TPTP Problem File: SLH0146^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : SCC_Bloemen_Sequential/0000_SCC_Bloemen_Sequential/prob_01046_036219__5853196_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1520 ( 614 unt; 231 typ; 0 def)
% Number of atoms : 3698 (1347 equ; 0 cnn)
% Maximal formula atoms : 10 ( 2 avg)
% Number of connectives : 12149 ( 442 ~; 54 |; 344 &;9736 @)
% ( 0 <=>;1573 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 7 avg)
% Number of types : 28 ( 27 usr)
% Number of type conns : 620 ( 620 >; 0 *; 0 +; 0 <<)
% Number of symbols : 207 ( 204 usr; 20 con; 0-9 aty)
% Number of variables : 3703 ( 132 ^;3394 !; 177 ?;3703 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 08:53:19.672
%------------------------------------------------------------------------------
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thf(sy_c_Set_Oinsert_001t__List__Olist_Itf__v_J,type,
insert_list_v: list_v > set_list_v > set_list_v ).
thf(sy_c_Set_Oinsert_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__v_Mtf__v_J_Mt__Product____Type__Oprod_Itf__v_Mtf__v_J_J,type,
insert5641704497130386615od_v_v: produc206430290419586791od_v_v > set_Pr2149350503807050951od_v_v > set_Pr2149350503807050951od_v_v ).
thf(sy_c_Set_Oinsert_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
insert1338601472111419319od_v_v: product_prod_v_v > set_Product_prod_v_v > set_Product_prod_v_v ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_It__Product____Type__Oprod_Itf__v_Mtf__v_J_J,type,
insert7504383016908236695od_v_v: set_Product_prod_v_v > set_se8455005133513928103od_v_v > set_se8455005133513928103od_v_v ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_Itf__v_J,type,
insert_set_v: set_v > set_set_v > set_set_v ).
thf(sy_c_Set_Oinsert_001tf__v,type,
insert_v2: v > set_v > set_v ).
thf(sy_c_Set_Ois__empty_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
is_emp8964507351669718201od_v_v: set_Product_prod_v_v > $o ).
thf(sy_c_Set_Ois__empty_001tf__v,type,
is_empty_v: set_v > $o ).
thf(sy_c_Set_Ois__singleton_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
is_sin9198872032823709915od_v_v: set_Product_prod_v_v > $o ).
thf(sy_c_Set_Ois__singleton_001tf__v,type,
is_singleton_v: set_v > $o ).
thf(sy_c_Set_Oremove_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
remove5001965847480235980od_v_v: product_prod_v_v > set_Product_prod_v_v > set_Product_prod_v_v ).
thf(sy_c_Set_Oremove_001tf__v,type,
remove_v: v > set_v > set_v ).
thf(sy_c_Set_Othe__elem_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
the_el5392834299063928540od_v_v: set_Product_prod_v_v > product_prod_v_v ).
thf(sy_c_Set_Othe__elem_001tf__v,type,
the_elem_v: set_v > v ).
thf(sy_c_member_001t__List__Olist_It__Product____Type__Oprod_Itf__v_Mtf__v_J_J,type,
member4190458934886417558od_v_v: list_P7986770385144383213od_v_v > set_li2323639185124838733od_v_v > $o ).
thf(sy_c_member_001t__List__Olist_Itf__v_J,type,
member_list_v: list_v > set_list_v > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_Itf__v_Mtf__v_J_J_Mt__List__Olist_It__Product____Type__Oprod_Itf__v_Mtf__v_J_J_J,type,
member6382463057129219728od_v_v: produc1504107476793160551od_v_v > set_Pr7499474215547700295od_v_v > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_Itf__v_J_Mt__List__Olist_Itf__v_J_J,type,
member418487059593946000list_v: produc1391462591744249447list_v > set_Pr6206931691796273479list_v > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__v_Mtf__v_J_Mt__Product____Type__Oprod_Itf__v_Mtf__v_J_J,type,
member3038538357316246288od_v_v: produc206430290419586791od_v_v > set_Pr2149350503807050951od_v_v > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__v_Mtf__v_J_Mtf__v_J,type,
member5544786109013881916_v_v_v: produc6517761641164185363_v_v_v > set_Pr7679524143894959091_v_v_v > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_Itf__v_Mt__Product____Type__Oprod_Itf__v_Mtf__v_J_J,type,
member5456077685714336484od_v_v: produc6429053217864639931od_v_v > set_Pr7862341151230101147od_v_v > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
member7453568604450474000od_v_v: product_prod_v_v > set_Product_prod_v_v > $o ).
thf(sy_c_member_001t__Set__Oset_It__Product____Type__Oprod_Itf__v_Mtf__v_J_J,type,
member8406446414694345712od_v_v: set_Product_prod_v_v > set_se8455005133513928103od_v_v > $o ).
thf(sy_c_member_001t__Set__Oset_Itf__v_J,type,
member_set_v: set_v > set_set_v > $o ).
thf(sy_c_member_001tf__v,type,
member_v: v > set_v > $o ).
thf(sy_v_e,type,
e: sCC_Bl1394983891496994913t_unit ).
thf(sy_v_e_H,type,
e2: sCC_Bl1394983891496994913t_unit ).
thf(sy_v_successors,type,
successors: v > set_v ).
thf(sy_v_v,type,
v2: v ).
thf(sy_v_vertices,type,
vertices: set_v ).
thf(sy_v_w,type,
w: v ).
% Relevant facts (1279)
thf(fact_0_pre,axiom,
sCC_Bl1748261141445803503t_unit @ successors @ v2 @ e ).
% pre
thf(fact_1_w_I1_J,axiom,
member_v @ w @ ( successors @ v2 ) ).
% w(1)
thf(fact_2_graph_Owf__env_Ocong,axiom,
sCC_Bl9196236973127232072t_unit = sCC_Bl9196236973127232072t_unit ).
% graph.wf_env.cong
thf(fact_3_S__reflexive,axiom,
! [E: sCC_Bl1394983891496994913t_unit,N: v] :
( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
=> ( member_v @ N @ ( sCC_Bl1280885523602775798t_unit @ E @ N ) ) ) ).
% S_reflexive
thf(fact_4_init__env__pre__dfs,axiom,
! [V: v] : ( sCC_Bl36166008131615352t_unit @ successors @ V @ ( sCC_Bl7693227186847904995_env_v @ V ) ) ).
% init_env_pre_dfs
thf(fact_5_w_I4_J,axiom,
~ ( member_v @ w @ ( sCC_Bl157864678168468314t_unit @ e ) ) ).
% w(4)
thf(fact_6_reachable__end_Ocases,axiom,
! [A1: v,A2: v] :
( ( sCC_Bl770211535891879572_end_v @ successors @ A1 @ A2 )
=> ( ( A2 != A1 )
=> ~ ! [Y: v] :
( ( sCC_Bl770211535891879572_end_v @ successors @ A1 @ Y )
=> ~ ( member_v @ A2 @ ( successors @ Y ) ) ) ) ) ).
% reachable_end.cases
thf(fact_7_re__refl,axiom,
! [X: v] : ( sCC_Bl770211535891879572_end_v @ successors @ X @ X ) ).
% re_refl
thf(fact_8_re__succ,axiom,
! [X: v,Y2: v,Z: v] :
( ( sCC_Bl770211535891879572_end_v @ successors @ X @ Y2 )
=> ( ( member_v @ Z @ ( successors @ Y2 ) )
=> ( sCC_Bl770211535891879572_end_v @ successors @ X @ Z ) ) ) ).
% re_succ
thf(fact_9_reachable__end_Osimps,axiom,
! [A1: v,A2: v] :
( ( sCC_Bl770211535891879572_end_v @ successors @ A1 @ A2 )
= ( ? [X2: v] :
( ( A1 = X2 )
& ( A2 = X2 ) )
| ? [X2: v,Y3: v,Z2: v] :
( ( A1 = X2 )
& ( A2 = Z2 )
& ( sCC_Bl770211535891879572_end_v @ successors @ X2 @ Y3 )
& ( member_v @ Z2 @ ( successors @ Y3 ) ) ) ) ) ).
% reachable_end.simps
thf(fact_10_succ__re,axiom,
! [Y2: v,X: v,Z: v] :
( ( member_v @ Y2 @ ( successors @ X ) )
=> ( ( sCC_Bl770211535891879572_end_v @ successors @ Y2 @ Z )
=> ( sCC_Bl770211535891879572_end_v @ successors @ X @ Z ) ) ) ).
% succ_re
thf(fact_11_w_I3_J,axiom,
member_v @ w @ ( sCC_Bl4645233313691564917t_unit @ e ) ).
% w(3)
thf(fact_12_e_H__def,axiom,
( e2
= ( sCC_Bloemen_unite_v @ v2 @ w @ e ) ) ).
% e'_def
thf(fact_13_w_I2_J,axiom,
~ ( member_v @ w @ ( sCC_Bl3795065053823578884t_unit @ e @ v2 ) ) ).
% w(2)
thf(fact_14_graph_Oreachable__end_Ocong,axiom,
sCC_Bl770211535891879572_end_v = sCC_Bl770211535891879572_end_v ).
% graph.reachable_end.cong
thf(fact_15_graph_Opre__dfs_Ocong,axiom,
sCC_Bl36166008131615352t_unit = sCC_Bl36166008131615352t_unit ).
% graph.pre_dfs.cong
thf(fact_16_graph_Opre__dfss_Ocong,axiom,
sCC_Bl1748261141445803503t_unit = sCC_Bl1748261141445803503t_unit ).
% graph.pre_dfss.cong
thf(fact_17_reachable__re,axiom,
! [X: v,Y2: v] :
( ( sCC_Bl649662514949026229able_v @ successors @ X @ Y2 )
=> ( sCC_Bl770211535891879572_end_v @ successors @ X @ Y2 ) ) ).
% reachable_re
thf(fact_18_re__reachable,axiom,
! [X: v,Y2: v] :
( ( sCC_Bl770211535891879572_end_v @ successors @ X @ Y2 )
=> ( sCC_Bl649662514949026229able_v @ successors @ X @ Y2 ) ) ).
% re_reachable
thf(fact_19_ra__trans,axiom,
! [X: v,Y2: v,E2: set_Product_prod_v_v,Z: v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Y2 @ E2 )
=> ( ( sCC_Bl4291963740693775144ding_v @ successors @ Y2 @ Z @ E2 )
=> ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Z @ E2 ) ) ) ).
% ra_trans
thf(fact_20_ra__refl,axiom,
! [X: v,E2: set_Product_prod_v_v] : ( sCC_Bl4291963740693775144ding_v @ successors @ X @ X @ E2 ) ).
% ra_refl
thf(fact_21_visited__unexplored,axiom,
! [E: sCC_Bl1394983891496994913t_unit,M: v] :
( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
=> ( ( member_v @ M @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ~ ( member_v @ M @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ~ ! [N2: v] :
( ( member_v @ N2 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) )
=> ~ ( member_v @ M @ ( sCC_Bl1280885523602775798t_unit @ E @ N2 ) ) ) ) ) ) ).
% visited_unexplored
thf(fact_22_graph__axioms,axiom,
sCC_Bloemen_graph_v @ vertices @ successors ).
% graph_axioms
thf(fact_23_succ__reachable,axiom,
! [X: v,Y2: v,Z: v] :
( ( sCC_Bl649662514949026229able_v @ successors @ X @ Y2 )
=> ( ( member_v @ Z @ ( successors @ Y2 ) )
=> ( sCC_Bl649662514949026229able_v @ successors @ X @ Z ) ) ) ).
% succ_reachable
thf(fact_24_reachable__trans,axiom,
! [X: v,Y2: v,Z: v] :
( ( sCC_Bl649662514949026229able_v @ successors @ X @ Y2 )
=> ( ( sCC_Bl649662514949026229able_v @ successors @ Y2 @ Z )
=> ( sCC_Bl649662514949026229able_v @ successors @ X @ Z ) ) ) ).
% reachable_trans
thf(fact_25_reachable__end__induct,axiom,
! [X: v,Y2: v,P: v > v > $o] :
( ( sCC_Bl649662514949026229able_v @ successors @ X @ Y2 )
=> ( ! [X3: v] : ( P @ X3 @ X3 )
=> ( ! [X3: v,Y: v,Z3: v] :
( ( P @ X3 @ Y )
=> ( ( member_v @ Z3 @ ( successors @ Y ) )
=> ( P @ X3 @ Z3 ) ) )
=> ( P @ X @ Y2 ) ) ) ) ).
% reachable_end_induct
thf(fact_26_reachable__edge,axiom,
! [Y2: v,X: v] :
( ( member_v @ Y2 @ ( successors @ X ) )
=> ( sCC_Bl649662514949026229able_v @ successors @ X @ Y2 ) ) ).
% reachable_edge
thf(fact_27_reachable_Osimps,axiom,
! [A1: v,A2: v] :
( ( sCC_Bl649662514949026229able_v @ successors @ A1 @ A2 )
= ( ? [X2: v] :
( ( A1 = X2 )
& ( A2 = X2 ) )
| ? [X2: v,Y3: v,Z2: v] :
( ( A1 = X2 )
& ( A2 = Z2 )
& ( member_v @ Y3 @ ( successors @ X2 ) )
& ( sCC_Bl649662514949026229able_v @ successors @ Y3 @ Z2 ) ) ) ) ).
% reachable.simps
thf(fact_28_reachable__succ,axiom,
! [Y2: v,X: v,Z: v] :
( ( member_v @ Y2 @ ( successors @ X ) )
=> ( ( sCC_Bl649662514949026229able_v @ successors @ Y2 @ Z )
=> ( sCC_Bl649662514949026229able_v @ successors @ X @ Z ) ) ) ).
% reachable_succ
thf(fact_29_reachable_Ocases,axiom,
! [A1: v,A2: v] :
( ( sCC_Bl649662514949026229able_v @ successors @ A1 @ A2 )
=> ( ( A2 != A1 )
=> ~ ! [Y: v] :
( ( member_v @ Y @ ( successors @ A1 ) )
=> ~ ( sCC_Bl649662514949026229able_v @ successors @ Y @ A2 ) ) ) ) ).
% reachable.cases
thf(fact_30_reachable__refl,axiom,
! [X: v] : ( sCC_Bl649662514949026229able_v @ successors @ X @ X ) ).
% reachable_refl
thf(fact_31_ra__reachable,axiom,
! [X: v,Y2: v,E2: set_Product_prod_v_v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Y2 @ E2 )
=> ( sCC_Bl649662514949026229able_v @ successors @ X @ Y2 ) ) ).
% ra_reachable
thf(fact_32_stack__visited,axiom,
! [E: sCC_Bl1394983891496994913t_unit,N: v] :
( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
=> ( ( member_v @ N @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) )
=> ( member_v @ N @ ( sCC_Bl4645233313691564917t_unit @ E ) ) ) ) ).
% stack_visited
thf(fact_33_stack__unexplored,axiom,
! [E: sCC_Bl1394983891496994913t_unit,N: v] :
( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
=> ( ( member_v @ N @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) )
=> ~ ( member_v @ N @ ( sCC_Bl157864678168468314t_unit @ E ) ) ) ) ).
% stack_unexplored
thf(fact_34_sccE,axiom,
! [S: set_v,X: v,Y2: v] :
( ( sCC_Bloemen_is_scc_v @ successors @ S )
=> ( ( member_v @ X @ S )
=> ( ( sCC_Bl649662514949026229able_v @ successors @ X @ Y2 )
=> ( ( sCC_Bl649662514949026229able_v @ successors @ Y2 @ X )
=> ( member_v @ Y2 @ S ) ) ) ) ) ).
% sccE
thf(fact_35_graph_Ora__refl,axiom,
! [Vertices: set_v,Successors: v > set_v,X: v,E2: set_Product_prod_v_v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ X @ E2 ) ) ).
% graph.ra_refl
thf(fact_36_graph_Oreachable__succ,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,Y2: product_prod_v_v,X: product_prod_v_v,Z: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( member7453568604450474000od_v_v @ Y2 @ ( Successors @ X ) )
=> ( ( sCC_Bl4981926079593201289od_v_v @ Successors @ Y2 @ Z )
=> ( sCC_Bl4981926079593201289od_v_v @ Successors @ X @ Z ) ) ) ) ).
% graph.reachable_succ
thf(fact_37_graph_Oreachable__succ,axiom,
! [Vertices: set_v,Successors: v > set_v,Y2: v,X: v,Z: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( member_v @ Y2 @ ( Successors @ X ) )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ Y2 @ Z )
=> ( sCC_Bl649662514949026229able_v @ Successors @ X @ Z ) ) ) ) ).
% graph.reachable_succ
thf(fact_38_graph_Oreachable__refl,axiom,
! [Vertices: set_v,Successors: v > set_v,X: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( sCC_Bl649662514949026229able_v @ Successors @ X @ X ) ) ).
% graph.reachable_refl
thf(fact_39_graph_Oreachable__avoiding_Ocong,axiom,
sCC_Bl4291963740693775144ding_v = sCC_Bl4291963740693775144ding_v ).
% graph.reachable_avoiding.cong
thf(fact_40_graph_Oreachable__end__induct,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,X: product_prod_v_v,Y2: product_prod_v_v,P: product_prod_v_v > product_prod_v_v > $o] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl4981926079593201289od_v_v @ Successors @ X @ Y2 )
=> ( ! [X3: product_prod_v_v] : ( P @ X3 @ X3 )
=> ( ! [X3: product_prod_v_v,Y: product_prod_v_v,Z3: product_prod_v_v] :
( ( P @ X3 @ Y )
=> ( ( member7453568604450474000od_v_v @ Z3 @ ( Successors @ Y ) )
=> ( P @ X3 @ Z3 ) ) )
=> ( P @ X @ Y2 ) ) ) ) ) ).
% graph.reachable_end_induct
thf(fact_41_graph_Oreachable__end__induct,axiom,
! [Vertices: set_v,Successors: v > set_v,X: v,Y2: v,P: v > v > $o] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ X @ Y2 )
=> ( ! [X3: v] : ( P @ X3 @ X3 )
=> ( ! [X3: v,Y: v,Z3: v] :
( ( P @ X3 @ Y )
=> ( ( member_v @ Z3 @ ( Successors @ Y ) )
=> ( P @ X3 @ Z3 ) ) )
=> ( P @ X @ Y2 ) ) ) ) ) ).
% graph.reachable_end_induct
thf(fact_42_mem__Collect__eq,axiom,
! [A: v,P: v > $o] :
( ( member_v @ A @ ( collect_v @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_43_mem__Collect__eq,axiom,
! [A: product_prod_v_v,P: product_prod_v_v > $o] :
( ( member7453568604450474000od_v_v @ A @ ( collec140062887454715474od_v_v @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_44_Collect__mem__eq,axiom,
! [A3: set_v] :
( ( collect_v
@ ^ [X2: v] : ( member_v @ X2 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_45_Collect__mem__eq,axiom,
! [A3: set_Product_prod_v_v] :
( ( collec140062887454715474od_v_v
@ ^ [X2: product_prod_v_v] : ( member7453568604450474000od_v_v @ X2 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_46_graph_Oreachable__trans,axiom,
! [Vertices: set_v,Successors: v > set_v,X: v,Y2: v,Z: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ X @ Y2 )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ Y2 @ Z )
=> ( sCC_Bl649662514949026229able_v @ Successors @ X @ Z ) ) ) ) ).
% graph.reachable_trans
thf(fact_47_graph_Oreachable_Osimps,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,A1: product_prod_v_v,A2: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl4981926079593201289od_v_v @ Successors @ A1 @ A2 )
= ( ? [X2: product_prod_v_v] :
( ( A1 = X2 )
& ( A2 = X2 ) )
| ? [X2: product_prod_v_v,Y3: product_prod_v_v,Z2: product_prod_v_v] :
( ( A1 = X2 )
& ( A2 = Z2 )
& ( member7453568604450474000od_v_v @ Y3 @ ( Successors @ X2 ) )
& ( sCC_Bl4981926079593201289od_v_v @ Successors @ Y3 @ Z2 ) ) ) ) ) ).
% graph.reachable.simps
thf(fact_48_graph_Oreachable_Osimps,axiom,
! [Vertices: set_v,Successors: v > set_v,A1: v,A2: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ A1 @ A2 )
= ( ? [X2: v] :
( ( A1 = X2 )
& ( A2 = X2 ) )
| ? [X2: v,Y3: v,Z2: v] :
( ( A1 = X2 )
& ( A2 = Z2 )
& ( member_v @ Y3 @ ( Successors @ X2 ) )
& ( sCC_Bl649662514949026229able_v @ Successors @ Y3 @ Z2 ) ) ) ) ) ).
% graph.reachable.simps
thf(fact_49_graph_Oreachable_Ocases,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,A1: product_prod_v_v,A2: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl4981926079593201289od_v_v @ Successors @ A1 @ A2 )
=> ( ( A2 != A1 )
=> ~ ! [Y: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ Y @ ( Successors @ A1 ) )
=> ~ ( sCC_Bl4981926079593201289od_v_v @ Successors @ Y @ A2 ) ) ) ) ) ).
% graph.reachable.cases
thf(fact_50_graph_Oreachable_Ocases,axiom,
! [Vertices: set_v,Successors: v > set_v,A1: v,A2: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ A1 @ A2 )
=> ( ( A2 != A1 )
=> ~ ! [Y: v] :
( ( member_v @ Y @ ( Successors @ A1 ) )
=> ~ ( sCC_Bl649662514949026229able_v @ Successors @ Y @ A2 ) ) ) ) ) ).
% graph.reachable.cases
thf(fact_51_graph_Osucc__reachable,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,X: product_prod_v_v,Y2: product_prod_v_v,Z: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl4981926079593201289od_v_v @ Successors @ X @ Y2 )
=> ( ( member7453568604450474000od_v_v @ Z @ ( Successors @ Y2 ) )
=> ( sCC_Bl4981926079593201289od_v_v @ Successors @ X @ Z ) ) ) ) ).
% graph.succ_reachable
thf(fact_52_graph_Osucc__reachable,axiom,
! [Vertices: set_v,Successors: v > set_v,X: v,Y2: v,Z: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ X @ Y2 )
=> ( ( member_v @ Z @ ( Successors @ Y2 ) )
=> ( sCC_Bl649662514949026229able_v @ Successors @ X @ Z ) ) ) ) ).
% graph.succ_reachable
thf(fact_53_graph_Oreachable__edge,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,Y2: product_prod_v_v,X: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( member7453568604450474000od_v_v @ Y2 @ ( Successors @ X ) )
=> ( sCC_Bl4981926079593201289od_v_v @ Successors @ X @ Y2 ) ) ) ).
% graph.reachable_edge
thf(fact_54_graph_Oreachable__edge,axiom,
! [Vertices: set_v,Successors: v > set_v,Y2: v,X: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( member_v @ Y2 @ ( Successors @ X ) )
=> ( sCC_Bl649662514949026229able_v @ Successors @ X @ Y2 ) ) ) ).
% graph.reachable_edge
thf(fact_55_graph_Oreachable_Ocong,axiom,
sCC_Bl649662514949026229able_v = sCC_Bl649662514949026229able_v ).
% graph.reachable.cong
thf(fact_56_graph_Ora__reachable,axiom,
! [Vertices: set_v,Successors: v > set_v,X: v,Y2: v,E2: set_Product_prod_v_v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ Y2 @ E2 )
=> ( sCC_Bl649662514949026229able_v @ Successors @ X @ Y2 ) ) ) ).
% graph.ra_reachable
thf(fact_57_graph_Ora__trans,axiom,
! [Vertices: set_v,Successors: v > set_v,X: v,Y2: v,E2: set_Product_prod_v_v,Z: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ Y2 @ E2 )
=> ( ( sCC_Bl4291963740693775144ding_v @ Successors @ Y2 @ Z @ E2 )
=> ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ Z @ E2 ) ) ) ) ).
% graph.ra_trans
thf(fact_58_graph_Oreachable__re,axiom,
! [Vertices: set_v,Successors: v > set_v,X: v,Y2: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ X @ Y2 )
=> ( sCC_Bl770211535891879572_end_v @ Successors @ X @ Y2 ) ) ) ).
% graph.reachable_re
thf(fact_59_graph_Ore__reachable,axiom,
! [Vertices: set_v,Successors: v > set_v,X: v,Y2: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl770211535891879572_end_v @ Successors @ X @ Y2 )
=> ( sCC_Bl649662514949026229able_v @ Successors @ X @ Y2 ) ) ) ).
% graph.re_reachable
thf(fact_60_graph_Ostack__visited,axiom,
! [Vertices: set_v,Successors: v > set_v,E: sCC_Bl1394983891496994913t_unit,N: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E )
=> ( ( member_v @ N @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) )
=> ( member_v @ N @ ( sCC_Bl4645233313691564917t_unit @ E ) ) ) ) ) ).
% graph.stack_visited
thf(fact_61_graph_Ostack__unexplored,axiom,
! [Vertices: set_v,Successors: v > set_v,E: sCC_Bl1394983891496994913t_unit,N: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E )
=> ( ( member_v @ N @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) )
=> ~ ( member_v @ N @ ( sCC_Bl157864678168468314t_unit @ E ) ) ) ) ) ).
% graph.stack_unexplored
thf(fact_62_graph_Ore__succ,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,X: product_prod_v_v,Y2: product_prod_v_v,Z: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl4714988717384592488od_v_v @ Successors @ X @ Y2 )
=> ( ( member7453568604450474000od_v_v @ Z @ ( Successors @ Y2 ) )
=> ( sCC_Bl4714988717384592488od_v_v @ Successors @ X @ Z ) ) ) ) ).
% graph.re_succ
thf(fact_63_graph_Ore__succ,axiom,
! [Vertices: set_v,Successors: v > set_v,X: v,Y2: v,Z: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl770211535891879572_end_v @ Successors @ X @ Y2 )
=> ( ( member_v @ Z @ ( Successors @ Y2 ) )
=> ( sCC_Bl770211535891879572_end_v @ Successors @ X @ Z ) ) ) ) ).
% graph.re_succ
thf(fact_64_graph_Ore__refl,axiom,
! [Vertices: set_v,Successors: v > set_v,X: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( sCC_Bl770211535891879572_end_v @ Successors @ X @ X ) ) ).
% graph.re_refl
thf(fact_65_graph_Oreachable__end_Osimps,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,A1: product_prod_v_v,A2: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl4714988717384592488od_v_v @ Successors @ A1 @ A2 )
= ( ? [X2: product_prod_v_v] :
( ( A1 = X2 )
& ( A2 = X2 ) )
| ? [X2: product_prod_v_v,Y3: product_prod_v_v,Z2: product_prod_v_v] :
( ( A1 = X2 )
& ( A2 = Z2 )
& ( sCC_Bl4714988717384592488od_v_v @ Successors @ X2 @ Y3 )
& ( member7453568604450474000od_v_v @ Z2 @ ( Successors @ Y3 ) ) ) ) ) ) ).
% graph.reachable_end.simps
thf(fact_66_graph_Oreachable__end_Osimps,axiom,
! [Vertices: set_v,Successors: v > set_v,A1: v,A2: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl770211535891879572_end_v @ Successors @ A1 @ A2 )
= ( ? [X2: v] :
( ( A1 = X2 )
& ( A2 = X2 ) )
| ? [X2: v,Y3: v,Z2: v] :
( ( A1 = X2 )
& ( A2 = Z2 )
& ( sCC_Bl770211535891879572_end_v @ Successors @ X2 @ Y3 )
& ( member_v @ Z2 @ ( Successors @ Y3 ) ) ) ) ) ) ).
% graph.reachable_end.simps
thf(fact_67_graph_Oreachable__end_Ocases,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,A1: product_prod_v_v,A2: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl4714988717384592488od_v_v @ Successors @ A1 @ A2 )
=> ( ( A2 != A1 )
=> ~ ! [Y: product_prod_v_v] :
( ( sCC_Bl4714988717384592488od_v_v @ Successors @ A1 @ Y )
=> ~ ( member7453568604450474000od_v_v @ A2 @ ( Successors @ Y ) ) ) ) ) ) ).
% graph.reachable_end.cases
thf(fact_68_graph_Oreachable__end_Ocases,axiom,
! [Vertices: set_v,Successors: v > set_v,A1: v,A2: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl770211535891879572_end_v @ Successors @ A1 @ A2 )
=> ( ( A2 != A1 )
=> ~ ! [Y: v] :
( ( sCC_Bl770211535891879572_end_v @ Successors @ A1 @ Y )
=> ~ ( member_v @ A2 @ ( Successors @ Y ) ) ) ) ) ) ).
% graph.reachable_end.cases
thf(fact_69_graph_Osucc__re,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,Y2: product_prod_v_v,X: product_prod_v_v,Z: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( member7453568604450474000od_v_v @ Y2 @ ( Successors @ X ) )
=> ( ( sCC_Bl4714988717384592488od_v_v @ Successors @ Y2 @ Z )
=> ( sCC_Bl4714988717384592488od_v_v @ Successors @ X @ Z ) ) ) ) ).
% graph.succ_re
thf(fact_70_graph_Osucc__re,axiom,
! [Vertices: set_v,Successors: v > set_v,Y2: v,X: v,Z: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( member_v @ Y2 @ ( Successors @ X ) )
=> ( ( sCC_Bl770211535891879572_end_v @ Successors @ Y2 @ Z )
=> ( sCC_Bl770211535891879572_end_v @ Successors @ X @ Z ) ) ) ) ).
% graph.succ_re
thf(fact_71_graph_Ovisited__unexplored,axiom,
! [Vertices: set_v,Successors: v > set_v,E: sCC_Bl1394983891496994913t_unit,M: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E )
=> ( ( member_v @ M @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ~ ( member_v @ M @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ~ ! [N2: v] :
( ( member_v @ N2 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) )
=> ~ ( member_v @ M @ ( sCC_Bl1280885523602775798t_unit @ E @ N2 ) ) ) ) ) ) ) ).
% graph.visited_unexplored
thf(fact_72_graph_OS__reflexive,axiom,
! [Vertices: set_v,Successors: v > set_v,E: sCC_Bl1394983891496994913t_unit,N: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E )
=> ( member_v @ N @ ( sCC_Bl1280885523602775798t_unit @ E @ N ) ) ) ) ).
% graph.S_reflexive
thf(fact_73_graph_Oinit__env__pre__dfs,axiom,
! [Vertices: set_v,Successors: v > set_v,V: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( sCC_Bl36166008131615352t_unit @ Successors @ V @ ( sCC_Bl7693227186847904995_env_v @ V ) ) ) ).
% graph.init_env_pre_dfs
thf(fact_74_is__subscc__def,axiom,
! [S: set_v] :
( ( sCC_Bl5398416737448265317bscc_v @ successors @ S )
= ( ! [X2: v] :
( ( member_v @ X2 @ S )
=> ! [Y3: v] :
( ( member_v @ Y3 @ S )
=> ( sCC_Bl649662514949026229able_v @ successors @ X2 @ Y3 ) ) ) ) ) ).
% is_subscc_def
thf(fact_75_unite__S__tl,axiom,
! [E: sCC_Bl1394983891496994913t_unit,W: v,V: v,N: v] :
( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
=> ( ( member_v @ W @ ( successors @ V ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl3795065053823578884t_unit @ E @ V ) )
=> ( ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ( ( member_v @ N @ ( set_v2 @ ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ ( sCC_Bloemen_unite_v @ V @ W @ E ) ) ) ) )
=> ( ( sCC_Bl1280885523602775798t_unit @ ( sCC_Bloemen_unite_v @ V @ W @ E ) @ N )
= ( sCC_Bl1280885523602775798t_unit @ E @ N ) ) ) ) ) ) ) ) ).
% unite_S_tl
thf(fact_76_stack__class,axiom,
! [E: sCC_Bl1394983891496994913t_unit,N: v,M: v] :
( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
=> ( ( member_v @ N @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) )
=> ( ( member_v @ M @ ( sCC_Bl1280885523602775798t_unit @ E @ N ) )
=> ( member_v @ M @ ( minus_minus_set_v @ ( sCC_Bl4645233313691564917t_unit @ E ) @ ( sCC_Bl157864678168468314t_unit @ E ) ) ) ) ) ) ).
% stack_class
thf(fact_77_reachable__visited,axiom,
! [E: sCC_Bl1394983891496994913t_unit,V: v,W: v] :
( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
=> ( ( member_v @ V @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ( sCC_Bl649662514949026229able_v @ successors @ V @ W )
=> ( ! [X3: v] :
( ( member_v @ X3 @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ! [Xa: v] :
( ( member_v @ Xa @ ( minus_minus_set_v @ ( successors @ X3 ) @ ( sCC_Bl3795065053823578884t_unit @ E @ X3 ) ) )
=> ( ( sCC_Bl649662514949026229able_v @ successors @ V @ X3 )
=> ~ ( sCC_Bl649662514949026229able_v @ successors @ Xa @ W ) ) ) )
=> ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) ) ) ) ) ) ).
% reachable_visited
thf(fact_78_ra__empty,axiom,
! [X: v,Y2: v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Y2 @ bot_bo723834152578015283od_v_v )
= ( sCC_Bl649662514949026229able_v @ successors @ X @ Y2 ) ) ).
% ra_empty
thf(fact_79_sclosed,axiom,
! [X4: v] :
( ( member_v @ X4 @ vertices )
=> ( ord_less_eq_set_v @ ( successors @ X4 ) @ vertices ) ) ).
% sclosed
thf(fact_80_pre__dfs__def,axiom,
! [V: v,E: sCC_Bl1394983891496994913t_unit] :
( ( sCC_Bl36166008131615352t_unit @ successors @ V @ E )
= ( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
& ~ ( member_v @ V @ ( sCC_Bl4645233313691564917t_unit @ E ) )
& ( sCC_Bl649662514949026229able_v @ successors @ ( sCC_Bl1090238580953940555t_unit @ E ) @ V )
& ( ( sCC_Bl3795065053823578884t_unit @ E @ V )
= bot_bot_set_v )
& ! [X2: v] :
( ( member_v @ X2 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) )
=> ( sCC_Bl649662514949026229able_v @ successors @ X2 @ V ) ) ) ) ).
% pre_dfs_def
thf(fact_81_ra__mono,axiom,
! [X: v,Y2: v,E2: set_Product_prod_v_v,E3: set_Product_prod_v_v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Y2 @ E2 )
=> ( ( ord_le7336532860387713383od_v_v @ E3 @ E2 )
=> ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Y2 @ E3 ) ) ) ).
% ra_mono
thf(fact_82_reachable__avoiding_Ocases,axiom,
! [A1: v,A2: v,A32: set_Product_prod_v_v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ A1 @ A2 @ A32 )
=> ( ( A2 != A1 )
=> ~ ! [Y: v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ A1 @ Y @ A32 )
=> ( ( member_v @ A2 @ ( successors @ Y ) )
=> ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ Y @ A2 ) @ A32 ) ) ) ) ) ).
% reachable_avoiding.cases
thf(fact_83_ra__succ,axiom,
! [X: v,Y2: v,E2: set_Product_prod_v_v,Z: v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Y2 @ E2 )
=> ( ( member_v @ Z @ ( successors @ Y2 ) )
=> ( ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ Y2 @ Z ) @ E2 )
=> ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Z @ E2 ) ) ) ) ).
% ra_succ
thf(fact_84_ra__cases,axiom,
! [X: v,Y2: v,E2: set_Product_prod_v_v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Y2 @ E2 )
=> ( ( X = Y2 )
| ? [Z3: v] :
( ( member_v @ Z3 @ ( successors @ X ) )
& ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X @ Z3 ) @ E2 )
& ( sCC_Bl4291963740693775144ding_v @ successors @ Z3 @ Y2 @ E2 ) ) ) ) ).
% ra_cases
thf(fact_85_edge__ra,axiom,
! [Y2: v,X: v,E2: set_Product_prod_v_v] :
( ( member_v @ Y2 @ ( successors @ X ) )
=> ( ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X @ Y2 ) @ E2 )
=> ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Y2 @ E2 ) ) ) ).
% edge_ra
thf(fact_86_reachable__avoiding_Osimps,axiom,
! [A1: v,A2: v,A32: set_Product_prod_v_v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ A1 @ A2 @ A32 )
= ( ? [X2: v,E4: set_Product_prod_v_v] :
( ( A1 = X2 )
& ( A2 = X2 )
& ( A32 = E4 ) )
| ? [X2: v,Y3: v,E4: set_Product_prod_v_v,Z2: v] :
( ( A1 = X2 )
& ( A2 = Z2 )
& ( A32 = E4 )
& ( sCC_Bl4291963740693775144ding_v @ successors @ X2 @ Y3 @ E4 )
& ( member_v @ Z2 @ ( successors @ Y3 ) )
& ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ Y3 @ Z2 ) @ E4 ) ) ) ) ).
% reachable_avoiding.simps
thf(fact_87_is__scc__def,axiom,
! [S: set_v] :
( ( sCC_Bloemen_is_scc_v @ successors @ S )
= ( ( S != bot_bot_set_v )
& ( sCC_Bl5398416737448265317bscc_v @ successors @ S )
& ! [S2: set_v] :
( ( ( ord_less_eq_set_v @ S @ S2 )
& ( sCC_Bl5398416737448265317bscc_v @ successors @ S2 ) )
=> ( S2 = S ) ) ) ) ).
% is_scc_def
thf(fact_88_graph_Ois__scc__def,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,S: set_Product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl6242042402218619277od_v_v @ Successors @ S )
= ( ( S != bot_bo723834152578015283od_v_v )
& ( sCC_Bl2301996248249672505od_v_v @ Successors @ S )
& ! [S2: set_Product_prod_v_v] :
( ( ( ord_le7336532860387713383od_v_v @ S @ S2 )
& ( sCC_Bl2301996248249672505od_v_v @ Successors @ S2 ) )
=> ( S2 = S ) ) ) ) ) ).
% graph.is_scc_def
thf(fact_89_graph_Ois__scc__def,axiom,
! [Vertices: set_v,Successors: v > set_v,S: set_v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bloemen_is_scc_v @ Successors @ S )
= ( ( S != bot_bot_set_v )
& ( sCC_Bl5398416737448265317bscc_v @ Successors @ S )
& ! [S2: set_v] :
( ( ( ord_less_eq_set_v @ S @ S2 )
& ( sCC_Bl5398416737448265317bscc_v @ Successors @ S2 ) )
=> ( S2 = S ) ) ) ) ) ).
% graph.is_scc_def
thf(fact_90_graph_Ois__subscc_Ocong,axiom,
sCC_Bl5398416737448265317bscc_v = sCC_Bl5398416737448265317bscc_v ).
% graph.is_subscc.cong
thf(fact_91_graph_Ois__scc_Ocong,axiom,
sCC_Bloemen_is_scc_v = sCC_Bloemen_is_scc_v ).
% graph.is_scc.cong
thf(fact_92_graph_Osclosed,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ! [X4: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X4 @ Vertices )
=> ( ord_le7336532860387713383od_v_v @ ( Successors @ X4 ) @ Vertices ) ) ) ).
% graph.sclosed
thf(fact_93_graph_Osclosed,axiom,
! [Vertices: set_v,Successors: v > set_v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ! [X4: v] :
( ( member_v @ X4 @ Vertices )
=> ( ord_less_eq_set_v @ ( Successors @ X4 ) @ Vertices ) ) ) ).
% graph.sclosed
thf(fact_94_graph_Oedge__ra,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,Y2: product_prod_v_v,X: product_prod_v_v,E2: set_Pr2149350503807050951od_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( member7453568604450474000od_v_v @ Y2 @ ( Successors @ X ) )
=> ( ~ ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ X @ Y2 ) @ E2 )
=> ( sCC_Bl5370300055464682748od_v_v @ Successors @ X @ Y2 @ E2 ) ) ) ) ).
% graph.edge_ra
thf(fact_95_graph_Oedge__ra,axiom,
! [Vertices: set_v,Successors: v > set_v,Y2: v,X: v,E2: set_Product_prod_v_v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( member_v @ Y2 @ ( Successors @ X ) )
=> ( ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X @ Y2 ) @ E2 )
=> ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ Y2 @ E2 ) ) ) ) ).
% graph.edge_ra
thf(fact_96_graph_Ora__cases,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,X: product_prod_v_v,Y2: product_prod_v_v,E2: set_Pr2149350503807050951od_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl5370300055464682748od_v_v @ Successors @ X @ Y2 @ E2 )
=> ( ( X = Y2 )
| ? [Z3: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ Z3 @ ( Successors @ X ) )
& ~ ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ X @ Z3 ) @ E2 )
& ( sCC_Bl5370300055464682748od_v_v @ Successors @ Z3 @ Y2 @ E2 ) ) ) ) ) ).
% graph.ra_cases
thf(fact_97_graph_Ora__cases,axiom,
! [Vertices: set_v,Successors: v > set_v,X: v,Y2: v,E2: set_Product_prod_v_v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ Y2 @ E2 )
=> ( ( X = Y2 )
| ? [Z3: v] :
( ( member_v @ Z3 @ ( Successors @ X ) )
& ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X @ Z3 ) @ E2 )
& ( sCC_Bl4291963740693775144ding_v @ Successors @ Z3 @ Y2 @ E2 ) ) ) ) ) ).
% graph.ra_cases
thf(fact_98_graph_Oreachable__avoiding_Ocases,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,A1: product_prod_v_v,A2: product_prod_v_v,A32: set_Pr2149350503807050951od_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl5370300055464682748od_v_v @ Successors @ A1 @ A2 @ A32 )
=> ( ( A2 != A1 )
=> ~ ! [Y: product_prod_v_v] :
( ( sCC_Bl5370300055464682748od_v_v @ Successors @ A1 @ Y @ A32 )
=> ( ( member7453568604450474000od_v_v @ A2 @ ( Successors @ Y ) )
=> ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ Y @ A2 ) @ A32 ) ) ) ) ) ) ).
% graph.reachable_avoiding.cases
thf(fact_99_graph_Oreachable__avoiding_Ocases,axiom,
! [Vertices: set_v,Successors: v > set_v,A1: v,A2: v,A32: set_Product_prod_v_v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl4291963740693775144ding_v @ Successors @ A1 @ A2 @ A32 )
=> ( ( A2 != A1 )
=> ~ ! [Y: v] :
( ( sCC_Bl4291963740693775144ding_v @ Successors @ A1 @ Y @ A32 )
=> ( ( member_v @ A2 @ ( Successors @ Y ) )
=> ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ Y @ A2 ) @ A32 ) ) ) ) ) ) ).
% graph.reachable_avoiding.cases
thf(fact_100_graph_Oreachable__avoiding_Osimps,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,A1: product_prod_v_v,A2: product_prod_v_v,A32: set_Pr2149350503807050951od_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl5370300055464682748od_v_v @ Successors @ A1 @ A2 @ A32 )
= ( ? [X2: product_prod_v_v,E4: set_Pr2149350503807050951od_v_v] :
( ( A1 = X2 )
& ( A2 = X2 )
& ( A32 = E4 ) )
| ? [X2: product_prod_v_v,Y3: product_prod_v_v,E4: set_Pr2149350503807050951od_v_v,Z2: product_prod_v_v] :
( ( A1 = X2 )
& ( A2 = Z2 )
& ( A32 = E4 )
& ( sCC_Bl5370300055464682748od_v_v @ Successors @ X2 @ Y3 @ E4 )
& ( member7453568604450474000od_v_v @ Z2 @ ( Successors @ Y3 ) )
& ~ ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ Y3 @ Z2 ) @ E4 ) ) ) ) ) ).
% graph.reachable_avoiding.simps
thf(fact_101_graph_Oreachable__avoiding_Osimps,axiom,
! [Vertices: set_v,Successors: v > set_v,A1: v,A2: v,A32: set_Product_prod_v_v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl4291963740693775144ding_v @ Successors @ A1 @ A2 @ A32 )
= ( ? [X2: v,E4: set_Product_prod_v_v] :
( ( A1 = X2 )
& ( A2 = X2 )
& ( A32 = E4 ) )
| ? [X2: v,Y3: v,E4: set_Product_prod_v_v,Z2: v] :
( ( A1 = X2 )
& ( A2 = Z2 )
& ( A32 = E4 )
& ( sCC_Bl4291963740693775144ding_v @ Successors @ X2 @ Y3 @ E4 )
& ( member_v @ Z2 @ ( Successors @ Y3 ) )
& ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ Y3 @ Z2 ) @ E4 ) ) ) ) ) ).
% graph.reachable_avoiding.simps
thf(fact_102_graph_Ora__succ,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,X: product_prod_v_v,Y2: product_prod_v_v,E2: set_Pr2149350503807050951od_v_v,Z: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl5370300055464682748od_v_v @ Successors @ X @ Y2 @ E2 )
=> ( ( member7453568604450474000od_v_v @ Z @ ( Successors @ Y2 ) )
=> ( ~ ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ Y2 @ Z ) @ E2 )
=> ( sCC_Bl5370300055464682748od_v_v @ Successors @ X @ Z @ E2 ) ) ) ) ) ).
% graph.ra_succ
thf(fact_103_graph_Ora__succ,axiom,
! [Vertices: set_v,Successors: v > set_v,X: v,Y2: v,E2: set_Product_prod_v_v,Z: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ Y2 @ E2 )
=> ( ( member_v @ Z @ ( Successors @ Y2 ) )
=> ( ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ Y2 @ Z ) @ E2 )
=> ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ Z @ E2 ) ) ) ) ) ).
% graph.ra_succ
thf(fact_104_graph_Ora__mono,axiom,
! [Vertices: set_v,Successors: v > set_v,X: v,Y2: v,E2: set_Product_prod_v_v,E3: set_Product_prod_v_v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ Y2 @ E2 )
=> ( ( ord_le7336532860387713383od_v_v @ E3 @ E2 )
=> ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ Y2 @ E3 ) ) ) ) ).
% graph.ra_mono
thf(fact_105_graph_Ounite__S__tl,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,E: sCC_Bl7326425374436813197t_unit,W: product_prod_v_v,V: product_prod_v_v,N: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl7798947040364291444t_unit @ Successors @ E )
=> ( ( member7453568604450474000od_v_v @ W @ ( Successors @ V ) )
=> ( ~ ( member7453568604450474000od_v_v @ W @ ( sCC_Bl3878977043676959280t_unit @ E @ V ) )
=> ( ( member7453568604450474000od_v_v @ W @ ( sCC_Bl5498988629518860705t_unit @ E ) )
=> ( ~ ( member7453568604450474000od_v_v @ W @ ( sCC_Bl5094201334446601350t_unit @ E ) )
=> ( ( member7453568604450474000od_v_v @ N @ ( set_Product_prod_v_v2 @ ( tl_Product_prod_v_v @ ( sCC_Bl2021302119412358655t_unit @ ( sCC_Bl4702006153222411093od_v_v @ V @ W @ E ) ) ) ) )
=> ( ( sCC_Bl8440648026628373538t_unit @ ( sCC_Bl4702006153222411093od_v_v @ V @ W @ E ) @ N )
= ( sCC_Bl8440648026628373538t_unit @ E @ N ) ) ) ) ) ) ) ) ) ).
% graph.unite_S_tl
thf(fact_106_graph_Ounite__S__tl,axiom,
! [Vertices: set_v,Successors: v > set_v,E: sCC_Bl1394983891496994913t_unit,W: v,V: v,N: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E )
=> ( ( member_v @ W @ ( Successors @ V ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl3795065053823578884t_unit @ E @ V ) )
=> ( ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ( ( member_v @ N @ ( set_v2 @ ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ ( sCC_Bloemen_unite_v @ V @ W @ E ) ) ) ) )
=> ( ( sCC_Bl1280885523602775798t_unit @ ( sCC_Bloemen_unite_v @ V @ W @ E ) @ N )
= ( sCC_Bl1280885523602775798t_unit @ E @ N ) ) ) ) ) ) ) ) ) ).
% graph.unite_S_tl
thf(fact_107_graph_Ora__empty,axiom,
! [Vertices: set_v,Successors: v > set_v,X: v,Y2: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ Y2 @ bot_bo723834152578015283od_v_v )
= ( sCC_Bl649662514949026229able_v @ Successors @ X @ Y2 ) ) ) ).
% graph.ra_empty
thf(fact_108_graph_Ois__subscc__def,axiom,
! [Vertices: set_v,Successors: v > set_v,S: set_v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl5398416737448265317bscc_v @ Successors @ S )
= ( ! [X2: v] :
( ( member_v @ X2 @ S )
=> ! [Y3: v] :
( ( member_v @ Y3 @ S )
=> ( sCC_Bl649662514949026229able_v @ Successors @ X2 @ Y3 ) ) ) ) ) ) ).
% graph.is_subscc_def
thf(fact_109_graph_OsccE,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,S: set_Product_prod_v_v,X: product_prod_v_v,Y2: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl6242042402218619277od_v_v @ Successors @ S )
=> ( ( member7453568604450474000od_v_v @ X @ S )
=> ( ( sCC_Bl4981926079593201289od_v_v @ Successors @ X @ Y2 )
=> ( ( sCC_Bl4981926079593201289od_v_v @ Successors @ Y2 @ X )
=> ( member7453568604450474000od_v_v @ Y2 @ S ) ) ) ) ) ) ).
% graph.sccE
thf(fact_110_graph_OsccE,axiom,
! [Vertices: set_v,Successors: v > set_v,S: set_v,X: v,Y2: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bloemen_is_scc_v @ Successors @ S )
=> ( ( member_v @ X @ S )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ X @ Y2 )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ Y2 @ X )
=> ( member_v @ Y2 @ S ) ) ) ) ) ) ).
% graph.sccE
thf(fact_111_graph_Oreachable__visited,axiom,
! [Vertices: set_v,Successors: v > set_v,E: sCC_Bl1394983891496994913t_unit,V: v,W: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E )
=> ( ( member_v @ V @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ V @ W )
=> ( ! [X3: v] :
( ( member_v @ X3 @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ! [Xa: v] :
( ( member_v @ Xa @ ( minus_minus_set_v @ ( Successors @ X3 ) @ ( sCC_Bl3795065053823578884t_unit @ E @ X3 ) ) )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ V @ X3 )
=> ~ ( sCC_Bl649662514949026229able_v @ Successors @ Xa @ W ) ) ) )
=> ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) ) ) ) ) ) ) ).
% graph.reachable_visited
thf(fact_112_graph_Opre__dfs__def,axiom,
! [Vertices: set_v,Successors: v > set_v,V: v,E: sCC_Bl1394983891496994913t_unit] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl36166008131615352t_unit @ Successors @ V @ E )
= ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E )
& ~ ( member_v @ V @ ( sCC_Bl4645233313691564917t_unit @ E ) )
& ( sCC_Bl649662514949026229able_v @ Successors @ ( sCC_Bl1090238580953940555t_unit @ E ) @ V )
& ( ( sCC_Bl3795065053823578884t_unit @ E @ V )
= bot_bot_set_v )
& ! [X2: v] :
( ( member_v @ X2 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) )
=> ( sCC_Bl649662514949026229able_v @ Successors @ X2 @ V ) ) ) ) ) ).
% graph.pre_dfs_def
thf(fact_113_graph_Ostack__class,axiom,
! [Vertices: set_v,Successors: v > set_v,E: sCC_Bl1394983891496994913t_unit,N: v,M: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E )
=> ( ( member_v @ N @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) )
=> ( ( member_v @ M @ ( sCC_Bl1280885523602775798t_unit @ E @ N ) )
=> ( member_v @ M @ ( minus_minus_set_v @ ( sCC_Bl4645233313691564917t_unit @ E ) @ ( sCC_Bl157864678168468314t_unit @ E ) ) ) ) ) ) ) ).
% graph.stack_class
thf(fact_114_Diff__eq__empty__iff,axiom,
! [A3: set_v,B: set_v] :
( ( ( minus_minus_set_v @ A3 @ B )
= bot_bot_set_v )
= ( ord_less_eq_set_v @ A3 @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_115_Diff__eq__empty__iff,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ( minus_4183494784930505774od_v_v @ A3 @ B )
= bot_bo723834152578015283od_v_v )
= ( ord_le7336532860387713383od_v_v @ A3 @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_116_Diff__empty,axiom,
! [A3: set_Product_prod_v_v] :
( ( minus_4183494784930505774od_v_v @ A3 @ bot_bo723834152578015283od_v_v )
= A3 ) ).
% Diff_empty
thf(fact_117_Diff__empty,axiom,
! [A3: set_v] :
( ( minus_minus_set_v @ A3 @ bot_bot_set_v )
= A3 ) ).
% Diff_empty
thf(fact_118_empty__Diff,axiom,
! [A3: set_Product_prod_v_v] :
( ( minus_4183494784930505774od_v_v @ bot_bo723834152578015283od_v_v @ A3 )
= bot_bo723834152578015283od_v_v ) ).
% empty_Diff
thf(fact_119_empty__Diff,axiom,
! [A3: set_v] :
( ( minus_minus_set_v @ bot_bot_set_v @ A3 )
= bot_bot_set_v ) ).
% empty_Diff
thf(fact_120_Diff__cancel,axiom,
! [A3: set_Product_prod_v_v] :
( ( minus_4183494784930505774od_v_v @ A3 @ A3 )
= bot_bo723834152578015283od_v_v ) ).
% Diff_cancel
thf(fact_121_Diff__cancel,axiom,
! [A3: set_v] :
( ( minus_minus_set_v @ A3 @ A3 )
= bot_bot_set_v ) ).
% Diff_cancel
thf(fact_122_subset__empty,axiom,
! [A3: set_v] :
( ( ord_less_eq_set_v @ A3 @ bot_bot_set_v )
= ( A3 = bot_bot_set_v ) ) ).
% subset_empty
thf(fact_123_subset__empty,axiom,
! [A3: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A3 @ bot_bo723834152578015283od_v_v )
= ( A3 = bot_bo723834152578015283od_v_v ) ) ).
% subset_empty
thf(fact_124_empty__subsetI,axiom,
! [A3: set_v] : ( ord_less_eq_set_v @ bot_bot_set_v @ A3 ) ).
% empty_subsetI
thf(fact_125_empty__subsetI,axiom,
! [A3: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ bot_bo723834152578015283od_v_v @ A3 ) ).
% empty_subsetI
thf(fact_126_subscc__add,axiom,
! [S: set_v,X: v,Y2: v] :
( ( sCC_Bl5398416737448265317bscc_v @ successors @ S )
=> ( ( member_v @ X @ S )
=> ( ( sCC_Bl649662514949026229able_v @ successors @ X @ Y2 )
=> ( ( sCC_Bl649662514949026229able_v @ successors @ Y2 @ X )
=> ( sCC_Bl5398416737448265317bscc_v @ successors @ ( insert_v2 @ Y2 @ S ) ) ) ) ) ) ).
% subscc_add
thf(fact_127_diff__shunt__var,axiom,
! [X: set_v,Y2: set_v] :
( ( ( minus_minus_set_v @ X @ Y2 )
= bot_bot_set_v )
= ( ord_less_eq_set_v @ X @ Y2 ) ) ).
% diff_shunt_var
thf(fact_128_diff__shunt__var,axiom,
! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v] :
( ( ( minus_4183494784930505774od_v_v @ X @ Y2 )
= bot_bo723834152578015283od_v_v )
= ( ord_le7336532860387713383od_v_v @ X @ Y2 ) ) ).
% diff_shunt_var
thf(fact_129_DiffI,axiom,
! [C: v,A3: set_v,B: set_v] :
( ( member_v @ C @ A3 )
=> ( ~ ( member_v @ C @ B )
=> ( member_v @ C @ ( minus_minus_set_v @ A3 @ B ) ) ) ) ).
% DiffI
thf(fact_130_DiffI,axiom,
! [C: product_prod_v_v,A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ A3 )
=> ( ~ ( member7453568604450474000od_v_v @ C @ B )
=> ( member7453568604450474000od_v_v @ C @ ( minus_4183494784930505774od_v_v @ A3 @ B ) ) ) ) ).
% DiffI
thf(fact_131_Diff__iff,axiom,
! [C: v,A3: set_v,B: set_v] :
( ( member_v @ C @ ( minus_minus_set_v @ A3 @ B ) )
= ( ( member_v @ C @ A3 )
& ~ ( member_v @ C @ B ) ) ) ).
% Diff_iff
thf(fact_132_Diff__iff,axiom,
! [C: product_prod_v_v,A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ ( minus_4183494784930505774od_v_v @ A3 @ B ) )
= ( ( member7453568604450474000od_v_v @ C @ A3 )
& ~ ( member7453568604450474000od_v_v @ C @ B ) ) ) ).
% Diff_iff
thf(fact_133_empty__Collect__eq,axiom,
! [P: product_prod_v_v > $o] :
( ( bot_bo723834152578015283od_v_v
= ( collec140062887454715474od_v_v @ P ) )
= ( ! [X2: product_prod_v_v] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_134_empty__Collect__eq,axiom,
! [P: v > $o] :
( ( bot_bot_set_v
= ( collect_v @ P ) )
= ( ! [X2: v] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_135_Collect__empty__eq,axiom,
! [P: product_prod_v_v > $o] :
( ( ( collec140062887454715474od_v_v @ P )
= bot_bo723834152578015283od_v_v )
= ( ! [X2: product_prod_v_v] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_136_Collect__empty__eq,axiom,
! [P: v > $o] :
( ( ( collect_v @ P )
= bot_bot_set_v )
= ( ! [X2: v] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_137_all__not__in__conv,axiom,
! [A3: set_Product_prod_v_v] :
( ( ! [X2: product_prod_v_v] :
~ ( member7453568604450474000od_v_v @ X2 @ A3 ) )
= ( A3 = bot_bo723834152578015283od_v_v ) ) ).
% all_not_in_conv
thf(fact_138_all__not__in__conv,axiom,
! [A3: set_v] :
( ( ! [X2: v] :
~ ( member_v @ X2 @ A3 ) )
= ( A3 = bot_bot_set_v ) ) ).
% all_not_in_conv
thf(fact_139_empty__iff,axiom,
! [C: product_prod_v_v] :
~ ( member7453568604450474000od_v_v @ C @ bot_bo723834152578015283od_v_v ) ).
% empty_iff
thf(fact_140_empty__iff,axiom,
! [C: v] :
~ ( member_v @ C @ bot_bot_set_v ) ).
% empty_iff
thf(fact_141_subset__antisym,axiom,
! [A3: set_v,B: set_v] :
( ( ord_less_eq_set_v @ A3 @ B )
=> ( ( ord_less_eq_set_v @ B @ A3 )
=> ( A3 = B ) ) ) ).
% subset_antisym
thf(fact_142_subset__antisym,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A3 @ B )
=> ( ( ord_le7336532860387713383od_v_v @ B @ A3 )
=> ( A3 = B ) ) ) ).
% subset_antisym
thf(fact_143_subsetI,axiom,
! [A3: set_v,B: set_v] :
( ! [X3: v] :
( ( member_v @ X3 @ A3 )
=> ( member_v @ X3 @ B ) )
=> ( ord_less_eq_set_v @ A3 @ B ) ) ).
% subsetI
thf(fact_144_subsetI,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ! [X3: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X3 @ A3 )
=> ( member7453568604450474000od_v_v @ X3 @ B ) )
=> ( ord_le7336532860387713383od_v_v @ A3 @ B ) ) ).
