produc3941937540969104678_nat_a @ A2 @ B6 ) @ R )
=> ~ ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ B6 @ C ) @ S ) ) ) ).
% relcompEpair
thf(fact_201_relcompEpair,axiom,
! [A2: nat > nat,C: nat,R: set_Pr9093778441882193744at_nat,S: set_Pr1261947904930325089at_nat] :
( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ A2 @ C ) @ ( relcom1859588284133128290at_nat @ R @ S ) )
=> ~ ! [B6: nat] :
( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ A2 @ B6 ) @ R )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B6 @ C ) @ S ) ) ) ).
% relcompEpair
thf(fact_202_relcompEpair,axiom,
! [A2: a,C: nat,R: set_Pr6841066879934740386at_nat,S: set_Pr9093778441882193744at_nat] :
( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ A2 @ C ) @ ( relcom8552328690651882190at_nat @ R @ S ) )
=> ~ ! [B6: nat > nat] :
( ( member5463269792551873667at_nat @ ( produc7514922136058452582at_nat @ A2 @ B6 ) @ R )
=> ~ ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ B6 @ C ) @ S ) ) ) ).
% relcompEpair
thf(fact_203_relcompEpair,axiom,
! [A2: a,C: nat,R: set_Product_prod_a_a,S: set_Pr4934435412358123699_a_nat] :
( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ A2 @ C ) @ ( relcomp_a_a_nat @ R @ S ) )
=> ~ ! [B6: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ B6 ) @ R )
=> ~ ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ B6 @ C ) @ S ) ) ) ).
% relcompEpair
thf(fact_204_relcompEpair,axiom,
! [A2: a,C: nat,R: set_Pr4934435412358123699_a_nat,S: set_Pr1261947904930325089at_nat] :
( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ A2 @ C ) @ ( relcomp_a_nat_nat @ R @ S ) )
=> ~ ! [B6: nat] :
( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ A2 @ B6 ) @ R )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B6 @ C ) @ S ) ) ) ).
% relcompEpair
thf(fact_205_relcompE,axiom,
! [Xz: produc357393685978478089rm_a_b,R: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ Xz @ ( relcom370159955682700863rm_a_b @ R @ S ) )
=> ~ ! [X: term_a_b,Y: term_a_b,Z3: term_a_b] :
( ( Xz
= ( produc7020197800436672577rm_a_b @ X @ Z3 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ R )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ Z3 ) @ S ) ) ) ) ).
% relcompE
thf(fact_206_relcompE,axiom,
! [Xz: produc8199716216217303280at_nat,R: set_Pr7682762132356531903at_nat,S: set_Pr9093778441882193744at_nat] :
( ( member7226740684066999833at_nat @ Xz @ ( relcom9054258243621454929at_nat @ R @ S ) )
=> ~ ! [X: nat > nat,Y: nat > nat,Z3: nat] :
( ( Xz
= ( produc72220940542539688at_nat @ X @ Z3 ) )
=> ( ( member1096249278104964872at_nat @ ( produc5770335208449155351at_nat @ X @ Y ) @ R )
=> ~ ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ Y @ Z3 ) @ S ) ) ) ) ).
% relcompE
thf(fact_207_relcompE,axiom,
! [Xz: produc8199716216217303280at_nat,R: set_Pr3254233319718257578_nat_a,S: set_Pr4934435412358123699_a_nat] :
( ( member7226740684066999833at_nat @ Xz @ ( relcom1163327185954966926_a_nat @ R @ S ) )
=> ~ ! [X: nat > nat,Y: a,Z3: nat] :
( ( Xz
= ( produc72220940542539688at_nat @ X @ Z3 ) )
=> ( ( member7754445317155421323_nat_a @ ( produc3941937540969104678_nat_a @ X @ Y ) @ R )
=> ~ ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ Y @ Z3 ) @ S ) ) ) ) ).
% relcompE
thf(fact_208_relcompE,axiom,
! [Xz: produc8199716216217303280at_nat,R: set_Pr9093778441882193744at_nat,S: set_Pr1261947904930325089at_nat] :
( ( member7226740684066999833at_nat @ Xz @ ( relcom1859588284133128290at_nat @ R @ S ) )
=> ~ ! [X: nat > nat,Y: nat,Z3: nat] :
( ( Xz
= ( produc72220940542539688at_nat @ X @ Z3 ) )
=> ( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ X @ Y ) @ R )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ Z3 ) @ S ) ) ) ) ).
% relcompE
thf(fact_209_relcompE,axiom,
! [Xz: product_prod_a_nat,R: set_Pr6841066879934740386at_nat,S: set_Pr9093778441882193744at_nat] :
( ( member5724188588386418708_a_nat @ Xz @ ( relcom8552328690651882190at_nat @ R @ S ) )
=> ~ ! [X: a,Y: nat > nat,Z3: nat] :
( ( Xz
= ( product_Pair_a_nat @ X @ Z3 ) )
=> ( ( member5463269792551873667at_nat @ ( produc7514922136058452582at_nat @ X @ Y ) @ R )
=> ~ ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ Y @ Z3 ) @ S ) ) ) ) ).
% relcompE
thf(fact_210_relcompE,axiom,
! [Xz: product_prod_a_nat,R: set_Product_prod_a_a,S: set_Pr4934435412358123699_a_nat] :
( ( member5724188588386418708_a_nat @ Xz @ ( relcomp_a_a_nat @ R @ S ) )
=> ~ ! [X: a,Y: a,Z3: nat] :
( ( Xz
= ( product_Pair_a_nat @ X @ Z3 ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R )
=> ~ ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ Y @ Z3 ) @ S ) ) ) ) ).
% relcompE
thf(fact_211_relcompE,axiom,
! [Xz: product_prod_a_nat,R: set_Pr4934435412358123699_a_nat,S: set_Pr1261947904930325089at_nat] :
( ( member5724188588386418708_a_nat @ Xz @ ( relcomp_a_nat_nat @ R @ S ) )
=> ~ ! [X: a,Y: nat,Z3: nat] :
( ( Xz
= ( product_Pair_a_nat @ X @ Z3 ) )
=> ( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ X @ Y ) @ R )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ Z3 ) @ S ) ) ) ) ).
% relcompE
thf(fact_212_relcomp_OrelcompI,axiom,
! [A2: nat > nat,B2: nat > nat,R: set_Pr7682762132356531903at_nat,C: nat,S: set_Pr9093778441882193744at_nat] :
( ( member1096249278104964872at_nat @ ( produc5770335208449155351at_nat @ A2 @ B2 ) @ R )
=> ( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ B2 @ C ) @ S )
=> ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ A2 @ C ) @ ( relcom9054258243621454929at_nat @ R @ S ) ) ) ) ).
% relcomp.relcompI
thf(fact_213_relcomp_OrelcompI,axiom,
! [A2: a,B2: nat > nat,R: set_Pr6841066879934740386at_nat,C: nat,S: set_Pr9093778441882193744at_nat] :
( ( member5463269792551873667at_nat @ ( produc7514922136058452582at_nat @ A2 @ B2 ) @ R )
=> ( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ B2 @ C ) @ S )
=> ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ A2 @ C ) @ ( relcom8552328690651882190at_nat @ R @ S ) ) ) ) ).
% relcomp.relcompI
thf(fact_214_relcomp_OrelcompI,axiom,
! [A2: nat > nat,B2: a,R: set_Pr3254233319718257578_nat_a,C: nat,S: set_Pr4934435412358123699_a_nat] :
( ( member7754445317155421323_nat_a @ ( produc3941937540969104678_nat_a @ A2 @ B2 ) @ R )
=> ( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ B2 @ C ) @ S )
=> ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ A2 @ C ) @ ( relcom1163327185954966926_a_nat @ R @ S ) ) ) ) ).
% relcomp.relcompI
thf(fact_215_relcomp_OrelcompI,axiom,
! [A2: a,B2: a,R: set_Product_prod_a_a,C: nat,S: set_Pr4934435412358123699_a_nat] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ B2 ) @ R )
=> ( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ B2 @ C ) @ S )
=> ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ A2 @ C ) @ ( relcomp_a_a_nat @ R @ S ) ) ) ) ).
% relcomp.relcompI
thf(fact_216_relcomp_OrelcompI,axiom,
! [A2: term_a_b,B2: term_a_b,R: set_Pr4386577575007340137rm_a_b,C: term_a_b,S: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ R )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ C ) @ S )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ C ) @ ( relcom370159955682700863rm_a_b @ R @ S ) ) ) ) ).
% relcomp.relcompI
thf(fact_217_relcomp_OrelcompI,axiom,
! [A2: nat > nat,B2: nat,R: set_Pr9093778441882193744at_nat,C: nat,S: set_Pr1261947904930325089at_nat] :
( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ A2 @ B2 ) @ R )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B2 @ C ) @ S )
=> ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ A2 @ C ) @ ( relcom1859588284133128290at_nat @ R @ S ) ) ) ) ).
% relcomp.relcompI
thf(fact_218_relcomp_OrelcompI,axiom,
! [A2: a,B2: nat,R: set_Pr4934435412358123699_a_nat,C: nat,S: set_Pr1261947904930325089at_nat] :
( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ A2 @ B2 ) @ R )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B2 @ C ) @ S )
=> ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ A2 @ C ) @ ( relcomp_a_nat_nat @ R @ S ) ) ) ) ).
% relcomp.relcompI
thf(fact_219_relcomp_Osimps,axiom,
! [A1: term_a_b,A22: term_a_b,R: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A22 ) @ ( relcom370159955682700863rm_a_b @ R @ S ) )
= ( ? [A5: term_a_b,B5: term_a_b,C5: term_a_b] :
( ( A1 = A5 )
& ( A22 = C5 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A5 @ B5 ) @ R )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B5 @ C5 ) @ S ) ) ) ) ).
% relcomp.simps
thf(fact_220_relcomp_Osimps,axiom,
! [A1: nat > nat,A22: nat,R: set_Pr7682762132356531903at_nat,S: set_Pr9093778441882193744at_nat] :
( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ A1 @ A22 ) @ ( relcom9054258243621454929at_nat @ R @ S ) )
= ( ? [A5: nat > nat,B5: nat > nat,C5: nat] :
( ( A1 = A5 )
& ( A22 = C5 )
& ( member1096249278104964872at_nat @ ( produc5770335208449155351at_nat @ A5 @ B5 ) @ R )
& ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ B5 @ C5 ) @ S ) ) ) ) ).
% relcomp.simps
thf(fact_221_relcomp_Osimps,axiom,
! [A1: nat > nat,A22: nat,R: set_Pr3254233319718257578_nat_a,S: set_Pr4934435412358123699_a_nat] :
( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ A1 @ A22 ) @ ( relcom1163327185954966926_a_nat @ R @ S ) )
= ( ? [A5: nat > nat,B5: a,C5: nat] :
( ( A1 = A5 )
& ( A22 = C5 )
& ( member7754445317155421323_nat_a @ ( produc3941937540969104678_nat_a @ A5 @ B5 ) @ R )
& ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ B5 @ C5 ) @ S ) ) ) ) ).
% relcomp.simps
thf(fact_222_relcomp_Osimps,axiom,
! [A1: nat > nat,A22: nat,R: set_Pr9093778441882193744at_nat,S: set_Pr1261947904930325089at_nat] :
( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ A1 @ A22 ) @ ( relcom1859588284133128290at_nat @ R @ S ) )
= ( ? [A5: nat > nat,B5: nat,C5: nat] :
( ( A1 = A5 )
& ( A22 = C5 )
& ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ A5 @ B5 ) @ R )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B5 @ C5 ) @ S ) ) ) ) ).
% relcomp.simps
thf(fact_223_relcomp_Osimps,axiom,
! [A1: a,A22: nat,R: set_Pr6841066879934740386at_nat,S: set_Pr9093778441882193744at_nat] :
( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ A1 @ A22 ) @ ( relcom8552328690651882190at_nat @ R @ S ) )
= ( ? [A5: a,B5: nat > nat,C5: nat] :
( ( A1 = A5 )
& ( A22 = C5 )
& ( member5463269792551873667at_nat @ ( produc7514922136058452582at_nat @ A5 @ B5 ) @ R )
& ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ B5 @ C5 ) @ S ) ) ) ) ).
% relcomp.simps
thf(fact_224_relcomp_Osimps,axiom,
! [A1: a,A22: nat,R: set_Product_prod_a_a,S: set_Pr4934435412358123699_a_nat] :
( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ A1 @ A22 ) @ ( relcomp_a_a_nat @ R @ S ) )
= ( ? [A5: a,B5: a,C5: nat] :
( ( A1 = A5 )
& ( A22 = C5 )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A5 @ B5 ) @ R )
& ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ B5 @ C5 ) @ S ) ) ) ) ).
% relcomp.simps
thf(fact_225_relcomp_Osimps,axiom,
! [A1: a,A22: nat,R: set_Pr4934435412358123699_a_nat,S: set_Pr1261947904930325089at_nat] :
( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ A1 @ A22 ) @ ( relcomp_a_nat_nat @ R @ S ) )
= ( ? [A5: a,B5: nat,C5: nat] :
( ( A1 = A5 )
& ( A22 = C5 )
& ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ A5 @ B5 ) @ R )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B5 @ C5 ) @ S ) ) ) ) ).
% relcomp.simps
thf(fact_226_relcomp_Ocases,axiom,
! [A1: term_a_b,A22: term_a_b,R: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A22 ) @ ( relcom370159955682700863rm_a_b @ R @ S ) )
=> ~ ! [B6: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ B6 ) @ R )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B6 @ A22 ) @ S ) ) ) ).
% relcomp.cases
thf(fact_227_relcomp_Ocases,axiom,
! [A1: nat > nat,A22: nat,R: set_Pr7682762132356531903at_nat,S: set_Pr9093778441882193744at_nat] :
( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ A1 @ A22 ) @ ( relcom9054258243621454929at_nat @ R @ S ) )
=> ~ ! [B6: nat > nat] :
( ( member1096249278104964872at_nat @ ( produc5770335208449155351at_nat @ A1 @ B6 ) @ R )
=> ~ ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ B6 @ A22 ) @ S ) ) ) ).
% relcomp.cases
thf(fact_228_relcomp_Ocases,axiom,
! [A1: nat > nat,A22: nat,R: set_Pr3254233319718257578_nat_a,S: set_Pr4934435412358123699_a_nat] :
( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ A1 @ A22 ) @ ( relcom1163327185954966926_a_nat @ R @ S ) )
=> ~ ! [B6: a] :
( ( member7754445317155421323_nat_a @ ( produc3941937540969104678_nat_a @ A1 @ B6 ) @ R )
=> ~ ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ B6 @ A22 ) @ S ) ) ) ).
% relcomp.cases
thf(fact_229_relcomp_Ocases,axiom,
! [A1: nat > nat,A22: nat,R: set_Pr9093778441882193744at_nat,S: set_Pr1261947904930325089at_nat] :
( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ A1 @ A22 ) @ ( relcom1859588284133128290at_nat @ R @ S ) )
=> ~ ! [B6: nat] :
( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ A1 @ B6 ) @ R )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B6 @ A22 ) @ S ) ) ) ).
% relcomp.cases
thf(fact_230_relcomp_Ocases,axiom,
! [A1: a,A22: nat,R: set_Pr6841066879934740386at_nat,S: set_Pr9093778441882193744at_nat] :
( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ A1 @ A22 ) @ ( relcom8552328690651882190at_nat @ R @ S ) )
=> ~ ! [B6: nat > nat] :
( ( member5463269792551873667at_nat @ ( produc7514922136058452582at_nat @ A1 @ B6 ) @ R )
=> ~ ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ B6 @ A22 ) @ S ) ) ) ).
% relcomp.cases
thf(fact_231_relcomp_Ocases,axiom,
! [A1: a,A22: nat,R: set_Product_prod_a_a,S: set_Pr4934435412358123699_a_nat] :
( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ A1 @ A22 ) @ ( relcomp_a_a_nat @ R @ S ) )
=> ~ ! [B6: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A1 @ B6 ) @ R )
=> ~ ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ B6 @ A22 ) @ S ) ) ) ).
% relcomp.cases
thf(fact_232_relcomp_Ocases,axiom,
! [A1: a,A22: nat,R: set_Pr4934435412358123699_a_nat,S: set_Pr1261947904930325089at_nat] :
( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ A1 @ A22 ) @ ( relcomp_a_nat_nat @ R @ S ) )
=> ~ ! [B6: nat] :
( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ A1 @ B6 ) @ R )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B6 @ A22 ) @ S ) ) ) ).
% relcomp.cases
thf(fact_233_relcomp__mono,axiom,
! [R6: set_Product_prod_a_a,R: set_Product_prod_a_a,S2: set_Pr4934435412358123699_a_nat,S: set_Pr4934435412358123699_a_nat] :
( ( ord_le746702958409616551od_a_a @ R6 @ R )
=> ( ( ord_le8666007276011122963_a_nat @ S2 @ S )
=> ( ord_le8666007276011122963_a_nat @ ( relcomp_a_a_nat @ R6 @ S2 ) @ ( relcomp_a_a_nat @ R @ S ) ) ) ) ).
% relcomp_mono
thf(fact_234_relcomp__mono,axiom,
! [R6: set_Pr3254233319718257578_nat_a,R: set_Pr3254233319718257578_nat_a,S2: set_Pr4934435412358123699_a_nat,S: set_Pr4934435412358123699_a_nat] :
( ( ord_le7986397757451867146_nat_a @ R6 @ R )
=> ( ( ord_le8666007276011122963_a_nat @ S2 @ S )
=> ( ord_le3678578370064672496at_nat @ ( relcom1163327185954966926_a_nat @ R6 @ S2 ) @ ( relcom1163327185954966926_a_nat @ R @ S ) ) ) ) ).
% relcomp_mono
thf(fact_235_relcomp__mono,axiom,
! [R6: set_Pr6841066879934740386at_nat,R: set_Pr6841066879934740386at_nat,S2: set_Pr9093778441882193744at_nat,S: set_Pr9093778441882193744at_nat] :
( ( ord_le2349859280813574146at_nat @ R6 @ R )
=> ( ( ord_le3678578370064672496at_nat @ S2 @ S )
=> ( ord_le8666007276011122963_a_nat @ ( relcom8552328690651882190at_nat @ R6 @ S2 ) @ ( relcom8552328690651882190at_nat @ R @ S ) ) ) ) ).
% relcomp_mono
thf(fact_236_relcomp__mono,axiom,
! [R6: set_Pr7682762132356531903at_nat,R: set_Pr7682762132356531903at_nat,S2: set_Pr9093778441882193744at_nat,S: set_Pr9093778441882193744at_nat] :
( ( ord_le7501158350158605919at_nat @ R6 @ R )
=> ( ( ord_le3678578370064672496at_nat @ S2 @ S )
=> ( ord_le3678578370064672496at_nat @ ( relcom9054258243621454929at_nat @ R6 @ S2 ) @ ( relcom9054258243621454929at_nat @ R @ S ) ) ) ) ).
% relcomp_mono
thf(fact_237_relcomp__mono,axiom,
! [R6: set_Pr4934435412358123699_a_nat,R: set_Pr4934435412358123699_a_nat,S2: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
( ( ord_le8666007276011122963_a_nat @ R6 @ R )
=> ( ( ord_le3146513528884898305at_nat @ S2 @ S )
=> ( ord_le8666007276011122963_a_nat @ ( relcomp_a_nat_nat @ R6 @ S2 ) @ ( relcomp_a_nat_nat @ R @ S ) ) ) ) ).
% relcomp_mono
thf(fact_238_relcomp__mono,axiom,
! [R6: set_Pr4386577575007340137rm_a_b,R: set_Pr4386577575007340137rm_a_b,S2: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ R6 @ R )
=> ( ( ord_le118470702582115849rm_a_b @ S2 @ S )
=> ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ R6 @ S2 ) @ ( relcom370159955682700863rm_a_b @ R @ S ) ) ) ) ).
% relcomp_mono
thf(fact_239_relcomp__mono,axiom,
! [R6: set_Pr9093778441882193744at_nat,R: set_Pr9093778441882193744at_nat,S2: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
( ( ord_le3678578370064672496at_nat @ R6 @ R )
=> ( ( ord_le3146513528884898305at_nat @ S2 @ S )
=> ( ord_le3678578370064672496at_nat @ ( relcom1859588284133128290at_nat @ R6 @ S2 ) @ ( relcom1859588284133128290at_nat @ R @ S ) ) ) ) ).
% relcomp_mono
thf(fact_240_Range__iff,axiom,
! [A2: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( member_term_a_b @ A2 @ ( range_8688746797885257759rm_a_b @ R ) )
= ( ? [Y5: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y5 @ A2 ) @ R ) ) ) ).
% Range_iff
thf(fact_241_Range__iff,axiom,
! [A2: nat,R: set_Pr9093778441882193744at_nat] :
( ( member_nat @ A2 @ ( range_nat_nat_nat @ R ) )
= ( ? [Y5: nat > nat] : ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ Y5 @ A2 ) @ R ) ) ) ).
% Range_iff
thf(fact_242_Range__iff,axiom,
! [A2: nat,R: set_Pr4934435412358123699_a_nat] :
( ( member_nat @ A2 @ ( range_a_nat @ R ) )
= ( ? [Y5: a] : ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ Y5 @ A2 ) @ R ) ) ) ).
% Range_iff
thf(fact_243_RangeE,axiom,
! [B2: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( member_term_a_b @ B2 @ ( range_8688746797885257759rm_a_b @ R ) )
=> ~ ! [A6: term_a_b] :
~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A6 @ B2 ) @ R ) ) ).
% RangeE
thf(fact_244_RangeE,axiom,
! [B2: nat,R: set_Pr9093778441882193744at_nat] :
( ( member_nat @ B2 @ ( range_nat_nat_nat @ R ) )
=> ~ ! [A6: nat > nat] :
~ ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ A6 @ B2 ) @ R ) ) ).
% RangeE
thf(fact_245_RangeE,axiom,
! [B2: nat,R: set_Pr4934435412358123699_a_nat] :
( ( member_nat @ B2 @ ( range_a_nat @ R ) )
=> ~ ! [A6: a] :
~ ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ A6 @ B2 ) @ R ) ) ).
% RangeE
thf(fact_246_Range_Ointros,axiom,
! [A2: term_a_b,B2: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ R )
=> ( member_term_a_b @ B2 @ ( range_8688746797885257759rm_a_b @ R ) ) ) ).
% Range.intros
thf(fact_247_Range_Ointros,axiom,
! [A2: nat > nat,B2: nat,R: set_Pr9093778441882193744at_nat] :
( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ A2 @ B2 ) @ R )
=> ( member_nat @ B2 @ ( range_nat_nat_nat @ R ) ) ) ).
% Range.intros
thf(fact_248_Range_Ointros,axiom,
! [A2: a,B2: nat,R: set_Pr4934435412358123699_a_nat] :
( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ A2 @ B2 ) @ R )
=> ( member_nat @ B2 @ ( range_a_nat @ R ) ) ) ).
% Range.intros
thf(fact_249_Range_Osimps,axiom,
! [A2: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( member_term_a_b @ A2 @ ( range_8688746797885257759rm_a_b @ R ) )
= ( ? [A5: term_a_b,B5: term_a_b] :
( ( A2 = B5 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A5 @ B5 ) @ R ) ) ) ) ).
% Range.simps
thf(fact_250_Range_Osimps,axiom,
! [A2: nat,R: set_Pr9093778441882193744at_nat] :
( ( member_nat @ A2 @ ( range_nat_nat_nat @ R ) )
= ( ? [A5: nat > nat,B5: nat] :
( ( A2 = B5 )
& ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ A5 @ B5 ) @ R ) ) ) ) ).
% Range.simps
thf(fact_251_Range_Osimps,axiom,
! [A2: nat,R: set_Pr4934435412358123699_a_nat] :
( ( member_nat @ A2 @ ( range_a_nat @ R ) )
= ( ? [A5: a,B5: nat] :
( ( A2 = B5 )
& ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ A5 @ B5 ) @ R ) ) ) ) ).
% Range.simps
thf(fact_252_Range_Ocases,axiom,
! [A2: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( member_term_a_b @ A2 @ ( range_8688746797885257759rm_a_b @ R ) )
=> ~ ! [A6: term_a_b] :
~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A6 @ A2 ) @ R ) ) ).
% Range.cases
thf(fact_253_Range_Ocases,axiom,
! [A2: nat,R: set_Pr9093778441882193744at_nat] :
( ( member_nat @ A2 @ ( range_nat_nat_nat @ R ) )
=> ~ ! [A6: nat > nat] :
~ ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ A6 @ A2 ) @ R ) ) ).
% Range.cases
thf(fact_254_Range_Ocases,axiom,
! [A2: nat,R: set_Pr4934435412358123699_a_nat] :
( ( member_nat @ A2 @ ( range_a_nat @ R ) )
=> ~ ! [A6: a] :
~ ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ A6 @ A2 ) @ R ) ) ).
% Range.cases
thf(fact_255_non__strict__into__strict,axiom,
! [NS: set_Product_prod_a_a,S5: set_Pr4934435412358123699_a_nat,S: a,T3: nat] :
( ( ord_le8666007276011122963_a_nat @ ( relcomp_a_a_nat @ NS @ S5 ) @ S5 )
=> ( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ S @ T3 ) @ ( relcomp_a_a_nat @ ( transitive_rtrancl_a @ NS ) @ S5 ) )
=> ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ S @ T3 ) @ S5 ) ) ) ).
% non_strict_into_strict
thf(fact_256_non__strict__into__strict,axiom,
! [NS: set_Pr4386577575007340137rm_a_b,S5: set_Pr4386577575007340137rm_a_b,S: term_a_b,T3: term_a_b] :
( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ NS @ S5 ) @ S5 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T3 ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ NS ) @ S5 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T3 ) @ S5 ) ) ) ).
% non_strict_into_strict
thf(fact_257_non__strict__into__strict,axiom,
! [NS: set_Pr7682762132356531903at_nat,S5: set_Pr9093778441882193744at_nat,S: nat > nat,T3: nat] :
( ( ord_le3678578370064672496at_nat @ ( relcom9054258243621454929at_nat @ NS @ S5 ) @ S5 )
=> ( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ S @ T3 ) @ ( relcom9054258243621454929at_nat @ ( transi3031656534272781564at_nat @ NS ) @ S5 ) )
=> ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ S @ T3 ) @ S5 ) ) ) ).
% non_strict_into_strict
thf(fact_258_comp__trancl,axiom,
! [R4: set_Pr4386577575007340137rm_a_b,S5: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ R4 @ S5 ) @ S5 )
=> ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ R4 @ ( transi7922773638565587891rm_a_b @ S5 ) ) @ ( transi7922773638565587891rm_a_b @ S5 ) ) ) ).
% comp_trancl
thf(fact_259_kleene__trancl__induct,axiom,
! [A: set_Pr4934435412358123699_a_nat,X6: set_Pr4934435412358123699_a_nat,B: set_Product_prod_a_a,C2: set_Pr1261947904930325089at_nat] :
( ( ord_le8666007276011122963_a_nat @ A @ X6 )
=> ( ( ord_le8666007276011122963_a_nat @ ( relcomp_a_a_nat @ B @ X6 ) @ X6 )
=> ( ( ord_le8666007276011122963_a_nat @ ( relcomp_a_nat_nat @ X6 @ C2 ) @ X6 )
=> ( ord_le8666007276011122963_a_nat @ ( relcomp_a_a_nat @ ( transitive_trancl_a @ B ) @ ( relcomp_a_nat_nat @ A @ ( transi6264000038957366511cl_nat @ C2 ) ) ) @ X6 ) ) ) ) ).
% kleene_trancl_induct
thf(fact_260_kleene__trancl__induct,axiom,
! [A: set_Pr4386577575007340137rm_a_b,X6: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b,C2: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A @ X6 )
=> ( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ B @ X6 ) @ X6 )
=> ( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ X6 @ C2 ) @ X6 )
=> ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ ( transi7922773638565587891rm_a_b @ B ) @ ( relcom370159955682700863rm_a_b @ A @ ( transi7922773638565587891rm_a_b @ C2 ) ) ) @ X6 ) ) ) ) ).
% kleene_trancl_induct
thf(fact_261_kleene__trancl__induct,axiom,
! [A: set_Pr9093778441882193744at_nat,X6: set_Pr9093778441882193744at_nat,B: set_Pr7682762132356531903at_nat,C2: set_Pr1261947904930325089at_nat] :
( ( ord_le3678578370064672496at_nat @ A @ X6 )
=> ( ( ord_le3678578370064672496at_nat @ ( relcom9054258243621454929at_nat @ B @ X6 ) @ X6 )
=> ( ( ord_le3678578370064672496at_nat @ ( relcom1859588284133128290at_nat @ X6 @ C2 ) @ X6 )
=> ( ord_le3678578370064672496at_nat @ ( relcom9054258243621454929at_nat @ ( transi2259355090335363038at_nat @ B ) @ ( relcom1859588284133128290at_nat @ A @ ( transi6264000038957366511cl_nat @ C2 ) ) ) @ X6 ) ) ) ) ).
% kleene_trancl_induct
thf(fact_262_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_263_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 ) )
=> ( ! [A6: term_a_b,B6: term_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ Ax @ Ay ) @ ( produc7020197800436672577rm_a_b @ A6 @ B6 ) ) @ R )
=> ( P @ A6 @ B6 ) )
=> ( ! [A6: term_a_b,B6: 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 @ A6 @ B6 ) ) @ ( transi9211502839322181930rm_a_b @ R ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ A6 @ B6 ) @ ( produc7020197800436672577rm_a_b @ Aa @ Ba ) ) @ R )
=> ( ( P @ A6 @ B6 )
=> ( P @ Aa @ Ba ) ) ) )
=> ( P @ Bx @ By ) ) ) ) ).
% trancl_induct2
thf(fact_264_trancl__induct2,axiom,
! [Ax: nat > nat,Ay: nat,Bx: nat > nat,By: nat,R: set_Pr7047708467701517959at_nat,P: ( nat > nat ) > nat > $o] :
( ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ ( produc72220940542539688at_nat @ Ax @ Ay ) @ ( produc72220940542539688at_nat @ Bx @ By ) ) @ ( transi3146596849628481425at_nat @ R ) )
=> ( ! [A6: nat > nat,B6: nat] :
( ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ ( produc72220940542539688at_nat @ Ax @ Ay ) @ ( produc72220940542539688at_nat @ A6 @ B6 ) ) @ R )
=> ( P @ A6 @ B6 ) )
=> ( ! [A6: nat > nat,B6: nat,Aa: nat > nat,Ba: nat] :
( ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ ( produc72220940542539688at_nat @ Ax @ Ay ) @ ( produc72220940542539688at_nat @ A6 @ B6 ) ) @ ( transi3146596849628481425at_nat @ R ) )
=> ( ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ ( produc72220940542539688at_nat @ A6 @ B6 ) @ ( produc72220940542539688at_nat @ Aa @ Ba ) ) @ R )
=> ( ( P @ A6 @ B6 )
=> ( P @ Aa @ Ba ) ) ) )
=> ( P @ Bx @ By ) ) ) ) ).
% trancl_induct2
thf(fact_265_trancl__induct2,axiom,
! [Ax: a,Ay: nat,Bx: a,By: nat,R: set_Pr1811044260758604347_a_nat,P: a > nat > $o] :
( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ ( product_Pair_a_nat @ Ax @ Ay ) @ ( product_Pair_a_nat @ Bx @ By ) ) @ ( transi6805586507763294364_a_nat @ R ) )
=> ( ! [A6: a,B6: nat] :
( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ ( product_Pair_a_nat @ Ax @ Ay ) @ ( product_Pair_a_nat @ A6 @ B6 ) ) @ R )
=> ( P @ A6 @ B6 ) )
=> ( ! [A6: a,B6: nat,Aa: a,Ba: nat] :
( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ ( product_Pair_a_nat @ Ax @ Ay ) @ ( product_Pair_a_nat @ A6 @ B6 ) ) @ ( transi6805586507763294364_a_nat @ R ) )
=> ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ ( product_Pair_a_nat @ A6 @ B6 ) @ ( product_Pair_a_nat @ Aa @ Ba ) ) @ R )
=> ( ( P @ A6 @ B6 )
=> ( P @ Aa @ Ba ) ) ) )
=> ( P @ Bx @ By ) ) ) ) ).
% trancl_induct2
thf(fact_266_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 )
=> ( ! [A6: term_a_b,B6: 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 @ A6 @ B6 ) ) @ ( transi2615809358984392588rm_a_b @ R ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ A6 @ B6 ) @ ( produc7020197800436672577rm_a_b @ Aa @ Ba ) ) @ R )
=> ( ( P @ A6 @ B6 )
=> ( P @ Aa @ Ba ) ) ) )
=> ( P @ Bx @ By ) ) ) ) ).
% rtrancl_induct2
thf(fact_267_rtrancl__induct2,axiom,
! [Ax: nat > nat,Ay: nat,Bx: nat > nat,By: nat,R: set_Pr7047708467701517959at_nat,P: ( nat > nat ) > nat > $o] :
( ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ ( produc72220940542539688at_nat @ Ax @ Ay ) @ ( produc72220940542539688at_nat @ Bx @ By ) ) @ ( transi3422749762518297843at_nat @ R ) )
=> ( ( P @ Ax @ Ay )
=> ( ! [A6: nat > nat,B6: nat,Aa: nat > nat,Ba: nat] :
( ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ ( produc72220940542539688at_nat @ Ax @ Ay ) @ ( produc72220940542539688at_nat @ A6 @ B6 ) ) @ ( transi3422749762518297843at_nat @ R ) )
=> ( ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ ( produc72220940542539688at_nat @ A6 @ B6 ) @ ( produc72220940542539688at_nat @ Aa @ Ba ) ) @ R )
=> ( ( P @ A6 @ B6 )
=> ( P @ Aa @ Ba ) ) ) )
=> ( P @ Bx @ By ) ) ) ) ).
% rtrancl_induct2
thf(fact_268_rtrancl__induct2,axiom,
! [Ax: a,Ay: nat,Bx: a,By: nat,R: set_Pr1811044260758604347_a_nat,P: a > nat > $o] :
( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ ( product_Pair_a_nat @ Ax @ Ay ) @ ( product_Pair_a_nat @ Bx @ By ) ) @ ( transi2726145917338391738_a_nat @ R ) )
=> ( ( P @ Ax @ Ay )
=> ( ! [A6: a,B6: nat,Aa: a,Ba: nat] :
( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ ( product_Pair_a_nat @ Ax @ Ay ) @ ( product_Pair_a_nat @ A6 @ B6 ) ) @ ( transi2726145917338391738_a_nat @ R ) )
=> ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ ( product_Pair_a_nat @ A6 @ B6 ) @ ( product_Pair_a_nat @ Aa @ Ba ) ) @ R )
=> ( ( P @ A6 @ B6 )
=> ( P @ Aa @ Ba ) ) ) )
=> ( P @ Bx @ By ) ) ) ) ).
% rtrancl_induct2
thf(fact_269_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 ) )
=> ~ ! [A6: term_a_b,B6: term_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ Xa @ Xb ) @ ( produc7020197800436672577rm_a_b @ A6 @ B6 ) ) @ R )
=> ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ A6 @ B6 ) @ ( produc7020197800436672577rm_a_b @ Za @ Zb ) ) @ ( transi2615809358984392588rm_a_b @ R ) ) ) ) ) ).
% converse_rtranclE2
thf(fact_270_converse__rtranclE2,axiom,
! [Xa: nat > nat,Xb: nat,Za: nat > nat,Zb: nat,R: set_Pr7047708467701517959at_nat] :
( ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ ( produc72220940542539688at_nat @ Xa @ Xb ) @ ( produc72220940542539688at_nat @ Za @ Zb ) ) @ ( transi3422749762518297843at_nat @ R ) )
=> ( ( ( produc72220940542539688at_nat @ Xa @ Xb )
!= ( produc72220940542539688at_nat @ Za @ Zb ) )
=> ~ ! [A6: nat > nat,B6: nat] :
( ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ ( produc72220940542539688at_nat @ Xa @ Xb ) @ ( produc72220940542539688at_nat @ A6 @ B6 ) ) @ R )
=> ~ ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ ( produc72220940542539688at_nat @ A6 @ B6 ) @ ( produc72220940542539688at_nat @ Za @ Zb ) ) @ ( transi3422749762518297843at_nat @ R ) ) ) ) ) ).
% converse_rtranclE2
thf(fact_271_converse__rtranclE2,axiom,
! [Xa: a,Xb: nat,Za: a,Zb: nat,R: set_Pr1811044260758604347_a_nat] :
( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ ( product_Pair_a_nat @ Xa @ Xb ) @ ( product_Pair_a_nat @ Za @ Zb ) ) @ ( transi2726145917338391738_a_nat @ R ) )
=> ( ( ( product_Pair_a_nat @ Xa @ Xb )
!= ( product_Pair_a_nat @ Za @ Zb ) )
=> ~ ! [A6: a,B6: nat] :
( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ ( product_Pair_a_nat @ Xa @ Xb ) @ ( product_Pair_a_nat @ A6 @ B6 ) ) @ R )
=> ~ ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ ( product_Pair_a_nat @ A6 @ B6 ) @ ( product_Pair_a_nat @ Za @ Zb ) ) @ ( transi2726145917338391738_a_nat @ R ) ) ) ) ) ).
% converse_rtranclE2
thf(fact_272_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 )
=> ( ! [A6: term_a_b,B6: term_a_b,Aa: term_a_b,Ba: term_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ A6 @ B6 ) @ ( 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 @ A6 @ B6 ) ) ) )
=> ( P @ Ax @ Ay ) ) ) ) ).
% converse_rtrancl_induct2
thf(fact_273_converse__rtrancl__induct2,axiom,
! [Ax: nat > nat,Ay: nat,Bx: nat > nat,By: nat,R: set_Pr7047708467701517959at_nat,P: ( nat > nat ) > nat > $o] :
( ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ ( produc72220940542539688at_nat @ Ax @ Ay ) @ ( produc72220940542539688at_nat @ Bx @ By ) ) @ ( transi3422749762518297843at_nat @ R ) )
=> ( ( P @ Bx @ By )
=> ( ! [A6: nat > nat,B6: nat,Aa: nat > nat,Ba: nat] :
( ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ ( produc72220940542539688at_nat @ A6 @ B6 ) @ ( produc72220940542539688at_nat @ Aa @ Ba ) ) @ R )
=> ( ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ ( produc72220940542539688at_nat @ Aa @ Ba ) @ ( produc72220940542539688at_nat @ Bx @ By ) ) @ ( transi3422749762518297843at_nat @ R ) )
=> ( ( P @ Aa @ Ba )
=> ( P @ A6 @ B6 ) ) ) )
=> ( P @ Ax @ Ay ) ) ) ) ).
% converse_rtrancl_induct2
thf(fact_274_converse__rtrancl__induct2,axiom,
! [Ax: a,Ay: nat,Bx: a,By: nat,R: set_Pr1811044260758604347_a_nat,P: a > nat > $o] :
( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ ( product_Pair_a_nat @ Ax @ Ay ) @ ( product_Pair_a_nat @ Bx @ By ) ) @ ( transi2726145917338391738_a_nat @ R ) )
=> ( ( P @ Bx @ By )
=> ( ! [A6: a,B6: nat,Aa: a,Ba: nat] :
( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ ( product_Pair_a_nat @ A6 @ B6 ) @ ( product_Pair_a_nat @ Aa @ Ba ) ) @ R )
=> ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ ( product_Pair_a_nat @ Aa @ Ba ) @ ( product_Pair_a_nat @ Bx @ By ) ) @ ( transi2726145917338391738_a_nat @ R ) )
=> ( ( P @ Aa @ Ba )
=> ( P @ A6 @ B6 ) ) ) )
=> ( P @ Ax @ Ay ) ) ) ) ).
% converse_rtrancl_induct2
thf(fact_275_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_276_rtrancl_Ocases,axiom,
! [A1: term_a_b,A22: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A22 ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ( ( A22 != A1 )
=> ~ ! [B6: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ B6 ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B6 @ A22 ) @ R ) ) ) ) ).
% rtrancl.cases
thf(fact_277_rtrancl_Osimps,axiom,
! [A1: term_a_b,A22: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A22 ) @ ( transi7742714808557438673rm_a_b @ R ) )
= ( ? [A5: term_a_b] :
( ( A1 = A5 )
& ( A22 = A5 ) )
| ? [A5: term_a_b,B5: term_a_b,C5: term_a_b] :
( ( A1 = A5 )
& ( A22 = C5 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A5 @ B5 ) @ ( transi7742714808557438673rm_a_b @ R ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B5 @ C5 ) @ R ) ) ) ) ).
% rtrancl.simps
thf(fact_278_rtrancl_Ortrancl__refl,axiom,
! [A2: term_a_b,R: set_Pr4386577575007340137rm_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ A2 ) @ ( transi7742714808557438673rm_a_b @ R ) ) ).
% rtrancl.rtrancl_refl
thf(fact_279_rtrancl_Ortrancl__into__rtrancl,axiom,
! [A2: term_a_b,B2: term_a_b,R: set_Pr4386577575007340137rm_a_b,C: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ C ) @ R )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ C ) @ ( transi7742714808557438673rm_a_b @ R ) ) ) ) ).
% rtrancl.rtrancl_into_rtrancl
thf(fact_280_rtranclE,axiom,
! [A2: term_a_b,B2: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ( ( A2 != B2 )
=> ~ ! [Y: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ Y ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ B2 ) @ R ) ) ) ) ).
% rtranclE
thf(fact_281_rtrancl__trans,axiom,
! [X3: term_a_b,Y3: term_a_b,R: set_Pr4386577575007340137rm_a_b,Z2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ Z2 ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Z2 ) @ ( transi7742714808557438673rm_a_b @ R ) ) ) ) ).
% rtrancl_trans
thf(fact_282_rtrancl__induct,axiom,
! [A2: term_a_b,B2: term_a_b,R: set_Pr4386577575007340137rm_a_b,P: term_a_b > $o] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ( ( P @ A2 )
=> ( ! [Y: term_a_b,Z3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ Y ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ Z3 ) @ R )
=> ( ( P @ Y )
=> ( P @ Z3 ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% rtrancl_induct
thf(fact_283_converse__rtranclE,axiom,
! [X3: term_a_b,Z2: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Z2 ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ( ( X3 != Z2 )
=> ~ ! [Y: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y ) @ R )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ Z2 ) @ ( transi7742714808557438673rm_a_b @ R ) ) ) ) ) ).
% converse_rtranclE
thf(fact_284_converse__rtrancl__induct,axiom,
! [A2: term_a_b,B2: term_a_b,R: set_Pr4386577575007340137rm_a_b,P: term_a_b > $o] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ( ( P @ B2 )
=> ( ! [Y: term_a_b,Z3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ Z3 ) @ R )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Z3 @ B2 ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ( ( P @ Z3 )
=> ( P @ Y ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% converse_rtrancl_induct
thf(fact_285_converse__rtrancl__into__rtrancl,axiom,
! [A2: term_a_b,B2: term_a_b,R: set_Pr4386577575007340137rm_a_b,C: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ R )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ C ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ C ) @ ( transi7742714808557438673rm_a_b @ R ) ) ) ) ).
% converse_rtrancl_into_rtrancl
thf(fact_286_relcomp3__I,axiom,
! [T3: nat,U: nat > nat,A: set_Pr6370437063884598352at_nat,S: nat > nat,B: set_Pr9093778441882193744at_nat,V: nat] :
( ( member8336108448496249625at_nat @ ( produc7839516862119294504at_nat @ T3 @ U ) @ A )
=> ( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ S @ T3 ) @ B )
=> ( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ U @ V ) @ B )
=> ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ S @ V ) @ ( relcom1859588284133128290at_nat @ B @ ( relcom6405311860805859042at_nat @ A @ B ) ) ) ) ) ) ).
% relcomp3_I
thf(fact_287_relcomp3__I,axiom,
! [T3: nat,U: a,A: set_Pr4193341848836149977_nat_a,S: a,B: set_Pr4934435412358123699_a_nat,V: nat] :
( ( member8962352052110095674_nat_a @ ( product_Pair_nat_a @ T3 @ U ) @ A )
=> ( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ S @ T3 ) @ B )
=> ( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ U @ V ) @ B )
=> ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ S @ V ) @ ( relcomp_a_nat_nat @ B @ ( relcomp_nat_a_nat @ A @ B ) ) ) ) ) ) ).
% relcomp3_I
thf(fact_288_relcomp3__I,axiom,
! [T3: term_a_b,U: term_a_b,A: set_Pr4386577575007340137rm_a_b,S: term_a_b,B: set_Pr4386577575007340137rm_a_b,V: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T3 @ U ) @ A )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T3 ) @ B )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ U @ V ) @ B )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ V ) @ ( relcom370159955682700863rm_a_b @ B @ ( relcom370159955682700863rm_a_b @ A @ B ) ) ) ) ) ) ).
% relcomp3_I
thf(fact_289_relcomp3__I,axiom,
! [T3: nat > nat,U: nat,A: set_Pr9093778441882193744at_nat,S: nat,B: set_Pr6370437063884598352at_nat,V: nat > nat] :
( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ T3 @ U ) @ A )
=> ( ( member8336108448496249625at_nat @ ( produc7839516862119294504at_nat @ S @ T3 ) @ B )
=> ( ( member8336108448496249625at_nat @ ( produc7839516862119294504at_nat @ U @ V ) @ B )
=> ( member8336108448496249625at_nat @ ( produc7839516862119294504at_nat @ S @ V ) @ ( relcom896424912281157969at_nat @ B @ ( relcom7598182128343433937at_nat @ A @ B ) ) ) ) ) ) ).
% relcomp3_I
thf(fact_290_relcomp3__I,axiom,
! [T3: a,U: nat,A: set_Pr4934435412358123699_a_nat,S: nat,B: set_Pr4193341848836149977_nat_a,V: a] :
( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ T3 @ U ) @ A )
=> ( ( member8962352052110095674_nat_a @ ( product_Pair_nat_a @ S @ T3 ) @ B )
=> ( ( member8962352052110095674_nat_a @ ( product_Pair_nat_a @ U @ V ) @ B )
=> ( member8962352052110095674_nat_a @ ( product_Pair_nat_a @ S @ V ) @ ( relcomp_nat_a_a @ B @ ( relcomp_a_nat_a @ A @ B ) ) ) ) ) ) ).
% relcomp3_I
thf(fact_291_trancl_Ocases,axiom,
! [A1: term_a_b,A22: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A22 ) @ ( transi7922773638565587891rm_a_b @ R ) )
=> ( ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A22 ) @ R )
=> ~ ! [B6: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ B6 ) @ ( transi7922773638565587891rm_a_b @ R ) )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B6 @ A22 ) @ R ) ) ) ) ).
% trancl.cases
thf(fact_292_trancl_Osimps,axiom,
! [A1: term_a_b,A22: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A22 ) @ ( transi7922773638565587891rm_a_b @ R ) )
= ( ? [A5: term_a_b,B5: term_a_b] :
( ( A1 = A5 )
& ( A22 = B5 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A5 @ B5 ) @ R ) )
| ? [A5: term_a_b,B5: term_a_b,C5: term_a_b] :
( ( A1 = A5 )
& ( A22 = C5 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A5 @ B5 ) @ ( transi7922773638565587891rm_a_b @ R ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B5 @ C5 ) @ R ) ) ) ) ).
% trancl.simps
thf(fact_293_trancl_Or__into__trancl,axiom,
! [A2: term_a_b,B2: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ R )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( transi7922773638565587891rm_a_b @ R ) ) ) ).
% trancl.r_into_trancl
thf(fact_294_trancl__map,axiom,
! [R: set_Pr4386577575007340137rm_a_b,F2: term_a_b > term_a_b,S: set_Pr4386577575007340137rm_a_b,X3: term_a_b,Y3: term_a_b] :
( ! [X: term_a_b,Y: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ R )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F2 @ X ) @ ( F2 @ Y ) ) @ S ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ ( transi7922773638565587891rm_a_b @ R ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) @ ( transi7922773638565587891rm_a_b @ S ) ) ) ) ).
% trancl_map
thf(fact_295_tranclE,axiom,
! [A2: term_a_b,B2: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( transi7922773638565587891rm_a_b @ R ) )
=> ( ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ R )
=> ~ ! [C6: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ C6 ) @ ( transi7922773638565587891rm_a_b @ R ) )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ C6 @ B2 ) @ R ) ) ) ) ).
% tranclE
thf(fact_296_trancl__trans,axiom,
! [X3: term_a_b,Y3: term_a_b,R: set_Pr4386577575007340137rm_a_b,Z2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ ( transi7922773638565587891rm_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ Z2 ) @ ( transi7922773638565587891rm_a_b @ R ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Z2 ) @ ( transi7922773638565587891rm_a_b @ R ) ) ) ) ).
% trancl_trans
thf(fact_297_trancl__induct,axiom,
! [A2: term_a_b,B2: term_a_b,R: set_Pr4386577575007340137rm_a_b,P: term_a_b > $o] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( transi7922773638565587891rm_a_b @ R ) )
=> ( ! [Y: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ Y ) @ R )
=> ( P @ Y ) )
=> ( ! [Y: term_a_b,Z3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ Y ) @ ( transi7922773638565587891rm_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ Z3 ) @ R )
=> ( ( P @ Y )
=> ( P @ Z3 ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% trancl_induct
thf(fact_298_r__r__into__trancl,axiom,
! [A2: term_a_b,B2: term_a_b,R4: set_Pr4386577575007340137rm_a_b,C: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ R4 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ C ) @ R4 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ C ) @ ( transi7922773638565587891rm_a_b @ R4 ) ) ) ) ).
% r_r_into_trancl
thf(fact_299_converse__tranclE,axiom,
! [X3: term_a_b,Z2: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Z2 ) @ ( transi7922773638565587891rm_a_b @ R ) )
=> ( ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Z2 ) @ R )
=> ~ ! [Y: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y ) @ R )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ Z2 ) @ ( transi7922773638565587891rm_a_b @ R ) ) ) ) ) ).
% converse_tranclE
thf(fact_300_irrefl__trancl__rD,axiom,
! [R: set_Pr4386577575007340137rm_a_b,X3: term_a_b,Y3: term_a_b] :
( ! [X: term_a_b] :
~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ X ) @ ( transi7922773638565587891rm_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ R )
=> ( X3 != Y3 ) ) ) ).
% irrefl_trancl_rD
thf(fact_301_Transitive__Closure_Otrancl__into__trancl,axiom,
! [A2: term_a_b,B2: term_a_b,R: set_Pr4386577575007340137rm_a_b,C: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( transi7922773638565587891rm_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ C ) @ R )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ C ) @ ( transi7922773638565587891rm_a_b @ R ) ) ) ) ).
% Transitive_Closure.trancl_into_trancl
thf(fact_302_trancl__into__trancl2,axiom,
! [A2: term_a_b,B2: term_a_b,R: set_Pr4386577575007340137rm_a_b,C: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ R )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ C ) @ ( transi7922773638565587891rm_a_b @ R ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ C ) @ ( transi7922773638565587891rm_a_b @ R ) ) ) ) ).
% trancl_into_trancl2
thf(fact_303_trancl__trans__induct,axiom,
! [X3: term_a_b,Y3: term_a_b,R: set_Pr4386577575007340137rm_a_b,P: term_a_b > term_a_b > $o] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ ( transi7922773638565587891rm_a_b @ R ) )
=> ( ! [X: term_a_b,Y: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ R )
=> ( P @ X @ Y ) )
=> ( ! [X: term_a_b,Y: term_a_b,Z3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7922773638565587891rm_a_b @ R ) )
=> ( ( P @ X @ Y )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ Z3 ) @ ( transi7922773638565587891rm_a_b @ R ) )
=> ( ( P @ Y @ Z3 )
=> ( P @ X @ Z3 ) ) ) ) )
=> ( P @ X3 @ Y3 ) ) ) ) ).
% trancl_trans_induct
thf(fact_304_converse__trancl__induct,axiom,
! [A2: term_a_b,B2: term_a_b,R: set_Pr4386577575007340137rm_a_b,P: term_a_b > $o] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( transi7922773638565587891rm_a_b @ R ) )
=> ( ! [Y: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ B2 ) @ R )
=> ( P @ Y ) )
=> ( ! [Y: term_a_b,Z3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ Z3 ) @ R )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Z3 @ B2 ) @ ( transi7922773638565587891rm_a_b @ R ) )
=> ( ( P @ Z3 )
=> ( P @ Y ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% converse_trancl_induct
thf(fact_305_rtrancl__mono,axiom,
! [R: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ R @ S )
=> ( ord_le118470702582115849rm_a_b @ ( transi7742714808557438673rm_a_b @ R ) @ ( transi7742714808557438673rm_a_b @ S ) ) ) ).
% rtrancl_mono
thf(fact_306_rtrancl__subset,axiom,
! [R4: set_Pr4386577575007340137rm_a_b,S5: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ R4 @ S5 )
=> ( ( ord_le118470702582115849rm_a_b @ S5 @ ( transi7742714808557438673rm_a_b @ R4 ) )
=> ( ( transi7742714808557438673rm_a_b @ S5 )
= ( transi7742714808557438673rm_a_b @ R4 ) ) ) ) ).
% rtrancl_subset
thf(fact_307_rtrancl__subset__rtrancl,axiom,
! [R: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ R @ ( transi7742714808557438673rm_a_b @ S ) )
=> ( ord_le118470702582115849rm_a_b @ ( transi7742714808557438673rm_a_b @ R ) @ ( transi7742714808557438673rm_a_b @ S ) ) ) ).
% rtrancl_subset_rtrancl
thf(fact_308_O__mono1,axiom,
! [R4: set_Pr4934435412358123699_a_nat,R7: set_Pr4934435412358123699_a_nat,S5: set_Product_prod_a_a] :
( ( ord_le8666007276011122963_a_nat @ R4 @ R7 )
=> ( ord_le8666007276011122963_a_nat @ ( relcomp_a_a_nat @ S5 @ R4 ) @ ( relcomp_a_a_nat @ S5 @ R7 ) ) ) ).
% O_mono1
thf(fact_309_O__mono1,axiom,
! [R4: set_Pr4934435412358123699_a_nat,R7: set_Pr4934435412358123699_a_nat,S5: set_Pr3254233319718257578_nat_a] :
( ( ord_le8666007276011122963_a_nat @ R4 @ R7 )
=> ( ord_le3678578370064672496at_nat @ ( relcom1163327185954966926_a_nat @ S5 @ R4 ) @ ( relcom1163327185954966926_a_nat @ S5 @ R7 ) ) ) ).
% O_mono1
thf(fact_310_O__mono1,axiom,
! [R4: set_Pr4386577575007340137rm_a_b,R7: set_Pr4386577575007340137rm_a_b,S5: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ R4 @ R7 )
=> ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ S5 @ R4 ) @ ( relcom370159955682700863rm_a_b @ S5 @ R7 ) ) ) ).
% O_mono1
thf(fact_311_O__mono1,axiom,
! [R4: set_Pr9093778441882193744at_nat,R7: set_Pr9093778441882193744at_nat,S5: set_Pr6841066879934740386at_nat] :
( ( ord_le3678578370064672496at_nat @ R4 @ R7 )
=> ( ord_le8666007276011122963_a_nat @ ( relcom8552328690651882190at_nat @ S5 @ R4 ) @ ( relcom8552328690651882190at_nat @ S5 @ R7 ) ) ) ).
% O_mono1
thf(fact_312_O__mono1,axiom,
! [R4: set_Pr9093778441882193744at_nat,R7: set_Pr9093778441882193744at_nat,S5: set_Pr7682762132356531903at_nat] :
( ( ord_le3678578370064672496at_nat @ R4 @ R7 )
=> ( ord_le3678578370064672496at_nat @ ( relcom9054258243621454929at_nat @ S5 @ R4 ) @ ( relcom9054258243621454929at_nat @ S5 @ R7 ) ) ) ).
% O_mono1
thf(fact_313_O__mono2,axiom,
! [R4: set_Pr4934435412358123699_a_nat,R7: set_Pr4934435412358123699_a_nat,T4: set_Pr1261947904930325089at_nat] :
( ( ord_le8666007276011122963_a_nat @ R4 @ R7 )
=> ( ord_le8666007276011122963_a_nat @ ( relcomp_a_nat_nat @ R4 @ T4 ) @ ( relcomp_a_nat_nat @ R7 @ T4 ) ) ) ).
% O_mono2
thf(fact_314_O__mono2,axiom,
! [R4: set_Pr4386577575007340137rm_a_b,R7: set_Pr4386577575007340137rm_a_b,T4: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ R4 @ R7 )
=> ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ R4 @ T4 ) @ ( relcom370159955682700863rm_a_b @ R7 @ T4 ) ) ) ).
% O_mono2
thf(fact_315_O__mono2,axiom,
! [R4: set_Pr9093778441882193744at_nat,R7: set_Pr9093778441882193744at_nat,T4: set_Pr1261947904930325089at_nat] :
( ( ord_le3678578370064672496at_nat @ R4 @ R7 )
=> ( ord_le3678578370064672496at_nat @ ( relcom1859588284133128290at_nat @ R4 @ T4 ) @ ( relcom1859588284133128290at_nat @ R7 @ T4 ) ) ) ).
% O_mono2
thf(fact_316_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_317_trancl__mono__set,axiom,
! [R: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ R @ S )
=> ( ord_le118470702582115849rm_a_b @ ( transi7922773638565587891rm_a_b @ R ) @ ( transi7922773638565587891rm_a_b @ S ) ) ) ).
% trancl_mono_set
thf(fact_318_steps__map,axiom,
! [P: term_a_b > $o,Q: set_Pr4386577575007340137rm_a_b > $o,F2: term_a_b > term_a_b,G2: set_Pr4386577575007340137rm_a_b > set_Pr4386577575007340137rm_a_b,T3: term_a_b,R4: set_Pr4386577575007340137rm_a_b,S5: set_Pr4386577575007340137rm_a_b,U: term_a_b] :
( ! [T2: term_a_b,U2: term_a_b,R8: set_Pr4386577575007340137rm_a_b] :
( ( P @ T2 )
=> ( ( Q @ R8 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T2 @ U2 ) @ R8 )
=> ( ( P @ U2 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F2 @ T2 ) @ ( F2 @ U2 ) ) @ ( G2 @ R8 ) ) ) ) ) )
=> ( ( P @ T3 )
=> ( ( Q @ R4 )
=> ( ( Q @ S5 )
=> ( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T3 @ U ) @ ( transi7742714808557438673rm_a_b @ R4 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F2 @ T3 ) @ ( F2 @ U ) ) @ ( transi7742714808557438673rm_a_b @ ( G2 @ R4 ) ) ) )
& ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T3 @ U ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ R4 ) @ ( relcom370159955682700863rm_a_b @ S5 @ ( transi7742714808557438673rm_a_b @ R4 ) ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F2 @ T3 ) @ ( F2 @ U ) ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( G2 @ R4 ) ) @ ( relcom370159955682700863rm_a_b @ ( G2 @ S5 ) @ ( transi7742714808557438673rm_a_b @ ( G2 @ R4 ) ) ) ) ) ) ) ) ) ) ) ).
% steps_map
thf(fact_319_trancl__rtrancl__trancl,axiom,
! [A2: term_a_b,B2: term_a_b,R: set_Pr4386577575007340137rm_a_b,C: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( transi7922773638565587891rm_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ C ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ C ) @ ( transi7922773638565587891rm_a_b @ R ) ) ) ) ).
% trancl_rtrancl_trancl
thf(fact_320_rtrancl__trancl__trancl,axiom,
! [X3: term_a_b,Y3: term_a_b,R: set_Pr4386577575007340137rm_a_b,Z2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ Z2 ) @ ( transi7922773638565587891rm_a_b @ R ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Z2 ) @ ( transi7922773638565587891rm_a_b @ R ) ) ) ) ).
