TPTP Problem File: SLH0958^1.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/0022_Ground_Reduction_on_LLRG/prob_00138_006883__14347826_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1557 ( 598 unt; 282 typ; 0 def)
% Number of atoms : 3587 (1129 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 12819 ( 293 ~; 47 |; 238 &;10786 @)
% ( 0 <=>;1455 =>; 0 <=; 0 <~>)
% Maximal formula depth : 22 ( 7 avg)
% Number of types : 34 ( 33 usr)
% Number of type conns : 611 ( 611 >; 0 *; 0 +; 0 <<)
% Number of symbols : 252 ( 249 usr; 33 con; 0-5 aty)
% Number of variables : 3777 ( 439 ^;3266 !; 72 ?;3777 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 14:19:44.892
%------------------------------------------------------------------------------
% Could-be-implicit typings (33)
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% Explicit typings (249)
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max_ext_nat: set_Pr1261947904930325089at_nat > set_Pr5488025237498180813et_nat ).
thf(sy_c_Wellfounded_Omin__ext_001t__Nat__Onat,type,
min_ext_nat: set_Pr1261947904930325089at_nat > set_Pr5488025237498180813et_nat ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
member8440522571783428010at_nat: product_prod_nat_nat > set_Pr1261947904930325089at_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mtf__a_J,type,
member8962352052110095674_nat_a: product_prod_nat_a > set_Pr4193341848836149977_nat_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_J,type,
member9062615507155100804_a_nat: produc4708774622424448987_a_nat > set_Pr1811044260758604347_a_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
member1876585039447381659rm_a_b: produc3105026996248004978rm_a_b > set_Pr7191425930741896914rm_a_b > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_J,type,
member4553550343464174107_a_nat: produc5781992300264797426_a_nat > set_Pr677149416131461714_a_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
member5869715511025134514rm_a_b: produc357393685978478089rm_a_b > set_Pr4386577575007340137rm_a_b > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J,type,
member5724188588386418708_a_nat: product_prod_a_nat > set_Pr4934435412358123699_a_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
member1426531477525435216od_a_a: product_prod_a_a > set_Product_prod_a_a > $o ).
thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
member_set_nat: set_nat > set_set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
member8823642877475723282rm_a_b: set_Pr4386577575007340137rm_a_b > set_se1467346398288745161rm_a_b > $o ).
thf(sy_c_member_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_J,type,
member8034581882086557258_a_nat: set_Pr4934435412358123699_a_nat > set_se6121441497158405097_a_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Term__Oterm_Itf__a_Mtf__b_J_J,type,
member_set_term_a_b: set_term_a_b > set_set_term_a_b > $o ).
thf(sy_c_member_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
member_term_a_b: term_a_b > set_term_a_b > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v__092_060F_062,type,
f: set_Pr4934435412358123699_a_nat ).
thf(sy_v__092_060R_062,type,
r: set_Pr4386577575007340137rm_a_b ).
thf(sy_v_c,type,
c: a ).
% Relevant facts (1274)
thf(fact_0_llrg,axiom,
rewrit5000278123254648670_b_a_b @ r ).
% llrg
thf(fact_1_UNIV__I,axiom,
! [X: set_Pr4934435412358123699_a_nat] : ( member8034581882086557258_a_nat @ X @ top_to4675147776276609689_a_nat ) ).
% UNIV_I
thf(fact_2_UNIV__I,axiom,
! [X: set_term_a_b] : ( member_set_term_a_b @ X @ top_to5470816188456996400rm_a_b ) ).
% UNIV_I
thf(fact_3_UNIV__I,axiom,
! [X: set_nat] : ( member_set_nat @ X @ top_top_set_set_nat ) ).
% UNIV_I
thf(fact_4_UNIV__I,axiom,
! [X: nat] : ( member_nat @ X @ top_top_set_nat ) ).
% UNIV_I
thf(fact_5_UNIV__I,axiom,
! [X: a] : ( member_a @ X @ top_top_set_a ) ).
% UNIV_I
thf(fact_6_UNIV__I,axiom,
! [X: product_prod_a_nat] : ( member5724188588386418708_a_nat @ X @ top_to3353692345378799459_a_nat ) ).
% UNIV_I
thf(fact_7_UNIV__I,axiom,
! [X: term_a_b] : ( member_term_a_b @ X @ top_top_set_term_a_b ) ).
% UNIV_I
thf(fact_8_iso__tuple__UNIV__I,axiom,
! [X: set_Pr4934435412358123699_a_nat] : ( member8034581882086557258_a_nat @ X @ top_to4675147776276609689_a_nat ) ).
% iso_tuple_UNIV_I
thf(fact_9_iso__tuple__UNIV__I,axiom,
! [X: set_term_a_b] : ( member_set_term_a_b @ X @ top_to5470816188456996400rm_a_b ) ).
% iso_tuple_UNIV_I
thf(fact_10_iso__tuple__UNIV__I,axiom,
! [X: set_nat] : ( member_set_nat @ X @ top_top_set_set_nat ) ).
% iso_tuple_UNIV_I
thf(fact_11_iso__tuple__UNIV__I,axiom,
! [X: nat] : ( member_nat @ X @ top_top_set_nat ) ).
% iso_tuple_UNIV_I
thf(fact_12_iso__tuple__UNIV__I,axiom,
! [X: a] : ( member_a @ X @ top_top_set_a ) ).
% iso_tuple_UNIV_I
thf(fact_13_iso__tuple__UNIV__I,axiom,
! [X: product_prod_a_nat] : ( member5724188588386418708_a_nat @ X @ top_to3353692345378799459_a_nat ) ).
% iso_tuple_UNIV_I
thf(fact_14_iso__tuple__UNIV__I,axiom,
! [X: term_a_b] : ( member_term_a_b @ X @ top_top_set_term_a_b ) ).
% iso_tuple_UNIV_I
thf(fact_15_top__apply,axiom,
( top_to1985590801843794146_nat_o
= ( ^ [X2: product_prod_a_nat] : top_top_o ) ) ).
% top_apply
thf(fact_16_top__apply,axiom,
( top_top_term_a_b_o
= ( ^ [X2: term_a_b] : top_top_o ) ) ).
% top_apply
thf(fact_17_top__apply,axiom,
( top_top_a_nat_o
= ( ^ [X2: a] : top_top_nat_o ) ) ).
% top_apply
thf(fact_18_sig,axiom,
ord_le8666007276011122963_a_nat @ ( terms_7988297476397195622_a_b_b @ r ) @ f ).
% sig
thf(fact_19_UNIV__eq__I,axiom,
! [A: set_Pr4934435412358123699_a_nat] :
( ! [X3: product_prod_a_nat] : ( member5724188588386418708_a_nat @ X3 @ A )
=> ( top_to3353692345378799459_a_nat = A ) ) ).
% UNIV_eq_I
thf(fact_20_UNIV__eq__I,axiom,
! [A: set_term_a_b] :
( ! [X3: term_a_b] : ( member_term_a_b @ X3 @ A )
=> ( top_top_set_term_a_b = A ) ) ).
% UNIV_eq_I
thf(fact_21_UNIV__witness,axiom,
? [X3: product_prod_a_nat] : ( member5724188588386418708_a_nat @ X3 @ top_to3353692345378799459_a_nat ) ).
% UNIV_witness
thf(fact_22_UNIV__witness,axiom,
? [X3: term_a_b] : ( member_term_a_b @ X3 @ top_top_set_term_a_b ) ).
% UNIV_witness
thf(fact_23_UNC__imp__UNF,axiom,
! [R: set_Pr1811044260758604347_a_nat] :
( ( abstra7179253994116303565_a_nat @ R )
=> ( abstra7833083522531381839_a_nat @ R @ top_to3353692345378799459_a_nat ) ) ).
% UNC_imp_UNF
thf(fact_24_UNC__imp__UNF,axiom,
! [R: set_Pr4386577575007340137rm_a_b] :
( ( abstra4649771476585718372rm_a_b @ R )
=> ( abstra3619855735679963878rm_a_b @ R @ top_top_set_term_a_b ) ) ).
% UNC_imp_UNF
thf(fact_25_fresh,axiom,
~ ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ c @ zero_zero_nat ) @ f ) ).
% fresh
thf(fact_26_unf,axiom,
( abstra3619855735679963878rm_a_b
@ ( inf_in1693821330460682071rm_a_b @ ( sig_step_a_b @ ( insert8054603423593749677_a_nat @ ( product_Pair_a_nat @ c @ zero_zero_nat ) @ f ) @ ( rstep_a_b @ r ) )
@ ( produc1953217797448512882rm_a_b @ ( collect_term_a_b @ term_ground_a_b )
@ ^ [Uu: term_a_b] : ( collect_term_a_b @ term_ground_a_b ) ) )
@ top_top_set_term_a_b ) ).
% unf
thf(fact_27_CR__imp__UNF,axiom,
! [R: set_Pr1811044260758604347_a_nat] :
( ( abstra7964079670522473415_a_nat @ R @ top_to3353692345378799459_a_nat )
=> ( abstra7833083522531381839_a_nat @ R @ top_to3353692345378799459_a_nat ) ) ).
% CR_imp_UNF
thf(fact_28_CR__imp__UNF,axiom,
! [R: set_Pr4386577575007340137rm_a_b] :
( ( abstra8448919418672941150rm_a_b @ R @ top_top_set_term_a_b )
=> ( abstra3619855735679963878rm_a_b @ R @ top_top_set_term_a_b ) ) ).
% CR_imp_UNF
thf(fact_29_Sup__UNIV,axiom,
( ( comple5064467279716934173rm_a_b @ top_to5470816188456996400rm_a_b )
= top_top_set_term_a_b ) ).
% Sup_UNIV
thf(fact_30_Sup__UNIV,axiom,
( ( comple3771923735958488838_a_nat @ top_to4675147776276609689_a_nat )
= top_to3353692345378799459_a_nat ) ).
% Sup_UNIV
thf(fact_31_CE__symmetric,axiom,
! [R: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
( ( rewrit660295756789745446rm_a_b @ R @ S @ top_top_set_term_a_b )
=> ( rewrit660295756789745446rm_a_b @ S @ R @ top_top_set_term_a_b ) ) ).
% CE_symmetric
thf(fact_32_CE__symmetric,axiom,
! [R: set_Pr1811044260758604347_a_nat,S: set_Pr1811044260758604347_a_nat] :
( ( rewrit1921941460311402127_a_nat @ R @ S @ top_to3353692345378799459_a_nat )
=> ( rewrit1921941460311402127_a_nat @ S @ R @ top_to3353692345378799459_a_nat ) ) ).
% CE_symmetric
thf(fact_33_order__refl,axiom,
! [X: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ X @ X ) ).
% order_refl
thf(fact_34_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_35_dual__order_Orefl,axiom,
! [A2: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_36_dual__order_Orefl,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_37_subsetI,axiom,
! [A: set_term_a_b,B: set_term_a_b] :
( ! [X3: term_a_b] :
( ( member_term_a_b @ X3 @ A )
=> ( member_term_a_b @ X3 @ B ) )
=> ( ord_le2705286416250468010rm_a_b @ A @ B ) ) ).
% subsetI
thf(fact_38_subsetI,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ! [X3: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ X3 @ A )
=> ( member5724188588386418708_a_nat @ X3 @ B ) )
=> ( ord_le8666007276011122963_a_nat @ A @ B ) ) ).
% subsetI
thf(fact_39_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_40_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_41_insertCI,axiom,
! [A2: term_a_b,B: set_term_a_b,B2: term_a_b] :
( ( ~ ( member_term_a_b @ A2 @ B )
=> ( A2 = B2 ) )
=> ( member_term_a_b @ A2 @ ( insert_term_a_b @ B2 @ B ) ) ) ).
% insertCI
thf(fact_42_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_43_insert__iff,axiom,
! [A2: term_a_b,B2: term_a_b,A: set_term_a_b] :
( ( member_term_a_b @ A2 @ ( insert_term_a_b @ B2 @ A ) )
= ( ( A2 = B2 )
| ( member_term_a_b @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_44_insert__absorb2,axiom,
! [X: product_prod_a_nat,A: set_Pr4934435412358123699_a_nat] :
( ( insert8054603423593749677_a_nat @ X @ ( insert8054603423593749677_a_nat @ X @ A ) )
= ( insert8054603423593749677_a_nat @ X @ A ) ) ).
% insert_absorb2
thf(fact_45_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_46_IntI,axiom,
! [C: term_a_b,A: set_term_a_b,B: set_term_a_b] :
( ( member_term_a_b @ C @ A )
=> ( ( member_term_a_b @ C @ B )
=> ( member_term_a_b @ C @ ( inf_inf_set_term_a_b @ A @ B ) ) ) ) ).
% IntI
thf(fact_47_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_48_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_49_Int__iff,axiom,
! [C: term_a_b,A: set_term_a_b,B: set_term_a_b] :
( ( member_term_a_b @ C @ ( inf_inf_set_term_a_b @ A @ B ) )
= ( ( member_term_a_b @ C @ A )
& ( member_term_a_b @ C @ B ) ) ) ).
% Int_iff
thf(fact_50_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_51_insert__subset,axiom,
! [X: term_a_b,A: set_term_a_b,B: set_term_a_b] :
( ( ord_le2705286416250468010rm_a_b @ ( insert_term_a_b @ X @ A ) @ B )
= ( ( member_term_a_b @ X @ B )
& ( ord_le2705286416250468010rm_a_b @ A @ B ) ) ) ).
% insert_subset
thf(fact_52_insert__subset,axiom,
! [X: product_prod_a_nat,A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ ( insert8054603423593749677_a_nat @ X @ A ) @ B )
= ( ( member5724188588386418708_a_nat @ X @ B )
& ( ord_le8666007276011122963_a_nat @ A @ B ) ) ) ).
% insert_subset
thf(fact_53_Int__UNIV,axiom,
! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ( ( inf_in1693821330460682071rm_a_b @ A @ B )
= top_to1314267278846557113rm_a_b )
= ( ( A = top_to1314267278846557113rm_a_b )
& ( B = top_to1314267278846557113rm_a_b ) ) ) ).
% Int_UNIV
thf(fact_54_Int__UNIV,axiom,
! [A: set_term_a_b,B: set_term_a_b] :
( ( ( inf_inf_set_term_a_b @ A @ B )
= top_top_set_term_a_b )
= ( ( A = top_top_set_term_a_b )
& ( B = top_top_set_term_a_b ) ) ) ).
% Int_UNIV
thf(fact_55_Int__UNIV,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( ( inf_in3813033988066259269_a_nat @ A @ B )
= top_to3353692345378799459_a_nat )
= ( ( A = top_to3353692345378799459_a_nat )
& ( B = top_to3353692345378799459_a_nat ) ) ) ).
% Int_UNIV
thf(fact_56_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_57_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_58_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_59_Int__insert__left__if0,axiom,
! [A2: term_a_b,C2: set_term_a_b,B: set_term_a_b] :
( ~ ( member_term_a_b @ A2 @ C2 )
=> ( ( inf_inf_set_term_a_b @ ( insert_term_a_b @ A2 @ B ) @ C2 )
= ( inf_inf_set_term_a_b @ B @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_60_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_61_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_62_Int__insert__left__if1,axiom,
! [A2: term_a_b,C2: set_term_a_b,B: set_term_a_b] :
( ( member_term_a_b @ A2 @ C2 )
=> ( ( inf_inf_set_term_a_b @ ( insert_term_a_b @ A2 @ B ) @ C2 )
= ( insert_term_a_b @ A2 @ ( inf_inf_set_term_a_b @ B @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_63_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_64_insert__inter__insert,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 ) @ ( insert8054603423593749677_a_nat @ A2 @ B ) )
= ( insert8054603423593749677_a_nat @ A2 @ ( inf_in3813033988066259269_a_nat @ A @ B ) ) ) ).
% insert_inter_insert
thf(fact_65_insert__inter__insert,axiom,
! [A2: produc357393685978478089rm_a_b,A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ( inf_in1693821330460682071rm_a_b @ ( insert7009541432154983385rm_a_b @ A2 @ A ) @ ( insert7009541432154983385rm_a_b @ A2 @ B ) )
= ( insert7009541432154983385rm_a_b @ A2 @ ( inf_in1693821330460682071rm_a_b @ A @ B ) ) ) ).
% insert_inter_insert
thf(fact_66_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_67_Int__insert__right__if0,axiom,
! [A2: term_a_b,A: set_term_a_b,B: set_term_a_b] :
( ~ ( member_term_a_b @ A2 @ A )
=> ( ( inf_inf_set_term_a_b @ A @ ( insert_term_a_b @ A2 @ B ) )
= ( inf_inf_set_term_a_b @ A @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_68_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_69_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_70_Int__insert__right__if1,axiom,
! [A2: term_a_b,A: set_term_a_b,B: set_term_a_b] :
( ( member_term_a_b @ A2 @ A )
=> ( ( inf_inf_set_term_a_b @ A @ ( insert_term_a_b @ A2 @ B ) )
= ( insert_term_a_b @ A2 @ ( inf_inf_set_term_a_b @ A @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_71_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_72_nle__le,axiom,
! [A2: nat,B2: nat] :
( ( ~ ( ord_less_eq_nat @ A2 @ B2 ) )
= ( ( ord_less_eq_nat @ B2 @ A2 )
& ( B2 != A2 ) ) ) ).
% nle_le
thf(fact_73_le__cases3,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ( ord_less_eq_nat @ X @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z ) )
=> ( ( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_eq_nat @ X @ Z ) )
=> ( ( ( ord_less_eq_nat @ X @ Z )
=> ~ ( ord_less_eq_nat @ Z @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z )
=> ~ ( ord_less_eq_nat @ Z @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z @ X )
=> ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_74_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y2: set_Pr4934435412358123699_a_nat,Z2: set_Pr4934435412358123699_a_nat] : ( Y2 = Z2 ) )
= ( ^ [X2: set_Pr4934435412358123699_a_nat,Y3: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X2 @ Y3 )
& ( ord_le8666007276011122963_a_nat @ Y3 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_75_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y2: nat,Z2: nat] : ( Y2 = Z2 ) )
= ( ^ [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
& ( ord_less_eq_nat @ Y3 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_76_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_77_ord__eq__le__trans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( A2 = B2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_78_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_79_ord__le__eq__trans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_80_order__antisym,axiom,
! [X: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X @ Y )
=> ( ( ord_le8666007276011122963_a_nat @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_81_order__antisym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_82_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_83_order_Otrans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% order.trans
thf(fact_84_order__trans,axiom,
! [X: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat,Z: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X @ Y )
=> ( ( ord_le8666007276011122963_a_nat @ Y @ Z )
=> ( ord_le8666007276011122963_a_nat @ X @ Z ) ) ) ).
% order_trans
thf(fact_85_order__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z )
=> ( ord_less_eq_nat @ X @ Z ) ) ) ).
% order_trans
thf(fact_86_linorder__wlog,axiom,
! [P: nat > nat > $o,A2: nat,B2: nat] :
( ! [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( P @ A3 @ B3 ) )
=> ( ! [A3: nat,B3: nat] :
( ( P @ B3 @ A3 )
=> ( P @ A3 @ B3 ) )
=> ( P @ A2 @ B2 ) ) ) ).
% linorder_wlog
thf(fact_87_dual__order_Oeq__iff,axiom,
( ( ^ [Y2: set_Pr4934435412358123699_a_nat,Z2: set_Pr4934435412358123699_a_nat] : ( Y2 = Z2 ) )
= ( ^ [A4: set_Pr4934435412358123699_a_nat,B4: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ B4 @ A4 )
& ( ord_le8666007276011122963_a_nat @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_88_dual__order_Oeq__iff,axiom,
( ( ^ [Y2: nat,Z2: nat] : ( Y2 = Z2 ) )
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ B4 @ A4 )
& ( ord_less_eq_nat @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
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: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( ord_less_eq_nat @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_91_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_92_dual__order_Otrans,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( ord_less_eq_nat @ C @ B2 )
=> ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_93_Sup__eqI,axiom,
! [A: set_se6121441497158405097_a_nat,X: set_Pr4934435412358123699_a_nat] :
( ! [Y4: set_Pr4934435412358123699_a_nat] :
( ( member8034581882086557258_a_nat @ Y4 @ A )
=> ( ord_le8666007276011122963_a_nat @ Y4 @ X ) )
=> ( ! [Y4: set_Pr4934435412358123699_a_nat] :
( ! [Z3: set_Pr4934435412358123699_a_nat] :
( ( member8034581882086557258_a_nat @ Z3 @ A )
=> ( ord_le8666007276011122963_a_nat @ Z3 @ Y4 ) )
=> ( ord_le8666007276011122963_a_nat @ X @ Y4 ) )
=> ( ( comple3771923735958488838_a_nat @ A )
= X ) ) ) ).
% Sup_eqI
thf(fact_94_Sup__mono,axiom,
! [A: set_se6121441497158405097_a_nat,B: set_se6121441497158405097_a_nat] :
( ! [A3: set_Pr4934435412358123699_a_nat] :
( ( member8034581882086557258_a_nat @ A3 @ A )
=> ? [X4: set_Pr4934435412358123699_a_nat] :
( ( member8034581882086557258_a_nat @ X4 @ B )
& ( ord_le8666007276011122963_a_nat @ A3 @ X4 ) ) )
=> ( ord_le8666007276011122963_a_nat @ ( comple3771923735958488838_a_nat @ A ) @ ( comple3771923735958488838_a_nat @ B ) ) ) ).
% Sup_mono
thf(fact_95_Sup__least,axiom,
! [A: set_se6121441497158405097_a_nat,Z: set_Pr4934435412358123699_a_nat] :
( ! [X3: set_Pr4934435412358123699_a_nat] :
( ( member8034581882086557258_a_nat @ X3 @ A )
=> ( ord_le8666007276011122963_a_nat @ X3 @ Z ) )
=> ( ord_le8666007276011122963_a_nat @ ( comple3771923735958488838_a_nat @ A ) @ Z ) ) ).
% Sup_least
thf(fact_96_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_97_mem__Collect__eq,axiom,
! [A2: term_a_b,P: term_a_b > $o] :
( ( member_term_a_b @ A2 @ ( collect_term_a_b @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_98_Collect__mem__eq,axiom,
! [A: set_Pr4934435412358123699_a_nat] :
( ( collec4464134535221767506_a_nat
@ ^ [X2: product_prod_a_nat] : ( member5724188588386418708_a_nat @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_99_Collect__mem__eq,axiom,
! [A: set_term_a_b] :
( ( collect_term_a_b
@ ^ [X2: term_a_b] : ( member_term_a_b @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_100_Collect__cong,axiom,
! [P: term_a_b > $o,Q: term_a_b > $o] :
( ! [X3: term_a_b] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_term_a_b @ P )
= ( collect_term_a_b @ Q ) ) ) ).
% Collect_cong
thf(fact_101_Sup__upper,axiom,
! [X: set_Pr4934435412358123699_a_nat,A: set_se6121441497158405097_a_nat] :
( ( member8034581882086557258_a_nat @ X @ A )
=> ( ord_le8666007276011122963_a_nat @ X @ ( comple3771923735958488838_a_nat @ A ) ) ) ).
% Sup_upper
thf(fact_102_Sup__le__iff,axiom,
! [A: set_se6121441497158405097_a_nat,B2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ ( comple3771923735958488838_a_nat @ A ) @ B2 )
= ( ! [X2: set_Pr4934435412358123699_a_nat] :
( ( member8034581882086557258_a_nat @ X2 @ A )
=> ( ord_le8666007276011122963_a_nat @ X2 @ B2 ) ) ) ) ).
% Sup_le_iff
thf(fact_103_Sup__upper2,axiom,
! [U: set_Pr4934435412358123699_a_nat,A: set_se6121441497158405097_a_nat,V: set_Pr4934435412358123699_a_nat] :
( ( member8034581882086557258_a_nat @ U @ A )
=> ( ( ord_le8666007276011122963_a_nat @ V @ U )
=> ( ord_le8666007276011122963_a_nat @ V @ ( comple3771923735958488838_a_nat @ A ) ) ) ) ).
% Sup_upper2
thf(fact_104_Sup__subset__mono,axiom,
! [A: set_se6121441497158405097_a_nat,B: set_se6121441497158405097_a_nat] :
( ( ord_le832892785549241417_a_nat @ A @ B )
=> ( ord_le8666007276011122963_a_nat @ ( comple3771923735958488838_a_nat @ A ) @ ( comple3771923735958488838_a_nat @ B ) ) ) ).
% Sup_subset_mono
thf(fact_105_Sup__inter__less__eq,axiom,
! [A: set_se1467346398288745161rm_a_b,B: set_se1467346398288745161rm_a_b] : ( ord_le118470702582115849rm_a_b @ ( comple4970752389892389974rm_a_b @ ( inf_in8882418861125483447rm_a_b @ A @ B ) ) @ ( inf_in1693821330460682071rm_a_b @ ( comple4970752389892389974rm_a_b @ A ) @ ( comple4970752389892389974rm_a_b @ B ) ) ) ).
% Sup_inter_less_eq
thf(fact_106_Sup__inter__less__eq,axiom,
! [A: set_se6121441497158405097_a_nat,B: set_se6121441497158405097_a_nat] : ( ord_le8666007276011122963_a_nat @ ( comple3771923735958488838_a_nat @ ( inf_in608777104516080763_a_nat @ A @ B ) ) @ ( inf_in3813033988066259269_a_nat @ ( comple3771923735958488838_a_nat @ A ) @ ( comple3771923735958488838_a_nat @ B ) ) ) ).
% Sup_inter_less_eq
thf(fact_107_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_108_IntE,axiom,
! [C: term_a_b,A: set_term_a_b,B: set_term_a_b] :
( ( member_term_a_b @ C @ ( inf_inf_set_term_a_b @ A @ B ) )
=> ~ ( ( member_term_a_b @ C @ A )
=> ~ ( member_term_a_b @ C @ B ) ) ) ).
% IntE
thf(fact_109_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_110_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_111_IntD1,axiom,
! [C: term_a_b,A: set_term_a_b,B: set_term_a_b] :
( ( member_term_a_b @ C @ ( inf_inf_set_term_a_b @ A @ B ) )
=> ( member_term_a_b @ C @ A ) ) ).
% IntD1
thf(fact_112_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_113_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_114_IntD2,axiom,
! [C: term_a_b,A: set_term_a_b,B: set_term_a_b] :
( ( member_term_a_b @ C @ ( inf_inf_set_term_a_b @ A @ B ) )
=> ( member_term_a_b @ C @ B ) ) ).
% IntD2
thf(fact_115_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_116_Int__def,axiom,
( inf_in3813033988066259269_a_nat
= ( ^ [A5: set_Pr4934435412358123699_a_nat,B5: set_Pr4934435412358123699_a_nat] :
( collec4464134535221767506_a_nat
@ ^ [X2: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ X2 @ A5 )
& ( member5724188588386418708_a_nat @ X2 @ B5 ) ) ) ) ) ).
% Int_def
thf(fact_117_Int__def,axiom,
( inf_inf_set_term_a_b
= ( ^ [A5: set_term_a_b,B5: set_term_a_b] :
( collect_term_a_b
@ ^ [X2: term_a_b] :
( ( member_term_a_b @ X2 @ A5 )
& ( member_term_a_b @ X2 @ B5 ) ) ) ) ) ).
% Int_def
thf(fact_118_Int__def,axiom,
( inf_in1693821330460682071rm_a_b
= ( ^ [A5: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
( collec99177395925924084rm_a_b
@ ^ [X2: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X2 @ A5 )
& ( member5869715511025134514rm_a_b @ X2 @ B5 ) ) ) ) ) ).
% Int_def
thf(fact_119_in__mono,axiom,
! [A: set_term_a_b,B: set_term_a_b,X: term_a_b] :
( ( ord_le2705286416250468010rm_a_b @ A @ B )
=> ( ( member_term_a_b @ X @ A )
=> ( member_term_a_b @ X @ B ) ) ) ).
% in_mono
thf(fact_120_in__mono,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat,X: product_prod_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A @ B )
=> ( ( member5724188588386418708_a_nat @ X @ A )
=> ( member5724188588386418708_a_nat @ X @ B ) ) ) ).
% in_mono
thf(fact_121_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_122_insertE,axiom,
! [A2: term_a_b,B2: term_a_b,A: set_term_a_b] :
( ( member_term_a_b @ A2 @ ( insert_term_a_b @ B2 @ A ) )
=> ( ( A2 != B2 )
=> ( member_term_a_b @ A2 @ A ) ) ) ).
% insertE
thf(fact_123_subsetD,axiom,
! [A: set_term_a_b,B: set_term_a_b,C: term_a_b] :
( ( ord_le2705286416250468010rm_a_b @ A @ B )
=> ( ( member_term_a_b @ C @ A )
=> ( member_term_a_b @ C @ B ) ) ) ).
% subsetD
thf(fact_124_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_125_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_126_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_127_UNIV__def,axiom,
( top_top_set_term_a_b
= ( collect_term_a_b
@ ^ [X2: term_a_b] : $true ) ) ).
% UNIV_def
thf(fact_128_UNIV__def,axiom,
( top_to3353692345378799459_a_nat
= ( collec4464134535221767506_a_nat
@ ^ [X2: product_prod_a_nat] : $true ) ) ).
% UNIV_def
thf(fact_129_insertI1,axiom,
! [A2: product_prod_a_nat,B: set_Pr4934435412358123699_a_nat] : ( member5724188588386418708_a_nat @ A2 @ ( insert8054603423593749677_a_nat @ A2 @ B ) ) ).
% insertI1
thf(fact_130_insertI1,axiom,
! [A2: term_a_b,B: set_term_a_b] : ( member_term_a_b @ A2 @ ( insert_term_a_b @ A2 @ B ) ) ).
% insertI1
thf(fact_131_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_132_insertI2,axiom,
! [A2: term_a_b,B: set_term_a_b,B2: term_a_b] :
( ( member_term_a_b @ A2 @ B )
=> ( member_term_a_b @ A2 @ ( insert_term_a_b @ B2 @ B ) ) ) ).
% insertI2
thf(fact_133_Int__assoc,axiom,
! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b,C2: set_Pr4386577575007340137rm_a_b] :
( ( inf_in1693821330460682071rm_a_b @ ( inf_in1693821330460682071rm_a_b @ A @ B ) @ C2 )
= ( inf_in1693821330460682071rm_a_b @ A @ ( inf_in1693821330460682071rm_a_b @ B @ C2 ) ) ) ).
% Int_assoc
thf(fact_134_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_135_subset__eq,axiom,
( ord_le2705286416250468010rm_a_b
= ( ^ [A5: set_term_a_b,B5: set_term_a_b] :
! [X2: term_a_b] :
( ( member_term_a_b @ X2 @ A5 )
=> ( member_term_a_b @ X2 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_136_subset__eq,axiom,
( ord_le8666007276011122963_a_nat
= ( ^ [A5: set_Pr4934435412358123699_a_nat,B5: set_Pr4934435412358123699_a_nat] :
! [X2: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ X2 @ A5 )
=> ( member5724188588386418708_a_nat @ X2 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_137_Int__absorb,axiom,
! [A: set_Pr4386577575007340137rm_a_b] :
( ( inf_in1693821330460682071rm_a_b @ A @ A )
= A ) ).
% Int_absorb
thf(fact_138_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_139_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_140_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_141_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_142_equalityD1,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( A = B )
=> ( ord_le8666007276011122963_a_nat @ A @ B ) ) ).
% equalityD1
thf(fact_143_equalityD2,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( A = B )
=> ( ord_le8666007276011122963_a_nat @ B @ A ) ) ).
% equalityD2
thf(fact_144_Set_Oset__insert,axiom,
! [X: product_prod_a_nat,A: set_Pr4934435412358123699_a_nat] :
( ( member5724188588386418708_a_nat @ X @ A )
=> ~ ! [B6: set_Pr4934435412358123699_a_nat] :
( ( A
= ( insert8054603423593749677_a_nat @ X @ B6 ) )
=> ( member5724188588386418708_a_nat @ X @ B6 ) ) ) ).
% Set.set_insert
thf(fact_145_Set_Oset__insert,axiom,
! [X: term_a_b,A: set_term_a_b] :
( ( member_term_a_b @ X @ A )
=> ~ ! [B6: set_term_a_b] :
( ( A
= ( insert_term_a_b @ X @ B6 ) )
=> ( member_term_a_b @ X @ B6 ) ) ) ).
% Set.set_insert
thf(fact_146_subset__iff,axiom,
( ord_le2705286416250468010rm_a_b
= ( ^ [A5: set_term_a_b,B5: set_term_a_b] :
! [T: term_a_b] :
( ( member_term_a_b @ T @ A5 )
=> ( member_term_a_b @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_147_subset__iff,axiom,
( ord_le8666007276011122963_a_nat
= ( ^ [A5: set_Pr4934435412358123699_a_nat,B5: set_Pr4934435412358123699_a_nat] :
! [T: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ T @ A5 )
=> ( member5724188588386418708_a_nat @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_148_Int__Collect,axiom,
! [X: product_prod_a_nat,A: set_Pr4934435412358123699_a_nat,P: product_prod_a_nat > $o] :
( ( member5724188588386418708_a_nat @ X @ ( inf_in3813033988066259269_a_nat @ A @ ( collec4464134535221767506_a_nat @ P ) ) )
= ( ( member5724188588386418708_a_nat @ X @ A )
& ( P @ X ) ) ) ).
% Int_Collect
thf(fact_149_Int__Collect,axiom,
! [X: term_a_b,A: set_term_a_b,P: term_a_b > $o] :
( ( member_term_a_b @ X @ ( inf_inf_set_term_a_b @ A @ ( collect_term_a_b @ P ) ) )
= ( ( member_term_a_b @ X @ A )
& ( P @ X ) ) ) ).
% Int_Collect
thf(fact_150_Int__Collect,axiom,
! [X: produc357393685978478089rm_a_b,A: set_Pr4386577575007340137rm_a_b,P: produc357393685978478089rm_a_b > $o] :
( ( member5869715511025134514rm_a_b @ X @ ( inf_in1693821330460682071rm_a_b @ A @ ( collec99177395925924084rm_a_b @ P ) ) )
= ( ( member5869715511025134514rm_a_b @ X @ A )
& ( P @ X ) ) ) ).
% Int_Collect
thf(fact_151_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_152_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_153_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_154_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_155_Int__commute,axiom,
( inf_in1693821330460682071rm_a_b
= ( ^ [A5: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] : ( inf_in1693821330460682071rm_a_b @ B5 @ A5 ) ) ) ).
% Int_commute
thf(fact_156_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_157_subset__refl,axiom,
! [A: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ A @ A ) ).
% subset_refl
thf(fact_158_Collect__mono,axiom,
! [P: term_a_b > $o,Q: term_a_b > $o] :
( ! [X3: term_a_b] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_le2705286416250468010rm_a_b @ ( collect_term_a_b @ P ) @ ( collect_term_a_b @ Q ) ) ) ).
% Collect_mono
thf(fact_159_Collect__mono,axiom,
! [P: product_prod_a_nat > $o,Q: product_prod_a_nat > $o] :
( ! [X3: product_prod_a_nat] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_le8666007276011122963_a_nat @ ( collec4464134535221767506_a_nat @ P ) @ ( collec4464134535221767506_a_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_160_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_161_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_162_insert__compr,axiom,
( insert8054603423593749677_a_nat
= ( ^ [A4: product_prod_a_nat,B5: set_Pr4934435412358123699_a_nat] :
( collec4464134535221767506_a_nat
@ ^ [X2: product_prod_a_nat] :
( ( X2 = A4 )
| ( member5724188588386418708_a_nat @ X2 @ B5 ) ) ) ) ) ).
% insert_compr
thf(fact_163_insert__compr,axiom,
( insert_term_a_b
= ( ^ [A4: term_a_b,B5: set_term_a_b] :
( collect_term_a_b
@ ^ [X2: term_a_b] :
( ( X2 = A4 )
| ( member_term_a_b @ X2 @ B5 ) ) ) ) ) ).
% insert_compr
thf(fact_164_insert__ident,axiom,
! [X: product_prod_a_nat,A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ~ ( member5724188588386418708_a_nat @ X @ A )
=> ( ~ ( member5724188588386418708_a_nat @ X @ B )
=> ( ( ( insert8054603423593749677_a_nat @ X @ A )
= ( insert8054603423593749677_a_nat @ X @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_165_insert__ident,axiom,
! [X: term_a_b,A: set_term_a_b,B: set_term_a_b] :
( ~ ( member_term_a_b @ X @ A )
=> ( ~ ( member_term_a_b @ X @ B )
=> ( ( ( insert_term_a_b @ X @ A )
= ( insert_term_a_b @ X @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_166_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_167_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_168_antisym,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_169_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_170_insert__absorb,axiom,
! [A2: term_a_b,A: set_term_a_b] :
( ( member_term_a_b @ A2 @ A )
=> ( ( insert_term_a_b @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_171_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 )
=> ? [C3: set_Pr4934435412358123699_a_nat] :
( ( A
= ( insert8054603423593749677_a_nat @ B2 @ C3 ) )
& ~ ( member5724188588386418708_a_nat @ B2 @ C3 )
& ( B
= ( insert8054603423593749677_a_nat @ A2 @ C3 ) )
& ~ ( member5724188588386418708_a_nat @ A2 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_172_insert__eq__iff,axiom,
! [A2: term_a_b,A: set_term_a_b,B2: term_a_b,B: set_term_a_b] :
( ~ ( member_term_a_b @ A2 @ A )
=> ( ~ ( member_term_a_b @ B2 @ B )
=> ( ( ( insert_term_a_b @ A2 @ A )
= ( insert_term_a_b @ B2 @ B ) )
= ( ( ( A2 = B2 )
=> ( A = B ) )
& ( ( A2 != B2 )
=> ? [C3: set_term_a_b] :
( ( A
= ( insert_term_a_b @ B2 @ C3 ) )
& ~ ( member_term_a_b @ B2 @ C3 )
& ( B
= ( insert_term_a_b @ A2 @ C3 ) )
& ~ ( member_term_a_b @ A2 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_173_set__eq__subset,axiom,
( ( ^ [Y2: set_Pr4934435412358123699_a_nat,Z2: set_Pr4934435412358123699_a_nat] : ( Y2 = Z2 ) )
= ( ^ [A5: set_Pr4934435412358123699_a_nat,B5: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A5 @ B5 )
& ( ord_le8666007276011122963_a_nat @ B5 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_174_subset__insert,axiom,
! [X: term_a_b,A: set_term_a_b,B: set_term_a_b] :
( ~ ( member_term_a_b @ X @ A )
=> ( ( ord_le2705286416250468010rm_a_b @ A @ ( insert_term_a_b @ X @ B ) )
= ( ord_le2705286416250468010rm_a_b @ A @ B ) ) ) ).
% subset_insert
thf(fact_175_subset__insert,axiom,
! [X: product_prod_a_nat,A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ~ ( member5724188588386418708_a_nat @ X @ A )
=> ( ( ord_le8666007276011122963_a_nat @ A @ ( insert8054603423593749677_a_nat @ X @ B ) )
= ( ord_le8666007276011122963_a_nat @ A @ B ) ) ) ).
% subset_insert
thf(fact_176_Collect__subset,axiom,
! [A: set_term_a_b,P: term_a_b > $o] :
( ord_le2705286416250468010rm_a_b
@ ( collect_term_a_b
@ ^ [X2: term_a_b] :
( ( member_term_a_b @ X2 @ A )
& ( P @ X2 ) ) )
@ A ) ).
% Collect_subset
thf(fact_177_Collect__subset,axiom,
! [A: set_Pr4934435412358123699_a_nat,P: product_prod_a_nat > $o] :
( ord_le8666007276011122963_a_nat
@ ( collec4464134535221767506_a_nat
@ ^ [X2: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ X2 @ A )
& ( P @ X2 ) ) )
@ A ) ).
% Collect_subset
thf(fact_178_insert__Collect,axiom,
! [A2: product_prod_a_nat,P: product_prod_a_nat > $o] :
( ( insert8054603423593749677_a_nat @ A2 @ ( collec4464134535221767506_a_nat @ P ) )
= ( collec4464134535221767506_a_nat
@ ^ [U2: product_prod_a_nat] :
( ( U2 != A2 )
=> ( P @ U2 ) ) ) ) ).
% insert_Collect
thf(fact_179_insert__Collect,axiom,
! [A2: term_a_b,P: term_a_b > $o] :
( ( insert_term_a_b @ A2 @ ( collect_term_a_b @ P ) )
= ( collect_term_a_b
@ ^ [U2: term_a_b] :
( ( U2 != A2 )
=> ( P @ U2 ) ) ) ) ).
% insert_Collect
thf(fact_180_insert__commute,axiom,
! [X: product_prod_a_nat,Y: product_prod_a_nat,A: set_Pr4934435412358123699_a_nat] :
( ( insert8054603423593749677_a_nat @ X @ ( insert8054603423593749677_a_nat @ Y @ A ) )
= ( insert8054603423593749677_a_nat @ Y @ ( insert8054603423593749677_a_nat @ X @ A ) ) ) ).
% insert_commute
thf(fact_181_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_182_Collect__conj__eq,axiom,
! [P: term_a_b > $o,Q: term_a_b > $o] :
( ( collect_term_a_b
@ ^ [X2: term_a_b] :
( ( P @ X2 )
& ( Q @ X2 ) ) )
= ( inf_inf_set_term_a_b @ ( collect_term_a_b @ P ) @ ( collect_term_a_b @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_183_Collect__conj__eq,axiom,
! [P: produc357393685978478089rm_a_b > $o,Q: produc357393685978478089rm_a_b > $o] :
( ( collec99177395925924084rm_a_b
@ ^ [X2: produc357393685978478089rm_a_b] :
( ( P @ X2 )
& ( Q @ X2 ) ) )
= ( inf_in1693821330460682071rm_a_b @ ( collec99177395925924084rm_a_b @ P ) @ ( collec99177395925924084rm_a_b @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_184_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_185_Int__insert__left,axiom,
! [A2: term_a_b,C2: set_term_a_b,B: set_term_a_b] :
( ( ( member_term_a_b @ A2 @ C2 )
=> ( ( inf_inf_set_term_a_b @ ( insert_term_a_b @ A2 @ B ) @ C2 )
= ( insert_term_a_b @ A2 @ ( inf_inf_set_term_a_b @ B @ C2 ) ) ) )
& ( ~ ( member_term_a_b @ A2 @ C2 )
=> ( ( inf_inf_set_term_a_b @ ( insert_term_a_b @ A2 @ B ) @ C2 )
= ( inf_inf_set_term_a_b @ B @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_186_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_187_Int__left__absorb,axiom,
! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ( inf_in1693821330460682071rm_a_b @ A @ ( inf_in1693821330460682071rm_a_b @ A @ B ) )
= ( inf_in1693821330460682071rm_a_b @ A @ B ) ) ).
% Int_left_absorb
thf(fact_188_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_189_Collect__mono__iff,axiom,
! [P: term_a_b > $o,Q: term_a_b > $o] :
( ( ord_le2705286416250468010rm_a_b @ ( collect_term_a_b @ P ) @ ( collect_term_a_b @ Q ) )
= ( ! [X2: term_a_b] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_190_Collect__mono__iff,axiom,
! [P: product_prod_a_nat > $o,Q: product_prod_a_nat > $o] :
( ( ord_le8666007276011122963_a_nat @ ( collec4464134535221767506_a_nat @ P ) @ ( collec4464134535221767506_a_nat @ Q ) )
= ( ! [X2: product_prod_a_nat] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_191_Int__Collect__mono,axiom,
! [A: set_term_a_b,B: set_term_a_b,P: term_a_b > $o,Q: term_a_b > $o] :
( ( ord_le2705286416250468010rm_a_b @ A @ B )
=> ( ! [X3: term_a_b] :
( ( member_term_a_b @ X3 @ A )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_le2705286416250468010rm_a_b @ ( inf_inf_set_term_a_b @ A @ ( collect_term_a_b @ P ) ) @ ( inf_inf_set_term_a_b @ B @ ( collect_term_a_b @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_192_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 )
=> ( ! [X3: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X3 @ A )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( 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_193_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 )
=> ( ! [X3: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ X3 @ A )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( 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_194_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_195_Int__insert__right,axiom,
! [A2: term_a_b,A: set_term_a_b,B: set_term_a_b] :
( ( ( member_term_a_b @ A2 @ A )
=> ( ( inf_inf_set_term_a_b @ A @ ( insert_term_a_b @ A2 @ B ) )
= ( insert_term_a_b @ A2 @ ( inf_inf_set_term_a_b @ A @ B ) ) ) )
& ( ~ ( member_term_a_b @ A2 @ A )
=> ( ( inf_inf_set_term_a_b @ A @ ( insert_term_a_b @ A2 @ B ) )
= ( inf_inf_set_term_a_b @ A @ B ) ) ) ) ).
% Int_insert_right
thf(fact_196_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_197_Int__left__commute,axiom,
! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b,C2: set_Pr4386577575007340137rm_a_b] :
( ( inf_in1693821330460682071rm_a_b @ A @ ( inf_in1693821330460682071rm_a_b @ B @ C2 ) )
= ( inf_in1693821330460682071rm_a_b @ B @ ( inf_in1693821330460682071rm_a_b @ A @ C2 ) ) ) ).
% Int_left_commute
thf(fact_198_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y2: set_Pr4934435412358123699_a_nat,Z2: set_Pr4934435412358123699_a_nat] : ( Y2 = Z2 ) )
= ( ^ [A4: set_Pr4934435412358123699_a_nat,B4: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A4 @ B4 )
& ( ord_le8666007276011122963_a_nat @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_199_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y2: nat,Z2: nat] : ( Y2 = Z2 ) )
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ A4 @ B4 )
& ( ord_less_eq_nat @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_200_order__subst1,axiom,
! [A2: set_Pr4934435412358123699_a_nat,F: set_Pr4934435412358123699_a_nat > set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat,C: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_le8666007276011122963_a_nat @ B2 @ C )
=> ( ! [X3: set_Pr4934435412358123699_a_nat,Y4: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X3 @ Y4 )
=> ( ord_le8666007276011122963_a_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le8666007276011122963_a_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_201_order__subst1,axiom,
! [A2: set_Pr4934435412358123699_a_nat,F: nat > set_Pr4934435412358123699_a_nat,B2: nat,C: nat] :
( ( ord_le8666007276011122963_a_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_le8666007276011122963_a_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le8666007276011122963_a_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_202_order__subst1,axiom,
! [A2: nat,F: set_Pr4934435412358123699_a_nat > nat,B2: set_Pr4934435412358123699_a_nat,C: set_Pr4934435412358123699_a_nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_le8666007276011122963_a_nat @ B2 @ C )
=> ( ! [X3: set_Pr4934435412358123699_a_nat,Y4: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_203_order__subst1,axiom,
! [A2: nat,F: nat > nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_204_order__subst2,axiom,
! [A2: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat,F: set_Pr4934435412358123699_a_nat > set_Pr4934435412358123699_a_nat,C: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A2 @ B2 )
=> ( ( ord_le8666007276011122963_a_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: set_Pr4934435412358123699_a_nat,Y4: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X3 @ Y4 )
=> ( ord_le8666007276011122963_a_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le8666007276011122963_a_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_205_order__subst2,axiom,
! [A2: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat,F: set_Pr4934435412358123699_a_nat > nat,C: nat] :
( ( ord_le8666007276011122963_a_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: set_Pr4934435412358123699_a_nat,Y4: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_206_order__subst2,axiom,
! [A2: nat,B2: nat,F: nat > set_Pr4934435412358123699_a_nat,C: set_Pr4934435412358123699_a_nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_le8666007276011122963_a_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_le8666007276011122963_a_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le8666007276011122963_a_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_207_order__subst2,axiom,
! [A2: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_208_mk__disjoint__insert,axiom,
! [A2: product_prod_a_nat,A: set_Pr4934435412358123699_a_nat] :
( ( member5724188588386418708_a_nat @ A2 @ A )
=> ? [B6: set_Pr4934435412358123699_a_nat] :
( ( A
= ( insert8054603423593749677_a_nat @ A2 @ B6 ) )
& ~ ( member5724188588386418708_a_nat @ A2 @ B6 ) ) ) ).
% mk_disjoint_insert
thf(fact_209_mk__disjoint__insert,axiom,
! [A2: term_a_b,A: set_term_a_b] :
( ( member_term_a_b @ A2 @ A )
=> ? [B6: set_term_a_b] :
( ( A
= ( insert_term_a_b @ A2 @ B6 ) )
& ~ ( member_term_a_b @ A2 @ B6 ) ) ) ).
% mk_disjoint_insert
thf(fact_210_order__eq__refl,axiom,
! [X: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat] :
( ( X = Y )
=> ( ord_le8666007276011122963_a_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_211_order__eq__refl,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_212_linorder__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_linear
thf(fact_213_ord__eq__le__subst,axiom,
! [A2: set_Pr4934435412358123699_a_nat,F: set_Pr4934435412358123699_a_nat > set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat,C: set_Pr4934435412358123699_a_nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_le8666007276011122963_a_nat @ B2 @ C )
=> ( ! [X3: set_Pr4934435412358123699_a_nat,Y4: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X3 @ Y4 )
=> ( ord_le8666007276011122963_a_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le8666007276011122963_a_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_214_ord__eq__le__subst,axiom,
! [A2: nat,F: set_Pr4934435412358123699_a_nat > nat,B2: set_Pr4934435412358123699_a_nat,C: set_Pr4934435412358123699_a_nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_le8666007276011122963_a_nat @ B2 @ C )
=> ( ! [X3: set_Pr4934435412358123699_a_nat,Y4: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_215_ord__eq__le__subst,axiom,
! [A2: set_Pr4934435412358123699_a_nat,F: nat > set_Pr4934435412358123699_a_nat,B2: nat,C: nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_le8666007276011122963_a_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le8666007276011122963_a_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_216_ord__eq__le__subst,axiom,
! [A2: nat,F: nat > nat,B2: nat,C: nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_217_ord__le__eq__subst,axiom,
! [A2: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat,F: set_Pr4934435412358123699_a_nat > set_Pr4934435412358123699_a_nat,C: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: set_Pr4934435412358123699_a_nat,Y4: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X3 @ Y4 )
=> ( ord_le8666007276011122963_a_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le8666007276011122963_a_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_218_ord__le__eq__subst,axiom,
! [A2: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat,F: set_Pr4934435412358123699_a_nat > nat,C: nat] :
( ( ord_le8666007276011122963_a_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: set_Pr4934435412358123699_a_nat,Y4: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_219_ord__le__eq__subst,axiom,
! [A2: nat,B2: nat,F: nat > set_Pr4934435412358123699_a_nat,C: set_Pr4934435412358123699_a_nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_le8666007276011122963_a_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le8666007276011122963_a_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_220_ord__le__eq__subst,axiom,
! [A2: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_221_linorder__le__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_222_Union__mono,axiom,
! [A: set_se6121441497158405097_a_nat,B: set_se6121441497158405097_a_nat] :
( ( ord_le832892785549241417_a_nat @ A @ B )
=> ( ord_le8666007276011122963_a_nat @ ( comple3771923735958488838_a_nat @ A ) @ ( comple3771923735958488838_a_nat @ B ) ) ) ).
% Union_mono
thf(fact_223_order__antisym__conv,axiom,
! [Y: set_Pr4934435412358123699_a_nat,X: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ Y @ X )
=> ( ( ord_le8666007276011122963_a_nat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_224_order__antisym__conv,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_225_Union__least,axiom,
! [A: set_se6121441497158405097_a_nat,C2: set_Pr4934435412358123699_a_nat] :
( ! [X5: set_Pr4934435412358123699_a_nat] :
( ( member8034581882086557258_a_nat @ X5 @ A )
=> ( ord_le8666007276011122963_a_nat @ X5 @ C2 ) )
=> ( ord_le8666007276011122963_a_nat @ ( comple3771923735958488838_a_nat @ A ) @ C2 ) ) ).
% Union_least
thf(fact_226_Union__upper,axiom,
! [B: set_Pr4934435412358123699_a_nat,A: set_se6121441497158405097_a_nat] :
( ( member8034581882086557258_a_nat @ B @ A )
=> ( ord_le8666007276011122963_a_nat @ B @ ( comple3771923735958488838_a_nat @ A ) ) ) ).
% Union_upper
thf(fact_227_Union__subsetI,axiom,
! [A: set_se6121441497158405097_a_nat,B: set_se6121441497158405097_a_nat] :
( ! [X3: set_Pr4934435412358123699_a_nat] :
( ( member8034581882086557258_a_nat @ X3 @ A )
=> ? [Y5: set_Pr4934435412358123699_a_nat] :
( ( member8034581882086557258_a_nat @ Y5 @ B )
& ( ord_le8666007276011122963_a_nat @ X3 @ Y5 ) ) )
=> ( ord_le8666007276011122963_a_nat @ ( comple3771923735958488838_a_nat @ A ) @ ( comple3771923735958488838_a_nat @ B ) ) ) ).
% Union_subsetI
thf(fact_228_llrg__CR,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,F2: set_Pr4934435412358123699_a_nat,C: a] :
( ( ord_le8666007276011122963_a_nat @ ( terms_7988297476397195622_a_b_b @ R2 ) @ F2 )
=> ( ~ ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ C @ zero_zero_nat ) @ F2 )
=> ( ( rewrit5000278123254648670_b_a_b @ R2 )
=> ( ( abstra8448919418672941150rm_a_b
@ ( inf_in1693821330460682071rm_a_b @ ( sig_step_a_b @ ( insert8054603423593749677_a_nat @ ( product_Pair_a_nat @ C @ zero_zero_nat ) @ F2 ) @ ( rstep_a_b @ R2 ) )
@ ( produc1953217797448512882rm_a_b @ ( collect_term_a_b @ term_ground_a_b )
@ ^ [Uu: term_a_b] : ( collect_term_a_b @ term_ground_a_b ) ) )
@ top_top_set_term_a_b )
=> ( abstra8448919418672941150rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) ) @ top_top_set_term_a_b ) ) ) ) ) ).
% llrg_CR
thf(fact_229_Union__Int__subset,axiom,
! [A: set_se1467346398288745161rm_a_b,B: set_se1467346398288745161rm_a_b] : ( ord_le118470702582115849rm_a_b @ ( comple4970752389892389974rm_a_b @ ( inf_in8882418861125483447rm_a_b @ A @ B ) ) @ ( inf_in1693821330460682071rm_a_b @ ( comple4970752389892389974rm_a_b @ A ) @ ( comple4970752389892389974rm_a_b @ B ) ) ) ).
% Union_Int_subset
thf(fact_230_Union__Int__subset,axiom,
! [A: set_se6121441497158405097_a_nat,B: set_se6121441497158405097_a_nat] : ( ord_le8666007276011122963_a_nat @ ( comple3771923735958488838_a_nat @ ( inf_in608777104516080763_a_nat @ A @ B ) ) @ ( inf_in3813033988066259269_a_nat @ ( comple3771923735958488838_a_nat @ A ) @ ( comple3771923735958488838_a_nat @ B ) ) ) ).