% subsetI
thf(fact_145_insert__absorb2,axiom,
! [X: v,A3: set_v] :
( ( insert_v2 @ X @ ( insert_v2 @ X @ A3 ) )
= ( insert_v2 @ X @ A3 ) ) ).
% insert_absorb2
thf(fact_146_insert__absorb2,axiom,
! [X: product_prod_v_v,A3: set_Product_prod_v_v] :
( ( insert1338601472111419319od_v_v @ X @ ( insert1338601472111419319od_v_v @ X @ A3 ) )
= ( insert1338601472111419319od_v_v @ X @ A3 ) ) ).
% insert_absorb2
thf(fact_147_insert__iff,axiom,
! [A: v,B2: v,A3: set_v] :
( ( member_v @ A @ ( insert_v2 @ B2 @ A3 ) )
= ( ( A = B2 )
| ( member_v @ A @ A3 ) ) ) ).
% insert_iff
thf(fact_148_insert__iff,axiom,
! [A: product_prod_v_v,B2: product_prod_v_v,A3: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ A @ ( insert1338601472111419319od_v_v @ B2 @ A3 ) )
= ( ( A = B2 )
| ( member7453568604450474000od_v_v @ A @ A3 ) ) ) ).
% insert_iff
thf(fact_149_insertCI,axiom,
! [A: v,B: set_v,B2: v] :
( ( ~ ( member_v @ A @ B )
=> ( A = B2 ) )
=> ( member_v @ A @ ( insert_v2 @ B2 @ B ) ) ) ).
% insertCI
thf(fact_150_insertCI,axiom,
! [A: product_prod_v_v,B: set_Product_prod_v_v,B2: product_prod_v_v] :
( ( ~ ( member7453568604450474000od_v_v @ A @ B )
=> ( A = B2 ) )
=> ( member7453568604450474000od_v_v @ A @ ( insert1338601472111419319od_v_v @ B2 @ B ) ) ) ).
% insertCI
thf(fact_151_singletonI,axiom,
! [A: product_prod_v_v] : ( member7453568604450474000od_v_v @ A @ ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) ) ).
% singletonI
thf(fact_152_singletonI,axiom,
! [A: v] : ( member_v @ A @ ( insert_v2 @ A @ bot_bot_set_v ) ) ).
% singletonI
thf(fact_153_insert__subset,axiom,
! [X: v,A3: set_v,B: set_v] :
( ( ord_less_eq_set_v @ ( insert_v2 @ X @ A3 ) @ B )
= ( ( member_v @ X @ B )
& ( ord_less_eq_set_v @ A3 @ B ) ) ) ).
% insert_subset
thf(fact_154_insert__subset,axiom,
! [X: product_prod_v_v,A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ ( insert1338601472111419319od_v_v @ X @ A3 ) @ B )
= ( ( member7453568604450474000od_v_v @ X @ B )
& ( ord_le7336532860387713383od_v_v @ A3 @ B ) ) ) ).
% insert_subset
thf(fact_155_insert__Diff1,axiom,
! [X: v,B: set_v,A3: set_v] :
( ( member_v @ X @ B )
=> ( ( minus_minus_set_v @ ( insert_v2 @ X @ A3 ) @ B )
= ( minus_minus_set_v @ A3 @ B ) ) ) ).
% insert_Diff1
thf(fact_156_insert__Diff1,axiom,
! [X: product_prod_v_v,B: set_Product_prod_v_v,A3: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X @ B )
=> ( ( minus_4183494784930505774od_v_v @ ( insert1338601472111419319od_v_v @ X @ A3 ) @ B )
= ( minus_4183494784930505774od_v_v @ A3 @ B ) ) ) ).
% insert_Diff1
thf(fact_157_Diff__insert0,axiom,
! [X: v,A3: set_v,B: set_v] :
( ~ ( member_v @ X @ A3 )
=> ( ( minus_minus_set_v @ A3 @ ( insert_v2 @ X @ B ) )
= ( minus_minus_set_v @ A3 @ B ) ) ) ).
% Diff_insert0
thf(fact_158_Diff__insert0,axiom,
! [X: product_prod_v_v,A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ~ ( member7453568604450474000od_v_v @ X @ A3 )
=> ( ( minus_4183494784930505774od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ X @ B ) )
= ( minus_4183494784930505774od_v_v @ A3 @ B ) ) ) ).
% Diff_insert0
thf(fact_159_singleton__insert__inj__eq_H,axiom,
! [A: v,A3: set_v,B2: v] :
( ( ( insert_v2 @ A @ A3 )
= ( insert_v2 @ B2 @ bot_bot_set_v ) )
= ( ( A = B2 )
& ( ord_less_eq_set_v @ A3 @ ( insert_v2 @ B2 @ bot_bot_set_v ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_160_singleton__insert__inj__eq_H,axiom,
! [A: product_prod_v_v,A3: set_Product_prod_v_v,B2: product_prod_v_v] :
( ( ( insert1338601472111419319od_v_v @ A @ A3 )
= ( insert1338601472111419319od_v_v @ B2 @ bot_bo723834152578015283od_v_v ) )
= ( ( A = B2 )
& ( ord_le7336532860387713383od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ B2 @ bot_bo723834152578015283od_v_v ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_161_singleton__insert__inj__eq,axiom,
! [B2: v,A: v,A3: set_v] :
( ( ( insert_v2 @ B2 @ bot_bot_set_v )
= ( insert_v2 @ A @ A3 ) )
= ( ( A = B2 )
& ( ord_less_eq_set_v @ A3 @ ( insert_v2 @ B2 @ bot_bot_set_v ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_162_singleton__insert__inj__eq,axiom,
! [B2: product_prod_v_v,A: product_prod_v_v,A3: set_Product_prod_v_v] :
( ( ( insert1338601472111419319od_v_v @ B2 @ bot_bo723834152578015283od_v_v )
= ( insert1338601472111419319od_v_v @ A @ A3 ) )
= ( ( A = B2 )
& ( ord_le7336532860387713383od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ B2 @ bot_bo723834152578015283od_v_v ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_163_insert__Diff__single,axiom,
! [A: product_prod_v_v,A3: set_Product_prod_v_v] :
( ( insert1338601472111419319od_v_v @ A @ ( minus_4183494784930505774od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) ) )
= ( insert1338601472111419319od_v_v @ A @ A3 ) ) ).
% insert_Diff_single
thf(fact_164_insert__Diff__single,axiom,
! [A: v,A3: set_v] :
( ( insert_v2 @ A @ ( minus_minus_set_v @ A3 @ ( insert_v2 @ A @ bot_bot_set_v ) ) )
= ( insert_v2 @ A @ A3 ) ) ).
% insert_Diff_single
thf(fact_165_mk__disjoint__insert,axiom,
! [A: v,A3: set_v] :
( ( member_v @ A @ A3 )
=> ? [B3: set_v] :
( ( A3
= ( insert_v2 @ A @ B3 ) )
& ~ ( member_v @ A @ B3 ) ) ) ).
% mk_disjoint_insert
thf(fact_166_mk__disjoint__insert,axiom,
! [A: product_prod_v_v,A3: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ A @ A3 )
=> ? [B3: set_Product_prod_v_v] :
( ( A3
= ( insert1338601472111419319od_v_v @ A @ B3 ) )
& ~ ( member7453568604450474000od_v_v @ A @ B3 ) ) ) ).
% mk_disjoint_insert
thf(fact_167_insert__commute,axiom,
! [X: v,Y2: v,A3: set_v] :
( ( insert_v2 @ X @ ( insert_v2 @ Y2 @ A3 ) )
= ( insert_v2 @ Y2 @ ( insert_v2 @ X @ A3 ) ) ) ).
% insert_commute
thf(fact_168_insert__commute,axiom,
! [X: product_prod_v_v,Y2: product_prod_v_v,A3: set_Product_prod_v_v] :
( ( insert1338601472111419319od_v_v @ X @ ( insert1338601472111419319od_v_v @ Y2 @ A3 ) )
= ( insert1338601472111419319od_v_v @ Y2 @ ( insert1338601472111419319od_v_v @ X @ A3 ) ) ) ).
% insert_commute
thf(fact_169_insert__eq__iff,axiom,
! [A: v,A3: set_v,B2: v,B: set_v] :
( ~ ( member_v @ A @ A3 )
=> ( ~ ( member_v @ B2 @ B )
=> ( ( ( insert_v2 @ A @ A3 )
= ( insert_v2 @ B2 @ B ) )
= ( ( ( A = B2 )
=> ( A3 = B ) )
& ( ( A != B2 )
=> ? [C2: set_v] :
( ( A3
= ( insert_v2 @ B2 @ C2 ) )
& ~ ( member_v @ B2 @ C2 )
& ( B
= ( insert_v2 @ A @ C2 ) )
& ~ ( member_v @ A @ C2 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_170_insert__eq__iff,axiom,
! [A: product_prod_v_v,A3: set_Product_prod_v_v,B2: product_prod_v_v,B: set_Product_prod_v_v] :
( ~ ( member7453568604450474000od_v_v @ A @ A3 )
=> ( ~ ( member7453568604450474000od_v_v @ B2 @ B )
=> ( ( ( insert1338601472111419319od_v_v @ A @ A3 )
= ( insert1338601472111419319od_v_v @ B2 @ B ) )
= ( ( ( A = B2 )
=> ( A3 = B ) )
& ( ( A != B2 )
=> ? [C2: set_Product_prod_v_v] :
( ( A3
= ( insert1338601472111419319od_v_v @ B2 @ C2 ) )
& ~ ( member7453568604450474000od_v_v @ B2 @ C2 )
& ( B
= ( insert1338601472111419319od_v_v @ A @ C2 ) )
& ~ ( member7453568604450474000od_v_v @ A @ C2 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_171_insert__absorb,axiom,
! [A: v,A3: set_v] :
( ( member_v @ A @ A3 )
=> ( ( insert_v2 @ A @ A3 )
= A3 ) ) ).
% insert_absorb
thf(fact_172_insert__absorb,axiom,
! [A: product_prod_v_v,A3: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ A @ A3 )
=> ( ( insert1338601472111419319od_v_v @ A @ A3 )
= A3 ) ) ).
% insert_absorb
thf(fact_173_insert__ident,axiom,
! [X: v,A3: set_v,B: set_v] :
( ~ ( member_v @ X @ A3 )
=> ( ~ ( member_v @ X @ B )
=> ( ( ( insert_v2 @ X @ A3 )
= ( insert_v2 @ X @ B ) )
= ( A3 = B ) ) ) ) ).
% insert_ident
thf(fact_174_insert__ident,axiom,
! [X: product_prod_v_v,A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ~ ( member7453568604450474000od_v_v @ X @ A3 )
=> ( ~ ( member7453568604450474000od_v_v @ X @ B )
=> ( ( ( insert1338601472111419319od_v_v @ X @ A3 )
= ( insert1338601472111419319od_v_v @ X @ B ) )
= ( A3 = B ) ) ) ) ).
% insert_ident
thf(fact_175_Set_Oset__insert,axiom,
! [X: v,A3: set_v] :
( ( member_v @ X @ A3 )
=> ~ ! [B3: set_v] :
( ( A3
= ( insert_v2 @ X @ B3 ) )
=> ( member_v @ X @ B3 ) ) ) ).
% Set.set_insert
thf(fact_176_Set_Oset__insert,axiom,
! [X: product_prod_v_v,A3: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X @ A3 )
=> ~ ! [B3: set_Product_prod_v_v] :
( ( A3
= ( insert1338601472111419319od_v_v @ X @ B3 ) )
=> ( member7453568604450474000od_v_v @ X @ B3 ) ) ) ).
% Set.set_insert
thf(fact_177_insertI2,axiom,
! [A: v,B: set_v,B2: v] :
( ( member_v @ A @ B )
=> ( member_v @ A @ ( insert_v2 @ B2 @ B ) ) ) ).
% insertI2
thf(fact_178_insertI2,axiom,
! [A: product_prod_v_v,B: set_Product_prod_v_v,B2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ A @ B )
=> ( member7453568604450474000od_v_v @ A @ ( insert1338601472111419319od_v_v @ B2 @ B ) ) ) ).
% insertI2
thf(fact_179_insertI1,axiom,
! [A: v,B: set_v] : ( member_v @ A @ ( insert_v2 @ A @ B ) ) ).
% insertI1
thf(fact_180_insertI1,axiom,
! [A: product_prod_v_v,B: set_Product_prod_v_v] : ( member7453568604450474000od_v_v @ A @ ( insert1338601472111419319od_v_v @ A @ B ) ) ).
% insertI1
thf(fact_181_insertE,axiom,
! [A: v,B2: v,A3: set_v] :
( ( member_v @ A @ ( insert_v2 @ B2 @ A3 ) )
=> ( ( A != B2 )
=> ( member_v @ A @ A3 ) ) ) ).
% insertE
thf(fact_182_insertE,axiom,
! [A: product_prod_v_v,B2: product_prod_v_v,A3: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ A @ ( insert1338601472111419319od_v_v @ B2 @ A3 ) )
=> ( ( A != B2 )
=> ( member7453568604450474000od_v_v @ A @ A3 ) ) ) ).
% insertE
thf(fact_183_bot__set__def,axiom,
( bot_bo723834152578015283od_v_v
= ( collec140062887454715474od_v_v @ bot_bo8461541820394803818_v_v_o ) ) ).
% bot_set_def
thf(fact_184_bot__set__def,axiom,
( bot_bot_set_v
= ( collect_v @ bot_bot_v_o ) ) ).
% bot_set_def
thf(fact_185_singleton__inject,axiom,
! [A: product_prod_v_v,B2: product_prod_v_v] :
( ( ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v )
= ( insert1338601472111419319od_v_v @ B2 @ bot_bo723834152578015283od_v_v ) )
=> ( A = B2 ) ) ).
% singleton_inject
thf(fact_186_singleton__inject,axiom,
! [A: v,B2: v] :
( ( ( insert_v2 @ A @ bot_bot_set_v )
= ( insert_v2 @ B2 @ bot_bot_set_v ) )
=> ( A = B2 ) ) ).
% singleton_inject
thf(fact_187_insert__not__empty,axiom,
! [A: product_prod_v_v,A3: set_Product_prod_v_v] :
( ( insert1338601472111419319od_v_v @ A @ A3 )
!= bot_bo723834152578015283od_v_v ) ).
% insert_not_empty
thf(fact_188_insert__not__empty,axiom,
! [A: v,A3: set_v] :
( ( insert_v2 @ A @ A3 )
!= bot_bot_set_v ) ).
% insert_not_empty
thf(fact_189_doubleton__eq__iff,axiom,
! [A: product_prod_v_v,B2: product_prod_v_v,C: product_prod_v_v,D: product_prod_v_v] :
( ( ( insert1338601472111419319od_v_v @ A @ ( insert1338601472111419319od_v_v @ B2 @ bot_bo723834152578015283od_v_v ) )
= ( insert1338601472111419319od_v_v @ C @ ( insert1338601472111419319od_v_v @ D @ bot_bo723834152578015283od_v_v ) ) )
= ( ( ( A = C )
& ( B2 = D ) )
| ( ( A = D )
& ( B2 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_190_doubleton__eq__iff,axiom,
! [A: v,B2: v,C: v,D: v] :
( ( ( insert_v2 @ A @ ( insert_v2 @ B2 @ bot_bot_set_v ) )
= ( insert_v2 @ C @ ( insert_v2 @ D @ bot_bot_set_v ) ) )
= ( ( ( A = C )
& ( B2 = D ) )
| ( ( A = D )
& ( B2 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_191_singleton__iff,axiom,
! [B2: product_prod_v_v,A: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ B2 @ ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) )
= ( B2 = A ) ) ).
% singleton_iff
thf(fact_192_singleton__iff,axiom,
! [B2: v,A: v] :
( ( member_v @ B2 @ ( insert_v2 @ A @ bot_bot_set_v ) )
= ( B2 = A ) ) ).
% singleton_iff
thf(fact_193_singletonD,axiom,
! [B2: product_prod_v_v,A: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ B2 @ ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) )
=> ( B2 = A ) ) ).
% singletonD
thf(fact_194_singletonD,axiom,
! [B2: v,A: v] :
( ( member_v @ B2 @ ( insert_v2 @ A @ bot_bot_set_v ) )
=> ( B2 = A ) ) ).
% singletonD
thf(fact_195_subset__insertI2,axiom,
! [A3: set_v,B: set_v,B2: v] :
( ( ord_less_eq_set_v @ A3 @ B )
=> ( ord_less_eq_set_v @ A3 @ ( insert_v2 @ B2 @ B ) ) ) ).
% subset_insertI2
thf(fact_196_subset__insertI2,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v,B2: product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A3 @ B )
=> ( ord_le7336532860387713383od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ B2 @ B ) ) ) ).
% subset_insertI2
thf(fact_197_subset__insertI,axiom,
! [B: set_v,A: v] : ( ord_less_eq_set_v @ B @ ( insert_v2 @ A @ B ) ) ).
% subset_insertI
thf(fact_198_subset__insertI,axiom,
! [B: set_Product_prod_v_v,A: product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ B @ ( insert1338601472111419319od_v_v @ A @ B ) ) ).
% subset_insertI
thf(fact_199_subset__insert,axiom,
! [X: v,A3: set_v,B: set_v] :
( ~ ( member_v @ X @ A3 )
=> ( ( ord_less_eq_set_v @ A3 @ ( insert_v2 @ X @ B ) )
= ( ord_less_eq_set_v @ A3 @ B ) ) ) ).
% subset_insert
thf(fact_200_subset__insert,axiom,
! [X: product_prod_v_v,A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ~ ( member7453568604450474000od_v_v @ X @ A3 )
=> ( ( ord_le7336532860387713383od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ X @ B ) )
= ( ord_le7336532860387713383od_v_v @ A3 @ B ) ) ) ).
% subset_insert
thf(fact_201_insert__mono,axiom,
! [C3: set_v,D2: set_v,A: v] :
( ( ord_less_eq_set_v @ C3 @ D2 )
=> ( ord_less_eq_set_v @ ( insert_v2 @ A @ C3 ) @ ( insert_v2 @ A @ D2 ) ) ) ).
% insert_mono
thf(fact_202_insert__mono,axiom,
! [C3: set_Product_prod_v_v,D2: set_Product_prod_v_v,A: product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ C3 @ D2 )
=> ( ord_le7336532860387713383od_v_v @ ( insert1338601472111419319od_v_v @ A @ C3 ) @ ( insert1338601472111419319od_v_v @ A @ D2 ) ) ) ).
% insert_mono
thf(fact_203_insert__Diff__if,axiom,
! [X: v,B: set_v,A3: set_v] :
( ( ( member_v @ X @ B )
=> ( ( minus_minus_set_v @ ( insert_v2 @ X @ A3 ) @ B )
= ( minus_minus_set_v @ A3 @ B ) ) )
& ( ~ ( member_v @ X @ B )
=> ( ( minus_minus_set_v @ ( insert_v2 @ X @ A3 ) @ B )
= ( insert_v2 @ X @ ( minus_minus_set_v @ A3 @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_204_insert__Diff__if,axiom,
! [X: product_prod_v_v,B: set_Product_prod_v_v,A3: set_Product_prod_v_v] :
( ( ( member7453568604450474000od_v_v @ X @ B )
=> ( ( minus_4183494784930505774od_v_v @ ( insert1338601472111419319od_v_v @ X @ A3 ) @ B )
= ( minus_4183494784930505774od_v_v @ A3 @ B ) ) )
& ( ~ ( member7453568604450474000od_v_v @ X @ B )
=> ( ( minus_4183494784930505774od_v_v @ ( insert1338601472111419319od_v_v @ X @ A3 ) @ B )
= ( insert1338601472111419319od_v_v @ X @ ( minus_4183494784930505774od_v_v @ A3 @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_205_graph_Odfss_Ocases,axiom,
! [Vertices: set_v,Successors: v > set_v,X: produc5741669702376414499t_unit] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ~ ! [V2: v,E5: sCC_Bl1394983891496994913t_unit] :
( X
!= ( produc3862955338007567901t_unit @ V2 @ E5 ) ) ) ).
% graph.dfss.cases
thf(fact_206_subset__singleton__iff,axiom,
! [X5: set_v,A: v] :
( ( ord_less_eq_set_v @ X5 @ ( insert_v2 @ A @ bot_bot_set_v ) )
= ( ( X5 = bot_bot_set_v )
| ( X5
= ( insert_v2 @ A @ bot_bot_set_v ) ) ) ) ).
% subset_singleton_iff
thf(fact_207_subset__singleton__iff,axiom,
! [X5: set_Product_prod_v_v,A: product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X5 @ ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) )
= ( ( X5 = bot_bo723834152578015283od_v_v )
| ( X5
= ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) ) ) ) ).
% subset_singleton_iff
thf(fact_208_subset__singletonD,axiom,
! [A3: set_v,X: v] :
( ( ord_less_eq_set_v @ A3 @ ( insert_v2 @ X @ bot_bot_set_v ) )
=> ( ( A3 = bot_bot_set_v )
| ( A3
= ( insert_v2 @ X @ bot_bot_set_v ) ) ) ) ).
% subset_singletonD
thf(fact_209_subset__singletonD,axiom,
! [A3: set_Product_prod_v_v,X: product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) )
=> ( ( A3 = bot_bo723834152578015283od_v_v )
| ( A3
= ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) ) ) ) ).
% subset_singletonD
thf(fact_210_Diff__insert__absorb,axiom,
! [X: product_prod_v_v,A3: set_Product_prod_v_v] :
( ~ ( member7453568604450474000od_v_v @ X @ A3 )
=> ( ( minus_4183494784930505774od_v_v @ ( insert1338601472111419319od_v_v @ X @ A3 ) @ ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) )
= A3 ) ) ).
% Diff_insert_absorb
thf(fact_211_Diff__insert__absorb,axiom,
! [X: v,A3: set_v] :
( ~ ( member_v @ X @ A3 )
=> ( ( minus_minus_set_v @ ( insert_v2 @ X @ A3 ) @ ( insert_v2 @ X @ bot_bot_set_v ) )
= A3 ) ) ).
% Diff_insert_absorb
thf(fact_212_Diff__insert2,axiom,
! [A3: set_Product_prod_v_v,A: product_prod_v_v,B: set_Product_prod_v_v] :
( ( minus_4183494784930505774od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ A @ B ) )
= ( minus_4183494784930505774od_v_v @ ( minus_4183494784930505774od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) ) @ B ) ) ).
% Diff_insert2
thf(fact_213_Diff__insert2,axiom,
! [A3: set_v,A: v,B: set_v] :
( ( minus_minus_set_v @ A3 @ ( insert_v2 @ A @ B ) )
= ( minus_minus_set_v @ ( minus_minus_set_v @ A3 @ ( insert_v2 @ A @ bot_bot_set_v ) ) @ B ) ) ).
% Diff_insert2
thf(fact_214_insert__Diff,axiom,
! [A: product_prod_v_v,A3: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ A @ A3 )
=> ( ( insert1338601472111419319od_v_v @ A @ ( minus_4183494784930505774od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) ) )
= A3 ) ) ).
% insert_Diff
thf(fact_215_insert__Diff,axiom,
! [A: v,A3: set_v] :
( ( member_v @ A @ A3 )
=> ( ( insert_v2 @ A @ ( minus_minus_set_v @ A3 @ ( insert_v2 @ A @ bot_bot_set_v ) ) )
= A3 ) ) ).
% insert_Diff
thf(fact_216_Diff__insert,axiom,
! [A3: set_Product_prod_v_v,A: product_prod_v_v,B: set_Product_prod_v_v] :
( ( minus_4183494784930505774od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ A @ B ) )
= ( minus_4183494784930505774od_v_v @ ( minus_4183494784930505774od_v_v @ A3 @ B ) @ ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) ) ) ).
% Diff_insert
thf(fact_217_Diff__insert,axiom,
! [A3: set_v,A: v,B: set_v] :
( ( minus_minus_set_v @ A3 @ ( insert_v2 @ A @ B ) )
= ( minus_minus_set_v @ ( minus_minus_set_v @ A3 @ B ) @ ( insert_v2 @ A @ bot_bot_set_v ) ) ) ).
% Diff_insert
thf(fact_218_subset__Diff__insert,axiom,
! [A3: set_v,B: set_v,X: v,C3: set_v] :
( ( ord_less_eq_set_v @ A3 @ ( minus_minus_set_v @ B @ ( insert_v2 @ X @ C3 ) ) )
= ( ( ord_less_eq_set_v @ A3 @ ( minus_minus_set_v @ B @ C3 ) )
& ~ ( member_v @ X @ A3 ) ) ) ).
% subset_Diff_insert
thf(fact_219_subset__Diff__insert,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v,X: product_prod_v_v,C3: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A3 @ ( minus_4183494784930505774od_v_v @ B @ ( insert1338601472111419319od_v_v @ X @ C3 ) ) )
= ( ( ord_le7336532860387713383od_v_v @ A3 @ ( minus_4183494784930505774od_v_v @ B @ C3 ) )
& ~ ( member7453568604450474000od_v_v @ X @ A3 ) ) ) ).
% subset_Diff_insert
thf(fact_220_Diff__single__insert,axiom,
! [A3: set_v,X: v,B: set_v] :
( ( ord_less_eq_set_v @ ( minus_minus_set_v @ A3 @ ( insert_v2 @ X @ bot_bot_set_v ) ) @ B )
=> ( ord_less_eq_set_v @ A3 @ ( insert_v2 @ X @ B ) ) ) ).
% Diff_single_insert
thf(fact_221_Diff__single__insert,axiom,
! [A3: set_Product_prod_v_v,X: product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ ( minus_4183494784930505774od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) ) @ B )
=> ( ord_le7336532860387713383od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ X @ B ) ) ) ).
% Diff_single_insert
thf(fact_222_subset__insert__iff,axiom,
! [A3: set_v,X: v,B: set_v] :
( ( ord_less_eq_set_v @ A3 @ ( insert_v2 @ X @ B ) )
= ( ( ( member_v @ X @ A3 )
=> ( ord_less_eq_set_v @ ( minus_minus_set_v @ A3 @ ( insert_v2 @ X @ bot_bot_set_v ) ) @ B ) )
& ( ~ ( member_v @ X @ A3 )
=> ( ord_less_eq_set_v @ A3 @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_223_subset__insert__iff,axiom,
! [A3: set_Product_prod_v_v,X: product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ X @ B ) )
= ( ( ( member7453568604450474000od_v_v @ X @ A3 )
=> ( ord_le7336532860387713383od_v_v @ ( minus_4183494784930505774od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) ) @ B ) )
& ( ~ ( member7453568604450474000od_v_v @ X @ A3 )
=> ( ord_le7336532860387713383od_v_v @ A3 @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_224_ex__in__conv,axiom,
! [A3: set_Product_prod_v_v] :
( ( ? [X2: product_prod_v_v] : ( member7453568604450474000od_v_v @ X2 @ A3 ) )
= ( A3 != bot_bo723834152578015283od_v_v ) ) ).
% ex_in_conv
thf(fact_225_ex__in__conv,axiom,
! [A3: set_v] :
( ( ? [X2: v] : ( member_v @ X2 @ A3 ) )
= ( A3 != bot_bot_set_v ) ) ).
% ex_in_conv
thf(fact_226_equals0I,axiom,
! [A3: set_Product_prod_v_v] :
( ! [Y: product_prod_v_v] :
~ ( member7453568604450474000od_v_v @ Y @ A3 )
=> ( A3 = bot_bo723834152578015283od_v_v ) ) ).
% equals0I
thf(fact_227_equals0I,axiom,
! [A3: set_v] :
( ! [Y: v] :
~ ( member_v @ Y @ A3 )
=> ( A3 = bot_bot_set_v ) ) ).
% equals0I
thf(fact_228_equals0D,axiom,
! [A3: set_Product_prod_v_v,A: product_prod_v_v] :
( ( A3 = bot_bo723834152578015283od_v_v )
=> ~ ( member7453568604450474000od_v_v @ A @ A3 ) ) ).
% equals0D
thf(fact_229_equals0D,axiom,
! [A3: set_v,A: v] :
( ( A3 = bot_bot_set_v )
=> ~ ( member_v @ A @ A3 ) ) ).
% equals0D
thf(fact_230_emptyE,axiom,
! [A: product_prod_v_v] :
~ ( member7453568604450474000od_v_v @ A @ bot_bo723834152578015283od_v_v ) ).
% emptyE
thf(fact_231_emptyE,axiom,
! [A: v] :
~ ( member_v @ A @ bot_bot_set_v ) ).
% emptyE
thf(fact_232_Collect__mono__iff,axiom,
! [P: v > $o,Q: v > $o] :
( ( ord_less_eq_set_v @ ( collect_v @ P ) @ ( collect_v @ Q ) )
= ( ! [X2: v] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_233_Collect__mono__iff,axiom,
! [P: product_prod_v_v > $o,Q: product_prod_v_v > $o] :
( ( ord_le7336532860387713383od_v_v @ ( collec140062887454715474od_v_v @ P ) @ ( collec140062887454715474od_v_v @ Q ) )
= ( ! [X2: product_prod_v_v] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_234_set__eq__subset,axiom,
( ( ^ [Y4: set_v,Z4: set_v] : ( Y4 = Z4 ) )
= ( ^ [A4: set_v,B4: set_v] :
( ( ord_less_eq_set_v @ A4 @ B4 )
& ( ord_less_eq_set_v @ B4 @ A4 ) ) ) ) ).
% set_eq_subset
thf(fact_235_set__eq__subset,axiom,
( ( ^ [Y4: set_Product_prod_v_v,Z4: set_Product_prod_v_v] : ( Y4 = Z4 ) )
= ( ^ [A4: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A4 @ B4 )
& ( ord_le7336532860387713383od_v_v @ B4 @ A4 ) ) ) ) ).
% set_eq_subset
thf(fact_236_subset__trans,axiom,
! [A3: set_v,B: set_v,C3: set_v] :
( ( ord_less_eq_set_v @ A3 @ B )
=> ( ( ord_less_eq_set_v @ B @ C3 )
=> ( ord_less_eq_set_v @ A3 @ C3 ) ) ) ).
% subset_trans
thf(fact_237_subset__trans,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v,C3: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A3 @ B )
=> ( ( ord_le7336532860387713383od_v_v @ B @ C3 )
=> ( ord_le7336532860387713383od_v_v @ A3 @ C3 ) ) ) ).
% subset_trans
thf(fact_238_Collect__mono,axiom,
! [P: v > $o,Q: v > $o] :
( ! [X3: v] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq_set_v @ ( collect_v @ P ) @ ( collect_v @ Q ) ) ) ).
% Collect_mono
thf(fact_239_Collect__mono,axiom,
! [P: product_prod_v_v > $o,Q: product_prod_v_v > $o] :
( ! [X3: product_prod_v_v] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_le7336532860387713383od_v_v @ ( collec140062887454715474od_v_v @ P ) @ ( collec140062887454715474od_v_v @ Q ) ) ) ).
% Collect_mono
thf(fact_240_subset__refl,axiom,
! [A3: set_v] : ( ord_less_eq_set_v @ A3 @ A3 ) ).
% subset_refl
thf(fact_241_subset__refl,axiom,
! [A3: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ A3 @ A3 ) ).
% subset_refl
thf(fact_242_subset__iff,axiom,
( ord_less_eq_set_v
= ( ^ [A4: set_v,B4: set_v] :
! [T: v] :
( ( member_v @ T @ A4 )
=> ( member_v @ T @ B4 ) ) ) ) ).
% subset_iff
thf(fact_243_subset__iff,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [A4: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
! [T: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ T @ A4 )
=> ( member7453568604450474000od_v_v @ T @ B4 ) ) ) ) ).
% subset_iff
thf(fact_244_equalityD2,axiom,
! [A3: set_v,B: set_v] :
( ( A3 = B )
=> ( ord_less_eq_set_v @ B @ A3 ) ) ).
% equalityD2
thf(fact_245_equalityD2,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( A3 = B )
=> ( ord_le7336532860387713383od_v_v @ B @ A3 ) ) ).
% equalityD2
thf(fact_246_equalityD1,axiom,
! [A3: set_v,B: set_v] :
( ( A3 = B )
=> ( ord_less_eq_set_v @ A3 @ B ) ) ).
% equalityD1
thf(fact_247_equalityD1,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( A3 = B )
=> ( ord_le7336532860387713383od_v_v @ A3 @ B ) ) ).
% equalityD1
thf(fact_248_subset__eq,axiom,
( ord_less_eq_set_v
= ( ^ [A4: set_v,B4: set_v] :
! [X2: v] :
( ( member_v @ X2 @ A4 )
=> ( member_v @ X2 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_249_subset__eq,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [A4: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
! [X2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ A4 )
=> ( member7453568604450474000od_v_v @ X2 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_250_equalityE,axiom,
! [A3: set_v,B: set_v] :
( ( A3 = B )
=> ~ ( ( ord_less_eq_set_v @ A3 @ B )
=> ~ ( ord_less_eq_set_v @ B @ A3 ) ) ) ).
% equalityE
thf(fact_251_equalityE,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( A3 = B )
=> ~ ( ( ord_le7336532860387713383od_v_v @ A3 @ B )
=> ~ ( ord_le7336532860387713383od_v_v @ B @ A3 ) ) ) ).
% equalityE
thf(fact_252_subsetD,axiom,
! [A3: set_v,B: set_v,C: v] :
( ( ord_less_eq_set_v @ A3 @ B )
=> ( ( member_v @ C @ A3 )
=> ( member_v @ C @ B ) ) ) ).
% subsetD
thf(fact_253_subsetD,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v,C: product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A3 @ B )
=> ( ( member7453568604450474000od_v_v @ C @ A3 )
=> ( member7453568604450474000od_v_v @ C @ B ) ) ) ).
% subsetD
thf(fact_254_in__mono,axiom,
! [A3: set_v,B: set_v,X: v] :
( ( ord_less_eq_set_v @ A3 @ B )
=> ( ( member_v @ X @ A3 )
=> ( member_v @ X @ B ) ) ) ).
% in_mono
thf(fact_255_in__mono,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v,X: product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A3 @ B )
=> ( ( member7453568604450474000od_v_v @ X @ A3 )
=> ( member7453568604450474000od_v_v @ X @ B ) ) ) ).
% in_mono
thf(fact_256_DiffD2,axiom,
! [C: v,A3: set_v,B: set_v] :
( ( member_v @ C @ ( minus_minus_set_v @ A3 @ B ) )
=> ~ ( member_v @ C @ B ) ) ).
% DiffD2
thf(fact_257_DiffD2,axiom,
! [C: product_prod_v_v,A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ ( minus_4183494784930505774od_v_v @ A3 @ B ) )
=> ~ ( member7453568604450474000od_v_v @ C @ B ) ) ).
% DiffD2
thf(fact_258_DiffD1,axiom,
! [C: v,A3: set_v,B: set_v] :
( ( member_v @ C @ ( minus_minus_set_v @ A3 @ B ) )
=> ( member_v @ C @ A3 ) ) ).
% DiffD1
thf(fact_259_DiffD1,axiom,
! [C: product_prod_v_v,A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ ( minus_4183494784930505774od_v_v @ A3 @ B ) )
=> ( member7453568604450474000od_v_v @ C @ A3 ) ) ).
% DiffD1
thf(fact_260_DiffE,axiom,
! [C: v,A3: set_v,B: set_v] :
( ( member_v @ C @ ( minus_minus_set_v @ A3 @ B ) )
=> ~ ( ( member_v @ C @ A3 )
=> ( member_v @ C @ B ) ) ) ).
% DiffE
thf(fact_261_DiffE,axiom,
! [C: product_prod_v_v,A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ ( minus_4183494784930505774od_v_v @ A3 @ B ) )
=> ~ ( ( member7453568604450474000od_v_v @ C @ A3 )
=> ( member7453568604450474000od_v_v @ C @ B ) ) ) ).
% DiffE
thf(fact_262_graph_Osubscc__add,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,S: set_Product_prod_v_v,X: product_prod_v_v,Y2: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl2301996248249672505od_v_v @ Successors @ S )
=> ( ( member7453568604450474000od_v_v @ X @ S )
=> ( ( sCC_Bl4981926079593201289od_v_v @ Successors @ X @ Y2 )
=> ( ( sCC_Bl4981926079593201289od_v_v @ Successors @ Y2 @ X )
=> ( sCC_Bl2301996248249672505od_v_v @ Successors @ ( insert1338601472111419319od_v_v @ Y2 @ S ) ) ) ) ) ) ) ).
% graph.subscc_add
thf(fact_263_graph_Osubscc__add,axiom,
! [Vertices: set_v,Successors: v > set_v,S: set_v,X: v,Y2: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl5398416737448265317bscc_v @ Successors @ S )
=> ( ( member_v @ X @ S )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ X @ Y2 )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ Y2 @ X )
=> ( sCC_Bl5398416737448265317bscc_v @ Successors @ ( insert_v2 @ Y2 @ S ) ) ) ) ) ) ) ).
% graph.subscc_add
thf(fact_264_double__diff,axiom,
! [A3: set_v,B: set_v,C3: set_v] :
( ( ord_less_eq_set_v @ A3 @ B )
=> ( ( ord_less_eq_set_v @ B @ C3 )
=> ( ( minus_minus_set_v @ B @ ( minus_minus_set_v @ C3 @ A3 ) )
= A3 ) ) ) ).
% double_diff
thf(fact_265_double__diff,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v,C3: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A3 @ B )
=> ( ( ord_le7336532860387713383od_v_v @ B @ C3 )
=> ( ( minus_4183494784930505774od_v_v @ B @ ( minus_4183494784930505774od_v_v @ C3 @ A3 ) )
= A3 ) ) ) ).
% double_diff
thf(fact_266_Diff__subset,axiom,
! [A3: set_v,B: set_v] : ( ord_less_eq_set_v @ ( minus_minus_set_v @ A3 @ B ) @ A3 ) ).
% Diff_subset
thf(fact_267_Diff__subset,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( minus_4183494784930505774od_v_v @ A3 @ B ) @ A3 ) ).
% Diff_subset
thf(fact_268_Diff__mono,axiom,
! [A3: set_v,C3: set_v,D2: set_v,B: set_v] :
( ( ord_less_eq_set_v @ A3 @ C3 )
=> ( ( ord_less_eq_set_v @ D2 @ B )
=> ( ord_less_eq_set_v @ ( minus_minus_set_v @ A3 @ B ) @ ( minus_minus_set_v @ C3 @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_269_Diff__mono,axiom,
! [A3: set_Product_prod_v_v,C3: set_Product_prod_v_v,D2: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A3 @ C3 )
=> ( ( ord_le7336532860387713383od_v_v @ D2 @ B )
=> ( ord_le7336532860387713383od_v_v @ ( minus_4183494784930505774od_v_v @ A3 @ B ) @ ( minus_4183494784930505774od_v_v @ C3 @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_270_scc__partition,axiom,
! [S: set_v,S3: set_v,X: v] :
( ( sCC_Bloemen_is_scc_v @ successors @ S )
=> ( ( sCC_Bloemen_is_scc_v @ successors @ S3 )
=> ( ( member_v @ X @ ( inf_inf_set_v @ S @ S3 ) )
=> ( S = S3 ) ) ) ) ).
% scc_partition
thf(fact_271_the__elem__eq,axiom,
! [X: product_prod_v_v] :
( ( the_el5392834299063928540od_v_v @ ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) )
= X ) ).
% the_elem_eq
thf(fact_272_the__elem__eq,axiom,
! [X: v] :
( ( the_elem_v @ ( insert_v2 @ X @ bot_bot_set_v ) )
= X ) ).
% the_elem_eq
thf(fact_273_old_Oprod_Oinject,axiom,
! [A: v,B2: v,A5: v,B5: v] :
( ( ( product_Pair_v_v @ A @ B2 )
= ( product_Pair_v_v @ A5 @ B5 ) )
= ( ( A = A5 )
& ( B2 = B5 ) ) ) ).
% old.prod.inject
thf(fact_274_prod_Oinject,axiom,
! [X1: v,X22: v,Y1: v,Y22: v] :
( ( ( product_Pair_v_v @ X1 @ X22 )
= ( product_Pair_v_v @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X22 = Y22 ) ) ) ).
% prod.inject
thf(fact_275_dual__order_Orefl,axiom,
! [A: set_v] : ( ord_less_eq_set_v @ A @ A ) ).
% dual_order.refl
thf(fact_276_dual__order_Orefl,axiom,
! [A: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ A @ A ) ).
% dual_order.refl
thf(fact_277_order__refl,axiom,
! [X: set_v] : ( ord_less_eq_set_v @ X @ X ) ).
% order_refl
thf(fact_278_order__refl,axiom,
! [X: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ X @ X ) ).
% order_refl
thf(fact_279_less__by__empty,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( A3 = bot_bo723834152578015283od_v_v )
=> ( ord_le7336532860387713383od_v_v @ A3 @ B ) ) ).
% less_by_empty
thf(fact_280_inf_Oidem,axiom,
! [A: set_v] :
( ( inf_inf_set_v @ A @ A )
= A ) ).
% inf.idem
thf(fact_281_inf__idem,axiom,
! [X: set_v] :
( ( inf_inf_set_v @ X @ X )
= X ) ).
% inf_idem
thf(fact_282_inf_Oleft__idem,axiom,
! [A: set_v,B2: set_v] :
( ( inf_inf_set_v @ A @ ( inf_inf_set_v @ A @ B2 ) )
= ( inf_inf_set_v @ A @ B2 ) ) ).
% inf.left_idem
thf(fact_283_inf__left__idem,axiom,
! [X: set_v,Y2: set_v] :
( ( inf_inf_set_v @ X @ ( inf_inf_set_v @ X @ Y2 ) )
= ( inf_inf_set_v @ X @ Y2 ) ) ).
% inf_left_idem
thf(fact_284_inf_Oright__idem,axiom,
! [A: set_v,B2: set_v] :
( ( inf_inf_set_v @ ( inf_inf_set_v @ A @ B2 ) @ B2 )
= ( inf_inf_set_v @ A @ B2 ) ) ).
% inf.right_idem
thf(fact_285_inf__right__idem,axiom,
! [X: set_v,Y2: set_v] :
( ( inf_inf_set_v @ ( inf_inf_set_v @ X @ Y2 ) @ Y2 )
= ( inf_inf_set_v @ X @ Y2 ) ) ).
% inf_right_idem
thf(fact_286_Int__iff,axiom,
! [C: product_prod_v_v,A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ ( inf_in6271465464967711157od_v_v @ A3 @ B ) )
= ( ( member7453568604450474000od_v_v @ C @ A3 )
& ( member7453568604450474000od_v_v @ C @ B ) ) ) ).
% Int_iff
thf(fact_287_Int__iff,axiom,
! [C: v,A3: set_v,B: set_v] :
( ( member_v @ C @ ( inf_inf_set_v @ A3 @ B ) )
= ( ( member_v @ C @ A3 )
& ( member_v @ C @ B ) ) ) ).
% Int_iff
thf(fact_288_IntI,axiom,
! [C: product_prod_v_v,A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ A3 )
=> ( ( member7453568604450474000od_v_v @ C @ B )
=> ( member7453568604450474000od_v_v @ C @ ( inf_in6271465464967711157od_v_v @ A3 @ B ) ) ) ) ).
% IntI
thf(fact_289_IntI,axiom,
! [C: v,A3: set_v,B: set_v] :
( ( member_v @ C @ A3 )
=> ( ( member_v @ C @ B )
=> ( member_v @ C @ ( inf_inf_set_v @ A3 @ B ) ) ) ) ).
% IntI
thf(fact_290_le__inf__iff,axiom,
! [X: set_v,Y2: set_v,Z: set_v] :
( ( ord_less_eq_set_v @ X @ ( inf_inf_set_v @ Y2 @ Z ) )
= ( ( ord_less_eq_set_v @ X @ Y2 )
& ( ord_less_eq_set_v @ X @ Z ) ) ) ).
% le_inf_iff
thf(fact_291_le__inf__iff,axiom,
! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X @ ( inf_in6271465464967711157od_v_v @ Y2 @ Z ) )
= ( ( ord_le7336532860387713383od_v_v @ X @ Y2 )
& ( ord_le7336532860387713383od_v_v @ X @ Z ) ) ) ).
% le_inf_iff
thf(fact_292_inf_Obounded__iff,axiom,
! [A: set_v,B2: set_v,C: set_v] :
( ( ord_less_eq_set_v @ A @ ( inf_inf_set_v @ B2 @ C ) )
= ( ( ord_less_eq_set_v @ A @ B2 )
& ( ord_less_eq_set_v @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_293_inf_Obounded__iff,axiom,
! [A: set_Product_prod_v_v,B2: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ ( inf_in6271465464967711157od_v_v @ B2 @ C ) )
= ( ( ord_le7336532860387713383od_v_v @ A @ B2 )
& ( ord_le7336532860387713383od_v_v @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_294_inf__bot__left,axiom,
! [X: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ bot_bo723834152578015283od_v_v @ X )
= bot_bo723834152578015283od_v_v ) ).
% inf_bot_left
thf(fact_295_inf__bot__left,axiom,
! [X: set_v] :
( ( inf_inf_set_v @ bot_bot_set_v @ X )
= bot_bot_set_v ) ).
% inf_bot_left
thf(fact_296_inf__bot__right,axiom,
! [X: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ X @ bot_bo723834152578015283od_v_v )
= bot_bo723834152578015283od_v_v ) ).
% inf_bot_right
thf(fact_297_inf__bot__right,axiom,
! [X: set_v] :
( ( inf_inf_set_v @ X @ bot_bot_set_v )
= bot_bot_set_v ) ).
% inf_bot_right
thf(fact_298_boolean__algebra_Oconj__zero__left,axiom,
! [X: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ bot_bo723834152578015283od_v_v @ X )
= bot_bo723834152578015283od_v_v ) ).
% boolean_algebra.conj_zero_left
thf(fact_299_boolean__algebra_Oconj__zero__left,axiom,
! [X: set_v] :
( ( inf_inf_set_v @ bot_bot_set_v @ X )
= bot_bot_set_v ) ).
% boolean_algebra.conj_zero_left
thf(fact_300_boolean__algebra_Oconj__zero__right,axiom,
! [X: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ X @ bot_bo723834152578015283od_v_v )
= bot_bo723834152578015283od_v_v ) ).
% boolean_algebra.conj_zero_right
thf(fact_301_boolean__algebra_Oconj__zero__right,axiom,
! [X: set_v] :
( ( inf_inf_set_v @ X @ bot_bot_set_v )
= bot_bot_set_v ) ).
% boolean_algebra.conj_zero_right
thf(fact_302_Int__subset__iff,axiom,
! [C3: set_v,A3: set_v,B: set_v] :
( ( ord_less_eq_set_v @ C3 @ ( inf_inf_set_v @ A3 @ B ) )
= ( ( ord_less_eq_set_v @ C3 @ A3 )
& ( ord_less_eq_set_v @ C3 @ B ) ) ) ).
% Int_subset_iff
thf(fact_303_Int__subset__iff,axiom,
! [C3: set_Product_prod_v_v,A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ C3 @ ( inf_in6271465464967711157od_v_v @ A3 @ B ) )
= ( ( ord_le7336532860387713383od_v_v @ C3 @ A3 )
& ( ord_le7336532860387713383od_v_v @ C3 @ B ) ) ) ).
% Int_subset_iff
thf(fact_304_Int__insert__left__if0,axiom,
! [A: product_prod_v_v,C3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ~ ( member7453568604450474000od_v_v @ A @ C3 )
=> ( ( inf_in6271465464967711157od_v_v @ ( insert1338601472111419319od_v_v @ A @ B ) @ C3 )
= ( inf_in6271465464967711157od_v_v @ B @ C3 ) ) ) ).
% Int_insert_left_if0
thf(fact_305_Int__insert__left__if0,axiom,
! [A: v,C3: set_v,B: set_v] :
( ~ ( member_v @ A @ C3 )
=> ( ( inf_inf_set_v @ ( insert_v2 @ A @ B ) @ C3 )
= ( inf_inf_set_v @ B @ C3 ) ) ) ).
% Int_insert_left_if0
thf(fact_306_Int__insert__left__if1,axiom,
! [A: product_prod_v_v,C3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ A @ C3 )
=> ( ( inf_in6271465464967711157od_v_v @ ( insert1338601472111419319od_v_v @ A @ B ) @ C3 )
= ( insert1338601472111419319od_v_v @ A @ ( inf_in6271465464967711157od_v_v @ B @ C3 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_307_Int__insert__left__if1,axiom,
! [A: v,C3: set_v,B: set_v] :
( ( member_v @ A @ C3 )
=> ( ( inf_inf_set_v @ ( insert_v2 @ A @ B ) @ C3 )
= ( insert_v2 @ A @ ( inf_inf_set_v @ B @ C3 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_308_insert__inter__insert,axiom,
! [A: product_prod_v_v,A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ ( insert1338601472111419319od_v_v @ A @ A3 ) @ ( insert1338601472111419319od_v_v @ A @ B ) )
= ( insert1338601472111419319od_v_v @ A @ ( inf_in6271465464967711157od_v_v @ A3 @ B ) ) ) ).
% insert_inter_insert
thf(fact_309_insert__inter__insert,axiom,
! [A: v,A3: set_v,B: set_v] :
( ( inf_inf_set_v @ ( insert_v2 @ A @ A3 ) @ ( insert_v2 @ A @ B ) )
= ( insert_v2 @ A @ ( inf_inf_set_v @ A3 @ B ) ) ) ).
% insert_inter_insert
thf(fact_310_Int__insert__right__if0,axiom,
! [A: product_prod_v_v,A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ~ ( member7453568604450474000od_v_v @ A @ A3 )
=> ( ( inf_in6271465464967711157od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ A @ B ) )
= ( inf_in6271465464967711157od_v_v @ A3 @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_311_Int__insert__right__if0,axiom,
! [A: v,A3: set_v,B: set_v] :
( ~ ( member_v @ A @ A3 )
=> ( ( inf_inf_set_v @ A3 @ ( insert_v2 @ A @ B ) )
= ( inf_inf_set_v @ A3 @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_312_Int__insert__right__if1,axiom,
! [A: product_prod_v_v,A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ A @ A3 )
=> ( ( inf_in6271465464967711157od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ A @ B ) )
= ( insert1338601472111419319od_v_v @ A @ ( inf_in6271465464967711157od_v_v @ A3 @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_313_Int__insert__right__if1,axiom,
! [A: v,A3: set_v,B: set_v] :
( ( member_v @ A @ A3 )
=> ( ( inf_inf_set_v @ A3 @ ( insert_v2 @ A @ B ) )
= ( insert_v2 @ A @ ( inf_inf_set_v @ A3 @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_314_insert__disjoint_I1_J,axiom,
! [A: product_prod_v_v,A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ( inf_in6271465464967711157od_v_v @ ( insert1338601472111419319od_v_v @ A @ A3 ) @ B )
= bot_bo723834152578015283od_v_v )
= ( ~ ( member7453568604450474000od_v_v @ A @ B )
& ( ( inf_in6271465464967711157od_v_v @ A3 @ B )
= bot_bo723834152578015283od_v_v ) ) ) ).
% insert_disjoint(1)
thf(fact_315_insert__disjoint_I1_J,axiom,
! [A: v,A3: set_v,B: set_v] :
( ( ( inf_inf_set_v @ ( insert_v2 @ A @ A3 ) @ B )
= bot_bot_set_v )
= ( ~ ( member_v @ A @ B )
& ( ( inf_inf_set_v @ A3 @ B )
= bot_bot_set_v ) ) ) ).
% insert_disjoint(1)
thf(fact_316_insert__disjoint_I2_J,axiom,
! [A: product_prod_v_v,A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( bot_bo723834152578015283od_v_v
= ( inf_in6271465464967711157od_v_v @ ( insert1338601472111419319od_v_v @ A @ A3 ) @ B ) )
= ( ~ ( member7453568604450474000od_v_v @ A @ B )
& ( bot_bo723834152578015283od_v_v
= ( inf_in6271465464967711157od_v_v @ A3 @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_317_insert__disjoint_I2_J,axiom,
! [A: v,A3: set_v,B: set_v] :
( ( bot_bot_set_v
= ( inf_inf_set_v @ ( insert_v2 @ A @ A3 ) @ B ) )
= ( ~ ( member_v @ A @ B )
& ( bot_bot_set_v
= ( inf_inf_set_v @ A3 @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_318_disjoint__insert_I1_J,axiom,
! [B: set_Product_prod_v_v,A: product_prod_v_v,A3: set_Product_prod_v_v] :
( ( ( inf_in6271465464967711157od_v_v @ B @ ( insert1338601472111419319od_v_v @ A @ A3 ) )
= bot_bo723834152578015283od_v_v )
= ( ~ ( member7453568604450474000od_v_v @ A @ B )
& ( ( inf_in6271465464967711157od_v_v @ B @ A3 )
= bot_bo723834152578015283od_v_v ) ) ) ).
% disjoint_insert(1)
thf(fact_319_disjoint__insert_I1_J,axiom,
! [B: set_v,A: v,A3: set_v] :
( ( ( inf_inf_set_v @ B @ ( insert_v2 @ A @ A3 ) )
= bot_bot_set_v )
= ( ~ ( member_v @ A @ B )
& ( ( inf_inf_set_v @ B @ A3 )
= bot_bot_set_v ) ) ) ).
% disjoint_insert(1)
thf(fact_320_disjoint__insert_I2_J,axiom,
! [A3: set_Product_prod_v_v,B2: product_prod_v_v,B: set_Product_prod_v_v] :
( ( bot_bo723834152578015283od_v_v
= ( inf_in6271465464967711157od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ B2 @ B ) ) )
= ( ~ ( member7453568604450474000od_v_v @ B2 @ A3 )
& ( bot_bo723834152578015283od_v_v
= ( inf_in6271465464967711157od_v_v @ A3 @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_321_disjoint__insert_I2_J,axiom,
! [A3: set_v,B2: v,B: set_v] :
( ( bot_bot_set_v
= ( inf_inf_set_v @ A3 @ ( insert_v2 @ B2 @ B ) ) )
= ( ~ ( member_v @ B2 @ A3 )
& ( bot_bot_set_v
= ( inf_inf_set_v @ A3 @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_322_Diff__disjoint,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ A3 @ ( minus_4183494784930505774od_v_v @ B @ A3 ) )
= bot_bo723834152578015283od_v_v ) ).
% Diff_disjoint
thf(fact_323_Diff__disjoint,axiom,
! [A3: set_v,B: set_v] :
( ( inf_inf_set_v @ A3 @ ( minus_minus_set_v @ B @ A3 ) )
= bot_bot_set_v ) ).
% Diff_disjoint
thf(fact_324_Int__left__commute,axiom,
! [A3: set_v,B: set_v,C3: set_v] :
( ( inf_inf_set_v @ A3 @ ( inf_inf_set_v @ B @ C3 ) )
= ( inf_inf_set_v @ B @ ( inf_inf_set_v @ A3 @ C3 ) ) ) ).
% Int_left_commute
thf(fact_325_Int__left__absorb,axiom,
! [A3: set_v,B: set_v] :
( ( inf_inf_set_v @ A3 @ ( inf_inf_set_v @ A3 @ B ) )
= ( inf_inf_set_v @ A3 @ B ) ) ).
% Int_left_absorb
thf(fact_326_Int__commute,axiom,
( inf_inf_set_v
= ( ^ [A4: set_v,B4: set_v] : ( inf_inf_set_v @ B4 @ A4 ) ) ) ).
% Int_commute
thf(fact_327_Int__absorb,axiom,
! [A3: set_v] :
( ( inf_inf_set_v @ A3 @ A3 )
= A3 ) ).
% Int_absorb
thf(fact_328_Int__assoc,axiom,
! [A3: set_v,B: set_v,C3: set_v] :
( ( inf_inf_set_v @ ( inf_inf_set_v @ A3 @ B ) @ C3 )
= ( inf_inf_set_v @ A3 @ ( inf_inf_set_v @ B @ C3 ) ) ) ).
% Int_assoc
thf(fact_329_IntD2,axiom,
! [C: product_prod_v_v,A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ ( inf_in6271465464967711157od_v_v @ A3 @ B ) )
=> ( member7453568604450474000od_v_v @ C @ B ) ) ).
% IntD2
thf(fact_330_IntD2,axiom,
! [C: v,A3: set_v,B: set_v] :
( ( member_v @ C @ ( inf_inf_set_v @ A3 @ B ) )
=> ( member_v @ C @ B ) ) ).
% IntD2
thf(fact_331_IntD1,axiom,
! [C: product_prod_v_v,A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ ( inf_in6271465464967711157od_v_v @ A3 @ B ) )
=> ( member7453568604450474000od_v_v @ C @ A3 ) ) ).
% IntD1
thf(fact_332_IntD1,axiom,
! [C: v,A3: set_v,B: set_v] :
( ( member_v @ C @ ( inf_inf_set_v @ A3 @ B ) )
=> ( member_v @ C @ A3 ) ) ).
% IntD1
thf(fact_333_IntE,axiom,
! [C: product_prod_v_v,A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ ( inf_in6271465464967711157od_v_v @ A3 @ B ) )
=> ~ ( ( member7453568604450474000od_v_v @ C @ A3 )
=> ~ ( member7453568604450474000od_v_v @ C @ B ) ) ) ).
% IntE
thf(fact_334_IntE,axiom,
! [C: v,A3: set_v,B: set_v] :
( ( member_v @ C @ ( inf_inf_set_v @ A3 @ B ) )
=> ~ ( ( member_v @ C @ A3 )
=> ~ ( member_v @ C @ B ) ) ) ).
% IntE
thf(fact_335_boolean__algebra__cancel_Oinf2,axiom,
! [B: set_v,K: set_v,B2: set_v,A: set_v] :
( ( B
= ( inf_inf_set_v @ K @ B2 ) )
=> ( ( inf_inf_set_v @ A @ B )
= ( inf_inf_set_v @ K @ ( inf_inf_set_v @ A @ B2 ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_336_boolean__algebra__cancel_Oinf1,axiom,
! [A3: set_v,K: set_v,A: set_v,B2: set_v] :
( ( A3
= ( inf_inf_set_v @ K @ A ) )
=> ( ( inf_inf_set_v @ A3 @ B2 )
= ( inf_inf_set_v @ K @ ( inf_inf_set_v @ A @ B2 ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_337_inf_OcoboundedI2,axiom,
! [B2: set_v,C: set_v,A: set_v] :
( ( ord_less_eq_set_v @ B2 @ C )
=> ( ord_less_eq_set_v @ ( inf_inf_set_v @ A @ B2 ) @ C ) ) ).
% inf.coboundedI2
thf(fact_338_inf_OcoboundedI2,axiom,
! [B2: set_Product_prod_v_v,C: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B2 @ C )
=> ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A @ B2 ) @ C ) ) ).
% inf.coboundedI2
thf(fact_339_inf_OcoboundedI1,axiom,
! [A: set_v,C: set_v,B2: set_v] :
( ( ord_less_eq_set_v @ A @ C )
=> ( ord_less_eq_set_v @ ( inf_inf_set_v @ A @ B2 ) @ C ) ) ).