% rtrancl_trancl_trancl
thf(fact_321_rtrancl__into__trancl2,axiom,
! [A2: term_a_b,B2: term_a_b,R: set_Pr4386577575007340137rm_a_b,C: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ R )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ C ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ C ) @ ( transi7922773638565587891rm_a_b @ R ) ) ) ) ).
% rtrancl_into_trancl2
thf(fact_322_rtrancl__into__trancl1,axiom,
! [A2: term_a_b,B2: term_a_b,R: set_Pr4386577575007340137rm_a_b,C: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ C ) @ R )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ C ) @ ( transi7922773638565587891rm_a_b @ R ) ) ) ) ).
% rtrancl_into_trancl1
thf(fact_323_rtrancl__eq__or__trancl,axiom,
! [X3: term_a_b,Y3: term_a_b,R4: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ ( transi7742714808557438673rm_a_b @ R4 ) )
= ( ( X3 = Y3 )
| ( ( X3 != Y3 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ ( transi7922773638565587891rm_a_b @ R4 ) ) ) ) ) ).
% rtrancl_eq_or_trancl
thf(fact_324_trancl__into__rtrancl,axiom,
! [A2: term_a_b,B2: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( transi7922773638565587891rm_a_b @ R ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( transi7742714808557438673rm_a_b @ R ) ) ) ).
% trancl_into_rtrancl
thf(fact_325_tranclD2,axiom,
! [X3: term_a_b,Y3: term_a_b,R4: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ ( transi7922773638565587891rm_a_b @ R4 ) )
=> ? [Z3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Z3 ) @ ( transi7742714808557438673rm_a_b @ R4 ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Z3 @ Y3 ) @ R4 ) ) ) ).
% tranclD2
thf(fact_326_rtranclD,axiom,
! [A2: term_a_b,B2: term_a_b,R4: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( transi7742714808557438673rm_a_b @ R4 ) )
=> ( ( A2 = B2 )
| ( ( A2 != B2 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( transi7922773638565587891rm_a_b @ R4 ) ) ) ) ) ).
% rtranclD
thf(fact_327_tranclD,axiom,
! [X3: term_a_b,Y3: term_a_b,R4: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ ( transi7922773638565587891rm_a_b @ R4 ) )
=> ? [Z3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Z3 ) @ R4 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Z3 @ Y3 ) @ ( transi7742714808557438673rm_a_b @ R4 ) ) ) ) ).
% tranclD
thf(fact_328_compat__tr__compat,axiom,
! [NS: set_Product_prod_a_a,S5: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ ( relcomp_a_a_nat @ NS @ S5 ) @ S5 )
=> ( ord_le8666007276011122963_a_nat @ ( relcomp_a_a_nat @ ( transitive_rtrancl_a @ NS ) @ S5 ) @ S5 ) ) ).
% compat_tr_compat
thf(fact_329_compat__tr__compat,axiom,
! [NS: set_Pr4386577575007340137rm_a_b,S5: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ NS @ S5 ) @ S5 )
=> ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ NS ) @ S5 ) @ S5 ) ) ).
% compat_tr_compat
thf(fact_330_compat__tr__compat,axiom,
! [NS: set_Pr7682762132356531903at_nat,S5: set_Pr9093778441882193744at_nat] :
( ( ord_le3678578370064672496at_nat @ ( relcom9054258243621454929at_nat @ NS @ S5 ) @ S5 )
=> ( ord_le3678578370064672496at_nat @ ( relcom9054258243621454929at_nat @ ( transi3031656534272781564at_nat @ NS ) @ S5 ) @ S5 ) ) ).
% compat_tr_compat
thf(fact_331_rtrancl__O__push,axiom,
! [S5: set_Pr4386577575007340137rm_a_b,R4: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ S5 @ R4 ) @ ( relcom370159955682700863rm_a_b @ R4 @ ( transi7742714808557438673rm_a_b @ S5 ) ) )
=> ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ S5 ) @ R4 ) @ ( relcom370159955682700863rm_a_b @ R4 @ ( transi7742714808557438673rm_a_b @ S5 ) ) ) ) ).
% rtrancl_O_push
thf(fact_332_kleene__induct,axiom,
! [A: set_Pr4934435412358123699_a_nat,X6: set_Pr4934435412358123699_a_nat,B: set_Product_prod_a_a,C2: set_Pr1261947904930325089at_nat] :
( ( ord_le8666007276011122963_a_nat @ A @ X6 )
=> ( ( ord_le8666007276011122963_a_nat @ ( relcomp_a_a_nat @ B @ X6 ) @ X6 )
=> ( ( ord_le8666007276011122963_a_nat @ ( relcomp_a_nat_nat @ X6 @ C2 ) @ X6 )
=> ( ord_le8666007276011122963_a_nat @ ( relcomp_a_a_nat @ ( transitive_rtrancl_a @ B ) @ ( relcomp_a_nat_nat @ A @ ( transi2905341329935302413cl_nat @ C2 ) ) ) @ X6 ) ) ) ) ).
% kleene_induct
thf(fact_333_kleene__induct,axiom,
! [A: set_Pr4386577575007340137rm_a_b,X6: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b,C2: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A @ X6 )
=> ( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ B @ X6 ) @ X6 )
=> ( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ X6 @ C2 ) @ X6 )
=> ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ B ) @ ( relcom370159955682700863rm_a_b @ A @ ( transi7742714808557438673rm_a_b @ C2 ) ) ) @ X6 ) ) ) ) ).
% kleene_induct
thf(fact_334_kleene__induct,axiom,
! [A: set_Pr9093778441882193744at_nat,X6: set_Pr9093778441882193744at_nat,B: set_Pr7682762132356531903at_nat,C2: set_Pr1261947904930325089at_nat] :
( ( ord_le3678578370064672496at_nat @ A @ X6 )
=> ( ( ord_le3678578370064672496at_nat @ ( relcom9054258243621454929at_nat @ B @ X6 ) @ X6 )
=> ( ( ord_le3678578370064672496at_nat @ ( relcom1859588284133128290at_nat @ X6 @ C2 ) @ X6 )
=> ( ord_le3678578370064672496at_nat @ ( relcom9054258243621454929at_nat @ ( transi3031656534272781564at_nat @ B ) @ ( relcom1859588284133128290at_nat @ A @ ( transi2905341329935302413cl_nat @ C2 ) ) ) @ X6 ) ) ) ) ).
% kleene_induct
thf(fact_335_meetI,axiom,
! [A2: term_a_b,B2: term_a_b,A: set_Pr4386577575007340137rm_a_b,C: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( transi7742714808557438673rm_a_b @ A ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ C ) @ ( transi7742714808557438673rm_a_b @ A ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ C ) @ ( abstra1093061187967292567rm_a_b @ A ) ) ) ) ).
% meetI
thf(fact_336_restrict__rtrancl,axiom,
! [R: set_Pr4386577575007340137rm_a_b,S5: set_term_a_b] : ( ord_le118470702582115849rm_a_b @ ( transi7742714808557438673rm_a_b @ ( abstra6662565036008789004rm_a_b @ R @ S5 ) ) @ ( transi7742714808557438673rm_a_b @ R ) ) ).
% restrict_rtrancl
thf(fact_337_comp__rtrancl__trancl,axiom,
! [R4: set_Pr4386577575007340137rm_a_b,S5: set_Pr4386577575007340137rm_a_b,S: term_a_b,T3: term_a_b] :
( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ R4 @ S5 ) @ S5 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T3 ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ R4 @ S5 ) ) @ S5 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T3 ) @ ( transi7922773638565587891rm_a_b @ S5 ) ) ) ) ).
% comp_rtrancl_trancl
thf(fact_338_meetE,axiom,
! [B2: term_a_b,C: term_a_b,A: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ C ) @ ( abstra1093061187967292567rm_a_b @ A ) )
=> ~ ! [A6: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A6 @ B2 ) @ ( transi7742714808557438673rm_a_b @ A ) )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A6 @ C ) @ ( transi7742714808557438673rm_a_b @ A ) ) ) ) ).
% meetE
thf(fact_339_meetD,axiom,
! [B2: term_a_b,C: term_a_b,A: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ C ) @ ( abstra1093061187967292567rm_a_b @ A ) )
=> ? [A6: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A6 @ B2 ) @ ( transi7742714808557438673rm_a_b @ A ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A6 @ C ) @ ( transi7742714808557438673rm_a_b @ A ) ) ) ) ).
% meetD
thf(fact_340_compatible__rtrancl__split,axiom,
! [NS: set_Pr4386577575007340137rm_a_b,S5: set_Pr4386577575007340137rm_a_b,X3: term_a_b,Y3: term_a_b] :
( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ NS @ S5 ) @ S5 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ NS @ S5 ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ ( sup_su6776935440552674877rm_a_b @ ( relcom370159955682700863rm_a_b @ S5 @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ S5 ) @ ( transi7742714808557438673rm_a_b @ NS ) ) ) @ ( transi7742714808557438673rm_a_b @ NS ) ) ) ) ) ).
% compatible_rtrancl_split
thf(fact_341_normalizability__I_H,axiom,
! [A2: term_a_b,B2: term_a_b,A: set_Pr4386577575007340137rm_a_b,C: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( transi7742714808557438673rm_a_b @ A ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ C ) @ ( abstra2972651231392952684rm_a_b @ A ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ C ) @ ( abstra2972651231392952684rm_a_b @ A ) ) ) ) ).
% normalizability_I'
thf(fact_342_Un__iff,axiom,
! [C: produc357393685978478089rm_a_b,A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ C @ ( sup_su6776935440552674877rm_a_b @ A @ B ) )
= ( ( member5869715511025134514rm_a_b @ C @ A )
| ( member5869715511025134514rm_a_b @ C @ B ) ) ) ).
% Un_iff
thf(fact_343_Un__iff,axiom,
! [C: product_prod_a_nat,A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( member5724188588386418708_a_nat @ C @ ( sup_su459911885395995103_a_nat @ A @ B ) )
= ( ( member5724188588386418708_a_nat @ C @ A )
| ( member5724188588386418708_a_nat @ C @ B ) ) ) ).
% Un_iff
thf(fact_344_Un__iff,axiom,
! [C: produc8199716216217303280at_nat,A: set_Pr9093778441882193744at_nat,B: set_Pr9093778441882193744at_nat] :
( ( member7226740684066999833at_nat @ C @ ( sup_su5262723077604177188at_nat @ A @ B ) )
= ( ( member7226740684066999833at_nat @ C @ A )
| ( member7226740684066999833at_nat @ C @ B ) ) ) ).
% Un_iff
thf(fact_345_UnCI,axiom,
! [C: produc357393685978478089rm_a_b,B: set_Pr4386577575007340137rm_a_b,A: set_Pr4386577575007340137rm_a_b] :
( ( ~ ( member5869715511025134514rm_a_b @ C @ B )
=> ( member5869715511025134514rm_a_b @ C @ A ) )
=> ( member5869715511025134514rm_a_b @ C @ ( sup_su6776935440552674877rm_a_b @ A @ B ) ) ) ).
% UnCI
thf(fact_346_UnCI,axiom,
! [C: product_prod_a_nat,B: set_Pr4934435412358123699_a_nat,A: set_Pr4934435412358123699_a_nat] :
( ( ~ ( member5724188588386418708_a_nat @ C @ B )
=> ( member5724188588386418708_a_nat @ C @ A ) )
=> ( member5724188588386418708_a_nat @ C @ ( sup_su459911885395995103_a_nat @ A @ B ) ) ) ).
% UnCI
thf(fact_347_UnCI,axiom,
! [C: produc8199716216217303280at_nat,B: set_Pr9093778441882193744at_nat,A: set_Pr9093778441882193744at_nat] :
( ( ~ ( member7226740684066999833at_nat @ C @ B )
=> ( member7226740684066999833at_nat @ C @ A ) )
=> ( member7226740684066999833at_nat @ C @ ( sup_su5262723077604177188at_nat @ A @ B ) ) ) ).
% UnCI
thf(fact_348_Un__subset__iff,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat,C2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ ( sup_su459911885395995103_a_nat @ A @ B ) @ C2 )
= ( ( ord_le8666007276011122963_a_nat @ A @ C2 )
& ( ord_le8666007276011122963_a_nat @ B @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_349_Un__subset__iff,axiom,
! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b,C2: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ ( sup_su6776935440552674877rm_a_b @ A @ B ) @ C2 )
= ( ( ord_le118470702582115849rm_a_b @ A @ C2 )
& ( ord_le118470702582115849rm_a_b @ B @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_350_Un__subset__iff,axiom,
! [A: set_Pr9093778441882193744at_nat,B: set_Pr9093778441882193744at_nat,C2: set_Pr9093778441882193744at_nat] :
( ( ord_le3678578370064672496at_nat @ ( sup_su5262723077604177188at_nat @ A @ B ) @ C2 )
= ( ( ord_le3678578370064672496at_nat @ A @ C2 )
& ( ord_le3678578370064672496at_nat @ B @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_351_UnI2,axiom,
! [C: produc357393685978478089rm_a_b,B: set_Pr4386577575007340137rm_a_b,A: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ C @ B )
=> ( member5869715511025134514rm_a_b @ C @ ( sup_su6776935440552674877rm_a_b @ A @ B ) ) ) ).
% UnI2
thf(fact_352_UnI2,axiom,
! [C: product_prod_a_nat,B: set_Pr4934435412358123699_a_nat,A: set_Pr4934435412358123699_a_nat] :
( ( member5724188588386418708_a_nat @ C @ B )
=> ( member5724188588386418708_a_nat @ C @ ( sup_su459911885395995103_a_nat @ A @ B ) ) ) ).
% UnI2
thf(fact_353_UnI2,axiom,
! [C: produc8199716216217303280at_nat,B: set_Pr9093778441882193744at_nat,A: set_Pr9093778441882193744at_nat] :
( ( member7226740684066999833at_nat @ C @ B )
=> ( member7226740684066999833at_nat @ C @ ( sup_su5262723077604177188at_nat @ A @ B ) ) ) ).
% UnI2
thf(fact_354_UnI1,axiom,
! [C: produc357393685978478089rm_a_b,A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ C @ A )
=> ( member5869715511025134514rm_a_b @ C @ ( sup_su6776935440552674877rm_a_b @ A @ B ) ) ) ).
% UnI1
thf(fact_355_UnI1,axiom,
! [C: product_prod_a_nat,A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( member5724188588386418708_a_nat @ C @ A )
=> ( member5724188588386418708_a_nat @ C @ ( sup_su459911885395995103_a_nat @ A @ B ) ) ) ).
% UnI1
thf(fact_356_UnI1,axiom,
! [C: produc8199716216217303280at_nat,A: set_Pr9093778441882193744at_nat,B: set_Pr9093778441882193744at_nat] :
( ( member7226740684066999833at_nat @ C @ A )
=> ( member7226740684066999833at_nat @ C @ ( sup_su5262723077604177188at_nat @ A @ B ) ) ) ).
% UnI1
thf(fact_357_UnE,axiom,
! [C: produc357393685978478089rm_a_b,A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ C @ ( sup_su6776935440552674877rm_a_b @ A @ B ) )
=> ( ~ ( member5869715511025134514rm_a_b @ C @ A )
=> ( member5869715511025134514rm_a_b @ C @ B ) ) ) ).
% UnE
thf(fact_358_UnE,axiom,
! [C: product_prod_a_nat,A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( member5724188588386418708_a_nat @ C @ ( sup_su459911885395995103_a_nat @ A @ B ) )
=> ( ~ ( member5724188588386418708_a_nat @ C @ A )
=> ( member5724188588386418708_a_nat @ C @ B ) ) ) ).
% UnE
thf(fact_359_UnE,axiom,
! [C: produc8199716216217303280at_nat,A: set_Pr9093778441882193744at_nat,B: set_Pr9093778441882193744at_nat] :
( ( member7226740684066999833at_nat @ C @ ( sup_su5262723077604177188at_nat @ A @ B ) )
=> ( ~ ( member7226740684066999833at_nat @ C @ A )
=> ( member7226740684066999833at_nat @ C @ B ) ) ) ).
% UnE
thf(fact_360_subset__Un__eq,axiom,
( ord_le8666007276011122963_a_nat
= ( ^ [A4: set_Pr4934435412358123699_a_nat,B4: set_Pr4934435412358123699_a_nat] :
( ( sup_su459911885395995103_a_nat @ A4 @ B4 )
= B4 ) ) ) ).
% subset_Un_eq
thf(fact_361_subset__Un__eq,axiom,
( ord_le118470702582115849rm_a_b
= ( ^ [A4: set_Pr4386577575007340137rm_a_b,B4: set_Pr4386577575007340137rm_a_b] :
( ( sup_su6776935440552674877rm_a_b @ A4 @ B4 )
= B4 ) ) ) ).
% subset_Un_eq
thf(fact_362_subset__Un__eq,axiom,
( ord_le3678578370064672496at_nat
= ( ^ [A4: set_Pr9093778441882193744at_nat,B4: set_Pr9093778441882193744at_nat] :
( ( sup_su5262723077604177188at_nat @ A4 @ B4 )
= B4 ) ) ) ).
% subset_Un_eq
thf(fact_363_subset__UnE,axiom,
! [C2: set_Pr4934435412358123699_a_nat,A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ C2 @ ( sup_su459911885395995103_a_nat @ A @ B ) )
=> ~ ! [A7: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A7 @ A )
=> ! [B7: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ B7 @ B )
=> ( C2
!= ( sup_su459911885395995103_a_nat @ A7 @ B7 ) ) ) ) ) ).
% subset_UnE
thf(fact_364_subset__UnE,axiom,
! [C2: set_Pr4386577575007340137rm_a_b,A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ C2 @ ( sup_su6776935440552674877rm_a_b @ A @ B ) )
=> ~ ! [A7: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A7 @ A )
=> ! [B7: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ B7 @ B )
=> ( C2
!= ( sup_su6776935440552674877rm_a_b @ A7 @ B7 ) ) ) ) ) ).
% subset_UnE
thf(fact_365_subset__UnE,axiom,
! [C2: set_Pr9093778441882193744at_nat,A: set_Pr9093778441882193744at_nat,B: set_Pr9093778441882193744at_nat] :
( ( ord_le3678578370064672496at_nat @ C2 @ ( sup_su5262723077604177188at_nat @ A @ B ) )
=> ~ ! [A7: set_Pr9093778441882193744at_nat] :
( ( ord_le3678578370064672496at_nat @ A7 @ A )
=> ! [B7: set_Pr9093778441882193744at_nat] :
( ( ord_le3678578370064672496at_nat @ B7 @ B )
=> ( C2
!= ( sup_su5262723077604177188at_nat @ A7 @ B7 ) ) ) ) ) ).
% subset_UnE
thf(fact_366_Un__absorb2,axiom,
! [B: set_Pr4934435412358123699_a_nat,A: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ B @ A )
=> ( ( sup_su459911885395995103_a_nat @ A @ B )
= A ) ) ).
% Un_absorb2
thf(fact_367_Un__absorb2,axiom,
! [B: set_Pr4386577575007340137rm_a_b,A: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ B @ A )
=> ( ( sup_su6776935440552674877rm_a_b @ A @ B )
= A ) ) ).
% Un_absorb2
thf(fact_368_Un__absorb2,axiom,
! [B: set_Pr9093778441882193744at_nat,A: set_Pr9093778441882193744at_nat] :
( ( ord_le3678578370064672496at_nat @ B @ A )
=> ( ( sup_su5262723077604177188at_nat @ A @ B )
= A ) ) ).
% Un_absorb2
thf(fact_369_Un__absorb1,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A @ B )
=> ( ( sup_su459911885395995103_a_nat @ A @ B )
= B ) ) ).
% Un_absorb1
thf(fact_370_Un__absorb1,axiom,
! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A @ B )
=> ( ( sup_su6776935440552674877rm_a_b @ A @ B )
= B ) ) ).
% Un_absorb1
thf(fact_371_Un__absorb1,axiom,
! [A: set_Pr9093778441882193744at_nat,B: set_Pr9093778441882193744at_nat] :
( ( ord_le3678578370064672496at_nat @ A @ B )
=> ( ( sup_su5262723077604177188at_nat @ A @ B )
= B ) ) ).
% Un_absorb1
thf(fact_372_Un__upper2,axiom,
! [B: set_Pr4934435412358123699_a_nat,A: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ B @ ( sup_su459911885395995103_a_nat @ A @ B ) ) ).
% Un_upper2
thf(fact_373_Un__upper2,axiom,
! [B: set_Pr4386577575007340137rm_a_b,A: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ B @ ( sup_su6776935440552674877rm_a_b @ A @ B ) ) ).
% Un_upper2
thf(fact_374_Un__upper2,axiom,
! [B: set_Pr9093778441882193744at_nat,A: set_Pr9093778441882193744at_nat] : ( ord_le3678578370064672496at_nat @ B @ ( sup_su5262723077604177188at_nat @ A @ B ) ) ).
% Un_upper2
thf(fact_375_Un__upper1,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ A @ ( sup_su459911885395995103_a_nat @ A @ B ) ) ).
% Un_upper1
thf(fact_376_Un__upper1,axiom,
! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ A @ ( sup_su6776935440552674877rm_a_b @ A @ B ) ) ).
% Un_upper1
thf(fact_377_Un__upper1,axiom,
! [A: set_Pr9093778441882193744at_nat,B: set_Pr9093778441882193744at_nat] : ( ord_le3678578370064672496at_nat @ A @ ( sup_su5262723077604177188at_nat @ A @ B ) ) ).
% Un_upper1
thf(fact_378_Un__least,axiom,
! [A: set_Pr4934435412358123699_a_nat,C2: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A @ C2 )
=> ( ( ord_le8666007276011122963_a_nat @ B @ C2 )
=> ( ord_le8666007276011122963_a_nat @ ( sup_su459911885395995103_a_nat @ A @ B ) @ C2 ) ) ) ).
% Un_least
thf(fact_379_Un__least,axiom,
! [A: set_Pr4386577575007340137rm_a_b,C2: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A @ C2 )
=> ( ( ord_le118470702582115849rm_a_b @ B @ C2 )
=> ( ord_le118470702582115849rm_a_b @ ( sup_su6776935440552674877rm_a_b @ A @ B ) @ C2 ) ) ) ).
% Un_least
thf(fact_380_Un__least,axiom,
! [A: set_Pr9093778441882193744at_nat,C2: set_Pr9093778441882193744at_nat,B: set_Pr9093778441882193744at_nat] :
( ( ord_le3678578370064672496at_nat @ A @ C2 )
=> ( ( ord_le3678578370064672496at_nat @ B @ C2 )
=> ( ord_le3678578370064672496at_nat @ ( sup_su5262723077604177188at_nat @ A @ B ) @ C2 ) ) ) ).
% Un_least
thf(fact_381_Un__mono,axiom,
! [A: set_Pr4934435412358123699_a_nat,C2: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat,D: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A @ C2 )
=> ( ( ord_le8666007276011122963_a_nat @ B @ D )
=> ( ord_le8666007276011122963_a_nat @ ( sup_su459911885395995103_a_nat @ A @ B ) @ ( sup_su459911885395995103_a_nat @ C2 @ D ) ) ) ) ).
% Un_mono
thf(fact_382_Un__mono,axiom,
! [A: set_Pr4386577575007340137rm_a_b,C2: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b,D: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A @ C2 )
=> ( ( ord_le118470702582115849rm_a_b @ B @ D )
=> ( ord_le118470702582115849rm_a_b @ ( sup_su6776935440552674877rm_a_b @ A @ B ) @ ( sup_su6776935440552674877rm_a_b @ C2 @ D ) ) ) ) ).
% Un_mono
thf(fact_383_Un__mono,axiom,
! [A: set_Pr9093778441882193744at_nat,C2: set_Pr9093778441882193744at_nat,B: set_Pr9093778441882193744at_nat,D: set_Pr9093778441882193744at_nat] :
( ( ord_le3678578370064672496at_nat @ A @ C2 )
=> ( ( ord_le3678578370064672496at_nat @ B @ D )
=> ( ord_le3678578370064672496at_nat @ ( sup_su5262723077604177188at_nat @ A @ B ) @ ( sup_su5262723077604177188at_nat @ C2 @ D ) ) ) ) ).
% Un_mono
thf(fact_384_in__rtrancl__UnI,axiom,
! [X3: produc357393685978478089rm_a_b,R4: set_Pr4386577575007340137rm_a_b,S5: set_Pr4386577575007340137rm_a_b] :
( ( ( member5869715511025134514rm_a_b @ X3 @ ( transi7742714808557438673rm_a_b @ R4 ) )
| ( member5869715511025134514rm_a_b @ X3 @ ( transi7742714808557438673rm_a_b @ S5 ) ) )
=> ( member5869715511025134514rm_a_b @ X3 @ ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ R4 @ S5 ) ) ) ) ).
% in_rtrancl_UnI
thf(fact_385_rtrancl__Un__separatorE,axiom,
! [A2: term_a_b,B2: term_a_b,P: set_Pr4386577575007340137rm_a_b,Q: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ P @ Q ) ) )
=> ( ! [X: term_a_b,Y: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ X ) @ ( transi7742714808557438673rm_a_b @ P ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ Q )
=> ( X = Y ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( transi7742714808557438673rm_a_b @ P ) ) ) ) ).
% rtrancl_Un_separatorE
thf(fact_386_rtrancl__Un__separator__converseE,axiom,
! [A2: term_a_b,B2: term_a_b,P: set_Pr4386577575007340137rm_a_b,Q: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ P @ Q ) ) )
=> ( ! [X: term_a_b,Y: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ B2 ) @ ( transi7742714808557438673rm_a_b @ P ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X ) @ Q )
=> ( Y = X ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( transi7742714808557438673rm_a_b @ P ) ) ) ) ).
% rtrancl_Un_separator_converseE
thf(fact_387_rtrancl__Un__subset,axiom,
! [R4: set_Pr4386577575007340137rm_a_b,S5: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ ( sup_su6776935440552674877rm_a_b @ ( transi7742714808557438673rm_a_b @ R4 ) @ ( transi7742714808557438673rm_a_b @ S5 ) ) @ ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ R4 @ S5 ) ) ) ).
% rtrancl_Un_subset
thf(fact_388_trancl__union__right,axiom,
! [R: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ ( transi7922773638565587891rm_a_b @ R ) @ ( transi7922773638565587891rm_a_b @ ( sup_su6776935440552674877rm_a_b @ S @ R ) ) ) ).
% trancl_union_right
thf(fact_389_restrict__SN__subset,axiom,
! [R4: set_Pr4386577575007340137rm_a_b,S5: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ ( abstra4933154510623179783rm_a_b @ R4 @ S5 ) @ R4 ) ).
% restrict_SN_subset
thf(fact_390_firstStep,axiom,
! [L2: set_Pr4386577575007340137rm_a_b,S5: set_Pr4386577575007340137rm_a_b,R4: set_Pr4386577575007340137rm_a_b,X3: term_a_b,Y3: term_a_b] :
( ( L2
= ( sup_su6776935440552674877rm_a_b @ S5 @ R4 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ ( transi7742714808557438673rm_a_b @ L2 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ ( transi7742714808557438673rm_a_b @ R4 ) )
| ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ R4 ) @ ( relcom370159955682700863rm_a_b @ S5 @ ( transi7742714808557438673rm_a_b @ L2 ) ) ) ) ) ) ) ).
% firstStep
thf(fact_391_first__step,axiom,
! [C2: set_Pr4386577575007340137rm_a_b,A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b,X3: term_a_b,Y3: term_a_b,Z2: term_a_b] :
( ( C2
= ( sup_su6776935440552674877rm_a_b @ A @ B ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ ( transi7742714808557438673rm_a_b @ C2 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ Z2 ) @ B )
=> ? [Y: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ A ) @ B ) ) ) ) ) ).
% first_step
thf(fact_392_first__step__O,axiom,
! [C2: set_Pr4386577575007340137rm_a_b,A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b,X3: term_a_b,Y3: term_a_b] :
( ( C2
= ( sup_su6776935440552674877rm_a_b @ A @ B ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ C2 ) @ B ) )
=> ? [Y: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ A ) @ B ) ) ) ) ).
% first_step_O
thf(fact_393_right__comp__S,axiom,
! [X3: term_a_b,Y3: term_a_b,S5: set_Pr4386577575007340137rm_a_b,NS: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ ( relcom370159955682700863rm_a_b @ S5 @ ( sup_su6776935440552674877rm_a_b @ ( relcom370159955682700863rm_a_b @ S5 @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ S5 ) @ ( transi7742714808557438673rm_a_b @ NS ) ) ) @ ( transi7742714808557438673rm_a_b @ NS ) ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ ( relcom370159955682700863rm_a_b @ S5 @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ S5 ) @ ( transi7742714808557438673rm_a_b @ NS ) ) ) ) ) ).
% right_comp_S
thf(fact_394_rtrancl__U__push,axiom,
! [S5: set_Pr4386577575007340137rm_a_b,R4: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ S5 @ R4 ) @ ( relcom370159955682700863rm_a_b @ R4 @ ( transi7742714808557438673rm_a_b @ S5 ) ) )
=> ( ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ S5 @ R4 ) )
= ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ R4 ) @ ( transi7742714808557438673rm_a_b @ S5 ) ) ) ) ).
% rtrancl_U_push
thf(fact_395_compatible__conv,axiom,
! [NS: set_Pr4386577575007340137rm_a_b,S5: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ NS @ S5 ) @ S5 )
=> ( ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ NS @ S5 ) ) @ ( relcom370159955682700863rm_a_b @ S5 @ ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ NS @ S5 ) ) ) )
= ( relcom370159955682700863rm_a_b @ S5 @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ S5 ) @ ( transi7742714808557438673rm_a_b @ NS ) ) ) ) ) ).
% compatible_conv
thf(fact_396_sup_Obounded__iff,axiom,
! [B2: set_Pr4934435412358123699_a_nat,C: set_Pr4934435412358123699_a_nat,A2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ ( sup_su459911885395995103_a_nat @ B2 @ C ) @ A2 )
= ( ( ord_le8666007276011122963_a_nat @ B2 @ A2 )
& ( ord_le8666007276011122963_a_nat @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_397_sup_Obounded__iff,axiom,
! [B2: set_Pr4386577575007340137rm_a_b,C: set_Pr4386577575007340137rm_a_b,A2: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ ( sup_su6776935440552674877rm_a_b @ B2 @ C ) @ A2 )
= ( ( ord_le118470702582115849rm_a_b @ B2 @ A2 )
& ( ord_le118470702582115849rm_a_b @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_398_sup_Obounded__iff,axiom,
! [B2: set_Pr9093778441882193744at_nat,C: set_Pr9093778441882193744at_nat,A2: set_Pr9093778441882193744at_nat] :
( ( ord_le3678578370064672496at_nat @ ( sup_su5262723077604177188at_nat @ B2 @ C ) @ A2 )
= ( ( ord_le3678578370064672496at_nat @ B2 @ A2 )
& ( ord_le3678578370064672496at_nat @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_399_le__sup__iff,axiom,
! [X3: set_Pr4934435412358123699_a_nat,Y3: set_Pr4934435412358123699_a_nat,Z2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ ( sup_su459911885395995103_a_nat @ X3 @ Y3 ) @ Z2 )
= ( ( ord_le8666007276011122963_a_nat @ X3 @ Z2 )
& ( ord_le8666007276011122963_a_nat @ Y3 @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_400_le__sup__iff,axiom,
! [X3: set_Pr4386577575007340137rm_a_b,Y3: set_Pr4386577575007340137rm_a_b,Z2: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ ( sup_su6776935440552674877rm_a_b @ X3 @ Y3 ) @ Z2 )
= ( ( ord_le118470702582115849rm_a_b @ X3 @ Z2 )
& ( ord_le118470702582115849rm_a_b @ Y3 @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_401_le__sup__iff,axiom,
! [X3: set_Pr9093778441882193744at_nat,Y3: set_Pr9093778441882193744at_nat,Z2: set_Pr9093778441882193744at_nat] :
( ( ord_le3678578370064672496at_nat @ ( sup_su5262723077604177188at_nat @ X3 @ Y3 ) @ Z2 )
= ( ( ord_le3678578370064672496at_nat @ X3 @ Z2 )
& ( ord_le3678578370064672496at_nat @ Y3 @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_402_quasi__commute__def,axiom,
( abstra4317384144805385751rm_a_b
= ( ^ [R2: set_Pr4386577575007340137rm_a_b,S4: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ S4 @ R2 ) @ ( relcom370159955682700863rm_a_b @ R2 @ ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ R2 @ S4 ) ) ) ) ) ) ).
% quasi_commute_def
thf(fact_403_sup_OcoboundedI2,axiom,
! [C: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat,A2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ C @ B2 )
=> ( ord_le8666007276011122963_a_nat @ C @ ( sup_su459911885395995103_a_nat @ A2 @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_404_sup_OcoboundedI2,axiom,
! [C: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b,A2: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ C @ B2 )
=> ( ord_le118470702582115849rm_a_b @ C @ ( sup_su6776935440552674877rm_a_b @ A2 @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_405_sup_OcoboundedI2,axiom,
! [C: set_Pr9093778441882193744at_nat,B2: set_Pr9093778441882193744at_nat,A2: set_Pr9093778441882193744at_nat] :
( ( ord_le3678578370064672496at_nat @ C @ B2 )
=> ( ord_le3678578370064672496at_nat @ C @ ( sup_su5262723077604177188at_nat @ A2 @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_406_sup_OcoboundedI1,axiom,
! [C: set_Pr4934435412358123699_a_nat,A2: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ C @ A2 )
=> ( ord_le8666007276011122963_a_nat @ C @ ( sup_su459911885395995103_a_nat @ A2 @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_407_sup_OcoboundedI1,axiom,
! [C: set_Pr4386577575007340137rm_a_b,A2: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ C @ A2 )
=> ( ord_le118470702582115849rm_a_b @ C @ ( sup_su6776935440552674877rm_a_b @ A2 @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_408_sup_OcoboundedI1,axiom,
! [C: set_Pr9093778441882193744at_nat,A2: set_Pr9093778441882193744at_nat,B2: set_Pr9093778441882193744at_nat] :
( ( ord_le3678578370064672496at_nat @ C @ A2 )
=> ( ord_le3678578370064672496at_nat @ C @ ( sup_su5262723077604177188at_nat @ A2 @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_409_sup_Oabsorb__iff2,axiom,
( ord_le8666007276011122963_a_nat
= ( ^ [A5: set_Pr4934435412358123699_a_nat,B5: set_Pr4934435412358123699_a_nat] :
( ( sup_su459911885395995103_a_nat @ A5 @ B5 )
= B5 ) ) ) ).
% sup.absorb_iff2
thf(fact_410_sup_Oabsorb__iff2,axiom,
( ord_le118470702582115849rm_a_b
= ( ^ [A5: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
( ( sup_su6776935440552674877rm_a_b @ A5 @ B5 )
= B5 ) ) ) ).
% sup.absorb_iff2
thf(fact_411_sup_Oabsorb__iff2,axiom,
( ord_le3678578370064672496at_nat
= ( ^ [A5: set_Pr9093778441882193744at_nat,B5: set_Pr9093778441882193744at_nat] :
( ( sup_su5262723077604177188at_nat @ A5 @ B5 )
= B5 ) ) ) ).
% sup.absorb_iff2
thf(fact_412_sup_Oabsorb__iff1,axiom,
( ord_le8666007276011122963_a_nat
= ( ^ [B5: set_Pr4934435412358123699_a_nat,A5: set_Pr4934435412358123699_a_nat] :
( ( sup_su459911885395995103_a_nat @ A5 @ B5 )
= A5 ) ) ) ).
% sup.absorb_iff1
thf(fact_413_sup_Oabsorb__iff1,axiom,
( ord_le118470702582115849rm_a_b
= ( ^ [B5: set_Pr4386577575007340137rm_a_b,A5: set_Pr4386577575007340137rm_a_b] :
( ( sup_su6776935440552674877rm_a_b @ A5 @ B5 )
= A5 ) ) ) ).
% sup.absorb_iff1
thf(fact_414_sup_Oabsorb__iff1,axiom,
( ord_le3678578370064672496at_nat
= ( ^ [B5: set_Pr9093778441882193744at_nat,A5: set_Pr9093778441882193744at_nat] :
( ( sup_su5262723077604177188at_nat @ A5 @ B5 )
= A5 ) ) ) ).
% sup.absorb_iff1
thf(fact_415_sup_Ocobounded2,axiom,
! [B2: set_Pr4934435412358123699_a_nat,A2: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ B2 @ ( sup_su459911885395995103_a_nat @ A2 @ B2 ) ) ).
% sup.cobounded2
thf(fact_416_sup_Ocobounded2,axiom,
! [B2: set_Pr4386577575007340137rm_a_b,A2: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ B2 @ ( sup_su6776935440552674877rm_a_b @ A2 @ B2 ) ) ).
% sup.cobounded2
thf(fact_417_sup_Ocobounded2,axiom,
! [B2: set_Pr9093778441882193744at_nat,A2: set_Pr9093778441882193744at_nat] : ( ord_le3678578370064672496at_nat @ B2 @ ( sup_su5262723077604177188at_nat @ A2 @ B2 ) ) ).
% sup.cobounded2
thf(fact_418_sup_Ocobounded1,axiom,
! [A2: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ A2 @ ( sup_su459911885395995103_a_nat @ A2 @ B2 ) ) ).
% sup.cobounded1
thf(fact_419_sup_Ocobounded1,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ A2 @ ( sup_su6776935440552674877rm_a_b @ A2 @ B2 ) ) ).
% sup.cobounded1
thf(fact_420_sup_Ocobounded1,axiom,
! [A2: set_Pr9093778441882193744at_nat,B2: set_Pr9093778441882193744at_nat] : ( ord_le3678578370064672496at_nat @ A2 @ ( sup_su5262723077604177188at_nat @ A2 @ B2 ) ) ).
% sup.cobounded1
thf(fact_421_sup_Oorder__iff,axiom,
( ord_le8666007276011122963_a_nat
= ( ^ [B5: set_Pr4934435412358123699_a_nat,A5: set_Pr4934435412358123699_a_nat] :
( A5
= ( sup_su459911885395995103_a_nat @ A5 @ B5 ) ) ) ) ).
% sup.order_iff
thf(fact_422_sup_Oorder__iff,axiom,
( ord_le118470702582115849rm_a_b
= ( ^ [B5: set_Pr4386577575007340137rm_a_b,A5: set_Pr4386577575007340137rm_a_b] :
( A5
= ( sup_su6776935440552674877rm_a_b @ A5 @ B5 ) ) ) ) ).
% sup.order_iff
thf(fact_423_sup_Oorder__iff,axiom,
( ord_le3678578370064672496at_nat
= ( ^ [B5: set_Pr9093778441882193744at_nat,A5: set_Pr9093778441882193744at_nat] :
( A5
= ( sup_su5262723077604177188at_nat @ A5 @ B5 ) ) ) ) ).
% sup.order_iff
thf(fact_424_sup_OboundedI,axiom,
! [B2: set_Pr4934435412358123699_a_nat,A2: set_Pr4934435412358123699_a_nat,C: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ B2 @ A2 )
=> ( ( ord_le8666007276011122963_a_nat @ C @ A2 )
=> ( ord_le8666007276011122963_a_nat @ ( sup_su459911885395995103_a_nat @ B2 @ C ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_425_sup_OboundedI,axiom,
! [B2: set_Pr4386577575007340137rm_a_b,A2: set_Pr4386577575007340137rm_a_b,C: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ B2 @ A2 )
=> ( ( ord_le118470702582115849rm_a_b @ C @ A2 )
=> ( ord_le118470702582115849rm_a_b @ ( sup_su6776935440552674877rm_a_b @ B2 @ C ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_426_sup_OboundedI,axiom,
! [B2: set_Pr9093778441882193744at_nat,A2: set_Pr9093778441882193744at_nat,C: set_Pr9093778441882193744at_nat] :
( ( ord_le3678578370064672496at_nat @ B2 @ A2 )
=> ( ( ord_le3678578370064672496at_nat @ C @ A2 )
=> ( ord_le3678578370064672496at_nat @ ( sup_su5262723077604177188at_nat @ B2 @ C ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_427_sup_OboundedE,axiom,
! [B2: set_Pr4934435412358123699_a_nat,C: set_Pr4934435412358123699_a_nat,A2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ ( sup_su459911885395995103_a_nat @ B2 @ C ) @ A2 )
=> ~ ( ( ord_le8666007276011122963_a_nat @ B2 @ A2 )
=> ~ ( ord_le8666007276011122963_a_nat @ C @ A2 ) ) ) ).
% sup.boundedE
thf(fact_428_sup_OboundedE,axiom,
! [B2: set_Pr4386577575007340137rm_a_b,C: set_Pr4386577575007340137rm_a_b,A2: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ ( sup_su6776935440552674877rm_a_b @ B2 @ C ) @ A2 )
=> ~ ( ( ord_le118470702582115849rm_a_b @ B2 @ A2 )
=> ~ ( ord_le118470702582115849rm_a_b @ C @ A2 ) ) ) ).
% sup.boundedE
thf(fact_429_sup_OboundedE,axiom,
! [B2: set_Pr9093778441882193744at_nat,C: set_Pr9093778441882193744at_nat,A2: set_Pr9093778441882193744at_nat] :
( ( ord_le3678578370064672496at_nat @ ( sup_su5262723077604177188at_nat @ B2 @ C ) @ A2 )
=> ~ ( ( ord_le3678578370064672496at_nat @ B2 @ A2 )
=> ~ ( ord_le3678578370064672496at_nat @ C @ A2 ) ) ) ).
% sup.boundedE
thf(fact_430_sup__absorb2,axiom,
! [X3: set_Pr4934435412358123699_a_nat,Y3: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X3 @ Y3 )
=> ( ( sup_su459911885395995103_a_nat @ X3 @ Y3 )
= Y3 ) ) ).
% sup_absorb2
thf(fact_431_sup__absorb2,axiom,
! [X3: set_Pr4386577575007340137rm_a_b,Y3: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ X3 @ Y3 )
=> ( ( sup_su6776935440552674877rm_a_b @ X3 @ Y3 )
= Y3 ) ) ).
% sup_absorb2
thf(fact_432_sup__absorb2,axiom,
! [X3: set_Pr9093778441882193744at_nat,Y3: set_Pr9093778441882193744at_nat] :
( ( ord_le3678578370064672496at_nat @ X3 @ Y3 )
=> ( ( sup_su5262723077604177188at_nat @ X3 @ Y3 )
= Y3 ) ) ).
% sup_absorb2
thf(fact_433_sup__absorb1,axiom,
! [Y3: set_Pr4934435412358123699_a_nat,X3: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ Y3 @ X3 )
=> ( ( sup_su459911885395995103_a_nat @ X3 @ Y3 )
= X3 ) ) ).
% sup_absorb1
thf(fact_434_sup__absorb1,axiom,
! [Y3: set_Pr4386577575007340137rm_a_b,X3: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ Y3 @ X3 )
=> ( ( sup_su6776935440552674877rm_a_b @ X3 @ Y3 )
= X3 ) ) ).
% sup_absorb1
thf(fact_435_sup__absorb1,axiom,
! [Y3: set_Pr9093778441882193744at_nat,X3: set_Pr9093778441882193744at_nat] :
( ( ord_le3678578370064672496at_nat @ Y3 @ X3 )
=> ( ( sup_su5262723077604177188at_nat @ X3 @ Y3 )
= X3 ) ) ).
% sup_absorb1
thf(fact_436_sup_Oabsorb2,axiom,
! [A2: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A2 @ B2 )
=> ( ( sup_su459911885395995103_a_nat @ A2 @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_437_sup_Oabsorb2,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A2 @ B2 )
=> ( ( sup_su6776935440552674877rm_a_b @ A2 @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_438_sup_Oabsorb2,axiom,
! [A2: set_Pr9093778441882193744at_nat,B2: set_Pr9093778441882193744at_nat] :
( ( ord_le3678578370064672496at_nat @ A2 @ B2 )
=> ( ( sup_su5262723077604177188at_nat @ A2 @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_439_sup_Oabsorb1,axiom,
! [B2: set_Pr4934435412358123699_a_nat,A2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ B2 @ A2 )
=> ( ( sup_su459911885395995103_a_nat @ A2 @ B2 )
= A2 ) ) ).
% sup.absorb1
thf(fact_440_sup_Oabsorb1,axiom,
! [B2: set_Pr4386577575007340137rm_a_b,A2: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ B2 @ A2 )
=> ( ( sup_su6776935440552674877rm_a_b @ A2 @ B2 )
= A2 ) ) ).
% sup.absorb1
thf(fact_441_sup_Oabsorb1,axiom,
! [B2: set_Pr9093778441882193744at_nat,A2: set_Pr9093778441882193744at_nat] :
( ( ord_le3678578370064672496at_nat @ B2 @ A2 )
=> ( ( sup_su5262723077604177188at_nat @ A2 @ B2 )
= A2 ) ) ).
% sup.absorb1
thf(fact_442_sup__unique,axiom,
! [F2: set_Pr4934435412358123699_a_nat > set_Pr4934435412358123699_a_nat > set_Pr4934435412358123699_a_nat,X3: set_Pr4934435412358123699_a_nat,Y3: set_Pr4934435412358123699_a_nat] :
( ! [X: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ X @ ( F2 @ X @ Y ) )
=> ( ! [X: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ Y @ ( F2 @ X @ Y ) )
=> ( ! [X: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat,Z3: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ Y @ X )
=> ( ( ord_le8666007276011122963_a_nat @ Z3 @ X )
=> ( ord_le8666007276011122963_a_nat @ ( F2 @ Y @ Z3 ) @ X ) ) )
=> ( ( sup_su459911885395995103_a_nat @ X3 @ Y3 )
= ( F2 @ X3 @ Y3 ) ) ) ) ) ).
% sup_unique
thf(fact_443_sup__unique,axiom,
! [F2: set_Pr4386577575007340137rm_a_b > set_Pr4386577575007340137rm_a_b > set_Pr4386577575007340137rm_a_b,X3: set_Pr4386577575007340137rm_a_b,Y3: set_Pr4386577575007340137rm_a_b] :
( ! [X: set_Pr4386577575007340137rm_a_b,Y: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ X @ ( F2 @ X @ Y ) )
=> ( ! [X: set_Pr4386577575007340137rm_a_b,Y: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ Y @ ( F2 @ X @ Y ) )
=> ( ! [X: set_Pr4386577575007340137rm_a_b,Y: set_Pr4386577575007340137rm_a_b,Z3: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ Y @ X )
=> ( ( ord_le118470702582115849rm_a_b @ Z3 @ X )
=> ( ord_le118470702582115849rm_a_b @ ( F2 @ Y @ Z3 ) @ X ) ) )
=> ( ( sup_su6776935440552674877rm_a_b @ X3 @ Y3 )
= ( F2 @ X3 @ Y3 ) ) ) ) ) ).
% sup_unique
thf(fact_444_sup__unique,axiom,
! [F2: set_Pr9093778441882193744at_nat > set_Pr9093778441882193744at_nat > set_Pr9093778441882193744at_nat,X3: set_Pr9093778441882193744at_nat,Y3: set_Pr9093778441882193744at_nat] :
( ! [X: set_Pr9093778441882193744at_nat,Y: set_Pr9093778441882193744at_nat] : ( ord_le3678578370064672496at_nat @ X @ ( F2 @ X @ Y ) )
=> ( ! [X: set_Pr9093778441882193744at_nat,Y: set_Pr9093778441882193744at_nat] : ( ord_le3678578370064672496at_nat @ Y @ ( F2 @ X @ Y ) )
=> ( ! [X: set_Pr9093778441882193744at_nat,Y: set_Pr9093778441882193744at_nat,Z3: set_Pr9093778441882193744at_nat] :
( ( ord_le3678578370064672496at_nat @ Y @ X )
=> ( ( ord_le3678578370064672496at_nat @ Z3 @ X )
=> ( ord_le3678578370064672496at_nat @ ( F2 @ Y @ Z3 ) @ X ) ) )
=> ( ( sup_su5262723077604177188at_nat @ X3 @ Y3 )
= ( F2 @ X3 @ Y3 ) ) ) ) ) ).
% sup_unique
thf(fact_445_sup_OorderI,axiom,
! [A2: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat] :
( ( A2
= ( sup_su459911885395995103_a_nat @ A2 @ B2 ) )
=> ( ord_le8666007276011122963_a_nat @ B2 @ A2 ) ) ).
% sup.orderI
thf(fact_446_sup_OorderI,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b] :
( ( A2
= ( sup_su6776935440552674877rm_a_b @ A2 @ B2 ) )
=> ( ord_le118470702582115849rm_a_b @ B2 @ A2 ) ) ).
% sup.orderI
thf(fact_447_sup_OorderI,axiom,
! [A2: set_Pr9093778441882193744at_nat,B2: set_Pr9093778441882193744at_nat] :
( ( A2
= ( sup_su5262723077604177188at_nat @ A2 @ B2 ) )
=> ( ord_le3678578370064672496at_nat @ B2 @ A2 ) ) ).
% sup.orderI
thf(fact_448_sup_OorderE,axiom,
! [B2: set_Pr4934435412358123699_a_nat,A2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ B2 @ A2 )
=> ( A2
= ( sup_su459911885395995103_a_nat @ A2 @ B2 ) ) ) ).
% sup.orderE
thf(fact_449_sup_OorderE,axiom,
! [B2: set_Pr4386577575007340137rm_a_b,A2: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ B2 @ A2 )
=> ( A2
= ( sup_su6776935440552674877rm_a_b @ A2 @ B2 ) ) ) ).
% sup.orderE
thf(fact_450_sup_OorderE,axiom,
! [B2: set_Pr9093778441882193744at_nat,A2: set_Pr9093778441882193744at_nat] :
( ( ord_le3678578370064672496at_nat @ B2 @ A2 )
=> ( A2
= ( sup_su5262723077604177188at_nat @ A2 @ B2 ) ) ) ).
% sup.orderE
thf(fact_451_le__iff__sup,axiom,
( ord_le8666007276011122963_a_nat
= ( ^ [X4: set_Pr4934435412358123699_a_nat,Y5: set_Pr4934435412358123699_a_nat] :
( ( sup_su459911885395995103_a_nat @ X4 @ Y5 )
= Y5 ) ) ) ).
% le_iff_sup
thf(fact_452_le__iff__sup,axiom,
( ord_le118470702582115849rm_a_b
= ( ^ [X4: set_Pr4386577575007340137rm_a_b,Y5: set_Pr4386577575007340137rm_a_b] :
( ( sup_su6776935440552674877rm_a_b @ X4 @ Y5 )
= Y5 ) ) ) ).
% le_iff_sup
thf(fact_453_le__iff__sup,axiom,
( ord_le3678578370064672496at_nat
= ( ^ [X4: set_Pr9093778441882193744at_nat,Y5: set_Pr9093778441882193744at_nat] :
( ( sup_su5262723077604177188at_nat @ X4 @ Y5 )
= Y5 ) ) ) ).
% le_iff_sup
thf(fact_454_sup__least,axiom,
! [Y3: set_Pr4934435412358123699_a_nat,X3: set_Pr4934435412358123699_a_nat,Z2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ Y3 @ X3 )
=> ( ( ord_le8666007276011122963_a_nat @ Z2 @ X3 )
=> ( ord_le8666007276011122963_a_nat @ ( sup_su459911885395995103_a_nat @ Y3 @ Z2 ) @ X3 ) ) ) ).
% sup_least
thf(fact_455_sup__least,axiom,
! [Y3: set_Pr4386577575007340137rm_a_b,X3: set_Pr4386577575007340137rm_a_b,Z2: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ Y3 @ X3 )
=> ( ( ord_le118470702582115849rm_a_b @ Z2 @ X3 )
=> ( ord_le118470702582115849rm_a_b @ ( sup_su6776935440552674877rm_a_b @ Y3 @ Z2 ) @ X3 ) ) ) ).
% sup_least
thf(fact_456_sup__least,axiom,
! [Y3: set_Pr9093778441882193744at_nat,X3: set_Pr9093778441882193744at_nat,Z2: set_Pr9093778441882193744at_nat] :
( ( ord_le3678578370064672496at_nat @ Y3 @ X3 )
=> ( ( ord_le3678578370064672496at_nat @ Z2 @ X3 )
=> ( ord_le3678578370064672496at_nat @ ( sup_su5262723077604177188at_nat @ Y3 @ Z2 ) @ X3 ) ) ) ).
% sup_least
thf(fact_457_sup__mono,axiom,
! [A2: set_Pr4934435412358123699_a_nat,C: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat,D2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A2 @ C )
=> ( ( ord_le8666007276011122963_a_nat @ B2 @ D2 )
=> ( ord_le8666007276011122963_a_nat @ ( sup_su459911885395995103_a_nat @ A2 @ B2 ) @ ( sup_su459911885395995103_a_nat @ C @ D2 ) ) ) ) ).
% sup_mono
thf(fact_458_sup__mono,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,C: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b,D2: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A2 @ C )
=> ( ( ord_le118470702582115849rm_a_b @ B2 @ D2 )
=> ( ord_le118470702582115849rm_a_b @ ( sup_su6776935440552674877rm_a_b @ A2 @ B2 ) @ ( sup_su6776935440552674877rm_a_b @ C @ D2 ) ) ) ) ).
% sup_mono
thf(fact_459_sup__mono,axiom,
! [A2: set_Pr9093778441882193744at_nat,C: set_Pr9093778441882193744at_nat,B2: set_Pr9093778441882193744at_nat,D2: set_Pr9093778441882193744at_nat] :
( ( ord_le3678578370064672496at_nat @ A2 @ C )
=> ( ( ord_le3678578370064672496at_nat @ B2 @ D2 )
=> ( ord_le3678578370064672496at_nat @ ( sup_su5262723077604177188at_nat @ A2 @ B2 ) @ ( sup_su5262723077604177188at_nat @ C @ D2 ) ) ) ) ).
% sup_mono
thf(fact_460_sup_Omono,axiom,
! [C: set_Pr4934435412358123699_a_nat,A2: set_Pr4934435412358123699_a_nat,D2: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ C @ A2 )
=> ( ( ord_le8666007276011122963_a_nat @ D2 @ B2 )
=> ( ord_le8666007276011122963_a_nat @ ( sup_su459911885395995103_a_nat @ C @ D2 ) @ ( sup_su459911885395995103_a_nat @ A2 @ B2 ) ) ) ) ).
% sup.mono
thf(fact_461_sup_Omono,axiom,
! [C: set_Pr4386577575007340137rm_a_b,A2: set_Pr4386577575007340137rm_a_b,D2: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ C @ A2 )
=> ( ( ord_le118470702582115849rm_a_b @ D2 @ B2 )
=> ( ord_le118470702582115849rm_a_b @ ( sup_su6776935440552674877rm_a_b @ C @ D2 ) @ ( sup_su6776935440552674877rm_a_b @ A2 @ B2 ) ) ) ) ).
% sup.mono
thf(fact_462_sup_Omono,axiom,
! [C: set_Pr9093778441882193744at_nat,A2: set_Pr9093778441882193744at_nat,D2: set_Pr9093778441882193744at_nat,B2: set_Pr9093778441882193744at_nat] :
( ( ord_le3678578370064672496at_nat @ C @ A2 )
=> ( ( ord_le3678578370064672496at_nat @ D2 @ B2 )
=> ( ord_le3678578370064672496at_nat @ ( sup_su5262723077604177188at_nat @ C @ D2 ) @ ( sup_su5262723077604177188at_nat @ A2 @ B2 ) ) ) ) ).
% sup.mono
thf(fact_463_le__supI2,axiom,
! [X3: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat,A2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X3 @ B2 )
=> ( ord_le8666007276011122963_a_nat @ X3 @ ( sup_su459911885395995103_a_nat @ A2 @ B2 ) ) ) ).
% le_supI2
thf(fact_464_le__supI2,axiom,
! [X3: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b,A2: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ X3 @ B2 )
=> ( ord_le118470702582115849rm_a_b @ X3 @ ( sup_su6776935440552674877rm_a_b @ A2 @ B2 ) ) ) ).
% le_supI2
thf(fact_465_le__supI2,axiom,
! [X3: set_Pr9093778441882193744at_nat,B2: set_Pr9093778441882193744at_nat,A2: set_Pr9093778441882193744at_nat] :
( ( ord_le3678578370064672496at_nat @ X3 @ B2 )
=> ( ord_le3678578370064672496at_nat @ X3 @ ( sup_su5262723077604177188at_nat @ A2 @ B2 ) ) ) ).
% le_supI2
thf(fact_466_le__supI1,axiom,
! [X3: set_Pr4934435412358123699_a_nat,A2: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X3 @ A2 )
=> ( ord_le8666007276011122963_a_nat @ X3 @ ( sup_su459911885395995103_a_nat @ A2 @ B2 ) ) ) ).
% le_supI1
thf(fact_467_le__supI1,axiom,
! [X3: set_Pr4386577575007340137rm_a_b,A2: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ X3 @ A2 )
=> ( ord_le118470702582115849rm_a_b @ X3 @ ( sup_su6776935440552674877rm_a_b @ A2 @ B2 ) ) ) ).
% le_supI1
thf(fact_468_le__supI1,axiom,
! [X3: set_Pr9093778441882193744at_nat,A2: set_Pr9093778441882193744at_nat,B2: set_Pr9093778441882193744at_nat] :
( ( ord_le3678578370064672496at_nat @ X3 @ A2 )
=> ( ord_le3678578370064672496at_nat @ X3 @ ( sup_su5262723077604177188at_nat @ A2 @ B2 ) ) ) ).
% le_supI1
thf(fact_469_sup__ge2,axiom,
! [Y3: set_Pr4934435412358123699_a_nat,X3: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ Y3 @ ( sup_su459911885395995103_a_nat @ X3 @ Y3 ) ) ).
% sup_ge2
thf(fact_470_sup__ge2,axiom,
! [Y3: set_Pr4386577575007340137rm_a_b,X3: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ Y3 @ ( sup_su6776935440552674877rm_a_b @ X3 @ Y3 ) ) ).
% sup_ge2
thf(fact_471_sup__ge2,axiom,
! [Y3: set_Pr9093778441882193744at_nat,X3: set_Pr9093778441882193744at_nat] : ( ord_le3678578370064672496at_nat @ Y3 @ ( sup_su5262723077604177188at_nat @ X3 @ Y3 ) ) ).
% sup_ge2
thf(fact_472_sup__ge1,axiom,
! [X3: set_Pr4934435412358123699_a_nat,Y3: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ X3 @ ( sup_su459911885395995103_a_nat @ X3 @ Y3 ) ) ).
% sup_ge1
thf(fact_473_sup__ge1,axiom,
! [X3: set_Pr4386577575007340137rm_a_b,Y3: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ X3 @ ( sup_su6776935440552674877rm_a_b @ X3 @ Y3 ) ) ).
% sup_ge1
thf(fact_474_sup__ge1,axiom,
! [X3: set_Pr9093778441882193744at_nat,Y3: set_Pr9093778441882193744at_nat] : ( ord_le3678578370064672496at_nat @ X3 @ ( sup_su5262723077604177188at_nat @ X3 @ Y3 ) ) ).
% sup_ge1
thf(fact_475_le__supI,axiom,
! [A2: set_Pr4934435412358123699_a_nat,X3: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A2 @ X3 )
=> ( ( ord_le8666007276011122963_a_nat @ B2 @ X3 )
=> ( ord_le8666007276011122963_a_nat @ ( sup_su459911885395995103_a_nat @ A2 @ B2 ) @ X3 ) ) ) ).