% Union_Int_subset
thf(fact_231_CR__imp__UNC,axiom,
! [R: set_Pr4386577575007340137rm_a_b] :
( ( abstra8448919418672941150rm_a_b @ R @ top_top_set_term_a_b )
=> ( abstra4649771476585718372rm_a_b @ R ) ) ).
% CR_imp_UNC
thf(fact_232_CR__imp__UNC,axiom,
! [R: set_Pr1811044260758604347_a_nat] :
( ( abstra7964079670522473415_a_nat @ R @ top_to3353692345378799459_a_nat )
=> ( abstra7179253994116303565_a_nat @ R ) ) ).
% CR_imp_UNC
thf(fact_233_top__set__def,axiom,
( top_top_set_term_a_b
= ( collect_term_a_b @ top_top_term_a_b_o ) ) ).
% top_set_def
thf(fact_234_top__set__def,axiom,
( top_to3353692345378799459_a_nat
= ( collec4464134535221767506_a_nat @ top_to1985590801843794146_nat_o ) ) ).
% top_set_def
thf(fact_235_Union__UNIV,axiom,
( ( comple5064467279716934173rm_a_b @ top_to5470816188456996400rm_a_b )
= top_top_set_term_a_b ) ).
% Union_UNIV
thf(fact_236_Union__UNIV,axiom,
( ( comple3771923735958488838_a_nat @ top_to4675147776276609689_a_nat )
= top_to3353692345378799459_a_nat ) ).
% Union_UNIV
thf(fact_237_Int__UNIV__right,axiom,
! [A: set_Pr4386577575007340137rm_a_b] :
( ( inf_in1693821330460682071rm_a_b @ A @ top_to1314267278846557113rm_a_b )
= A ) ).
% Int_UNIV_right
thf(fact_238_Int__UNIV__right,axiom,
! [A: set_term_a_b] :
( ( inf_inf_set_term_a_b @ A @ top_top_set_term_a_b )
= A ) ).
% Int_UNIV_right
thf(fact_239_Int__UNIV__right,axiom,
! [A: set_Pr4934435412358123699_a_nat] :
( ( inf_in3813033988066259269_a_nat @ A @ top_to3353692345378799459_a_nat )
= A ) ).
% Int_UNIV_right
thf(fact_240_Int__UNIV__left,axiom,
! [B: set_Pr4386577575007340137rm_a_b] :
( ( inf_in1693821330460682071rm_a_b @ top_to1314267278846557113rm_a_b @ B )
= B ) ).
% Int_UNIV_left
thf(fact_241_Int__UNIV__left,axiom,
! [B: set_term_a_b] :
( ( inf_inf_set_term_a_b @ top_top_set_term_a_b @ B )
= B ) ).
% Int_UNIV_left
thf(fact_242_Int__UNIV__left,axiom,
! [B: set_Pr4934435412358123699_a_nat] :
( ( inf_in3813033988066259269_a_nat @ top_to3353692345378799459_a_nat @ B )
= B ) ).
% Int_UNIV_left
thf(fact_243_insert__UNIV,axiom,
! [X: term_a_b] :
( ( insert_term_a_b @ X @ top_top_set_term_a_b )
= top_top_set_term_a_b ) ).
% insert_UNIV
thf(fact_244_insert__UNIV,axiom,
! [X: product_prod_a_nat] :
( ( insert8054603423593749677_a_nat @ X @ top_to3353692345378799459_a_nat )
= top_to3353692345378799459_a_nat ) ).
% insert_UNIV
thf(fact_245_top_Oextremum__uniqueI,axiom,
! [A2: set_term_a_b] :
( ( ord_le2705286416250468010rm_a_b @ top_top_set_term_a_b @ A2 )
=> ( A2 = top_top_set_term_a_b ) ) ).
% top.extremum_uniqueI
thf(fact_246_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_247_top_Oextremum__unique,axiom,
! [A2: set_term_a_b] :
( ( ord_le2705286416250468010rm_a_b @ top_top_set_term_a_b @ A2 )
= ( A2 = top_top_set_term_a_b ) ) ).
% top.extremum_unique
thf(fact_248_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_249_top__greatest,axiom,
! [A2: set_term_a_b] : ( ord_le2705286416250468010rm_a_b @ A2 @ top_top_set_term_a_b ) ).
% top_greatest
thf(fact_250_top__greatest,axiom,
! [A2: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ A2 @ top_to3353692345378799459_a_nat ) ).
% top_greatest
thf(fact_251_subset__UNIV,axiom,
! [A: set_term_a_b] : ( ord_le2705286416250468010rm_a_b @ A @ top_top_set_term_a_b ) ).
% subset_UNIV
thf(fact_252_subset__UNIV,axiom,
! [A: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ A @ top_to3353692345378799459_a_nat ) ).
% subset_UNIV
thf(fact_253_llrg__SCR,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,F2: set_Pr4934435412358123699_a_nat,C: a] :
( ( ord_le8666007276011122963_a_nat @ ( terms_7988297476397195622_a_b_b @ R2 ) @ F2 )
=> ( ~ ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ C @ zero_zero_nat ) @ F2 )
=> ( ( rewrit5000278123254648670_b_a_b @ R2 )
=> ( ( rewrit4966517460798004134rm_a_b
@ ( inf_in1693821330460682071rm_a_b @ ( sig_step_a_b @ ( insert8054603423593749677_a_nat @ ( product_Pair_a_nat @ C @ zero_zero_nat ) @ F2 ) @ ( rstep_a_b @ R2 ) )
@ ( produc1953217797448512882rm_a_b @ ( collect_term_a_b @ term_ground_a_b )
@ ^ [Uu: term_a_b] : ( collect_term_a_b @ term_ground_a_b ) ) )
@ top_top_set_term_a_b )
=> ( rewrit4966517460798004134rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) ) @ top_top_set_term_a_b ) ) ) ) ) ).
% llrg_SCR
thf(fact_254_llrg__commute,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,F2: set_Pr4934435412358123699_a_nat,S2: set_Pr4386577575007340137rm_a_b,C: a] :
( ( ord_le8666007276011122963_a_nat @ ( terms_7988297476397195622_a_b_b @ R2 ) @ F2 )
=> ( ( ord_le8666007276011122963_a_nat @ ( terms_7988297476397195622_a_b_b @ S2 ) @ F2 )
=> ( ~ ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ C @ zero_zero_nat ) @ F2 )
=> ( ( rewrit5000278123254648670_b_a_b @ R2 )
=> ( ( rewrit5000278123254648670_b_a_b @ S2 )
=> ( ( abstra455195320138238452rm_a_b
@ ( inf_in1693821330460682071rm_a_b @ ( sig_step_a_b @ ( insert8054603423593749677_a_nat @ ( product_Pair_a_nat @ C @ zero_zero_nat ) @ F2 ) @ ( rstep_a_b @ R2 ) )
@ ( produc1953217797448512882rm_a_b @ ( collect_term_a_b @ term_ground_a_b )
@ ^ [Uu: term_a_b] : ( collect_term_a_b @ term_ground_a_b ) ) )
@ ( inf_in1693821330460682071rm_a_b @ ( sig_step_a_b @ ( insert8054603423593749677_a_nat @ ( product_Pair_a_nat @ C @ zero_zero_nat ) @ F2 ) @ ( rstep_a_b @ S2 ) )
@ ( produc1953217797448512882rm_a_b @ ( collect_term_a_b @ term_ground_a_b )
@ ^ [Uu: term_a_b] : ( collect_term_a_b @ term_ground_a_b ) ) ) )
=> ( abstra455195320138238452rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) ) @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ S2 ) ) ) ) ) ) ) ) ) ).
% llrg_commute
thf(fact_255_llrg__WCR,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,F2: set_Pr4934435412358123699_a_nat,C: a] :
( ( ord_le8666007276011122963_a_nat @ ( terms_7988297476397195622_a_b_b @ R2 ) @ F2 )
=> ( ~ ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ C @ zero_zero_nat ) @ F2 )
=> ( ( rewrit5000278123254648670_b_a_b @ R2 )
=> ( ( abstra5026849107050860991rm_a_b
@ ( inf_in1693821330460682071rm_a_b @ ( sig_step_a_b @ ( insert8054603423593749677_a_nat @ ( product_Pair_a_nat @ C @ zero_zero_nat ) @ F2 ) @ ( rstep_a_b @ R2 ) )
@ ( produc1953217797448512882rm_a_b @ ( collect_term_a_b @ term_ground_a_b )
@ ^ [Uu: term_a_b] : ( collect_term_a_b @ term_ground_a_b ) ) )
@ top_top_set_term_a_b )
=> ( abstra5026849107050860991rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) ) @ top_top_set_term_a_b ) ) ) ) ) ).
% llrg_WCR
thf(fact_256_sig__step__grstep__dist,axiom,
! [F2: set_Pr4934435412358123699_a_nat,R2: set_Pr4386577575007340137rm_a_b] :
( ( inf_in1693821330460682071rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) )
@ ( produc1953217797448512882rm_a_b @ ( collect_term_a_b @ term_ground_a_b )
@ ^ [Uu: term_a_b] : ( collect_term_a_b @ term_ground_a_b ) ) )
= ( sig_step_a_b @ F2
@ ( inf_in1693821330460682071rm_a_b @ ( rstep_a_b @ R2 )
@ ( produc1953217797448512882rm_a_b @ ( collect_term_a_b @ term_ground_a_b )
@ ^ [Uu: term_a_b] : ( collect_term_a_b @ term_ground_a_b ) ) ) ) ) ).
% sig_step_grstep_dist
thf(fact_257_UNIV__Times__UNIV,axiom,
( ( product_Sigma_a_nat @ top_top_set_a
@ ^ [Uu: a] : top_top_set_nat )
= top_to3353692345378799459_a_nat ) ).
% UNIV_Times_UNIV
thf(fact_258_UNIV__Times__UNIV,axiom,
( ( produc1953217797448512882rm_a_b @ top_top_set_term_a_b
@ ^ [Uu: term_a_b] : top_top_set_term_a_b )
= top_to1314267278846557113rm_a_b ) ).
% UNIV_Times_UNIV
thf(fact_259_UNIV__Times__UNIV,axiom,
( ( produc18138527422250715_a_nat @ top_top_set_term_a_b
@ ^ [Uu: term_a_b] : top_to3353692345378799459_a_nat )
= top_to8175914192959557538_a_nat ) ).
% UNIV_Times_UNIV
thf(fact_260_UNIV__Times__UNIV,axiom,
( ( produc1406664433328107739rm_a_b @ top_to3353692345378799459_a_nat
@ ^ [Uu: product_prod_a_nat] : top_top_set_term_a_b )
= top_to5466818670715216930rm_a_b ) ).
% UNIV_Times_UNIV
thf(fact_261_UNIV__Times__UNIV,axiom,
( ( produc3832836940518797764_a_nat @ top_to3353692345378799459_a_nat
@ ^ [Uu: product_prod_a_nat] : top_to3353692345378799459_a_nat )
= top_to1901438589647237003_a_nat ) ).
% UNIV_Times_UNIV
thf(fact_262_SigmaI,axiom,
! [A2: product_prod_a_nat,A: set_Pr4934435412358123699_a_nat,B2: product_prod_a_nat,B: product_prod_a_nat > set_Pr4934435412358123699_a_nat] :
( ( member5724188588386418708_a_nat @ A2 @ A )
=> ( ( member5724188588386418708_a_nat @ B2 @ ( B @ A2 ) )
=> ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A2 @ B2 ) @ ( produc3832836940518797764_a_nat @ A @ B ) ) ) ) ).
% SigmaI
thf(fact_263_SigmaI,axiom,
! [A2: product_prod_a_nat,A: set_Pr4934435412358123699_a_nat,B2: term_a_b,B: product_prod_a_nat > set_term_a_b] :
( ( member5724188588386418708_a_nat @ A2 @ A )
=> ( ( member_term_a_b @ B2 @ ( B @ A2 ) )
=> ( member1876585039447381659rm_a_b @ ( produc4170664577406169130rm_a_b @ A2 @ B2 ) @ ( produc1406664433328107739rm_a_b @ A @ B ) ) ) ) ).
% SigmaI
thf(fact_264_SigmaI,axiom,
! [A2: term_a_b,A: set_term_a_b,B2: product_prod_a_nat,B: term_a_b > set_Pr4934435412358123699_a_nat] :
( ( member_term_a_b @ A2 @ A )
=> ( ( member5724188588386418708_a_nat @ B2 @ ( B @ A2 ) )
=> ( member4553550343464174107_a_nat @ ( produc2782138671500312106_a_nat @ A2 @ B2 ) @ ( produc18138527422250715_a_nat @ A @ B ) ) ) ) ).
% SigmaI
thf(fact_265_SigmaI,axiom,
! [A2: a,A: set_a,B2: nat,B: a > set_nat] :
( ( member_a @ A2 @ A )
=> ( ( member_nat @ B2 @ ( B @ A2 ) )
=> ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ A2 @ B2 ) @ ( product_Sigma_a_nat @ A @ B ) ) ) ) ).
% SigmaI
thf(fact_266_SigmaI,axiom,
! [A2: term_a_b,A: set_term_a_b,B2: term_a_b,B: term_a_b > set_term_a_b] :
( ( member_term_a_b @ A2 @ A )
=> ( ( member_term_a_b @ B2 @ ( B @ A2 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( produc1953217797448512882rm_a_b @ A @ B ) ) ) ) ).
% SigmaI
thf(fact_267_mem__Sigma__iff,axiom,
! [A2: product_prod_a_nat,B2: product_prod_a_nat,A: set_Pr4934435412358123699_a_nat,B: product_prod_a_nat > set_Pr4934435412358123699_a_nat] :
( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A2 @ B2 ) @ ( produc3832836940518797764_a_nat @ A @ B ) )
= ( ( member5724188588386418708_a_nat @ A2 @ A )
& ( member5724188588386418708_a_nat @ B2 @ ( B @ A2 ) ) ) ) ).
% mem_Sigma_iff
thf(fact_268_mem__Sigma__iff,axiom,
! [A2: product_prod_a_nat,B2: term_a_b,A: set_Pr4934435412358123699_a_nat,B: product_prod_a_nat > set_term_a_b] :
( ( member1876585039447381659rm_a_b @ ( produc4170664577406169130rm_a_b @ A2 @ B2 ) @ ( produc1406664433328107739rm_a_b @ A @ B ) )
= ( ( member5724188588386418708_a_nat @ A2 @ A )
& ( member_term_a_b @ B2 @ ( B @ A2 ) ) ) ) ).
% mem_Sigma_iff
thf(fact_269_mem__Sigma__iff,axiom,
! [A2: term_a_b,B2: product_prod_a_nat,A: set_term_a_b,B: term_a_b > set_Pr4934435412358123699_a_nat] :
( ( member4553550343464174107_a_nat @ ( produc2782138671500312106_a_nat @ A2 @ B2 ) @ ( produc18138527422250715_a_nat @ A @ B ) )
= ( ( member_term_a_b @ A2 @ A )
& ( member5724188588386418708_a_nat @ B2 @ ( B @ A2 ) ) ) ) ).
% mem_Sigma_iff
thf(fact_270_mem__Sigma__iff,axiom,
! [A2: a,B2: nat,A: set_a,B: a > set_nat] :
( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ A2 @ B2 ) @ ( product_Sigma_a_nat @ A @ B ) )
= ( ( member_a @ A2 @ A )
& ( member_nat @ B2 @ ( B @ A2 ) ) ) ) ).
% mem_Sigma_iff
thf(fact_271_mem__Sigma__iff,axiom,
! [A2: term_a_b,B2: term_a_b,A: set_term_a_b,B: term_a_b > set_term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( produc1953217797448512882rm_a_b @ A @ B ) )
= ( ( member_term_a_b @ A2 @ A )
& ( member_term_a_b @ B2 @ ( B @ A2 ) ) ) ) ).
% mem_Sigma_iff
thf(fact_272_inf__top_Oright__neutral,axiom,
! [A2: set_Pr4386577575007340137rm_a_b] :
( ( inf_in1693821330460682071rm_a_b @ A2 @ top_to1314267278846557113rm_a_b )
= A2 ) ).
% inf_top.right_neutral
thf(fact_273_inf__top_Oright__neutral,axiom,
! [A2: set_term_a_b] :
( ( inf_inf_set_term_a_b @ A2 @ top_top_set_term_a_b )
= A2 ) ).
% inf_top.right_neutral
thf(fact_274_inf__top_Oright__neutral,axiom,
! [A2: set_Pr4934435412358123699_a_nat] :
( ( inf_in3813033988066259269_a_nat @ A2 @ top_to3353692345378799459_a_nat )
= A2 ) ).
% inf_top.right_neutral
thf(fact_275_inf__top_Oneutr__eq__iff,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b] :
( ( top_to1314267278846557113rm_a_b
= ( inf_in1693821330460682071rm_a_b @ A2 @ B2 ) )
= ( ( A2 = top_to1314267278846557113rm_a_b )
& ( B2 = top_to1314267278846557113rm_a_b ) ) ) ).
% inf_top.neutr_eq_iff
thf(fact_276_inf__top_Oneutr__eq__iff,axiom,
! [A2: set_term_a_b,B2: set_term_a_b] :
( ( top_top_set_term_a_b
= ( inf_inf_set_term_a_b @ A2 @ B2 ) )
= ( ( A2 = top_top_set_term_a_b )
& ( B2 = top_top_set_term_a_b ) ) ) ).
% inf_top.neutr_eq_iff
thf(fact_277_inf__top_Oneutr__eq__iff,axiom,
! [A2: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat] :
( ( top_to3353692345378799459_a_nat
= ( inf_in3813033988066259269_a_nat @ A2 @ B2 ) )
= ( ( A2 = top_to3353692345378799459_a_nat )
& ( B2 = top_to3353692345378799459_a_nat ) ) ) ).
% inf_top.neutr_eq_iff
thf(fact_278_inf__top_Oleft__neutral,axiom,
! [A2: set_Pr4386577575007340137rm_a_b] :
( ( inf_in1693821330460682071rm_a_b @ top_to1314267278846557113rm_a_b @ A2 )
= A2 ) ).
% inf_top.left_neutral
thf(fact_279_inf__top_Oleft__neutral,axiom,
! [A2: set_term_a_b] :
( ( inf_inf_set_term_a_b @ top_top_set_term_a_b @ A2 )
= A2 ) ).
% inf_top.left_neutral
thf(fact_280_inf__top_Oleft__neutral,axiom,
! [A2: set_Pr4934435412358123699_a_nat] :
( ( inf_in3813033988066259269_a_nat @ top_to3353692345378799459_a_nat @ A2 )
= A2 ) ).
% inf_top.left_neutral
thf(fact_281_inf__top_Oeq__neutr__iff,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b] :
( ( ( inf_in1693821330460682071rm_a_b @ A2 @ B2 )
= top_to1314267278846557113rm_a_b )
= ( ( A2 = top_to1314267278846557113rm_a_b )
& ( B2 = top_to1314267278846557113rm_a_b ) ) ) ).
% inf_top.eq_neutr_iff
thf(fact_282_inf__top_Oeq__neutr__iff,axiom,
! [A2: set_term_a_b,B2: set_term_a_b] :
( ( ( inf_inf_set_term_a_b @ A2 @ B2 )
= top_top_set_term_a_b )
= ( ( A2 = top_top_set_term_a_b )
& ( B2 = top_top_set_term_a_b ) ) ) ).
% inf_top.eq_neutr_iff
thf(fact_283_inf__top_Oeq__neutr__iff,axiom,
! [A2: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat] :
( ( ( inf_in3813033988066259269_a_nat @ A2 @ B2 )
= top_to3353692345378799459_a_nat )
= ( ( A2 = top_to3353692345378799459_a_nat )
& ( B2 = top_to3353692345378799459_a_nat ) ) ) ).
% inf_top.eq_neutr_iff
thf(fact_284_top__eq__inf__iff,axiom,
! [X: set_Pr4386577575007340137rm_a_b,Y: set_Pr4386577575007340137rm_a_b] :
( ( top_to1314267278846557113rm_a_b
= ( inf_in1693821330460682071rm_a_b @ X @ Y ) )
= ( ( X = top_to1314267278846557113rm_a_b )
& ( Y = top_to1314267278846557113rm_a_b ) ) ) ).
% top_eq_inf_iff
thf(fact_285_top__eq__inf__iff,axiom,
! [X: set_term_a_b,Y: set_term_a_b] :
( ( top_top_set_term_a_b
= ( inf_inf_set_term_a_b @ X @ Y ) )
= ( ( X = top_top_set_term_a_b )
& ( Y = top_top_set_term_a_b ) ) ) ).
% top_eq_inf_iff
thf(fact_286_top__eq__inf__iff,axiom,
! [X: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat] :
( ( top_to3353692345378799459_a_nat
= ( inf_in3813033988066259269_a_nat @ X @ Y ) )
= ( ( X = top_to3353692345378799459_a_nat )
& ( Y = top_to3353692345378799459_a_nat ) ) ) ).
% top_eq_inf_iff
thf(fact_287_old_Oprod_Oinject,axiom,
! [A2: a,B2: nat,A6: a,B7: nat] :
( ( ( product_Pair_a_nat @ A2 @ B2 )
= ( product_Pair_a_nat @ A6 @ B7 ) )
= ( ( A2 = A6 )
& ( B2 = B7 ) ) ) ).
% old.prod.inject
thf(fact_288_prod_Oinject,axiom,
! [X1: a,X22: nat,Y1: a,Y22: nat] :
( ( ( product_Pair_a_nat @ X1 @ X22 )
= ( product_Pair_a_nat @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X22 = Y22 ) ) ) ).
% prod.inject
thf(fact_289_inf__right__idem,axiom,
! [X: set_Pr4386577575007340137rm_a_b,Y: set_Pr4386577575007340137rm_a_b] :
( ( inf_in1693821330460682071rm_a_b @ ( inf_in1693821330460682071rm_a_b @ X @ Y ) @ Y )
= ( inf_in1693821330460682071rm_a_b @ X @ Y ) ) ).
% inf_right_idem
thf(fact_290_inf_Oright__idem,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b] :
( ( inf_in1693821330460682071rm_a_b @ ( inf_in1693821330460682071rm_a_b @ A2 @ B2 ) @ B2 )
= ( inf_in1693821330460682071rm_a_b @ A2 @ B2 ) ) ).
% inf.right_idem
thf(fact_291_inf__left__idem,axiom,
! [X: set_Pr4386577575007340137rm_a_b,Y: set_Pr4386577575007340137rm_a_b] :
( ( inf_in1693821330460682071rm_a_b @ X @ ( inf_in1693821330460682071rm_a_b @ X @ Y ) )
= ( inf_in1693821330460682071rm_a_b @ X @ Y ) ) ).
% inf_left_idem
thf(fact_292_inf_Oleft__idem,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b] :
( ( inf_in1693821330460682071rm_a_b @ A2 @ ( inf_in1693821330460682071rm_a_b @ A2 @ B2 ) )
= ( inf_in1693821330460682071rm_a_b @ A2 @ B2 ) ) ).
% inf.left_idem
thf(fact_293_inf__idem,axiom,
! [X: set_Pr4386577575007340137rm_a_b] :
( ( inf_in1693821330460682071rm_a_b @ X @ X )
= X ) ).
% inf_idem
thf(fact_294_inf_Oidem,axiom,
! [A2: set_Pr4386577575007340137rm_a_b] :
( ( inf_in1693821330460682071rm_a_b @ A2 @ A2 )
= A2 ) ).
% inf.idem
thf(fact_295_UnionI,axiom,
! [X6: set_Pr4934435412358123699_a_nat,C2: set_se6121441497158405097_a_nat,A: product_prod_a_nat] :
( ( member8034581882086557258_a_nat @ X6 @ C2 )
=> ( ( member5724188588386418708_a_nat @ A @ X6 )
=> ( member5724188588386418708_a_nat @ A @ ( comple3771923735958488838_a_nat @ C2 ) ) ) ) ).
% UnionI
thf(fact_296_UnionI,axiom,
! [X6: set_term_a_b,C2: set_set_term_a_b,A: term_a_b] :
( ( member_set_term_a_b @ X6 @ C2 )
=> ( ( member_term_a_b @ A @ X6 )
=> ( member_term_a_b @ A @ ( comple5064467279716934173rm_a_b @ C2 ) ) ) ) ).
% UnionI
thf(fact_297_Union__iff,axiom,
! [A: product_prod_a_nat,C2: set_se6121441497158405097_a_nat] :
( ( member5724188588386418708_a_nat @ A @ ( comple3771923735958488838_a_nat @ C2 ) )
= ( ? [X2: set_Pr4934435412358123699_a_nat] :
( ( member8034581882086557258_a_nat @ X2 @ C2 )
& ( member5724188588386418708_a_nat @ A @ X2 ) ) ) ) ).
% Union_iff
thf(fact_298_Union__iff,axiom,
! [A: term_a_b,C2: set_set_term_a_b] :
( ( member_term_a_b @ A @ ( comple5064467279716934173rm_a_b @ C2 ) )
= ( ? [X2: set_term_a_b] :
( ( member_set_term_a_b @ X2 @ C2 )
& ( member_term_a_b @ A @ X2 ) ) ) ) ).
% Union_iff
thf(fact_299_rstep__ruleI,axiom,
! [L: term_a_b,R: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ L @ R ) @ R2 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ L @ R ) @ ( rstep_a_b @ R2 ) ) ) ).
% rstep_ruleI
thf(fact_300_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_301_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_302_inf_Obounded__iff,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( inf_inf_nat @ B2 @ C ) )
= ( ( ord_less_eq_nat @ A2 @ B2 )
& ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_303_le__inf__iff,axiom,
! [X: set_Pr4386577575007340137rm_a_b,Y: set_Pr4386577575007340137rm_a_b,Z: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ X @ ( inf_in1693821330460682071rm_a_b @ Y @ Z ) )
= ( ( ord_le118470702582115849rm_a_b @ X @ Y )
& ( ord_le118470702582115849rm_a_b @ X @ Z ) ) ) ).
% le_inf_iff
thf(fact_304_le__inf__iff,axiom,
! [X: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat,Z: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X @ ( inf_in3813033988066259269_a_nat @ Y @ Z ) )
= ( ( ord_le8666007276011122963_a_nat @ X @ Y )
& ( ord_le8666007276011122963_a_nat @ X @ Z ) ) ) ).
% le_inf_iff
thf(fact_305_le__inf__iff,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ ( inf_inf_nat @ Y @ Z ) )
= ( ( ord_less_eq_nat @ X @ Y )
& ( ord_less_eq_nat @ X @ Z ) ) ) ).
% le_inf_iff
thf(fact_306_inf__top__left,axiom,
! [X: set_Pr4386577575007340137rm_a_b] :
( ( inf_in1693821330460682071rm_a_b @ top_to1314267278846557113rm_a_b @ X )
= X ) ).
% inf_top_left
thf(fact_307_inf__top__left,axiom,
! [X: set_term_a_b] :
( ( inf_inf_set_term_a_b @ top_top_set_term_a_b @ X )
= X ) ).
% inf_top_left
thf(fact_308_inf__top__left,axiom,
! [X: set_Pr4934435412358123699_a_nat] :
( ( inf_in3813033988066259269_a_nat @ top_to3353692345378799459_a_nat @ X )
= X ) ).
% inf_top_left
thf(fact_309_inf__top__right,axiom,
! [X: set_Pr4386577575007340137rm_a_b] :
( ( inf_in1693821330460682071rm_a_b @ X @ top_to1314267278846557113rm_a_b )
= X ) ).
% inf_top_right
thf(fact_310_inf__top__right,axiom,
! [X: set_term_a_b] :
( ( inf_inf_set_term_a_b @ X @ top_top_set_term_a_b )
= X ) ).
% inf_top_right
thf(fact_311_inf__top__right,axiom,
! [X: set_Pr4934435412358123699_a_nat] :
( ( inf_in3813033988066259269_a_nat @ X @ top_to3353692345378799459_a_nat )
= X ) ).
% inf_top_right
thf(fact_312_inf__eq__top__iff,axiom,
! [X: set_Pr4386577575007340137rm_a_b,Y: set_Pr4386577575007340137rm_a_b] :
( ( ( inf_in1693821330460682071rm_a_b @ X @ Y )
= top_to1314267278846557113rm_a_b )
= ( ( X = top_to1314267278846557113rm_a_b )
& ( Y = top_to1314267278846557113rm_a_b ) ) ) ).
% inf_eq_top_iff
thf(fact_313_inf__eq__top__iff,axiom,
! [X: set_term_a_b,Y: set_term_a_b] :
( ( ( inf_inf_set_term_a_b @ X @ Y )
= top_top_set_term_a_b )
= ( ( X = top_top_set_term_a_b )
& ( Y = top_top_set_term_a_b ) ) ) ).
% inf_eq_top_iff
thf(fact_314_inf__eq__top__iff,axiom,
! [X: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat] :
( ( ( inf_in3813033988066259269_a_nat @ X @ Y )
= top_to3353692345378799459_a_nat )
= ( ( X = top_to3353692345378799459_a_nat )
& ( Y = top_to3353692345378799459_a_nat ) ) ) ).
% inf_eq_top_iff
thf(fact_315_UnionE,axiom,
! [A: product_prod_a_nat,C2: set_se6121441497158405097_a_nat] :
( ( member5724188588386418708_a_nat @ A @ ( comple3771923735958488838_a_nat @ C2 ) )
=> ~ ! [X5: set_Pr4934435412358123699_a_nat] :
( ( member5724188588386418708_a_nat @ A @ X5 )
=> ~ ( member8034581882086557258_a_nat @ X5 @ C2 ) ) ) ).
% UnionE
thf(fact_316_UnionE,axiom,
! [A: term_a_b,C2: set_set_term_a_b] :
( ( member_term_a_b @ A @ ( comple5064467279716934173rm_a_b @ C2 ) )
=> ~ ! [X5: set_term_a_b] :
( ( member_term_a_b @ A @ X5 )
=> ~ ( member_set_term_a_b @ X5 @ C2 ) ) ) ).
% UnionE
thf(fact_317_sig__step__mono2,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,L2: set_Pr4386577575007340137rm_a_b,F2: set_Pr4934435412358123699_a_nat] :
( ( ord_le118470702582115849rm_a_b @ R2 @ L2 )
=> ( ord_le118470702582115849rm_a_b @ ( sig_step_a_b @ F2 @ R2 ) @ ( sig_step_a_b @ F2 @ L2 ) ) ) ).
% sig_step_mono2
thf(fact_318_inf__set__def,axiom,
( inf_in3813033988066259269_a_nat
= ( ^ [A5: set_Pr4934435412358123699_a_nat,B5: set_Pr4934435412358123699_a_nat] :
( collec4464134535221767506_a_nat
@ ( inf_in4234122607708886400_nat_o
@ ^ [X2: product_prod_a_nat] : ( member5724188588386418708_a_nat @ X2 @ A5 )
@ ^ [X2: product_prod_a_nat] : ( member5724188588386418708_a_nat @ X2 @ B5 ) ) ) ) ) ).
% inf_set_def
thf(fact_319_inf__set__def,axiom,
( inf_inf_set_term_a_b
= ( ^ [A5: set_term_a_b,B5: set_term_a_b] :
( collect_term_a_b
@ ( inf_inf_term_a_b_o
@ ^ [X2: term_a_b] : ( member_term_a_b @ X2 @ A5 )
@ ^ [X2: term_a_b] : ( member_term_a_b @ X2 @ B5 ) ) ) ) ) ).
% inf_set_def
thf(fact_320_inf__set__def,axiom,
( inf_in1693821330460682071rm_a_b
= ( ^ [A5: set_Pr4386577575007340137rm_a_b,B5: set_Pr4386577575007340137rm_a_b] :
( collec99177395925924084rm_a_b
@ ( inf_in7523581892642934342_a_b_o
@ ^ [X2: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ X2 @ A5 )
@ ^ [X2: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ X2 @ B5 ) ) ) ) ) ).
% inf_set_def
thf(fact_321_less__eq__set__def,axiom,
( ord_le2705286416250468010rm_a_b
= ( ^ [A5: set_term_a_b,B5: set_term_a_b] :
( ord_le3490609322949561435_a_b_o
@ ^ [X2: term_a_b] : ( member_term_a_b @ X2 @ A5 )
@ ^ [X2: term_a_b] : ( member_term_a_b @ X2 @ B5 ) ) ) ) ).
% less_eq_set_def
thf(fact_322_less__eq__set__def,axiom,
( ord_le8666007276011122963_a_nat
= ( ^ [A5: set_Pr4934435412358123699_a_nat,B5: set_Pr4934435412358123699_a_nat] :
( ord_le8950881081082477362_nat_o
@ ^ [X2: product_prod_a_nat] : ( member5724188588386418708_a_nat @ X2 @ A5 )
@ ^ [X2: product_prod_a_nat] : ( member5724188588386418708_a_nat @ X2 @ B5 ) ) ) ) ).
% less_eq_set_def
thf(fact_323_sig__step__mono,axiom,
! [F2: set_Pr4934435412358123699_a_nat,G: set_Pr4934435412358123699_a_nat,R2: set_Pr4386577575007340137rm_a_b] :
( ( ord_le8666007276011122963_a_nat @ F2 @ G )
=> ( ord_le118470702582115849rm_a_b @ ( sig_step_a_b @ F2 @ R2 ) @ ( sig_step_a_b @ G @ R2 ) ) ) ).
% sig_step_mono
thf(fact_324_srstep__monp,axiom,
! [F2: set_Pr4934435412358123699_a_nat,G: set_Pr4934435412358123699_a_nat,R2: set_Pr4386577575007340137rm_a_b] :
( ( ord_le8666007276011122963_a_nat @ F2 @ G )
=> ( ord_le118470702582115849rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) ) @ ( sig_step_a_b @ G @ ( rstep_a_b @ R2 ) ) ) ) ).
% srstep_monp
thf(fact_325_Pair__inject,axiom,
! [A2: a,B2: nat,A6: a,B7: nat] :
( ( ( product_Pair_a_nat @ A2 @ B2 )
= ( product_Pair_a_nat @ A6 @ B7 ) )
=> ~ ( ( A2 = A6 )
=> ( B2 != B7 ) ) ) ).
% Pair_inject
thf(fact_326_prod__cases,axiom,
! [P: product_prod_a_nat > $o,P2: product_prod_a_nat] :
( ! [A3: a,B3: nat] : ( P @ ( product_Pair_a_nat @ A3 @ B3 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_327_surj__pair,axiom,
! [P2: product_prod_a_nat] :
? [X3: a,Y4: nat] :
( P2
= ( product_Pair_a_nat @ X3 @ Y4 ) ) ).
% surj_pair
thf(fact_328_old_Oprod_Oexhaust,axiom,
! [Y: product_prod_a_nat] :
~ ! [A3: a,B3: nat] :
( Y
!= ( product_Pair_a_nat @ A3 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_329_inf__left__commute,axiom,
! [X: set_Pr4386577575007340137rm_a_b,Y: set_Pr4386577575007340137rm_a_b,Z: set_Pr4386577575007340137rm_a_b] :
( ( inf_in1693821330460682071rm_a_b @ X @ ( inf_in1693821330460682071rm_a_b @ Y @ Z ) )
= ( inf_in1693821330460682071rm_a_b @ Y @ ( inf_in1693821330460682071rm_a_b @ X @ Z ) ) ) ).
% inf_left_commute
thf(fact_330_inf_Oleft__commute,axiom,
! [B2: set_Pr4386577575007340137rm_a_b,A2: set_Pr4386577575007340137rm_a_b,C: set_Pr4386577575007340137rm_a_b] :
( ( inf_in1693821330460682071rm_a_b @ B2 @ ( inf_in1693821330460682071rm_a_b @ A2 @ C ) )
= ( inf_in1693821330460682071rm_a_b @ A2 @ ( inf_in1693821330460682071rm_a_b @ B2 @ C ) ) ) ).
% inf.left_commute
thf(fact_331_inf__commute,axiom,
( inf_in1693821330460682071rm_a_b
= ( ^ [X2: set_Pr4386577575007340137rm_a_b,Y3: set_Pr4386577575007340137rm_a_b] : ( inf_in1693821330460682071rm_a_b @ Y3 @ X2 ) ) ) ).
% inf_commute
thf(fact_332_inf_Ocommute,axiom,
( inf_in1693821330460682071rm_a_b
= ( ^ [A4: set_Pr4386577575007340137rm_a_b,B4: set_Pr4386577575007340137rm_a_b] : ( inf_in1693821330460682071rm_a_b @ B4 @ A4 ) ) ) ).
% inf.commute
thf(fact_333_inf__assoc,axiom,
! [X: set_Pr4386577575007340137rm_a_b,Y: set_Pr4386577575007340137rm_a_b,Z: set_Pr4386577575007340137rm_a_b] :
( ( inf_in1693821330460682071rm_a_b @ ( inf_in1693821330460682071rm_a_b @ X @ Y ) @ Z )
= ( inf_in1693821330460682071rm_a_b @ X @ ( inf_in1693821330460682071rm_a_b @ Y @ Z ) ) ) ).
% inf_assoc
thf(fact_334_inf_Oassoc,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b,C: set_Pr4386577575007340137rm_a_b] :
( ( inf_in1693821330460682071rm_a_b @ ( inf_in1693821330460682071rm_a_b @ A2 @ B2 ) @ C )
= ( inf_in1693821330460682071rm_a_b @ A2 @ ( inf_in1693821330460682071rm_a_b @ B2 @ C ) ) ) ).
% inf.assoc
thf(fact_335_inf__sup__aci_I1_J,axiom,
( inf_in1693821330460682071rm_a_b
= ( ^ [X2: set_Pr4386577575007340137rm_a_b,Y3: set_Pr4386577575007340137rm_a_b] : ( inf_in1693821330460682071rm_a_b @ Y3 @ X2 ) ) ) ).
% inf_sup_aci(1)
thf(fact_336_inf__sup__aci_I2_J,axiom,
! [X: set_Pr4386577575007340137rm_a_b,Y: set_Pr4386577575007340137rm_a_b,Z: set_Pr4386577575007340137rm_a_b] :
( ( inf_in1693821330460682071rm_a_b @ ( inf_in1693821330460682071rm_a_b @ X @ Y ) @ Z )
= ( inf_in1693821330460682071rm_a_b @ X @ ( inf_in1693821330460682071rm_a_b @ Y @ Z ) ) ) ).
% inf_sup_aci(2)
thf(fact_337_inf__sup__aci_I3_J,axiom,
! [X: set_Pr4386577575007340137rm_a_b,Y: set_Pr4386577575007340137rm_a_b,Z: set_Pr4386577575007340137rm_a_b] :
( ( inf_in1693821330460682071rm_a_b @ X @ ( inf_in1693821330460682071rm_a_b @ Y @ Z ) )
= ( inf_in1693821330460682071rm_a_b @ Y @ ( inf_in1693821330460682071rm_a_b @ X @ Z ) ) ) ).
% inf_sup_aci(3)
thf(fact_338_inf__sup__aci_I4_J,axiom,
! [X: set_Pr4386577575007340137rm_a_b,Y: set_Pr4386577575007340137rm_a_b] :
( ( inf_in1693821330460682071rm_a_b @ X @ ( inf_in1693821330460682071rm_a_b @ X @ Y ) )
= ( inf_in1693821330460682071rm_a_b @ X @ Y ) ) ).
% inf_sup_aci(4)
thf(fact_339_Times__eq__cancel2,axiom,
! [X: term_a_b,C2: set_term_a_b,A: set_term_a_b,B: set_term_a_b] :
( ( member_term_a_b @ X @ C2 )
=> ( ( ( produc1953217797448512882rm_a_b @ A
@ ^ [Uu: term_a_b] : C2 )
= ( produc1953217797448512882rm_a_b @ B
@ ^ [Uu: term_a_b] : C2 ) )
= ( A = B ) ) ) ).
% Times_eq_cancel2
thf(fact_340_Sigma__cong,axiom,
! [A: set_term_a_b,B: set_term_a_b,C2: term_a_b > set_term_a_b,D: term_a_b > set_term_a_b] :
( ( A = B )
=> ( ! [X3: term_a_b] :
( ( member_term_a_b @ X3 @ B )
=> ( ( C2 @ X3 )
= ( D @ X3 ) ) )
=> ( ( produc1953217797448512882rm_a_b @ A @ C2 )
= ( produc1953217797448512882rm_a_b @ B @ D ) ) ) ) ).
% Sigma_cong
thf(fact_341_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_342_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_343_inf_OcoboundedI2,axiom,
! [B2: nat,C: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI2
thf(fact_344_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_345_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_346_inf_OcoboundedI1,axiom,
! [A2: nat,C: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ C )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI1
thf(fact_347_inf_Oabsorb__iff2,axiom,
( ord_le118470702582115849rm_a_b
= ( ^ [B4: set_Pr4386577575007340137rm_a_b,A4: set_Pr4386577575007340137rm_a_b] :
( ( inf_in1693821330460682071rm_a_b @ A4 @ B4 )
= B4 ) ) ) ).
% inf.absorb_iff2
thf(fact_348_inf_Oabsorb__iff2,axiom,
( ord_le8666007276011122963_a_nat
= ( ^ [B4: set_Pr4934435412358123699_a_nat,A4: set_Pr4934435412358123699_a_nat] :
( ( inf_in3813033988066259269_a_nat @ A4 @ B4 )
= B4 ) ) ) ).
% inf.absorb_iff2
thf(fact_349_inf_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [B4: nat,A4: nat] :
( ( inf_inf_nat @ A4 @ B4 )
= B4 ) ) ) ).
% inf.absorb_iff2
thf(fact_350_inf_Oabsorb__iff1,axiom,
( ord_le118470702582115849rm_a_b
= ( ^ [A4: set_Pr4386577575007340137rm_a_b,B4: set_Pr4386577575007340137rm_a_b] :
( ( inf_in1693821330460682071rm_a_b @ A4 @ B4 )
= A4 ) ) ) ).
% inf.absorb_iff1
thf(fact_351_inf_Oabsorb__iff1,axiom,
( ord_le8666007276011122963_a_nat
= ( ^ [A4: set_Pr4934435412358123699_a_nat,B4: set_Pr4934435412358123699_a_nat] :
( ( inf_in3813033988066259269_a_nat @ A4 @ B4 )
= A4 ) ) ) ).
% inf.absorb_iff1
thf(fact_352_inf_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B4: nat] :
( ( inf_inf_nat @ A4 @ B4 )
= A4 ) ) ) ).
% inf.absorb_iff1
thf(fact_353_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_354_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_355_inf_Ocobounded2,axiom,
! [A2: nat,B2: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_356_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_357_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_358_inf_Ocobounded1,axiom,
! [A2: nat,B2: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ A2 ) ).
% inf.cobounded1
thf(fact_359_inf_Oorder__iff,axiom,
( ord_le118470702582115849rm_a_b
= ( ^ [A4: set_Pr4386577575007340137rm_a_b,B4: set_Pr4386577575007340137rm_a_b] :
( A4
= ( inf_in1693821330460682071rm_a_b @ A4 @ B4 ) ) ) ) ).
% inf.order_iff
thf(fact_360_inf_Oorder__iff,axiom,
( ord_le8666007276011122963_a_nat
= ( ^ [A4: set_Pr4934435412358123699_a_nat,B4: set_Pr4934435412358123699_a_nat] :
( A4
= ( inf_in3813033988066259269_a_nat @ A4 @ B4 ) ) ) ) ).
% inf.order_iff
thf(fact_361_inf_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B4: nat] :
( A4
= ( inf_inf_nat @ A4 @ B4 ) ) ) ) ).
% inf.order_iff
thf(fact_362_inf__greatest,axiom,
! [X: set_Pr4386577575007340137rm_a_b,Y: set_Pr4386577575007340137rm_a_b,Z: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ X @ Y )
=> ( ( ord_le118470702582115849rm_a_b @ X @ Z )
=> ( ord_le118470702582115849rm_a_b @ X @ ( inf_in1693821330460682071rm_a_b @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_363_inf__greatest,axiom,
! [X: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat,Z: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X @ Y )
=> ( ( ord_le8666007276011122963_a_nat @ X @ Z )
=> ( ord_le8666007276011122963_a_nat @ X @ ( inf_in3813033988066259269_a_nat @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_364_inf__greatest,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Z )
=> ( ord_less_eq_nat @ X @ ( inf_inf_nat @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_365_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_366_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_367_inf_OboundedI,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ A2 @ C )
=> ( ord_less_eq_nat @ A2 @ ( inf_inf_nat @ B2 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_368_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_369_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_370_inf_OboundedE,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( inf_inf_nat @ B2 @ C ) )
=> ~ ( ( ord_less_eq_nat @ A2 @ B2 )
=> ~ ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% inf.boundedE
thf(fact_371_inf__absorb2,axiom,
! [Y: set_Pr4386577575007340137rm_a_b,X: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ Y @ X )
=> ( ( inf_in1693821330460682071rm_a_b @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_372_inf__absorb2,axiom,
! [Y: set_Pr4934435412358123699_a_nat,X: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ Y @ X )
=> ( ( inf_in3813033988066259269_a_nat @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_373_inf__absorb2,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( inf_inf_nat @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_374_inf__absorb1,axiom,
! [X: set_Pr4386577575007340137rm_a_b,Y: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ X @ Y )
=> ( ( inf_in1693821330460682071rm_a_b @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_375_inf__absorb1,axiom,
! [X: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X @ Y )
=> ( ( inf_in3813033988066259269_a_nat @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_376_inf__absorb1,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( inf_inf_nat @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_377_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_378_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_379_inf_Oabsorb2,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( inf_inf_nat @ A2 @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_380_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_381_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_382_inf_Oabsorb1,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( inf_inf_nat @ A2 @ B2 )
= A2 ) ) ).
% inf.absorb1
thf(fact_383_le__iff__inf,axiom,
( ord_le118470702582115849rm_a_b
= ( ^ [X2: set_Pr4386577575007340137rm_a_b,Y3: set_Pr4386577575007340137rm_a_b] :
( ( inf_in1693821330460682071rm_a_b @ X2 @ Y3 )
= X2 ) ) ) ).
% le_iff_inf
thf(fact_384_le__iff__inf,axiom,
( ord_le8666007276011122963_a_nat
= ( ^ [X2: set_Pr4934435412358123699_a_nat,Y3: set_Pr4934435412358123699_a_nat] :
( ( inf_in3813033988066259269_a_nat @ X2 @ Y3 )
= X2 ) ) ) ).
% le_iff_inf
thf(fact_385_le__iff__inf,axiom,
( ord_less_eq_nat
= ( ^ [X2: nat,Y3: nat] :
( ( inf_inf_nat @ X2 @ Y3 )
= X2 ) ) ) ).
% le_iff_inf
thf(fact_386_inf__unique,axiom,
! [F: set_Pr4386577575007340137rm_a_b > set_Pr4386577575007340137rm_a_b > set_Pr4386577575007340137rm_a_b,X: set_Pr4386577575007340137rm_a_b,Y: set_Pr4386577575007340137rm_a_b] :
( ! [X3: set_Pr4386577575007340137rm_a_b,Y4: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ ( F @ X3 @ Y4 ) @ X3 )
=> ( ! [X3: set_Pr4386577575007340137rm_a_b,Y4: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ ( F @ X3 @ Y4 ) @ Y4 )
=> ( ! [X3: set_Pr4386577575007340137rm_a_b,Y4: set_Pr4386577575007340137rm_a_b,Z4: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ X3 @ Y4 )
=> ( ( ord_le118470702582115849rm_a_b @ X3 @ Z4 )
=> ( ord_le118470702582115849rm_a_b @ X3 @ ( F @ Y4 @ Z4 ) ) ) )
=> ( ( inf_in1693821330460682071rm_a_b @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_387_inf__unique,axiom,
! [F: set_Pr4934435412358123699_a_nat > set_Pr4934435412358123699_a_nat > set_Pr4934435412358123699_a_nat,X: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat] :
( ! [X3: set_Pr4934435412358123699_a_nat,Y4: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ ( F @ X3 @ Y4 ) @ X3 )
=> ( ! [X3: set_Pr4934435412358123699_a_nat,Y4: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ ( F @ X3 @ Y4 ) @ Y4 )
=> ( ! [X3: set_Pr4934435412358123699_a_nat,Y4: set_Pr4934435412358123699_a_nat,Z4: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X3 @ Y4 )
=> ( ( ord_le8666007276011122963_a_nat @ X3 @ Z4 )
=> ( ord_le8666007276011122963_a_nat @ X3 @ ( F @ Y4 @ Z4 ) ) ) )
=> ( ( inf_in3813033988066259269_a_nat @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_388_inf__unique,axiom,
! [F: nat > nat > nat,X: nat,Y: nat] :
( ! [X3: nat,Y4: nat] : ( ord_less_eq_nat @ ( F @ X3 @ Y4 ) @ X3 )
=> ( ! [X3: nat,Y4: nat] : ( ord_less_eq_nat @ ( F @ X3 @ Y4 ) @ Y4 )
=> ( ! [X3: nat,Y4: nat,Z4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ( ord_less_eq_nat @ X3 @ Z4 )
=> ( ord_less_eq_nat @ X3 @ ( F @ Y4 @ Z4 ) ) ) )
=> ( ( inf_inf_nat @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_389_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_390_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_391_inf_OorderI,axiom,
! [A2: nat,B2: nat] :
( ( A2
= ( inf_inf_nat @ A2 @ B2 ) )
=> ( ord_less_eq_nat @ A2 @ B2 ) ) ).
% inf.orderI
thf(fact_392_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_393_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_394_inf_OorderE,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( A2
= ( inf_inf_nat @ A2 @ B2 ) ) ) ).
% inf.orderE
thf(fact_395_le__infI2,axiom,
! [B2: set_Pr4386577575007340137rm_a_b,X: set_Pr4386577575007340137rm_a_b,A2: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ B2 @ X )
=> ( ord_le118470702582115849rm_a_b @ ( inf_in1693821330460682071rm_a_b @ A2 @ B2 ) @ X ) ) ).
% le_infI2
thf(fact_396_le__infI2,axiom,
! [B2: set_Pr4934435412358123699_a_nat,X: set_Pr4934435412358123699_a_nat,A2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ B2 @ X )
=> ( ord_le8666007276011122963_a_nat @ ( inf_in3813033988066259269_a_nat @ A2 @ B2 ) @ X ) ) ).
% le_infI2
thf(fact_397_le__infI2,axiom,
! [B2: nat,X: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ X )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ X ) ) ).
% le_infI2
thf(fact_398_le__infI1,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,X: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A2 @ X )
=> ( ord_le118470702582115849rm_a_b @ ( inf_in1693821330460682071rm_a_b @ A2 @ B2 ) @ X ) ) ).
% le_infI1
thf(fact_399_le__infI1,axiom,
! [A2: set_Pr4934435412358123699_a_nat,X: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A2 @ X )
=> ( ord_le8666007276011122963_a_nat @ ( inf_in3813033988066259269_a_nat @ A2 @ B2 ) @ X ) ) ).
% le_infI1
thf(fact_400_le__infI1,axiom,
! [A2: nat,X: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ X )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ X ) ) ).
% le_infI1
thf(fact_401_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_402_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_403_inf__mono,axiom,
! [A2: nat,C: nat,B2: nat,D2: nat] :
( ( ord_less_eq_nat @ A2 @ C )
=> ( ( ord_less_eq_nat @ B2 @ D2 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ ( inf_inf_nat @ C @ D2 ) ) ) ) ).
% inf_mono
thf(fact_404_le__infI,axiom,
! [X: set_Pr4386577575007340137rm_a_b,A2: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ X @ A2 )
=> ( ( ord_le118470702582115849rm_a_b @ X @ B2 )
=> ( ord_le118470702582115849rm_a_b @ X @ ( inf_in1693821330460682071rm_a_b @ A2 @ B2 ) ) ) ) ).
% le_infI
thf(fact_405_le__infI,axiom,
! [X: set_Pr4934435412358123699_a_nat,A2: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X @ A2 )
=> ( ( ord_le8666007276011122963_a_nat @ X @ B2 )
=> ( ord_le8666007276011122963_a_nat @ X @ ( inf_in3813033988066259269_a_nat @ A2 @ B2 ) ) ) ) ).
% le_infI
thf(fact_406_le__infI,axiom,
! [X: nat,A2: nat,B2: nat] :
( ( ord_less_eq_nat @ X @ A2 )
=> ( ( ord_less_eq_nat @ X @ B2 )
=> ( ord_less_eq_nat @ X @ ( inf_inf_nat @ A2 @ B2 ) ) ) ) ).
% le_infI
thf(fact_407_le__infE,axiom,
! [X: set_Pr4386577575007340137rm_a_b,A2: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ X @ ( inf_in1693821330460682071rm_a_b @ A2 @ B2 ) )
=> ~ ( ( ord_le118470702582115849rm_a_b @ X @ A2 )
=> ~ ( ord_le118470702582115849rm_a_b @ X @ B2 ) ) ) ).
% le_infE
thf(fact_408_le__infE,axiom,
! [X: set_Pr4934435412358123699_a_nat,A2: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X @ ( inf_in3813033988066259269_a_nat @ A2 @ B2 ) )
=> ~ ( ( ord_le8666007276011122963_a_nat @ X @ A2 )
=> ~ ( ord_le8666007276011122963_a_nat @ X @ B2 ) ) ) ).
% le_infE
thf(fact_409_le__infE,axiom,
! [X: nat,A2: nat,B2: nat] :
( ( ord_less_eq_nat @ X @ ( inf_inf_nat @ A2 @ B2 ) )
=> ~ ( ( ord_less_eq_nat @ X @ A2 )
=> ~ ( ord_less_eq_nat @ X @ B2 ) ) ) ).
% le_infE
thf(fact_410_inf__le2,axiom,
! [X: set_Pr4386577575007340137rm_a_b,Y: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ ( inf_in1693821330460682071rm_a_b @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_411_inf__le2,axiom,
! [X: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ ( inf_in3813033988066259269_a_nat @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_412_inf__le2,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_413_inf__le1,axiom,
! [X: set_Pr4386577575007340137rm_a_b,Y: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ ( inf_in1693821330460682071rm_a_b @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_414_inf__le1,axiom,
! [X: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ ( inf_in3813033988066259269_a_nat @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_415_inf__le1,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_416_inf__sup__ord_I1_J,axiom,
! [X: set_Pr4386577575007340137rm_a_b,Y: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ ( inf_in1693821330460682071rm_a_b @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_417_inf__sup__ord_I1_J,axiom,
! [X: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ ( inf_in3813033988066259269_a_nat @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_418_inf__sup__ord_I1_J,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_419_inf__sup__ord_I2_J,axiom,
! [X: set_Pr4386577575007340137rm_a_b,Y: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ ( inf_in1693821330460682071rm_a_b @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_420_inf__sup__ord_I2_J,axiom,
! [X: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ ( inf_in3813033988066259269_a_nat @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_421_inf__sup__ord_I2_J,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_422_SigmaE2,axiom,
! [A2: product_prod_a_nat,B2: product_prod_a_nat,A: set_Pr4934435412358123699_a_nat,B: product_prod_a_nat > set_Pr4934435412358123699_a_nat] :
( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A2 @ B2 ) @ ( produc3832836940518797764_a_nat @ A @ B ) )
=> ~ ( ( member5724188588386418708_a_nat @ A2 @ A )
=> ~ ( member5724188588386418708_a_nat @ B2 @ ( B @ A2 ) ) ) ) ).