% inf.coboundedI1
thf(fact_340_inf_OcoboundedI1,axiom,
! [A: set_Product_prod_v_v,C: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ C )
=> ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A @ B2 ) @ C ) ) ).
% inf.coboundedI1
thf(fact_341_inf_Oabsorb__iff2,axiom,
( ord_less_eq_set_v
= ( ^ [B6: set_v,A6: set_v] :
( ( inf_inf_set_v @ A6 @ B6 )
= B6 ) ) ) ).
% inf.absorb_iff2
thf(fact_342_inf_Oabsorb__iff2,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [B6: set_Product_prod_v_v,A6: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ A6 @ B6 )
= B6 ) ) ) ).
% inf.absorb_iff2
thf(fact_343_inf_Oabsorb__iff1,axiom,
( ord_less_eq_set_v
= ( ^ [A6: set_v,B6: set_v] :
( ( inf_inf_set_v @ A6 @ B6 )
= A6 ) ) ) ).
% inf.absorb_iff1
thf(fact_344_inf_Oabsorb__iff1,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [A6: set_Product_prod_v_v,B6: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ A6 @ B6 )
= A6 ) ) ) ).
% inf.absorb_iff1
thf(fact_345_inf_Ocobounded2,axiom,
! [A: set_v,B2: set_v] : ( ord_less_eq_set_v @ ( inf_inf_set_v @ A @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_346_inf_Ocobounded2,axiom,
! [A: set_Product_prod_v_v,B2: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_347_inf_Ocobounded1,axiom,
! [A: set_v,B2: set_v] : ( ord_less_eq_set_v @ ( inf_inf_set_v @ A @ B2 ) @ A ) ).
% inf.cobounded1
thf(fact_348_inf_Ocobounded1,axiom,
! [A: set_Product_prod_v_v,B2: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A @ B2 ) @ A ) ).
% inf.cobounded1
thf(fact_349_inf_Oorder__iff,axiom,
( ord_less_eq_set_v
= ( ^ [A6: set_v,B6: set_v] :
( A6
= ( inf_inf_set_v @ A6 @ B6 ) ) ) ) ).
% inf.order_iff
thf(fact_350_inf_Oorder__iff,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [A6: set_Product_prod_v_v,B6: set_Product_prod_v_v] :
( A6
= ( inf_in6271465464967711157od_v_v @ A6 @ B6 ) ) ) ) ).
% inf.order_iff
thf(fact_351_inf__greatest,axiom,
! [X: set_v,Y2: set_v,Z: set_v] :
( ( ord_less_eq_set_v @ X @ Y2 )
=> ( ( ord_less_eq_set_v @ X @ Z )
=> ( ord_less_eq_set_v @ X @ ( inf_inf_set_v @ Y2 @ Z ) ) ) ) ).
% inf_greatest
thf(fact_352_inf__greatest,axiom,
! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X @ Y2 )
=> ( ( ord_le7336532860387713383od_v_v @ X @ Z )
=> ( ord_le7336532860387713383od_v_v @ X @ ( inf_in6271465464967711157od_v_v @ Y2 @ Z ) ) ) ) ).
% inf_greatest
thf(fact_353_inf_OboundedI,axiom,
! [A: set_v,B2: set_v,C: set_v] :
( ( ord_less_eq_set_v @ A @ B2 )
=> ( ( ord_less_eq_set_v @ A @ C )
=> ( ord_less_eq_set_v @ A @ ( inf_inf_set_v @ B2 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_354_inf_OboundedI,axiom,
! [A: set_Product_prod_v_v,B2: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B2 )
=> ( ( ord_le7336532860387713383od_v_v @ A @ C )
=> ( ord_le7336532860387713383od_v_v @ A @ ( inf_in6271465464967711157od_v_v @ B2 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_355_inf_OboundedE,axiom,
! [A: set_v,B2: set_v,C: set_v] :
( ( ord_less_eq_set_v @ A @ ( inf_inf_set_v @ B2 @ C ) )
=> ~ ( ( ord_less_eq_set_v @ A @ B2 )
=> ~ ( ord_less_eq_set_v @ A @ C ) ) ) ).
% inf.boundedE
thf(fact_356_inf_OboundedE,axiom,
! [A: set_Product_prod_v_v,B2: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ ( inf_in6271465464967711157od_v_v @ B2 @ C ) )
=> ~ ( ( ord_le7336532860387713383od_v_v @ A @ B2 )
=> ~ ( ord_le7336532860387713383od_v_v @ A @ C ) ) ) ).
% inf.boundedE
thf(fact_357_inf__absorb2,axiom,
! [Y2: set_v,X: set_v] :
( ( ord_less_eq_set_v @ Y2 @ X )
=> ( ( inf_inf_set_v @ X @ Y2 )
= Y2 ) ) ).
% inf_absorb2
thf(fact_358_inf__absorb2,axiom,
! [Y2: set_Product_prod_v_v,X: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ Y2 @ X )
=> ( ( inf_in6271465464967711157od_v_v @ X @ Y2 )
= Y2 ) ) ).
% inf_absorb2
thf(fact_359_inf__absorb1,axiom,
! [X: set_v,Y2: set_v] :
( ( ord_less_eq_set_v @ X @ Y2 )
=> ( ( inf_inf_set_v @ X @ Y2 )
= X ) ) ).
% inf_absorb1
thf(fact_360_inf__absorb1,axiom,
! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X @ Y2 )
=> ( ( inf_in6271465464967711157od_v_v @ X @ Y2 )
= X ) ) ).
% inf_absorb1
thf(fact_361_inf_Oabsorb2,axiom,
! [B2: set_v,A: set_v] :
( ( ord_less_eq_set_v @ B2 @ A )
=> ( ( inf_inf_set_v @ A @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_362_inf_Oabsorb2,axiom,
! [B2: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B2 @ A )
=> ( ( inf_in6271465464967711157od_v_v @ A @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_363_inf_Oabsorb1,axiom,
! [A: set_v,B2: set_v] :
( ( ord_less_eq_set_v @ A @ B2 )
=> ( ( inf_inf_set_v @ A @ B2 )
= A ) ) ).
% inf.absorb1
thf(fact_364_inf_Oabsorb1,axiom,
! [A: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B2 )
=> ( ( inf_in6271465464967711157od_v_v @ A @ B2 )
= A ) ) ).
% inf.absorb1
thf(fact_365_le__iff__inf,axiom,
( ord_less_eq_set_v
= ( ^ [X2: set_v,Y3: set_v] :
( ( inf_inf_set_v @ X2 @ Y3 )
= X2 ) ) ) ).
% le_iff_inf
thf(fact_366_le__iff__inf,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [X2: set_Product_prod_v_v,Y3: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ X2 @ Y3 )
= X2 ) ) ) ).
% le_iff_inf
thf(fact_367_inf__unique,axiom,
! [F: set_v > set_v > set_v,X: set_v,Y2: set_v] :
( ! [X3: set_v,Y: set_v] : ( ord_less_eq_set_v @ ( F @ X3 @ Y ) @ X3 )
=> ( ! [X3: set_v,Y: set_v] : ( ord_less_eq_set_v @ ( F @ X3 @ Y ) @ Y )
=> ( ! [X3: set_v,Y: set_v,Z3: set_v] :
( ( ord_less_eq_set_v @ X3 @ Y )
=> ( ( ord_less_eq_set_v @ X3 @ Z3 )
=> ( ord_less_eq_set_v @ X3 @ ( F @ Y @ Z3 ) ) ) )
=> ( ( inf_inf_set_v @ X @ Y2 )
= ( F @ X @ Y2 ) ) ) ) ) ).
% inf_unique
thf(fact_368_inf__unique,axiom,
! [F: set_Product_prod_v_v > set_Product_prod_v_v > set_Product_prod_v_v,X: set_Product_prod_v_v,Y2: set_Product_prod_v_v] :
( ! [X3: set_Product_prod_v_v,Y: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( F @ X3 @ Y ) @ X3 )
=> ( ! [X3: set_Product_prod_v_v,Y: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( F @ X3 @ Y ) @ Y )
=> ( ! [X3: set_Product_prod_v_v,Y: set_Product_prod_v_v,Z3: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X3 @ Y )
=> ( ( ord_le7336532860387713383od_v_v @ X3 @ Z3 )
=> ( ord_le7336532860387713383od_v_v @ X3 @ ( F @ Y @ Z3 ) ) ) )
=> ( ( inf_in6271465464967711157od_v_v @ X @ Y2 )
= ( F @ X @ Y2 ) ) ) ) ) ).
% inf_unique
thf(fact_369_inf_OorderI,axiom,
! [A: set_v,B2: set_v] :
( ( A
= ( inf_inf_set_v @ A @ B2 ) )
=> ( ord_less_eq_set_v @ A @ B2 ) ) ).
% inf.orderI
thf(fact_370_inf_OorderI,axiom,
! [A: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( A
= ( inf_in6271465464967711157od_v_v @ A @ B2 ) )
=> ( ord_le7336532860387713383od_v_v @ A @ B2 ) ) ).
% inf.orderI
thf(fact_371_inf_OorderE,axiom,
! [A: set_v,B2: set_v] :
( ( ord_less_eq_set_v @ A @ B2 )
=> ( A
= ( inf_inf_set_v @ A @ B2 ) ) ) ).
% inf.orderE
thf(fact_372_inf_OorderE,axiom,
! [A: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B2 )
=> ( A
= ( inf_in6271465464967711157od_v_v @ A @ B2 ) ) ) ).
% inf.orderE
thf(fact_373_le__infI2,axiom,
! [B2: set_v,X: set_v,A: set_v] :
( ( ord_less_eq_set_v @ B2 @ X )
=> ( ord_less_eq_set_v @ ( inf_inf_set_v @ A @ B2 ) @ X ) ) ).
% le_infI2
thf(fact_374_le__infI2,axiom,
! [B2: set_Product_prod_v_v,X: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B2 @ X )
=> ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A @ B2 ) @ X ) ) ).
% le_infI2
thf(fact_375_le__infI1,axiom,
! [A: set_v,X: set_v,B2: set_v] :
( ( ord_less_eq_set_v @ A @ X )
=> ( ord_less_eq_set_v @ ( inf_inf_set_v @ A @ B2 ) @ X ) ) ).
% le_infI1
thf(fact_376_le__infI1,axiom,
! [A: set_Product_prod_v_v,X: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ X )
=> ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A @ B2 ) @ X ) ) ).
% le_infI1
thf(fact_377_inf__mono,axiom,
! [A: set_v,C: set_v,B2: set_v,D: set_v] :
( ( ord_less_eq_set_v @ A @ C )
=> ( ( ord_less_eq_set_v @ B2 @ D )
=> ( ord_less_eq_set_v @ ( inf_inf_set_v @ A @ B2 ) @ ( inf_inf_set_v @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_378_inf__mono,axiom,
! [A: set_Product_prod_v_v,C: set_Product_prod_v_v,B2: set_Product_prod_v_v,D: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ C )
=> ( ( ord_le7336532860387713383od_v_v @ B2 @ D )
=> ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A @ B2 ) @ ( inf_in6271465464967711157od_v_v @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_379_le__infI,axiom,
! [X: set_v,A: set_v,B2: set_v] :
( ( ord_less_eq_set_v @ X @ A )
=> ( ( ord_less_eq_set_v @ X @ B2 )
=> ( ord_less_eq_set_v @ X @ ( inf_inf_set_v @ A @ B2 ) ) ) ) ).
% le_infI
thf(fact_380_le__infI,axiom,
! [X: set_Product_prod_v_v,A: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X @ A )
=> ( ( ord_le7336532860387713383od_v_v @ X @ B2 )
=> ( ord_le7336532860387713383od_v_v @ X @ ( inf_in6271465464967711157od_v_v @ A @ B2 ) ) ) ) ).
% le_infI
thf(fact_381_le__infE,axiom,
! [X: set_v,A: set_v,B2: set_v] :
( ( ord_less_eq_set_v @ X @ ( inf_inf_set_v @ A @ B2 ) )
=> ~ ( ( ord_less_eq_set_v @ X @ A )
=> ~ ( ord_less_eq_set_v @ X @ B2 ) ) ) ).
% le_infE
thf(fact_382_le__infE,axiom,
! [X: set_Product_prod_v_v,A: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X @ ( inf_in6271465464967711157od_v_v @ A @ B2 ) )
=> ~ ( ( ord_le7336532860387713383od_v_v @ X @ A )
=> ~ ( ord_le7336532860387713383od_v_v @ X @ B2 ) ) ) ).
% le_infE
thf(fact_383_inf__le2,axiom,
! [X: set_v,Y2: set_v] : ( ord_less_eq_set_v @ ( inf_inf_set_v @ X @ Y2 ) @ Y2 ) ).
% inf_le2
thf(fact_384_inf__le2,axiom,
! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ X @ Y2 ) @ Y2 ) ).
% inf_le2
thf(fact_385_inf__le1,axiom,
! [X: set_v,Y2: set_v] : ( ord_less_eq_set_v @ ( inf_inf_set_v @ X @ Y2 ) @ X ) ).
% inf_le1
thf(fact_386_inf__le1,axiom,
! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ X @ Y2 ) @ X ) ).
% inf_le1
thf(fact_387_inf__sup__ord_I1_J,axiom,
! [X: set_v,Y2: set_v] : ( ord_less_eq_set_v @ ( inf_inf_set_v @ X @ Y2 ) @ X ) ).
% inf_sup_ord(1)
thf(fact_388_inf__sup__ord_I1_J,axiom,
! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ X @ Y2 ) @ X ) ).
% inf_sup_ord(1)
thf(fact_389_inf__sup__ord_I2_J,axiom,
! [X: set_v,Y2: set_v] : ( ord_less_eq_set_v @ ( inf_inf_set_v @ X @ Y2 ) @ Y2 ) ).
% inf_sup_ord(2)
thf(fact_390_inf__sup__ord_I2_J,axiom,
! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ X @ Y2 ) @ Y2 ) ).
% inf_sup_ord(2)
thf(fact_391_inf__sup__aci_I4_J,axiom,
! [X: set_v,Y2: set_v] :
( ( inf_inf_set_v @ X @ ( inf_inf_set_v @ X @ Y2 ) )
= ( inf_inf_set_v @ X @ Y2 ) ) ).
% inf_sup_aci(4)
thf(fact_392_inf__sup__aci_I3_J,axiom,
! [X: set_v,Y2: set_v,Z: set_v] :
( ( inf_inf_set_v @ X @ ( inf_inf_set_v @ Y2 @ Z ) )
= ( inf_inf_set_v @ Y2 @ ( inf_inf_set_v @ X @ Z ) ) ) ).
% inf_sup_aci(3)
thf(fact_393_inf__sup__aci_I2_J,axiom,
! [X: set_v,Y2: set_v,Z: set_v] :
( ( inf_inf_set_v @ ( inf_inf_set_v @ X @ Y2 ) @ Z )
= ( inf_inf_set_v @ X @ ( inf_inf_set_v @ Y2 @ Z ) ) ) ).
% inf_sup_aci(2)
thf(fact_394_inf__sup__aci_I1_J,axiom,
( inf_inf_set_v
= ( ^ [X2: set_v,Y3: set_v] : ( inf_inf_set_v @ Y3 @ X2 ) ) ) ).
% inf_sup_aci(1)
thf(fact_395_inf_Oassoc,axiom,
! [A: set_v,B2: set_v,C: set_v] :
( ( inf_inf_set_v @ ( inf_inf_set_v @ A @ B2 ) @ C )
= ( inf_inf_set_v @ A @ ( inf_inf_set_v @ B2 @ C ) ) ) ).
% inf.assoc
thf(fact_396_inf__assoc,axiom,
! [X: set_v,Y2: set_v,Z: set_v] :
( ( inf_inf_set_v @ ( inf_inf_set_v @ X @ Y2 ) @ Z )
= ( inf_inf_set_v @ X @ ( inf_inf_set_v @ Y2 @ Z ) ) ) ).
% inf_assoc
thf(fact_397_inf_Ocommute,axiom,
( inf_inf_set_v
= ( ^ [A6: set_v,B6: set_v] : ( inf_inf_set_v @ B6 @ A6 ) ) ) ).
% inf.commute
thf(fact_398_inf__commute,axiom,
( inf_inf_set_v
= ( ^ [X2: set_v,Y3: set_v] : ( inf_inf_set_v @ Y3 @ X2 ) ) ) ).
% inf_commute
thf(fact_399_inf_Oleft__commute,axiom,
! [B2: set_v,A: set_v,C: set_v] :
( ( inf_inf_set_v @ B2 @ ( inf_inf_set_v @ A @ C ) )
= ( inf_inf_set_v @ A @ ( inf_inf_set_v @ B2 @ C ) ) ) ).
% inf.left_commute
thf(fact_400_inf__left__commute,axiom,
! [X: set_v,Y2: set_v,Z: set_v] :
( ( inf_inf_set_v @ X @ ( inf_inf_set_v @ Y2 @ Z ) )
= ( inf_inf_set_v @ Y2 @ ( inf_inf_set_v @ X @ Z ) ) ) ).
% inf_left_commute
thf(fact_401_Int__emptyI,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ! [X3: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X3 @ A3 )
=> ~ ( member7453568604450474000od_v_v @ X3 @ B ) )
=> ( ( inf_in6271465464967711157od_v_v @ A3 @ B )
= bot_bo723834152578015283od_v_v ) ) ).
% Int_emptyI
thf(fact_402_Int__emptyI,axiom,
! [A3: set_v,B: set_v] :
( ! [X3: v] :
( ( member_v @ X3 @ A3 )
=> ~ ( member_v @ X3 @ B ) )
=> ( ( inf_inf_set_v @ A3 @ B )
= bot_bot_set_v ) ) ).
% Int_emptyI
thf(fact_403_disjoint__iff,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ( inf_in6271465464967711157od_v_v @ A3 @ B )
= bot_bo723834152578015283od_v_v )
= ( ! [X2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ A3 )
=> ~ ( member7453568604450474000od_v_v @ X2 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_404_disjoint__iff,axiom,
! [A3: set_v,B: set_v] :
( ( ( inf_inf_set_v @ A3 @ B )
= bot_bot_set_v )
= ( ! [X2: v] :
( ( member_v @ X2 @ A3 )
=> ~ ( member_v @ X2 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_405_Int__empty__left,axiom,
! [B: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ bot_bo723834152578015283od_v_v @ B )
= bot_bo723834152578015283od_v_v ) ).
% Int_empty_left
thf(fact_406_Int__empty__left,axiom,
! [B: set_v] :
( ( inf_inf_set_v @ bot_bot_set_v @ B )
= bot_bot_set_v ) ).
% Int_empty_left
thf(fact_407_Int__empty__right,axiom,
! [A3: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ A3 @ bot_bo723834152578015283od_v_v )
= bot_bo723834152578015283od_v_v ) ).
% Int_empty_right
thf(fact_408_Int__empty__right,axiom,
! [A3: set_v] :
( ( inf_inf_set_v @ A3 @ bot_bot_set_v )
= bot_bot_set_v ) ).
% Int_empty_right
thf(fact_409_disjoint__iff__not__equal,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ( inf_in6271465464967711157od_v_v @ A3 @ B )
= bot_bo723834152578015283od_v_v )
= ( ! [X2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ A3 )
=> ! [Y3: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ Y3 @ B )
=> ( X2 != Y3 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_410_disjoint__iff__not__equal,axiom,
! [A3: set_v,B: set_v] :
( ( ( inf_inf_set_v @ A3 @ B )
= bot_bot_set_v )
= ( ! [X2: v] :
( ( member_v @ X2 @ A3 )
=> ! [Y3: v] :
( ( member_v @ Y3 @ B )
=> ( X2 != Y3 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_411_Int__mono,axiom,
! [A3: set_v,C3: set_v,B: set_v,D2: set_v] :
( ( ord_less_eq_set_v @ A3 @ C3 )
=> ( ( ord_less_eq_set_v @ B @ D2 )
=> ( ord_less_eq_set_v @ ( inf_inf_set_v @ A3 @ B ) @ ( inf_inf_set_v @ C3 @ D2 ) ) ) ) ).
% Int_mono
thf(fact_412_Int__mono,axiom,
! [A3: set_Product_prod_v_v,C3: set_Product_prod_v_v,B: set_Product_prod_v_v,D2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A3 @ C3 )
=> ( ( ord_le7336532860387713383od_v_v @ B @ D2 )
=> ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A3 @ B ) @ ( inf_in6271465464967711157od_v_v @ C3 @ D2 ) ) ) ) ).
% Int_mono
thf(fact_413_Int__lower1,axiom,
! [A3: set_v,B: set_v] : ( ord_less_eq_set_v @ ( inf_inf_set_v @ A3 @ B ) @ A3 ) ).
% Int_lower1
thf(fact_414_Int__lower1,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A3 @ B ) @ A3 ) ).
% Int_lower1
thf(fact_415_Int__lower2,axiom,
! [A3: set_v,B: set_v] : ( ord_less_eq_set_v @ ( inf_inf_set_v @ A3 @ B ) @ B ) ).
% Int_lower2
thf(fact_416_Int__lower2,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A3 @ B ) @ B ) ).
% Int_lower2
thf(fact_417_Int__absorb1,axiom,
! [B: set_v,A3: set_v] :
( ( ord_less_eq_set_v @ B @ A3 )
=> ( ( inf_inf_set_v @ A3 @ B )
= B ) ) ).
% Int_absorb1
thf(fact_418_Int__absorb1,axiom,
! [B: set_Product_prod_v_v,A3: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B @ A3 )
=> ( ( inf_in6271465464967711157od_v_v @ A3 @ B )
= B ) ) ).
% Int_absorb1
thf(fact_419_Int__absorb2,axiom,
! [A3: set_v,B: set_v] :
( ( ord_less_eq_set_v @ A3 @ B )
=> ( ( inf_inf_set_v @ A3 @ B )
= A3 ) ) ).
% Int_absorb2
thf(fact_420_Int__absorb2,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A3 @ B )
=> ( ( inf_in6271465464967711157od_v_v @ A3 @ B )
= A3 ) ) ).
% Int_absorb2
thf(fact_421_Int__greatest,axiom,
! [C3: set_v,A3: set_v,B: set_v] :
( ( ord_less_eq_set_v @ C3 @ A3 )
=> ( ( ord_less_eq_set_v @ C3 @ B )
=> ( ord_less_eq_set_v @ C3 @ ( inf_inf_set_v @ A3 @ B ) ) ) ) ).
% Int_greatest
thf(fact_422_Int__greatest,axiom,
! [C3: set_Product_prod_v_v,A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ C3 @ A3 )
=> ( ( ord_le7336532860387713383od_v_v @ C3 @ B )
=> ( ord_le7336532860387713383od_v_v @ C3 @ ( inf_in6271465464967711157od_v_v @ A3 @ B ) ) ) ) ).
% Int_greatest
thf(fact_423_Int__Collect__mono,axiom,
! [A3: set_v,B: set_v,P: v > $o,Q: v > $o] :
( ( ord_less_eq_set_v @ A3 @ B )
=> ( ! [X3: v] :
( ( member_v @ X3 @ A3 )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_less_eq_set_v @ ( inf_inf_set_v @ A3 @ ( collect_v @ P ) ) @ ( inf_inf_set_v @ B @ ( collect_v @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_424_Int__Collect__mono,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v,P: product_prod_v_v > $o,Q: product_prod_v_v > $o] :
( ( ord_le7336532860387713383od_v_v @ A3 @ B )
=> ( ! [X3: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X3 @ A3 )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A3 @ ( collec140062887454715474od_v_v @ P ) ) @ ( inf_in6271465464967711157od_v_v @ B @ ( collec140062887454715474od_v_v @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_425_Int__insert__left,axiom,
! [A: product_prod_v_v,C3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ( member7453568604450474000od_v_v @ A @ C3 )
=> ( ( inf_in6271465464967711157od_v_v @ ( insert1338601472111419319od_v_v @ A @ B ) @ C3 )
= ( insert1338601472111419319od_v_v @ A @ ( inf_in6271465464967711157od_v_v @ B @ C3 ) ) ) )
& ( ~ ( member7453568604450474000od_v_v @ A @ C3 )
=> ( ( inf_in6271465464967711157od_v_v @ ( insert1338601472111419319od_v_v @ A @ B ) @ C3 )
= ( inf_in6271465464967711157od_v_v @ B @ C3 ) ) ) ) ).
% Int_insert_left
thf(fact_426_Int__insert__left,axiom,
! [A: v,C3: set_v,B: set_v] :
( ( ( member_v @ A @ C3 )
=> ( ( inf_inf_set_v @ ( insert_v2 @ A @ B ) @ C3 )
= ( insert_v2 @ A @ ( inf_inf_set_v @ B @ C3 ) ) ) )
& ( ~ ( member_v @ A @ C3 )
=> ( ( inf_inf_set_v @ ( insert_v2 @ A @ B ) @ C3 )
= ( inf_inf_set_v @ B @ C3 ) ) ) ) ).
% Int_insert_left
thf(fact_427_Int__insert__right,axiom,
! [A: product_prod_v_v,A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ( member7453568604450474000od_v_v @ A @ A3 )
=> ( ( inf_in6271465464967711157od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ A @ B ) )
= ( insert1338601472111419319od_v_v @ A @ ( inf_in6271465464967711157od_v_v @ A3 @ B ) ) ) )
& ( ~ ( member7453568604450474000od_v_v @ A @ A3 )
=> ( ( inf_in6271465464967711157od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ A @ B ) )
= ( inf_in6271465464967711157od_v_v @ A3 @ B ) ) ) ) ).
% Int_insert_right
thf(fact_428_Int__insert__right,axiom,
! [A: v,A3: set_v,B: set_v] :
( ( ( member_v @ A @ A3 )
=> ( ( inf_inf_set_v @ A3 @ ( insert_v2 @ A @ B ) )
= ( insert_v2 @ A @ ( inf_inf_set_v @ A3 @ B ) ) ) )
& ( ~ ( member_v @ A @ A3 )
=> ( ( inf_inf_set_v @ A3 @ ( insert_v2 @ A @ B ) )
= ( inf_inf_set_v @ A3 @ B ) ) ) ) ).
% Int_insert_right
thf(fact_429_Int__Diff,axiom,
! [A3: set_v,B: set_v,C3: set_v] :
( ( minus_minus_set_v @ ( inf_inf_set_v @ A3 @ B ) @ C3 )
= ( inf_inf_set_v @ A3 @ ( minus_minus_set_v @ B @ C3 ) ) ) ).
% Int_Diff
thf(fact_430_Diff__Int2,axiom,
! [A3: set_v,C3: set_v,B: set_v] :
( ( minus_minus_set_v @ ( inf_inf_set_v @ A3 @ C3 ) @ ( inf_inf_set_v @ B @ C3 ) )
= ( minus_minus_set_v @ ( inf_inf_set_v @ A3 @ C3 ) @ B ) ) ).
% Diff_Int2
thf(fact_431_Diff__Diff__Int,axiom,
! [A3: set_v,B: set_v] :
( ( minus_minus_set_v @ A3 @ ( minus_minus_set_v @ A3 @ B ) )
= ( inf_inf_set_v @ A3 @ B ) ) ).
% Diff_Diff_Int
thf(fact_432_Diff__Int__distrib,axiom,
! [C3: set_v,A3: set_v,B: set_v] :
( ( inf_inf_set_v @ C3 @ ( minus_minus_set_v @ A3 @ B ) )
= ( minus_minus_set_v @ ( inf_inf_set_v @ C3 @ A3 ) @ ( inf_inf_set_v @ C3 @ B ) ) ) ).
% Diff_Int_distrib
thf(fact_433_Diff__Int__distrib2,axiom,
! [A3: set_v,B: set_v,C3: set_v] :
( ( inf_inf_set_v @ ( minus_minus_set_v @ A3 @ B ) @ C3 )
= ( minus_minus_set_v @ ( inf_inf_set_v @ A3 @ C3 ) @ ( inf_inf_set_v @ B @ C3 ) ) ) ).
% Diff_Int_distrib2
thf(fact_434_Diff__triv,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ( inf_in6271465464967711157od_v_v @ A3 @ B )
= bot_bo723834152578015283od_v_v )
=> ( ( minus_4183494784930505774od_v_v @ A3 @ B )
= A3 ) ) ).
% Diff_triv
thf(fact_435_Diff__triv,axiom,
! [A3: set_v,B: set_v] :
( ( ( inf_inf_set_v @ A3 @ B )
= bot_bot_set_v )
=> ( ( minus_minus_set_v @ A3 @ B )
= A3 ) ) ).
% Diff_triv
thf(fact_436_Int__Diff__disjoint,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ ( inf_in6271465464967711157od_v_v @ A3 @ B ) @ ( minus_4183494784930505774od_v_v @ A3 @ B ) )
= bot_bo723834152578015283od_v_v ) ).
% Int_Diff_disjoint
thf(fact_437_Int__Diff__disjoint,axiom,
! [A3: set_v,B: set_v] :
( ( inf_inf_set_v @ ( inf_inf_set_v @ A3 @ B ) @ ( minus_minus_set_v @ A3 @ B ) )
= bot_bot_set_v ) ).
% Int_Diff_disjoint
thf(fact_438_graph_Oscc__partition,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,S: set_Product_prod_v_v,S3: set_Product_prod_v_v,X: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl6242042402218619277od_v_v @ Successors @ S )
=> ( ( sCC_Bl6242042402218619277od_v_v @ Successors @ S3 )
=> ( ( member7453568604450474000od_v_v @ X @ ( inf_in6271465464967711157od_v_v @ S @ S3 ) )
=> ( S = S3 ) ) ) ) ) ).
% graph.scc_partition
thf(fact_439_graph_Oscc__partition,axiom,
! [Vertices: set_v,Successors: v > set_v,S: set_v,S3: set_v,X: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bloemen_is_scc_v @ Successors @ S )
=> ( ( sCC_Bloemen_is_scc_v @ Successors @ S3 )
=> ( ( member_v @ X @ ( inf_inf_set_v @ S @ S3 ) )
=> ( S = S3 ) ) ) ) ) ).
% graph.scc_partition
thf(fact_440_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_v,Z4: set_v] : ( Y4 = Z4 ) )
= ( ^ [X2: set_v,Y3: set_v] :
( ( ord_less_eq_set_v @ X2 @ Y3 )
& ( ord_less_eq_set_v @ Y3 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_441_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_Product_prod_v_v,Z4: set_Product_prod_v_v] : ( Y4 = Z4 ) )
= ( ^ [X2: set_Product_prod_v_v,Y3: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X2 @ Y3 )
& ( ord_le7336532860387713383od_v_v @ Y3 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_442_ord__eq__le__trans,axiom,
! [A: set_v,B2: set_v,C: set_v] :
( ( A = B2 )
=> ( ( ord_less_eq_set_v @ B2 @ C )
=> ( ord_less_eq_set_v @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_443_ord__eq__le__trans,axiom,
! [A: set_Product_prod_v_v,B2: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( A = B2 )
=> ( ( ord_le7336532860387713383od_v_v @ B2 @ C )
=> ( ord_le7336532860387713383od_v_v @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_444_ord__le__eq__trans,axiom,
! [A: set_v,B2: set_v,C: set_v] :
( ( ord_less_eq_set_v @ A @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_set_v @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_445_ord__le__eq__trans,axiom,
! [A: set_Product_prod_v_v,B2: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B2 )
=> ( ( B2 = C )
=> ( ord_le7336532860387713383od_v_v @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_446_order__antisym,axiom,
! [X: set_v,Y2: set_v] :
( ( ord_less_eq_set_v @ X @ Y2 )
=> ( ( ord_less_eq_set_v @ Y2 @ X )
=> ( X = Y2 ) ) ) ).
% order_antisym
thf(fact_447_order__antisym,axiom,
! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X @ Y2 )
=> ( ( ord_le7336532860387713383od_v_v @ Y2 @ X )
=> ( X = Y2 ) ) ) ).
% order_antisym
thf(fact_448_order_Otrans,axiom,
! [A: set_v,B2: set_v,C: set_v] :
( ( ord_less_eq_set_v @ A @ B2 )
=> ( ( ord_less_eq_set_v @ B2 @ C )
=> ( ord_less_eq_set_v @ A @ C ) ) ) ).
% order.trans
thf(fact_449_order_Otrans,axiom,
! [A: set_Product_prod_v_v,B2: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B2 )
=> ( ( ord_le7336532860387713383od_v_v @ B2 @ C )
=> ( ord_le7336532860387713383od_v_v @ A @ C ) ) ) ).
% order.trans
thf(fact_450_order__trans,axiom,
! [X: set_v,Y2: set_v,Z: set_v] :
( ( ord_less_eq_set_v @ X @ Y2 )
=> ( ( ord_less_eq_set_v @ Y2 @ Z )
=> ( ord_less_eq_set_v @ X @ Z ) ) ) ).
% order_trans
thf(fact_451_order__trans,axiom,
! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X @ Y2 )
=> ( ( ord_le7336532860387713383od_v_v @ Y2 @ Z )
=> ( ord_le7336532860387713383od_v_v @ X @ Z ) ) ) ).
% order_trans
thf(fact_452_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_v,Z4: set_v] : ( Y4 = Z4 ) )
= ( ^ [A6: set_v,B6: set_v] :
( ( ord_less_eq_set_v @ B6 @ A6 )
& ( ord_less_eq_set_v @ A6 @ B6 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_453_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_Product_prod_v_v,Z4: set_Product_prod_v_v] : ( Y4 = Z4 ) )
= ( ^ [A6: set_Product_prod_v_v,B6: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B6 @ A6 )
& ( ord_le7336532860387713383od_v_v @ A6 @ B6 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_454_dual__order_Oantisym,axiom,
! [B2: set_v,A: set_v] :
( ( ord_less_eq_set_v @ B2 @ A )
=> ( ( ord_less_eq_set_v @ A @ B2 )
=> ( A = B2 ) ) ) ).
% dual_order.antisym
thf(fact_455_dual__order_Oantisym,axiom,
! [B2: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B2 @ A )
=> ( ( ord_le7336532860387713383od_v_v @ A @ B2 )
=> ( A = B2 ) ) ) ).
% dual_order.antisym
thf(fact_456_dual__order_Otrans,axiom,
! [B2: set_v,A: set_v,C: set_v] :
( ( ord_less_eq_set_v @ B2 @ A )
=> ( ( ord_less_eq_set_v @ C @ B2 )
=> ( ord_less_eq_set_v @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_457_dual__order_Otrans,axiom,
! [B2: set_Product_prod_v_v,A: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B2 @ A )
=> ( ( ord_le7336532860387713383od_v_v @ C @ B2 )
=> ( ord_le7336532860387713383od_v_v @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_458_antisym,axiom,
! [A: set_v,B2: set_v] :
( ( ord_less_eq_set_v @ A @ B2 )
=> ( ( ord_less_eq_set_v @ B2 @ A )
=> ( A = B2 ) ) ) ).
% antisym
thf(fact_459_antisym,axiom,
! [A: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B2 )
=> ( ( ord_le7336532860387713383od_v_v @ B2 @ A )
=> ( A = B2 ) ) ) ).
% antisym
thf(fact_460_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_v,Z4: set_v] : ( Y4 = Z4 ) )
= ( ^ [A6: set_v,B6: set_v] :
( ( ord_less_eq_set_v @ A6 @ B6 )
& ( ord_less_eq_set_v @ B6 @ A6 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_461_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_Product_prod_v_v,Z4: set_Product_prod_v_v] : ( Y4 = Z4 ) )
= ( ^ [A6: set_Product_prod_v_v,B6: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A6 @ B6 )
& ( ord_le7336532860387713383od_v_v @ B6 @ A6 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_462_order__subst1,axiom,
! [A: set_v,F: set_v > set_v,B2: set_v,C: set_v] :
( ( ord_less_eq_set_v @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_v @ B2 @ C )
=> ( ! [X3: set_v,Y: set_v] :
( ( ord_less_eq_set_v @ X3 @ Y )
=> ( ord_less_eq_set_v @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_v @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_463_order__subst1,axiom,
! [A: set_v,F: set_Product_prod_v_v > set_v,B2: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_less_eq_set_v @ A @ ( F @ B2 ) )
=> ( ( ord_le7336532860387713383od_v_v @ B2 @ C )
=> ( ! [X3: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X3 @ Y )
=> ( ord_less_eq_set_v @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_v @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_464_order__subst1,axiom,
! [A: set_Product_prod_v_v,F: set_v > set_Product_prod_v_v,B2: set_v,C: set_v] :
( ( ord_le7336532860387713383od_v_v @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_v @ B2 @ C )
=> ( ! [X3: set_v,Y: set_v] :
( ( ord_less_eq_set_v @ X3 @ Y )
=> ( ord_le7336532860387713383od_v_v @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_le7336532860387713383od_v_v @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_465_order__subst1,axiom,
! [A: set_Product_prod_v_v,F: set_Product_prod_v_v > set_Product_prod_v_v,B2: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ ( F @ B2 ) )
=> ( ( ord_le7336532860387713383od_v_v @ B2 @ C )
=> ( ! [X3: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X3 @ Y )
=> ( ord_le7336532860387713383od_v_v @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_le7336532860387713383od_v_v @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_466_order__subst2,axiom,
! [A: set_v,B2: set_v,F: set_v > set_v,C: set_v] :
( ( ord_less_eq_set_v @ A @ B2 )
=> ( ( ord_less_eq_set_v @ ( F @ B2 ) @ C )
=> ( ! [X3: set_v,Y: set_v] :
( ( ord_less_eq_set_v @ X3 @ Y )
=> ( ord_less_eq_set_v @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_v @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_467_order__subst2,axiom,
! [A: set_v,B2: set_v,F: set_v > set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_less_eq_set_v @ A @ B2 )
=> ( ( ord_le7336532860387713383od_v_v @ ( F @ B2 ) @ C )
=> ( ! [X3: set_v,Y: set_v] :
( ( ord_less_eq_set_v @ X3 @ Y )
=> ( ord_le7336532860387713383od_v_v @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_le7336532860387713383od_v_v @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_468_order__subst2,axiom,
! [A: set_Product_prod_v_v,B2: set_Product_prod_v_v,F: set_Product_prod_v_v > set_v,C: set_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B2 )
=> ( ( ord_less_eq_set_v @ ( F @ B2 ) @ C )
=> ( ! [X3: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X3 @ Y )
=> ( ord_less_eq_set_v @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_v @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_469_order__subst2,axiom,
! [A: set_Product_prod_v_v,B2: set_Product_prod_v_v,F: set_Product_prod_v_v > set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B2 )
=> ( ( ord_le7336532860387713383od_v_v @ ( F @ B2 ) @ C )
=> ( ! [X3: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X3 @ Y )
=> ( ord_le7336532860387713383od_v_v @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_le7336532860387713383od_v_v @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_470_order__eq__refl,axiom,
! [X: set_v,Y2: set_v] :
( ( X = Y2 )
=> ( ord_less_eq_set_v @ X @ Y2 ) ) ).
% order_eq_refl
thf(fact_471_order__eq__refl,axiom,
! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v] :
( ( X = Y2 )
=> ( ord_le7336532860387713383od_v_v @ X @ Y2 ) ) ).
% order_eq_refl
thf(fact_472_ord__eq__le__subst,axiom,
! [A: set_v,F: set_v > set_v,B2: set_v,C: set_v] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_set_v @ B2 @ C )
=> ( ! [X3: set_v,Y: set_v] :
( ( ord_less_eq_set_v @ X3 @ Y )
=> ( ord_less_eq_set_v @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_v @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_473_ord__eq__le__subst,axiom,
! [A: set_Product_prod_v_v,F: set_v > set_Product_prod_v_v,B2: set_v,C: set_v] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_set_v @ B2 @ C )
=> ( ! [X3: set_v,Y: set_v] :
( ( ord_less_eq_set_v @ X3 @ Y )
=> ( ord_le7336532860387713383od_v_v @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_le7336532860387713383od_v_v @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_474_ord__eq__le__subst,axiom,
! [A: set_v,F: set_Product_prod_v_v > set_v,B2: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_le7336532860387713383od_v_v @ B2 @ C )
=> ( ! [X3: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X3 @ Y )
=> ( ord_less_eq_set_v @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_v @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_475_ord__eq__le__subst,axiom,
! [A: set_Product_prod_v_v,F: set_Product_prod_v_v > set_Product_prod_v_v,B2: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_le7336532860387713383od_v_v @ B2 @ C )
=> ( ! [X3: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X3 @ Y )
=> ( ord_le7336532860387713383od_v_v @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_le7336532860387713383od_v_v @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_476_ord__le__eq__subst,axiom,
! [A: set_v,B2: set_v,F: set_v > set_v,C: set_v] :
( ( ord_less_eq_set_v @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: set_v,Y: set_v] :
( ( ord_less_eq_set_v @ X3 @ Y )
=> ( ord_less_eq_set_v @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_v @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_477_ord__le__eq__subst,axiom,
! [A: set_v,B2: set_v,F: set_v > set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_less_eq_set_v @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: set_v,Y: set_v] :
( ( ord_less_eq_set_v @ X3 @ Y )
=> ( ord_le7336532860387713383od_v_v @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_le7336532860387713383od_v_v @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_478_ord__le__eq__subst,axiom,
! [A: set_Product_prod_v_v,B2: set_Product_prod_v_v,F: set_Product_prod_v_v > set_v,C: set_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X3 @ Y )
=> ( ord_less_eq_set_v @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_v @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_479_ord__le__eq__subst,axiom,
! [A: set_Product_prod_v_v,B2: set_Product_prod_v_v,F: set_Product_prod_v_v > set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X3 @ Y )
=> ( ord_le7336532860387713383od_v_v @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_le7336532860387713383od_v_v @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_480_order__antisym__conv,axiom,
! [Y2: set_v,X: set_v] :
( ( ord_less_eq_set_v @ Y2 @ X )
=> ( ( ord_less_eq_set_v @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_481_order__antisym__conv,axiom,
! [Y2: set_Product_prod_v_v,X: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ Y2 @ X )
=> ( ( ord_le7336532860387713383od_v_v @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_482_old_Oprod_Oexhaust,axiom,
! [Y2: product_prod_v_v] :
~ ! [A7: v,B7: v] :
( Y2
!= ( product_Pair_v_v @ A7 @ B7 ) ) ).
% old.prod.exhaust
thf(fact_483_surj__pair,axiom,
! [P2: product_prod_v_v] :
? [X3: v,Y: v] :
( P2
= ( product_Pair_v_v @ X3 @ Y ) ) ).
% surj_pair
thf(fact_484_prod__cases,axiom,
! [P: product_prod_v_v > $o,P2: product_prod_v_v] :
( ! [A7: v,B7: v] : ( P @ ( product_Pair_v_v @ A7 @ B7 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_485_Pair__inject,axiom,
! [A: v,B2: v,A5: v,B5: v] :
( ( ( product_Pair_v_v @ A @ B2 )
= ( product_Pair_v_v @ A5 @ B5 ) )
=> ~ ( ( A = A5 )
=> ( B2 != B5 ) ) ) ).
% Pair_inject
thf(fact_486_bot_Oextremum,axiom,
! [A: set_v] : ( ord_less_eq_set_v @ bot_bot_set_v @ A ) ).
% bot.extremum
thf(fact_487_bot_Oextremum,axiom,
! [A: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ bot_bo723834152578015283od_v_v @ A ) ).
% bot.extremum
thf(fact_488_bot_Oextremum__unique,axiom,
! [A: set_v] :
( ( ord_less_eq_set_v @ A @ bot_bot_set_v )
= ( A = bot_bot_set_v ) ) ).
% bot.extremum_unique
thf(fact_489_bot_Oextremum__unique,axiom,
! [A: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ bot_bo723834152578015283od_v_v )
= ( A = bot_bo723834152578015283od_v_v ) ) ).
% bot.extremum_unique
thf(fact_490_bot_Oextremum__uniqueI,axiom,
! [A: set_v] :
( ( ord_less_eq_set_v @ A @ bot_bot_set_v )
=> ( A = bot_bot_set_v ) ) ).
% bot.extremum_uniqueI
thf(fact_491_bot_Oextremum__uniqueI,axiom,
! [A: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ bot_bo723834152578015283od_v_v )
=> ( A = bot_bo723834152578015283od_v_v ) ) ).
% bot.extremum_uniqueI
thf(fact_492_avoiding__explored,axiom,
! [E: sCC_Bl1394983891496994913t_unit,X: v,Y2: v,E2: set_Product_prod_v_v,W: v,V: v] :
( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
=> ( ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Y2 @ E2 )
=> ( ~ ( member_v @ Y2 @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ( ( member_v @ W @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Y2 @ ( sup_su414716646722978715od_v_v @ E2 @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ V @ W ) @ bot_bo723834152578015283od_v_v ) ) ) ) ) ) ) ).
% avoiding_explored
thf(fact_493_ra__add__edge,axiom,
! [X: v,Y2: v,E2: set_Product_prod_v_v,V: v,W: v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Y2 @ E2 )
=> ( ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Y2 @ ( sup_su414716646722978715od_v_v @ E2 @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ V @ W ) @ bot_bo723834152578015283od_v_v ) ) )
| ( ( sCC_Bl4291963740693775144ding_v @ successors @ X @ V @ ( sup_su414716646722978715od_v_v @ E2 @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ V @ W ) @ bot_bo723834152578015283od_v_v ) ) )
& ( sCC_Bl4291963740693775144ding_v @ successors @ W @ Y2 @ ( sup_su414716646722978715od_v_v @ E2 @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ V @ W ) @ bot_bo723834152578015283od_v_v ) ) ) ) ) ) ).
% ra_add_edge
thf(fact_494_bot__empty__eq,axiom,
( bot_bo8461541820394803818_v_v_o
= ( ^ [X2: product_prod_v_v] : ( member7453568604450474000od_v_v @ X2 @ bot_bo723834152578015283od_v_v ) ) ) ).
% bot_empty_eq
thf(fact_495_bot__empty__eq,axiom,
( bot_bot_v_o
= ( ^ [X2: v] : ( member_v @ X2 @ bot_bot_set_v ) ) ) ).
% bot_empty_eq
thf(fact_496_Collect__empty__eq__bot,axiom,
! [P: product_prod_v_v > $o] :
( ( ( collec140062887454715474od_v_v @ P )
= bot_bo723834152578015283od_v_v )
= ( P = bot_bo8461541820394803818_v_v_o ) ) ).
% Collect_empty_eq_bot
thf(fact_497_Collect__empty__eq__bot,axiom,
! [P: v > $o] :
( ( ( collect_v @ P )
= bot_bot_set_v )
= ( P = bot_bot_v_o ) ) ).
% Collect_empty_eq_bot
thf(fact_498_is__singleton__the__elem,axiom,
( is_sin9198872032823709915od_v_v
= ( ^ [A4: set_Product_prod_v_v] :
( A4
= ( insert1338601472111419319od_v_v @ ( the_el5392834299063928540od_v_v @ A4 ) @ bot_bo723834152578015283od_v_v ) ) ) ) ).
% is_singleton_the_elem
thf(fact_499_is__singleton__the__elem,axiom,
( is_singleton_v
= ( ^ [A4: set_v] :
( A4
= ( insert_v2 @ ( the_elem_v @ A4 ) @ bot_bot_set_v ) ) ) ) ).
% is_singleton_the_elem
thf(fact_500_is__singletonI,axiom,
! [X: product_prod_v_v] : ( is_sin9198872032823709915od_v_v @ ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) ) ).
% is_singletonI
thf(fact_501_is__singletonI,axiom,
! [X: v] : ( is_singleton_v @ ( insert_v2 @ X @ bot_bot_set_v ) ) ).
% is_singletonI
thf(fact_502_vfin,axiom,
finite_finite_v @ vertices ).
% vfin
thf(fact_503_sup_Oidem,axiom,
! [A: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A @ A )
= A ) ).
% sup.idem
thf(fact_504_sup__idem,axiom,
! [X: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ X @ X )
= X ) ).
% sup_idem
thf(fact_505_sup_Oleft__idem,axiom,
! [A: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A @ ( sup_su414716646722978715od_v_v @ A @ B2 ) )
= ( sup_su414716646722978715od_v_v @ A @ B2 ) ) ).
% sup.left_idem
thf(fact_506_sup__left__idem,axiom,
! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ X @ ( sup_su414716646722978715od_v_v @ X @ Y2 ) )
= ( sup_su414716646722978715od_v_v @ X @ Y2 ) ) ).
% sup_left_idem
thf(fact_507_sup_Oright__idem,axiom,
! [A: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( sup_su414716646722978715od_v_v @ A @ B2 ) @ B2 )
= ( sup_su414716646722978715od_v_v @ A @ B2 ) ) ).
% sup.right_idem
thf(fact_508_UnCI,axiom,
! [C: v,B: set_v,A3: set_v] :
( ( ~ ( member_v @ C @ B )
=> ( member_v @ C @ A3 ) )
=> ( member_v @ C @ ( sup_sup_set_v @ A3 @ B ) ) ) ).
% UnCI
thf(fact_509_UnCI,axiom,
! [C: product_prod_v_v,B: set_Product_prod_v_v,A3: set_Product_prod_v_v] :
( ( ~ ( member7453568604450474000od_v_v @ C @ B )
=> ( member7453568604450474000od_v_v @ C @ A3 ) )
=> ( member7453568604450474000od_v_v @ C @ ( sup_su414716646722978715od_v_v @ A3 @ B ) ) ) ).
% UnCI
thf(fact_510_Un__iff,axiom,
! [C: v,A3: set_v,B: set_v] :
( ( member_v @ C @ ( sup_sup_set_v @ A3 @ B ) )
= ( ( member_v @ C @ A3 )
| ( member_v @ C @ B ) ) ) ).
% Un_iff
thf(fact_511_Un__iff,axiom,
! [C: product_prod_v_v,A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ ( sup_su414716646722978715od_v_v @ A3 @ B ) )
= ( ( member7453568604450474000od_v_v @ C @ A3 )
| ( member7453568604450474000od_v_v @ C @ B ) ) ) ).
% Un_iff
thf(fact_512_sup_Obounded__iff,axiom,
! [B2: set_v,C: set_v,A: set_v] :
( ( ord_less_eq_set_v @ ( sup_sup_set_v @ B2 @ C ) @ A )
= ( ( ord_less_eq_set_v @ B2 @ A )
& ( ord_less_eq_set_v @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_513_sup_Obounded__iff,axiom,
! [B2: set_Product_prod_v_v,C: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ B2 @ C ) @ A )
= ( ( ord_le7336532860387713383od_v_v @ B2 @ A )
& ( ord_le7336532860387713383od_v_v @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_514_le__sup__iff,axiom,
! [X: set_v,Y2: set_v,Z: set_v] :
( ( ord_less_eq_set_v @ ( sup_sup_set_v @ X @ Y2 ) @ Z )
= ( ( ord_less_eq_set_v @ X @ Z )
& ( ord_less_eq_set_v @ Y2 @ Z ) ) ) ).
% le_sup_iff
thf(fact_515_le__sup__iff,axiom,
! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ X @ Y2 ) @ Z )
= ( ( ord_le7336532860387713383od_v_v @ X @ Z )
& ( ord_le7336532860387713383od_v_v @ Y2 @ Z ) ) ) ).
% le_sup_iff
thf(fact_516_sup__bot_Oright__neutral,axiom,
! [A: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A @ bot_bo723834152578015283od_v_v )
= A ) ).
% sup_bot.right_neutral
thf(fact_517_sup__bot_Oright__neutral,axiom,
! [A: set_v] :
( ( sup_sup_set_v @ A @ bot_bot_set_v )
= A ) ).
% sup_bot.right_neutral
thf(fact_518_sup__bot_Oneutr__eq__iff,axiom,
! [A: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( bot_bo723834152578015283od_v_v
= ( sup_su414716646722978715od_v_v @ A @ B2 ) )
= ( ( A = bot_bo723834152578015283od_v_v )
& ( B2 = bot_bo723834152578015283od_v_v ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_519_sup__bot_Oneutr__eq__iff,axiom,
! [A: set_v,B2: set_v] :
( ( bot_bot_set_v
= ( sup_sup_set_v @ A @ B2 ) )
= ( ( A = bot_bot_set_v )
& ( B2 = bot_bot_set_v ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_520_sup__bot_Oleft__neutral,axiom,
! [A: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ bot_bo723834152578015283od_v_v @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_521_sup__bot_Oleft__neutral,axiom,
! [A: set_v] :
( ( sup_sup_set_v @ bot_bot_set_v @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_522_sup__bot_Oeq__neutr__iff,axiom,
! [A: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ( sup_su414716646722978715od_v_v @ A @ B2 )
= bot_bo723834152578015283od_v_v )
= ( ( A = bot_bo723834152578015283od_v_v )
& ( B2 = bot_bo723834152578015283od_v_v ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_523_sup__bot_Oeq__neutr__iff,axiom,
! [A: set_v,B2: set_v] :
( ( ( sup_sup_set_v @ A @ B2 )
= bot_bot_set_v )
= ( ( A = bot_bot_set_v )
& ( B2 = bot_bot_set_v ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_524_sup__eq__bot__iff,axiom,
! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v] :
( ( ( sup_su414716646722978715od_v_v @ X @ Y2 )
= bot_bo723834152578015283od_v_v )
= ( ( X = bot_bo723834152578015283od_v_v )
& ( Y2 = bot_bo723834152578015283od_v_v ) ) ) ).
% sup_eq_bot_iff
thf(fact_525_sup__eq__bot__iff,axiom,
! [X: set_v,Y2: set_v] :
( ( ( sup_sup_set_v @ X @ Y2 )
= bot_bot_set_v )
= ( ( X = bot_bot_set_v )
& ( Y2 = bot_bot_set_v ) ) ) ).
% sup_eq_bot_iff
thf(fact_526_bot__eq__sup__iff,axiom,
! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v] :
( ( bot_bo723834152578015283od_v_v
= ( sup_su414716646722978715od_v_v @ X @ Y2 ) )
= ( ( X = bot_bo723834152578015283od_v_v )
& ( Y2 = bot_bo723834152578015283od_v_v ) ) ) ).
% bot_eq_sup_iff
thf(fact_527_bot__eq__sup__iff,axiom,
! [X: set_v,Y2: set_v] :
( ( bot_bot_set_v
= ( sup_sup_set_v @ X @ Y2 ) )
= ( ( X = bot_bot_set_v )
& ( Y2 = bot_bot_set_v ) ) ) ).
% bot_eq_sup_iff
thf(fact_528_sup__bot__right,axiom,
! [X: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ X @ bot_bo723834152578015283od_v_v )
= X ) ).
% sup_bot_right
thf(fact_529_sup__bot__right,axiom,
! [X: set_v] :
( ( sup_sup_set_v @ X @ bot_bot_set_v )
= X ) ).
% sup_bot_right
thf(fact_530_sup__bot__left,axiom,
! [X: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ bot_bo723834152578015283od_v_v @ X )
= X ) ).
% sup_bot_left
thf(fact_531_sup__bot__left,axiom,
! [X: set_v] :
( ( sup_sup_set_v @ bot_bot_set_v @ X )
= X ) ).
% sup_bot_left
thf(fact_532_sup__inf__absorb,axiom,
! [X: set_v,Y2: set_v] :
( ( sup_sup_set_v @ X @ ( inf_inf_set_v @ X @ Y2 ) )
= X ) ).
% sup_inf_absorb
thf(fact_533_sup__inf__absorb,axiom,
! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ X @ ( inf_in6271465464967711157od_v_v @ X @ Y2 ) )
= X ) ).
% sup_inf_absorb
thf(fact_534_inf__sup__absorb,axiom,
! [X: set_v,Y2: set_v] :
( ( inf_inf_set_v @ X @ ( sup_sup_set_v @ X @ Y2 ) )
= X ) ).
% inf_sup_absorb
thf(fact_535_inf__sup__absorb,axiom,
! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ X @ ( sup_su414716646722978715od_v_v @ X @ Y2 ) )
= X ) ).
% inf_sup_absorb
thf(fact_536_Un__empty,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ( sup_su414716646722978715od_v_v @ A3 @ B )
= bot_bo723834152578015283od_v_v )
= ( ( A3 = bot_bo723834152578015283od_v_v )
& ( B = bot_bo723834152578015283od_v_v ) ) ) ).
% Un_empty
thf(fact_537_Un__empty,axiom,
! [A3: set_v,B: set_v] :
( ( ( sup_sup_set_v @ A3 @ B )
= bot_bot_set_v )
= ( ( A3 = bot_bot_set_v )
& ( B = bot_bot_set_v ) ) ) ).
% Un_empty
thf(fact_538_Un__subset__iff,axiom,
! [A3: set_v,B: set_v,C3: set_v] :
( ( ord_less_eq_set_v @ ( sup_sup_set_v @ A3 @ B ) @ C3 )
= ( ( ord_less_eq_set_v @ A3 @ C3 )
& ( ord_less_eq_set_v @ B @ C3 ) ) ) ).
% Un_subset_iff
thf(fact_539_Un__subset__iff,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v,C3: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ A3 @ B ) @ C3 )
= ( ( ord_le7336532860387713383od_v_v @ A3 @ C3 )
& ( ord_le7336532860387713383od_v_v @ B @ C3 ) ) ) ).
% Un_subset_iff
thf(fact_540_Un__insert__right,axiom,
! [A3: set_v,A: v,B: set_v] :
( ( sup_sup_set_v @ A3 @ ( insert_v2 @ A @ B ) )
= ( insert_v2 @ A @ ( sup_sup_set_v @ A3 @ B ) ) ) ).
% Un_insert_right
thf(fact_541_Un__insert__right,axiom,
! [A3: set_Product_prod_v_v,A: product_prod_v_v,B: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ A @ B ) )
= ( insert1338601472111419319od_v_v @ A @ ( sup_su414716646722978715od_v_v @ A3 @ B ) ) ) ).
% Un_insert_right
thf(fact_542_Un__insert__left,axiom,
! [A: v,B: set_v,C3: set_v] :
( ( sup_sup_set_v @ ( insert_v2 @ A @ B ) @ C3 )
= ( insert_v2 @ A @ ( sup_sup_set_v @ B @ C3 ) ) ) ).
% Un_insert_left
thf(fact_543_Un__insert__left,axiom,
! [A: product_prod_v_v,B: set_Product_prod_v_v,C3: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( insert1338601472111419319od_v_v @ A @ B ) @ C3 )
= ( insert1338601472111419319od_v_v @ A @ ( sup_su414716646722978715od_v_v @ B @ C3 ) ) ) ).
% Un_insert_left
thf(fact_544_Int__Un__eq_I4_J,axiom,
! [T2: set_v,S: set_v] :
( ( sup_sup_set_v @ T2 @ ( inf_inf_set_v @ S @ T2 ) )
= T2 ) ).
% Int_Un_eq(4)
thf(fact_545_Int__Un__eq_I4_J,axiom,
! [T2: set_Product_prod_v_v,S: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ T2 @ ( inf_in6271465464967711157od_v_v @ S @ T2 ) )
= T2 ) ).
% Int_Un_eq(4)
thf(fact_546_Int__Un__eq_I3_J,axiom,
! [S: set_v,T2: set_v] :
( ( sup_sup_set_v @ S @ ( inf_inf_set_v @ S @ T2 ) )
= S ) ).