% le_supI
thf(fact_476_le__supI,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,X3: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A2 @ X3 )
=> ( ( ord_le118470702582115849rm_a_b @ B2 @ X3 )
=> ( ord_le118470702582115849rm_a_b @ ( sup_su6776935440552674877rm_a_b @ A2 @ B2 ) @ X3 ) ) ) ).
% le_supI
thf(fact_477_le__supI,axiom,
! [A2: set_Pr9093778441882193744at_nat,X3: set_Pr9093778441882193744at_nat,B2: set_Pr9093778441882193744at_nat] :
( ( ord_le3678578370064672496at_nat @ A2 @ X3 )
=> ( ( ord_le3678578370064672496at_nat @ B2 @ X3 )
=> ( ord_le3678578370064672496at_nat @ ( sup_su5262723077604177188at_nat @ A2 @ B2 ) @ X3 ) ) ) ).
% le_supI
thf(fact_478_le__supE,axiom,
! [A2: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat,X3: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ ( sup_su459911885395995103_a_nat @ A2 @ B2 ) @ X3 )
=> ~ ( ( ord_le8666007276011122963_a_nat @ A2 @ X3 )
=> ~ ( ord_le8666007276011122963_a_nat @ B2 @ X3 ) ) ) ).
% le_supE
thf(fact_479_le__supE,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b,X3: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ ( sup_su6776935440552674877rm_a_b @ A2 @ B2 ) @ X3 )
=> ~ ( ( ord_le118470702582115849rm_a_b @ A2 @ X3 )
=> ~ ( ord_le118470702582115849rm_a_b @ B2 @ X3 ) ) ) ).
% le_supE
thf(fact_480_le__supE,axiom,
! [A2: set_Pr9093778441882193744at_nat,B2: set_Pr9093778441882193744at_nat,X3: set_Pr9093778441882193744at_nat] :
( ( ord_le3678578370064672496at_nat @ ( sup_su5262723077604177188at_nat @ A2 @ B2 ) @ X3 )
=> ~ ( ( ord_le3678578370064672496at_nat @ A2 @ X3 )
=> ~ ( ord_le3678578370064672496at_nat @ B2 @ X3 ) ) ) ).
% le_supE
thf(fact_481_inf__sup__ord_I3_J,axiom,
! [X3: set_Pr4934435412358123699_a_nat,Y3: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ X3 @ ( sup_su459911885395995103_a_nat @ X3 @ Y3 ) ) ).
% inf_sup_ord(3)
thf(fact_482_inf__sup__ord_I3_J,axiom,
! [X3: set_Pr4386577575007340137rm_a_b,Y3: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ X3 @ ( sup_su6776935440552674877rm_a_b @ X3 @ Y3 ) ) ).
% inf_sup_ord(3)
thf(fact_483_inf__sup__ord_I3_J,axiom,
! [X3: set_Pr9093778441882193744at_nat,Y3: set_Pr9093778441882193744at_nat] : ( ord_le3678578370064672496at_nat @ X3 @ ( sup_su5262723077604177188at_nat @ X3 @ Y3 ) ) ).
% inf_sup_ord(3)
thf(fact_484_inf__sup__ord_I4_J,axiom,
! [Y3: set_Pr4934435412358123699_a_nat,X3: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ Y3 @ ( sup_su459911885395995103_a_nat @ X3 @ Y3 ) ) ).
% inf_sup_ord(4)
thf(fact_485_inf__sup__ord_I4_J,axiom,
! [Y3: set_Pr4386577575007340137rm_a_b,X3: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ Y3 @ ( sup_su6776935440552674877rm_a_b @ X3 @ Y3 ) ) ).
% inf_sup_ord(4)
thf(fact_486_inf__sup__ord_I4_J,axiom,
! [Y3: set_Pr9093778441882193744at_nat,X3: set_Pr9093778441882193744at_nat] : ( ord_le3678578370064672496at_nat @ Y3 @ ( sup_su5262723077604177188at_nat @ X3 @ Y3 ) ) ).
% inf_sup_ord(4)
thf(fact_487_UNF__onI,axiom,
! [A: set_Pr4386577575007340137rm_a_b,R: set_Pr2972776593051762503rm_a_b] :
( ! [A6: produc357393685978478089rm_a_b,B6: produc357393685978478089rm_a_b,C6: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ A6 @ A )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A6 @ B6 ) @ ( abstra1496011390211842225rm_a_b @ R ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A6 @ C6 ) @ ( abstra1496011390211842225rm_a_b @ R ) )
=> ( B6 = C6 ) ) ) )
=> ( abstra1303798319023342519rm_a_b @ R @ A ) ) ).
% UNF_onI
thf(fact_488_UNF__onI,axiom,
! [A: set_Pr4934435412358123699_a_nat,R: set_Pr1811044260758604347_a_nat] :
( ! [A6: product_prod_a_nat,B6: product_prod_a_nat,C6: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ A6 @ A )
=> ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A6 @ B6 ) @ ( abstra3832961281062565333_a_nat @ R ) )
=> ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A6 @ C6 ) @ ( abstra3832961281062565333_a_nat @ R ) )
=> ( B6 = C6 ) ) ) )
=> ( abstra7833083522531381839_a_nat @ R @ A ) ) ).
% UNF_onI
thf(fact_489_UNF__onI,axiom,
! [A: set_Pr9093778441882193744at_nat,R: set_Pr7047708467701517959at_nat] :
( ! [A6: produc8199716216217303280at_nat,B6: produc8199716216217303280at_nat,C6: produc8199716216217303280at_nat] :
( ( member7226740684066999833at_nat @ A6 @ A )
=> ( ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ A6 @ B6 ) @ ( abstra1436239171852513176at_nat @ R ) )
=> ( ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ A6 @ C6 ) @ ( abstra1436239171852513176at_nat @ R ) )
=> ( B6 = C6 ) ) ) )
=> ( abstra4852570278320572318at_nat @ R @ A ) ) ).
% UNF_onI
thf(fact_490_UNF__onI,axiom,
! [A: set_term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ! [A6: term_a_b,B6: term_a_b,C6: term_a_b] :
( ( member_term_a_b @ A6 @ A )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A6 @ B6 ) @ ( abstra2972651231392952684rm_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A6 @ C6 ) @ ( abstra2972651231392952684rm_a_b @ R ) )
=> ( B6 = C6 ) ) ) )
=> ( abstra3619855735679963878rm_a_b @ R @ A ) ) ).
% UNF_onI
thf(fact_491_WN__onI,axiom,
! [A: set_Pr4386577575007340137rm_a_b,R: set_Pr2972776593051762503rm_a_b] :
( ! [A6: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ A6 @ A )
=> ? [B8: produc357393685978478089rm_a_b] : ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A6 @ B8 ) @ ( abstra1496011390211842225rm_a_b @ R ) ) )
=> ( abstra2721873542170697047rm_a_b @ R @ A ) ) ).
% WN_onI
thf(fact_492_WN__onI,axiom,
! [A: set_Pr4934435412358123699_a_nat,R: set_Pr1811044260758604347_a_nat] :
( ! [A6: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ A6 @ A )
=> ? [B8: product_prod_a_nat] : ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A6 @ B8 ) @ ( abstra3832961281062565333_a_nat @ R ) ) )
=> ( abstra4582696096554303663_a_nat @ R @ A ) ) ).
% WN_onI
thf(fact_493_WN__onI,axiom,
! [A: set_Pr9093778441882193744at_nat,R: set_Pr7047708467701517959at_nat] :
( ! [A6: produc8199716216217303280at_nat] :
( ( member7226740684066999833at_nat @ A6 @ A )
=> ? [B8: produc8199716216217303280at_nat] : ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ A6 @ B8 ) @ ( abstra1436239171852513176at_nat @ R ) ) )
=> ( abstra7972952937207517502at_nat @ R @ A ) ) ).
% WN_onI
thf(fact_494_WN__onI,axiom,
! [A: set_term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ! [A6: term_a_b] :
( ( member_term_a_b @ A6 @ A )
=> ? [B8: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A6 @ B8 ) @ ( abstra2972651231392952684rm_a_b @ R ) ) )
=> ( abstra4644735066934440262rm_a_b @ R @ A ) ) ).
% WN_onI
thf(fact_495_normalizability__I,axiom,
! [A2: produc357393685978478089rm_a_b,B2: produc357393685978478089rm_a_b,A: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A2 @ B2 ) @ ( transi2615809358984392588rm_a_b @ A ) )
=> ( ( member5869715511025134514rm_a_b @ B2 @ ( abstra1902783451776770197rm_a_b @ A ) )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A2 @ B2 ) @ ( abstra1496011390211842225rm_a_b @ A ) ) ) ) ).
% normalizability_I
thf(fact_496_normalizability__I,axiom,
! [A2: product_prod_a_nat,B2: product_prod_a_nat,A: set_Pr1811044260758604347_a_nat] :
( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A2 @ B2 ) @ ( transi2726145917338391738_a_nat @ A ) )
=> ( ( member5724188588386418708_a_nat @ B2 @ ( abstra2685462378305987313_a_nat @ A ) )
=> ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A2 @ B2 ) @ ( abstra3832961281062565333_a_nat @ A ) ) ) ) ).
% normalizability_I
thf(fact_497_normalizability__I,axiom,
! [A2: produc8199716216217303280at_nat,B2: produc8199716216217303280at_nat,A: set_Pr7047708467701517959at_nat] :
( ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ A2 @ B2 ) @ ( transi3422749762518297843at_nat @ A ) )
=> ( ( member7226740684066999833at_nat @ B2 @ ( abstra8406325002171223420at_nat @ A ) )
=> ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ A2 @ B2 ) @ ( abstra1436239171852513176at_nat @ A ) ) ) ) ).
% normalizability_I
thf(fact_498_normalizability__I,axiom,
! [A2: term_a_b,B2: term_a_b,A: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( transi7742714808557438673rm_a_b @ A ) )
=> ( ( member_term_a_b @ B2 @ ( abstract_NF_term_a_b @ A ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( abstra2972651231392952684rm_a_b @ A ) ) ) ) ).
% normalizability_I
thf(fact_499_UNF__onD,axiom,
! [R: set_Pr2972776593051762503rm_a_b,A: set_Pr4386577575007340137rm_a_b,A2: produc357393685978478089rm_a_b,B2: produc357393685978478089rm_a_b,C: produc357393685978478089rm_a_b] :
( ( abstra1303798319023342519rm_a_b @ R @ A )
=> ( ( member5869715511025134514rm_a_b @ A2 @ A )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A2 @ B2 ) @ ( abstra1496011390211842225rm_a_b @ R ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A2 @ C ) @ ( abstra1496011390211842225rm_a_b @ R ) )
=> ( B2 = C ) ) ) ) ) ).
% UNF_onD
thf(fact_500_UNF__onD,axiom,
! [R: set_Pr1811044260758604347_a_nat,A: set_Pr4934435412358123699_a_nat,A2: product_prod_a_nat,B2: product_prod_a_nat,C: product_prod_a_nat] :
( ( abstra7833083522531381839_a_nat @ R @ A )
=> ( ( member5724188588386418708_a_nat @ A2 @ A )
=> ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A2 @ B2 ) @ ( abstra3832961281062565333_a_nat @ R ) )
=> ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A2 @ C ) @ ( abstra3832961281062565333_a_nat @ R ) )
=> ( B2 = C ) ) ) ) ) ).
% UNF_onD
thf(fact_501_UNF__onD,axiom,
! [R: set_Pr7047708467701517959at_nat,A: set_Pr9093778441882193744at_nat,A2: produc8199716216217303280at_nat,B2: produc8199716216217303280at_nat,C: produc8199716216217303280at_nat] :
( ( abstra4852570278320572318at_nat @ R @ A )
=> ( ( member7226740684066999833at_nat @ A2 @ A )
=> ( ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ A2 @ B2 ) @ ( abstra1436239171852513176at_nat @ R ) )
=> ( ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ A2 @ C ) @ ( abstra1436239171852513176at_nat @ R ) )
=> ( B2 = C ) ) ) ) ) ).
% UNF_onD
thf(fact_502_UNF__onD,axiom,
! [R: set_Pr4386577575007340137rm_a_b,A: set_term_a_b,A2: term_a_b,B2: term_a_b,C: term_a_b] :
( ( abstra3619855735679963878rm_a_b @ R @ A )
=> ( ( member_term_a_b @ A2 @ A )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( abstra2972651231392952684rm_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ C ) @ ( abstra2972651231392952684rm_a_b @ R ) )
=> ( B2 = C ) ) ) ) ) ).
% UNF_onD
thf(fact_503_UNF__onE,axiom,
! [R: set_Pr2972776593051762503rm_a_b,A: set_Pr4386577575007340137rm_a_b,A2: produc357393685978478089rm_a_b,B2: produc357393685978478089rm_a_b,C: produc357393685978478089rm_a_b] :
( ( abstra1303798319023342519rm_a_b @ R @ A )
=> ( ( member5869715511025134514rm_a_b @ A2 @ A )
=> ( ( B2 != C )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A2 @ B2 ) @ ( abstra1496011390211842225rm_a_b @ R ) )
=> ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A2 @ C ) @ ( abstra1496011390211842225rm_a_b @ R ) ) ) ) ) ) ).
% UNF_onE
thf(fact_504_UNF__onE,axiom,
! [R: set_Pr1811044260758604347_a_nat,A: set_Pr4934435412358123699_a_nat,A2: product_prod_a_nat,B2: product_prod_a_nat,C: product_prod_a_nat] :
( ( abstra7833083522531381839_a_nat @ R @ A )
=> ( ( member5724188588386418708_a_nat @ A2 @ A )
=> ( ( B2 != C )
=> ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A2 @ B2 ) @ ( abstra3832961281062565333_a_nat @ R ) )
=> ~ ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A2 @ C ) @ ( abstra3832961281062565333_a_nat @ R ) ) ) ) ) ) ).
% UNF_onE
thf(fact_505_UNF__onE,axiom,
! [R: set_Pr7047708467701517959at_nat,A: set_Pr9093778441882193744at_nat,A2: produc8199716216217303280at_nat,B2: produc8199716216217303280at_nat,C: produc8199716216217303280at_nat] :
( ( abstra4852570278320572318at_nat @ R @ A )
=> ( ( member7226740684066999833at_nat @ A2 @ A )
=> ( ( B2 != C )
=> ( ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ A2 @ B2 ) @ ( abstra1436239171852513176at_nat @ R ) )
=> ~ ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ A2 @ C ) @ ( abstra1436239171852513176at_nat @ R ) ) ) ) ) ) ).
% UNF_onE
thf(fact_506_UNF__onE,axiom,
! [R: set_Pr4386577575007340137rm_a_b,A: set_term_a_b,A2: term_a_b,B2: term_a_b,C: term_a_b] :
( ( abstra3619855735679963878rm_a_b @ R @ A )
=> ( ( member_term_a_b @ A2 @ A )
=> ( ( B2 != C )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( abstra2972651231392952684rm_a_b @ R ) )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ C ) @ ( abstra2972651231392952684rm_a_b @ R ) ) ) ) ) ) ).
% UNF_onE
thf(fact_507_UNF__on__def,axiom,
( abstra3619855735679963878rm_a_b
= ( ^ [R2: set_Pr4386577575007340137rm_a_b,A4: set_term_a_b] :
! [X4: term_a_b] :
( ( member_term_a_b @ X4 @ A4 )
=> ! [B5: term_a_b,C5: term_a_b] :
( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X4 @ B5 ) @ ( abstra2972651231392952684rm_a_b @ R2 ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X4 @ C5 ) @ ( abstra2972651231392952684rm_a_b @ R2 ) ) )
=> ( B5 = C5 ) ) ) ) ) ).
% UNF_on_def
thf(fact_508_same__fstI,axiom,
! [P: ( nat > nat ) > $o,X3: nat > nat,Y8: nat,Y3: nat,R4: ( nat > nat ) > set_Pr1261947904930325089at_nat] :
( ( P @ X3 )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y8 @ Y3 ) @ ( R4 @ X3 ) )
=> ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ ( produc72220940542539688at_nat @ X3 @ Y8 ) @ ( produc72220940542539688at_nat @ X3 @ Y3 ) ) @ ( same_fst_nat_nat_nat @ P @ R4 ) ) ) ) ).
% same_fstI
thf(fact_509_same__fstI,axiom,
! [P: a > $o,X3: a,Y8: nat,Y3: nat,R4: a > set_Pr1261947904930325089at_nat] :
( ( P @ X3 )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y8 @ Y3 ) @ ( R4 @ X3 ) )
=> ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ ( product_Pair_a_nat @ X3 @ Y8 ) @ ( product_Pair_a_nat @ X3 @ Y3 ) ) @ ( same_fst_a_nat @ P @ R4 ) ) ) ) ).
% same_fstI
thf(fact_510_same__fstI,axiom,
! [P: term_a_b > $o,X3: term_a_b,Y8: term_a_b,Y3: term_a_b,R4: term_a_b > set_Pr4386577575007340137rm_a_b] :
( ( P @ X3 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y8 @ Y3 ) @ ( R4 @ X3 ) )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y8 ) @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) ) @ ( same_f843731131676418198rm_a_b @ P @ R4 ) ) ) ) ).
% same_fstI
thf(fact_511_NF__I,axiom,
! [A2: produc357393685978478089rm_a_b,A: set_Pr2972776593051762503rm_a_b] :
( ! [B6: produc357393685978478089rm_a_b] :
~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A2 @ B6 ) @ A )
=> ( member5869715511025134514rm_a_b @ A2 @ ( abstra1902783451776770197rm_a_b @ A ) ) ) ).
% NF_I
thf(fact_512_NF__I,axiom,
! [A2: product_prod_a_nat,A: set_Pr1811044260758604347_a_nat] :
( ! [B6: product_prod_a_nat] :
~ ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A2 @ B6 ) @ A )
=> ( member5724188588386418708_a_nat @ A2 @ ( abstra2685462378305987313_a_nat @ A ) ) ) ).
% NF_I
thf(fact_513_NF__I,axiom,
! [A2: produc8199716216217303280at_nat,A: set_Pr7047708467701517959at_nat] :
( ! [B6: produc8199716216217303280at_nat] :
~ ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ A2 @ B6 ) @ A )
=> ( member7226740684066999833at_nat @ A2 @ ( abstra8406325002171223420at_nat @ A ) ) ) ).
% NF_I
thf(fact_514_NF__I,axiom,
! [A2: term_a_b,A: set_Pr4386577575007340137rm_a_b] :
( ! [B6: term_a_b] :
~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B6 ) @ A )
=> ( member_term_a_b @ A2 @ ( abstract_NF_term_a_b @ A ) ) ) ).
% NF_I
thf(fact_515_NF__iff__no__step,axiom,
! [A2: produc357393685978478089rm_a_b,A: set_Pr2972776593051762503rm_a_b] :
( ( member5869715511025134514rm_a_b @ A2 @ ( abstra1902783451776770197rm_a_b @ A ) )
= ( ! [B5: produc357393685978478089rm_a_b] :
~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A2 @ B5 ) @ A ) ) ) ).
% NF_iff_no_step
thf(fact_516_NF__iff__no__step,axiom,
! [A2: product_prod_a_nat,A: set_Pr1811044260758604347_a_nat] :
( ( member5724188588386418708_a_nat @ A2 @ ( abstra2685462378305987313_a_nat @ A ) )
= ( ! [B5: product_prod_a_nat] :
~ ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A2 @ B5 ) @ A ) ) ) ).
% NF_iff_no_step
thf(fact_517_NF__iff__no__step,axiom,
! [A2: produc8199716216217303280at_nat,A: set_Pr7047708467701517959at_nat] :
( ( member7226740684066999833at_nat @ A2 @ ( abstra8406325002171223420at_nat @ A ) )
= ( ! [B5: produc8199716216217303280at_nat] :
~ ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ A2 @ B5 ) @ A ) ) ) ).
% NF_iff_no_step
thf(fact_518_NF__iff__no__step,axiom,
! [A2: term_a_b,A: set_Pr4386577575007340137rm_a_b] :
( ( member_term_a_b @ A2 @ ( abstract_NF_term_a_b @ A ) )
= ( ! [B5: term_a_b] :
~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B5 ) @ A ) ) ) ).
% NF_iff_no_step
thf(fact_519_NF__no__step,axiom,
! [A2: produc357393685978478089rm_a_b,A: set_Pr2972776593051762503rm_a_b] :
( ( member5869715511025134514rm_a_b @ A2 @ ( abstra1902783451776770197rm_a_b @ A ) )
=> ! [B8: produc357393685978478089rm_a_b] :
~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A2 @ B8 ) @ A ) ) ).
% NF_no_step
thf(fact_520_NF__no__step,axiom,
! [A2: product_prod_a_nat,A: set_Pr1811044260758604347_a_nat] :
( ( member5724188588386418708_a_nat @ A2 @ ( abstra2685462378305987313_a_nat @ A ) )
=> ! [B8: product_prod_a_nat] :
~ ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A2 @ B8 ) @ A ) ) ).
% NF_no_step
thf(fact_521_NF__no__step,axiom,
! [A2: produc8199716216217303280at_nat,A: set_Pr7047708467701517959at_nat] :
( ( member7226740684066999833at_nat @ A2 @ ( abstra8406325002171223420at_nat @ A ) )
=> ! [B8: produc8199716216217303280at_nat] :
~ ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ A2 @ B8 ) @ A ) ) ).
% NF_no_step
thf(fact_522_NF__no__step,axiom,
! [A2: term_a_b,A: set_Pr4386577575007340137rm_a_b] :
( ( member_term_a_b @ A2 @ ( abstract_NF_term_a_b @ A ) )
=> ! [B8: term_a_b] :
~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B8 ) @ A ) ) ).
% NF_no_step
thf(fact_523_NF__E,axiom,
! [A2: produc357393685978478089rm_a_b,A: set_Pr2972776593051762503rm_a_b,B2: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ A2 @ ( abstra1902783451776770197rm_a_b @ A ) )
=> ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A2 @ B2 ) @ A ) ) ).
% NF_E
thf(fact_524_NF__E,axiom,
! [A2: product_prod_a_nat,A: set_Pr1811044260758604347_a_nat,B2: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ A2 @ ( abstra2685462378305987313_a_nat @ A ) )
=> ~ ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A2 @ B2 ) @ A ) ) ).
% NF_E
thf(fact_525_NF__E,axiom,
! [A2: produc8199716216217303280at_nat,A: set_Pr7047708467701517959at_nat,B2: produc8199716216217303280at_nat] :
( ( member7226740684066999833at_nat @ A2 @ ( abstra8406325002171223420at_nat @ A ) )
=> ~ ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ A2 @ B2 ) @ A ) ) ).
% NF_E
thf(fact_526_NF__E,axiom,
! [A2: term_a_b,A: set_Pr4386577575007340137rm_a_b,B2: term_a_b] :
( ( member_term_a_b @ A2 @ ( abstract_NF_term_a_b @ A ) )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ A ) ) ).
% NF_E
thf(fact_527_NF__not__suc,axiom,
! [X3: produc357393685978478089rm_a_b,Y3: produc357393685978478089rm_a_b,R: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X3 @ Y3 ) @ ( transi2615809358984392588rm_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ X3 @ ( abstra1902783451776770197rm_a_b @ R ) )
=> ( X3 = Y3 ) ) ) ).
% NF_not_suc
thf(fact_528_NF__not__suc,axiom,
! [X3: product_prod_a_nat,Y3: product_prod_a_nat,R: set_Pr1811044260758604347_a_nat] :
( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ X3 @ Y3 ) @ ( transi2726145917338391738_a_nat @ R ) )
=> ( ( member5724188588386418708_a_nat @ X3 @ ( abstra2685462378305987313_a_nat @ R ) )
=> ( X3 = Y3 ) ) ) ).
% NF_not_suc
thf(fact_529_NF__not__suc,axiom,
! [X3: produc8199716216217303280at_nat,Y3: produc8199716216217303280at_nat,R: set_Pr7047708467701517959at_nat] :
( ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ X3 @ Y3 ) @ ( transi3422749762518297843at_nat @ R ) )
=> ( ( member7226740684066999833at_nat @ X3 @ ( abstra8406325002171223420at_nat @ R ) )
=> ( X3 = Y3 ) ) ) ).
% NF_not_suc
thf(fact_530_NF__not__suc,axiom,
! [X3: term_a_b,Y3: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ( ( member_term_a_b @ X3 @ ( abstract_NF_term_a_b @ R ) )
=> ( X3 = Y3 ) ) ) ).
% NF_not_suc
thf(fact_531_NF__anti__mono,axiom,
! [A: set_Pr1811044260758604347_a_nat,B: set_Pr1811044260758604347_a_nat] :
( ( ord_le1018141841545957851_a_nat @ A @ B )
=> ( ord_le8666007276011122963_a_nat @ ( abstra2685462378305987313_a_nat @ B ) @ ( abstra2685462378305987313_a_nat @ A ) ) ) ).
% NF_anti_mono
thf(fact_532_NF__anti__mono,axiom,
! [A: set_Pr2972776593051762503rm_a_b,B: set_Pr2972776593051762503rm_a_b] :
( ( ord_le2593958535671617255rm_a_b @ A @ B )
=> ( ord_le118470702582115849rm_a_b @ ( abstra1902783451776770197rm_a_b @ B ) @ ( abstra1902783451776770197rm_a_b @ A ) ) ) ).
% NF_anti_mono
thf(fact_533_NF__anti__mono,axiom,
! [A: set_Pr7047708467701517959at_nat,B: set_Pr7047708467701517959at_nat] :
( ( ord_le1281664332108154407at_nat @ A @ B )
=> ( ord_le3678578370064672496at_nat @ ( abstra8406325002171223420at_nat @ B ) @ ( abstra8406325002171223420at_nat @ A ) ) ) ).
% NF_anti_mono
thf(fact_534_NF__anti__mono,axiom,
! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A @ B )
=> ( ord_le2705286416250468010rm_a_b @ ( abstract_NF_term_a_b @ B ) @ ( abstract_NF_term_a_b @ A ) ) ) ).
% NF_anti_mono
thf(fact_535_NF__no__trancl__step,axiom,
! [A2: produc357393685978478089rm_a_b,A: set_Pr2972776593051762503rm_a_b] :
( ( member5869715511025134514rm_a_b @ A2 @ ( abstra1902783451776770197rm_a_b @ A ) )
=> ! [B8: produc357393685978478089rm_a_b] :
~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A2 @ B8 ) @ ( transi9211502839322181930rm_a_b @ A ) ) ) ).
% NF_no_trancl_step
thf(fact_536_NF__no__trancl__step,axiom,
! [A2: product_prod_a_nat,A: set_Pr1811044260758604347_a_nat] :
( ( member5724188588386418708_a_nat @ A2 @ ( abstra2685462378305987313_a_nat @ A ) )
=> ! [B8: product_prod_a_nat] :
~ ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A2 @ B8 ) @ ( transi6805586507763294364_a_nat @ A ) ) ) ).
% NF_no_trancl_step
thf(fact_537_NF__no__trancl__step,axiom,
! [A2: produc8199716216217303280at_nat,A: set_Pr7047708467701517959at_nat] :
( ( member7226740684066999833at_nat @ A2 @ ( abstra8406325002171223420at_nat @ A ) )
=> ! [B8: produc8199716216217303280at_nat] :
~ ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ A2 @ B8 ) @ ( transi3146596849628481425at_nat @ A ) ) ) ).
% NF_no_trancl_step
thf(fact_538_NF__no__trancl__step,axiom,
! [A2: term_a_b,A: set_Pr4386577575007340137rm_a_b] :
( ( member_term_a_b @ A2 @ ( abstract_NF_term_a_b @ A ) )
=> ! [B8: term_a_b] :
~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B8 ) @ ( transi7922773638565587891rm_a_b @ A ) ) ) ).
% NF_no_trancl_step
thf(fact_539_normalizability__E,axiom,
! [A2: produc357393685978478089rm_a_b,B2: produc357393685978478089rm_a_b,A: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A2 @ B2 ) @ ( abstra1496011390211842225rm_a_b @ A ) )
=> ~ ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A2 @ B2 ) @ ( transi2615809358984392588rm_a_b @ A ) )
=> ~ ( member5869715511025134514rm_a_b @ B2 @ ( abstra1902783451776770197rm_a_b @ A ) ) ) ) ).
% normalizability_E
thf(fact_540_normalizability__E,axiom,
! [A2: product_prod_a_nat,B2: product_prod_a_nat,A: set_Pr1811044260758604347_a_nat] :
( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A2 @ B2 ) @ ( abstra3832961281062565333_a_nat @ A ) )
=> ~ ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A2 @ B2 ) @ ( transi2726145917338391738_a_nat @ A ) )
=> ~ ( member5724188588386418708_a_nat @ B2 @ ( abstra2685462378305987313_a_nat @ A ) ) ) ) ).
% normalizability_E
thf(fact_541_normalizability__E,axiom,
! [A2: produc8199716216217303280at_nat,B2: produc8199716216217303280at_nat,A: set_Pr7047708467701517959at_nat] :
( ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ A2 @ B2 ) @ ( abstra1436239171852513176at_nat @ A ) )
=> ~ ( ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ A2 @ B2 ) @ ( transi3422749762518297843at_nat @ A ) )
=> ~ ( member7226740684066999833at_nat @ B2 @ ( abstra8406325002171223420at_nat @ A ) ) ) ) ).
% normalizability_E
thf(fact_542_normalizability__E,axiom,
! [A2: term_a_b,B2: term_a_b,A: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( abstra2972651231392952684rm_a_b @ A ) )
=> ~ ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( transi7742714808557438673rm_a_b @ A ) )
=> ~ ( member_term_a_b @ B2 @ ( abstract_NF_term_a_b @ A ) ) ) ) ).
% normalizability_E
thf(fact_543_WN__on__def,axiom,
( abstra4644735066934440262rm_a_b
= ( ^ [R2: set_Pr4386577575007340137rm_a_b,A4: set_term_a_b] :
! [X4: term_a_b] :
( ( member_term_a_b @ X4 @ A4 )
=> ? [B5: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X4 @ B5 ) @ ( abstra2972651231392952684rm_a_b @ R2 ) ) ) ) ) ).
% WN_on_def
thf(fact_544_WN__onE,axiom,
! [R: set_Pr2972776593051762503rm_a_b,A: set_Pr4386577575007340137rm_a_b,A2: produc357393685978478089rm_a_b] :
( ( abstra2721873542170697047rm_a_b @ R @ A )
=> ( ( member5869715511025134514rm_a_b @ A2 @ A )
=> ~ ! [B6: produc357393685978478089rm_a_b] :
~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A2 @ B6 ) @ ( abstra1496011390211842225rm_a_b @ R ) ) ) ) ).
% WN_onE
thf(fact_545_WN__onE,axiom,
! [R: set_Pr1811044260758604347_a_nat,A: set_Pr4934435412358123699_a_nat,A2: product_prod_a_nat] :
( ( abstra4582696096554303663_a_nat @ R @ A )
=> ( ( member5724188588386418708_a_nat @ A2 @ A )
=> ~ ! [B6: product_prod_a_nat] :
~ ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A2 @ B6 ) @ ( abstra3832961281062565333_a_nat @ R ) ) ) ) ).
% WN_onE
thf(fact_546_WN__onE,axiom,
! [R: set_Pr7047708467701517959at_nat,A: set_Pr9093778441882193744at_nat,A2: produc8199716216217303280at_nat] :
( ( abstra7972952937207517502at_nat @ R @ A )
=> ( ( member7226740684066999833at_nat @ A2 @ A )
=> ~ ! [B6: produc8199716216217303280at_nat] :
~ ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ A2 @ B6 ) @ ( abstra1436239171852513176at_nat @ R ) ) ) ) ).
% WN_onE
thf(fact_547_WN__onE,axiom,
! [R: set_Pr4386577575007340137rm_a_b,A: set_term_a_b,A2: term_a_b] :
( ( abstra4644735066934440262rm_a_b @ R @ A )
=> ( ( member_term_a_b @ A2 @ A )
=> ~ ! [B6: term_a_b] :
~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B6 ) @ ( abstra2972651231392952684rm_a_b @ R ) ) ) ) ).
% WN_onE
thf(fact_548_WN__onD,axiom,
! [R: set_Pr2972776593051762503rm_a_b,A: set_Pr4386577575007340137rm_a_b,A2: produc357393685978478089rm_a_b] :
( ( abstra2721873542170697047rm_a_b @ R @ A )
=> ( ( member5869715511025134514rm_a_b @ A2 @ A )
=> ? [B6: produc357393685978478089rm_a_b] : ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A2 @ B6 ) @ ( abstra1496011390211842225rm_a_b @ R ) ) ) ) ).
% WN_onD
thf(fact_549_WN__onD,axiom,
! [R: set_Pr1811044260758604347_a_nat,A: set_Pr4934435412358123699_a_nat,A2: product_prod_a_nat] :
( ( abstra4582696096554303663_a_nat @ R @ A )
=> ( ( member5724188588386418708_a_nat @ A2 @ A )
=> ? [B6: product_prod_a_nat] : ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A2 @ B6 ) @ ( abstra3832961281062565333_a_nat @ R ) ) ) ) ).
% WN_onD
thf(fact_550_WN__onD,axiom,
! [R: set_Pr7047708467701517959at_nat,A: set_Pr9093778441882193744at_nat,A2: produc8199716216217303280at_nat] :
( ( abstra7972952937207517502at_nat @ R @ A )
=> ( ( member7226740684066999833at_nat @ A2 @ A )
=> ? [B6: produc8199716216217303280at_nat] : ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ A2 @ B6 ) @ ( abstra1436239171852513176at_nat @ R ) ) ) ) ).
% WN_onD
thf(fact_551_WN__onD,axiom,
! [R: set_Pr4386577575007340137rm_a_b,A: set_term_a_b,A2: term_a_b] :
( ( abstra4644735066934440262rm_a_b @ R @ A )
=> ( ( member_term_a_b @ A2 @ A )
=> ? [B6: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B6 ) @ ( abstra2972651231392952684rm_a_b @ R ) ) ) ) ).
% WN_onD
thf(fact_552_NF__join__imp__reach,axiom,
! [Y3: produc357393685978478089rm_a_b,A: set_Pr2972776593051762503rm_a_b,X3: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ Y3 @ ( abstra1902783451776770197rm_a_b @ A ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X3 @ Y3 ) @ ( abstra4889682940153293507rm_a_b @ A ) )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X3 @ Y3 ) @ ( transi2615809358984392588rm_a_b @ A ) ) ) ) ).
% NF_join_imp_reach
thf(fact_553_NF__join__imp__reach,axiom,
! [Y3: product_prod_a_nat,A: set_Pr1811044260758604347_a_nat,X3: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ Y3 @ ( abstra2685462378305987313_a_nat @ A ) )
=> ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ X3 @ Y3 ) @ ( abstra881512052393830979_a_nat @ A ) )
=> ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ X3 @ Y3 ) @ ( transi2726145917338391738_a_nat @ A ) ) ) ) ).
% NF_join_imp_reach
thf(fact_554_NF__join__imp__reach,axiom,
! [Y3: produc8199716216217303280at_nat,A: set_Pr7047708467701517959at_nat,X3: produc8199716216217303280at_nat] :
( ( member7226740684066999833at_nat @ Y3 @ ( abstra8406325002171223420at_nat @ A ) )
=> ( ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ X3 @ Y3 ) @ ( abstra8195730983011915946at_nat @ A ) )
=> ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ X3 @ Y3 ) @ ( transi3422749762518297843at_nat @ A ) ) ) ) ).
% NF_join_imp_reach
thf(fact_555_NF__join__imp__reach,axiom,
! [Y3: term_a_b,A: set_Pr4386577575007340137rm_a_b,X3: term_a_b] :
( ( member_term_a_b @ Y3 @ ( abstract_NF_term_a_b @ A ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ ( abstra4096080454567261402rm_a_b @ A ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ ( transi7742714808557438673rm_a_b @ A ) ) ) ) ).
% NF_join_imp_reach
thf(fact_556_empty__iff,axiom,
! [C: produc357393685978478089rm_a_b] :
~ ( member5869715511025134514rm_a_b @ C @ bot_bo197521221353338581rm_a_b ) ).
% empty_iff
thf(fact_557_empty__iff,axiom,
! [C: product_prod_a_nat] :
~ ( member5724188588386418708_a_nat @ C @ bot_bo9049108969261143879_a_nat ) ).
% empty_iff
thf(fact_558_empty__iff,axiom,
! [C: produc8199716216217303280at_nat] :
~ ( member7226740684066999833at_nat @ C @ bot_bo8000532843592726972at_nat ) ).
% empty_iff
thf(fact_559_all__not__in__conv,axiom,
! [A: set_Pr4386577575007340137rm_a_b] :
( ( ! [X4: produc357393685978478089rm_a_b] :
~ ( member5869715511025134514rm_a_b @ X4 @ A ) )
= ( A = bot_bo197521221353338581rm_a_b ) ) ).
% all_not_in_conv
thf(fact_560_all__not__in__conv,axiom,
! [A: set_Pr4934435412358123699_a_nat] :
( ( ! [X4: product_prod_a_nat] :
~ ( member5724188588386418708_a_nat @ X4 @ A ) )
= ( A = bot_bo9049108969261143879_a_nat ) ) ).
% all_not_in_conv
thf(fact_561_all__not__in__conv,axiom,
! [A: set_Pr9093778441882193744at_nat] :
( ( ! [X4: produc8199716216217303280at_nat] :
~ ( member7226740684066999833at_nat @ X4 @ A ) )
= ( A = bot_bo8000532843592726972at_nat ) ) ).
% all_not_in_conv
thf(fact_562_subset__empty,axiom,
! [A: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A @ bot_bo9049108969261143879_a_nat )
= ( A = bot_bo9049108969261143879_a_nat ) ) ).
% subset_empty
thf(fact_563_subset__empty,axiom,
! [A: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A @ bot_bo197521221353338581rm_a_b )
= ( A = bot_bo197521221353338581rm_a_b ) ) ).
% subset_empty
thf(fact_564_subset__empty,axiom,
! [A: set_Pr9093778441882193744at_nat] :
( ( ord_le3678578370064672496at_nat @ A @ bot_bo8000532843592726972at_nat )
= ( A = bot_bo8000532843592726972at_nat ) ) ).
% subset_empty
thf(fact_565_empty__subsetI,axiom,
! [A: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ bot_bo9049108969261143879_a_nat @ A ) ).
% empty_subsetI
thf(fact_566_empty__subsetI,axiom,
! [A: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ bot_bo197521221353338581rm_a_b @ A ) ).
% empty_subsetI
thf(fact_567_empty__subsetI,axiom,
! [A: set_Pr9093778441882193744at_nat] : ( ord_le3678578370064672496at_nat @ bot_bo8000532843592726972at_nat @ A ) ).
% empty_subsetI
thf(fact_568_converse__iff,axiom,
! [A2: nat,B2: nat > nat,R: set_Pr9093778441882193744at_nat] :
( ( member8336108448496249625at_nat @ ( produc7839516862119294504at_nat @ A2 @ B2 ) @ ( converse_nat_nat_nat @ R ) )
= ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ B2 @ A2 ) @ R ) ) ).
% converse_iff
thf(fact_569_converse__iff,axiom,
! [A2: nat,B2: a,R: set_Pr4934435412358123699_a_nat] :
( ( member8962352052110095674_nat_a @ ( product_Pair_nat_a @ A2 @ B2 ) @ ( converse_a_nat @ R ) )
= ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ B2 @ A2 ) @ R ) ) ).
% converse_iff
thf(fact_570_converse__iff,axiom,
! [A2: term_a_b,B2: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( conver7608559976937845093rm_a_b @ R ) )
= ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ A2 ) @ R ) ) ).
% converse_iff
thf(fact_571_converse__iff,axiom,
! [A2: nat > nat,B2: nat,R: set_Pr6370437063884598352at_nat] :
( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ A2 @ B2 ) @ ( converse_nat_nat_nat2 @ R ) )
= ( member8336108448496249625at_nat @ ( produc7839516862119294504at_nat @ B2 @ A2 ) @ R ) ) ).
% converse_iff
thf(fact_572_converse__iff,axiom,
! [A2: a,B2: nat,R: set_Pr4193341848836149977_nat_a] :
( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ A2 @ B2 ) @ ( converse_nat_a @ R ) )
= ( member8962352052110095674_nat_a @ ( product_Pair_nat_a @ B2 @ A2 ) @ R ) ) ).
% converse_iff
thf(fact_573_IdI,axiom,
! [A2: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ A2 ) @ id_term_a_b ) ).
% IdI
thf(fact_574_pair__in__Id__conv,axiom,
! [A2: term_a_b,B2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ id_term_a_b )
= ( A2 = B2 ) ) ).
% pair_in_Id_conv
thf(fact_575_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_576_converse__mono,axiom,
! [R: set_Pr9093778441882193744at_nat,S: set_Pr9093778441882193744at_nat] :
( ( ord_le955236992067077104at_nat @ ( converse_nat_nat_nat @ R ) @ ( converse_nat_nat_nat @ S ) )
= ( ord_le3678578370064672496at_nat @ R @ S ) ) ).
% converse_mono
thf(fact_577_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_578_converse__mono,axiom,
! [R: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ ( conver7608559976937845093rm_a_b @ R ) @ ( conver7608559976937845093rm_a_b @ S ) )
= ( ord_le118470702582115849rm_a_b @ R @ S ) ) ).
% converse_mono
thf(fact_579_converse__mono,axiom,
! [R: set_Pr6370437063884598352at_nat,S: set_Pr6370437063884598352at_nat] :
( ( ord_le3678578370064672496at_nat @ ( converse_nat_nat_nat2 @ R ) @ ( converse_nat_nat_nat2 @ S ) )
= ( ord_le955236992067077104at_nat @ R @ S ) ) ).
% converse_mono
thf(fact_580_joinI__right,axiom,
! [B2: term_a_b,A2: term_a_b,A: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ A2 ) @ ( transi7742714808557438673rm_a_b @ A ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( abstra4096080454567261402rm_a_b @ A ) ) ) ).
% joinI_right
thf(fact_581_joinI__left,axiom,
! [A2: term_a_b,B2: term_a_b,A: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( transi7742714808557438673rm_a_b @ A ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( abstra4096080454567261402rm_a_b @ A ) ) ) ).
% joinI_left
thf(fact_582_joinI,axiom,
! [A2: term_a_b,C: term_a_b,A: set_Pr4386577575007340137rm_a_b,B2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ C ) @ ( transi7742714808557438673rm_a_b @ A ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ C ) @ ( transi7742714808557438673rm_a_b @ A ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( abstra4096080454567261402rm_a_b @ A ) ) ) ) ).
% joinI
thf(fact_583_emptyE,axiom,
! [A2: produc357393685978478089rm_a_b] :
~ ( member5869715511025134514rm_a_b @ A2 @ bot_bo197521221353338581rm_a_b ) ).
% emptyE
thf(fact_584_emptyE,axiom,
! [A2: product_prod_a_nat] :
~ ( member5724188588386418708_a_nat @ A2 @ bot_bo9049108969261143879_a_nat ) ).
% emptyE
thf(fact_585_emptyE,axiom,
! [A2: produc8199716216217303280at_nat] :
~ ( member7226740684066999833at_nat @ A2 @ bot_bo8000532843592726972at_nat ) ).
% emptyE
thf(fact_586_equals0D,axiom,
! [A: set_Pr4386577575007340137rm_a_b,A2: produc357393685978478089rm_a_b] :
( ( A = bot_bo197521221353338581rm_a_b )
=> ~ ( member5869715511025134514rm_a_b @ A2 @ A ) ) ).
% equals0D
thf(fact_587_equals0D,axiom,
! [A: set_Pr4934435412358123699_a_nat,A2: product_prod_a_nat] :
( ( A = bot_bo9049108969261143879_a_nat )
=> ~ ( member5724188588386418708_a_nat @ A2 @ A ) ) ).
% equals0D
thf(fact_588_equals0D,axiom,
! [A: set_Pr9093778441882193744at_nat,A2: produc8199716216217303280at_nat] :
( ( A = bot_bo8000532843592726972at_nat )
=> ~ ( member7226740684066999833at_nat @ A2 @ A ) ) ).
% equals0D
thf(fact_589_equals0I,axiom,
! [A: set_Pr4386577575007340137rm_a_b] :
( ! [Y: produc357393685978478089rm_a_b] :
~ ( member5869715511025134514rm_a_b @ Y @ A )
=> ( A = bot_bo197521221353338581rm_a_b ) ) ).
% equals0I
thf(fact_590_equals0I,axiom,
! [A: set_Pr4934435412358123699_a_nat] :
( ! [Y: product_prod_a_nat] :
~ ( member5724188588386418708_a_nat @ Y @ A )
=> ( A = bot_bo9049108969261143879_a_nat ) ) ).
% equals0I
thf(fact_591_equals0I,axiom,
! [A: set_Pr9093778441882193744at_nat] :
( ! [Y: produc8199716216217303280at_nat] :
~ ( member7226740684066999833at_nat @ Y @ A )
=> ( A = bot_bo8000532843592726972at_nat ) ) ).
% equals0I
thf(fact_592_ex__in__conv,axiom,
! [A: set_Pr4386577575007340137rm_a_b] :
( ( ? [X4: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ X4 @ A ) )
= ( A != bot_bo197521221353338581rm_a_b ) ) ).
% ex_in_conv
thf(fact_593_ex__in__conv,axiom,
! [A: set_Pr4934435412358123699_a_nat] :
( ( ? [X4: product_prod_a_nat] : ( member5724188588386418708_a_nat @ X4 @ A ) )
= ( A != bot_bo9049108969261143879_a_nat ) ) ).
% ex_in_conv
thf(fact_594_ex__in__conv,axiom,
! [A: set_Pr9093778441882193744at_nat] :
( ( ? [X4: produc8199716216217303280at_nat] : ( member7226740684066999833at_nat @ X4 @ A ) )
= ( A != bot_bo8000532843592726972at_nat ) ) ).
% ex_in_conv
thf(fact_595_converse_Ocases,axiom,
! [A1: nat,A22: nat > nat,R: set_Pr9093778441882193744at_nat] :
( ( member8336108448496249625at_nat @ ( produc7839516862119294504at_nat @ A1 @ A22 ) @ ( converse_nat_nat_nat @ R ) )
=> ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ A22 @ A1 ) @ R ) ) ).
% converse.cases
thf(fact_596_converse_Ocases,axiom,
! [A1: nat,A22: a,R: set_Pr4934435412358123699_a_nat] :
( ( member8962352052110095674_nat_a @ ( product_Pair_nat_a @ A1 @ A22 ) @ ( converse_a_nat @ R ) )
=> ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ A22 @ A1 ) @ R ) ) ).
% converse.cases
thf(fact_597_converse_Ocases,axiom,
! [A1: term_a_b,A22: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A22 ) @ ( conver7608559976937845093rm_a_b @ R ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A22 @ A1 ) @ R ) ) ).
% converse.cases
thf(fact_598_converse_Ocases,axiom,
! [A1: nat > nat,A22: nat,R: set_Pr6370437063884598352at_nat] :
( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ A1 @ A22 ) @ ( converse_nat_nat_nat2 @ R ) )
=> ( member8336108448496249625at_nat @ ( produc7839516862119294504at_nat @ A22 @ A1 ) @ R ) ) ).
% converse.cases
thf(fact_599_converse_Ocases,axiom,
! [A1: a,A22: nat,R: set_Pr4193341848836149977_nat_a] :
( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ A1 @ A22 ) @ ( converse_nat_a @ R ) )
=> ( member8962352052110095674_nat_a @ ( product_Pair_nat_a @ A22 @ A1 ) @ R ) ) ).
% converse.cases
thf(fact_600_converse_Osimps,axiom,
! [A1: nat,A22: nat > nat,R: set_Pr9093778441882193744at_nat] :
( ( member8336108448496249625at_nat @ ( produc7839516862119294504at_nat @ A1 @ A22 ) @ ( converse_nat_nat_nat @ R ) )
= ( ? [A5: nat > nat,B5: nat] :
( ( A1 = B5 )
& ( A22 = A5 )
& ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ A5 @ B5 ) @ R ) ) ) ) ).
% converse.simps
thf(fact_601_converse_Osimps,axiom,
! [A1: nat,A22: a,R: set_Pr4934435412358123699_a_nat] :
( ( member8962352052110095674_nat_a @ ( product_Pair_nat_a @ A1 @ A22 ) @ ( converse_a_nat @ R ) )
= ( ? [A5: a,B5: nat] :
( ( A1 = B5 )
& ( A22 = A5 )
& ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ A5 @ B5 ) @ R ) ) ) ) ).
% converse.simps
thf(fact_602_converse_Osimps,axiom,
! [A1: term_a_b,A22: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A22 ) @ ( conver7608559976937845093rm_a_b @ R ) )
= ( ? [A5: term_a_b,B5: term_a_b] :
( ( A1 = B5 )
& ( A22 = A5 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A5 @ B5 ) @ R ) ) ) ) ).
% converse.simps
thf(fact_603_converse_Osimps,axiom,
! [A1: nat > nat,A22: nat,R: set_Pr6370437063884598352at_nat] :
( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ A1 @ A22 ) @ ( converse_nat_nat_nat2 @ R ) )
= ( ? [A5: nat,B5: nat > nat] :
( ( A1 = B5 )
& ( A22 = A5 )
& ( member8336108448496249625at_nat @ ( produc7839516862119294504at_nat @ A5 @ B5 ) @ R ) ) ) ) ).
% converse.simps
thf(fact_604_converse_Osimps,axiom,
! [A1: a,A22: nat,R: set_Pr4193341848836149977_nat_a] :
( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ A1 @ A22 ) @ ( converse_nat_a @ R ) )
= ( ? [A5: nat,B5: a] :
( ( A1 = B5 )
& ( A22 = A5 )
& ( member8962352052110095674_nat_a @ ( product_Pair_nat_a @ A5 @ B5 ) @ R ) ) ) ) ).
% converse.simps
thf(fact_605_converseD,axiom,
! [A2: nat,B2: nat > nat,R: set_Pr9093778441882193744at_nat] :
( ( member8336108448496249625at_nat @ ( produc7839516862119294504at_nat @ A2 @ B2 ) @ ( converse_nat_nat_nat @ R ) )
=> ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ B2 @ A2 ) @ R ) ) ).
% converseD
thf(fact_606_converseD,axiom,
! [A2: nat,B2: a,R: set_Pr4934435412358123699_a_nat] :
( ( member8962352052110095674_nat_a @ ( product_Pair_nat_a @ A2 @ B2 ) @ ( converse_a_nat @ R ) )
=> ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ B2 @ A2 ) @ R ) ) ).
% converseD
thf(fact_607_converseD,axiom,
! [A2: term_a_b,B2: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( conver7608559976937845093rm_a_b @ R ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ A2 ) @ R ) ) ).
% converseD
thf(fact_608_converseD,axiom,
! [A2: nat > nat,B2: nat,R: set_Pr6370437063884598352at_nat] :
( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ A2 @ B2 ) @ ( converse_nat_nat_nat2 @ R ) )
=> ( member8336108448496249625at_nat @ ( produc7839516862119294504at_nat @ B2 @ A2 ) @ R ) ) ).
% converseD
thf(fact_609_converseD,axiom,
! [A2: a,B2: nat,R: set_Pr4193341848836149977_nat_a] :
( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ A2 @ B2 ) @ ( converse_nat_a @ R ) )
=> ( member8962352052110095674_nat_a @ ( product_Pair_nat_a @ B2 @ A2 ) @ R ) ) ).
% converseD
thf(fact_610_converseE,axiom,
! [Yx: produc85711943791777264at_nat,R: set_Pr9093778441882193744at_nat] :
( ( member8336108448496249625at_nat @ Yx @ ( converse_nat_nat_nat @ R ) )
=> ~ ! [X: nat > nat,Y: nat] :
( ( Yx
= ( produc7839516862119294504at_nat @ Y @ X ) )
=> ~ ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ X @ Y ) @ R ) ) ) ).
% converseE
thf(fact_611_converseE,axiom,
! [Yx: product_prod_nat_a,R: set_Pr4934435412358123699_a_nat] :
( ( member8962352052110095674_nat_a @ Yx @ ( converse_a_nat @ R ) )
=> ~ ! [X: a,Y: nat] :
( ( Yx
= ( product_Pair_nat_a @ Y @ X ) )
=> ~ ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ X @ Y ) @ R ) ) ) ).
% converseE
thf(fact_612_converseE,axiom,
! [Yx: produc357393685978478089rm_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ Yx @ ( conver7608559976937845093rm_a_b @ R ) )
=> ~ ! [X: term_a_b,Y: term_a_b] :
( ( Yx
= ( produc7020197800436672577rm_a_b @ Y @ X ) )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ R ) ) ) ).
% converseE
thf(fact_613_converseE,axiom,
! [Yx: produc8199716216217303280at_nat,R: set_Pr6370437063884598352at_nat] :
( ( member7226740684066999833at_nat @ Yx @ ( converse_nat_nat_nat2 @ R ) )
=> ~ ! [X: nat,Y: nat > nat] :
( ( Yx
= ( produc72220940542539688at_nat @ Y @ X ) )
=> ~ ( member8336108448496249625at_nat @ ( produc7839516862119294504at_nat @ X @ Y ) @ R ) ) ) ).
% converseE
thf(fact_614_converseE,axiom,
! [Yx: product_prod_a_nat,R: set_Pr4193341848836149977_nat_a] :
( ( member5724188588386418708_a_nat @ Yx @ ( converse_nat_a @ R ) )
=> ~ ! [X: nat,Y: a] :
( ( Yx
= ( product_Pair_a_nat @ Y @ X ) )
=> ~ ( member8962352052110095674_nat_a @ ( product_Pair_nat_a @ X @ Y ) @ R ) ) ) ).
% converseE
thf(fact_615_converseI,axiom,
! [A2: nat,B2: nat > nat,R: set_Pr6370437063884598352at_nat] :
( ( member8336108448496249625at_nat @ ( produc7839516862119294504at_nat @ A2 @ B2 ) @ R )
=> ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ B2 @ A2 ) @ ( converse_nat_nat_nat2 @ R ) ) ) ).
% converseI
thf(fact_616_converseI,axiom,
! [A2: nat,B2: a,R: set_Pr4193341848836149977_nat_a] :
( ( member8962352052110095674_nat_a @ ( product_Pair_nat_a @ A2 @ B2 ) @ R )
=> ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ B2 @ A2 ) @ ( converse_nat_a @ R ) ) ) ).
% converseI
thf(fact_617_converseI,axiom,
! [A2: term_a_b,B2: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ R )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ A2 ) @ ( conver7608559976937845093rm_a_b @ R ) ) ) ).
% converseI
thf(fact_618_converseI,axiom,
! [A2: nat > nat,B2: nat,R: set_Pr9093778441882193744at_nat] :
( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ A2 @ B2 ) @ R )
=> ( member8336108448496249625at_nat @ ( produc7839516862119294504at_nat @ B2 @ A2 ) @ ( converse_nat_nat_nat @ R ) ) ) ).
% converseI
thf(fact_619_converseI,axiom,
! [A2: a,B2: nat,R: set_Pr4934435412358123699_a_nat] :
( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ A2 @ B2 ) @ R )
=> ( member8962352052110095674_nat_a @ ( product_Pair_nat_a @ B2 @ A2 ) @ ( converse_a_nat @ R ) ) ) ).
% converseI
thf(fact_620_bot_Oextremum,axiom,
! [A2: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ bot_bo9049108969261143879_a_nat @ A2 ) ).
% bot.extremum
thf(fact_621_bot_Oextremum,axiom,
! [A2: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ bot_bo197521221353338581rm_a_b @ A2 ) ).
% bot.extremum
thf(fact_622_bot_Oextremum,axiom,
! [A2: set_Pr9093778441882193744at_nat] : ( ord_le3678578370064672496at_nat @ bot_bo8000532843592726972at_nat @ A2 ) ).
% bot.extremum
thf(fact_623_bot_Oextremum__unique,axiom,
! [A2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A2 @ bot_bo9049108969261143879_a_nat )
= ( A2 = bot_bo9049108969261143879_a_nat ) ) ).
% bot.extremum_unique
thf(fact_624_bot_Oextremum__unique,axiom,
! [A2: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A2 @ bot_bo197521221353338581rm_a_b )
= ( A2 = bot_bo197521221353338581rm_a_b ) ) ).
% bot.extremum_unique
thf(fact_625_bot_Oextremum__unique,axiom,
! [A2: set_Pr9093778441882193744at_nat] :
( ( ord_le3678578370064672496at_nat @ A2 @ bot_bo8000532843592726972at_nat )
= ( A2 = bot_bo8000532843592726972at_nat ) ) ).
% bot.extremum_unique
thf(fact_626_bot_Oextremum__uniqueI,axiom,
! [A2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A2 @ bot_bo9049108969261143879_a_nat )
=> ( A2 = bot_bo9049108969261143879_a_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_627_bot_Oextremum__uniqueI,axiom,
! [A2: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A2 @ bot_bo197521221353338581rm_a_b )
=> ( A2 = bot_bo197521221353338581rm_a_b ) ) ).
% bot.extremum_uniqueI
thf(fact_628_bot_Oextremum__uniqueI,axiom,
! [A2: set_Pr9093778441882193744at_nat] :
( ( ord_le3678578370064672496at_nat @ A2 @ bot_bo8000532843592726972at_nat )
=> ( A2 = bot_bo8000532843592726972at_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_629_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_630_converse__subset__swap,axiom,
! [R: set_Pr6370437063884598352at_nat,S: set_Pr9093778441882193744at_nat] :
( ( ord_le955236992067077104at_nat @ R @ ( converse_nat_nat_nat @ S ) )
= ( ord_le3678578370064672496at_nat @ ( converse_nat_nat_nat2 @ R ) @ S ) ) ).
% converse_subset_swap
thf(fact_631_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_632_converse__subset__swap,axiom,
! [R: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ R @ ( conver7608559976937845093rm_a_b @ S ) )
= ( ord_le118470702582115849rm_a_b @ ( conver7608559976937845093rm_a_b @ R ) @ S ) ) ).
% converse_subset_swap
thf(fact_633_converse__subset__swap,axiom,
! [R: set_Pr9093778441882193744at_nat,S: set_Pr6370437063884598352at_nat] :
( ( ord_le3678578370064672496at_nat @ R @ ( converse_nat_nat_nat2 @ S ) )
= ( ord_le955236992067077104at_nat @ ( converse_nat_nat_nat @ R ) @ S ) ) ).
% converse_subset_swap
thf(fact_634_IdE,axiom,
! [P2: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ P2 @ id_term_a_b )
=> ~ ! [X: term_a_b] :
( P2
!= ( produc7020197800436672577rm_a_b @ X @ X ) ) ) ).
% IdE
thf(fact_635_subset__emptyI,axiom,
! [A: set_Pr4934435412358123699_a_nat] :
( ! [X: product_prod_a_nat] :
~ ( member5724188588386418708_a_nat @ X @ A )
=> ( ord_le8666007276011122963_a_nat @ A @ bot_bo9049108969261143879_a_nat ) ) ).
% subset_emptyI
thf(fact_636_subset__emptyI,axiom,
! [A: set_Pr4386577575007340137rm_a_b] :
( ! [X: produc357393685978478089rm_a_b] :
~ ( member5869715511025134514rm_a_b @ X @ A )
=> ( ord_le118470702582115849rm_a_b @ A @ bot_bo197521221353338581rm_a_b ) ) ).