% SigmaE2
thf(fact_423_SigmaE2,axiom,
! [A2: product_prod_a_nat,B2: term_a_b,A: set_Pr4934435412358123699_a_nat,B: product_prod_a_nat > set_term_a_b] :
( ( member1876585039447381659rm_a_b @ ( produc4170664577406169130rm_a_b @ A2 @ B2 ) @ ( produc1406664433328107739rm_a_b @ A @ B ) )
=> ~ ( ( member5724188588386418708_a_nat @ A2 @ A )
=> ~ ( member_term_a_b @ B2 @ ( B @ A2 ) ) ) ) ).
% SigmaE2
thf(fact_424_SigmaE2,axiom,
! [A2: term_a_b,B2: product_prod_a_nat,A: set_term_a_b,B: term_a_b > set_Pr4934435412358123699_a_nat] :
( ( member4553550343464174107_a_nat @ ( produc2782138671500312106_a_nat @ A2 @ B2 ) @ ( produc18138527422250715_a_nat @ A @ B ) )
=> ~ ( ( member_term_a_b @ A2 @ A )
=> ~ ( member5724188588386418708_a_nat @ B2 @ ( B @ A2 ) ) ) ) ).
% SigmaE2
thf(fact_425_SigmaE2,axiom,
! [A2: a,B2: nat,A: set_a,B: a > set_nat] :
( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ A2 @ B2 ) @ ( product_Sigma_a_nat @ A @ B ) )
=> ~ ( ( member_a @ A2 @ A )
=> ~ ( member_nat @ B2 @ ( B @ A2 ) ) ) ) ).
% SigmaE2
thf(fact_426_SigmaE2,axiom,
! [A2: term_a_b,B2: term_a_b,A: set_term_a_b,B: term_a_b > set_term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( produc1953217797448512882rm_a_b @ A @ B ) )
=> ~ ( ( member_term_a_b @ A2 @ A )
=> ~ ( member_term_a_b @ B2 @ ( B @ A2 ) ) ) ) ).
% SigmaE2
thf(fact_427_SigmaD2,axiom,
! [A2: a,B2: nat,A: set_a,B: a > set_nat] :
( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ A2 @ B2 ) @ ( product_Sigma_a_nat @ A @ B ) )
=> ( member_nat @ B2 @ ( B @ A2 ) ) ) ).
% SigmaD2
thf(fact_428_SigmaD2,axiom,
! [A2: term_a_b,B2: term_a_b,A: set_term_a_b,B: term_a_b > set_term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( produc1953217797448512882rm_a_b @ A @ B ) )
=> ( member_term_a_b @ B2 @ ( B @ A2 ) ) ) ).
% SigmaD2
thf(fact_429_SigmaD1,axiom,
! [A2: a,B2: nat,A: set_a,B: a > set_nat] :
( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ A2 @ B2 ) @ ( product_Sigma_a_nat @ A @ B ) )
=> ( member_a @ A2 @ A ) ) ).
% SigmaD1
thf(fact_430_SigmaD1,axiom,
! [A2: term_a_b,B2: term_a_b,A: set_term_a_b,B: term_a_b > set_term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( produc1953217797448512882rm_a_b @ A @ B ) )
=> ( member_term_a_b @ A2 @ A ) ) ).
% SigmaD1
thf(fact_431_SigmaE,axiom,
! [C: produc4708774622424448987_a_nat,A: set_Pr4934435412358123699_a_nat,B: product_prod_a_nat > set_Pr4934435412358123699_a_nat] :
( ( member9062615507155100804_a_nat @ C @ ( produc3832836940518797764_a_nat @ A @ B ) )
=> ~ ! [X3: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ X3 @ A )
=> ! [Y4: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ Y4 @ ( B @ X3 ) )
=> ( C
!= ( produc2026711137822539155_a_nat @ X3 @ Y4 ) ) ) ) ) ).
% SigmaE
thf(fact_432_SigmaE,axiom,
! [C: produc3105026996248004978rm_a_b,A: set_Pr4934435412358123699_a_nat,B: product_prod_a_nat > set_term_a_b] :
( ( member1876585039447381659rm_a_b @ C @ ( produc1406664433328107739rm_a_b @ A @ B ) )
=> ~ ! [X3: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ X3 @ A )
=> ! [Y4: term_a_b] :
( ( member_term_a_b @ Y4 @ ( B @ X3 ) )
=> ( C
!= ( produc4170664577406169130rm_a_b @ X3 @ Y4 ) ) ) ) ) ).
% SigmaE
thf(fact_433_SigmaE,axiom,
! [C: produc5781992300264797426_a_nat,A: set_term_a_b,B: term_a_b > set_Pr4934435412358123699_a_nat] :
( ( member4553550343464174107_a_nat @ C @ ( produc18138527422250715_a_nat @ A @ B ) )
=> ~ ! [X3: term_a_b] :
( ( member_term_a_b @ X3 @ A )
=> ! [Y4: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ Y4 @ ( B @ X3 ) )
=> ( C
!= ( produc2782138671500312106_a_nat @ X3 @ Y4 ) ) ) ) ) ).
% SigmaE
thf(fact_434_SigmaE,axiom,
! [C: product_prod_a_nat,A: set_a,B: a > set_nat] :
( ( member5724188588386418708_a_nat @ C @ ( product_Sigma_a_nat @ A @ B ) )
=> ~ ! [X3: a] :
( ( member_a @ X3 @ A )
=> ! [Y4: nat] :
( ( member_nat @ Y4 @ ( B @ X3 ) )
=> ( C
!= ( product_Pair_a_nat @ X3 @ Y4 ) ) ) ) ) ).
% SigmaE
thf(fact_435_SigmaE,axiom,
! [C: produc357393685978478089rm_a_b,A: set_term_a_b,B: term_a_b > set_term_a_b] :
( ( member5869715511025134514rm_a_b @ C @ ( produc1953217797448512882rm_a_b @ A @ B ) )
=> ~ ! [X3: term_a_b] :
( ( member_term_a_b @ X3 @ A )
=> ! [Y4: term_a_b] :
( ( member_term_a_b @ Y4 @ ( B @ X3 ) )
=> ( C
!= ( produc7020197800436672577rm_a_b @ X3 @ Y4 ) ) ) ) ) ).
% SigmaE
thf(fact_436_CR__iff__self__commute,axiom,
! [R: set_Pr4386577575007340137rm_a_b] :
( ( abstra8448919418672941150rm_a_b @ R @ top_top_set_term_a_b )
= ( abstra455195320138238452rm_a_b @ R @ R ) ) ).
% CR_iff_self_commute
thf(fact_437_CR__iff__self__commute,axiom,
! [R: set_Pr1811044260758604347_a_nat] :
( ( abstra7964079670522473415_a_nat @ R @ top_to3353692345378799459_a_nat )
= ( abstra5452889001094163037_a_nat @ R @ R ) ) ).
% CR_iff_self_commute
thf(fact_438_Sigma__mono,axiom,
! [A: set_term_a_b,C2: set_term_a_b,B: term_a_b > set_term_a_b,D: term_a_b > set_term_a_b] :
( ( ord_le2705286416250468010rm_a_b @ A @ C2 )
=> ( ! [X3: term_a_b] :
( ( member_term_a_b @ X3 @ A )
=> ( ord_le2705286416250468010rm_a_b @ ( B @ X3 ) @ ( D @ X3 ) ) )
=> ( ord_le118470702582115849rm_a_b @ ( produc1953217797448512882rm_a_b @ A @ B ) @ ( produc1953217797448512882rm_a_b @ C2 @ D ) ) ) ) ).
% Sigma_mono
thf(fact_439_Sigma__mono,axiom,
! [A: set_a,C2: set_a,B: a > set_nat,D: a > set_nat] :
( ( ord_less_eq_set_a @ A @ C2 )
=> ( ! [X3: a] :
( ( member_a @ X3 @ A )
=> ( ord_less_eq_set_nat @ ( B @ X3 ) @ ( D @ X3 ) ) )
=> ( ord_le8666007276011122963_a_nat @ ( product_Sigma_a_nat @ A @ B ) @ ( product_Sigma_a_nat @ C2 @ D ) ) ) ) ).
% Sigma_mono
thf(fact_440_Sigma__mono,axiom,
! [A: set_term_a_b,C2: set_term_a_b,B: term_a_b > set_Pr4934435412358123699_a_nat,D: term_a_b > set_Pr4934435412358123699_a_nat] :
( ( ord_le2705286416250468010rm_a_b @ A @ C2 )
=> ( ! [X3: term_a_b] :
( ( member_term_a_b @ X3 @ A )
=> ( ord_le8666007276011122963_a_nat @ ( B @ X3 ) @ ( D @ X3 ) ) )
=> ( ord_le4886990623789914610_a_nat @ ( produc18138527422250715_a_nat @ A @ B ) @ ( produc18138527422250715_a_nat @ C2 @ D ) ) ) ) ).
% Sigma_mono
thf(fact_441_Sigma__mono,axiom,
! [A: set_Pr4934435412358123699_a_nat,C2: set_Pr4934435412358123699_a_nat,B: product_prod_a_nat > set_Pr4934435412358123699_a_nat,D: product_prod_a_nat > set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A @ C2 )
=> ( ! [X3: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ X3 @ A )
=> ( ord_le8666007276011122963_a_nat @ ( B @ X3 ) @ ( D @ X3 ) ) )
=> ( ord_le1018141841545957851_a_nat @ ( produc3832836940518797764_a_nat @ A @ B ) @ ( produc3832836940518797764_a_nat @ C2 @ D ) ) ) ) ).
% Sigma_mono
thf(fact_442_Sigma__Int__distrib1,axiom,
! [I: set_term_a_b,J: set_term_a_b,C2: term_a_b > set_term_a_b] :
( ( produc1953217797448512882rm_a_b @ ( inf_inf_set_term_a_b @ I @ J ) @ C2 )
= ( inf_in1693821330460682071rm_a_b @ ( produc1953217797448512882rm_a_b @ I @ C2 ) @ ( produc1953217797448512882rm_a_b @ J @ C2 ) ) ) ).
% Sigma_Int_distrib1
thf(fact_443_Times__subset__cancel2,axiom,
! [X: term_a_b,C2: set_term_a_b,A: set_term_a_b,B: set_term_a_b] :
( ( member_term_a_b @ X @ C2 )
=> ( ( ord_le118470702582115849rm_a_b
@ ( produc1953217797448512882rm_a_b @ A
@ ^ [Uu: term_a_b] : C2 )
@ ( produc1953217797448512882rm_a_b @ B
@ ^ [Uu: term_a_b] : C2 ) )
= ( ord_le2705286416250468010rm_a_b @ A @ B ) ) ) ).
% Times_subset_cancel2
thf(fact_444_Times__subset__cancel2,axiom,
! [X: product_prod_a_nat,C2: set_Pr4934435412358123699_a_nat,A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( member5724188588386418708_a_nat @ X @ C2 )
=> ( ( ord_le1018141841545957851_a_nat
@ ( produc3832836940518797764_a_nat @ A
@ ^ [Uu: product_prod_a_nat] : C2 )
@ ( produc3832836940518797764_a_nat @ B
@ ^ [Uu: product_prod_a_nat] : C2 ) )
= ( ord_le8666007276011122963_a_nat @ A @ B ) ) ) ).
% Times_subset_cancel2
thf(fact_445_Times__subset__cancel2,axiom,
! [X: term_a_b,C2: set_term_a_b,A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( member_term_a_b @ X @ C2 )
=> ( ( ord_le2177895101545574002rm_a_b
@ ( produc1406664433328107739rm_a_b @ A
@ ^ [Uu: product_prod_a_nat] : C2 )
@ ( produc1406664433328107739rm_a_b @ B
@ ^ [Uu: product_prod_a_nat] : C2 ) )
= ( ord_le8666007276011122963_a_nat @ A @ B ) ) ) ).
% Times_subset_cancel2
thf(fact_446_Times__subset__cancel2,axiom,
! [X: nat,C2: set_nat,A: set_a,B: set_a] :
( ( member_nat @ X @ C2 )
=> ( ( ord_le8666007276011122963_a_nat
@ ( product_Sigma_a_nat @ A
@ ^ [Uu: a] : C2 )
@ ( product_Sigma_a_nat @ B
@ ^ [Uu: a] : C2 ) )
= ( ord_less_eq_set_a @ A @ B ) ) ) ).
% Times_subset_cancel2
thf(fact_447_Times__Int__distrib1,axiom,
! [A: set_term_a_b,B: set_term_a_b,C2: set_term_a_b] :
( ( produc1953217797448512882rm_a_b @ ( inf_inf_set_term_a_b @ A @ B )
@ ^ [Uu: term_a_b] : C2 )
= ( inf_in1693821330460682071rm_a_b
@ ( produc1953217797448512882rm_a_b @ A
@ ^ [Uu: term_a_b] : C2 )
@ ( produc1953217797448512882rm_a_b @ B
@ ^ [Uu: term_a_b] : C2 ) ) ) ).
% Times_Int_distrib1
thf(fact_448_Sigma__Int__distrib2,axiom,
! [I: set_term_a_b,A: term_a_b > set_term_a_b,B: term_a_b > set_term_a_b] :
( ( produc1953217797448512882rm_a_b @ I
@ ^ [I2: term_a_b] : ( inf_inf_set_term_a_b @ ( A @ I2 ) @ ( B @ I2 ) ) )
= ( inf_in1693821330460682071rm_a_b @ ( produc1953217797448512882rm_a_b @ I @ A ) @ ( produc1953217797448512882rm_a_b @ I @ B ) ) ) ).
% Sigma_Int_distrib2
thf(fact_449_Times__Int__Times,axiom,
! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b,C2: set_Pr4386577575007340137rm_a_b,D: set_Pr4386577575007340137rm_a_b] :
( ( inf_in7724013137991509301rm_a_b
@ ( produc5745020764710804872rm_a_b @ A
@ ^ [Uu: produc357393685978478089rm_a_b] : B )
@ ( produc5745020764710804872rm_a_b @ C2
@ ^ [Uu: produc357393685978478089rm_a_b] : D ) )
= ( produc5745020764710804872rm_a_b @ ( inf_in1693821330460682071rm_a_b @ A @ C2 )
@ ^ [Uu: produc357393685978478089rm_a_b] : ( inf_in1693821330460682071rm_a_b @ B @ D ) ) ) ).
% Times_Int_Times
thf(fact_450_Times__Int__Times,axiom,
! [A: set_term_a_b,B: set_term_a_b,C2: set_term_a_b,D: set_term_a_b] :
( ( inf_in1693821330460682071rm_a_b
@ ( produc1953217797448512882rm_a_b @ A
@ ^ [Uu: term_a_b] : B )
@ ( produc1953217797448512882rm_a_b @ C2
@ ^ [Uu: term_a_b] : D ) )
= ( produc1953217797448512882rm_a_b @ ( inf_inf_set_term_a_b @ A @ C2 )
@ ^ [Uu: term_a_b] : ( inf_inf_set_term_a_b @ B @ D ) ) ) ).
% Times_Int_Times
thf(fact_451_Restr__mono,axiom,
! [R: set_Pr4386577575007340137rm_a_b,A: set_term_a_b] :
( ord_le118470702582115849rm_a_b
@ ( inf_in1693821330460682071rm_a_b @ R
@ ( produc1953217797448512882rm_a_b @ A
@ ^ [Uu: term_a_b] : A ) )
@ R ) ).
% Restr_mono
thf(fact_452_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_453_Restr__subset,axiom,
! [A: set_term_a_b,B: set_term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( ord_le2705286416250468010rm_a_b @ A @ B )
=> ( ( inf_in1693821330460682071rm_a_b
@ ( inf_in1693821330460682071rm_a_b @ R
@ ( produc1953217797448512882rm_a_b @ B
@ ^ [Uu: term_a_b] : B ) )
@ ( produc1953217797448512882rm_a_b @ A
@ ^ [Uu: term_a_b] : A ) )
= ( inf_in1693821330460682071rm_a_b @ R
@ ( produc1953217797448512882rm_a_b @ A
@ ^ [Uu: term_a_b] : A ) ) ) ) ).
% Restr_subset
thf(fact_454_Restr__subset,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat,R: set_Pr1811044260758604347_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A @ B )
=> ( ( inf_in8678195602585455913_a_nat
@ ( inf_in8678195602585455913_a_nat @ R
@ ( produc3832836940518797764_a_nat @ B
@ ^ [Uu: product_prod_a_nat] : B ) )
@ ( produc3832836940518797764_a_nat @ A
@ ^ [Uu: product_prod_a_nat] : A ) )
= ( inf_in8678195602585455913_a_nat @ R
@ ( produc3832836940518797764_a_nat @ A
@ ^ [Uu: product_prod_a_nat] : A ) ) ) ) ).
% Restr_subset
thf(fact_455_weak__diamond__imp__CR,axiom,
! [R: set_Pr4386577575007340137rm_a_b] :
( ( abstra5190899124789103926rm_a_b @ R )
=> ( abstra8448919418672941150rm_a_b @ R @ top_top_set_term_a_b ) ) ).
% weak_diamond_imp_CR
thf(fact_456_weak__diamond__imp__CR,axiom,
! [R: set_Pr1811044260758604347_a_nat] :
( ( abstra8414283475881616543_a_nat @ R )
=> ( abstra7964079670522473415_a_nat @ R @ top_to3353692345378799459_a_nat ) ) ).
% weak_diamond_imp_CR
thf(fact_457_top__empty__eq2,axiom,
( top_top_a_nat_o
= ( ^ [X2: a,Y3: nat] : ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ X2 @ Y3 ) @ top_to3353692345378799459_a_nat ) ) ) ).
% top_empty_eq2
thf(fact_458_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_459_bot__nat__0_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A2 ) ).
% bot_nat_0.extremum
thf(fact_460_inf__Int__eq2,axiom,
! [R3: set_Pr4934435412358123699_a_nat,S3: set_Pr4934435412358123699_a_nat] :
( ( inf_inf_a_nat_o
@ ^ [X2: a,Y3: nat] : ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ X2 @ Y3 ) @ R3 )
@ ^ [X2: a,Y3: nat] : ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ X2 @ Y3 ) @ S3 ) )
= ( ^ [X2: a,Y3: nat] : ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ X2 @ Y3 ) @ ( inf_in3813033988066259269_a_nat @ R3 @ S3 ) ) ) ) ).
% inf_Int_eq2
thf(fact_461_inf__Int__eq2,axiom,
! [R3: set_Pr4386577575007340137rm_a_b,S3: set_Pr4386577575007340137rm_a_b] :
( ( inf_in2005149299588960588_a_b_o
@ ^ [X2: term_a_b,Y3: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y3 ) @ R3 )
@ ^ [X2: term_a_b,Y3: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y3 ) @ S3 ) )
= ( ^ [X2: term_a_b,Y3: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y3 ) @ ( inf_in1693821330460682071rm_a_b @ R3 @ S3 ) ) ) ) ).
% inf_Int_eq2
thf(fact_462_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_463_le__trans,axiom,
! [I3: nat,J2: nat,K: nat] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ( ord_less_eq_nat @ J2 @ K )
=> ( ord_less_eq_nat @ I3 @ K ) ) ) ).
% le_trans
thf(fact_464_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_465_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( M = N ) ) ) ).
% le_antisym
thf(fact_466_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
| ( ord_less_eq_nat @ N @ M ) ) ).
% nat_le_linear
thf(fact_467_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B2: nat] :
( ( P @ K )
=> ( ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ B2 ) )
=> ? [X3: nat] :
( ( P @ X3 )
& ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_468_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_469_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_470_bot__nat__0_Oextremum__unique,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
= ( A2 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_471_bot__nat__0_Oextremum__uniqueI,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( A2 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_472_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_473_pred__equals__eq2,axiom,
! [R3: set_Pr4934435412358123699_a_nat,S3: set_Pr4934435412358123699_a_nat] :
( ( ( ^ [X2: a,Y3: nat] : ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ X2 @ Y3 ) @ R3 ) )
= ( ^ [X2: a,Y3: nat] : ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ X2 @ Y3 ) @ S3 ) ) )
= ( R3 = S3 ) ) ).
% pred_equals_eq2
thf(fact_474_pred__subset__eq,axiom,
! [R3: set_term_a_b,S3: set_term_a_b] :
( ( ord_le3490609322949561435_a_b_o
@ ^ [X2: term_a_b] : ( member_term_a_b @ X2 @ R3 )
@ ^ [X2: term_a_b] : ( member_term_a_b @ X2 @ S3 ) )
= ( ord_le2705286416250468010rm_a_b @ R3 @ S3 ) ) ).
% pred_subset_eq
thf(fact_475_pred__subset__eq,axiom,
! [R3: set_Pr4934435412358123699_a_nat,S3: set_Pr4934435412358123699_a_nat] :
( ( ord_le8950881081082477362_nat_o
@ ^ [X2: product_prod_a_nat] : ( member5724188588386418708_a_nat @ X2 @ R3 )
@ ^ [X2: product_prod_a_nat] : ( member5724188588386418708_a_nat @ X2 @ S3 ) )
= ( ord_le8666007276011122963_a_nat @ R3 @ S3 ) ) ).
% pred_subset_eq
thf(fact_476_inf__Int__eq,axiom,
! [R3: set_Pr4934435412358123699_a_nat,S3: set_Pr4934435412358123699_a_nat] :
( ( inf_in4234122607708886400_nat_o
@ ^ [X2: product_prod_a_nat] : ( member5724188588386418708_a_nat @ X2 @ R3 )
@ ^ [X2: product_prod_a_nat] : ( member5724188588386418708_a_nat @ X2 @ S3 ) )
= ( ^ [X2: product_prod_a_nat] : ( member5724188588386418708_a_nat @ X2 @ ( inf_in3813033988066259269_a_nat @ R3 @ S3 ) ) ) ) ).
% inf_Int_eq
thf(fact_477_inf__Int__eq,axiom,
! [R3: set_term_a_b,S3: set_term_a_b] :
( ( inf_inf_term_a_b_o
@ ^ [X2: term_a_b] : ( member_term_a_b @ X2 @ R3 )
@ ^ [X2: term_a_b] : ( member_term_a_b @ X2 @ S3 ) )
= ( ^ [X2: term_a_b] : ( member_term_a_b @ X2 @ ( inf_inf_set_term_a_b @ R3 @ S3 ) ) ) ) ).
% inf_Int_eq
thf(fact_478_inf__Int__eq,axiom,
! [R3: set_Pr4386577575007340137rm_a_b,S3: set_Pr4386577575007340137rm_a_b] :
( ( inf_in7523581892642934342_a_b_o
@ ^ [X2: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ X2 @ R3 )
@ ^ [X2: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ X2 @ S3 ) )
= ( ^ [X2: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ X2 @ ( inf_in1693821330460682071rm_a_b @ R3 @ S3 ) ) ) ) ).
% inf_Int_eq
thf(fact_479_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_480_subrelI,axiom,
! [R: set_Pr4934435412358123699_a_nat,S: set_Pr4934435412358123699_a_nat] :
( ! [X3: a,Y4: nat] :
( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ X3 @ Y4 ) @ R )
=> ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ X3 @ Y4 ) @ S ) )
=> ( ord_le8666007276011122963_a_nat @ R @ S ) ) ).
% subrelI
thf(fact_481_pred__subset__eq2,axiom,
! [R3: set_Pr4934435412358123699_a_nat,S3: set_Pr4934435412358123699_a_nat] :
( ( ord_less_eq_a_nat_o
@ ^ [X2: a,Y3: nat] : ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ X2 @ Y3 ) @ R3 )
@ ^ [X2: a,Y3: nat] : ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ X2 @ Y3 ) @ S3 ) )
= ( ord_le8666007276011122963_a_nat @ R3 @ S3 ) ) ).
% pred_subset_eq2
thf(fact_482_top__empty__eq,axiom,
( top_top_term_a_b_o
= ( ^ [X2: term_a_b] : ( member_term_a_b @ X2 @ top_top_set_term_a_b ) ) ) ).
% top_empty_eq
thf(fact_483_top__empty__eq,axiom,
( top_to1985590801843794146_nat_o
= ( ^ [X2: product_prod_a_nat] : ( member5724188588386418708_a_nat @ X2 @ top_to3353692345378799459_a_nat ) ) ) ).
% top_empty_eq
thf(fact_484_llrg__ground__rhs,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,L: term_a_b,R: term_a_b] :
( ( rewrit5000278123254648670_b_a_b @ R2 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ L @ R ) @ R2 )
=> ( term_ground_a_b @ R ) ) ) ).
% llrg_ground_rhs
thf(fact_485_insert__subsetI,axiom,
! [X: term_a_b,A: set_term_a_b,X6: set_term_a_b] :
( ( member_term_a_b @ X @ A )
=> ( ( ord_le2705286416250468010rm_a_b @ X6 @ A )
=> ( ord_le2705286416250468010rm_a_b @ ( insert_term_a_b @ X @ X6 ) @ A ) ) ) ).
% insert_subsetI
thf(fact_486_insert__subsetI,axiom,
! [X: product_prod_a_nat,A: set_Pr4934435412358123699_a_nat,X6: set_Pr4934435412358123699_a_nat] :
( ( member5724188588386418708_a_nat @ X @ A )
=> ( ( ord_le8666007276011122963_a_nat @ X6 @ A )
=> ( ord_le8666007276011122963_a_nat @ ( insert8054603423593749677_a_nat @ X @ X6 ) @ A ) ) ) ).
% insert_subsetI
thf(fact_487_boolean__algebra_Oconj__one__right,axiom,
! [X: set_Pr4386577575007340137rm_a_b] :
( ( inf_in1693821330460682071rm_a_b @ X @ top_to1314267278846557113rm_a_b )
= X ) ).
% boolean_algebra.conj_one_right
thf(fact_488_boolean__algebra_Oconj__one__right,axiom,
! [X: set_term_a_b] :
( ( inf_inf_set_term_a_b @ X @ top_top_set_term_a_b )
= X ) ).
% boolean_algebra.conj_one_right
thf(fact_489_boolean__algebra_Oconj__one__right,axiom,
! [X: set_Pr4934435412358123699_a_nat] :
( ( inf_in3813033988066259269_a_nat @ X @ top_to3353692345378799459_a_nat )
= X ) ).
% boolean_algebra.conj_one_right
thf(fact_490_WN__UNF__imp__CR,axiom,
! [R: set_Pr1811044260758604347_a_nat] :
( ( abstra4582696096554303663_a_nat @ R @ top_to3353692345378799459_a_nat )
=> ( ( abstra7833083522531381839_a_nat @ R @ top_to3353692345378799459_a_nat )
=> ( abstra7964079670522473415_a_nat @ R @ top_to3353692345378799459_a_nat ) ) ) ).
% WN_UNF_imp_CR
thf(fact_491_WN__UNF__imp__CR,axiom,
! [R: set_Pr4386577575007340137rm_a_b] :
( ( abstra4644735066934440262rm_a_b @ R @ top_top_set_term_a_b )
=> ( ( abstra3619855735679963878rm_a_b @ R @ top_top_set_term_a_b )
=> ( abstra8448919418672941150rm_a_b @ R @ top_top_set_term_a_b ) ) ) ).
% WN_UNF_imp_CR
thf(fact_492_cSup__eq__maximum,axiom,
! [Z: set_Pr4934435412358123699_a_nat,X6: set_se6121441497158405097_a_nat] :
( ( member8034581882086557258_a_nat @ Z @ X6 )
=> ( ! [X3: set_Pr4934435412358123699_a_nat] :
( ( member8034581882086557258_a_nat @ X3 @ X6 )
=> ( ord_le8666007276011122963_a_nat @ X3 @ Z ) )
=> ( ( comple3771923735958488838_a_nat @ X6 )
= Z ) ) ) ).
% cSup_eq_maximum
thf(fact_493_cSup__eq__maximum,axiom,
! [Z: nat,X6: set_nat] :
( ( member_nat @ Z @ X6 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ X6 )
=> ( ord_less_eq_nat @ X3 @ Z ) )
=> ( ( complete_Sup_Sup_nat @ X6 )
= Z ) ) ) ).
% cSup_eq_maximum
thf(fact_494_ssubst__Pair__rhs,axiom,
! [R: a,S: nat,R3: set_Pr4934435412358123699_a_nat,S4: nat] :
( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ R @ S ) @ R3 )
=> ( ( S4 = S )
=> ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ R @ S4 ) @ R3 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_495_boolean__algebra__cancel_Oinf2,axiom,
! [B: set_Pr4386577575007340137rm_a_b,K: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b,A2: set_Pr4386577575007340137rm_a_b] :
( ( B
= ( inf_in1693821330460682071rm_a_b @ K @ B2 ) )
=> ( ( inf_in1693821330460682071rm_a_b @ A2 @ B )
= ( inf_in1693821330460682071rm_a_b @ K @ ( inf_in1693821330460682071rm_a_b @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_496_boolean__algebra__cancel_Oinf1,axiom,
! [A: set_Pr4386577575007340137rm_a_b,K: set_Pr4386577575007340137rm_a_b,A2: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b] :
( ( A
= ( inf_in1693821330460682071rm_a_b @ K @ A2 ) )
=> ( ( inf_in1693821330460682071rm_a_b @ A @ B2 )
= ( inf_in1693821330460682071rm_a_b @ K @ ( inf_in1693821330460682071rm_a_b @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_497_Collect__restrict,axiom,
! [X6: set_term_a_b,P: term_a_b > $o] :
( ord_le2705286416250468010rm_a_b
@ ( collect_term_a_b
@ ^ [X2: term_a_b] :
( ( member_term_a_b @ X2 @ X6 )
& ( P @ X2 ) ) )
@ X6 ) ).
% Collect_restrict
thf(fact_498_Collect__restrict,axiom,
! [X6: set_Pr4934435412358123699_a_nat,P: product_prod_a_nat > $o] :
( ord_le8666007276011122963_a_nat
@ ( collec4464134535221767506_a_nat
@ ^ [X2: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ X2 @ X6 )
& ( P @ X2 ) ) )
@ X6 ) ).
% Collect_restrict
thf(fact_499_prop__restrict,axiom,
! [X: term_a_b,Z5: set_term_a_b,X6: set_term_a_b,P: term_a_b > $o] :
( ( member_term_a_b @ X @ Z5 )
=> ( ( ord_le2705286416250468010rm_a_b @ Z5
@ ( collect_term_a_b
@ ^ [X2: term_a_b] :
( ( member_term_a_b @ X2 @ X6 )
& ( P @ X2 ) ) ) )
=> ( P @ X ) ) ) ).
% prop_restrict
thf(fact_500_prop__restrict,axiom,
! [X: product_prod_a_nat,Z5: set_Pr4934435412358123699_a_nat,X6: set_Pr4934435412358123699_a_nat,P: product_prod_a_nat > $o] :
( ( member5724188588386418708_a_nat @ X @ Z5 )
=> ( ( ord_le8666007276011122963_a_nat @ Z5
@ ( collec4464134535221767506_a_nat
@ ^ [X2: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ X2 @ X6 )
& ( P @ X2 ) ) ) )
=> ( P @ X ) ) ) ).
% prop_restrict
thf(fact_501_llrg__monadic__rstep__pres__groundness,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,F2: set_Pr4934435412358123699_a_nat,S: term_a_b,T2: term_a_b] :
( ( rewrit5000278123254648670_b_a_b @ R2 )
=> ( ( rewrit3916899077429870363adic_a @ F2 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T2 ) @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) ) )
=> ( term_ground_a_b @ T2 ) ) ) ) ).
% llrg_monadic_rstep_pres_groundness
thf(fact_502_ground__srsteps__eq__gsrsteps__eq,axiom,
! [S: term_a_b,T2: term_a_b,F2: set_Pr4934435412358123699_a_nat,R2: set_Pr4386577575007340137rm_a_b] :
( ( term_ground_a_b @ S )
=> ( ( term_ground_a_b @ T2 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T2 ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T2 )
@ ( transi7742714808557438673rm_a_b
@ ( inf_in1693821330460682071rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) )
@ ( produc1953217797448512882rm_a_b @ ( collect_term_a_b @ term_ground_a_b )
@ ^ [Uu: term_a_b] : ( collect_term_a_b @ term_ground_a_b ) ) ) ) ) ) ) ) ).
% ground_srsteps_eq_gsrsteps_eq
thf(fact_503_gsrsteps__eq__to__srsteps__eq,axiom,
! [S: term_a_b,T2: term_a_b,F2: set_Pr4934435412358123699_a_nat,R2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T2 )
@ ( transi7742714808557438673rm_a_b
@ ( inf_in1693821330460682071rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) )
@ ( produc1953217797448512882rm_a_b @ ( collect_term_a_b @ term_ground_a_b )
@ ^ [Uu: term_a_b] : ( collect_term_a_b @ term_ground_a_b ) ) ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T2 ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) ) ) ) ) ).
% gsrsteps_eq_to_srsteps_eq
thf(fact_504_gsrsteps__eq__to__rsteps__eq,axiom,
! [S: term_a_b,T2: term_a_b,F2: set_Pr4934435412358123699_a_nat,R2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T2 )
@ ( transi7742714808557438673rm_a_b
@ ( inf_in1693821330460682071rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) )
@ ( produc1953217797448512882rm_a_b @ ( collect_term_a_b @ term_ground_a_b )
@ ^ [Uu: term_a_b] : ( collect_term_a_b @ term_ground_a_b ) ) ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T2 ) @ ( transi7742714808557438673rm_a_b @ ( rstep_a_b @ R2 ) ) ) ) ).
% gsrsteps_eq_to_rsteps_eq
thf(fact_505_rtrancl__Restr,axiom,
! [X: term_a_b,Y: term_a_b,R: set_Pr4386577575007340137rm_a_b,A: set_term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y )
@ ( transi7742714808557438673rm_a_b
@ ( inf_in1693821330460682071rm_a_b @ R
@ ( produc1953217797448512882rm_a_b @ A
@ ^ [Uu: term_a_b] : A ) ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ R ) ) ) ).
% rtrancl_Restr
thf(fact_506_srsteps__eq__monp,axiom,
! [F2: set_Pr4934435412358123699_a_nat,G: set_Pr4934435412358123699_a_nat,R2: set_Pr4386577575007340137rm_a_b] :
( ( ord_le8666007276011122963_a_nat @ F2 @ G )
=> ( ord_le118470702582115849rm_a_b @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) ) ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ G @ ( rstep_a_b @ R2 ) ) ) ) ) ).
% srsteps_eq_monp
thf(fact_507_llrg__rsteps__pres__groundness,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,S: term_a_b,T2: term_a_b,F2: set_Pr4934435412358123699_a_nat] :
( ( rewrit5000278123254648670_b_a_b @ R2 )
=> ( ( term_ground_a_b @ S )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T2 ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) ) ) )
=> ( term_ground_a_b @ T2 ) ) ) ) ).
% llrg_rsteps_pres_groundness
thf(fact_508_trancl__subset__Sigma__aux,axiom,
! [A2: product_prod_a_nat,B2: product_prod_a_nat,R: set_Pr1811044260758604347_a_nat,A: set_Pr4934435412358123699_a_nat] :
( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A2 @ B2 ) @ ( transi2726145917338391738_a_nat @ R ) )
=> ( ( ord_le1018141841545957851_a_nat @ R
@ ( produc3832836940518797764_a_nat @ A
@ ^ [Uu: product_prod_a_nat] : A ) )
=> ( ( A2 = B2 )
| ( member5724188588386418708_a_nat @ A2 @ A ) ) ) ) ).
% trancl_subset_Sigma_aux
thf(fact_509_trancl__subset__Sigma__aux,axiom,
! [A2: term_a_b,B2: term_a_b,R: set_Pr4386577575007340137rm_a_b,A: set_term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ( ( ord_le118470702582115849rm_a_b @ R
@ ( produc1953217797448512882rm_a_b @ A
@ ^ [Uu: term_a_b] : A ) )
=> ( ( A2 = B2 )
| ( member_term_a_b @ A2 @ A ) ) ) ) ).
% trancl_subset_Sigma_aux
thf(fact_510_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_511_diamond__imp__CR_H,axiom,
! [S: set_Pr1811044260758604347_a_nat,R: set_Pr1811044260758604347_a_nat] :
( ( abstra1356934219170352343_a_nat @ S )
=> ( ( ord_le1018141841545957851_a_nat @ R @ S )
=> ( ( ord_le1018141841545957851_a_nat @ S @ ( transi2726145917338391738_a_nat @ R ) )
=> ( abstra7964079670522473415_a_nat @ R @ top_to3353692345378799459_a_nat ) ) ) ) ).
% diamond_imp_CR'
thf(fact_512_srsteps__eq__relcomp__gsrsteps__relcomp,axiom,
! [S: term_a_b,T2: term_a_b,F2: set_Pr4934435412358123699_a_nat,R2: set_Pr4386577575007340137rm_a_b,S2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T2 ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) ) ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ S2 ) ) ) ) )
=> ( ( term_ground_a_b @ S )
=> ( ( term_ground_a_b @ T2 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T2 )
@ ( relcom370159955682700863rm_a_b
@ ( transi7742714808557438673rm_a_b
@ ( inf_in1693821330460682071rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) )
@ ( produc1953217797448512882rm_a_b @ ( collect_term_a_b @ term_ground_a_b )
@ ^ [Uu: term_a_b] : ( collect_term_a_b @ term_ground_a_b ) ) ) )
@ ( transi7742714808557438673rm_a_b
@ ( inf_in1693821330460682071rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ S2 ) )
@ ( produc1953217797448512882rm_a_b @ ( collect_term_a_b @ term_ground_a_b )
@ ^ [Uu: term_a_b] : ( collect_term_a_b @ term_ground_a_b ) ) ) ) ) ) ) ) ) ).
% srsteps_eq_relcomp_gsrsteps_relcomp
thf(fact_513_gsrsteps__eq__relcomp__srsteps__relcompD,axiom,
! [S: term_a_b,T2: term_a_b,F2: set_Pr4934435412358123699_a_nat,R2: set_Pr4386577575007340137rm_a_b,S2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T2 )
@ ( relcom370159955682700863rm_a_b
@ ( transi7742714808557438673rm_a_b
@ ( inf_in1693821330460682071rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) )
@ ( produc1953217797448512882rm_a_b @ ( collect_term_a_b @ term_ground_a_b )
@ ^ [Uu: term_a_b] : ( collect_term_a_b @ term_ground_a_b ) ) ) )
@ ( transi7742714808557438673rm_a_b
@ ( inf_in1693821330460682071rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ S2 ) )
@ ( produc1953217797448512882rm_a_b @ ( collect_term_a_b @ term_ground_a_b )
@ ^ [Uu: term_a_b] : ( collect_term_a_b @ term_ground_a_b ) ) ) ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T2 ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) ) ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ S2 ) ) ) ) ) ) ).
% gsrsteps_eq_relcomp_srsteps_relcompD
thf(fact_514_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 )
=> ( ! [A3: a,B3: nat,Aa: a,Ba: nat] :
( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ ( product_Pair_a_nat @ Ax @ Ay ) @ ( product_Pair_a_nat @ A3 @ B3 ) ) @ ( transi2726145917338391738_a_nat @ R ) )
=> ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ ( product_Pair_a_nat @ A3 @ B3 ) @ ( product_Pair_a_nat @ Aa @ Ba ) ) @ R )
=> ( ( P @ A3 @ B3 )
=> ( P @ Aa @ Ba ) ) ) )
=> ( P @ Bx @ By ) ) ) ) ).
% rtrancl_induct2
thf(fact_515_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 ) )
=> ~ ! [A3: a,B3: nat] :
( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ ( product_Pair_a_nat @ Xa @ Xb ) @ ( product_Pair_a_nat @ A3 @ B3 ) ) @ R )
=> ~ ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ ( product_Pair_a_nat @ A3 @ B3 ) @ ( product_Pair_a_nat @ Za @ Zb ) ) @ ( transi2726145917338391738_a_nat @ R ) ) ) ) ) ).
% converse_rtranclE2
thf(fact_516_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 )
=> ( ! [A3: a,B3: nat,Aa: a,Ba: nat] :
( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ ( product_Pair_a_nat @ A3 @ B3 ) @ ( 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 @ A3 @ B3 ) ) ) )
=> ( P @ Ax @ Ay ) ) ) ) ).
% converse_rtrancl_induct2
thf(fact_517_relcomp3__I,axiom,
! [T2: nat,U: a,A: set_Pr4193341848836149977_nat_a,S: a,B: set_Pr4934435412358123699_a_nat,V: nat] :
( ( member8962352052110095674_nat_a @ ( product_Pair_nat_a @ T2 @ U ) @ A )
=> ( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ S @ T2 ) @ 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_518_relcomp3__I,axiom,
! [T2: a,U: nat,A: set_Pr4934435412358123699_a_nat,S: nat,B: set_Pr4193341848836149977_nat_a,V: a] :
( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ T2 @ U ) @ A )
=> ( ( member8962352052110095674_nat_a @ ( product_Pair_nat_a @ S @ T2 ) @ 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_519_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 ) )
=> ~ ! [B3: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A1 @ B3 ) @ R )
=> ~ ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ B3 @ A22 ) @ S ) ) ) ).
% relcomp.cases
thf(fact_520_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 ) )
=> ~ ! [B3: nat] :
( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ A1 @ B3 ) @ R )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B3 @ A22 ) @ S ) ) ) ).
% relcomp.cases
thf(fact_521_relcomp_Osimps,axiom,
! [A1: a,A22: nat,R: set_Product_prod_a_a,S: set_Pr4934435412358123699_a_nat] :
( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ A1 @ A22 ) @ ( relcomp_a_a_nat @ R @ S ) )
= ( ? [A4: a,B4: a,C4: nat] :
( ( A1 = A4 )
& ( A22 = C4 )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A4 @ B4 ) @ R )
& ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ B4 @ C4 ) @ S ) ) ) ) ).
% relcomp.simps
thf(fact_522_relcomp_Osimps,axiom,
! [A1: a,A22: nat,R: set_Pr4934435412358123699_a_nat,S: set_Pr1261947904930325089at_nat] :
( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ A1 @ A22 ) @ ( relcomp_a_nat_nat @ R @ S ) )
= ( ? [A4: a,B4: nat,C4: nat] :
( ( A1 = A4 )
& ( A22 = C4 )
& ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ A4 @ B4 ) @ R )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B4 @ C4 ) @ S ) ) ) ) ).
% relcomp.simps
thf(fact_523_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_524_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_525_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 ) )
=> ~ ! [X3: a,Y4: a,Z4: nat] :
( ( Xz
= ( product_Pair_a_nat @ X3 @ Z4 ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y4 ) @ R )
=> ~ ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ Y4 @ Z4 ) @ S ) ) ) ) ).
% relcompE
thf(fact_526_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 ) )
=> ~ ! [X3: a,Y4: nat,Z4: nat] :
( ( Xz
= ( product_Pair_a_nat @ X3 @ Z4 ) )
=> ( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ X3 @ Y4 ) @ R )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y4 @ Z4 ) @ S ) ) ) ) ).
% relcompE
thf(fact_527_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 ) )
=> ~ ! [B3: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ B3 ) @ R )
=> ~ ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ B3 @ C ) @ S ) ) ) ).
% relcompEpair
thf(fact_528_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 ) )
=> ~ ! [B3: nat] :
( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ A2 @ B3 ) @ R )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B3 @ C ) @ S ) ) ) ).
% relcompEpair
thf(fact_529_O__mono2,axiom,
! [R3: set_Pr4934435412358123699_a_nat,R4: set_Pr4934435412358123699_a_nat,T3: set_Pr1261947904930325089at_nat] :
( ( ord_le8666007276011122963_a_nat @ R3 @ R4 )
=> ( ord_le8666007276011122963_a_nat @ ( relcomp_a_nat_nat @ R3 @ T3 ) @ ( relcomp_a_nat_nat @ R4 @ T3 ) ) ) ).
% O_mono2
thf(fact_530_O__mono1,axiom,
! [R3: set_Pr4934435412358123699_a_nat,R4: set_Pr4934435412358123699_a_nat,S3: set_Product_prod_a_a] :
( ( ord_le8666007276011122963_a_nat @ R3 @ R4 )
=> ( ord_le8666007276011122963_a_nat @ ( relcomp_a_a_nat @ S3 @ R3 ) @ ( relcomp_a_a_nat @ S3 @ R4 ) ) ) ).
% O_mono1
thf(fact_531_relcomp__mono,axiom,
! [R5: set_Product_prod_a_a,R: set_Product_prod_a_a,S4: set_Pr4934435412358123699_a_nat,S: set_Pr4934435412358123699_a_nat] :
( ( ord_le746702958409616551od_a_a @ R5 @ R )
=> ( ( ord_le8666007276011122963_a_nat @ S4 @ S )
=> ( ord_le8666007276011122963_a_nat @ ( relcomp_a_a_nat @ R5 @ S4 ) @ ( relcomp_a_a_nat @ R @ S ) ) ) ) ).
% relcomp_mono
thf(fact_532_relcomp__mono,axiom,
! [R5: set_Pr4934435412358123699_a_nat,R: set_Pr4934435412358123699_a_nat,S4: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
( ( ord_le8666007276011122963_a_nat @ R5 @ R )
=> ( ( ord_le3146513528884898305at_nat @ S4 @ S )
=> ( ord_le8666007276011122963_a_nat @ ( relcomp_a_nat_nat @ R5 @ S4 ) @ ( relcomp_a_nat_nat @ R @ S ) ) ) ) ).
% relcomp_mono
thf(fact_533_compat__tr__compat,axiom,
! [NS: set_Product_prod_a_a,S3: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ ( relcomp_a_a_nat @ NS @ S3 ) @ S3 )
=> ( ord_le8666007276011122963_a_nat @ ( relcomp_a_a_nat @ ( transitive_rtrancl_a @ NS ) @ S3 ) @ S3 ) ) ).
% compat_tr_compat
thf(fact_534_relcomp__subset__Sigma,axiom,
! [R: set_Pr4386577575007340137rm_a_b,A: set_term_a_b,B: set_term_a_b,S: set_Pr4386577575007340137rm_a_b,C2: set_term_a_b] :
( ( ord_le118470702582115849rm_a_b @ R
@ ( produc1953217797448512882rm_a_b @ A
@ ^ [Uu: term_a_b] : B ) )
=> ( ( ord_le118470702582115849rm_a_b @ S
@ ( produc1953217797448512882rm_a_b @ B
@ ^ [Uu: term_a_b] : C2 ) )
=> ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ R @ S )
@ ( produc1953217797448512882rm_a_b @ A
@ ^ [Uu: term_a_b] : C2 ) ) ) ) ).
% relcomp_subset_Sigma
thf(fact_535_relcomp__subset__Sigma,axiom,
! [R: set_Product_prod_a_a,A: set_a,B: set_a,S: set_Pr4934435412358123699_a_nat,C2: set_nat] :
( ( ord_le746702958409616551od_a_a @ R
@ ( product_Sigma_a_a @ A
@ ^ [Uu: a] : B ) )
=> ( ( ord_le8666007276011122963_a_nat @ S
@ ( product_Sigma_a_nat @ B
@ ^ [Uu: a] : C2 ) )
=> ( ord_le8666007276011122963_a_nat @ ( relcomp_a_a_nat @ R @ S )
@ ( product_Sigma_a_nat @ A
@ ^ [Uu: a] : C2 ) ) ) ) ).
% relcomp_subset_Sigma
thf(fact_536_relcomp__subset__Sigma,axiom,
! [R: set_Pr4934435412358123699_a_nat,A: set_a,B: set_nat,S: set_Pr1261947904930325089at_nat,C2: set_nat] :
( ( ord_le8666007276011122963_a_nat @ R
@ ( product_Sigma_a_nat @ A
@ ^ [Uu: a] : B ) )
=> ( ( ord_le3146513528884898305at_nat @ S
@ ( produc457027306803732586at_nat @ B
@ ^ [Uu: nat] : C2 ) )
=> ( ord_le8666007276011122963_a_nat @ ( relcomp_a_nat_nat @ R @ S )
@ ( product_Sigma_a_nat @ A
@ ^ [Uu: a] : C2 ) ) ) ) ).
% relcomp_subset_Sigma
thf(fact_537_non__strict__into__strict,axiom,
! [NS: set_Product_prod_a_a,S3: set_Pr4934435412358123699_a_nat,S: a,T2: nat] :
( ( ord_le8666007276011122963_a_nat @ ( relcomp_a_a_nat @ NS @ S3 ) @ S3 )
=> ( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ S @ T2 ) @ ( relcomp_a_a_nat @ ( transitive_rtrancl_a @ NS ) @ S3 ) )
=> ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ S @ T2 ) @ S3 ) ) ) ).
% non_strict_into_strict
thf(fact_538_diamond__imp__CR,axiom,
! [R: set_Pr4386577575007340137rm_a_b] :
( ( abstra8151087076625346926rm_a_b @ R )
=> ( abstra8448919418672941150rm_a_b @ R @ top_top_set_term_a_b ) ) ).
% diamond_imp_CR
thf(fact_539_diamond__imp__CR,axiom,
! [R: set_Pr1811044260758604347_a_nat] :
( ( abstra1356934219170352343_a_nat @ R )
=> ( abstra7964079670522473415_a_nat @ R @ top_to3353692345378799459_a_nat ) ) ).
% diamond_imp_CR
thf(fact_540_gsrsteps__eq__relcomp__to__rsteps__relcomp,axiom,
! [S: term_a_b,T2: term_a_b,F2: set_Pr4934435412358123699_a_nat,R2: set_Pr4386577575007340137rm_a_b,S2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T2 )
@ ( relcom370159955682700863rm_a_b
@ ( transi7742714808557438673rm_a_b
@ ( inf_in1693821330460682071rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) )
@ ( produc1953217797448512882rm_a_b @ ( collect_term_a_b @ term_ground_a_b )
@ ^ [Uu: term_a_b] : ( collect_term_a_b @ term_ground_a_b ) ) ) )
@ ( transi7742714808557438673rm_a_b
@ ( inf_in1693821330460682071rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ S2 ) )
@ ( produc1953217797448512882rm_a_b @ ( collect_term_a_b @ term_ground_a_b )
@ ^ [Uu: term_a_b] : ( collect_term_a_b @ term_ground_a_b ) ) ) ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T2 ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( rstep_a_b @ R2 ) ) @ ( transi7742714808557438673rm_a_b @ ( rstep_a_b @ S2 ) ) ) ) ) ).
% gsrsteps_eq_relcomp_to_rsteps_relcomp
thf(fact_541_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_542_rsteps__eq__relcomp__srsteps__eq__relcompI,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,F2: set_Pr4934435412358123699_a_nat,S2: set_Pr4386577575007340137rm_a_b,S: term_a_b,T2: term_a_b] :
( ( ord_le8666007276011122963_a_nat @ ( terms_7988297476397195622_a_b_b @ R2 ) @ F2 )
=> ( ( ord_le8666007276011122963_a_nat @ ( terms_7988297476397195622_a_b_b @ S2 ) @ F2 )
=> ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ S ) @ F2 )
=> ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ T2 ) @ F2 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T2 ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( rstep_a_b @ R2 ) ) @ ( transi7742714808557438673rm_a_b @ ( rstep_a_b @ S2 ) ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T2 ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) ) ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ S2 ) ) ) ) ) ) ) ) ) ) ).
% rsteps_eq_relcomp_srsteps_eq_relcompI
thf(fact_543_rsteps__eq__srsteps__eqI,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,F2: set_Pr4934435412358123699_a_nat,S: term_a_b,T2: term_a_b] :
( ( ord_le8666007276011122963_a_nat @ ( terms_7988297476397195622_a_b_b @ R2 ) @ F2 )
=> ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ S ) @ F2 )
=> ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ T2 ) @ F2 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T2 ) @ ( transi7742714808557438673rm_a_b @ ( rstep_a_b @ R2 ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T2 ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) ) ) ) ) ) ) ) ).
% rsteps_eq_srsteps_eqI
thf(fact_544_ground__srsteps__gsrsteps,axiom,
! [S: term_a_b,T2: term_a_b,F2: set_Pr4934435412358123699_a_nat,R2: set_Pr4386577575007340137rm_a_b] :
( ( term_ground_a_b @ S )
=> ( ( term_ground_a_b @ T2 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T2 ) @ ( transi7922773638565587891rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T2 )
@ ( transi7922773638565587891rm_a_b
@ ( inf_in1693821330460682071rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) )
@ ( produc1953217797448512882rm_a_b @ ( collect_term_a_b @ term_ground_a_b )
@ ^ [Uu: term_a_b] : ( collect_term_a_b @ term_ground_a_b ) ) ) ) ) ) ) ) ).
% ground_srsteps_gsrsteps
thf(fact_545_sig__stepI,axiom,
! [S: term_a_b,F2: set_Pr4934435412358123699_a_nat,T2: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ S ) @ F2 )
=> ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ T2 ) @ F2 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T2 ) @ R2 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T2 ) @ ( sig_step_a_b @ F2 @ R2 ) ) ) ) ) ).
% sig_stepI
thf(fact_546_rstep__srstepI,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,F2: set_Pr4934435412358123699_a_nat,S: term_a_b,T2: term_a_b] :
( ( ord_le8666007276011122963_a_nat @ ( terms_7988297476397195622_a_b_b @ R2 ) @ F2 )
=> ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ S ) @ F2 )
=> ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ T2 ) @ F2 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T2 ) @ ( rstep_a_b @ R2 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T2 ) @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) ) ) ) ) ) ) ).
% rstep_srstepI
thf(fact_547_rsteps__srstepsI,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,F2: set_Pr4934435412358123699_a_nat,S: term_a_b,T2: term_a_b] :
( ( ord_le8666007276011122963_a_nat @ ( terms_7988297476397195622_a_b_b @ R2 ) @ F2 )
=> ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ S ) @ F2 )
=> ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ T2 ) @ F2 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T2 ) @ ( transi7922773638565587891rm_a_b @ ( rstep_a_b @ R2 ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T2 ) @ ( transi7922773638565587891rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) ) ) ) ) ) ) ) ).
% rsteps_srstepsI
thf(fact_548_trancl__full__on,axiom,
! [X6: set_term_a_b] :
( ( transi7922773638565587891rm_a_b
@ ( produc1953217797448512882rm_a_b @ X6
@ ^ [Uu: term_a_b] : X6 ) )
= ( produc1953217797448512882rm_a_b @ X6
@ ^ [Uu: term_a_b] : X6 ) ) ).
% trancl_full_on
thf(fact_549_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_550_srstepsD,axiom,
! [S: term_a_b,T2: term_a_b,F2: set_Pr4934435412358123699_a_nat,R2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T2 ) @ ( transi7922773638565587891rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) ) ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T2 ) @ ( transi7922773638565587891rm_a_b @ ( rstep_a_b @ R2 ) ) )
& ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ S ) @ F2 )
& ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ T2 ) @ F2 ) ) ) ).