% Int_Un_eq(3)
thf(fact_547_Int__Un__eq_I3_J,axiom,
! [S: set_Product_prod_v_v,T2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ S @ ( inf_in6271465464967711157od_v_v @ S @ T2 ) )
= S ) ).
% Int_Un_eq(3)
thf(fact_548_Int__Un__eq_I2_J,axiom,
! [S: set_v,T2: set_v] :
( ( sup_sup_set_v @ ( inf_inf_set_v @ S @ T2 ) @ T2 )
= T2 ) ).
% Int_Un_eq(2)
thf(fact_549_Int__Un__eq_I2_J,axiom,
! [S: set_Product_prod_v_v,T2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ S @ T2 ) @ T2 )
= T2 ) ).
% Int_Un_eq(2)
thf(fact_550_Int__Un__eq_I1_J,axiom,
! [S: set_v,T2: set_v] :
( ( sup_sup_set_v @ ( inf_inf_set_v @ S @ T2 ) @ S )
= S ) ).
% Int_Un_eq(1)
thf(fact_551_Int__Un__eq_I1_J,axiom,
! [S: set_Product_prod_v_v,T2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ S @ T2 ) @ S )
= S ) ).
% Int_Un_eq(1)
thf(fact_552_Un__Int__eq_I4_J,axiom,
! [T2: set_v,S: set_v] :
( ( inf_inf_set_v @ T2 @ ( sup_sup_set_v @ S @ T2 ) )
= T2 ) ).
% Un_Int_eq(4)
thf(fact_553_Un__Int__eq_I4_J,axiom,
! [T2: set_Product_prod_v_v,S: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ T2 @ ( sup_su414716646722978715od_v_v @ S @ T2 ) )
= T2 ) ).
% Un_Int_eq(4)
thf(fact_554_Un__Int__eq_I3_J,axiom,
! [S: set_v,T2: set_v] :
( ( inf_inf_set_v @ S @ ( sup_sup_set_v @ S @ T2 ) )
= S ) ).
% Un_Int_eq(3)
thf(fact_555_Un__Int__eq_I3_J,axiom,
! [S: set_Product_prod_v_v,T2: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ S @ ( sup_su414716646722978715od_v_v @ S @ T2 ) )
= S ) ).
% Un_Int_eq(3)
thf(fact_556_Un__Int__eq_I2_J,axiom,
! [S: set_v,T2: set_v] :
( ( inf_inf_set_v @ ( sup_sup_set_v @ S @ T2 ) @ T2 )
= T2 ) ).
% Un_Int_eq(2)
thf(fact_557_Un__Int__eq_I2_J,axiom,
! [S: set_Product_prod_v_v,T2: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ S @ T2 ) @ T2 )
= T2 ) ).
% Un_Int_eq(2)
thf(fact_558_Un__Int__eq_I1_J,axiom,
! [S: set_v,T2: set_v] :
( ( inf_inf_set_v @ ( sup_sup_set_v @ S @ T2 ) @ S )
= S ) ).
% Un_Int_eq(1)
thf(fact_559_Un__Int__eq_I1_J,axiom,
! [S: set_Product_prod_v_v,T2: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ S @ T2 ) @ S )
= S ) ).
% Un_Int_eq(1)
thf(fact_560_Un__Diff__cancel2,axiom,
! [B: set_Product_prod_v_v,A3: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( minus_4183494784930505774od_v_v @ B @ A3 ) @ A3 )
= ( sup_su414716646722978715od_v_v @ B @ A3 ) ) ).
% Un_Diff_cancel2
thf(fact_561_Un__Diff__cancel,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A3 @ ( minus_4183494784930505774od_v_v @ B @ A3 ) )
= ( sup_su414716646722978715od_v_v @ A3 @ B ) ) ).
% Un_Diff_cancel
thf(fact_562_inf__sup__aci_I8_J,axiom,
! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ X @ ( sup_su414716646722978715od_v_v @ X @ Y2 ) )
= ( sup_su414716646722978715od_v_v @ X @ Y2 ) ) ).
% inf_sup_aci(8)
thf(fact_563_inf__sup__aci_I7_J,axiom,
! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ X @ ( sup_su414716646722978715od_v_v @ Y2 @ Z ) )
= ( sup_su414716646722978715od_v_v @ Y2 @ ( sup_su414716646722978715od_v_v @ X @ Z ) ) ) ).
% inf_sup_aci(7)
thf(fact_564_inf__sup__aci_I6_J,axiom,
! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( sup_su414716646722978715od_v_v @ X @ Y2 ) @ Z )
= ( sup_su414716646722978715od_v_v @ X @ ( sup_su414716646722978715od_v_v @ Y2 @ Z ) ) ) ).
% inf_sup_aci(6)
thf(fact_565_inf__sup__aci_I5_J,axiom,
( sup_su414716646722978715od_v_v
= ( ^ [X2: set_Product_prod_v_v,Y3: set_Product_prod_v_v] : ( sup_su414716646722978715od_v_v @ Y3 @ X2 ) ) ) ).
% inf_sup_aci(5)
thf(fact_566_sup_Oassoc,axiom,
! [A: set_Product_prod_v_v,B2: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( sup_su414716646722978715od_v_v @ A @ B2 ) @ C )
= ( sup_su414716646722978715od_v_v @ A @ ( sup_su414716646722978715od_v_v @ B2 @ C ) ) ) ).
% sup.assoc
thf(fact_567_sup__assoc,axiom,
! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( sup_su414716646722978715od_v_v @ X @ Y2 ) @ Z )
= ( sup_su414716646722978715od_v_v @ X @ ( sup_su414716646722978715od_v_v @ Y2 @ Z ) ) ) ).
% sup_assoc
thf(fact_568_sup_Ocommute,axiom,
( sup_su414716646722978715od_v_v
= ( ^ [A6: set_Product_prod_v_v,B6: set_Product_prod_v_v] : ( sup_su414716646722978715od_v_v @ B6 @ A6 ) ) ) ).
% sup.commute
thf(fact_569_sup__commute,axiom,
( sup_su414716646722978715od_v_v
= ( ^ [X2: set_Product_prod_v_v,Y3: set_Product_prod_v_v] : ( sup_su414716646722978715od_v_v @ Y3 @ X2 ) ) ) ).
% sup_commute
thf(fact_570_sup_Oleft__commute,axiom,
! [B2: set_Product_prod_v_v,A: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ B2 @ ( sup_su414716646722978715od_v_v @ A @ C ) )
= ( sup_su414716646722978715od_v_v @ A @ ( sup_su414716646722978715od_v_v @ B2 @ C ) ) ) ).
% sup.left_commute
thf(fact_571_sup__left__commute,axiom,
! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ X @ ( sup_su414716646722978715od_v_v @ Y2 @ Z ) )
= ( sup_su414716646722978715od_v_v @ Y2 @ ( sup_su414716646722978715od_v_v @ X @ Z ) ) ) ).
% sup_left_commute
thf(fact_572_boolean__algebra__cancel_Osup1,axiom,
! [A3: set_Product_prod_v_v,K: set_Product_prod_v_v,A: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( A3
= ( sup_su414716646722978715od_v_v @ K @ A ) )
=> ( ( sup_su414716646722978715od_v_v @ A3 @ B2 )
= ( sup_su414716646722978715od_v_v @ K @ ( sup_su414716646722978715od_v_v @ A @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_573_boolean__algebra__cancel_Osup2,axiom,
! [B: set_Product_prod_v_v,K: set_Product_prod_v_v,B2: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( B
= ( sup_su414716646722978715od_v_v @ K @ B2 ) )
=> ( ( sup_su414716646722978715od_v_v @ A @ B )
= ( sup_su414716646722978715od_v_v @ K @ ( sup_su414716646722978715od_v_v @ A @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_574_UnE,axiom,
! [C: v,A3: set_v,B: set_v] :
( ( member_v @ C @ ( sup_sup_set_v @ A3 @ B ) )
=> ( ~ ( member_v @ C @ A3 )
=> ( member_v @ C @ B ) ) ) ).
% UnE
thf(fact_575_UnE,axiom,
! [C: product_prod_v_v,A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ ( sup_su414716646722978715od_v_v @ A3 @ B ) )
=> ( ~ ( member7453568604450474000od_v_v @ C @ A3 )
=> ( member7453568604450474000od_v_v @ C @ B ) ) ) ).
% UnE
thf(fact_576_UnI1,axiom,
! [C: v,A3: set_v,B: set_v] :
( ( member_v @ C @ A3 )
=> ( member_v @ C @ ( sup_sup_set_v @ A3 @ B ) ) ) ).
% UnI1
thf(fact_577_UnI1,axiom,
! [C: product_prod_v_v,A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ A3 )
=> ( member7453568604450474000od_v_v @ C @ ( sup_su414716646722978715od_v_v @ A3 @ B ) ) ) ).
% UnI1
thf(fact_578_UnI2,axiom,
! [C: v,B: set_v,A3: set_v] :
( ( member_v @ C @ B )
=> ( member_v @ C @ ( sup_sup_set_v @ A3 @ B ) ) ) ).
% UnI2
thf(fact_579_UnI2,axiom,
! [C: product_prod_v_v,B: set_Product_prod_v_v,A3: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ B )
=> ( member7453568604450474000od_v_v @ C @ ( sup_su414716646722978715od_v_v @ A3 @ B ) ) ) ).
% UnI2
thf(fact_580_bex__Un,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v,P: product_prod_v_v > $o] :
( ( ? [X2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ ( sup_su414716646722978715od_v_v @ A3 @ B ) )
& ( P @ X2 ) ) )
= ( ? [X2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ A3 )
& ( P @ X2 ) )
| ? [X2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ B )
& ( P @ X2 ) ) ) ) ).
% bex_Un
thf(fact_581_ball__Un,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v,P: product_prod_v_v > $o] :
( ( ! [X2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ ( sup_su414716646722978715od_v_v @ A3 @ B ) )
=> ( P @ X2 ) ) )
= ( ! [X2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ A3 )
=> ( P @ X2 ) )
& ! [X2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ B )
=> ( P @ X2 ) ) ) ) ).
% ball_Un
thf(fact_582_Un__assoc,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v,C3: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( sup_su414716646722978715od_v_v @ A3 @ B ) @ C3 )
= ( sup_su414716646722978715od_v_v @ A3 @ ( sup_su414716646722978715od_v_v @ B @ C3 ) ) ) ).
% Un_assoc
thf(fact_583_Un__absorb,axiom,
! [A3: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A3 @ A3 )
= A3 ) ).
% Un_absorb
thf(fact_584_Un__commute,axiom,
( sup_su414716646722978715od_v_v
= ( ^ [A4: set_Product_prod_v_v,B4: set_Product_prod_v_v] : ( sup_su414716646722978715od_v_v @ B4 @ A4 ) ) ) ).
% Un_commute
thf(fact_585_Un__left__absorb,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A3 @ ( sup_su414716646722978715od_v_v @ A3 @ B ) )
= ( sup_su414716646722978715od_v_v @ A3 @ B ) ) ).
% Un_left_absorb
thf(fact_586_Un__left__commute,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v,C3: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A3 @ ( sup_su414716646722978715od_v_v @ B @ C3 ) )
= ( sup_su414716646722978715od_v_v @ B @ ( sup_su414716646722978715od_v_v @ A3 @ C3 ) ) ) ).
% Un_left_commute
thf(fact_587_inf__sup__ord_I4_J,axiom,
! [Y2: set_v,X: set_v] : ( ord_less_eq_set_v @ Y2 @ ( sup_sup_set_v @ X @ Y2 ) ) ).
% inf_sup_ord(4)
thf(fact_588_inf__sup__ord_I4_J,axiom,
! [Y2: set_Product_prod_v_v,X: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ Y2 @ ( sup_su414716646722978715od_v_v @ X @ Y2 ) ) ).
% inf_sup_ord(4)
thf(fact_589_inf__sup__ord_I3_J,axiom,
! [X: set_v,Y2: set_v] : ( ord_less_eq_set_v @ X @ ( sup_sup_set_v @ X @ Y2 ) ) ).
% inf_sup_ord(3)
thf(fact_590_inf__sup__ord_I3_J,axiom,
! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ X @ ( sup_su414716646722978715od_v_v @ X @ Y2 ) ) ).
% inf_sup_ord(3)
thf(fact_591_le__supE,axiom,
! [A: set_v,B2: set_v,X: set_v] :
( ( ord_less_eq_set_v @ ( sup_sup_set_v @ A @ B2 ) @ X )
=> ~ ( ( ord_less_eq_set_v @ A @ X )
=> ~ ( ord_less_eq_set_v @ B2 @ X ) ) ) ).
% le_supE
thf(fact_592_le__supE,axiom,
! [A: set_Product_prod_v_v,B2: set_Product_prod_v_v,X: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ A @ B2 ) @ X )
=> ~ ( ( ord_le7336532860387713383od_v_v @ A @ X )
=> ~ ( ord_le7336532860387713383od_v_v @ B2 @ X ) ) ) ).
% le_supE
thf(fact_593_le__supI,axiom,
! [A: set_v,X: set_v,B2: set_v] :
( ( ord_less_eq_set_v @ A @ X )
=> ( ( ord_less_eq_set_v @ B2 @ X )
=> ( ord_less_eq_set_v @ ( sup_sup_set_v @ A @ B2 ) @ X ) ) ) ).
% le_supI
thf(fact_594_le__supI,axiom,
! [A: set_Product_prod_v_v,X: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ X )
=> ( ( ord_le7336532860387713383od_v_v @ B2 @ X )
=> ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ A @ B2 ) @ X ) ) ) ).
% le_supI
thf(fact_595_sup__ge1,axiom,
! [X: set_v,Y2: set_v] : ( ord_less_eq_set_v @ X @ ( sup_sup_set_v @ X @ Y2 ) ) ).
% sup_ge1
thf(fact_596_sup__ge1,axiom,
! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ X @ ( sup_su414716646722978715od_v_v @ X @ Y2 ) ) ).
% sup_ge1
thf(fact_597_sup__ge2,axiom,
! [Y2: set_v,X: set_v] : ( ord_less_eq_set_v @ Y2 @ ( sup_sup_set_v @ X @ Y2 ) ) ).
% sup_ge2
thf(fact_598_sup__ge2,axiom,
! [Y2: set_Product_prod_v_v,X: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ Y2 @ ( sup_su414716646722978715od_v_v @ X @ Y2 ) ) ).
% sup_ge2
thf(fact_599_le__supI1,axiom,
! [X: set_v,A: set_v,B2: set_v] :
( ( ord_less_eq_set_v @ X @ A )
=> ( ord_less_eq_set_v @ X @ ( sup_sup_set_v @ A @ B2 ) ) ) ).
% le_supI1
thf(fact_600_le__supI1,axiom,
! [X: set_Product_prod_v_v,A: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X @ A )
=> ( ord_le7336532860387713383od_v_v @ X @ ( sup_su414716646722978715od_v_v @ A @ B2 ) ) ) ).
% le_supI1
thf(fact_601_le__supI2,axiom,
! [X: set_v,B2: set_v,A: set_v] :
( ( ord_less_eq_set_v @ X @ B2 )
=> ( ord_less_eq_set_v @ X @ ( sup_sup_set_v @ A @ B2 ) ) ) ).
% le_supI2
thf(fact_602_le__supI2,axiom,
! [X: set_Product_prod_v_v,B2: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X @ B2 )
=> ( ord_le7336532860387713383od_v_v @ X @ ( sup_su414716646722978715od_v_v @ A @ B2 ) ) ) ).
% le_supI2
thf(fact_603_sup_Omono,axiom,
! [C: set_v,A: set_v,D: set_v,B2: set_v] :
( ( ord_less_eq_set_v @ C @ A )
=> ( ( ord_less_eq_set_v @ D @ B2 )
=> ( ord_less_eq_set_v @ ( sup_sup_set_v @ C @ D ) @ ( sup_sup_set_v @ A @ B2 ) ) ) ) ).
% sup.mono
thf(fact_604_sup_Omono,axiom,
! [C: set_Product_prod_v_v,A: set_Product_prod_v_v,D: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ C @ A )
=> ( ( ord_le7336532860387713383od_v_v @ D @ B2 )
=> ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ C @ D ) @ ( sup_su414716646722978715od_v_v @ A @ B2 ) ) ) ) ).
% sup.mono
thf(fact_605_sup__mono,axiom,
! [A: set_v,C: set_v,B2: set_v,D: set_v] :
( ( ord_less_eq_set_v @ A @ C )
=> ( ( ord_less_eq_set_v @ B2 @ D )
=> ( ord_less_eq_set_v @ ( sup_sup_set_v @ A @ B2 ) @ ( sup_sup_set_v @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_606_sup__mono,axiom,
! [A: set_Product_prod_v_v,C: set_Product_prod_v_v,B2: set_Product_prod_v_v,D: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ C )
=> ( ( ord_le7336532860387713383od_v_v @ B2 @ D )
=> ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ A @ B2 ) @ ( sup_su414716646722978715od_v_v @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_607_sup__least,axiom,
! [Y2: set_v,X: set_v,Z: set_v] :
( ( ord_less_eq_set_v @ Y2 @ X )
=> ( ( ord_less_eq_set_v @ Z @ X )
=> ( ord_less_eq_set_v @ ( sup_sup_set_v @ Y2 @ Z ) @ X ) ) ) ).
% sup_least
thf(fact_608_sup__least,axiom,
! [Y2: set_Product_prod_v_v,X: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ Y2 @ X )
=> ( ( ord_le7336532860387713383od_v_v @ Z @ X )
=> ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ Y2 @ Z ) @ X ) ) ) ).
% sup_least
thf(fact_609_le__iff__sup,axiom,
( ord_less_eq_set_v
= ( ^ [X2: set_v,Y3: set_v] :
( ( sup_sup_set_v @ X2 @ Y3 )
= Y3 ) ) ) ).
% le_iff_sup
thf(fact_610_le__iff__sup,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [X2: set_Product_prod_v_v,Y3: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ X2 @ Y3 )
= Y3 ) ) ) ).
% le_iff_sup
thf(fact_611_sup_OorderE,axiom,
! [B2: set_v,A: set_v] :
( ( ord_less_eq_set_v @ B2 @ A )
=> ( A
= ( sup_sup_set_v @ A @ B2 ) ) ) ).
% sup.orderE
thf(fact_612_sup_OorderE,axiom,
! [B2: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B2 @ A )
=> ( A
= ( sup_su414716646722978715od_v_v @ A @ B2 ) ) ) ).
% sup.orderE
thf(fact_613_sup_OorderI,axiom,
! [A: set_v,B2: set_v] :
( ( A
= ( sup_sup_set_v @ A @ B2 ) )
=> ( ord_less_eq_set_v @ B2 @ A ) ) ).
% sup.orderI
thf(fact_614_sup_OorderI,axiom,
! [A: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( A
= ( sup_su414716646722978715od_v_v @ A @ B2 ) )
=> ( ord_le7336532860387713383od_v_v @ B2 @ A ) ) ).
% sup.orderI
thf(fact_615_sup__unique,axiom,
! [F: set_v > set_v > set_v,X: set_v,Y2: set_v] :
( ! [X3: set_v,Y: set_v] : ( ord_less_eq_set_v @ X3 @ ( F @ X3 @ Y ) )
=> ( ! [X3: set_v,Y: set_v] : ( ord_less_eq_set_v @ Y @ ( F @ X3 @ Y ) )
=> ( ! [X3: set_v,Y: set_v,Z3: set_v] :
( ( ord_less_eq_set_v @ Y @ X3 )
=> ( ( ord_less_eq_set_v @ Z3 @ X3 )
=> ( ord_less_eq_set_v @ ( F @ Y @ Z3 ) @ X3 ) ) )
=> ( ( sup_sup_set_v @ X @ Y2 )
= ( F @ X @ Y2 ) ) ) ) ) ).
% sup_unique
thf(fact_616_sup__unique,axiom,
! [F: set_Product_prod_v_v > set_Product_prod_v_v > set_Product_prod_v_v,X: set_Product_prod_v_v,Y2: set_Product_prod_v_v] :
( ! [X3: set_Product_prod_v_v,Y: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ X3 @ ( F @ X3 @ Y ) )
=> ( ! [X3: set_Product_prod_v_v,Y: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ Y @ ( F @ X3 @ Y ) )
=> ( ! [X3: set_Product_prod_v_v,Y: set_Product_prod_v_v,Z3: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ Y @ X3 )
=> ( ( ord_le7336532860387713383od_v_v @ Z3 @ X3 )
=> ( ord_le7336532860387713383od_v_v @ ( F @ Y @ Z3 ) @ X3 ) ) )
=> ( ( sup_su414716646722978715od_v_v @ X @ Y2 )
= ( F @ X @ Y2 ) ) ) ) ) ).
% sup_unique
thf(fact_617_sup_Oabsorb1,axiom,
! [B2: set_v,A: set_v] :
( ( ord_less_eq_set_v @ B2 @ A )
=> ( ( sup_sup_set_v @ A @ B2 )
= A ) ) ).
% sup.absorb1
thf(fact_618_sup_Oabsorb1,axiom,
! [B2: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B2 @ A )
=> ( ( sup_su414716646722978715od_v_v @ A @ B2 )
= A ) ) ).
% sup.absorb1
thf(fact_619_sup_Oabsorb2,axiom,
! [A: set_v,B2: set_v] :
( ( ord_less_eq_set_v @ A @ B2 )
=> ( ( sup_sup_set_v @ A @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_620_sup_Oabsorb2,axiom,
! [A: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B2 )
=> ( ( sup_su414716646722978715od_v_v @ A @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_621_sup__absorb1,axiom,
! [Y2: set_v,X: set_v] :
( ( ord_less_eq_set_v @ Y2 @ X )
=> ( ( sup_sup_set_v @ X @ Y2 )
= X ) ) ).
% sup_absorb1
thf(fact_622_sup__absorb1,axiom,
! [Y2: set_Product_prod_v_v,X: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ Y2 @ X )
=> ( ( sup_su414716646722978715od_v_v @ X @ Y2 )
= X ) ) ).
% sup_absorb1
thf(fact_623_sup__absorb2,axiom,
! [X: set_v,Y2: set_v] :
( ( ord_less_eq_set_v @ X @ Y2 )
=> ( ( sup_sup_set_v @ X @ Y2 )
= Y2 ) ) ).
% sup_absorb2
thf(fact_624_sup__absorb2,axiom,
! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X @ Y2 )
=> ( ( sup_su414716646722978715od_v_v @ X @ Y2 )
= Y2 ) ) ).
% sup_absorb2
thf(fact_625_sup_OboundedE,axiom,
! [B2: set_v,C: set_v,A: set_v] :
( ( ord_less_eq_set_v @ ( sup_sup_set_v @ B2 @ C ) @ A )
=> ~ ( ( ord_less_eq_set_v @ B2 @ A )
=> ~ ( ord_less_eq_set_v @ C @ A ) ) ) ).
% sup.boundedE
thf(fact_626_sup_OboundedE,axiom,
! [B2: set_Product_prod_v_v,C: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ B2 @ C ) @ A )
=> ~ ( ( ord_le7336532860387713383od_v_v @ B2 @ A )
=> ~ ( ord_le7336532860387713383od_v_v @ C @ A ) ) ) ).
% sup.boundedE
thf(fact_627_sup_OboundedI,axiom,
! [B2: set_v,A: set_v,C: set_v] :
( ( ord_less_eq_set_v @ B2 @ A )
=> ( ( ord_less_eq_set_v @ C @ A )
=> ( ord_less_eq_set_v @ ( sup_sup_set_v @ B2 @ C ) @ A ) ) ) ).
% sup.boundedI
thf(fact_628_sup_OboundedI,axiom,
! [B2: set_Product_prod_v_v,A: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B2 @ A )
=> ( ( ord_le7336532860387713383od_v_v @ C @ A )
=> ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ B2 @ C ) @ A ) ) ) ).
% sup.boundedI
thf(fact_629_sup_Oorder__iff,axiom,
( ord_less_eq_set_v
= ( ^ [B6: set_v,A6: set_v] :
( A6
= ( sup_sup_set_v @ A6 @ B6 ) ) ) ) ).
% sup.order_iff
thf(fact_630_sup_Oorder__iff,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [B6: set_Product_prod_v_v,A6: set_Product_prod_v_v] :
( A6
= ( sup_su414716646722978715od_v_v @ A6 @ B6 ) ) ) ) ).
% sup.order_iff
thf(fact_631_sup_Ocobounded1,axiom,
! [A: set_v,B2: set_v] : ( ord_less_eq_set_v @ A @ ( sup_sup_set_v @ A @ B2 ) ) ).
% sup.cobounded1
thf(fact_632_sup_Ocobounded1,axiom,
! [A: set_Product_prod_v_v,B2: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ A @ ( sup_su414716646722978715od_v_v @ A @ B2 ) ) ).
% sup.cobounded1
thf(fact_633_sup_Ocobounded2,axiom,
! [B2: set_v,A: set_v] : ( ord_less_eq_set_v @ B2 @ ( sup_sup_set_v @ A @ B2 ) ) ).
% sup.cobounded2
thf(fact_634_sup_Ocobounded2,axiom,
! [B2: set_Product_prod_v_v,A: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ B2 @ ( sup_su414716646722978715od_v_v @ A @ B2 ) ) ).
% sup.cobounded2
thf(fact_635_sup_Oabsorb__iff1,axiom,
( ord_less_eq_set_v
= ( ^ [B6: set_v,A6: set_v] :
( ( sup_sup_set_v @ A6 @ B6 )
= A6 ) ) ) ).
% sup.absorb_iff1
thf(fact_636_sup_Oabsorb__iff1,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [B6: set_Product_prod_v_v,A6: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A6 @ B6 )
= A6 ) ) ) ).
% sup.absorb_iff1
thf(fact_637_sup_Oabsorb__iff2,axiom,
( ord_less_eq_set_v
= ( ^ [A6: set_v,B6: set_v] :
( ( sup_sup_set_v @ A6 @ B6 )
= B6 ) ) ) ).
% sup.absorb_iff2
thf(fact_638_sup_Oabsorb__iff2,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [A6: set_Product_prod_v_v,B6: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A6 @ B6 )
= B6 ) ) ) ).
% sup.absorb_iff2
thf(fact_639_sup_OcoboundedI1,axiom,
! [C: set_v,A: set_v,B2: set_v] :
( ( ord_less_eq_set_v @ C @ A )
=> ( ord_less_eq_set_v @ C @ ( sup_sup_set_v @ A @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_640_sup_OcoboundedI1,axiom,
! [C: set_Product_prod_v_v,A: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ C @ A )
=> ( ord_le7336532860387713383od_v_v @ C @ ( sup_su414716646722978715od_v_v @ A @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_641_sup_OcoboundedI2,axiom,
! [C: set_v,B2: set_v,A: set_v] :
( ( ord_less_eq_set_v @ C @ B2 )
=> ( ord_less_eq_set_v @ C @ ( sup_sup_set_v @ A @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_642_sup_OcoboundedI2,axiom,
! [C: set_Product_prod_v_v,B2: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ C @ B2 )
=> ( ord_le7336532860387713383od_v_v @ C @ ( sup_su414716646722978715od_v_v @ A @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_643_boolean__algebra_Odisj__zero__right,axiom,
! [X: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ X @ bot_bo723834152578015283od_v_v )
= X ) ).
% boolean_algebra.disj_zero_right
thf(fact_644_boolean__algebra_Odisj__zero__right,axiom,
! [X: set_v] :
( ( sup_sup_set_v @ X @ bot_bot_set_v )
= X ) ).
% boolean_algebra.disj_zero_right
thf(fact_645_sup__inf__distrib2,axiom,
! [Y2: set_v,Z: set_v,X: set_v] :
( ( sup_sup_set_v @ ( inf_inf_set_v @ Y2 @ Z ) @ X )
= ( inf_inf_set_v @ ( sup_sup_set_v @ Y2 @ X ) @ ( sup_sup_set_v @ Z @ X ) ) ) ).
% sup_inf_distrib2
thf(fact_646_sup__inf__distrib2,axiom,
! [Y2: set_Product_prod_v_v,Z: set_Product_prod_v_v,X: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ Y2 @ Z ) @ X )
= ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ Y2 @ X ) @ ( sup_su414716646722978715od_v_v @ Z @ X ) ) ) ).
% sup_inf_distrib2
thf(fact_647_sup__inf__distrib1,axiom,
! [X: set_v,Y2: set_v,Z: set_v] :
( ( sup_sup_set_v @ X @ ( inf_inf_set_v @ Y2 @ Z ) )
= ( inf_inf_set_v @ ( sup_sup_set_v @ X @ Y2 ) @ ( sup_sup_set_v @ X @ Z ) ) ) ).
% sup_inf_distrib1
thf(fact_648_sup__inf__distrib1,axiom,
! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ X @ ( inf_in6271465464967711157od_v_v @ Y2 @ Z ) )
= ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ X @ Y2 ) @ ( sup_su414716646722978715od_v_v @ X @ Z ) ) ) ).
% sup_inf_distrib1
thf(fact_649_inf__sup__distrib2,axiom,
! [Y2: set_v,Z: set_v,X: set_v] :
( ( inf_inf_set_v @ ( sup_sup_set_v @ Y2 @ Z ) @ X )
= ( sup_sup_set_v @ ( inf_inf_set_v @ Y2 @ X ) @ ( inf_inf_set_v @ Z @ X ) ) ) ).
% inf_sup_distrib2
thf(fact_650_inf__sup__distrib2,axiom,
! [Y2: set_Product_prod_v_v,Z: set_Product_prod_v_v,X: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ Y2 @ Z ) @ X )
= ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ Y2 @ X ) @ ( inf_in6271465464967711157od_v_v @ Z @ X ) ) ) ).
% inf_sup_distrib2
thf(fact_651_inf__sup__distrib1,axiom,
! [X: set_v,Y2: set_v,Z: set_v] :
( ( inf_inf_set_v @ X @ ( sup_sup_set_v @ Y2 @ Z ) )
= ( sup_sup_set_v @ ( inf_inf_set_v @ X @ Y2 ) @ ( inf_inf_set_v @ X @ Z ) ) ) ).
% inf_sup_distrib1
thf(fact_652_inf__sup__distrib1,axiom,
! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ X @ ( sup_su414716646722978715od_v_v @ Y2 @ Z ) )
= ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ X @ Y2 ) @ ( inf_in6271465464967711157od_v_v @ X @ Z ) ) ) ).
% inf_sup_distrib1
thf(fact_653_distrib__imp2,axiom,
! [X: set_v,Y2: set_v,Z: set_v] :
( ! [X3: set_v,Y: set_v,Z3: set_v] :
( ( sup_sup_set_v @ X3 @ ( inf_inf_set_v @ Y @ Z3 ) )
= ( inf_inf_set_v @ ( sup_sup_set_v @ X3 @ Y ) @ ( sup_sup_set_v @ X3 @ Z3 ) ) )
=> ( ( inf_inf_set_v @ X @ ( sup_sup_set_v @ Y2 @ Z ) )
= ( sup_sup_set_v @ ( inf_inf_set_v @ X @ Y2 ) @ ( inf_inf_set_v @ X @ Z ) ) ) ) ).
% distrib_imp2
thf(fact_654_distrib__imp2,axiom,
! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
( ! [X3: set_Product_prod_v_v,Y: set_Product_prod_v_v,Z3: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ X3 @ ( inf_in6271465464967711157od_v_v @ Y @ Z3 ) )
= ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ X3 @ Y ) @ ( sup_su414716646722978715od_v_v @ X3 @ Z3 ) ) )
=> ( ( inf_in6271465464967711157od_v_v @ X @ ( sup_su414716646722978715od_v_v @ Y2 @ Z ) )
= ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ X @ Y2 ) @ ( inf_in6271465464967711157od_v_v @ X @ Z ) ) ) ) ).
% distrib_imp2
thf(fact_655_distrib__imp1,axiom,
! [X: set_v,Y2: set_v,Z: set_v] :
( ! [X3: set_v,Y: set_v,Z3: set_v] :
( ( inf_inf_set_v @ X3 @ ( sup_sup_set_v @ Y @ Z3 ) )
= ( sup_sup_set_v @ ( inf_inf_set_v @ X3 @ Y ) @ ( inf_inf_set_v @ X3 @ Z3 ) ) )
=> ( ( sup_sup_set_v @ X @ ( inf_inf_set_v @ Y2 @ Z ) )
= ( inf_inf_set_v @ ( sup_sup_set_v @ X @ Y2 ) @ ( sup_sup_set_v @ X @ Z ) ) ) ) ).
% distrib_imp1
thf(fact_656_distrib__imp1,axiom,
! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
( ! [X3: set_Product_prod_v_v,Y: set_Product_prod_v_v,Z3: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ X3 @ ( sup_su414716646722978715od_v_v @ Y @ Z3 ) )
= ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ X3 @ Y ) @ ( inf_in6271465464967711157od_v_v @ X3 @ Z3 ) ) )
=> ( ( sup_su414716646722978715od_v_v @ X @ ( inf_in6271465464967711157od_v_v @ Y2 @ Z ) )
= ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ X @ Y2 ) @ ( sup_su414716646722978715od_v_v @ X @ Z ) ) ) ) ).
% distrib_imp1
thf(fact_657_boolean__algebra_Oconj__disj__distrib,axiom,
! [X: set_v,Y2: set_v,Z: set_v] :
( ( inf_inf_set_v @ X @ ( sup_sup_set_v @ Y2 @ Z ) )
= ( sup_sup_set_v @ ( inf_inf_set_v @ X @ Y2 ) @ ( inf_inf_set_v @ X @ Z ) ) ) ).
% boolean_algebra.conj_disj_distrib
thf(fact_658_boolean__algebra_Oconj__disj__distrib,axiom,
! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ X @ ( sup_su414716646722978715od_v_v @ Y2 @ Z ) )
= ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ X @ Y2 ) @ ( inf_in6271465464967711157od_v_v @ X @ Z ) ) ) ).
% boolean_algebra.conj_disj_distrib
thf(fact_659_boolean__algebra_Odisj__conj__distrib,axiom,
! [X: set_v,Y2: set_v,Z: set_v] :
( ( sup_sup_set_v @ X @ ( inf_inf_set_v @ Y2 @ Z ) )
= ( inf_inf_set_v @ ( sup_sup_set_v @ X @ Y2 ) @ ( sup_sup_set_v @ X @ Z ) ) ) ).
% boolean_algebra.disj_conj_distrib
thf(fact_660_boolean__algebra_Odisj__conj__distrib,axiom,
! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ X @ ( inf_in6271465464967711157od_v_v @ Y2 @ Z ) )
= ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ X @ Y2 ) @ ( sup_su414716646722978715od_v_v @ X @ Z ) ) ) ).
% boolean_algebra.disj_conj_distrib
thf(fact_661_boolean__algebra_Oconj__disj__distrib2,axiom,
! [Y2: set_v,Z: set_v,X: set_v] :
( ( inf_inf_set_v @ ( sup_sup_set_v @ Y2 @ Z ) @ X )
= ( sup_sup_set_v @ ( inf_inf_set_v @ Y2 @ X ) @ ( inf_inf_set_v @ Z @ X ) ) ) ).
% boolean_algebra.conj_disj_distrib2
thf(fact_662_boolean__algebra_Oconj__disj__distrib2,axiom,
! [Y2: set_Product_prod_v_v,Z: set_Product_prod_v_v,X: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ Y2 @ Z ) @ X )
= ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ Y2 @ X ) @ ( inf_in6271465464967711157od_v_v @ Z @ X ) ) ) ).
% boolean_algebra.conj_disj_distrib2
thf(fact_663_boolean__algebra_Odisj__conj__distrib2,axiom,
! [Y2: set_v,Z: set_v,X: set_v] :
( ( sup_sup_set_v @ ( inf_inf_set_v @ Y2 @ Z ) @ X )
= ( inf_inf_set_v @ ( sup_sup_set_v @ Y2 @ X ) @ ( sup_sup_set_v @ Z @ X ) ) ) ).
% boolean_algebra.disj_conj_distrib2
thf(fact_664_boolean__algebra_Odisj__conj__distrib2,axiom,
! [Y2: set_Product_prod_v_v,Z: set_Product_prod_v_v,X: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ Y2 @ Z ) @ X )
= ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ Y2 @ X ) @ ( sup_su414716646722978715od_v_v @ Z @ X ) ) ) ).
% boolean_algebra.disj_conj_distrib2
thf(fact_665_Un__empty__right,axiom,
! [A3: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A3 @ bot_bo723834152578015283od_v_v )
= A3 ) ).
% Un_empty_right
thf(fact_666_Un__empty__right,axiom,
! [A3: set_v] :
( ( sup_sup_set_v @ A3 @ bot_bot_set_v )
= A3 ) ).
% Un_empty_right
thf(fact_667_Un__empty__left,axiom,
! [B: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ bot_bo723834152578015283od_v_v @ B )
= B ) ).
% Un_empty_left
thf(fact_668_Un__empty__left,axiom,
! [B: set_v] :
( ( sup_sup_set_v @ bot_bot_set_v @ B )
= B ) ).
% Un_empty_left
thf(fact_669_subset__Un__eq,axiom,
( ord_less_eq_set_v
= ( ^ [A4: set_v,B4: set_v] :
( ( sup_sup_set_v @ A4 @ B4 )
= B4 ) ) ) ).
% subset_Un_eq
thf(fact_670_subset__Un__eq,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [A4: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A4 @ B4 )
= B4 ) ) ) ).
% subset_Un_eq
thf(fact_671_subset__UnE,axiom,
! [C3: set_v,A3: set_v,B: set_v] :
( ( ord_less_eq_set_v @ C3 @ ( sup_sup_set_v @ A3 @ B ) )
=> ~ ! [A8: set_v] :
( ( ord_less_eq_set_v @ A8 @ A3 )
=> ! [B8: set_v] :
( ( ord_less_eq_set_v @ B8 @ B )
=> ( C3
!= ( sup_sup_set_v @ A8 @ B8 ) ) ) ) ) ).
% subset_UnE
thf(fact_672_subset__UnE,axiom,
! [C3: set_Product_prod_v_v,A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ C3 @ ( sup_su414716646722978715od_v_v @ A3 @ B ) )
=> ~ ! [A8: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A8 @ A3 )
=> ! [B8: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B8 @ B )
=> ( C3
!= ( sup_su414716646722978715od_v_v @ A8 @ B8 ) ) ) ) ) ).
% subset_UnE
thf(fact_673_Un__absorb2,axiom,
! [B: set_v,A3: set_v] :
( ( ord_less_eq_set_v @ B @ A3 )
=> ( ( sup_sup_set_v @ A3 @ B )
= A3 ) ) ).
% Un_absorb2
thf(fact_674_Un__absorb2,axiom,
! [B: set_Product_prod_v_v,A3: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B @ A3 )
=> ( ( sup_su414716646722978715od_v_v @ A3 @ B )
= A3 ) ) ).
% Un_absorb2
thf(fact_675_Un__absorb1,axiom,
! [A3: set_v,B: set_v] :
( ( ord_less_eq_set_v @ A3 @ B )
=> ( ( sup_sup_set_v @ A3 @ B )
= B ) ) ).
% Un_absorb1
thf(fact_676_Un__absorb1,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A3 @ B )
=> ( ( sup_su414716646722978715od_v_v @ A3 @ B )
= B ) ) ).
% Un_absorb1
thf(fact_677_Un__upper2,axiom,
! [B: set_v,A3: set_v] : ( ord_less_eq_set_v @ B @ ( sup_sup_set_v @ A3 @ B ) ) ).
% Un_upper2
thf(fact_678_Un__upper2,axiom,
! [B: set_Product_prod_v_v,A3: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ B @ ( sup_su414716646722978715od_v_v @ A3 @ B ) ) ).
% Un_upper2
thf(fact_679_Un__upper1,axiom,
! [A3: set_v,B: set_v] : ( ord_less_eq_set_v @ A3 @ ( sup_sup_set_v @ A3 @ B ) ) ).
% Un_upper1
thf(fact_680_Un__upper1,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ A3 @ ( sup_su414716646722978715od_v_v @ A3 @ B ) ) ).
% Un_upper1
thf(fact_681_Un__least,axiom,
! [A3: set_v,C3: set_v,B: set_v] :
( ( ord_less_eq_set_v @ A3 @ C3 )
=> ( ( ord_less_eq_set_v @ B @ C3 )
=> ( ord_less_eq_set_v @ ( sup_sup_set_v @ A3 @ B ) @ C3 ) ) ) ).
% Un_least
thf(fact_682_Un__least,axiom,
! [A3: set_Product_prod_v_v,C3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A3 @ C3 )
=> ( ( ord_le7336532860387713383od_v_v @ B @ C3 )
=> ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ A3 @ B ) @ C3 ) ) ) ).
% Un_least
thf(fact_683_Un__mono,axiom,
! [A3: set_v,C3: set_v,B: set_v,D2: set_v] :
( ( ord_less_eq_set_v @ A3 @ C3 )
=> ( ( ord_less_eq_set_v @ B @ D2 )
=> ( ord_less_eq_set_v @ ( sup_sup_set_v @ A3 @ B ) @ ( sup_sup_set_v @ C3 @ D2 ) ) ) ) ).
% Un_mono
thf(fact_684_Un__mono,axiom,
! [A3: set_Product_prod_v_v,C3: set_Product_prod_v_v,B: set_Product_prod_v_v,D2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A3 @ C3 )
=> ( ( ord_le7336532860387713383od_v_v @ B @ D2 )
=> ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ A3 @ B ) @ ( sup_su414716646722978715od_v_v @ C3 @ D2 ) ) ) ) ).
% Un_mono
thf(fact_685_Un__Int__crazy,axiom,
! [A3: set_v,B: set_v,C3: set_v] :
( ( sup_sup_set_v @ ( sup_sup_set_v @ ( inf_inf_set_v @ A3 @ B ) @ ( inf_inf_set_v @ B @ C3 ) ) @ ( inf_inf_set_v @ C3 @ A3 ) )
= ( inf_inf_set_v @ ( inf_inf_set_v @ ( sup_sup_set_v @ A3 @ B ) @ ( sup_sup_set_v @ B @ C3 ) ) @ ( sup_sup_set_v @ C3 @ A3 ) ) ) ).
% Un_Int_crazy
thf(fact_686_Un__Int__crazy,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v,C3: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ A3 @ B ) @ ( inf_in6271465464967711157od_v_v @ B @ C3 ) ) @ ( inf_in6271465464967711157od_v_v @ C3 @ A3 ) )
= ( inf_in6271465464967711157od_v_v @ ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ A3 @ B ) @ ( sup_su414716646722978715od_v_v @ B @ C3 ) ) @ ( sup_su414716646722978715od_v_v @ C3 @ A3 ) ) ) ).
% Un_Int_crazy
thf(fact_687_Int__Un__distrib,axiom,
! [A3: set_v,B: set_v,C3: set_v] :
( ( inf_inf_set_v @ A3 @ ( sup_sup_set_v @ B @ C3 ) )
= ( sup_sup_set_v @ ( inf_inf_set_v @ A3 @ B ) @ ( inf_inf_set_v @ A3 @ C3 ) ) ) ).
% Int_Un_distrib
thf(fact_688_Int__Un__distrib,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v,C3: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ A3 @ ( sup_su414716646722978715od_v_v @ B @ C3 ) )
= ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ A3 @ B ) @ ( inf_in6271465464967711157od_v_v @ A3 @ C3 ) ) ) ).
% Int_Un_distrib
thf(fact_689_Un__Int__distrib,axiom,
! [A3: set_v,B: set_v,C3: set_v] :
( ( sup_sup_set_v @ A3 @ ( inf_inf_set_v @ B @ C3 ) )
= ( inf_inf_set_v @ ( sup_sup_set_v @ A3 @ B ) @ ( sup_sup_set_v @ A3 @ C3 ) ) ) ).
% Un_Int_distrib
thf(fact_690_Un__Int__distrib,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v,C3: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A3 @ ( inf_in6271465464967711157od_v_v @ B @ C3 ) )
= ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ A3 @ B ) @ ( sup_su414716646722978715od_v_v @ A3 @ C3 ) ) ) ).
% Un_Int_distrib
thf(fact_691_Int__Un__distrib2,axiom,
! [B: set_v,C3: set_v,A3: set_v] :
( ( inf_inf_set_v @ ( sup_sup_set_v @ B @ C3 ) @ A3 )
= ( sup_sup_set_v @ ( inf_inf_set_v @ B @ A3 ) @ ( inf_inf_set_v @ C3 @ A3 ) ) ) ).
% Int_Un_distrib2
thf(fact_692_Int__Un__distrib2,axiom,
! [B: set_Product_prod_v_v,C3: set_Product_prod_v_v,A3: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ B @ C3 ) @ A3 )
= ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ B @ A3 ) @ ( inf_in6271465464967711157od_v_v @ C3 @ A3 ) ) ) ).
% Int_Un_distrib2
thf(fact_693_Un__Int__distrib2,axiom,
! [B: set_v,C3: set_v,A3: set_v] :
( ( sup_sup_set_v @ ( inf_inf_set_v @ B @ C3 ) @ A3 )
= ( inf_inf_set_v @ ( sup_sup_set_v @ B @ A3 ) @ ( sup_sup_set_v @ C3 @ A3 ) ) ) ).
% Un_Int_distrib2
thf(fact_694_Un__Int__distrib2,axiom,
! [B: set_Product_prod_v_v,C3: set_Product_prod_v_v,A3: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ B @ C3 ) @ A3 )
= ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ B @ A3 ) @ ( sup_su414716646722978715od_v_v @ C3 @ A3 ) ) ) ).
% Un_Int_distrib2
thf(fact_695_Un__Diff,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v,C3: set_Product_prod_v_v] :
( ( minus_4183494784930505774od_v_v @ ( sup_su414716646722978715od_v_v @ A3 @ B ) @ C3 )
= ( sup_su414716646722978715od_v_v @ ( minus_4183494784930505774od_v_v @ A3 @ C3 ) @ ( minus_4183494784930505774od_v_v @ B @ C3 ) ) ) ).
% Un_Diff
thf(fact_696_graph_Ovfin,axiom,
! [Vertices: set_v,Successors: v > set_v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( finite_finite_v @ Vertices ) ) ).
% graph.vfin
thf(fact_697_distrib__sup__le,axiom,
! [X: set_v,Y2: set_v,Z: set_v] : ( ord_less_eq_set_v @ ( sup_sup_set_v @ X @ ( inf_inf_set_v @ Y2 @ Z ) ) @ ( inf_inf_set_v @ ( sup_sup_set_v @ X @ Y2 ) @ ( sup_sup_set_v @ X @ Z ) ) ) ).
% distrib_sup_le
thf(fact_698_distrib__sup__le,axiom,
! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v,Z: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ X @ ( inf_in6271465464967711157od_v_v @ Y2 @ Z ) ) @ ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ X @ Y2 ) @ ( sup_su414716646722978715od_v_v @ X @ Z ) ) ) ).
% distrib_sup_le
thf(fact_699_distrib__inf__le,axiom,
! [X: set_v,Y2: set_v,Z: set_v] : ( ord_less_eq_set_v @ ( sup_sup_set_v @ ( inf_inf_set_v @ X @ Y2 ) @ ( inf_inf_set_v @ X @ Z ) ) @ ( inf_inf_set_v @ X @ ( sup_sup_set_v @ Y2 @ Z ) ) ) ).
% distrib_inf_le
thf(fact_700_distrib__inf__le,axiom,
! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v,Z: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ X @ Y2 ) @ ( inf_in6271465464967711157od_v_v @ X @ Z ) ) @ ( inf_in6271465464967711157od_v_v @ X @ ( sup_su414716646722978715od_v_v @ Y2 @ Z ) ) ) ).
% distrib_inf_le
thf(fact_701_insert__is__Un,axiom,
( insert1338601472111419319od_v_v
= ( ^ [A6: product_prod_v_v] : ( sup_su414716646722978715od_v_v @ ( insert1338601472111419319od_v_v @ A6 @ bot_bo723834152578015283od_v_v ) ) ) ) ).
% insert_is_Un
thf(fact_702_insert__is__Un,axiom,
( insert_v2
= ( ^ [A6: v] : ( sup_sup_set_v @ ( insert_v2 @ A6 @ bot_bot_set_v ) ) ) ) ).
% insert_is_Un
thf(fact_703_Un__singleton__iff,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v,X: product_prod_v_v] :
( ( ( sup_su414716646722978715od_v_v @ A3 @ B )
= ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) )
= ( ( ( A3 = bot_bo723834152578015283od_v_v )
& ( B
= ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) ) )
| ( ( A3
= ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) )
& ( B = bot_bo723834152578015283od_v_v ) )
| ( ( A3
= ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) )
& ( B
= ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_704_Un__singleton__iff,axiom,
! [A3: set_v,B: set_v,X: v] :
( ( ( sup_sup_set_v @ A3 @ B )
= ( insert_v2 @ X @ bot_bot_set_v ) )
= ( ( ( A3 = bot_bot_set_v )
& ( B
= ( insert_v2 @ X @ bot_bot_set_v ) ) )
| ( ( A3
= ( insert_v2 @ X @ bot_bot_set_v ) )
& ( B = bot_bot_set_v ) )
| ( ( A3
= ( insert_v2 @ X @ bot_bot_set_v ) )
& ( B
= ( insert_v2 @ X @ bot_bot_set_v ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_705_singleton__Un__iff,axiom,
! [X: product_prod_v_v,A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v )
= ( sup_su414716646722978715od_v_v @ A3 @ B ) )
= ( ( ( A3 = bot_bo723834152578015283od_v_v )
& ( B
= ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) ) )
| ( ( A3
= ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) )
& ( B = bot_bo723834152578015283od_v_v ) )
| ( ( A3
= ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) )
& ( B
= ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_706_singleton__Un__iff,axiom,
! [X: v,A3: set_v,B: set_v] :
( ( ( insert_v2 @ X @ bot_bot_set_v )
= ( sup_sup_set_v @ A3 @ B ) )
= ( ( ( A3 = bot_bot_set_v )
& ( B
= ( insert_v2 @ X @ bot_bot_set_v ) ) )
| ( ( A3
= ( insert_v2 @ X @ bot_bot_set_v ) )
& ( B = bot_bot_set_v ) )
| ( ( A3
= ( insert_v2 @ X @ bot_bot_set_v ) )
& ( B
= ( insert_v2 @ X @ bot_bot_set_v ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_707_Un__Int__assoc__eq,axiom,
! [A3: set_v,B: set_v,C3: set_v] :
( ( ( sup_sup_set_v @ ( inf_inf_set_v @ A3 @ B ) @ C3 )
= ( inf_inf_set_v @ A3 @ ( sup_sup_set_v @ B @ C3 ) ) )
= ( ord_less_eq_set_v @ C3 @ A3 ) ) ).
% Un_Int_assoc_eq
thf(fact_708_Un__Int__assoc__eq,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v,C3: set_Product_prod_v_v] :
( ( ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ A3 @ B ) @ C3 )
= ( inf_in6271465464967711157od_v_v @ A3 @ ( sup_su414716646722978715od_v_v @ B @ C3 ) ) )
= ( ord_le7336532860387713383od_v_v @ C3 @ A3 ) ) ).
% Un_Int_assoc_eq
thf(fact_709_Diff__subset__conv,axiom,
! [A3: set_v,B: set_v,C3: set_v] :
( ( ord_less_eq_set_v @ ( minus_minus_set_v @ A3 @ B ) @ C3 )
= ( ord_less_eq_set_v @ A3 @ ( sup_sup_set_v @ B @ C3 ) ) ) ).
% Diff_subset_conv
thf(fact_710_Diff__subset__conv,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v,C3: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ ( minus_4183494784930505774od_v_v @ A3 @ B ) @ C3 )
= ( ord_le7336532860387713383od_v_v @ A3 @ ( sup_su414716646722978715od_v_v @ B @ C3 ) ) ) ).
% Diff_subset_conv
thf(fact_711_Diff__partition,axiom,
! [A3: set_v,B: set_v] :
( ( ord_less_eq_set_v @ A3 @ B )
=> ( ( sup_sup_set_v @ A3 @ ( minus_minus_set_v @ B @ A3 ) )
= B ) ) ).
% Diff_partition
thf(fact_712_Diff__partition,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A3 @ B )
=> ( ( sup_su414716646722978715od_v_v @ A3 @ ( minus_4183494784930505774od_v_v @ B @ A3 ) )
= B ) ) ).
% Diff_partition
thf(fact_713_Un__Diff__Int,axiom,
! [A3: set_v,B: set_v] :
( ( sup_sup_set_v @ ( minus_minus_set_v @ A3 @ B ) @ ( inf_inf_set_v @ A3 @ B ) )
= A3 ) ).
% Un_Diff_Int
thf(fact_714_Un__Diff__Int,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( minus_4183494784930505774od_v_v @ A3 @ B ) @ ( inf_in6271465464967711157od_v_v @ A3 @ B ) )
= A3 ) ).
% Un_Diff_Int
thf(fact_715_Int__Diff__Un,axiom,
! [A3: set_v,B: set_v] :
( ( sup_sup_set_v @ ( inf_inf_set_v @ A3 @ B ) @ ( minus_minus_set_v @ A3 @ B ) )
= A3 ) ).
% Int_Diff_Un
thf(fact_716_Int__Diff__Un,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ A3 @ B ) @ ( minus_4183494784930505774od_v_v @ A3 @ B ) )
= A3 ) ).
% Int_Diff_Un
thf(fact_717_Diff__Int,axiom,
! [A3: set_v,B: set_v,C3: set_v] :
( ( minus_minus_set_v @ A3 @ ( inf_inf_set_v @ B @ C3 ) )
= ( sup_sup_set_v @ ( minus_minus_set_v @ A3 @ B ) @ ( minus_minus_set_v @ A3 @ C3 ) ) ) ).
% Diff_Int
thf(fact_718_Diff__Int,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v,C3: set_Product_prod_v_v] :
( ( minus_4183494784930505774od_v_v @ A3 @ ( inf_in6271465464967711157od_v_v @ B @ C3 ) )
= ( sup_su414716646722978715od_v_v @ ( minus_4183494784930505774od_v_v @ A3 @ B ) @ ( minus_4183494784930505774od_v_v @ A3 @ C3 ) ) ) ).
% Diff_Int
thf(fact_719_Diff__Un,axiom,
! [A3: set_v,B: set_v,C3: set_v] :
( ( minus_minus_set_v @ A3 @ ( sup_sup_set_v @ B @ C3 ) )
= ( inf_inf_set_v @ ( minus_minus_set_v @ A3 @ B ) @ ( minus_minus_set_v @ A3 @ C3 ) ) ) ).
% Diff_Un
thf(fact_720_Diff__Un,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v,C3: set_Product_prod_v_v] :
( ( minus_4183494784930505774od_v_v @ A3 @ ( sup_su414716646722978715od_v_v @ B @ C3 ) )
= ( inf_in6271465464967711157od_v_v @ ( minus_4183494784930505774od_v_v @ A3 @ B ) @ ( minus_4183494784930505774od_v_v @ A3 @ C3 ) ) ) ).
% Diff_Un
thf(fact_721_SCC__Bloemen__Sequential_Ograph__def,axiom,
( sCC_Bl8307124943676871238od_v_v
= ( ^ [Vertices2: set_Product_prod_v_v,Successors2: product_prod_v_v > set_Product_prod_v_v] :
( ( finite3348123685078250256od_v_v @ Vertices2 )
& ! [X2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ Vertices2 )
=> ( ord_le7336532860387713383od_v_v @ ( Successors2 @ X2 ) @ Vertices2 ) ) ) ) ) ).
% SCC_Bloemen_Sequential.graph_def
thf(fact_722_SCC__Bloemen__Sequential_Ograph__def,axiom,
( sCC_Bloemen_graph_v
= ( ^ [Vertices2: set_v,Successors2: v > set_v] :
( ( finite_finite_v @ Vertices2 )
& ! [X2: v] :
( ( member_v @ X2 @ Vertices2 )
=> ( ord_less_eq_set_v @ ( Successors2 @ X2 ) @ Vertices2 ) ) ) ) ) ).
% SCC_Bloemen_Sequential.graph_def
thf(fact_723_graph_Ointro,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v] :
( ( finite3348123685078250256od_v_v @ Vertices )
=> ( ! [X3: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X3 @ Vertices )
=> ( ord_le7336532860387713383od_v_v @ ( Successors @ X3 ) @ Vertices ) )
=> ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors ) ) ) ).
% graph.intro
thf(fact_724_graph_Ointro,axiom,
! [Vertices: set_v,Successors: v > set_v] :
( ( finite_finite_v @ Vertices )
=> ( ! [X3: v] :
( ( member_v @ X3 @ Vertices )
=> ( ord_less_eq_set_v @ ( Successors @ X3 ) @ Vertices ) )
=> ( sCC_Bloemen_graph_v @ Vertices @ Successors ) ) ) ).
% graph.intro
thf(fact_725_is__singletonI_H,axiom,
! [A3: set_Product_prod_v_v] :
( ( A3 != bot_bo723834152578015283od_v_v )
=> ( ! [X3: product_prod_v_v,Y: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X3 @ A3 )
=> ( ( member7453568604450474000od_v_v @ Y @ A3 )
=> ( X3 = Y ) ) )
=> ( is_sin9198872032823709915od_v_v @ A3 ) ) ) ).
% is_singletonI'
thf(fact_726_is__singletonI_H,axiom,
! [A3: set_v] :
( ( A3 != bot_bot_set_v )
=> ( ! [X3: v,Y: v] :
( ( member_v @ X3 @ A3 )
=> ( ( member_v @ Y @ A3 )
=> ( X3 = Y ) ) )
=> ( is_singleton_v @ A3 ) ) ) ).
% is_singletonI'
thf(fact_727_graph_Ora__add__edge,axiom,
! [Vertices: set_v,Successors: v > set_v,X: v,Y2: v,E2: set_Product_prod_v_v,V: v,W: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ Y2 @ E2 )
=> ( ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ Y2 @ ( sup_su414716646722978715od_v_v @ E2 @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ V @ W ) @ bot_bo723834152578015283od_v_v ) ) )
| ( ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ V @ ( sup_su414716646722978715od_v_v @ E2 @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ V @ W ) @ bot_bo723834152578015283od_v_v ) ) )
& ( sCC_Bl4291963740693775144ding_v @ Successors @ W @ Y2 @ ( sup_su414716646722978715od_v_v @ E2 @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ V @ W ) @ bot_bo723834152578015283od_v_v ) ) ) ) ) ) ) ).
% graph.ra_add_edge
thf(fact_728_is__singletonE,axiom,
! [A3: set_Product_prod_v_v] :
( ( is_sin9198872032823709915od_v_v @ A3 )
=> ~ ! [X3: product_prod_v_v] :
( A3
!= ( insert1338601472111419319od_v_v @ X3 @ bot_bo723834152578015283od_v_v ) ) ) ).
% is_singletonE
thf(fact_729_is__singletonE,axiom,
! [A3: set_v] :
( ( is_singleton_v @ A3 )
=> ~ ! [X3: v] :
( A3
!= ( insert_v2 @ X3 @ bot_bot_set_v ) ) ) ).