% subset_emptyI
thf(fact_637_subset__emptyI,axiom,
! [A: set_Pr9093778441882193744at_nat] :
( ! [X: produc8199716216217303280at_nat] :
~ ( member7226740684066999833at_nat @ X @ A )
=> ( ord_le3678578370064672496at_nat @ A @ bot_bo8000532843592726972at_nat ) ) ).
% subset_emptyI
thf(fact_638_join__sym,axiom,
! [S: term_a_b,T3: term_a_b,A: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T3 ) @ ( abstra4096080454567261402rm_a_b @ A ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T3 @ S ) @ ( abstra4096080454567261402rm_a_b @ A ) ) ) ).
% join_sym
thf(fact_639_join__mono,axiom,
! [R: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ R @ S )
=> ( ord_le118470702582115849rm_a_b @ ( abstra4096080454567261402rm_a_b @ R ) @ ( abstra4096080454567261402rm_a_b @ S ) ) ) ).
% join_mono
thf(fact_640_rtrancl__converseI,axiom,
! [Y3: term_a_b,X3: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ X3 ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ ( transi7742714808557438673rm_a_b @ ( conver7608559976937845093rm_a_b @ R ) ) ) ) ).
% rtrancl_converseI
thf(fact_641_rtrancl__converseD,axiom,
! [X3: term_a_b,Y3: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ ( transi7742714808557438673rm_a_b @ ( conver7608559976937845093rm_a_b @ R ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ X3 ) @ ( transi7742714808557438673rm_a_b @ R ) ) ) ).
% rtrancl_converseD
thf(fact_642_peak__iff,axiom,
! [X3: nat,Y3: nat,A: set_Pr9093778441882193744at_nat,B: set_Pr9093778441882193744at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ ( relcom6405311860805859042at_nat @ ( converse_nat_nat_nat @ A ) @ B ) )
= ( ? [U3: nat > nat] :
( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ U3 @ X3 ) @ A )
& ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ U3 @ Y3 ) @ B ) ) ) ) ).
% peak_iff
thf(fact_643_peak__iff,axiom,
! [X3: nat,Y3: nat,A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ ( relcomp_nat_a_nat @ ( converse_a_nat @ A ) @ B ) )
= ( ? [U3: a] :
( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ U3 @ X3 ) @ A )
& ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ U3 @ Y3 ) @ B ) ) ) ) ).
% peak_iff
thf(fact_644_peak__iff,axiom,
! [X3: term_a_b,Y3: term_a_b,A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ ( relcom370159955682700863rm_a_b @ ( conver7608559976937845093rm_a_b @ A ) @ B ) )
= ( ? [U3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ U3 @ X3 ) @ A )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ U3 @ Y3 ) @ B ) ) ) ) ).
% peak_iff
thf(fact_645_peak__iff,axiom,
! [X3: nat > nat,Y3: nat,A: set_Pr7682762132356531903at_nat,B: set_Pr9093778441882193744at_nat] :
( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ X3 @ Y3 ) @ ( relcom9054258243621454929at_nat @ ( conver1636425912871821883at_nat @ A ) @ B ) )
= ( ? [U3: nat > nat] :
( ( member1096249278104964872at_nat @ ( produc5770335208449155351at_nat @ U3 @ X3 ) @ A )
& ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ U3 @ Y3 ) @ B ) ) ) ) ).
% peak_iff
thf(fact_646_peak__iff,axiom,
! [X3: nat > nat,Y3: nat,A: set_Pr6841066879934740386at_nat,B: set_Pr4934435412358123699_a_nat] :
( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ X3 @ Y3 ) @ ( relcom1163327185954966926_a_nat @ ( converse_a_nat_nat @ A ) @ B ) )
= ( ? [U3: a] :
( ( member5463269792551873667at_nat @ ( produc7514922136058452582at_nat @ U3 @ X3 ) @ A )
& ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ U3 @ Y3 ) @ B ) ) ) ) ).
% peak_iff
thf(fact_647_peak__iff,axiom,
! [X3: a,Y3: nat,A: set_Pr3254233319718257578_nat_a,B: set_Pr9093778441882193744at_nat] :
( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ X3 @ Y3 ) @ ( relcom8552328690651882190at_nat @ ( converse_nat_nat_a @ A ) @ B ) )
= ( ? [U3: nat > nat] :
( ( member7754445317155421323_nat_a @ ( produc3941937540969104678_nat_a @ U3 @ X3 ) @ A )
& ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ U3 @ Y3 ) @ B ) ) ) ) ).
% peak_iff
thf(fact_648_peak__iff,axiom,
! [X3: a,Y3: nat,A: set_Product_prod_a_a,B: set_Pr4934435412358123699_a_nat] :
( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ X3 @ Y3 ) @ ( relcomp_a_a_nat @ ( converse_a_a @ A ) @ B ) )
= ( ? [U3: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ U3 @ X3 ) @ A )
& ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ U3 @ Y3 ) @ B ) ) ) ) ).
% peak_iff
thf(fact_649_trancl__converseI,axiom,
! [X3: term_a_b,Y3: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ ( conver7608559976937845093rm_a_b @ ( transi7922773638565587891rm_a_b @ R ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ ( transi7922773638565587891rm_a_b @ ( conver7608559976937845093rm_a_b @ R ) ) ) ) ).
% trancl_converseI
thf(fact_650_trancl__converseD,axiom,
! [X3: term_a_b,Y3: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ ( transi7922773638565587891rm_a_b @ ( conver7608559976937845093rm_a_b @ R ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ ( conver7608559976937845093rm_a_b @ ( transi7922773638565587891rm_a_b @ R ) ) ) ) ).
% trancl_converseD
thf(fact_651_rtrancl__join__join,axiom,
! [A2: term_a_b,B2: term_a_b,A: set_Pr4386577575007340137rm_a_b,C: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( transi7742714808557438673rm_a_b @ A ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ C ) @ ( abstra4096080454567261402rm_a_b @ A ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ C ) @ ( abstra4096080454567261402rm_a_b @ A ) ) ) ) ).
% rtrancl_join_join
thf(fact_652_join__rtrancl__join,axiom,
! [A2: term_a_b,B2: term_a_b,A: set_Pr4386577575007340137rm_a_b,C: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( abstra4096080454567261402rm_a_b @ A ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ C @ B2 ) @ ( transi7742714808557438673rm_a_b @ A ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ C ) @ ( abstra4096080454567261402rm_a_b @ A ) ) ) ) ).
% join_rtrancl_join
thf(fact_653_joinE,axiom,
! [A2: term_a_b,B2: term_a_b,A: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( abstra4096080454567261402rm_a_b @ A ) )
=> ~ ! [C6: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ C6 ) @ ( transi7742714808557438673rm_a_b @ A ) )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ C6 ) @ ( transi7742714808557438673rm_a_b @ A ) ) ) ) ).
% joinE
thf(fact_654_joinD,axiom,
! [A2: term_a_b,B2: term_a_b,A: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( abstra4096080454567261402rm_a_b @ A ) )
=> ? [C6: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ C6 ) @ ( transi7742714808557438673rm_a_b @ A ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ C6 ) @ ( transi7742714808557438673rm_a_b @ A ) ) ) ) ).
% joinD
thf(fact_655_join__NF__imp__eq,axiom,
! [X3: produc357393685978478089rm_a_b,Y3: produc357393685978478089rm_a_b,R: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X3 @ Y3 ) @ ( abstra4889682940153293507rm_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ X3 @ ( abstra1902783451776770197rm_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ Y3 @ ( abstra1902783451776770197rm_a_b @ R ) )
=> ( X3 = Y3 ) ) ) ) ).
% join_NF_imp_eq
thf(fact_656_join__NF__imp__eq,axiom,
! [X3: product_prod_a_nat,Y3: product_prod_a_nat,R: set_Pr1811044260758604347_a_nat] :
( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ X3 @ Y3 ) @ ( abstra881512052393830979_a_nat @ R ) )
=> ( ( member5724188588386418708_a_nat @ X3 @ ( abstra2685462378305987313_a_nat @ R ) )
=> ( ( member5724188588386418708_a_nat @ Y3 @ ( abstra2685462378305987313_a_nat @ R ) )
=> ( X3 = Y3 ) ) ) ) ).
% join_NF_imp_eq
thf(fact_657_join__NF__imp__eq,axiom,
! [X3: produc8199716216217303280at_nat,Y3: produc8199716216217303280at_nat,R: set_Pr7047708467701517959at_nat] :
( ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ X3 @ Y3 ) @ ( abstra8195730983011915946at_nat @ R ) )
=> ( ( member7226740684066999833at_nat @ X3 @ ( abstra8406325002171223420at_nat @ R ) )
=> ( ( member7226740684066999833at_nat @ Y3 @ ( abstra8406325002171223420at_nat @ R ) )
=> ( X3 = Y3 ) ) ) ) ).
% join_NF_imp_eq
thf(fact_658_join__NF__imp__eq,axiom,
! [X3: term_a_b,Y3: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ ( abstra4096080454567261402rm_a_b @ R ) )
=> ( ( member_term_a_b @ X3 @ ( abstract_NF_term_a_b @ R ) )
=> ( ( member_term_a_b @ Y3 @ ( abstract_NF_term_a_b @ R ) )
=> ( X3 = Y3 ) ) ) ) ).
% join_NF_imp_eq
thf(fact_659_commute__def,axiom,
( abstra455195320138238452rm_a_b
= ( ^ [R2: set_Pr4386577575007340137rm_a_b,S4: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( conver7608559976937845093rm_a_b @ R2 ) ) @ ( transi7742714808557438673rm_a_b @ S4 ) ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ S4 ) @ ( transi7742714808557438673rm_a_b @ ( conver7608559976937845093rm_a_b @ R2 ) ) ) ) ) ) ).
% commute_def
thf(fact_660_diamond__imp__semi__confluence,axiom,
! [R: set_Pr4386577575007340137rm_a_b] :
( ( abstra8151087076625346926rm_a_b @ R )
=> ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ ( conver7608559976937845093rm_a_b @ R ) @ ( transi7742714808557438673rm_a_b @ R ) ) @ ( abstra4096080454567261402rm_a_b @ R ) ) ) ).
% diamond_imp_semi_confluence
thf(fact_661_diamond__I,axiom,
! [R: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ ( conver7608559976937845093rm_a_b @ R ) @ R ) @ ( relcom370159955682700863rm_a_b @ R @ ( conver7608559976937845093rm_a_b @ R ) ) )
=> ( abstra8151087076625346926rm_a_b @ R ) ) ).
% diamond_I
thf(fact_662_strongly__confluent__on__def,axiom,
( abstra5664096641628173427rm_a_b
= ( ^ [R2: set_Pr4386577575007340137rm_a_b,A4: set_term_a_b] :
! [X4: term_a_b] :
( ( member_term_a_b @ X4 @ A4 )
=> ! [Y5: term_a_b,Z4: term_a_b] :
( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X4 @ Y5 ) @ R2 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X4 @ Z4 ) @ R2 ) )
=> ? [U3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y5 @ U3 ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Z4 @ U3 ) @ ( sup_su6776935440552674877rm_a_b @ R2 @ id_term_a_b ) ) ) ) ) ) ) ).
% strongly_confluent_on_def
thf(fact_663_bot__empty__eq,axiom,
( bot_bo23760184229683272_a_b_o
= ( ^ [X4: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ X4 @ bot_bo197521221353338581rm_a_b ) ) ) ).
% bot_empty_eq
thf(fact_664_bot__empty__eq,axiom,
( bot_bo1388487281569869310_nat_o
= ( ^ [X4: product_prod_a_nat] : ( member5724188588386418708_a_nat @ X4 @ bot_bo9049108969261143879_a_nat ) ) ) ).
% bot_empty_eq
thf(fact_665_bot__empty__eq,axiom,
( bot_bo2705904528838883361_nat_o
= ( ^ [X4: produc8199716216217303280at_nat] : ( member7226740684066999833at_nat @ X4 @ bot_bo8000532843592726972at_nat ) ) ) ).
% bot_empty_eq
thf(fact_666_diamond__E,axiom,
! [R: set_Pr4386577575007340137rm_a_b] :
( ( abstra8151087076625346926rm_a_b @ R )
=> ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ ( conver7608559976937845093rm_a_b @ R ) @ R ) @ ( relcom370159955682700863rm_a_b @ R @ ( conver7608559976937845093rm_a_b @ R ) ) ) ) ).
% diamond_E
thf(fact_667_diamond__def,axiom,
( abstra8151087076625346926rm_a_b
= ( ^ [R2: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ ( conver7608559976937845093rm_a_b @ R2 ) @ R2 ) @ ( relcom370159955682700863rm_a_b @ R2 @ ( conver7608559976937845093rm_a_b @ R2 ) ) ) ) ) ).
% diamond_def
thf(fact_668_less__by__empty,axiom,
! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ( A = bot_bo197521221353338581rm_a_b )
=> ( ord_le118470702582115849rm_a_b @ A @ B ) ) ).
% less_by_empty
thf(fact_669_strongly__confluent__on__E11,axiom,
! [R: set_Pr2972776593051762503rm_a_b,A: set_Pr4386577575007340137rm_a_b,X3: produc357393685978478089rm_a_b,Y3: produc357393685978478089rm_a_b,Z2: produc357393685978478089rm_a_b] :
( ( abstra837640964651051114rm_a_b @ R @ A )
=> ( ( member5869715511025134514rm_a_b @ X3 @ A )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X3 @ Y3 ) @ R )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X3 @ Z2 ) @ R )
=> ? [U2: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y3 @ U2 ) @ ( transi2615809358984392588rm_a_b @ R ) )
& ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Z2 @ U2 ) @ ( sup_su1845815533690797339rm_a_b @ R @ id_Pro3488709451141449061rm_a_b ) ) ) ) ) ) ) ).
% strongly_confluent_on_E11
thf(fact_670_strongly__confluent__on__E11,axiom,
! [R: set_Pr1811044260758604347_a_nat,A: set_Pr4934435412358123699_a_nat,X3: product_prod_a_nat,Y3: product_prod_a_nat,Z2: product_prod_a_nat] :
( ( abstra5315942894939654492_a_nat @ R @ A )
=> ( ( member5724188588386418708_a_nat @ X3 @ A )
=> ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ X3 @ Y3 ) @ R )
=> ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ X3 @ Z2 ) @ R )
=> ? [U2: product_prod_a_nat] :
( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ Y3 @ U2 ) @ ( transi2726145917338391738_a_nat @ R ) )
& ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ Z2 @ U2 ) @ ( sup_su958340139570419215_a_nat @ R @ id_Pro5207055338379199009_a_nat ) ) ) ) ) ) ) ).
% strongly_confluent_on_E11
thf(fact_671_strongly__confluent__on__E11,axiom,
! [R: set_Pr7047708467701517959at_nat,A: set_Pr9093778441882193744at_nat,X3: produc8199716216217303280at_nat,Y3: produc8199716216217303280at_nat,Z2: produc8199716216217303280at_nat] :
( ( abstra3669381686128266961at_nat @ R @ A )
=> ( ( member7226740684066999833at_nat @ X3 @ A )
=> ( ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ X3 @ Y3 ) @ R )
=> ( ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ X3 @ Z2 ) @ R )
=> ? [U2: produc8199716216217303280at_nat] :
( ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ Y3 @ U2 ) @ ( transi3422749762518297843at_nat @ R ) )
& ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ Z2 @ U2 ) @ ( sup_su2948762899108616283at_nat @ R @ id_Pro414489527043525196at_nat ) ) ) ) ) ) ) ).
% strongly_confluent_on_E11
thf(fact_672_strongly__confluent__on__E11,axiom,
! [R: set_Pr4386577575007340137rm_a_b,A: set_term_a_b,X3: term_a_b,Y3: term_a_b,Z2: term_a_b] :
( ( abstra5664096641628173427rm_a_b @ R @ A )
=> ( ( member_term_a_b @ X3 @ A )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ R )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Z2 ) @ R )
=> ? [U2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ U2 ) @ ( transi7742714808557438673rm_a_b @ R ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Z2 @ U2 ) @ ( sup_su6776935440552674877rm_a_b @ R @ id_term_a_b ) ) ) ) ) ) ) ).
% strongly_confluent_on_E11
thf(fact_673_strongly__confluentI,axiom,
! [R: set_Pr4386577575007340137rm_a_b] :
( ! [X: term_a_b,Y: term_a_b,Z3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ R )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z3 ) @ R )
=> ? [U4: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ U4 ) @ ( transi7742714808557438673rm_a_b @ R ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Z3 @ U4 ) @ ( sup_su6776935440552674877rm_a_b @ R @ id_term_a_b ) ) ) ) )
=> ( abstra5664096641628173427rm_a_b @ R @ top_top_set_term_a_b ) ) ).
% strongly_confluentI
thf(fact_674_IdD,axiom,
! [A2: term_a_b,B2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ id_term_a_b )
=> ( A2 = B2 ) ) ).
% IdD
thf(fact_675_conversionI,axiom,
! [A2: term_a_b,B2: term_a_b,A: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ A @ ( conver7608559976937845093rm_a_b @ A ) ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( abstra1130568886744591910rm_a_b @ A ) ) ) ).
% conversionI
thf(fact_676_WCR__onI,axiom,
! [A: set_Pr4386577575007340137rm_a_b,R: set_Pr2972776593051762503rm_a_b] :
( ! [A6: produc357393685978478089rm_a_b,B6: produc357393685978478089rm_a_b,C6: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ A6 @ A )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A6 @ B6 ) @ R )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A6 @ C6 ) @ R )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B6 @ C6 ) @ ( abstra4889682940153293507rm_a_b @ R ) ) ) ) )
=> ( abstra875157153669124638rm_a_b @ R @ A ) ) ).
% WCR_onI
thf(fact_677_WCR__onI,axiom,
! [A: set_Pr4934435412358123699_a_nat,R: set_Pr1811044260758604347_a_nat] :
( ! [A6: product_prod_a_nat,B6: product_prod_a_nat,C6: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ A6 @ A )
=> ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A6 @ B6 ) @ R )
=> ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A6 @ C6 ) @ R )
=> ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ B6 @ C6 ) @ ( abstra881512052393830979_a_nat @ R ) ) ) ) )
=> ( abstra6648775111202918568_a_nat @ R @ A ) ) ).
% WCR_onI
thf(fact_678_WCR__onI,axiom,
! [A: set_Pr9093778441882193744at_nat,R: set_Pr7047708467701517959at_nat] :
( ! [A6: produc8199716216217303280at_nat,B6: produc8199716216217303280at_nat,C6: produc8199716216217303280at_nat] :
( ( member7226740684066999833at_nat @ A6 @ A )
=> ( ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ A6 @ B6 ) @ R )
=> ( ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ A6 @ C6 ) @ R )
=> ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ B6 @ C6 ) @ ( abstra8195730983011915946at_nat @ R ) ) ) ) )
=> ( abstra880406612099378821at_nat @ R @ A ) ) ).
% WCR_onI
thf(fact_679_WCR__onI,axiom,
! [A: set_term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ! [A6: term_a_b,B6: term_a_b,C6: term_a_b] :
( ( member_term_a_b @ A6 @ A )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A6 @ B6 ) @ R )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A6 @ C6 ) @ R )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B6 @ C6 ) @ ( abstra4096080454567261402rm_a_b @ R ) ) ) ) )
=> ( abstra5026849107050860991rm_a_b @ R @ A ) ) ).
% WCR_onI
thf(fact_680_UNIV__I,axiom,
! [X3: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ X3 @ top_to1314267278846557113rm_a_b ) ).
% UNIV_I
thf(fact_681_UNIV__I,axiom,
! [X3: product_prod_a_nat] : ( member5724188588386418708_a_nat @ X3 @ top_to3353692345378799459_a_nat ) ).
% UNIV_I
thf(fact_682_UNIV__I,axiom,
! [X3: produc8199716216217303280at_nat] : ( member7226740684066999833at_nat @ X3 @ top_to7836785446628990112at_nat ) ).
% UNIV_I
thf(fact_683_conversionI_H,axiom,
! [A2: term_a_b,B2: term_a_b,A: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( transi7742714808557438673rm_a_b @ A ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( abstra1130568886744591910rm_a_b @ A ) ) ) ).
% conversionI'
thf(fact_684_top_Oextremum__uniqueI,axiom,
! [A2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ top_to3353692345378799459_a_nat @ A2 )
=> ( A2 = top_to3353692345378799459_a_nat ) ) ).
% top.extremum_uniqueI
thf(fact_685_top_Oextremum__uniqueI,axiom,
! [A2: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ top_to1314267278846557113rm_a_b @ A2 )
=> ( A2 = top_to1314267278846557113rm_a_b ) ) ).
% top.extremum_uniqueI
thf(fact_686_top_Oextremum__uniqueI,axiom,
! [A2: set_Pr9093778441882193744at_nat] :
( ( ord_le3678578370064672496at_nat @ top_to7836785446628990112at_nat @ A2 )
=> ( A2 = top_to7836785446628990112at_nat ) ) ).
% top.extremum_uniqueI
thf(fact_687_top_Oextremum__unique,axiom,
! [A2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ top_to3353692345378799459_a_nat @ A2 )
= ( A2 = top_to3353692345378799459_a_nat ) ) ).
% top.extremum_unique
thf(fact_688_top_Oextremum__unique,axiom,
! [A2: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ top_to1314267278846557113rm_a_b @ A2 )
= ( A2 = top_to1314267278846557113rm_a_b ) ) ).
% top.extremum_unique
thf(fact_689_top_Oextremum__unique,axiom,
! [A2: set_Pr9093778441882193744at_nat] :
( ( ord_le3678578370064672496at_nat @ top_to7836785446628990112at_nat @ A2 )
= ( A2 = top_to7836785446628990112at_nat ) ) ).
% top.extremum_unique
thf(fact_690_top__greatest,axiom,
! [A2: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ A2 @ top_to3353692345378799459_a_nat ) ).
% top_greatest
thf(fact_691_top__greatest,axiom,
! [A2: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ A2 @ top_to1314267278846557113rm_a_b ) ).
% top_greatest
thf(fact_692_top__greatest,axiom,
! [A2: set_Pr9093778441882193744at_nat] : ( ord_le3678578370064672496at_nat @ A2 @ top_to7836785446628990112at_nat ) ).
% top_greatest
thf(fact_693_UNIV__eq__I,axiom,
! [A: set_Pr4386577575007340137rm_a_b] :
( ! [X: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ X @ A )
=> ( top_to1314267278846557113rm_a_b = A ) ) ).
% UNIV_eq_I
thf(fact_694_UNIV__eq__I,axiom,
! [A: set_Pr4934435412358123699_a_nat] :
( ! [X: product_prod_a_nat] : ( member5724188588386418708_a_nat @ X @ A )
=> ( top_to3353692345378799459_a_nat = A ) ) ).
% UNIV_eq_I
thf(fact_695_UNIV__eq__I,axiom,
! [A: set_Pr9093778441882193744at_nat] :
( ! [X: produc8199716216217303280at_nat] : ( member7226740684066999833at_nat @ X @ A )
=> ( top_to7836785446628990112at_nat = A ) ) ).
% UNIV_eq_I
thf(fact_696_UNIV__witness,axiom,
? [X: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ X @ top_to1314267278846557113rm_a_b ) ).
% UNIV_witness
thf(fact_697_UNIV__witness,axiom,
? [X: product_prod_a_nat] : ( member5724188588386418708_a_nat @ X @ top_to3353692345378799459_a_nat ) ).
% UNIV_witness
thf(fact_698_UNIV__witness,axiom,
? [X: produc8199716216217303280at_nat] : ( member7226740684066999833at_nat @ X @ top_to7836785446628990112at_nat ) ).
% UNIV_witness
thf(fact_699_subset__UNIV,axiom,
! [A: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ A @ top_to3353692345378799459_a_nat ) ).
% subset_UNIV
thf(fact_700_subset__UNIV,axiom,
! [A: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ A @ top_to1314267278846557113rm_a_b ) ).
% subset_UNIV
thf(fact_701_subset__UNIV,axiom,
! [A: set_Pr9093778441882193744at_nat] : ( ord_le3678578370064672496at_nat @ A @ top_to7836785446628990112at_nat ) ).
% subset_UNIV
thf(fact_702_conversion__inv,axiom,
! [X3: term_a_b,Y3: term_a_b,R4: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ ( abstra1130568886744591910rm_a_b @ R4 ) )
= ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ X3 ) @ ( abstra1130568886744591910rm_a_b @ R4 ) ) ) ).
% conversion_inv
thf(fact_703_conversion__refl,axiom,
! [A2: term_a_b,A: set_Pr4386577575007340137rm_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ A2 ) @ ( abstra1130568886744591910rm_a_b @ A ) ) ).
% conversion_refl
thf(fact_704_conversion__mono,axiom,
! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A @ B )
=> ( ord_le118470702582115849rm_a_b @ ( abstra1130568886744591910rm_a_b @ A ) @ ( abstra1130568886744591910rm_a_b @ B ) ) ) ).
% conversion_mono
thf(fact_705_join__imp__conversion,axiom,
! [R: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ ( abstra4096080454567261402rm_a_b @ R ) @ ( abstra1130568886744591910rm_a_b @ R ) ) ).
% join_imp_conversion
thf(fact_706_WCR__alt__def,axiom,
! [A: set_Pr4386577575007340137rm_a_b] :
( ( abstra5026849107050860991rm_a_b @ A @ top_top_set_term_a_b )
= ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ ( conver7608559976937845093rm_a_b @ A ) @ A ) @ ( abstra4096080454567261402rm_a_b @ A ) ) ) ).
% WCR_alt_def
thf(fact_707_meet__imp__conversion,axiom,
! [R: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ ( abstra1093061187967292567rm_a_b @ R ) @ ( abstra1130568886744591910rm_a_b @ R ) ) ).
% meet_imp_conversion
thf(fact_708_WCR__on__def,axiom,
( abstra5026849107050860991rm_a_b
= ( ^ [R2: set_Pr4386577575007340137rm_a_b,A4: set_term_a_b] :
! [X4: term_a_b] :
( ( member_term_a_b @ X4 @ A4 )
=> ! [B5: term_a_b,C5: term_a_b] :
( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X4 @ B5 ) @ R2 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X4 @ C5 ) @ R2 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B5 @ C5 ) @ ( abstra4096080454567261402rm_a_b @ R2 ) ) ) ) ) ) ).
% WCR_on_def
thf(fact_709_WCR__onE,axiom,
! [R: set_Pr2972776593051762503rm_a_b,A: set_Pr4386577575007340137rm_a_b,A2: produc357393685978478089rm_a_b,B2: produc357393685978478089rm_a_b,C: produc357393685978478089rm_a_b] :
( ( abstra875157153669124638rm_a_b @ R @ A )
=> ( ( member5869715511025134514rm_a_b @ A2 @ A )
=> ( ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B2 @ C ) @ ( abstra4889682940153293507rm_a_b @ R ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A2 @ B2 ) @ R )
=> ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A2 @ C ) @ R ) ) ) ) ) ).
% WCR_onE
thf(fact_710_WCR__onE,axiom,
! [R: set_Pr1811044260758604347_a_nat,A: set_Pr4934435412358123699_a_nat,A2: product_prod_a_nat,B2: product_prod_a_nat,C: product_prod_a_nat] :
( ( abstra6648775111202918568_a_nat @ R @ A )
=> ( ( member5724188588386418708_a_nat @ A2 @ A )
=> ( ~ ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ B2 @ C ) @ ( abstra881512052393830979_a_nat @ R ) )
=> ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A2 @ B2 ) @ R )
=> ~ ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A2 @ C ) @ R ) ) ) ) ) ).
% WCR_onE
thf(fact_711_WCR__onE,axiom,
! [R: set_Pr7047708467701517959at_nat,A: set_Pr9093778441882193744at_nat,A2: produc8199716216217303280at_nat,B2: produc8199716216217303280at_nat,C: produc8199716216217303280at_nat] :
( ( abstra880406612099378821at_nat @ R @ A )
=> ( ( member7226740684066999833at_nat @ A2 @ A )
=> ( ~ ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ B2 @ C ) @ ( abstra8195730983011915946at_nat @ R ) )
=> ( ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ A2 @ B2 ) @ R )
=> ~ ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ A2 @ C ) @ R ) ) ) ) ) ).
% WCR_onE
thf(fact_712_WCR__onE,axiom,
! [R: set_Pr4386577575007340137rm_a_b,A: set_term_a_b,A2: term_a_b,B2: term_a_b,C: term_a_b] :
( ( abstra5026849107050860991rm_a_b @ R @ A )
=> ( ( member_term_a_b @ A2 @ A )
=> ( ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ C ) @ ( abstra4096080454567261402rm_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ R )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ C ) @ R ) ) ) ) ) ).
% WCR_onE
thf(fact_713_WCR__onD,axiom,
! [R: set_Pr2972776593051762503rm_a_b,A: set_Pr4386577575007340137rm_a_b,A2: produc357393685978478089rm_a_b,B2: produc357393685978478089rm_a_b,C: produc357393685978478089rm_a_b] :
( ( abstra875157153669124638rm_a_b @ R @ A )
=> ( ( member5869715511025134514rm_a_b @ A2 @ A )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A2 @ B2 ) @ R )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A2 @ C ) @ R )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B2 @ C ) @ ( abstra4889682940153293507rm_a_b @ R ) ) ) ) ) ) ).
% WCR_onD
thf(fact_714_WCR__onD,axiom,
! [R: set_Pr1811044260758604347_a_nat,A: set_Pr4934435412358123699_a_nat,A2: product_prod_a_nat,B2: product_prod_a_nat,C: product_prod_a_nat] :
( ( abstra6648775111202918568_a_nat @ R @ A )
=> ( ( member5724188588386418708_a_nat @ A2 @ A )
=> ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A2 @ B2 ) @ R )
=> ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A2 @ C ) @ R )
=> ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ B2 @ C ) @ ( abstra881512052393830979_a_nat @ R ) ) ) ) ) ) ).
% WCR_onD
thf(fact_715_WCR__onD,axiom,
! [R: set_Pr7047708467701517959at_nat,A: set_Pr9093778441882193744at_nat,A2: produc8199716216217303280at_nat,B2: produc8199716216217303280at_nat,C: produc8199716216217303280at_nat] :
( ( abstra880406612099378821at_nat @ R @ A )
=> ( ( member7226740684066999833at_nat @ A2 @ A )
=> ( ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ A2 @ B2 ) @ R )
=> ( ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ A2 @ C ) @ R )
=> ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ B2 @ C ) @ ( abstra8195730983011915946at_nat @ R ) ) ) ) ) ) ).
% WCR_onD
thf(fact_716_WCR__onD,axiom,
! [R: set_Pr4386577575007340137rm_a_b,A: set_term_a_b,A2: term_a_b,B2: term_a_b,C: term_a_b] :
( ( abstra5026849107050860991rm_a_b @ R @ A )
=> ( ( member_term_a_b @ A2 @ A )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ R )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ C ) @ R )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ C ) @ ( abstra4096080454567261402rm_a_b @ R ) ) ) ) ) ) ).
% WCR_onD
thf(fact_717_conversionE,axiom,
! [A2: term_a_b,B2: term_a_b,A: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( abstra1130568886744591910rm_a_b @ A ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ A @ ( conver7608559976937845093rm_a_b @ A ) ) ) ) ) ).
% conversionE
thf(fact_718_some__NF__UNF,axiom,
! [R: set_Pr2972776593051762503rm_a_b,X3: produc357393685978478089rm_a_b,Y3: produc357393685978478089rm_a_b] :
( ( abstra1303798319023342519rm_a_b @ R @ top_to1314267278846557113rm_a_b )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X3 @ Y3 ) @ ( transi2615809358984392588rm_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ Y3 @ ( abstra1902783451776770197rm_a_b @ R ) )
=> ( Y3
= ( abstra8203652949708803681rm_a_b @ R @ X3 ) ) ) ) ) ).
% some_NF_UNF
thf(fact_719_some__NF__UNF,axiom,
! [R: set_Pr1811044260758604347_a_nat,X3: product_prod_a_nat,Y3: product_prod_a_nat] :
( ( abstra7833083522531381839_a_nat @ R @ top_to3353692345378799459_a_nat )
=> ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ X3 @ Y3 ) @ ( transi2726145917338391738_a_nat @ R ) )
=> ( ( member5724188588386418708_a_nat @ Y3 @ ( abstra2685462378305987313_a_nat @ R ) )
=> ( Y3
= ( abstra5913684503434863653_a_nat @ R @ X3 ) ) ) ) ) ).
% some_NF_UNF
thf(fact_720_some__NF__UNF,axiom,
! [R: set_Pr7047708467701517959at_nat,X3: produc8199716216217303280at_nat,Y3: produc8199716216217303280at_nat] :
( ( abstra4852570278320572318at_nat @ R @ top_to7836785446628990112at_nat )
=> ( ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ X3 @ Y3 ) @ ( transi3422749762518297843at_nat @ R ) )
=> ( ( member7226740684066999833at_nat @ Y3 @ ( abstra8406325002171223420at_nat @ R ) )
=> ( Y3
= ( abstra5373487907108472136at_nat @ R @ X3 ) ) ) ) ) ).
% some_NF_UNF
thf(fact_721_some__NF__UNF,axiom,
! [R: set_Pr4386577575007340137rm_a_b,X3: term_a_b,Y3: term_a_b] :
( ( abstra3619855735679963878rm_a_b @ R @ top_top_set_term_a_b )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ( ( member_term_a_b @ Y3 @ ( abstract_NF_term_a_b @ R ) )
=> ( Y3
= ( abstra2589370493620291516rm_a_b @ R @ X3 ) ) ) ) ) ).
% some_NF_UNF
thf(fact_722_semi__complete__imp__conversionIff__same__NF,axiom,
! [R: set_Pr4386577575007340137rm_a_b,X3: term_a_b,Y3: term_a_b] :
( ( abstra7162307272211697846rm_a_b @ R @ top_top_set_term_a_b )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ ( abstra1130568886744591910rm_a_b @ R ) )
= ( ! [U3: term_a_b,V2: term_a_b] :
( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ U3 ) @ ( abstra2972651231392952684rm_a_b @ R ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ V2 ) @ ( abstra2972651231392952684rm_a_b @ R ) ) )
=> ( U3 = V2 ) ) ) ) ) ).
% semi_complete_imp_conversionIff_same_NF
thf(fact_723_UNC__E,axiom,
! [A: set_Pr2972776593051762503rm_a_b,A2: produc357393685978478089rm_a_b,B2: produc357393685978478089rm_a_b] :
( ( abstra8480497335821202873rm_a_b @ A )
=> ( ( A2 != B2 )
=> ( ( member5869715511025134514rm_a_b @ A2 @ ( abstra1902783451776770197rm_a_b @ A ) )
=> ( ( member5869715511025134514rm_a_b @ B2 @ ( abstra1902783451776770197rm_a_b @ A ) )
=> ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A2 @ B2 ) @ ( abstra6125656530262095991rm_a_b @ A ) ) ) ) ) ) ).
% UNC_E
thf(fact_724_UNC__E,axiom,
! [A: set_Pr1811044260758604347_a_nat,A2: product_prod_a_nat,B2: product_prod_a_nat] :
( ( abstra7179253994116303565_a_nat @ A )
=> ( ( A2 != B2 )
=> ( ( member5724188588386418708_a_nat @ A2 @ ( abstra2685462378305987313_a_nat @ A ) )
=> ( ( member5724188588386418708_a_nat @ B2 @ ( abstra2685462378305987313_a_nat @ A ) )
=> ~ ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A2 @ B2 ) @ ( abstra3054561309482028431_a_nat @ A ) ) ) ) ) ) ).
% UNC_E
thf(fact_725_UNC__E,axiom,
! [A: set_Pr7047708467701517959at_nat,A2: produc8199716216217303280at_nat,B2: produc8199716216217303280at_nat] :
( ( abstra8684767081465667744at_nat @ A )
=> ( ( A2 != B2 )
=> ( ( member7226740684066999833at_nat @ A2 @ ( abstra8406325002171223420at_nat @ A ) )
=> ( ( member7226740684066999833at_nat @ B2 @ ( abstra8406325002171223420at_nat @ A ) )
=> ~ ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ A2 @ B2 ) @ ( abstra1617541938821389406at_nat @ A ) ) ) ) ) ) ).
% UNC_E
thf(fact_726_UNC__E,axiom,
! [A: set_Pr4386577575007340137rm_a_b,A2: term_a_b,B2: term_a_b] :
( ( abstra4649771476585718372rm_a_b @ A )
=> ( ( A2 != B2 )
=> ( ( member_term_a_b @ A2 @ ( abstract_NF_term_a_b @ A ) )
=> ( ( member_term_a_b @ B2 @ ( abstract_NF_term_a_b @ A ) )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( abstra1130568886744591910rm_a_b @ A ) ) ) ) ) ) ).
% UNC_E
thf(fact_727_UNC__I,axiom,
! [A: set_Pr2972776593051762503rm_a_b] :
( ! [A6: produc357393685978478089rm_a_b,B6: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ A6 @ ( abstra1902783451776770197rm_a_b @ A ) )
=> ( ( member5869715511025134514rm_a_b @ B6 @ ( abstra1902783451776770197rm_a_b @ A ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A6 @ B6 ) @ ( abstra6125656530262095991rm_a_b @ A ) )
=> ( A6 = B6 ) ) ) )
=> ( abstra8480497335821202873rm_a_b @ A ) ) ).
% UNC_I
thf(fact_728_UNC__I,axiom,
! [A: set_Pr1811044260758604347_a_nat] :
( ! [A6: product_prod_a_nat,B6: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ A6 @ ( abstra2685462378305987313_a_nat @ A ) )
=> ( ( member5724188588386418708_a_nat @ B6 @ ( abstra2685462378305987313_a_nat @ A ) )
=> ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A6 @ B6 ) @ ( abstra3054561309482028431_a_nat @ A ) )
=> ( A6 = B6 ) ) ) )
=> ( abstra7179253994116303565_a_nat @ A ) ) ).
% UNC_I
thf(fact_729_UNC__I,axiom,
! [A: set_Pr7047708467701517959at_nat] :
( ! [A6: produc8199716216217303280at_nat,B6: produc8199716216217303280at_nat] :
( ( member7226740684066999833at_nat @ A6 @ ( abstra8406325002171223420at_nat @ A ) )
=> ( ( member7226740684066999833at_nat @ B6 @ ( abstra8406325002171223420at_nat @ A ) )
=> ( ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ A6 @ B6 ) @ ( abstra1617541938821389406at_nat @ A ) )
=> ( A6 = B6 ) ) ) )
=> ( abstra8684767081465667744at_nat @ A ) ) ).
% UNC_I
thf(fact_730_UNC__I,axiom,
! [A: set_Pr4386577575007340137rm_a_b] :
( ! [A6: term_a_b,B6: term_a_b] :
( ( member_term_a_b @ A6 @ ( abstract_NF_term_a_b @ A ) )
=> ( ( member_term_a_b @ B6 @ ( abstract_NF_term_a_b @ A ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A6 @ B6 ) @ ( abstra1130568886744591910rm_a_b @ A ) )
=> ( A6 = B6 ) ) ) )
=> ( abstra4649771476585718372rm_a_b @ A ) ) ).
% UNC_I
thf(fact_731_UNC__def,axiom,
( abstra8480497335821202873rm_a_b
= ( ^ [A4: set_Pr2972776593051762503rm_a_b] :
! [A5: produc357393685978478089rm_a_b,B5: produc357393685978478089rm_a_b] :
( ( ( member5869715511025134514rm_a_b @ A5 @ ( abstra1902783451776770197rm_a_b @ A4 ) )
& ( member5869715511025134514rm_a_b @ B5 @ ( abstra1902783451776770197rm_a_b @ A4 ) )
& ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A5 @ B5 ) @ ( abstra6125656530262095991rm_a_b @ A4 ) ) )
=> ( A5 = B5 ) ) ) ) ).
% UNC_def
thf(fact_732_UNC__def,axiom,
( abstra7179253994116303565_a_nat
= ( ^ [A4: set_Pr1811044260758604347_a_nat] :
! [A5: product_prod_a_nat,B5: product_prod_a_nat] :
( ( ( member5724188588386418708_a_nat @ A5 @ ( abstra2685462378305987313_a_nat @ A4 ) )
& ( member5724188588386418708_a_nat @ B5 @ ( abstra2685462378305987313_a_nat @ A4 ) )
& ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A5 @ B5 ) @ ( abstra3054561309482028431_a_nat @ A4 ) ) )
=> ( A5 = B5 ) ) ) ) ).
% UNC_def
thf(fact_733_UNC__def,axiom,
( abstra8684767081465667744at_nat
= ( ^ [A4: set_Pr7047708467701517959at_nat] :
! [A5: produc8199716216217303280at_nat,B5: produc8199716216217303280at_nat] :
( ( ( member7226740684066999833at_nat @ A5 @ ( abstra8406325002171223420at_nat @ A4 ) )
& ( member7226740684066999833at_nat @ B5 @ ( abstra8406325002171223420at_nat @ A4 ) )
& ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ A5 @ B5 ) @ ( abstra1617541938821389406at_nat @ A4 ) ) )
=> ( A5 = B5 ) ) ) ) ).
% UNC_def
thf(fact_734_UNC__def,axiom,
( abstra4649771476585718372rm_a_b
= ( ^ [A4: set_Pr4386577575007340137rm_a_b] :
! [A5: term_a_b,B5: term_a_b] :
( ( ( member_term_a_b @ A5 @ ( abstract_NF_term_a_b @ A4 ) )
& ( member_term_a_b @ B5 @ ( abstract_NF_term_a_b @ A4 ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A5 @ B5 ) @ ( abstra1130568886744591910rm_a_b @ A4 ) ) )
=> ( A5 = B5 ) ) ) ) ).
% UNC_def
thf(fact_735_top__empty__eq,axiom,
( top_to1262317115727556708_a_b_o
= ( ^ [X4: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ X4 @ top_to1314267278846557113rm_a_b ) ) ) ).
% top_empty_eq
thf(fact_736_top__empty__eq,axiom,
( top_to1985590801843794146_nat_o
= ( ^ [X4: product_prod_a_nat] : ( member5724188588386418708_a_nat @ X4 @ top_to3353692345378799459_a_nat ) ) ) ).
% top_empty_eq
thf(fact_737_top__empty__eq,axiom,
( top_to2374939387204954685_nat_o
= ( ^ [X4: produc8199716216217303280at_nat] : ( member7226740684066999833at_nat @ X4 @ top_to7836785446628990112at_nat ) ) ) ).
% top_empty_eq
thf(fact_738_semi__confluence__imp__CR,axiom,
! [R: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ ( conver7608559976937845093rm_a_b @ R ) @ ( transi7742714808557438673rm_a_b @ R ) ) @ ( abstra4096080454567261402rm_a_b @ R ) )
=> ( abstra8448919418672941150rm_a_b @ R @ top_top_set_term_a_b ) ) ).
% semi_confluence_imp_CR
thf(fact_739_strongly__confluent__E1n,axiom,
! [R: set_Pr4386577575007340137rm_a_b,X3: term_a_b,Y3: term_a_b,Z2: term_a_b,N: nat] :
( ( abstra5664096641628173427rm_a_b @ R @ top_top_set_term_a_b )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ ( sup_su6776935440552674877rm_a_b @ R @ id_term_a_b ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Z2 ) @ ( compow4057154403645558940rm_a_b @ N @ R ) )
=> ? [U2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ U2 ) @ ( transi7742714808557438673rm_a_b @ R ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Z2 @ U2 ) @ ( sup_su6776935440552674877rm_a_b @ R @ id_term_a_b ) ) ) ) ) ) ).
% strongly_confluent_E1n
thf(fact_740_CR__NF__conv,axiom,
! [R: set_Pr2972776593051762503rm_a_b,T3: produc357393685978478089rm_a_b,U: produc357393685978478089rm_a_b] :
( ( abstra7359694574113289279rm_a_b @ R @ top_to1314267278846557113rm_a_b )
=> ( ( member5869715511025134514rm_a_b @ T3 @ ( abstra1902783451776770197rm_a_b @ R ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ U @ T3 ) @ ( abstra6125656530262095991rm_a_b @ R ) )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ U @ T3 ) @ ( abstra1496011390211842225rm_a_b @ R ) ) ) ) ) ).
% CR_NF_conv
thf(fact_741_CR__NF__conv,axiom,
! [R: set_Pr1811044260758604347_a_nat,T3: product_prod_a_nat,U: product_prod_a_nat] :
( ( abstra7964079670522473415_a_nat @ R @ top_to3353692345378799459_a_nat )
=> ( ( member5724188588386418708_a_nat @ T3 @ ( abstra2685462378305987313_a_nat @ R ) )
=> ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ U @ T3 ) @ ( abstra3054561309482028431_a_nat @ R ) )
=> ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ U @ T3 ) @ ( abstra3832961281062565333_a_nat @ R ) ) ) ) ) ).
% CR_NF_conv
thf(fact_742_CR__NF__conv,axiom,
! [R: set_Pr7047708467701517959at_nat,T3: produc8199716216217303280at_nat,U: produc8199716216217303280at_nat] :
( ( abstra8052594214376182822at_nat @ R @ top_to7836785446628990112at_nat )
=> ( ( member7226740684066999833at_nat @ T3 @ ( abstra8406325002171223420at_nat @ R ) )
=> ( ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ U @ T3 ) @ ( abstra1617541938821389406at_nat @ R ) )
=> ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ U @ T3 ) @ ( abstra1436239171852513176at_nat @ R ) ) ) ) ) ).
% CR_NF_conv
thf(fact_743_CR__NF__conv,axiom,
! [R: set_Pr4386577575007340137rm_a_b,T3: term_a_b,U: term_a_b] :
( ( abstra8448919418672941150rm_a_b @ R @ top_top_set_term_a_b )
=> ( ( member_term_a_b @ T3 @ ( abstract_NF_term_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ U @ T3 ) @ ( abstra1130568886744591910rm_a_b @ R ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ U @ T3 ) @ ( abstra2972651231392952684rm_a_b @ R ) ) ) ) ) ).
% CR_NF_conv
thf(fact_744_CR__iff__meet__subset__join,axiom,
! [R: set_Pr4386577575007340137rm_a_b] :
( ( abstra8448919418672941150rm_a_b @ R @ top_top_set_term_a_b )
= ( ord_le118470702582115849rm_a_b @ ( abstra1093061187967292567rm_a_b @ R ) @ ( abstra4096080454567261402rm_a_b @ R ) ) ) ).
% CR_iff_meet_subset_join
thf(fact_745_diamond__imp__CR_H,axiom,
! [S: set_Pr4386577575007340137rm_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( abstra8151087076625346926rm_a_b @ S )
=> ( ( ord_le118470702582115849rm_a_b @ R @ S )
=> ( ( ord_le118470702582115849rm_a_b @ S @ ( transi7742714808557438673rm_a_b @ R ) )
=> ( abstra8448919418672941150rm_a_b @ R @ top_top_set_term_a_b ) ) ) ) ).
% diamond_imp_CR'
thf(fact_746_CR__onI,axiom,
! [A: set_Pr4386577575007340137rm_a_b,R: set_Pr2972776593051762503rm_a_b] :
( ! [A6: produc357393685978478089rm_a_b,B6: produc357393685978478089rm_a_b,C6: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ A6 @ A )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A6 @ B6 ) @ ( transi2615809358984392588rm_a_b @ R ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A6 @ C6 ) @ ( transi2615809358984392588rm_a_b @ R ) )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B6 @ C6 ) @ ( abstra4889682940153293507rm_a_b @ R ) ) ) ) )
=> ( abstra7359694574113289279rm_a_b @ R @ A ) ) ).
% CR_onI
thf(fact_747_CR__onI,axiom,
! [A: set_Pr4934435412358123699_a_nat,R: set_Pr1811044260758604347_a_nat] :
( ! [A6: product_prod_a_nat,B6: product_prod_a_nat,C6: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ A6 @ A )
=> ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A6 @ B6 ) @ ( transi2726145917338391738_a_nat @ R ) )
=> ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A6 @ C6 ) @ ( transi2726145917338391738_a_nat @ R ) )
=> ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ B6 @ C6 ) @ ( abstra881512052393830979_a_nat @ R ) ) ) ) )
=> ( abstra7964079670522473415_a_nat @ R @ A ) ) ).
% CR_onI
thf(fact_748_CR__onI,axiom,
! [A: set_Pr9093778441882193744at_nat,R: set_Pr7047708467701517959at_nat] :
( ! [A6: produc8199716216217303280at_nat,B6: produc8199716216217303280at_nat,C6: produc8199716216217303280at_nat] :
( ( member7226740684066999833at_nat @ A6 @ A )
=> ( ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ A6 @ B6 ) @ ( transi3422749762518297843at_nat @ R ) )
=> ( ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ A6 @ C6 ) @ ( transi3422749762518297843at_nat @ R ) )
=> ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ B6 @ C6 ) @ ( abstra8195730983011915946at_nat @ R ) ) ) ) )
=> ( abstra8052594214376182822at_nat @ R @ A ) ) ).
% CR_onI
thf(fact_749_CR__onI,axiom,
! [A: set_term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ! [A6: term_a_b,B6: term_a_b,C6: term_a_b] :
( ( member_term_a_b @ A6 @ A )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A6 @ B6 ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A6 @ C6 ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B6 @ C6 ) @ ( abstra4096080454567261402rm_a_b @ R ) ) ) ) )
=> ( abstra8448919418672941150rm_a_b @ R @ A ) ) ).
% CR_onI
thf(fact_750_rtrancl__power,axiom,
! [P2: produc357393685978478089rm_a_b,R4: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ P2 @ ( transi7742714808557438673rm_a_b @ R4 ) )
= ( ? [N2: nat] : ( member5869715511025134514rm_a_b @ P2 @ ( compow4057154403645558940rm_a_b @ N2 @ R4 ) ) ) ) ).
% rtrancl_power
thf(fact_751_relpow__imp__rtrancl,axiom,
! [P2: produc357393685978478089rm_a_b,N: nat,R4: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ P2 @ ( compow4057154403645558940rm_a_b @ N @ R4 ) )
=> ( member5869715511025134514rm_a_b @ P2 @ ( transi7742714808557438673rm_a_b @ R4 ) ) ) ).
% relpow_imp_rtrancl
thf(fact_752_rtrancl__imp__relpow,axiom,
! [P2: produc357393685978478089rm_a_b,R4: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ P2 @ ( transi7742714808557438673rm_a_b @ R4 ) )
=> ? [N3: nat] : ( member5869715511025134514rm_a_b @ P2 @ ( compow4057154403645558940rm_a_b @ N3 @ R4 ) ) ) ).
% rtrancl_imp_relpow
thf(fact_753_relpow__mono,axiom,
! [R: set_Pr4386577575007340137rm_a_b,R6: set_Pr4386577575007340137rm_a_b,N: nat] :
( ( ord_le118470702582115849rm_a_b @ R @ R6 )
=> ( ord_le118470702582115849rm_a_b @ ( compow4057154403645558940rm_a_b @ N @ R ) @ ( compow4057154403645558940rm_a_b @ N @ R6 ) ) ) ).
% relpow_mono
thf(fact_754_relpow__refl__mono,axiom,
! [Rel: set_Pr4386577575007340137rm_a_b,M: nat,N: nat,A2: term_a_b,B2: term_a_b] :
( ! [X: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ X ) @ Rel )
=> ( ( ord_less_eq_nat @ M @ N )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( compow4057154403645558940rm_a_b @ M @ Rel ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( compow4057154403645558940rm_a_b @ N @ Rel ) ) ) ) ) ).
% relpow_refl_mono
thf(fact_755_relpow__Suc__D2_H,axiom,
! [N: nat,R4: set_Pr4386577575007340137rm_a_b,X7: term_a_b,Y6: term_a_b,Z5: term_a_b] :
( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X7 @ Y6 ) @ ( compow4057154403645558940rm_a_b @ N @ R4 ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y6 @ Z5 ) @ R4 ) )
=> ? [W: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X7 @ W ) @ R4 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ W @ Z5 ) @ ( compow4057154403645558940rm_a_b @ N @ R4 ) ) ) ) ).
% relpow_Suc_D2'
thf(fact_756_relpow__image,axiom,
! [R: set_Pr4386577575007340137rm_a_b,F2: term_a_b > term_a_b,R6: set_Pr4386577575007340137rm_a_b,S: term_a_b,T3: term_a_b,N: nat] :
( ! [S3: term_a_b,T2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S3 @ T2 ) @ R )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F2 @ S3 ) @ ( F2 @ T2 ) ) @ R6 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T3 ) @ ( compow4057154403645558940rm_a_b @ N @ R ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F2 @ S ) @ ( F2 @ T3 ) ) @ ( compow4057154403645558940rm_a_b @ N @ R6 ) ) ) ) ).
% relpow_image
thf(fact_757_rtrancl__len__E,axiom,
! [X3: term_a_b,Y3: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ~ ! [N3: nat] :
~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ ( compow4057154403645558940rm_a_b @ N3 @ R ) ) ) ).
% rtrancl_len_E
thf(fact_758_trancl__steps__relpow,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b,X3: term_a_b,Y3: term_a_b,N: nat] :
( ( ord_le118470702582115849rm_a_b @ A2 @ ( transi7922773638565587891rm_a_b @ B2 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ ( compow4057154403645558940rm_a_b @ N @ A2 ) )
=> ? [M2: nat] :
( ( ord_less_eq_nat @ N @ M2 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ ( compow4057154403645558940rm_a_b @ M2 @ B2 ) ) ) ) ) ).
% trancl_steps_relpow
thf(fact_759_CR__on__def,axiom,
( abstra8448919418672941150rm_a_b
= ( ^ [R2: set_Pr4386577575007340137rm_a_b,A4: set_term_a_b] :
! [X4: term_a_b] :
( ( member_term_a_b @ X4 @ A4 )
=> ! [B5: term_a_b,C5: term_a_b] :
( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X4 @ B5 ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X4 @ C5 ) @ ( transi7742714808557438673rm_a_b @ R2 ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B5 @ C5 ) @ ( abstra4096080454567261402rm_a_b @ R2 ) ) ) ) ) ) ).
% CR_on_def
thf(fact_760_CR__onE,axiom,
! [R: set_Pr2972776593051762503rm_a_b,A: set_Pr4386577575007340137rm_a_b,A2: produc357393685978478089rm_a_b,B2: produc357393685978478089rm_a_b,C: produc357393685978478089rm_a_b] :
( ( abstra7359694574113289279rm_a_b @ R @ A )
=> ( ( member5869715511025134514rm_a_b @ A2 @ A )
=> ( ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B2 @ C ) @ ( abstra4889682940153293507rm_a_b @ R ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A2 @ B2 ) @ ( transi2615809358984392588rm_a_b @ R ) )
=> ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A2 @ C ) @ ( transi2615809358984392588rm_a_b @ R ) ) ) ) ) ) ).
% CR_onE
thf(fact_761_CR__onE,axiom,
! [R: set_Pr1811044260758604347_a_nat,A: set_Pr4934435412358123699_a_nat,A2: product_prod_a_nat,B2: product_prod_a_nat,C: product_prod_a_nat] :
( ( abstra7964079670522473415_a_nat @ R @ A )
=> ( ( member5724188588386418708_a_nat @ A2 @ A )
=> ( ~ ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ B2 @ C ) @ ( abstra881512052393830979_a_nat @ R ) )
=> ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A2 @ B2 ) @ ( transi2726145917338391738_a_nat @ R ) )
=> ~ ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A2 @ C ) @ ( transi2726145917338391738_a_nat @ R ) ) ) ) ) ) ).
% CR_onE
thf(fact_762_CR__onE,axiom,
! [R: set_Pr7047708467701517959at_nat,A: set_Pr9093778441882193744at_nat,A2: produc8199716216217303280at_nat,B2: produc8199716216217303280at_nat,C: produc8199716216217303280at_nat] :
( ( abstra8052594214376182822at_nat @ R @ A )
=> ( ( member7226740684066999833at_nat @ A2 @ A )
=> ( ~ ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ B2 @ C ) @ ( abstra8195730983011915946at_nat @ R ) )
=> ( ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ A2 @ B2 ) @ ( transi3422749762518297843at_nat @ R ) )
=> ~ ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ A2 @ C ) @ ( transi3422749762518297843at_nat @ R ) ) ) ) ) ) ).
% CR_onE
thf(fact_763_CR__onE,axiom,
! [R: set_Pr4386577575007340137rm_a_b,A: set_term_a_b,A2: term_a_b,B2: term_a_b,C: term_a_b] :
( ( abstra8448919418672941150rm_a_b @ R @ A )
=> ( ( member_term_a_b @ A2 @ A )
=> ( ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ C ) @ ( abstra4096080454567261402rm_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ C ) @ ( transi7742714808557438673rm_a_b @ R ) ) ) ) ) ) ).
% CR_onE
thf(fact_764_CR__onD,axiom,
! [R: set_Pr2972776593051762503rm_a_b,A: set_Pr4386577575007340137rm_a_b,A2: produc357393685978478089rm_a_b,B2: produc357393685978478089rm_a_b,C: produc357393685978478089rm_a_b] :
( ( abstra7359694574113289279rm_a_b @ R @ A )
=> ( ( member5869715511025134514rm_a_b @ A2 @ A )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A2 @ B2 ) @ ( transi2615809358984392588rm_a_b @ R ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A2 @ C ) @ ( transi2615809358984392588rm_a_b @ R ) )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B2 @ C ) @ ( abstra4889682940153293507rm_a_b @ R ) ) ) ) ) ) ).
% CR_onD
thf(fact_765_CR__onD,axiom,
! [R: set_Pr1811044260758604347_a_nat,A: set_Pr4934435412358123699_a_nat,A2: product_prod_a_nat,B2: product_prod_a_nat,C: product_prod_a_nat] :
( ( abstra7964079670522473415_a_nat @ R @ A )
=> ( ( member5724188588386418708_a_nat @ A2 @ A )
=> ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A2 @ B2 ) @ ( transi2726145917338391738_a_nat @ R ) )
=> ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A2 @ C ) @ ( transi2726145917338391738_a_nat @ R ) )
=> ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ B2 @ C ) @ ( abstra881512052393830979_a_nat @ R ) ) ) ) ) ) ).
% CR_onD
thf(fact_766_CR__onD,axiom,
! [R: set_Pr7047708467701517959at_nat,A: set_Pr9093778441882193744at_nat,A2: produc8199716216217303280at_nat,B2: produc8199716216217303280at_nat,C: produc8199716216217303280at_nat] :
( ( abstra8052594214376182822at_nat @ R @ A )
=> ( ( member7226740684066999833at_nat @ A2 @ A )
=> ( ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ A2 @ B2 ) @ ( transi3422749762518297843at_nat @ R ) )
=> ( ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ A2 @ C ) @ ( transi3422749762518297843at_nat @ R ) )
=> ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ B2 @ C ) @ ( abstra8195730983011915946at_nat @ R ) ) ) ) ) ) ).
% CR_onD
thf(fact_767_CR__onD,axiom,
! [R: set_Pr4386577575007340137rm_a_b,A: set_term_a_b,A2: term_a_b,B2: term_a_b,C: term_a_b] :
( ( abstra8448919418672941150rm_a_b @ R @ A )
=> ( ( member_term_a_b @ A2 @ A )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ C ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ C ) @ ( abstra4096080454567261402rm_a_b @ R ) ) ) ) ) ) ).
% CR_onD
thf(fact_768_CR__divergence__imp__join,axiom,
! [R: set_Pr4386577575007340137rm_a_b,X3: term_a_b,Y3: term_a_b,Z2: term_a_b] :
( ( abstra8448919418672941150rm_a_b @ R @ top_top_set_term_a_b )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Z2 ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ Z2 ) @ ( abstra4096080454567261402rm_a_b @ R ) ) ) ) ) ).