% srstepsD
thf(fact_551_trancl__subset__Sigma,axiom,
! [R: set_Pr4386577575007340137rm_a_b,A: set_term_a_b] :
( ( ord_le118470702582115849rm_a_b @ R
@ ( produc1953217797448512882rm_a_b @ A
@ ^ [Uu: term_a_b] : A ) )
=> ( ord_le118470702582115849rm_a_b @ ( transi7922773638565587891rm_a_b @ R )
@ ( produc1953217797448512882rm_a_b @ A
@ ^ [Uu: term_a_b] : A ) ) ) ).
% trancl_subset_Sigma
thf(fact_552_sig__stepE,axiom,
! [S: term_a_b,T2: term_a_b,F2: set_Pr4934435412358123699_a_nat,R2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T2 ) @ ( sig_step_a_b @ F2 @ R2 ) )
=> ~ ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T2 ) @ R2 )
=> ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ S ) @ F2 )
=> ~ ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ T2 ) @ F2 ) ) ) ) ).
% sig_stepE
thf(fact_553_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_554_Restr__trancl__mono__set,axiom,
! [R: set_Pr4386577575007340137rm_a_b,A: set_term_a_b] :
( ord_le118470702582115849rm_a_b
@ ( transi7922773638565587891rm_a_b
@ ( inf_in1693821330460682071rm_a_b @ R
@ ( produc1953217797448512882rm_a_b @ A
@ ^ [Uu: term_a_b] : A ) ) )
@ ( produc1953217797448512882rm_a_b @ A
@ ^ [Uu: term_a_b] : A ) ) ).
% Restr_trancl_mono_set
thf(fact_555_srstepD,axiom,
! [S: term_a_b,T2: term_a_b,F2: set_Pr4934435412358123699_a_nat,R2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T2 ) @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T2 ) @ ( rstep_a_b @ R2 ) )
& ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ S ) @ F2 )
& ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ T2 ) @ F2 ) ) ) ).
% srstepD
thf(fact_556_sig__step__rsteps__dist,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,F2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ ( terms_7988297476397195622_a_b_b @ R2 ) @ F2 )
=> ( ( sig_step_a_b @ F2 @ ( transi7922773638565587891rm_a_b @ ( rstep_a_b @ R2 ) ) )
= ( transi7922773638565587891rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) ) ) ) ) ).
% sig_step_rsteps_dist
thf(fact_557_srsteps__monp,axiom,
! [F2: set_Pr4934435412358123699_a_nat,G: set_Pr4934435412358123699_a_nat,R2: set_Pr4386577575007340137rm_a_b] :
( ( ord_le8666007276011122963_a_nat @ F2 @ G )
=> ( ord_le118470702582115849rm_a_b @ ( transi7922773638565587891rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) ) ) @ ( transi7922773638565587891rm_a_b @ ( sig_step_a_b @ G @ ( rstep_a_b @ R2 ) ) ) ) ) ).
% srsteps_monp
thf(fact_558_sig__step__def,axiom,
( sig_step_a_b
= ( ^ [F3: set_Pr4934435412358123699_a_nat,R6: set_Pr4386577575007340137rm_a_b] :
( inf_in1693821330460682071rm_a_b @ R6
@ ( produc1953217797448512882rm_a_b
@ ( collect_term_a_b
@ ^ [S5: term_a_b] : ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ S5 ) @ F3 ) )
@ ^ [Uu: term_a_b] :
( collect_term_a_b
@ ^ [S5: term_a_b] : ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ S5 ) @ F3 ) ) ) ) ) ) ).
% sig_step_def
thf(fact_559_sigstep__trancl__funas_I2_J,axiom,
! [S: term_a_b,T2: term_a_b,F2: set_Pr4934435412358123699_a_nat,S2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T2 ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ S2 ) ) ) )
=> ( ( S != T2 )
=> ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ T2 ) @ F2 ) ) ) ).
% sigstep_trancl_funas(2)
thf(fact_560_sigstep__trancl__funas_I1_J,axiom,
! [S: term_a_b,T2: term_a_b,F2: set_Pr4934435412358123699_a_nat,S2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T2 ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ S2 ) ) ) )
=> ( ( S != T2 )
=> ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ S ) @ F2 ) ) ) ).
% sigstep_trancl_funas(1)
thf(fact_561_llrg__monadic__rsteps__groundness,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,F2: set_Pr4934435412358123699_a_nat,S: term_a_b,T2: term_a_b] :
( ( rewrit5000278123254648670_b_a_b @ R2 )
=> ( ( rewrit3916899077429870363adic_a @ F2 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T2 ) @ ( transi7922773638565587891rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) ) ) )
=> ( term_ground_a_b @ T2 ) ) ) ) ).
% llrg_monadic_rsteps_groundness
thf(fact_562_gsrsteps__to__rsteps,axiom,
! [S: term_a_b,T2: term_a_b,F2: set_Pr4934435412358123699_a_nat,R2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T2 )
@ ( transi7922773638565587891rm_a_b
@ ( inf_in1693821330460682071rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) )
@ ( produc1953217797448512882rm_a_b @ ( collect_term_a_b @ term_ground_a_b )
@ ^ [Uu: term_a_b] : ( collect_term_a_b @ term_ground_a_b ) ) ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T2 ) @ ( transi7922773638565587891rm_a_b @ ( rstep_a_b @ R2 ) ) ) ) ).
% gsrsteps_to_rsteps
thf(fact_563_gsrsteps__to__srsteps,axiom,
! [S: term_a_b,T2: term_a_b,F2: set_Pr4934435412358123699_a_nat,R2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T2 )
@ ( transi7922773638565587891rm_a_b
@ ( inf_in1693821330460682071rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) )
@ ( produc1953217797448512882rm_a_b @ ( collect_term_a_b @ term_ground_a_b )
@ ^ [Uu: term_a_b] : ( collect_term_a_b @ term_ground_a_b ) ) ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T2 ) @ ( transi7922773638565587891rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) ) ) ) ) ).
% gsrsteps_to_srsteps
thf(fact_564_rstep__trancl__sig__step__l,axiom,
! [S: term_a_b,T2: term_a_b,R2: set_Pr4386577575007340137rm_a_b,F2: set_Pr4934435412358123699_a_nat,V: b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T2 ) @ ( transi7922773638565587891rm_a_b @ ( rstep_a_b @ R2 ) ) )
=> ( ( ord_le8666007276011122963_a_nat @ ( terms_7988297476397195622_a_b_b @ R2 ) @ F2 )
=> ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ T2 ) @ F2 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( terms_8519481630511763164ig_a_b @ F2 @ V @ S ) @ T2 ) @ ( transi7922773638565587891rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) ) ) ) ) ) ) ).
% rstep_trancl_sig_step_l
thf(fact_565_rstep__trancl__sig__step__r,axiom,
! [S: term_a_b,T2: term_a_b,R2: set_Pr4386577575007340137rm_a_b,F2: set_Pr4934435412358123699_a_nat,V: b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T2 ) @ ( transi7922773638565587891rm_a_b @ ( rstep_a_b @ R2 ) ) )
=> ( ( ord_le8666007276011122963_a_nat @ ( terms_7988297476397195622_a_b_b @ R2 ) @ F2 )
=> ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ S ) @ F2 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ ( terms_8519481630511763164ig_a_b @ F2 @ V @ T2 ) ) @ ( transi7922773638565587891rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) ) ) ) ) ) ) ).
% rstep_trancl_sig_step_r
thf(fact_566_llrg__funas__term__steps__pres,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,S: term_a_b,T2: term_a_b] :
( ( rewrit5000278123254648670_b_a_b @ R2 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T2 ) @ ( transi7742714808557438673rm_a_b @ ( rstep_a_b @ R2 ) ) )
=> ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ T2 ) @ ( sup_su459911885395995103_a_nat @ ( terms_7988297476397195622_a_b_b @ R2 ) @ ( term_funas_term_a_b @ S ) ) ) ) ) ).
% llrg_funas_term_steps_pres
thf(fact_567_srsteps__eq__subst__relcomp__closed,axiom,
! [S: term_a_b,T2: term_a_b,F2: set_Pr4934435412358123699_a_nat,R2: set_Pr4386577575007340137rm_a_b,S2: set_Pr4386577575007340137rm_a_b,Sigma: b > term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T2 ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) ) ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ S2 ) ) ) ) )
=> ( ! [X3: b] : ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ ( Sigma @ X3 ) ) @ F2 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( subst_7999470309526761004_a_b_b @ S @ Sigma ) @ ( subst_7999470309526761004_a_b_b @ T2 @ Sigma ) ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) ) ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ S2 ) ) ) ) ) ) ) ).
% srsteps_eq_subst_relcomp_closed
thf(fact_568_llrg__funas__term__step__pres,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,S: term_a_b,T2: term_a_b] :
( ( rewrit5000278123254648670_b_a_b @ R2 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T2 ) @ ( rstep_a_b @ R2 ) )
=> ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ T2 ) @ ( sup_su459911885395995103_a_nat @ ( terms_7988297476397195622_a_b_b @ R2 ) @ ( term_funas_term_a_b @ S ) ) ) ) ) ).
% llrg_funas_term_step_pres
thf(fact_569_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_570_UnCI,axiom,
! [C: term_a_b,B: set_term_a_b,A: set_term_a_b] :
( ( ~ ( member_term_a_b @ C @ B )
=> ( member_term_a_b @ C @ A ) )
=> ( member_term_a_b @ C @ ( sup_sup_set_term_a_b @ A @ B ) ) ) ).
% UnCI
thf(fact_571_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_572_Un__iff,axiom,
! [C: term_a_b,A: set_term_a_b,B: set_term_a_b] :
( ( member_term_a_b @ C @ ( sup_sup_set_term_a_b @ A @ B ) )
= ( ( member_term_a_b @ C @ A )
| ( member_term_a_b @ C @ B ) ) ) ).
% Un_iff
thf(fact_573_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_574_sup_Obounded__iff,axiom,
! [B2: nat,C: nat,A2: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B2 @ C ) @ A2 )
= ( ( ord_less_eq_nat @ B2 @ A2 )
& ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_575_le__sup__iff,axiom,
! [X: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat,Z: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ ( sup_su459911885395995103_a_nat @ X @ Y ) @ Z )
= ( ( ord_le8666007276011122963_a_nat @ X @ Z )
& ( ord_le8666007276011122963_a_nat @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_576_le__sup__iff,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ X @ Y ) @ Z )
= ( ( ord_less_eq_nat @ X @ Z )
& ( ord_less_eq_nat @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_577_boolean__algebra_Odisj__one__right,axiom,
! [X: set_term_a_b] :
( ( sup_sup_set_term_a_b @ X @ top_top_set_term_a_b )
= top_top_set_term_a_b ) ).
% boolean_algebra.disj_one_right
thf(fact_578_boolean__algebra_Odisj__one__right,axiom,
! [X: set_Pr4934435412358123699_a_nat] :
( ( sup_su459911885395995103_a_nat @ X @ top_to3353692345378799459_a_nat )
= top_to3353692345378799459_a_nat ) ).
% boolean_algebra.disj_one_right
thf(fact_579_boolean__algebra_Odisj__one__left,axiom,
! [X: set_term_a_b] :
( ( sup_sup_set_term_a_b @ top_top_set_term_a_b @ X )
= top_top_set_term_a_b ) ).
% boolean_algebra.disj_one_left
thf(fact_580_boolean__algebra_Odisj__one__left,axiom,
! [X: set_Pr4934435412358123699_a_nat] :
( ( sup_su459911885395995103_a_nat @ top_to3353692345378799459_a_nat @ X )
= top_to3353692345378799459_a_nat ) ).
% boolean_algebra.disj_one_left
thf(fact_581_sup__top__left,axiom,
! [X: set_term_a_b] :
( ( sup_sup_set_term_a_b @ top_top_set_term_a_b @ X )
= top_top_set_term_a_b ) ).
% sup_top_left
thf(fact_582_sup__top__left,axiom,
! [X: set_Pr4934435412358123699_a_nat] :
( ( sup_su459911885395995103_a_nat @ top_to3353692345378799459_a_nat @ X )
= top_to3353692345378799459_a_nat ) ).
% sup_top_left
thf(fact_583_sup__top__right,axiom,
! [X: set_term_a_b] :
( ( sup_sup_set_term_a_b @ X @ top_top_set_term_a_b )
= top_top_set_term_a_b ) ).
% sup_top_right
thf(fact_584_sup__top__right,axiom,
! [X: set_Pr4934435412358123699_a_nat] :
( ( sup_su459911885395995103_a_nat @ X @ top_to3353692345378799459_a_nat )
= top_to3353692345378799459_a_nat ) ).
% sup_top_right
thf(fact_585_sup__inf__absorb,axiom,
! [X: set_Pr4386577575007340137rm_a_b,Y: set_Pr4386577575007340137rm_a_b] :
( ( sup_su6776935440552674877rm_a_b @ X @ ( inf_in1693821330460682071rm_a_b @ X @ Y ) )
= X ) ).
% sup_inf_absorb
thf(fact_586_inf__sup__absorb,axiom,
! [X: set_Pr4386577575007340137rm_a_b,Y: set_Pr4386577575007340137rm_a_b] :
( ( inf_in1693821330460682071rm_a_b @ X @ ( sup_su6776935440552674877rm_a_b @ X @ Y ) )
= X ) ).
% inf_sup_absorb
thf(fact_587_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_588_Un__insert__right,axiom,
! [A: set_Pr4934435412358123699_a_nat,A2: product_prod_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( sup_su459911885395995103_a_nat @ A @ ( insert8054603423593749677_a_nat @ A2 @ B ) )
= ( insert8054603423593749677_a_nat @ A2 @ ( sup_su459911885395995103_a_nat @ A @ B ) ) ) ).
% Un_insert_right
thf(fact_589_Un__insert__left,axiom,
! [A2: product_prod_a_nat,B: set_Pr4934435412358123699_a_nat,C2: set_Pr4934435412358123699_a_nat] :
( ( sup_su459911885395995103_a_nat @ ( insert8054603423593749677_a_nat @ A2 @ B ) @ C2 )
= ( insert8054603423593749677_a_nat @ A2 @ ( sup_su459911885395995103_a_nat @ B @ C2 ) ) ) ).
% Un_insert_left
thf(fact_590_Un__Int__eq_I1_J,axiom,
! [S3: set_Pr4386577575007340137rm_a_b,T3: set_Pr4386577575007340137rm_a_b] :
( ( inf_in1693821330460682071rm_a_b @ ( sup_su6776935440552674877rm_a_b @ S3 @ T3 ) @ S3 )
= S3 ) ).
% Un_Int_eq(1)
thf(fact_591_Un__Int__eq_I2_J,axiom,
! [S3: set_Pr4386577575007340137rm_a_b,T3: set_Pr4386577575007340137rm_a_b] :
( ( inf_in1693821330460682071rm_a_b @ ( sup_su6776935440552674877rm_a_b @ S3 @ T3 ) @ T3 )
= T3 ) ).
% Un_Int_eq(2)
thf(fact_592_Un__Int__eq_I3_J,axiom,
! [S3: set_Pr4386577575007340137rm_a_b,T3: set_Pr4386577575007340137rm_a_b] :
( ( inf_in1693821330460682071rm_a_b @ S3 @ ( sup_su6776935440552674877rm_a_b @ S3 @ T3 ) )
= S3 ) ).
% Un_Int_eq(3)
thf(fact_593_Un__Int__eq_I4_J,axiom,
! [T3: set_Pr4386577575007340137rm_a_b,S3: set_Pr4386577575007340137rm_a_b] :
( ( inf_in1693821330460682071rm_a_b @ T3 @ ( sup_su6776935440552674877rm_a_b @ S3 @ T3 ) )
= T3 ) ).
% Un_Int_eq(4)
thf(fact_594_Int__Un__eq_I1_J,axiom,
! [S3: set_Pr4386577575007340137rm_a_b,T3: set_Pr4386577575007340137rm_a_b] :
( ( sup_su6776935440552674877rm_a_b @ ( inf_in1693821330460682071rm_a_b @ S3 @ T3 ) @ S3 )
= S3 ) ).
% Int_Un_eq(1)
thf(fact_595_Int__Un__eq_I2_J,axiom,
! [S3: set_Pr4386577575007340137rm_a_b,T3: set_Pr4386577575007340137rm_a_b] :
( ( sup_su6776935440552674877rm_a_b @ ( inf_in1693821330460682071rm_a_b @ S3 @ T3 ) @ T3 )
= T3 ) ).
% Int_Un_eq(2)
thf(fact_596_Int__Un__eq_I3_J,axiom,
! [S3: set_Pr4386577575007340137rm_a_b,T3: set_Pr4386577575007340137rm_a_b] :
( ( sup_su6776935440552674877rm_a_b @ S3 @ ( inf_in1693821330460682071rm_a_b @ S3 @ T3 ) )
= S3 ) ).
% Int_Un_eq(3)
thf(fact_597_Int__Un__eq_I4_J,axiom,
! [T3: set_Pr4386577575007340137rm_a_b,S3: set_Pr4386577575007340137rm_a_b] :
( ( sup_su6776935440552674877rm_a_b @ T3 @ ( inf_in1693821330460682071rm_a_b @ S3 @ T3 ) )
= T3 ) ).
% Int_Un_eq(4)
thf(fact_598_rstep__substI,axiom,
! [S: term_a_b,T2: term_a_b,R2: set_Pr4386577575007340137rm_a_b,Sigma: b > term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T2 ) @ ( rstep_a_b @ R2 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( subst_7999470309526761004_a_b_b @ S @ Sigma ) @ ( subst_7999470309526761004_a_b_b @ T2 @ Sigma ) ) @ ( rstep_a_b @ R2 ) ) ) ).
% rstep_substI
thf(fact_599_all__ctxt__closed__sig__rsteps,axiom,
! [F2: set_Pr4934435412358123699_a_nat,R2: set_Pr4386577575007340137rm_a_b] : ( terms_5226143800768910156ed_a_b @ F2 @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) ) ) ) ).
% all_ctxt_closed_sig_rsteps
thf(fact_600_insert__Times__insert,axiom,
! [A2: product_prod_a_nat,A: set_Pr4934435412358123699_a_nat,B2: product_prod_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( produc3832836940518797764_a_nat @ ( insert8054603423593749677_a_nat @ A2 @ A )
@ ^ [Uu: product_prod_a_nat] : ( insert8054603423593749677_a_nat @ B2 @ B ) )
= ( insert6796211413315061163_a_nat @ ( produc2026711137822539155_a_nat @ A2 @ B2 )
@ ( sup_su958340139570419215_a_nat
@ ( produc3832836940518797764_a_nat @ A
@ ^ [Uu: product_prod_a_nat] : ( insert8054603423593749677_a_nat @ B2 @ B ) )
@ ( produc3832836940518797764_a_nat @ ( insert8054603423593749677_a_nat @ A2 @ A )
@ ^ [Uu: product_prod_a_nat] : B ) ) ) ) ).
% insert_Times_insert
thf(fact_601_insert__Times__insert,axiom,
! [A2: a,A: set_a,B2: nat,B: set_nat] :
( ( product_Sigma_a_nat @ ( insert_a @ A2 @ A )
@ ^ [Uu: a] : ( insert_nat @ B2 @ B ) )
= ( insert8054603423593749677_a_nat @ ( product_Pair_a_nat @ A2 @ B2 )
@ ( sup_su459911885395995103_a_nat
@ ( product_Sigma_a_nat @ A
@ ^ [Uu: a] : ( insert_nat @ B2 @ B ) )
@ ( product_Sigma_a_nat @ ( insert_a @ A2 @ A )
@ ^ [Uu: a] : B ) ) ) ) ).
% insert_Times_insert
thf(fact_602_insert__Times__insert,axiom,
! [A2: term_a_b,A: set_term_a_b,B2: term_a_b,B: set_term_a_b] :
( ( produc1953217797448512882rm_a_b @ ( insert_term_a_b @ A2 @ A )
@ ^ [Uu: term_a_b] : ( insert_term_a_b @ B2 @ B ) )
= ( insert7009541432154983385rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 )
@ ( sup_su6776935440552674877rm_a_b
@ ( produc1953217797448512882rm_a_b @ A
@ ^ [Uu: term_a_b] : ( insert_term_a_b @ B2 @ B ) )
@ ( produc1953217797448512882rm_a_b @ ( insert_term_a_b @ A2 @ A )
@ ^ [Uu: term_a_b] : B ) ) ) ) ).
% insert_Times_insert
thf(fact_603_Sigma__Un__distrib2,axiom,
! [I: set_term_a_b,A: term_a_b > set_term_a_b,B: term_a_b > set_term_a_b] :
( ( produc1953217797448512882rm_a_b @ I
@ ^ [I2: term_a_b] : ( sup_sup_set_term_a_b @ ( A @ I2 ) @ ( B @ I2 ) ) )
= ( sup_su6776935440552674877rm_a_b @ ( produc1953217797448512882rm_a_b @ I @ A ) @ ( produc1953217797448512882rm_a_b @ I @ B ) ) ) ).
% Sigma_Un_distrib2
thf(fact_604_Times__Un__distrib1,axiom,
! [A: set_term_a_b,B: set_term_a_b,C2: set_term_a_b] :
( ( produc1953217797448512882rm_a_b @ ( sup_sup_set_term_a_b @ A @ B )
@ ^ [Uu: term_a_b] : C2 )
= ( sup_su6776935440552674877rm_a_b
@ ( produc1953217797448512882rm_a_b @ A
@ ^ [Uu: term_a_b] : C2 )
@ ( produc1953217797448512882rm_a_b @ B
@ ^ [Uu: term_a_b] : C2 ) ) ) ).
% Times_Un_distrib1
thf(fact_605_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_606_UnE,axiom,
! [C: term_a_b,A: set_term_a_b,B: set_term_a_b] :
( ( member_term_a_b @ C @ ( sup_sup_set_term_a_b @ A @ B ) )
=> ( ~ ( member_term_a_b @ C @ A )
=> ( member_term_a_b @ C @ B ) ) ) ).
% UnE
thf(fact_607_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_608_UnI1,axiom,
! [C: term_a_b,A: set_term_a_b,B: set_term_a_b] :
( ( member_term_a_b @ C @ A )
=> ( member_term_a_b @ C @ ( sup_sup_set_term_a_b @ A @ B ) ) ) ).
% UnI1
thf(fact_609_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_610_UnI2,axiom,
! [C: term_a_b,B: set_term_a_b,A: set_term_a_b] :
( ( member_term_a_b @ C @ B )
=> ( member_term_a_b @ C @ ( sup_sup_set_term_a_b @ A @ B ) ) ) ).
% UnI2
thf(fact_611_Un__def,axiom,
( sup_su459911885395995103_a_nat
= ( ^ [A5: set_Pr4934435412358123699_a_nat,B5: set_Pr4934435412358123699_a_nat] :
( collec4464134535221767506_a_nat
@ ^ [X2: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ X2 @ A5 )
| ( member5724188588386418708_a_nat @ X2 @ B5 ) ) ) ) ) ).
% Un_def
thf(fact_612_Un__def,axiom,
( sup_sup_set_term_a_b
= ( ^ [A5: set_term_a_b,B5: set_term_a_b] :
( collect_term_a_b
@ ^ [X2: term_a_b] :
( ( member_term_a_b @ X2 @ A5 )
| ( member_term_a_b @ X2 @ B5 ) ) ) ) ) ).
% Un_def
thf(fact_613_Collect__disj__eq,axiom,
! [P: term_a_b > $o,Q: term_a_b > $o] :
( ( collect_term_a_b
@ ^ [X2: term_a_b] :
( ( P @ X2 )
| ( Q @ X2 ) ) )
= ( sup_sup_set_term_a_b @ ( collect_term_a_b @ P ) @ ( collect_term_a_b @ Q ) ) ) ).
% Collect_disj_eq
thf(fact_614_Sigma__Un__distrib1,axiom,
! [I: set_term_a_b,J: set_term_a_b,C2: term_a_b > set_term_a_b] :
( ( produc1953217797448512882rm_a_b @ ( sup_sup_set_term_a_b @ I @ J ) @ C2 )
= ( sup_su6776935440552674877rm_a_b @ ( produc1953217797448512882rm_a_b @ I @ C2 ) @ ( produc1953217797448512882rm_a_b @ J @ C2 ) ) ) ).
% Sigma_Un_distrib1
thf(fact_615_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_616_sup_OcoboundedI2,axiom,
! [C: nat,B2: nat,A2: nat] :
( ( ord_less_eq_nat @ C @ B2 )
=> ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_617_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_618_sup_OcoboundedI1,axiom,
! [C: nat,A2: nat,B2: nat] :
( ( ord_less_eq_nat @ C @ A2 )
=> ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_619_sup_Oabsorb__iff2,axiom,
( ord_le8666007276011122963_a_nat
= ( ^ [A4: set_Pr4934435412358123699_a_nat,B4: set_Pr4934435412358123699_a_nat] :
( ( sup_su459911885395995103_a_nat @ A4 @ B4 )
= B4 ) ) ) ).
% sup.absorb_iff2
thf(fact_620_sup_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B4: nat] :
( ( sup_sup_nat @ A4 @ B4 )
= B4 ) ) ) ).
% sup.absorb_iff2
thf(fact_621_sup_Oabsorb__iff1,axiom,
( ord_le8666007276011122963_a_nat
= ( ^ [B4: set_Pr4934435412358123699_a_nat,A4: set_Pr4934435412358123699_a_nat] :
( ( sup_su459911885395995103_a_nat @ A4 @ B4 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_622_sup_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [B4: nat,A4: nat] :
( ( sup_sup_nat @ A4 @ B4 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_623_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_624_sup_Ocobounded2,axiom,
! [B2: nat,A2: nat] : ( ord_less_eq_nat @ B2 @ ( sup_sup_nat @ A2 @ B2 ) ) ).
% sup.cobounded2
thf(fact_625_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_626_sup_Ocobounded1,axiom,
! [A2: nat,B2: nat] : ( ord_less_eq_nat @ A2 @ ( sup_sup_nat @ A2 @ B2 ) ) ).
% sup.cobounded1
thf(fact_627_sup_Oorder__iff,axiom,
( ord_le8666007276011122963_a_nat
= ( ^ [B4: set_Pr4934435412358123699_a_nat,A4: set_Pr4934435412358123699_a_nat] :
( A4
= ( sup_su459911885395995103_a_nat @ A4 @ B4 ) ) ) ) ).
% sup.order_iff
thf(fact_628_sup_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [B4: nat,A4: nat] :
( A4
= ( sup_sup_nat @ A4 @ B4 ) ) ) ) ).
% sup.order_iff
thf(fact_629_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_630_sup_OboundedI,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( ord_less_eq_nat @ C @ A2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ B2 @ C ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_631_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_632_sup_OboundedE,axiom,
! [B2: nat,C: nat,A2: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B2 @ C ) @ A2 )
=> ~ ( ( ord_less_eq_nat @ B2 @ A2 )
=> ~ ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% sup.boundedE
thf(fact_633_sup__absorb2,axiom,
! [X: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X @ Y )
=> ( ( sup_su459911885395995103_a_nat @ X @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_634_sup__absorb2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( sup_sup_nat @ X @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_635_sup__absorb1,axiom,
! [Y: set_Pr4934435412358123699_a_nat,X: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ Y @ X )
=> ( ( sup_su459911885395995103_a_nat @ X @ Y )
= X ) ) ).
% sup_absorb1
thf(fact_636_sup__absorb1,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( sup_sup_nat @ X @ Y )
= X ) ) ).
% sup_absorb1
thf(fact_637_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_638_sup_Oabsorb2,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( sup_sup_nat @ A2 @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_639_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_640_sup_Oabsorb1,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( sup_sup_nat @ A2 @ B2 )
= A2 ) ) ).
% sup.absorb1
thf(fact_641_sup__unique,axiom,
! [F: set_Pr4934435412358123699_a_nat > set_Pr4934435412358123699_a_nat > set_Pr4934435412358123699_a_nat,X: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat] :
( ! [X3: set_Pr4934435412358123699_a_nat,Y4: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ X3 @ ( F @ X3 @ Y4 ) )
=> ( ! [X3: set_Pr4934435412358123699_a_nat,Y4: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ Y4 @ ( F @ X3 @ Y4 ) )
=> ( ! [X3: set_Pr4934435412358123699_a_nat,Y4: set_Pr4934435412358123699_a_nat,Z4: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ Y4 @ X3 )
=> ( ( ord_le8666007276011122963_a_nat @ Z4 @ X3 )
=> ( ord_le8666007276011122963_a_nat @ ( F @ Y4 @ Z4 ) @ X3 ) ) )
=> ( ( sup_su459911885395995103_a_nat @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_642_sup__unique,axiom,
! [F: nat > nat > nat,X: nat,Y: nat] :
( ! [X3: nat,Y4: nat] : ( ord_less_eq_nat @ X3 @ ( F @ X3 @ Y4 ) )
=> ( ! [X3: nat,Y4: nat] : ( ord_less_eq_nat @ Y4 @ ( F @ X3 @ Y4 ) )
=> ( ! [X3: nat,Y4: nat,Z4: nat] :
( ( ord_less_eq_nat @ Y4 @ X3 )
=> ( ( ord_less_eq_nat @ Z4 @ X3 )
=> ( ord_less_eq_nat @ ( F @ Y4 @ Z4 ) @ X3 ) ) )
=> ( ( sup_sup_nat @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_643_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_644_sup_OorderI,axiom,
! [A2: nat,B2: nat] :
( ( A2
= ( sup_sup_nat @ A2 @ B2 ) )
=> ( ord_less_eq_nat @ B2 @ A2 ) ) ).
% sup.orderI
thf(fact_645_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_646_sup_OorderE,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( A2
= ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% sup.orderE
thf(fact_647_le__iff__sup,axiom,
( ord_le8666007276011122963_a_nat
= ( ^ [X2: set_Pr4934435412358123699_a_nat,Y3: set_Pr4934435412358123699_a_nat] :
( ( sup_su459911885395995103_a_nat @ X2 @ Y3 )
= Y3 ) ) ) ).
% le_iff_sup
thf(fact_648_le__iff__sup,axiom,
( ord_less_eq_nat
= ( ^ [X2: nat,Y3: nat] :
( ( sup_sup_nat @ X2 @ Y3 )
= Y3 ) ) ) ).
% le_iff_sup
thf(fact_649_sup__least,axiom,
! [Y: set_Pr4934435412358123699_a_nat,X: set_Pr4934435412358123699_a_nat,Z: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ Y @ X )
=> ( ( ord_le8666007276011122963_a_nat @ Z @ X )
=> ( ord_le8666007276011122963_a_nat @ ( sup_su459911885395995103_a_nat @ Y @ Z ) @ X ) ) ) ).
% sup_least
thf(fact_650_sup__least,axiom,
! [Y: nat,X: nat,Z: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_less_eq_nat @ Z @ X )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ Y @ Z ) @ X ) ) ) ).
% sup_least
thf(fact_651_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_652_sup__mono,axiom,
! [A2: nat,C: nat,B2: nat,D2: nat] :
( ( ord_less_eq_nat @ A2 @ C )
=> ( ( ord_less_eq_nat @ B2 @ D2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B2 ) @ ( sup_sup_nat @ C @ D2 ) ) ) ) ).
% sup_mono
thf(fact_653_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_654_sup_Omono,axiom,
! [C: nat,A2: nat,D2: nat,B2: nat] :
( ( ord_less_eq_nat @ C @ A2 )
=> ( ( ord_less_eq_nat @ D2 @ B2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ C @ D2 ) @ ( sup_sup_nat @ A2 @ B2 ) ) ) ) ).
% sup.mono
thf(fact_655_le__supI2,axiom,
! [X: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat,A2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X @ B2 )
=> ( ord_le8666007276011122963_a_nat @ X @ ( sup_su459911885395995103_a_nat @ A2 @ B2 ) ) ) ).
% le_supI2
thf(fact_656_le__supI2,axiom,
! [X: nat,B2: nat,A2: nat] :
( ( ord_less_eq_nat @ X @ B2 )
=> ( ord_less_eq_nat @ X @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% le_supI2
thf(fact_657_le__supI1,axiom,
! [X: set_Pr4934435412358123699_a_nat,A2: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X @ A2 )
=> ( ord_le8666007276011122963_a_nat @ X @ ( sup_su459911885395995103_a_nat @ A2 @ B2 ) ) ) ).
% le_supI1
thf(fact_658_le__supI1,axiom,
! [X: nat,A2: nat,B2: nat] :
( ( ord_less_eq_nat @ X @ A2 )
=> ( ord_less_eq_nat @ X @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% le_supI1
thf(fact_659_sup__ge2,axiom,
! [Y: set_Pr4934435412358123699_a_nat,X: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ Y @ ( sup_su459911885395995103_a_nat @ X @ Y ) ) ).
% sup_ge2
thf(fact_660_sup__ge2,axiom,
! [Y: nat,X: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X @ Y ) ) ).
% sup_ge2
thf(fact_661_sup__ge1,axiom,
! [X: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ X @ ( sup_su459911885395995103_a_nat @ X @ Y ) ) ).
% sup_ge1
thf(fact_662_sup__ge1,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ X @ ( sup_sup_nat @ X @ Y ) ) ).
% sup_ge1
thf(fact_663_le__supI,axiom,
! [A2: set_Pr4934435412358123699_a_nat,X: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A2 @ X )
=> ( ( ord_le8666007276011122963_a_nat @ B2 @ X )
=> ( ord_le8666007276011122963_a_nat @ ( sup_su459911885395995103_a_nat @ A2 @ B2 ) @ X ) ) ) ).
% le_supI
thf(fact_664_le__supI,axiom,
! [A2: nat,X: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ X )
=> ( ( ord_less_eq_nat @ B2 @ X )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B2 ) @ X ) ) ) ).
% le_supI
thf(fact_665_le__supE,axiom,
! [A2: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat,X: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ ( sup_su459911885395995103_a_nat @ A2 @ B2 ) @ X )
=> ~ ( ( ord_le8666007276011122963_a_nat @ A2 @ X )
=> ~ ( ord_le8666007276011122963_a_nat @ B2 @ X ) ) ) ).
% le_supE
thf(fact_666_le__supE,axiom,
! [A2: nat,B2: nat,X: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B2 ) @ X )
=> ~ ( ( ord_less_eq_nat @ A2 @ X )
=> ~ ( ord_less_eq_nat @ B2 @ X ) ) ) ).
% le_supE
thf(fact_667_inf__sup__ord_I3_J,axiom,
! [X: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ X @ ( sup_su459911885395995103_a_nat @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_668_inf__sup__ord_I3_J,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ X @ ( sup_sup_nat @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_669_inf__sup__ord_I4_J,axiom,
! [Y: set_Pr4934435412358123699_a_nat,X: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ Y @ ( sup_su459911885395995103_a_nat @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_670_inf__sup__ord_I4_J,axiom,
! [Y: nat,X: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_671_boolean__algebra_Odisj__conj__distrib2,axiom,
! [Y: set_Pr4386577575007340137rm_a_b,Z: set_Pr4386577575007340137rm_a_b,X: set_Pr4386577575007340137rm_a_b] :
( ( sup_su6776935440552674877rm_a_b @ ( inf_in1693821330460682071rm_a_b @ Y @ Z ) @ X )
= ( inf_in1693821330460682071rm_a_b @ ( sup_su6776935440552674877rm_a_b @ Y @ X ) @ ( sup_su6776935440552674877rm_a_b @ Z @ X ) ) ) ).
% boolean_algebra.disj_conj_distrib2
thf(fact_672_boolean__algebra_Oconj__disj__distrib2,axiom,
! [Y: set_Pr4386577575007340137rm_a_b,Z: set_Pr4386577575007340137rm_a_b,X: set_Pr4386577575007340137rm_a_b] :
( ( inf_in1693821330460682071rm_a_b @ ( sup_su6776935440552674877rm_a_b @ Y @ Z ) @ X )
= ( sup_su6776935440552674877rm_a_b @ ( inf_in1693821330460682071rm_a_b @ Y @ X ) @ ( inf_in1693821330460682071rm_a_b @ Z @ X ) ) ) ).
% boolean_algebra.conj_disj_distrib2
thf(fact_673_boolean__algebra_Odisj__conj__distrib,axiom,
! [X: set_Pr4386577575007340137rm_a_b,Y: set_Pr4386577575007340137rm_a_b,Z: set_Pr4386577575007340137rm_a_b] :
( ( sup_su6776935440552674877rm_a_b @ X @ ( inf_in1693821330460682071rm_a_b @ Y @ Z ) )
= ( inf_in1693821330460682071rm_a_b @ ( sup_su6776935440552674877rm_a_b @ X @ Y ) @ ( sup_su6776935440552674877rm_a_b @ X @ Z ) ) ) ).
% boolean_algebra.disj_conj_distrib
thf(fact_674_boolean__algebra_Oconj__disj__distrib,axiom,
! [X: set_Pr4386577575007340137rm_a_b,Y: set_Pr4386577575007340137rm_a_b,Z: set_Pr4386577575007340137rm_a_b] :
( ( inf_in1693821330460682071rm_a_b @ X @ ( sup_su6776935440552674877rm_a_b @ Y @ Z ) )
= ( sup_su6776935440552674877rm_a_b @ ( inf_in1693821330460682071rm_a_b @ X @ Y ) @ ( inf_in1693821330460682071rm_a_b @ X @ Z ) ) ) ).
% boolean_algebra.conj_disj_distrib
thf(fact_675_sup__inf__distrib2,axiom,
! [Y: set_Pr4386577575007340137rm_a_b,Z: set_Pr4386577575007340137rm_a_b,X: set_Pr4386577575007340137rm_a_b] :
( ( sup_su6776935440552674877rm_a_b @ ( inf_in1693821330460682071rm_a_b @ Y @ Z ) @ X )
= ( inf_in1693821330460682071rm_a_b @ ( sup_su6776935440552674877rm_a_b @ Y @ X ) @ ( sup_su6776935440552674877rm_a_b @ Z @ X ) ) ) ).
% sup_inf_distrib2
thf(fact_676_sup__inf__distrib1,axiom,
! [X: set_Pr4386577575007340137rm_a_b,Y: set_Pr4386577575007340137rm_a_b,Z: set_Pr4386577575007340137rm_a_b] :
( ( sup_su6776935440552674877rm_a_b @ X @ ( inf_in1693821330460682071rm_a_b @ Y @ Z ) )
= ( inf_in1693821330460682071rm_a_b @ ( sup_su6776935440552674877rm_a_b @ X @ Y ) @ ( sup_su6776935440552674877rm_a_b @ X @ Z ) ) ) ).
% sup_inf_distrib1
thf(fact_677_inf__sup__distrib2,axiom,
! [Y: set_Pr4386577575007340137rm_a_b,Z: set_Pr4386577575007340137rm_a_b,X: set_Pr4386577575007340137rm_a_b] :
( ( inf_in1693821330460682071rm_a_b @ ( sup_su6776935440552674877rm_a_b @ Y @ Z ) @ X )
= ( sup_su6776935440552674877rm_a_b @ ( inf_in1693821330460682071rm_a_b @ Y @ X ) @ ( inf_in1693821330460682071rm_a_b @ Z @ X ) ) ) ).
% inf_sup_distrib2
thf(fact_678_inf__sup__distrib1,axiom,
! [X: set_Pr4386577575007340137rm_a_b,Y: set_Pr4386577575007340137rm_a_b,Z: set_Pr4386577575007340137rm_a_b] :
( ( inf_in1693821330460682071rm_a_b @ X @ ( sup_su6776935440552674877rm_a_b @ Y @ Z ) )
= ( sup_su6776935440552674877rm_a_b @ ( inf_in1693821330460682071rm_a_b @ X @ Y ) @ ( inf_in1693821330460682071rm_a_b @ X @ Z ) ) ) ).
% inf_sup_distrib1
thf(fact_679_distrib__imp2,axiom,
! [X: set_Pr4386577575007340137rm_a_b,Y: set_Pr4386577575007340137rm_a_b,Z: set_Pr4386577575007340137rm_a_b] :
( ! [X3: set_Pr4386577575007340137rm_a_b,Y4: set_Pr4386577575007340137rm_a_b,Z4: set_Pr4386577575007340137rm_a_b] :
( ( sup_su6776935440552674877rm_a_b @ X3 @ ( inf_in1693821330460682071rm_a_b @ Y4 @ Z4 ) )
= ( inf_in1693821330460682071rm_a_b @ ( sup_su6776935440552674877rm_a_b @ X3 @ Y4 ) @ ( sup_su6776935440552674877rm_a_b @ X3 @ Z4 ) ) )
=> ( ( inf_in1693821330460682071rm_a_b @ X @ ( sup_su6776935440552674877rm_a_b @ Y @ Z ) )
= ( sup_su6776935440552674877rm_a_b @ ( inf_in1693821330460682071rm_a_b @ X @ Y ) @ ( inf_in1693821330460682071rm_a_b @ X @ Z ) ) ) ) ).
% distrib_imp2
thf(fact_680_distrib__imp1,axiom,
! [X: set_Pr4386577575007340137rm_a_b,Y: set_Pr4386577575007340137rm_a_b,Z: set_Pr4386577575007340137rm_a_b] :
( ! [X3: set_Pr4386577575007340137rm_a_b,Y4: set_Pr4386577575007340137rm_a_b,Z4: set_Pr4386577575007340137rm_a_b] :
( ( inf_in1693821330460682071rm_a_b @ X3 @ ( sup_su6776935440552674877rm_a_b @ Y4 @ Z4 ) )
= ( sup_su6776935440552674877rm_a_b @ ( inf_in1693821330460682071rm_a_b @ X3 @ Y4 ) @ ( inf_in1693821330460682071rm_a_b @ X3 @ Z4 ) ) )
=> ( ( sup_su6776935440552674877rm_a_b @ X @ ( inf_in1693821330460682071rm_a_b @ Y @ Z ) )
= ( inf_in1693821330460682071rm_a_b @ ( sup_su6776935440552674877rm_a_b @ X @ Y ) @ ( sup_su6776935440552674877rm_a_b @ X @ Z ) ) ) ) ).
% distrib_imp1
thf(fact_681_Un__UNIV__left,axiom,
! [B: set_term_a_b] :
( ( sup_sup_set_term_a_b @ top_top_set_term_a_b @ B )
= top_top_set_term_a_b ) ).
% Un_UNIV_left
thf(fact_682_Un__UNIV__left,axiom,
! [B: set_Pr4934435412358123699_a_nat] :
( ( sup_su459911885395995103_a_nat @ top_to3353692345378799459_a_nat @ B )
= top_to3353692345378799459_a_nat ) ).
% Un_UNIV_left
thf(fact_683_Un__UNIV__right,axiom,
! [A: set_term_a_b] :
( ( sup_sup_set_term_a_b @ A @ top_top_set_term_a_b )
= top_top_set_term_a_b ) ).
% Un_UNIV_right
thf(fact_684_Un__UNIV__right,axiom,
! [A: set_Pr4934435412358123699_a_nat] :
( ( sup_su459911885395995103_a_nat @ A @ top_to3353692345378799459_a_nat )
= top_to3353692345378799459_a_nat ) ).
% Un_UNIV_right
thf(fact_685_subset__Un__eq,axiom,
( ord_le8666007276011122963_a_nat
= ( ^ [A5: set_Pr4934435412358123699_a_nat,B5: set_Pr4934435412358123699_a_nat] :
( ( sup_su459911885395995103_a_nat @ A5 @ B5 )
= B5 ) ) ) ).
% subset_Un_eq
thf(fact_686_subset__UnE,axiom,
! [C2: set_Pr4934435412358123699_a_nat,A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ C2 @ ( sup_su459911885395995103_a_nat @ A @ B ) )
=> ~ ! [A7: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A7 @ A )
=> ! [B8: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ B8 @ B )
=> ( C2
!= ( sup_su459911885395995103_a_nat @ A7 @ B8 ) ) ) ) ) ).
% subset_UnE
thf(fact_687_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_688_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_689_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_690_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_691_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_692_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_693_Un__Int__distrib2,axiom,
! [B: set_Pr4386577575007340137rm_a_b,C2: set_Pr4386577575007340137rm_a_b,A: set_Pr4386577575007340137rm_a_b] :
( ( sup_su6776935440552674877rm_a_b @ ( inf_in1693821330460682071rm_a_b @ B @ C2 ) @ A )
= ( inf_in1693821330460682071rm_a_b @ ( sup_su6776935440552674877rm_a_b @ B @ A ) @ ( sup_su6776935440552674877rm_a_b @ C2 @ A ) ) ) ).
% Un_Int_distrib2
thf(fact_694_Int__Un__distrib2,axiom,
! [B: set_Pr4386577575007340137rm_a_b,C2: set_Pr4386577575007340137rm_a_b,A: set_Pr4386577575007340137rm_a_b] :
( ( inf_in1693821330460682071rm_a_b @ ( sup_su6776935440552674877rm_a_b @ B @ C2 ) @ A )
= ( sup_su6776935440552674877rm_a_b @ ( inf_in1693821330460682071rm_a_b @ B @ A ) @ ( inf_in1693821330460682071rm_a_b @ C2 @ A ) ) ) ).
% Int_Un_distrib2
thf(fact_695_Un__Int__distrib,axiom,
! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b,C2: set_Pr4386577575007340137rm_a_b] :
( ( sup_su6776935440552674877rm_a_b @ A @ ( inf_in1693821330460682071rm_a_b @ B @ C2 ) )
= ( inf_in1693821330460682071rm_a_b @ ( sup_su6776935440552674877rm_a_b @ A @ B ) @ ( sup_su6776935440552674877rm_a_b @ A @ C2 ) ) ) ).
% Un_Int_distrib
thf(fact_696_Int__Un__distrib,axiom,
! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b,C2: set_Pr4386577575007340137rm_a_b] :
( ( inf_in1693821330460682071rm_a_b @ A @ ( sup_su6776935440552674877rm_a_b @ B @ C2 ) )
= ( sup_su6776935440552674877rm_a_b @ ( inf_in1693821330460682071rm_a_b @ A @ B ) @ ( inf_in1693821330460682071rm_a_b @ A @ C2 ) ) ) ).
% Int_Un_distrib
thf(fact_697_Un__Int__crazy,axiom,
! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b,C2: set_Pr4386577575007340137rm_a_b] :
( ( sup_su6776935440552674877rm_a_b @ ( sup_su6776935440552674877rm_a_b @ ( inf_in1693821330460682071rm_a_b @ A @ B ) @ ( inf_in1693821330460682071rm_a_b @ B @ C2 ) ) @ ( inf_in1693821330460682071rm_a_b @ C2 @ A ) )
= ( inf_in1693821330460682071rm_a_b @ ( inf_in1693821330460682071rm_a_b @ ( sup_su6776935440552674877rm_a_b @ A @ B ) @ ( sup_su6776935440552674877rm_a_b @ B @ C2 ) ) @ ( sup_su6776935440552674877rm_a_b @ C2 @ A ) ) ) ).
% Un_Int_crazy
thf(fact_698_insert__def,axiom,
( insert8054603423593749677_a_nat
= ( ^ [A4: product_prod_a_nat] :
( sup_su459911885395995103_a_nat
@ ( collec4464134535221767506_a_nat
@ ^ [X2: product_prod_a_nat] : ( X2 = A4 ) ) ) ) ) ).
% insert_def
thf(fact_699_insert__def,axiom,
( insert_term_a_b
= ( ^ [A4: term_a_b] :
( sup_sup_set_term_a_b
@ ( collect_term_a_b
@ ^ [X2: term_a_b] : ( X2 = A4 ) ) ) ) ) ).
% insert_def
thf(fact_700_distrib__sup__le,axiom,
! [X: set_Pr4386577575007340137rm_a_b,Y: set_Pr4386577575007340137rm_a_b,Z: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ ( sup_su6776935440552674877rm_a_b @ X @ ( inf_in1693821330460682071rm_a_b @ Y @ Z ) ) @ ( inf_in1693821330460682071rm_a_b @ ( sup_su6776935440552674877rm_a_b @ X @ Y ) @ ( sup_su6776935440552674877rm_a_b @ X @ Z ) ) ) ).
% distrib_sup_le
thf(fact_701_distrib__sup__le,axiom,
! [X: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat,Z: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ ( sup_su459911885395995103_a_nat @ X @ ( inf_in3813033988066259269_a_nat @ Y @ Z ) ) @ ( inf_in3813033988066259269_a_nat @ ( sup_su459911885395995103_a_nat @ X @ Y ) @ ( sup_su459911885395995103_a_nat @ X @ Z ) ) ) ).
% distrib_sup_le
thf(fact_702_distrib__sup__le,axiom,
! [X: nat,Y: nat,Z: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ X @ ( inf_inf_nat @ Y @ Z ) ) @ ( inf_inf_nat @ ( sup_sup_nat @ X @ Y ) @ ( sup_sup_nat @ X @ Z ) ) ) ).
% distrib_sup_le
thf(fact_703_distrib__inf__le,axiom,
! [X: set_Pr4386577575007340137rm_a_b,Y: set_Pr4386577575007340137rm_a_b,Z: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ ( sup_su6776935440552674877rm_a_b @ ( inf_in1693821330460682071rm_a_b @ X @ Y ) @ ( inf_in1693821330460682071rm_a_b @ X @ Z ) ) @ ( inf_in1693821330460682071rm_a_b @ X @ ( sup_su6776935440552674877rm_a_b @ Y @ Z ) ) ) ).
% distrib_inf_le
thf(fact_704_distrib__inf__le,axiom,
! [X: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat,Z: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ ( sup_su459911885395995103_a_nat @ ( inf_in3813033988066259269_a_nat @ X @ Y ) @ ( inf_in3813033988066259269_a_nat @ X @ Z ) ) @ ( inf_in3813033988066259269_a_nat @ X @ ( sup_su459911885395995103_a_nat @ Y @ Z ) ) ) ).
% distrib_inf_le
thf(fact_705_distrib__inf__le,axiom,
! [X: nat,Y: nat,Z: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ ( inf_inf_nat @ X @ Y ) @ ( inf_inf_nat @ X @ Z ) ) @ ( inf_inf_nat @ X @ ( sup_sup_nat @ Y @ Z ) ) ) ).
% distrib_inf_le
thf(fact_706_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 ) )
=> ( ! [A3: a,B3: nat] :
( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ ( product_Pair_a_nat @ Ax @ Ay ) @ ( product_Pair_a_nat @ A3 @ B3 ) ) @ R )
=> ( P @ A3 @ B3 ) )
=> ( ! [A3: a,B3: nat,Aa: a,Ba: nat] :
( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ ( product_Pair_a_nat @ Ax @ Ay ) @ ( product_Pair_a_nat @ A3 @ B3 ) ) @ ( transi6805586507763294364_a_nat @ R ) )
=> ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ ( product_Pair_a_nat @ A3 @ B3 ) @ ( product_Pair_a_nat @ Aa @ Ba ) ) @ R )
=> ( ( P @ A3 @ B3 )
=> ( P @ Aa @ Ba ) ) ) )
=> ( P @ Bx @ By ) ) ) ) ).
% trancl_induct2
thf(fact_707_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_708_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_709_srstep__subst__closed,axiom,
! [S: term_a_b,T2: term_a_b,F2: set_Pr4934435412358123699_a_nat,R2: set_Pr4386577575007340137rm_a_b,Sigma: b > term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T2 ) @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) ) )
=> ( ! [X3: b] : ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ ( Sigma @ X3 ) ) @ F2 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( subst_7999470309526761004_a_b_b @ S @ Sigma ) @ ( subst_7999470309526761004_a_b_b @ T2 @ Sigma ) ) @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) ) ) ) ) ).
% srstep_subst_closed
thf(fact_710_srsteps__eq__subst__closed,axiom,
! [S: term_a_b,T2: term_a_b,F2: set_Pr4934435412358123699_a_nat,R2: set_Pr4386577575007340137rm_a_b,Sigma: b > term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T2 ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) ) ) )
=> ( ! [X3: b] : ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ ( Sigma @ X3 ) ) @ F2 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( subst_7999470309526761004_a_b_b @ S @ Sigma ) @ ( subst_7999470309526761004_a_b_b @ T2 @ Sigma ) ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) ) ) ) ) ) ).
% srsteps_eq_subst_closed
thf(fact_711_srsteps__subst__closed,axiom,
! [S: term_a_b,T2: term_a_b,F2: set_Pr4934435412358123699_a_nat,R2: set_Pr4386577575007340137rm_a_b,Sigma: b > term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T2 ) @ ( transi7922773638565587891rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) ) ) )
=> ( ! [X3: b] : ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ ( Sigma @ X3 ) ) @ F2 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( subst_7999470309526761004_a_b_b @ S @ Sigma ) @ ( subst_7999470309526761004_a_b_b @ T2 @ Sigma ) ) @ ( transi7922773638565587891rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) ) ) ) ) ) ).
% srsteps_subst_closed
thf(fact_712_rstep__term__to__sig__r,axiom,
! [S: term_a_b,T2: term_a_b,R2: set_Pr4386577575007340137rm_a_b,F2: set_Pr4934435412358123699_a_nat,V: b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T2 ) @ ( rstep_a_b @ R2 ) )
=> ( ( ord_le8666007276011122963_a_nat @ ( terms_7988297476397195622_a_b_b @ R2 ) @ F2 )
=> ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ S ) @ F2 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ ( terms_8519481630511763164ig_a_b @ F2 @ V @ T2 ) ) @ ( rstep_a_b @ R2 ) ) ) ) ) ).
% rstep_term_to_sig_r
thf(fact_713_rstep__term__to__sig__l,axiom,
! [S: term_a_b,T2: term_a_b,R2: set_Pr4386577575007340137rm_a_b,F2: set_Pr4934435412358123699_a_nat,V: b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T2 ) @ ( rstep_a_b @ R2 ) )
=> ( ( ord_le8666007276011122963_a_nat @ ( terms_7988297476397195622_a_b_b @ R2 ) @ F2 )
=> ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ T2 ) @ F2 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( terms_8519481630511763164ig_a_b @ F2 @ V @ S ) @ T2 ) @ ( rstep_a_b @ R2 ) ) ) ) ) ).
% rstep_term_to_sig_l
thf(fact_714_ground__subst__apply,axiom,
! [T2: term_a_b,Sigma: b > term_a_b] :
( ( term_ground_a_b @ T2 )
=> ( ( subst_7999470309526761004_a_b_b @ T2 @ Sigma )
= T2 ) ) ).
% ground_subst_apply
thf(fact_715_sup__set__def,axiom,
( sup_su459911885395995103_a_nat
= ( ^ [A5: set_Pr4934435412358123699_a_nat,B5: set_Pr4934435412358123699_a_nat] :
( collec4464134535221767506_a_nat
@ ( sup_su3231633873331431782_nat_o
@ ^ [X2: product_prod_a_nat] : ( member5724188588386418708_a_nat @ X2 @ A5 )
@ ^ [X2: product_prod_a_nat] : ( member5724188588386418708_a_nat @ X2 @ B5 ) ) ) ) ) ).