% is_singletonE
thf(fact_730_is__singleton__def,axiom,
( is_sin9198872032823709915od_v_v
= ( ^ [A4: set_Product_prod_v_v] :
? [X2: product_prod_v_v] :
( A4
= ( insert1338601472111419319od_v_v @ X2 @ bot_bo723834152578015283od_v_v ) ) ) ) ).
% is_singleton_def
thf(fact_731_is__singleton__def,axiom,
( is_singleton_v
= ( ^ [A4: set_v] :
? [X2: v] :
( A4
= ( insert_v2 @ X2 @ bot_bot_set_v ) ) ) ) ).
% is_singleton_def
thf(fact_732_graph_Oavoiding__explored,axiom,
! [Vertices: set_v,Successors: v > set_v,E: sCC_Bl1394983891496994913t_unit,X: v,Y2: v,E2: set_Product_prod_v_v,W: v,V: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E )
=> ( ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ Y2 @ E2 )
=> ( ~ ( member_v @ Y2 @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ( ( member_v @ W @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ Y2 @ ( sup_su414716646722978715od_v_v @ E2 @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ V @ W ) @ bot_bo723834152578015283od_v_v ) ) ) ) ) ) ) ) ).
% graph.avoiding_explored
thf(fact_733_subrelI,axiom,
! [R: set_Product_prod_v_v,S4: set_Product_prod_v_v] :
( ! [X3: v,Y: v] :
( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X3 @ Y ) @ R )
=> ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X3 @ Y ) @ S4 ) )
=> ( ord_le7336532860387713383od_v_v @ R @ S4 ) ) ).
% subrelI
thf(fact_734_finite__Diff__insert,axiom,
! [A3: set_Product_prod_v_v,A: product_prod_v_v,B: set_Product_prod_v_v] :
( ( finite3348123685078250256od_v_v @ ( minus_4183494784930505774od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ A @ B ) ) )
= ( finite3348123685078250256od_v_v @ ( minus_4183494784930505774od_v_v @ A3 @ B ) ) ) ).
% finite_Diff_insert
thf(fact_735_finite__Diff__insert,axiom,
! [A3: set_v,A: v,B: set_v] :
( ( finite_finite_v @ ( minus_minus_set_v @ A3 @ ( insert_v2 @ A @ B ) ) )
= ( finite_finite_v @ ( minus_minus_set_v @ A3 @ B ) ) ) ).
% finite_Diff_insert
thf(fact_736_finite__Diff2,axiom,
! [B: set_v,A3: set_v] :
( ( finite_finite_v @ B )
=> ( ( finite_finite_v @ ( minus_minus_set_v @ A3 @ B ) )
= ( finite_finite_v @ A3 ) ) ) ).
% finite_Diff2
thf(fact_737_finite__Diff,axiom,
! [A3: set_v,B: set_v] :
( ( finite_finite_v @ A3 )
=> ( finite_finite_v @ ( minus_minus_set_v @ A3 @ B ) ) ) ).
% finite_Diff
thf(fact_738_finite__Int,axiom,
! [F2: set_v,G: set_v] :
( ( ( finite_finite_v @ F2 )
| ( finite_finite_v @ G ) )
=> ( finite_finite_v @ ( inf_inf_set_v @ F2 @ G ) ) ) ).
% finite_Int
thf(fact_739_finite__remove__induct,axiom,
! [B: set_v,P: set_v > $o] :
( ( finite_finite_v @ B )
=> ( ( P @ bot_bot_set_v )
=> ( ! [A9: set_v] :
( ( finite_finite_v @ A9 )
=> ( ( A9 != bot_bot_set_v )
=> ( ( ord_less_eq_set_v @ A9 @ B )
=> ( ! [X4: v] :
( ( member_v @ X4 @ A9 )
=> ( P @ ( minus_minus_set_v @ A9 @ ( insert_v2 @ X4 @ bot_bot_set_v ) ) ) )
=> ( P @ A9 ) ) ) ) )
=> ( P @ B ) ) ) ) ).
% finite_remove_induct
thf(fact_740_finite__remove__induct,axiom,
! [B: set_Product_prod_v_v,P: set_Product_prod_v_v > $o] :
( ( finite3348123685078250256od_v_v @ B )
=> ( ( P @ bot_bo723834152578015283od_v_v )
=> ( ! [A9: set_Product_prod_v_v] :
( ( finite3348123685078250256od_v_v @ A9 )
=> ( ( A9 != bot_bo723834152578015283od_v_v )
=> ( ( ord_le7336532860387713383od_v_v @ A9 @ B )
=> ( ! [X4: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X4 @ A9 )
=> ( P @ ( minus_4183494784930505774od_v_v @ A9 @ ( insert1338601472111419319od_v_v @ X4 @ bot_bo723834152578015283od_v_v ) ) ) )
=> ( P @ A9 ) ) ) ) )
=> ( P @ B ) ) ) ) ).
% finite_remove_induct
thf(fact_741_remove__induct,axiom,
! [P: set_v > $o,B: set_v] :
( ( P @ bot_bot_set_v )
=> ( ( ~ ( finite_finite_v @ B )
=> ( P @ B ) )
=> ( ! [A9: set_v] :
( ( finite_finite_v @ A9 )
=> ( ( A9 != bot_bot_set_v )
=> ( ( ord_less_eq_set_v @ A9 @ B )
=> ( ! [X4: v] :
( ( member_v @ X4 @ A9 )
=> ( P @ ( minus_minus_set_v @ A9 @ ( insert_v2 @ X4 @ bot_bot_set_v ) ) ) )
=> ( P @ A9 ) ) ) ) )
=> ( P @ B ) ) ) ) ).
% remove_induct
thf(fact_742_remove__induct,axiom,
! [P: set_Product_prod_v_v > $o,B: set_Product_prod_v_v] :
( ( P @ bot_bo723834152578015283od_v_v )
=> ( ( ~ ( finite3348123685078250256od_v_v @ B )
=> ( P @ B ) )
=> ( ! [A9: set_Product_prod_v_v] :
( ( finite3348123685078250256od_v_v @ A9 )
=> ( ( A9 != bot_bo723834152578015283od_v_v )
=> ( ( ord_le7336532860387713383od_v_v @ A9 @ B )
=> ( ! [X4: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X4 @ A9 )
=> ( P @ ( minus_4183494784930505774od_v_v @ A9 @ ( insert1338601472111419319od_v_v @ X4 @ bot_bo723834152578015283od_v_v ) ) ) )
=> ( P @ A9 ) ) ) ) )
=> ( P @ B ) ) ) ) ).
% remove_induct
thf(fact_743_finite__insert,axiom,
! [A: product_prod_v_v,A3: set_Product_prod_v_v] :
( ( finite3348123685078250256od_v_v @ ( insert1338601472111419319od_v_v @ A @ A3 ) )
= ( finite3348123685078250256od_v_v @ A3 ) ) ).
% finite_insert
thf(fact_744_finite__insert,axiom,
! [A: v,A3: set_v] :
( ( finite_finite_v @ ( insert_v2 @ A @ A3 ) )
= ( finite_finite_v @ A3 ) ) ).
% finite_insert
thf(fact_745_finite__has__minimal2,axiom,
! [A3: set_set_v,A: set_v] :
( ( finite_finite_set_v @ A3 )
=> ( ( member_set_v @ A @ A3 )
=> ? [X3: set_v] :
( ( member_set_v @ X3 @ A3 )
& ( ord_less_eq_set_v @ X3 @ A )
& ! [Xa2: set_v] :
( ( member_set_v @ Xa2 @ A3 )
=> ( ( ord_less_eq_set_v @ Xa2 @ X3 )
=> ( X3 = Xa2 ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_746_finite__has__minimal2,axiom,
! [A3: set_se8455005133513928103od_v_v,A: set_Product_prod_v_v] :
( ( finite6084192165098772208od_v_v @ A3 )
=> ( ( member8406446414694345712od_v_v @ A @ A3 )
=> ? [X3: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X3 @ A3 )
& ( ord_le7336532860387713383od_v_v @ X3 @ A )
& ! [Xa2: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ Xa2 @ A3 )
=> ( ( ord_le7336532860387713383od_v_v @ Xa2 @ X3 )
=> ( X3 = Xa2 ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_747_finite__has__maximal2,axiom,
! [A3: set_set_v,A: set_v] :
( ( finite_finite_set_v @ A3 )
=> ( ( member_set_v @ A @ A3 )
=> ? [X3: set_v] :
( ( member_set_v @ X3 @ A3 )
& ( ord_less_eq_set_v @ A @ X3 )
& ! [Xa2: set_v] :
( ( member_set_v @ Xa2 @ A3 )
=> ( ( ord_less_eq_set_v @ X3 @ Xa2 )
=> ( X3 = Xa2 ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_748_finite__has__maximal2,axiom,
! [A3: set_se8455005133513928103od_v_v,A: set_Product_prod_v_v] :
( ( finite6084192165098772208od_v_v @ A3 )
=> ( ( member8406446414694345712od_v_v @ A @ A3 )
=> ? [X3: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X3 @ A3 )
& ( ord_le7336532860387713383od_v_v @ A @ X3 )
& ! [Xa2: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ Xa2 @ A3 )
=> ( ( ord_le7336532860387713383od_v_v @ X3 @ Xa2 )
=> ( X3 = Xa2 ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_749_infinite__imp__nonempty,axiom,
! [S: set_Product_prod_v_v] :
( ~ ( finite3348123685078250256od_v_v @ S )
=> ( S != bot_bo723834152578015283od_v_v ) ) ).
% infinite_imp_nonempty
thf(fact_750_infinite__imp__nonempty,axiom,
! [S: set_v] :
( ~ ( finite_finite_v @ S )
=> ( S != bot_bot_set_v ) ) ).
% infinite_imp_nonempty
thf(fact_751_finite_OemptyI,axiom,
finite3348123685078250256od_v_v @ bot_bo723834152578015283od_v_v ).
% finite.emptyI
thf(fact_752_finite_OemptyI,axiom,
finite_finite_v @ bot_bot_set_v ).
% finite.emptyI
thf(fact_753_rev__finite__subset,axiom,
! [B: set_v,A3: set_v] :
( ( finite_finite_v @ B )
=> ( ( ord_less_eq_set_v @ A3 @ B )
=> ( finite_finite_v @ A3 ) ) ) ).
% rev_finite_subset
thf(fact_754_rev__finite__subset,axiom,
! [B: set_Product_prod_v_v,A3: set_Product_prod_v_v] :
( ( finite3348123685078250256od_v_v @ B )
=> ( ( ord_le7336532860387713383od_v_v @ A3 @ B )
=> ( finite3348123685078250256od_v_v @ A3 ) ) ) ).
% rev_finite_subset
thf(fact_755_infinite__super,axiom,
! [S: set_v,T2: set_v] :
( ( ord_less_eq_set_v @ S @ T2 )
=> ( ~ ( finite_finite_v @ S )
=> ~ ( finite_finite_v @ T2 ) ) ) ).
% infinite_super
thf(fact_756_infinite__super,axiom,
! [S: set_Product_prod_v_v,T2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ S @ T2 )
=> ( ~ ( finite3348123685078250256od_v_v @ S )
=> ~ ( finite3348123685078250256od_v_v @ T2 ) ) ) ).
% infinite_super
thf(fact_757_finite__subset,axiom,
! [A3: set_v,B: set_v] :
( ( ord_less_eq_set_v @ A3 @ B )
=> ( ( finite_finite_v @ B )
=> ( finite_finite_v @ A3 ) ) ) ).
% finite_subset
thf(fact_758_finite__subset,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A3 @ B )
=> ( ( finite3348123685078250256od_v_v @ B )
=> ( finite3348123685078250256od_v_v @ A3 ) ) ) ).
% finite_subset
thf(fact_759_finite_OinsertI,axiom,
! [A3: set_Product_prod_v_v,A: product_prod_v_v] :
( ( finite3348123685078250256od_v_v @ A3 )
=> ( finite3348123685078250256od_v_v @ ( insert1338601472111419319od_v_v @ A @ A3 ) ) ) ).
% finite.insertI
thf(fact_760_finite_OinsertI,axiom,
! [A3: set_v,A: v] :
( ( finite_finite_v @ A3 )
=> ( finite_finite_v @ ( insert_v2 @ A @ A3 ) ) ) ).
% finite.insertI
thf(fact_761_Diff__infinite__finite,axiom,
! [T2: set_v,S: set_v] :
( ( finite_finite_v @ T2 )
=> ( ~ ( finite_finite_v @ S )
=> ~ ( finite_finite_v @ ( minus_minus_set_v @ S @ T2 ) ) ) ) ).
% Diff_infinite_finite
thf(fact_762_finite__has__maximal,axiom,
! [A3: set_set_v] :
( ( finite_finite_set_v @ A3 )
=> ( ( A3 != bot_bot_set_set_v )
=> ? [X3: set_v] :
( ( member_set_v @ X3 @ A3 )
& ! [Xa2: set_v] :
( ( member_set_v @ Xa2 @ A3 )
=> ( ( ord_less_eq_set_v @ X3 @ Xa2 )
=> ( X3 = Xa2 ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_763_finite__has__maximal,axiom,
! [A3: set_se8455005133513928103od_v_v] :
( ( finite6084192165098772208od_v_v @ A3 )
=> ( ( A3 != bot_bo3497076220358800403od_v_v )
=> ? [X3: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X3 @ A3 )
& ! [Xa2: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ Xa2 @ A3 )
=> ( ( ord_le7336532860387713383od_v_v @ X3 @ Xa2 )
=> ( X3 = Xa2 ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_764_finite__has__minimal,axiom,
! [A3: set_set_v] :
( ( finite_finite_set_v @ A3 )
=> ( ( A3 != bot_bot_set_set_v )
=> ? [X3: set_v] :
( ( member_set_v @ X3 @ A3 )
& ! [Xa2: set_v] :
( ( member_set_v @ Xa2 @ A3 )
=> ( ( ord_less_eq_set_v @ Xa2 @ X3 )
=> ( X3 = Xa2 ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_765_finite__has__minimal,axiom,
! [A3: set_se8455005133513928103od_v_v] :
( ( finite6084192165098772208od_v_v @ A3 )
=> ( ( A3 != bot_bo3497076220358800403od_v_v )
=> ? [X3: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X3 @ A3 )
& ! [Xa2: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ Xa2 @ A3 )
=> ( ( ord_le7336532860387713383od_v_v @ Xa2 @ X3 )
=> ( X3 = Xa2 ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_766_infinite__finite__induct,axiom,
! [P: set_Product_prod_v_v > $o,A3: set_Product_prod_v_v] :
( ! [A9: set_Product_prod_v_v] :
( ~ ( finite3348123685078250256od_v_v @ A9 )
=> ( P @ A9 ) )
=> ( ( P @ bot_bo723834152578015283od_v_v )
=> ( ! [X3: product_prod_v_v,F3: set_Product_prod_v_v] :
( ( finite3348123685078250256od_v_v @ F3 )
=> ( ~ ( member7453568604450474000od_v_v @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert1338601472111419319od_v_v @ X3 @ F3 ) ) ) ) )
=> ( P @ A3 ) ) ) ) ).
% infinite_finite_induct
thf(fact_767_infinite__finite__induct,axiom,
! [P: set_v > $o,A3: set_v] :
( ! [A9: set_v] :
( ~ ( finite_finite_v @ A9 )
=> ( P @ A9 ) )
=> ( ( P @ bot_bot_set_v )
=> ( ! [X3: v,F3: set_v] :
( ( finite_finite_v @ F3 )
=> ( ~ ( member_v @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_v2 @ X3 @ F3 ) ) ) ) )
=> ( P @ A3 ) ) ) ) ).
% infinite_finite_induct
thf(fact_768_finite__ne__induct,axiom,
! [F2: set_Product_prod_v_v,P: set_Product_prod_v_v > $o] :
( ( finite3348123685078250256od_v_v @ F2 )
=> ( ( F2 != bot_bo723834152578015283od_v_v )
=> ( ! [X3: product_prod_v_v] : ( P @ ( insert1338601472111419319od_v_v @ X3 @ bot_bo723834152578015283od_v_v ) )
=> ( ! [X3: product_prod_v_v,F3: set_Product_prod_v_v] :
( ( finite3348123685078250256od_v_v @ F3 )
=> ( ( F3 != bot_bo723834152578015283od_v_v )
=> ( ~ ( member7453568604450474000od_v_v @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert1338601472111419319od_v_v @ X3 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_769_finite__ne__induct,axiom,
! [F2: set_v,P: set_v > $o] :
( ( finite_finite_v @ F2 )
=> ( ( F2 != bot_bot_set_v )
=> ( ! [X3: v] : ( P @ ( insert_v2 @ X3 @ bot_bot_set_v ) )
=> ( ! [X3: v,F3: set_v] :
( ( finite_finite_v @ F3 )
=> ( ( F3 != bot_bot_set_v )
=> ( ~ ( member_v @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_v2 @ X3 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_770_finite__induct,axiom,
! [F2: set_Product_prod_v_v,P: set_Product_prod_v_v > $o] :
( ( finite3348123685078250256od_v_v @ F2 )
=> ( ( P @ bot_bo723834152578015283od_v_v )
=> ( ! [X3: product_prod_v_v,F3: set_Product_prod_v_v] :
( ( finite3348123685078250256od_v_v @ F3 )
=> ( ~ ( member7453568604450474000od_v_v @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert1338601472111419319od_v_v @ X3 @ F3 ) ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% finite_induct
thf(fact_771_finite__induct,axiom,
! [F2: set_v,P: set_v > $o] :
( ( finite_finite_v @ F2 )
=> ( ( P @ bot_bot_set_v )
=> ( ! [X3: v,F3: set_v] :
( ( finite_finite_v @ F3 )
=> ( ~ ( member_v @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_v2 @ X3 @ F3 ) ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% finite_induct
thf(fact_772_finite_Osimps,axiom,
( finite3348123685078250256od_v_v
= ( ^ [A6: set_Product_prod_v_v] :
( ( A6 = bot_bo723834152578015283od_v_v )
| ? [A4: set_Product_prod_v_v,B6: product_prod_v_v] :
( ( A6
= ( insert1338601472111419319od_v_v @ B6 @ A4 ) )
& ( finite3348123685078250256od_v_v @ A4 ) ) ) ) ) ).
% finite.simps
thf(fact_773_finite_Osimps,axiom,
( finite_finite_v
= ( ^ [A6: set_v] :
( ( A6 = bot_bot_set_v )
| ? [A4: set_v,B6: v] :
( ( A6
= ( insert_v2 @ B6 @ A4 ) )
& ( finite_finite_v @ A4 ) ) ) ) ) ).
% finite.simps
thf(fact_774_finite_Ocases,axiom,
! [A: set_Product_prod_v_v] :
( ( finite3348123685078250256od_v_v @ A )
=> ( ( A != bot_bo723834152578015283od_v_v )
=> ~ ! [A9: set_Product_prod_v_v] :
( ? [A7: product_prod_v_v] :
( A
= ( insert1338601472111419319od_v_v @ A7 @ A9 ) )
=> ~ ( finite3348123685078250256od_v_v @ A9 ) ) ) ) ).
% finite.cases
thf(fact_775_finite_Ocases,axiom,
! [A: set_v] :
( ( finite_finite_v @ A )
=> ( ( A != bot_bot_set_v )
=> ~ ! [A9: set_v] :
( ? [A7: v] :
( A
= ( insert_v2 @ A7 @ A9 ) )
=> ~ ( finite_finite_v @ A9 ) ) ) ) ).
% finite.cases
thf(fact_776_finite__subset__induct,axiom,
! [F2: set_v,A3: set_v,P: set_v > $o] :
( ( finite_finite_v @ F2 )
=> ( ( ord_less_eq_set_v @ F2 @ A3 )
=> ( ( P @ bot_bot_set_v )
=> ( ! [A7: v,F3: set_v] :
( ( finite_finite_v @ F3 )
=> ( ( member_v @ A7 @ A3 )
=> ( ~ ( member_v @ A7 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_v2 @ A7 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_777_finite__subset__induct,axiom,
! [F2: set_Product_prod_v_v,A3: set_Product_prod_v_v,P: set_Product_prod_v_v > $o] :
( ( finite3348123685078250256od_v_v @ F2 )
=> ( ( ord_le7336532860387713383od_v_v @ F2 @ A3 )
=> ( ( P @ bot_bo723834152578015283od_v_v )
=> ( ! [A7: product_prod_v_v,F3: set_Product_prod_v_v] :
( ( finite3348123685078250256od_v_v @ F3 )
=> ( ( member7453568604450474000od_v_v @ A7 @ A3 )
=> ( ~ ( member7453568604450474000od_v_v @ A7 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert1338601472111419319od_v_v @ A7 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_778_finite__subset__induct_H,axiom,
! [F2: set_v,A3: set_v,P: set_v > $o] :
( ( finite_finite_v @ F2 )
=> ( ( ord_less_eq_set_v @ F2 @ A3 )
=> ( ( P @ bot_bot_set_v )
=> ( ! [A7: v,F3: set_v] :
( ( finite_finite_v @ F3 )
=> ( ( member_v @ A7 @ A3 )
=> ( ( ord_less_eq_set_v @ F3 @ A3 )
=> ( ~ ( member_v @ A7 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_v2 @ A7 @ F3 ) ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_779_finite__subset__induct_H,axiom,
! [F2: set_Product_prod_v_v,A3: set_Product_prod_v_v,P: set_Product_prod_v_v > $o] :
( ( finite3348123685078250256od_v_v @ F2 )
=> ( ( ord_le7336532860387713383od_v_v @ F2 @ A3 )
=> ( ( P @ bot_bo723834152578015283od_v_v )
=> ( ! [A7: product_prod_v_v,F3: set_Product_prod_v_v] :
( ( finite3348123685078250256od_v_v @ F3 )
=> ( ( member7453568604450474000od_v_v @ A7 @ A3 )
=> ( ( ord_le7336532860387713383od_v_v @ F3 @ A3 )
=> ( ~ ( member7453568604450474000od_v_v @ A7 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert1338601472111419319od_v_v @ A7 @ F3 ) ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_780_infinite__remove,axiom,
! [S: set_Product_prod_v_v,A: product_prod_v_v] :
( ~ ( finite3348123685078250256od_v_v @ S )
=> ~ ( finite3348123685078250256od_v_v @ ( minus_4183494784930505774od_v_v @ S @ ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) ) ) ) ).
% infinite_remove
thf(fact_781_infinite__remove,axiom,
! [S: set_v,A: v] :
( ~ ( finite_finite_v @ S )
=> ~ ( finite_finite_v @ ( minus_minus_set_v @ S @ ( insert_v2 @ A @ bot_bot_set_v ) ) ) ) ).
% infinite_remove
thf(fact_782_infinite__coinduct,axiom,
! [X5: set_Product_prod_v_v > $o,A3: set_Product_prod_v_v] :
( ( X5 @ A3 )
=> ( ! [A9: set_Product_prod_v_v] :
( ( X5 @ A9 )
=> ? [X4: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X4 @ A9 )
& ( ( X5 @ ( minus_4183494784930505774od_v_v @ A9 @ ( insert1338601472111419319od_v_v @ X4 @ bot_bo723834152578015283od_v_v ) ) )
| ~ ( finite3348123685078250256od_v_v @ ( minus_4183494784930505774od_v_v @ A9 @ ( insert1338601472111419319od_v_v @ X4 @ bot_bo723834152578015283od_v_v ) ) ) ) ) )
=> ~ ( finite3348123685078250256od_v_v @ A3 ) ) ) ).
% infinite_coinduct
thf(fact_783_infinite__coinduct,axiom,
! [X5: set_v > $o,A3: set_v] :
( ( X5 @ A3 )
=> ( ! [A9: set_v] :
( ( X5 @ A9 )
=> ? [X4: v] :
( ( member_v @ X4 @ A9 )
& ( ( X5 @ ( minus_minus_set_v @ A9 @ ( insert_v2 @ X4 @ bot_bot_set_v ) ) )
| ~ ( finite_finite_v @ ( minus_minus_set_v @ A9 @ ( insert_v2 @ X4 @ bot_bot_set_v ) ) ) ) ) )
=> ~ ( finite_finite_v @ A3 ) ) ) ).
% infinite_coinduct
thf(fact_784_finite__empty__induct,axiom,
! [A3: set_Product_prod_v_v,P: set_Product_prod_v_v > $o] :
( ( finite3348123685078250256od_v_v @ A3 )
=> ( ( P @ A3 )
=> ( ! [A7: product_prod_v_v,A9: set_Product_prod_v_v] :
( ( finite3348123685078250256od_v_v @ A9 )
=> ( ( member7453568604450474000od_v_v @ A7 @ A9 )
=> ( ( P @ A9 )
=> ( P @ ( minus_4183494784930505774od_v_v @ A9 @ ( insert1338601472111419319od_v_v @ A7 @ bot_bo723834152578015283od_v_v ) ) ) ) ) )
=> ( P @ bot_bo723834152578015283od_v_v ) ) ) ) ).
% finite_empty_induct
thf(fact_785_finite__empty__induct,axiom,
! [A3: set_v,P: set_v > $o] :
( ( finite_finite_v @ A3 )
=> ( ( P @ A3 )
=> ( ! [A7: v,A9: set_v] :
( ( finite_finite_v @ A9 )
=> ( ( member_v @ A7 @ A9 )
=> ( ( P @ A9 )
=> ( P @ ( minus_minus_set_v @ A9 @ ( insert_v2 @ A7 @ bot_bot_set_v ) ) ) ) ) )
=> ( P @ bot_bot_set_v ) ) ) ) ).
% finite_empty_induct
thf(fact_786_List_Ofinite__set,axiom,
! [Xs: list_v] : ( finite_finite_v @ ( set_v2 @ Xs ) ) ).
% List.finite_set
thf(fact_787_subset__code_I1_J,axiom,
! [Xs: list_v,B: set_v] :
( ( ord_less_eq_set_v @ ( set_v2 @ Xs ) @ B )
= ( ! [X2: v] :
( ( member_v @ X2 @ ( set_v2 @ Xs ) )
=> ( member_v @ X2 @ B ) ) ) ) ).
% subset_code(1)
thf(fact_788_subset__code_I1_J,axiom,
! [Xs: list_P7986770385144383213od_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ ( set_Product_prod_v_v2 @ Xs ) @ B )
= ( ! [X2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ ( set_Product_prod_v_v2 @ Xs ) )
=> ( member7453568604450474000od_v_v @ X2 @ B ) ) ) ) ).
% subset_code(1)
thf(fact_789_insert__subsetI,axiom,
! [X: v,A3: set_v,X5: set_v] :
( ( member_v @ X @ A3 )
=> ( ( ord_less_eq_set_v @ X5 @ A3 )
=> ( ord_less_eq_set_v @ ( insert_v2 @ X @ X5 ) @ A3 ) ) ) ).
% insert_subsetI
thf(fact_790_insert__subsetI,axiom,
! [X: product_prod_v_v,A3: set_Product_prod_v_v,X5: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X @ A3 )
=> ( ( ord_le7336532860387713383od_v_v @ X5 @ A3 )
=> ( ord_le7336532860387713383od_v_v @ ( insert1338601472111419319od_v_v @ X @ X5 ) @ A3 ) ) ) ).
% insert_subsetI
thf(fact_791_ssubst__Pair__rhs,axiom,
! [R: v,S4: v,R2: set_Product_prod_v_v,S5: v] :
( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ R @ S4 ) @ R2 )
=> ( ( S5 = S4 )
=> ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ R @ S5 ) @ R2 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_792_subset__emptyI,axiom,
! [A3: set_v] :
( ! [X3: v] :
~ ( member_v @ X3 @ A3 )
=> ( ord_less_eq_set_v @ A3 @ bot_bot_set_v ) ) ).
% subset_emptyI
thf(fact_793_subset__emptyI,axiom,
! [A3: set_Product_prod_v_v] :
( ! [X3: product_prod_v_v] :
~ ( member7453568604450474000od_v_v @ X3 @ A3 )
=> ( ord_le7336532860387713383od_v_v @ A3 @ bot_bo723834152578015283od_v_v ) ) ).
% subset_emptyI
thf(fact_794_finite__list,axiom,
! [A3: set_v] :
( ( finite_finite_v @ A3 )
=> ? [Xs2: list_v] :
( ( set_v2 @ Xs2 )
= A3 ) ) ).
% finite_list
thf(fact_795_set__union,axiom,
! [Xs: list_v,Ys: list_v] :
( ( set_v2 @ ( union_v @ Xs @ Ys ) )
= ( sup_sup_set_v @ ( set_v2 @ Xs ) @ ( set_v2 @ Ys ) ) ) ).
% set_union
thf(fact_796_set__union,axiom,
! [Xs: list_P7986770385144383213od_v_v,Ys: list_P7986770385144383213od_v_v] :
( ( set_Product_prod_v_v2 @ ( union_4602324378607836129od_v_v @ Xs @ Ys ) )
= ( sup_su414716646722978715od_v_v @ ( set_Product_prod_v_v2 @ Xs ) @ ( set_Product_prod_v_v2 @ Ys ) ) ) ).
% set_union
thf(fact_797_set__removeAll,axiom,
! [X: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ( set_Product_prod_v_v2 @ ( remove481895986417801203od_v_v @ X @ Xs ) )
= ( minus_4183494784930505774od_v_v @ ( set_Product_prod_v_v2 @ Xs ) @ ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) ) ) ).
% set_removeAll
thf(fact_798_set__removeAll,axiom,
! [X: v,Xs: list_v] :
( ( set_v2 @ ( removeAll_v @ X @ Xs ) )
= ( minus_minus_set_v @ ( set_v2 @ Xs ) @ ( insert_v2 @ X @ bot_bot_set_v ) ) ) ).
% set_removeAll
thf(fact_799_dfs__S__tl__stack_I2_J,axiom,
! [V: v,E: sCC_Bl1394983891496994913t_unit,E6: sCC_Bl1394983891496994913t_unit] :
( ( sCC_Bl8953792750115413617t_unit @ successors @ V @ E @ E6 )
=> ( ( ( sCC_Bl8828226123343373779t_unit @ E )
!= nil_v )
=> ! [X4: v] :
( ( member_v @ X4 @ ( set_v2 @ ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ E6 ) ) ) )
=> ( ( sCC_Bl1280885523602775798t_unit @ E6 @ X4 )
= ( sCC_Bl1280885523602775798t_unit @ E @ X4 ) ) ) ) ) ).
% dfs_S_tl_stack(2)
thf(fact_800_dfs__S__tl__stack_I1_J,axiom,
! [V: v,E: sCC_Bl1394983891496994913t_unit,E6: sCC_Bl1394983891496994913t_unit] :
( ( sCC_Bl8953792750115413617t_unit @ successors @ V @ E @ E6 )
=> ( ( ( sCC_Bl8828226123343373779t_unit @ E )
!= nil_v )
=> ( ( sCC_Bl8828226123343373779t_unit @ E6 )
!= nil_v ) ) ) ).
% dfs_S_tl_stack(1)
thf(fact_801_removeAll__id,axiom,
! [X: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ~ ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ Xs ) )
=> ( ( remove481895986417801203od_v_v @ X @ Xs )
= Xs ) ) ).
% removeAll_id
thf(fact_802_removeAll__id,axiom,
! [X: v,Xs: list_v] :
( ~ ( member_v @ X @ ( set_v2 @ Xs ) )
=> ( ( removeAll_v @ X @ Xs )
= Xs ) ) ).
% removeAll_id
thf(fact_803_set__empty2,axiom,
! [Xs: list_P7986770385144383213od_v_v] :
( ( bot_bo723834152578015283od_v_v
= ( set_Product_prod_v_v2 @ Xs ) )
= ( Xs = nil_Product_prod_v_v ) ) ).
% set_empty2
thf(fact_804_set__empty2,axiom,
! [Xs: list_v] :
( ( bot_bot_set_v
= ( set_v2 @ Xs ) )
= ( Xs = nil_v ) ) ).
% set_empty2
thf(fact_805_set__empty,axiom,
! [Xs: list_P7986770385144383213od_v_v] :
( ( ( set_Product_prod_v_v2 @ Xs )
= bot_bo723834152578015283od_v_v )
= ( Xs = nil_Product_prod_v_v ) ) ).
% set_empty
thf(fact_806_set__empty,axiom,
! [Xs: list_v] :
( ( ( set_v2 @ Xs )
= bot_bot_set_v )
= ( Xs = nil_v ) ) ).
% set_empty
thf(fact_807_list_Osel_I2_J,axiom,
( ( tl_v @ nil_v )
= nil_v ) ).
% list.sel(2)
thf(fact_808_graph_Odfs__S__tl__stack_I1_J,axiom,
! [Vertices: set_v,Successors: v > set_v,V: v,E: sCC_Bl1394983891496994913t_unit,E6: sCC_Bl1394983891496994913t_unit] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl8953792750115413617t_unit @ Successors @ V @ E @ E6 )
=> ( ( ( sCC_Bl8828226123343373779t_unit @ E )
!= nil_v )
=> ( ( sCC_Bl8828226123343373779t_unit @ E6 )
!= nil_v ) ) ) ) ).
% graph.dfs_S_tl_stack(1)
thf(fact_809_empty__set,axiom,
( bot_bo723834152578015283od_v_v
= ( set_Product_prod_v_v2 @ nil_Product_prod_v_v ) ) ).
% empty_set
thf(fact_810_empty__set,axiom,
( bot_bot_set_v
= ( set_v2 @ nil_v ) ) ).
% empty_set
thf(fact_811_list_Oset__sel_I2_J,axiom,
! [A: list_P7986770385144383213od_v_v,X: product_prod_v_v] :
( ( A != nil_Product_prod_v_v )
=> ( ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ ( tl_Product_prod_v_v @ A ) ) )
=> ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ A ) ) ) ) ).
% list.set_sel(2)
thf(fact_812_list_Oset__sel_I2_J,axiom,
! [A: list_v,X: v] :
( ( A != nil_v )
=> ( ( member_v @ X @ ( set_v2 @ ( tl_v @ A ) ) )
=> ( member_v @ X @ ( set_v2 @ A ) ) ) ) ).
% list.set_sel(2)
thf(fact_813_graph_Odfs__S__tl__stack_I2_J,axiom,
! [Vertices: set_v,Successors: v > set_v,V: v,E: sCC_Bl1394983891496994913t_unit,E6: sCC_Bl1394983891496994913t_unit] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl8953792750115413617t_unit @ Successors @ V @ E @ E6 )
=> ( ( ( sCC_Bl8828226123343373779t_unit @ E )
!= nil_v )
=> ! [X4: v] :
( ( member_v @ X4 @ ( set_v2 @ ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ E6 ) ) ) )
=> ( ( sCC_Bl1280885523602775798t_unit @ E6 @ X4 )
= ( sCC_Bl1280885523602775798t_unit @ E @ X4 ) ) ) ) ) ) ).
% graph.dfs_S_tl_stack(2)
thf(fact_814_dfs__S__hd__stack_I1_J,axiom,
! [E: sCC_Bl1394983891496994913t_unit,V: v,E6: sCC_Bl1394983891496994913t_unit,N: v] :
( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
=> ( ( sCC_Bl8953792750115413617t_unit @ successors @ V @ E @ E6 )
=> ( ( ( sCC_Bl8828226123343373779t_unit @ E )
!= nil_v )
=> ( ( member_v @ N @ ( sCC_Bl1280885523602775798t_unit @ E @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E ) ) ) )
=> ( ( sCC_Bl8828226123343373779t_unit @ E6 )
!= nil_v ) ) ) ) ) ).
% dfs_S_hd_stack(1)
thf(fact_815_dfs__S__hd__stack_I2_J,axiom,
! [E: sCC_Bl1394983891496994913t_unit,V: v,E6: sCC_Bl1394983891496994913t_unit,N: v] :
( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
=> ( ( sCC_Bl8953792750115413617t_unit @ successors @ V @ E @ E6 )
=> ( ( ( sCC_Bl8828226123343373779t_unit @ E )
!= nil_v )
=> ( ( member_v @ N @ ( sCC_Bl1280885523602775798t_unit @ E @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E ) ) ) )
=> ( member_v @ N @ ( sCC_Bl1280885523602775798t_unit @ E6 @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E6 ) ) ) ) ) ) ) ) ).
% dfs_S_hd_stack(2)
thf(fact_816_post__dfs__def,axiom,
! [V: v,E: sCC_Bl1394983891496994913t_unit,E6: sCC_Bl1394983891496994913t_unit] :
( ( sCC_Bl8953792750115413617t_unit @ successors @ V @ E @ E6 )
= ( ( sCC_Bl9196236973127232072t_unit @ successors @ E6 )
& ( member_v @ V @ ( sCC_Bl4645233313691564917t_unit @ E6 ) )
& ( sCC_Bl5768913643336123637t_unit @ E @ E6 )
& ( ( sCC_Bl3795065053823578884t_unit @ E6 @ V )
= ( successors @ V ) )
& ! [X2: v] :
( ( member_v @ X2 @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ( sCC_Bl3795065053823578884t_unit @ E6 @ X2 )
= ( sCC_Bl3795065053823578884t_unit @ E @ X2 ) ) )
& ! [X2: v] :
( ( member_v @ X2 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E6 ) ) )
=> ( sCC_Bl649662514949026229able_v @ successors @ X2 @ V ) )
& ? [Ns: list_v] :
( ( sCC_Bl8828226123343373779t_unit @ E )
= ( append_v @ Ns @ ( sCC_Bl8828226123343373779t_unit @ E6 ) ) )
& ( ( ( member_v @ V @ ( sCC_Bl157864678168468314t_unit @ E6 ) )
& ( ( sCC_Bl8828226123343373779t_unit @ E6 )
= ( sCC_Bl8828226123343373779t_unit @ E ) )
& ! [X2: v] :
( ( member_v @ X2 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E6 ) ) )
=> ( ( sCC_Bl1280885523602775798t_unit @ E6 @ X2 )
= ( sCC_Bl1280885523602775798t_unit @ E @ X2 ) ) ) )
| ( ( ( sCC_Bl8828226123343373779t_unit @ E6 )
!= nil_v )
& ( member_v @ V @ ( sCC_Bl1280885523602775798t_unit @ E6 @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E6 ) ) ) )
& ! [X2: v] :
( ( member_v @ X2 @ ( set_v2 @ ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ E6 ) ) ) )
=> ( ( sCC_Bl1280885523602775798t_unit @ E6 @ X2 )
= ( sCC_Bl1280885523602775798t_unit @ E @ X2 ) ) ) ) )
& ( ( sCC_Bl9201514103433284750t_unit @ E6 )
= ( sCC_Bl9201514103433284750t_unit @ E ) ) ) ) ).
% post_dfs_def
thf(fact_817_post__dfss__def,axiom,
! [V: v,E: sCC_Bl1394983891496994913t_unit,E6: sCC_Bl1394983891496994913t_unit] :
( ( sCC_Bl6082031138996704384t_unit @ successors @ V @ E @ E6 )
= ( ( sCC_Bl9196236973127232072t_unit @ successors @ E6 )
& ( ( sCC_Bl3795065053823578884t_unit @ E6 @ V )
= ( successors @ V ) )
& ! [X2: v] :
( ( member_v @ X2 @ ( minus_minus_set_v @ ( sCC_Bl4645233313691564917t_unit @ E ) @ ( insert_v2 @ V @ bot_bot_set_v ) ) )
=> ( ( sCC_Bl3795065053823578884t_unit @ E6 @ X2 )
= ( sCC_Bl3795065053823578884t_unit @ E @ X2 ) ) )
& ( sCC_Bl5768913643336123637t_unit @ E @ E6 )
& ! [X2: v] :
( ( member_v @ X2 @ ( successors @ V ) )
=> ( member_v @ X2 @ ( sup_sup_set_v @ ( sCC_Bl157864678168468314t_unit @ E6 ) @ ( sCC_Bl1280885523602775798t_unit @ E6 @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E6 ) ) ) ) ) )
& ! [X2: v] :
( ( member_v @ X2 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E6 ) ) )
=> ( sCC_Bl649662514949026229able_v @ successors @ X2 @ V ) )
& ( ( sCC_Bl8828226123343373779t_unit @ E6 )
!= nil_v )
& ? [Ns: list_v] :
( ( sCC_Bl8828226123343373779t_unit @ E )
= ( append_v @ Ns @ ( sCC_Bl8828226123343373779t_unit @ E6 ) ) )
& ( member_v @ V @ ( sCC_Bl1280885523602775798t_unit @ E6 @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E6 ) ) ) )
& ! [X2: v] :
( ( member_v @ X2 @ ( set_v2 @ ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ E6 ) ) ) )
=> ( ( sCC_Bl1280885523602775798t_unit @ E6 @ X2 )
= ( sCC_Bl1280885523602775798t_unit @ E @ X2 ) ) )
& ( ( ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E6 ) )
= V )
=> ! [X2: v] :
( ( member_v @ X2 @ ( set_v2 @ ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ E6 ) ) ) )
=> ~ ( sCC_Bl649662514949026229able_v @ successors @ V @ X2 ) ) )
& ( ( sCC_Bl9201514103433284750t_unit @ E6 )
= ( sCC_Bl9201514103433284750t_unit @ E ) ) ) ) ).
% post_dfss_def
thf(fact_818_set__append,axiom,
! [Xs: list_v,Ys: list_v] :
( ( set_v2 @ ( append_v @ Xs @ Ys ) )
= ( sup_sup_set_v @ ( set_v2 @ Xs ) @ ( set_v2 @ Ys ) ) ) ).
% set_append
thf(fact_819_set__append,axiom,
! [Xs: list_P7986770385144383213od_v_v,Ys: list_P7986770385144383213od_v_v] :
( ( set_Product_prod_v_v2 @ ( append2138873909117096322od_v_v @ Xs @ Ys ) )
= ( sup_su414716646722978715od_v_v @ ( set_Product_prod_v_v2 @ Xs ) @ ( set_Product_prod_v_v2 @ Ys ) ) ) ).
% set_append
thf(fact_820_tl__append2,axiom,
! [Xs: list_v,Ys: list_v] :
( ( Xs != nil_v )
=> ( ( tl_v @ ( append_v @ Xs @ Ys ) )
= ( append_v @ ( tl_v @ Xs ) @ Ys ) ) ) ).
% tl_append2
thf(fact_821_list_Oset__sel_I1_J,axiom,
! [A: list_P7986770385144383213od_v_v] :
( ( A != nil_Product_prod_v_v )
=> ( member7453568604450474000od_v_v @ ( hd_Product_prod_v_v @ A ) @ ( set_Product_prod_v_v2 @ A ) ) ) ).
% list.set_sel(1)
thf(fact_822_list_Oset__sel_I1_J,axiom,
! [A: list_v] :
( ( A != nil_v )
=> ( member_v @ ( hd_v @ A ) @ ( set_v2 @ A ) ) ) ).
% list.set_sel(1)
thf(fact_823_hd__in__set,axiom,
! [Xs: list_P7986770385144383213od_v_v] :
( ( Xs != nil_Product_prod_v_v )
=> ( member7453568604450474000od_v_v @ ( hd_Product_prod_v_v @ Xs ) @ ( set_Product_prod_v_v2 @ Xs ) ) ) ).
% hd_in_set
thf(fact_824_hd__in__set,axiom,
! [Xs: list_v] :
( ( Xs != nil_v )
=> ( member_v @ ( hd_v @ Xs ) @ ( set_v2 @ Xs ) ) ) ).
% hd_in_set
thf(fact_825_tl__append__if,axiom,
! [Xs: list_v,Ys: list_v] :
( ( ( Xs = nil_v )
=> ( ( tl_v @ ( append_v @ Xs @ Ys ) )
= ( tl_v @ Ys ) ) )
& ( ( Xs != nil_v )
=> ( ( tl_v @ ( append_v @ Xs @ Ys ) )
= ( append_v @ ( tl_v @ Xs ) @ Ys ) ) ) ) ).
% tl_append_if
thf(fact_826_list_Oexpand,axiom,
! [List: list_v,List2: list_v] :
( ( ( List = nil_v )
= ( List2 = nil_v ) )
=> ( ( ( List != nil_v )
=> ( ( List2 != nil_v )
=> ( ( ( hd_v @ List )
= ( hd_v @ List2 ) )
& ( ( tl_v @ List )
= ( tl_v @ List2 ) ) ) ) )
=> ( List = List2 ) ) ) ).
% list.expand
thf(fact_827_graph_Odfs__S__hd__stack_I1_J,axiom,
! [Vertices: set_v,Successors: v > set_v,E: sCC_Bl1394983891496994913t_unit,V: v,E6: sCC_Bl1394983891496994913t_unit,N: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E )
=> ( ( sCC_Bl8953792750115413617t_unit @ Successors @ V @ E @ E6 )
=> ( ( ( sCC_Bl8828226123343373779t_unit @ E )
!= nil_v )
=> ( ( member_v @ N @ ( sCC_Bl1280885523602775798t_unit @ E @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E ) ) ) )
=> ( ( sCC_Bl8828226123343373779t_unit @ E6 )
!= nil_v ) ) ) ) ) ) ).
% graph.dfs_S_hd_stack(1)
thf(fact_828_graph_Odfs__S__hd__stack_I2_J,axiom,
! [Vertices: set_v,Successors: v > set_v,E: sCC_Bl1394983891496994913t_unit,V: v,E6: sCC_Bl1394983891496994913t_unit,N: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E )
=> ( ( sCC_Bl8953792750115413617t_unit @ Successors @ V @ E @ E6 )
=> ( ( ( sCC_Bl8828226123343373779t_unit @ E )
!= nil_v )
=> ( ( member_v @ N @ ( sCC_Bl1280885523602775798t_unit @ E @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E ) ) ) )
=> ( member_v @ N @ ( sCC_Bl1280885523602775798t_unit @ E6 @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E6 ) ) ) ) ) ) ) ) ) ).
% graph.dfs_S_hd_stack(2)
thf(fact_829_pre__dfss__def,axiom,
! [V: v,E: sCC_Bl1394983891496994913t_unit] :
( ( sCC_Bl1748261141445803503t_unit @ successors @ V @ E )
= ( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
& ( member_v @ V @ ( sCC_Bl4645233313691564917t_unit @ E ) )
& ( ( sCC_Bl8828226123343373779t_unit @ E )
!= nil_v )
& ( member_v @ V @ ( sCC_Bl1280885523602775798t_unit @ E @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E ) ) ) )
& ! [X2: v] :
( ( member_v @ X2 @ ( sCC_Bl3795065053823578884t_unit @ E @ V ) )
=> ( member_v @ X2 @ ( sup_sup_set_v @ ( sCC_Bl157864678168468314t_unit @ E ) @ ( sCC_Bl1280885523602775798t_unit @ E @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E ) ) ) ) ) )
& ! [X2: v] :
( ( member_v @ X2 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) )
=> ( sCC_Bl649662514949026229able_v @ successors @ X2 @ V ) )
& ? [Ns: list_v] :
( ( sCC_Bl9201514103433284750t_unit @ E )
= ( cons_v @ V @ Ns ) ) ) ) ).
% pre_dfss_def
thf(fact_830_graph_Opre__dfss__def,axiom,
! [Vertices: set_v,Successors: v > set_v,V: v,E: sCC_Bl1394983891496994913t_unit] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl1748261141445803503t_unit @ Successors @ V @ E )
= ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E )
& ( member_v @ V @ ( sCC_Bl4645233313691564917t_unit @ E ) )
& ( ( sCC_Bl8828226123343373779t_unit @ E )
!= nil_v )
& ( member_v @ V @ ( sCC_Bl1280885523602775798t_unit @ E @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E ) ) ) )
& ! [X2: v] :
( ( member_v @ X2 @ ( sCC_Bl3795065053823578884t_unit @ E @ V ) )
=> ( member_v @ X2 @ ( sup_sup_set_v @ ( sCC_Bl157864678168468314t_unit @ E ) @ ( sCC_Bl1280885523602775798t_unit @ E @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E ) ) ) ) ) )
& ! [X2: v] :
( ( member_v @ X2 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) )
=> ( sCC_Bl649662514949026229able_v @ Successors @ X2 @ V ) )
& ? [Ns: list_v] :
( ( sCC_Bl9201514103433284750t_unit @ E )
= ( cons_v @ V @ Ns ) ) ) ) ) ).
% graph.pre_dfss_def
thf(fact_831_equality,axiom,
! [R: sCC_Bl1394983891496994913t_unit,R3: sCC_Bl1394983891496994913t_unit] :
( ( ( sCC_Bl1090238580953940555t_unit @ R )
= ( sCC_Bl1090238580953940555t_unit @ R3 ) )
=> ( ( ( sCC_Bl1280885523602775798t_unit @ R )
= ( sCC_Bl1280885523602775798t_unit @ R3 ) )
=> ( ( ( sCC_Bl157864678168468314t_unit @ R )
= ( sCC_Bl157864678168468314t_unit @ R3 ) )
=> ( ( ( sCC_Bl4645233313691564917t_unit @ R )
= ( sCC_Bl4645233313691564917t_unit @ R3 ) )
=> ( ( ( sCC_Bl3795065053823578884t_unit @ R )
= ( sCC_Bl3795065053823578884t_unit @ R3 ) )
=> ( ( ( sCC_Bl2536197123907397897t_unit @ R )
= ( sCC_Bl2536197123907397897t_unit @ R3 ) )
=> ( ( ( sCC_Bl8828226123343373779t_unit @ R )
= ( sCC_Bl8828226123343373779t_unit @ R3 ) )
=> ( ( ( sCC_Bl9201514103433284750t_unit @ R )
= ( sCC_Bl9201514103433284750t_unit @ R3 ) )
=> ( ( ( sCC_Bl3567736435408124606t_unit @ R )
= ( sCC_Bl3567736435408124606t_unit @ R3 ) )
=> ( R = R3 ) ) ) ) ) ) ) ) ) ) ).
% equality
thf(fact_832_list_Osimps_I15_J,axiom,
! [X21: product_prod_v_v,X222: list_P7986770385144383213od_v_v] :
( ( set_Product_prod_v_v2 @ ( cons_P4120604216776828829od_v_v @ X21 @ X222 ) )
= ( insert1338601472111419319od_v_v @ X21 @ ( set_Product_prod_v_v2 @ X222 ) ) ) ).
% list.simps(15)
thf(fact_833_list_Osimps_I15_J,axiom,
! [X21: v,X222: list_v] :
( ( set_v2 @ ( cons_v @ X21 @ X222 ) )
= ( insert_v2 @ X21 @ ( set_v2 @ X222 ) ) ) ).
% list.simps(15)
thf(fact_834_list_Ocollapse,axiom,
! [List: list_v] :
( ( List != nil_v )
=> ( ( cons_v @ ( hd_v @ List ) @ ( tl_v @ List ) )
= List ) ) ).
% list.collapse
thf(fact_835_hd__Cons__tl,axiom,
! [Xs: list_v] :
( ( Xs != nil_v )
=> ( ( cons_v @ ( hd_v @ Xs ) @ ( tl_v @ Xs ) )
= Xs ) ) ).
% hd_Cons_tl
thf(fact_836_list_Oset__intros_I2_J,axiom,
! [Y2: product_prod_v_v,X222: list_P7986770385144383213od_v_v,X21: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ Y2 @ ( set_Product_prod_v_v2 @ X222 ) )
=> ( member7453568604450474000od_v_v @ Y2 @ ( set_Product_prod_v_v2 @ ( cons_P4120604216776828829od_v_v @ X21 @ X222 ) ) ) ) ).
% list.set_intros(2)
thf(fact_837_list_Oset__intros_I2_J,axiom,
! [Y2: v,X222: list_v,X21: v] :
( ( member_v @ Y2 @ ( set_v2 @ X222 ) )
=> ( member_v @ Y2 @ ( set_v2 @ ( cons_v @ X21 @ X222 ) ) ) ) ).
% list.set_intros(2)
thf(fact_838_list_Oset__intros_I1_J,axiom,
! [X21: product_prod_v_v,X222: list_P7986770385144383213od_v_v] : ( member7453568604450474000od_v_v @ X21 @ ( set_Product_prod_v_v2 @ ( cons_P4120604216776828829od_v_v @ X21 @ X222 ) ) ) ).
% list.set_intros(1)
thf(fact_839_list_Oset__intros_I1_J,axiom,
! [X21: v,X222: list_v] : ( member_v @ X21 @ ( set_v2 @ ( cons_v @ X21 @ X222 ) ) ) ).
% list.set_intros(1)
thf(fact_840_list_Oset__cases,axiom,
! [E: product_prod_v_v,A: list_P7986770385144383213od_v_v] :
( ( member7453568604450474000od_v_v @ E @ ( set_Product_prod_v_v2 @ A ) )
=> ( ! [Z22: list_P7986770385144383213od_v_v] :
( A
!= ( cons_P4120604216776828829od_v_v @ E @ Z22 ) )
=> ~ ! [Z1: product_prod_v_v,Z22: list_P7986770385144383213od_v_v] :
( ( A
= ( cons_P4120604216776828829od_v_v @ Z1 @ Z22 ) )
=> ~ ( member7453568604450474000od_v_v @ E @ ( set_Product_prod_v_v2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_841_list_Oset__cases,axiom,
! [E: v,A: list_v] :
( ( member_v @ E @ ( set_v2 @ A ) )
=> ( ! [Z22: list_v] :
( A
!= ( cons_v @ E @ Z22 ) )
=> ~ ! [Z1: v,Z22: list_v] :
( ( A
= ( cons_v @ Z1 @ Z22 ) )
=> ~ ( member_v @ E @ ( set_v2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_842_set__ConsD,axiom,
! [Y2: product_prod_v_v,X: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ( member7453568604450474000od_v_v @ Y2 @ ( set_Product_prod_v_v2 @ ( cons_P4120604216776828829od_v_v @ X @ Xs ) ) )
=> ( ( Y2 = X )
| ( member7453568604450474000od_v_v @ Y2 @ ( set_Product_prod_v_v2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_843_set__ConsD,axiom,
! [Y2: v,X: v,Xs: list_v] :
( ( member_v @ Y2 @ ( set_v2 @ ( cons_v @ X @ Xs ) ) )
=> ( ( Y2 = X )
| ( member_v @ Y2 @ ( set_v2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_844_list_Osel_I3_J,axiom,
! [X21: v,X222: list_v] :
( ( tl_v @ ( cons_v @ X21 @ X222 ) )
= X222 ) ).
% list.sel(3)
thf(fact_845_set__subset__Cons,axiom,
! [Xs: list_v,X: v] : ( ord_less_eq_set_v @ ( set_v2 @ Xs ) @ ( set_v2 @ ( cons_v @ X @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_846_set__subset__Cons,axiom,
! [Xs: list_P7986770385144383213od_v_v,X: product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( set_Product_prod_v_v2 @ Xs ) @ ( set_Product_prod_v_v2 @ ( cons_P4120604216776828829od_v_v @ X @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_847_split__list__first__prop__iff,axiom,
! [Xs: list_v,P: v > $o] :
( ( ? [X2: v] :
( ( member_v @ X2 @ ( set_v2 @ Xs ) )
& ( P @ X2 ) ) )
= ( ? [Ys2: list_v,X2: v] :
( ? [Zs: list_v] :
( Xs
= ( append_v @ Ys2 @ ( cons_v @ X2 @ Zs ) ) )
& ( P @ X2 )
& ! [Y3: v] :
( ( member_v @ Y3 @ ( set_v2 @ Ys2 ) )
=> ~ ( P @ Y3 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_848_split__list__last__prop__iff,axiom,
! [Xs: list_v,P: v > $o] :
( ( ? [X2: v] :
( ( member_v @ X2 @ ( set_v2 @ Xs ) )
& ( P @ X2 ) ) )
= ( ? [Ys2: list_v,X2: v,Zs: list_v] :
( ( Xs
= ( append_v @ Ys2 @ ( cons_v @ X2 @ Zs ) ) )
& ( P @ X2 )
& ! [Y3: v] :
( ( member_v @ Y3 @ ( set_v2 @ Zs ) )
=> ~ ( P @ Y3 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_849_in__set__conv__decomp__first,axiom,
! [X: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ Xs ) )
= ( ? [Ys2: list_P7986770385144383213od_v_v,Zs: list_P7986770385144383213od_v_v] :
( ( Xs
= ( append2138873909117096322od_v_v @ Ys2 @ ( cons_P4120604216776828829od_v_v @ X @ Zs ) ) )
& ~ ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ Ys2 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_850_in__set__conv__decomp__first,axiom,
! [X: v,Xs: list_v] :
( ( member_v @ X @ ( set_v2 @ Xs ) )
= ( ? [Ys2: list_v,Zs: list_v] :
( ( Xs
= ( append_v @ Ys2 @ ( cons_v @ X @ Zs ) ) )
& ~ ( member_v @ X @ ( set_v2 @ Ys2 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_851_in__set__conv__decomp__last,axiom,
! [X: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ Xs ) )
= ( ? [Ys2: list_P7986770385144383213od_v_v,Zs: list_P7986770385144383213od_v_v] :
( ( Xs
= ( append2138873909117096322od_v_v @ Ys2 @ ( cons_P4120604216776828829od_v_v @ X @ Zs ) ) )
& ~ ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ Zs ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_852_in__set__conv__decomp__last,axiom,
! [X: v,Xs: list_v] :
( ( member_v @ X @ ( set_v2 @ Xs ) )
= ( ? [Ys2: list_v,Zs: list_v] :
( ( Xs
= ( append_v @ Ys2 @ ( cons_v @ X @ Zs ) ) )
& ~ ( member_v @ X @ ( set_v2 @ Zs ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_853_split__list__first__propE,axiom,
! [Xs: list_v,P: v > $o] :
( ? [X4: v] :
( ( member_v @ X4 @ ( set_v2 @ Xs ) )
& ( P @ X4 ) )
=> ~ ! [Ys3: list_v,X3: v] :
( ? [Zs2: list_v] :
( Xs
= ( append_v @ Ys3 @ ( cons_v @ X3 @ Zs2 ) ) )
=> ( ( P @ X3 )
=> ~ ! [Xa2: v] :
( ( member_v @ Xa2 @ ( set_v2 @ Ys3 ) )
=> ~ ( P @ Xa2 ) ) ) ) ) ).
% split_list_first_propE
thf(fact_854_split__list__last__propE,axiom,
! [Xs: list_v,P: v > $o] :
( ? [X4: v] :
( ( member_v @ X4 @ ( set_v2 @ Xs ) )
& ( P @ X4 ) )
=> ~ ! [Ys3: list_v,X3: v,Zs2: list_v] :
( ( Xs
= ( append_v @ Ys3 @ ( cons_v @ X3 @ Zs2 ) ) )
=> ( ( P @ X3 )
=> ~ ! [Xa2: v] :
( ( member_v @ Xa2 @ ( set_v2 @ Zs2 ) )
=> ~ ( P @ Xa2 ) ) ) ) ) ).
% split_list_last_propE
thf(fact_855_split__list__first__prop,axiom,
! [Xs: list_v,P: v > $o] :
( ? [X4: v] :
( ( member_v @ X4 @ ( set_v2 @ Xs ) )
& ( P @ X4 ) )
=> ? [Ys3: list_v,X3: v] :
( ? [Zs2: list_v] :
( Xs
= ( append_v @ Ys3 @ ( cons_v @ X3 @ Zs2 ) ) )
& ( P @ X3 )
& ! [Xa2: v] :
( ( member_v @ Xa2 @ ( set_v2 @ Ys3 ) )
=> ~ ( P @ Xa2 ) ) ) ) ).