% CR_divergence_imp_join
thf(fact_769_partially__localize__CR,axiom,
! [R: set_Pr4386577575007340137rm_a_b] :
( ( abstra8448919418672941150rm_a_b @ R @ top_top_set_term_a_b )
= ( ! [X4: term_a_b,Y5: term_a_b,Z4: term_a_b] :
( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X4 @ Y5 ) @ R )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X4 @ Z4 ) @ ( transi7742714808557438673rm_a_b @ R ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y5 @ Z4 ) @ ( abstra4096080454567261402rm_a_b @ R ) ) ) ) ) ).
% partially_localize_CR
thf(fact_770_CR__join__right__I,axiom,
! [R: set_Pr4386577575007340137rm_a_b,X3: term_a_b,Y3: term_a_b,Z2: term_a_b] :
( ( abstra8448919418672941150rm_a_b @ R @ top_top_set_term_a_b )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ ( abstra4096080454567261402rm_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ Z2 ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Z2 ) @ ( abstra4096080454567261402rm_a_b @ R ) ) ) ) ) ).
% CR_join_right_I
thf(fact_771_CR__join__left__I,axiom,
! [R: set_Pr4386577575007340137rm_a_b,X3: term_a_b,Y3: term_a_b,Z2: term_a_b] :
( ( abstra8448919418672941150rm_a_b @ R @ top_top_set_term_a_b )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Z2 ) @ ( abstra4096080454567261402rm_a_b @ R ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ Z2 ) @ ( abstra4096080454567261402rm_a_b @ R ) ) ) ) ) ).
% CR_join_left_I
thf(fact_772_CR__iff__conversion__imp__join,axiom,
! [R: set_Pr4386577575007340137rm_a_b] :
( ( abstra8448919418672941150rm_a_b @ R @ top_top_set_term_a_b )
= ( ord_le118470702582115849rm_a_b @ ( abstra1130568886744591910rm_a_b @ R ) @ ( abstra4096080454567261402rm_a_b @ R ) ) ) ).
% CR_iff_conversion_imp_join
thf(fact_773_CR__on__singletonI,axiom,
! [A2: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ! [B6: term_a_b,C6: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B6 ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ C6 ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B6 @ C6 ) @ ( abstra4096080454567261402rm_a_b @ R ) ) ) )
=> ( abstra8448919418672941150rm_a_b @ R @ ( insert_term_a_b @ A2 @ bot_bot_set_term_a_b ) ) ) ).
% CR_on_singletonI
thf(fact_774_irrefl__tranclI,axiom,
! [R: set_Pr4386577575007340137rm_a_b,X3: term_a_b] :
( ( ( inf_in1693821330460682071rm_a_b @ ( conver7608559976937845093rm_a_b @ R ) @ ( transi7742714808557438673rm_a_b @ R ) )
= bot_bo197521221353338581rm_a_b )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ X3 ) @ ( transi7922773638565587891rm_a_b @ R ) ) ) ).
% irrefl_tranclI
thf(fact_775_insert__iff,axiom,
! [A2: produc357393685978478089rm_a_b,B2: produc357393685978478089rm_a_b,A: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ A2 @ ( insert7009541432154983385rm_a_b @ B2 @ A ) )
= ( ( A2 = B2 )
| ( member5869715511025134514rm_a_b @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_776_insert__iff,axiom,
! [A2: product_prod_a_nat,B2: product_prod_a_nat,A: set_Pr4934435412358123699_a_nat] :
( ( member5724188588386418708_a_nat @ A2 @ ( insert8054603423593749677_a_nat @ B2 @ A ) )
= ( ( A2 = B2 )
| ( member5724188588386418708_a_nat @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_777_insert__iff,axiom,
! [A2: produc8199716216217303280at_nat,B2: produc8199716216217303280at_nat,A: set_Pr9093778441882193744at_nat] :
( ( member7226740684066999833at_nat @ A2 @ ( insert891772969674204864at_nat @ B2 @ A ) )
= ( ( A2 = B2 )
| ( member7226740684066999833at_nat @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_778_insertCI,axiom,
! [A2: produc357393685978478089rm_a_b,B: set_Pr4386577575007340137rm_a_b,B2: produc357393685978478089rm_a_b] :
( ( ~ ( member5869715511025134514rm_a_b @ A2 @ B )
=> ( A2 = B2 ) )
=> ( member5869715511025134514rm_a_b @ A2 @ ( insert7009541432154983385rm_a_b @ B2 @ B ) ) ) ).
% insertCI
thf(fact_779_insertCI,axiom,
! [A2: product_prod_a_nat,B: set_Pr4934435412358123699_a_nat,B2: product_prod_a_nat] :
( ( ~ ( member5724188588386418708_a_nat @ A2 @ B )
=> ( A2 = B2 ) )
=> ( member5724188588386418708_a_nat @ A2 @ ( insert8054603423593749677_a_nat @ B2 @ B ) ) ) ).
% insertCI
thf(fact_780_insertCI,axiom,
! [A2: produc8199716216217303280at_nat,B: set_Pr9093778441882193744at_nat,B2: produc8199716216217303280at_nat] :
( ( ~ ( member7226740684066999833at_nat @ A2 @ B )
=> ( A2 = B2 ) )
=> ( member7226740684066999833at_nat @ A2 @ ( insert891772969674204864at_nat @ B2 @ B ) ) ) ).
% insertCI
thf(fact_781_Int__iff,axiom,
! [C: produc357393685978478089rm_a_b,A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ C @ ( inf_in1693821330460682071rm_a_b @ A @ B ) )
= ( ( member5869715511025134514rm_a_b @ C @ A )
& ( member5869715511025134514rm_a_b @ C @ B ) ) ) ).
% Int_iff
thf(fact_782_Int__iff,axiom,
! [C: product_prod_a_nat,A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( member5724188588386418708_a_nat @ C @ ( inf_in3813033988066259269_a_nat @ A @ B ) )
= ( ( member5724188588386418708_a_nat @ C @ A )
& ( member5724188588386418708_a_nat @ C @ B ) ) ) ).
% Int_iff
thf(fact_783_Int__iff,axiom,
! [C: produc8199716216217303280at_nat,A: set_Pr9093778441882193744at_nat,B: set_Pr9093778441882193744at_nat] :
( ( member7226740684066999833at_nat @ C @ ( inf_in907833683887804734at_nat @ A @ B ) )
= ( ( member7226740684066999833at_nat @ C @ A )
& ( member7226740684066999833at_nat @ C @ B ) ) ) ).
% Int_iff
thf(fact_784_IntI,axiom,
! [C: produc357393685978478089rm_a_b,A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ C @ A )
=> ( ( member5869715511025134514rm_a_b @ C @ B )
=> ( member5869715511025134514rm_a_b @ C @ ( inf_in1693821330460682071rm_a_b @ A @ B ) ) ) ) ).
% IntI
thf(fact_785_IntI,axiom,
! [C: product_prod_a_nat,A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( member5724188588386418708_a_nat @ C @ A )
=> ( ( member5724188588386418708_a_nat @ C @ B )
=> ( member5724188588386418708_a_nat @ C @ ( inf_in3813033988066259269_a_nat @ A @ B ) ) ) ) ).
% IntI
thf(fact_786_IntI,axiom,
! [C: produc8199716216217303280at_nat,A: set_Pr9093778441882193744at_nat,B: set_Pr9093778441882193744at_nat] :
( ( member7226740684066999833at_nat @ C @ A )
=> ( ( member7226740684066999833at_nat @ C @ B )
=> ( member7226740684066999833at_nat @ C @ ( inf_in907833683887804734at_nat @ A @ B ) ) ) ) ).
% IntI
thf(fact_787_le__inf__iff,axiom,
! [X3: set_Pr4934435412358123699_a_nat,Y3: set_Pr4934435412358123699_a_nat,Z2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X3 @ ( inf_in3813033988066259269_a_nat @ Y3 @ Z2 ) )
= ( ( ord_le8666007276011122963_a_nat @ X3 @ Y3 )
& ( ord_le8666007276011122963_a_nat @ X3 @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_788_le__inf__iff,axiom,
! [X3: set_Pr4386577575007340137rm_a_b,Y3: set_Pr4386577575007340137rm_a_b,Z2: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ X3 @ ( inf_in1693821330460682071rm_a_b @ Y3 @ Z2 ) )
= ( ( ord_le118470702582115849rm_a_b @ X3 @ Y3 )
& ( ord_le118470702582115849rm_a_b @ X3 @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_789_le__inf__iff,axiom,
! [X3: set_Pr9093778441882193744at_nat,Y3: set_Pr9093778441882193744at_nat,Z2: set_Pr9093778441882193744at_nat] :
( ( ord_le3678578370064672496at_nat @ X3 @ ( inf_in907833683887804734at_nat @ Y3 @ Z2 ) )
= ( ( ord_le3678578370064672496at_nat @ X3 @ Y3 )
& ( ord_le3678578370064672496at_nat @ X3 @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_790_inf_Obounded__iff,axiom,
! [A2: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat,C: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A2 @ ( inf_in3813033988066259269_a_nat @ B2 @ C ) )
= ( ( ord_le8666007276011122963_a_nat @ A2 @ B2 )
& ( ord_le8666007276011122963_a_nat @ A2 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_791_inf_Obounded__iff,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b,C: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A2 @ ( inf_in1693821330460682071rm_a_b @ B2 @ C ) )
= ( ( ord_le118470702582115849rm_a_b @ A2 @ B2 )
& ( ord_le118470702582115849rm_a_b @ A2 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_792_inf_Obounded__iff,axiom,
! [A2: set_Pr9093778441882193744at_nat,B2: set_Pr9093778441882193744at_nat,C: set_Pr9093778441882193744at_nat] :
( ( ord_le3678578370064672496at_nat @ A2 @ ( inf_in907833683887804734at_nat @ B2 @ C ) )
= ( ( ord_le3678578370064672496at_nat @ A2 @ B2 )
& ( ord_le3678578370064672496at_nat @ A2 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_793_singletonI,axiom,
! [A2: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ A2 @ ( insert7009541432154983385rm_a_b @ A2 @ bot_bo197521221353338581rm_a_b ) ) ).
% singletonI
thf(fact_794_singletonI,axiom,
! [A2: product_prod_a_nat] : ( member5724188588386418708_a_nat @ A2 @ ( insert8054603423593749677_a_nat @ A2 @ bot_bo9049108969261143879_a_nat ) ) ).
% singletonI
thf(fact_795_singletonI,axiom,
! [A2: produc8199716216217303280at_nat] : ( member7226740684066999833at_nat @ A2 @ ( insert891772969674204864at_nat @ A2 @ bot_bo8000532843592726972at_nat ) ) ).
% singletonI
thf(fact_796_insert__subset,axiom,
! [X3: product_prod_a_nat,A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ ( insert8054603423593749677_a_nat @ X3 @ A ) @ B )
= ( ( member5724188588386418708_a_nat @ X3 @ B )
& ( ord_le8666007276011122963_a_nat @ A @ B ) ) ) ).
% insert_subset
thf(fact_797_insert__subset,axiom,
! [X3: produc357393685978478089rm_a_b,A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ ( insert7009541432154983385rm_a_b @ X3 @ A ) @ B )
= ( ( member5869715511025134514rm_a_b @ X3 @ B )
& ( ord_le118470702582115849rm_a_b @ A @ B ) ) ) ).
% insert_subset
thf(fact_798_insert__subset,axiom,
! [X3: produc8199716216217303280at_nat,A: set_Pr9093778441882193744at_nat,B: set_Pr9093778441882193744at_nat] :
( ( ord_le3678578370064672496at_nat @ ( insert891772969674204864at_nat @ X3 @ A ) @ B )
= ( ( member7226740684066999833at_nat @ X3 @ B )
& ( ord_le3678578370064672496at_nat @ A @ B ) ) ) ).
% insert_subset
thf(fact_799_Int__subset__iff,axiom,
! [C2: set_Pr4934435412358123699_a_nat,A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ C2 @ ( inf_in3813033988066259269_a_nat @ A @ B ) )
= ( ( ord_le8666007276011122963_a_nat @ C2 @ A )
& ( ord_le8666007276011122963_a_nat @ C2 @ B ) ) ) ).
% Int_subset_iff
thf(fact_800_Int__subset__iff,axiom,
! [C2: set_Pr4386577575007340137rm_a_b,A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ C2 @ ( inf_in1693821330460682071rm_a_b @ A @ B ) )
= ( ( ord_le118470702582115849rm_a_b @ C2 @ A )
& ( ord_le118470702582115849rm_a_b @ C2 @ B ) ) ) ).
% Int_subset_iff
thf(fact_801_Int__subset__iff,axiom,
! [C2: set_Pr9093778441882193744at_nat,A: set_Pr9093778441882193744at_nat,B: set_Pr9093778441882193744at_nat] :
( ( ord_le3678578370064672496at_nat @ C2 @ ( inf_in907833683887804734at_nat @ A @ B ) )
= ( ( ord_le3678578370064672496at_nat @ C2 @ A )
& ( ord_le3678578370064672496at_nat @ C2 @ B ) ) ) ).
% Int_subset_iff
thf(fact_802_Int__insert__right__if1,axiom,
! [A2: produc357393685978478089rm_a_b,A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ A2 @ A )
=> ( ( inf_in1693821330460682071rm_a_b @ A @ ( insert7009541432154983385rm_a_b @ A2 @ B ) )
= ( insert7009541432154983385rm_a_b @ A2 @ ( inf_in1693821330460682071rm_a_b @ A @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_803_Int__insert__right__if1,axiom,
! [A2: product_prod_a_nat,A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( member5724188588386418708_a_nat @ A2 @ A )
=> ( ( inf_in3813033988066259269_a_nat @ A @ ( insert8054603423593749677_a_nat @ A2 @ B ) )
= ( insert8054603423593749677_a_nat @ A2 @ ( inf_in3813033988066259269_a_nat @ A @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_804_Int__insert__right__if1,axiom,
! [A2: produc8199716216217303280at_nat,A: set_Pr9093778441882193744at_nat,B: set_Pr9093778441882193744at_nat] :
( ( member7226740684066999833at_nat @ A2 @ A )
=> ( ( inf_in907833683887804734at_nat @ A @ ( insert891772969674204864at_nat @ A2 @ B ) )
= ( insert891772969674204864at_nat @ A2 @ ( inf_in907833683887804734at_nat @ A @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_805_Int__insert__right__if0,axiom,
! [A2: produc357393685978478089rm_a_b,A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ~ ( member5869715511025134514rm_a_b @ A2 @ A )
=> ( ( inf_in1693821330460682071rm_a_b @ A @ ( insert7009541432154983385rm_a_b @ A2 @ B ) )
= ( inf_in1693821330460682071rm_a_b @ A @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_806_Int__insert__right__if0,axiom,
! [A2: product_prod_a_nat,A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ~ ( member5724188588386418708_a_nat @ A2 @ A )
=> ( ( inf_in3813033988066259269_a_nat @ A @ ( insert8054603423593749677_a_nat @ A2 @ B ) )
= ( inf_in3813033988066259269_a_nat @ A @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_807_Int__insert__right__if0,axiom,
! [A2: produc8199716216217303280at_nat,A: set_Pr9093778441882193744at_nat,B: set_Pr9093778441882193744at_nat] :
( ~ ( member7226740684066999833at_nat @ A2 @ A )
=> ( ( inf_in907833683887804734at_nat @ A @ ( insert891772969674204864at_nat @ A2 @ B ) )
= ( inf_in907833683887804734at_nat @ A @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_808_Int__insert__left__if1,axiom,
! [A2: produc357393685978478089rm_a_b,C2: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ A2 @ C2 )
=> ( ( inf_in1693821330460682071rm_a_b @ ( insert7009541432154983385rm_a_b @ A2 @ B ) @ C2 )
= ( insert7009541432154983385rm_a_b @ A2 @ ( inf_in1693821330460682071rm_a_b @ B @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_809_Int__insert__left__if1,axiom,
! [A2: product_prod_a_nat,C2: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( member5724188588386418708_a_nat @ A2 @ C2 )
=> ( ( inf_in3813033988066259269_a_nat @ ( insert8054603423593749677_a_nat @ A2 @ B ) @ C2 )
= ( insert8054603423593749677_a_nat @ A2 @ ( inf_in3813033988066259269_a_nat @ B @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_810_Int__insert__left__if1,axiom,
! [A2: produc8199716216217303280at_nat,C2: set_Pr9093778441882193744at_nat,B: set_Pr9093778441882193744at_nat] :
( ( member7226740684066999833at_nat @ A2 @ C2 )
=> ( ( inf_in907833683887804734at_nat @ ( insert891772969674204864at_nat @ A2 @ B ) @ C2 )
= ( insert891772969674204864at_nat @ A2 @ ( inf_in907833683887804734at_nat @ B @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_811_Int__insert__left__if0,axiom,
! [A2: produc357393685978478089rm_a_b,C2: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ~ ( member5869715511025134514rm_a_b @ A2 @ C2 )
=> ( ( inf_in1693821330460682071rm_a_b @ ( insert7009541432154983385rm_a_b @ A2 @ B ) @ C2 )
= ( inf_in1693821330460682071rm_a_b @ B @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_812_Int__insert__left__if0,axiom,
! [A2: product_prod_a_nat,C2: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ~ ( member5724188588386418708_a_nat @ A2 @ C2 )
=> ( ( inf_in3813033988066259269_a_nat @ ( insert8054603423593749677_a_nat @ A2 @ B ) @ C2 )
= ( inf_in3813033988066259269_a_nat @ B @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_813_Int__insert__left__if0,axiom,
! [A2: produc8199716216217303280at_nat,C2: set_Pr9093778441882193744at_nat,B: set_Pr9093778441882193744at_nat] :
( ~ ( member7226740684066999833at_nat @ A2 @ C2 )
=> ( ( inf_in907833683887804734at_nat @ ( insert891772969674204864at_nat @ A2 @ B ) @ C2 )
= ( inf_in907833683887804734at_nat @ B @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_814_singleton__insert__inj__eq_H,axiom,
! [A2: product_prod_a_nat,A: set_Pr4934435412358123699_a_nat,B2: product_prod_a_nat] :
( ( ( insert8054603423593749677_a_nat @ A2 @ A )
= ( insert8054603423593749677_a_nat @ B2 @ bot_bo9049108969261143879_a_nat ) )
= ( ( A2 = B2 )
& ( ord_le8666007276011122963_a_nat @ A @ ( insert8054603423593749677_a_nat @ B2 @ bot_bo9049108969261143879_a_nat ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_815_singleton__insert__inj__eq_H,axiom,
! [A2: produc357393685978478089rm_a_b,A: set_Pr4386577575007340137rm_a_b,B2: produc357393685978478089rm_a_b] :
( ( ( insert7009541432154983385rm_a_b @ A2 @ A )
= ( insert7009541432154983385rm_a_b @ B2 @ bot_bo197521221353338581rm_a_b ) )
= ( ( A2 = B2 )
& ( ord_le118470702582115849rm_a_b @ A @ ( insert7009541432154983385rm_a_b @ B2 @ bot_bo197521221353338581rm_a_b ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_816_singleton__insert__inj__eq_H,axiom,
! [A2: produc8199716216217303280at_nat,A: set_Pr9093778441882193744at_nat,B2: produc8199716216217303280at_nat] :
( ( ( insert891772969674204864at_nat @ A2 @ A )
= ( insert891772969674204864at_nat @ B2 @ bot_bo8000532843592726972at_nat ) )
= ( ( A2 = B2 )
& ( ord_le3678578370064672496at_nat @ A @ ( insert891772969674204864at_nat @ B2 @ bot_bo8000532843592726972at_nat ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_817_singleton__insert__inj__eq,axiom,
! [B2: product_prod_a_nat,A2: product_prod_a_nat,A: set_Pr4934435412358123699_a_nat] :
( ( ( insert8054603423593749677_a_nat @ B2 @ bot_bo9049108969261143879_a_nat )
= ( insert8054603423593749677_a_nat @ A2 @ A ) )
= ( ( A2 = B2 )
& ( ord_le8666007276011122963_a_nat @ A @ ( insert8054603423593749677_a_nat @ B2 @ bot_bo9049108969261143879_a_nat ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_818_singleton__insert__inj__eq,axiom,
! [B2: produc357393685978478089rm_a_b,A2: produc357393685978478089rm_a_b,A: set_Pr4386577575007340137rm_a_b] :
( ( ( insert7009541432154983385rm_a_b @ B2 @ bot_bo197521221353338581rm_a_b )
= ( insert7009541432154983385rm_a_b @ A2 @ A ) )
= ( ( A2 = B2 )
& ( ord_le118470702582115849rm_a_b @ A @ ( insert7009541432154983385rm_a_b @ B2 @ bot_bo197521221353338581rm_a_b ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_819_singleton__insert__inj__eq,axiom,
! [B2: produc8199716216217303280at_nat,A2: produc8199716216217303280at_nat,A: set_Pr9093778441882193744at_nat] :
( ( ( insert891772969674204864at_nat @ B2 @ bot_bo8000532843592726972at_nat )
= ( insert891772969674204864at_nat @ A2 @ A ) )
= ( ( A2 = B2 )
& ( ord_le3678578370064672496at_nat @ A @ ( insert891772969674204864at_nat @ B2 @ bot_bo8000532843592726972at_nat ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_820_disjoint__insert_I2_J,axiom,
! [A: set_Pr4386577575007340137rm_a_b,B2: produc357393685978478089rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ( bot_bo197521221353338581rm_a_b
= ( inf_in1693821330460682071rm_a_b @ A @ ( insert7009541432154983385rm_a_b @ B2 @ B ) ) )
= ( ~ ( member5869715511025134514rm_a_b @ B2 @ A )
& ( bot_bo197521221353338581rm_a_b
= ( inf_in1693821330460682071rm_a_b @ A @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_821_disjoint__insert_I2_J,axiom,
! [A: set_Pr4934435412358123699_a_nat,B2: product_prod_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( bot_bo9049108969261143879_a_nat
= ( inf_in3813033988066259269_a_nat @ A @ ( insert8054603423593749677_a_nat @ B2 @ B ) ) )
= ( ~ ( member5724188588386418708_a_nat @ B2 @ A )
& ( bot_bo9049108969261143879_a_nat
= ( inf_in3813033988066259269_a_nat @ A @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_822_disjoint__insert_I2_J,axiom,
! [A: set_Pr9093778441882193744at_nat,B2: produc8199716216217303280at_nat,B: set_Pr9093778441882193744at_nat] :
( ( bot_bo8000532843592726972at_nat
= ( inf_in907833683887804734at_nat @ A @ ( insert891772969674204864at_nat @ B2 @ B ) ) )
= ( ~ ( member7226740684066999833at_nat @ B2 @ A )
& ( bot_bo8000532843592726972at_nat
= ( inf_in907833683887804734at_nat @ A @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_823_disjoint__insert_I1_J,axiom,
! [B: set_Pr4386577575007340137rm_a_b,A2: produc357393685978478089rm_a_b,A: set_Pr4386577575007340137rm_a_b] :
( ( ( inf_in1693821330460682071rm_a_b @ B @ ( insert7009541432154983385rm_a_b @ A2 @ A ) )
= bot_bo197521221353338581rm_a_b )
= ( ~ ( member5869715511025134514rm_a_b @ A2 @ B )
& ( ( inf_in1693821330460682071rm_a_b @ B @ A )
= bot_bo197521221353338581rm_a_b ) ) ) ).
% disjoint_insert(1)
thf(fact_824_disjoint__insert_I1_J,axiom,
! [B: set_Pr4934435412358123699_a_nat,A2: product_prod_a_nat,A: set_Pr4934435412358123699_a_nat] :
( ( ( inf_in3813033988066259269_a_nat @ B @ ( insert8054603423593749677_a_nat @ A2 @ A ) )
= bot_bo9049108969261143879_a_nat )
= ( ~ ( member5724188588386418708_a_nat @ A2 @ B )
& ( ( inf_in3813033988066259269_a_nat @ B @ A )
= bot_bo9049108969261143879_a_nat ) ) ) ).
% disjoint_insert(1)
thf(fact_825_disjoint__insert_I1_J,axiom,
! [B: set_Pr9093778441882193744at_nat,A2: produc8199716216217303280at_nat,A: set_Pr9093778441882193744at_nat] :
( ( ( inf_in907833683887804734at_nat @ B @ ( insert891772969674204864at_nat @ A2 @ A ) )
= bot_bo8000532843592726972at_nat )
= ( ~ ( member7226740684066999833at_nat @ A2 @ B )
& ( ( inf_in907833683887804734at_nat @ B @ A )
= bot_bo8000532843592726972at_nat ) ) ) ).
% disjoint_insert(1)
thf(fact_826_insert__disjoint_I2_J,axiom,
! [A2: produc357393685978478089rm_a_b,A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ( bot_bo197521221353338581rm_a_b
= ( inf_in1693821330460682071rm_a_b @ ( insert7009541432154983385rm_a_b @ A2 @ A ) @ B ) )
= ( ~ ( member5869715511025134514rm_a_b @ A2 @ B )
& ( bot_bo197521221353338581rm_a_b
= ( inf_in1693821330460682071rm_a_b @ A @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_827_insert__disjoint_I2_J,axiom,
! [A2: product_prod_a_nat,A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( bot_bo9049108969261143879_a_nat
= ( inf_in3813033988066259269_a_nat @ ( insert8054603423593749677_a_nat @ A2 @ A ) @ B ) )
= ( ~ ( member5724188588386418708_a_nat @ A2 @ B )
& ( bot_bo9049108969261143879_a_nat
= ( inf_in3813033988066259269_a_nat @ A @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_828_insert__disjoint_I2_J,axiom,
! [A2: produc8199716216217303280at_nat,A: set_Pr9093778441882193744at_nat,B: set_Pr9093778441882193744at_nat] :
( ( bot_bo8000532843592726972at_nat
= ( inf_in907833683887804734at_nat @ ( insert891772969674204864at_nat @ A2 @ A ) @ B ) )
= ( ~ ( member7226740684066999833at_nat @ A2 @ B )
& ( bot_bo8000532843592726972at_nat
= ( inf_in907833683887804734at_nat @ A @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_829_insert__disjoint_I1_J,axiom,
! [A2: produc357393685978478089rm_a_b,A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ( ( inf_in1693821330460682071rm_a_b @ ( insert7009541432154983385rm_a_b @ A2 @ A ) @ B )
= bot_bo197521221353338581rm_a_b )
= ( ~ ( member5869715511025134514rm_a_b @ A2 @ B )
& ( ( inf_in1693821330460682071rm_a_b @ A @ B )
= bot_bo197521221353338581rm_a_b ) ) ) ).
% insert_disjoint(1)
thf(fact_830_insert__disjoint_I1_J,axiom,
! [A2: product_prod_a_nat,A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( ( inf_in3813033988066259269_a_nat @ ( insert8054603423593749677_a_nat @ A2 @ A ) @ B )
= bot_bo9049108969261143879_a_nat )
= ( ~ ( member5724188588386418708_a_nat @ A2 @ B )
& ( ( inf_in3813033988066259269_a_nat @ A @ B )
= bot_bo9049108969261143879_a_nat ) ) ) ).
% insert_disjoint(1)
thf(fact_831_insert__disjoint_I1_J,axiom,
! [A2: produc8199716216217303280at_nat,A: set_Pr9093778441882193744at_nat,B: set_Pr9093778441882193744at_nat] :
( ( ( inf_in907833683887804734at_nat @ ( insert891772969674204864at_nat @ A2 @ A ) @ B )
= bot_bo8000532843592726972at_nat )
= ( ~ ( member7226740684066999833at_nat @ A2 @ B )
& ( ( inf_in907833683887804734at_nat @ A @ B )
= bot_bo8000532843592726972at_nat ) ) ) ).
% insert_disjoint(1)
thf(fact_832_Range__insert,axiom,
! [A2: term_a_b,B2: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( range_8688746797885257759rm_a_b @ ( insert7009541432154983385rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ R ) )
= ( insert_term_a_b @ B2 @ ( range_8688746797885257759rm_a_b @ R ) ) ) ).
% Range_insert
thf(fact_833_Range__insert,axiom,
! [A2: nat > nat,B2: nat,R: set_Pr9093778441882193744at_nat] :
( ( range_nat_nat_nat @ ( insert891772969674204864at_nat @ ( produc72220940542539688at_nat @ A2 @ B2 ) @ R ) )
= ( insert_nat @ B2 @ ( range_nat_nat_nat @ R ) ) ) ).
% Range_insert
thf(fact_834_Range__insert,axiom,
! [A2: a,B2: nat,R: set_Pr4934435412358123699_a_nat] :
( ( range_a_nat @ ( insert8054603423593749677_a_nat @ ( product_Pair_a_nat @ A2 @ B2 ) @ R ) )
= ( insert_nat @ B2 @ ( range_a_nat @ R ) ) ) ).
% Range_insert
thf(fact_835_Int__emptyI,axiom,
! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ! [X: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X @ A )
=> ~ ( member5869715511025134514rm_a_b @ X @ B ) )
=> ( ( inf_in1693821330460682071rm_a_b @ A @ B )
= bot_bo197521221353338581rm_a_b ) ) ).
% Int_emptyI
thf(fact_836_Int__emptyI,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ! [X: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ X @ A )
=> ~ ( member5724188588386418708_a_nat @ X @ B ) )
=> ( ( inf_in3813033988066259269_a_nat @ A @ B )
= bot_bo9049108969261143879_a_nat ) ) ).
% Int_emptyI
thf(fact_837_Int__emptyI,axiom,
! [A: set_Pr9093778441882193744at_nat,B: set_Pr9093778441882193744at_nat] :
( ! [X: produc8199716216217303280at_nat] :
( ( member7226740684066999833at_nat @ X @ A )
=> ~ ( member7226740684066999833at_nat @ X @ B ) )
=> ( ( inf_in907833683887804734at_nat @ A @ B )
= bot_bo8000532843592726972at_nat ) ) ).
% Int_emptyI
thf(fact_838_disjoint__iff,axiom,
! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ( ( inf_in1693821330460682071rm_a_b @ A @ B )
= bot_bo197521221353338581rm_a_b )
= ( ! [X4: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X4 @ A )
=> ~ ( member5869715511025134514rm_a_b @ X4 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_839_disjoint__iff,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( ( inf_in3813033988066259269_a_nat @ A @ B )
= bot_bo9049108969261143879_a_nat )
= ( ! [X4: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ X4 @ A )
=> ~ ( member5724188588386418708_a_nat @ X4 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_840_disjoint__iff,axiom,
! [A: set_Pr9093778441882193744at_nat,B: set_Pr9093778441882193744at_nat] :
( ( ( inf_in907833683887804734at_nat @ A @ B )
= bot_bo8000532843592726972at_nat )
= ( ! [X4: produc8199716216217303280at_nat] :
( ( member7226740684066999833at_nat @ X4 @ A )
=> ~ ( member7226740684066999833at_nat @ X4 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_841_mk__disjoint__insert,axiom,
! [A2: produc357393685978478089rm_a_b,A: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ A2 @ A )
=> ? [B9: set_Pr4386577575007340137rm_a_b] :
( ( A
= ( insert7009541432154983385rm_a_b @ A2 @ B9 ) )
& ~ ( member5869715511025134514rm_a_b @ A2 @ B9 ) ) ) ).
% mk_disjoint_insert
thf(fact_842_mk__disjoint__insert,axiom,
! [A2: product_prod_a_nat,A: set_Pr4934435412358123699_a_nat] :
( ( member5724188588386418708_a_nat @ A2 @ A )
=> ? [B9: set_Pr4934435412358123699_a_nat] :
( ( A
= ( insert8054603423593749677_a_nat @ A2 @ B9 ) )
& ~ ( member5724188588386418708_a_nat @ A2 @ B9 ) ) ) ).
% mk_disjoint_insert
thf(fact_843_mk__disjoint__insert,axiom,
! [A2: produc8199716216217303280at_nat,A: set_Pr9093778441882193744at_nat] :
( ( member7226740684066999833at_nat @ A2 @ A )
=> ? [B9: set_Pr9093778441882193744at_nat] :
( ( A
= ( insert891772969674204864at_nat @ A2 @ B9 ) )
& ~ ( member7226740684066999833at_nat @ A2 @ B9 ) ) ) ).
% mk_disjoint_insert
thf(fact_844_Int__insert__right,axiom,
! [A2: produc357393685978478089rm_a_b,A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ( ( member5869715511025134514rm_a_b @ A2 @ A )
=> ( ( inf_in1693821330460682071rm_a_b @ A @ ( insert7009541432154983385rm_a_b @ A2 @ B ) )
= ( insert7009541432154983385rm_a_b @ A2 @ ( inf_in1693821330460682071rm_a_b @ A @ B ) ) ) )
& ( ~ ( member5869715511025134514rm_a_b @ A2 @ A )
=> ( ( inf_in1693821330460682071rm_a_b @ A @ ( insert7009541432154983385rm_a_b @ A2 @ B ) )
= ( inf_in1693821330460682071rm_a_b @ A @ B ) ) ) ) ).
% Int_insert_right
thf(fact_845_Int__insert__right,axiom,
! [A2: product_prod_a_nat,A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( ( member5724188588386418708_a_nat @ A2 @ A )
=> ( ( inf_in3813033988066259269_a_nat @ A @ ( insert8054603423593749677_a_nat @ A2 @ B ) )
= ( insert8054603423593749677_a_nat @ A2 @ ( inf_in3813033988066259269_a_nat @ A @ B ) ) ) )
& ( ~ ( member5724188588386418708_a_nat @ A2 @ A )
=> ( ( inf_in3813033988066259269_a_nat @ A @ ( insert8054603423593749677_a_nat @ A2 @ B ) )
= ( inf_in3813033988066259269_a_nat @ A @ B ) ) ) ) ).
% Int_insert_right
thf(fact_846_Int__insert__right,axiom,
! [A2: produc8199716216217303280at_nat,A: set_Pr9093778441882193744at_nat,B: set_Pr9093778441882193744at_nat] :
( ( ( member7226740684066999833at_nat @ A2 @ A )
=> ( ( inf_in907833683887804734at_nat @ A @ ( insert891772969674204864at_nat @ A2 @ B ) )
= ( insert891772969674204864at_nat @ A2 @ ( inf_in907833683887804734at_nat @ A @ B ) ) ) )
& ( ~ ( member7226740684066999833at_nat @ A2 @ A )
=> ( ( inf_in907833683887804734at_nat @ A @ ( insert891772969674204864at_nat @ A2 @ B ) )
= ( inf_in907833683887804734at_nat @ A @ B ) ) ) ) ).
% Int_insert_right
thf(fact_847_Int__insert__left,axiom,
! [A2: produc357393685978478089rm_a_b,C2: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ( ( member5869715511025134514rm_a_b @ A2 @ C2 )
=> ( ( inf_in1693821330460682071rm_a_b @ ( insert7009541432154983385rm_a_b @ A2 @ B ) @ C2 )
= ( insert7009541432154983385rm_a_b @ A2 @ ( inf_in1693821330460682071rm_a_b @ B @ C2 ) ) ) )
& ( ~ ( member5869715511025134514rm_a_b @ A2 @ C2 )
=> ( ( inf_in1693821330460682071rm_a_b @ ( insert7009541432154983385rm_a_b @ A2 @ B ) @ C2 )
= ( inf_in1693821330460682071rm_a_b @ B @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_848_Int__insert__left,axiom,
! [A2: product_prod_a_nat,C2: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( ( member5724188588386418708_a_nat @ A2 @ C2 )
=> ( ( inf_in3813033988066259269_a_nat @ ( insert8054603423593749677_a_nat @ A2 @ B ) @ C2 )
= ( insert8054603423593749677_a_nat @ A2 @ ( inf_in3813033988066259269_a_nat @ B @ C2 ) ) ) )
& ( ~ ( member5724188588386418708_a_nat @ A2 @ C2 )
=> ( ( inf_in3813033988066259269_a_nat @ ( insert8054603423593749677_a_nat @ A2 @ B ) @ C2 )
= ( inf_in3813033988066259269_a_nat @ B @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_849_Int__insert__left,axiom,
! [A2: produc8199716216217303280at_nat,C2: set_Pr9093778441882193744at_nat,B: set_Pr9093778441882193744at_nat] :
( ( ( member7226740684066999833at_nat @ A2 @ C2 )
=> ( ( inf_in907833683887804734at_nat @ ( insert891772969674204864at_nat @ A2 @ B ) @ C2 )
= ( insert891772969674204864at_nat @ A2 @ ( inf_in907833683887804734at_nat @ B @ C2 ) ) ) )
& ( ~ ( member7226740684066999833at_nat @ A2 @ C2 )
=> ( ( inf_in907833683887804734at_nat @ ( insert891772969674204864at_nat @ A2 @ B ) @ C2 )
= ( inf_in907833683887804734at_nat @ B @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_850_insert__eq__iff,axiom,
! [A2: produc357393685978478089rm_a_b,A: set_Pr4386577575007340137rm_a_b,B2: produc357393685978478089rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ~ ( member5869715511025134514rm_a_b @ A2 @ A )
=> ( ~ ( member5869715511025134514rm_a_b @ B2 @ B )
=> ( ( ( insert7009541432154983385rm_a_b @ A2 @ A )
= ( insert7009541432154983385rm_a_b @ B2 @ B ) )
= ( ( ( A2 = B2 )
=> ( A = B ) )
& ( ( A2 != B2 )
=> ? [C4: set_Pr4386577575007340137rm_a_b] :
( ( A
= ( insert7009541432154983385rm_a_b @ B2 @ C4 ) )
& ~ ( member5869715511025134514rm_a_b @ B2 @ C4 )
& ( B
= ( insert7009541432154983385rm_a_b @ A2 @ C4 ) )
& ~ ( member5869715511025134514rm_a_b @ A2 @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_851_insert__eq__iff,axiom,
! [A2: product_prod_a_nat,A: set_Pr4934435412358123699_a_nat,B2: product_prod_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ~ ( member5724188588386418708_a_nat @ A2 @ A )
=> ( ~ ( member5724188588386418708_a_nat @ B2 @ B )
=> ( ( ( insert8054603423593749677_a_nat @ A2 @ A )
= ( insert8054603423593749677_a_nat @ B2 @ B ) )
= ( ( ( A2 = B2 )
=> ( A = B ) )
& ( ( A2 != B2 )
=> ? [C4: set_Pr4934435412358123699_a_nat] :
( ( A
= ( insert8054603423593749677_a_nat @ B2 @ C4 ) )
& ~ ( member5724188588386418708_a_nat @ B2 @ C4 )
& ( B
= ( insert8054603423593749677_a_nat @ A2 @ C4 ) )
& ~ ( member5724188588386418708_a_nat @ A2 @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_852_insert__eq__iff,axiom,
! [A2: produc8199716216217303280at_nat,A: set_Pr9093778441882193744at_nat,B2: produc8199716216217303280at_nat,B: set_Pr9093778441882193744at_nat] :
( ~ ( member7226740684066999833at_nat @ A2 @ A )
=> ( ~ ( member7226740684066999833at_nat @ B2 @ B )
=> ( ( ( insert891772969674204864at_nat @ A2 @ A )
= ( insert891772969674204864at_nat @ B2 @ B ) )
= ( ( ( A2 = B2 )
=> ( A = B ) )
& ( ( A2 != B2 )
=> ? [C4: set_Pr9093778441882193744at_nat] :
( ( A
= ( insert891772969674204864at_nat @ B2 @ C4 ) )
& ~ ( member7226740684066999833at_nat @ B2 @ C4 )
& ( B
= ( insert891772969674204864at_nat @ A2 @ C4 ) )
& ~ ( member7226740684066999833at_nat @ A2 @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_853_insert__absorb,axiom,
! [A2: produc357393685978478089rm_a_b,A: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ A2 @ A )
=> ( ( insert7009541432154983385rm_a_b @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_854_insert__absorb,axiom,
! [A2: product_prod_a_nat,A: set_Pr4934435412358123699_a_nat] :
( ( member5724188588386418708_a_nat @ A2 @ A )
=> ( ( insert8054603423593749677_a_nat @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_855_insert__absorb,axiom,
! [A2: produc8199716216217303280at_nat,A: set_Pr9093778441882193744at_nat] :
( ( member7226740684066999833at_nat @ A2 @ A )
=> ( ( insert891772969674204864at_nat @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_856_insert__ident,axiom,
! [X3: produc357393685978478089rm_a_b,A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ~ ( member5869715511025134514rm_a_b @ X3 @ A )
=> ( ~ ( member5869715511025134514rm_a_b @ X3 @ B )
=> ( ( ( insert7009541432154983385rm_a_b @ X3 @ A )
= ( insert7009541432154983385rm_a_b @ X3 @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_857_insert__ident,axiom,
! [X3: product_prod_a_nat,A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ~ ( member5724188588386418708_a_nat @ X3 @ A )
=> ( ~ ( member5724188588386418708_a_nat @ X3 @ B )
=> ( ( ( insert8054603423593749677_a_nat @ X3 @ A )
= ( insert8054603423593749677_a_nat @ X3 @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_858_insert__ident,axiom,
! [X3: produc8199716216217303280at_nat,A: set_Pr9093778441882193744at_nat,B: set_Pr9093778441882193744at_nat] :
( ~ ( member7226740684066999833at_nat @ X3 @ A )
=> ( ~ ( member7226740684066999833at_nat @ X3 @ B )
=> ( ( ( insert891772969674204864at_nat @ X3 @ A )
= ( insert891772969674204864at_nat @ X3 @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_859_Set_Oset__insert,axiom,
! [X3: produc357393685978478089rm_a_b,A: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ X3 @ A )
=> ~ ! [B9: set_Pr4386577575007340137rm_a_b] :
( ( A
= ( insert7009541432154983385rm_a_b @ X3 @ B9 ) )
=> ( member5869715511025134514rm_a_b @ X3 @ B9 ) ) ) ).
% Set.set_insert
thf(fact_860_Set_Oset__insert,axiom,
! [X3: product_prod_a_nat,A: set_Pr4934435412358123699_a_nat] :
( ( member5724188588386418708_a_nat @ X3 @ A )
=> ~ ! [B9: set_Pr4934435412358123699_a_nat] :
( ( A
= ( insert8054603423593749677_a_nat @ X3 @ B9 ) )
=> ( member5724188588386418708_a_nat @ X3 @ B9 ) ) ) ).
% Set.set_insert
thf(fact_861_Set_Oset__insert,axiom,
! [X3: produc8199716216217303280at_nat,A: set_Pr9093778441882193744at_nat] :
( ( member7226740684066999833at_nat @ X3 @ A )
=> ~ ! [B9: set_Pr9093778441882193744at_nat] :
( ( A
= ( insert891772969674204864at_nat @ X3 @ B9 ) )
=> ( member7226740684066999833at_nat @ X3 @ B9 ) ) ) ).
% Set.set_insert
thf(fact_862_insertI2,axiom,
! [A2: produc357393685978478089rm_a_b,B: set_Pr4386577575007340137rm_a_b,B2: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ A2 @ B )
=> ( member5869715511025134514rm_a_b @ A2 @ ( insert7009541432154983385rm_a_b @ B2 @ B ) ) ) ).
% insertI2
thf(fact_863_insertI2,axiom,
! [A2: product_prod_a_nat,B: set_Pr4934435412358123699_a_nat,B2: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ A2 @ B )
=> ( member5724188588386418708_a_nat @ A2 @ ( insert8054603423593749677_a_nat @ B2 @ B ) ) ) ).
% insertI2
thf(fact_864_insertI2,axiom,
! [A2: produc8199716216217303280at_nat,B: set_Pr9093778441882193744at_nat,B2: produc8199716216217303280at_nat] :
( ( member7226740684066999833at_nat @ A2 @ B )
=> ( member7226740684066999833at_nat @ A2 @ ( insert891772969674204864at_nat @ B2 @ B ) ) ) ).
% insertI2
thf(fact_865_insertI1,axiom,
! [A2: produc357393685978478089rm_a_b,B: set_Pr4386577575007340137rm_a_b] : ( member5869715511025134514rm_a_b @ A2 @ ( insert7009541432154983385rm_a_b @ A2 @ B ) ) ).
% insertI1
thf(fact_866_insertI1,axiom,
! [A2: product_prod_a_nat,B: set_Pr4934435412358123699_a_nat] : ( member5724188588386418708_a_nat @ A2 @ ( insert8054603423593749677_a_nat @ A2 @ B ) ) ).
% insertI1
thf(fact_867_insertI1,axiom,
! [A2: produc8199716216217303280at_nat,B: set_Pr9093778441882193744at_nat] : ( member7226740684066999833at_nat @ A2 @ ( insert891772969674204864at_nat @ A2 @ B ) ) ).
% insertI1
thf(fact_868_insertE,axiom,
! [A2: produc357393685978478089rm_a_b,B2: produc357393685978478089rm_a_b,A: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ A2 @ ( insert7009541432154983385rm_a_b @ B2 @ A ) )
=> ( ( A2 != B2 )
=> ( member5869715511025134514rm_a_b @ A2 @ A ) ) ) ).
% insertE
thf(fact_869_insertE,axiom,
! [A2: product_prod_a_nat,B2: product_prod_a_nat,A: set_Pr4934435412358123699_a_nat] :
( ( member5724188588386418708_a_nat @ A2 @ ( insert8054603423593749677_a_nat @ B2 @ A ) )
=> ( ( A2 != B2 )
=> ( member5724188588386418708_a_nat @ A2 @ A ) ) ) ).
% insertE
thf(fact_870_insertE,axiom,
! [A2: produc8199716216217303280at_nat,B2: produc8199716216217303280at_nat,A: set_Pr9093778441882193744at_nat] :
( ( member7226740684066999833at_nat @ A2 @ ( insert891772969674204864at_nat @ B2 @ A ) )
=> ( ( A2 != B2 )
=> ( member7226740684066999833at_nat @ A2 @ A ) ) ) ).
% insertE
thf(fact_871_IntD2,axiom,
! [C: produc357393685978478089rm_a_b,A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ C @ ( inf_in1693821330460682071rm_a_b @ A @ B ) )
=> ( member5869715511025134514rm_a_b @ C @ B ) ) ).
% IntD2
thf(fact_872_IntD2,axiom,
! [C: product_prod_a_nat,A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( member5724188588386418708_a_nat @ C @ ( inf_in3813033988066259269_a_nat @ A @ B ) )
=> ( member5724188588386418708_a_nat @ C @ B ) ) ).
% IntD2
thf(fact_873_IntD2,axiom,
! [C: produc8199716216217303280at_nat,A: set_Pr9093778441882193744at_nat,B: set_Pr9093778441882193744at_nat] :
( ( member7226740684066999833at_nat @ C @ ( inf_in907833683887804734at_nat @ A @ B ) )
=> ( member7226740684066999833at_nat @ C @ B ) ) ).
% IntD2
thf(fact_874_IntD1,axiom,
! [C: produc357393685978478089rm_a_b,A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ C @ ( inf_in1693821330460682071rm_a_b @ A @ B ) )
=> ( member5869715511025134514rm_a_b @ C @ A ) ) ).
% IntD1
thf(fact_875_IntD1,axiom,
! [C: product_prod_a_nat,A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( member5724188588386418708_a_nat @ C @ ( inf_in3813033988066259269_a_nat @ A @ B ) )
=> ( member5724188588386418708_a_nat @ C @ A ) ) ).
% IntD1
thf(fact_876_IntD1,axiom,
! [C: produc8199716216217303280at_nat,A: set_Pr9093778441882193744at_nat,B: set_Pr9093778441882193744at_nat] :
( ( member7226740684066999833at_nat @ C @ ( inf_in907833683887804734at_nat @ A @ B ) )
=> ( member7226740684066999833at_nat @ C @ A ) ) ).
% IntD1
thf(fact_877_IntE,axiom,
! [C: produc357393685978478089rm_a_b,A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ C @ ( inf_in1693821330460682071rm_a_b @ A @ B ) )
=> ~ ( ( member5869715511025134514rm_a_b @ C @ A )
=> ~ ( member5869715511025134514rm_a_b @ C @ B ) ) ) ).
% IntE
thf(fact_878_IntE,axiom,
! [C: product_prod_a_nat,A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( member5724188588386418708_a_nat @ C @ ( inf_in3813033988066259269_a_nat @ A @ B ) )
=> ~ ( ( member5724188588386418708_a_nat @ C @ A )
=> ~ ( member5724188588386418708_a_nat @ C @ B ) ) ) ).
% IntE
thf(fact_879_IntE,axiom,
! [C: produc8199716216217303280at_nat,A: set_Pr9093778441882193744at_nat,B: set_Pr9093778441882193744at_nat] :
( ( member7226740684066999833at_nat @ C @ ( inf_in907833683887804734at_nat @ A @ B ) )
=> ~ ( ( member7226740684066999833at_nat @ C @ A )
=> ~ ( member7226740684066999833at_nat @ C @ B ) ) ) ).
% IntE
thf(fact_880_inf_OcoboundedI2,axiom,
! [B2: set_Pr4934435412358123699_a_nat,C: set_Pr4934435412358123699_a_nat,A2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ B2 @ C )
=> ( ord_le8666007276011122963_a_nat @ ( inf_in3813033988066259269_a_nat @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI2
thf(fact_881_inf_OcoboundedI2,axiom,
! [B2: set_Pr4386577575007340137rm_a_b,C: set_Pr4386577575007340137rm_a_b,A2: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ B2 @ C )
=> ( ord_le118470702582115849rm_a_b @ ( inf_in1693821330460682071rm_a_b @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI2
thf(fact_882_inf_OcoboundedI2,axiom,
! [B2: set_Pr9093778441882193744at_nat,C: set_Pr9093778441882193744at_nat,A2: set_Pr9093778441882193744at_nat] :
( ( ord_le3678578370064672496at_nat @ B2 @ C )
=> ( ord_le3678578370064672496at_nat @ ( inf_in907833683887804734at_nat @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI2
thf(fact_883_inf_OcoboundedI1,axiom,
! [A2: set_Pr4934435412358123699_a_nat,C: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A2 @ C )
=> ( ord_le8666007276011122963_a_nat @ ( inf_in3813033988066259269_a_nat @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI1
thf(fact_884_inf_OcoboundedI1,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,C: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A2 @ C )
=> ( ord_le118470702582115849rm_a_b @ ( inf_in1693821330460682071rm_a_b @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI1
thf(fact_885_inf_OcoboundedI1,axiom,
! [A2: set_Pr9093778441882193744at_nat,C: set_Pr9093778441882193744at_nat,B2: set_Pr9093778441882193744at_nat] :
( ( ord_le3678578370064672496at_nat @ A2 @ C )
=> ( ord_le3678578370064672496at_nat @ ( inf_in907833683887804734at_nat @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI1
thf(fact_886_inf_Oabsorb__iff2,axiom,
( ord_le8666007276011122963_a_nat
= ( ^ [B5: set_Pr4934435412358123699_a_nat,A5: set_Pr4934435412358123699_a_nat] :
( ( inf_in3813033988066259269_a_nat @ A5 @ B5 )
= B5 ) ) ) ).
% inf.absorb_iff2
thf(fact_887_inf_Oabsorb__iff2,axiom,
( ord_le118470702582115849rm_a_b
= ( ^ [B5: set_Pr4386577575007340137rm_a_b,A5: set_Pr4386577575007340137rm_a_b] :
( ( inf_in1693821330460682071rm_a_b @ A5 @ B5 )
= B5 ) ) ) ).
% inf.absorb_iff2
thf(fact_888_inf_Oabsorb__iff2,axiom,
( ord_le3678578370064672496at_nat
= ( ^ [B5: set_Pr9093778441882193744at_nat,A5: set_Pr9093778441882193744at_nat] :
( ( inf_in907833683887804734at_nat @ A5 @ B5 )
= B5 ) ) ) ).
% inf.absorb_iff2
thf(fact_889_inf_Oabsorb__iff1,axiom,
( ord_le8666007276011122963_a_nat
= ( ^ [A5: set_Pr4934435412358123699_a_nat,B5: set_Pr4934435412358123699_a_nat] :
( ( inf_in3813033988066259269_a_nat @ A5 @ B5 )
= A5 ) ) ) ).
% inf.absorb_iff1
thf(fact_890_inf_Oabsorb__iff1,axiom,
( ord_le118470702582115849rm_a_b
= ( ^ [A5: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
( ( inf_in1693821330460682071rm_a_b @ A5 @ B5 )
= A5 ) ) ) ).
% inf.absorb_iff1
thf(fact_891_inf_Oabsorb__iff1,axiom,
( ord_le3678578370064672496at_nat
= ( ^ [A5: set_Pr9093778441882193744at_nat,B5: set_Pr9093778441882193744at_nat] :
( ( inf_in907833683887804734at_nat @ A5 @ B5 )
= A5 ) ) ) ).
% inf.absorb_iff1
thf(fact_892_inf_Ocobounded2,axiom,
! [A2: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ ( inf_in3813033988066259269_a_nat @ A2 @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_893_inf_Ocobounded2,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ ( inf_in1693821330460682071rm_a_b @ A2 @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_894_inf_Ocobounded2,axiom,
! [A2: set_Pr9093778441882193744at_nat,B2: set_Pr9093778441882193744at_nat] : ( ord_le3678578370064672496at_nat @ ( inf_in907833683887804734at_nat @ A2 @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_895_inf_Ocobounded1,axiom,
! [A2: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ ( inf_in3813033988066259269_a_nat @ A2 @ B2 ) @ A2 ) ).
% inf.cobounded1
thf(fact_896_inf_Ocobounded1,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ ( inf_in1693821330460682071rm_a_b @ A2 @ B2 ) @ A2 ) ).
% inf.cobounded1
thf(fact_897_inf_Ocobounded1,axiom,
! [A2: set_Pr9093778441882193744at_nat,B2: set_Pr9093778441882193744at_nat] : ( ord_le3678578370064672496at_nat @ ( inf_in907833683887804734at_nat @ A2 @ B2 ) @ A2 ) ).
% inf.cobounded1
thf(fact_898_inf_Oorder__iff,axiom,
( ord_le8666007276011122963_a_nat
= ( ^ [A5: set_Pr4934435412358123699_a_nat,B5: set_Pr4934435412358123699_a_nat] :
( A5
= ( inf_in3813033988066259269_a_nat @ A5 @ B5 ) ) ) ) ).
% inf.order_iff
thf(fact_899_inf_Oorder__iff,axiom,
( ord_le118470702582115849rm_a_b
= ( ^ [A5: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
( A5
= ( inf_in1693821330460682071rm_a_b @ A5 @ B5 ) ) ) ) ).
% inf.order_iff
thf(fact_900_inf_Oorder__iff,axiom,
( ord_le3678578370064672496at_nat
= ( ^ [A5: set_Pr9093778441882193744at_nat,B5: set_Pr9093778441882193744at_nat] :
( A5
= ( inf_in907833683887804734at_nat @ A5 @ B5 ) ) ) ) ).
% inf.order_iff
thf(fact_901_inf__greatest,axiom,
! [X3: set_Pr4934435412358123699_a_nat,Y3: set_Pr4934435412358123699_a_nat,Z2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X3 @ Y3 )
=> ( ( ord_le8666007276011122963_a_nat @ X3 @ Z2 )
=> ( ord_le8666007276011122963_a_nat @ X3 @ ( inf_in3813033988066259269_a_nat @ Y3 @ Z2 ) ) ) ) ).
% inf_greatest
thf(fact_902_inf__greatest,axiom,
! [X3: set_Pr4386577575007340137rm_a_b,Y3: set_Pr4386577575007340137rm_a_b,Z2: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ X3 @ Y3 )
=> ( ( ord_le118470702582115849rm_a_b @ X3 @ Z2 )
=> ( ord_le118470702582115849rm_a_b @ X3 @ ( inf_in1693821330460682071rm_a_b @ Y3 @ Z2 ) ) ) ) ).
% inf_greatest
thf(fact_903_inf__greatest,axiom,
! [X3: set_Pr9093778441882193744at_nat,Y3: set_Pr9093778441882193744at_nat,Z2: set_Pr9093778441882193744at_nat] :
( ( ord_le3678578370064672496at_nat @ X3 @ Y3 )
=> ( ( ord_le3678578370064672496at_nat @ X3 @ Z2 )
=> ( ord_le3678578370064672496at_nat @ X3 @ ( inf_in907833683887804734at_nat @ Y3 @ Z2 ) ) ) ) ).
% inf_greatest
thf(fact_904_inf_OboundedI,axiom,
! [A2: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat,C: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A2 @ B2 )
=> ( ( ord_le8666007276011122963_a_nat @ A2 @ C )
=> ( ord_le8666007276011122963_a_nat @ A2 @ ( inf_in3813033988066259269_a_nat @ B2 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_905_inf_OboundedI,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b,C: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A2 @ B2 )
=> ( ( ord_le118470702582115849rm_a_b @ A2 @ C )
=> ( ord_le118470702582115849rm_a_b @ A2 @ ( inf_in1693821330460682071rm_a_b @ B2 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_906_inf_OboundedI,axiom,
! [A2: set_Pr9093778441882193744at_nat,B2: set_Pr9093778441882193744at_nat,C: set_Pr9093778441882193744at_nat] :
( ( ord_le3678578370064672496at_nat @ A2 @ B2 )
=> ( ( ord_le3678578370064672496at_nat @ A2 @ C )
=> ( ord_le3678578370064672496at_nat @ A2 @ ( inf_in907833683887804734at_nat @ B2 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_907_inf_OboundedE,axiom,
! [A2: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat,C: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A2 @ ( inf_in3813033988066259269_a_nat @ B2 @ C ) )
=> ~ ( ( ord_le8666007276011122963_a_nat @ A2 @ B2 )
=> ~ ( ord_le8666007276011122963_a_nat @ A2 @ C ) ) ) ).
% inf.boundedE
thf(fact_908_inf_OboundedE,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b,C: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A2 @ ( inf_in1693821330460682071rm_a_b @ B2 @ C ) )
=> ~ ( ( ord_le118470702582115849rm_a_b @ A2 @ B2 )
=> ~ ( ord_le118470702582115849rm_a_b @ A2 @ C ) ) ) ).
% inf.boundedE
thf(fact_909_inf_OboundedE,axiom,
! [A2: set_Pr9093778441882193744at_nat,B2: set_Pr9093778441882193744at_nat,C: set_Pr9093778441882193744at_nat] :
( ( ord_le3678578370064672496at_nat @ A2 @ ( inf_in907833683887804734at_nat @ B2 @ C ) )
=> ~ ( ( ord_le3678578370064672496at_nat @ A2 @ B2 )
=> ~ ( ord_le3678578370064672496at_nat @ A2 @ C ) ) ) ).
% inf.boundedE
thf(fact_910_inf__absorb2,axiom,
! [Y3: set_Pr4934435412358123699_a_nat,X3: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ Y3 @ X3 )
=> ( ( inf_in3813033988066259269_a_nat @ X3 @ Y3 )
= Y3 ) ) ).
% inf_absorb2
thf(fact_911_inf__absorb2,axiom,
! [Y3: set_Pr4386577575007340137rm_a_b,X3: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ Y3 @ X3 )
=> ( ( inf_in1693821330460682071rm_a_b @ X3 @ Y3 )
= Y3 ) ) ).
% inf_absorb2
thf(fact_912_inf__absorb2,axiom,
! [Y3: set_Pr9093778441882193744at_nat,X3: set_Pr9093778441882193744at_nat] :
( ( ord_le3678578370064672496at_nat @ Y3 @ X3 )
=> ( ( inf_in907833683887804734at_nat @ X3 @ Y3 )
= Y3 ) ) ).