% sup_set_def
thf(fact_716_sup__set__def,axiom,
( sup_sup_set_term_a_b
= ( ^ [A5: set_term_a_b,B5: set_term_a_b] :
( collect_term_a_b
@ ( sup_sup_term_a_b_o
@ ^ [X2: term_a_b] : ( member_term_a_b @ X2 @ A5 )
@ ^ [X2: term_a_b] : ( member_term_a_b @ X2 @ B5 ) ) ) ) ) ).
% sup_set_def
thf(fact_717_sup__Un__eq,axiom,
! [R3: set_Pr4934435412358123699_a_nat,S3: set_Pr4934435412358123699_a_nat] :
( ( sup_su3231633873331431782_nat_o
@ ^ [X2: product_prod_a_nat] : ( member5724188588386418708_a_nat @ X2 @ R3 )
@ ^ [X2: product_prod_a_nat] : ( member5724188588386418708_a_nat @ X2 @ S3 ) )
= ( ^ [X2: product_prod_a_nat] : ( member5724188588386418708_a_nat @ X2 @ ( sup_su459911885395995103_a_nat @ R3 @ S3 ) ) ) ) ).
% sup_Un_eq
thf(fact_718_sup__Un__eq,axiom,
! [R3: set_term_a_b,S3: set_term_a_b] :
( ( sup_sup_term_a_b_o
@ ^ [X2: term_a_b] : ( member_term_a_b @ X2 @ R3 )
@ ^ [X2: term_a_b] : ( member_term_a_b @ X2 @ S3 ) )
= ( ^ [X2: term_a_b] : ( member_term_a_b @ X2 @ ( sup_sup_set_term_a_b @ R3 @ S3 ) ) ) ) ).
% sup_Un_eq
thf(fact_719_sup__Un__eq2,axiom,
! [R3: set_Pr4934435412358123699_a_nat,S3: set_Pr4934435412358123699_a_nat] :
( ( sup_sup_a_nat_o
@ ^ [X2: a,Y3: nat] : ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ X2 @ Y3 ) @ R3 )
@ ^ [X2: a,Y3: nat] : ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ X2 @ Y3 ) @ S3 ) )
= ( ^ [X2: a,Y3: nat] : ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ X2 @ Y3 ) @ ( sup_su459911885395995103_a_nat @ R3 @ S3 ) ) ) ) ).
% sup_Un_eq2
thf(fact_720_sig__step__rsteps__eq__dist,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,F2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ ( terms_7988297476397195622_a_b_b @ R2 ) @ F2 )
=> ( ( sup_su6776935440552674877rm_a_b @ ( sig_step_a_b @ F2 @ ( transi7922773638565587891rm_a_b @ ( rstep_a_b @ R2 ) ) ) @ id_term_a_b )
= ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) ) ) ) ) ).
% sig_step_rsteps_eq_dist
thf(fact_721_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_722_ground__NF__srstep__gsrstep,axiom,
! [S: term_a_b,F2: set_Pr4934435412358123699_a_nat,R2: set_Pr4386577575007340137rm_a_b] :
( ( term_ground_a_b @ S )
=> ( ( member_term_a_b @ S @ ( abstract_NF_term_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) ) ) )
=> ( member_term_a_b @ S
@ ( abstract_NF_term_a_b
@ ( inf_in1693821330460682071rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) )
@ ( produc1953217797448512882rm_a_b @ ( collect_term_a_b @ term_ground_a_b )
@ ^ [Uu: term_a_b] : ( collect_term_a_b @ term_ground_a_b ) ) ) ) ) ) ) ).
% ground_NF_srstep_gsrstep
thf(fact_723_strong__confluence__imp__CR,axiom,
! [R: set_Pr4386577575007340137rm_a_b] :
( ( abstra5664096641628173427rm_a_b @ R @ top_top_set_term_a_b )
=> ( abstra8448919418672941150rm_a_b @ R @ top_top_set_term_a_b ) ) ).
% strong_confluence_imp_CR
thf(fact_724_strong__confluence__imp__CR,axiom,
! [R: set_Pr1811044260758604347_a_nat] :
( ( abstra5315942894939654492_a_nat @ R @ top_to3353692345378799459_a_nat )
=> ( abstra7964079670522473415_a_nat @ R @ top_to3353692345378799459_a_nat ) ) ).
% strong_confluence_imp_CR
thf(fact_725_NF__I,axiom,
! [A2: product_prod_a_nat,A: set_Pr1811044260758604347_a_nat] :
( ! [B3: product_prod_a_nat] :
~ ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A2 @ B3 ) @ A )
=> ( member5724188588386418708_a_nat @ A2 @ ( abstra2685462378305987313_a_nat @ A ) ) ) ).
% NF_I
thf(fact_726_NF__I,axiom,
! [A2: term_a_b,A: set_Pr4386577575007340137rm_a_b] :
( ! [B3: term_a_b] :
~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B3 ) @ A )
=> ( member_term_a_b @ A2 @ ( abstract_NF_term_a_b @ A ) ) ) ).
% NF_I
thf(fact_727_strongly__confluentI,axiom,
! [R: set_Pr4386577575007340137rm_a_b] :
( ! [X3: term_a_b,Y4: term_a_b,Z4: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y4 ) @ R )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Z4 ) @ R )
=> ? [U3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y4 @ U3 ) @ ( transi7742714808557438673rm_a_b @ R ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Z4 @ U3 ) @ ( sup_su6776935440552674877rm_a_b @ R @ id_term_a_b ) ) ) ) )
=> ( abstra5664096641628173427rm_a_b @ R @ top_top_set_term_a_b ) ) ).
% strongly_confluentI
thf(fact_728_strongly__confluentI,axiom,
! [R: set_Pr1811044260758604347_a_nat] :
( ! [X3: product_prod_a_nat,Y4: product_prod_a_nat,Z4: product_prod_a_nat] :
( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ X3 @ Y4 ) @ R )
=> ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ X3 @ Z4 ) @ R )
=> ? [U3: product_prod_a_nat] :
( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ Y4 @ U3 ) @ ( transi2726145917338391738_a_nat @ R ) )
& ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ Z4 @ U3 ) @ ( sup_su958340139570419215_a_nat @ R @ id_Pro5207055338379199009_a_nat ) ) ) ) )
=> ( abstra5315942894939654492_a_nat @ R @ top_to3353692345378799459_a_nat ) ) ).
% strongly_confluentI
thf(fact_729_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_730_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_731_NF__no__step,axiom,
! [A2: product_prod_a_nat,A: set_Pr1811044260758604347_a_nat] :
( ( member5724188588386418708_a_nat @ A2 @ ( abstra2685462378305987313_a_nat @ A ) )
=> ! [B9: product_prod_a_nat] :
~ ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A2 @ B9 ) @ A ) ) ).
% NF_no_step
thf(fact_732_NF__no__step,axiom,
! [A2: term_a_b,A: set_Pr4386577575007340137rm_a_b] :
( ( member_term_a_b @ A2 @ ( abstract_NF_term_a_b @ A ) )
=> ! [B9: term_a_b] :
~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B9 ) @ A ) ) ).
% NF_no_step
thf(fact_733_NF__iff__no__step,axiom,
! [A2: product_prod_a_nat,A: set_Pr1811044260758604347_a_nat] :
( ( member5724188588386418708_a_nat @ A2 @ ( abstra2685462378305987313_a_nat @ A ) )
= ( ! [B4: product_prod_a_nat] :
~ ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A2 @ B4 ) @ A ) ) ) ).
% NF_iff_no_step
thf(fact_734_NF__iff__no__step,axiom,
! [A2: term_a_b,A: set_Pr4386577575007340137rm_a_b] :
( ( member_term_a_b @ A2 @ ( abstract_NF_term_a_b @ A ) )
= ( ! [B4: term_a_b] :
~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B4 ) @ A ) ) ) ).
% NF_iff_no_step
thf(fact_735_strongly__confluent__on__E11,axiom,
! [R: set_Pr1811044260758604347_a_nat,A: set_Pr4934435412358123699_a_nat,X: product_prod_a_nat,Y: product_prod_a_nat,Z: product_prod_a_nat] :
( ( abstra5315942894939654492_a_nat @ R @ A )
=> ( ( member5724188588386418708_a_nat @ X @ A )
=> ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ X @ Y ) @ R )
=> ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ X @ Z ) @ R )
=> ? [U4: product_prod_a_nat] :
( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ Y @ U4 ) @ ( transi2726145917338391738_a_nat @ R ) )
& ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ Z @ U4 ) @ ( sup_su958340139570419215_a_nat @ R @ id_Pro5207055338379199009_a_nat ) ) ) ) ) ) ) ).
% strongly_confluent_on_E11
thf(fact_736_strongly__confluent__on__E11,axiom,
! [R: set_Pr4386577575007340137rm_a_b,A: set_term_a_b,X: term_a_b,Y: term_a_b,Z: term_a_b] :
( ( abstra5664096641628173427rm_a_b @ R @ A )
=> ( ( member_term_a_b @ X @ A )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ R )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z ) @ R )
=> ? [U4: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ U4 ) @ ( transi7742714808557438673rm_a_b @ R ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Z @ U4 ) @ ( sup_su6776935440552674877rm_a_b @ R @ id_term_a_b ) ) ) ) ) ) ) ).
% strongly_confluent_on_E11
thf(fact_737_NF__not__suc,axiom,
! [X: product_prod_a_nat,Y: product_prod_a_nat,R: set_Pr1811044260758604347_a_nat] :
( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ X @ Y ) @ ( transi2726145917338391738_a_nat @ R ) )
=> ( ( member5724188588386418708_a_nat @ X @ ( abstra2685462378305987313_a_nat @ R ) )
=> ( X = Y ) ) ) ).
% NF_not_suc
thf(fact_738_NF__not__suc,axiom,
! [X: term_a_b,Y: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ( ( member_term_a_b @ X @ ( abstract_NF_term_a_b @ R ) )
=> ( X = Y ) ) ) ).
% NF_not_suc
thf(fact_739_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_740_NF__no__trancl__step,axiom,
! [A2: product_prod_a_nat,A: set_Pr1811044260758604347_a_nat] :
( ( member5724188588386418708_a_nat @ A2 @ ( abstra2685462378305987313_a_nat @ A ) )
=> ! [B9: product_prod_a_nat] :
~ ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A2 @ B9 ) @ ( transi6805586507763294364_a_nat @ A ) ) ) ).
% NF_no_trancl_step
thf(fact_741_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 ) )
=> ! [B9: term_a_b] :
~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B9 ) @ ( transi7922773638565587891rm_a_b @ A ) ) ) ).
% NF_no_trancl_step
thf(fact_742_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_743_NFP__redp__def,axiom,
( rewrit232733863534887208dp_a_b
= ( ^ [F3: set_Pr4934435412358123699_a_nat,R6: set_Pr4386577575007340137rm_a_b,S5: term_a_b,T: term_a_b] :
( ( member_term_a_b @ T @ ( abstract_NF_term_a_b @ ( sig_step_a_b @ F3 @ ( rstep_a_b @ R6 ) ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S5 @ T ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F3 @ ( rstep_a_b @ R6 ) ) ) ) ) ) ) ).
% NFP_redp_def
thf(fact_744_some__NF__UNF,axiom,
! [R: set_Pr1811044260758604347_a_nat,X: product_prod_a_nat,Y: product_prod_a_nat] :
( ( abstra7833083522531381839_a_nat @ R @ top_to3353692345378799459_a_nat )
=> ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ X @ Y ) @ ( transi2726145917338391738_a_nat @ R ) )
=> ( ( member5724188588386418708_a_nat @ Y @ ( abstra2685462378305987313_a_nat @ R ) )
=> ( Y
= ( abstra5913684503434863653_a_nat @ R @ X ) ) ) ) ) ).
% some_NF_UNF
thf(fact_745_some__NF__UNF,axiom,
! [R: set_Pr4386577575007340137rm_a_b,X: term_a_b,Y: term_a_b] :
( ( abstra3619855735679963878rm_a_b @ R @ top_top_set_term_a_b )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ( ( member_term_a_b @ Y @ ( abstract_NF_term_a_b @ R ) )
=> ( Y
= ( abstra2589370493620291516rm_a_b @ R @ X ) ) ) ) ) ).
% some_NF_UNF
thf(fact_746_NFP__stepD,axiom,
! [R: set_Pr4386577575007340137rm_a_b,A2: term_a_b,B2: term_a_b,C: term_a_b] :
( ( rewrit6304709981509430748rm_a_b @ R @ top_top_set_term_a_b )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ C ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ( ( member_term_a_b @ C @ ( abstract_NF_term_a_b @ R ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ C ) @ ( transi7742714808557438673rm_a_b @ R ) ) ) ) ) ) ).
% NFP_stepD
thf(fact_747_NFP__stepD,axiom,
! [R: set_Pr1811044260758604347_a_nat,A2: product_prod_a_nat,B2: product_prod_a_nat,C: product_prod_a_nat] :
( ( rewrit1859573128460188229_a_nat @ R @ top_to3353692345378799459_a_nat )
=> ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A2 @ B2 ) @ ( transi2726145917338391738_a_nat @ R ) )
=> ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A2 @ C ) @ ( transi2726145917338391738_a_nat @ R ) )
=> ( ( member5724188588386418708_a_nat @ C @ ( abstra2685462378305987313_a_nat @ R ) )
=> ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ B2 @ C ) @ ( transi2726145917338391738_a_nat @ R ) ) ) ) ) ) ).
% NFP_stepD
thf(fact_748_NE__redp__def,axiom,
( rewrit3451484132325884455dp_a_b
= ( ^ [F3: set_Pr4934435412358123699_a_nat,R6: set_Pr4386577575007340137rm_a_b,S6: set_Pr4386577575007340137rm_a_b,S5: term_a_b,T: term_a_b] :
( ( member_term_a_b @ T @ ( abstract_NF_term_a_b @ ( sig_step_a_b @ F3 @ ( rstep_a_b @ R6 ) ) ) )
=> ( ( member_term_a_b @ T @ ( abstract_NF_term_a_b @ ( sig_step_a_b @ F3 @ ( rstep_a_b @ R6 ) ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S5 @ T ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F3 @ ( rstep_a_b @ S6 ) ) ) ) ) ) ) ) ).
% NE_redp_def
thf(fact_749_SCR__rrstep__intro,axiom,
! [F2: set_Pr4934435412358123699_a_nat,R2: set_Pr4386577575007340137rm_a_b] :
( ! [S7: term_a_b,T4: term_a_b,U4: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S7 @ T4 ) @ ( sig_step_a_b @ F2 @ ( rrstep_a_b @ R2 ) ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S7 @ U4 ) @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) ) )
=> ? [V2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T4 @ V2 ) @ ( sup_su6776935440552674877rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) ) @ id_term_a_b ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ U4 @ V2 ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) ) ) ) ) ) )
=> ( ! [S7: term_a_b,T4: term_a_b,U4: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S7 @ T4 ) @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S7 @ U4 ) @ ( sig_step_a_b @ F2 @ ( rrstep_a_b @ R2 ) ) )
=> ? [V2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T4 @ V2 ) @ ( sup_su6776935440552674877rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) ) @ id_term_a_b ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ U4 @ V2 ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) ) ) ) ) ) )
=> ( rewrit4966517460798004134rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) ) @ top_top_set_term_a_b ) ) ) ).
% SCR_rrstep_intro
thf(fact_750_SCE__to__rrstep_I1_J,axiom,
! [F2: set_Pr4934435412358123699_a_nat,R2: set_Pr4386577575007340137rm_a_b,S: term_a_b,T2: term_a_b,U: term_a_b] :
( ( rewrit4966517460798004134rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) ) @ top_top_set_term_a_b )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T2 ) @ ( sig_step_a_b @ F2 @ ( rrstep_a_b @ R2 ) ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ U ) @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) ) )
=> ? [V3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T2 @ V3 ) @ ( sup_su6776935440552674877rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) ) @ id_term_a_b ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ U @ V3 ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) ) ) ) ) ) ) ) ).
% SCE_to_rrstep(1)
thf(fact_751_SCE__to__rrstep_I2_J,axiom,
! [F2: set_Pr4934435412358123699_a_nat,R2: set_Pr4386577575007340137rm_a_b,S: term_a_b,T2: term_a_b,U: term_a_b] :
( ( rewrit4966517460798004134rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) ) @ top_top_set_term_a_b )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T2 ) @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ U ) @ ( sig_step_a_b @ F2 @ ( rrstep_a_b @ R2 ) ) )
=> ? [V3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T2 @ V3 ) @ ( sup_su6776935440552674877rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) ) @ id_term_a_b ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ U @ V3 ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) ) ) ) ) ) ) ) ).
% SCE_to_rrstep(2)
thf(fact_752_rrstep__rstep__mono,axiom,
! [R2: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ ( rrstep_a_b @ R2 ) @ ( rstep_a_b @ R2 ) ) ).
% rrstep_rstep_mono
thf(fact_753_srrstep__to__srestep,axiom,
! [S: term_a_b,T2: term_a_b,F2: set_Pr4934435412358123699_a_nat,R2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T2 ) @ ( sig_step_a_b @ F2 @ ( rrstep_a_b @ R2 ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T2 ) @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) ) ) ) ).
% srrstep_to_srestep
thf(fact_754_srrstep__monp,axiom,
! [F2: set_Pr4934435412358123699_a_nat,G: set_Pr4934435412358123699_a_nat,R2: set_Pr4386577575007340137rm_a_b] :
( ( ord_le8666007276011122963_a_nat @ F2 @ G )
=> ( ord_le118470702582115849rm_a_b @ ( sig_step_a_b @ F2 @ ( rrstep_a_b @ R2 ) ) @ ( sig_step_a_b @ G @ ( rrstep_a_b @ R2 ) ) ) ) ).
% srrstep_monp
thf(fact_755_srrstepD,axiom,
! [S: term_a_b,T2: term_a_b,F2: set_Pr4934435412358123699_a_nat,R2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T2 ) @ ( sig_step_a_b @ F2 @ ( rrstep_a_b @ R2 ) ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T2 ) @ ( rrstep_a_b @ R2 ) )
& ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ S ) @ F2 )
& ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ T2 ) @ F2 ) ) ) ).
% srrstepD
thf(fact_756_llrg__rrsteps__groundness,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,S: term_a_b,T2: term_a_b,F2: set_Pr4934435412358123699_a_nat] :
( ( rewrit5000278123254648670_b_a_b @ R2 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T2 ) @ ( sig_step_a_b @ F2 @ ( rrstep_a_b @ R2 ) ) )
=> ( term_ground_a_b @ T2 ) ) ) ).
% llrg_rrsteps_groundness
thf(fact_757_lv__root__step__idep__subst,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,S: term_a_b,T2: term_a_b,F2: set_Pr4934435412358123699_a_nat,Sigma: b > term_a_b,Tau: b > term_a_b] :
( ( rewrit2869827417384121323_a_b_a @ R2 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T2 ) @ ( sig_step_a_b @ F2 @ ( rrstep_a_b @ R2 ) ) )
=> ( ! [X3: b] : ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ ( Sigma @ X3 ) ) @ F2 )
=> ( ! [X3: b] : ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ ( Tau @ X3 ) ) @ F2 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( subst_7999470309526761004_a_b_b @ S @ Sigma ) @ ( subst_7999470309526761004_a_b_b @ T2 @ Tau ) ) @ ( sig_step_a_b @ F2 @ ( rrstep_a_b @ R2 ) ) ) ) ) ) ) ).
% lv_root_step_idep_subst
thf(fact_758_WCR__rrstep__intro,axiom,
! [F2: set_Pr4934435412358123699_a_nat,R2: set_Pr4386577575007340137rm_a_b] :
( ! [S7: term_a_b,T4: term_a_b,U4: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S7 @ T4 ) @ ( sig_step_a_b @ F2 @ ( rrstep_a_b @ R2 ) ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S7 @ U4 ) @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T4 @ U4 ) @ ( abstra4096080454567261402rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) ) ) ) ) )
=> ( abstra5026849107050860991rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) ) @ top_top_set_term_a_b ) ) ).
% WCR_rrstep_intro
thf(fact_759_UN__redp__def,axiom,
( rewrit983855061078472681dp_a_b
= ( ^ [F3: set_Pr4934435412358123699_a_nat,R6: set_Pr4386577575007340137rm_a_b,S5: term_a_b,T: term_a_b] :
( ( ( member_term_a_b @ S5 @ ( abstract_NF_term_a_b @ ( sig_step_a_b @ F3 @ ( rstep_a_b @ R6 ) ) ) )
& ( member_term_a_b @ T @ ( abstract_NF_term_a_b @ ( sig_step_a_b @ F3 @ ( rstep_a_b @ R6 ) ) ) ) )
=> ( S5 = T ) ) ) ) ).
% UN_redp_def
thf(fact_760_WCR__onI,axiom,
! [A: set_Pr4934435412358123699_a_nat,R: set_Pr1811044260758604347_a_nat] :
( ! [A3: product_prod_a_nat,B3: product_prod_a_nat,C5: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ A3 @ A )
=> ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A3 @ B3 ) @ R )
=> ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A3 @ C5 ) @ R )
=> ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ B3 @ C5 ) @ ( abstra881512052393830979_a_nat @ R ) ) ) ) )
=> ( abstra6648775111202918568_a_nat @ R @ A ) ) ).
% WCR_onI
thf(fact_761_WCR__onI,axiom,
! [A: set_term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ! [A3: term_a_b,B3: term_a_b,C5: term_a_b] :
( ( member_term_a_b @ A3 @ A )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B3 ) @ R )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ C5 ) @ R )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B3 @ C5 ) @ ( abstra4096080454567261402rm_a_b @ R ) ) ) ) )
=> ( abstra5026849107050860991rm_a_b @ R @ A ) ) ).
% WCR_onI
thf(fact_762_CR__onI,axiom,
! [A: set_Pr4934435412358123699_a_nat,R: set_Pr1811044260758604347_a_nat] :
( ! [A3: product_prod_a_nat,B3: product_prod_a_nat,C5: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ A3 @ A )
=> ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A3 @ B3 ) @ ( transi2726145917338391738_a_nat @ R ) )
=> ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A3 @ C5 ) @ ( transi2726145917338391738_a_nat @ R ) )
=> ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ B3 @ C5 ) @ ( abstra881512052393830979_a_nat @ R ) ) ) ) )
=> ( abstra7964079670522473415_a_nat @ R @ A ) ) ).
% CR_onI
thf(fact_763_CR__onI,axiom,
! [A: set_term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ! [A3: term_a_b,B3: term_a_b,C5: term_a_b] :
( ( member_term_a_b @ A3 @ A )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B3 ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ C5 ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B3 @ C5 ) @ ( abstra4096080454567261402rm_a_b @ R ) ) ) ) )
=> ( abstra8448919418672941150rm_a_b @ R @ A ) ) ).
% CR_onI
thf(fact_764_join__NF__imp__eq,axiom,
! [X: product_prod_a_nat,Y: product_prod_a_nat,R: set_Pr1811044260758604347_a_nat] :
( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ X @ Y ) @ ( abstra881512052393830979_a_nat @ R ) )
=> ( ( member5724188588386418708_a_nat @ X @ ( abstra2685462378305987313_a_nat @ R ) )
=> ( ( member5724188588386418708_a_nat @ Y @ ( abstra2685462378305987313_a_nat @ R ) )
=> ( X = Y ) ) ) ) ).
% join_NF_imp_eq
thf(fact_765_join__NF__imp__eq,axiom,
! [X: term_a_b,Y: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( abstra4096080454567261402rm_a_b @ R ) )
=> ( ( member_term_a_b @ X @ ( abstract_NF_term_a_b @ R ) )
=> ( ( member_term_a_b @ Y @ ( abstract_NF_term_a_b @ R ) )
=> ( X = Y ) ) ) ) ).
% join_NF_imp_eq
thf(fact_766_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_767_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_768_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_769_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_770_NF__join__imp__reach,axiom,
! [Y: product_prod_a_nat,A: set_Pr1811044260758604347_a_nat,X: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ Y @ ( abstra2685462378305987313_a_nat @ A ) )
=> ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ X @ Y ) @ ( abstra881512052393830979_a_nat @ A ) )
=> ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ X @ Y ) @ ( transi2726145917338391738_a_nat @ A ) ) ) ) ).
% NF_join_imp_reach
thf(fact_771_NF__join__imp__reach,axiom,
! [Y: term_a_b,A: set_Pr4386577575007340137rm_a_b,X: term_a_b] :
( ( member_term_a_b @ Y @ ( abstract_NF_term_a_b @ A ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( abstra4096080454567261402rm_a_b @ A ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ A ) ) ) ) ).
% NF_join_imp_reach
thf(fact_772_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_773_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_774_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_775_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_776_CR__join__left__I,axiom,
! [R: set_Pr4386577575007340137rm_a_b,X: term_a_b,Y: term_a_b,Z: term_a_b] :
( ( abstra8448919418672941150rm_a_b @ R @ top_top_set_term_a_b )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z ) @ ( abstra4096080454567261402rm_a_b @ R ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ Z ) @ ( abstra4096080454567261402rm_a_b @ R ) ) ) ) ) ).
% CR_join_left_I
thf(fact_777_CR__join__left__I,axiom,
! [R: set_Pr1811044260758604347_a_nat,X: product_prod_a_nat,Y: product_prod_a_nat,Z: product_prod_a_nat] :
( ( abstra7964079670522473415_a_nat @ R @ top_to3353692345378799459_a_nat )
=> ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ X @ Y ) @ ( transi2726145917338391738_a_nat @ R ) )
=> ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ X @ Z ) @ ( abstra881512052393830979_a_nat @ R ) )
=> ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ Y @ Z ) @ ( abstra881512052393830979_a_nat @ R ) ) ) ) ) ).
% CR_join_left_I
thf(fact_778_CR__join__right__I,axiom,
! [R: set_Pr4386577575007340137rm_a_b,X: term_a_b,Y: term_a_b,Z: term_a_b] :
( ( abstra8448919418672941150rm_a_b @ R @ top_top_set_term_a_b )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( abstra4096080454567261402rm_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ Z ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z ) @ ( abstra4096080454567261402rm_a_b @ R ) ) ) ) ) ).
% CR_join_right_I
thf(fact_779_CR__join__right__I,axiom,
! [R: set_Pr1811044260758604347_a_nat,X: product_prod_a_nat,Y: product_prod_a_nat,Z: product_prod_a_nat] :
( ( abstra7964079670522473415_a_nat @ R @ top_to3353692345378799459_a_nat )
=> ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ X @ Y ) @ ( abstra881512052393830979_a_nat @ R ) )
=> ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ Y @ Z ) @ ( transi2726145917338391738_a_nat @ R ) )
=> ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ X @ Z ) @ ( abstra881512052393830979_a_nat @ R ) ) ) ) ) ).
% CR_join_right_I
thf(fact_780_partially__localize__CR,axiom,
! [R: set_Pr4386577575007340137rm_a_b] :
( ( abstra8448919418672941150rm_a_b @ R @ top_top_set_term_a_b )
= ( ! [X2: term_a_b,Y3: term_a_b,Z6: term_a_b] :
( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y3 ) @ R )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Z6 ) @ ( transi7742714808557438673rm_a_b @ R ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ Z6 ) @ ( abstra4096080454567261402rm_a_b @ R ) ) ) ) ) ).
% partially_localize_CR
thf(fact_781_partially__localize__CR,axiom,
! [R: set_Pr1811044260758604347_a_nat] :
( ( abstra7964079670522473415_a_nat @ R @ top_to3353692345378799459_a_nat )
= ( ! [X2: product_prod_a_nat,Y3: product_prod_a_nat,Z6: product_prod_a_nat] :
( ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ X2 @ Y3 ) @ R )
& ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ X2 @ Z6 ) @ ( transi2726145917338391738_a_nat @ R ) ) )
=> ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ Y3 @ Z6 ) @ ( abstra881512052393830979_a_nat @ R ) ) ) ) ) ).
% partially_localize_CR
thf(fact_782_CR__divergence__imp__join,axiom,
! [R: set_Pr4386577575007340137rm_a_b,X: term_a_b,Y: term_a_b,Z: term_a_b] :
( ( abstra8448919418672941150rm_a_b @ R @ top_top_set_term_a_b )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ Z ) @ ( abstra4096080454567261402rm_a_b @ R ) ) ) ) ) ).
% CR_divergence_imp_join
thf(fact_783_CR__divergence__imp__join,axiom,
! [R: set_Pr1811044260758604347_a_nat,X: product_prod_a_nat,Y: product_prod_a_nat,Z: product_prod_a_nat] :
( ( abstra7964079670522473415_a_nat @ R @ top_to3353692345378799459_a_nat )
=> ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ X @ Y ) @ ( transi2726145917338391738_a_nat @ R ) )
=> ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ X @ Z ) @ ( transi2726145917338391738_a_nat @ R ) )
=> ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ Y @ Z ) @ ( abstra881512052393830979_a_nat @ R ) ) ) ) ) ).
% CR_divergence_imp_join
thf(fact_784_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_785_CR__iff__meet__subset__join,axiom,
! [R: set_Pr1811044260758604347_a_nat] :
( ( abstra7964079670522473415_a_nat @ R @ top_to3353692345378799459_a_nat )
= ( ord_le1018141841545957851_a_nat @ ( abstra5218407276271306624_a_nat @ R ) @ ( abstra881512052393830979_a_nat @ R ) ) ) ).
% CR_iff_meet_subset_join
thf(fact_786_CR__on__singletonI,axiom,
! [A2: product_prod_a_nat,R: set_Pr1811044260758604347_a_nat] :
( ! [B3: product_prod_a_nat,C5: product_prod_a_nat] :
( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A2 @ B3 ) @ ( transi2726145917338391738_a_nat @ R ) )
=> ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A2 @ C5 ) @ ( transi2726145917338391738_a_nat @ R ) )
=> ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ B3 @ C5 ) @ ( abstra881512052393830979_a_nat @ R ) ) ) )
=> ( abstra7964079670522473415_a_nat @ R @ ( insert8054603423593749677_a_nat @ A2 @ bot_bo9049108969261143879_a_nat ) ) ) ).
% CR_on_singletonI
thf(fact_787_CR__on__singletonI,axiom,
! [A2: nat,R: set_Pr1261947904930325089at_nat] :
( ! [B3: nat,C5: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B3 ) @ ( transi2905341329935302413cl_nat @ R ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ C5 ) @ ( transi2905341329935302413cl_nat @ R ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B3 @ C5 ) @ ( abstract_join_nat @ R ) ) ) )
=> ( abstract_CR_on_nat @ R @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) ) ).
% CR_on_singletonI
thf(fact_788_fresh__const__subst__NF__pres,axiom,
! [C: a,R2: set_Pr4386577575007340137rm_a_b,F2: set_Pr4934435412358123699_a_nat,S: term_a_b,H: set_Pr4934435412358123699_a_nat] :
( ~ ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ C @ zero_zero_nat ) @ ( terms_7988297476397195622_a_b_b @ R2 ) )
=> ( ( ord_le8666007276011122963_a_nat @ ( terms_7988297476397195622_a_b_b @ R2 ) @ F2 )
=> ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ S ) @ F2 )
=> ( ( ord_le8666007276011122963_a_nat @ F2 @ H )
=> ( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ C @ zero_zero_nat ) @ H )
=> ( ( member_term_a_b
@ ( subst_7999470309526761004_a_b_b @ S
@ ^ [X2: b] : ( fun_a_b @ C @ nil_term_a_b ) )
@ ( abstract_NF_term_a_b
@ ( inf_in1693821330460682071rm_a_b @ ( sig_step_a_b @ H @ ( rstep_a_b @ R2 ) )
@ ( produc1953217797448512882rm_a_b @ ( collect_term_a_b @ term_ground_a_b )
@ ^ [Uu: term_a_b] : ( collect_term_a_b @ term_ground_a_b ) ) ) ) )
=> ( member_term_a_b @ S @ ( abstract_NF_term_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) ) ) ) ) ) ) ) ) ) ).
% fresh_const_subst_NF_pres
thf(fact_789_empty__iff,axiom,
! [C: product_prod_a_nat] :
~ ( member5724188588386418708_a_nat @ C @ bot_bo9049108969261143879_a_nat ) ).
% empty_iff
thf(fact_790_empty__iff,axiom,
! [C: term_a_b] :
~ ( member_term_a_b @ C @ bot_bot_set_term_a_b ) ).
% empty_iff
thf(fact_791_empty__iff,axiom,
! [C: nat] :
~ ( member_nat @ C @ bot_bot_set_nat ) ).
% empty_iff
thf(fact_792_all__not__in__conv,axiom,
! [A: set_Pr4934435412358123699_a_nat] :
( ( ! [X2: product_prod_a_nat] :
~ ( member5724188588386418708_a_nat @ X2 @ A ) )
= ( A = bot_bo9049108969261143879_a_nat ) ) ).
% all_not_in_conv
thf(fact_793_all__not__in__conv,axiom,
! [A: set_term_a_b] :
( ( ! [X2: term_a_b] :
~ ( member_term_a_b @ X2 @ A ) )
= ( A = bot_bot_set_term_a_b ) ) ).
% all_not_in_conv
thf(fact_794_all__not__in__conv,axiom,
! [A: set_nat] :
( ( ! [X2: nat] :
~ ( member_nat @ X2 @ A ) )
= ( A = bot_bot_set_nat ) ) ).
% all_not_in_conv
thf(fact_795_Collect__empty__eq,axiom,
! [P: term_a_b > $o] :
( ( ( collect_term_a_b @ P )
= bot_bot_set_term_a_b )
= ( ! [X2: term_a_b] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_796_Collect__empty__eq,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( ! [X2: nat] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_797_empty__Collect__eq,axiom,
! [P: term_a_b > $o] :
( ( bot_bot_set_term_a_b
= ( collect_term_a_b @ P ) )
= ( ! [X2: term_a_b] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_798_empty__Collect__eq,axiom,
! [P: nat > $o] :
( ( bot_bot_set_nat
= ( collect_nat @ P ) )
= ( ! [X2: nat] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_799_Sup__nat__empty,axiom,
( ( complete_Sup_Sup_nat @ bot_bot_set_nat )
= zero_zero_nat ) ).
% Sup_nat_empty
thf(fact_800_subset__empty,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
= ( A = bot_bot_set_nat ) ) ).
% subset_empty
thf(fact_801_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_802_empty__subsetI,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A ) ).
% empty_subsetI
thf(fact_803_empty__subsetI,axiom,
! [A: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ bot_bo9049108969261143879_a_nat @ A ) ).
% empty_subsetI
thf(fact_804_sup__bot__left,axiom,
! [X: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ X )
= X ) ).
% sup_bot_left
thf(fact_805_sup__bot__right,axiom,
! [X: set_nat] :
( ( sup_sup_set_nat @ X @ bot_bot_set_nat )
= X ) ).
% sup_bot_right
thf(fact_806_bot__eq__sup__iff,axiom,
! [X: set_nat,Y: set_nat] :
( ( bot_bot_set_nat
= ( sup_sup_set_nat @ X @ Y ) )
= ( ( X = bot_bot_set_nat )
& ( Y = bot_bot_set_nat ) ) ) ).
% bot_eq_sup_iff
thf(fact_807_sup__eq__bot__iff,axiom,
! [X: set_nat,Y: set_nat] :
( ( ( sup_sup_set_nat @ X @ Y )
= bot_bot_set_nat )
= ( ( X = bot_bot_set_nat )
& ( Y = bot_bot_set_nat ) ) ) ).
% sup_eq_bot_iff
thf(fact_808_sup__bot_Oeq__neutr__iff,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ( sup_sup_set_nat @ A2 @ B2 )
= bot_bot_set_nat )
= ( ( A2 = bot_bot_set_nat )
& ( B2 = bot_bot_set_nat ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_809_sup__bot_Oleft__neutral,axiom,
! [A2: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_810_sup__bot_Oneutr__eq__iff,axiom,
! [A2: set_nat,B2: set_nat] :
( ( bot_bot_set_nat
= ( sup_sup_set_nat @ A2 @ B2 ) )
= ( ( A2 = bot_bot_set_nat )
& ( B2 = bot_bot_set_nat ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_811_sup__bot_Oright__neutral,axiom,
! [A2: set_nat] :
( ( sup_sup_set_nat @ A2 @ bot_bot_set_nat )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_812_inf__bot__left,axiom,
! [X: set_Pr4386577575007340137rm_a_b] :
( ( inf_in1693821330460682071rm_a_b @ bot_bo197521221353338581rm_a_b @ X )
= bot_bo197521221353338581rm_a_b ) ).
% inf_bot_left
thf(fact_813_inf__bot__left,axiom,
! [X: set_nat] :
( ( inf_inf_set_nat @ bot_bot_set_nat @ X )
= bot_bot_set_nat ) ).
% inf_bot_left
thf(fact_814_inf__bot__right,axiom,
! [X: set_Pr4386577575007340137rm_a_b] :
( ( inf_in1693821330460682071rm_a_b @ X @ bot_bo197521221353338581rm_a_b )
= bot_bo197521221353338581rm_a_b ) ).
% inf_bot_right
thf(fact_815_inf__bot__right,axiom,
! [X: set_nat] :
( ( inf_inf_set_nat @ X @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% inf_bot_right
thf(fact_816_boolean__algebra_Oconj__zero__right,axiom,
! [X: set_Pr4386577575007340137rm_a_b] :
( ( inf_in1693821330460682071rm_a_b @ X @ bot_bo197521221353338581rm_a_b )
= bot_bo197521221353338581rm_a_b ) ).
% boolean_algebra.conj_zero_right
thf(fact_817_boolean__algebra_Oconj__zero__right,axiom,
! [X: set_nat] :
( ( inf_inf_set_nat @ X @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% boolean_algebra.conj_zero_right
thf(fact_818_boolean__algebra_Oconj__zero__left,axiom,
! [X: set_Pr4386577575007340137rm_a_b] :
( ( inf_in1693821330460682071rm_a_b @ bot_bo197521221353338581rm_a_b @ X )
= bot_bo197521221353338581rm_a_b ) ).
% boolean_algebra.conj_zero_left
thf(fact_819_boolean__algebra_Oconj__zero__left,axiom,
! [X: set_nat] :
( ( inf_inf_set_nat @ bot_bot_set_nat @ X )
= bot_bot_set_nat ) ).
% boolean_algebra.conj_zero_left
thf(fact_820_singletonI,axiom,
! [A2: product_prod_a_nat] : ( member5724188588386418708_a_nat @ A2 @ ( insert8054603423593749677_a_nat @ A2 @ bot_bo9049108969261143879_a_nat ) ) ).
% singletonI
thf(fact_821_singletonI,axiom,
! [A2: term_a_b] : ( member_term_a_b @ A2 @ ( insert_term_a_b @ A2 @ bot_bot_set_term_a_b ) ) ).
% singletonI
thf(fact_822_singletonI,axiom,
! [A2: nat] : ( member_nat @ A2 @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) ).
% singletonI
thf(fact_823_Sup__bot__conv_I2_J,axiom,
! [A: set_set_nat] :
( ( bot_bot_set_nat
= ( comple7399068483239264473et_nat @ A ) )
= ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A )
=> ( X2 = bot_bot_set_nat ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_824_Sup__bot__conv_I1_J,axiom,
! [A: set_set_nat] :
( ( ( comple7399068483239264473et_nat @ A )
= bot_bot_set_nat )
= ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A )
=> ( X2 = bot_bot_set_nat ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_825_Un__empty,axiom,
! [A: set_nat,B: set_nat] :
( ( ( sup_sup_set_nat @ A @ B )
= bot_bot_set_nat )
= ( ( A = bot_bot_set_nat )
& ( B = bot_bot_set_nat ) ) ) ).
% Un_empty
thf(fact_826_Sigma__empty1,axiom,
! [B: term_a_b > set_term_a_b] :
( ( produc1953217797448512882rm_a_b @ bot_bot_set_term_a_b @ B )
= bot_bo197521221353338581rm_a_b ) ).
% Sigma_empty1
thf(fact_827_NF__empty,axiom,
( ( abstract_NF_term_a_b @ bot_bo197521221353338581rm_a_b )
= top_top_set_term_a_b ) ).
% NF_empty
thf(fact_828_NF__empty,axiom,
( ( abstra2685462378305987313_a_nat @ bot_bo4243320254410501799_a_nat )
= top_to3353692345378799459_a_nat ) ).
% NF_empty
thf(fact_829_Collect__const,axiom,
! [P: $o] :
( ( P
=> ( ( collect_nat
@ ^ [S5: nat] : P )
= top_top_set_nat ) )
& ( ~ P
=> ( ( collect_nat
@ ^ [S5: nat] : P )
= bot_bot_set_nat ) ) ) ).
% Collect_const
thf(fact_830_Collect__const,axiom,
! [P: $o] :
( ( P
=> ( ( collect_term_a_b
@ ^ [S5: term_a_b] : P )
= top_top_set_term_a_b ) )
& ( ~ P
=> ( ( collect_term_a_b
@ ^ [S5: term_a_b] : P )
= bot_bot_set_term_a_b ) ) ) ).
% Collect_const
thf(fact_831_Collect__const,axiom,
! [P: $o] :
( ( P
=> ( ( collec4464134535221767506_a_nat
@ ^ [S5: product_prod_a_nat] : P )
= top_to3353692345378799459_a_nat ) )
& ( ~ P
=> ( ( collec4464134535221767506_a_nat
@ ^ [S5: product_prod_a_nat] : P )
= bot_bo9049108969261143879_a_nat ) ) ) ).
% Collect_const
thf(fact_832_singleton__conv2,axiom,
! [A2: product_prod_a_nat] :
( ( collec4464134535221767506_a_nat
@ ( ^ [Y2: product_prod_a_nat,Z2: product_prod_a_nat] : ( Y2 = Z2 )
@ A2 ) )
= ( insert8054603423593749677_a_nat @ A2 @ bot_bo9049108969261143879_a_nat ) ) ).
% singleton_conv2
thf(fact_833_singleton__conv2,axiom,
! [A2: term_a_b] :
( ( collect_term_a_b
@ ( ^ [Y2: term_a_b,Z2: term_a_b] : ( Y2 = Z2 )
@ A2 ) )
= ( insert_term_a_b @ A2 @ bot_bot_set_term_a_b ) ) ).
% singleton_conv2
thf(fact_834_singleton__conv2,axiom,
! [A2: nat] :
( ( collect_nat
@ ( ^ [Y2: nat,Z2: nat] : ( Y2 = Z2 )
@ A2 ) )
= ( insert_nat @ A2 @ bot_bot_set_nat ) ) ).
% singleton_conv2
thf(fact_835_singleton__conv,axiom,
! [A2: product_prod_a_nat] :
( ( collec4464134535221767506_a_nat
@ ^ [X2: product_prod_a_nat] : ( X2 = A2 ) )
= ( insert8054603423593749677_a_nat @ A2 @ bot_bo9049108969261143879_a_nat ) ) ).
% singleton_conv
thf(fact_836_singleton__conv,axiom,
! [A2: term_a_b] :
( ( collect_term_a_b
@ ^ [X2: term_a_b] : ( X2 = A2 ) )
= ( insert_term_a_b @ A2 @ bot_bot_set_term_a_b ) ) ).
% singleton_conv
thf(fact_837_singleton__conv,axiom,
! [A2: nat] :
( ( collect_nat
@ ^ [X2: nat] : ( X2 = A2 ) )
= ( insert_nat @ A2 @ bot_bot_set_nat ) ) ).
% singleton_conv
thf(fact_838_Times__empty,axiom,
! [A: set_term_a_b,B: set_term_a_b] :
( ( ( produc1953217797448512882rm_a_b @ A
@ ^ [Uu: term_a_b] : B )
= bot_bo197521221353338581rm_a_b )
= ( ( A = bot_bot_set_term_a_b )
| ( B = bot_bot_set_term_a_b ) ) ) ).
% Times_empty
thf(fact_839_Times__empty,axiom,
! [A: set_nat,B: set_nat] :
( ( ( produc457027306803732586at_nat @ A
@ ^ [Uu: nat] : B )
= bot_bo2099793752762293965at_nat )
= ( ( A = bot_bot_set_nat )
| ( B = bot_bot_set_nat ) ) ) ).
% Times_empty
thf(fact_840_Sigma__empty2,axiom,
! [A: set_term_a_b] :
( ( produc1953217797448512882rm_a_b @ A
@ ^ [Uu: term_a_b] : bot_bot_set_term_a_b )
= bot_bo197521221353338581rm_a_b ) ).
% Sigma_empty2
thf(fact_841_Sup__empty,axiom,
( ( comple7399068483239264473et_nat @ bot_bot_set_set_nat )
= bot_bot_set_nat ) ).
% Sup_empty
thf(fact_842_singleton__insert__inj__eq_H,axiom,
! [A2: nat,A: set_nat,B2: nat] :
( ( ( insert_nat @ A2 @ A )
= ( insert_nat @ B2 @ bot_bot_set_nat ) )
= ( ( A2 = B2 )
& ( ord_less_eq_set_nat @ A @ ( insert_nat @ B2 @ bot_bot_set_nat ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_843_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_844_singleton__insert__inj__eq,axiom,
! [B2: nat,A2: nat,A: set_nat] :
( ( ( insert_nat @ B2 @ bot_bot_set_nat )
= ( insert_nat @ A2 @ A ) )
= ( ( A2 = B2 )
& ( ord_less_eq_set_nat @ A @ ( insert_nat @ B2 @ bot_bot_set_nat ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_845_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_846_cSup__singleton,axiom,
! [X: nat] :
( ( complete_Sup_Sup_nat @ ( insert_nat @ X @ bot_bot_set_nat ) )
= X ) ).
% cSup_singleton
thf(fact_847_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_848_insert__disjoint_I1_J,axiom,
! [A2: term_a_b,A: set_term_a_b,B: set_term_a_b] :
( ( ( inf_inf_set_term_a_b @ ( insert_term_a_b @ A2 @ A ) @ B )
= bot_bot_set_term_a_b )
= ( ~ ( member_term_a_b @ A2 @ B )
& ( ( inf_inf_set_term_a_b @ A @ B )
= bot_bot_set_term_a_b ) ) ) ).
% insert_disjoint(1)
thf(fact_849_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_850_insert__disjoint_I1_J,axiom,
! [A2: nat,A: set_nat,B: set_nat] :
( ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ A ) @ B )
= bot_bot_set_nat )
= ( ~ ( member_nat @ A2 @ B )
& ( ( inf_inf_set_nat @ A @ B )
= bot_bot_set_nat ) ) ) ).
% insert_disjoint(1)
thf(fact_851_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_852_insert__disjoint_I2_J,axiom,
! [A2: term_a_b,A: set_term_a_b,B: set_term_a_b] :
( ( bot_bot_set_term_a_b
= ( inf_inf_set_term_a_b @ ( insert_term_a_b @ A2 @ A ) @ B ) )
= ( ~ ( member_term_a_b @ A2 @ B )
& ( bot_bot_set_term_a_b
= ( inf_inf_set_term_a_b @ A @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_853_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_854_insert__disjoint_I2_J,axiom,
! [A2: nat,A: set_nat,B: set_nat] :
( ( bot_bot_set_nat
= ( inf_inf_set_nat @ ( insert_nat @ A2 @ A ) @ B ) )
= ( ~ ( member_nat @ A2 @ B )
& ( bot_bot_set_nat
= ( inf_inf_set_nat @ A @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_855_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_856_disjoint__insert_I1_J,axiom,
! [B: set_term_a_b,A2: term_a_b,A: set_term_a_b] :
( ( ( inf_inf_set_term_a_b @ B @ ( insert_term_a_b @ A2 @ A ) )
= bot_bot_set_term_a_b )
= ( ~ ( member_term_a_b @ A2 @ B )
& ( ( inf_inf_set_term_a_b @ B @ A )
= bot_bot_set_term_a_b ) ) ) ).
% disjoint_insert(1)
thf(fact_857_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_858_disjoint__insert_I1_J,axiom,
! [B: set_nat,A2: nat,A: set_nat] :
( ( ( inf_inf_set_nat @ B @ ( insert_nat @ A2 @ A ) )
= bot_bot_set_nat )
= ( ~ ( member_nat @ A2 @ B )
& ( ( inf_inf_set_nat @ B @ A )
= bot_bot_set_nat ) ) ) ).
% disjoint_insert(1)
thf(fact_859_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_860_disjoint__insert_I2_J,axiom,
! [A: set_term_a_b,B2: term_a_b,B: set_term_a_b] :
( ( bot_bot_set_term_a_b
= ( inf_inf_set_term_a_b @ A @ ( insert_term_a_b @ B2 @ B ) ) )
= ( ~ ( member_term_a_b @ B2 @ A )
& ( bot_bot_set_term_a_b
= ( inf_inf_set_term_a_b @ A @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_861_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_862_disjoint__insert_I2_J,axiom,
! [A: set_nat,B2: nat,B: set_nat] :
( ( bot_bot_set_nat
= ( inf_inf_set_nat @ A @ ( insert_nat @ B2 @ B ) ) )
= ( ~ ( member_nat @ B2 @ A )
& ( bot_bot_set_nat
= ( inf_inf_set_nat @ A @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_863_Collect__conv__if,axiom,
! [P: product_prod_a_nat > $o,A2: product_prod_a_nat] :
( ( ( P @ A2 )
=> ( ( collec4464134535221767506_a_nat
@ ^ [X2: product_prod_a_nat] :
( ( X2 = A2 )
& ( P @ X2 ) ) )
= ( insert8054603423593749677_a_nat @ A2 @ bot_bo9049108969261143879_a_nat ) ) )
& ( ~ ( P @ A2 )
=> ( ( collec4464134535221767506_a_nat
@ ^ [X2: product_prod_a_nat] :
( ( X2 = A2 )
& ( P @ X2 ) ) )
= bot_bo9049108969261143879_a_nat ) ) ) ).
% Collect_conv_if
thf(fact_864_Collect__conv__if,axiom,
! [P: term_a_b > $o,A2: term_a_b] :
( ( ( P @ A2 )
=> ( ( collect_term_a_b
@ ^ [X2: term_a_b] :
( ( X2 = A2 )
& ( P @ X2 ) ) )
= ( insert_term_a_b @ A2 @ bot_bot_set_term_a_b ) ) )
& ( ~ ( P @ A2 )
=> ( ( collect_term_a_b
@ ^ [X2: term_a_b] :
( ( X2 = A2 )
& ( P @ X2 ) ) )
= bot_bot_set_term_a_b ) ) ) ).
% Collect_conv_if
thf(fact_865_Collect__conv__if,axiom,
! [P: nat > $o,A2: nat] :
( ( ( P @ A2 )
=> ( ( collect_nat
@ ^ [X2: nat] :
( ( X2 = A2 )
& ( P @ X2 ) ) )
= ( insert_nat @ A2 @ bot_bot_set_nat ) ) )
& ( ~ ( P @ A2 )
=> ( ( collect_nat
@ ^ [X2: nat] :
( ( X2 = A2 )
& ( P @ X2 ) ) )
= bot_bot_set_nat ) ) ) ).
% Collect_conv_if
thf(fact_866_Collect__conv__if2,axiom,
! [P: product_prod_a_nat > $o,A2: product_prod_a_nat] :
( ( ( P @ A2 )
=> ( ( collec4464134535221767506_a_nat
@ ^ [X2: product_prod_a_nat] :
( ( A2 = X2 )
& ( P @ X2 ) ) )
= ( insert8054603423593749677_a_nat @ A2 @ bot_bo9049108969261143879_a_nat ) ) )
& ( ~ ( P @ A2 )
=> ( ( collec4464134535221767506_a_nat
@ ^ [X2: product_prod_a_nat] :
( ( A2 = X2 )
& ( P @ X2 ) ) )
= bot_bo9049108969261143879_a_nat ) ) ) ).
% Collect_conv_if2
thf(fact_867_Collect__conv__if2,axiom,
! [P: term_a_b > $o,A2: term_a_b] :
( ( ( P @ A2 )
=> ( ( collect_term_a_b
@ ^ [X2: term_a_b] :
( ( A2 = X2 )
& ( P @ X2 ) ) )
= ( insert_term_a_b @ A2 @ bot_bot_set_term_a_b ) ) )
& ( ~ ( P @ A2 )
=> ( ( collect_term_a_b
@ ^ [X2: term_a_b] :
( ( A2 = X2 )
& ( P @ X2 ) ) )
= bot_bot_set_term_a_b ) ) ) ).
% Collect_conv_if2
thf(fact_868_Collect__conv__if2,axiom,
! [P: nat > $o,A2: nat] :
( ( ( P @ A2 )
=> ( ( collect_nat
@ ^ [X2: nat] :
( ( A2 = X2 )
& ( P @ X2 ) ) )
= ( insert_nat @ A2 @ bot_bot_set_nat ) ) )
& ( ~ ( P @ A2 )
=> ( ( collect_nat
@ ^ [X2: nat] :
( ( A2 = X2 )
& ( P @ X2 ) ) )
= bot_bot_set_nat ) ) ) ).
% Collect_conv_if2
thf(fact_869_times__eq__iff,axiom,
! [A: set_term_a_b,B: set_term_a_b,C2: set_term_a_b,D: set_term_a_b] :
( ( ( produc1953217797448512882rm_a_b @ A
@ ^ [Uu: term_a_b] : B )
= ( produc1953217797448512882rm_a_b @ C2
@ ^ [Uu: term_a_b] : D ) )
= ( ( ( A = C2 )
& ( B = D ) )
| ( ( ( A = bot_bot_set_term_a_b )
| ( B = bot_bot_set_term_a_b ) )
& ( ( C2 = bot_bot_set_term_a_b )
| ( D = bot_bot_set_term_a_b ) ) ) ) ) ).
% times_eq_iff
thf(fact_870_times__eq__iff,axiom,
! [A: set_nat,B: set_nat,C2: set_nat,D: set_nat] :
( ( ( produc457027306803732586at_nat @ A
@ ^ [Uu: nat] : B )
= ( produc457027306803732586at_nat @ C2
@ ^ [Uu: nat] : D ) )
= ( ( ( A = C2 )
& ( B = D ) )
| ( ( ( A = bot_bot_set_nat )
| ( B = bot_bot_set_nat ) )
& ( ( C2 = bot_bot_set_nat )
| ( D = bot_bot_set_nat ) ) ) ) ) ).
% times_eq_iff
thf(fact_871_empty__not__UNIV,axiom,
bot_bot_set_nat != top_top_set_nat ).
% empty_not_UNIV
thf(fact_872_empty__not__UNIV,axiom,
bot_bot_set_term_a_b != top_top_set_term_a_b ).
% empty_not_UNIV
thf(fact_873_empty__not__UNIV,axiom,
bot_bo9049108969261143879_a_nat != top_to3353692345378799459_a_nat ).
% empty_not_UNIV
thf(fact_874_subset__emptyI,axiom,
! [A: set_term_a_b] :
( ! [X3: term_a_b] :
~ ( member_term_a_b @ X3 @ A )
=> ( ord_le2705286416250468010rm_a_b @ A @ bot_bot_set_term_a_b ) ) ).
% subset_emptyI
thf(fact_875_subset__emptyI,axiom,
! [A: set_nat] :
( ! [X3: nat] :
~ ( member_nat @ X3 @ A )
=> ( ord_less_eq_set_nat @ A @ bot_bot_set_nat ) ) ).