% split_list_first_prop
thf(fact_856_split__list__last__prop,axiom,
! [Xs: list_v,P: v > $o] :
( ? [X4: v] :
( ( member_v @ X4 @ ( set_v2 @ Xs ) )
& ( P @ X4 ) )
=> ? [Ys3: list_v,X3: v,Zs2: list_v] :
( ( Xs
= ( append_v @ Ys3 @ ( cons_v @ X3 @ Zs2 ) ) )
& ( P @ X3 )
& ! [Xa2: v] :
( ( member_v @ Xa2 @ ( set_v2 @ Zs2 ) )
=> ~ ( P @ Xa2 ) ) ) ) ).
% split_list_last_prop
thf(fact_857_in__set__conv__decomp,axiom,
! [X: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ Xs ) )
= ( ? [Ys2: list_P7986770385144383213od_v_v,Zs: list_P7986770385144383213od_v_v] :
( Xs
= ( append2138873909117096322od_v_v @ Ys2 @ ( cons_P4120604216776828829od_v_v @ X @ Zs ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_858_in__set__conv__decomp,axiom,
! [X: v,Xs: list_v] :
( ( member_v @ X @ ( set_v2 @ Xs ) )
= ( ? [Ys2: list_v,Zs: list_v] :
( Xs
= ( append_v @ Ys2 @ ( cons_v @ X @ Zs ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_859_append__Cons__eq__iff,axiom,
! [X: product_prod_v_v,Xs: list_P7986770385144383213od_v_v,Ys: list_P7986770385144383213od_v_v,Xs3: list_P7986770385144383213od_v_v,Ys4: list_P7986770385144383213od_v_v] :
( ~ ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ Xs ) )
=> ( ~ ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ Ys ) )
=> ( ( ( append2138873909117096322od_v_v @ Xs @ ( cons_P4120604216776828829od_v_v @ X @ Ys ) )
= ( append2138873909117096322od_v_v @ Xs3 @ ( cons_P4120604216776828829od_v_v @ X @ Ys4 ) ) )
= ( ( Xs = Xs3 )
& ( Ys = Ys4 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_860_append__Cons__eq__iff,axiom,
! [X: v,Xs: list_v,Ys: list_v,Xs3: list_v,Ys4: list_v] :
( ~ ( member_v @ X @ ( set_v2 @ Xs ) )
=> ( ~ ( member_v @ X @ ( set_v2 @ Ys ) )
=> ( ( ( append_v @ Xs @ ( cons_v @ X @ Ys ) )
= ( append_v @ Xs3 @ ( cons_v @ X @ Ys4 ) ) )
= ( ( Xs = Xs3 )
& ( Ys = Ys4 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_861_split__list__propE,axiom,
! [Xs: list_v,P: v > $o] :
( ? [X4: v] :
( ( member_v @ X4 @ ( set_v2 @ Xs ) )
& ( P @ X4 ) )
=> ~ ! [Ys3: list_v,X3: v] :
( ? [Zs2: list_v] :
( Xs
= ( append_v @ Ys3 @ ( cons_v @ X3 @ Zs2 ) ) )
=> ~ ( P @ X3 ) ) ) ).
% split_list_propE
thf(fact_862_split__list__first,axiom,
! [X: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ Xs ) )
=> ? [Ys3: list_P7986770385144383213od_v_v,Zs2: list_P7986770385144383213od_v_v] :
( ( Xs
= ( append2138873909117096322od_v_v @ Ys3 @ ( cons_P4120604216776828829od_v_v @ X @ Zs2 ) ) )
& ~ ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ Ys3 ) ) ) ) ).
% split_list_first
thf(fact_863_split__list__first,axiom,
! [X: v,Xs: list_v] :
( ( member_v @ X @ ( set_v2 @ Xs ) )
=> ? [Ys3: list_v,Zs2: list_v] :
( ( Xs
= ( append_v @ Ys3 @ ( cons_v @ X @ Zs2 ) ) )
& ~ ( member_v @ X @ ( set_v2 @ Ys3 ) ) ) ) ).
% split_list_first
thf(fact_864_split__list__prop,axiom,
! [Xs: list_v,P: v > $o] :
( ? [X4: v] :
( ( member_v @ X4 @ ( set_v2 @ Xs ) )
& ( P @ X4 ) )
=> ? [Ys3: list_v,X3: v] :
( ? [Zs2: list_v] :
( Xs
= ( append_v @ Ys3 @ ( cons_v @ X3 @ Zs2 ) ) )
& ( P @ X3 ) ) ) ).
% split_list_prop
thf(fact_865_split__list__last,axiom,
! [X: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ Xs ) )
=> ? [Ys3: list_P7986770385144383213od_v_v,Zs2: list_P7986770385144383213od_v_v] :
( ( Xs
= ( append2138873909117096322od_v_v @ Ys3 @ ( cons_P4120604216776828829od_v_v @ X @ Zs2 ) ) )
& ~ ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ Zs2 ) ) ) ) ).
% split_list_last
thf(fact_866_split__list__last,axiom,
! [X: v,Xs: list_v] :
( ( member_v @ X @ ( set_v2 @ Xs ) )
=> ? [Ys3: list_v,Zs2: list_v] :
( ( Xs
= ( append_v @ Ys3 @ ( cons_v @ X @ Zs2 ) ) )
& ~ ( member_v @ X @ ( set_v2 @ Zs2 ) ) ) ) ).
% split_list_last
thf(fact_867_split__list,axiom,
! [X: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ Xs ) )
=> ? [Ys3: list_P7986770385144383213od_v_v,Zs2: list_P7986770385144383213od_v_v] :
( Xs
= ( append2138873909117096322od_v_v @ Ys3 @ ( cons_P4120604216776828829od_v_v @ X @ Zs2 ) ) ) ) ).
% split_list
thf(fact_868_split__list,axiom,
! [X: v,Xs: list_v] :
( ( member_v @ X @ ( set_v2 @ Xs ) )
=> ? [Ys3: list_v,Zs2: list_v] :
( Xs
= ( append_v @ Ys3 @ ( cons_v @ X @ Zs2 ) ) ) ) ).
% split_list
thf(fact_869_tl__Nil,axiom,
! [Xs: list_v] :
( ( ( tl_v @ Xs )
= nil_v )
= ( ( Xs = nil_v )
| ? [X2: v] :
( Xs
= ( cons_v @ X2 @ nil_v ) ) ) ) ).
% tl_Nil
thf(fact_870_Nil__tl,axiom,
! [Xs: list_v] :
( ( nil_v
= ( tl_v @ Xs ) )
= ( ( Xs = nil_v )
| ? [X2: v] :
( Xs
= ( cons_v @ X2 @ nil_v ) ) ) ) ).
% Nil_tl
thf(fact_871_list_Oexhaust__sel,axiom,
! [List: list_v] :
( ( List != nil_v )
=> ( List
= ( cons_v @ ( hd_v @ List ) @ ( tl_v @ List ) ) ) ) ).
% list.exhaust_sel
thf(fact_872_the__elem__set,axiom,
! [X: v] :
( ( the_elem_v @ ( set_v2 @ ( cons_v @ X @ nil_v ) ) )
= X ) ).
% the_elem_set
thf(fact_873_surjective,axiom,
! [R: sCC_Bl1394983891496994913t_unit] :
( R
= ( sCC_Bl8064756265740546429t_unit @ ( sCC_Bl1090238580953940555t_unit @ R ) @ ( sCC_Bl1280885523602775798t_unit @ R ) @ ( sCC_Bl157864678168468314t_unit @ R ) @ ( sCC_Bl4645233313691564917t_unit @ R ) @ ( sCC_Bl3795065053823578884t_unit @ R ) @ ( sCC_Bl2536197123907397897t_unit @ R ) @ ( sCC_Bl8828226123343373779t_unit @ R ) @ ( sCC_Bl9201514103433284750t_unit @ R ) @ ( sCC_Bl3567736435408124606t_unit @ R ) ) ) ).
% surjective
thf(fact_874_rotate1__hd__tl,axiom,
! [Xs: list_v] :
( ( Xs != nil_v )
=> ( ( rotate1_v @ Xs )
= ( append_v @ ( tl_v @ Xs ) @ ( cons_v @ ( hd_v @ Xs ) @ nil_v ) ) ) ) ).
% rotate1_hd_tl
thf(fact_875_set__rotate1,axiom,
! [Xs: list_v] :
( ( set_v2 @ ( rotate1_v @ Xs ) )
= ( set_v2 @ Xs ) ) ).
% set_rotate1
thf(fact_876_select__convs_I5_J,axiom,
! [Root: v,S6: v > set_v,Explored: set_v,Visited: set_v,Vsuccs: v > set_v,Sccs: set_set_v,Stack: list_v,Cstack: list_v,More: product_unit] :
( ( sCC_Bl3795065053823578884t_unit @ ( sCC_Bl8064756265740546429t_unit @ Root @ S6 @ Explored @ Visited @ Vsuccs @ Sccs @ Stack @ Cstack @ More ) )
= Vsuccs ) ).
% select_convs(5)
thf(fact_877_select__convs_I3_J,axiom,
! [Root: v,S6: v > set_v,Explored: set_v,Visited: set_v,Vsuccs: v > set_v,Sccs: set_set_v,Stack: list_v,Cstack: list_v,More: product_unit] :
( ( sCC_Bl157864678168468314t_unit @ ( sCC_Bl8064756265740546429t_unit @ Root @ S6 @ Explored @ Visited @ Vsuccs @ Sccs @ Stack @ Cstack @ More ) )
= Explored ) ).
% select_convs(3)
thf(fact_878_select__convs_I2_J,axiom,
! [Root: v,S6: v > set_v,Explored: set_v,Visited: set_v,Vsuccs: v > set_v,Sccs: set_set_v,Stack: list_v,Cstack: list_v,More: product_unit] :
( ( sCC_Bl1280885523602775798t_unit @ ( sCC_Bl8064756265740546429t_unit @ Root @ S6 @ Explored @ Visited @ Vsuccs @ Sccs @ Stack @ Cstack @ More ) )
= S6 ) ).
% select_convs(2)
thf(fact_879_select__convs_I7_J,axiom,
! [Root: v,S6: v > set_v,Explored: set_v,Visited: set_v,Vsuccs: v > set_v,Sccs: set_set_v,Stack: list_v,Cstack: list_v,More: product_unit] :
( ( sCC_Bl8828226123343373779t_unit @ ( sCC_Bl8064756265740546429t_unit @ Root @ S6 @ Explored @ Visited @ Vsuccs @ Sccs @ Stack @ Cstack @ More ) )
= Stack ) ).
% select_convs(7)
thf(fact_880_select__convs_I4_J,axiom,
! [Root: v,S6: v > set_v,Explored: set_v,Visited: set_v,Vsuccs: v > set_v,Sccs: set_set_v,Stack: list_v,Cstack: list_v,More: product_unit] :
( ( sCC_Bl4645233313691564917t_unit @ ( sCC_Bl8064756265740546429t_unit @ Root @ S6 @ Explored @ Visited @ Vsuccs @ Sccs @ Stack @ Cstack @ More ) )
= Visited ) ).
% select_convs(4)
thf(fact_881_Field__insert,axiom,
! [A: product_prod_v_v,B2: product_prod_v_v,R: set_Pr2149350503807050951od_v_v] :
( ( field_7153129647634986036od_v_v @ ( insert5641704497130386615od_v_v @ ( produc4031800376763917143od_v_v @ A @ B2 ) @ R ) )
= ( sup_su414716646722978715od_v_v @ ( insert1338601472111419319od_v_v @ A @ ( insert1338601472111419319od_v_v @ B2 @ bot_bo723834152578015283od_v_v ) ) @ ( field_7153129647634986036od_v_v @ R ) ) ) ).
% Field_insert
thf(fact_882_Field__insert,axiom,
! [A: v,B2: v,R: set_Product_prod_v_v] :
( ( field_v @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ A @ B2 ) @ R ) )
= ( sup_sup_set_v @ ( insert_v2 @ A @ ( insert_v2 @ B2 @ bot_bot_set_v ) ) @ ( field_v @ R ) ) ) ).
% Field_insert
thf(fact_883_remove__def,axiom,
( remove5001965847480235980od_v_v
= ( ^ [X2: product_prod_v_v,A4: set_Product_prod_v_v] : ( minus_4183494784930505774od_v_v @ A4 @ ( insert1338601472111419319od_v_v @ X2 @ bot_bo723834152578015283od_v_v ) ) ) ) ).
% remove_def
thf(fact_884_remove__def,axiom,
( remove_v
= ( ^ [X2: v,A4: set_v] : ( minus_minus_set_v @ A4 @ ( insert_v2 @ X2 @ bot_bot_set_v ) ) ) ) ).
% remove_def
thf(fact_885_Sup__fin_Oremove,axiom,
! [A3: set_se8455005133513928103od_v_v,X: set_Product_prod_v_v] :
( ( finite6084192165098772208od_v_v @ A3 )
=> ( ( member8406446414694345712od_v_v @ X @ A3 )
=> ( ( ( ( minus_7679383599658060814od_v_v @ A3 @ ( insert7504383016908236695od_v_v @ X @ bot_bo3497076220358800403od_v_v ) )
= bot_bo3497076220358800403od_v_v )
=> ( ( lattic5151207300795964030od_v_v @ A3 )
= X ) )
& ( ( ( minus_7679383599658060814od_v_v @ A3 @ ( insert7504383016908236695od_v_v @ X @ bot_bo3497076220358800403od_v_v ) )
!= bot_bo3497076220358800403od_v_v )
=> ( ( lattic5151207300795964030od_v_v @ A3 )
= ( sup_su414716646722978715od_v_v @ X @ ( lattic5151207300795964030od_v_v @ ( minus_7679383599658060814od_v_v @ A3 @ ( insert7504383016908236695od_v_v @ X @ bot_bo3497076220358800403od_v_v ) ) ) ) ) ) ) ) ) ).
% Sup_fin.remove
thf(fact_886_member__remove,axiom,
! [X: v,Y2: v,A3: set_v] :
( ( member_v @ X @ ( remove_v @ Y2 @ A3 ) )
= ( ( member_v @ X @ A3 )
& ( X != Y2 ) ) ) ).
% member_remove
thf(fact_887_member__remove,axiom,
! [X: product_prod_v_v,Y2: product_prod_v_v,A3: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X @ ( remove5001965847480235980od_v_v @ Y2 @ A3 ) )
= ( ( member7453568604450474000od_v_v @ X @ A3 )
& ( X != Y2 ) ) ) ).
% member_remove
thf(fact_888_Field__empty,axiom,
( ( field_7153129647634986036od_v_v @ bot_bo3282589961317712691od_v_v )
= bot_bo723834152578015283od_v_v ) ).
% Field_empty
thf(fact_889_Field__empty,axiom,
( ( field_v @ bot_bo723834152578015283od_v_v )
= bot_bot_set_v ) ).
% Field_empty
thf(fact_890_inf__Sup__absorb,axiom,
! [A3: set_set_v,A: set_v] :
( ( finite_finite_set_v @ A3 )
=> ( ( member_set_v @ A @ A3 )
=> ( ( inf_inf_set_v @ A @ ( lattic2918178447194608042_set_v @ A3 ) )
= A ) ) ) ).
% inf_Sup_absorb
thf(fact_891_Sup__fin_Oinsert,axiom,
! [A3: set_se8455005133513928103od_v_v,X: set_Product_prod_v_v] :
( ( finite6084192165098772208od_v_v @ A3 )
=> ( ( A3 != bot_bo3497076220358800403od_v_v )
=> ( ( lattic5151207300795964030od_v_v @ ( insert7504383016908236695od_v_v @ X @ A3 ) )
= ( sup_su414716646722978715od_v_v @ X @ ( lattic5151207300795964030od_v_v @ A3 ) ) ) ) ) ).
% Sup_fin.insert
thf(fact_892_FieldI2,axiom,
! [I: product_prod_v_v,J: product_prod_v_v,R2: set_Pr2149350503807050951od_v_v] :
( ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ I @ J ) @ R2 )
=> ( member7453568604450474000od_v_v @ J @ ( field_7153129647634986036od_v_v @ R2 ) ) ) ).
% FieldI2
thf(fact_893_FieldI2,axiom,
! [I: v,J: v,R2: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ I @ J ) @ R2 )
=> ( member_v @ J @ ( field_v @ R2 ) ) ) ).
% FieldI2
thf(fact_894_FieldI1,axiom,
! [I: product_prod_v_v,J: product_prod_v_v,R2: set_Pr2149350503807050951od_v_v] :
( ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ I @ J ) @ R2 )
=> ( member7453568604450474000od_v_v @ I @ ( field_7153129647634986036od_v_v @ R2 ) ) ) ).
% FieldI1
thf(fact_895_FieldI1,axiom,
! [I: v,J: v,R2: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ I @ J ) @ R2 )
=> ( member_v @ I @ ( field_v @ R2 ) ) ) ).
% FieldI1
thf(fact_896_mono__Field,axiom,
! [R: set_Pr2149350503807050951od_v_v,S4: set_Pr2149350503807050951od_v_v] :
( ( ord_le6241436655786843239od_v_v @ R @ S4 )
=> ( ord_le7336532860387713383od_v_v @ ( field_7153129647634986036od_v_v @ R ) @ ( field_7153129647634986036od_v_v @ S4 ) ) ) ).
% mono_Field
thf(fact_897_mono__Field,axiom,
! [R: set_Product_prod_v_v,S4: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ R @ S4 )
=> ( ord_less_eq_set_v @ ( field_v @ R ) @ ( field_v @ S4 ) ) ) ).
% mono_Field
thf(fact_898_Sup__fin_OcoboundedI,axiom,
! [A3: set_set_v,A: set_v] :
( ( finite_finite_set_v @ A3 )
=> ( ( member_set_v @ A @ A3 )
=> ( ord_less_eq_set_v @ A @ ( lattic2918178447194608042_set_v @ A3 ) ) ) ) ).
% Sup_fin.coboundedI
thf(fact_899_Sup__fin_OcoboundedI,axiom,
! [A3: set_se8455005133513928103od_v_v,A: set_Product_prod_v_v] :
( ( finite6084192165098772208od_v_v @ A3 )
=> ( ( member8406446414694345712od_v_v @ A @ A3 )
=> ( ord_le7336532860387713383od_v_v @ A @ ( lattic5151207300795964030od_v_v @ A3 ) ) ) ) ).
% Sup_fin.coboundedI
thf(fact_900_Sup__fin_Obounded__iff,axiom,
! [A3: set_set_v,X: set_v] :
( ( finite_finite_set_v @ A3 )
=> ( ( A3 != bot_bot_set_set_v )
=> ( ( ord_less_eq_set_v @ ( lattic2918178447194608042_set_v @ A3 ) @ X )
= ( ! [X2: set_v] :
( ( member_set_v @ X2 @ A3 )
=> ( ord_less_eq_set_v @ X2 @ X ) ) ) ) ) ) ).
% Sup_fin.bounded_iff
thf(fact_901_Sup__fin_Obounded__iff,axiom,
! [A3: set_se8455005133513928103od_v_v,X: set_Product_prod_v_v] :
( ( finite6084192165098772208od_v_v @ A3 )
=> ( ( A3 != bot_bo3497076220358800403od_v_v )
=> ( ( ord_le7336532860387713383od_v_v @ ( lattic5151207300795964030od_v_v @ A3 ) @ X )
= ( ! [X2: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X2 @ A3 )
=> ( ord_le7336532860387713383od_v_v @ X2 @ X ) ) ) ) ) ) ).
% Sup_fin.bounded_iff
thf(fact_902_Sup__fin_OboundedI,axiom,
! [A3: set_set_v,X: set_v] :
( ( finite_finite_set_v @ A3 )
=> ( ( A3 != bot_bot_set_set_v )
=> ( ! [A7: set_v] :
( ( member_set_v @ A7 @ A3 )
=> ( ord_less_eq_set_v @ A7 @ X ) )
=> ( ord_less_eq_set_v @ ( lattic2918178447194608042_set_v @ A3 ) @ X ) ) ) ) ).
% Sup_fin.boundedI
thf(fact_903_Sup__fin_OboundedI,axiom,
! [A3: set_se8455005133513928103od_v_v,X: set_Product_prod_v_v] :
( ( finite6084192165098772208od_v_v @ A3 )
=> ( ( A3 != bot_bo3497076220358800403od_v_v )
=> ( ! [A7: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ A7 @ A3 )
=> ( ord_le7336532860387713383od_v_v @ A7 @ X ) )
=> ( ord_le7336532860387713383od_v_v @ ( lattic5151207300795964030od_v_v @ A3 ) @ X ) ) ) ) ).
% Sup_fin.boundedI
thf(fact_904_Sup__fin_OboundedE,axiom,
! [A3: set_set_v,X: set_v] :
( ( finite_finite_set_v @ A3 )
=> ( ( A3 != bot_bot_set_set_v )
=> ( ( ord_less_eq_set_v @ ( lattic2918178447194608042_set_v @ A3 ) @ X )
=> ! [A10: set_v] :
( ( member_set_v @ A10 @ A3 )
=> ( ord_less_eq_set_v @ A10 @ X ) ) ) ) ) ).
% Sup_fin.boundedE
thf(fact_905_Sup__fin_OboundedE,axiom,
! [A3: set_se8455005133513928103od_v_v,X: set_Product_prod_v_v] :
( ( finite6084192165098772208od_v_v @ A3 )
=> ( ( A3 != bot_bo3497076220358800403od_v_v )
=> ( ( ord_le7336532860387713383od_v_v @ ( lattic5151207300795964030od_v_v @ A3 ) @ X )
=> ! [A10: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ A10 @ A3 )
=> ( ord_le7336532860387713383od_v_v @ A10 @ X ) ) ) ) ) ).
% Sup_fin.boundedE
thf(fact_906_Sup__fin_Osubset__imp,axiom,
! [A3: set_set_v,B: set_set_v] :
( ( ord_le5216385588623774835_set_v @ A3 @ B )
=> ( ( A3 != bot_bot_set_set_v )
=> ( ( finite_finite_set_v @ B )
=> ( ord_less_eq_set_v @ ( lattic2918178447194608042_set_v @ A3 ) @ ( lattic2918178447194608042_set_v @ B ) ) ) ) ) ).
% Sup_fin.subset_imp
thf(fact_907_Sup__fin_Osubset__imp,axiom,
! [A3: set_se8455005133513928103od_v_v,B: set_se8455005133513928103od_v_v] :
( ( ord_le4714265922333009223od_v_v @ A3 @ B )
=> ( ( A3 != bot_bo3497076220358800403od_v_v )
=> ( ( finite6084192165098772208od_v_v @ B )
=> ( ord_le7336532860387713383od_v_v @ ( lattic5151207300795964030od_v_v @ A3 ) @ ( lattic5151207300795964030od_v_v @ B ) ) ) ) ) ).
% Sup_fin.subset_imp
thf(fact_908_Sup__fin_Osubset,axiom,
! [A3: set_se8455005133513928103od_v_v,B: set_se8455005133513928103od_v_v] :
( ( finite6084192165098772208od_v_v @ A3 )
=> ( ( B != bot_bo3497076220358800403od_v_v )
=> ( ( ord_le4714265922333009223od_v_v @ B @ A3 )
=> ( ( sup_su414716646722978715od_v_v @ ( lattic5151207300795964030od_v_v @ B ) @ ( lattic5151207300795964030od_v_v @ A3 ) )
= ( lattic5151207300795964030od_v_v @ A3 ) ) ) ) ) ).
% Sup_fin.subset
thf(fact_909_Sup__fin_Oinsert__not__elem,axiom,
! [A3: set_se8455005133513928103od_v_v,X: set_Product_prod_v_v] :
( ( finite6084192165098772208od_v_v @ A3 )
=> ( ~ ( member8406446414694345712od_v_v @ X @ A3 )
=> ( ( A3 != bot_bo3497076220358800403od_v_v )
=> ( ( lattic5151207300795964030od_v_v @ ( insert7504383016908236695od_v_v @ X @ A3 ) )
= ( sup_su414716646722978715od_v_v @ X @ ( lattic5151207300795964030od_v_v @ A3 ) ) ) ) ) ) ).
% Sup_fin.insert_not_elem
thf(fact_910_Sup__fin_Oclosed,axiom,
! [A3: set_se8455005133513928103od_v_v] :
( ( finite6084192165098772208od_v_v @ A3 )
=> ( ( A3 != bot_bo3497076220358800403od_v_v )
=> ( ! [X3: set_Product_prod_v_v,Y: set_Product_prod_v_v] : ( member8406446414694345712od_v_v @ ( sup_su414716646722978715od_v_v @ X3 @ Y ) @ ( insert7504383016908236695od_v_v @ X3 @ ( insert7504383016908236695od_v_v @ Y @ bot_bo3497076220358800403od_v_v ) ) )
=> ( member8406446414694345712od_v_v @ ( lattic5151207300795964030od_v_v @ A3 ) @ A3 ) ) ) ) ).
% Sup_fin.closed
thf(fact_911_Sup__fin_Ounion,axiom,
! [A3: set_se8455005133513928103od_v_v,B: set_se8455005133513928103od_v_v] :
( ( finite6084192165098772208od_v_v @ A3 )
=> ( ( A3 != bot_bo3497076220358800403od_v_v )
=> ( ( finite6084192165098772208od_v_v @ B )
=> ( ( B != bot_bo3497076220358800403od_v_v )
=> ( ( lattic5151207300795964030od_v_v @ ( sup_su335656005089752955od_v_v @ A3 @ B ) )
= ( sup_su414716646722978715od_v_v @ ( lattic5151207300795964030od_v_v @ A3 ) @ ( lattic5151207300795964030od_v_v @ B ) ) ) ) ) ) ) ).
% Sup_fin.union
thf(fact_912_remove__code_I1_J,axiom,
! [X: v,Xs: list_v] :
( ( remove_v @ X @ ( set_v2 @ Xs ) )
= ( set_v2 @ ( removeAll_v @ X @ Xs ) ) ) ).
% remove_code(1)
thf(fact_913_Sup__fin_Oinsert__remove,axiom,
! [A3: set_se8455005133513928103od_v_v,X: set_Product_prod_v_v] :
( ( finite6084192165098772208od_v_v @ A3 )
=> ( ( ( ( minus_7679383599658060814od_v_v @ A3 @ ( insert7504383016908236695od_v_v @ X @ bot_bo3497076220358800403od_v_v ) )
= bot_bo3497076220358800403od_v_v )
=> ( ( lattic5151207300795964030od_v_v @ ( insert7504383016908236695od_v_v @ X @ A3 ) )
= X ) )
& ( ( ( minus_7679383599658060814od_v_v @ A3 @ ( insert7504383016908236695od_v_v @ X @ bot_bo3497076220358800403od_v_v ) )
!= bot_bo3497076220358800403od_v_v )
=> ( ( lattic5151207300795964030od_v_v @ ( insert7504383016908236695od_v_v @ X @ A3 ) )
= ( sup_su414716646722978715od_v_v @ X @ ( lattic5151207300795964030od_v_v @ ( minus_7679383599658060814od_v_v @ A3 @ ( insert7504383016908236695od_v_v @ X @ bot_bo3497076220358800403od_v_v ) ) ) ) ) ) ) ) ).
% Sup_fin.insert_remove
thf(fact_914_Inf__fin_Oinsert__remove,axiom,
! [A3: set_set_v,X: set_v] :
( ( finite_finite_set_v @ A3 )
=> ( ( ( ( minus_7228012346218142266_set_v @ A3 @ ( insert_set_v @ X @ bot_bot_set_set_v ) )
= bot_bot_set_set_v )
=> ( ( lattic8209813555532694032_set_v @ ( insert_set_v @ X @ A3 ) )
= X ) )
& ( ( ( minus_7228012346218142266_set_v @ A3 @ ( insert_set_v @ X @ bot_bot_set_set_v ) )
!= bot_bot_set_set_v )
=> ( ( lattic8209813555532694032_set_v @ ( insert_set_v @ X @ A3 ) )
= ( inf_inf_set_v @ X @ ( lattic8209813555532694032_set_v @ ( minus_7228012346218142266_set_v @ A3 @ ( insert_set_v @ X @ bot_bot_set_set_v ) ) ) ) ) ) ) ) ).
% Inf_fin.insert_remove
thf(fact_915_Inf__fin_Oremove,axiom,
! [A3: set_set_v,X: set_v] :
( ( finite_finite_set_v @ A3 )
=> ( ( member_set_v @ X @ A3 )
=> ( ( ( ( minus_7228012346218142266_set_v @ A3 @ ( insert_set_v @ X @ bot_bot_set_set_v ) )
= bot_bot_set_set_v )
=> ( ( lattic8209813555532694032_set_v @ A3 )
= X ) )
& ( ( ( minus_7228012346218142266_set_v @ A3 @ ( insert_set_v @ X @ bot_bot_set_set_v ) )
!= bot_bot_set_set_v )
=> ( ( lattic8209813555532694032_set_v @ A3 )
= ( inf_inf_set_v @ X @ ( lattic8209813555532694032_set_v @ ( minus_7228012346218142266_set_v @ A3 @ ( insert_set_v @ X @ bot_bot_set_set_v ) ) ) ) ) ) ) ) ) ).
% Inf_fin.remove
thf(fact_916_Inf__fin_Oinsert,axiom,
! [A3: set_set_v,X: set_v] :
( ( finite_finite_set_v @ A3 )
=> ( ( A3 != bot_bot_set_set_v )
=> ( ( lattic8209813555532694032_set_v @ ( insert_set_v @ X @ A3 ) )
= ( inf_inf_set_v @ X @ ( lattic8209813555532694032_set_v @ A3 ) ) ) ) ) ).
% Inf_fin.insert
thf(fact_917_Inf__fin_OcoboundedI,axiom,
! [A3: set_set_v,A: set_v] :
( ( finite_finite_set_v @ A3 )
=> ( ( member_set_v @ A @ A3 )
=> ( ord_less_eq_set_v @ ( lattic8209813555532694032_set_v @ A3 ) @ A ) ) ) ).
% Inf_fin.coboundedI
thf(fact_918_Inf__fin_OcoboundedI,axiom,
! [A3: set_se8455005133513928103od_v_v,A: set_Product_prod_v_v] :
( ( finite6084192165098772208od_v_v @ A3 )
=> ( ( member8406446414694345712od_v_v @ A @ A3 )
=> ( ord_le7336532860387713383od_v_v @ ( lattic4767070952889939172od_v_v @ A3 ) @ A ) ) ) ).
% Inf_fin.coboundedI
thf(fact_919_Inf__fin_Oin__idem,axiom,
! [A3: set_set_v,X: set_v] :
( ( finite_finite_set_v @ A3 )
=> ( ( member_set_v @ X @ A3 )
=> ( ( inf_inf_set_v @ X @ ( lattic8209813555532694032_set_v @ A3 ) )
= ( lattic8209813555532694032_set_v @ A3 ) ) ) ) ).
% Inf_fin.in_idem
thf(fact_920_Inf__fin_OboundedE,axiom,
! [A3: set_set_v,X: set_v] :
( ( finite_finite_set_v @ A3 )
=> ( ( A3 != bot_bot_set_set_v )
=> ( ( ord_less_eq_set_v @ X @ ( lattic8209813555532694032_set_v @ A3 ) )
=> ! [A10: set_v] :
( ( member_set_v @ A10 @ A3 )
=> ( ord_less_eq_set_v @ X @ A10 ) ) ) ) ) ).
% Inf_fin.boundedE
thf(fact_921_Inf__fin_OboundedE,axiom,
! [A3: set_se8455005133513928103od_v_v,X: set_Product_prod_v_v] :
( ( finite6084192165098772208od_v_v @ A3 )
=> ( ( A3 != bot_bo3497076220358800403od_v_v )
=> ( ( ord_le7336532860387713383od_v_v @ X @ ( lattic4767070952889939172od_v_v @ A3 ) )
=> ! [A10: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ A10 @ A3 )
=> ( ord_le7336532860387713383od_v_v @ X @ A10 ) ) ) ) ) ).
% Inf_fin.boundedE
thf(fact_922_Inf__fin_OboundedI,axiom,
! [A3: set_set_v,X: set_v] :
( ( finite_finite_set_v @ A3 )
=> ( ( A3 != bot_bot_set_set_v )
=> ( ! [A7: set_v] :
( ( member_set_v @ A7 @ A3 )
=> ( ord_less_eq_set_v @ X @ A7 ) )
=> ( ord_less_eq_set_v @ X @ ( lattic8209813555532694032_set_v @ A3 ) ) ) ) ) ).
% Inf_fin.boundedI
thf(fact_923_Inf__fin_OboundedI,axiom,
! [A3: set_se8455005133513928103od_v_v,X: set_Product_prod_v_v] :
( ( finite6084192165098772208od_v_v @ A3 )
=> ( ( A3 != bot_bo3497076220358800403od_v_v )
=> ( ! [A7: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ A7 @ A3 )
=> ( ord_le7336532860387713383od_v_v @ X @ A7 ) )
=> ( ord_le7336532860387713383od_v_v @ X @ ( lattic4767070952889939172od_v_v @ A3 ) ) ) ) ) ).
% Inf_fin.boundedI
thf(fact_924_Inf__fin_Obounded__iff,axiom,
! [A3: set_set_v,X: set_v] :
( ( finite_finite_set_v @ A3 )
=> ( ( A3 != bot_bot_set_set_v )
=> ( ( ord_less_eq_set_v @ X @ ( lattic8209813555532694032_set_v @ A3 ) )
= ( ! [X2: set_v] :
( ( member_set_v @ X2 @ A3 )
=> ( ord_less_eq_set_v @ X @ X2 ) ) ) ) ) ) ).
% Inf_fin.bounded_iff
thf(fact_925_Inf__fin_Obounded__iff,axiom,
! [A3: set_se8455005133513928103od_v_v,X: set_Product_prod_v_v] :
( ( finite6084192165098772208od_v_v @ A3 )
=> ( ( A3 != bot_bo3497076220358800403od_v_v )
=> ( ( ord_le7336532860387713383od_v_v @ X @ ( lattic4767070952889939172od_v_v @ A3 ) )
= ( ! [X2: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X2 @ A3 )
=> ( ord_le7336532860387713383od_v_v @ X @ X2 ) ) ) ) ) ) ).
% Inf_fin.bounded_iff
thf(fact_926_Inf__fin_Osubset__imp,axiom,
! [A3: set_set_v,B: set_set_v] :
( ( ord_le5216385588623774835_set_v @ A3 @ B )
=> ( ( A3 != bot_bot_set_set_v )
=> ( ( finite_finite_set_v @ B )
=> ( ord_less_eq_set_v @ ( lattic8209813555532694032_set_v @ B ) @ ( lattic8209813555532694032_set_v @ A3 ) ) ) ) ) ).
% Inf_fin.subset_imp
thf(fact_927_Inf__fin_Osubset__imp,axiom,
! [A3: set_se8455005133513928103od_v_v,B: set_se8455005133513928103od_v_v] :
( ( ord_le4714265922333009223od_v_v @ A3 @ B )
=> ( ( A3 != bot_bo3497076220358800403od_v_v )
=> ( ( finite6084192165098772208od_v_v @ B )
=> ( ord_le7336532860387713383od_v_v @ ( lattic4767070952889939172od_v_v @ B ) @ ( lattic4767070952889939172od_v_v @ A3 ) ) ) ) ) ).
% Inf_fin.subset_imp
thf(fact_928_Inf__fin_Osubset,axiom,
! [A3: set_set_v,B: set_set_v] :
( ( finite_finite_set_v @ A3 )
=> ( ( B != bot_bot_set_set_v )
=> ( ( ord_le5216385588623774835_set_v @ B @ A3 )
=> ( ( inf_inf_set_v @ ( lattic8209813555532694032_set_v @ B ) @ ( lattic8209813555532694032_set_v @ A3 ) )
= ( lattic8209813555532694032_set_v @ A3 ) ) ) ) ) ).
% Inf_fin.subset
thf(fact_929_Inf__fin_Oinsert__not__elem,axiom,
! [A3: set_set_v,X: set_v] :
( ( finite_finite_set_v @ A3 )
=> ( ~ ( member_set_v @ X @ A3 )
=> ( ( A3 != bot_bot_set_set_v )
=> ( ( lattic8209813555532694032_set_v @ ( insert_set_v @ X @ A3 ) )
= ( inf_inf_set_v @ X @ ( lattic8209813555532694032_set_v @ A3 ) ) ) ) ) ) ).
% Inf_fin.insert_not_elem
thf(fact_930_Inf__fin_Oclosed,axiom,
! [A3: set_set_v] :
( ( finite_finite_set_v @ A3 )
=> ( ( A3 != bot_bot_set_set_v )
=> ( ! [X3: set_v,Y: set_v] : ( member_set_v @ ( inf_inf_set_v @ X3 @ Y ) @ ( insert_set_v @ X3 @ ( insert_set_v @ Y @ bot_bot_set_set_v ) ) )
=> ( member_set_v @ ( lattic8209813555532694032_set_v @ A3 ) @ A3 ) ) ) ) ).
% Inf_fin.closed
thf(fact_931_Inf__fin_Ounion,axiom,
! [A3: set_set_v,B: set_set_v] :
( ( finite_finite_set_v @ A3 )
=> ( ( A3 != bot_bot_set_set_v )
=> ( ( finite_finite_set_v @ B )
=> ( ( B != bot_bot_set_set_v )
=> ( ( lattic8209813555532694032_set_v @ ( sup_sup_set_set_v @ A3 @ B ) )
= ( inf_inf_set_v @ ( lattic8209813555532694032_set_v @ A3 ) @ ( lattic8209813555532694032_set_v @ B ) ) ) ) ) ) ) ).
% Inf_fin.union
thf(fact_932_Inf__fin__le__Sup__fin,axiom,
! [A3: set_set_v] :
( ( finite_finite_set_v @ A3 )
=> ( ( A3 != bot_bot_set_set_v )
=> ( ord_less_eq_set_v @ ( lattic8209813555532694032_set_v @ A3 ) @ ( lattic2918178447194608042_set_v @ A3 ) ) ) ) ).
% Inf_fin_le_Sup_fin
thf(fact_933_Inf__fin__le__Sup__fin,axiom,
! [A3: set_se8455005133513928103od_v_v] :
( ( finite6084192165098772208od_v_v @ A3 )
=> ( ( A3 != bot_bo3497076220358800403od_v_v )
=> ( ord_le7336532860387713383od_v_v @ ( lattic4767070952889939172od_v_v @ A3 ) @ ( lattic5151207300795964030od_v_v @ A3 ) ) ) ) ).
% Inf_fin_le_Sup_fin
thf(fact_934_refl__on__singleton,axiom,
! [X: product_prod_v_v] : ( refl_o4548774019903118566od_v_v @ ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) @ ( insert5641704497130386615od_v_v @ ( produc4031800376763917143od_v_v @ X @ X ) @ bot_bo3282589961317712691od_v_v ) ) ).
% refl_on_singleton
thf(fact_935_refl__on__singleton,axiom,
! [X: v] : ( refl_on_v @ ( insert_v2 @ X @ bot_bot_set_v ) @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ X @ X ) @ bot_bo723834152578015283od_v_v ) ) ).
% refl_on_singleton
thf(fact_936_snoc__listrel1__snoc__iff,axiom,
! [Xs: list_v,X: v,Ys: list_v,Y2: v,R: set_Product_prod_v_v] :
( ( member418487059593946000list_v @ ( produc6795410681906604247list_v @ ( append_v @ Xs @ ( cons_v @ X @ nil_v ) ) @ ( append_v @ Ys @ ( cons_v @ Y2 @ nil_v ) ) ) @ ( listrel1_v @ R ) )
= ( ( ( member418487059593946000list_v @ ( produc6795410681906604247list_v @ Xs @ Ys ) @ ( listrel1_v @ R ) )
& ( X = Y2 ) )
| ( ( Xs = Ys )
& ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X @ Y2 ) @ R ) ) ) ) ).
% snoc_listrel1_snoc_iff
thf(fact_937_split__list__precedes,axiom,
! [Y2: product_prod_v_v,Ys: list_P7986770385144383213od_v_v,X: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ( member7453568604450474000od_v_v @ Y2 @ ( set_Product_prod_v_v2 @ ( append2138873909117096322od_v_v @ Ys @ ( cons_P4120604216776828829od_v_v @ X @ nil_Product_prod_v_v ) ) ) )
=> ( sCC_Bl2026170059108282219od_v_v @ Y2 @ X @ ( append2138873909117096322od_v_v @ Ys @ ( cons_P4120604216776828829od_v_v @ X @ Xs ) ) ) ) ).
% split_list_precedes
thf(fact_938_split__list__precedes,axiom,
! [Y2: v,Ys: list_v,X: v,Xs: list_v] :
( ( member_v @ Y2 @ ( set_v2 @ ( append_v @ Ys @ ( cons_v @ X @ nil_v ) ) ) )
=> ( sCC_Bl4022239298816431255edes_v @ Y2 @ X @ ( append_v @ Ys @ ( cons_v @ X @ Xs ) ) ) ) ).
% split_list_precedes
thf(fact_939_precedes__refl,axiom,
! [X: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ( sCC_Bl2026170059108282219od_v_v @ X @ X @ Xs )
= ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ Xs ) ) ) ).
% precedes_refl
thf(fact_940_precedes__refl,axiom,
! [X: v,Xs: list_v] :
( ( sCC_Bl4022239298816431255edes_v @ X @ X @ Xs )
= ( member_v @ X @ ( set_v2 @ Xs ) ) ) ).
% precedes_refl
thf(fact_941_Cons__listrel1__Cons,axiom,
! [X: v,Xs: list_v,Y2: v,Ys: list_v,R: set_Product_prod_v_v] :
( ( member418487059593946000list_v @ ( produc6795410681906604247list_v @ ( cons_v @ X @ Xs ) @ ( cons_v @ Y2 @ Ys ) ) @ ( listrel1_v @ R ) )
= ( ( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X @ Y2 ) @ R )
& ( Xs = Ys ) )
| ( ( X = Y2 )
& ( member418487059593946000list_v @ ( produc6795410681906604247list_v @ Xs @ Ys ) @ ( listrel1_v @ R ) ) ) ) ) ).
% Cons_listrel1_Cons
thf(fact_942_refl__onD2,axiom,
! [A3: set_Product_prod_v_v,R: set_Pr2149350503807050951od_v_v,X: product_prod_v_v,Y2: product_prod_v_v] :
( ( refl_o4548774019903118566od_v_v @ A3 @ R )
=> ( ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ X @ Y2 ) @ R )
=> ( member7453568604450474000od_v_v @ Y2 @ A3 ) ) ) ).
% refl_onD2
thf(fact_943_refl__onD2,axiom,
! [A3: set_v,R: set_Product_prod_v_v,X: v,Y2: v] :
( ( refl_on_v @ A3 @ R )
=> ( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X @ Y2 ) @ R )
=> ( member_v @ Y2 @ A3 ) ) ) ).
% refl_onD2
thf(fact_944_refl__onD1,axiom,
! [A3: set_Product_prod_v_v,R: set_Pr2149350503807050951od_v_v,X: product_prod_v_v,Y2: product_prod_v_v] :
( ( refl_o4548774019903118566od_v_v @ A3 @ R )
=> ( ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ X @ Y2 ) @ R )
=> ( member7453568604450474000od_v_v @ X @ A3 ) ) ) ).
% refl_onD1
thf(fact_945_refl__onD1,axiom,
! [A3: set_v,R: set_Product_prod_v_v,X: v,Y2: v] :
( ( refl_on_v @ A3 @ R )
=> ( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X @ Y2 ) @ R )
=> ( member_v @ X @ A3 ) ) ) ).
% refl_onD1
thf(fact_946_refl__onD,axiom,
! [A3: set_Product_prod_v_v,R: set_Pr2149350503807050951od_v_v,A: product_prod_v_v] :
( ( refl_o4548774019903118566od_v_v @ A3 @ R )
=> ( ( member7453568604450474000od_v_v @ A @ A3 )
=> ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ A @ A ) @ R ) ) ) ).
% refl_onD
thf(fact_947_refl__onD,axiom,
! [A3: set_v,R: set_Product_prod_v_v,A: v] :
( ( refl_on_v @ A3 @ R )
=> ( ( member_v @ A @ A3 )
=> ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ A @ A ) @ R ) ) ) ).
% refl_onD
thf(fact_948_precedes__mem_I1_J,axiom,
! [X: product_prod_v_v,Y2: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ( sCC_Bl2026170059108282219od_v_v @ X @ Y2 @ Xs )
=> ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ Xs ) ) ) ).
% precedes_mem(1)
thf(fact_949_precedes__mem_I1_J,axiom,
! [X: v,Y2: v,Xs: list_v] :
( ( sCC_Bl4022239298816431255edes_v @ X @ Y2 @ Xs )
=> ( member_v @ X @ ( set_v2 @ Xs ) ) ) ).
% precedes_mem(1)
thf(fact_950_precedes__mem_I2_J,axiom,
! [X: product_prod_v_v,Y2: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ( sCC_Bl2026170059108282219od_v_v @ X @ Y2 @ Xs )
=> ( member7453568604450474000od_v_v @ Y2 @ ( set_Product_prod_v_v2 @ Xs ) ) ) ).
% precedes_mem(2)
thf(fact_951_precedes__mem_I2_J,axiom,
! [X: v,Y2: v,Xs: list_v] :
( ( sCC_Bl4022239298816431255edes_v @ X @ Y2 @ Xs )
=> ( member_v @ Y2 @ ( set_v2 @ Xs ) ) ) ).
% precedes_mem(2)
thf(fact_952_listrel1__mono,axiom,
! [R: set_Product_prod_v_v,S4: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ R @ S4 )
=> ( ord_le791731619978752231list_v @ ( listrel1_v @ R ) @ ( listrel1_v @ S4 ) ) ) ).
% listrel1_mono
thf(fact_953_refl__on__Int,axiom,
! [A3: set_v,R: set_Product_prod_v_v,B: set_v,S4: set_Product_prod_v_v] :
( ( refl_on_v @ A3 @ R )
=> ( ( refl_on_v @ B @ S4 )
=> ( refl_on_v @ ( inf_inf_set_v @ A3 @ B ) @ ( inf_in6271465464967711157od_v_v @ R @ S4 ) ) ) ) ).
% refl_on_Int
thf(fact_954_refl__on__empty,axiom,
refl_o4548774019903118566od_v_v @ bot_bo723834152578015283od_v_v @ bot_bo3282589961317712691od_v_v ).
% refl_on_empty
thf(fact_955_refl__on__empty,axiom,
refl_on_v @ bot_bot_set_v @ bot_bo723834152578015283od_v_v ).
% refl_on_empty
thf(fact_956_tail__not__precedes,axiom,
! [Y2: product_prod_v_v,X: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ( sCC_Bl2026170059108282219od_v_v @ Y2 @ X @ ( cons_P4120604216776828829od_v_v @ X @ Xs ) )
=> ( ~ ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ Xs ) )
=> ( X = Y2 ) ) ) ).
% tail_not_precedes
thf(fact_957_tail__not__precedes,axiom,
! [Y2: v,X: v,Xs: list_v] :
( ( sCC_Bl4022239298816431255edes_v @ Y2 @ X @ ( cons_v @ X @ Xs ) )
=> ( ~ ( member_v @ X @ ( set_v2 @ Xs ) )
=> ( X = Y2 ) ) ) ).
% tail_not_precedes
thf(fact_958_head__precedes,axiom,
! [Y2: product_prod_v_v,X: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ( member7453568604450474000od_v_v @ Y2 @ ( set_Product_prod_v_v2 @ ( cons_P4120604216776828829od_v_v @ X @ Xs ) ) )
=> ( sCC_Bl2026170059108282219od_v_v @ X @ Y2 @ ( cons_P4120604216776828829od_v_v @ X @ Xs ) ) ) ).
% head_precedes
thf(fact_959_head__precedes,axiom,
! [Y2: v,X: v,Xs: list_v] :
( ( member_v @ Y2 @ ( set_v2 @ ( cons_v @ X @ Xs ) ) )
=> ( sCC_Bl4022239298816431255edes_v @ X @ Y2 @ ( cons_v @ X @ Xs ) ) ) ).
% head_precedes
thf(fact_960_precedes__append__right__iff,axiom,
! [Y2: product_prod_v_v,Ys: list_P7986770385144383213od_v_v,X: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ~ ( member7453568604450474000od_v_v @ Y2 @ ( set_Product_prod_v_v2 @ Ys ) )
=> ( ( sCC_Bl2026170059108282219od_v_v @ X @ Y2 @ ( append2138873909117096322od_v_v @ Xs @ Ys ) )
= ( sCC_Bl2026170059108282219od_v_v @ X @ Y2 @ Xs ) ) ) ).
% precedes_append_right_iff
thf(fact_961_precedes__append__right__iff,axiom,
! [Y2: v,Ys: list_v,X: v,Xs: list_v] :
( ~ ( member_v @ Y2 @ ( set_v2 @ Ys ) )
=> ( ( sCC_Bl4022239298816431255edes_v @ X @ Y2 @ ( append_v @ Xs @ Ys ) )
= ( sCC_Bl4022239298816431255edes_v @ X @ Y2 @ Xs ) ) ) ).
% precedes_append_right_iff
thf(fact_962_precedes__append__left__iff,axiom,
! [X: product_prod_v_v,Ys: list_P7986770385144383213od_v_v,Y2: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ~ ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ Ys ) )
=> ( ( sCC_Bl2026170059108282219od_v_v @ X @ Y2 @ ( append2138873909117096322od_v_v @ Ys @ Xs ) )
= ( sCC_Bl2026170059108282219od_v_v @ X @ Y2 @ Xs ) ) ) ).
% precedes_append_left_iff
thf(fact_963_precedes__append__left__iff,axiom,
! [X: v,Ys: list_v,Y2: v,Xs: list_v] :
( ~ ( member_v @ X @ ( set_v2 @ Ys ) )
=> ( ( sCC_Bl4022239298816431255edes_v @ X @ Y2 @ ( append_v @ Ys @ Xs ) )
= ( sCC_Bl4022239298816431255edes_v @ X @ Y2 @ Xs ) ) ) ).
% precedes_append_left_iff
thf(fact_964_Cons__listrel1E2,axiom,
! [Xs: list_v,Y2: v,Ys: list_v,R: set_Product_prod_v_v] :
( ( member418487059593946000list_v @ ( produc6795410681906604247list_v @ Xs @ ( cons_v @ Y2 @ Ys ) ) @ ( listrel1_v @ R ) )
=> ( ! [X3: v] :
( ( Xs
= ( cons_v @ X3 @ Ys ) )
=> ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X3 @ Y2 ) @ R ) )
=> ~ ! [Zs2: list_v] :
( ( Xs
= ( cons_v @ Y2 @ Zs2 ) )
=> ~ ( member418487059593946000list_v @ ( produc6795410681906604247list_v @ Zs2 @ Ys ) @ ( listrel1_v @ R ) ) ) ) ) ).
% Cons_listrel1E2
thf(fact_965_Cons__listrel1E1,axiom,
! [X: v,Xs: list_v,Ys: list_v,R: set_Product_prod_v_v] :
( ( member418487059593946000list_v @ ( produc6795410681906604247list_v @ ( cons_v @ X @ Xs ) @ Ys ) @ ( listrel1_v @ R ) )
=> ( ! [Y: v] :
( ( Ys
= ( cons_v @ Y @ Xs ) )
=> ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X @ Y ) @ R ) )
=> ~ ! [Zs2: list_v] :
( ( Ys
= ( cons_v @ X @ Zs2 ) )
=> ~ ( member418487059593946000list_v @ ( produc6795410681906604247list_v @ Xs @ Zs2 ) @ ( listrel1_v @ R ) ) ) ) ) ).
% Cons_listrel1E1
thf(fact_966_listrel1I1,axiom,
! [X: v,Y2: v,R: set_Product_prod_v_v,Xs: list_v] :
( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X @ Y2 ) @ R )
=> ( member418487059593946000list_v @ ( produc6795410681906604247list_v @ ( cons_v @ X @ Xs ) @ ( cons_v @ Y2 @ Xs ) ) @ ( listrel1_v @ R ) ) ) ).
% listrel1I1
thf(fact_967_precedes__def,axiom,
( sCC_Bl2026170059108282219od_v_v
= ( ^ [X2: product_prod_v_v,Y3: product_prod_v_v,Xs4: list_P7986770385144383213od_v_v] :
? [L: list_P7986770385144383213od_v_v,R4: list_P7986770385144383213od_v_v] :
( ( Xs4
= ( append2138873909117096322od_v_v @ L @ ( cons_P4120604216776828829od_v_v @ X2 @ R4 ) ) )
& ( member7453568604450474000od_v_v @ Y3 @ ( set_Product_prod_v_v2 @ ( cons_P4120604216776828829od_v_v @ X2 @ R4 ) ) ) ) ) ) ).
% precedes_def
thf(fact_968_precedes__def,axiom,
( sCC_Bl4022239298816431255edes_v
= ( ^ [X2: v,Y3: v,Xs4: list_v] :
? [L: list_v,R4: list_v] :
( ( Xs4
= ( append_v @ L @ ( cons_v @ X2 @ R4 ) ) )
& ( member_v @ Y3 @ ( set_v2 @ ( cons_v @ X2 @ R4 ) ) ) ) ) ) ).
% precedes_def
thf(fact_969_listrel1E,axiom,
! [Xs: list_v,Ys: list_v,R: set_Product_prod_v_v] :
( ( member418487059593946000list_v @ ( produc6795410681906604247list_v @ Xs @ Ys ) @ ( listrel1_v @ R ) )
=> ~ ! [X3: v,Y: v] :
( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X3 @ Y ) @ R )
=> ! [Us: list_v,Vs: list_v] :
( ( Xs
= ( append_v @ Us @ ( cons_v @ X3 @ Vs ) ) )
=> ( Ys
!= ( append_v @ Us @ ( cons_v @ Y @ Vs ) ) ) ) ) ) ).
% listrel1E
thf(fact_970_listrel1I,axiom,
! [X: v,Y2: v,R: set_Product_prod_v_v,Xs: list_v,Us2: list_v,Vs2: list_v,Ys: list_v] :
( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X @ Y2 ) @ R )
=> ( ( Xs
= ( append_v @ Us2 @ ( cons_v @ X @ Vs2 ) ) )
=> ( ( Ys
= ( append_v @ Us2 @ ( cons_v @ Y2 @ Vs2 ) ) )
=> ( member418487059593946000list_v @ ( produc6795410681906604247list_v @ Xs @ Ys ) @ ( listrel1_v @ R ) ) ) ) ) ).
% listrel1I
thf(fact_971_refl__on__domain,axiom,
! [A3: set_Product_prod_v_v,R: set_Pr2149350503807050951od_v_v,A: product_prod_v_v,B2: product_prod_v_v] :
( ( refl_o4548774019903118566od_v_v @ A3 @ R )
=> ( ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ A @ B2 ) @ R )
=> ( ( member7453568604450474000od_v_v @ A @ A3 )
& ( member7453568604450474000od_v_v @ B2 @ A3 ) ) ) ) ).
% refl_on_domain
thf(fact_972_refl__on__domain,axiom,
! [A3: set_v,R: set_Product_prod_v_v,A: v,B2: v] :
( ( refl_on_v @ A3 @ R )
=> ( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ A @ B2 ) @ R )
=> ( ( member_v @ A @ A3 )
& ( member_v @ B2 @ A3 ) ) ) ) ).
% refl_on_domain
thf(fact_973_partition__set,axiom,
! [P: v > $o,Xs: list_v,Yes: list_v,No: list_v] :
( ( ( partition_v @ P @ Xs )
= ( produc6795410681906604247list_v @ Yes @ No ) )
=> ( ( sup_sup_set_v @ ( set_v2 @ Yes ) @ ( set_v2 @ No ) )
= ( set_v2 @ Xs ) ) ) ).
% partition_set
thf(fact_974_partition__set,axiom,
! [P: product_prod_v_v > $o,Xs: list_P7986770385144383213od_v_v,Yes: list_P7986770385144383213od_v_v,No: list_P7986770385144383213od_v_v] :
( ( ( partit5288610572509583718od_v_v @ P @ Xs )
= ( produc674067373767953879od_v_v @ Yes @ No ) )
=> ( ( sup_su414716646722978715od_v_v @ ( set_Product_prod_v_v2 @ Yes ) @ ( set_Product_prod_v_v2 @ No ) )
= ( set_Product_prod_v_v2 @ Xs ) ) ) ).
% partition_set
thf(fact_975_partition__P,axiom,
! [P: v > $o,Xs: list_v,Yes: list_v,No: list_v] :
( ( ( partition_v @ P @ Xs )
= ( produc6795410681906604247list_v @ Yes @ No ) )
=> ( ! [X4: v] :
( ( member_v @ X4 @ ( set_v2 @ Yes ) )
=> ( P @ X4 ) )
& ! [X4: v] :
( ( member_v @ X4 @ ( set_v2 @ No ) )
=> ~ ( P @ X4 ) ) ) ) ).
% partition_P
thf(fact_976_linear__order__on__singleton,axiom,
! [X: product_prod_v_v] : ( order_6462556390437124636od_v_v @ ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) @ ( insert5641704497130386615od_v_v @ ( produc4031800376763917143od_v_v @ X @ X ) @ bot_bo3282589961317712691od_v_v ) ) ).
% linear_order_on_singleton
thf(fact_977_linear__order__on__singleton,axiom,
! [X: v] : ( order_8768733634509060168r_on_v @ ( insert_v2 @ X @ bot_bot_set_v ) @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ X @ X ) @ bot_bo723834152578015283od_v_v ) ) ).
% linear_order_on_singleton
thf(fact_978_in__set__remove1,axiom,
! [A: product_prod_v_v,B2: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ( A != B2 )
=> ( ( member7453568604450474000od_v_v @ A @ ( set_Product_prod_v_v2 @ ( remove333779696311199107od_v_v @ B2 @ Xs ) ) )
= ( member7453568604450474000od_v_v @ A @ ( set_Product_prod_v_v2 @ Xs ) ) ) ) ).
% in_set_remove1
thf(fact_979_in__set__remove1,axiom,
! [A: v,B2: v,Xs: list_v] :
( ( A != B2 )
=> ( ( member_v @ A @ ( set_v2 @ ( remove1_v @ B2 @ Xs ) ) )
= ( member_v @ A @ ( set_v2 @ Xs ) ) ) ) ).
% in_set_remove1
thf(fact_980_remove1__idem,axiom,
! [X: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ~ ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ Xs ) )
=> ( ( remove333779696311199107od_v_v @ X @ Xs )
= Xs ) ) ).
% remove1_idem
thf(fact_981_remove1__idem,axiom,
! [X: v,Xs: list_v] :
( ~ ( member_v @ X @ ( set_v2 @ Xs ) )
=> ( ( remove1_v @ X @ Xs )
= Xs ) ) ).