% inf_absorb2
thf(fact_913_inf__absorb1,axiom,
! [X3: set_Pr4934435412358123699_a_nat,Y3: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X3 @ Y3 )
=> ( ( inf_in3813033988066259269_a_nat @ X3 @ Y3 )
= X3 ) ) ).
% inf_absorb1
thf(fact_914_inf__absorb1,axiom,
! [X3: set_Pr4386577575007340137rm_a_b,Y3: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ X3 @ Y3 )
=> ( ( inf_in1693821330460682071rm_a_b @ X3 @ Y3 )
= X3 ) ) ).
% inf_absorb1
thf(fact_915_inf__absorb1,axiom,
! [X3: set_Pr9093778441882193744at_nat,Y3: set_Pr9093778441882193744at_nat] :
( ( ord_le3678578370064672496at_nat @ X3 @ Y3 )
=> ( ( inf_in907833683887804734at_nat @ X3 @ Y3 )
= X3 ) ) ).
% inf_absorb1
thf(fact_916_inf_Oabsorb2,axiom,
! [B2: set_Pr4934435412358123699_a_nat,A2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ B2 @ A2 )
=> ( ( inf_in3813033988066259269_a_nat @ A2 @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_917_inf_Oabsorb2,axiom,
! [B2: set_Pr4386577575007340137rm_a_b,A2: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ B2 @ A2 )
=> ( ( inf_in1693821330460682071rm_a_b @ A2 @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_918_inf_Oabsorb2,axiom,
! [B2: set_Pr9093778441882193744at_nat,A2: set_Pr9093778441882193744at_nat] :
( ( ord_le3678578370064672496at_nat @ B2 @ A2 )
=> ( ( inf_in907833683887804734at_nat @ A2 @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_919_inf_Oabsorb1,axiom,
! [A2: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A2 @ B2 )
=> ( ( inf_in3813033988066259269_a_nat @ A2 @ B2 )
= A2 ) ) ).
% inf.absorb1
thf(fact_920_inf_Oabsorb1,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A2 @ B2 )
=> ( ( inf_in1693821330460682071rm_a_b @ A2 @ B2 )
= A2 ) ) ).
% inf.absorb1
thf(fact_921_inf_Oabsorb1,axiom,
! [A2: set_Pr9093778441882193744at_nat,B2: set_Pr9093778441882193744at_nat] :
( ( ord_le3678578370064672496at_nat @ A2 @ B2 )
=> ( ( inf_in907833683887804734at_nat @ A2 @ B2 )
= A2 ) ) ).
% inf.absorb1
thf(fact_922_le__iff__inf,axiom,
( ord_le8666007276011122963_a_nat
= ( ^ [X4: set_Pr4934435412358123699_a_nat,Y5: set_Pr4934435412358123699_a_nat] :
( ( inf_in3813033988066259269_a_nat @ X4 @ Y5 )
= X4 ) ) ) ).
% le_iff_inf
thf(fact_923_le__iff__inf,axiom,
( ord_le118470702582115849rm_a_b
= ( ^ [X4: set_Pr4386577575007340137rm_a_b,Y5: set_Pr4386577575007340137rm_a_b] :
( ( inf_in1693821330460682071rm_a_b @ X4 @ Y5 )
= X4 ) ) ) ).
% le_iff_inf
thf(fact_924_le__iff__inf,axiom,
( ord_le3678578370064672496at_nat
= ( ^ [X4: set_Pr9093778441882193744at_nat,Y5: set_Pr9093778441882193744at_nat] :
( ( inf_in907833683887804734at_nat @ X4 @ Y5 )
= X4 ) ) ) ).
% le_iff_inf
thf(fact_925_inf__unique,axiom,
! [F2: set_Pr4934435412358123699_a_nat > set_Pr4934435412358123699_a_nat > set_Pr4934435412358123699_a_nat,X3: set_Pr4934435412358123699_a_nat,Y3: set_Pr4934435412358123699_a_nat] :
( ! [X: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ ( F2 @ X @ Y ) @ X )
=> ( ! [X: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ ( F2 @ X @ Y ) @ Y )
=> ( ! [X: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat,Z3: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X @ Y )
=> ( ( ord_le8666007276011122963_a_nat @ X @ Z3 )
=> ( ord_le8666007276011122963_a_nat @ X @ ( F2 @ Y @ Z3 ) ) ) )
=> ( ( inf_in3813033988066259269_a_nat @ X3 @ Y3 )
= ( F2 @ X3 @ Y3 ) ) ) ) ) ).
% inf_unique
thf(fact_926_inf__unique,axiom,
! [F2: set_Pr4386577575007340137rm_a_b > set_Pr4386577575007340137rm_a_b > set_Pr4386577575007340137rm_a_b,X3: set_Pr4386577575007340137rm_a_b,Y3: set_Pr4386577575007340137rm_a_b] :
( ! [X: set_Pr4386577575007340137rm_a_b,Y: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ ( F2 @ X @ Y ) @ X )
=> ( ! [X: set_Pr4386577575007340137rm_a_b,Y: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ ( F2 @ X @ Y ) @ Y )
=> ( ! [X: set_Pr4386577575007340137rm_a_b,Y: set_Pr4386577575007340137rm_a_b,Z3: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ X @ Y )
=> ( ( ord_le118470702582115849rm_a_b @ X @ Z3 )
=> ( ord_le118470702582115849rm_a_b @ X @ ( F2 @ Y @ Z3 ) ) ) )
=> ( ( inf_in1693821330460682071rm_a_b @ X3 @ Y3 )
= ( F2 @ X3 @ Y3 ) ) ) ) ) ).
% inf_unique
thf(fact_927_inf__unique,axiom,
! [F2: set_Pr9093778441882193744at_nat > set_Pr9093778441882193744at_nat > set_Pr9093778441882193744at_nat,X3: set_Pr9093778441882193744at_nat,Y3: set_Pr9093778441882193744at_nat] :
( ! [X: set_Pr9093778441882193744at_nat,Y: set_Pr9093778441882193744at_nat] : ( ord_le3678578370064672496at_nat @ ( F2 @ X @ Y ) @ X )
=> ( ! [X: set_Pr9093778441882193744at_nat,Y: set_Pr9093778441882193744at_nat] : ( ord_le3678578370064672496at_nat @ ( F2 @ X @ Y ) @ Y )
=> ( ! [X: set_Pr9093778441882193744at_nat,Y: set_Pr9093778441882193744at_nat,Z3: set_Pr9093778441882193744at_nat] :
( ( ord_le3678578370064672496at_nat @ X @ Y )
=> ( ( ord_le3678578370064672496at_nat @ X @ Z3 )
=> ( ord_le3678578370064672496at_nat @ X @ ( F2 @ Y @ Z3 ) ) ) )
=> ( ( inf_in907833683887804734at_nat @ X3 @ Y3 )
= ( F2 @ X3 @ Y3 ) ) ) ) ) ).
% inf_unique
thf(fact_928_inf_OorderI,axiom,
! [A2: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat] :
( ( A2
= ( inf_in3813033988066259269_a_nat @ A2 @ B2 ) )
=> ( ord_le8666007276011122963_a_nat @ A2 @ B2 ) ) ).
% inf.orderI
thf(fact_929_inf_OorderI,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b] :
( ( A2
= ( inf_in1693821330460682071rm_a_b @ A2 @ B2 ) )
=> ( ord_le118470702582115849rm_a_b @ A2 @ B2 ) ) ).
% inf.orderI
thf(fact_930_inf_OorderI,axiom,
! [A2: set_Pr9093778441882193744at_nat,B2: set_Pr9093778441882193744at_nat] :
( ( A2
= ( inf_in907833683887804734at_nat @ A2 @ B2 ) )
=> ( ord_le3678578370064672496at_nat @ A2 @ B2 ) ) ).
% inf.orderI
thf(fact_931_inf_OorderE,axiom,
! [A2: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A2 @ B2 )
=> ( A2
= ( inf_in3813033988066259269_a_nat @ A2 @ B2 ) ) ) ).
% inf.orderE
thf(fact_932_inf_OorderE,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A2 @ B2 )
=> ( A2
= ( inf_in1693821330460682071rm_a_b @ A2 @ B2 ) ) ) ).
% inf.orderE
thf(fact_933_inf_OorderE,axiom,
! [A2: set_Pr9093778441882193744at_nat,B2: set_Pr9093778441882193744at_nat] :
( ( ord_le3678578370064672496at_nat @ A2 @ B2 )
=> ( A2
= ( inf_in907833683887804734at_nat @ A2 @ B2 ) ) ) ).
% inf.orderE
thf(fact_934_le__infI2,axiom,
! [B2: set_Pr4934435412358123699_a_nat,X3: set_Pr4934435412358123699_a_nat,A2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ B2 @ X3 )
=> ( ord_le8666007276011122963_a_nat @ ( inf_in3813033988066259269_a_nat @ A2 @ B2 ) @ X3 ) ) ).
% le_infI2
thf(fact_935_le__infI2,axiom,
! [B2: set_Pr4386577575007340137rm_a_b,X3: set_Pr4386577575007340137rm_a_b,A2: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ B2 @ X3 )
=> ( ord_le118470702582115849rm_a_b @ ( inf_in1693821330460682071rm_a_b @ A2 @ B2 ) @ X3 ) ) ).
% le_infI2
thf(fact_936_le__infI2,axiom,
! [B2: set_Pr9093778441882193744at_nat,X3: set_Pr9093778441882193744at_nat,A2: set_Pr9093778441882193744at_nat] :
( ( ord_le3678578370064672496at_nat @ B2 @ X3 )
=> ( ord_le3678578370064672496at_nat @ ( inf_in907833683887804734at_nat @ A2 @ B2 ) @ X3 ) ) ).
% le_infI2
thf(fact_937_le__infI1,axiom,
! [A2: set_Pr4934435412358123699_a_nat,X3: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A2 @ X3 )
=> ( ord_le8666007276011122963_a_nat @ ( inf_in3813033988066259269_a_nat @ A2 @ B2 ) @ X3 ) ) ).
% le_infI1
thf(fact_938_le__infI1,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,X3: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A2 @ X3 )
=> ( ord_le118470702582115849rm_a_b @ ( inf_in1693821330460682071rm_a_b @ A2 @ B2 ) @ X3 ) ) ).
% le_infI1
thf(fact_939_le__infI1,axiom,
! [A2: set_Pr9093778441882193744at_nat,X3: set_Pr9093778441882193744at_nat,B2: set_Pr9093778441882193744at_nat] :
( ( ord_le3678578370064672496at_nat @ A2 @ X3 )
=> ( ord_le3678578370064672496at_nat @ ( inf_in907833683887804734at_nat @ A2 @ B2 ) @ X3 ) ) ).
% le_infI1
thf(fact_940_inf__mono,axiom,
! [A2: set_Pr4934435412358123699_a_nat,C: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat,D2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A2 @ C )
=> ( ( ord_le8666007276011122963_a_nat @ B2 @ D2 )
=> ( ord_le8666007276011122963_a_nat @ ( inf_in3813033988066259269_a_nat @ A2 @ B2 ) @ ( inf_in3813033988066259269_a_nat @ C @ D2 ) ) ) ) ).
% inf_mono
thf(fact_941_inf__mono,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,C: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b,D2: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A2 @ C )
=> ( ( ord_le118470702582115849rm_a_b @ B2 @ D2 )
=> ( ord_le118470702582115849rm_a_b @ ( inf_in1693821330460682071rm_a_b @ A2 @ B2 ) @ ( inf_in1693821330460682071rm_a_b @ C @ D2 ) ) ) ) ).
% inf_mono
thf(fact_942_inf__mono,axiom,
! [A2: set_Pr9093778441882193744at_nat,C: set_Pr9093778441882193744at_nat,B2: set_Pr9093778441882193744at_nat,D2: set_Pr9093778441882193744at_nat] :
( ( ord_le3678578370064672496at_nat @ A2 @ C )
=> ( ( ord_le3678578370064672496at_nat @ B2 @ D2 )
=> ( ord_le3678578370064672496at_nat @ ( inf_in907833683887804734at_nat @ A2 @ B2 ) @ ( inf_in907833683887804734at_nat @ C @ D2 ) ) ) ) ).
% inf_mono
thf(fact_943_le__infI,axiom,
! [X3: set_Pr4934435412358123699_a_nat,A2: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X3 @ A2 )
=> ( ( ord_le8666007276011122963_a_nat @ X3 @ B2 )
=> ( ord_le8666007276011122963_a_nat @ X3 @ ( inf_in3813033988066259269_a_nat @ A2 @ B2 ) ) ) ) ).
% le_infI
thf(fact_944_le__infI,axiom,
! [X3: set_Pr4386577575007340137rm_a_b,A2: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ X3 @ A2 )
=> ( ( ord_le118470702582115849rm_a_b @ X3 @ B2 )
=> ( ord_le118470702582115849rm_a_b @ X3 @ ( inf_in1693821330460682071rm_a_b @ A2 @ B2 ) ) ) ) ).
% le_infI
thf(fact_945_le__infI,axiom,
! [X3: set_Pr9093778441882193744at_nat,A2: set_Pr9093778441882193744at_nat,B2: set_Pr9093778441882193744at_nat] :
( ( ord_le3678578370064672496at_nat @ X3 @ A2 )
=> ( ( ord_le3678578370064672496at_nat @ X3 @ B2 )
=> ( ord_le3678578370064672496at_nat @ X3 @ ( inf_in907833683887804734at_nat @ A2 @ B2 ) ) ) ) ).
% le_infI
thf(fact_946_le__infE,axiom,
! [X3: set_Pr4934435412358123699_a_nat,A2: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X3 @ ( inf_in3813033988066259269_a_nat @ A2 @ B2 ) )
=> ~ ( ( ord_le8666007276011122963_a_nat @ X3 @ A2 )
=> ~ ( ord_le8666007276011122963_a_nat @ X3 @ B2 ) ) ) ).
% le_infE
thf(fact_947_le__infE,axiom,
! [X3: set_Pr4386577575007340137rm_a_b,A2: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ X3 @ ( inf_in1693821330460682071rm_a_b @ A2 @ B2 ) )
=> ~ ( ( ord_le118470702582115849rm_a_b @ X3 @ A2 )
=> ~ ( ord_le118470702582115849rm_a_b @ X3 @ B2 ) ) ) ).
% le_infE
thf(fact_948_le__infE,axiom,
! [X3: set_Pr9093778441882193744at_nat,A2: set_Pr9093778441882193744at_nat,B2: set_Pr9093778441882193744at_nat] :
( ( ord_le3678578370064672496at_nat @ X3 @ ( inf_in907833683887804734at_nat @ A2 @ B2 ) )
=> ~ ( ( ord_le3678578370064672496at_nat @ X3 @ A2 )
=> ~ ( ord_le3678578370064672496at_nat @ X3 @ B2 ) ) ) ).
% le_infE
thf(fact_949_inf__le2,axiom,
! [X3: set_Pr4934435412358123699_a_nat,Y3: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ ( inf_in3813033988066259269_a_nat @ X3 @ Y3 ) @ Y3 ) ).
% inf_le2
thf(fact_950_inf__le2,axiom,
! [X3: set_Pr4386577575007340137rm_a_b,Y3: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ ( inf_in1693821330460682071rm_a_b @ X3 @ Y3 ) @ Y3 ) ).
% inf_le2
thf(fact_951_inf__le2,axiom,
! [X3: set_Pr9093778441882193744at_nat,Y3: set_Pr9093778441882193744at_nat] : ( ord_le3678578370064672496at_nat @ ( inf_in907833683887804734at_nat @ X3 @ Y3 ) @ Y3 ) ).
% inf_le2
thf(fact_952_inf__le1,axiom,
! [X3: set_Pr4934435412358123699_a_nat,Y3: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ ( inf_in3813033988066259269_a_nat @ X3 @ Y3 ) @ X3 ) ).
% inf_le1
thf(fact_953_inf__le1,axiom,
! [X3: set_Pr4386577575007340137rm_a_b,Y3: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ ( inf_in1693821330460682071rm_a_b @ X3 @ Y3 ) @ X3 ) ).
% inf_le1
thf(fact_954_inf__le1,axiom,
! [X3: set_Pr9093778441882193744at_nat,Y3: set_Pr9093778441882193744at_nat] : ( ord_le3678578370064672496at_nat @ ( inf_in907833683887804734at_nat @ X3 @ Y3 ) @ X3 ) ).
% inf_le1
thf(fact_955_inf__sup__ord_I1_J,axiom,
! [X3: set_Pr4934435412358123699_a_nat,Y3: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ ( inf_in3813033988066259269_a_nat @ X3 @ Y3 ) @ X3 ) ).
% inf_sup_ord(1)
thf(fact_956_inf__sup__ord_I1_J,axiom,
! [X3: set_Pr4386577575007340137rm_a_b,Y3: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ ( inf_in1693821330460682071rm_a_b @ X3 @ Y3 ) @ X3 ) ).
% inf_sup_ord(1)
thf(fact_957_inf__sup__ord_I1_J,axiom,
! [X3: set_Pr9093778441882193744at_nat,Y3: set_Pr9093778441882193744at_nat] : ( ord_le3678578370064672496at_nat @ ( inf_in907833683887804734at_nat @ X3 @ Y3 ) @ X3 ) ).
% inf_sup_ord(1)
thf(fact_958_inf__sup__ord_I2_J,axiom,
! [X3: set_Pr4934435412358123699_a_nat,Y3: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ ( inf_in3813033988066259269_a_nat @ X3 @ Y3 ) @ Y3 ) ).
% inf_sup_ord(2)
thf(fact_959_inf__sup__ord_I2_J,axiom,
! [X3: set_Pr4386577575007340137rm_a_b,Y3: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ ( inf_in1693821330460682071rm_a_b @ X3 @ Y3 ) @ Y3 ) ).
% inf_sup_ord(2)
thf(fact_960_inf__sup__ord_I2_J,axiom,
! [X3: set_Pr9093778441882193744at_nat,Y3: set_Pr9093778441882193744at_nat] : ( ord_le3678578370064672496at_nat @ ( inf_in907833683887804734at_nat @ X3 @ Y3 ) @ Y3 ) ).
% inf_sup_ord(2)
thf(fact_961_insert__subsetI,axiom,
! [X3: product_prod_a_nat,A: set_Pr4934435412358123699_a_nat,X6: set_Pr4934435412358123699_a_nat] :
( ( member5724188588386418708_a_nat @ X3 @ A )
=> ( ( ord_le8666007276011122963_a_nat @ X6 @ A )
=> ( ord_le8666007276011122963_a_nat @ ( insert8054603423593749677_a_nat @ X3 @ X6 ) @ A ) ) ) ).
% insert_subsetI
thf(fact_962_insert__subsetI,axiom,
! [X3: produc357393685978478089rm_a_b,A: set_Pr4386577575007340137rm_a_b,X6: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ X3 @ A )
=> ( ( ord_le118470702582115849rm_a_b @ X6 @ A )
=> ( ord_le118470702582115849rm_a_b @ ( insert7009541432154983385rm_a_b @ X3 @ X6 ) @ A ) ) ) ).
% insert_subsetI
thf(fact_963_insert__subsetI,axiom,
! [X3: produc8199716216217303280at_nat,A: set_Pr9093778441882193744at_nat,X6: set_Pr9093778441882193744at_nat] :
( ( member7226740684066999833at_nat @ X3 @ A )
=> ( ( ord_le3678578370064672496at_nat @ X6 @ A )
=> ( ord_le3678578370064672496at_nat @ ( insert891772969674204864at_nat @ X3 @ X6 ) @ A ) ) ) ).
% insert_subsetI
thf(fact_964_insert__mono,axiom,
! [C2: set_Pr4934435412358123699_a_nat,D: set_Pr4934435412358123699_a_nat,A2: product_prod_a_nat] :
( ( ord_le8666007276011122963_a_nat @ C2 @ D )
=> ( ord_le8666007276011122963_a_nat @ ( insert8054603423593749677_a_nat @ A2 @ C2 ) @ ( insert8054603423593749677_a_nat @ A2 @ D ) ) ) ).
% insert_mono
thf(fact_965_insert__mono,axiom,
! [C2: set_Pr4386577575007340137rm_a_b,D: set_Pr4386577575007340137rm_a_b,A2: produc357393685978478089rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ C2 @ D )
=> ( ord_le118470702582115849rm_a_b @ ( insert7009541432154983385rm_a_b @ A2 @ C2 ) @ ( insert7009541432154983385rm_a_b @ A2 @ D ) ) ) ).
% insert_mono
thf(fact_966_insert__mono,axiom,
! [C2: set_Pr9093778441882193744at_nat,D: set_Pr9093778441882193744at_nat,A2: produc8199716216217303280at_nat] :
( ( ord_le3678578370064672496at_nat @ C2 @ D )
=> ( ord_le3678578370064672496at_nat @ ( insert891772969674204864at_nat @ A2 @ C2 ) @ ( insert891772969674204864at_nat @ A2 @ D ) ) ) ).
% insert_mono
thf(fact_967_subset__insert,axiom,
! [X3: product_prod_a_nat,A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ~ ( member5724188588386418708_a_nat @ X3 @ A )
=> ( ( ord_le8666007276011122963_a_nat @ A @ ( insert8054603423593749677_a_nat @ X3 @ B ) )
= ( ord_le8666007276011122963_a_nat @ A @ B ) ) ) ).
% subset_insert
thf(fact_968_subset__insert,axiom,
! [X3: produc357393685978478089rm_a_b,A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ~ ( member5869715511025134514rm_a_b @ X3 @ A )
=> ( ( ord_le118470702582115849rm_a_b @ A @ ( insert7009541432154983385rm_a_b @ X3 @ B ) )
= ( ord_le118470702582115849rm_a_b @ A @ B ) ) ) ).
% subset_insert
thf(fact_969_subset__insert,axiom,
! [X3: produc8199716216217303280at_nat,A: set_Pr9093778441882193744at_nat,B: set_Pr9093778441882193744at_nat] :
( ~ ( member7226740684066999833at_nat @ X3 @ A )
=> ( ( ord_le3678578370064672496at_nat @ A @ ( insert891772969674204864at_nat @ X3 @ B ) )
= ( ord_le3678578370064672496at_nat @ A @ B ) ) ) ).
% subset_insert
thf(fact_970_subset__insertI,axiom,
! [B: set_Pr4934435412358123699_a_nat,A2: product_prod_a_nat] : ( ord_le8666007276011122963_a_nat @ B @ ( insert8054603423593749677_a_nat @ A2 @ B ) ) ).
% subset_insertI
thf(fact_971_subset__insertI,axiom,
! [B: set_Pr4386577575007340137rm_a_b,A2: produc357393685978478089rm_a_b] : ( ord_le118470702582115849rm_a_b @ B @ ( insert7009541432154983385rm_a_b @ A2 @ B ) ) ).
% subset_insertI
thf(fact_972_subset__insertI,axiom,
! [B: set_Pr9093778441882193744at_nat,A2: produc8199716216217303280at_nat] : ( ord_le3678578370064672496at_nat @ B @ ( insert891772969674204864at_nat @ A2 @ B ) ) ).
% subset_insertI
thf(fact_973_subset__insertI2,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat,B2: product_prod_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A @ B )
=> ( ord_le8666007276011122963_a_nat @ A @ ( insert8054603423593749677_a_nat @ B2 @ B ) ) ) ).
% subset_insertI2
thf(fact_974_subset__insertI2,axiom,
! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b,B2: produc357393685978478089rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A @ B )
=> ( ord_le118470702582115849rm_a_b @ A @ ( insert7009541432154983385rm_a_b @ B2 @ B ) ) ) ).
% subset_insertI2
thf(fact_975_subset__insertI2,axiom,
! [A: set_Pr9093778441882193744at_nat,B: set_Pr9093778441882193744at_nat,B2: produc8199716216217303280at_nat] :
( ( ord_le3678578370064672496at_nat @ A @ B )
=> ( ord_le3678578370064672496at_nat @ A @ ( insert891772969674204864at_nat @ B2 @ B ) ) ) ).
% subset_insertI2
thf(fact_976_Int__mono,axiom,
! [A: set_Pr4934435412358123699_a_nat,C2: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat,D: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A @ C2 )
=> ( ( ord_le8666007276011122963_a_nat @ B @ D )
=> ( ord_le8666007276011122963_a_nat @ ( inf_in3813033988066259269_a_nat @ A @ B ) @ ( inf_in3813033988066259269_a_nat @ C2 @ D ) ) ) ) ).
% Int_mono
thf(fact_977_Int__mono,axiom,
! [A: set_Pr4386577575007340137rm_a_b,C2: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b,D: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A @ C2 )
=> ( ( ord_le118470702582115849rm_a_b @ B @ D )
=> ( ord_le118470702582115849rm_a_b @ ( inf_in1693821330460682071rm_a_b @ A @ B ) @ ( inf_in1693821330460682071rm_a_b @ C2 @ D ) ) ) ) ).
% Int_mono
thf(fact_978_Int__mono,axiom,
! [A: set_Pr9093778441882193744at_nat,C2: set_Pr9093778441882193744at_nat,B: set_Pr9093778441882193744at_nat,D: set_Pr9093778441882193744at_nat] :
( ( ord_le3678578370064672496at_nat @ A @ C2 )
=> ( ( ord_le3678578370064672496at_nat @ B @ D )
=> ( ord_le3678578370064672496at_nat @ ( inf_in907833683887804734at_nat @ A @ B ) @ ( inf_in907833683887804734at_nat @ C2 @ D ) ) ) ) ).
% Int_mono
thf(fact_979_Int__lower1,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ ( inf_in3813033988066259269_a_nat @ A @ B ) @ A ) ).
% Int_lower1
thf(fact_980_Int__lower1,axiom,
! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ ( inf_in1693821330460682071rm_a_b @ A @ B ) @ A ) ).
% Int_lower1
thf(fact_981_Int__lower1,axiom,
! [A: set_Pr9093778441882193744at_nat,B: set_Pr9093778441882193744at_nat] : ( ord_le3678578370064672496at_nat @ ( inf_in907833683887804734at_nat @ A @ B ) @ A ) ).
% Int_lower1
thf(fact_982_Int__lower2,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ ( inf_in3813033988066259269_a_nat @ A @ B ) @ B ) ).
% Int_lower2
thf(fact_983_Int__lower2,axiom,
! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ ( inf_in1693821330460682071rm_a_b @ A @ B ) @ B ) ).
% Int_lower2
thf(fact_984_Int__lower2,axiom,
! [A: set_Pr9093778441882193744at_nat,B: set_Pr9093778441882193744at_nat] : ( ord_le3678578370064672496at_nat @ ( inf_in907833683887804734at_nat @ A @ B ) @ B ) ).
% Int_lower2
thf(fact_985_Int__absorb1,axiom,
! [B: set_Pr4934435412358123699_a_nat,A: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ B @ A )
=> ( ( inf_in3813033988066259269_a_nat @ A @ B )
= B ) ) ).
% Int_absorb1
thf(fact_986_Int__absorb1,axiom,
! [B: set_Pr4386577575007340137rm_a_b,A: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ B @ A )
=> ( ( inf_in1693821330460682071rm_a_b @ A @ B )
= B ) ) ).
% Int_absorb1
thf(fact_987_Int__absorb1,axiom,
! [B: set_Pr9093778441882193744at_nat,A: set_Pr9093778441882193744at_nat] :
( ( ord_le3678578370064672496at_nat @ B @ A )
=> ( ( inf_in907833683887804734at_nat @ A @ B )
= B ) ) ).
% Int_absorb1
thf(fact_988_Int__absorb2,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A @ B )
=> ( ( inf_in3813033988066259269_a_nat @ A @ B )
= A ) ) ).
% Int_absorb2
thf(fact_989_Int__absorb2,axiom,
! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A @ B )
=> ( ( inf_in1693821330460682071rm_a_b @ A @ B )
= A ) ) ).
% Int_absorb2
thf(fact_990_Int__absorb2,axiom,
! [A: set_Pr9093778441882193744at_nat,B: set_Pr9093778441882193744at_nat] :
( ( ord_le3678578370064672496at_nat @ A @ B )
=> ( ( inf_in907833683887804734at_nat @ A @ B )
= A ) ) ).
% Int_absorb2
thf(fact_991_Int__greatest,axiom,
! [C2: set_Pr4934435412358123699_a_nat,A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ C2 @ A )
=> ( ( ord_le8666007276011122963_a_nat @ C2 @ B )
=> ( ord_le8666007276011122963_a_nat @ C2 @ ( inf_in3813033988066259269_a_nat @ A @ B ) ) ) ) ).
% Int_greatest
thf(fact_992_Int__greatest,axiom,
! [C2: set_Pr4386577575007340137rm_a_b,A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ C2 @ A )
=> ( ( ord_le118470702582115849rm_a_b @ C2 @ B )
=> ( ord_le118470702582115849rm_a_b @ C2 @ ( inf_in1693821330460682071rm_a_b @ A @ B ) ) ) ) ).
% Int_greatest
thf(fact_993_Int__greatest,axiom,
! [C2: set_Pr9093778441882193744at_nat,A: set_Pr9093778441882193744at_nat,B: set_Pr9093778441882193744at_nat] :
( ( ord_le3678578370064672496at_nat @ C2 @ A )
=> ( ( ord_le3678578370064672496at_nat @ C2 @ B )
=> ( ord_le3678578370064672496at_nat @ C2 @ ( inf_in907833683887804734at_nat @ A @ B ) ) ) ) ).
% Int_greatest
thf(fact_994_Int__Collect__mono,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat,P: product_prod_a_nat > $o,Q: product_prod_a_nat > $o] :
( ( ord_le8666007276011122963_a_nat @ A @ B )
=> ( ! [X: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ X @ A )
=> ( ( P @ X )
=> ( Q @ X ) ) )
=> ( ord_le8666007276011122963_a_nat @ ( inf_in3813033988066259269_a_nat @ A @ ( collec4464134535221767506_a_nat @ P ) ) @ ( inf_in3813033988066259269_a_nat @ B @ ( collec4464134535221767506_a_nat @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_995_Int__Collect__mono,axiom,
! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b,P: produc357393685978478089rm_a_b > $o,Q: produc357393685978478089rm_a_b > $o] :
( ( ord_le118470702582115849rm_a_b @ A @ B )
=> ( ! [X: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X @ A )
=> ( ( P @ X )
=> ( Q @ X ) ) )
=> ( ord_le118470702582115849rm_a_b @ ( inf_in1693821330460682071rm_a_b @ A @ ( collec99177395925924084rm_a_b @ P ) ) @ ( inf_in1693821330460682071rm_a_b @ B @ ( collec99177395925924084rm_a_b @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_996_Int__Collect__mono,axiom,
! [A: set_Pr9093778441882193744at_nat,B: set_Pr9093778441882193744at_nat,P: produc8199716216217303280at_nat > $o,Q: produc8199716216217303280at_nat > $o] :
( ( ord_le3678578370064672496at_nat @ A @ B )
=> ( ! [X: produc8199716216217303280at_nat] :
( ( member7226740684066999833at_nat @ X @ A )
=> ( ( P @ X )
=> ( Q @ X ) ) )
=> ( ord_le3678578370064672496at_nat @ ( inf_in907833683887804734at_nat @ A @ ( collec9201399625632817755at_nat @ P ) ) @ ( inf_in907833683887804734at_nat @ B @ ( collec9201399625632817755at_nat @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_997_singletonD,axiom,
! [B2: produc357393685978478089rm_a_b,A2: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ B2 @ ( insert7009541432154983385rm_a_b @ A2 @ bot_bo197521221353338581rm_a_b ) )
=> ( B2 = A2 ) ) ).
% singletonD
thf(fact_998_singletonD,axiom,
! [B2: product_prod_a_nat,A2: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ B2 @ ( insert8054603423593749677_a_nat @ A2 @ bot_bo9049108969261143879_a_nat ) )
=> ( B2 = A2 ) ) ).
% singletonD
thf(fact_999_singletonD,axiom,
! [B2: produc8199716216217303280at_nat,A2: produc8199716216217303280at_nat] :
( ( member7226740684066999833at_nat @ B2 @ ( insert891772969674204864at_nat @ A2 @ bot_bo8000532843592726972at_nat ) )
=> ( B2 = A2 ) ) ).
% singletonD
thf(fact_1000_singleton__iff,axiom,
! [B2: produc357393685978478089rm_a_b,A2: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ B2 @ ( insert7009541432154983385rm_a_b @ A2 @ bot_bo197521221353338581rm_a_b ) )
= ( B2 = A2 ) ) ).
% singleton_iff
thf(fact_1001_singleton__iff,axiom,
! [B2: product_prod_a_nat,A2: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ B2 @ ( insert8054603423593749677_a_nat @ A2 @ bot_bo9049108969261143879_a_nat ) )
= ( B2 = A2 ) ) ).
% singleton_iff
thf(fact_1002_singleton__iff,axiom,
! [B2: produc8199716216217303280at_nat,A2: produc8199716216217303280at_nat] :
( ( member7226740684066999833at_nat @ B2 @ ( insert891772969674204864at_nat @ A2 @ bot_bo8000532843592726972at_nat ) )
= ( B2 = A2 ) ) ).
% singleton_iff
thf(fact_1003_distrib__inf__le,axiom,
! [X3: set_Pr4934435412358123699_a_nat,Y3: set_Pr4934435412358123699_a_nat,Z2: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ ( sup_su459911885395995103_a_nat @ ( inf_in3813033988066259269_a_nat @ X3 @ Y3 ) @ ( inf_in3813033988066259269_a_nat @ X3 @ Z2 ) ) @ ( inf_in3813033988066259269_a_nat @ X3 @ ( sup_su459911885395995103_a_nat @ Y3 @ Z2 ) ) ) ).
% distrib_inf_le
thf(fact_1004_distrib__inf__le,axiom,
! [X3: set_Pr4386577575007340137rm_a_b,Y3: set_Pr4386577575007340137rm_a_b,Z2: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ ( sup_su6776935440552674877rm_a_b @ ( inf_in1693821330460682071rm_a_b @ X3 @ Y3 ) @ ( inf_in1693821330460682071rm_a_b @ X3 @ Z2 ) ) @ ( inf_in1693821330460682071rm_a_b @ X3 @ ( sup_su6776935440552674877rm_a_b @ Y3 @ Z2 ) ) ) ).
% distrib_inf_le
thf(fact_1005_distrib__inf__le,axiom,
! [X3: set_Pr9093778441882193744at_nat,Y3: set_Pr9093778441882193744at_nat,Z2: set_Pr9093778441882193744at_nat] : ( ord_le3678578370064672496at_nat @ ( sup_su5262723077604177188at_nat @ ( inf_in907833683887804734at_nat @ X3 @ Y3 ) @ ( inf_in907833683887804734at_nat @ X3 @ Z2 ) ) @ ( inf_in907833683887804734at_nat @ X3 @ ( sup_su5262723077604177188at_nat @ Y3 @ Z2 ) ) ) ).
% distrib_inf_le
thf(fact_1006_distrib__sup__le,axiom,
! [X3: set_Pr4934435412358123699_a_nat,Y3: set_Pr4934435412358123699_a_nat,Z2: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ ( sup_su459911885395995103_a_nat @ X3 @ ( inf_in3813033988066259269_a_nat @ Y3 @ Z2 ) ) @ ( inf_in3813033988066259269_a_nat @ ( sup_su459911885395995103_a_nat @ X3 @ Y3 ) @ ( sup_su459911885395995103_a_nat @ X3 @ Z2 ) ) ) ).
% distrib_sup_le
thf(fact_1007_distrib__sup__le,axiom,
! [X3: set_Pr4386577575007340137rm_a_b,Y3: set_Pr4386577575007340137rm_a_b,Z2: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ ( sup_su6776935440552674877rm_a_b @ X3 @ ( inf_in1693821330460682071rm_a_b @ Y3 @ Z2 ) ) @ ( inf_in1693821330460682071rm_a_b @ ( sup_su6776935440552674877rm_a_b @ X3 @ Y3 ) @ ( sup_su6776935440552674877rm_a_b @ X3 @ Z2 ) ) ) ).
% distrib_sup_le
thf(fact_1008_distrib__sup__le,axiom,
! [X3: set_Pr9093778441882193744at_nat,Y3: set_Pr9093778441882193744at_nat,Z2: set_Pr9093778441882193744at_nat] : ( ord_le3678578370064672496at_nat @ ( sup_su5262723077604177188at_nat @ X3 @ ( inf_in907833683887804734at_nat @ Y3 @ Z2 ) ) @ ( inf_in907833683887804734at_nat @ ( sup_su5262723077604177188at_nat @ X3 @ Y3 ) @ ( sup_su5262723077604177188at_nat @ X3 @ Z2 ) ) ) ).
% distrib_sup_le
thf(fact_1009_subset__singleton__iff,axiom,
! [X6: set_Pr4934435412358123699_a_nat,A2: product_prod_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X6 @ ( insert8054603423593749677_a_nat @ A2 @ bot_bo9049108969261143879_a_nat ) )
= ( ( X6 = bot_bo9049108969261143879_a_nat )
| ( X6
= ( insert8054603423593749677_a_nat @ A2 @ bot_bo9049108969261143879_a_nat ) ) ) ) ).
% subset_singleton_iff
thf(fact_1010_subset__singleton__iff,axiom,
! [X6: set_Pr4386577575007340137rm_a_b,A2: produc357393685978478089rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ X6 @ ( insert7009541432154983385rm_a_b @ A2 @ bot_bo197521221353338581rm_a_b ) )
= ( ( X6 = bot_bo197521221353338581rm_a_b )
| ( X6
= ( insert7009541432154983385rm_a_b @ A2 @ bot_bo197521221353338581rm_a_b ) ) ) ) ).
% subset_singleton_iff
thf(fact_1011_subset__singleton__iff,axiom,
! [X6: set_Pr9093778441882193744at_nat,A2: produc8199716216217303280at_nat] :
( ( ord_le3678578370064672496at_nat @ X6 @ ( insert891772969674204864at_nat @ A2 @ bot_bo8000532843592726972at_nat ) )
= ( ( X6 = bot_bo8000532843592726972at_nat )
| ( X6
= ( insert891772969674204864at_nat @ A2 @ bot_bo8000532843592726972at_nat ) ) ) ) ).
% subset_singleton_iff
thf(fact_1012_subset__singletonD,axiom,
! [A: set_Pr4934435412358123699_a_nat,X3: product_prod_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A @ ( insert8054603423593749677_a_nat @ X3 @ bot_bo9049108969261143879_a_nat ) )
=> ( ( A = bot_bo9049108969261143879_a_nat )
| ( A
= ( insert8054603423593749677_a_nat @ X3 @ bot_bo9049108969261143879_a_nat ) ) ) ) ).
% subset_singletonD
thf(fact_1013_subset__singletonD,axiom,
! [A: set_Pr4386577575007340137rm_a_b,X3: produc357393685978478089rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A @ ( insert7009541432154983385rm_a_b @ X3 @ bot_bo197521221353338581rm_a_b ) )
=> ( ( A = bot_bo197521221353338581rm_a_b )
| ( A
= ( insert7009541432154983385rm_a_b @ X3 @ bot_bo197521221353338581rm_a_b ) ) ) ) ).
% subset_singletonD
thf(fact_1014_subset__singletonD,axiom,
! [A: set_Pr9093778441882193744at_nat,X3: produc8199716216217303280at_nat] :
( ( ord_le3678578370064672496at_nat @ A @ ( insert891772969674204864at_nat @ X3 @ bot_bo8000532843592726972at_nat ) )
=> ( ( A = bot_bo8000532843592726972at_nat )
| ( A
= ( insert891772969674204864at_nat @ X3 @ bot_bo8000532843592726972at_nat ) ) ) ) ).
% subset_singletonD
thf(fact_1015_Un__Int__assoc__eq,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat,C2: set_Pr4934435412358123699_a_nat] :
( ( ( sup_su459911885395995103_a_nat @ ( inf_in3813033988066259269_a_nat @ A @ B ) @ C2 )
= ( inf_in3813033988066259269_a_nat @ A @ ( sup_su459911885395995103_a_nat @ B @ C2 ) ) )
= ( ord_le8666007276011122963_a_nat @ C2 @ A ) ) ).
% Un_Int_assoc_eq
thf(fact_1016_Un__Int__assoc__eq,axiom,
! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b,C2: set_Pr4386577575007340137rm_a_b] :
( ( ( sup_su6776935440552674877rm_a_b @ ( inf_in1693821330460682071rm_a_b @ A @ B ) @ C2 )
= ( inf_in1693821330460682071rm_a_b @ A @ ( sup_su6776935440552674877rm_a_b @ B @ C2 ) ) )
= ( ord_le118470702582115849rm_a_b @ C2 @ A ) ) ).
% Un_Int_assoc_eq
thf(fact_1017_Un__Int__assoc__eq,axiom,
! [A: set_Pr9093778441882193744at_nat,B: set_Pr9093778441882193744at_nat,C2: set_Pr9093778441882193744at_nat] :
( ( ( sup_su5262723077604177188at_nat @ ( inf_in907833683887804734at_nat @ A @ B ) @ C2 )
= ( inf_in907833683887804734at_nat @ A @ ( sup_su5262723077604177188at_nat @ B @ C2 ) ) )
= ( ord_le3678578370064672496at_nat @ C2 @ A ) ) ).
% Un_Int_assoc_eq
thf(fact_1018_trancl__Int__subset,axiom,
! [R: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ R @ S )
=> ( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ ( inf_in1693821330460682071rm_a_b @ ( transi7922773638565587891rm_a_b @ R ) @ S ) @ R ) @ S )
=> ( ord_le118470702582115849rm_a_b @ ( transi7922773638565587891rm_a_b @ R ) @ S ) ) ) ).
% trancl_Int_subset
thf(fact_1019_rtrancl__Int__subset,axiom,
! [S: set_Pr4386577575007340137rm_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ id_term_a_b @ S )
=> ( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ ( inf_in1693821330460682071rm_a_b @ ( transi7742714808557438673rm_a_b @ R ) @ S ) @ R ) @ S )
=> ( ord_le118470702582115849rm_a_b @ ( transi7742714808557438673rm_a_b @ R ) @ S ) ) ) ).
% rtrancl_Int_subset
thf(fact_1020_min__ext__compat,axiom,
! [R4: set_Pr4386577575007340137rm_a_b,S5: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ R4 @ S5 ) @ R4 )
=> ( ord_le2663425029446080117rm_a_b @ ( relcom3320975603336820193rm_a_b @ ( min_ext_term_a_b @ R4 ) @ ( sup_su1246761269818691241rm_a_b @ ( min_ext_term_a_b @ S5 ) @ ( insert807748097412076101rm_a_b @ ( produc8517751122659631533rm_a_b @ bot_bot_set_term_a_b @ bot_bot_set_term_a_b ) @ bot_bo6544111711586975553rm_a_b ) ) ) @ ( min_ext_term_a_b @ R4 ) ) ) ).
% min_ext_compat
thf(fact_1021_weak__diamond__def,axiom,
( abstra5190899124789103926rm_a_b
= ( ^ [R2: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ ( minus_5192120951422937424rm_a_b @ ( relcom370159955682700863rm_a_b @ ( conver7608559976937845093rm_a_b @ R2 ) @ R2 ) @ id_term_a_b ) @ ( relcom370159955682700863rm_a_b @ R2 @ ( conver7608559976937845093rm_a_b @ R2 ) ) ) ) ) ).
% weak_diamond_def
thf(fact_1022_DiffI,axiom,
! [C: produc357393685978478089rm_a_b,A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ C @ A )
=> ( ~ ( member5869715511025134514rm_a_b @ C @ B )
=> ( member5869715511025134514rm_a_b @ C @ ( minus_5192120951422937424rm_a_b @ A @ B ) ) ) ) ).
% DiffI
thf(fact_1023_DiffI,axiom,
! [C: product_prod_a_nat,A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( member5724188588386418708_a_nat @ C @ A )
=> ( ~ ( member5724188588386418708_a_nat @ C @ B )
=> ( member5724188588386418708_a_nat @ C @ ( minus_8630901905721465484_a_nat @ A @ B ) ) ) ) ).
% DiffI
thf(fact_1024_DiffI,axiom,
! [C: produc8199716216217303280at_nat,A: set_Pr9093778441882193744at_nat,B: set_Pr9093778441882193744at_nat] :
( ( member7226740684066999833at_nat @ C @ A )
=> ( ~ ( member7226740684066999833at_nat @ C @ B )
=> ( member7226740684066999833at_nat @ C @ ( minus_3544293778650172087at_nat @ A @ B ) ) ) ) ).
% DiffI
thf(fact_1025_Diff__iff,axiom,
! [C: produc357393685978478089rm_a_b,A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ C @ ( minus_5192120951422937424rm_a_b @ A @ B ) )
= ( ( member5869715511025134514rm_a_b @ C @ A )
& ~ ( member5869715511025134514rm_a_b @ C @ B ) ) ) ).
% Diff_iff
thf(fact_1026_Diff__iff,axiom,
! [C: product_prod_a_nat,A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( member5724188588386418708_a_nat @ C @ ( minus_8630901905721465484_a_nat @ A @ B ) )
= ( ( member5724188588386418708_a_nat @ C @ A )
& ~ ( member5724188588386418708_a_nat @ C @ B ) ) ) ).
% Diff_iff
thf(fact_1027_Diff__iff,axiom,
! [C: produc8199716216217303280at_nat,A: set_Pr9093778441882193744at_nat,B: set_Pr9093778441882193744at_nat] :
( ( member7226740684066999833at_nat @ C @ ( minus_3544293778650172087at_nat @ A @ B ) )
= ( ( member7226740684066999833at_nat @ C @ A )
& ~ ( member7226740684066999833at_nat @ C @ B ) ) ) ).
% Diff_iff
thf(fact_1028_Diff__insert0,axiom,
! [X3: produc357393685978478089rm_a_b,A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ~ ( member5869715511025134514rm_a_b @ X3 @ A )
=> ( ( minus_5192120951422937424rm_a_b @ A @ ( insert7009541432154983385rm_a_b @ X3 @ B ) )
= ( minus_5192120951422937424rm_a_b @ A @ B ) ) ) ).
% Diff_insert0
thf(fact_1029_Diff__insert0,axiom,
! [X3: product_prod_a_nat,A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ~ ( member5724188588386418708_a_nat @ X3 @ A )
=> ( ( minus_8630901905721465484_a_nat @ A @ ( insert8054603423593749677_a_nat @ X3 @ B ) )
= ( minus_8630901905721465484_a_nat @ A @ B ) ) ) ).
% Diff_insert0
thf(fact_1030_Diff__insert0,axiom,
! [X3: produc8199716216217303280at_nat,A: set_Pr9093778441882193744at_nat,B: set_Pr9093778441882193744at_nat] :
( ~ ( member7226740684066999833at_nat @ X3 @ A )
=> ( ( minus_3544293778650172087at_nat @ A @ ( insert891772969674204864at_nat @ X3 @ B ) )
= ( minus_3544293778650172087at_nat @ A @ B ) ) ) ).
% Diff_insert0
thf(fact_1031_insert__Diff1,axiom,
! [X3: produc357393685978478089rm_a_b,B: set_Pr4386577575007340137rm_a_b,A: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ X3 @ B )
=> ( ( minus_5192120951422937424rm_a_b @ ( insert7009541432154983385rm_a_b @ X3 @ A ) @ B )
= ( minus_5192120951422937424rm_a_b @ A @ B ) ) ) ).
% insert_Diff1
thf(fact_1032_insert__Diff1,axiom,
! [X3: product_prod_a_nat,B: set_Pr4934435412358123699_a_nat,A: set_Pr4934435412358123699_a_nat] :
( ( member5724188588386418708_a_nat @ X3 @ B )
=> ( ( minus_8630901905721465484_a_nat @ ( insert8054603423593749677_a_nat @ X3 @ A ) @ B )
= ( minus_8630901905721465484_a_nat @ A @ B ) ) ) ).
% insert_Diff1
thf(fact_1033_insert__Diff1,axiom,
! [X3: produc8199716216217303280at_nat,B: set_Pr9093778441882193744at_nat,A: set_Pr9093778441882193744at_nat] :
( ( member7226740684066999833at_nat @ X3 @ B )
=> ( ( minus_3544293778650172087at_nat @ ( insert891772969674204864at_nat @ X3 @ A ) @ B )
= ( minus_3544293778650172087at_nat @ A @ B ) ) ) ).
% insert_Diff1
thf(fact_1034_Diff__eq__empty__iff,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( ( minus_8630901905721465484_a_nat @ A @ B )
= bot_bo9049108969261143879_a_nat )
= ( ord_le8666007276011122963_a_nat @ A @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_1035_Diff__eq__empty__iff,axiom,
! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ( ( minus_5192120951422937424rm_a_b @ A @ B )
= bot_bo197521221353338581rm_a_b )
= ( ord_le118470702582115849rm_a_b @ A @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_1036_Diff__eq__empty__iff,axiom,
! [A: set_Pr9093778441882193744at_nat,B: set_Pr9093778441882193744at_nat] :
( ( ( minus_3544293778650172087at_nat @ A @ B )
= bot_bo8000532843592726972at_nat )
= ( ord_le3678578370064672496at_nat @ A @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_1037_insert__Diff__if,axiom,
! [X3: produc357393685978478089rm_a_b,B: set_Pr4386577575007340137rm_a_b,A: set_Pr4386577575007340137rm_a_b] :
( ( ( member5869715511025134514rm_a_b @ X3 @ B )
=> ( ( minus_5192120951422937424rm_a_b @ ( insert7009541432154983385rm_a_b @ X3 @ A ) @ B )
= ( minus_5192120951422937424rm_a_b @ A @ B ) ) )
& ( ~ ( member5869715511025134514rm_a_b @ X3 @ B )
=> ( ( minus_5192120951422937424rm_a_b @ ( insert7009541432154983385rm_a_b @ X3 @ A ) @ B )
= ( insert7009541432154983385rm_a_b @ X3 @ ( minus_5192120951422937424rm_a_b @ A @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_1038_insert__Diff__if,axiom,
! [X3: product_prod_a_nat,B: set_Pr4934435412358123699_a_nat,A: set_Pr4934435412358123699_a_nat] :
( ( ( member5724188588386418708_a_nat @ X3 @ B )
=> ( ( minus_8630901905721465484_a_nat @ ( insert8054603423593749677_a_nat @ X3 @ A ) @ B )
= ( minus_8630901905721465484_a_nat @ A @ B ) ) )
& ( ~ ( member5724188588386418708_a_nat @ X3 @ B )
=> ( ( minus_8630901905721465484_a_nat @ ( insert8054603423593749677_a_nat @ X3 @ A ) @ B )
= ( insert8054603423593749677_a_nat @ X3 @ ( minus_8630901905721465484_a_nat @ A @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_1039_insert__Diff__if,axiom,
! [X3: produc8199716216217303280at_nat,B: set_Pr9093778441882193744at_nat,A: set_Pr9093778441882193744at_nat] :
( ( ( member7226740684066999833at_nat @ X3 @ B )
=> ( ( minus_3544293778650172087at_nat @ ( insert891772969674204864at_nat @ X3 @ A ) @ B )
= ( minus_3544293778650172087at_nat @ A @ B ) ) )
& ( ~ ( member7226740684066999833at_nat @ X3 @ B )
=> ( ( minus_3544293778650172087at_nat @ ( insert891772969674204864at_nat @ X3 @ A ) @ B )
= ( insert891772969674204864at_nat @ X3 @ ( minus_3544293778650172087at_nat @ A @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_1040_Diff__mono,axiom,
! [A: set_Pr4934435412358123699_a_nat,C2: set_Pr4934435412358123699_a_nat,D: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A @ C2 )
=> ( ( ord_le8666007276011122963_a_nat @ D @ B )
=> ( ord_le8666007276011122963_a_nat @ ( minus_8630901905721465484_a_nat @ A @ B ) @ ( minus_8630901905721465484_a_nat @ C2 @ D ) ) ) ) ).
% Diff_mono
thf(fact_1041_Diff__mono,axiom,
! [A: set_Pr4386577575007340137rm_a_b,C2: set_Pr4386577575007340137rm_a_b,D: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A @ C2 )
=> ( ( ord_le118470702582115849rm_a_b @ D @ B )
=> ( ord_le118470702582115849rm_a_b @ ( minus_5192120951422937424rm_a_b @ A @ B ) @ ( minus_5192120951422937424rm_a_b @ C2 @ D ) ) ) ) ).
% Diff_mono
thf(fact_1042_Diff__mono,axiom,
! [A: set_Pr9093778441882193744at_nat,C2: set_Pr9093778441882193744at_nat,D: set_Pr9093778441882193744at_nat,B: set_Pr9093778441882193744at_nat] :
( ( ord_le3678578370064672496at_nat @ A @ C2 )
=> ( ( ord_le3678578370064672496at_nat @ D @ B )
=> ( ord_le3678578370064672496at_nat @ ( minus_3544293778650172087at_nat @ A @ B ) @ ( minus_3544293778650172087at_nat @ C2 @ D ) ) ) ) ).
% Diff_mono
thf(fact_1043_Diff__subset,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ ( minus_8630901905721465484_a_nat @ A @ B ) @ A ) ).
% Diff_subset
thf(fact_1044_Diff__subset,axiom,
! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ ( minus_5192120951422937424rm_a_b @ A @ B ) @ A ) ).
% Diff_subset
thf(fact_1045_Diff__subset,axiom,
! [A: set_Pr9093778441882193744at_nat,B: set_Pr9093778441882193744at_nat] : ( ord_le3678578370064672496at_nat @ ( minus_3544293778650172087at_nat @ A @ B ) @ A ) ).
% Diff_subset
thf(fact_1046_double__diff,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat,C2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A @ B )
=> ( ( ord_le8666007276011122963_a_nat @ B @ C2 )
=> ( ( minus_8630901905721465484_a_nat @ B @ ( minus_8630901905721465484_a_nat @ C2 @ A ) )
= A ) ) ) ).
% double_diff
thf(fact_1047_double__diff,axiom,
! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b,C2: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A @ B )
=> ( ( ord_le118470702582115849rm_a_b @ B @ C2 )
=> ( ( minus_5192120951422937424rm_a_b @ B @ ( minus_5192120951422937424rm_a_b @ C2 @ A ) )
= A ) ) ) ).
% double_diff
thf(fact_1048_double__diff,axiom,
! [A: set_Pr9093778441882193744at_nat,B: set_Pr9093778441882193744at_nat,C2: set_Pr9093778441882193744at_nat] :
( ( ord_le3678578370064672496at_nat @ A @ B )
=> ( ( ord_le3678578370064672496at_nat @ B @ C2 )
=> ( ( minus_3544293778650172087at_nat @ B @ ( minus_3544293778650172087at_nat @ C2 @ A ) )
= A ) ) ) ).
% double_diff
thf(fact_1049_DiffE,axiom,
! [C: produc357393685978478089rm_a_b,A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ C @ ( minus_5192120951422937424rm_a_b @ A @ B ) )
=> ~ ( ( member5869715511025134514rm_a_b @ C @ A )
=> ( member5869715511025134514rm_a_b @ C @ B ) ) ) ).
% DiffE
thf(fact_1050_DiffE,axiom,
! [C: product_prod_a_nat,A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( member5724188588386418708_a_nat @ C @ ( minus_8630901905721465484_a_nat @ A @ B ) )
=> ~ ( ( member5724188588386418708_a_nat @ C @ A )
=> ( member5724188588386418708_a_nat @ C @ B ) ) ) ).
% DiffE
thf(fact_1051_DiffE,axiom,
! [C: produc8199716216217303280at_nat,A: set_Pr9093778441882193744at_nat,B: set_Pr9093778441882193744at_nat] :
( ( member7226740684066999833at_nat @ C @ ( minus_3544293778650172087at_nat @ A @ B ) )
=> ~ ( ( member7226740684066999833at_nat @ C @ A )
=> ( member7226740684066999833at_nat @ C @ B ) ) ) ).
% DiffE
thf(fact_1052_DiffD1,axiom,
! [C: produc357393685978478089rm_a_b,A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ C @ ( minus_5192120951422937424rm_a_b @ A @ B ) )
=> ( member5869715511025134514rm_a_b @ C @ A ) ) ).
% DiffD1
thf(fact_1053_DiffD1,axiom,
! [C: product_prod_a_nat,A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( member5724188588386418708_a_nat @ C @ ( minus_8630901905721465484_a_nat @ A @ B ) )
=> ( member5724188588386418708_a_nat @ C @ A ) ) ).
% DiffD1
thf(fact_1054_DiffD1,axiom,
! [C: produc8199716216217303280at_nat,A: set_Pr9093778441882193744at_nat,B: set_Pr9093778441882193744at_nat] :
( ( member7226740684066999833at_nat @ C @ ( minus_3544293778650172087at_nat @ A @ B ) )
=> ( member7226740684066999833at_nat @ C @ A ) ) ).
% DiffD1
thf(fact_1055_DiffD2,axiom,
! [C: produc357393685978478089rm_a_b,A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ C @ ( minus_5192120951422937424rm_a_b @ A @ B ) )
=> ~ ( member5869715511025134514rm_a_b @ C @ B ) ) ).
% DiffD2
thf(fact_1056_DiffD2,axiom,
! [C: product_prod_a_nat,A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( member5724188588386418708_a_nat @ C @ ( minus_8630901905721465484_a_nat @ A @ B ) )
=> ~ ( member5724188588386418708_a_nat @ C @ B ) ) ).
% DiffD2
thf(fact_1057_DiffD2,axiom,
! [C: produc8199716216217303280at_nat,A: set_Pr9093778441882193744at_nat,B: set_Pr9093778441882193744at_nat] :
( ( member7226740684066999833at_nat @ C @ ( minus_3544293778650172087at_nat @ A @ B ) )
=> ~ ( member7226740684066999833at_nat @ C @ B ) ) ).
% DiffD2
thf(fact_1058_diff__shunt__var,axiom,
! [X3: set_Pr4934435412358123699_a_nat,Y3: set_Pr4934435412358123699_a_nat] :
( ( ( minus_8630901905721465484_a_nat @ X3 @ Y3 )
= bot_bo9049108969261143879_a_nat )
= ( ord_le8666007276011122963_a_nat @ X3 @ Y3 ) ) ).
% diff_shunt_var
thf(fact_1059_diff__shunt__var,axiom,
! [X3: set_Pr4386577575007340137rm_a_b,Y3: set_Pr4386577575007340137rm_a_b] :
( ( ( minus_5192120951422937424rm_a_b @ X3 @ Y3 )
= bot_bo197521221353338581rm_a_b )
= ( ord_le118470702582115849rm_a_b @ X3 @ Y3 ) ) ).
% diff_shunt_var
thf(fact_1060_diff__shunt__var,axiom,
! [X3: set_Pr9093778441882193744at_nat,Y3: set_Pr9093778441882193744at_nat] :
( ( ( minus_3544293778650172087at_nat @ X3 @ Y3 )
= bot_bo8000532843592726972at_nat )
= ( ord_le3678578370064672496at_nat @ X3 @ Y3 ) ) ).
% diff_shunt_var
thf(fact_1061_Diff__insert__absorb,axiom,
! [X3: produc357393685978478089rm_a_b,A: set_Pr4386577575007340137rm_a_b] :
( ~ ( member5869715511025134514rm_a_b @ X3 @ A )
=> ( ( minus_5192120951422937424rm_a_b @ ( insert7009541432154983385rm_a_b @ X3 @ A ) @ ( insert7009541432154983385rm_a_b @ X3 @ bot_bo197521221353338581rm_a_b ) )
= A ) ) ).