% subset_emptyI
thf(fact_876_subset__emptyI,axiom,
! [A: set_Pr4934435412358123699_a_nat] :
( ! [X3: product_prod_a_nat] :
~ ( member5724188588386418708_a_nat @ X3 @ A )
=> ( ord_le8666007276011122963_a_nat @ A @ bot_bo9049108969261143879_a_nat ) ) ).
% subset_emptyI
thf(fact_877_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_878_singletonD,axiom,
! [B2: term_a_b,A2: term_a_b] :
( ( member_term_a_b @ B2 @ ( insert_term_a_b @ A2 @ bot_bot_set_term_a_b ) )
=> ( B2 = A2 ) ) ).
% singletonD
thf(fact_879_singletonD,axiom,
! [B2: nat,A2: nat] :
( ( member_nat @ B2 @ ( insert_nat @ A2 @ bot_bot_set_nat ) )
=> ( B2 = A2 ) ) ).
% singletonD
thf(fact_880_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_881_singleton__iff,axiom,
! [B2: term_a_b,A2: term_a_b] :
( ( member_term_a_b @ B2 @ ( insert_term_a_b @ A2 @ bot_bot_set_term_a_b ) )
= ( B2 = A2 ) ) ).
% singleton_iff
thf(fact_882_singleton__iff,axiom,
! [B2: nat,A2: nat] :
( ( member_nat @ B2 @ ( insert_nat @ A2 @ bot_bot_set_nat ) )
= ( B2 = A2 ) ) ).
% singleton_iff
thf(fact_883_doubleton__eq__iff,axiom,
! [A2: product_prod_a_nat,B2: product_prod_a_nat,C: product_prod_a_nat,D2: product_prod_a_nat] :
( ( ( insert8054603423593749677_a_nat @ A2 @ ( insert8054603423593749677_a_nat @ B2 @ bot_bo9049108969261143879_a_nat ) )
= ( insert8054603423593749677_a_nat @ C @ ( insert8054603423593749677_a_nat @ D2 @ bot_bo9049108969261143879_a_nat ) ) )
= ( ( ( A2 = C )
& ( B2 = D2 ) )
| ( ( A2 = D2 )
& ( B2 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_884_doubleton__eq__iff,axiom,
! [A2: nat,B2: nat,C: nat,D2: nat] :
( ( ( insert_nat @ A2 @ ( insert_nat @ B2 @ bot_bot_set_nat ) )
= ( insert_nat @ C @ ( insert_nat @ D2 @ bot_bot_set_nat ) ) )
= ( ( ( A2 = C )
& ( B2 = D2 ) )
| ( ( A2 = D2 )
& ( B2 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_885_insert__not__empty,axiom,
! [A2: product_prod_a_nat,A: set_Pr4934435412358123699_a_nat] :
( ( insert8054603423593749677_a_nat @ A2 @ A )
!= bot_bo9049108969261143879_a_nat ) ).
% insert_not_empty
thf(fact_886_insert__not__empty,axiom,
! [A2: nat,A: set_nat] :
( ( insert_nat @ A2 @ A )
!= bot_bot_set_nat ) ).
% insert_not_empty
thf(fact_887_singleton__inject,axiom,
! [A2: product_prod_a_nat,B2: product_prod_a_nat] :
( ( ( insert8054603423593749677_a_nat @ A2 @ bot_bo9049108969261143879_a_nat )
= ( insert8054603423593749677_a_nat @ B2 @ bot_bo9049108969261143879_a_nat ) )
=> ( A2 = B2 ) ) ).
% singleton_inject
thf(fact_888_singleton__inject,axiom,
! [A2: nat,B2: nat] :
( ( ( insert_nat @ A2 @ bot_bot_set_nat )
= ( insert_nat @ B2 @ bot_bot_set_nat ) )
=> ( A2 = B2 ) ) ).
% singleton_inject
thf(fact_889_Int__emptyI,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ! [X3: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ X3 @ A )
=> ~ ( member5724188588386418708_a_nat @ X3 @ B ) )
=> ( ( inf_in3813033988066259269_a_nat @ A @ B )
= bot_bo9049108969261143879_a_nat ) ) ).
% Int_emptyI
thf(fact_890_Int__emptyI,axiom,
! [A: set_term_a_b,B: set_term_a_b] :
( ! [X3: term_a_b] :
( ( member_term_a_b @ X3 @ A )
=> ~ ( member_term_a_b @ X3 @ B ) )
=> ( ( inf_inf_set_term_a_b @ A @ B )
= bot_bot_set_term_a_b ) ) ).
% Int_emptyI
thf(fact_891_Int__emptyI,axiom,
! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ! [X3: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X3 @ A )
=> ~ ( member5869715511025134514rm_a_b @ X3 @ B ) )
=> ( ( inf_in1693821330460682071rm_a_b @ A @ B )
= bot_bo197521221353338581rm_a_b ) ) ).
% Int_emptyI
thf(fact_892_Int__emptyI,axiom,
! [A: set_nat,B: set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ~ ( member_nat @ X3 @ B ) )
=> ( ( inf_inf_set_nat @ A @ B )
= bot_bot_set_nat ) ) ).
% Int_emptyI
thf(fact_893_disjoint__iff,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( ( inf_in3813033988066259269_a_nat @ A @ B )
= bot_bo9049108969261143879_a_nat )
= ( ! [X2: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ X2 @ A )
=> ~ ( member5724188588386418708_a_nat @ X2 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_894_disjoint__iff,axiom,
! [A: set_term_a_b,B: set_term_a_b] :
( ( ( inf_inf_set_term_a_b @ A @ B )
= bot_bot_set_term_a_b )
= ( ! [X2: term_a_b] :
( ( member_term_a_b @ X2 @ A )
=> ~ ( member_term_a_b @ X2 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_895_disjoint__iff,axiom,
! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ( ( inf_in1693821330460682071rm_a_b @ A @ B )
= bot_bo197521221353338581rm_a_b )
= ( ! [X2: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X2 @ A )
=> ~ ( member5869715511025134514rm_a_b @ X2 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_896_disjoint__iff,axiom,
! [A: set_nat,B: set_nat] :
( ( ( inf_inf_set_nat @ A @ B )
= bot_bot_set_nat )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ~ ( member_nat @ X2 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_897_Int__empty__left,axiom,
! [B: set_Pr4386577575007340137rm_a_b] :
( ( inf_in1693821330460682071rm_a_b @ bot_bo197521221353338581rm_a_b @ B )
= bot_bo197521221353338581rm_a_b ) ).
% Int_empty_left
thf(fact_898_Int__empty__left,axiom,
! [B: set_nat] :
( ( inf_inf_set_nat @ bot_bot_set_nat @ B )
= bot_bot_set_nat ) ).
% Int_empty_left
thf(fact_899_Int__empty__right,axiom,
! [A: set_Pr4386577575007340137rm_a_b] :
( ( inf_in1693821330460682071rm_a_b @ A @ bot_bo197521221353338581rm_a_b )
= bot_bo197521221353338581rm_a_b ) ).
% Int_empty_right
thf(fact_900_Int__empty__right,axiom,
! [A: set_nat] :
( ( inf_inf_set_nat @ A @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% Int_empty_right
thf(fact_901_disjoint__iff__not__equal,axiom,
! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ( ( inf_in1693821330460682071rm_a_b @ A @ B )
= bot_bo197521221353338581rm_a_b )
= ( ! [X2: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X2 @ A )
=> ! [Y3: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ Y3 @ B )
=> ( X2 != Y3 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_902_disjoint__iff__not__equal,axiom,
! [A: set_nat,B: set_nat] :
( ( ( inf_inf_set_nat @ A @ B )
= bot_bot_set_nat )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ! [Y3: nat] :
( ( member_nat @ Y3 @ B )
=> ( X2 != Y3 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_903_empty__Union__conv,axiom,
! [A: set_set_nat] :
( ( bot_bot_set_nat
= ( comple7399068483239264473et_nat @ A ) )
= ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A )
=> ( X2 = bot_bot_set_nat ) ) ) ) ).
% empty_Union_conv
thf(fact_904_Union__empty__conv,axiom,
! [A: set_set_nat] :
( ( ( comple7399068483239264473et_nat @ A )
= bot_bot_set_nat )
= ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A )
=> ( X2 = bot_bot_set_nat ) ) ) ) ).
% Union_empty_conv
thf(fact_905_empty__def,axiom,
( bot_bot_set_term_a_b
= ( collect_term_a_b
@ ^ [X2: term_a_b] : $false ) ) ).
% empty_def
thf(fact_906_empty__def,axiom,
( bot_bot_set_nat
= ( collect_nat
@ ^ [X2: nat] : $false ) ) ).
% empty_def
thf(fact_907_Sigma__empty__iff,axiom,
! [I: set_term_a_b,X6: term_a_b > set_term_a_b] :
( ( ( produc1953217797448512882rm_a_b @ I @ X6 )
= bot_bo197521221353338581rm_a_b )
= ( ! [X2: term_a_b] :
( ( member_term_a_b @ X2 @ I )
=> ( ( X6 @ X2 )
= bot_bot_set_term_a_b ) ) ) ) ).
% Sigma_empty_iff
thf(fact_908_emptyE,axiom,
! [A2: product_prod_a_nat] :
~ ( member5724188588386418708_a_nat @ A2 @ bot_bo9049108969261143879_a_nat ) ).
% emptyE
thf(fact_909_emptyE,axiom,
! [A2: term_a_b] :
~ ( member_term_a_b @ A2 @ bot_bot_set_term_a_b ) ).
% emptyE
thf(fact_910_emptyE,axiom,
! [A2: nat] :
~ ( member_nat @ A2 @ bot_bot_set_nat ) ).
% emptyE
thf(fact_911_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_912_equals0D,axiom,
! [A: set_term_a_b,A2: term_a_b] :
( ( A = bot_bot_set_term_a_b )
=> ~ ( member_term_a_b @ A2 @ A ) ) ).
% equals0D
thf(fact_913_equals0D,axiom,
! [A: set_nat,A2: nat] :
( ( A = bot_bot_set_nat )
=> ~ ( member_nat @ A2 @ A ) ) ).
% equals0D
thf(fact_914_equals0I,axiom,
! [A: set_Pr4934435412358123699_a_nat] :
( ! [Y4: product_prod_a_nat] :
~ ( member5724188588386418708_a_nat @ Y4 @ A )
=> ( A = bot_bo9049108969261143879_a_nat ) ) ).
% equals0I
thf(fact_915_equals0I,axiom,
! [A: set_term_a_b] :
( ! [Y4: term_a_b] :
~ ( member_term_a_b @ Y4 @ A )
=> ( A = bot_bot_set_term_a_b ) ) ).
% equals0I
thf(fact_916_equals0I,axiom,
! [A: set_nat] :
( ! [Y4: nat] :
~ ( member_nat @ Y4 @ A )
=> ( A = bot_bot_set_nat ) ) ).
% equals0I
thf(fact_917_ex__in__conv,axiom,
! [A: set_Pr4934435412358123699_a_nat] :
( ( ? [X2: product_prod_a_nat] : ( member5724188588386418708_a_nat @ X2 @ A ) )
= ( A != bot_bo9049108969261143879_a_nat ) ) ).
% ex_in_conv
thf(fact_918_ex__in__conv,axiom,
! [A: set_term_a_b] :
( ( ? [X2: term_a_b] : ( member_term_a_b @ X2 @ A ) )
= ( A != bot_bot_set_term_a_b ) ) ).
% ex_in_conv
thf(fact_919_ex__in__conv,axiom,
! [A: set_nat] :
( ( ? [X2: nat] : ( member_nat @ X2 @ A ) )
= ( A != bot_bot_set_nat ) ) ).
% ex_in_conv
thf(fact_920_Union__empty,axiom,
( ( comple7399068483239264473et_nat @ bot_bot_set_set_nat )
= bot_bot_set_nat ) ).
% Union_empty
thf(fact_921_bot_Oextremum,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A2 ) ).
% bot.extremum
thf(fact_922_bot_Oextremum,axiom,
! [A2: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ bot_bo9049108969261143879_a_nat @ A2 ) ).
% bot.extremum
thf(fact_923_bot_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A2 ) ).
% bot.extremum
thf(fact_924_bot_Oextremum__unique,axiom,
! [A2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat )
= ( A2 = bot_bot_set_nat ) ) ).
% bot.extremum_unique
thf(fact_925_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_926_bot_Oextremum__unique,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ bot_bot_nat )
= ( A2 = bot_bot_nat ) ) ).
% bot.extremum_unique
thf(fact_927_bot_Oextremum__uniqueI,axiom,
! [A2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat )
=> ( A2 = bot_bot_set_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_928_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_929_bot_Oextremum__uniqueI,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ bot_bot_nat )
=> ( A2 = bot_bot_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_930_Un__empty__left,axiom,
! [B: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ B )
= B ) ).
% Un_empty_left
thf(fact_931_Un__empty__right,axiom,
! [A: set_nat] :
( ( sup_sup_set_nat @ A @ bot_bot_set_nat )
= A ) ).
% Un_empty_right
thf(fact_932_boolean__algebra_Odisj__zero__right,axiom,
! [X: set_nat] :
( ( sup_sup_set_nat @ X @ bot_bot_set_nat )
= X ) ).
% boolean_algebra.disj_zero_right
thf(fact_933_cSup__eq__non__empty,axiom,
! [X6: set_se6121441497158405097_a_nat,A2: set_Pr4934435412358123699_a_nat] :
( ( X6 != bot_bo2146796114724504189_a_nat )
=> ( ! [X3: set_Pr4934435412358123699_a_nat] :
( ( member8034581882086557258_a_nat @ X3 @ X6 )
=> ( ord_le8666007276011122963_a_nat @ X3 @ A2 ) )
=> ( ! [Y4: set_Pr4934435412358123699_a_nat] :
( ! [X4: set_Pr4934435412358123699_a_nat] :
( ( member8034581882086557258_a_nat @ X4 @ X6 )
=> ( ord_le8666007276011122963_a_nat @ X4 @ Y4 ) )
=> ( ord_le8666007276011122963_a_nat @ A2 @ Y4 ) )
=> ( ( comple3771923735958488838_a_nat @ X6 )
= A2 ) ) ) ) ).
% cSup_eq_non_empty
thf(fact_934_cSup__eq__non__empty,axiom,
! [X6: set_nat,A2: nat] :
( ( X6 != bot_bot_set_nat )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ X6 )
=> ( ord_less_eq_nat @ X3 @ A2 ) )
=> ( ! [Y4: nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ X6 )
=> ( ord_less_eq_nat @ X4 @ Y4 ) )
=> ( ord_less_eq_nat @ A2 @ Y4 ) )
=> ( ( complete_Sup_Sup_nat @ X6 )
= A2 ) ) ) ) ).
% cSup_eq_non_empty
thf(fact_935_cSup__least,axiom,
! [X6: set_se6121441497158405097_a_nat,Z: set_Pr4934435412358123699_a_nat] :
( ( X6 != bot_bo2146796114724504189_a_nat )
=> ( ! [X3: set_Pr4934435412358123699_a_nat] :
( ( member8034581882086557258_a_nat @ X3 @ X6 )
=> ( ord_le8666007276011122963_a_nat @ X3 @ Z ) )
=> ( ord_le8666007276011122963_a_nat @ ( comple3771923735958488838_a_nat @ X6 ) @ Z ) ) ) ).
% cSup_least
thf(fact_936_cSup__least,axiom,
! [X6: set_nat,Z: nat] :
( ( X6 != bot_bot_set_nat )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ X6 )
=> ( ord_less_eq_nat @ X3 @ Z ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ X6 ) @ Z ) ) ) ).
% cSup_least
thf(fact_937_less__eq__Sup,axiom,
! [A: set_se6121441497158405097_a_nat,U: set_Pr4934435412358123699_a_nat] :
( ! [V3: set_Pr4934435412358123699_a_nat] :
( ( member8034581882086557258_a_nat @ V3 @ A )
=> ( ord_le8666007276011122963_a_nat @ U @ V3 ) )
=> ( ( A != bot_bo2146796114724504189_a_nat )
=> ( ord_le8666007276011122963_a_nat @ U @ ( comple3771923735958488838_a_nat @ A ) ) ) ) ).
% less_eq_Sup
thf(fact_938_subset__singleton__iff,axiom,
! [X6: set_nat,A2: nat] :
( ( ord_less_eq_set_nat @ X6 @ ( insert_nat @ A2 @ bot_bot_set_nat ) )
= ( ( X6 = bot_bot_set_nat )
| ( X6
= ( insert_nat @ A2 @ bot_bot_set_nat ) ) ) ) ).
% subset_singleton_iff
thf(fact_939_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_940_subset__singletonD,axiom,
! [A: set_nat,X: nat] :
( ( ord_less_eq_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) )
=> ( ( A = bot_bot_set_nat )
| ( A
= ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ).
% subset_singletonD
thf(fact_941_subset__singletonD,axiom,
! [A: set_Pr4934435412358123699_a_nat,X: product_prod_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A @ ( insert8054603423593749677_a_nat @ X @ bot_bo9049108969261143879_a_nat ) )
=> ( ( A = bot_bo9049108969261143879_a_nat )
| ( A
= ( insert8054603423593749677_a_nat @ X @ bot_bo9049108969261143879_a_nat ) ) ) ) ).
% subset_singletonD
thf(fact_942_insert__is__Un,axiom,
( insert8054603423593749677_a_nat
= ( ^ [A4: product_prod_a_nat] : ( sup_su459911885395995103_a_nat @ ( insert8054603423593749677_a_nat @ A4 @ bot_bo9049108969261143879_a_nat ) ) ) ) ).
% insert_is_Un
thf(fact_943_insert__is__Un,axiom,
( insert_nat
= ( ^ [A4: nat] : ( sup_sup_set_nat @ ( insert_nat @ A4 @ bot_bot_set_nat ) ) ) ) ).
% insert_is_Un
thf(fact_944_Un__singleton__iff,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat,X: product_prod_a_nat] :
( ( ( sup_su459911885395995103_a_nat @ A @ B )
= ( insert8054603423593749677_a_nat @ X @ bot_bo9049108969261143879_a_nat ) )
= ( ( ( A = bot_bo9049108969261143879_a_nat )
& ( B
= ( insert8054603423593749677_a_nat @ X @ bot_bo9049108969261143879_a_nat ) ) )
| ( ( A
= ( insert8054603423593749677_a_nat @ X @ bot_bo9049108969261143879_a_nat ) )
& ( B = bot_bo9049108969261143879_a_nat ) )
| ( ( A
= ( insert8054603423593749677_a_nat @ X @ bot_bo9049108969261143879_a_nat ) )
& ( B
= ( insert8054603423593749677_a_nat @ X @ bot_bo9049108969261143879_a_nat ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_945_Un__singleton__iff,axiom,
! [A: set_nat,B: set_nat,X: nat] :
( ( ( sup_sup_set_nat @ A @ B )
= ( insert_nat @ X @ bot_bot_set_nat ) )
= ( ( ( A = bot_bot_set_nat )
& ( B
= ( insert_nat @ X @ bot_bot_set_nat ) ) )
| ( ( A
= ( insert_nat @ X @ bot_bot_set_nat ) )
& ( B = bot_bot_set_nat ) )
| ( ( A
= ( insert_nat @ X @ bot_bot_set_nat ) )
& ( B
= ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_946_singleton__Un__iff,axiom,
! [X: product_prod_a_nat,A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( ( insert8054603423593749677_a_nat @ X @ bot_bo9049108969261143879_a_nat )
= ( sup_su459911885395995103_a_nat @ A @ B ) )
= ( ( ( A = bot_bo9049108969261143879_a_nat )
& ( B
= ( insert8054603423593749677_a_nat @ X @ bot_bo9049108969261143879_a_nat ) ) )
| ( ( A
= ( insert8054603423593749677_a_nat @ X @ bot_bo9049108969261143879_a_nat ) )
& ( B = bot_bo9049108969261143879_a_nat ) )
| ( ( A
= ( insert8054603423593749677_a_nat @ X @ bot_bo9049108969261143879_a_nat ) )
& ( B
= ( insert8054603423593749677_a_nat @ X @ bot_bo9049108969261143879_a_nat ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_947_singleton__Un__iff,axiom,
! [X: nat,A: set_nat,B: set_nat] :
( ( ( insert_nat @ X @ bot_bot_set_nat )
= ( sup_sup_set_nat @ A @ B ) )
= ( ( ( A = bot_bot_set_nat )
& ( B
= ( insert_nat @ X @ bot_bot_set_nat ) ) )
| ( ( A
= ( insert_nat @ X @ bot_bot_set_nat ) )
& ( B = bot_bot_set_nat ) )
| ( ( A
= ( insert_nat @ X @ bot_bot_set_nat ) )
& ( B
= ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_948_Union__disjoint,axiom,
! [C2: set_se1467346398288745161rm_a_b,A: set_Pr4386577575007340137rm_a_b] :
( ( ( inf_in1693821330460682071rm_a_b @ ( comple4970752389892389974rm_a_b @ C2 ) @ A )
= bot_bo197521221353338581rm_a_b )
= ( ! [X2: set_Pr4386577575007340137rm_a_b] :
( ( member8823642877475723282rm_a_b @ X2 @ C2 )
=> ( ( inf_in1693821330460682071rm_a_b @ X2 @ A )
= bot_bo197521221353338581rm_a_b ) ) ) ) ).
% Union_disjoint
thf(fact_949_Union__disjoint,axiom,
! [C2: set_set_nat,A: set_nat] :
( ( ( inf_inf_set_nat @ ( comple7399068483239264473et_nat @ C2 ) @ A )
= bot_bot_set_nat )
= ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ C2 )
=> ( ( inf_inf_set_nat @ X2 @ A )
= bot_bot_set_nat ) ) ) ) ).
% Union_disjoint
thf(fact_950_boolean__algebra_Ocomplement__unique,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,X: set_Pr4386577575007340137rm_a_b,Y: set_Pr4386577575007340137rm_a_b] :
( ( ( inf_in1693821330460682071rm_a_b @ A2 @ X )
= bot_bo197521221353338581rm_a_b )
=> ( ( ( sup_su6776935440552674877rm_a_b @ A2 @ X )
= top_to1314267278846557113rm_a_b )
=> ( ( ( inf_in1693821330460682071rm_a_b @ A2 @ Y )
= bot_bo197521221353338581rm_a_b )
=> ( ( ( sup_su6776935440552674877rm_a_b @ A2 @ Y )
= top_to1314267278846557113rm_a_b )
=> ( X = Y ) ) ) ) ) ).
% boolean_algebra.complement_unique
thf(fact_951_boolean__algebra_Ocomplement__unique,axiom,
! [A2: set_nat,X: set_nat,Y: set_nat] :
( ( ( inf_inf_set_nat @ A2 @ X )
= bot_bot_set_nat )
=> ( ( ( sup_sup_set_nat @ A2 @ X )
= top_top_set_nat )
=> ( ( ( inf_inf_set_nat @ A2 @ Y )
= bot_bot_set_nat )
=> ( ( ( sup_sup_set_nat @ A2 @ Y )
= top_top_set_nat )
=> ( X = Y ) ) ) ) ) ).
% boolean_algebra.complement_unique
thf(fact_952_boolean__algebra_Ocomplement__unique,axiom,
! [A2: set_term_a_b,X: set_term_a_b,Y: set_term_a_b] :
( ( ( inf_inf_set_term_a_b @ A2 @ X )
= bot_bot_set_term_a_b )
=> ( ( ( sup_sup_set_term_a_b @ A2 @ X )
= top_top_set_term_a_b )
=> ( ( ( inf_inf_set_term_a_b @ A2 @ Y )
= bot_bot_set_term_a_b )
=> ( ( ( sup_sup_set_term_a_b @ A2 @ Y )
= top_top_set_term_a_b )
=> ( X = Y ) ) ) ) ) ).
% boolean_algebra.complement_unique
thf(fact_953_boolean__algebra_Ocomplement__unique,axiom,
! [A2: set_Pr4934435412358123699_a_nat,X: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat] :
( ( ( inf_in3813033988066259269_a_nat @ A2 @ X )
= bot_bo9049108969261143879_a_nat )
=> ( ( ( sup_su459911885395995103_a_nat @ A2 @ X )
= top_to3353692345378799459_a_nat )
=> ( ( ( inf_in3813033988066259269_a_nat @ A2 @ Y )
= bot_bo9049108969261143879_a_nat )
=> ( ( ( sup_su459911885395995103_a_nat @ A2 @ Y )
= top_to3353692345378799459_a_nat )
=> ( X = Y ) ) ) ) ) ).
% boolean_algebra.complement_unique
thf(fact_954_times__subset__iff,axiom,
! [A: set_term_a_b,C2: set_term_a_b,B: set_term_a_b,D: set_term_a_b] :
( ( ord_le118470702582115849rm_a_b
@ ( produc1953217797448512882rm_a_b @ A
@ ^ [Uu: term_a_b] : C2 )
@ ( produc1953217797448512882rm_a_b @ B
@ ^ [Uu: term_a_b] : D ) )
= ( ( A = bot_bot_set_term_a_b )
| ( C2 = bot_bot_set_term_a_b )
| ( ( ord_le2705286416250468010rm_a_b @ A @ B )
& ( ord_le2705286416250468010rm_a_b @ C2 @ D ) ) ) ) ).
% times_subset_iff
thf(fact_955_times__subset__iff,axiom,
! [A: set_nat,C2: set_nat,B: set_nat,D: set_nat] :
( ( ord_le3146513528884898305at_nat
@ ( produc457027306803732586at_nat @ A
@ ^ [Uu: nat] : C2 )
@ ( produc457027306803732586at_nat @ B
@ ^ [Uu: nat] : D ) )
= ( ( A = bot_bot_set_nat )
| ( C2 = bot_bot_set_nat )
| ( ( ord_less_eq_set_nat @ A @ B )
& ( ord_less_eq_set_nat @ C2 @ D ) ) ) ) ).
% times_subset_iff
thf(fact_956_times__subset__iff,axiom,
! [A: set_nat,C2: set_Pr4934435412358123699_a_nat,B: set_nat,D: set_Pr4934435412358123699_a_nat] :
( ( ord_le5367656705095804078_a_nat
@ ( produc4304465545695482135_a_nat @ A
@ ^ [Uu: nat] : C2 )
@ ( produc4304465545695482135_a_nat @ B
@ ^ [Uu: nat] : D ) )
= ( ( A = bot_bot_set_nat )
| ( C2 = bot_bo9049108969261143879_a_nat )
| ( ( ord_less_eq_set_nat @ A @ B )
& ( ord_le8666007276011122963_a_nat @ C2 @ D ) ) ) ) ).
% times_subset_iff
thf(fact_957_times__subset__iff,axiom,
! [A: set_Pr4934435412358123699_a_nat,C2: set_nat,B: set_Pr4934435412358123699_a_nat,D: set_nat] :
( ( ord_le3546921724939684910at_nat
@ ( produc8935565175082432023at_nat @ A
@ ^ [Uu: product_prod_a_nat] : C2 )
@ ( produc8935565175082432023at_nat @ B
@ ^ [Uu: product_prod_a_nat] : D ) )
= ( ( A = bot_bo9049108969261143879_a_nat )
| ( C2 = bot_bot_set_nat )
| ( ( ord_le8666007276011122963_a_nat @ A @ B )
& ( ord_less_eq_set_nat @ C2 @ D ) ) ) ) ).
% times_subset_iff
thf(fact_958_times__subset__iff,axiom,
! [A: set_Pr4934435412358123699_a_nat,C2: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat,D: set_Pr4934435412358123699_a_nat] :
( ( ord_le1018141841545957851_a_nat
@ ( produc3832836940518797764_a_nat @ A
@ ^ [Uu: product_prod_a_nat] : C2 )
@ ( produc3832836940518797764_a_nat @ B
@ ^ [Uu: product_prod_a_nat] : D ) )
= ( ( A = bot_bo9049108969261143879_a_nat )
| ( C2 = bot_bo9049108969261143879_a_nat )
| ( ( ord_le8666007276011122963_a_nat @ A @ B )
& ( ord_le8666007276011122963_a_nat @ C2 @ D ) ) ) ) ).
% times_subset_iff
thf(fact_959_times__subset__iff,axiom,
! [A: set_a,C2: set_nat,B: set_a,D: set_nat] :
( ( ord_le8666007276011122963_a_nat
@ ( product_Sigma_a_nat @ A
@ ^ [Uu: a] : C2 )
@ ( product_Sigma_a_nat @ B
@ ^ [Uu: a] : D ) )
= ( ( A = bot_bot_set_a )
| ( C2 = bot_bot_set_nat )
| ( ( ord_less_eq_set_a @ A @ B )
& ( ord_less_eq_set_nat @ C2 @ D ) ) ) ) ).
% times_subset_iff
thf(fact_960_subset__Collect__iff,axiom,
! [B: set_term_a_b,A: set_term_a_b,P: term_a_b > $o] :
( ( ord_le2705286416250468010rm_a_b @ B @ A )
=> ( ( ord_le2705286416250468010rm_a_b @ B
@ ( collect_term_a_b
@ ^ [X2: term_a_b] :
( ( member_term_a_b @ X2 @ A )
& ( P @ X2 ) ) ) )
= ( ! [X2: term_a_b] :
( ( member_term_a_b @ X2 @ B )
=> ( P @ X2 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_961_subset__Collect__iff,axiom,
! [B: set_Pr4934435412358123699_a_nat,A: set_Pr4934435412358123699_a_nat,P: product_prod_a_nat > $o] :
( ( ord_le8666007276011122963_a_nat @ B @ A )
=> ( ( ord_le8666007276011122963_a_nat @ B
@ ( collec4464134535221767506_a_nat
@ ^ [X2: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ X2 @ A )
& ( P @ X2 ) ) ) )
= ( ! [X2: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ X2 @ B )
=> ( P @ X2 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_962_subset__CollectI,axiom,
! [B: set_term_a_b,A: set_term_a_b,Q: term_a_b > $o,P: term_a_b > $o] :
( ( ord_le2705286416250468010rm_a_b @ B @ A )
=> ( ! [X3: term_a_b] :
( ( member_term_a_b @ X3 @ B )
=> ( ( Q @ X3 )
=> ( P @ X3 ) ) )
=> ( ord_le2705286416250468010rm_a_b
@ ( collect_term_a_b
@ ^ [X2: term_a_b] :
( ( member_term_a_b @ X2 @ B )
& ( Q @ X2 ) ) )
@ ( collect_term_a_b
@ ^ [X2: term_a_b] :
( ( member_term_a_b @ X2 @ A )
& ( P @ X2 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_963_subset__CollectI,axiom,
! [B: set_Pr4934435412358123699_a_nat,A: set_Pr4934435412358123699_a_nat,Q: product_prod_a_nat > $o,P: product_prod_a_nat > $o] :
( ( ord_le8666007276011122963_a_nat @ B @ A )
=> ( ! [X3: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ X3 @ B )
=> ( ( Q @ X3 )
=> ( P @ X3 ) ) )
=> ( ord_le8666007276011122963_a_nat
@ ( collec4464134535221767506_a_nat
@ ^ [X2: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ X2 @ B )
& ( Q @ X2 ) ) )
@ ( collec4464134535221767506_a_nat
@ ^ [X2: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ X2 @ A )
& ( P @ X2 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_964_remove__const__lv__mondaic__step__rhs,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,C: a,F2: set_Pr4934435412358123699_a_nat,S: term_a_b,T2: term_a_b] :
( ( rewrit2869827417384121323_a_b_a @ R2 )
=> ( ~ ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ C @ zero_zero_nat ) @ ( terms_7988297476397195622_a_b_b @ R2 ) )
=> ( ( rewrit3916899077429870363adic_a @ F2 )
=> ( ( member5869715511025134514rm_a_b
@ ( produc7020197800436672577rm_a_b @ S
@ ( subst_7999470309526761004_a_b_b @ T2
@ ^ [X2: b] : ( fun_a_b @ C @ nil_term_a_b ) ) )
@ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T2 ) @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) ) ) ) ) ) ) ).
% remove_const_lv_mondaic_step_rhs
thf(fact_965_remove__const__lv__mondaic__step__lhs,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,C: a,F2: set_Pr4934435412358123699_a_nat,S: term_a_b,T2: term_a_b] :
( ( rewrit2869827417384121323_a_b_a @ R2 )
=> ( ~ ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ C @ zero_zero_nat ) @ ( terms_7988297476397195622_a_b_b @ R2 ) )
=> ( ( rewrit3916899077429870363adic_a @ F2 )
=> ( ( member5869715511025134514rm_a_b
@ ( produc7020197800436672577rm_a_b
@ ( subst_7999470309526761004_a_b_b @ S
@ ^ [X2: b] : ( fun_a_b @ C @ nil_term_a_b ) )
@ T2 )
@ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T2 ) @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) ) ) ) ) ) ) ).
% remove_const_lv_mondaic_step_lhs
thf(fact_966_remove__const__lv__mondaic__steps__rhs,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,C: a,F2: set_Pr4934435412358123699_a_nat,S: term_a_b,T2: term_a_b] :
( ( rewrit2869827417384121323_a_b_a @ R2 )
=> ( ~ ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ C @ zero_zero_nat ) @ ( terms_7988297476397195622_a_b_b @ R2 ) )
=> ( ( rewrit3916899077429870363adic_a @ F2 )
=> ( ( member5869715511025134514rm_a_b
@ ( produc7020197800436672577rm_a_b @ S
@ ( subst_7999470309526761004_a_b_b @ T2
@ ^ [X2: b] : ( fun_a_b @ C @ nil_term_a_b ) ) )
@ ( transi7922773638565587891rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T2 ) @ ( transi7922773638565587891rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) ) ) ) ) ) ) ) ).
% remove_const_lv_mondaic_steps_rhs
thf(fact_967_remove__const__lv__mondaic__steps__lhs,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,C: a,F2: set_Pr4934435412358123699_a_nat,S: term_a_b,T2: term_a_b] :
( ( rewrit2869827417384121323_a_b_a @ R2 )
=> ( ~ ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ C @ zero_zero_nat ) @ ( terms_7988297476397195622_a_b_b @ R2 ) )
=> ( ( rewrit3916899077429870363adic_a @ F2 )
=> ( ( member5869715511025134514rm_a_b
@ ( produc7020197800436672577rm_a_b
@ ( subst_7999470309526761004_a_b_b @ S
@ ^ [X2: b] : ( fun_a_b @ C @ nil_term_a_b ) )
@ T2 )
@ ( transi7922773638565587891rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T2 ) @ ( transi7922773638565587891rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) ) ) ) ) ) ) ) ).
% remove_const_lv_mondaic_steps_lhs
thf(fact_968_remove__const__lv__mondaic__steps,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,C: a,F2: set_Pr4934435412358123699_a_nat,S: term_a_b,T2: term_a_b] :
( ( rewrit2869827417384121323_a_b_a @ R2 )
=> ( ~ ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ C @ zero_zero_nat ) @ ( terms_7988297476397195622_a_b_b @ R2 ) )
=> ( ( rewrit3916899077429870363adic_a @ F2 )
=> ( ( member5869715511025134514rm_a_b
@ ( produc7020197800436672577rm_a_b
@ ( subst_7999470309526761004_a_b_b @ S
@ ^ [X2: b] : ( fun_a_b @ C @ nil_term_a_b ) )
@ ( subst_7999470309526761004_a_b_b @ T2
@ ^ [X2: b] : ( fun_a_b @ C @ nil_term_a_b ) ) )
@ ( transi7922773638565587891rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T2 ) @ ( transi7922773638565587891rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) ) ) ) ) ) ) ) ).
% remove_const_lv_mondaic_steps
thf(fact_969_insert__partition,axiom,
! [X: set_Pr4386577575007340137rm_a_b,F4: set_se1467346398288745161rm_a_b] :
( ~ ( member8823642877475723282rm_a_b @ X @ F4 )
=> ( ! [X3: set_Pr4386577575007340137rm_a_b] :
( ( member8823642877475723282rm_a_b @ X3 @ ( insert4425293090001427513rm_a_b @ X @ F4 ) )
=> ! [Xa2: set_Pr4386577575007340137rm_a_b] :
( ( member8823642877475723282rm_a_b @ Xa2 @ ( insert4425293090001427513rm_a_b @ X @ F4 ) )
=> ( ( X3 != Xa2 )
=> ( ( inf_in1693821330460682071rm_a_b @ X3 @ Xa2 )
= bot_bo197521221353338581rm_a_b ) ) ) )
=> ( ( inf_in1693821330460682071rm_a_b @ X @ ( comple4970752389892389974rm_a_b @ F4 ) )
= bot_bo197521221353338581rm_a_b ) ) ) ).
% insert_partition
thf(fact_970_insert__partition,axiom,
! [X: set_nat,F4: set_set_nat] :
( ~ ( member_set_nat @ X @ F4 )
=> ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ ( insert_set_nat @ X @ F4 ) )
=> ! [Xa2: set_nat] :
( ( member_set_nat @ Xa2 @ ( insert_set_nat @ X @ F4 ) )
=> ( ( X3 != Xa2 )
=> ( ( inf_inf_set_nat @ X3 @ Xa2 )
= bot_bot_set_nat ) ) ) )
=> ( ( inf_inf_set_nat @ X @ ( comple7399068483239264473et_nat @ F4 ) )
= bot_bot_set_nat ) ) ) ).
% insert_partition
thf(fact_971_bot__empty__eq2,axiom,
( bot_bot_a_nat_o
= ( ^ [X2: a,Y3: nat] : ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ X2 @ Y3 ) @ bot_bo9049108969261143879_a_nat ) ) ) ).
% bot_empty_eq2
thf(fact_972_bot__empty__eq,axiom,
( bot_bo1388487281569869310_nat_o
= ( ^ [X2: product_prod_a_nat] : ( member5724188588386418708_a_nat @ X2 @ bot_bo9049108969261143879_a_nat ) ) ) ).
% bot_empty_eq
thf(fact_973_bot__empty__eq,axiom,
( bot_bot_term_a_b_o
= ( ^ [X2: term_a_b] : ( member_term_a_b @ X2 @ bot_bot_set_term_a_b ) ) ) ).
% bot_empty_eq
thf(fact_974_bot__empty__eq,axiom,
( bot_bot_nat_o
= ( ^ [X2: nat] : ( member_nat @ X2 @ bot_bot_set_nat ) ) ) ).
% bot_empty_eq
thf(fact_975_bot__set__def,axiom,
( bot_bot_set_term_a_b
= ( collect_term_a_b @ bot_bot_term_a_b_o ) ) ).
% bot_set_def
thf(fact_976_bot__set__def,axiom,
( bot_bot_set_nat
= ( collect_nat @ bot_bot_nat_o ) ) ).
% bot_set_def
thf(fact_977_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
% bot_nat_def
thf(fact_978_Sup__inf__eq__bot__iff,axiom,
! [B: set_se1467346398288745161rm_a_b,A2: set_Pr4386577575007340137rm_a_b] :
( ( ( inf_in1693821330460682071rm_a_b @ ( comple4970752389892389974rm_a_b @ B ) @ A2 )
= bot_bo197521221353338581rm_a_b )
= ( ! [X2: set_Pr4386577575007340137rm_a_b] :
( ( member8823642877475723282rm_a_b @ X2 @ B )
=> ( ( inf_in1693821330460682071rm_a_b @ X2 @ A2 )
= bot_bo197521221353338581rm_a_b ) ) ) ) ).
% Sup_inf_eq_bot_iff
thf(fact_979_Sup__inf__eq__bot__iff,axiom,
! [B: set_set_nat,A2: set_nat] :
( ( ( inf_inf_set_nat @ ( comple7399068483239264473et_nat @ B ) @ A2 )
= bot_bot_set_nat )
= ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ B )
=> ( ( inf_inf_set_nat @ X2 @ A2 )
= bot_bot_set_nat ) ) ) ) ).
% Sup_inf_eq_bot_iff
thf(fact_980_the__elem__eq,axiom,
! [X: product_prod_a_nat] :
( ( the_el7394697526727337544_a_nat @ ( insert8054603423593749677_a_nat @ X @ bot_bo9049108969261143879_a_nat ) )
= X ) ).
% the_elem_eq
thf(fact_981_the__elem__eq,axiom,
! [X: nat] :
( ( the_elem_nat @ ( insert_nat @ X @ bot_bot_set_nat ) )
= X ) ).
% the_elem_eq
thf(fact_982_is__singleton__the__elem,axiom,
( is_sin7638514652509285001_a_nat
= ( ^ [A5: set_Pr4934435412358123699_a_nat] :
( A5
= ( insert8054603423593749677_a_nat @ ( the_el7394697526727337544_a_nat @ A5 ) @ bot_bo9049108969261143879_a_nat ) ) ) ) ).
% is_singleton_the_elem
thf(fact_983_is__singleton__the__elem,axiom,
( is_singleton_nat
= ( ^ [A5: set_nat] :
( A5
= ( insert_nat @ ( the_elem_nat @ A5 ) @ bot_bot_set_nat ) ) ) ) ).
% is_singleton_the_elem
thf(fact_984_some__NF__WCR,axiom,
! [R: set_Pr1811044260758604347_a_nat,X: product_prod_a_nat,Y: product_prod_a_nat] :
( ( abstra8653715922312955827_a_nat @ R @ ( insert8054603423593749677_a_nat @ X @ bot_bo9049108969261143879_a_nat ) )
=> ( ( abstra6648775111202918568_a_nat @ R
@ ( collec4464134535221767506_a_nat
@ ^ [X2: product_prod_a_nat] : ( abstra8653715922312955827_a_nat @ R @ ( insert8054603423593749677_a_nat @ X2 @ bot_bo9049108969261143879_a_nat ) ) ) )
=> ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ X @ Y ) @ ( transi2726145917338391738_a_nat @ R ) )
=> ( ( member5724188588386418708_a_nat @ Y @ ( abstra2685462378305987313_a_nat @ R ) )
=> ( Y
= ( abstra5913684503434863653_a_nat @ R @ X ) ) ) ) ) ) ).
% some_NF_WCR
thf(fact_985_some__NF__WCR,axiom,
! [R: set_Pr4386577575007340137rm_a_b,X: term_a_b,Y: term_a_b] :
( ( abstra4720023341729745482rm_a_b @ R @ ( insert_term_a_b @ X @ bot_bot_set_term_a_b ) )
=> ( ( abstra5026849107050860991rm_a_b @ R
@ ( collect_term_a_b
@ ^ [X2: term_a_b] : ( abstra4720023341729745482rm_a_b @ R @ ( insert_term_a_b @ X2 @ bot_bot_set_term_a_b ) ) ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ( ( member_term_a_b @ Y @ ( abstract_NF_term_a_b @ R ) )
=> ( Y
= ( abstra2589370493620291516rm_a_b @ R @ X ) ) ) ) ) ) ).
% some_NF_WCR
thf(fact_986_some__NF__WCR,axiom,
! [R: set_Pr1261947904930325089at_nat,X: nat,Y: nat] :
( ( abstract_SN_on_nat @ R @ ( insert_nat @ X @ bot_bot_set_nat ) )
=> ( ( abstract_WCR_on_nat @ R
@ ( collect_nat
@ ^ [X2: nat] : ( abstract_SN_on_nat @ R @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( transi2905341329935302413cl_nat @ R ) )
=> ( ( member_nat @ Y @ ( abstract_NF_nat @ R ) )
=> ( Y
= ( abstract_some_NF_nat @ R @ X ) ) ) ) ) ) ).
% some_NF_WCR
thf(fact_987_is__singletonI,axiom,
! [X: product_prod_a_nat] : ( is_sin7638514652509285001_a_nat @ ( insert8054603423593749677_a_nat @ X @ bot_bo9049108969261143879_a_nat ) ) ).
% is_singletonI
thf(fact_988_is__singletonI,axiom,
! [X: nat] : ( is_singleton_nat @ ( insert_nat @ X @ bot_bot_set_nat ) ) ).
% is_singletonI
thf(fact_989_SN__empty,axiom,
abstra4720023341729745482rm_a_b @ bot_bo197521221353338581rm_a_b @ top_top_set_term_a_b ).
% SN_empty
thf(fact_990_SN__empty,axiom,
abstra8653715922312955827_a_nat @ bot_bo4243320254410501799_a_nat @ top_to3353692345378799459_a_nat ).
% SN_empty
thf(fact_991_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_992_SN__on__weakening,axiom,
! [R1: set_Pr4386577575007340137rm_a_b,A: set_term_a_b,R22: set_Pr4386577575007340137rm_a_b] :
( ( abstra4720023341729745482rm_a_b @ R1 @ A )
=> ( abstra4720023341729745482rm_a_b @ ( inf_in1693821330460682071rm_a_b @ R1 @ R22 ) @ A ) ) ).
% SN_on_weakening
thf(fact_993_SN__on__induct,axiom,
! [R3: set_Pr1811044260758604347_a_nat,A: set_Pr4934435412358123699_a_nat,S: product_prod_a_nat,P: product_prod_a_nat > $o] :
( ( abstra8653715922312955827_a_nat @ R3 @ A )
=> ( ( member5724188588386418708_a_nat @ S @ A )
=> ( ! [T4: product_prod_a_nat] :
( ! [U3: product_prod_a_nat] :
( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ T4 @ U3 ) @ R3 )
=> ( P @ U3 ) )
=> ( P @ T4 ) )
=> ( P @ S ) ) ) ) ).
% SN_on_induct
thf(fact_994_SN__on__induct,axiom,
! [R3: set_Pr4386577575007340137rm_a_b,A: set_term_a_b,S: term_a_b,P: term_a_b > $o] :
( ( abstra4720023341729745482rm_a_b @ R3 @ A )
=> ( ( member_term_a_b @ S @ A )
=> ( ! [T4: term_a_b] :
( ! [U3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T4 @ U3 ) @ R3 )
=> ( P @ U3 ) )
=> ( P @ T4 ) )
=> ( P @ S ) ) ) ) ).
% SN_on_induct
thf(fact_995_SN__trancl__SN__conv,axiom,
! [R3: set_Pr4386577575007340137rm_a_b] :
( ( abstra4720023341729745482rm_a_b @ ( transi7922773638565587891rm_a_b @ R3 ) @ top_top_set_term_a_b )
= ( abstra4720023341729745482rm_a_b @ R3 @ top_top_set_term_a_b ) ) ).
% SN_trancl_SN_conv
thf(fact_996_SN__trancl__SN__conv,axiom,
! [R3: set_Pr1811044260758604347_a_nat] :
( ( abstra8653715922312955827_a_nat @ ( transi6805586507763294364_a_nat @ R3 ) @ top_to3353692345378799459_a_nat )
= ( abstra8653715922312955827_a_nat @ R3 @ top_to3353692345378799459_a_nat ) ) ).
% SN_trancl_SN_conv
thf(fact_997_SN__trancl__imp__SN,axiom,
! [R3: set_Pr4386577575007340137rm_a_b] :
( ( abstra4720023341729745482rm_a_b @ ( transi7922773638565587891rm_a_b @ R3 ) @ top_top_set_term_a_b )
=> ( abstra4720023341729745482rm_a_b @ R3 @ top_top_set_term_a_b ) ) ).
% SN_trancl_imp_SN
thf(fact_998_SN__trancl__imp__SN,axiom,
! [R3: set_Pr1811044260758604347_a_nat] :
( ( abstra8653715922312955827_a_nat @ ( transi6805586507763294364_a_nat @ R3 ) @ top_to3353692345378799459_a_nat )
=> ( abstra8653715922312955827_a_nat @ R3 @ top_to3353692345378799459_a_nat ) ) ).
% SN_trancl_imp_SN
thf(fact_999_SN__imp__SN__trancl,axiom,
! [R3: set_Pr4386577575007340137rm_a_b] :
( ( abstra4720023341729745482rm_a_b @ R3 @ top_top_set_term_a_b )
=> ( abstra4720023341729745482rm_a_b @ ( transi7922773638565587891rm_a_b @ R3 ) @ top_top_set_term_a_b ) ) ).
% SN_imp_SN_trancl
thf(fact_1000_SN__imp__SN__trancl,axiom,
! [R3: set_Pr1811044260758604347_a_nat] :
( ( abstra8653715922312955827_a_nat @ R3 @ top_to3353692345378799459_a_nat )
=> ( abstra8653715922312955827_a_nat @ ( transi6805586507763294364_a_nat @ R3 ) @ top_to3353692345378799459_a_nat ) ) ).
% SN_imp_SN_trancl
thf(fact_1001_SN__O__comm,axiom,
! [R3: set_Pr4386577575007340137rm_a_b,S3: set_Pr4386577575007340137rm_a_b] :
( ( abstra4720023341729745482rm_a_b @ ( relcom370159955682700863rm_a_b @ R3 @ S3 ) @ top_top_set_term_a_b )
= ( abstra4720023341729745482rm_a_b @ ( relcom370159955682700863rm_a_b @ S3 @ R3 ) @ top_top_set_term_a_b ) ) ).
% SN_O_comm
thf(fact_1002_SN__O__comm,axiom,
! [R3: set_Pr677149416131461714_a_nat,S3: set_Pr7191425930741896914rm_a_b] :
( ( abstra4720023341729745482rm_a_b @ ( relcom8386039462915235624rm_a_b @ R3 @ S3 ) @ top_top_set_term_a_b )
= ( abstra8653715922312955827_a_nat @ ( relcom1082227207227705233_a_nat @ S3 @ R3 ) @ top_to3353692345378799459_a_nat ) ) ).
% SN_O_comm
thf(fact_1003_SN__O__comm,axiom,
! [R3: set_Pr7191425930741896914rm_a_b,S3: set_Pr677149416131461714_a_nat] :
( ( abstra8653715922312955827_a_nat @ ( relcom1082227207227705233_a_nat @ R3 @ S3 ) @ top_to3353692345378799459_a_nat )
= ( abstra4720023341729745482rm_a_b @ ( relcom8386039462915235624rm_a_b @ S3 @ R3 ) @ top_top_set_term_a_b ) ) ).
% SN_O_comm
thf(fact_1004_SN__O__comm,axiom,
! [R3: set_Pr1811044260758604347_a_nat,S3: set_Pr1811044260758604347_a_nat] :
( ( abstra8653715922312955827_a_nat @ ( relcom3132138491631691642_a_nat @ R3 @ S3 ) @ top_to3353692345378799459_a_nat )
= ( abstra8653715922312955827_a_nat @ ( relcom3132138491631691642_a_nat @ S3 @ R3 ) @ top_to3353692345378799459_a_nat ) ) ).
% SN_O_comm
thf(fact_1005_SN__subset,axiom,
! [R3: set_Pr4386577575007340137rm_a_b,R4: set_Pr4386577575007340137rm_a_b] :
( ( abstra4720023341729745482rm_a_b @ R3 @ top_top_set_term_a_b )
=> ( ( ord_le118470702582115849rm_a_b @ R4 @ R3 )
=> ( abstra4720023341729745482rm_a_b @ R4 @ top_top_set_term_a_b ) ) ) ).
% SN_subset
thf(fact_1006_SN__subset,axiom,
! [R3: set_Pr1811044260758604347_a_nat,R4: set_Pr1811044260758604347_a_nat] :
( ( abstra8653715922312955827_a_nat @ R3 @ top_to3353692345378799459_a_nat )
=> ( ( ord_le1018141841545957851_a_nat @ R4 @ R3 )
=> ( abstra8653715922312955827_a_nat @ R4 @ top_to3353692345378799459_a_nat ) ) ) ).
% SN_subset
thf(fact_1007_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 )
=> ( ! [A3: term_a_b] :
( ! [B9: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B9 ) @ R )
=> ( P @ B9 ) )
=> ( P @ A3 ) )
=> ( P @ A2 ) ) ) ).
% SN_induct_rule
thf(fact_1008_SN__induct__rule,axiom,
! [R: set_Pr1811044260758604347_a_nat,P: product_prod_a_nat > $o,A2: product_prod_a_nat] :
( ( abstra8653715922312955827_a_nat @ R @ top_to3353692345378799459_a_nat )
=> ( ! [A3: product_prod_a_nat] :
( ! [B9: product_prod_a_nat] :
( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A3 @ B9 ) @ R )
=> ( P @ B9 ) )
=> ( P @ A3 ) )
=> ( P @ A2 ) ) ) ).
% SN_induct_rule
thf(fact_1009_SN__imp__minimal,axiom,
! [A: set_Pr4386577575007340137rm_a_b] :
( ( abstra4720023341729745482rm_a_b @ A @ top_top_set_term_a_b )
=> ! [Q2: set_term_a_b] :
( ? [X4: term_a_b] : ( member_term_a_b @ X4 @ Q2 )
=> ? [X3: term_a_b] :
( ( member_term_a_b @ X3 @ Q2 )
& ! [Y5: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y5 ) @ A )
=> ~ ( member_term_a_b @ Y5 @ Q2 ) ) ) ) ) ).
% SN_imp_minimal
thf(fact_1010_SN__imp__minimal,axiom,
! [A: set_Pr1811044260758604347_a_nat] :
( ( abstra8653715922312955827_a_nat @ A @ top_to3353692345378799459_a_nat )
=> ! [Q2: set_Pr4934435412358123699_a_nat] :
( ? [X4: product_prod_a_nat] : ( member5724188588386418708_a_nat @ X4 @ Q2 )
=> ? [X3: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ X3 @ Q2 )
& ! [Y5: product_prod_a_nat] :
( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ X3 @ Y5 ) @ A )
=> ~ ( member5724188588386418708_a_nat @ Y5 @ Q2 ) ) ) ) ) ).
% SN_imp_minimal
thf(fact_1011_refl__not__SN,axiom,
! [X: term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ X ) @ R3 )
=> ~ ( abstra4720023341729745482rm_a_b @ R3 @ top_top_set_term_a_b ) ) ).
% refl_not_SN
thf(fact_1012_refl__not__SN,axiom,
! [X: product_prod_a_nat,R3: set_Pr1811044260758604347_a_nat] :
( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ X @ X ) @ R3 )
=> ~ ( abstra8653715922312955827_a_nat @ R3 @ top_to3353692345378799459_a_nat ) ) ).
% refl_not_SN
thf(fact_1013_SN__imp__WN,axiom,
! [A: set_Pr4386577575007340137rm_a_b] :
( ( abstra4720023341729745482rm_a_b @ A @ top_top_set_term_a_b )
=> ( abstra4644735066934440262rm_a_b @ A @ top_top_set_term_a_b ) ) ).
% SN_imp_WN
thf(fact_1014_SN__imp__WN,axiom,
! [A: set_Pr1811044260758604347_a_nat] :
( ( abstra8653715922312955827_a_nat @ A @ top_to3353692345378799459_a_nat )
=> ( abstra4582696096554303663_a_nat @ A @ top_to3353692345378799459_a_nat ) ) ).
% SN_imp_WN
thf(fact_1015_is__singletonI_H,axiom,
! [A: set_Pr4934435412358123699_a_nat] :
( ( A != bot_bo9049108969261143879_a_nat )
=> ( ! [X3: product_prod_a_nat,Y4: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ X3 @ A )
=> ( ( member5724188588386418708_a_nat @ Y4 @ A )
=> ( X3 = Y4 ) ) )
=> ( is_sin7638514652509285001_a_nat @ A ) ) ) ).