% remove1_idem
thf(fact_982_notin__set__remove1,axiom,
! [X: product_prod_v_v,Xs: list_P7986770385144383213od_v_v,Y2: product_prod_v_v] :
( ~ ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ Xs ) )
=> ~ ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ ( remove333779696311199107od_v_v @ Y2 @ Xs ) ) ) ) ).
% notin_set_remove1
thf(fact_983_notin__set__remove1,axiom,
! [X: v,Xs: list_v,Y2: v] :
( ~ ( member_v @ X @ ( set_v2 @ Xs ) )
=> ~ ( member_v @ X @ ( set_v2 @ ( remove1_v @ Y2 @ Xs ) ) ) ) ).
% notin_set_remove1
thf(fact_984_set__remove1__subset,axiom,
! [X: v,Xs: list_v] : ( ord_less_eq_set_v @ ( set_v2 @ ( remove1_v @ X @ Xs ) ) @ ( set_v2 @ Xs ) ) ).
% set_remove1_subset
thf(fact_985_set__remove1__subset,axiom,
! [X: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] : ( ord_le7336532860387713383od_v_v @ ( set_Product_prod_v_v2 @ ( remove333779696311199107od_v_v @ X @ Xs ) ) @ ( set_Product_prod_v_v2 @ Xs ) ) ).
% set_remove1_subset
thf(fact_986_remove1__append,axiom,
! [X: product_prod_v_v,Xs: list_P7986770385144383213od_v_v,Ys: list_P7986770385144383213od_v_v] :
( ( ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ Xs ) )
=> ( ( remove333779696311199107od_v_v @ X @ ( append2138873909117096322od_v_v @ Xs @ Ys ) )
= ( append2138873909117096322od_v_v @ ( remove333779696311199107od_v_v @ X @ Xs ) @ Ys ) ) )
& ( ~ ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ Xs ) )
=> ( ( remove333779696311199107od_v_v @ X @ ( append2138873909117096322od_v_v @ Xs @ Ys ) )
= ( append2138873909117096322od_v_v @ Xs @ ( remove333779696311199107od_v_v @ X @ Ys ) ) ) ) ) ).
% remove1_append
thf(fact_987_remove1__append,axiom,
! [X: v,Xs: list_v,Ys: list_v] :
( ( ( member_v @ X @ ( set_v2 @ Xs ) )
=> ( ( remove1_v @ X @ ( append_v @ Xs @ Ys ) )
= ( append_v @ ( remove1_v @ X @ Xs ) @ Ys ) ) )
& ( ~ ( member_v @ X @ ( set_v2 @ Xs ) )
=> ( ( remove1_v @ X @ ( append_v @ Xs @ Ys ) )
= ( append_v @ Xs @ ( remove1_v @ X @ Ys ) ) ) ) ) ).
% remove1_append
thf(fact_988_lnear__order__on__empty,axiom,
order_6462556390437124636od_v_v @ bot_bo723834152578015283od_v_v @ bot_bo3282589961317712691od_v_v ).
% lnear_order_on_empty
thf(fact_989_lnear__order__on__empty,axiom,
order_8768733634509060168r_on_v @ bot_bot_set_v @ bot_bo723834152578015283od_v_v ).
% lnear_order_on_empty
thf(fact_990_remove1__split,axiom,
! [A: product_prod_v_v,Xs: list_P7986770385144383213od_v_v,Ys: list_P7986770385144383213od_v_v] :
( ( member7453568604450474000od_v_v @ A @ ( set_Product_prod_v_v2 @ Xs ) )
=> ( ( ( remove333779696311199107od_v_v @ A @ Xs )
= Ys )
= ( ? [Ls: list_P7986770385144383213od_v_v,Rs: list_P7986770385144383213od_v_v] :
( ( Xs
= ( append2138873909117096322od_v_v @ Ls @ ( cons_P4120604216776828829od_v_v @ A @ Rs ) ) )
& ~ ( member7453568604450474000od_v_v @ A @ ( set_Product_prod_v_v2 @ Ls ) )
& ( Ys
= ( append2138873909117096322od_v_v @ Ls @ Rs ) ) ) ) ) ) ).
% remove1_split
thf(fact_991_remove1__split,axiom,
! [A: v,Xs: list_v,Ys: list_v] :
( ( member_v @ A @ ( set_v2 @ Xs ) )
=> ( ( ( remove1_v @ A @ Xs )
= Ys )
= ( ? [Ls: list_v,Rs: list_v] :
( ( Xs
= ( append_v @ Ls @ ( cons_v @ A @ Rs ) ) )
& ~ ( member_v @ A @ ( set_v2 @ Ls ) )
& ( Ys
= ( append_v @ Ls @ Rs ) ) ) ) ) ) ).
% remove1_split
thf(fact_992_set__remove1__eq,axiom,
! [Xs: list_P7986770385144383213od_v_v,X: product_prod_v_v] :
( ( distin6159370996967099744od_v_v @ Xs )
=> ( ( set_Product_prod_v_v2 @ ( remove333779696311199107od_v_v @ X @ Xs ) )
= ( minus_4183494784930505774od_v_v @ ( set_Product_prod_v_v2 @ Xs ) @ ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) ) ) ) ).
% set_remove1_eq
thf(fact_993_set__remove1__eq,axiom,
! [Xs: list_v,X: v] :
( ( distinct_v @ Xs )
=> ( ( set_v2 @ ( remove1_v @ X @ Xs ) )
= ( minus_minus_set_v @ ( set_v2 @ Xs ) @ ( insert_v2 @ X @ bot_bot_set_v ) ) ) ) ).
% set_remove1_eq
thf(fact_994_subset__code_I3_J,axiom,
~ ( ord_less_eq_set_v @ ( coset_v @ nil_v ) @ ( set_v2 @ nil_v ) ) ).
% subset_code(3)
thf(fact_995_subset__code_I3_J,axiom,
~ ( ord_le7336532860387713383od_v_v @ ( coset_766761627116920666od_v_v @ nil_Product_prod_v_v ) @ ( set_Product_prod_v_v2 @ nil_Product_prod_v_v ) ) ).
% subset_code(3)
thf(fact_996_distinct__append,axiom,
! [Xs: list_P7986770385144383213od_v_v,Ys: list_P7986770385144383213od_v_v] :
( ( distin6159370996967099744od_v_v @ ( append2138873909117096322od_v_v @ Xs @ Ys ) )
= ( ( distin6159370996967099744od_v_v @ Xs )
& ( distin6159370996967099744od_v_v @ Ys )
& ( ( inf_in6271465464967711157od_v_v @ ( set_Product_prod_v_v2 @ Xs ) @ ( set_Product_prod_v_v2 @ Ys ) )
= bot_bo723834152578015283od_v_v ) ) ) ).
% distinct_append
thf(fact_997_distinct__append,axiom,
! [Xs: list_v,Ys: list_v] :
( ( distinct_v @ ( append_v @ Xs @ Ys ) )
= ( ( distinct_v @ Xs )
& ( distinct_v @ Ys )
& ( ( inf_inf_set_v @ ( set_v2 @ Xs ) @ ( set_v2 @ Ys ) )
= bot_bot_set_v ) ) ) ).
% distinct_append
thf(fact_998_distinct__tl,axiom,
! [Xs: list_v] :
( ( distinct_v @ Xs )
=> ( distinct_v @ ( tl_v @ Xs ) ) ) ).
% distinct_tl
thf(fact_999_distinct_Osimps_I2_J,axiom,
! [X: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ( distin6159370996967099744od_v_v @ ( cons_P4120604216776828829od_v_v @ X @ Xs ) )
= ( ~ ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ Xs ) )
& ( distin6159370996967099744od_v_v @ Xs ) ) ) ).
% distinct.simps(2)
thf(fact_1000_distinct_Osimps_I2_J,axiom,
! [X: v,Xs: list_v] :
( ( distinct_v @ ( cons_v @ X @ Xs ) )
= ( ~ ( member_v @ X @ ( set_v2 @ Xs ) )
& ( distinct_v @ Xs ) ) ) ).
% distinct.simps(2)
thf(fact_1001_finite__distinct__list,axiom,
! [A3: set_v] :
( ( finite_finite_v @ A3 )
=> ? [Xs2: list_v] :
( ( ( set_v2 @ Xs2 )
= A3 )
& ( distinct_v @ Xs2 ) ) ) ).
% finite_distinct_list
thf(fact_1002_not__distinct__conv__prefix,axiom,
! [As: list_P7986770385144383213od_v_v] :
( ( ~ ( distin6159370996967099744od_v_v @ As ) )
= ( ? [Xs4: list_P7986770385144383213od_v_v,Y3: product_prod_v_v,Ys2: list_P7986770385144383213od_v_v] :
( ( member7453568604450474000od_v_v @ Y3 @ ( set_Product_prod_v_v2 @ Xs4 ) )
& ( distin6159370996967099744od_v_v @ Xs4 )
& ( As
= ( append2138873909117096322od_v_v @ Xs4 @ ( cons_P4120604216776828829od_v_v @ Y3 @ Ys2 ) ) ) ) ) ) ).
% not_distinct_conv_prefix
thf(fact_1003_not__distinct__conv__prefix,axiom,
! [As: list_v] :
( ( ~ ( distinct_v @ As ) )
= ( ? [Xs4: list_v,Y3: v,Ys2: list_v] :
( ( member_v @ Y3 @ ( set_v2 @ Xs4 ) )
& ( distinct_v @ Xs4 )
& ( As
= ( append_v @ Xs4 @ ( cons_v @ Y3 @ Ys2 ) ) ) ) ) ) ).
% not_distinct_conv_prefix
thf(fact_1004_subset__code_I2_J,axiom,
! [A3: set_v,Ys: list_v] :
( ( ord_less_eq_set_v @ A3 @ ( coset_v @ Ys ) )
= ( ! [X2: v] :
( ( member_v @ X2 @ ( set_v2 @ Ys ) )
=> ~ ( member_v @ X2 @ A3 ) ) ) ) ).
% subset_code(2)
thf(fact_1005_subset__code_I2_J,axiom,
! [A3: set_Product_prod_v_v,Ys: list_P7986770385144383213od_v_v] :
( ( ord_le7336532860387713383od_v_v @ A3 @ ( coset_766761627116920666od_v_v @ Ys ) )
= ( ! [X2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ ( set_Product_prod_v_v2 @ Ys ) )
=> ~ ( member7453568604450474000od_v_v @ X2 @ A3 ) ) ) ) ).
% subset_code(2)
thf(fact_1006_insert__code_I2_J,axiom,
! [X: v,Xs: list_v] :
( ( insert_v2 @ X @ ( coset_v @ Xs ) )
= ( coset_v @ ( removeAll_v @ X @ Xs ) ) ) ).
% insert_code(2)
thf(fact_1007_insert__code_I2_J,axiom,
! [X: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ( insert1338601472111419319od_v_v @ X @ ( coset_766761627116920666od_v_v @ Xs ) )
= ( coset_766761627116920666od_v_v @ ( remove481895986417801203od_v_v @ X @ Xs ) ) ) ).
% insert_code(2)
thf(fact_1008_distinct__concat__iff,axiom,
! [Xs: list_l4795378083388841843od_v_v] :
( ( distin6159370996967099744od_v_v @ ( concat2875663619778446888od_v_v @ Xs ) )
= ( ( distin913317783593574886od_v_v @ ( remove5095778601549809401od_v_v @ nil_Product_prod_v_v @ Xs ) )
& ! [Ys2: list_P7986770385144383213od_v_v] :
( ( member4190458934886417558od_v_v @ Ys2 @ ( set_li2340707408155270402od_v_v @ Xs ) )
=> ( distin6159370996967099744od_v_v @ Ys2 ) )
& ! [Ys2: list_P7986770385144383213od_v_v,Zs: list_P7986770385144383213od_v_v] :
( ( ( member4190458934886417558od_v_v @ Ys2 @ ( set_li2340707408155270402od_v_v @ Xs ) )
& ( member4190458934886417558od_v_v @ Zs @ ( set_li2340707408155270402od_v_v @ Xs ) )
& ( Ys2 != Zs ) )
=> ( ( inf_in6271465464967711157od_v_v @ ( set_Product_prod_v_v2 @ Ys2 ) @ ( set_Product_prod_v_v2 @ Zs ) )
= bot_bo723834152578015283od_v_v ) ) ) ) ).
% distinct_concat_iff
thf(fact_1009_distinct__concat__iff,axiom,
! [Xs: list_list_v] :
( ( distinct_v @ ( concat_v @ Xs ) )
= ( ( distinct_list_v @ ( removeAll_list_v @ nil_v @ Xs ) )
& ! [Ys2: list_v] :
( ( member_list_v @ Ys2 @ ( set_list_v2 @ Xs ) )
=> ( distinct_v @ Ys2 ) )
& ! [Ys2: list_v,Zs: list_v] :
( ( ( member_list_v @ Ys2 @ ( set_list_v2 @ Xs ) )
& ( member_list_v @ Zs @ ( set_list_v2 @ Xs ) )
& ( Ys2 != Zs ) )
=> ( ( inf_inf_set_v @ ( set_v2 @ Ys2 ) @ ( set_v2 @ Zs ) )
= bot_bot_set_v ) ) ) ) ).
% distinct_concat_iff
thf(fact_1010_List_Oset__insert,axiom,
! [X: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ( set_Product_prod_v_v2 @ ( insert4539780211034306307od_v_v @ X @ Xs ) )
= ( insert1338601472111419319od_v_v @ X @ ( set_Product_prod_v_v2 @ Xs ) ) ) ).
% List.set_insert
thf(fact_1011_List_Oset__insert,axiom,
! [X: v,Xs: list_v] :
( ( set_v2 @ ( insert_v @ X @ Xs ) )
= ( insert_v2 @ X @ ( set_v2 @ Xs ) ) ) ).
% List.set_insert
thf(fact_1012_not__in__set__insert,axiom,
! [X: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ~ ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ Xs ) )
=> ( ( insert4539780211034306307od_v_v @ X @ Xs )
= ( cons_P4120604216776828829od_v_v @ X @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_1013_not__in__set__insert,axiom,
! [X: v,Xs: list_v] :
( ~ ( member_v @ X @ ( set_v2 @ Xs ) )
=> ( ( insert_v @ X @ Xs )
= ( cons_v @ X @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_1014_in__set__insert,axiom,
! [X: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ Xs ) )
=> ( ( insert4539780211034306307od_v_v @ X @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_1015_in__set__insert,axiom,
! [X: v,Xs: list_v] :
( ( member_v @ X @ ( set_v2 @ Xs ) )
=> ( ( insert_v @ X @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_1016_List_Oinsert__def,axiom,
( insert4539780211034306307od_v_v
= ( ^ [X2: product_prod_v_v,Xs4: list_P7986770385144383213od_v_v] : ( if_lis7521272669439687347od_v_v @ ( member7453568604450474000od_v_v @ X2 @ ( set_Product_prod_v_v2 @ Xs4 ) ) @ Xs4 @ ( cons_P4120604216776828829od_v_v @ X2 @ Xs4 ) ) ) ) ).
% List.insert_def
thf(fact_1017_List_Oinsert__def,axiom,
( insert_v
= ( ^ [X2: v,Xs4: list_v] : ( if_list_v @ ( member_v @ X2 @ ( set_v2 @ Xs4 ) ) @ Xs4 @ ( cons_v @ X2 @ Xs4 ) ) ) ) ).
% List.insert_def
thf(fact_1018_distinct__concat,axiom,
! [Xs: list_l4795378083388841843od_v_v] :
( ( distin913317783593574886od_v_v @ Xs )
=> ( ! [Ys3: list_P7986770385144383213od_v_v] :
( ( member4190458934886417558od_v_v @ Ys3 @ ( set_li2340707408155270402od_v_v @ Xs ) )
=> ( distin6159370996967099744od_v_v @ Ys3 ) )
=> ( ! [Ys3: list_P7986770385144383213od_v_v,Zs2: list_P7986770385144383213od_v_v] :
( ( member4190458934886417558od_v_v @ Ys3 @ ( set_li2340707408155270402od_v_v @ Xs ) )
=> ( ( member4190458934886417558od_v_v @ Zs2 @ ( set_li2340707408155270402od_v_v @ Xs ) )
=> ( ( Ys3 != Zs2 )
=> ( ( inf_in6271465464967711157od_v_v @ ( set_Product_prod_v_v2 @ Ys3 ) @ ( set_Product_prod_v_v2 @ Zs2 ) )
= bot_bo723834152578015283od_v_v ) ) ) )
=> ( distin6159370996967099744od_v_v @ ( concat2875663619778446888od_v_v @ Xs ) ) ) ) ) ).
% distinct_concat
thf(fact_1019_distinct__concat,axiom,
! [Xs: list_list_v] :
( ( distinct_list_v @ Xs )
=> ( ! [Ys3: list_v] :
( ( member_list_v @ Ys3 @ ( set_list_v2 @ Xs ) )
=> ( distinct_v @ Ys3 ) )
=> ( ! [Ys3: list_v,Zs2: list_v] :
( ( member_list_v @ Ys3 @ ( set_list_v2 @ Xs ) )
=> ( ( member_list_v @ Zs2 @ ( set_list_v2 @ Xs ) )
=> ( ( Ys3 != Zs2 )
=> ( ( inf_inf_set_v @ ( set_v2 @ Ys3 ) @ ( set_v2 @ Zs2 ) )
= bot_bot_set_v ) ) ) )
=> ( distinct_v @ ( concat_v @ Xs ) ) ) ) ) ).
% distinct_concat
thf(fact_1020_Set_Ois__empty__def,axiom,
( is_emp8964507351669718201od_v_v
= ( ^ [A4: set_Product_prod_v_v] : ( A4 = bot_bo723834152578015283od_v_v ) ) ) ).
% Set.is_empty_def
thf(fact_1021_Set_Ois__empty__def,axiom,
( is_empty_v
= ( ^ [A4: set_v] : ( A4 = bot_bot_set_v ) ) ) ).
% Set.is_empty_def
thf(fact_1022_Linear__order__Well__order__iff,axiom,
! [R: set_Product_prod_v_v] :
( ( order_8768733634509060168r_on_v @ ( field_v @ R ) @ R )
=> ( ( order_6972113574731384241r_on_v @ ( field_v @ R ) @ R )
= ( ! [A4: set_v] :
( ( ord_less_eq_set_v @ A4 @ ( field_v @ R ) )
=> ( ( A4 != bot_bot_set_v )
=> ? [X2: v] :
( ( member_v @ X2 @ A4 )
& ! [Y3: v] :
( ( member_v @ Y3 @ A4 )
=> ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X2 @ Y3 ) @ R ) ) ) ) ) ) ) ) ).
% Linear_order_Well_order_iff
thf(fact_1023_Linear__order__Well__order__iff,axiom,
! [R: set_Pr2149350503807050951od_v_v] :
( ( order_6462556390437124636od_v_v @ ( field_7153129647634986036od_v_v @ R ) @ R )
=> ( ( order_7541072052284126853od_v_v @ ( field_7153129647634986036od_v_v @ R ) @ R )
= ( ! [A4: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A4 @ ( field_7153129647634986036od_v_v @ R ) )
=> ( ( A4 != bot_bo723834152578015283od_v_v )
=> ? [X2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ A4 )
& ! [Y3: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ Y3 @ A4 )
=> ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ X2 @ Y3 ) @ R ) ) ) ) ) ) ) ) ).
% Linear_order_Well_order_iff
thf(fact_1024_min__bot,axiom,
! [X: set_Product_prod_v_v] :
( ( ord_mi6996445931809003310od_v_v @ bot_bo723834152578015283od_v_v @ X )
= bot_bo723834152578015283od_v_v ) ).
% min_bot
thf(fact_1025_min__bot,axiom,
! [X: set_v] :
( ( ord_min_set_v @ bot_bot_set_v @ X )
= bot_bot_set_v ) ).
% min_bot
thf(fact_1026_min__bot2,axiom,
! [X: set_Product_prod_v_v] :
( ( ord_mi6996445931809003310od_v_v @ X @ bot_bo723834152578015283od_v_v )
= bot_bo723834152578015283od_v_v ) ).
% min_bot2
thf(fact_1027_min__bot2,axiom,
! [X: set_v] :
( ( ord_min_set_v @ X @ bot_bot_set_v )
= bot_bot_set_v ) ).
% min_bot2
thf(fact_1028_well__order__on__domain,axiom,
! [A3: set_Product_prod_v_v,R: set_Pr2149350503807050951od_v_v,A: product_prod_v_v,B2: product_prod_v_v] :
( ( order_7541072052284126853od_v_v @ A3 @ R )
=> ( ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ A @ B2 ) @ R )
=> ( ( member7453568604450474000od_v_v @ A @ A3 )
& ( member7453568604450474000od_v_v @ B2 @ A3 ) ) ) ) ).
% well_order_on_domain
thf(fact_1029_well__order__on__domain,axiom,
! [A3: set_v,R: set_Product_prod_v_v,A: v,B2: v] :
( ( order_6972113574731384241r_on_v @ A3 @ R )
=> ( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ A @ B2 ) @ R )
=> ( ( member_v @ A @ A3 )
& ( member_v @ B2 @ A3 ) ) ) ) ).
% well_order_on_domain
thf(fact_1030_min__def,axiom,
( ord_min_set_v
= ( ^ [A6: set_v,B6: set_v] : ( if_set_v @ ( ord_less_eq_set_v @ A6 @ B6 ) @ A6 @ B6 ) ) ) ).
% min_def
thf(fact_1031_min__def,axiom,
( ord_mi6996445931809003310od_v_v
= ( ^ [A6: set_Product_prod_v_v,B6: set_Product_prod_v_v] : ( if_set4279007504652509325od_v_v @ ( ord_le7336532860387713383od_v_v @ A6 @ B6 ) @ A6 @ B6 ) ) ) ).
% min_def
thf(fact_1032_min__absorb1,axiom,
! [X: set_v,Y2: set_v] :
( ( ord_less_eq_set_v @ X @ Y2 )
=> ( ( ord_min_set_v @ X @ Y2 )
= X ) ) ).
% min_absorb1
thf(fact_1033_min__absorb1,axiom,
! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X @ Y2 )
=> ( ( ord_mi6996445931809003310od_v_v @ X @ Y2 )
= X ) ) ).
% min_absorb1
thf(fact_1034_min__absorb2,axiom,
! [Y2: set_v,X: set_v] :
( ( ord_less_eq_set_v @ Y2 @ X )
=> ( ( ord_min_set_v @ X @ Y2 )
= Y2 ) ) ).
% min_absorb2
thf(fact_1035_min__absorb2,axiom,
! [Y2: set_Product_prod_v_v,X: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ Y2 @ X )
=> ( ( ord_mi6996445931809003310od_v_v @ X @ Y2 )
= Y2 ) ) ).
% min_absorb2
thf(fact_1036_well__order__on__empty,axiom,
order_7541072052284126853od_v_v @ bot_bo723834152578015283od_v_v @ bot_bo3282589961317712691od_v_v ).
% well_order_on_empty
thf(fact_1037_well__order__on__empty,axiom,
order_6972113574731384241r_on_v @ bot_bot_set_v @ bot_bo723834152578015283od_v_v ).
% well_order_on_empty
thf(fact_1038_is__empty__set,axiom,
! [Xs: list_v] :
( ( is_empty_v @ ( set_v2 @ Xs ) )
= ( null_v @ Xs ) ) ).
% is_empty_set
thf(fact_1039_lexord__same__pref__iff,axiom,
! [Xs: list_v,Ys: list_v,Zs3: list_v,R: set_Product_prod_v_v] :
( ( member418487059593946000list_v @ ( produc6795410681906604247list_v @ ( append_v @ Xs @ Ys ) @ ( append_v @ Xs @ Zs3 ) ) @ ( lexord_v @ R ) )
= ( ? [X2: v] :
( ( member_v @ X2 @ ( set_v2 @ Xs ) )
& ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X2 @ X2 ) @ R ) )
| ( member418487059593946000list_v @ ( produc6795410681906604247list_v @ Ys @ Zs3 ) @ ( lexord_v @ R ) ) ) ) ).
% lexord_same_pref_iff
thf(fact_1040_lexord__cons__cons,axiom,
! [A: v,X: list_v,B2: v,Y2: list_v,R: set_Product_prod_v_v] :
( ( member418487059593946000list_v @ ( produc6795410681906604247list_v @ ( cons_v @ A @ X ) @ ( cons_v @ B2 @ Y2 ) ) @ ( lexord_v @ R ) )
= ( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ A @ B2 ) @ R )
| ( ( A = B2 )
& ( member418487059593946000list_v @ ( produc6795410681906604247list_v @ X @ Y2 ) @ ( lexord_v @ R ) ) ) ) ) ).
% lexord_cons_cons
thf(fact_1041_lexord__irreflexive,axiom,
! [R: set_Product_prod_v_v,Xs: list_v] :
( ! [X3: v] :
~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X3 @ X3 ) @ R )
=> ~ ( member418487059593946000list_v @ ( produc6795410681906604247list_v @ Xs @ Xs ) @ ( lexord_v @ R ) ) ) ).
% lexord_irreflexive
thf(fact_1042_lexord__linear,axiom,
! [R: set_Product_prod_v_v,X: list_v,Y2: list_v] :
( ! [A7: v,B7: v] :
( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ A7 @ B7 ) @ R )
| ( A7 = B7 )
| ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ B7 @ A7 ) @ R ) )
=> ( ( member418487059593946000list_v @ ( produc6795410681906604247list_v @ X @ Y2 ) @ ( lexord_v @ R ) )
| ( X = Y2 )
| ( member418487059593946000list_v @ ( produc6795410681906604247list_v @ Y2 @ X ) @ ( lexord_v @ R ) ) ) ) ).
% lexord_linear
thf(fact_1043_lexord__partial__trans,axiom,
! [Xs: list_P7986770385144383213od_v_v,R: set_Pr2149350503807050951od_v_v,Ys: list_P7986770385144383213od_v_v,Zs3: list_P7986770385144383213od_v_v] :
( ! [X3: product_prod_v_v,Y: product_prod_v_v,Z3: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X3 @ ( set_Product_prod_v_v2 @ Xs ) )
=> ( ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ X3 @ Y ) @ R )
=> ( ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ Y @ Z3 ) @ R )
=> ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ X3 @ Z3 ) @ R ) ) ) )
=> ( ( member6382463057129219728od_v_v @ ( produc674067373767953879od_v_v @ Xs @ Ys ) @ ( lexord8601710409828808922od_v_v @ R ) )
=> ( ( member6382463057129219728od_v_v @ ( produc674067373767953879od_v_v @ Ys @ Zs3 ) @ ( lexord8601710409828808922od_v_v @ R ) )
=> ( member6382463057129219728od_v_v @ ( produc674067373767953879od_v_v @ Xs @ Zs3 ) @ ( lexord8601710409828808922od_v_v @ R ) ) ) ) ) ).
% lexord_partial_trans
thf(fact_1044_lexord__partial__trans,axiom,
! [Xs: list_v,R: set_Product_prod_v_v,Ys: list_v,Zs3: list_v] :
( ! [X3: v,Y: v,Z3: v] :
( ( member_v @ X3 @ ( set_v2 @ Xs ) )
=> ( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X3 @ Y ) @ R )
=> ( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ Y @ Z3 ) @ R )
=> ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X3 @ Z3 ) @ R ) ) ) )
=> ( ( member418487059593946000list_v @ ( produc6795410681906604247list_v @ Xs @ Ys ) @ ( lexord_v @ R ) )
=> ( ( member418487059593946000list_v @ ( produc6795410681906604247list_v @ Ys @ Zs3 ) @ ( lexord_v @ R ) )
=> ( member418487059593946000list_v @ ( produc6795410681906604247list_v @ Xs @ Zs3 ) @ ( lexord_v @ R ) ) ) ) ) ).
% lexord_partial_trans
thf(fact_1045_lexord__append__leftD,axiom,
! [X: list_v,U: list_v,V: list_v,R: set_Product_prod_v_v] :
( ( member418487059593946000list_v @ ( produc6795410681906604247list_v @ ( append_v @ X @ U ) @ ( append_v @ X @ V ) ) @ ( lexord_v @ R ) )
=> ( ! [A7: v] :
~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ A7 @ A7 ) @ R )
=> ( member418487059593946000list_v @ ( produc6795410681906604247list_v @ U @ V ) @ ( lexord_v @ R ) ) ) ) ).
% lexord_append_leftD
thf(fact_1046_lexord__append__left__rightI,axiom,
! [A: v,B2: v,R: set_Product_prod_v_v,U: list_v,X: list_v,Y2: list_v] :
( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ A @ B2 ) @ R )
=> ( member418487059593946000list_v @ ( produc6795410681906604247list_v @ ( append_v @ U @ ( cons_v @ A @ X ) ) @ ( append_v @ U @ ( cons_v @ B2 @ Y2 ) ) ) @ ( lexord_v @ R ) ) ) ).
% lexord_append_left_rightI
thf(fact_1047_underS__incl__iff,axiom,
! [R: set_Product_prod_v_v,A: v,B2: v] :
( ( order_8768733634509060168r_on_v @ ( field_v @ R ) @ R )
=> ( ( member_v @ A @ ( field_v @ R ) )
=> ( ( member_v @ B2 @ ( field_v @ R ) )
=> ( ( ord_less_eq_set_v @ ( order_underS_v @ R @ A ) @ ( order_underS_v @ R @ B2 ) )
= ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ A @ B2 ) @ R ) ) ) ) ) ).
% underS_incl_iff
thf(fact_1048_underS__incl__iff,axiom,
! [R: set_Pr2149350503807050951od_v_v,A: product_prod_v_v,B2: product_prod_v_v] :
( ( order_6462556390437124636od_v_v @ ( field_7153129647634986036od_v_v @ R ) @ R )
=> ( ( member7453568604450474000od_v_v @ A @ ( field_7153129647634986036od_v_v @ R ) )
=> ( ( member7453568604450474000od_v_v @ B2 @ ( field_7153129647634986036od_v_v @ R ) )
=> ( ( ord_le7336532860387713383od_v_v @ ( order_5211820470575790509od_v_v @ R @ A ) @ ( order_5211820470575790509od_v_v @ R @ B2 ) )
= ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ A @ B2 ) @ R ) ) ) ) ) ).
% underS_incl_iff
thf(fact_1049_underS__I,axiom,
! [I: product_prod_v_v,J: product_prod_v_v,R2: set_Pr2149350503807050951od_v_v] :
( ( I != J )
=> ( ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ I @ J ) @ R2 )
=> ( member7453568604450474000od_v_v @ I @ ( order_5211820470575790509od_v_v @ R2 @ J ) ) ) ) ).
% underS_I
thf(fact_1050_underS__I,axiom,
! [I: v,J: v,R2: set_Product_prod_v_v] :
( ( I != J )
=> ( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ I @ J ) @ R2 )
=> ( member_v @ I @ ( order_underS_v @ R2 @ J ) ) ) ) ).
% underS_I
thf(fact_1051_underS__E,axiom,
! [I: product_prod_v_v,R2: set_Pr2149350503807050951od_v_v,J: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ I @ ( order_5211820470575790509od_v_v @ R2 @ J ) )
=> ( ( I != J )
& ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ I @ J ) @ R2 ) ) ) ).
% underS_E
thf(fact_1052_underS__E,axiom,
! [I: v,R2: set_Product_prod_v_v,J: v] :
( ( member_v @ I @ ( order_underS_v @ R2 @ J ) )
=> ( ( I != J )
& ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ I @ J ) @ R2 ) ) ) ).
% underS_E
thf(fact_1053_underS__empty,axiom,
! [A: product_prod_v_v,R: set_Pr2149350503807050951od_v_v] :
( ~ ( member7453568604450474000od_v_v @ A @ ( field_7153129647634986036od_v_v @ R ) )
=> ( ( order_5211820470575790509od_v_v @ R @ A )
= bot_bo723834152578015283od_v_v ) ) ).
% underS_empty
thf(fact_1054_underS__empty,axiom,
! [A: v,R: set_Product_prod_v_v] :
( ~ ( member_v @ A @ ( field_v @ R ) )
=> ( ( order_underS_v @ R @ A )
= bot_bot_set_v ) ) ).
% underS_empty
thf(fact_1055_Order__Relation_OunderS__Field,axiom,
! [R: set_Product_prod_v_v,A: v] : ( ord_less_eq_set_v @ ( order_underS_v @ R @ A ) @ ( field_v @ R ) ) ).
% Order_Relation.underS_Field
thf(fact_1056_Order__Relation_OunderS__Field,axiom,
! [R: set_Pr2149350503807050951od_v_v,A: product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( order_5211820470575790509od_v_v @ R @ A ) @ ( field_7153129647634986036od_v_v @ R ) ) ).
% Order_Relation.underS_Field
thf(fact_1057_listrel__mono,axiom,
! [R: set_Product_prod_v_v,S4: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ R @ S4 )
=> ( ord_le791731619978752231list_v @ ( listrel_v_v @ R ) @ ( listrel_v_v @ S4 ) ) ) ).
% listrel_mono
thf(fact_1058_listrel_OCons,axiom,
! [X: v,Y2: v,R: set_Product_prod_v_v,Xs: list_v,Ys: list_v] :
( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X @ Y2 ) @ R )
=> ( ( member418487059593946000list_v @ ( produc6795410681906604247list_v @ Xs @ Ys ) @ ( listrel_v_v @ R ) )
=> ( member418487059593946000list_v @ ( produc6795410681906604247list_v @ ( cons_v @ X @ Xs ) @ ( cons_v @ Y2 @ Ys ) ) @ ( listrel_v_v @ R ) ) ) ) ).
% listrel.Cons
thf(fact_1059_listrel__Cons1,axiom,
! [Y2: v,Ys: list_v,Xs: list_v,R: set_Product_prod_v_v] :
( ( member418487059593946000list_v @ ( produc6795410681906604247list_v @ ( cons_v @ Y2 @ Ys ) @ Xs ) @ ( listrel_v_v @ R ) )
=> ~ ! [Y: v,Ys3: list_v] :
( ( Xs
= ( cons_v @ Y @ Ys3 ) )
=> ( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ Y2 @ Y ) @ R )
=> ~ ( member418487059593946000list_v @ ( produc6795410681906604247list_v @ Ys @ Ys3 ) @ ( listrel_v_v @ R ) ) ) ) ) ).
% listrel_Cons1
thf(fact_1060_listrel__Cons2,axiom,
! [Xs: list_v,Y2: v,Ys: list_v,R: set_Product_prod_v_v] :
( ( member418487059593946000list_v @ ( produc6795410681906604247list_v @ Xs @ ( cons_v @ Y2 @ Ys ) ) @ ( listrel_v_v @ R ) )
=> ~ ! [X3: v,Xs2: list_v] :
( ( Xs
= ( cons_v @ X3 @ Xs2 ) )
=> ( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X3 @ Y2 ) @ R )
=> ~ ( member418487059593946000list_v @ ( produc6795410681906604247list_v @ Xs2 @ Ys ) @ ( listrel_v_v @ R ) ) ) ) ) ).
% listrel_Cons2
thf(fact_1061_listrel_Ocases,axiom,
! [A1: list_v,A2: list_v,R: set_Product_prod_v_v] :
( ( member418487059593946000list_v @ ( produc6795410681906604247list_v @ A1 @ A2 ) @ ( listrel_v_v @ R ) )
=> ( ( ( A1 = nil_v )
=> ( A2 != nil_v ) )
=> ~ ! [X3: v,Y: v,Xs2: list_v] :
( ( A1
= ( cons_v @ X3 @ Xs2 ) )
=> ! [Ys3: list_v] :
( ( A2
= ( cons_v @ Y @ Ys3 ) )
=> ( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X3 @ Y ) @ R )
=> ~ ( member418487059593946000list_v @ ( produc6795410681906604247list_v @ Xs2 @ Ys3 ) @ ( listrel_v_v @ R ) ) ) ) ) ) ) ).
% listrel.cases
thf(fact_1062_listrel_Osimps,axiom,
! [A1: list_v,A2: list_v,R: set_Product_prod_v_v] :
( ( member418487059593946000list_v @ ( produc6795410681906604247list_v @ A1 @ A2 ) @ ( listrel_v_v @ R ) )
= ( ( ( A1 = nil_v )
& ( A2 = nil_v ) )
| ? [X2: v,Y3: v,Xs4: list_v,Ys2: list_v] :
( ( A1
= ( cons_v @ X2 @ Xs4 ) )
& ( A2
= ( cons_v @ Y3 @ Ys2 ) )
& ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X2 @ Y3 ) @ R )
& ( member418487059593946000list_v @ ( produc6795410681906604247list_v @ Xs4 @ Ys2 ) @ ( listrel_v_v @ R ) ) ) ) ) ).
% listrel.simps
thf(fact_1063_Refl__under__underS,axiom,
! [R: set_Pr2149350503807050951od_v_v,A: product_prod_v_v] :
( ( refl_o4548774019903118566od_v_v @ ( field_7153129647634986036od_v_v @ R ) @ R )
=> ( ( member7453568604450474000od_v_v @ A @ ( field_7153129647634986036od_v_v @ R ) )
=> ( ( order_6892855479609198156od_v_v @ R @ A )
= ( sup_su414716646722978715od_v_v @ ( order_5211820470575790509od_v_v @ R @ A ) @ ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) ) ) ) ) ).
% Refl_under_underS
thf(fact_1064_Refl__under__underS,axiom,
! [R: set_Product_prod_v_v,A: v] :
( ( refl_on_v @ ( field_v @ R ) @ R )
=> ( ( member_v @ A @ ( field_v @ R ) )
=> ( ( order_under_v @ R @ A )
= ( sup_sup_set_v @ ( order_underS_v @ R @ A ) @ ( insert_v2 @ A @ bot_bot_set_v ) ) ) ) ) ).
% Refl_under_underS
thf(fact_1065_listrel1__subset__listrel,axiom,
! [R: set_Product_prod_v_v,R3: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ R @ R3 )
=> ( ( refl_on_v @ top_top_set_v @ R3 )
=> ( ord_le791731619978752231list_v @ ( listrel1_v @ R ) @ ( listrel_v_v @ R3 ) ) ) ) ).
% listrel1_subset_listrel
thf(fact_1066_UNIV__I,axiom,
! [X: v] : ( member_v @ X @ top_top_set_v ) ).
% UNIV_I
thf(fact_1067_UNIV__I,axiom,
! [X: product_prod_v_v] : ( member7453568604450474000od_v_v @ X @ top_to5429829297380968215od_v_v ) ).
% UNIV_I
thf(fact_1068_inf__top__left,axiom,
! [X: set_v] :
( ( inf_inf_set_v @ top_top_set_v @ X )
= X ) ).
% inf_top_left
thf(fact_1069_inf__top__right,axiom,
! [X: set_v] :
( ( inf_inf_set_v @ X @ top_top_set_v )
= X ) ).
% inf_top_right
thf(fact_1070_inf__eq__top__iff,axiom,
! [X: set_v,Y2: set_v] :
( ( ( inf_inf_set_v @ X @ Y2 )
= top_top_set_v )
= ( ( X = top_top_set_v )
& ( Y2 = top_top_set_v ) ) ) ).
% inf_eq_top_iff
thf(fact_1071_top__eq__inf__iff,axiom,
! [X: set_v,Y2: set_v] :
( ( top_top_set_v
= ( inf_inf_set_v @ X @ Y2 ) )
= ( ( X = top_top_set_v )
& ( Y2 = top_top_set_v ) ) ) ).
% top_eq_inf_iff
thf(fact_1072_inf__top_Oeq__neutr__iff,axiom,
! [A: set_v,B2: set_v] :
( ( ( inf_inf_set_v @ A @ B2 )
= top_top_set_v )
= ( ( A = top_top_set_v )
& ( B2 = top_top_set_v ) ) ) ).
% inf_top.eq_neutr_iff
thf(fact_1073_inf__top_Oleft__neutral,axiom,
! [A: set_v] :
( ( inf_inf_set_v @ top_top_set_v @ A )
= A ) ).
% inf_top.left_neutral
thf(fact_1074_inf__top_Oneutr__eq__iff,axiom,
! [A: set_v,B2: set_v] :
( ( top_top_set_v
= ( inf_inf_set_v @ A @ B2 ) )
= ( ( A = top_top_set_v )
& ( B2 = top_top_set_v ) ) ) ).
% inf_top.neutr_eq_iff
thf(fact_1075_inf__top_Oright__neutral,axiom,
! [A: set_v] :
( ( inf_inf_set_v @ A @ top_top_set_v )
= A ) ).
% inf_top.right_neutral
thf(fact_1076_boolean__algebra_Odisj__one__right,axiom,
! [X: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ X @ top_to5429829297380968215od_v_v )
= top_to5429829297380968215od_v_v ) ).
% boolean_algebra.disj_one_right
thf(fact_1077_boolean__algebra_Odisj__one__left,axiom,
! [X: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ top_to5429829297380968215od_v_v @ X )
= top_to5429829297380968215od_v_v ) ).
% boolean_algebra.disj_one_left
thf(fact_1078_sup__top__right,axiom,
! [X: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ X @ top_to5429829297380968215od_v_v )
= top_to5429829297380968215od_v_v ) ).
% sup_top_right
thf(fact_1079_sup__top__left,axiom,
! [X: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ top_to5429829297380968215od_v_v @ X )
= top_to5429829297380968215od_v_v ) ).
% sup_top_left
thf(fact_1080_Int__UNIV,axiom,
! [A3: set_v,B: set_v] :
( ( ( inf_inf_set_v @ A3 @ B )
= top_top_set_v )
= ( ( A3 = top_top_set_v )
& ( B = top_top_set_v ) ) ) ).
% Int_UNIV
thf(fact_1081_Diff__UNIV,axiom,
! [A3: set_Product_prod_v_v] :
( ( minus_4183494784930505774od_v_v @ A3 @ top_to5429829297380968215od_v_v )
= bot_bo723834152578015283od_v_v ) ).
% Diff_UNIV
thf(fact_1082_Diff__UNIV,axiom,
! [A3: set_v] :
( ( minus_minus_set_v @ A3 @ top_top_set_v )
= bot_bot_set_v ) ).
% Diff_UNIV
thf(fact_1083_top_Oextremum__uniqueI,axiom,
! [A: set_v] :
( ( ord_less_eq_set_v @ top_top_set_v @ A )
=> ( A = top_top_set_v ) ) ).
% top.extremum_uniqueI
thf(fact_1084_top_Oextremum__uniqueI,axiom,
! [A: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ top_to5429829297380968215od_v_v @ A )
=> ( A = top_to5429829297380968215od_v_v ) ) ).
% top.extremum_uniqueI
thf(fact_1085_top_Oextremum__unique,axiom,
! [A: set_v] :
( ( ord_less_eq_set_v @ top_top_set_v @ A )
= ( A = top_top_set_v ) ) ).
% top.extremum_unique
thf(fact_1086_top_Oextremum__unique,axiom,
! [A: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ top_to5429829297380968215od_v_v @ A )
= ( A = top_to5429829297380968215od_v_v ) ) ).
% top.extremum_unique
thf(fact_1087_top__greatest,axiom,
! [A: set_v] : ( ord_less_eq_set_v @ A @ top_top_set_v ) ).
% top_greatest
thf(fact_1088_top__greatest,axiom,
! [A: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ A @ top_to5429829297380968215od_v_v ) ).
% top_greatest
thf(fact_1089_boolean__algebra_Oconj__one__right,axiom,
! [X: set_v] :
( ( inf_inf_set_v @ X @ top_top_set_v )
= X ) ).
% boolean_algebra.conj_one_right
thf(fact_1090_subset__UNIV,axiom,
! [A3: set_v] : ( ord_less_eq_set_v @ A3 @ top_top_set_v ) ).
% subset_UNIV
thf(fact_1091_subset__UNIV,axiom,
! [A3: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ A3 @ top_to5429829297380968215od_v_v ) ).
% subset_UNIV
thf(fact_1092_Int__UNIV__right,axiom,
! [A3: set_v] :
( ( inf_inf_set_v @ A3 @ top_top_set_v )
= A3 ) ).
% Int_UNIV_right
thf(fact_1093_Int__UNIV__left,axiom,
! [B: set_v] :
( ( inf_inf_set_v @ top_top_set_v @ B )
= B ) ).
% Int_UNIV_left
thf(fact_1094_Un__UNIV__right,axiom,
! [A3: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A3 @ top_to5429829297380968215od_v_v )
= top_to5429829297380968215od_v_v ) ).
% Un_UNIV_right
thf(fact_1095_Un__UNIV__left,axiom,
! [B: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ top_to5429829297380968215od_v_v @ B )
= top_to5429829297380968215od_v_v ) ).
% Un_UNIV_left
thf(fact_1096_UNIV__witness,axiom,
? [X3: v] : ( member_v @ X3 @ top_top_set_v ) ).
% UNIV_witness
thf(fact_1097_UNIV__witness,axiom,
? [X3: product_prod_v_v] : ( member7453568604450474000od_v_v @ X3 @ top_to5429829297380968215od_v_v ) ).
% UNIV_witness
thf(fact_1098_UNIV__eq__I,axiom,
! [A3: set_v] :
( ! [X3: v] : ( member_v @ X3 @ A3 )
=> ( top_top_set_v = A3 ) ) ).
% UNIV_eq_I
thf(fact_1099_UNIV__eq__I,axiom,
! [A3: set_Product_prod_v_v] :
( ! [X3: product_prod_v_v] : ( member7453568604450474000od_v_v @ X3 @ A3 )
=> ( top_to5429829297380968215od_v_v = A3 ) ) ).
% UNIV_eq_I
thf(fact_1100_insert__UNIV,axiom,
! [X: v] :
( ( insert_v2 @ X @ top_top_set_v )
= top_top_set_v ) ).
% insert_UNIV
thf(fact_1101_insert__UNIV,axiom,
! [X: product_prod_v_v] :
( ( insert1338601472111419319od_v_v @ X @ top_to5429829297380968215od_v_v )
= top_to5429829297380968215od_v_v ) ).
% insert_UNIV
thf(fact_1102_empty__not__UNIV,axiom,
bot_bo723834152578015283od_v_v != top_to5429829297380968215od_v_v ).
% empty_not_UNIV
thf(fact_1103_empty__not__UNIV,axiom,
bot_bot_set_v != top_top_set_v ).
% empty_not_UNIV
thf(fact_1104_reflI,axiom,
! [R: set_Product_prod_v_v] :
( ! [X3: v] : ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X3 @ X3 ) @ R )
=> ( refl_on_v @ top_top_set_v @ R ) ) ).
% reflI
thf(fact_1105_reflD,axiom,
! [R: set_Product_prod_v_v,A: v] :
( ( refl_on_v @ top_top_set_v @ R )
=> ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ A @ A ) @ R ) ) ).
% reflD
thf(fact_1106_under__Field,axiom,
! [R: set_Product_prod_v_v,A: v] : ( ord_less_eq_set_v @ ( order_under_v @ R @ A ) @ ( field_v @ R ) ) ).
% under_Field
thf(fact_1107_under__Field,axiom,
! [R: set_Pr2149350503807050951od_v_v,A: product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( order_6892855479609198156od_v_v @ R @ A ) @ ( field_7153129647634986036od_v_v @ R ) ) ).
% under_Field
thf(fact_1108_underS__subset__under,axiom,
! [R: set_Product_prod_v_v,A: v] : ( ord_less_eq_set_v @ ( order_underS_v @ R @ A ) @ ( order_under_v @ R @ A ) ) ).
% underS_subset_under
thf(fact_1109_underS__subset__under,axiom,
! [R: set_Pr2149350503807050951od_v_v,A: product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( order_5211820470575790509od_v_v @ R @ A ) @ ( order_6892855479609198156od_v_v @ R @ A ) ) ).
% underS_subset_under
thf(fact_1110_boolean__algebra_Ocomplement__unique,axiom,
! [A: set_Product_prod_v_v,X: set_Product_prod_v_v,Y2: set_Product_prod_v_v] :
( ( ( inf_in6271465464967711157od_v_v @ A @ X )
= bot_bo723834152578015283od_v_v )
=> ( ( ( sup_su414716646722978715od_v_v @ A @ X )
= top_to5429829297380968215od_v_v )
=> ( ( ( inf_in6271465464967711157od_v_v @ A @ Y2 )
= bot_bo723834152578015283od_v_v )
=> ( ( ( sup_su414716646722978715od_v_v @ A @ Y2 )
= top_to5429829297380968215od_v_v )
=> ( X = Y2 ) ) ) ) ) ).
% boolean_algebra.complement_unique
thf(fact_1111_boolean__algebra_Ocomplement__unique,axiom,
! [A: set_v,X: set_v,Y2: set_v] :
( ( ( inf_inf_set_v @ A @ X )
= bot_bot_set_v )
=> ( ( ( sup_sup_set_v @ A @ X )
= top_top_set_v )
=> ( ( ( inf_inf_set_v @ A @ Y2 )
= bot_bot_set_v )
=> ( ( ( sup_sup_set_v @ A @ Y2 )
= top_top_set_v )
=> ( X = Y2 ) ) ) ) ) ).
% boolean_algebra.complement_unique
thf(fact_1112_cofinite__bot,axiom,
( ( cofinite_v = bot_bot_filter_v )
= ( finite_finite_v @ top_top_set_v ) ) ).
% cofinite_bot
thf(fact_1113_irrefl__onD,axiom,
! [A3: set_Product_prod_v_v,R: set_Pr2149350503807050951od_v_v,A: product_prod_v_v] :
( ( irrefl758561187244522973od_v_v @ A3 @ R )
=> ( ( member7453568604450474000od_v_v @ A @ A3 )
=> ~ ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ A @ A ) @ R ) ) ) ).
% irrefl_onD
thf(fact_1114_irrefl__onD,axiom,
! [A3: set_v,R: set_Product_prod_v_v,A: v] :
( ( irrefl_on_v @ A3 @ R )
=> ( ( member_v @ A @ A3 )
=> ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ A @ A ) @ R ) ) ) ).
% irrefl_onD
thf(fact_1115_irrefl__onI,axiom,
! [A3: set_Product_prod_v_v,R: set_Pr2149350503807050951od_v_v] :
( ! [A7: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ A7 @ A3 )
=> ~ ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ A7 @ A7 ) @ R ) )
=> ( irrefl758561187244522973od_v_v @ A3 @ R ) ) ).
% irrefl_onI
thf(fact_1116_irrefl__onI,axiom,
! [A3: set_v,R: set_Product_prod_v_v] :
( ! [A7: v] :
( ( member_v @ A7 @ A3 )
=> ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ A7 @ A7 ) @ R ) )
=> ( irrefl_on_v @ A3 @ R ) ) ).
% irrefl_onI
thf(fact_1117_irrefl__on__def,axiom,
( irrefl_on_v
= ( ^ [A4: set_v,R4: set_Product_prod_v_v] :
! [X2: v] :
( ( member_v @ X2 @ A4 )
=> ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X2 @ X2 ) @ R4 ) ) ) ) ).
% irrefl_on_def
thf(fact_1118_irrefl__on__subset,axiom,
! [A3: set_v,R: set_Product_prod_v_v,B: set_v] :
( ( irrefl_on_v @ A3 @ R )
=> ( ( ord_less_eq_set_v @ B @ A3 )
=> ( irrefl_on_v @ B @ R ) ) ) ).
% irrefl_on_subset
thf(fact_1119_irrefl__on__subset,axiom,
! [A3: set_Product_prod_v_v,R: set_Pr2149350503807050951od_v_v,B: set_Product_prod_v_v] :
( ( irrefl758561187244522973od_v_v @ A3 @ R )
=> ( ( ord_le7336532860387713383od_v_v @ B @ A3 )
=> ( irrefl758561187244522973od_v_v @ B @ R ) ) ) ).
% irrefl_on_subset
thf(fact_1120_irreflD,axiom,
! [R: set_Product_prod_v_v,X: v] :
( ( irrefl_on_v @ top_top_set_v @ R )
=> ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X @ X ) @ R ) ) ).
% irreflD
thf(fact_1121_irreflI,axiom,
! [R: set_Product_prod_v_v] :
( ! [A7: v] :
~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ A7 @ A7 ) @ R )
=> ( irrefl_on_v @ top_top_set_v @ R ) ) ).
% irreflI
thf(fact_1122_lenlex__irreflexive,axiom,
! [R: set_Product_prod_v_v,Xs: list_v] :
( ! [X3: v] :
~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X3 @ X3 ) @ R )
=> ~ ( member418487059593946000list_v @ ( produc6795410681906604247list_v @ Xs @ Xs ) @ ( lenlex_v @ R ) ) ) ).
% lenlex_irreflexive
thf(fact_1123_Cons__in__shuffles__iff,axiom,
! [Z: v,Zs3: list_v,Xs: list_v,Ys: list_v] :
( ( member_list_v @ ( cons_v @ Z @ Zs3 ) @ ( shuffles_v @ Xs @ Ys ) )
= ( ( ( Xs != nil_v )
& ( ( hd_v @ Xs )
= Z )
& ( member_list_v @ Zs3 @ ( shuffles_v @ ( tl_v @ Xs ) @ Ys ) ) )
| ( ( Ys != nil_v )
& ( ( hd_v @ Ys )
= Z )
& ( member_list_v @ Zs3 @ ( shuffles_v @ Xs @ ( tl_v @ Ys ) ) ) ) ) ) ).
% Cons_in_shuffles_iff
thf(fact_1124_Range__insert,axiom,
! [A: v,B2: v,R: set_Product_prod_v_v] :
( ( range_v_v @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ A @ B2 ) @ R ) )
= ( insert_v2 @ B2 @ ( range_v_v @ R ) ) ) ).
% Range_insert
thf(fact_1125_Range__empty,axiom,
( ( range_v_v @ bot_bo723834152578015283od_v_v )
= bot_bot_set_v ) ).
% Range_empty
thf(fact_1126_Range_Ocases,axiom,
! [A: v,R: set_Product_prod_v_v] :
( ( member_v @ A @ ( range_v_v @ R ) )
=> ~ ! [A7: v] :
~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ A7 @ A ) @ R ) ) ).
% Range.cases
thf(fact_1127_Range_Osimps,axiom,
! [A: v,R: set_Product_prod_v_v] :
( ( member_v @ A @ ( range_v_v @ R ) )
= ( ? [A6: v,B6: v] :
( ( A = B6 )
& ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ A6 @ B6 ) @ R ) ) ) ) ).
% Range.simps
thf(fact_1128_Range_Ointros,axiom,
! [A: v,B2: v,R: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ A @ B2 ) @ R )
=> ( member_v @ B2 @ ( range_v_v @ R ) ) ) ).
% Range.intros
thf(fact_1129_RangeE,axiom,
! [B2: v,R: set_Product_prod_v_v] :
( ( member_v @ B2 @ ( range_v_v @ R ) )
=> ~ ! [A7: v] :
~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ A7 @ B2 ) @ R ) ) ).
% RangeE
thf(fact_1130_Range__iff,axiom,
! [A: v,R: set_Product_prod_v_v] :
( ( member_v @ A @ ( range_v_v @ R ) )
= ( ? [Y3: v] : ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ Y3 @ A ) @ R ) ) ) ).
% Range_iff
thf(fact_1131_set__shuffles,axiom,
! [Zs3: list_v,Xs: list_v,Ys: list_v] :
( ( member_list_v @ Zs3 @ ( shuffles_v @ Xs @ Ys ) )
=> ( ( set_v2 @ Zs3 )
= ( sup_sup_set_v @ ( set_v2 @ Xs ) @ ( set_v2 @ Ys ) ) ) ) ).
% set_shuffles
thf(fact_1132_set__shuffles,axiom,
! [Zs3: list_P7986770385144383213od_v_v,Xs: list_P7986770385144383213od_v_v,Ys: list_P7986770385144383213od_v_v] :
( ( member4190458934886417558od_v_v @ Zs3 @ ( shuffl71542398924059522od_v_v @ Xs @ Ys ) )
=> ( ( set_Product_prod_v_v2 @ Zs3 )
= ( sup_su414716646722978715od_v_v @ ( set_Product_prod_v_v2 @ Xs ) @ ( set_Product_prod_v_v2 @ Ys ) ) ) ) ).
% set_shuffles
thf(fact_1133_Range__empty__iff,axiom,
! [R: set_Product_prod_v_v] :
( ( ( range_v_v @ R )
= bot_bot_set_v )
= ( R = bot_bo723834152578015283od_v_v ) ) ).
% Range_empty_iff
thf(fact_1134_Range__mono,axiom,
! [R: set_Product_prod_v_v,S4: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ R @ S4 )
=> ( ord_less_eq_set_v @ ( range_v_v @ R ) @ ( range_v_v @ S4 ) ) ) ).
% Range_mono
thf(fact_1135_distinct__disjoint__shuffles,axiom,
! [Xs: list_P7986770385144383213od_v_v,Ys: list_P7986770385144383213od_v_v,Zs3: list_P7986770385144383213od_v_v] :
( ( distin6159370996967099744od_v_v @ Xs )
=> ( ( distin6159370996967099744od_v_v @ Ys )
=> ( ( ( inf_in6271465464967711157od_v_v @ ( set_Product_prod_v_v2 @ Xs ) @ ( set_Product_prod_v_v2 @ Ys ) )
= bot_bo723834152578015283od_v_v )
=> ( ( member4190458934886417558od_v_v @ Zs3 @ ( shuffl71542398924059522od_v_v @ Xs @ Ys ) )
=> ( distin6159370996967099744od_v_v @ Zs3 ) ) ) ) ) ).
% distinct_disjoint_shuffles
thf(fact_1136_distinct__disjoint__shuffles,axiom,
! [Xs: list_v,Ys: list_v,Zs3: list_v] :
( ( distinct_v @ Xs )
=> ( ( distinct_v @ Ys )
=> ( ( ( inf_inf_set_v @ ( set_v2 @ Xs ) @ ( set_v2 @ Ys ) )
= bot_bot_set_v )
=> ( ( member_list_v @ Zs3 @ ( shuffles_v @ Xs @ Ys ) )
=> ( distinct_v @ Zs3 ) ) ) ) ) ).
% distinct_disjoint_shuffles
thf(fact_1137_Domain__insert,axiom,
! [A: v,B2: v,R: set_Product_prod_v_v] :
( ( domain_v_v @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ A @ B2 ) @ R ) )
= ( insert_v2 @ A @ ( domain_v_v @ R ) ) ) ).
% Domain_insert
thf(fact_1138_Domain__empty,axiom,
( ( domain_v_v @ bot_bo723834152578015283od_v_v )
= bot_bot_set_v ) ).
% Domain_empty
thf(fact_1139_Domain__mono,axiom,
! [R: set_Product_prod_v_v,S4: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ R @ S4 )
=> ( ord_less_eq_set_v @ ( domain_v_v @ R ) @ ( domain_v_v @ S4 ) ) ) ).
% Domain_mono
thf(fact_1140_Domain__empty__iff,axiom,
! [R: set_Product_prod_v_v] :
( ( ( domain_v_v @ R )
= bot_bot_set_v )
= ( R = bot_bo723834152578015283od_v_v ) ) ).
% Domain_empty_iff
thf(fact_1141_Domain__iff,axiom,
! [A: v,R: set_Product_prod_v_v] :
( ( member_v @ A @ ( domain_v_v @ R ) )
= ( ? [Y3: v] : ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ A @ Y3 ) @ R ) ) ) ).