% Diff_insert_absorb
thf(fact_1062_Diff__insert__absorb,axiom,
! [X3: product_prod_a_nat,A: set_Pr4934435412358123699_a_nat] :
( ~ ( member5724188588386418708_a_nat @ X3 @ A )
=> ( ( minus_8630901905721465484_a_nat @ ( insert8054603423593749677_a_nat @ X3 @ A ) @ ( insert8054603423593749677_a_nat @ X3 @ bot_bo9049108969261143879_a_nat ) )
= A ) ) ).
% Diff_insert_absorb
thf(fact_1063_Diff__insert__absorb,axiom,
! [X3: produc8199716216217303280at_nat,A: set_Pr9093778441882193744at_nat] :
( ~ ( member7226740684066999833at_nat @ X3 @ A )
=> ( ( minus_3544293778650172087at_nat @ ( insert891772969674204864at_nat @ X3 @ A ) @ ( insert891772969674204864at_nat @ X3 @ bot_bo8000532843592726972at_nat ) )
= A ) ) ).
% Diff_insert_absorb
thf(fact_1064_insert__Diff,axiom,
! [A2: produc357393685978478089rm_a_b,A: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ A2 @ A )
=> ( ( insert7009541432154983385rm_a_b @ A2 @ ( minus_5192120951422937424rm_a_b @ A @ ( insert7009541432154983385rm_a_b @ A2 @ bot_bo197521221353338581rm_a_b ) ) )
= A ) ) ).
% insert_Diff
thf(fact_1065_insert__Diff,axiom,
! [A2: product_prod_a_nat,A: set_Pr4934435412358123699_a_nat] :
( ( member5724188588386418708_a_nat @ A2 @ A )
=> ( ( insert8054603423593749677_a_nat @ A2 @ ( minus_8630901905721465484_a_nat @ A @ ( insert8054603423593749677_a_nat @ A2 @ bot_bo9049108969261143879_a_nat ) ) )
= A ) ) ).
% insert_Diff
thf(fact_1066_insert__Diff,axiom,
! [A2: produc8199716216217303280at_nat,A: set_Pr9093778441882193744at_nat] :
( ( member7226740684066999833at_nat @ A2 @ A )
=> ( ( insert891772969674204864at_nat @ A2 @ ( minus_3544293778650172087at_nat @ A @ ( insert891772969674204864at_nat @ A2 @ bot_bo8000532843592726972at_nat ) ) )
= A ) ) ).
% insert_Diff
thf(fact_1067_subset__Diff__insert,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat,X3: product_prod_a_nat,C2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A @ ( minus_8630901905721465484_a_nat @ B @ ( insert8054603423593749677_a_nat @ X3 @ C2 ) ) )
= ( ( ord_le8666007276011122963_a_nat @ A @ ( minus_8630901905721465484_a_nat @ B @ C2 ) )
& ~ ( member5724188588386418708_a_nat @ X3 @ A ) ) ) ).
% subset_Diff_insert
thf(fact_1068_subset__Diff__insert,axiom,
! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b,X3: produc357393685978478089rm_a_b,C2: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A @ ( minus_5192120951422937424rm_a_b @ B @ ( insert7009541432154983385rm_a_b @ X3 @ C2 ) ) )
= ( ( ord_le118470702582115849rm_a_b @ A @ ( minus_5192120951422937424rm_a_b @ B @ C2 ) )
& ~ ( member5869715511025134514rm_a_b @ X3 @ A ) ) ) ).
% subset_Diff_insert
thf(fact_1069_subset__Diff__insert,axiom,
! [A: set_Pr9093778441882193744at_nat,B: set_Pr9093778441882193744at_nat,X3: produc8199716216217303280at_nat,C2: set_Pr9093778441882193744at_nat] :
( ( ord_le3678578370064672496at_nat @ A @ ( minus_3544293778650172087at_nat @ B @ ( insert891772969674204864at_nat @ X3 @ C2 ) ) )
= ( ( ord_le3678578370064672496at_nat @ A @ ( minus_3544293778650172087at_nat @ B @ C2 ) )
& ~ ( member7226740684066999833at_nat @ X3 @ A ) ) ) ).
% subset_Diff_insert
thf(fact_1070_Diff__partition,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A @ B )
=> ( ( sup_su459911885395995103_a_nat @ A @ ( minus_8630901905721465484_a_nat @ B @ A ) )
= B ) ) ).
% Diff_partition
thf(fact_1071_Diff__partition,axiom,
! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A @ B )
=> ( ( sup_su6776935440552674877rm_a_b @ A @ ( minus_5192120951422937424rm_a_b @ B @ A ) )
= B ) ) ).
% Diff_partition
thf(fact_1072_Diff__partition,axiom,
! [A: set_Pr9093778441882193744at_nat,B: set_Pr9093778441882193744at_nat] :
( ( ord_le3678578370064672496at_nat @ A @ B )
=> ( ( sup_su5262723077604177188at_nat @ A @ ( minus_3544293778650172087at_nat @ B @ A ) )
= B ) ) ).
% Diff_partition
thf(fact_1073_Diff__subset__conv,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat,C2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ ( minus_8630901905721465484_a_nat @ A @ B ) @ C2 )
= ( ord_le8666007276011122963_a_nat @ A @ ( sup_su459911885395995103_a_nat @ B @ C2 ) ) ) ).
% Diff_subset_conv
thf(fact_1074_Diff__subset__conv,axiom,
! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b,C2: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ ( minus_5192120951422937424rm_a_b @ A @ B ) @ C2 )
= ( ord_le118470702582115849rm_a_b @ A @ ( sup_su6776935440552674877rm_a_b @ B @ C2 ) ) ) ).
% Diff_subset_conv
thf(fact_1075_Diff__subset__conv,axiom,
! [A: set_Pr9093778441882193744at_nat,B: set_Pr9093778441882193744at_nat,C2: set_Pr9093778441882193744at_nat] :
( ( ord_le3678578370064672496at_nat @ ( minus_3544293778650172087at_nat @ A @ B ) @ C2 )
= ( ord_le3678578370064672496at_nat @ A @ ( sup_su5262723077604177188at_nat @ B @ C2 ) ) ) ).
% Diff_subset_conv
thf(fact_1076_is__singletonI_H,axiom,
! [A: set_Pr4386577575007340137rm_a_b] :
( ( A != bot_bo197521221353338581rm_a_b )
=> ( ! [X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X @ A )
=> ( ( member5869715511025134514rm_a_b @ Y @ A )
=> ( X = Y ) ) )
=> ( is_sin332545968047684349rm_a_b @ A ) ) ) ).
% is_singletonI'
thf(fact_1077_is__singletonI_H,axiom,
! [A: set_Pr4934435412358123699_a_nat] :
( ( A != bot_bo9049108969261143879_a_nat )
=> ( ! [X: product_prod_a_nat,Y: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ X @ A )
=> ( ( member5724188588386418708_a_nat @ Y @ A )
=> ( X = Y ) ) )
=> ( is_sin7638514652509285001_a_nat @ A ) ) ) ).
% is_singletonI'
thf(fact_1078_is__singletonI_H,axiom,
! [A: set_Pr9093778441882193744at_nat] :
( ( A != bot_bo8000532843592726972at_nat )
=> ( ! [X: produc8199716216217303280at_nat,Y: produc8199716216217303280at_nat] :
( ( member7226740684066999833at_nat @ X @ A )
=> ( ( member7226740684066999833at_nat @ Y @ A )
=> ( X = Y ) ) )
=> ( is_sin1382905349753836516at_nat @ A ) ) ) ).
% is_singletonI'
thf(fact_1079_Diff__single__insert,axiom,
! [A: set_Pr4934435412358123699_a_nat,X3: product_prod_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ ( minus_8630901905721465484_a_nat @ A @ ( insert8054603423593749677_a_nat @ X3 @ bot_bo9049108969261143879_a_nat ) ) @ B )
=> ( ord_le8666007276011122963_a_nat @ A @ ( insert8054603423593749677_a_nat @ X3 @ B ) ) ) ).
% Diff_single_insert
thf(fact_1080_Diff__single__insert,axiom,
! [A: set_Pr4386577575007340137rm_a_b,X3: produc357393685978478089rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ ( minus_5192120951422937424rm_a_b @ A @ ( insert7009541432154983385rm_a_b @ X3 @ bot_bo197521221353338581rm_a_b ) ) @ B )
=> ( ord_le118470702582115849rm_a_b @ A @ ( insert7009541432154983385rm_a_b @ X3 @ B ) ) ) ).
% Diff_single_insert
thf(fact_1081_Diff__single__insert,axiom,
! [A: set_Pr9093778441882193744at_nat,X3: produc8199716216217303280at_nat,B: set_Pr9093778441882193744at_nat] :
( ( ord_le3678578370064672496at_nat @ ( minus_3544293778650172087at_nat @ A @ ( insert891772969674204864at_nat @ X3 @ bot_bo8000532843592726972at_nat ) ) @ B )
=> ( ord_le3678578370064672496at_nat @ A @ ( insert891772969674204864at_nat @ X3 @ B ) ) ) ).
% Diff_single_insert
thf(fact_1082_subset__insert__iff,axiom,
! [A: set_Pr4934435412358123699_a_nat,X3: product_prod_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A @ ( insert8054603423593749677_a_nat @ X3 @ B ) )
= ( ( ( member5724188588386418708_a_nat @ X3 @ A )
=> ( ord_le8666007276011122963_a_nat @ ( minus_8630901905721465484_a_nat @ A @ ( insert8054603423593749677_a_nat @ X3 @ bot_bo9049108969261143879_a_nat ) ) @ B ) )
& ( ~ ( member5724188588386418708_a_nat @ X3 @ A )
=> ( ord_le8666007276011122963_a_nat @ A @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_1083_subset__insert__iff,axiom,
! [A: set_Pr4386577575007340137rm_a_b,X3: produc357393685978478089rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A @ ( insert7009541432154983385rm_a_b @ X3 @ B ) )
= ( ( ( member5869715511025134514rm_a_b @ X3 @ A )
=> ( ord_le118470702582115849rm_a_b @ ( minus_5192120951422937424rm_a_b @ A @ ( insert7009541432154983385rm_a_b @ X3 @ bot_bo197521221353338581rm_a_b ) ) @ B ) )
& ( ~ ( member5869715511025134514rm_a_b @ X3 @ A )
=> ( ord_le118470702582115849rm_a_b @ A @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_1084_subset__insert__iff,axiom,
! [A: set_Pr9093778441882193744at_nat,X3: produc8199716216217303280at_nat,B: set_Pr9093778441882193744at_nat] :
( ( ord_le3678578370064672496at_nat @ A @ ( insert891772969674204864at_nat @ X3 @ B ) )
= ( ( ( member7226740684066999833at_nat @ X3 @ A )
=> ( ord_le3678578370064672496at_nat @ ( minus_3544293778650172087at_nat @ A @ ( insert891772969674204864at_nat @ X3 @ bot_bo8000532843592726972at_nat ) ) @ B ) )
& ( ~ ( member7226740684066999833at_nat @ X3 @ A )
=> ( ord_le3678578370064672496at_nat @ A @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_1085_rtrancl__diff__decomp,axiom,
! [X3: term_a_b,Y3: term_a_b,A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ ( minus_5192120951422937424rm_a_b @ ( transi7742714808557438673rm_a_b @ A ) @ ( transi7742714808557438673rm_a_b @ B ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ A ) @ ( relcom370159955682700863rm_a_b @ ( minus_5192120951422937424rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ A ) ) ) ) ) ).
% rtrancl_diff_decomp
thf(fact_1086_max__ext__compat,axiom,
! [R4: set_Pr4386577575007340137rm_a_b,S5: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ R4 @ S5 ) @ R4 )
=> ( ord_le2663425029446080117rm_a_b @ ( relcom3320975603336820193rm_a_b @ ( max_ext_term_a_b @ R4 ) @ ( sup_su1246761269818691241rm_a_b @ ( max_ext_term_a_b @ S5 ) @ ( insert807748097412076101rm_a_b @ ( produc8517751122659631533rm_a_b @ bot_bot_set_term_a_b @ bot_bot_set_term_a_b ) @ bot_bo6544111711586975553rm_a_b ) ) ) @ ( max_ext_term_a_b @ R4 ) ) ) ).
% max_ext_compat
thf(fact_1087_some__NF,axiom,
! [R: set_Pr2972776593051762503rm_a_b,X3: produc357393685978478089rm_a_b] :
( ( abstra2398554102055911763rm_a_b @ R @ ( insert7009541432154983385rm_a_b @ X3 @ bot_bo197521221353338581rm_a_b ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X3 @ ( abstra8203652949708803681rm_a_b @ R @ X3 ) ) @ ( transi2615809358984392588rm_a_b @ R ) )
& ( member5869715511025134514rm_a_b @ ( abstra8203652949708803681rm_a_b @ R @ X3 ) @ ( abstra1902783451776770197rm_a_b @ R ) ) ) ) ).
% some_NF
thf(fact_1088_some__NF,axiom,
! [R: set_Pr1811044260758604347_a_nat,X3: product_prod_a_nat] :
( ( abstra8653715922312955827_a_nat @ R @ ( insert8054603423593749677_a_nat @ X3 @ bot_bo9049108969261143879_a_nat ) )
=> ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ X3 @ ( abstra5913684503434863653_a_nat @ R @ X3 ) ) @ ( transi2726145917338391738_a_nat @ R ) )
& ( member5724188588386418708_a_nat @ ( abstra5913684503434863653_a_nat @ R @ X3 ) @ ( abstra2685462378305987313_a_nat @ R ) ) ) ) ).
% some_NF
thf(fact_1089_some__NF,axiom,
! [R: set_Pr7047708467701517959at_nat,X3: produc8199716216217303280at_nat] :
( ( abstra7460355483315988282at_nat @ R @ ( insert891772969674204864at_nat @ X3 @ bot_bo8000532843592726972at_nat ) )
=> ( ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ X3 @ ( abstra5373487907108472136at_nat @ R @ X3 ) ) @ ( transi3422749762518297843at_nat @ R ) )
& ( member7226740684066999833at_nat @ ( abstra5373487907108472136at_nat @ R @ X3 ) @ ( abstra8406325002171223420at_nat @ R ) ) ) ) ).
% some_NF
thf(fact_1090_some__NF,axiom,
! [R: set_Pr4386577575007340137rm_a_b,X3: term_a_b] :
( ( abstra4720023341729745482rm_a_b @ R @ ( insert_term_a_b @ X3 @ bot_bot_set_term_a_b ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ ( abstra2589370493620291516rm_a_b @ R @ X3 ) ) @ ( transi7742714808557438673rm_a_b @ R ) )
& ( member_term_a_b @ ( abstra2589370493620291516rm_a_b @ R @ X3 ) @ ( abstract_NF_term_a_b @ R ) ) ) ) ).
% some_NF
thf(fact_1091_Field__insert,axiom,
! [A2: term_a_b,B2: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( field_term_a_b @ ( insert7009541432154983385rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ R ) )
= ( sup_sup_set_term_a_b @ ( insert_term_a_b @ A2 @ ( insert_term_a_b @ B2 @ bot_bot_set_term_a_b ) ) @ ( field_term_a_b @ R ) ) ) ).
% Field_insert
thf(fact_1092_SN__induct__rule,axiom,
! [R: set_Pr4386577575007340137rm_a_b,P: term_a_b > $o,A2: term_a_b] :
( ( abstra4720023341729745482rm_a_b @ R @ top_top_set_term_a_b )
=> ( ! [A6: term_a_b] :
( ! [B8: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A6 @ B8 ) @ R )
=> ( P @ B8 ) )
=> ( P @ A6 ) )
=> ( P @ A2 ) ) ) ).
% SN_induct_rule
thf(fact_1093_SN__imp__minimal,axiom,
! [A: set_Pr2972776593051762503rm_a_b] :
( ( abstra2398554102055911763rm_a_b @ A @ top_to1314267278846557113rm_a_b )
=> ! [Q2: set_Pr4386577575007340137rm_a_b] :
( ? [X7: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ X7 @ Q2 )
=> ? [X: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X @ Q2 )
& ! [Y6: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y6 ) @ A )
=> ~ ( member5869715511025134514rm_a_b @ Y6 @ Q2 ) ) ) ) ) ).
% SN_imp_minimal
thf(fact_1094_SN__imp__minimal,axiom,
! [A: set_Pr1811044260758604347_a_nat] :
( ( abstra8653715922312955827_a_nat @ A @ top_to3353692345378799459_a_nat )
=> ! [Q2: set_Pr4934435412358123699_a_nat] :
( ? [X7: product_prod_a_nat] : ( member5724188588386418708_a_nat @ X7 @ Q2 )
=> ? [X: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ X @ Q2 )
& ! [Y6: product_prod_a_nat] :
( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ X @ Y6 ) @ A )
=> ~ ( member5724188588386418708_a_nat @ Y6 @ Q2 ) ) ) ) ) ).
% SN_imp_minimal
thf(fact_1095_SN__imp__minimal,axiom,
! [A: set_Pr7047708467701517959at_nat] :
( ( abstra7460355483315988282at_nat @ A @ top_to7836785446628990112at_nat )
=> ! [Q2: set_Pr9093778441882193744at_nat] :
( ? [X7: produc8199716216217303280at_nat] : ( member7226740684066999833at_nat @ X7 @ Q2 )
=> ? [X: produc8199716216217303280at_nat] :
( ( member7226740684066999833at_nat @ X @ Q2 )
& ! [Y6: produc8199716216217303280at_nat] :
( ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ X @ Y6 ) @ A )
=> ~ ( member7226740684066999833at_nat @ Y6 @ Q2 ) ) ) ) ) ).
% SN_imp_minimal
thf(fact_1096_SN__imp__minimal,axiom,
! [A: set_Pr4386577575007340137rm_a_b] :
( ( abstra4720023341729745482rm_a_b @ A @ top_top_set_term_a_b )
=> ! [Q2: set_term_a_b] :
( ? [X7: term_a_b] : ( member_term_a_b @ X7 @ Q2 )
=> ? [X: term_a_b] :
( ( member_term_a_b @ X @ Q2 )
& ! [Y6: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y6 ) @ A )
=> ~ ( member_term_a_b @ Y6 @ Q2 ) ) ) ) ) ).
% SN_imp_minimal
thf(fact_1097_refl__not__SN,axiom,
! [X3: term_a_b,R4: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ X3 ) @ R4 )
=> ~ ( abstra4720023341729745482rm_a_b @ R4 @ top_top_set_term_a_b ) ) ).
% refl_not_SN
thf(fact_1098_FieldI1,axiom,
! [I2: produc357393685978478089rm_a_b,J2: produc357393685978478089rm_a_b,R4: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ I2 @ J2 ) @ R4 )
=> ( member5869715511025134514rm_a_b @ I2 @ ( field_6884932134483023318rm_a_b @ R4 ) ) ) ).
% FieldI1
thf(fact_1099_FieldI1,axiom,
! [I2: product_prod_a_nat,J2: product_prod_a_nat,R4: set_Pr1811044260758604347_a_nat] :
( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ I2 @ J2 ) @ R4 )
=> ( member5724188588386418708_a_nat @ I2 @ ( field_8954927560578634480_a_nat @ R4 ) ) ) ).
% FieldI1
thf(fact_1100_FieldI1,axiom,
! [I2: produc8199716216217303280at_nat,J2: produc8199716216217303280at_nat,R4: set_Pr7047708467701517959at_nat] :
( ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ I2 @ J2 ) @ R4 )
=> ( member7226740684066999833at_nat @ I2 @ ( field_718184728585074237at_nat @ R4 ) ) ) ).
% FieldI1
thf(fact_1101_FieldI1,axiom,
! [I2: term_a_b,J2: term_a_b,R4: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ I2 @ J2 ) @ R4 )
=> ( member_term_a_b @ I2 @ ( field_term_a_b @ R4 ) ) ) ).
% FieldI1
thf(fact_1102_FieldI2,axiom,
! [I2: produc357393685978478089rm_a_b,J2: produc357393685978478089rm_a_b,R4: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ I2 @ J2 ) @ R4 )
=> ( member5869715511025134514rm_a_b @ J2 @ ( field_6884932134483023318rm_a_b @ R4 ) ) ) ).
% FieldI2
thf(fact_1103_FieldI2,axiom,
! [I2: product_prod_a_nat,J2: product_prod_a_nat,R4: set_Pr1811044260758604347_a_nat] :
( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ I2 @ J2 ) @ R4 )
=> ( member5724188588386418708_a_nat @ J2 @ ( field_8954927560578634480_a_nat @ R4 ) ) ) ).
% FieldI2
thf(fact_1104_FieldI2,axiom,
! [I2: produc8199716216217303280at_nat,J2: produc8199716216217303280at_nat,R4: set_Pr7047708467701517959at_nat] :
( ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ I2 @ J2 ) @ R4 )
=> ( member7226740684066999833at_nat @ J2 @ ( field_718184728585074237at_nat @ R4 ) ) ) ).
% FieldI2
thf(fact_1105_FieldI2,axiom,
! [I2: term_a_b,J2: term_a_b,R4: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ I2 @ J2 ) @ R4 )
=> ( member_term_a_b @ J2 @ ( field_term_a_b @ R4 ) ) ) ).
% FieldI2
thf(fact_1106_SN__on__irrefl,axiom,
! [R: set_Pr4386577575007340137rm_a_b,A: set_term_a_b] :
( ( abstra4720023341729745482rm_a_b @ R @ A )
=> ! [X7: term_a_b] :
( ( member_term_a_b @ X7 @ A )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X7 @ X7 ) @ R ) ) ) ).
% SN_on_irrefl
thf(fact_1107_SN__on__induct,axiom,
! [R4: set_Pr2972776593051762503rm_a_b,A: set_Pr4386577575007340137rm_a_b,S: produc357393685978478089rm_a_b,P: produc357393685978478089rm_a_b > $o] :
( ( abstra2398554102055911763rm_a_b @ R4 @ A )
=> ( ( member5869715511025134514rm_a_b @ S @ A )
=> ( ! [T2: produc357393685978478089rm_a_b] :
( ! [U4: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ T2 @ U4 ) @ R4 )
=> ( P @ U4 ) )
=> ( P @ T2 ) )
=> ( P @ S ) ) ) ) ).
% SN_on_induct
thf(fact_1108_SN__on__induct,axiom,
! [R4: set_Pr1811044260758604347_a_nat,A: set_Pr4934435412358123699_a_nat,S: product_prod_a_nat,P: product_prod_a_nat > $o] :
( ( abstra8653715922312955827_a_nat @ R4 @ A )
=> ( ( member5724188588386418708_a_nat @ S @ A )
=> ( ! [T2: product_prod_a_nat] :
( ! [U4: product_prod_a_nat] :
( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ T2 @ U4 ) @ R4 )
=> ( P @ U4 ) )
=> ( P @ T2 ) )
=> ( P @ S ) ) ) ) ).
% SN_on_induct
thf(fact_1109_SN__on__induct,axiom,
! [R4: set_Pr7047708467701517959at_nat,A: set_Pr9093778441882193744at_nat,S: produc8199716216217303280at_nat,P: produc8199716216217303280at_nat > $o] :
( ( abstra7460355483315988282at_nat @ R4 @ A )
=> ( ( member7226740684066999833at_nat @ S @ A )
=> ( ! [T2: produc8199716216217303280at_nat] :
( ! [U4: produc8199716216217303280at_nat] :
( ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ T2 @ U4 ) @ R4 )
=> ( P @ U4 ) )
=> ( P @ T2 ) )
=> ( P @ S ) ) ) ) ).
% SN_on_induct
thf(fact_1110_SN__on__induct,axiom,
! [R4: set_Pr4386577575007340137rm_a_b,A: set_term_a_b,S: term_a_b,P: term_a_b > $o] :
( ( abstra4720023341729745482rm_a_b @ R4 @ A )
=> ( ( member_term_a_b @ S @ A )
=> ( ! [T2: term_a_b] :
( ! [U4: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T2 @ U4 ) @ R4 )
=> ( P @ U4 ) )
=> ( P @ T2 ) )
=> ( P @ S ) ) ) ) ).
% SN_on_induct
thf(fact_1111_SN__on__subset2,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat,R: set_Pr1811044260758604347_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A @ B )
=> ( ( abstra8653715922312955827_a_nat @ R @ B )
=> ( abstra8653715922312955827_a_nat @ R @ A ) ) ) ).
% SN_on_subset2
thf(fact_1112_SN__on__subset2,axiom,
! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b,R: set_Pr2972776593051762503rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A @ B )
=> ( ( abstra2398554102055911763rm_a_b @ R @ B )
=> ( abstra2398554102055911763rm_a_b @ R @ A ) ) ) ).
% SN_on_subset2
thf(fact_1113_SN__on__subset2,axiom,
! [A: set_Pr9093778441882193744at_nat,B: set_Pr9093778441882193744at_nat,R: set_Pr7047708467701517959at_nat] :
( ( ord_le3678578370064672496at_nat @ A @ B )
=> ( ( abstra7460355483315988282at_nat @ R @ B )
=> ( abstra7460355483315988282at_nat @ R @ A ) ) ) ).
% SN_on_subset2
thf(fact_1114_SN__on__subset1,axiom,
! [R: set_Pr4386577575007340137rm_a_b,A: set_term_a_b,S: set_Pr4386577575007340137rm_a_b] :
( ( abstra4720023341729745482rm_a_b @ R @ A )
=> ( ( ord_le118470702582115849rm_a_b @ S @ R )
=> ( abstra4720023341729745482rm_a_b @ S @ A ) ) ) ).
% SN_on_subset1
thf(fact_1115_SN__subset,axiom,
! [R4: set_Pr4386577575007340137rm_a_b,R7: set_Pr4386577575007340137rm_a_b] :
( ( abstra4720023341729745482rm_a_b @ R4 @ top_top_set_term_a_b )
=> ( ( ord_le118470702582115849rm_a_b @ R7 @ R4 )
=> ( abstra4720023341729745482rm_a_b @ R7 @ top_top_set_term_a_b ) ) ) ).
% SN_subset
thf(fact_1116_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_1117_mono__Field,axiom,
! [R: set_Pr2972776593051762503rm_a_b,S: set_Pr2972776593051762503rm_a_b] :
( ( ord_le2593958535671617255rm_a_b @ R @ S )
=> ( ord_le118470702582115849rm_a_b @ ( field_6884932134483023318rm_a_b @ R ) @ ( field_6884932134483023318rm_a_b @ S ) ) ) ).
% mono_Field
thf(fact_1118_mono__Field,axiom,
! [R: set_Pr7047708467701517959at_nat,S: set_Pr7047708467701517959at_nat] :
( ( ord_le1281664332108154407at_nat @ R @ S )
=> ( ord_le3678578370064672496at_nat @ ( field_718184728585074237at_nat @ R ) @ ( field_718184728585074237at_nat @ S ) ) ) ).
% mono_Field
thf(fact_1119_mono__Field,axiom,
! [R: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ R @ S )
=> ( ord_le2705286416250468010rm_a_b @ ( field_term_a_b @ R ) @ ( field_term_a_b @ S ) ) ) ).
% mono_Field
thf(fact_1120_SN__on__all__reducts__SN__on__conv,axiom,
! [R: set_Pr4386577575007340137rm_a_b,A2: term_a_b] :
( ( abstra4720023341729745482rm_a_b @ R @ ( insert_term_a_b @ A2 @ bot_bot_set_term_a_b ) )
= ( ! [B5: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B5 ) @ R )
=> ( abstra4720023341729745482rm_a_b @ R @ ( insert_term_a_b @ B5 @ bot_bot_set_term_a_b ) ) ) ) ) ).
% SN_on_all_reducts_SN_on_conv
thf(fact_1121_lower__set__imp__not__SN__on,axiom,
! [S: produc357393685978478089rm_a_b,X6: set_Pr4386577575007340137rm_a_b,R4: set_Pr2972776593051762503rm_a_b] :
( ( member5869715511025134514rm_a_b @ S @ X6 )
=> ( ! [X: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X @ X6 )
=> ? [Xa2: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ Xa2 @ X6 )
& ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Xa2 ) @ R4 ) ) )
=> ~ ( abstra2398554102055911763rm_a_b @ R4 @ ( insert7009541432154983385rm_a_b @ S @ bot_bo197521221353338581rm_a_b ) ) ) ) ).
% lower_set_imp_not_SN_on
thf(fact_1122_lower__set__imp__not__SN__on,axiom,
! [S: product_prod_a_nat,X6: set_Pr4934435412358123699_a_nat,R4: set_Pr1811044260758604347_a_nat] :
( ( member5724188588386418708_a_nat @ S @ X6 )
=> ( ! [X: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ X @ X6 )
=> ? [Xa2: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ Xa2 @ X6 )
& ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ X @ Xa2 ) @ R4 ) ) )
=> ~ ( abstra8653715922312955827_a_nat @ R4 @ ( insert8054603423593749677_a_nat @ S @ bot_bo9049108969261143879_a_nat ) ) ) ) ).
% lower_set_imp_not_SN_on
thf(fact_1123_lower__set__imp__not__SN__on,axiom,
! [S: produc8199716216217303280at_nat,X6: set_Pr9093778441882193744at_nat,R4: set_Pr7047708467701517959at_nat] :
( ( member7226740684066999833at_nat @ S @ X6 )
=> ( ! [X: produc8199716216217303280at_nat] :
( ( member7226740684066999833at_nat @ X @ X6 )
=> ? [Xa2: produc8199716216217303280at_nat] :
( ( member7226740684066999833at_nat @ Xa2 @ X6 )
& ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ X @ Xa2 ) @ R4 ) ) )
=> ~ ( abstra7460355483315988282at_nat @ R4 @ ( insert891772969674204864at_nat @ S @ bot_bo8000532843592726972at_nat ) ) ) ) ).
% lower_set_imp_not_SN_on
thf(fact_1124_lower__set__imp__not__SN__on,axiom,
! [S: term_a_b,X6: set_term_a_b,R4: set_Pr4386577575007340137rm_a_b] :
( ( member_term_a_b @ S @ X6 )
=> ( ! [X: term_a_b] :
( ( member_term_a_b @ X @ X6 )
=> ? [Xa2: term_a_b] :
( ( member_term_a_b @ Xa2 @ X6 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Xa2 ) @ R4 ) ) )
=> ~ ( abstra4720023341729745482rm_a_b @ R4 @ ( insert_term_a_b @ S @ bot_bot_set_term_a_b ) ) ) ) ).
% lower_set_imp_not_SN_on
thf(fact_1125_SN__on__induct__acc__style,axiom,
! [R4: set_Pr4386577575007340137rm_a_b,A2: term_a_b,P: term_a_b > $o] :
( ( abstra4720023341729745482rm_a_b @ R4 @ ( insert_term_a_b @ A2 @ bot_bot_set_term_a_b ) )
=> ( ! [X: term_a_b] :
( ( abstra4720023341729745482rm_a_b @ R4 @ ( insert_term_a_b @ X @ bot_bot_set_term_a_b ) )
=> ( ! [Y6: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y6 ) @ R4 )
=> ( P @ Y6 ) )
=> ( P @ X ) ) )
=> ( P @ A2 ) ) ) ).
% SN_on_induct_acc_style
thf(fact_1126_step__preserves__SN__on,axiom,
! [A2: term_a_b,B2: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ R )
=> ( ( abstra4720023341729745482rm_a_b @ R @ ( insert_term_a_b @ A2 @ bot_bot_set_term_a_b ) )
=> ( abstra4720023341729745482rm_a_b @ R @ ( insert_term_a_b @ B2 @ bot_bot_set_term_a_b ) ) ) ) ).
% step_preserves_SN_on
thf(fact_1127_SN__on__imp__on__minimal,axiom,
! [R: set_Pr2972776593051762503rm_a_b,X3: produc357393685978478089rm_a_b] :
( ( abstra2398554102055911763rm_a_b @ R @ ( insert7009541432154983385rm_a_b @ X3 @ bot_bo197521221353338581rm_a_b ) )
=> ! [Q2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ X3 @ Q2 )
=> ? [X: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X @ Q2 )
& ! [Y6: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y6 ) @ R )
=> ~ ( member5869715511025134514rm_a_b @ Y6 @ Q2 ) ) ) ) ) ).
% SN_on_imp_on_minimal
thf(fact_1128_SN__on__imp__on__minimal,axiom,
! [R: set_Pr1811044260758604347_a_nat,X3: product_prod_a_nat] :
( ( abstra8653715922312955827_a_nat @ R @ ( insert8054603423593749677_a_nat @ X3 @ bot_bo9049108969261143879_a_nat ) )
=> ! [Q2: set_Pr4934435412358123699_a_nat] :
( ( member5724188588386418708_a_nat @ X3 @ Q2 )
=> ? [X: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ X @ Q2 )
& ! [Y6: product_prod_a_nat] :
( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ X @ Y6 ) @ R )
=> ~ ( member5724188588386418708_a_nat @ Y6 @ Q2 ) ) ) ) ) ).
% SN_on_imp_on_minimal
thf(fact_1129_SN__on__imp__on__minimal,axiom,
! [R: set_Pr7047708467701517959at_nat,X3: produc8199716216217303280at_nat] :
( ( abstra7460355483315988282at_nat @ R @ ( insert891772969674204864at_nat @ X3 @ bot_bo8000532843592726972at_nat ) )
=> ! [Q2: set_Pr9093778441882193744at_nat] :
( ( member7226740684066999833at_nat @ X3 @ Q2 )
=> ? [X: produc8199716216217303280at_nat] :
( ( member7226740684066999833at_nat @ X @ Q2 )
& ! [Y6: produc8199716216217303280at_nat] :
( ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ X @ Y6 ) @ R )
=> ~ ( member7226740684066999833at_nat @ Y6 @ Q2 ) ) ) ) ) ).
% SN_on_imp_on_minimal
thf(fact_1130_SN__on__imp__on__minimal,axiom,
! [R: set_Pr4386577575007340137rm_a_b,X3: term_a_b] :
( ( abstra4720023341729745482rm_a_b @ R @ ( insert_term_a_b @ X3 @ bot_bot_set_term_a_b ) )
=> ! [Q2: set_term_a_b] :
( ( member_term_a_b @ X3 @ Q2 )
=> ? [X: term_a_b] :
( ( member_term_a_b @ X @ Q2 )
& ! [Y6: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y6 ) @ R )
=> ~ ( member_term_a_b @ Y6 @ Q2 ) ) ) ) ) ).
% SN_on_imp_on_minimal
thf(fact_1131_step__reflects__SN__on,axiom,
! [A2: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ! [B6: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B6 ) @ R )
=> ( abstra4720023341729745482rm_a_b @ R @ ( insert_term_a_b @ B6 @ bot_bot_set_term_a_b ) ) )
=> ( abstra4720023341729745482rm_a_b @ R @ ( insert_term_a_b @ A2 @ bot_bot_set_term_a_b ) ) ) ).
% step_reflects_SN_on
thf(fact_1132_SN__on__induct_H,axiom,
! [R4: set_Pr4386577575007340137rm_a_b,S: term_a_b,P: term_a_b > $o] :
( ( abstra4720023341729745482rm_a_b @ R4 @ ( insert_term_a_b @ S @ bot_bot_set_term_a_b ) )
=> ( ! [T2: term_a_b] :
( ! [U4: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T2 @ U4 ) @ R4 )
=> ( P @ U4 ) )
=> ( P @ T2 ) )
=> ( P @ S ) ) ) ).
% SN_on_induct'
thf(fact_1133_SN__on__O__push,axiom,
! [S5: set_Pr4386577575007340137rm_a_b,R4: set_Pr4386577575007340137rm_a_b,X6: set_term_a_b] :
( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ S5 @ R4 ) @ ( relcom370159955682700863rm_a_b @ R4 @ ( transi7742714808557438673rm_a_b @ S5 ) ) )
=> ( ( abstra4720023341729745482rm_a_b @ R4 @ X6 )
=> ( abstra4720023341729745482rm_a_b @ ( relcom370159955682700863rm_a_b @ R4 @ ( transi7742714808557438673rm_a_b @ S5 ) ) @ X6 ) ) ) ).
% SN_on_O_push
thf(fact_1134_NF__imp__SN__on,axiom,
! [A2: produc357393685978478089rm_a_b,R4: set_Pr2972776593051762503rm_a_b] :
( ( member5869715511025134514rm_a_b @ A2 @ ( abstra1902783451776770197rm_a_b @ R4 ) )
=> ( abstra2398554102055911763rm_a_b @ R4 @ ( insert7009541432154983385rm_a_b @ A2 @ bot_bo197521221353338581rm_a_b ) ) ) ).
% NF_imp_SN_on
thf(fact_1135_NF__imp__SN__on,axiom,
! [A2: product_prod_a_nat,R4: set_Pr1811044260758604347_a_nat] :
( ( member5724188588386418708_a_nat @ A2 @ ( abstra2685462378305987313_a_nat @ R4 ) )
=> ( abstra8653715922312955827_a_nat @ R4 @ ( insert8054603423593749677_a_nat @ A2 @ bot_bo9049108969261143879_a_nat ) ) ) ).
% NF_imp_SN_on
thf(fact_1136_NF__imp__SN__on,axiom,
! [A2: produc8199716216217303280at_nat,R4: set_Pr7047708467701517959at_nat] :
( ( member7226740684066999833at_nat @ A2 @ ( abstra8406325002171223420at_nat @ R4 ) )
=> ( abstra7460355483315988282at_nat @ R4 @ ( insert891772969674204864at_nat @ A2 @ bot_bo8000532843592726972at_nat ) ) ) ).
% NF_imp_SN_on
thf(fact_1137_steps__preserve__SN__on,axiom,
! [A2: term_a_b,B2: term_a_b,A: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( transi7742714808557438673rm_a_b @ A ) )
=> ( ( abstra4720023341729745482rm_a_b @ A @ ( insert_term_a_b @ A2 @ bot_bot_set_term_a_b ) )
=> ( abstra4720023341729745482rm_a_b @ A @ ( insert_term_a_b @ B2 @ bot_bot_set_term_a_b ) ) ) ) ).
% steps_preserve_SN_on
thf(fact_1138_steps__reflect__SN__on,axiom,
! [R: set_Pr4386577575007340137rm_a_b,B2: term_a_b,A2: term_a_b] :
( ~ ( abstra4720023341729745482rm_a_b @ R @ ( insert_term_a_b @ B2 @ bot_bot_set_term_a_b ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ~ ( abstra4720023341729745482rm_a_b @ R @ ( insert_term_a_b @ A2 @ bot_bot_set_term_a_b ) ) ) ) ).
% steps_reflect_SN_on
thf(fact_1139_compatible__SN,axiom,
! [S5: set_Pr4386577575007340137rm_a_b,NS: set_Pr4386577575007340137rm_a_b] :
( ( abstra4720023341729745482rm_a_b @ S5 @ top_top_set_term_a_b )
=> ( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ NS @ S5 ) @ S5 )
=> ( abstra4720023341729745482rm_a_b @ ( relcom370159955682700863rm_a_b @ S5 @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ S5 ) @ ( transi7742714808557438673rm_a_b @ NS ) ) ) @ top_top_set_term_a_b ) ) ) ).
% compatible_SN
thf(fact_1140_SN__on__imp__normalizability,axiom,
! [R: set_Pr4386577575007340137rm_a_b,A2: term_a_b] :
( ( abstra4720023341729745482rm_a_b @ R @ ( insert_term_a_b @ A2 @ bot_bot_set_term_a_b ) )
=> ? [B6: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B6 ) @ ( abstra2972651231392952684rm_a_b @ R ) ) ) ).
% SN_on_imp_normalizability
thf(fact_1141_SN__reaches__NF,axiom,
! [R: set_Pr2972776593051762503rm_a_b,X3: produc357393685978478089rm_a_b] :
( ( abstra2398554102055911763rm_a_b @ R @ ( insert7009541432154983385rm_a_b @ X3 @ bot_bo197521221353338581rm_a_b ) )
=> ? [Y: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X3 @ Y ) @ ( transi2615809358984392588rm_a_b @ R ) )
& ( member5869715511025134514rm_a_b @ Y @ ( abstra1902783451776770197rm_a_b @ R ) ) ) ) ).
% SN_reaches_NF
thf(fact_1142_SN__reaches__NF,axiom,
! [R: set_Pr1811044260758604347_a_nat,X3: product_prod_a_nat] :
( ( abstra8653715922312955827_a_nat @ R @ ( insert8054603423593749677_a_nat @ X3 @ bot_bo9049108969261143879_a_nat ) )
=> ? [Y: product_prod_a_nat] :
( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ X3 @ Y ) @ ( transi2726145917338391738_a_nat @ R ) )
& ( member5724188588386418708_a_nat @ Y @ ( abstra2685462378305987313_a_nat @ R ) ) ) ) ).
% SN_reaches_NF
thf(fact_1143_SN__reaches__NF,axiom,
! [R: set_Pr7047708467701517959at_nat,X3: produc8199716216217303280at_nat] :
( ( abstra7460355483315988282at_nat @ R @ ( insert891772969674204864at_nat @ X3 @ bot_bo8000532843592726972at_nat ) )
=> ? [Y: produc8199716216217303280at_nat] :
( ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ X3 @ Y ) @ ( transi3422749762518297843at_nat @ R ) )
& ( member7226740684066999833at_nat @ Y @ ( abstra8406325002171223420at_nat @ R ) ) ) ) ).
% SN_reaches_NF
thf(fact_1144_SN__reaches__NF,axiom,
! [R: set_Pr4386577575007340137rm_a_b,X3: term_a_b] :
( ( abstra4720023341729745482rm_a_b @ R @ ( insert_term_a_b @ X3 @ bot_bot_set_term_a_b ) )
=> ? [Y: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y ) @ ( transi7742714808557438673rm_a_b @ R ) )
& ( member_term_a_b @ Y @ ( abstract_NF_term_a_b @ R ) ) ) ) ).
% SN_reaches_NF
thf(fact_1145_qc__SN__relto__iff,axiom,
! [R: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ R @ S ) @ ( relcom370159955682700863rm_a_b @ S @ ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ S @ R ) ) ) )
=> ( ( abstra4720023341729745482rm_a_b @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ R ) @ ( relcom370159955682700863rm_a_b @ S @ ( transi7742714808557438673rm_a_b @ R ) ) ) @ top_top_set_term_a_b )
= ( abstra4720023341729745482rm_a_b @ S @ top_top_set_term_a_b ) ) ) ).
% qc_SN_relto_iff
thf(fact_1146_compatible__SN_H,axiom,
! [NS: set_Pr4386577575007340137rm_a_b,S5: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ NS @ S5 ) @ S5 )
=> ( ( abstra4720023341729745482rm_a_b @ S5 @ top_top_set_term_a_b )
=> ( abstra4720023341729745482rm_a_b @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ NS @ S5 ) ) @ ( relcom370159955682700863rm_a_b @ S5 @ ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ NS @ S5 ) ) ) ) @ top_top_set_term_a_b ) ) ) ).
% compatible_SN'
thf(fact_1147_ideriv__SN,axiom,
! [S5: set_Pr4386577575007340137rm_a_b,NS: set_Pr4386577575007340137rm_a_b,R4: set_Pr4386577575007340137rm_a_b,As2: nat > term_a_b] :
( ( abstra4720023341729745482rm_a_b @ S5 @ top_top_set_term_a_b )
=> ( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ NS @ S5 ) @ S5 )
=> ( ( ord_le118470702582115849rm_a_b @ R4 @ ( sup_su6776935440552674877rm_a_b @ NS @ S5 ) )
=> ~ ( abstra5386884192082944997rm_a_b @ ( inf_in1693821330460682071rm_a_b @ S5 @ R4 ) @ ( minus_5192120951422937424rm_a_b @ R4 @ S5 ) @ As2 ) ) ) ) ).
% ideriv_SN
thf(fact_1148_the__NF,axiom,
! [A: set_Pr4386577575007340137rm_a_b,A2: term_a_b] :
( ( abstra4720023341729745482rm_a_b @ A @ top_top_set_term_a_b )
=> ( ( abstra8448919418672941150rm_a_b @ A @ top_top_set_term_a_b )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ ( abstra2242782261322123075rm_a_b @ A @ A2 ) ) @ ( abstra2972651231392952684rm_a_b @ A ) ) ) ) ).
% the_NF
thf(fact_1149_ideriv__mono,axiom,
! [R4: set_Pr4386577575007340137rm_a_b,R7: set_Pr4386577575007340137rm_a_b,S5: set_Pr4386577575007340137rm_a_b,S6: set_Pr4386577575007340137rm_a_b,As2: nat > term_a_b] :
( ( ord_le118470702582115849rm_a_b @ R4 @ R7 )
=> ( ( ord_le118470702582115849rm_a_b @ S5 @ S6 )
=> ( ( abstra5386884192082944997rm_a_b @ R4 @ S5 @ As2 )
=> ( abstra5386884192082944997rm_a_b @ R7 @ S6 @ As2 ) ) ) ) ).
% ideriv_mono
thf(fact_1150_the__NF__step,axiom,
! [A: set_Pr4386577575007340137rm_a_b,A2: term_a_b,B2: term_a_b] :
( ( abstra4720023341729745482rm_a_b @ A @ top_top_set_term_a_b )
=> ( ( abstra8448919418672941150rm_a_b @ A @ top_top_set_term_a_b )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ A )
=> ( ( abstra2242782261322123075rm_a_b @ A @ A2 )
= ( abstra2242782261322123075rm_a_b @ A @ B2 ) ) ) ) ) ).
% the_NF_step
thf(fact_1151_the__NF__NF,axiom,
! [A: set_Pr2972776593051762503rm_a_b,A2: produc357393685978478089rm_a_b] :
( ( abstra2398554102055911763rm_a_b @ A @ top_to1314267278846557113rm_a_b )
=> ( ( abstra7359694574113289279rm_a_b @ A @ top_to1314267278846557113rm_a_b )
=> ( member5869715511025134514rm_a_b @ ( abstra8972549633426078106rm_a_b @ A @ A2 ) @ ( abstra1902783451776770197rm_a_b @ A ) ) ) ) ).
% the_NF_NF
thf(fact_1152_the__NF__NF,axiom,
! [A: set_Pr1811044260758604347_a_nat,A2: product_prod_a_nat] :
( ( abstra8653715922312955827_a_nat @ A @ top_to3353692345378799459_a_nat )
=> ( ( abstra7964079670522473415_a_nat @ A @ top_to3353692345378799459_a_nat )
=> ( member5724188588386418708_a_nat @ ( abstra6900011479857116204_a_nat @ A @ A2 ) @ ( abstra2685462378305987313_a_nat @ A ) ) ) ) ).
% the_NF_NF
thf(fact_1153_the__NF__NF,axiom,
! [A: set_Pr7047708467701517959at_nat,A2: produc8199716216217303280at_nat] :
( ( abstra7460355483315988282at_nat @ A @ top_to7836785446628990112at_nat )
=> ( ( abstra8052594214376182822at_nat @ A @ top_to7836785446628990112at_nat )
=> ( member7226740684066999833at_nat @ ( abstra7000268279564191745at_nat @ A @ A2 ) @ ( abstra8406325002171223420at_nat @ A ) ) ) ) ).
% the_NF_NF
thf(fact_1154_the__NF__steps,axiom,
! [A: set_Pr4386577575007340137rm_a_b,A2: term_a_b,B2: term_a_b] :
( ( abstra4720023341729745482rm_a_b @ A @ top_top_set_term_a_b )
=> ( ( abstra8448919418672941150rm_a_b @ A @ top_top_set_term_a_b )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( transi7742714808557438673rm_a_b @ A ) )
=> ( ( abstra2242782261322123075rm_a_b @ A @ A2 )
= ( abstra2242782261322123075rm_a_b @ A @ B2 ) ) ) ) ) ).
% the_NF_steps
thf(fact_1155_the__NF__conv,axiom,
! [A: set_Pr4386577575007340137rm_a_b,A2: term_a_b,B2: term_a_b] :
( ( abstra4720023341729745482rm_a_b @ A @ top_top_set_term_a_b )
=> ( ( abstra8448919418672941150rm_a_b @ A @ top_top_set_term_a_b )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( abstra1130568886744591910rm_a_b @ A ) )
=> ( ( abstra2242782261322123075rm_a_b @ A @ A2 )
= ( abstra2242782261322123075rm_a_b @ A @ B2 ) ) ) ) ) ).
% the_NF_conv
thf(fact_1156_cofinal__def,axiom,
( bNF_Ca4789820777293777539rm_a_b
= ( ^ [A4: set_term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
! [X4: term_a_b] :
( ( member_term_a_b @ X4 @ ( field_term_a_b @ R2 ) )
=> ? [Y5: term_a_b] :
( ( member_term_a_b @ Y5 @ A4 )
& ( X4 != Y5 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X4 @ Y5 ) @ R2 ) ) ) ) ) ).
% cofinal_def
thf(fact_1157_member__remove,axiom,
! [X3: produc357393685978478089rm_a_b,Y3: produc357393685978478089rm_a_b,A: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ X3 @ ( remove8523906646656265582rm_a_b @ Y3 @ A ) )
= ( ( member5869715511025134514rm_a_b @ X3 @ A )
& ( X3 != Y3 ) ) ) ).
% member_remove
thf(fact_1158_member__remove,axiom,
! [X3: product_prod_a_nat,Y3: product_prod_a_nat,A: set_Pr4934435412358123699_a_nat] :
( ( member5724188588386418708_a_nat @ X3 @ ( remove5305571144230967640_a_nat @ Y3 @ A ) )
= ( ( member5724188588386418708_a_nat @ X3 @ A )
& ( X3 != Y3 ) ) ) ).
% member_remove
thf(fact_1159_member__remove,axiom,
! [X3: produc8199716216217303280at_nat,Y3: produc8199716216217303280at_nat,A: set_Pr9093778441882193744at_nat] :
( ( member7226740684066999833at_nat @ X3 @ ( remove8285889648388334037at_nat @ Y3 @ A ) )
= ( ( member7226740684066999833at_nat @ X3 @ A )
& ( X3 != Y3 ) ) ) ).
% member_remove
thf(fact_1160_acyclic__insert,axiom,
! [Y3: term_a_b,X3: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( transi5314701259734769157rm_a_b @ ( insert7009541432154983385rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ X3 ) @ R ) )
= ( ( transi5314701259734769157rm_a_b @ R )
& ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ ( transi7742714808557438673rm_a_b @ R ) ) ) ) ).
% acyclic_insert
thf(fact_1161_wo__rel_Ocases__Total3,axiom,
! [R: set_Pr4386577575007340137rm_a_b,A2: term_a_b,B2: term_a_b,Phi: term_a_b > term_a_b > $o] :
( ( bNF_We251836928773027112rm_a_b @ R )
=> ( ( ord_le2705286416250468010rm_a_b @ ( insert_term_a_b @ A2 @ ( insert_term_a_b @ B2 @ bot_bot_set_term_a_b ) ) @ ( field_term_a_b @ R ) )
=> ( ( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( minus_5192120951422937424rm_a_b @ R @ id_term_a_b ) )
| ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ A2 ) @ ( minus_5192120951422937424rm_a_b @ R @ id_term_a_b ) ) )
=> ( Phi @ A2 @ B2 ) )
=> ( ( ( A2 = B2 )
=> ( Phi @ A2 @ B2 ) )
=> ( Phi @ A2 @ B2 ) ) ) ) ) ).
% wo_rel.cases_Total3
thf(fact_1162_wo__rel_Ocases__Total3,axiom,
! [R: set_Pr1811044260758604347_a_nat,A2: product_prod_a_nat,B2: product_prod_a_nat,Phi: product_prod_a_nat > product_prod_a_nat > $o] :
( ( bNF_We4298404315170990993_a_nat @ R )
=> ( ( ord_le8666007276011122963_a_nat @ ( insert8054603423593749677_a_nat @ A2 @ ( insert8054603423593749677_a_nat @ B2 @ bot_bo9049108969261143879_a_nat ) ) @ ( field_8954927560578634480_a_nat @ R ) )
=> ( ( ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A2 @ B2 ) @ ( minus_1341117088723627554_a_nat @ R @ id_Pro5207055338379199009_a_nat ) )
| ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ B2 @ A2 ) @ ( minus_1341117088723627554_a_nat @ R @ id_Pro5207055338379199009_a_nat ) ) )
=> ( Phi @ A2 @ B2 ) )
=> ( ( ( A2 = B2 )
=> ( Phi @ A2 @ B2 ) )
=> ( Phi @ A2 @ B2 ) ) ) ) ) ).
% wo_rel.cases_Total3
thf(fact_1163_wo__rel_Ocases__Total3,axiom,
! [R: set_Pr2972776593051762503rm_a_b,A2: produc357393685978478089rm_a_b,B2: produc357393685978478089rm_a_b,Phi: produc357393685978478089rm_a_b > produc357393685978478089rm_a_b > $o] :
( ( bNF_We1688910813892497397rm_a_b @ R )
=> ( ( ord_le118470702582115849rm_a_b @ ( insert7009541432154983385rm_a_b @ A2 @ ( insert7009541432154983385rm_a_b @ B2 @ bot_bo197521221353338581rm_a_b ) ) @ ( field_6884932134483023318rm_a_b @ R ) )
=> ( ( ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A2 @ B2 ) @ ( minus_45395518312058030rm_a_b @ R @ id_Pro3488709451141449061rm_a_b ) )
| ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B2 @ A2 ) @ ( minus_45395518312058030rm_a_b @ R @ id_Pro3488709451141449061rm_a_b ) ) )
=> ( Phi @ A2 @ B2 ) )
=> ( ( ( A2 = B2 )
=> ( Phi @ A2 @ B2 ) )
=> ( Phi @ A2 @ B2 ) ) ) ) ) ).
% wo_rel.cases_Total3
thf(fact_1164_wo__rel_Ocases__Total3,axiom,
! [R: set_Pr7047708467701517959at_nat,A2: produc8199716216217303280at_nat,B2: produc8199716216217303280at_nat,Phi: produc8199716216217303280at_nat > produc8199716216217303280at_nat > $o] :
( ( bNF_We4809617423727447260at_nat @ R )
=> ( ( ord_le3678578370064672496at_nat @ ( insert891772969674204864at_nat @ A2 @ ( insert891772969674204864at_nat @ B2 @ bot_bo8000532843592726972at_nat ) ) @ ( field_718184728585074237at_nat @ R ) )
=> ( ( ( ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ A2 @ B2 ) @ ( minus_7913753035872161006at_nat @ R @ id_Pro414489527043525196at_nat ) )
| ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ B2 @ A2 ) @ ( minus_7913753035872161006at_nat @ R @ id_Pro414489527043525196at_nat ) ) )
=> ( Phi @ A2 @ B2 ) )
=> ( ( ( A2 = B2 )
=> ( Phi @ A2 @ B2 ) )
=> ( Phi @ A2 @ B2 ) ) ) ) ) ).
% wo_rel.cases_Total3
thf(fact_1165_refl__on__singleton,axiom,
! [X3: term_a_b] : ( refl_on_term_a_b @ ( insert_term_a_b @ X3 @ bot_bot_set_term_a_b ) @ ( insert7009541432154983385rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ X3 ) @ bot_bo197521221353338581rm_a_b ) ) ).
% refl_on_singleton
thf(fact_1166_wo__rel_OTOTALS,axiom,
! [R: set_Pr4386577575007340137rm_a_b] :
( ( bNF_We251836928773027112rm_a_b @ R )
=> ! [X7: term_a_b] :
( ( member_term_a_b @ X7 @ ( field_term_a_b @ R ) )
=> ! [Xa2: term_a_b] :
( ( member_term_a_b @ Xa2 @ ( field_term_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X7 @ Xa2 ) @ R )
| ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Xa2 @ X7 ) @ R ) ) ) ) ) ).
% wo_rel.TOTALS
thf(fact_1167_wo__rel_Oin__notinI,axiom,
! [R: set_Pr2972776593051762503rm_a_b,J2: produc357393685978478089rm_a_b,I2: produc357393685978478089rm_a_b] :
( ( bNF_We1688910813892497397rm_a_b @ R )
=> ( ( ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ J2 @ I2 ) @ R )
| ( J2 = I2 ) )
=> ( ( member5869715511025134514rm_a_b @ I2 @ ( field_6884932134483023318rm_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ J2 @ ( field_6884932134483023318rm_a_b @ R ) )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ I2 @ J2 ) @ R ) ) ) ) ) ).
% wo_rel.in_notinI
thf(fact_1168_wo__rel_Oin__notinI,axiom,
! [R: set_Pr1811044260758604347_a_nat,J2: product_prod_a_nat,I2: product_prod_a_nat] :
( ( bNF_We4298404315170990993_a_nat @ R )
=> ( ( ~ ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ J2 @ I2 ) @ R )
| ( J2 = I2 ) )
=> ( ( member5724188588386418708_a_nat @ I2 @ ( field_8954927560578634480_a_nat @ R ) )
=> ( ( member5724188588386418708_a_nat @ J2 @ ( field_8954927560578634480_a_nat @ R ) )
=> ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ I2 @ J2 ) @ R ) ) ) ) ) ).
% wo_rel.in_notinI
thf(fact_1169_wo__rel_Oin__notinI,axiom,
! [R: set_Pr7047708467701517959at_nat,J2: produc8199716216217303280at_nat,I2: produc8199716216217303280at_nat] :
( ( bNF_We4809617423727447260at_nat @ R )
=> ( ( ~ ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ J2 @ I2 ) @ R )
| ( J2 = I2 ) )
=> ( ( member7226740684066999833at_nat @ I2 @ ( field_718184728585074237at_nat @ R ) )
=> ( ( member7226740684066999833at_nat @ J2 @ ( field_718184728585074237at_nat @ R ) )
=> ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ I2 @ J2 ) @ R ) ) ) ) ) ).
% wo_rel.in_notinI
thf(fact_1170_wo__rel_Oin__notinI,axiom,
! [R: set_Pr4386577575007340137rm_a_b,J2: term_a_b,I2: term_a_b] :
( ( bNF_We251836928773027112rm_a_b @ R )
=> ( ( ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ J2 @ I2 ) @ R )
| ( J2 = I2 ) )
=> ( ( member_term_a_b @ I2 @ ( field_term_a_b @ R ) )
=> ( ( member_term_a_b @ J2 @ ( field_term_a_b @ R ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ I2 @ J2 ) @ R ) ) ) ) ) ).
% wo_rel.in_notinI
thf(fact_1171_refl__onD2,axiom,
! [A: set_Pr4386577575007340137rm_a_b,R: set_Pr2972776593051762503rm_a_b,X3: produc357393685978478089rm_a_b,Y3: produc357393685978478089rm_a_b] :
( ( refl_o5344404488616963464rm_a_b @ A @ R )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X3 @ Y3 ) @ R )
=> ( member5869715511025134514rm_a_b @ Y3 @ A ) ) ) ).
% refl_onD2
thf(fact_1172_refl__onD2,axiom,
! [A: set_Pr4934435412358123699_a_nat,R: set_Pr1811044260758604347_a_nat,X3: product_prod_a_nat,Y3: product_prod_a_nat] :
( ( refl_o8237544196271505854_a_nat @ A @ R )
=> ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ X3 @ Y3 ) @ R )
=> ( member5724188588386418708_a_nat @ Y3 @ A ) ) ) ).
% refl_onD2
thf(fact_1173_refl__onD2,axiom,
! [A: set_Pr9093778441882193744at_nat,R: set_Pr7047708467701517959at_nat,X3: produc8199716216217303280at_nat,Y3: produc8199716216217303280at_nat] :
( ( refl_o7598126563300187375at_nat @ A @ R )
=> ( ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ X3 @ Y3 ) @ R )
=> ( member7226740684066999833at_nat @ Y3 @ A ) ) ) ).