% is_singletonI'
thf(fact_1016_is__singletonI_H,axiom,
! [A: set_term_a_b] :
( ( A != bot_bot_set_term_a_b )
=> ( ! [X3: term_a_b,Y4: term_a_b] :
( ( member_term_a_b @ X3 @ A )
=> ( ( member_term_a_b @ Y4 @ A )
=> ( X3 = Y4 ) ) )
=> ( is_sin5099058466283458592rm_a_b @ A ) ) ) ).
% is_singletonI'
thf(fact_1017_is__singletonI_H,axiom,
! [A: set_nat] :
( ( A != bot_bot_set_nat )
=> ( ! [X3: nat,Y4: nat] :
( ( member_nat @ X3 @ A )
=> ( ( member_nat @ Y4 @ A )
=> ( X3 = Y4 ) ) )
=> ( is_singleton_nat @ A ) ) ) ).
% is_singletonI'
thf(fact_1018_SN__on__induct_H,axiom,
! [R3: set_Pr1811044260758604347_a_nat,S: product_prod_a_nat,P: product_prod_a_nat > $o] :
( ( abstra8653715922312955827_a_nat @ R3 @ ( insert8054603423593749677_a_nat @ S @ bot_bo9049108969261143879_a_nat ) )
=> ( ! [T4: product_prod_a_nat] :
( ! [U3: product_prod_a_nat] :
( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ T4 @ U3 ) @ R3 )
=> ( P @ U3 ) )
=> ( P @ T4 ) )
=> ( P @ S ) ) ) ).
% SN_on_induct'
thf(fact_1019_SN__on__induct_H,axiom,
! [R3: set_Pr1261947904930325089at_nat,S: nat,P: nat > $o] :
( ( abstract_SN_on_nat @ R3 @ ( insert_nat @ S @ bot_bot_set_nat ) )
=> ( ! [T4: nat] :
( ! [U3: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ T4 @ U3 ) @ R3 )
=> ( P @ U3 ) )
=> ( P @ T4 ) )
=> ( P @ S ) ) ) ).
% SN_on_induct'
thf(fact_1020_step__reflects__SN__on,axiom,
! [A2: product_prod_a_nat,R: set_Pr1811044260758604347_a_nat] :
( ! [B3: product_prod_a_nat] :
( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A2 @ B3 ) @ R )
=> ( abstra8653715922312955827_a_nat @ R @ ( insert8054603423593749677_a_nat @ B3 @ bot_bo9049108969261143879_a_nat ) ) )
=> ( abstra8653715922312955827_a_nat @ R @ ( insert8054603423593749677_a_nat @ A2 @ bot_bo9049108969261143879_a_nat ) ) ) ).
% step_reflects_SN_on
thf(fact_1021_step__reflects__SN__on,axiom,
! [A2: nat,R: set_Pr1261947904930325089at_nat] :
( ! [B3: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B3 ) @ R )
=> ( abstract_SN_on_nat @ R @ ( insert_nat @ B3 @ bot_bot_set_nat ) ) )
=> ( abstract_SN_on_nat @ R @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) ) ).
% step_reflects_SN_on
thf(fact_1022_SN__on__imp__on__minimal,axiom,
! [R: set_Pr1811044260758604347_a_nat,X: product_prod_a_nat] :
( ( abstra8653715922312955827_a_nat @ R @ ( insert8054603423593749677_a_nat @ X @ bot_bo9049108969261143879_a_nat ) )
=> ! [Q2: set_Pr4934435412358123699_a_nat] :
( ( member5724188588386418708_a_nat @ X @ Q2 )
=> ? [X3: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ X3 @ Q2 )
& ! [Y5: product_prod_a_nat] :
( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ X3 @ Y5 ) @ R )
=> ~ ( member5724188588386418708_a_nat @ Y5 @ Q2 ) ) ) ) ) ).
% SN_on_imp_on_minimal
thf(fact_1023_SN__on__imp__on__minimal,axiom,
! [R: set_Pr4386577575007340137rm_a_b,X: term_a_b] :
( ( abstra4720023341729745482rm_a_b @ R @ ( insert_term_a_b @ X @ bot_bot_set_term_a_b ) )
=> ! [Q2: set_term_a_b] :
( ( member_term_a_b @ X @ Q2 )
=> ? [X3: term_a_b] :
( ( member_term_a_b @ X3 @ Q2 )
& ! [Y5: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y5 ) @ R )
=> ~ ( member_term_a_b @ Y5 @ Q2 ) ) ) ) ) ).
% SN_on_imp_on_minimal
thf(fact_1024_SN__on__imp__on__minimal,axiom,
! [R: set_Pr1261947904930325089at_nat,X: nat] :
( ( abstract_SN_on_nat @ R @ ( insert_nat @ X @ bot_bot_set_nat ) )
=> ! [Q2: set_nat] :
( ( member_nat @ X @ Q2 )
=> ? [X3: nat] :
( ( member_nat @ X3 @ Q2 )
& ! [Y5: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y5 ) @ R )
=> ~ ( member_nat @ Y5 @ Q2 ) ) ) ) ) ).
% SN_on_imp_on_minimal
thf(fact_1025_step__preserves__SN__on,axiom,
! [A2: product_prod_a_nat,B2: product_prod_a_nat,R: set_Pr1811044260758604347_a_nat] :
( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A2 @ B2 ) @ R )
=> ( ( abstra8653715922312955827_a_nat @ R @ ( insert8054603423593749677_a_nat @ A2 @ bot_bo9049108969261143879_a_nat ) )
=> ( abstra8653715922312955827_a_nat @ R @ ( insert8054603423593749677_a_nat @ B2 @ bot_bo9049108969261143879_a_nat ) ) ) ) ).
% step_preserves_SN_on
thf(fact_1026_step__preserves__SN__on,axiom,
! [A2: nat,B2: nat,R: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B2 ) @ R )
=> ( ( abstract_SN_on_nat @ R @ ( insert_nat @ A2 @ bot_bot_set_nat ) )
=> ( abstract_SN_on_nat @ R @ ( insert_nat @ B2 @ bot_bot_set_nat ) ) ) ) ).
% step_preserves_SN_on
thf(fact_1027_SN__on__induct__acc__style,axiom,
! [R3: set_Pr1811044260758604347_a_nat,A2: product_prod_a_nat,P: product_prod_a_nat > $o] :
( ( abstra8653715922312955827_a_nat @ R3 @ ( insert8054603423593749677_a_nat @ A2 @ bot_bo9049108969261143879_a_nat ) )
=> ( ! [X3: product_prod_a_nat] :
( ( abstra8653715922312955827_a_nat @ R3 @ ( insert8054603423593749677_a_nat @ X3 @ bot_bo9049108969261143879_a_nat ) )
=> ( ! [Y5: product_prod_a_nat] :
( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ X3 @ Y5 ) @ R3 )
=> ( P @ Y5 ) )
=> ( P @ X3 ) ) )
=> ( P @ A2 ) ) ) ).
% SN_on_induct_acc_style
thf(fact_1028_SN__on__induct__acc__style,axiom,
! [R3: set_Pr1261947904930325089at_nat,A2: nat,P: nat > $o] :
( ( abstract_SN_on_nat @ R3 @ ( insert_nat @ A2 @ bot_bot_set_nat ) )
=> ( ! [X3: nat] :
( ( abstract_SN_on_nat @ R3 @ ( insert_nat @ X3 @ bot_bot_set_nat ) )
=> ( ! [Y5: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y5 ) @ R3 )
=> ( P @ Y5 ) )
=> ( P @ X3 ) ) )
=> ( P @ A2 ) ) ) ).
% SN_on_induct_acc_style
thf(fact_1029_lower__set__imp__not__SN__on,axiom,
! [S: product_prod_a_nat,X6: set_Pr4934435412358123699_a_nat,R3: set_Pr1811044260758604347_a_nat] :
( ( member5724188588386418708_a_nat @ S @ X6 )
=> ( ! [X3: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ X3 @ X6 )
=> ? [Xa3: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ Xa3 @ X6 )
& ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ X3 @ Xa3 ) @ R3 ) ) )
=> ~ ( abstra8653715922312955827_a_nat @ R3 @ ( insert8054603423593749677_a_nat @ S @ bot_bo9049108969261143879_a_nat ) ) ) ) ).
% lower_set_imp_not_SN_on
thf(fact_1030_lower__set__imp__not__SN__on,axiom,
! [S: term_a_b,X6: set_term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
( ( member_term_a_b @ S @ X6 )
=> ( ! [X3: term_a_b] :
( ( member_term_a_b @ X3 @ X6 )
=> ? [Xa3: term_a_b] :
( ( member_term_a_b @ Xa3 @ X6 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Xa3 ) @ R3 ) ) )
=> ~ ( abstra4720023341729745482rm_a_b @ R3 @ ( insert_term_a_b @ S @ bot_bot_set_term_a_b ) ) ) ) ).
% lower_set_imp_not_SN_on
thf(fact_1031_lower__set__imp__not__SN__on,axiom,
! [S: nat,X6: set_nat,R3: set_Pr1261947904930325089at_nat] :
( ( member_nat @ S @ X6 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ X6 )
=> ? [Xa3: nat] :
( ( member_nat @ Xa3 @ X6 )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Xa3 ) @ R3 ) ) )
=> ~ ( abstract_SN_on_nat @ R3 @ ( insert_nat @ S @ bot_bot_set_nat ) ) ) ) ).
% lower_set_imp_not_SN_on
thf(fact_1032_SN__on__all__reducts__SN__on__conv,axiom,
! [R: set_Pr1811044260758604347_a_nat,A2: product_prod_a_nat] :
( ( abstra8653715922312955827_a_nat @ R @ ( insert8054603423593749677_a_nat @ A2 @ bot_bo9049108969261143879_a_nat ) )
= ( ! [B4: product_prod_a_nat] :
( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A2 @ B4 ) @ R )
=> ( abstra8653715922312955827_a_nat @ R @ ( insert8054603423593749677_a_nat @ B4 @ bot_bo9049108969261143879_a_nat ) ) ) ) ) ).
% SN_on_all_reducts_SN_on_conv
thf(fact_1033_SN__on__all__reducts__SN__on__conv,axiom,
! [R: set_Pr1261947904930325089at_nat,A2: nat] :
( ( abstract_SN_on_nat @ R @ ( insert_nat @ A2 @ bot_bot_set_nat ) )
= ( ! [B4: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B4 ) @ R )
=> ( abstract_SN_on_nat @ R @ ( insert_nat @ B4 @ bot_bot_set_nat ) ) ) ) ) ).
% SN_on_all_reducts_SN_on_conv
thf(fact_1034_SN__I,axiom,
! [A: set_Pr1261947904930325089at_nat] :
( ! [A3: nat] : ( abstract_SN_on_nat @ A @ ( insert_nat @ A3 @ bot_bot_set_nat ) )
=> ( abstract_SN_on_nat @ A @ top_top_set_nat ) ) ).
% SN_I
thf(fact_1035_SN__I,axiom,
! [A: set_Pr4386577575007340137rm_a_b] :
( ! [A3: term_a_b] : ( abstra4720023341729745482rm_a_b @ A @ ( insert_term_a_b @ A3 @ bot_bot_set_term_a_b ) )
=> ( abstra4720023341729745482rm_a_b @ A @ top_top_set_term_a_b ) ) ).
% SN_I
thf(fact_1036_SN__I,axiom,
! [A: set_Pr1811044260758604347_a_nat] :
( ! [A3: product_prod_a_nat] : ( abstra8653715922312955827_a_nat @ A @ ( insert8054603423593749677_a_nat @ A3 @ bot_bo9049108969261143879_a_nat ) )
=> ( abstra8653715922312955827_a_nat @ A @ top_to3353692345378799459_a_nat ) ) ).
% SN_I
thf(fact_1037_SN__def,axiom,
! [R: set_Pr1261947904930325089at_nat] :
( ( abstract_SN_on_nat @ R @ top_top_set_nat )
= ( ! [X2: nat] : ( abstract_SN_on_nat @ R @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ) ).
% SN_def
thf(fact_1038_SN__def,axiom,
! [R: set_Pr4386577575007340137rm_a_b] :
( ( abstra4720023341729745482rm_a_b @ R @ top_top_set_term_a_b )
= ( ! [X2: term_a_b] : ( abstra4720023341729745482rm_a_b @ R @ ( insert_term_a_b @ X2 @ bot_bot_set_term_a_b ) ) ) ) ).
% SN_def
thf(fact_1039_SN__def,axiom,
! [R: set_Pr1811044260758604347_a_nat] :
( ( abstra8653715922312955827_a_nat @ R @ top_to3353692345378799459_a_nat )
= ( ! [X2: product_prod_a_nat] : ( abstra8653715922312955827_a_nat @ R @ ( insert8054603423593749677_a_nat @ X2 @ bot_bo9049108969261143879_a_nat ) ) ) ) ).
% SN_def
thf(fact_1040_NF__imp__SN__on,axiom,
! [A2: product_prod_a_nat,R3: set_Pr1811044260758604347_a_nat] :
( ( member5724188588386418708_a_nat @ A2 @ ( abstra2685462378305987313_a_nat @ R3 ) )
=> ( abstra8653715922312955827_a_nat @ R3 @ ( insert8054603423593749677_a_nat @ A2 @ bot_bo9049108969261143879_a_nat ) ) ) ).
% NF_imp_SN_on
thf(fact_1041_NF__imp__SN__on,axiom,
! [A2: term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
( ( member_term_a_b @ A2 @ ( abstract_NF_term_a_b @ R3 ) )
=> ( abstra4720023341729745482rm_a_b @ R3 @ ( insert_term_a_b @ A2 @ bot_bot_set_term_a_b ) ) ) ).
% NF_imp_SN_on
thf(fact_1042_NF__imp__SN__on,axiom,
! [A2: nat,R3: set_Pr1261947904930325089at_nat] :
( ( member_nat @ A2 @ ( abstract_NF_nat @ R3 ) )
=> ( abstract_SN_on_nat @ R3 @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) ) ).
% NF_imp_SN_on
thf(fact_1043_SN__on__subset__SN__terms,axiom,
! [R3: set_Pr4386577575007340137rm_a_b,X6: set_term_a_b] :
( ( abstra4720023341729745482rm_a_b @ R3 @ X6 )
=> ( ord_le2705286416250468010rm_a_b @ X6
@ ( collect_term_a_b
@ ^ [X2: term_a_b] : ( abstra4720023341729745482rm_a_b @ R3 @ ( insert_term_a_b @ X2 @ bot_bot_set_term_a_b ) ) ) ) ) ).
% SN_on_subset_SN_terms
thf(fact_1044_SN__on__subset__SN__terms,axiom,
! [R3: set_Pr1261947904930325089at_nat,X6: set_nat] :
( ( abstract_SN_on_nat @ R3 @ X6 )
=> ( ord_less_eq_set_nat @ X6
@ ( collect_nat
@ ^ [X2: nat] : ( abstract_SN_on_nat @ R3 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ) ) ).
% SN_on_subset_SN_terms
thf(fact_1045_SN__on__subset__SN__terms,axiom,
! [R3: set_Pr1811044260758604347_a_nat,X6: set_Pr4934435412358123699_a_nat] :
( ( abstra8653715922312955827_a_nat @ R3 @ X6 )
=> ( ord_le8666007276011122963_a_nat @ X6
@ ( collec4464134535221767506_a_nat
@ ^ [X2: product_prod_a_nat] : ( abstra8653715922312955827_a_nat @ R3 @ ( insert8054603423593749677_a_nat @ X2 @ bot_bo9049108969261143879_a_nat ) ) ) ) ) ).
% SN_on_subset_SN_terms
thf(fact_1046_Newman,axiom,
! [R: set_Pr4386577575007340137rm_a_b] :
( ( abstra4720023341729745482rm_a_b @ R @ top_top_set_term_a_b )
=> ( ( abstra5026849107050860991rm_a_b @ R @ top_top_set_term_a_b )
=> ( abstra8448919418672941150rm_a_b @ R @ top_top_set_term_a_b ) ) ) ).
% Newman
thf(fact_1047_Newman,axiom,
! [R: set_Pr1811044260758604347_a_nat] :
( ( abstra8653715922312955827_a_nat @ R @ top_to3353692345378799459_a_nat )
=> ( ( abstra6648775111202918568_a_nat @ R @ top_to3353692345378799459_a_nat )
=> ( abstra7964079670522473415_a_nat @ R @ top_to3353692345378799459_a_nat ) ) ) ).
% Newman
thf(fact_1048_WCR__SN__on__imp__CR__on,axiom,
! [R: set_Pr4386577575007340137rm_a_b,A: set_term_a_b] :
( ( abstra5026849107050860991rm_a_b @ R @ top_top_set_term_a_b )
=> ( ( abstra4720023341729745482rm_a_b @ R @ A )
=> ( abstra8448919418672941150rm_a_b @ R @ A ) ) ) ).
% WCR_SN_on_imp_CR_on
thf(fact_1049_WCR__SN__on__imp__CR__on,axiom,
! [R: set_Pr1811044260758604347_a_nat,A: set_Pr4934435412358123699_a_nat] :
( ( abstra6648775111202918568_a_nat @ R @ top_to3353692345378799459_a_nat )
=> ( ( abstra8653715922312955827_a_nat @ R @ A )
=> ( abstra7964079670522473415_a_nat @ R @ A ) ) ) ).
% WCR_SN_on_imp_CR_on
thf(fact_1050_quasi__commute__imp__SN,axiom,
! [R: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
( ( abstra4720023341729745482rm_a_b @ R @ top_top_set_term_a_b )
=> ( ( abstra4720023341729745482rm_a_b @ S @ top_top_set_term_a_b )
=> ( ( abstra4317384144805385751rm_a_b @ R @ S )
=> ( abstra4720023341729745482rm_a_b @ ( sup_su6776935440552674877rm_a_b @ R @ S ) @ top_top_set_term_a_b ) ) ) ) ).
% quasi_commute_imp_SN
thf(fact_1051_quasi__commute__imp__SN,axiom,
! [R: set_Pr1811044260758604347_a_nat,S: set_Pr1811044260758604347_a_nat] :
( ( abstra8653715922312955827_a_nat @ R @ top_to3353692345378799459_a_nat )
=> ( ( abstra8653715922312955827_a_nat @ S @ top_to3353692345378799459_a_nat )
=> ( ( abstra4664366908292794624_a_nat @ R @ S )
=> ( abstra8653715922312955827_a_nat @ ( sup_su958340139570419215_a_nat @ R @ S ) @ top_to3353692345378799459_a_nat ) ) ) ) ).
% quasi_commute_imp_SN
thf(fact_1052_steps__preserve__SN__on,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 ) )
=> ( ( abstra8653715922312955827_a_nat @ A @ ( insert8054603423593749677_a_nat @ A2 @ bot_bo9049108969261143879_a_nat ) )
=> ( abstra8653715922312955827_a_nat @ A @ ( insert8054603423593749677_a_nat @ B2 @ bot_bo9049108969261143879_a_nat ) ) ) ) ).
% steps_preserve_SN_on
thf(fact_1053_steps__preserve__SN__on,axiom,
! [A2: nat,B2: nat,A: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B2 ) @ ( transi2905341329935302413cl_nat @ A ) )
=> ( ( abstract_SN_on_nat @ A @ ( insert_nat @ A2 @ bot_bot_set_nat ) )
=> ( abstract_SN_on_nat @ A @ ( insert_nat @ B2 @ bot_bot_set_nat ) ) ) ) ).
% steps_preserve_SN_on
thf(fact_1054_steps__reflect__SN__on,axiom,
! [R: set_Pr1811044260758604347_a_nat,B2: product_prod_a_nat,A2: product_prod_a_nat] :
( ~ ( abstra8653715922312955827_a_nat @ R @ ( insert8054603423593749677_a_nat @ B2 @ bot_bo9049108969261143879_a_nat ) )
=> ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A2 @ B2 ) @ ( transi2726145917338391738_a_nat @ R ) )
=> ~ ( abstra8653715922312955827_a_nat @ R @ ( insert8054603423593749677_a_nat @ A2 @ bot_bo9049108969261143879_a_nat ) ) ) ) ).
% steps_reflect_SN_on
thf(fact_1055_steps__reflect__SN__on,axiom,
! [R: set_Pr1261947904930325089at_nat,B2: nat,A2: nat] :
( ~ ( abstract_SN_on_nat @ R @ ( insert_nat @ B2 @ bot_bot_set_nat ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B2 ) @ ( transi2905341329935302413cl_nat @ R ) )
=> ~ ( abstract_SN_on_nat @ R @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) ) ) ).
% steps_reflect_SN_on
thf(fact_1056_compatible__SN,axiom,
! [S3: set_Pr4386577575007340137rm_a_b,NS: set_Pr4386577575007340137rm_a_b] :
( ( abstra4720023341729745482rm_a_b @ S3 @ top_top_set_term_a_b )
=> ( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ NS @ S3 ) @ S3 )
=> ( abstra4720023341729745482rm_a_b @ ( relcom370159955682700863rm_a_b @ S3 @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ S3 ) @ ( transi7742714808557438673rm_a_b @ NS ) ) ) @ top_top_set_term_a_b ) ) ) ).
% compatible_SN
thf(fact_1057_compatible__SN,axiom,
! [S3: set_Pr1811044260758604347_a_nat,NS: set_Pr1811044260758604347_a_nat] :
( ( abstra8653715922312955827_a_nat @ S3 @ top_to3353692345378799459_a_nat )
=> ( ( ord_le1018141841545957851_a_nat @ ( relcom3132138491631691642_a_nat @ NS @ S3 ) @ S3 )
=> ( abstra8653715922312955827_a_nat @ ( relcom3132138491631691642_a_nat @ S3 @ ( relcom3132138491631691642_a_nat @ ( transi2726145917338391738_a_nat @ S3 ) @ ( transi2726145917338391738_a_nat @ NS ) ) ) @ top_to3353692345378799459_a_nat ) ) ) ).
% compatible_SN
thf(fact_1058_Newman__local,axiom,
! [R: set_Pr1811044260758604347_a_nat,X6: set_Pr4934435412358123699_a_nat] :
( ( abstra8653715922312955827_a_nat @ R @ X6 )
=> ( ( abstra6648775111202918568_a_nat @ R
@ ( collec4464134535221767506_a_nat
@ ^ [X2: product_prod_a_nat] : ( abstra8653715922312955827_a_nat @ R @ ( insert8054603423593749677_a_nat @ X2 @ bot_bo9049108969261143879_a_nat ) ) ) )
=> ( abstra7964079670522473415_a_nat @ R @ X6 ) ) ) ).
% Newman_local
thf(fact_1059_Newman__local,axiom,
! [R: set_Pr4386577575007340137rm_a_b,X6: set_term_a_b] :
( ( abstra4720023341729745482rm_a_b @ R @ X6 )
=> ( ( abstra5026849107050860991rm_a_b @ R
@ ( collect_term_a_b
@ ^ [X2: term_a_b] : ( abstra4720023341729745482rm_a_b @ R @ ( insert_term_a_b @ X2 @ bot_bot_set_term_a_b ) ) ) )
=> ( abstra8448919418672941150rm_a_b @ R @ X6 ) ) ) ).
% Newman_local
thf(fact_1060_Newman__local,axiom,
! [R: set_Pr1261947904930325089at_nat,X6: set_nat] :
( ( abstract_SN_on_nat @ R @ X6 )
=> ( ( abstract_WCR_on_nat @ R
@ ( collect_nat
@ ^ [X2: nat] : ( abstract_SN_on_nat @ R @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) )
=> ( abstract_CR_on_nat @ R @ X6 ) ) ) ).
% Newman_local
thf(fact_1061_SN__reaches__NF,axiom,
! [R: set_Pr1811044260758604347_a_nat,X: product_prod_a_nat] :
( ( abstra8653715922312955827_a_nat @ R @ ( insert8054603423593749677_a_nat @ X @ bot_bo9049108969261143879_a_nat ) )
=> ? [Y4: product_prod_a_nat] :
( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ X @ Y4 ) @ ( transi2726145917338391738_a_nat @ R ) )
& ( member5724188588386418708_a_nat @ Y4 @ ( abstra2685462378305987313_a_nat @ R ) ) ) ) ).
% SN_reaches_NF
thf(fact_1062_SN__reaches__NF,axiom,
! [R: set_Pr4386577575007340137rm_a_b,X: term_a_b] :
( ( abstra4720023341729745482rm_a_b @ R @ ( insert_term_a_b @ X @ bot_bot_set_term_a_b ) )
=> ? [Y4: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y4 ) @ ( transi7742714808557438673rm_a_b @ R ) )
& ( member_term_a_b @ Y4 @ ( abstract_NF_term_a_b @ R ) ) ) ) ).
% SN_reaches_NF
thf(fact_1063_SN__reaches__NF,axiom,
! [R: set_Pr1261947904930325089at_nat,X: nat] :
( ( abstract_SN_on_nat @ R @ ( insert_nat @ X @ bot_bot_set_nat ) )
=> ? [Y4: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y4 ) @ ( transi2905341329935302413cl_nat @ R ) )
& ( member_nat @ Y4 @ ( abstract_NF_nat @ R ) ) ) ) ).
% SN_reaches_NF
thf(fact_1064_is__singletonE,axiom,
! [A: set_Pr4934435412358123699_a_nat] :
( ( is_sin7638514652509285001_a_nat @ A )
=> ~ ! [X3: product_prod_a_nat] :
( A
!= ( insert8054603423593749677_a_nat @ X3 @ bot_bo9049108969261143879_a_nat ) ) ) ).
% is_singletonE
thf(fact_1065_is__singletonE,axiom,
! [A: set_nat] :
( ( is_singleton_nat @ A )
=> ~ ! [X3: nat] :
( A
!= ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ).
% is_singletonE
thf(fact_1066_is__singleton__def,axiom,
( is_sin7638514652509285001_a_nat
= ( ^ [A5: set_Pr4934435412358123699_a_nat] :
? [X2: product_prod_a_nat] :
( A5
= ( insert8054603423593749677_a_nat @ X2 @ bot_bo9049108969261143879_a_nat ) ) ) ) ).
% is_singleton_def
thf(fact_1067_is__singleton__def,axiom,
( is_singleton_nat
= ( ^ [A5: set_nat] :
? [X2: nat] :
( A5
= ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ) ).
% is_singleton_def
thf(fact_1068_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_1069_qc__SN__relto__iff,axiom,
! [R: set_Pr1811044260758604347_a_nat,S: set_Pr1811044260758604347_a_nat] :
( ( ord_le1018141841545957851_a_nat @ ( relcom3132138491631691642_a_nat @ R @ S ) @ ( relcom3132138491631691642_a_nat @ S @ ( transi2726145917338391738_a_nat @ ( sup_su958340139570419215_a_nat @ S @ R ) ) ) )
=> ( ( abstra8653715922312955827_a_nat @ ( relcom3132138491631691642_a_nat @ ( transi2726145917338391738_a_nat @ R ) @ ( relcom3132138491631691642_a_nat @ S @ ( transi2726145917338391738_a_nat @ R ) ) ) @ top_to3353692345378799459_a_nat )
= ( abstra8653715922312955827_a_nat @ S @ top_to3353692345378799459_a_nat ) ) ) ).
% qc_SN_relto_iff
thf(fact_1070_compatible__SN_H,axiom,
! [NS: set_Pr4386577575007340137rm_a_b,S3: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ NS @ S3 ) @ S3 )
=> ( ( abstra4720023341729745482rm_a_b @ S3 @ top_top_set_term_a_b )
=> ( abstra4720023341729745482rm_a_b @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ NS @ S3 ) ) @ ( relcom370159955682700863rm_a_b @ S3 @ ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ NS @ S3 ) ) ) ) @ top_top_set_term_a_b ) ) ) ).
% compatible_SN'
thf(fact_1071_compatible__SN_H,axiom,
! [NS: set_Pr1811044260758604347_a_nat,S3: set_Pr1811044260758604347_a_nat] :
( ( ord_le1018141841545957851_a_nat @ ( relcom3132138491631691642_a_nat @ NS @ S3 ) @ S3 )
=> ( ( abstra8653715922312955827_a_nat @ S3 @ top_to3353692345378799459_a_nat )
=> ( abstra8653715922312955827_a_nat @ ( relcom3132138491631691642_a_nat @ ( transi2726145917338391738_a_nat @ ( sup_su958340139570419215_a_nat @ NS @ S3 ) ) @ ( relcom3132138491631691642_a_nat @ S3 @ ( transi2726145917338391738_a_nat @ ( sup_su958340139570419215_a_nat @ NS @ S3 ) ) ) ) @ top_to3353692345378799459_a_nat ) ) ) ).
% compatible_SN'
thf(fact_1072_some__NF,axiom,
! [R: set_Pr1811044260758604347_a_nat,X: product_prod_a_nat] :
( ( abstra8653715922312955827_a_nat @ R @ ( insert8054603423593749677_a_nat @ X @ bot_bo9049108969261143879_a_nat ) )
=> ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ X @ ( abstra5913684503434863653_a_nat @ R @ X ) ) @ ( transi2726145917338391738_a_nat @ R ) )
& ( member5724188588386418708_a_nat @ ( abstra5913684503434863653_a_nat @ R @ X ) @ ( abstra2685462378305987313_a_nat @ R ) ) ) ) ).
% some_NF
thf(fact_1073_some__NF,axiom,
! [R: set_Pr4386577575007340137rm_a_b,X: term_a_b] :
( ( abstra4720023341729745482rm_a_b @ R @ ( insert_term_a_b @ X @ bot_bot_set_term_a_b ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ ( abstra2589370493620291516rm_a_b @ R @ X ) ) @ ( transi7742714808557438673rm_a_b @ R ) )
& ( member_term_a_b @ ( abstra2589370493620291516rm_a_b @ R @ X ) @ ( abstract_NF_term_a_b @ R ) ) ) ) ).
% some_NF
thf(fact_1074_some__NF,axiom,
! [R: set_Pr1261947904930325089at_nat,X: nat] :
( ( abstract_SN_on_nat @ R @ ( insert_nat @ X @ bot_bot_set_nat ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ ( abstract_some_NF_nat @ R @ X ) ) @ ( transi2905341329935302413cl_nat @ R ) )
& ( member_nat @ ( abstract_some_NF_nat @ R @ X ) @ ( abstract_NF_nat @ R ) ) ) ) ).
% some_NF
thf(fact_1075_SN__WCR__reaches__NF,axiom,
! [R: set_Pr1811044260758604347_a_nat,X: product_prod_a_nat] :
( ( abstra8653715922312955827_a_nat @ R @ ( insert8054603423593749677_a_nat @ X @ bot_bo9049108969261143879_a_nat ) )
=> ( ( abstra6648775111202918568_a_nat @ R
@ ( collec4464134535221767506_a_nat
@ ^ [X2: product_prod_a_nat] : ( abstra8653715922312955827_a_nat @ R @ ( insert8054603423593749677_a_nat @ X2 @ bot_bo9049108969261143879_a_nat ) ) ) )
=> ? [X3: product_prod_a_nat] :
( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ X @ X3 ) @ ( transi2726145917338391738_a_nat @ R ) )
& ( member5724188588386418708_a_nat @ X3 @ ( abstra2685462378305987313_a_nat @ R ) )
& ! [Y5: product_prod_a_nat] :
( ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ X @ Y5 ) @ ( transi2726145917338391738_a_nat @ R ) )
& ( member5724188588386418708_a_nat @ Y5 @ ( abstra2685462378305987313_a_nat @ R ) ) )
=> ( Y5 = X3 ) ) ) ) ) ).
% SN_WCR_reaches_NF
thf(fact_1076_SN__WCR__reaches__NF,axiom,
! [R: set_Pr4386577575007340137rm_a_b,X: term_a_b] :
( ( abstra4720023341729745482rm_a_b @ R @ ( insert_term_a_b @ X @ bot_bot_set_term_a_b ) )
=> ( ( abstra5026849107050860991rm_a_b @ R
@ ( collect_term_a_b
@ ^ [X2: term_a_b] : ( abstra4720023341729745482rm_a_b @ R @ ( insert_term_a_b @ X2 @ bot_bot_set_term_a_b ) ) ) )
=> ? [X3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ X3 ) @ ( transi7742714808557438673rm_a_b @ R ) )
& ( member_term_a_b @ X3 @ ( abstract_NF_term_a_b @ R ) )
& ! [Y5: term_a_b] :
( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y5 ) @ ( transi7742714808557438673rm_a_b @ R ) )
& ( member_term_a_b @ Y5 @ ( abstract_NF_term_a_b @ R ) ) )
=> ( Y5 = X3 ) ) ) ) ) ).
% SN_WCR_reaches_NF
thf(fact_1077_SN__WCR__reaches__NF,axiom,
! [R: set_Pr1261947904930325089at_nat,X: nat] :
( ( abstract_SN_on_nat @ R @ ( insert_nat @ X @ bot_bot_set_nat ) )
=> ( ( abstract_WCR_on_nat @ R
@ ( collect_nat
@ ^ [X2: nat] : ( abstract_SN_on_nat @ R @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) )
=> ? [X3: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ X3 ) @ ( transi2905341329935302413cl_nat @ R ) )
& ( member_nat @ X3 @ ( abstract_NF_nat @ R ) )
& ! [Y5: nat] :
( ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y5 ) @ ( transi2905341329935302413cl_nat @ R ) )
& ( member_nat @ Y5 @ ( abstract_NF_nat @ R ) ) )
=> ( Y5 = X3 ) ) ) ) ) ).
% SN_WCR_reaches_NF
thf(fact_1078_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_1079_the__NF__steps,axiom,
! [A: set_Pr1811044260758604347_a_nat,A2: product_prod_a_nat,B2: product_prod_a_nat] :
( ( abstra8653715922312955827_a_nat @ A @ top_to3353692345378799459_a_nat )
=> ( ( abstra7964079670522473415_a_nat @ A @ top_to3353692345378799459_a_nat )
=> ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A2 @ B2 ) @ ( transi2726145917338391738_a_nat @ A ) )
=> ( ( abstra6900011479857116204_a_nat @ A @ A2 )
= ( abstra6900011479857116204_a_nat @ A @ B2 ) ) ) ) ) ).
% the_NF_steps
thf(fact_1080_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_1081_the__NF__step,axiom,
! [A: set_Pr1811044260758604347_a_nat,A2: product_prod_a_nat,B2: product_prod_a_nat] :
( ( abstra8653715922312955827_a_nat @ A @ top_to3353692345378799459_a_nat )
=> ( ( abstra7964079670522473415_a_nat @ A @ top_to3353692345378799459_a_nat )
=> ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A2 @ B2 ) @ A )
=> ( ( abstra6900011479857116204_a_nat @ A @ A2 )
= ( abstra6900011479857116204_a_nat @ A @ B2 ) ) ) ) ) ).
% the_NF_step
thf(fact_1082_the__NF__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 )
=> ( member_term_a_b @ ( abstra2242782261322123075rm_a_b @ A @ A2 ) @ ( abstract_NF_term_a_b @ A ) ) ) ) ).
% the_NF_NF
thf(fact_1083_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_1084_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_1085_the__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 )
=> ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A2 @ ( abstra6900011479857116204_a_nat @ A @ A2 ) ) @ ( abstra3832961281062565333_a_nat @ A ) ) ) ) ).
% the_NF
thf(fact_1086_UNF__onI,axiom,
! [A: set_Pr4934435412358123699_a_nat,R: set_Pr1811044260758604347_a_nat] :
( ! [A3: product_prod_a_nat,B3: product_prod_a_nat,C5: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ A3 @ A )
=> ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A3 @ B3 ) @ ( abstra3832961281062565333_a_nat @ R ) )
=> ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A3 @ C5 ) @ ( abstra3832961281062565333_a_nat @ R ) )
=> ( B3 = C5 ) ) ) )
=> ( abstra7833083522531381839_a_nat @ R @ A ) ) ).
% UNF_onI
thf(fact_1087_UNF__onI,axiom,
! [A: set_term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ! [A3: term_a_b,B3: term_a_b,C5: term_a_b] :
( ( member_term_a_b @ A3 @ A )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B3 ) @ ( abstra2972651231392952684rm_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ C5 ) @ ( abstra2972651231392952684rm_a_b @ R ) )
=> ( B3 = C5 ) ) ) )
=> ( abstra3619855735679963878rm_a_b @ R @ A ) ) ).
% UNF_onI
thf(fact_1088_WN__onI,axiom,
! [A: set_Pr4934435412358123699_a_nat,R: set_Pr1811044260758604347_a_nat] :
( ! [A3: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ A3 @ A )
=> ? [B9: product_prod_a_nat] : ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A3 @ B9 ) @ ( abstra3832961281062565333_a_nat @ R ) ) )
=> ( abstra4582696096554303663_a_nat @ R @ A ) ) ).
% WN_onI
thf(fact_1089_WN__onI,axiom,
! [A: set_term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ! [A3: term_a_b] :
( ( member_term_a_b @ A3 @ A )
=> ? [B9: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B9 ) @ ( abstra2972651231392952684rm_a_b @ R ) ) )
=> ( abstra4644735066934440262rm_a_b @ R @ A ) ) ).
% WN_onI
thf(fact_1090_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_1091_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_1092_UNF__on__def,axiom,
( abstra3619855735679963878rm_a_b
= ( ^ [R7: set_Pr4386577575007340137rm_a_b,A5: set_term_a_b] :
! [X2: term_a_b] :
( ( member_term_a_b @ X2 @ A5 )
=> ! [B4: term_a_b,C4: term_a_b] :
( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ B4 ) @ ( abstra2972651231392952684rm_a_b @ R7 ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ C4 ) @ ( abstra2972651231392952684rm_a_b @ R7 ) ) )
=> ( B4 = C4 ) ) ) ) ) ).
% UNF_on_def
thf(fact_1093_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_1094_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_1095_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_1096_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_1097_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 )
=> ~ ! [B3: product_prod_a_nat] :
~ ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A2 @ B3 ) @ ( abstra3832961281062565333_a_nat @ R ) ) ) ) ).
% WN_onE
thf(fact_1098_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 )
=> ~ ! [B3: term_a_b] :
~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B3 ) @ ( abstra2972651231392952684rm_a_b @ R ) ) ) ) ).
% WN_onE
thf(fact_1099_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 )
=> ? [B3: product_prod_a_nat] : ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A2 @ B3 ) @ ( abstra3832961281062565333_a_nat @ R ) ) ) ) ).
% WN_onD
thf(fact_1100_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 )
=> ? [B3: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B3 ) @ ( abstra2972651231392952684rm_a_b @ R ) ) ) ) ).
% WN_onD
thf(fact_1101_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_1102_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_1103_SN__on__imp__normalizability,axiom,
! [R: set_Pr1811044260758604347_a_nat,A2: product_prod_a_nat] :
( ( abstra8653715922312955827_a_nat @ R @ ( insert8054603423593749677_a_nat @ A2 @ bot_bo9049108969261143879_a_nat ) )
=> ? [B3: product_prod_a_nat] : ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A2 @ B3 ) @ ( abstra3832961281062565333_a_nat @ R ) ) ) ).
% SN_on_imp_normalizability
thf(fact_1104_SN__on__imp__normalizability,axiom,
! [R: set_Pr1261947904930325089at_nat,A2: nat] :
( ( abstract_SN_on_nat @ R @ ( insert_nat @ A2 @ bot_bot_set_nat ) )
=> ? [B3: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B3 ) @ ( abstra7312431243108365096ty_nat @ R ) ) ) ).
% SN_on_imp_normalizability
thf(fact_1105_min__ext__compat,axiom,
! [R3: set_Pr1261947904930325089at_nat,S3: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ ( relcomp_nat_nat_nat @ R3 @ S3 ) @ R3 )
=> ( ord_le4402255868550298733et_nat @ ( relcom236561585145426837et_nat @ ( min_ext_nat @ R3 ) @ ( sup_su4251893418135750817et_nat @ ( min_ext_nat @ S3 ) @ ( insert3810226134351308605et_nat @ ( produc4532415448927165861et_nat @ bot_bot_set_nat @ bot_bot_set_nat ) @ bot_bo4952753636535259449et_nat ) ) ) @ ( min_ext_nat @ R3 ) ) ) ).
% min_ext_compat
thf(fact_1106_max__ext__compat,axiom,
! [R3: set_Pr1261947904930325089at_nat,S3: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ ( relcomp_nat_nat_nat @ R3 @ S3 ) @ R3 )
=> ( ord_le4402255868550298733et_nat @ ( relcom236561585145426837et_nat @ ( max_ext_nat @ R3 ) @ ( sup_su4251893418135750817et_nat @ ( max_ext_nat @ S3 ) @ ( insert3810226134351308605et_nat @ ( produc4532415448927165861et_nat @ bot_bot_set_nat @ bot_bot_set_nat ) @ bot_bo4952753636535259449et_nat ) ) ) @ ( max_ext_nat @ R3 ) ) ) ).
% max_ext_compat
thf(fact_1107_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_1108_the__NF__conv,axiom,
! [A: set_Pr1811044260758604347_a_nat,A2: product_prod_a_nat,B2: product_prod_a_nat] :
( ( abstra8653715922312955827_a_nat @ A @ top_to3353692345378799459_a_nat )
=> ( ( abstra7964079670522473415_a_nat @ A @ top_to3353692345378799459_a_nat )
=> ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A2 @ B2 ) @ ( abstra3054561309482028431_a_nat @ A ) )
=> ( ( abstra6900011479857116204_a_nat @ A @ A2 )
= ( abstra6900011479857116204_a_nat @ A @ B2 ) ) ) ) ) ).
% the_NF_conv
thf(fact_1109_all__ctxt__closed__srstep__conversions,axiom,
! [F2: set_Pr4934435412358123699_a_nat,R2: set_Pr4386577575007340137rm_a_b] : ( terms_5226143800768910156ed_a_b @ F2 @ ( abstra1130568886744591910rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) ) ) ) ).
% all_ctxt_closed_srstep_conversions
thf(fact_1110_CR__imp__conversionIff__join,axiom,
! [R: set_Pr4386577575007340137rm_a_b] :
( ( abstra8448919418672941150rm_a_b @ R @ top_top_set_term_a_b )
=> ( ( abstra1130568886744591910rm_a_b @ R )
= ( abstra4096080454567261402rm_a_b @ R ) ) ) ).
% CR_imp_conversionIff_join
thf(fact_1111_CR__imp__conversionIff__join,axiom,
! [R: set_Pr1811044260758604347_a_nat] :
( ( abstra7964079670522473415_a_nat @ R @ top_to3353692345378799459_a_nat )
=> ( ( abstra3054561309482028431_a_nat @ R )
= ( abstra881512052393830979_a_nat @ R ) ) ) ).
% CR_imp_conversionIff_join
thf(fact_1112_UNC__def,axiom,
( abstra7179253994116303565_a_nat
= ( ^ [A5: set_Pr1811044260758604347_a_nat] :
! [A4: product_prod_a_nat,B4: product_prod_a_nat] :
( ( ( member5724188588386418708_a_nat @ A4 @ ( abstra2685462378305987313_a_nat @ A5 ) )
& ( member5724188588386418708_a_nat @ B4 @ ( abstra2685462378305987313_a_nat @ A5 ) )
& ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A4 @ B4 ) @ ( abstra3054561309482028431_a_nat @ A5 ) ) )
=> ( A4 = B4 ) ) ) ) ).
% UNC_def
thf(fact_1113_UNC__def,axiom,
( abstra4649771476585718372rm_a_b
= ( ^ [A5: set_Pr4386577575007340137rm_a_b] :
! [A4: term_a_b,B4: term_a_b] :
( ( ( member_term_a_b @ A4 @ ( abstract_NF_term_a_b @ A5 ) )
& ( member_term_a_b @ B4 @ ( abstract_NF_term_a_b @ A5 ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A4 @ B4 ) @ ( abstra1130568886744591910rm_a_b @ A5 ) ) )
=> ( A4 = B4 ) ) ) ) ).
% UNC_def
thf(fact_1114_UNC__I,axiom,
! [A: set_Pr1811044260758604347_a_nat] :
( ! [A3: product_prod_a_nat,B3: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ A3 @ ( abstra2685462378305987313_a_nat @ A ) )
=> ( ( member5724188588386418708_a_nat @ B3 @ ( abstra2685462378305987313_a_nat @ A ) )
=> ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A3 @ B3 ) @ ( abstra3054561309482028431_a_nat @ A ) )
=> ( A3 = B3 ) ) ) )
=> ( abstra7179253994116303565_a_nat @ A ) ) ).
% UNC_I
thf(fact_1115_UNC__I,axiom,
! [A: set_Pr4386577575007340137rm_a_b] :
( ! [A3: term_a_b,B3: term_a_b] :
( ( member_term_a_b @ A3 @ ( abstract_NF_term_a_b @ A ) )
=> ( ( member_term_a_b @ B3 @ ( abstract_NF_term_a_b @ A ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B3 ) @ ( abstra1130568886744591910rm_a_b @ A ) )
=> ( A3 = B3 ) ) ) )
=> ( abstra4649771476585718372rm_a_b @ A ) ) ).
% UNC_I
thf(fact_1116_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_1117_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_1118_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_1119_CR__iff__conversion__imp__join,axiom,
! [R: set_Pr1811044260758604347_a_nat] :
( ( abstra7964079670522473415_a_nat @ R @ top_to3353692345378799459_a_nat )
= ( ord_le1018141841545957851_a_nat @ ( abstra3054561309482028431_a_nat @ R ) @ ( abstra881512052393830979_a_nat @ R ) ) ) ).
% CR_iff_conversion_imp_join
thf(fact_1120_semi__complete__imp__conversionIff__same__NF,axiom,
! [R: set_Pr4386577575007340137rm_a_b,X: term_a_b,Y: term_a_b] :
( ( abstra7162307272211697846rm_a_b @ R @ top_top_set_term_a_b )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( abstra1130568886744591910rm_a_b @ R ) )
= ( ! [U2: term_a_b,V4: term_a_b] :
( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ U2 ) @ ( abstra2972651231392952684rm_a_b @ R ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ V4 ) @ ( abstra2972651231392952684rm_a_b @ R ) ) )
=> ( U2 = V4 ) ) ) ) ) ).
% semi_complete_imp_conversionIff_same_NF
thf(fact_1121_semi__complete__imp__conversionIff__same__NF,axiom,
! [R: set_Pr1811044260758604347_a_nat,X: product_prod_a_nat,Y: product_prod_a_nat] :
( ( abstra4162078339941820447_a_nat @ R @ top_to3353692345378799459_a_nat )
=> ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ X @ Y ) @ ( abstra3054561309482028431_a_nat @ R ) )
= ( ! [U2: product_prod_a_nat,V4: product_prod_a_nat] :
( ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ X @ U2 ) @ ( abstra3832961281062565333_a_nat @ R ) )
& ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ Y @ V4 ) @ ( abstra3832961281062565333_a_nat @ R ) ) )
=> ( U2 = V4 ) ) ) ) ) ).
% semi_complete_imp_conversionIff_same_NF
thf(fact_1122_CR__NF__conv,axiom,
! [R: set_Pr4386577575007340137rm_a_b,T2: term_a_b,U: term_a_b] :
( ( abstra8448919418672941150rm_a_b @ R @ top_top_set_term_a_b )
=> ( ( member_term_a_b @ T2 @ ( abstract_NF_term_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ U @ T2 ) @ ( abstra1130568886744591910rm_a_b @ R ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ U @ T2 ) @ ( abstra2972651231392952684rm_a_b @ R ) ) ) ) ) ).
% CR_NF_conv
thf(fact_1123_CR__NF__conv,axiom,
! [R: set_Pr1811044260758604347_a_nat,T2: product_prod_a_nat,U: product_prod_a_nat] :
( ( abstra7964079670522473415_a_nat @ R @ top_to3353692345378799459_a_nat )
=> ( ( member5724188588386418708_a_nat @ T2 @ ( abstra2685462378305987313_a_nat @ R ) )
=> ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ U @ T2 ) @ ( abstra3054561309482028431_a_nat @ R ) )
=> ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ U @ T2 ) @ ( abstra3832961281062565333_a_nat @ R ) ) ) ) ) ).
% CR_NF_conv
thf(fact_1124_gsrstep__conversion__dist,axiom,
! [F2: set_Pr4934435412358123699_a_nat,R2: set_Pr4386577575007340137rm_a_b] :
( ( abstra1130568886744591910rm_a_b
@ ( inf_in1693821330460682071rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) )
@ ( produc1953217797448512882rm_a_b @ ( collect_term_a_b @ term_ground_a_b )
@ ^ [Uu: term_a_b] : ( collect_term_a_b @ term_ground_a_b ) ) ) )
= ( transi7742714808557438673rm_a_b
@ ( inf_in1693821330460682071rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ ( sup_su6776935440552674877rm_a_b @ R2 @ ( conver7608559976937845093rm_a_b @ R2 ) ) ) )
@ ( produc1953217797448512882rm_a_b @ ( collect_term_a_b @ term_ground_a_b )
@ ^ [Uu: term_a_b] : ( collect_term_a_b @ term_ground_a_b ) ) ) ) ) ).
% gsrstep_conversion_dist
thf(fact_1125_strongly__confluent__E1n,axiom,
! [R: set_Pr4386577575007340137rm_a_b,X: term_a_b,Y: term_a_b,Z: term_a_b,N: nat] :
( ( abstra5664096641628173427rm_a_b @ R @ top_top_set_term_a_b )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( sup_su6776935440552674877rm_a_b @ R @ id_term_a_b ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z ) @ ( compow4057154403645558940rm_a_b @ N @ R ) )
=> ? [U4: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ U4 ) @ ( transi7742714808557438673rm_a_b @ R ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Z @ U4 ) @ ( sup_su6776935440552674877rm_a_b @ R @ id_term_a_b ) ) ) ) ) ) ).
% strongly_confluent_E1n
thf(fact_1126_strongly__confluent__E1n,axiom,
! [R: set_Pr1811044260758604347_a_nat,X: product_prod_a_nat,Y: product_prod_a_nat,Z: product_prod_a_nat,N: nat] :
( ( abstra5315942894939654492_a_nat @ R @ top_to3353692345378799459_a_nat )
=> ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ X @ Y ) @ ( sup_su958340139570419215_a_nat @ R @ id_Pro5207055338379199009_a_nat ) )
=> ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ X @ Z ) @ ( compow3129981181699418478_a_nat @ N @ R ) )
=> ? [U4: product_prod_a_nat] :
( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ Y @ U4 ) @ ( transi2726145917338391738_a_nat @ R ) )
& ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ Z @ U4 ) @ ( sup_su958340139570419215_a_nat @ R @ id_Pro5207055338379199009_a_nat ) ) ) ) ) ) ).
% strongly_confluent_E1n
thf(fact_1127_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_1128_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_1129_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_1130_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_1131_converse__UNIV,axiom,
( ( converse_nat_a @ top_to2612598781856825737_nat_a )
= top_to3353692345378799459_a_nat ) ).
% converse_UNIV
thf(fact_1132_converse__UNIV,axiom,
( ( converse_a_nat @ top_to3353692345378799459_a_nat )
= top_to2612598781856825737_nat_a ) ).
% converse_UNIV
thf(fact_1133_sig__step__converse__rstep,axiom,
! [F2: set_Pr4934435412358123699_a_nat,R2: set_Pr4386577575007340137rm_a_b] :
( ( conver7608559976937845093rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) ) )
= ( sig_step_a_b @ F2 @ ( conver7608559976937845093rm_a_b @ ( rstep_a_b @ R2 ) ) ) ) ).
% sig_step_converse_rstep
thf(fact_1134_srstep__converse__dist,axiom,
! [F2: set_Pr4934435412358123699_a_nat,R2: set_Pr4386577575007340137rm_a_b] :
( ( conver7608559976937845093rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) ) )
= ( sig_step_a_b @ F2 @ ( rstep_a_b @ ( conver7608559976937845093rm_a_b @ R2 ) ) ) ) ).
% srstep_converse_dist
thf(fact_1135_converse__Int,axiom,
! [R: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
( ( conver7608559976937845093rm_a_b @ ( inf_in1693821330460682071rm_a_b @ R @ S ) )
= ( inf_in1693821330460682071rm_a_b @ ( conver7608559976937845093rm_a_b @ R ) @ ( conver7608559976937845093rm_a_b @ S ) ) ) ).
% converse_Int
thf(fact_1136_rstep__converse__dist,axiom,
! [R2: set_Pr4386577575007340137rm_a_b] :
( ( conver7608559976937845093rm_a_b @ ( rstep_a_b @ R2 ) )
= ( rstep_a_b @ ( conver7608559976937845093rm_a_b @ R2 ) ) ) ).
% rstep_converse_dist
thf(fact_1137_converse__Times,axiom,
! [A: set_term_a_b,B: set_term_a_b] :
( ( conver7608559976937845093rm_a_b
@ ( produc1953217797448512882rm_a_b @ A
@ ^ [Uu: term_a_b] : B ) )
= ( produc1953217797448512882rm_a_b @ B
@ ^ [Uu: term_a_b] : A ) ) ).
% converse_Times
thf(fact_1138_srrstep__converse__dist,axiom,
! [F2: set_Pr4934435412358123699_a_nat,R2: set_Pr4386577575007340137rm_a_b] :
( ( conver7608559976937845093rm_a_b @ ( sig_step_a_b @ F2 @ ( rrstep_a_b @ R2 ) ) )
= ( sig_step_a_b @ F2 @ ( rrstep_a_b @ ( conver7608559976937845093rm_a_b @ R2 ) ) ) ) ).
% srrstep_converse_dist
thf(fact_1139_Restr__converse,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,A: set_term_a_b] :
( ( conver7608559976937845093rm_a_b
@ ( inf_in1693821330460682071rm_a_b @ R2
@ ( produc1953217797448512882rm_a_b @ A
@ ^ [Uu: term_a_b] : A ) ) )
= ( inf_in1693821330460682071rm_a_b @ ( conver7608559976937845093rm_a_b @ R2 )
@ ( produc1953217797448512882rm_a_b @ A
@ ^ [Uu: term_a_b] : A ) ) ) ).
% Restr_converse
thf(fact_1140_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_1141_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_1142_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_1143_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_1144_converse_Osimps,axiom,
! [A1: nat,A22: a,R: set_Pr4934435412358123699_a_nat] :
( ( member8962352052110095674_nat_a @ ( product_Pair_nat_a @ A1 @ A22 ) @ ( converse_a_nat @ R ) )
= ( ? [A4: a,B4: nat] :
( ( A1 = B4 )
& ( A22 = A4 )
& ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ A4 @ B4 ) @ R ) ) ) ) ).
% converse.simps
thf(fact_1145_converse_Osimps,axiom,
! [A1: a,A22: nat,R: set_Pr4193341848836149977_nat_a] :
( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ A1 @ A22 ) @ ( converse_nat_a @ R ) )
= ( ? [A4: nat,B4: a] :
( ( A1 = B4 )
& ( A22 = A4 )
& ( member8962352052110095674_nat_a @ ( product_Pair_nat_a @ A4 @ B4 ) @ R ) ) ) ) ).