% Domain_iff
thf(fact_1142_DomainE,axiom,
! [A: v,R: set_Product_prod_v_v] :
( ( member_v @ A @ ( domain_v_v @ R ) )
=> ~ ! [B7: v] :
~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ A @ B7 ) @ R ) ) ).
% DomainE
thf(fact_1143_Domain_ODomainI,axiom,
! [A: v,B2: v,R: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ A @ B2 ) @ R )
=> ( member_v @ A @ ( domain_v_v @ R ) ) ) ).
% Domain.DomainI
thf(fact_1144_Domain_Osimps,axiom,
! [A: v,R: set_Product_prod_v_v] :
( ( member_v @ A @ ( domain_v_v @ R ) )
= ( ? [A6: v,B6: v] :
( ( A = A6 )
& ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ A6 @ B6 ) @ R ) ) ) ) ).
% Domain.simps
thf(fact_1145_Domain_Ocases,axiom,
! [A: v,R: set_Product_prod_v_v] :
( ( member_v @ A @ ( domain_v_v @ R ) )
=> ~ ! [B7: v] :
~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ A @ B7 ) @ R ) ) ).
% Domain.cases
thf(fact_1146_lex__append__left__iff,axiom,
! [R: set_Product_prod_v_v,Xs: list_v,Ys: list_v,Zs3: list_v] :
( ! [X3: v] :
~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X3 @ X3 ) @ R )
=> ( ( member418487059593946000list_v @ ( produc6795410681906604247list_v @ ( append_v @ Xs @ Ys ) @ ( append_v @ Xs @ Zs3 ) ) @ ( lex_v @ R ) )
= ( member418487059593946000list_v @ ( produc6795410681906604247list_v @ Ys @ Zs3 ) @ ( lex_v @ R ) ) ) ) ).
% lex_append_left_iff
thf(fact_1147_lex__append__leftD,axiom,
! [R: set_Product_prod_v_v,Xs: list_v,Ys: list_v,Zs3: list_v] :
( ! [X3: v] :
~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X3 @ X3 ) @ R )
=> ( ( member418487059593946000list_v @ ( produc6795410681906604247list_v @ ( append_v @ Xs @ Ys ) @ ( append_v @ Xs @ Zs3 ) ) @ ( lex_v @ R ) )
=> ( member418487059593946000list_v @ ( produc6795410681906604247list_v @ Ys @ Zs3 ) @ ( lex_v @ R ) ) ) ) ).
% lex_append_leftD
thf(fact_1148_lists__empty,axiom,
( ( lists_5865669170805476827od_v_v @ bot_bo723834152578015283od_v_v )
= ( insert4087971119735676093od_v_v @ nil_Product_prod_v_v @ bot_bo54012148666785209od_v_v ) ) ).
% lists_empty
thf(fact_1149_lists__empty,axiom,
( ( lists_v @ bot_bot_set_v )
= ( insert_list_v @ nil_v @ bot_bot_set_list_v ) ) ).
% lists_empty
thf(fact_1150_in__listsI,axiom,
! [Xs: list_P7986770385144383213od_v_v,A3: set_Product_prod_v_v] :
( ! [X3: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X3 @ ( set_Product_prod_v_v2 @ Xs ) )
=> ( member7453568604450474000od_v_v @ X3 @ A3 ) )
=> ( member4190458934886417558od_v_v @ Xs @ ( lists_5865669170805476827od_v_v @ A3 ) ) ) ).
% in_listsI
thf(fact_1151_in__listsI,axiom,
! [Xs: list_v,A3: set_v] :
( ! [X3: v] :
( ( member_v @ X3 @ ( set_v2 @ Xs ) )
=> ( member_v @ X3 @ A3 ) )
=> ( member_list_v @ Xs @ ( lists_v @ A3 ) ) ) ).
% in_listsI
thf(fact_1152_lists__Int__eq,axiom,
! [A3: set_v,B: set_v] :
( ( lists_v @ ( inf_inf_set_v @ A3 @ B ) )
= ( inf_inf_set_list_v @ ( lists_v @ A3 ) @ ( lists_v @ B ) ) ) ).
% lists_Int_eq
thf(fact_1153_in__listsD,axiom,
! [Xs: list_P7986770385144383213od_v_v,A3: set_Product_prod_v_v] :
( ( member4190458934886417558od_v_v @ Xs @ ( lists_5865669170805476827od_v_v @ A3 ) )
=> ! [X4: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X4 @ ( set_Product_prod_v_v2 @ Xs ) )
=> ( member7453568604450474000od_v_v @ X4 @ A3 ) ) ) ).
% in_listsD
thf(fact_1154_in__listsD,axiom,
! [Xs: list_v,A3: set_v] :
( ( member_list_v @ Xs @ ( lists_v @ A3 ) )
=> ! [X4: v] :
( ( member_v @ X4 @ ( set_v2 @ Xs ) )
=> ( member_v @ X4 @ A3 ) ) ) ).
% in_listsD
thf(fact_1155_in__lists__conv__set,axiom,
! [Xs: list_P7986770385144383213od_v_v,A3: set_Product_prod_v_v] :
( ( member4190458934886417558od_v_v @ Xs @ ( lists_5865669170805476827od_v_v @ A3 ) )
= ( ! [X2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ ( set_Product_prod_v_v2 @ Xs ) )
=> ( member7453568604450474000od_v_v @ X2 @ A3 ) ) ) ) ).
% in_lists_conv_set
thf(fact_1156_in__lists__conv__set,axiom,
! [Xs: list_v,A3: set_v] :
( ( member_list_v @ Xs @ ( lists_v @ A3 ) )
= ( ! [X2: v] :
( ( member_v @ X2 @ ( set_v2 @ Xs ) )
=> ( member_v @ X2 @ A3 ) ) ) ) ).
% in_lists_conv_set
thf(fact_1157_lists__mono,axiom,
! [A3: set_v,B: set_v] :
( ( ord_less_eq_set_v @ A3 @ B )
=> ( ord_le1129530298279361049list_v @ ( lists_v @ A3 ) @ ( lists_v @ B ) ) ) ).
% lists_mono
thf(fact_1158_lists__mono,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A3 @ B )
=> ( ord_le5393391283775026413od_v_v @ ( lists_5865669170805476827od_v_v @ A3 ) @ ( lists_5865669170805476827od_v_v @ B ) ) ) ).
% lists_mono
thf(fact_1159_lists__IntI,axiom,
! [L2: list_v,A3: set_v,B: set_v] :
( ( member_list_v @ L2 @ ( lists_v @ A3 ) )
=> ( ( member_list_v @ L2 @ ( lists_v @ B ) )
=> ( member_list_v @ L2 @ ( lists_v @ ( inf_inf_set_v @ A3 @ B ) ) ) ) ) ).
% lists_IntI
thf(fact_1160_image__eqI,axiom,
! [B2: v,F: v > v,X: v,A3: set_v] :
( ( B2
= ( F @ X ) )
=> ( ( member_v @ X @ A3 )
=> ( member_v @ B2 @ ( image_v_v2 @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_1161_image__eqI,axiom,
! [B2: product_prod_v_v,F: v > product_prod_v_v,X: v,A3: set_v] :
( ( B2
= ( F @ X ) )
=> ( ( member_v @ X @ A3 )
=> ( member7453568604450474000od_v_v @ B2 @ ( image_9222788639401671577od_v_v @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_1162_image__eqI,axiom,
! [B2: v,F: product_prod_v_v > v,X: product_prod_v_v,A3: set_Product_prod_v_v] :
( ( B2
= ( F @ X ) )
=> ( ( member7453568604450474000od_v_v @ X @ A3 )
=> ( member_v @ B2 @ ( image_6152814753742948081_v_v_v @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_1163_image__eqI,axiom,
! [B2: product_prod_v_v,F: product_prod_v_v > product_prod_v_v,X: product_prod_v_v,A3: set_Product_prod_v_v] :
( ( B2
= ( F @ X ) )
=> ( ( member7453568604450474000od_v_v @ X @ A3 )
=> ( member7453568604450474000od_v_v @ B2 @ ( image_781944334261467077od_v_v @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_1164_image__is__empty,axiom,
! [F: product_prod_v_v > product_prod_v_v,A3: set_Product_prod_v_v] :
( ( ( image_781944334261467077od_v_v @ F @ A3 )
= bot_bo723834152578015283od_v_v )
= ( A3 = bot_bo723834152578015283od_v_v ) ) ).
% image_is_empty
thf(fact_1165_image__is__empty,axiom,
! [F: v > product_prod_v_v,A3: set_v] :
( ( ( image_9222788639401671577od_v_v @ F @ A3 )
= bot_bo723834152578015283od_v_v )
= ( A3 = bot_bot_set_v ) ) ).
% image_is_empty
thf(fact_1166_image__is__empty,axiom,
! [F: product_prod_v_v > v,A3: set_Product_prod_v_v] :
( ( ( image_6152814753742948081_v_v_v @ F @ A3 )
= bot_bot_set_v )
= ( A3 = bot_bo723834152578015283od_v_v ) ) ).
% image_is_empty
thf(fact_1167_image__is__empty,axiom,
! [F: v > v,A3: set_v] :
( ( ( image_v_v2 @ F @ A3 )
= bot_bot_set_v )
= ( A3 = bot_bot_set_v ) ) ).
% image_is_empty
thf(fact_1168_empty__is__image,axiom,
! [F: product_prod_v_v > product_prod_v_v,A3: set_Product_prod_v_v] :
( ( bot_bo723834152578015283od_v_v
= ( image_781944334261467077od_v_v @ F @ A3 ) )
= ( A3 = bot_bo723834152578015283od_v_v ) ) ).
% empty_is_image
thf(fact_1169_empty__is__image,axiom,
! [F: v > product_prod_v_v,A3: set_v] :
( ( bot_bo723834152578015283od_v_v
= ( image_9222788639401671577od_v_v @ F @ A3 ) )
= ( A3 = bot_bot_set_v ) ) ).
% empty_is_image
thf(fact_1170_empty__is__image,axiom,
! [F: product_prod_v_v > v,A3: set_Product_prod_v_v] :
( ( bot_bot_set_v
= ( image_6152814753742948081_v_v_v @ F @ A3 ) )
= ( A3 = bot_bo723834152578015283od_v_v ) ) ).
% empty_is_image
thf(fact_1171_empty__is__image,axiom,
! [F: v > v,A3: set_v] :
( ( bot_bot_set_v
= ( image_v_v2 @ F @ A3 ) )
= ( A3 = bot_bot_set_v ) ) ).
% empty_is_image
thf(fact_1172_image__empty,axiom,
! [F: product_prod_v_v > product_prod_v_v] :
( ( image_781944334261467077od_v_v @ F @ bot_bo723834152578015283od_v_v )
= bot_bo723834152578015283od_v_v ) ).
% image_empty
thf(fact_1173_image__empty,axiom,
! [F: product_prod_v_v > v] :
( ( image_6152814753742948081_v_v_v @ F @ bot_bo723834152578015283od_v_v )
= bot_bot_set_v ) ).
% image_empty
thf(fact_1174_image__empty,axiom,
! [F: v > product_prod_v_v] :
( ( image_9222788639401671577od_v_v @ F @ bot_bot_set_v )
= bot_bo723834152578015283od_v_v ) ).
% image_empty
thf(fact_1175_image__empty,axiom,
! [F: v > v] :
( ( image_v_v2 @ F @ bot_bot_set_v )
= bot_bot_set_v ) ).
% image_empty
thf(fact_1176_image__insert,axiom,
! [F: v > v,A: v,B: set_v] :
( ( image_v_v2 @ F @ ( insert_v2 @ A @ B ) )
= ( insert_v2 @ ( F @ A ) @ ( image_v_v2 @ F @ B ) ) ) ).
% image_insert
thf(fact_1177_image__insert,axiom,
! [F: v > product_prod_v_v,A: v,B: set_v] :
( ( image_9222788639401671577od_v_v @ F @ ( insert_v2 @ A @ B ) )
= ( insert1338601472111419319od_v_v @ ( F @ A ) @ ( image_9222788639401671577od_v_v @ F @ B ) ) ) ).
% image_insert
thf(fact_1178_image__insert,axiom,
! [F: product_prod_v_v > v,A: product_prod_v_v,B: set_Product_prod_v_v] :
( ( image_6152814753742948081_v_v_v @ F @ ( insert1338601472111419319od_v_v @ A @ B ) )
= ( insert_v2 @ ( F @ A ) @ ( image_6152814753742948081_v_v_v @ F @ B ) ) ) ).
% image_insert
thf(fact_1179_image__insert,axiom,
! [F: product_prod_v_v > product_prod_v_v,A: product_prod_v_v,B: set_Product_prod_v_v] :
( ( image_781944334261467077od_v_v @ F @ ( insert1338601472111419319od_v_v @ A @ B ) )
= ( insert1338601472111419319od_v_v @ ( F @ A ) @ ( image_781944334261467077od_v_v @ F @ B ) ) ) ).
% image_insert
thf(fact_1180_insert__image,axiom,
! [X: v,A3: set_v,F: v > v] :
( ( member_v @ X @ A3 )
=> ( ( insert_v2 @ ( F @ X ) @ ( image_v_v2 @ F @ A3 ) )
= ( image_v_v2 @ F @ A3 ) ) ) ).
% insert_image
thf(fact_1181_insert__image,axiom,
! [X: v,A3: set_v,F: v > product_prod_v_v] :
( ( member_v @ X @ A3 )
=> ( ( insert1338601472111419319od_v_v @ ( F @ X ) @ ( image_9222788639401671577od_v_v @ F @ A3 ) )
= ( image_9222788639401671577od_v_v @ F @ A3 ) ) ) ).
% insert_image
thf(fact_1182_insert__image,axiom,
! [X: product_prod_v_v,A3: set_Product_prod_v_v,F: product_prod_v_v > v] :
( ( member7453568604450474000od_v_v @ X @ A3 )
=> ( ( insert_v2 @ ( F @ X ) @ ( image_6152814753742948081_v_v_v @ F @ A3 ) )
= ( image_6152814753742948081_v_v_v @ F @ A3 ) ) ) ).
% insert_image
thf(fact_1183_insert__image,axiom,
! [X: product_prod_v_v,A3: set_Product_prod_v_v,F: product_prod_v_v > product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X @ A3 )
=> ( ( insert1338601472111419319od_v_v @ ( F @ X ) @ ( image_781944334261467077od_v_v @ F @ A3 ) )
= ( image_781944334261467077od_v_v @ F @ A3 ) ) ) ).
% insert_image
thf(fact_1184_ImageI,axiom,
! [A: v,B2: product_prod_v_v,R: set_Pr7862341151230101147od_v_v,A3: set_v] :
( ( member5456077685714336484od_v_v @ ( produc2254008198234949931od_v_v @ A @ B2 ) @ R )
=> ( ( member_v @ A @ A3 )
=> ( member7453568604450474000od_v_v @ B2 @ ( image_4621343323592381799od_v_v @ R @ A3 ) ) ) ) ).
% ImageI
thf(fact_1185_ImageI,axiom,
! [A: product_prod_v_v,B2: v,R: set_Pr7679524143894959091_v_v_v,A3: set_Product_prod_v_v] :
( ( member5544786109013881916_v_v_v @ ( produc8407406349431002243_v_v_v @ A @ B2 ) @ R )
=> ( ( member7453568604450474000od_v_v @ A @ A3 )
=> ( member_v @ B2 @ ( image_1551369437933658303_v_v_v @ R @ A3 ) ) ) ) ).
% ImageI
thf(fact_1186_ImageI,axiom,
! [A: product_prod_v_v,B2: product_prod_v_v,R: set_Pr2149350503807050951od_v_v,A3: set_Product_prod_v_v] :
( ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ A @ B2 ) @ R )
=> ( ( member7453568604450474000od_v_v @ A @ A3 )
=> ( member7453568604450474000od_v_v @ B2 @ ( image_5221874569633403795od_v_v @ R @ A3 ) ) ) ) ).
% ImageI
thf(fact_1187_ImageI,axiom,
! [A: v,B2: v,R: set_Product_prod_v_v,A3: set_v] :
( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ A @ B2 ) @ R )
=> ( ( member_v @ A @ A3 )
=> ( member_v @ B2 @ ( image_v_v @ R @ A3 ) ) ) ) ).
% ImageI
thf(fact_1188_Image__empty2,axiom,
! [R2: set_Pr2149350503807050951od_v_v] :
( ( image_5221874569633403795od_v_v @ R2 @ bot_bo723834152578015283od_v_v )
= bot_bo723834152578015283od_v_v ) ).
% Image_empty2
thf(fact_1189_Image__empty2,axiom,
! [R2: set_Pr7679524143894959091_v_v_v] :
( ( image_1551369437933658303_v_v_v @ R2 @ bot_bo723834152578015283od_v_v )
= bot_bot_set_v ) ).
% Image_empty2
thf(fact_1190_Image__empty2,axiom,
! [R2: set_Pr7862341151230101147od_v_v] :
( ( image_4621343323592381799od_v_v @ R2 @ bot_bot_set_v )
= bot_bo723834152578015283od_v_v ) ).
% Image_empty2
thf(fact_1191_Image__empty2,axiom,
! [R2: set_Product_prod_v_v] :
( ( image_v_v @ R2 @ bot_bot_set_v )
= bot_bot_set_v ) ).
% Image_empty2
thf(fact_1192_Image__empty1,axiom,
! [X5: set_v] :
( ( image_v_v @ bot_bo723834152578015283od_v_v @ X5 )
= bot_bot_set_v ) ).
% Image_empty1
thf(fact_1193_Image__singleton__iff,axiom,
! [B2: v,R: set_Pr7679524143894959091_v_v_v,A: product_prod_v_v] :
( ( member_v @ B2 @ ( image_1551369437933658303_v_v_v @ R @ ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) ) )
= ( member5544786109013881916_v_v_v @ ( produc8407406349431002243_v_v_v @ A @ B2 ) @ R ) ) ).
% Image_singleton_iff
thf(fact_1194_Image__singleton__iff,axiom,
! [B2: product_prod_v_v,R: set_Pr2149350503807050951od_v_v,A: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ B2 @ ( image_5221874569633403795od_v_v @ R @ ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) ) )
= ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ A @ B2 ) @ R ) ) ).
% Image_singleton_iff
thf(fact_1195_Image__singleton__iff,axiom,
! [B2: product_prod_v_v,R: set_Pr7862341151230101147od_v_v,A: v] :
( ( member7453568604450474000od_v_v @ B2 @ ( image_4621343323592381799od_v_v @ R @ ( insert_v2 @ A @ bot_bot_set_v ) ) )
= ( member5456077685714336484od_v_v @ ( produc2254008198234949931od_v_v @ A @ B2 ) @ R ) ) ).
% Image_singleton_iff
thf(fact_1196_Image__singleton__iff,axiom,
! [B2: v,R: set_Product_prod_v_v,A: v] :
( ( member_v @ B2 @ ( image_v_v @ R @ ( insert_v2 @ A @ bot_bot_set_v ) ) )
= ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ A @ B2 ) @ R ) ) ).
% Image_singleton_iff
thf(fact_1197_Image__mono,axiom,
! [R3: set_Pr7862341151230101147od_v_v,R: set_Pr7862341151230101147od_v_v,A11: set_v,A3: set_v] :
( ( ord_le2447141079412579899od_v_v @ R3 @ R )
=> ( ( ord_less_eq_set_v @ A11 @ A3 )
=> ( ord_le7336532860387713383od_v_v @ ( image_4621343323592381799od_v_v @ R3 @ A11 ) @ ( image_4621343323592381799od_v_v @ R @ A3 ) ) ) ) ).
% Image_mono
thf(fact_1198_Image__mono,axiom,
! [R3: set_Pr7679524143894959091_v_v_v,R: set_Pr7679524143894959091_v_v_v,A11: set_Product_prod_v_v,A3: set_Product_prod_v_v] :
( ( ord_le2264324072077437843_v_v_v @ R3 @ R )
=> ( ( ord_le7336532860387713383od_v_v @ A11 @ A3 )
=> ( ord_less_eq_set_v @ ( image_1551369437933658303_v_v_v @ R3 @ A11 ) @ ( image_1551369437933658303_v_v_v @ R @ A3 ) ) ) ) ).
% Image_mono
thf(fact_1199_Image__mono,axiom,
! [R3: set_Pr2149350503807050951od_v_v,R: set_Pr2149350503807050951od_v_v,A11: set_Product_prod_v_v,A3: set_Product_prod_v_v] :
( ( ord_le6241436655786843239od_v_v @ R3 @ R )
=> ( ( ord_le7336532860387713383od_v_v @ A11 @ A3 )
=> ( ord_le7336532860387713383od_v_v @ ( image_5221874569633403795od_v_v @ R3 @ A11 ) @ ( image_5221874569633403795od_v_v @ R @ A3 ) ) ) ) ).
% Image_mono
thf(fact_1200_Image__mono,axiom,
! [R3: set_Product_prod_v_v,R: set_Product_prod_v_v,A11: set_v,A3: set_v] :
( ( ord_le7336532860387713383od_v_v @ R3 @ R )
=> ( ( ord_less_eq_set_v @ A11 @ A3 )
=> ( ord_less_eq_set_v @ ( image_v_v @ R3 @ A11 ) @ ( image_v_v @ R @ A3 ) ) ) ) ).
% Image_mono
thf(fact_1201_all__finite__subset__image,axiom,
! [F: v > v,A3: set_v,P: set_v > $o] :
( ( ! [B4: set_v] :
( ( ( finite_finite_v @ B4 )
& ( ord_less_eq_set_v @ B4 @ ( image_v_v2 @ F @ A3 ) ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_v] :
( ( ( finite_finite_v @ B4 )
& ( ord_less_eq_set_v @ B4 @ A3 ) )
=> ( P @ ( image_v_v2 @ F @ B4 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_1202_all__finite__subset__image,axiom,
! [F: product_prod_v_v > v,A3: set_Product_prod_v_v,P: set_v > $o] :
( ( ! [B4: set_v] :
( ( ( finite_finite_v @ B4 )
& ( ord_less_eq_set_v @ B4 @ ( image_6152814753742948081_v_v_v @ F @ A3 ) ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_Product_prod_v_v] :
( ( ( finite3348123685078250256od_v_v @ B4 )
& ( ord_le7336532860387713383od_v_v @ B4 @ A3 ) )
=> ( P @ ( image_6152814753742948081_v_v_v @ F @ B4 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_1203_all__finite__subset__image,axiom,
! [F: v > product_prod_v_v,A3: set_v,P: set_Product_prod_v_v > $o] :
( ( ! [B4: set_Product_prod_v_v] :
( ( ( finite3348123685078250256od_v_v @ B4 )
& ( ord_le7336532860387713383od_v_v @ B4 @ ( image_9222788639401671577od_v_v @ F @ A3 ) ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_v] :
( ( ( finite_finite_v @ B4 )
& ( ord_less_eq_set_v @ B4 @ A3 ) )
=> ( P @ ( image_9222788639401671577od_v_v @ F @ B4 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_1204_all__finite__subset__image,axiom,
! [F: product_prod_v_v > product_prod_v_v,A3: set_Product_prod_v_v,P: set_Product_prod_v_v > $o] :
( ( ! [B4: set_Product_prod_v_v] :
( ( ( finite3348123685078250256od_v_v @ B4 )
& ( ord_le7336532860387713383od_v_v @ B4 @ ( image_781944334261467077od_v_v @ F @ A3 ) ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_Product_prod_v_v] :
( ( ( finite3348123685078250256od_v_v @ B4 )
& ( ord_le7336532860387713383od_v_v @ B4 @ A3 ) )
=> ( P @ ( image_781944334261467077od_v_v @ F @ B4 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_1205_ex__finite__subset__image,axiom,
! [F: v > v,A3: set_v,P: set_v > $o] :
( ( ? [B4: set_v] :
( ( finite_finite_v @ B4 )
& ( ord_less_eq_set_v @ B4 @ ( image_v_v2 @ F @ A3 ) )
& ( P @ B4 ) ) )
= ( ? [B4: set_v] :
( ( finite_finite_v @ B4 )
& ( ord_less_eq_set_v @ B4 @ A3 )
& ( P @ ( image_v_v2 @ F @ B4 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_1206_ex__finite__subset__image,axiom,
! [F: product_prod_v_v > v,A3: set_Product_prod_v_v,P: set_v > $o] :
( ( ? [B4: set_v] :
( ( finite_finite_v @ B4 )
& ( ord_less_eq_set_v @ B4 @ ( image_6152814753742948081_v_v_v @ F @ A3 ) )
& ( P @ B4 ) ) )
= ( ? [B4: set_Product_prod_v_v] :
( ( finite3348123685078250256od_v_v @ B4 )
& ( ord_le7336532860387713383od_v_v @ B4 @ A3 )
& ( P @ ( image_6152814753742948081_v_v_v @ F @ B4 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_1207_ex__finite__subset__image,axiom,
! [F: v > product_prod_v_v,A3: set_v,P: set_Product_prod_v_v > $o] :
( ( ? [B4: set_Product_prod_v_v] :
( ( finite3348123685078250256od_v_v @ B4 )
& ( ord_le7336532860387713383od_v_v @ B4 @ ( image_9222788639401671577od_v_v @ F @ A3 ) )
& ( P @ B4 ) ) )
= ( ? [B4: set_v] :
( ( finite_finite_v @ B4 )
& ( ord_less_eq_set_v @ B4 @ A3 )
& ( P @ ( image_9222788639401671577od_v_v @ F @ B4 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_1208_ex__finite__subset__image,axiom,
! [F: product_prod_v_v > product_prod_v_v,A3: set_Product_prod_v_v,P: set_Product_prod_v_v > $o] :
( ( ? [B4: set_Product_prod_v_v] :
( ( finite3348123685078250256od_v_v @ B4 )
& ( ord_le7336532860387713383od_v_v @ B4 @ ( image_781944334261467077od_v_v @ F @ A3 ) )
& ( P @ B4 ) ) )
= ( ? [B4: set_Product_prod_v_v] :
( ( finite3348123685078250256od_v_v @ B4 )
& ( ord_le7336532860387713383od_v_v @ B4 @ A3 )
& ( P @ ( image_781944334261467077od_v_v @ F @ B4 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_1209_finite__subset__image,axiom,
! [B: set_v,F: v > v,A3: set_v] :
( ( finite_finite_v @ B )
=> ( ( ord_less_eq_set_v @ B @ ( image_v_v2 @ F @ A3 ) )
=> ? [C4: set_v] :
( ( ord_less_eq_set_v @ C4 @ A3 )
& ( finite_finite_v @ C4 )
& ( B
= ( image_v_v2 @ F @ C4 ) ) ) ) ) ).
% finite_subset_image
thf(fact_1210_finite__subset__image,axiom,
! [B: set_v,F: product_prod_v_v > v,A3: set_Product_prod_v_v] :
( ( finite_finite_v @ B )
=> ( ( ord_less_eq_set_v @ B @ ( image_6152814753742948081_v_v_v @ F @ A3 ) )
=> ? [C4: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ C4 @ A3 )
& ( finite3348123685078250256od_v_v @ C4 )
& ( B
= ( image_6152814753742948081_v_v_v @ F @ C4 ) ) ) ) ) ).
% finite_subset_image
thf(fact_1211_finite__subset__image,axiom,
! [B: set_Product_prod_v_v,F: v > product_prod_v_v,A3: set_v] :
( ( finite3348123685078250256od_v_v @ B )
=> ( ( ord_le7336532860387713383od_v_v @ B @ ( image_9222788639401671577od_v_v @ F @ A3 ) )
=> ? [C4: set_v] :
( ( ord_less_eq_set_v @ C4 @ A3 )
& ( finite_finite_v @ C4 )
& ( B
= ( image_9222788639401671577od_v_v @ F @ C4 ) ) ) ) ) ).
% finite_subset_image
thf(fact_1212_finite__subset__image,axiom,
! [B: set_Product_prod_v_v,F: product_prod_v_v > product_prod_v_v,A3: set_Product_prod_v_v] :
( ( finite3348123685078250256od_v_v @ B )
=> ( ( ord_le7336532860387713383od_v_v @ B @ ( image_781944334261467077od_v_v @ F @ A3 ) )
=> ? [C4: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ C4 @ A3 )
& ( finite3348123685078250256od_v_v @ C4 )
& ( B
= ( image_781944334261467077od_v_v @ F @ C4 ) ) ) ) ) ).
% finite_subset_image
thf(fact_1213_finite__surj,axiom,
! [A3: set_v,B: set_v,F: v > v] :
( ( finite_finite_v @ A3 )
=> ( ( ord_less_eq_set_v @ B @ ( image_v_v2 @ F @ A3 ) )
=> ( finite_finite_v @ B ) ) ) ).
% finite_surj
thf(fact_1214_finite__surj,axiom,
! [A3: set_v,B: set_Product_prod_v_v,F: v > product_prod_v_v] :
( ( finite_finite_v @ A3 )
=> ( ( ord_le7336532860387713383od_v_v @ B @ ( image_9222788639401671577od_v_v @ F @ A3 ) )
=> ( finite3348123685078250256od_v_v @ B ) ) ) ).
% finite_surj
thf(fact_1215_image__Int__subset,axiom,
! [F: v > v,A3: set_v,B: set_v] : ( ord_less_eq_set_v @ ( image_v_v2 @ F @ ( inf_inf_set_v @ A3 @ B ) ) @ ( inf_inf_set_v @ ( image_v_v2 @ F @ A3 ) @ ( image_v_v2 @ F @ B ) ) ) ).
% image_Int_subset
thf(fact_1216_image__Int__subset,axiom,
! [F: v > product_prod_v_v,A3: set_v,B: set_v] : ( ord_le7336532860387713383od_v_v @ ( image_9222788639401671577od_v_v @ F @ ( inf_inf_set_v @ A3 @ B ) ) @ ( inf_in6271465464967711157od_v_v @ ( image_9222788639401671577od_v_v @ F @ A3 ) @ ( image_9222788639401671577od_v_v @ F @ B ) ) ) ).
% image_Int_subset
thf(fact_1217_all__subset__image,axiom,
! [F: v > v,A3: set_v,P: set_v > $o] :
( ( ! [B4: set_v] :
( ( ord_less_eq_set_v @ B4 @ ( image_v_v2 @ F @ A3 ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_v] :
( ( ord_less_eq_set_v @ B4 @ A3 )
=> ( P @ ( image_v_v2 @ F @ B4 ) ) ) ) ) ).
% all_subset_image
thf(fact_1218_all__subset__image,axiom,
! [F: product_prod_v_v > v,A3: set_Product_prod_v_v,P: set_v > $o] :
( ( ! [B4: set_v] :
( ( ord_less_eq_set_v @ B4 @ ( image_6152814753742948081_v_v_v @ F @ A3 ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B4 @ A3 )
=> ( P @ ( image_6152814753742948081_v_v_v @ F @ B4 ) ) ) ) ) ).
% all_subset_image
thf(fact_1219_all__subset__image,axiom,
! [F: v > product_prod_v_v,A3: set_v,P: set_Product_prod_v_v > $o] :
( ( ! [B4: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B4 @ ( image_9222788639401671577od_v_v @ F @ A3 ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_v] :
( ( ord_less_eq_set_v @ B4 @ A3 )
=> ( P @ ( image_9222788639401671577od_v_v @ F @ B4 ) ) ) ) ) ).
% all_subset_image
thf(fact_1220_all__subset__image,axiom,
! [F: product_prod_v_v > product_prod_v_v,A3: set_Product_prod_v_v,P: set_Product_prod_v_v > $o] :
( ( ! [B4: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B4 @ ( image_781944334261467077od_v_v @ F @ A3 ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B4 @ A3 )
=> ( P @ ( image_781944334261467077od_v_v @ F @ B4 ) ) ) ) ) ).
% all_subset_image
thf(fact_1221_image__mono,axiom,
! [A3: set_v,B: set_v,F: v > v] :
( ( ord_less_eq_set_v @ A3 @ B )
=> ( ord_less_eq_set_v @ ( image_v_v2 @ F @ A3 ) @ ( image_v_v2 @ F @ B ) ) ) ).
% image_mono
thf(fact_1222_image__mono,axiom,
! [A3: set_v,B: set_v,F: v > product_prod_v_v] :
( ( ord_less_eq_set_v @ A3 @ B )
=> ( ord_le7336532860387713383od_v_v @ ( image_9222788639401671577od_v_v @ F @ A3 ) @ ( image_9222788639401671577od_v_v @ F @ B ) ) ) ).
% image_mono
thf(fact_1223_image__mono,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v,F: product_prod_v_v > v] :
( ( ord_le7336532860387713383od_v_v @ A3 @ B )
=> ( ord_less_eq_set_v @ ( image_6152814753742948081_v_v_v @ F @ A3 ) @ ( image_6152814753742948081_v_v_v @ F @ B ) ) ) ).
% image_mono
thf(fact_1224_image__mono,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v,F: product_prod_v_v > product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A3 @ B )
=> ( ord_le7336532860387713383od_v_v @ ( image_781944334261467077od_v_v @ F @ A3 ) @ ( image_781944334261467077od_v_v @ F @ B ) ) ) ).
% image_mono
thf(fact_1225_image__subsetI,axiom,
! [A3: set_v,F: v > v,B: set_v] :
( ! [X3: v] :
( ( member_v @ X3 @ A3 )
=> ( member_v @ ( F @ X3 ) @ B ) )
=> ( ord_less_eq_set_v @ ( image_v_v2 @ F @ A3 ) @ B ) ) ).
% image_subsetI
thf(fact_1226_image__subsetI,axiom,
! [A3: set_Product_prod_v_v,F: product_prod_v_v > v,B: set_v] :
( ! [X3: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X3 @ A3 )
=> ( member_v @ ( F @ X3 ) @ B ) )
=> ( ord_less_eq_set_v @ ( image_6152814753742948081_v_v_v @ F @ A3 ) @ B ) ) ).
% image_subsetI
thf(fact_1227_image__subsetI,axiom,
! [A3: set_v,F: v > product_prod_v_v,B: set_Product_prod_v_v] :
( ! [X3: v] :
( ( member_v @ X3 @ A3 )
=> ( member7453568604450474000od_v_v @ ( F @ X3 ) @ B ) )
=> ( ord_le7336532860387713383od_v_v @ ( image_9222788639401671577od_v_v @ F @ A3 ) @ B ) ) ).
% image_subsetI
thf(fact_1228_image__subsetI,axiom,
! [A3: set_Product_prod_v_v,F: product_prod_v_v > product_prod_v_v,B: set_Product_prod_v_v] :
( ! [X3: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X3 @ A3 )
=> ( member7453568604450474000od_v_v @ ( F @ X3 ) @ B ) )
=> ( ord_le7336532860387713383od_v_v @ ( image_781944334261467077od_v_v @ F @ A3 ) @ B ) ) ).
% image_subsetI
thf(fact_1229_subset__imageE,axiom,
! [B: set_v,F: v > v,A3: set_v] :
( ( ord_less_eq_set_v @ B @ ( image_v_v2 @ F @ A3 ) )
=> ~ ! [C4: set_v] :
( ( ord_less_eq_set_v @ C4 @ A3 )
=> ( B
!= ( image_v_v2 @ F @ C4 ) ) ) ) ).
% subset_imageE
thf(fact_1230_subset__imageE,axiom,
! [B: set_v,F: product_prod_v_v > v,A3: set_Product_prod_v_v] :
( ( ord_less_eq_set_v @ B @ ( image_6152814753742948081_v_v_v @ F @ A3 ) )
=> ~ ! [C4: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ C4 @ A3 )
=> ( B
!= ( image_6152814753742948081_v_v_v @ F @ C4 ) ) ) ) ).
% subset_imageE
thf(fact_1231_subset__imageE,axiom,
! [B: set_Product_prod_v_v,F: v > product_prod_v_v,A3: set_v] :
( ( ord_le7336532860387713383od_v_v @ B @ ( image_9222788639401671577od_v_v @ F @ A3 ) )
=> ~ ! [C4: set_v] :
( ( ord_less_eq_set_v @ C4 @ A3 )
=> ( B
!= ( image_9222788639401671577od_v_v @ F @ C4 ) ) ) ) ).
% subset_imageE
thf(fact_1232_subset__imageE,axiom,
! [B: set_Product_prod_v_v,F: product_prod_v_v > product_prod_v_v,A3: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B @ ( image_781944334261467077od_v_v @ F @ A3 ) )
=> ~ ! [C4: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ C4 @ A3 )
=> ( B
!= ( image_781944334261467077od_v_v @ F @ C4 ) ) ) ) ).
% subset_imageE
thf(fact_1233_subset__image__iff,axiom,
! [B: set_v,F: v > v,A3: set_v] :
( ( ord_less_eq_set_v @ B @ ( image_v_v2 @ F @ A3 ) )
= ( ? [AA: set_v] :
( ( ord_less_eq_set_v @ AA @ A3 )
& ( B
= ( image_v_v2 @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_1234_subset__image__iff,axiom,
! [B: set_v,F: product_prod_v_v > v,A3: set_Product_prod_v_v] :
( ( ord_less_eq_set_v @ B @ ( image_6152814753742948081_v_v_v @ F @ A3 ) )
= ( ? [AA: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ AA @ A3 )
& ( B
= ( image_6152814753742948081_v_v_v @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_1235_subset__image__iff,axiom,
! [B: set_Product_prod_v_v,F: v > product_prod_v_v,A3: set_v] :
( ( ord_le7336532860387713383od_v_v @ B @ ( image_9222788639401671577od_v_v @ F @ A3 ) )
= ( ? [AA: set_v] :
( ( ord_less_eq_set_v @ AA @ A3 )
& ( B
= ( image_9222788639401671577od_v_v @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_1236_subset__image__iff,axiom,
! [B: set_Product_prod_v_v,F: product_prod_v_v > product_prod_v_v,A3: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B @ ( image_781944334261467077od_v_v @ F @ A3 ) )
= ( ? [AA: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ AA @ A3 )
& ( B
= ( image_781944334261467077od_v_v @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_1237_Image__Int__subset,axiom,
! [R2: set_Product_prod_v_v,A3: set_v,B: set_v] : ( ord_less_eq_set_v @ ( image_v_v @ R2 @ ( inf_inf_set_v @ A3 @ B ) ) @ ( inf_inf_set_v @ ( image_v_v @ R2 @ A3 ) @ ( image_v_v @ R2 @ B ) ) ) ).
% Image_Int_subset
thf(fact_1238_Image__Int__subset,axiom,
! [R2: set_Pr7862341151230101147od_v_v,A3: set_v,B: set_v] : ( ord_le7336532860387713383od_v_v @ ( image_4621343323592381799od_v_v @ R2 @ ( inf_inf_set_v @ A3 @ B ) ) @ ( inf_in6271465464967711157od_v_v @ ( image_4621343323592381799od_v_v @ R2 @ A3 ) @ ( image_4621343323592381799od_v_v @ R2 @ B ) ) ) ).
% Image_Int_subset
thf(fact_1239_ImageE,axiom,
! [B2: v,R: set_Pr7679524143894959091_v_v_v,A3: set_Product_prod_v_v] :
( ( member_v @ B2 @ ( image_1551369437933658303_v_v_v @ R @ A3 ) )
=> ~ ! [X3: product_prod_v_v] :
( ( member5544786109013881916_v_v_v @ ( produc8407406349431002243_v_v_v @ X3 @ B2 ) @ R )
=> ~ ( member7453568604450474000od_v_v @ X3 @ A3 ) ) ) ).
% ImageE
thf(fact_1240_ImageE,axiom,
! [B2: product_prod_v_v,R: set_Pr7862341151230101147od_v_v,A3: set_v] :
( ( member7453568604450474000od_v_v @ B2 @ ( image_4621343323592381799od_v_v @ R @ A3 ) )
=> ~ ! [X3: v] :
( ( member5456077685714336484od_v_v @ ( produc2254008198234949931od_v_v @ X3 @ B2 ) @ R )
=> ~ ( member_v @ X3 @ A3 ) ) ) ).
% ImageE
thf(fact_1241_ImageE,axiom,
! [B2: product_prod_v_v,R: set_Pr2149350503807050951od_v_v,A3: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ B2 @ ( image_5221874569633403795od_v_v @ R @ A3 ) )
=> ~ ! [X3: product_prod_v_v] :
( ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ X3 @ B2 ) @ R )
=> ~ ( member7453568604450474000od_v_v @ X3 @ A3 ) ) ) ).
% ImageE
thf(fact_1242_ImageE,axiom,
! [B2: v,R: set_Product_prod_v_v,A3: set_v] :
( ( member_v @ B2 @ ( image_v_v @ R @ A3 ) )
=> ~ ! [X3: v] :
( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X3 @ B2 ) @ R )
=> ~ ( member_v @ X3 @ A3 ) ) ) ).
% ImageE
thf(fact_1243_Image__iff,axiom,
! [B2: v,R: set_Product_prod_v_v,A3: set_v] :
( ( member_v @ B2 @ ( image_v_v @ R @ A3 ) )
= ( ? [X2: v] :
( ( member_v @ X2 @ A3 )
& ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X2 @ B2 ) @ R ) ) ) ) ).
% Image_iff
thf(fact_1244_rev__ImageI,axiom,
! [A: v,A3: set_v,B2: product_prod_v_v,R: set_Pr7862341151230101147od_v_v] :
( ( member_v @ A @ A3 )
=> ( ( member5456077685714336484od_v_v @ ( produc2254008198234949931od_v_v @ A @ B2 ) @ R )
=> ( member7453568604450474000od_v_v @ B2 @ ( image_4621343323592381799od_v_v @ R @ A3 ) ) ) ) ).
% rev_ImageI
thf(fact_1245_rev__ImageI,axiom,
! [A: product_prod_v_v,A3: set_Product_prod_v_v,B2: v,R: set_Pr7679524143894959091_v_v_v] :
( ( member7453568604450474000od_v_v @ A @ A3 )
=> ( ( member5544786109013881916_v_v_v @ ( produc8407406349431002243_v_v_v @ A @ B2 ) @ R )
=> ( member_v @ B2 @ ( image_1551369437933658303_v_v_v @ R @ A3 ) ) ) ) ).
% rev_ImageI
thf(fact_1246_rev__ImageI,axiom,
! [A: product_prod_v_v,A3: set_Product_prod_v_v,B2: product_prod_v_v,R: set_Pr2149350503807050951od_v_v] :
( ( member7453568604450474000od_v_v @ A @ A3 )
=> ( ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ A @ B2 ) @ R )
=> ( member7453568604450474000od_v_v @ B2 @ ( image_5221874569633403795od_v_v @ R @ A3 ) ) ) ) ).
% rev_ImageI
thf(fact_1247_rev__ImageI,axiom,
! [A: v,A3: set_v,B2: v,R: set_Product_prod_v_v] :
( ( member_v @ A @ A3 )
=> ( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ A @ B2 ) @ R )
=> ( member_v @ B2 @ ( image_v_v @ R @ A3 ) ) ) ) ).
% rev_ImageI
thf(fact_1248_image__Un,axiom,
! [F: product_prod_v_v > product_prod_v_v,A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( image_781944334261467077od_v_v @ F @ ( sup_su414716646722978715od_v_v @ A3 @ B ) )
= ( sup_su414716646722978715od_v_v @ ( image_781944334261467077od_v_v @ F @ A3 ) @ ( image_781944334261467077od_v_v @ F @ B ) ) ) ).
% image_Un
thf(fact_1249_rev__image__eqI,axiom,
! [X: v,A3: set_v,B2: v,F: v > v] :
( ( member_v @ X @ A3 )
=> ( ( B2
= ( F @ X ) )
=> ( member_v @ B2 @ ( image_v_v2 @ F @ A3 ) ) ) ) ).
% rev_image_eqI
thf(fact_1250_rev__image__eqI,axiom,
! [X: v,A3: set_v,B2: product_prod_v_v,F: v > product_prod_v_v] :
( ( member_v @ X @ A3 )
=> ( ( B2
= ( F @ X ) )
=> ( member7453568604450474000od_v_v @ B2 @ ( image_9222788639401671577od_v_v @ F @ A3 ) ) ) ) ).
% rev_image_eqI
thf(fact_1251_rev__image__eqI,axiom,
! [X: product_prod_v_v,A3: set_Product_prod_v_v,B2: v,F: product_prod_v_v > v] :
( ( member7453568604450474000od_v_v @ X @ A3 )
=> ( ( B2
= ( F @ X ) )
=> ( member_v @ B2 @ ( image_6152814753742948081_v_v_v @ F @ A3 ) ) ) ) ).
% rev_image_eqI
thf(fact_1252_rev__image__eqI,axiom,
! [X: product_prod_v_v,A3: set_Product_prod_v_v,B2: product_prod_v_v,F: product_prod_v_v > product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X @ A3 )
=> ( ( B2
= ( F @ X ) )
=> ( member7453568604450474000od_v_v @ B2 @ ( image_781944334261467077od_v_v @ F @ A3 ) ) ) ) ).
% rev_image_eqI
thf(fact_1253_imageI,axiom,
! [X: v,A3: set_v,F: v > v] :
( ( member_v @ X @ A3 )
=> ( member_v @ ( F @ X ) @ ( image_v_v2 @ F @ A3 ) ) ) ).
% imageI
thf(fact_1254_imageI,axiom,
! [X: v,A3: set_v,F: v > product_prod_v_v] :
( ( member_v @ X @ A3 )
=> ( member7453568604450474000od_v_v @ ( F @ X ) @ ( image_9222788639401671577od_v_v @ F @ A3 ) ) ) ).
% imageI
thf(fact_1255_imageI,axiom,
! [X: product_prod_v_v,A3: set_Product_prod_v_v,F: product_prod_v_v > v] :
( ( member7453568604450474000od_v_v @ X @ A3 )
=> ( member_v @ ( F @ X ) @ ( image_6152814753742948081_v_v_v @ F @ A3 ) ) ) ).
% imageI
thf(fact_1256_imageI,axiom,
! [X: product_prod_v_v,A3: set_Product_prod_v_v,F: product_prod_v_v > product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X @ A3 )
=> ( member7453568604450474000od_v_v @ ( F @ X ) @ ( image_781944334261467077od_v_v @ F @ A3 ) ) ) ).
% imageI
thf(fact_1257_Inf__fin_Ohom__commute,axiom,
! [H: set_v > set_v,N3: set_set_v] :
( ! [X3: set_v,Y: set_v] :
( ( H @ ( inf_inf_set_v @ X3 @ Y ) )
= ( inf_inf_set_v @ ( H @ X3 ) @ ( H @ Y ) ) )
=> ( ( finite_finite_set_v @ N3 )
=> ( ( N3 != bot_bot_set_set_v )
=> ( ( H @ ( lattic8209813555532694032_set_v @ N3 ) )
= ( lattic8209813555532694032_set_v @ ( image_set_v_set_v @ H @ N3 ) ) ) ) ) ) ).
% Inf_fin.hom_commute
thf(fact_1258_Sup__fin_Ohom__commute,axiom,
! [H: set_Product_prod_v_v > set_Product_prod_v_v,N3: set_se8455005133513928103od_v_v] :
( ! [X3: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
( ( H @ ( sup_su414716646722978715od_v_v @ X3 @ Y ) )
= ( sup_su414716646722978715od_v_v @ ( H @ X3 ) @ ( H @ Y ) ) )
=> ( ( finite6084192165098772208od_v_v @ N3 )
=> ( ( N3 != bot_bo3497076220358800403od_v_v )
=> ( ( H @ ( lattic5151207300795964030od_v_v @ N3 ) )
= ( lattic5151207300795964030od_v_v @ ( image_5212826947168092101od_v_v @ H @ N3 ) ) ) ) ) ) ).
% Sup_fin.hom_commute
thf(fact_1259_in__image__insert__iff,axiom,
! [B: set_se8455005133513928103od_v_v,X: product_prod_v_v,A3: set_Product_prod_v_v] :
( ! [C4: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ C4 @ B )
=> ~ ( member7453568604450474000od_v_v @ X @ C4 ) )
=> ( ( member8406446414694345712od_v_v @ A3 @ ( image_5212826947168092101od_v_v @ ( insert1338601472111419319od_v_v @ X ) @ B ) )
= ( ( member7453568604450474000od_v_v @ X @ A3 )
& ( member8406446414694345712od_v_v @ ( minus_4183494784930505774od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) ) @ B ) ) ) ) ).
% in_image_insert_iff
thf(fact_1260_in__image__insert__iff,axiom,
! [B: set_set_v,X: v,A3: set_v] :
( ! [C4: set_v] :
( ( member_set_v @ C4 @ B )
=> ~ ( member_v @ X @ C4 ) )
=> ( ( member_set_v @ A3 @ ( image_set_v_set_v @ ( insert_v2 @ X ) @ B ) )
= ( ( member_v @ X @ A3 )
& ( member_set_v @ ( minus_minus_set_v @ A3 @ ( insert_v2 @ X @ bot_bot_set_v ) ) @ B ) ) ) ) ).
% in_image_insert_iff
thf(fact_1261_subset__Image1__Image1__iff,axiom,
! [R: set_Product_prod_v_v,A: v,B2: v] :
( ( order_preorder_on_v @ ( field_v @ R ) @ R )
=> ( ( member_v @ A @ ( field_v @ R ) )
=> ( ( member_v @ B2 @ ( field_v @ R ) )
=> ( ( ord_less_eq_set_v @ ( image_v_v @ R @ ( insert_v2 @ A @ bot_bot_set_v ) ) @ ( image_v_v @ R @ ( insert_v2 @ B2 @ bot_bot_set_v ) ) )
= ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ B2 @ A ) @ R ) ) ) ) ) ).
% subset_Image1_Image1_iff
thf(fact_1262_subset__Image1__Image1__iff,axiom,
! [R: set_Pr2149350503807050951od_v_v,A: product_prod_v_v,B2: product_prod_v_v] :
( ( order_5932439346107134od_v_v @ ( field_7153129647634986036od_v_v @ R ) @ R )
=> ( ( member7453568604450474000od_v_v @ A @ ( field_7153129647634986036od_v_v @ R ) )
=> ( ( member7453568604450474000od_v_v @ B2 @ ( field_7153129647634986036od_v_v @ R ) )
=> ( ( ord_le7336532860387713383od_v_v @ ( image_5221874569633403795od_v_v @ R @ ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) ) @ ( image_5221874569633403795od_v_v @ R @ ( insert1338601472111419319od_v_v @ B2 @ bot_bo723834152578015283od_v_v ) ) )
= ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ B2 @ A ) @ R ) ) ) ) ) ).
% subset_Image1_Image1_iff
thf(fact_1263_Refl__antisym__eq__Image1__Image1__iff,axiom,
! [R: set_Pr2149350503807050951od_v_v,A: product_prod_v_v,B2: product_prod_v_v] :
( ( refl_o4548774019903118566od_v_v @ ( field_7153129647634986036od_v_v @ R ) @ R )
=> ( ( antisy5811833586610396106od_v_v @ top_to5429829297380968215od_v_v @ R )
=> ( ( member7453568604450474000od_v_v @ A @ ( field_7153129647634986036od_v_v @ R ) )
=> ( ( member7453568604450474000od_v_v @ B2 @ ( field_7153129647634986036od_v_v @ R ) )
=> ( ( ( image_5221874569633403795od_v_v @ R @ ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) )
= ( image_5221874569633403795od_v_v @ R @ ( insert1338601472111419319od_v_v @ B2 @ bot_bo723834152578015283od_v_v ) ) )
= ( A = B2 ) ) ) ) ) ) ).
% Refl_antisym_eq_Image1_Image1_iff
thf(fact_1264_Refl__antisym__eq__Image1__Image1__iff,axiom,
! [R: set_Product_prod_v_v,A: v,B2: v] :
( ( refl_on_v @ ( field_v @ R ) @ R )
=> ( ( antisym_on_v @ top_top_set_v @ R )
=> ( ( member_v @ A @ ( field_v @ R ) )
=> ( ( member_v @ B2 @ ( field_v @ R ) )
=> ( ( ( image_v_v @ R @ ( insert_v2 @ A @ bot_bot_set_v ) )
= ( image_v_v @ R @ ( insert_v2 @ B2 @ bot_bot_set_v ) ) )
= ( A = B2 ) ) ) ) ) ) ).
% Refl_antisym_eq_Image1_Image1_iff
thf(fact_1265_antisym__on__def,axiom,
( antisym_on_v
= ( ^ [A4: set_v,R4: set_Product_prod_v_v] :
! [X2: v] :
( ( member_v @ X2 @ A4 )
=> ! [Y3: v] :
( ( member_v @ Y3 @ A4 )
=> ( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X2 @ Y3 ) @ R4 )
=> ( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ Y3 @ X2 ) @ R4 )
=> ( X2 = Y3 ) ) ) ) ) ) ) ).
% antisym_on_def
thf(fact_1266_antisym__onI,axiom,
! [A3: set_Product_prod_v_v,R: set_Pr2149350503807050951od_v_v] :
( ! [X3: product_prod_v_v,Y: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X3 @ A3 )
=> ( ( member7453568604450474000od_v_v @ Y @ A3 )
=> ( ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ X3 @ Y ) @ R )
=> ( ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ Y @ X3 ) @ R )
=> ( X3 = Y ) ) ) ) )
=> ( antisy5811833586610396106od_v_v @ A3 @ R ) ) ).
% antisym_onI
thf(fact_1267_antisym__onI,axiom,
! [A3: set_v,R: set_Product_prod_v_v] :
( ! [X3: v,Y: v] :
( ( member_v @ X3 @ A3 )
=> ( ( member_v @ Y @ A3 )
=> ( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X3 @ Y ) @ R )
=> ( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ Y @ X3 ) @ R )
=> ( X3 = Y ) ) ) ) )
=> ( antisym_on_v @ A3 @ R ) ) ).
% antisym_onI
thf(fact_1268_antisym__onD,axiom,
! [A3: set_Product_prod_v_v,R: set_Pr2149350503807050951od_v_v,X: product_prod_v_v,Y2: product_prod_v_v] :
( ( antisy5811833586610396106od_v_v @ A3 @ R )
=> ( ( member7453568604450474000od_v_v @ X @ A3 )
=> ( ( member7453568604450474000od_v_v @ Y2 @ A3 )
=> ( ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ X @ Y2 ) @ R )
=> ( ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ Y2 @ X ) @ R )
=> ( X = Y2 ) ) ) ) ) ) ).
% antisym_onD
thf(fact_1269_antisym__onD,axiom,
! [A3: set_v,R: set_Product_prod_v_v,X: v,Y2: v] :
( ( antisym_on_v @ A3 @ R )
=> ( ( member_v @ X @ A3 )
=> ( ( member_v @ Y2 @ A3 )
=> ( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X @ Y2 ) @ R )
=> ( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ Y2 @ X ) @ R )
=> ( X = Y2 ) ) ) ) ) ) ).
% antisym_onD
thf(fact_1270_antisymI,axiom,
! [R: set_Product_prod_v_v] :
( ! [X3: v,Y: v] :
( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X3 @ Y ) @ R )
=> ( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ Y @ X3 ) @ R )
=> ( X3 = Y ) ) )
=> ( antisym_on_v @ top_top_set_v @ R ) ) ).
% antisymI
thf(fact_1271_antisymD,axiom,
! [R: set_Product_prod_v_v,X: v,Y2: v] :
( ( antisym_on_v @ top_top_set_v @ R )
=> ( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X @ Y2 ) @ R )
=> ( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ Y2 @ X ) @ R )
=> ( X = Y2 ) ) ) ) ).
% antisymD
thf(fact_1272_antisym__empty,axiom,
antisym_on_v @ top_top_set_v @ bot_bo723834152578015283od_v_v ).
% antisym_empty
thf(fact_1273_antisym__subset,axiom,
! [R: set_Product_prod_v_v,S4: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ R @ S4 )
=> ( ( antisym_on_v @ top_top_set_v @ S4 )
=> ( antisym_on_v @ top_top_set_v @ R ) ) ) ).
% antisym_subset
thf(fact_1274_antisym__on__subset,axiom,
! [A3: set_v,R: set_Product_prod_v_v,B: set_v] :
( ( antisym_on_v @ A3 @ R )
=> ( ( ord_less_eq_set_v @ B @ A3 )
=> ( antisym_on_v @ B @ R ) ) ) ).
% antisym_on_subset
thf(fact_1275_antisym__on__subset,axiom,
! [A3: set_Product_prod_v_v,R: set_Pr2149350503807050951od_v_v,B: set_Product_prod_v_v] :
( ( antisy5811833586610396106od_v_v @ A3 @ R )
=> ( ( ord_le7336532860387713383od_v_v @ B @ A3 )
=> ( antisy5811833586610396106od_v_v @ B @ R ) ) ) ).
% antisym_on_subset
thf(fact_1276_preorder__on__empty,axiom,
order_5932439346107134od_v_v @ bot_bo723834152578015283od_v_v @ bot_bo3282589961317712691od_v_v ).
% preorder_on_empty
thf(fact_1277_preorder__on__empty,axiom,
order_preorder_on_v @ bot_bot_set_v @ bot_bo723834152578015283od_v_v ).
% preorder_on_empty
thf(fact_1278_antisym__singleton,axiom,
! [X: product_prod_v_v] : ( antisym_on_v @ top_top_set_v @ ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) ) ).
% antisym_singleton
% Helper facts (9)
thf(help_If_2_1_If_001t__Set__Oset_Itf__v_J_T,axiom,
! [X: set_v,Y2: set_v] :
( ( if_set_v @ $false @ X @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__Set__Oset_Itf__v_J_T,axiom,
! [X: set_v,Y2: set_v] :
( ( if_set_v @ $true @ X @ Y2 )
= X ) ).
thf(help_If_2_1_If_001t__List__Olist_Itf__v_J_T,axiom,
! [X: list_v,Y2: list_v] :
( ( if_list_v @ $false @ X @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__List__Olist_Itf__v_J_T,axiom,
! [X: list_v,Y2: list_v] :
( ( if_list_v @ $true @ X @ Y2 )
= X ) ).
thf(help_If_2_1_If_001t__Set__Oset_It__Product____Type__Oprod_Itf__v_Mtf__v_J_J_T,axiom,
! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v] :
( ( if_set4279007504652509325od_v_v @ $false @ X @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__Set__Oset_It__Product____Type__Oprod_Itf__v_Mtf__v_J_J_T,axiom,
! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v] :
( ( if_set4279007504652509325od_v_v @ $true @ X @ Y2 )
= X ) ).
thf(help_If_3_1_If_001t__List__Olist_It__Product____Type__Oprod_Itf__v_Mtf__v_J_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__List__Olist_It__Product____Type__Oprod_Itf__v_Mtf__v_J_J_T,axiom,
! [X: list_P7986770385144383213od_v_v,Y2: list_P7986770385144383213od_v_v] :
( ( if_lis7521272669439687347od_v_v @ $false @ X @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Product____Type__Oprod_Itf__v_Mtf__v_J_J_T,axiom,
! [X: list_P7986770385144383213od_v_v,Y2: list_P7986770385144383213od_v_v] :
( ( if_lis7521272669439687347od_v_v @ $true @ X @ Y2 )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
sCC_Bl9196236973127232072t_unit @ successors @ e ).
%------------------------------------------------------------------------------