% refl_onD2
thf(fact_1174_refl__onD2,axiom,
! [A: set_term_a_b,R: set_Pr4386577575007340137rm_a_b,X3: term_a_b,Y3: term_a_b] :
( ( refl_on_term_a_b @ A @ R )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ R )
=> ( member_term_a_b @ Y3 @ A ) ) ) ).
% refl_onD2
thf(fact_1175_refl__onD1,axiom,
! [A: set_Pr4386577575007340137rm_a_b,R: set_Pr2972776593051762503rm_a_b,X3: produc357393685978478089rm_a_b,Y3: produc357393685978478089rm_a_b] :
( ( refl_o5344404488616963464rm_a_b @ A @ R )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X3 @ Y3 ) @ R )
=> ( member5869715511025134514rm_a_b @ X3 @ A ) ) ) ).
% refl_onD1
thf(fact_1176_refl__onD1,axiom,
! [A: set_Pr4934435412358123699_a_nat,R: set_Pr1811044260758604347_a_nat,X3: product_prod_a_nat,Y3: product_prod_a_nat] :
( ( refl_o8237544196271505854_a_nat @ A @ R )
=> ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ X3 @ Y3 ) @ R )
=> ( member5724188588386418708_a_nat @ X3 @ A ) ) ) ).
% refl_onD1
thf(fact_1177_refl__onD1,axiom,
! [A: set_Pr9093778441882193744at_nat,R: set_Pr7047708467701517959at_nat,X3: produc8199716216217303280at_nat,Y3: produc8199716216217303280at_nat] :
( ( refl_o7598126563300187375at_nat @ A @ R )
=> ( ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ X3 @ Y3 ) @ R )
=> ( member7226740684066999833at_nat @ X3 @ A ) ) ) ).
% refl_onD1
thf(fact_1178_refl__onD1,axiom,
! [A: set_term_a_b,R: set_Pr4386577575007340137rm_a_b,X3: term_a_b,Y3: term_a_b] :
( ( refl_on_term_a_b @ A @ R )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ R )
=> ( member_term_a_b @ X3 @ A ) ) ) ).
% refl_onD1
thf(fact_1179_refl__onD,axiom,
! [A: set_Pr4386577575007340137rm_a_b,R: set_Pr2972776593051762503rm_a_b,A2: produc357393685978478089rm_a_b] :
( ( refl_o5344404488616963464rm_a_b @ A @ R )
=> ( ( member5869715511025134514rm_a_b @ A2 @ A )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A2 @ A2 ) @ R ) ) ) ).
% refl_onD
thf(fact_1180_refl__onD,axiom,
! [A: set_Pr4934435412358123699_a_nat,R: set_Pr1811044260758604347_a_nat,A2: product_prod_a_nat] :
( ( refl_o8237544196271505854_a_nat @ A @ R )
=> ( ( member5724188588386418708_a_nat @ A2 @ A )
=> ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A2 @ A2 ) @ R ) ) ) ).
% refl_onD
thf(fact_1181_refl__onD,axiom,
! [A: set_Pr9093778441882193744at_nat,R: set_Pr7047708467701517959at_nat,A2: produc8199716216217303280at_nat] :
( ( refl_o7598126563300187375at_nat @ A @ R )
=> ( ( member7226740684066999833at_nat @ A2 @ A )
=> ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ A2 @ A2 ) @ R ) ) ) ).
% refl_onD
thf(fact_1182_refl__onD,axiom,
! [A: set_term_a_b,R: set_Pr4386577575007340137rm_a_b,A2: term_a_b] :
( ( refl_on_term_a_b @ A @ R )
=> ( ( member_term_a_b @ A2 @ A )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ A2 ) @ R ) ) ) ).
% refl_onD
thf(fact_1183_wo__rel_Owell__order__induct,axiom,
! [R: set_Pr4386577575007340137rm_a_b,P: term_a_b > $o,A2: term_a_b] :
( ( bNF_We251836928773027112rm_a_b @ R )
=> ( ! [X: term_a_b] :
( ! [Y6: term_a_b] :
( ( ( Y6 != X )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y6 @ X ) @ R ) )
=> ( P @ Y6 ) )
=> ( P @ X ) )
=> ( P @ A2 ) ) ) ).
% wo_rel.well_order_induct
thf(fact_1184_acyclic__subset,axiom,
! [S: set_Pr4386577575007340137rm_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( transi5314701259734769157rm_a_b @ S )
=> ( ( ord_le118470702582115849rm_a_b @ R @ S )
=> ( transi5314701259734769157rm_a_b @ R ) ) ) ).
% acyclic_subset
thf(fact_1185_reflI,axiom,
! [R: set_Pr4386577575007340137rm_a_b] :
( ! [X: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ X ) @ R )
=> ( refl_on_term_a_b @ top_top_set_term_a_b @ R ) ) ).
% reflI
thf(fact_1186_reflD,axiom,
! [R: set_Pr4386577575007340137rm_a_b,A2: term_a_b] :
( ( refl_on_term_a_b @ top_top_set_term_a_b @ R )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ A2 ) @ R ) ) ).
% reflD
thf(fact_1187_well__order__induct__imp,axiom,
! [R: set_Pr2972776593051762503rm_a_b,P: produc357393685978478089rm_a_b > $o,A2: produc357393685978478089rm_a_b] :
( ( bNF_We1688910813892497397rm_a_b @ R )
=> ( ! [X: produc357393685978478089rm_a_b] :
( ! [Y6: produc357393685978478089rm_a_b] :
( ( ( Y6 != X )
& ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y6 @ X ) @ R ) )
=> ( ( member5869715511025134514rm_a_b @ Y6 @ ( field_6884932134483023318rm_a_b @ R ) )
=> ( P @ Y6 ) ) )
=> ( ( member5869715511025134514rm_a_b @ X @ ( field_6884932134483023318rm_a_b @ R ) )
=> ( P @ X ) ) )
=> ( ( member5869715511025134514rm_a_b @ A2 @ ( field_6884932134483023318rm_a_b @ R ) )
=> ( P @ A2 ) ) ) ) ).
% well_order_induct_imp
thf(fact_1188_well__order__induct__imp,axiom,
! [R: set_Pr1811044260758604347_a_nat,P: product_prod_a_nat > $o,A2: product_prod_a_nat] :
( ( bNF_We4298404315170990993_a_nat @ R )
=> ( ! [X: product_prod_a_nat] :
( ! [Y6: product_prod_a_nat] :
( ( ( Y6 != X )
& ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ Y6 @ X ) @ R ) )
=> ( ( member5724188588386418708_a_nat @ Y6 @ ( field_8954927560578634480_a_nat @ R ) )
=> ( P @ Y6 ) ) )
=> ( ( member5724188588386418708_a_nat @ X @ ( field_8954927560578634480_a_nat @ R ) )
=> ( P @ X ) ) )
=> ( ( member5724188588386418708_a_nat @ A2 @ ( field_8954927560578634480_a_nat @ R ) )
=> ( P @ A2 ) ) ) ) ).
% well_order_induct_imp
thf(fact_1189_well__order__induct__imp,axiom,
! [R: set_Pr7047708467701517959at_nat,P: produc8199716216217303280at_nat > $o,A2: produc8199716216217303280at_nat] :
( ( bNF_We4809617423727447260at_nat @ R )
=> ( ! [X: produc8199716216217303280at_nat] :
( ! [Y6: produc8199716216217303280at_nat] :
( ( ( Y6 != X )
& ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ Y6 @ X ) @ R ) )
=> ( ( member7226740684066999833at_nat @ Y6 @ ( field_718184728585074237at_nat @ R ) )
=> ( P @ Y6 ) ) )
=> ( ( member7226740684066999833at_nat @ X @ ( field_718184728585074237at_nat @ R ) )
=> ( P @ X ) ) )
=> ( ( member7226740684066999833at_nat @ A2 @ ( field_718184728585074237at_nat @ R ) )
=> ( P @ A2 ) ) ) ) ).
% well_order_induct_imp
thf(fact_1190_well__order__induct__imp,axiom,
! [R: set_Pr4386577575007340137rm_a_b,P: term_a_b > $o,A2: term_a_b] :
( ( bNF_We251836928773027112rm_a_b @ R )
=> ( ! [X: term_a_b] :
( ! [Y6: term_a_b] :
( ( ( Y6 != X )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y6 @ X ) @ R ) )
=> ( ( member_term_a_b @ Y6 @ ( field_term_a_b @ R ) )
=> ( P @ Y6 ) ) )
=> ( ( member_term_a_b @ X @ ( field_term_a_b @ R ) )
=> ( P @ X ) ) )
=> ( ( member_term_a_b @ A2 @ ( field_term_a_b @ R ) )
=> ( P @ A2 ) ) ) ) ).
% well_order_induct_imp
thf(fact_1191_acyclicI,axiom,
! [R: set_Pr4386577575007340137rm_a_b] :
( ! [X: term_a_b] :
~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ X ) @ ( transi7922773638565587891rm_a_b @ R ) )
=> ( transi5314701259734769157rm_a_b @ R ) ) ).
% acyclicI
thf(fact_1192_acyclic__def,axiom,
( transi5314701259734769157rm_a_b
= ( ^ [R2: set_Pr4386577575007340137rm_a_b] :
! [X4: term_a_b] :
~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X4 @ X4 ) @ ( transi7922773638565587891rm_a_b @ R2 ) ) ) ) ).
% acyclic_def
thf(fact_1193_refl__O__iff,axiom,
! [A: set_Pr4386577575007340137rm_a_b] :
( ( refl_on_term_a_b @ top_top_set_term_a_b @ A )
= ( ord_le118470702582115849rm_a_b @ id_term_a_b @ A ) ) ).
% refl_O_iff
thf(fact_1194_wo__rel_Ocases__Total,axiom,
! [R: set_Pr4386577575007340137rm_a_b,A2: term_a_b,B2: term_a_b,Phi: term_a_b > term_a_b > $o] :
( ( bNF_We251836928773027112rm_a_b @ R )
=> ( ( ord_le2705286416250468010rm_a_b @ ( insert_term_a_b @ A2 @ ( insert_term_a_b @ B2 @ bot_bot_set_term_a_b ) ) @ ( field_term_a_b @ R ) )
=> ( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ R )
=> ( Phi @ A2 @ B2 ) )
=> ( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ A2 ) @ R )
=> ( Phi @ A2 @ B2 ) )
=> ( Phi @ A2 @ B2 ) ) ) ) ) ).
% wo_rel.cases_Total
thf(fact_1195_wo__rel_Ocases__Total,axiom,
! [R: set_Pr1811044260758604347_a_nat,A2: product_prod_a_nat,B2: product_prod_a_nat,Phi: product_prod_a_nat > product_prod_a_nat > $o] :
( ( bNF_We4298404315170990993_a_nat @ R )
=> ( ( ord_le8666007276011122963_a_nat @ ( insert8054603423593749677_a_nat @ A2 @ ( insert8054603423593749677_a_nat @ B2 @ bot_bo9049108969261143879_a_nat ) ) @ ( field_8954927560578634480_a_nat @ R ) )
=> ( ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A2 @ B2 ) @ R )
=> ( Phi @ A2 @ B2 ) )
=> ( ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ B2 @ A2 ) @ R )
=> ( Phi @ A2 @ B2 ) )
=> ( Phi @ A2 @ B2 ) ) ) ) ) ).
% wo_rel.cases_Total
thf(fact_1196_wo__rel_Ocases__Total,axiom,
! [R: set_Pr2972776593051762503rm_a_b,A2: produc357393685978478089rm_a_b,B2: produc357393685978478089rm_a_b,Phi: produc357393685978478089rm_a_b > produc357393685978478089rm_a_b > $o] :
( ( bNF_We1688910813892497397rm_a_b @ R )
=> ( ( ord_le118470702582115849rm_a_b @ ( insert7009541432154983385rm_a_b @ A2 @ ( insert7009541432154983385rm_a_b @ B2 @ bot_bo197521221353338581rm_a_b ) ) @ ( field_6884932134483023318rm_a_b @ R ) )
=> ( ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A2 @ B2 ) @ R )
=> ( Phi @ A2 @ B2 ) )
=> ( ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B2 @ A2 ) @ R )
=> ( Phi @ A2 @ B2 ) )
=> ( Phi @ A2 @ B2 ) ) ) ) ) ).
% wo_rel.cases_Total
thf(fact_1197_wo__rel_Ocases__Total,axiom,
! [R: set_Pr7047708467701517959at_nat,A2: produc8199716216217303280at_nat,B2: produc8199716216217303280at_nat,Phi: produc8199716216217303280at_nat > produc8199716216217303280at_nat > $o] :
( ( bNF_We4809617423727447260at_nat @ R )
=> ( ( ord_le3678578370064672496at_nat @ ( insert891772969674204864at_nat @ A2 @ ( insert891772969674204864at_nat @ B2 @ bot_bo8000532843592726972at_nat ) ) @ ( field_718184728585074237at_nat @ R ) )
=> ( ( ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ A2 @ B2 ) @ R )
=> ( Phi @ A2 @ B2 ) )
=> ( ( ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ B2 @ A2 ) @ R )
=> ( Phi @ A2 @ B2 ) )
=> ( Phi @ A2 @ B2 ) ) ) ) ) ).
% wo_rel.cases_Total
thf(fact_1198_refl__on__comp__subset,axiom,
! [A: set_term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( refl_on_term_a_b @ A @ R )
=> ( ord_le118470702582115849rm_a_b @ R @ ( relcom370159955682700863rm_a_b @ ( conver7608559976937845093rm_a_b @ R ) @ R ) ) ) ).
% refl_on_comp_subset
thf(fact_1199_wo__rel_OWell__order__isMinim__exists,axiom,
! [R: set_Pr1811044260758604347_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( bNF_We4298404315170990993_a_nat @ R )
=> ( ( ord_le8666007276011122963_a_nat @ B @ ( field_8954927560578634480_a_nat @ R ) )
=> ( ( B != bot_bo9049108969261143879_a_nat )
=> ? [X_1: product_prod_a_nat] : ( bNF_We4498669257762480967_a_nat @ R @ B @ X_1 ) ) ) ) ).
% wo_rel.Well_order_isMinim_exists
thf(fact_1200_wo__rel_OWell__order__isMinim__exists,axiom,
! [R: set_Pr2972776593051762503rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ( bNF_We1688910813892497397rm_a_b @ R )
=> ( ( ord_le118470702582115849rm_a_b @ B @ ( field_6884932134483023318rm_a_b @ R ) )
=> ( ( B != bot_bo197521221353338581rm_a_b )
=> ? [X_1: produc357393685978478089rm_a_b] : ( bNF_We8430659725849320639rm_a_b @ R @ B @ X_1 ) ) ) ) ).
% wo_rel.Well_order_isMinim_exists
thf(fact_1201_wo__rel_OWell__order__isMinim__exists,axiom,
! [R: set_Pr7047708467701517959at_nat,B: set_Pr9093778441882193744at_nat] :
( ( bNF_We4809617423727447260at_nat @ R )
=> ( ( ord_le3678578370064672496at_nat @ B @ ( field_718184728585074237at_nat @ R ) )
=> ( ( B != bot_bo8000532843592726972at_nat )
=> ? [X_1: produc8199716216217303280at_nat] : ( bNF_We1085401387027938470at_nat @ R @ B @ X_1 ) ) ) ) ).
% wo_rel.Well_order_isMinim_exists
thf(fact_1202_wo__rel_Ominim__in,axiom,
! [R: set_Pr1811044260758604347_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( bNF_We4298404315170990993_a_nat @ R )
=> ( ( ord_le8666007276011122963_a_nat @ B @ ( field_8954927560578634480_a_nat @ R ) )
=> ( ( B != bot_bo9049108969261143879_a_nat )
=> ( member5724188588386418708_a_nat @ ( bNF_We3895015349946260913_a_nat @ R @ B ) @ B ) ) ) ) ).
% wo_rel.minim_in
thf(fact_1203_wo__rel_Ominim__in,axiom,
! [R: set_Pr2972776593051762503rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ( bNF_We1688910813892497397rm_a_b @ R )
=> ( ( ord_le118470702582115849rm_a_b @ B @ ( field_6884932134483023318rm_a_b @ R ) )
=> ( ( B != bot_bo197521221353338581rm_a_b )
=> ( member5869715511025134514rm_a_b @ ( bNF_We1338740440619677141rm_a_b @ R @ B ) @ B ) ) ) ) ).
% wo_rel.minim_in
thf(fact_1204_wo__rel_Ominim__in,axiom,
! [R: set_Pr7047708467701517959at_nat,B: set_Pr9093778441882193744at_nat] :
( ( bNF_We4809617423727447260at_nat @ R )
=> ( ( ord_le3678578370064672496at_nat @ B @ ( field_718184728585074237at_nat @ R ) )
=> ( ( B != bot_bo8000532843592726972at_nat )
=> ( member7226740684066999833at_nat @ ( bNF_We4166897169537008316at_nat @ R @ B ) @ B ) ) ) ) ).
% wo_rel.minim_in
thf(fact_1205_wo__rel_Ominim__isMinim,axiom,
! [R: set_Pr1811044260758604347_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( bNF_We4298404315170990993_a_nat @ R )
=> ( ( ord_le8666007276011122963_a_nat @ B @ ( field_8954927560578634480_a_nat @ R ) )
=> ( ( B != bot_bo9049108969261143879_a_nat )
=> ( bNF_We4498669257762480967_a_nat @ R @ B @ ( bNF_We3895015349946260913_a_nat @ R @ B ) ) ) ) ) ).
% wo_rel.minim_isMinim
thf(fact_1206_wo__rel_Ominim__isMinim,axiom,
! [R: set_Pr2972776593051762503rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ( bNF_We1688910813892497397rm_a_b @ R )
=> ( ( ord_le118470702582115849rm_a_b @ B @ ( field_6884932134483023318rm_a_b @ R ) )
=> ( ( B != bot_bo197521221353338581rm_a_b )
=> ( bNF_We8430659725849320639rm_a_b @ R @ B @ ( bNF_We1338740440619677141rm_a_b @ R @ B ) ) ) ) ) ).
% wo_rel.minim_isMinim
thf(fact_1207_wo__rel_Ominim__isMinim,axiom,
! [R: set_Pr7047708467701517959at_nat,B: set_Pr9093778441882193744at_nat] :
( ( bNF_We4809617423727447260at_nat @ R )
=> ( ( ord_le3678578370064672496at_nat @ B @ ( field_718184728585074237at_nat @ R ) )
=> ( ( B != bot_bo8000532843592726972at_nat )
=> ( bNF_We1085401387027938470at_nat @ R @ B @ ( bNF_We4166897169537008316at_nat @ R @ B ) ) ) ) ) ).
% wo_rel.minim_isMinim
thf(fact_1208_wo__rel_OisMinim__def,axiom,
! [R: set_Pr2972776593051762503rm_a_b,A: set_Pr4386577575007340137rm_a_b,B2: produc357393685978478089rm_a_b] :
( ( bNF_We1688910813892497397rm_a_b @ R )
=> ( ( bNF_We8430659725849320639rm_a_b @ R @ A @ B2 )
= ( ( member5869715511025134514rm_a_b @ B2 @ A )
& ! [X4: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X4 @ A )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B2 @ X4 ) @ R ) ) ) ) ) ).
% wo_rel.isMinim_def
thf(fact_1209_wo__rel_OisMinim__def,axiom,
! [R: set_Pr1811044260758604347_a_nat,A: set_Pr4934435412358123699_a_nat,B2: product_prod_a_nat] :
( ( bNF_We4298404315170990993_a_nat @ R )
=> ( ( bNF_We4498669257762480967_a_nat @ R @ A @ B2 )
= ( ( member5724188588386418708_a_nat @ B2 @ A )
& ! [X4: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ X4 @ A )
=> ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ B2 @ X4 ) @ R ) ) ) ) ) ).
% wo_rel.isMinim_def
thf(fact_1210_wo__rel_OisMinim__def,axiom,
! [R: set_Pr7047708467701517959at_nat,A: set_Pr9093778441882193744at_nat,B2: produc8199716216217303280at_nat] :
( ( bNF_We4809617423727447260at_nat @ R )
=> ( ( bNF_We1085401387027938470at_nat @ R @ A @ B2 )
= ( ( member7226740684066999833at_nat @ B2 @ A )
& ! [X4: produc8199716216217303280at_nat] :
( ( member7226740684066999833at_nat @ X4 @ A )
=> ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ B2 @ X4 ) @ R ) ) ) ) ) ).
% wo_rel.isMinim_def
thf(fact_1211_wo__rel_OisMinim__def,axiom,
! [R: set_Pr4386577575007340137rm_a_b,A: set_term_a_b,B2: term_a_b] :
( ( bNF_We251836928773027112rm_a_b @ R )
=> ( ( bNF_We695392074589568990rm_a_b @ R @ A @ B2 )
= ( ( member_term_a_b @ B2 @ A )
& ! [X4: term_a_b] :
( ( member_term_a_b @ X4 @ A )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ X4 ) @ R ) ) ) ) ) ).
% wo_rel.isMinim_def
thf(fact_1212_wo__rel_Ominim__least,axiom,
! [R: set_Pr4386577575007340137rm_a_b,B: set_term_a_b,B2: term_a_b] :
( ( bNF_We251836928773027112rm_a_b @ R )
=> ( ( ord_le2705286416250468010rm_a_b @ B @ ( field_term_a_b @ R ) )
=> ( ( member_term_a_b @ B2 @ B )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( bNF_We6258903063523145544rm_a_b @ R @ B ) @ B2 ) @ R ) ) ) ) ).
% wo_rel.minim_least
thf(fact_1213_wo__rel_Ominim__least,axiom,
! [R: set_Pr1811044260758604347_a_nat,B: set_Pr4934435412358123699_a_nat,B2: product_prod_a_nat] :
( ( bNF_We4298404315170990993_a_nat @ R )
=> ( ( ord_le8666007276011122963_a_nat @ B @ ( field_8954927560578634480_a_nat @ R ) )
=> ( ( member5724188588386418708_a_nat @ B2 @ B )
=> ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ ( bNF_We3895015349946260913_a_nat @ R @ B ) @ B2 ) @ R ) ) ) ) ).
% wo_rel.minim_least
thf(fact_1214_wo__rel_Ominim__least,axiom,
! [R: set_Pr2972776593051762503rm_a_b,B: set_Pr4386577575007340137rm_a_b,B2: produc357393685978478089rm_a_b] :
( ( bNF_We1688910813892497397rm_a_b @ R )
=> ( ( ord_le118470702582115849rm_a_b @ B @ ( field_6884932134483023318rm_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ B2 @ B )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( bNF_We1338740440619677141rm_a_b @ R @ B ) @ B2 ) @ R ) ) ) ) ).
% wo_rel.minim_least
thf(fact_1215_wo__rel_Ominim__least,axiom,
! [R: set_Pr7047708467701517959at_nat,B: set_Pr9093778441882193744at_nat,B2: produc8199716216217303280at_nat] :
( ( bNF_We4809617423727447260at_nat @ R )
=> ( ( ord_le3678578370064672496at_nat @ B @ ( field_718184728585074237at_nat @ R ) )
=> ( ( member7226740684066999833at_nat @ B2 @ B )
=> ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ ( bNF_We4166897169537008316at_nat @ R @ B ) @ B2 ) @ R ) ) ) ) ).
% wo_rel.minim_least
thf(fact_1216_wo__rel_Oequals__minim,axiom,
! [R: set_Pr4386577575007340137rm_a_b,B: set_term_a_b,A2: term_a_b] :
( ( bNF_We251836928773027112rm_a_b @ R )
=> ( ( ord_le2705286416250468010rm_a_b @ B @ ( field_term_a_b @ R ) )
=> ( ( member_term_a_b @ A2 @ B )
=> ( ! [B6: term_a_b] :
( ( member_term_a_b @ B6 @ B )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B6 ) @ R ) )
=> ( A2
= ( bNF_We6258903063523145544rm_a_b @ R @ B ) ) ) ) ) ) ).
% wo_rel.equals_minim
thf(fact_1217_wo__rel_Oequals__minim,axiom,
! [R: set_Pr1811044260758604347_a_nat,B: set_Pr4934435412358123699_a_nat,A2: product_prod_a_nat] :
( ( bNF_We4298404315170990993_a_nat @ R )
=> ( ( ord_le8666007276011122963_a_nat @ B @ ( field_8954927560578634480_a_nat @ R ) )
=> ( ( member5724188588386418708_a_nat @ A2 @ B )
=> ( ! [B6: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ B6 @ B )
=> ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A2 @ B6 ) @ R ) )
=> ( A2
= ( bNF_We3895015349946260913_a_nat @ R @ B ) ) ) ) ) ) ).
% wo_rel.equals_minim
thf(fact_1218_wo__rel_Oequals__minim,axiom,
! [R: set_Pr2972776593051762503rm_a_b,B: set_Pr4386577575007340137rm_a_b,A2: produc357393685978478089rm_a_b] :
( ( bNF_We1688910813892497397rm_a_b @ R )
=> ( ( ord_le118470702582115849rm_a_b @ B @ ( field_6884932134483023318rm_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ A2 @ B )
=> ( ! [B6: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ B6 @ B )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A2 @ B6 ) @ R ) )
=> ( A2
= ( bNF_We1338740440619677141rm_a_b @ R @ B ) ) ) ) ) ) ).
% wo_rel.equals_minim
thf(fact_1219_wo__rel_Oequals__minim,axiom,
! [R: set_Pr7047708467701517959at_nat,B: set_Pr9093778441882193744at_nat,A2: produc8199716216217303280at_nat] :
( ( bNF_We4809617423727447260at_nat @ R )
=> ( ( ord_le3678578370064672496at_nat @ B @ ( field_718184728585074237at_nat @ R ) )
=> ( ( member7226740684066999833at_nat @ A2 @ B )
=> ( ! [B6: produc8199716216217303280at_nat] :
( ( member7226740684066999833at_nat @ B6 @ B )
=> ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ A2 @ B6 ) @ R ) )
=> ( A2
= ( bNF_We4166897169537008316at_nat @ R @ B ) ) ) ) ) ) ).
% wo_rel.equals_minim
thf(fact_1220_wo__rel_Ominim__inField,axiom,
! [R: set_Pr1811044260758604347_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( bNF_We4298404315170990993_a_nat @ R )
=> ( ( ord_le8666007276011122963_a_nat @ B @ ( field_8954927560578634480_a_nat @ R ) )
=> ( ( B != bot_bo9049108969261143879_a_nat )
=> ( member5724188588386418708_a_nat @ ( bNF_We3895015349946260913_a_nat @ R @ B ) @ ( field_8954927560578634480_a_nat @ R ) ) ) ) ) ).
% wo_rel.minim_inField
thf(fact_1221_wo__rel_Ominim__inField,axiom,
! [R: set_Pr2972776593051762503rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ( bNF_We1688910813892497397rm_a_b @ R )
=> ( ( ord_le118470702582115849rm_a_b @ B @ ( field_6884932134483023318rm_a_b @ R ) )
=> ( ( B != bot_bo197521221353338581rm_a_b )
=> ( member5869715511025134514rm_a_b @ ( bNF_We1338740440619677141rm_a_b @ R @ B ) @ ( field_6884932134483023318rm_a_b @ R ) ) ) ) ) ).
% wo_rel.minim_inField
thf(fact_1222_wo__rel_Ominim__inField,axiom,
! [R: set_Pr7047708467701517959at_nat,B: set_Pr9093778441882193744at_nat] :
( ( bNF_We4809617423727447260at_nat @ R )
=> ( ( ord_le3678578370064672496at_nat @ B @ ( field_718184728585074237at_nat @ R ) )
=> ( ( B != bot_bo8000532843592726972at_nat )
=> ( member7226740684066999833at_nat @ ( bNF_We4166897169537008316at_nat @ R @ B ) @ ( field_718184728585074237at_nat @ R ) ) ) ) ) ).
% wo_rel.minim_inField
thf(fact_1223_wo__rel_Omax2__greater__among,axiom,
! [R: set_Pr2972776593051762503rm_a_b,A2: produc357393685978478089rm_a_b,B2: produc357393685978478089rm_a_b] :
( ( bNF_We1688910813892497397rm_a_b @ R )
=> ( ( member5869715511025134514rm_a_b @ A2 @ ( field_6884932134483023318rm_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ B2 @ ( field_6884932134483023318rm_a_b @ R ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A2 @ ( bNF_We1369222903935127127rm_a_b @ R @ A2 @ B2 ) ) @ R )
& ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B2 @ ( bNF_We1369222903935127127rm_a_b @ R @ A2 @ B2 ) ) @ R )
& ( member5869715511025134514rm_a_b @ ( bNF_We1369222903935127127rm_a_b @ R @ A2 @ B2 ) @ ( insert7009541432154983385rm_a_b @ A2 @ ( insert7009541432154983385rm_a_b @ B2 @ bot_bo197521221353338581rm_a_b ) ) ) ) ) ) ) ).
% wo_rel.max2_greater_among
thf(fact_1224_wo__rel_Omax2__greater__among,axiom,
! [R: set_Pr1811044260758604347_a_nat,A2: product_prod_a_nat,B2: product_prod_a_nat] :
( ( bNF_We4298404315170990993_a_nat @ R )
=> ( ( member5724188588386418708_a_nat @ A2 @ ( field_8954927560578634480_a_nat @ R ) )
=> ( ( member5724188588386418708_a_nat @ B2 @ ( field_8954927560578634480_a_nat @ R ) )
=> ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A2 @ ( bNF_We7882469860449324975_a_nat @ R @ A2 @ B2 ) ) @ R )
& ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ B2 @ ( bNF_We7882469860449324975_a_nat @ R @ A2 @ B2 ) ) @ R )
& ( member5724188588386418708_a_nat @ ( bNF_We7882469860449324975_a_nat @ R @ A2 @ B2 ) @ ( insert8054603423593749677_a_nat @ A2 @ ( insert8054603423593749677_a_nat @ B2 @ bot_bo9049108969261143879_a_nat ) ) ) ) ) ) ) ).
% wo_rel.max2_greater_among
thf(fact_1225_wo__rel_Omax2__greater__among,axiom,
! [R: set_Pr7047708467701517959at_nat,A2: produc8199716216217303280at_nat,B2: produc8199716216217303280at_nat] :
( ( bNF_We4809617423727447260at_nat @ R )
=> ( ( member7226740684066999833at_nat @ A2 @ ( field_718184728585074237at_nat @ R ) )
=> ( ( member7226740684066999833at_nat @ B2 @ ( field_718184728585074237at_nat @ R ) )
=> ( ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ A2 @ ( bNF_We8299386922674438718at_nat @ R @ A2 @ B2 ) ) @ R )
& ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ B2 @ ( bNF_We8299386922674438718at_nat @ R @ A2 @ B2 ) ) @ R )
& ( member7226740684066999833at_nat @ ( bNF_We8299386922674438718at_nat @ R @ A2 @ B2 ) @ ( insert891772969674204864at_nat @ A2 @ ( insert891772969674204864at_nat @ B2 @ bot_bo8000532843592726972at_nat ) ) ) ) ) ) ) ).
% wo_rel.max2_greater_among
thf(fact_1226_wo__rel_Omax2__greater__among,axiom,
! [R: set_Pr4386577575007340137rm_a_b,A2: term_a_b,B2: term_a_b] :
( ( bNF_We251836928773027112rm_a_b @ R )
=> ( ( member_term_a_b @ A2 @ ( field_term_a_b @ R ) )
=> ( ( member_term_a_b @ B2 @ ( field_term_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ ( bNF_We4618759378340593734rm_a_b @ R @ A2 @ B2 ) ) @ R )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ ( bNF_We4618759378340593734rm_a_b @ R @ A2 @ B2 ) ) @ R )
& ( member_term_a_b @ ( bNF_We4618759378340593734rm_a_b @ R @ A2 @ B2 ) @ ( insert_term_a_b @ A2 @ ( insert_term_a_b @ B2 @ bot_bot_set_term_a_b ) ) ) ) ) ) ) ).
% wo_rel.max2_greater_among
thf(fact_1227_refl__on__domain,axiom,
! [A: set_Pr4386577575007340137rm_a_b,R: set_Pr2972776593051762503rm_a_b,A2: produc357393685978478089rm_a_b,B2: produc357393685978478089rm_a_b] :
( ( refl_o5344404488616963464rm_a_b @ A @ R )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A2 @ B2 ) @ R )
=> ( ( member5869715511025134514rm_a_b @ A2 @ A )
& ( member5869715511025134514rm_a_b @ B2 @ A ) ) ) ) ).
% refl_on_domain
thf(fact_1228_refl__on__domain,axiom,
! [A: set_Pr4934435412358123699_a_nat,R: set_Pr1811044260758604347_a_nat,A2: product_prod_a_nat,B2: product_prod_a_nat] :
( ( refl_o8237544196271505854_a_nat @ A @ R )
=> ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A2 @ B2 ) @ R )
=> ( ( member5724188588386418708_a_nat @ A2 @ A )
& ( member5724188588386418708_a_nat @ B2 @ A ) ) ) ) ).
% refl_on_domain
thf(fact_1229_refl__on__domain,axiom,
! [A: set_Pr9093778441882193744at_nat,R: set_Pr7047708467701517959at_nat,A2: produc8199716216217303280at_nat,B2: produc8199716216217303280at_nat] :
( ( refl_o7598126563300187375at_nat @ A @ R )
=> ( ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ A2 @ B2 ) @ R )
=> ( ( member7226740684066999833at_nat @ A2 @ A )
& ( member7226740684066999833at_nat @ B2 @ A ) ) ) ) ).
% refl_on_domain
thf(fact_1230_refl__on__domain,axiom,
! [A: set_term_a_b,R: set_Pr4386577575007340137rm_a_b,A2: term_a_b,B2: term_a_b] :
( ( refl_on_term_a_b @ A @ R )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ R )
=> ( ( member_term_a_b @ A2 @ A )
& ( member_term_a_b @ B2 @ A ) ) ) ) ).
% refl_on_domain
thf(fact_1231_wo__rel_Omax2__def,axiom,
! [R: set_Pr4386577575007340137rm_a_b,A2: term_a_b,B2: term_a_b] :
( ( bNF_We251836928773027112rm_a_b @ R )
=> ( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ R )
=> ( ( bNF_We4618759378340593734rm_a_b @ R @ A2 @ B2 )
= B2 ) )
& ( ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ R )
=> ( ( bNF_We4618759378340593734rm_a_b @ R @ A2 @ B2 )
= A2 ) ) ) ) ).
% wo_rel.max2_def
thf(fact_1232_wo__rel_Omax2__equals1,axiom,
! [R: set_Pr2972776593051762503rm_a_b,A2: produc357393685978478089rm_a_b,B2: produc357393685978478089rm_a_b] :
( ( bNF_We1688910813892497397rm_a_b @ R )
=> ( ( member5869715511025134514rm_a_b @ A2 @ ( field_6884932134483023318rm_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ B2 @ ( field_6884932134483023318rm_a_b @ R ) )
=> ( ( ( bNF_We1369222903935127127rm_a_b @ R @ A2 @ B2 )
= A2 )
= ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B2 @ A2 ) @ R ) ) ) ) ) ).
% wo_rel.max2_equals1
thf(fact_1233_wo__rel_Omax2__equals1,axiom,
! [R: set_Pr1811044260758604347_a_nat,A2: product_prod_a_nat,B2: product_prod_a_nat] :
( ( bNF_We4298404315170990993_a_nat @ R )
=> ( ( member5724188588386418708_a_nat @ A2 @ ( field_8954927560578634480_a_nat @ R ) )
=> ( ( member5724188588386418708_a_nat @ B2 @ ( field_8954927560578634480_a_nat @ R ) )
=> ( ( ( bNF_We7882469860449324975_a_nat @ R @ A2 @ B2 )
= A2 )
= ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ B2 @ A2 ) @ R ) ) ) ) ) ).
% wo_rel.max2_equals1
thf(fact_1234_wo__rel_Omax2__equals1,axiom,
! [R: set_Pr7047708467701517959at_nat,A2: produc8199716216217303280at_nat,B2: produc8199716216217303280at_nat] :
( ( bNF_We4809617423727447260at_nat @ R )
=> ( ( member7226740684066999833at_nat @ A2 @ ( field_718184728585074237at_nat @ R ) )
=> ( ( member7226740684066999833at_nat @ B2 @ ( field_718184728585074237at_nat @ R ) )
=> ( ( ( bNF_We8299386922674438718at_nat @ R @ A2 @ B2 )
= A2 )
= ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ B2 @ A2 ) @ R ) ) ) ) ) ).
% wo_rel.max2_equals1
thf(fact_1235_wo__rel_Omax2__equals1,axiom,
! [R: set_Pr4386577575007340137rm_a_b,A2: term_a_b,B2: term_a_b] :
( ( bNF_We251836928773027112rm_a_b @ R )
=> ( ( member_term_a_b @ A2 @ ( field_term_a_b @ R ) )
=> ( ( member_term_a_b @ B2 @ ( field_term_a_b @ R ) )
=> ( ( ( bNF_We4618759378340593734rm_a_b @ R @ A2 @ B2 )
= A2 )
= ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ A2 ) @ R ) ) ) ) ) ).
% wo_rel.max2_equals1
thf(fact_1236_wo__rel_Omax2__equals2,axiom,
! [R: set_Pr2972776593051762503rm_a_b,A2: produc357393685978478089rm_a_b,B2: produc357393685978478089rm_a_b] :
( ( bNF_We1688910813892497397rm_a_b @ R )
=> ( ( member5869715511025134514rm_a_b @ A2 @ ( field_6884932134483023318rm_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ B2 @ ( field_6884932134483023318rm_a_b @ R ) )
=> ( ( ( bNF_We1369222903935127127rm_a_b @ R @ A2 @ B2 )
= B2 )
= ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A2 @ B2 ) @ R ) ) ) ) ) ).
% wo_rel.max2_equals2
thf(fact_1237_wo__rel_Omax2__equals2,axiom,
! [R: set_Pr1811044260758604347_a_nat,A2: product_prod_a_nat,B2: product_prod_a_nat] :
( ( bNF_We4298404315170990993_a_nat @ R )
=> ( ( member5724188588386418708_a_nat @ A2 @ ( field_8954927560578634480_a_nat @ R ) )
=> ( ( member5724188588386418708_a_nat @ B2 @ ( field_8954927560578634480_a_nat @ R ) )
=> ( ( ( bNF_We7882469860449324975_a_nat @ R @ A2 @ B2 )
= B2 )
= ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A2 @ B2 ) @ R ) ) ) ) ) ).
% wo_rel.max2_equals2
thf(fact_1238_wo__rel_Omax2__equals2,axiom,
! [R: set_Pr7047708467701517959at_nat,A2: produc8199716216217303280at_nat,B2: produc8199716216217303280at_nat] :
( ( bNF_We4809617423727447260at_nat @ R )
=> ( ( member7226740684066999833at_nat @ A2 @ ( field_718184728585074237at_nat @ R ) )
=> ( ( member7226740684066999833at_nat @ B2 @ ( field_718184728585074237at_nat @ R ) )
=> ( ( ( bNF_We8299386922674438718at_nat @ R @ A2 @ B2 )
= B2 )
= ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ A2 @ B2 ) @ R ) ) ) ) ) ).
% wo_rel.max2_equals2
thf(fact_1239_wo__rel_Omax2__equals2,axiom,
! [R: set_Pr4386577575007340137rm_a_b,A2: term_a_b,B2: term_a_b] :
( ( bNF_We251836928773027112rm_a_b @ R )
=> ( ( member_term_a_b @ A2 @ ( field_term_a_b @ R ) )
=> ( ( member_term_a_b @ B2 @ ( field_term_a_b @ R ) )
=> ( ( ( bNF_We4618759378340593734rm_a_b @ R @ A2 @ B2 )
= B2 )
= ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ R ) ) ) ) ) ).
% wo_rel.max2_equals2
thf(fact_1240_wo__rel_Omax2__greater,axiom,
! [R: set_Pr2972776593051762503rm_a_b,A2: produc357393685978478089rm_a_b,B2: produc357393685978478089rm_a_b] :
( ( bNF_We1688910813892497397rm_a_b @ R )
=> ( ( member5869715511025134514rm_a_b @ A2 @ ( field_6884932134483023318rm_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ B2 @ ( field_6884932134483023318rm_a_b @ R ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A2 @ ( bNF_We1369222903935127127rm_a_b @ R @ A2 @ B2 ) ) @ R )
& ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B2 @ ( bNF_We1369222903935127127rm_a_b @ R @ A2 @ B2 ) ) @ R ) ) ) ) ) ).
% wo_rel.max2_greater
thf(fact_1241_wo__rel_Omax2__greater,axiom,
! [R: set_Pr1811044260758604347_a_nat,A2: product_prod_a_nat,B2: product_prod_a_nat] :
( ( bNF_We4298404315170990993_a_nat @ R )
=> ( ( member5724188588386418708_a_nat @ A2 @ ( field_8954927560578634480_a_nat @ R ) )
=> ( ( member5724188588386418708_a_nat @ B2 @ ( field_8954927560578634480_a_nat @ R ) )
=> ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A2 @ ( bNF_We7882469860449324975_a_nat @ R @ A2 @ B2 ) ) @ R )
& ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ B2 @ ( bNF_We7882469860449324975_a_nat @ R @ A2 @ B2 ) ) @ R ) ) ) ) ) ).
% wo_rel.max2_greater
thf(fact_1242_wo__rel_Omax2__greater,axiom,
! [R: set_Pr7047708467701517959at_nat,A2: produc8199716216217303280at_nat,B2: produc8199716216217303280at_nat] :
( ( bNF_We4809617423727447260at_nat @ R )
=> ( ( member7226740684066999833at_nat @ A2 @ ( field_718184728585074237at_nat @ R ) )
=> ( ( member7226740684066999833at_nat @ B2 @ ( field_718184728585074237at_nat @ R ) )
=> ( ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ A2 @ ( bNF_We8299386922674438718at_nat @ R @ A2 @ B2 ) ) @ R )
& ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ B2 @ ( bNF_We8299386922674438718at_nat @ R @ A2 @ B2 ) ) @ R ) ) ) ) ) ).
% wo_rel.max2_greater
thf(fact_1243_wo__rel_Omax2__greater,axiom,
! [R: set_Pr4386577575007340137rm_a_b,A2: term_a_b,B2: term_a_b] :
( ( bNF_We251836928773027112rm_a_b @ R )
=> ( ( member_term_a_b @ A2 @ ( field_term_a_b @ R ) )
=> ( ( member_term_a_b @ B2 @ ( field_term_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ ( bNF_We4618759378340593734rm_a_b @ R @ A2 @ B2 ) ) @ R )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ ( bNF_We4618759378340593734rm_a_b @ R @ A2 @ B2 ) ) @ R ) ) ) ) ) ).
% wo_rel.max2_greater
thf(fact_1244_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 )
| ? [A6: term_a_b] :
( R
= ( insert7009541432154983385rm_a_b @ ( produc7020197800436672577rm_a_b @ A6 @ A6 ) @ bot_bo197521221353338581rm_a_b ) ) ) ) ) ).
% Total_subset_Id
thf(fact_1245_linear__order__on__singleton,axiom,
! [X3: term_a_b] : ( order_5388802246213473311rm_a_b @ ( insert_term_a_b @ X3 @ bot_bot_set_term_a_b ) @ ( insert7009541432154983385rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ X3 ) @ bot_bo197521221353338581rm_a_b ) ) ).
% linear_order_on_singleton
thf(fact_1246_total__on__subset,axiom,
! [A: set_Pr4934435412358123699_a_nat,R: set_Pr1811044260758604347_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( total_9110697612923228665_a_nat @ A @ R )
=> ( ( ord_le8666007276011122963_a_nat @ B @ A )
=> ( total_9110697612923228665_a_nat @ B @ R ) ) ) ).
% total_on_subset
thf(fact_1247_total__on__subset,axiom,
! [A: set_Pr4386577575007340137rm_a_b,R: set_Pr2972776593051762503rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ( total_4010809703584253837rm_a_b @ A @ R )
=> ( ( ord_le118470702582115849rm_a_b @ B @ A )
=> ( total_4010809703584253837rm_a_b @ B @ R ) ) ) ).
% total_on_subset
thf(fact_1248_total__on__subset,axiom,
! [A: set_Pr9093778441882193744at_nat,R: set_Pr7047708467701517959at_nat,B: set_Pr9093778441882193744at_nat] :
( ( total_9210148749937563764at_nat @ A @ R )
=> ( ( ord_le3678578370064672496at_nat @ B @ A )
=> ( total_9210148749937563764at_nat @ B @ R ) ) ) ).
% total_on_subset
thf(fact_1249_total__onI,axiom,
! [A: set_Pr4386577575007340137rm_a_b,R: set_Pr2972776593051762503rm_a_b] :
( ! [X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X @ A )
=> ( ( member5869715511025134514rm_a_b @ Y @ A )
=> ( ( X != Y )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ R )
| ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y @ X ) @ R ) ) ) ) )
=> ( total_4010809703584253837rm_a_b @ A @ R ) ) ).
% total_onI
thf(fact_1250_total__onI,axiom,
! [A: set_Pr4934435412358123699_a_nat,R: set_Pr1811044260758604347_a_nat] :
( ! [X: product_prod_a_nat,Y: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ X @ A )
=> ( ( member5724188588386418708_a_nat @ Y @ A )
=> ( ( X != Y )
=> ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ X @ Y ) @ R )
| ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ Y @ X ) @ R ) ) ) ) )
=> ( total_9110697612923228665_a_nat @ A @ R ) ) ).
% total_onI
thf(fact_1251_total__onI,axiom,
! [A: set_Pr9093778441882193744at_nat,R: set_Pr7047708467701517959at_nat] :
( ! [X: produc8199716216217303280at_nat,Y: produc8199716216217303280at_nat] :
( ( member7226740684066999833at_nat @ X @ A )
=> ( ( member7226740684066999833at_nat @ Y @ A )
=> ( ( X != Y )
=> ( ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ X @ Y ) @ R )
| ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ Y @ X ) @ R ) ) ) ) )
=> ( total_9210148749937563764at_nat @ A @ R ) ) ).
% total_onI
thf(fact_1252_total__onI,axiom,
! [A: set_term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ! [X: term_a_b,Y: term_a_b] :
( ( member_term_a_b @ X @ A )
=> ( ( member_term_a_b @ Y @ A )
=> ( ( X != Y )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ R )
| ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X ) @ R ) ) ) ) )
=> ( total_on_term_a_b @ A @ R ) ) ).
% total_onI
thf(fact_1253_total__on__def,axiom,
( total_on_term_a_b
= ( ^ [A4: set_term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
! [X4: term_a_b] :
( ( member_term_a_b @ X4 @ A4 )
=> ! [Y5: term_a_b] :
( ( member_term_a_b @ Y5 @ A4 )
=> ( ( X4 != Y5 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X4 @ Y5 ) @ R2 )
| ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y5 @ X4 ) @ R2 ) ) ) ) ) ) ) ).
% total_on_def
thf(fact_1254_totalI,axiom,
! [R: set_Pr4386577575007340137rm_a_b] :
( ! [X: term_a_b,Y: term_a_b] :
( ( X != Y )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ R )
| ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X ) @ R ) ) )
=> ( total_on_term_a_b @ top_top_set_term_a_b @ R ) ) ).
% totalI
thf(fact_1255_Total__Id__Field,axiom,
! [R: set_Pr4386577575007340137rm_a_b] :
( ( total_on_term_a_b @ ( field_term_a_b @ R ) @ R )
=> ( ~ ( ord_le118470702582115849rm_a_b @ R @ id_term_a_b )
=> ( ( field_term_a_b @ R )
= ( field_term_a_b @ ( minus_5192120951422937424rm_a_b @ R @ id_term_a_b ) ) ) ) ) ).
% Total_Id_Field
thf(fact_1256_Linear__order__in__diff__Id,axiom,
! [R: set_Pr2972776593051762503rm_a_b,A2: produc357393685978478089rm_a_b,B2: produc357393685978478089rm_a_b] :
( ( order_681589137112254398rm_a_b @ ( field_6884932134483023318rm_a_b @ R ) @ R )
=> ( ( member5869715511025134514rm_a_b @ A2 @ ( field_6884932134483023318rm_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ B2 @ ( field_6884932134483023318rm_a_b @ R ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A2 @ B2 ) @ R )
= ( ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B2 @ A2 ) @ ( minus_45395518312058030rm_a_b @ R @ id_Pro3488709451141449061rm_a_b ) ) ) ) ) ) ) ).
% Linear_order_in_diff_Id
thf(fact_1257_Linear__order__in__diff__Id,axiom,
! [R: set_Pr1811044260758604347_a_nat,A2: product_prod_a_nat,B2: product_prod_a_nat] :
( ( order_6388386362885710600_a_nat @ ( field_8954927560578634480_a_nat @ R ) @ R )
=> ( ( member5724188588386418708_a_nat @ A2 @ ( field_8954927560578634480_a_nat @ R ) )
=> ( ( member5724188588386418708_a_nat @ B2 @ ( field_8954927560578634480_a_nat @ R ) )
=> ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A2 @ B2 ) @ R )
= ( ~ ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ B2 @ A2 ) @ ( minus_1341117088723627554_a_nat @ R @ id_Pro5207055338379199009_a_nat ) ) ) ) ) ) ) ).
% Linear_order_in_diff_Id
thf(fact_1258_Linear__order__in__diff__Id,axiom,
! [R: set_Pr7047708467701517959at_nat,A2: produc8199716216217303280at_nat,B2: produc8199716216217303280at_nat] :
( ( order_3740343183938982949at_nat @ ( field_718184728585074237at_nat @ R ) @ R )
=> ( ( member7226740684066999833at_nat @ A2 @ ( field_718184728585074237at_nat @ R ) )
=> ( ( member7226740684066999833at_nat @ B2 @ ( field_718184728585074237at_nat @ R ) )
=> ( ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ A2 @ B2 ) @ R )
= ( ~ ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ B2 @ A2 ) @ ( minus_7913753035872161006at_nat @ R @ id_Pro414489527043525196at_nat ) ) ) ) ) ) ) ).
% Linear_order_in_diff_Id
thf(fact_1259_Linear__order__in__diff__Id,axiom,
! [R: set_Pr4386577575007340137rm_a_b,A2: term_a_b,B2: term_a_b] :
( ( order_5388802246213473311rm_a_b @ ( field_term_a_b @ R ) @ R )
=> ( ( member_term_a_b @ A2 @ ( field_term_a_b @ R ) )
=> ( ( member_term_a_b @ B2 @ ( field_term_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ R )
= ( ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ A2 ) @ ( minus_5192120951422937424rm_a_b @ R @ id_term_a_b ) ) ) ) ) ) ) ).
% Linear_order_in_diff_Id
thf(fact_1260_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 ) )
= ( ! [A4: set_term_a_b] :
( ( ord_le2705286416250468010rm_a_b @ A4 @ ( field_term_a_b @ R ) )
=> ( ( A4 != bot_bot_set_term_a_b )
=> ? [X4: term_a_b] :
( ( member_term_a_b @ X4 @ A4 )
& ! [Y5: term_a_b] :
( ( member_term_a_b @ Y5 @ A4 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X4 @ Y5 ) @ R ) ) ) ) ) ) ) ) ).
% Linear_order_wf_diff_Id
thf(fact_1261_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 ) )
= ( ! [A4: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A4 @ ( field_8954927560578634480_a_nat @ R ) )
=> ( ( A4 != bot_bo9049108969261143879_a_nat )
=> ? [X4: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ X4 @ A4 )
& ! [Y5: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ Y5 @ A4 )
=> ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ X4 @ Y5 ) @ R ) ) ) ) ) ) ) ) ).
% Linear_order_wf_diff_Id
thf(fact_1262_Linear__order__wf__diff__Id,axiom,
! [R: set_Pr2972776593051762503rm_a_b] :
( ( order_681589137112254398rm_a_b @ ( field_6884932134483023318rm_a_b @ R ) @ R )
=> ( ( wf_Pro2335863617654816626rm_a_b @ ( minus_45395518312058030rm_a_b @ R @ id_Pro3488709451141449061rm_a_b ) )
= ( ! [A4: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A4 @ ( field_6884932134483023318rm_a_b @ R ) )
=> ( ( A4 != bot_bo197521221353338581rm_a_b )
=> ? [X4: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X4 @ A4 )
& ! [Y5: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ Y5 @ A4 )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X4 @ Y5 ) @ R ) ) ) ) ) ) ) ) ).
% Linear_order_wf_diff_Id
thf(fact_1263_Linear__order__wf__diff__Id,axiom,
! [R: set_Pr7047708467701517959at_nat] :
( ( order_3740343183938982949at_nat @ ( field_718184728585074237at_nat @ R ) @ R )
=> ( ( wf_Pro1646224750900445657at_nat @ ( minus_7913753035872161006at_nat @ R @ id_Pro414489527043525196at_nat ) )
= ( ! [A4: set_Pr9093778441882193744at_nat] :
( ( ord_le3678578370064672496at_nat @ A4 @ ( field_718184728585074237at_nat @ R ) )
=> ( ( A4 != bot_bo8000532843592726972at_nat )
=> ? [X4: produc8199716216217303280at_nat] :
( ( member7226740684066999833at_nat @ X4 @ A4 )
& ! [Y5: produc8199716216217303280at_nat] :
( ( member7226740684066999833at_nat @ Y5 @ A4 )
=> ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ X4 @ Y5 ) @ R ) ) ) ) ) ) ) ) ).
% Linear_order_wf_diff_Id
thf(fact_1264_in__lex__prod,axiom,
! [A2: term_a_b,B2: term_a_b,A3: term_a_b,B3: term_a_b,R: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( produc7020197800436672577rm_a_b @ A3 @ B3 ) ) @ ( lex_pr6163557265797435481rm_a_b @ R @ S ) )
= ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ A3 ) @ R )
| ( ( A2 = A3 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ B3 ) @ S ) ) ) ) ).
% in_lex_prod
thf(fact_1265_in__lex__prod,axiom,
! [A2: nat > nat,B2: nat,A3: nat > nat,B3: nat,R: set_Pr7682762132356531903at_nat,S: set_Pr1261947904930325089at_nat] :
( ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ ( produc72220940542539688at_nat @ A2 @ B2 ) @ ( produc72220940542539688at_nat @ A3 @ B3 ) ) @ ( lex_prod_nat_nat_nat @ R @ S ) )
= ( ( member1096249278104964872at_nat @ ( produc5770335208449155351at_nat @ A2 @ A3 ) @ R )
| ( ( A2 = A3 )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B2 @ B3 ) @ S ) ) ) ) ).
% in_lex_prod
thf(fact_1266_in__lex__prod,axiom,
! [A2: a,B2: nat,A3: a,B3: nat,R: set_Product_prod_a_a,S: set_Pr1261947904930325089at_nat] :
( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ ( product_Pair_a_nat @ A2 @ B2 ) @ ( product_Pair_a_nat @ A3 @ B3 ) ) @ ( lex_prod_a_nat @ R @ S ) )
= ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ A3 ) @ R )
| ( ( A2 = A3 )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B2 @ B3 ) @ S ) ) ) ) ).
% in_lex_prod
thf(fact_1267_wf__insert,axiom,
! [Y3: term_a_b,X3: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( wf_term_a_b @ ( insert7009541432154983385rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ X3 ) @ R ) )
= ( ( wf_term_a_b @ R )
& ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ ( transi7742714808557438673rm_a_b @ R ) ) ) ) ).
% wf_insert
thf(fact_1268_wf__if__convertible__to__wf,axiom,
! [S: set_Pr4386577575007340137rm_a_b,R: set_Pr4386577575007340137rm_a_b,F2: term_a_b > term_a_b] :
( ( wf_term_a_b @ S )
=> ( ! [X: term_a_b,Y: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ R )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F2 @ X ) @ ( F2 @ Y ) ) @ S ) )
=> ( wf_term_a_b @ R ) ) ) ).
% wf_if_convertible_to_wf
thf(fact_1269_wf__induct__rule,axiom,
! [R: set_Pr4386577575007340137rm_a_b,P: term_a_b > $o,A2: term_a_b] :
( ( wf_term_a_b @ R )
=> ( ! [X: term_a_b] :
( ! [Y6: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y6 @ X ) @ R )
=> ( P @ Y6 ) )
=> ( P @ X ) )
=> ( P @ A2 ) ) ) ).
% wf_induct_rule
thf(fact_1270_wf__eq__minimal,axiom,
( wf_Pro2335863617654816626rm_a_b
= ( ^ [R2: set_Pr2972776593051762503rm_a_b] :
! [Q3: set_Pr4386577575007340137rm_a_b] :
( ? [X4: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ X4 @ Q3 )
=> ? [X4: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X4 @ Q3 )
& ! [Y5: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y5 @ X4 ) @ R2 )
=> ~ ( member5869715511025134514rm_a_b @ Y5 @ Q3 ) ) ) ) ) ) ).
% wf_eq_minimal
thf(fact_1271_wf__eq__minimal,axiom,
( wf_Pro8109884346892000340_a_nat
= ( ^ [R2: set_Pr1811044260758604347_a_nat] :
! [Q3: set_Pr4934435412358123699_a_nat] :
( ? [X4: product_prod_a_nat] : ( member5724188588386418708_a_nat @ X4 @ Q3 )
=> ? [X4: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ X4 @ Q3 )
& ! [Y5: product_prod_a_nat] :
( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ Y5 @ X4 ) @ R2 )
=> ~ ( member5724188588386418708_a_nat @ Y5 @ Q3 ) ) ) ) ) ) ).
% wf_eq_minimal
thf(fact_1272_wf__eq__minimal,axiom,
( wf_Pro1646224750900445657at_nat
= ( ^ [R2: set_Pr7047708467701517959at_nat] :
! [Q3: set_Pr9093778441882193744at_nat] :
( ? [X4: produc8199716216217303280at_nat] : ( member7226740684066999833at_nat @ X4 @ Q3 )
=> ? [X4: produc8199716216217303280at_nat] :
( ( member7226740684066999833at_nat @ X4 @ Q3 )
& ! [Y5: produc8199716216217303280at_nat] :
( ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ Y5 @ X4 ) @ R2 )
=> ~ ( member7226740684066999833at_nat @ Y5 @ Q3 ) ) ) ) ) ) ).
% wf_eq_minimal
thf(fact_1273_wf__eq__minimal,axiom,
( wf_term_a_b
= ( ^ [R2: set_Pr4386577575007340137rm_a_b] :
! [Q3: set_term_a_b] :
( ? [X4: term_a_b] : ( member_term_a_b @ X4 @ Q3 )
=> ? [X4: term_a_b] :
( ( member_term_a_b @ X4 @ Q3 )
& ! [Y5: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y5 @ X4 ) @ R2 )
=> ~ ( member_term_a_b @ Y5 @ Q3 ) ) ) ) ) ) ).
% wf_eq_minimal
thf(fact_1274_inf__concat__simple_Ocases,axiom,
! [X3: produc8199716216217303280at_nat] :
( ! [F4: nat > nat] :
( X3
!= ( produc72220940542539688at_nat @ F4 @ zero_zero_nat ) )
=> ~ ! [F4: nat > nat,N3: nat] :
( X3
!= ( produc72220940542539688at_nat @ F4 @ ( suc @ N3 ) ) ) ) ).
% inf_concat_simple.cases
% Conjectures (3)
thf(conj_0,hypothesis,
member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ s @ t ) @ ( sig_step_a_b @ f @ r ) ).
thf(conj_1,hypothesis,
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ s @ t ) @ r )
=> ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ s ) @ f )
=> ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ t ) @ f )
=> p ) ) ) ).
thf(conj_2,conjecture,
p ).
%------------------------------------------------------------------------------