% converse.simps
thf(fact_1146_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_1147_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_1148_converseE,axiom,
! [Yx: product_prod_nat_a,R: set_Pr4934435412358123699_a_nat] :
( ( member8962352052110095674_nat_a @ Yx @ ( converse_a_nat @ R ) )
=> ~ ! [X3: a,Y4: nat] :
( ( Yx
= ( product_Pair_nat_a @ Y4 @ X3 ) )
=> ~ ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ X3 @ Y4 ) @ R ) ) ) ).
% converseE
thf(fact_1149_converseE,axiom,
! [Yx: product_prod_a_nat,R: set_Pr4193341848836149977_nat_a] :
( ( member5724188588386418708_a_nat @ Yx @ ( converse_nat_a @ R ) )
=> ~ ! [X3: nat,Y4: a] :
( ( Yx
= ( product_Pair_a_nat @ Y4 @ X3 ) )
=> ~ ( member8962352052110095674_nat_a @ ( product_Pair_nat_a @ X3 @ Y4 ) @ R ) ) ) ).
% converseE
thf(fact_1150_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_1151_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_1152_sig__step__symcl__dist,axiom,
! [F2: set_Pr4934435412358123699_a_nat,R2: set_Pr4386577575007340137rm_a_b] :
( ( sup_su6776935440552674877rm_a_b @ ( sig_step_a_b @ F2 @ R2 ) @ ( conver7608559976937845093rm_a_b @ ( sig_step_a_b @ F2 @ R2 ) ) )
= ( sig_step_a_b @ F2 @ ( sup_su6776935440552674877rm_a_b @ R2 @ ( conver7608559976937845093rm_a_b @ R2 ) ) ) ) ).
% sig_step_symcl_dist
thf(fact_1153_rstep__smycl__dist,axiom,
! [R2: set_Pr4386577575007340137rm_a_b] :
( ( sup_su6776935440552674877rm_a_b @ ( rstep_a_b @ R2 ) @ ( conver7608559976937845093rm_a_b @ ( rstep_a_b @ R2 ) ) )
= ( rstep_a_b @ ( sup_su6776935440552674877rm_a_b @ R2 @ ( conver7608559976937845093rm_a_b @ R2 ) ) ) ) ).
% rstep_smycl_dist
thf(fact_1154_funas__rel__converse,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,F2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ ( terms_7988297476397195622_a_b_b @ R2 ) @ F2 )
=> ( ord_le8666007276011122963_a_nat @ ( terms_7988297476397195622_a_b_b @ ( conver7608559976937845093rm_a_b @ R2 ) ) @ F2 ) ) ).
% funas_rel_converse
thf(fact_1155_peak__iff,axiom,
! [X: nat,Y: nat,A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( relcomp_nat_a_nat @ ( converse_a_nat @ A ) @ B ) )
= ( ? [U2: a] :
( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ U2 @ X ) @ A )
& ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ U2 @ Y ) @ B ) ) ) ) ).
% peak_iff
thf(fact_1156_peak__iff,axiom,
! [X: a,Y: nat,A: set_Product_prod_a_a,B: set_Pr4934435412358123699_a_nat] :
( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ X @ Y ) @ ( relcomp_a_a_nat @ ( converse_a_a @ A ) @ B ) )
= ( ? [U2: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ U2 @ X ) @ A )
& ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ U2 @ Y ) @ B ) ) ) ) ).
% peak_iff
thf(fact_1157_srstep__symcl__dist,axiom,
! [F2: set_Pr4934435412358123699_a_nat,R2: set_Pr4386577575007340137rm_a_b] :
( ( sup_su6776935440552674877rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) ) @ ( conver7608559976937845093rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) ) ) )
= ( sig_step_a_b @ F2 @ ( rstep_a_b @ ( sup_su6776935440552674877rm_a_b @ R2 @ ( conver7608559976937845093rm_a_b @ R2 ) ) ) ) ) ).
% srstep_symcl_dist
thf(fact_1158_Restr__smycl__dist,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,A8: set_term_a_b] :
( ( sup_su6776935440552674877rm_a_b
@ ( inf_in1693821330460682071rm_a_b @ R2
@ ( produc1953217797448512882rm_a_b @ A8
@ ^ [Uu: term_a_b] : A8 ) )
@ ( conver7608559976937845093rm_a_b
@ ( inf_in1693821330460682071rm_a_b @ R2
@ ( produc1953217797448512882rm_a_b @ A8
@ ^ [Uu: term_a_b] : A8 ) ) ) )
= ( inf_in1693821330460682071rm_a_b @ ( sup_su6776935440552674877rm_a_b @ R2 @ ( conver7608559976937845093rm_a_b @ R2 ) )
@ ( produc1953217797448512882rm_a_b @ A8
@ ^ [Uu: term_a_b] : A8 ) ) ) ).
% Restr_smycl_dist
thf(fact_1159_symcl__srstep__conversion,axiom,
! [S: term_a_b,T2: term_a_b,F2: set_Pr4934435412358123699_a_nat,R2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T2 ) @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ ( sup_su6776935440552674877rm_a_b @ R2 @ ( conver7608559976937845093rm_a_b @ R2 ) ) ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T2 ) @ ( abstra1130568886744591910rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) ) ) ) ) ).
% symcl_srstep_conversion
thf(fact_1160_sig__step__conversion__dist,axiom,
! [F2: set_Pr4934435412358123699_a_nat,R2: set_Pr4386577575007340137rm_a_b] :
( ( abstra1130568886744591910rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) ) )
= ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ ( sup_su6776935440552674877rm_a_b @ R2 @ ( conver7608559976937845093rm_a_b @ R2 ) ) ) ) ) ) ).
% sig_step_conversion_dist
thf(fact_1161_irrefl__tranclI,axiom,
! [R: set_Pr4386577575007340137rm_a_b,X: term_a_b] :
( ( ( inf_in1693821330460682071rm_a_b @ ( conver7608559976937845093rm_a_b @ R ) @ ( transi7742714808557438673rm_a_b @ R ) )
= bot_bo197521221353338581rm_a_b )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ X ) @ ( transi7922773638565587891rm_a_b @ R ) ) ) ).
% irrefl_tranclI
thf(fact_1162_symcl__srsteps__conversion,axiom,
! [S: term_a_b,T2: term_a_b,F2: set_Pr4934435412358123699_a_nat,R2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T2 ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ ( sup_su6776935440552674877rm_a_b @ R2 @ ( conver7608559976937845093rm_a_b @ R2 ) ) ) ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T2 ) @ ( abstra1130568886744591910rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) ) ) ) ) ).
% symcl_srsteps_conversion
thf(fact_1163_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_1164_WCR__alt__def,axiom,
! [A: set_Pr1811044260758604347_a_nat] :
( ( abstra6648775111202918568_a_nat @ A @ top_to3353692345378799459_a_nat )
= ( ord_le1018141841545957851_a_nat @ ( relcom3132138491631691642_a_nat @ ( conver4024954229950350007_a_nat @ A ) @ A ) @ ( abstra881512052393830979_a_nat @ A ) ) ) ).
% WCR_alt_def
thf(fact_1165_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_1166_semi__confluence__imp__CR,axiom,
! [R: set_Pr1811044260758604347_a_nat] :
( ( ord_le1018141841545957851_a_nat @ ( relcom3132138491631691642_a_nat @ ( conver4024954229950350007_a_nat @ R ) @ ( transi2726145917338391738_a_nat @ R ) ) @ ( abstra881512052393830979_a_nat @ R ) )
=> ( abstra7964079670522473415_a_nat @ R @ top_to3353692345378799459_a_nat ) ) ).
% semi_confluence_imp_CR
thf(fact_1167_NFP__lift__to__conversion,axiom,
! [R: set_Pr4386577575007340137rm_a_b,S: term_a_b,T2: term_a_b] :
( ( rewrit6304709981509430748rm_a_b @ R @ top_top_set_term_a_b )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T2 ) @ ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ R @ ( conver7608559976937845093rm_a_b @ R ) ) ) )
=> ( ( member_term_a_b @ T2 @ ( abstract_NF_term_a_b @ R ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T2 ) @ ( transi7742714808557438673rm_a_b @ R ) ) ) ) ) ).
% NFP_lift_to_conversion
thf(fact_1168_NFP__lift__to__conversion,axiom,
! [R: set_Pr1811044260758604347_a_nat,S: product_prod_a_nat,T2: product_prod_a_nat] :
( ( rewrit1859573128460188229_a_nat @ R @ top_to3353692345378799459_a_nat )
=> ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ S @ T2 ) @ ( transi2726145917338391738_a_nat @ ( sup_su958340139570419215_a_nat @ R @ ( conver4024954229950350007_a_nat @ R ) ) ) )
=> ( ( member5724188588386418708_a_nat @ T2 @ ( abstra2685462378305987313_a_nat @ R ) )
=> ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ S @ T2 ) @ ( transi2726145917338391738_a_nat @ R ) ) ) ) ) ).
% NFP_lift_to_conversion
thf(fact_1169_UNF__rrstep__intro,axiom,
! [F2: set_Pr4934435412358123699_a_nat,R2: set_Pr4386577575007340137rm_a_b] :
( ! [T4: term_a_b,U4: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T4 @ U4 ) @ ( sup_su6776935440552674877rm_a_b @ ( relcom370159955682700863rm_a_b @ ( srstep7844470518422762656ep_a_b @ F2 @ ( conver7608559976937845093rm_a_b @ R2 ) ) @ ( transi7922773638565587891rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) ) ) ) @ ( relcom370159955682700863rm_a_b @ ( transi7922773638565587891rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ ( conver7608559976937845093rm_a_b @ R2 ) ) ) ) @ ( srstep7844470518422762656ep_a_b @ F2 @ R2 ) ) ) )
=> ( rewrit983855061078472681dp_a_b @ F2 @ R2 @ T4 @ U4 ) )
=> ( abstra3619855735679963878rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) ) @ top_top_set_term_a_b ) ) ).
% UNF_rrstep_intro
thf(fact_1170_UNF__to__rrstep,axiom,
! [F2: set_Pr4934435412358123699_a_nat,R2: set_Pr4386577575007340137rm_a_b,S: term_a_b,T2: term_a_b] :
( ( abstra3619855735679963878rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) ) @ top_top_set_term_a_b )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T2 ) @ ( sup_su6776935440552674877rm_a_b @ ( relcom370159955682700863rm_a_b @ ( srstep7844470518422762656ep_a_b @ F2 @ ( conver7608559976937845093rm_a_b @ R2 ) ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) ) ) ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ ( conver7608559976937845093rm_a_b @ R2 ) ) ) ) @ ( srstep7844470518422762656ep_a_b @ F2 @ R2 ) ) ) )
=> ( rewrit983855061078472681dp_a_b @ F2 @ R2 @ S @ T2 ) ) ) ).
% UNF_to_rrstep
thf(fact_1171_llrg__srsteps__with__root__step__ground,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,S: term_a_b,T2: term_a_b,F2: set_Pr4934435412358123699_a_nat] :
( ( rewrit5000278123254648670_b_a_b @ R2 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T2 ) @ ( srstep7844470518422762656ep_a_b @ F2 @ R2 ) )
=> ( term_ground_a_b @ T2 ) ) ) ).
% llrg_srsteps_with_root_step_ground
thf(fact_1172_srsteps__with__root__step__sresteps__eqD,axiom,
! [S: term_a_b,T2: term_a_b,F2: set_Pr4934435412358123699_a_nat,R2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T2 ) @ ( srstep7844470518422762656ep_a_b @ F2 @ R2 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T2 ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) ) ) ) ) ).
% srsteps_with_root_step_sresteps_eqD
thf(fact_1173_srsteps__with__root__step__srstepsD,axiom,
! [S: term_a_b,T2: term_a_b,F2: set_Pr4934435412358123699_a_nat,R2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T2 ) @ ( srstep7844470518422762656ep_a_b @ F2 @ R2 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T2 ) @ ( transi7922773638565587891rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) ) ) ) ) ).
% srsteps_with_root_step_srstepsD
thf(fact_1174_llrg__srsteps__with__root__step__inv__ground,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,S: term_a_b,T2: term_a_b,F2: set_Pr4934435412358123699_a_nat] :
( ( rewrit5000278123254648670_b_a_b @ R2 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T2 ) @ ( srstep7844470518422762656ep_a_b @ F2 @ ( conver7608559976937845093rm_a_b @ R2 ) ) )
=> ( term_ground_a_b @ S ) ) ) ).
% llrg_srsteps_with_root_step_inv_ground
thf(fact_1175_srsteps__with__root__step__def,axiom,
( srstep7844470518422762656ep_a_b
= ( ^ [F3: set_Pr4934435412358123699_a_nat,R6: set_Pr4386577575007340137rm_a_b] : ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F3 @ ( rstep_a_b @ R6 ) ) ) @ ( relcom370159955682700863rm_a_b @ ( sig_step_a_b @ F3 @ ( rrstep_a_b @ R6 ) ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F3 @ ( rstep_a_b @ R6 ) ) ) ) ) ) ) ).
% srsteps_with_root_step_def
thf(fact_1176_UNC__rrstep__intro,axiom,
! [F2: set_Pr4934435412358123699_a_nat,R2: set_Pr4386577575007340137rm_a_b] :
( ! [S7: term_a_b,T4: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S7 @ T4 ) @ ( srstep7844470518422762656ep_a_b @ F2 @ ( sup_su6776935440552674877rm_a_b @ R2 @ ( conver7608559976937845093rm_a_b @ R2 ) ) ) )
=> ( rewrit983855061078472681dp_a_b @ F2 @ R2 @ S7 @ T4 ) )
=> ( abstra4649771476585718372rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) ) ) ) ).
% UNC_rrstep_intro
thf(fact_1177_UNC__to__rrstep,axiom,
! [F2: set_Pr4934435412358123699_a_nat,R2: set_Pr4386577575007340137rm_a_b,S: term_a_b,T2: term_a_b] :
( ( abstra4649771476585718372rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T2 ) @ ( srstep7844470518422762656ep_a_b @ F2 @ ( sup_su6776935440552674877rm_a_b @ R2 @ ( conver7608559976937845093rm_a_b @ R2 ) ) ) )
=> ( rewrit983855061078472681dp_a_b @ F2 @ R2 @ S @ T2 ) ) ) ).
% UNC_to_rrstep
thf(fact_1178_CE__to__rrstep_I2_J,axiom,
! [F2: set_Pr4934435412358123699_a_nat,R2: set_Pr4386577575007340137rm_a_b,S2: set_Pr4386577575007340137rm_a_b,S: term_a_b,T2: term_a_b] :
( ( rewrit660295756789745446rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) ) @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ S2 ) ) @ top_top_set_term_a_b )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T2 ) @ ( srstep7844470518422762656ep_a_b @ F2 @ ( sup_su6776935440552674877rm_a_b @ S2 @ ( conver7608559976937845093rm_a_b @ S2 ) ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T2 ) @ ( abstra1130568886744591910rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) ) ) ) ) ) ).
% CE_to_rrstep(2)
thf(fact_1179_CE__to__rrstep_I1_J,axiom,
! [F2: set_Pr4934435412358123699_a_nat,R2: set_Pr4386577575007340137rm_a_b,S2: set_Pr4386577575007340137rm_a_b,S: term_a_b,T2: term_a_b] :
( ( rewrit660295756789745446rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) ) @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ S2 ) ) @ top_top_set_term_a_b )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T2 ) @ ( srstep7844470518422762656ep_a_b @ F2 @ ( sup_su6776935440552674877rm_a_b @ R2 @ ( conver7608559976937845093rm_a_b @ R2 ) ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T2 ) @ ( abstra1130568886744591910rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ S2 ) ) ) ) ) ) ).
% CE_to_rrstep(1)
thf(fact_1180_CE__rrstep__intro,axiom,
! [F2: set_Pr4934435412358123699_a_nat,R2: set_Pr4386577575007340137rm_a_b,S2: set_Pr4386577575007340137rm_a_b] :
( ! [S7: term_a_b,T4: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S7 @ T4 ) @ ( srstep7844470518422762656ep_a_b @ F2 @ ( sup_su6776935440552674877rm_a_b @ R2 @ ( conver7608559976937845093rm_a_b @ R2 ) ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S7 @ T4 ) @ ( abstra1130568886744591910rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ S2 ) ) ) ) )
=> ( ! [S7: term_a_b,T4: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S7 @ T4 ) @ ( srstep7844470518422762656ep_a_b @ F2 @ ( sup_su6776935440552674877rm_a_b @ S2 @ ( conver7608559976937845093rm_a_b @ S2 ) ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S7 @ T4 ) @ ( abstra1130568886744591910rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) ) ) ) )
=> ( rewrit660295756789745446rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) ) @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ S2 ) ) @ top_top_set_term_a_b ) ) ) ).
% CE_rrstep_intro
thf(fact_1181_NFP__to__rrstep,axiom,
! [F2: set_Pr4934435412358123699_a_nat,R2: set_Pr4386577575007340137rm_a_b,S: term_a_b,T2: term_a_b] :
( ( rewrit6304709981509430748rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) ) @ top_top_set_term_a_b )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T2 ) @ ( srstep7844470518422762656ep_a_b @ F2 @ ( sup_su6776935440552674877rm_a_b @ R2 @ ( conver7608559976937845093rm_a_b @ R2 ) ) ) )
=> ( rewrit232733863534887208dp_a_b @ F2 @ R2 @ S @ T2 ) ) ) ).
% NFP_to_rrstep
thf(fact_1182_NFP__rrstep__intro,axiom,
! [F2: set_Pr4934435412358123699_a_nat,R2: set_Pr4386577575007340137rm_a_b] :
( ! [S7: term_a_b,T4: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S7 @ T4 ) @ ( sup_su6776935440552674877rm_a_b @ ( relcom370159955682700863rm_a_b @ ( srstep7844470518422762656ep_a_b @ F2 @ ( conver7608559976937845093rm_a_b @ R2 ) ) @ ( transi7922773638565587891rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) ) ) ) @ ( relcom370159955682700863rm_a_b @ ( transi7922773638565587891rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ ( conver7608559976937845093rm_a_b @ R2 ) ) ) ) @ ( srstep7844470518422762656ep_a_b @ F2 @ R2 ) ) ) )
=> ( rewrit232733863534887208dp_a_b @ F2 @ R2 @ S7 @ T4 ) )
=> ( rewrit6304709981509430748rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) ) @ top_top_set_term_a_b ) ) ).
% NFP_rrstep_intro
thf(fact_1183_CR__to__rrstep,axiom,
! [F2: set_Pr4934435412358123699_a_nat,R2: set_Pr4386577575007340137rm_a_b,S: term_a_b,T2: term_a_b] :
( ( abstra8448919418672941150rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) ) @ top_top_set_term_a_b )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T2 ) @ ( sup_su6776935440552674877rm_a_b @ ( relcom370159955682700863rm_a_b @ ( srstep7844470518422762656ep_a_b @ F2 @ ( conver7608559976937845093rm_a_b @ R2 ) ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) ) ) ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ ( conver7608559976937845093rm_a_b @ R2 ) ) ) ) @ ( srstep7844470518422762656ep_a_b @ F2 @ R2 ) ) ) )
=> ( rewrit8865997592796547114dp_a_b @ F2 @ R2 @ R2 @ S @ T2 ) ) ) ).
% CR_to_rrstep
thf(fact_1184_commute__rrstep__intro,axiom,
! [F2: set_Pr4934435412358123699_a_nat,R2: set_Pr4386577575007340137rm_a_b,S2: set_Pr4386577575007340137rm_a_b] :
( ! [S7: term_a_b,T4: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S7 @ T4 ) @ ( sup_su6776935440552674877rm_a_b @ ( relcom370159955682700863rm_a_b @ ( srstep7844470518422762656ep_a_b @ F2 @ ( conver7608559976937845093rm_a_b @ R2 ) ) @ ( transi7922773638565587891rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ S2 ) ) ) ) @ ( relcom370159955682700863rm_a_b @ ( transi7922773638565587891rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ ( conver7608559976937845093rm_a_b @ R2 ) ) ) ) @ ( srstep7844470518422762656ep_a_b @ F2 @ S2 ) ) ) )
=> ( rewrit8865997592796547114dp_a_b @ F2 @ R2 @ S2 @ S7 @ T4 ) )
=> ( abstra455195320138238452rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) ) @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ S2 ) ) ) ) ).
% commute_rrstep_intro
thf(fact_1185_commute__redp__def,axiom,
( rewrit8865997592796547114dp_a_b
= ( ^ [F3: set_Pr4934435412358123699_a_nat,R6: set_Pr4386577575007340137rm_a_b,S6: set_Pr4386577575007340137rm_a_b,S5: term_a_b,T: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S5 @ T ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F3 @ ( rstep_a_b @ S6 ) ) ) @ ( transi7742714808557438673rm_a_b @ ( conver7608559976937845093rm_a_b @ ( sig_step_a_b @ F3 @ ( rstep_a_b @ R6 ) ) ) ) ) ) ) ) ).
% commute_redp_def
thf(fact_1186_CR__Aux,axiom,
! [F2: set_Pr4934435412358123699_a_nat,R2: set_Pr4386577575007340137rm_a_b,S: term_a_b,T2: term_a_b] :
( ! [S7: term_a_b,T4: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S7 @ T4 ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ ( conver7608559976937845093rm_a_b @ R2 ) ) ) ) @ ( srstep7844470518422762656ep_a_b @ F2 @ R2 ) ) )
=> ( rewrit8865997592796547114dp_a_b @ F2 @ R2 @ R2 @ S7 @ T4 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T2 ) @ ( sup_su6776935440552674877rm_a_b @ ( relcom370159955682700863rm_a_b @ ( srstep7844470518422762656ep_a_b @ F2 @ ( conver7608559976937845093rm_a_b @ R2 ) ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) ) ) ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ ( conver7608559976937845093rm_a_b @ R2 ) ) ) ) @ ( srstep7844470518422762656ep_a_b @ F2 @ R2 ) ) ) )
=> ( rewrit8865997592796547114dp_a_b @ F2 @ R2 @ R2 @ S @ T2 ) ) ) ).
% CR_Aux
thf(fact_1187_commute__to__rrstep,axiom,
! [F2: set_Pr4934435412358123699_a_nat,R2: set_Pr4386577575007340137rm_a_b,S2: set_Pr4386577575007340137rm_a_b,S: term_a_b,T2: term_a_b] :
( ( abstra455195320138238452rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) ) @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ S2 ) ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T2 ) @ ( sup_su6776935440552674877rm_a_b @ ( relcom370159955682700863rm_a_b @ ( srstep7844470518422762656ep_a_b @ F2 @ ( conver7608559976937845093rm_a_b @ R2 ) ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ S2 ) ) ) ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ ( conver7608559976937845093rm_a_b @ R2 ) ) ) ) @ ( srstep7844470518422762656ep_a_b @ F2 @ S2 ) ) ) )
=> ( rewrit8865997592796547114dp_a_b @ F2 @ R2 @ S2 @ S @ T2 ) ) ) ).
% commute_to_rrstep
thf(fact_1188_CR__rrstep__intro,axiom,
! [F2: set_Pr4934435412358123699_a_nat,R2: set_Pr4386577575007340137rm_a_b] :
( ! [S7: term_a_b,T4: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S7 @ T4 ) @ ( relcom370159955682700863rm_a_b @ ( transi7922773638565587891rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ ( conver7608559976937845093rm_a_b @ R2 ) ) ) ) @ ( srstep7844470518422762656ep_a_b @ F2 @ R2 ) ) )
=> ( rewrit8865997592796547114dp_a_b @ F2 @ R2 @ R2 @ S7 @ T4 ) )
=> ( abstra8448919418672941150rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) ) @ top_top_set_term_a_b ) ) ).
% CR_rrstep_intro
thf(fact_1189_NE__rrstep__intro,axiom,
! [F2: set_Pr4934435412358123699_a_nat,R2: set_Pr4386577575007340137rm_a_b,S2: set_Pr4386577575007340137rm_a_b] :
( ! [S7: term_a_b,T4: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S7 @ T4 ) @ ( srstep7844470518422762656ep_a_b @ F2 @ R2 ) )
=> ( rewrit3451484132325884455dp_a_b @ F2 @ R2 @ S2 @ S7 @ T4 ) )
=> ( ! [S7: term_a_b,T4: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S7 @ T4 ) @ ( srstep7844470518422762656ep_a_b @ F2 @ S2 ) )
=> ( rewrit3451484132325884455dp_a_b @ F2 @ S2 @ R2 @ S7 @ T4 ) )
=> ( ( ( abstract_NF_term_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) ) )
= ( abstract_NF_term_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ S2 ) ) ) )
=> ( rewrit7370782028743737947rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) ) @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ S2 ) ) @ top_top_set_term_a_b ) ) ) ) ).
% NE_rrstep_intro
thf(fact_1190_NE__to__rrstep_I2_J,axiom,
! [F2: set_Pr4934435412358123699_a_nat,R2: set_Pr4386577575007340137rm_a_b,S2: set_Pr4386577575007340137rm_a_b,S: term_a_b,T2: term_a_b] :
( ( rewrit7370782028743737947rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) ) @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ S2 ) ) @ top_top_set_term_a_b )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T2 ) @ ( srstep7844470518422762656ep_a_b @ F2 @ S2 ) )
=> ( rewrit3451484132325884455dp_a_b @ F2 @ S2 @ R2 @ S @ T2 ) ) ) ).
% NE_to_rrstep(2)
thf(fact_1191_NE__symmetric,axiom,
! [R: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
( ( rewrit7370782028743737947rm_a_b @ R @ S @ top_top_set_term_a_b )
=> ( rewrit7370782028743737947rm_a_b @ S @ R @ top_top_set_term_a_b ) ) ).
% NE_symmetric
thf(fact_1192_NE__symmetric,axiom,
! [R: set_Pr1811044260758604347_a_nat,S: set_Pr1811044260758604347_a_nat] :
( ( rewrit2255851985543578436_a_nat @ R @ S @ top_to3353692345378799459_a_nat )
=> ( rewrit2255851985543578436_a_nat @ S @ R @ top_to3353692345378799459_a_nat ) ) ).
% NE_symmetric
thf(fact_1193_NE__NF__eq,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,S2: set_Pr4386577575007340137rm_a_b] :
( ( rewrit7370782028743737947rm_a_b @ R2 @ S2 @ top_top_set_term_a_b )
=> ( ( abstract_NF_term_a_b @ R2 )
= ( abstract_NF_term_a_b @ S2 ) ) ) ).
% NE_NF_eq
thf(fact_1194_NE__NF__eq,axiom,
! [R2: set_Pr1811044260758604347_a_nat,S2: set_Pr1811044260758604347_a_nat] :
( ( rewrit2255851985543578436_a_nat @ R2 @ S2 @ top_to3353692345378799459_a_nat )
=> ( ( abstra2685462378305987313_a_nat @ R2 )
= ( abstra2685462378305987313_a_nat @ S2 ) ) ) ).
% NE_NF_eq
thf(fact_1195_NE__to__rrstep_I1_J,axiom,
! [F2: set_Pr4934435412358123699_a_nat,R2: set_Pr4386577575007340137rm_a_b,S2: set_Pr4386577575007340137rm_a_b,S: term_a_b,T2: term_a_b] :
( ( rewrit7370782028743737947rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) ) @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ S2 ) ) @ top_top_set_term_a_b )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T2 ) @ ( srstep7844470518422762656ep_a_b @ F2 @ R2 ) )
=> ( rewrit3451484132325884455dp_a_b @ F2 @ R2 @ S2 @ S @ T2 ) ) ) ).
% NE_to_rrstep(1)
thf(fact_1196_reduction__join__relations__to__root__step,axiom,
! [F2: set_Pr4934435412358123699_a_nat,R2: set_Pr4386577575007340137rm_a_b,S2: set_Pr4386577575007340137rm_a_b,P: term_a_b > term_a_b > $o,S: term_a_b,T2: term_a_b] :
( ! [S7: term_a_b,T4: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S7 @ T4 ) @ ( sup_su6776935440552674877rm_a_b @ ( relcom370159955682700863rm_a_b @ ( srstep7844470518422762656ep_a_b @ F2 @ R2 ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ S2 ) ) ) ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) ) ) @ ( srstep7844470518422762656ep_a_b @ F2 @ S2 ) ) ) )
=> ( P @ S7 @ T4 ) )
=> ( ( terms_5226143800768910156ed_a_b @ F2
@ ( collec99177395925924084rm_a_b
@ ^ [Uu: produc357393685978478089rm_a_b] :
? [S5: term_a_b,T: term_a_b] :
( ( Uu
= ( produc7020197800436672577rm_a_b @ S5 @ T ) )
& ( P @ S5 @ T ) ) ) )
=> ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ S ) @ F2 )
=> ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ T2 ) @ F2 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T2 ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) ) ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ S2 ) ) ) ) )
=> ( P @ S @ T2 ) ) ) ) ) ) ).
% reduction_join_relations_to_root_step
thf(fact_1197_reduction__relations__to__root__step,axiom,
! [F2: set_Pr4934435412358123699_a_nat,R2: set_Pr4386577575007340137rm_a_b,P: term_a_b > term_a_b > $o,S: term_a_b,T2: term_a_b] :
( ! [S7: term_a_b,T4: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S7 @ T4 ) @ ( srstep7844470518422762656ep_a_b @ F2 @ R2 ) )
=> ( P @ S7 @ T4 ) )
=> ( ( terms_5226143800768910156ed_a_b @ F2
@ ( collec99177395925924084rm_a_b
@ ^ [Uu: produc357393685978478089rm_a_b] :
? [S5: term_a_b,T: term_a_b] :
( ( Uu
= ( produc7020197800436672577rm_a_b @ S5 @ T ) )
& ( P @ S5 @ T ) ) ) )
=> ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ S ) @ F2 )
=> ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ T2 ) @ F2 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T2 ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R2 ) ) ) )
=> ( P @ S @ T2 ) ) ) ) ) ) ).
% reduction_relations_to_root_step
thf(fact_1198_restrict__SN__def,axiom,
( abstra7212103836473851632_a_nat
= ( ^ [R7: set_Pr1811044260758604347_a_nat,S5: set_Pr1811044260758604347_a_nat] :
( collec925066218191019974_a_nat
@ ^ [Uu: produc4708774622424448987_a_nat] :
? [A4: product_prod_a_nat,B4: product_prod_a_nat] :
( ( Uu
= ( produc2026711137822539155_a_nat @ A4 @ B4 ) )
& ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ A4 @ B4 ) @ R7 )
& ( abstra8653715922312955827_a_nat @ S5 @ ( insert8054603423593749677_a_nat @ A4 @ bot_bo9049108969261143879_a_nat ) ) ) ) ) ) ).
% restrict_SN_def
thf(fact_1199_restrict__SN__def,axiom,
( abstra2540617779794743107SN_nat
= ( ^ [R7: set_Pr1261947904930325089at_nat,S5: set_Pr1261947904930325089at_nat] :
( collec3392354462482085612at_nat
@ ^ [Uu: product_prod_nat_nat] :
? [A4: nat,B4: nat] :
( ( Uu
= ( product_Pair_nat_nat @ A4 @ B4 ) )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A4 @ B4 ) @ R7 )
& ( abstract_SN_on_nat @ S5 @ ( insert_nat @ A4 @ bot_bot_set_nat ) ) ) ) ) ) ).
% restrict_SN_def
thf(fact_1200_image2__def,axiom,
( bNF_Gr2779003347868658743_a_nat
= ( ^ [A5: set_Pr4934435412358123699_a_nat,F5: product_prod_a_nat > a,G2: product_prod_a_nat > nat] :
( collec4464134535221767506_a_nat
@ ^ [Uu: product_prod_a_nat] :
? [A4: product_prod_a_nat] :
( ( Uu
= ( product_Pair_a_nat @ ( F5 @ A4 ) @ ( G2 @ A4 ) ) )
& ( member5724188588386418708_a_nat @ A4 @ A5 ) ) ) ) ) ).
% image2_def
thf(fact_1201_image2__def,axiom,
( bNF_Gr2166505328656471904_a_nat
= ( ^ [A5: set_term_a_b,F5: term_a_b > a,G2: term_a_b > nat] :
( collec4464134535221767506_a_nat
@ ^ [Uu: product_prod_a_nat] :
? [A4: term_a_b] :
( ( Uu
= ( product_Pair_a_nat @ ( F5 @ A4 ) @ ( G2 @ A4 ) ) )
& ( member_term_a_b @ A4 @ A5 ) ) ) ) ) ).
% image2_def
thf(fact_1202_image2__eqI,axiom,
! [B2: a,F: product_prod_a_nat > a,X: product_prod_a_nat,C: nat,G3: product_prod_a_nat > nat,A: set_Pr4934435412358123699_a_nat] :
( ( B2
= ( F @ X ) )
=> ( ( C
= ( G3 @ X ) )
=> ( ( member5724188588386418708_a_nat @ X @ A )
=> ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ B2 @ C ) @ ( bNF_Gr2779003347868658743_a_nat @ A @ F @ G3 ) ) ) ) ) ).
% image2_eqI
thf(fact_1203_image2__eqI,axiom,
! [B2: a,F: term_a_b > a,X: term_a_b,C: nat,G3: term_a_b > nat,A: set_term_a_b] :
( ( B2
= ( F @ X ) )
=> ( ( C
= ( G3 @ X ) )
=> ( ( member_term_a_b @ X @ A )
=> ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ B2 @ C ) @ ( bNF_Gr2166505328656471904_a_nat @ A @ F @ G3 ) ) ) ) ) ).
% image2_eqI
thf(fact_1204_SN__restrict__SN__idemp,axiom,
! [A: set_Pr4386577575007340137rm_a_b] : ( abstra4720023341729745482rm_a_b @ ( abstra4933154510623179783rm_a_b @ A @ A ) @ top_top_set_term_a_b ) ).
% SN_restrict_SN_idemp
thf(fact_1205_SN__restrict__SN__idemp,axiom,
! [A: set_Pr1811044260758604347_a_nat] : ( abstra8653715922312955827_a_nat @ ( abstra7212103836473851632_a_nat @ A @ A ) @ top_to3353692345378799459_a_nat ) ).
% SN_restrict_SN_idemp
thf(fact_1206_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_1207_DiffI,axiom,
! [C: term_a_b,A: set_term_a_b,B: set_term_a_b] :
( ( member_term_a_b @ C @ A )
=> ( ~ ( member_term_a_b @ C @ B )
=> ( member_term_a_b @ C @ ( minus_4613853266328121507rm_a_b @ A @ B ) ) ) ) ).
% DiffI
thf(fact_1208_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_1209_Diff__iff,axiom,
! [C: term_a_b,A: set_term_a_b,B: set_term_a_b] :
( ( member_term_a_b @ C @ ( minus_4613853266328121507rm_a_b @ A @ B ) )
= ( ( member_term_a_b @ C @ A )
& ~ ( member_term_a_b @ C @ B ) ) ) ).
% Diff_iff
thf(fact_1210_zero__diff,axiom,
! [A2: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A2 )
= zero_zero_nat ) ).
% zero_diff
thf(fact_1211_diff__zero,axiom,
! [A2: nat] :
( ( minus_minus_nat @ A2 @ zero_zero_nat )
= A2 ) ).
% diff_zero
thf(fact_1212_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A2: nat] :
( ( minus_minus_nat @ A2 @ A2 )
= zero_zero_nat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_1213_Diff__empty,axiom,
! [A: set_nat] :
( ( minus_minus_set_nat @ A @ bot_bot_set_nat )
= A ) ).
% Diff_empty
thf(fact_1214_empty__Diff,axiom,
! [A: set_nat] :
( ( minus_minus_set_nat @ bot_bot_set_nat @ A )
= bot_bot_set_nat ) ).
% empty_Diff
thf(fact_1215_Diff__cancel,axiom,
! [A: set_nat] :
( ( minus_minus_set_nat @ A @ A )
= bot_bot_set_nat ) ).
% Diff_cancel
thf(fact_1216_insert__Diff1,axiom,
! [X: product_prod_a_nat,B: set_Pr4934435412358123699_a_nat,A: set_Pr4934435412358123699_a_nat] :
( ( member5724188588386418708_a_nat @ X @ B )
=> ( ( minus_8630901905721465484_a_nat @ ( insert8054603423593749677_a_nat @ X @ A ) @ B )
= ( minus_8630901905721465484_a_nat @ A @ B ) ) ) ).
% insert_Diff1
thf(fact_1217_insert__Diff1,axiom,
! [X: term_a_b,B: set_term_a_b,A: set_term_a_b] :
( ( member_term_a_b @ X @ B )
=> ( ( minus_4613853266328121507rm_a_b @ ( insert_term_a_b @ X @ A ) @ B )
= ( minus_4613853266328121507rm_a_b @ A @ B ) ) ) ).
% insert_Diff1
thf(fact_1218_Diff__insert0,axiom,
! [X: product_prod_a_nat,A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ~ ( member5724188588386418708_a_nat @ X @ A )
=> ( ( minus_8630901905721465484_a_nat @ A @ ( insert8054603423593749677_a_nat @ X @ B ) )
= ( minus_8630901905721465484_a_nat @ A @ B ) ) ) ).
% Diff_insert0
thf(fact_1219_Diff__insert0,axiom,
! [X: term_a_b,A: set_term_a_b,B: set_term_a_b] :
( ~ ( member_term_a_b @ X @ A )
=> ( ( minus_4613853266328121507rm_a_b @ A @ ( insert_term_a_b @ X @ B ) )
= ( minus_4613853266328121507rm_a_b @ A @ B ) ) ) ).
% Diff_insert0
thf(fact_1220_Diff__UNIV,axiom,
! [A: set_nat] :
( ( minus_minus_set_nat @ A @ top_top_set_nat )
= bot_bot_set_nat ) ).
% Diff_UNIV
thf(fact_1221_Diff__UNIV,axiom,
! [A: set_term_a_b] :
( ( minus_4613853266328121507rm_a_b @ A @ top_top_set_term_a_b )
= bot_bot_set_term_a_b ) ).
% Diff_UNIV
thf(fact_1222_Diff__UNIV,axiom,
! [A: set_Pr4934435412358123699_a_nat] :
( ( minus_8630901905721465484_a_nat @ A @ top_to3353692345378799459_a_nat )
= bot_bo9049108969261143879_a_nat ) ).
% Diff_UNIV
thf(fact_1223_Diff__eq__empty__iff,axiom,
! [A: set_nat,B: set_nat] :
( ( ( minus_minus_set_nat @ A @ B )
= bot_bot_set_nat )
= ( ord_less_eq_set_nat @ A @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_1224_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_1225_insert__Diff__single,axiom,
! [A2: product_prod_a_nat,A: set_Pr4934435412358123699_a_nat] :
( ( insert8054603423593749677_a_nat @ A2 @ ( minus_8630901905721465484_a_nat @ A @ ( insert8054603423593749677_a_nat @ A2 @ bot_bo9049108969261143879_a_nat ) ) )
= ( insert8054603423593749677_a_nat @ A2 @ A ) ) ).
% insert_Diff_single
thf(fact_1226_insert__Diff__single,axiom,
! [A2: nat,A: set_nat] :
( ( insert_nat @ A2 @ ( minus_minus_set_nat @ A @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) )
= ( insert_nat @ A2 @ A ) ) ).
% insert_Diff_single
thf(fact_1227_Diff__disjoint,axiom,
! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ( inf_in1693821330460682071rm_a_b @ A @ ( minus_5192120951422937424rm_a_b @ B @ A ) )
= bot_bo197521221353338581rm_a_b ) ).
% Diff_disjoint
thf(fact_1228_Diff__disjoint,axiom,
! [A: set_nat,B: set_nat] :
( ( inf_inf_set_nat @ A @ ( minus_minus_set_nat @ B @ A ) )
= bot_bot_set_nat ) ).
% Diff_disjoint
thf(fact_1229_Sigma__Diff__distrib2,axiom,
! [I: set_term_a_b,A: term_a_b > set_term_a_b,B: term_a_b > set_term_a_b] :
( ( produc1953217797448512882rm_a_b @ I
@ ^ [I2: term_a_b] : ( minus_4613853266328121507rm_a_b @ ( A @ I2 ) @ ( B @ I2 ) ) )
= ( minus_5192120951422937424rm_a_b @ ( produc1953217797448512882rm_a_b @ I @ A ) @ ( produc1953217797448512882rm_a_b @ I @ B ) ) ) ).
% Sigma_Diff_distrib2
thf(fact_1230_Times__Diff__distrib1,axiom,
! [A: set_term_a_b,B: set_term_a_b,C2: set_term_a_b] :
( ( produc1953217797448512882rm_a_b @ ( minus_4613853266328121507rm_a_b @ A @ B )
@ ^ [Uu: term_a_b] : C2 )
= ( minus_5192120951422937424rm_a_b
@ ( produc1953217797448512882rm_a_b @ A
@ ^ [Uu: term_a_b] : C2 )
@ ( produc1953217797448512882rm_a_b @ B
@ ^ [Uu: term_a_b] : C2 ) ) ) ).
% Times_Diff_distrib1
thf(fact_1231_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_1232_DiffE,axiom,
! [C: term_a_b,A: set_term_a_b,B: set_term_a_b] :
( ( member_term_a_b @ C @ ( minus_4613853266328121507rm_a_b @ A @ B ) )
=> ~ ( ( member_term_a_b @ C @ A )
=> ( member_term_a_b @ C @ B ) ) ) ).
% DiffE
thf(fact_1233_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_1234_DiffD1,axiom,
! [C: term_a_b,A: set_term_a_b,B: set_term_a_b] :
( ( member_term_a_b @ C @ ( minus_4613853266328121507rm_a_b @ A @ B ) )
=> ( member_term_a_b @ C @ A ) ) ).
% DiffD1
thf(fact_1235_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_1236_DiffD2,axiom,
! [C: term_a_b,A: set_term_a_b,B: set_term_a_b] :
( ( member_term_a_b @ C @ ( minus_4613853266328121507rm_a_b @ A @ B ) )
=> ~ ( member_term_a_b @ C @ B ) ) ).
% DiffD2
thf(fact_1237_set__diff__eq,axiom,
( minus_8630901905721465484_a_nat
= ( ^ [A5: set_Pr4934435412358123699_a_nat,B5: set_Pr4934435412358123699_a_nat] :
( collec4464134535221767506_a_nat
@ ^ [X2: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ X2 @ A5 )
& ~ ( member5724188588386418708_a_nat @ X2 @ B5 ) ) ) ) ) ).
% set_diff_eq
thf(fact_1238_set__diff__eq,axiom,
( minus_4613853266328121507rm_a_b
= ( ^ [A5: set_term_a_b,B5: set_term_a_b] :
( collect_term_a_b
@ ^ [X2: term_a_b] :
( ( member_term_a_b @ X2 @ A5 )
& ~ ( member_term_a_b @ X2 @ B5 ) ) ) ) ) ).
% set_diff_eq
thf(fact_1239_Sigma__Diff__distrib1,axiom,
! [I: set_term_a_b,J: set_term_a_b,C2: term_a_b > set_term_a_b] :
( ( produc1953217797448512882rm_a_b @ ( minus_4613853266328121507rm_a_b @ I @ J ) @ C2 )
= ( minus_5192120951422937424rm_a_b @ ( produc1953217797448512882rm_a_b @ I @ C2 ) @ ( produc1953217797448512882rm_a_b @ J @ C2 ) ) ) ).
% Sigma_Diff_distrib1
thf(fact_1240_Int__Diff,axiom,
! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b,C2: set_Pr4386577575007340137rm_a_b] :
( ( minus_5192120951422937424rm_a_b @ ( inf_in1693821330460682071rm_a_b @ A @ B ) @ C2 )
= ( inf_in1693821330460682071rm_a_b @ A @ ( minus_5192120951422937424rm_a_b @ B @ C2 ) ) ) ).
% Int_Diff
thf(fact_1241_Diff__Int2,axiom,
! [A: set_Pr4386577575007340137rm_a_b,C2: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ( minus_5192120951422937424rm_a_b @ ( inf_in1693821330460682071rm_a_b @ A @ C2 ) @ ( inf_in1693821330460682071rm_a_b @ B @ C2 ) )
= ( minus_5192120951422937424rm_a_b @ ( inf_in1693821330460682071rm_a_b @ A @ C2 ) @ B ) ) ).
% Diff_Int2
thf(fact_1242_Diff__Diff__Int,axiom,
! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ( minus_5192120951422937424rm_a_b @ A @ ( minus_5192120951422937424rm_a_b @ A @ B ) )
= ( inf_in1693821330460682071rm_a_b @ A @ B ) ) ).
% Diff_Diff_Int
thf(fact_1243_Diff__Int__distrib,axiom,
! [C2: set_Pr4386577575007340137rm_a_b,A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ( inf_in1693821330460682071rm_a_b @ C2 @ ( minus_5192120951422937424rm_a_b @ A @ B ) )
= ( minus_5192120951422937424rm_a_b @ ( inf_in1693821330460682071rm_a_b @ C2 @ A ) @ ( inf_in1693821330460682071rm_a_b @ C2 @ B ) ) ) ).
% Diff_Int_distrib
thf(fact_1244_Diff__Int__distrib2,axiom,
! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b,C2: set_Pr4386577575007340137rm_a_b] :
( ( inf_in1693821330460682071rm_a_b @ ( minus_5192120951422937424rm_a_b @ A @ B ) @ C2 )
= ( minus_5192120951422937424rm_a_b @ ( inf_in1693821330460682071rm_a_b @ A @ C2 ) @ ( inf_in1693821330460682071rm_a_b @ B @ C2 ) ) ) ).
% Diff_Int_distrib2
thf(fact_1245_diff__shunt__var,axiom,
! [X: set_nat,Y: set_nat] :
( ( ( minus_minus_set_nat @ X @ Y )
= bot_bot_set_nat )
= ( ord_less_eq_set_nat @ X @ Y ) ) ).
% diff_shunt_var
thf(fact_1246_diff__shunt__var,axiom,
! [X: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat] :
( ( ( minus_8630901905721465484_a_nat @ X @ Y )
= bot_bo9049108969261143879_a_nat )
= ( ord_le8666007276011122963_a_nat @ X @ Y ) ) ).
% diff_shunt_var
thf(fact_1247_Diff__insert,axiom,
! [A: set_Pr4934435412358123699_a_nat,A2: product_prod_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( minus_8630901905721465484_a_nat @ A @ ( insert8054603423593749677_a_nat @ A2 @ B ) )
= ( minus_8630901905721465484_a_nat @ ( minus_8630901905721465484_a_nat @ A @ B ) @ ( insert8054603423593749677_a_nat @ A2 @ bot_bo9049108969261143879_a_nat ) ) ) ).
% Diff_insert
thf(fact_1248_Diff__insert,axiom,
! [A: set_nat,A2: nat,B: set_nat] :
( ( minus_minus_set_nat @ A @ ( insert_nat @ A2 @ B ) )
= ( minus_minus_set_nat @ ( minus_minus_set_nat @ A @ B ) @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) ) ).
% Diff_insert
thf(fact_1249_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_1250_insert__Diff,axiom,
! [A2: term_a_b,A: set_term_a_b] :
( ( member_term_a_b @ A2 @ A )
=> ( ( insert_term_a_b @ A2 @ ( minus_4613853266328121507rm_a_b @ A @ ( insert_term_a_b @ A2 @ bot_bot_set_term_a_b ) ) )
= A ) ) ).
% insert_Diff
thf(fact_1251_insert__Diff,axiom,
! [A2: nat,A: set_nat] :
( ( member_nat @ A2 @ A )
=> ( ( insert_nat @ A2 @ ( minus_minus_set_nat @ A @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) )
= A ) ) ).
% insert_Diff
thf(fact_1252_Diff__insert2,axiom,
! [A: set_Pr4934435412358123699_a_nat,A2: product_prod_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( minus_8630901905721465484_a_nat @ A @ ( insert8054603423593749677_a_nat @ A2 @ B ) )
= ( minus_8630901905721465484_a_nat @ ( minus_8630901905721465484_a_nat @ A @ ( insert8054603423593749677_a_nat @ A2 @ bot_bo9049108969261143879_a_nat ) ) @ B ) ) ).
% Diff_insert2
thf(fact_1253_Diff__insert2,axiom,
! [A: set_nat,A2: nat,B: set_nat] :
( ( minus_minus_set_nat @ A @ ( insert_nat @ A2 @ B ) )
= ( minus_minus_set_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) @ B ) ) ).
% Diff_insert2
thf(fact_1254_Diff__insert__absorb,axiom,
! [X: product_prod_a_nat,A: set_Pr4934435412358123699_a_nat] :
( ~ ( member5724188588386418708_a_nat @ X @ A )
=> ( ( minus_8630901905721465484_a_nat @ ( insert8054603423593749677_a_nat @ X @ A ) @ ( insert8054603423593749677_a_nat @ X @ bot_bo9049108969261143879_a_nat ) )
= A ) ) ).
% Diff_insert_absorb
thf(fact_1255_Diff__insert__absorb,axiom,
! [X: term_a_b,A: set_term_a_b] :
( ~ ( member_term_a_b @ X @ A )
=> ( ( minus_4613853266328121507rm_a_b @ ( insert_term_a_b @ X @ A ) @ ( insert_term_a_b @ X @ bot_bot_set_term_a_b ) )
= A ) ) ).
% Diff_insert_absorb
thf(fact_1256_Diff__insert__absorb,axiom,
! [X: nat,A: set_nat] :
( ~ ( member_nat @ X @ A )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X @ A ) @ ( insert_nat @ X @ bot_bot_set_nat ) )
= A ) ) ).
% Diff_insert_absorb
thf(fact_1257_Diff__triv,axiom,
! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ( ( inf_in1693821330460682071rm_a_b @ A @ B )
= bot_bo197521221353338581rm_a_b )
=> ( ( minus_5192120951422937424rm_a_b @ A @ B )
= A ) ) ).
% Diff_triv
thf(fact_1258_Diff__triv,axiom,
! [A: set_nat,B: set_nat] :
( ( ( inf_inf_set_nat @ A @ B )
= bot_bot_set_nat )
=> ( ( minus_minus_set_nat @ A @ B )
= A ) ) ).
% Diff_triv
thf(fact_1259_Int__Diff__disjoint,axiom,
! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ( inf_in1693821330460682071rm_a_b @ ( inf_in1693821330460682071rm_a_b @ A @ B ) @ ( minus_5192120951422937424rm_a_b @ A @ B ) )
= bot_bo197521221353338581rm_a_b ) ).
% Int_Diff_disjoint
thf(fact_1260_Int__Diff__disjoint,axiom,
! [A: set_nat,B: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ A @ B ) @ ( minus_minus_set_nat @ A @ B ) )
= bot_bot_set_nat ) ).
% Int_Diff_disjoint
thf(fact_1261_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_1262_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_1263_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_1264_insert__Diff__if,axiom,
! [X: product_prod_a_nat,B: set_Pr4934435412358123699_a_nat,A: set_Pr4934435412358123699_a_nat] :
( ( ( member5724188588386418708_a_nat @ X @ B )
=> ( ( minus_8630901905721465484_a_nat @ ( insert8054603423593749677_a_nat @ X @ A ) @ B )
= ( minus_8630901905721465484_a_nat @ A @ B ) ) )
& ( ~ ( member5724188588386418708_a_nat @ X @ B )
=> ( ( minus_8630901905721465484_a_nat @ ( insert8054603423593749677_a_nat @ X @ A ) @ B )
= ( insert8054603423593749677_a_nat @ X @ ( minus_8630901905721465484_a_nat @ A @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_1265_insert__Diff__if,axiom,
! [X: term_a_b,B: set_term_a_b,A: set_term_a_b] :
( ( ( member_term_a_b @ X @ B )
=> ( ( minus_4613853266328121507rm_a_b @ ( insert_term_a_b @ X @ A ) @ B )
= ( minus_4613853266328121507rm_a_b @ A @ B ) ) )
& ( ~ ( member_term_a_b @ X @ B )
=> ( ( minus_4613853266328121507rm_a_b @ ( insert_term_a_b @ X @ A ) @ B )
= ( insert_term_a_b @ X @ ( minus_4613853266328121507rm_a_b @ A @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_1266_Un__Diff__Int,axiom,
! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ( sup_su6776935440552674877rm_a_b @ ( minus_5192120951422937424rm_a_b @ A @ B ) @ ( inf_in1693821330460682071rm_a_b @ A @ B ) )
= A ) ).
% Un_Diff_Int
thf(fact_1267_Int__Diff__Un,axiom,
! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b] :
( ( sup_su6776935440552674877rm_a_b @ ( inf_in1693821330460682071rm_a_b @ A @ B ) @ ( minus_5192120951422937424rm_a_b @ A @ B ) )
= A ) ).
% Int_Diff_Un
thf(fact_1268_Diff__Int,axiom,
! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b,C2: set_Pr4386577575007340137rm_a_b] :
( ( minus_5192120951422937424rm_a_b @ A @ ( inf_in1693821330460682071rm_a_b @ B @ C2 ) )
= ( sup_su6776935440552674877rm_a_b @ ( minus_5192120951422937424rm_a_b @ A @ B ) @ ( minus_5192120951422937424rm_a_b @ A @ C2 ) ) ) ).
% Diff_Int
thf(fact_1269_Diff__Un,axiom,
! [A: set_Pr4386577575007340137rm_a_b,B: set_Pr4386577575007340137rm_a_b,C2: set_Pr4386577575007340137rm_a_b] :
( ( minus_5192120951422937424rm_a_b @ A @ ( sup_su6776935440552674877rm_a_b @ B @ C2 ) )
= ( inf_in1693821330460682071rm_a_b @ ( minus_5192120951422937424rm_a_b @ A @ B ) @ ( minus_5192120951422937424rm_a_b @ A @ C2 ) ) ) ).
% Diff_Un
thf(fact_1270_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_1271_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_1272_subset__Diff__insert,axiom,
! [A: set_term_a_b,B: set_term_a_b,X: term_a_b,C2: set_term_a_b] :
( ( ord_le2705286416250468010rm_a_b @ A @ ( minus_4613853266328121507rm_a_b @ B @ ( insert_term_a_b @ X @ C2 ) ) )
= ( ( ord_le2705286416250468010rm_a_b @ A @ ( minus_4613853266328121507rm_a_b @ B @ C2 ) )
& ~ ( member_term_a_b @ X @ A ) ) ) ).
% subset_Diff_insert
thf(fact_1273_subset__Diff__insert,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat,X: product_prod_a_nat,C2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A @ ( minus_8630901905721465484_a_nat @ B @ ( insert8054603423593749677_a_nat @ X @ C2 ) ) )
= ( ( ord_le8666007276011122963_a_nat @ A @ ( minus_8630901905721465484_a_nat @ B @ C2 ) )
& ~ ( member5724188588386418708_a_nat @ X @ A ) ) ) ).
% subset_Diff_insert
% Conjectures (1)
thf(conj_0,conjecture,
abstra3619855735679963878rm_a_b @ ( sig_step_a_b @ f @ ( rstep_a_b @ r ) ) @ top_top_set_term_a_b ).
%------------------------------------------------------------------------------